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http://unapologetic.wordpress.com/category/analysis/calculus/page/3/	# The Unapologetic Mathematician ## Oscillation Oscillation in a function is sort of a local and non-directional version of variation. If $f:X\rightarrow\mathbb{R}$ is a bounded function on some region $X\subseteq\mathbb{R}^n$, and if $T$ is a nonempty subset of $X$, then we define the oscillation of $f$ on $T$ by the formula $\displaystyle\Omega_f(T)=\sup\limits_{x,y\in T}\left\{f(y)-f(x)\right\}$ measuring the greatest difference in values of $f$ on $T$. We also want a version that’s localized to a single point $x\in X$. To do this, we first note that the collection of all subsets $N$ of $X$ which contain $x$ form a poset as usual by inclusion. But we want to reverse this order and say that $M\preceq N$ if and only if $N\subseteq M$. Now for any two subsets $x\in N_1\subseteq X$ and $x\in N_2\subseteq X$, their intersection $N_1\cap N_2$ is another such subset containing $x$. And since it’s contained in both $N_1$ and $N_2$, it’s above both of them in our partial order, which makes this poset a directed set, and the oscillation of $f$ is a net. In fact, it’s easy to see that if $N\subseteq M$ then $\Omega_f(N)\leq\Omega_f(M)$, so this net is monotonically decreasing as the subset gets smaller and smaller. Further, we can see that $\Omega_f(N)\geq0$, since if we can always consider the difference $f(t)-f(t)=0$, the supremum must be at least this big. Anyhow, now we know that the net has a limit, and we define $\displaystyle\omega_f(x)=\lim\Omega_f(N)$ where $N$ is a subset of $X$ containing $x$, and we take the limit as $N$ gets smaller and smaller. In fact, this is slightly overdoing it. Our domain is a topological subspace of $\mathbb{R}^n$, and is thus a metric space. If we want we can just work with metric balls and define $\displaystyle\omega_f(x)=\lim\limits_{r\rightarrow0^+}\Omega_f(N(x;r)\cap X)$ where $N(x;r)$ is the ball of radius $r$ around $x$. These definitions are exactly equivalent in metric spaces, but the net definition works in more general topological spaces, and it’s extremely useful in its own right later, so it’s worth thinking about now. Oscillation provides a nice way to restate our condition for continuity, and it works either using the metric space definition or the neighborhood definition of continuity. I’ll work it out in the latter case for generality, but it’s worth writing out the parallel proof for the $\epsilon$-$\delta$ definition. Our assertion is that $f$ is continuous at a point $x$ if and only if $\omega_f(x)=0$. If $f$ is continuous, then for every $\epsilon$ there is some neighborhood $N$ of $x$ so that $\lvert f(y)-f(x)\rvert<\frac{\epsilon}{3}$ for all $y\in N$. Then we can check that $f(y_2)-f(y_1)=\left(f(y_2)-f(x)\right)+\left(f(x)-f(y_1)\right)<\frac{\epsilon}{3}+\frac{\epsilon}{3}<\frac{2}{3}\epsilon$ for all $y_1$ and $y_2$ in $N$, and so $\Omega_f(N)<\epsilon$. Further, any smaller neighborhood of $x$ will also satisfy this inequality, so the net is eventually within $\epsilon$ of ${0}$. Since this holds for any $\epsilon$, we find that the net has limit ${0}$. Conversely, let’s assume that the oscillation of $f$ at $x$ is zero. That is, for any $\epsilon$ we have some neighborhood $N$ of $x$ so that $\Omega_f(N)<\frac{\epsilon}{2}$, and the same will automatically hold for smaller neighborhoods. This tells us that $f(y)-f(x)<\epsilon$ for all $y\in N$, and also $f(x)-f(y)<\epsilon$. Together, these tell us that $\lvert f(y)-f(x)\rvert<\epsilon$, and so $f$ is continuous at $x$. December 7, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Upper and Lower Integrals and Riemann’s Condition Yesterday we defined the Riemann integral of a multivariable function defined on an interval $[a,b]\subseteq\mathbb{R}^n$. We want to move towards some understanding of when a given function $f$ is integrable on a given interval $[a,b]$. First off, we remember how we set up Darboux sums. These were given by prescribing specific methods of tagging a given partition $P$. In one, we always picked the point in the subinterval where $f$ attained its maximum within that subinterval, and in the other we always picked the point where $f$ attained its minimum. We even extended these to the Riemann-Stieltjes case and built up upper and lower integrals. And we can do the same thing again. Given a partition $P$ of $[a,b]$ and a function defined on $[a,b]$, we define the upper Riemann sum by $\displaystyle U_P(f)=\sum\limits_{i_1=1}^{m_1}\dots\sum\limits_{i_n=1}^{m_n}\max\limits_{t\in I_{i_1\dots i_n}}\{f(t)\}\mathrm{vol}(I_{i_1\dots i_n})$ In each subinterval we pick a sample point which gives the largest possible sample function value in that subinterval. We similarly define a lower Riemann sum by $\displaystyle L_P(f)=\sum\limits_{i_1=1}^{m_1}\dots\sum\limits_{i_n=1}^{m_n}\min\limits_{t\in I_{i_1\dots i_n}}\{f(t)\}\mathrm{vol}(I_{i_1\dots i_n})$ As before, any Riemann sum must fall between these upper and lower sums, since the value of the function on each subinterval is somewhere between its maximum and minimum. Just like when we did this for single-variable Riemann-Stieltjes integrals, we find that these nets are monotonic. That is, if $P'$ is a refinement of $P$, then $U_{P'}(f)\leq U_P(f)$ and $L_{P'}(f)\geq L_P(f)$. As we refine the partition, the upper sum can only get smaller and smaller, while the lower sum can only get larger and larger. And so we define \displaystyle\begin{aligned}\overline{I}_{[a,b]}(f)&=\inf\limits_{P\in\mathcal{P}[a,b]}\{U_P(f)\}\\\underline{I}_{[a,b]}(f)&=\sup\limits_{P\in\mathcal{P}[a,b]}\{L_P(f)\}\end{aligned} The upper integral is the infimum of the upper sums, while the lower integral is the supremum of the lower sums. Again, as before we find that the upper integral is convex over its integrand, while the lower integral is concave \displaystyle\begin{aligned}\overline{I}_{[a,b]}(f+g)&\leq\overline{I}_{[a,b]}(f)+\overline{I}_{[a,b]}(g)\\\underline{I}_{[a,b]}(f+g)&\geq\underline{I}_{[a,b]}(f)+\underline{I}_{[a,b]}(g)\end{aligned} and if we break up an interval into a collection of nonoverlapping subintervals, the upper and lower integrals over the large interval are the sums of the upper and lower integrals over each of the subintervals, respectively. And, finally, we have Riemann’s condition. The function $f$ satisfies Riemann’s condition on $[a,b]$ we can make upper and lower sums arbitrarily close. That is, if for every $\epsilon>0$ there is some partition $P_\epsilon$ so that $U_{P_\epsilon}(f)-L_{P_\epsilon}(f)<\epsilon$. In this case, the upper and lower integrals will coincide, and we can show that $f$ is actually integrable over $[a,b]$. The proof is almost exactly the same one we gave before, and so I’ll just refer you back there. December 2, 2009 Posted by \| Analysis, Calculus \| 9 Comments ## Higher-Dimensional Riemann Integrals Our coverage of multiple integrals will actually parallel our earlier coverage of Riemann integrals pretty closely. Only now we have to change our notion of “interval” to a higher-dimensional version. For the moment, the only shapes we’ll integrate over will be closed rectangular $n$-dimensional parallelepipeds with sides parallel to the coordinate axes in $\mathbb{R}^n$. In one dimension, these are just coordinate intervals, but there aren’t many other obvious shapes to use in that case. In two dimensions, we are (for the moment) throwing out circles, ellipses, triangles, and anything else that’s not a rectangle; and even rectangles that are tilted with respect to the coordinate axes! Don’t worry, we’ll get them back later. Anyway, such a shape in $n$-dimensional space is the product of $n$ closed intervals: $\displaystyle[a^1,b^1]\times\dots[a^n,b^n]=\{(x^1,\dots,x^n)\in\mathbb{R}^n\vert a^1\leq x^1\leq b^1,\dots,a^n\leq x^n\leq b^n\}$ In this context, we’ll write this as $[a,b]$ for $a=(a^1,\dots,a^n)$ and $b=(b^1,\dots,b^n)$. This looks identical to, but is not to be confused with, the closed line segment from $a$ to $b$ we used in the mean value theorem. I’ll try to keep them separate by always referring to this rectangular parallelepiped as an “interval” and never using that term for a line segment. Of course, in one dimension the two are exactly the same. Now we define a partition of an interval. We’ll do this just by partitioning each side of the interval. That is, we pick points \displaystyle\begin{aligned}a^1={x^1}_0<&\dots<{x^1}_{m_1}=b^1\\&\vdots\\a^n={x^n}_0<&\dots<{x^n}_{m_n}=b^n\end{aligned} where we cut the $i$th side into $m_i$ pieces, and each of the $m_i$ could be different. If we index the subintervals of the partition by $1\leq i_1\leq m_1$ through $1\leq i_n\leq m_n$, we can write $\displaystyle I_{i_1\dots i_n}=[{x^1}_{i_1},{x^1}_{i_1+1}]\times\dots\times[{x^n}_{i_n},{x^n}_{i_n+1}]$ picking out the $i_k$th slice of the $k$th side. This cuts the whole interval up into a bunch of smaller pieces, which are themselves rectangular parallelepipeds. Again, we have the notion of a tagged partition, which picks out a sample point $t_{i_1\dots i_n}\in I_{i_1\dots i_n}$ for each subinterval. And again we say that one tagged partition is a “refinement” of another one if every partition point of the coarser partition is also one in the finer partition on the same side, and if every sample point in the coarser partition is one in the finer partition as well. There’s a lot of numbers here and a lot of notation, but relax: most of it is going to go away very quickly. Go back and look at how partitions, tagged partitions, and refinements were defined in one variable, and just think of partitioning (and refining) the one-dimensional interval that makes up each side of the $n$-dimensional interval. Of course, like I said before, the number of parts on each side might have nothing to do with each other. Further, there’s no reason for the tags to have anything to do with each other. One might think of tagging the partition on each side of the interval, and then letting the sample points in the subintervals have those tags as coordinates. This is a perfectly valid way to come up with a collection of sample points, but the sample points don’t have to arise in this manner at all, so long as there’s exactly one within each subinterval. So, just like in the one-dimensional case, the collection $\mathcal{P}[a,b]$ of all tagged partitions of an interval $[a,b]$ form a directed set, where we say that $P\preceq P'$ if $P'$ is a refinement of $P$. And again we define nets on this directed set; given a function $f$ defined on the interval $[a,b]$ and a partition $P\in\mathcal{P}[a,b]$, we define the Riemann sum $\displaystyle f_P=\sum\limits_{i_1=1}^{m_1}\dots\sum\limits_{i_n=1}^{m_n}f(t_{i_1\dots i_n})\mathrm{vol}(I_{i_1\dots i_n})$ by sampling the function $f$ at the specified point $t_{i_1\dots i_n}$ in the subinterval $I_{i_1\dots i_n}$, multiplying by the $n$-dimensional volume of the subinterval (which is just the product of its side-lengths), and summing this over all the subintervals in $[a,b]$. And again, as before, if this net converges to a limiting value $s$, we say that $f$ is Riemann integrable on the interval $[a,b]$ and we write $\displaystyle\int\limits_{[a,b]}f\,dx=\int\limits_{[a,b]}f(x)\,dx=\int\limits_{[a,b]}f(x^1,\dots,x^n)\,d(x^1,\dots,x^n)=s$ in three different notations, depending on what we want to emphasize. The first emphasizes the function as an object in and of itself; the second, the function’s dependence on the vector variable $x$; and the third, the function’s dependence on each individual real variable. In two and three dimensions, we often write the “double integral” and “triple integral” as \displaystyle\begin{aligned}\iint\limits_{[a,b]}&f(x)\,dx\\\iiint\limits_{[a,b]}&f(x)\,dx\end{aligned} as visual reminders that these are multiple integrals. This gets unwieldy very quickly, and so we usually just write one integral sign for each integration. December 1, 2009 Posted by \| Analysis, Calculus \| 10 Comments ## Extrema with Constraints II As we said last time, we have an idea for a necessary condition for finding local extrema subject to constraints of a certain form. To be explicit, we assume that $f:X\rightarrow\mathbb{R}$ is continuously differentiable on an open region $X\subset\mathbb{R}^n$, and we also assume that $g:X\rightarrow\mathbb{R}^m$ is a continuously differentiable vector-valued function on the same region (and that $m). We use $g$ to define the region $A\subseteq X$ consisting of those points $x\in X$ so that $g(x)=0$. Now if $a\in A$ is a point with a neighborhood $N$ so that for all $x\in N\cap A$ we have $f(x)\leq f(a)$ (or $f(x)\geq f(a)$ for all such $x$). And, finally, we assume that the $m\times m$ determinant $\displaystyle\det\left(\frac{\partial g^i}{\partial x^j}\right)$ is nonzero at $a$. Then we have reason to believe that there will exist real numbers $\lambda_1,\dots,\lambda_m$, one for each component of the constraint function, so that the $n$ equations $\displaystyle\frac{\partial f}{\partial x^j}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^j}=0\qquad(j=1,\dots,n)$ are satisfied at $a$. Well, first off we can solve the first $m$ equations and determine the $\lambda_j$ right off. Rewrite them as $\displaystyle\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^j}\bigg\vert_a=-\frac{\partial f}{\partial x^j}\bigg\vert_a\qquad(j=1,\dots,m)$ a system of $m$ equations in the $m$ unknowns $\lambda_i$. Since the matrix has a nonzero determinant (by assumption) we can solve this system uniquely to determine the $\lambda_i$. What’s left is to verify that this choice of the $\lambda_i$ also satisfies the remaining $n-m$ equations. To take care of this, we’ll write $t^k=x^{m+k}$, so we can write the point $x=(x^1,\dots,x^n)$ as $(x';t)=(x^1,\dots,x^m;t^1,\dots,t^{n-m})$ and particularly $a=(a';b)$. Now we can invoke the implicit function theorem! We find an $n-m$-dimensional neighborhood $T$ of $b$ and a unique continuously differentiable function $h:T\rightarrow\mathbb{R}^m$ so that $h(b)=a'$ and $g(h(t);t)=0$ for all $t\in T$. Without loss of generality, we can choose $T$ so that $h(t)\in N\cap A$, where $N$ is the neighborhood from the assumptions above. This is the parameterization we discussed last time, and we can now substitute these functions into the function $f$. That is, we can define \displaystyle\begin{aligned}F(t^1,\dots,t^{n-m})&=f\left(h^1(t^1,\dots,t^{n-m}),\dots,h^m(t^1,\dots,t^{n-m});t^1,\dots,t^{n-m}\right)\\G^i(t^1,\dots,t^{n-m})&=g^i\left(h^1(t^1,\dots,t^{n-m}),\dots,h^m(t^1,\dots,t^{n-m});t^1,\dots,t^{n-m}\right)\qquad(i=1,\dots,m)\end{aligned} Or if we define $H(t)=(h(t);t)$ we can say this more succinctly as $F(t)=f(H(t))$ and $G^i(t)=g^i(H(t))$. Anyhow, now all of these $G^i$ are identically zero on $T$ as a consequence of the implicit function theorem, and so each partial derivative $\frac{\partial G^i}{\partial t^j}$ is identically zero as well. But since the $G^i$ are composite functions we can also use the chain rule to evaluate these partial derivatives. We find \displaystyle\begin{aligned}0&=\frac{\partial G^i}{\partial t^j}\\&=\sum\limits_{k=1}^n\frac{\partial g^i}{\partial x^k}\frac{\partial H^k}{\partial t^j}\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\sum\limits_{k=m+1}^n\frac{\partial g^i}{\partial x^k}\frac{\partial t^{k-m}}{\partial t^j}\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\sum\limits_{k=1}^{n-m}\frac{\partial g^i}{\partial x^{k+m}}\delta_j^k\\&=\sum\limits_{k=1}^m\frac{\partial g^i}{\partial x^k}\frac{\partial h^k}{\partial t^j}+\frac{\partial g^i}{\partial x^{j+m}}\end{aligned} Similarly, since $F$ has a local minimum (as a function of the $t^j$) at $b$ we must find its partial derivatives zero at that point. That is $\displaystyle0=\frac{\partial F}{\partial t^j}\bigg\vert_{t=b}=\sum\limits_{k=1}^m\frac{\partial f}{\partial x^k}\bigg\vert_{x=H(b)}\frac{\partial h^k}{\partial t^j}\bigg\vert_{t=b}+\frac{\partial f}{\partial x^{j+m}}\bigg\vert_{x=H(b)}$ Now let’s take the previous equation involving $g^i$, evaluate it at $t=b$, multiply it by $\lambda_i$, sum over $i$, and add it to this latest equation. We find $\displaystyle0=\sum\limits_{k=1}^m\left[\frac{\partial f}{\partial x^k}\bigg\vert_{x=H(b)}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^k}\bigg\vert_{x=H(b)}\right]\frac{\partial h^k}{\partial j}\bigg\vert_{t=b}+\frac{\partial f}{\partial x^{j+m}}\bigg\vert_{x=H(b)}+\sum\limits_{i=1}^m\lambda_i\frac{\partial g^i}{\partial x^{j+m}}\bigg\vert_{x=H(b)}$ Now, the expression in brackets is zero because that’s actually how we defined the $\lambda_i$ way back at the start of the proof! And then what remains is exactly the equations we need to complete the proof. November 27, 2009 Posted by \| Analysis, Calculus \| 2 Comments ## Extrema with Constraints I We can consider the problem of maximizing or minimizing a function, as we have been, but insisting that our solution satisfy some constraint. For instance, we might have a function $f:\mathbb{R}^3\rightarrow\mathbb{R}$ to maximize, but we’re only concerned with unit-length vectors on $S^2\subseteq\mathbb{R}^3$. More generally, we’ll be concerned with constraints imposed by setting a function $g$ equal to zero. In the example, we might set $g(x,y,z)=x^2+y^2+z^2-1$. If we want to impose more conditions, we can make $g$ a vector-valued function with as many components as constraint functions we want to set equal to zero. Now, we might be able to parameterize the collection of points satisfying $g(x)=0$. In the example, we could use the usual parameterization of the sphere by latitude and longitude, writing \displaystyle\begin{aligned}x&=\sin(\theta)\cos(\phi)\\y&=\sin(\theta)\sin(\phi)\\z&=\cos(\theta)\end{aligned} where I’ve used the physicists’ convention on the variables instead of the common one in multivariable calculus classes. Then we could plug these expressions for $x$, $y$, and $z$ into our function $f$, and get a composite function of the variables $\phi$ and $\theta$, which we can then attack with the tools from the last couple days, being careful about when we can and can’t trust Cauchy’s invariant rule, since the second differential can transform oddly. Besides even that care that must be taken, it may not even be possible to parameterize the surface, or it may be extremely difficult. At least we do know that such a parameterization will often exist. Indeed, the implicit function theorem tells us that if we have $m$ continuously differentiable constraint functions $g^i$ whose zeroes describe a collection of points in an $n$-dimensional space $\mathbb{R}^n$, and the $m\times m$ determinant $\displaystyle\det\left(\frac{\partial g^i}{\partial x^j}\right)$ is nonzero at some point $a\in\mathbb{R}^n$ satisfying $g(a)=0$, then we can “solve” these equations for the first $m$ variables as functions of the last $n-m$. This gives us exactly such a parameterization, and in principle we could use it. But the calculations get amazingly painful. Instead, we want to think about this problem another way. We want to consider a point $a\in\mathbb{R}^n$ is a point satisfying $g(a)=0$ which has a neighborhood $N$ so that for all $x\in N$ satisfying $g(x)=0$ we have $f(a)\geq f(x)$. This does not say that there are no nearby points to $a$ where $f$ takes on larger values, but it does say that to reach any such point we must leave the region described by $g(x)=0$. Now, let’s think about this sort of heuristically. As we look in various directions $v$ from $a$, some of them are tangent to the region described by $g(x)=0$. These are the directions satisfying $[D_vg](a)=dg(a;v)=0$ — where to first order the value of $g$ is not changing in the direction of $v$. I say that in none of these directions can $f$ change (again, to first order) either. For if it did, either $f$ would increase in that direction or not. If it did, then we could find a path in the region where $g(x)=0$ along which $f$ was increasing, contradicting our assertion that we’d have to leave the region for this to happen. But if $f$ decreased to first order, then it would increase to first order in the opposite direction, and we’d have the same problem. That is, we must have $df(a;v)=0$ whenever $dg(a;v)=0$. And so we find that $\mathrm{Ker}(dg(a))=\bigcap\limits_{i=1}^m\mathrm{Ker}(dg^i(a))\subseteq\mathrm{Ker}(df(a))$ The kernel of $df(a)$ consists of all vectors orthogonal to the gradient vector $\nabla f(a)$, and the line it spans is the orthogonal complement to the kernel. Similarly, the kernel of $dg(a)$ consists of all vectors orthogonal to each of the gradient vectors $\nabla g^i(a)$, and is thus the orthogonal complement to the entire subspace they span. The kernel of $dg(a)$ is contained in the kernel of $df(a)$, and orthogonal complements are order-reversing, which means that $\nabla f(a)$ must lie within the span of the $\nabla g^i(a)$. That is, there must be real numbers $\lambda_i$ so that $\displaystyle\nabla f(a)=\sum\limits_{i=1}^m\lambda_i\nabla g^i(a)$ or, passing back to differentials $\displaystyle df(a)=\sum\limits_{i=1}^m\lambda_idg^i(a)$ So in the presence of constraints we replace the condition $df(a)=0$ by this one. We call the $\lambda_i$ “Lagrange multipliers”, and for every one of these variables we add to the system of equations, we also add the constraint equation $g^i(a)=0$, so we should still get an isolated collection of points. Now, we reached this conclusion by a rather handwavy argument about being able to find increasing directions and so on within the region $g(x)=0$. This line of reasoning could possibly be firmed up, but we’ll find our proof next time in a slightly different approach. November 25, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Classifying Critical Points So let’s say we’ve got a critical point of a multivariable function $f:X\rightarrow\mathbb{R}$. That is, a point $a\in X$ where the differential $df(x)$ vanishes. We want something like the second derivative test that might tell us more about the behavior of the function near that point, and to identify (some) local maxima and minima. We’ll assume here that $f$ is twice continuously differentiable in some region $S$ around $a$. The analogue of the second derivative for multivariable functions is the second differential $d^2f(x)$. This function assigns to every point a bilinear function of two displacement vectors $u$ and $v$, and it measures the rate at which the directional derivative in the direction of $v$ is changing as we move in the direction of $u$. That is, $\displaystyle d^2f(x;u,v)=\left[D_u\left(D_vf\right)\right](x)$ If we choose coordinates on $X$ given by an orthonormal basis $\{e_i\}_{i=1}^n$, we can write the second differential in terms of coordinates $\displaystyle d^2f(x)=\frac{\partial^2f}{\partial x^i\partial x^j}dx^idx^j$ This matrix is often called the “Hessian” of $f$ at the point $x$. As I said above, this is a bilinear form. Further, Clairaut’s theorem tells us that it’s a symmetric form. Then the spectral theorem tells us that we can find an orthonormal basis with respect to which the Hessian is actually diagonal, and the diagonal entries are the eigenvalues of the matrix. So let’s go back and assume we’re working with such a basis. This means that our second partial derivatives are particularly simple. We find that for $i\neq j$ we have $\displaystyle\frac{\partial^2f}{\partial x^i\partial x^j}=0$ and for $i=j$, the second partial derivative is an eigenvalue $\displaystyle\frac{\partial^2f}{{\partial x^i}^2}=\lambda_i$ which we can assume (without loss of generality) are nondecreasing. That is, $\lambda_1\leq\lambda_2\leq\dots\leq\lambda_n$. Now, if all of these eigenvalues are positive at a critical point $a$, then the Hessian is positive-definite. That is, given any direction $v$ we have $d^2f(a;v,v)>0$. On the other hand, if all of the eigenvalues are negative, the Hessian is negative definite; given any direction $v$ we have $d^2f(a;v,v)<0$. In the former case, we’ll find that $f$ has a local minimum in a neighborhood of $a$, and in the latter case we’ll find that $f$ has a local maximum there. If some eigenvalues are negative and others are positive, then the function has a mixed behavior at $a$ we’ll call a “saddle” (sketch the graph of $f(x,y)=xy$ near $(0,0)$ to see why). And if any eigenvalues are zero, all sorts of weird things can happen, though at least if we can find one positive and one negative eigenvalue we know that the critical point can’t be a local extremum. We remember that the determinant of a diagonal matrix is the product of its eigenvalues, so if the determinant of the Hessian is nonzero then either we have a local maximum, we have a local minimum, or we have some form of well-behaved saddle. These behaviors we call “generic” critical points, since if we “wiggle” the function a bit (while maintaining a critical point at $a$) the Hessian determinant will stay nonzero. If the Hessian determinant is zero, wiggling the function a little will make it nonzero, and so this sort of critical point is not generic. This is the sort of unstable situation analogous to a failure of the second derivative test. Unfortunately, the analogy doesn’t extent, in that the sign of the Hessian determinant isn’t instantly meaningful. In two dimensions a positive determinant means both eigenvalues have the same sign — denoting a local maximum or a local minimum — while a negative determinant denotes eigenvalues of different signs — denoting a saddle. This much is included in multivariable calculus courses, although usually without a clear explanation why it works. So, given a direction vector $v$ so that $d^2f(a;v,v)>0$, then since $f$ is in $C^2(S)$, there will be some neighborhood $N$ of $a$ so that $d^2f(x;v,v)>0$ for all $x\in N$. In particular, there will be some range of $t$ so that $b=a+tv\in N$. For any such point we can use Taylor’s theorem with $m=2$ to tell us that $\displaystyle f(b)-f(a)=\frac{1}{2}d^2f(\xi;tv,tv)=\frac{t^2}{2}d^2f(\xi;v,v)$ for some $\xi\in[a,b]\subseteq N$. And from this we see that $f(b)>f(a)$ for every $b\in N$ so that $b-a=tv$. A similar argument shows that if $d^2f(a;v,v)<0$ then $f(b) for any $b$ near $a$ in the direction of $v$. Now if the Hessian is positive-definite then every direction $v$ from $a$ gives us $d^2f(a;v,v)>0$, and so every point $b$ near $a$ satisfies $f(b)>f(a)$. If the Hessian is negative-definite, then every point $b$ near $a$ satisfies $f(b). And if the Hessian has both positive and negative eigenvalues then within any neighborhood we can find some directions in which $f(b)>f(a)$ and some in which $f(b). November 24, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## Local Extrema in Multiple Variables Just like in one variable, we’re interested in local maxima and minima of a function $f:X\rightarrow\mathbb{R}$, where $X$ is an open region in $\mathbb{R}^n$. Again, we say that $f$ has a local minimum at a point $a\in X$ if there is some neighborhood $N$ of $a$ so that $f(a)\leq f(x)$ for all $x\in N$. A maximum is similarly defined, except that we require $f(a)\geq f(x)$ in the neighborhood. As I alluded to recently, we can bring Fermat’s theorem to bear to determine a necessary condition. Specifically, if we have coordinates on $\mathbb{R}^n$ given by a basis $\{e_i\}_{i=1}^n$, we can regard $f$ as a function of the $n$ variables $x^i$. We can fix $n-1$ of these variables $x^i=a^i$ for $i\neq k$ and let $x^k$ vary in a neighborhood of $a^k$. If $f$ has a local extremum at $x=a$, then in particular it has a local extremum along this coordinate line at $x^k=a^k$. And so we can use Fermat’s theorem to draw conclusions about the derivative of this restricted function at $x^k=a^k$, which of course is the partial derivative $\frac{\partial f}{\partial x^k}\big\vert_{x=a}$. So what can we say? For each variable $x^k$, the partial derivative $\frac{\partial f}{\partial x^k}$ either does not exist or is equal to zero at $x=a$. And because the differential subsumes the partial derivatives, if any of them fail to exist the differential must fail to exist as well. On the other hand, if they all exist they’re all zero, and so $df(a)=0$ as well. Incidentally, we can again make the connection to the usual coverage in a multivariable calculus course by remembering that the gradient $\nabla f(a)$ is the vector that corresponds to the linear functional of the differential $df(a)$. So at a local extremum we must have $\nabla f(a)=0$. As was the case with Fermat’s theorem, this provides a necessary, but not a sufficient condition to have a local extremum. Anything that can go wrong in one dimension can be copied here. For instance, we could define $f(x,y)=x^2+y^3$. Then we find $df=2x\,dx+3y^2\,dy$, which is zero at $(0,0)$. But any neighborhood of this point will contain points $(0,t)$ and $(0,-t)$ for small enough $t>0$, and we see that $f(0,t)>f(0,0)>f(0,-t)$, so the origin cannot be a local extremum. But weirder things can happen. We might ask that $f$ have a local minimum at $a$ along any line, like we tried with directional derivatives. But even this can go wrong. If we define $\displaystyle f(x,y)=(y-x^2)(y-3x^2)=y^2-4x^2y+3x^4$ we can calculate $\displaystyle df=\left(-8xy+12x^3\right)dx+\left(2y-4x^2\right)dy$ which again is zero at $(0,0)$. Along any slanted line through the origin $y=kx$ we find \displaystyle\begin{aligned}f(t,kt)&=3t^4-4kt^3+k^2t^2\\\frac{d}{dt}f(t,kt)&=12t^3-12kt^2+2k^2t\\\frac{d^2}{dt^2}f(t,kt)&=36t^2-24kt+2k^2\end{aligned} and so the second derivative is always positive at the origin, except along the $x$-axis. For the vertical line, we find \displaystyle\begin{aligned}f(0,t)&=t^2\\\frac{d}{dt}f(t,kt)&=2t\\\frac{d^2}{dt^2}f(t,kt)&=2\end{aligned} so along all of these lines we have a local minimum at the origin by the second derivative test. And along the $x$-axis, we have $f(x,0)=3x^4$, which has the origin as a local minimum. Unfortunately, it’s still not a local minimum in the plane, since any neighborhood of the origin must contain points of the form $(t,2t^2)$ for small enough $t$. For these points we find $\displaystyle f(t,2t^2)=-t^4<0=f(0,0)$ and so $f$ cannot have a local minimum at the origin. What we’ll do is content ourselves with this analogue and extension of Fermat’s theorem as a necessary condition, and then develop tools that can distinguish the common behaviors near such critical points, analogous to the second derivative test. November 23, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## The Implicit Function Theorem II Okay, today we’re going to prove the implicit function theorem. We’re going to think of our function $f$ as taking an $n$-dimensional vector $x$ and a $m$-dimensional vector $t$ and giving back an $n$-dimensional vector $f(x;t)$. In essence, what we want to do is see how this output vector must change as we change $t$, and then undo that by making a corresponding change in $x$. And to do that, we need to know how changing the output changes $x$, at least in a neighborhood of $f(x;t)=0$. That is, we’ve got to invert a function, and we’ll need to use the inverse function theorem. But we’re not going to apply it directly as the above heuristic suggests. Instead, we’re going to “puff up” the function $f:S\rightarrow\mathbb{R}^n$ into a bigger function $F:S\rightarrow\mathbb{R}^{n+m}$ that will give us some room to maneuver. For $1\leq i\leq n$ we define $\displaystyle F^i(x;t)=f^i(x;t)$ just copying over our original function. Then we continue by defining for $1\leq j\leq m$ $\displaystyle F^{n+j}(x;t)=t^j$ That is, the new $m$ component functions are just the coordinate functions $t^j$. We can easily calculate the Jacobian matrix $\displaystyle dF=\begin{pmatrix}\frac{\partial f^i}{\partial x^j}&\frac{\partial f^i}{\partial t^j}\\{0}&I_m\end{pmatrix}$ where ${0}$ is the $m\times n$ zero matrix and $I_m$ is the $m\times m$ identity matrix. From here it’s straightforward to find the Jacobian determinant $\displaystyle J_F(x;t)=\det\left(dF\right)=\det\left(\frac{\partial f^i}{\partial x^j}\right)$ which is exactly the determinant we assert to be nonzero at $(a;b)$. We also easily see that $F(a;b)=(0;b)$. And so the inverse function theorem tells us that there are neighborhoods $X$ of $(a;b)$ and $Y$ of $(0;b)$ so that $F$ is injective on $X$ and $Y=F(X)$, and that there is a continuously differentiable inverse function $G:Y\rightarrow X$ so that $G(F(x;t))=(x;t)$ for all $(x;t)\in X$. We want to study this inverse function to recover our implicit function from it. First off, we can write $G(y;s)=(v(y;s);w(y;s))$ for two functions: $v$ which takes $n$-dimensional vector values, and $w$ which takes $m$-dimensional vector values. Our inverse relation tells us that \displaystyle\begin{aligned}v(F(x;t))&=x\\w(F(x;t))&=t\end{aligned} But since $F$ is injective from $X$ onto $Y$, we can write any point $(y;s)\in Y$ as $(y;s)=F(x;t)$, and in this case we must have $s=t$ by the definition of $s$. That is, we have \displaystyle\begin{aligned}v(y;t)&=v(F(x;t))=x\\w(y;t)&=w(F(x;t))=t\end{aligned} And so we see that $G(y;t)=(x;t)$, where $x$ is the $n$-dimensional vector so that $y=f(x;t)$. We thus have $f(v(y;t);t)=y$ for every $(y;t)\in Y$. Now define $T\subseteq\mathbb{R}^m$ be the collection of vectors $t$ so that $(0;t)\in Y$, and for each such $t\in T$ define $g(t)=v(0;t)$, so $F(g(t);t)=0$. As a slice of the open set $Y$ in the product topology on $\mathbb{R}^n\times\mathbb{R}^m$, the set $T$ is open in $\mathbb{R}^m$. Further, $g$ is continuously differentiable on $T$ since $G$ is continuously differentiable on $Y$, and the components of $g$ are taken directly from those of $G$. Finally, $b$ is in $T$ since $(a;b)\in X$, and $F(a;b)=(0;b)\in Y$ by assumption. This also shows that $g(b)=a$. The only thing left is to show that $g$ is uniquely defined. But there can only be one such function, by the injectivity of $f$. If there were another such function $h$ then we’d have $f(g(t);t)=f(h(t);t)$, and thus $(g(t);t)=(h(t);t)$, or $g(t)=h(t)$ for every $t\in T$. November 20, 2009 Posted by \| Analysis, Calculus \| 1 Comment ## The Implicit Function Theorem I Let’s consider the function $F(x,y)=x^2+y^2-1$. The collection of points $(x,y)$ so that $F(x,y)=0$ defines a curve in the plane: the unit circle. Unfortunately, this relation is not a function. Neither is $y$ defined as a function of $x$, nor is $x$ defined as a function of $y$ by this curve. However, if we consider a point $(a,b)$ on the curve (that is, with $F(a,b)=0$), then near this point we usually do have a graph of $x$ as a function of $y$ (except for a few isolated points). That is, as we move $y$ near the value $b$ then we have to adjust $x$ to maintain the relation $F(x,y)=0$. There is some function $f(y)$ defined “implicitly” in a neighborhood of $b$ satisfying the relation $F(f(y),y)=0$. We want to generalize this situation. Given a system of $n$ functions of $n+m$ variables $\displaystyle f^i(x;t)=f^i(x^1,\dots,x^n;t^1,\dots,t^m)$ we consider the collection of points $(x;t)$ in $n+m$-dimensional space satisfying $f(x;t)=0$. If this were a linear system, the rank-nullity theorem would tell us that our solution space is (generically) $m$ dimensional. Indeed, we could use Gauss-Jordan elimination to put the system into reduced row echelon form, and (usually) find the resulting matrix starting with an $n\times n$ identity matrix, like $\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}$ This makes finding solutions to the system easy. We put our $n+m$ variables into a column vector and write $\displaystyle\begin{pmatrix}1&0&0&2&1\\{0}&1&0&3&0\\{0}&0&1&-1&1\end{pmatrix}\begin{pmatrix}x^1\\x^2\\x^3\\t^1\\t^2\end{pmatrix}=\begin{pmatrix}x^1+2t^1+t^2\\x^2+3t^1\\x^3-t^1+t^2\end{pmatrix}=\begin{pmatrix}0\\{0}\\{0}\end{pmatrix}$ and from this we find \displaystyle\begin{aligned}x^1&=-2t^1-t^2\\x^2&=-3t^1\\x^3&=t^1-t^2\end{aligned} Thus we can use the $m$ variables $t^j$ as parameters on the space of solutions, and define each of the $x^i$ as a function of the $t^j$. But in general we don’t have a linear system. Still, we want to know some circumstances under which we can do something similar and write each of the $x^i$ as a function of the other variables $t^j$, at least near some known point $(a;b)$. The key observation is that we can perform the Gauss-Jordan elimination above and get a matrix with rank $n$ if and only if the leading $n\times n$ matrix is invertible. And this is generalized to asking that some Jacobian determinant of our system of functions is nonzero. Specifically, let’s assume that all of the $f^i$ are continuously differentiable on some region $S$ in $n+m$-dimensional space, and that $(a;b)$ is some point in $S$ where $f(a;b)=0$, and at which the determinant $\displaystyle\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{(a;t)}\right)\neq0$ where both indices $i$ and $j$ run from $1$ to $n$ to make a square matrix. Then I assert that there is some $k$-dimensional neighborhood $T$ of $b$ and a uniquely defined, continuously differentiable, vector-valued function $g:T\rightarrow\mathbb{R}^n$ so that $g(b)=a$ and $f(g(t);t)=0$. That is, near $(a;b)$ we can use the variables $t^j$ as parameters on the space of solutions to our system of equations. Near this point, the solution set looks like the graph of the function $x=g(t)$, which is implicitly defined by the need to stay on the solution set as we vary $t$. This is the implicit function theorem, and we will prove it next time. November 19, 2009 Posted by \| Analysis, Calculus \| 4 Comments ## The Inverse Function Theorem At last we come to the theorem that I promised. Let $f:S\rightarrow\mathbb{R}^n$ be continuously differentiable on an open region $S\subseteq\mathbb{R}^n$, and $T=f(S)$. If the Jacobian determinant $J_f(a)\neq0$ at some point $a\in S$, then there is a uniquely determined function $g$ and two open sets $X\subseteq S$ and $Y\subseteq T$ so that • $a\in X$, and $f(a)\in Y$ • $Y=f(X)$ • $f$ is injective on $X$ • $g$ is defined on $Y$, $g(Y)=X$, and $g(f(x))=x$ for all $x\in X$ • $g$ is continuously differentiable on $Y$ The Jacobian determinant $J_f(x)$ is continuous as a function of $x$, so there is some neighborhood $N_1$ of $a$ so that the Jacobian is nonzero within $N_1$. Our second lemma tells us that there is a smaller neighborhood $N\subseteq N_1$ on which $f$ is injective. We pick some closed ball $\overline{K}\subseteq N$ centered at $a$, and use our first lemma to find that $f(K)$ must contain an open neighborhood $Y$ of $f(a)$. Then we define $X=f^{-1}(Y)\cap K$, which is open since both $K$ and $f^{-1}(Y)$ are (the latter by the continuity of $f$). Since $f$ is injective on the compact set $\overline{K}\subseteq N$, it has a uniquely-defined continuous inverse $g$ on $Y\subseteq f(\overline{K})$. This establishes the first four of the conditions of the theorem. Now the hard part is showing that $g$ is continuously differentiable on $Y$. To this end, like we did in our second lemma, we define the function $\displaystyle h(z_1,\dots,z_n)=\det\left(\frac{\partial f^i}{\partial x^j}\bigg\vert_{x=z_i}\right)$ along with a neighborhood $N_2$ of $a$ so that as long as all the $z_i$ are within $N_2$ this function is nonzero. Without loss of generality we can go back and choose our earlier neighborhood $N$ so that $N\subseteq N_2$, and thus that $\overline{K}\subseteq N_2$. To show that the partial derivative $\frac{\partial g^i}{\partial y^j}$ exists at a point $y\in Y$, we consider the difference quotient $\displaystyle\frac{g^i(y+\lambda e_j)-g^i(y)}{\lambda}$ with $y+\lambda e_j$ also in $Y$ for sufficiently small $\lvert\lambda\rvert$. Then writing $x_1=g(y)$ and $x_2=g(y+\lambda e_j)$ we find $f(x_2)-f(x_1)=\lambda e_j$. The mean value theorem then tells us that \displaystyle\begin{aligned}\delta_j^k&=\frac{f^k(x_2)-f^k(x_1)}{\lambda}\\&=df^k(\xi_k)\left(\frac{1}{\lambda}(x_2-x_1)\right)\\&=\frac{\partial f^k}{\partial x^i}\bigg\vert_{x=\xi_k}\frac{x_2^i-x_1^i}{\lambda}\\&=\frac{\partial f^k}{\partial x^i}\bigg\vert_{x=\xi_k}\frac{g^i(y+\lambda e_j)-g^i(y)}{\lambda}\end{aligned} for some $\xi_k\in[x_1,x_2]\subseteq K$ (no summation on $k$). As usual, $\delta_j^k$ is the Kronecker delta. This is a linear system of equations, which has a unique solution since the determinant of its matrix is $h(\xi_1,\dots,\xi_n)\neq0$. We use Cramer’s rule to solve it, and get an expression for our difference quotient as a quotient of two determinants. This is why we want the form of the solution given by Cramer’s rule, and not by a more computationally-efficient method like Gaussian elimination. As $\lambda$ approaches zero, continuity of $g$ tells us that $x_2$ approaches $x_1$, and thus so do all of the $\xi_k$. Therefore the determinant in the denominator of Cramer’s rule is in the limit $h(x,\dots,x)=J_f(x)\neq0$, and thus limits of the solutions given by Cramer’s rule actually do exist. This establishes that the partial derivative $\frac{\partial g^i}{\partial y^j}$ exists at each $y\in Y$. Further, since we found the limit of the difference quotient by Cramer’s rule, we have an expression given by the quotient of two determinants, each of which only involves the partial derivatives of $f$, which are themselves all continuous. Therefore the partial derivatives of $g$ not only exist but are in fact continuous. 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http://openstudy.com/updates/50da51c7e4b069916c85617a	Here's the question you clicked on: 55 members online • 0 viewing ## anonymous 3 years ago How can I simplify this: $\left(P\wedge\neg R\right)\vee\left(P\wedge\neg Q\right)\vee\left(Q\wedge\neg R\right)$ Delete Cancel Submit • This Question is Closed 1. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I've been working on this but it seem I can't do anymore. =/ 2. anonymous • 3 years ago Best Response You've already chosen the best response. 0 The original expression is this: $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$ 3. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I see that it is like prove the transitivy propertie for the implication operator. I made a truth table for this and it turned out to be false, but I want to get to the same conclussion without using truth tables. 4. anonymous • 3 years ago Best Response You've already chosen the best response. 0 This is the work I've done so far. ##### 1 Attachment 5. abb0t • 3 years ago Best Response You've already chosen the best response. 0 Well, tautology is a proposition that is always true. That's all I know. That's all I know. 6. anonymous • 3 years ago Best Response You've already chosen the best response. 0 good reasoning @No-data 7. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Thank you @Rohangrr 8. anonymous • 3 years ago Best Response You've already chosen the best response. 0 @brinethery Do you have any idea? 9. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Are you sure it's true? It looks like the converse is true. 10. anonymous • 3 years ago Best Response You've already chosen the best response. 0 $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$Suppose $$P \gets T\quad Q\gets F\quad R\gets T$$ $(T\implies T)\implies [(T\implies F)\wedge (F \implies T)] \\ T\implies (F\wedge T) \\ T\implies F \\ F$ 11. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I feel like q is irrelvent and its just If P then R 12. anonymous • 3 years ago Best Response You've already chosen the best response. 0 $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$I think it should be $\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right] \Rightarrow \left(P\Rightarrow R\right)$ 13. anonymous • 3 years ago Best Response You've already chosen the best response. 0 ^^ makes alot of sense 14. anonymous • 3 years ago Best Response You've already chosen the best response. 0 There is nothing wrong with using a counter example to show something is false. Trying to prove it without a counter example seems pointless to me. 15. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I did not say it was true 16. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Mmm I don't understand the expression after "Suppose" what those arrows mean? 17. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Ok I got the counter example, is a row on my truth table and its correct. Thank you @wio 18. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I was just trying to show that the expression is not a tautology. 19. anonymous • 3 years ago Best Response You've already chosen the best response. 0 The expression you wrote saying you thought it should be is, in fact, a tautology. 20. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Imagine if I have you the proposition $$P$$ and claimed it was a tautology. How would you prove it is wrong? 21. anonymous • 3 years ago Best Response You've already chosen the best response. 0 There is no boolean algebra to prove it, you just give a counter example, right? 22. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Do you think that is not possible to find a way to show this using only theorems and definitions? 23. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Which theorems? 24. anonymous • 3 years ago Best Response You've already chosen the best response. 0 How would you show that $$P$$ in and of itself is not a tautology? 25. anonymous • 3 years ago Best Response You've already chosen the best response. 0 There is no way to know. Unless you tell me that P can take the values T and F. 26. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I mean, I have nothing against trying to do things a particular way, but what are the rules really? How would you show $$P$$ is not a tautology without a counter example. 27. anonymous • 3 years ago Best Response You've already chosen the best response. 0 and then, by definition of P. P is not a tautology 28. anonymous • 3 years ago Best Response You've already chosen the best response. 0 So you'd say by definition of proposition, a proposition is not a tautology? Then how about $$P\wedge Q$$? 29. anonymous • 3 years ago Best Response You've already chosen the best response. 0 You meant that proposition is not a tautology. 30. anonymous • 3 years ago Best Response You've already chosen the best response. 0 As far as I know a tautology is a predicate that is always true for any value of their parameters. 31. anonymous • 3 years ago Best Response You've already chosen the best response. 0 $P\wedge Q$ is a predicate that can be true or false. So it's not a tautology. 32. anonymous • 3 years ago Best Response You've already chosen the best response. 0 33. anonymous • 3 years ago Best Response You've already chosen the best response. 0 You're claiming it can be false... The point is to get you to prove it without counterexample. 34. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I'm just curious as to the rules of the game. 35. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I think I'm not using a counter example and that is a valid proof. 36. anonymous • 3 years ago Best Response You've already chosen the best response. 0 A tautology is a restatement within the same premiss like all trees are made of wood 37. anonymous • 3 years ago Best Response You've already chosen the best response. 0 What if I say "is a predicate that can be true or false.", about the original predicate? Said it about $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$Would simply saying that be a valid proof that it is not a tautology? 38. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I don't think saying "is a predicate that can be true or false." is valid without a counter example. 39. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Or maybe there is another way, but I'm not sure 40. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Is not evident that that predicate can be true or false. Its not a valid proof. 41. anonymous • 3 years ago Best Response You've already chosen the best response. 0 and $P\wedge Q$is by definition true or false. 42. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Sure but then what makes something 'evident'? 43. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Is there some standard form you want it to be in? 44. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I wanted to say that there is a definition by saying "evident" 45. anonymous • 3 years ago Best Response You've already chosen the best response. 0 What about $$P\wedge Q \wedge R$$? 46. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I accepted you method buy I feel like you're trying to say that there is no other method to prove this problem. 47. anonymous • 3 years ago Best Response You've already chosen the best response. 0 What I am saying is that if there is another way to prove this problem, then there needs to be a way to prove very simple predicates. 48. anonymous • 3 years ago Best Response You've already chosen the best response. 0 let $P\wedge Q\Leftrightarrow S$$S\wedge R$ 49. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I mean prove without counter example 50. anonymous • 3 years ago Best Response You've already chosen the best response. 0 so it's not a tautology. 51. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I know that, I already proved it using other method. I just wanted to know if it was possible to do it the way I'm trying. 52. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Using your logic that $$\wedge$$ doesn't give a tautology, then couldn't we say $$\neg P\vee P$$ isn't a tautology? 53. anonymous • 3 years ago Best Response You've already chosen the best response. 0 ITs a Syllogism if the expanded form is the antecedent, but because its the consequent it is not tautological 54. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Well it's just an interesting thing to think about. I would like to find a way to prove false without counterexample but it seems hard to understand what is allowed. 55. anonymous • 3 years ago Best Response You've already chosen the best response. 0 q's relation to p and r 56. anonymous • 3 years ago Best Response You've already chosen the best response. 0 This is not a tautology:$P\wedge Q$, not just the $\wedge$ symbol. 57. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Sorry I did not understand that @Edutopia 58. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I'm just beginning with this logic things lol. 59. anonymous • 3 years ago Best Response You've already chosen the best response. 0 my comp is sllooow, i ment to say: q's relation to p and r would have to be in the premiss for it to be tautological 60. anonymous • 3 years ago Best Response You've already chosen the best response. 0 ahh that is right Edutopia. 61. anonymous • 3 years ago Best Response You've already chosen the best response. 0 You know what's interesting about proving something is true, is that you are showing it is true for all cases. If you are proving something is false, you only need to show one case is false. To prove that it is false for all cases can't always be done and doesn't need to be done. I think to even say $$P$$ is not a tautology, you are implicitly calling upon the counter example just to claim "$$P$$ can be false". 62. anonymous • 3 years ago Best Response You've already chosen the best response. 0 But maybe that is a semantic argument? It's just my thought. 63. anonymous • 3 years ago Best Response You've already chosen the best response. 0 in IF (IF P THEN R) THEN BOTH (IF P THEN Q) AND ( IF Q THEN R) the relation of Q to P and R is not established, imagine the argument where P and R have to do with Physics and Q is someones opinion on abortion. 64. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Yeah it's interesting @wio and I agree your counterexamples are a good tool. I just think they are not elegant. 65. anonymous • 3 years ago Best Response You've already chosen the best response. 0 When I first saw $\left(P\Rightarrow R\right)\Rightarrow\left[\left(P\Rightarrow Q\right)\wedge\left(Q\Rightarrow R\right)\right]$The $$Q$$ made me think it's likely not a tautology right away. Since it wasn't something inconsequential like $$Q\vee \neg Q$$. 66. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I'm not sure, but I think it's valid in math.If$x=y$then$x+a=y+a$ 67. anonymous • 3 years ago Best Response You've already chosen the best response. 0 That is the addition property of equality. $$Q$$ was not being appended to both sides of the implication. 68. anonymous • 3 years ago Best Response You've already chosen the best response. 0 yes, but in math its all a valid argument because you are using numbers 69. anonymous • 3 years ago Best Response You've already chosen the best response. 0 I know it just look similar. Thank you guys. It is nice to be able to discuss this kind of things with other people. Thank you for you r time. 70. anonymous • 3 years ago Best Response You've already chosen the best response. 0 a=a and so forth 71. anonymous • 3 years ago Best Response You've already chosen the best response. 0 @No-data I have no idea :-). I'm so stupid. 72. anonymous • 3 years ago Best Response You've already chosen the best response. 0 Is this for linear algebra? 73. anonymous • 3 years ago Best Response You've already chosen the best response. 0 It is propositional logic. You're not stupid at all @brinethery. I was curious about how databases work so I picked a book about math applied to databases and this is a problem from the first chapter. Logic and Set Theory are the foundation of DB systems. 74. Not the answer you are looking for? Search for more explanations. • Attachments: Find more explanations on OpenStudy ##### spraguer (Moderator) 5→ View Detailed Profile 23 • Teamwork 19 Teammate • Problem Solving 19 Hero • You have blocked this person. • ✔ You're a fan Checking fan status... Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998667240142822, "perplexity": 4090.37302610115}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738660342.42/warc/CC-MAIN-20160924173740-00141-ip-10-143-35-109.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/77786-arc-length.html	1. ## Arc Length Consider the path r(t) = (10t, 5t^2, 5lnt) defined for t > 0. Find the length of the curve between the points (10, 5, 0) and (20, 20, 5 ln(2)). Thanks 2. Originally Posted by jffyx Consider the path r(t) = (10t, 5t^2, 5lnt) defined for t > 0. Find the length of the curve between the points (10, 5, 0) and (20, 20, 5 ln(2)). Thanks $L = \int_{t=1}^{t=2} \sqrt{100 + 100t^2 + \frac{25}{t^2}} \, dt = \int_{t=1}^{t=2} \frac{1}{t} \sqrt{(10t^2 + 5)^2} \, dt = \, ....$ 3. How did you determine the interval of integration? 4. the path r(t) = (10t, 5t^2, 5lnt) defined for t > 0. Find the length of the curve between the points (10, 5, 0) and (20, 20, 5 ln(2)). What value of t makes (10t, 5t^2, 5ln t) equal to (10, 5, 0)? What value of t makes (10t, 5t^2, 5 ln t) equal to (20, 20, 5 ln 2)?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9579236507415771, "perplexity": 1812.4960611003132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661778.22/warc/CC-MAIN-20160924173741-00284-ip-10-143-35-109.ec2.internal.warc.gz"}
https://asone.ai/polymath/index.php?title=Finding_primes&diff=prev&oldid=2067	# Difference between revisions of "Finding primes" This is the main blog page for the "Deterministic way to find primes" project, which will be started in within a few weeks. The main aim of the project is as follows: Problem. Find a deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in k. You may assume as many standard conjectures in number theory (e.g. the generalised Riemann hypothesis) as necessary, but avoid powerful conjectures in complexity theory (e.g. P=BPP) if possible. Here is the proposal for the project, which is also the de facto research thread for that project. Here is the discussion thread for the project. Please add and develop this wiki; in order to maximise participation in the project when it launches, we will need as much expository material on this project as we can manage. ## Partial results 1. [There is a deterministic algorithm to find primes if P=NP] 2. Problem is true if Cramer's conjecture holds (explain!) 3. Problem is true if strong pseudorandom number generators exist (explain!) 4. k-digit primes can be found with high probability using O(k) random bits (explain!) ## Easier versions of the problem 1. Subexponential time rather than polynomial 1. Equivalently, find primes with superlogarithmically many digits ($\gg \log k$) in poly(k) time 2. Find k-digit primes using o(k) random bits 3. Assume complexity conjectures such as P=BPP, P=BQP, etc. ## Relevant concepts 1. Complexity classes 2. Pseudo-random number generators (PRG) 3. Expander graphs 4. Cramer's random model for the primes 5. Prime gaps ## Relevant conjectures 1. P=NP 2. P=BPP 3. P=promise-BPP 4. P=BQP 5. existence of PRG 6. existence of one-way functions 7. whether DTIME(2^n) has subexponential circuits 8. GRH 9. the Hardy-Littlewood prime tuples conjecture 10. the ABC conjecture 11. Cramer’s conjecture	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8525272011756897, "perplexity": 3563.9971975700337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401643509.96/warc/CC-MAIN-20200929123413-20200929153413-00043.warc.gz"}
http://math.stackexchange.com/questions/119860/what-operations-are-preserved-by-the-canonical-map-to-quotient-rings	What operations are preserved by the canonical map to quotient rings Let $R$ be a commutative ring and $I,J,K$ be ideals such that $I\subseteq J$ and $I\subseteq K$. Let $\pi: R \to R/I$ be the canonical map. I am able to prove $\pi$ preserves sum and products. Unsure about intersections. $J/I + K/I = (J+K)/I$ $J/I \cdot K/I = (J\cdot K)/I$ $J/I \cap K/I = (J\cap K)/I$? - Remember that the quotient map induces a lattice isomorphism between the ideals of $R/I$ and the ideals of $R$ that contain $I$; in particular, since $J\cap K$ is the largest ideal of $R$ that contains $I$ and is contained in both $J$ and $K$, then it follows that $(J\cap K)/I$ is the largest ideal of $R/I$ that is contained in both $J/I$ and $K/I$, hence must equal $(J/I) \cap (K/I)$. Though I think that is by far the best way to go, to prove it explicitly, note that if $x\in J\cap K$, then $x+I\in J/I$ and $x+I\in K/I$, so you get $(J\cap K)/I\subseteq (J/I)\cap(K/I)$. For the converse inclusion, let $a+I\in (J/I)\cap(K/I)$. Then there exists $j\in J$ and $k\in K$ with $j+I = k+I = a+I$. In particular, $j-k\in I$. Therefore, $k=j-(j-k)\in J$, because $j\in J$ and $j-k\in I\subseteq J$, hence $k\in J\cap K$, thus, $a+I = k+I\in (J\cap K)/I$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868558645248413, "perplexity": 30.56340287140894}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645366585.94/warc/CC-MAIN-20150827031606-00316-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.ti.inf.ethz.ch/ew/mise/mittagssem.html?action=show&what=abstract&id=c897612a7d5645a4c61b93dc3b1659089d72bc11	## Theory of Combinatorial Algorithms Prof. Emo Welzl and Prof. Bernd Gärtner # Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov) Mittagsseminar Talk Information Date and Time: Tuesday, December 09, 2014, 12:15 pm Duration: 30 minutes Location: CAB G51 Speaker: Roman Glebov ## A probem of Erdos and Sos on 3-graphs We show that for every epsilon > 0 there exist delta > 0 and natural number n_0 such that every 3-uniform hypergraph on n >= n_0 vertices with the property that every k-vertex subset, where k is at least deltan, induces at least (1/4 + epsilon)(k choose 3) edges, contains K4- as a subgraph, where K4- is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible. This is a joint work with Dan Kral and Jan Volec. Previous talks by year: 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 Information for students and suggested topics for student talks	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8073939681053162, "perplexity": 1626.0547131222988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864354.27/warc/CC-MAIN-20180622045658-20180622065658-00488.warc.gz"}
http://zbmath.org/?q=an:1185.62147	# zbMATH — the first resource for mathematics ##### Examples Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev \| Tschebyscheff The or-operator \| allows to search for Chebyshev or Tschebyscheff. "Quasi* map" py: 1989 The resulting documents have publication year 1989. so: Eur J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A \| 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used. ##### Operators a & b logic and a \| b logic or !ab logic not abc right wildcard "ab c" phrase (ab c) parentheses ##### Fields any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article) Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes. (English) Zbl 1185.62147 Summary: First we consider a process ${\left({X}_{t}^{\left(\alpha \right)}\right)}_{t\in \left[0,T\right)}$ given by a SDE $d{X}_{t}^{\left(\alpha \right)}=\alpha b\left(t\right){X}_{t}^{\left(\alpha \right)}\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{t},\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right),$ with parameter $\alpha \in ℝ$, where $T\in \left(0,\infty \right]$ and ${\left(Bt\right)}_{t\in \left[0,T\right)}$ is a standard Wiener process. We study the asymptotic behavior of the MLE ${\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}$ of $\alpha$ based on the observation ${\left({X}_{s}^{\left(\alpha \right)}\right)}_{s\in \left[0,T\right]}$ as $t↑T$. We formulate sufficient conditions under which $\sqrt{{I}_{{X}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}-\alpha \right)$ converges to the distribution of $c{\int }_{0}^{1}{W}_{s}\phantom{\rule{0.166667em}{0ex}}d{W}_{s}/{\int }_{0}^{1}{\left({W}_{s}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}ds$, where ${I}_{{X}^{\left(\alpha \right)}}\left(t\right)$ denotes the Fisher information for $\alpha$ contained in the sample ${\left({X}_{s}^{\left(\alpha \right)}\right)}_{s\in \left[0,t\right]},{\left({W}_{s}\right)}_{s\in \left[0,1\right]}$ is a standard Wiener process, and $c=1/\sqrt{2}$ or $c=-1/\sqrt{2}$. We also weaken the sufficient conditions due to H. Luschgy [Probab. Theory Relat. Fields 92, No. 2, 151–176 (1992; Zbl 0768.62067), Section 4.2)] under which $\sqrt{{I}_{{X}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}-\alpha \right)$ converges to a Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of $\alpha$ is asymptotically normal with some appropriate random normalizing factor. Next we study a SDE $d{Y}_{t}^{\left(\alpha \right)}=\alpha b\left(t\right)a\left({Y}_{t}^{\left(\alpha \right)}\right)\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{t},\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right),$ with a perturbed drift satisfying $a\left(x\right)=x+O\left(1+\|x{\|}^{\gamma }\right)$ with some $\gamma \in \left[0,1\right)$. We give again sufficient conditions under which $\sqrt{{I}_{{Y}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({Y}^{\left(\alpha \right)}\right)}-\alpha \right)$ converges to the distribution of $c{\int }_{0}^{1}{W}_{s}\phantom{\rule{0.166667em}{0ex}}{W}_{s}/{\int }_{0}^{1}{\left({W}_{s}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}ds$. We emphasize that our results are valid in both cases $T\in \left(0,\infty \right)$ and $T=\infty$, and we develop a unified approach to handle these cases. ##### MSC: 62M05 Markov processes: estimation 62F12 Asymptotic properties of parametric estimators 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 25, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8882200121879578, "perplexity": 2560.5276450485962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011042531/warc/CC-MAIN-20140305091722-00057-ip-10-183-142-35.ec2.internal.warc.gz"}
https://hal.inria.fr/hal-00789444	Assessing the Impact and Limits of Steady-State Scheduling for Mixed Task and Data Parallelism on Heterogeneous Platforms 2 SCALAPPLIX - Algorithms and high performance computing for grand challenge applications INRIA Futurs, Université Bordeaux Segalen - Bordeaux 2, Université Sciences et Technologies - Bordeaux 1, École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), CNRS - Centre National de la Recherche Scientifique : UMR5800 3 GRAAL - Algorithms and Scheduling for Distributed Heterogeneous Platforms Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme 5 REMAP - Regularity and massive parallel computing Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme Abstract : In this paper, we consider steady-state scheduling techniques for mapping a collection of task graphs onto heterogeneous systems, such as clusters and grids. We advocate the use of steady-state scheduling to solve this difficult problem. Due to space limitations, we concentrate on complexity results. We show that the problem of optimizing the steady-state throughput is NP-complete in the general case. We formulate a compact version of the problem that belongs to the NP complexity class but which does not restrict the optimality of the solution. We provide many positive results in the extended version (Beaumont et al., 2004). Indeed, we show how to determine in polynomial time the best steady-state scheduling strategy for a large class of application graphs and for an arbitrary platform graphs, using a linear programming approach. Type de document : Communication dans un congrès HeteroPar\'2004: International Conference on Heterogeneous Computing, Jointly Published with ISPDC\'2004: International Symposium on Parallel and Distributed Computing, 2004, Unknown, IEEE Computer Society Press, pp.296―302, 2004, <10.1109/ISPDC.2004.12> Liste complète des métadonnées https://hal.inria.fr/hal-00789444 Contributeur : Arnaud Legrand <> Soumis le : lundi 18 février 2013 - 11:51:35 Dernière modification le : vendredi 11 septembre 2015 - 01:06:01 Citation Olivier Beaumont, Arnaud Legrand, Loris Marchal, Yves Robert. Assessing the Impact and Limits of Steady-State Scheduling for Mixed Task and Data Parallelism on Heterogeneous Platforms. HeteroPar\'2004: International Conference on Heterogeneous Computing, Jointly Published with ISPDC\'2004: International Symposium on Parallel and Distributed Computing, 2004, Unknown, IEEE Computer Society Press, pp.296―302, 2004, <10.1109/ISPDC.2004.12>. <hal-00789444> Métriques Consultations de la notice	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8192971348762512, "perplexity": 3947.8447494145867}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320887.15/warc/CC-MAIN-20170627013832-20170627033832-00485.warc.gz"}
https://www.physicsforums.com/threads/integrating-factor-of-a-1-ydx-b-1-xdy-0.92048/	# Homework Help: Integrating factor of (a+1)ydx + (b+1)xdy = 0 1. Oct 3, 2005 ### asdf1 for the equation, (a+1)ydx+(b+1)xdy=0, i am wondering how to get (x^a)(y^b) as an integrating factor~ the following is my work: (1/F)(dF/dx)=(a-b)/[(b+1)x] => F=cx^[(a-b)/(b+1)] why doesn't that method work? 2. Oct 3, 2005 ### HallsofIvy It's not clear to me what you are doing. What happened to y? The simplest way to get the integrating factor is just to try an integrating factor of the form xαyβ. The equation becomes $$((a+1)x^{\alpha}y^{\beta+1})dx+ ((b+1)x^{\alpha+1}y^{\beta})dy$$ In order for that to be exact, we must have $$((a+1)x^{\alpha}y^{\beta+1})_y= ((b+1)x^{\alpha+1}y^{\beta})_x$$ or $$(a+1)(\beta+1)x^{\alpha}y^{\beta}= (b+1)(\alpha+1)xx^{\alpha}y^{\beta}$$ That will clearly be true if α= a and β= b. Therefore xayb is an integrating factor. 3. Oct 3, 2005 ### asdf1 wow! how'd you think of trying the integrating factor of the form (x^α)(y^β)? 4. Oct 4, 2005 ### saltydog Well, shouldn't that be: $$(a+1)(\beta+1)x^{\alpha}y^{\beta}= (b+1)(\alpha+1)x^{\alpha}y^{\beta}$$ Just want to be precise that's all. 5. Oct 5, 2005 ### asdf1 hmmm... i looked up an edition of advanced engineering mathematics, and i saw a little description of that kind of substition, but it didn't explain why... :P	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9135774374008179, "perplexity": 3162.6853093549253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156311.20/warc/CC-MAIN-20180919220117-20180920000117-00040.warc.gz"}
https://charlieduclut.github.io/publication/2021-11-19-duclut2021active	# Active T1 transitions in cellular networks Preprint, 2021 ### Abstract: In amorphous solids as in tissues, neighbour exchanges can relax local stresses and allow the material to flow. In this paper, we use an anisotropic vertex model to study T1 rearrangements in polygonal cellular networks. We consider two different physical realization of the active anisotropic stresses: (i) anisotropic bond tension and (ii) anisotropic cell stress. Interestingly, the two types of active stress lead to patterns of oriented T1 transitions that are different. We describe and explain these observations through the lens of a continuum description of the tissue as an anisotropic active material. We furthermore discuss the energetics of the tissue and express the energy balance in terms of internal elastic energy, mechanical work, chemical work and heat. This allows us to define active T1 transitions that can perform mechanical work while consuming chemical energy. Recommended citation: "Active T1 transitions in cellular networks", C. Duclut, J. Paijmans, M. M. Inamdar, C. D. Modes, and F. Jülicher, arXiv:2111.10327 (2021). https://arxiv.org/abs/2111.10327	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8388981819152832, "perplexity": 2425.793726708392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00560.warc.gz"}
https://cyril-imbert.blog/2019/06/14/new-preprint-partial-regularity-in-time-for-the-landau-coulomb-equation/	## New preprint: partial regularity in time for the Landau-Coulomb equation In collaboration with F. Golse, M. P. Gualdani and A. F. Vasseur. It can be found in the hal and also in arxiv. Abstract: We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2. Publicités	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9035508632659912, "perplexity": 890.9051421197425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027317037.24/warc/CC-MAIN-20190822084513-20190822110513-00176.warc.gz"}
http://mathoverflow.net/questions/22309/is-a-contraction-space-always-complete/22314	## Is a “contraction space” always complete? Some of the fundamental results in analysis (inverse function theorem, existence and uniqueness of solutions to ODEs) have slick proofs using the idea of a contraction. So, it seems plausible to me that one might be motivated to study a "contraction space": I'll define a contraction space as a metric space $(X,d)$ such that there is at least one fixed point of any function $f:X\rightarrow X$ with the property that for all $x,y$ we have $d(f(x),f(y)) \le \frac12 d(x,y)$. Here's the question: is every contraction space complete? I feel like the answer should be yes. The only approach I can see for proving this, though, is to find some way to take a Cauchy sequence $(a_n)$ and construct a contraction on the whole space taking each $a_i$ to some $a_j$ with $j > i$, but even for the case of Cauchy sequences in the rationals I don't see an obvious approach. A related question: suppose we take an arbitrary metric space, and add a fixed point for every contraction the same way we add limits to Cauchy sequences. Is the resulting space then a contraction space? - There is a "contraction space" which is not complete. For example, consider a metric $d$ on $[1,+\infty)$ such that for $x,y\in[n,n+1]$ where $n\in\mathbb N$ one has $d(x,y)=2^{-n}\|x-y\|^{1/n}$ (other distances are defined by gluing the segments together). The completion is obtained by adding one point at $+\infty$. But there is no map contracting this space to $+\infty$. Indeed, any Lipschitz map $f$ with $f(x)>n+1$ for some $x\in[n,n+1]$ must be a constant on $[n,n+1]$, so there will be a finite fixed point. Added. Here is a counter-example to the second question. Let $X=(\mathbb R,d)$ from the above example. Define $Y=(\mathbb R,d')$ similarly by setting $d'(x,y)=2^{-n}\|x-y\|$ for $x,y\in[n,n+1]$. Then $Y$ is isometric to $[0,1)$. Consider the disjoint union of $X$ and $Y$. For $t\in [0,+\infty)$, denote by $t_X$ and $t_Y$ the copies of $t$ in $X$ and $Y$. For every $t$, attach an arc $\gamma_t$ of length $\ell_t:=10\cdot 2^{-t}$ connecting $t_X$ to $t_Y$. Let $Z$ denote the union of $X$, $Y$ and all these arcs. We have yet defined distances on $X$, on $Y$ and on every arc $\gamma_t$. The metric on $Z$ is defined as the maximal metric bounded above by these metrics on these subsets. It is easy to see that $X$ and $Y$ are embedded into $Z$ isometrically, as well as sufficiently short intervals of the arcs $\gamma_t$ (e.g. their halves are sufficiently short). The counter-example is the space $Z\setminus Y$. Its completion is $Z\cup\{+\infty\}$. The space cannot be contracted to $+\infty$ for the same reasons as above: the arcs $\gamma_t$ added to $X$ form a tree and don't help to go around weird parts of $X$. On the other hand, $Z\setminus Y$ can easily be contracted to any point of $Y$ because $\gamma_t$ nearby $t_Y$ is isometric to a straight line segment $(0,\epsilon_t)$. Just map every point of $Z\setminus X$ to a point in this segment in such a way that the distance to $t_Y$ gets multiplied by a small positive constant (like $\epsilon_t/100$). Thus adding "contraction points" to $Z\setminus Y$ yields $Z$. But $Z$ can be contracted to $+\infty$ - just project everything to $Y$ and contract there. - It seems that a circle without a point is also a counter-example to the first question. – Petya Apr 23 2010 at 19:38 No, fold it twice and contract the half-circle. – Sergei Ivanov Apr 23 2010 at 19:43 @Petya In fact, a (planar) Cook continuum without a point would be such a counterexample. – Ady Apr 23 2010 at 22:29 I think http://www.renyi.hu/~emarci/fix2.ps should be helpful. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9811328649520874, "perplexity": 101.27253843251715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703489876/warc/CC-MAIN-20130516112449-00063-ip-10-60-113-184.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/find-the-net-torque-acting-on-the-system.254561/	# Find the net torque acting on the system 1. Sep 8, 2008 ### kehler 1. The problem statement, all variables and given/known data In the figure below, the incline is frictionless and the string passes through the centre of mass of each block. The pulley has moment of inertia I and radius R. a) Find the net torque acting on the system (the two masses, string and pulley) about the centre of the pulley b) Write an expression for the total angular momentum of the system about the center of the pulley when the masses are moving with a speed v c) Find the acceleration of the masses from your results for a and b by setting the net torque equal to the rate of change of the angular momentum of the system 2. Relevant equations torque = r x F angular momentum = r x p or Iw 3. The attempt at a solution a) I got an answer for this but I dunno if its right F on m2 in direction parallel to incline = m2 x g sin theta F on m1 = m1 x g So torque about centre of pulley due to force on m2 is R x m2 x g sin theta, out of the page and torque about centre of pulley due to force on m1 is R x m x g, into the page Net torque is (R x m2 x g sin theta) - (R x m x g), out of the page b) I'm not sure how to do this. Do I calculate the angular momentum of each of the masses about the centre of the pulley and add it with the angular momentum of the pulley? But if this is the case, I don't know what the angular velocity of the pulley is :S Angular momentum of m1 about centre = R x m1 v, into the page Angular momentum of m2 about centre = R x m2 v, into the page Angular momentum of pulley about it's centre = Iw = 0.5MR^2 w but we don't know w c)No idea. Do I just set the answer for a to be equal to the differential of the answer to b??? Once again, any help would be much appreciated :) Last edited: Sep 8, 2008 2. Sep 8, 2008 ### Staff: Mentor This is good, except that you seem to have the directions reversed. Using the right hand rule for cross products, r X F for the force on m2 should be into the page. Yes. There's a simple relationship between angular speed of the pulley and the linear speed of the masses. OK, but again, the directions are off. Let's assume that m2 moves up, m1 down. Thus the angular momentum of m2, r X m2v, should be out of the page. Express w in terms of v. Yes, "rate of change" means take the derivative. What's the derivative of v? 3. Sep 8, 2008 ### kehler Thanks Doc Al. Oops, I don't know what I was thinking while typing out my solution to part (a)! I got the directions right on the piece of paper I was writing on. For part (b), I intended for m2 to move down and m1 to move up. I'm pretty sure the angular momentum for both masses should be into the page if I do it like that. Is the total angular momentum (R x m2 v) + (R x m1 v) + I(v/R), into the page (based on the assumption that m2 moves down and m1 up) ? So for (c), is the equation I'm meant to get (R x m2 a) + (R x m2 a) + (I/R)a = (R x m2 x g sin theta) - (R x m x g)? Which would mean a = (M2 x gsintheta - M1g)/(M1 + M2 + (I/R^2)) Did I get that right? :) 4. Sep 8, 2008 ### Staff: Mentor Absolutely. You were right the first time. Looks good. Yes, except for a few typos. (Make sure your m's are properly labeled.) Looks good to me. 5. Sep 8, 2008 ### kehler Awesome :). Thanks again. 6. Sep 8, 2008 ### Staff: Mentor You are most welcome. Just for fun, I recommend that you solve for the acceleration the "usual" way (by applying Newton's 2nd law to each block and the pulley and then combining the resulting equations) and confirm that you get the same answer. Similar Discussions: Find the net torque acting on the system	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9482319951057434, "perplexity": 807.1038305272024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109682.23/warc/CC-MAIN-20170821232346-20170822012346-00657.warc.gz"}
https://www.math.princeton.edu/events/divisor-matrix-dirichlet-series-and-sl2z-2008-05-08t203005	# The Divisor Matrix, Dirichlet Series and SL(2,Z) - P. Sin, University of Florida IAS - Simonyi Hall Seminar Room SH-101 The divisor matrix is indexed by the natural numbers with $(i,j)$ entry equal to $1$ if $i$ divides $j$ and $0$ otherwise. The convolution of a Dirichlet series with the Riemann zeta function corresponds to multiplication of the sequence of coefficients by the divisor matrix. In this talk, we consider groups which contain the divisor matrix and preserve the space of convergent Dirichlet series. A reduction step is to show that the divisor matrix can be brought to a Jordan normal form by transition matrices which preserve the above space. We then construct an representation of $SL(2,Z)$ on the space of convergent Dirichlet series in which the standard unipotent element is represented by the divisor matrix. Finally, we discuss the relation between the zeta function and the Dirichlet series arising from other elements of $SL(2,Z)$ in this representation. Joint work with John G. Thompson.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9903073906898499, "perplexity": 181.42389097384438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738425.43/warc/CC-MAIN-20200809043422-20200809073422-00069.warc.gz"}
http://math.stackexchange.com/questions/294943/equivalent-definition-of-algebraically-closed	# Equivalent definition of algebraically closed In Hungerford's Algebra text, it is stated that a field $K$ is algebraically closed iff there exists a subfield $F$ such that $K$ is algebraic over $F$ and all polynomials in $F[x]$ split in $K[x]$. This seems to be a much weaker condition that $K$ being algebraically closed. I don't see why it is true (it is not proven in the text) Is there a particular reason why this is true? - If $\alpha$ is algebraic over $K$, then $\alpha$ is algebraic over $F$. The minimal polynomial splits over $K$. Thus, $\alpha \in K$. So this is quite immediate. However, there is even a weaker condition: Every non-constant polynomial in $F[x]$ has some root in $K$. Then it also follows that $K$ is algebraically closed; but this is a nontrivial theorem (Isaacs, Roots of polynomials in algebraic extensions of fields; or: Gilmer, A Note on the Algebraic Closure of a Field). Let $\bar{K}$ be the algebraic closure of $K$ and fix $\alpha\in \bar{K}$. Then $\alpha$ is algebraic over $F$ because algebraic extensions are transitive. In particular this implies that the minimum polynomial of $\alpha$ over $F$ splits in $K[x]$ so that $\alpha \in K$. So we have $K=\bar{K}$ and $K$ is algebraically closed.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.975082516670227, "perplexity": 40.59274077366825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011041911/warc/CC-MAIN-20140305091721-00074-ip-10-183-142-35.ec2.internal.warc.gz"}
http://www.physicsforums.com/showpost.php?p=2841016&postcount=6	View Single Post P: 161 Quote by ismaili I reconsidered this recently. I think for my first question, the reason that reduces the independent components $$2m^2$$ into $$m^2$$ is the (anti-)hermitian properties of the gamma matrices, not the Clifford algebra. Then, the antisymmetrized products of gamma matrices form a basis for the algebra, hence, by matching independent components $$m^2$$ and the number of basis $$2^d$$, we see that $$m = 2^{d/2}$$ for even dimension $$d$$. The solution to the second equation is due to the role of $$\gamma_5$$. That's why only in odd dimension, those basis are related by Levi-Civita tensor, and the number of basis is reduced to $$2^{(d-1)/2}$$ for odd $$d$$. But at least those $$B,C$$ matrices are not suddenly popped.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9044010043144226, "perplexity": 279.5903487201107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535917663.12/warc/CC-MAIN-20140901014517-00058-ip-10-180-136-8.ec2.internal.warc.gz"}
http://justinbangerter.com/posts/imaginary_numbers.html	# Justin's Blog ## February 19, 2012 The imaginary numbers: i, j, and k behave like so when multiplied: $$\hat{i}^{2} = \hat{j}^{2} = \hat{k}^{2} = -1$$ $$\hat{i}\hat{j} = -\hat{j}\hat{i} = \hat{k}$$ $$\hat{j}\hat{k} = -\hat{k}\hat{j} = \hat{i}$$ $$\hat{k}\hat{i} = -\hat{i}\hat{k} = \hat{j}$$ Note that imaginary multiplication is commutative. I found this graphic in Griffith's Electrodynamics. You can use it to help you remember the rule. Going clockwise, multiplying two units will create the next unit. Going counter-clockwise, multiplying two units will make the negative of the next unit. Think ijk... alphabetical... if you go backwards, it's negative. Sir William Rowan Hamilton chose these rules so quaternions would multiply in a way that made sense. He thought it was so important that he carved them into Broom Bridge on his way home. He wanted to make sure they were written down in case he died. You can't see the markings today, but it is a very famous story. Back to the top	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807602524757385, "perplexity": 1095.5147799845067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320666.34/warc/CC-MAIN-20170626013946-20170626033946-00483.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/calculus-3rd-edition/chapter-13-vector-geometry-13-1-vectors-in-the-plane-exercises-page-650/32	## Calculus (3rd Edition) Since $\overrightarrow {AB}$ and $\overrightarrow {PQ}$ have the same components, they are equivalent. We have $A = \left( {5,8} \right)$ and $B = \left( {1,8} \right)$. So, $\overrightarrow {AB} = \left( {1,8} \right) - \left( {5,8} \right) = \left( { - 4,0} \right)$. We have $P = \left( {1,8} \right)$ and $Q = \left( { - 3,8} \right)$. So, $\overrightarrow {PQ} = \left( { - 3,8} \right) - \left( {1,8} \right) = \left( { - 4,0} \right)$. Since $\overrightarrow {AB}$ and $\overrightarrow {PQ}$ have the same components, they are equivalent.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9523117542266846, "perplexity": 97.77787600012987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991537.32/warc/CC-MAIN-20210513045934-20210513075934-00529.warc.gz"}
https://www.qb365.in/materials/stateboard/11th-chemistry-basic-concepts-of-chemistry-and-chemical-calculations-one-mark-question-paper-6472.html	" /> --> #### Basic Concepts of Chemistry and Chemical Calculations One Mark Questions 11th Standard Reg.No. : • • • • • • Chemistry Time : 01:00:00 Hrs Total Marks : 20 20 x 1 = 20 1. 40 ml of methane is completely burnt using 80 ml of oxygen at room temperature The volume of gas left. after cooling to room temperature is (a) 40 ml CO2 gas (b) 40 ml CO2 gas and 80 ml H2O gas (c) 60 ml CO2 gas and 60 ml H2O gas (d) 120 ml CO2 gas 2. The number of water molecules in a drop of water weighing 0.018 g is (a) 6.022 x 1026 (b) 6.022 x1023 (c) 6.022 x 1020 (d) 9.9 x 1022 3. Fe2 + $\longrightarrow$ Fe3+ + e- is a ________ reaction. (a) redox (b) reduction (c) oxidation (d) decomposition 4. The oxidation number of hydrogen in LiH is _____________ (a) +1 (b) -1 (c) +2 (d) -2 5. Which one of the following represents 180g of water? (a) 5 Moles of water (b) 90 moles of water (c) $\frac { 6.022\times { 10 }^{ 23 } }{ 180 }$ molecules of water (d) 6.022 x 1024molecules of water 6. Rusting of iron articles is an example of ___________ reaction (a) Combustion (b) decomposition (c) redox (d) hydrolysis 7. Two 22.4 litre containers A and B contains 8 g of O2 and 8 g of SO2 respectively at 273 K and 1 atm pressure, then (a) Number of molecules inA and B are same (b) Number of molecules in B is more than that in A. (c) The ratio between the number of molecules in A= to number of molecules in B is 2:1 (d) Number of molecules in B is three times greater than the number of molecules in A 8. Which one of the following is used as a standard for atomic mass. (a) 6C12 (b) 7C12 (c) 6C13 (d) 6C14 9. Maximum oxidation state is present in the central metal atom of which compound (a) CrO2Cl2 (b) MnO2 (c) [Fe(CN)6]3- (d) MnO 10. In which of the following reactions, hydrogen peroxide acts as an oxidising agent? (a) I2+ H2O2 + 20H- $\longrightarrow$ 21- + 2H2O + O2 (b) PbS + 4H2O2 $\longrightarrow$ PbSO4 + 4H2O (c) 2MnO4-+ 3H2O2 $\longrightarrow$ 2MnO2 + 3O2 + 2H2O + 2OH- (d) HOCI + H2O2 $\longrightarrow$ H2O+ + Cl- + O2 11. The solid state of matter is converted into gas by (a) sublimation (b) deposition (c) freezing (d) condensation 12. Which form of based on physical characteristics possess neither definite volume nor definite shape? (a) Solids (b) Liquids (c) Gases (d) Both (a) and (b) 13. Atomicity of nitrogen is (a) 1 (b) 2 (c) 3 (d) Zero 14. Calculate the percentage of N in ammonia molecule. (a) 121.42% (b) 28.35% (c) 82.35% (d) 28.53% 15. The equivalent mass of potassium permanganate in alkaline medium is: MnO4- + 2H2O + 3e-$\rightarrow$ MnO2 + 4OH- (a) 31.6 (b) 52.7 (c) 79 (d) None of these 16. Which of the following has the highest mass? (a) 1 g atom of C (b) 1/2 mole of CH4 (c) 10 ml of water (d) 3.011 x 1023 atoms of oxygen 17. The mass of an atom of nitrogen is _______________. (a) $\frac { 14 }{ 6.023\times { 10 }^{ 23 } }$ (b) $\frac { 28 }{ 6.023\times { 10 }^{ 23 } }$ (c) $\frac { 1 }{ 6.023\times { 10 }^{ 23 } }$ (d) 14 amu 18. Which of the following halogens do not exhibit positive oxidation number in its compounds? (a) Fluorine (b) Chlorine (c) Iodine (d) Bromine 19. Which of the following is the most powerful oxidising agent? (a) KMnO4 (b) K2Cr2O7 (c) O3 (d) H2O2 20. On the reaction 2Ag + H2SO4 $\rightarrow$ Ag2SO4 + 2H2O + SO2.Sulphuric acid acts as _______________. (a) oxidising agent (b) reducing agent (c) a catalyst (d) an acid as well as an oxidant ### TN 11th Standard Chemistry free Online practice tests #### 02-Sep-2019 11 chemistry 1st volume quartearly exam model question papers	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8183862566947937, "perplexity": 4415.705917381659}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370519111.47/warc/CC-MAIN-20200404011558-20200404041558-00294.warc.gz"}
https://www.physicsforums.com/threads/magnetic-field-homework-question.112678/	# Homework Help: Magnetic Field Homework Question 1. Mar 1, 2006 ### wr1015 Two long, straight wires are oriented perpendicular to the computer screen, as shown in Figure 22-43, in which L = 6.0 cm. The current in one wire is I1 = 3.7 A, pointing into the screen, and the current in the other wire is I2 = 4.0 A, pointing out of the screen. Find the magnitude and direction of the net magnetic field at point P. http://server5.ihostphotos.com/show.php?id=1fZ6ef1d174f6G10F7f41d4e154C3d9c [Broken] ok so first off i used pythagorean theorem to find the straight line distance from point to the wire coming out of the page and used that as $$r_{2}$$. then, i used the formual for magnetic force to find the force on point P from the two wires $$B = \mu_{0}I_{1}/2 \pi r_{1}$$ and $$B= \mu_{0}I_{2}/2 \pi r_{2}$$ and added the results. The answer i'm getting is wrong.. any suggestions?? i know the current in each of the wires are going in opposite directions so the forces repel but how does that relate to a point?? Last edited by a moderator: May 2, 2017 2. Mar 1, 2006 ### wr1015 for a more detailed description of what i did: $$B= (4 \pi x 10^-7)(3.7A)/(2 \pi (.06m) + (4 \pi x 10^-7)(4.0A)/(2 \pi (.0848m))$$ which gave $$2.1767x10^-5 T$$ Last edited: Mar 1, 2006 3. Mar 1, 2006 ### Meir Achuz You have to add the two B's vectorially. B1 is horizontal. B2 makes and angle of 45 degrees. 4. Mar 1, 2006 ### wr1015 like this??: $$I_{1}_{y}$$= (4$$\pi$$x 10$$^-7$$)(3.7)/(2$$\pi$$(.06)) which gives 1.233x10$$^-5$$ $$I_{1}_{x}$$= 0 then, $$I_{2}_{y}$$= (4$$\pi$$x 10 $$^-7$$)(4.0)/(2$$\pi$$(.0848)(sin 45$$^0$$) $$I_{2}_{x}$$= (4$$\pi$$^-7[/tex](4.0)/(2$$\pi$$(.0848)(cos 45$$^0$$) then use pythagorean theorem again to find the total force?? Last edited: Mar 1, 2006 5. Mar 2, 2006 ### Meir Achuz I_1y=0. I_1x=what you have for I_1y. I_2x and I_2y are both negative. Then use Phyth for magniktude of B. You should know the right hand rule to give you the correct direction of B from each wire.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8766462206840515, "perplexity": 1658.7354810007894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676596336.96/warc/CC-MAIN-20180723110342-20180723130342-00435.warc.gz"}
http://mfleck.cs.illinois.edu/study-problems/two-way-bounding/bounding-3-solx.html	# Two-way bounding problem 3 Here are the set definitions again for reference: • $$A = \{(x,y) \in \mathbb{R}^2 : y = 3x + 7\}$$ • $$B = \{\lambda(-2,1) + (1-\lambda)(1,10) : \lambda \in \mathbb{R}\}$$ ### Solution Lemma: $$\lambda(-2,1) + (1-\lambda)(1,10) = (-2\lambda + (1 - \lambda), \lambda + (10 - 10\lambda)) = (1 - 3\lambda, 10 - 9\lambda)$$ $$A \subseteq B$$ Let (x,y) be an element of A. Then $$y = 3x + 7$$ by the definition of A. Consider $$\lambda = \frac{1-x}{3}$$. Then, using our lemma above, we can calculate: $$\lambda(-2,1) + (1-\lambda)(1,10) = (1 - 3\lambda, 10 - 9\lambda) = (1 - 3\frac{1-x}{3}, 10 - 9\frac{1-x}{3}) = (1 - (1-x), 10 - 3(1-x)) = (x, 3x+7) = (x,y)$$ So we've shown that (x,y) is an element of B. Since every element of A is also an element of B, $$A \subseteq B$$. $$B \subseteq A$$ Let (x,y) be an element of B. By the definition of B, we know that $$(x,y) = \lambda(-2,1) + (1-\lambda)(1,10)$$ for some real number $$\lambda$$. So, using our lemma above, $$(x,y) = (1 - 3\lambda, 10 - 9\lambda)$$. Then $$3x+7 = 3(1 - 3\lambda) + 7 = 3 - 9\lambda + 7 = 10 - 9\lambda = y$$. So 3x+7 = y, which means that (x,y) is an element of A. Since every element of B is also an element of A, $$B \subseteq A$$. Since $$A \subseteq B$$ and $$B \subseteq A$$, A=B, which is what we needed to show.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.985305666923523, "perplexity": 422.7671352056903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890928.82/warc/CC-MAIN-20180121234728-20180122014728-00036.warc.gz"}
http://math.stackexchange.com/questions/96332/optimal-polyomino-induced-coloring	# Optimal polyomino induced coloring Which polyominos (with orientation) of $n$ squares, requires the least number of different colors, $c(n)$, such that if this polyomino is placed anywhere on an optimally colored infinite square grid of $c(n)$ colors, it will have all its squares placed on different colors? And which polyomino(s) require the most colors, and how many colors as a function of $n$? Also: How to show that a finite number of colors (depending on n), suffices for any subset of n unit-m-cubes, in Z^m? - Maybe the more natural question is for which not necessarily connected set of n squares – user1708 Jan 4 '12 at 12:43 – joriki Jan 4 '12 at 13:20 Brooks' theorem says that the chromatic number of a graph is at most $\Delta + 1$, where $\Delta$ is the maximum vertex degree (and is only $\Delta + 1$ if the graph is complete or an odd cycle). A set of $n$ points in $\mathbb{Z}^m$ has at most $n(n-1)$ different vector separations between distinct points, so the induced graph (which is regular) has degree at most $n(n-1)$, and can't be either complete or an odd cycle; so $n(n-1)$ colors will always suffice. – mjqxxxx Jan 4 '12 at 19:06 Rectangular $n$-ominoes have $c(n)=n$, the minimal possible value. Let’s call polyominoes for which $c(n)=n$, “optimal”. Colouring the grid with diagonal stripes shows that thin snake-like directed polyominoes such as the following are also optimal. $\hspace{2.5in}$ Optimal polyominoes can’t be arch-shaped like the following; each of the cells in the ‘gap’ have to differ in colour from all the cells in the arch shape. $\hspace{2.75in}$ This is a particular case of the following condition on optimal polyominoes. If $P=\{(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n)\}$ is an $n$-omino with the $(x_i,y_i)$ being the cell coordinates, then let $P+v$ be $P$ translated by vector $v$, and let $$D(P)=\{(x_i-x_j,y_i-y_j):(x_i,y_i)\in P,(x_j,y_j)\in P\}$$ be the “distance polyomino” of $P$. For example, the distance polyomino of the ‘F’ pentomino (on it’s ‘back’) looks like this (distance polyominoes are always centrally symmetric): $\hspace{2.75in}$ Now if for any cell $v\notin P$, we have $P\subset D(P)+v$, then $P$ is not optimal, because there is no way to colour $v$. As well as excluding arch-shapes, this condition excludes others from being optimal, such as this hexomino: $\hspace{2.75in}$ This condition can be generalised as follows: If for any $S\subseteq P$ and any cell $v$ such that $S+v\cap P=\varnothing$, we have $$\left\|\;\bigcup_{u\in S}\;P\;\setminus (D(P)+u+v)\;\right\|<\|S\|,$$ then $P$ is not optimal, because there is no way to colour the cells in $S$. This generalised condition excludes the ‘T’ pentomino from being optimal (the two cells on one side of the stroke cannot be coloured). However, it fails to exclude the ‘F’ pentomino (which is also non-optimal). Many polyominoes can be seen to be optimal simply by (rotating and) numbering the cells from left-to-right, top-to-bottom in the natural manner. If in the result, the numbers down the columns form arithmetic progressions with the same difference in each column, then the polyomino is optimal (cycle all the colours on every row). This works for rectangles, snake-like directed polyominoes, all row-convex polyominoes with only two rows, and all pentominoes except the ‘U’, ‘T’ and ‘F’ (which are all non-optimal). It also works for some non-convex polyominoes, like this octomino: $\hspace{3.1in}$ Also, any row-convex polyomino in which each row is the same length is also optimal (cycle the colours on each row). Thus the following (non-convex) hexomino is optimal: $\hspace{3in}$ Are there any optimal polyominoes that are neither row- nor column-convex? - By "As well as excluding non-convex polyominoes", I think you mean "As well as excluding some non-convex polyominoes"? Otherwise your last two examples would be counterexamples. – joriki Jan 5 '12 at 11:18 @joriki: Corrected; thanks. – David Bevan Jan 5 '12 at 12:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9116530418395996, "perplexity": 976.381271964319}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644060413.1/warc/CC-MAIN-20150827025420-00089-ip-10-171-96-226.ec2.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/1109.0354/	arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org. # Derived splinters in positive characteristic Bhargav Bhatt ###### Abstract. This paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic , this condition characterises rational singularities by a result of Kovács. Our main theorem asserts that over a field of characteristic , derived splinters are the same as (underived) splinters, i.e., as schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning extending Serre/Kodaira type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove “up to finite cover” analogues in characteristic of many vanishing theorems one knows in characteristic . All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjecture. The goal of this paper is to introduce the notion of derived splinters and prove some basic results about them. Since derived splinters are analogues of splinters, we review the definition of the latter first. ###### Definition 0.1. A scheme is called a splinter if for any finite surjective map , the pullback map is split in the category of coherent sheaves on . This term was coined in [Sin99], though the idea is much older. In characteristic , splinters are exactly normal schemes (see Example 1.1). Away from characteristic , however, splinters become much more interesting, and are the subject of numerous results and questions in commutative algebra. In particular, the still open direct summand conjecture [Hoc73] posits that all regular rings are splinters; this conjecture is known in equicharacteristic by [Hoc73] or in mixed characteristic for dimensions by [Hei02], but is unknown in general, and is a fundamental open problem in the subject. Our definition of derived splinters is inspired by that of splinters and the philosophy that proper maps provide robust derived analogues of finite maps, at least for coherent sheaf theory. More precisely, we have: ###### Definition 0.2. A scheme is called a derived splinter, or simply a D-splinter, if for any proper surjective map , the pullback map is split in derived category of coherent sheaves on . Like splinters, the idea of D-splinters is not new. In fact, a theorem of Kovács [Kov00] identifies D-splinters in characteristic with schemes whose singularities are at worst rational. Since splinters in characteristic are precisely normal schemes, one notices immediately that splinters and D-splinters define extremely different classes of singularities in characteristic . In characteristic however, we discover a remarkably different picture; one of the main theorems of this paper is the following: ###### Theorem 0.3. A noetherian -scheme is a splinter if and only if it is a D-splinter. The main tool used to prove Theorem 0.3 is a cohomology-annihilation result that is of independent interest: we show that the higher cohomology of the structure sheaf on a projective variety in characteristic can always be killed by a finite cover. In fact, we prove the following stronger relative statement: ###### Theorem 0.4. Let be a proper morphism of noetherian -schemes. Then there exists a proper morphism and a finite surjective morphism such that the pullback map is . The proof of Theorem 0.4 is inspired by the paper [HH92] of M. Hochster and C. Huneke proving the existence of big Cohen-Macaulay algebras in positive characteristic (and also [HL07]). The same paper led K. Smith to ask certain questions concerning extensions of the vanishing theorems of Serre and Kodaira beyond the ample cone (see §6). Using our methods, we are able to answer these questions. The negative answers are recorded in the form of counterexamples at the end of §6, while the affirmative answers are summarised below in Theorem 0.5. We refer the reader to Propositions 6.2 and 6.3 for more precise statements. ###### Theorem 0.5. Let be a proper variety over a field of positive characteristic, and let be a semiample line bundle on . Then can be killed by finite covers for . If is big as well, then can be killed by finite covers for . It seems worthwhile to remark at this point that the results mentioned above, in conjunction with those proven in §6, have applications unrelated to splinters or D-splinters: these results suggest that numerous vanishing theorems that are true in characteristic have analogues in characteristic provided one works “up to finite covers.” This idea has been pursued in much more depth in the recent work [BST11], where “up to finite cover” analogues of the Nadel vanishing have been established. Returning to affine D-splinters, we note that in positive characteristic , by Theorem 0.3, this class of singularities is closely related to other classes of singularities, the so-called F-singularities, defined using the Frobenius action. For example, locally excellent affine -Gorenstein splinters are F-regular by [Sin99] which builds on the Gorenstein case proven in [HH94]; see also Example 1.4 below. In contrast, in the case of mixed characteristic, our knowledge about either splinters or D-splinters is minimal, primarily because the direct summand conjecture is unknown. For progress towards establishing an analogue of Theorem 0.3, and especially a weak mixed characteristic analogue of Theorem 0.4, we refer the reader to [Bhab]. #### Organisation of this paper The purpose of §1 is to collect some examples and non-examples of splinters and D-splinters; the goal here is to describe some of the geometry underlying these definitions. In §2 we review some general notation and results about derived categories used in this paper; the key result is a method for passing from conclusions at the level of cohomology groups to those at the level of complexes. Theorems 0.3 and 0.4 are proven in §3, and some refinements are proven in §4; the method here is inspired by that of [HL07] and, by transitivity, by that of [HH92]. Moving to applications, we discuss some purely algebraic applications of the preceding theorems in §5. In §6, we review some questions raised by Karen Smith in the wake of [HH92], and then discuss both positive and negative answers we can provide; the highlights here are the “up to finite covers” version of Kodaira vanishing in Proposition 6.3, and some of the counterexamples, especially Example 6.10. Finally, in §7 we use Theorem 0.4 to show that the complete flag variety for is a D-splinter, thereby providing the first non-toric projective example of one. #### Acknowledgements This paper forms a part of the author’s doctoral dissertation written under Aise Johan de Jong, and would not have been possible without his consistent support. In particular, the author would like to thank de Jong for suggesting some of the questions addressed in this paper, and for generously sharing ideas related to many parts of this paper. In addition, the author would also like to thank Karl Schwede and, especially, Anurag Singh for many conversations about derived splinters. ## 1. Some examples The goal of this section is collect some examples of splinters and D-splinters. Since the notions in characteristic are quite well understood, we focus mainly on the case of characteristic . Moreover, it is typically non-trivial to prove that any given ring is a splinter or D-splinter. Hence, we freely use results in the literature or elsewhere in this paper in our proofs; we hope that despite the resulting non-elementary nature of the examples, the reader will be convinced that splinters and D-splinters are geometrically interesting. We dispose of the characteristic case. ###### Example 1.1 (Splinters in characteristic 0). A connected noetherian -scheme is a splinter if and only if it is normal. For the forward direction, note that the map from the disjoint union of the irreducible components of to immediately shows that is forced to be a domain if it is a splinter. The desired claim now follows from the following ring-theoretic fact: if is an integral domain with integral over , then is not split unless . To prove this, we simply observe that if were split, then the quotient would be a torsion free -module with generic rank , which can only happen when the quotient is trivial. For the converse implication, we need to show that if is a finite surjective morphism, and is normal and connected, then has a section in . After replacing with an irreducible component dominating , we may assume that is integral. Let denote the degree of the map induced by at the level of function fields. Then the map provides a canonical splitting for the map (here we use that the trace map on function fields preserves integrality). ###### Example 1.2 (D-splinters in characteristic 0). Let be a variety over . Then is D-splinter if and only if has rational singularities, i.e., if for some (equivalently, every) resolution of singularities . We refer the reader to [Kov00] for a proof. A splinter in positive characteristic is subtler than its characteristic avatar, as being a splinter imposes some kind of positivity (both local and global) on the variety. In fact, in view of Theorem 0.3, over , being a splinter is equivalent to being a D-splinter, a condition that is a priori much more restrictive. Nevertheless, large classes of examples of splinters (or, equivalently, D-splinters) over do exist, and are catalogued below. The intuition informing most of these examples is that splinters should be analogous to rational singularities in characteristic . ###### Example 1.3 (Smooth affines are splinters). All regular affine -schemes are splinters; this is a result of Mel Hochster (see [Hoc73]), and we record a proof of Hochster’s theorem below for the convenience of the reader. The proof given below is cohomological in nature, and different from Hochster’s. We first explain the idea informally. Let be a finite surjective map. Using the fact that is Gorenstein, an elementary duality argument will reduce us to showing that is injective. The kernel of this map is a Frobenius stable proper submodule of of finite length by an inductive argument due to Grothendieck (see [Gro68, Exposé VIII, Théorème 2.1]). The regularity of is regular will then imply that this is impossible for length reasons. Now for the details. After localising and completing, we may assume that is a complete regular local -algebra of dimension . By the Cohen structure theorem (see [Mat80, §28, Theorem 28.J and Corollary 2]), we know that . Since field extensions split as -modules, we may pass to the algebraic closure of the coefficient field to assume that is algebraically closed. In particular, the Frobenius map is finite. Given a finite extension , we need to show that the natural map is surjective. By induction, we may assume that the cokernel is supported only at the closed point . In particular, the cokernel has finite length. Now the fact that is Gorenstein implies that . Thus, the map is isomorphic to the trace map , which is dual to the canonical pullback map . As local duality interchanges kernels and cokernels while preserving lengths, it follows that the kernel has the same length as ; this kernel is also Frobenius-stable by construction. Now consider the diagram The map is injective since is exact (by regularity of ), while the map is an isomorphism by the flat base change isomorphism (see [BS98, §4.3.2]). The diagram then shows that is also injective, and thus the length of is bounded above by that of . The claim now follows from the elementary observation that multiplies length by . Following the proof of Example 1.3 leads to a much larger class of splinters, defined in terms of F-rational rings. We remind the reader that a noetherian local -algebra of dimension is called F-rational if it is Cohen-Macaulay, normal, and has the property that has no proper Frobenius-stable submodules except . This is not the original definition of F-rationality, but equivalent to it by work of Karen Smith, see [Smi94, Smi97b]. ###### Example 1.4 (F-rational Gorenstein rings are splinters). Let be a noetherian excellent local -algebra admitting a dualising complex. Assume that is Gorenstein. If is F-rational, then is a splinter. This follows from the proof given in Example 1.3. In more detail, we may assume without loss of generality that that is an F-rational Gorenstein complete noetherian local ring of dimension . Given a finite extension , we need to verify that is surjective. By the Gorenstein assumption, we can identify this map with . The image of this last map is a Frobenius-stable submodule . Moreover, since the formation of commutes with localisation, we know that is generically non-zero. The definition of F-rationality then shows that as desired. ###### Remark 1.5. One may show a converse to Example 1.4 as follows: any excellent splinter is -rational. To see this, note that the argument in Example 1.1 shows that is normal, while Corollary 5.3 below shows that is Cohen-Macaulay. To show that is -rational, one can then use [Smi97b, Theorem 2.6] and [Smi94, Theorem 5.4]. Together, these theorems imply that it is enough to check that for all ideals generated by a system of parameters, we have for all finite extensions . The splinter property implies that , which easily shows that . We work out a special case of Example 1.4, to give an idea of the relevant geometry. ###### Example 1.6 (The quadric cone). We claim that is a splinter for provided . By Example 1.4, it suffices to show that is -rational. By [Hun96, Theorem 4.2], it suffices to show that is -rational. Thus, we can set up an induction once we settle the case. This case follows from [Hoc73, Example 3]. Alternately, in the case, we may identify with (completion at the origin of) the affine cone on a smooth conic . Since is a hypersurface, the scheme has an isolated hypersurface singularity at , and is thus Cohen-Macaulay and normal. Moreover, identifying with the total space of the complement of the section in shows that H2m(R)≃⊕n∈ZH1(C,OP2(−n)\|C)≃⊕n∈ZH1(P1,O(2n)). The preceding presentation is Frobenius equivariant, where Frobenius acts on the grading on the right by multiplying the weights by . By inspection, it easily follows then that has no Frobenius-stable proper non-zero submodules, proving F-rationality. Next, we show that certain quotient singularities are splinters. ###### Example 1.7 (Quotient singularities are often splinters). Let be a field, and let be a regular -algebra. Let be a linearly reductive group acting on . Then is a splinter. Indeed, the inclusion has an -linear section given by the Reynolds operator, and so the splinter property for follows from that of . More generally, the same argument shows that any subring of a regular ring that splits off as an -linear summand is a splinter. In particular, if is a reductive group over acting on an affine algebraic -scheme , then almost all positive characteristic reductions of are splinters. We next list a large class of non-examples. ###### Example 1.8 (General type cones are not splinters). Let be a hypersurface of degree over a perfect field of characteristic , and let be the affine cone on . Then is not a splinter. To see this, note that as in Example 1.6, we have an identification Hnm(S)≃⊕i∈ZHn−1(X,OX(i)) that is Frobenius equivariant, where Frobenius acts on the right by scaling the weights by . Now by adjunction. One then easily computes that , and thus has a non-trivial kernel. It follows that also has a non-trivial kernel, and so is not split. Lastly, we discuss a non-example due to Hochster: a hypersurface singularity of dimension in characteristic that is not a splinter. Aside from its intrinsic interest, this example is meant to caution the reader as the standard lift of this hypersurface to characteristic has rational singularities. ###### Example 1.9. Let be a field of characteristic . Let be a polynomial ring, and let . Since , admits the presentation where , , and . In particular, is a hypersurface singularity of dimension . Since the singularity is isolated, is even normal. On the other hand, is not a splinter because the natural map is a finite surjective map such that has no section: identifying sheaves with modules and applying such a section to would give us which is false. The same example can be adapted to arbitrary positive characteristic by setting for some . The examples discussed hitherto were all affine. Requiring a projective variety over a positive characteristic field to be a splinter leads to questions of a very different flavour as the geometry of is heavily constrained. For example, Theorem 0.4 shows that for all . In fact, the same theorem applied to a high iterate of Frobenius shows that for whenever is an ample line bundle. Thus, projective examples are harder to find; nevertheless, they do exist, as we show below. We will discuss such examples further in §7. ###### Example 1.10 (Toric varieties are often splinters). Any toric variety that is projective over an affine is a splinter. To see this, note that any such can be obtained as a quotient (see [MS05, Theorem 10.27]), where is an open subscheme, and is an algebraic subgroup preserving . As is linearly reductive, so is (see [AOV08, Proposition 2.5]). In particular, we see that is a direct summand. The result now follows from the fact that is a splinter, which in turn follows from Example 1.3 and the fact any finite cover of comes from a finite cover of (by normalisation, for example). Lastly, we record an elementary example showing that not all smooth projective varieties are splinters. ###### Example 1.11. Let be an elliptic curve over an algebraically closed field of positive characteristic . We will show that is not a splinter. Consider the multiplication by map . The induced map on can be easily seen to be ; for example, one can show that induces multiplication by on for any abelian variety . It follows that is not split. ## 2. Some facts about derived categories The purpose of this section is to record some notation and results about triangulated categories for later use. As a general reference for triangulated categories and -structures, we suggest [BBD82]. For the convenience of the reader, we first recall some notation regarding truncations. ###### Notation 2.1. Let be a triangulated category with a -structure given by a pair of full subcategories satisfying the usual axioms. For each integer , we let (respectively, ); this can be thought of as the fullsubcategory spanned by objects with cohomology only in degree at least (respectively, at most) . Moreover, there exist truncation functors: for each integer , there exist endofunctors and of which are retractions of onto the fullsubcategories and . We let , and . These truncation functors are not exact, but they sit in an exact triangle . Moreover, they satisfy the adjunctions HomD≤n(K,τ≤nL)≃HomD(K,τ≤nL)≃HomD(K,L)forK∈D≤nandL∈D and dually HomD≥n(τ≥nK,L)≃HomD(τ≥nK,L)≃HomD(K,L)forK∈DandL∈D≥n. These adjunctions can be remembered as algebraic analogues of the fact that all maps are nullhomotopic if is an -connected CW complex, and is an -truncated one. Let us fix a triangulated category , with a -structure . The main question that arises repeatedly in the sequel is the following: given a morphism in such that , when can we conclude that ? As the non-trivial extension in the derived category of abelian groups shows, the short answer is “not always”. To understand this phenomenon better, fix a test object , and consider the associated map of abelian groups Hom(M,f):Hom(M,K)→Hom(M,L) The chosen -structure gives rise to a functorial filtration on the morphism spaces of (via the filtration by cohomology groups of the target). Thus, the preceding map is a filtered map of filtered abelian groups. The assumption that implies that this filtered map induces the map on the associated graded pieces. In other words, moves the filtration one level down. This simple analysis suggests that under certain boundedness hypotheses, we may be able to salvage an implication of the form “” at the expense of iterating a map like a few times. This idea is formalised in the following lemma: ###### Lemma 2.2. Let be a triangulated category with -structure whose heart is . Assume that for a fixed integer , we are given objects and maps such that for all . Then the composite map is the map. ###### Proof. Consider the exact triangle τ≤d−1K2→K2→Hd(K2)[−d]→τ≤d−1K2[1]. Applying and using the formula (coming from adjunction) HomD(K1,Hd(K2)[−d]) = HomD≥d(τ≥dK1,Hd(K2)[−d]) = HomD≥d(Hd(K1)[−d],Hd(K2)[−d]) = HomA(Hd(K1),Hd(K2)), we see that the map factors through and is thus by hypothesis. We may therefore choose a (non-unique) factorisation of of the form . The same method shows that the morphism factors through . Thus, we obtain a diagram of morphisms: As , we see that . Thus, the composite vertical morphism on the left is zero, which implies that the one on the right is as well. ∎ ## 3. The main theorem This section is dedicated to the proof of Theorems 0.4 and 0.3. In fact, the bulk of the work involves proving Theorem 0.4 as Theorem 0.3 then follows by a fairly formal argument. The proof we give here draws on ideas whose origin can be traced back to Hochster and Huneke’s work [HH92] on big Cohen-Macaulay algebras in positive characteristic. We begin with a rather elementary result on extending covers of schemes. ###### Proposition 3.1. Fix a noetherian scheme . Given an open dense subscheme and a finite (surjective) morphism , there exists a finite (surjective) morphism such that is isomorphic to . Given a Zariski open cover with a finite index set, and finite (surjective) morphisms , there exists a finite (surjective) morphism such that factors through . The same claims hold if “finite (surjective)” is replaced by “proper (surjective)” everywhere. ###### Proof. We first explain how to deal with the claims for finite morphisms. For the first part, Zariski’s main theorem [Gro66, Théorème 8.12.6] applied to the morphism gives a factorisation where is an open immersion, and is a finite morphism. The scheme-theoretic closure of in provides the required compactification in view of the fact that finite morphisms are closed. For the second part, by the first part, we may extend each to a finite surjective morphism such that restricts to over . Setting to be the fibre product over of all the is then seen to solve the problem. To deal with the case of proper (surjective) morphisms instead of finite (surjective) , we repeat the same argument as above replacing the reference to Zariski’s main theorem by one to Nagata’s compactification theorem (see [Con07, Theorem 4.1]). ∎ Next, we present the primary ingredient in the present proof of Theorem 0.4: a general technique for constructing covers to annihilate coherent cohomology of -schemes under suitable finiteness assumptions. The method of construction is essentially borrowed from [HL07] where it is used to reinterpret and simplify the “Equational Lemma,” one of the main ingrendients in the proof of the existence of big Cohen-Macaulay algebras in positive characteristic (see [HH92]). ###### Proposition 3.2. Let be a noetherian -scheme with finite over a ring . Given an -finite Frobenius-stable submodule for , there exists a finite surjective morphism such that ###### Proof. We first explain the idea informally. As is -finite, it suffices to work one cohomology class at a time. If , then the Frobenius-stability of gives us a monic additive polynomial such that where acts by Frobenius. After adjoining -th roots of certain local functions representing a coboundary, we can promote the preceding equation in cohomology to an equation of cocycles, i.e, we find where is a cocycle of local functions that represents , and the displayed equality is an equality of functions on the nose, not simply up to coboundaries. Since is monic, such functions are forced to be globally defined (after normalisation), and this gives the desired result; the details follow. Fix a finite affine open cover of , and consider the cosimplicial -algebra as a model for the -algebra . The Frobenius action is modelled by the actual Frobenius map on each term of this -algebra. This gives the structure of an -module, where is the non-commutative polynomial ring on one generator over satisfying the commutation relation (see [Lau96, §1.1] for more details on this ring). In more concrete terms, at the level of cohomology, we see the following: for each polynomial , classes , and a scalar , we have , and . The -finiteness of the Frobenius stable module ensures that for any class , there exists a monic polynomial such that . If we pick representatives in for this equation, we obtain an equation in of the form g(~m)=d(n) where is a cocycle lifting and . As is a monic equation, we can find a finite surjective morphism such that for some . For example, we could do the following: for each component of (where is a multi-index), the scheme is a quasi-finite -scheme such that the equation admits a solution in . Using Proposition 3.1, we find a and with the desired properties. The additivity of Frobenius now tells us that we obtain an equation in of the form g(~m−d(n′))=0. The monicity of implies that the components of are integral over . Setting to be an irreducible component of that dominates under the natural map, we find a finite surjective morphism . The pullback of in is a vector of local functions whose components satisfy the monic polynomial over . As is integral and already contains roots of , it follows that these functions are globally defined. Thus, they lie in the image of the natural map where is viewed as a constant cosimplicial algebra. As the complex underlying the former cosimplicial algebra has cohomology only in degree , it follows that is a coboundary, which implies that is a coboundary on , which shows that satisfies the required conditions. ∎ ###### Remark 3.3. One may wonder whether Proposition 3.2 can be refined to show the existence of generically separable finite surjective maps that kill the relevant cohomology groups. In the local algebra setting, one can indeed do so by [SS10, Theorem 1.3]. Globally, however, requiring separability is too strong. For example, if is a smooth projective variety over a perfect field with a non-zero class killed by , then for any finite surjective generically separable map , one has ; see [Mum67, Lemma 5] for a proof. ###### Remark 3.4. Proposition 3.2 is proven by above by mimicking the cocycle-theoretic methods of [HL07]. It is also possible to give more conceptual proofs of this result. We refer the reader to [Bhaa] for a proof based on general results on finite flat group schemes, and [Bhab] for a geometric proof based on curve fibrations which has the advantage of generalising to mixed characteristic. As a corollary of Proposition 3.2 and the finiteness properties enjoyed by proper morphisms, we arrive at the following result: ###### Corollary 3.5. Let be a proper morphism of -schemes, with noetherian and affine. Then there exists a finite surjective morphism such that is for . ###### Proof. The properness of over an affine implies that is a finite -module and that for sufficiently large (see [Gro61, Corollaire 3.2.3]). Proposition 3.2 then finishes the proof. ∎ We will now finish the proof of Theorem 0.4. To pass from the conclusion of Corollary 3.5 to the general statement of Theorem 0.4, the obvious strategy is to cover with affines, construct covers that work over the affines, and take the normalisation of in the fibre product of all of these. When carried out, this process produces a finite cover such that, with , the maps are for . This is not quite enough to prove the theorem: a map in that induces the map on cohomology sheaves is not necessarily zero. However, with the boundedness conditions enforced by properness, a sufficiently high iteration of this process turns out to be enough. ###### Proof of Theorem 0.4. Fix a finite affine covering of , and denote by . Using Corollary 3.5, we can find finite surjective maps such that the induced map is for each . Using Proposition 3.1, we may find a finite surjective morphism such that factors through . This implies that is for each (as vanishing is a local statement on ). Iterating this construction times and using Lemma 2.2, we obtain a proper -scheme and a finite surjective -morphism such the natural pullback map is , thereby proving the theorem. ∎ Finally, having proven Theorem 0.4, we point out how Theorem 0.3 follows. ###### Proof of Theorem 0.3. It is clear that all D-splinters are also splinters. Conversely, let be splinter over , and let be a proper surjective morphism. By Theorem 0.4, there exists a finite surjective morphism such that, with , the pullback map is . By applying to the exact triangle g∗OY→Rg∗OY→τ≥1Rg∗OY→g∗OY[1] we see that the natural pullback map factors through ; choose some factorisation . As is a proper surjective morphism, the algebra is a coherent sheaf of algebras corresponding to the structure sheaf of a finite surjective morphism. By assumption, the natural map has a splitting , and thus the map splits . ∎ ## 4. Some refinements Roughly speaking, Theorem 0.4 says that proper morphisms behave like finite morphisms after passage to finite covers, at least as far as theorems concerning the annihilation of coherent sheaf cohomology. In the following proposition, we formalise this intuition, extract a kind of “converse” to this statement, and work with non-trivial coefficients. These results will be useful in the sequel when we prove vanishing results. ###### Proposition 4.1. Let be a noetherian -scheme, and be a proper surjective morphism. Then we can find a diagram with and finite surjective morphisms such that for every locally free sheaf on and every , we have: 1. The morphism factors through . 2. The morphism factors through . ###### Proof. Theorem 0.4 gives a finite surjective morphism such that, with , we have a map and the following diagram: We claim that this is a commutative diagram. The triangle based at commutes by construction. Given this commutativity, to see that the triangle based at commutes, it suffices to show that is injective. This injectivity (and, in fact, bijectivity) follows from adjunction for . Thus, the preceding diagram is a commutative diagram in . Applying , setting to be the Stein factorisation of , and using the projection formula now gives the desired result. ∎ We have not strived to find the most general setting for Theorem 0.4. For example, one can easily extend the theorem to algebraic spaces or even Deligne-Mumford stacks. On the other hand, the properness hypothesis seems essential as the example below shows. In fact, the method of the proof shows that the essential property we use is that the relative cohomology classes of the structure sheaf for are annihilated by a monic polynomial in Frobenius. We do not know if there is a better characterisation of this class of maps. ###### Example 4.2. Fix a base field . Let , and . The quotient map gives a natural identification . We claim that the non-zero classes in this group cannot be killed by a finite cover of . To see this, note that one may view as the local cohomology group , where is the coordinate ring of and is the maximal ideal corresponding to the origin. Given a finite surjective morphism , we may normalise in to obtain a finite surjective morphism which realises as the fibre over . As before, the cohomology group can be viewed as which, in turn, may be viewed as , where is the coordinate ring of considered as an -module in the natural way. Under these identifications, the pullback map corresponds to the morphism induced by the inclusion coming from . By Example 1.3, the inclusion is a direct summand as an -module map. In particular, the map is injective, which shows that the non-zero classes in persist after passage to finite covers. ## 5. Application: A result in commutative algebra We discuss some applications of Proposition 3.2 to commutative algebra. Most of these applications are implicit in [HL07]. The first result we want to dicuss is an analogue of Proposition 3.2 for local cohomology. ###### Proposition 5.1. Let be an excellent local noetherian -algebra such that is finite over some ring . For any -finite Frobenius-stable submodule with , there exists a finite surjective morphism such that . ###### Proof. Since is excellent, we may pass to the normalisation and assume that is normal. In particular, for . For , we have an Frobenius equivariant identification , where is the punctured spectrum of . Since , Proposition 3.2 gives us a finite surjective morphism such that . Setting to be the normalisation of in is then easily seen to do job. ∎ Next, we dualise the Proposition 5.1 to obtain a global result in terms of dualising sheaves. ###### Proposition 5.2. Let be an excellent noetherian -scheme of equidimension that admits a dualising complex . Then there exists a finite surjective morphism such that , where is the trace map . ###### Proof. Fix an integer . We will prove by induction on the dimension that there exists a finite surjective morphism such that ; this is enough by virture of Lemma 2.2 and the fact that the dualising complexes appearing have bounded amplitude. We may assume that as the there is nothing to prove when the dimension is . By passing to irreducible components, we may even assume that is integral. As vanishing of a map of sheaves is a local statement, we reduce to the case that is an excellent noetherian local -domain admitting a dualising complex. For each non-maximal , we can inductively find a finite morphism such that is the map. By duality formalism, the -module localises to at . Hence, the normalisation induces the map on when localised at . Finding such a cover for each non-maximal prime in the finite set of associated primes of and normalising in the fibre product of the resulting collection, we find a cover such that has an image supported only at the closed point. Setting , duality tells us that the image of is a finite length Frobenius-stable -submodule. Proposition 5.1 then allows us to find a finite surjective morphism such that . It follows that the composite map induces the map on . By duality, we see that as desired. ∎ Using Proposition 5.1, we discover that splinters are automatically Cohen-Macaulay. ###### Corollary 5.3. Let is an excellent noetherian local -algebra that is a splinter. Assume that admits a dualising complex. Then is a normal Cohen-Macaulay domain. ###### Proof. The normality of follows from the argument in Example 1.1. To verify that is Cohen-Macaulay, it suffices to show that is concenctrated in degree where , i.e., that for . By Proposition 5.2, we can find a finite surjective morphism such that , where . Since is a splinter, the inclusion is a direct summand. Applying , we see that the trace map is the projection onto a summand. Hence, the assumption that implies that , as desired. ∎ ###### Remark 5.4. One key ingredient in the proof of Proposition 5.2 is the good behaviour of local cohomology and dualising sheaves with respect to localisation. This behaviour seems to first have been observed by Grothendieck in [Gro68, Exposé VIII, Théorème 2.1] where it is used to show the following: a noetherian local ring of dimension that is Cohen-Macaulay outside the closed point and admits a dualising complex has the property that has finite length for . This argument can also be found in the main theorem [HL07]. ## 6. Application: A question of Karen Smith The main result of Hochster-Huneke [HH92] is a result in commutative algebra. While geometrising it in [Smi97c], K. Smith arrived at the following question (see [Smi97a]): ###### Question 6.1. Let be a projective variety over a field of characteristic , and let be a “weakly positive” line bundle on . For any and any , does there exist a finite surjective morphism such that is ? Using the algebraic result of Hochster-Huneke [HH92], one can show that if we take “weakly positive” to mean ample, then Question 6.1 has an affirmative answer (see Remark 6.4). Smith had originally hoped that “weakly positive” could be taken to mean nef. We give some examples in the sequel to show that this cannot be the case. However, first, we prove some positive results using the theorems above. ### 6.1. Positive results We first examine Question 6.1 in the case of positive twists. It is clear that being ample is a sufficiently positive condition for the required vanishing statement to be true: Frobenius twisting can be realised by pulling back along a finite morphism and has the effect of changing by , whence Serre vanishing shows the desired result. It is natural to wonder if the result passes to the closure of the ample cone, i.e., the nef cone. We show in Example 6.10 that this is not the case: there exist non-torsion degree line bundles on surfaces whose middle cohomology cannot be killed by finite covers. On the other hand, Corollary 3.5 coupled with the fact that torsion line bundles can be replaced with on passage to a finite cover ensures that Question 6.1 has a positve answer for torsion line bundles. The necessity of the non-torsion requirement and the observation that torsion line bundles are semiample suggests the following proposition. ###### Proposition 6.2. Let be a proper variety over a field of characteristic , and let be a semiample line bundle on . For any , there exists a finite surjective morphism such that the induced map is . ###### Proof. As is a semiample bundle, there exists some positive integer such that is globally generated. If we fix a basis for , then the cyclic covering trick (see [Laz04a, Proposition 4.1.3]) ensures that there’s a finite flat cover such that admits an -th root in and, consequently, is globally generated. In particular, as semiamplitude is preserved under pullbacks, we may replace with and assume that arises as the pullback of an ample bundle under a proper surjective morphism . Furthermore, once is fixed, to show the required vanishing statement, we may always replace by for because the Frobenius morphism is finite surjective with . Now the projection formula for implies that . Using Theorem 0.4, we may find a finite surjective morphism such that, with , we have a factorisation of the natural map . Applying to the composite morphism gives us the desired morphism. Thus, to show the required statement, it suffices to show that	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9811004996299744, "perplexity": 379.99739726256087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400203096.42/warc/CC-MAIN-20200922031902-20200922061902-00511.warc.gz"}
http://algebra2014.wikidot.com/hw10-problem-9	Hw10 Problem 9 ### Problem 9 (+) Let $G$ be a group that is generated by the set $\{a_i\|i\in I\}$ where $I$ is some indexing set and $a_i\in G$ for all $i\in I$. Let $\phi:G\rightarrow G'$ and $\mu: G\rightarrow G'$ be two homomorphisms from $G$ into a group $G'$ such that $\phi(a_i)=\mu(a_i)$ for all $i\in I$.Prove that $\phi=\mu$. (This shows that any homomorphism of $G$ is determined by its action on the generators of $G$.) Solution Let $G=\{a_1^{n_1}a_2^{n_2}\dots a_i^{n_i}\|i\in I,n_i\in\mathbb Z^+\cup\{0\}\}$.(Since $G$ is generated by $\{a_i\|i\in I\}$). Because $\phi: G\rightarrow G'$ and $\mu:G\rightarrow G'$ are two homomorphisms,then $\forall x\in G$,we have $\phi(x)=\phi(a_1^{n_1}a_2^{n_2}\dots a_i^{n_i})$ $=\phi(a_1^{n_1})\phi(a_2^{n_2})\dots \phi(a_i^{n_i})$ $=[\phi(a_1)]^{n_1}[\phi(a_2)]^{n_2}\dots [\phi(a_i)]^{n_i}$ and $\mu(x)=\mu(a_1^{n_1}a_2^{n_2}\dots a_i^{n_i})$ $=\mu(a_1^{n_1})\mu(a_2^{n_2})\dots \mu(a_i^{n_i})$ $=[\mu(a_1)]^{n_1}[\mu(a_2)]^{n_2}\dots [\mu(a_i)]^{n_i}$ by the homomorphism property. Because $\phi(a_i)=\mu(a_i)$ for all $i\in I$, $\phi(x)=\phi(a_1^{n_1}a_2^{n_2}\dots a_i^{n_i})$=$\mu(x)=\mu(a_1^{n_1}a_2^{n_2}\dots a_i^{n_i})$ That is,$\forall x\in G$, $\phi(x)=\mu(x)$. Hence,$\phi=\mu$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.986945390701294, "perplexity": 136.283936740555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125945493.69/warc/CC-MAIN-20180422041610-20180422061610-00101.warc.gz"}
http://math.stackexchange.com/questions/157713/rank-for-the-matrix-of-concatenating-all-n-times-n-permutation-matrics	# rank for the matrix of concatenating all $N \times N$ permutation matrics Consider all $N\times N$ permutation matrix $\{M_1,M_2,\ldots,M_{N!}\}$ Define $S_N$ as concatenating each $\operatorname{vec}(M_i)$ as $S_N$'s $i$th column Is there any convenient way to calculate $\operatorname{rank}(S_N)$ ? Take $N=3$ for example. $$M_1=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ $$M_2=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}\right]$$ $$M_3=\left[\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}\right]$$ ... then $$S_3=\left[\begin{matrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \end{matrix}\right]$$ and $\operatorname{rank}(S_3)=5$ Thanks for any suggestions! - In general this should be 1 plus the dimension $(n-1)^2$ of the Birkhoff polytope. The extra $1$ just comes from it lying in an affine, rather than linear, subspace. Thanks, I found a similar problem, which might be more general(in some aspects) here. mt_ even gives the $(1+(n-1)^2)$ basis for this space. – Benson Jun 13 '12 at 11:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9804106950759888, "perplexity": 813.3555517457789}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257824185.14/warc/CC-MAIN-20160723071024-00106-ip-10-185-27-174.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/82804/determining-if-a-matrix-of-linear-forms-represents-a-non-degenerate-matrix?sort=newest	# determining if a matrix of linear forms represents a non-degenerate matrix Let $k$ be a field with $p$ elements. Consider the following computational problem Input: a natural number $n$, $n^2$ linear forms $M_{ij}$, $i,j=1,\ldots n$ in $n^2$ variables $X_{11}, \ldots X_{nn}$. Problem: Is there an assignement of values to the variables $X_{ij}$ so that the matrix $M_{ij}$ is invertible? ${}$ Question: What is known about algorithms for this problem? As usually, let's assume the addition and multiplication in the field to have computational cost $1$. The naive algorithm of checking each assignment of the variables $X_{ij}$ takes time bounded by a polynomial in $p^{n^2}$. I'd be interested to know if there is an improvement to polynomial in $p^n$ (or better). EDIT: Below Emil Jeřábek shows that the problem is NP-complete, but the reduction from 3-SAT is done in such a way that it still could be that there is an improvement to $p^n$ without proving anything unexpected about 3-SAT. EDIT: The special case when each $M_{ij}$ is equal either to $0$ or to $X_{ij}$ is solved below by Emil Jeřábek. EDIT: I've decided to ask a more specific follow-up question. - I've posted it also to cstheory.stackexchange.com/questions/9316/… – Łukasz Grabowski Dec 6 '11 at 17:52 As an experienced user, you are aware that simultaneous cross-posting is discouraged on both these sites? – Emil Jeřábek Dec 6 '11 at 18:42 no, thanks for letting me know, i'll keep it mind – Łukasz Grabowski Dec 7 '11 at 0:09 Trying to solve $M_{ij}=c_{ij}$ is a linear system in n^2 unknowns and n^2 equations. If it has a solution for suitable $c_{ij}$ you are done, but the matrix might not be full rank. – joro Dec 7 '11 at 16:08 @joro: this is just another naive algorithm which takes time bounded by a polynomial in $p^{n^2}$, as this is how many possibilities for $c_{ij}$ there are, and it's not difficult to find examples where only $p-1$ choices for $c_{ij}$ lead to an invertible matrix. – Łukasz Grabowski Dec 7 '11 at 16:31 The determinant of $M$, considered as a matrix over the polynomial ring $R=k[X_{11},\dots,X_{nn}]$, is a polynomial $f\in R$, and your problem is to determine whether $f$ defines the constant $0$ function over $k$. There are several division-free algorithms for computation of determinant in any commutative ring using polynomially many ring operations. In principle, you can use such an algorithm to compute $f$, but this may result in a long expression, since $f$ may have exponentially many terms. It is better to combine this with a polynomial identity testing algorithm: since $f$ has degree $n$, the Schwartz–Zippel lemma tells you that $f(a_{11},\dots,a_{nn})\ne0$ for randomly chosen $a_{ij}\in k$ with probability at least $1-n/p$, as long as $f\ne0$ and $p>n$. Thus, if (say) $p>2n$, you don’t actually have to evaluate $f$, you have a simple probabilistic polynomial-time algorithm: choose random assignment to your variables, and test whether the resulting matrix is nonsingular. In the special case where each $M_{ij}$ is either $0$ or $X_{ij}$, things are much simpler: it is easy to see that there is an assignment making the matrix nonsingular if and only if there is a permutation $\pi$ such that $M_{i\pi(i)}\ne0$ for each $i$, in other words, if and only if the bipartite graph whose biadjacency matrix is defined from $M$ by replacing every $X_{ij}$ with $1$ has a perfect matching. There are various efficient algorithm for finding perfect matchings and/or checking their existence, see e.g. http://en.wikipedia.org/wiki/Perfect_matching#Algorithms_and_computational_complexity . EDIT: For a fixed $p$, the full problem is NP-complete. The reduction is simplest for $\mathbb F_2$. Assume we are given a $3$-CNF $C(x_1,\dots,x_n)=\bigwedge_{i< m}\bigvee_{j< 3}l_{ij}$, where each $l_{ij}$ is either some $x_k$ or $\neg x_k$. We identify $\neg x_k$ with the linear polynomial $1+x_k$. Then $C$ is satisfiable if and only if the polynomial $f(x_1,\dots,x_n)=\prod_i(1+\prod_jl_{ij})$ assumes a nonzero value for some assignment over $\mathbb F_2$. Let $M$ be the block diagonal matrix consisting of $m$ blocks $M_i$, where $$M_i=\begin{pmatrix}1&l_{i0}&0\\\0&1&l_{i1}\\\\l_{i2}&0&1\end{pmatrix}.$$ Then $\det(M)=f$, thus there is an assignment in $\mathbb F_2$ making $M$ invertible iff $C$ is satisfiable. One can get rid of the constants $1$ in the matrix as follows: we include in $M$ the identity matrix as yet another block, and then we replace $1$ with a new variable $x_0$ everywhere (i.e., in the $1$ entries as well as in the entries of the form $1+x_k$). Any assignment making the new matrix invertible must have $x_0=1$, and then it works as before. The new matrix has only entries of the form $0$, $x_k$, $x_0+x_k$. - Thanks, this solves the special problem. I'll reformulate the question to make clear that I mean $p$ to be fixed; in particular the probabilistic polynomial-time algorithm you suggest is of not much use. – Łukasz Grabowski Dec 7 '11 at 0:11 After thinking about it, I don't understand your solution to the special problem. I agree that it's equivalen to checking if there is a permutaion $\pi$ such that $M_{i\pi(i)}\neq 0$ for all $i$. But what doe it have to do with perfect matchings? Could you give a reference or expand the argument? – Łukasz Grabowski Dec 7 '11 at 13:03 Also, if you conider the case when $M_{ij}$ is non-zero for $(i,j)= (1,2), (2,3), (3,1)$, there's no perfect matching. Perhaps you meant that vertiices lie in a union of disjoint cycles? (mathings would be then "cycles of length 2", but then the problem dangerously reminds me of finding a hamiltonian cycle) – Łukasz Grabowski Dec 7 '11 at 13:06 I’m talking about bipartite graphs. Your matrix corresponds to the graph with vertex set $U\cup V$, $U=\{u_1,u_2,u_3\}$, $V=\{v_1,v_2,v_3\}$, and edge set $E=\{(u_1,v_2),(u_2,v_3),(u_3,v_1)\}$. This graph has a perfect matching, in fact, it is a perfect matching. – Emil Jeřábek Dec 7 '11 at 13:12 It's useful to know it's NP complete, but note that you encode a 3-sat instance of $m$ clauses with $n$ variables into a $3m \times 3m$ matrix. I believe this can be solved in a number of steps which is polynomial in $2^m$, so this doesn't seem to directly answer my question whether you can go from $p^{n^2}$ to $p^n$. – Łukasz Grabowski Dec 7 '11 at 14:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9210348725318909, "perplexity": 162.1494007389663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207928423.12/warc/CC-MAIN-20150521113208-00038-ip-10-180-206-219.ec2.internal.warc.gz"}
https://tex.stackexchange.com/questions/180724/bad-rendering-of-parentheses-of-binomial-coefficient-in-beamer/180742	# Bad rendering of parentheses of binomial coefficient in Beamer I am working with documentclass Beamer version 3.10 and I have a problem with the rendering of the parentheses of the binomial coefficient. Here the code: \documentclass[aspectratio=169, 8pt]{beamer} \usefonttheme{serif} %required for article-like displaying of math \usepackage{amsmath,amssymb} \begin{document} \begin{frame}[plain,t] $\dbinom{n}{k}$ \end{frame} \end{document} And this is the output: The weird thing is that if I change the documentclass to article and comment the line \usefonttheme{serif}, then I get which is the desired rendering. Any idea of how to correct the printing of the binomial coefficient in beamer? Why this strange behavior in Beamer? Note: I have tried with \choose, with $\displaystyle \binom{n}{k}$ and putting the expression within an equation environment, but without success. • Thanks to @StevenBSegletes and Svend Tveskaeg by their replies, all of their solutions work, I just add an additional question in my original post about why that behavior in Beamer. – Carlos Mendoza May 29 '14 at 11:02 \documentclass[aspectratio=169, 8pt]{beamer} \usefonttheme{serif} %required for article-like displaying of math \usepackage{amsmath,amssymb,scalerel} \usepackage[usestackEOL]{stackengine} \stackMath \begin{document} \begin{frame}[plain,t] $\dbinom{n}{k}$ DBINOM $\scaleleftright[1.7ex]{(}{\Centerstack{n\\k}}{)}$ SCALEREL + STACKENGINE $\parenVectorstack{n\\k}$ JUST STACKENGINE \end{frame} \end{document} If you don't have to use 8pt font size, 10pt does the trick: \documentclass[10pt]{beamer} \usefonttheme{serif} \usepackage{amsmath,amssymb} \begin{document} \begin{frame} $\dbinom{n}{k}$ \end{frame} \end{document} • With global font size of 8 pt and sorrounding the math expression with \large also works: {\large [\binom{n}{k}]} In any case I still do not understand why is all this necessary in Beamer. – Carlos Mendoza May 29 '14 at 12:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8055357336997986, "perplexity": 2771.845266042409}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487626122.27/warc/CC-MAIN-20210616220531-20210617010531-00002.warc.gz"}
http://mathhelpforum.com/advanced-statistics/19981-urgent-help-joint-distributions-random-vectors-weiner-processes.html	## Urgent Help With Joint Distributions of Random Vectors (Weiner Processes) Hi, I really need help with this problem, any help would be greatly appreciated.. Compute the joint density of the random vector (W_2 , 2W_3 , W_5) Now i have calculated the joint density of (W_2 , 2W_3) and it comes to 1/(4pisqrt2) exp ( -1/8y^2 + 1/2xy - 3/4*x^2) Does anyone have any idea how i can use this information to calculate the density specified above??? Thanks Ps. W is the standard Weiner process	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9689371585845947, "perplexity": 787.874600489753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257829970.64/warc/CC-MAIN-20160723071029-00188-ip-10-185-27-174.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/2610/neukirchs-class-field-axiom-and-cohomology-of-units-for-unramified-extension	# Neukirch's class field axiom and cohomology of units for unramified extension This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the tate cohomology groups H^n(G(L\|K),UL) for n=0,-1 vanish for finite unramified extensions L\|K, where UL is the group of units. He mentions in the proof that every element a \in AL can be written as a = \epsilon * \piK^m, where \epsilon \in UL and \piK is a prime element in AK. Why does this work? I absolutely understand this argument, when the image of the valuation just lies in \ZZ! But how does this work for a valuation whose image is \widehat{\ZZ}? Unless A is not a profinite module, I don't know what \piK^m is for some general m \in \widehat{\ZZ}. Unfortunately this has to work in this generality for global class field theory. (\ZZ denotes the integers of course, sorry for my personal notation.) - You dont need to make sense out of piKm for a general m in Z-hat. All you really need to know for his argument is that vK(AK) = vL(AL) as subgroups of Z-hat. I didn't think this through but I think it should be pretty easy to establish from the fact that piK is prime for both valuations. All he really uses is that the Galois group fixes piK. - The problem is somehow that if m \in \ZZ, then v<sub>L</sub>(a) = v<sub>K</sub>(\epsilon * \pi<sub>K</sub>^m) = m and therefore a would be a very special element of A<sub>L</sub>. I don't know why the valuation of a should lie in \ZZ and not somewhere in \widehat{Z}... – user717 Oct 26 '09 at 16:48 Ah, no HTML available in comments. Sorry. – user717 Oct 26 '09 at 16:50 Again, you don't need to show that you can take m in Z. Forget about pi as well for the moment. It would suffice to show that a=epsilon.x where v(epsilon)=0 and x is fixed by the Galois group. Show that the two valuations take the same values and you're done. – Joel Dodge Oct 26 '09 at 19:20 Oh, that works of course! But then this looks like a mistake there!? Anyway. – user717 Oct 27 '09 at 9:13	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9509528279304504, "perplexity": 698.1405199659931}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860122501.61/warc/CC-MAIN-20160428161522-00014-ip-10-239-7-51.ec2.internal.warc.gz"}
https://www.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=&number=2254	WIAS Preprint No. 2254, (2016) # Thin-film electrodes for high-capacity lithium-ion batteries: Influence of phase transformations on stress Authors • Meca Álvarez, Esteban • Münch, Andreas • Wagner, Barbara 2010 Mathematics Subject Classification • 74N20 35Q74 74S10 Keywords • Phase-field Model, Interface Dynamics, Numerical Methods Abstract In this study we revisit experiments by Sethuraman et al. [J. Power Sources, 195, 5062 (2010)] on the stress evolution during the lithiation/delithiation cycle of a thin film of amorphous silicon. Based on recent work that show a two-phase process of lithiation of amorphous silicon, we formulate a phase-field model coupled to elasticity in the framework of Larché-Cahn. Using an adaptive nonlinear multigrid algorithm for the finite-volume discretization of this model, our two-dimensional numerical simulations show the formation of a sharp phase boundary between the lithiated and the amorphous silicon that continues to move as a front through the thin layer. We show that our model captures the non-monotone stress loading curve and rate dependence, as observed in experiments and connects characteristic features of the curve with the stucture formation within the layer. We take advantage of the thin film geometry and study the corresponding one-dimensional model to establish the dependence on the material parameters and obtain a comprehensive picture of the behaviour of the system. Appeared in • Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 472 (2016) pp. 20160093/1--20160093/15.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8010641932487488, "perplexity": 1961.4828078608575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574159.19/warc/CC-MAIN-20190921001810-20190921023810-00061.warc.gz"}
https://www.legisquebec.gouv.qc.ca/en/version/cs/R-3.1?code=se:10_1&history=20220920	### R-3.1 - Act to promote the reform of the cadastre in Québec 10.1. The Minister shall give notice of his intention to renew the cadastre to the Land Registry Office and the municipality concerned; he shall also give such notice to the owner of each lot affected by the renewal, at the address appearing on the assessment roll. The notice shall indicate, in particular, the object, procedure and consequences of the renewal. 1992, c. 29, s. 5; 1993, c. 52, s. 24; 2000, c. 42, s. 212; 2020, c. 17, s. 93. 10.1. The Minister shall give notice of his intention to renew the cadastre to the registry office and the municipality concerned; he shall also give such notice to the owner of each lot affected by the renewal, at the address appearing on the assessment roll. The notice shall indicate, in particular, the object, procedure and consequences of the renewal; it shall be posted at the registry office of the registration division concerned, by the registrar. 1992, c. 29, s. 5; 1993, c. 52, s. 24; 2000, c. 42, s. 212. 10.1. The Minister shall give notice of his intention to renew the cadastre to the registry office of the registration division and the municipality affected; he shall also give such notice to the owner of each lot affected by the renewal, at the address appearing on the assessment roll. The notice shall indicate, in particular, the object, procedure and consequences of the renewal; it shall be posted at the registry office of the registration division by the registrar. 1992, c. 29, s. 5; 1993, c. 52, s. 24. 10.1. The Minister shall give notice of his intention to renew the cadastre to the registry office of the registration division and the municipality concerned; he shall also give such notice to the owner of each lot affected by the renewal, at the address appearing on the assessment roll. The notice shall indicate, in particular, the object, procedure and consequences of the renewal; it shall be posted at the registry office of the registration division by the registrar. 1992, c. 29, s. 5.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9491490125656128, "perplexity": 4849.90456587693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710869.86/warc/CC-MAIN-20221201185801-20221201215801-00121.warc.gz"}
