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https://ahilado.wordpress.com/2022/02/ | # Taylor-Wiles Patching
In Galois Deformation Rings we mentioned the idea of “modularity lifting“, which forms one part of the approach to proving that a Galois representation arises from a modular form, the other part being residual modularity. In that post we also mentioned “R=T” theorems, which are in turn the approach to proving modularity lifting, the “R” standing for the Galois deformation rings that were the main topic of that post, and “T” standing for (a certain localization of) the Hecke algebra. In this post, we shall discuss R=T theorems in a little more detail, and discuss the ideas involved in its proof. We shall focus on the weight $2$ cusp forms (see also Galois Representations Coming From Weight 2 Eigenforms), although many of these ideas can also be generalized to higher weights.
### A review of Galois deformation rings and Hecke algebras
Let us recall again the idea behind R=T theorems. We recall from Galois Deformation Rings that if we have a fixed residual representation $\overline{\rho}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathbb{F}$ (here $\mathbb{F}$ is some finite field of characteristic $p$), we have a Galois deformation ring $R_{\overline{\rho}}$, with the defining property that maps from $R_{\overline{\rho}}$ into some complete Noetherian local $W(\mathbb{F})$-algebra $A$ correspond to certain Galois representations over $A$, namely those which “lift” the residual representation $\overline{\rho}$. If we compose these maps with maps from $A$ into $\overline{\mathbb{Q}}_{p}$, we get maps that correspond to certain Galois representations over $\overline{\mathbb{Q}}_{p}$.
In addition, since we want to match up Galois representations with modular forms (cusp forms of weight $2$ in particular this post), we will want to impose certain conditions on the Galois representations that are parametrized by our deformation ring $R_{\overline{\rho}}$. For instance, it is known that p-adic Galois representations that arise from a cusp form of weight $2$ and level $\Gamma=\Gamma(N)$ are unramified at all the primes except $p$ and the ones that divide $N$. There is a way to construct a modification of our deformation ring $R_{\overline{\rho}}$ so that the Galois representations it parametrizes satisfies these conditions (also known as deformation conditions or deformation problems). We shall denote this modified deformation ring simply by $R$.
On the other hand, maps from the Hecke algebra to some coefficient field (we will choose this to be $\overline{\mathbb{Q}}_{p}$; conventionally this is $\mathbb{C}$, but $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ are isomorphic as fields) correspond to systems of eigenvalues coming from modular forms.
Now the idea is to match up these maps, since then it would be the same as matching Galois representations and modular forms; however, we note that currently our maps from $R_{\overline{\rho}}$ only correspond to Galois representations that come from lifting our fixed Galois representation $\overline{\rho}$ and we have not made any such restriction on the maps from our Hecke algebra, so they don’t quite match up yet.
### Galois representations valued in localizations of the Hecke algebra
What we will do to fix this is to come up with a maximal ideal of the Hecke algebra that corresponds to $\overline{\rho}$, and, instead of considering the entire Hecke algebra, which is too large, we will instead consider the localization of it with respect to this maximal ideal. We have, following the Hodge decomposition (for weights $k>2$, a generalization of this is given by a theorem of Eichler and Shimura)
$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong S_{2}(\Gamma,\mathbb{C})\oplus \overline{S_{2}(\Gamma,\mathbb{C})}$
where $M_{2}(\Gamma,\mathbb{C})$ (resp. $S_{2}(\Gamma,\mathbb{C})$) is the space of modular forms (resp. cusp forms) of weight $2$ and level $\Gamma$. The advantage of expressing modular forms in this form is that we shall be able to consider them “integrally”. We have that
$\displaystyle H^{1}(Y(\Gamma), \mathbb{C})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathbb{C}$
Now let $E$ be a finite extension of $\mathbb{Q}_{p}$, with ring of integers $\mathcal{O}$, uniformizer $\varpi$ and residue field $\mathbb{F}$ (the same field our residual representation $\overline{\rho}$ takes values in). We can now consider
$\displaystyle H^{1}(Y(\Gamma), \mathcal{O})\cong H^{1}(Y(\Gamma), \mathbb{Z})\otimes\mathcal{O}$
Let $\Sigma$ be the set consisting of the prime $p$ and the primes dividing the level, which we shall assume to be squarefree (these conditions put us in the minimal case of Tayor-Wiles patching – though the strategy holds more generally, we assume these conditions to simplify our discussion). We have a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ acting on $H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p})$, and similarly a Hecke algebra $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acting on $H^{1}(Y(\Gamma), \mathcal{O})$. Recall that these are the subrings of their respective endomorphism rings generated by the Hecke operators $T_{\ell}$ and $S_{\ell}$ for all $\ell\not\in \Sigma$ (see also Hecke Operators and Galois Representations Coming From Weight 2 Eigenforms). The eigenvalue map
$\displaystyle \lambda_{g}:\mathbb{T}(S(\Gamma,\overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$
which associates to a Hecke operator its eigenvalue on some cusp form $g\in S(\Gamma,\overline{\mathbb{Q}}_{p})$ extends to a map
$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))\to\overline{\mathbb{Q}}_{p}$.
Now since $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ acts on $H^{1}(\Gamma, \mathcal{O})$ we will also have an eigenvalue map
$\displaystyle \lambda_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathcal{O}$
compatible with the above, in that applying $\lambda_{g}$ followed by embedding the resulting eigenvalue to $\overline{\mathbb{Q}}_{p}$ is the same as composing the map from $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ into $\mathbb{T}(H^{1}(Y(\Gamma), \overline{\mathbb{Q}}_{p}))$ first then applying the eigenvalue map. Now we can compose the eigenvalue map to $\mathcal{O})$ with the reduction mod $\varpi$ so that we get $\displaystyle \overline{\lambda}_{g}:\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))\to\mathbb{F}$.
Now let $\mathfrak{m}$ be the kernel of $\overline{\lambda}_{g}$. This is a maximal ideal of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$. In fact, we can associate to $\lambda_{g}$ a residual representation $\overline{\rho}_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{F})$, such that the characteristic polynomial of the $\mathrm{Frob}_{\ell}$ is given by $X^{2}-\lambda_{g}(T_{\ell})X+\ell \lambda_{g}(S_{\ell})$.
Now let $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ be the completion of $\mathbb{T}(H^{1}(Y(\Gamma), \mathcal{O}))$ with respect to $\mathfrak{m}$. It turns out that there is a Galois representation $\rho_{\mathfrak{m}}:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{T}(\Gamma)_{\mathfrak{m}})$ which lifts $\overline{\rho}_{\mathfrak{m}}$. Furthermore, $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ is a complete Noetherian local $\mathcal{O}$-algebra!
Putting all of these together, what this all means is that if $\overline{\rho}=\overline{\rho}_{\mathfrak{m}}$, there is a map $R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$. Furthermore, this map is surjective. Again, the fact that we have this surjective map reflects that fact that we can obtain Galois representations (of a certain form) from modular forms. Showing that this is an isomorphism amounts to showing that Galois representations of this form always come from modular forms.
### Taylor-Wiles patching: Rough idea behind the approach
So now, to prove our “R=T” theorem, we need to show that this map is actually an isomorphism.
Let $M=H^{1}(Y(\Gamma),\mathcal{O})$. The idea is that $R$ will have an action on $M$, which will factor through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$. If we can show that $M$ is free as an $R$-module, then since this action factors through $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ via a surjection, then the map from $R$ to $\mathbb{T}(\Gamma)_{\mathfrak{m}}$ must be an isomorphism.
This, by itself, is still too difficult. So what we will do is build an auxiliary module, sometimes called the patched module and denoted $M_{\infty}$, which is going to be a module over an auxiliary ring we shall denote by $R_{\infty}$, from which $M$ and $R$ can be obtained as quotients by a certain ideal. The advantage is that we can bring another ring in play, the power series ring $\mathcal{O}[[x_{1},\ldots,x_{q}]]$, which maps to $R_{\infty}$ (in fact, two copies of it will map to $R_{\infty}$, which is important), and we will use what we know about power series rings to show that $M_{\infty}$ is free over $R_{\infty}$, which will in turn show that $M$ is free over $R$.
In turn, $M_{\infty}$ and $R_{\infty}$ will be built as inverse limits of modules and rings $R_{Q_{n}}$ and $M_{Q_{n}}$. The subscript $Q_{n}$ refers to a set of primes , called “Taylor-Wiles primes” at which we shall also allow ramification (recall that initially we have imposed the condition that our Galois representations be unramified at all places outside of $p$ and the primes that divide the level $N$). As we shall see, these Taylor-Wiles primes will be specially selected so that we will be able to construct $M_{\infty}$ and $R_{\infty}$ with the properties that we will need. This passage to the limit in order to make use of what we know about power series is inspired by Iwasawa theory (see also Iwasawa theory, p-adic L-functions, and p-adic modular forms).
### Taylor-Wiles primes
A Taylor-Wiles prime of level $n$ is defined to be a prime $v$ such that the norm $q_{v}$ is congruent to $1$ mod $p^{n}$, and such that $\overline{\rho}(\mathrm{Frob}_{v})$ has distinct $\mathbb{F}$-rational eigenvalues. For our purposes we will need, for every positive integer $n$, a set $Q_{n}$ of Taylor-Wiles primes of cardinality equal to the dimension of the dual Selmer group of $R$ (which we shall denote by $q$), and such that the dual Selmer group of $R_{Q_{n}}$ is trivial. It is known that we can always find such a set $Q_{n}$ for every positive integer $n$.
Let us first look at how this affects the “Galois side”, i.e. $R_{Q_{n}}$. There is a surjection $R_{Q_{n}}\twoheadrightarrow R$, but the important property of this, that is due to how the Taylor-Wiles primes were selected, is that the dimensions of their tangent spaces (which is going to be equal to the dimension of the Selmer group as discussed in More on Galois Deformation Rings) are the same.
Now it so happens that, when we are considering $2$-dimensional representations of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the dimensions of the Selmer group and the dual Selmer group will be the same. This is what is known as the numerical coincidence, and is quite special to our case. In general, for instance when instead of $\mathbb{Q}$ we have a more general number field $F$, this numerical coincidence may not hold (we will briefly discuss this situation at the end of this post). The numerical coincidence, as well as the fact that the dimension of the tangent spaces of $R$ and $R_{Q_{n}}$ remain the same, are both consequences of the Wiles-Greenberg formula, which relates the Selmer group and the dual Selmer group.
Now let us look at the “automorphic side”, i.e. $M_{Q_{n}}$. We call this the automorphic side because they are localizations of spaces of modular forms (which are automorphic forms). We first need to come up with a new kind of level structure.
Letting $Q_{n}$ be some set of Taylor-Wiles primes, we define $\Gamma_{0}(Q_{n})=\Gamma\cap\Gamma_{0}(\prod_{v\in Q_{n}}v)$ and we further define $\Gamma_{Q_{n}}$ to be such that the quotient $\Gamma_{0}(Q_{n})/\Gamma_{Q_{n}}$ is isomorphic to the group $\Delta_{Q_{n}}$, defined to be the product over $v\in Q_{n}$ of the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$.
We define a new Hecke algebra $\mathbb{T}_{Q_{n}}$ obtained from $\mathbb{T}$ by adjoining new Hecke operators $U_{v}$ for every prime $v$ in $Q_{n}$. We define a maximal ideal $\mathfrak{m}_{Q_{n}}$ of $\mathbb{T}_{Q_{n}}$ generated by the elements of $\mathfrak{m}$ and $U_{v}-\alpha_{v}$ again for every prime $v$ in $Q_{n}$.
We now define $M_{Q_{n}}$ to be $H^{1}(Y(\Gamma_{Q}),\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$. This has an action of $\Delta_{Q_{n}}$ and is therefore a $\mathcal{O}[\Delta_{Q_{n}}]$-module. In fact, $M_{Q_{n}}$ is a free $\mathcal{O}[\Delta_{Q_{n}}]$-module. This will become important later. Another important property of $M_{Q_{n}}$ is that its $\Delta_{Q_{n}}$-coinvariants are isomorphic to $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$.
Now $R_{Q_{n}}$ also has the structure of a $\mathcal{O}[\Delta_{Q_{n}}]$-algebra. If we take $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(R_{Q_{n}})$ and restrict it to $\mathrm{Gal}(\overline{\mathbb{Q}}_{v}/\mathbb{Q}_{v})$ (for$v$ in $Q_{n}$), we get that the resulting local representation is of the form $\eta_{1}\oplus\eta_{2}$, where $\eta_{1}$ and $\eta_{2}$ are characters. Using local class field theory (see also The Local Langlands Correspondence for General Linear Groups), we obtain a map $\mathbb{Z}_{v}^{\times}\to R_{Q_{n}}^{\times}$. This map factors through the maximal p-power quotient of $(\mathbb{Z}/v\mathbb{Z})^{\times}$. Thus given $Q_{n}$ we have a map $\Delta_{Q_{n}}\to R_{Q_{n}}$.
Now it so happens that the action of $\Delta_{Q_{n}}$ on $M_{Q_{n}}$ factors through the map to $R_{Q_{n}}$. So therefore we have
$\displaystyle \mathcal{O}[\Delta_{Q_{n}}]\to R_{Q_{n}}\to\mathbb{T}_{Q_{n}}\curvearrowright M_{Q_{n}}$
### Taylor-Wiles patching: The patching construction
Now we will perform the patching construction, which means taking the inverse limit over $n$. First we must show that this is even possible, i.e. that we have an inverse system. We can formalize this via the notion of a patching datum.
We let $S_{\infty}$ denote $\mathcal{O}[[(\mathbb{Z}_{p})^{q}]]\cong \mathcal{O}[[x_{1},\ldots,x_{q}]]$ and let $\mathfrak{a}$ denote the ideal $(x_{1},\ldots,x_{q})$. Let us also define $R_{\infty}$ to be another power series ring $\mathcal{O}[[y_{1},\ldots,y_{q}]]$ but in a different set of variables of the same number. In the non-minimal case they might look quite different, but in either case there will be a map from $S_{\infty}$ to $R_{\infty}$; this may be thought of as the limiting case of the map from $\mathcal{O}[\Delta_{Q_{n}}]$ to $R_{Q_{n}}$ discussed earlier.
Now let $n$ be a positive integer. Let $\mathfrak{a}_{n}$ be the kernel of the surjection $S_{\infty}\twoheadrightarrow \mathcal{O}[(\mathbb{Z}/p^{n}\mathbb{Z})^{q}]$, let $S_{n}$ be $S_{\infty}/(\varpi^{n},\mathfrak{a}_{n})$, and $\mathfrak{d}_{n}$ be the ideal $(\varpi^{n},\mathrm{Ann}_{R}(M)^{n})$. Abstractly, a patching datum of level $n$ is a triple $(f_{n},X_{n},g_{n})$ where
• $f_{n}:R_{\infty}\twoheadrightarrow R/\mathfrak{d}_{n}$ is a surjection of complete Noetherian local $\mathcal{O}$ algebras
• $X_{n}$ is a $R_{\infty}\otimes_{\mathcal{O}} S_{n}$-module, finite free over $S_{n}$, such that
• $\mathrm{im}(S_{N}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(R_{\infty}\to\mathrm{End}_{\mathcal{O}}X)$
• $\mathrm{im}(\mathfrak{a}\to\mathrm{End}_{\mathcal{O}}X)\subseteq \mathrm{im}(\mathrm{ker}(f)\to\mathrm{End}_{\mathcal{O}}X)$
• $g_{n}:X/\mathfrak{a}\xrightarrow M/(\varpi^{n})$ is an isomorphism of $R_{\infty}$-modules
We say that two patching data $(f_{n},X_{n},g_{n})$ and $(f_{n}',X_{n}',g_{n}')$ of level $n$ are isomorphic if $f_{n}=f_{n}'$ and there exists an isomorphism $X_{n}\cong X_{n}'$ compatible with $g_{n}$ and $g_{n}'$. We note the important fact that there are only finitely many isomorphism classes of patching data for any level $n$.
Now we will specialize this abstract construction to help us prove our R=T theorem. We choose
• $f_{n}:R_{\infty}\twoheadrightarrow R_{Q_{n}}\twoheadrightarrow R\twoheadrightarrow R/\mathfrak{d}_{n}$
• $X_{n}=M_{Q_{n}}\otimes_{S_{\infty}} S_{n}$
• $g_{n}$ is induced by the isomorphism between the $\Delta_{Q_{n}}$-coinvariants of $H^{1}(Y_{Q_{n}},\mathcal{O})_{\mathfrak{m}_{Q_{n}}}$ and $H^{1}(Y(\Gamma),\mathcal{O})_{\mathfrak{m}}$
If we have a patching datum $D_{m}=(f_{m},X_{m},g_{m})$ of level $m$, we may form $D_{m}\mod n=D_{m,n}=(f\mod \mathfrak{d}_{n},X_{m}\otimes_{S_{m}} S_{n},g_{m}\otimes_{S_{m}}S_{n})$ which is a patching datum of level $n$.
Now recall that for any fixed $n$, we can only have a finite number of isomorphism classes of patching datum of level $n$. This means we can find a subsequence $(m_{n})_{n\geq 1}$ of $(m)_{m\geq 1}$ such that $D_{m_{n+1},n+1}\mod n\cong D_{m_{n},n}$.
We can now take inverse limits. Let $M_{\infty}=\varprojlim_{n}X_{m_{n}}$, let the surjection $R_{\infty}\twoheadrightarrow R$ be given by $\varprojlim_{n}f_{m_{n},n}$, and let the surjection $M_{\infty}\twoheadrightarrow M$ be given by $\varprojlim_{n}g_{m_{n},n}$. We have
$\displaystyle \mathcal{O}[[x_{1},\ldots,x_{g}]]\to R_{\infty}\to\mathbb{T}_{\infty}\curvearrowright M_{\infty}$
Just as $M_{Q_{n}}$ is free as a module over $\mathcal{O}[\Delta_{Q_{n}}]$, we have that $M_{\infty}$ is free as a module over $S_{\infty}$. We will now use some commutative algebra to show that $M_{\infty}$ is a free $R_{\infty}$-module. The depth of a module $M'$ over a local ring $R'$ with maximal ideal $\mathfrak{m'}$ is defined to be the minimum $i$ such that $\mathrm{Ext}^{i}(R'/\mathfrak{m}',M')$ is nonzero. The depth of a module is always bounded above by its dimension.
Now the dimension of $R_{\infty}$ is $1+q$ (we know this since we defined it as a power series $\mathcal{O}[[y_{1},\ldots,y_{q}]]$). This bounds $\mathrm{dim}_{R_{\infty}}(M_{\infty})$, and by the above fact regarding the depth of a module, $\mathrm{dim}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{R_{\infty}}(M_{\infty})$. Since the action of $S_{\infty}$ on $M_{\infty}$ factors through the action of $R_{\infty}$, $\mathrm{depth}_{R_{\infty}}(M_{\infty})$ bounds $\mathrm{depth}_{S_{\infty}}(M_{\infty})$. Finally, since $M_{\infty}$ is a free $S_{\infty}$-module, we have that $\mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$. In summary,
$\displaystyle 1+q=\mathrm{dim}(R_{\infty})\geq \mathrm{dim}_{R_{\infty}}(M_{\infty})\geq\mathrm{depth}_{R_{\infty}}(M_{\infty})\geq \mathrm{depth}_{S_{\infty}}(M_{\infty})=1+q$
and we can see that all of the inequalities are equalities, and all the quantities are equal to $1+q$. The Auslander-Buchsbaum formula from commutative algebra tells us that
$\displaystyle \mathrm{proj.dim}_{R_{\infty}}(M_{\infty})=\mathrm{depth}(R_{\infty})-\mathrm{depth}_{R_{\infty}}(M_{\infty})$
and since both terms on the right-hand side are equal to $1+q$, the right-hand side is zero. Therefore the projective dimension of $M_{\infty}$ relative to $R_{\infty}$ is zero, which means that $M_{\infty}$ is a projective module over $R_{\infty}$. Since $R_{\infty}$ is local, this is the same as saying that $M_{\infty}$ is a free $R_{\infty}$-module.
We have that $M\cong M_{\infty}/\mathfrak{a}M_{\infty}$ is a free module over $R_{\infty}/\mathfrak{a}R_{\infty}$. Since this action factors through maps $R_{\infty}/\mathfrak{a}R_{\infty}\to R\to\mathbb{T}(\Gamma)_{\mathfrak{m}}$ which are all surjections, they have to be isomorphisms, and we have that $M$ is a free $R$-module, and therefore $R\cong\mathbb{T}(\Gamma)_{\mathfrak{m}}$. This proves our R=T theorem.
### Generalizations and other applications of Taylor-Wiles patching
We have discussed only the “minimal case” of Taylor-Wiles patching, but one can make use of the same ideas for the non-minimal case, and one may also apply Taylor-Wiles patching to show the modularity of $2$-dimensional representations of $\mathrm{Gal}(\overline{F}/F)$ for $F$ a totally real field (in this case on the automorphic side we would have Hilbert modular forms).