https://advances.sciencemag.org/content/4/6/eaat0346.full	Research ArticleMATERIALS SCIENCE Higher-order topological insulators See allHide authors and affiliations Vol. 4, no. 6, eaat0346 Abstract Three-dimensional topological (crystalline) insulators are materials with an insulating bulk but conducting surface states that are topologically protected by time-reversal (or spatial) symmetries. We extend the notion of three-dimensional topological insulators to systems that host no gapless surface states but exhibit topologically protected gapless hinge states. Their topological character is protected by spatiotemporal symmetries of which we present two cases: (i) Chiral higher-order topological insulators protected by the combination of time-reversal and a fourfold rotation symmetry. Their hinge states are chiral modes, and the bulk topology is Z2-classified. (ii) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs, and the bulk topology is Z-classified. We provide the topological invariants for both cases. Furthermore, we show that SnTe as well as surface-modified Bi2TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states. INTRODUCTION The bulk-boundary correspondence is often taken as a defining property of topological insulators (TIs) (13): If a d-dimensional system with given symmetry is insulating in the bulk but supports gapless boundary excitations that cannot be removed by local boundary perturbations without breaking the symmetry, then the system is called TI. The electric multipole insulators in the study of Benalcazar et al. (4) generalize this bulk-boundary correspondence: In two and three dimensions, these insulators exhibit no edge or surface states, respectively, but feature gapless, topological corner excitations corresponding to quantized higher electric multipole moments. Here, we introduce a new class of three-dimensional (3D) topological phases to which the usual form of the bulk-boundary correspondence also does not apply. The topology of the bulk protects gapless states on the hinges, while the surfaces are gapped. Both systems, with gapless corner and hinge states, respectively, can be subsumed under the notion of higher-order TIs (HOTI): An nth-order TI has protected gapless modes at a boundary of the system of codimension n. Following this terminology, we introduce second-order 3D TIs in this work, while the study of Benalcazar et al. (4) has introduced second-order 2D TIs and third-order 3D TIs. The important aspect of 3D HOTIs is that they exhibit protected hinge states with (spectral) flow between the valence and conduction bands, whereas the corner states have no spectral flow. The topological properties of HOTIs are protected by symmetries that involve spatial transformations, possibly augmented by time reversal. They thus generalize topological crystalline insulators (5, 6), which have been encompassed in a recent exhaustive classification of TIs in the study of Bradlyn et al. (7). Here, we propose two cases: (i) chiral HOTIs with hinge modes that propagate unidirectionally, akin to the edge states of a 2D quantum Hall effect (8), or Chern insulator (9). We show that chiral HOTIs may be protected by the product C^4T^ of time reversal T^ and a C^4 rotation symmetry. The existence of these hinge modes—but not the direction in which they propagate—is determined by the topology of the 3D bulk. By a C^4T^-respecting surface manipulation, the direction of all hinge modes can be reversed, but they cannot be removed. This constitutes a bulk Z2 topological classification. We also show that chiral HOTIs may have a bulk Z topological classification protected by mirror symmetries that leave the hinges invariant when time-reversal symmetry T^ is broken. (ii) helical HOTIs with Kramers pairs of counterpropagating hinge modes, akin to the edge states of a 2D quantum spin Hall effect (1, 1012). We show that helical HOTIs may occur when a system is invariant under time reversal T^ and a C^4 rotation symmetry. We further show that helical HOTIs can also be protected by T^ and mirror symmetries that leave the hinges invariant. Any integer number of Kramers pairs is topologically protected against symmetry-preserving surface manipulations, yielding a Z classification. For both cases, we show the topological bulk-surface-hinge correspondence, provide concrete lattice-model realizations, and provide expressions for the bulk topological invariants. The latter are given by the magnetoelectric polarizability and mirror Chern numbers (6, 13), for chiral and helical HOTIs, respectively. For the case where a chiral HOTI also respects the product of inversion times time-reversal symmetry I^T^, we formulate a simplified topological index akin to the Fu-Kane formula for inversion symmetric TIs (2). Finally, on the basis of tight-binding and ab initio calculations, we propose SnTe as a material realization for helical HOTIs. We also propose an explicit experimental setup to cleanly create hinge states in a topological SnTe coaxial cable. In contrast, chiral HOTIs may arise in 3D TI materials that exhibit noncollinear antiferromagnetic order at low temperatures. Our work is complemented by two related articles: The study of Langbehn et al. (14) provides a general classification of second-order phases with reflection symmetry for all 10 Altland-Zirnbauer symmetry classes, and the study of Benalcazar et al. (15) establishes a physical interpretation of the topological invariants of higher-order phases in terms of electric multipole moments. RESULTS Chiral HOTI We first give an intuitive argument for the topological nature of a chiral 3D HOTI. We consider a hypothetical but realizable electronic structure where gapless degrees of freedom are only found on the hinge. For concreteness, let us consider a system with a square cross section, periodic boundary conditions in z direction, and C^4zT^ symmetry that has a single chiral mode at each hinge, as sketched in Fig. 1A. For these modes to be a feature associated with the 3D bulk topology of the system, they should be protected against any C^4zT^-preserving surface or hinge perturbation of the system. The minimal relevant surface perturbation of that kind is the addition of an integer quantum Hall (or Chern insulator) layer with Hall conductivity σxy = e2/h and σxy = − e2/h on the (100) surfaces and the (010) surfaces, respectively, which respects C^4zT^. As seen from Fig. 1C, this adds to each hinge two chiral hinge channels. Repeating this procedure, we can change—via a pure surface manipulation—the number of chiral channels on each hinge by any even number. Hence, only the Z2 parity of hinge channels can be a topological property protected by the system’s 3D bulk. A concrete model of this phase is defined via the four-band Bloch HamiltonianHc(k)=(M+ticos ki)τzσ0+Δ1isin kiτxσi+Δ2(cos kxcos ky)τyσ0(1)where σi and τi, i = x, y, z, are the three Pauli matrices acting on spin and orbital degree of freedoms, respectively (see Methods for a real-space representation of the model). For Δ2 = 0, Hc(k) represents the well-known 3D TI if 1 < \|M\| < 3. Time-reversal symmetry is represented by THc(k)T1=Hc(k), with T ≡ τ0σyK, where K denotes complex conjugation. For Δ2 = 0, Hamiltonian (1) has a C^4z rotation symmetry C4zHc(k)(C4z)1=Hc(DC^4zk), where C4zτ0eiπ4σz and DC^4zk=(ky,kx,kz). The term proportional to Δ2 breaks both T^ and C^4z individually but respects the antiunitary combination C^4zT^, which means that(C4zT)Hc(k)(C4zT)1=Hc(DC^4zT^k),DC^4zT^k=(ky,kx,kz)(2)is a symmetry of the Hamiltonian also when Δ2 ≠ 0. Because [C^4z,T^]=0, we have (C^4zT^)4=1, independent of the choice of representation. The phase diagram of Hamiltonian (1) is shown in Fig. 2A. For 1 < \|M/t\| < 3 and Δ1, Δ2 ≠ 0, the system is a chiral 3D HOTI. The spectrum in the case of open boundary conditions in x and y directions is presented in Fig. 2C, where the chiral hinge modes (each twofold degenerate) are seen to traverse the bulk gap. Physically, the term multiplied by Δ2 corresponds to orbital currents that break time-reversal symmetry oppositely in the x and y directions. When infinitesimally small, its main effect is thus to open gaps with alternating signs for the surface Dirac electrons of the 3D TI on the (100) and (010) surfaces. The four hinges are then domain walls at which the Dirac mass changes sign. It is well known (16, 17) that such a domain wall on the surface of a 3D TI binds a gapless chiral mode, which, in the case at hand, is reinterpreted as the hinge mode of the HOTI. Another physical mechanism that breaks time-reversal symmetry and preserves C^4zT^ would be (π, π, 0) noncollinear antiferromagnetic order with a unit cell as shown in Fig. 2B. Note that even with finite Δ2, the (001) surface of the model remains gapless, because its Dirac cone is protected by the C^4zT^ symmetry that leaves the surface invariant and enforces a Kramers-like degeneracy discussed in the Supplementary Materials. The gapless nature of the (001) surface in the geometry of Fig. 1B is also required by current conservation because the chiral hinge currents cannot terminate in a gapped region of the sample. A current-conserving geometry with gapped surfaces is given in the Supplementary Materials. We turn to the bulk topological invariant that describes the Z2 topology. The topological invariant of 3D TIs is the theta angle or Chern Simons invariant θ (see Methods for its definition), which is quantized by time-reversal symmetry to be θ = 0, π mod 2π, with θ = π being the nontrivial case (18). The very same quantity θ is the topological invariant of chiral HOTIs. What changes is that its quantization to values 0, π is not enforced by T^ but by C^4T^ symmetry in this case. θ attains a new meaning in the second-order picture: It uniquely characterizes a different symmetry-protected topological phase that exhibits T^-breaking but C^4T^-preserving hinge currents instead of T^-preserving gapless surface excitations. In the Supplementary Materials, we show the quantization of θ enforced by C^4T^ symmetry and explicitly evaluate θ = π for the model (1). We furthermore note that for a nontrivial θ in the presence of C^4T^ symmetry to uniquely characterize the presence of gapless hinge excitations, the bulk and the surfaces of the material that adjoin the hinge are required to be insulating. This constitutes the bulk-surface-hinge correspondence of chiral HOTIs. The explicit evaluation of θ is impractical for ab initio computations in generic insulators. This motivates the discussion of alternative forms of the topological invariant. The Pfaffian invariant (1) used to define first-order 3D TIs rests on the group relation T^2=1, and it fails in our case where (C^4T^)4=1. We may instead use a non-Abelian Wilson loop characterization of the topology, as presented in the Supplementary Materials (19, 20). There, we also provide two further topological characterizations, one based on so-called nested Wilson loop (4) and entanglement spectra (2123) and one applicable to systems that are in addition invariant under the product I^T^ of inversion symmetry I^ and T^ (3). Helical HOTI Helical HOTIs feature Kramers pairs of counterpropagating hinge modes. They are protected by time-reversal symmetry and a spatial symmetry. For concreteness, let us consider a system with a square (or rhombic) cross section, periodic boundary conditions in the z direction, and two mirror symmetries M^xy and M^xy¯ that leave, respectively, the x = −y and the x = y planes invariant and with it a pair of hinges each (as sketched in Fig. 1B). We consider a hypothetical but realizable electronic structure where gapless degrees of freedom are only found on the hinge. At a given hinge, for instance, one that is invariant under M^xy, we can choose all hinge modes as eigenstates of M^xy. We denote the number of modes that propagate parallel, R, (antiparallel, L) to the z direction and have M^xy eigenvalue iλ, λ = ± 1, by NR,λ (NL,λ). We argue that the net number of helical hinge pairs nNR,+NL,+ (which by time-reversal symmetry is equal to NL,−NR,−) is topologically protected. In particular, n cannot be changed by any surface or hinge manipulation that respects both T^ and M^xy. First, note that if both NR,+ and NL,+ are nonzero (assuming from now on that NR,+ > NR,−), then we can always hybridize NL,+ right-moving modes with all NL,+ left-moving modes within the λ = + subspace without breaking any symmetry. Therefore, only the difference n is well defined and corresponds to the number of remaining pairs of modes. The argument for their topological protection proceeds similar to the chiral HOTI case by considering a minimal symmetry-preserving surface perturbation. It consists of a layer of a 2D time-reversal symmetric TI and its mirror-conjugated partner added to surfaces that border the hinge under consideration. Each of the TIs contributes a single Kramers pair of boundary modes to the hinge so that (NL,− + NL,+) and (NR,− + NR,+) each increase by 2 (see Fig. 3A). Because mirror symmetry maps the right-moving modes of the two Kramers pairs onto one another (and the same for the two left-moving modes), we can form a “bonding” and “antibonding” superposition with mirror eigenvalues + i and − i out of each pair. Thus, each of NL,+, NL,−, NR,+, and NR,− increases by 1 because of this minimal surface manipulation. This leaves n invariant, suggesting a Z classification of the helical HOTI for each pair of mirror-invariant hinges. The case depicted in Fig. 1B with two mirror symmetries is then Z×Z–classified. A more rigorous version of this argument can be found in the Supplementary Materials. The topological invariant for the Z×Z classification of the helical HOTI is the set of mirror Chern numbers (6, 13) Cm/2 on the M^xy and M^xy¯ mirror planes (see Methods for the definition of Cm). First, observe that if Cm were odd, then the system would be a strong 3D TI: The M^xy mirror planes in momentum space include all time-reversal invariant momenta in the (110) surface Brillouin zone. Thus, if Cm is odd, then there is an odd number of Dirac cones on the (110) surface, and time-reversal symmetry implies that such a system is a strong 3D TI. As the surfaces of a strong 3D TI cannot be gapped out with a time-reversal symmetric perturbation, we cannot construct a helical HOTI from it. We conclude that Cm is even for all systems of interest to us. We now discuss the correspondence between the bulk topological invariant Cm/2 and the existence of Kramers-paired hinge modes. For this, we first consider the electronic structure of the (110) surface, which is invariant under M^xy and then that of a pair of surfaces with a normal n± = (1 ± α, 1 ∓ α, 0) for small α, which are mapped into each other under M^xy and form a hinge at their interface (see Fig. 3, B to D). A nonzero bulk mirror Chern number Cm with respect to the M^xy symmetry enforces the existence of gapless Dirac cones on the (110) surface. These Dirac cones are pinned to the mirror invariant lines k1 = 0, π in the surface Brillouin zone of the (110) surface, where k1 is the momentum along the direction with the unit vector e^1=(e^xe^y)/2. If we consider the electronic structure along these lines in momentum space (see Fig. 3B), then each Dirac cone has an effective Hamiltonian HD=v1σz(k1k1(0))+vzσx(kzkz(0)) when expanded around a Dirac point at (k1,kz)=(k1(0),kz(0)) for k1(0)=0 or k1(0)=π. The mirror symmetry is represented by Mxy = iσx, preventing mass terms of the form mσy from appearing. The sign of vz is tied to the M^xy eigenvalue (i sgn vz) of the eigenstate with a positive group velocity in the z direction (at k1k1(0)=0). Denoting the total number of Dirac cones with vz > 0 (vz < 0) by n+ (n), the bulk-boundary correspondence of a topological crystalline insulator (5) impliesCm=n+n(3)Consider now a pair of surfaces with slightly tilted normals n+ and n, which are not invariant under the mirror symmetry but map into each other. Mass terms are allowed, and the Hamiltonians on the surfaces with normal n± readHD,±=v1σz(k1k1(0)±κα)+vzσx(kzkz(0))±mασy(4)to linear order in α, (k1k1(0)) and (kzkz(0)) with m and κ real parameters. The two surfaces with normals n+ and n meet in a hinge (see Fig. 3B). Equation 4 describes a Dirac fermion with a mass of opposite sign on the two surfaces. The hinge therefore forms a domain wall in the Dirac mass from which a single chiral channel connecting valence and conduction bands arises (24). As we show in the Supplementary Materials, this domain wall either binds an R moving mode with Mxy mirror eigenvalue iλ = i sgn(vz) or an L moving mode with mirror eigenvalue − i sgn(vz). The equality nsgn(vz)=NR,sgn(vz)+NL,sgn(vz) follows, which connects the number of hinge modes NL/R,± we had introduced before to the mirror-graded numbers of Dirac cones on the (110) surface n±. From Eq. 3, we obtainCm=(NR,+NR,+NL,NL,+)2n(5)relating the 3D bulk invariant Cm to the number of protected helical hinge pairs n of the HOTI. Notice that by time-reversal symmetry, NR,+NR,− = NL,−NL,+, so that n in Eq. 5 is an integer. Cm is even as aforementioned. Note that the above deformation of the surfaces can be extended to nonperturbative angles α, until, for example, the (100) and (010) surface orientations are reached. The surfaces on either side of the hinge may undergo gap-closing transitions as α is increased. But as we argued at the beginning of the section, surface transitions of this kind may not change the net number of helical hinge states with a given mirror eigenvalue, if they occur in a mirror-symmetric manner. We remark that an equation similar to Eq. 5 also holds in the absence of time-reversal symmetry for each mirror subspace. Then, the Chern number in each mirror subspace is an independent topological invariant, which gives rise to a Z×Z classification on each hinge (as opposed to Z with time-reversal symmetry). This case corresponds to chiral HOTIs protected by mirror symmetries instead of the C^4T^ symmetry used in Eq. 2. Conversely, we show in the Supplementary Materials that a helical HOTI protected by C^4 and T^ exists and has a Z2 classification. Material candidates and experimental setup We propose that SnTe realizes a helical HOTI. In its cubic rocksalt structure, SnTe is known to be a topological crystalline insulator (5, 6). This crystal structure has mirror symmetries M^xy [acting as (x, y, z) → (y, x, z)] as well as its partners under cubic symmetry (M^xy¯, M^xz, M^xz¯, M^yz, and M^yz¯). Further spatial symmetries irrelevant to the discussion are not mentioned. The bulk electronic structure of SnTe is insulating and topologically characterized by a mirror Chern number Cm = 2 with respect to the mirror symmetries on the mirror planes that include the Γ point in momentum space. All other mirror planes have Cm = 0. As a result, cubic SnTe has mirror-symmetry protected Dirac cones on specific surfaces. We consider the geometry of Fig. 1B with open boundary conditions in the x and y directions and periodic boundary conditions in the z direction. The M^xz, M^xz¯, M^yz, and M^yz¯ symmetries along with their mirror Chern numbers protect either four Dirac cones at generic surface momenta or two at the surface Brillouin zone Kramers points on the (100) as well as the (010) surfaces (see Fig. 4B). In the case at hand, the former possibility is realized. We now discuss two distortions of the crystal structure that turn SnTe into a HOTI. (1) At about 98 K, SnTe undergoes a structural distortion into a low-temperature rhombohedral phase via a relative displacement of the two sublattices along the (111) direction (25, 26). This breaks the mirror symmetries M^xz¯, M^yz¯, and M^xy¯ but preserves M^xz, M^yz, and M^xy. On the (100) surface in the geometry in Fig. 1B, for instance, the two Dirac cones protected by M^yz¯ can thus be gapped out, while the two Dirac cones protected by M^yz remain [and similarly for the (010) surface]. Therefore, the (100) and (010) surfaces remain gapless, and the geometry of Fig. 1B cannot be used to expose the HOTI nature of SnTe with (111) uniaxial displacement. For that reason, we instead consider the (1¯01) and (01¯1) surfaces, which are not invariant under any mirror symmetry of SnTe with (111) uniaxial displacement. The spectral function focused on the hinge weight of a semi-infinite geometry with a single hinge formed between the (1¯01) and (01¯1) surfaces is shown in Fig. 4E. This tight-binding calculation, based on density functional theory (DFT)–derived Wannier functions (WFs) (see Methods), demonstrates the existence of this single Kramers pair of states on the two hinges invariant under M^xy, in line with the prediction of Eq. 5 for Cm = 2. (2) If uniaxial strain along the (110) direction is applied to SnTe, then M^xz, M^xz¯, M^yz, and M^yz¯ symmetries are broken, but M^xy and M^xy¯ are preserved. This gaps the (100) and (010) surfaces in the geometry in Fig. 1B completely. We calculated the surface states by using a slab geometry along the (100) direction with DFT. Because of the smallness of the bandgap induced by strain, we needed to achieve a negligible interaction between the surface states from both sides of the slab. To reduce the overlap between top and bottom surface states, we considered a slab of 45 layers and 1 nm vacuum thickness, artificially localized the states on one of the surfaces, and added one layer of hydrogen on one of the surfaces. The evolution of the surface gap size with strain is shown in Fig. 4D (see the Supplementary Materials for more details). Figure 4F is the spectrum of a tight-binding calculation (6) with (110) strain, demonstrating that there exists one Kramers pair of hinge modes on all four hinges in the geometry of Fig. 1B. We propose to physically realize the (110) uniaxial strain in SnTe with a topological coaxial cable geometry, which would enable the use of its protected hinge states as quasi-1D dissipationless conduction channels (see Fig. 4F). The starting point is an insulating nanowire substrate made from Si or SiO, with a slightly rhombohedral cross section imprinted by anisotropic etching. SnTe is grown in layers on the surfaces by using molecular beam epitaxy, with a thickness of about 10 layers. SnTe will experience the uniaxial strain to gap out its surfaces and protect the helical HOTI phase. The hinge states can be studied by scanning tunneling microscopy and transport experiments with contacts applied through electronic-beam lithography. Note that in the process of growth, regions with step edges are likely to form on the surfaces and should be avoided in measurements, as they may carry their own gapless modes (27). Alternatively, we propose to use a superconducting substrate to study proximity-induced superconductivity on the helical hinge states. In addition to the topological crystalline insulator SnTe, we propose weak TIs with nonvanishing mirror Chern numbers as possible avenues to realize helical HOTIs. We computed the relevant mirror Chern numbers for the weak TIs Bi2TeI (28), BiSe (29), and BiTe (30), which all turn out to be 2. These materials are therefore dual TIs, in the sense that they carry nontrivial weak and crystalline topological invariants. Their surface Dirac cones are protected by a nontrivial weak index, that is, by time reversal together with translation symmetry. To gap them, it is necessary to break at least one of these symmetries, which is possible by inducing magnetic or charge density wave order. DISCUSSION We have introduced 3D HOTIs, which have gapped surfaces but gapless hinge modes, as intrinsically 3D topological phases of matter. Both time-reversal symmetry breaking and time-reversal symmetric systems were explored, which support hinge states akin to those of the integer quantum Hall effect and 2D time-reversal symmetric TIs, respectively. The former may be realized in magnetically ordered TIs; we propose the naturally occurring rhombohedral or a uniaxially distorted phase of SnTe as a material realization for the latter. Despite their global topological characterization based on spatial symmetries, the hinge states are as robust against local perturbations as quantum (spin) Hall edge modes. The concepts introduced here can be extended to define novel topological superconductors with chiral and helical Majorana modes at their hinges and may further be transferred to strongly interacting, possibly topologically ordered, states of matter and to mechanical (31), electrical (32), and photonic analogs of Bloch Hamiltonians. METHODS First-principles calculations We used DFT as implemented in the Vienna Ab initio Simulation Package (33). The exchange correlation term was described according to the Perdew-Burke-Ernzerhof prescription together with projected augmented-wave pseudopotentials (34). For the autoconsistent calculations, we used a 12 × 12 × 12 k-point mesh for the bulk and 7 × 7 × 1 for the slab calculations. For the electronic structure of SnTe with (110) distortion, the kinetic energy cutoff was set to 400 eV. We calculated the surface states by using a slab geometry along the (100) direction. Because of the smallness of the bandgap induced by strain, we needed to achieve a negligible interaction between the surface states from both sides of the slab (to avoid a spurious gap opened by the creation of bonding and antibonding states from the top and bottom surface states). To reduce the overlap between top and bottom surface states, we considered a slab of 45 layers and 1 nm vacuum thickness and artificially localized the states on one of the surfaces. The latter was performed by adding one layer of hydrogen to one of the surfaces. To obtain the electronic structure of bulk SnTe with (111) ferroelectric distortion, we set the cutoff energy for wave-function expansion to 500 eV. We used the parameter λ introduced in the study of Plekhanov et al. (35) to parameterize a path linearly connecting the cubic structure (space group Fm3m) to the rhombohedral structure (space group R3m). Our calculations are focused on the λ = 0.1 structure. Then, to obtain the hinge electronic structure, we first constructed the maximally localized WFs from the bulk ab initio calculations. These WFs were used in a Green’s function calculation for a system finite in a direction, semi-infinite in b direction, and periodic in c direction (a, b, and c are the conventional lattice vectors in the space group R3m). The hinge state spectrum was obtained by projecting on the atoms at the corner, which preserve the mirror symmetry M^xy. Chiral HOTI tight-binding model We considered a model on a simple cubic lattice spanned by the basis vectors e^i, i = x, y, z, with two orbitals dx2y2 (denoted α = 0 below) and fz(x2y2) (α = 1) on each site, which is populated by spin 1/2 electrons. It is defined by the tight-binding HamiltonianHc=M2r,α(1)αcr,αcr,α+t2r,αi=x,y,z(1)αcr+e^i,αcr,α+Δ12r,αi=x,y,zcr+e^i,α+1σicr,αΔ22ir,αi=x,y,z(1)αnicr+e^i,α+1cr,α+h.c.(6)where α is defined modulo 2, n^=(1,1,0), and cr,α=(cr,α,cr,α,) creates a spinor in orbital α at lattice site r. We denoted by σ0 and σi, i = x, y, z, respectively, the 2 × 2 identity matrix and the three Pauli matrices acting on the spin 1/2 degree of freedom. Chern-Simons topological invariant The invariant for chiral HOTIs with C^4T^ symmetry is given byθ=14πd3kabctr[AabAc+i23AaAbAc](7)written in terms of the Berry gauge field Aa;n,n=iun\|a\|un, where \|un〉 are the Bloch eigenstates of the Bloch Hamiltonian and n, n′ are running over the occupied bands of the insulator. ∂a is the partial derivative with respect to the momentum component ka, a = x, z, y. The trace is performed with respect to band indices. Mirror Chern number The topological invariant of a 3D helical HOTI is the mirror Chern number Cm. Because for a spinful system a mirror symmetry M^ satisfies M^2=1, its representation M has eigenvalues ± i. Given a surface ∑ in the Brillouin zone, which is left invariant under the action of M^, the eigenstates \|un〉 of the Bloch Hamiltonian on ∑ can be decomposed into two groups, {\|ul+} and {\|ul}, with mirror eigenvalue ±i, respectively. Time-reversal symmetry maps one mirror eigenspace into the other; if time-reversal symmetry is present, then the two mirror eigenspaces are of the same dimension. We can define the Chern number in each mirror subspace asC±=12πΣdkxdkyFxy±(k)(8)HereFab±(k)=aAb+(k)bAa+(k)+i[Aa+(k),Ab+(k)](9)is the non-Abelian Berry curvature field in the ± i mirror subspace, with Aa;l,l±=iul±\|a\|ul±, and matrix multiplication is implied in the expressions. Note that in time-reversal symmetric systems, C+ = − C, and we define the mirror Chern numberCm(C+C)/2(10) SUPPLEMENTARY MATERIALS Supplementary Text fig. S1. Nested entanglement and Wilson loop spectra for the second-order 3D chiral TI model defined in Eq. 1 in the main text with M/t = 2 and Δ1/t = Δ2/t = 1. fig. S2. Real-space hopping picture for the optical lattice model with H4=ij(x,y)(tij(x,y)cz,icz,j+tijzcz,icz+1,j). fig. S3. Real-space structure for a chiral HOTI. fig. S4. Constraints on mirror-symmetric domain wall modes in two dimensions. fig. S5. Wilson loop characterization of helical HOTIs. fig. S6. Band structure of the surface Dirac cones of the topological crystalline insulator SnTe calculated in a slab geometry. fig. S7. High-symmetry points in the BZ of SnTe. References (3640) This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited. REFERENCES AND NOTES Acknowledgments: B.A.B. wishes to thank Ecole Normale Superieure, Université Pierre et Marie Curie, Paris, and Donostia International Physics Center for their sabbatical hosting during some of the stages of this work. Funding: F.S. and T.N. acknowledge support from the Swiss National Science Foundation (grant number: 200021_169061) and from the European Union’s Horizon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP). A.M.C. wishes to thank the Aspen Center for Physics, which is supported by NSF grant PHY-1066293, for hosting during some stages of this work. M.G.V. was supported by FIS2016-75862-P national projects of the Spanish Ministry of Economy and Competitiveness. B.A.B. acknowledges support for the analytic work from the Department of Energy (de-sc0016239), Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. The computational part of the Princeton work was performed under NSF Early-Concept Grants for Exploratory Research grant DMR-1643312, ONR-N00014-14-1-0330, ARO MURI W911NF-12-1-0461, and NSF-MRSEC DMR-1420541. Author contributions: F.S., A.M.C., B.A.B., and T.N. worked out the theoretical results presented here, M.G.V. and Z.W. performed first-principles calculations, and S.S.P.P. contributed the experimental proposal for SnTe nanowires. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors. 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https://en.wikipedia.org/wiki/Triangulation_(computer_vision)	# Triangulation (computer vision) In computer vision, triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function from 3D to 2D for the cameras involved, in the simplest case represented by the camera matrices. Triangulation is sometimes also referred to as reconstruction or intersection. The triangulation problem is in principle trivial. Since each point in an image corresponds to a line in 3D space, all points on the line in 3D are projected to the point in the image. If a pair of corresponding points in two, or more images, can be found it must be the case that they are the projection of a common 3D point x. The set of lines generated by the image points must intersect at x (3D point) and the algebraic formulation of the coordinates of x (3D point) can be computed in a variety of ways, as is presented below. In practice, however, the coordinates of image points cannot be measured with arbitrary accuracy. Instead, various types of noise, such as geometric noise from lens distortion or interest point detection error, lead to inaccuracies in the measured image coordinates. As a consequence, the lines generated by the corresponding image points do not always intersect in 3D space. The problem, then, is to find a 3D point which optimally fits the measured image points. In the literature there are multiple proposals for how to define optimality and how to find the optimal 3D point. Since they are based on different optimality criteria, the various methods produce different estimates of the 3D point x when noise is involved. ## Introduction In the following, it is assumed that triangulation is made on corresponding image points from two views generated by pinhole cameras. Generalization from these assumptions are discussed here. The ideal case of epipolar geometry. A 3D point x is projected onto two camera images through lines (green) which intersect with each camera's focal point, O1 and O2. The resulting image points are y1 and y2. The green lines intersect at x. In practice, the image points y1 and y2 cannot be measured with arbitrary accuracy. Instead points y'1 and y'2 are detected and used for the triangulation. The corresponding projection lines (blue) do not, in general, intersect in 3D space and may also not intersect with point x. The image to the left illustrates the epipolar geometry of a pair of stereo cameras of pinhole model. A point x (3D point) in 3D space is projected onto the respective image plane along a line (green) which goes through the camera's focal point, ${\displaystyle \mathbf {O} _{1}}$ and ${\displaystyle \mathbf {O} _{2}}$, resulting in the two corresponding image points ${\displaystyle \mathbf {y} _{1}}$ and ${\displaystyle \mathbf {y} _{2}}$. If ${\displaystyle \mathbf {y} _{1}}$ and ${\displaystyle \mathbf {y} _{2}}$ are given and the geometry of the two cameras are known, the two projection lines (green lines) can be determined and it must be the case that they intersect at point x (3D point). Using basic linear algebra that intersection point can be determined in a straightforward way. The image to the right shows the real case. The position of the image points ${\displaystyle \mathbf {y} _{1}}$ and ${\displaystyle \mathbf {y} _{2}}$ cannot be measured exactly. The reason is a combination of factors such as • Geometric distortion, for example lens distortion, which means that the 3D to 2D mapping of the camera deviates from the pinhole camera model. To some extent these errors can be compensated for, leaving a residual geometric error. • A single ray of light from x (3D point) is dispersed in the lens system of the cameras according to a point spread function. The recovery of the corresponding image point from measurements of the dispersed intensity function in the images gives errors. • In a digital camera, the image intensity function is only measured in discrete sensor elements. Inexact interpolation of the discrete intensity function have to be used to recover the true one. • The image points y1' and y2' used for triangulation are often found using various types of feature extractors, for example of corners or interest points in general. There is an inherent localization error for any type of feature extraction based on neighborhood operations. As a consequence, the measured image points are ${\displaystyle \mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} '_{2}}$ instead of ${\displaystyle \mathbf {y} _{1}}$ and ${\displaystyle \mathbf {y} _{2}}$. However, their projection lines (blue) do not have to intersect in 3D space or come close to x. In fact, these lines intersect if and only if ${\displaystyle \mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} '_{2}}$ satisfy the epipolar constraint defined by the fundamental matrix. Given the measurement noise in ${\displaystyle \mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} '_{2}}$ it is rather likely that the epipolar constraint is not satisfied and the projection lines do not intersect. This observation leads to the problem which is solved in triangulation. Which 3D point xest is the best estimate of x given ${\displaystyle \mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} '_{2}}$ and the geometry of the cameras? The answer is often found by defining an error measure which depends on xest and then minimizing this error. In the following sections, some of the various methods for computing xest presented in the literature are briefly described. All triangulation methods produce xest = x in the case that ${\displaystyle \mathbf {y} _{1}=\mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} _{2}=\mathbf {y} '_{2}}$, that is, when the epipolar constraint is satisfied (except for singular points, see below). It is what happens when the constraint is not satisfied which differs between the methods. ## Properties A triangulation method can be described in terms of a function ${\displaystyle \tau \,}$ such that ${\displaystyle \mathbf {x} \sim \tau (\mathbf {y} '_{1},\mathbf {y} '_{2},\mathbf {C} _{1},\mathbf {C} _{2})}$ where ${\displaystyle \mathbf {y} '_{1},\mathbf {y} '_{2}}$ are the homogeneous coordinates of the detected image points and ${\displaystyle \mathbf {C} _{1},\mathbf {C} _{2}}$ are the camera matrices. x (3D point) is the homogeneous representation of the resulting 3D point. The ${\displaystyle \sim \,}$ sign implies that ${\displaystyle \tau \,}$ is only required to produce a vector which is equal to x up to a multiplication by a non-zero scalar since homogeneous vectors are involved. Before looking at the specific methods, that is, specific functions ${\displaystyle \tau \,}$, there are some general concepts related to the methods that need to be explained. Which triangulation method is chosen for a particular problem depends to some extent on these characteristics. ### Singularities Some of the methods fail to correctly compute an estimate of x (3D point) if it lies in a certain subset of the 3D space, corresponding to some combination of ${\displaystyle \mathbf {y} '_{1},\mathbf {y} '_{2},\mathbf {C} _{1},\mathbf {C} _{2}}$. A point in this subset is then a singularity of the triangulation method. The reason for the failure can be that some equation system to be solved is under-determined or that the projective representation of xest becomes the zero vector for the singular points. ### Invariance In some applications, it is desirable that the triangulation is independent of the coordinate system used to represent 3D points; if the triangulation problem is formulated in one coordinate system and then transformed into another the resulting estimate xest should transform in the same way. This property is commonly referred to as invariance. Not every triangulation method assures invariance, at least not for general types of coordinate transformations. For a homogeneous representation of 3D coordinates, the most general transformation is a projective transformation, represented by a ${\displaystyle 4\times 4}$ matrix ${\displaystyle \mathbf {T} }$. If the homogeneous coordinates are transformed according to ${\displaystyle \mathbf {\bar {x}} \sim \mathbf {T} \,\mathbf {x} }$ then the camera matrices must transform as (Ck) ${\displaystyle \mathbf {\bar {C}} _{k}\sim \mathbf {C} _{k}\,\mathbf {T} ^{-1}}$ to produce the same homogeneous image coordinates (yk) ${\displaystyle \mathbf {y} _{k}\sim \mathbf {\bar {C}} _{k}\,\mathbf {\bar {x}} =\mathbf {C} _{k}\,\mathbf {x} }$ If the triangulation function ${\displaystyle \tau }$ is invariant to ${\displaystyle \mathbf {T} }$ then the following relation must be valid ${\displaystyle \mathbf {\bar {x}} _{\rm {est}}\sim \mathbf {T} \,\mathbf {x} _{\rm {est}}}$ from which follows that ${\displaystyle \tau (\mathbf {y} '_{1},\mathbf {y} '_{2},\mathbf {C} _{1},\mathbf {C} _{2})\sim \mathbf {T} ^{-1}\,\tau (\mathbf {y} '_{1},\mathbf {y} '_{2},\mathbf {C} _{1}\,\mathbf {T} ^{-1},\mathbf {C} _{2}\,\mathbf {T} ^{-1}),}$ for all ${\displaystyle \mathbf {y} '_{1},\mathbf {y} '_{2}}$ For each triangulation method, it can be determined if this last relation is valid. If it is, it may be satisfied only for a subset of the projective transformations, for example, rigid or affine transformations. ### Computational complexity The function ${\displaystyle \tau }$ is only an abstract representation of a computation which, in practice, may be relatively complex. Some methods result in a ${\displaystyle \tau }$ which is a closed-form continuous function while others need to be decomposed into a series of computational steps involving, for example, SVD or finding the roots of a polynomial. Yet another class of methods results in ${\displaystyle \tau }$ which must rely on iterative estimation of some parameters. This means that both the computation time and the complexity of the operations involved may vary between the different methods. ## Methods ### Mid-point method Each of the two image points ${\displaystyle \mathbf {y} '_{1}}$ and ${\displaystyle \mathbf {y} '_{2}}$ has a corresponding projection line (blue in the right image above), here denoted as ${\displaystyle \mathbf {L} '_{1}}$ and ${\displaystyle \mathbf {L} '_{2}}$, which can be determined given the camera matrices ${\displaystyle \mathbf {C} _{1},\mathbf {C} _{2}}$. Let ${\displaystyle d\,}$ be a distance function between a (3D line) L and a x (3D point) such that ${\displaystyle d(\mathbf {L} ,\mathbf {x} )}$ is the Euclidean distance between ${\displaystyle \mathbf {L} }$ and ${\displaystyle \mathbf {x} }$. The midpoint method finds the point xest which minimizes ${\displaystyle d(\mathbf {L} '_{1},\mathbf {x} )^{2}+d(\mathbf {L} '_{2},\mathbf {x} )^{2}}$ It turns out that xest lies exactly at the middle of the shortest line segment which joins the two projection lines. ### Via the essential matrix The problem to be solved there is how to compute ${\displaystyle (x_{1},x_{2},x_{3})}$ given corresponding normalized image coordinates ${\displaystyle (y_{1},y_{2})}$ and ${\displaystyle (y'_{1},y'_{2})}$. If the essential matrix is known and the corresponding rotation and translation transformations have been determined, this algorithm (described in Longuet-Higgins' paper) provides a solution. Let ${\displaystyle \mathbf {r} _{k}}$ denote row k of the rotation matrix ${\displaystyle \mathbf {R} }$: ${\displaystyle \mathbf {R} ={\begin{pmatrix}-\mathbf {r} _{1}-\\-\mathbf {r} _{2}-\\-\mathbf {r} _{3}-\end{pmatrix}}}$ Combining the above relations between 3D coordinates in the two coordinate systems and the mapping between 3D and 2D points described earlier gives ${\displaystyle y'_{1}={\frac {x'_{1}}{x'_{3}}}={\frac {\mathbf {r} _{1}\cdot ({\tilde {\mathbf {x} }}-\mathbf {t} )}{\mathbf {r} _{3}\cdot ({\tilde {\mathbf {x} }}-\mathbf {t} )}}={\frac {\mathbf {r} _{1}\cdot (\mathbf {y} -\mathbf {t} /x_{3})}{\mathbf {r} _{3}\cdot (\mathbf {y} -\mathbf {t} /x_{3})}}}$ or ${\displaystyle x_{3}={\frac {(\mathbf {r} _{1}-y'_{1}\,\mathbf {r} _{3})\cdot \mathbf {t} }{(\mathbf {r} _{1}-y'_{1}\,\mathbf {r} _{3})\cdot \mathbf {y} }}}$ Once ${\displaystyle x_{3}}$ is determined, the other two coordinates can be computed as ${\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}}=x_{3}{\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}}$ The above derivation is not unique. It is also possible to start with an expression for ${\displaystyle y'_{2}}$ and derive an expression for ${\displaystyle x_{3}}$ according to ${\displaystyle x_{3}={\frac {(\mathbf {r} _{2}-y'_{2}\,\mathbf {r} _{3})\cdot \mathbf {t} }{(\mathbf {r} _{2}-y'_{2}\,\mathbf {r} _{3})\cdot \mathbf {y} }}}$ In the ideal case, when the camera maps the 3D points according to a perfect pinhole camera and the resulting 2D points can be detected without any noise, the two expressions for ${\displaystyle x_{3}}$ are equal. In practice, however, they are not and it may be advantageous to combine the two estimates of ${\displaystyle x_{3}}$, for example, in terms of some sort of average. There are also other types of extensions of the above computations which are possible. They started with an expression of the primed image coordinates and derived 3D coordinates in the unprimed system. It is also possible to start with unprimed image coordinates and obtain primed 3D coordinates, which finally can be transformed into unprimed 3D coordinates. Again, in the ideal case the result should be equal to the above expressions, but in practice they may deviate. A final remark relates to the fact that if the essential matrix is determined from corresponding image coordinate, which often is the case when 3D points are determined in this way, the translation vector ${\displaystyle \mathbf {t} }$ is known only up to an unknown positive scaling. As a consequence, the reconstructed 3D points, too, are undetermined with respect to a positive scaling.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 67, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8522668480873108, "perplexity": 280.461665259489}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711232.54/warc/CC-MAIN-20221208014204-20221208044204-00541.warc.gz"}
https://homework.cpm.org/category/CCI_CT/textbook/pc/chapter/13/lesson/13.1.3/problem/13H-3	### Home > PC > Chapter 13 > Lesson 13.1.3 > Problem13H-3 13H-3. Refer to Example 2 in this section to assist you with this problem.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9288932681083679, "perplexity": 3666.780899610722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896374.33/warc/CC-MAIN-20200708031342-20200708061342-00348.warc.gz"}
https://bookdown.org/pkaldunn/Book/CIpKnownp.html	## 20.2 Sampling intervals: Known proportion The possible values of the sample proportions $$\hat{p}$$ can be described by an approximate normal distribution, as just discussed. This enables the 68–95–99.7 rule to be applied; for example, about 68% of the time with sets of 25 rolls, the sample proportion of even rolls will be between $$0.5$$ give-or-take one standard deviation (that is, give-or-take 0.1). So, about 68% of the time, the proportion of even rolls in a set of 25 rolls will be between: • $$0.5 - 0.1 = 0.4$$ and • $$0.5 + 0.1 = 0.6$$. Similarly, about 95% of the time, the proportion of even rolls will be between $$0.5$$ give-or-take two standard deviations, or between: • $$0.5 - (2\times0.1) = 0.3$$ and • $$0.5 + (2\times0.1) = 0.7$$. This interval tell us what values of $$\hat{p}$$ are likely to be observed in samples of size 25. Most of the time (i.e., approximately 95% of the time), the value of $$\hat{p}$$ is expected to be between 0.30 and 0.70. (For instance, in the animation above, all ten sets of 25 rolls (or 100%) had a sample proportion betweeen 0.30 and 0.70.) More formally, the sample proportion $$\hat{p}$$ is likely to lie within the interval $p \pm (\text{multiplier} \times \text{s.e.}(\hat{p})),$ where $$\text{s.e.}(\hat{p})$$ is the standard error of the sample proportion (calculated using Eq. (20.1)). The symbol ‘$$\pm$$’ means ‘plus or minus,’ or ‘give-or-take.’ The multiplier depends on how confident we wish to be that the interval contains the value of $$\hat{p}$$. For a 95% interval—the most common level of confidence—the multiplier is approximately 2, based on the 68–95–99.7 rule: Approximately 95% of observations are within two standard deviations of the value of $$p$$ (the mean of the normal distribution in Fig. 20.1). That is, the approximate 95% interval is: $p \pm (2 \times \text{s.e.}(\hat{p}) ).$ For a 90% interval, either tables or a computer would be used to find the correct multiplier, since the 68–95–99.7 rule isn’t helpful. In practice, 95% intervals are the most common, and we’ll use a multiplier of $$2$$ to find an approximate 95% interval when computing the interval without using software. Software can be used for any other percentage interval (or for an exact 95% interval). In general, higher confidence means wider intervals (Fig. 20.2), since wider intervals are needed to be more certain that the interval contains $$\hat{p}$$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9752914309501648, "perplexity": 445.3239109289386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057366.40/warc/CC-MAIN-20210922132653-20210922162653-00574.warc.gz"}
https://www.physicsforums.com/threads/simple-conservation-of-momentum-question.127194/	# Simple conservation of momentum question 1. Jul 26, 2006 ### delfick hello i have a question.... I have two objects going towards each other (in 2 dimensions) I know the mass, initial speed and direction.... I want to know how to figure out what the final speed and direction of the two objects will be after a collision assuming there is not friction.... thnakyou for any help... 2. Jul 26, 2006 ### Hootenanny Staff Emeritus A good place to start would be writing an equation for the conservation of momentum; $$m_{1}\vec{v_{1i}} + m_{2}\vec{v_{2i}} = m_{1}\vec{v_{1f}} + m_{2}\vec{v_{2f}}$$ Perhaps if you post the full question and outline the steps you have taken and where your difficulties lie, we could be of more use. 3. Jul 26, 2006 ### Hawknc I'm assuming it's an elastic collision, i.e. they don't stick together and there's no loss of energy due to deformation? 4. Jul 26, 2006 ### delfick well....The question is for a friend who is doing a simple simulation/program thing. during the course of the simulation he has two objects travelling towards each other...and wants a way of figuring out what direction and speed the objects move in after they hit.... so if it makes it easier...assume this is an elastic collision it is just a simple program and so it doesn't need to be super realistic... (also i knew the equation Hootenanny posted above here already....but i only know how to get the final velocity of one of the objects (using this equation) if i either know the final velocity of the other or if i know they will both have the same final velocity, neither which occur here) ... 5. Jul 26, 2006 ### Hootenanny Staff Emeritus If we assume that the collision is elastic, then we can also use conservation of energy thus; $$\frac{1}{2}m_{1}v_{1i}^{2} + \frac{1}{2}m_{2}v_{2i}^{2} = \frac{1}{2}m_{1}v_{1f}^{2} + \frac{1}{2}m_{2}v_{2f}^{2}$$ Now, we have two equations with two unknowns, namely $v_{1f}$ and $v_{2f}$. The two equations (conservation of energy and momentum) can be solved simultaneously to yield the values of the unknowns. Do you (and your friend) follow? 6. Jul 26, 2006 ### delfick my friend left school in year 10 so didn't do physics...but i stayed at school and did physics in year 11 and currently in year 12...hence he asked for my help.... so my friend is completely lost :D but to me, what u just said makes sense.... thankyou very much for that....i will tell you if i have any troubles... 7. Jul 26, 2006 ### PPonte Synthetizing: 1. An elastic collision is one where both momentum and kinetic energy are conserved (e.g., billiard balls, air molecules, etc.). To solve a collision problem with this situation we use both equations: $$m_{1}\vec{v_{1i}} + m_{2}\vec{v_{2i}} = m_{1}\vec{v_{1f}} + m_{2}\vec{v_{2f}}$$ $$\frac{1}{2}m_{1}v_{1i}^{2} + \frac{1}{2}m_{2}v_{2i}^{2} = \frac{1}{2}m_{1}v_{1f}^{2} + \frac{1}{2}m_{2}v_{2f}^{2}$$ 2. An inelastic collision is one in which momentum is conserved but kinetic energy is not. Then, to solve problems with such collisions you just use: $$m_{1}\vec{v_{1i}} + m_{2}\vec{v_{2i}} = m_{1}\vec{v_{1f}} + m_{2}\vec{v_{2f}}$$ 3. If the two objects stick together during a collision, it is called a perfectly inelastic collision. For such a collision, the conservation of momentum, becomes: $$m_{1}\vec{v_{1i}} + m_{2}\vec{v_{2i}} = (m_{1} + m_{2} )\vec{v_{f}}$$ 8. Jul 26, 2006 ### delfick we're gonna use an elastic collision but i am at a loss as to how to do this....:( especially since the two objects are not likely to be in line with each other.... are u able to explain how to come up with the final velocities a bit more in depth? that would be really cool if u could.....:D thnx 9. Jul 26, 2006 ### PPonte • Define a 2-D coordinate system and identify the masses and velocities of the objects. I advise you to make your x-axis coincide with one of the initial velocities direction. • Make sketches of the situations for both before and after the collision with respect to your coordinate systhem. • Write the equations for the total momentum before and after the collision for both coordinates (x and y), since the momentum is conserved in both directions. NOTE:Decompose the velocities given in their x and y components. • Then, when you find the x and y components of the final velocities, to calculate the magnitude of the velocities use the Pythagorean Theorem: $$v = \sqrt{{v_x}^2 + {v_y}^2}$$ To calculate the angle the velocity makes with the horizontal use: $$\alpha = \tan^{-1}(\frac{v_y}{v_x})$$ These are the main points, the rest depends on the data given and the variables to determine. Hope I could help. 10. Jul 26, 2006 ### Staff: Mentor Since this is a two-dimensional problem, you need more than conservation of energy and conservation of momentum to determine a unique solution. To see this in a real-world situation, consider shooting a billiard ball against a stationary one. Even if you use the same initial velocity for the moving ball, you can get the balls to recoil at different directions and speeds, by aiming for a glancing collision, or a head-on collision, or something in between. To see this mathematically, note that conservation of momentum gives you two equations (one for x-components and one for y-components). Conservation of energy gives you a third equation. So you have three equations that you can solve together. But you have four unknown quantities: the x and y components of velocity of each of the two balls, after the collision. You need as many equations as you have unknowns. For spherical objects, if you can ignore their spin, you can get a fourth equation involving the "impact parameter" that describes how glancing the collision is. If you Google around with search phrases like "two dimensional collision impact parameter" you can probably find lecture notes with the detailed solution. 11. Jul 26, 2006 ### Hootenanny Staff Emeritus Ahh, thanks jtbell, I didn't spot that the OP needed a direction also, my bad. 12. Jul 26, 2006 ### PPonte Thank you jtbell! Sorry, delfick for my incomplete explanation. 13. Jul 27, 2006 ### delfick thankyou all! i won't be able to think about this properly to see if i really understand it for a few days from now due to school and such..... (i think it makes sense...probably won't :p) so basically expect my cries for help in a few days ! (please note that i will try hard to understand it myself first :D) thnx 14. Jul 31, 2006 ### Bashinerox What we need Hello, this is Delficks friend. I'll try to explain the problem as best i can. We have two 2D objects, which in order to make it easier for the computers running this simulation, are treated as circles. We have the velocity and mass of each object, and can add any other data needed. As this is a computer simulation, I am not limited by what goes into the equation, so if we need to for instance know the friction, i can always go back and make up values for the objects. Although in that case, friction will probably just complicate the situation. Ok, so the two objects collide, and we can know any information we want about the objects before the collision. Then i want to be able to calculate the new velocity of both objects, at the very minimum taking into account the mass of each object. So, i have this: x, y value of each object at the time of collision. direction each object is heading. This is in this format: 0 = right, 90 = down, and so on velocity of the object in meters per second mass of the object in kilograms I obviously know the mass, velocity and direction of each object at the collision, as they are stored as variables in my program (becoming the known values?) It doesnt matter how many equations i have to run, but as im not that good with maths, it would be helpfull if you could help me (translate?) the equation so that only one variable is on one side of the equation e.g. Velocity Of Object 1 = ... Some variable needed for Direction calulation = Direction of Object 1 = ... Velocity Of Object 2 = ... Direction of Object 2 = ... P.S. Oh, i can determine the angle of the collision easily (good ol' trigonometry) 15. Feb 8, 2007 ### MrLobster collision problem I came across this thread while trying to do basically the same kind of computer simulation. I found an algorithm that seems to work for billiard balls that have the same mass and am trying to figure out how to modify it to take differing masses into account. I understand that impact parameter basically gives an angle to the glancing collision which I think would be useful to know that precise angle for attempting to solve these kinds of problems using trigonometry but not so useful for a vector solution which would be preferable for a computer solution. The half-solution I found essentially defines a collision plane as the plane tangent to both spheres on impact and from that gives the spheres a normal component (to the plane) and tangential components (to each other). Below is the algorithm that appears to work for objects of the same mass. Can anyone help modify it to take differing masses into account? This isn't a homework problem, it is just something I'm attempting for my amusement. # mag() is a function to calculate magnitude of a vector # dot() is a function to calculate the dot product of two vectors # pos stands for a position vector # anything with velocity stands for a velocity vector # Step 1: Find the unit normal to the collision plane. unitNormal = (ball1.pos - ball2.pos) / mag(ball1.pos - ball2.pos) # Step 2: Find the normal component and tangential component of each velocity vector. ball1.normalVelocity = -1 * unitNormal * dot(ball1.velocity,-unitNormal) ball2.normalVelocity = unitNormal * dot(ball2.velocity,unitNormal) ball1.tangentialVelocity = ball1.velocity - ball1.normalVelocity ball2.tangentialVelocity = ball2.velocity - ball2.normalVelocity # Step3: Calculate New Velocities ball1.velocity = ball1.tangentialVelocity + ball2.normalVelocity ball2.velocity = ball2.tangentialVelocity + ball1.normalVelocity	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8252042531967163, "perplexity": 595.8791258089058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719155.26/warc/CC-MAIN-20161020183839-00412-ip-10-171-6-4.ec2.internal.warc.gz"}
http://www.ck12.org/book/CK-12-Middle-School-Math-Concepts-Grade-8/r9/section/3.3/	<meta http-equiv="refresh" content="1; url=/nojavascript/"> Solve Equations Involving Inverse Properties of Subtraction and Multiplication \| CK-12 Foundation You are reading an older version of this FlexBook® textbook: CK-12 Middle School Math Concepts - Grade 8 Go to the latest version. # 3.3: Solve Equations Involving Inverse Properties of Subtraction and Multiplication Created by: CK-12 % Best Score Practice Two-Step Equations with Subtraction and Multiplication... Best Score % Have you ever looked at a homework problem and wondered how to solve it? Look at this situation that Henry faced. Henry looked at the first problem on his homework page. $14x - 9 = 19$ Even though he'd been paying attention in class, Henry had no idea how to solve this problem. Do you know how to solve it? This is a two-step equation involving subtraction and multiplication. This Concept will teach you the steps for solving equations like this one. ### Guidance You are going to learn how to solve two step equations with subtraction and multiplication in them. Let’s begin. To solve a two-step equation, we will need to use more than one inverse operation. Let's take a look at how to solve a two-step equation now. When we perform inverse operations to find the value of a variable, we work to get the variable alone on one side of the equals. This is called isolating the variable. It is one strategy for solving equations. You can use isolating the variable whether you are solving one step or two step equations. Solve for $x$ : $2x - 9 = 17$ . Notice that there are two terms on the left side of the equation, $2x$ and 9. Our first step should be to use inverse operations to get the term that includes a variable, $2x$ , by itself on one side of the equal (=) sign. In the equation, 9 is subtracted from $2x$ . So, we can use the inverse of subtraction—addition. We can subtract 9 from both sides of the equation. $2x - 9 & = 17\\2x(-9 + 9) & = 17 + 9\\2x & = 26$ Notice how we rewrote the problem above. Since we are adding a positive number, 9, to a number that is being subtracted from $2x$ , we can represent this as adding 9 to -9 as we did above: (-9 + 9). The number 9 is the additive inverse , or opposite, of -9. We can now use inverse operations to get the $x$ by itself. Since $2x$ means $2 \cdot x$ , we can use the inverse of multiplication—division. We can divide both sides of the equation by 2. $2x & = 26\\\frac{2x}{2} & = \frac{26}{2}\\x & = 13$ The value of $x$ is 13. Let’s review our steps to solving this two step equation. Take a few minutes to write these steps in your notebook. #### Example A $9x - 5 = 40$ Solution: $x = 5$ #### Example B $9y - 6 = 66$ Solution: $y = 8$ #### Example C $12a - 4 = 44$ Solution: $a = 4$ Now let's go back to the dilemma from the beginning of the Concept. Here is the problem that Henry saw on his page. $14x - 9 = 19$ To solve this problem, we can first add nine to both sides of the equation. $14x - 9 + 9 = 19 + 9$ $14x = 28$ Now Henry can solve this as a one-step equation by dividing both sides by 14. $x = 2$ This is the answer to this problem. ### Vocabulary Equation a mathematical statement with an equal sign where the quantity on one side of the equation is equal to the quantity on the other side. Variable a letter used to represent an unknown quantity. Algebraic Equation An equation with at least one variable in it. One Step Equation An algebraic equation with one operation in it. Two Step Equation An algebraic equation with two operations in it. ### Guided Practice Here is one for you to try on your own. Eight times a number minus four is equal to ninety - two. Write a two-step equation and solve for the missing variable. Solution First, walk through the words to write the equation. $8x - 4 = 92$ Now solve the for the variable. First, add four to both sides of the equation. $8x - 4 + 4 = 92 + 4$ $8x = 96$ Now divide both sides by 8. $x = 12$ ### Practice Directions: Solve each two step equation that has multiplication and subtraction in it. 1. $4x - 3 = 13$ 2. $5y - 8 = 22$ 3. $7x - 11 = 31$ 4. $8y - 15 = 25$ 5. $9x - 12 = 42$ 6. $12y - 9 = 99$ 7. $2y - 3 = 23$ 8. $3x - 8 = 19$ 9. $5y - 2 = 28$ 10. $7x - 11 = 38$ 11. $5y - 9 = 51$ 12. $6a - 12 = 30$ 13. $9x - 14 = 13$ 14. $12x - 23 = 49$ 15. $13y - 3 = 23$ 16. $18x - 12 = 42$ Basic Dec 19, 2012 Aug 21, 2014	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 43, "texerror": 0, "math_score": 0.8166905641555786, "perplexity": 533.3440411615859}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830074.72/warc/CC-MAIN-20140820021350-00381-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/measuring-the-relative-velocity-of-light.30767/	# Measuring The Relative Velocity Of Light 1. Jun 13, 2004 ### grounded Anyone attempting to argue the Special Theory of Relativity needs to understand the basics of how light travels, and how we perceive it. Einstein wrote that the speed of light does not depend on the speed of the object emitting the light. To prove this, Einstein referred to De Sitter’s observation of the binary stars, which are two stars that are orbiting each other. De Sitter concluded that if the speed of light were dependant on the speed of the star, then the light emitted from the star as it is traveling towards us would eventually catch up to the light that was emitted from the same star when it was traveling away from us. That logic is incorrect since relative to the binary stars, they are not moving and we are orbiting the binary stars. By viewing the stars as motionless, it becomes clear that while we orbit the binary stars, we are running into the light of one star as we are running away from the light from the other star. Relative to the binary stars, their light is not approaching us at different speeds; we are approaching the light at different speeds. This proves that the speed of light can be based off the speed of the star without disturbing our perceptual view of the orbits. Maxwell stated that all types of light would have a frequency that is inversely proportionate to its wavelength. Einstein believed that an increase in frequency caused by traveling towards the light source would cause an inversely proportionate change in the wavelength. What Maxwell meant was that since all types of light travel from the source at the same speed, than while at rest relative to the source, any light with a high frequency will have a short wavelength, and any light with a low frequency will have a long wavelength since multiplying them together must equal the speed of light. He did not mean that a perceptual change in frequency caused by the observer’s speed would change the wavelength. The wavelength of light is not a relative measurement; it is the distance that the light has to travel away from the source in order to complete one wave. That distance is not determined by the observer’s speed, it is the same for all observers traveling at any speed or direction. The frequency of light is a relative measurement; it is the number of wavelengths the observer passes in one second. This number is determined by the speed of the observation and will be different between observers traveling at different speeds relative to the source. The wavelength of light is unaffected by the observers speed, any measured change in wavelength is an error that is caused by not including the distance the observer has traveled relative to the source. When calculating the wavelength, the distance that the light travels from the source in one second must be added to the distance the observer has traveled relative to the source in one second, and then divided by the measured frequency. If the distance the observer has traveled is not included, then the relative speed will never change since the total distance traveled would only include the distance the light has traveled. In order to accurately measure the relative speed between two objects, the distance traveled by both objects in the same amount of time must be included. Interferometers and oscilloscopes only account for the distance that the light has traveled, both need to be adjusted to include the distance traveled by the observer relative to the source. An observer using an interferometer moves a mirror a specific amount of distance while counting the number of changes in the pattern of interference fringes. When used to measure wavelengths while in motion relative to the light source, the scale used to measure the distance that the mirror has moved must be adjusted to include the distance the observer has traveled relative to the source. If the observer is traveling towards the source, the same amount of movement of the mirror will represent a larger distance since it now includes the distance the observer has traveled. If the observer’s distance is not included, any increase in frequency caused by the observer’s speed will appear to decrease the wavelength causing the speed to remain unchanged. Traveling towards the source will increase the number of waves displayed on the screen of an oscilloscope. Displaying more waves in the same amount of space means the length of each wave displayed on the screen will be reduced. This does not mean that traveling towards the source will reduce the actual length of the waves. The oscilloscope shows the waves closer together because the total distance that the screen represents has been increased to include the distance the observer has traveled relative to the source. Traveling towards the source causes the oscilloscope to use a smaller amount of the screen to represent the same amount of distance. If the distance is not included, any increase in frequency caused by the observer’s speed will appear to decrease the wavelength causing the speed to remain unchanged. While at rest relative to the source, a one second screen of an oscilloscope will represent 186,000 miles. If the oscilloscope is traveling 1,000 miles per second towards the source, then the screen of the oscilloscope must represent 187,000 miles. Traveling towards the light does not change the distance that the light has to travel to complete one wave, just as traveling towards an oncoming train does not reduce the length of the boxcars. Traveling towards the train will increase the number of boxcars that are passed and it will increase the relative speed between the observer and the train, but it will not change the length of the boxcars. If the observer plotted the number of boxcars that passed in one minute on a four-inch line, and then did the same thing after increasing speed towards the train, the second experiment would have more marks on the four-inch line and they would be closer together. This does not mean the length of the boxcars have gotten shorter, it means that the four-inch line represents a greater distance while traveling towards the source than it does when not moving relative to the source. The increase in measured frequency caused by the observer’s speed is equal to the distance the observer has traveled (in one-second) towards the source, divided by the known wavelength. When calculating the wavelength using the measured frequency, it must be divided into the sum of “the distance light has traveled away from the source in one second” plus “the distance the observer has traveled towards the source in one second”. When measuring the wavelength, the scale of the tool used to measure the length must account for the distance the observer has traveled relative to the source. While in motion relative to the source, the wavelength or frequency will always be divided into a number that is greater than or less than 186,000 miles, but never equal to 186,000 miles. The frequency multiplied by the wavelength must equal the sum of “the distance that the observer has traveled relative to the source in one second” plus “the distance the light has traveled relative to the source in one second”. The speed of light is not constant to all observers, and it is not the universal speed limit. Traveling at relativistic speeds will not alter time, lengths, or mass. The Doppler effect is not a stretching or compressing of the wavelengths; it is an increase or decrease in frequency and relative speed. The only way the speed of light can be measured constant between observers traveling at different speeds is to measure a change in the length of the wave. The only way to measure a change in wavelength caused by the observer’s speed is by not including the distance the observer has traveled relative to the source. If the distance the observer has traveled is not included when measuring the speed of the train, then the speed of the train will never change. If the distance the observer has traveled is not included when measuring the speed of the light, then the speed of the light will never change. The Special Theory of Relativity is interesting, but incorrect. In my opinion, Einstein created the Special Theory of Relativity because he misunderstood the following facts. Frequency and wavelength are only inversely proportionate when measured at rest relative to the source. When measuring the relative speed of light, the distance the observer has traveled relative to the source must be included with the distance that the light has traveled away from the source in the same amount of time. Light travels at about 186,000 miles per second relative to the source. Relative to the orbiting binary stars, we are circling them and are running into the light at different speeds (actually different distances), which explains why we don’t see multiple images of the same star. The wavelength, or the distance light travels away from the source in order to complete one cycle, is not a relative measurement and it cannot be altered by changing speed or direction. Traveling past a wavelength at a faster rate does not mean the light has traveled a shorter distance from the source to complete one cycle. Changing speed relative to the source can only change the number of wavelengths passed and the relative speed of light, not the distance the light has traveled relative to the source. It is not the speed of light that remains constant it’s the wavelength. 2. Jun 13, 2004 ### Tom Mattson Staff Emeritus Anyone attempting to argue the Special Theory of Relativity does understand those things, because they are required to study classical electrodynamics in the process. That's not all. Einstein opened his case by referring to the observation that, if Galileo's relativity were correct, then the electrodynamics of moving bodies should result in "asymmetries which do not appear to be inherent in the phenomena". That is, Galileo and Maxwell could not both be right. With you so far. No, the logic is just fine. The binary stars are orbiting each other, and it is this relative motion that DeSitter had in mind. You can't simply transform that relative motion away with a change of reference frames, because the motion is accelerated. No, it doesn't, because you can't fix both stars simultaneously. He didn't just "believe" it, it is a consequence of his postulates, which by the way are required to explain the apparent paradoxes in Maxwell's EM theory. One such paradox is that, if Galilean relativity were correct, EM waves would not even appear as EM waves in any frame moving relative to the source. Everyone knows that Einstein's view was an extrapolation of Maxwell's. That is only true if you assume Galilean relativity in the first place. Since that model is now long defunct, there is no reason to assume it. SR has successfully overthrown it, on both theoretical and experimental grounds. The next 6 paragraphs are based on the same faulty assumption, so I'm going to skip them. In my opinion, you reject relativity because you either misunderstand or are simply unaware of the following facts: 1. The speed of light has actually been measured to be independent of the speed of the source in pion decay experiments. 2. Time dilation has actually been measured in muon decay experiments. 3. Rejecting SR necessarily means rejecting Maxwell's electrodynamics, because Maxwell is only frame-independent under SR. 4. Injecting SR into quantum theory makes it much more accurate, not less so. Indeed, QED (the marriage of quantum theory and SR) is the most accurate theory ever devised by man. You need to take some courses in physics, especially electrodynamics. 3. Jun 14, 2004 ### geistkiesel Can you estimate the correction to the MM experiments when the earth velocity is added? 4. Jun 14, 2004 ### grounded Geistkiesel Wrote: Can you estimate the correction to the MM experiments when the earth velocity is added? What type of correction are you looking for? Since there is no ether (at least none with resistance), the velocity of the earth or the source is not important. Light travels in all directions at the same speed. The Michelson-Morley experiment was designed to show the resistance light encountered while traveling with or against the ether. Since there is no resistance, the experiment failed to detect it. The only reason a correction would be needed is to sustain the belief in the resistance, such as the Lorentz-Fitzgerald contraction. No matter how fast the earth is traveling, any ray of light used in the experiment will travel the same speed going straight as it would if reflected 90-degrees. 5. Jun 14, 2004 ### grounded TOM MATTSON Can I pick your brain again? Is it that you don’t believe the number of cycle on the screen of an oscilloscope will increase as you increase speed towards the source? Or is it that you don’t believe the distance the observer has traveled has to be included when making relative measurements? 6. Jun 14, 2004 ### Staff: Mentor An oscilloscope has nothing to do with the speed of light. When moving toward a source, there is a red-shift, and if its enough, you would see more cycles on the oscilloscope, but that doesn't tell you anything about C. 7. Jun 14, 2004 ### ram2048 your hypothesis about light and wavelengths is incorrect i think. i'm not going to go into what "I" personally believe on the matter, because i already have alot of stuff on my plate at the moment <grin> the frequency of the wave is not merely the wave length, although that is one way to look at it. it is how often the wave completes a cycle in a given time, say 1 second. now suppose you're stationary looking towards a light source with your eyes open. you're taking in that light as a specific frequency meaning let's say arbitrarily 5 waves per second. moving TOWARDS the light, you're catching them as you approach, meaning at extremely high speeds you're catching more waves per second. if you moved at a velocity equal to 1/2 lightspeed second towards the source you'd catch 1/2 as many more waves than you would if you were stationary. of course you wouldn't KNOW you were moving towards the lightsource in an SR relativistic frame, so you would just "perceive" light as being a different wavelength/frequency in that "reality" 8. Jun 15, 2004 ### Tom Mattson Staff Emeritus Not at all. If you move towards a source, you will definitely see that the frequency of the radiation is Doppler shifted such that the frequency increases. The distance traveled by the observer only has to be taken into account when we want to translate the data taken from measurements to actual coordinates of events in spacetime. This is because it takes a finite time for information to propagate from an event to an observer. 9. Jun 15, 2004 ### grounded When measuring the relative speed between two objects wouldn't the distance traveled by both objects in the same amount of time always have to be included? An observer traveling towards a source of light will measure an increase in frequency. We call this Doppler shift, which is cause by passing the wavelengths (the full lengths) at a faster rate. We also measure a decrease in wavelength, which is said to be an effect of the relativity. What I see is that the only reason we measure a decrease in the wavelength is because we are not including the distance the observer has traveled. If the distance the observer has traveled relative to the source is included, then the wavelength will not change and the speed of light will not be constant. The only reason we measure the speed of light to be constant is because we measure a change in the wavelength. Run some real or theoretical experiments with the formulas below and see what you find. Change in frequency: The amount of change in the measured frequency caused by the observer’s speed relative to the source is equal to the distance the observer has traveled relative to the source in one second (“positive when traveling towards the source” “negative when traveling away from the source”), divided by the known wavelength. Observer’s speed: The speed of the observer (relative to the source) equals the measured frequency multiplied by the known wavelength, minus the speed of light. Measured frequency: The measured frequency equals the speed of light added to the speed of the observer relative to the source (“positive when traveling towards the source” “negative when traveling away from the source”), divided by the known wavelength. True wavelength: The wavelength (relative to everyone) equals the speed of the observer relative to the source (“positive when traveling towards the source” “negative when traveling away from the source”) added to the speed of light, divided by the measured frequency. 10. Jun 15, 2004 ### Tom Mattson Staff Emeritus I was taking the observer's speed to be an independent variable. If you determine the speed by determining the distance traveled and the time elapsed, then yes you need to know the distance. OK But from the observer's point of view (the one who is measuring the wavelength), he hasn't moved at all. It's the source that is moving towards him. But the speed of light will be constant. It has been measured to be independent of the motion of the source in measurements from the decay of fast particles. This is nonsense. A change in frequency cannot be equal to the change in the observer's distance. The two quantities don't even have the same units. This is also nonsense. The frequency times the wavelength is the speed of light. Your formula says that, no matter what, the speed of the observer relative to the source is zero. Which "known wavelength" is that? The one measured by the observer, or the one measured by someone at rest relative to the source? There is no "true wavelength, relative to everyone". 11. Jun 15, 2004 ### grounded TOM MATTSON wrote: Then you must include the distance the source has moved towards the observer, it doesn't matter. But did they include the distance that our measuring equipment has traveled relative to the source of the light? Or put another way, did they include the distance the source traveled relative to the test equipment? The amount of change in the measured frequency caused by the observer’s speed relative to the source is equal to the distance the observer has traveled relative to the source in one second (“positive when traveling towards the source” “negative when traveling away from the source”), divided by the known wavelength. The answer is the number of additional cycle measured per second, or the reduction. The speed of the observer (relative to the source) equals the measured frequency (measured by the observer) multiplied by the known wavelength, minus the speed of light. The known wavelength is the wavelength measured while at rest compared to the source. The only way to measure a changing wavelength is by not including the distance the observer has traveled towards the source, or the distance the source has traveled towards the observer. 12. Jun 15, 2004 ### Tom Mattson Staff Emeritus Even so, there is still a measurable change in wavelength. Yes. Right, I hit "send" before going back to complete that sentence. So, it's a distance divided by a distance, which sets a frequency equal to a quantity with no units at all. Unless, of course, you mean to say that it is not the distance divided by a wavelength, but a speed divided by a wavelength. Is that it? But why should we combine those quantities to form a speed? A main point of SR is that we cannot take it for granted that we can combine quantities taken from different inertial frames. No, you measure a changing wavelength simply by measuring the wavelength of light when you are in different states of motion relative to the source. Taking the motion of the source relative to the observer into account doesn't give you the "real" wavelength in your frame, it gives you the wavelength in the rest frame of the source. Last edited: Jun 15, 2004 13. Jun 15, 2004 ### Tom Mattson Staff Emeritus I'm guessing that the following idea of yours is what is most preventing you from accepting SR. And I responded thusly: To that I will add that you could make your claim about any measurable quantitiy that varies from frame to frame. You could say that the only way that time dilates is because we don't include the relative motion between muons and our laboratory. Thus, they don't really take longer to decay, and we can prove that if we just calculate back to what the muon lifetime is in its own frame. Well no kidding! Yes, you can always calculate proper times, lengths, wavelengths and frequencies by transforming back to the rest frame of the object under study, be it a muon, a meter stick, or a light source. But just because we can go back and calculate the values of those quantities, it does not imply that those values are somehow more "real" than the ones we measure. You seem to be trying to get me to consider a universe in which SR is false. There's no need: I've already done it (I had to, as part of my studies). What you don't understand here is that you need to consider such a universe. A universe in which Maxwell's equations look different in every frame. A universe in which the momentum of a photon (which is inversely proportional to its wavelength) is independent of the state of motion of particles with which it collides. A universe in which measurements of microscopic systems agree with relativistic qunantum theories vastly better than they do with nonrelativistic qunantum theories. The longer you look at a "Universe without SR", the more you'll see that it is an illusion, and it is most definitely not the universe we live in. Do yourself a favor: Study some physics. You'll be the better for it. 14. Jun 15, 2004 ### grounded The reason I believe that is because of the following: If the width of the screen of an oscilloscope represents a one-second-time period, then it will also represent 186,000 miles since that is the distance light travels in ones second. Right? The number of cycles displayed on the screen is the number of cycles that has passed by in one second. If you take 186,000 miles and divide it by the number of cycles on the screen, it will equal the length of each cycle. Right? If the width of the screen is six inches long, then six inches represents 186,000 miles, and 3 inches would represent 93,000 miles. Doing this we can apply a scale to the screen and measure the length of each cycle. Right? Although it would be difficult. If the oscilloscope increases speed towards the source, the number of cycles displayed on the screen (relative frequency) will increase, and the length of each wave (relative wavelength) will decrease. And this is why you believe the wavelength has changed right? The frequency increases because traveling towards the source has increased the total relative distance traveled in one second. The new frequency must be divided into the sum of 186,000 miles plus the distance the oscilloscope has traveled. Right? Relative speed divided by relative frequency equals relative wavelength, right? The cycles appear closer together on the screen because the screen represents a greater area(the relative speed, or distance per second). It’s only by not adjusting the scale of the screen to represent the total distance traveled by the light and the observer that a change in wavelength can occur. If we do adjust the scale, the wavelength never changes. Do you think the scale of an oscilloscope needs to be adjusted when measuring light while in motion relative to the source? 15. Jun 15, 2004 ### grounded Tom If you used an oscilloscope to measure a pulse emitted from the front of each boxcar as a train passes by you, the number of pulses on the screen will increase if the train speeds up or if you increase speed towards the train. The individual pulses will also be closer together on the screen, this does not mean the length of the boxcars has changed; it means the distance that the screen represents has been increased. If you do not adjust the scale of the screen, then the relative speed of the train will never change. 16. Jun 15, 2004 ### Staff: Mentor What, you've never heard of length contraction? Well, if you don't buy time dilation, I doubt you'll buy length contraction either, but.... ...an observer in a moving boxcar will indeed measure the boxcar to be a different length than an observer on the ground. Like Tom said, what you are doing is assigning a "proper" length in order to calculate speed. Sorry, but you can't do that. For the person in the boxcar, time and distance are what he measures them to be and for the person on the ground, time and distance are what he measures them to be. And they will not necessarily agree on the time and distance involved with the same event. I must emphasize again that though it may seem counterintuitive to you, these phenomena have actually been observed to occur. They are reality. 17. Jun 15, 2004 ### Tom Mattson Staff Emeritus The reason you believe that is because you don't know anything about physics. Take a few courses, and try to learn something. Either that, or open up to learning something about real physics here at Physics Forums, whether or not it defies your intuition. If not, then you're just going to be another crackpot around here. Last edited: Jun 15, 2004 18. Jun 16, 2004 ### geistkiesel The rejection of SR You use the lack of formally gathered knowledge "about the real physics" as a priest uses his call to faith. You are only superior in your own perception, a legend in your own mind only. I reject SR for philosophic reasons, primarily becasuse SR and SR spokes agents such as yourself, talk silly, deferentially,with the superiority of mathematical supremeacy. I reject SR and refuse to study the essence of the insanity for the same reason I don't put my finger in fires tio find out how hot it may be. Do you understand? All of your references to experimental proof of SR are understoof by myself as mantras of idiots. Do you understand? I consider you a scientific clown. I want not what you have to offer. I do not need your pemission to scrutinize any offered theory, nor do I need you permission, or support, to offer my own ideas, even though you and I both know, my ideas are guaranteed to remain unambiguosly contradictory to your ideas. I reject your totality of expression, your scientific methods, your authoritative persona, I reject your smug educator's personality, I reject your mentorial status as having any useful significant additions to the exchange of ideas and the progress of science, I reject your every manifest breath, I reject your efforts to maintain the status quo in a state of blind obedient sleep-walking darkness. I reject your contnued insistance that I have something owed to you. Your reference to the "proof" that mass moving at .99c measured a value of c for the velocity difference of itself and light was ignored, unread. You have a huge theoretical abyss to consider: 1. Either pull up the drawbridge and defend the castle, or []demonstrate how moving toward oncoming photons shortens the wave length of the light, by demonstrating that the increase in frequency is other than your eye moving faster over the oncoming wave lengths and, [] justification of suppressing the addition of the distance traveled by an observer to the wave length of the oncoming photons in the analysis of data from experiments measuring the constancy of the speed of light, and [] direct experimental results disproving the Grounded claim in 2.) and 3.) above. See Grounded's Post #1 in this thread. 19. Jun 16, 2004 ### Hurkyl Staff Emeritus Pot to kettle What about your own faith, Geistkiesel? It must be pretty strong for you to continue to have your unwavering belief in the incorrectness of Special Relativity. I mean, if I had a strong conviction against learning something, I would resign myself to the fact that I don't know that thing. But here you are, steadfastly refusing to learn Special Relativity, refusing to hear evidence confirming it, and you are able to stick to your conviction that it cannot possibly be anywhere close to valid! Wait, I'm wrong, you're not so much sticking to your convictions; you never face any scrutiny because you denounce anyone who would scrutinize you as being in a "state of blind obedient sleep-walking darkness". You still have your blind faith, but you don't have the confidence in your faith, so you need to belittle those who might erode your faith in order to maintain it. You liken learning SR to putting a finger in a fire. Why? Putting your finger in a fire can cause physical pain and injury; are you suggesting that you will suffer mental pain and injury if you tried to learn SR? Are you really that afraid you might be wrong? 20. Jun 16, 2004 ### Tom Mattson Staff Emeritus That's just it: Nothing about SR is "understood by yourself". The experimental and theoretical arguments in support of SR are overwhelming, and furthermore they are available for anyone to examine. How you can say, on the one hand, that I lack the formal knowledge of physics, and then on the other hand that you refuse even to look at SR seriously, is hypocritical beyond belief. You're just another idiot with an internet connection, a bad attitude, and too much time on his hands. Similar Discussions: Measuring The Relative Velocity Of Light	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8565033674240112, "perplexity": 481.9383611286028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689874.50/warc/CC-MAIN-20170924044206-20170924064206-00485.warc.gz"}
https://www.esaral.com/q/if-tan-125-find-the-value-of-1-sin-1-sin-56844	# If tan θ=125, find the value of 1+sin θ1−sin θ. Question: If $\tan \theta=\frac{12}{5}$, find the value of $\frac{1+\sin \theta}{1-\sin \theta}$. Solution: Given: $\tan \theta=\frac{12}{5}$ We have to find the value of the expression $\frac{1+\sin \theta}{1-\sin \theta}$. $A C=\sqrt{A B^{2}+B C^{2}}$ $=\sqrt{12^{2}+5^{2}}$ $=13$ $\Rightarrow \sin \theta=\frac{12}{13}$ Therefore, $\frac{1+\sin \theta}{1-\sin \theta}=\frac{1+\frac{12}{13}}{1-\frac{12}{13}}$ $=25$ Hence, the value of the given expression is 25.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8488420248031616, "perplexity": 904.9956892292397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949387.98/warc/CC-MAIN-20230330194843-20230330224843-00095.warc.gz"}
https://en.wikipedia.org/wiki/Planck_scale	# Planck length (Redirected from Planck scale) Planck length Unit system Planck units Unit of length Symbol P Unit conversions 1 P in ... ... is equal to ... SI units 1.616229(38)×10−35 m natural units 11.706 S 3.0542×10−25 a0 imperial/US units 6.3631×10−34 in In physics, the Planck length, denoted P, is a unit of length, equal to 1.616229(38)×10−35 metres. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant. ## Value The Planck length P is defined as ${\displaystyle \ell _{\mathrm {P} }={\sqrt {\frac {\hbar G}{c^{3}}}}\approx 1.616\;229(38)\times 10^{-35}\ \mathrm {m} }$ where ${\displaystyle c}$ is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant. The two digits enclosed by parentheses are the estimated standard error associated with the reported numerical value.[1][2] The Planck length is about 10−20 times the diameter of a proton. ## Theoretical significance There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research. Since the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is no way of examining it directly. According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.[citation needed] In some forms of quantum gravity, the Planck length is the length scale at which the structure of spacetime becomes dominated by quantum effects, and it is impossible to determine the difference between two locations less than one Planck length apart. The precise effects of quantum gravity are unknown; it is often guessed that spacetime might have a discrete or foamy structure at a Planck length scale.[citation needed] The Planck area, equal to the square of the Planck length, plays a role in black hole entropy. The value of this entropy, in units of the Boltzmann constant, is known to be given by ${\displaystyle {\frac {A}{4\ell _{\mathrm {P} }^{2}}}}$, where A is the area of the event horizon. The Planck area is the area by which the surface of a spherical black hole increases when the black hole swallows one bit of information, as was proven by Jacob Bekenstein.[3] If large extra dimensions exist, the measured strength of gravity may be much smaller than its true (small-scale) value. In this case the Planck length would have no fundamental physical significance, and quantum gravitational effects would appear at other scales. In string theory, the Planck length is the order of magnitude of the oscillating strings that form elementary particles, and shorter lengths do not make physical sense.[4] The string scale ls is related to the Planck scale by P = gs1/4ls, where gs is the string coupling constant. Contrary to what the name suggests, the string coupling constant is not constant, but depends on the value of a scalar field known as the dilaton. In loop quantum gravity, area is quantized, and the Planck area is, within a factor of 10, the smallest possible area value. In doubly special relativity, the Planck length is observer-invariant. The search for the laws of physics valid at the Planck length is a part of the search for the theory of everything.[clarification needed] ## Visualization The size of the Planck length can be visualized as follows: if a particle or dot about 0.1 mm in size (which is approximately the smallest the unaided human eye can see) were magnified in size to be as large as the observable universe, then inside that universe-sized "dot", the Planck length would be roughly the size of an actual 0.1 mm dot. In other words, a 0.1 mm dot is halfway between the Planck length and the size of the observable universe on a logarithmic scale.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9541761875152588, "perplexity": 397.0667092154606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542687.37/warc/CC-MAIN-20161202170902-00292-ip-10-31-129-80.ec2.internal.warc.gz"}
http://xrpp.iucr.org/Ba/ch5o1v0001/sec5o1o3o3/	International Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli International Tables for Crystallography (2006). Vol. B, ch. 5.1, p. 539 \| 1 \| 2 \| ## Section 5.1.3.3. Middle of the reflection domain A. Authiera* aLaboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France Correspondence e-mail: [email protected] #### 5.1.3.3. Middle of the reflection domain \| top \| pdf \| It will be apparent from the equations given later that the incident wavevector corresponding to the middle of the reflection domain is, in both cases, OI, where I is the intersection of the normal to the crystal surface drawn from the Lorentz point, , with (Figs. 5.1.3.4 and 5.1.3.5), while, according to Bragg's law, it should be . The angle Δθ between the incident wavevectors and OI, corresponding to the middle of the reflecting domain according to the geometrical and dynamical theories, respectively, is Figure 5.1.3.4 \| top \| pdf \|Boundary conditions at the entrance surface for transmission geometry. Figure 5.1.3.5 \| top \| pdf \|Boundary conditions at the entrance surface for reflection geometry. (a) Reciprocal space; (b) direct space. In the Bragg case, the asymmetry ratio γ is negative and is never equal to zero. This difference in Bragg angle between the two theories is due to the refraction effect, which is neglected in geometrical theory. In the Laue case, is equal to zero for symmetric reflections .	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9196910858154297, "perplexity": 2578.6763906391748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575751.84/warc/CC-MAIN-20190922221623-20190923003623-00412.warc.gz"}
http://math.stackexchange.com/questions/730995/iterated-dual-vector-spaces	# iterated dual vector spaces Let $K$ be a field and $\mathcal U$ a universe such that $K\in\mathcal U$. (Here, "universe" means "uncountable Grothendieck universe".) Let $\mathcal C$ be the category of $K$-vector spaces belonging to $\mathcal U$, and let $i\ge0$ be an integer. If $i$ is even, put $\mathcal C_i:=\mathcal C$; if $i$ is odd, put $\mathcal C_i:=\mathcal C^\text{op}$. Let $F_i:\mathcal C_i\to\mathcal C$ be the $i$-th dual functor. For integers $i,j\ge0$ of same parity, $\operatorname{Hom}(F_i,F_j)$ is a $K$-vector space. In particular $$d(K,\mathcal U,i,j):=\dim\operatorname{Hom}(F_i,F_j)$$ is a well-defined cardinal. Can one compute this cardinal? Does $d(K,\mathcal U,i,j)$ depend on $K$ and $\mathcal U$? Is $d(K,\mathcal U,i,j)$ finite? Have these questions been asked before? Edit 1. As an illustration, here is a proof of the equality $d(K,\mathcal U,2,0)=0$. I can't believe that this observation had not been made before. This case can be handled without universes. Assume that, for each vector space $V$ (over the field $K$ chosen once and for all), we have a linear map $\theta_V:V^{}\to V$, and suppose that, for each linear map $f:V\to W$, we have $$f\circ\theta_V=\theta_W\circ f^{}.\qquad()$$ We claim: $\theta_V=0$ for all $V$. Proof. As a general notation, put $V_1:=V^,V_2:=V^{*},f_1:=f^,f_2:=f^{*}$, and, for each vector space $V$, let $\varepsilon_V:V\to V_2$ be the natural map. It is easy to see that there is a scalar $\lambda\in K$ such that $\theta_V\circ\varepsilon_V=\lambda\operatorname{id}_V$ for all $V$, and that we can assume either $\lambda=0$ or $\lambda=1$. Case $\lambda=0$. By $()$ we have $v_1(\theta_V(v_2))=0$ for all $v_1$ in $V_1$ and all $v_2$ in $V_2$. This implies $\theta_V=0$. Case $\lambda=1$. We are seeking a contradiction. Let $v_1$ be in $V_1$. We have $v_{12}=\varepsilon_K\circ v_1\circ\theta_V$ by $()$, that is, for $v_2$ in $V_2$ and $k_1$ in $K_1$, we have $$v_{12}(v_2)(k_1)=\varepsilon_K(v_1(\theta_V(v_2)))(k_1),$$ or $$v_2(k_1\circ v_1)=k_1(v_1(\theta_V(v_2))).$$ Taking the identity of $K$ as $k_1$, we get $$v_2(v_1)=v_1(\theta_V(v_2)).$$ Remember that this holds for all $v_1$ in $V_1$ and all $v_2$ in $V_2$. Let $V$ be infinite dimensional. By the Erdos-Kaplansky Theorem, there is a nonzero $v_2$ in $V_2$ such that $\theta_V(v_2)=0$. Since there is obviously a $v_1$ in $V_1$ such that $v_2(v_1)\neq0$, we are done. Q.E.D. I tried unsuccessfully to prove $d(K,\mathcal U,2,2)=1$. Here is an elementary formulation. Instead of $\theta_V:V_2\to V$ we start with $\theta_V:V_2\to V_2$, we assume that the analog of $()$ holds, and that we have $\theta_V\circ\varepsilon_V=0$ for all $V$. Then the question is: does this imply $\theta_V=0$ for all $V$? Edit 2. Here is a proof of the equality $d(K,\mathcal U,2,2)=1$. Let us use the same notation as in Edit 1. Assume that we have, for each vector space $V$, an endomorphism $\theta_V:V_2\to V_2$ of the second dual $V_2$ of $V$. Suppose also that we have $$\theta_W\circ f_2=f_2\circ\theta_V\qquad()$$ for all linear map $f:V\to W$, and that $\theta_V$ vanishes on $V$ when $V$ is viewed as a subspace of $V_2$. We claim: $\theta_V=0$ for all $V$. It is easy to see that this implies $d(K,\mathcal U,2,2)=1$. Proof: Let $v_1$ be in $V_1$, that is, $v_1$ is a linear map $v_1:V\to K$. Using $()$ and the assumption that $\theta_K=0$, we get $$0=v_{12}(\theta_V(v_2))=\theta_V(v_2)\circ v_{11}$$ for all $v_2$ in $V_2$, or $$0=\theta_V(v_2)(v_{11}(k_1))=\theta_V(v_2)(k_1\circ v_1)$$ for all $v_2$ in $V_2$ and all $k_1$ in $K_1$. If $k_1$ is the identity of $K$, this gives $$\theta_V(v_2)(v_1)=0.$$ As $v_1$ is arbitrary, the proof is complete. - I asked a related question on Mathoverflow. – Pierre-Yves Gaillard Apr 2 '14 at 6:59 This is only a very partial answer. I hope there will be more complete answers in the future. Let $V$ be a vector space, let $V_i$ be its $i$-th dual, let $\varepsilon_i:V_i\to V_{i+2}$ be the natural morphism, and let $\varepsilon_{i1}:V_{i+3}\to V_{i+1}$ be the dual of $\varepsilon_i$. We claim $$\varepsilon_{01}\circ\varepsilon_1=\operatorname{id}_{V_1}.$$ Indeed, for $v_1$ in $V_1$ and $v$ in $V$ we get $$\varepsilon_{01}(\varepsilon_1(v_1))(v)=\varepsilon_1(v_1)(\varepsilon_0(v))=\varepsilon_0(v)(v_1)=v_1(v).$$ This shows that, in the notation of the question, there is a subfunctor $G_3$ of $F_3$ such that $F_3\simeq F_1\oplus G_3$, and, more generally, that there are subfunctors $G_i$ of $F_i$ for $i\ge3$ such that, if we put $G_1:=F_1,G_2:=F_2$, we have $$F_i\simeq G_1\oplus G_3\oplus\cdots\oplus G_i$$ for $i$ odd, and $$F_i\simeq G_2\oplus G_4\oplus\cdots\oplus G_i$$ for $i$ even, $i\ge2$. Then the question comes down to computing the cardinals $\dim(G_i,G_j)$. The most naive hope would be to have $\dim(G_i,G_j)=\delta_{ij}$. One can also wonder if $G_i$ has nontrivial subfunctors. By the way, wouldn't it be convenient to decree that the expressions "$1$-variant" and "$(-1)$-variant" are synonyms for "covariant" and "contravariant"? Then $F_i$ and $G_i$ would be $(-1)^i$-variant. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9968828558921814, "perplexity": 59.86960569767774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783396872.10/warc/CC-MAIN-20160624154956-00181-ip-10-164-35-72.ec2.internal.warc.gz"}
https://chemistry.stackexchange.com/questions/68719/syntax-and-typography-of-atomic-orbitals	# Syntax and Typography of Atomic Orbitals For mhchem (i.e. the \ce command you use here), I want to improve the support for atomic orbitals. (Reason: I see it quite often used here.) But only a few things about orbitals are mentioned in IUPAC documents. Many more seem to be "unwritten law". As I am not a chemist I need your help. ## 1 The Green Book p. 32 defines the letters s, p, d, f, g, h, i, k ... (omitting j). They have to be typeset upright. Question a: How far does this "alphabet" reasonably go? (Wikipedia goes up to i). ## 2 On the same Greek Book page, electron configurations are explained, e.g. $\mathrm{(1s)^2 (2s)^2 (2p)^1}$. Many other sources write it without parenthesis: $\mathrm{[Kr]\mskip2mu4d^{10}\mskip2mu5s^1\mskip2mu5p^3\mskip2mu5d^4}$ Question a: What values can the left-side number take? (At Wikipedia, I see 1 to 7.) Can it be 2-digit? Question b: What values can the right-side exponent take? (At Wikipedia, I see 1 to 10.) ## 3 I think I did not come across the configuration with one-electron orbitals (in Greek). $\mathrm{(1\sigma)^2 (2\sigma)^2 (3\sigma)^2 (1\pi)^1}$. (The Greek letters should be upright, but that's a drawback of MathJax.) Would this be useful? Or not needed? ## 4 The examples introduce another notation, "electronic states" $\mathrm{\cdot\cdot\cdot (2a_1)^2 (1b_2)^2 (3a_1)^2}$ a: What values could the letter take? b: The left-side number? c: The subscripts? d: The right-side exponent? ## 5 From Wikipedia, I learned that there are notations IUPAC does not mention: $\mathrm{p_x}$, $\mathrm{p_y}$ and $\mathrm{p_z}$. (All subscripts should be upright, because they are labels, no variables.) But it does not end there. Here and here I saw: $\mathrm{d_{xy}, d_{xz}, d_{yx}, d_{x^2{-}y^2}, d_{z^2}}$ This was confirmed by this book. There are alternative styles here ($\mathrm{{p_x}^1}$). and here ($\mathrm{p_{x1}}$). a: I guess the $\mathrm{d_{z^2}}$ notation (with index z-square) is correct? Oh, I just found this book that also includes $\mathrm{4f_{x(x^2{-}3y^2)}}$ and $\mathrm{4f_{y(3x^2{-}y^2)}}$. b: Are there other indexes possible than x, y, z? c: Can it be just squares? Or are other index-superscripts possible? d: What about the 3 in the last examples? Could this also be any other numer? (You see, I just look at the typography of things and don't know their meaning.) e: Could there be any other forms of indexes? f: Typographically, what does this dash stand for? Is it a hyphen or a "transition between a higher energy state and a lower energy state"? ## 6 Finally, there is orbital hybridization, which will lead to notations like $\mathrm{sp, 2sp, sp^2, 2sp^3, s^{0.5}p^3, s{-}s, sp^3{-}sp^3}$. a: Can the superscript be any other value than 2 and 3? Can the "base" be any other thing than s, p and sp? b: Is $\mathrm{2sp^3}$, is it $\mathrm{2s}$ and $\mathrm{p^3}$ or is sp one ... thing with a 2 before and a 3 in superscript. c: Is it a hyphen or a "transition between a higher energy state and a lower energy state"? • Wow, tempted to flag as too broad. – Mithoron Feb 17 '17 at 21:05 • Question 1 remember there are also the corresponding term symbols en.wikipedia.org/wiki/Term_symbol – MaxW Feb 17 '17 at 21:27 • @Mithoron Yes, it got rather long. However, I assume that there is a high correlation between the answers (e.g. "left-side number can be ..., everywhere") and a comprehensive answer would be less than 10 lines. – mhchem Feb 20 '17 at 22:23 Question 1. The Green Book p. 32 defines the letters s, p, d, f, g, h, i, k ... (omitting j). They have to be typeset upright. How far does this "alphabet" reasonably go? (Wikipedia goes up to i). 1. Section 2.6.3 of that version of the Green Book shows the term symbols (Wikipedia) which are capital letters S,P,D,F,G,H,I,K,... and used to indicate the electronic configuration of the valence electrons and for excited states. 2. The lower case letters are used for the azimuthal quantum number (ℓ) which is one of the four quantum numbers for an electron in an atom. You can typeset $99.9~\%$ of chemistry of even more using only the letters s, p, d and f, corresponding to the left-hand main group elements, the right-hand main group elements, the transition metals and the lanthanides/actinides, respectively. Everything above f is and will be for quite some time not relevant to chemistry outside a very small circle that would know how to deal with stuff. Question 2. On the same Greek Book page, electron configurations are explained, e.g. $\mathrm{(1s)^2 (2s)^2 (2p)^1}$. Many other sources write it without parenthesis: $\mathrm{[Kr]\mskip2mu4d^{10}\mskip2mu5s^1\mskip2mu5p^3\mskip2mu5d^4}$ Question a: What values can the left-side number take? (At Wikipedia, I see 1 to 7.) Can it be 2-digit? Question b: What values can the right-side exponent take? (At Wikipedia, I see 1 to 10.) 1. This has two sub-questions. 1. By a similar reasoning as presented in the answer to 1, I think 8 will be the limit until the not-so-close future. But there is nothing intrinsically preventing it from becoming two-digit at some distant point. 2. The superscript can take values of up to $4l+2$, where $l$ can be considered ‘computer science counting’ of the letter sequence s, p, d, …; so for s $l = 0$ and thus $4l+2=2$; likewise for p ($l=2$) the value is maximum $4l+2=6$ etc. Note that $4l+2$ is actually $2(2l+1)$: each subshell gives up to $2l+1$ orientation possibilities that each can be populated with two electrons. Question 3. I think I did not come across the configuration with one-electron orbitals (in Greek). $\mathrm{(1\sigma)^2 (2\sigma)^2 (3\sigma)^2 (1\pi)^1}$. (The Greek letters should be upright, but that's a drawback of MathJax.) Would this be useful? Or not needed? 1. Yes, this is useful, especially for discussing molecular orbitals. The Greek letters you want to be aware of are σ, π and δ. Maybe a fourth will be added soon corresponding to f orbitals (υ?) but I don’t see that coming anytime soon. Question 4. The examples introduce another notation, "electronic states" $\mathrm{\cdot\cdot\cdot (2a_1)^2 (1b_2)^2 (3a_1)^2}$ a: What values could the letter take? b: The left-side number? c: The subscripts? d: The right-side exponent? 1. I’m going to answer that as one. This notation uses irreducible representations to define orbitals. The bit between the initial number and the superscript can be any of the symbols that appear in Little Bobby Tables (courtesy of Orthocresol). They each represent a type of orbital with a certain symmetry. Orbitals of the same irreducible representation (i.e. the same centre) are numbered from lowest to highest energy — so in principle the sky is the limit. Practically, however, you will be hard-pressed finding three-digit examples. The exponent can at maximum be $2n$, where $n$ is the number in the first column of the respective character table (the E column). This number corresponds to a certain symmetry’s degeneracy (i.e. so many orbitals of the same energy) and each orbital can be populated by two electrons. Question 5. From Wikipedia, I learned that there are notations IUPAC does not mention: $\mathrm{p_x}$, $\mathrm{p_y}$ and $\mathrm{p_z}$. (All subscripts should be upright, because they are labels, no variables.) But it does not end there. Here and here I saw: $\mathrm{d_{xy}, d_{xz}, d_{yx}, d_{x^2{-}y^2}, d_{z^2}}$ This was confirmed by this book. There are alternative styles here ($\mathrm{{p_x}^1}$). and here ($\mathrm{p_{x1}}$). a: I guess the $\mathrm{d_{z^2}}$ notation (with index z-square) is correct? Oh, I just found this book that also includes $\mathrm{4f_{x(x^2{-}3y^2)}}$ and $\mathrm{4f_{y(3x^2{-}y^2)}}$. b: Are there other indexes possible than x, y, z? c: Can it be just squares? Or are other index-superscripts possible? d: What about the 3 in the last examples? Could this also be any other numer? e: Could there be any other forms of indexes? f: Typographically, what does this dash stand for? Is it a hyphen or a "transition between a higher energy state and a lower energy state"? It was my assumption that the axis directions should be italicised but I don’t know. To discuss this, please go to the break-out discussion for this question. p orbitals will always have $x$, $y$ or $z$. d orbitals will have one of the five you noted ($\mathrm{d}_{z^2}$ is correct — as I said, I am not sure on the upright versus italic question but I would have assumed italics). The f orbitals themselves can be labelled by various different polynomials in $x$, $y$, $z$, and $r$. See these two web pages: 1, 2. In general the subscript is just a polynomial. Which symbols and exponents can appear here is fixed already by the results of the hydrogen-like orbitals — again, for most of chemistry d orbitals are sufficient and you can cover $99.9~\%$ by including f orbitals. The ‘dash’ is a mathematical minus-sign operator. Yes, it truly is $x^2$ minus $y^2$. Question 6. Finally, there is orbital hybridization, which will lead to notations like $\mathrm{sp, 2sp, sp^2, 2sp^3, s^{0.5}p^3, s{-}s, sp^3{-}sp^3}$. a: Can the superscript be any other value than 2 and 3? Can the "base" be any other thing than s, p and sp? b: Is $\mathrm{2sp^3}$, is it $\mathrm{2s}$ and $\mathrm{p^3}$ or is sp one ... thing with a 2 before and a 3 in superscript. c: Is it a hyphen or a "transition between a higher energy state and a lower energy state"? 1. Sub questions. 1. Yes. You have $0.5$ as a superscript in your own examples. But, as posts by Martin and Ron somewhere point out, in $\mathrm{sp}^x$ the superscript $x$ can, in principle, be any number. 2. $\mathrm{2\,sp^3}$ is $\mathrm{\{2\}\{sp^3\}}$ — and there could potentially be another superscript making it $\mathrm{(2sp^3)^2}$. 3. I don’t know what that hyphen is supposed to be. • More question than a comment: do we use italics at all? I see sometimes s,p... italicized, and frankly, I use it often, too, because I found texts more readable when orbital notations pop up. – Greg Feb 21 '17 at 1:13 • @Greg The orbital designation letters should be in upright type as they are not variables. So that bit at least is clear. – Jan Feb 21 '17 at 1:15 • Great! I went through it step by step. At question 4, I am stuck. These tables surpass my brain capacity. Starting with the fact that everything is captical letters, there. Is it the left-most column you are referring to? So, the middle part could be anything like $\mathrm{e_1, d_{4h}, a_2''}$? – mhchem Feb 21 '17 at 9:26 • @mhchem Sort the answers by "active", and ignore the last two posts on "direct product tables" and "tables of descent in symmetry" - those are different things. Apart from those two posts, it can be anything in the left-most column, except for the header row. So, $D_\mathrm{4h}$ does not count, but $\mathrm{e_1}$ and $\mathrm{a_2''}$ are both possible. If you are referring to orbitals, then the capital letters will become small letters. The capital letters in the tables are referring to irreducible representations, which are sets of matrices describing the behaviour of the orbitals. – orthocresol Feb 21 '17 at 12:25 • For question 1: if you take the orbital letters up to i you'll be able to handle ~100% of chemistry -- computational heavy-element chemistry needs to go higher than a lot of chemistry because they need to add on higher-level functions to allow molecular orbitals to be more finely shaped. – Aesin Feb 21 '17 at 17:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.847647488117218, "perplexity": 1059.385355218689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999800.5/warc/CC-MAIN-20190625051950-20190625073950-00535.warc.gz"}
https://socratic.org/questions/how-do-you-graph-y-abs-x-6-1	Algebra Topics # How do you graph y = -abs(x+6)? ##### 1 Answer Jul 5, 2015 This is an upside-down V shape with arms having slope $\pm 1$ and vertex at $\left(- 6 , 0\right)$ #### Explanation: $\left\mid x + 6 \right\mid = x + 6$ when $x + 6 \ge 0$, that is when $x \ge - 6$ So $y = - \left\mid x + 6 \right\mid = - x - 6$ has slope $- 1$ when $x \ge - 6$ $\left\mid x + 6 \right\mid = - \left(x + 6\right)$ when $x + 6 < 0$, that is when $x < - 6$ So $y = - \left\mid x + 6 \right\mid = x + 6$ has slope $1$ when $x < - 6$ The vertex is at the point where $\left(x + 6\right) = 0$, giving $x = - 6$ and $y = 0$. So the vertex is at $\left(- 6 , 0\right)$ The intersection with the $y$ axis will be where $x = 0$. Substituting $x = 0$ into the equation, we get $y = - \left\mid 6 \right\mid = - 6$. So the intersection is at $\left(0 , - 6\right)$ graph{-abs(x+6) [-13.63, 6.37, -6.96, 3.04]} ##### Impact of this question 243 views around the world You can reuse this answer Creative Commons License	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 23, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.948331356048584, "perplexity": 624.8697862600221}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575541307797.77/warc/CC-MAIN-20191215070636-20191215094636-00343.warc.gz"}
http://www.gradesaver.com/textbooks/science/physics/conceptual-physics-12th-edition/chapter-31-think-and-discuss-page-599/71	# Chapter 31 - Think and Discuss: 71 It becomes shorter. #### Work Step by Step Momentum is p = mv. As the proton's velocity increases, its momentum also increases. The de Broglie wavelength is h/p, so as momentum p increases, the proton's wavelength becomes shorter. After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9474976062774658, "perplexity": 2685.088260819626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917123097.48/warc/CC-MAIN-20170423031203-00531-ip-10-145-167-34.ec2.internal.warc.gz"}
https://electronics.stackexchange.com/questions/305251/24vac-vs-220vac-inductive-voltage-transient	24VAC vs 220VAC inductive voltage transient I'm designing a PCB with a PIC micro-controller that has to switch some contactors. This PIC drives some NPN BJT transistors and these transistors drive some Relays that switch the contactors. I'm trying to minimize EMI interference and sparks in my circuit and for thus i have to decide what coils to use on the contactors (and snubbers but thats not related to the question). Seaching for contactors I've got two models (for the coil): 1) 7VA, cos(phi)=0.3 (50Hz), 24VAC. 2) 7VA, cos(phi)=0.3 (50Hz), 220VAC. What's the comparison between both voltage spikes when the coil is suddenly opened ? (How much higher, how much longer, frequency, etc) 7 VA at 24 V AC means a current of 7/24 amps or 292 mA. From this, the impedance is 24/0.292 = 82.3 ohms. From the cos(phi) bit you can then calculate the series inductive reactance and resistance that add up to the 82.3 ohms. So, assuming you can do this (I get XL = 78.5 ohms), convert this to inductance and you have everything needed to compute the stored energy ($\dfrac{I^2L}{2}$). Compare the energy in both scenarios and the one with the highest energy storage capability is the one that could generate the bigger spark. What's the comparison between both voltage spikes when the coil is suddenly opened ? (How much higher, how much longer, frequency, etc) Higher and longer is about energy and control of that discharge of energy. Frequency is much harder to predict because you haven't stated what the parasitic components (or sunubber) are. • Thanks, just one question, when calculating the Energy is $I^2 = \|I\|^2$ ? or is $I$ the real part of the current ? – Tomás Arturo Herrera Castro May 14 '17 at 17:31 • As far as pure inductor is concerned, there is no real or imaginary current so, any current flowing in the inductor at any moment in time represents stored energy. – Andy aka May 14 '17 at 17:52 • So, you will liberate the most energy when the current is maximimum or, for an inductor fed from a sinewave, when voltage passes through 0 volts. – Andy aka May 14 '17 at 17:54 • No need for thanks. The normal thing is to up-vote an answer or, in your case (the guy making the question), to formally accept the answer. – Andy aka May 15 '17 at 7:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8490760922431946, "perplexity": 1984.9507325215998}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00244.warc.gz"}
http://pub.acta.hu/acta/showCustomerArticle.action?id=29999&dataObjectType=article&returnAction=showCustomerVolume&sessionDataSetId=4445ebf262c875c&style=	ACTA issues ## Generalized Weyl's theorem and spectral continuity for quasi-class $(A,k)$ operators Fugen Gao, Xiaochun Fang Acta Sci. Math. (Szeged) 78:1-2(2012), 241-250 69/2010 Abstract. If $T$ or $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator acting on an infinite-dimensional separable Hilbert space, then we prove that generalized Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$, where $H(\sigma(T))$ denotes the set of all analytic functions in a neighborhood of $\sigma(T)$. Moreover, if $T^{\ast }$ is an algebraically quasi-class $(A, k)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. Also, we prove that the spectrum, Weyl spectrum and Browder spectrum are continuous on the class of all quasi-class $(A, k)$ operators. AMS Subject Classification (1991): 47A10, 47A53, 47B20 Keyword(s): algebraically quasi-class $(A, k)$ operator, generalized Weyl's theorem, generalized $a$-Weyl's theorem, continuity of the spectrum Received September 24, 2010, and in final form January 26, 2011. (Registered under 69/2010.)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9961875677108765, "perplexity": 810.1568707478724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202781.83/warc/CC-MAIN-20190323101107-20190323123107-00383.warc.gz"}
http://www.chegg.com/homework-help/questions-and-answers/energy-released-fission-100-kg-uranium-thathas-enriched-30-percent-isotope-235-u-answer-ju-q137940	## enrichment Find the energy released in the fission of 1.00 kg of uranium thathas been enriched to 3.0 percent in the isotope 235 U? Is the answer just 3 percent of the final Q value? Whichnuclear reaction do they want manipulated?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8923879861831665, "perplexity": 3597.832616314251}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368701943764/warc/CC-MAIN-20130516105903-00009-ip-10-60-113-184.ec2.internal.warc.gz"}
http://stats.stackexchange.com/questions/105941/proper-number-of-clicks-for-conversion-rate-testing	# Proper Number of Clicks for Conversion Rate Testing I need to track the conversion rate of a landing page. I want to figure out how many clicks I need to get before I can be sure that I have an accurate conversion rate calculation. I first figured I would calculate the confidence interval using the appropriate formula for a proportion, and find the number of clicks needed to make the interval sufficiently tight. However, according to this link you need at least 1,000 clicks before you can assume the distribution is normal. Unfortunately I can't get that many clicks for all of my landing pages. (I'm using landing pages as a parallel for what I'm really doing - in reality I'm testing hundreds of CPA advertisements for their conversion rates which prevents me from giving each thousands of clicks. This detail doesn't change the problem, though, so it can be ignored). I then had a different idea - I'd go through the click/conversion data that I do have so far for previous landing pages using different "window sizes", such as a window of 10 clicks, 50 clicks, 100 clicks, 200 clicks, etc. For each window size, I'd go through each click in sequential order and calculate the conversion rate using the next x clicks, starting from the one I'm inspecting. I'd then calculate the variance of the conversion rates I found. I'd then find a window size that gives me a desired amount of minimal variance across the majority of the landing pages, and then use that as the number of clicks to test for for future tests. Does this approach seem to make sense / has anyone heard of something similar? Probably more importantly, what is the normal way to accomplish this goal? Thanks! - Paul. A conversion rate is defined as: $\textrm{Conversion rate} = r =\frac{\textrm{Number of goal achievements}}{\textrm{Visits}} = \frac{s}{n}$. Assuming that the number of goal achievements is basically the number of success $s$ out of the number of trials $v$ (rather than the number of events per unit time or space), then what you are trying to do with your conversion rate data is estimate the underlying but unknown probability of conversion $\kappa$. There is absolutely no need to make a normality assumption in estimating the rate or its uncertainty. Instead, you could use the Bayesian Beta-binomial model for estimating the probability distribution of an unknown proportion. In the Beta-binomial model, your conversion rate data $v$ follows a binomial distribution with size $n$ and probability $\kappa$: $s \sim \textrm{Bin}(n,\kappa)$ Of course, you don't know what $\kappa$ is, so you are going to use Bayes' theorem to estimate it by combining your prior beliefs about what the probability might be and your conversion rate data. It turns out that a very useful model for the distribution of your priors beliefs about the probability in this case is the Beta distribution with concentration parameters $\alpha$ and $\beta$. $\kappa \sim \textrm{Beta}(\alpha, \beta)$ In the Beta distribution, the parameters \alpha and \beta represent your prior beliefs about the concentration of successes and failures. The greater one concentration parameter is relative to another, the greater your belief that the probability favors that event. Also, the greater the sum of the concentration parameters $\alpha + \beta$, the more prior information you have about the conversion rate (say, from previous experiments using the same landing page), and the more certain you are in the expected probability $\frac{\alpha}{\alpha + \beta}$. There is a nifty mathematical result here. It turns out that the posterior distribution of the conversion probability $\kappa$ follows a Beta distribution with the concentration parameters being a simply modification of those in your prior. The posterior distribution of the conversion probability is: $\textrm{Pr}(\kappa\|s,n,\alpha,\beta) \sim \textrm{Beta}(\alpha + s, \beta + n - s)$ That is, you just add the counts in your data to the appropriate concentration parameter (\alpha being the concentration of successes and \beta that of failures), and voila! But how should you set the values of $\alpha$ and $\beta$? If you have prior information about the conversion rate for that landing page, perhaps you could set the concentration parameters to be the counts of those prior experiments. But be careful: the bigger the prior sample size, the more new conversion data you will need to overwhelm your prior beliefs. Then again, you could set the parameters so that the prior expected value $\frac{\alpha}{\alpha + \beta}$ is equal to the conversion probability from your prior experiments, but choose the parameters also so that their sum is low, reflecting your lack of information about the present landing page, say, because this landing page is much different from your previous ones. How you set the prior depends on your circumstances and your beliefs and how strong those beliefs are. Another option is to claim ignorance about what the value of $\kappa$ might be. In this case, you could set $\alpha = \beta = 1$, which is equivalent to a continuous uniform (i.e., flat) prior distribution. Some suggest that you should use the Jeffrey's prior distribution, wherein $\alpha = \beta = 1/2$ instead, which as a U-shape with modes at 0 and 1. Regardless of what prior distribution you choose, now you can estimate the expected value of the conversion probability, which is: $\textrm{E}(\kappa\|s,n,\alpha,\beta) = \frac{\alpha + s}{\alpha + \beta + n}$ You could also estimate the posterior variance using the formula for the Beta distribution variance, or the posteiror median, or the posteiror kurtosis, or the posterior skewness, or what have you. You could use a computer program such as R to estimate the credible range for conversion rate given your data. For example, you could estimate the posterior 95% confidence interval by firing up R and running the following code (assuming that you've defined $\alpha$, $\beta$, $s$, and $n$ in your code previously): post_CI <- qbeta(c(0.025, 0.975), alpha, beta) You could also use simulations from the posterior distribution to compute any number of alternative credible intervals, such as the highest posterior density interval or lowest posterior loss interval. You should seriously look into highest posterior density interval because the Beta distribution can be quite skewed, causing the traditional quantile interval approach to allow values into the interval that have lower posterior probability than values that are not in the interval. Below is the code for computing the highest posterior density interval, assuming you've defined everything as before: sims <- rbeta(1000000, alpha, beta) require(coda) \|\| install.packages("coda") post_HDI <- HPDinterval(as.mcmc(sims), prob=0.95) - I must say that your suggesting here a beysian approach to some one who might be looking for a frequentist answer, the standard of the industry (which is frequentistic!) is to use the binomial distribution without the prior, see here an example r-bloggers.com/… – Yehoshaphat Schellekens Jul 6 at 5:39 I must say that I've seen plenty of people in the industry use Bayesian methods for computing conversion rates, and I also must say that I don't know what the value of $\kappa$ is, and I don't trust the maximum likelihood estimate of its expectation if I don't have that much data. Of course, if I do have lots of data, the two approaches give the same answer, so who cares? Well, I do, because it turns out at least for me that the Bayesian approach is easier to understand. To each their own I guess. – Brash Equilibrium Jul 6 at 5:49 I like your attitude, and ive been tring to use the beysian approach as well (see stats.stackexchange.com/questions/105260/…) but there's seem to be some resistance to it, out there. – Yehoshaphat Schellekens Jul 6 at 5:53 In the post you link to, I see you comparing two posterior Beta distributions that have improper priors with concentration parameters set to zero. So long as your posterior distributions are proper, that is a perfectly legitimate way to compare two proportions. In fact, I prefer it, because it doesn't rely on normal approximations. Just be careful, because if you ever got a zero count for one of your categories, your posterior would no longer be proper. – Brash Equilibrium Jul 6 at 6:11 Thanks for the help! – Yehoshaphat Schellekens Jul 6 at 6:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9221410155296326, "perplexity": 464.07699463481043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770043.48/warc/CC-MAIN-20141217075250-00007-ip-10-231-17-201.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/103542/changing-the-basis-of-vector-space-in-terms-of-a-linear-mapping-and-of-a-vector	# Changing the basis of vector space in terms of a linear mapping and of a vector I know that left multiplication of a vector $x$ by a orthonormal matrix $U$ is just changing the basis, but I'm not so sure what $U^TAU$ means. It seems that they have some relations. Let $A$ be a symmetric matrix, and I take the symmetric matrix $A$ as a hyperellipsoid (in terms of $x^TAx=1$), $U^TAU$ seems to be just changing the basis to the direction of eigenvectors of $A$. In this way, $Ux$ is changing basis in terms of vector $x$ and $U^TAU$ is changing basis in terms of $A$. If $B$ is any matrix viewed as a linear transformation, can I safely regard $U^TBU$ as changing the basis in terms of this linear transformation? How to verify this? For example, will a horizontal shear mapping $C$ be change into a shear mapping in other direction if I use $U^TCU$? And will a scaling mapping in the direction of standard basis be change into a scaling mapping in other direction? If this is true, a symmetric matrix $A$ can be viewed as a scaling transformation in the directions of eigenvector of $A$. (https://math.stackexchange.com/a/103053/21762) So, if $B$ is any matrix viewed as a linear transformation, can I safely regard $U^TBU$ as changing the basis in terms of this linear transformation? The reason is that for orthogonal matrices $U^T= U^{-1}$. If you have a linear transformation $$x \mapsto Bx =y.$$ Then changing the basis (in the primed coordinates) with $U x'= x$ and $Uy'=y$, we have $$Uy'= BUx' \Rightarrow y'= U^T B U x' = B' x'$$ such that $$B' = U^T B U$$ is the matrix of the linear transformation in the new coordinate system.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9505526423454285, "perplexity": 98.68063751588367}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573832.23/warc/CC-MAIN-20190920050858-20190920072858-00251.warc.gz"}
https://iwaponline.com/wst/article/32/4/139/4611/Horizontal-distribution-patterns-during-a	A hierarchical sampling was performed in order to give a picture of the horizontal distribution of cyanobacterial colonies. It showed how the importance of different scales in distance changed during the summer bloom of Gloeotrichia echinulata in the moderately eutrophic and stratified Lake Erken in southeastern Sweden. G. echinulata colonies occurred in patches at a distance of ∼1000 m and the variance at smaller scales was low. For colony content of phosphorus, chlorophyll a and size, however, the largest variances were found at the 20 m (and error) scale if the variances between dates are not taken into account. It was concluded that in order to estimate the lake population of cyanobacterial colonies properly these variances must be considered when deciding the sampling program.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9279831647872925, "perplexity": 2106.0974030634547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744513.64/warc/CC-MAIN-20181118155835-20181118181835-00457.warc.gz"}
http://www.scottaaronson.com/blog/?p=401	## The QIS workshop As promised, here’s my report on the Quantum Information Science Workshop in Virginia, only a week or so behind schedule. I tried to be cynical—really I did. But despite my best efforts, somehow I went home more excited about quantum than I’ve been in a long time. The highlight of the workshop was of course the closed, invitation-only, late-night meeting in the basement of NSF headquarters, at which a group of us hidebound quantum computing reactionaries plotted to keep the field focused on irrelevant mathematical abstractions, and to ostracize the paradigm-smashing entrepreneurial innovators who threaten our status. I don’t think I’ve ever heard so much cackling in the space of a single evening, or so much clinking of bone goblets. Stuff like that is why I entered the field in the first place. But there were some other highlights as well: [Full list of talks iz heer] 1. In his talk on quantum algorithms with polynomial speedups, Andris Ambainis called attention to a spectacular recent paper by Ben Reichardt, which characterizes the quantum query complexity of any partial or total Boolean function f (up to a logarithmic factor) as the optimal witness size of a span program for f, and also as the negative-weight quantum adversary lower bound for f. Assuming this result is correct, it seems possible that the remaining open problems in quantum query complexity will be pulverized, one after another, by solving the associated SDPs for the optimal span programs. (Incidentally, using Reichardt’s result, it must be possible to prove, e.g., a Ω(n1/3/log(n)) lower bound for the quantum query complexity of the collision problem using the adversary method. This was a longstanding open problem. Can one say, explicitly, what the adversary matrices are in this case?) Alas, it also seems possible that span programs will turn out to be almost as hard to analyze as quantum algorithms were… (1+√5)/2. Despite the obvious danger to the future funding of the entire field, by some clerical error I was released from my padded cell to speak about “Quantum Complexity and Fundamental Physics”. My “talk,” if it can be called that, was in my opinion neither rational nor integral to the workshop. 2. In her talk on blind quantum computation, Anne Broadbent (who’s also visiting MIT this week) described some beautiful new results that partly answer my Aaronson $25.00 Challenge from a year and a half ago. The Challenge, if you recall, was whether a quantum computer can always “prove its work” to a classical skeptic who doesn’t believe quantum mechanics—or more formally, whether every problem in BQP admits an interactive protocol where the prover in BQP and the verifier is in BPP. Anne, Joe Fitzsimons, and Elham Kashefi haven’t quite answered this question, but in a recent paper they’ve come close: they’ve shown that a quantum computer can prove its work to someone who’s almost completely classical, her only “quantum” power being to prepare individual polarized photons and send them over to the quantum computer. Furthermore, their protocol has the amazing property that the quantum computer learns nothing whatsoever about which particular quantum computation it’s performing! (Aharonov, Ben-Or, and Eban independently gave a protocol with the same amazing properties, except theirs requires the “classical” verifier to have a constant-sized quantum computer.) Anne et al. also show that two quantum computers, who share entanglement but can’t communicate with each other, can prove their work to a completely classical verifier (while, again, remaining completely oblivious to what they computed). On top of everything else, these results appear to be the first complexity-theoretic application of the measurement-based quantum computing paradigm, as well as the first “inherently quantum” non-relativizing results. (Admittedly, we don’t yet have an oracle relative to which the blind quantum computing protocols don’t work—but the protocols rely essentially on the gate structure of the quantum circuits, and I conjecture that such an oracle exists.) Rereading my Challenge, I noticed that “the [one-member] Committee may also choose to award smaller prizes for partial results.” And thus, yesterday I had the pleasure of awarding Anne a crumpled$10 bill, with an additional $5 contributed by Seth Lloyd, for a grand total of$15.00 to be shared equally among Anne, Joe, and Elham. (Update: Since I wrote that, Anne has elected to trade in for three signed and doodled-upon $5 bills.) (Another Update: A$12, or $15-$O(1), prize shall be awarded to Dorit Aharonov, Michael Ben-Or, and Elad Eban the next time I see them.) This is, I believe, the first time a monetary reward offered on Shtetl-Optimized has actually been paid out. 3. In a talk that was so good, you almost forgot it involved chemistry, Alán Aspuru-Guzik discussed applications of quantum complexity theory to understanding photosynthesis and the design of efficient solar cells (!). To give you a sense of how mindblowing that is, it briefly made me wonder whether I should reread some of John Sidles’ cheerful ramblings about the coming merger of quantum systems engineering with biology in the 21st century (of which, I predict, this very sentence will inspire dozens more). So what then is the connection between quantum complexity theory and photosynthesis? Well, a few of you might remember my post “Low-Hanging Fruit from Two Conjoined Trees” from years ago, which discussed the lovely result of Childs et al. that a quantum walk on two conjoined binary trees can reach a designated end vertex exponentially faster than a classical walk on the same graph. That result interested me, among other things, because it can be shown to lead to an oracle relative to which BQPSZK, which at the time I didn’t know how to find otherwise. But especially given the bizarre nature of the graph needed to produce the oracle separation, I thought of this result as pretty much the prototype of an irrelevant complexity-theoretic curiosity (which, naturally, made me like it all the more). You can probably guess where this is going. Shown above is a light-harvesting molecule (image snagged from Alán’s slides), which apparently is efficient at concentrating light at its center for essentially the same reason the Childs et al. quantum walk reaches the target vertex exponentially faster than a classical walk: namely, because of destructive interference between the paths that point backward, toward the leaves. According to Alán, what plants do to harvest sunlight is not entirely unrelated either (it also involves quantum coherence), and fully understanding these mechanisms in quantum information terms might conceivably be useful in designing better solar cells. To be fair, a part of me always did suspect that quantum oracle separations would turn out to be the key to solving the world energy crisis. I’ll point you here or here if you want to know more. Incidentally, Alán’s talk had another, also extremely interesting part, which was about coming up with precise numerical estimates of the number of qubits you’d need to simulate the wavefunctions of (say) benzene, caffeine, and cholesterol. (Many of us have long thought that simulating physics and chemistry will be the real application for scalable quantum computers if we ever build them, practical long before breaking RSA and ultimately more useful too. But it’s not something we often talk about—ostensibly for lack of meaty things to say, really because we don’t know chemistry.) 4. In her talk, Dorit Aharonov posed an open problem that I now have no choice but to inflict on others, if I don’t want to feel forced to think about it myself. So here’s her problem: how hard is it to find the ground state of a local Hamiltonian H=H1+…+Hm (that is, a sum of k-qubit interactions, for some constant k), if we impose the constraint that the Hi‘s all commute with each other? Clearly it’s somewhere between NP and QMA. It might seem obvious that this problem should be in NP—to which I can only respond, prove it! There were also lots of great talks by the experimentalists. Having attended them, I can report with confidence that (1) they’re still trying to build a quantum computer but (2) decoherence is still a big problem. If you want to know even more detail than I’ve just provided—or you want to know about the theory talks I didn’t mention, or more about the ones I did mention—ask away in the comments. I can’t promise that no one will know the answer. ### 57 Responses to “The QIS workshop” 1. rrtucci Says: That molecule looks suspiciously like a parabolic mirror. It does not look, at least to me, like the Childs et al graph. Could you please explain your statement “which apparently is efficient at concentrating light at its center for essentially the same reason the Childs et al. quantum walk reaches the target vertex exponentially faster than a classical walk:” 2. matt Says: I think Dorit’s problem is really a decision problem: “does there exist a ground state which minimizes the energy of every H_i separately?” 3. Kevin Young Says: matt: I think the question of finding the groundstate is much harder. As to your question, “does there exist a ground state which minimizes the energy of every H_i separately?”, the answer is known to be, “Yes.” Because the H_i’s commute with each other and with H, a common eigenbasis can be found. The ground state will then be an eigenstate of each H_i with eigenvalue e_i, so the energy of the groundstate is e=e_1+e_2+…+e_m. The lowest possible value of e is achieved when the e_i’s are all as low as they can be. So, the groundstate of H is also the groundstate of each H_i. 4. Scott Says: Matt: Yes, formally it’s a decision problem (just like the ordinary local Hamiltonians problem). Kevin: Can’t we encode 3SAT using commuting local Hamiltonians? If so, then finding the energy of the ground state can’t be that easy… 5. Scott Says: rrtucci: More accurately, it looks isomorphic to the right half of the Childs et al. graph–that is to say, it’s a tree. 🙂 6. Dave Bacon Says: What, no experimental details? Were there many experimentalists, because from a glance at the schedule the workshop seem heavily theory oriented (nothing wrong with that, just saying 🙂 ) I’m also missing why the commuting Hamiltonian problem isn’t obviously in NP. How are the H_i’s given to you? Diagonalize each H_i, it has basis (v_i)_j with eigenvalue (E_i)_j (diagonalize only over the k qubits where H_i acts nontrivially). One is tempted to define variables (x_i) \in {0,1}^k and then say the total energy is the sum of these (x_i)’s, but this is not true since there are dependencies between the x_i’s. Further the (E_i)_j’s might be degenerate (in which case the basis is not completely defined). But isn’t there any easy algorithm for resolving this? Start with H_1 and H_2. If H_1 and H_2 do not act on the same qubits, ignore them. If they do overlap, then simulatenously diagonalize H_1 and H_2 (max size of matrix 2^(2k)). From this simulatenous diagonalization you can match the different energy values, i.e. you can put a constraint between x_i and x_j. Further you should need to deal with degeneracy. But you can keep track of this locally: for degenerate energy levels, you can use the simultaenous diagonalization to say whether this degeneracy is split or not and if it is split, keep track of the new basis under which it is split (i.e. you will update your local basis (v_1)_j and (v_2)_j appropriately.) Proceed to do this for every pair H_i and H_j. At the end of the day you will have a bunch of x_i’s with associated energies and a bunch of contraints between the x_i’s. I must be missing something. I did just eat lunch so all the blood is in my tummy. 7. rrtucci Says: rrtucci: More accurately, it looks isomorphic to the right half of the Childs et al. graph–that is to say, it’s a tree. 🙂 Doh! But Childs et al are looking for a traveling wave solution , so their wave bounces only once off the perimeter of that molecule. Besides, I never had to use complexity theory to understand a parabolic mirror (or a round trampoline.) How is the exponentially faster result of Childs et al relevant? Does their model predict the data? 8. matt Says: Dave: here is an example will show why this commuting problem is tricky. Consider a toric code Hamiltonian on a sphere. There is a ground state (in fact, 1 such state). Now change the Hamiltonian so that on some given plaquette the system wants \pi flux instead of zero flux. Now there are no ground states. Even though locally everything can be satisfied, and in fact every subsystem of size 9. matt Says: Oops, part of my comment got dropped cause I used a less-than symbol. continued: everything can be satisfied on every system of size less than L, but there is no global solution. This example is easy since we can count whether there are an even or odd number of such plaquettes, but maybe there are harder examples. 10. rrtucci Says: More to the point: Are you claiming that there is quantum coherence across this huge molecule that encompasses thousands of atoms and is embedded in a thermal environment? I can show you a tree in back yard. Does it also illustrate the Childs et al result? 11. Joe Fitzsimons Says: Scott: Thanks for bigging up the bling QC protocol (and for my share of the $15). By the way, blind quantum computing in the sense we describe seems to become impossible if quantum mechanics is non-linear (or if you have closed time-like curves which follow Deutsch’s model). I’ve been working on the communications complexity of blind QIP, but I am limited to processing actual quantum states rather than dealing with a BQP oracle, since the structure of BQP seems to make it impossible to exploit Feigenbaum’s no-go results, so I don’t think there is much chance of proving BQP != IP_BQP directly. 12. Dorit Says: Hey! Scott told me about this discussion going on about the commuting Hamiltonians problem, so I am chipping in. Me and Itai Arad kind of bumped into this problem when trying to prove quantum PCP theorem for the non-commuting case, thinking that the commuting case is o b v i o u s l y true because it is, well, classical…. But we should have been smarter… That the commuting Hamiltonians case cannot be viewed as simply classical is known to anyone familiar with Toric codes. There we get groundstates of local commuting stabilizers which are highly entangled! Which also explains why the problem of approximating the ground energy of the commuting Hamiltonian problem is not “obviously” in NP – the witnesses may be terribly complicated! One might think that since all terms in the Hamiltonian are simultanuously diagonalizable, we can rotate the basis of eigenstates to the standard computational basis by _local_ transformations, and so everything is locally equivalent to classical. well, we do not know how to do this… At this point we do not know how to give an efficient classical description of ground states of commuting Hamiltonians, which we can also use for efficient classical estimation of the energy. I think this is a very interesting problem, the computational power of commuting Hamiltonians. Espically if it turns out that the commuting Hamiltonian case is in fact much harder than it seems at first sight… It might be as hard as the non-commuting case! (which might seem unlikely, but who knows…) And on a different note – congratulations to Anne, Joe and Elham for recieving the honarable 10$(+5!) first Shtetl award for their progress on the beautiful question of proving BQP to a BPP verifier! However, I have to express my protest. How come we, the other team, (that is Elad and Michael as well as myself) who heroically and independently attacked this question and arrived at similar results, did not get even one dollar? I think the Shtetl-Optimized prize committee should correct this injustice. Actually, while we are at it, it might be interesting to stir up the discussion here with the provocative motivations we had in mind when we approached the problem. Our team attacked this question of proving quantum dynamics to a classical observer not from the cryptographic angle, but rather we were motivated by more foundational questions: the question of testing quantum mecahanics. It all starts with the following striking observation about experiments on quantum mechanics (striking to me, at least): If we view physical dynamics as computations – inputs lead to outputs, then one can talk about quantum dynamics that are within BPP. But general quantum dynamics are, as we now believe, not simulatable in BPP. Of course, our interest in quantum computation is due to the fact that we believe that such dynamics that are outside of BPP are realizable in nature. However, when looking for experimental evidence for quantum dynamics outside of BPP, you will find that they are impossible to find! (I discovered this when, during a conversation with oded Goldreich, I wanted to come up with such evidence to why we so strongly believe in quantum computation…) In fact, no property of quantum mechanics that is outside of BPP, was ever tested experimentally! The reason is simply that such properties _cannot_ be tested by the usual scientific paradigm. The usual paradigm is this: peform an experiment and compare its results to the predictions of the theory, which are computed on the side. However, if we assume the Modern Church Turing thesis, all such predictions must be computed in BPP, and so if we are talking about dynamics outside of BPP, we simply have no predictions to compare to… …so, as far as we know, all experiments on quantum mechanics could be fully consistent with an alternative theory to QM which lies within BPP! Shor’s algorithm is striking for many reasons, but one of them is that it is the first example of a physical experiment out of the usual experimental paradigm: it is able to test quantum mechanicat dynamics outside of BPP! this is because it uses _verification_, rather than predictions, to compare its results to the theory. It verifies that the product of the outcomes is the integer it needs to factor – though the classical computer cannot predict what the factors (the outcome of the experiment) would be. So we asked whether it is even possible to test _general_ quantum dynamics despite the fact that in general they cannot be simulated in BPP, and thus there are no predictions for them. And the result is: yes. We can test general dynamics of QM experimentally. But we need to change the notion of a physical experiment: we allow interactions between the verifier and the experimentalist. Our answer is partial – our interactions must involve sending few qubits at a time, and in particular, they involve quantum interactions. So the verifier is not entirely classical – it should be able to handle a few qubits at a time. Whether or not general quantum behavior can be tested using solely classical interactions, so that the verifier can be fully classical, is exactly Scott’s original question. Which I think is a great question! dorit 13. Kevin Says: Matt, Scott, Dave: I see that I was completely neglecting the constraints on the states. Matt, very good example, it helped a lot. 14. Joe Fitzsimons Says: Hi Dorit, Since the Hamiltonians are simultaneously diagonalizable, in the case where all terms can be simultaneously minimized the problem would seem to essentially be a SAT instance. If the terms cannot simultaneously be minimized then the problem would seem to be in coNP rather than NP (though I could easily be wrong). 15. Dave Bacon Says: matt: I think I’m missing your point. Classically I can change a local term in a 3-SAT problem and it no longer no longer be satisfied. I guess I’m missing what is different between your example and the following. Take the Ising model with ferromagnetic interactions and an external magnetic field. There is one ground state it is (say) the all zeros vector. Now change one of the interacts to antiferromagnetic. Of course the model does not have a way to satisfy all interactions, but this doesn’t have anything to do with quantum it just has to do with properties of changing one of the energy terms (I guess I also lose you when you say “doesn’t have a ground state” Don’t all quantum systems have a ground state.) If a Hamiltonian is made up of a sum of stabilizer elements from a stabilizer group there is an even simpler way to look at the energy levels. Let S_1….S_k be the stabilizer generators. Then there is a basis \|s_1,…s_k,X> where s_i \in \{+1,-1}. Every term in your Hamiltonian will be a possible product of stabilizer generators (since it is a sum of stabilizer elements.) Thus the energy can be expressed as a sum of products of s_i. Finding the minimal value for the energy is then a classic minimizing problem which is in NP. For the toric code this just means that E= \sum_{p \neq p’} s_p + \sum_{v \neq v’} s_v + \prod_{p \neq p’} s_p + \prod_{v \neq v’} v_p. Each s_p is associated with a plaquette and each s_v is associated with a vertex. A special plaquette p’ and special vertex v’ is singled out, because these the product of the remaining stabilizer generators (changing which one of these you use doesn’t change the energy function’s form.) (Of course the space of a logical qubit doesn’t enter into the energy function but tells you that each energy level is at least this degenerate.) Now I have no excuse about my tummy just my dense head! 16. Greg Kuperberg Says: However, when looking for experimental evidence for quantum dynamics outside of BPP, you will find that they are impossible to find! I don’t entirely agree. You can confirm that a system performed a calculation that exceeds its computational resources, even if some larger classical computer could also perform that calculation. Of course a skeptic could argue that your bounds on the system’s computational resources are not true. Maybe you think that the system only has room for 4 qubits (say), but it actually has room for 200 classical bits. But if you accept quantum dynamics itself, you can then conclude that the system computed more than it has a “right” to compute. The demonstration is perhaps clearer with communication complexity than with computational complexity. I suspect that it is already feasible to demonstrate violation of classical communication complexity bounds. The demonstration is clearer still if, in the spirit of communication complexity, you violate statistical bounds that hold classically with zero allowed communication. I.e., if you violate Bell-type inequalities. This is already related to communication complexity and communication complexity is related to computational complexity. As I see it, once you witness violation of Bell-type inequalities, the strong Church-Turing thesis is already on shaky ground as a scientific hypothesis. Why wouldn’t such a violation lead to quantum accelerations? At best, there would have to be a new reason for it not to happen. I think perhaps the phrase quantum “dynamics” is a bit misleading. I like to call the precepts of quantum computation quantum probability. Quantum probability is clearly different from classical probability, and quantum probability can be tested and has been amply verified. Either quantum probability is not completely true (which would be new science), or there is some statistical censorship principle that prevents full access to quantum probability (which would also be new science), or quantum computation is possible in principle. 17. Scott Says: Dorit: Thanks for the interesting comment! The reason why you haven’t received a share of the prize is extremely simple: because I haven’t seen you (or Elad, or Michael) since remembering about my offer of smaller prizes for partial results. But don’t worry, I haven’t forgotten you! Next we meet, you shall receive the proud sum of $12.00 ($12 being $15-O(1), representing the O(1) qubits held by the verifier in your protocol … but$12 also being divisible by 3, so that you and your two coauthors can share the winnings equally). 18. Joe Fitzsimons Says: Dorit, I gave a talk on blind quantum computing in Bristol a month or so ago, and was asked whether it was possible to perform blind computation with a constant number of rounds of computation. I’ve been thinking about this, and the answer would seem to be that it is possible. As far as I can see we can reduce our protocol to only two rounds of communication. Have you done any thinking about the number of rounds? I am wondering whether there exists a blind QC protocol which requires only one round. 19. Scott Says: Are you claiming that there is quantum coherence across this huge molecule that encompasses thousands of atoms and is embedded in a thermal environment? I can show you a tree in back yard. Does it also illustrate the Childs et al result? rr: Firstly, I’m not claiming it, Alán was—I’m just reporting what he said and that it seemed pretty exciting to me. But yes, I did ask around among the physicists, and they told me it’s a well-accepted fact that chloroplasts do use quantum coherence to harvest light—indeed, that the disagreement is just about whether that’s already so well known that quantum information has little more to contribute. Apparently, what makes it possible—despite the fact that chloroplasts are relatively “large,” “thermal” systems—is that the coherence is extremely short-lived. Again, I’ll refer you to the papers for details, or maybe I can get Alán to contribute to this thread… 20. Alan Aspuru-Guzik Says: Hi all! FIrst of all, I am honored to be part of the Shtetl, being a happy Guzik, whose great grandparents moved from the Polish Shtetls all the way to Mexico City a few generations ago. Now, off to science. I am going to answer R. Tucci’s questions: – The theory that we are working on is based on recent ultrafast experiments carried out by the Fleming group at UC Berkeley on photosynthetic complexes. These experiments show long-lived (for molecules, ~600 fs) coherent energy transfer amongst the chlorophyll molecules. The paper of Engel et. al (Nature, 2007) is noteworthy and interested us originally, as it suggested that the system was carrying out a quantum search. http://www.lbl.gov/Science-Articles/Archive/PBD-quantum-secrets.html We showed that this was not the case, but we still wanted to analyze the role of coherence further. Masoud Mohseni, Patrick Rebentrost, Seth Lloyd and myself started thinking about this in the context of quantum walks. We extended the idea of quantum walks to open systems (as most of the action happens while the system decoheres to the ground state after the initial light pulse). We proposed the idea of environment-assisted quantum walks to explain the role of the environment in this context. – The next problem to tackle was understanding what was the role of coherence on the EFFICIENCY of energy transfer of the system. It is not enough to see nice coherent oscillations; now, you want to see if one actually wants to partition the quantum process in its different components (dephasing, relaxation, dissipation, coherent evolution), and what percentage of the process is due to each one of them. Our results show that the role of coherence in the efficiency of the complex is between 10-30% for the Fenna-Mathews-Olson complex (the molecule studied by Engel et. al). 3) Finally, we thought of the Child’s et al exponential speedup (a lovely paper) and wondered if it was possible to see this exponential speedup in a chemical structure. Dendrimers (branching molecules) come to mind, as it is not hard to imagine having an excitation start at the leaves of the tree and go to the root. (Imagine just carrying out the final half of Childs’ algorithm) Of course, I told you that things decohere quickly, so you could ask yourself the question: Does one gain any speedup under decoherence? In the third paper (Environment-Assisted Quantum Transport), we found that indeed if the dendrimer (or tree structure for the conjoined tree case) had static disorder (mismatch in the site energies of the different nodes of the graph) the destructive interference that resulted from these imperfections could be “fought” by introducing dephasing. The interplay between dephasing and static disorder can lead to a regime where the dendrimer or tree “works again” (albeit less efficiently than if the evolution were fully coherent). – The story became very exciting recently: Greg Scholes’s lab in Toronto found long-lived coherent transfer in a conducting polymer at room temperature (Science, 2009). This basically tells us that this is more ubiquitous than one thought, in the sense that even a polymer in solution can have these hundreds of femtosecond coherent effects. There are many open questions still: – To what extent can one “engineer” physical systems to benefit from these short-lived coherent effects. – Can we learn from quantum algorithms, quantum information, etc. to help have for example, better light-harvesting systems, such as the dendrimer that Scott pasted here (the image is originally from Professor Takuzo Aida, from Japan)? – We need better understanding of the reasons for these long-lived coherences. (The main consensus right now is correlated protein environments). Many quantum info. people are working on this now, such as Martin Plenio, Alexandra Olaya-Castro, etc. You can check out a conference that Masoud and Yasser Omar are organizing in Portugal about the subject: http://sqig.math.ist.utl.pt/lqcil/quebs09/home/ To get our papers on this, look at #24, 25, 27 on my publications page: http://aspuru.chem.harvard.edu/Publications/ Finally, if you are up to it, you can read a recent Discover article. It was fun to appear there, but more fun to appear here. http://discovermagazine.com/2009/feb/13-is-quantum-mechanics-controlling-your-thoughts/article_view?b_start:int=1&-C Well, off to sleep guys. I will check tomorrow to see if there are more comments. 21. rrtucci Says: Scott, your answer is a total red herring. It’s no great revelation that chemistry is based on quantum mechanics You said: “which apparently is efficient at concentrating light at its center for essentially the same reason the Childs et al. quantum walk reaches the target vertex exponentially faster than a classical walk:” That is totally false. The Childs et al quantum walk result is for a pure state with quantum coherence over all nodes. There is no way that there is quantum coherence (for any significant amount of time) across thousands of atoms sitting inside a stew at room temperature or hotter. The fact that a moron like me has to point this out, that nobody at the QIS conference figured this out or you would have never said it here, speaks volumes about the competence of the QIS scientists. 22. Andrew Hunter Says: Scott, As a side note, why don’t you use Beamer? Just curious as to your opinion. 23. asdf Says: Scott, the vienna.ppt presentation you linked to doesn’t open in Open Office 3. It says something like “general i/o error” suggesting that the file is corrupted somehow. The file I downloaded (twice) has sha1 checksum: be21b36b83ef25f57a6f4ca561a6bfb7aca94c44 24. KaoriBlue Says: Scott, Thanks for posting the link to Alán Aspuru-Guzik slides – looks like a pretty excellent talk. As a side note, these molecular graphs, and ones considerably more complex than what he’s presenting, are easily within synthetic reach of modern dendrimer chemistry. I’d bet the turn-around time for experimental verification of a particularly intriguing or efficient structure would be no more than a year or two. 25. KaoriBlue Says: Where does his classical cost for factoring come from though?? Shouldn’t it be O(e^((ln(n))^(1/3)(ln(ln(n)))^(2/3)) for the number field sieve algorithm? 26. Gil Says: Very nice post. Can you say something on Leggett’s lecture and , in particular: What is the definition of the “disconnectivity” measure D; and what is the mysterious K defined in the last slide. And a technical question. How to open Freedman’s presentation. 27. Scott Says: Sorry, everyone! My presentation file was indeed corrupted. I can’t figure out what the problem is: it opens fine on my computer, then I upload it and download it again and it doesn’t work. Anyway, I changed the link to point to the version of my presentation on Caltech’s site, which works fine. 28. Scott Says: KaoriBlue: In his chart, n is the number of bits in the number; it’s not the number itself. 29. matt Says: Dave, everywhere I said “ground state”, I meant “zero energy ground state”. The classical example you mention shows a case where again you can satisfy the problem locally, but not globally. The difference, though, is that in the classical case there is the very important property that the set of all the local reduced density matrices is globally consistent, and that this global consistency of density matrices is easy to check since the local reduced density matrices are antiferromagnets. In the quantum cases, the local reduced density matrices are not globally consistent and this is harder to check. That is, you can’t just give the local reduced density matrices as a witness. The case I mention is too easy, of course, as would be any stabilizer Hamiltonian, but maybe there are harder cases. 30. matt Says: sorry, total typo in that…replace “are antiferromagnets” with “are pure” 31. Scott Says: As a side note, why don’t you use Beamer? Just curious as to your opinion. Can Beamer make a giant hammer fall onto the slide, smashing papers that violate the BBBV theorem? Can it make a baseball encyclopedia fall into a black hole and then bounce back out? Or make the black hole expand until it swallows up the slide? Can it make an Uncle Sam with a ψ in his hat rise onto the page, saying “That’s why YOU should care about quantum computing”? Does Beamer easily let you create a binary tree with copies of Gwyneth Paltrow at the bottom? What about a spiky-tailed hydra whose two heads are Razborov and Rudich? If Beamer can do all these things, then yes, I’ll consider switching from PowerPoint. 32. KaoriBlue Says: Ah, stupid mistake – I should have thought before posting on that. 33. fernando brandao Says: Concerning the local Hamiltonian problem for commuting Hamiltonians, it was actually studied some time ago by Bravyi and Vyalyi in quant-ph/0308021. They not only mentioned that the problem might not be in NP in general, but actually were able to show that this is not the case for 2-local Hamiltonians: they proved that there is always an effcient classical desciption for the ground state of a 2-local commuting Hamiltonian (see corollary 1 of their paper). That’s of course in contrast to the general case, by the QMA-completeness of 2-local Hamiltonians proved by Kempe, Kitaev and Regev. It would be great to find out what happens in the general case! Changing topic, I would like to add a remark to the discussion about interactive proofs for quantum computation. In my opinion there are two main reasons why such kind of result would be extremely interesting (not considering the cryptographic applications): First, we, believers of quantum computation and hence of the correctness of quantum theory in the large scale, would like to have an efficent way to test if a very large quantum system (which we cannot simulate on our desktop computer) behaves in the way we think it does. Here we are happy to assume that quantum mechanics is correct, but are skeptical about whether e.g. we have full knowledge of the system’s Hamiltonian. In this setting – which would be very relevant if we manage to build a quantum computer – the two great papers by Broadbent/Fitzsimons/Kashefi and Aharonov/Ben-Or/Eban essentially solve the question (positively!). We could of course ask for a protocol with a classical verifier, but in my opinion that’s a not so important detail… But there is also another reason why interactive proof systems for quantum computation would be great, more in the spirit of Scott’s $25 question: can we convice a skeptical of quantum mechanics that we are able to solve any problem in BQP, if we only have access to BQP? In this situation, a protocol whose soundness proof assumes the correctness of quantum theory is not good enough: the prover might think you have access to more than what is possible by quantum mechanics, an extra power which you could use to cheat him. In this context, the interactive proofs presented in the two papers are not good enough, as they assume the correctness of quantum theory (please correct me if I’m wrong :-). Of course, the full$25 shtetl question would be satisfactory also in this context. What I think is a nice question is whether there are intermediate settings: for example, could the soundness proofs of any of the two papers (which assume quantum theory as correct) be generalized to a broader class of theories (of course I’m being too sloopy, I guess that to find a meaningful set of such theories would also be part of the problem :-)? 34. Pascal Koiran Says: Scott writes: >Assuming this result is correct, it seems possible that the >remaining open problems in quantum query complexity will >be pulverized, one after another, by solving the associated >SDPs for the optimal span programs. I would love to see whether this method yields nontrivial lower bounds for hidden subgroup problems. As of now, such bounds could be derived for Abelian groups only (using the polynomial method, as in the collision lower bound). 35. Gil Kalai Says: Greg: “I suspect that it is already feasible to demonstrate violation of classical communication complexity bounds.” Greg, This is very interesting and I would be quite surprised if it is true. What would be the simplest way to demonstrate it? 36. Andrew Landahl Says: Just a quick comment on Alán Aspuru-Guzik’s talk, which was indeed fantastic and inspirational. A common misconception is that the graph on which a (continuous-time) quantum walk occurs has a direct realization in physical space. In fact, the graph on which the walk occurs is an abstract graph, whose vertices are labeled by states in a Hilbert space, not locations in physical space. From a complexity point of view, quantum walks are most interesting when they occur on an exponentially large graph realized using only polynomially many qubits. Certainly (continuous-time) random walks also occur on exponentially large graphs realized by only polynomially many “pbits,” but such walks can’t exhibit interference effects. On the other hand, classical wave dynamics can exhibit interference effects, but not efficiently on exponentially large graphs. Alán’s picture of a light-harvesting molecule has the unfortunate side-effect of potentially of luring folks into the state graph = physical layout fallacy. I think the main point that Alán and his collaborators are making is that quantum coherence plays a non-negligible a role in transport in such structures, contrary to (at least my) intuition that decoherence would totally wash out quantum coherence effects. I think it’s too strong to say that Alán and his collaborators have found evidence that Nature has found a way to harness exponential speedups in the quantum query model á la the welded binary tree problem. Instead, they’ve pointed out the (surprising) importance of quantum coherence in a room-temperature process. That said, perhaps a sufficiently advanced civilization will figure out a way to leverage exponential quantum speedups to improve device design, such as for photovoltaics. The jury is still definitely out, in my opinion. I’m waiting for the day when we find a natural process that implements quantum error correction!!! 37. Greg Kuperberg Says: I would love to see whether this method yields nontrivial lower bounds for hidden subgroup problems. Unfortunately no, because it is already known (Ettinger, Hoyer, Knill, quant-ph/0401083) that the query complexity is very low. 38. Greg Kuperberg Says: Greg: “I suspect that it is already feasible to demonstrate violation of classical communication complexity bounds.” Greg, This is very interesting and I would be quite surprised if it is true. What would be the simplest way to demonstrate it? Gil: At the moment this is no more than a suspicion, and I think that it would be good to figure out how to test any such communication bounds, or determine that implicitly they have already been tested. Again, my intuition is that the violation of Bell’s inequalities is similar in spirit to the violation of communication complexity bounds. But note that for the latter there is a certain apples-vs-oranges aspect, because quantum communication complexity is based on sending qubits, while classical communication complexity is based on sending bits. Although you can reduce the former to the latter (with dense coding) if you allow shared prior entanglement. In fact — I need to think this through better — I wonder if the known violation of Bell’s inequalities is already a violation of a classical bound in the “refereed simultaneous messages” model. It seems quite possible that if Alice and Bob share prior entanglement, they can reduce the amount that they tell the referee, and that there are examples that can already be demonstrated with existing quantum devices. 39. Scott Says: Greg: No, he wants a (polynomial, not exponential) lower bound on the query complexity anyway. We actually have an Ω(n) lower bound on the quantum query complexity of Simon’s problem, and it’s quite beautiful (an unexpected use of the polynomial method for a highly non-symmetric problem). 40. Pascal Koiran Says: Thank you Scott. May I add that the same beautiful method 🙂 yields a tight lower bound for all Abelian groups. Basically no lower bound is known for non-Abelian groups. In particular, the method seems to be break down completely for dihedral groups. There are some lovers of dihedral groups among the regular readers of this blog so my challenge is : prove a nonconstant lower bound on the query complexity of the hidden subgroup problem in dihedral groups! Note that I’m not even asking about an optimal lower bound. By the way, it looks like it might be possible to adapt the Abelian lower bound method to some extraspecial groups (see quant-ph/0701235 for some upper bounds). But I’ve never taken the time to think about it carefully. 41. Scott Says: Gil: Sorry for the delay! Can you say something on Leggett’s lecture and , in particular: What is the definition of the “disconnectivity” measure D; and what is the mysterious K defined in the last slide. Leggett’s lecture was extremely interesting as usual (I’ve heard many previous lectures by him on similar themes). Alas, I don’t remember the definition of “disconnectivity,” but I do remember Leggett saying in the lecture that he’s no longer happy with it. (Nor, alas, do I remember what K is.) And a technical question. How to open Freedman’s presentation. I was able to open it—but it’s a pptx file, which means that it’s not going to open in older versions of PowerPoint. (Complain to Freedman’s employer about that!) Special for you, I’ve converted his presentation into the old PowerPoint format and uploaded it here. 42. Gil Kalai Says: Thanks a lot Scott & Greg. 43. dorit Says: Hi! Sorry for the delay, was out of the loop for a few days… Answering Fernando first. My own motivation for thinking about those interactive proofs with quantum computers was really the question of whether and to what extent it is possible to test quantum mechanics if we don’t (yet?) believe in it fully (and it _is_ a question of “belief” and “religion” at this stage, regarding the aspects of quantum mechanics that are outside of BPP, given the (striking!) realization that these non-BPP aspects were never in fact tested, and we have no direct experimental evidence for them – except for the fact that the theory extrapolates nicely the things we do have direct experimental evidence for.). Obviously physically realizing Shor’s algorithm would be some kind of such a test for non-BPP aspects of QM, and quite a convincing one. But it seems that it will take some time to get such experiments actually performed…One can ask whether we can run tests on smaller physical systems, tests that we can already run today. Also, we can ask whether we could, in the future, run tests that would really test the full quantumness of quantum mechanics – its BQP nature – after all, Shor’s algorithm is not quantum-complete. And it turns out that the protocols we suggest (and those of Broadbent-Fitzimons-Kashefi too) in fact do provide quite satisfying though still somewhat partial positive answer to this quesiton. Yes, we can test quantum mechanics while assuming its correctness only on very small systems, plus some very weak assumptions on larger systems. So you do not have to believe in quantum mechanics to get convinced – but you do have to assume SOMETHING – you do have to assume something that generalizes both known quantum _and_ classical dynamics. And then you can get convinced that the system is really quantum. I am in fact writing an article to a philosophy of science onthology which explains this whole line of arguments. Starting from what I mean by the fact that the non-BPP aspects of quantum mechanics were not tested so far, and how (and what aspect of them and in what sense and under what assumptions) we can in fact test them if we use interactive experiments – namely, interactive proofs with Nature. It is very interesting to try to say these things more formally, though the essential ideas are explained already in the original A-Ben-Or-Eban paper. dorit 44. dorit Says: Hey, Now about Greg’s comment – I am not sure I agree with your disagreement… 🙂 “I don’t entirely agree. You can confirm that a system performed a calculation that exceeds its computational resources, even if some larger classical computer could also perform that calculation.” The point is that you cannot confirm it in the _usual paradigm_ (i.e, without interactions or vierfications, but just by comparing predictions to experimental outcomes), _unless_ a classical computer can compute it… Perhaps I am missing your point, in this argument? anyway, I think I disagree with what I thought was your main point: “I think perhaps the phrase quantum “dynamics” is a bit misleading. I like to call the precepts of quantum computation quantum probability. Quantum probability is clearly different from classical probability, and quantum probability can be tested and has been amply verified. Either quantum probability is not completely true (which would be new science), or there is some statistical censorship principle that prevents full access to quantum probability (which would also be new science), or quantum computation is possible in principle.” You are right in the three possibilities. The point is that though quantum probability has been amply tested and verified, it was NOT tested, and in principle CANNOT be tested in its non-BPP aspects, using the usual predict-and-compare scientific paradigm. So there may still be a theory which is alternative to QM – new science indeed – that is simulatable within BPP and which agrees with all known experiments. BTW – I am not sure why you moved to the static probabilities point of view rather than talking about dynamics. After all, we are talking about computations, which have everything to do with dynamics. The quesiton of which states can exist in quantum mechanics is a related though somewhat different question… How do you translate BPP to your probability language? dorit 45. dorit Says: Fernando, Thanks for pointing out to that paper of Bravyi and Vyalyi (quant-ph 0308021) which defined the commuting Hamiltonian problem. It looks beautiful, I was not aware of it! I think this is a beautiful and very counter intuitive problem… I am wondering if there would be some brave people here who would be willing to bet regarding its computational complexity for large constant k, QMA-complete or in NP or somewhere in between??? dorit 46. matt Says: Sure, Dorit. My bet: the commuting Hamiltonian problem is the same complexity class as quantum PCP, and they both are NP^{DQC_1}-complete. 47. asdf Says: Hey Scott (or anyone): do physics whizzes these days believe (and does QC require) that there is such a thing as true randomness in the physical universe? That is, suppose I flip a coin 2 million times and interpret the result as a bit string X. Since it’s a real physical coin, there’s likely to be some small sources of bias and correlation in the flips, and there’s a tiny chance that I might get a low complexity string like two million zeros, but can I generally say with very high confidence that X has Kolmogorov complexity at least 1 million? (I.e. X cannot be compressed to less than half its length). I’m asking this because if X is really random, the arithmetic proposition “K(X) > 10^6” (let’s call this proposition P) is almost certainly true, and yet (per Chaitin’s version of Gödel’s incompleteness theorem), P is surely independent of any reasonable axiomatic theory of arithmetic comprehensible by humans. So where does the universe get access to these extremely complex and utterly unprovable arithmetic truths? Is that question interesting from a logic point of view? It sounds that way to me, but maybe I’m wrong. Also, am I correct that P is a Pi-0-1 proposition? I think it says: “for all n, the Turing machine T1 has not halted leaving X on the output tape, same for T2, same for T3, …” where T1,T2,… is the (finite) enumeration of all Turing machines with low enough complexity to refute P. I think P being Pi-0-1 would put it in reach of some reflected version of Peano arithmetic. This question was inspired by an article by Leonid Levin: http://www.cs.bu.edu/fac/lnd/expo/gdl.htm 48. dorit Says: Matt, seems you have been working too much with computational complexity scientists lately! be careful, you have no idea what dark allies it can take you to. BTW, DQC is already taken… by too many: DQC Digital Quaker Collection DQC Data Quality Control DQC Double Quarter Column (print advertising) DQC Double Quantum Coherence DQC Design Quality Control DQC Data Quality Center DQC Delaware Quality Conference DQC Definite Quantity Contract DQC Directional Quasi-Convexity DQC Dame Quaker’s Club (gaming) DQC Dream Quest Communications DQC Dosimetric Quality Control DQC Domain Quick Connect (Apollo) DQC Disk Quota Control (Microsoft Windows NT) DQC Deemed QC DQC Division Quality Council DQC Dose and Quality Control DQC Departmental Qualification Course DQC Doomed Quantum Computation 49. Greg Kuperberg Says: Dorit: Well, I am not sure whether we truly disagree or not. I could say more, but maybe we would partly be arguing over semantics. It is difficult to clear up a semantic confusion in a blog thread; but maybe when we next meet in person…? Okay, I will say a little more, and try to steer the discussion away from semantics. My position is that yes, while technically you cannot test whether any dynamics exists outside of BPP without actually building a quantum computer, such a test is only impossible if you are partly skeptical of quantum mechanics itself. If you believe quantum mechanics, then you can do a test in which a very small physical system computed more than is classically possible using its own limited computational resources, even if it can be simulated by some other, much larger computer. But again, if you do not entirely believe quantum mechanics, then who is to say how much computational power is available to ten trapped ions, say. Maybe they have twenty megabytes of storage and the equivalent of a megahertz processor. That would be plenty to simulate ten qubits, and it would also be easy to simulate with modern computers, so it would prove nothing that the ten ions can do a Grover search. do physics whizzes these days believe (and does QC require) that there is such a thing as true randomness in the physical universe? Quantum computation does not separately require “true randomness”. However, true randomness is a consequence (not an assumption) of quantum probability, which is my name for the axiomatic foundation of quantum mechanics. In other words, you can construct a coin flip using a measurement protocol with some qubits. That coin flip may or may not be “random” in some philosophical sense; but if you or any other physical actor can predict this quantumly produced coin flip, even with the aid of a large computer, then quantum mechanics is wrong. 50. fernando brandao Says: Dorit, thanks for the explanations! Yes, we can test quantum mechanics while assuming its correctness only on very small systems, plus some very weak assumptions on larger systems. So you do not have to believe in quantum mechanics to get convinced – but you do have to assume SOMETHING – you do have to assume something that generalizes both known quantum _and_ classical dynamics. That’s really cool. Is that in your paper (perhaps it’s implicit and I missed it)? If it’s not too complicated, could you tell what would be the assumptions you need on larger systems? I’d bet the commuting problem is in NP, or at most in QCMA: this paper arXiv:0803.1447 has some nice results concerning frustation free commuting Hamiltonains which seems to point in this direction, modulo the fact that I don’t understand the argument 🙂 51. asdf Says: Correction, meant to say “for all n, after running for n steps, Turing machine T1 has not halted…” It also turns out that Levin has an arxiv paper linked from the article I linked to, where he considers a question sort of like mine. He hypothesizes that these physics experiments won’t reveal anything mathematically interesting, and presents that a bit more formally, as sort of an extended Church-Turing thesis. 52. asdf Says: Greg, thanks for the response (I didn’t notice it at first, deep down in your comment). I’m not sure what it means to be random in a “philosophical sense”. Certainly, the proposition that a string is compressible to half its length is a purely mathematical property of the string, which is provable if true (though maybe not by any feasible computation). If the proposition is false, its falsehood is generally unprovable using any reasonable mathematical axioms. It is algorithmically undecidable whether a given string is compressible like that or not. So the rather curious thing is that if there is randomness, our creaky old physical universe is able to supply us with endless supplies of almost-certain mathematical truths that we could never generate with any recursive algorithms. 53. Sensational Says: I wonder if you have any idea how much transistors is in simplest calculator (I mean without sinus and cosine and so on, where only +, -, /, * and with precision of about 6-8 decimal digits). How many NAND gates (which consist of about 4 or 6 transistors) or some over need to build such simplest calculator and maybe there is some drowed scheme on internet somthing, because it seems very interesting and usefull). Why you don’t analizing even and not teaching this in university with such great examples (like simplest calculator scheme must be in each textbook)? 54. Greg Kuperberg Says: I’m not sure what it means to be random in a “philosophical sense”. Certainly, the proposition that a string is compressible to half its length is a purely mathematical property of the string, which is provable if true (though maybe not by any feasible computation). All I meant by “philosophical” was that a quantumly generated coin flip might possibly not be random to “God”. But it cannot be predicted by observers that live within the universe. 55. Joe Fitzsimons Says: Fernando, I’m not sure exactly what assumptions are made in Dorit’s paper, but in ours the only necessary requirement for the larger system is that operations are linear. If there is a way to perform non-linear operations deterministically then the privacy theorem does not hold. 56. Gil Kalai Says: There is another point worth mentioning about cases that digital computation is computationally sufficient for simulating quantum physics. The difference between simulating quantum physics with digital computers compared to simulating it with a quantum computers is like the difference between a computer simulation of a human heart and an operating artificial heart. Creating an EPR pair is not the same as describing its properties on your lap top. Another thought about it is that many physics experiments can be regarded as “depth one quantum circuits” so that moving from depth one to depth two can already be powerful. (Again I am not talking about computational complexity but about creating interesting quantum states. 57. Elias Kai Says: Thank you for the PPT	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8023993372917175, "perplexity": 931.0879727813324}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549424645.77/warc/CC-MAIN-20170724002355-20170724022355-00066.warc.gz"}
http://math.stackexchange.com/questions/349773/sequential-allocation-of-n-balls-into-n-urns/351739	# Sequential allocation of $n$ balls into $n$ urns Assume that there are $n$ balls (numbered from $1$ to $n$) and $n$ urns (numbered from $1$ to $n$). At the beginning no ball is placed in any urn. At $t=1$, each ball is randomly put into an urn (no restriction on how many balls an urn can contain, each ball can be placed into one urn for example) Check each urn and if there is more than one ball, randomly choose one of the balls and keep it in the urn and remove all the other balls. (do this for each urn) At $t=2$, take the balls removed from some urn at $t=1$ and again randomly place into an urn (except the urn that the ball was removed from at $t=1$, say ball 1 was removed from urn 1, then ball 1 can be thrown at urns 2,3,...,n; same for other balls). Again, check each urn and if there is an urn that contains more than one ball and if there was no ball placed to that urn at $t=1$ choose one of them randomly and remove others. If there was a ball placed into that urn at $t=1$ remove all the new balls placed at $t=2$ from that urn. At $t=k$, take the balls removed from some urn at $t=(k-1)$ and again randomly place into an urn ( except the urn that the ball was removed from at $t=1,...,(k-1)$). Again, check each urn and if there is an urn that contains more than one ball, choose one of them randomly if there was no ball placed to that urn at $t=1,\ldots,(k-1)$. If there was a ball placed into that urn at $t=1,\ldots,(k-1)$ remove all the new balls placed at $t=k$ from that urn. Continue in this manner and stop when each urn has only one ball. Although it should be clear, just to emphasize, if a ball is chosen to be placed into an urn at the end of $t=k$, it remains in that urn afterwards, that is, it can not be removed in the later stages. Also, a ball can not be thrown into an urn it was removed in earlier stages. What is the probability that a certain ball, say Ball 1, is placed into some urn at exactly $t=k$ for each $k=1,\ldots,n$? Any suggestions for the solution or even references on similar problems would be appreciated. The answer for $t=1$ is easy: For example, consider Ball 1, and assume it is thrown into Urn 1 at $t=1$ and let $j$ be the number of balls that are placed into urn 1 at $t=1$ among the remaining $n-1$ balls. Then, probability that ball 1 is placed into urn 1 is: $$\sum_{j=0}^{n-1}\binom{n-1}{j}\left( \frac{1}{n}\right) ^{j}\left( \frac{n-1}{n% }\right) ^{n-j-1}\left( \frac{1}{j+1% }\right)$$ - I am a little unclear; at step t=1.5 when we are removing balls are these being removed from all urns with more than 1 ball or just 1? – Dale M Apr 3 '13 at 9:22 Do this for each urn That has at least one ball: if there is only one ball, the ball is kept and if there are say 2 balls, choose one of them and remove so that there is only one ball left in the urn, if there are three balls remove 2 balls and so on. – Emre Per Apr 3 '13 at 14:30 Your formula for $\mathbb{P}(t=1)$ simplifies to $1-\left({n-1\over n}\right)^n$. Otherwise, I don't believe you will get simple formulas. But I'll keep looking. – Byron Schmuland Apr 5 '13 at 16:30 Thanks for looking. Yeah, I knew that but wanted to put this to give an idea how I proceed for solving the problem. I got the solution for $t=2$ also but stuck for above – Emre Per Apr 5 '13 at 16:36 An approach: the thing to think about is how many urns are unoccupied after each run. The first ball that is put in an urn sticks and is never removed. After the first toss, on average $\frac 1e$ of the urns will be empty. That is also the number of balls thrown at urns in the second toss. The no repeat rule makes this a little messier as the empty urns are slightly more probable to receive a ball than the full ones. Ignoring that, each urn receives on average $\frac 1{e}$ balls, so the fraction of newly filled urns is $\frac 1e\exp(-\frac 1e)$, leaving about $0.255$ of the urns empty. After three tosses, the number of empty urns is $\frac 1e\exp(-\frac 1e)\exp(-\frac 1e\exp(-\frac 1e))\approx 0.197$. The chance that ball $1$ goes into some urn at turn $k$ is precisely the number of empty urns after turn $k-1$ as that is the number of balls that are not stuck. Again ignoring the no repeat rule, the chance ball $1$ goes in urn 1 at turn $k$ is then $\frac 1n$ (empty urns at turn $k-1$). - The approach is exactly what I was trying to follow but it is not easy to compute the expected number of empty urns. I do not think how you compute is true for the expected number of empty urns. – Emre Per Apr 3 '13 at 18:22 @Emre Per: I am using the Poisson approximation, valid for lage $n$. Each ball arrives in a given urn with probability $\frac 1n$ and the probability of an urn being empty is $\exp(-\lambda)$ – Ross Millikan Apr 3 '13 at 18:39 Let $\mathbb{U}_j$ be the random variable that a given ball that is "in play" (see below) $b$ is placed into urn $u$ at step $j$. In the first instance, the probability mass function of $\mathbb{U}_1$ is $$f_{\mathbb{U}_1}(k)=\begin{cases}\frac{1}{n} & \text{for k \in\{1,2,\ldots ,n\}}\\ 0 & \text{otherwise} \\\end{cases}$$ $$\mathbb{P}(\text{exactly } k \text{ urns are empty})={{n\choose n-k}{n \brace n-k}(n-k)!\over n^n}.$$ The notation ${n \brace n-k}$ refers to Stirling numbers of the second kind. This allows you to create a random variable $\mathbb{X}_1$ that represents the number of empty urns at step $j=1$ and, importantly, the number of balls "in play" for step $j=2$. $$f_{\mathbb{X}_1}(k)=\begin{cases}{{{n\choose n-k}{n \brace n-k}(n-k)!}\over {n^n}} & \text{for k \in\{1,2,\ldots ,n\}}\\ 0 & \text{otherwise} \\\end{cases}$$ So, $\frac{f_{\mathbb{X}_1}(k)}{n}$ is the probability a given ball is "in play" , and the probability that that ball will hit a given urn is $\frac{1}{n}$, so \begin{align}f_{\mathbb{U}_2}(k)&=\frac{f_{\mathbb{X}_1}(k)(1-f_{\mathbb{U}_1}(k))}{n^2}\\ &=\begin{cases}{{n\choose n-k}{n \brace n-k}(n-k)!(n-1)\over n^{(n+3)}}& \text{for k \in\{1,2,\ldots ,n\}, k\ne u_1}\\ 0 & \text{otherwise} \\\end{cases} \end{align} Removing the restriction that a ball cannot be placed in the same urn from which it was removed (which would be a fair approximation for large $n$), simplifies this to \begin{align}f_{\mathbb{U}_2}(k)&=\frac{f_{\mathbb{X}_1}(k)}{n^2}\\ &=\begin{cases}{{n\choose n-k}{n \brace n-k}(n-k)!\over n^{(n+2)}}& \text{for k \in\{1,2,\ldots ,n\}}\\ 0 & \text{otherwise} \\\end{cases} \end{align} The next step is to determine $\mathbb{X}_2$ It is clear that the number of balls in play is a monotonically decreasing function as once an urn has a ball it can never be empty again and this will end when each urn has one ball. Now the $\mathbb{X}_1$ balls that are "in play" can be divided into $\mathbb{S}_1$ subsets with \begin{align} f_{\mathbb{S}_1}(k)&= {{f_{\mathbb{X}_1}(k) \brace f_{\mathbb{X}_1}(k)-s}} \\ &=\begin{cases}{{n\choose n-k}{n \brace n-k}(n-k)!\over n^{n}} \brace {{{n\choose n-k}{n \brace n-k}(n-k)!\over n^{n}}-s}& \text{for k \in\{1,2,\ldots ,n\}, s\le k}\\ 0 & \text{otherwise} \end{cases}\end{align} These subsets can be divided among the urns but now I'm stuck. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9232267141342163, "perplexity": 278.91596337702225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765002.8/warc/CC-MAIN-20141217075245-00162-ip-10-231-17-201.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/polar-equations-of-conics-question.319484/	# Polar equations of conics question 1. Jun 11, 2009 ### demonelite123 for this problem, i've been given the vertices of the hyperbola as (4, pi/2) and (-1, 3pi/2). the question asks to find the polar equation of this hyperbola. so what i did was do a quick sketch of the graph. (4, pi/2) is essentially (0,4) and (-1, 3pi/2) is essentially (0,1). the midpoint of the two vertices is (0,2.5) so that's the center. since the focus is at (0,0), c = 2.5 and a = 1.5 therefore e = c/a and e = 5/3 so i wrote out the equation r = (5/3)p / 1 + (5/3)sin& since the directrix is above the x axis or polar axis between the two vertices. (by the way i'm using "&" to stand for the greek letter theta.) then i just used one of the points they gave me (4, pi/2) and plugged it into the equation to solve for p. 4 = (5/3)p / (1 + (5/3)) when i solve for p, i get that p = 32/5. however when i plug in the other point they gave me for the hyperbola (-1, 3pi/2), i get a different p value. -1 = (5/3)p / (1 - (5/3)) and when i solve for p i get that p = 2/5. what's going on? how come the p values are different? both points are vertices of the hyperbola so i don't know why i'm getting different answers. the back of my textbook says that p = 8/5. how did the book get that? and how am i getting two different p values? please help me! i'm so confused. 2. Jun 12, 2009 ### HallsofIvy Staff Emeritus What are you using for the eccentricity? Assuming the vertices are also given in polar coordinates, the vertices are at (0, 4) and (0, -1) so the center is at (0,3/2) but there are an infinite number of hyperboles that fit that. 3. Jun 12, 2009 ### demonelite123 the vertices they have given me are (4, pi/2) and (-1, 3pi/2) which in rectangular coordinates is (0,4) and (0,1) so the center of the hyperbola is (0,5/2) and (0,0) is one focus of the hyperbola. so using that info i determined that c = 2.5 (the distance from center to focus) and a = 1.5 (distance from center to vertex). and i know that e = c/a, so i did e = 2.5 / 1.5 = 5/3 so my eccentricity of this hyperbola is 5/3. 4. Jun 12, 2009 ### gabbagabbahey Hmmm... shouldn't this be: $$r=\frac{\left(\frac{5}{3}\right)p}{-1+\left(\frac{5}{3}\right)\sin(\theta)}$$ ??? 5. Jun 18, 2009 ### demonelite123 hm in my textbook, they only show the formulas r = ep / (1 +/- cos(&)) and r = ep / (1 +/- sin(&)) (& stands for "theta") how did you get the -1 in your equation? Similar Discussions: Polar equations of conics question	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8674060702323914, "perplexity": 745.1060608779255}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103270.12/warc/CC-MAIN-20170817111816-20170817131816-00647.warc.gz"}
https://www.mathdoubts.com/algebra-numerical-factor/	# Numerical factor of algebraic terms A number that multiplies at least a literal to form an algebraic term is called the numerical factor of an algebraic term. ## Introduction An algebraic term is formed by the product of a number and at least a literal to represent a quantity in mathematical form. The number multiplies the literal in the algebraic term. Hence, it is known as a factor basically but it is a numeral. Therefore, it is known as a numerical factor. ### Example $6xy$ is an algebraic term. In this algebraic term, the number $6$ and the literals $x$ and $y$ are multiplying each other to represent a quantity in product form. Mathematically, a multiplying element is called a factor in a term. Hence, all three of them are factors but $6$ is a factor in numerical form. Hence, the number $6$ is called a numerical factor. ### More Examples Look at the following examples to know how to determine a numerical factor in every algebraic term. $(1)\,\,\,\,\,\,$ $-4a$ In this example, $-4$ is a number and multiplying the literal $a$. Hence, $-4$ is called a numerical factor. $(2)\,\,\,\,\,\,$ $7b^2c$ $7$ is called the numerical factor. $(3)\,\,\,\,\,\,$ $0.07gh^3$ $0.07$ is a decimal number. Hence, it is called the numerical factor. $(4)\,\,\,\,\,\,$ $\dfrac{7e^4}{5}$ The algebraic term $\dfrac{7e^4}{5}$ can be written as $\dfrac{7}{5}e^4$. Therefore, $\dfrac{7}{5}$ is called the numerical factor. $(5)\,\,\,\,\,\,$ $\dfrac{-j^4}{4k}$ The algebraic term can be written as $-\dfrac{1}{4} \times \dfrac{j^4}{k}$. Therefore, the numerical factor is $-\dfrac{1}{4}$. A best free mathematics education website for students, teachers and researchers. ###### Maths Topics Learn each topic of the mathematics easily with understandable proofs and visual animation graphics. ###### Maths Problems Learn how to solve the maths problems in different methods with understandable steps. Learn solutions ###### Subscribe us You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8146008849143982, "perplexity": 679.8011434468199}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00112.warc.gz"}
http://scholarship.rice.edu/handle/1911/16252	# Random process simulation for stochastic fatigue analysis ### Files in this item Files Size Format View 9012821.PDF 2.947Mb application/pdf Title: Random process simulation for stochastic fatigue analysis Larsen, Curtis Eliot Lutes, Loren D. A simulation technique is described which directly synthesizes the extrema of a random process and is more efficient than the Gaussian simulation method. Such a technique is particularly useful in stochastic fatigue analysis because the required stress range moment, E(R$\sp{\rm m}$), is a function only of the extrema of the random stress process. The family of autoregressive moving average (ARMA) models is reviewed and an autoregressive model is presented for modeling the extrema of any random process which has a unimodal power spectral density (psd). The proposed autoregressive technique is found to produce rainflow stress range moments which compare favorably with those computed by the Gaussian technique and to average 11.7 times faster than the Gaussian technique. The autoregressive technique is also adapted for processes having bimodal psd's. The adaptation involves using two autoregressive processes to simulate the extrema due to each mode and the superposition of these two extrema sequences. The proposed autoregressive superposition technique is found to be 9 to 13 times faster than the Gaussian technique and to produce comparable values for E(R$\sp{\rm m}$) for bimodal psd's having the frequency of one mode at least 2.5 times that of the other mode. A key parameter in the autoregressive model is the correlation coefficient $\rho\sb1$ between adjacent extrema. A linear regression of $\rho\sb1$ on Vanmarcke's bandwidth parameter is presented as a practical description of $\rho\sb1$'s dependence on bandwidth for both unimodal and bimodal psd's. The effect of psd shape on the expected fatigue damage rate is also investigated. For bimodal psd's, the contribution of the two frequency components to the damage rate is determined for frequency ratios from 1.5 to 15. The relative contribution of the two modes is measured by a parameter b which is the ratio of the mean squared value of the high frequency component to that of the other component. It is found that both components must be considered for b values from 0.01 to 10. The effect of high frequency truncation of the psd on the expected damage rate is also studied for two unimodal psd's. Larsen, Curtis Eliot. (1988) "Random process simulation for stochastic fatigue analysis." Doctoral Thesis, Rice University. http://hdl.handle.net/1911/16252. http://hdl.handle.net/1911/16252 1988	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8984350562095642, "perplexity": 919.3037558677586}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609524259.30/warc/CC-MAIN-20140416005204-00409-ip-10-147-4-33.ec2.internal.warc.gz"}
https://faculty.ucr.edu/~legneref/medical/Anophelini-Genera.htm	File: <Anophelini-Genera.htm> <Medical Index> <General Index> Site Description <Navigate to Home> Insecta: Diptera (Mosquitoes) ANOPHELINAE (Genera & Common Subgenera) (Contact) ADULTS 1. The scutellum has three slight lobes (occur in South America) _ _ _ _ _ _ Cruz Scutellum is shaped like a crescent and is evenly rounded _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 The head is round. There are anterior prenotal lobes or "collar" behind the head (Fig. 1) _ _ _ 6 2. The stem of the wing's 2nd fork cell is wavy _ _ _ _ _ _ _ _ _ _ _ _ _ Bironella spp. Theobald The stem of the wing's 2nd fork cell is straight (Fig. 2) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Anopheles spp. Meigen 3 3. The thorax is more or less black with a broad gray line from the neck to the scutellum _ _ _ Subgenus: Stethomyia Theobald Designs on the thorax are not like previously described _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 4. Wings seldom have more than two pale spots on the costa (Fig. 3). The sidepiece of male genitalis has 1-2 tough basal spines on tubercles _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Anopheles spp. Meigen Wings have 4 or more pale costal spots (Fig. 4) _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 5. The sidepiece of male genitalis has one spine at its base and two beyond (American species) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Subgenus: Nyssorhynchus Blanchard Male genitalis sidepiece has a few weak spines near its base that are not set on tubercles (Old World species) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Subgenus Myzomyia Blanchard 6. The head is round. There are prenotal lobes or collar behind the head. The hind leg has a paddle (Fig. 1) (South American species)_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Sabethes spp. LARVAE 7. The larva's body is heavily clothed with short hairs in addition to the regular number of hairs. There are leaflets of palmate tufts considerably expanded apically and each ending in a long central hair. The anterior flap of the spiracular apparatus is extended into a long, heavy bristlelike structure _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Chagasia spp. Cruz a. Larval body is not covered with fine hairs. The leaflets of palmate tufts are different from that described previously. There is no elongation of the anterior flap of the spiracular apparatus_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 8 b. A dark sclerotized boss occurs at the base of the ventral brush or hairs on the siphon occur singly_ _ _ _ _ _ _ _ _ _ _ _ _ _ 9 8. There are 2 pairs of feathery hairs on the thorax_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ Bironella spp. Theobald Feathery hairs on the thorax do not exceed one-pair (Fig. 5 & Fig. 6) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Anopheles spp. Meigen 9. A Pecten is present on the siphon. The ventral brush has more than one pair of hairs and a dark boss occurs at its base. Hairs on he siphon are scarce or absent _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (Fig. 7) (South American species) There are no pecten or dark boss present, & ventral brush has only one pair of hairs, and hairs on the siphon occur singly (Fig. 8) (South American species) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Sabethes spp. = = = = = = = = = = = = = = = = = = EXPANDED CULICIDAE KEYS - - - - - - - - - - - - - - - - - - - - Key References: Matheson, R. 1950. Medical Entomology. Comstock Publ. Co, Inc. 610 p. Service, M. 2008. Medical Entomology For Students. Cambridge Univ. Press. 289 p Legner, E. F. 1995. Biological control of Diptera of medical and veterinary importance. J. Vector Ecology 20(1): 59_120. Legner, E. F.. 2000. Biological control of aquatic Diptera. p. 847_870. Contributions to a Manual of Palaearctic Diptera, Vol. 1, Science Herald, Budapest. 978 p.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9959501028060913, "perplexity": 398.6680769921087}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499768.15/warc/CC-MAIN-20230129211612-20230130001612-00179.warc.gz"}
https://cms.math.ca/cjm/kw/Fourier%20multiplier	location: Publications → journals Search results Search: All articles in the CJM digital archive with keyword Fourier multiplier Expand all Collapse all Results 1 - 6 of 6 1. CJM 2016 (vol 69 pp. 284) Chen, Xianghong; Seeger, Andreas Convolution Powers of Salem Measures with Applications We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of KÃ¶rner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$. Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz typeCategories:42A85, 42B99, 42B15, 42A61 2. CJM 2014 (vol 66 pp. 1358) Sharp Localized Inequalities for Fourier Multipliers In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of LÃ©vy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator . Keywords:Fourier multiplier, martingale, laminateCategories:42B15, 60G44, 42B20 3. CJM 2012 (vol 65 pp. 510) Blasco de la Cruz, Oscar; Villarroya Alvarez, Paco Transference of vector-valued multipliers on weighted $L^p$-spaces We prove restriction and extension of multipliers between weighted Lebesgue spaces with two different weights, which belong to a class more general than periodic weights, and two different exponents of integrability which can be below one. We also develop some ad-hoc methods which apply to weights defined by the product of periodic weights with functions of power type. Our vector-valued approach allow us to extend results to transference of maximal multipliers and provide transference of Littlewood-Paley inequalities. Keywords:Fourier multipliers, restriction theorems, weighted spacesCategories:42B15, 42B35 4. CJM 2012 (vol 65 pp. 299) Grafakos, Loukas; Miyachi, Akihiko; Tomita, Naohito On Multilinear Fourier Multipliers of Limited Smoothness In this paper, we prove certain $L^2$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of CalderÃ³n and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed. Keywords:multilinear Fourier multipliers, HÃ¶rmander multiplier theorem, Hardy spacesCategories:42B15, 42B20 5. CJM 2011 (vol 63 pp. 1161) Neuwirth, Stefan; Ricard, Éric Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1. Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projectionCategories:47B49, 43A22, 43A46, 46B28 6. CJM 1998 (vol 50 pp. 897) Bloom, Walter R.; Xu, Zengfu Fourier multipliers for local hardy spaces on ChÃ©bli-TrimÃ¨che hypergroups In this paper we consider Fourier multipliers on local Hardy spaces $\qin$ \$(0 Keywords:Fourier multipliers, Hardy spaces, hypergroupCategories:43A62, 43A15, 43A32 top of page \| contact us \| privacy \| site map \|	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.93061363697052, "perplexity": 2529.4357410431057}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806422.29/warc/CC-MAIN-20171121185236-20171121205236-00526.warc.gz"}
http://mca.ignougroup.com/2017/05/meaning-proof-of-these-regex.html	World's most popular travel blog for travel bloggers. Meaning / proof of these regex , , Problem Detail: The regex associative laws are: $$(L+M)+N=L+(M+N)$$ $$(LM)N=L(MN)$$ Some important implications out of associative laws are: $$r(sr)^=(rs)^r$$ $$(rs+r)^r=r(sr+r)^$$ $$s(rs+s)^r=(sr+s)^sr$$ $$(LM)^N\neq L(MN)$$ The issue is that • I don't find the implications much intuitive as the identities themselves are. How can I understand the implications intuitively? I can always form a strings belonging to left hand side regex and check whether it can be accepted by other regex. The first implication is very simple to test this way. However how can I make them more intuitive?? • Are these implications simply made up expressions which are tested rigorously to hold true and they don't have any specific expression as we can form many such expressions? • I am unable to get the point behind stating these implications. I dont think of any problem in which I can use these regexes straight / immediately. It may be because I am not able to get intuition behind these implications so that it may strike in my head immediately when to use these implications. For all three of your statements, the answer is that, generally speaking, the implications simply aren't as intuitive as, say, the associative laws. Look at the analogous problem with algebraic expressions: we have associative laws for addition and multiplication and we also have a distributive law that states that for any expressions $p,q,r$ we have $$p\cdot(q+r)=(p\cdot q)+(p\cdot r)$$ Eventually, these rules should be intuitively obvious. However, one implication of these rules, that $$(p+q)^3=p^3+3p^2q+3pq^2+q^3$$ is probably not as intuitively obvious as the rules used to derive that identity. The utility of the result above is that it can be used as a tool to simplify other more complicated problems. It's the same for regular expressions: the fact that, say, $r(sr)^=(rs)^r$, is correct can be proven rigorously, but having done that you can use it as a tool to show that $$aa+(aab)^aa=aa+aa(baa)^=aa(\epsilon+(baa)^*)$$ should you ever need to. For example, there is a handy technique, involving what's known as Arden's lemma that can be used to produce a regular expression describing the language accepted by a finite automaton. Depending on how it's applied, this can produce several regular expressions from the same FA, so it might fall to you to show that the expressions are indeed equivalent, in which case the "implications" you listed might be handy. The upshot (here's the tl;dr part), is that the implications you mentioned are simply tools you can use when needed: they've beed proven to hold, but there's no reason why they should be intuitively obvious.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8165764212608337, "perplexity": 353.9227998686284}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657172545.84/warc/CC-MAIN-20200716153247-20200716183247-00275.warc.gz"}
http://michaelnielsen.org/polymath1/index.php?title=List_of_results_implied_by_the_Riemann_Hypothesis	List of results implied by the Riemann Hypothesis This is a partial list of potentially useful results conditional on the Riemann Hypothesis (RH) or the Generalized Riemann Hypothesis (GRH). Conditional on RH All of the results listed below are known to be true assuming the Riemann hypothesis. It should be stressed that none of these results are known to be true unconditionally, although weaker versions of most of them are known. Prime gaps. Let $p_n$ denote the nth prime. Then $p_{n+1}-p_n = O(\sqrt{p_n} log(p_n))$. Primes in intervals. Related to the above result. There is always a prime in the interval $[n, n + O(\sqrt{n} log(n))]$. Strong-form algorithm. As a result of the above, there is an algorithm with running time $O(k(10^k)^{1/2})$ that locates a prime number with at least k digits (with an oracle for testing primality). Distribution of prime logarithms. Let K be the set of logarithms of the primes in the interval $[S, 2S]$. Then the Fourier coefficients of K have size at most about $S^{-1/2}$ up to logarithmic factors. Distribution of square-free numbers. Let $Q(n)$ be the number of square-free numbers less than or equal to n. Then $Q(n) - \frac{6}{\pi^2}n = O(n^{17/45 + \epsilon})$ for all $\epsilon \gt 0$. Conditional on GRH All of the results listed below are known to be true assuming the generalized Riemann hypothesis. Again, we stress that none of these results are known to be true unconditionally, nor even assuming the (standard) Riemann hypothesis. Strong-form algorithm. It may be possible, using the W-trick, to obtain a slightly faster algorithm that locates a prime number of at least k digits than the one assuming the standard RH. Primitive roots. Let p be an odd prime. Then the smallest primitive root (mod p) has order $O(log^6(p))$. Generating the multiplicative group (mod p). Let p be an odd prime. Then the multiplicative group (mod p) $(Z/pZ)*$ is generated by its elements of size $O(log^2(n))$. If the logarithm is natural, we can take 2 as the implied constant. Primality testing. As a result of the above, there is a deterministic polynomial-time form of the Miller-Rabin primality test. Until the AKS algorithm was discovered, this was the best known candidate for a deterministic polynomial-time algorithm to test primality. Square roots mod p. Let p be an odd prime and n be a quadratic residue (mod p). Then the Shanks-Tonelli algorithm is a deterministic polynomial-time algorithm to compute a solution to $x^2 \equiv n \pmod{p}$. References 1. Bach, Eric; Shallit, Jeffrey (1996), Algorithmic Number Theory (Vol I: Efficient Algorithms), Cambridge: The MIT Press, ISBN 0-262-02045-5 2. Jia, Chao Hua. "The distribution of square-free numbers", Science in China Series A: Mathematics 36:2 (1993), pp. 154–169.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888848066329956, "perplexity": 275.1519940872078}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812306.16/warc/CC-MAIN-20180219012716-20180219032716-00377.warc.gz"}
http://www.physicsformulae.com/	# Physics FormulaeDonate ## Related physics websites ### Physics Unit Conversions Physics Unit Conversions is a fast and minimalistic web-app for converting units in physics. ### LaTeX Mathematics Symbols www.latexmathematicssymbols.com is a simple web app that you can use to look up the LaTeX commands for various mathematical symbols. ### Ultraphysics Ultraphysics is a small website where you can download lecture notes on various topics in physics.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9868444800376892, "perplexity": 2908.5801123708316}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107911792.65/warc/CC-MAIN-20201030212708-20201031002708-00702.warc.gz"}
http://math.stackexchange.com/questions/123239/dirac-delta-function?answertab=votes	# Dirac Delta function I know that $\int_{-\epsilon}^\infty f(x)\delta(x)dx=f(0)$ but what about $\int_0^\infty f(x)\delta(x)dx$? I suppose we have to do this by definition since the lower limit is bang on $0$? - – joriki Mar 22 '12 at 11:55 @joriki: Thanks! So it is just 1? – Ringo Mar 22 '12 at 12:15 As Carl's answer and the (partly contradictory) answers to the question I linked to indicate, it's not well-defined. If you tell us the context in which you're trying to use this, more might be said about what the appropriate definition might be for that context. – joriki Mar 22 '12 at 12:33 Joriki's comment is right. My answer uses the measure-theoretic interpretation of $\delta$ as a point mass measure. But if you used the interpretation via test functions instead, you should get $f(0)/2$ instead of $f(0)$. – Carl Mummert Mar 22 '12 at 12:42 See this answer. – Did Mar 24 '12 at 13:01 The dirac delta function is a strange beast. The entire notation "$\delta(x)dx$" is a little bit of a lie, because there is not actually a function that has the properties that the $\delta$ function is axiomatized to have, which are: • $\delta(x) = 0$ when $x \not = 0$ • $\int_{-\infty}^\infty \delta(x)\, dx= 1$ Using these, and the usual properties of the integral, it is not hard to prove that $\int_0^\infty f(x)\delta(x)\,dx = f(0)$ for every function $f$. The main point is that $$\int_{-\infty}^\infty f(x)\delta(x)\,dx = \int_{(-\infty,0)} f(x)\delta(x)\,dx + \int_{[0,\infty)}f(x)\delta(x)\,dx = 0 + \int_0^\infty f(x)\delta(x)\,dx$$ and $$\int_{-\infty}^\infty f(x)\delta(x)\,dx = \int_{-\infty}^\infty f(0)\delta(x)\,dx = f(0)\int_{-\infty}^\infty \delta(x)\,dx = f(0)$$ Of course you can show by similar methods that $\int_0^0 f(x)\delta(x)\,dx= f(0)$, although this integral should always be 0. This is one way to see that no $\delta$ with the two properties listed above really exists as a function. A way to make sense of this is to use measure-theoretic techniques and Lebesgue integration. We can treat the $\delta$ function as a measure; it puts 1 unit of "mass" at the origin and no mass anywhere else. If we call that measure $\phi$ and we integrate a function $f$ with respect to that measure, we really do get $\int_{-\infty}^\infty f(x) \,d\phi = \int_0^0 f(x)\,d\phi = f(0)$. The problem is that this measure does not have a density function (a Radon-Nikodym derivative) with respect to Lebesgue measure, and so we cannot replace the $d\phi$ with anything of the form $\delta(x)dx$ where $dx$ indicates Lebesgue measure. In practice, people use the notation $\delta(x)dx$ as a purely formal device for calculations. That works out OK as long as only sound methods used for manipulating integrals involving $\delta(x)$. In introductory books, the entire issue is usually glossed over, with just a sentence of two saying "this is formally wrong but it works as long as you do it in the way shown here". - Thank you, Carl! – Ringo Mar 22 '12 at 15:00 What a nice answer! + 1 – Rudy the Reindeer Mar 25 '12 at 10:54 You are something of a victim of ambiguity in notation. The definition of the integral $$\int_A f \; da$$ is taken over the open or closed (or neither) object $A$. As Carl M notes, we have that $$\int_{\{0\}} f(x) \delta(x) \; dx = f(0)$$ by construction of the delta functional, while $$\int_{(0,\infty)} f(x) \delta(x) \; dx = \int_{(-\infty,0)} f(x) \delta(x) \; dx = 0.$$ More generally, $$\int_A f(x) \delta(x) \; dx = \begin{cases}f(0)\qquad&0 \in A\\0\qquad&\text{otherwise.}\end{cases}$$ When someone writes $$\int_0^\infty g(x) \; dx$$ they are not being clear what the domain of integration really is. - This is certainly an ambiguity, but I think there is another one as well. If we let $(g_n)$ be a sequence of even test functions such that $\lim_{n \to \infty} \int_{(-\infty,\infty)} f(x)g_n(x)\,dx = f(0)$ then, because each $g_n$ is even, if $f$ is also even we have $\lim_{n \to \infty} 2\int_{[0,\infty)} f(x)g_n(x)\,dx = f(0)$ and so $\lim_{n \to \infty} \int_{[0,\infty)} f(x)g_n(x)\,dx = f(0)/2$. Some people would write $\int_{[0,\infty)} f(x)\delta(x)\,dx$ to stand for the latter limit, and in particular they would have $\int_{[0,\infty)}\delta(x)\,dx = 1/2$ because of evenness. – Carl Mummert Mar 22 '12 at 13:21 Thank you, Brian! – Ringo Mar 22 '12 at 15:00 @Carl: Agreed. The original dirac delta was a limit of gaussians, defined in such a way. It is possible to define a point mass functional as a limit of 1-sided kernels as well. I think of these things in functional analytic terms: the $\delta$ defines a functional taking $f$ to its value at the location of the point mass. A limit of even functions is an odder beast, taking $f$ to $f(0)/2$ if the point is on the boundary and $f(0)$ or $0$ otherwise. Note that defining $\delta(x)$ as a limit of even functions has the annoying property that $\int_{(-\infty,0)} f(x) \delta(x) dx = f(0)/2$. – Brian B Mar 22 '12 at 16:07	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9523969292640686, "perplexity": 258.7281872862374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246651727.46/warc/CC-MAIN-20150417045731-00192-ip-10-235-10-82.ec2.internal.warc.gz"}
https://www.photard.fr/iwp/bkh84n/x1odc.php?tag=coefficient-binomial-latex-225be4	b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. 分式和二项式系数是非常常见的数学元素，它们有着一些共同的特点：一个数字位于另外一个数字的上方。本篇文章解释如何在 LaTeX 中输入它们。 Expanding a combinatorial argument involving permutation coefficients. \] And of course this command can be included in the normal text flow \ (\binom{n} {k}\). b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. (adsbygoogle = window.adsbygoogle \|\| []).push({}); Binomial coefficient denoted as c(n,k) or n c r is defined as coefficient of x k in the binomial expansion of (1+X) n. The Binomial coefficient also gives the value of the number of ways in which k items are chosen from among n objects i.e. \boxed, How to write table in Latex ? Where $\text{m}$ is the mean of the binomial distribution. Le coefficient binomial (n k) est le nombre de possibilités de choisir k élément dans un ensemble de n éléments. More generally, for a real or complex number $\alpha$ and an integer $k$ , the (generalized) binomial coefficient[note 1]is defined by the product representation 1. Section 4.1 Binomial Coeff Identities 3. Knowledge base dedicated to Linux and applied mathematics. ここでは (LaTeX) で「二項係数」を出力する方法を紹介します。数式 - 二項係数二項定理の一般項の係数である二項係数を $$binom{n}{k} = {}_n C_{k} = frac{n!}{(n-k)!k!} Visualisation of binomial expansion up to the 4th power. Using fractions and binomial coefficients in an expression is straightforward. k!(n−k)! Binomial coefficients have been known for centuries, but they're best known from Blaise Pascal's work circa 1640. Binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. 3. An example of a binomial coefficient is (5 2) = C(5, 2) = 10. The usage of fractions is quite flexible, they can be nested to obtain more complex expressions. matrix, pmatrix, bmatrix, vmatrix, Vmatrix, Latex horizontal space: qquad,hspace, thinspace,enspace, Horizontal and vertical curly Latex braces: \left\{,\right\},\underbrace{} and \overbrace{}. is called a binomial coefficient. Sign up to join this community. On the other side, \textstyle will change the style of the fraction as if it were part of the text. Number of prime divisors with multiplicity in a sum of Gaussian binomial coefficients. As you see, the command \binom{}{} will print the binomial coefficient using the parameters passed inside the braces. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. 1. Home > Latex > FAQ > Latex - FAQ > Latex binomial coefficient. The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. The Texworks shows … 2. binomial coefficient: A coefficient of any of the terms in the expansion of the binomial power $(x+y)^n$. 原文：Fractions and Binomials 译者：Xovee 翻译时间：2020年6月24日. Below is a construction of the first 11 rows of Pascal's triangle. Der Binomialkoeffizient ist eine mathematische Funktion, mit der sich eine der Grundaufgaben der Kombinatorik lösen lässt. An inequality with binomial coefficients . C — All combinations of v matrix. 4 Chapter 4 Binomial Coef Þcients Combinatorial vs. Alg ebraic Pr oofs Symmetr y. . Januar 2013 um 14:14 Uhr bearbeitet. 2. This could take hours! integers which are sums of binomial coefficients: \sum_i {n \choose k_i} 6. è il fattoriale di .Può essere calcolato anche facendo ricorso al triangolo di Tartaglia.Esso fornisce il numero delle combinazioni semplici di elementi di classe .. … Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. Binomial coefficient real life example. Einzelheiten sind in den Nutzungsbedingungen beschrieben. You can set this manually if you want. If we wanted to expand ${\left(x+y\right)}^{52}$, we might multiply $\left(x+y\right)$ by itself fifty-two times. As you may have guessed, the command \frac{1}{2} is the one that displays the fraction. = (n k) = nCk = Ck n n! The coin was tossed 12 times, so $\text{N}=12$. 2. Zusätzliche Bedingungen können gelten. When we expand ${\left(x+y\right)}^{n}$ by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. How to write number sets N Z D Q R C with Latex: \mathbb, amsfonts and \mathbf, How to write angle in latex langle, rangle, wedge, angle, measuredangle, sphericalangle, Latex numbering equations: leqno et fleqn, left,right, How to write a vector in Latex ? If we wanted to expand ${\left(x+y\right)}^{52}$, we might multiply $\left(x+y\right)$ by itself fifty-two times. Then it's a good reason to buy me a coffee. 4.1 Binomial Coef Þ cient Identities 4.2 Binomial In ver sion Operation 4.3 Applications to Statistics 4.4 The Catalan Recurrence 1. = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k n! Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. For example, $5! (n − k)! }}{{k!\left({n - k} \right)! \binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k(k-1)\cdots1}=\prod_{j=1}^k\frac{\alpha-j+1}{j}\quad\text{if }k\ge0\qquad(1b) … Thank you ! Vaccin Bcg Pus, Ou Acheter Dans Le Var, Légende Autochtone Ours, Affouiller 6 Lettres, Recette Escalope De Dinde, Accord Guitare Alunisson Nekfeu, " /> # coefficient binomial latex Inequality involving binomial coefficients. begin{tabular}...end{tabular}, How to write matrices in Latex ? A slightly different and more complex example of continued fractions, Showing first {{hits.length}} results of {{hits_total}} for {{searchQueryText}}, {{hits.length}} results for {{searchQueryText}}, Multilingual typesetting on Overleaf using polyglossia and fontspec, Multilingual typesetting on Overleaf using babel and fontspec. 1. 5. summation of binomial coefficients with squares. Recall that the binomial theorem is an algebraic method of expanding a binomial that is raised to a certain power, such as [latex](4x+y)^7$. Inequality with two binomial coefficients. ⋅ (−)!,, ∈, ≤ ≤,dove ! The order of selection of items not considered. C — All combinations of v matrix. Le coefficient binomial est très utilisé en probabilité, et permet notamment de résoudre des problèmes sans faire d’arbre pondéré (qui peuvent atteindre des tailles très grandes). Dieser Fall tritt auf beim -fachen Münzwurf mit einer fairen Münze (Wahrscheinlichkeit für Kopf gleich der für Zahl, also gleich 1/2).Die erste Abbildung zeigt die Binomialverteilung für =, und für verschiedene Werte von als Funktion von .Diese Binomialverteilungen sind spiegelsymmetrisch um den Wert = /: LaTeX Basics Creating your first LaTeX document Choosing a LaTeX Compiler Paragraphs and new lines Bold, italics and underlining Lists ... Fractions and binomial coefficients are common mathematical elements with similar characteristics, one number goes on top of other. Le coefficient binomial est défini comme le nombre de chemins conduisant à k succès. Specially useful for continued fractions. Diese Seite wurde zuletzt am 14. How to write it in Latex ? They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk. k!) \vec,\overrightarrow, Latex how to insert a blank or empty page with or without numbering \thispagestyle,\newpage,\usepackage{afterpage}, LateX Derivatives, Limits, Sums, Products and Integrals, How to get dots in Latex \ldots,\cdots,\vdots and \ddots, How to write algorithm and pseudocode in Latex ?\usepackage{algorithm},\usepackage{algorithmic}, How to display formulas inside a box or frame in Latex ? Cette définition donne une valeur infinie au coefficient binomial dans le cas où s est un entier négatif et t n'est pas un entier (ce qui n'est pas en contradiction avec la définition précédente puisqu'elle ne prenait pas en compte ce cas là). The binomial coefficient is defined by the next expression: \ [ \binom{n} {k} = \frac{n!} The binomial coefficient \binom{n}{k} can be interpreted as the number of ways to choose k elements from an n-element set. Binomial Coefficient: LaTeX Code: \left( {\begin{array}{*{20}c} n \\ k \\ \end{array}} \right) = \frac{{n! En Latex, on doit utiliser la fonction \binom comme suit : \frac{n!}{k! {k! (n-k)!} This website was useful to you? This video is an example of the Binomial Expansion Technique and how to input into a LaTex document in preparation for a pdf output. Binomial coefficient, returned as a nonnegative scalar value. 2. If we examine some simple binomial expansions, we can find patterns that will … k!) }}{{k!\left( {n - k} \right)!}} The command \displaystyle will format the fraction as if it were in mathematical display mode. = (n k) = n C k = C n k In latex mode we must use \binom fonction as follows: \frac{n!}{k! In matematica, il coefficiente binomiale (che si legge "su ") è un numero intero non negativo definito dalla seguente formula = (;) =!! If we examine some simple binomial expansions, we can find patterns that will … Any coefficient $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient. The text inside the first pair of braces is the numerator and the text inside the second pair is the denominator. Formules faisant intervenir les coefficients binomiaux. Binomial coefficient, returned as a nonnegative scalar value. Blog template built with Bootstrap and Spip by Nadir Soualem @mathlinux. It only takes a minute to sign up. Binomial Coefficients Inequality 3. This could take hours! Der Text ist unter der Lizenz Creative Commons Namensnennung – Weitergabe unter gleichen Bedingungen verfügbar. ; Datenschutz Do Order of the Scribes wizards have reduced spell learning GP costs? Binomial coefficient : According to Wikipedia - In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 = 120[/latex]. \endgroup – Giuseppe Negro Sep 30 '15 at 18:21 Latex Inequality with Sum of Binomial Coefficients. How to prove an elementary functional equation for polylogarithms? binomial All combinations of v, returned as a matrix of the same type as v. Matrix C has k columns and n!/((n–k)! (n - k)!} = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k$$, $$\binom{n}{k} = \binom{n-1}{k-1} +\binom{n-1}{k}$$. (n - k)!} }}. 0. k! The variance of the binomial distribution is: $\text{s}^2 = \text{Np}(1-\text{p})$, where $\text{s}^2$ is the variance of the binomial distribution. k-combinations of n-element set. rows, where n is length(v). Er gibt an, auf wie viele verschiedene Arten man {\displaystyle k} bestimmte Objekte aus einer Menge von {\displaystyle n} verschiedenen Objekten auswählen kann (ohne Zurücklegen, ohne Beachtung der Reihenfolge). 2 Chapter 4 Binomial Coef Þcients 4.1 BINOMIAL COEFF IDENTITIES T a b le 4.1.1. On suppose que k, n sont des entiers ; x, y, z, z′ des complexes. (n - k)!} Intuitive explanation of binomial coefficient formula. This article explains how to typeset them in LaTeX. This coefficient, G, of a data set or income distribution curve has a range between 0 and 1, 0 being where wealth Any given (x, y) point on thisWhile the inferred coefficients may differ between the tasks, they are constrained to agree on the features that are selected (non-zero coefficients). Fractions and Binomials. The second fraction displayed in the previous example uses the command \cfrac{}{} provided by the package amsmath (see the introduction), this command displays nested fractions without changing the size of the font. By Stirling's theorem your approximation is off by a factor of $\sqrt{n}$, (which later cancels in the fraction expressing the binomial coefficients). Hot Network Questions Was there a US state where video games were banned by accident? Also, the text size of the fraction changes according to the text around it. Why is it not possible to calculate the amount of combinations for a byte using the Binomial coefficient. coefficient Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Jobs; Binomial coefficient parentheses are … $$(LaTeX) で出力する方法をみていき … The binomial coefficient (n k) (n k) can be interpreted as the number of ways to choose k elements from an n-element set. In latex mode we must use \binom fonction as follows:$$\frac{n!}{k! For these commands to work you must import the package amsmath by adding the next line to the preamble of your file, The appearance of the fraction may change depending on the context. = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. All combinations of v, returned as a matrix of the same type as v. Matrix C has k columns and n!/((n–k)! Probability of a contiguous sub-sequence with different elements. Fractions and binomial coefficients are common mathematical elements with similar characteristics - one number goes on top of another. rows, where n is length(v). It will give me the energy and motivation to continue this development. 14. When we expand ${\left(x+y\right)}^{n}$ by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. (adsbygoogle = window.adsbygoogle \|\| []).push({}); All the versions of this article: b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. 分式和二项式系数是非常常见的数学元素，它们有着一些共同的特点：一个数字位于另外一个数字的上方。本篇文章解释如何在 LaTeX 中输入它们。 Expanding a combinatorial argument involving permutation coefficients. \] And of course this command can be included in the normal text flow \ (\binom{n} {k}\). b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. (adsbygoogle = window.adsbygoogle \|\| []).push({}); Binomial coefficient denoted as c(n,k) or n c r is defined as coefficient of x k in the binomial expansion of (1+X) n. The Binomial coefficient also gives the value of the number of ways in which k items are chosen from among n objects i.e. \boxed, How to write table in Latex ? Where $\text{m}$ is the mean of the binomial distribution. Le coefficient binomial (n k) est le nombre de possibilités de choisir k élément dans un ensemble de n éléments. More generally, for a real or complex number $\alpha$ and an integer $k$ , the (generalized) binomial coefficient[note 1]is defined by the product representation 1. Section 4.1 Binomial Coeff Identities 3. Knowledge base dedicated to Linux and applied mathematics. ここでは (LaTeX) で「二項係数」を出力する方法を紹介します。数式 - 二項係数二項定理の一般項の係数である二項係数を binom{n}{k} = {}_n C_{k} = frac{n!}{(n-k)!k!} Visualisation of binomial expansion up to the 4th power. Using fractions and binomial coefficients in an expression is straightforward. k!(n−k)! Binomial coefficients have been known for centuries, but they're best known from Blaise Pascal's work circa 1640. Binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. 3. An example of a binomial coefficient is (5 2) = C(5, 2) = 10. The usage of fractions is quite flexible, they can be nested to obtain more complex expressions. matrix, pmatrix, bmatrix, vmatrix, Vmatrix, Latex horizontal space: qquad,hspace, thinspace,enspace, Horizontal and vertical curly Latex braces: \left\{,\right\},\underbrace{} and \overbrace{}. is called a binomial coefficient. Sign up to join this community. On the other side, \textstyle will change the style of the fraction as if it were part of the text. Number of prime divisors with multiplicity in a sum of Gaussian binomial coefficients. As you see, the command \binom{}{} will print the binomial coefficient using the parameters passed inside the braces. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. 1. Home > Latex > FAQ > Latex - FAQ > Latex binomial coefficient. The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. The Texworks shows … 2. binomial coefficient: A coefficient of any of the terms in the expansion of the binomial power $(x+y)^n$. 原文：Fractions and Binomials 译者：Xovee 翻译时间：2020年6月24日. Below is a construction of the first 11 rows of Pascal's triangle. Der Binomialkoeffizient ist eine mathematische Funktion, mit der sich eine der Grundaufgaben der Kombinatorik lösen lässt. An inequality with binomial coefficients . C — All combinations of v matrix. 4 Chapter 4 Binomial Coef Þcients Combinatorial vs. Alg ebraic Pr oofs Symmetr y. . Januar 2013 um 14:14 Uhr bearbeitet. 2. This could take hours! integers which are sums of binomial coefficients: $\sum_i {n \choose k_i}$ 6. è il fattoriale di .Può essere calcolato anche facendo ricorso al triangolo di Tartaglia.Esso fornisce il numero delle combinazioni semplici di elementi di classe .. … Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. Binomial coefficient real life example. Einzelheiten sind in den Nutzungsbedingungen beschrieben. You can set this manually if you want. If we wanted to expand ${\left(x+y\right)}^{52}$, we might multiply $\left(x+y\right)$ by itself fifty-two times. As you may have guessed, the command \frac{1}{2} is the one that displays the fraction. = (n k) = nCk = Ck n n! The coin was tossed 12 times, so $\text{N}=12$. 2. Zusätzliche Bedingungen können gelten. When we expand ${\left(x+y\right)}^{n}$ by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. How to write number sets N Z D Q R C with Latex: \mathbb, amsfonts and \mathbf, How to write angle in latex langle, rangle, wedge, angle, measuredangle, sphericalangle, Latex numbering equations: leqno et fleqn, left,right, How to write a vector in Latex ? If we wanted to expand ${\left(x+y\right)}^{52}$, we might multiply $\left(x+y\right)$ by itself fifty-two times. Then it's a good reason to buy me a coffee. 4.1 Binomial Coef Þ cient Identities 4.2 Binomial In ver sion Operation 4.3 Applications to Statistics 4.4 The Catalan Recurrence 1. = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k n! Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. For example, [latex]5! (n − k)! }}{{k!\left({n - k} \right)! $\binom{\alpha}{k}=\frac{\alpha(\alpha-1)\cdots(\alpha-k+1)}{k(k-1)\cdots1}=\prod_{j=1}^k\frac{\alpha-j+1}{j}\quad\text{if }k\ge0\qquad(1b)$ … Thank you ! Ce site utilise Akismet pour réduire les indésirables. 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https://ada.silverchair.com/view-large/3837221	Skip to Main Content Table 1 Rating scale for the quality of evidence High: Further research is very unlikely to change our confidence in the estimate of effect. The body of evidence comes from high-level individual studies that are sufficiently powered and provide precise, consistent, and directly applicable results in a relevant population. Moderate: Further research is likely to have an important impact on our confidence in the estimate of effect and may change the estimate and the recommendation. The body of evidence comes from high-/moderate-level individual studies that are sufficient to determine effects, but the strength of the evidence is limited by the number, quality, or consistency of the included studies; generalizability of results to routine practice; or indirect nature of the evidence. Low: Further research is very likely to have an important impact on our confidence in the estimate of effect and is likely to change the estimate and the recommendation. The body of evidence is of low level and comes from studies with serious design flaws, or evidence is indirect. Very low: Any estimate of effect is very uncertain. Recommendation may change when higher-quality evidence becomes available. Evidence is insufficient to assess the effects on health outcomes because of limited number or power of studies, important flaws in their design or conduct, gaps in the chain of evidence, or lack of information. High: Further research is very unlikely to change our confidence in the estimate of effect. The body of evidence comes from high-level individual studies that are sufficiently powered and provide precise, consistent, and directly applicable results in a relevant population. Moderate: Further research is likely to have an important impact on our confidence in the estimate of effect and may change the estimate and the recommendation. The body of evidence comes from high-/moderate-level individual studies that are sufficient to determine effects, but the strength of the evidence is limited by the number, quality, or consistency of the included studies; generalizability of results to routine practice; or indirect nature of the evidence. Low: Further research is very likely to have an important impact on our confidence in the estimate of effect and is likely to change the estimate and the recommendation. The body of evidence is of low level and comes from studies with serious design flaws, or evidence is indirect. Very low: Any estimate of effect is very uncertain. Recommendation may change when higher-quality evidence becomes available. Evidence is insufficient to assess the effects on health outcomes because of limited number or power of studies, important flaws in their design or conduct, gaps in the chain of evidence, or lack of information. Close Modal	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.850823700428009, "perplexity": 905.5011814341066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337421.33/warc/CC-MAIN-20221003133425-20221003163425-00784.warc.gz"}
https://stats.stackexchange.com/questions/280297/dropout-effectiveness-on-small-neural-networks	# Dropout effectiveness on small neural networks I implemented dropout on my neural networks. I tried to train the neural network to act as the f(x) = sin(x) function. During normal backpropagation without dropout regularization, it literally needed less than 10 iterations to reach an extremely small error. However, when I activated dropout, it got stuck at a reasonably high MSE error (0.05) and there was no improvement over time any more. I used two hidden units. But asides from that, are there any examples of dropout regularization on small neural networks. Or any papers describing the effectiveness on small neural networks? That way I can test if my implementation was done correctly. ## 2 Answers Have you tried early stopping? Usually early stopping is enough for small neural network architecture over dropout. Dropout is most widely used for hundreds and thousands of parameters, usually it is the employ to prevent over-fitting in high dimensional models. In other words, you can use early stopping first, then you can tweak your architecture for more improvements like weight decay and other things. Dropout is when, with a certain probability, we remove units from the network entirely. If you're approximating a sine, then your $x$ is one-dimensional and your $f(x)$ is also one dimensional, correct? So if you remove one of your two hidden units, then you're trying to approximate a sine using the nonlinearity you've chosen, which can only ever be so good. Generally speaking, dropout is better in situations when you can overfit more. The larger the network, the more expressive it is, and so one would expect it to be less useful in the case of a very small network where it is already hard to overfit.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8266533613204956, "perplexity": 647.1709663299062}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232262029.97/warc/CC-MAIN-20190527065651-20190527091651-00195.warc.gz"}
https://www.physicsforums.com/threads/velocity-and-acceleration-question.34539/	# Velocity and Acceleration Question 1. Jul 10, 2004 ### faust9 OK, got a test Monday, and my professor e-mailed us our review for siad test this past Friday. His tests are very similar to his reviews so did I do this right? Question: The motion of a particle is described as $v=2x-\frac{9}{x} \ {\rm for}\ (0 \leq t \leq 10)s\ {\rm where}\ t_0=0,x_0=0$ Ok, this is what I did: a) Determine the distance traveled in t=4 seconds $$v=2x-\frac{9}{x}=\frac{dx}{dt}$$ $$\frac{2x^2-9}{x}dx=dt$$ $$\int \frac{2x^2-9}{x}dx=\int dt$$ $$\int \frac{1}{4U}dU=t+c$$ $$\frac{1}{4}\ln (2x^2-9)=t+c$$ $${\rm let }\ t=0,\ {\rm and }\ x=3$$ $$c=\frac{ \ln (9)}{16}$$ thus: $$x^2=\frac{9}{2}e^{4t}+\frac{9}{2}$$ $$x(4)\approx 6323.567 m$$ b)How long will it take the particle to reach 0.02c? $$0.02c \approx 1.8 \times 10^8$$ so I plugged that into the original equation $$1.8 \times 10^8 =\frac{2x^2-9}{x}$$ $$2x^2-1.8 \times 10^8x-9=0$$ $$x=\{ -5 \times 10^{-8},9.0 \times 10^7\}$$ the first was a trivial solution so: $$t=\frac{1}{4}\ln (2(9.0\times 10^7)^2-9)-\frac{\ln 9}{16}$$ $$t=8.7816s$$ c) Express acceleration as a function of x $$a=v\frac{dv}{dx}$$ $$a=(2x-\frac{9}{x})(2x-\frac{9}{x})\frac{d}{dx}$$ $$a=(2x-\frac{9}{x})(2+\frac{9}{x^2})=4x-\frac{81}{x^3}$$ d) Determine the acceleration at 0.02c $$a=4(9.0\times 10^7)-\frac{81}{(9.0\times 10^7)^3}=3.6\times 10^8$$ Is the above correct? I was talking to another student who did it differently and got a different answer. I want to make sure the above is correct so I don't miss this question on the test. Any insight would be greatly appreciated. Thanks. Last edited: Jul 10, 2004 2. Jul 10, 2004 ### AKG You did it wrong. It seems like a weird question though. Notice that at t=0, x=0, and velocity is undefined (it approaches $-\infty$). $$v = 2x - \frac{9}{x}$$ $$\frac{dx}{dt} = 2x - \frac{9}{x}$$ $$\frac{dx}{2x - \frac{9}{x}} = dt$$ $$\frac{x}{2x^2 - 9}dx = dt$$ $$\int _{x(0)} ^{x(4)} \frac{x}{2x^2 - 9}dx = \int _0 ^4 dt$$ Now look at the function on the left that we'll be integrating. It is not defined at $x = \sqrt{4.5}$, however, we don't even know if this thing shows up. Assuming it doesn't, we can proceed as follows: $$u = 2x^2 - 9$$ $$du = 4xdx$$ $$dx = \frac{du}{4x}$$ $$\frac{1}{4} \int _{-9} ^{2[x(4)]^2 - 9} \frac{du}{u} = 4$$ $$\ln \frac{2[x(4)]^2 - 9}{-9} = 16$$ $$x(4) = \sqrt{\frac{-9e^{16} + 9}{2}}$$ The above is undefined. This question is very strange. Normally, you have velocity given as a function of time, and you can integrate that to find x(t), but given v(x), it seems you have to solve some sort of differential equation. I tried this integration, doubtful it would work, just to see what it would give. I might have done something wrong, but in general this question seem weird. Are you sure the question you've given us is the actual question? 3. Jul 10, 2004 ### faust9 Oh I'm sorry, $x_0=3$ not $x_0=0$. Other than that, the question was posed as I described. Velocity was initially given as a function of x. 4. Jul 10, 2004 ### AKG I think I know what's going on here I think you meant $x_0 = 3$, not $x_0 = 0$. That would explain one of your lines of work, and the problem seems to make a little more sense. Also, you mean $c = \frac{\ln (9)}{4}$. Try to take a little more care, no need to rush, because it's too confusing otherwise, and makes it difficult to help you. Now, although you got the answer that you did for (a), and it seems right, you've used indefinite integration, and I don't know if that is "allowed." I suppose you have to assume that it is, however, if $x = \sqrt{4.5}$ on the interval $0 \leq t \leq 4$, you wouldn't be allowed to integrate like that, I believe. (c) seems okay, as long as you did the final simplification right (I didn't check). As long as you plugged stuff in right (for b and d), and are allowed to do that integration for (a), everything seems fine. Well, your answers seem fine, you made some strange mistakes along the way though: * I think you meant that x(0) = 3 * The integration you started to do was upside down, however, midway through your work you flipped it around again. In this case, two wrongs made a right, but I don't know if your markers will like that * c = ln(9)/4 5. Jul 10, 2004 ### AKG Looking back, with x(0) = 3, my approach gives the same answer as yours. I would suggest that in situations like these, it is better to use definite integration, not indefinite. For one, it eliminates the need to solve explicitly for "c." Even if you just want a generally formula for x(t), and not specifically x(4), for example, all you need to do is replace the limits of integration x(4) with x(t) and 4 with t (on the right side, where you're integrating 1dt. Also, I don't know if it is necessary, but the equation you arrived at for x never gets $\sqrt{4.5}$ as it's value, and given that x, dx/dt is indeed the given v, so there is no inconsistency in using that x, so it's safe to use it. 6. Jul 10, 2004 ### faust9 Thank you. I'll go over it once again and see if there's another way. The flipped integral--I did it correclty on paper; however, I inadvertently flipped it when I was typing it in. I'm not a profecient keyboard artist. Cool beans (in regards to your post immediatly above). Again thanks. Last edited: Jul 10, 2004 7. Jul 10, 2004 ### AKG You've confused me again. You know that $$c = \frac{\ln 3}{2}$$ ... is wrong also, right? And beans??? You're welcome. 8. Jul 10, 2004 ### faust9 I know I fixed it. I redid it using definate integration and saw my faux pas. 9. Jul 10, 2004 ### AKG Beans? I don't get it. 10. Jul 10, 2004 ### faust9 Its a colloquialism. It's kind of like aloha in that it has many meanings depending on the context used. In this case it ment I got it and thanks. Similar Discussions: Velocity and Acceleration Question	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9622074961662292, "perplexity": 659.5634897174989}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103891.56/warc/CC-MAIN-20170817170613-20170817190613-00710.warc.gz"}
https://www.physicsforums.com/threads/luminosity-flux-equations-confusion.691498/	# Luminosity & flux equations confusion 1. May 13, 2013 ### Flucky Hi folks First off I have only just figured out that v is used as bloody frequency in loads of astrophysics equations. Not fun. I've got monochromatic flux as: Fv = dE / dt·dA·dv now I'm happy with this. But now the lecture notes I'm looking at describes monochromatic luminosity as "the energy emitted by the source in unit time, per unit wavelength" but then gives the equation as: Lv = dE / dt·dv so surely it should be ""the energy emitted by the source in unit time, per unit frequency" BUT monochromatic means the energy at a given wavelength doesn't it, however there is no wavelength in the equation. Could somebody please clearly state the definition and equation, saying what each letter in the equation represents, of monochromatic/bolometric flux/luminosity as I'm really getting stressed over it and can't revise anything else until I've got my head wrapped round it. 2. May 13, 2013 ### phyzguy I think you're right, it should be per unit frequency - the definition you have given is inconsistent between the word and the equation. And, since the relation between wavelength and frequency is inverse, they will not be the same. Different sources use different definitions, so you need to make sure the definitions you are using are consistent (as you are doing). I strongly suggest Rybicki and Lightman, "Radiative Processes in Astrophysics" - the first few chapters spell these things out in careful detail. 3. May 13, 2013 ### Flucky Seems to be going for £88 on Amazon so I might be giving that a miss. Am I right in saying the following: --------------------------------------- Monochromatic Flux is the energy falling on a unit area, per unit time, at a given frequency. As no light is emitted at a single frequency instead we use how much light is emitted in an infinitesimally small range of frequency, dv. Fv = ΔE / Δt·ΔA·Δv Bolometric Flux is the amount of energy across all frequencies. Fbol = ∫ Fv dv --------------------------------------- Monochromatic Luminosity is the energy emitted by the source in unit time, per unit frequency. Lv = ΔE / Δt·Δv Bolometric Luminosity is the amount of energy across all frequencies. Lbol = ∫ Lv dv 4. May 13, 2013 ### SteamKing Staff Emeritus The 'v' should be the lower case Greek letter 'nu'. What did you think it meant before you found out it was frequency? 5. May 13, 2013 ### Flucky I didn't really think about it so automatically assumed "ah it must be velocity" 6. May 13, 2013 ### phyzguy I think what you have is correct. Draft saved Draft deleted Similar Discussions: Luminosity & flux equations confusion	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9356928467750549, "perplexity": 1529.1797464347676}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886112533.84/warc/CC-MAIN-20170822162608-20170822182608-00004.warc.gz"}
http://www.ck12.org/book/Basic-Probability-and-Statistics-A-Full-Course/r1/section/6.4/	<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" /> 6.4: Review Questions Difficulty Level: At Grade Created by: CK-12 Part A. For each question, circle the most appropriate answer. 1. If the standard deviation of a population is 6, the population variance is: 1. 2.44 2. 3 3. 6 4. 36 2. What is the sample standard deviation of the following data values? \begin{align}3.3 \qquad 2.9 \qquad 8.5 \qquad 11.5\end{align} 1. 3.61 2. 4.17 3. 6.55 4. 13.52 3. Suppose data are normally distributed, with a mean of 100 and a standard deviation of 20. Between what 2 values will approximately 68% of the data fall? 1. 60 and 140 2. 80 and 120 3. 20 and 100 4. 100 and 125 4. The sum of all of the deviations about the mean of a set of data is always going to be equal to: 1. positive 2. the mode 3. the standard deviation total 4. 0 5. What is the population variance of the following data values? \begin{align}40 \qquad 38 \qquad 42 \qquad 47 \qquad 35\end{align} 1. 4.03 2. 4.51 3. 15.24 4. 20.34 6. Suppose data are normally distributed, with a mean of 50 and a standard deviation of 10. Between what 2 values will approximately 95% of the data fall? 1. 40 and 60 2. 30 and 70 3. 20 and 80 4. 10 and 95 7. Suppose data are normally distributed, with a mean of 50 and a standard deviation of 10. What would be the variance? 1. 10 2. 40 3. 50 4. 100 8. If data are normally distributed, what percentage of the data should lie within the range of \begin{align}\mu \pm 3\sigma\end{align}? 1. 34% 2. 68% 3. 95% 4. 99.7% 9. If a normally distributed population has a mean of 75 and a standard deviation of 15, what proportion of the values would be expected to lie between 45 and 105? 1. 34% 2. 68% 3. 95% 4. 99.7% 10. If a normally distributed population has a mean of 25 and a standard deviation of 5.5, what proportion of the values would be expected to lie between 19.5 and 30.5? 1. 34% 2. 68% 3. 95% 4. 99.7% Part B. Answer the following questions and show all work (including diagrams) to create a complete answer. 1. In the United States, cola can normally be bought in 8 oz cans. A survey was conducted where 250 cans of cola were taken from a manufacturing warehouse and the volumes were measured. It was found that the mean volume was 7.5 oz, and the standard deviation was 0.1 oz. Draw a normal distribution curve to represent this data and then answer the following questions. (b) 68% of the volumes can be found between __________ and _________. (c) 95% of the volumes can be found between __________ and _________. (d) 99.7% of the volumes can be found between __________ and _________. 2. The mean height of the fourth graders in a local elementary school was found to be 4’8”, or 56”. The standard deviation was found to be 5”. Draw a normal distribution curve to represent this data and then answer the following questions. (b) 68% of the heights can be found between __________ and _________. (c) 95% of the heights can be found between __________ and _________. (d) 99.7% of the heights can be found between __________ and _________. 3. The following data was collected: \begin{align}5 \qquad 8 \qquad 9 \qquad 10 \qquad 4 \qquad 3 \qquad 7 \qquad 5\end{align} Fill in the chart below and calculate the standard deviation and the variance. Data \begin{align}(x)\end{align} Mean \begin{align}(\mu)\end{align} Mean \begin{align}-\end{align} Data \begin{align}(\mu - x)\end{align} Square of Mean \begin{align}-\end{align} Data \begin{align}(\mu-x)^2\end{align} \begin{align}\sum\end{align} 1. The following data was collected. \begin{align}11 \qquad 15 \qquad 16 \qquad 12 \qquad 19 \qquad 17 \qquad 14 \qquad 18 \qquad 15 \qquad 10\end{align} Fill in the chart below and calculate the standard deviation and the variance. Data \begin{align}(x)\end{align} Mean \begin{align}(\mu)\end{align} Mean \begin{align}-\end{align} Data \begin{align}(\mu - x)\end{align} Square of Mean \begin{align}-\end{align} Data \begin{align}(\mu-x)^2\end{align} \begin{align}\sum\end{align} 1. Mrs. Meery has recorded her exam results for the current mathematics exam. The results are shown below: \begin{align}& 64 \quad 98 \quad 78 \quad 76 \quad 56 \quad 48 \quad 89 \quad 78 \quad 69 \quad 90 \quad 89\\ & 97 \quad 67 \quad 58 \quad 59 \quad 50 \quad 78 \quad 89 \quad 68 \quad 83 \quad 72 \quad 91\end{align} (b) Determine the mean for this data. (c) Determine the standard deviation for this data. (d) Determine the variance for this data. (e) Draw a normal distribution curve to represent the data Mrs. Meery found in her class. 2. Mrs. Landry decided to do the same analysis as Mrs. Meery for her math class. She has recorded her exam results for the current mathematics exam. The results are shown below: \begin{align}& 89 \quad 87 \quad 81 \quad 84 \quad 76 \quad 72 \quad 67 \quad 49 \quad 55 \quad 38 \quad 67 \quad 90 \quad 59\\ & 87 \quad 89 \quad 69 \quad 92 \quad 90 \quad 79 \quad 84 \quad 69 \quad 93 \quad 85 \quad 70 \quad 87 \quad 80\end{align} (b) Determine the mean for this data. (c) Determine the standard deviation for this data. (d) Determine the variance for this data. (e) Draw a normal distribution curve to represent the data Mrs. Landry found in her class. 3. 200 senior high students were asked how long they had to wait in the cafeteria line for lunch. Their responses were found to be normally distributed, with a mean of 15 minutes and a standard deviation of 3.5 minutes. Copy the following bell curve onto your paper and answer the questions below. (b) How many students would you expect to wait more than 11.5 minutes? (c) How many students would you expect to wait more than 18.5 minutes? (d) How many students would you expect to wait between 11.5 and 18.5 minutes? 4. 350 babies were born at Neo Hospital in the past 6 months. The average weight for the babies was found to be 6.8 lbs, with a standard deviation of 0.5 lbs. Copy the following bell curve onto your paper and answer the questions below. (b) How many babies would you expect to weigh more than 7.3 lbs? (c) How many babies would you expect to weigh more than 7.8 lbs? (d) How many babies would you expect to weigh between 6.3 and 7.8 lbs? 5. Sudoku is a very popular logic game of number combinations. It originated in the late 1800's by the French press, Le Siècle. The average times (in minutes) it takes those in a senior math class to complete a Sudoku puzzle are found below. Draw a normal distribution curve to represent this data. Determine what time a student must complete a Sudoku puzzle in to be in the top 0.13%. \begin{align}& 20 \quad 15 \quad 21 \quad 24 \quad 7 \quad \ 19 \quad 10 \quad 17 \quad 15 \quad 22 \quad 31 \quad 19 \quad 20 \quad 21\\ & 21 \quad \ 9 \quad 12 \quad 26 \quad 24 \quad 28 \quad 19 \quad 16 \quad 24 \quad 11 \quad 17 \quad 31 \quad 25 \quad 13\\ & 16 \quad 18 \quad 22 \quad 32 \quad 9 \quad \ 15 \quad 19 \quad 27 \quad 14 \quad 25 \quad 32 \quad 29 \quad \quad \end{align} 6. Sheldon has planted seedlings in his garden in the back yard. After 30 days, he measures the heights of the seedlings to determine how much they have grown. The differences in heights can be seen in the table below. The heights are measured in inches. Draw a normal distribution curve to represent the data. Determine what the differences in heights of the seedlings are for 68% of the data. \begin{align}& 10 \quad 3 \quad 8 \quad 4 \quad \ \ 7 \quad 12 \quad 8 \quad \ 5 \quad 4 \quad \ 9 \quad \ 3 \quad 8\\ & 6 \quad 10 \quad 7 \quad 10 \quad 11 \quad 8 \quad 12 \quad 9 \quad 10 \quad 7 \quad 8 \quad 11\end{align} Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes Show Hide Details Description Tags: Subjects:	{"extraction_info": {"found_math": true, "script_math_tex": 23, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991357922554016, "perplexity": 446.3251702513064}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982292659.26/warc/CC-MAIN-20160823195812-00108-ip-10-153-172-175.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/appendix-a-numbers-inequalities-and-absolute-values-a-exercises-page-a9/6	Calculus 8th Edition $1$ Start with absolute value expression: $\|\|-2\|-\|-3\|\|$ Find inner absolute values: $\|2-3\|$ Since $2-3=-1$ is negative, we must find the negative of it to find the absolute value. $\|-1\|=-(-1)=1$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9224460124969482, "perplexity": 423.90270168141063}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540533401.22/warc/CC-MAIN-20191211212657-20191212000657-00425.warc.gz"}
https://chemistry.stackexchange.com/questions/37833/which-metal-hydroxides-are-soluble	# Which metal hydroxides are soluble? Alkali metals (Li, Na, K, Rb, Cs) and the alkaline earth metals (Ca, Sr, Ba - sometimes Mg is included also) form hydroxides that are well-known to be soluble or at least slightly soluble and create strongly basic solutions. Apparently these solutions of hydroxides are also known as alkali. Since the more well-known d metals (Cr, Fe, Mn, Cu, Zn, Ni, Co, Ti etc.) forms no soluble hydroxide, I used to assume that apart from the 9 metals stated at the top, there is no more metals that corresponds to soluble hydroxides. However, while searching my (non-English) version of Constants of Inorganic Substances: A Handbook by Lidin, Andreyeva and Molochko, I discovered that Europium (II) hydroxide was written as if it is soluble in a chemical equation in the Europium section. Later, I did search for information on the compound with the keywords ""Europium II hydroxide" soluble" , and I found some results: europium (II) sulfate is scarcely soluble in water, while europium (II) hydroxide dissolves readily and gives an alkaline reaction Is there any other research on the compound Europium (II) hydroxide? Is it really soluble? Which metal hydroxides, apart from those of those group 1 and 2 metals, are soluble? Is there any specific reason why europium (II) is such an anomaly? If there are other such cations, is there a specific reason for that particular cation to form such a soluble hydroxide? • Thallium hydroxide is one example because the size of $\ce{Tl+}$ is appox. same to size of $\ce{K+}$ and thus thallium hydroxide dissolve in water easily and form a strong basic solution. – Nilay Ghosh Sep 30 '16 at 12:16 • Thallium (I) hydroxide, of course. – Oscar Lanzi Mar 11 '17 at 13:02 Europium (II) is an anomaly because it is, well, "II". Most metals in period 4 or higher, apart from the alkali and alkaline earth metals, favor higher oxidation states which lead to highly insoluble hydroxides or oxides. Europium is different because the $+2$ ion is stabilized by having a spherically symmetric electron configuration $[\ce{Xe}]\mathrm{4f^7}$. So we have a $+2$ ion similar in size and spherically symmetric electron configuration to the heavier alkaline earth metals, and $\ce{Eu(II)}$ compound tend to resemble their alkaline earth counterparts -- including the solubility of the hydroxide.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8239926695823669, "perplexity": 3599.791407232684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250608062.57/warc/CC-MAIN-20200123011418-20200123040418-00319.warc.gz"}
http://greenprecalc.com/precalc1-MHCC/activity-logarithmic-functions.html	$\newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$ ## Section4.1Logarithmic Functions Logarithms can sometimes be the source of much turmoil and angst to students. To some members of the English department, the word “logarithms” is used as a joke to respond to something that is confusing, “Is that like logarithms?” (laughter ensues). But you will not have these issues, because logarithms are not bad, they're just spelled that way. Remember this: A logarithm is an exponent. This activity is an introduction to logarithms and the logarithmic function. Since logarithms are exponents, it is a good idea to review some key properties of exponents, particularly negative and fractional exponents. ### Subsection4.1.1Review of Exponents Remember that negative exponents mean “reciprocal” and fractional exponents mean “roots”. ###### Example4.1.1 \begin{align} 9^{-1}\amp= \frac{1}{9}\amp 9^{1/2}= \sqrt{9} = 3 \end{align} In the next set of exercises, practice the idea of using the power (exponent) of a number (the base) to get a particular result. WeBWorK Exercise ### Subsection4.1.2The Meaning of a Logarithm Earlier in the study of exponential functions, we solved exponential equations graphically. Another way to solve an exponential equation is to “guess-and-check”. It is not a very efficient method, but in this case it may help clarify what a logarithm does. ###### Exercise4.1.3Powers of $10$ WeBWorK Exercise Notice that for each ordered pair, we could say “$10$ raised to the (input) power is (output)”. For instance, $10$ raised to the $-1$ power is $\frac{1}{10}=0.1$ In the following exercise, our goal is to find the power of $10$ that equals $36\text{.}$ In other words, to solve the equation $10^x=36$ in which we want to know “What power of $10$ gives us $36\text{.}$” From the results in the previous table we realize the exponent is not going to be an integer like $0$ or $1$ or $2\text{.}$ It will have to be some decimal value. ###### Exercise4.1.4Guess-and-Check WeBWorK Exercise Next, we can try a graphical approach. ###### Exercise4.1.5Solving Graphically So far the Geogebra graphing tool has been sufficient in helping us create and manipulate graphs. However, to get more accuracy in our solution we can use another free, online graphing tool, Desmos. Although you may also use your graphing calculator. WeBWorK Exercise As you experienced in the last two exercises, different powers of $10$ give us different results. In fact, we can get any positive result we want using only powers of $10$ (or any positive base for that matter). Since logarithms are exponents, then finding a logarithm is like asking: “What power of (choose a base) do I need to get (whatever result you want)?” ###### Example4.1.6 In English we can ask an exponential question and get a logarithm answer: Exponential question, What power of $3$ gives you $81\text{?}$ Logarithm answer, The power of $3$ that gives you $81$ is the $4$th power. (Because $3^{4}=81$) In Math that same question and answer look like this: Exponential question, $3^{x}=81$ Logarithm answer, $x= \log_{3}(81)=4$ $log_a(b)$ represents the power we raise the base $a$ to, in order to get $b\text{.}$ The structure of a logarithm is that it must tell the reader the base number and it has to state what value you are trying to get. Base $10$ means you are using powers of $10$ as in $10^{x}\text{.}$ Base $3$ means you are using powers of $3$ as in $3^{x}$ etc. Below is a table that shows how Exercise 4.1.5, which used base $10\text{,}$ looks when written in both English and Math. English Math How you “say the math” The power of $10$ that gives you $36$ $\log_{10}(36)$ log base $10$ of $36$ The power of $10$ that gives you $0.6$ $\log_{10}(0.6)$ log base $10$ of $0.6$ For example, $\log_{10}(100)$ means “The power of $10$ that gives you $100\text{.}$” ###### Exercise4.1.7 WeBWorK Exercise The math word log is short for logarithm. It means “log base $10$”. But we can use any positive number as a base, so $\log_b$ means “log base $b$”. Since $\log$ is base $10$ (powers of $10$) we technically should write $\log_{10}\text{,}$ but base $10$ is so often used, we leave out the $10$ for convenience. Another commonly used logarithm is base $e\text{.}$ It's called the natural logarithm and is written $\ln\text{.}$ Again, technically we should write $\log_e\text{,}$ but we write $\ln$ for convenience. All other logarithms are written in standard format. The base, $b\text{,}$ indicates what number is being multiplied over and over. So if $2$ is the base, we write $\log_2$ which indicates powers of $2\text{.}$ WeBWorK Exercise WeBWorK Exercise ###### Exercise4.1.10 WeBWorK Exercise At this point of the activity we should be able to estimate the value of a given logarithmic expression with the understanding that a logarithm is just an exponent. WeBWorK Exercise ### Subsection4.1.3Domain of a Logarithmic Function Try to evaluate $\log(-100)$ on your calculator. Your calculator should read error or quit or something indicating it cannot complete the calculation. To understand why this is happening, we could try writing out the meaning of $\log(-100)$ in English. ###### Exercise4.1.12 WeBWorK Exercise If you have a bunch of positive $10$s then how many do you have to multiply to get a negative $100\text{?}$ Of course, you can't get a negative result if you only have positive $10$s! You would need at least one negative $10$ and you don't have any. Try to evaluate $\log(0)$ on your calculator. Your calculator should read error or quit or something indicating it cannot complete the calculation. To understand why this is happening, we could again write out the meaning of $\log(0)$ in English. ###### Exercise4.1.13 WeBWorK Exercise If you have a bunch of positive $10$s then how many do you have to multiply to get $0$ as a result? Of course, you can't get $0$ as the result if you only have positive $10$s! At least one factor would have to be $0\text{,}$ but all you have is $10$'s. It turns out that the $\log$ function (with any base $b \gt 0$) is not defined for inputs that are less than or equal to $0$. Therefore the domain of the $\log$ function (with any base $b \gt 0$) is only positive numbers. WeBWorK Exercise ### Subsection4.1.4The Range of a Logarithmic Function Earlier we used graphs to answer the questions: 1. What power of $10$ gives you $36\text{?}$ 2. What power of $10$ gives you $0.6\text{?}$ We answered the questions using logarithm statements: “$\log(36) \approx 1.556$” and “$\log(0.6) \approx -0.222$” Notice with base $10\text{,}$ we can use positive or negative exponents to get any positive result we want. Do you want to get a result of $36\text{?}$ Use a power of about $1.556$ as in $10^{1.556} \approx 36$ Do you want to get a result of $0.6\text{?}$ Use a power of about $-0.222$ as in $10^{-0.222} \approx 0.6\text{.}$ Remember that logarithms are exponents, and clearly an exponent can be positive, negative or zero. Logarithms tell you the power you need to get a particular result. You input the result you want, the function tells you the exponent you need. The exponent is the output. ###### Exercise4.1.15 WeBWorK Exercise Given an input from the domain of a logarithmic function, the range can be positive, negative or zero. In fact, the range is all real numbers. If $\log_{b}(x)=y$ then $y$ can be any real number. ### Subsection4.1.5The Graph of a Logarithmic Function Our discoveries about the domain and range of a logarithmic function lead us to wonder what the graph might look like. So far we know: • The domain is limited to only positive input values. So there should be some kind of boundary that separates the inputs the function can use from the ones the function cannot use. • For a base greater than $1\text{,}$ the bigger the result we want to get, the bigger the exponent needs to be. So the logarithm should be an increasing function. • The output can be any real number, so we expect the graph to exist both above and below the horizontal axis as well as having a horizontal intercept. WeBWorK Exercise WeBWorK Exercise WeBWorK Exercise ### Subsection4.1.6Rules of Logarithms In order for this part of the activity to make sense, we must remember what a base and an exponent are. Remember in the expression $b^x\text{,}$ $b$ is the base and $x$ is the exponent. The exponent literally counts how many times you multiplied the base. For instance $b^7$ means $\overbrace{b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot b}^{7\text{ times}}\text{.}$ #### Subsubsection4.1.6.1Definition: A logarithm is an exponent An exponential equation \begin{equation} b^{y}=x \end{equation} can be rewritten as a logarithmic equation. \begin{equation} \log_b(x)=y \end{equation} WeBWorK Exercise #### Subsubsection4.1.6.2Rule: The Log of a product We start with a very simple property of exponents: $x^a\cdot x^b=x^{a+b}\text{.}$ When you multiply numbers of the same base you add the exponents. ###### Exercise4.1.22 WeBWorK Exercise Now let's take some obvious information that you already concluded above to discover a not-so-obvious property of logarithms. \begin{equation} 8\cdot16=128\text{.} \end{equation} Apply the $\log_2(\phantom{x})$ function to both sides of the equation to get \begin{equation} \log_2(8\cdot16)=\log_2(128)\text{.} \end{equation} Notice on the right side we already know the answer (i.e. the power of $2$ that gives you $128$) ###### Exercise4.1.23 WeBWorK Exercise Now you can write the equation $\log_{2}(8\cdot16)=\log_{2}(128)$ as \begin{equation} \log_{2}(8\cdot16)=7\text{.} \end{equation} Notice on the left side of the equation: • There are $3$ factors of $2$ in $8\text{.}$ That's why $\log_{2}(8)=3\text{.}$ • There are $4$ factors of $2$ in $16\text{.}$ That's why $\log_{2}(8)=4\text{.}$ So the simple equation $7=3+4\text{,}$ can be written using logarithms \begin{gather} 7=3+4\\ 7=\log_{2}(8)+\log_{2}(16)\\ \log_{2}(128)=\log_{2}(8)+\log_{2}(16)\\ \log_{2}(8 \cdot 16)=\log_{2}(8)+\log_{2}(16) \end{gather} What??? Yes, you just discovered that \begin{equation} \log_2(8\cdot16)=\log_2(8)+\log_2(16)\text{.} \end{equation} In fact this property applies to all $\log_{b}(x)\text{.}$ ###### Exercise4.1.24 WeBWorK Exercise In general, for any base \begin{equation} \log_b(A\cdot B)=\log_b(A)+\log_b(B)\text{.} \end{equation} #### Subsubsection4.1.6.3Rule: Bring Down the Exponent We start with something obvious. You already know that instead of writing $8+8+8+8$ you could just write $4\cdot8\text{.}$ Maybe you never thought about it, but multiplication is just repeated addition. It's a shortcut way to write the same thing added over and over. On the other hand, exponents are just repeated multiplication. Exponents are a shortcut way to write the same thing multiplied over and over. Consider the expression $\log(8\cdot8\cdot8\cdot8)\text{.}$ We can use some shortcuts to write the expression in at least two different ways. \begin{equation} \log(8\cdot8\cdot8\cdot8)=\log\mathopen{}\left(8^4\right)\mathclose{} \end{equation} and using the last rule we discovered about the log of a product \begin{equation} \log(8\cdot8\cdot8\cdot8)=\log(8)+\log(8)+\log(8)+\log(8)\text{.} \end{equation} Since both shortcuts must give the same answer, we get \begin{equation} \log\mathopen{}\left(8^4\right)\mathclose{}=\log(8)+\log(8)+\log(8)+\log(8) \end{equation} ###### Exercise4.1.25 WeBWorK Exercise A very important rule: \begin{equation} \log_{b}(N^{x})=x \log_{b}(N) \end{equation} #### Subsubsection4.1.6.4Rule: Log of a Quotient You will discover this rule by using the two rules you already know. Remember that a negative exponent means “reciprocal” so a quotient can be rewritten as a product \begin{equation} \frac{Q}{P}=Q \cdot P^{-1} \end{equation} ###### Exercise4.1.26 WeBWorK Exercise \begin{equation} \log_{b}\mathopen{}\left(\frac{A}{B}\right)\mathclose{}=\log_{b}(A)-\log_{b}(B) \end{equation}	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.882964015007019, "perplexity": 584.3970274136543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00212.warc.gz"}
https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=26&document_srl=787894&sort_index=room&order_type=desc	A class of decomposition of Green's functions for the compressilbe Navier-Stokes linearized around a constant state is introduced. The singular structures of the Green's functions are developed as essential devices to use the nonlinearity directly to covert the 2nd order quasi-linear PDE into a system of zero-th order integral equation with regular integral kernels. The system of integrable equations allows a wider class of functions such as BV solutions. We have shown global existence and well-posedness of the compressible Navier-Stokes equation for isentropic gas with the gas constant $$extract_itex$$\gamma \in (0,e)$$/extract_itex$$ in the Lagrangian coordinate for the class of the BV functions and point wise $$extract_itex$$L^\infty$$/extract_itex$$ around a constant state; and the underline pointwise structure of the solutions is constructed. It is also shown that for the class of BV solution the solution is at most piecewise $$extract_itex$$C^2$$/extract_itex$$-solution even though the initial data is piecewise C^infty.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739846587181091, "perplexity": 1325.6370407076977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572908.71/warc/CC-MAIN-20220817122626-20220817152626-00103.warc.gz"}
http://mathhelpforum.com/math-puzzles/198126-problem-solving-question-print.html	# Problem solving question • April 30th 2012, 12:53 AM HerroDere Problem solving question Hi guys, I am at a loss when it comes to these questions, so I was wondering if somebody would help me out. Identify the numbers represented by letters. P, Q, R are all different numbers and not zero. QP0 x PQR 3RR0 + 49Q00 QP000 P242R0 • May 2nd 2012, 01:16 PM Soroban Re: Problem solving question Hello, HerroDere! This is called an "Alphametic". Quote: $\text{Identify the digits represented by letters. }\;P, Q, R\text{ are different nonzero digits.}$ . . $\begin{array}{ccccccc} ^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\ &&& Q & P & 0 \\ && \times & P & Q & R \\ \hline && 3 & R & R & 0 \\ & 4 & 9 & Q & 0 \\ & Q & P & 0 \\ \hline P & 2 & 4 & 2 & R & 0 \end{array}$ In column-1, we see that $P = 1.$ $\text{We have: }\;\begin{array}{ccccccc} ^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\ &&& Q & 1 & 0 \\ && \times & 1 & Q & R \\ \hline && 3 & R & R & 0 \\ & 4 & 9 & Q & 0 \\ & Q & 1 & 0 \\ \hline 1 & 2 & 4 & 2 & R & 0 \end{array}$ In column-3, we see that the sum is 14. Hence, column-2 is: . $4 + Q + 1 \:=\:12 \quad\Rightarrow\quad Q = 7$ $\text{We have: }\;\begin{array}{ccccccc} ^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\ &&& 7 & 1 & 0 \\ && \times & 1 & 7 & R \\ \hline && 3 & R & R & 0 \\ & 4 & 9 & 7 & 0 \\ & 7 & 1 & 0 \\ \hline 1 & 2 & 4 & 2 & R & 0 \end{array}$ In column-4: . $R + 7 \,=\,12 \quad\Rightarrow\quad R \,=\,5$ $\text{Therefore: }\;\begin{array}{ccccccc} ^1 & ^2 & ^3 & ^4 & ^5 & ^6 \\ &&& 7 & 1 & 0 \\ && \times & 1 & 7 & 5 \\ \hline && 3 & 5 & 5 & 0 \\ & 4 & 9 & 7 & 0 \\ & 7 & 1 & 0 \\ \hline 1 & 2 & 4 & 2 &7 & 0 \end{array}$ • May 11th 2012, 03:10 AM HerroDere Re: Problem solving question thanks for that :) your explanation really helped	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9523018002510071, "perplexity": 4372.965455252433}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982294097.9/warc/CC-MAIN-20160823195814-00112-ip-10-153-172-175.ec2.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/1311.7131/	# Determining the structure of dark-matter couplings at the LHC Ulrich Haisch Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom Anthony Hibbs Anthony.H Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom Emanuele Re Emanuele.R Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom ###### Abstract The latest LHC mono-jet searches place stringent bounds on the cross section of dark matter. Further properties such as the dark matter mass or the precise structure of the interactions between dark matter and the standard model can however not be determined in this manner. We point out that measurements of the azimuthal angle correlations between the two jets in events may be used to disentangle whether dark matter pair production proceeds dominantly through tree or loop diagrams. Our general observation is illustrated by considering theories in which dark matter interacts predominantly with the top quark. We show explicitly that in this case the jet-jet azimuthal angle difference is a gold-plated observable to probe the Lorentz structure of the couplings of dark matter to top quarks, thus testing the CP nature of the particle mediating these interactions. preprint: OUTP-13-23P ## I Introduction The minimal experimental signature of dark matter (DM) pair production at the LHC would be an excess of events with a single jet in association with large amounts of missing transverse energy (). The experimental search for events provides bounds on the interaction strength of DM with quarks and gluons, constraining the same parameters as direct detection experiments (see e.g. ATLAS ; CMS ). These measurements place the leading (and in some cases only) limits on models of DM over certain regions of parameter space. While the channel can be used to constrain the cross section, it provides insufficient information to determine additional DM properties such as its mass or the precise nature of its interactions with the standard model (SM). In fact, the transverse momentum () spectrum of the signal is essentially featureless and almost independent of the chirality and/or the CP properties of the DM couplings to quarks.111For instance, the spectra corresponding to effective vector and axial-vector DM-quark interactions are within the uncertainties present at the next-to-leading order (NLO) plus parton-shower (PS) level Haisch:2013ata indistinguishable. This suggests that while ATLAS and CMS are well suited to discover light DM, the LHC prospects of using this channel to make more definitive statements about specific DM properties seem to be slim. In this letter we observe that this unsatisfactory situation may be remedied by studying two-jet final states involving . In particular, we will argue that measurements of the azimuthal angle difference in events can possibly show a strong cosine-like or sine-like correlation only if DM pair production is loop induced, whereas tree-level interactions result in a distribution of a quite different shape. In order to illustrate our general observation we will consider DM models that generate the effective operators OS=mtΛ3S¯tt¯χχ,OP=mtΛ3P¯tγ5t¯χγ5χ. (1) Examples of Feynman diagrams with an insertion of that give rise to a signal are displayed in Fig. 1. For this well-motivated case we will explicitly show that the Lorentz structure of the DM top-quark interactions — and in consequence the CP nature of the mediator inducing (1) — can be disentangled by measuring the normalised distribution. After a discovery of an enhanced mono-jet signal, combining the measurements of the top-loop induced cross section Haisch:2012kf ; Fox:2012ru and the spectrum of the jet-jet azimuthal angle difference would hence not only allow to determine the suppression scales in (1) but also whether the scalar operator or the pseudo-scalar operator is responsible for the observed excess of events. Other constraints on effective interactions between DM and top quarks have been discussed for example in Haisch:2013uaa ; Lin:2013sca . Our work is organised as follows: in Sec. II we introduce the DM interactions which we intend to examine. In Sec. III we calculate the azimuthal angle correlations of the two jets in production induced by the operators , including the full top-quark mass dependence of the squared matrix elements. Our calculation is performed at the leading order (LO) in QCD. We will also comment on the applicability of the heavy top-quark approximation and the impact of higher-order QCD effects. In Sec. IV we discuss the case where the mediator can be resonantly produced, before concluding in Sec. V. ## Ii DM interactions In the following we are interested in DM pair production from quark or gluon initial states. We will restrict our discussion to the case where the production proceeds via the exchange of a spin- -channel mediator. We consider the following interactions between DM and top quarks involving a colourless scalar () or pseudo-scalar () mediator222LHC constraints on the scalar and pseudo-scalar DM-quark interactions involving the light flavours have been discussed in Haisch:2013ata ; Fox:2012ru . LS=gSχ(¯χχ)S+gStmtv(¯tt)S,LP=igPχ(¯χγ5χ)P+igPtmtv(¯tγ5t)P, (2) where is the Higgs vacuum expectation value. Notice that we have assumed that the couplings of the mediators to top quarks are proportional to the associated SM Yukawa coupling. This is motivated by the hypothesis of minimal flavour violation (MFV), which curbs the size of dangerous flavour-changing neutral current processes and automatically leads to a stable DM candidate Batell:2011tc . While the DM particle in (2) is understood to be a Dirac fermion, extending our discussion to Majorana DM or the case of a complex/real scalar is straightforward (see Haisch:2012kf for details). If the mediator masses are large compared to the invariant mass of the DM pair, we can describe production by means of an effective field theory (EFT). Integrating out the scalar and pseudo-scalar mediator then gives rise to (1) as well as composite operators consisting of four top-quark fields, which we do not consider further.333Unlike the operator , which is strongly constrained because it contributes to the electric dipole moment of the neutron Brod:2013cka , the purely scalar or pseudo-scalar four-top operators resulting from (2) are experimentally not well bounded. The appearance of the operator can be avoided by taking the spin-0 mediators to be CP eigenstates. In the case of the suppression scale is related to the mediator mass and the fundamental couplings by ΛS=(vM2SgSχgSt)1/3, (3) and an analogous expression with holds for . With the current Haisch:2012kf and Lin:2013sca data, one can exclude values of the suppression scale below roughly () in the scalar (pseudo-scalar) case for light DM, which is small compared to typical LHC energies. In order to discuss the validity of the EFT approach (see also Fox:2011pm ; Shoemaker:2011vi ; Fox:2012ee ; Busoni:2013lha ; Profumo:2013hqa ; Buchmueller:2013dya ), we will consider in Sec. IV also the simplest ultraviolet (UV) completion, where (1) arises from the full theory (2) after integrating out the fields and . We will see that in this case the analysis becomes more model-dependent, because the predictions now depend on and as well as the masses and the decay widths of the mediators. Apart from these minor complications our general conclusions will however also hold in the case where the -channel resonances can be directly produced in collisions. ## Iii DM production with two jets In our analysis we consider production at the LHC with center-of-mass (CM) energy. We adopt event selection criteria corresponding to the latest CMS mono-jet search CMS .444The cuts imposed in the existing ATLAS and CMS analyses will not be suitable for DM searches at the LHC due to triggering limitations Schramm . Our work should hence only be considered as a proof of concept. A more realistic study, including NLO corrections, PS effects and hadronisation corrections for both the DM signal and the SM backgrounds, will be presented elsewhere inpreparation . In this search events of more than two jets with pseudo-rapidity below 4.5 and transverse momentum above are rejected. In order to suppress QCD di-jet events, CMS puts an angular requirement on the azimuthal distance between the two tagging jets of . Our reference signal region is defined by , and , but we will comment on the sensitivity of the signal on the and cuts. To improve the separation between the azimuthal angle distribution of the SM background and the signal, we also impose a cut of on the invariant mass of the di-jet system. The calculation of the azimuthal distance of the signal events is performed with the help of GGFLO which is part of VBFNLO Arnold:2012xn , modifying the process appropriately. The GGFLO implementation of the production process is based on the analytical LO results of DelDuca:2001eu ; DelDuca:2001fn for the scalar Higgs () case and of Campanario:2010mi for the pseudo-scalar Higgs () case. Our simulations utilise MSTW2008LO parton distributions Martin:2009iq and jets are constructed according to the anti- algorithm Cacciari:2008gp with a radius parameter of , which corresponds to the value used in the CMS analysis CMS . We start our numerical analysis by showing results obtained for , a DM mass of , employing the reference cuts described above. Our choice of parameters will be motivated in Sec. IV. The central values of the corresponding and signal cross sections are and ( and ) for (), while the SM background predictions amount to and . To put these numbers into perspective we recall that the latest CMS analysis CMS excludes excesses in the mono-jet cross section with signal-over-background ratios of at 95% confidence level. Given these numbers the mono-jet signals corresponding to and should be easily detectable at the LHC. The normalised distributions associated to the operators are displayed in Fig. 2. From the figure it is evident that the scalar operator produces a strong correlation between the two jets, with a distribution that is peaked at and heavily suppressed at (red solid curve). In the case of the pseudo-scalar operator the position of the peak and trough is instead reversed (blue solid curve). The cosine-like (sine-like) modulation in the azimuthal angle distribution corresponding to () should be contrasted with the spectrum of the dominant SM background process, , which has a minimum at and a maximum in the vicinity of (green solid curve). We simulate the background at LO using the POWHEG BOX Alioli:2010xd ; Re:2012zi . PS effects or hadronisation corrections are not included in our SM prediction. To assess the significance of our findings, we study the scale uncertainties of the results. As advocated in DelDuca:2001fn , we identify the factorisation scale as and replace the overall factor entering the cross section by . We evaluate these quantities for every event generated by our Monte Carlo (MC) and vary in the range . In the total cross sections the induced scale uncertainties are around , while the relative shifts in the normalised differential azimuthal angle distributions do not exceed the level of . We conclude from this that even after considering scale ambiguities, the normalised distribution for is different than that of , and both spectra are clearly distinguishable from the SM background. The distinction in the radiation pattern of and can be most easily understood by employing the heavy top-quark mass limit. In fact, in this approximation the effect of top-quark loops in production can be described in terms of the following two effective operators OG=αs12πΛ3SGaμνGa,μν¯χχ,O˜G=αs8πΛ3PGaμν˜Ga,μν¯χγ5χ, (4) where denotes the gluon field strength tensor and its dual. In the limit that the external partons only experience a small energy loss and that the momentum components of the tagging jets in the beam direction are much greater than those in the transverse plane, the structure of the matrix element of and is easy to work out Plehn:2001nj . Denoting the currents and momenta of the gluons that initiate the scattering by and , one finds in the case of the result . This implies that the spectrum corresponding to should be enhanced for collinear tagging jets, , while for it should show an approximate zero. In the case of one obtains instead . It follows that the distribution for should have a dip if the two jets are collinear, , or back-to-back, , as the Levi-Civita tensor forces the result to zero. These features are clearly visible in Fig. 2. The above discussion also implies that in any theory in which one of the loop-induced operators in (4) is generated, the azimuthal angle difference in events will show a strong cosine-like or sine-like correlation. In theories in which DM pair production proceeds dominantly via tree-level graphs this will not be the case. Measurements of the spectrum are thus in principle sensitive to the quantum structure of the DM interactions with the SM. The lower part of Fig. 2 also shows that while the predictions obtained in the heavy top-quark mass approximation (dotted red and blue curves) describe the full results (solid red and blue curves) within an accuracy of or better, taking this limit always reduces the amplitude of the cosine-like and sine-like modulations. The behaviour found for is in clear contrast to that obtained in the case of the loop-induced mono-jet cross section for which the limit is not a good approximation Haisch:2012kf , because the high- jet is able to resolve the sub-structure of the top-quark loop. In fact, also in the case of , we find that the EFT predictions and the exact results are vastly different. For our standard cuts the infinite top-quark mass approximation overestimates the cross section by a factor of around () in the case of the operator (). In order to further illustrate this point we show in Fig. 3 the normalised distributions for using again and , but applying the stronger signal cuts and . The corresponding and cross sections read and (), and () and and (SM). One first observes that the infinite top-quark mass limit still furnishes an acceptable description of the full results in this case. Second, the cosine-like and sine-like modulations of the spectra are less pronounced if the requirements on and are more exclusive. This feature can be understood by recalling that a pure cosine-like or sine-like spectrum requires that the transverse momenta of the jets are much smaller than the momentum components along the beam direction. For harder cuts this approximation is not as good and as a result the strong jet-jet correlation is less marked. We conclude from this that in order to maximise the power of the distribution in determining the Lorentz structure of the DM top-quark interactions the and cuts should be as loose as possible. Making this statement more precise would require to perform a dedicated analysis of the cut dependencies of both the signal and the background. While such a study is beyond the scope of this letter, we plan to return to this question in a future publication inpreparation . Another important and related issue is the question whether higher-order QCD effects can potentially wash out the observed strong correlations between the two jets. This question can be addressed by relying again on the similarities of the signal process and its QCD analog . In the latter case it has been shown by explicit calculations (see e.g. DelDuca:2006hk ; Campbell:2006xx ; Andersen:2010zx ) that the shape of the lowest order distributions are unchanged and that therefore the jet-jet correlations survive the addition of NLO QCD corrections as well as PS and hadronisation effects. We verified that the latter feature is also present in the case of by showering our LO results with PYTHIA 6.4 Sjostrand:2006za . We find that PS effects result in relative shifts of maximal in the distributions and slightly reduce the amplitudes of the cosine-like and sine-like modulations, but do not distort the spectra. Given its stability under radiative corrections, we believe that the normalised spectrum of the azimuthal angle difference in production is a gold-plated observable for determining the structure of the couplings of DM to top quarks. ## Iv Discussion Until now we have considered an EFT framework to interpret a hypothetical mono-jet signal. This is particularly simple because in such a case the complete information is encoded in the scales that suppress the effective couplings (1), making it unnecessary to specify details of the particle mediating the interactions. Given the weakness of the bounds on Haisch:2012kf ; Lin:2013sca , there are however serious concerns regarding the validity of the EFT approach (see also Fox:2011pm ; Shoemaker:2011vi ; Fox:2012ee ; Busoni:2013lha ; Profumo:2013hqa ; Buchmueller:2013dya for similar discussions). In this section, we will therefore quantify when the simple-minded limits on the scale of the scalar and pseudo-scalar interactions apply and under which circumstances the EFT framework breaks down. In order to go beyond the effective description, one has to specify a concrete UV completion. In the following, we will assume that the full theory is provided by (2), which implies that the effective interactions (1) are generated by the -channel exchange of the colourless spin-0 states . We will not discuss the case of -channel exchange of coloured spin-0 mediators, which is interesting in its own right and has been utilised in Kumar:2013hfa ; Batell:2013zwa to construct MFV DM models where the relic carries top flavour. We follow Buchmueller:2013dya to determine the minimum value of the couplings for which the EFT approach is applicable. First, we derive the limits on the suppression scales as a function of the DM mass . For concreteness, our analysis is based on the most recent mono-jet search by CMS CMS with an integrated luminosity of at , utilising our standard event selection criteria. Second, we calculate in the full theory as a function of both and . The actual computation of the top-loop induced cross sections is performed by means of the MC codes developed in Haisch:2012kf ; Fox:2012ru , which give identical results. For each DM mass, the minimum value of consistent with an EFT description is then found from the requirement that the full theory calculation of agrees with the corresponding EFT result to better than . In the whole procedure, we take into account that and are related via (3). The minimal coupling strengths determined in this manner are indicated by the red solid curves and bands in Fig. 4. The width of the bands reflects the dependence of the predictions on the relative width of the mediators, which we vary in the range to obtain the shown results. We see that for the EFT to work the couplings of the -channel mediators to DM and top quarks have to be strong and that increasingly larger values of are needed for an accurate description, if the DM mass lies at or above the weak scale. In fact, in the case of () the theory becomes necessarily non-perturbative for () as indicated by the blue dashed curves in the plots. It is important to realise that the values for which the EFT is applicable are below a TeV if DM is light. To give an example, for the displayed EFT limits correspond to and , respectively, if one assumes that the relative widths are . The DM relic abundance also depends on the couplings and the masses . However, this observable is sensitive to the full particle content of the underlying UV theory, because the mass spectrum determines the number and the strengths of the DM annihilation channels. This feature makes the prediction for more model-dependent than the mono-jet cross sections analysed above. For simplicity, we will assume that the couplings and the particle content are completely specified by (2), meaning that only annihilation processes with top quarks and gluon pairs in the final state are possible. We also allow for either scalar or pseudo-scalar interactions but not both. Using the relevant formulas for the annihilation cross sections given in Haisch:2012kf and requiring that the relic abundance saturates the observed value Ade:2013zuv , we find the green dotted curves in the panels of Fig. 4. The parameter regions to the left and right of the curves correspond to DM overproduction and underproduction in the early universe. From the intersections of the non-perturbativity bounds and the relic density constraints, we obtain the following limit () in the case of the operator (). Combining all constraints we then find the yellow coloured wedges, which correspond to strongly-coupled theories with weak scale DM masses. Numerically, we arrive at and and . The parameters and used in Sec. III to simulate the distributions have hence been specifically chosen so that the EFT approach applies and the universe is not over closed. We emphasise that while large regions of parameter space are excluded due to DM overproduction, these bounds can be ameliorated if DM has large annihilation cross sections to other SM particles or (in particular) new hidden sector states. Such additional annihilation channels can reduce the tension between the LHC mono-jet limits and the relic density constraints significantly. The preceding discussion should have made clear that the applicability of the LHC mono-jet limits on is limited. This raises the question of whether the jet-jet azimuthal angle difference in remains a good observable to probe the structure of the DM top-quark interactions also beyond the EFT framework. To answer this question we study a simplified -channel model described by (2), fixing the relevant parameters to , and . Notice that for these parameter choices the DM relic constraints are satisfied. We furthermore verified that our DM models do not lead to an observable signal in existing and future LHC resonance searches in (di-jet) final states. Numerically, we find that including the one-loop process changes the total cross section by . A di-jet signal arises in the simplified models (2) first via the two-loop amplitude , which renders the contributions of exchange to di-jet production utterly small. The signal strength in production depends sensitively also on the total widths of the mediators . In the case of the scalar mediator, we obtain the following results for the partial decay widths Γ(S→¯tt)=(mtvgSt)238πMS(1−4m2tM2S)3/2,Γ(S→¯χχ)=(gSχ)218πMS(1−4m2χM2S)3/2,Γ(S→gg)=(mtvgSt)2α2s2π3m2tMS∣∣ ∣∣FS(4m2tM2S)∣∣ ∣∣2, (5) where FS(τ)=1+(1−τ)arctan2(1√τ−1). (6) The analog expressions for the pseudo-scalar mediator are obtained from (5) by the replacements and in the exponents, and the relevant form factor reads FP(τ)=arctan2(1√τ−1). (7) Using the above values for the couplings and masses, we arrive at and , which implies that we are dealing with narrow resonances. The corresponding values of the mono-jet cross sections at the LHC are and , if our standard signal cuts are applied. For the signal cross sections we find instead and , respectively.555We recall that the dominant SM backgrounds due to and have cross sections of and , respectively. At the LHC with an integrated luminosity of one hence expects to see more than 1000 signal events, which should allow for a measurement of the distribution in the sample. In Fig. 5 we show the normalised azimuthal angle distributions corresponding to our explicit DM models. We see that the strong cosine-like (sine-like) correlation between the two tagging jets in survives in the full theory with resonant scalar (pseudo-scalar) exchange. This shows that, unlike the mono-jet cross section, which depends strongly to the exact model realisation, the normalised distribution is rather insensitive to the precise structure of the underlying theory, and therefore provides a unique way to probe the anatomy of possible couplings between DM and top quarks. As in the case of the EFT calculations, we also see from the latter figure that the approximations of the spectra describe the exact results reasonably well. We furthermore find that the heavy top-quark mass limit describes the total cross sections much better in the full theory than in the EFT framework. Numerically, we obtain for the standard cuts that the ratio of EFT to exact cross sections is around 1.4 for both scalar and pseudo-scalar interactions. The observed feature is explained by the fact that in the full theory the cross section is dominated by invariant masses close to , while in the EFT calculation the momentum transfer to the DM pair can be (and is on average) much larger. The quality of the heavy top-quark mass approximation however degrades rapidly with the amount of off-shellness Haisch:2012kf , which explains why for the total cross sections the limit works fairly well in the case of the simplified model, while it fails badly in the EFT approach. ## V Conclusions While mono-jet searches provide already stringent constraints on the pair-production cross sections of DM and may lead to a future discovery at the LHC, even the observation of an unambiguous signal will not be enough to determine details of the nature of DM such as the mass of the DM candidate or the structure of its couplings to quarks and gluons. This is due to the fact that while the spectrum of the signal is somewhat harder than that of the background, the enhancement of the high- tail is fairly universal, in the sense that it is independent of the type of interactions that lead to the events. In this letter we have pointed out that some of the limitations of the LHC DM searches can be overcome by studying the jet-jet azimuthal angle difference in final states with two jets and a large amount of missing transverse energy. We showed in particular that if the signal arises from Feynman diagrams involving top-quark loops, measurements of the normalised distribution would provide a powerful handle to disentangle whether the DM top-quark interactions are of scalar or pseudo-scalar type. In contrast to the prediction of the mono-jet cross section that is highly model-dependent, we emphasised that the strong angular correlation between the two tagging jets is present irrespectively of whether the calculation is performed in an EFT or in a simplified DM model with scalar and pseudo-scalar exchange in the -channel. This feature combined with the stability of the suggested observable under QCD corrections, makes a gold-plated observable to determine the Lorentz structure of the DM top-quark couplings and/or to test the CP properties of the associated mediators. The method outlined in our work is more general as, after a DM discovery through a signal at the LHC, it can in principle be used to tell apart whether DM pair production proceeds dominantly via tree or loop graphs. Only in the latter case, measurements of the azimuthal angle difference in events can potentially show a strong cosine-like or sine-like modulation, while tree-level exchange of spin-0 and spin-1 mediators will lead to a distribution with a rather different dependence. In the case of discovery, it is hence imperative that ATLAS and CMS study the differential distributions of final states beyond . ###### Acknowledgements. We would like to thank David Berge, Felix Kahlhoefer, Steven Schramm and Giulia Zanderighi for useful discussions and their valuable comments on the manuscript. Helpful correspondence with Francisco Campanario, Barbara Jaeger and Michael Kubocz concerning GGFLO and VBFNLO is acknowledged. The research of A. H. is supported by an STFC Postgraduate Studentship.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.962293267250061, "perplexity": 780.7993807036348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038064898.14/warc/CC-MAIN-20210411174053-20210411204053-00194.warc.gz"}