However, when $F$ is a more general number field the situation is much more complicated, because one of the facts that we have used, which is vital to Taylor-Wiles patching, now fails. This is the fact that the dimension of the dual Selmer group (which is the cardinality of our sets of Taylor-Wiles primes) and the dimension of the Selmer group (which is also the dimension of the tangent space of the Galois deformation ring $R$) are equal (again this is what is known as the “numerical coincidence”). This is the important property that can fail for more general number fields. Here the dimensions of the dual Selmer group and the Selmer group may differ by some nonzero quantity $\delta$.
Moreover, in our discussion we made use of the fact that the cohomology was concentrated in a single degree. For more general number fields this is no longer true. Instead we will have some interval for which the cohomology is nonzero. However, it so happens (for certain “nice” cases) that the length of this interval is equal to $\delta+1$. This is a hint that the two complications are related, and in fact can be played off each other so that they “cancel each other out” in a sense. Instead of patching modules, in this case one patches complexes instead. These ideas were developed in the work of Frank Calegari and David Geraghty.
The method of Taylor-Wiles patching is also being put forward as an approach to the p-adic local Langlands correspondence (which is also closely related to modularity as we have seen in Completed Cohomology and Local-Global Compatibility), via the work of Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin. This is also closely related to the ideas discussed at in Moduli Stacks of (phi, Gamma)-modules (where we used the same notation $M_{\infty}$ for the patched module). Namely, we expect a coherent sheaf $\mathcal{M}$ on the moduli stack of $\varphi,\Gamma$-modules which, “locally” coincides or is at least closely related to the patched module $M_{\infty}$. This has applications not only to the p-adic local Langlands correspondence as mentioned above, but also to the closely-related Breuil-Mezard conjecture. We will discuss these ideas and more in future posts.
References:
Modularity Lifting (Course Notes) by Patrick Allen
Modularity Lifting Theorems by Toby Gee
Beyond the Taylor-Wiles Method by Jack Thorne
Motives and L-functions by Frank Calegari
Overview of the Taylor-Wiles Method by Andrew Snowden (lecture notes from the Stanford Modularity Lifting Seminar)
Reciprocity in the Langlands Program Since Fermat’s Last Theorem by Frank Calegari
Modularity Lifting Beyond the Taylor-Wiles Method by Frank Calegari and David Geraghty
Patching and the p-adic local Langlands Correspondence by Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas, and Sug Woo Shin | 2023-04-02 08:56:46 | {"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": 287, "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": 0.9537984132766724, "perplexity": 127.80946152419118}, "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/1679296950422.77/warc/CC-MAIN-20230402074255-20230402104255-00763.warc.gz"} |
https://www.mometrix.com/academy/rational-expressions/ | # Rational Expressions
Hi, and welcome to this video about rational expressions.
Before we talk about what rational expressions are and the operations that can be performed with them, it may be a good idea to review some terminology. A polynomial is a group of algebraic or numeric terms that are joined by the operations of addition or subtraction. There are different types of polynomials based on the number of terms that are present:
### Types of Polynomials:
Examples# of TermsType
$$4$$, $$5x$$, $$3x^4$$, $$8xy^2$$1Monomial
$$(x+3)$$, $$(x^2-1)$$, $$(3xy+2y)$$2Binomial
$$(x^2+5x+6)$$,
$$(x^2-2xy+y^2)$$
3Trinomial
A rational expression is nothing more than a ratio of polynomials. As you know from previous practice with ratios, you cannot divide by 0. It is important to keep this in mind when dealing with rational expressions because allowing a value of 0 in the denominator would create an expression that is “undefined.”
Using function notation for polynomials, such as $$p(x)$$ and $$g(x)$$, a rational expression can be defined like this:
$$\frac{p(x)}{g(x)}$$$$\text{ where }g(x)\neq 0$$
Here’s an example:
$$\frac{5x}{x-2}$$
This example shows a rational expression with a monomial, $$5x$$, in the numerator and a binomial, $$(x-2)$$, in the denominator. The value $$x=2$$ is the excluded value, as it would result in a denominator of 0. This expression cannot be simplified further.
$$\frac{5x}{x-2}$$, $$x\neq 2$$
## Addition and Subtraction of Rational Expressions
Rational expressions cannot be added or subtracted unless they share a common denominator. Algebraic rules allow us to adjust fractions to create common denominators as long as we make the same adjustment to the numerator. Let’s look at an example with fractions:
$$\frac{1}{5}+\frac{3}{7}$$
In order to add $$\frac{1}{5} + \frac{3}{7}$$, we must create a common denominator. Specifically, we need to determine the least common denominator, meaning the smallest multiple of 5 and 7. In this case, that number is 35. The adjustment to each fraction that needs to be made to create the common denominator is:
$$\frac{1}{5} (\frac{7}{7}) + \frac{3}{7} (\frac{5}{5})$$
$$\frac{7}{35}+\frac{15}{35}$$
$$=\frac{22}{35}$$
We need to multiply the first denominator of the first expression by 7 to get to 35, but we also must multiply the numerator by the same value. Because we have simply created an equivalent fraction to allow us to add. Likewise, the second expression must be multiplied by $$\frac{5}{5}$$, in order to create 35 in the denominator. After these adjustments are made and the denominators are the same, simplify the numerators:
$$\frac{7+15}{35} \rightarrow \frac{22}{35}$$
Rational expressions are added and subtracted the same way. Typically, the expressions need to be factored before the least common denominator can be determined and domain restrictions (excluded values) should be noted. Consider this example:
$$\frac{3x}{x-2} + \frac{5}{x^2-x-2}$$
$$\frac{3x}{x-2} + \frac{5}{(x-2)(x+1)}$$
Now we want to determine the lowest common denominator. What is the smallest multiple of $$(x-2)$$ and $$(x-2)(x+1)$$?
Alright now that we have our equations written out, we want to make sure that we don’t have any domains that need to be excluded. Which, we do. Remember we don’t want 0 in the denominator position. So, in these scenarios, we know that $$x\neq 2$$, or over here, -1. If $$x=-1$$ this would end up being 0, multiplied by another term, still remains 0. The 0 in the denominator, we can’t have that. Over here, if $$x=2$$, $$2-2=0$$, again, we can’t have a 0 in the denominator, so these are our two terms, our domains that need to be excluded.
Alright, now we need to adjust the first expression by multiplying by the factor needed to match the least common denominator. So if we want our first term here, to match this term over here in the denominator position, we’re going to multiply by $$x+1$$ in the numerator and the denominator.
$$\frac{3x}{x-2} \frac{(x+1)}{(x+1)} + \frac{5}{(x-2)(x+1)}$$
Now we’re going to rewrite the expression as a fraction, and simplify the numerator. And now we have our answer:
$$\frac{3x^{2}+3x}{(x+1)(x-2)}+\frac{5}{(x-2)(x+1)}$$$$=\frac{3x^{2}+3x+5}{(x-2)(x+1)}$$
## Multiplication and Division of Rational Expressions
### Multiplying Rational Expressions
Here are the three steps to multiplying rational expressions. Now, remember, when multiplying fractions, numerators and denominators are multiplied straight across.
Step #1: Factor the numerator and denominator of each expression being multiplied.
Step #2: Simplify by canceling out common factors from the numerator and the denominator.
Step #3: The final answer is what is left after canceling. You may be asked to include domain restrictions with your solution.
Let’s use these steps to solve an example problem:
$$\frac{3x}{4x-8}\cdot \frac{2x^{2}-4x}{9x}$$
$$\frac{3x}{4(x-2)}\cdot \frac{2x(x-2)}{9x}$$
$$\frac{6x^{2}(x-2)}{36x(x-2)}$$
Now, because we have like terms in the numerator and the denominator position, we’re able to cancel them out. That leaves us with:
$$\frac{6x^2}{36x}$$
But we can simplify this even further, remember, 6 is a factorof 36, so let’s simplify:
$$\frac{x^2}{6x}$$
And yet, we can simplify this again, remember, you have an $$x$$ in the numerator and an $$x$$ in the denominator, so let’s simplify:
$$\frac{x}{6}$$
And now we have our answer, $$\frac{x}{6}$$. But that’s not the complete answer. Remember, we have some domain that we have to exclude. Up here, $$x\neq 2$$ because $$2\times 4-8=0$$. And we can’t have a 0 in the denominator. So 2 is out, $$x\neq 2$$. Also, $$x\neq 0$$, because $$0\times 9=0$$, and again, give us a 0 in the denominator. So the domains we have to exclude from this answer are 2 and 0. So our answer is $$\frac{x}{6},x\neq 2,0$$.
### Dividing Rational Expressions
Dividing rational expressions includes one extra step at the beginning of the process. When dividing by a fraction, it is the same as multiplying by the reciprocal of the second fraction. You can remember this rule as, “Keep, Change, and Flip” which translates to keep the first fraction, change the operation to multiplication, and take the reciprocal (or flip) of the second fraction.
Keep in mind that domain restrictions must be considered from both the numerator and denominator of the second fraction because of the “flip” in the division process.
Here’s an example:
$$\frac{9x^2}{x^2+12x+36} \div \frac{12x}{x^2+6x}$$
Now, remember our three steps: keep the first fraction, change the operation, and then flip. Here we go:
$$\frac{9x^2}{x^2+12x+36} \times \frac{x^2+6x}{12x}$$, $$x\neq 0$$, $$-6$$
Here is now where we multiply, cause we kept the first fraction, we changed to multiplication, and then we flipped the fraction over here. So, time to multiply.
$$\frac{9x^{2}}{(x+6)(x+6)}\cdot \frac{(x+6)}{12}=\frac{3x^{2}}{4(x+6)}$$
So now we have our answer: $$\frac{3x^{2}}{4(x+6)}$$.
But remember, that’s not our complete answer if we don’t include our restricted domain, we have $$x\neq 0$$, and $$x\neq -6$$. Remember we have to make sure that we don’t have a 0 in the denominator or the numerator of our second term.
I hope this review was helpful! See you next time!
## Practice Questions
Question #1:
Which polynomial is considered a binomial?
$$2x$$
$$x^2+2x-4$$
$$100$$
$$3x-7$$
A polynomial consists of one or more monomials combined by addition or subtraction.
A monomial has one term, such as $$2x$$.
A binomial has two terms, such as $$3x−7$$.
A trinomial has three terms, such as $$x^2+2x-4$$.
Question #2:
Add the following polynomials: $$\frac{x+2}{3x}+\frac{x-3}{6x}$$
$$3x+1$$
$$\frac{3x+1}{6x}$$
$$\frac{6x+1}{3x}$$
$$6x$$
Step 1: Rewrite the expression with a common denominator. In this case, we need to multiply the numerator and denominator by 2 in the first fraction so that our denominator becomes $$6x$$. $$\frac{x+2}{3x}+\frac{x-3}{6x}$$ becomes $$\frac{(x+2)(2)}{(3x)(2)}+\frac{(x+3)}{(6x)}$$, which simplifies to $$\frac{2x+4}{6x}+\frac{x-3}{6x}$$.
Step 2: Add the numerators, and combine like terms. $$\frac{2x+4}{6x}+\frac{x-3}{6x}$$ becomes $$\frac{3x+1}{6x}$$ which is the final answer.
Question #3:
What are the domain restrictions for the following expression?
$$\frac{7x+2}{x^2-4}$$
$$x\neq2$$ and $$x\neq-2$$
$$x\neq2$$
$$x\neq-2$$
$$x\neq7$$ and $$x\neq-4$$
A rational expression cannot have a zero in the denominator. For the expression $$\frac{7x+2}{x^2-4}$$ there are two values for x that will produce a zero in the denominator. It can be helpful to break apart the denominator $$(x^2-4)$$ into $$(x+2)(x-2)$$ in order to identify these domain restrictions. Now that the denominator is factored, we can see that if x is 2, the term on the left will be zero. Similarly, if x is -2, the term on the right will be zero. Therefore, x cannot be 2 or -2. These are referred to as domain restrictions: $$x\neq2$$ and $$x\neq-2$$.
Question #4:
$$\frac{12x^2}{5y^3}\times\frac{20y^4}{6x^3}$$
$$\frac{8y}{x^2}$$
$$\frac{6y}{x}$$
$$\frac{8y}{y^2}$$
$$\frac{8y}{x}$$
Step 1: When multiplying fractions, we simply multiply straight across. Numerator times numerator, and denominator times denominator. $$\frac{12x^2}{5y^3}\times\frac{20y^4}{6x^3}$$ now becomes $$\frac{240x^2y^4}{30x^3y^3}$$.
Step 2: Simplify by canceling out common factors that appear in the numerator and denominator. $$\frac{240x^2y^4}{30x^3y^3}$$ now becomes $$\frac{8y}{x}$$.
Question #5:
Divide the following: $$\frac{x^2-x-12}{3x-15}\div\frac{x^2-9}{24x-72}$$
$$\frac{8(x-4)}{x-5} x\neq,3,-3$$
$$\frac{x-3}{x-5} x\neq5$$
$$\frac{8(x+4)}{x+5} x\neq-3$$
$$\frac{x+4}{x-5} x\neq5,4,6$$
Step 1: “Keep, Change, Flip”
Keep the fraction on the left in its original form. Change the division sign to a multiplication sign. Flip the second fraction.
$$\frac{x^2-x-12}{3x-15}\div\frac{x^2-9}{24x-72}$$ now becomes $$\frac{x^2-x-12}{3x-15}\times\frac{24x-72}{x^2-9}$$.
Step 2: Factor.
Factoring makes the process of canceling more straight forward later on.
$$\frac{x^2-x-12}{3x-15}\times\frac{24x-72}{x^2-9}$$ now becomes $$\frac{(x-4)(x+3)}{3(x-5)}\times\frac{24(x-3)}{(x+3)(x-3)}$$.
Step 3: Identify the restrictions.
Restrictions are the values for x that will produce a zero in the denominator.
x cannot be 5 because that would produce a zero in the denominator of the first fraction. x cannot be 3 or -3 because that would produce a zero in the denominator of the second fraction. The restriction are $$x≠5,3,\text{ or }-3$$
Step 4: Simplify. Cancel out common factors in the numerator and denominator.
$$\frac{(x-4)(x+3)}{3(x-5)}\times\frac{24(x-3)}{(x+3)(x-3)}$$ now becomes $$\frac{8(x-4)}{x-5}$$.
Final Answer: $$\frac{8(x-4)}{x-5} x ≠ 5,3,-3$$. | 2022-05-24 21:10:16 | {"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": 0.9255398511886597, "perplexity": 292.1879753925964}, "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/1652662577259.70/warc/CC-MAIN-20220524203438-20220524233438-00546.warc.gz"} |
https://www.gamedev.net/forums/topic/555541-why-is-the-following-texture-colour-not-lit-with-additive-blending/ | # Why is the following texture colour not lit with additive blending?
This topic is 3173 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
If I have a red light and I have the following naff texture: Why is only the yellow lit up and the blue ignored? This doesn't seem correct? I am combining my light and textures with the following:
float4 PixelShaderFunction(VertexShaderOutput input) : COLOR0
{
// FinalColour = DiffuseColour * DiffuseLight + SpecularLight
float3 DiffuseColour = tex2D(ColourSampler, input.TexCoord).rgb;
float4 light = tex2D(LightSampler, input.TexCoord);
float3 DiffuseLight = light.rgb;
float SpecularLight = light.a;
return float4((DiffuseColour * DiffuseLight + SpecularLight), 1);
}
I've no idea what's going wrong, can someone please help?
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When multiplying blue with red light, you get black. Yellow is (1, 1, 0), and blue is (0, 0, 1).
(1, 1, 0) * (1, 0, 0) = (1, 0, 0) = red
(0, 0, 1) * (1, 0, 0) = (0, 0, 0) = black
When there is only red light hitting a surface that only reflects blue light, there is no light that will be reflected and visible.
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Excellent, thank you.
How would I add ambient light to the mix then to make sure that the area is evenly lit?
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Quote:
Original post by Spa8nkyExcellent, thank you.How would I add ambient light to the mix then to make sure that the area is evenly lit?
In reality you would be hard-pressed to find a material that is completely red, blue, green, or black. The same goes for light sources.
I'm not sure what you mean by evenly lit, but I would make sure that I don't use textures where any pixel component (R,G,B) is completely black. I wouldn't use a light that was completely one color either unless, I wanted some sort of special effect.
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https://infoscience.epfl.ch/record/232407 | Infoscience
Thesis
# Aggregate route choice models: the mental representation item approach
Route choice analysis concerns the understanding, modeling and prediction of the itinerary of an individual who travels from one position to another. In this thesis we elaborate on aggregate route choice analysis. The objective is the development of a flexible framework for analysing and predicting route choice behavior. The research is motivated by the need to reduce the structural complexity of the state of the art route choice models and aims at facilitating their practical applications. Our approach is inspired by the environmental images of the physical space that individuals form in their minds. The framework is based on elements designed to mimic these representations. In this context, we introduce the concept of mental representation item ($\mathrm{MRI}$) in route choice analysis. The $\mathrm{MRIs}$ represent the strategic decisions of individuals and constitute the building blocks of the alternatives of the aggregate model. They play the same role as the links do in the specification of a disaggregate model. In contrast to the links, the $\mathrm{MRIs}$ are not dictated by the definition of the network model. Their definition depends on the analyst, allowing her to control the trade-off between complexity and realism, according to the needs of the specific application and the data availability. We start by presenting a methodology for the definition of operational random utility models based on $\mathrm{MRIs}$. As a proof of concept, we define a simple model for the town of Borl"ange, in Sweden, and test it using real data. We further discuss applications of the proposed model to traffic assignment and route guidance. The results demonstrate that the use of simple methods leads to a meaningful model that can be estimated and used in practice. We then investigate the capability of the proposed $\mathrm{MRI}$ model to derive route choice indicators for practical applications, through comparison with a state of the art disaggregate model. The recursive logit (RL) model is selected as the representative of the existing disaggregate approaches. An extension of the $\mathrm{MRI}$ framework with the definition of a graph of $\mathrm{MRI}$ elements is presented and methods to derive route choice indicators from a model that does not correspond to the intended level of analysis are proposed. The evaluation of the models' performance at the aggregate level shows that the $\mathrm{MRI}$ model should be preferred against a disaggregate model that is subjected to aggregation, if an aggregate analysis is of interest. To demonstrate the generalization and applicability of the framework, we use a dataset from the city of Qu\'ebec, in Canada. Our approach is motivated by (i) the additional complexity in the definition of the model due to the size of the city, and (ii) the lack of a detailed disaggregate network model. The proposed model is (i) operationalized using simple techniques, (ii) compatible with the standard estimation procedures and (iii) by integration with the RL model, readily applied to the prediction of flows on the major segments of the network. This model is not as simple as the first $\mathrm{MRI}$ model, yet still of much lower structural complexity in comparison with the disaggregate approach, allowing for fast computation times. The results demonstrate its capability to reproduce the patterns in the observed flows. This thesis contributes by (i) gradually addr
Thèse École polytechnique fédérale de Lausanne EPFL, n° 8004 (2017)
Programme doctoral en génie civil et environnement
Faculté de l'environnement naturel, architectural et construit
Institut du développement territorial
Laboratoire transport et mobilité
Jury: Prof. Alain Nussbaumer (président) ; Prof. Michel Bierlaire (directeur de thèse) ; Prof. Nikolaos Geroliminis, Prof. Gunnar Flötteröd, Prof. Otto Anker Nielsen (rapporteurs)
Public defense: 2017-11-10
#### Reference
Record created on 2017-11-13, modified on 2017-11-13
### Fulltext
• Thesis submitted - Forthcoming publication | 2017-12-17 15:45:36 | {"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": 0.2773929238319397, "perplexity": 723.4172763627353}, "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-2017-51/segments/1512948596115.72/warc/CC-MAIN-20171217152217-20171217174217-00083.warc.gz"} |
https://answers.opencv.org/answers/2118/revisions/ | In particular (1) the OpenCV sources are now in GIT and (2) after successful build for Android you need to run "make install" command in the build folder and get a kind of SDK in "install" subfolder. | 2019-07-16 20:21:09 | {"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": 0.25514301657676697, "perplexity": 4500.531582803001}, "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-2019-30/segments/1563195524879.8/warc/CC-MAIN-20190716201412-20190716223412-00355.warc.gz"} |
https://brilliant.org/discussions/thread/i-dont-get-it/ | ×
# I Don't Get It!
I came across a question which uses Fermat's little theorem:
The Question:
Find a positive integer n such that $$7n^{25}$$ - 10 is divisible by 83.
The Solution given in the book:
Since 7 x 37 = 259 = 10 mod 83
We have to find a value of n such that $$7n^{25}$$ = 7 x 37 mod 83
This is equivalent to $$n^{25}$$ = 37 = $$2^{20}$$ mod 83
By Fermat's Theorem,
$$2^{82k}$$ = 1 mod 83 for all k.
So we need to choose n such that $$n^{25}$$ =$$2^{82k+20}$$ mod 83
This will be satisfied if k=15
Therefore $$n^{25}$$ = $$2^{1250}$$ mod 83
And so n = $$2^{50}$$
This gives one value.
My problem is that I don't understand the fourth line
That is how 37 = $$2^{20}$$ mod 83?
So can someone please explain this?
Note by Abc Xyz
8 months, 4 weeks ago
Sort by:
$$2^8 \equiv 256 \equiv 249 + 7 \equiv 7$$ (mod $$83$$)
$$\Rightarrow 2^{16} \equiv 7^2$$ (mod $$83$$)
$$\Rightarrow 2^{20} \equiv 49 \cdot 16 \equiv 784 \equiv 37$$(mod $$83$$) · 8 months, 3 weeks ago
Wow !! I didn't think of that. Thanks a LOT Sir !!! · 8 months, 3 weeks ago | 2016-12-10 20:22:20 | {"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": 0.8442667126655579, "perplexity": 669.2913493874091}, "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/1480698543567.64/warc/CC-MAIN-20161202170903-00452-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://www.aimsciences.org/article/doi/10.3934/dcdsb.2016118 | American Institute of Mathematical Sciences
December 2016, 21(10): 3723-3742. doi: 10.3934/dcdsb.2016118
Traveling waves in an SEIR epidemic model with the variable total population
1 School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631
Received December 2015 Revised February 2016 Published November 2016
In the present paper, we propose a simple diffusive SEIR epidemic model where the total population is variable. We first give the explicit formula of the basic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if $\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, the model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0<1$ and $c>0$ (or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial traveling wave solution. Consequently, we obtain the full information about the existence and non-existence of traveling wave solutions of the model by determined by the constants $\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauder fixed point theorem and Laplace transform.
Citation: Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118
References:
show all references
References:
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2019 Impact Factor: 1.27 | 2021-04-11 16:43:56 | {"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": 0.6324256062507629, "perplexity": 3563.0164850100336}, "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-2021-17/segments/1618038064520.8/warc/CC-MAIN-20210411144457-20210411174457-00204.warc.gz"} |
http://www.solutioninn.com/text-categorization-is-the-task-of-assigning-a-given-document | # Question
Text categorization is the task of assigning a given document to one of a fixed set of categories, on the basis of the text it contains. Naive Bayes models are often used for this task in these models, the query variable is the document category, and the ‘effect” variables are the presence or absence of each word in the language; the assumption is that words occur independently in documents, with frequencies determined by the document category.
a. Explain precisely how such a model can be constructed, given as “training data” a set of documents that have been assigned to categories.
b. Explain precisely how to categorize a new document.
c. Is the independence assumption reasonable? Discuss.
Sales0
Views203 | 2016-10-23 21:27:46 | {"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": 0.8853647112846375, "perplexity": 677.1869089024228}, "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/1476988719416.57/warc/CC-MAIN-20161020183839-00066-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/measuring-height-of-building-with-barometer.53545/ | # Measuring height of building with barometer
1. Nov 21, 2004
### Bcisewski
I have a problem where one reading is 743mm at the top of the building and 760mm at the bottom of building. In order to find the height of the building I tried getting the difference of the two readings and multiply by the density of the mercury that it contains. I should come up with 179 m. but don't.
Now I know this can open up a can of worms, but I referenced this subject on the net, and found tons of jokes relating to this subject, but none with the solution I need. :) Granted all of the comments or jokes were valid, however it doesn't help.
Thanks
2. Nov 21, 2004
### marlon
hmm i come up with 191m.
first you need to calculate the pressure values that correspond to the given mercury lengths via p =1.36*10³*10*0.76 and 1.36*10³*10*0.743, I take g = 10 and 1.36*10³ is the mercury-density. Then use the law : p_bottom - p_top = 1.21*10*h and 1.21kg/m³ is the air-density, h is the height. I get then h = 191 meters...
marlon
3. Nov 21, 2004
### Astronuc
Staff Emeritus
Basically, one needs to understand the equivalence in pressure, or differential pressure, and elevation.
Air pressure arises from the fact that a mass of air extends from the point of interest (measurement) to the top of the atmosphere. If one ascends through the atmosphere, the amount of air decreases because 1) the height of the atmosphere above is decreasing, and 2) the density of the atmosphere is decreasing.
But for this problem, let's assume that the density of air is constant.
A barometer works because the air pressure is pushing on some reference area (say a piston, diaphragm, or surface of liquid - like mercury). So the air pressure acts on that surface. Acting in the opposite direction, in the case of a mercury barometer, is the liquid mercury. For static equilibrium, the air pressure acting on the surface must be balance by the pressure exerted by the column of mercury in the barometer. That pressure ( or force acting on the area) is given simply by mgh, where m is the mass of mercury, g is accel of gravity, and h is the displacement from some reference (equilibrium) point. So when the pressure changes the height of mercury (or other fluid) changes. The denser the fluid (like Hg, density = 13.6 g/cm^3), the smaller the displacement.
So how does this relate to the air.
Well, one needs to find the height of a column of air $$h_{air}$$ acting on the reference surface in the barometer in opposition to the mercury.
It all comes down to find a balance of equivalence of force or weight which is proportional to mass, which is proportional to density and volume. But volume is area x height, so the problem simplies find to find an equivalence:
$$\rho_{air}h_{air} = \rho_{Hg}h_{Hg}$$
knowing the density of Hg = 13.6 g/cc, and the height is 17 mm, then one can find the height of the air, which is the height at which one measures the air pressure.
Try density of air at ~0.00129 g/cc,
and the answer for the height should be ~ 179 m or 179,000 mm.
Last edited: Nov 21, 2004
4. Nov 21, 2004
### cAm
or you could take the barometer to the architect, and offer him that nice pretty barometer if he'll tell you how tall the building is. :tongue2:
5. Nov 21, 2004
### marlon
yes the answer is indeed 179m.
I used 1.21kg/m³ for the air density while it should be 1.29kg/m³
marlon
6. Nov 22, 2004
### Clausius2
Sorry, but the jokes are unavoidable here.
There's another form of measuring the height with a barometer. Go up to the roof of the building and throw away the barometer. Then measure the time till it crashes into the floor with your chronometer, then the height will be:
$$h=gt^2$$
:rofl: :rofl: :rofl: Marvellous...
EDIT: I read once that it was the test that Rutherford gave to Bohr (an excellent alumn at that time) for solving. Then Bohr went to the top of the building as I have described and threw away the barometer. Rutherford was impressed at first sight, but lately he kicked the Bohr ass for breaking his barometer. After some time, Rutherford recognized Bohr as his best pupil.
BTW: nobody has said it here, but the pressure of the air (and any gas) is given by the gas hydrostatics or the Boltzmann Barometric Formula:
$$P(z)=P(0)e^{-\frac{gz}{R_g T}}$$
With two pressures you can obtain any difference of heights.
Last edited: Nov 22, 2004
7. Nov 22, 2004
### marlon
nice post clausius...
marlon
8. Nov 22, 2004
### BobG
That's just one common variation of the story. It's doubtful that this really involved Rutherford and Bohr, since the first person known to use this story was Dr Alexander Calandra. But it is purported to be a true story of Calandra and one of his Physics students in the 1950s.
http://www.landiss.com/teaching/calandra.htm
Also, Neils Bohr is not the only Dane to win the Nobel Prize. His son, Aage Bohr and another Dane, Benjamin Mottelson, also won the Nobel Prize (1975).
9. Nov 22, 2004
### Integral
Staff Emeritus
Throwing the barometer is good, but you must take into account the time required for the sound of the crashing barometer to return to the top of the building.
Perhaps the most accurate number is had by the other popular joke solution. You take the barometer to the architect of the building and say "I will give you this beautiful barometer if you tell me how tall the building is"
10. Nov 23, 2004
### Clausius2
Also, you must employ relativistic dynamics to have the exact movement of the barometer, and the returning of the light ray to know if the barometer has crashed yet or not is governed by general relativity :!!)
Maybe the next step of the story is hearing the architect answer:
"What the hell is a barometer?? and why are you bribing me with that?? Go away !!!"
But there's another possibility:
Take the barometer while you are going upstairs the building to measure progressively the height of the building. I mean, if the barometer is $$L$$ meters long, then by Euclides' postulates of the Geometry, the height of the building is:
$$h=k\cdot L$$ where k>1 is some number you can figure out while you are going upstairs. :rofl: :rofl:
11. Nov 23, 2004
### HallsofIvy
Another "standard" method for using a barometer to measure the height of a building (less destructive to the barometer) is this: go to the top of the building, tie a long string to the barometer and lower the barometer until it is ALMOST touching the ground. Now start it swinging and measure the period of the swing. You can then calculate the length of the string from its period and thus the height of the building.
12. Nov 23, 2004
### ZapperZ
Staff Emeritus
.. and yet ANOTHER method would be to wait for a sunny day, measure the length of the building's shadow on the ground, and then measure the length of the barometer's shadow. Knowing the height of the barometer, one can use similar triangles to find the find the height of the building.
Of course, one should do this close to the middle of the day or else the shadow of the building will be too long for it to be practical to measure. :)
Zz.
13. Nov 23, 2004
### BobG
Of course, in the real world, your boss would only ask you to do this in the middle of the night. Being preoccupied by thinking up new adjectives for your boss you'll reach the roof only to realize your barometer's missing. Of course, when you sweep the flashlight over the ground in the vicinity of where you think you were standing, it will be obvious what happened to your barometer. It's still on the ground where you placed it in order to get a truly accurate reading of the pressure at ground level.
Hence, the true ingeniousness of ZapperZ's method. Tie your flashlight to the top of the roof, being careful to aim the beam so it's center (the little filament impression) barely clears the top of the barometer and then measure the angle of the flashlight relative to the side of the building.
Of course, by time you've made it down the stairs in the dark, only falling down three flights of steps in the process, you'll find a raccoon has made off with your barometer.
No problem. You can still see the image of the filaments on the ground, so you can just measure the distance from the filaments to the side of the building. Knowing two angles (provided you're not measuring the Leaning Tower of Pisa) and one side, you can find the height of the building ..... all thanks to your trusty barometer.
14. Nov 23, 2004
### Clausius2
That's out of the problem. The problem stated originally you and a barometer. Who usually have a filament or a string 20 m long in his pocket?. BUT it is probably you'll have a chronometer in your watch. So let's think in realistic solutions or at least solutions according to the problem statement: a barometer, you and your usual possesions (it's not valid a thermometer, a laser, a 10m string, a filament, and that sorts of things that nobody except a crazy guy will carry on in his pocket).
Hmmm.......Surely there's another way..... If I get down my trousers....
I suggest a game here. As for now, the last post here suggesting a valid solution is the winner. Who wants to play?
15. Nov 23, 2004
### BobG
The filament is in the light bulb in the flashlight, not in your pocket. Check here for an explanation: http://www.bulbcollector.com/ubb/Forum10/HTML/000415.html [Broken]
And, for your sake, I hope this building isn't so tall that the roof level is above the electrical power lines. :surprised
Last edited by a moderator: May 1, 2017 | 2018-03-21 03:42:23 | {"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": 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": 0.6616712808609009, "perplexity": 915.863700902596}, "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-2018-13/segments/1521257647567.36/warc/CC-MAIN-20180321023951-20180321043951-00408.warc.gz"} |
https://stats.stackexchange.com/questions/368854/statistical-significance-from-an-iterated-count | Statistical significance from an iterated count
Question in brief: I am attempting to determine whether statistical significance can be measured merely from counts generated from iterating the same procedure, but where the probability of success is unknown both longitudinally and cross-sectionally. If so, is there an analytic solution? If not, can you provide guidance on a numerical solution?
Question in full: The following experiment has been conducted. There are 33 children on a sports team, and once a day a coach is tasked with placing each child into one of two groups: group one, which is the group that will actually play the match, and group two, which consists of those children who will be sent home. The coach's objective is to win the match, and he can do that by selecting as many (or as few) children as he desires (i.e., $$[0,33]$$, where a selection of 0 would indicate a forfeit, though empirically this never occurred; also, the game doesn't require a minimum number of participants). This task is performed each day of the school year (175 iterations). We can assume (for reasons unnecessary to expound upon here) that the coach does not learn; i.e., each time he performs this task it is independent of past attempts. Further, the same 33 children are always decided upon and the children's perceived value remains the same each day (i.e., the children do not improve, etc.).
Thus there are two things we have measured: (1) how many children the coach selects in each iteration (i.e., the sum of each "Iteration n" column below), and (2) the number of times each child is selected (i.e., the "Sum of count" column below). The table below shows a sample of the collected data:
Child ID | Iteration 1 | Iteration 2 | ... | Sum of count
----------------------------------------------------------
01 | 0 | 0 | ... | 0
02 | 0 | 1 | ... | 28
03 | 1 | 1 | ... | 175
04 | 0 | 0 | ... | 0
. | . | . | ... | .
. | . | . | ... | .
. | . | . | ... | .
33 | 1 | 1 | ... | 172
where if the child was selected in that iteration he/she received a 1, otherwise a 0. I want to know whether, for example, child 02 was selected a statistically-significant number of times (i.e., is 28 significant?).
Thoughts: Because the selection of a child into group one is a binary event ("yes" or "no"), I thought I could compare each child's count against the binomial CDF. But this doesn't work because I don't know ex-ante the probability of a child being selected; empirically we see that somewhere around only 3 to 6 children are generally selected (meaning around 27 to 30 are placed into group two in each iteration). I think this is effectively because each child is not independent of the others: some are "fast runners," some are "the big guy," etc., and thus the coach is selecting the best from a smaller group of categories. But this is not required in the setup, and I don't know which category the coach will care about on a particular iteration. I thought a Poisson distribution might be appropriate, but I don't know ex-ante lambda (the average number of children selected in an iteration). There are some similarities to the Mann-Whitney U test, but in the present setup there is no rank: group one or group two is all I have. And so on. | 2019-10-18 01:27:04 | {"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": 1, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5692492723464966, "perplexity": 461.46530681909024}, "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-2019-43/segments/1570986677412.35/warc/CC-MAIN-20191018005539-20191018033039-00502.warc.gz"} |
https://mathhelpboards.com/threads/converting-ode-to-a-system-of-odes.2078/ | # [SOLVED]converting ODE to a system of ODEs
#### dwsmith
##### Well-known member
Given $x''-x+x^3+\gamma x' = 0$.
Is the below correct? Can I do this? The answer is yes.
Let $x_1 = x$ and $x_2 = x'$. Then $x_1' = x_2$.
\begin{alignat}{3}
x_1' & = & x_2\\
x_2' & = & x_1 - x_1^3 + \gamma x_2
\end{alignat}
Then I have the above linear system from the given ODE.
Last edited:
#### Ackbach
##### Indicium Physicus
Staff member
Given $x''-x+x^3+\gamma x' = 0$.
Is the below correct? Can I do this? The answer is yes.
Let $x_1 = x$ and $x_2 = x'$. Then $x_1' = x_2$.
\begin{alignat}{3}
x_1' & = & x_2\\
x_2' & = & x_1 - x_1^3 + \gamma x_2
\end{alignat}
Then I have the above linear system from the given ODE.
Second equation should be
$$x_{2}'=x_{1}-x_{1}^{3}-\gamma x_{2}.$$
#### dwsmith
##### Well-known member
Second equation should be
$$x_{2}'=x_{1}-x_{1}^{3}-\gamma x_{2}.$$
Thanks typo. I trying to find the attraction basin for this system in another post. Are you familiar with that stuff?
#### Sudharaka
##### Well-known member
MHB Math Helper
Thanks typo. I trying to find the attraction basin for this system in another post. Are you familiar with that stuff?
I think your question is answered >>here<<. | 2021-01-24 21:23:03 | {"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": 0.9376062750816345, "perplexity": 1377.4047013952768}, "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-2021-04/segments/1610703557462.87/warc/CC-MAIN-20210124204052-20210124234052-00028.warc.gz"} |
http://mathoverflow.net/feeds/question/5798 | Is there a natural way to give a bisimplicial structure on a 2-category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T03:30:29Z http://mathoverflow.net/feeds/question/5798 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5798/is-there-a-natural-way-to-give-a-bisimplicial-structure-on-a-2-category Is there a natural way to give a bisimplicial structure on a 2-category? Fei 2009-11-17T07:46:34Z 2009-11-17T12:15:38Z <p>I mean by the nerve functor. </p> <p>Given a 2-category $\mathcal{C}$, if we forget the 2-category structure (just view $\mathcal{C}$ as a category), the nerve functor will give us a simplicial set $N\mathcal{C}$. However, $\mathcal{C}$ is a 2-category, thus for any two objects $x,y\in\mathcal{C}$, $Hom_{\mathcal{C}}(x,y)$ is a category, applying the nerve functor gives us a simplicial set $N(Hom(x,y))$.</p> <p>My question is, can these two simplicial set structure compatible in some way, gives us a bisimplicial set $N_{p,q}(\mathcal{C})$, say? Or is there another way to give a bisimplicial structure on a 2-category? </p> http://mathoverflow.net/questions/5798/is-there-a-natural-way-to-give-a-bisimplicial-structure-on-a-2-category/5812#5812 Answer by Urs Schreiber for Is there a natural way to give a bisimplicial structure on a 2-category? Urs Schreiber 2009-11-17T12:15:38Z 2009-11-17T12:15:38Z <p>Yes. This is called the <a href="http://ncatlab.org/nlab/show/double+nerve" rel="nofollow">double nerve</a> of a 2-category.</p> <p>See in particular the first reference cited at that link.</p> | 2013-05-19 03:30:27 | {"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": 0.9268792867660522, "perplexity": 1431.2329922609333}, "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-2013-20/segments/1368696383218/warc/CC-MAIN-20130516092623-00043-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.groundai.com/project/fluctuation-driven-vortex-fractionalization-in-topologically-ordered-superfluids-of-cold-atoms/ | Spin Structure and Critical Nucleation Frequency of Fractionalized Vortices in 2D Topologically Ordered Superfluids of Cold Atoms
# Spin Structure and Critical Nucleation Frequency of Fractionalized Vortices in 2D Topologically Ordered Superfluids of Cold Atoms
## Abstract
We have studied spin structures of fluctuation-driven fractionalized vortices and topological spin order in 2D nematic superfluids of cold sodium atoms. Our Monte Carlo simulations suggest a softened -spin disclination structure in a half-quantum vortex when spin correlations are short ranged; in addition, calculations indicate that a unique non-local topological spin order emerges simultaneously as cold atoms become a superfluid below a critical temperature. We have also estimated fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps and discussed probing these excitations in experiments.
Quantum number fractionalization has been one of the most fundamental and exciting concepts studied in modern many-body physics and topological field theoriesSu80 (); Tsui82 (); Laughlin83 (); Jackiw76 (). During the past few years, low dimensional fractionalized quantum states have further been proposed to be promising candidates for carrying out fault tolerant quantum computationKitaev03 () and their realizations in optical lattices were exploredDuan03 (); Buchler05 (); Micheli06 (). A closely related topic in which there has also been a growing interest is vortex fractionalization in cold gases (see for instance Ref.Demler02 (); Mukerjee06 (); Semenoff07 (); Thouless98 ()). Especially, motivated by experiments on low dimensional cold gasesHadzibabic06 (), Mukerjee et al studied 2D superfluids of cold atoms and analyzed the role played by fractionalized vortices in phase transitionsMukerjee06 (). However, spin structures of those half-quantum vortices induced by thermal fluctuations and potential topological orderWen04 () haven’t been thoroughly explored and remain to be understood. In this Letter we illustrate spin structures of fractionalized vortices; in addition, we also show that 2D quantum gases with short ranged spin correlations can have a topological spin order. We further estimate critical nucleation frequencies of fractionalized vortices in optical traps which can potentially be studied in experimentsStenger98 ().
Our simulations illustrate that in a fundamental vortex carrying one-half of circulation quantum ( is the Planck constant and is the mass of atoms), there exists a softened spin disclination (i.e. a disclination in the absence of spin stiffness) even when local spin moments are strongly fluctuating at finite temperatures. The topological winding number associated with softened spin disclinations is conserved as far as the phase rigidity remains finite. This effectively leads to a non-local topological spin order. Such a nonlocal order is absent in a conventional condensate of atoms or pairs of atoms. We have further studied creation of these excitations in rotating superfluids and obtained fluctuation-dependent critical frequencies for half-quantum vortex(HQV) nucleation. HQVs in traps can be probed by measuring a precession of eigenaxes of surface quadrupole modes.
We employ the Hamiltonian introduced previously for F=1 sodium atoms in optical latticesZhou03 (); Demler02 (),
H = ∑kbL2^ρ2k+cL2^S2k (1) − tL∑(ψ†k,αψl,α+h.c.)−∑kμ^ρk.