https://socratic.org/questions/542b66c7581e2a1bd455e9c7#110099	Chemistry Topics # Question 5e9c7 Oct 1, 2014 The mass ratio of zinc to iodine is 1 to 3.882. #### Explanation: The mass ratio of two elements in a compound is the mass of one element in a given amount of the compound divided by the mass of the other element. Let's calculate the mass of $\text{Zn}$ in 1 mol ${\text{ZnI}}_{2}$. $\text{Mass of Zn" = 1 color(red)(cancel(color(black)("mol ZnI"_2))) × (1 color(red)(cancel(color(black)("mol Zn"))))/(1 color(red)(cancel(color(black)("mol ZnI"_2)))) × (65.38 "g Zn")/(1 color(red)(cancel(color(black)("mol Zn")))) = "65.38 g Zn}$ Also, the mass of $\text{I}$ in 1 mol of ${\text{ZnI}}_{2}$ is $\text{Mass of I" = 1 color(red)(cancel(color(black)("mol ZnI"_2))) × (2 color(red)(cancel(color(black)("mol I"))))/(1 color(red)(cancel(color(black)("mol ZnI"_2)))) × (126.9 "g I")/(1 color(red)(cancel(color(black)("mol I")))) = "253.8 g I}$ $\text{Mass of Zn"/"Mass of I" = "65.38 g"/"253.8 g}$ To get the ratio in its smallest terms, we divide the numerator and the denominator by the smaller number ("65.38 g")#. Then $\text{Mass of Zn"/"Mass of I} = \frac{1}{3.882}$ ##### Impact of this question 2976 views around the world	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 9, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9796918630599976, "perplexity": 853.2149745387999}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00446.warc.gz"}
http://tex.stackexchange.com/questions/87523/control-space-between-paragraph-and-example-environment-amsthm-package	# Control space between paragraph and example environment - amsthm package Which parameter sets the spacing before and after of the example environment from amsthm library? MWE: \documentclass{article} \usepackage{lipsum} \usepackage{amsthm} \newtheorem{example}{Exam}[section] \begin{document} \section{My section} \lipsum[2] % Dummy text \begin{example} \lipsum[1] % Dummy text $$x^2 + y^2 = 1$$ \end{example} \lipsum[2] % Dummy text \end{document} - amsthm has three different theorem styles, one of which -- remark -- has no extra space above and below. it would probably be a good idea to investigate this before trying to change the predefined spacing. – barbara beeton Dec 18 '12 at 16:48 The amsthm package offers three predefined styles for the theorem-like structures: plain (the default style), definition and remark; the remark style leaves less space before and after the structure than the other two styles, as can be seen in the following example: \documentclass{article} \usepackage{amsthm} \usepackage[hmargin=2cm]{geometry}% just for the example \usepackage{lipsum}% just to generate text for the example \newtheorem{exai}{Plain} \theoremstyle{definition} \newtheorem{exaii}{Definition} \theoremstyle{remark} \newtheorem{exaiii}{Remark} \begin{document} \lipsum[4] \begin{exai} \lipsum[4] \end{exai} \lipsum[4] \begin{exaii} \lipsum[4] \end{exaii} \lipsum[4] \begin{exaiii} \lipsum[4] \end{exaiii} \lipsum[4] \end{document} The internal lengths controlling the spacing before and after the structures are \thm@preskip (for the space before) and \thm@postskip (for the space after); in amsthm.sty one finds: \def\thm@space@setup{% \thm@preskip=\topsep \thm@postskip=\thm@preskip So the value of \topsep will be used for the structures built with the plain and definition styles; on the other hand, for the remark style, amsthm.sty has: \def\th@remark{% \normalfont % body font \thm@preskip\topsep \divide\thm@preskip\tw@ \thm@postskip\thm@preskip } so, for the remark style, half the value of \topsep will be used for the structures built with this style. If you want to change the default value for the plain and definition styles, you could then use in the preamble something like \makeatletter \def\thm@space@setup{% \thm@preskip=.5\topsep \thm@postskip=\thm@preskip } \makeatother A complete example: \documentclass{article} \usepackage{lipsum} \usepackage{amsthm} \makeatletter \def\thm@space@setup{% \thm@preskip=.5\topsep \thm@postskip=\thm@preskip } \makeatother \newtheorem{example}{Exam}[section] \begin{document} \section{My section} \lipsum[2] % Dummy text \begin{example} \lipsum[1] % Dummy text $$x^2 + y^2 = 1$$ \end{example} \lipsum[2] % Dummy text \end{document} Of course, this will affect all theorem-like structures for the two styles plain and definition, and it will also affect the spacing after the structure, since \thm@postskip=\thm@preskip; if you want the change only for a particular structure, you can use \newtheoremstyle (refer to the documentation for amsthm) to define the settings appropriately. A little example (using a space that is too small. but it is just for example purposes): \documentclass{article} \usepackage{lipsum} \usepackage{amsthm} \newtheoremstyle{example}% ⟨name⟩ {3pt}%⟨Space above⟩ {3pt}%⟨Space below⟩ {\itshape}%⟨Body font⟩ {}%⟨Indent amount⟩ {}%⟨Theorem head spec (can be left empty, meaning ‘normal’)⟩ \theoremstyle{example} \newtheorem{example}{Exam}[section] \begin{document} \section{My section} \lipsum[2] % Dummy text \begin{example} \lipsum[1] % Dummy text $$x^2 + y^2 = 1$$ \end{example} \lipsum[2] % Dummy text \end{document} - When I use library \usepackage{parskip} the \thm@preskip=.5\topsep \thm@postskip=\thm@preskip doesn't work. – jafan Dec 19 '12 at 11:41 @jafan I did some tests and it still works when the parskip package is loaded; if it doesn't for you, you should consider opening a new question with this new issue. – Gonzalo Medina Dec 19 '12 at 14:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9503927230834961, "perplexity": 3625.1988615062132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736676033.18/warc/CC-MAIN-20151001215756-00114-ip-10-137-6-227.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/465056/question-of-hartshorne-proposition-ii6-6	# Question of Hartshorne proposition II6.6 Let $X$ be a scheme which is a noetherian integral separated. In hartshorne's book, $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ is also a noetherian integral separated. I understand $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ is a noetherian and separated. But I don't know that $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ is integral... • Usually to check integral, you just check irreducible and reduced. $X \times_\mathbb{Z}\mathbb{A}_\mathbb{Z}$ can be seen to be reduced since if $R$ is a reduced ring, then $R\otimes_{\Bbb Z}\Bbb Z[x] \cong R[x]$ which is reduced and reduced is a local property. I think $X\times Y$ irreducible if $X$ and $Y$ are irreducible is a general fact but I have been unable to find a proof for it. – PVAL-inactive Aug 11 '13 at 21:27 • Thanks PVAL. I got it. I try to find a proof... – Sang Cheol Lee Aug 12 '13 at 0:34 • @PVAL: It does not hold in general. But it is true over alg. closed fields. See math.stackexchange.com/a/356169/1650 for instance. – Martin Brandenburg Aug 12 '13 at 8:28 1. If $R$ is an integral domain, then $R[T]$ is an integral domain. 2. If $R \to S$ is an injective homomorphism, then also $R[T] \to S[T]$ is injective. Now let $X$ be an integral scheme, i.e. reduced and irreducible. If $X$ is affine, then 1. shows that $X[T] := X \times \mathbb{A}^1$ is also integral. In general, let $\emptyset \neq U \subseteq X$ be an open affine subset. Then $U$ is integral, hence $U[T]$ is integral. Being reduced is a local property, so we already know that $X[T]$ is reduced. Now we have to prove that the generic points of $U[T]$, where $\emptyset \neq U \subseteq X$ is open affine, are all the same in $X[T]$. It suffices to check that if $\emptyset \neq V \subseteq U$ is another open affine, then $V[T] \to U[T]$ preserves the generic points. But this is precisely 2. Thus there is a unique point in $X[T]$ which is the generic point in $U[T]$ for every open affine $\emptyset \neq U \subseteq X$. Since these $U[T]$ cover $X[T]$, we see that this point is generic. Hence $X[T]$ is irreducible.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9942808151245117, "perplexity": 110.22133628349644}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540484815.34/warc/CC-MAIN-20191206050236-20191206074236-00252.warc.gz"}
https://www.physicsforums.com/threads/lift-drag-normal-and-axial-forces-on-a-flat-plate.527385/	# Lift, Drag, Normal, and Axial Forces on a Flat Plate 1. Sep 5, 2011 ### twmggc 1. The problem statement, all variables and given/known data Consider an infinite thin flat plate with a 1m chord at an angle of attack of 10 degrees in a super sonic flow. Describe the angle theta between the Lift and Normal Force, is it the same as the Drag and Axial? 2. Relevant equations L=Ncos(a)-Asin(a) D=Nsin(a)+Acos(a) 3. The attempt at a solution I thought the theta would be alpha (angle of attack) but since we have a flat plate I know that something is going to be different. I emailed my prof and he mentioned that it is going to be some constant directly related to geometry. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution Can you offer guidance or do you also need help? Draft saved Draft deleted Similar Discussions: Lift, Drag, Normal, and Axial Forces on a Flat Plate	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8321146368980408, "perplexity": 609.9811542527518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806124.52/warc/CC-MAIN-20171120183849-20171120203849-00775.warc.gz"}
https://ch.mathworks.com/help/radar/ref/rcssignature.html	# rcsSignature ## Description `rcsSignature` creates a radar cross-section (RCS) signature object. You can use this object to model an angle-dependent and frequency-dependent radar cross-section pattern. The radar cross-section determines the intensity of reflected radar signal power from a target. The object models only non-polarized signals. ## Creation ### Syntax ``rcssig = rcsSignature`` ``rcssig = rcsSignature(Name,Value)`` ### Description ``` `rcssig = rcsSignature` creates an `rcsSignature` object with default property values.``` example ````rcssig = rcsSignature(Name,Value)` sets object properties using one or more `Name,Value` pair arguments. `Name` is a property name and `Value` is the corresponding value. `Name` must appear inside single quotes (`''`). You can specify several name-value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`. Any unspecified properties take default values.``` Note You can only set property values of `rcsSignature` when constructing the object. The property values are not changeable after construction. ## Properties expand all Sampled radar cross-section (RCS) pattern, specified as a scalar, a Q-by-P real-valued matrix, or a Q-by-P-by-K real-valued array. The pattern is an array of RCS values defined on a grid of elevation angles, azimuth angles, and frequencies. Azimuth and elevation are defined in the body frame of the target. • Q is the number of RCS samples in elevation. • P is the number of RCS samples in azimuth. • K is the number of RCS samples in frequency. Q, P, and K usually match the length of the vectors defined in the `Elevation`, `Azimuth`, and `Frequency` properties, respectively, with these exceptions: • To model an RCS pattern for an elevation cut (constant azimuth), you can specify the RCS pattern as a Q-by-1 vector or a 1-by-Q-by-K matrix. Then, the elevation vector specified in the `Elevation` property must have length 2. • To model an RCS pattern for an azimuth cut (constant elevation), you can specify the RCS pattern as a 1-by-P vector or a 1-by-P-by-K matrix. Then, the azimuth vector specified in the `Azimuth` property must have length 2. • To model an RCS pattern for one frequency, you can specify the RCS pattern as a Q-by-P matrix. Then, the frequency vector specified in the `Frequency` property must have length 2. Example: `[10,0;0,-5]` Data Types: `double` Azimuth angles used to define the angular coordinates of each column of the matrix or array, specified by the `Pattern` property. Specify the azimuth angles as a length-P vector. P must be greater than two. Angle units are in degrees. Example: `[-45:0.5:45]` Data Types: `double` Elevation angles used to define the coordinates of each row of the matrix or array, specified by the `Pattern` property. Specify the elevation angles as a length-Q vector. Q must be greater than two. Angle units are in degrees. Example: `[-30:0.5:30]` Data Types: `double` Frequencies used to define the applicable RCS for each page of the `Pattern` property, specified as a K-element vector of positive scalars. K is the number of RCS samples in frequency. K must be no less than two. Frequency units are in hertz. Example: `[0:0.1:30]` Data Types: `double` ## Object Functions `value` Radar cross-section at specified angle and frequency `toStruct` Convert to structure ## Examples collapse all Specify the radar cross-section (RCS) of a triaxial ellipsoid and plot RCS values along an azimuth cut. Specify the lengths of the axes of the ellipsoid. Units are in meters. ```a = 0.15; b = 0.20; c = 0.95;``` Create an RCS array. Specify the range of azimuth and elevation angles over which RCS is defined. Then, use an analytical model to compute the radar cross-section of the ellipsoid. Create an image of the RCS. ```az = [-180:1:180]; el = [-90:1:90]; rcs = rcs_ellipsoid(a,b,c,az,el); rcsdb = 10log10(rcs); imagesc(az,el,rcsdb) title('Radar Cross-Section') xlabel('Azimuth (deg)') ylabel('Elevation (deg)') colorbar``` Create an `rcsSignature` object and plot an elevation cut at $3{0}^{\circ }$ azimuth. ```rcssig = rcsSignature('Pattern',rcsdb,'Azimuth',az,'Elevation',el,'Frequency',[300e6 300e6]); rcsdb1 = value(rcssig,30,el,300e6); plot(el,rcsdb1) grid title('Elevation Profile of Radar Cross-Section') xlabel('Elevation (deg)') ylabel('RCS (dBsm)')``` ```function rcs = rcs_ellipsoid(a,b,c,az,el) sinaz = sind(az); cosaz = cosd(az); sintheta = sind(90 - el); costheta = cosd(90 - el); denom = (a^2(sintheta'.^2)cosaz.^2 + b^2(sintheta'.^2)sinaz.^2 + c^2(costheta'.^2)ones(size(cosaz))).^2; rcs = (pia^2b^2c^2)./denom; end``` ## References [1] Richards, Mark A. Fundamentals of Radar Signal Processing. New York, McGraw-Hill, 2005. ## Extended Capabilities ### C/C++ Code GenerationGenerate C and C++ code using MATLAB® Coder™. Introduced in R2021a	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8413608074188232, "perplexity": 2599.371206103955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057039.7/warc/CC-MAIN-20210920131052-20210920161052-00075.warc.gz"}
http://mathhelpforum.com/calculus/2254-calculus-problem.html	# Math Help - Calculus Problem 1. ## Calculus Problem 83. What is the area of the region in the first quadrant enclosed by the graphs of y = cosx, y = x,and the y-axis? (A) 0.127 (B) 0.385 (C) 0.400 (D) 0.600 (E) 0.947 2. Originally Posted by frozenflames 83. What is the area of the region in the first quadrant enclosed by the graphs of y = cosx, y = x,and the y-axis? (A) 0.127 (B) 0.385 (C) 0.400 (D) 0.600 (E) 0.947 First sketch the required area, see attachment. The required area is the shaded region in the figure. Just counting squares indicates that the area is close to 0.4, but we will find this area by integration. This is: $ \int_0^{x_1} \cos(x)-x\ dx $ where $x_1$ is the solution of $\cos(x)=x$ in $[0,1]$. Now using the bisection method gives that $x_1\approx 0.739 $ So we seek: $ \int_0^{0.739} \cos(x)-x\ dx=\left[\sin(x)-x^2/2\right]_0^{0.739}=0.400 $ RonL Attached Thumbnails 3. I dont quite understand 4. Originally Posted by frozenflames I dont quite understand I had not finished typing the solution so look again RonL	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8304411768913269, "perplexity": 1660.5949564673542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413558067114.72/warc/CC-MAIN-20141017150107-00128-ip-10-16-133-185.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/170224-linear-approximation.html	1. ## Linear approximation Let $f:\mathbb R^2\to\mathbb R$ so that $f(x,y)=(s,t)=\left( x+\dfrac{1}{2}\arctan y,y+\dfrac{1}{2}\arctan x \right)$ Find the linear approximation of $f^{-1}$ on a neighborhood of $(s_0,t_0)=f(0,1).$ Don't worry about the invertibility, that was another question which I solved, but now I need help with this one, I don't get how to solve it. 2. By the chain rule, we get that $Df(0,1)\circ Df^{-1}(s_0,t_0)=I$. 3. Okay... but, I don't know how to relate it to find the linear approximation. If you could help me a bit more. 4. It's pretty straightforward: Since that relation gives that the derivatives involved are invertible we get $Df^{-1}(s_0,t_0)=Df(0,1)^{-1}$ 5. Oh, now I get, so I just need to compute the inverse but do I need to multiply it by a vector or something? 6. The linear approximation to g(x, y) at (x_0, y_0) is g(x_0, y_0)+ D_g(x_0, y_0)(x, y) where the last term is the product of the matrix D_g(x_0, y_0) with the vector (x, y).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9712311625480652, "perplexity": 297.3018205829974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345777160/warc/CC-MAIN-20131218054937-00099-ip-10-33-133-15.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/129726/sign-error-in-calculating-the-electric-field-of-point-charge	Sign error in calculating the electric field of point charge I have to determine the electric field of a point charge, I get a true result except for a sign. Here is my passages. $$\nabla \cdot e = \frac{\rho}{\epsilon}$$ $$e = - \nabla u$$ $$\nabla^2 u = - \frac{\rho}{\epsilon}$$ $$\frac{1}{r^2}\ \frac{\partial}{\partial r}\ \lgroup r^2 \frac{\partial u}{\partial r}\ \rgroup = -\frac{q}{\epsilon}\ \delta(r)$$ $$r^2 \frac{\partial u}{\partial r}\ = 0 \Rightarrow u(r) = \frac{f(r)}{r} \Rightarrow \frac{\partial u}{\partial r} = \frac{f'(r)}{r}-\frac{f(r)}{r^2}$$ $$\frac{\partial}{\partial r}\ \left( r^2 \frac{\partial u}{\partial r}\ \right) = \frac{\partial}{\partial r}\ \left( f'(r)r - f(r) \right( = rf''(r) = 0 \Rightarrow f(r) = Ar + B$$ $$u(r) = A + \frac{B}{r}$$ Now I set $A = 0$ to find $B$. $$\int _V \nabla \cdot \nabla u \,\,\, dV = \int _{\partial V} \nabla u\,\, i_n dS = \int _{\partial V} - \frac{B}{r^2} \,\, dS = - \frac{B}{r^2} 4\pi r^2 = - \frac{q}{\epsilon}$$ $$B = \frac{q}{4 \pi \epsilon} \Rightarrow u(r) = \frac{q}{4 \pi \epsilon r} \Rightarrow e(r) = - \frac{q}{4 \pi \epsilon r^2}$$ How is possible that the module of the electric field is negative? Thanks to Ruslan I get the answer. In Spherical coordinates, if we consider only $r$ $$\nabla u = \partial _r u(r) \hat{r}$$ Could you expand on how you calculated $e(r)$ from $u(r)$? Also, you seem to have missed $r$ in denominator of $u(r)$ in the last equation. Currently $e(r)$ you show is just a $\partial_r u(r)$ instead of the $\|-\nabla u\|$ calculated in spherical coordinates. – Ruslan Aug 6 '14 at 9:35 Yes, I missed r in denominator of u(r); I have seen on wikipedia that in spherical coordinates $$-\nabla u = \partial _r u(r)$$ if we consider only the r. Here's the link en.wikipedia.org/wiki/… – Gnamm Aug 6 '14 at 9:41 There's no minus in wikipedia in "Gradient" row. You should have $-\nabla u=-\partial_r u(r)\hat{\boldsymbol r}$ instead. – Ruslan Aug 6 '14 at 9:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9514879584312439, "perplexity": 338.2457322660848}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701169261.41/warc/CC-MAIN-20160205193929-00231-ip-10-236-182-209.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-equations/158269-first-order-non-linear-differential-equation.html	# Math Help - First Order Non-Linear Differential Equation 1. ## First Order Non-Linear Differential Equation Question: Solve the Initial value problem: $(5t^4y^{-3}+7t^6y^7)dt + (-3t^5y^{-4} + 7t^7y^6)dy = 0, y(1) = 1$. Express your answer in the form $F(t,y) = 2$, where $F(t,y)$ has no constant term. $F(t,y)=$ __________________ $= 2$ Attempt at the question: First of all I checked the terms to see if they were exact and they are not. I took the integral with respect to $t$ for $M(t,y)$ which is the first part of the equation and it came out to be $t^5y^{-3} + t^7y^7 + k(y)$. Afterwards I took the partial with respect to y of that integral we just got which was: $-3t^5y^{-4} + 7t^6y^7 + k'(y)$. I now would equate both these equations together and would cancel terms. The result gives me $k'(y) = 0$. So, my equation $F(t,y) = t^5y^{-3} + t^7y^7$. But my question now is, how do I solve using the initial values since my equation has both terms $t$ and $y$? Thanks for everyone who contributes! 2. This is an implicitly defined curve. You are told that the solution should look like $F(y,t)=2$ this must hold for all y and t You have $F(y,t)=\frac{t^5}{y^3}+t^7y^7=2=F(1,1)$ so the implict solution is $\frac{t^5}{y^3}+t^7y^7=2$ we can check this by taking the derivative $5t^4y^{-3}-(3t^5y^{-4})\frac{dy}{dt}+7t^6y^7+(7t^7y^6)\frac{dy}{dt}=0$ $(-3t^5y^{-4}+7t^7y^6)dy+(5t^4y^{-3}+7t^6y^7)dt=0$ 3. Ok, so my answer that i reached is correct. Essentially, what does no constant term mean? I just want to clarify that for my understanding!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 18, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9660763144493103, "perplexity": 200.1995084262692}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246636650.1/warc/CC-MAIN-20150417045716-00015-ip-10-235-10-82.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/solution-for-sylvester-equation.590375/	# Solution for Sylvester Equation 1. Mar 26, 2012 ### matematikawan Given n by n matrices A, B, C. I know how to solve the Sylvester equation AX + XB + C = 0 using the matlab command >> X=lyap(A,B,C) But how do we solve the extended Sylvester equation AX + XB + CXD + E = 0 ? Either numerical or analytical method I'm willing to learn. 2. Mar 26, 2012 ### chiro Hey matematikawan. Have you tried just expanding out the system, collecting the terms and getting a form of AX = B? In other words you get a matrix corresponding to ZX = F and then apply the formula X = Z^-1 x F. For the Z matrix you will need to do some algebra to get this and in this particular example, F = -E. 3. Mar 26, 2012 ### matematikawan I don't think it is possible to express it as AX=B. Even to solve the Sylvester equation you have to diagonalize the matrices. 4. Mar 26, 2012 ### chiro Try pen and paper first instead of using a computer. What will happen is that when you collect everything together you should get a linear system in terms of your X and some matrix that is premultiplied by it. Once you have separated the matrix from your X by specifying what that matrix is then you can do normal inversion techniques. You might have to write the algorithm yourself after doing a pen and paper derivation, but the idea doesn't change. Also when you expand out everything using algebra, I'm sure you'll find conditions for when this does not hold, possibly even as a function of A, B, and C. Again I urge you to do the pen and paper algebraic computation if you can't use any other known results. Similar Discussions: Solution for Sylvester Equation	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8259888291358948, "perplexity": 584.5650600112509}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824820.28/warc/CC-MAIN-20171021152723-20171021172723-00562.warc.gz"}
http://mathhelpforum.com/calculus/196970-power-series-respresentation-function.html	# Math Help - power series respresentation of a function 1. ## power series respresentation of a function powerseries respresentation of the function and the interval of convergance. was not given an "a" by which to rotate the function. f(x)=lnx 2. ## Re: power series respresentation of a function Originally Posted by bryonsingletary powerseries respresentation of the function and the interval of convergance. was not given an "a" by which to rotate the function. f(x)=lnx Try centring the function at x = 1... 3. ## Re: power series respresentation of a function Originally Posted by Prove It Try centring the function at x = 1... I can do it with a taylor series around 1 but I think I need a general sumation and interval of convergence to solve the problem. thank you for the tip thought I always welcome suggestions.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9224278330802917, "perplexity": 1553.201899159453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447729.93/warc/CC-MAIN-20151124205407-00051-ip-10-71-132-137.ec2.internal.warc.gz"}
http://www.maa.org/press/maa-reviews/elementary-topology	# Elementary Topology ###### Michael C. Gemignani Publisher: Dover Publications Publication Date: 1990 Number of Pages: 288 Format: Paperback Edition: 2 Price: 11.95 ISBN: 9780486665221 Category: Textbook BLL Rating: The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. [Reviewed by Mark Hunacek , on 04/15/2015 ] Michael Gemignani has led an interesting life — he is an Episcopal priest, lawyer and mathematician, and has written books in all three areas. Back in the late 1960s and 1970s he wrote quite a few books in mathematics (including ones on topology, real analysis, axiomatic geometry, probability and calculus); in the 1980s he wrote several law-related books, including at least one on computer law; more recently he has written on spiritual topics, both fiction (What Really Happened to Harry?) and nonfiction (Paths to Contemplation). He has even written a book (unseen by me), Farnsworth, about a near-bankrupt university facing an offer from Satan. The book now under review is a Dover reprint of the second (1972) edition of his topology book; Satan does not appear, but lots of topics do that have bedeviled many students over the years. I recall flipping through the first edition of this book (then published by Addison-Wesley) as supplementary reading in an undergraduate topology course that I was taking around 1970; I thought at the time that it was a good book, and still do, although, as explained below, I do have some concerns about a couple of points. The topics covered here are, for the most part, ones that are fairly standard in introductory topology courses: after a preliminary chapter on sets and cardinality (and also a brief discussion of groups, in anticipation of the final chapter on the fundamental group), Gemignani offers a chapter on metric, and then two on topological, spaces, each one covering the basics. (I strongly believe that this is the proper order in which to do things; metric spaces provide a natural generalization of the student’s work in calculus and analysis, and topological spaces are a natural generalization of metric spaces; this provides a well-motivated path to what might otherwise be a strange concept.) This is then followed by a chapter on the separation axioms. The next chapter (“Convergence”) is a particularly nice, well-motivated account of general notions of convergence (filters and nets), beginning with a few pages explaining why sequential convergence is not terribly useful in general topological spaces. The author could also have pointed out a much simpler example: in an indiscrete topological space, every sequence converges to every point in the space, so for that reason alone the notion of sequential convergence seems to be of limited utility. One standard application of these ideas is a proof of Tychonoff’s theorem on products of compact spaces, and the author gives this as part of a discussion, comprising the next three chapters of the text, on compactness and connectedness. The version of Tychonoff’s theorem presented in the main body of the text is for countable products only, and the author also gives an alternative proof of this theorem that does not involve filters and nets. In an Appendix, he discusses arbitrary products and proves the result in that generality. The final two chapters of the book address, first, metrizability and complete metric spaces, and then homotopy theory. The first of these chapters, among other things, proves the Urysohn metrization theorem and discusses compactness in metric spaces. In the second of these chapters, the author defines the fundamental group, provides some simple examples, and proves some basic theorems (e.g., the fundamental group of a product space, the mapping on fundamental groups induced by a continuous function between spaces). This chapter stops short of being a full-blown discussion of this topic; covering spaces are not mentioned, for example, and neither is the Seifert van Kampen theorem. Gemignani’s decision not to discuss these topics seems entirely reasonable for a first course in basic topology that is not intended to go very far into algebraic topology. There are some books, like Munkres’ Topology, that go deeper into the fundamental group, as do books like Singh’s Elements of Topology and Introduction to Topology by Gamelin and Greene; these texts all contain enough material for a two semester course, however. There are, however, some other omitted topics that I would prefer to have seen included: Although quotient spaces are defined (the author calls them identification spaces, which seems a bit old-fashioned to me), familiar spaces like the Moebius strip and Klein bottle are not mentioned at all. It seems hard to imagine an introductory text on topology that does not have a picture of a Moebius strip in it, but this is by no means an isolated phenomenon; the book I learned this material from, Simmons’ Introduction to Topology and Modern Analysis omits it, as does the much more recent A Course in Point-Set Topology by Conway. In addition, and this may simply reflect my own personal prejudices, I like the idea of discussing, at least briefly, the topology of some of the more basic matrix groups, but these are also not mentioned; perhaps, when the book was originally written, they were not viewed as being as important as they are now. I have a few other quibbles, the first of which is not the fault of the author: perhaps the fault lies with my increasingly elderly eyes, but I thought the typeface in the book was rather small, particularly with regard to page numbers; on several occasions I had trouble determining what number a particular page was. On a more substantive level, there are two issues that I think merit discussion. First, the author is sometimes a bit loose with terminology. The term “path in X”, where X is a topological space, is first defined (page 111) as a continuous function from [0 ,1] into X; later, however, on page 189, the exact same phrase is defined anew to mean that X is the image of such a function. It is this latter definition that the author uses when he later gives, without proof, this statement of the Hahn-Mazurkiewicz theorem: “A metric space is compact, connected, and locally connected if and only if it is a path.” Another example along these lines occurs in section 3.5, where the term “derived set” is given both its usual meaning in point-set topology (the set of limit points of a set) and is also used in the section heading to refer to sets “which are topologically related to or ‘derived’” from a given set A. While these examples of linguistic imprecision are probably easily fixed in lecture, there is a second issue that I think is more problematic. The important concepts of compactness and connectedness are not introduced until quite late in the text, following the chapters on separation axioms and convergence; it seems clear to me, though, that it is more important for a beginning student in topology to know what a connected or compact space is, then it is for him or her to know the difference between a T3 and T4 space. In fact, I think that the concepts of connectedness and compactness are so fundamental to a good understanding of elementary topology that they should have been introduced initially in the metric space chapter, thus giving the reader adequate time in which to absorb and assimilate these ideas. Likewise, the important concept of a complete metric space is not discussed until very near the end of the book; this is another topic that could (and in my opinion should) have been brought up earlier, when metric spaces were first discussed. On the plus side, there are certainly aspects of this book that I liked. The author’s writing style is clear and generally well motivated, and the exercise collection is a good one, containing quite a lot of exercises, many of which are just challenging enough to be interesting but not unreasonable. Working through them would be a valuable activity for students. (Solutions are not provided in the text, and I see no indication of a solutions manual being available.) And of course the very inexpensive price of this book (about seven dollars on amazon, as I write this) is a huge inducement to use this as a text instead of a book that costs twenty times that price. Whether these positive aspects of the book outweigh some of the concerns expressed previously is, of course, a matter of personal taste; in any event, I think this text definitely merits serious consideration as a possible text for an undergraduate-level text in topology. And, at this price, it is certainly a book that a student or instructor might just want to buy as a reference. Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8352558612823486, "perplexity": 458.53355663168566}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737913406.61/warc/CC-MAIN-20151001221833-00016-ip-10-137-6-227.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/help-set-theory-proving.275801/	# Help, set theory proving 1. Nov 29, 2008 ### mbcsantin -------------------------------------------------------------------------------- I tried to do the questions but im just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks the symbol "n" means "intersect" U for Union 1. The problem statement, all variables and given/known data 1) Prove A contained B iff A n B = A 2) Prove the following: For any sets A, B, C in a universe U: A n B = Universe iff A = Universe and B = Universe 3) Prove or find counterexamples. For any sets A, B, C in a universe U: if A union C contained B union C then A contained B 2. Relevant equations none. 3. The attempt at a solution 1) (=>) Assume A contained B Let x is an element of A, since A n A = A, x is an element of A and x is an element of B Case 1: x is an element of A: Since A contained B, x is an element of B so x is an element of A n B Case 2: x is an element of B: If x is an element of B then x is an element of (A n B) Hence x is an element of A n B This shows A contained A n B (<=) Assume A n B = A then A’=A’UA = A’ U (A n B) =(A’UA) n (A’U B) = empty set n A’ U B = A’ U B Hence Universe = A’ U B 2) Suppose A n B = U and suppose that A is a proper subset of U then x is an element of B but x is not an element of A n B since x is not an element of A 3) Let A be the empty set, and let B = C Then A union C = B and B union C = B so, A union C contains B union C, but A does not contain B because A is the empty set and B is not. 1. The problem statement, all variables and given/known data 2. Relevant equations 3. The attempt at a solution Can you offer guidance or do you also need help? Similar Discussions: Help, set theory proving	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8782921433448792, "perplexity": 788.2104833292966}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109803.8/warc/CC-MAIN-20170822011838-20170822031838-00304.warc.gz"}
https://stacks.math.columbia.edu/tag/00VZ	\begin{equation} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation} 7.9 The example of G-sets As an example, consider the site $\mathcal{T}_ G$ of Example 7.6.5. We will describe the category of sheaves on $\mathcal{T}_ G$. The answer will turn out to be independent of the choices made in defining $\mathcal{T}_ G$. In fact, during the proof we will need only the following properties of the site $\mathcal{T}_ G$: 1. $\mathcal{T}_ G$ is a full subcategory of $G\textit{-Sets}$, 2. $\mathcal{T}_ G$ contains the $G$-set ${}_ GG$, 3. $\mathcal{T}_ G$ has fibre products and they are the same as in $G\textit{-Sets}$, 4. given $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T}_ G)$ and a $G$-invariant subset $O \subset U$, there exists an object of $\mathcal{T}_ G$ isomorphic to $O$, and 5. any surjective family of maps $\{ U_ i \to U\} _{i \in I}$, with $U, U_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T}_ G)$ is combinatorially equivalent to a covering of $\mathcal{T}_ G$. These properties hold by Sets, Lemmas 3.10.2 and 3.11.1. Remark that the map $\mathop{\mathrm{Hom}}\nolimits _ G({}_ GG, {}_ GG) \longrightarrow G^{opp}, \varphi \longmapsto \varphi (1)$ is an isomorphism of groups. The inverse map sends $g \in G$ to the map $R_ g : s \mapsto sg$ (i.e. right multiplication). Note that $R_{g_1g_2} = R_{g_2} \circ R_{g_1}$ so the opposite is necessary. This implies that for every presheaf $\mathcal{F}$ on $\mathcal{T}_ G$ the value $\mathcal{F}({}_ GG)$ inherits the structure of a $G$-set as follows: $g \cdot s$ for $g \in G$ and $s \in \mathcal{F}({}_ GG)$ defined by $\mathcal{F}(R_ g)(s)$. This is a left action because $(g_1g_2) \cdot s = \mathcal{F}(R_{g_1g_2})(s) = \mathcal{F}(R_{g_2}\circ R_{g_1})(s) = \mathcal{F}(R_{g_1})( \mathcal{F}(R_{g_2})(s)) = g_1 \cdot (g_2 \cdot s).$ Here we've used that $\mathcal{F}$ is contravariant. Note that if $\mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets on $\mathcal{T}_ G$ then we get a map $\mathcal{F}({}_ GG) \to \mathcal{G}({}_ GG)$ which is compatible with the $G$-actions we have just defined. All in all we have constructed a functor $\textit{PSh}(\mathcal{T}_ G) \longrightarrow G\textit{-Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}({}_ GG).$ We leave it to the reader to verify that this construction has the pleasing property that the representable presheaf $h_ U$ is mapped to something canonically isomorphic to $U$. In a formula $h_ U({}_ GG) = \mathop{\mathrm{Hom}}\nolimits _ G({}_ GG, U) \cong U$. Suppose that $S$ is a $G$-set. We define a presheaf $\mathcal{F}_ S$ by the formula1 $\mathcal{F}_ S(U) = \mathop{Mor}\nolimits _{G\textit{-Sets}}(U, S).$ This is clearly a presheaf. On the other hand, suppose that $\{ U_ i \to U\} _{i\in I}$ is a covering in $\mathcal{T}_ G$. This implies that $\coprod _ i U_ i \to U$ is surjective. Thus it is clear that the map $\mathcal{F}_ S(U) = \mathop{Mor}\nolimits _{G\textit{-Sets}}(U, S) \longrightarrow \prod \mathcal{F}_ S(U_ i) = \prod \mathop{Mor}\nolimits _{G\textit{-Sets}}(U_ i, S)$ is injective. And, given a family of $G$-equivariant maps $s_ i : U_ i \to S$, such that all the diagrams $\xymatrix{ U_ i \times _ U U_ j \ar[d] \ar[r] & U_ j \ar[d]^{s_ j} \\ U_ i \ar[r]^{s_ i} & S }$ commute, there is a unique $G$-equivariant map $s : U \to S$ such that $s_ i$ is the composition $U_ i \to U \to S$. Namely, we just define $s(u) = s_ i(u_ i)$ where $i\in I$ is any index such that there exists some $u_ i \in U_ i$ mapping to $u$ under the map $U_ i \to U$. The commutativity of the diagrams above implies exactly that this construction is well defined. All in all we have constructed a functor $G\textit{-Sets} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G), \quad S \longmapsto \mathcal{F}_ S .$ We now have the following diagram of categories and functors $\xymatrix{ \textit{PSh}(\mathcal{T}_ G) \ar[rr]^{\mathcal{F} \mapsto \mathcal{F}({}_ GG)} & & G\textit{-Sets} \ar[ld]_{S \mapsto \mathcal{F}_ S} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) \ar[lu] & }$ It is immediate from the definitions that $\mathcal{F}_ S({}_ GG) = \mathop{Mor}\nolimits _ G({}_ GG, S) = S$, the last equality by evaluation at $1$. This almost proves the following. Proposition 7.9.1. The functors $\mathcal{F} \mapsto \mathcal{F}({}_ GG)$ and $S \mapsto \mathcal{F}_ S$ define quasi-inverse equivalences between $\mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G)$ and $G\textit{-Sets}$. Proof. We have already seen that composing the functors one way around is isomorphic to the identity functor. In the other direction, for any sheaf $\mathcal{H}$ there is a natural map of sheaves $can : \mathcal{H} \longrightarrow \mathcal{F}_{\mathcal{H}({}_ GG)}.$ Namely, for any object $U$ of $\mathcal{T}_ G$ we let $can_ U$ be the map $\begin{matrix} \mathcal{H}(U) & \longrightarrow & \mathcal{F}_{\mathcal{H}({}_ GG)}(U) = \mathop{Mor}\nolimits _ G(U, \mathcal{H}({}_ GG)) \\ s & \longmapsto & (u \mapsto \alpha _ u^s). \end{matrix}$ Here $\alpha _ u : {}_ GG \to U$ is the map $\alpha _ u(g) = gu$ and $\alpha _ u^ : \mathcal{H}(U) \to \mathcal{H}({}_ GG)$ is the pullback map. A trivial but confusing verification shows that this is indeed a map of presheaves. We have to show that $can$ is an isomorphism. We do this by showing $can_ U$ is an isomorphism for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{T}_ G)$. We leave the (important but easy) case that $U = {}_ GG$ to the reader. A general object $U$ of $\mathcal{T}_ G$ is a disjoint union of $G$-orbits: $U = \coprod _{i\in I} O_ i$. The family of maps $\{ O_ i \to U\} _{i \in I}$ is tautologically equivalent to a covering in $\mathcal{T}_ G$ (by the properties of $\mathcal{T}_ G$ listed at the beginning of this section). Hence by Lemma 7.8.4 the sheaf $\mathcal{H}$ satisfies the sheaf property with respect to $\{ O_ i \to U\} _{i \in I}$. The sheaf property for this covering implies $\mathcal{H}(U) = \prod _ i \mathcal{H}(O_ i)$. Hence it suffices to show that $can_ U$ is an isomorphism when $U$ consists of a single $G$-orbit. Let $u \in U$ and let $H \subset G$ be its stabilizer. Clearly, $\mathop{Mor}\nolimits _ G(U, \mathcal{H}({}_ GG)) = \mathcal{H}({}_ GG)^ H$ equals the subset of $H$-invariant elements. On the other hand consider the covering $\{ {}_ GG \to U\}$ given by $g \mapsto gu$ (again it is just combinatorially equivalent to some covering of $\mathcal{T}_ G$, and again this doesn't matter). Note that the fibre product $({}_ GG)\times _ U ({}_ GG)$ is equal to $\{ (g, gh), g\in G, h\in H\} \cong \coprod _{h\in H} {}_ GG$. Hence the sheaf property for this covering reads as $\xymatrix{ \mathcal{H}(U) \ar[r] & \mathcal{H}({}_ GG) \ar@<1ex>[r]^-{\text{pr}_0^} \ar@<-1ex>[r]_-{\text{pr}_1^} & \prod _{h \in H} \mathcal{H}({}_ GG). }$ The two maps $\text{pr}_ i^*$ into the factor $\mathcal{H}({}_ GG)$ differ by multiplication by $h$. Now the result follows from this and the fact that $can$ is an isomorphism for $U = {}_ GG$. $\square$ [1] It may appear this is the representable presheaf defined by $S$. This may not be the case because $S$ may not be an object of $\mathcal{T}_ G$ which was chosen to be a sufficiently large set of $G$-sets. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.987876296043396, "perplexity": 131.81376809072094}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201996.61/warc/CC-MAIN-20190319143502-20190319165502-00027.warc.gz"}
http://math.stackexchange.com/questions/422474/does-sum-k-1-n-x-1kn-xnn-have-integer-solution-when-n-ge-4	# Does $\sum_{k=1 }^ n (x-1+k)^n=(x+n)^n$ have integer solution when $n\ge 4$? from a post in SE, one says $3^2+4^2=5^2,3^3+4^3+5^3=6^3$,that is interesting for me. so I begin to explore further, the general equation is $\sum_{k=1}^n (x-1+k)^n=(x+n)^n$, from $n \ge 4$ to $n=41$, there is no integer solution for $x$. for $n>41$,I can't get result as Walframalpha doesn't work. I doubt if there is any integer solution for $n \ge 4$, when $n$ is bigger,the $x$ is close to $\dfrac{n}{2}$. Can some one have an answer? thanks! - To me it seems clear that there are no solutions, because the difference between those two formulas is just going to get bigger. I am thinking about how to prove this. – Nick Thomas Jun 17 '13 at 4:31 Are you interested just in positive integer solutions, or in arbitrary integer solutions? To me the latter question seems substantially harder. – Nick Thomas Jun 17 '13 at 5:00 my interesting is positive integer. But i think there may be negative solution when n is odd in case there is no positive solution. – chenbai Jun 17 '13 at 5:24 @chenbai : If you want to check some cases higher than $n=41$, I think you can just check until (right side) - (sum on left) becomes positive. After that there won't be solutions. – coffeemath Jun 17 '13 at 12:27 Your problem has been studied, and it is conjectured that only $3,4,5$ for squares and $3,4,5,6$ for cubes are such that all the numbers are consecutive, and the $k$th power of the last is the sum of the $k$th powers of the others ($k>1$). I've spent a lot of time on this question, and no wonder did not get anything final on a proof, since apparently nobody else has succeeded in proving it. The website calls it "Cyprian's Last Theorem", arguing that it seems so likely true but nobody has shown it yet, just like for Fermat for so many years. The page reference I found: http://www.nugae.com/mathematics/cyprian.htm There may be other links there to further get ideas... - thanks a lot. it is lucky to know it to avoid many calculations. – chenbai Jun 17 '13 at 23:30 Here is an argument that there are no solutions when $x$ is positive and $n$ and $x$ are sufficiently large. What "sufficiently large" means, I do not know exactly. I am making this answer community wiki; maybe others can improve on it. $$(\sum_{k=x}^{x+n-1} k^n) - (x+n)^n \geq \int_x^{x+n-1} t^n dt - (x+n)^n = \frac{1}{n+1}(x^{x+n-1} - x^{n+1}) - (x+n)^n,$$ which is greater than zero if $x$ and $n$ are sufficiently large. (Note that I wrote your initial sum a little differently than you did.) - but there is always two $x$s, the integers can let RHS or LHS >0 and <0, when n is bigger. – chenbai Jun 17 '13 at 5:30 That's right; but if $x$ and $n$ are big enough, the term $x^{x+n-1}$ overtakes $x^{n+1}$ and $(x+n)^n$. In a polynomial $p(x)$, when $x$ gets big, the term with the largest exponent ends up being the only one that "matters." – Nick Thomas Jun 17 '13 at 5:33 For any specific $n$, the function on the left side of the inequality is negative at $x=1$. Then as the real variable $x$ increases, your argument shows the left side eventually becomes positive. So for any fixed $n$ there is a value (or values) of $x>1$ for which the left side is $0$. The point is to somehow show such $x$ cannot be integral. So I don't think an argument based on an inequality can work. – coffeemath Jun 17 '13 at 5:54 If by some careful approximations one could trap the real $x$ where the sign change occurs, and trap it between integers, it would be done. Experimentally there are very large jumps as one goes (using integer $x$) from negative to positive. – coffeemath Jun 17 '13 at 6:04 Added to last comment-- it's only necessary to trap it relatively near the middle of a gap between adjacent $n$th powers. – coffeemath Jun 17 '13 at 7:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8196107745170593, "perplexity": 396.61199447197583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119658961.52/warc/CC-MAIN-20141024030058-00042-ip-10-16-133-185.ec2.internal.warc.gz"}
https://actuarialmodelingtopics.wordpress.com/tag/inverse-distributions/	# Transformed Pareto distribution One way to generate new probability distributions from old ones is to raise a distribution to a power. Two previous posts are devoted on this topic – raising exponential distribution to a power and raising a gamma distribution to a power. Many familiar and useful models can be generated in this fashion. For example, Weibull distribution is generated by raising an exponential distribution to a positive power. This post discusses the raising of a Pareto distribution to a power, as a result generating Burr distribution and inverse Burr distribution. Raising to a Power Let $X$ be a random variable. Let $\tau$ be a positive constant. The random variables $Y=X^{1/\tau}$, $Y=X^{-1}$ and $Y=X^{-1/\tau}$ are called transformed, inverse and inverse transformed, respectively. Let $f_X(x)$, $F_X(x)$ and $S_X(x)=1-F_X(x)$ be the probability density function (PDF), the cumulative distribution function (CDF) and the survival function of the random variable $X$ (the base distribution). The goal is to express the CDFs of the “transformed” variables in terms of the base CDF $F_X(x)$. The following table shows how. Name of Distribution Random Variable CDF Transformed $Y=X^{1 / \tau}, \ \tau >0$ $F_Y(y)=F_X(y^\tau)$ Inverse $Y=X^{-1}$ $F_Y(y)=1-F_X(y^{-1})$ Inverse Transformed $Y=X^{-1 / \tau}, \ \tau >0$ $F_Y(y)=1-F_X(y^{-\tau})$ If the CDF of the base distribution, as represented by the random variable $X$, is known, then the CDF of the “transformed” distribution can be derived using $F_X(x)$ as shown in this table. Thus the CDF, in many cases, is a good entry point of the transformed distribution. Pareto Information Before the transformation, we first list out the information on the Pareto distribution. The Pareto distribution of interest here is the Type II Lomax distribution (discussed here). The following table gives several distributional quantities for a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta$. Pareto Type II Lomax Survival Function $S(x)=\displaystyle \biggl( \frac{\theta}{x+\theta} \biggr)^\alpha \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$ Cumulative Distribution Function $F(x)=1-\displaystyle \biggl( \frac{\theta}{x+\theta} \biggr)^\alpha \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$ Probability Density Function $\displaystyle f(x)=\frac{\alpha \ \theta^\alpha}{(x+\theta)^{\alpha+1}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x >0$ Mean $\displaystyle E(X)=\frac{\theta}{\alpha-1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \alpha>1$ Median $\displaystyle \theta \ 2^{\frac{\alpha}{2}}-\theta$ Mode 0 Variance $\displaystyle Var(X)=\frac{\theta^2 \ \alpha}{(\alpha-1)^2 \ (\alpha-2)} \ \ \ \ \ \ \ \alpha>2$ Higher Moments $\displaystyle E(X^k)=\frac{k! \ \theta^k}{(\alpha-1) \cdots (\alpha-k)} \ \ \ \ \ \ \alpha>k \ \ \ k$ is integer Higher Moments $\displaystyle E(X^k)=\frac{\theta^k \ \Gamma(k+1) \Gamma(\alpha-k)}{\Gamma(\alpha)} \ \ \ \ \alpha>k$ The higher moments in the general case use $\Gamma(\cdot)$, which is the gamma function. The Distributions Derived from Pareto Let $X$ be a random variable that has a Pareto distribution (as described in the table in the preceding section). Assume that $X$ has a shape parameter $\alpha$ and scale parameter $\theta$. Let $\tau$ be a positive number. When raising $X$ to the power $1/\tau$, the resulting distribution is a transformed Pareto distribution and is also called a Burr distribution, which then is a distribution with three parameters – $\alpha$, $\theta$ and $\tau$. When raising $X$ to the power $-1/\tau$, the resulting distribution is an inverse transformed Pareto distribution and it is also called an inverse Burr distribution. When raising $X$ to the power -1, the resulting distribution is an inverse Pareto distribution (it does not have a special name other than inverse Pareto). The paralogistic family of distributions is created from the Burr distribution by collapsing two of the parameters into one. Let $\alpha$, $\theta$ and $\tau$ be the parameters of a Burr distribution. By equating $\tau=\alpha$, the resulting distribution is a paralogistic distribution. By equating $\tau=\alpha$ in the corresponding inverse Burr distribution, the resulting distribution is an inverse paralogistic distribution. Transformed Pareto = Burr There are two ways to create the transformed Pareto distribution. One is to start with a base Pareto with shape parameter $\alpha$ and scale parameter 1 and then raise it to $1/\tau$. The scale parameter $\theta$ is added at the end. Another way is to start with a base Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^\tau$ and then raise it to the power $1/\tau$. Both ways would generate the same CDF. We take the latter approach since it generates both the CDF and moments quite conveniently. Let $X$ be a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^\tau$. The following table gives the distribution information on $Y^{1/\tau}$. Burr Distribution CDF $F_Y(y)=\displaystyle 1-\biggl( \frac{1}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$ Survival Function $S_Y(x)=\displaystyle \biggl( \frac{1}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$ Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \tau \ (y/\theta)^\tau}{y \ [(y/\theta)^\tau+1 ]^{\alpha+1}}$ $y >0$ Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1/\tau+1) \Gamma(\alpha-1/\tau)}{\Gamma(\alpha)}$ $1 <\alpha \ \tau$ Median $\displaystyle \theta \ (2^{1/\alpha}-1)^{1/\tau}$ Mode $\displaystyle \theta \ \biggl(\frac{\tau-1}{\alpha \tau+1} \biggr)^{1/\tau}$ $\tau >1$, else 0 Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(k/\tau+1) \Gamma(\alpha-k/\tau)}{\Gamma(\alpha)}$ $-\tau The distribution displayed in the above table is a three-parameter distribution. It is called the Burr distribution with parameters $\alpha$ (shape), $\theta$ (scale) and $\tau$ (power). To obtain the moments, note that $E(Y^k)=E(X^{k/\tau})$, which is derived using the Pareto moments. The Burr CDF has a closed form that is relatively easy to compute. Thus percentiles are very accessible. The moments rely on the gamma function and are usually calculated by software. Inverse Transformed Pareto = Inverse Burr One way to generate inverse transformed Pareto distribution is to raise a Pareto distribution with shape parameter $\alpha$ and scale parameter 1 to the power of -1 and then add the scale parameter. Another way is to raise a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-\tau}$. Both ways derive the same CDF. As in the preceding case, we take the latter approach. Let $X$ be a Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-\tau}$. The following table gives the distribution information on $Y^{-1/\tau}$. Inverse Burr Distribution CDF $F_Y(y)=\displaystyle \biggl( \frac{(y/\theta)^\tau}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$ Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{(y/\theta)^\tau}{(y/\theta )^\tau+1} \biggr)^\alpha$ $y >0$ Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \tau \ (y/\theta)^{\tau \alpha}}{y \ [1+(y/\theta)^\tau]^{\alpha+1}}$ $y >0$ Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1-1/\tau) \Gamma(\alpha+1/\tau)}{\Gamma(\alpha)}$ $1 <\tau$ Median $\displaystyle \theta \ \biggl[\frac{1}{ 2^{1/\alpha}-1} \biggr]^{1/\tau}$ Mode $\displaystyle \theta \ \biggl(\frac{\alpha \tau-1}{\tau+1} \biggr)^{1/\tau}$ $\alpha \tau >1$, else 0 Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k/\tau) \Gamma(\alpha+k/\tau)}{\Gamma(\alpha)}$ $-\alpha \tau The distribution displayed in the above table is a three-parameter distribution. It is called the Inverse Burr distribution with parameters $\alpha$ (shape), $\theta$ (scale) and $\tau$ (power). Note that both the moments for Burr and inverse Burr distributions are limited, the Burr limited by the product of the parameters $\alpha$ and $\tau$ and the inverse Burr limited by the parameter $\tau$. This is not surprising since the base Pareto distribution has limited moments. This is one indication that all of these distributions have a heavy right tail. The Paralogistic Family With the facts of the Burr distribution and the inverse Burr distribution established, paralogistic and inverse paralogistic distributions can now be obtained. A paralogistic distribution is simply a Burr distribution with $\tau=\alpha$. An inverse paralogistic distribution is simply an inverse Burr distribution with $\tau=\alpha$. In the above tables for Burr and inverse Burr, replacing $\tau$ by $\alpha$ gives the following table. Paralogistic Distribution CDF $F_Y(y)=\displaystyle 1-\biggl( \frac{1}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$ Survival Function $S_Y(x)=\displaystyle \biggl( \frac{1}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$ Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha^2 \ \ (y/\theta)^\alpha}{y \ [(y/\theta)^\alpha+1 ]^{\alpha+1}}$ $y >0$ Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1/\alpha+1) \Gamma(\alpha-1/\alpha)}{\Gamma(\alpha)}$ $1 <\alpha^2$ Median $\displaystyle \theta \ (2^{1/\alpha}-1)^{1/\alpha}$ Mode $\displaystyle \theta \ \biggl(\frac{\alpha-1}{\alpha^2+1} \biggr)^{1/\alpha}$ $\alpha >1$, else 0 Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(k/\alpha+1) \Gamma(\alpha-k/\alpha)}{\Gamma(\alpha)}$ $-\alpha Inverse Paralogistic Distribution CDF $F_Y(y)=\displaystyle \biggl( \frac{(y/\theta)^\alpha}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$ Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{(y/\theta)^\alpha}{(y/\theta )^\alpha+1} \biggr)^\alpha$ $y >0$ Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha^2 \ (y/\theta)^{\alpha^2}}{y \ [1+(y/\theta)^\alpha]^{\alpha+1}}$ $y >0$ Mean $\displaystyle E(Y)=\frac{\theta \ \Gamma(1-1/\alpha) \Gamma(\alpha+1/\alpha)}{\Gamma(\alpha)}$ $1 <\alpha$ Median $\displaystyle \theta \ \biggl[\frac{1}{ 2^{1/\alpha}-1} \biggr]^{1/\alpha}$ Mode $\displaystyle \theta \ (\alpha-1)^{1/\alpha}$ $\alpha^2 >1$, else 0 Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k/\alpha) \Gamma(\alpha+k/\alpha)}{\Gamma(\alpha)}$ $-\alpha^2 Inverse Pareto Distribution The distribution that has not been discussed is the inverse Pareto. Again, we have the option of deriving it by raising to a base Pareto with just the shape parameter to -1 and then add the scale parameter. We take the approach of raising a base Pareto distribution with shape parameter $\alpha$ and scale parameter $\theta^{-1}$. Both approaches lead to the same CDF. Inverse Pareto Distribution CDF $F_Y(y)=\displaystyle \biggl( \frac{y}{\theta+y} \biggr)^\alpha$ $y >0$ Survival Function $S_Y(x)=\displaystyle 1-\biggl( \frac{y}{\theta+y} \biggr)^\alpha$ $y >0$ Probability Density Function $\displaystyle f_Y(y)=\frac{\alpha \ \theta \ y^{\alpha-1}}{[\theta+y ]^{\alpha+1}}$ $y >0$ Median $\displaystyle \frac{\theta}{2^{1/\alpha}-1}$ Mode $\displaystyle \theta \ \frac{\alpha-1}{2}$ $\alpha >1$, else 0 Higher Moments $\displaystyle E(Y^k)=\frac{\theta^k \ \Gamma(1-k) \Gamma(\alpha+k)}{\Gamma(\alpha)}$ $-\alpha The distribution described in the above table is an inverse Pareto distribution with parameters $\alpha$ (shape) and $\theta$ (scale). Note that the moments are even more limited than the Burr and inverse Burr distributions. For inverse Pareto, even the mean $E(Y)$ is nonexistent. Remarks The Burr and paralogistic families of distributions are derived from the Pareto family (Pareto Type II Lomax). The Pareto connection helps put Burr and paralogistic distributions in perspective. The Pareto distribution itself can be generated as a mixture of exponential distributions with gamma mixing weight (see here). Thus from basic building blocks (exponential and gamma), vast families of distributions can be created, thus expanding the toolkit for modeling. The distributions discussed here are found in the appendix that is found in this link. $\copyright$ 2017 – Dan Ma # Transformed gamma distribution The previous post opens up a discussion on generating distributions by raising an existing distribution to a power. The previous post focuses on the example of raising an exponential distribution to a power. This post focuses on the distributions generated by raising a gamma distribution to a power. Raising to a Power Let $X$ be a random variable. Let $\tau$ be a positive constant. The random variables $Y=X^{1/\tau}$, $Y=X^{-1}$ and $Y=X^{-1/\tau}$ are called transformed, inverse and inverse transformed, respectively. Let $f_X(x)$, $F_X(x)$ and $S_X(x)=1-F_X(x)$ be the probability density function (PDF), the cumulative distribution function (CDF) and the survival function of the random variable $X$ (the base distribution). The following derivation seeks to express the CDFs of the “transformed” variables in terms of the base CDF $F_X(x)$. $Y=X^{1/\tau}$ (Transformed): \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{1/\tau} \le y) \\&=P(X \le y^\tau) \\&=F_X(y^\tau) \end{aligned} $Y=X^{-1}$ (Inverse): \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{-1} \le y) \\&=P(X \ge y^{-1}) \\&=S_X(y^{-1})=1-F_X(y^{-1}) \end{aligned} $Y=X^{-1/\tau}$ (Inverse Transformed): \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{-1/\tau} \le y)\\&=P(X^{-1} \le y^\tau) \\&=P(X \ge y^{-\tau)} \\&=S_X(y^{-\tau})=1-F_X(y^{-\tau}) \end{aligned} Gamma CDF Unlike the exponential distribution, the CDF of the gamma distribution does not have a closed form. Suppose that $X$ is a random variable that has a gamma distribution with shape parameter $\alpha$ and scale parameter $\theta$. The following is the expression of the gamma CDF. $\displaystyle F_X(x)=\int_0^x \frac{1}{\Gamma(\alpha)} \ \frac{1}{\theta^\alpha} \ t^{\alpha-1} \ e^{- t/\theta} \ dt \ \ \ \ \ \ \ \ \ x>0$ By a change of variable, the CDF can be expressed as the following integral. \displaystyle \begin{aligned} F_X(x)&=\int_0^{x/\theta} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du \ \ \ \ \ \ \ \ \ x>0 \\&=\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; x/\theta) \end{aligned} Note that $\Gamma(\cdot)$ and $\gamma(\beta;\cdot)$ are the gamma function and incomplete gamma function, respectively, defined as follows. $\displaystyle \Gamma(\beta)=\int_0^\infty \ t^{\beta-1} \ e^{- t} \ dt$ $\displaystyle \gamma(\beta; w)=\int_0^w \ t^{\beta-1} \ e^{- t} \ dt$ The CDF $F_X(x)$ can be evaluated numerically using software. When the gamma distribution is raised to a power, the resulting CDF will be defined as a function of $F_X(x)$. “Transformed” Gamma CDFs The CDF of the “transformed” gamma distributions does not have a closed form. Thus the CDFs are to be defined based on an integral or the incomplete gamma function, shown in the preceding section. We still use the two-step approach – first deriving the CDF without the scale parameter and then add it at the end. Based on the two preceding sections, the following shows the CDFs of the three different cases. Step 1. “Transformed” Gamma CDF (without scale parameter) Transformed $Y=X^{1 / \tau}$ $\tau >0$ $F_Y(y)=F_X(y^\tau)$ $\displaystyle F_Y(y)=\int_0^{y^\tau} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; y^\tau)$ $y >0$ Inverse $Y=X^{1 / \tau}$ $\tau=-1$ $F_Y(y)=S_X(y^{-1})=1-F_X(y^{-1})$ $\displaystyle F_Y(y)=1-\int_0^{y^{-1}} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=1-\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; y^{-1})$ $y >0$ Inverse Transformed $Y=X^{-1 / \tau}$ $\tau >0$ $F_Y(y)=S_X(y^{-\tau})=1-F_X(y^{-\tau})$ $\displaystyle F_Y(y)=1-\int_0^{y^{-\tau}} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=1-\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; y^{-\tau})$ $y >0$ Step 2. “Transformed” Gamma CDF (with scale parameter) Transformed $Y=X^{1 / \tau}$ $\tau >0$ $\displaystyle F_Y(y)=\int_0^{(y/\theta)^\tau} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; (y/\theta)^\tau)$ $y >0$ Inverse $Y=X^{1 / \tau}$ $\tau=-1$ $\displaystyle F_Y(y)=1-\int_0^{(y/\theta)^{-1}} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=1-\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; (y/\theta)^{-1})$ $y >0$ Inverse Transformed $Y=X^{-1 / \tau}$ $\tau >0$ $\displaystyle F_Y(y)=1-\int_0^{(y/\theta)^{-\tau}} \frac{1}{\Gamma(\alpha)} \ u^{\alpha-1} \ e^{- u} \ du$ $y >0$ $\displaystyle F_Y(y)=1-\frac{1}{\Gamma(\alpha)} \ \gamma(\alpha; (y/\theta)^{-\tau})$ $y >0$ The transformed gamma distribution and the inverse transformed gamma distribution are three-parameter distributions with $\tau$ being the shape parameter, $\theta$ being the scale parameter and $\tau$ being in the power to which the base gamma distribution is raised. The inverse gamma distribution has two parameters with $\theta$ being the scale parameter and $\alpha$ being shape parameter (the same two parameters in the base gamma distribution). Computation of these CDFs would require the use of software that can evaluate incomplete gamma function. Another Way to Work with “Transformed” Gamma The CDFs derived in the preceding section is a two-step approach – first raising a gamma distribution with scale parameter equals to 1 to a power and then adding a scale parameter. The end result gives CDFs that are a function of the incomplete gamma function. Calculating a CDF would require using a software that has the capability of evaluating incomplete gamma function (or evaluating an equivalent integral). If the software that is used does not have the incomplete gamma function but has gamma CDF (e.g. Excel), then there is another way of generating the “transformed” gamma CDF. Observe that the CDFs in the last section are the results of raising a base gamma distribution with shape parameter $\alpha$ and $\theta^\tau$ (transformed), shape parameter $\alpha$ and $\theta^{-1}$ (inverse) and shape parameter $\alpha$ and $\theta^{-\tau}$ (inverse transformed). Based on this observation, we can evaluate the CDFs and the moments. This section shows how to evaluate CDFs using this approach. The next section shows how to evaluate the moments. Transformed Gamma Distribution Given a transformed gamma random variable $Y$ with parameters $\tau$, $\alpha$ (shape) and $\theta$ (scale), know that $Y=X^{1/\tau}$ where $X$ gas a gamma distribution with parameters $\alpha$ (shape) and $\theta^\tau$ (scale). Then $F_Y(y)=F_X(y^\tau)$ such that $F_X(y^\tau)$ is evaluated using a software with the capability of evaluating gamma CDF (e.g. Excel). Inverse Gamma Distribution Given an inverse gamma random variable $Y$ with parameters $\tau$ and $\theta$ (scale), know that $Y=X^{-1}$ where $X$ gas a gamma distribution with parameters $\alpha$ (shape) and $\theta^{-1}$ (scale). Then $F_Y(y)=S_X(y^{-1})=1-F_X(y^{-1})$ such that $F_X(y^{-1})$ is evaluated using a software with the capability of evaluating gamma CDF (e.g. Excel). Inverse Transformed Gamma Distribution Given an inverse transformed gamma random variable $Y$ with parameters $\tau$, $\alpha$ (shape) and $\theta$ (scale), know that $Y=X^{-1/\tau}$ where $X$ gas a gamma distribution with parameters $\alpha$ (shape) and $\theta^{-\tau}$ (scale). Then $F_Y(y)=S_X(y^{-\tau})=1-F_X(y^{-\tau})$ such that $F_X(y^{-\tau})$ is evaluated using a software with the capability of evaluating gamma CDF (e.g. Excel). Example 1 below uses Excel to compute the transformed gamma CDF. “Transformed” Gamma Moments Following the idea in the preceding section, the moments for the “transformed” gamma moments can be derived using gamma moments with the appropriate parameters (see here for the gamma moments). The following table shows the results. “Transformed Gamma Moments Base Gamma Parameters Moments Transformed $\alpha$ and $\theta^\tau$ $\displaystyle E(Y^k)=E(X^{k/\tau})=\frac{\theta^k \Gamma(\alpha+k/\tau)}{\Gamma(\alpha)}$ $k>- \alpha \ \tau$ Inverse $\alpha$ and $\theta^{-1}$ $\displaystyle E(Y^k)=E(X^{-k})=\frac{\theta^k \Gamma(\alpha-k)}{\Gamma(\alpha)}$ $k<\alpha$ Inverse Transformed $\alpha$ and $\theta^{-\tau}$ $\displaystyle E(Y^k)=E(X^{-k/\tau})=\frac{\theta^k \Gamma(\alpha-k/\tau)}{\Gamma(\alpha)}$ $k<\alpha \ \tau$ Note that the moments for the transformed gamma distribution exists for all positive $k$. The moments for inverse gamma are limited (capped by the shape parameter $\alpha$). The moments for inverse transformed gamma are also limited, this time limited by both parameters. The computation for these moments is usually done by software, except when the argument of the gamma function is a positive integer. “Transformed” Gamma PDFs Once the CDFs are known, the PDFs are derived by taking derivative. The following table gives the three PDFs. “Transformed” Gamma PDFs Transformed $\displaystyle f_Y(y)=\frac{\tau}{\Gamma(\alpha)} \ \biggl( \frac{y}{\theta} \biggr)^{\tau \alpha} \ \frac{1}{y} \ \exp(-(y/\theta)^\tau)$ $y>0$ $\text{ }$ Inverse $\displaystyle f_Y(y)=\frac{1}{\Gamma(\alpha)} \ \biggl( \frac{\theta}{y} \biggr)^{\alpha} \ \frac{1}{y} \ \exp(-\theta/y)$ $y>0$ $\text{ }$ Inverse Transformed $\displaystyle f_Y(y)=\frac{\tau}{\Gamma(\alpha)} \ \biggl( \frac{\theta}{y} \biggr)^{\tau \alpha} \ \frac{1}{y} \ \exp(-(\theta/y)^\tau)$ $y>0$ Each PDF is obtained by taking derivative of the integral for the corresponding CDF. Example The post is concluded with one example demonstrating the calculation for CDF and percentiles using the gamma distribution function in Excel. Example 1 The size of a collision claim from a large pool of auto insurance policies has a transformed gamma distribution with parameters $\tau=2$, $\alpha=2.5$ and $\theta=4$. Determine the following. • The probability that a randomly selected claim is greater than than 6.75. • The probability that a randomly selected claim is between 4.25 and 8.25. • The median size of a claim from this pool of insurance policies. • The mean and variance of a claim from this pool of insurance policies. • The probability that a randomly selected claim is within one standard deviation of the mean claim size. • The probability that a randomly selected claim is within two standard deviations of the mean claim size. Since $\tau=2$, this is a transformed gamma distribution. There are two ways to get a handle on this distribution. One way is to plug in the three parameters $\tau=2$, $\alpha=2.5$ and $\theta=4$ into the transformed gamma CDF (in the table Step 2. “Transformed” Gamma CDF (with scale parameter)). This would require using a software that can evaluate the incomplete gamma function or its equivalent integrals. The other way is to know that this distribution is the result of raising the gamma distribution with shape parameter $\alpha=2.5$ and scale parameter $4^2=16$ to the power of 1/2. We take the latter approach. In Excel, =GAMMA.DIST(A1,B1,C1,TRUE) is the function that produces the gamma CDF $F_X(x)$, assuming that the value $x$ is in cell A1, the shape parameter is in cell B1 and the scale parameter is in cell C1. When the last parameter is TRUE, it gives the CDF. If it is FALSE, it gives the PDF. For example, =GAMMA.DIST(2.5, 2, 1, TRUE) gives the value of 0.712702505. The transformed gamma distribution in question (random variable $Y$) is the result of raising gamma $\alpha=2.5$ and scale parameter 16 (random variable $X$) to 1/2. Thus the CDF $F_Y(y)$ is obtained from evaluating the gamma CDF $F_X(y^2)$, which is =GAMMA.DIST($y^2$, 2.5, 16, TRUE). As a result, the following gives the answers for the first two bullet points. $P(Y>6.75)=1-F_X(6.75)=1-0.663=0.337$ =GAMMA.DIST($6.75^2$, 2.5, 16, TRUE) $P(4.25 \le Y \le 8.25)=F_X(8.25)-F_X(4.25)=0.8696-0.1876=0.6820$ The median or other percentiles of transformed gamma distribution are obtained by a trial and error approach, i.e. by plugging in values of $y$ into the gamma CDF =GAMMA.DIST($y^2$, 2.5, 16, TRUE). After performing the trial and error process, we see that $F_Y(5.9001425)=0.5$ and $F_Y(5.9101425)=0.502020715$. Thus we take the median to be 5.9. So the median size of a claim is around 5.9. Computation of the moments of a gamma distribution requires the evaluation of the gamma function $\Gamma(\cdot)$. Excel does not have an explicit function for gamma function. Since $\Gamma(\cdot)$ is in the gamma PDF, we can derive the gamma function from the gamma PDF in Excel =GAMMA.DIST(1, a, 1, FALSE). The value of this gamma PDF is $e^{-1}/\Gamma(a)$. This the value of $\Gamma(a)$ is $e^{-1}$ divided by the value of this gamma PDF value. For example, the following Excel formulas give $\Gamma(5)$ and $\Gamma(2.5)$. =EXP(-1)/GAMMA.DIST(1, 5, 1, FALSE) = 24 =EXP(-1)/GAMMA.DIST(1, 2.5, 1, FALSE) = 1.329340388 The moments $E(Y^k)$ is $E(X^{k/2})$. Based on the results in the section on “Transformed” gamma moments, the following gives the mean and variance. $\displaystyle E(Y)=E(X^{1/2})=\frac{16^{1/2} \ \Gamma(2.5+1/2)}{\Gamma(2.5)}=6.018022225$ $\displaystyle E(Y^2)=E(X)=2.5 \cdot 16=40$ $Var(Y)=40-6.018022225^2=3.783408505$ $\sigma_Y=3.783408505^{0.5}=1.945098585$ The following gives the probability of a claim within one or two standard deviations of the mean. \displaystyle \begin{aligned} P(\mu-\sigma \displaystyle \begin{aligned} P(\mu-2 \sigma $\text{ }$ $\text{ }$ $\text{ }$ $\copyright$ 2017 – Dan Ma # Transformed exponential distributions The processes of creating distributions from existing ones are an important topic in the study of probability models. Such processes expand the tool kit in the modeling process. Two examples: new distributions can be generated by taking independent sum of old ones or by mixing distributions (the result would be called a mixture). Another way to generate distributions is through raising a distribution to a power, which is the subject of this post. Start with a random variable $X$ (the base distribution). Then raising it to a constant generates a new distribution. In this post, the base distribution is the exponential distribution. The next post discusses transforming the gamma distribution. ______________________________________________________________________________ Raising to a Power Let $X$ be a random variable. Let $\tau$ be a nonzero constant. The new distribution is generated when $X$ is raised to the power of $1 / \tau$. Thus the random variable $Y=X^{1 / \tau}$ is the subject of the discussion in this post. When $\tau >0$, the distribution for $Y=X^{1 / \tau}$ is called transformed. When $\tau=-1$, the distribution for $Y=X^{1 / \tau}$ is called inverse. When $\tau <0$ and $\tau \ne -1$, the distribution for $Y=X^{1 / \tau}$ is called inverse transformed. If the base distribution is exponential, then raising it to $1 / \tau$ would produce a transformed exponential distribution for the case of $\tau >0$, an inverse exponential distribution for the case of $\tau=-1$ and an inverse transformed exponential distribution for the case $\tau <0$ with $\tau \ne -1$. If the base distribution is a gamma distribution, the three new distributions would be transformed gamma distribution, inverse gamma distribution and inverse transformed gamma distribution. For the case of inverse transformed, we make the random variable $Y=X^{-1 / \tau}$ by letting $\tau >0$. The following summarizes the definition. Name of Distribution Parameter $\tau$ Random Variable Transformed $\tau >0$ $Y=X^{1 / \tau}$ Inverse $\tau=-1$ $Y=X^{1 / \tau}$ Inverse Transformed $\tau >0$ $Y=X^{-1 / \tau}$ ______________________________________________________________________________ Transforming Exponential The “transformed” distributions discussed here have two parameters, $\tau$ and $\theta$ ($\tau=1$ for inverse exponential). The parameter $\tau$ is the shape parameter, which comes from the exponent $1 / \tau$. The scale parameter $\theta$ is added after raising the base distribution to a power. Let $X$ be the random variable for the base exponential distribution. The following shows the information on the base exponential distribution. Base Exponential Density Function $f_X(x)=e^{-x} \ \ \ \ \ \ \ \ \ \ x>0$ CDF $F_X(x)=1-e^{-x} \ \ \ \ x>0$ Survival Function $S_X(x)=e^{-x} \ \ \ \ \ \ \ \ \ \ x>0$ Note that the above density function and CDF do not have the scale parameter. Once the base distribution is raised to a power, the scale parameter will be added to the newly created distribution. The following gives the CDF and the density function of the transformed exponential distribution. The density function is obtained by taking the derivative of the CDF. \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{1 / \tau} \le y) \\&=P(X \le y^\tau)\\&=F_X(y^\tau) \\&=1-e^{- y^\tau} \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned} $\displaystyle f_Y(y)=\tau \ y^{\tau-1} \ e^{- y^\tau} \ \ \ \ \ \ \ \ \ y>0$ The following gives the CDF and the density function of the inverse exponential distribution. \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{-1} \le y) \\&=P(X \ge 1/y)\\&=S_X(1/y) \\&=e^{- 1/y} \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned} $\displaystyle f_Y(y)=\frac{1}{y^2} \ e^{- 1/y} \ \ \ \ \ \ \ \ \ y>0$ The following gives the CDF and the density function of the inverse transformed exponential distribution. \displaystyle \begin{aligned} F_Y(y)&=P(Y \le y) \\&=P(X^{- 1 / \tau} \le y) \\&=P(X \ge y^{- \tau})\\&=S_X(y^{- \tau}) \\&=e^{- y^{- \tau}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y>0 \end{aligned} $\displaystyle f_Y(y)= \frac{\tau}{y^{\tau+1}} \ e^{- 1/y^\tau} \ \ \ \ \ \ \ \ \ y>0$ The above derivation does not involve the scale parameter. Now it is added to the results. Transformed Distribution Transformed Exponential CDF $F_Y(y)=1-e^{- (y/\theta)^\tau}$ $y>0$ Survival Function $S_Y(y)=e^{- (y/\theta)^\tau}$ $y>0$ Density Function $f_Y(y)=(\tau / \theta) \ (y/\theta)^{\tau-1} \ e^{- (y/\theta)^\tau}$ $y>0$ Inverse Exponential CDF $F_Y(y)=e^{- \theta/y}$ $y>0$ Survival Function $S_Y(y)=1-e^{- \theta/y}$ $y>0$ Density Function $f_Y(y)=\frac{\theta}{y^2} \ e^{- \theta/y}$ $y>0$ Inverse Transformed Exponential CDF $F_Y(y)=e^{- (\theta/y)^{\tau}}$ $y>0$ Survival Function $S_Y(y)=1-e^{- (\theta/y)^{\tau}}$ $y>0$ Density Function $f_Y(y)=\tau ( \theta / y )^\tau \ (1/y) \ e^{- (\theta/y)^{\tau}}$ $y>0$ The transformed exponential distribution and the inverse transformed distribution have two parameters $\tau$ and $\theta$. The inverse exponential distribution has only one parameter $\theta$. The parameter $\theta$ is the scale parameter. The parameter $\tau$, when there is one, is the shape parameter and it comes from the exponent when the exponential is raised to a power. The above transformation starts with the exponential distribution with mean 1 (without the scale parameter) and the scale parameter $\theta$ is added back in at the end. We can also accomplish the same result by starting with an exponential variable $X$ with mean (scale parameter) $\theta^\tau$. Then raising $X$ to $1/\tau$, -1, and $-1/\tau$ would generate the three distributions described in the above table. In this process, the scale parameter $\theta$ is baked into the base distribution. This makes it easier to obtain the moments of the “transformed” exponential distributions since the moments would be derived from exponential moments. ______________________________________________________________________________ Connection with Weibull Distribution Compare the density function for the transformed exponential distribution with the density of the Weibull distribution discussed here. Note that the two are identical. Thus raising an exponential distribution to $1 / \tau$ where $\tau >0$ produces a Weibull distribution. On the other hand, raising a Weillbull distribution to -1 produces an inverse Weillbull distribution (by definition). Let $F_X(x)=1-e^{- x^\tau}$ be the CDF of the base Weibull distribution where $\tau >0$. Let’s find the CDF of $Y=X^{-1}$. Then add the scale parameter. $\displaystyle F_Y(y)=P(Y \le y)=P(X \ge 1 / y)=e^{- (1/y)^\tau}$ $\displaystyle F_Y(y)=e^{- (\theta/y)^\tau}$ (scale parameter added) Note the the CDF of inverse Weibull distribution is identical to the one for inverse transformed exponential distribution. Thus transformed exponential distribution is identical to a Weibull distribution and inverse transformed exponential distribution is identical to an inverse Weibull distribution. Since Weibull distribution is the same as transformed exponential distribution, the previous post on Weibull distribution can inform us on transformed exponential distribution. For example, assuming that the Weibull distribution (or transformed exponential) is a model for the time until death of a life, varying the shape parameter $\tau$ yields different mortality patterns. The following are two graphics from the proevious post. Figure 1 Figure 2 Figure 1 shows the Weibull density functions for different values of the shape parameter (the scale parameter $\theta$ is fixed at 1). The curve for $\tau=1$ is the exponential density curve. It is clear that the green density curve ($\tau=2$) approaches the x-axis at a faster rate then the other two curves and thus has a lighter tail than the other two density curves. In general, the Weibull (transformed exponential) distribution with shape parameter $\tau >1$ has a lighter tail than the Weibull with shape parameter $0<\tau <1$. Figure 2 shows the failure rates for the Weibull (transformed exponential) distributions with the same three values of $\tau$. Note that the failure rate for $\tau=0.5$ (blue) decreases over time and the failure rate for $\tau=2$ increases over time. The failure rate for $\tau=1$ is constant since it is the exponential distribution. What is being displayed in Figure 2 describes a general pattern. When the shape parameter is $0<\tau<0.5$, the failure rate decreases as time increases and the Weibull (transformed exponential) distribution is a model for infant mortality, or early-life failures. Hence these Weibull distributions have a thicker tail as shown in Figure 1. When the shape parameter is $\tau >1$, the failure rate increases as time increases and the Weibull (transformed exponential) distribution is a model for wear-out failures. As times go by, the lives are fatigued and “die off.” Hence these Weibull distributions have a lighter tail as shown in Figure 1. When $\tau=1$, the resulting Weibull (transformed exponential) distribution is exponential. The failure rate is constant and it is a model for random failures (failures that are independent of age). Thus the transformed exponential family has a great deal of flexibility for modeling the failures of objects (machines, devices). ______________________________________________________________________________ Moments and Other Distributional Quantities The moments for the three “transformed” exponential distributions are based on the gamma function. The two inverse distributions have limited moments. Since the transformed exponential distribution is identical to Weibull, its moments are identical to that of the Weibull distribution. The moments of the “transformed” exponential distributions are $E(Y)=E(X^{1 / \tau})$ where $X$ has an exponential distribution with mean (scale parameter) $\theta^\tau$. See here for the information on exponential moments. The following shows the moments of the “transformed” exponential distributions. Name of Distribution Moment Transformed Exponential $E(Y^k)=\theta^k \Gamma(1+k/\tau)$ $k >- \tau$ Inverse Exponential $E(Y^k)=\theta^k \Gamma(1-k)$ $k <1$ Inverse Transformed Exponential $E(Y^k)=\theta^k \Gamma(1-k/\tau)$ $k <\tau$ The function $\Gamma(\cdot)$ is the Gamma function. The transformed exponential moment $E(Y^k)$ exists for all $k >- \tau$. The moments are limited for the other two distributions. The first moment $E(Y)$ does not exist for the inverse exponential distribution. The inverse transformed exponential moment $E(Y^k)$ exist only for $k<\tau$. Thus the inverse transformed exponential mean and variance exist only if the shape parameter $\tau$ is larger than 2. The distributional quantities that are based on moments can be calculated (e.g. variance, skewness and kurtosis) when the moments are available. For all three "transformed" exponential distributions, percentiles are easily computed since the CDFs contain only one instance of the unknown $y$. The following gives the mode of the three distributions. Name of Distribution Mode Transformed Exponential $\displaystyle \theta \biggl(\frac{\tau-1}{\tau} \biggr)^{1/\tau}$ for $\tau >1$, else 0 Inverse Exponential $\theta / 2$ Inverse Transformed Exponential $\displaystyle \theta \biggl(\frac{\tau}{\tau+1} \biggr)^{1/\tau}$ $\text{ }$ $\text{ }$ $\text{ }$ $\copyright$ 2017 – Dan Ma	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 436, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9920789003372192, "perplexity": 3232.6910420941635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330800.17/warc/CC-MAIN-20190825194252-20190825220252-00422.warc.gz"}
https://en.wikipedia.org/wiki/Four-potential	Electromagnetic four-potential (Redirected from Four-potential) An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.[1] As measured in a given frame of reference, and for a given gauge, the first component of the electromagnetic four-potential is conventionally taken to be the electric scalar potential, and the other three components make up the magnetic vector potential. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant. Like other potentials, many different electromagnetic four-potentials correspond to the same electromagnetic field, depending upon the choice of gauge. This article uses tensor index notation and the Minkowski metric sign convention (+ − − −). See also covariance and contravariance of vectors and raising and lowering indices for more details on notation. Formulae are given in SI units and Gaussian-cgs units. Definition The electromagnetic four-potential can be defined as:[2] SI units Gaussian units ${\displaystyle A^{\alpha }=\left(\phi /c,\mathbf {A} \right)\,\!}$ ${\displaystyle A^{\alpha }=(\phi ,\mathbf {A} )}$ in which ϕ is the electric potential, and A is the magnetic potential (a vector potential). The units of Aα are V·s·m−1 in SI, and Mx·cm−1 in Gaussian-cgs. The electric and magnetic fields associated with these four-potentials are:[3] SI units Gaussian units ${\displaystyle \mathbf {E} =-\mathbf {\nabla } \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$ ${\displaystyle \mathbf {E} =-\mathbf {\nabla } \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}}$ ${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}$ ${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} .}$ In special relativity, the electric and magnetic fields transform under Lorentz transformations. This can be written in the form of a tensor-the electromagnetic tensor. This is written in terms of the electromagnetic four-potential and the four-gradient as: ${\displaystyle F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$ This essentially defines the four-potential in terms of physically observable quantities, as well as reducing to the above definition. In the Lorenz gauge Often, the Lorenz gauge condition ${\displaystyle \partial _{\alpha }A^{\alpha }=0}$ in an inertial frame of reference is employed to simplify Maxwell's equations as:[4] SI units Gaussian units ${\displaystyle \Box A^{\alpha }=\mu _{0}J^{\alpha }}$ ${\displaystyle \Box A^{\alpha }={\frac {4\pi }{c}}J^{\alpha }}$ where Jα are the components of the four-current, and ${\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}}$ is the d'Alembertian operator. In terms of the scalar and vector potentials, this last equation becomes: SI units Gaussian units ${\displaystyle \Box \phi ={\frac {\rho }{\epsilon _{0}}}}$ ${\displaystyle \Box \phi =4\pi \rho }$ ${\displaystyle \Box \mathbf {A} =\mu _{0}\mathbf {j} }$ ${\displaystyle \Box \mathbf {A} ={\frac {4\pi }{c}}\mathbf {j} }$ For a given charge and current distribution, ρ(r, t) and j(r, t), the solutions to these equations in SI units are:[5] ${\displaystyle \phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \mathrm {d} ^{3}x^{\prime }{\frac {\rho (\mathbf {r} ^{\prime },t_{r})}{\left\|\mathbf {r} -\mathbf {r} ^{\prime }\right\|}}}$ ${\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \mathrm {d} ^{3}x^{\prime }{\frac {\mathbf {j} (\mathbf {r} ^{\prime },t_{r})}{\left\|\mathbf {r} -\mathbf {r} ^{\prime }\right\|}},}$ where ${\displaystyle t_{r}=t-{\frac {\left\|\mathbf {r} -\mathbf {r} '\right\|}{c}}}$ is the retarded time. This is sometimes also expressed with ${\displaystyle \rho (\mathbf {r} ',t_{r})=[\rho (\mathbf {r} ',t)],}$ where the square brackets are meant to indicate that the time should be evaluated at the retarded time. Of course, since the above equations are simply the solution to an inhomogeneous differential equation, any solution to the homogeneous equation can be added to these to satisfy the boundary conditions. These homogeneous solutions in general represent waves propagating from sources outside the boundary. When the integrals above are evaluated for typical cases, e.g. of an oscillating current (or charge), they are found to give both a magnetic field component varying according to r−2 (the induction field) and a component decreasing as r−1 (the radiation field).[clarification needed]	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 19, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9742399454116821, "perplexity": 523.9995858270893}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934807089.35/warc/CC-MAIN-20171124051000-20171124071000-00703.warc.gz"}
https://pos.sissa.it/363/033/	Volume 363 - 37th International Symposium on Lattice Field Theory (LATTICE2019) - Main session Tensor network approach to real-time path integral S. Takeda Full text: pdf Pre-published on: December 05, 2019 Published on: August 27, 2020 Abstract We present a tensor network representation of the path integral for the one-component real scalar field theory in 1+1 dimensional Minkowski space-time. It is numerically verified by comparing with the exact result in the non-interacting case. DOI: https://doi.org/10.22323/1.363.0033 How to cite Metadata are provided both in "article" format (very similar to INSPIRE) as this helps creating very compact bibliographies which can be beneficial to authors and readers, and in "proceeding" format which is more detailed and complete. Open Access	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8535493612289429, "perplexity": 2680.5784695934826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141747774.97/warc/CC-MAIN-20201205104937-20201205134937-00293.warc.gz"}
https://books.physics.oregonstate.edu/GMM/eigenvector.html	Section5.3Finding Eigenvectors Having found the eigenvalues of the example matrix $$A=\begin{pmatrix}1\amp 2\\9\amp 4 \end{pmatrix}$$ in the last section to be $$7$$ and $$-2\text{,}$$ we can now ask what the corresponding eigenvectors are. We must therefore solve the equation $$A \left\|v\right> = \lambda \left\|v\right>\tag{5.3.1}$$ in the two cases $$\lambda=7$$ and $$\lambda=-2\text{.}$$ In the first case, we have $$\begin{pmatrix}1\amp 2\\9\amp 4 \end{pmatrix} \begin{pmatrix}x\\ y \end{pmatrix} = 7\begin{pmatrix}x\\ y \end{pmatrix}\tag{5.3.2}$$ or equivalently $$\begin{pmatrix}x+2y\\ 9x+4y \end{pmatrix} = \begin{pmatrix}7x\\ 7y \end{pmatrix}\text{.}\tag{5.3.3}$$ Thus, we must solve this system of two equations. But we quickly discover that these equations are reduntant, since the first implies $$2y=6x\text{,}$$ while the second implies $$3y=9x\text{,}$$ which is the same condition. This is a good thing! (Why?) We conclude that any vector $$\begin{pmatrix}x\\ y \end{pmatrix}$$ with $$y=3x$$ is an eigenvector of $$A$$ with eigenvalue $$7\text{.}$$ To check an explicit example, choose any value for $$x\text{,}$$ such as $$x=1\text{,}$$ yielding the vector $$v_7 = \begin{pmatrix}1\\ 3 \end{pmatrix}\tag{5.3.4}$$ and check by explicit computation that $$Av_7=7v_7\text{,}$$ as expected. Turning to the case $$\lambda=-2\text{,}$$ a similar construction yields $$\begin{pmatrix}1\amp 2\\9\amp 4 \end{pmatrix} \begin{pmatrix}x\\ y \end{pmatrix} = -2\begin{pmatrix}x\\ y \end{pmatrix}\tag{5.3.5}$$ or equivalently $$\begin{pmatrix}x+2y\\ 9x+4y \end{pmatrix} = \begin{pmatrix}-2x\\ -2y \end{pmatrix}\text{,}\tag{5.3.6}$$ and the first equation now yields $$2y=-3x\text{,}$$ while the second yields $$6y=-9x\text{.}$$ Again, these two equations are redundant; the eigenvectors with eigenvalue $$-2$$ satisfy $$y=-\frac32\text{.}$$ An explicit example is $$v_{-2} = \begin{pmatrix}2\\ -3 \end{pmatrix}\tag{5.3.7}$$ and you should again check by explicit computation that $$Av_{-2}=-2v_{-2}\text{.}$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 7, "x-ck12": 0, "texerror": 0, "math_score": 0.9874565601348877, "perplexity": 183.61900566248235}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943746.73/warc/CC-MAIN-20230321193811-20230321223811-00295.warc.gz"}
http://heasarc.gsfc.nasa.gov/docs/asca/ao6/abs/61193.html	## SUMMARY OF PROPOSAL 61193 #### DR. YASUO TANAKA [email protected] ISAS KANAGAWA JAPAN 229 SAGAMIHARA 3-1-1, YOSHINODAI Tel: 0427-51-3911 Fax: 0427-59-4253 Medium: 2.5GB8MM Subject: CLUSTERS OF GALAXIES AND SUPERCLUSTERS #### ABSTRACT Heavy elements in the hot intracluster medium (ICM) of distant clusters are an important tool to study cosmological evolution as well as the early phase of the member galaxies, since these heavy elements maintain the history of the metal enrichment processes. We observed earlier the distant (redshift \$z\$=0.402) cluster A851 and obtained an iron abundance upper limit of \$\sim\$0.2 solar, substantially lower than that for more recent clusters. However, the statistical uncertainty is still too large to conclude the low abundance. Because of the scarcity and crucial importance of the data of such high-redshifted clusters, we propose an additional observation of this cluster. #### TARGETS 1)A851 2 145.7500 46.9917 40 ksN N P 0.02000 1	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9486396908760071, "perplexity": 3473.0929759270225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386165158218/warc/CC-MAIN-20131204135238-00097-ip-10-33-133-15.ec2.internal.warc.gz"}
https://lpsa.swarthmore.edu/Analogs/RelElecMech1.html	# Electrical to Mechanical 1 with Ungrounded Capacitors The diagram shown below is the same as the previous example for Electrical to Mechanical 1, but two of the branches have been switched (since the branches are in parallel, this is an equivalent circuit). The currents are now chosen such that two loop currents flow through the inductor. We can write the node equations. We can rewrite the equations using analogous quantities. This set of equations presents a problem because the acceleration of the mass is no relative to a fixed reference, but is relative. We can fix this by defining a new set of positions. Now we can rewrite the equations of motion using these new variables. After a little thought, and the realization that: the analogous system can be built. The dimension z3 is the extension of the spring. The position left side of the spring and friction element (B2) is only known relative to z2, the position of the mass. Note that this is the same system (with different coordinates) that was created originally when the circuit parameters were defined such that only one current passed through the inductor. References	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9488256573677063, "perplexity": 557.642117274362}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00634.warc.gz"}
https://www.science.gov/topicpages/l/low-mass+eclipsing+binaries.html	#### Sample records for low-mass eclipsing binaries 1. Low-mass eclipsing binaries in the WFCAM Transit Survey Cruz, Patricia; Diaz, Marcos; Barrado, David; Birkby, Jayne 2017-10-01 The characterization of short-period detached low-mass binaries, by the determination of their physical and orbital parameters, reveal the most precise basic parameters of low-mass stars. Particularly, when photometric and spectroscopic data of eclipsing binaries (EBs) are combined. Recently, 16 new low-mass EBs were discovered by the WFCAM Transit Survey (WTS), however, only three of them were fully characterized. Therefore, new spectroscopic data were already acquired with the objective to characterize five new detached low-mass EBs discovered in the WTS, with short periods between 0.59 and 1.72 days. A preliminary analysis of the radial velocity and light curves was performed, where we have derived orbital separations of 2.88 to 6.69 R ⊙, and considering both components, we have found stellar radii ranging from 0.40 to 0.80 R ⊙, and masses between 0.24 and 0.71 M ⊙. In addition to the determination of the orbital parameters of these systems, the relation between mass, radius and orbital period of these objects can be investigated in order to study the mass-radius relationship and the radius anomaly in the low main-sequence. 2. Five New Low-Mass Eclipsing Binary Systems Coughlin, Jeffrey L.; López-Morales, M.; Shaw, J. S. 2006-12-01 We present the discovery of five new low-mass eclipsing binaries with masses between 0.54 and 0.95 M⊙, their photometric light curves, and preliminary models. This is part of a continuing campaign to increase the available data on these interesting systems. Once radial-velocity curves are completed, physical parameters will be determined with an error of less than 2-3%, thus allowing for a rigorous examination of stellar models in the lower-main sequence. Our initial analysis seems to support the current findings that low-mass stars have greater radii than models predict, most likely due to the presence of strong magnetic fields. This work is funded by a partnership between the National Science Foundation (NSF AST-0552798) Research Experiences for Undergraduates (REU) and the Department of Defense (DoD) ASSURE (Awards to Stimulate and Support Undergraduate Research Experiences) programs. 3. Flare activity on low-mass eclipsing binary GJ 3236* Šmelcer, L.; Wolf, M.; Kučáková, H.; Bílek, F.; Dubovský, P.; Hoňková, K.; Vraštil, J. 2017-04-01 We report the discovery of optical flares on the very low-mass red-dwarf eclipsing binary GJ 3236 and the results of our 2014-2016 photometric campaign. In total, this binary was monitored photometrically in all filters for about 900 h, which has revealed a flare rate of about 0.06 flares per hour. The amplitude of its flares is the largest among those detected in the V band (∼1.3 mag), R band (∼0.8 mag), I band (∼0.2 mag) and clear band (∼0.5 mag). The light curves of GJ 3236 were analysed and the statistics of detected flare events are presented. The energy released during individual flares was calculated as up to 2.4 × 1027 J and compared with other known active stars. The cumulative distribution of flare energies appears to follow a broken power law. The flare activity of this binary also plays an important role in the precise determination of its physical parameters and evolutionary status. 4. LOW-MASS ECLIPSING BINARIES IN THE INITIAL KEPLER DATA RELEASE SciTech Connect Coughlin, J. L.; Harrison, T. E.; Ule, N.; Lopez-Morales, M.; Hoffman, D. I. 2011-03-15 We identify 231 objects in the newly released Cycle 0 data set from the Kepler Mission as double-eclipse, detached eclipsing binary systems with T{sub eff} < 5500 K and orbital periods shorter than {approx}32 days. We model each light curve using the JKTEBOP code with a genetic algorithm to obtain precise values for each system. We identify 95 new systems with both components below 1.0 M{sub sun} and eclipses of at least 0.1 mag, suitable for ground-based follow-up. Of these, 14 have periods less than 1.0 day, 52 have periods between 1.0 and 10.0 days, and 29 have periods greater than 10.0 days. This new sample of main-sequence, low-mass, double-eclipse, detached eclipsing binary candidates more than doubles the number of previously known systems and extends the sample into the completely heretofore unexplored P > 10.0 day period regime. We find preliminary evidence from these systems that the radii of low-mass stars in binary systems decrease with period. This supports the theory that binary spin-up is the primary cause of inflated radii in low-mass binary systems, although a full analysis of each system with radial-velocity and multi-color light curves is needed to fully explore this hypothesis. Also, we present seven new transiting planet candidates that do not appear among the list of 706 candidates recently released by the Kepler team, or in the Kepler False Positive Catalog, along with several other new and interesting systems. We also present novel techniques for the identification, period analysis, and modeling of eclipsing binaries. 5. Testing low-mass stellar models with M-dwarf eclipsing binaries from SDSS Stripe 82 Bhatti, Waqas A. Large astronomical surveys such as the Sloan Digital Sky Survey (SDSS) have revolutionized ensemble studies of stellar populations in the Galaxy. Modern and upcoming synoptic surveys extend this concept to the time-domain, by covering large areas of the sky to a faint magnitude limit, and at observing cadences optimized for a large range in variability. In this thesis, we explore methods of efficiently analyzing a large synoptic survey dataset and its application to stellar astronomy, specifically focusing on the discovery and characterization of low-mass star eclipsing binaries. Eclipsing binaries (EBs) provide direct measurements of the absolute masses and radii of the component stars. Recent observations of EBs composed of low-mass stars (< 0.7 M⊙ ) indicate that the measured radii of the component stars are systematically 10-15% larger than those predicted by stellar models. Tidally induced magnetic fields that arise in these close binaries may be responsible for this discrepancy. The small number of fully characterized low-mass EBs, however, makes any hypothesis for this discrepancy difficult to verify. These objects are difficult to detect because of the intrinsic faintness of low-mass stars, in addition to the already low probability of favorable orbital alignment for eclipse observation. Fortunately, both of these problems can be overcome by a large-area and deep time-domain survey. We describe a search for periodic variables carried out using multi-band timeseries photometry from SDSS Stripe 82 focused on identifying a large sample of EBs to help resolve this issue. We outline the construction of our light-curve catalog and the methodology for extracting variable point sources. We discuss the classification of the ˜1100 periodic variables found in these data, and the subsequent discovery of ˜211 EB candidates with securely determined periods. For ˜90 EBs with suitable light-curves, we fit binary models and estimate parameters for the binary components 6. Testing Low-Mass Stellar Models: Three New Detached Eclipsing Binaries below 0.75Msun López-Morales, M.; Shaw, J. S. 2007-06-01 Full tests of stellar models below 1Msun have been hindered until now by the scarce number of precise measurements of the stars' most fundamental parameters: their masses and radii. With current observational techniques, the required precision to distinguish between different models (errors < 2-3 %) can only be achieved using detached eclipsing binaries where 1) both stars are similar in mass, i.e. q = M1/M2 ˜ 1.0, and 2) each star is a main sequence object below 1Msun. Until 2003 only three such binaries had been found and analyzed in detail. Two new systems were published in 2005 (Creevey et al.; López-Morales & Ribas), almost doubling the previous number of data points. Here we present preliminary results for 3 new low-mass detached eclipsing binaries. These are the first studied systems from our sample of 41 new binaries (Shaw & López-Morales, this proceedings). We also provide an updated comparison between the Mass-Radius and the Mass-Teff relations predicted by the models and the observational data from detached eclipsing binaries. 7. A New γ-Ray Loud, Eclipsing Low-mass X-Ray Binary Strader, Jay; Li, Kwan-Lok; Chomiuk, Laura; Heinke, Craig O.; Udalski, Andrzej; Peacock, Mark; Shishkovsky, Laura; Tremou, Evangelia 2016-11-01 We report the discovery of an eclipsing low-mass X-ray binary at the center of the 3FGL error ellipse of the unassociated Fermi/Large Area Telescope γ-ray source 3FGL J0427.9-6704. Photometry from OGLE and the SMARTS 1.3 m telescope and spectroscopy from the SOAR telescope have allowed us to classify the system as an eclipsing low-mass X-ray binary (P = 8.8 hr) with a main-sequence donor and a neutron-star accretor. Broad double-peaked H and He emission lines suggest the ongoing presence of an accretion disk. Remarkably, the system shows separate sets of absorption lines associated with the accretion disk and the secondary, and we use their radial velocities to find evidence for a massive (˜1.8-1.9 M ⊙) neutron-star primary. In addition to a total X-ray eclipse with a duration of ˜2200 s observed with NuSTAR, the X-ray light curve also shows properties similar to those observed among known transitional millisecond pulsars: short-term variability, a hard power-law spectrum ({{Γ }}˜ 1.7), and a comparable 0.5-10 keV luminosity (˜ 2.4× {10}33 erg s-1). We find tentative evidence for a partial (˜ 60 % ) γ-ray eclipse at the same phase as the X-ray eclipse, suggesting the γ-ray emission may not be confined to the immediate region of the compact object. The favorable inclination of this binary is promising for future efforts to determine the origin of γ-rays among accreting neutron stars. 8. Substellar companions in low-mass eclipsing binaries. NSVS 01286630, NSVS 02502726, and NSVS 07453183 Wolf, M.; Zasche, P.; Kučáková, H.; Vraštil, J.; Hornoch, K.; Šmelcer, L.; Bílek, F.; Pilarčík, L.; Chrastina, M. 2016-03-01 Aims: As part of our long-term observational project we aim to measure very precise mid-eclipse times for low-mass eclipsing binaries, which are needed to accurately determine their period changes. Over two hundred new precise times of minimum light recorded with CCD were obtained for three eclipsing binaries with short orbital periods: NSVS 01286630 (P = 0.38°), NSVS 02502726 (0.56°), and NSVS 07453183 (0.37°). Methods: O-C diagrams of studied stars were analysed using all reliable timings, and new parameters of the light-time effect were obtained. Results: We derived for the first time or improved the very short orbital periods of third bodies of between one and seven years for all measured low-mass systems. We calculated that the lowest masses of the third components are between those of red and brown dwarfs. The multiplicity of these systems also plays an important role in the precise determination of their physical parameters. This research is part of an ongoing collaboration between professional astronomers and the Czech Astronomical Society, Variable Star and Exoplanet Section. 9. Identification and Follow-Up Observations of Low-Mass Eclipsing Binaries from Kepler Coughlin, Jeffrey; Lopez-Morales, M.; Marzoa, R. I.; Harrison, T.; Ule, N.; Hoffman, D. 2011-01-01 An outstanding problem in Astronomy for the past 15+ years has been that the radii of low-mass, (M < 1.0 M⊙), main-sequence stars in eclipsing binary systems are consistently about 15% larger than predicted by theoretical models. The main cause is hypothesized to be rapid rotation due to binary spin-up, as all but one of the currently known systems have P < 3.0 days. We present 100+ new low-mass, main-sequence, double-lined eclipsing binaries (LMMS DDEBs) from both our Kepler Guest Observer Program, as well as the initial Kepler public data release. We identify over 25 new systems with P > 10 days, extending the sample of LMMS DDEBs into this completely heretofore unexplored period range. We present the initial results of our intensive observing campaign to obtain ground-based radial-velocity and multi-color photometry follow-up of these long-period systems, in order to determine precise masses and radii. We thank all the hard-working members of the Kepler team, and acknowledge support from the Kepler Guest Observer Program, the New Mexico Space Grant Consortium, and a NSF Graduate Research Fellowship. 10. 2MASS J05162881+2607387: A New Low-mass Double-lined Eclipsing Binary Bayless, Amanda J.; Orosz, Jerome A. 2006-11-01 We show that the star known as 2MASS J05162881+2607387 (hereafter J0516) is a double-lined eclipsing binary with nearly identical low-mass components. The spectroscopic elements derived from 18 spectra obtained with the High Resolution Spectrograph on the Hobby-Eberly Telescope during the fall of 2005 are K1=88.45+/-0.48 and K2=90.43+/-0.60 km s-1, resulting in a mass ratio of q=K1/K2=0.978+/-0.018 and minimum masses of M1sin3i=0.775+/-0.016 Msolar and M2sin3i=0.759+/-0.012 Msolar, respectively. We have extensive differential photometry of J0516 obtained over several nights between 2004 January and March (epoch 1) and between 2004 October and 2005 January plus 2006 January (epoch 2) using the 1 m telescope at the Mount Laguna Observatory. The source was roughly 0.1 mag brighter in all three bandpasses during epoch 1 when compared to epoch 2. Also, phased light curves from epoch 1 show considerable out-of-eclipse variability, presumably due to bright spots on one or both stars. In contrast, the phased light curves from epoch 2 show little out-of-eclipse variability. The light curves from epoch 2 and the radial velocity curves were analyzed using our ELC code with updated model atmospheres for low-mass stars. We find the following: M1=0.787+/-0.012 Msolar, R1=0.788+/-0.015 Rsolar, M2=0.770+/-0.009 Msolar, and R2=0.817+/-0.010 Rsolar. The stars in J0516 have radii that are significantly larger than model predictions for their masses, similar to what is seen in a handful of other well-studied low-mass double-lined eclipsing binaries. We compiled all recent mass and radius determinations from low-mass binaries and determine an empirical mass-radius relation of the form R(Rsolar)=0.0324+0.9343M(Msolar)+0.0374M2(Msolar). Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg 11. GJ 3236: A NEW BRIGHT, VERY LOW MASS ECLIPSING BINARY SYSTEM DISCOVERED BY THE MEARTH OBSERVATORY SciTech Connect Irwin, Jonathan; Charbonneau, David; Berta, Zachory K.; Quinn, Samuel N.; Latham, David W.; Torres, Guillermo; Blake, Cullen H.; Burke, Christopher J.; Esquerdo, Gilbert A.; Fueresz, Gabor; Mink, Douglas J.; Nutzman, Philip; Szentgyorgyi, Andrew H.; Calkins, Michael L.; Falco, Emilio E.; Bloom, Joshua S.; Starr, Dan L. 2009-08-20 We report the detection of eclipses in GJ 3236, a bright (I = 11.6), very low mass binary system with an orbital period of 0.77 days. Analysis of light and radial velocity curves of the system yielded component masses of 0.38 {+-} 0.02 M{sub sun} and 0.28 {+-} 0.02 M{sub sun}. The central values for the stellar radii are larger than the theoretical models predict for these masses, in agreement with the results for existing eclipsing binaries, although the present 5% observational uncertainties limit the significance of the larger radii to approximately 1{sigma}. Degeneracies in the light curve models resulting from the unknown configuration of surface spots on the components of GJ 3236 currently dominate the uncertainties in the radii, and could be reduced by obtaining precise, multiband photometry covering the full orbital period. The system appears to be tidally synchronized and shows signs of high activity levels as expected for such a short orbital period, evidenced by strong H{alpha} emission lines in the spectra of both components. These observations probe an important region of mass-radius parameter space around the predicted transition to fully convective stellar interiors, where there are a limited number of precise measurements available in the literature. 12. PHYSICAL PROPERTIES OF THE LOW-MASS ECLIPSING BINARY NSVS 02502726 SciTech Connect Lee, Jae Woo; Youn, Jae-Hyuck; Kim, Seung-Lee; Lee, Chung-Uk E-mail: [email protected] E-mail: [email protected] 2013-01-01 NSVS 02502726 has been known as a double-lined, detached eclipsing binary that consists of two low-mass stars. We obtained BVRI photometric follow-up observations in 2009 and 2011 to measure improved physical properties of the binary star. Each set of light curves, including the 2008 data given by Cakirli et al., was simultaneously analyzed with the previously published radial velocity curves using the Wilson-Devinney binary code. The conspicuous seasonal light variations of the system are satisfactorily modeled by a two-spot model with one starspot on each component and by changes of the spot parameters with time. Based on 23 eclipse timings calculated from the synthetic model and one ephemeris epoch, an orbital period study of NSVS 02502726 reveals that the period has experienced a continuous decrease of -5.9 Multiplication-Sign 10{sup -7} day yr{sup -1} or a sinusoidal variation with a period and semi-amplitude of 2.51 yr and 0.0011 days, respectively. The timing variations could be interpreted as either the light-travel-time effect due to the presence of an unseen third body, or as the combination of this effect and angular momentum loss via magnetic stellar wind braking. Individual masses and radii of both components are determined to be M{sub 1} = 0.689 {+-} 0.016 M{sub Sun }, M{sub 2} = 0.341 {+-} 0.009 M{sub Sun }, R{sub 1} = 0.707 {+-} 0.007 R{sub Sun }, and R{sub 2} = 0.657 {+-} 0.008 R{sub Sun }. The results are very different from those of Cakirli et al. with the primary's radius (0.674 {+-} 0.006 R{sub Sun }) smaller the secondary's (0.763 {+-} 0.007 R{sub Sun }). We compared the physical parameters presented in this paper with current low-mass stellar models and found that the measured values of the primary star are best fitted to a 79 Myr isochrone. The primary is in good agreement with the empirical mass-radius relation from low-mass binaries, but the secondary is oversized by about 85%. 13. DETECTABILITY OF TRANSITING JUPITERS AND LOW-MASS ECLIPSING BINARIES IN SPARSELY SAMPLED PAN-STARRS-1 SURVEY DATA SciTech Connect Dupuy, Trent J.; Liu, Michael C. 2009-10-20 We present detailed simulations of the Pan-STARRS-1 (PS1) multi-epoch, multiband 3pi Survey in order to assess its potential yield of transiting planets and eclipsing binaries. This survey differs from dedicated transit surveys in that it will cover the entire northern sky but provide only sparsely sampled light curves. Since most eclipses would be detected at only a single epoch, the 3pi Survey will be most sensitive to deep eclipses (approx>0.10 mag) caused by Jupiters transiting M dwarfs and eclipsing stellar/substellar binaries. The survey will measure parallaxes for the approx4 x 10{sup 5} stars within 100 pc, which will enable a volume-limited eclipse search, reducing the number of astrophysical false positives compared with previous magnitude-limited searches. Using the best available empirical data, we constructed a model of the extended solar neighborhood that includes stars, brown dwarfs, and a realistic binary population. We computed the yield of deeply eclipsing systems using both a semianalytic and a full Monte Carlo approach. We examined statistical tests for detecting single-epoch eclipses in sparsely sampled data and assessed their vulnerability to false positives due to stellar variability. Assuming a short-period planet frequency of 0.5% for M dwarfs, our simulations predict that about a dozen transiting Jupiters around low-mass stars (M {sub } < 0.3 M {sub sun}) within 100 pc are potentially detectable in the PS1 3pi Survey, along with approx300 low-mass eclipsing binaries (both component masses <0.5 M {sub sun}), including approx10 eclipsing field brown dwarfs. Extensive follow-up observations would be required to characterize these candidate eclipsing systems, thereby enabling comprehensive tests of structural models and novel insights into the planetary architecture of low-mass stars. 14. A SuperWASP Benchmark Eclipsing Binary with a Very Low-Mass Secondary in the Brown Dwarf Desert Gomez Maqueo Chew, Yilen; Garcia-Melendo, Enrique; Hebb, Leslie; Faedi, Francesca; Lopez-Morales, Mercedes; Pollacco, Don 2012-08-01 We will obtain eclipse light curves of a newly discovered eclipsing binary composed of a Sun-like primary with a secondary companion which can be either a very low mass M-dwarf (less than ~0.15 Msun) or a brown dwarf. The objects orbit each other with a period of ~14.3 days in an eccentric orbit, which as been confirmed with a high- precision radial velocity curve for the system. Therefore, these eclipse light curves will allow us to constrain the radii of the eclipsing components and orbital inclination of the system. Furthermore, the depth of the secondary eclipse which can only be observed in the near-infrared, directly constrains the temperature ratio between the components. In combination with the the masses derived from the radial velocity curve, our light curve analysis will unveil the true nature of the secondary. Whether it is a very-low mass star or a brown dwarf, direct measurements of the fundamental properties (masses, radii and temperatures) of such objects are very scarce and will provide key tests to current evolutionary models. Thus, we request two nights with FLAMINGOS at the KPNO 2.1m to observe a complete secondary eclipse of the system at near-infrared wavelengths in order to fully characterize the very low-mass component of the system. 15. Detection of a very low mass star in an eclipsing binary system Chaturvedi, Priyanka; Chakraborty, Abhijit; Anandarao, B. G.; Roy, Arpita; Mahadevan, Suvrath 2016-10-01 We report the detection of a very low mass star (VLMS) companion to the primary star 1SWASP J234318.41+295556.5A (J2343+29A), using radial velocity (RV) measurements from the PARAS (PRL Advanced Radial-velocity Abu-sky Search) high-resolution echelle spectrograph. The periodicity of the single-lined eclipsing binary (SB1) system, as determined from 20 sets of RV observations from PARAS and 6 supporting sets of observations from SOPHIE data, is found to be 16.953 d as against the 4.24 d period reported from SuperWASP photometry. It is likely that inadequate phase coverage of the transit with SuperWASP photometry led to the incorrect determination of the period for this system. We derive the spectral properties of the primary star from the observed stellar spectra: Teff = 5125 ± 67 K, [Fe/H] = 0.1 ± 0.14 and logg = 4.6 ± 0.14, indicating a K1V primary. Applying the Torres relation to the derived stellar parameters, we estimate a primary mass 0.864_{-0.098}^{+0.097} M⊙ and a radius of 0.854_{-0.060}^{+0.050} R⊙. We combine RV data with SuperWASP photometry to estimate the mass of the secondary, MB = 0.098 ± 0.007 M⊙, and its radius, RB = 0.127 ± 0.007 R⊙, with an accuracy of ˜7 per cent. Although the observed radius is found to be consistent with the Baraffe's theoretical models, the uncertainties on the mass and radius of the secondary reported here are model dependent and should be used with discretion. Here, we establish this system as a potential benchmark for the study of VLMS objects, worthy of both photometric follow-up and the investment of time on high-resolution spectrographs paired with large-aperture telescopes. 16. Photometric monitoring of open clusters: Low-mass eclipsing binary stars and the stellar mass-luminosity-radius relation Hebb, Leslie 2006-06-01 This thesis describes a photometric monitoring survey of Galactic star clusters designed to detect low-mass eclipsing binary star systems through variations in their relative lightcurves. The aim is to use cluster eclipsing binaries to measure the masses and radii of M-dwarf stars with ages and metallicities known from studies of brighter cluster stars. This information will provide an improved calibration of the mass-luminosity-radius relation for low-mass stars, be used to test stellar structure and evolution models, and help quantify the contribution of low-mass stars to the global mass census in the Galaxy. The survey is designed to detect eclipse events in stars of ~0.3 M_sun and consists of 600 Gbytes of raw imaging data on six open clusters with a range of ages (~ 0.15 - 4 Gyr) and metallicites (~ -0.2 - 0.0 dex). The clusters NGC 1647 and M 35 contain excellent candidate systems showing eclipse like variations in brightness and photometry consistent with cluster membership. The analysis of these clusters and the eclipsing M-dwarf stars detected in them are presented. Analysis of the candidate system in NGC 1647 confirms the object as a newly discovered M-dwarf eclipsing binary in the cluster with compenent masses of M 1 = 0.47 ± 0.05[Special characters omitted.] and M 2 = 0.19 ± 0.02[Special characters omitted.] . The small mass ratio ( M 2 / M 1 ) and low secondary mass of this object provide an unprecedented opportunity to test stellar models. We find that no stellar evolution models are consistent with all the properties of both M-dwarf stars in the eclipsing binary. The candidate in M 35 has been confirmed as an M-dwarf eclipsing binary, and the masses of the individual components are estimated to be M 1 ~ 0.25 M_sun and M 2 ~ 0.15 M_sun . Additional high resolution spectroscopic and photometric observations, for which we have applied and been awarded time, are necessary to accurately derive the intrinsic properties of the individual stellar 17. NSVS 06507557: a low-mass double-lined eclipsing binary Çakırlı, Ö.; Ibanoǧlu, C. 2010-01-01 In this paper, we present the results of a detailed spectroscopic and photometric analysis of the V = 13.4 mag low-mass eclipsing binary NSVS 06507557 with an orbital period of 0.515d. We have obtained a series of mid-resolution spectra covering nearly the entire orbit of the system. In addition, we have obtained simultaneous VRI broad-band photometry using a small aperture telescope. From these spectroscopic and photometric data, we have derived the system's orbital parameters and we have determined the fundamental stellar parameters of the two components. Our results indicate that NSVS 06507557 consists of a K9 pre-main-sequence star and an M3 pre-main-sequence star. These have masses of 0.66 +/- 0.09 Msolar and 0.28 +/- 0.05 Msolar and radii of 0.60 +/- 0.03 and 0.44 +/- 0.02 Rsolar, respectively, and are located at a distance of 111 +/- 9 pc. The radius of the less massive secondary component is larger than that of a zero-age main-sequence (ZAMS) star having the same mass. While the radius of the primary component is in agreement with ZAMS, the secondary component appears to be larger by about 35 per cent with respect to its ZAMS counterpart. Night-to-night intrinsic light variations up to 0.2 mag have been observed. In addition, the Hα and Hβ lines and the forbidden line of [OI] are seen in emission. The LiI 6708 Å absorption line is seen in most of the spectra. These features are taken to be signs of the characteristics of classic T Tauri stars. The parameters we have derived are consistent with an age of about 20 Myr, according to stellar evolutionary models. The spectroscopic and photometric results are in agreement with those obtained using theoretical predictions. Based on spectroscopic observations collected at TÜBİTAK (Turkey). E-mail: [email protected] 18. ECLIPSE TIMINGS OF THE TRANSIENT LOW-MASS X-RAY BINARY EXO 0748-676. IV. THE ROSSI X-RAY TIMING EXPLORER ECLIPSES SciTech Connect Wolff, Michael T.; Ray, Paul S.; Wood, Kent S.; Hertz, Paul L. E-mail: [email protected] E-mail: [email protected] 2009-07-01 We report our complete database of X-ray eclipse timings of the low-mass X-ray binary EXO 0748-676 observed by the Rossi X-Ray Timing Explorer (RXTE) satellite. As of this writing we have accumulated 443 full X-ray eclipses, 392 of which have been observed with the Proportional Counter Array on RXTE. These include both observations where an eclipse was specifically targeted and those eclipses found in the RXTE data archive. Eclipse cycle count has been maintained since the discovery of the EXO 0748-676 system in 1985 February. We describe our observing and analysis techniques for each eclipse and describe improvements we have made since the last compilation by Wolff et al. The principal result of this paper is the database containing the timing results from a seven-parameter fit to the X-ray light curve for each observed eclipse along with the associated errors in the fitted parameters. Based on the standard O - C analysis, EXO 0748-676 has undergone four distinct orbital period epochs since its discovery. In addition, EXO 0748-676 shows small-scale events in the O - C curve that are likely due to short-lived changes in the secondary star. 19. The Interior Structure Constants as an Age Diagnostic for Low-mass, Pre-main-sequence Detached Eclipsing Binary Stars Feiden, Gregory A.; Dotter, Aaron 2013-03-01 We propose a novel method for determining the ages of low-mass, pre-main-sequence stellar systems using the apsidal motion of low-mass detached eclipsing binaries. The apsidal motion of a binary system with an eccentric orbit provides information regarding the interior structure constants of the individual stars. These constants are related to the normalized stellar interior density distribution and can be extracted from the predictions of stellar evolution models. We demonstrate that low-mass, pre-main-sequence stars undergoing radiative core contraction display rapidly changing interior structure constants (greater than 5% per 10 Myr) that, when combined with observational determinations of the interior structure constants (with 5%-10% precision), allow for a robust age estimate. This age estimate, unlike those based on surface quantities, is largely insensitive to the surface layer where effects of magnetic activity are likely to be most pronounced. On the main sequence, where age sensitivity is minimal, the interior structure constants provide a valuable test of the physics used in stellar structure models of low-mass stars. There are currently no known systems where this technique is applicable. Nevertheless, the emphasis on time domain astronomy with current missions, such as Kepler, and future missions, such as LSST, has the potential to discover systems where the proposed method will be observationally feasible. 20. Precise orbit solution of MML 53, a low-mass, pre-main sequence eclipsing binary in Upper Centaurus Lupus Hebb, L.; Cegla, H. M.; Stassun, K. G.; Stempels, H. C.; Cargile, P. A.; Palladino, L. E. 2011-07-01 Aims: We present a double-lined orbit solution for MML 53, the recently discovered low-mass pre-main sequence eclipsing binary. Methods: Using high-resolution spectra from the SMARTS 1.5 m echelle spectrograph, we measure precise radial velocities and derive the orbital parameters of the system. Results: The 2.1 d orbit of the eclipsing pair is circular, and we find the minimum masses of the eclipsing components to be M1sin3i = 0.97 M⊙ and M2sin3i = 0.84 M⊙, with formal uncertainties of 2.0% and an additional systematic uncertainty of ≈2.5% most likely caused by large star spots on the primary star. MML 53 has been previously identified as a member of the Upper Centaurus Lupus (UCL) star forming region (age ~15 Myr). The systemic radial velocity from our orbit solution, vγ = +1.4 ± 0.3 ± 0.8 km s-1 (statistical and systematic), is also consistent with kinematic membership in this association. In addition, we detect a change in vγ between 2006 and 2009 providing further evidence for the presence of a the third body in a wide (several year) orbit. 1. The orbital ephemeris and eclipse transitions of the low-mass X-ray binary EXO 0748 - 676 SciTech Connect Parmar, A.N.; Smale, A.P.; Verbunt, F.; Corbet, R.H.D. NASA, Goddard Space Flight Center, Greenbelt, MD Utrecht Rijksuniversitet Institute of Space and Astronautical Science, Sagamihara ) 1991-01-01 Using the eclipses as fiducial markers, an updated ephemeris for EXO 0748 - 676 is derived and evidence is found that between February 1985 and March 1989 the 3.82-h orbital period of EXO 0748 - 676 decreased with a time scale of -5 x 10 to the 6th yr. The sense of this change is the same as that predicted by simple models for the evolution of low-mass X-ray binaries containing main-sequence companions, but is a factor about 100 faster than expected. This rapid change in orbital period could result from the expansion of the companion due to the effects of X-ray heating. The eclipse transition durations are variable, with the shortest observed taking 1.5 s and the longest 40 s. This latter figure is about an order of magnitude too large to be due to absorption effects in the atmosphere of the secondary assuming a Roche geometry and likely stellar temperature. Either flaring activity or the presence of an X-ray heated evaporative wind or a corona may enhance the scale height of the companion's atmosphere producing the extended eclipse transitions. 38 refs. 2. The orbital ephemeris and eclipse transitions of the low-mass X-ray binary EXO 0748 - 676 NASA Technical Reports Server (NTRS) Parmar, A. N.; Smale, A. P.; Verbunt, F.; Corbet, R. H. D. 1991-01-01 Using the eclipses as fiducial markers, an updated ephemeris for EXO 0748 - 676 is derived and evidence is found that between February 1985 and March 1989 the 3.82-h orbital period of EXO 0748 - 676 decreased with a time scale of -5 x 10 to the 6th yr. The sense of this change is the same as that predicted by simple models for the evolution of low-mass X-ray binaries containing main-sequence companions, but is a factor about 100 faster than expected. This rapid change in orbital period could result from the expansion of the companion due to the effects of X-ray heating. The eclipse transition durations are variable, with the shortest observed taking 1.5 s and the longest 40 s. This latter figure is about an order of magnitude too large to be due to absorption effects in the atmosphere of the secondary assuming a Roche geometry and likely stellar temperature. Either flaring activity or the presence of an X-ray heated evaporative wind or a corona may enhance the scale height of the companion's atmosphere producing the extended eclipse transitions. 3. Absolute parameter determination in low-mass eclipsing binaries - Radiative parameters for BH Vir, ZZ UMA and CR CAS Clement, R.; Reglero, V.; Garcia, M.; Fabregat, J.; Bravo, A.; Suso, J. 1993-01-01 A new uvby and H-beta monitoring program of low mass eclipsing binaries is currently being carried out in the framework of a 5-yr observational program which also involves radial velocity determinations. The scope of this work is to provide very accurate absolute astrophysical parameters: mass, radius, and effective temperatures, for main-sequence late-type stars. One of the main goals is to improve the mass-luminosity relation in the low and intermediate mass range. A second objective is to perform accurate tests of the most recent grids of evolutionary models. This program is complementary to that currently being implemented by the Copenhagen group. In this contribution we present the photometric preliminary results obtained for three of the systems included in our long-term survey: BH Vir, ZZ UMa, and CR Cas for which primary eclipses have been observed. Particular attention is paid to the determination of reddening, distances, and radiative properties. A more detailed study will be carried out when the light curves and radial velocity measurements are completed. 4. A refined analysis of the low-mass eclipsing binary system T-Cyg1-12664 Iglesias-Marzoa, Ramón; López-Morales, Mercedes; Arévalo, María J.; Coughlin, Jeffrey L.; Lázaro, Carlos 2017-03-01 Context. The observational mass-radius relation of main sequence stars with masses between 0.3 and 1.0 M⊙ reveals deviations between the stellar radii predicted by models and the observed radii of stars in detached binaries. Aims: We generate an accurate physical model of the low-mass eclipsing binary T-Cyg1-12664 in the Kepler mission field to measure the physical parameters of its components and to compare them with the prediction of theoretical stellar evolution models. Methods: We analyze the Kepler mission light curve of T-Cyg1-12664 to accurately measure the times and phases of the primary and secondary eclipse. In addition, we measure the rotational period of the primary component by analyzing the out-of-eclipse oscillations that are due to spots. We accurately constrain the effective temperature of the system using ground-based absolute photometry in B, V, RC, and IC. We also obtain and analyze VRCIC differential light curves to measure the eccentricity and the orbital inclination of the system, and a precise Teff ratio. From the joint analysis of new radial velocities and those in the literature we measure the individual masses of the stars. Finally, we use the PHOEBE code to generate a physical model of the system. Results: T-Cyg1-12664 is a low eccentricity system, located d = 360 ± 22 pc away from us, with an orbital period of P = 4.1287955(4) days, and an orbital inclination i = 86.969 ± 0.056 degrees. It is composed of two very different stars with an active G6 primary with Teff1 = 5560 ± 160 K, M1 = 0.680 ± 0.045 M⊙, R1 = 0.799 ± 0.017 R⊙, and a M3V secondary star with Teff2 = 3460 ± 210 K, M2 = 0.376 ± 0.017 M⊙, and R2 = 0.3475 ± 0.0081 R⊙. Conclusions: The primary star is an oversized and spotted active star, hotter than the stars in its mass range. The secondary is a cool star near the mass boundary for fully convective stars (M 0.35 M⊙), whose parameters appear to be in agreement with low-mass stellar model. Full Tables 1 5. MML 53: a new low-mass, pre-main sequence eclipsing binary in the Upper Centaurus-Lupus region discovered by SuperWASP Hebb, L.; Stempels, H. C.; Aigrain, S.; Collier-Cameron, A.; Hodgkin, S. T.; Irwin, J. M.; Maxted, P. F. L.; Pollacco, D.; Street, R. A.; Wilson, D. M.; Stassun, K. G. 2010-11-01 We announce the discovery of a new low-mass, pre-main sequence eclipsing binary, MML 53. Previous observations of MML 53 found it to be a pre-main sequence spectroscopic multiple associated with the 15-22 Myr Upper Centaurus-Lupus cluster. We identify the object as an eclipsing binary for the first time through the analysis of multiple seasons of time series photometry from the SuperWASP transiting planet survey. Re-analysis of a single archive spectrum shows MML 53 to be a spatially unresolved triple system of young stars which all exhibit significant lithium absorption. Two of the components comprise an eclipsing binary with period, P = 2.097891(6) ± 0.000005 and mass ratio, q ~ 0.8. Here, we present the analysis of the discovery data. 6. MML 53: A New Low-Mass, Pre-Main Sequence Eclipsing Binary in the Lupus Cloud Discovered By SuperWASP Cegla, Heather; Hebb, L.; Stassun, K. G.; Stempels, H. C.; Cargile, P. A.; Palladino, L. E.; SuperWASP Consortium 2010-01-01 We announce the discovery of a new low-mass, pre-main sequence (PMS) eclipsing binary star in the Lupus Cloud, MML 53. This is only the 6th sub-solar mass PMS eclipsing binary known. Previous observations of MML 53 found it to be a spectroscopic multiple associated with the 15-22 Myr Upper Centaurus Lupus (UCL) cluster. Here, we identify the object as an eclipsing binary for the first time through the analysis of photometric time series photometry from the SuperWASP transiting planet survey. We derive an accurate ephemeris for the system and present a double-lined orbit solution based on high resolution spectra. The spectra confirm MML 53 to be a triple system of young stars composed of an eclipsing binary and a stationary third component all of which show strong lithium absorption as expected for low-mass, PMS stars. The 2.1 d orbit of the eclipsing pair is circular, and we find the minimum masses of M1 sin3 i = 0.94 M⊙ and M2 sin3 i = 0.81 M⊙ with formal uncertainties of 2.0 %.We find a systemic radial velocity, vγ = +1.00 ± 0.33 ± 0.81 km s-1, consistent with kinematic membership in the UCL association, and the radii of the component stars are 30 % larger than expected for main sequence stars. Follow-up modeling of high cadence, multi-band light-curve data will provide precise fundamental properties of the components of the system which will ultimately be used to place constraints on theoretical models of PMS stellar evolution. This research is supported by an NSF REU grant to the Vanderbilt Physics & Astronomy Department, and by an NSF PAARE grant to Fisk University. 7. The Effect of Starspots on Accurate Radius Determination of the Low-Mass Double-Lined Eclipsing Binary Gu Boo Windmiller, G.; Orosz, J. A.; Etzel, P. B. 2010-04-01 GU Boo is one of only a relatively small number of well-studied double-lined eclipsing binaries that contain low-mass stars. López-Morales & Ribas present a comprehensive analysis of multi-color light and radial velocity curves for this system. The GU Boo light curves presented by López-Morales & Ribas had substantial asymmetries, which were attributed to large spots. In spite of the asymmetry, López-Morales & Ribas derived masses and radii accurate to sime2%. We obtained additional photometry of GU Boo using both a CCD and a single-channel photometer and modeled the light curves with the ELC software to determine if the large spots in the light curves give rise to systematic errors at the few percent level. We also modeled the original light curves from the work of López-Morales & Ribas using models with and without spots. We derived a radius of the primary of 0.6329 ± 0.0026 R sun, 0.6413 ± 0.0049 R sun, and 0.6373 ± 0.0029 R sun from the CCD, photoelectric, and López-Morales & Ribas data, respectively. Each of these measurements agrees with the value reported by López-Morales & Ribas (R 1 = 0.623 ± 0.016 R sun) at the level of ≈2%. In addition, the spread in these values is ≈1%-2% from the mean. For the secondary, we derive radii of 0.6074 ± 0.0035 R sun, 0.5944 ± 0.0069 R sun, and 0.5976 ± 0.0059 R sun from the three respective data sets. The López-Morales & Ribas value is R 2 = 0.620 ± 0.020 R sun, which is ≈2%-3% larger than each of the three values we found. The spread in these values is ≈2% from the mean. The systematic difference between our three determinations of the secondary radius and that of López-Morales & Ribas might be attributed to differences in the modeling process and codes used. Our own fits suggest that, for GU Boo at least, using accurate spot modeling of a single set of multi-color light curves results in radii determinations accurate at the ≈2% level. 8. Eclipse timings of the low-mass X-ray binary EXO 0748-676: Statistical arguments against orbital period changes NASA Technical Reports Server (NTRS) Hertz, Paul; Wood, Kent S.; Cominsky, Lynn 1995-01-01 EXO 0748-676, an eclipsing low-mass X-ray binary, is one of only about four or five low-mass X-ray binaries for which orbital period evolution has been reported. We observed a single eclipse egress with ROSAT . The time of this egress is consistent with the apparent increase in P(sub orb) previously reported on the basis of EXOSAT and Ginga observations. Standard analysis, in which O-C (observed minus calculated) timing residuals are examined for deviations from a constant period, implicitly assume that the only uncertainty in each residual is measurement error and that these errors are independent. We argue that the variable eclipse durations and profiles observed in EXO 0748-676 imply that there is an additional source of uncertainty in timing measurements, that this uncertainty is intrinsic to the binary system, and that it is correlated from observation to observation with a variance which increases as a function of the number of binary cycles between observations. This intrinsic variability gives rise to spurious trends in O-C residuals which are misinterpreted as changes in the orbital period. We describe several statistics tests which can be used to test for the presence of intrinsic variability. We apply those statistical tests which are suitable to the EXO 0748-676 observations. The apparent changes in the orbital period of EXO 0748-676 can be completely accounted for by intrinsic variability with an rms variability of approximately 0.35 s per orbital cycle. The variability appears to be correlated from cycle-to-cycle on timescales of less than 1 yr. We suggest that the intrinsic variability is related to slow changes in either the source's X-ray luminosity or the structure of the companion star's atmosphere. We note that several other X-ray binaries and cataclysmic variables have previously reported orbital period changes which may also be due to intrinsic variability rather than orbital period evolution. 9. Photometric Observations of the low mass Eclipsing Binaries GU Boo and TrES-Her0-07621 from Mount Laguna Observatory Rosenfield, P. A.; Huk, L.; Garcia, D.; Downum, C.; Etzel, P. B.; Orosz, J. A. 2005-12-01 Low mass double-lined eclipsing binaries are of general interest because precise masses and radii can be derived for the component stars, and these data can in turn be used to test evolutionary models on the lower main sequence. The low mass double-lined eclipsing binaries GU Boo and TrES-Her0-07621 were observed in several bandpasses from the Mount Laguna Observatory (MLO) 2005 May - June with the goal of obtaining precise light curves that can be used to derive accurate radii. Lopez-Morales & Ribas (2005) present light curves of GU Boo covering both eclipses and radial velocity curves for each component and derive component masses and radii accurate to 1.3% and 2.3%, respectively. Their light curves were not symmetric about the primary eclipse, and Lopez-Morales & Ribas modelled the asymmetry using two dark spots on the primary. Our light curves of GU Boo from MLO show only a slight asymmetry, which indicates the spotted region is much smaller. The radii we derive for the component stars are consistent with the measurements of Lopez-Morales & Ribas, showing that the presence of large star spots did not lead to any systematic errors in the derived radii. TrES-Her0-07621 was discovered by the Trans-Atlantic Exoplanet Survey (TrES, e.g. Alonso et al. 2004). Creevey et al. (2005) obtained radial velocity curves for both components and derive component masses and radii accurate to 0.6% and 13%, respectively. The radii are poorly constrained since the light curve presented in Creevey et al. is somewhat noisy. Our light curves from MLO have a high signal-to-noise ratio, but unfortunately only cover the secondary eclipse. We see evidence of a sinusoidal variation in the out-of-eclipse phases (also noted by Creevey et al.). More observations will be needed to cover the primary eclipse and to establish the nature of the out-of-eclipse variation. 10. Discovery and Precise Characterization by the MEarth Project of LP 661-13, an Eclipsing Binary Consisting of Two Fully Convective Low-mass Stars Dittmann, Jason A.; Irwin, Jonathan M.; Charbonneau, David; Berta-Thompson, Zachory K.; Newton, Elisabeth R.; Latham, David W.; Latham, Christian A.; Esquerdo, Gilbert; Berlind, Perry; Calkins, Michael L. 2017-02-01 We report the detection of stellar eclipses in the LP 661-13 system. We present the discovery and characterization of this system, including high-resolution spectroscopic radial velocities and a photometric solution spanning two observing seasons. LP 661-13 is a low-mass binary system with an orbital period of {4.7043512}-0.0000010+0.0000013 days at a distance of 24.9 ± 1.3 parsecs. LP 661-13A is a 0.30795 ± 0.00084 M ⊙ star, while LP 661-13B is a 0.19400 ± 0.00034 M ⊙ star. The radius of each component is 0.3226 ± 0.0033 R ⊙ and 0.2174 ± 0.0023 R ⊙, respectively. We detect out-of-eclipse modulations at a period slightly shorter than the orbital period, implying that at least one of the components is not rotating synchronously. We find that each component is slightly inflated compared to stellar models, and that this cannot be reconciled through age or metallicity effects. As a nearby eclipsing binary system, where both components are near or below the full-convection limit, LP 661-13 will be a valuable test of models for the structure of cool dwarf stars. 11. The Brown Dwarf Eclipsing Binary 2M0535-05: A Case Study for Activity Effects on Physical Properties of Low-Mass Stars and Brown Dwarfs Stassun, K. G. 2013-02-01 2M0535-05 is a one-of-a-kind eclipsing binary (EB) comprising two brown dwarfs (BDs), and is an important benchmark for understanding the fundamental properties of BDs and low-mass stars. Because 2M0535-05 presents a peculiar reversal of temperatures with mass (the higher mass, magnetically active BD in the system is cooler than the lower mass companion BD), 2M0535-05 is particularly important as a case study for the effects of magnetic activity on the properties of low-mass objects. Using a large number of low-mass M-dwarfs and EBs in the field, we have developed empirical relations for determining the amount by which the temperatures and radii-and therefore the estimated masses-of low-mass stars and BDs are altered due to chromospheric activity. The relations link the amount by which an active object's temperature is suppressed, and its radius inflated, to the strength of its Hα emission. These relations are found to approximately preserve bolometric luminosity. Applying these relations to 2M0535-05 brings the activity-corrected radii and temperatures of 2M0535-05 into precise agreement with theoretical isochrones for inactive stars. The relations that we present are applicable to BDs and low-mass stars with masses below 0.8 M⊙ and for which the activity, as measured by Hα, is in the range - 4.6 < log LHα/Lbol < -3.3. We discuss implications of this work for determinations of young cluster IMFs, and discuss competing ideas for the physical mechanism by which magnetic fields alter the temperatures and radii of low-mass stars. 12. Calibrating convective-core overshooting with eclipsing binary systems. The case of low-mass main-sequence stars Valle, G.; Dell'Omodarme, M.; Prada Moroni, P. G.; Degl'Innocenti, S. 2016-03-01 Context. Double-lined eclipsing binaries have often been adopted in literature to calibrate the extension of the convective-core overshooting beyond the border defined by the Schwarzschild criterion. Aims: In a robust statistical way, we quantify the magnitude of the uncertainty that affects the calibration of the overshooting efficiency parameter β that is owing to the uncertainty on the observational data. We also quantify the biases on the β determination that is caused by the lack of constraints on the initial helium content and on the efficiencies of the superadiabatic convection and microscopic diffusion. Methods: We adopted a modified grid-based SCEPtER pipeline to recover the β parameter from synthetic stellar data. Our grid spans the mass range [1.1; 1.6] M⊙ and evolutionary stages from the zero-age main sequence (MS) to the central hydrogen depletion. The β estimates were obtained by generalising the maximum likelihood technique described in our previous works. As observational constraint, we adopted the effective temperatures, [Fe/H], masses, and radii of the two stars. Results: By means of Monte Carlo simulations, adopting a reference scenario of mild overshooting β = 0.2 for the synthetic data, and taking typical observational errors into account, we found both large statistical uncertainties and biases on the estimated values of β. For the first 80% of the MS evolution, β is biased by about -0.04, with the 1σ error practically unconstrained in the whole explored range [0.0; 0.4]. In the last 5% of the evolution the bias vanishes and the 1σ error is about 0.05. The 1σ errors are similar when adopting different reference values of β. Interestingly, for synthetic data computed without convective-core overshooting, the estimated β is biased by about 0.12 in the first 80% of the MS evolution, and by 0.05 afterwards. Assuming an uncertainty of ±1 in the helium-to-metal enrichment ratio ΔY/ ΔZ, we found a large systematic uncertainty in the 13. CoRoT 223992193: Investigating the variability in a low-mass, pre-main sequence eclipsing binary with evidence of a circumbinary disk Gillen, E.; Aigrain, S.; Terquem, C.; Bouvier, J.; Alencar, S. H. P.; Gandolfi, D.; Stauffer, J.; Cody, A.; Venuti, L.; Almeida, P. Viana; Micela, G.; Favata, F.; Deeg, H. J. 2017-02-01 CoRoT 223992193 is the only known low-mass, pre-main sequence eclipsing binary that shows evidence of a circumbinary disk. The system displays complex photometric and spectroscopic variability over a range of timescales and wavelengths. Using two optical CoRoT runs from 2008 and 2011/2012 (spanning 23 and 39 days), along with infrared Spitzer 3.6 and 4.5 μm observations (spanning 21 and 29 days, and simultaneous with the second CoRoT run), we model the out-of-eclipse light curves, finding that the large scale structure in both CoRoT light curves is consistent with the constructive and destructive interference of starspot signals at two slightly different periods. Using the vsini of both stars, we interpret this as the two stars having slightly different rotation periods: the primary is consistent with synchronisation and the secondary rotates slightly supersynchronously. Comparison of the raw 2011/2012 light curve data to the residuals of our spot model in colour-magnitude space indicates additional contributions consistent with a combination of variable dust emission and obscuration. There appears to be a tentative correlation between this additional variability and the binary orbital phase, with the system displaying increases in its infrared flux around primary and secondary eclipse. We also identify short-duration flux dips preceding secondary eclipse in all three CoRoT and Spitzer bands. We construct a model of the inner regions of the binary and propose that these dips could be caused by partial occultation of the central binary by the accretion stream onto the primary star. Analysis of 15 Hα profiles obtained with the FLAMES instrument on the Very Large Telescope reveal an emission profile associated with each star. The majority of this is consistent with chromospheric emission but additional higher velocity emission is also seen, which could be due to prominences. However, half of the secondary star's emission profiles display full widths at 10% intensity 14. The EBLM project. I. Physical and orbital parameters, including spin-orbit angles, of two low-mass eclipsing binaries on opposite sides of the brown dwarf limit Triaud, A. H. M. J.; Hebb, L.; Anderson, D. R.; Cargile, P.; Collier Cameron, A.; Doyle, A. P.; Faedi, F.; Gillon, M.; Gomez Maqueo Chew, Y.; Hellier, C.; Jehin, E.; Maxted, P.; Naef, D.; Pepe, F.; Pollacco, D.; Queloz, D.; Ségransan, D.; Smalley, B.; Stassun, K.; Udry, S.; West, R. G. 2013-01-01 This paper introduces a series of papers aiming to study the dozens of low-mass eclipsing binaries (EBLM), with F, G, K primaries, that have been discovered in the course of the WASP survey. Our objects are mostly single-line binaries whose eclipses have been detected by WASP and were initially followed up as potential planetary transit candidates. These have bright primaries, which facilitates spectroscopic observations during transit and allows the study of the spin-orbit distribution of F, G, K+M eclipsing binaries through the Rossiter-McLaughlin effect. Here we report on the spin-orbit angle of WASP-30b, a transiting brown dwarf, and improve its orbital parameters. We also present the mass, radius, spin-orbit angle and orbital parameters of a new eclipsing binary, J1219-39b (1SWAPJ121921.03-395125.6, TYC 7760-484-1), which, with a mass of 95 ± 2 Mjup, is close to the limit between brown dwarfs and stars. We find that both objects have projected spin-orbit angles aligned with their primaries' rotation. Neither primaries are synchronous. J1219-39b has a modestly eccentric orbit and is in agreement with the theoretical mass-radius relationship, whereas WASP-30b lies above it. Using WASP-South photometric observations (Sutherland, South Africa) confirmed with radial velocity measurement from the CORALIE spectrograph, photometry from the EulerCam camera (both mounted on the Swiss 1.2 m Euler Telescope), radial velocities from the HARPS spectrograph on the ESO's 3.6 m Telescope (prog ID 085.C-0393), and photometry from the robotic 60 cm TRAPPIST telescope, all located at ESO, La Silla, Chile. The data is publicly available at the CDS Strasbourg and on demand to the main author.Tables A.1-A.3 are available in electronic form at http://www.aanda.orgPhotometry tables are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/549/A18 15. Orbital and Spin Parameter Variations of Partial Eclipsing Low Mass X-Ray Binary X 1822-371 Chou, Yi; Hsieh, Hung-En; Hu, Chin-Ping; Yang, Ting-Chang; Su, Yi-Hao 2016-11-01 We report our measurements for the orbital and spin parameters of X 1822-371 using its X-ray partial eclipsing profile and pulsar timing from data collected by the Rossi X-ray Timing Explorer (RXTE). Four more X-ray eclipse times obtained by the RXTE 2011 observations were combined with historical records to trace the evolution of the orbital period. We found that a cubic ephemeris likely better describes the evolution of the X-ray eclipse times during a time span of about 34 years with a marginal second order derivative of {\\ddot{P}}{orb}=(-1.05+/- 0.59)× {10}-19 s-1. Using the pulse arrival time delay technique, the orbital and spin parameters were obtained from RXTE observations from 1998 to 2011. The detected pulse periods show that the neutron star in X 1822-371 is continuously spun-up with a rate of {\\dot{P}}s=(-2.6288+/- 0.0095)× {10}-12 s s-1. Although the evolution of the epoch of the mean longitude l = π/2 (i.e., T π/2) gives an orbital period derivative value consistent with that obtained from the quadratic ephemeris evaluated from the X-ray eclipse, the detected T π/2 values are significantly and systematically earlier than the corresponding expected X-ray eclipse times by 90 ± 11 s. This deviation is probably caused by asymmetric X-ray emissions. We also attempted to constrain the mass and radius of the neutron star using the spin period change rate and concluded that the intrinsic luminosity of X 1822-371 is likely more than 1038 erg s-1. 