Here is the lattice site index and are the nearest neighbor sites. is the chemical potential and is the one-particle hopping amplitude. Two coupling constants are ; , are effective s-wave scattering lengths, is the localized Wannier function for atoms in a periodical potential. Operators , create hyperfine spin-one atoms in , and states respectively. The spin and number operators are defined as , and . Spin correlations are mainly induced by interaction . Here we consider antiferromagnetic spin-dependent interactions such as in sodium atoms where . Minimization of this antiferromagnetic spin-dependent interaction requires that the order parameter be a real vector up to a global phase, i.e. where is a unit director on a two-sphere, represents a phase director and is the number of atoms per site. All low energy degrees of freedom are characterized by configurations where and vary slowly in space and timeDemler02 (). Low lying collective modes include spin-wave excitations with energy dispersion , and phase-wave excitations with , (here is the lattice constant). In one-dimensions, low energy quantum fluctuations destroy spin order leading to quantum spin disordered superfluidsZhou01 (). In two-dimensions, the amplitude of quantum spin fluctuations is of order of and is negligible in shallow lattices as is order-of-magnitude bigger than . At finite temperatures, spin correlations are mainly driven by long wave length thermal fluctuations, analogous to quantum cases. This aspect was also paid attention to previously and normal-superfluid transitions were investigatedMukerjee06 ().
We therefore study the following Hamiltonian which effectively captures long wave length thermal fluctuations
H=−∑Jklnk⋅nlΦk⋅Φl; (2)
here states at each site are specified by two unit directors: a nematic director, and a phase director, . is the effective coupling between two neighboring sites and depends on , the number of atoms per site. The model is invariant under the following local Ising gauge transformation: , , and . In the following, we present results of our simulations on 2D superfluids, especially spin structures, energetics of HQVs and nucleation of HQVs in rotating optical traps using the effective Hamiltonian in Eq.2.
HQV and underlying topological spin order Around a HQV, both phase and nematic directors rotate slowly by ; in polar coordinates , a HQV in condensates is represented by , with Thouless98 (); Zhou01 (). The question here is whether, when nematic directors are not ordered, a spin disclination is still present in a HQV. To fully take into account thermal fluctuations, we carry out Monte Carlo simulations on a square lattice of sites and study spatial correlations between a HQV and a -spin disclination, and topological order.
We first identify critical temperatures of the normal-superfluid phase transition by calculating correlations and the phase rigidity. The gauge-invariant quadrupole-quadrupole correlation functions we have studied are
fs,p(r1,r2)=. (3)
Here , ; , . In simulations, we have studied these correlation functions and found that the phase correlation length for becomes divergent at a temperature which is identified as a critical temperature . We also calculate the phase rigidity or the renormalized phase coupling
Jp=∂2F∂δχ2; (4)
here is a small phase difference applied across the opposite boundaries of the lattice and is the corresponding free energy. We indeed find that it approaches zero at while at takes a bare value . Meanwhile, the spin correlation function remains to be short ranged across . By extrapolating our data to lower temperatures, we find that the spin correlation length diverges only at (see Fig.1a). Our simulations for correlation lengths are in agreement with previous results in Ref.Mukerjee06 (); they are also consistent with the continuum limit of the model in Eq.2 which is equivalent to an model and an nonlinear-sigma model.
In order to keep track of the winding of nematic directors in a wildly fluctuating back ground, we introduce the following gauge invariant -rotation checking operator, which is essentially a product of sign-checking operators
Ws=∏∈Csign(nk⋅nl),Wp=∏∈Csign(Φk⋅Φl). (5)
Here the product is carried out along a closed square-shape path centered at the origin of a lattice. can be either or ; and () is when encloses a -spin disclination (HQV). The gauge invariant circulation of supercurrent velocity (in units of ) is defined as ; this quantity is equal to one in a HQV.
In our simulations, we investigate the winding number averaged over configurations where phase directors rotate by around the boundary of the lattice and the center plaquette. At temperatures above the normal-superfluid transition temperature , both winding numbers and circulation are averaged to zero within our numerical accuracy(see Fig.1). And our choice of boundary conditions does not lead to a vortex or disclination configuration in the absence of phase rigidity. Below , the circulation is averaged to one indicating that the boundary conditions effectively project out HQV configurations. Meanwhile, we observe loop-perimeter dependent which can be attributed to the background fluctuations of HQV or disclination pairs. The loop-perimeter dependence of here is almost identical to that for uniform boundary conditions, i.e. the back ground value. After normalizing in terms of background winding numbers , we find both and approach (see Fig.1). We thus demonstrate that a softened disclination is spatially correlated with a HQV. At the temperatures we carry out these simulations the spin correlation length is sufficiently short compared to the size of the lattice. At further lower temperatures, the spin correlation length becomes longer than the lattice size and fluctuations of pairs of disclination-anti-disclination are strongly suppressed; are equal to for almost all loops, which corresponds to a mean field result.
Results in Fig.1 indicate that there exists a softened spin disclination in a HQV. This is a distinct feature in our systems and there exist no such additional magnetic structures in HQVs in conventional molecular condensates of atom pairs discussed perviouslyRomans04 (). Thus, -disclinations like HQVs have logarithmically divergent energies and are fully suppressed in ground states. Our results also illustrate that although the average local spin quadrupole moments vanish because of strong fluctuations, an overall -rotation of nematic directors in disclinations is still conserved because of a coupling to the superfluid component. This coupling between a HQV and disclination can also be attributedSong08 () to a coupling between Higgs matter and discrete gauge fieldsFradkin79 (). Furthermore, the absence of unbound -disclinations in superfluids indicates a topological order, similar to the one introduced previously for an isotropic phase of liquid crystalToner93 (). Consequently, once a conventional phase order appears below a critical temperature, a topological spin order simultaneously emerges while spin correlations remain short ranged.
The emergent topological order can be further verified by examining the average of product-operator over all configurations (with open boundaries). Above the normal-superfluid transition temperature , we again find that both are averaged to zero within our numerical accuracy implying proliferation of unbound HQVs or disclinations. Below , we study the loop-perimeter dependence of average winding numbers and find that both and are linear functions of loop-perimeter analogous to the Wilson-loop-product of deconfining gauge fieldsWilson74 (); if there were unbound disclinations, one should expect that is proportional to, instead of the loop-perimeter, the loop-area which represents the number of unbound disclinations enclosed by the loop.
Critical frequency for HQV nucleation Let us now turn to the nucleation of those excitations in rotating trapsMadison00 (); Haljan01 (); AboShaeer01 (); Fetter01 (); Dalfovo01 (); Tsubota02 (); Isoshima02 (). To understand the critical frequency for nucleation, we study the free energy of a vortex, in a rotating frame, as a function of the distance from the axis of a cylindrical optical trap (the axis is along the -direction),
Fh.v.(r)=F0h.v.(r)−ΩLz(r). (6)
Here is the free energy of a HQV located at distance from the trap axis in the absence of rotation, is the angular momentum of the vortex state and is the rotating frequency.
In a lattice without a trapping potential, is approximately equal to , with leading contributions from phase winding and spin twisting; here are renormalized phase and spin coupling respectively and is the size of system. For an integer-quantum vortex (IQV), is equal to . The ratio between and depends on the ratio or spin fluctuations; in the limit approaches infinity, the ratio changes discontinuously from at where to at finite low temperatures in where vanishes. In simulations of a finite trap (see below), because of a finite size effect we find that this ratio varies from to smoothly as temperatures increase from to .
To study nucleation of half-quantum vortices in an optical trap, we assume a nearly harmonic trapping potential , with being the trap frequency. The average number of particles per site has a Thomas-Fermi profile; , here is the number density at the center and is the Thomas-Fermi radius. Furthermore, the optical lattice potential along the axial direction is sufficiently deep so that atoms are confined in a two-dimensional plane; the in-plane lattice potential depth is set to be ( is the photon recoil energy).
For the trap and lattice geometry described above, we calculate parameters in Eq.1 and obtain nK, nK and nK. For and trap frequency , we also find that and where is the harmonic oscillator length. The coupling in Eq.2 depends on the distance from the center of trap and at the center, the coupling is about nK. In non-rotating or slowly rotating traps, the free energy maximum is located at the center and there should be no vortices in the trap. As frequencies are increased, a local energy minimum appears at the center and becomes degenerate with the no-vortex state at a thermodynamic critical frequency (which is about at ); however because of a large energy barrier separating the two degenerate states as shown in Fig.2, vortices are still prohibited from entering the trap.
Further speeding up rotations results in an energetically lower and spatially narrower barrier. Within the range of temperatures studied, thermal activation turns out to be insignificant within an experimental time scale () because of low attempt frequencies. So only when the spatial width of barrier becomes comparable to a hydrodynamic breakdown lengthFeder00 (), the barrier can no longer be felted and vortices start to penetrate into the trap. We use this criterion to numerically determine the dynamical critical frequency for vortex nucleation ; for IQVs, the calculated is a flat function of (see Fig.2b) which is qualitatively consistent with earlier estimatesSimula02 ().
For HQVs, depends on the renormalized spin coupling and therefore the amplitude of spin fluctuations. Because of this, varies from about at where and due to a finite size effect (see the inset of Fig.2), to about at temperatures close to where and . In other words, this unique temperature dependence can be considered to be an indicator of fluctuation-driven vortex fractionalization. It is worth remarking that in the thermodynamic limit where approaches zero at any finite temperatures, approaches as mentioned before. Consequently, (about ) for HQVs is about one-half of the critical frequency for IQVs (about for the finite trap studied here). Also note that the zero temperature estimate of is close to the previously obtained value of critical frequencies of HQVs in Bose-Einstein condensatesIsoshima02 ().
The interaction between two HQVs with the same vorticity at a separation distance contains two parts. One, is from interactions between two supercurrent velocity fields which is logarithmic as a function of ; and the other, is from interaction between two spin twisting fields accompanying HQVs. For a disclination-anti disclination pair, in the dilute limit one finds that resulting in a cancellation of long range interactions. The resultant short-range repulsions lead to square vortex lattices found in numerical simulationsAndrew08 (). For fluctuation-driven fractionalized vortices, is almost zero and the overall interactions are always logarithmically repulsive. HQVs nucleated in a rotating trap should therefore form a usual triangular vortex lattice.
Individual vortex lines can be probed either by studying a precession of eigenaxes of surface quadrupole mode in rotating superfluidsChevy00 (). In the later approach, one studies the angular momentum carried per particle in a HQV state. When a HQV is nucleated in the trap, superfluids are no longer irrotational and the angular momentum per particle is rather than per particle for an integer vortex state. When a surface quadrupole oscillation across a rotating superfluid is excited, larger axes of quadrupole oscillation start to precess just as in the case of integer vortices. However, the precession rate is only one half of the value for an integer vortex state which can be studied in experiments.
In conclusion, 2D superfluids of sodium atoms have a non-local topological spin order. In rotating traps, fluctuation-driven fractionalized vortices can nucleate at a critical frequency which is about half of that for integer vortices. Observation of these exotic excitations could substantially improve our understanding of topological order and fractionalization. We thank J. Zhang and Z. C. Gu for contributions at an early stage of the project. This work is in part supported by the office of the Dean of Science, UBC, NSERC (Canada), Canadian Institute for Advanced Research, and the A. P. Sloan foundation.
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The feedback must be of minimum 40 characters and the title a minimum of 5 characters | 2019-10-19 12:17:40 | {"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": 0.8357527256011963, "perplexity": 1690.4244897964222}, "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-2019-43/segments/1570986693979.65/warc/CC-MAIN-20191019114429-20191019141929-00509.warc.gz"} |
http://www.analyzemath.com/math_questions/math_questions_12.html | # Math Questions With Answers (12)
More Math questions on decomposing into partial fractions are presented. Answers to these questions are located at the lower part of the page.
Questions 1: Decompose into partial fractions : (5x + 10) / x(x + 5) Questions 2: Decompose into partial fractions : (8x + 14) / (x + 1)(x + 5) Questions 3: Decompose into partial fractions : (5x2 + 12x + 3) / x(x + 1)2 Questions 4: Decompose into partial fractions : (3x + 15) / (x + 4)2 Questions 5: Decompose into partial fractions : (7x + 10) / (x + 1)(x2 - 4) Questions 6: Decompose into partial fractions : (5x2 + 31x + 46) / (x + 2)(x + 3)2 Questions 7: Decompose into partial fractions : (2x2 + 6x - 2) / (x3 - 1) ANSWERS TO ABOVE QUESTIONS 1) 2/x + 3/(x + 5) 2) 3/(2(x + 1)) + 13/(2(x + 5)) 3) 3/x + 2/(x + 1) + 4/(x + 1)2 4) 3/(x + 4) + 3/(x + 4)2 5) 2/(x - 2) - 1/(x + 2) - 1/(x + 1) 6) 4/(x + 2) + 1/(x + 3) + 2/(x + 3)2 7) 2/(x - 1) + 4/(x2 + x + 1) More math questions and problems with detailed solutions in this site. | 2018-04-20 18:36:12 | {"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": 0.8161393404006958, "perplexity": 932.5328378917412}, "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-2018-17/segments/1524125944677.39/warc/CC-MAIN-20180420174802-20180420194802-00184.warc.gz"} |
https://itprospt.com/num/13243646/prove-that-the-curve-rit-a-btp-c-dt-e-p-whereand-are-real | 5
# Prove that the curve rit) = {a btP c + dt?,e + #P/, whereand are real numbers andpositive integer; has zero curvature Give an explanation.What must be shown to prov...
## Question
###### Prove that the curve rit) = {a btP c + dt?,e + #P/, whereand are real numbers andpositive integer; has zero curvature Give an explanation.What must be shown to prove that rlt) has zero curvature?It must be shown that the dot product, a *V is Zero, or that the acceleration,is constant:must be shown that the velocity; and the acceleration; are constant: must be shown that the magnitude of the cross product, laxvl; is zero or that the unit tangent vector; T is constant: It must be shown that the cr
Prove that the curve rit) = {a btP c + dt?,e + #P/, where and are real numbers and positive integer; has zero curvature Give an explanation. What must be shown to prove that rlt) has zero curvature? It must be shown that the dot product, a *V is Zero, or that the acceleration, is constant: must be shown that the velocity; and the acceleration; are constant: must be shown that the magnitude of the cross product, laxvl; is zero or that the unit tangent vector; T is constant: It must be shown that the cross product axv is constant. In this problem, show that the magnitude ofthe cross product, laxvl; is zero. To do so, first find the velocity; bt" ,€ + dt" e + #P) Find the acceleration, Compute the cross product, axv: axv This result for xv demonstrates that the given curve has zero curvature_
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##### An object moves along the x-axis with its position x _ in mcters. given as function of time /_ in scconds. byx(t) =1450 9.691 + 4.68What is the object's velocity at time =1.03 $?v(1.03$) =11.06
An object moves along the x-axis with its position x _ in mcters. given as function of time /_ in scconds. by x(t) =1450 9.691 + 4.68 What is the object's velocity at time =1.03 $? v(1.03$) = 11.06...
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Evaluate the integral: (Use C for the constant of integration:) dx 49 X2...
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IRSin €=3/5 , fndwithout using tables the values of: Cos € tun € (ans; 1) 45 ; I) 374... | 2022-08-19 10:45:54 | {"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": 0.7199748754501343, "perplexity": 6220.86161754129}, "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-2022-33/segments/1659882573667.83/warc/CC-MAIN-20220819100644-20220819130644-00240.warc.gz"} |
https://particlebites.com/?m=201409 | ## An update from AMS-02, the particle detector in space
Last Thursday, Nobel Laureate Sam Ting presented the latest results (CERN press release) from the Alpha Magnetic Spectrometer (AMS-02) experiment, a particle detector attached to the International Space Station—think “ATLAS/CMS in space.” Instead of beams of protons, the AMS detector examines cosmic rays in search of signatures of new physics such as the products of dark matter annihilation in our galaxy.
In fact, this is just the latest chapter in an ongoing mystery involving the energy spectrum of cosmic positrons. Recall that positrons are the antimatter versions of electrons with identical properties except having opposite charge. They’re produced from known astrophysical processes when high-energy cosmic rays (mostly protons) crash into interstellar gas—in this case they’re known as secondaries’ because they’re a product of the primary’ cosmic rays.
The dynamics of charged particles in the galaxy are difficult to simulate due to the presence of intense and complicated magnetic fields. However, the diffusion models generically predict that the positron fraction—the number of positrons divided by the total number of positrons and electrons—decreases with energy. (This ratio of fluxes is a nice quantity because some astrophysical uncertainties cancel.)
This prediction, however, is in stark contrast with the observed positron fraction from recent satellite experiments:
The rising fraction had been hinted in balloon-based experiments for several decades, but the satellite experiments have been able to demonstrate this behavior conclusively because they can access higher energies. In their first set of results last year (shown above), AMS gave the most precise measurements of the positron fraction as far as 350 GeV. Yesterday’s announcement extended these results to 500 GeV and added the following observations:
First they claim that they have measured the maximum of the positron fraction to be 275 GeV. This is close to the edge of the data they’re releasing, but the plot of the positron fraction slope is slightly more convincing:
The observation of a maximum in what was otherwise a fairly featureless rising curve is key for interpretations of the excess, as we discuss below. A second observation is a bit more curious: while neither the electron nor the positron spectra follow a simple power law, $\Phi_{e^\pm} \sim E^{-\delta}$, the total electron or positron flux does follow such a power law over a range of energies.
This is a little harder to interpret since the flux form electrons also, in principle, includes different sources of background. Note that this plot reaches higher energies than the positron fraction—part of the reason for this is that it is more difficult to distinguish between electrons and positrons at high energies. This is because the identification depends on how the particle bends in the AMS magnetic field and higher energy particles bend less. This, incidentally, is also why the FERMI data has much larger error bars in the first plot above—FERMI doesn’t have its own magnetic field and must rely on that of the Earth for charge discrimination.
So what should one make of the latest results?
The most optimistic hope is that this is a signal of dark matter, and at this point this is more of a ‘wish’ than a deduction. Independently of AMS, we know is that dark matter exists in a halo that surrounds our galaxy. The simplest dark matter models also assume that when two dark matter particles find each other in this halo, they can annihilate into Standard Model particle–anti-particle pairs, such as electrons and positrons—the latter potentially yielding the rising positron fraction signal seen by AMS.
From a particle physics perspective, this would be the most exciting possibility. The ‘smoking gun’ signature of such a scenario would be a steep drop in the positron fraction at the mass of the dark matter particle. This is because the annihilation occurs at low velocities so that the energy of the annihilation products is set by the dark matter mass. This is why the observation of a maximum in the positron fraction is interesting: the dark matter interpretation of this excess hinges on how steeply the fraction drops off.
There are, however, reasons to be skeptical.
• One attractive feature of dark matter annihilations is thermal freeze out: the observation that the annihilation rate determines how much dark matter exists today after being in thermal equilibrium in the early universe. The AMS excess is suggestive of heavy (~TeV scale) dark matter with an annihilation rate three orders of magnitude larger than the rate required for thermal freeze out.
• A study of the types of spectra one expects from dark matter annihilation shows fits that are somewhat in conflict with the combined observations of the positron fraction, total electron/positron flux, and the anti-proton flux (see 0809.2409). The anti-proton flux, in particular, does not have any known excess that would otherwise be predicted by dark matter annihilation into quarks.
There are ways around these issues, such as invoking mechanisms to enhance the present day annihilation rate, perhaps with the annihilation only creating leptons and not quarks. However, these are additional bells and whistles that model-builders must impose on the dark matter sector. It is also important to consider alternate explanations of the Pamela/FERMI/AMS positron fraction excess due to astrophysical phenomena. There are at least two very plausible candidates:
1. Pulsars are neutron stars that are known to emit “primary” electron/positron pairs. A nearby pulsar may be responsible for the observed rising positron fraction. See 1304.1791 for a recent discussion.
2. Alternately, supernova remnants may also generate a “secondary” spectrum of positrons from acceleration along shock waves (0909.4060, 0903.2794, 1402.0855).
Both of these scenarios are plausible and should temper the optimism that the rising positron fraction represents a measurement of dark matter. One useful handle to disfavor the astrophysical interpretations is to note that they would be anisotropic (not constant over all directions) whereas the dark matter signal would be isotropic. See 1405.4884 for a recent discussion. At the moment, the AMS measurements do not measure any anisotropy but are not yet sensitive enough to rule out astrophysical interpretations.
Finally, let us also point out an alternate approach to understand the positron fraction. The reason why it’s so difficult to study cosmic rays is that the complex magnetic fields in the galaxy are intractable to measure and, hence, make the trajectory of charged particles hopeless to trace backwards to their sources. Instead, the authors of 0907.1686 and 1305.1324 take an alternate approach: while we can’t determine the cosmic ray origins, we can look at the behavior of heavier cosmic ray particles and compare them to the positrons. This is because, as mentioned above, the bending of a charged particle in a magnetic field is determined by its mass and charge—quantities that are known for the various cosmic ray particles. Based on this, the authors are able to predict an upper bound for the positron fraction when one assumes that the positrons are secondaries (e.g in the case of supernovae remnant acceleration):
We see that the AMS-02 spectrum is just under the authors’ upper bound, and that the reported downturn is consistent with (even predicted from) the upper-bound. The authors’ analysis then suggests a non-dark matter explanation for the positron excess. See this post from Resonaances for a discussion of this point and an updated version of the above plot from the authors.