16. Magnetic Inflation and Stellar Mass. I. Revised Parameters for the Component Stars of the Kepler Low-mass Eclipsing Binary T-Cyg1-12664 Han, Eunkyu; Muirhead, Philip S.; Swift, Jonathan J.; Baranec, Christoph; Law, Nicholas M.; Riddle, Reed; Atkinson, Dani; Mace, Gregory N.; DeFelippis, Daniel 2017-09-01 Several low-mass eclipsing binary stars show larger than expected radii for their measured mass, metallicity, and age. One proposed mechanism for this radius inflation involves inhibited internal convection and starspots caused by strong magnetic fields. One particular eclipsing binary, T-Cyg1-12664, has proven confounding to this scenario. Çakırlı et al. measured a radius for the secondary component that is twice as large as model predictions for stars with the same mass and age, but a primary mass that is consistent with predictions. Iglesias-Marzoa et al. independently measured the radii and masses of the component stars and found that the radius of the secondary is not in fact inflated with respect to models, but that the primary is, which is consistent with the inhibited convection scenario. However, in their mass determinations, Iglesias-Marzoa et al. lacked independent radial velocity measurements for the secondary component due to the star’s faintness at optical wavelengths. The secondary component is especially interesting, as its purported mass is near the transition from partially convective to a fully convective interior. In this article, we independently determined the masses and radii of the component stars of T-Cyg1-12664 using archival Kepler data and radial velocity measurements of both component stars obtained with IGRINS on the Discovery Channel Telescope and NIRSPEC and HIRES on the Keck Telescopes. We show that neither of the component stars is inflated with respect to models. Our results are broadly consistent with modern stellar evolutionary models for main-sequence M dwarf stars and do not require inhibited convection by magnetic fields to account for the stellar radii. 17. THIRTY NEW LOW-MASS SPECTROSCOPIC BINARIES SciTech Connect Shkolnik, Evgenya L.; Hebb, Leslie; Cameron, Andrew C.; Liu, Michael C.; Neill Reid, I. E-mail: [email protected] E-mail: [email protected] 2010-06-20 As part of our search for young M dwarfs within 25 pc, we acquired high-resolution spectra of 185 low-mass stars compiled by the NStars project that have strong X-ray emission. By cross-correlating these spectra with radial velocity standard stars, we are sensitive to finding multi-lined spectroscopic binaries. We find a low-mass spectroscopic binary fraction of 16% consisting of 27 SB2s, 2 SB3s, and 1 SB4, increasing the number of known low-mass spectroscopic binaries (SBs) by 50% and proving that strong X-ray emission is an extremely efficient way to find M-dwarf SBs. WASP photometry of 23 of these systems revealed two low-mass eclipsing binaries (EBs), bringing the count of known M-dwarf EBs to 15. BD-22 5866, the ESB4, was fully described in 2008 by Shkolnik et al. and CCDM J04404+3127 B consists of two mid-M stars orbiting each other every 2.048 days. WASP also provided rotation periods for 12 systems, and in the cases where the synchronization time scales are short, we used P{sub rot} to determine the true orbital parameters. For those with no P{sub rot}, we used differential radial velocities to set upper limits on orbital periods and semimajor axes. More than half of our sample has near-equal-mass components (q > 0.8). This is expected since our sample is biased toward tight orbits where saturated X-ray emission is due to tidal spin-up rather than stellar youth. Increasing the samples of M-dwarf SBs and EBs is extremely valuable in setting constraints on current theories of stellar multiplicity and evolution scenarios for low-mass multiple systems. 18. Eclipsing Binaries as Astrophysical Laboratories: CM Draconis - Accurate Absolute Physical Properties of Low Mass Stars and an Independent Estimate of the Primordial Helium Abundance McCook, G. P.; Guinan, E. F.; Saumon, D.; Kang, Y. W. 1997-05-01 CM Draconis (Gl 630.1; Vmax = +12.93) is an important eclipsing binary consisting of two dM4.5e stars with an orbital period of 1.2684 days. This binary is a high velocity star (s= 164 km/s) and the brighter member of a common proper motion pair with a cool faint white dwarf companion (LP 101-16). CM Dra and its white dwarf companion were once considered by Zwicky to belong to a class of "pygmy stars", but they turned out to be ordinary old, cool white dwarfs or faint red dwarfs. Lacy (ApJ 218,444L) determined the first orbital and physical properties of CM Dra from the analysis of his light and radial velocity curves. In addition to providing directly measured masses, radii, and luminosities for low mass stars, CM Dra was also recognized by Lacy and later by Paczynski and Sienkiewicz (ApJ 286,332) as an important laboratory for cosmology, as a possible old Pop II object where it may be possible to determine the primordial helium abundance. Recently, Metcalfe et al.(ApJ 456,356) obtained accurate RV measures for CM Dra and recomputed refined elements along with its helium abundance. Starting in 1995, we have been carrying out intensive RI photoelectric photometry of CM Dra to obtain well defined, accurate light curves so that its fundamental properties can be improved, and at the same time, to search for evidence of planets around the binary from planetary transit eclipses. During 1996 and 1997 well defined light curves were secured and these were combined with the RV measures of Metcalfe et al. (1996) to determine the orbital and physical parameters of the system, including a refined orbital period. A recent version of the Wilson-Devinney program was used to analyze the data. New radii, masses, mean densities, Teff, and luminosities were found as well as a re-determination of the helium abundance (Y). The results of the recent analyses of the light and RV curves will be presented and modelling results discussed. This research is supported by NSF grants AST-9315365 19. CoRoT 223992193: A new, low-mass, pre-main sequence eclipsing binary with evidence of a circumbinary disk Gillen, E.; Aigrain, S.; McQuillan, A.; Bouvier, J.; Hodgkin, S.; Alencar, S. H. P.; Terquem, C.; Southworth, J.; Gibson, N. P.; Cody, A.; Lendl, M.; Morales-Calderón, M.; Favata, F.; Stauffer, J.; Micela, G. 2014-02-01 We present the discovery of CoRoT 223992193, a double-lined, detached eclipsing binary, comprising two pre-main sequence M dwarfs, discovered by the CoRoT space mission during a 23-day observation of the 3 Myr old NGC 2264 star-forming region. Using multi-epoch optical and near-IR follow-up spectroscopy with FLAMES on the Very Large Telescope and ISIS on the William Herschel Telescope we obtain a full orbital solution and derive the fundamental parameters of both stars by modelling the light curve and radial velocity data. The orbit is circular and has a period of 3.8745745 ± 0.0000014 days. The masses and radii of the two stars are 0.67 ± 0.01 and 0.495 ± 0.007 M⊙ and 1.30 ± 0.04 and 1.11-0.05+0.04 R⊙, respectively. This system is a useful test of evolutionary models of young low-mass stars, as it lies in a region of parameter space where observational constraints are scarce; comparison with these models indicates an apparent age of ~3.5-6 Myr. The systemic velocity is within 1σ of the cluster value which, along with the presence of lithium absorption, strongly indicates cluster membership. The CoRoT light curve also contains large-amplitude, rapidly evolving out-of-eclipse variations, which are difficult to explain using starspots alone. The system's spectral energy distribution reveals a mid-infrared excess, which we model as thermal emission from a small amount of dust located in the inner cavity of a circumbinary disk. In turn, this opens up the possibility that some of the out-of-eclipse variability could be due to occultations of the central stars by material located at the inner edge or in the central cavity of the circumbinary disk. The CoRoT space mission was developed and is operated by the French space agency CNES, with participation of ESAs RSSD and Science Programmes, Austria, Belgium, Brazil, Germany, and Spain. 20. KEPLER CYCLE 1 OBSERVATIONS OF LOW-MASS STARS: NEW ECLIPSING BINARIES, SINGLE STAR ROTATION RATES, AND THE NATURE AND FREQUENCY OF STARSPOTS SciTech Connect Harrison, T. E.; Coughlin, J. L.; Ule, N. M.; Lopez-Morales, M. E-mail: [email protected] E-mail: [email protected] 2012-01-15 We have analyzed Kepler light curves for 849 stars with T{sub eff} {<=} 5200 K from our Cycle 1 Guest Observer program. We identify six new eclipsing binaries, one of which has an orbital period of 29.91 days and two of which are probably W UMa variables. In addition, we identify a candidate 'warm Jupiter' exoplanet. We further examine a subset of 670 sources for variability. Of these objects, 265 stars clearly show periodic variability that we assign to rotation of the low-mass star. At the photometric precision level provided by Kepler, 251 of our objects showed no evidence for variability. We were unable to determine periods for 154 variable objects. We find that 79% of stars with T{sub eff} {<=} 5200 K are variable. The rotation periods we derive for the periodic variables span the range 0.31 days {<=} P{sub rot} {<=} 126.5 days. A considerable number of stars with rotation periods similar to the solar value show activity levels that are 100 times higher than the Sun. This is consistent with results for solar-like field stars. As has been found in previous studies, stars with shorter rotation periods generally exhibit larger modulations. This trend flattens beyond P{sub rot} = 25 days, demonstrating that even long-period binaries may still have components with high levels of activity and investigating whether the masses and radii of the stellar components in these systems are consistent with stellar models could remain problematic. Surprisingly, our modeling of the light curves suggests that the active regions on these cool stars are either preferentially located near the rotational poles, or that there are two spot groups located at lower latitudes, but in opposing hemispheres. 1. Orbital and physical parameters of eclipsing binaries from the All-Sky Automated Survey catalogue. III. Two new low-mass systems with rapidly evolving spots Hełminiak, K. G.; Konacki, M.; Złoczewski, K.; Ratajczak, M.; Reichart, D. E.; Ivarsen, K. M.; Haislip, J. B.; Crain, J. A.; Foster, A. C.; Nysewander, M. C.; Lacluyze, A. P. 2011-03-01 Aims: We present the results of our spectroscopic and photometric analysis of two newly discovered low-mass detached eclipsing binaries found in the All-Sky Automated Survey (ASAS) catalogue: ASAS J093814-0104.4 and ASAS J212954-5620.1. Methods: Using the Grating Instrument for Radiation Analysis with a Fibre-Fed Echelle (GIRAFFE) on the 1.9-m Radcliffe telescope at the South African Astronomical Observatory (SAAO) and the University College London Echelle Spectrograph (UCLES) on the 3.9-m Anglo-Australian Telescope, we obtained high-resolution spectra of both objects and derived their radial velocities (RVs) at various orbital phases. The RVs of both objects were measured with the two-dimensional cross-correlation technique (TODCOR) using synthetic template spectra as references. We also obtained V and I band photometry using the 1.0-m Elizabeth telescope at SAAO and the 0.4-m Panchromatic Robotic Optical Monitoring and Polarimetry Telescopes (PROMPT) located at the Cerro Tololo Inter-American Observatory (CTIO). The orbital and physical parameters of the systems were derived with PHOEBE and JKTEBOP codes. We compared our results with several sets of widely-used isochrones. Results: Our multi-epoch photometric observations demonstrate that both objects show significant out-of-eclipse modulations, which vary in time. We believe that this effect is caused by stellar spots, which evolve on time scales of tens of days. For this reason, we constructed our models on the basis of photometric observations spanning short time scales (less than a month). Our modeling indicates that (1) ASAS J093814-0104.04 is a main sequence active system with nearly-twin components with masses of M1 = 0.771 ± 0.033 M⊙, M2 = 0.768 ± 0.021 M⊙ and radii of R1 = 0.772 ± 0.012 R⊙ and R2 = 0.769 ± 0.013 R⊙. (2) ASAS J212954-5620.1 is a main sequence active binary with component masses of M1 = 0.833 ± 0.017 M⊙, M2 = 0.703 ± 0.013 M⊙ and radii of R1 = 0.845 ± 0.012 R⊙ and R2 2. The G+M eclipsing binary V530 Orionis: a stringent test of magnetic stellar evolution models for low-mass stars SciTech Connect Torres, Guillermo; Lacy, Claud H. Sandberg; Pavlovski, Krešimir; Feiden, Gregory A.; Sabby, Jeffrey A.; Bruntt, Hans; Clausen, Jens Viggo 2014-12-10 We report extensive photometric and spectroscopic observations of the 6.1 day period, G+M-type detached double-lined eclipsing binary V530 Ori, an important new benchmark system for testing stellar evolution models for low-mass stars. We determine accurate masses and radii for the components with errors of 0.7% and 1.3%, as follows: M {sub A} = 1.0038 ± 0.0066 M {sub ☉}, M {sub B} = 0.5955 ± 0.0022 M {sub ☉}, R {sub A} = 0.980 ± 0.013 R {sub ☉}, and R {sub B} = 0.5873 ± 0.0067 R {sub ☉}. The effective temperatures are 5890 ± 100 K (G1 V) and 3880 ± 120 K (M1 V), respectively. A detailed chemical analysis probing more than 20 elements in the primary spectrum shows the system to have a slightly subsolar abundance, with [Fe/H] = –0.12 ± 0.08. A comparison with theory reveals that standard models underpredict the radius and overpredict the temperature of the secondary, as has been found previously for other M dwarfs. On the other hand, models from the Dartmouth series incorporating magnetic fields are able to match the observations of the secondary star at the same age as the primary (∼3 Gyr) with a surface field strength of 2.1 ± 0.4 kG when using a rotational dynamo prescription, or 1.3 ± 0.4 kG with a turbulent dynamo approach, not far from our empirical estimate for this star of 0.83 ± 0.65 kG. The observations are most consistent with magnetic fields playing only a small role in changing the global properties of the primary. The V530 Ori system thus provides an important demonstration that recent advances in modeling appear to be on the right track to explain the long-standing problem of radius inflation and temperature suppression in low-mass stars. 3. Cyclic period changes and the light-time effect in eclipsing binaries: A low-mass companion around the system VV Ursae Majoris Tanrıver, Mehmet 2015-04-01 In this article, a period analysis of the late-type eclipsing binary VV UMa is presented. This work is based on the periodic variation of eclipse timings of the VV UMa binary. We determined the orbital properties and mass of a third orbiting body in the system by analyzing the light-travel time effect. The O-C diagram constructed for all available minima times of VV UMa exhibits a cyclic character superimposed on a linear variation. This variation includes three maxima and two minima within approximately 28,240 orbital periods of the system, which can be explained as the light-travel time effect (LITE) because of an unseen third body in a triple system that causes variations of the eclipse arrival times. New parameter values of the light-time travel effect because of the third body were computed with a period of 23.22 ± 0.17 years in the system. The cyclic-variation analysis produces a value of 0.0139 day as the semi-amplitude of the light-travel time effect and 0.35 as the orbital eccentricity of the third body. The mass of the third body that orbits the eclipsing binary stars is 0.787 ± 0.02 M⊙, and the semi-major axis of its orbit is 10.75 AU. 4. Eclipsing Binary Update, No. 2. Williams, D. B. 1996-01-01 Contents: 1. Wrong again! The elusive period of DHK 41. 2. Stars observed and not observed. 3. Eclipsing binary chart information. 4. Eclipsing binary news and notes. 5. A note on SS Arietis. 6. Featured star: TX Ursae Majoris. 5. The Factory and the Beehive. III. PTFEB132.707+19.810, A Low-mass Eclipsing Binary in Praesepe Observed by PTF and K2 Kraus, Adam L.; Douglas, Stephanie T.; Mann, Andrew W.; Agüeros, Marcel A.; Law, Nicholas M.; Covey, Kevin R.; Feiden, Gregory A.; Rizzuto, Aaron C.; Howard, Andrew W.; Isaacson, Howard; Gaidos, Eric; Torres, Guillermo; Bakos, Gaspar 2017-08-01 Theoretical models of stars constitute the fundamental bedrock upon which much of astrophysics is built, but large swaths of model parameter space remain uncalibrated by observations. The best calibrators are eclipsing binaries in clusters, allowing measurement of masses, radii, luminosities, and temperatures for stars of known metallicity and age. We present the discovery and detailed characterization of PTFEB132.707+19.810, a P = 6.0 day eclipsing binary in the Praesepe cluster (τ ˜ 600-800 Myr [Fe/H] = 0.14 ± 0.04). The system contains two late-type stars (SpT P = M3.5 ± 0.2; SpT S = M4.3 ± 0.7) with precise masses ({M}p=0.3953+/- 0.0020 M ⊙ {M}s=0.2098 +/- 0.0014 M ⊙) and radii ({R}p=0.363+/- 0.008 R ⊙ {R}s=0.272+/- 0.012 R ⊙). Neither star meets the predictions of stellar evolutionary models. The primary has the expected radius but is cooler and less luminous, while the secondary has the expected luminosity but is cooler and substantially larger (by 20%). The system is not tidally locked or circularized. Exploiting a fortuitous 4:5 commensurability between P orb and {P}{rot,{prim}}, we demonstrate that fitting errors from the unknown spot configuration only change the inferred radii by ≲1%-2%. We also analyze subsets of data to test the robustness of radius measurements; the radius sum is more robust to systematic errors and preferable for model comparisons. We also test plausible changes in limb darkening and find corresponding uncertainties of ˜1%. Finally, we validate our pipeline using extant data for GU Boo, finding that our independent results match previous radii to within the mutual uncertainties (2%-3%). We therefore suggest that the substantial discrepancies are astrophysical; since they are larger than those for old field stars, they may be tied to the intermediate age of PTFEB132.707+19.810. 6. Spectroscopic Orbits for Kepler FOV Eclipsing Binaries Matson, Rachel A.; Gies, Douglas R.; Williams, Stephen J.; Guo, Zhao 2013-02-01 We are currently involved in a four year program of precise eclipsing binary photometry with the NASA Kepler Observatory. Our goal is to search for variations in minimum light timing for intermediate mass eclipsing binaries. Such periodic variations will reveal the reflex motion caused by any distant, low mass object that orbits the close binary. it Kepler's unprecedented accuracy and continuous observations provide a unique opportunity to detect the low mass companions that are predicted to result from the angular momentum of the natal cloud. The goal of this proposal is to obtain blue spectra of short period (0.9-6d) eclipsing binaries, derive radial velocities, and produce a double-lined spectroscopic orbit (as well as estimates of the stellar effective temperatures, gravities, and metallicities). Combined with the it Kepler light curve, we will determine very accurate masses and radii for the members of the close binary, which will yield the mass-inclination product M_3 sin i for any companions detected by light travel time or other effects. An extended sample of eclipsing binaries with longer periods (up to 50d) is now being investigated to test whether the presence of a tertiary companion declines with increasing period. We propose to obtain a single spectrum at quadrature for the brightest 48 stars in this expanded sample to characterize the effective temperatures and total mass contained in these systems. 7. Double Eclipsing Binary Fitting Cagas, P.; Pejcha, O. 2012-06-01 The parameters of the mutual orbit of eclipsing binaries that are physically connected can be obtained by precision timing of minima over time through light travel time effect, apsidal motion or orbital precession. This, however, requires joint analysis of data from different sources obtained through various techniques and with insufficiently quantified uncertainties. In particular, photometric uncertainties are often underestimated, which yields too small uncertainties in minima timings if determined through analysis of a χ2 surface. The task is even more difficult for double eclipsing binaries, especially those with periods close to a resonance such as CzeV344, where minima get often blended with each other. This code solves the double binary parameters simultaneously and then uses these parameters to determine minima timings (or more specifically O-C values) for individual datasets. In both cases, the uncertainties (or more precisely confidence intervals) are determined through bootstrap resampling of the original data. This procedure to a large extent alleviates the common problem with underestimated photometric uncertainties and provides a check on possible degeneracies in the parameters and the stability of the results. While there are shortcomings to this method as well when compared to Markov Chain Monte Carlo methods, the ease of the implementation of bootstrapping is a significant advantage. 8. Photometric Study of Fourteen Low-mass Binaries Korda, D.; Zasche, P.; Wolf, M.; Kučáková, H.; Hoňková, K.; Vraštil, J. 2017-07-01 New CCD photometric observations of fourteen short-period low-mass eclipsing binaries (LMBs) in the photometric filters I, R, and V were used for a light curve analysis. A discrepancy remains between observed radii and those derived from the theoretical modeling for LMBs, in general. Mass calibration of all observed LMBs was performed using only the photometric indices. The light curve modeling of these selected systems was completed, yielding the new derived masses and radii for both components. We compared these systems with the compilation of other known double-lined LMB systems with uncertainties of masses and radii less then 5%, which includes 66 components of binaries where both spectroscopy and photometry were combined together. All of our systems are circular short-period binaries, and for some of them, the photospheric spots were also used. A purely photometric study of the light curves without spectroscopy seems unable to achieve high enough precision and accuracy in the masses and radii to act as meaningful test of the M-R relation for low-mass stars. Based on observations collected via the 65 cm telescope at the Ondřejov observatory in Czech Republic. 9. PHOEBE: PHysics Of Eclipsing BinariEs Prsa, Andrej; Matijevic, Gal; Latkovic, Olivera; Vilardell, Francesc; Wils, Patrick 2011-06-01 PHOEBE (PHysics Of Eclipsing BinariEs) is a modeling package for eclipsing binary stars, built on top of the widely used WD program (Wilson & Devinney 1971). This introductory paper overviews most important scientific extensions (incorporating observational spectra of eclipsing binaries into the solution-seeking process, extracting individual temperatures from observed color indices, main-sequence constraining and proper treatment of the reddening), numerical innovations (suggested improvements to WD's Differential Corrections method, the new Nelder & Mead's downhill Simplex method) and technical aspects (back-end scripter structure, graphical user interface). While PHOEBE retains 100% WD compatibility, its add-ons are a powerful way to enhance WD by encompassing even more physics and solution reliability. 10. The low-mass X-ray binary LMC X-2 SciTech Connect Crampton, D.; Hutchings, J.B.; Cowley, A.P.; Schmidtke, P.C.; Thompson, I.B. Arizona State Univ., Tempe Mount Wilson and Las Campanas Observatories, Pasadena, CA ) 1990-06-01 Spectroscopic and photometric observations of LMC X-2 reveal the source to be an X-ray binary with a relatively long orbital period, probably 12.5 days. It appears to be a partially eclipsing system. It is one of a small subclass of low-mass X-ray binaries with longer orbital periods and higher X-ray luminosity than average, which contain a compact object accreting material from an evolving giant companion. 26 refs. 11. Eclipsing binaries - selection of targets Zasche, P. 2017-04-01 Are the ground-based observations still needed in the era of robotic all-sky surveys? There were highlighted several fields in the eclipsing binary research, where also the amateur photometry would be very fruitful with also a few suitable systems where the monitoring is needed also using the smaller telescopes. 12. How I Learned to Stop Worrying and Love Eclipsing Binaries Relatively massive B-type stars with closely orbiting stellar companions can evolve to produce Type Ia supernovae, X-ray binaries, millisecond pulsars, mergers of neutron stars, gamma ray bursts, and sources of gravitational waves. However, the formation mechanism, intrinsic frequency, and evolutionary processes of B-type binaries are poorly understood. As of 2012, the binary statistics of massive stars had not been measured at low metallicities, extreme mass ratios, or intermediate orbital periods. This thesis utilizes large data sets of eclipsing binaries to measure the physical properties of B-type binaries in these previously unexplored portions of the parameter space. The updated binary statistics provide invaluable insight into the formation of massive stars and binaries as well as reliable initial conditions for population synthesis studies of binary star evolution. We first compare the properties of B-type eclipsing binaries in our Milky Way Galaxy and the nearby Magellanic Cloud Galaxies. We model the eclipsing binary light curves and perform detailed Monte Carlo simulations to recover the intrinsic properties and distributions of the close binary population. We find the frequency, period distribution, and mass-ratio distribution of close B-type binaries do not significantly depend on metallicity or environment. These results indicate the formation of massive binaries are relatively insensitive to their chemical abundances or immediate surroundings. Second, we search for low-mass eclipsing companions to massive B-type stars in the Large Magellanic Cloud Galaxy. In addition to finding such extreme mass-ratio binaries, we serendipitously discover a new class of eclipsing binaries. Each system comprises a massive B-type star that is fully formed and a nascent low-mass companion that is still contracting toward its normal phase of evolution. The large low-mass secondaries discernibly reflect much of the light they intercept from the hot B-type stars, thereby 13. Close Binaries, Triples, and Eclipses Sanborn, Jason; Zavala, R. T. 2013-01-01 Observations of the variable radio source b Per (HR1324) are part of an ongoing survey of close binary systems using the Navy Precision Optical Interferometer. Historical observations of b Per include sparse photometric and spectroscopic observations dating back to 1923, clearly showing this object to be a non-eclipsing, single-lined ellipsoidal variable. This is where the story for b Per stopped until recent inclusion of optical interferometric data which led to the detection of a third, long-period component. As the interferometric observations continue to build up so to is the understanding of this binary system, with the modeled orbital parameters pointing to an edge-on orientation that may allow for the detection of an eclipse by the long-period component. These types of eclipse events are quite rare for long-period binaries due to the nearly edge-on orientation required for their detection, leaving open the opportunity for more traditional methods of observation to add to the body of knowledge concerning this understudied system. Here we present the latest observational data of the b Per system along with an introduction to the best fit orbital parameters governing the eclipsing nature of this complex triple-system. 14. KEPLER ECLIPSING BINARIES WITH STELLAR COMPANIONS SciTech Connect Gies, D. R.; Matson, R. A.; Guo, Z.; Lester, K. V.; Orosz, J. A.; Peters, G. J. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] 2015-12-15 Many short-period binary stars have distant orbiting companions that have played a role in driving the binary components into close separation. Indirect detection of a tertiary star is possible by measuring apparent changes in eclipse times of eclipsing binaries as the binary orbits the common center of mass. Here we present an analysis of the eclipse timings of 41 eclipsing binaries observed throughout the NASA Kepler mission of long duration and precise photometry. This subset of binaries is characterized by relatively deep and frequent eclipses of both stellar components. We present preliminary orbital elements for seven probable triple stars among this sample, and we discuss apparent period changes in seven additional eclipsing binaries that may be related to motion about a tertiary in a long period orbit. The results will be used in ongoing investigations of the spectra and light curves of these binaries for further evidence of the presence of third stars. 15. KOI-126: a triply eclipsing hierarchical triple with two low-mass stars. PubMed Carter, Joshua A; Fabrycky, Daniel C; Ragozzine, Darin; Holman, Matthew J; Quinn, Samuel N; Latham, David W; Buchhave, Lars A; Van Cleve, Jeffrey; Cochran, William D; Cote, Miles T; Endl, Michael; Ford, Eric B; Haas, Michael R; Jenkins, Jon M; Koch, David G; Li, Jie; Lissauer, Jack J; MacQueen, Phillip J; Middour, Christopher K; Orosz, Jerome A; Rowe, Jason F; Steffen, Jason H; Welsh, William F 2011-02-04 The Kepler spacecraft has been monitoring the light from 150,000 stars in its primary quest to detect transiting exoplanets. Here, we report on the detection of an eclipsing stellar hierarchical triple, identified in the Kepler photometry. KOI-126 [A, (B, C)], is composed of a low-mass binary [masses M(B) = 0.2413 ± 0.0030 solar mass (M(⊙)), M(C) = 0.2127 ± 0.0026 M(⊙); radii R(B) = 0.2543 ± 0.0014 solar radius (R(⊙)), R(C) = 0.2318 ± 0.0013 R(⊙); orbital period P(1) = 1.76713 ± 0.00019 days] on an eccentric orbit about a third star (mass M(A) = 1.347 ± 0.032 M(⊙); radius R(A) = 2.0254 ± 0.0098 R(⊙); period of orbit around the low-mass binary P(2) = 33.9214 ± 0.0013 days; eccentricity of that orbit e(2) = 0.3043 ± 0.0024). The low-mass pair probe the poorly sampled fully convective stellar domain offering a crucial benchmark for theoretical stellar models. 16. MML 53 - The Brightest Pre-Main-Sequence Eclipsing Binary Stempels, H. C.; Hebb, L. 2011-12-01 MML 53 is a newly discovered and bright (V=10.8m) pre-main-sequence eclipsing binary located in the 15-22 Myr old Upper-Centaur-Lupus (UCL) star forming region, with component masses of ˜1.02 M⊙ and ˜0.88 M⊙. This system is the first low-mass pre-main-sequence eclipsing binary discovered outside the ˜10 Myr old Orion star-forming region, and samples a slightly older age. A closer examination of MML 53 reveals that this is a three-component system, where the primary and secondary form a close eclipsing binary. Here we present preliminary results from our recent high-resolution spectroscopic study of this object, including estimates of the individual component masses, radii and temperatures. In addition we find indications that the tertiary in MML 53 interacts gravitationally with the eclipsing components. 17. Record-Breaking Eclipsing Binary Kohler, Susanna 2016-05-01 A new record holder exists for the longest-period eclipsing binary star system: TYC-2505-672-1. This intriguing system contains a primary star that is eclipsed by its companion once every 69 years with each eclipse lasting several years!120 Years of ObservationsIn a recent study, a team of scientists led by Joseph Rodriguez (Vanderbilt University) characterizes the components of TYC-2505-672-1. This binary star system consists of an M-type red giant star that undergoes a ~3.45-year-long, near-total eclipse with a period of ~69.1 years. This period is more than double that of the previous longest-period eclipsing binary!Rodriguez and collaborators combined photometric observations of TYC-2505-672-1 by the Kilodegree Extremely Little Telescope (KELT) with a variety of archival data, including observations by the American Association of Variable Star Observers (AAVSO) network and historical data from the Digital Access to a Sky Century @ Harvard (DASCH) program.In the 120 years spanned by these observations, two eclipses are detected: one in 1942-1945 and one in 2011-2015. The authors use the observations to analyze the components of the system and attempt to better understand what causes its unusual light curve.Characterizing an Unusual SystemObservations of TYC-2505-672-1 plotted from 1890 to 2015 reveal two eclipses. (The blue KELT observations during the eclipse show upper limits only.) [Rodriguez et al. 2016]By modeling the systems emission, Rodriguez and collaborators establish that TYC-2505-672-1 consists of a 3600-K primary star thats the M giant orbited by a small, hot, dim companion thats a toasty 8000 K. But if the companion is small, why does the eclipse last several years?The authors argue that the best model of TYC-2505-672-1 is one in which the small companion star is surrounded by a large, opaque circumstellar disk. Rodriguez and collaborators suggest that the companion could be a former red giant whose atmosphere was stripped from it, leaving behind 18. Radius constraints from high-speed photometry of 20 low-mass white dwarf binaries SciTech Connect Hermes, J. J.; Brown, Warren R.; Kilic, Mukremin; Gianninas, A.; Chote, Paul; Sullivan, D. J.; Winget, D. E.; Bell, Keaton J.; Falcon, R. E.; Winget, K. I.; Harrold, Samuel T.; Montgomery, M. H.; Mason, Paul A. 2014-09-01 We carry out high-speed photometry on 20 of the shortest-period, detached white dwarf binaries known and discover systems with eclipses, ellipsoidal variations (due to tidal deformations of the visible white dwarf), and Doppler beaming. All of the binaries contain low-mass white dwarfs with orbital periods of less than four hr. Our observations identify the first eight tidally distorted white dwarfs, four of which are reported for the first time here. We use these observations to place empirical constraints on the mass-radius relationship for extremely low-mass (≤0.30 M {sub ☉}) white dwarfs. We also detect Doppler beaming in several of these binaries, which confirms their high-amplitude radial-velocity variability. All of these systems are strong sources of gravitational radiation, and long-term monitoring of those that display ellipsoidal variations can be used to detect spin-up of the tidal bulge due to orbital decay. 19. The neutron-star low-mass X-ray binary H 1658-298 back in quiescence Parikh, Aastha; Wijnands, Rudy; Bahramian, Arash; Degenaar, Nathalie; Heinke, Craig 2017-03-01 The transient and eclipsing neutron-star low-mass X-ray binary H 1658-298 began its most recent outburst in August 2015 as determined using MAXI (ATel #7943) and we continued to monitor the outburst using Swift/XRT (e.g., ATel #7957, #8046). 20. Stripped red giant cores in eclipsing binary star systems Maxted, P. F. L.; Heber, U.; Smalley, B.; Marsh, T. R. 2013-02-01 Red giant stars can be stripped of their outer layers by stellar collisions or mass transfer in binary star systems such as low mass X-ray binaries. If the star is stripped on or before its first ascent of the red giant branch it will eventually become a very low mass white dwarf composed almost entirely of helium. Very low mass white dwarfs are well known in binary milli-second pulsars and many have recently been found in surveys such as the Sloan Digital Sky Survey, but the precursor phase during which the remnant evolves to higher effective temperature at nearly constant luminosity has rarely been observed. The cooling timescale for very low mass white dwarfs is very uncertain because they are thought to be born with thick hydrogen envelopes which can sustain weak but stable p-p shell burning, but unstable phases of CNO burning (shell flashes) can remove this hydrogen envelope. The predicted number of shell flashes (if any) is dependent on the mass and composition of the star and other details of the models used. In this talk I present new observations of a bright eclipsing binary star recently discovered in the WASP archive in which a stripped red giant is eclipsed by an A-type dwarf star. These observations were used to derive precise masses and radii for both stars and have be used to test the formation scenario outlined above. In addition, I present the main characteristics of 17 new eclipsing binary stars that are also likely to contain the precursors of very low mass white dwarfs. 1. ROTATIONAL DOPPLER BEAMING IN ECLIPSING BINARIES SciTech Connect Groot, Paul J. 2012-01-20 In eclipsing binaries the stellar rotation of the two components will cause a rotational Doppler beaming during eclipse ingress and egress when only part of the eclipsed component is covered. For eclipsing binaries with fast spinning components this photometric analog of the well-known spectroscopic Rossiter-McLaughlin effect can exceed the strength of the orbital effect. Example light curves are shown for a detached double white dwarf binary, a massive O-star binary and a transiting exoplanet case, similar to WASP-33b. Inclusion of the rotational Doppler beaming in eclipsing systems is a prerequisite for deriving the correct stellar parameters from fitting high-quality photometric light curves and can be used to determine stellar obliquities as well as, e.g., an independent measure of the rotational velocity in those systems that may be expected to be fully synchronized. 2. DETECTION OF LOW-MASS-RATIO STELLAR BINARY SYSTEMS SciTech Connect Gullikson, Kevin; Dodson-Robinson, Sarah 2013-01-01 O- and B-type stars are often found in binary systems, but the low binary mass-ratio regime is relatively unexplored due to observational difficulties. Binary systems with low mass ratios may have formed through fragmentation of the circumstellar disk rather than molecular cloud core fragmentation. We describe a new technique sensitive to G- and K-type companions to early B stars, a mass ratio of roughly 0.1, using high-resolution, high signal-to-noise spectra. We apply this technique to a sample of archived VLT/CRIRES observations of nearby B stars in the CO bandhead near 2300 nm. While there are no unambiguous binary detections in our sample, we identify HIP 92855 and HIP 26713 as binary candidates warranting follow-up observations. We use our non-detections to determine upper limits to the frequency of FGK stars orbiting early B-type primaries. 3. Coevality in Young Eclipsing Binaries 2017-06-01 The ages of the components in very short period pre-main-sequence (PMS) binaries are essential to an understanding of their formation. We considered a sample of seven PMS eclipsing binaries (EBs) with ages 1-6.3 MY and component masses 0.2-1.4 {M}⊙ . The very high precision with which their masses and radii have been measured and the capability provided by the Modules for Experiments in Stellar Astrophysics to calculate their evolutionary tracks at exactly the measured masses allows the determination of age differences of the components independent of their luminosities and effective temperatures. We found that the components of five EBs, ASAS J052821+0338.5, Parenago 1802, JW 380, CoRoT 223992193, and UScoCTIO 5, formed within 0.3 MY of each other. The parameters for the components of V1174 Ori imply an implausible large age difference of 2.7 MY and should be reconsidered. The seventh EB in our sample, RX J0529.4+0041 fell outside the applicability of our analysis. 4. Recent Minima of 171 Eclipsing Binary Stars Samolyk, G. 2015-12-01 This paper continues the publication of times of minima for 171 eclipsing binary stars from observations reported to the AAVSO EB section. Times of minima from observations received by the author from March 2015 thru October 2015 are presented. 5. Orbital period change of the low-mass X-ray binary EXO 0748-676 NASA Technical Reports Server (NTRS) Asai, Kazumi; Dotani, Tadayasu; Nagase, Fumiaki; Corbet, Robin H. D.; Shaham, Jacob 1992-01-01 The transient low-mass X-ray binary, EXO 0748-676, discovered with EXOSAT, is known to exhibit eclipses of a 492-s duration with a 3.82-hr period, intensity dips at pre-eclipse phases and type-I X-ray bursts. We observed this source with Ginga in 1989 March, 1990 December, 1991 January, and 1991 August and determined nine eclipse center times. Combining these eclipse center times with the previous result of the EXOSAT observations, we find that the orbital period of this source is not decaying monotonically, contrary to the previously reported suggestion. Instead, it shows a more complex behavior. A quadratic fit to the eclipse data yields a positive rate of change in orbital period with an approximate rate of 0.9 x 10 exp 7/yr, although the EXOSAT observations made in 1985 do not fit this trend. A sinusoidal function gives a better fit to the observed orbital period changes with a period of about 12 yr and an amplitude of about 44 lt-s, although the period is much longer than the observation interval of about 6.5 yr. Possible mechanisms for the orbital period change are discussed. 6. Evolution of Low-mass X-Ray Binaries: The Effect of Donor Evaporation Jia, Kun; Li, Xiang-Dong 2016-10-01 Millisecond pulsars (MSPs) are thought to originate from low-mass X-ray binaries (LMXBs). The discovery of eclipsing radio MSPs, including redbacks and black widows, indicates that evaporation of the donor star by the MSP’s irradiation takes place during the LMXB evolution. In this work, we investigate the effect of donor evaporation on the secular evolution of LMXBs, considering different evaporation efficiencies and related angular momentum loss. We find that for widening LMXBs, the donor star leaves a less massive white dwarf than without evaporation; for contracting systems, evaporation can speed up the evolution, resulting in dynamically unstable mass transfer and possibly the formation of isolated MSPs. 7. The low-mass classic Algol-type binary UU Leo revisited Yang, Yuan-Gui 2013-12-01 New multi-color photometry of the eclipsing binary UU Leo, acquired from 2010 to 2013, was carried out by using the 60-cm and 85-cm telescopes at the Xinglong station, which is administered by National Astronomical Observatories, Chinese Academy of Sciences. With the updated Wilson-Devinney code, the photometric solution was derived from BVR light curves. The results imply that UU Leo is a semi-detached Algol-type binary, with a mass ratio of q = 0.100(±0.002). The change in orbital period was reanalyzed based on all available eclipsing times. The O - C curve could be described by an upward parabola superimposed on a quasi-sinusoidal curve. The period and semi-amplitudes are Pmod = 54.5(±1.1) yr and A = 0.0273d(±0.0015d), which may be attributed to the light-time effect via the presence of an invisible third body. The long-term period increases at a rate of dP/dt = +4.64(±0.14) × 10-7d yr-1, which may be interpreted by the conserved mass being transferred from the secondary to the primary. With mass being transferred, the low-mass Algol-type binary UU Leo may evolve into a binary system with a main sequence star and a helium white dwarf. 8. The Kepler Mission and Eclipsing Binaries DTIC Science & Technology 2006-01-01 Opportunities to Participate, 2005, in A Decade of Extrasolar Planets around Normal Stars (ed. M. Livio), Cambridge: Cambridge University Press, in...smaller planets in the habitable zone of solar-like stars. In the process, many eclipsing binaries (EB) will also be detected and light curves pro- duced...determine the component masses and thereby separate eclipses caused by stellar companions from transits caused by planets . The result will be a rich 9. ROTATIONAL VELOCITIES OF INDIVIDUAL COMPONENTS IN VERY LOW MASS BINARIES SciTech Connect Konopacky, Q. M.; Macintosh, B. A.; Ghez, A. M.; Fabrycky, D. C.; White, R. J.; Barman, T. S.; Rice, E. L.; Hallinan, G.; Duchene, G. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] 2012-05-01 We present rotational velocities for individual components of 11 very low mass (VLM) binaries with spectral types between M7 and L7.5. These results are based on observations taken with the near-infrared spectrograph, NIRSPEC, and the Keck II laser guide star adaptive optics system. We find that the observed sources tend to be rapid rotators (v sin i > 10 km s{sup -1}), consistent with previous seeing-limited measurements of VLM objects. The two sources with the largest v sin i, LP 349-25B and HD 130948C, are rotating at {approx}30% of their break-up speed, and are among the most rapidly rotating VLM objects known. Furthermore, five binary systems, all with orbital semimajor axes {approx}<3.5 AU, have component v sin i values that differ by greater than 3{sigma}. To bring the binary components with discrepant rotational velocities into agreement would require the rotational axes to be inclined with respect to each other, and that at least one component is inclined with respect to the orbital plane. Alternatively, each component could be rotating at a different rate, even though they have similar spectral types. Both differing rotational velocities and inclinations have implications for binary star formation and evolution. We also investigate possible dynamical evolution in the triple system HD 130948A-BC. The close binary brown dwarfs B and C have significantly different v sin i values. We demonstrate that components B and C could have been torqued into misalignment by the primary star, A, via orbital precession. Such a scenario can also be applied to another triple system in our sample, GJ 569A-Bab. Interactions such as these may play an important role in the dynamical evolution of VLM binaries. Finally, we note that two of the binaries with large differences in component v sin i, LP 349-25AB and 2MASS 0746+20AB, are also known radio sources. 10. Rotational Velocities of Individual Components in Very Low Mass Binaries Konopacky, Q. M.; Ghez, A. M.; Fabrycky, D. C.; Macintosh, B. A.; White, R. J.; Barman, T. S.; Rice, E. L.; Hallinan, G.; Duchêne, G. 2012-05-01 We present rotational velocities for individual components of 11 very low mass (VLM) binaries with spectral types between M7 and L7.5. These results are based on observations taken with the near-infrared spectrograph, NIRSPEC, and the Keck II laser guide star adaptive optics system. We find that the observed sources tend to be rapid rotators (v sin i > 10 km s-1), consistent with previous seeing-limited measurements of VLM objects. The two sources with the largest v sin i, LP 349-25B and HD 130948C, are rotating at ~30% of their break-up speed, and are among the most rapidly rotating VLM objects known. Furthermore, five binary systems, all with orbital semimajor axes lsim3.5 AU, have component v sin i values that differ by greater than 3σ. To bring the binary components with discrepant rotational velocities into agreement would require the rotational axes to be inclined with respect to each other, and that at least one component is inclined with respect to the orbital plane. Alternatively, each component could be rotating at a different rate, even though they have similar spectral types. Both differing rotational velocities and inclinations have implications for binary star formation and evolution. We also investigate possible dynamical evolution in the triple system HD 130948A-BC. The close binary brown dwarfs B and C have significantly different v sin i values. We demonstrate that components B and C could have been torqued into misalignment by the primary star, A, via orbital precession. Such a scenario can also be applied to another triple system in our sample, GJ 569A-Bab. Interactions such as these may play an important role in the dynamical evolution of VLM binaries. Finally, we note that two of the binaries with large differences in component v sin i, LP 349-25AB and 2MASS 0746+20AB, are also known radio sources. 11. Periodicity of Eclipsing Binary Star GK Cepheus 2001-10-01 Eclipsing Binary stars are stars in which there is some mass exchange taking place between two main bodies. This mass exchange produces a change in the magnitude or “brightness” of the star. The star known as GK Cephius has been listed as an eclipsing binary in number of publications, journal articles, and data tables. If the light curve is examined carefully, it exhibits some behavior that is not typical of simple eclipsing binary stars. A study of this light curve is underway to examine the possibility of another gravitational influence being at work in the region of this star. In this paper we will report on the predictions concerning an additional candidate that may be influencing the light curves of the GK Cephius system. 12. The Low-mass Astrometric Binary LSR 1610-0040 Koren, Seth C.; Blake, Cullen H.; Dahn, Conard C.; Harris, Hugh C. 2016-03-01 Even though it was discovered more than a decade ago, LSR 1610-0040 remains an enigma. This object has a peculiar spectrum that exhibits some features typically found in L subdwarfs, and others common in the spectra of more massive M dwarf stars. It is also a binary system with a known astrometric orbital solution. Given the available data, it remains a challenge to reconcile the observed properties of the combined light of LSR 1610-0040AB with current theoretical models of low-mass stars and brown dwarfs. We present the results of a joint fit to both astrometric and radial velocity measurements of this unresolved, low-mass binary. We find that the photocentric orbit has a period P=633.0+/- 1.7 days, somewhat longer than previous results, eccentricity of e=0.42+/- 0.03, and we estimate that the semimajor axis of the orbit of the primary is {a}1≈ 0.32 {{AU}}, consistent with previous results. While a complete characterization of the system is limited by our small number of radial velocity measurements, we establish a likely primary mass range of 0.09-0.10 {M}⊙ from photometric and color-magnitude data. For a primary mass in this range, the secondary is constrained to be 0.06-0.075 {M}⊙ , making a negligible contribution to the total I-band luminosity. This effectively rules out the possibility of the secondary being a compact object such as an old, low-mass white dwarf. Based on our analysis, we predict a likely angular separation at apoapsis comparable to the resolution limits of current high-resolution imaging systems. Measuring the angular separation of the A and B components would finally enable a full, unambiguous solution for the masses of the components of this system. 13. Survey of Candidate Pulsating Eclipsing Binaries - I Dvorak, S. 2009-08-01 Initial results from a photometric survey of stars selected from the list of eclipsing binaries that may contain a pulsating component by Soydugan et al. (2006) are reported. A minimum of two nights of CCD observations with V and/or B filters of each of the 35 stars from this list was collected. Of the 35 stars stud- ied, a pulsating component was detected in three of the systems. Pulsations were also serendiptiously detected in the eclipsing binary RR Leporis, which is not on the candidate list. 14. Eclipsing Binaries in the 21st Century—Opportunities for Amateur Astronomers Guinan, E. F.; Engle, S. G.; Devinney, E. J. 2012-06-01 Eclipsing binaries play major roles in modern astrophysical research. These stars provide fundamental data on the masses, radii, ages, atmospheres, and interiors of stars as well as serving as test beds for stellar structure and evolution models. The study of eclipsing binaries also returns vital information about the formation and evolution of close binaries themselves. Studying the changes in their periods from the observations of eclipse timings provides insights into evolution of close binaries, mass exchange and loss, apsidal motion for eccentric systems, as well as the discovery of the low mass (unseen) third bodies. Moreover eclipsing binaries in clusters and other galaxies can provide accurate distances to the star clusters and galaxies in which they reside. More recently observations of eclipsing exoplanet-star systems (that is, transiting exoplanets) when coupled with spectroscopy are yielding fundamental information about the frequency and the physical properties of planets orbiting other stars. For the reasons discussed above, observations of eclipsing binary systems have been popular for AAVSO observers and many papers have been published (see Williams et al. 2012, JAAVSO, 40, No. 1). A recent example is the highly successful AAVSO’s Citizen Sky Project focused on the enigmatic long-period eclipsing binary ɛ Aur. Building on the success of the AAVSO during the last century, this paper explores the present and future prospects for research in eclipsing binaries. We focus on what can be done by AAVSO members and other amateur astronomers in the study of eclipsing binaries. Several examples of observing strategies and interesting (and scientifically valuable) projects are discussed as well as future prospects. As discussed, there are many opportunities for AAVSO members to contribute to study of eclipsing binary stars and an increasing variety of objects to observe. 15. THE RADIUS DISCREPANCY IN LOW-MASS STARS: SINGLE VERSUS BINARIES SciTech Connect Spada, F.; Demarque, P.; Kim, Y.-C.; Sills, A. 2013-10-20 A long-standing issue in the theory of low-mass stars is the discrepancy between predicted and observed radii and effective temperatures. In spite of the increasing availability of very precise radius determinations from eclipsing binaries and interferometric measurements of radii of single stars, there is no unanimous consensus on the extent (or even the existence) of the discrepancy and on its connection with other stellar properties (e.g., metallicity, magnetic activity). We investigate the radius discrepancy phenomenon using the best data currently available (accuracy ∼< 5%). We have constructed a grid of stellar models covering the entire range of low-mass stars (0.1-1.25 M{sub ☉}) and various choices of the metallicity and mixing length parameter, α. We used an improved version of the Yale Rotational stellar Evolution Code, implementing surface boundary conditions based on the most up-to-date PHOENIX atmosphere models. Our models are in good agreement with others in the literature and improve and extend the low mass end of the Yale-Yonsei isochrones. Our calculations include rotation-related quantities, such as moments of inertia and convective turnover timescales, useful in studies of magnetic activity and rotational evolution of solar-like stars. Consistent with previous works, we find that both binaries and single stars have radii inflated by about 3% with respect to the theoretical models; among binaries, the components of short orbital period systems are found to be the most deviant. We conclude that both binaries and single stars are comparably affected by the radius discrepancy phenomenon. 16. New Eclipsing Contact Binary System in Auriga Austin, S. J.; Robertson, J. W.; Justice, C.; Campbell, R. T.; Hoskins, J. 2004-05-01 We present data on a newly discovered eclipsing binary system. The serendipitous discovery of this variable star was made by J.W. Robertson analyzing inhomogeneous ensemble photometry of stars in the field of the cataclysmic variable FS Aurigae from Indiana University RoboScope data. We obtained differential time-series BVR photometry during 2003 of this field variable using an ensemble of telescopes including the university observatories at ATU, UCA and joint ventures with amateur observatories in the state of Arkansas (Whispering Pines Observatory and Nubbin Ridge Observatory). The orbital period of this eclipsing system is 0.2508 days. The B-V light curve indicates colors of 1.2 around quadrature, to nearly 1.4 at primary eclipse. Binary star light curve models that best fit the BVR differential photometry suggest that the system is a contact binary overfilling the inner Roche Lobe by 12%, a primary component with a temperature of 4350K, a secondary component with a temperature of 3500K, a mass ratio of 0.37, and an inclination of 83 degrees. We present BVR light curves, an ephemeris, and best fit model parameters for the physical characteristics of this new eclipsing binary system. 17. Infrared spectroscopy of low-mass X-ray binaries Bandyopadhyay, R.; Shahbaz, T.; Charles, P. A.; van Kerkwijk, M. H.; Naylor, T. 1997-03-01 Using CGS4 on UKIRT, we have obtained the first 2.05-2.45 μm infrared spectra of the Galactic bulge low-mass X-ray binaries (LMXBs) GX1+4 and GX13+1. We report the detection of Brackett gamma emission from the accretion disc in both systems, confirming the identification of the IR counterpart to GX13+1. In addition, both spectra show CO molecular bands and metal lines in absorption, representing the first infrared spectroscopic detection of the secondary in a heavily obscured bulge source. We also present a JHK spectrum of the LMXB ScoX-1, which shows strong Hi, Hei and HeII emission. 18. Fundamental Calibrators for Stellar Evolution Models: New Eclipsing Binaries in Young Clusters Identified by K2 David, Trevor 2016-07-01 Double-lined eclipsing binaries serve as fundamental calibrators for stellar evolution models. Benchmark grade calibrators (with mass and radius uncertainties of 3%) having component masses below 1 solar mass are rare, particularly at pre-main sequence stages. We present the discovery and characterization of new eclipsing binaries in young stellar clusters, all identified by K2. In the 5-10 Myr old Upper Scorpius region, the nearest OB association, we present the lowest mass stellar eclipsing binary to date, with both components close to the hydrogen burning limit. Also in Upper Scorpius, we present evidence for a hierarchical triple with an eclipsing pair of brown dwarfs, only the second eclipsing brown dwarf pair known to date. In the 110-125 Myr old Pleiades cluster, only one eclipsing binary was known prior to the K2 mission. We present three new Pleiades eclipsing binaries, all with system masses less than 1 solar mass. We use these systems to critically assess stellar evolution models at low masses and young ages. K2 data in hand has also revealed new eclipsing and transiting systems in the moderately older (600-800 Myr) Hyades and Praesepe clusters. 19. The Galactic Distribution of Contact Eclipsing Binaries Castelaz, Michael W.; Dorn, Leah; Breitfeld, Abby; Mies, Regan; Avery, Tess 2017-01-01 The number of eclipsing contact binaries in different galactic latitudes and longitudes show peak distributions in the number per square degree in two latitudinal zones (-30 degrees to -25 degrees and +25 degrees to +30 degrees) and large fluctuations in longitude (Huang and Wade 1966, ApJ, 143, 146). Semi-detached or detached binaries are largely concentrated in the galactic plane as shown by Paczynski et al. (MNRAS, 368, 1311), different from the distribution of contact eclipsing binaries. The differences in distributions of different types of eclipsing binaries may be related to either distances or interstellar reddening. We will present a method to calculate photometric distances of W Urase Majoris systems (W UMa; used as a proxy for contact binaries) from 2MASS J and K magnitudes and interstellar reddening models (Schlafly and Finkbeiner 2011, ApJ. 737, 103). We compare the distances to those calculated from the period-luminosity-color relationship described by Rucinski (2004, NewAR, 48, 703). The W UMa systems are taken from the General Catalog of Variable Stars. 20. The low mass ratio contact binary system V728 Herculis Erkan, N.; Ulaş, B. 2016-07-01 We present the orbital period study and the photometric analysis of the contact binary system V728 Her. Our orbital period analysis shows that the period of the system increases (dP / dt = 1.92 ×10-7 dyr-1) and the mass transfer rate from the less massive component to more massive one is 2.51 ×10-8 M⊙y-1 . In addition, an advanced sinusoidal variation in period can be attributed to the light-time effect by a tertiary component or the Applegate mechanism triggered by the secondary component. The simultaneous multicolor BVR light and radial velocity curves solution indicates that the physical parameters of the system are M1 = 1.8M⊙ , M2 = 0.28M⊙ , R1 = 1.87R⊙ , R2 = 0.82R⊙ , L1 = 5.9L⊙ , and L2 = 1.2L⊙ . We discuss the evolutionary status and conclude that V728 Her is a deep (f = 81%), low mass ratio (q = 0.16) contact binary system. 1. Infrared Spectroscopy of Low-mass X-ray Binaries Bandyopadhyay, R. M.; Shahbaz, T.; Charles, P. A.; Naylor, T. 1999-04-01 Using CGS4 on UKIRT, we have obtained 2.00--2.45 mu m infrared spectra of a number of low-mass X-ray binaries including Sco X-1, Sco X-2, and GX13+1. Sco X-1 shows emission lines only, supporting our previous conclusion that the spectral type of the evolved secondary must be earlier than G5. Emission lines are also seen in the spectrum of Sco X-2, confirming the identity of the IR counterpart. We report the detection of CO bands in GX13+1 and estimate the most likely spectral type of the secondary to be K5 sc iii. We also find P Cygni type profiles in the Brackett gamma lines of Sco X-1 and GX13+1, indicating the presence of high velocity outflows in these systems. We present spectra of candidate IR counterparts to several other elusive X-ray binaries. Finally, implications for the nature and classification of these systems are discussed. 2. Observations of the eclipsing binary b Persei Templeton, Matthew R. 2015-01-01 Dr. Robert Zavala (USNO-Flagstaff) et al. request V time-series observations of the bright variable star b Persei 7-21 January 2015 UT, in hopes of catching a predicted eclipse on January 15. This is a follow-up to the February 2013 campaign announced in Alert Notice 476, and will be used as a photometric comparison for upcoming interferometric observations with the Navy Precision Optical Interferometer (NPOI) in Arizona. b Per (V=4.598, B-V=0.054) is ideal for photoelectric photometers or DSLR cameras. Telescopic CCD observers may observe by stopping down larger apertures. Comparison and check stars assigned by PI: Comp: SAO 24412, V=4.285, B-V = -0.013; Check: SAO 24512, V=5.19, B-V = -0.05. From the PI: "[W]e wanted to try and involve AAVSO observers in a follow up to our successful detection of the b Persei eclipse of Feb 2013, AAVSO Alert Notice 476 and Special Notice 333. Our goal now is to get good time resolution photometry as the third star passes in front of the close ellipsoidal binary. The potential for multiple eclipses exists. The close binary has a 1.5 day orbital period, and the eclipsing C component requires about 4 days to pass across the close binary pair. The primary eclipse depth is 0.15 magnitude. Photometry to 0.02 or 0.03 mags would be fine to detect this eclipse. Eclipse prediction date (JD 2457033.79 = 2015 01 11 UT, ~+/- 1 day) is based on one orbital period from the 2013 eclipse." More information is available at PI's b Persei eclipse web page: http://inside.warren-wilson.edu/~dcollins/bPersei/. Finder charts with sequence may be created using the AAVSO Variable Star Plotter (https://www.aavso.org/vsp). Observations should be submitted to the AAVSO International Database. See full Alert Notice for more details and information on the targets. 3. HIGH-PRECISION DYNAMICAL MASSES OF VERY LOW MASS BINARIES SciTech Connect Konopacky, Q. M.; Ghez, A. M.; McLean, I. S.; Barman, T. S.; Rice, E. L.; Bailey, J. I.; White, R. J.; Duchene, G. E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] 2010-03-10 We present the results of a three year monitoring program of a sample of very low mass (VLM) field binaries using both astrometric and spectroscopic data obtained in conjunction with the laser guide star adaptive optics system on the W. M. Keck II 10 m telescope. Among the 24 systems studied, 15 have undergone sufficient orbital motion, allowing us to derive their relative orbital parameters and hence their total system mass. These measurements more than double the number of mass measurements for VLM objects, and include the most precise mass measurement to date (<2%). Among the 11 systems with both astrometric and spectroscopic measurements, six have sufficient radial velocity variations to allow us to obtain individual component masses. This is the first derivation of the component masses for five of these systems. Altogether, the orbital solutions of these low mass systems show a correlation between eccentricity and orbital period, consistent with their higher mass counterparts. In our primary analysis, we find that there are systematic discrepancies between our dynamical mass measurements and the predictions of theoretical evolutionary models (TUCSON and LYON) with both models either underpredicting or overpredicting the most precisely determined dynamical masses. These discrepancies are a function of spectral type, with late-M through mid-L systems tending to have their masses underpredicted, while one T-type system has its mass overpredicted. These discrepancies imply that either the temperatures predicted by evolutionary and atmosphere models are inconsistent for an object of a given mass, or the mass-radius relationship or cooling timescales predicted by the evolutionary models are incorrect. If these spectral-type trends are correct and hold into the planetary mass regime, the implication is that the masses of directly imaged extrasolar planets are overpredicted by the evolutionary models. 4. KEPLER ECLIPSING BINARY STARS. II. 2165 ECLIPSING BINARIES IN THE SECOND DATA RELEASE SciTech Connect Slawson, Robert W.; Doyle, Laurance R.; Prsa, Andrej; Engle, Scott G.; Conroy, Kyle; Coughlin, Jared; Welsh, William F.; Orosz, Jerome A.; Gregg, Trevor A.; Fetherolf, Tara; Short, Donald R.; Windmiller, Gur; Rucker, Michael; Batalha, Natalie; Fabrycky, Daniel C.; Jenkins, Jon M.; Mullally, F.; Seader, Shawn E. 2011-11-15 The Kepler Mission provides nearly continuous monitoring of {approx}156,000 objects with unprecedented photometric precision. Coincident with the first data release, we presented a catalog of 1879 eclipsing binary systems identified within the 115 deg{sup 2} Kepler field of view (FOV). Here, we provide an updated catalog augmented with the second Kepler data release which increases the baseline nearly fourfold to 125 days. Three hundred and eighty-six new systems have been added, ephemerides and principal parameters have been recomputed. We have removed 42 previously cataloged systems that are now clearly recognized as short-period pulsating variables and another 58 blended systems where we have determined that the Kepler target object is not itself the eclipsing binary. A number of interesting objects are identified. We present several exemplary cases: four eclipsing binaries that exhibit extra (tertiary) eclipse events; and eight systems that show clear eclipse timing variations indicative of the presence of additional bodies bound in the system. We have updated the period and galactic latitude distribution diagrams. With these changes, the total number of identified eclipsing binary systems in the Kepler FOV has increased to 2165, 1.4% of the Kepler target stars. An online version of this catalog is maintained at http://keplerEBs.villanova.edu. 5. Eclipsing Binary B-Star Mass Determinations Townsend, Amanda; Eikenberry, Stephen S. 2016-01-01 B-stars in binary pairs provide a laboratory for key astrophysical measurements of massive stars, including key insights for the formation of compact objects (neutron stars and black holes). In their paper, Martayan et al (2004) find 23 Be binary star pairs in NGC2004 in the Large Magellanic Cloud, five of which are both eclipsing and spectroscopic binaries with archival data from VLT-Giraffe and photometric data from MACHO. By using the Wilson eclipsing binary code (e.g., Wilson, 1971), we can determine preliminary stellar masses of the binary components. We present the first results from this analysis. This study also serves as proof-of-concept for future observations with the Photonic Synthesis Telescope Array (Eikenberry et al., in prep) that we are currently building for low-cost, precision spectroscopic observations. With higher resolution and dedicated time for observations, we can follow-up observations of these Be stars as well as Be/X-ray binaries, for improved mass measurements of neutron stars and black holes and better constraints on their origin/formation. 6. A Unified Model of Low Mass X-ray Binaries Balucinska-Church, M.; Church, M. 2014-07-01 We present a unified physical model of Low Mass X-ray Binaries explaining the basic Atoll and Z-track types of source. In all LMXB with luminosity above 1-2.10^{37} erg/s, we have a new fundamental result that the temperature of the Comptonizing ADC corona equals that of the neutron star, i.e. there is thermal equilibrium. This equilibrium explains the properties of the basic Banana State of Atoll sources. Below this luminosity, equilibrium breaks down, T_ADC rising towards 100 keV by an unknown heating mechanism, explaining the Island State. Above 5.10^{37} erg/s flaring begins in the GX-Atolls which we show is unstable nuclear burning. Above 1.10^{38} erg/s, LMXB are seen as Z-track sources. Flaring in these and the GX-Atolls occurs when the mass accretion rate to the neutron star falls to the critical value for unstable nuclear burning on the star. Below 2.10^{37} erg/s, a different unstable burning: X-ray bursting, takes over. We show that the Normal Branch of the Z-track consists simply of increasing mass accretion rate, as is the Banana State in Atolls. In the Horizontal Branch, a measured, strongly increasing radiation pressure of the neutron star disrupts the inner disk launching the relativistic jets seen on this branch. 7. EVOLUTION OF TRANSIENT LOW-MASS X-RAY BINARIES TO REDBACK MILLISECOND PULSARS SciTech Connect Jia, Kun; Li, Xiang-Dong 2015-11-20 Redback millisecond pulsars (MSPs; hereafter redbacks) are a subpopulation of eclipsing MSPs in close binaries. The formation processes of these systems are not clear. The three pulsars showing transitions between rotation- and accretion-powered states belong to both redbacks and transient low-mass X-ray binaries (LMXBs), suggesting a possible evolutionary link between them. Through binary evolution calculations, we show that the accretion disks in almost all LMXBs are subject to the thermal-viscous instability during certain evolutionary stages, and the parameter space for the disk instability covers the distribution of known redbacks in the orbital period—companion mass plane. We accordingly suggest that the abrupt reduction of the mass accretion rate during quiescence of transient LMXBs provides a plausible way to switch on the pulsar activity, leading to the formation of redbacks, if the neutron star has been spun up to be an energetic MSP. We investigate the evolution of redbacks, taking into account the evaporation feedback, and discuss its possible influence on the formation of black widow MSPs. 8. Apsidal motion in eclipsing binary GG Orionis Yilan, E.; Bulut, I. 2016-03-01 The study of apsidal motion in binary stars with eccentric orbit is well known as an important source of information for the stellar internal structure as well as the possibility of verification of general relativity. In this study, the apsidal motion of the eccentric eclipsing binary GG Ori (P = 6.631 days, e = 0.22) has been analyzed using the times of minimum light taken from the literature and databases and the elements of apsidal motion have been computed. The method described by Giménez and García-Pelayo (1983) has been used for the apsidal motion analysis. 9. EL CVn-type binaries - discovery of 17 helium white dwarf precursors in bright eclipsing binary star systems Maxted, P. F. L.; Bloemen, S.; Heber, U.; Geier, S.; Wheatley, P. J.; Marsh, T. R.; Breedt, E.; Sebastian, D.; Faillace, G.; Owen, C.; Pulley, D.; Smith, D.; Kolb, U.; Haswell, C. A.; Southworth, J.; Anderson, D. R.; Smalley, B.; Collier Cameron, A.; Hebb, L.; Simpson, E. K.; West, R. G.; Bochinski, J.; Busuttil, R.; Hadigal, S. 2014-01-01 The star 1SWASP J024743.37-251549.2 was recently discovered to be a binary star in which an A-type dwarf star eclipses the remnant of a disrupted red giant star (WASP 0247-25 B). The remnant is in a rarely observed state evolving to higher effective temperatures at nearly constant luminosity prior to becoming a very low mass white dwarf composed almost entirely of helium, i.e. it is a pre-helium white dwarf (pre-He-WD). We have used the photometric database from the Wide Angle Search for Planets (WASP) to find 17 eclipsing binary stars with orbital periods P = 0.7-2.2 d with similar light curves to 1SWASP J024743.37-251549.2. The only star in this group previously identified as a variable star is the brightest one, EL CVn, which we adopt as the prototype for this class of eclipsing binary star. The characteristic light curves of EL CVn-type stars show a total eclipse by an A-type dwarf star of a smaller, hotter star and a secondary eclipse of comparable depth to the primary eclipse. We have used new spectroscopic observations for six of these systems to confirm that the companions to the A-type stars in these binaries have very low masses ({≈ } 0.2{ M_{⊙}}). This includes the companion to EL CVn which was not previously known to be a pre-He-WD. EL CVn-type binary star systems will enable us to study the formation of very low mass white dwarfs in great detail, particularly in those cases where the pre-He-WD star shows non-radial pulsations similar to those recently discovered in WASP0247-25 B. 10. Kepler Eclipsing Binary Stars. VIII. Identification of False Positive Eclipsing Binaries and Re-extraction of New Light Curves Abdul-Masih, Michael; Prša, Andrej; Conroy, Kyle; Bloemen, Steven; Boyajian, Tabetha; Doyle, Laurance R.; Johnston, Cole; Kostov, Veselin; Latham, David W.; Matijevič, Gal; Shporer, Avi; Southworth, John 2016-04-01 The Kepler mission has provided unprecedented, nearly continuous photometric data of ∼200,000 objects in the ∼105 deg2 field of view (FOV) from the beginning of science operations in May of 2009 until the loss of the second reaction wheel in May of 2013. The Kepler Eclipsing Binary Catalog contains information including but not limited to ephemerides, stellar parameters, and analytical approximation fits for every known eclipsing binary system in the Kepler FOV. Using target pixel level data collected from Kepler in conjunction with the Kepler Eclipsing Binary Catalog, we identify false positives among eclipsing binaries, i.e., targets that are not eclipsing binaries themselves, but are instead contaminated by eclipsing binary sources nearby on the sky and show eclipsing binary signatures in their light curves. We present methods for identifying these false positives and for extracting new light curves for the true source of the observed binary signal. For each source, we extract three separate light curves for each quarter of available data by optimizing the signal-to-noise ratio, the relative percent eclipse depth, and the flux eclipse depth. We present 289 new eclipsing binaries in the Kepler FOV that were not targets for observation, and these have been added to the catalog. An online version of this catalog with downloadable content and visualization tools is maintained at http://keplerEBs.villanova.edu. 11. Phenomenological modelling of eclipsing binary stars Andronov, I. L.; Tkachenko, M. G.; Chinarova, L. L. 2016-03-01 We review the method NAV ("New Algol Variable") first introduced in (2012Ap.....55..536A) which uses the locally-dependent shapes of eclipses in an addition to the trigonometric polynomial of the second order (which typically describes the "out-of-eclipse" part of the light curve with effects of reflection, ellipticity and O'Connell). Eclipsing binary stars are believed to show distinct eclipses only if belonging to the EA (Algol) type. With a decreasing eclipse width, the statistically optimal value of the trigonometric polynomial s(2003ASPC..292..391A) drastically increases from ~2 for elliptic (EL) variables without eclipses, ~6-8 for EW and up to ~30-50 for some EA with narrow eclipses. In this case of large number of parameters, the smoothing curve becomes very noisy and apparent waves (the Gibbs phenomenon) may be seen. The NAV set of the parameters may be used for classification in the GCVS, VSX and similar catalogs. The maximal number of parameters is m=12, which corresponds to s=5, if correcting both the period and the initial epoch. We have applied the method to few stars, also in a case of multi-color photometry (2015JASS...32..127A), when it is possible to use the phenomenological parameters from the NAV fit to estimate physical parameters using statistical dependencies. For the one-color observations, one may estimate the ratio of the surface brightnesses of the components. We compiled a catalog of phenomenological characteristics based on published observations. We conclude that the NAV approximation is better than the TP one even for the case of EW-type stars with much wider eclipses. It may also be used to determine timings (see 2005ASPC..335...37A for a review of methods) or to determine parameters in the case of variable period, using a complete light curve modeling the phase variations. The method is illustrated on 2MASS J11080447-6143290 (EA-type), USNO-B1.0 1265-0306001 and USNO-B1.01266-0313413 (EW-type) and compared to various other methods 12. DEEP, LOW MASS RATIO OVERCONTACT BINARY SYSTEMS. XIII. DZ PISCIUM WITH INTRINSIC LIGHT VARIABILITY SciTech Connect Yang, Y.-G.; Dai, H.-F.; Qian, S.-B.; Soonthornthum, B. E-mail: [email protected] 2013-08-01 New multi-color photometry for the eclipsing binary DZ Psc was performed in 2011 and 2012 using the 85 cm telescope at the Xinglong Station of the National Astronomical Observatories of China. Using the updated Wilson-Devinney (W-D) code, we deduced two sets of photometric solutions. The overcontact degree is f = 89.7({+-} 1.0)%, identifying DZ Psc as a deep, low mass ratio overcontact binary. The asymmetric light curves (i.e., LC{sub 2} in 2012) were modeled by a hot spot on the primary star. Based on all of the available light minimum times, we discovered that the orbital period of DZ Psc may be undergoing a secular period increase with a cyclic variation. The modulated period and semi-amplitude of this oscillation are P{sub mod} = 11.89({+-} 0.19) yr and A = 0.0064({+-} 0.0006) days, which may be possibly attributed to either cyclic magnetic activity or light-time effect due to the third body. The long-term period increases at a rate of dP/dt=+7.43({+-}0.17) Multiplication-Sign 10{sup -7} days yr{sup -1}, which may be interpreted as conserved mass transfer from the less massive component to the more massive one. With mass transferring, DZ Psc will finally merge into a rapid-rotation single star when J{sub spin}/J{sub orb} > 1/3. 13. Deep, Low Mass Ratio Overcontact Binary Systems. XIII. DZ Piscium with Intrinsic Light Variability Yang, Y.-G.; Qian, S.-B.; Zhang, L.-Y.; Dai, H.-F.; Soonthornthum, B. 2013-08-01 New multi-color photometry for the eclipsing binary DZ Psc was performed in 2011 and 2012 using the 85 cm telescope at the Xinglong Station of the National Astronomical Observatories of China. Using the updated Wilson-Devinney (W-D) code, we deduced two sets of photometric solutions. The overcontact degree is f = 89.7(± 1.0)%, identifying DZ Psc as a deep, low mass ratio overcontact binary. The asymmetric light curves (i.e., LC2 in 2012) were modeled by a hot spot on the primary star. Based on all of the available light minimum times, we discovered that the orbital period of DZ Psc may be undergoing a secular period increase with a cyclic variation. The modulated period and semi-amplitude of this oscillation are P mod = 11.89(± 0.19) yr and A = 0.0064(± 0.0006) days, which may be possibly attributed to either cyclic magnetic activity or light-time effect due to the third body. The long-term period increases at a rate of dP/dt=+7.43(+/- 0.17)\\times 10^{-7}{\\,days\\, yr^{-1}}, which may be interpreted as conserved mass transfer from the less massive component to the more massive one. With mass transferring, DZ Psc will finally merge into a rapid-rotation single star when J spin/J orb > 1/3. 14. Photometry of Two Intense Low Mass X-Ray Binaries Wachter, S.; Margon, B.; Anderson, S. 1995-12-01 The intense galactic X-ray source GX349+2 (Sco X-2) belongs to the class of persistently bright low-mass X-ray binaries called Z-sources. GX349+2 has only recently been optically identified with a 19th mag star. Of the six known Z-sources, only two (Sco X-1 and Cyg X-2) have been studied in the optical. It has been suggested that Z-sources as a group are characterized by evolved companions and correspondingly long orbital periods (Sco X-1, P=0.8d; Cyg X-2, P=9.8d). Recently Southwell et al. have presented spectroscopic observations of GX349+2 suggesting a 14d orbital period. We have obtained broadband photometry of the system on six consecutive nights in May 1995, and find evidence for a 21.7 +/- 0.3hr period of 0.14 mag half-amplitude, superposed on erratic flickering typical of Sco X-1 type objects. As with other Z-sources, caution will be needed to insure that the variations are truly periodic, and not simply due to chaotic variability observed over a relatively short time span. If our period is confirmed, then the nature of the 14d spectroscopic variation found by Southwell et al. is unclear. GX13+1 is a bright X-ray burst source, located in the galactic bulge. Due to heavy obscuration, no optical counterpart brighter than R ~ 22 has been detected, but an infrared counterpart (K=12) has recently been identified by Naylor et al. (1991) based on spatial coincidence with an accurate radio position. GX13+1 is unusual as there is a disagreement over its classification. Studies of the X-ray time variability place it among the Atoll-sources. However, there is some evidence that the system contains a giant companion (Garcia et al. 1992) which would place it among the Z-sources. In an attempt to determine the period of the system, we observed GX13+1 for 9 days in May -- July 1995. Preliminary photometry confirms variability of ~ 0.4 mag on a timescale of several days, as previously discovered by Charles & Naylor (1992). If GX13+1 is found to have a large orbital period 15. Properties of an eclipsing double white dwarf binary NLTT 11748 SciTech Connect Kaplan, David L.; Walker, Arielle N.; Marsh, Thomas R.; Bours, Madelon C. P.; Breedt, Elmé; Bildsten, Lars; Copperwheat, Chris M.; Dhillon, Vik S.; Littlefair, Stuart P.; Howell, Steve B.; Shporer, Avi; Steinfadt, Justin D. R. 2014-01-10 We present high-quality ULTRACAM photometry of the eclipsing detached double white dwarf binary NLTT 11748. This system consists of a carbon/oxygen white dwarf and an extremely low mass (<0.2 M {sub ☉}) helium-core white dwarf in a 5.6 hr orbit. To date, such extremely low-mass white dwarfs, which can have thin, stably burning outer layers, have been modeled via poorly constrained atmosphere and cooling calculations where uncertainties in the detailed structure can strongly influence the eventual fates of these systems when mass transfer begins. With precise (individual precision ≈1%), high-cadence (≈2 s), multicolor photometry of multiple primary and secondary eclipses spanning >1.5 yr, we constrain the masses and radii of both objects in the NLTT 11748 system to a statistical uncertainty of a few percent. However, we find that overall uncertainty in the thickness of the envelope of the secondary carbon/oxygen white dwarf leads to a larger (≈13%) systematic uncertainty in the primary He WD's mass. Over the full range of possible envelope thicknesses, we find that our primary mass (0.136-0.162 M {sub ☉}) and surface gravity (log (g) = 6.32-6.38; radii are 0.0423-0.0433 R {sub ☉}) constraints do not agree with previous spectroscopic determinations. We use precise eclipse timing to detect the Rømer delay at 7σ significance, providing an additional weak constraint on the masses and limiting the eccentricity to ecos ω = (– 4 ± 5) × 10{sup –5}. Finally, we use multicolor data to constrain the secondary's effective temperature (7600 ± 120 K) and cooling age (1.6-1.7 Gyr). 16. Deep, Low Mass Ratio Overcontact Binary Systems. III. CU Tauri and TV Muscae Qian, S.-B.; Yang, Y.-G.; Soonthornthum, B.; Zhu, L.-Y.; He, J.-J.; Yuan, J.-Z. 2005-07-01 New CCD photometric light curves in the B and V bands of the neglected W UMa-type eclipsing variable star CU Tauri are presented. The O'Connell effect in the V light curve obtained in 2001 by Yang and Liu was about ΔV=+0.015, but it vanished in our 2004 observations. The variations in the levels of both minima were seen. Our two epochs of light minimum and others compiled from the literature were used for the period study. It is shown that the types of some eclipse times were incorrect and the values of the period obtained by previous investigators were aliases that prevented formation of a plausible O-C curve. A new linear ephemeris was derived, and it is discovered that the orbital period of CU Tau shows a continuous decrease at a rate of dP/dt=-1.81×10-6 days yr-1. The present symmetric light curves were solved with the 2003 version of the Wilson-Devinney (W-D) code. Both our solutions and those derived by Yang and Liu reveal that CU Tau is a deep (f=50.1%+/-3.2%), low mass ratio (q=0.1770+/-0.0017) overcontact binary system. Meanwhile, the photoelectric light curves in the B, V, R, and I bands of TV Muscae published by Hilditch and coworkers were reanalyzed with the 2003 version of the W-D code. It is shown that the low mass ratio binary turns out to be a deep overcontact system with f=74.3%+/-11.3%. A period analysis with all collected times of light minimum revealed a combination of a long-term period decrease (dP/dt=-2.16×10-7 days yr-1) and a possible cyclic change with a period of 29.1 yr. The rapid long-term period decreases of both systems can be explained as a combination of the mass transfer from the more massive component to the less massive one and the angular momentum loss due to mass outflow from the L2 point. In that way, the overcontact degrees of the two systems will become deeper as their periods decrease, and finally they will evolve into a single rapid-rotation star. However, for CU Tau, the rate of the secular period decrease is very 17. Light curve analysis of southern eclipsing binary EM Car Ćiçek, C.; Bulut, I.; Bulut, A. 2017-02-01 In this study, ASAS light curve of the eclipsing binary EM Car (Sp = O8V, P = 3.4 days) has been analyzed using the Wilson-Devinney method. The light curve analyses have found that EM Car is a detached eclipsing binary system with small eccentric orbit 18. A Photometric Study of the Eclipsing Binary Star BN Ari Michaels, E. J. 2015-12-01 Presented are a set of multi-band light curves, synthetic light curve solutions, and period study for the eclipsing binary star BN Ari. The orbital period was found to be decreasing the past 8 years (~8,200 orbits). The observed light curves were analyzed with the Wilson-Devinney program. The resulting synthetic light curve solution showed the system to be a contact eclipsing binary with total eclipses. 19. Studies of Long Period Eclipsing Binaries Ratajczak, M.; Hełminiak, K. G.; Konacki, M. 2015-07-01 The survey of long period eclipsing binaries from the All Sky Automated Survey (ASAS) catalog aims at searching for and characterizing subgiants and red giants in double-lined detached binary systems. Absolute physical and orbital parameters are presented based on radial velocities from high-quality optical spectra obtained with the following telescope/instrument combinations: 8.2 m Subaru/HDS, ESO 3.6 m/HARPS, 1.9 m Radcliffe/GIRAFFE, CTIO 1.5 m/CHIRON, and 1.2 m Euler/CORALIE. Photometric data from ASAS, SuperWASP, and the Solaris Project were also used. We discuss the derived uncertainties for the individual masses and radii of the components (better than 3% for several systems), as well as results from the spectral analysis performed for components of systems whose spectra we disentangled. 20. Kepler eclipsing binary stars. IV. Precise eclipse times for close binaries and identification of candidate three-body systems SciTech Connect Conroy, Kyle E.; Stassun, Keivan G.; Prša, Andrej; Orosz, Jerome A.; Welsh, William F.; Fabrycky, Daniel C. 2014-02-01 We present a catalog of precise eclipse times and analysis of third-body signals among 1279 close binaries in the latest Kepler Eclipsing Binary Catalog. For these short-period binaries, Kepler's 30 minute exposure time causes significant smearing of light curves. In addition, common astrophysical phenomena such as chromospheric activity, as well as imperfections in the light curve detrending process, can create systematic artifacts that may produce fictitious signals in the eclipse timings. We present a method to measure precise eclipse times in the presence of distorted light curves, such as in contact and near-contact binaries which exhibit continuously changing light levels in and out of eclipse. We identify 236 systems for which we find a timing variation signal compatible with the presence of a third body. These are modeled for the light travel time effect and the basic properties of the third body are derived. This study complements J. A. Orosz et al. (in preparation), which focuses on eclipse timing variations of longer period binaries with flat out-of-eclipse regions. Together, these two papers provide comprehensive eclipse timings for all binaries in the Kepler Eclipsing Binary Catalog, as an ongoing resource freely accessible online to the community. 1. Light Curve Modeling of Eclipsing Binary Stars Milone, E. F. In the two decades since the development of the first eclipsing-binary modeling code, new analytic techniques and the availability of powerful, sometimes dedicated computing facilities have made possible vastly improved determinations of fundamental and even transient stellar parameters. The scale of these developments, of course, raises questions about modeling tools, techniques, and philosophies, such as: Who will maintain and upgrade the codes? Will the codes be open to improvement by outsiders, and if so, how? And, indeed, what should be the goals of a modeling program? Such questions had not been aired for a long time and, for this reason alone, deserved to be discussed in as general a forum as the community provides. This volume contains material presented by Commission 42 (Close Binary Stars) during the International Astronomical Union's XXI General Assembly in Argentina, July 1991, and during IAU Colloquium 151, Cordoba, Argentina, August 1991. The techniques discussed include simulations of stellar bright and dark spots, streams, partial and complete stellar disks, prominences, and other features characterizing active stars; modeling of polarization parameters; models that use radial velocities as well as line profile simulations to model velocity field variation across stellar disks; the weighted effects of brightness asymmetries; and models for translucent eclipsing agents such as stellar winds. 2. Astrometric and Light-Travel Time Orbits to Detect Low-Mass Companions: A Case Study of the Eclipsing System R Canis Majoris Ribas, Ignasi; Arenou, Frédéric; Guinan, Edward F. 2002-04-01 We discuss a method to determine orbital properties and masses of low-mass bodies orbiting eclipsing binaries. The analysis combines long-term eclipse timing modulations (the light-travel time [LTT] effect) with short-term, high-accuracy astrometry. As an illustration of the method, the results of a comprehensive study of Hipparcos astrometry and over 100 years of eclipse timings of the Algol-type eclipsing binary R Canis Majoris are presented. A simultaneous solution of the astrometry and the LTTs yields an orbital period of P12=92.8+/-1.3 yr, an LTT semiamplitude of 2574+/-57 s, an angular semimajor axis of a12=117+/-5 mas, and values of the orbital eccentricity and inclination of e12=0.49+/-0.05 and i12=91.7d+/-4.7d, respectively. Adopting the total mass of R CMa of M12=1.24+/-0.05 Msolar, the mass of the third body is M3=0.34+/-0.02 Msolar, and the semimajor axis of its orbit is a3=18.7+/-1.7 AU. From its mass, the third body is either a dM3-4 star or, more unlikely, a white dwarf. With the upcoming microarcsecond-level astrometric missions, the technique that we discuss can be successfully applied to detect and characterize long-period planetary-size objects and brown dwarfs around eclipsing binaries. Possibilities for extending the method to pulsating variables or stars with transiting planets are briefly addressed. 3. The Behavior of Accretion Disks in Low Mass X-ray Binaries: Disk Winds and Alpha Model Bayless, Amanda J. 2010-01-01 This dissertation presents research on two low mass X-ray binaries. The eclipsing low-mass X-ray binary 4U 1822-371 is the prototypical accretion disk corona (ADC) system. We have obtained new time-resolved UV spectroscopy with the ACS/SBC on the Hubble Space Telescope and new V- and J-band photometry with the 1.3-m SMARTS telescope at CTIO. We show that the accretion disk in the system has a strong wind with projected velocities up to 4000 km/s as determined from the Doppler width of the C IV emission line. The broad and shallow eclipse indicates that the disk has a vertically-extended, optically-thick component at optical wavelengths. This component extends almost to the edge of the disk and has a height equal to 50% of the disk radius. As it has a low brightness temperature, we identify it as the optically-thick base of the disk wind. V1408 Aql (=4U 1957+115) is a low mass X-ray binary which continues to be a black hole candidate. We have new photometric data of this system from the Otto Struve 2.1-m telescope's high speed CCD photometer at McDonald Observatory. The light curve is largely sinusoidal which we model with two components: a constant light source from the disk and a sinusoidal modulation at the orbital period from the irradiated face of the companion star. This is a radical re-interpretation of the orbital light curve. We do not require a large or asymmetric disk rim to account for the modulation in the light curve. Thus, the orbital inclination is unconstrained in our new model, removing the foundation for any claims of the compact object being a black hole. 4. Long-term orbital period behaviour of low mass ratio contact binaries GR Vir and FP Boo Ćetinkaya, Halil; Soydugan, Faruk 2017-02-01 In this study, we investigated orbital period variations of two low mass ratio contact binaries GR Vir and FP Boo based on published minima times. From the O-C analysis, it was found that FP Boo indicates orbital period decrease while the period of GR Vir is increasing. Mass transfer process was used to explain increase and decrease in the orbital periods. In the O-C diagrams of both systems periodic variations also exist. Cyclic changes can be explained as being the result of a light-travel time effect via a third component around the eclipsing binaries. In order to interpret of cyclic orbital period changes for GR Vir, which has late-type components, possible magnetic activity cycles of the components have been also considered. 5. Eclipsing binary stars with a δ Scuti component Kahraman Aliçavuş, F.; Soydugan, E.; Smalley, B.; Kubát, J. 2017-09-01 Eclipsing binaries with a δ Sct component are powerful tools to derive the fundamental parameters and probe the internal structure of stars. In this study, spectral analysis of six primary δ Sct components in eclipsing binaries has been performed. Values of Teff, v sin i, and metallicity for the stars have been derived from medium-resolution spectroscopy. Additionally, a revised list of δ Sct stars in eclipsing binaries is presented. In this list, we have only given the δ Sct stars in eclipsing binaries to show the effects of the secondary components and tidal-locking on the pulsations of primary δ Sct components. The stellar pulsation, atmospheric and fundamental parameters (e.g. mass, radius) of 92 δ Sct stars in eclipsing binaries have been gathered. Comparison of the properties of single and eclipsing binary member δ Sct stars has been made. We find that single δ Sct stars pulsate in longer periods and with higher amplitudes than the primary δ Sct components in eclipsing binaries. The v sin i of δ Sct components is found to be significantly lower than that of single δ Sct stars. Relationships between the pulsation periods, amplitudes and stellar parameters in our list have been examined. Significant correlations between the pulsation periods and the orbital periods, Teff, log g, radius, mass ratio, v sin i and the filling factor have been found. 6. The hot subdwarf in the eclipsing binary HD 185510 NASA Technical Reports Server (NTRS) Jeffery, C. S.; Simon, Theodore; Evans, T. L. 1992-01-01 High-resolution spectroscopic measurements of radial velocity are employed to characterize the eclipsing binary HD 185510 in terms of masses and evolutionary status. The IUE is used to obtain the radial velocities which indicate a large mass ratio Mp/Ms of 7.45 +/- 0.15, and Teff is given at 25,000 +/- 1000 K based on Ly alpha and UV spectrophotometry. Photometric observations are used to give an orbital inclination of between 90 and 70 deg inclusive, leading to masses of 0.31-0.37 and 2.3-2.8 solar mass for the hot star and the K star, respectively. The surface gravity of HD 185510B is shown to be higher than those values for sdB stars suggesting that the object is a low-mass white dwarf that has not reached its fully degenerate configuration. The object is theorized to be a low-mass helium main-sequence star or a nascent helium degenerate in a post-Algol system. 7. The hot subdwarf in the eclipsing binary HD 185510 NASA Technical Reports Server (NTRS) Jeffery, C. S.; Simon, Theodore; Evans, T. L. 1992-01-01 High-resolution spectroscopic measurements of radial velocity are employed to characterize the eclipsing binary HD 185510 in terms of masses and evolutionary status. The IUE is used to obtain the radial velocities which indicate a large mass ratio Mp/Ms of 7.45 +/- 0.15, and Teff is given at 25,000 +/- 1000 K based on Ly alpha and UV spectrophotometry. Photometric observations are used to give an orbital inclination of between 90 and 70 deg inclusive, leading to masses of 0.31-0.37 and 2.3-2.8 solar mass for the hot star and the K star, respectively. The surface gravity of HD 185510B is shown to be higher than those values for sdB stars suggesting that the object is a low-mass white dwarf that has not reached its fully degenerate configuration. The object is theorized to be a low-mass helium main-sequence star or a nascent helium degenerate in a post-Algol system. 8. The Kepler Mission and Eclipsing Binaries NASA Technical Reports Server (NTRS) Koch, David; Borucki, William; Lissauer, J.; Basri, Gibor; Brown, Timothy; Caldwell, Douglas; Cochran, William; Jenkins, Jon; Dunham, Edward; Gautier, Nick 2006-01-01 The Kepler Mission is a photometric mission with a precision of 14 ppm (at R=12) that is designed to continuously observe a single field of view (FOV) of greater 100 sq deg in the Cygnus-Lyra region for four or more years. The primary goal of the mission is to monitor greater than 100,000 stars for transits of Earth-size and smaller planets in the habitable zone of solar-like stars. In the process, many eclipsing binaries (EB) will also be detected and light curves produced. To enhance and optimize the mission results, the stellar characteristics for all the stars in the FOV with R less than 16 will have been determined prior to launch. As part of the verification process, stars with transit candidates will have radial velocity follow-up observations performed to determine the component masses and thereby separate eclipses caused by stellar companions from transits caused by planets. The result will be a rich database on EBs. The community will have access to the archive for further analysis, such as, for EB modeling of the high-precision light curves. A guest observer program is also planned to allow for photometric observations of objects not on the target list but within the FOV, since only the pixels of interest from those stars monitored will be transmitted to the ground. 9. The Kepler Mission and Eclipsing Binaries NASA Technical Reports Server (NTRS) Koch, David; Borucki, William; Lissauer, J.; Basri, Gibor; Brown, Timothy; Caldwell, Douglas; Cochran, William; Jenkins, Jon; Dunham, Edward; Gautier, Nick 2006-01-01 The Kepler Mission is a photometric mission with a precision of 14 ppm (at R=12) that is designed to continuously observe a single field of view (FOV) of greater 100 sq deg in the Cygnus-Lyra region for four or more years. The primary goal of the mission is to monitor greater than 100,000 stars for transits of Earth-size and smaller planets in the habitable zone of solar-like stars. In the process, many eclipsing binaries (EB) will also be detected and light curves produced. To enhance and optimize the mission results, the stellar characteristics for all the stars in the FOV with R less than 16 will have been determined prior to launch. As part of the verification process, stars with transit candidates will have radial velocity follow-up observations performed to determine the component masses and thereby separate eclipses caused by stellar companions from transits caused by planets. The result will be a rich database on EBs. The community will have access to the archive for further analysis, such as, for EB modeling of the high-precision light curves. A guest observer program is also planned to allow for photometric observations of objects not on the target list but within the FOV, since only the pixels of interest from those stars monitored will be transmitted to the ground. 10. The M Dwarf Eclipsing Binary CU Cancri Wilson, R. E.; Pilachowski, C. A.; Terrell, Dirk 2017-02-01 Spectral features, radial velocities, elemental abundance estimates, other spectral data, and BVIC light curves are reported for the double-M dwarf eclipsing binary CU Cancri—a good target for a radius check versus the Zero Age Main Sequence (ZAMS) due to the low component masses and corresponding very slow evolutionary expansion. The estimate of [Fe/H] is about 0.4, although continuum placement and other difficulties due to line crowding introduce the usual uncertainties for red dwarfs. Detection of the Li i λ6707 line was attempted, with an estimated upper limit of 50 mÅ. Spectral and photometric indicators of stellar activity are described and illustrated. Other objectives were to measure the stellar radii via simultaneous velocity and light-curve solutions of earlier and new data while also improving the ephemeris by filling gaps in timewise coverage with the new velocities and eclipse data from the new light curves. The radii from our solutions agree within about 2% with those from Ribas, being slightly larger than expected for most estimates of the ZAMS. Some aspects of the red dwarf radius anomaly are briefly discussed. Evolution tracks show only very slight age-related expansion for masses near those in CU Cnc. Such expansion could be significant if CU Cnc were similar in age to the Galaxy, but then its Galactic velocity components should be representative of Population II, and they are not. 11. Photometric orbits of seven detached eclipsing binaries Popper, D. M.; Etzel, P. B. 1981-01-01 Photoelectric light curves of the detached eclipsing binaries V805 Aql, TV Cet, MY Cyg, V478 Cyg, V1143 Cyg, BS Dra, and BK Peg are analyzed. The systems are among those with good spectrographic orbits of both components that are in need of good photometric solutions in order to obtain the absolute properties of the components. The analyses are carried out with a computer program valid for detached systems of spherical or slightly oblate stars in orbits of arbitrary eccentricity. A range of solutions much greater than implied by the internal mean errors of the parameters, is found to give satisfactory fits to the observations. Some of the fits are displayed for a variety of solutions for each system. For the three systems with measurable light variation between eclipses - V478 Cyg, V805 Aql, and MY Cyg - the effect of reflection appears to be substantially less than predicted. Very small, but nonzero, orbital eccentricities are found for four of the systems. The variations of limb darkening with wavelength and with spectral type are found to be in reasonable agreement with predictions from atmospheric theory. 12. Evolution of Intermediate and Low Mass Binary Systems SciTech Connect Eggleton, P P 2005-10-25 There are a number of binaries, fairly wide and with one or even two evolved giant components, that do not agree very well with conventional stellar evolution: the secondaries are substantially larger (oversized) than they should be because their masses are quite low compared with the primaries. I discuss the possibility that these binaries are former triples, in which a merger has occurred fairly recently in a short-period binary sub-component. Some mergers are expected, and may follow a phase of contact evolution. I suggest that in contact there is substantial transfer of luminosity between the components due to differential rotation, of the character observed by helioseismology in the Sun's surface convection zone. 13. Eclipsing Binary Star Detection Using Kepler Vydra, Ekaterina; Buzasi, Derek L. 2017-01-01 Eclipsing binaries (EBs) are laboratories for precision astrophysics, because use of the orbital information of the system allows the determination of the physical parameters of the stars to a much higher degree of precision than is possible for single stars. The Kepler Space Telescope, while designed to hunt for planets, has also been a valuable tool in detecting and characterizing EBs and has already observed over 2200 specimens. Kepler suffered a failure in 2013 that affected its pointing ability, but some ingenious engineering adjustments have allowed it to continue collecting photometric data from new fields of view. Our goals were to develop an algorithm for EB detection using Kepler data, and then with the help of FGCU's K2 Aperture Photometry Pipeline to extend that algorithm to discover new EBs in the K2 fields. Here we report on our progess to date as well as future plans. 14. B-type stars in eclipsing binaries Ratajczak, Milena; Pigulski, Andrzej 2016-07-01 B-type stars in eclipsing binary systems are unique astrophysical tools to test several aspects of stellar evolution. Such objects can be used e.g. to determine the masses of Beta Cephei variable stars, as well as help to place tighter constraints on the value of the convective core overshooting parameter α. Both precise photometry and high-resolution spectroscopy with high SNR are required to achieve these goals, but since many of the targets are bright enough, the challenge is fair. Following this assumption, we shall explain how we plan to examine both the aforementioned aspects of stellar evolution using observations of B-type stars obtained with a wide range of spectrographs, as well as BRITE-Constellation satellites. 15. Nucleosynthesis of Binary low mass zero-metallicity stars Lau, Ho Bun Herbert; Stancliffe, R. J.; Tout, C. A. The Cambridge STARS code is used to model the evolution and nucleosynthesis of binary zero- metallicity low to intermediate mass stars. The surfaces of these stars are enriched in CNO ele- ments after second dredge up. During binary interaction metals can be released from these stars and the secondary enriched in CNO. The observed abundances of HE 0107-5240 can be repro- duced from enhanced wind accretion from a 7 M after second dredge up. HE 1327-2326, richer in nitrogen and Sr, can similarly be formed by wind accretion in a later AGB phase after third dredge up. 16. Double-lined M dwarf eclipsing binaries from Catalina Sky Survey and LAMOST Lee, Chien-Hsiu; Lin, Chien-Cheng 2017-02-01 Eclipsing binaries provide a unique opportunity to determine fundamental stellar properties. In the era of wide-field cameras and all-sky imaging surveys, thousands of eclipsing binaries have been reported through light curve classification, yet their basic properties remain unexplored due to the extensive efforts needed to follow them up spectroscopically. In this paper we investigate three M2-M3 type double-lined eclipsing binaries discovered by cross-matching eclipsing binaries from the Catalina Sky Survey with spectroscopically classified M dwarfs from the Large Sky Area Multi-Object Fiber Spectroscopic Telescope survey data release one and two. Because these three M dwarf binaries are faint, we further acquire radial velocity measurements using GMOS on the Gemini North telescope with R∼ 4000, enabling us to determine the mass and radius of individual stellar components. By jointly fitting the light and radial velocity curves of these systems, we derive the mass and radius of the primary and secondary components of these three systems, in the range between 0.28-0.42M_ȯ and 0.29-0.67R_ȯ, respectively. Future observations with a high resolution spectrograph will help us pin down the uncertainties in their stellar parameters, and render these systems benchmarks to study M dwarfs, providing inputs to improving stellar models in the low mass regime, or establishing an empirical mass-radius relation for M dwarf stars. 17. The solar-type eclipsing binary system LL Aquarii Southworth, J. 2013-09-01 The eclipsing binary LL Aqr consists of two late-type stars in an eccentric orbit with a period of 20.17 d. We use an extensive light curve from the SuperWASP survey augmented by published radial velocities and UBV light curves to measure the physical properties of the system. The primary star has a mass of 1.167 ± 0.009 M⊙ and a radius of 1.305 ± 0.007 R⊙. The secondary star is an analogue of the Sun, with a mass and radius of 1.014 ± 0.006 M⊙ and 0.990 ± 0.008 R⊙ respectively. The system shows no signs of stellar activity: the upper limit on spot-induced rotational modulation is 3 mmag, it is slowly rotating, has not been detected at X-ray wavelengths, and the calcium H and K lines exhibit no emission. Theoretical stellar models provide a good match to its properties for a sub-solar metal abundance of Z = 0.008 and an age of 2.5 Gyr. Most low-mass eclipsing binary systems are found to have radii larger than expected from theoretical predictions, blamed on tidally-enhanced magnetic fields in these short-period systems. The properties of LL Aqr support this scenario: it exhibits negligible tidal effects, shows no signs of magnetic activity, and matches theoretical models well. Full Tables 1 and 7 are available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A119 18. MARVELS Radial Velocity Solutions to Seven Kepler Eclipsing Binaries Heslar, Michael Francis; Thomas, Neil B.; Ge, Jian; Ma, Bo; Herczeg, Alec; Reyes, Alan; SDSS-III MARVELS Team 2016-01-01 Eclipsing binaries serve momentous purposes to improve the basis of understanding aspects of stellar astrophysics, such as the accurate calculation of the physical parameters of stars and the enigmatic mass-radius relationship of M and K dwarfs. We report the investigation results of 7 eclipsing binary candidates, initially identified by the Kepler mission, overlapped with the radial velocity observations from the SDSS-III Multi-Object APO Radial-Velocity Exoplanet Large-Area Survey (MARVELS). The RV extractions and spectroscopic solutions of these eclipsing binaries were generated by the University of Florida's 1D data pipeline with a median RV precision of ~60-100 m/s, which was utilized for the DR12 data release. We performed the cross-reference fitting of the MARVELS RV data and the Kepler photometric fluxes obtained from the Kepler Eclipsing Binary Catalog (V2) and modelled the 7 eclipsing binaries in the BinaryMaker3 and PHOEBE programs. This analysis accurately determined the absolute physical and orbital parameters of each binary. Most of the companion stars were determined to have masses of K and M dwarf stars (0.3-0.8 M⊙), and allowed for an investigation into the mass-radius relationship of M and K dwarfs. Among the cases are KIC 9163796, a 122.2 day period "heartbeat star", a recently-discovered class of eccentric binaries known for tidal distortions and pulsations, with a high eccentricity (e~0.75) and KIC 11244501, a 0.29 day period, contact binary with a double-lined spectrum and mass ratio (q~0.45). We also report on the possible reclassification of 2 Kepler eclipsing binary candidates as background eclipsing binaries based on the analysis of the flux measurements, flux ratios of the spectroscopic and photometric solutions, the differences in the FOVs, the image processing of Kepler, and RV and spectral analysis of MARVELS. 19. CCD Photometry of Five Neglected Eclipsing Binary Stars Cook, Stephen P. Differential V-magnitude CCD photometric data are presented for five neglected eclipsing binary stars with shallow eclipses. An improved period is derived for SV Equ, past O-C trends are confirmed for AN And and DL Vir, and an unexpectedly large O-C values are found for BW DEL nad CS Lac. 20. Indoor Astronomy: A Model Eclipsing Binary Star System. ERIC Educational Resources Information Center Bloomer, Raymond H., Jr. 1979-01-01 Describes a two-hour physics laboratory experiment modeling the phenomena of eclipsing binary stars developed by the Air Force Academy as part of a week-long laboratory-oriented experience for visiting high school students. (BT) 1. On the eclipsing binary nature of a nearby ultracool dwarf Gillon, Michael; Jehin, Emmanuel; de Wit, Julien; Demory, Brice-Olivier; Burgasser, Adam; Van Grootel, Valerie; Lederer, Susan; Triaud, Amaury; Delrez, Laetitia; Burdanov, Artem; Queloz, Didier; Magain, Pierre 2016-02-01 The eclipsing binary nature of a nearby ultracool dwarf has just been revealed. The aim of this DDT is to investigate this nearby system further through high-precision infrared time-series photometry. 2. On the eclipsing binary nature of a nearby ultracool dwarf Gillon, Michael; de Wit, Julien; Jehin, Emmanuel; Burdanov, Artem; Van Grootel, valerie. vangrootel@ulg. ac. be; Delrez, Laetitia; Magain, Pierre; Burgasser, Adam; Demory, Brice-Olivier; Triaud, Amaury; Queloz, Didier; Lederer, Sue 2016-01-01 The eclipsing binary nature of a nearby ultracool dwarf has just been revealed. The aim of this DDT is to investigate this nearby system further through high-precision infrared time-series photometry. 3. CCD Times of Minima of Selected Eclipsing Binaries Zejda, Miloslav 2004-12-01 682 CCD minima observations of 259 eclipsing binaries made mainly by author are presented. The observed stars were chosen mainly from catalogue BRKA of observing programme of BRNO-Variable Star Section of CAS. 4. Properties OF M31. V. 298 eclipsing binaries from PAndromeda SciTech Connect Lee, C.-H.; Koppenhoefer, J.; Seitz, S.; Bender, R.; Riffeser, A.; Kodric, M.; Hopp, U.; Snigula, J.; Gössl, C.; Kudritzki, R.-P.; Burgett, W.; Chambers, K.; Hodapp, K.; Kaiser, N.; Waters, C. 2014-12-10 The goal of this work is to conduct a photometric study of eclipsing binaries in M31. We apply a modified box-fitting algorithm to search for eclipsing binary candidates and determine their period. We classify these candidates into detached, semi-detached, and contact systems using the Fourier decomposition method. We cross-match the position of our detached candidates with the photometry from Local Group Survey and select 13 candidates brighter than 20.5 mag in V. The relative physical parameters of these detached candidates are further characterized with the Detached Eclipsing Binary Light curve fitter (DEBiL) by Devor. We will follow up the detached eclipsing binaries spectroscopically and determine the distance to M31. 5. STATE TRANSITIONS IN LOW-MASS X-RAY BINARIES SciTech Connect 2009-10-10 We investigate the model of disk/coronal accretion into a black hole. We show that the inner regions of an accretion disk in X-ray binaries can transform from a cool standard disk to an advection-dominated flow through the known properties of Coulomb interaction in a two-temperature plasma, viscous heating, radiative processes, and thermal conduction. A hot, diffuse corona covering the disk is powered by accretion, but it exchanges mass with the underlying cold disk. If the accretion rate in the system is low enough, we show that the corona evaporates the disk away, leaving an advective flow to continue toward the hole. In the soft/hard transition commonly seen in X-ray binaries, we show that this advective flow can recondense back onto the underlying disk if the change in the system's accretion rate is slow enough due to thermal conduction. Unabsorbed spectra are produced to test against observations as well as prediction of the location of truncation radii of the accretion disk. 6. A Study of the Low Mass Binary Star Ross 614 Gatewood, G.; Han, I.; Tangren, W. 2001-12-01 We have combined photograph, MAP, interferometric, and spectroscopic data to determine the orbital characteristics and masses of the Ross 614 binary star system. Attention was first drawn to the star by Frank E. Ross (1927, AJ 37, 193) who noticed its high proper motion in a comparison of new plates with those taken at the Yerkes Observatory by E.E. Barnard. The Binary nature of the star was recognized from accelerations in the star's proper motion (D. Reuyl 1936, AJ 55, 236) and the mass of the companion was first estimated by combining measurements of McCormick and Sproul plates with a separation measured by Walter Baade at the Hale 5-m reflector (S.L. Lippincott 1955, AJ 60, 379). In her paper Lippincott notes the companion's significance as defining the lower end of the observational main sequence. Fifty six years later the star still holds that honor. With a wealth of new data spanning more than 3 additional orbits, we find her value of 0.08 solar masses to be within our error of our value. 7. High ionisation absorption in low mass X-ray binaries Ponti, G.; Bianchi, S.; Muñoz-Darias, T.; De, K.; Fender, R.; Merloni, A. 2016-05-01 The advent of the new generation of X-ray telescopes yielded a significant step forward in our understanding of ionised absorption generated in the accretion discs of X-ray binaries. It has become evident that these relatively weak and narrow absorption features, sporadically present in the X-ray spectra of some systems, are actually the signature of equatorial outflows, which might carry away more matter than that being accreted. Therefore, they play a major role in the accretion phenomenon. These outflows (or ionised atmospheres) are ubiquitous during the softer states but absent during the power-law dominated, hard states, suggesting a strong link with the state of the inner accretion disc, presence of the radio-jet and the properties of the central source. Here, we discuss the current understanding of this field. 8. Pro-Am Collaborations on Eclipsing Binary Star Problems Terrell, D. 2004-05-01 I discuss the fruits of a decade of amateur-professional collaboration on eclipsing binary stars. Our team consists of a mix of visual, photoelectric and CCD observers that use the strengths of each observing approach to study newly discovered and neglected eclipsing binary systems. We have active programs on time of minimum measurements and high-precision photometry that results in detailed analysis of the binaries to find fundamental parameters such as masses and radii. We have also discovered and/or characterized several unusal binaries that have had an influence on stellar evolution theory. 9. APSIDAL MOTION IN ECCENTRIC ECLIPSING BINARY WW CAMELOPARDALIS SciTech Connect Wolf, M.; Kotkova, L.; Kocian, R.; Dreveny, R.; Hanzl, D. 2010-03-15 WW Camelopardalis is a relatively bright eclipsing binary system with a slightly eccentric orbit. A dozen of its new eclipse times were measured as part of our long-term observational project of eccentric eclipsing binaries. Based on a new solution of the current O - C diagram, we found for the first time an apsidal motion in good agreement with theory. Its period is about 370 {+-} 50 years. The determined internal structure constant is close to the theoretically expected value. The relativistic effect is significant, being about 13% of the total apsidal motion rate. 10. Catalogue of cataclysmic binaries, low-mass X-ray binaries and related objects (Seventh edition) Ritter, H.; Kolb, U. 2003-06-01 The catalogue lists coordinates, apparent magnitudes, orbital parameters, and stellar parameters of the components and other characteristc properties of 472 cataclysmic binaries, 71 low-mass X-ray binaries and 113 related objects with known or suspected orbital periods together with a comprehensive selection of the relevant recent literature. In addition, the catalogue contains a list of references to published finding charts for 635 of the 656 objects, and a cross-reference list of alias object designations. Literature published before 1 January 2003 has, as far as possible, been taken into account. All data can be accessed via the dedicated catalogue webpage at http://www.mpa-garching.mpg.de/RKcat/ and http://physics.open.ac.uk/RKcat/ and at CDS via anonymous ftp to cdsarc.u-strasbg.fr (30.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/404/301. We will update the information given on the catalogue webpage regularly, initially every six months. 11. Investigating the Properties of Low-Mass Stars Using Spectra of Wide Binaries Schluns, Kyle; West, A. A.; Dhital, S.; Massey, A. P. 2013-01-01 We present results from a study designed to characterize wide, low-mass (< 0.5 M_Sun) binaries identified in the Sloan Digital Sky Survey (SDSS). We examine the SDSS database level completeness (for identifying visual binaries) and analyze the pairs that both have individual SDSS spectra. A comprehensive by-eye examination reveals that a significant fraction of the sources within 1" of the primary stellar source are misclassified as duplicate detections and, hence, are omitted from the photometric primary catalog in the SDSS database. This discrepancy has a noticeable effect on estimates of the binary fraction, mass function, luminosity function, and other studies that rely on large, photometric samples of low-mass stars. Due to their coeval nature, binaries with at least one low-mass component are important for calibrating the age-activity relation and the relative metallicity scales. Better defined stellar ages and metallicities allow for a proper analysis of stellar and Galactic evolution using ubiquitous low-mass stars. We constructed a spectroscopic sample of wide binaries, for which there is at least one low-mass component and an individual spectrum for each star. Each binary was verified using measurements of their common proper motions and a chance alignment probability calculated from a six-dimensional model of the Milky Way. The orbital separation of the binary components provides an extra age constraint due to mechanisms that destroy wide binaries during thin-disk dynamical heating. We evaluate the behavior of the magnetic activity in coeval systems, with a specific focus on the dependence of activity on orbital separation and location in the Galactic disk. The preliminary results of our analysis will help calibrate the age-activity relation in M dwarfs. In addition, we calibrate the relative metallicity scale for metal poor K and M dwarfs using a modified index based on TiO and CaH molecular band features. 12. A systematic study of magnetic braking in low-mass binaries NASA Technical Reports Server (NTRS) Verbunt, F.; Rappaport, S.; Joss, P. C. 1985-01-01 A short summary is given of Rappaport et al. (1983) which described results of extending a simplified stellar evolution code covering the evolution of low-mass compact binaries. Magnetic braking is probably an important process in the evolution of such binaries (such as cataclysmic variables and low-mass X-ray sources). The initial simplified code describes the mass-losing star as an n = 3/2 polytrope and was developed to study the evolution of binaries with a secondary of low mass (between 0.01 and 0.4 solar mass) when the angular momentum losses are due to gravitational radiation. In the extended code, a composite polytrope model is used for the secondary, wherein the structure of the radiative core is described by an n = 3 polytrope and the convective envelope by an n = 3/2 polytrope. 13. Eclipsing Binaries from the Kepler Mission NASA Technical Reports Server (NTRS) Koch, David; Borucki, William; Lissauer, J.; Basri, Gibor; Brown, Timothy; Caldwell, Douglas; Cochran, William; Jenkins, Jon; Dunham, Edward; Gautier, Nick 2005-01-01 The Kepler Mission is a photometric space mission that will continuously observe a single 100 sq deg field of view (FOV) of greater than 100,000 stars in the Cygnus-Lyra region for 4 or more years with a precision of 14 ppm (R=12). The primary goal of the mission is to detect Earth-size planets in the habitable zone of solar-like stars. In the process, many eclipsing binaries (EB) will also be detected. Prior to launch, the stellar characteristics will have been detennined for all the stars in the FOV with R<16. As part of the verification process, stars with transits <5% will need to have follow-up radial velocity observations performed to determine the component masses and thereby separate transits caused by stellar companions from those caused by planets. The result will be a rich database on EBs. The community will have access to the archive for uses such as for EB modeling of the high-precision light curves. A guest observer program is also planned for objects not already on the target list. 14. A NEW CLASS OF NASCENT ECLIPSING BINARIES WITH EXTREME MASS RATIOS SciTech Connect Moe, Maxwell; Stefano, Rosanne Di 2015-03-10 Early B-type main-sequence (MS) stars (M {sub 1} ≈ 5-16 M {sub ☉}) with closely orbiting low-mass stellar companions (q = M {sub 2}/M {sub 1} < 0.25) can evolve to produce Type Ia supernovae, low-mass X-ray binaries, and millisecond pulsars. However, the formation mechanism and intrinsic frequency of such close extreme mass-ratio binaries have been debated, especially considering none have hitherto been detected. Utilizing observations of the Large Magellanic Cloud galaxy conducted by the Optical Gravitational Lensing Experiment, we have discovered a new class of eclipsing binaries in which a luminous B-type MS star irradiates a closely orbiting low-mass pre-MS companion that has not yet fully formed. The primordial pre-MS companions have large radii and discernibly reflect much of the light they intercept from the B-type MS primaries (ΔI {sub refl} ≈ 0.02-0.14 mag). For the 18 definitive MS + pre-MS eclipsing binaries in our sample with good model fits to the observed light-curves, we measure short orbital periods P = 3.0-8.5 days, young ages τ ≈ 0.6-8 Myr, and small secondary masses M {sub 2} ≈ 0.8-2.4 M {sub ☉} (q ≈ 0.07-0.36). The majority of these nascent eclipsing binaries are still associated with stellar nurseries, e.g., the system with the deepest eclipse ΔI {sub 1} = 2.8 mag and youngest age τ = 0.6 ± 0.4 Myr is embedded in the bright H II region 30 Doradus. After correcting for selection effects, we find that (2.0 ± 0.6)% of B-type MS stars have companions with short orbital periods P = 3.0-8.5 days and extreme mass ratios q ≈ 0.06-0.25. This is ≈10 times greater than that observed for solar-type MS primaries. We discuss how these new eclipsing binaries provide invaluable insights, diagnostics, and challenges for the formation and evolution of stars, binaries, and H II regions. 15. Studies of low-mass interacting binary stars Rainger, Paul P. 1990-01-01 Spectroscopic and photometric observations of eight contact/near-contact binaries are presented and analysed. Spectroscopic observations were obtained at 4200 A (radial velocity spectra) and 6563 A (hydrogen-alpha line profiles). New photometric observations were obtained at visual and infrared wavelengths, and other previously published light curves are also re-analysed. Absolute dimensions have been obtained for five systems; TY Boo, VW Boo, BX And, SS Ari and AG Vir, and their evolutionary positions discussed. Four of the systems are found to be in marginal but poor thermal contact, exhibiting regions of apparent "excess luminosity" in their light curves. A qualitative analysis of these "hot spot" regions has been attempted for the first time using spot models now incorporated into a light curve synthesis programme. Substantial time for this project was awarded on telescopes funded by the United Kingdom Science and Engineering Research Council (SERC), comprising 14 nights at the Issac Newton Telescope (INT) on La Palma, and 4 nights at the United Kingdom Infrared Telescope (UKIRT) on Mauna Kea. Additional observations were made during an 8 night commissioning run on the Jacobus Kapteyn Telescope (JKT) on La Palma, and extensive observations were made with the Twin Photometric Telescope (TPT) at St Andrews University Observatory between 1985 and 1989. These resulted in over 100 spectra at 4200 A and over 50 spectra at 6563 A (INT and JKT observations), over 300 infrared photometric observations (UKIRT), and over 3500 visual photometric observations (TPT). Of the five systems analysed in detail in this work, TY Boo appears to be a normal shallow-contact W-type system. Both VW Boo and BX And exhibit regions of "excess luminosity" around the ingress and egress of secondary minimum which are well modelled by a warm spot on the cooler component sitting symmetrically around the neck joining the pair. Such a phenomenon may be expected to arise naturally in systems which 16. The ELM Survey. VII. Orbital Properties of Low-Mass White Dwarf Binaries Brown, Warren R.; Gianninas, A.; Kilic, Mukremin; Kenyon, Scott J.; Allende Prieto, Carlos 2016-02-01 We present the discovery of 15 extremely low-mass (5\\lt {log}g\\lt 7) white dwarf (WD) candidates, 9 of which are in ultra-compact double-degenerate binaries. Our targeted extremely low-mass Survey sample now includes 76 binaries. The sample has a lognormal distribution of orbital periods with a median period of 5.4 hr. The velocity amplitudes imply that the binary companions have a normal distribution of mass with 0.76 M⊙ mean and 0.25 M⊙ dispersion. Thus extremely low-mass WDs are found in binaries with a typical mass ratio of 1:4. Statistically speaking, 95% of the WD binaries have a total mass below the Chandrasekhar mass, and thus are not type Ia supernova progenitors. Yet half of the observed binaries will merge in less than 6 Gyr due to gravitational wave radiation; probable outcomes include single massive WDs and stable mass transfer AM CVn binaries. Based on observations obtained at the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona. 17. Fundamental parameters of four massive eclipsing binaries in Westerlund 1 Koumpia, E.; Bonanos, A. Z. 2012-11-01 Context. Only a small number of high mass stars (>30 M⊙) have fundamental parameters (i.e. masses and radii) measured with high enough accuracy from eclipsing binaries to constrain formation and evolutionary models of massive stars. Aims: This work aims to increase this limited sample, by studying the four massive eclipsing binary candidates discovered by Bonanos in the young massive cluster Westerlund 1. Methods: We present new follow-up echelle spectroscopy of these binaries and models of their light and radial velocity curves. Results: We obtain fundamental parameters for the eight component stars, finding masses that span a range of 10-40 M⊙, and contributing accurate fundamental parameters for one additional very massive star, the 33 M⊙ component of W13. WR77o is found to have a ~40 M⊙ companion, which provides a second dynamical constraint on the mass of the progenitor of the magnetar known in the cluster. We also use W13 to estimate the first, direct, eclipsing binary distance to Westerlund 1 and therefore the magnetar and find it to be at 3.7 ± 0.6 kpc. Conclusions: Our results confirm previous evidence for a high mass for the progenitor of the magnetar. In addition, the availability of eclipsing binaries with accurate parameters opens the way for direct, independent, high precision eclipsing binary distance measurements to Westerlund 1. 18. Eclipsing Binaries through the Double Looking Glass of Kepler and Keck Weiss, Lauren M.; Marcy, G.; Orosz, J.; Welsh, W.; Prša, A.; Richards, J.; Gegenheimer, S.; Bloom, J. S. 2012-05-01 The Kepler Space Telescope has detected a panoply of physical effects in binary star systems to unprecedented precision, in particular the relativistic Doppler beaming signature. We present Keck/HIRES radial velocity measurements of four Kepler Input Catalog (KIC) eclipsing binaries. We compute self-consistent solutions to the eclipsing binaries by fitting the light curve and radial velocities simultaneously with the legacy code ELC. We also attempt a novel two-dimensional radial velocity fitting technique to solve for the velocity of the primary and secondary star in each system. Our velocity fitting procedure draws from a library of over 700 Keck/HIRES spectra of stars with known effective temperature, surface gravity, and metallicity to construct a composite spectral template from which we can measure the velocities of both stars. We discuss the orbital dynamics and stellar physics of these four KIC systems, including ellipsoidal variations, relativistic Doppler beaming, spot activity, pulsations, and third-body, low-mass companions of the binary systems. In particular, we note that long-lived spot activity in tidally synchronous binary systems can produce long-lived asymmetries in the light curve before and after the primary eclipse. These asymmetries can masquerade as the relativistic Doppler beaming signature, and we caution beaming enthusiasts to consider spot activity before attributing light curve asymmetries to relativistic beaming. 19. What we learn from eclipsing binaries in the ultraviolet NASA Technical Reports Server (NTRS) Guinan, Edward F. 1990-01-01 Recent results on stars and stellar physics from IUE (International Ultraviolet Explorer) observations of eclipsing binaries are discussed. Several case studies are presented, including V 444 Cyg, Aur stars, V 471 Tau and AR Lac. Topics include stellar winds and mass loss, stellar atmospheres, stellar dynamos, and surface activity. Studies of binary star dynamics and evolution are discussed. The progress made with IUE in understanding the complex dynamical and evolutionary processes taking place in W UMa-type binaries and Algol systems is highlighted. The initial results of intensive studies of the W UMa star VW Cep and three representative Algol-type binaries (in different stages of evolution) focused on gas flows and accretion, are included. The future prospects of eclipsing binary research are explored. Remaining problems are surveyed and the next challenges are presented. The roles that eclipsing binaries could play in studies of stellar evolution, cluster dynamics, galactic structure, mass luminosity relations for extra galactic systems, cosmology, and even possible detection of extra solar system planets using eclipsing binaries are discussed. 20. PERIOD ERROR ESTIMATION FOR THE KEPLER ECLIPSING BINARY CATALOG SciTech Connect Mighell, Kenneth J.; Plavchan, Peter 2013-06-15 The Kepler Eclipsing Binary Catalog (KEBC) describes 2165 eclipsing binaries identified in the 115 deg{sup 2} Kepler Field based on observations from Kepler quarters Q0, Q1, and Q2. The periods in the KEBC are given in units of days out to six decimal places but no period errors are provided. We present the PEC (Period Error Calculator) algorithm, which can be used to estimate the period errors of strictly periodic variables observed by the Kepler Mission. The PEC algorithm is based on propagation of error theory and assumes that observation of every light curve peak/minimum in a long time-series observation can be unambiguously identified. The PEC algorithm can be efficiently programmed using just a few lines of C computer language code. The PEC algorithm was used to develop a simple model that provides period error estimates for eclipsing binaries in the KEBC with periods less than 62.5 days: log {sigma}{sub P} Almost-Equal-To - 5.8908 + 1.4425(1 + log P), where P is the period of an eclipsing binary in the KEBC in units of days. KEBC systems with periods {>=}62.5 days have KEBC period errors of {approx}0.0144 days. Periods and period errors of seven eclipsing binary systems in the KEBC were measured using the NASA Exoplanet Archive Periodogram Service and compared to period errors estimated using the PEC algorithm. 1. A DEEPLY ECLIPSING DETACHED DOUBLE HELIUM WHITE DWARF BINARY SciTech Connect Parsons, S. G.; Marsh, T. R.; Gaensicke, B. T.; Drake, A. J.; Koester, D. 2011-07-10 Using Liverpool Telescope+RISE photometry we identify the 2.78 hr period binary star CSS 41177 as a detached eclipsing double white dwarf binary with a 21,100 K primary star and a 10,500 K secondary star. This makes CSS 41177 only the second known eclipsing double white dwarf binary after NLTT 11748. The 2 minute long primary eclipse is 40% deep and the secondary eclipse 10% deep. From Gemini+GMOS spectroscopy, we measure the radial velocities of both components of the binary from the H{alpha} absorption line cores. These measurements, combined with the light curve information, yield white dwarf masses of M{sub 1} = 0.283 {+-} 0.064 M{sub sun} and M{sub 2} = 0.274 {+-} 0.034 M{sub sun}, making them both helium core white dwarfs. As an eclipsing, double-lined spectroscopic binary, CSS 41177 is ideally suited to measuring precise, model-independent masses and radii. The two white dwarfs will merge in roughly 1.1 Gyr to form a single sdB star. 2. Relativistic Astrophysics in Black Hole and Low-Mass Neutron Star X-ray Binaries NASA Technical Reports Server (NTRS) 2000-01-01 During the five-year period, our study of "Relativistic Astrophysics in Black Hole and Low-Mass Neutron Star X-ray Binaries" has been focused on the following aspects: observations, data analysis, Monte-Carlo simulations, numerical calculations, and theoretical modeling. Most of the results of our study have been published in refereed journals and conference presentations. 3. FORMATION OF MILLISECOND PULSARS FROM INTERMEDIATE- AND LOW-MASS X-RAY BINARIES SciTech Connect Shao Yong; Li Xiangdong 2012-09-01 We present a systematic study of the evolution of intermediate- and low-mass X-ray binaries consisting of an accreting neutron star of mass 1.0-1.8 M{sub Sun} and a donor star of mass 1.0-6.0 M{sub Sun }. In our calculations we take into account physical processes such as unstable disk accretion, radio ejection, bump-induced detachment, and outflow from the L{sub 2} point. Comparing the calculated results with the observations of binary radio pulsars, we report the following results. (1) The allowed parameter space for forming binary pulsars in the initial orbital period-donor mass plane increases with increasing neutron star mass. This may help explain why some millisecond pulsars with orbital periods longer than {approx}60 days seem to have less massive white dwarfs than expected. Alternatively, some of these wide binary pulsars may be formed through mass transfer driven by planet/brown-dwarf-involved common envelope evolution. (2) Some of the pulsars in compact binaries might have evolved from intermediate-mass X-ray binaries with anomalous magnetic braking. (3) The equilibrium spin periods of neutron stars in low-mass X-ray binaries are in general shorter than the observed spin periods of binary pulsars by more than one order of magnitude, suggesting that either the simple equilibrium spin model does not apply or there are other mechanisms/processes spinning down the neutron stars. 4. A LONG-PERIOD TOTALLY ECLIPSING BINARY STAR AT THE TURNOFF OF THE OPEN CLUSTER NGC 6819 DISCOVERED WITH KEPLER SciTech Connect Sandquist, Eric L.; Orosz, Jerome A.; Jeffries, Mark W. Jr.; Brewer, Lauren N. E-mail: [email protected]; and others 2013-01-01 We present the discovery of the totally eclipsing long-period (P = 771.8 days) binary system WOCS 23009 in the old open cluster NGC 6819 that contains both an evolved star near central hydrogen exhaustion and a low-mass (0.45 M {sub Sun }) star. This system was previously known to be a single-lined spectroscopic binary, but the discovery of an eclipse near apastron using data from the Kepler space telescope makes it clear that the system has an inclination that is very close to 90 Degree-Sign . Although the secondary star has not been identified in spectra, the mass of the primary star can be constrained using other eclipsing binaries in the cluster. The combination of the total eclipses and a mass constraint for the primary star allows us to determine a reliable mass for the secondary star and radii for both stars, and to constrain the cluster age. Unlike well-measured stars of similar mass in field binaries, the low-mass secondary is not significantly inflated in radius compared to model predictions. The primary star characteristics, in combination with cluster photometry and masses from other cluster binaries, indicate a best age of 2.62 {+-} 0.25 Gyr, although stellar model physics may introduce systematic uncertainties at the {approx}10% level. We find preliminary evidence that the asteroseismic predictions for red giant masses in this cluster are systematically too high by as much as 8%. 5. Selection effects on the orbital period distribution of Low Mass X-ray Binaries Arur, Kavitha; Maccarone, Tom 2017-01-01 Observations show a lack of Low Mass Black Hole Binaries with orbital periods below 4 hours. While it is known that Black Hole Binaries (BHBs) tend to have lower peak luminosities in outburst compared to their Neutron Star counterparts, it is unclear if selection effects can account for the difference in the numbers. Studying the effect of these selection biases is important for binary population studies. Here we report on the implications for the inferred orbital period distribution of these BHBs after a simulation that accounts for extinction of the optical counterpart, absorption of X-ray counts and detectability of the outburst. 6. How eclipse time variations, eclipse duration variations, and radial velocities can reveal S-type planets in close eclipsing binaries Oshagh, M.; Heller, R.; Dreizler, S. 2016-12-01 While about a dozen transiting planets have been found in wide orbits around an inner, close stellar binary (so-called "P-type planets"), no planet has yet been detected orbiting only one star (a so-called "S-type planet") in an eclipsing binary. This is despite a large number of eclipsing binary systems discovered with the Kepler telescope. Here we propose a new detection method for these S-type planets, which uses a correlation between the stellar radial velocities (RVs), eclipse timing variations (ETVs), and eclipse duration variations (EDVs). We test the capability of this technique by simulating a realistic benchmark system and demonstrate its detectability with existing high-accuracy RV and photometry instruments. We illustrate that, with a small number of RV observations, the RV-ETV diagrams allows us to distinguish between prograde and retrograde planetary orbits and also the planetary mass can be estimated if the stellar cross-correlation functions can be disentangled. We also identify a new (though minimal) contribution of S-type planets to the Rossiter-McLaughlin effect in eclipsing stellar binaries. We finally explore possible detection of exomoons around transiting luminous giant planets and find that the precision required to detect moons in the RV curves of their host planets is of the order of cm s-1 and therefore not accessible with current instruments. 7. How eclipse time variations, eclipse duration variations, and radial velocities can reveal S-type planets in close eclipsing binaries Oshagh, M.; Heller, R.; Dreizler, S. 2017-04-01 While about a dozen transiting planets have been found in wide orbits around an inner, close stellar binary (so-called P-type planets), no planet has yet been detected orbiting only one star (a so-called S-type planet) in an eclipsing binary. This is despite a large number of eclipsing binary systems discovered with the Kepler telescope. Here we propose a new detection method for these S-type planets, which uses a correlation between the stellar radial velocities (RVs), eclipse timing variations (ETVs) and eclipse duration variations (EDVs). We test the capability of this technique by simulating a realistic benchmark system and demonstrate its detectability with existing high-accuracy RV and photometry instruments. We illustrate that with a small number of RV observations, the RV-ETV diagrams allows us to distinguish between prograde and retrograde planetary orbits and also the planetary mass can be estimated if the stellar cross-correlation functions can be disentangled. We also identify a new (though minimal) contribution of S-type planets to the Rossiter-McLaughlin effect in eclipsing stellar binaries. We finally explore possible detection of exomoons around transiting luminous giant planets and find that the precision required to detect moons in the RV curves of their host planets is of the order of cm s-1 and therefore not accessible with current instruments. 8. The dynamical mass of a classical Cepheid variable star in an eclipsing binary system. PubMed Pietrzyński, G; Thompson, I B; Gieren, W; Graczyk, D; Bono, G; Udalski, A; Soszyński, I; Minniti, D; Pilecki, B 2010-11-25 Stellar pulsation theory provides a means of determining the masses of pulsating classical Cepheid supergiants-it is the pulsation that causes their luminosity to vary. Such pulsational masses are found to be smaller than the masses derived from stellar evolution theory: this is the Cepheid mass discrepancy problem, for which a solution is missing. An independent, accurate dynamical mass determination for a classical Cepheid variable star (as opposed to type-II Cepheids, low-mass stars with a very different evolutionary history) in a binary system is needed in order to determine which is correct. The accuracy of previous efforts to establish a dynamical Cepheid mass from Galactic single-lined non-eclipsing binaries was typically about 15-30% (refs 6, 7), which is not good enough to resolve the mass discrepancy problem. In spite of many observational efforts, no firm detection of a classical Cepheid in an eclipsing double-lined binary has hitherto been reported. Here we report the discovery of a classical Cepheid in a well detached, double-lined eclipsing binary in the Large Magellanic Cloud. We determine the mass to a precision of 1% and show that it agrees with its pulsation mass, providing strong evidence that pulsation theory correctly and precisely predicts the masses of classical Cepheids. 9. Spectroscopy of Low Mass X-Ray Binaries: New Insights into Accretion NASA Technical Reports Server (NTRS) DilVrtilek, Saeqa; Mushotsky, Richard (Technical Monitor) 2004-01-01 This project is to observe two low mass X-ray binaries, chosen for their X-ray brightness, low column density, and diversity of accretion behavior. The high spectral resolution of the RGS, the broad energy range and tremendous collecting power of EPIC, and simultaneous optical monitoring with the OM are particularly well-suited to these studies. The second of two objects was observed on September of 2002. Data analysis for both observation has been completed: an investigation of the physical conditions of the emitting gas using emission and recombination line diagnostics to determine temperatures, densities, elemental abundances, and ionization structure. A study of behavior of the emission features as a function of binary orbit shows modulated behavior in one of the systems. A paper on "High-resolution observations of low-mass X-ray binaries" is near completion. The paper includes observations with the Chandra HETG that are not yet completed. 10. The doubly eclipsing quintuple low-mass star system 1SWASP J093010.78+533859.5 Lohr, M. E.; Norton, A. J.; Gillen, E.; Busuttil, R.; Kolb, U. C.; Aigrain, S.; McQuillan, A.; Hodgkin, S. T.; González, E. 2015-06-01 Our discovery of 1SWASP J093010.78+533859.5 as a probable doubly eclipsing quadruple system, containing a contact binary with P ~ 0.23 d and a detached binary with P ~ 1.31 d, was announced in 2013. Subsequently, Koo and collaborators confirmed the detached binary spectroscopically, and identified a fifth set of static spectral lines at its location, corresponding to an additional non-eclipsing component of the system. Here we present new spectroscopic and photometric observations, allowing confirmation of the contact binary and improved modelling of all four eclipsing components. The detached binary is found to contain components of masses 0.837 ± 0.008 and 0.674 ± 0.007M⊙, with radii of 0.832 ± 0.018 and 0.669 ± 0.018R⊙ and effective temperatures of and K, respectively; the contact system has masses 0.86 ± 0.02 and 0.341 ± 0.011M⊙, radii of 0.79 ± 0.04 and 0.52 ± 0.05R⊙, respectively, and a common effective temperature of 4700 ± 50 K. The fifth star is of similar temperature and spectral type to the primaries in the two binaries. Long-term photometric observations indicate the presence of a spot on one component of the detached binary, moving at an apparent rate of approximately one rotation every two years. Both binaries have consistent system velocities around -11 to -12 km s-1, which match the average radial velocity of the fifth star; consistent distance estimates for both subsystems of d = 78 ± 3 and d = 73 ± 4 pc are also found, and, with some further assumptions, of d = 83 ± 9 pc for the fifth star. These findings strongly support the claim that both binaries - and very probably all five stars - are gravitationally bound in a single system. The consistent angles of inclination found for the two binaries (88.2 ± 0.3°and 86 ± 4°) may also indicate that they originally formed by fragmentation (around 9-10 Gyr ago) from a single protostellar disk, and subsequently remained in the same orbital plane. Table 1 is available in electronic 11. On the Distribution of Orbital Eccentricities for Very Low-mass Binaries Dupuy, Trent J.; Liu, Michael C. 2011-06-01 We have compiled a sample of 16 orbits for very low-mass stellar (<0.1 M sun) and brown dwarf binaries, including updated orbits for HD 130948BC and LP 415-20AB. This sample enables the first comprehensive study of the eccentricity distribution for such objects. We find that very low-mass binaries span a broad range of eccentricities from near-circular to highly eccentric (e ≈ 0.8), with a median eccentricity of 0.34. We have examined potential observational biases in this sample, and for visual binaries we show through Monte Carlo simulations that if we choose appropriate selection criteria then all eccentricities are equally represented (lsim 5% difference between input and output eccentricity distributions). The orbits of this sample of very low-mass binaries show some significant differences from their solar-type counterparts. They lack a correlation between orbital period and eccentricity, and display a much higher fraction of near-circular orbits (e < 0.1) than solar-type stars, which together may suggest a different formation mechanism or dynamical history for these two populations. Very low-mass binaries also do not follow the e 2 distribution of Ambartsumian, which would be expected if their orbits were distributed in phase space according to a function of energy alone (e.g., the Boltzmann distribution). We find that current numerical simulations of very low-mass star formation do not completely reproduce the observed properties of our binary sample. The cluster formation model of Bate agrees very well with the overall e distribution, but the lack of any high-e (>0.6) binaries at orbital periods comparable to our sample suggests that tidal damping due to gas disks may play too large of a role in the simulations. In contrast, the circumstellar disk fragmentation model of Stamatellos & Whitworth predicts only high-e binaries and thus is highly inconsistent with our sample. These discrepancies could be explained if multiple formation processes are 12. On the formation of galactic black hole low-mass X-ray binaries Wang, Chen; Jia, Kun; Li, Xiang-Dong 2016-03-01 Currently, there are 24 black hole (BH) X-ray binary systems that have been dynamically confirmed in the Galaxy. Most of them are low-mass X-ray binaries (LMXBs) comprised of a stellar-mass BH and a low-mass donor star. Although the formation of these systems has been extensively investigated, some crucial issues remain unresolved. The most noticeable one is that, the low-mass companion has difficulties in ejecting the tightly bound envelope of the massive primary during the spiral-in process. While initially intermediate-mass binaries are more likely to survive the common envelope (CE) evolution, the resultant BH LMXBs mismatch the observations. In this paper, we use both stellar evolution and binary population synthesis to study the evolutionary history of BH LMXBs. We test various assumptions and prescriptions for the supernova mechanisms that produce BHs, the binding energy parameter, the CE efficiency and the initial mass distributions of the companion stars. We obtain the birthrate and the distributions of the donor mass, effective temperature and orbital period for the BH LMXBs in each case. By comparing the calculated results with the observations, we put useful constraints on the aforementioned parameters. In particular, we show that it is possible to form BH LMXBs with the standard CE scenario if most BHs are born through failed supernovae. 13. The Eclipsing Binary On-Line Atlas (EBOLA) Bradstreet, D. H.; Steelman, D. P.; Sanders, S. J.; Hargis, J. R. 2004-05-01 In conjunction with the upcoming release of \\it Binary Maker 3.0, an extensive on-line database of eclipsing binaries is being made available. The purposes of the atlas are: \\begin {enumerate} Allow quick and easy access to information on published eclipsing binaries. Amass a consistent database of light and radial velocity curve solutions to aid in solving new systems. Provide invaluable querying capabilities on all of the parameters of the systems so that informative research can be quickly accomplished on a multitude of published results. Aid observers in establishing new observing programs based upon stars needing new light and/or radial velocity curves. Encourage workers to submit their published results so that others may have easy access to their work. Provide a vast but easily accessible storehouse of information on eclipsing binaries to accelerate the process of understanding analysis techniques and current work in the field. \\end {enumerate} The database will eventually consist of all published eclipsing binaries with light curve solutions. The following information and data will be supplied whenever available for each binary: original light curves in all bandpasses, original radial velocity observations, light curve parameters, RA and Dec, V-magnitudes, spectral types, color indices, periods, binary type, 3D representation of the system near quadrature, plots of the original light curves and synthetic models, plots of the radial velocity observations with theoretical models, and \\it Binary Maker 3.0 data files (parameter, light curve, radial velocity). The pertinent references for each star are also given with hyperlinks directly to the papers via the NASA Abstract website for downloading, if available. In addition the Atlas has extensive searching options so that workers can specifically search for binaries with specific characteristics. The website has more than 150 systems already uploaded. The URL for the site is http://ebola.eastern.edu/. 14. MM Herculis - An eclipsing binary of the RS CVn NASA Technical Reports Server (NTRS) Sowell, J. R.; Hall, D. S.; Henry, G. W.; Burke, E. W., Jr.; Milone, E. F. 1983-01-01 V, B and U differential photoelectric photometry has been obtained for the RS Canum Venaticorum-class eclipsing binary star MM Her, with the light outside the eclipse being Fourier-analyzed to study wave migration and amplitude. These, together with the mean light level of the system, have been monitored from 1976 through 1980. Observations within the eclipse have revealed eclipses to be partial, rather than total as previously thought. The geometric elements of the presently rectified light curve are forced on the pre-1980 light curves and found to be compatible. With these elements, and previously obtained double line radial velocity curves, new absolute dimensions of 1.18 solar masses and 1.58 solar radii are calculated for the hotter star and 1.27 solar masses and 2.83 solar radii for the cooler star. The plotting of color indices on the color-color curve indicates G2V and K2IV spectral types. 15. Photoelectric Photometry of the Eclipsing Binary V505-MONOCEROTIS Chochol, D.; Bakos, G. A.; Bartolini, C.; Guarnieri, A.; Dapergolas, A.; Szabados, L. Photoelectric U, B, V observations of the eclipsing binary V 505 Mon, performed at the observatories in Skalnate Pleso, Budapest, Bologna and Waterloo in the years 1972-1984 are presented. The following ephemeris has been derived, using all these data: prim.min. Indications of ongoing mass transfer in a semidetached binary configuration are presented. The possible causes of the observed short-term changes of brightness are discussed. 16. Photometric Analysis and Period Investigation of the EW Type Eclipsing Binary V441 Lac Li, K.; Hu, S.-M.; Guo, D.-F.; Jiang, Y.-G.; Gao, D.-Y.; Chen, X. 2016-09-01 Four color light curves of the EW type eclipsing binary V441 Lac were presented and analyzed by the W-D code. It is found that V441 Lac is an extremely low mass ratio ( q = 0.093±0.001) semi-detached binary with the less massive secondary component filling the inner Roche lobe. Two dark spots on the primary component were introduced to explain the asymmetric light curves. By analyzing all times of light minimum, we determined that the orbital period of V441 Lac is continuously increasing at a rate of d P/d t = 5.874(±0.007) × 10-7 d yr-1. The semi-detached Algol type configuration of V441 Lac is possibly formed by a contact configuration destroyed shallow contact binary due to mass transfer from the less massive component to the more massive one predicted by the thermal relaxation oscillation theory. 17. A model of V356 Sagittarii. [eclipsing binary star NASA Technical Reports Server (NTRS) Wilson, R. E.; Caldwell, C. N. 1978-01-01 It is pointed out that V356 Sgr is an abnormal member of the Algol class of binaries. According to Popper (1955), the primary component is of spectral type B3V and is rotating rapidly, while the secondary is of type A2II and is rotating at least approximately in synchronism with the orbital motion. The system is either semidetached or quite near to being semidetached. The main anomalies are related to the ratio of eclipse depths, the very small reflection effect of the light curves, differences between the duration of the primary and the secondary eclipse, and the unusual characteristics of the primary eclipse. It is concluded that the lack of agreement between theory and observation can be due only to an important attribute of the binary which has not yet been incorporated into the theory. The peculiarities can most reasonably be explained in terms of a geometrically and optically thick disk which surrounds the primary component. 18. Long-term variability of low-mass X-ray binaries Filippova, E.; Revnivtsev, M.; Parkin, E. R. 2014-01-01 We consider modulations of mass captured by the compact object from the companion star's stellar wind in Low Mass X-ray Binaries with late type giants. Based on 3D simulations with two different hydrodynamic codes used Lagrangian and Eulerian approaches - the SPH code GADGET and the Eulerian code PLUTO, we conclude that a hydrodynamical interaction of the wind matter within a binary system even without eccentricity results in variability of the mass accretion rate with characteristic time-scales close to the orbital period. Observational appearances of this wind might be similar to that of an accretion disc corona/wind. 19. Towards a Fundamental Understanding of Short Period Eclipsing Binary Systems Using Kepler Data Prsa, Andrej Kepler's ultra-high precision photometry is revolutionizing stellar astrophysics. We are seeing intrinsic phenomena on an unprecedented scale, and interpreting them is both a challenge and an exciting privilege. Eclipsing binary stars are of particular significance for stellar astrophysics because precise modeling leads to fundamental parameters of the orbiting components: masses, radii, temperatures and luminosities to better than 1-2%. On top of that, eclipsing binaries are ideal physical laboratories for studying other physical phenomena, such as asteroseismic properties, chromospheric activity, proximity effects, mass transfer in close binaries, etc. Because of the eclipses, the basic geometry is well constrained, but a follow-up spectroscopy is required to get the dynamical masses and the absolute scale of the system. A conjunction of Kepler photometry and ground- based spectroscopy is a treasure trove for eclipsing binary star astrophysics. This proposal focuses on a carefully selected set of 100 short period eclipsing binary stars. The fundamental goal of the project is to study the intrinsic astrophysical effects typical of short period binaries in great detail, utilizing Kepler photometry and follow-up spectroscopy to devise a robust and consistent set of modeling results. The complementing spectroscopy is being secured from 3 approved and fully funded programs: the NOAO 4-m echelle spectroscopy at Kitt Peak (30 nights; PI Prsa), the 10- m Hobby-Eberly Telescope high-resolution spectroscopy (PI Mahadevan), and the 2.5-m Sloan Digital Sky Survey III spectroscopy (PI Mahadevan). The targets are prioritized by the projected scientific yield. Short period detached binaries host low-mass (K- and M- type) components for which the mass-radius relationship is sparsely populated and still poorly understood, as the radii appear up to 20% larger than predicted by the population models. We demonstrate the spectroscopic detection viability in the secondary 20. A New Orbit for the Eclipsing Binary V577 Oph Jeffery, Elizabeth J.; Barnes, Thomas G., III; Skillen, Ian; Montemayor, Thomas J. 2017-09-01 Pulsating stars in eclipsing binary systems are unique objects for providing constraints on stellar models. To fully leverage the information available from the binary system, full orbital radial velocity curves must be obtained. We report 23 radial velocities for components of the eclipsing binary V577 Oph, whose primary star is a δ Sct variable. The velocities cover a nearly complete orbit and a time base of 20 years. We computed orbital elements for the binary and compared them to the ephemeris computed by Creevey et al. The comparison shows marginally different results. In particular, a change in the systemic velocity by ‑2 km s‑1 is suggested by our results. We compare this systemic velocity difference to that expected due to reflex motion of the binary in response to the third body in the system. The systemic velocity difference is consistent with reflex motion, given our mass determination for the eclipsing binary and the orbital parameters determined by Volkov & Volkova for the three-body orbit. We see no evidence for the third body in our spectra, but we do see strong interstellar Na D lines that are consistent in strength with the direction and expected distance of V577 Oph. 1. Kepler Eclipsing Binary Stars. VII. The Catalog of Eclipsing Binaries Found in the Entire Kepler Data Set Kirk, Brian; Conroy, Kyle; Prša, Andrej; Abdul-Masih, Michael; Kochoska, Angela; Matijevič, Gal; Hambleton, Kelly; Barclay, Thomas; Bloemen, Steven; Boyajian, Tabetha; Doyle, Laurance R.; Fulton, B. J.; Hoekstra, Abe Johannes; Jek, Kian; Kane, Stephen R.; Kostov, Veselin; Latham, David; Mazeh, Tsevi; Orosz, Jerome A.; Pepper, Joshua; Quarles, Billy; Ragozzine, Darin; Shporer, Avi; Southworth, John; Stassun, Keivan; Thompson, Susan E.; Welsh, William F.; Agol, Eric; Derekas, Aliz; Devor, Jonathan; Fischer, Debra; Green, Gregory; Gropp, Jeff; Jacobs, Tom; Johnston, Cole; LaCourse, Daryll Matthew; Saetre, Kristian; Schwengeler, Hans; Toczyski, Jacek; Werner, Griffin; Garrett, Matthew; Gore, Joanna; Martinez, Arturo O.; Spitzer, Isaac; Stevick, Justin; Thomadis, Pantelis C.; Vrijmoet, Eliot Halley; Yenawine, Mitchell; Batalha, Natalie; Borucki, William 2016-03-01 The primary Kepler Mission provided nearly continuous monitoring of ∼200,000 objects with unprecedented photometric precision. We present the final catalog of eclipsing binary systems within the 105 deg2 Kepler field of view. This release incorporates the full extent of the data from the primary mission (Q0-Q17 Data Release). As a result, new systems have been added, additional false positives have been removed, ephemerides and principal parameters have been recomputed, classifications have been revised to rely on analytical models, and eclipse timing variations have been computed for each system. We identify several classes of systems including those that exhibit tertiary eclipse events, systems that show clear evidence of additional bodies, heartbeat systems, systems with changing eclipse depths, and systems exhibiting only one eclipse event over the duration of the mission. We have updated the period and galactic latitude distribution diagrams and included a catalog completeness evaluation. The total number of identified eclipsing and ellipsoidal binary systems in the Kepler field of view has increased to 2878, 1.3% of all observed Kepler targets. An online version of this catalog with downloadable content and visualization tools is maintained at http://keplerEBs.villanova.edu. 2. Eclipsing Binaries From the CSTAR Project at Dome A, Antarctica Yang, Ming; Zhang, Hui; Wang, Songhu; Zhou, Ji-Lin; Zhou, Xu; Wang, Lingzhi; Wang, Lifan; Wittenmyer, R. A.; Liu, Hui-Gen; Meng, Zeyang; Ashley, M. C. B.; Storey, J. W. V.; Bayliss, D.; Tinney, Chris; Wang, Ying; Wu, Donghong; Liang, Ensi; Yu, Zhouyi; Fan, Zhou; Feng, Long-Long; Gong, Xuefei; Lawrence, J. S.; Liu, Qiang; Luong-Van, D. M.; Ma, Jun; Wu, Zhenyu; Yan, Jun; Yang, Huigen; Yang, Ji; Yuan, Xiangyan; Zhang, Tianmeng; Zhu, Zhenxi; Zou, Hu 2015-04-01 The Chinese Small Telescope ARray (CSTAR) has observed an area around the Celestial South Pole at Dome A since 2008. About 20,000 light curves in the i band were obtained during the observation season lasting from 2008 March to July. The photometric precision achieves about 4 mmag at i = 7.5 and 20 mmag at i = 12 within a 30 s exposure time. These light curves are analyzed using Lomb-Scargle, Phase Dispersion Minimization, and Box Least Squares methods to search for periodic signals. False positives may appear as a variable signature caused by contaminating stars and the observation mode of CSTAR. Therefore, the period and position of each variable candidate are checked to eliminate false positives. Eclipsing binaries are removed by visual inspection, frequency spectrum analysis, and a locally linear embedding technique. We identify 53 eclipsing binaries in the field of view of CSTAR, containing 24 detached binaries, 8 semi-detached binaries, 18 contact binaries, and 3 ellipsoidal variables. To derive the parameters of these binaries, we use the Eclipsing Binaries via Artificial Intelligence method. The primary and secondary eclipse timing variations (ETVs) for semi-detached and contact systems are analyzed. Correlated primary and secondary ETVs confirmed by false alarm tests may indicate an unseen perturbing companion. Through ETV analysis, we identify two triple systems (CSTAR J084612.64-883342.9 and CSTAR J220502.55-895206.7). The orbital parameters of the third body in CSTAR J220502.55-895206.7 are derived using a simple dynamical model. 3. Eclipsing Binary Science through the Monocle of Kepler Prsa, Andrej; Eclipsing Binary Working Group 2013-07-01 The notable success of space-borne missions such as MOST, CoRoT and Kepler triggered a surge of exciting new results in stellar astrophysics, ranging from asteroseismology, discoveries of new subclasses of objects such as heartbeat stars, to the literal firehose of extrasolar planets. The nearly continuous observing mode and an unprecedented photometric precision provide us with data that challenge even the most sophisticated models. Eclipsing binary stars play a major role since their accurate modeling provides fundamental stellar parameters (masses, radii, temperatures and luminosities) across the H-R diagram by relying on the uniquely favorable geometry that alleviates the need for any calibrations. NASA's Kepler mission is particularly well suited for the study of binaries; the ~10-ppm precision and the ~105-square degree field of view yield a sample of ~2500 eclipsing systems of varying types and morphologies, that have been observed uninterruptedly for 4 years in a row. I will present statistical results of the complete set of Kepler eclipsing binaries, including the distributions of the periods, galactic latitudes, morphologies, orbital properties and fundamental stellar parameters. The mission provided us with ground-breaking observations of multiple components through the measurements of eclipse timing variations. I will emphasize the pioneering efforts to detect and analyze stellar and substellar tertiaries orbiting binary stars and explore the implications of multiplicity on the evolution of these systems. Several theoretical aspects of reliable modeling still elude our grasp, and I will provide a theorist's perspective of the direction that our field might take in the next several years. Lastly, I will focus on a few notable "head-scratchers", systems that deserve special attention because of their uniqueness and/or general importance to astrophysics. This presentation will encapsulate the results based on the work and dedication of the entire Kepler 4. Surface imaging of eclipsing binary stars. 1: Techniques Vincent, A.; Piskunov, N. E.; Tuominen, I. 1993-11-01 Surface (Doppler) imaging techniques for mapping the temperature distribution of a single star are generalized to the case of an eclipsing spectroscopic binary. In this paper we study three main questions, crucial for further application of the techniques. We found that the method described in this paper can be successfully used for imaging eclipsing binary systems. The resulting map is more sensitive to the errors in the parameters of the system than is the case of a single star. Characteristic distortions of the map can be used as indicators for fine tuning of some of the parameters. We also found that a good phase coverage of the observations is most important for reducing the artificial equatorial symmetry, typical for the line profile inversion when used for high inclination binary systems. 5. Photometric study of the pulsating, eclipsing binary OO DRA SciTech Connect Zhang, X. B.; Deng, L. C.; Tian, J. F.; Wang, K.; Yan, Z. Z.; Luo, C. Q.; Sun, J. J.; Liu, Q. L.; Xin, H. Q.; Zhou, Q.; Luo, Z. Q. 2014-12-01 We present a comprehensive photometric study of the pulsating, eclipsing binary OO Dra. Simultaneous B- and V-band photometry of the star was carried out on 14 nights. A revised orbital period and a new ephemeris were derived from the data. The first photometric solution of the binary system and the physical parameters of the component stars are determined. They reveal that OO Dra could be a detached system with a less-massive secondary component nearly filling its Roche lobe. By subtracting the eclipsing light changes from the data, we obtained the intrinsic pulsating light curves of the hotter, massive primary component. A frequency analysis of the residual light yields two confident pulsation modes in both B- and V-band data with the dominant frequency detected at 41.865 c/d. A brief discussion concerning the evolutionary status and the pulsation nature of the binary system is finally given. 6. Formation of Galactic Black Hole Low-Mass X-ray Binaries Li, Xiangdong 2016-07-01 Most of the Galactic black hole (BH) X-ray binary systems are low-mass X-ray binaries (LMXBs). Although the formation of these systems has been extensively investigated, some crucial issues remain unresolved. The most noticeable one is that, the low-mass companion has difficulties in ejecting the tightly bound envelope of the massive primary during the spiral-in process. While initially intermediate-mass binaries are more likely to survive the common envelope (CE) evolution, the resultant BH LMXBs mismatch the observations. Here we use both stellar evolution and binary population synthesis to study the evolutionary history of BH LMXBs. We test various assumptions and prescriptions for the supernova mechanisms that produce BHs, the binding energy parameter, the CE efficiency, and the initial mass distributions of the companion stars. We obtain the birthrate and the distributions of the donor mass, effective temperature and orbital period for the BH LMXBs in each case. By comparing the calculated results with the observations, we put useful constraints on the aforementioned parameters. In particular, we show that it is possible to form BH LMXBs with the standard CE scenario if most BHs are born through failed supernovae. 7. Formation of millisecond pulsars with low-mass helium white dwarf companions in very compact binaries SciTech Connect Jia, Kun; Li, X.-D. 2014-08-20 Binary millisecond pulsars (BMSPs) are thought to have evolved from low-mass X-ray binaries (LMXBs). If the mass transfer in LMXBs is driven by nuclear evolution of the donor star, the final orbital period is predicted to be well correlated with the mass of the white dwarf (WD), which is the degenerate He core of the donor. Here we show that this relation can be extended to very small WD mass (∼0.14-0.17 M {sub ☉}) and narrow orbital period (about a few hours), depending mainly on the metallicities of the donor stars. There is also discontinuity in the relation, which is due to the temporary contraction of the donor when the H-burning shell crosses the hydrogen discontinuity. BMSPs with low-mass He WD companions in very compact binaries can be accounted for if the progenitor binary experienced very late Case A mass transfer. The WD companion of PSR J1738+0333 is likely to evolve from a Pop II star. For PSR J0348+0432, to explain its extreme compact orbit in the Roche-lobe-decoupling phase, even lower metallicity (Z = 0.0001) is required. 8. KEPLER ECLIPSING BINARY STARS. III. CLASSIFICATION OF KEPLER ECLIPSING BINARY LIGHT CURVES WITH LOCALLY LINEAR EMBEDDING SciTech Connect Matijevic, Gal; Prsa, Andrej; Orosz, Jerome A.; Welsh, William F.; Bloemen, Steven; Barclay, Thomas E-mail: [email protected] 2012-05-15 We present an automated classification of 2165 Kepler eclipsing binary (EB) light curves that accompanied the second Kepler data release. The light curves are classified using locally linear embedding, a general nonlinear dimensionality reduction tool, into morphology types (detached, semi-detached, overcontact, ellipsoidal). The method, related to a more widely used principal component analysis, produces a lower-dimensional representation of the input data while preserving local geometry and, consequently, the similarity between neighboring data points. We use this property to reduce the dimensionality in a series of steps to a one-dimensional manifold and classify light curves with a single parameter that is a measure of 'detachedness' of the system. This fully automated classification correlates well with the manual determination of morphology from the data release, and also efficiently highlights any misclassified objects. Once a lower-dimensional projection space is defined, the classification of additional light curves runs in a negligible time and the method can therefore be used as a fully automated classifier in pipeline structures. The classifier forms a tier of the Kepler EB pipeline that pre-processes light curves for the artificial intelligence based parameter estimator. 9. Searching Planets Around Some Selected Eclipsing Close Binary Stars Systems Nasiroglu, Ilham; Slowikowska, Agnieszka; Krzeszowski, Krzysztof; Zejmo, M. Michal; Er, Hüseyin; Goździewski, Krzysztof; Zola, Stanislaw; Koziel-Wierzbowska, Dorota; Debski, Bartholomew; Ogloza, Waldemar; Drozdz, Marek 2016-07-01 We present updated O-C diagrams of selected short period eclipsing binaries observed since 2009 with the T100 Telescope at the TUBITAK National Observatory (Antalya, Turkey), the T60 Telescope at the Adiyaman University Observatory (Adiyaman, Turkey), the 60cm at the Mt. Suhora Observatory of the Pedagogical University (Poland) and the 50cm Cassegrain telescope at the Fort Skala Astronomical Observatory of the Jagiellonian University in Krakow, Poland. All four telescopes are equipped with sensitive, back-illuminated CCD cameras and sets of wide band filters. One of the targets in our sample is a post-common envelope eclipsing binary NSVS 14256825. We collected more than 50 new eclipses for this system that together with the literature data gives more than 120 eclipse timings over the time span of 8.5 years. The obtained O-C diagram shows quasi-periodic variations that can be well explained by the existence of the third body on Jupiter-like orbit. We also present new results indicating a possible light time travel effect inferred from the O-C diagrams of two other binary systems: HU Aqr and V470 Cam. 10. Modeling the luminosity function of galactic low-mass X-ray binaries Kuranov, A. G.; Postnov, K. A.; Revnivtsev, M. G. 2014-01-01 The evolution of the family of binaries with a low-mass star and a compact neutron star companion (low-mass X-ray binaries (LMXBs) with neutron stars) ismodeled by the method of population synthesis. Continuous Roche-lobe filling by the optical star in LMXBs is assumed to be maintained by the removal of orbital angular momentum from the binary by a magnetic stellar wind from the optical star and the radiation of gravitational waves by the binary. The developed model of LMXB evolution has the following significant distinctions: (1) allowance for the effect of the rotational evolution of a magnetized compact remnant on themass transfer scenario in the binary, (2) amore accurate allowance for the response of the donor star to mass loss at the Roche-lobe filling stage. The results of theoretical calculations are shown to be in good agreement with the observed orbital period-X-ray luminosity diagrams for persistent Galactic LMXBs and their X-ray luminosity function. This suggests that the main elements of binary evolution, on the whole, are correctly reflected in the developed code. It is shown that most of the Galactic bulge LMXBs at luminosities L x > 1037 erg s-1 should have a post-main-sequence Roche-lobe-filling secondary component (low-mass giants). Almost all of the models considered predict a deficit of LMXBs at X-ray luminosities near ˜1036.5 erg s-1 due to the transition of the binary from the regime of angular momentum removal by a magnetic stellar wind to the regime of gravitational waves (analogous to the widely known period gap in cataclysmic variables, accreting white dwarfs). At low luminosities, the shape of the model luminosity function for LMXBs is affected significantly by their transient behavior-the accretion rate onto the compact companion is not always equal to the mass transfer rate due to instabilities in the accretion disk around the compact object. The best agreement with observed binaries is achieved in the models suggesting that heavy 11. New Pleiades Eclipsing Binaries and a Hyades Transiting System Identified by K2 David, Trevor J.; Conroy, Kyle E.; Hillenbrand, Lynne A.; Stassun, Keivan G.; Stauffer, John; Rebull, Luisa M.; Cody, Ann Marie; Isaacson, Howard; Howard, Andrew W.; Aigrain, Suzanne 2016-05-01 We present the discovery in Kepler’s K2 mission observations and our follow-up radial velocity (RV) observations from Keck/HIRES for four eclipsing binary (EB) star systems in the young benchmark Pleiades and Hyades clusters. Based on our modeling results, we announce two new low mass ({M}{tot}\\lt 0.6 {M}⊙ ) EBs among Pleiades members (HCG 76 and MHO 9) and we report on two previously known Pleiades binaries that are also found to be EB systems (HII 2407 and HD 23642). We measured the masses of the binary HCG 76 to ≲2.5% precision, and the radii to ≲4.5% precision, which together with the precise effective temperatures yield an independent Pleiades distance of 132 ± 5 pc. We discuss another EB toward the Pleiades that is a possible but unlikely Pleiades cluster member (AK II 465). The two new confirmed Pleiades systems extend the mass range of Pleiades EB components to 0.2-2 {M}⊙ . Our initial measurements of the fundamental stellar parameters for the Pleiades EBs are discussed in the context of the current stellar models and the nominal cluster isochrone, finding good agreement with the stellar models of Baraffe et al. at the nominal Pleiades age of 120 Myr. Finally, in the Hyades, we report a new low mass eclipsing system (vA 50) that was concurrently discovered and studied by Mann et al. We confirm that the eclipse is likely caused by a Neptune-sized transiting planet, and with the additional RV constraints presented here we improve the constraint on the maximum mass of the planet to be ≲1.2 MJup. 12. Stellar and Circumstellar Properties of Low-Mass, Young, Subarcsecond Binaries Bruhns, Sara; Prato, L. A. 2014-01-01 We present a study of the stellar and circumstellar characteristics of close (< 1''), young (< 2 to 3 Myr), low-mass (<1 solar mass) binary stars in the Taurus star forming region. Low-resolution (R ~ 2000) spectra were taken in the K-band using adaptive optics to separate the observations for each component and identify the individual spectral types, extinction, and K-band excess. Combining these data with stellar luminosities allows us to estimate the stellar masses and ages. We also measured equivalent widths of the hydrogen Brackett gamma line in order to estimate the strength of gas accretion. We obtained spectra for six binary systems with separations from 1'' down to 0.3''. In the CZ Tau binary we found that the fainter secondary star spectrum appears to be of earlier spectral type than the primary; we speculate on the origin of this inversion. 13. Solar-Type Eclipsing Binary Systems with Impacting Gas Streams Samec, Ronald G.; Hube, Doug; Faulkner, Danny R.; van Hamme, W. 2002-08-01 Our quest is the recovery of near contact solar type eclipsing binaries with evidence for stream impacts. Their existence will provide strong support of dynamic mass transfer leading to coalescence into a state of contact. This will lend strong support to the theoretical scenarios of 1) angular momentum loss(AML)via magnetic breaking scenario and 2)Thermal Relaxation Oscillations (TRO)or oscillations between a nearÂcontact and shallow contact modes. We hypothesize that many F to early K spectral type binaries formerly classified as ''thermally decoupled'' contact binaries and other binaries with large differences in eclipse depths formerly classified as contact binaries in the 0.33 to 0.5d period range will reveal evidence for stream impacts when they are subjected to precision UBVRI multiÂband photometry, since these fall in the preÂcontact period range for F to K dwarf binaries. Modern light curve synthesis techniques will be used to simultaneously model the multiÂband light curves. Impact spots will be adjusted numerically along with the stellar atmosphere parameters. Spectroscopic work will follow to verify stream activity and to obtain fundamental physical characteristics. Our larger goal is to understand close binary evolution in general. This study could supply an important piece to the puzzle. We now have found four candidates, CN And, BE Cep, ZZ Eri and V343 Cen giving us an encouraging 40 percent recovery thus far. 14. Southern Very Low Mass Stars and Brown Dwarfs in Wide Binary and Multiple Systems Caballero, José Antonio 2007-09-01 The results of the Königstuhl survey in the Southern Hemisphere are presented. I have searched for common proper motion companions to 173 field very low mass stars and brown dwarfs with spectral types >M5.0 V and magnitudes J<~14.5 mag. I have measured for the first time the common proper motion of two new wide systems containing very low mass components, Königstuhl 2 AB and 3 A-BC. Together with Königstuhl 1 AB and 2M 0126-50 AB, they are among the widest systems in their respective classes (r=450-11,900 AU). I have determined the minimum frequency of field wide multiples (r>100 AU) with late-type components at 5.0%+/-1.8% and the frequency of field wide late-type binaries with mass ratios q>0.5 at 1.2%+/-0.9%. These values represent a key diagnostic of evolution history and low-mass star and brown dwarf formation scenarios. In addition, the proper motions of 62 field very low mass dwarfs are measured here for the first time. 15. NIP of Stars: early results and new eclipsing binaries Jaque Arancibia, M.; Barba, R.; Morrell, N.; Roman Lopes, A.; Torres Robledo, S.; Gunthardt, G.; Soto, M.; Ferrero, G.; Arias, J. I.; Gamen, R.; Fernadez Lajus, E. 2014-10-01 We have performed a near-infrared photometric monitoring of 39 galactic young star clusters and star-forming regions, known as NIP of Stars, between the years 2009-2011, using the Swope telescope at Las Campanas Observatory (Chile) and the RetroCam camera, in H- and Y-bands. This monitoring program is complementary to the Vista Variables in the Via Láctea (VVV), as the brightest sources observed in NIP of Stars are saturated in VVV. The aim of this campaign is to perform a census of photometric variability of such clusters and star-forming regions, with the main goal of discovering massive eclipsing binary stars. In this work, we present a preliminary analysis of this photometric monitoring program with the discovery of tens of candidates for variable stars, among them candidates for massive eclipsing binaries. We included also to the analysis of variability, a small set of images obtained in the Ks with the VISTA telescope in the framework of VVV survey (Minniti et al. 2010). In special, we announce the infrared discovering of four massive eclipsing binaries in the massive young cluster NGC 3603. The stars have been classified spectroscopically as O-type stars, and one of them, MTT 58, has a rare star with a spectral type of O2 If/WN6, as one of its components. We present a preliminary analysis of the light-curves of these binaries. 16. Physics Of Eclipsing Binaries. II. Toward the Increased Model Fidelity Prša, A.; Conroy, K. E.; Horvat, M.; Pablo, H.; Kochoska, A.; Bloemen, S.; Giammarco, J.; Hambleton, K. M.; Degroote, P. 2016-12-01 The precision of photometric and spectroscopic observations has been systematically improved in the last decade, mostly thanks to space-borne photometric missions and ground-based spectrographs dedicated to finding exoplanets. The field of eclipsing binary stars strongly benefited from this development. Eclipsing binaries serve as critical tools for determining fundamental stellar properties (masses, radii, temperatures, and luminosities), yet the models are not capable of reproducing observed data well, either because of the missing physics or because of insufficient precision. This led to a predicament where radiative and dynamical effects, insofar buried in noise, started showing up routinely in the data, but were not accounted for in the models. PHOEBE (PHysics Of Eclipsing BinariEs; http://phoebe-project.org) is an open source modeling code for computing theoretical light and radial velocity curves that addresses both problems by incorporating missing physics and by increasing the computational fidelity. In particular, we discuss triangulation as a superior surface discretization algorithm, meshing of rotating single stars, light travel time effects, advanced phase computation, volume conservation in eccentric orbits, and improved computation of local intensity across the stellar surfaces that includes the photon-weighted mode, the enhanced limb darkening treatment, the better reflection treatment, and Doppler boosting. Here we present the concepts on which PHOEBE is built and proofs of concept that demonstrate the increased model fidelity. 17. Gaia eclipsing binary and multiple systems. A study of detectability and classification of eclipsing binaries with Gaia Kochoska, A.; Mowlavi, N.; Prša, A.; Lecoeur-Taïbi, I.; Holl, B.; Rimoldini, L.; Süveges, M.; Eyer, L. 2017-06-01 Context. In the new era of large-scale astronomical surveys, automated methods of analysis and classification of bulk data are a fundamental tool for fast and efficient production of deliverables. This becomes ever more important as we enter the Gaia era. Aims: We investigate the potential detectability of eclipsing binaries with Gaia using a data set of all Kepler eclipsing binaries sampled with Gaia cadence and folded with the Kepler period. The performance of fitting methods is evaluated in comparison to real Kepler data parameters and a classification scheme is proposed for the potentially detectable sources based on the geometry of the light curve fits. Methods: The polynomial chain (polyfit) and two-Gaussian models are used for light curve fitting of the data set. Classification is performed with a combination of the t-distributed stochastic neighbor embedding (t-SNE) and density-based spatial clustering of applications with noise (DBSCAN) algorithms. Results: We find that 68% of the Kepler Eclipsing Binary Catalog sources are potentially detectable by Gaia when folded with the Kepler period; we propose a classification scheme of the detectable sources based on the morphological type indicative of the light curve with subclasses that reflect the properties of the fitted model (presence and visibility of eclipses, their width, depth, etc.). 18. The Palomar Transient Factory Orion Project: Eclipsing Binaries and Young Stellar Objects van Eyken, Julian C.; Ciardi, David R.; Rebull, Luisa M.; Stauffer, John R.; Akeson, Rachel L.; Beichman, Charles A.; Boden, Andrew F.; von Braun, Kaspar; Gelino, Dawn M.; Hoard, D. W.; Howell, Steve B.; Kane, Stephen R.; Plavchan, Peter; Ramírez, Solange V.; Bloom, Joshua S.; Cenko, S. Bradley; Kasliwal, Mansi M.; Kulkarni, Shrinivas R.; Law, Nicholas M.; Nugent, Peter E.; Ofek, Eran O.; Poznanski, Dovi; Quimby, Robert M.; Grillmair, Carl J.; Laher, Russ; Levitan, David; Mattingly, Sean; Surace, Jason A. 2011-08-01 The Palomar Transient Factory (PTF) Orion project is one of the experiments within the broader PTF survey, a systematic automated exploration of the sky for optical transients. Taking advantage of the wide (3fdg5 × 2fdg3) field of view available using the PTF camera installed at the Palomar 48 inch telescope, 40 nights were dedicated in 2009 December to 2010 January to perform continuous high-cadence differential photometry on a single field containing the young (7-10 Myr) 25 Ori association. Little is known empirically about the formation of planets at these young ages, and the primary motivation for the project is to search for planets around young stars in this region. The unique data set also provides for much ancillary science. In this first paper, we describe the survey and the data reduction pipeline, and present some initial results from an inspection of the most clearly varying stars relating to two of the ancillary science objectives: detection of eclipsing binaries and young stellar objects. We find 82 new eclipsing binary systems, 9 of which are good candidate 25 Ori or Orion OB1a association members. Of these, two are potential young W UMa type systems. We report on the possible low-mass (M-dwarf primary) eclipsing systems in the sample, which include six of the candidate young systems. Forty-five of the binary systems are close (mainly contact) systems, and one of these shows an orbital period among the shortest known for W UMa binaries, at 0.2156509 ± 0.0000071 days, with flat-bottomed primary eclipses, and a derived distance that appears consistent with membership in the general Orion association. One of the candidate young systems presents an unusual light curve, perhaps representing a semi-detached binary system with an inflated low-mass primary or a star with a warped disk, and may represent an additional young Orion member. Finally, we identify 14 probable new classical T-Tauri stars in our data, along with one previously known (CVSO 35) and 19. THE PALOMAR TRANSIENT FACTORY ORION PROJECT: ECLIPSING BINARIES AND YOUNG STELLAR OBJECTS SciTech Connect Van Eyken, Julian C.; Ciardi, David R.; Akeson, Rachel L.; Beichman, Charles A.; Von Braun, Kaspar; Gelino, Dawn M.; Kane, Stephen R.; Plavchan, Peter; RamIrez, Solange V.; Rebull, Luisa M.; Stauffer, John R.; Hoard, D. W.; Howell, Steve B.; Bloom, Joshua S.; Cenko, S. Bradley; Kasliwal, Mansi M.; Kulkarni, Shrinivas R.; Law, Nicholas M.; Nugent, Peter E. 2011-08-15 The Palomar Transient Factory (PTF) Orion project is one of the experiments within the broader PTF survey, a systematic automated exploration of the sky for optical transients. Taking advantage of the wide (3.{sup 0}5 x 2.{sup 0}3) field of view available using the PTF camera installed at the Palomar 48 inch telescope, 40 nights were dedicated in 2009 December to 2010 January to perform continuous high-cadence differential photometry on a single field containing the young (7-10 Myr) 25 Ori association. Little is known empirically about the formation of planets at these young ages, and the primary motivation for the project is to search for planets around young stars in this region. The unique data set also provides for much ancillary science. In this first paper, we describe the survey and the data reduction pipeline, and present some initial results from an inspection of the most clearly varying stars relating to two of the ancillary science objectives: detection of eclipsing binaries and young stellar objects. We find 82 new eclipsing binary systems, 9 of which are good candidate 25 Ori or Orion OB1a association members. Of these, two are potential young W UMa type systems. We report on the possible low-mass (M-dwarf primary) eclipsing systems in the sample, which include six of the candidate young systems. Forty-five of the binary systems are close (mainly contact) systems, and one of these shows an orbital period among the shortest known for W UMa binaries, at 0.2156509 {+-} 0.0000071 days, with flat-bottomed primary eclipses, and a derived distance that appears consistent with membership in the general Orion association. One of the candidate young systems presents an unusual light curve, perhaps representing a semi-detached binary system with an inflated low-mass primary or a star with a warped disk, and may represent an additional young Orion member. Finally, we identify 14 probable new classical T-Tauri stars in our data, along with one previously known 20. Period change investigation of the low mass ratio contact binary BO Ari Kriwattanawong, W.; Tasuya, O.; Poojon, P. 2016-04-01 A photometric study and period change analysis for the A-type low mass ratio contact binary BO Ari is presented. The BVR light curves were fitted by using the Wilson-Devinney method. The photometric solution yields a low mass ratio of q = 0.1754(±0.0016) with a contact degree of f = 27.72%(±2.37%). We found a long-term orbital period decrease at a rate of dPdt = - 3.49 ×10-7 d yr-1. This result indicates that the system is undergoing mass transfer from the primary component to the secondary with a mass transfer rate of m˙1m1 = - 7.77 ×10-8 yr-1. With the period decrease, the inner and outer critical Roche surfaces will tighten and cause the degree of contact to increase. Therefore, BO Ari may evolve into a deeper contact system. 1. KIC 11401845: An Eclipsing Binary with Multiperiodic Pulsations and Light-travel Time Lee, Jae Woo; Hong, Kyeongsoo; Kim, Seung-Lee; Koo, Jae-Rim 2017-02-01 We report the {\\text{}}{Kepler} photometry of KIC 11401845 displaying multiperiodic pulsations, superimposed on binary effects. Light-curve synthesis shows that the binary star is a short-period detached system with a very low mass ratio of q = 0.070 and filling factors of F1 = 45% and F2 = 99%. Multiple-frequency analyses were applied to the light residuals after subtracting the synthetic eclipsing curve from the observed data. We detected 23 frequencies with signal-to-noise ratios larger than 4.0, of which the orbital harmonics (f4, f6, f9, f15) in the low-frequency domain may originate from tidally excited modes. For the high frequencies of 13.7–23.8 day‑1, the period ratios and pulsation constants are in the ranges of {P}{pul}/{P}{orb}=0.020{--}0.034 and Q = 0.018–0.031 days, respectively. These values and the position on the Hertzsprung–Russell diagram demonstrate that the primary component is a δ Sct pulsating star. We examined the eclipse timing variation of KIC 11401845 from the pulsation-subtracted data and found a delay of 56 ± 17 s in the arrival times of the secondary eclipses relative to the primary eclipses. A possible explanation of the time shift may be some combination of a light-travel-time delay of about 34 s and a very small eccentricity of e\\cos ω < 0.0002. This result represents the first measurement of the Rømer delay in noncompact binaries. 2. DISCOVERY OF THE ECLIPSING DETACHED DOUBLE WHITE DWARF BINARY NLTT 11748 SciTech Connect Steinfadt, Justin D. R.; Shporer, Avi; Bildsten, Lars; Kaplan, David L.; Howell, Steve B. 2010-06-20 We report the discovery of the first eclipsing detached double white dwarf (WD) binary. In a pulsation search, the low-mass helium core WD NLTT 11748 was targeted for fast ({approx}1 minute) differential photometry with the Las Cumbres Observatory's Faulkes Telescope North. Rather than pulsations, we discovered {approx}180 s 3%-6% dips in the photometry. Subsequent radial velocity measurements of the primary white dwarf from the Keck telescope found variations with a semi-amplitude K{sub 1} = 271 {+-} 3 km s{sup -1} and confirmed the dips as eclipses caused by an orbiting WD with a mass M{sub 2} = 0.648-0.771 M{sub sun} for M{sub 1} = 0.1-0.2 M{sub sun}. We detect both the primary and secondary eclipses during the P{sub orb} = 5.64 hr orbit and measure the secondary's brightness to be 3.5% {+-} 0.3% of the primary at SDSS-g'. Assuming that the secondary follows the mass-radius relation of a cold C/O WD and including the effects of microlensing in the binary, the primary eclipse yields a primary radius of R{sub 1} = 0.043-0.039 R{sub sun} for M{sub 1} = 0.1-0.2 M{sub sun}, consistent with the theoretically expected values for a helium core WD with a thick, stably burning hydrogen envelope. Though nearby (at {approx}150 pc), the gravitational wave strain from NLTT 11748 is likely not adequate for direct detection by the Laser Interferometer Space Antenna. Future observational efforts will determine M{sub 1}, yielding accurate WD mass-radius measurement of both components, as well as a clearer indication of the binary's fate once contact is reached. 3. Orbital Circularization of Hot and Cool Kepler Eclipsing Binaries Van Eylen, Vincent; Winn, Joshua N.; Albrecht, Simon 2016-06-01 The rate of tidal circularization is predicted to be faster for relatively cool stars with convective outer layers, compared to hotter stars with radiative outer layers. Observing this effect is challenging because it requires large and well-characterized samples that include both hot and cool stars. Here we seek evidence of the predicted dependence of circularization upon stellar type, using a sample of 945 eclipsing binaries observed by Kepler. This sample complements earlier studies of this effect, which employed smaller samples of better-characterized stars. For each Kepler binary we measure e cos ω based on the relative timing of the primary and secondary eclipses. We examine the distribution of e cos ω as a function of period for binaries composed of hot stars, cool stars, and mixtures of the two types. At the shortest periods, hot-hot binaries are most likely to be eccentric; for periods shorter than four days, significant eccentricities occur frequently for hot-hot binaries, but not for hot-cool or cool-cool binaries. This is in qualitative agreement with theoretical expectations based on the slower dissipation rates of hot stars. However, the interpretation of our results is complicated by the largely unknown ages and evolutionary states of the stars in our sample. 4. A quintuple star system containing two eclipsing binaries Rappaport, S.; Lehmann, H.; Kalomeni, B.; Borkovits, T.; Latham, D.; Bieryla, A.; Ngo, H.; Mawet, D.; Howell, S.; Horch, E.; Jacobs, T. L.; LaCourse, D.; Sódor, Á.; Vanderburg, A.; Pavlovski, K. 2016-10-01 We present a quintuple star system that contains two eclipsing binaries. The unusual architecture includes two stellar images separated by 11 arcsec on the sky: EPIC 212651213 and EPIC 212651234. The more easterly image (212651213) actually hosts both eclipsing binaries which are resolved within that image at 0.09 arcsec, while the westerly image (212651234) appears to be single in adaptive optics (AO), speckle imaging, and radial velocity (RV) studies. The A' binary is circular with a 5.1-d period, while the B' binary is eccentric with a 13.1-d period. The γ velocities of the A and B binaries are different by ˜10 km s-1. That, coupled with their resolved projected separation of 0.09 arcsec, indicates that the orbital period and separation of the C' binary (consisting of A orbiting B) are ≃65 yr and ≃25 au, respectively, under the simplifying assumption of a circular orbit. Motion within the C orbit should be discernible via future RV, AO, and speckle imaging studies within a couple of years. The C system (i.e. 212651213) has an RV and proper motion that differ from that of 212651234 by only ˜1.4 km s-1 and ˜3 mas yr-1. This set of similar space velocities in three dimensions strongly implies that these two objects are also physically bound, making this at least a quintuple star system. 5. Fundamental Parameters of Four Massive Eclipsing Binaries in Westerlund 1 Koumpia, E.; Bonanos, A. Z. 2012-04-01 We present fundamental parameters of four massive eclipsing binaries in the young massive cluster Westerlund 1. The goal is to measure accurate masses and radii of their component stars, which provide much needed constraints for evolutionary models of massive stars. Accurate parameters can further be used to determine a dynamical lower limit for the magnetar progenitor and to obtain an independent distance to the cluster. Our results confirm and extend the evidence for a high mass for the progenitor of the magnetar. 6. Fundamental Parameters of Four Massive Eclipsing Binaries in Westerlund 1 Bonanos, A.; Koumpia, E. 2012-01-01 We present fundamental parameters of four massive eclipsing binaries in the young massive cluster Westerlund 1. The goal is to measure accurate masses and radii of their component stars, which provide much needed constraints for evolutionary models of massive stars. Accurate parameters can further be used to determine a dynamical lower limit for the magnetar progenitor and to obtain an independent distance to the cluster. Our results confirm and extend the evidence for a high mass for the progenitor of the magnetar. 7. Sigma observations of the low mass X-ray binaries of the galactic bulge NASA Technical Reports Server (NTRS) Goldwurm, A.; Denis, M.; Paul, J.; Faisse, S.; Roques, J. P.; Bouchet, L.; Vedrenne, G.; Mandrou, P.; Sunyaev, R.; Churazov, E. 1995-01-01 The soft gamma-ray telescope (35-1300 keV) SIGMA aboard the high energy GRANAT space observatory has been monitoring the Galactic Bulge region for more than 2000 h of effective time since March 1990. In the resulting average 35-75 keV image we detected ten sources at a level of greater than 5 standard deviations, 6 of which can be identified with low mass X-ray binaries (LMXB). Among them, one is the 1993 X-ray nova in Ophiuchus (GRS 1726-249), one is an X-ray pulsar (GX 1+4), two are associated with X-ray bursters (GX 354-0 and A 1742-294) and two with bursting X-ray binaries in the globular clusters Terzan 2 and Terzan 1. Their spectral and long term variability behavior as measured by SIGMMA are presented and discussed. 8. Spectroscopy of Low Mass X-Ray Binaries: New Insights into Accretion. Revised NASA Technical Reports Server (NTRS) DilVrtilek, Saeqa; Mushotzky, Richard (Technical Monitor) 2001-01-01 This project is to observe two low mass X-ray binaries, chosen for their X-ray brightness, low column density, and diversity of accretion behavior. The high spectral resolution of the RGS, the broad energy range and tremendous collecting power of EPIC, and simultaneous optical monitoring with the OM are particularly well-suited to these studies. observation of one of the two objects has taken place and the data were received in late November. The second object is yet to be observed. Over the next year we will: investigate the physical conditions of the emitting gas using emission and recombination line diagnostics to determine temperatures, densities, elemental abundances, and ionization structure; study the behavior of emission features as a function of binary orbit; and test and improve models of X-ray line emission developed by us over the past decade. We will gain insight on both the geometry of the accretion flow and on the evolutionary history of LMXBs. 9. Spectroscopy of Low Mass X-Ray Binaries: New Insights into Accretion NASA Technical Reports Server (NTRS) Vrtilek, Saeqa Dil; Mushotzky, Richard F. (Technical Monitor) 2002-01-01 This project is to observe two low mass X-ray binaries, chosen for their X-ray brightness, low column density, and diversity of accretion behavior. The high spectral resolution of the RGS, the broad energy range and tremendous collecting power of EPIC, and simultaneous optical monitoring with the OM are particularly well-suited to these studies. The second of two objects was not observed until September of 2002. Data analysis for the new observation is underway. over the next year we will: investigate the physical conditions of the emitting gas using emission and recombination line diagnostics to determine temperatures, densities, elemental abundances, and ionization structure; study the behavior of emission features as a function of binary orbit; and test and improve models of X-ray line emission developed by us over the past decade. We will gain insight on both the geometry of the accretion flow and on the evolutionary history of LMXBs. 10. Sigma observations of the low mass X-ray binaries of the galactic bulge NASA Technical Reports Server (NTRS) Goldwurm, A.; Denis, M.; Paul, J.; Faisse, S.; Roques, J. P.; Bouchet, L.; Vedrenne, G.; Mandrou, P.; Sunyaev, R.; Churazov, E. 1995-01-01 The soft gamma-ray telescope (35-1300 keV) SIGMA aboard the high energy GRANAT space observatory has been monitoring the Galactic Bulge region for more than 2000 h of effective time since March 1990. In the resulting average 35-75 keV image we detected ten sources at a level of greater than 5 standard deviations, 6 of which can be identified with low mass X-ray binaries (LMXB). Among them, one is the 1993 X-ray nova in Ophiuchus (GRS 1726-249), one is an X-ray pulsar (GX 1+4), two are associated with X-ray bursters (GX 354-0 and A 1742-294) and two with bursting X-ray binaries in the globular clusters Terzan 2 and Terzan 1. Their spectral and long term variability behavior as measured by SIGMMA are presented and discussed. 11. Searching Kepler Variable Stars with the Eclipsing Binary Factory Pipeline Parvizi, Mahmoud; Paegert, M. 2014-01-01 Repositories of large survey data, such as the Mikulski Archive for Space Telescopes, provide an ideally sized sample from which to identify astrophysically interesting eclipsing binary systems (EBs). However, constraints on the rate of human analysis in solving for the characteristic parameters make mining this data using classical techniques prohibitive. The Kepler data set provides both the high precision simple aperture photometry necessary to detect EBs and a corresponding Kepler Eclipsing Binary Catalog - V3 (KEBC3) of 2,406 EBs in the Kepler filed of view (FoV) as a benchmark. We developed a fully automated end-to-end computational pipeline known as the Eclipsing Binary Factory (EBF) that employs pre-classification data processing modules, a feed-forward single layer perception neural network classifier (NNC), and a subsequent neural network solution estimator (NNSE). This paper focuses on the EBF component modules to include NNC, but excludes the NNSE, as a precursor to a fully automated pipeline that uses solution estimates of characteristic parameters to identify astrophysically interesting EBs. The EBF was found to recover ~94% of KEBC3 EBs contained in the Kepler “Q3” data release where the period is less than thirty days. 12. Tidally Induced Pulsations in Kepler Eclipsing Binary KIC 3230227 Guo, Zhao; Gies, Douglas R.; Fuller, Jim 2017-01-01 KIC 3230227 is a short period (P ≈ 7.0 days) eclipsing binary with a very eccentric orbit (e = 0.6). From combined analysis of radial velocities and Kepler light curves, this system is found to be composed of two A-type stars, with masses of M1 = 1.84 ± 0.18 M⊙, M2 = 1.73 ± 0.17 M⊙ and radii of R1 = 2.01 ± 0.09 R⊙, R2 = 1.68 ± 0.08 R⊙ for the primary and secondary, respectively. In addition to an eclipse, the binary light curve shows a brightening and dimming near periastron, making this a somewhat rare eclipsing heartbeat star system. After removing the binary light curve model, more than 10 pulsational frequencies are present in the Fourier spectrum of the residuals, and most of them are integer multiples of the orbital frequency. These pulsations are tidally driven, and both the amplitudes and phases are in agreement with predictions from linear tidal theory for l = 2, m = ‑2 prograde modes. 13. Fundamental Parameters of Kepler Eclipsing Binaries. I. KIC 5738698 Matson, Rachel A.; Gies, Douglas R.; Guo, Zhao; Orosz, Jerome A. 2016-06-01 Eclipsing binaries serve as a valuable source of stellar masses and radii that inform stellar evolutionary models and provide insight into additional astrophysical processes. The exquisite light curves generated by space-based missions such as Kepler offer the most stringent tests to date. We use the Kepler light curve of the 4.8 day eclipsing binary KIC 5739896 with ground based optical spectra to derive fundamental parameters for the system. We reconstruct the component spectra to determine the individual atmospheric parameters, and model the Kepler photometry with the binary synthesis code Eclipsing Light Curve to obtain accurate masses and radii. The two components of KIC 5738698 are F-type stars with {M}1\\=\\1.39+/- 0.04 {M}⊙ , {M}2\\=\\1.34+/- 0.06 {M}⊙ , and {R}1\\=\\1.84+/- 0.03 {R}⊙ , {R}2\\=\\1.72+/- 0.03 {R}⊙ . We also report a small eccentricity (e≲ 0.0017) and unusual albedo values that are required to match the detailed shape of the Kepler light curve. Comparison with evolutionary models indicate an approximate age of 2.3 Gyr for the system. 14. FORMATION OF BLACK WIDOWS AND REDBACKS—TWO DISTINCT POPULATIONS OF ECLIPSING BINARY MILLISECOND PULSARS SciTech Connect Chen, Hai-Liang; Chen, Xuefei; Han, Zhanwen; Tauris, Thomas M. 2013-09-20 Eclipsing binary millisecond pulsars (MSPs; the so-called black widows and redbacks) can provide important information about accretion history, pulsar irradiation of their companion stars, and the evolutionary link between accreting X-ray pulsars and isolated MSPs. However, the formation of such systems is not well understood, nor the difference in progenitor evolution between the two populations of black widows and redbacks. Whereas both populations have orbital periods between 0.1 and 1.0 days, their companion masses differ by an order of magnitude. In this paper, we investigate the formation of these systems via the evolution of converging low-mass X-ray binaries by employing the MESA stellar evolution code. Our results confirm that one can explain the formation of most of these eclipsing binary MSPs using this scenario. More notably, we find that the determining factor for producing either black widows or redbacks is the efficiency of the irradiation process, such that the redbacks absorb a larger fraction of the emitted spin-down energy of the radio pulsar (resulting in more efficient mass loss via evaporation) compared to that of the black widow systems. We argue that geometric effects (beaming) are responsible for the strong bimodality of these two populations. Finally, we conclude that redback systems do not evolve into black widow systems with time. 15. ASAS J083241+2332.4: A New Extreme Low Mass Ratio Overcontact Binary System Sriram, K.; Malu, S.; Choi, C. S.; Vivekananda Rao, P. 2016-03-01 We present the R- and V-band CCD photometry and Hα line studies of an overcontact binary ASAS J083241+2332.4. The light curves exhibit totality along with a trace of the O’Connell effect. The photometric solution indicates that this system falls into the category of extreme low-mass ratio overcontact binaries with a mass ratio, q ˜ 0.06. Although a trace of the O’ Connell effect is observed, constancy of the Hα line along various phases suggest that a relatively higher magnetic activity is needed for it to show a prominent fill-in effect. The study of O-C variations reveals that the period of the binary shows a secular increase at the rate of dP/dt ˜ 0.0765 s years-1, which is superimposed by a low, but significant, sinusoidal modulation with a period of ˜8.25 years. Assuming that the sinusoidal variation is due to the presence of a third body, orbital elements have been derived. There exist three other similar systems, SX Crv, V857 Her, and E53, which have extremely low mass ratios and we conclude that ASAS J083241+2332.4 resembles SX Crv in many respects. Theoretical studies indicate that at a critical mass ratio range, qcritical = 0.07-0.09, overcontact binaries should merge and form a fast rotating star, but it has been suggested that qcritical can continue to fall up to 0.05 depending on the primary's mass and structure. Moreover, the obtained fill-out factors (50%-70%) indicate that mass loss is considerable and hydrodynamical simulations advocate that mass loss from L2 is mandatory for a successful merging process. Comprehensively, the results indicate that ASAS J083241+2332.4 is at a stage of merger. The pivotal role played by the subtle nature of the derived mass ratio in forming a rapidly rotating star has been discussed. 16. POTENTIAL GAMMA-RAY EMISSIONS FROM LOW-MASS X-RAY BINARY JETS SciTech Connect Zhang, Jian-Fu; Gu, Wei-Min; Liu, Tong; Xue, Li; Lu, Ju-Fu E-mail: [email protected] 2015-06-20 By proposing a pure leptonic radiation model, we study the potential gamma-ray emissions from the jets of low-mass X-ray binaries. In this model, the relativistic electrons that are accelerated in the jets are responsible for radiative outputs. Nevertheless, jet dynamics are dominated by magnetic and proton–matter kinetic energies. The model involves all kinds of related radiative processes and considers the evolution of relativistic electrons along the jet by numerically solving the kinetic equation. Numerical results show that the spectral energy distributions can extend up to TeV bands, in which synchrotron radiation and synchrotron self-Compton scattering are dominant components. As an example, we apply the model to the low-mass X-ray binary GX 339–4. The results not only can reproduce the currently available observations from GX 339–4, but also predict detectable radiation at GeV and TeV bands by the Fermi and CTA telescopes. Future observations with Fermi and CTA can be used to test our model, which could be employed to distinguish the origin of X-ray emissions. 17. ORBITAL SOLUTIONS FOR TWO YOUNG, LOW-MASS SPECTROSCOPIC BINARIES IN OPHIUCHUS SciTech Connect Rosero, V.; Prato, L.; Wasserman, L. H.; Rodgers, B. E-mail: [email protected] E-mail: [email protected] 2011-01-15 We report the orbital parameters for ROXR1 14 and RX J1622.7-2325Nw, two young, low-mass, and double-lined spectroscopic binaries recently discovered in the Ophiuchus star-forming region. Accurate orbital solutions were determined from over a dozen high-resolution spectra taken with the Keck II and Gemini South telescopes. These objects are T Tauri stars with mass ratios close to unity and periods of {approx}5 and {approx}3 days, respectively. In particular, RX J1622.7-2325Nw shows a non-circularized orbit with an eccentricity of 0.30, higher than any other short-period pre-main-sequence (PMS) spectroscopic binary known to date. We speculate that the orbit of RX J1622.7-2325Nw has not yet circularized because of the perturbing action of a {approx}1'' companion, itself a close visual pair. A comparison of known young spectroscopic binaries (SBs) and main-sequence (MS) SBs in the eccentricity-period plane shows an indistinguishable distribution of the two populations, implying that orbital circularization occurs in the first 1 Myr of a star's lifetime. With the results presented in this paper we increase by {approx}4% the small sample of PMS spectroscopic binary stars with known orbital elements. 18. On the formation of low-mass black holes in massive binary stars SciTech Connect Brown, G.E.; Weingartner, J.C.; Wijers, R.A. \| 1996-05-01 Recently, Brown & Bethe suggested that most stars with main-sequence mass in the range of {approximately}18{minus}30 {ital M}{sub {circle_dot}} explode, returning matter to the Galaxy, and then go into low-mass ({ge}1.5 {ital M}{sub {circle_dot}}) black holes. Even more massive main-sequence stars would chiefly go into high-mass ({approximately}10 {ital M}{sub {circle_dot}}) black holes. The Brown-Bethe estimates gave {approximately}5{times}10{sup 8} low-mass black holes in the Galaxy. We here address why none of these have been seen, with the possible exception of the compact objects in SN 1987A and 4U 1700-37. Our main point is that the primary star in a binary loses its hydrogen envelope by transfer of matter to the secondary and loss into space, and the resulting {open_quote}{open_quote}naked{close_quote}{close_quote} helium star evolves differently than a helium core, which is at least initially covered by the hydrogen envelope in a massive main-sequence star. We show that primary stars in binaries can end up as neutron stars even if their initial mass substantially exceeds the mass limit for neutron star formation from single stars ({approximately}18 {ital M}{sub {circle_dot}}). An example is 4U 1223{endash}62, in which we suggest that the initial primary mass exceeded 35 {ital M}{sub {circle_dot}}, yet X-ray pulsations show a neutron star to be present. We also discuss some individual systems and argue that 4U 1700{endash}37, the only example of a well-studied high-mass X-ray binary that does not pulse, could well contain a low-mass black hole. The statistical composition of the X-ray binary population is consistent with our scenario, but due to the paucity of systems it is consistent with more traditional models as well. {copyright} {ital 1996 The American Astronomical Society.} 19. Low-mass spectroscopic binaries in the Hyades: a candidate brown dwarf companion Reid, I. Neill; Mahoney, S. 2000-08-01 We have used the HIRES echelle spectrograph on the Keck I telescope to obtain high-resolution spectroscopy of 51 late-type M dwarfs in the Hyades cluster. Cross-correlating the calibrated data against spectra of white dwarfs allows us to determine heliocentric velocities with an accuracy of +/-0.3kms-1. 27 stars were observed at two epochs in 1997; two stars, RHy 42 and RHy 403, are confirmed spectroscopic binaries. RHy 42 is a double-lined, equal-mass system; RHy 403 is a single-lined, short-period binary, P~1.275d. RHy 403A has an absolute magnitude of MI=10.85, consistent with a mass of 0.15Msolar. The systemic mass function has a value M2sin(i)]3/(M1+M2)2 =0.0085, which, combined with the non-detection of a secondary peak in the cross-correlation function, implies 0.095>M2>0.07Msolar, and the strong possibility that the companion is the first Hyades brown dwarf to be identified. Unfortunately, the maximum expected angular separation in the system is only ~0.25mas. Five other low-mass Hyads are identified as possible spectroscopic binaries, based either on repeat observations or on a comparison between the observed radial velocity and the value expected for Hyades cluster members. Combined with HST imaging data, we infer a binary fraction between 23 and 30per cent. All of the stars are chromospherically active. RHy 281 was caught in mid-flare and, based on that detection, we estimate a flaring frequency of ~2.5per cent for low-mass Hyades stars. Nine stars have rotational velocities, vsin(i), exceeding 20kms-1, and most of the sample have detectable rotation. We examine the H&alpha emission characteristics of low-mass cluster members, and show that there is no evidence for a correlation with rotation. 20. Nova Scorpii and Coalescing Low-Mass Black Hole Binaries as LIGO Sources Sipior, Michael S.; Sigurdsson, Steinn 2002-06-01 Double neutron star (NS-NS) binaries, analogous to the well-known Hulse-Taylor pulsar PSR 1913+16 (documented by Hulse & Taylor in 1974), are guaranteed-to-exist sources of high-frequency gravitational radiation detectable by LIGO. There is considerable uncertainty in the estimated rate of coalescence of such systems (see the work of Phinney in 1991, Narayan and coworkers in 1991, and Kalogera and coworkers in 2001), with conservative estimates of ~1 per 106 yr per galaxy, and optimistic theoretical estimates 1 or more mag larger. Formation rates of low-mass black hole (BH)-neutron star binaries may be higher than those of NS-NS binaries and may dominate the detectable LIGO signal rate. Rate estimates for such binaries are plagued by severe model uncertainties. Recent estimates by Portegies Zwart & Yungelson in 1998 and De Donder & Vanbeveren in 1998 suggest that BH-BH binaries do not coalesce at significant rates despite being formed at high rates. We estimate the enhanced coalescence rate for BH-BH binaries due to weak asymmetric kicks during the formation of low-mass black holes like Nova Sco (see the work of Brandt, Podsiadlowski, & Sigurdsson in 1995) and find they may contribute significantly to the LIGO signal rate, possibly dominating the phase I detectable signals if the range of black hole masses for which there is significant kick is broad enough. For a standard Salpeter initial mass function, assuming mild natal kicks, we project that the R6 merger rate (the rate of mergers per 106 yr in a Milky Way-like galaxy) of BH-BH systems is ~0.5, smaller than that of NS-NS systems. However, the higher chirp mass of these systems produces a signal nearly 4 times greater, on average, with a commensurate increase in search volume, hence, our claim that BH-BH mergers (and, to a lesser extent, BH-NS coalescence) should comprise a significant fraction of the signal seen by LIGO. The BH-BH coalescence channel considered here also predicts that a substantial fraction of 1. Determination of Individual Temperatures and Luminosities in Eclipsing Binary Star Systems. DTIC Science & Technology 1983-06-20 REPORT 1NO. W DETERMINATION OF INDIVIDUAL TEMPERATURES AND LUMINOSITIES IN ECLIPSING BINARY STAR SYSTEMS UNITED STATES NAVAL ACADEMY ANNAPOLIS, MARYLAND...U.S.N.A. - Trident Scholar project report; no. 122 (1983) DETERMINATION OF INDIVIDUAL TEMPERATURES AND LUMINOSITIES IN ECLIPSING BINARY STAR SYSTEMS A...the temperatures and luminosities of the individual components of eclipsing binary star systems. -’r. Richard L. Walker of the U.S. Naval Observatory 2. Discovery of the Partially Eclipsing White Dwarf Binary SDSS J143547.87+373338.5 Steinfadt, Justin D. R.; Bildsten, Lars; Howell, Steve B. 2008-04-01 We have discovered a partially eclipsing white dwarf, low-mass M dwarf binary (3.015114 hr orbital period), SDSS J143547.87+373338.5, from 2007 May observations at the WIYN telescope. Here we present blue-band photometry of three eclipses. Eclipse fitting gives main-sequence solutions to the M dwarf companion of MS = 0.15-0.35 M⊙ and RS = 0.17-0.32 R⊙. Analysis of the SDSS spectrum constrains the M dwarf further to be of type M4-M6 with MS = 0.11-0.20 M⊙. Once full radial velocity curves are measured, high-precision determinations of the masses and radii of both components will be easily obtained without any knowledge of stellar structure or evolution. ZZ Ceti pulsations from the white dwarf were not found at our 4 mmag detection limit. The WIYN Observatory is a joint facility of the University of Wisconsin-Madison, Indiana University, Yale University, and the National Optical Astronomy Observatory. 3. KEPLER ECLIPSING BINARY STARS. I. CATALOG AND PRINCIPAL CHARACTERIZATION OF 1879 ECLIPSING BINARIES IN THE FIRST DATA RELEASE SciTech Connect Prsa, Andrej; Engle, Scott G.; Conroy, Kyle; Batalha, Natalie; Rucker, Michael; Mjaseth, Kimberly; Slawson, Robert W.; Doyle, Laurance R.; Welsh, William F.; Orosz, Jerome A.; Seager, Sara; Jenkins, Jon; Caldwell, Douglas 2011-03-15 The Kepler space mission is devoted to finding Earth-size planets orbiting other stars in their habitable zones. Its large, 105 deg{sup 2} field of view features over 156,000 stars that are observed continuously to detect and characterize planet transits. Yet, this high-precision instrument holds great promise for other types of objects as well. Here we present a comprehensive catalog of eclipsing binary stars observed by Kepler in the first 44 days of operation, the data being publicly available through MAST as of 2010 June 15. The catalog contains 1879 unique objects. For each object, we provide its Kepler ID (KID), ephemeris (BJD{sub 0}, P{sub 0}), morphology type, physical parameters (T{sub eff}, log g, E(B - V)), the estimate of third light contamination (crowding), and principal parameters (T{sub 2}/T{sub 1}, q, fillout factor, and sin i for overcontacts, and T{sub 2}/T{sub 1}, (R{sub 1} + R{sub 2})/a, esin {omega}, ecos {omega}, and sin i for detached binaries). We present statistics based on the determined periods and measure the average occurrence rate of eclipsing binaries to be {approx}1.2% across the Kepler field. We further discuss the distribution of binaries as a function of galactic latitude and thoroughly explain the application of artificial intelligence to obtain principal parameters in a matter of seconds for the whole sample. The catalog was envisioned to serve as a bridge between the now public Kepler data and the scientific community interested in eclipsing binary stars. 4. THE ORIGIN OF BLACK HOLE SPIN IN GALACTIC LOW-MASS X-RAY BINARIES SciTech Connect Fragos, T.; McClintock, J. E. 2015-02-10 Galactic field black hole (BH) low-mass X-ray binaries (LMXBs) are believed to form in situ via the evolution of isolated binaries. In the standard formation channel, these systems survived a common envelope phase, after which the remaining helium core of the primary star and the subsequently formed BH are not expected to be highly spinning. However, the measured spins of BHs in LMXBs cover the whole range of spin parameters. We propose here that the BH spin in LMXBs is acquired through accretion onto the BH after its formation. In order to test this hypothesis, we calculated extensive grids of detailed binary mass-transfer sequences. For each sequence, we examined whether, at any point in time, the calculated binary properties are in agreement with their observationally inferred counterparts of 16 Galactic LMXBs. The ''successful'' sequences give estimates of the mass that the BH has accreted since the onset of Roche-Lobe overflow. We find that in all Galactic LMXBs with measured BH spin, the origin of the spin can be accounted for by the accreted matter, and we make predictions about the maximum BH spin in LMXBs where no measurement is yet available. Furthermore, we derive limits on the maximum spin that any BH can have depending on current properties of the binary it resides in. Finally we discuss the implication that our findings have on the BH birth-mass distribution, which is shifted by ∼1.5 M {sub ☉} toward lower masses, compared to the currently observed one. 5. The origin of Black-Hole Spin in Galactic Low-Mass X-ray Binaries Fragos, Tassos; McClintock, Jeffrey 2015-08-01 Galactic field low-mass X-ray binaries (LMXBs), like the ones for which black hole (BH) spin measurements are available, are believed to form in situ via the evolution of isolated binaries. In the standard formation channel, these systems survived a common envelope phase, after which the remaining helium core of the primary star and the subsequently formed BH are not expected to be highly spinning. However, the measured spins of BHs in LMXBs cover the whole range of spin parameters from a~0 to a1. In this talk I propose that the BH spin in LMXBs is acquired through accretion onto the BH during its long stable accretion phase. In order to test this hypothesis, I calculated extensive grids of binary evolutionary sequences in which a BH accretes matter from a close companion. For each evolutionary sequence, I examined whether, at any point in time, the calculated binary properties are in agreement with their observationally inferred counterparts of observed Galactic LMXBs with BH spin measurements. Mass-transfer sequences that simultaneously satisfy all observational constraints represent possible progenitors of the considered LMXBs and thus give estimates of the amount of matter that the BH has accreted since the onset of Roche-Lobe overflow. I find that in all Galactic LMXBs with measured BH spin, the origin of the spin can be accounted by the accreted matter. Furthermore, based on this hypothesis, I derive limits on the maximum spin that a BH can have depending on the orbital period of the binary it resides in, and give predictions on the maximum possible BH spin of Galactic LMXBs where a BH spin measurement is not yet available. Finally I will discuss the implication that our findings have on the birth black hole mass distribution. 6. FORMATION AND EVOLUTION OF GALACTIC INTERMEDIATE/LOW-MASS X-RAY BINARIES SciTech Connect Shao, Yong; Li, Xiang-Dong 2015-08-10 We investigate the formation and evolutionary sequences of Galactic intermediate- and low-mass X-ray binaries (I/LMXBs) by combining binary population synthesis (BPS) and detailed stellar evolutionary calculations. Using an updated BPS code we compute the evolution of massive binaries that leads to the formation of incipient I/LMXBs and present their distribution in the initial donor mass versus initial orbital period diagram. We then follow the evolution of the I/LMXBs until the formation of binary millisecond pulsars (BMSPs). We find that the birthrate of the I/LMXB population is in the range of 9 × 10{sup −6}–3.4 × 10{sup −5} yr{sup −1}, compatible with that of BMSPs that are thought to descend from I/LMXBs. We show that during the evolution of I/LMXBs they are likely to be observed as relatively compact binaries with orbital periods ≲1 day and donor masses ≲0.3M{sub ⊙}. The resultant BMSPs have orbital periods ranging from less than 1 day to a few hundred days. These features are consistent with observations of LMXBs and BMSPs. We also confirm the discrepancies between theoretical predictions and observations mentioned in the literature, that is, the theoretical average mass transfer rates (∼10{sup −10} M{sub ⊙} yr{sup −1}) of LMXBs are considerably lower than observed, and the number of BMSPs with orbital periods ∼0.1–10 days is severely underestimated. These discrepancies imply that something is missing in the modeling of LMXBs, which is likely to be related to the mechanisms of the orbital angular momentum loss. 7. Physical parameters of neglected southern eclipsing binary IL Lib Özkardeş, B. 2017-07-01 This paper presents results from the combined analysis of light curve (from the All Sky Automated Survey) and radial velocity curves (from Nordström et al., 1997) of the eclipsing binary IL Lib. The final solution describes the system as a detached binary. Absolute parameters of this southern detached binary were calculated as follows: M1 = 1.49 ± 0.12 M⊙, M2 = 1.31 ± 0.14 M⊙, R1 = 1.52 ± 0.23 R⊙, R2 = 1.52 ± 0.23 R⊙, L1 = 4.99 ± 2.07 L⊙ and L2 = 3.65 ± 1.55 L⊙. The distance to IL Lib was computed as 103 ± 20 pc using the distance modulus with corrections for interstellar extinction. The positions of the components of IL Lib in the HR diagram are also discussed. 8. Searches for millisecond pulsations in low-mass X-ray binaries, 2 NASA Technical Reports Server (NTRS) Vaughan, B. A.; Van Der Klis, M.; Wood, K. S.; Norris, J. P.; Hertz, P.; Michelson, P. F.; Paradijs, J. Van; Lewin, W. H. G.; Mitsuda, K.; Penninx, W. 1994-01-01 Coherent millisecond X-ray pulsations are expected from low-mass X-ray binaries (LMXBs), but remain undetected. Using the single-parameter Quadratic Coherence Recovery Technique (QCRT) to correct for unknown binary orbit motion, we have performed Fourier transform searches for coherent oscillations in all long, continuous segments of data obtained at 1 ms time resolution during Ginga observations of LMXB. We have searched the six known Z sources (GX 5-1, Cyg X-2, Sco X-1, GX 17+2, GX 340+0, and GX 349+2), seven of the 14 known atoll sources (GX 3+1. GX 9+1, GX 9+9, 1728-33. 1820-30, 1636-53 and 1608-52), the 'peculiar' source Cir X-1, and the high-mass binary Cyg X-3. We find no evidence for coherent pulsations in any of these sources, with 99% confidence limits on the pulsed fraction between 0.3% and 5.0% at frequencies below the Nyquist frequency of 512 Hz. A key assumption made in determining upper limits in previous searches is shown to be incorrect. We provide a recipe for correctly setting upper limits and detection thresholds. Finally we discuss and apply two strategies to improve sensitivity by utilizing multiple, independent, continuous segments of data with comparable count rates. 9. WISE detection of the galactic low-mass X-ray binaries SciTech Connect Wang, Xuebing; Wang, Zhongxiang 2014-06-20 We report on the results from our search for the Wide-field Infrared Survey Explorer (WISE) detection of the Galactic low-mass X-ray binaries (LMXBs). Among 187 binaries cataloged in Liu et al., we find 13 counterparts and 2 candidate counterparts. For the 13 counterparts, 2 (4U 0614+091 and GX 339–4) have already been confirmed by previous studies to have a jet and 1 (GRS 1915+105) to have a candidate circumbinary disk, from which the detected infrared emission arose. Having collected the broadband optical and near-infrared data in the literature and constructed flux density spectra for the other 10 binaries, we identify that 3 (A0620–00, XTE J1118+480, and GX 1+4) are candidate circumbinary disk systems, 4 (Cen X-4, 4U 1700+24, 3A 1954+319, and Cyg X-2) had thermal emission from their companion stars, and 3 (Sco X-1, Her X-1, and Swift J1753.5–0127) are peculiar systems with the origin of their infrared emission rather uncertain. We discuss the results and WISE counterparts' brightness distribution among the known LMXBs, and suggest that more than half of the LMXBs would have a jet, a circumbinary disk, or both. 10. The evolution of cataclysmic and low-mass X-ray binaries NASA Technical Reports Server (NTRS) Patterson, J. 1984-01-01 The observational data for the 124 cataclysmic and low-mass X-ray binaries of known orbital period are compiled. It is found that the eruption properties are very well correlated with the orbital periods, in a manner that suggests that the mass transfer is the all-important determinant of evolution and of the eruptive behavior. Transfer rates of 10 to the -11th to 10 to the -7th solar mass/year are found, which are well correlated with orbital period. It is concluded that the mechanism which drives these systems is the magnetic braking of the secondary's rotation by its own stellar wind, coupled with the enforcement of synchronous rotation by tidal friction. This permits CVs to lose sufficient angular momentum to begin mass transfer in a reasonable time and drives the mass transfer at a high rate once it begins. An account of the long-lived phases of CV evolution is given. 11. Evolution of the Spin Periods of Neutron Stars in Low-mass X-ray Binaries Xu, X. T.; Zhu, Z. L. 2016-11-01 We present numerical analysis of the spin evolution of the neutron stars in low-mass X-ray binaries, trying to explain the discrepancy in the spin period distribution between observations of millisecond pulsars and theoretical results. In our calculations, we take account of possible effect of radiation pressure, and irradiation-induced instability on the structure of the disk, and the evolution of the mass transfer rate, respectively. We report the following results: (1) Radiation pressure leads to a slight increase of the spin periods, and irradiation-induced mass transfer cycles can shorten the spin-down phase of evolution. (2) The calculated results in the model combining radiation pressure and irradiation-induced mass transfer cycles show that accretion is strongly limited by radiation pressure in high mass transfer phase. (3) The accreted mass and the critical fastness parameter can affect the number of systems in equilibrium state. 12. V404 Cyg - an Interacting Black-Hole Low-Mass X-ray Binary Fox, Ori; Mauerhan, Jon; Graham, Melissa 2015-07-01 This DDT proposal is prompted by the June 15, 2015 outburst of V404 Cyg, a black-hole (BH) low-mass X-ray binary (LMXB). This outburst stands out since it is the first black hole system with a measured parallax, lying at a distance of only 2.39+/-0.14 kpc. An extensive and loosely organized multi-wavelength campaign is already underway by the astronomical community. One of the missing pieces of the puzzle is the mid-infrared (IR). Combined with radio, optical, and X-ray data, the mid-IR will help to discriminate discriminate between an accretion disk, jet emission, or circumstellar dust scenarios. Spitzer offers a unique opportunity to observe at these wavelengths. Here we propose 4 very short (5-minutes at 3.6 and 4.5 micron) observations of IRAC hotometry to search for the presence of warm dust and, if present, constrain the heating mechanism. 13. Theoretical spectra of nonmagnetized low-mass X-ray binaries NASA Technical Reports Server (NTRS) Czerny, Bozena; Czerny, Michal; Grindlay, Jonathan E. 1986-01-01 Theoretical X-ray spectra of low-mass X-ray binaries with negligible magnetic fields are presented. The geometry of the X-ray emitting region, the energetic efficiency of the accretion in the disk and in the boundary layer which leads to a relation between the disk and the boundary layer luminosities, and the irradiation of the disk by the boundary layer are studied. The model of the radiation spectrum emerging from the neutron star and the innermost part of the disk is presented. The relativistic and Doppler effects and their influence on the spectrum as a function of inclination angle are discussed. A simple method for comparing the spectrum model with observations by studying the hardness ratio is given, and the results for three X-ray sources in globular clusters observed by the Einstein satellite are presented. The range of applicability of the spectrum models is also discussed. 14. An Update to the Kepler Eclipsing Binary Catalog: the use of Pixel Time Series to Identify Blended Eclipsing Binary Systems Rucker, Michael; Batalha, N. M.; Prsa, A.; Bryson, S. T.; Doyle, L. R.; Slawson, R. W.; Welsh, W. F.; Orosz, J. A. 2011-01-01 The Kepler telescope is providing a nearly seamless stream of photometric data of approximately 150,000 stars with unprecedented precision. The Kepler Eclipsing Binary (EB) catalog (based on the first 43 days of data; arXiv:1006.2815) is being continuously augmented as more data are collected and EBs are detected at longer periods. The catalog is expected to contain a small fraction of blends - cases where the eclipse signature is from a nearby source in the photometric aperture. In constructing the original catalog, obvious blends were identified and removed and/or reassigned to the appropriate point source. We build upon this work by performing pixel-level tests similar to those used to identify false positives amongst the Kepler exoplanet candidates. We summarize these tests here and provide examples that illustrate the types of blend scenarios that we have identified. Where appropriate and possible, we modified Kepler's target list with the newly found Kepler star identification numbers. The changes reported here will affect the target lists which will go into effect on December 23, 2010 (start of Quarter 8). An updated version of the Kepler Eclipsing Binary catalog is available online at NASA's Multimission Archive at STSci (MAST) website (http://archive.stsci.edu/kepler). 15. Kepler eclipsing binary stars - VI. Identification of eclipsing binaries in the K2 Campaign 0 data set LaCourse, Daryll M.; Jek, Kian J.; Jacobs, Thomas L.; Winarski, Troy; Boyajian, Tabetha S.; Rappaport, Saul A.; Sanchis-Ojeda, Roberto; Conroy, Kyle E.; Nelson, Lorne; Barclay, Tom; Fischer, Debra A.; Schmitt, Joseph R.; Wang, Ji; Stassun, Keivan G.; Pepper, Joshua; Coughlin, Jeffrey L.; Shporer, Avi; Prša, Andrej 2015-10-01 The original Kepler mission observed and characterized over 2400 eclipsing binaries (EBs) in addition to its prolific exoplanet detections. Despite the mechanical malfunction and subsequent non-recovery of two reaction wheels used to stabilize the instrument, the Kepler satellite continues collecting data in its repurposed K2 mission surveying a series of fields along the ecliptic plane. Here, we present an analysis of the first full baseline K2 data release: the Campaign 0 data set. In the 7761 light curves we have identified a total of 207 EBs. Of these, 97 are new discoveries that were not previously identified. Our pixel-level analysis of these objects has also resulted in identification of several false positives (observed targets contaminated by neighbouring EBs), as well as the serendipitous discovery of two short-period exoplanet candidates. We provide catalogue cross-matched source identifications, orbital periods, morphologies and ephemerides for these eclipsing systems. We also describe the incorporation of the K2 sample into the Kepler Eclipsing Binary Catalog,§ present spectroscopic follow-up observations for a limited selection of nine systems and discuss prospects for upcoming K2 campaigns. 16. ASAS J083241+2332.4: A NEW EXTREME LOW MASS RATIO OVERCONTACT BINARY SYSTEM SciTech Connect Sriram, K.; Malu, S.; Vivekananda Rao, P.; Choi, C. S. 2016-03-15 We present the R- and V-band CCD photometry and Hα line studies of an overcontact binary ASAS J083241+2332.4. The light curves exhibit totality along with a trace of the O’Connell effect. The photometric solution indicates that this system falls into the category of extreme low-mass ratio overcontact binaries with a mass ratio, q ∼ 0.06. Although a trace of the O’ Connell effect is observed, constancy of the Hα line along various phases suggest that a relatively higher magnetic activity is needed for it to show a prominent fill-in effect. The study of O–C variations reveals that the period of the binary shows a secular increase at the rate of dP/dt ∼ 0.0765 s years{sup −1}, which is superimposed by a low, but significant, sinusoidal modulation with a period of ∼8.25 years. Assuming that the sinusoidal variation is due to the presence of a third body, orbital elements have been derived. There exist three other similar systems, SX Crv, V857 Her, and E53, which have extremely low mass ratios and we conclude that ASAS J083241+2332.4 resembles SX Crv in many respects. Theoretical studies indicate that at a critical mass ratio range, q{sub critical} = 0.07–0.09, overcontact binaries should merge and form a fast rotating star, but it has been suggested that q{sub critical} can continue to fall up to 0.05 depending on the primary's mass and structure. Moreover, the obtained fill-out factors (50%–70%) indicate that mass loss is considerable and hydrodynamical simulations advocate that mass loss from L{sub 2} is mandatory for a successful merging process. Comprehensively, the results indicate that ASAS J083241+2332.4 is at a stage of merger. The pivotal role played by the subtle nature of the derived mass ratio in forming a rapidly rotating star has been discussed. 17. Discovery of wide low and very low-mass binary systems using Virtual Observatory tools Gálvez-Ortiz, M. C.; Solano, E.; Lodieu, N.; Aberasturi, M. 2017-04-01 The frequency of multiple systems and their properties are key constraints of stellar formation and evolution. Formation mechanisms of very low-mass (VLM) objects are still under considerable debate, and an accurate assessment of their multiplicity and orbital properties is essential for constraining current theoretical models. Taking advantage of the virtual observatory capabilities, we looked for comoving low and VLM binary (or multiple) systems using the Large Area Survey of the UKIDSS LAS DR10, SDSS DR9 and the 2MASS Catalogues. Other catalogues (WISE, GLIMPSE, SuperCosmos, etc.) were used to derive the physical parameters of the systems. We report the identification of 36 low and VLM (˜M0-L0 spectral types) candidates to binary/multiple system (separations between 200 and 92 000 au), whose physical association is confirmed through common proper motion, distance and low probability of chance alignment. This new system list notably increases the previous sampling in their mass-separation parameter space (˜100). We have also found 50 low-mass objects that we can classify as ˜L0-T2 according to their photometric information. Only one of these objects presents a common proper motion high-mass companion. Although we could not constrain the age of the majority of the candidates, probably most of them are still bound except four that may be under disruption processes. We suggest that our sample could be divided in two populations: one tightly bound wide VLM systems that are expected to last more than 10 Gyr, and other formed by weak bound wide VLM systems that will dissipate within a few Gyr. 18. Investigating Low-Mass Binary Stars And Brown Dwarfs with Near-Infrared Spectroscopy Mace, Gregory Nathan The mass of a star at formation determines its subsequent evolution and demise. Low-mass stars are the most common products of star formation and their long main-sequence lifetimes cause them to accumulate over time. Star formation also produces many substellar-mass objects known as brown dwarfs, which emerge from their natal molecular clouds and continually cool as they age, pervading the Milky Way. Low-mass stars and brown dwarfs exhibit a wide range of physical characteristics and their abundance make them ideal subjects for testing formation and evolution models. I have examined a pair of pre-main sequence spectroscopic binaries and used radial velocity variations to determine orbital solutions and mass ratios. Additionally, I have employed synthetic spectra to estimate their effective temperatures and place them on theoretical Hertzsprung-Russell diagrams. From this analysis I discuss the formation and evolution of young binary systems and place bounds on absolute masses and radii. I have also studied the late-type T dwarfs revealed by the Wide-field Infrared Survey Explorer (WISE). This includes the exemplar T8 subdwarf Wolf 1130C, which has the lowest inferred metallicity in the literature and spectroscopic traits consistent with old age. Comparison to synthetic spectra implies that the dispersion in near-infrared colors of late-type T dwarfs is a result of age and/or thin sulfide clouds. With the updated census of the L, T, and Y dwarfs we can now study specific brown dwarf subpopulations. Finally, I present a number of future studies that would develop our understanding of the physical qualities of T dwarf color outliers and disentangle the tracers of age and atmospheric properties. 19. Hystereses in dwarf nova outbursts and low-mass X-ray binaries Hameury, J.-M.; Lasota, J.-P.; Knigge, C.; Körding, E. G. 2017-04-01 Context. The disc instability model (DIM) successfully explains why many accreting compact binary systems exhibit outbursts during which their luminosity increases by orders of magnitude. The DIM correctly predicts which systems should be transient and works regardless of whether the accretor is a black hole, a neutron star, or a white dwarf. However, it has been known for some time that the outbursts of X-ray binaries, which contain neutron-star or black-hole accretors, exhibit hysteresis in the X-ray hardness-intensity diagram (HID). More recently, it has been shown that the outbursts of accreting white dwarfs also show hysteresis, but in a diagram combining optical, EUV, and X-ray fluxes. Aims: We examine the nature of the hysteresis observed in cataclysmic variables and low-mass X-ray binaries. Methods: We used our disc evolution code for modelling dwarf nova outbursts, and constructed the hardness intensity diagram as predicted by the disc instability model. Results: We show explicitly that the standard DIM, modified only to account for disc truncation, can explain the hysteresis observed in accreting white dwarfs, but cannot explain that observed in X-ray binaries. Conclusions: The spectral evidence for the existence of different accretion regimes or components (disc, corona, jets, etc.) should only be based on wavebands that are specific to the innermost parts of the discs, i.e. EUV and X-rays; this task is difficult because of interstellar absorption. The existing data, however, indicate that a hysteresis is in the EUV - X-ray domain is present in SS Cyg. 20. Fundamental Parameters of 4 Massive Eclipsing Binaries in Westerlund 1 Bonanos, Alceste Z.; Koumpia, E. 2011-05-01 We present fundamental parameters of 4 massive eclipsing binaries in the young massive cluster Westerlund 1. The goal is to measure accurate masses and radii of their component stars, which provide much needed constraints for evolutionary models of massive stars. Accurate parameters can further be used to determine a dynamical lower limit for the magnetar progenitor and to obtain an independent distance to the cluster. Our results confirm and extend the evidence for a high mass for the progenitor of the magnetar. The authors acknowledge research and travel support from the European Commission Framework Program Seven under the Marie Curie International Reintegration Grant PIRG04-GA-2008-239335. 1. KIC11560447: An Active Eclipsing Binary From the Kepler Field Ozavci, Ibrahim; Hussain, Gaitee; Yılmaz, Mesut; O'Neal, Douglas; osman Selam, Selim; Şenavcı, Hakan Volkan 2016-07-01 We performed spectroscopic and photometric analysis of the detached eclipsing binary KIC11560447, in order to investigate the spot activity of the system. In this context, we reconstructed the surface maps with the help of the code DoTS, using time series spectra obtained at the 2.1m Otto Struve Telescope of the McDonald Observatory. We also analysed high precision Kepler light curves of the system simultaneously with the code DoTS to reveal the spot migration and activity behaviour. 2. Neutron Star Mass-Radius Constraints of the Quiescent Low-mass X-Ray Binaries X7 and X5 in the Globular Cluster 47 Tuc Bogdanov, Slavko; Heinke, Craig O.; Özel, Feryal; Güver, Tolga 2016-11-01 We present Chandra/ACIS-S subarray observations of the quiescent neutron star (NS) low-mass X-ray binaries X7 and X5 in the globular cluster 47 Tuc. The large reduction in photon pile-up compared to previous deep exposures enables a substantial improvement in the spectroscopic determination of the NS radius and mass of these NSs. Modeling the thermal emission from the NS surface with a non-magnetized hydrogen atmosphere and accounting for numerous sources of uncertainties, we obtain for the NS in X7 a radius of R={11.1}-0.7+0.8 km for an assumed stellar mass of M = 1.4 M ⊙ (68% confidence level). We argue, based on astrophysical grounds, that the presence of a He atmosphere is unlikely for this source. Due to the excision of data affected by eclipses and variable absorption, the quiescent low-mass X-ray binary X5 provides less stringent constraints, leading to a radius of R={9.6}-1.1+0.9 km, assuming a hydrogen atmosphere and a mass of M = 1.4 M ⊙. When combined with all existing spectroscopic radius measurements from other quiescent low-mass X-ray binaries and Type I X-ray bursts, these measurements strongly favor radii in the 9.9-11.2 km range for a ˜1.5 M ⊙ NS and point to a dense matter equation of state that is somewhat softer than the nucleonic ones that are consistent with laboratory experiments at low densities. 3. Discovery and Characterization of Wide Binary Systems with a Very Low Mass Component Baron, Frédérique; Lafrenière, David; Artigau, Étienne; Doyon, René; Gagné, Jonathan; Davison, Cassy L.; Malo, Lison; Robert, Jasmin; Nadeau, Daniel; Reylé, Céline 2015-03-01 We report the discovery of 14 low-mass binary systems containing mid-M to mid-L dwarf companions with separations larger than 250 AU. We also report the independent discovery of nine other systems with similar characteristics that were recently discovered in other studies. We have identified these systems by searching for common proper motion sources in the vicinity of known high proper motion stars, based on a cross-correlation of wide area near-infrared surveys (2MASS, SDSS, and SIMP). An astrometric follow-up, for common proper motion confirmation, was made with SIMON and/or CPAPIR at the Observatoire du Mont Mégantic 1.6 m and CTIO 1.5 m telescopes for all the candidates identified. A spectroscopic follow-up was also made with GMOS or GNIRS at Gemini to determine the spectral types of 11 of our newly identified companions and 10 of our primaries. Statistical arguments are provided to show that all of the systems we report here are very likely to be physical binaries. One of the new systems reported features a brown dwarf companion: LSPM J1259+1001 (M5) has an L4.5 (2M1259+1001) companion at ˜340 AU. This brown dwarf was previously unknown. Seven other systems have a companion of spectral type L0-L1 at a separation in the 250-7500 AU range. Our sample includes 14 systems with a mass ratio below 0.3. 4. Shapiro Delay in the Low Mass Binary Millisecond Pulsar J1713+0747 Camilo, F.; Foster, R. S.; Wolszczan, A. 1993-12-01 The binary millisecond pulsar J1713+0747 (P=4.57 ms;P_b=67.8 d) was discovered in a systematic continuing survey for millisecond pulsars with the Arecibo radio telescope (Foster, Wolszczan & Camilo 1993, ApJ, 410, L91). We have carried out multi-frequency observations of this object at approximately bi-weekly intervals. With an rms residual in the predicted vs. observed times-of-arrival (TOAs) of <0.5 mu sec, and a large characteristic age, tau_c ~ 10(10) yr, this object is one of the most precise celestial clocks among all known pulsars. We detect a signature in the TOA residuals which is most naturally interpreted in terms of a general relativistic Shapiro Delay'', caused as the pulsar signals traverse the gravitational potential well of its ~ 0.2 M_sun companion, with the orbital angular momentum of the system lying nearly parallel to the plane of the sky. With this information we can determine the mass of the (presumed) white dwarf companion star, and the inclination angle of the orbit. Knowing the pulsar mass function (0.0079 M_sun), we can in turn determine the mass of the pulsar itself. This measurement is important, among other reasons, for comparisons against the evolutionary scenarios that predict substantial mass accretion by the pulsar as it is spun up to millisecond periods by mass transfer from its companion in a low mass x-ray binary phase. 5. ORBITAL PERIOD AND OUTBURST LUMINOSITY OF TRANSIENT LOW MASS X-RAY BINARIES SciTech Connect Wu, Y. X.; Yu, W.; Li, T. P.; Maccarone, T. J.; Li, X. D. 2010-08-01 In this paper, we investigate the relationship between the maximal luminosity of X-ray outburst and the orbital period in transient low mass X-ray binaries (or soft X-ray transients) observed by the Rossi X-ray Timing Explorer (RXTE) in the past decade. We find that the maximal luminosity (3-200 keV) in Eddington units generally increases with increasing orbital period, which does not show a luminosity saturation but in general agrees with theoretical prediction. The peak luminosities in ultra-compact binaries might be higher than those with an orbital period of 2-4 hr, but more data are needed to make this claim. We also find that there is no significant difference in the 3-200 keV peak outburst luminosity between neutron star (NS) systems and black hole (BH) systems with orbital periods above 4 hr; however, there might be a significant difference at smaller orbital periods where only NS systems are observed and radiatively inefficient accretion flow is expected to work at low luminosities for BH accreters. 6. Phenomenological Modelling of a Group of Eclipsing Binary Stars Andronov, Ivan L.; Tkachenko, Mariia G.; Chinarova, Lidia L. 2016-03-01 Phenomenological modeling of variable stars allows determination of a set of the parameters, which are needed for classification in the "General Catalogue of Variable Stars" and similar catalogs. We apply a recent method NAV ("New Algol Variable") to eclipsing binary stars of different types. Although all periodic functions may be represented as Fourier series with an infinite number of coefficients, this is impossible for a finite number of the observations. Thus one may use a restricted Fourier series, i.e. a trigonometric polynomial (TP) of order s either for fitting the light curve, or to make a periodogram analysis. However, the number of parameters needed drastically increases with decreasing width of minimum. In the NAV algorithm, the special shape of minimum is used, so the number of parameters is limited to 10 (if the period and initial epoch are fixed) or 12 (not fixed). We illustrate the NAV method by application to a recently discovered Algol-type eclipsing variable 2MASS J11080308-6145589 (in the field of previously known variable star RS Car) and compare results to that obtained using the TP fits. For this system, the statistically optimal number of parameters is 44, but the fit is still worse than that of the NAV fit. Application to the system GSC 3692-00624 argues that the NAV fit is better than the TP one even for the case of EW-type stars with much wider eclipses. Model parameters are listed. 7. Timing of Eclipses of Binary Stars from the ASAS Catalog Kozlowski, S. K.; Konacki, M.; Sybilski, P. 2011-09-01 Light was thought of as something infinite and transcendent till 1676 when Olaus Roemer carried out precise measurements of the times of eclipses of Jovian moons. Roemer's scrupulous observations led him to a qualitative conclusion that light travels at a finite speed, at the same time providing scientists with the basics of the Light-Time Effect (LTE). LTE is observed whenever the distance between the observer and any kind of periodic event changes in time. The usual cause of this distance change is the reflex motion about the system's barycenter due to the gravitational influence of one or more additional bodies. We present results of the analysis of 5032 eclipsing contact and detached binaries from the All Sky Automated Survey (ASAS) catalogue for variations in the times of eclipses. We use an approach known from the radio pulsar timing where a template radio pulse of a pulsar is used as a reference to measure the times of arrivals of the collected pulses. Most of the variations we detect in O--Cs correspond to a linear period change, but three show evidence of more than one complete LTE-orbit. For these objects we present preliminary orbital solutions. Our results demonstrate that the timing analysis employed in radio pulsar timing can be effectively used to study large data sets from photometric surveys. This is the prelude to the analysis of data gathered by the Solaris Project which aims at the search for circumbinary planets. 8. Short Period Eclipsing Binaries In The Field Of The Young Cluster NGC 2362 Hamilton, Catrina; Brinckerhoff, Matthew; Richey-Yowell, Tyler; James, David; Cargile, Phillip 2016-08-01 In a study of rotation periods in the young (t 5 Myr) cluster NGC 2362 (Hamilton et al. 2009), several new eclipsing systems were discovered. In this poster, we present photometric observations of these systems and separate them into likely eclipsing binaries and interesting puzzles. The binaries are most likely field stars and are not a part of the cluster. 9. VizieR Online Data Catalog: Bolometric fluxes of eclipsing binaries in Tycho-2 (Stassun+, 2016) Stassun, K. G.; Torres, G. 2017-03-01 We present fits to the broadband photometric Spectral Energy Distributions (SEDs) of 158 eclipsing binaries in the Tycho-2 catalog. The complete list of 158 eclipsing binaries is given in Table1, sorted by Tycho number. The results of the SED fitting procedure are summarized in Table2. (3 data files). 10. A1540-53, an eclipsing X-ray binary pulsator NASA Technical Reports Server (NTRS) Becker, R. H.; Swank, J. H.; Boldt, E. A.; Holt, S. S.; Pravdo, S. H.; Saba, J. R.; Serlemitsos, P. J. 1977-01-01 An eclipsing X-ray binary pulsator consistent with the location of A1540-53 was observed. The source pulse period was 528.93 plus or minus 0.10 seconds. The binary nature is confirmed by a Doppler curve for the pulsation period. The eclipse angle of 30.5 deg plus or minus 3 deg and the 4 h transition to and from eclipse suggest an early type, giant or supergiant, primary star. 11. A Photometric Study of Three Eclipsing Binary Stars (Poster abstract) Ryan, A. 2016-12-01 (Abstract only) As part of a program to study eclipsing binary stars that exhibit the O'Connell Effect (OCE) we are observing a selection of binary stars in a long term study. The OCE is a difference in maximum light across the ligthcurve possibly cause by starspots. We observed for 7 nights at McDonald Observatory using the 30-inch telescope in July 2015, and used the same telescope remotely for a total of 20 additional nights in August, October, December, and January. We will present lightcurves for three stars from this study, characterize the OCE for these stars, and present our model results for the physical parameters of the star making up each of these systems. 12. Period Variations in Three Eclipsing Binaries in Vulpecula Hanna, M. A. 2015-07-01 We investigate the period variations for three eclipsing binaries showing strong evidence of period changes, by means of the O-C residual diagram. Two of them, BK Vul and KN Vul, are W UMa-type contact binaries, while the third, V467 Vul, is a β Lyr semi-detached system. All three exhibit period changes dP/dt (decreases) over the long term, which are usually interpreted as being due to mass transfer from the primary to the secondary component, or mass loss from L2. The period variability of KN Vul also shows a sine-like variation superimposed on the parabolic behavior, which may be interpreted in terms of either magnetic activity or the light time effect. 13. Inferred Eccentricity and Period Distributions of Kepler Eclipsing Binaries Prsa, Andrej; Matijevic, G. 2014-01-01 Determining the underlying eccentricity and orbital period distributions from an observed sample of eclipsing binary stars is not a trivial task. Shen and Turner (2008) have shown that the commonly used maximum likelihood estimators are biased to larger eccentricities and they do not describe the underlying distribution correctly; orbital periods suffer from a similar bias. Hogg, Myers and Bovy (2010) proposed a hierarchical probabilistic method for inferring the true eccentricity distribution of exoplanet orbits that uses the likelihood functions for individual star eccentricities. The authors show that proper inference outperforms the simple histogramming of the best-fit eccentricity values. We apply this method to the complete sample of eclipsing binary stars observed by the Kepler mission (Prsa et al. 2011) to derive the unbiased underlying eccentricity and orbital period distributions. These distributions can be used for the studies of multiple star formation, dynamical evolution, and they can serve as a drop-in replacement to prior, ad-hoc distributions used in the exoplanet field for determining false positive occurrence rates. 14. MASS CONSTRAINTS FROM ECLIPSE TIMING IN DOUBLE WHITE DWARF BINARIES SciTech Connect Kaplan, David L. 2010-07-10 I demonstrate that an effect similar to the Roemer delay, familiar from timing radio pulsars, should be detectable in the first eclipsing double white dwarf (WD) binary, NLTT 11748. By measuring the difference of the time between the secondary and primary eclipses from one-half period (4.6 s), one can determine the physical size of the orbit and hence constrain the masses of the individual WDs. A measurement with uncertainty <0.1 s-possible with modern large telescopes-will determine the individual masses to {+-}0.02 M{sub sun} when combined with good-quality (<1 km s{sup -1}) radial velocity data, although the eccentricity must also be known to high accuracy ({+-}10{sup -3}). Mass constraints improve as P {sup -1/2} (where P is the orbital period), so this works best in wide binaries and should be detectable even for non-degenerate stars, but such constraints require the mass ratio to differ from 1, as well as undistorted orbits. 15. Absolute Dimensions of the Eccentric Eclipsing Binary V541 Cygni Torres, Guillermo; McGruder, Chima D.; Siverd, Robert J.; Rodriguez, Joseph E.; Pepper, Joshua; Stevens, Daniel J.; Stassun, Keivan G.; Lund, Michael B.; James, David 2017-02-01 We report new spectroscopic and photometric observations of the main-sequence, detached, eccentric, double-lined eclipsing binary V541 Cyg (P = 15.34 days, e = 0.468). Using these observations together with existing measurements, we determine the component masses and radii to better than 1% precision: {M}1={2.335}-0.013+0.017 {M}ȯ , {M}2={2.260}-0.013+0.016 {M}ȯ , {R}1={1.859}-0.009+0.012 {R}ȯ , and {R}2={1.808}-0.013+0.015 {R}ȯ . The nearly identical B9.5 stars have estimated effective temperatures of 10650 ± 200 K and 10350 ± 200 K. A comparison of these properties with current stellar evolution models shows excellent agreement at an age of about 190 Myr and [Fe/H] ≈ ‑0.18. Both components are found to be rotating at the pseudo-synchronous rate. The system displays a slow periastron advance that is dominated by general relativity (GR), and has previously been claimed to be slower than predicted by theory. Our new measurement, \\dot{ω }={0.859}-0.017+0.042 deg century‑1, has an 88% contribution from GR and agrees with the expected rate within the uncertainties. We also clarify the use of the gravity darkening coefficients in the light-curve fitting Eclipsing Binary Orbit Program (EBOP), a version of which we use here. 16. Early-type Eclipsing Binaries with Intermediate Orbital Periods Moe, Maxwell; Di Stefano, Rosanne 2015-09-01 We analyze 221 eclipsing binaries (EBs) in the Large Magellanic Cloud with B-type main-sequence (MS) primaries (M1 ≈ 4-14 {M}⊙ ) and orbital periods P = 20-50 days that were photometrically monitored by the Optical Gravitational Lensing Experiment. We utilize our three-stage automated pipeline to (1) classify all 221 EBs, (2) fit physical models to the light curves of 130 detached well-defined EBs from which unique parameters can be determined, and (3) recover the intrinsic binary statistics by correcting for selection effects. We uncover two statistically significant trends with age. First, younger EBs tend to reside in dustier environments with larger photometric extinctions, an empirical relation that can be implemented when modeling stellar populations. Second, younger EBs generally have large eccentricities. This demonstrates that massive binaries at moderate orbital periods are born with a Maxwellian “thermal” orbital velocity distribution, which indicates they formed via dynamical interactions. In addition, the age-eccentricity anticorrelation provides a direct constraint for tidal evolution in highly eccentric binaries containing hot MS stars with radiative envelopes. The intrinsic fraction of B-type MS stars with stellar companions q = M2/M1 > 0.2 and orbital periods P = 20-50 days is (7 ± 2)%. We find early-type binaries at P = 20-50 days are weighted significantly toward small mass ratios q ≈ 0.2-0.3, which is different than the results from previous observations of closer binaries with P < 20 days. This indicates that early-type binaries at slightly wider orbital separations have experienced substantially less competitive accretion and coevolution during their formation in the circumbinary disk. 17. DISCOVERY OF PSR J1227−4853: A TRANSITION FROM A LOW-MASS X-RAY BINARY TO A REDBACK MILLISECOND PULSAR SciTech Connect Roy, Jayanta; Bhattacharyya, Bhaswati; Stappers, Ben; Ray, Paul S.; Wolff, Michael; Wood, Kent S.; Chengalur, Jayaram N.; Deneva, Julia; Camilo, Fernando; Johnson, Tyrel J.; Hessels, Jason W. T.; Bassa, Cees G.; Keane, Evan F.; Ferrara, Elizabeth C.; Harding, Alice K. 2015-02-10 XSS J12270−4859 is an X-ray binary associated with the Fermi Large Area Telescope gamma-ray source 1FGL J1227.9−4852. In 2012 December, this source underwent a transition where the X-ray and optical luminosity dropped and the spectral signatures of an accretion disk disappeared. We report the discovery of a 1.69 millisecond pulsar (MSP), PSR J1227−4853, at a dispersion measure of 43.4 pc cm{sup −3} associated with this source, using the Giant Metrewave Radio Telescope (GMRT) at 607 MHz. This demonstrates that, post-transition, the system hosts an active radio MSP. This is the third system after PSR J1023+0038 and PSR J1824−2452I showing evidence of state switching between radio MSP and low-mass X-ray binary states. We report timing observations of PSR J1227−4853 with the GMRT and Parkes, which give a precise determination of the rotational and orbital parameters of the system. The companion mass measurement of 0.17–0.46 M{sub ⊙} suggests that this is a redback system. PSR J1227−4853 is eclipsed for about 40% of its orbit at 607 MHz with additional short-duration eclipses at all orbital phases. We also find that the pulsar is very energetic, with a spin-down luminosity of ∼10{sup 35} erg s{sup −1}. We report simultaneous imaging and timing observations with the GMRT, which suggests that eclipses are caused by absorption rather than dispersion smearing or scattering. 18. Oscillations of red dwarfs in evolved low-mass binaries with neutron stars NASA Technical Reports Server (NTRS) Sarna, Marek J.; Lee, Umin; Muslimov, Alexander G. 1994-01-01 We investigate a novel aspect of a problem related to the properties of low-mass binaries (LMBs) with millisecond pulsars: the pulsations of the red dwarf (donor) companion of the neutron star (NS). The illumination of the donor star by the pulsar's high-energy nonthermal radiation and relativistic wind may substantially affect its structure. We present a quantitative analysis of the oscillation spectrum of a red dwarf which has evolved in an LMB and has undergone the stage of evaporation. We calculate the p- and g-modes for red dwarfs with masses in the interval (0.2-0.6) stellar mass. For comparison, similar calculations are presented for zero age main-sequence (ZAMS) stars of the same masses. For less massive donor stars (approximately 0.2 stellar mass) the oscillation spectrum becomes quantitatively different from that of their ZAMS counterparts. The differnce is due to the fact that a ZAMS star of 0.2 stellar mass is fully convective, while the donor star in an LMB is expected to be far from thermal equilibrium and not fully convective. As a result, in contrast to a low-mass ZAMS star, a red dwarf of the same mass in an LMB allows the existence of g-modes. We also consider tidally forced g-modes, and perform a linear analysis of these oscillations for different degrees of nonsynchronism between the orbital and spin rotation of the red dwarf component. We demonstrate the existence of a series of reasonances for the low-order g-modes which may occur in LMBs at a late stage of their evolution. We discuss the possibility that these oscillations may trigger Roche lobe overflow and sudden mass loss by the donor star. Further implications of this effect for gamma- and X-ray burst phenomena are outlined. 19. Formation of Black Hole Low-mass X-Ray Binaries in Hierarchical Triple Systems Naoz, Smadar; Fragos, Tassos; Geller, Aaron; Stephan, Alexander P.; Rasio, Frederic A. 2016-05-01 The formation of black hole (BH) low-mass X-ray binaries (LMXB) poses a theoretical challenge, as low-mass companions are not expected to survive the common-envelope scenario with the BH progenitor. Here we propose a formation mechanism that skips the common-envelope scenario and relies on triple-body dynamics. We study the evolution of hierarchical triples following the secular dynamical evolution up to the octupole-level of approximation, including general relativity, tidal effects, and post-main-sequence evolution such as mass loss, changes to stellar radii, and supernovae. During the dynamical evolution of the triple system the “eccentric Kozai-Lidov” mechanism can cause large eccentricity excitations in the LMXB progenitor, resulting in three main BH-LMXB formation channels. Here we define BH-LMXB candidates as systems where the inner BH-companion star crosses its Roche limit. In the “eccentric” channel (˜81% of the LMXBs in our simulations) the donor star crosses its Roche limit during an extreme eccentricity excitation while still on a wide orbit. Second, we find a “giant” LMXB channel (˜11%), where a system undergoes only moderate eccentricity excitations but the donor star fills its Roche-lobe after evolving toward the giant branch. Third, we identify a “classical” channel (˜8%), where tidal forces and magnetic braking shrink and circularize the orbit to short periods, triggering mass-transfer. Finally, for the giant channel we predict an eccentric (˜0.3-0.6) preferably inclined (˜40°, ˜140°) tertiary, typically on a wide enough orbit (˜104 au) to potentially become unbound later in the triple evolution. While this initial study considers only one representative system and neglects BH natal kicks, we expect our scenario to apply across a broad region of parameter space for triple-star systems. 20. VX Her: Eclipsing Binary System or Single Variable Star Perry, Kathleen; Castelaz, Michael; Henson, Gary; Boghozian, Andrew 2015-01-01 VX Her is a pulsating variable star with a period of .4556504 days. It is believed to be part of an eclipsing binary system (Fitch et al. 1966). This hypothesis originated from Fitch seeing VX Her's minimum point on its light curve reaching a 0.7 magnitude fainter than normal and remaining that way for nearly two hours. If VX Her were indeed a binary system, I would expect to see similar results with a fainter minimum and a broader, more horizontal dip. Having reduced and analyzed images from the Southeastern Association for Research in Astronomy Observatory in Chile and Kitt Peak, as well as images from a 0.15m reflector at East Tennessee State University, I found that VX Her has the standard light curve of the prototype variable star, RR Lyrae. Using photometry, I found no differing features in its light curve to suggest that it is indeed a binary system. However, more observations are needed in case VX Her is a wide binary. 1. Low mass binary neutron star mergers: Gravitational waves and neutrino emission Foucart, Francois; Haas, Roland; Duez, Matthew D.; O'Connor, Evan; Ott, Christian D.; Roberts, Luke; Kidder, Lawrence E.; Lippuner, Jonas; Pfeiffer, Harald P.; Scheel, Mark A. 2016-02-01 Neutron star mergers are among the most promising sources of gravitational waves for advanced ground-based detectors. These mergers are also expected to power bright electromagnetic signals, in the form of short gamma-ray bursts, infrared/optical transients powered by r-process nucleosynthesis in neutron-rich material ejected by the merger, and radio emission from the interaction of that ejecta with the interstellar medium. Simulations of these mergers with fully general relativistic codes are critical to understand the merger and postmerger gravitational wave signals and their neutrinos and electromagnetic counterparts. In this paper, we employ the Spectral Einstein Code to simulate the merger of low mass neutron star binaries (two 1.2 M⊙ neutron stars) for a set of three nuclear-theory-based, finite temperature equations of state. We show that the frequency peaks of the postmerger gravitational wave signal are in good agreement with predictions obtained from recent simulations using a simpler treatment of gravity. We find, however, that only the fundamental mode of the remnant is excited for long periods of time: emission at the secondary peaks is damped on a millisecond time scale in the simulated binaries. For such low mass systems, the remnant is a massive neutron star which, depending on the equation of state, is either permanently stable or long lived (i.e. rapid uniform rotation is sufficient to prevent its collapse). We observe strong excitations of l =2 , m =2 modes, both in the massive neutron star and in the form of hot, shocked tidal arms in the surrounding accretion torus. We estimate the neutrino emission of the remnant using a neutrino leakage scheme and, in one case, compare these results with a gray two-moment neutrino transport scheme. We confirm the complex geometry of the neutrino emission, also observed in previous simulations with neutrino leakage, and show explicitly the presence of important differences in the neutrino luminosity, disk 2. Aperiodic variability of low-mass X-ray binaries at very low frequencies Reig, P.; Papadakis, I.; Kylafis, N. D. 2003-02-01 We have obtained discrete Fourier power spectra of a sample of persistent low-mass neutron-star X-ray binaries using long-term light curves from the All Sky Monitor on board the Rossi X-ray Timing Explorer. Our aim is to investigate their aperiodic variability at frequencies in the range 1 x 10-7-5 x 10-6 Hz and compare their properties with those of the black-hole source Cyg X-1. We find that the classification scheme that divides LMXBs into Z and atoll sources blurs at very low frequencies. Based on the long-term ( ~ years) pattern of variability and the results of power-law fits (P(nu ) ~ nu -alpha ) to the 1 x 10-7-5 x 10-6 Hz power density spectra, low-mass neutron-star binaries fall into three categories. Type I includes all Z sources, except Cyg X-2, and the atoll sources GX9+1 and GX13+1. They show relatively flat power spectra (alpha <~ 0.9) and low variability (rms <~ 20%). Type II systems comprise 4U 1636-53, 4U 1735-44 and GX3+1. They are more variable (20% larm rms <~ 30%) and display steeper power spectra (0.9 <~ alpha <~ 1.2) than type I sources. Type III systems are the most variable (rms > 30%) and exhibit the steepest power spectra (alpha > 1.2). The sources 4U 1705-44, GX354-0 and 4U 1820-30 belong to this group. GX9+9 and Cyg X-2 appear as intermediate systems in between type I and II and type II and III sources, respectively. We speculate that the differences in these systems may be caused by the presence of different types of mass-donor companions. Other factors, like the size of the accretion disc and/or the presence of weak magnetic fields, are also expected to affect their low-frequency X-ray aperiodic varibility. 3. The first eclipsing binary catalogue from the MOA-II data base Li, M. C. A.; Rattenbury, N. J.; Bond, I. A.; Sumi, T.; Bennett, D. P.; Koshimoto, N.; Abe, F.; Asakura, Y.; Barry, R.; Bhattacharya, A.; Donachie, M.; Evans, P.; Freeman, M.; Fukui, A.; Hirao, Y.; Itow, Y.; Ling, C. H.; Masuda, K.; Matsubara, Y.; Muraki, Y.; Nagakane, M.; Ohnishi, K.; Saito, To.; Sharan, A.; Sullivan, D. J.; Suzuki, D.; Tristram, P. J.; Yonehara, A. 2017-09-01 We present the first catalogue of eclipsing binaries in two MOA (Microlensing Observations in Astrophysics) fields towards the Galactic bulge, in which over 8000 candidates, mostly contact and semidetached binaries of periods <1 d, were identified. In this paper, the light curves of a small number of interesting candidates, including eccentric binaries, binaries with noteworthy phase modulations and eclipsing RS Canum Venaticorum type stars, are shown as examples. In addition, we identified three triple object candidates by detecting the light-travel-time effect in their eclipse time variation curves. 4. Red Giants in Eclipsing Binaries as a Benchmark for Asteroseismology Rawls, Meredith L. 2016-04-01 Red giants with solar-like oscillations are astrophysical laboratories for probing the Milky Way. The Kepler Space Telescope revolutionized asteroseismology by consistently monitoring thousands of targets, including several red giants in eclipsing binaries. Binarity allows us to directly measure stellar properties independently of asteroseismology. In this dissertation, we study a subset of eight red giant eclipsing binaries observed by Kepler with a range of orbital periods, oscillation behavior, and stellar activity. Two of the systems do not show solar-like oscillations at all. We use a suite of modeling tools to combine photometry and spectroscopy into a comprehensive picture of each star's life. One noteworthy case is a double red giant binary. The two stars are nearly twins, but have one main set of solar-like oscillations with unusually low-amplitude, wide modes, likely due to stellar activity and modest tidal forces acting over the 171 day eccentric orbit. Mixed modes indicate the main oscillating star is on the secondary red clump (a core-He-burning star), and stellar evolution modeling supports this with a coeval history for a pair of red clump stars. The other seven systems are all red giant branch stars (shell-H-burning) with main sequence companions. The two non-oscillators have the strongest magnetic signatures and some of the strongest lifetime tidal forces with nearly-circular 20-34 day orbits. One system defies this trend with oscillations and a 19 day orbit. The four long-period systems (>100 days) have oscillations, more eccentric orbits, and less stellar activity. They are all detached binaries consistent with coevolution. We find the asteroseismic scaling laws are approximately correct, but fail the most for stars that are least like the Sun by systematically overestimating both mass and radius. Strong magnetic activity and tidal effects often occur in tandem and act to suppress solar-like oscillations. These red giant binaries offer an 5. DISCOVERY AND CHARACTERIZATION OF WIDE BINARY SYSTEMS WITH A VERY LOW MASS COMPONENT SciTech Connect Baron, Frédérique; Lafrenière, David; Artigau, Étienne; Doyon, René; Gagné, Jonathan; Robert, Jasmin; Nadeau, Daniel; Davison, Cassy L.; Malo, Lison; Reylé, Céline 2015-03-20 We report the discovery of 14 low-mass binary systems containing mid-M to mid-L dwarf companions with separations larger than 250 AU. We also report the independent discovery of nine other systems with similar characteristics that were recently discovered in other studies. We have identified these systems by searching for common proper motion sources in the vicinity of known high proper motion stars, based on a cross-correlation of wide area near-infrared surveys (2MASS, SDSS, and SIMP). An astrometric follow-up, for common proper motion confirmation, was made with SIMON and/or CPAPIR at the Observatoire du Mont Mégantic 1.6 m and CTIO 1.5 m telescopes for all the candidates identified. A spectroscopic follow-up was also made with GMOS or GNIRS at Gemini to determine the spectral types of 11 of our newly identified companions and 10 of our primaries. Statistical arguments are provided to show that all of the systems we report here are very likely to be physical binaries. One of the new systems reported features a brown dwarf companion: LSPM J1259+1001 (M5) has an L4.5 (2M1259+1001) companion at ∼340 AU. This brown dwarf was previously unknown. Seven other systems have a companion of spectral type L0–L1 at a separation in the 250–7500 AU range. Our sample includes 14 systems with a mass ratio below 0.3. 6. LSPM J1112+7626: Detection of a 41 Day M-dwarf Eclipsing Binary from the MEarth Transit Survey Irwin, Jonathan M.; Quinn, Samuel N.; Berta, Zachory K.; Latham, David W.; Torres, Guillermo; Burke, Christopher J.; Charbonneau, David; Dittmann, Jason; Esquerdo, Gilbert A.; Stefanik, Robert P.; Oksanen, Arto; Buchhave, Lars A.; Nutzman, Philip; Berlind, Perry; Calkins, Michael L.; Falco, Emilio E. 2011-12-01 We report the detection of eclipses in LSPM J1112+7626, which we find to be a moderately bright (IC = 12.14 ± 0.05) very low mass binary system with an orbital period of 41.03236 ± 0.00002 days, and component masses M 1 = 0.395 ± 0.002 M ⊙ and M 2 = 0.275 ± 0.001 M ⊙ in an eccentric (e = 0.239 ± 0.002) orbit. A 65 day out-of-eclipse modulation of approximately 2% peak-to-peak amplitude is seen in I-band, which is probably due to rotational modulation of photospheric spots on one of the binary components. This paper presents the discovery and characterization of the object, including radial velocities sufficient to determine both component masses to better than 1% precision, and a photometric solution. We find that the sum of the component radii, which is much better determined than the individual radii, is inflated by 3.8+0.9 -0.5% compared to the theoretical model predictions, depending on the age and metallicity assumed. These results demonstrate that the difficulties in reproducing observed M-dwarf eclipsing binary radii with theoretical models are not confined to systems with very short orbital periods. This object promises to be a fruitful testing ground for the hypothesized link between inflated radii in M-dwarfs and activity. 7. LSPM J1112+7626: DETECTION OF A 41 DAY M-DWARF ECLIPSING BINARY FROM THE MEARTH TRANSIT SURVEY SciTech Connect Irwin, Jonathan M.; Quinn, Samuel N.; Berta, Zachory K.; Latham, David W.; Torres, Guillermo; Burke, Christopher J.; Charbonneau, David; Dittmann, Jason; Esquerdo, Gilbert A.; Stefanik, Robert P.; Oksanen, Arto; Buchhave, Lars A.; Nutzman, Philip; Berlind, Perry; Calkins, Michael L.; Falco, Emilio E. 2011-12-01 We report the detection of eclipses in LSPM J1112+7626, which we find to be a moderately bright (I{sub C} = 12.14 {+-} 0.05) very low mass binary system with an orbital period of 41.03236 {+-} 0.00002 days, and component masses M{sub 1} = 0.395 {+-} 0.002 M{sub Sun} and M{sub 2} = 0.275 {+-} 0.001 M{sub Sun} in an eccentric (e = 0.239 {+-} 0.002) orbit. A 65 day out-of-eclipse modulation of approximately 2% peak-to-peak amplitude is seen in I-band, which is probably due to rotational modulation of photospheric spots on one of the binary components. This paper presents the discovery and characterization of the object, including radial velocities sufficient to determine both component masses to better than 1% precision, and a photometric solution. We find that the sum of the component radii, which is much better determined than the individual radii, is inflated by 3.8{sup +0.9}{sub -0.5}% compared to the theoretical model predictions, depending on the age and metallicity assumed. These results demonstrate that the difficulties in reproducing observed M-dwarf eclipsing binary radii with theoretical models are not confined to systems with very short orbital periods. This object promises to be a fruitful testing ground for the hypothesized link between inflated radii in M-dwarfs and activity. 8. Two white dwarfs in ultrashort binaries with detached, eclipsing, likely sub-stellar companions detected by K2 Parsons, S. G.; Hermes, J. J.; Marsh, T. R.; Gänsicke, B. T.; Tremblay, P.-E.; Littlefair, S. P.; Sahman, D. I.; Ashley, R. P.; Green, M.; Rattanasoon, S.; Dhillon, V. S.; Burleigh, M. R.; Casewell, S. L.; Buckley, D. A. H.; Braker, I. P.; Irawati, P.; Dennihy, E.; Rodríguez-Gil, P.; Winget, D. E.; Winget, K. I.; Bell, Keaton J.; Kilic, Mukremin 2017-10-01 Using data from the extended Kepler mission in K2 Campaign 10, we identify two eclipsing binaries containing white dwarfs with cool companions that have extremely short orbital periods of only 71.2 min (SDSS J1205-0242, a.k.a. EPIC 201283111) and 72.5 min (SDSS J1231+0041, a.k.a. EPIC 248368963). Despite their short periods, both systems are detached with small, low-mass companions, in one case a brown dwarf and in the other case either a brown dwarf or a low-mass star. We present follow-up photometry and spectroscopy of both binaries, as well as phase-resolved spectroscopy of the brighter system, and use these data to place preliminary estimates on the physical and binary parameters. SDSS J1205-0242 is composed of a 0.39 ± 0.02 M⊙ helium-core white dwarf that is totally eclipsed by a 0.049 ± 0.006 M⊙ (51 ± 6MJ) brown-dwarf companion, while SDSS J1231+0041 is composed of a 0.56 ± 0.07 M⊙ white dwarf that is partially eclipsed by a companion of mass ≲0.095 M⊙. In the case of SDSS J1205-0242, we look at the combined constraints from common-envelope evolution and brown-dwarf models; the system is compatible with similar constraints from other post-common-envelope binaries, given the current parameter uncertainties, but has potential for future refinement. 9. Indication of a massive circumbinary planet orbiting the low-mass X-ray binary MXB 1658-298 Jain, Chetana; Paul, Biswajit; Sharma, Rahul; Jaleel, Abdul; Dutta, Anjan 2017-06-01 We present an X-ray timing analysis of the transient X-ray binary MXB 1658-298, using data obtained from the RXTE and XMM-Newton observatories. We have made 27 new mid-eclipse time measurements from observations made during the two outbursts of the source. These new measurements have been combined with the previously known values to study long-term changes in orbital period of the binary system. We have found that the mid-eclipse timing record of MXB 1658-298 is quite unusual. The long-term evolution of mid-eclipse times indicates an overall orbital period decay with a time-scale of -6.5(7) × 107 yr. Over and above this orbital period decay, the O-C residual curve also shows a periodic residual on shorter time-scales. This sinusoidal variation has an amplitude of ˜9 lt-s and a period of ˜760 d. This is indicative of the presence of a third body around the compact X-ray binary. The mass and orbital radius of the third body are estimated to lie in the ranges 20.5-26.9 Jupiter mass and 750-860 lt-s, respectively. If true, then it will be the most massive circumbinary planet and also the smallest period binary known to host a planet. 10. A state change in the low-mass X-ray binary XSS J12270-4859 Bassa, C. G.; Patruno, A.; Hessels, J. W. T.; Keane, E. F.; Monard, B.; Mahony, E. K.; Bogdanov, S.; Corbel, S.; Edwards, P. G.; Archibald, A. M.; Janssen, G. H.; Stappers, B. W.; Tendulkar, S. 2014-06-01 Millisecond radio pulsars acquire their rapid rotation rates through mass and angular momentum transfer in a low-mass X-ray binary system. Recent studies of PSR J1824-2452I and PSR J1023+0038 have observationally demonstrated this link, and they have also shown that such systems can repeatedly transition back-and-forth between the radio millisecond pulsar and low-mass X-ray binary states. This also suggests that a fraction of such systems are not newly born radio millisecond pulsars but are rather suspended in a back-and-forth, state-switching phase, perhaps for gigayears. XSS J12270-4859 has been previously suggested to be a low-mass X-ray binary, and until recently the only such system to be seen at MeV-GeV energies. We present radio, optical and X-ray observations that offer compelling evidence that XSS J12270-4859 is a low-mass X-ray binary which transitioned to a radio millisecond pulsar state between 2012 November 14 and December 21. We use optical and X-ray photometry/spectroscopy to show that the system has undergone a sudden dimming and no longer shows evidence for an accretion disc. The optical observations constrain the orbital period to 6.913 ± 0.002 h. 11. Low Mass X-ray Binary 4U1705-44 Exiting an Extended High X-ray State Phillipson, Rebecca; Boyd, Patricia T.; Smale, Alan P. 2017-09-01 The neutron-star low-mass X-ray binary 4U1705-44, which exhibited high amplitude long-term X-ray variability on the order of hundreds of days during the 16-year continuous monitoring by the RXTE ASM (1995-2012), entered an anomalously long high state in July 2012 as observed by MAXI (2009-present). 12. Evidence for Simultaneous Jets and Disk Winds in Luminous Low-mass X-Ray Binaries Homan, Jeroen; Neilsen, Joseph; Allen, Jessamyn L.; Chakrabarty, Deepto; Fender, Rob; Fridriksson, Joel K.; Remillard, Ronald A.; Schulz, Norbert 2016-10-01 Recent work on jets and disk winds in low-mass X-ray binaries (LMXBs) suggests that they are to a large extent mutually exclusive, with jets observed in spectrally hard states and disk winds observed in spectrally soft states. In this paper we use existing literature on jets and disk winds in the luminous neutron star (NS) LMXB GX 13+1, in combination with archival Rossi X-ray Timing Explorer data, to show that this source is likely able to produce jets and disk winds simultaneously. We find that jets and disk winds occur in the same location on the source’s track in its X-ray color-color diagram. A further study of literature on other luminous LMXBs reveals that this behavior is more common, with indications for simultaneous jets and disk winds in the black hole LMXBs V404 Cyg and GRS 1915+105 and the NS LMXBs Sco X-1 and Cir X-1. For the three sources for which we have the necessary spectral information, we find that simultaneous jets/winds all occur in their spectrally hardest states. Our findings indicate that in LMXBs with luminosities above a few tens of percent of the Eddington luminosity, jets and disk winds are not mutually exclusive, and the presence of disk winds does not necessarily result in jet suppression. 13. Low-mass X-ray binaries and globular clusters streamers and arcs in NGC 4278 SciTech Connect D'Abrusco, R.; Fabbiano, G.; Brassington, N. J. 2014-03-01 We report significant inhomogeneities in the projected two-dimensional spatial distributions of low-mass X-ray binaries (LMXBs) and globular clusters (GCs) of the intermediate mass elliptical galaxy NGC 4278. In the inner region of NGC 4278, a significant arc-like excess of LMXBs extending south of the center at ∼50'' in the western side of the galaxy can be associated with a similar overdensity of the spatial distribution of red GCs from Brassington et al. Using a recent catalog of GCs produced by Usher et al. and covering the whole field of the NGC 4278 galaxy, we have discovered two other significant density structures outside the D {sub 25} isophote to the W and E of the center of NGC 4278, associated with an overdensity and an underdensity, respectively. We discuss the nature of these structures in the context of the similar spatial inhomogeneities discovered in the LMXBs and GCs populations of NGC 4649 and NGC 4261, respectively. These features suggest streamers from disrupted and accreted dwarf companions. 14. Chandra Observations of the Faintest Low-Mass X-ray Binaries NASA Technical Reports Server (NTRS) Wilson, Colleen A.; Patel, Sandeep K.; Kouveliotou, Chryssa; Jonker, Peter G.; vanderKlis, Michiel; Lewin, Walter H. G.; Belloni, Tomaso 2003-01-01 There exists a group of persistently faint galactic X-ray sources that, based on their location in the galaxy, high L(sub X)/L(sub opt), association with X-ray bursts, and absence of low frequency X-ray pulsations, are thought to be low-mass X-ray binaries (LMXBs). We present results from Chandra observations for 8 of these systems: 4U 1708-408, 2S 1711-339, KS 1739-304, SLX 1735-269, GRS 1736-297, SLX 1746-331, 1E 1746.7-3224, and 4U 1812-12. Locations for all sources, excluding GRS 1736-297, SLX 1746-331, and KS 1739-304 (which were not detected) were improved to 0.6 sec error circles (90% confidence). Our observations support earlier findings of transient behavior of GRS 1736-297, KS 1739-304, SLX 1746-331, and 2S 1711-339 (which we detect in one of two observations). Energy spectra for 4U 1708-408,2S 1711-339, SLX 1735-269, 1E 1746.7-3224, and 4U 1812-12 are hard, with power law indices typically 1.4-2.1, which are consistent with typical faint LMXB spectra. 15. Chandra Observations of the Faintest Low-Mass X-Ray Binaries NASA Technical Reports Server (NTRS) Wilson, Colleen A.; Patel, Sandeep K.; Kouveliotou, Chryssa; Jonker, Peter G.; vanderKlis, Michiel; Lewin, Walter H. G.; Belloni, Tomaso; Mendez, Mariano 2003-01-01 A group of persistently faint Galactic X-ray sources exist that, based on their location in the Galaxy, high L(sub X)/L(sub opt), association with X-ray bursts, and absence of low-frequency X-ray pulsations, are thought to be low-mass X-ray binaries (LMXBs). We present results from Chandra observations for eight of these systems: 4U 1708-408, 2S 1711-339, KS 1739-304, SLX 1735-269, GRS 1736-297, SLX 1746-331, 1E 1746.7-3224, and 4U 1812-12. Locations for all these sources, excluding GRS 1736-297, SLX 1746-331, and KS 1739-304 (which were not detected), were improved to 0.6 sec error circles (90% confidence). Our observations support earlier findings of transient behavior of GRS 1736-297, KS 1739-304, SLX 1746-331, and 2S 1711-339 (which we detect in one of two observations). Energy spectra for 4U 1708-408, 2S 1711-339, SLX 1735-269, 1E 1746.7-3224, and 4U 1812-12 are hard, with power-law indices typically 1.4-2.1, which is consistent with typical faint LMXB spectra. 16. General Relativistic Simulations of Low-Mass Magnetized Binary Neutron Star Mergers Giacomazzo, Bruno 2017-01-01 We will present general relativistic magnetohydrodynamic (GRMHD) simulations of binary neutron star (BNS) systems that produce long-lived neutron stars (NSs) after merger. While the standard scenario for short gamma-ray bursts (SGRBs) requires the formation after merger of a spinning black hole surrounded by an accretion disk, other theoretical models, such as the time-reversal scenario, predict the formation of a long-lived magnetar. The formation of a long-lived magnetar could in particular explain the X-ray plateaus that have been observed in some SGRBs. Moreover, observations of NSs with masses of 2 solar masses indicate that the equation of state of NS matter should support masses larger than that. Therefore a significant fraction of BNS mergers will produce long-lived NSs. This has important consequences both on the emission of gravitational wave signals and on their electromagnetic counterparts. We will discuss GRMHD simulations of low-mass'' magnetized BNS systems with different equations of state and mass ratios. We will describe the properties of their post-merger remnants and of their gravitational and electromagnetic emission. 17. Low mass binary neutron star mergers : gravitational waves and neutrino emission Foucart, Francois; SXS Collaboration Collaboration 2016-03-01 We present numerical simulations of low mass binary neutron star mergers (1 . 2M⊙ - 1 . 2M⊙) with the SpEC code for a set of three nuclear-theory based, finite temperature equations of state. The merger remnant is a massive neutron star which is either permanently stable or long-lived. We focus on the post-merger gravitational wave signal, and on neutrino-matter interactions in the merger remnant. We show that the frequency peaks of the post-merger gravitational wave signal are in good agreement with predictions obtained from simulations using a simpler treatment of gravity. We then estimate the neutrino emission of the remnant using a neutrino leakage scheme and, in one case, compare these results with a gray two-moment neutrino transport scheme. We confirm the complex geometry of the neutrino emission, also observed in previous simulations with neutrino leakage, and show explicitly the presence of important differences in the neutrino luminosity, disk composition, and outflow properties between the neutrino leakage and transport schemes. We discuss the impact of our results on our ability to measure the neutron star equation of state, and on the post-merger electromagnetic signal and r-process nucleosynthesis in neutron star mergers. Einstein Fellow. 18. Chandra Observations of the Faintest Low-Mass X-Ray Binaries NASA Technical Reports Server (NTRS) Wilson, Colleen A.; Patel, Sandeep K.; Kouveliotou, Chryssa; Jonker, Peter G.; vanderKlis, Michiel; Lewin, Walter H. G.; Belloni, Tomaso; Mendez, Mariano 2003-01-01 A group of persistently faint Galactic X-ray sources exist that, based on their location in the Galaxy, high L(sub X)/L(sub opt), association with X-ray bursts, and absence of low-frequency X-ray pulsations, are thought to be low-mass X-ray binaries (LMXBs). We present results from Chandra observations for eight of these systems: 4U 1708-408, 2S 1711-339, KS 1739-304, SLX 1735-269, GRS 1736-297, SLX 1746-331, 1E 1746.7-3224, and 4U 1812-12. Locations for all these sources, excluding GRS 1736-297, SLX 1746-331, and KS 1739-304 (which were not detected), were improved to 0.6 sec error circles (90% confidence). Our observations support earlier findings of transient behavior of GRS 1736-297, KS 1739-304, SLX 1746-331, and 2S 1711-339 (which we detect in one of two observations). Energy spectra for 4U 1708-408, 2S 1711-339, SLX 1735-269, 1E 1746.7-3224, and 4U 1812-12 are hard, with power-law indices typically 1.4-2.1, which is consistent with typical faint LMXB spectra. 19. Population synthesis of classical low-mass X-ray binaries in the Galactic Bulge van Haaften, L. M.; Nelemans, G.; Voss, R.; van der Sluys, M. V.; Toonen, S. 2015-07-01 Aims: We model the present-day population of classical low-mass X-ray binaries (LMXBs) with neutron star accretors, which have hydrogen-rich donor stars. Their population is compared with that of hydrogen-deficient LMXBs, known as ultracompact X-ray binaries (UCXBs). We model the observable LMXB population and compare it to observations. We model the Galactic Bulge because it contains a well-observed population and it is the target of the Galactic Bulge Survey. Methods: We combine the binary population synthesis code SeBa with detailed LMXB evolutionary tracks to model the size and properties of the present-day LMXB population in the Galactic Bulge. Whether sources are persistent or transient, and what their instantaneous X-ray luminosities are, is predicted using the thermal-viscous disk instability model. Results: We find a population of ~2.1 × 103 LMXBs with neutron star accretors. Of these about 15-40 are expected to be persistent (depending on model assumptions), with luminosities higher than 1035 erg s-1. About 7-20 transient sources are expected to be in outburst at any given time. Within a factor of two these numbers are consistent with the observed population of bright LMXBs in the Bulge. This gives credence to our prediction of the existence of a population of ~1.6 × 103 LMXBs with low donor masses that have gone through the period minimum, and have present-day mass transfer rates below 10-11 M⊙ yr-1. Conclusions: Even though the observed population of hydrogen-rich LMXBs in the Bulge is larger than the observed population of (hydrogen-deficient) UCXBs, the latter have a higher formation rate. While UCXBs may dominate the total LMXB population at the present time, the majority would be very faint or may have become detached and produced millisecond radio pulsars. In that case UCXBs would contribute significantly more to the formation of millisecond radio pulsars than hydrogen-rich LMXBs. 20. The X-ray eclipse of the LMC binary CAL 87 NASA Technical Reports Server (NTRS) Schmidtke, P. C.; Mcgrath, T. K.; Cowley, A. P.; Frattare, L. M. 1993-01-01 ROSAT-PSPC observations of the LMC eclipsing binary CAL 87 show a short-duration, shallow X-ray eclipse which coincides in phase with the primary optical minimum. Characteristics of the eclipse suggest the X-ray emitting region is only partially occulted. Similarities with the eclipse of the accretion-disk corona in X 1822-37 are discussed. However, no temperature variation through eclipse is found for CAL 87. A revised orbital period, combining published data and recent optical photometry, is given. 1. ON THE GEOMETRIC NATURE OF LOW-FREQUENCY QUASI-PERIODIC OSCILLATIONS IN NEUTRON-STAR LOW-MASS X-RAY BINARIES SciTech Connect Homan, Jeroen; Remillard, Ronald A.; Fridriksson, Joel K. 2015-10-10 We report on a detailed analysis of the so-called ∼1 Hz quasi-periodic oscillation (QPO) in the eclipsing and dipping neutron-star low-mass X-ray binary EXO 0748–676. This type of QPO has previously been shown to have a geometric origin. Our study focuses on the evolution of the QPO as the source moves through the color–color diagram in which it traces out an atoll-source-like track. The QPO frequency increases from ∼0.4 Hz in the hard state to ∼25 Hz as the source approaches the soft state. Combining power spectra based on QPO frequency reveals additional features that strongly resemble those seen in non-dipping/eclipsing atoll sources. We show that the low-frequency QPOs in atoll sources and the ∼1 Hz QPO in EXO 0748–676 follow similar relations with respect to the noise components in their power spectra. We conclude that the frequencies of both types of QPOs are likely set by (the same) precession of a misaligned inner accretion disk. For high-inclination systems like EXO 0748–676 this results in modulations of the neutron-star emission due to obscuration or scattering, while for lower-inclination systems the modulations likely arise from relativistic Doppler-boosting and light-bending effects. 