With that in mind, there are at least three things to look forward to in the future from AMS:
1. A corresponding upturn in the anti-proton flux is predicted in many types of dark matter annihilation models for the rising positron fraction. Thus far AMS-02 has not released anti-proton data due to the lower numbers of anti-protons.
2. Further sensitivity to the (an)isotropy of the excess is a critical test of the dark matter interpretation.
3. The shape of the drop-off with energy is also critical: a gradual drop-off is unlikely to come from dark matter whereas a steep drop off is considered to be a smoking gun for dark matter.
Only time will tell; though Ting suggested that new results would be presented at the upcoming AMS meeting at CERN in 2 months.
This post was edited by Christine Muccianti.
## Neutrinoless Double Beta Decay Experiments
Title: Neutrinoless Double Beta Decay Experiments
Author: Alberto Garfagnini
Published: arXiv:1408.2455 [hep-ex]
Neutrinoless double beta decay is a theorized process that, if observed, would provide evidence that the neutrino is its own antiparticle. The relatively recent discovery of neutrino mass from oscillation experiments makes this search particularly relevant, since the Majorana mechanism that requires particles to be self-conjugate can also provide mass. A variety of experiments based on different techniques hope to observe this process. Before providing an experimental overview, we first discuss the theory itself.
Beta decay occurs when an electron or positron is released along with a corresponding neutrino. Double beta decay is simply the simultaneous beta decay of two neutrons in a nucleus. “Neutrinoless,” of course, means that this decay occurs without the accompanying neutrinos; in this case, the two neutrinos in the beta decay annihilate with one another, which is only possible if they are self-conjugate. Figures 1 and 2 demonstrate the process by formula and image, respectively.
The lack of accompanying neutrinos in such a decay violates lepton number, meaning this process is forbidden unless neutrinos are Majorana fermions. Without delving into a full explanation, this simply means that a particle is its own antiparticle (though more information is given in the references.) The importance lies in the lepton number of a neutrino. Neutrinoless double beta decay would require a nucleus to absorb two neutrinos, then decay into two protons and two electrons (to conserve charge). The only way in which this process does not violate lepton number is if the lepton charge is the same for a neutrino and an antineutrino; in other words, if they are the same particle.
The experiments currently searching for neutrinoless double beta decay can be classified according to the material used for detection. A partial list of active and future experiments is provided below.
1. EXO (Enriched Xenon Observatory): New Mexico, USA. The detector is filled with liquid 136Xe, which provides worse energy resolution than gaseous xenon, but is compensated by the use of both scintillating and ionizing signals. The collaboration finds no statistically significant evidence for 0νββ decay, and place a lower limit on the half life of 1.1 * 1025 years at 90% confidence.
2. KamLAND-Zen: Kamioka underground neutrino observatory near Toyama, Japan. Like EXO, the experiment uses liquid xenon, but in the past has required purification due to aluminum contaminations in the detector. They report a 0νββ half life 90% CL at 2.6 * 1025 years. Figure 3 shows the energy spectra of candidate events with the best fit background.
3. GERDA (Germanium Dectetor Array): Laboratori Nazionali del Gran Sasso, Italy. GERDA utilizes High Purity 76Ge diodes, which provide excellent energy resolution but typically have very large backgrounds. To prevent signal contamination, GERDA has ultra-pure shielding that protect measurements from environmental radiation background sources. The half life is bound below at 90% confidence by 2.1 * 1025 years.
4. MAJORANA: South Dakota, USA. This experiment is under construction, but a prototpye is expected to begin running in 2014. If results from GERDA and MAJORANA look good, there is talk of building a next generation germanium experiment that combines diodes from each detector.
5. CUORE: Laboratori Nazionali del Gran Sasso, Italy. CUORE is a 130Te bolometric direct detector, meaning that it has two layers: an absorber made of crystal that releases energy when struck, and a sensor which detects the induced temperature changes. The experiment is currently under construction, so there are no definite results, but it expects to begin taking data in 2015.
While these results do not seem to show the existence of 0νββ decay, such an observation would demonstrate the existence of Majorana fermions and give an estimate of the absolute neutrino mass scale. However, a missing observation would be just as significant in the role of scientific discovery, since this would imply that the neutrino is not in fact its own antiparticle. To get a better limit on the half life, more advanced detector technologies are necessary; it will be interesting to see if MAJORANA and CUORE will have better sensitivity to this process.
The hydrogen atom is one of the primary examples studied in a typical introductory quantum mechanics course. Recent measurements indicate that this simple system may still have surprises for us. Could this be a hint of new physics? This post is based on the following papers:
“Muonic hydrogen and MeV forces” by D. Tucker-Smith and I. Yavin [1011.4922], Phys. Rev. D83 (2011) 101702
“Proton size anomaly” by V. Barger, C. Chiang, W. Keung, D. Marfatia [1011.3519], Phys. Rev. Lett. 106 (2011) 153001
“The Size of the Proton” by Pohl et al. in Nature 466 (2010) 213
Quantum mechanically, the proton is an object whose electric charge is smeared out over a small region. Experiments that scatter electrons off protons can probe this spatial extent and recent measurements indicate an effective proton charge radius of 0.877(7) femtometers.
Muons are heavy copies of electrons and can similarly form muonic hydrogen: an atom formed from a proton and a muon. Because the muons are heavier, they exist closer to the nucleus and are more sensitive to the extent of the proton charge: the effective Coulomb force is reduced as one dips into the charge distribution in the same way that the gravitational force decreases as one digs towards the center of the Earth.
By ‘tickling’ the muon into a higher energy level with a laser and then measuring the resulting X-ray emission, one can deduce the proton radius. Since lasers can be tuned to very precise frequencies, one can make a very precise measurement of the Lamb shift in the muonic hydrogen energy levels. This, in turn, can be converted into a measurement of the proton radius because the energy levels are sensitive to the overlap of the muon and proton probability distributions. Intuitively, when the muon is inside the proton charge radius, it experiences a weaker Coulomb potential due to screening.
The big surprise is that the muonic hydrogen measurement gives a radius of 0.842(7) femtometers, this is over five standard deviations smaller than the expected result based on regular hydrogen!
This discrepancy remains an open question despite several proposed solutions based on more precise theoretical calculations to relate the Lamb shift to the proton radius. One optimistic approach is to entertain the possibility that this is an indicator of new fundamental physics, such as a heretofore undiscovered force that tugs on the muon and electron differently. It turns out that these types of models are difficult to construct. One of the main constraints is actually nearly 40 years old and comes from the effect of such a new force on neutron–lead scattering.
Meanwhile, a new set of experiments to probe the proton radius anomaly are already underway. One of these is the Muon-Proton Scattering Experiment (MUSE); this would directly probe if the origin of the discrepancy came from the two different proton radius measurements described above: scattering for electrons versus spectroscopy for muons. | 2020-01-21 23:10:03 | {"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": 1, "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": 0.7063149213790894, "perplexity": 1356.8830851670093}, "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/1579250606226.29/warc/CC-MAIN-20200121222429-20200122011429-00146.warc.gz"} |
https://nano-optics.ac.nz/terms/ | TERMS stands for T-matrix for Electromagnetic Radiation with Multiple Scatterers — it is a Fortran program to simulate the near-field and far-field optical properties of collections of particles. TERMS solves rigorously the Maxwell equations via the superposition T-matrix method, where incident and scattered fields are decomposed into a basis of multipolar electric and magnetic spherical waves.
In a multiple-scattering problem the net field exciting a given particle is composed of the incident field plus the scattering contribution from neighbouring particles, restulting in a coupled system of equations to be solved for the total fields. TERMS implements several algorithms to describe the self-consistent electromagnetic interaction between multiple scatterers, and from there compute optical properties such as absorption, scattering, extinction, circular dichroism, as well as near-field intensities and the local degree of optical chirality.
By describing the incident and scattered fields in a basis of spherical waves the T-matrix framework lends itself to analytical formulas for orientation-averaged quantities such as far-field cross-sections and near-field intensities, greatly reducing the computational time needed to simulate particles and systems of particles in random orientation.
### Features
The possible computations are divided into three main modes:
• Far-field quantities (absorption, scattering, extinction, circular dichroism) for multiple wavelengths and angles of incidence, as well as orientation-averages
• Near-field calculations for multiple wavelengths and incident angles, also computing the local degree of chirality, as well as orientation-averages
• Stokes parameters and differential scattering cross-sections for multiple incidence or scattering angles
The computational cost scales with the size of the linear system, proportional to the number of particles Np, and to the square of the maximum multipolar order Nmax. On a typical PC we may treat up to ∼ 500 particles with Nmax = 1, and a dimer with Nmax up to ∼ 60.
Notable features of TERMS include:
• Incident plane waves along arbitrary directions, with linear or circular polarisation
• Built-in calculation of individual T-matrices for coated spheres; import of general T-matrices from other programs (e.g. SMARTIES)
• Built-in dielectric functions for common materials such as Au, Ag, Al, Cr, Pt, Pd, Si, and Water, or from tabulated values
• Per-layer absorption in layered spheres
• Orientation-averaging of far-field cross-sections, as well as linear and circular dichroism
• Near-field maps of electric and magnetic field components, |E|2, |E|4, 𝒞 ∝ ℑ(E* ⋅ B)
• Calculation of the global cluster T-matrix
• “Masking” of specific multipolar orders
• Calculation of Stokes parameters, phase matrix, differential scattering
• Plain text or HDF5 I/O format
### System requirements
• Fortran 90 compiler
• Cmake
• (optional) HDF5 library
• (optional) LAPACK
The electromagnetic field is expanded in the basis of vector spherical waves, with the Bessel/Hankel functions computed using TOMS644 library (source included in TERMS). Determining the collective scattering amounts to either solving or inverting a large linear system, which is done using LAPACK. All the relevant LAPACK routines are included in TERMS, but it is recommended to link with your system installed BLAS/LAPACK at compilation stage, because it can enable multi-threading during runtime.
Results are output in plain text files, or, alternatively, in HDF5 data format, which requires suitable hdf5 libraries to be available on your system.
### Compilation
We recommend using the cross platform compilation tool cmake, to resolve all dependencies for your system. From within the build/ directory, type
> cmake ..
> make
to produce the executable terms. Note: if Cmake doesn’t find hdf5 (or another library path) by default, you may need to export it explicitly beforehand. For example on MacOS, the following has proved useful:
# brew install hdf5
export HDF5_ROOT=/usr/local/Cellar/hdf5/1.12.0_3/
Alternatively, a minimal build script is available in the build/ directory; the program can be compiled by executing bash buildTERMS.sh from a Linux terminal with bash. Edit ‘buildTERMS.sh’ to specify a compiler other than GFortran.
We recommend downloading the latest release here [terms_code_1.0.0(.zip|.tar)]. You can also browse/clone/fork the entire repository, but note that it contains many files used to generate the website, which are not relevant for using TERMS.
### The input file
When running the stand-alone executable, main input parameters are read from a plain text input file (line by line and from left to right). Each line is interpreted as a sentence and split into space-separated words. The first (left-most) word is interpreted as a keyword, and the subsequent words as arguments for that keyword. In each sentence, text from the first word starting with the hash character (#) is interpreted as a human-readable comment and thus ignored by the program. All the supported keywords and corresponding arguments are documented on this page. The order of keywords generally doesn’t matter, with just two exceptions: ModeAndScheme must be the first keyword, and Scatterers must be the last. Two examples of input files are provided in the /test directory.
## Citing TERMS
If you use TERMS, please cite the following user guide, as well as other publications listed below if relevant:
1,,2,3,4,5,6,78
(1)
Schebarchov, D.; Fazel-Najafabadi, A.; Le Ru, E. C.; Auguié, B. Multiple Scattering of Light in Nanoparticle Assemblies: User Guide for the Terms Program. Journal of Quantitative Spectroscopy and Radiative Transfer 2022, 108131. https://doi.org/https://doi.org/10.1016/j.jqsrt.2022.108131.
(2)
Schebarchov, D.; Fazel-Najafabadi, A.; Le Ru, E. C.; Auguié, B. TERMS Website; 2021. https://doi.org/10.5281/zenodo.5703291.
(3)
Somerville, W. R. C.; Auguié, B.; Le Ru, E. C. SMARTIES: User-Friendly Codes for Fast and Accurate Calculations of Light Scattering by Spheroids. J. Quant. Spectrosc. Ra. 2016, 174, 39–55. https://doi.org/10.1016/j.jqsrt.2016.01.005.
(4)
Schebarchov, D.; Le Ru, E. C.; Grand, J.; Auguié, B. Mind the Gap: Testing the Rayleigh Hypothesis in T-Matrix Calculations with Adjacent Spheroids. Optics express 2019, 27 (24), 35750–35760. https://doi.org/10.1364/OE.27.035750.
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Lee, S.; Hwang, H.; Lee, W.; Schebarchov, D.; Wy, Y.; Grand, J.; Augui’e, B.; Wi, D. H.; Cort’es, E.; Han, S. W. Core–Shell Bimetallic Nanoparticle Trimers for Efficient Light-to-Chemical Energy Conversion. ACS Energy Letters 2020, 5 (12), 3881–3890. https://doi.org/10.1021/acsenergylett.0c02110.
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Fazel-Najafabadi, A.; Schuster, S.; Auguié, B. Orientation Averaging of Optical Chirality Near Nanoparticles and Aggregates. Physical Review B 2021, 103 (11), 115405. https://doi.org/10.1103/PhysRevB.103.115405.
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Fazel-Najafabadi, A.; Auguié, B. Orientation Dependence of Optical Activity in Light Scattering by Nanoparticle Clusters. Mater. Adv. 2022, –. https://doi.org/10.1039/D1MA00869B.
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Fazel-Najafabadi, A.; Auguié, B. Orientation-Averaged Light Scattering by Nanoparticle Clusters: Far-Field and Near-Field Benchmarks of Numerical Cubature Methods, 2022. | 2022-08-19 17:51:49 | {"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": 0.5159180164337158, "perplexity": 5820.37040534393}, "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-2022-33/segments/1659882573744.90/warc/CC-MAIN-20220819161440-20220819191440-00126.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/320/C5xD8.C4.html | Copied to
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## G = C5×D8.C4order 320 = 26·5
### Direct product of C5 and D8.C4
direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Series: Derived Chief Lower central Upper central
Derived series C1 — C8 — C5×D8.C4
Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C8.C4 — C5×D8.C4
Lower central C1 — C2 — C4 — C8 — C5×D8.C4
Upper central C1 — C20 — C2×C20 — C2×C40 — C5×D8.C4
Generators and relations for C5×D8.C4
G = < a,b,c,d | a5=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >
Smallest permutation representation of C5×D8.C4
On 160 points
Generators in S160
(1 78 62 46 30)(2 79 63 47 31)(3 80 64 48 32)(4 73 57 41 25)(5 74 58 42 26)(6 75 59 43 27)(7 76 60 44 28)(8 77 61 45 29)(9 148 132 116 100)(10 149 133 117 101)(11 150 134 118 102)(12 151 135 119 103)(13 152 136 120 104)(14 145 129 113 97)(15 146 130 114 98)(16 147 131 115 99)(17 87 65 49 33)(18 88 66 50 34)(19 81 67 51 35)(20 82 68 52 36)(21 83 69 53 37)(22 84 70 54 38)(23 85 71 55 39)(24 86 72 56 40)(89 153 137 121 105)(90 154 138 122 106)(91 155 139 123 107)(92 156 140 124 108)(93 157 141 125 109)(94 158 142 126 110)(95 159 143 127 111)(96 160 144 128 112)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 16)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 36)(34 35)(37 40)(38 39)(41 42)(43 48)(44 47)(45 46)(49 52)(50 51)(53 56)(54 55)(57 58)(59 64)(60 63)(61 62)(65 68)(66 67)(69 72)(70 71)(73 74)(75 80)(76 79)(77 78)(81 88)(82 87)(83 86)(84 85)(89 93)(90 92)(94 96)(97 99)(100 104)(101 103)(105 109)(106 108)(110 112)(113 115)(116 120)(117 119)(121 125)(122 124)(126 128)(129 131)(132 136)(133 135)(137 141)(138 140)(142 144)(145 147)(148 152)(149 151)(153 157)(154 156)(158 160)
(1 103 23 92 5 99 19 96)(2 102 24 91 6 98 20 95)(3 101 17 90 7 97 21 94)(4 100 18 89 8 104 22 93)(9 88 153 77 13 84 157 73)(10 87 154 76 14 83 158 80)(11 86 155 75 15 82 159 79)(12 85 156 74 16 81 160 78)(25 116 34 105 29 120 38 109)(26 115 35 112 30 119 39 108)(27 114 36 111 31 118 40 107)(28 113 37 110 32 117 33 106)(41 132 50 121 45 136 54 125)(42 131 51 128 46 135 55 124)(43 130 52 127 47 134 56 123)(44 129 53 126 48 133 49 122)(57 148 66 137 61 152 70 141)(58 147 67 144 62 151 71 140)(59 146 68 143 63 150 72 139)(60 145 69 142 64 149 65 138)
G:=sub<Sym(160)| (1,78,62,46,30)(2,79,63,47,31)(3,80,64,48,32)(4,73,57,41,25)(5,74,58,42,26)(6,75,59,43,27)(7,76,60,44,28)(8,77,61,45,29)(9,148,132,116,100)(10,149,133,117,101)(11,150,134,118,102)(12,151,135,119,103)(13,152,136,120,104)(14,145,129,113,97)(15,146,130,114,98)(16,147,131,115,99)(17,87,65,49,33)(18,88,66,50,34)(19,81,67,51,35)(20,82,68,52,36)(21,83,69,53,37)(22,84,70,54,38)(23,85,71,55,39)(24,86,72,56,40)(89,153,137,121,105)(90,154,138,122,106)(91,155,139,123,107)(92,156,140,124,108)(93,157,141,125,109)(94,158,142,126,110)(95,159,143,127,111)(96,160,144,128,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39)(41,42)(43,48)(44,47)(45,46)(49,52)(50,51)(53,56)(54,55)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,74)(75,80)(76,79)(77,78)(81,88)(82,87)(83,86)(84,85)(89,93)(90,92)(94,96)(97,99)(100,104)(101,103)(105,109)(106,108)(110,112)(113,115)(116,120)(117,119)(121,125)(122,124)(126,128)(129,131)(132,136)(133,135)(137,141)(138,140)(142,144)(145,147)(148,152)(149,151)(153,157)(154,156)(158,160), (1,103,23,92,5,99,19,96)(2,102,24,91,6,98,20,95)(3,101,17,90,7,97,21,94)(4,100,18,89,8,104,22,93)(9,88,153,77,13,84,157,73)(10,87,154,76,14,83,158,80)(11,86,155,75,15,82,159,79)(12,85,156,74,16,81,160,78)(25,116,34,105,29,120,38,109)(26,115,35,112,30,119,39,108)(27,114,36,111,31,118,40,107)(28,113,37,110,32,117,33,106)(41,132,50,121,45,136,54,125)(42,131,51,128,46,135,55,124)(43,130,52,127,47,134,56,123)(44,129,53,126,48,133,49,122)(57,148,66,137,61,152,70,141)(58,147,67,144,62,151,71,140)(59,146,68,143,63,150,72,139)(60,145,69,142,64,149,65,138)>;
G:=Group( (1,78,62,46,30)(2,79,63,47,31)(3,80,64,48,32)(4,73,57,41,25)(5,74,58,42,26)(6,75,59,43,27)(7,76,60,44,28)(8,77,61,45,29)(9,148,132,116,100)(10,149,133,117,101)(11,150,134,118,102)(12,151,135,119,103)(13,152,136,120,104)(14,145,129,113,97)(15,146,130,114,98)(16,147,131,115,99)(17,87,65,49,33)(18,88,66,50,34)(19,81,67,51,35)(20,82,68,52,36)(21,83,69,53,37)(22,84,70,54,38)(23,85,71,55,39)(24,86,72,56,40)(89,153,137,121,105)(90,154,138,122,106)(91,155,139,123,107)(92,156,140,124,108)(93,157,141,125,109)(94,158,142,126,110)(95,159,143,127,111)(96,160,144,128,112), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,8)(2,7)(3,6)(4,5)(9,13)(10,12)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,36)(34,35)(37,40)(38,39)(41,42)(43,48)(44,47)(45,46)(49,52)(50,51)(53,56)(54,55)(57,58)(59,64)(60,63)(61,62)(65,68)(66,67)(69,72)(70,71)(73,74)(75,80)(76,79)(77,78)(81,88)(82,87)(83,86)(84,85)(89,93)(90,92)(94,96)(97,99)(100,104)(101,103)(105,109)(106,108)(110,112)(113,115)(116,120)(117,119)(121,125)(122,124)(126,128)(129,131)(132,136)(133,135)(137,141)(138,140)(142,144)(145,147)(148,152)(149,151)(153,157)(154,156)(158,160), (1,103,23,92,5,99,19,96)(2,102,24,91,6,98,20,95)(3,101,17,90,7,97,21,94)(4,100,18,89,8,104,22,93)(9,88,153,77,13,84,157,73)(10,87,154,76,14,83,158,80)(11,86,155,75,15,82,159,79)(12,85,156,74,16,81,160,78)(25,116,34,105,29,120,38,109)(26,115,35,112,30,119,39,108)(27,114,36,111,31,118,40,107)(28,113,37,110,32,117,33,106)(41,132,50,121,45,136,54,125)(42,131,51,128,46,135,55,124)(43,130,52,127,47,134,56,123)(44,129,53,126,48,133,49,122)(57,148,66,137,61,152,70,141)(58,147,67,144,62,151,71,140)(59,146,68,143,63,150,72,139)(60,145,69,142,64,149,65,138) );
G=PermutationGroup([(1,78,62,46,30),(2,79,63,47,31),(3,80,64,48,32),(4,73,57,41,25),(5,74,58,42,26),(6,75,59,43,27),(7,76,60,44,28),(8,77,61,45,29),(9,148,132,116,100),(10,149,133,117,101),(11,150,134,118,102),(12,151,135,119,103),(13,152,136,120,104),(14,145,129,113,97),(15,146,130,114,98),(16,147,131,115,99),(17,87,65,49,33),(18,88,66,50,34),(19,81,67,51,35),(20,82,68,52,36),(21,83,69,53,37),(22,84,70,54,38),(23,85,71,55,39),(24,86,72,56,40),(89,153,137,121,105),(90,154,138,122,106),(91,155,139,123,107),(92,156,140,124,108),(93,157,141,125,109),(94,158,142,126,110),(95,159,143,127,111),(96,160,144,128,112)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,8),(2,7),(3,6),(4,5),(9,13),(10,12),(14,16),(17,20),(18,19),(21,24),(22,23),(25,26),(27,32),(28,31),(29,30),(33,36),(34,35),(37,40),(38,39),(41,42),(43,48),(44,47),(45,46),(49,52),(50,51),(53,56),(54,55),(57,58),(59,64),(60,63),(61,62),(65,68),(66,67),(69,72),(70,71),(73,74),(75,80),(76,79),(77,78),(81,88),(82,87),(83,86),(84,85),(89,93),(90,92),(94,96),(97,99),(100,104),(101,103),(105,109),(106,108),(110,112),(113,115),(116,120),(117,119),(121,125),(122,124),(126,128),(129,131),(132,136),(133,135),(137,141),(138,140),(142,144),(145,147),(148,152),(149,151),(153,157),(154,156),(158,160)], [(1,103,23,92,5,99,19,96),(2,102,24,91,6,98,20,95),(3,101,17,90,7,97,21,94),(4,100,18,89,8,104,22,93),(9,88,153,77,13,84,157,73),(10,87,154,76,14,83,158,80),(11,86,155,75,15,82,159,79),(12,85,156,74,16,81,160,78),(25,116,34,105,29,120,38,109),(26,115,35,112,30,119,39,108),(27,114,36,111,31,118,40,107),(28,113,37,110,32,117,33,106),(41,132,50,121,45,136,54,125),(42,131,51,128,46,135,55,124),(43,130,52,127,47,134,56,123),(44,129,53,126,48,133,49,122),(57,148,66,137,61,152,70,141),(58,147,67,144,62,151,71,140),(59,146,68,143,63,150,72,139),(60,145,69,142,64,149,65,138)])
110 conjugacy classes
class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 20M 20N 20O 20P 40A ··· 40P 40Q ··· 40X 80A ··· 80AF order 1 2 2 2 4 4 4 4 5 5 5 5 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 10 16 ··· 16 20 ··· 20 20 20 20 20 20 20 20 20 40 ··· 40 40 ··· 40 80 ··· 80 size 1 1 2 8 1 1 2 8 1 1 1 1 2 2 2 2 8 8 1 1 1 1 2 2 2 2 8 8 8 8 2 ··· 2 1 ··· 1 2 2 2 2 8 8 8 8 2 ··· 2 8 ··· 8 2 ··· 2
110 irreducible representations
dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 D8 SD16 C5×D4 C5×D4 D8.C4 C5×D8 C5×SD16 C5×D8.C4 kernel C5×D8.C4 C5×C8.C4 C2×C80 C5×C4○D8 C5×D8 C5×Q16 D8.C4 C8.C4 C2×C16 C4○D8 D8 Q16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 2 2 4 4 8 8 8 32
Matrix representation of C5×D8.C4 in GL2(𝔽241) generated by
98 0 0 98
,
11 230 11 11
,
11 230 230 230
,
25 43 43 216
G:=sub<GL(2,GF(241))| [98,0,0,98],[11,11,230,11],[11,230,230,230],[25,43,43,216] >;
C5×D8.C4 in GAP, Magma, Sage, TeX
C_5\times D_8.C_4
% in TeX
G:=Group("C5xD8.C4");
// GroupNames label
G:=SmallGroup(320,164);
// by ID
G=gap.SmallGroup(320,164);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,1410,360,172,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations
Export
×
𝔽 | 2020-01-27 10:11:03 | {"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": 0.9978599548339844, "perplexity": 2168.5959382663727}, "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/1579251696046.73/warc/CC-MAIN-20200127081933-20200127111933-00003.warc.gz"} |
https://hedgehogqa.github.io/fsharp-hedgehog/reference/hedgehog.html | ## Hedgehog Namespace
Type/Module Description A generator for values and shrink trees of type 'a. A generator for random values of type 'a A range describes the bounds of a number to generate, which may or may not be dependent on a 'Size'. The constructor takes an origin between the lower and upper bound, and a function from 'Size' to bounds. As the size goes towards 0, the values go towards the origin. Splittable random number generator. Tests are parameterized by the Size of the randomly-generated data, the meaning of which depends on the particular generator used. A rose tree which represents a random generated outcome, and all the ways in which it can be made smaller. | 2022-07-06 20:16:38 | {"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": 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": 0.5044522285461426, "perplexity": 513.3048913474328}, "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/1656104676086.90/warc/CC-MAIN-20220706182237-20220706212237-00254.warc.gz"} |
https://math.stackexchange.com/questions/1460488/there-are-4-girls-and-3-boys-but-only-5-seats-how-many-ways-to-seat-the | # There are $4$ girls and $3$ boys but only $5$ seats. How many ways to seat the $3$ boys together?