2. Eclipse timing variations to detect exoplanets in binary star systems Schwarz, Richard; Funk, Barbara; Bazso, Akos; Zechner, Renate 2016-02-01 This work is devoted to study the circumstances favorable to detect planets in S- or P-Type orbits in close binary star systems by the help of eclipse timing variations (ETVs). A planet in S-Type motion orbits one of the two stars while a planet in P-Type Motion orbits both stars. One can detect ETV signals with the help of former (CoRoT and Kepler) and future space missions Plato, Tess and Cheops). To determine the probability of the detection of such ETV signals with ground based and space telescopes we investigated the dynamics of close binary star systems (stars separated by 0.5 to 3 AU). Therefore we did numerical simulations by using the full three-body problem as dynamical model. The stability and the ETVs are investigated by computing ETV maps for different masses of the secondary star and the exoplanet (Earth, Neptune and Jupiter mass). In addition we changed the planets eccentricity. We can conclude that many ETV amplitudes are large enough to detect planets in S- or P-Type orbits in binary star systems. 3. Distances to Four Solar Neighborhood Eclipsing Binaries from Absolute Fluxes Wilson, R. E.; Van Hamme, W. 2009-07-01 Eclipsing binary (EB)-based distances are estimated for four solar neighborhood EBs by means of the Direct Distance Estimation (DDE) algorithm. Results are part of a project to map the solar neighborhood EBs in three dimensions, independently of parallaxes, and provide statistical comparisons between EB and parallax distances. Apart from judgments on adopted temperature and interstellar extinction, DDE's simultaneous light-velocity solutions are essentially objective and work as well for semidetached (SD) and overcontact binaries as for detached systems. Here, we analyze two detached and two SD binaries, all double lined. RS Chamaeleontis is a pre-main-sequence (MS), detached EB with weak δ Scuti variations. WW Aurigae is detached and uncomplicated, except for having high metallicity. RZ Cassiopeiae is SD and has very clear δ Scuti variations and several peculiarities. R Canis Majoris (R CMa) is an apparently simple but historically problematic SD system, also with weak δ Scuti variations. Discussions include solution rules and strategies, weighting, convergence, and third light problems. So far there is no indication of systematic band dependence among the derived distances, so the adopted band-calibration ratios seem consistent. Agreement of EB-based and parallax distances is typically within the overlapped uncertainties, with minor exceptions. We also suggest an explanation for the long-standing undermassiveness problem of R CMa's hotter component, in terms of a fortuitous combination of low metallicity and evolution slightly beyond the MS. 4. Absolute dimensions of eclipsing binaries. XI - V 451 Ophiuchi Clausen, J. V.; Gimenez, A.; Scarfe, C. 1986-10-01 V451 Oph is a detached eclipsing binary with B9 - A0 main sequence components in a slightly eccentric orbit, and this paper presents accurate absolute dimensions for the system: masses 2.78±0.06 and 2.36±0.05 M_sun;; radii 2.64±0.03 and 2.03±0.05 R_sun;; effective temperatures 10800±800 and 9800±500K. An orbital eccentricity e = 0.0125±0.0015 is obtained and the period of periastron revolution is 180±30 yr. The orbital inclination is 85°.9±0°.5, the relative radii are 0.2155±0.0020 and 0.1655±0.0020, respectively, and secondary eclipse is close to being total. The luminosity ratio between the components is found to be 0.489±0.015 (y), slightly lower than the spectroscopic result 0.60±0.05, which cannot be reproduced from the available photometric information. The components are well-represented by the theoretical evolutionary models by Hejlesen (1980) for an initial chemical composition of (X, Z) = (0.70, 0.02) and a common age of 2×108yr. 5. CzeV615 - a new eclipsing binary Liska, J.; Liskova, Z. 2014-12-01 Discovery of a new eclipsing binary system CzeV615 = BD+09 3111 in the vicinity of an RR Lyrae star, AT Ser, is presented. The variability of the star was detected from CCD measurements with a small telescope. Observations from ASAS-3, NSVS and WISE sky-surveys were used for follow-up analysis. A brightness variation with a period of 1.4869803 d and an amplitude of about 0.1 mag was found. The shapes of light curves in the optical and infrared bands indicate a similar surface temperature for both components. Nevertheless, a period of half the value of the presented period can not be ruled out using the mentioned data. 6. ABSOLUTE PROPERTIES OF THE ECLIPSING BINARY STAR V335 SERPENTIS SciTech Connect Lacy, Claud H. Sandberg; Fekel, Francis C.; Claret, Antonio E-mail: [email protected] 2012-08-15 V335 Ser is now known to be an eccentric double-lined A1+A3 binary star with fairly deep (0.5 mag) partial eclipses. Previous studies of the system are improved with 7456 differential photometric observations from the URSA WebScope and 5666 from the NFO WebScope, and 67 high-resolution spectroscopic observations from the Tennessee State University 2 m automatic spectroscopic telescope. From dates of minima, the apsidal period is about 880 years. Accurate (better than 2%) masses and radii are determined from analysis of the two new light curves and the radial velocity curve. Theoretical models match the absolute properties of the stars at an age of about 380 Myr, though the age agreement for the two components is poor. Tidal theory correctly confirms that the orbit should still be eccentric, but we find that standard tidal theory is unable to match the observed asynchronous rotation rates of the components' surface layers. 7. ABSOLUTE PROPERTIES OF THE ECLIPSING BINARY STAR HY VIRGINIS SciTech Connect Sandberg Lacy, Claud H.; Fekel, Francis C. E-mail: [email protected] 2011-12-15 HY Vir is found to be a double-lined F0m+F5 binary star with relatively shallow (0.3 mag) partial eclipses. Previous studies of the system are improved with 7509 differential photometric observations from the URSA WebScope and 8862 from the NFO WebScope, and 68 high-resolution spectroscopic observations from the Tennessee State University 2 m automatic spectroscopic telescope, and the 1 m coude-feed spectrometer at Kitt Peak National Observatory. Very accurate (better than 0.5%) masses and radii are determined from analysis of the new light curves and radial velocity curves. Theoretical models match the absolute properties of the stars at an age of about 1.35 Gy. 8. Absolute properties of the eclipsing binary star IM Persei SciTech Connect Lacy, Claud H. Sandberg; Torres, Guillermo; Fekel, Francis C.; Muterspaugh, Matthew W.; Southworth, John E-mail: [email protected] E-mail: [email protected] 2015-01-01 IM Per is a detached A7 eccentric eclipsing binary star. We have obtained extensive measurements of the light curve (28,225 differential magnitude observations) and radial velocity curve (81 spectroscopic observations) which allow us to fit orbits and determine the absolute properties of the components very accurately: masses of 1.7831 ± 0.0094 and 1.7741 ± 0.0097 solar masses, and radii of 2.409 ± 0.018 and 2.366 ± 0.017 solar radii. The orbital period is 2.25422694(15) days and the eccentricity is 0.0473(26). A faint third component was detected in the analysis of the light curves, and also directly observed in the spectra. The observed rate of apsidal motion is consistent with theory (U = 151.4 ± 8.4 year). We determine a distance to the system of 566 ± 46 pc. 9. Eclipsing binary stars in the era of massive surveys First results and future prospects Papageorgiou, Athanasios; Catelan, Márcio; Ramos, Rodrigo Contreras; Drake, Andrew J. 2017-09-01 Our thinking about eclipsing binary stars has undergone a tremendous change in the last decade. Eclipsing binary stars are one of nature's best laboratories for determining the fundamental physical properties of stars and thus for testing the predictions of theoretical models. Some of the largest ongoing variable star surveys include the Catalina Real-time Transient Survey (CRTS) and the VISTA Variables in the Vía Láctea survey (VVV). They both contain a large amount of photometric data and plenty of information about eclipsing binaries that wait to be extracted and exploited. Here we briefly describe our efforts in this direction. 10. Dip Spectroscopy of the Low Mass X-Ray Binary XB 1254-690 NASA Technical Reports Server (NTRS) Smale, Alan P.; Church, M. J.; BalucinskaChurch, M.; White, Nicholas E. (Technical Monitor) 2002-01-01 We observed the low mass X-ray binary XB 1254-690 with the Rossi X-ray Timing Explorer in 2001 May and December. During the first observation strong dipping on the 3.9-hr orbital period and a high degree of variability were observed, along with "shoulders" approx. 15% deep during extended intervals on each side of the main dips. The first observation also included pronounced flaring activity. The non-dip spectrum obtained using the PCA instrument was well-described by a two-component model consisting of a blackbody with kT = 1.30 +/- 0.10 keV plus a cut-off power law representation of Comptonized emission with power law photon index 1.10 +/- 0.46 and a cut-off energy of 5.9(sup +3.0, sub -1.4) keV. The intensity decrease in the shoulders of dipping is energy-independent, consistent with electron scattering in the outer ionized regions of the absorber. In deep dipping the depth of dipping reached 100%, in the energy band below 5 keV, indicating that all emitting regions were covered by absorber. Intensity-selected dip spectra were well-fit by a model in which the point-like blackbody is rapidly covered, while the extended Comptonized emission is progressively overlapped by the absorber, with the, covering fraction rising to 95% in the deepest portion of the dip. The intensity of this component in the dip spectra could be modeled by a combination of electron scattering and photoelectric absorption. Dipping did not occur during the 2001 December observation, but remarkably, both bursting and flaring were observed contemporaneously. 11. Interstellar medium composition through X-ray spectroscopy of low-mass X-ray binaries Pinto, C.; Kaastra, J. S.; Costantini, E.; de Vries, C. 2013-03-01 Context. The diffuse interstellar medium (ISM) is an integral part of the evolution of the entire Galaxy. Metals are produced by stars and their abundances are the direct testimony of the history of stellar evolution. However, the interstellar dust composition is not well known and the total abundances are yet to be accurately determined. Aims: We probe ISM dust composition, total abundances, and abundance gradients through the study of interstellar absorption features in the high-resolution X-ray spectra of Galactic low-mass X-ray binaries (LMXBs). Methods: We used high-quality grating spectra of nine LMXBs taken with XMM-Newton. We measured the column densities of O, Ne, Mg, and Fe with an empirical model and estimated the Galactic abundance gradients. Results: The column densities of the neutral gas species are in agreement with those found in the literature. Solids are a significant reservoir of metals like oxygen and iron. Respectively, 15-25% and 65-90% of the total amount of O i and Fe i is found in dust. The dust amount and mixture seem to be consistent along all the lines-of-sight (LOS). Our estimates of abundance gradients and predictions of local interstellar abundances are in agreement with those measured at longer wavelengths. Conclusions: Our work shows that X-ray spectroscopy is a very powerful method to probe the ISM. For instance, on a large scale the ISM appears to be chemically homogeneous showing similar gas ionization ratios and dust mixtures. The agreement between the abundances of the ISM and the stellar objects suggests that the local Galaxy is also chemically homogeneous. 12. REJECTING PROPOSED DENSE MATTER EQUATIONS OF STATE WITH QUIESCENT LOW-MASS X-RAY BINARIES SciTech Connect Guillot, Sebastien; Rutledge, Robert E. E-mail: [email protected] 2014-11-20 Neutrons stars are unique laboratories for discriminating between the various proposed equations of state of matter at and above nuclear density. One sub-class of neutron stars—those inside quiescent low-mass X-ray binaries (qLMXBs)—produce a thermal surface emission from which the neutron star radius (R {sub NS}) can be measured, using the widely accepted observational scenario for qLMXBs, assuming unmagnetized H atmospheres. In a combined spectral analysis, this work first reproduces a previously published measurement of the R {sub NS}, assumed to be the same for all neutron stars, using a slightly expanded data set. The radius measured is R{sub NS}=9.4±1.2 km. On the basis of spectral analysis alone, this measured value is not affected by imposing an assumption of causality in the core. However, the assumptions underlying this R {sub NS} measurement would be falsified by the observation of any neutron star with a mass >2.6 M {sub ☉}, since radii <11 km would be rejected if causality is assumed, which would exclude most of the R {sub NS} parameter space obtained in this analysis. Finally, this work directly tests a selection of dense matter equations of state: WFF1, AP4, MPA1, PAL1, MS0, and three versions of equations of state produced through chiral effective theory. Two of those, MS0 and PAL1, are rejected at the 99% confidence level, accounting for all quantifiable uncertainties, while the other cannot be excluded at >99% certainty. 13. Explaining observations of rapidly rotating neutron stars in low-mass x-ray binaries Gusakov, Mikhail E.; Chugunov, Andrey I.; Kantor, Elena M. 2014-09-01 In a previous paper [M. E. Gusakov, A. I. Chugunov, and E. M. Kantor, Phys. Rev. Lett. 112, 151101 (2014)], we introduced a new scenario that explains the existence of rapidly rotating warm neutron stars (NSs) observed in low-mass x-ray binaries (LMXBs). Here it is described in more detail. The scenario takes into account the interaction between superfluid inertial modes and the normal (quadrupole) m=2 r mode, which can be driven unstable by the Chandrasekhar-Friedman-Schutz (CFS) mechanism. This interaction can only occur at some fixed "resonance" stellar temperatures; it leads to formation of the "stability peaks" which stabilize a star in the vicinity of these temperatures. We demonstrate that a NS in LMXB spends a substantial fraction of time on the stability peak, that is, in the region of stellar temperatures and spin frequencies that has been previously thought to be CFS unstable with respect to excitation of r modes. We also find that the spin frequencies of NSs are limited by the CFS instability of normal (octupole) m=3 r mode rather than by m=2 r mode. This result agrees with the predicted value of the cutoff spin frequency ˜730 Hz in the spin distribution of accreting millisecond x-ray pulsars. In addition, we analyze evolution of a NS after the end of the accretion phase and demonstrate that millisecond pulsars can be born in LMXBs within our scenario. Besides millisecond pulsars, our scenario also predicts a new class of LMXB descendants—hot and rapidly rotating nonaccreting NSs ("hot widows"/HOFNARs). Further comparison of the proposed theory with observations of rotating NSs can impose new important constraints on the properties of superdense matter. 14. Neutron star masses and radii from quiescent low-mass x-ray binaries SciTech Connect Lattimer, James M.; Steiner, Andrew W. E-mail: [email protected] 2014-04-01 We perform a systematic analysis of neutron star radius constraints from five quiescent low-mass X-ray binaries and examine how they depend on measurements of their distances and amounts of intervening absorbing material, as well as their assumed atmospheric compositions. We construct and calibrate to published results a semi-analytic model of the neutron star atmosphere which approximates these effects for the predicted masses and radii. Starting from mass and radius probability distributions established from hydrogen-atmosphere spectral fits of quiescent sources, we apply this model to compute alternate sets of probability distributions. We perform Bayesian analyses to estimate neutron star mass-radius curves and equation of state (EOS) parameters that best-fit each set of distributions, assuming the existence of a known low-density neutron star crustal EOS, a simple model for the high-density EOS, causality, and the observation that the neutron star maximum mass exceeds 2 M {sub ☉}. We compute the posterior probabilities for each set of distance measurements and assumptions about absorption and composition. We find that, within the context of our assumptions and our parameterized EOS models, some absorption models are disfavored. We find that neutron stars composed of hadrons are favored relative to those with exotic matter with strong phase transitions. In addition, models in which all five stars have hydrogen atmospheres are found to be weakly disfavored. Our most likely models predict neutron star radii that are consistent with current experimental results concerning the nature of the nucleon-nucleon interaction near the nuclear saturation density. 15. Very hard states in neutron star low-mass X-ray binaries Parikh, A. S.; Wijnands, R.; Degenaar, N.; Altamirano, D.; Patruno, A.; Gusinskaia, N. V.; Hessels, J. W. T. 2017-07-01 We report on unusually very hard spectral states in three confirmed neutron-star low-mass X-ray binaries (1RXS J180408.9-342058, EXO 1745-248 and IGR J18245-2452) at a luminosity between ˜1036 and 1037 erg s-1. When fitting the Swift X-ray spectra (0.5-10 keV) in those states with an absorbed power-law model, we found photon indices of Γ ˜ 1, significantly lower than the Γ = 1.5-2.0 typically seen when such systems are in their so called hard state. For individual sources, very hard spectra were already previously identified, but here we show for the first time that likely our sources were in a distinct spectral state (i.e. different from the hard state) when they exhibited such very hard spectra. It is unclear how such very hard spectra can be formed; if the emission mechanism is similar to that operating in their hard states (i.e. up-scattering of soft photons due to hot electrons), then the electrons should have higher temperatures or a higher optical depth in the very hard state compared to those observed in the hard state. By using our obtained Γ as a tracer for the spectral evolution with luminosity, we have compared our results with those obtained by Wijnands et al. Our sample of sources follows the same track as the other neutron star systems in Wijnands et al., confirming their general results. However, we do not find that the accreting millisecond pulsars are systematically harder than the non-pulsating systems. 16. Discovery and Characterization of Eclipsing Binary Stars and Transiting Planets in Young Benchmark Clusters: The Pleiades and Hyades Stassun, Keivan; David, Trevor J.; Conroy, Kyle E.; Hillenbrand, Lynne; Stauffer, John R.; Pepper, Joshua; Rebull, Luisa M.; Cody, Ann Marie 2016-06-01 Prior to K2, only one eclipsing binary in the Pleiades was known (HD 23642). We present the discovery and characterization of three additional eclipsing binaries (EBs) in this ~120 Myr old benchmark open cluster. Unlike HD 23642, all three of the new EBs are low mass (Mtot < 1 M⊙) and thus their components are still undergoing pre-main-sequence contraction at the Pleiades age. Low mass EBs are rare, especially in the pre-main-sequence phase, and thus these systems are valuable for constraining theoretical stellar evolution models. One of the three new EBs is single-lined with a K-type primary (HII 2407). The second (HCG 76) comprises two nearly equal-mass 0.3 M⊙ stars, with masses and radii measured with precisions of better than 3% and 5%, respectively. The third (MHO 9) has an M-type primary with a secondary that is possibly quite close to the hydrogen-burning limit, but needs additional follow-up observations to better constrain its parameters. We use the precise parameters of HCG 76 to test the predictions of stellar evolution models, and to derive an independent distance to the Pleiades of 132±5 pc. Finally, we present tentative evidence for differential rotation in the primary component of the newly discovered Pleiades EB HII 2407, and we also characterize a newly discovered transiting Neptune-sized planet orbiting an M-dwarf in the Hyades. 17. Discovery of a Stripped Red-giant Core in a Bright Eclipsing Binary Star Maxted, P. F. L.; Anderson, D. R.; Burleigh, M. R.; Collier Cameron, A.; Heber, U.; Gänsicke, B. T.; Geier, S.; Kupfer, T.; Marsh, T. R.; Nelemans, G.; O'Toole, S. J.; Østensen, R. H.; Smalley, B.; West, R. G.; Bloemen, S. 2012-03-01 We report the serendipitous discovery from WASP archive photometry of a binary star in which an apparently normal A-type star (J0247-25 A) eclipses a smaller, hotter subdwarf star (J0247-25 B). The kinematics of J0247-25 A show that it is a blue-straggler member of the Galactic thick-disk. We present follow-up photometry and spectroscopy from which we derive approximate values for the mass, radius and luminosity for J0247-25 B assuming that J0247-25 A has the mass appropriate for a normal thick-disk star. We find that the properties of J0247-25 B are well matched by models for a red giant stripped of its outer layers and currently in a shell hydrogen-burning stage. In this scenario, J0247-25 B will go on to become a low mass white dwarf (M ˜ 0.25 M⊙) composed mostly of helium. J0247-25 B can be studied in much greater detail than the handful of pre helium white dwarfs (pre-He-WD) identified to-date. These results have been published by Maxted et al. (2011). We also present a preliminary analysis of more recent observations of J0247-25 with the UVES spectrograph, from which we derive much improved masses for both stars in the binary. We find that both stars are more massive than expected and that J0247-25 A rotates sub-synchronously by a factor of about 2. We also present lightcurves for 5 new eclipsing pre-He-WD subsequently identified from the WASP archive photometry, 4 of which have mass estimates for the subdwarf companion based on a pair of radial velocity measurements. 18. Orbital Period Variation Study of the Algol Eclipsing Binary DI Pegasi Hanna, M. A.; Amin, S. M. 2013-08-01 We discuss the orbital period changes of the Algol semi-detached eclipsing binary DI Peg by constructing the (O-C) residual diagram via using all the available precise minima times. We conclude that the period variation can be explained by a sine-like variation due to the presence of a third body orbiting the binary, together with a long-term orbital period increase ( dP/dt=0.17 sec/century) that can be interpreted to be due to mass transfer from the evolved secondary component (of rate 1.52×10^{-8} M_{⊙}/ yr) to the primary one. The detected low-mass third body (M_{3 min.}=0.22±0.0006 M_{⊙}) is responsible for a periodic variation of about 55 years light time effect. We have determined the orbital parameters of the third component which show a considerable eccentricity e_{3}= 0.77±0.07 together with a longitude of periastron ω_{3}=300° ± 10°. 19. Physical parameters and multiplicity of five southern close eclipsing binaries Szalai, T.; Kiss, L. L.; Mészáros, Sz.; Vinkó, J.; Csizmadia, Sz. 2007-04-01 Aims:We detected tertiary components of close binaries from spectroscopy and light curve modelling, investigated the light-travel time effect and the possibility of magnetic activity cycles, measured mass ratios for unstudied systems, and derived absolute parameters. Methods: We carried out new photometric and spectroscopic observations of five bright (< V >< 10.5 mag) close eclipsing binaries, predominantly in the southern skies. We obtained full Johnson BV light curves, which were modelled with the Wilson-Devinney code. Radial velocities were measured with the cross-correlation method using IAU radial velocity standards as spectral templates. Period changes were studied with the O-C method, utilising published epochs of minimum light (XY Leo) and ASAS photometry (VZ Lib). Results: For three objects (DX Tuc, QY Hya, V870 Ara), absolute parameters have been determined for the first time. We spectroscopically detected the tertiary components in XY Leo and VZ Lib and discovered one in QY Hya. For XY Leo we updated the light-time effect parameters and detected a secondary periodicity of about 5100 d in the O-C diagram that may hint at the existence of short-period magnetic cycles. A combination of recent photometric data shows that the orbital period of the tertiary star in VZ Lib is likely to be over 1500 d. QY Hya is a semi-detached X-ray active binary in a triple system with K and M-type components, while V870 Ara is a contact binary with the third smallest spectroscopic mass ratio for a W UMa star to date (q = 0.082 ± 0.030). Being close to the theoretical minimum for contact binaries, this small mass ratio suggests that V870 Ara has the potential of constraining evolutionary scenarios of binary mergers. The inferred distances to these systems are compatible with the Hipparcos parallaxes. Based on observations made at the Siding Spring Observatory, Australia. Light curves and radial velocity data are only available in electronic form at the CDS via anonymous ftp 20. The first Doppler images of the eclipsing binary SZ Piscium Xiang, Yue; Gu, Shenghong; Cameron, A. Collier; Barnes, J. R.; Zhang, Liyun 2016-02-01 We present the first Doppler images of the active eclipsing binary system SZ Psc, based on the high-resolution spectral data sets obtained in 2004 November and 2006 September-December. The least-squares deconvolution technique was applied to derive high signal-to-noise profiles from the observed spectra of SZ Psc. Absorption features contributed by a third component of the system were detected in the LSD profiles at all observed phases. We estimated the mass and period of the third component to be about 0.9 M⊙ and 1283 ± 10 d, respectively. After removing the contribution of the third body from the least-squares deconvolved profiles, we derived the surface maps of SZ Psc. The resulting Doppler images indicate significant star-spot activities on the surface of the K subgiant component. The distributions of star-spots are more complex than that revealed by previous photometric studies. The cooler K component exhibited pronounced high-latitude spots as well as numerous low- and intermediate-latitude spot groups during the entire observing seasons, but did not show any large, stable polar cap, different from many other active RS CVn-type binaries. 1. The First Photometric Study of NSVS 1461538: A New W-subtype Contact Binary with a Low Mass Ratio and Moderate Fill-out Factor Kim, Hyoun-Woo; Kim, Chun-Hwey; Song, Mi-Hwa; Jeong, Min-Ji; Kim, Hye-Young 2016-09-01 New multiband BVRI light curves of NSVS 1461538 were obtained as a byproduct during the photometric observations of our program star PV Cas for three years from 2011 to 2013. The light curves indicate characteristics of a typical W-subtype W UMa eclipsing system, displaying a flat bottom at primary eclipse and the O’Connell effect, rather than those of an Algol/ b Lyrae eclipsing variable classified by the northern sky variability survey (NSVS). A total of 35 times of minimum lights were determined from our observations (20 timings) and the SuperWASP measurements (15 ones). A period study with all the timings shows that the orbital period may vary in a sinusoidal manner with a period of about 5.6 yr and a small semiamplitude of about 0.008 day. The cyclical period variation can be interpreted as a light-time effect due to a tertiary body with a minimum mass of 0.71 M⊙. Simultaneous analysis of the multiband light curves using the 2003 version of the WilsonDevinney binary model shows that NSVS 1461538 is a genuine W-subtype W UMa contact binary with the hotter primary component being less massive and the system shows a low mass ratio of q(mc/mh)=3.51, a high orbital inclination of 88.7°, a moderate fill-out factor of 30 %, and a temperature difference of ΔT=412 K. The O’Connell effect can be similarly explained by cool spots on either the hotter primary star or the cool secondary star. A small third-light corresponding to about 5 % and 2 % of the total systemic light in the B and V bandpasses, respectively, supports the third-body hypothesis proposed by the period study. Preliminary absolute dimensions of the system were derived and used to look into its evolutionary status with other W UMa binaries in the mass-radius and mass-luminosity diagrams. A possible evolution scenario of the system was also discussed in the context of the mass vs mass ratio diagram. 2. Collective properties of neutron-star X-ray binary populations of galaxies. II. Pre-low-mass X-ray binary properties, formation rates, and constraints SciTech Connect 2014-04-01 We continue our exploration of the collective properties of neutron-star X-ray binaries in the stellar fields (i.e., outside globular clusters) of normal galaxies. In Paper I of this series, we considered high-mass X-ray binaries (HMXBs). In this paper (Paper II), we consider low-mass X-ray binaries (LMXBs), whose evolutionary scenario is very different from that of HMXBs. We consider the evolution of primordial binaries up to the stage where the neutron star just formed in the supernova explosion of the primary is in a binary with its low-mass, unevolved companion, and this binary has circularized tidally, producing what we call a pre-low-mass X-ray binary (pre-LMXB). We study the constraints on the formation of such pre-LMXBs in detail (since these are low-probability events), and calculate their collective properties and formation rates. To this end, we first consider the changes in the binary parameters in the various steps involved, viz., the common-envelope phase, the supernova, and the tidal evolution. This naturally leads to a clarification of the constraints. We then describe our calculation of the evolution of the distributions of primordial binary parameters into those of pre-LMXB parameters, following the standard evolutionary scenario for individual binaries. We display the latter as both bivariate and monovariate distributions, discuss their essential properties, and indicate the influences of some essential factors on these. Finally, we calculate the formation rate of these pre-LMXBs. The results of this paper will be used in a subsequent one to compute the expected X-ray luminosity function of LMXBs. 3. The Masses and Radii of the Eclipsing Binary zeta Aurigae Bennett, Philip D.; Harper, Graham M.; Brown, Alexander; Hummel, Christian A. 1996-11-01 We present a full determination of the fundamental stellar and orbital parameters of the eclipsing binary ζ Aurigae (K4 Ib + BS V) using recent observations with the Hubble Space Telescope Goddard High Resolution Spectrograph (GHRS) and the Mark III long-baseline optical interferometer. The information obtained from spectroscopic and interferometric measurements is complementary, and the combination permits a complete determination of the stellar masses, the absolute semimajor axis of the orbit, and the distance. A complete solution requires that both components be visible spectroscopically, and this has always been difficult for the ζ Aur systems. The ζ Aur K star primary presents no difficulty, and accurate radial velocities are readily obtainable in the optical. However, the B star secondary is more problematic. Ground-based radial velocity measurements are hampered by the difficulty of working with the composite spectrum in the blue-violet region, the small number of suitable lines in the generally featureless optical spectrum of the B star, and the great width of the few available lines (the Balmer lines of hydrogen and a few weak He I lines) due to rapid rotation. We avoid the worst of these problems by using GHRS observations in the ultraviolet, where the K star flux is negligible and the intrinsic B star spectrum is more distinctive, and obtain the most accurate determination of the B star radial velocity amplitude to date. We also analyze published photometry of previous eclipses and near-eclipse phases of ζ Aur in order to obtain eclipse durations, which fix the length of the eclipse chord and therefore determine the orbit inclination. The long-baseline interferometry (LBI) yields, in conjunction with the spectroscopic solution, the distance to the system and thus the absolute stellar radius of the resolved K supergiant primary star, ζ Aur A. The secondary is not resolved by LBI, but its angular (and absolute) radius is found by fitting the model 4. Suzaku spectra of the neutron-star low-mass X-ray binary 4U 1608-52 Lei, Yajuan; Zhang, Haotong; zhang, Yanxia 2015-08-01 We present the spectral analysis of the neutron-star low-mass X-ray binary 4U 1608-52 using data from four Suzaku observations in 2010 March. 4U 1608-52 is a transient atoll source, and the analyzed observations contain the “island” and “banana” states, corresponding transitional, and soft states. The spectra are fitted with the hybrid model for the soft states, which consists of two thermal components (a multicolor accretion disk and a single-temperature blackbody) plus a broken power law. The fitting results show that the continuum spectra evolve during the different states. Fe emission line is often detected in low-mass X-ray binary, however, no obviously Fe line is detected in the four observations of 4U 1608-52. 5. Eclipsing binary stars with extreme light curve asymmetries mined from large astronomical surveys Papageorgiou, Athanasios; Kleftogiannis, Georgios; Christopoulou, Panagiota-Eleftheria 2017-09-01 The O'Connell effect is one of the most perplexing challenges in binary studies as it has not been convincingly explained. Furthermore, a simple method to obtain essential parameters for eclipsing binaries exhibiting this effect and to extract information describing the asymmetry in the light curve maxima is needed. We have developed an automated program that characterizes the morphology of light curves by depth of both minima, height of both maxima and curvature outside the eclipses. 6. Coordinated X-Ray, Ultraviolet, Optical, and Radio Observations of the PSR J1023+0038 System in a Low-mass X-Ray Binary State Bogdanov, Slavko; Archibald, Anne M.; Bassa, Cees; Deller, Adam T.; Halpern, Jules P.; Heald, George; Hessels, Jason W. T.; Janssen, Gemma H.; Lyne, Andrew G.; Moldón, Javier; Paragi, Zsolt; Patruno, Alessandro; Perera, Benetge B. P.; Stappers, Ben W.; Tendulkar, Shriharsh P.; D'Angelo, Caroline R.; Wijnands, Rudy 2015-06-01 The PSR J1023+0038 binary system hosts a neutron star and a low-mass, main-sequence-like star. It switches on year timescales between states as an eclipsing radio millisecond pulsar and a low-mass X-ray binary (LMXB). We present a multi-wavelength observational campaign of PSR J1023+0038 in its most recent LMXB state. Two long XMM-Newton observations reveal that the system spends ˜70% of the time in a ≈3 × 1033 erg s-1 X-ray luminosity mode, which, as shown in Archibald et al., exhibits coherent X-ray pulsations. This emission is interspersed with frequent lower flux mode intervals with ≈ 5× {10}32 erg s-1 and sporadic flares reaching up to ≈1034 erg s-1, with neither mode showing significant X-ray pulsations. The switches between the three flux modes occur on timescales of order 10 s. In the UV and optical, we observe occasional intense flares coincident with those observed in X-rays. Our radio timing observations reveal no pulsations at the pulsar period during any of the three X-ray modes, presumably due to complete quenching of the radio emission mechanism by the accretion flow. Radio imaging detects highly variable, flat-spectrum continuum radiation from PSR J1023+0038, consistent with an origin in a weak jet-like outflow. Our concurrent X-ray and radio continuum data sets do not exhibit any correlated behavior. The observational evidence we present bears qualitative resemblance to the behavior predicted by some existing “propeller” and “trapped” disk accretion models although none can account for key aspects of the rich phenomenology of this system. 7. ROSAT Energy Spectra of Low-Mass X-Ray Binaries Schulz, N. S. 1999-01-01 The 0.1-2.4 keV bandpass of the ROSAT Position Sensitive Proportional Counter (PSPC) offers an opportunity to study the very soft X-ray continuum of bright low-mass X-ray binaries (LMXBs). In 46 pointed observations, 23 LMXBs were observed with count rates between 0.4 and 165.4 counts s-1. The survey identified a total of 29 different luminosity levels, which are compared with observations and identified spectral states from other missions. The atoll source 4U 1705-44 was observed near Eddington luminosities in an unusually high intensity state. Spectral analysis provided a measure of the interstellar column density for all 49 observations. The sensitivity of spectral fits depends strongly on column density. Fits to highly absorbed spectra are merely insensitive toward any particular spectral model. Sources with column densities well below 1022 cm-2 are best fitted by power laws, while the blackbody model gives clearly worse fits to the data. Most single-component fits from sources with low column densities, however, are not acceptable at all. The inclusion of a blackbody component in eight sources can improve the fits significantly. The obtained emission radii of less than 5 km suggest emission from the neutron star surface. In 10 sources acceptable fits can only be achieved by including soft-line components. With a spectral resolution of the PSPC of 320-450 eV, between 0.6 and 1.2 keV unresolved broad-line features were detected around 0.65, 0.85, and 1.0 keV. The line fluxes range within 10-11 and 10-12 ergs cm-2 s-1, with equivalent widths between 24 and 210 eV. In LMC X-2, 2S 0918-549, and 4U 1254-690, line emission is indicated for the first time. The soft emission observed in 4U 0614+091 compares with recent ASCA results, with a new feature indicated at 1.31 keV. The deduced line fluxes in 4U 1820-30 and Cyg X-2 showed variability of a factor of 2 within timescales of 1-2 days. Average fluxes of line components in 4U 1820-30 varied by the same factor over a 8. A solar twin in the eclipsing binary LL Aquarii Graczyk, D.; Smolec, R.; Pavlovski, K.; Southworth, J.; Pietrzyński, G.; Maxted, P. F. L.; Konorski, P.; Gieren, W.; Pilecki, B.; Taormina, M.; Suchomska, K.; Karczmarek, P.; Górski, M.; Wielgórski, P.; Anderson, R. I. 2016-10-01 Aims: In the course of a project to study eclipsing binary stars in vinicity of the Sun, we found that the cooler component of LL Aqr is a solar twin candidate. This is the first known star with properties of a solar twin existing in a non-interacting eclipsing binary, offering an excellent opportunity to fully characterise its physical properties with very high precision. Methods: We used extensive multi-band, archival photometry and the Super-WASP project and high-resolution spectroscopy obtained from the HARPS and CORALIE spectrographs. The spectra of both components were decomposed and a detailed LTE abundance analysis was performed. The light and radial velocity curves were simultanously analysed with the Wilson-Devinney code. The resulting highly precise stellar parameters were used for a detailed comparison with PARSEC, MESA, and GARSTEC stellar evolution models. Results: LL Aqr consists of two main-sequence stars (F9 V + G3 V) with masses of M1 = 1.1949 ± 0.0007 and M2 = 1.0337 ± 0.0007 M⊙, radii R1 = 1.321 ± 0.006 and R2 = 1.002 ± 0.005 R⊙, temperatures T1 = 6080 ± 45 and T2 = 5703 ± 50 K and solar chemical composition [M/H] = 0.02 ± 0.05. The absolute dimensions, radiative and photometric properties, and atmospheric abundances of the secondary are all fully consistent with being a solar twin. Both stars are cooler by about 3.5σ or less metal abundant by 5σ than predicted by standard sets of stellar evolution models. When advanced modelling was performed, we found that full agreement with observations can only be obtained for values of the mixing length and envelope overshooting parameters that are hard to accept. The most reasonable and physically justified model fits found with MESA and GARSTEC codes still have discrepancies with observations but only at the level of 1σ. The system is significantly younger that the Sun, with an age between 2.3 Gyr and 2.7 Gyr, which agrees well with the relatively high lithium abundance of the secondary, A 9. Discovery of a 3.6-hr Eclipsing Luminous X-Ray Binary in the Galaxy NGC 4214 NASA Technical Reports Server (NTRS) Ghosh, Kajal K.; Rappaport, Saul; Tennant, Allyn F.; Swartz, Douglas A.; Pooley, David; Madhusudhan, N. 2006-01-01 We report the discovery of an eclipsing X-ray binary with a 3.62-hr period within 24 arcsec of the center of the dwarf starburst galaxy NGC 4214. The orbital period places interesting constraints on the nature of the binary, and allows for a few very different interpretations. The most likely possibility is that the source lies within NGC 4214 and has an X-ray luminosity of up to 7e38. In this case the binary may well be comprised of a naked He-burning donor star with a neutron-star accretor, though a stellar-mass black-hole accretor cannot be completely excluded. There is no obvious evidence for a strong stellar wind in the X-ray orbital light curve that would be expected from a massive He star; thus, the mass of the He star should be <3-4 solar masses. If correct, this would represent a new class of very luminous X-ray binary----perhaps related to Cyg X-3. Other less likely possibilities include a conventional low-mass X-ray binary that somehow manages to produce such a high X-ray luminosity and is apparently persistent over an interval of years; or a foreground AM Her binary of much lower luminosity that fortuitously lies in the direction of NGC 4214. Any model for this system must accommodate the lack of an optical counterpart down to a limiting magnitude of 22.6 in the visible. 10. Discovery of a 3.6-hr Eclipsing Luminous X-Ray Binary in the Galaxy NGC 4214 NASA Technical Reports Server (NTRS) Ghosh, Kajal K.; Rappaport, Saul; Tennant, Allyn F.; Swartz, Douglas A.; Pooley, David; Madhusudhan, N. 2006-01-01 We report the discovery of an eclipsing X-ray binary with a 3.62-hr period within 24 arcsec of the center of the dwarf starburst galaxy NGC 4214. The orbital period places interesting constraints on the nature of the binary, and allows for a few very different interpretations. The most likely possibility is that the source lies within NGC 4214 and has an X-ray luminosity of up to 7e38. In this case the binary may well be comprised of a naked He-burning donor star with a neutron-star accretor, though a stellar-mass black-hole accretor cannot be completely excluded. There is no obvious evidence for a strong stellar wind in the X-ray orbital light curve that would be expected from a massive He star; thus, the mass of the He star should be <3-4 solar masses. If correct, this would represent a new class of very luminous X-ray binary----perhaps related to Cyg X-3. Other less likely possibilities include a conventional low-mass X-ray binary that somehow manages to produce such a high X-ray luminosity and is apparently persistent over an interval of years; or a foreground AM Her binary of much lower luminosity that fortuitously lies in the direction of NGC 4214. Any model for this system must accommodate the lack of an optical counterpart down to a limiting magnitude of 22.6 in the visible. 11. DE Canum Venaticorum: a bright, eclipsing red dwarf-white dwarf binary van den Besselaar, E. J. M.; Greimel, R.; Morales-Rueda, L.; Nelemans, G.; Thorstensen, J. R.; Marsh, T. R.; Dhillon, V. S.; Robb, R. M.; Balam, D. D.; Guenther, E. W.; Kemp, J.; Augusteijn, T.; Groot, P. J. 2007-05-01 Context: Close white dwarf-red dwarf binaries must have gone through a common-envelope phase during their evolution. DE CVn is a detached white dwarf-red dwarf binary with a relatively short (~8.7 h) orbital period. Its brightness and the presence of eclipses makes this system ideal for a more detailed study. Aims: From a study of photometric and spectroscopic observations of DE CVn we derive the system parameters that we discuss in the framework of common-envelope evolution. Methods: Photometric observations of the eclipses are used to determine an accurate ephemeris. From a model fit to an average low-resolution spectrum of DE CVn, we constrain the temperature of the white dwarf and the spectral type of the red dwarf. The eclipse light curve is analysed and combined with the radial velocity curve of the red dwarf determined from time-resolved spectroscopy to derive constraints on the inclination and the masses of the components in the system. Results: The derived ephemeris is HJDmin = 2 452 784.5533(1) + 0.3641394(2) × E. The red dwarf in DE CVn has a spectral type of M3V and the white dwarf has an effective temperature of 8 000 K. The inclination of the system is 86+3°-2 and the mass and radius of the red dwarf are 0.41± 0.06 M⊙ and 0.37+0.06-0.007 R⊙, respectively, and the mass and radius of the white dwarf are 0.51+0.06-0.02 M⊙ and 0.0136+0.0008-0.0002 R⊙, respectively. Conclusions: We found that the white dwarf has a hydrogen-rich atmosphere (DA-type). Given that DE CVn has experienced a common-envelope phase, we can reconstruct its evolution and we find that the progenitor of the white dwarf was a relatively low-mass star (M≤ 1.6~M⊙). The current age of this system is 3.3-7.3× 109 years, while it will take longer than the Hubble time for DE CVn to evolve into a semi-detached system. 12. Analysis of the Extreme Mass Ratio, High Contact Eclipsing Binary, V802 Aquilae Samec, R. G.; Martin, M. W.; Faulkner, D. R. 2004-05-01 We present our observations and initial analysis of BVRI light curves of the solar type, high contact binary, V802 Aquilae [GSC 5119 948, α (2000) = 18h 58m 54.82s, δ (2000) = -03° 01' 11.5"]. The observations were taken on the evenings of 5, 6 and 8 June 2002, by RGS and DRF with the 0.9-m reflector at CTIO. Standard UBVRcIc filters were used. We took from 138 to 148 observations in each BVRI pass band and about 100 in U. Mean epochs of minimum light for one primary eclipse, HJD = 2452431.82156 (81) as well as two secondary eclipses 2452434.89764 (11) and 2452432.75617 (21) were calculated. We calculated the following linear ephemeris: J.D. Hel Min I = 2450300.43417 (69) + 0.26769479 (11) dE. (2) The light curves are shallow (0.35 mag in V) yet show a broad time of constant light (width about 0.1 phase) in the secondary eclipse. Its depressed primary maxima (about 0.06 mag in B) suggest the presence of heavy spot activity. Our Wilson code BVRI simultaneous solution of the instrumental magnitude light curves yields a mass ratio of M2/M1 = 0.16, and a fill-out 32.7 %. The temperature difference is T2-T1 = 136 K with the tiny secondary component having the higher mean surface temperature. A 20.2° cool spot was modeled on the primary component. Its longitude, co-latitude and temperature factor were 281° , 67° , and 0.915 respectively. Further results are presented. The system is a part of a rare group of binaries with a very low mass secondary and high mass ratio that are near a phase of final coalescence into an FK Comae type star. Much of the work was done by an undergraduate student, MWM. We wish to thank Cerro Tololo InterAmerican Observatory for their allocation of observing time, and the grant from NASA administered by the American Astronomical Society. 13. VizieR Online Data Catalog: Kepler Mission. VII. Eclipsing binaries in DR3 (Kirk+, 2016) Kirk, B.; Conroy, K.; Prsa, A.; Abdul-Masih, M.; Kochoska, A.; Matijevic, G.; Hambleton, K.; Barclay, T.; Bloemen, S.; Boyajian, T.; Doyle, L. R.; Fulton, B. J.; Hoekstra, A. J.; Jek, K.; Kane, S. R.; Kostov, V.; Latham, D.; Mazeh, T.; Orosz, J. A.; Pepper, J.; Quarles, B.; Ragozzine, D.; Shporer, A.; Southworth, J.; Stassun, K.; Thompson, S. E.; Welsh, W. F.; Agol, E.; Derekas, A.; Devor, J.; Fischer, D.; Green, G.; Gropp, J.; Jacobs, T.; Johnston, C.; Lacourse, D. M.; Saetre, K.; Schwengeler, H.; Toczyski, J.; Werner, G.; Garrett, M.; Gore, J.; Martinez, A. O.; Spitzer, I.; Stevick, J.; Thomadis, P. C.; Vrijmoet, E. H.; Yenawine, M.; Batalha, N.; Borucki, W. 2016-07-01 The Kepler Eclipsing Binary Catalog lists the stellar parameters from the Kepler Input Catalog (KIC) augmented by: primary and secondary eclipse depth, eclipse width, separation of eclipse, ephemeris, morphological classification parameter, and principal parameters determined by geometric analysis of the phased light curve. The previous release of the Catalog (Paper II; Slawson et al. 2011, cat. J/AJ/142/160) contained 2165 objects, through the second Kepler data release (Q0-Q2). In this release, 2878 objects are identified and analyzed from the entire data set of the primary Kepler mission (Q0-Q17). The online version of the Catalog is currently maintained at http://keplerEBs.villanova.edu/. A static version of the online Catalog associated with this paper is maintained at MAST https://archive.stsci.edu/kepler/eclipsing_binaries.html. (10 data files). 14. Multiwavelength Observations of the Eclipsing Binary NSV 03438 Between January 2013 and March 2016 Becker, Carter M. 2017-06-01 The eclipsing binary NSV 03438 in Canis Minor consists of two M-type stars having approximate effective temperatures of 3235K (M4V) and 2898K (M6V). The period for a cycle during this study was 1.535 days, essentially unchanged from that reported in 1996. A modification of the bisected chord method provides estimates of mid-eclipse Julian Dates with 95% confidence limits for 22 primary and 29 secondary eclipses. The mean depths of primary and secondary eclipses with filter B are 0.69 and 0.62 magnitude, respectively, and 0.65 and 0.61 magnitude, respectively for filter V. APASS standard stars closely associated with NSV 03438 provide a means of determining the magnitude of NSV 03438. In addition, B - V color indexes and effective temperatures of the binary can be assessed at critical stages throughout the eclipse cycle. 15. Artificial Intelligence and the Brave New World of Eclipsing Binaries Devinney, E.; Guinan, E.; Bradstreet, D.; DeGeorge, M.; Giammarco, J.; Alcock, C.; Engle, S. 2005-12-01 The explosive growth of observational capabilities and information technology over the past decade has brought astronomy to a tipping point - we are going to be deluged by a virtual fire hose (more like Niagara Falls!) of data. An important component of this deluge will be newly discovered eclipsing binary stars (EBs) and other valuable variable stars. As exploration of the Local Group Galaxies grows via current and new ground-based and satellite programs, the number of EBs is expected to grow explosively from some 10,000 today to 8 million as GAIA comes online. These observational advances will present a unique opportunity to study the properties of EBs formed in galaxies with vastly different dynamical, star formation, and chemical histories than our home Galaxy. Thus the study of these binaries (e.g., from light curve analyses) is expected to provide clues about the star formation rates and dynamics of their host galaxies as well as the possible effects of varying chemical abundance on stellar evolution and structure. Additionally, minimal-assumption-based distances to Local Group objects (and possibly 3-D mapping within these objects) shall be returned. These huge datasets of binary stars will provide tests of current theories (or suggest new theories) regarding binary star formation and evolution. However, these enormous data will far exceed the capabilities of analysis via human examination. To meet the daunting challenge of successfully mining this vast potential of EBs and variable stars for astrophysical results with minimum human intervention, we are developing new data processing techniques and methodologies. Faced with an overwhelming volume of data, our goal is to integrate technologies of Machine Learning and Pattern Processing (Artificial Intelligence [AI]) into the data processing pipelines of the major current and future ground- and space-based observational programs. Data pipelines of the future will have to carry us from observations to 16. Photometric Analysis of the Eclipsing Binary: DE Canis Venatici (RX J1326.9+4532) Goker, U. D.; Tas, G. 2007-08-01 White dwarfs and red dwarfs represent two different evolutionary stages of low-mass stars. In our Galaxy, the low-mass stars form the most numerous group of objects. For members of binary systems among them, one can derive their physical parameters like mass and radius. In addition, they include valuable information about the mass distribution of our galaxy. Different evolution phases of the binary stars consisting of white dwarfs and red dwarfs are very important for the astronomy because they allow us to test the theories of the stellar evolution. In this study, a literature survey about the structure and evolution of these systems is done and theoretical and observational results for DE CVn are presented. After obtaining new light curves, we derived the geometrical and physical parameters of the eclipsing binary DE CVn consisting of a white dwarf and a red dwarf. We also discuss the problems of both DE CVn and related systems. DE CVn was observed with 3 different telescopes and 2 different receivers through the Johnson B, V, R filters in 2002-2003. Since the clearest variations were seen in the B filter, the B light curve was analysed using the Wilson-Devinney method with Mode 2 designed to solve detached binaries. The mass ratio q=1.1 was found. The visual magnitude of the white dwarf is 13.04 mag. in 0.0 phase and orbital period of the system is 0.364077 days. The DE CVn system consists of a DA-DB white dwarf (He-WDs) and a M1-M2 red dwarf according to our solution. The system conforms to the classical cataclysmic-variable definitions, but the P-M and P-R relation of cataclysmic variables which results from the light curve differs from that obtained from Patterson's P-T relation (1984). The latter indicates a different spectral class for the red dwarf. It is not well known whether the second companion of the system is in post-evolution phase or is not conformed to standard ZAMS M-R relation. 17. The MACHO Project LMC variable star inventory. V. Classification and orbits of 611 eclipsing binary stars SciTech Connect The MACHO Collaboration 1997-07-01 We report the characteristics of 611 eclipsing binary stars in the Large Megallanic Cloud found by using the MACHO Project photometry database. The sample is magnitude limited, and extends down the main sequence to about spectral type A0. Many evolved binaries are also included. Each eclipsing binary is classified according to the traditional scheme of the {ital General Catalogue of Variable Stars} (EA and EB), and also according to a new decimal classification scheme defined in this paper. The new scheme is sensitive to the two major sources of variance in eclipsing binary star light curves{emdash}the sum of radii, and the surface-brightness ratio, and allow greater precision in characterizing the light curves. Examples of each type of light curve and their variations are given. Sixty-four of the eclipsing binaries have eccentric, rather than circular, orbits. The ephemeris and principal photometric characteristics of each eclipsing binary are listed in a table. Photometric orbits based on the Nelson{endash}Davis{endash}Etzel model have been fitted to all light curves. These data will be useful for planning future observations of these binaries. Plots of all data and fitted orbits and a table of the fitted orbital parameters are available on the AAS CD-ROM series, Vol. 9, 1997. These data are also available at the MACHO home page (http://wwwmacho.mcmaster.ca/). {copyright} {ital 1997 American Astronomical Society.} 18. K2 Discovery of Young Eclipsing Binaries in Upper Scorpius: Direct Mass and Radius Determinations for the Lowest Mass Stars and Initial Characterization of an Eclipsing Brown Dwarf Binary David, Trevor J.; Hillenbrand, Lynne A.; Cody, Ann Marie; Carpenter, John M.; Howard, Andrew W. 2016-01-01 We report the discovery of three low-mass double-lined eclipsing binaries in the pre-main sequence Upper Scorpius association, revealed by K2 photometric monitoring of the region over ˜78 days. The orbital periods of all three systems are <5 days. We use the K2 photometry plus multiple Keck/HIRES radial velocities (RVs) and spectroscopic flux ratios to determine fundamental stellar parameters for both the primary and secondary components of each system, along with the orbital parameters. We present tentative evidence that EPIC 203868608 is a hierarchical triple system comprised of an eclipsing pair of ˜25 MJup brown dwarfs with a wide M-type companion. If confirmed, it would constitute only the second double-lined eclipsing brown dwarf binary system discovered to date. The double-lined system EPIC 203710387 is composed of nearly identical M4.5-M5 stars with fundamentally determined masses and radii measured to better than 3% precision ({M}1=0.1183+/- 0.0028{M}⊙ , {M}2=0.1076+/- 0.0031{M}⊙ and {R}1=0.417+/- 0.010{R}⊙ , {R}2=0.450+/- 0.012{R}⊙ ) from combination of the light curve and RV time series. These stars have the lowest masses of any stellar mass double-lined eclipsing binary to date. Comparing our derived stellar parameters with evolutionary models, we suggest an age of ˜10-11 Myr for this system, in contrast to the canonical age of 3-5 Myr for the association. Finally, EPIC 203476597 is a compact single-lined system with a G8-K0 primary and a likely mid-K secondary whose lines are revealed in spectral ratios. Continued measurement of RVs and spectroscopic flux ratios will better constrain fundamental parameters and should elevate the objects to benchmark status. We also present revised parameters for the double-lined eclipsing binary UScoCTIO 5 ({M}1=0.3336+/- 0.0022{M}⊙ , {M}2=0.3200+/- 0.0022{M}⊙ and {R}1=0.862+/- 0.012, {R}2=0.852+/- 0.013{R}⊙ ), which are suggestive of a system age younger than previously reported. We discuss the 19. KIC 7385478: An Eclipsing Binary with a γ Doradus Component Özdarcan, Orkun; Dal, Hasan Ali 2017-04-01 We present spectroscopic and photometric analysis of the eclipsing binary KIC 7385478. We find that the system is formed by F1V + K4III-IV components. Combining results from analysis of spectroscopic data and Kepler photometry, we calculate masses and radii of the primary and the secondary components as M 1 = 1.71 ± 0.08 M⊙, M 2 = 0.37 ± 0.04 M⊙ and R 1 = 1.59 ± 0.03 R⊙, R 2 = 1.90 ± 0.03 R⊙, respectively. Position of the primary component in HR diagram is in the region of γ Doradus type pulsators and residuals from light curve modelling exhibit additional light variation with a dominant period of 0.5 d. These are clear evidences of the γ Doradus type pulsations on the primary component. We also observe occasional increase in amplitude of the residuals, where the orbital period becomes the most dominant period. These may be attributed to the cool star activity originating from the secondary component. 20. Absolute properties of the eclipsing binary VV CORVI SciTech Connect Fekel, Francis C.; Henry, Gregory W.; Sowell, James R. E-mail: [email protected] 2013-12-01 We have obtained red-wavelength spectroscopy and Johnson B and V differential photoelectric photometry of the eclipsing binary VV Crv = HR 4821. The system is the secondary of the common proper motion double star ADS 8627, which has a separation of 5.''2. VV Crv has an orbital period of 3.144536 days and a low but non-zero eccentricity of 0.085. With the Wilson-Devinney program we have determined a simultaneous solution of our spectroscopic and photometric observations. Those orbital elements produce masses of M {sub 1} = 1.978 ± 0.010 M {sub ☉} and M {sub 2} = 1.513 ± 0.008 M {sub ☉}, and radii of R {sub 1} = 3.375 ± 0.010 R {sub ☉} and R {sub 2} = 1.650 ± 0.008 R {sub ☉} for the primary and secondary, respectively. The effective temperatures of the two components are 6500 K (fixed) and 6638 K, so the star we call the primary is the more massive but cooler and larger component. A comparison with evolutionary tracks indicates that the components are metal rich with [Fe/H] = 0.3, and the system has an age of 1.2 Gyr. The primary is near the end of its main-sequence lifetime and is rotating significantly faster than its pseudosynchronous velocity. The secondary is still well ensconced on the main sequence and is rotating more slowly than its pseudosynchronous rate. 1. Absolute properties of the eclipsing binary star AP Andromedae SciTech Connect Sandberg Lacy, Claud H.; Torres, Guillermo; Fekel, Francis C.; Muterspaugh, Matthew W. E-mail: [email protected] E-mail: [email protected] 2014-06-01 AP And is a well-detached F5 eclipsing binary star for which only a very limited amount of information was available before this publication. We have obtained very extensive measurements of the light curve (19,097 differential V magnitude observations) and a radial velocity curve (83 spectroscopic observations) which allow us to fit orbits and determine the absolute properties of the components very accurately: masses of 1.277 ± 0.004 and 1.251 ± 0.004 M {sub ☉}, radii of 1.233 ± 0.006 and 1.1953 ± 0.005 R {sub ☉}, and temperatures of 6565 ± 150 K and 6495 ± 150 K. The distance to the system is about 400 ± 30 pc. Comparison with the theoretical properties of the stellar evolutionary models of the Yonsei-Yale series of Yi et al. shows good agreement between the observations and the theory at an age of about 500 Myr and a slightly sub-solar metallicity. 2. Absolute properties of the eclipsing binary star V501 Herculis SciTech Connect Lacy, Claud H. Sandberg; Fekel, Francis C. E-mail: [email protected] 2014-10-01 V501 Her is a well detached G3 eclipsing binary star with a period of 8.597687 days for which we have determined very accurate light and radial-velocity curves using robotic telescopes. Results of these data indicate that the component stars have masses of 1.269 ± 0.004 and 1.211 ± 0.003 solar masses, radii of 2.001 ± 0.003 and 1.511 ± 0.003 solar radii, and temperatures of 5683 ± 100 K and 5720 ± 100 K, respectively. Comparison with the Yonsei-Yale series of evolutionary models results in good agreement at an age of about 5.1 Gyr for a somewhat metal-rich composition. Those models indicate that the more massive, larger, slightly cooler star is just beyond core hydrogen exhaustion while the less massive, smaller, slightly hotter star has not quite reached core hydrogen exhaustion. The orbit is not yet circularized, and the components are rotating at or near their pseudosynchronous velocities. The distance to the system is 420 ± 30 pc. 3. Fundamental Parameters of the Eclipsing Binary TU Canis Majoris Garcés L., J.; Mennickent, R. E.; Zharikov, S. 2017-04-01 We present a spectroscopic and photometric study of the eclipsing binary TU Canis Majoris aimed to obtain their fundamental stellar parameters and evolutionary stage. Our results indicate that the masses, radii, temperatures, and luminosities for the primary and secondary stars are: {M}1=1.761+/- 0.012 {M}ȯ , {M}2=1.144+/- 0.010 {M}ȯ , {R}1=1.553+/- 0.002 {R}ȯ , {R}2=1.075+/- 0.002 {R}ȯ , {T}1=8014+/- 151 K, {T}2=6060+/- 100 K, {L}1=8.913 +/- 0.695 {L}ȯ , and {L}2=1.396+/- 0.097 {L}ȯ . We estimate an age for the system of τ =2.11+/- 0.24× {10}8 years, and a distance of d = 324.81+/- 12.86 pc. In addition, we note that none of the components has filled their respective Roche lobe and both are on the main sequence. 4. ABSOLUTE PROPERTIES OF THE ECLIPSING BINARY STAR BF DRACONIS SciTech Connect Sandberg Lacy, Claud H.; Torres, Guillermo; Fekel, Francis C.; Sabby, Jeffrey A.; Claret, Antonio E-mail: [email protected] E-mail: [email protected] 2012-06-15 BF Dra is now known to be an eccentric double-lined F6+F6 binary star with relatively deep (0.7 mag) partial eclipses. Previous studies of the system are improved with 7494 differential photometric observations from the URSA WebScope and 9700 from the NFO WebScope, 106 high-resolution spectroscopic observations from the Tennessee State University 2 m automatic spectroscopic telescope and the 1 m coude-feed spectrometer at Kitt Peak National Observatory, and 31 accurate radial velocities from the CfA. Very accurate (better than 0.6%) masses and radii are determined from analysis of the two new light curves and four radial velocity curves. Theoretical models match the absolute properties of the stars at an age of about 2.72 Gyr and [Fe/H] = -0.17, and tidal theory correctly confirms that the orbit should still be eccentric. Our observations of BF Dra constrain the convective core overshooting parameter to be larger than about 0.13 H{sub p}. We find, however, that standard tidal theory is unable to match the observed slow rotation rates of the components' surface layers. 5. A 12 MINUTE ORBITAL PERIOD DETACHED WHITE DWARF ECLIPSING BINARY SciTech Connect Brown, Warren R.; Kilic, Mukremin; Kenyon, Scott J.; Hermes, J. J.; Winget, D. E.; Prieto, Carlos Allende E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] 2011-08-10 We have discovered a detached pair of white dwarfs (WDs) with a 12.75 minute orbital period and a 1315 km s{sup -1} radial velocity amplitude. We measure the full orbital parameters of the system using its light curve, which shows ellipsoidal variations, Doppler boosting, and primary and secondary eclipses. The primary is a 0.25 M{sub sun} tidally distorted helium WD, only the second tidally distorted WD known. The unseen secondary is a 0.55 M{sub sun} carbon-oxygen WD. The two WDs will come into contact in 0.9 Myr due to loss of energy and angular momentum via gravitational wave radiation. Upon contact the systems may merge (yielding a rapidly spinning massive WD), form a stable interacting binary, or possibly explode as an underluminous Type Ia supernova. The system currently has a gravitational wave strain of 10{sup -22}, about 10,000 times larger than the Hulse-Taylor pulsar; this system would be detected by the proposed Laser Interferometer Space Antenna gravitational wave mission in the first week of operation. This system's rapid change in orbital period will provide a fundamental test of general relativity. 6. Doppler Imaging with FUSE: The Partially Eclipsing Binary VW Cep NASA Technical Reports Server (NTRS) Sonneborn, George (Technical Monitor); Brickhouse, Nancy 2003-01-01 This report covers the FUSE Guest Observer program. This project involves the study of emission line profiles for the partially eclipsing, rapidly rotating binary system VW Cep. Active regions on the surface of the star(s) produce observable line shifts as the stars move with respect to the observer. By studying the time-dependence of the line profile changes and centroid shifts, one can determine the location of the activity. FUSE spectra were obtained by the P.I. 27 Sept 2002 and data reduction is in progress. Since we are interested in line profile analysis, we are now investigating the wavelength scale calibration in some detail. We have also obtained and are analyzing Chandra data in order to compare the X-ray velocities with the FUV velocities. A complementary project comparing X-ray and Far UltraViolet (FUV) emission for the similar system 44i Boo is also underway. Postdoctoral fellow Ronnie Hoogerwerf has joined the investigation team and will perform the data analysis, once the calibration is optimized. 7. Discovery of a stripped red giant core in a bright eclipsing binary system Maxted, P. F. L.; Anderson, D. R.; Burleigh, M. R.; Collier Cameron, A.; Heber, U.; Gänsicke, B. T.; Geier, S.; Kupfer, T.; Marsh, T. R.; Nelemans, G.; O'Toole, S. J.; Østensen, R. H.; Smalley, B.; West, R. G. 2011-12-01 We have identified a star in the Wide Angle Search for Planets (WASP) archive photometry with an unusual light curve due to the total eclipse of a small, hot star by an apparently normal A-type star and with an orbital period of only 0.668 d. From an analysis of the WASP light curve together with V-band and IC-band photometry of the eclipse and a spectroscopic orbit for the A-type star we estimate that the companion star has a mass of 0.23 ± 0.03 M⊙ and a radius of 0.33 ± 0.01 R⊙, assuming that the A-type star is a main-sequence star with the metallicity appropriate for a thick-disc star. The effective temperature of the companion is 13 400 ± 1200 K from which we infer a luminosity of 3 ± 1 L⊙. From a comparison of these parameters to various models we conclude that the companion is most likely to be the remnant of a red giant star that has been very recently stripped of its outer layers by mass transfer on to the A-type star. In this scenario, the companion is currently in a shell hydrogen-burning phase of its evolution, evolving at nearly constant luminosity to hotter effective temperatures prior to ceasing hydrogen burning and fading to become a low-mass white dwarf composed of helium (He-WD). The system will then resemble the pre-He-WD/He-WD companions to A- and B-type stars recently identified from their Kepler satellite light curves (KOI-74, KOI-81 and KIC 10657664). This newly discovered binary offers the opportunity to study the evolution of a stripped red giant star through the pre-He-WD stage in great detail. Based on observations made with ESO Telescopes at the La Silla Observatory under programme ID 084.D-0348(A). 8. PSR J1723–2837: AN ECLIPSING BINARY RADIO MILLISECOND PULSAR SciTech Connect Crawford, Fronefield; Lyne, Andrew G.; Stairs, Ingrid H.; Kaplan, David L.; McLaughlin, Maura A.; Lorimer, Duncan R.; Freire, Paulo C. C.; Kramer, Michael; Burgay, Marta; D'Amico, Nichi; Possenti, Andrea; Camilo, Fernando; Faulkner, Andrew; Manchester, Richard N.; Steeghs, Danny 2013-10-10 We present a study of PSR J1723–2837, an eclipsing, 1.86 ms millisecond binary radio pulsar discovered in the Parkes Multibeam survey. Radio timing indicates that the pulsar has a circular orbit with a 15 hr orbital period, a low-mass companion, and a measurable orbital period derivative. The eclipse fraction of ∼15% during the pulsar's orbit is twice the Roche lobe size inferred for the companion. The timing behavior is significantly affected by unmodeled systematics of astrophysical origin, and higher-order orbital period derivatives are needed in the timing solution to account for these variations. We have identified the pulsar's (non-degenerate) companion using archival ultraviolet, optical, and infrared survey data and new optical photometry. Doppler shifts from optical spectroscopy confirm the star's association with the pulsar and indicate a pulsar-to-companion mass ratio of 3.3 ± 0.5, corresponding to a companion mass range of 0.4 to 0.7 M{sub ☉} and an orbital inclination angle range of between 30° and 41°, assuming a pulsar mass range of 1.4-2.0 M{sub ☉}. Spectroscopy indicates a spectral type of G for the companion and an inferred Roche-lobe-filling distance that is consistent with the distance estimated from radio dispersion. The features of PSR J1723–2837 indicate that it is likely a 'redback' system. Unlike the five other Galactic redbacks discovered to date, PSR J1723–2837 has not been detected as a γ-ray source with Fermi. This may be due to an intrinsic spin-down luminosity that is much smaller than the measured value if the unmeasured contribution from proper motion is large. 9. PSR J1723-2837: An Eclipsing Binary Radio Millisecond Pulsar Crawford, Fronefield; Lyne, Andrew G.; Stairs, Ingrid H.; Kaplan, David L.; McLaughlin, Maura A.; Freire, Paulo C. C.; Burgay, Marta; Camilo, Fernando; D'Amico, Nichi; Faulkner, Andrew; Kramer, Michael; Lorimer, Duncan R.; Manchester, Richard N.; Possenti, Andrea; Steeghs, Danny 2013-10-01 We present a study of PSR J1723-2837, an eclipsing, 1.86 ms millisecond binary radio pulsar discovered in the Parkes Multibeam survey. Radio timing indicates that the pulsar has a circular orbit with a 15 hr orbital period, a low-mass companion, and a measurable orbital period derivative. The eclipse fraction of ~15% during the pulsar's orbit is twice the Roche lobe size inferred for the companion. The timing behavior is significantly affected by unmodeled systematics of astrophysical origin, and higher-order orbital period derivatives are needed in the timing solution to account for these variations. We have identified the pulsar's (non-degenerate) companion using archival ultraviolet, optical, and infrared survey data and new optical photometry. Doppler shifts from optical spectroscopy confirm the star's association with the pulsar and indicate a pulsar-to-companion mass ratio of 3.3 ± 0.5, corresponding to a companion mass range of 0.4 to 0.7 M ⊙ and an orbital inclination angle range of between 30° and 41°, assuming a pulsar mass range of 1.4-2.0 M ⊙. Spectroscopy indicates a spectral type of G for the companion and an inferred Roche-lobe-filling distance that is consistent with the distance estimated from radio dispersion. The features of PSR J1723-2837 indicate that it is likely a "redback" system. Unlike the five other Galactic redbacks discovered to date, PSR J1723-2837 has not been detected as a γ-ray source with Fermi. This may be due to an intrinsic spin-down luminosity that is much smaller than the measured value if the unmeasured contribution from proper motion is large. 10. Candidates of eclipsing multiples based on extraneous eclipses on binary light curves: KIC 7622486, KIC 7668648, KIC 7670485 and KIC 8938628 Zhang, Jia; Qian, Sheng-Bang; He, Jian-Duo 2017-02-01 Four candidates of eclipsing multiples, based on new extraneous eclipses found on Kepler binary light curves, are presented and studied. KIC 7622486 is a double eclipsing binary candidate with orbital periods of 2.2799960 d and 40.246503 d. The two binary systems do not eclipse each other in the line of sight, but there is mutual gravitational influence between them which leads to the small but definite eccentricity of 0.0035(0.0022) associated with the short 2.2799960 d period orbit. KIC 7668648 is a hierarchical quadruple system candidate, with two sets of solid 203 ± 5 d period extraneous eclipses and another independent set of extraneous eclipses. A clear and credible extraneous eclipse is found on the binary light curve of KIC 7670485 which makes it a triple system candidate. Two sets of extraneous eclipses with periods of about 390 d and 220 d are found on KIC 8938628 binary curves, which not only confirm the previous conclusion of the 388.5 ± 0.3 triple system, but also indicate new additional objects that make KIC 8938628 a hierarchical quadruple system candidate. The results from these four candidates will contribute to the field of eclipsing multiples. 11. Spectroscopic observations of V443 Herculis - A symbiotic binary with a low mass white dwarf NASA Technical Reports Server (NTRS) Dobrzycka, Danuta; Kenyon, Scott J.; Mikolajewska, Joanna 1993-01-01 We present an analysis of new and existing photometric and spectroscopic observations of the symbiotic binary V443 Herculis. This binary system consists of a normal M5 giant and a hot compact star. These two objects have comparable luminosities: about 1500 solar for the M5 giant and about 1000 solar for the compact star. We identify three nebular regions in this binary: a small, highly ionized volume surrounding the hot component, a modestly ionized shell close to the red giant photosphere, and a less dense region of intermediate ionization encompassing both binary components. The system parameters for V443 Her suggest the hot component currently declines from a symbiotic nova eruption. 12. Search for gravitational waves from low mass binary coalescences in the first year of LIGO's S5 data Abbott, B. P.; Abbott, R.; Adhikari, R.; Ajith, P.; Allen, B.; Allen, G.; Amin, R. S.; Anderson, S. B.; Anderson, W. G.; Arain, M. A.; Araya, M.; Armandula, H.; Armor, P.; Aso, Y.; Aston, S.; Aufmuth, P.; Aulbert, C.; Babak, S.; Baker, P.; Ballmer, S.; Barker, C.; Barker, D.; Barr, B.; Barriga, P.; Barsotti, L.; Barton, M. A.; Bartos, I.; Bassiri, R.; Bastarrika, M.; Behnke, B.; Benacquista, M.; Betzwieser, J.; Beyersdorf, P. T.; Bilenko, I. A.; Billingsley, G.; Biswas, R.; Black, E.; Blackburn, J. K.; Blackburn, L.; Blair, D.; Bland, B.; Bodiya, T. P.; Bogue, L.; Bork, R.; Boschi, V.; Bose, S.; Brady, P. R.; Braginsky, V. B.; Brau, J. E.; Bridges, D. O.; Brinkmann, M.; Brooks, A. F.; Brown, D. A.; Brummit, A.; Brunet, G.; Bullington, A.; Buonanno, A.; Burmeister, O.; Byer, R. L.; Cadonati, L.; Camp, J. B.; Cannizzo, J.; Cannon, K. C.; Cao, J.; Capano, C. D.; Cardenas, L.; Caride, S.; Castaldi, G.; Caudill, S.; Cavaglià, M.; Cepeda, C.; Chalermsongsak, T.; Chalkley, E.; Charlton, P.; Chatterji, S.; Chelkowski, S.; Chen, Y.; Christensen, N.; Chung, C. T. Y.; Clark, D.; Clark, J.; Clayton, J. H.; Cokelaer, T.; Colacino, C. N.; Conte, R.; Cook, D.; Corbitt, T. R. C.; Cornish, N.; Coward, D.; Coyne, D. C.; Creighton, J. D. E.; Creighton, T. D.; Cruise, A. M.; Culter, R. M.; Cumming, A.; Cunningham, L.; Danilishin, S. L.; Danzmann, K.; Daudert, B.; Davies, G.; Daw, E. J.; Debra, D.; Degallaix, J.; Dergachev, V.; Desai, S.; Desalvo, R.; Dhurandhar, S.; Díaz, M.; Dietz, A.; Donovan, F.; Dooley, K. L.; Doomes, E. E.; Drever, R. W. P.; Dueck, J.; Duke, I.; Dumas, J.-C.; Dwyer, J. G.; Echols, C.; Edgar, M.; Effler, A.; Ehrens, P.; Ely, G.; Espinoza, E.; Etzel, T.; Evans, M.; Evans, T.; Fairhurst, S.; Faltas, Y.; Fan, Y.; Fazi, D.; Fehrmann, H.; Finn, L. S.; Flasch, K.; Foley, S.; Forrest, C.; Fotopoulos, N.; Franzen, A.; Frede, M.; Frei, M.; Frei, Z.; Freise, A.; Frey, R.; Fricke, T.; Fritschel, P.; Frolov, V. V.; Fyffe, M.; Galdi, V.; Garofoli, J. A.; Gholami, I.; Giaime, J. A.; Giampanis, S.; Giardina, K. D.; Goda, K.; Goetz, E.; Goggin, L. M.; González, G.; Gorodetsky, M. L.; Goßler, S.; Gouaty, R.; Grant, A.; Gras, S.; Gray, C.; Gray, M.; Greenhalgh, R. J. S.; Gretarsson, A. M.; Grimaldi, F.; Grosso, R.; Grote, H.; Grunewald, S.; Guenther, M.; Gustafson, E. K.; Gustafson, R.; Hage, B.; Hallam, J. M.; Hammer, D.; Hammond, G. D.; Hanna, C.; Hanson, J.; Harms, J.; Harry, G. M.; Harry, I. W.; Harstad, E. D.; Haughian, K.; Hayama, K.; Heefner, J.; Heng, I. S.; Heptonstall, A.; Hewitson, M.; Hild, S.; Hirose, E.; Hoak, D.; Hodge, K. A.; Holt, K.; Hosken, D. J.; Hough, J.; Hoyland, D.; Hughey, B.; Huttner, S. H.; Ingram, D. R.; Isogai, T.; Ito, M.; Ivanov, A.; Johnson, B.; Johnson, W. W.; Jones, D. I.; Jones, G.; Jones, R.; Ju, L.; Kalmus, P.; Kalogera, V.; Kandhasamy, S.; Kanner, J.; Kasprzyk, D.; Katsavounidis, E.; Kawabe, K.; Kawamura, S.; Kawazoe, F.; Kells, W.; Keppel, D. G.; Khalaidovski, A.; Khalili, F. Y.; Khan, R.; Khazanov, E.; King, P.; Kissel, J. S.; Klimenko, S.; Kokeyama, K.; Kondrashov, V.; Kopparapu, R.; Koranda, S.; Kozak, D.; Krishnan, B.; Kumar, R.; Kwee, P.; Laljani, V.; Lam, P. K.; Landry, M.; Lantz, B.; Lazzarini, A.; Lei, H.; Lei, M.; Leindecker, N.; Leonor, I.; Li, C.; Lin, H.; Lindquist, P. E.; Littenberg, T. B.; Lockerbie, N. A.; Lodhia, D.; Longo, M.; Lormand, M.; Lu, P.; Lubiński, M.; Lucianetti, A.; Lück, H.; Lundgren, A.; Machenschalk, B.; Macinnis, M.; Mageswaran, M.; Mailand, K.; Mandel, I.; Mandic, V.; Márka, S.; Márka, Z.; Markosyan, A.; Markowitz, J.; Maros, E.; Martin, I. W.; Martin, R. M.; Marx, J. N.; Mason, K.; Matichard, F.; Matone, L.; Matzner, R. A.; Mavalvala, N.; McCarthy, R.; McClelland, D. E.; McGuire, S. C.; McHugh, M.; McIntyre, G.; McKechan, D. J. A.; McKenzie, K.; Mehmet, M.; Melatos, A.; Melissinos, A. C.; Menéndez, D. F.; Mendell, G.; Mercer, R. A.; Meshkov, S.; Messenger, C.; Meyer, M. S.; Miller, J.; Minelli, J.; Mino, Y.; Mitrofanov, V. P.; Mitselmakher, G.; Mittleman, R.; Miyakawa, O.; Moe, B.; Mohanty, S. D.; Mohapatra, S. R. P.; Moreno, G.; Morioka, T.; Mors, K.; Mossavi, K.; Mowlowry, C.; Mueller, G.; Müller-Ebhardt, H.; Muhammad, D.; Mukherjee, S.; Mukhopadhyay, H.; Mullavey, A.; Munch, J.; Murray, P. G.; Myers, E.; Myers, J.; Nash, T.; Nelson, J.; Newton, G.; Nishizawa, A.; Numata, K.; O'Dell, J.; O'Reilly, B.; O'Shaughnessy, R.; Ochsner, E.; Ogin, G. H.; Ottaway, D. J.; Ottens, R. S.; Overmier, H.; Owen, B. J.; Pan, Y.; Pankow, C.; Papa, M. A.; Parameshwaraiah, V.; Patel, P.; Pedraza, M.; Penn, S.; Perraca, A.; Pierro, V.; Pinto, I. M.; Pitkin, M.; Pletsch, H. J.; Plissi, M. V.; Postiglione, F.; Principe, M.; Prix, R.; Prokhorov, L.; Punken, O.; Quetschke, V.; Raab, F. J.; Rabeling, D. S.; Radkins, H.; Raffai, P.; Raics, Z.; Rainer, N.; Rakhmanov, M.; Raymond, V.; Reed, C. M.; Reed, T.; Rehbein, H.; Reid, S.; Reitze, D. H.; Riesen, R.; Riles, K.; Rivera, B.; Roberts, P.; Robertson, N. A.; Robinson, C.; Robinson, E. L.; Roddy, S.; Röver, C.; Rollins, J.; Romano, J. D.; Romie, J. H.; Rowan, S.; Rüdiger, A.; Russell, P.; Ryan, K.; Sakata, S.; de La Jordana, L. Sancho; Sandberg, V.; Sannibale, V.; Santamaría, L.; Saraf, S.; Sarin, P.; Sathyaprakash, B. S.; Sato, S.; Satterthwaite, M.; Saulson, P. R.; Savage, R.; Savov, P.; Scanlan, M.; Schilling, R.; Schnabel, R.; Schofield, R.; Schulz, B.; Schutz, B. F.; Schwinberg, P.; Scott, J.; Scott, S. M.; Searle, A. C.; Sears, B.; Seifert, F.; Sellers, D.; Sengupta, A. S.; Sergeev, A.; Shapiro, B.; Shawhan, P.; Shoemaker, D. H.; Sibley, A.; Siemens, X.; Sigg, D.; Sinha, S.; Sintes, A. M.; Slagmolen, B. J. J.; Slutsky, J.; Smith, J. R.; Smith, M. R.; Smith, N. D.; Somiya, K.; Sorazu, B.; Stein, A.; Stein, L. C.; Steplewski, S.; Stochino, A.; Stone, R.; Strain, K. A.; Strigin, S.; Stroeer, A.; Stuver, A. L.; Summerscales, T. Z.; Sun, K.-X.; Sung, M.; Sutton, P. J.; Szokoly, G. P.; Talukder, D.; Tang, L.; Tanner, D. B.; Tarabrin, S. P.; Taylor, J. R.; Taylor, R.; Thacker, J.; Thorne, K. A.; Thorne, K. S.; Thüring, A.; Tokmakov, K. V.; Torres, C.; Torrie, C.; Traylor, G.; Trias, M.; Ugolini, D.; Ulmen, J.; Urbanek, K.; Vahlbruch, H.; Vallisneri, M.; van den Broeck, C.; van der Sluys, M. V.; van Veggel, A. A.; Vass, S.; Vaulin, R.; Vecchio, A.; Veitch, J.; Veitch, P.; Veltkamp, C.; Villar, A.; Vorvick, C.; Vyachanin, S. P.; Waldman, S. J.; Wallace, L.; Ward, R. L.; Weidner, A.; Weinert, M.; Weinstein, A. J.; Weiss, R.; Wen, L.; Wen, S.; Wette, K.; Whelan, J. T.; Whitcomb, S. E.; Whiting, B. F.; Wilkinson, C.; Willems, P. A.; Williams, H. R.; Williams, L.; Willke, B.; Wilmut, I.; Winkelmann, L.; Winkler, W.; Wipf, C. C.; Wiseman, A. G.; Woan, G.; Wooley, R.; Worden, J.; Wu, W.; Yakushin, I.; Yamamoto, H.; Yan, Z.; Yoshida, S.; Zanolin, M.; Zhang, J.; Zhang, L.; Zhao, C.; Zotov, N.; Zucker, M. E.; Zur Mühlen, H.; Zweizig, J. 2009-06-01 We have searched for gravitational waves from coalescing low mass compact binary systems with a total mass between 2M⊙ and 35M⊙ and a minimum component mass of 1M⊙ using data from the first year of the fifth science run of the three LIGO detectors, operating at design sensitivity. Depending on the mass, we are sensitive to coalescences as far as 150 Mpc from the Earth. No gravitational-wave signals were observed above the expected background. Assuming a population of compact binary objects with a Gaussian mass distribution representing binary neutron star systems, black hole-neutron star binary systems, and binary black hole systems, we calculate the 90% confidence upper limit on the rate of coalescences to be 3.9×10-2yr-1L10-1, 1.1×10-2yr-1L10-1, and 2.5×10-3yr-1L10-1, respectively, where L10 is 1010 times the blue solar luminosity. We also set improved upper limits on the rate of compact binary coalescences per unit blue-light luminosity, as a function of mass. 13. Constraining the formation of black holes in short-period black hole low-mass X-ray binaries Repetto, Serena; Nelemans, Gijs 2015-11-01 The formation of stellar-mass black holes (BHs) is still very uncertain. Two main uncertainties are the amount of mass ejected in the supernova (SN) event (if any) and the magnitude of the natal kick (NK) the BH receives at birth (if any). Repetto et al., studying the position of Galactic X-ray binaries containing BHs, found evidence for BHs receiving high NKs at birth. In this paper, we extend that study, taking into account the previous binary evolution of the sources as well. The seven short-period BH X-ray binaries that we use are compact binaries consisting of a low-mass star orbiting a BH in a period less than 1 d. We trace their binary evolution backwards in time, from the current observed state of mass transfer, to the moment the BH was formed, and we add the extra information on the kinematics of the binaries. We find that several systems could be explained by no NK, just mass ejection, while for two systems (and possibly more) a high kick is required. So unless the latter have an alternative formation, such as within a globular cluster, we conclude that at least some BHs get high kicks. This challenges the standard picture that BH kicks would be scaled down from neutron star kicks. Furthermore, we find that five systems could have formed with a non-zero NK but zero mass ejected (i.e. no SN) at formation, as predicted by neutrino-driven NKs. 14. The population of low-mass X-ray binaries ejected from black-hole retaining globular clusters Giesler, Matthew; Clausen, Drew; Ott, Christian 2017-01-01 The fate of stellar-mass black holes (BHs) formed in globular clusters (GCs) is still widely uncertain; recent studies suggest that GCs may retain a substantial population of BHs, in contrast to the long held belief of a few to zero BHs. We model the population of BH low-mass X-ray binaries (BH-LMXB) ejected from GCs that are representative of Milky Way GCs with variable BH populations. We simulate the formation of BH-binaries in GCs through exchange interactions between binary and single stars in the company of tens to hundreds of BHs. We construct Monte Carlo realizations of the present day BH-LMXB population that account for both the binary evolution of the ejected systems and the dynamical evolution of these binaries in the Milky Way potential. We find that the orbital parameters of the ejected binaries are sensitive to both the GC's observable structural parameters and its unobservable BH population. Our results suggest that these dynamically formed BH-LMXBs will be easily distinguishable, by their distinctive kinematic properties and larger BH masses, from those produced in the field. Identifying this population of BH-LMXBs, an ideal observable proxy for elusive single BHs, would provide observational constraints on the GC BH retention fraction. 15. Search for gravitational waves from low mass binary coalescences in the first year of LIGO's S5 data SciTech Connect Abbott, B. P.; Abbott, R.; Adhikari, R.; Anderson, S. B.; Araya, M.; Armandula, H.; Aso, Y.; Ballmer, S.; Barton, M. A.; Betzwieser, J.; Billingsley, G.; Black, E.; Blackburn, J. K.; Bork, R.; Boschi, V.; Brooks, A. F.; Cannon, K. C.; Cardenas, L.; Cepeda, C.; Chalermsongsak, T. 2009-06-15 We have searched for gravitational waves from coalescing low mass compact binary systems with a total mass between 2M{sub {center_dot}} and 35M{sub {center_dot}} and a minimum component mass of 1M{sub {center_dot}} using data from the first year of the fifth science run of the three LIGO detectors, operating at design sensitivity. Depending on the mass, we are sensitive to coalescences as far as 150 Mpc from the Earth. No gravitational-wave signals were observed above the expected background. Assuming a population of compact binary objects with a Gaussian mass distribution representing binary neutron star systems, black hole-neutron star binary systems, and binary black hole systems, we calculate the 90% confidence upper limit on the rate of coalescences to be 3.9x10{sup -2} yr{sup -1}L{sub 10}{sup -1}, 1.1x10{sup -2} yr{sup -1}L{sub 10}{sup -1}, and 2.5x10{sup -3} yr{sup -1}L{sub 10}{sup -1}, respectively, where L{sub 10} is 10{sup 10} times the blue solar luminosity. We also set improved upper limits on the rate of compact binary coalescences per unit blue-light luminosity, as a function of mass. 16. First photometric study of two southern eclipsing binaries IS Tel and DW Aps Özer, S.; Sürgit, D.; Erdem, A.; Öztürk, O. 2017-02-01 The paper presents the first photometric analysis of two southern eclipsing binary stars, IS Tel and DW Aps. Their V light curves from the All Sky Automated Survey were modelled by using Wilson-Devinney method. The final models give these two Algol-like binary stars as having detached configurations. Absolute parameters of the components of the systems were also estimated. 17. Discovery of two young brown dwarfs in an eclipsing binary system. PubMed Stassun, Keivan G; Mathieu, Robert D; Valenti, Jeff A 2006-03-16 Brown dwarfs are considered to be 'failed stars' in the sense that they are born with masses between the least massive stars (0.072 solar masses, M(o)) and the most massive planets (approximately 0.013M(o)); they therefore serve as a critical link in our understanding of the formation of both stars and planets. Even the most fundamental physical properties of brown dwarfs remain, however, largely unconstrained by direct measurement. Here we report the discovery of a brown-dwarf eclipsing binary system, in the Orion Nebula star-forming region, from which we obtain direct measurements of mass and radius for these newly formed brown dwarfs. Our mass measurements establish both objects as brown dwarfs, with masses of 0.054 +/- 0.005M(o) and 0.034 +/- 0.003M(o). At the same time, with radii relative to the Sun's of 0.669 +/- 0.034R(o) and 0.511 +/- 0.026R(o), these brown dwarfs are more akin to low-mass stars in size. Such large radii are generally consistent with theoretical predictions for young brown dwarfs in the earliest stages of gravitational contraction. Surprisingly, however, we find that the less-massive brown dwarf is the hotter of the pair; this result is contrary to the predictions of all current theoretical models of coeval brown dwarfs. 18. Dust Scattering Halo from an Eclipsing X-ray Binary at 1.5 arcmin from Sgr A Jin, Chichuan; Haberl, Frank; Ponti, Gabriele 2016-07-01 AX J1745.6-2901 is an eclipsing neutron star low mass X-ray binary. This source is bright in X-rays and it has a high column density of absorbing gas along the line of sight, showcasing a strong dust scattering halo. Moreover, the dust scattering halo shows time evolution during the eclipsing phase. The combination of these phenomena can provide important information about the location of the neutron star and the dust properties along the line of sight. In this talk, I will show that based on a large set of XMM-Newton and Chandra data, we can conduct, for the first time, a powerful combined analysis of the radial profile of the dust scattering halo and the time evolution of the halo during the eclipsing phase. Our study can put constraints on the location of the source, the distribution and composition of the dust, and the metal abundance towards the source. Due to the proximity of the source to Sgr A* (only 1.5 arcmin), these properties are highly relevant to the dust in the Galactic centre, and are likely to be similar as the dust properties on the line of sight towards Sgr A*. 19. Introducing Adapted Nelder & Mead's Downhill Simplex Method to a Fully Automated Analysis of Eclipsing Binaries Prsa, A.; Zwitter, T. 2005-01-01 Eclipsing binaries are extremely attractive objects because absolute physical parameters (masses, luminosities, radii) of both components may be determined from observations. Since most efforts to extract these parameters were based on dedicated observing programmes, existing modelling code is based on interactivity. Gaia will make a revolutionary advance in shear number of observed eclipsing binaries and new methods for automatic handling must be introduced and thoroughly tested. This paper focuses on Nelder & Mead's downhill simplex method applied to a synthetically created test binary as it will be observed by Gaia. 20. Binaries discovered by the MUCHFUSS project. SDSS J162256.66+473051.1: An eclipsing subdwarf B binary with a brown dwarf companion Schaffenroth, V.; Geier, S.; Heber, U.; Kupfer, T.; Ziegerer, E.; Heuser, C.; Classen, L.; Cordes, O. 2014-04-01 Hot subdwarf B stars (sdBs) are core helium-burning stars located on the extreme horizontal branch. About half of the known sdB stars are found in close binaries. Their short orbital periods of 1.2 h to a few days suggest that they are post common-envelope systems. Eclipsing hot subdwarf binaries are rare but are important in determining the fundamental stellar parameters. Low-mass companions are identified by the reflection effect. In most cases, the companion is a main sequence star near the stellar mass limit. Here, we report the discovery of an eclipsing hot subdwarf binary SDSS J162256.66+473051.1 (J1622) with very short orbital period (0.0697 d), which has been found in the course of the MUCHFUSS project. The lightcurve shows grazing eclipses and a prominent reflection effect. An analysis of the light- and radial velocity (RV) curves indicated a mass ratio of q = 0.1325, an RV semi-amplitude K = 47.2 km s-1, and an inclination of i = 72.33°. We show that a companion mass of 0.064 M⊙, which is well below the hydrogen-burning limit, is the most plausible solution, which implies a mass close to the canonical mass (0.47 M⊙) of the sdB star. Therefore, the companion is a brown dwarf, which has not only survived the engulfment by the red-giant envelope but also triggered its ejection and enabled the sdB star to form. The rotation of J1622 is expected to be tidally locked to the orbit. However, J1622 rotates too slowly (vrot = 74.5 ± 7 km s-1) to be synchronized, challenging tidal interaction models. Appendix A is available in electronic form at http://www.aanda.org 1. VizieR Online Data Catalog: Kepler Mission. II. Eclipsing binaries in DR2 (Slawson+, 2011) Slawson, R. W.; Prsa, A.; Welsh, W. F.; Orosz, J. A.; Rucker, M.; Batalha, N.; Doyle, L. R.; Engle, S. G.; Conroy, K.; Coughlin, J.; Gregg, T. A.; Fetherolf, T.; Short, D. R.; Windmiller, G.; Fabrycky, D. C.; Howell, S. B.; Jenkins, J. M.; Uddin, K.; Mullally, F.; Seader, S. E.; Thompson, S. E.; Sanderfer, D. T.; Borucki, W.; Koch, D. 2013-03-01 The Kepler Mission (launched in 2009 March) provides nearly continuous monitoring of ~156000 objects with unprecedented photometric precision. Coincident with the first data release, we presented a catalog of 1879 eclipsing binary systems identified within the 115deg2 Kepler field of view (FOV). Here, we provide an updated catalog from paper I (Prsa et al. 2011, Cat. J/AJ/141/83) augmented with the second Kepler data release which increases the baseline nearly fourfold to 125 days. Three hundred and eighty-six new systems have been added, ephemerides and principal parameters have been recomputed. We have removed 42 previously cataloged systems that are now clearly recognized as short-period pulsating variables and another 58 blended systems where we have determined that the Kepler target object is not itself the eclipsing binary. A number of interesting objects are identified. We present several exemplary cases: four eclipsing binaries that exhibit extra (tertiary) eclipse events; and eight systems that show clear eclipse timing variations indicative of the presence of additional bodies bound in the system. We have updated the period and galactic latitude distribution diagrams. With these changes, the total number of identified eclipsing binary systems in the Kepler FOV has increased to 2165, 1.4% of the Kepler target stars. (4 data files). 2. Implementation of the frequency-modulated sideband search method for gravitational waves from low mass x-ray binaries Sammut, L.; Messenger, C.; Melatos, A.; Owen, B. J. 2014-02-01 We describe the practical implementation of the sideband search, a search for periodic gravitational waves from neutron stars in binary systems. The orbital motion of the source in its binary system causes frequency modulation in the combination of matched filters known as the F-statistic. The sideband search is based on the incoherent summation of these frequency-modulated F-statistic sidebands. It provides a new detection statistic for sources in binary systems, called the C-statistic. The search is well suited to low-mass x-ray binaries, the brightest of which, called Sco X-1, is an ideal target candidate. For sources like Sco X-1, with well-constrained orbital parameters, a slight variation on the search is possible. The extra orbital information can be used to approximately demodulate the data from the binary orbital motion in the coherent stage, before incoherently summing the now reduced number of sidebands. We investigate this approach and show that it improves the sensitivity of the standard Sco X-1 directed sideband search. Prior information on the neutron star inclination and gravitational wave polarization can also be used to improve upper limit sensitivity. We estimate the sensitivity of a Sco X-1 directed sideband search on ten days of LIGO data and show that it can beat previous upper limits in current LIGO data, with a possibility of constraining theoretical upper limits using future advanced instruments. 3. Eclipsing Binary Modeling Advances - Recent and On the Way Wilson, R. E.; van Hamme, W. 2010-12-01 Several recent departures from tradition in EB models demonstrate increased capability to extract information from observations. In one area, model light curves in absolute flux are matched to observed absolute flux curves to find distances and their uncertainties in a straightforward, one step process called Direct Distance Estimation (DDE) that works well for all morphological types. Eclipsing binaries (EB's) can yield distances as accurate as the better parallax distances for much more remote objects. Comparisons with Hipparcos parallax distances are illustrated for 10 binaries. Published absolute flux calibrations can be supplemented by inverse application of the DDE algorithm to EB's with accurately known distances, and weak EB and ellipsoidal variable solutions can be strengthened in inverse DDE solutions where distances and extinctions are accurately known. A temperature-distance (T-d) theorem guards against over-determined and under-determined solutions and has been generalized to include interstellar extinction (T-d-A theorem). Dependence of derived distance on calibration accuracy, in the presence of interstellar extinction, is investigated here. Simulations showed fast and reliable convergence while recovering known extinction from synthetic light curves in widely separated bands. Conversion from spectroscopic to mean global temperature is now rigorously computable if the spectroscopic observation time is known. We derive even greater distance to D33 J013346.2+304439.9 in M33 than did Bonanos, which already was greater than 11 previous estimates from various indicators, and discuss possible reasons for the disparity. In another area, pulsation and EB analyses are being unified to exploit the growing lists of EB’s that show pulsations. Our interest is in coherent pulsation/EB models in which stars actually pulsate geometrically and thermally, and in impersonal solutions that yield standard errors. For now we apply only a phenomenological model so as to 4. THE PHASES DIFFERENTIAL ASTROMETRY DATA ARCHIVE. II. UPDATED BINARY STAR ORBITS AND A LONG PERIOD ECLIPSING BINARY SciTech Connect Muterspaugh, Matthew W.; O'Connell, J.; Hartkopf, William I.; Lane, Benjamin F.; Williamson, M.; Kulkarni, S. R.; Konacki, Maciej; Burke, Bernard F.; Colavita, M. M.; Shao, M.; Wiktorowicz, Sloane J. E-mail: [email protected] E-mail: [email protected] 2010-12-15 Differential astrometry measurements from the Palomar High-precision Astrometric Search for Exoplanet Systems have been combined with lower precision single-aperture measurements covering a much longer timespan (from eyepiece measurements, speckle interferometry, and adaptive optics) to determine improved visual orbits for 20 binary stars. In some cases, radial velocity observations exist to constrain the full three-dimensional orbit and determine component masses. The visual orbit of one of these binaries-{alpha} Com (HD 114378)-shows that the system is likely to have eclipses, despite its very long period of 26 years. The next eclipse is predicted to be within a week of 2015 January 24. 5. APOGEE/Kepler Overlap Yields Orbital Solutions for a Variety of Eclipsing Binaries Clark Cunningham, Joni Marie; Windemuth, Diana; Ali, Aleezah; Rawls, Meredith L.; Jackiewicz, Jason 2017-01-01 We present orbital solutions, masses, and radii for a set of eclipsing spectroscopic binaries observed by both Kepler and APOGEE. Kepler’s primary mission is to find earth-like planets, but several of the observed stars are instead eclipsing binaries with a range of properties. The Apache Point Observatory Galactic Evolution Experiment (APOGEE) has observed many of these same systems during its near-infrared spectroscopic survey. In this work, we combine Kepler light curves and radial velocities extracted from APOGEE spectra to yield binary orbital solutions, stellar masses, and stellar radii. We select binaries that have at least three good-quality APOGEE visits, are sufficiently bright, are listed in the Kepler Eclipsing Binary Catalog (Kirk et al. 2016), show both a primary and a secondary eclipse, and have well- or semi-detached light curve morphologies. We identify a total of 50 promising targets, and present results for a subset of these. Once radial velocity solutions for both stars in each system are found, we combine them with Kepler light curves to solve for mass and radius. These inferences are especially rare for longer-period binaries, and will contribute to our knowledge of fundamental stellar parameters and binary star statistics. This work is supported the SDSS Faculty and Student (FAST) initiative. 6. TIME-SERIES SPECTROSCOPY OF THE ECLIPSING BINARY Y CAM WITH A PULSATING COMPONENT SciTech Connect Hong, Kyeongsoo; Lee, Jae Woo; Kim, Seung-Lee; Koo, Jae-Rim; Lee, Chung-Uk; Yushchenko, Alexander V.; Kang, Young-Woon 2015-10-15 We present the physical properties of the semi-detached Algol-type eclipsing binary Y Cam based on high resolution spectra obtained using the Bohyunsan Optical Echelle Spectrograph. This is the first spectroscopic monitoring data obtained for this interesting binary system, which has a δ Sct-type pulsating component. We obtained a total of 59 spectra over 14 nights from 2009 December to 2011 March. Double-lined spectral features from the hot primary and cool secondary components were well identified. We determined the effective temperatures of the two stars to be T{sub eff,1} = 8000 ± 250 K and T{sub eff,2} = 4629 ± 150 K. The projected rotational velocities are v{sub 1}sin i{sub 1} = 51 ± 4 km s{sup −1} and v{sub 2}sin i{sub 2} = 50 ± 10 km s{sup −1}, which are very similar to a synchronous rotation with the orbital motion. Physical parameters of each component were derived by analyzing our radial velocity data together with previous photometric light curves from the literature. The masses and radii are M{sub 1} = 2.08 ± 0.09 M{sub ⊙}, M{sub 2} = 0.48 ± 0.03 M{sub ⊙}, R{sub 1} = 3.14 ± 0.05 R{sub ⊙}, and R{sub 2} = 3.33 ± 0.05 R{sub ⊙}, respectively. A comparison of these parameters with the theoretical evolution tracks showed that the primary component is located between the zero-age main sequence and the terminal-age main sequence, while the low-mass secondary is noticeably evolved. This indicates that the two components have experienced mass exchange with each other and the primary has undergone an evolution process different from that of single δ Sct-type pulsators. 7. WASP 1628+10 - an EL CVn-type binary with a very low mass stripped red giant star and multiperiodic pulsations Maxted, P. F. L.; Serenelli, A. M.; Marsh, T. R.; Catalán, S.; Mahtani, D. P.; Dhillon, V. S. 2014-10-01 The star 1SWASP J162842.31+101416.7 (WASP 1628+10) is one of several EL CVn-type stars recently identified using the Wide Angle Search for Planets (WASP) data base, i.e. an eclipsing binary star in which an A-type dwarf star (WASP 1628+10 A) eclipses the remnant of a disrupted red giant star (WASP 1628+10 B). We have measured the masses, radii and luminosities of the stars in WASP 1628+10 using photometry obtained in three bands (u', g', r') with the ULTRACAM instrument and medium-resolution spectroscopy. The properties of the remnant are well matched by models for stars in a rarely observed state evolving to higher effective temperatures at nearly constant luminosity prior to becoming a very low mass white dwarf composed almost entirely of helium, i.e. we confirm that WASP 1628+10 B is a precursor of a helium white dwarf (pre-He-WD). WASP 1628+10 A appears to be a normal A2 V star with a mass of 1.36 ± 0.05 M⊙. By fitting models to the spectrum of this star around the Hγ line we find that it has an effective temperature Teff, A = 7500 ± 200 K and a metallicity [Fe/H] = -0.3 ± 0.3. The mass of WASP 1628+10 B is only 0.135 ± 0.02 M⊙. The effective temperature of this pre-He-WD is approximately 9200 K. The ULTRACAM photometry of WASP 1628+10 shows variability at several frequencies around 40 cycles d-1, which is typical for δ Sct-type pulsations often observed in early A-type stars like WASP 1628+10 A. We also observe frequencies near 114 and 129 cycles d-1, much higher than the frequencies normally seen in δ Sct stars. Additional photometry through the primary eclipse will be required to confirm that these higher frequencies are due to pulsations in WASP 1628+10 B. If confirmed, this would be only the second known example of a pre-He-WD showing high-frequency pulsations. 8. The changing photoionized plasma in the bright Low-Mass X-ray binary GX 13+1 Diaz Trigo, Maria 2006-10-01 We propose five 10 ks XMM-Newton observations of GX13+1, the brightest low-mass X-ray binary (LMXB) which exhibits strong X-ray absorption features. Such features have been observed in a number of LMXBs and are identified with ions such as Fe XXV and Fe XXVI. GX13+1 is the best source to study the variations in the lines with the intensity of the source due to its both high and strongly variable luminosity. We will test whether the absorption lines detected in GX13+1 by XMM-Newton and Chandra are consistent with the presence of a highly-ionized absorber located around the accretion disk, similar to other LMXBs and study the changes of such absorber. This reveals dynamics of the disk wind, which commonly exists in high luminosity accretion-disk systems as galactic binaries and AGNs. 9. Apsidal motion of two eclipsing binaries: V796 Cyg and V2783 Ori Bulut, A.; Bulut, I.; ćiçek, C.; Erdem, A. 2017-02-01 In this study, the orbital period variations of two eclipsing binary systems showing apsidal motion were studied. Their O - C diagrams were analysed using all reliable eclipse timings and the elements of apsidal motion of two systems were improved. We found periods of apsidal motion of V796 Cyg and V2783 Ori to be 32.7 ± 0.2 years and 415 ± 50 years, respectively. 10. Estimation of the accuracy of methods for determining component masses for low-mass X-ray binary systems Antokhina, E. A.; Petrov, V. S.; Cherepashchuk, A. M. 2017-01-01 Modern modeling of the population of low-mass X-ray binary systems containing black holes applying standard assumptions leads to a lack of agreement between the modeled and observed mass distributions for the optical components, with the observed masses being lower. This makes the task of estimating the systematic errors in the derived component masses due to imperfect models relevant. To estimate the influence of systematic errors in the derived masses of stars in X-ray binary systems, we considered two approximations for the tidally deformed star in a Roche model. Approximating the star as a sphere with a volume equal to that of the Roche lobe leads to slight overestimation of the equatorial rotational velocity V rot sin i, and hence to slight underestimation of the mass ratio q = M x / M v . Approximating the star as a flat, circular disk with constant local line profiles and a linear limb-darkening law (a classical rotational broadeningmodel) is an appreciably cruder approach, and leads to overestimation of V rot sin i by about 20%. In the case of high values of q = M x / M v , this approximation leads to substantial underestimation of the mass ratio q, which can reach several tens of percent. The mass of the optical star is overestimated by a factor of 1.5 in this case, while the mass of the black hole is changed only slightly. Since most estimates of component mass ratios for X-ray binary systems are carried out using a classical rotational broadening model for the lines, this leads to the need for appreciable corrections to (reductions of) previously published masses for the optical stars, which enhances the contradiction with the standard evolutionary scenario for low-mass X-ray binaries containing black holes. 11. Calibrating the updated overshoot mixing model on eclipsing binary stars: HY Vir, YZ Cas, χ{sup 2} Hya, and VV Crv SciTech Connect Meng, Y.; Zhang, Q. S. 2014-06-01 Detached eclipsing binary stars with convective cores provide a good tool to investigate convective core overshoot. It has been performed on some binary stars to restrict the classical overshoot model which simply extends the boundary of the fully mixed region. However, the classical overshoot model is physically unreasonable and inconsistent with helioseismic investigations. An updated model of overshoot mixing was established recently. There is a key parameter in the model. In this paper, we use observations of four eclipsing binary stars, i.e., HY Vir, YZ Cas, χ{sup 2} Hya, and VV Crv, to investigate a suitable value for the parameter. It is found that the value suggested by calibrations on eclipsing binary stars is the same as the value recommended by other methods. In addition, we have studied the effects of the updated overshoot model on the stellar structure. The diffusion coefficient of convective/overshoot mixing is very high in the convection zone, then quickly decreases near the convective boundary, and exponentially decreases in the overshoot region. The low value of the diffusion coefficient in the overshoot region leads to weak mixing and a partially mixed overshoot region. Semi-convection, which appears in the standard stellar models of low-mass stars with convective cores, is removed by partial overshoot mixing. 12. Did the Ancient Egyptians Record the Period of the Eclipsing Binary Algol—The Raging One? Jetsu, L.; Porceddu, S.; Lyytinen, J.; Kajatkari, P.; Lehtinen, J.; Markkanen, T.; Toivari-Viitala, J. 2013-08-01 The eclipses in binary stars give precise information of orbital period changes. Goodricke discovered the 2.867 day period in the eclipses of Algol in the year 1783. The irregular orbital period changes of this longest known eclipsing binary continue to puzzle astronomers. The mass transfer between the two members of this binary should cause a long-term increase of the orbital period, but observations over two centuries have not confirmed this effect. Here, we present evidence indicating that the period of Algol was 2.850 days three millennia ago. For religious reasons, the ancient Egyptians have recorded this period into the Cairo Calendar (CC), which describes the repetitive changes of the Raging one. CC may be the oldest preserved historical document of the discovery of a variable star. 13. DID THE ANCIENT EGYPTIANS RECORD THE PERIOD OF THE ECLIPSING BINARY ALGOL-THE RAGING ONE? SciTech Connect Jetsu, L.; Porceddu, S.; Lyytinen, J.; Kajatkari, P.; Lehtinen, J.; Markkanen, T.; Toivari-Viitala, J. 2013-08-10 The eclipses in binary stars give precise information of orbital period changes. Goodricke discovered the 2.867 day period in the eclipses of Algol in the year 1783. The irregular orbital period changes of this longest known eclipsing binary continue to puzzle astronomers. The mass transfer between the two members of this binary should cause a long-term increase of the orbital period, but observations over two centuries have not confirmed this effect. Here, we present evidence indicating that the period of Algol was 2.850 days three millennia ago. For religious reasons, the ancient Egyptians have recorded this period into the Cairo Calendar (CC), which describes the repetitive changes of the Raging one. CC may be the oldest preserved historical document of the discovery of a variable star. 14. Asiago eclipsing binaries program. III. V570 Persei Tomasella, L.; Munari, U.; Cassisi, S.; Siviero, A.; Dallaporta, S.; Sordo, R.; Zwitter, T. 2008-05-01 The orbit and physical parameters of the previously unsolved double-lined eclipsing binary V570 Per, discovered by the Hipparcos satellite, were derived using high-resolution Echelle spectroscopy and B, V photoelectric photometry. The metallicity from χ2 analysis of the spectra is [ M/H]=+0.02±0.03, and reddening from interstellar NaI and KI absorption lines is EB-V=0.023±0.007. V570 Per is a well-detached system, with shallow eclipses (due to low orbital inclination) and no sign of chromospheric activity. The two components have masses of 1.449±0.006 and 1.350±0.006~M_⊙ and spectral types F3 and F5, respectively. They are both still within the main sequence band (T_1=6842±25 K, T_2=6562± 25 K from χ2 analysis, R_1=1.523±0.030, R_2=1.388± 0.019 R_⊙ derived by forcing the orbital solution to conform to the spectroscopic light ratio) and are dynamically relaxed to co-rotation with the orbital motion (V_rot,1,2 sin i=40 and 36 (±1) km s-1). The distance to V570 Per obtained from the orbital solution is 123 ±2 pc, in excellent agreement with the revised Hipparcos distance of 123±11 pc. The observed properties of V570 Per components were compared to available families of stellar evolutionary tracks and, in particular, to BaSTI models computed on purpose for exactly the observed masses and varied chemical compositions. This system is interesting since both components have their masses in the range where the efficiency of convective core overshooting has to decrease with the total mass as a consequence of the decreasing size of the convective core during the central H-burning stage. Our numerical simulations show that, in order to match all empirical constraints, a small but not null overshooting is required, with efficiencies of λ_OV=0.14 and 0.11 for the 1.449 and 1.350 M_⊙ components, respectively. This confirms the finding of Paper II on the similar system V505 Per. At the ≈0.8 Gyr age of the system, the element diffusion has reduced the surface 15. LOW-MASS TERTIARY COMPANIONS TO SPECTROSCOPIC BINARIES. I. COMMON PROPER MOTION SURVEY FOR WIDE COMPANIONS USING 2MASS SciTech Connect Allen, Peter R.; Burgasser, Adam J.; Faherty, Jacqueline K.; Kirkpatrick, J. Davy 2012-08-15 We report the first results of a multi-epoch search for wide (separations greater than a few tens of AU), low-mass tertiary companions of a volume-limited sample of 118 known spectroscopic binaries within 30 pc of the Sun, using the Two Micron All Sky Survey Point Source Catalog and follow-up observations with the KPNO and CTIO 4 m telescopes. Note that this sample is not volume complete but volume limited, and, thus, there is incompleteness in our reported companion rates. We are sensitive to common proper motion companions with separations from roughly 200 AU to 10,000 AU ({approx}10'' {yields} {approx} 10'). From 77 sources followed-up to date, we recover 11 previously known tertiaries, 3 previously known candidate tertiaries, of which 2 are spectroscopically confirmed and 1 rejected, and 3 new candidates, of which 2 are confirmed and 1 rejected. This yields an estimated wide tertiary fraction of 19.5{sup +5.2}{sub -3.7}%. This observed fraction is consistent with predictions set out in star formation simulations where the fraction of wide, low-mass companions to spectroscopic binaries is >10%. 16. HIGH RESOLUTION H{alpha} IMAGES OF THE BINARY LOW-MASS PROPLYD LV 1 WITH THE MAGELLAN AO SYSTEM SciTech Connect Wu, Y.-L.; Close, L. M.; Males, J. R.; Follette, K.; Morzinski, K.; Kopon, D.; Rodigas, T. J.; Hinz, P.; Puglisi, A.; Esposito, S.; Pinna, E.; Riccardi, A.; Xompero, M.; Briguglio, R. 2013-09-01 We utilize the new Magellan adaptive optics system (MagAO) to image the binary proplyd LV 1 in the Orion Trapezium at H{alpha}. This is among the first AO results in visible wavelengths. The H{alpha} image clearly shows the ionization fronts, the interproplyd shell, and the cometary tails. Our astrometric measurements find no significant relative motion between components over {approx}18 yr, implying that LV 1 is a low-mass system. We also analyze Large Binocular Telescope AO observations, and find a point source which may be the embedded protostar's photosphere in the continuum. Converting the H magnitudes to mass, we show that the LV 1 binary may consist of one very-low-mass star with a likely brown dwarf secondary, or even plausibly a double brown dwarf. Finally, the magnetopause of the minor proplyd is estimated to have a radius of 110 AU, consistent with the location of the bow shock seen in H{alpha}. 17. REFINED METALLICITY INDICES FOR M DWARFS USING THE SLoWPoKES CATALOG OF WIDE, LOW-MASS BINARIES SciTech Connect Dhital, Saurav; Stassun, Keivan G.; Bastien, Fabienne A.; West, Andrew A.; Massey, Angela P.; Bochanski, John J. 2012-03-15 We report the results from spectroscopic observations of 113 ultra-wide, low-mass binary systems, largely composed of M0-M3 dwarfs, from the SLoWPoKES catalog of common proper motion pairs identified in the Sloan Digital Sky Survey. Radial velocities of each binary member were used to confirm that they are comoving and, consequently, to further validate the high fidelity of the SLoWPoKES catalog. Ten stars appear to be spectroscopic binaries based on broad or split spectral features, supporting previous findings that wide binaries are likely to be hierarchical systems. We measured the H{alpha} equivalent width of the stars in our sample and found that components of 81% of the observed pairs have similar H{alpha} levels. The difference in H{alpha} equivalent width among components with similar masses was smaller than the range of H{alpha} variability for individual objects. We confirm that the Lepine et al. {zeta}-index traces iso-metallicity loci for most of our sample of M dwarfs. However, we find a small systematic bias in {zeta}, especially in the early-type M dwarfs. We use our sample to recalibrate the definition of {zeta}. While representing a small change in the definition, the new {zeta} is a significantly better predictor of iso-metallicity for the higher-mass M dwarfs. 18. Evolutionary and pulsational properties of low-mass white dwarf stars with oxygen cores resulting from close binary evolution Althaus, L. G.; Córsico, A. H.; Gautschy, A.; Han, Z.; Serenelli, A. M.; Panei, J. A. 2004-01-01 The present work is designed to explore the evolutionary and pulsational properties of low-mass white dwarfs with carbon/oxygen cores. In particular, we follow the evolution of a 0.33-Msolar white dwarf remnant in a self-consistent way with the predictions of nuclear burning, element diffusion and the history of the white dwarf progenitor. Attention is focused on the occurrence of hydrogen shell flashes induced by diffusion processes during cooling phases. The evolutionary stages prior to the white dwarf formation are also fully accounted for by computing the conservative binary evolution of an initially 2.5-Msolar Population I star with a 1.25-Msolar companion, and with period Pi= 3 d. Evolution is followed down to the domain of the ZZ Ceti stars on the white dwarf cooling branch. We find that chemical diffusion induces the occurrence of an additional hydrogen thermonuclear flash, which leads to stellar models with thin hydrogen envelopes. As a result, a fast cooling is encountered at advanced stages of evolution. In addition, we explore the adiabatic pulsational properties of the resulting white dwarf models. As compared with their helium-core counterparts, low-mass oxygen-core white dwarfs are characterized by a pulsational spectrum much more featured, an aspect which could eventually be used for distinguishing both types of stars, if low-mass white dwarfs were in fact found to pulsate as ZZ Ceti-type variables. Finally, we perform a non-adiabatic pulsational analysis on the resulting carbon/oxygen low-mass white dwarf models. 19. The distance to the Andromeda galaxy from eclipsing binaries Vilardell, F.; Ribas, I.; Jordi, C.; Fitzpatrick, E. L.; Guinan, E. F. 2010-01-01 The cosmic distance scale largely depends on distance determinations to galaxies of the Local Group. In this sense, the Andromeda galaxy (M 31) is a key rung to better constrain the cosmic distance ladder. A project was started in 1999 to firmly establish a direct and accurate distance to M 31 using eclipsing binaries (EBs). After the determination of the first direct distance to M 31 from EBs, the second direct distance to an EB system is presented: M31V J00443610+4129194. Light and radial velocity curves were obtained and fitted to derive the masses and radii of the components. The acquired spectra were combined and disentangled to determine the temperature of the components. The analysis of the studied EB resulted in a distance determination to M 31 of (m-M)0 = 24.30 ± 0.11 mag. This result, when combined with the previous distance determination to M 31, results in a distance modulus of (m-M)0 = 24.36 ± 0.08 mag (744 ± 33 kpc), fully compatible with other distance determinations to M 31. With an error of only 4%, the obtained value firmly establishes the distance to this important galaxy and represents the fulfillment of the main goal of our project. Based on observations made with the Isaac Newton Telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina)Original data are only available in 20. The Effect of Micro-lensing in Eclipsing Binary-star Systems Hoffman, Kelsey L.; Rowe, J.; Hansen, B. 2013-04-01 Using photometric observations from the Kepler Space Telescope of eclipsing binary star systems where one component is a white dwarf we have investigated the strength of the micro-lensing effect. We have examined the stellar binary KOI-81 from the Kepler mission. KOI-81 is composed of a white dwarf and a A-type main-sequence star in a 24 day circular orbit and have found that micro-lensing is detectable. We use our lightcurve models to measure the strength of the micro-lensing signal and refine the radius of the eclipsing white dwarf. 1. Search with UVES and X-Shooter for signatures of the low-mass secondary in the post common-envelope binary AA Doradus Hoyer, D.; Rauch, T.; Werner, K.; Hauschildt, P. H.; Kruk, J. W. 2015-06-01 Context. AA Dor is a close, totally eclipsing, post common-envelope binary with an sdOB-type primary star and an extremely low-mass secondary star, located close to the mass limit of stable central hydrogen burning. Within error limits, it may either be a brown dwarf or a late M-type dwarf. Aims: We aim to extract the secondary's contribution to the phase-dependent composite spectra. The spectrum and identified lines of the secondary decide on its nature. Methods: In January 2014, we measured the phase-dependent spectrum of AA Dor with X-Shooter over one complete orbital period. Since the secondary's rotation is presumable synchronized with the orbital period, its surface strictly divides into a day and night side. Therefore, we may obtain the spectrum of its cool side during its transit and of its hot, irradiated side close to its occultation. We developed the Virtual Observatory (VO) tool TLISA to search for weak lines of a faint companion in a binary system. We successfully applied it to the observations of AA Dor. Results: We identified 53 spectral lines of the secondary in the ultraviolet-blue, visual, and near-infrared X-Shooter spectra that are strongest close to its occultation. We identified 57 (20 additional) lines in available Ultraviolet and Visual Echelle Spectrograph (UVES) spectra from 2001. The lines are mostly from C ii-iii and O ii, typical for a low-mass star that is irradiated and heated by the primary. We verified the orbital period of P = 22 597.033201 ± 0.00007 s and determined the orbital velocity K_sec = 232.9+16.6-6.5 km s-1 of the secondary. The mass of the secondary is M_sec = 0.081+0.018-0.010 M_⊙ and, hence, it is not possible to reliably determine a brown dwarf or an M-type dwarf nature. Conclusions: Although we identified many emission lines of the secondary's irradiated surface, the resolution and signal-to-noise ratio of our UVES and X-Shooter spectra are not good enough to extract a good spectrum of the secondary 2. Determination of Low-Mass Star Multiplicity. Detection of Star J1158+4239 Binary Nature Kulikova, A. M.; Khovritchev, M. Yu.; Sokov, E. N.; Dyachenko, V. V.; Rastegaev, D. A.; Beskakotov, A. S.; Balega, Yu. Yu.; Safonov, B. S.; Dodin, A. V.; Vozyakova, O. V. 2017-06-01 The search for binaries among low-massive stars is one of the main goals of the Pulkovo program of investigation of stars with large proper motions. The motions of 1308 stars with proper motions larger than 300 mas·yr-1 down to magnitude 17 were studied. While analysing, we used Pulkovo Normal Astrograph observations in 2008-2015 and different digital surveys (DSS, 2MASS, SDSS DR12, and WISE). The main idea of the search for binary stars is reduced to comparing the quasi-mean (POSS2-POSS1; an epoch difference of ≍50 yr) and quasi-instantaneous (2MASS, SDSS, WISE, Pulkovo; an epoch difference of ≍ 10 yr) proper motions. If the difference is statistically significant compared to the proper motion errors, then the object may be considered as a Δμ -binary candidate. As a result, 121 stars have been qualified as astrometric binary candidates. The brightest of them (12 stars) have been included in the program of speckle observations with the BTA (SAO RAS) and the 2.5-m telescope of CMO (SAI MSU). The binarity of the brightest of these stars, J1158+4239 (GJ 3697), has been confirmed. The weighted mean estimates of the pair parameters are ρ=286.5±1.2 mas and θ=230.24±0.16° at the epoch B2015.88248. The magnitude difference between the pair stars is Δ m=0.55±0.03 (the filter with a central wavelength of 800 nm and a FWHM of 100 nm) and Δ m=0.9±0.1 (the R filter). 3. A New, Young, Low-Mass Spectroscopic Binary Without a Home Flagg, Laura S.; Shkolnik, Evgenya L.; Weinberger, Alycia J.; Bowler, Brendan P.; Kraus, Adam L.; Liu, Michael C. 2016-01-01 We have discovered that 2MASS 08355977-3042306 is an accreting K7, double-lined, spectroscopic binary younger than ~20 Myr. The age of a dispersed young star can best be determined if it is a member of a known young moving group. However, the three dimensional space velocities (UVW) we calculate using radial velocity measurements, proper motions, and plausible photometric distances make membership in any known young moving group unlikely. 4. High Resolution Imaging of Very Low Mass Spectral Binaries: Three Resolved Systems and Detection of Orbital Motion in an L/T Transition Binary Bardalez Gagliuffi, Daniella C.; Gelino, Christopher R.; Burgasser, Adam J. 2015-11-01 We present high resolution Laser Guide Star Adaptive Optics imaging of 43 late-M, L and T dwarf systems with Keck/NIRC2. These include 17 spectral binary candidates, systems whose spectra suggest the presence of a T dwarf secondary. We resolve three systems: 2MASS J1341-3052, SDSS J1511+0607 and SDSS J2052-1609 the first two are resolved for the first time. All three have projected separations <8 AU and estimated periods of 14-80 years. We also report a preliminary orbit determination for SDSS J2052-1609 based on six epochs of resolved astrometry between 2005 and 2010. Among the 14 unresolved spectral binaries, 5 systems were confirmed binaries but remained unresolved, implying a minimum binary fraction of {47}-11+12% for this sample. Our inability to resolve most of the spectral binaries, including the confirmed binaries, supports the hypothesis that a large fraction of very low mass systems have relatively small separations and are missed with direct imaging. Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. 5. SpeX spectroscopy of unresolved very low mass binaries. II. Identification of 14 candidate binaries with late-M/early-L and T dwarf components SciTech Connect Bardalez Gagliuffi, Daniella C.; Burgasser, Adam J.; Nicholls, Christine P.; Gelino, Christopher R.; Looper, Dagny L.; Schmidt, Sarah J.; Cruz, Kelle; West, Andrew A.; Gizis, John E.; Metchev, Stanimir 2014-10-20 Multiplicity is a key statistic for understanding the formation of very low mass (VLM) stars and brown dwarfs. Currently, the separation distribution of VLM binaries remains poorly constrained at small separations (≤1 AU), leading to uncertainty in the overall binary fraction. We approach this problem by searching for late-M/early-L plus T dwarf spectral binaries whose combined light spectra exhibit distinct peculiarities, allowing for separation-independent identification. We define a set of spectral indices designed to identify these systems, and we use a spectral template fitting method to confirm and characterize spectral binary candidates from a library of 815 spectra from the SpeX Prism Spectral Libraries. We present 11 new binary candidates, confirm 3 previously reported candidates, and rule out 2 previously identified candidates, all with primary and secondary spectral types in the range M7-L7 and T1-T8, respectively. We find that subdwarfs and blue L dwarfs are the primary contaminants in our sample and propose a method for segregating these sources. If confirmed by follow-up observations, these systems may add to the growing list of tight separation binaries, whose orbital properties may yield further insight into brown dwarf formation scenarios. 6. EXPECTED LARGE SYNOPTIC SURVEY TELESCOPE (LSST) YIELD OF ECLIPSING BINARY STARS SciTech Connect Prsa, Andrej; Pepper, Joshua; Stassun, Keivan G. 2011-08-15 In this paper, we estimate the Large Synoptic Survey Telescope (LSST) yield of eclipsing binary stars, which will survey {approx}20,000 deg{sup 2} of the southern sky during a period of 10 years in six photometric passbands to r {approx} 24.5. We generate a set of 10,000 eclipsing binary light curves sampled to the LSST time cadence across the whole sky, with added noise as a function of apparent magnitude. This set is passed to the analysis-of-variance period finder to assess the recoverability rate for the periods, and the successfully phased light curves are passed to the artificial-intelligence-based pipeline ebai to assess the recoverability rate in terms of the eclipsing binaries' physical and geometric parameters. We find that, out of {approx}24 million eclipsing binaries observed by LSST with a signal-to-noise ratio >10 in mission lifetime, {approx}28% or 6.7 million can be fully characterized by the pipeline. Of those, {approx}25% or 1.7 million will be double-lined binaries, a true treasure trove for stellar astrophysics. 7. Dynamical masses of the low-mass stellar binary AB Doradus B Azulay, R.; Guirado, J. C.; Marcaide, J. M.; Martí-Vidal, I.; Ros, E.; Jauncey, D. L.; Lestrade, J.-F.; Preston, R. A.; Reynolds, J. E.; Tognelli, E.; Ventura, P. 2015-06-01 Context. AB Doradus is the main system of the AB Doradus moving group. It is a quadruple system formed by two widely separated binaries of pre-main-sequence (PMS) stars: AB Dor A/C and AB Dor Ba/Bb. The pair AB Dor A/C has been extensively studied and its dynamical masses have been determined with high precision, thus making AB Dor C a benchmark for calibrating PMS stellar models. If the orbit and dynamical masses of the pair AB Dor Ba/Bb could be determined, they could play a similar role to that of AB Dor C in calibrating PMS models, and would also help to better understand the dynamics of the whole AB Doradus system. Aims: We aim to determine the individual masses of the pair AB Dor Ba/Bb using VLBI observations and archive infrared data as part of a larger program that monitors binary systems in the AB Doradus moving group. Methods: We observed the system AB Dor B between 2007 and 2013 with the Australian Long Baseline Array (LBA) at a frequency of 8.4 GHz in phase-reference mode. Results: We detected, for the first time, compact radio emission from both stars in the binary, AB Dor Ba and AB Dor Bb. This result allowed us to determine the orbital parameters of both the relative and absolute orbits and, consequently, their individual dynamical masses: 0.28 ± 0.05 M⊙ and 0.25 ± 0.05 M⊙, respectively. Conclusions: Comparisons of the dynamical masses with the prediction of PMS evolutionary models show that the models underpredict the dynamical masses of the binary components Ba and Bb by 10-30% and 10-40%, respectively, although they still agree at the 2σ level. Some of the stellar models considered favor an age between 50 and 100 Myr for this system, while others predict older ages. We also discuss the evolutionary status of AB Dor Ba/Bb in terms of an earlier double-double star scenario that might explain the strong radio emission detected in both components. 8. Searches for millisecond pulsations in low-mass X-ray binaries NASA Technical Reports Server (NTRS) Wood, K. S.; Hertz, P.; Norris, J. P.; Vaughan, B. A.; Michelson, P. F.; Mitsuda, K.; Lewin, W. H. G.; Van Paradijs, J.; Penninx, W.; Van Der Klis, M. 1991-01-01 High-sensitivity search techniques for millisecond periods are presented and applied to data from the Japanese satellite Ginga and HEAO 1. The search is optimized for pulsed signals whose period, drift rate, and amplitude conform with what is expected for low-class X-ray binary (LMXB) sources. Consideration is given to how the current understanding of LMXBs guides the search strategy and sets these parameter limits. An optimized one-parameter coherence recovery technique (CRT) developed for recovery of phase coherence is presented. This technique provides a large increase in sensitivity over the method of incoherent summation of Fourier power spectra. The range of spin periods expected from LMXB phenomenology is discussed, the necessary constraints on the application of CRT are described in terms of integration time and orbital parameters, and the residual power unrecovered by the quadratic approximation for realistic cases is estimated. 9. Searches for millisecond pulsations in low-mass X-ray binaries NASA Technical Reports Server (NTRS) Wood, K. S.; Hertz, P.; Norris, J. P.; Vaughan, B. A.; Michelson, P. F.; Mitsuda, K.; Lewin, W. H. G.; Van Paradijs, J.; Penninx, W.; Van Der Klis, M. 1991-01-01 High-sensitivity search techniques for millisecond periods are presented and applied to data from the Japanese satellite Ginga and HEAO 1. The search is optimized for pulsed signals whose period, drift rate, and amplitude conform with what is expected for low-class X-ray binary (LMXB) sources. Consideration is given to how the current understanding of LMXBs guides the search strategy and sets these parameter limits. An optimized one-parameter coherence recovery technique (CRT) developed for recovery of phase coherence is presented. This technique provides a large increase in sensitivity over the method of incoherent summation of Fourier power spectra. The range of spin periods expected from LMXB phenomenology is discussed, the necessary constraints on the application of CRT are described in terms of integration time and orbital parameters, and the residual power unrecovered by the quadratic approximation for realistic cases is estimated. 10. Data Mining Analysis for Eclipsing Binary TrES-Cyg3-04450 Hinzel, D. H. 2015-12-01 A data mining algorithm was utilized to analyze Johnson V-band charge-coupled device (CCD) photometric data of an object that were taken during a wide field survey of a region in the constellation Cygnus. That algorithm was the Date Compensated Discrete Fourier Transform (DC DFT) which is part of the AAVSO VSTAR applications software. This analysis clearly indicated that the object under study is a detached eclipsing binary, specifically an EA β Persei-type (Algol) eclipsing system, with an orbital period of 2.0664 days. Neither the type nor period of this eclipsing binary had been characterized up to this point. This object has been given the AAVSO designation TrES-Cyg3-04450 and the AUID 000-BLL-484. 11. Binaries among low-mass stars in nearby young moving groups Janson, Markus; Durkan, Stephen; Hippler, Stefan; Dai, Xiaolin; Brandner, Wolfgang; Schlieder, Joshua; Bonnefoy, Mickaël; Henning, Thomas 2017-03-01 The solar galactic neighborhood contains a number of young co-moving associations of stars (known as young moving groups) with ages of 10-150 Myr, which are prime targets for a range of scientific studies, including direct imaging planet searches. The late-type stellar populations of such groups still remain in their pre-main sequence phase, and are thus well suited for purposes such as isochronal dating. Close binaries are particularly useful in this regard since they allow for a model-independent dynamical mass determination. Here we present a dedicated effort to identify new close binaries in nearby young moving groups, through high-resolution imaging with the AstraLux Sur Lucky Imaging camera. We surveyed 181 targets, resulting in the detection of 61 companions or candidates, of which 38 are new discoveries. An interesting example of such a case is 2MASS J00302572-6236015 AB, which is a high-probability member of the Tucana-Horologium moving group, and has an estimated orbital period of less than 10 yr. Among the previously known objects is a serendipitous detection of the deuterium burning boundary circumbinary companion 2MASS J01033563-5515561 (AB)b in the z' band, thereby extending the spectral coverage for this object down to near-visible wavelengths. Based on observations collected at the European Southern Observatory, Chile (Programs 096.C-0243 and 097.C-0135).Tables 1-3 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/599/A70 12. Connections between X-ray and optical variability in the low mass X-ray binary 1735-444 NASA Technical Reports Server (NTRS) Corbet, R. H. D.; Smale, A. P.; Charles, P. A.; Lewin, W. H. G.; Menzies, J. W. 1989-01-01 The results of a long duration (4 day) simultaneous optical and X-ray observation of the low mass X-ray binary 1735-444 are presented. The observed X-ray and optical fluxes are correlated; the strength of this correlation is increased when allowance is made for the relatively large orbital modulation of the optical light. A simple interpretation of the optical radiation as reprocessed X-rays in a blackbody disk leads to an implausibly low disk temperature if the disk is assumed to have constant geometry. 1735-444 exhibits bimodal behavior having an X-ray spectral hardness ratio versus source intensity which is similar to that previously seen in sources such as Cyg X-2. 13. The Motif of Globular Clusters and Low Mass X-ray Binaries in Ellipticals: a Tale of Three Galaxies D'Abrusco, Raffaele; Fabbiano, Giuseppina; Mineo, Stefano; Strader, Jay; Fragos, Tassos; Kim, Dong-Woo; Luo, Bin; Zezas, Andreas 2014-06-01 I will discuss significant inhomogeneities in the projected two-dimensional spatial distributions of Globular Clusters and Low Mass X-Ray Binaries observed in three elliptical galaxies with extensive spatial coverage in the optical and X-ray: NGC4261, NGC4649 and NGC4278. The spatial structures in the distributions of GCs and LMXBs have been detected with a new method based on the K-Nearest Neighbor density estimator of Dressler (1980), complemented by MonteCarlo simulations to establish the statistical significance of the results. I will present the spatial structures as a function of the color and luminosity of the GCs, and will compare their shape and significance with the spatial distribution of field LMXBs. I will then examine the nature of these structures in the context of the evolution history of the host galaxies. 14. Asiago eclipsing binaries program. II. V505 Persei Tomasella, L.; Munari, U.; Siviero, A.; Cassisi, S.; Dallaporta, S.; Zwitter, T.; Sordo, R. 2008-03-01 The orbit and fundamental physical parameters of the double-lined eclipsing binary V505 Per are derived by means of Echelle high-resolution and high S/N spectroscopy, and B, V photometry. In addition, effective temperatures, gravities, rotational velocities, and metallicities of both components are also obtained from atmospheric χ2 analysis, showing an excellent match with the results of the orbital solution. An EB-V ≤ 0.01 mag upper limit to the reddening is derived from intensity analysis of interstellar NaI (5890.0 & 5895.9 Å) and KI (7699.0 Å) lines. The distance to the system computed from orbital parameters (60.6 ± 1 pc) is identical to the newly re-reduced Hipparcos parallax (61.5 ± 1.9 pc). The masses of the two components (M1 = 1.2693±0.0011 and M2 = 1.2514±0.0012 M_⊙) place them in the transition region between convective and radiative stellar cores of the HR diagram, with the more massive of the two already showing the effect of evolution within the main sequence band (T1 = 6512±21 K, T2 = 6462±12 K, R1 = 1.287±0.014, R2 = 1.266±0.013 R_⊙). This makes this system particularly relevant to theoretical stellar models as a test of the overshooting. We compare the firm observational results for V505 Per component stars with the predictions of various libraries of theoretical stellar models (BaSTI, Padova, Granada, Yonsei-Yale, Victoria-Regina), as well as with BaSTI models computed specifically for the masses and chemical abundances of V505 Per. We find that the overshooting at the masses of V505 Per component stars is already pretty low, but not nil, and it is described by efficiencies λ_OV = 0.093 and 0.087 for the 1.27 and 1.25 M_⊙ components, respectively. According to the computed BaSTI models, the age of the system is ~0.9 Gyr, and the element diffusion during this time has reduced the surface metallicity from the initial [M/H]= -0.03 to the current [M/H] = -0.13, in excellent agreement with the observed [M/H] = -0.12±0.03. Based 15. Search for A-F Spectral type pulsating components in Algol-type eclipsing binary systems Kim, S.-L.; Lee, J. W.; Kwon, S.-G.; Youn, J.-H.; Mkrtichian, D. E.; Kim, C. 2003-07-01 We present the results of a systematic search for pulsating components in Algol-type eclipsing binary systems. A total number of 14 eclipsing binaries with A-F spectral type primary components were observed for 22 nights. We confirmed small-amplitude oscillating features of a recently detected pulsator TW Dra, which has a pulsating period of 0.053 day and a semi-amplitude of about 5 mmag in B-passband. We discovered new pulsating components in two eclipsing binaries of RX Hya and AB Per. The primary component of RX Hya is pulsating with a dominant period of 0.052 day and a semi-amplitude of about 7 mmag. AB Per has also a pulsating component with a period of 0.196 day and a semi-amplitude of about 10 mmag in B-passband. We suggest that these two new pulsators are members of the newly introduced group of mass-accreting pulsating stars in semi-detached Algol-type eclipsing binary systems. Table 4 is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/405/231 16. HIDES spectroscopy of bright detached eclipsing binaries from the Kepler field - I. Single-lined objects Hełminiak, K. G.; Ukita, N.; Kambe, E.; Kozłowski, S. K.; Sybilski, P.; Ratajczak, M.; Maehara, H.; Konacki, M. 2016-09-01 We present results of our spectroscopic observations of nine detached eclipsing binaries (DEBs), selected from the Kepler Eclipsing Binary Catalog, that only show one set of spectral lines. Radial velocities (RVs) were calculated from the high-resolution spectra obtained with the HIgh-Dispersion Echelle Spectrograph (HIDES) instrument, attached to the 1.88-m telescope at the Okayama Astrophysical Observatory, and from the public Apache Point Observatory Galactic Evolution Experiment archive. In our sample, we found five single-lined binaries, with one component dominating the spectrum. The orbital and light-curve solutions were found for four of them, and compared with isochrones, in order to estimate absolute physical parameters and evolutionary status of the components. For the fifth case, we only update the orbital parameters, and estimate the properties of the unseen star. Two other systems show orbital motion with a period known from the eclipse timing variations (ETVs). For these we obtained parameters of outer orbits, by translating the ETVs to RVs of the centre of mass of the eclipsing binary, and combining with the RVs of the outer star. Of the two remaining ones, one is most likely a blend of a faint background DEB with a bright foreground star, which lines we see in the spectra, and the last case is possibly a quadruple bearing a sub-stellar mass object. Where possible, we compare our results with literature, especially with results from asteroseismology. We also report possible detections of solar-like oscillations in our RVs. 17. VizieR Online Data Catalog: Near-IR spectroscopy of low-mass binaries and brown dwarfs (Mace, 2014) Mace, G. N. 2014-05-01 The mass of a star at formation determines its subsequent evolution and demise. Low-mass stars are the most common products of star formation and their long main-sequence lifetimes cause them to accumulate over time. Star formation also produces many substellar-mass objects known as brown dwarfs, which emerge from their natal molecular clouds and continually cool as they age, pervading the Milky Way. Low-mass stars and brown dwarfs exhibit a wide range of physical characteristics and their abundance make them ideal subjects for testing formation and evolution models. I have examined a pair of pre-main sequence spectroscopic binaries and used radial velocity variations to determine orbital solutions and mass ratios. Additionally, I have employed synthetic spectra to estimate their effective temperatures and place them on theoretical Hertzsprung-Russell diagrams. From this analysis I discuss the formation and evolution of young binary systems and place bounds on absolute masses and radii. I have also studied the late-type T dwarfs revealed by the Wide-field Infrared Survey Explorer (WISE). This includes the exemplar T8 subdwarf Wolf 1130C, which has the lowest inferred metallicity in the literature and spectroscopic traits consistent with old age. Comparison to synthetic spectra implies that the dispersion in near-infrared colors of late-type T dwarfs is a result of age and/or thin sul de clouds. With the updated census of the L, T, and Y dwarfs we can now study specific brown dwarf subpopulations. Finally, I present a number of future studies that would develop our understanding of the physical qualities of T dwarf color outliers and disentangle the tracers of age and atmospheric properties. The thesis is available at: http://www.astro.ucla.edu/~gmace/thesis.html (7 data files). 18. DEEP, LOW-MASS RATIO OVERCONTACT BINARY SYSTEMS. XII. CK BOOTIS WITH POSSIBLE CYCLIC MAGNETIC ACTIVITY AND ADDITIONAL COMPANION SciTech Connect Yang, Y.-G.; Qian, S.-B.; Soonthornthum, B. E-mail: [email protected] 2012-05-15 We present precision CCD photometry, a period study, and a two-color simultaneous Wilson code solution of the short-period contact binary CK Bootis. The asymmetric light curves were modeled by a dark spot on the primary component. The result identifies that CK Boo is an A-type W UMa binary with a high fillout of f = 71.7({+-} 4.4)%. From the O - C curve, it is found that the orbital period changes in a complicated mode, i.e., a long-term increase with two sinusoidal variations. One cyclic oscillation with a period of 10.67({+-} 0.20) yr may result from magnetic activity cycles, which are identified by the variability of Max. I - Max. II. Another sinusoidal variation (i.e., A = 0.0131 days({+-} 0.0009 days) and P{sub 3} = 24.16({+-} 0.64) yr) may be attributed to the light-time effect due to a third body. This kind of additional companion can extract angular momentum from the central binary system. The orbital period secularly increases at a rate of dP/dt = +9.79 ({+-}0.80) Multiplication-Sign 10{sup -8} days yr{sup -1}, which may be interpreted by conservative mass transfer from the secondary to the primary. This kind of deep, low-mass ratio overcontact binaries may evolve into a rapid-rotating single star, only if the contact configuration do not break down at J{sub spin} > (1/3)J{sub orb}. 19. The Reverberation Lag in the Low-mass X-ray Binary H1743-322 De Marco, Barbara; Ponti, Gabriele 2016-07-01 The evolution of the inner accretion flow of a black hole X-ray binary during an outburst is still a matter of active research. X-ray reverberation lags are powerful tools for constraining disk-corona geometry. We present a study of X-ray lags in the black hole transient H1743-322. We compared the results obtained from analysis of all the publicly available XMM-Newton observations. These observations were carried out during two different outbursts that occurred in 2008 and 2014. During all the observations the source was caught in the hard state and at similar luminosities ({L}3-10{keV}/{L}{Edd}˜ 0.004). We detected a soft X-ray lag of ˜60 ms, most likely due to thermal reverberation. We did not detect any significant change of the lag amplitude among the different observations, indicating a similar disk-corona geometry at the same luminosity in the hard state. On the other hand, we observe significant differences between the reverberation lag detected in H1743-322 and in GX 339-4 (at similar luminosities in the hard state), which might indicate variations of the geometry from source to source. 20. Dynamical Formation of Low-mass Merging Black Hole Binaries like GW151226 Chatterjee, Sourav; Rodriguez, Carl L.; Kalogera, Vicky; Rasio, Frederic A. 2017-02-01 Using numerical models for star clusters spanning a wide range in ages and metallicities (Z) we study the masses of binary black holes (BBHs) produced dynamically and merging in the local universe (z ≲ 0.2). After taking into account cosmological constraints on star formation rate and metallicity evolution, which realistically relate merger delay times obtained from models with merger redshifts, we show here for the first time that while old, metal-poor globular clusters can naturally produce merging BBHs with heavier components, as observed in GW150914, lower-mass BBHs like GW151226 are easily formed dynamically in younger, higher-metallicity clusters. More specifically, we show that the mass of GW151226 is well within 1σ of the mass distribution obtained from our models for clusters with Z/Z⊙ ≳ 0.5. Indeed, dynamical formation of a system like GW151226 likely requires a cluster that is younger and has a higher metallicity than typical Galactic globular clusters. The LVT151012 system, if real, could have been created in any cluster with Z/Z⊙ ≲ 0.25. On the other hand, GW150914 is more massive (beyond 1σ) than typical BBHs from even the lowest-metallicity (Z/Z⊙ = 0.005) clusters we consider, but is within 2σ of the intrinsic mass distribution from our cluster models with Z/Z⊙ ≲ 0.05 of course, detection biases also push the observed distributions toward higher masses. 1. Multi-periodic pulsations of a stripped red-giant star in an eclipsing binary system. PubMed Maxted, Pierre F L; Serenelli, Aldo M; Miglio, Andrea; Marsh, Thomas R; Heber, Ulrich; Dhillon, Vikram S; Littlefair, Stuart; Copperwheat, Chris; Smalley, Barry; Breedt, Elmé; Schaffenroth, Veronika 2013-06-27 Low-mass white-dwarf stars are the remnants of disrupted red-giant stars in binary millisecond pulsars and other exotic binary star systems. Some low-mass white dwarfs cool rapidly, whereas others stay bright for millions of years because of stable fusion in thick surface hydrogen layers. This dichotomy is not well understood, so the potential use of low-mass white dwarfs as independent clocks with which to test the spin-down ages of pulsars or as probes of the extreme environments in which low-mass white dwarfs form cannot fully be exploited. Here we report precise mass and radius measurements for the precursor to a low-mass white dwarf. We find that only models in which this disrupted red-giant star has a thick hydrogen envelope can match the strong constraints provided by our data. Very cool low-mass white dwarfs must therefore have lost their thick hydrogen envelopes by irradiation from pulsar companions or by episodes of unstable hydrogen fusion (shell flashes). We also find that this low-mass white-dwarf precursor is a type of pulsating star not hitherto seen. The observed pulsation frequencies are sensitive to internal processes that determine whether this star will undergo shell flashes. 2. ON THE ORIGIN OF THE METALLICITY DEPENDENCE IN DYNAMICALLY FORMED EXTRAGALACTIC LOW-MASS X-RAY BINARIES SciTech Connect Ivanova, N.; Avendano Nandez, J. L.; Sivakoff, G. R.; Fragos, T.; Kim, D.-W.; Fabbiano, G.; Lombardi, J. C.; Voss, R. 2012-12-01 Globular clusters (GCs) effectively produce dynamically formed low-mass X-ray binaries (LMXBs). Observers detect {approx}100 times more LMXBs per stellar mass in GCs compared to stars in the fields of galaxies. Observationally, metal-rich GCs are about three times more likely to contain an X-ray source than their metal-poor counterparts. Recent observations have shown that this ratio holds in extragalactic GCs for all bright X-ray sources with L{sub X} between 2 Multiplication-Sign 10{sup 37} and 5 Multiplication-Sign 10{sup 38} erg s{sup -1}. In this Letter, we propose that the observed metallicity dependence of LMXBs in extragalactic GCs can be explained by the differences in the number densities and average masses of red giants in populations of different metallicities. Red giants serve as seeds for the dynamical production of bright LMXBs via two channels-binary exchanges and physical collisions-and the increase of the number densities and masses of red giants boost LMXB production, leading to the observed difference. We also discuss a possible effect of the age difference in stellar populations of different metallicities. 