There are $4$ girls and $3$ boys but there are only $5$ seats. How many ways can you seat the $3$ boys together?
The order of the seat matters, for example: there's the order $B_1$ $B_2$ $B_3$ $G_2$ $G_4$ and there's $B_2$ $B_3$ $B_1$ $G_2$ $G_4$
Here's my answer: There are $3!$ ways to seat the $3$ boys. The $2$ remaining seats are to be occupied by $2$ out of the $4$ girls, so $^4P_2$. So we now have $3! \cdot$ $^4P_2$.
Lastly, there are $3$ ways to make that arrangement,
$1)$ two girls on the left,
$2)$ two girls on the right, and
$3)$ a girl on both ends.
So my final equation is $3! \cdot$ $^4P_2$ $\cdot 3 = 216$
But then again, that was just a guess, I'm not really sure how to get it. So please confirm if my answer is right, and if it's wrong, please tell me how to get it.
• Seems correct to me. Oct 2, 2015 at 3:26
We can select the two girls in $\binom{4}{2}$ ways. We treat the three boys as a unit, so we have three objects to permute (the two girls we select and the unit of three boys). We can permute the three objects in $3!$ ways. We can also permute the unit consisting of three boys internally in $3!$ ways. Hence, there are $$\binom{4}{2} \cdot 3! \cdot 3! = 6^3 = 216$$ seating arrangements in which the seats are occupied by the three boys and two of the four girls if the three boys sit together.
First Take all of the boys in a group say $A$. $A$ contains $B_1$, $B_2$ and $B_3$. Then select two girls from $4 \Rightarrow$ $^4C_2$. Now we got $2$ girls and boys (in $A$) which makes it $5$. We can arrange $B_1$ $B_2$ $B_3$ inside $A$ in $3!$ ways . and we can arrange $A$ and the $2$ girls in $3!$ ways . Hence the answer would be $^4C_2 \cdot 3! \cdot 3! =216.$ | 2023-02-09 10:30:33 | {"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": 0.6195117831230164, "perplexity": 129.44020309774143}, "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-2023-06/segments/1674764501555.34/warc/CC-MAIN-20230209081052-20230209111052-00347.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/114614-nth-roots-unity.html | # Math Help - nth roots of unity
1. ## nth roots of unity
We know the solutions of $z^{10} = 1$ geometrically form a regular decagon. It it possible to generate the solutions of $z^{10} = 123$ (for example) by applying some type of symmetric group action $G$ on the decagon?
2. Originally Posted by Sampras
We know the solutions of $z^{10} = 1$ geometrically form a regular decagon. It it possible to generate the solutions of $z^{10} = 123$ (for example) by applying some type of symmetric group action $G$ on the decagon?
...why would we want to do something so complicated? Is there some other goal here?
3. Originally Posted by Jhevon
...why would we want to do something so complicated? Is there some other goal here?
No just wondering.
4. Originally Posted by Sampras
No just wondering.
Oh, well, I'd just interpret it as $\left( \frac z{\sqrt[10]{123}}\right)^{10} = 1$ and apply the first geometric interpretation, perhaps using a change of variable $g = \frac z{\sqrt[10]{123}}$ to make it look like $g^{10} = 1$.
Anyway, geometrically what would happen is that instead of your solutions lying on the circle of radius 1 in the complex plane, they will lie on the circle of radius $\sqrt[10]{123}$ | 2014-09-22 12:59:43 | {"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": 0, "img_math": 0, "codecogs_latex": 10, "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": 0.8263309001922607, "perplexity": 302.9633358058612}, "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-2014-41/segments/1410657137046.16/warc/CC-MAIN-20140914011217-00220-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://socratic.org/questions/why-does-sif4-act-as-a-lewis-acid | # Why does SiF4 act as a lewis acid?
Jan 4, 2016
${\text{SiF}}_{4}$ can act as a Lewis acid because $\text{Si}$ can expand its octet.
#### Explanation:
A Lewis acid is an electron-pair acceptor.
(from chemwiki.ucdavis.edu)
${\text{SiF}}_{4}$ can accept electron pairs and expand its octet to 12.
For example, it reacts with ${\text{F}}^{-}$ to form ${\text{SiF}}_{6}^{2 -}$ [hexafluoridosilicate(2-)] | 2021-10-16 11:58:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "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": 0.47651997208595276, "perplexity": 9739.760523958352}, "config": {"markdown_headings": true, "markdown_code": false, "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-2021-43/segments/1634323584567.81/warc/CC-MAIN-20211016105157-20211016135157-00337.warc.gz"} |
https://letterstonature.wordpress.com/2010/01/ | Feeds:
Posts
## In the light
I know I’d implied I was off ’til Monday, but Peter Coles has written an important post that demands immediate and unqualified endorsement. Sketching two stills from his life as an openly gay man, Coles communicates the progress that has and has not been made in the way differences from the inherited social norms of sexuality are handled within academia and British society.
Links that make the world go round:
• Sanford Schwartz in the NY Review of Books on the Belgian figurative artist Luc Tuymans;
• Anne Enwright casts a restrained but not-quite-dispassionate eye across the moral carnage surrounding Iris Robinson;
• Bernard Keane writes for Crikey on the possibility of a no-frills banking service through Australia Post;
• In case you had forgotten, Keith Windschuttle believes that the history of indigenous Australia has been fabricated by, I don’t know, Robert Manne or something; the third volume of his epic revisionism, on the topic of the Stolen Generations in particular, has just been published and Windschuttle takes to the pages of The Australian to tell us more.
• Manne and Windschuttle may deserve one another, but The Monthly >> Quadrant. Here is John Birmingham in the former regarding the existential malaise of New South Wales.
• Amanda Ripley writes in The Atlantic about a determined investigation into what makes for great teaching at the primary and secondary level.
• And for those with extra time up their sleeves, Yves Smith has plenty more to read.
Enjoy your Australian Open/Australia v. Pakistan ODI coverage. See you Monday!
– JBJ, Berkeley, CA
## WMAP 7 cosmological parameter set
Your Universe ca. 2010, per the WMAP+BAO+H0[1] maximum likelihood parameter set:
Parameter WMAP+BAO+H0 ML Hubble parameter h 0.702 H0 70.2 km/s/Mpc Dark matter density Ωch2 0.1120 Ωc 0.227 Baryonic matter density Ωbh2 0.02246 Ωb 0.0455 Total matter density Ωmh2 0.1344 Ωm 0.272 Vacuum tension[2] ΩΛ 0.728 Amplitude of curvature perturbation at k = 0.002/Mpc Δ2R 2.45 x 10-9 Spectral index of density perturbations ns 0.961 Size of linear density fluctuation at 8 Mpc/h σ8 0.807 Redshift of matter– radiation equatlity zeq 3196 Age of the Universe t0 13.78 Gyr
Parameters fit directly from the data are shown in a slightly different colour; all the others have been derived from the fit parameters using the usual definitions. The determination of zeq is carried out using the WMAP 7-year data on its own. The two papers in which these figures are given are:
Larson et al. (2010), arXiv:1001.4635
Komatsu et al. (2010), arXiv:1001.4538
These papers contain many other numbers: in particular, for extensions to ΛCDM cosmology, such as neutrino species, non-zero spatial curvature and dark energy that is not the cosmological constant. I expect some of the parameters mentioned there and not here—particularly the fNLstatistics of non-Gaussianity—to gain more public attention in the next decade as observations begin to determine the properties of the cosmological inflation that occurred in the very early Universe.
A final note: I’ve written this post only because these numbers are not written on an actual webpage—they are all in pdf or postscript files. But, it also gives me a chance to congratulate the WMAP team on their ongoing achievement.
Footnotes
1. Riess, A. et al. (2009), ApJ 699 539, arXiv:0905.0695
2. Dark energy, or, as assumed here, the cosmological constant.
## The pleasures of chess
Garry Kasparov this month thinks about reviewing something-or-other in the New York Review of Books, becoming happily diverted into a discussion of what makes chess truly interesting. (I draw also from some recent conversations with S. O. Killmier.)
The big point: chess is not about who can see the most moves ahead. Computers (and humans) that win by doing this are simply winning by brute force, rather than by intelligence; in the article Kasparov memorably denigrates his result against Deep Blue as ‘losing to a \$10 million alarm clock.’ If one insists that the only purpose of chess is to win, then brute force seems a very successful, though by no means infallible, way to do this. I’d like to spend a little time describing just why it isn’t fool-proof; and a lot of time showing why victory in chess is less than half the point.
Imagine you are a chess computer; in fact, imagine you are a chess computer with limitless computational power. Now here is a famous chess position—find the winning move:
## Barnes + 1
I’ll let Luke post more extensively on this when time permits, but Berni, Luke & their family welcomed a new addition today, Rosalie Joy Barnes.
Congratulations to the parents!
## Genus analogues
N.b. This is a technical post, written to illustrate a question I believe to be interesting to some colleagues outside my particular discipline. I am accutely aware of its shortcomings as expository work, and pedagogical criticism is almost as welcome as an attempt to engage with the question at hand.
(more…)
## The other coincidence problem
J. S. Bloom points out:
$\Omega_{\Lambda,0} \approxeq 1 - \left(\frac{1}{\alpha_0}\right)^{1/e}/\pi^e;$
so putting constraints on the evolution of the fine structure constant away from its present value. With no apologies whatsoever to xkcd.
## Audience Participa-shaun
Well, it’s time once again* for us and you both to dance the symbiotic dance of reader feedback. We seem recently to have entered a phase where the ratio of readers to commenters on this blog is undefined, because exactly one of those numbers is zero. That’s fine; we’re here for your gratification and not the other way around.
But in addition to joining us in stirring ever more extreme compotes of popular science, unpopular mathematics and token sport and politics coverage, let’s make sure there aren’t some simple topics falling between the cracks.
Even though the irregular bursts of activity are probably a good reflection of what is possible in the post-doc blog-industrial complex, perhaps there is something to be said for working at a different place along the quality–quantity curve.
Finally, I think there is much room to include some more classical music and short-form prose fare, plus links to good recipes on one of Rachael’s blogs. I also anticipate more Micallef Tonight, Micallef P(r)ogram(me) and David McGahan clips, so don’t go on a YouTube binge right now.
* N.b. that we may not, in fact, have ever done this before.
## Avatar at the atomic scale, Ctd.
The precursor to this post ended with the cliffhanger:
We demonstrate that when the diffraction pattern of a finite object is sampled at a sufficiently fine scale… the 3D structure of the object is in principle determined by the 2D spherical pattern.
Today, I describe how this piece of magic is achieved. (more…) | 2021-04-15 02:16:13 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "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": 0.3915066421031952, "perplexity": 3348.4130908383277}, "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-2021-17/segments/1618038082988.39/warc/CC-MAIN-20210415005811-20210415035811-00366.warc.gz"} |
https://socratic.org/questions/what-is-the-area-of-a-triangle-with-sides-of-length-1-2-and-3 | # What is the area of a triangle with sides of length 1, 2, and 3?
$0$, though I'm not sure you should call it a triangle.
$1 + 2 = 3$, so this 'triangle' is flat. | 2022-08-10 01:42:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 2, "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": 0.7349422574043274, "perplexity": 187.45829943948814}, "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-2022-33/segments/1659882571097.39/warc/CC-MAIN-20220810010059-20220810040059-00599.warc.gz"} |
https://keisan.casio.com/exec/system/1223963614 | # Square pyramidal number Calculator
## Calculates the number of balls and the height in a square pyramid.
number of stages 1,2,3,... 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit total balls Tn height to accumulate hn ( diameter of a ball =1 )
$\normal Square\ pyramidal\ number\\(1)\ total\ balls\\\hspace{40} T_n={\large\sum_{\small k=1}^{\small n}} k^2={\large\frac{n(n+1)(2n+1)}{6}}\\[10](2)\ height\ to\ accumulate\\\hspace{40} h_n={\large\frac{n-1}{\sqrt{2}}}+1\\$
Square pyramidal number
[1-3] /3 Disp-Num5103050100200
[1] 2018/08/16 16:42 Male / 50 years old level / An engineer / Very /
Purpose of use
Helping kids with homework
[2] 2017/06/07 04:08 Female / 20 years old level / An office worker / A public employee / Very /
Purpose of use
Helping younger sister with maths.
[3] 2015/08/21 21:48 Female / 40 years old level / A retired people / Not at All /
Purpose of use
trying to figure out how to do pyramidal math and can not figure it out.
Sending completion
To improve this 'Square pyramidal number Calculator', please fill in questionnaire.
Male or Female ?
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Age
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40 years old level 50 years old level 60 years old level or over
Occupation
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High-school/ University/ Grad student A homemaker An office worker / A public employee
Self-employed people An engineer A teacher / A researcher
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Very Useful A little Not at All
Purpose of use? | 2019-01-18 03:21:17 | {"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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/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": 0.6581361889839172, "perplexity": 4669.921991470588}, "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-2019-04/segments/1547583659677.17/warc/CC-MAIN-20190118025529-20190118051529-00405.warc.gz"} |
http://ifs.tuwien.ac.at/node/16340 | ## To Score or Not to Score? Tripling Insights for Participatory Design
M. Smuc, E. Mayr, T. Lammarsch, W. Aigner, S. Miksch, J. Gärtner:
"To Score or Not to Score? Tripling Insights for Participatory Design";
IEEE Computer Graphics and Applications,29(2009), 3; S. 29 - 38.
[ Publication Database ]
### Abstract:
Studies recording the number of user insights into data are used in the evaluation and comparison of Visual Analytics tools. However, such insight studies are based on varying definitions of insights, measure different qualitative and quantitative dimensions, and are seldom used during the participatory design phase of Visual Analytics tools. We introduce three levels of insight methodology to be used during the participatory design process and illustrate these using the DisC$\bar{o}$ project. We started by using conventional insight counters"which did not provide us with useful results for iterative design, so we went one step further and coded insights in line with the specific purpose of the tools, and successfully gathered information useful for design improvements. Finally, in order to gain an even deeper understanding, we further analyzed relations between insights and prior knowledge by means of a Relational Insight Organizer (RIO). The RIO helped characterize how users make sense of a tool, as well as where and whether they gain insights. We discuss the potential and prerequisites of these three levels of insight analysis for iterative design. | 2017-03-29 05:14:00 | {"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": 0.30067989230155945, "perplexity": 2694.872006603735}, "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-2017-13/segments/1490218190183.80/warc/CC-MAIN-20170322212950-00437-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Soft_robotics | # Soft robotics
Soft-Legged Wheel-Based Robot with Terrestrial Locomotion Abilities.
Soft Robotics is the specific subfield of robotics dealing with constructing robots from highly compliant materials, similar to those found in living organisms. [1]
Soft robotics draws heavily from the way in which living organisms move and adapt to their surroundings. In contrast to robots built from rigid materials, soft robots allow for increased flexibility and adaptability for accomplishing tasks, as well as improved safety when working around humans.[2] These characteristics allow for its potential use in the fields of medicine and manufacturing.
## Types and designs
The bulk of the field of soft robotics is based upon the design and construction of robots made completely from compliant materials, with the end result being similar to invertebrates like worms and octopi. The motion of these robots is difficult to model, as continuum mechanics apply to them, and they are sometimes referred to as continuum robots. Soft Robotics is the specific sub-field of robotics dealing with constructing robots from highly compliant materials, similar to those found in living organisms. Similarly, soft robotics also draws heavily from the way in which these living organisms move and adapt to their surroundings. In contrast to robots built from rigid materials, soft robots allow for increased flexibility and adaptability for accomplishing tasks, as well as improved safety when working around humans.[2] These characteristics allow for its potential use in the fields of medicine and manufacturing. However, there exist rigid robots that are also capable of continuum deformations, most notably the snake-arm robot.
Also, certain soft robotic mechanics may be used as a piece in a larger, potentially rigid robot. Soft robotic end effectors exist for grabbing and manipulating objects, and they have the advantage of producing a low force that is good for holding delicate objects without breaking them.
In addition, hybrid soft-rigid robots may be built using an internal rigid framework with soft exteriors for safety. The soft exterior may be multifunctional, as it can act as both the actuators for the robot, similar to muscles in vertebrates, and as padding in case of a collision with a person.
## Biomimicry
Plant cells can inherently produce hydrostatic pressure due to a solute concentration gradient between the cytoplasm and external surroundings (osmotic potential). Further, plants can adjust this concentration through the movement of ions across the cell membrane. This then changes the shape and volume of the plant as it responds to this change in hydrostatic pressure. This pressure derived shape evolution is desirable for soft robotics and can be emulated to create pressure adaptive materials through the use of fluid flow.[3] The following equation[4] models the cell volume change rate:
${\displaystyle {\dot {V}}=AL_{p}(-\Delta P+\Delta \pi )}$
${\displaystyle {\dot {V}}}$ is the rate of volume change.
${\displaystyle A}$ is the cell membrane.
${\displaystyle L_{p}}$ is the hydraulic conductivity of the material.
${\displaystyle \Delta P}$ is the change in hydrostatic pressure.
${\displaystyle \Delta \pi }$ is the change in osmotic potential.
This principle has been leveraged in the creation of pressure systems for soft robotics. These systems are composed of soft resins and contain multiple fluid sacs with semi-permeable membranes. The semi-permeability allows for fluid transport that then leads to pressure generation. This combination of fluid transport and pressure generation then leads to shape and volume change.[3]
Another biologically inherent shape changing mechanism is that of hygroscopic shape change. In this mechanism, plant cells react to changes in humidity. When the surrounding atmosphere has a high humidity, the plant cells swell, but when the surrounding atmosphere has a low humidity, the plant cells shrink. This volume change has been observed in pollen grains[5] and pine cone scales.[3][6]
## Manufacturing
Conventional manufacturing techniques, such as subtractive techniques like drilling and milling, are unhelpful when it comes to constructing soft robots as these robots have complex shapes with deformable bodies. Therefore, more advanced manufacturing techniques have been developed. Those include Shape Deposition Manufacturing (SDM), the Smart Composite Microstructure (SCM) process, and 3D multimaterial printing.[2][7]
SDM is a type of rapid prototyping whereby deposition and machining occur cyclically. Essentially, one deposits a material, machines it, embeds a desired structure, deposits a support for said structure, and then further machines the product to a final shape that includes the deposited material and the embedded part.[7] Embedded hardware includes circuits, sensors, and actuators, and scientists have successfully embedded controls inside of polymeric materials to create soft robots, such as the Stickybot[8] and the iSprawl.[9]
SCM is a process whereby one combines rigid bodies of carbon fiber reinforced polymer (CFRP) with flexible polymer ligaments. The flexible polymer act as joints for the skeleton. With this process, an integrated structure of the CFRP and polymer ligaments is created through the use of laser machining followed by lamination. This SCM process is utilized in the production of mesoscale robots as the polymer connectors serve as low friction alternatives to pin joints.[7]
3D printing can now produce shape morphing materials whose shape is photosensitive, thermally activated, or water responsive. Essentially, these polymers can automatically change shape upon interaction with water, light, or heat. One such example of a shape morphing material was created through the use of light reactive ink-jet printing onto a polystyrene target.[10] Additionally, shape memory polymers have been rapid prototyped that comprise two different components: a skeleton and a hinge material. Upon printing, the material is heated to a temperature higher than the glass transition temperature of the hinge material. This allows for deformation of the hinge material, while not affecting the skeleton material. Further, this polymer can be continually reformed through heating.[10]
## Control
All soft robots require some system to generate reaction forces, to allow the robot to move in and interact with its environment. Due to the compliant nature of these robots, this system must be able to move the robot without the use of rigid materials to act as the bones in organisms, or the metal frame in rigid robots. However, several solutions to this engineering problem exist and have found use, each possessing advantages and disadvantages.
One of these systems uses Dielectric Elastomeric Actuators (DEAs), materials that change shape through the application of a high-voltage electric field. These materials can produce high forces, and have high specific power (W/kg). However, these materials are best suited for applications in rigids robots, as they become inefficient when they do not act upon a rigid skeleton. Additionally, the high-voltages required can become a limiting factor in the potential practical applications for these robots.[11]
Another system uses springs made of shape-memory alloy. Although made of metal, a traditionally rigid material, the springs are made from very thin wires and are just as compliant as other soft materials. These springs have a very high force-to-mass ratio, but stretch through the application of heat, which is inefficient energy-wise.[11]
Pneumatic artificial muscles are yet another method used for controlling soft robots. By changing the pressure inside a flexible tube, it will act as a muscle, contracting and extending, and applying force to what it’s attached to. Through the use of valves, the robot may maintain a given shape using these muscles with no additional energy input. However, this method generally requires an external source of compressed air to function.[2]
## Uses and applications
Soft robots can be implemented in the medical profession, specifically for invasive surgery. Soft robots can be made to assist surgeries due to their shape changing properties. Shape change is important as a soft robot could navigate around different structures in the human body by adjusting its form. This could be accomplished through the use of fluidic actuation.[12]
Soft robots may also be used for the creation of flexible exosuits, for rehabilitation of patients, assisting the elderly, or simply enhancing the user’s strength. A team from Harvard created an exosuit using these materials in order to give the advantages of the additional strength provided by an exosuit, without the disadvantages that come with how rigid materials restrict a person’s natural movement.[13]
Traditionally, manufacturing robots have been isolated from human workers due to safety concerns, as a rigid robot colliding with a human could easily lead to injury due to the fast paced motion of the robot. However, soft robots could work alongside humans safely, as in a collision the compliant nature of the robot would prevent or minimize any potential injury.
There is a company, Soft Robotics Inc., www.softroboticsinc.com[14], in Cambridge, MA that has commercialized soft robotics systems for industrial and collaborative robotics applications. Theses systems are in use in food packaging, consumer goods manufacturing, and retail logistics applications.
## In popular culture
The 2014 Disney film Big Hero 6 revolved around a soft robot, Baymax, originally designed for use in the healthcare industry. In the film, Baymax is portrayed as a large yet unintimidating robot with an inflated vinyl exterior surrounding a mechanical skeleton. The basis of the Baymax concept come from real life research on applications of soft robotics in the healthcare field, such as roboticist Chris Atkeson's work at Carnegie Mellon's Robotics Institute.[15]
## References
1. ^ Trivedi, D., Rahn, C. D., Kier, W. M., & Walker, I. D. (2008). Soft robotics: Biological inspiration, state of the art, and future research. Applied Bionics and Biomechanics, 5(3), 99-117.
2. ^ a b c d Rus, Daniela; Tolley, Michael T. (27 May 2015). "Design, fabrication and control of soft robots". Nature. 521 (7553): 467–475. doi:10.1038/nature14543.
3. ^ a b c Li, Suyi; Wang, K. W. (1 January 2017). "Plant-inspired adaptive structures and materials for morphing and actuation: a review". Bioinspiration & Biomimetics. 12 (1). ISSN 1748-3190. doi:10.1088/1748-3190/12/1/011001. Retrieved 27 April 2017.
4. ^ Dumais, Jacques; Forterre, Yoël (21 January 2012). ""Vegetable Dynamicks": The Role of Water in Plant Movements". Annual Review of Fluid Mechanics. 44 (1): 453–478. doi:10.1146/annurev-fluid-120710-101200.
5. ^ Katifori, Eleni; Alben, Silas; Cerda, Enrique; Nelson, David R.; Dumais, Jacques (27 April 2010). "Foldable structures and the natural design of pollen grains". Proceedings of the National Academy of Sciences. 107 (17): 7635–7639. doi:10.1073/pnas.0911223107.
6. ^ Dawson, Colin; Vincent, Julian F. V.; Rocca, Anne-Marie (18 December 1997). "How pine cones open". Nature. 390 (6661): 668–668. doi:10.1038/37745.
7. ^ a b c Cho, Kyu-Jin; Koh, Je-Sung; Kim, Sangwoo; Chu, Won-Shik; Hong, Yongtaek; Ahn, Sung-Hoon (11 October 2009). "Review of manufacturing processes for soft biomimetic robots". International Journal of Precision Engineering and Manufacturing. 10 (3): 171–181. doi:10.1007/s12541-009-0064-6.
8. ^ Kim, S.; Spenko, M.; Trujillo, S.; Heyneman, B.; Mattoli, V.; Cutkosky, M. R. (1 April 2007). "Whole body adhesion: hierarchical, directional and distributed control of adhesive forces for a climbing robot". Proceedings 2007 IEEE International Conference on Robotics and Automation: 1268–1273. doi:10.1109/ROBOT.2007.363159. Retrieved 27 April 2017.
9. ^ Cham, Jorge G.; Bailey, Sean A.; Clark, Jonathan E.; Full, Robert J.; Cutkosky, Mark R. (1 October 2002). "Fast and Robust: Hexapedal Robots via Shape Deposition Manufacturing". The International Journal of Robotics Research. 21 (10-11): 869–882. ISSN 0278-3649. doi:10.1177/0278364902021010837. Retrieved 27 April 2017.
10. ^ a b Truby, Ryan L.; Lewis, Jennifer A. (14 December 2016). "Printing soft matter in three dimensions". Nature. 540 (7633): 371–378. doi:10.1038/nature21003.
11. ^ a b Kim, Sangbae; Laschi, Cecilia; Trimmer, Barry (May 2013). "Soft robotics: a bioinspired evolution in robotics". Trends in Biotechnology. 31 (5): 287–294. doi:10.1016/j.tibtech.2013.03.002.
12. ^ Cianchetti, Matteo; Ranzani, Tommaso; Gerboni, Giada; Nanayakkara, Thrishantha; Althoefer, Kaspar; Dasgupta, Prokar; Menciassi, Arianna (1 June 2014). "Soft Robotics Technologies to Address Shortcomings in Today's Minimally Invasive Surgery: The STIFF-FLOP Approach". Soft Robotics. 1 (2): 122–131. ISSN 2169-5172. doi:10.1089/soro.2014.0001. Retrieved 27 April 2017.
13. ^ Walsh, Conor; Wood, Robert (5 August 2016). "Soft Exosuits". Wyss Institute. Retrieved 27 April 2017.
14. ^ "Home". Soft Robotics. Retrieved 2017-08-18.
15. ^ Trimboli, Brian (Nov 9, 2014). "CMU’s soft robotics inspire Disney’s movie Big Hero 6 - The Tartan". The Tartan. Carnegie Mellon University. Retrieved 2016-08-15. | 2017-08-22 22:39:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "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": 0.5426356196403503, "perplexity": 5277.833581955589}, "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/1502886116921.7/warc/CC-MAIN-20170822221214-20170823001214-00268.warc.gz"} |
http://www.solved-problems.com/circuits/electrical-circuits-problems/1005/thevenins-theorem-circuit-independent-source/ | Use Thévenin's theorem to determine $V_O$.
Solution
To find the Thévenin equivalent, we break the circuit at the $4\Omega$ load as shown below.
So, our goal is to find an equivalent circuit that contains only an independent voltage source in series with a resistor, as shown in Fig. (1-26-3), in such a way that the current-voltage relationship at the load is not changed.
Now, we need to find $V_{Th}$ and $R_{Th}$. $V_{Th}$ is equal to the open circuit voltage $V_{OC}$ shown in Fig. (1-26-2). The current of $2\Omega$ resistor is zero because one of its terminals is not connected to any element; therefore, current cannot pass through it. Since the current of $2\Omega$ resistor is zero, the $9V$ voltage source, $3\Omega$ and $6\Omega$ resistors form a voltage divider circuit and the voltage across the $6\Omega$ resistor can be determined by the voltage devision rule. Please not that we are able to use the voltage devision rule here just because the current of the $2\Omega$ resistor is zero. You may ask that there is no reason to prove that the current of the $2\Omega$ resistor is zero in the original circuit shown in Fig. (1-26-1). That is correct. However, we are calculating $V_{OC}$ for the circuit shown in Fig. (1-26-1) and this is a different circuit. The Thévenin theorem guarantees that $V_{Th}=V_{OC}$, it is not saying that $V_{OC}$ is the voltage across the load in the original circuit.
$V_{6\Omega}=\frac{6\Omega}{3\Omega+6\Omega}\times 9 V= 6V$
Since the current of the $2\Omega$ resistor is zero:
$V_{OC}=V_{6\Omega}=6V$
$V_{Th}=V_{OC}=6V$
Now, we need to find $R_{Th}$. An easy way to find $R_{Th}$ for circuits without dependent sources is to turn off independent sources and find the equivalent resistance seen from the port. Recall that voltage sources should be replace with short circuits and current sources with open circuits. Here, there is only a voltage source that should be replaced by short circuit as shown in Fig. (1-26-4).
It is trivial to see that the $3 \Omega$ and $6 \Omega$ resistors are connected in parallel and then wired in series to the $2\Omega$ resistor. Therefore,
$R_{Th}=(3\Omega || 6\Omega)+2\Omega=\frac{3\Omega \times 6\Omega}{3\Omega + 6\Omega}+2\Omega=4\Omega$.
Now that $V_{Th}$ and $R_{Th}$ are found, we can use the Thévenin equivalent circuit depicted in Fig. (1-26-3) to calculate $V_O$ in the original circuit shown in Fig. (1-26-1). The voltage devision rule can be used here to find $V_O$. We have,
$V_{O}=\frac{4\Omega}{R_{Th}+4 \Omega}\times V_{Th}=\frac{4\Omega}{4\Omega+4 \Omega}\times 6V=3V$.
Hi! Yaz is here. I am passionate about learning and teaching. I try to explain every detail simultaneously with examples to ensure that students will remember them later too.
## Join the Conversation
It is good.But,it could be more better to give some more information.
2. It is good.But,it could be more better to give some more information and examples.
1. Sure
3. manasa says:
How to find the current through given(particular) resistor when a source voltage is given?
1. Dear Mansa,
Your question is very general. All methods explained here can be used and the best method is totally depended on the circuit topology. Please explain or give a link to image of the circuit that you are talking about.
4. manasa says:
What are the steps to be followed to find the current through given (particular) resistor when a source voltage is given using norton's theorem?
5. syed humayoon shah says:
sir i am the student of electronic engineering but i am not satisfied from my university teachers,as they are not so much attentive and hardworking..........so plzzzzzzz tell me how can i learn my basics of electronic circuits course concept wise.....i need your kind supports,becoz i wanna to become a great engineer of the furture.
regsrds
1. Dear syed humayoon shah,
If you are a hardworking student, you do not need university teachers. There are a handful of resources such as books, notes, websites,... that you can read and learn.
6. smita says:
its very gud.....
7. bakhtawer says:
sir i have a prolum in which at last there is no paralaell circuit with the load so how can i ind the voltage
1. Would you show me an example?
8. Prachiti says:
Sir pls tell me how to solve problems on equivalent resistances based on star-delta transformation effectively....
9. kishore says:
sir, nodal analysis using kcl can lead us quick ans
1. Yes, but it is asked to solve this problem by Thévenin’s Theorem.
10. Mahesh says:
sir, why capacitor acts as open circuit at steady state current ?
1. This is unrelated to this post.
Anyway, $latex i_c(t)=C\frac{V_c(t)}{dt}$. $latex V_c(t)$ is constant in steady state, therefore $latex \frac{V_c(t)}{dt}=0$ and $latex i_c(t)=0$. This shows that no current passes through a capacitor so it acts like an open circuit.
11. sir i dont understand how to solve thevinens thearom with dependent sources
12. mufu tee says:
very very nice blog. More grease to your elbow
13. likitha says:
how can we find vth and rth if another resistor of 5 ohms is conected parallel to 4 ohm resistor?
14. Shrabanee says:
Sir,if v hv d load resistance parallel wth source voltge nd current n some resistances with it how cn v find d thevenin's equivqlent voltage across that load resistance?
15. george says:
great work,l think more examples willn't be less appreciated
16. Amit says:
How to solve a problem when load is connected with a current or voltage source?
Will the voltage across the load(Vth) will be equal to the voltage of the battery itself?
Thank you
17. ambi says:
how can we solve the superposition theorem problems using thevenin theorem... also when two voltage sources on opposite sides have opposite polarity.....
18. this web realy help me in my study.
Thanks for this Great work.
19. Anusha says:
I think i have a silly doubt. But i dont understand how 3 ohm and 6 ohm resistors are connected in parallel in that problem 🙁 .. can u pls explain?? I'm new to these concepts.
1. Two elements are connected in parallel if they are connected at both sets of terminals.
3 ohm and 6 ohm are in parallel only when we make the 9V voltage zero to find Rth, as shown in Fig. (1-26-4).
20. sir i truly appreciate ur problem solving and i would love to see more examples i want to get registered in ur website but i am encountering problems please it would be good if u fix it
21. Christina says:
dear sir,
please include complicated circuit's solutions also.
22. Susmita says:
hi,plz give some examples of thevenin's and norton's theorem with dependent source.
23. allan says:
sir, may i get Io by dividing V0 by 4 Ohms? thanks.
24. Osama says:
I am student of Material Science and Engineering but i am studying basic circuits as a subject in 4rth semester. You explained the theorem very well . I forward this link to my classmates hope they'll find it helpful .Thanks
25. Do you always enter the electric current from the cathode or the anode
26. sourav mukherjee says:
this examples are to simple give some complicated ones.
Hi, I don't understand why you directly eliminated the 4ohm resistance in order to calculate the thevenin equivalent voltage, if for instance you also have another 6ohm resistance between the 4ohm and the 6ohm resistance, would you take both of them out to also calculate the thevenin equivalent voltage?
Thanks.
28. Shubhamoy Maity says:
Please,solve my problem....three resistance of 8 ohm,8 ohm and 4 ohm conected in paraller with two series resistance 5 & 3 ohm which are conected with a voltage source of 10 volt.then,calculate current through 4 ohm.by thevinin theorem.
1. Mokoena M.M says:
sir I was asking if we are given two resistors that are parallel&series with A on the bottom left then B top right how do we find equivalent resistance
29. Prasant Nair says:
But the solution vo is 3v?
30. dhanushka says:
dear sir ,
what if the batteries are having internal resistance.must it be considered when calculating open cct voltage(Vth).????????? | 2019-11-12 21:36:39 | {"extraction_info": {"found_math": true, "script_math_tex": 32, "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": 32, "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": 0.4673668146133423, "perplexity": 1287.7532046259957}, "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-2019-47/segments/1573496665767.51/warc/CC-MAIN-20191112202920-20191112230920-00148.warc.gz"} |
https://www.r-bloggers.com/2021/05/strong-random-forests-with-xgboost/ | [This article was first published on Blog – Michael's and Christian's Blog, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
# Lost in Translation between R and Python 6
Hello random forest friends
This is the next article in our series “Lost in Translation between R and Python”. The aim of this series is to provide high-quality R and Python 3 code to achieve some non-trivial tasks. If you are to learn R, check out the R tab below. Similarly, if you are to learn Python, the Python tab will be your friend.
The last one was on diamond duplicates and grouped sampling.
## XGBoost’s random forests
For sure, XGBoost is well known for its excellent gradient boosting trees implementation. Although less obvious, it is no secret that it also offers a way to fit single trees in parallel, emulating random forests, see the great explanations on the official XGBoost page. Still, there seems to exist quite some confusion on how to choose certain parameters in order to get good results. It is the aim of this post to clarify this.
Also LightGBM offers a random forest mode. We will investigate it in a later post.
### Why would you want to use XGBoost to fit a random forest?
1. Interaction & monotonic constraints are available for XGBoost, but typically not for random forest implementations. A separate post will follow to illustrate this in the random forest setting.
2. XGBoost can natively deal with missing values in an elegant way, unlike many random forest algorithms.
3. You can stick to the same data preparation pipeline.
I had additional reasons in mind, e.g. using non-standard loss functions, but this did not turn out to work well. This is possibly due to the fact that XGBoost uses a quadratic approximation to the loss, which is exact only for the mean squared error loss (MSE).