3. ANALYSIS OF DETACHED ECLIPSING BINARIES NEAR THE TURNOFF OF THE OPEN CLUSTER NGC 7142 SciTech Connect Sandquist, Eric L.; Serio, Andrew W.; Orosz, Jerome; Shetrone, Matthew E-mail: [email protected] E-mail: [email protected] 2013-08-01 We analyze extensive BVR{sub C}I{sub C} photometry and radial velocity measurements for three double-lined deeply eclipsing binary stars in the field of the old open cluster NGC 7142. The short period (P = 1.9096825 days) detached binary V375 Cep is a high probability cluster member, and has a total eclipse of the secondary star. The characteristics of the primary star (M = 1.288 {+-} 0.017 M{sub Sun }) at the cluster turnoff indicate an age of 3.6 Gyr (with a random uncertainty of 0.25 Gyr), consistent with earlier analysis of the color-magnitude diagram. The secondary star (M = 0.871 {+-} 0.008 M{sub Sun }) is not expected to have evolved significantly, but its radius is more than 10% larger than predicted by models. Because this binary system has a known age, it is useful for testing the idea that radius inflation can occur in short period binaries for stars with significant convective envelopes due to the inhibition of energy transport by magnetic fields. The brighter star in the binary also produces a precision estimate of the distance modulus, independent of reddening estimates: (m - M){sub V} = 12.86 {+-} 0.07. The other two eclipsing binary systems are not cluster members, although one of the systems (V2) could only be conclusively ruled out as a present or former member once the stellar characteristics were determined. That binary is within 0. Degree-Sign 5 of edge-on, is in a fairly long-period eccentric binary, and contains two almost indistinguishable stars. The other binary (V1) has a small but nonzero eccentricity (e = 0.038) in spite of having an orbital period under 5 days. 4. Determination of individual temperatures and luminosities in eclipsing binary star systems Campbell, R. M. 1983-06-01 The purpose of this project was to determine the temperatures and luminosities of the individual components of eclipsing binary star systems. The information was gained by UBV photometry of a system at total eclipse and at a time outside eclipse. The light at totality is due entirely to the occulting star, and outside eclipse, both stars contribute fully. A method is derived for subtracting out the light of the occulting star to obtain measurements of the occulted. Systems for which a complete solution (temperature and luminosity of both components) was reached include: TU Camelopardi, TW Draconis, AK Herculis, V566 Ophiuchi, W Ursae Majoris, and AG Virginis. Systems observed only during totality, thus solving only the occulting star, include alpha Corona Borealis and AM Leonis. RS Canes Venatici and TZ Bootes were observed only out of eclipse, and must await further study. Once a solution for a system was obtained, it was presented graphically on a Hertzsprung-Russell diagram, and was examined from the viewpoint of binary evolution. 5. Thermal inertia of eclipsing binary asteroids: the role of component shape Mueller, Michael; van de Weijgaert, Marlies 2015-11-01 Thermal inertia controls the temperature distribution on asteroid surfaces. This is of crucial importance to the Yarkovsky effect and for the planning of spacecraft operations on or near the surface. Additionally, thermal inertia is a sensitive indicator for regolith structure.A uniquely direct way of measuring thermal inertia is through observations of the thermal response to an eclipse in a binary system, when one component shadows the other. This method was pioneered by Mueller et al. (2010), who observed eclipses in (617) Patroclus using Spitzer IRS. Buie et al. (2015) report observations of a stellar occultation by Patroclus. Their estimate for the system's projected size agrees well with the Spitzer result. However, the occultation revealed that the components are much more oblately shaped than was assumed by Mueller et al.This prompted us to study the role of component shape in the analysis of thermal eclipse data. Conceivably, the global shape can have a significant impact on the shape and size of the eclipsed area and therefore on its thermal emission. So far, this has not been studied in a systematic way. Using Patroclus and the existing Spitzer data as our test case, we vary the ellipsoidal component shape and determine the resulting best-fit thermal inertia. This will lead to an updated estimate of Patroclus' thermal inertia, along with a potentially more realistic estimate of its uncertainty. Beyond that, our results will inform ongoing and future thermal studies of other eclipsing binary asteroids. 6. A Long Period Eclipsing Binary Project - Five Years of Observations at ESO Ahlin, P.; Sundman, A. 1982-06-01 The star HO 161387 first caught our eyes when we were reading an article on ~ Aurigae stars by K. O. Wright in Vistas in Astronomy No. 12. This was some 8 or 9 years ago. Aurigae stars are eclipsing binaries formed by a cool supergiant K star and a very much smaller and holter mainsequence (more or less normal) B star. Out of eclipse the B star dominates the blue spectral region, but a pure K-type spectrum is found in eclipse. The drastic spectral changes lor HO 161387 can be seen in Fig 1c and 1d. Periods for these binaries are in the range of 2 to 10 years. The general benefit 01 ~ Aurigae star studies is the possibility of direct determination 01 physical parameters of the components such as masses and radii. In practice, what one does observe is the change in radial velocity of the stars as they orbit around their common centre 01 gravity and the change in magnitude as the light from the B star is eclipsed by the K supergiant. There is also the possibility of studying the structure of the atmosphere of a K supergiant manifested by spectral changes occurring as the point light of the B star shines through the outer parts of the K star c1ose to the total eclipse. Besides Aurigae itsell only the stars 31 and 32 Cygni have been studied in greater detail. 7. Initial Estimates on the Performance of the LSST on the Detection of Eclipsing Binaries Wells, Mark; Prs̆a, Andrej; Jones, Lynne; Yoachim, Peter 2017-06-01 In this work, we quantify the performance of the Large Synoptic Survey Telescope (LSST) on the detection of eclipsing binaries. We use Kepler observed binaries to create a large sample of simulated pseudo-LSST binary light curves. From these light curves, we attempt to recover the known binary signal. The success rate of period recovery from the pseudo-LSST light curves is indicative of LSST's expected performance. Using an off-the-shelf analysis of variance routine, we successfully recover 71% of the targets in our sample. We examine how the binary period impacts recovery success and see that for periods longer than 10 days, the chance of successful binary recovery drops below 50%. 8. An automated search of O'Connell effect for Large Numbers of Eclipsing Binaries Papageorgiou, A.; Kleftogiannis, G.; Christopoulou, P. E. 2013-09-01 The O'Connell effect in eclipsing binary systems (unequally high maxima) has stood for many decades as one of the most perplexing challenges in binary studies. So far, this simple asymmetry has not been convincingly explained, but most theories attribute the effect to dynamic phenomena such as migrating star-spots or swirling circumstellar gas and dust. Nevertheless there has been no clear demonstration of a correlation between the assumptions of any one theory and the morphology of physical parameters of binary systems that exhibit O'Connell effect. We have developed an automated program that characterizes the morphology of light curves by depth of both minima, height of both maxima and curvature outside the eclipses. In terms of programming it is being developed in FORTRAN and PYTHON. This project results from realization of two needs, both related to recent discoveries of large number of contact binaries. Thus the first need is of a simple method to obtain essential parameters for these systems, without the necessity of full light-curve synthesis solution. The second is a statistical one: we would like to extract information from light curves with the use of coefficients that describe the asymmetry in the light curve maxima and the overall shape in the growing observational data of eclipsing binaries (OGLE, ASAS, KEPLER, GAIA). Before applying the automated program several complications must be addressed, as eccentricity, quality of data with many outlying points, limitations to the classification method already applied. 9. A Computational Guide to Physics of Eclipsing Binaries. I. Demonstrations and Perspectives Prša, A.; Zwitter, T. 2005-07-01 PHOEBE (PHysics Of Eclipsing BinariEs) is a modeling package for eclipsing binary stars, built on top of the widely used WD program of Wilson & Devinney. This introductory paper gives an overview of the most important scientific extensions (incorporating observational spectra of eclipsing binaries into the solution-seeking process, extracting individual temperatures from observed color indices, main-sequence constraining, and proper treatment of the reddening), numerical innovations (suggested improvements to WD's differential corrections method, the new Nelder & Mead downhill simplex method), and technical aspects (back-end scripter structure, graphical user interface). While PHOEBE retains 100% WD compatibility, its add-ons are a powerful way to enhance WD by encompassing even more physics and solution reliability. The operability of all these extensions is demonstrated on a synthetic main-sequence test binary; applications to real data will be published in follow-up papers. PHOEBE is released under the GNU General Public License, which guarantees it to be free and open to anyone interested in joining in on future development. 10. Eclipsing binary stars in the Large and Small Magellanic Clouds from the MACHO project: The Sample SciTech Connect Faccioli, L; Alcock, C; Cook, K; Prochter, G; Protopapas, P; Syphers, D 2007-03-29 We present a new sample of 4634 eclipsing binary stars in the Large Magellanic Cloud (LMC), expanding on a previous sample of 611 objects and a new sample of 1509 eclipsing binary stars in the Small Magellanic Cloud (SMC), that were identified in the light curve database of the MACHO project. We perform a cross correlation with the OGLE-II LMC sample, finding 1236 matches. A cross correlation with the OGLE-II SMC sample finds 698 matches. We then compare the LMC subsamples corresponding to center and the periphery of the LMC and find only minor differences between the two populations. These samples are sufficiently large and complete that statistical studies of the binary star populations are possible. 11. HII 2407: AN ECLIPSING BINARY REVEALED BY K2 OBSERVATIONS OF THE PLEIADES SciTech Connect David, Trevor J.; Hillenbrand, Lynne A.; Zhang, Celia; Riddle, Reed L.; Stauffer, John; Rebull, L. M.; Cody, Ann Marie; Conroy, Kyle; Stassun, Keivan G.; Pope, Benjamin; Aigrain, Suzanne; Gillen, Ed; Cameron, Andrew Collier; Barrado, David; Isaacson, Howard; Marcy, Geoffrey W.; Ziegler, Carl; Law, Nicholas M.; Baranec, Christoph 2015-11-20 The star HII 2407 is a member of the relatively young Pleiades star cluster and was previously discovered to be a single-lined spectroscopic binary. It is newly identified here within Kepler/K2 photometric time series data as an eclipsing binary system. Mutual fitting of the radial velocity and photometric data leads to an orbital solution and constraints on fundamental stellar parameters. While the primary has arrived on the main sequence, the secondary is still pre-main sequence and we compare our results for the M/M{sub ⊙} and R/R{sub ⊙} values with stellar evolutionary models. We also demonstrate that the system is likely to be tidally synchronized. Follow-up infrared spectroscopy is likely to reveal the lines of the secondary, allowing for dynamically measured masses and elevating the system to benchmark eclipsing binary status. 12. Doubled-lined eclipsing binary system KIC~2306740 with pulsating component discovered from Kepler space photometry 2015-08-01 We present a detailed study of KIC 2306740, an eccentric double-lined eclipsing binary system with a pulsating component.Archive Kepler satellite data were combined with newly obtained spectroscopic data with 4.2\\,m William Herschel Telescope(WHT). This allowed us to determine rather precise orbital and physical parameters of this long period, slightly eccentric, pulsating binary system. Duplicity effects are extracted from the light curve in order to estimate pulsation frequencies from the residuals.We modelled the detached binary system assuming non-conservative evolution models with the Cambridge STARS(TWIN) code. 13. Phenomenological modeling of the light curves of algol-type eclipsing binary stars Andronov, I. L. 2012-12-01 We propose a special class of functions for mathematical modeling of periodic signals of a special type with a nonuniform distribution of the arguments. This method has been developed for determining the phenomenological characteristics of light curves required for listing in the "General Catalog of Variable Stars" (GCVS) and other data bases. For eclipsing binary stars with smooth light curves (types EB and EW) a trigonometric polynomial of optimal degree in a complete or symmetric form is recommended. For eclipsing binary systems with relatively narrow minima, approximating the light curves by a class of nonpolynomial spline functions is statistically optimal. A combination of a second order trigonometric polynomial (TP2, which describes "reflection", ellipsoidal" and "spotting" effects) and localized contributions of the minima (parametrized with respect to depth and profile separately for the primary and secondary minima) is used. This approach is characterized by a statistical accuracy of the smoothing curve that is a factor of ~1.5-2 times better than for a trigonometric polynomial of statistically optimal degree, and by the absence of false "waves" in the light curve associated with the Gibbs effect. Besides finding the width of the minimum, which cannot be determined using a trigonometric polynomial approximation, this method can be used to determine its depth with better accuracy, and to separate the effects of the eclipse and the part outside the eclipse. For multicolor observations, the improved accuracy of the smoothing curve for each filter makes it possible to obtain more accurate plots of the variation in the color index. The efficiency of the proposed method increases as the width of the eclipse becomes smaller. This method supplements the trigonometric polynomial approximation. The method, referred to as the NAV (New Algol Variable) method, is illustrated by applying it to the eclipsing binary systems VSX J022427.8-104034=USNO-B1.0 0793-0023471 and 14. A precision study of two eclipsing white dwarf plus M dwarf binaries Parsons, S. G.; Marsh, T. R.; Gänsicke, B. T.; Rebassa-Mansergas, A.; Dhillon, V. S.; Littlefair, S. P.; Copperwheat, C. M.; Hickman, R. D. G.; Burleigh, M. R.; Kerry, P.; Koester, D.; Nebot Gómez-Morán, A.; Pyrzas, S.; Savoury, C. D. J.; Schreiber, M. R.; Schmidtobreick, L.; Schwope, A. D.; Steele, P. R.; Tappert, C. 2012-03-01 We use a combination of X-shooter spectroscopy, ULTRACAM high-speed photometry and SOFI near-infrared photometry to measure the masses and radii of both components of the eclipsing post common envelope binaries SDSS J121258.25-012310.1 and GK Vir. For both systems, we measure the gravitational redshift of the white dwarf (WD) and combine it with light-curve model fits to determine the inclinations, masses and radii. For SDSS J1212-0123, we find an inclination of i= 85?7 ± 0?5, masses of MWD= 0.439 ± 0.002 M⊙ and Msec= 0.273 ± 0.002 M⊙, and radii RWD= 0.0168 ± 0.0003 R⊙ and Rsec= 0.306 ± 0.007 R⊙. For GK Vir, we find an inclination of i= 89?5°± 0?6, masses of MWD= 0.564 ± 0.014 M⊙ and Msec= 0.116 ± 0.003 M⊙ and radii RWD= 0.0170 ± 0.0004 R⊙ and Rsec= 0.155 ± 0.003 R⊙. The mass and radius of the WD in GK Vir are consistent with evolutionary models for a 50 000 K carbon-oxygen (CO) core WD. Although the mass and radius of the WD in SDSS J1212-0123 are consistent with CO core models, evolutionary models imply that a WD with such a low mass and in a short period binary must have a helium core. The mass and radius measurements are consistent with helium core models but only if the WD has a very thin hydrogen envelope (MH/MWD≤ 10-6). Such a thin envelope has not been predicted by any evolutionary models. The mass and radius of the secondary star in GK Vir are consistent with evolutionary models after correcting for the effects of irradiation by the WD. The secondary star in SDSS J1212-0123 has a radius ˜9 per cent larger than predicted. 15. A Time Series of BV photometry and Halpha Emission Fluxes of the Eclipsing Binary VV Cep Pollmann, E.; Vollmann, W.; Bennett, P. D. 2017-02-01 VV Cephei (= HR 8383 = HD 208816) is the brightest eclipsing M supergiant binary (M2 Iab + B0-2? V) in the sky, and is a massive binary with one of the longest known orbital periods (7430 days = 20.34 years) of any eclipsing system. With the next eclipse beginning in August 2017, and lasting nearly two years (650 days) from 1st to 4th contact, an extensive observational campaign is planned. This paper focusses on the photometric variability of the system out of eclipse, and presents and photometry and H emission fluxes observed over the period from mid-2008 through late 2016. The and light curves show correlated variability with peak power of 145 days, presumably due to low-amplitude pulsations of the M supergiant. The H emission fluxes show a short-term variability, but the sampling is not sufficient to permit quantitative analysis. However, the H fluxes also appear to show a long-term modulation related to the variable orbital separation of this eccentric binary. 16. Erratum: Some Constraints on the Effect of Age and Metallicity on the Low-Mass X-Ray Binary Formation Rate'' (ApJ, 589, L81 [2003]) Kundu, Arunav; Maccarone, Thomas J.; Zepf, Stephen E.; Puzia, Thomas H. 2004-01-01 The number of low-mass X-ray binaries in NGC 4365 that are within 0.5" of a globular cluster and considered to be matches is 18, not 23 as mistakenly reported in the first paragraph of § 3. The correct number is used elsewhere in the above Letter. The fraction of low-mass X-ray binaries that are associated with globular clusters in NGC 4365 is indeed 40% as noted later in the same paragraph. We thank W. Lewin for alerting us to this error. 17. Observations of candidate oscillating eclipsing binaries and two newly discovered pulsating variables Liakos, A.; Niarchos, P. 2009-03-01 CCD observations of 24 eclipsing binary systems with spectral types ranging between A0-F0, candidate for containing pulsating components, were obtained. Appropriate exposure times in one or more photometric filters were used so that short-periodic pulsations could be detected. Their light curves were analyzed using the Period04 software in order to search for pulsational behaviour. Two new variable stars, namely GSC 2673-1583 and GSC 3641-0359, were discov- ered as by-product during the observations of eclipsing variables. The Fourier analysis of the observations of each star, the dominant pulsation frequencies and the derived frequency spectra are also presented. 18. Further Constraints on Variations in the Initial Mass Function from Low-mass X-ray Binary Populations Peacock, Mark B.; Zepf, Stephen E.; Kundu, Arunav; Maccarone, Thomas J.; Lehmer, Bret D.; Maraston, Claudia; Gonzalez, Anthony H.; Eufrasio, Rafael T.; Coulter, David A. 2017-05-01 We present constraints on variations in the initial mass function (IMF) of nine local early-type galaxies based on their low-mass X-ray binary (LMXB) populations. Comprised of accreting black holes and neutron stars, these LMXBs can be used to constrain the important high-mass end of the IMF. We consider LMXB populations beyond the cores of the galaxies (>0.2R e; covering 75%-90% of their stellar light) and find no evidence for systematic variations of the IMF with velocity dispersion (σ). We reject IMFs which become increasingly bottom-heavy with σ, up to steep power laws (exponent, α > 2.8) in massive galaxies (σ > 300 {km} {{{s}}}-1), for galactocentric radii >1/4 R e. Previously proposed IMFs that become increasingly bottom-heavy with σ are consistent with these data if only the number of low-mass stars (<0.5 M ⊙) varies. We note that our results are consistent with some recent work which proposes that extreme IMFs are only present in the central regions of these galaxies. We also consider IMFs that become increasingly top-heavy with σ, resulting in significantly more LMXBs. Such a model is consistent with these observations, but additional data are required to significantly distinguish between this and an invariant IMF. For six of these galaxies, we directly compare with published “IMF mismatch” parameters from the Atlas3D survey, α dyn. We find good agreement with the LMXB population if galaxies with higher α dyn have more top-heavy IMFs—although we caution that our sample is quite small. Future LMXB observations can provide further insights into the origin of α dyn variations. 19. Spectral-Timing Analysis of Kilohetrz Quasi-Periodic Osciallations in Neutron Star Low-Mass X-ray Binaries Troyer, Jon; Peille, Philippe; Cackett, Edward; Barret, Didier 2017-08-01 Kilohertz quasi-periodic oscillations or kHz QPOs are intensity variations that occur in the X-ray band observed in neutron star low-mass X-ray binary (LMXB) systems. In such systems, matter is transferred from a secondary low-mass star to a neutron star via the process of accretion. kHz QPOs occur on the timescale of the inner accretion flow and may carry signatures of the physics of strong gravity (c2 ~ GM/R) and possibly clues to constraining the neutron star equation of state (EOS). Both the timing behavior of kHz QPOs and the time-averaged spectra of these systems have been studied extensively. No model derived from these techniques has been able to illuminate the origin of kHz QPOs. Spectral-timing is an analysis technique that can be used to derive information about the nature of physical processes occurring within the accretion flow on the timescale of the kHz QPO. To date, kHz QPOs of (4) neutron star LMXB systems have been studied with spectral-timing techniques. We present a comprehensive study of spectral-timing products of kHz QPOs from systems where data is available in the RXTE archive to demonstrate the promise of this technique to gain insights regarding the origin of kHz QPOs. Specifically, we show correlated time-lags as a function of QPO frequency and energy for the various LMXB systems where kHz QPOs are detected. 20. The EBAI Project: Neural Network/Artificial Intelligence Approaches to Solve Automatically Large Numbers of Light Curves of Eclipsing Binaries Guinan, E. F.; Prša, A.; Devinney, E. J.; Engle, S. G. 2009-08-01 Major advances in observing technology promise to greatly increase discovery rates of eclipsing binaries (EBs). For example, missions such as the Large Synoptic Survey Telescope (LSST), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) and Gaia are expected to yield hundreds of thousands (even millions) of new variable stars and eclipsing binaries. Current personal interactive (and time consuming) methods of determining the physical and orbital parameters of eclipsing binaries from the current practice of analyzing their light curves will be inadequate to keep up with the overwhelming influx of new data. At present, the currently used methods require significant technical skill and experience; it typically takes 2-3 weeks to model a single binary. We are therefore developing an Artificial Intelligence / Neural Network system with the hope of creating a fully automated, high throughput process for gleaning the orbital and physical properties of EB systems from the observations of tens of thousands of eclipsing binaries at a time. The EBAI project -- Eclipsing Binaries with Artificial Intelligence -- aims to provide estimates of principal parameters for thousands of eclipsing binaries in a matter of seconds. Initial tests of the neural network's performance and reliability have been conducted and are presented here. 1. THE ANTICORRELATED NATURE OF THE PRIMARY AND SECONDARY ECLIPSE TIMING VARIATIONS FOR THE KEPLER CONTACT BINARIES SciTech Connect Tran, K.; Rappaport, S.; Levine, A.; Borkovits, T.; Csizmadia, Sz.; Kalomeni, B. E-mail: [email protected] E-mail: [email protected] 2013-09-01 We report a study of the eclipse timing variations in contact binary systems, using long-cadence lightcurves from the Kepler archive. As a first step, observed minus calculated (O - C) curves were produced for both the primary and secondary eclipses of some 2000 Kepler binaries. We find {approx}390 short-period binaries with O - C curves that exhibit (1) random walk-like variations or quasi-periodicities, with typical amplitudes of {+-}200-300 s, and (2) anticorrelations between the primary and secondary eclipse timing variations. We present a detailed analysis and results for 32 of these binaries with orbital periods in the range of 0.35 {+-} 0.05 days. The anticorrelations observed in their O - C curves cannot be explained by a model involving mass transfer, which, among other things, requires implausibly high rates of {approx}0.01 M{sub Sun} yr{sup -1}. We show that the anticorrelated behavior, the amplitude of the O - C delays, and the overall random walk-like behavior can be explained by the presence of a starspot that is continuously visible around the orbit and slowly changes its longitude on timescales of weeks to months. The quasi-periods of {approx}50-200 days observed in the O - C curves suggest values for k, the coefficient of the latitude dependence of the stellar differential rotation, of {approx}0.003-0.013. 2. USNO-A2.0 1200-1153830 is a binary star with a total eclipse with sharp transitions Roy, Rene; Behrend, Raoul 2017-02-01 Based on their photometric observations, R. Roy (Blauvac, France) and R. Behrend (Geneva Observatory) found that USNO-A2.0 1200-1153830 is a binary star for which the lightcurve is characterized by a 0.4mag total eclipse and a rather soft secondary eclipse. 3. Undergraduate Research to Obtain Preliminary Solutions for New Eclipsing Binary Systems Wetterer, Charles J.; Walker, A. C.; Izzo, D. M.; Bloomer, R. H. 2009-01-01 An ongoing research program using the 0.61-m telescope at the US Air Force Academy (AFA) Observatory strives to identify, conduct multi-filter photometry, and obtain preliminary model solutions to new eclipsing binary systems. The new candidate systems currently come from the recent list of over 1800 suspected variable stars in the original CCD/Transit Instrument (CTI-I) survey. The undergraduate students involved in the program are AFA cadets in academic research courses and visiting students from the Appalachian College Association's Consortium for Astronomy Research and Teaching (CART) who conduct observations over the summer at the AFA Observatory and continue collaborative interactions with AFA cadets during the following academic year. The goal is to increase the number of known eclipsing variable star systems and identify interesting systems for follow-up research. This is an ideal small telescope research program in which to involve undergraduate students. Hundreds of potential eclipsing systems remaining to be explored. 4. WOCS 40007: A DETACHED ECLIPSING BINARY NEAR THE TURNOFF OF THE OPEN CLUSTER NGC 6819 SciTech Connect Jeffries, Mark W. Jr.; Sandquist, Eric L.; Orosz, Jerome A.; Brewer, Lauren N. E-mail: [email protected] E-mail: [email protected]; and others 2013-09-15 We analyze extensive BVR{sub c}I{sub c} time-series photometry and radial-velocity measurements for WOCS 40007 (Auner 259; KIC 5113053), a double-lined detached eclipsing binary and a member of the open cluster NGC 6819. Utilizing photometric observations from the 1 m telescope at Mount Laguna Observatory and spectra from the WIYN 3.5 m telescope, we measure precise and accurate masses ({approx}1.6% uncertainty) and radii ({approx}0.5%) for the binary components. In addition, we discover a third star orbiting the binary with a period greater than 3000 days using radial velocities and Kepler eclipse timings. Because the stars in the eclipsing binary are near the cluster turnoff, they are evolving rapidly in size and are sensitive to age. With a metallicity of [Fe/H] = +0.09 {+-} 0.03, we find the age of NGC 6819 to be about 2.4 Gyr from color-magnitude diagram (CMD) isochrone fitting and 3.1 {+-} 0.4 Gyr by analyzing the mass-radius (M-R) data for this binary. The M-R age is above previous determinations for this cluster, but consistent within 1{sigma} uncertainties. When the M-R data for the primary star of the additional cluster binary WOCS 23009 is included, the weighted age estimate drops to 2.5 {+-} 0.2 Gyr, with a systematic uncertainty of at least 0.2 Gyr. The age difference between our CMD and M-R findings may be the result of systematic error in the metallicity or helium abundance used in models, or due to slight radius inflation of one or both stars in the WOCS 40007 binary. 5. Adaptive Optics imaging of VHS 1256-1257: A Low Mass Companion to a Brown Dwarf Binary System Stone, Jordan M.; Skemer, Andrew J.; Kratter, Kaitlin M.; Dupuy, Trent J.; Close, Laird M.; Eisner, Josh A.; Fortney, Jonathan J.; Hinz, Philip M.; Males, Jared R.; Morley, Caroline V.; Morzinski, Katie M.; Ward-Duong, Kimberly 2016-02-01 Recently, Gauza et al. reported the discovery of a companion to the late M-dwarf, VHS J125601.92-125723.9 (VHS 1256-1257). The companion’s absolute photometry suggests its mass and atmosphere are similar to the HR 8799 planets. However, as a wide companion to a late-type star, it is more accessible to spectroscopic characterization. We discovered that the primary of this system is an equal-magnitude binary. For an age ˜300 Myr the A and B components each have a mass of {64.6}-2.0+0.8 {M}{Jup}, and the b component has a mass of {11.2}-1.8+9.7, making VHS 1256-1257 only the third brown dwarf triple system. There exists some tension between the spectrophotometric distance of 17.2 ± 2.6 pc and the parallax distance of 12.7 ± 1.0 pc. At 12.7 pc VHS 1256-1257 A and B would be the faintest known M7.5 objects, and are even faint outliers among M8 types. If the larger spectrophotmetric distance is more accurate than the parallax, then the mass of each component increases. In particular, the mass of the b component increases well above the deuterium burning limit to ˜ 35 {M}{Jup} and the mass of each binary component increases to {73}-17+20 {M}{Jup}. At 17.1 pc, the UVW kinematics of the system are consistent with membership in the AB Dor moving group. The architecture of the system resembles a hierarchical stellar multiple suggesting it formed via an extension of the star formation process to low masses. Continued astrometric monitoring will resolve this distance uncertainty and will provide dynamical masses for a new benchmark system. 6. COMMON PATTERNS IN THE EVOLUTION BETWEEN THE LUMINOUS NEUTRON STAR LOW-MASS X-RAY BINARY SUBCLASSES SciTech Connect Fridriksson, Joel K.; Homan, Jeroen; Remillard, Ronald A. 2015-08-10 The X-ray transient XTE J1701–462 was the first source observed to evolve through all known subclasses of low-magnetic-field neutron star low-mass X-ray binaries (NS-LMXBs), as a result of large changes in its mass accretion rate. To investigate to what extent similar evolution is seen in other NS-LMXBs we have performed a detailed study of the color–color and hardness–intensity diagrams (CDs and HIDs) of Cyg X-2, Cir X-1, and GX 13+1—three luminous X-ray binaries, containing weakly magnetized neutron stars, known to exhibit strong secular changes in their CD/HID tracks. Using the full set of Rossi X-ray Timing Explorer Proportional Counter Array data collected for the sources over the 16 year duration of the mission, we show that Cyg X-2 and Cir X-1 display CD/HID evolution with close similarities to XTE J1701–462. Although GX 13+1 shows behavior that is in some ways unique, it also exhibits similarities to XTE J1701–462, and we conclude that its overall CD/HID properties strongly indicate that it should be classified as a Z source, rather than as an atoll source. We conjecture that the secular evolution of Cyg X-2, Cir X-1, and GX 13+1—illustrated by sequences of CD/HID tracks we construct—arises from changes in the mass accretion rate. Our results strengthen previous suggestions that within single sources Cyg-like Z source behavior takes place at higher luminosities and mass accretion rates than Sco-like Z behavior, and lend support to the notion that the mass accretion rate is the primary physical parameter distinguishing the various NS-LMXB subclasses. 7. The eclipsing, double-lined, Of supergiant binary Cygnus OB2-B17 Stroud, V. E.; Clark, J. S.; Negueruela, I.; Roche, P.; Norton, A. J.; Vilardell, F. 2010-02-01 Context. Massive, eclipsing, double-lined, spectroscopic binaries are not common but are necessary to understand the evolution of massive stars as they are the only direct way to determine stellar masses. They are also the progenitors of energetic phenomena such as X-ray binaries and γ-ray bursts. Aims: We present a photometric and spectroscopic analysis of the candidate binary system Cyg OB2-B17 to show that it is indeed a massive evolved binary. Methods: We utilise V band and white-light photometry to obtain a light curve and period of the system, and spectra at different resolutions to calculate preliminary orbital parameters and spectral classes for the components. Results: Our results suggest that B17 is an eclipsing, double-lined, spectroscopic binary with a period of 4.0217±0.0004 days, with two massive evolved components with preliminary classifications of O7 and O9 supergiants. The radial velocity and light curves are consistent with a massive binary containing components with similar luminosities, and in turn with the preliminary spectral types and age of the association. 8. Radio luminosity upper limits of the transient neutron star low-mass X-ray binary GRO J1744-28 Russell, Thomas; Degenaar, Nathalie; Miller-Jones, James; Tudor, Vlad 2017-02-01 Following the new outburst of the Galactic neutron star low-mass X-ray binary and 2.1 Hz X-ray pulsar GRO J1744-28 (ATels #10073, #10079), we performed target of opportunity observations of this source with the Australia Telescope Compact Array (ATCA). 9. The magnetohydrodynamical model of kilohertz quasi-periodic oscillations in neutron star low-mass X-ray binaries (II) SciTech Connect Shi, Chang-Sheng; Zhang, Shuang-Nan; Li, Xiang-Dong 2014-08-10 We study the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low-mass X-ray binaries (LMXBs) with a new magnetohydrodynamics (MHD) model, in which the compressed magnetosphere is considered. The previous MHD model is reexamined and the relation between the frequencies of the kHz QPOs and the accretion rate in LMXBs is obtained. Our result agrees with the observations of six sources (4U 0614+09, 4U 1636-53, 4U 1608-52, 4U 1915-15, 4U 1728-34, and XTE 1807-294) with measured spins. In this model, the kHz QPOs originate from the MHD waves in the compressed magnetosphere. The single kHz QPOs and twin kHz QPOs are produced in two different parts of the accretion disk and the boundary is close to the corotation radius. The lower QPO frequency in a frequency-accretion rate diagram is cut off at a low accretion rate and the twin kHz QPOs encounter a top ceiling at a high accretion rate due to the restriction of the innermost stable circular orbit. 10. The Discovery of a Second Luminous Low Mass X-Ray Binary System in the Globular Cluster M15 NASA Technical Reports Server (NTRS) White, Nicholas E.; Angelini, Lorella 2001-01-01 Using the Chandra X-ray Observatory we have discovered a second bright X-ray source in the globular cluster M15 that is 2.7" to the west of AC211, the previously known low mass X-ray binary (LMXB) in this system. Prior to the 0.5" imaging capability of Chandra this second source could not have been resolved from AC211. The luminosity and spectrum of this new source, which we call M15-X2, are consistent with it also being a LMXB system. This is the first time that two LMXBs have been seen to be simultaneously active in a globular cluster. The new source, M15-X2, is coincident with a 18th U magnitude very blue star. The discovery of a second LMXB in M15 clears up a long standing puzzle where the X-ray and optical properties of AC211 appear consistent with the central source being hidden behind an accretion disk corona, and yet also showed a luminous X-ray burst suggesting the neutron star is directly visible. This discovery suggests instead that the X-ray burst did not come from AC211, but rather from the newly discovered X-ray source. We discuss the implications of this discovery for X-ray observations of globular clusters in nearby galaxies. 11. Neutron star crustal plate tectonics. I. Magnetic dipole evolution in millisecond pulsars and low-mass X-ray binaries SciTech Connect Ruderman, M. ) 1991-01-01 Crust lattices in spinning-up or spinning-down neutron stars have growing shear stresses caused by neutron superfluid vortex lines pinned to lattice nuclei. For the most rapidly spinning stars, this stress will break and move the crust before vortex unpinning occurs. In spinning-down neutron stars, crustal plates will move an equatorial subduction zone in which the plates are forced into the stellar core below the crust. The opposite plate motion occurs in spinning-up stars. Magnetic fields which pass through the crust or have sources in it move with the crust. Spun-up neutron stars in accreting low-mass X-ray binaries LMXBs should then have almost axially symmetric magnetic fields. Spun-down ones with very weak magnetic fields should have external magnetic fields which enter and leave the neutron star surface only near its equator. The lowest field millisecond radiopulsars seem to be orthogonal rotators implying that they have not previously been spun-up in LMXBs but are neutron stars initially formed with periods near 0.001 s that subsequently spin down to their present periods. Accretion-induced white dwarf collapse is then the most plausible genesis for them. 29 refs. 12. The Magnetohydrodynamical Model of Kilohertz Quasi-periodic Oscillations in Neutron Star Low-mass X-Ray Binaries (II) Shi, Chang-Sheng; Zhang, Shuang-Nan; Li, Xiang-Dong 2014-08-01 We study the kilohertz quasi-periodic oscillations (kHz QPOs) in neutron star low-mass X-ray binaries (LMXBs) with a new magnetohydrodynamics (MHD) model, in which the compressed magnetosphere is considered. The previous MHD model is reexamined and the relation between the frequencies of the kHz QPOs and the accretion rate in LMXBs is obtained. Our result agrees with the observations of six sources (4U 0614+09, 4U 1636-53, 4U 1608-52, 4U 1915-15, 4U 1728-34, and XTE 1807-294) with measured spins. In this model, the kHz QPOs originate from the MHD waves in the compressed magnetosphere. The single kHz QPOs and twin kHz QPOs are produced in two different parts of the accretion disk and the boundary is close to the corotation radius. The lower QPO frequency in a frequency-accretion rate diagram is cut off at a low accretion rate and the twin kHz QPOs encounter a top ceiling at a high accretion rate due to the restriction of the innermost stable circular orbit. 13. A unified model of accretion flows and X ray emission in low mass X ray binary systems NASA Technical Reports Server (NTRS) Lamb, F. K. 1989-01-01 Recent work on a unified model of accretion flows and X-ray emission in low mass X-ray binaries is summarized. In this model, a weakly magnetic neutron star accretes gas simultaneously from a Keplerian disk and a corona above the inner part of the disk. Photons are produced and escape through an approximately radial inflow of gas captured from the inner disk corona. Changes in the optical depths of the central corona and the radial flow may explain the Z-shaped hardness-intensity and color-color tracks observed in the most luminous sources. Numerical simulations show that the radial flow oscillates when the luminosity rises to within a few percent of the Eddington critical luminosity L sub E, and that the oscillation frequency is approximately 5 to 10 Hz if the radial flow develops approximately 300 km from the neutron star. The 10 to 20 Hz oscillations observed in Sco X-1 when it is on the flaring branch are discussed. 14. A COMPARISON OF BROAD IRON EMISSION LINES IN ARCHIVAL DATA OF NEUTRON STAR LOW-MASS X-RAY BINARIES SciTech Connect Cackett, Edward M.; Miller, Jon M.; Reis, Rubens C.; Fabian, Andrew C.; Barret, Didier 2012-08-10 Relativistic X-ray disklines have been found in multiple neutron star low-mass X-ray binaries, in close analogy with black holes across the mass scale. These lines have tremendous diagnostic power and have been used to constrain stellar radii and magnetic fields, often finding values that are consistent with independent timing techniques. Here, we compare CCD-based data from Suzaku with Fe K line profiles from archival data taken with gas-based spectrometers. In general, we find good consistency between the gas-based line profiles from EXOSAT, BeppoSAX, and RXTE and the CCD data from Suzaku, demonstrating that the broad profiles seen are intrinsic to the line and not broad due to instrumental issues. However, we do find that when fitting with a Gaussian line profile, the width of the Gaussian can depend on the continuum model in instruments with low spectral resolution, though when the different models fit equally well the line widths generally agree. We also demonstrate that three BeppoSAX observations show evidence for asymmetric lines, with a relativistic diskline model providing a significantly better fit than a Gaussian. We test this by using the posterior predictive p-value method, and bootstrapping of the spectra to show that such deviations from a Gaussian are unlikely to be observed by chance. 15. Jet quenching in the neutron star low-mass X-ray binary 1RXS J180408.9-342058 Gusinskaia, N. V.; Deller, A. T.; Hessels, J. W. T.; Degenaar, N.; Miller-Jones, J. C. A.; Wijnands, R.; Parikh, A. S.; Russell, T. D.; Altamirano, D. 2017-09-01 We present quasi-simultaneous radio (VLA) and X-ray (Swift) observations of the neutron star low-mass X-ray binary (NS-LMXB) 1RXS J180408.9-342058 (J1804) during its 2015 outburst. We found that the radio jet of J1804 was bright (232 ± 4 μJy at 10 GHz) during the initial hard X-ray state, before being quenched by more than an order of magnitude during the soft X-ray state (19 ± 4 μJy). The source then was undetected in radio (<13 μJy) as it faded to quiescence. In NS-LMXBs, possible jet quenching has been observed in only three sources and the J1804 jet quenching we show here is the deepest and clearest example to date. Radio observations when the source was fading towards quiescence (LX = 1034-35 erg s-1) show that J1804 must follow a steep track in the radio/X-ray luminosity plane with β > 0.7 (where L_R ∝ L_X^{β }). Few other sources have been studied in this faint regime, but a steep track is inconsistent with the suggested behaviour for the recently identified class of transitional millisecond pulsars. J1804 also shows fainter radio emission at LX < 1035 erg s-1 than what is typically observed for accreting millisecond pulsars. This suggests that J1804 is likely not an accreting X-ray or transitional millisecond pulsar. 16. CONTINUED COOLING OF THE CRUST IN THE NEUTRON STAR LOW-MASS X-RAY BINARY KS 1731-260 SciTech Connect Cackett, Edward M.; Miller, Jon M.; Brown, Edward F.; Cumming, Andrew; Degenaar, Nathalie; Wijnands, Rudy 2010-10-20 Some neutron star low-mass X-ray binaries have very long outbursts (lasting several years) which can generate a significant amount of heat in the neutron star crust. After the system has returned to quiescence, the crust then thermally relaxes. This provides a rare opportunity to study the thermal properties of neutron star crusts, putting constraints on the thermal conductivity and hence the structure and composition of the crust. KS 1731-260 is one of only four systems where this crustal cooling has been observed. Here, we present a new Chandra observation of this source approximately eight years after the end of the last outburst and four years since the last observation. We find that the source has continued to cool, with the cooling curve displaying a simple power-law decay. This suggests that the crust has not fully thermally relaxed yet and may continue to cool further. A simple power-law decay is in contrast to theoretical cooling models of the crust, which predict that the crust should now have cooled to the same temperature as the neutron star core. 17. AbsParEB and InPeVEB: Software for the Calculation of Absolute and Orbital Period Changes Parameters of Eclipsing Binaries Liakos, A. 2015-07-01 The software ABSPAREB (Absolute Parameters of Eclipsing Binaries) calculates the absolute parameters and their formal errors for three different modes: a) double-lined spectroscopic eclipsing binary, b) single-lined spectroscopic eclipsing binary, and c) for an eclipsing binary for which there is no spectroscopic information. In addition, the positions of the binary's members on the mass-radius and color-magnitude diagrams can be also plotted. INPEVEB (Interpretation of Period Variations of Eclipsing Binaries) calculates the parameters as well as their formal errors for several orbital period modulating mechanisms in eclipsing binaries (i.e., LITE, the Applegate mechanism, mass transfer/loss, apsidal motion, magnetic braking) using from an analysis of their O-C diagrams. Both programs are available online (free of charge) in Graphical User Interface form and were written in PYTHON. 18. Phenomenological Parameters of the Prototype Eclipsing Binaries Algol, β Lyrae and W UMa Tkachenko, Mariia G.; Andronov, Ivan L.; Chinarova, Lidia L. 2016-12-01 The phenomenological parameters of eclipsing binary stars, which are the prototypes of the EA, EB and EW systems are determined using the expert complex of computer programs, which realizes the NAV ("New Algol Variable") algorithm (Andronov 2010, 2012) and its possible modifications are discussed, as well as constrains for estimates of some physical parameters of the systems in a case of photometric observations only, such as the degree of eclipse, ratio of the mean surface brightnesses of the components. The half-duration of the eclipse is 0.0617(7), 0.1092(18) and 0.1015(7) for Algol, β Lyrae and W UMa, respectively. The brightness ratio is 6.8±1.0, 4.9±1.0 and 1.15±0.13. These results show that the eclipses have distinct begin and end not only in EA (as generally assumed), but also in EB and EW - type systems as well. The algorithm may be applied to classification and study of the newly discovered (or poorly studied) eclipsing variables based on own observations or that obtained using photometric surveys. 19. New Developments in Eclipsing Binary Light Curve Modeling Milone, E. F.; Stagg, C. R. 1994-03-01 The light curve modeling of binary stars has continued to evolve since its founding by Henry Norris Russell (see Russell and Merrill 1952 and citations therein) nearly a century ago, accelerated in the 1950s by Kopal's introduction of Roche geometry into models and by the development of synthetic light curve computer code in the 1970's. Improved physics and the use of more kinds of observational input are providing another round of important advances that promise to enlarge our knowledge of both binary stars and ensembles containing them. Here we discuss the newer horizons of light curve modeling and the steps being taken toward them. 20. Orbital Parameters of the Eclipsing Detached Kepler Binaries with Eccentric Orbits Kjurkchieva, Diana; Vasileva, Doroteya; Atanasova, Teodora 2017-09-01 We present precise values of the eccentricity and periastron angle of 529 detached, eccentric, eclipsing stars from the Kepler Eclipsing Binary catalog that were determined by modeling their long cadence data. The temperatures and relative radii of their components as well as their mass ratios were calculated based on approximate values of the empirical relations of MS stars. Around one-third of the secondary components were revealed to be very late dwarfs, some of them possible brown dwarf candidates. Most of our targets fall below the envelope P(1 - e 2)3/2 = 5 days. The (e, P) distribution of the known eccentric binaries exhibits a rough trend of increasing eccentricity with the period. The prolonged and continuous Kepler observations allowed us to identify 60 new highly eccentric targets with e > 0.5. 1. An eclipsing double-line spectroscopic binary at the stellar/substellar boundary in the Upper Scorpius OB association Lodieu, N.; Alonso, R.; González Hernández, J. I.; Sanchis-Ojeda, R.; Narita, N.; Kawashima, Y.; Kawauchi, K.; Suárez Mascareño, A.; Deeg, H.; Prieto Arranz, J.; Rebolo, R.; Pallé, E.; Béjar, V. J. S.; Ferragamo, A.; Rubiño-Martín, J. A. 2015-12-01 Aims: We aim at constraining evolutionary models at low mass and young ages by identifying interesting transiting system members of the nearest OB association to the Sun, Upper Scorpius (USco), which has been targeted by the Kepler mission. Methods: We produced light curves for M-dwarf members of the USco region that has been surveyed during the second campaign of the Kepler K2 mission. We identified by eye a transiting system, USco J161630.68-251220.1 (=EPIC 203710387) with a combined spectral type of M5.25, whose photometric, astrometric, and spectroscopic properties makes it a member of USco. We conducted an extensive photometric and spectroscopic follow-up of this transiting system with a suite of telescopes and instruments to characterise the properties of each component of the system. Results: We calculated a transit duration of about 2.42 h that occurs every 2.88 days with a slight difference in transit depth and phase between the two components. We estimated a mass ratio of 0.922 ± 0.015 from the semi-amplitudes of the radial velocity curves for each component. We derived masses of 0.091 ± 0.005M⊙ and 0.084 ± 0.004M⊙, radii of 0.388 ± 0.008R⊙ and 0.380 ± 0.008R⊙, luminosities of log (L/L⊙) = -2.020-0.121+0.099 dex and -2.032-0.121+0.099 dex, and effective temperatures of 2901-172+199 K and 2908-172+199 K for the primary and secondary, respectively. Conclusions: We present a complete photometric and radial velocity characterisation of the least massive double-line eclipsing binary system in the young USco association with two components close to the stellar/substellar limit. This system falls in a gap between the least massive eclipsing binaries in the low-mass and substellar regimes at young ages and represents an important addition to constraining evolutionary models at young ages. Based on observations made with telescopes (GTC, WHT) installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de 2. Using Gaussian Processes to Model Noise in Eclipsing Binary Light Curves Prsa, Andrej; Hambleton, Kelly M. 2017-01-01 The most precise data we have at hand arguably comes from NASA's Kepler mission, for which there is no good flux calibration available since it was designed to measure relative flux changes down to ~20ppm level. Instrumental artifacts thus abound in the data, and they vary with the module, location on the CCD, target brightness, electronic cross-talk, etc. In addition, Kepler's near-uninterrupted mode of observation reveals astrophysical signals and transient phenomena (i.e. spots, flares, protuberances, pulsations, magnetic field features, etc) that are not accounted for in the models. These "nuisance" signals, along with instrumental artifacts, are considered noise when modeling light curves; this noise is highly correlated and it cannot be considered poissonian or gaussian. Detrending non-white noise from light curve data has been an ongoing challenge in modeling eclipsing binary star and exoplanet transit light curves. Here we present an approach using Gaussian Processes (GP) to model noise as part of the overall likelihood function. The likelihood function consists of the eclipsing binary light curve generator PHOEBE, correlated noise model using GP, and a poissonian (shot) noise attributed to the actual stochastic component of the entire noise model. We consider GP parameters and poissonian noise amplitude as free parameters that are being sampled within the likelihood function, so the end result is the posterior probability not only for eclipsing binary model parameters, but for the noise parameters as well. We show that the posteriors of principal parameters are significantly more robust when noise is modeled rigorously compared to modeling detrended data with an eclipsing binary model alone. This work has been funded by NSF grant #1517460. 3. VizieR Online Data Catalog: Eclipsing binary parallaxes with Gaia data (Stassun+, 2016) Stassun, K. G.; Torres, G. 2017-02-01 We adopted the predicted parallaxes for the 158 eclipsing binaries (EBs) included in the study of Stassun & Torres (2016, arXiv:1609.02579). Of these, 116 had parallaxes available in the Gaia first data release (see I/337). We excluded from our analysis any EBs identified as potentially problematic in Stassun & Torres (2016). This left 111 EBs with good parallaxes from both the EB analysis and from Gaia. (1 data file). 4. Data Mining the Ogle-II I-band Database for Eclipsing Binary Stars Ciocca, M. 2013-08-01 The OGLE I-band database is a searchable database of quality photometric data available to the public. During Phase 2 of the experiment, known as "OGLE-II", I-band observations were made over a period of approximately 1,000 days, resulting in over 1010 measurements of more than 40 million stars. This was accomplished by using a filter with a passband near the standard Cousins Ic. The database of these observations is fully searchable using the mysql database engine, and provides the magnitude measurements and their uncertainties. In this work, a program of data mining the OGLE I-band database was performed, resulting in the discovery of 42 previously unreported eclipsing binaries. Using the software package Peranso (Vanmuster 2011) to analyze the light curves obtained from OGLE-II, the eclipsing types, the epochs and the periods of these eclipsing variables were determined, to one part in 106. A preliminary attempt to model the physical parameters of these binaries was also performed, using the Binary Maker 3 software (Bradstreet and Steelman 2004). 5. GSC 4232.2850, a new eclipsing binary with elliptical orbit Goranskij, V.; Shugarov, S.; Kroll, P.; Golovin, A. 2005-04-01 GSC 4232.2830 (20h 01m 28s.407, +61? 10' 17".18, 2000.0, v=12m.1) was suspected to be an eclipsing binary by VPG in the routine overview of photographical plates taken with 40-cm astrograph of SAI Crimean station. To define orbital elements of the binary, we searched for observations in Sonneberg Observatory plate collection, NSVS database (Wozniak et al., 2004), and carried out visual monitoring with a small telescope equipped with an electronic image tube, an analogue of a night vision device. Later, when we had found a preliminary solution, we carried out accurate CCD photometry to improve the orbital elements. We should note, that the depths of eclipses in the NSVS database do not exceed 0m.2, what contradicts to other observations. We suppose that NSVS measurements concern to integral light of two stars, a variable star, and a nearby brighter star, GSC 4232.2395, due to low resolution of this survey, 72". Using all the available observations we found the single orbital solution with an elliptical orbit and the period of 11,6 day. The center of the secondary minimum occurs at the orbital phase 0.69835 or 8.1 day after the primary minimum. The improved ephemeris derived using accurate CCD observations is following: HJD Min I = 2453278,3185(2) + 11.628188 (5) x E. O-C analysis does not show orbital period variations during the time interval of observations, or any evidence of apsidal motion. The observations show that both eclipses have about equal depth 0m.60, but essentially different duration, 0p.028 (7 h.8) for Min I, and 0 p.0175 (4 h.9) for Min II. The eclipses are partial. CCD photometry gives mean colors U-B = -0 m.06, B-V = 0 m.57, and V-R = 0 m.50 without notable color variations in the eclipse phases. Old Sonneberg photographic observations indicate that the eclipses were shallower in the middle of the past century than in the present time! Such contradictions may suggest that the depth of eclipses varied, as in the well-known system SSLac 6. WW Geminorum: An early B-type eclipsing binary evolving into the contact phase SciTech Connect Yang, Y.-G.; Dai, H.-F.; Yin, X.-G.; Yang, Y. E-mail: [email protected] 2014-11-01 WW Gem is a B-type eclipsing binary with a period of 1.2378 days. The CCD photometry of this binary was performed in 2013 December using the 85 cm telescope at the Xinglong Stations of the National Astronomical Observatories of China. Using the updated W-D program, the photometric model was deduced from the VRI light curves. The results imply that WW Gem is a near-contact eclipsing binary whose primary component almost fills its Roche lobe. The photometric mass ratio is q {sub ph} = 0.48(± 0.05). All collected times of minimum light, including two new ones, were used for the period studies. The orbital period changes of WW Gem could be described by an upward parabola, possibly overlaid by a light-time orbit with a period of P {sub mod} = 7.41(± 0.04) yr and a semi-amplitude of A = 0.0079 days(± 0.0005 days), respectively. This kind of cyclic oscillation may be attributed to the light-travel time effect via the third body. The long-term period increases at a rate of dP/dt = +3.47(±0.04) × 10{sup –8} day yr{sup –1}, which may be explained by the conserved mass transfer from the less massive component to the more massive one. With mass transfer, the massive binary WW Gem may be evolving into a contact binary. 7. Light Curve Solutions of Eclipsing Binaries in the Large Magellanic Cloud Rawls, Meredith L.; Rao, M. S. 2012-01-01 We present model light curves for nine eclipsing binary stars in the Large Magellanic Cloud (LMC). These systems are detached binaries with nearly circular orbits, and were pseudorandomly selected from three of 21 LMC regions in the Optical Gravitational Lensing Experiment II (OGLE-II) survey. We make use of light curves, orbital periods, and binary classification as reported in Wyrzykowski et al. (2003). We present light curve solutions created with the software PHysics Of Eclipsing BinariEs (PHOEBE, Prsa & Zwitter 2005). Each solution has the best-fit mass ratio q, system inclination i, component temperatures T1 and T2, and modified Kopal potentials Ω1 and Ω2. PHOEBE employs a Nelder & Mead's Simplex fitting method that adjusts all the input parameters to find the best fit to the light curve. Many of the light curves have significant scatter, which can lead to multiple degenerate best-fit solutions, and we discuss what can be done in the future to refine our results, derive global stellar parameters, and place these nine systems in a larger context. We acknowledge the support of the International Research Experience for Students (IRES) program, which is sponsored by the NSF and administered by NSO/GONG. 8. WW Geminorum: An Early B-type Eclipsing Binary Evolving into the Contact Phase Yang, Y.-G.; Yang, Y.; Dai, H.-F.; Yin, X.-G. 2014-11-01 WW Gem is a B-type eclipsing binary with a period of 1.2378 days. The CCD photometry of this binary was performed in 2013 December using the 85 cm telescope at the Xinglong Stations of the National Astronomical Observatories of China. Using the updated W-D program, the photometric model was deduced from the VRI light curves. The results imply that WW Gem is a near-contact eclipsing binary whose primary component almost fills its Roche lobe. The photometric mass ratio is q ph = 0.48(± 0.05). All collected times of minimum light, including two new ones, were used for the period studies. The orbital period changes of WW Gem could be described by an upward parabola, possibly overlaid by a light-time orbit with a period of P mod = 7.41(± 0.04) yr and a semi-amplitude of A = 0.0079 days(± 0.0005 days), respectively. This kind of cyclic oscillation may be attributed to the light-travel time effect via the third body. The long-term period increases at a rate of dP/dt = +3.47(±0.04) × 10-8 day yr-1, which may be explained by the conserved mass transfer from the less massive component to the more massive one. With mass transfer, the massive binary WW Gem may be evolving into a contact binary. 9. UBVRI analysis of the totally eclipsing extreme mass ratio W UMa binary, GSC 3208 1986 SciTech Connect Samec, R. G.; Kring, J. D.; Robb, Russell; Van Hamme, W.; Faulkner, D. R. 2015-03-01 GSC 3208 1986 is an NSVS and TYCHO binary, first observed from 1999 to 2000. It is a W UMa binary with a period of 0.405 days. The present observations were taken in 2012 September and are of high precision, averaging a standard deviation of better than 5 mmag. The amplitude of the light curve is very nearly 0.5 mag yet it undergoes total eclipses. Dominion Astrophysical Observatory spectra give an F3V type (T∼6900 K) for the system, the earliest of the extreme mass ratio W UMa binaries. The linear period determination of 0.4045672 days was calculated with the two sets of epochs available. An early NSVS light curve reveals that the period has been smoothly decreasing over its past 12,000 orbits. The binary may be undergoing sinusoidal oscillations due to the presence of a third body, possibly with a period of 23±3 years. The high inclination of 85° results in a long duration secondary total eclipse, lasting some 49.5 minutes. Findings indicate that GSC 3208 1986 is an immaculate extreme mass ratio, q(m{sub 2}/m{sub 1}) = 0.24, A-type W UMa binary. 10. ASCA Observation of MS 1603.6+2600 (=UW Coronae Borealis): A Dipping Low-Mass X-ray Binary in the Outer Halo? NASA Technical Reports Server (NTRS) Mukai, Koji; Smale, Alan; Stahle, Caroline K.; Schlegel, Eric M.; Wijnands, Rudy; White, Nicholas E. (Technical Monitor) 2001-01-01 MS 1603.6+2600 is a high-latitude X-ray binary with a 111 min orbital period, thought to be either an unusual cataclysmic variable or an unusual low-mass X-ray binary. In an ASCA observation in 1997 August, we find a burst whose light curve suggests a Type 1 (thermonuclear flash) origin. We also find an orbital X-ray modulation in MS 1603.6+2600, which is likely to be periodic dips, presumably due to azimuthal structure in the accretion disk. Both are consistent with this system being a normal low-mass X-ray binary harboring a neutron star, but at a great distance. We tentatively suggest that MS 1603.6+2600 is located in the outer halo of the Milky Way, perhaps associated with the globular cluster Palomar 14, 11 deg away from MS 1603.6+2600 on the sky at an estimated distance of 73.8 kpc. 11. Accurate Parameters for the Most Massive Stars in the Local Universe: the Brightest Eclipsing Binaries in M33 Prieto, José L.; Bonanos, Alceste; Stanek, Krzysztof 2007-08-01 Eclipsing binaries are the only systems that provide accurate fundamental parameters of distant stars. Currently, only a handful of accurate measurements of stars with masses between 40-80 Msun have been made. We propose to make accurate measurements of the masses, radii and luminosities of the most massive eclipsing binaries in M33. The results of this study will provide much needed constraints on theories that model the formation and evolution of massive stars and binary systems. Furthermore, it will provide vital statistics on the occurrence of massive binary twins, like the 80+80 solar masses WR 20a system and the 30+30 solar masses detached eclipsing binary in M33. 12. Photometric Study of Extreme Long Period Eclipsing Binary 31 CYG Jeong, Jang Hae; Lee, Yong-Sam; Kim, Ho-Il 1991-12-01 The UBV light curves of an extreme long period binary star 31 Cyg are made with the observations obtained at Yonsei University Observatory for three seasons from 1988 to 1991 and the RI light curves are also made for one season in 1990-1991. The new combined UBV light curves of 31 Cyg are constructed with YOU's and collected data. A preliminary solution of the light curves of 31 Cygis made using Wilson-Devinney codes. 13. Studying Low Mass X-Ray Binaries: Revealing the Optical Counterpart in 1747-214 and Measuring the Masses of the Black Holes in 1859+226 and 1009-45 Gelino, Dawn M.; Tomsick, John A. 2003-02-01 Low mass x-ray binaries (LMXBs) contain compact, black hole (BH) or neutron star (NS) primaries, and cool, low-mass secondary stars. A limited number of BHs and NSs have accurate mass measurements. It is important to determine the primary mass of the LMXBs to better understand how BH masses influence their outburst behavior, and to better constrain the NS equations of state. To determine the mass of the primary object we need to measure the orbital inclination, i. We propose to determine i for two BH LMXBs, XTE J1859+226 and GRS 1009-45 (=N Vel 93) through modeling of their ellipsoidal variations. Because most LMXBs are not eclipsing, modeling their light curves is currently the only feasible method for determining the inclination. We will model the light curves with WD98. We also propose to identify the optical counterpart to the NS system EXO 1747-214, in order to begin the process of measuring the NS mass. We have successfully used NOAO facilities and this modeling technique to find accurate BH masses in four LMXBs. In order to expand the sample of known BH and NS systems, we request seven nights on the KPNO and CTIO 4m to obtain optical and infrared data on XTE J1859+226, GRS 1009-45, and EXO 1747-214. 14. A Double-line M-dwarf Eclipsing Binary from CSS x SDSS Lee, Chien-Hsiu 2017-03-01 Eclipsing binaries offer a unique opportunity to determine basic stellar properties. With the advent of wide-field camera and all-sky time-domain surveys, thousands of eclipsing binaries have been charted via light curve classification, yet their fundamental properties remain unexplored mainly due to the extensive efforts needed for spectroscopic follow-ups. In this paper, we present the discovery of a short-period (P = 0.313 day), double-lined M-dwarf eclipsing binary, CSSJ114804.3+255132/SDSSJ114804.35+255132.6, by cross-matching binary light curves from the Catalina Sky Survey and spectroscopically classified M dwarfs from the Sloan Digital Sky Survey. We obtain follow-up spectra using the Gemini telescope, enabling us to determine the mass, radius, and temperature of the primary and secondary component to be M 1 = 0.47 ± 0.03(statistic) ± 0.03(systematic) M ⊙, M 2 = 0.46 ± 0.03(statistic) ± 0.03(systematic) M ⊙, R 1 = 0.52 ± 0.08(statistic) ± 0.07(systematic) R ⊙, R 2 =0.60 ± 0.08(statistic) ± 0.08(systematic) R ⊙, T 1 = 3560 ± 100 K, and T 2 = 3040 ± 100 K, respectively. The systematic error was estimated using the difference between eccentric and non-eccentric fits. Our analysis also indicates that there is definitively third-light contamination (66%) in the CSS photometry. The secondary star seems inflated, probably due to tidal locking of the close secondary companion, which is common for very short-period binary systems. Future spectroscopic observations with high resolution will narrow down the uncertainties of stellar parameters for both components, rendering this system as a benchmark for studying fundamental properties of M dwarfs. 15. A New Comptonization Model for Weakly Magnetized, Accreting Neutron Stars in Low-Mass X-Ray Binaries Farinelli, Ruben; Titarchuk, Lev; Paizis, Ada; Frontera, Filippo 2008-06-01 We have developed a new model for the X-ray spectral fitting package XSPEC that takes into account the effects of both thermal and dynamical (i.e., bulk) Comptonization. The model consists of two components: one is the direct blackbody-like emission due to seed photons that are not subjected to effective Compton scattering, while the other is a convolution of the Green's function of the energy operator with a blackbody-like seed photon spectrum. When combined thermal and bulk effects are considered, the analytical form of the Green's function may be obtained as a solution of the diffusion equation describing Comptonization. Using data from the BeppoSAX, INTEGRAL, and RXTE satellites, we test our model on the spectra of a sample of six bright neutron star low-mass X-ray binaries with low magnetic fields, covering three different spectral states. Particular attention is given to the transient power-law-like hard X-ray (gtrsim30 keV) tails, which we interpret in the framework of the bulk motion Comptonization process. We show that the values of the best-fit δ-parameter, which represents the importance of bulk with respect to thermal Comptonization, can be physically meaningful and can at least qualitatively describe the physical conditions of the environment in the innermost part of the system. Moreover, we show that in fitting the thermal Comptonization spectra to the X-ray spectra of these systems, the best-fit parameters of our model are in excellent agreement with those from compTT, a broadly used and well-established XSPEC model. 16. Timing Observations of PSR J1023+0038 During a Low-mass X-Ray Binary State Jaodand, Amruta; Archibald, Anne M.; Hessels, Jason W. T.; Bogdanov, Slavko; D'Angelo, Caroline R.; Patruno, Alessandro; Bassa, Cees; Deller, Adam T. 2016-10-01 Transitional millisecond pulsars (tMSPs) switch, on roughly multi-year timescales, between rotation-powered radio millisecond pulsar (RMSP) and accretion-powered low-mass X-ray binary (LMXB) states. The tMSPs have raised several questions related to the nature of accretion flow in their LMXB state and the mechanism that causes the state switch. The discovery of coherent X-ray pulsations from PSR J1023+0038 (while in the LMXB state) provides us with the first opportunity to perform timing observations and to compare the neutron star’s spin variation during this state to the measured spin-down in the RMSP state. Whereas the X-ray pulsations in the LMXB state likely indicate that some material is accreting onto the neutron star’s magnetic polar caps, radio continuum observations indicate the presence of an outflow. The fraction of the inflowing material being ejected is not clear, but it may be much larger than that reaching the neutron star’s surface. Timing observations can measure the total torque on the neutron star. We have phase-connected nine XMM-Newton observations of PSR J1023+0038 over the last 2.5 years of the LMXB state to establish a precise measurement of spin evolution. We find that the average spin-down rate as an LMXB is 26.8 ± 0.4% faster than the rate (-2.39 × 10-15 Hz s-1) determined during the RMSP state. This shows that negative angular momentum contributions (dipolar magnetic braking, and outflow) exceed positive ones (accreted material), and suggests that the pulsar wind continues to operate at a largely unmodified level. We discuss implications of this tight observational constraint in the context of possible accretion models. 17. The Discovery of a Second Luminous Low-Mass X-Ray Binary in the Globular Cluster M15 NASA Technical Reports Server (NTRS) White, Nicholas E.; Angelini, Lorella 2001-01-01 We report an observation by the Chandra X-Ray Observatory of 4U 2127+119, the X-ray source identified with the globular cluster M15. The Chandra observation reveals that 4U 2127+119 is in fact two bright sources, separated by 2.7 arcsec. One source is associated with AC 211, the previously identified optical counterpart to 4U 2127+119, a low-mass X-ray binary (LMXB). The second source, M15 X-2, is coincident with a 19th U magnitude blue star that is 3.3 arcsec from the cluster core. The Chandra count rate of M15 X-2 is 2.5 times higher than that of AC 211. Prior to the 0.5 arcsec imaging capability of Chandra, the presence of two so closely separated bright sources would not have been resolved. The optical counterpart, X-ray luminosity, and spectrum of M15 X-2 are consistent with it also being an LMXB system. This is the first time that two LMXBs have been seen to be simultaneously active in a globular cluster. The discovery of a second active LMXB in M15 solves a long-standing puzzle where the properties of AC 211 appear consistent with it being dominated by an extended accretion disk corona, and yet 4U 2127+119 also shows luminous X-ray bursts requiring that the neutron star be directly visible. The resolution of 4U 2127+119 into two sources suggests that the X-ray bursts did not come from AC 211 but rather from M15 X-2. We discuss the implications of this discovery for understanding the origin and evolution of LMXBs in globular clusters as well as X-ray observations of globular clusters in nearby galaxies. 18. The Discovery of a Second Luminous Low Mass X-ray Binary in the Globular Cluster M15 NASA Technical Reports Server (NTRS) White, Nicholas E.; Angelini, Lorella 2001-01-01 We report an observation by the Chandra X-ray Observatory of 4U2127+119, the X-ray source identified with the globular cluster M15. The Chandra observation reveals that 4U2127+119 is in fact two bright sources, separated by 2.7". One source is associated with AC21 1, the previously identified optical counterpart to 4U2127+119, a low mass X-ray binary (LMXB). The second source, M15-X2, is coincident with a 19th U magnitude blue star that is 3.3" from the cluster core. The Chandra count rate of M15-X2 is 2.5 times higher than that of AC211. Prior to the 0.5" imaging capability of Chandra the presence of two so closely separated bright sources would not have been resolved, The optical counterpart, X-ray luminosity and spectrum of M15-X2 are consistent with it also being an LMXB system. This is the first time that two LMXBS have been seen to be simultaneously active in a globular cluster. The discovery of a second active LMXB in M15 solves a long standing puzzle where the properties of AC211 appear consistent with it being dominated by an extended accretion disk corona, and yet 4U2127+119 also shows luminous X-ray bursts requiring that the neutron star be directly visible. The resolution of 4U2127+119 into two sources suggests that the X-ray bursts did not come from AC211, but rather from M15X2. We discuss the implications of this discovery for understanding the origin and evolution of LMXBs in GCs as well as X-ray observations of globular clusters in nearby galaxies. 19. X-RAY OUTBURSTS OF LOW-MASS X-RAY BINARY TRANSIENTS OBSERVED IN THE RXTE ERA SciTech Connect Yan, Zhen; Yu, Wenfei E-mail: [email protected] 2015-06-01 We have performed a statistical study of the properties of 110 bright X-ray outbursts in 36 low-mass X-ray binary transients (LMXBTs) seen with the All-Sky Monitor (2–12 keV) on board the Rossi X-ray Timing Explorer (RXTE) in 1996–2011. We have measured a number of outburst properties, including peak X-ray luminosity, rate of change of luminosity on a daily timescale, e-folding rise and decay timescales, outburst duration, and total radiated energy. We found that the average properties, such as peak X-ray luminosity, rise and decay timescales, outburst duration, and total radiated energy of black hole LMXBTs, are at least two times larger than those of neutron star LMXBTs, implying that the measurements of these properties may provide preliminary clues to the nature of the compact object of a newly discovered LMXBT. We also found that the outburst peak X-ray luminosity is correlated with the rate of change of X-ray luminosity in both the rise and decay phases, which is consistent with our previous studies. Positive correlations between total radiated energy and peak X-ray luminosity, and between total radiated energy and the e-folding rise or decay timescale, are also found in the outbursts. These correlations suggest that the mass stored in the disk before an outburst is the primary initial condition that sets up the outburst properties seen later. We also found that the outbursts of two transient stellar-mass ultraluminous X-ray sources in M31 also roughly follow the correlations, which indicate that the same outburst mechanism works for the brighter outbursts of these two sources in M31 that reached the Eddington luminosity. 20. Partial Accretion in the Propeller Stage of Low-mass X-Ray Binary Aql X–1 Güngör, C.; Ekşi˙, K. Y.; Göğüş, E.; Güver, T. 2017-10-01 Aql X–1 is one of the most prolific low-mass X-ray binary transients (LMXBTs) showing outbursts almost annually. We present the results of our spectral analyses of Rossi X-Ray Timing Explorer/proportional counter-array observations of the 2000 and 2011 outbursts. We investigate the spectral changes related to the changing disk-magnetosphere interaction modes of Aql X–1. The X-ray light curves of the outbursts of LMXBTs typically show phases of fast rise and exponential decay. The decay phase shows a “knee” where the flux goes from the slow-decay to the rapid-decay stage. We assume that the rapid decay corresponds to a weak propeller stage at which a fraction of the inflowing matter in the disk accretes onto the star. We introduce a novel method for inferring, from the light curve, the fraction of the inflowing matter in the disk that accretes onto the neutron star depending on the fastness parameter. We determine the fastness parameter range within which the transition from the accretion to the partial propeller stage is realized. This fastness parameter range is a measure of the scale height of the disk in units of the inner disk radius. We applied the method to a sample of outbursts of Aql X–1 with different maximum flux and duration times. We show that different outbursts with different maximum luminosity and duration follow a similar path in the parameter space of accreted/inflowing mass flux fraction versus fastness parameter. 1. Upper Bounds on r-Mode Amplitudes from Observations of Low-Mass X-Ray Binary Neutron Stars NASA Technical Reports Server (NTRS) Mahmoodifar, Simin; Strohmayer, Tod 2013-01-01 We present upper limits on the amplitude of r-mode oscillations and gravitational-radiation-induced spin-down rates in low-mass X-ray binary neutron stars, under the assumption that the quiescent neutron star luminosity is powered by dissipation from a steady-state r-mode. For masses <2M solar mass we find dimensionless r-mode amplitudes in the range from about 1×10(exp-8) to 1.5×10(exp-6). For the accreting millisecond X-ray pulsar sources with known quiescent spin-down rates, these limits suggest that approx. less than 1% of the observed rate can be due to an unstable r-mode. Interestingly, the source with the highest amplitude limit, NGC 6440, could have an r-mode spin-down rate comparable to the observed, quiescent rate for SAX J1808-3658. Thus, quiescent spin-down measurements for this source would be particularly interesting. For all sources considered here, our amplitude limits suggest that gravitational wave signals are likely too weak for detection with Advanced LIGO. Our highest mass model (2.21M solar mass) can support enhanced, direct Urca neutrino emission in the core and thus can have higher r-mode amplitudes. Indeed, the inferred r-mode spin-down rates at these higher amplitudes are inconsistent with the observed spin-down rates for some of the sources, such as IGR J00291+5934 and XTE J1751-305. In the absence of other significant sources of internal heat, these results could be used to place an upper limit on the masses of these sources if they were made of hadronic matter, or alternatively it could be used to probe the existence of exotic matter in them if their masses were known. 2. The effects of thermodynamic stability on wind properties in different low-mass black hole binary states Chakravorty, Susmita; Lee, Julia C.; Neilsen, Joseph 2013-11-01 We present a systematic theory-motivated study of the thermodynamic stability condition as an explanation for the observed accretion disc wind signatures in different states of low-mass black hole binaries (BHB). The variability in observed ions is conventionally explained either by variations in the driving mechanisms or by the changes in the ionizing flux or due to density effects, whilst thermodynamic stability considerations have been largely ignored. It would appear that the observability of particular ions in different BHB states can be accounted for through simple thermodynamic considerations in the static limit. Our calculations predict that in the disc-dominated soft thermal and intermediate states, the wind should be thermodynamically stable and hence observable. On the other hand, in the power-law-dominated spectrally hard state the wind is found to be thermodynamically unstable for a certain range of 3.55 ≤ log ξ ≤ 4.20. In the spectrally hard state, a large number of the He-like and H-like ions (including e.g. Fe XXV, Ar XVIII and S XV) have peak ion fractions in the unstable ionization parameter (ξ) range, making these ions undetectable. Our theoretical predictions have clear corroboration in the literature reporting differences in wind ion observability as the BHBs transition through the accretion states While this effect may not be the only one responsible for the observed gradient in the wind properties as a function of the accretion state in BHBs, it is clear that its inclusion in the calculations is crucial for understanding the link between the environment of the compact object and its accretion processes. 3. Links between quasi-periodic oscillations and accretion states in neutron star low-mass X-ray binaries Motta, S. E.; Rouco-Escorial, A.; Kuulkers, E.; Muñoz-Darias, T.; Sanna, A. 2017-06-01 We analysed the Rossi X-ray Timing Explorer data from a sample of bright accreting neutron star (NS) low-mass X-ray binaries (LMXBs). With the aim of studying the quasi-periodic variability as a function of the accretion regime, we carried out a systematic search of the quasi-periodic oscillations (QPOs) in the X-ray time series of these systems, using the integrated fractional variability as a tracker for the accretion states. We found that the three QPO types originally identified in the 1980s for the brightest LMXBs, the so-called Z sources, i.e. horizontal, normal and flaring branch oscillations (HBOs, NBOs and FBOs, respectively), are also identified in the slightly less bright NS LMXBs, the so-called atoll sources, where we see QPOs with a behaviour consistent with the HBOs and FBOs. We compared the quasi-periodic variability properties of our NS sample with those of a sample of black hole (BH) LMXBs. We confirm the association between HBOs, NBOs and FBOs observed in Z sources, with the type-C, type-B and type-A QPOs, respectively, observed in BH systems, and we extended the comparison to the HBO-like and FBO-like QPOs seen in atoll sources. We conclude that the variability properties of BH and weakly magnetized NS LMXBs show strong similarities, with QPOs only weakly sensitive to the nature of the central compact object in both classes of systems. We find that the historical association between kHz QPOs and high-frequency QPOs, seen around NSs and BHs, respectively, is not obvious when comparing similar accretion states in the two kinds of systems. 4. Testing the validity of the phenomenological gravitational waveform models for nonspinning binary black hole searches at low masses Cho, Hee-Suk 2015-11-01 The phenomenological gravitational waveform models, which we refer to as PhenomA, PhenomB, and PhenomC, generate full inspiral, merger, and ringdown (IMR) waveforms of coalescing binary back holes (BBHs). These models are defined in the Fourier domain, thus can be used for fast matched filtering in the gravitational wave search. PhenomA has been developed for nonspinning BBH waveforms, while PhenomB and PhenomC were designed to model the waveforms of BBH systems with nonprecessing (aligned) spins, but can also be used for nonspinning systems. In this work, we study the validity of the phenomenological models for nonspinning BBH searches at low masses, {m}{1,2}≥slant 4{M}⊙ and {m}1+{m}2\\equiv M≤slant 30{M}⊙ , with Advanced LIGO. As our complete signal waveform model, we adopt EOBNRv2, which is a time-domain IMR waveform model. To investigate the search efficiency of the phenomenological template models, we calculate fitting factors (FFs) by exploring overlap surfaces. We find that only PhenomC is valid to obtain FFs better than 0.97 in the mass range of M\\lt 15{M}⊙ . Above 15{M}⊙ , PhenomA is most efficient in symmetric mass region, PhenomB is most efficient in highly asymmetric mass region, and PhenomC is most efficient in the intermediate region. Specifically, we propose an effective phenomenological template family that can be constructed by employing the phenomenological models in four subregions individually. We find that FFs of the effective templates are better than 0.97 in our entire mass region and mostly greater than 0.99. 5. UPPER BOUNDS ON r-MODE AMPLITUDES FROM OBSERVATIONS OF LOW-MASS X-RAY BINARY NEUTRON STARS SciTech Connect Mahmoodifar, Simin; Strohmayer, Tod 2013-08-20 We present upper limits on the amplitude of r-mode oscillations and gravitational-radiation-induced spin-down rates in low-mass X-ray binary neutron stars, under the assumption that the quiescent neutron star luminosity is powered by dissipation from a steady-state r-mode. For masses <2 M{sub Sun} we find dimensionless r-mode amplitudes in the range from about 1 Multiplication-Sign 10{sup -8} to 1.5 Multiplication-Sign 10{sup -6}. For the accreting millisecond X-ray pulsar sources with known quiescent spin-down rates, these limits suggest that {approx}< 1% of the observed rate can be due to an unstable r-mode. Interestingly, the source with the highest amplitude limit, NGC 6440, could have an r-mode spin-down rate comparable to the observed, quiescent rate for SAX J1808-3658. Thus, quiescent spin-down measurements for this source would be particularly interesting. For all sources considered here, our amplitude limits suggest that gravitational wave signals are likely too weak for detection with Advanced LIGO. Our highest mass model (2.21 M{sub Sun }) can support enhanced, direct Urca neutrino emission in the core and thus can have higher r-mode amplitudes. Indeed, the inferred r-mode spin-down rates at these higher amplitudes are inconsistent with the observed spin-down rates for some of the sources, such as IGR J00291+5934 and XTE J1751-305. In the absence of other significant sources of internal heat, these results could be used to place an upper limit on the masses of these sources if they were made of hadronic matter, or alternatively it could be used to probe the existence of exotic matter in them if their masses were known. 6. CIRCUMSTELLAR ENVIRONMENT AND EFFECTIVE TEMPERATURE OF THE YOUNG SUBSTELLAR ECLIPSING BINARY 2MASS J05352184-0546085 SciTech Connect Mohanty, Subhanjoy; Stassun, Keivan G.; Mathieu, Robert D. 2009-05-20 We present new Spitzer IRAC/PU/MIPS photometry from 3.6 to 24 {mu}m, and new Gemini GMOS photometry at 0.48 {mu}m, of the young brown dwarf eclipsing binary 2MASS J05352184-0546085, located in the Orion Nebula Cluster. No excess disk emission is detected. The measured fluxes at {lambda} {<=} 8 {mu}m are within 1{sigma} ({approx}<0.1 mJy) of a bare photosphere, and the 3{sigma} upper limit at 16 {mu}m is a mere 0.04 mJy above the bare photospheric level. Together with the known properties of the system, this implies the absence of optically thick disks around the individual components. It also implies that if any circumbinary disk is present, it must either be optically thin and extremely tenuous (10{sup -10} M {sub sun}) if it extends in to within {approx}0.1 AU of the binary (the approximate tidal truncation radius), or it must be optically thick with a large inner hole, >0.6-10 AU in radius depending on degree of flaring. The consequence in all cases is that disk accretion is likely to be negligible or absent. This supports the recent proposal that the strong H{alpha} emission in the primary (more massive) brown dwarf results from chromospheric activity, and thereby bolsters the hypothesis that the surprising T {sub eff} inversion observed between the components is due to strong magnetic fields on the primary. Our data also set constraints on the T {sub eff} of the components independent of spectral type, and thereby on models of the aforementioned magnetic field effects. We discuss the consequences for the derived fundamental properties of young brown dwarfs and very low mass stars in general. Specifically, if very active isolated young brown dwarfs and very low mass stars suffer the same activity/field related effects as the 2M0535-05 primary, the low-mass stellar/substellar initial mass function currently derived from standard evolutionary tracks may be substantially in error. 7. γ Doradus Pulsations in the Eclipsing Binary Star KIC 6048106 Lee, Jae Woo 2016-12-01 We present the Kepler photometry of KIC 6048106, which is exhibiting the O’Connell effect and multiperiodic pulsations. Including a starspot on either of the components, light-curve synthesis indicates that this system is a semi-detached Algol with a mass ratio of 0.211, an orbital inclination of 73.°9, and a large temperature difference of 2534 K. To examine in detail both the spot variations and pulsations, we separately analyzed the Kepler time-series data at the interval of an orbital period in an iterative way. The results reveal that the variable asymmetries of the light maxima can be interpreted as the changes with time of a magnetic cool spot on the secondary component. Multiple frequency analyses were performed in the outside-eclipse light residuals after removal of the binarity effects from the observed Kepler data. We detected 30 frequencies with signal to noise amplitude ratios larger than 4.0, of which six (f 2-f 6 and f 10) can be identified as high-order (17 ≤ n ≤ 25) low-degree (ℓ = 2) gravity-mode pulsations that were stable during the observing run of 200 days. In contrast, the other frequencies may be harmonic and combination terms. For the six frequencies, the pulsation periods and pulsation constants are in the ranges of 0.352-0.506 days and 0.232-0.333 days, respectively. These values and the position on the Hertzsprung-Russell diagram demonstrate that the primary star is a γ Dor variable. The evolutionary status and the pulsation nature of KIC 6048106 are discussed. 8. AN X-RAY AND OPTICAL LIGHT CURVE MODEL OF THE ECLIPSING SYMBIOTIC BINARY SMC3 SciTech Connect Kato, Mariko; Hachisu, Izumi; Mikolajewska, Joanna 2013-01-20 Some binary evolution scenarios for Type Ia supernovae (SNe Ia) include long-period binaries that evolve to symbiotic supersoft X-ray sources in their late stage of evolution. However, symbiotic stars with steady hydrogen burning on the white dwarf's (WD) surface are very rare, and the X-ray characteristics are not well known. SMC3 is one such rare example and a key object for understanding the evolution of symbiotic stars to SNe Ia. SMC3 is an eclipsing symbiotic binary, consisting of a massive WD and red giant (RG), with an orbital period of 4.5 years in the Small Magellanic Cloud. The long-term V light curve variations are reproduced as orbital variations in the irradiated RG, whose atmosphere fills its Roche lobe, thus supporting the idea that the RG supplies matter to the WD at rates high enough to maintain steady hydrogen burning on the WD. We also present an eclipse model in which an X-ray-emitting region around the WD is almost totally occulted by the RG swelling over the Roche lobe on the trailing side, although it is always partly obscured by a long spiral tail of neutral hydrogen surrounding the binary in the orbital plane. 9. The Araucaria Project. The Distance to the Small Magellanic Cloud from Late-type Eclipsing Binaries Graczyk, Dariusz; Pietrzyński, Grzegorz; Thompson, Ian B.; Gieren, Wolfgang; Pilecki, Bogumił; Konorski, Piotr; Udalski, Andrzej; Soszyński, Igor; Villanova, Sandro; Górski, Marek; Suchomska, Ksenia; Karczmarek, Paulina; Kudritzki, Rolf-Peter; Bresolin, Fabio; Gallenne, Alexandre 2014-01-01 We present a distance determination to the Small Magellanic Cloud (SMC) based on an analysis of four detached, long-period, late-type eclipsing binaries discovered by the Optical Gravitational Lensing Experiment (OGLE) survey. The components of the binaries show negligible intrinsic variability. A consistent set of stellar parameters was derived with low statistical and systematic uncertainty. The absolute dimensions of the stars are calculated with a precision of better than 3%. The surface brightness-infrared color relation was used to derive the distance to each binary. The four systems clump around a distance modulus of (m - M) = 18.99 with a dispersion of only 0.05 mag. Combining these results with the distance published by Graczyk et al. for the eclipsing binary OGLE SMC113.3 4007, we obtain a mean distance modulus to the SMC of 18.965 ± 0.025 (stat.) ± 0.048 (syst.) mag. This corresponds to a distance of 62.1 ± 1.9 kpc, where the error includes both uncertainties. Taking into account other recent published determinations of the SMC distance we calculated the distance modulus difference between the SMC and the Large Magellanic Cloud equal to 0.458 ± 0.068 mag. Finally, we advocate μSMC = 18.95 ± 0.07 as a new "canonical" value of the distance modulus to this galaxy. 10. The Araucaria project. The distance to the small Magellanic Cloud from late-type eclipsing binaries SciTech Connect Graczyk, Dariusz; Pietrzyński, Grzegorz; Gieren, Wolfgang; Pilecki, Bogumił; Villanova, Sandro; Gallenne, Alexandre; Thompson, Ian B.; Konorski, Piotr; Udalski, Andrzej; Soszyński, Igor; Górski, Marek; Suchomska, Ksenia; Karczmarek, Paulina; Kudritzki, Rolf-Peter; Bresolin, Fabio 2014-01-01 We present a distance determination to the Small Magellanic Cloud (SMC) based on an analysis of four detached, long-period, late-type eclipsing binaries discovered by the Optical Gravitational Lensing Experiment (OGLE) survey. The components of the binaries show negligible intrinsic variability. A consistent set of stellar parameters was derived with low statistical and systematic uncertainty. The absolute dimensions of the stars are calculated with a precision of better than 3%. The surface brightness-infrared color relation was used to derive the distance to each binary. The four systems clump around a distance modulus of (m – M) = 18.99 with a dispersion of only 0.05 mag. Combining these results with the distance published by Graczyk et al. for the eclipsing binary OGLE SMC113.3 4007, we obtain a mean distance modulus to the SMC of 18.965 ± 0.025 (stat.) ± 0.048 (syst.) mag. This corresponds to a distance of 62.1 ± 1.9 kpc, where the error includes both uncertainties. Taking into account other recent published determinations of the SMC distance we calculated the distance modulus difference between the SMC and the Large Magellanic Cloud equal to 0.458 ± 0.068 mag. Finally, we advocate μ{sub SMC} = 18.95 ± 0.07 as a new 'canonical' value of the distance modulus to this galaxy. 11. A period study and revised photometric model for the eclipsing binary ZZ Cyg Yang, Yuangui; Zhang, Liyun; Dai, Haifeng; Li, Huali 2015-05-01 We present new photometry of the eclipsing binary ZZ Cyg. From all accumulated eclipsing times, we constructed the (O-C) curve, which can be described by a downward parabola with a possible light-time orbit. The period decrease rate is dP / dt = - 5.73 (± 0.18) dyr-1 . The modulated period, semi-amplitude and eccentricity for the light-time orbit are Pmod = 71.4 (± 1.1)yr , A = 0.0071 (± 0.0005)day and e = 0.420 (± 0.053) , respectively. After removing effects of the magnetic activity, this kind of cyclic oscillation may be attributed to light-time effect via an additional companion. By using the W-D code, the photometric model was updated, which identified that ZZ Cyg is a near-contact binary. We find that a hot spot may occur on the primary that explains the asymmetric light curve. The secular period decrease may possibly cause the fill-out factor of the primary to increase. Finally it will finally fill its Roche lobe. This kind of binary, ZZ Cyg, may evolve into contact binary star. 12. 3D modelling of accretion disc in eclipsing binary system V1239 Her Lukin, V. V.; Malanchev, K. L.; Shakura, N. I.; Postnov, K. A.; Chechetkin, V. M.; Utrobin, V. P. 2017-05-01 We present the results of 3D-hydrodynamical simulations of accretion flow in the eclipsing dwarf nova V1239 Her in quiescence. The model includes the optical star filling its Roche lobe, a gas stream emanating from the inner Lagrangian point of the binary system, and the accretion disc structure. A cold hydrogen gas stream is initially emitted towards a point-like gravitational centre. A stationary accretion disc is formed in about 15 orbital periods after the beginning of accretion. The model takes into account partial ionization of hydrogen and uses realistic cooling function for hydrogen. The light curve of the system is calculated as the volume emission of optically thin layers along the line of sight up to the optical depth τ = 2/3 calculated using Planck-averaged opacities. The calculated eclipse light curves show good agreement with observations, with the changing shape of pre-eclipse and post-eclipse light curves being explained entirely due to ˜50 per cent variations in the mass accretion rate through the gas stream. 13. Eclipsing binary stars as tests of gravity theories - The apsidal motion of AS Camelopardalis NASA Technical Reports Server (NTRS) Maloney, Frank P.; Guinan, Edward F.; Boyd, Patricia T. 1989-01-01 AS Camelopardalis is an 8th-magnitude eclipsing binary that consists of two main-sequence (B8 V and a B9.5 V) components in an eccentric orbit (e = 0.17) with an orbital period of 3.43 days. Like the eccentric eclipsing system DI Herculis, and a few other systems, AS Cam is an important test case for studying relativistic apsidal motion. In these systems, the theoretical general relativistic apsidal motion is comparable to that expected from classical effects arising from tidal and rotational deformation of the stellar components. Accurate determinations of the orbital and stellar properties of AS Cam have been made by Hilditch (1972) and Khalliulin and Kozyreva (1983) that permit the theoretical relativistic and classical contributions to the apsidal motion to be determined reasonably well. All the published timings of primary and secondary minima have been gathered and supplemented with eclipse timings from 1899 to 1920 obtained from the Harvard plate collection. Least-squares solutions of the eclipse timings extending over an 80 yr interval yield a smaller than expected apsidal motion, in agreement with that found by Khalliulin and Kozyreva from a smaller set of data. The observed apsidal motion for AS Cam is about one-third that expected from the combined relativistic and classical effects. Thus, AS Cam joins DI Her in having an observed apsidal motion significantly less than that predicted from theory. 14. ABSOLUTE DIMENSIONS OF THE G7+K7 ECLIPSING BINARY STAR IM VIRGINIS: DISCREPANCIES WITH STELLAR EVOLUTION MODELS SciTech Connect Morales, Juan Carlos; Marschall, Laurence A.; Brehm, William 2009-12-10 We report extensive spectroscopic and differential photometric BVRI observations of the active, detached, 1.309-day double-lined eclipsing binary IM Vir, composed of a G7-type primary and a K7 secondary. With these observations, we derive accurate absolute masses and radii of M {sub 1} = 0.981 +- 0.012 M {sub sun}, M {sub 2} = 0.6644 +- 0.0048 M {sub sun}, R {sub 1} = 1.061 +- 0.016 R {sub sun}, and R {sub 2} = 0.681 +- 0.013 R {sub sun} for the primary and secondary, with relative errors under 2%. The effective temperatures are 5570 +- 100 K and 4250 +- 130 K, respectively. The significant difference in mass makes this a favorable case for comparison with stellar evolution theory. We find that both stars are larger than the models predict, by 3.7% for the primary and 7.5% for the secondary, as well as cooler than expected, by 100 K and 150 K, respectively. These discrepancies are in line with previously reported differences in low-mass stars, and are believed to be caused by chromospheric activity, which is not accounted for in current models. The effect is not confined to low-mass stars: the rapidly rotating primary of IM Vir joins the growing list of objects of near-solar mass (but still with convective envelopes) that show similar anomalies. The comparison with the models suggests an age of 2.4 Gyr for the system, and a metallicity of [Fe/H] approx-0.3 that is consistent with other indications, but requires confirmation. 15. THE SDSS-HET SURVEY OF KEPLER ECLIPSING BINARIES: SPECTROSCOPIC DYNAMICAL MASSES OF THE KEPLER-16 CIRCUMBINARY PLANET HOSTS SciTech Connect Bender, Chad F.; Mahadevan, Suvrath; Deshpande, Rohit; Wright, Jason T.; Roy, Arpita; Terrien, Ryan C.; Sigurdsson, Steinn; Ramsey, Lawrence W.; Schneider, Donald P.; Fleming, Scott W. 2012-06-01 We have used high-resolution spectroscopy to observe the Kepler-16 eclipsing binary as a double-lined system and measure precise radial velocities for both stellar components. These velocities yield a dynamical mass ratio of q = 0.2994 {+-} 0.0031. When combined with the inclination, i 90.{sup 0}3401{sup +0.0016}{sub -0.0019}, measured from the Kepler photometric data by Doyle et al. (D11), we derive dynamical masses for the Kepler-16 components of M{sub A} = 0.654 {+-} 0.017 M{sub Sun} and M{sub B} = 0.1959 {+-} 0.0031 M{sub Sun }, a precision of 2.5% and 1.5%, respectively. Our results confirm at the {approx}2% level the mass-ratio derived by D11 with their photometric-dynamical model (PDM), q = 0.2937 {+-} 0.0006. These are among the most precise spectroscopic dynamical masses ever measured for low-mass stars and provide an important direct test of the results from the PDM technique. 16. VizieR Online Data Catalog: OGLE II SMC eclipsing binaries (Wyrzykowski+, 2004) Wyrzykowski, L.; Udalski, A.; Kubiak, M.; Szymanski, M. K.; Zebrun, K.; Soszinski, I.; Wozniak, P. R.; Pietrzynski, G.; Szewczyk, O. 2009-03-01 We present new version of the OGLE-II catalog of eclipsing binary stars detected in the Small Magellanic Cloud, based on Difference Image Analysis catalog of variable stars in the Magellanic Clouds containing data collected from 1997 to 2000. We found 1351 eclipsing binary stars in the central 2.4 square degree area of the SMC. 455 stars are newly discovered objects, not found in the previous release of the catalog. The eclipsing objects were selected with the automatic search algorithm based on the artificial neural network. The full catalog with individual photometry is accessible from the OGLE INTERNET archive, at ftp://sirius.astrouw.edu.pl/ogle/ogle2/var_stars/smc/ecl . Regular observations of the SMC fields started on June 26, 1997 and covered about 2.4 square degrees of central parts of the SMC. Reductions of the photometric data collected up to the end of May 2000 were performed with the Difference Image Analysis (DIA) package. (1 data file). 17. High-Resolution Infrared Spectroscopic Observations of the Upper Scorpius Eclipsing Binary EPIC 203868608 Johnson, Marshall C.; Mace, Gregory N.; Kim, Hwihyun; Kaplan, Kyle; McLane, Jacob; Sokal, Kimberly R. 2017-06-01 EPIC 203868608 is a source in the ~10 Myr old Upper Scorpius OB association. Using K2 photometry and ground-based follow-up observations, David et al. (2016) found that it consists of two brown dwarfs with a tertiary object at a projected separation of ~20 AU; the former objects appear to be a double-lined eclipsing binary with a period of 4.5 days. This is one of only two known eclipsing SB2s where both components are below the hydrogen-burning limit. We present additional follow-up observations of this system from the IGRINS high-resolution near-infrared spectrograph at McDonald Observatory. Our measured radial velocities do not follow the orbital solution presented by David et al. (2016). Instead, our combined IGRINS plus literature radial velocity dataset appears to indicate a period significantly different than that of the eclipsing binary obvious from the K2 light curve. We will discuss possible scenarios to account for the conflicting observations of this system. 18. A photometric study of the eclipsing binary RX Hercules NASA Technical Reports Server (NTRS) Jeffreys, K. W. 1980-01-01 A new photoelectric light curve of RX Hercules, a binary system with similar components, has been analyzed using Wood's computer model. RX Her, using Popper's spectroscopic mass ratio of q = 0.8472, turned out to be composed of a dimmer AO component and a larger B9.5 component. This detached system, upon analysis of the residuals in secondary minimum, shows some asymmetry during ingress which then disappears just before secondary minimum. The eccentricity e = 0.022 determined in this study is a little larger than previously published values of e = 0.018. In combination with the spectroscopic analysis of Popper, and ubvy data of Olson and Hill and Hilditch new photometric elements for RX Her were found. 19. RED GIANTS IN ECLIPSING BINARY AND MULTIPLE-STAR SYSTEMS: MODELING AND ASTEROSEISMIC ANALYSIS OF 70 CANDIDATES FROM KEPLER DATA SciTech Connect Gaulme, P.; McKeever, J.; Rawls, M. L.; Jackiewicz, J.; Mosser, B.; Guzik, J. A. 2013-04-10 Red giant stars are proving to be an incredible source of information for testing models of stellar evolution, as asteroseismology has opened up a window into their interiors. Such insights are a direct result of the unprecedented data from space missions CoRoT and Kepler as well as recent theoretical advances. Eclipsing binaries are also fundamental astrophysical objects, and when coupled with asteroseismology, binaries provide two independent methods to obtain masses and radii and exciting opportunities to develop highly constrained stellar models. The possibility of discovering pulsating red giants in eclipsing binary systems is therefore an important goal that could potentially offer very robust characterization of these systems. Until recently, only one case has been discovered with Kepler. We cross-correlate the detected red giant and eclipsing-binary catalogs from Kepler data to find possible candidate systems. Light-curve modeling and mean properties measured from asteroseismology are combined to yield specific measurements of periods, masses, radii, temperatures, eclipse timing variations, core rotation rates, and red giant evolutionary state. After using three different techniques to eliminate false positives, out of the 70 systems common to the red giant and eclipsing-binary catalogs we find 13 strong candidates (12 previously unknown) to be eclipsing binaries, one to be a non-eclipsing binary with tidally induced oscillations, and 10 more to be hierarchical triple systems, all of which include a pulsating red giant. The systems span a range of orbital eccentricities, periods, and spectral types F, G, K, and M for the companion of the red giant. One case even suggests an eclipsing binary composed of two red giant stars and another of a red giant with a {delta}-Scuti star. The discovery of multiple pulsating red giants in eclipsing binaries provides an exciting test bed for precise astrophysical modeling, and follow-up spectroscopic observations of many 20. Testing the Asteroseismic Scaling Relations for Red Giants with Eclipsing Binaries Observed by Kepler Gaulme, P.; McKeever, J.; Jackiewicz, J.; Rawls, M. L.; Corsaro, E.; Mosser, B.; Southworth, J.; Mahadevan, S.; Bender, C.; Deshpande, R. 2016-12-01 Given the potential of ensemble asteroseismology for understanding fundamental properties of large numbers of stars, it is critical to determine the accuracy of the scaling relations on which these measurements are based. From several powerful validation techniques, all indications so far show that stellar radius estimates from the asteroseismic scaling relations are accurate to within a few percent. Eclipsing binary systems hosting at least one star with detectable solar-like oscillations constitute the ideal test objects for validating asteroseismic radius and mass inferences. By combining radial velocity (RV) measurements and photometric time series of eclipses, it is possible to determine the masses and radii of each component of a double-lined spectroscopic binary. We report the results of a four-year RV survey performed with the échelle spectrometer of the Astrophysical Research Consortium’s 3.5 m telescope and the APOGEE spectrometer at Apache Point Observatory. We compare the masses and radii of 10 red giants (RGs) obtained by combining radial velocities and eclipse photometry with the estimates from the asteroseismic scaling relations. We find that the asteroseismic scaling relations overestimate RG radii by about 5% on average and masses by about 15% for stars at various stages of RG evolution. Systematic overestimation of mass leads to underestimation of stellar age, which can have important implications for ensemble asteroseismology used for Galactic studies. As part of a second objective, where asteroseismology is used for understanding binary systems, we confirm that oscillations of RGs in close binaries can be suppressed enough to be undetectable, a hypothesis that was proposed in a previous work. 1. Orbital period variation study of massive Beta-Lyrae eclipsing binary IU Auriga Yilan, Erkan; Bulut, İbrahim 2016-07-01 The system IU Aur is a semi-detached close binary system with an orbital period of 1.81 days, containing a massive star. The O-C diagram of this binary was analyzed with the least-squares method by using all available times of minima. We have found a periodic change of orbital period of IU Aur. This change has been explained by the gravitational effects of a third companion on the binay star. The orbit Parameters of the third body have been derived from the analysis of the O-C curve. The analysis indicates that the eclipsing binary revolves around a third-body with a mass of about M_{3}>10M_{⊙} in a highly eccentric orbit. 2. A study of the EB-type eclipsing binary GR Tauri with mass transfer Gu, Sheng-hong; Chen, Pei-sheng; Choy, Yu-kou; Leung, Kam-cheung; Chung, Wai-keung; Poon, Tak-sun 2004-08-01 In this paper, new CCD BV light curves of the EB-type eclipsing binary GR Tau, which were obtained in 1999, are analyzed by means of the Wilson-Devinney program. The photometric solution of this system is obtained, and its absolute parameters are also derived. Our study has demonstrated that GR Tau is a near-contact binary system with an almost-contact semidetached configuration in which the primary fills its Roche lobe, and both components are main-sequence stars. The asymmetric shape of the light curves can be explained by a hot spot on the surface of the secondary, which is created by the mass transfer from the primary to the secondary. GR Tau belongs to the V1010 Oph subclass of near-contact binaries, and is a good example of a system in the broken-contact phase predicted by the TRO theory. 3. Gaia eclipsing binary and multiple systems. Supervised classification and self-organizing maps Süveges, M.; Barblan, F.; Lecoeur-Taïbi, I.; Prša, A.; Holl, B.; Eyer, L.; Kochoska, A.; Mowlavi, N.; Rimoldini, L. 2017-07-01 Context. Large surveys producing tera- and petabyte-scale databases require machine-learning and knowledge discovery methods to deal with the overwhelming quantity of data and the difficulties of extracting concise, meaningful information with reliable assessment of its uncertainty. This study investigates the potential of a few machine-learning methods for the automated analysis of eclipsing binaries in the data of such surveys. Aims: We aim to aid the extraction of samples of eclipsing binaries from such databases and to provide basic information about the objects. We intend to estimate class labels according to two different, well-known classification systems, one based on the light curve morphology (EA/EB/EW classes) and the other based on the physical characteristics of the binary system (system morphology classes; detached through overcontact systems). Furthermore, we explore low-dimensional surfaces along which the light curves of eclipsing binaries are concentrated, and consider their use in the characterization of the binary systems and in the exploration of biases of the full unknown Gaia data with respect to the training sets. Methods: We have explored the performance of principal component analysis (PCA), linear discriminant analysis (LDA), Random Forest classification and self-organizing maps (SOM) for the above aims. We pre-processed the photometric time series by combining a double Gaussian profile fit and a constrained smoothing spline, in order to de-noise and interpolate the observed light curves. We achieved further denoising, and selected the most important variability elements from the light curves using PCA. Supervised classification was performed using Random Forest and LDA based on the PC decomposition, while SOM gives a continuous 2-dimensional manifold of the light curves arranged by a few important features. We estimated the uncertainty of the supervised methods due to the specific finite training set using ensembles of models constructed 4. Observations of Mutual Eclipses by the Binary Kuiper Belt Object Manwe-Thorondor Rabinowitz, David L.; Benecchi, Susan D.; Grundy, William M.; Thirouin, Audrey; Verbiscer, Anne J. 2016-10-01 The binary Kuiper Belt Object (385446) Manwe-Thorondor (aka 2003 QW111) is currently undergoing mutual events whereby the two ~100-km bodies alternately eclipse and occult each other as seen from Earth [1]. Such events are extremely rare among KBOs (Pluto-Charon and Sila-Nunam being notable exceptions). For Manwe-Thorondor, the events occur over ~0.5-d periods 4 to 5 times per year until the end of 2019. Here we report the results of observations to be made with the Soar 4m telescope at Cerro Pachon, Chile on 2016 Aug 25 and 26 UT, covering one of the deepest predicted eclipses. We use these observations to constrain the rotational variability of the two bodies, determine their physical properties (size, shape, albedo, density), and set limits on the presence of any prominent surface features.[1] Grundy, W. et al. 2012, Icarus, 220, 74 5. The apsidal motion of the eccentric eclipsing binary DI Herculis - An apparent discrepancy with general relativity NASA Technical Reports Server (NTRS) Guinan, E. F.; Maloney, F. P. 1985-01-01 The apsidal motion of the eccentric eclipsing binary DI Herculis (HD 175227) is determined from an analysis of the available observations and eclipse timings from 1959 to 1984. Least squares solutions to the primary and secondary minima extending over an 84-yr interval yielded a small advance of periastron omega dot of 0.65 deg/100 yr + or - 0.18/100 yr. The observed advance of the periastron is about one seventh of the theoretical value of 4.27 deg/100 yr that is expected from the combined relativistic and classical effects. The discrepancy is about -3.62 deg/100 yr, or a magnitude of about 20 sigma. Classical mechanisms which explain the discrepancy are discussed, together with the possibility that there may be problems with general relativity itself. 6. The apsidal motion of the eccentric eclipsing binary DI Herculis - An apparent discrepancy with general relativity NASA Technical Reports Server (NTRS) Guinan, E. F.; Maloney, F. P. 1985-01-01 The apsidal motion of the eccentric eclipsing binary DI Herculis (HD 175227) is determined from an analysis of the available observations and eclipse timings from 1959 to 1984. Least squares solutions to the primary and secondary minima extending over an 84-yr interval yielded a small advance of periastron omega dot of 0.65 deg/100 yr + or - 0.18/100 yr. The observed advance of the periastron is about one seventh of the theoretical value of 4.27 deg/100 yr that is expected from the combined relativistic and classical effects. The discrepancy is about -3.62 deg/100 yr, or a magnitude of about 20 sigma. Classical mechanisms which explain the discrepancy are discussed, together with the possibility that there may be problems with general relativity itself. 7. DISCOVERY OF A SECOND TRANSIENT LOW-MASS X-RAY BINARY IN THE GLOBULAR CLUSTER NGC 6440 SciTech Connect Heinke, C. O.; Budac, S. A.; Altamirano, D.; Linares, M.; Wijnands, R.; Cohn, H. N.; Lugger, P. M.; Servillat, M.; Grindlay, J. E.; Strohmayer, T. E.; Markwardt, C. B.; Swank, J. H.; Bailyn, C. 2010-05-01 We have discovered a new transient low-mass X-ray binary, NGC 6440 X-2, with Chandra/ACIS, RXTE/PCA, and Swift/XRT observations of the globular cluster NGC 6440. The discovery outburst (2009 July 28-31) peaked at L{sub X} {approx} 1.5 x 10{sup 36} erg s{sup -1} and lasted for <4 days above L{sub X} = 10{sup 35} erg s{sup -1}. Four other outbursts (2009 May 29-June 4, August 29-September 1, October 1-3, and October 28-31) have been observed with RXTE/PCA (identifying millisecond pulsations) and Swift/XRT (confirming a positional association with NGC 6440 X-2), with similar peak luminosities and decay times. Optical and infrared imaging did not detect a clear counterpart, with best limits of V>21, B>22 in quiescence from archival Hubble Space Telescope imaging, g'>22 during the August outburst from Gemini-South GMOS imaging, and J {approx_gt} 18.5 and K {approx_gt} 17 during the July outburst from CTIO 4 m ISPI imaging. Archival Chandra X-ray images of the core do not detect the quiescent counterpart (L{sub X} < (1-2) x 10{sup 31} erg s{sup -1}) and place a bolometric luminosity limit of L{sub NS} < 6 x 10{sup 31} erg s{sup -1} (one of the lowest measured) for a hydrogen atmosphere neutron star. A short Chandra observation 10 days into quiescence found two photons at NGC 6440 X-2's position, suggesting enhanced quiescent emission at L{sub X} {approx} 6 x 10{sup 31} erg s{sup -1}. NGC 6440 X-2 currently shows the shortest recurrence time ({approx}31 days) of any known X-ray transient, although regular outbursts were not visible in the bulge scans before early 2009. Fast, low-luminosity transients like NGC 6440 X-2 may be easily missed by current X-ray monitoring. 8. A First Robust Measurement of the Aging of Field Low Mass X-ray Binary Populations from Hubble and Chandra Lehmer, Bret Our understanding of X-ray binary (XRB) formation and evolution have been revolutionized by HST and Chandra by allowing us to study in detail XRBs in extragalactic environments. Theoretically, XRB formation is sensitive to parent stellar population properties like metallicity and stellar age. These dependencies not only make XRBs promising populations for aiding in the measurement of galaxy properties themselves, but also have important astrophysical implications. For example, due to the relatively young stellar ages and primordial metallicities in the early Universe (z > 3), it is predicted that XRBs were more luminous than today and played a significant role in the heating of the intergalactic medium. Unlocking the potential of XRBs as useful probes of galaxy properties and understanding in detail their evolutionary pathways critically requires empirical constraints using well-studied galaxies that span a variety of evolutionary stages. In this ADAP, we will use the combined power of archival observations from Hubble and Chandra data of 16 nearby early-type galaxies to study how low-mass XRBs (LMXBs) populations evolve with age. LMXBs are critically important since they are the most numerous XRBs in the MW and are expected to dominate the normal galaxy Xray emissivity of the Universe out to z ~ 2. Understanding separately LMXBs that form via dynamical interactions (e.g., in globular clusters; GCs) versus those that form in-situ in galactic fields is an important poorly constrained area of XRB astrophysics. We are guided by the following key questions: 1. How does the shape and normalization of the field LMXB X-ray luminosity function (XLF) evolve as parent stellar populations age? Using theoretical population synthesis models, what can we learn about the evolution of contributions from various LMXB donor stars (e.g., red-giant, main-sequence, and white dwarf donors)? 2. Is there any evidence that globular cluster (GC) LMXBs seeded field LMXB populations through 9. Precise analysis of two Kepler detached eclipsing binary stars KIC 3327980 and KIC 10156064 Aliçavuş, Fahri; Soydugan, Faruk 2017-02-01 Stars are one of the most important objects to understand how the galaxies are formed, shaped and evolved. Hence, the determination of the absolute parameters of stars plays a crucial role. The absolute parameters (e.g. mass and radii) of the detached eclipsing binary stars could be determined with well accuracy. These accurate parameters could be used for understanding of the evolutional status of single stars in detailed. In this study, we carried out light curve solutions of two well-detached binaries KIC 3327980 and KIC 10156064 which were observed by Kepler space telescope. As a result, mass and radii of primary and secondary components were derived as M1 = 1.64M⊙, M2 = 1.42M⊙, R1 = 2.08R⊙, R2 = 1.66R⊙ for KIC 3327980 and M1 = 1.67M⊙, M2 = 1.05M⊙, R1 = 1.92R⊙, R2 = 1.06R⊙ for KIC 10156064. Additionally, the evolutionary status of the components of the systems were discussed and compared with the evolutional status of the other detached eclipsing binaries. 10. Physical Properties and Evolution of the Eclipsing Binary System XZ Canis Minoris Poochaum, R.; Komonjinda, S.; Soonthornthum, B.; Rattanasoon, S. 2010-07-01 This research aims to study the eclipse binary system so that its physical properties and evolution can be determined and used as an example to teach high school astronomy. The study of an eclipsing binary system XZ Canis Minoris (XZ CMi) was done at Sirindhorn Observatory, Chiang Mai University using a 0.5-meter reflecting telescope with CCD photometric system (2184×1417 pixel) in B V and R bands of UVB System. The data obtained were used to construct the light curve for each wavelength band and to compute the times of its light minima. New elements were derived using observations with linear to all available minima. As a result, linear ephemeris is HDJmin I = .578 808 948+/-0.000 000 121+2450 515.321 26+/-0.001 07 E, and the new orbital period of XZ CMi is 0.578 808 948+/-0.000 000 121 day. The values obtained were used with the previously published times of minima to get O-C curve of XZ CMi. The result revealed that the orbital period of XZ CMi is continuously decreased at a rate of 0.007 31+/-0.000 57 sec/year. This result indicates that the binary stars are moving closer continuously. From the O-C residuals, there is significant change to indicate the existence of the third body or magnetic activity cycle on the star. However, further analysis of the physical properties of XZ CMi is required. 11. The Eclipsing Binary Pulsar PSR B1718-19: a Clean RS CVN System? Kaspi, Victoria 1996-07-01 We request WFPC2 observations of the eclipsing binary pulsarPSR B1718-19. This slowly-rotating pulsar lies in thedirection of the globular cluster NGC 6342, and defiesstandard binary pulsar formation models in which the pulsar isspun-up'' via mass accretion. Furthermore, the observedeclipses cannot be explained with standard models. Thispulsar's unusual properties can be explained elegantly if itscompanion is an active, non-degenerate star like thoseobserved in RS CVn systems, but in this case, subject only togravity. Following Keck observations of the field, we proposeHST observations of PSR B1718-19 to detect and studyvariability in the companion, in order to answer the followingquestions. 1- Are the eclipses seen in PSR B1718-19 indeed aresult of RS CVn-type activity in the companion, and what isthe evolutionary history of the binary? 2- Is the activity inRS CVn systems purely a tidal effect? 3- How are mass loss,rotation, and surface activity related in RS CVn stars? 4- IsPSR B1718-19 in NGC 6342? 12. Period Discovery and Light Curve Analysis of the Young 25 Ori Association Eclipsing Binary GSC 118-199 Bradstreet, David H.; Sanders, S. J.; Regi, A. 2014-01-01 GSC 118-199 (5-6432) was discovered to be an eclipsing binary by Van Eyken et al. (2011) in their intensive monitoring of the young (7-25 Myr) 25 Ori association as part of the Palomar Transient Factory (PTF) Orion project. However, because of the brightness of the system (Rc = 12.7 mag), their instruments could only accurately monitor the binary during its deep (0.6 mag in Rc) total eclipse. They did not observe enough eclipses to reliably determine a period and no secondary eclipse was ever detected. GSC 118-199 was observed at Eastern University from Feb 2011 - Mar 2013 in order to discover the period and subsequently analyze the light curve of this presumably very young totally eclipsing system. More than 3500 observations in V and Rc were obtained and several primary and secondary eclipses were secured and an accurate period determined. The secondary eclipse depth was measured to be ~0.1 mag in Rc centered at 0.496P in phase. The ephemeris for the system has been determined to be 2455189.72682(5) + 6.185181(4) E. Preliminary light curve analyses indicate that the system is a detached, totally eclipsing binary, where the larger but less massive component is the cooler star by ~3000 K. The system also exhibits a slightly eccentric 0.02) orbit, a typical indicator of youth in a binary system. This poster will investigate the possible ramifications of the unusual nature of this young system with the hope that additional observations obtained in late 2013 will provide additional weight and clarity to the preliminary solution. 13. A Catalog of Eclipsing Binaries and Variable Stars Observed with ASTEP 400 from Dome C, Antarctica Chapellier, E.; Mékarnia, D.; Abe, L.; Guillot, T.; Agabi, K.; Rivet, J.-P.; Schmider, F.-X.; Crouzet, N.; Aristidi, E. 2016-10-01 We used the large photometric database of the ASTEP program, whose primary goal was to detect exoplanets in the southern hemisphere from Antarctica, to search for eclipsing binaries (EcBs) and variable stars. 673 EcBs and 1166 variable stars were detected, including 31 previously known stars. The resulting online catalogs give the identification, the classification, the period, and the depth or semi-amplitude of each star. Data and light curves for each object are available at http://astep-vo.oca.eu. 14. Light-time effect in two eclipsing binaries: NO Vul and EW Lyr Bulut, A.; Bulut, I.; ćiçek, C.; Erdem, A. 2017-02-01 In this study, orbital period variations of two eclipsing binary systems (NO Vul and EW Lyr) were discussed. Possible light time effects due to third bodies in these systems were re-examined. The mass function and orbital period of hypothetical third bodies were calculated to be 0.000627 ± 0.000003 M⊙, 26.17 ± 0.05 years and 0.12682 ± 0.00003 M⊙, 77.23 ± 0.72 years for NO Vul and EW Lyr, respectively. 15. Investigation of Cyclic O-C Changes in a Sample of Eclipsing Binaries Jableka, D.; Zola, S.; Riddle, R.; Baranec, C.; Law, N. 2015-07-01 In this work we present an analysis of cyclic or pseudo-cyclic O-C behavior in a sample of 29 eclipsing binaries, selected to exhibit large-amplitude changes in O-C. We attempt to explain the period variations by: 1) the light time travel effect due to an unseen third body orbiting a system; 2) a sudden jump in the linear ephemeris caused by either variations in the mass transfer rate or CME ejections. A search for tertiary components was carried out with adaptive optics imaging for six systems exhibiting the highest amplitude in their O-C diagrams. 16. First Spectroscopic Solutions of Two Southern Eclipsing Binaries: HO Tel and QY Tel Sürgit, D.; Erdem, A.; Engelbrecht, C. A.; van Heerden, P.; Manick, R. 2015-07-01 We present preliminary results from the analysis of spectroscopic observations of two southern eclipsing binary stars, HO Tel and QY Tel. The grating spectra of these two systems were obtained at the Sutherland Station of the South African Astronomical Observatory in 2013. Radial velocities of the components were determined by the Fourier disentangling technique. Keplerian radial velocity models of HO Tel and QY Tel give their mass ratio as 0.921±0.005 and 1.089±0.007, respectively. 17. Finding False Positives Planet Candidates Due To Background Eclipsing Binaries in K2 Mullally, Fergal; Thompson, Susan E.; Coughlin, Jeffrey; DAVE Team 2016-06-01 We adapt the difference image centroid approach, used for finding background eclipsing binaries, to vet K2 planet candidates. Difference image centroids were used with great success to vet planet candidates in the original Kepler mission, where the source of a transit could be identified by subtracting images of out-of-transit cadences from in-transit cadences. To account for K2's roll pattern, we reconstruct out-of-transit images from cadences that are nearby in both time and spacecraft roll angle. We describe the method and discuss some K2 planet candidates which this method suggests are false positives. 18. The Wide Brown Dwarf Binary Oph 1622-2405 and Discovery of a Wide, Low-Mass Binary in Ophiuchus (Oph 1623-2402): A New Class of Young Evaporating Wide Binaries? Close, Laird M.; Zuckerman, B.; Song, Inseok; Barman, Travis; Marois, Christian; Rice, Emily L.; Siegler, Nick; Macintosh, Bruce; Becklin, E. E.; Campbell, Randy; Lyke, James E.; Conrad, Al; Le Mignant, David 2007-05-01 We imaged five objects near the star-forming clouds of Ophiuchus with the Keck Laser Guide Star AO system. We resolved sources 11 (Oph 16222-2405) and 16 (Oph 16233-2402) from Allers and coworkers into binary systems. Source 11 is resolved into a 243 AU binary, the widest known for a very low mass (VLM) binary. The binary nature of source 11 was discovered first by Allers and independently here, during which we obtained the first spatially resolved R~2000 near-infrared (J and K) spectra, mid-IR photometry, and orbital motion estimates. We estimate for 11A and 11B gravities (logg>3.75), ages (5+/-2 Myr), luminosities [log(L/Lsolar)=-2.77+/-0.10 and -2.96+/-0.10], and temperatures (Teff=2375+/-175 K and 2175+/-175 K). We find self-consistent DUSTY evolutionary model (Chabrier and coworkers) masses of 17+4-5 MJ and 14+6-5 MJ, for 11A and 11B, respectively. Our masses are higher than those previously reported (13-15 MJ and 7-8 MJ) by Jayawardhana & Ivanov. Hence, we find that the system is unlikely a `planetary mass binary,'' as do Luhman and coworkers, but it has the second lowest mass and lowest binding energy of any known binary. Oph 11 and Oph 16 belong to a newly recognized population of wide (>~100 AU), young (<10 Myr), roughly equal mass, VLM stellar and brown dwarf binaries. We deduce that ~6%+/-3% of young (<10 Myr) VLM objects are in such wide systems. However, only 0.3%+/-0.1% of old field VLM objects are found in such wide systems. Thus, young, wide, VLM binary populations may be evaporating, due to stellar encounters in their natal clusters, leading to a field population depleted in wide VLM systems. Based on observations made with the Keck and Gemini North telescopes. 19. The substellar companion in the eclipsing white dwarf binary SDSS J141126.20+200911.1 Littlefair, S. P.; Casewell, S. L.; Parsons, S. G.; Dhillon, V. S.; Marsh, T. R.; Gänsicke, B. T.; Bloemen, S.; Catalan, S.; Irawati, P.; Hardy, L. K.; Mcallister, M.; Bours, M. C. P.; Richichi, Andrea; Burleigh, M. R.; Burningham, B.; Breedt, E.; Kerry, P. 2014-12-01 We present high time resolution SDSS-g' and SDSS-z' light curves of the primary eclipse in SDSS J141126.20+200911.1, together with time-resolved X-Shooter spectroscopy and near-infrared (NIR) JHKs photometry. Our observations confirm the substellar nature of the companion, making SDSS J141126.20+200911.1 the first eclipsing white dwarf/brown dwarf binary known. We measure a (white dwarf model dependent) mass and radius for the brown dwarf companion of M2 = 0.050 ± 0.002 M⊙ and R2 = 0.072 ± 0.004 M⊙, respectively. The lack of a robust detection of the companion light in the z'-band eclipse constrains the spectral type of the companion to be later than L5. Comparing the NIR photometry to the expected white dwarf flux reveals a clear Ks-band excess, suggesting a spectral type in the range L7-T1. The radius measurement is consistent with the predictions of evolutionary models, and suggests a system age in excess of 3 Gyr. The low companion mass is inconsistent with the inferred spectral type of L7-T1, instead predicting a spectral type nearer T5. This indicates that irradiation of the companion in SDSS J141126.20+200911.1 could be causing a significant temperature increase, at least on one hemisphere. 20. The chromospherically active, triple, ellipsoidal, and eclipsing binary HD 6286 = BE Piscium: a laboratory for binary evolution Strassmeier, K. G.; Bartus, J.; Fekel, F. C.; Henry, G. W. 2008-07-01 Aims: We present a detailed analysis of the star HD 6286 = BE Psc from 16 years of spectroscopic observations and 18 seasons of photometric ones. The star is an evolved, chromospherically active, eclipsing binary, consisting of a K1 giant plus an F6 dwarf/subgiant in a circular orbit with a period of 35.671 days. A faint, close visual companion of spectral type ≈G0 makes the system triple. The orbital inclination of the eclipsing pair is 81.8deg. Methods: We have obtained simultaneous solutions with our extensive set of radial velocities and BVI light curves that include the star spot variability of the K giant, the ellipticity of the K giant, and the eclipses of the spectroscopic binary system. Results: Our spot solutions suggest persistent polar spots, one in each hemisphere, that are cooler than the surrounding photosphere by 810±150 K over the timespan of our observations. The K giant and the F6 dwarf/subgiant have masses of 1.56 M⊙ and 1.31 M⊙ and mean radii of 12.0 and 1.9 R⊙, respectively. The masses have uncertainties of just ≈1.5%. No irradiation effect was detected. We compared our results to theoretical evolutionary tracks that suggest an age for the system of ≈2.7 Gyr. The modest logarithmic lithium abundance of the primary of 1.30 (upper limit) indicates that the star may have already experienced its first dredge up. The rotation period of the primary is 35.49 ± 0.01 days and appears to be synchronized with the orbital period of the eclipsing pair to within 0.5%. Our data are inconclusive as to whether the secondary is synchronized. Conclusions: Circularization of the orbit has taken place, and we conclude that the rapid increase in the size of the K giant, as it evolved across the Hertzsprung gap and up the base of the giant branch, likely caused the orbit to become circular. 1. Period Variations of the Eclipsing Binary Systems T LMi and VX Lac Yılmaz, M.; İzci, D. D.; Gümüş, D.; Özavci, İ.; Selam, S. O. 2015-07-01 We present a period analysis of the two Algol-type eclipsing binary systems T LMi and VX Lac using all available times of minimum in the literature, as well as new minima obtained at the Ankara University Kreiken Observatory. The period analysis of T LMi suggests mass transfer between the components and also a third body that is dynamically bound to the binary system. The analysis of VX Lac also suggests mass transfer between the components, and the presence of a third and a fourth body under the assumption of a Light-Time Effect. In addition, the periodic variation of VX Lac was examined under the hypothesis of magnetic activity, and the corresponding parameters were derived. We report here the orbital parameters for both systems, along with the ones related to mass transfer, and those for the third and fourth bodies. 2. Age and Metallicity Estimates for Moderate-Mass Stars in Eclipsing Binaries Kovaleva, D. A. 2001-12-01 We estimate the ages and metallicities for the components of 43 binary systems using a compilation of accurate observational data on eclipsing binaries for which lines of both components are visible in their spectra, together with two independent modern sets of stellar evolution models computed for a wide range of masses and chemical abundances. The uncertainties of the resulting values are computed, and their stability is demonstrated. The ages and metallicity are compared with those derived in other studies using different methods, as well as with independent estimates from photometric observations and observations of clusters. These comparisons con firm the reliability of our age estimates. The resulting metallicities depend significantly on the choice of theoretical model. Comparison with independent estimates favors the estimates based on the evolutionary tracks of the Geneva group. 3. The unique eclipsing binary system V541 Cygni with relativistic apsidal motion Khaliullin, K. F. 1985-12-01 The first photoelectric light curve has been obtained for the binary star V541 Cygni (B8.5 V+B8.7 V, P = 15d.34) discovered by Kulikowski (1948). The light curve exhibits extremely narrow and deep minima of almost equal depth. Photometric elements are determined. Very small relative radii, orbital inclination close to π/2, high eccentricity (e = 0.474), and favorable orientation of the line of apsides with respect to an observer, as well as close similarity of the components, render the V541 Cyg system unique among eclipsing binaries. The apsidal motion in this system has been detected. The observed rotation rate of the line of apsides, ωobs = 0°.090±0°.013 yr-1, agrees within the errors with the prediction by general relativity. 4. ASAS LIGHT CURVES OF INTERMEDIATE-MASS ECLIPSING BINARY STARS AND THE PARAMETERS OF HI Mon SciTech Connect Williams, S. J.; Gies, D. R.; Matson, R. A.; Caballero-Nieves, S.; Helsel, J. W. E-mail: [email protected] E-mail: [email protected] 2011-07-15 We present a catalog of 56 candidate intermediate-mass eclipsing binary systems extracted from the third data release of the All Sky Automated Survey. We gather pertinent observational data and derive orbital properties, including ephemerides, for these systems as a prelude to anticipated spectroscopic observations. We find that 37 of the 56, or {approx}66%, of the systems are not identified in the Simbad Astronomical Database as known binaries. As a specific example, we show spectroscopic data obtained for the system HI Mon (B0 V + B0.5 V) observed at key orbital phases based on the computed ephemeris, present a combined spectroscopic and photometric solution for the system, and give stellar parameters for each component. 5. ASAS Light Curves of Intermediate-mass Eclipsing Binary Stars and the Parameters of HI Mon Williams, S. J.; Gies, D. R.; Helsel, J. W.; Matson, R. A.; Caballero-Nieves, S. 2011-07-01 We present a catalog of 56 candidate intermediate-mass eclipsing binary systems extracted from the third data release of the All Sky Automated Survey. We gather pertinent observational data and derive orbital properties, including ephemerides, for these systems as a prelude to anticipated spectroscopic observations. We find that 37 of the 56, or ~66%, of the systems are not identified in the Simbad Astronomical Database as known binaries. As a specific example, we show spectroscopic data obtained for the system HI Mon (B0 V + B0.5 V) observed at key orbital phases based on the computed ephemeris, present a combined spectroscopic and photometric solution for the system, and give stellar parameters for each component. 6. Multi-band photometric study of the short-period eclipsing binary GR Boo Wang, Daimei; Zhang, Liyun; Han, Xianming L.; Lu, Hongpeng 2017-05-01 We present BVRI light curves with complete phase coverage for the short-period (p = 0.377day) eclipsing binary star GR Boo. We carried out the observations using the SARA 90 cm telescope located at Kitt Peak National Observatory. We obtained six new light curve minimum times. By fitting all of the available O-C minimum times, we obtained an updated ephemeris that shows the orbital period of GR Boo is decreasing at a rate of P˙ = - 2.36 ×10-7 days/year. This decrease in its period can be explained by either mass transfer from the more massive component to the less massive one, or angular momentum exchange due to magnetic activities. We also obtained a set of revised orbital parameters using the Wilson & Devinney program. And finally, we concluded that GR Boo is a contact binary with a dark spot. 7. RXJ2130.6+4710 - an eclipsing white dwarf-M-dwarf binary star Maxted, P. F. L.; Marsh, T. R.; Morales-Rueda, L.; Barstow, M. A.; Dobbie, P. D.; Schreiber, M. R.; Dhillon, V. S.; Brinkworth, C. S. 2004-12-01 We report the detection of eclipses in the close white-dwarf-M-dwarf binary star RXJ2130.6+4710. We present light curves in the B, V and I bands and fast photometry obtained with the three-channel CCD photometer Ultracam of the eclipse in the u', g' and r' bands. The depth of the eclipse varies from 3.0 mag in the u' band to less than 0.1 mag in the I band. The times of mid-eclipse are given by the ephemeris BJD(mid-eclipse) = 2452785.681876(2) + 0.521035625(3) E, where figures in parentheses denote uncertainties in the final digit. We present medium-resolution spectroscopy from which we have measured the spectroscopic orbits of the M dwarf and white dwarf. We estimate that the spectral type of the M dwarf is M3.5Ve or M4Ve, although the data on which this is based are not ideal for spectral classification. We have compared the spectra of the white dwarf with synthetic spectra from pure hydrogen model atmospheres to estimate that the effective temperature of the white dwarf is Teff= 18000 +/- 1000 K. We have used the width of the primary eclipse and duration of totality measured precisely from the Ultracam u' data combined with the amplitude of the ellipsoidal effect in the I band and the semi-amplitudes of the spectroscopic orbits to derive masses and radii for the M dwarf and white dwarf. The M dwarf has a mass of 0.555 +/- 0.023 Msolar and a radius of 0.534 +/- 0.053 Rsolar, which is a typical radius for stars of this mass. The mass of the white dwarf is 0.554 +/- 0.017 Msolar and its radius is 0.0137 +/- 0.0014 Rsolar, which is the radius expected for a carbon-oxygen white dwarf of this mass and effective temperature. The light curves are affected by frequent flares from the M dwarf and the associated dark spots on its surface can be detected from the distortions to the light curves and radial velocities. RXJ2130.6+4710 is a rare example of a pre-cataclysmic variable star that will start mass transfer at a period above the period gap for cataclysmic variables. 8. Detection of the Second Eclipsing High-Mass X-Ray Binary in M 33 Pietsch, Wolfgang; Haberl, Frank; Gaetz, Terrance J.; Hartman, Joel D.; Plucinsky, Paul P.; Tüllmann, Ralph; Williams, Benjamin F.; Shporer, Avi; Mazeh, Tsevi; Pannuti, Thomas G. 2009-03-01 Chandra data of the X-ray source [PMH2004] 47 were obtained in the ACIS Survey of M 33 (ChASeM33) in 2006. During one of the observations, the source varied from a high state to a low state and back, in two other observations it varied from a low state to respectively intermediate states. These transitions are interpreted as eclipse ingresses and egresses of a compact object in a high-mass X-ray binary (HMXB) system. The phase of mideclipse is given by HJD 245 3997.476 ± 0.006, the eclipse half angle is 30fdg6 ± 1fdg2. Adding XMM-Newton observations of [PMH2004] 47 in 2001 we determine the binary period to be 1.732479 ± 0.000027 days. This period is also consistent with ROSAT HRI observations of the source in 1994. No short-term periodicity compatible with a rotation period of the compact object is detected. There are indications for a long-term variability similar to that detected for Her X-1. During the high state the spectrum of the source is hard (power-law spectrum with photon index ~0.85) with an unabsorbed luminosity of 2 ×1037 erg s-1 (0.2-4.5 keV). We identify as an optical counterpart a V ~ 21.0 mag star with T eff>19000 K, log(g)>2.5. The Canada-France-Hawaii Telescope optical light curves for this star show an ellipsoidal variation with the same period as the X-ray light curve. The optical light curve together with the X-ray eclipse can be modeled by a compact object with a mass consistent with a neutron star or a black hole in an HMXB. However, the hard power-law X-ray spectrum favors a neutron star as the compact object in this second eclipsing X-ray binary in M 33. Assuming a neutron star with a canonical mass of 1.4 M sun and the best-fit companion temperature of 33,000 K, a system inclination i = 72° and a companion mass of 10.9 M sun are implied. 9. The atmospheric structures of the companion stars of eclipsing binary x ray sources NASA Technical Reports Server (NTRS) Clark, George W. 1992-01-01 This investigation was aimed at determining structural features of the atmospheres of the massive early-type companion stars of eclipse x-ray pulsars by measurement of the attenuation of the x-ray spectrum during eclipse transitions and in deep eclipse. Several extended visits were made to ISAS in Japan by G. Clark and his graduate student, Jonathan Woo to coordinate the Ginga observations and preliminary data reduction, and to work with the Japanese host scientist, Fumiaki Nagase, in the interpretation of the data. At MIT extensive developments were made in software systems for data interpretation. In particular, a Monte Carlo code was developed for a 3-D simulation of the propagation of x-rays from the neutron star through the ionized atmosphere of the companion. With this code it was possible to determine the spectrum of Compton-scattered x-rays in deep eclipse and to subtract that component from the observed spectra, thereby isolating the software component that is attributable in large measure to x-rays that have been scattered by interstellar grains. This research has culminated in the submission of paper to the Astrophysical Journal on the determination of properties of the atmosphere of QV Nor, the BOI companion of 4U 1538-52, and the properties of interstellar dust grains along the line of sight from the source. The latter results were an unanticipated byproduct of the investigation. Data from Ginga observations of the Magellanic binaries SMC X-1 and LMC X-4 are currently under investigation as the PhD thesis project of Jonathan Woo who anticipated completion in the spring of 1993. 10. KIC 6220497: a new Algol-type eclipsing binary with multiperiodic pulsations	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8812805414199829, "perplexity": 4461.211709006053}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809695.97/warc/CC-MAIN-20171125071427-20171125091427-00704.warc.gz"}
https://math.stackexchange.com/questions/412130/optimization-of-entropy-for-fixed-distance-to-uniform	# Optimization of entropy for fixed distance to uniform Suppose that I know that a probability distribution with $n$ outcomes is very close to being uniform (that is: $\forall i,p_i=\frac{1}{n}$), and in particular for $n\epsilon\ll 1$ the distribution verifies $$\sum_{i=1}^n\|\frac{1}{n}-p_i\|=\epsilon$$ Now consider the Shannon entropy function: $$H(p_1,\ldots,p_n)=-\sum_{i=1}^np_i\log_2p_i$$ My question is: what are the probability distributions that, verifying the constraint, maximize and minimize the entropy? My guess is that $(\underbrace{\frac{1}{n},\ldots,\frac{1}{n}}_{n-2\text{ times}},\frac{1}{n}+\frac{\epsilon}{2},\frac{1}{n}-\frac{\epsilon}{2})$ minimizes the entropy and $(\underbrace{\frac{1}{n}-\frac{\epsilon}{n},\ldots,\frac{1}{n}-\frac{\epsilon}{n}}_{n/2\text{ times}},\underbrace{\frac{1}{n}+\frac{\epsilon}{n},\ldots,\frac{1}{n}+\frac{\epsilon}{n}}_{n/2\text{ times}})$ maximizes the entropy but I have not been able to prove any of them. For the first, you have $H=-(n-2)\frac 1n \log_2 \frac 1n-(\frac 1n + \frac {\epsilon}2)\log_2 (\frac 1n + \frac {\epsilon}2)- (\frac 1n - \frac {\epsilon}2)\log_2 (\frac 1n - \frac {\epsilon}2) \\=-(n-2)\frac 1n \log_2 \frac 1n-(\frac 1n + \frac {\epsilon}2)[\log_2 \frac 1n +\log_2 (1+\frac {n\epsilon}2)]- (\frac 1n - \frac {\epsilon}2)[\log_2 \frac 1n - \log_2 (1-\frac {n\epsilon}2)] \\ \approx-\log_2\frac 1n-\frac {n\epsilon^2}{2 \ln 2}$ For the second you have $H=-\frac n2(\frac {1-\epsilon}n \log_2 \frac {1-\epsilon}n + \frac {1+\epsilon}n \log_2 \frac {1+\epsilon}n) \\=-\frac n2(\frac {1-\epsilon}n (\log_2 \frac 1n + \log_2(1-\epsilon)) + \frac {1+\epsilon}n (\log_2 \frac 1n + \log_2(1+\epsilon)) \\\approx-\log_2 \frac 1n-\frac {\epsilon^2}{2 \ln 2}$ So the second has lower change in entropy by a factor $n$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9877145290374756, "perplexity": 165.5537442587936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488539764.83/warc/CC-MAIN-20210623165014-20210623195014-00254.warc.gz"}
http://mathhelpforum.com/differential-equations/159213-solve-following-ode-4-a.html	# Math Help - Solve the following ODE .. #4 1. ## Solve the following ODE .. #4 Problem: Solve the following equation: $(1+e^y \cdot e^{\left( e^x \right)}) dx - dy = 0$ Solution: I multiply the equation by $e^x$, to get: $(e^x+e^y \, e^x \cdot e^{\left( e^x \right)}) dx - e^x dy = 0$ $e^xdx + e^y e^x e^{\left( e^x \right)} dx - e^x dy = 0$ Let $t=e^x \implies dt=e^xdx$ the equation will be: $dt+e^ye^tdt-tdy=0$ Now, I stopped! 2. Multipy by $e^{-y}.$ It should be exact then. 3. Thanks. I forgot the method of determinating the integrating factor .	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9979463815689087, "perplexity": 2920.845292859264}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609535095.7/warc/CC-MAIN-20140416005215-00177-ip-10-147-4-33.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/147098/do-there-exist-non-isomorphic-groups-with-the-same-cohomology	# Do there exist non-isomorphic groups with the same cohomology? For any group $G$, cohomology can be viewed as a functor $$H^\ast(G,-): G{\sf\text{-}mod}\to {\sf GrAbGrp},$$ where $G{\sf\text{-}mod}$ denotes the category of (left) $\mathbb{Z}[G]$-modules and ${\sf GrAbGrp}$ denotes the category of (non-negatively) graded abelian groups. It seems that there are non-isomorphic groups $G_1$ and $G_2$ whose integral group rings $\mathbb{Z}[G_1]$ and $\mathbb{Z}[G_2]$ are Morita equivalent (this is a result of Roggenkamp and Zimmermann). One could then ask the following question: Do there exist non-isomorphic groups $G_1$ and $G_2$ for which there is an equivalence of categories $F: G_1{\sf\text{-}mod}\to G_2{\sf\text{-}mod}$ such that the functors $H^\ast(G_1,-)$ and $H^\ast(G_2,-)\circ F$ from $G_1{\sf\text{-}mod}$ to ${\sf GrAbGrp}$ are naturally isomorphic? This is my (possibly naive) attempt to formalise the question in the title. It may be that the answer is trivially "no" by looking at $H^0$, ie the functor of coinvariants. In that case, I would like to know if there are other formulations for which the question becomes interesting. • A minor formatting suggestion: I would recommend replacing {\sf-mod} with {\sf\text{-}mod}. Even when using the sans-serif font, the symbol - in math mode is interpreted as a minus sign. Compare $G{\sf-mod}$ and $G{\sf\text{-}mod}$. – Zev Chonoles Nov 6 '13 at 14:32 The answer would seem to be yes if I understood your question properly. The paper http://www.m-hikari.com/ija/ija-password-2009/ija-password9-12-2009/ladraIJA9-12-2009.pdf shows that if $\mathbb ZG$ is isomorphic to $\mathbb ZH$, then one can choose an isomorphism which is augmentation preserving. They then show that if $f\colon \mathbb ZG\to \mathbb ZH$ is the augmentation-preserving isomorphism, then viewing a $\mathbb ZH$-module as a $\mathbb ZG$-module gives isomorphisms in homology and cohomology. I didn't check if their proof gives naturality of the isomorphisms, but I would be surprised if it didn't.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9857698082923889, "perplexity": 94.67369159439903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143805.13/warc/CC-MAIN-20200218180919-20200218210919-00364.warc.gz"}
http://mathhelpforum.com/differential-geometry/124577-mobius-n-manifolds-within-complex-n-manifolds.html	## Mobius n-manifolds within Complex n manifolds General apologies for newbiness. I have arrived at this idea from considering the proposition that an electron is a photon looping on itself in one wavelength. Immediately one sees that if the whole loop was within our space the angular momentum would be hbar rather than the hbar/2 we know the electron to have. Indeed hbar/2 implies a radius of lambda/4*pi. But the photon has its own spin that we are apparently looping on itself like a smoke-ring so that all the spin's momentum must go through the tiny area of the loop. This high momentum flux through a tiny loop should set Einstein's stress/energy tensor of General Relativity into odd behaviour. While considering the "inside" and "outside" of a loop it seemed to me that it is inevitable in connected space. However a Mobius manifold doesn't have an inside and outside and seems to achieve this trick by rotating each dimension through its complex plane. I think this leads to a space that is internally orthogonal and repeating through a single real tangent point. Although I can see clearly what I mean I'm aware I don't have the terminology to explain it well, any help appreciated.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8988742232322693, "perplexity": 716.9681946205454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997858581.26/warc/CC-MAIN-20140722025738-00248-ip-10-33-131-23.ec2.internal.warc.gz"}

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