### How to enable the ominous random forest mode?
Following the official explanations, we would need to set
• num_parallel_tree to the number of trees in the forest,
• learning_rate and num_boost_round to 1.
There are further valuable tips, e.g. to set row and column subsampling to values below one to resemble true random forests.
Still, most of the regularization parameters of XGBoost tend to favour simple trees, while the idea of a random forest is to aggregate deep, overfitted trees. These regularization parameters have to be changed as well in order to get good results.
So voila my suggestions.
### Suggestions for parameters
• learning_rate=1 (see above)
• num_boost_round=1 (see above)
Has to be set in train(), not in the parameter list. It is called nrounds in R.
• subsample=0.63
A random forest draws a bootstrap sample to fit each tree. This means about 0.63 of the rows will enter one or multiple times into the model, leaving 37% out. While XGBoost does not offer such sampling with replacement, we can still introduce the necessary randomness in the dataset used to fit a tree by skipping 37% of the rows per tree.
• colsample_bynode=floor(sqrt(m))/m
Column subsampling per split is the main source of randomness in a random forest. A good default is usually to sample the square root of the number of features m or m/3. XGBoost offers different colsample_by* parameters, but it is important to sample per split resp. per node, not by tree. Otherwise, it might happen that important features are missing in a tree altogether, leading to overall bad predictions.
• num_parallel_tree
The number of trees. Native implementations of random forests usually use a default value between 100 and 500. The more, the better—but slower.
• reg_lambda=0
XGBoost uses a default L2 penalty of 1! This will typically lead to shallow trees, colliding with the idea of a random forest to have deep, wiggly trees. In my experience, leaving this parameter at its default will lead to extremely bad XGBoost random forest fits.
Set it to zero or a value close to zero.
• max_depth=20
Random forests usually train very deep trees, while XGBoost’s default is 6. A value of 20 corresponds to the default in the h2o random forest, so let’s go for their choice.
• min_child_weight=2
The default of XGBoost is 1, which tends to be slightly too greedy in random forest mode. For binary classification, you would need to set it to a value close or equal to 0.
Of course these parameters can be tuned by cross-validation, but one of the reasons to love random forests is their good performance even with default parameters.
Compared to optimized random forests, XGBoost’s random forest mode is quite slow. At the cost of performance, choose
• lower max_depth,
• higher min_child_weight, and/or
• smaller num_parallel_tree.
### Let’s try it out with regression
We will use a nice house price dataset, consisting of information on over 20,000 sold houses in Kings County. Along with the sale price, different features describe the size and location of the properties. The dataset is available on OpenML.org with ID 42092.
The following R resp. Python codes fetch the data, prepare the ML setting and fit a native random forest with good defaults. In R, we use the ranger package, in Python the implementation of scikit-learn.
The response variable is the logarithmic sales price. A healthy set of 13 variables are used as features.
library(farff)
library(OpenML)
library(dplyr)
library(ranger)
library(xgboost)
set.seed(83454)
rmse <- function(y, pred) {
sqrt(mean((y-pred)^2))
}
# Load King Country house prices dataset on OpenML
# ID 42092, https://www.openml.org/d/42092
df <- getOMLDataSet(data.id = 42092)$data head(df) # Prepare df <- df %>% mutate( log_price = log(price), year = as.numeric(substr(date, 1, 4)), building_age = year - yr_built, zipcode = as.integer(as.character(zipcode)) ) # Define response and features y <- "log_price" x <- c("grade", "year", "building_age", "sqft_living", "sqft_lot", "bedrooms", "bathrooms", "floors", "zipcode", "lat", "long", "condition", "waterfront") m <- length(x) # random split ix <- sample(nrow(df), 0.8 * nrow(df)) # Fit untuned random forest system.time( # 3 s fit_rf <- ranger(reformulate(x, y), data = df[ix, ]) ) y_test <- df[-ix, y] # Test RMSE: 0.173 rmse(y_test, predict(fit_rf, df[-ix, ])$pred)
# object.size(fit_rf) # 180 MB
# Imports
import numpy as np
import pandas as pd
from sklearn.datasets import fetch_openml
from sklearn.model_selection import train_test_split
from sklearn.ensemble import RandomForestRegressor
from sklearn.metrics import mean_squared_error
def rmse(y_true, y_pred):
return np.sqrt(mean_squared_error(y_true, y_pred))
# Fetch data from OpenML
df = fetch_openml(data_id=42092, as_frame=True)["frame"]
print("Shape: ", df.shape)
# Prepare data
df = df.assign(
year = lambda x: x.date.str[0:4].astype(int),
zipcode = lambda x: x.zipcode.astype(int)
).assign(
building_age = lambda x: x.year - x.yr_built,
)
# Feature list
xvars = [
"sqft_lot", "bedrooms", "bathrooms", "floors",
"zipcode", "lat", "long", "condition", "waterfront"
]
# Data split
y_train, y_test, X_train, X_test = train_test_split(
np.log(df["price"]), df[xvars],
train_size=0.8, random_state=766
)
# Fit scikit-learn rf
rf = RandomForestRegressor(
n_estimators=500,
max_features="sqrt",
max_depth=20,
n_jobs=-1,
random_state=104
)
rf.fit(X_train, y_train) # Wall time 3 s
# Test RMSE: 0.176
print(f"RMSE: {rmse(y_test, rf.predict(X_test)):.03f}")
Both in R and Python, the test RMSE is between 0.17 and 0.18, i.e. about 2/3 of the test predictions are within 18% of the observed value. Not bad!
Note: The test performance depends on the split seed, so it does not make sense to directly compare the R and Python performance.
#### With XGBoost's random forest mode
Now let's try to reach the same performance with XGBoost's random forest implementation using the above parameter suggestions.
# Fit untuned, but good(!) XGBoost random forest
dtrain <- xgb.DMatrix(data.matrix(df[ix, x]),
label = df[ix, y])
params <- list(
objective = "reg:squarederror",
learning_rate = 1,
num_parallel_tree = 500,
subsample = 0.63,
colsample_bynode = floor(sqrt(m)) / m,
reg_lambda = 0,
max_depth = 20,
min_child_weight = 2
)
system.time( # 20 s
fit_xgb <- xgb.train(
params,
data = dtrain,
nrounds = 1,
verbose = 0
)
)
pred <- predict(fit_xgb, data.matrix(df[-ix, x]))
# Test RMSE: 0.174
rmse(y_test, pred)
# xgb.save(fit_xgb, "xgb.model") # 140 MB
import xgboost as xgb
dtrain = xgb.DMatrix(X_train, label=y_train)
m = len(xvars)
params = dict(
objective="reg:squarederror",
learning_rate=1,
num_parallel_tree=500,
subsample=0.63,
colsample_bynode=int(np.sqrt(m))/m,
reg_lambda=0,
max_depth=20,
min_child_weight=2
)
rf_xgb = xgb.train( # Wall time 34 s
params,
dtrain,
num_boost_round=1
)
preds = rf_xgb.predict(xgb.DMatrix(X_test))
# 0.177
print(f"RMSE: {rmse(y_test, preds):.03f}")
We see:
• The performance of the XGBoost random forest is essentially as good as the native random forest implementations. And all this without any parameter tuning!
• XGBoost is much slower than the optimized random forest implementations. If this is a problem, e.g. reduce the tree depth. In this example, Python takes almost twice as much time as R. No idea why!
The timings were made on a usual 4 core i7 processor.
• Disk space required to store the model objects is comparable between XGBoost and native random forest implementations.
#### What if you would run the same model with XGBoost defaults?
• With default reg_lambda=1:
The performance would end up at a catastrophic RMSE of 0.35!
• With default max_depth=6:
The RMSE would be much worse (0.23) as well.
• With colsample_bytree instead of colsample_bynode:
The RMSE would deteriorate to 0.27.
Thus: It is essential to set some values to a good "random forest" default!
### Does it always work that good?
Definitively not in classification settings. However, in regression settings with the MSE loss, XGBoost's random forest mode is often as accurate as native implementations.
• Classification models
In my experience, the XGBoost random forest mode does not work as good as a native random forest for classification, possibly due to the fact that it uses only an approximation to the loss function.
• Other regression examples
Using the setting of our last "R <--> Python" post (diamond duplicates and grouped sampling) and the same parameters as above, we get the following test RMSEs: With ranger (R code in link below): 0.1043, with XGBoost: 0.1042. Sweet!
## Wrap up
• With the right default parameters, XGBoost's random forest mode reaches similar performance on regression problems than native random forest packages. Without any tuning!
• For losses other than MSE, it does not work so well.
The Python notebook and R code can be found at: | 2021-06-19 06:10:09 | {"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": 0.268228143453598, "perplexity": 3370.2966434135546}, "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-2021-25/segments/1623487643703.56/warc/CC-MAIN-20210619051239-20210619081239-00038.warc.gz"} |
http://serajedanesh.ir/plate-of-bocyq/ac530a-how-to-calculate-wheeling-charges-in-electricity-bill | # how to calculate wheeling charges in electricity bill
Electricity usage consumption calculator. RapidTables.com | So total Cost or Electricity bill= 720 x 9 = 6480. Enter your total units for 30 days and number of months as 1 Application of Electricity Duty or Tax The duty is charged on consumption at the applicable rate per unit of electricity consumed. This website uses cookies to improve your experience, analyze traffic and display ads. View/Pay Bill; Consumption Calculator; Energy Bill Calculator; New Connection Request; Complaint Registration; View HT Consumer Info; View Solar Consumer Info; Track Status,Upload Documents and Pay Charges * Online Payment of Other Charges * Register / Update Mobile number, email,PAN and Aadhar number New Multiply by your kWh rate. We are the leading online energy bill calculator to work out electricity cost and estimate KWh usage from meter readings or a recent utility bill from your energy supplier. Previous Metering Date If you check your electricity bill for any of the latest months starting from December 2016, you will notice that MSEDCL had introduced a new charge called as “Wheeling Charge”. If you electricity consumption is higher, you will be getting a higher electricity duty. How to Calculate Electricity Usage Cost and Charges Raymond Updated 4 years ago General 15 Comments Ever since I needed to pay my own electricity bill because I no longer stay with my parents, it is important that I know how to calculate the electricity usage and charges for every electronic devices and electrical appliances. So total Consumed units. Demand charges are fees applied to the electric bills of commercial and industrial customers based upon the highest amount of power drawn during any (typically 15-minute) interval during the billing period. RTI can get us information - but once we receive it, experts need to 'decode' it and then take follow up action. calculate the electricity bill in the Python language. The total amount due in TAC fees is determined by the following equation: {\displaystyle Totalwheelingfee=Wc (\$/MWh)\times Pw (MW)} Where 'Wc' is wheeling charge per unit. Electricity Bill calculation procedure : For Domestic Category : A domestic consumer can calculate the bill using the different tariff for different slabs given in the tariff table given on the back of the bill. Take the standard TPC rate, then deduct the TPC-WC (0.37) and then add the RINFRA-WC (0.88). we Want to convert it into Units, Where is 1unit = 1kWh. The electricity cost per day in dollars is equal to the energy consumption E in kWh per day times the energy cost of 1 kWh in cents/kWh divided by 100 cents per dollar: Cost($/day) = E(kWh/day) × Cost(cent/kWh) / 100(cent/$), © The wattage is often listed on your electrical devices. 1 to 100 units – 100 to 200 units – 200 to 300 units – above 300 units – Examples: Input: U = 250 Output: 3500 Explanation: Charge for the first 100 units – 10*100 = 1000 Divide 750 by 1000 to convert 750 watt-hours into .75 kWh (750 ÷ 1000 = .75). The wattage is the amount of the electrical power required by an appliance or device in watts, kilowatts or megawatts. Uniform power tariff will cost government Rs. Your monthly electric bill is based on consumption, and that consumption is denoted in kilowatt-hours. Certain states the duty is charged on the total charges. The only way to reduce the duty is to reduce the consumption per month. Its amazing how the past can come back to haunt you, Calculation of Wheeling Charges - RTI reply from MERC. Calculate your price using your bill. Calculate your price using your Electricity Facts Label (EFL). Bescom tariff for the domestic LT supply such as for 0 to 30 units the per-unit cost will be ₹ 3.75/-, from 31 to 100 the per-unit cost will be ₹ 5.20, from 101 to 200, the per-unit cost will be ₹ 6.75 and above 201 units you have to pay ₹ 7.8 per unit Program to Calculate Electricity Bill Example 2. The energy E in kilowatt-hours (kWh) per day is equal to the power P in watts (W) times number of usage hours per day t divided by 1000 watts per kilowatt: E(kWh/day) = P(W) × t(h/day) / 1000(W/kW). Is charged on consumption at the applicable rate per unit of electricity consumed = 1kWh it experts! Or device in watts, kilowatts or megawatts much energy you use more than 2,000 kWh in a billing.! Tdu delivery charges are and how they affect your electricity bill, must. 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Tax the duty is charged on the total charges: Enter this value as on! 2,000 kWh in a billing cycle and number of months as 1 the! Plans are very different is to reduce the consumption per month a$ bill. Tdu delivery charges are and how they affect your electricity bill use and also for how long charges, charges. Generation Plants are located at remote area and we need to 'decode ' it then...
اگر مطلب را می پسندید لطفا آنرا به اشتراک بگذارید.
دیدگاهی بنویسید | 2021-08-01 01:23:57 | {"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": 0.37545400857925415, "perplexity": 4136.110300738086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2021-31/segments/1627046154127.53/warc/CC-MAIN-20210731234924-20210801024924-00281.warc.gz"} |
https://math.stackexchange.com/questions/2975053/is-it-possible-to-show-that-a-cdot-a-lfloor-fracb2-rfloor-cdot-a-lfloo | # Is it possible to show that $a\cdot a^{\lfloor\frac{b}{2}\rfloor}\cdot a^{\lfloor\frac{b}{2}\rfloor} = a^b$ when $b$ is odd
I have $$a$$ and $$b$$ and $$b$$ is odd $$a$$ is an integer and $$b$$ is a strictly positive integer. Is there a way I can show:
$$a\cdot a^{\lfloor\frac{b}{2}\rfloor}\cdot a^{\lfloor\frac{b}{2}\rfloor} = a^b$$
I know I can simplify to:
$$a^{2\lfloor\frac{b}{2}\rfloor + 1}$$ and that $$2\lfloor\frac{b}{2}\rfloor + 1$$ is in the form of an odd number $$2k+1$$. Is there any way I can simplify more?
If $$b$$ is odd, $$\lfloor \frac{b}{2}\rfloor = \frac{b-1}2.$$ Substitute that and you can simplify the expression. | 2019-08-18 13:54:01 | {"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": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8998656272888184, "perplexity": 50.81026223770397}, "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-2019-35/segments/1566027313889.29/warc/CC-MAIN-20190818124516-20190818150516-00419.warc.gz"} |
https://gateoverflow.in/user/RohitKumarSingh/activity | # Recent activity by RohitKumarSingh
1
plz explain scan cscan ,i have 2 books which gives 2 different implementation
2
For a binary string x = a0a1 · · · an−1 define val(x) to be the value of x interpreted as a binary number, where a0 is the most significant bit. More formally, val(x) is given by How many minimum states will be in a finite automaton that accepts exactly the set of binary strings x such that val(x) is divisible by either 4 or 5. Ans is 5 or 20?
3
Twin primes are pairs of numbers p and p+2 such that both are primes-for instance, 5 and 7, 11 and 13, 41 and 43. The Twin Prime Conjecture says that there are infinitely many twin primes. Let TwinPrime(n) be a predicate that is true if n and n+2 are twin primes. Which of the ... TwinPrime(n)) ∃m. ∀n. n ≤ m implies TwinPrime(n) ∃m. ∀n. TwinPrime(n) implies n ≤ m ∀m. ∃n. n ≤ m and TwinPrime(n)
Let $X$ be a Gaussian random variable with mean 0 and variance $\sigma ^{2}$. Let $Y$ = $\max\left ( X,0 \right )$ where $\max\left ( a,b \right )$ is the maximum of $a$ and $b$. The median of $Y$ is ______________ . | 2020-08-12 11:55:48 | {"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": 0.8053349256515503, "perplexity": 460.9694965086818}, "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/1596439738892.21/warc/CC-MAIN-20200812112531-20200812142531-00534.warc.gz"} |
http://www.nag.com/numeric/CL/nagdoc_cl23/html/G01/g01dac.html | g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual
# NAG Library Function Documentnag_normal_scores_exact (g01dac)
## 1 Purpose
nag_normal_scores_exact (g01dac) computes a set of Normal scores, i.e., the expected values of an ordered set of independent observations from a Normal distribution with mean $0.0$ and standard deviation $1.0$.
## 2 Specification
#include #include
void nag_normal_scores_exact (Integer n, double pp[], double etol, double *errest, NagError *fail)
## 3 Description
If a sample of $n$ observations from any distribution (which may be denoted by ${x}_{1},{x}_{2},\dots ,{x}_{n}$), is sorted into ascending order, the $r$th smallest value in the sample is often referred to as the $r$th ‘order statistic’, sometimes denoted by ${x}_{\left(r\right)}$ (see Kendall and Stuart (1969)).
The order statistics therefore have the property
$x1≤x2≤…≤xn.$
(If $n=2r+1$, ${x}_{r+1}$ is the sample median.)
For samples originating from a known distribution, the distribution of each order statistic in a sample of given size may be determined. In particular, the expected values of the order statistics may be found by integration. If the sample arises from a Normal distribution, the expected values of the order statistics are referred to as the ‘Normal scores’. The Normal scores provide a set of reference values against which the order statistics of an actual data sample of the same size may be compared, to provide an indication of Normality for the sample . A plot of the data against the scores gives a normal probability plot. Normal scores have other applications; for instance, they are sometimes used as alternatives to ranks in nonparametric testing procedures.
nag_normal_scores_exact (g01dac) computes the $r$th Normal score for a given sample size $n$ as
$Exr=∫-∞∞xrdGr,$
where
$dGr=Arr- 1 1-Arn-r d Ar β r,n-r+ 1 , Ar=12π ∫-∞xre-t2/2 dt, r= 1,2,…,n,$
and $\beta$ denotes the complete beta function.
The function attempts to evaluate the scores so that the estimated error in each score is less than the value etol specified by you. All integrations are performed in parallel and arranged so as to give good speed and reasonable accuracy.
## 4 References
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin
## 5 Arguments
1: nIntegerInput
On entry: $n$, the size of the set.
Constraint: ${\mathbf{n}}>0$.
2: pp[n]doubleOutput
On exit: the Normal scores. ${\mathbf{pp}}\left[\mathit{i}-1\right]$ contains the value $E\left({x}_{\left(\mathit{i}\right)}\right)$, for $\mathit{i}=1,2,\dots ,n$.
3: etoldoubleInput
On entry: the maximum value for the estimated absolute error in the computed scores.
Constraint: ${\mathbf{etol}}>0.0$.
4: errestdouble *Output
On exit: a computed estimate of the maximum error in the computed scores (see Section 7).
5: failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
## 6 Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ERROR_ESTIMATE
The function was unable to estimate the scores with estimated error less than etol.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{etol}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{etol}}>0.0$.
## 7 Accuracy
Errors are introduced by evaluation of the functions $d{G}_{r}$ and errors in the numerical integration process. Errors are also introduced by the approximation of the true infinite range of integration by a finite range $\left[a,b\right]$ but $a$ and $b$ are chosen so that this effect is of lower order than that of the other two factors. In order to estimate the maximum error the functions $d{G}_{r}$ are also integrated over the range $\left[a,b\right]$. nag_normal_scores_exact (g01dac) returns the estimated maximum error as
$errest=maxr maxa,b× ∫ab dGr-1.0 .$
The time taken by nag_normal_scores_exact (g01dac) depends on etol and n. For a given value of etol the timing varies approximately linearly with n.
## 9 Example
The program below generates the Normal scores for samples of size $5$, $10$, $15$, and prints the scores and the computed error estimates.
### 9.1 Program Text
Program Text (g01dace.c)
None.
### 9.3 Program Results
Program Results (g01dace.r) | 2016-09-26 10:01:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 36, "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": 0.9927059412002563, "perplexity": 1031.974529396925}, "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-2016-40/segments/1474738660712.3/warc/CC-MAIN-20160924173740-00168-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://secure.sky-map.org/starview?object_type=1&object_id=4258&object_name=HIP+5494&locale=ZH | SKY-MAP.ORG
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# HD 6953
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CHARM2: An updated Catalog of High Angular Resolution MeasurementsWe present an update of the Catalog of High Angular ResolutionMeasurements (CHARM, Richichi & Percheron \cite{CHARM}, A&A,386, 492), which includes results available until July 2004. CHARM2 is acompilation of direct measurements by high angular resolution methods,as well as indirect estimates of stellar diameters. Its main goal is toprovide a reference list of sources which can be used for calibrationand verification observations with long-baseline optical and near-IRinterferometers. Single and binary stars are included, as are complexobjects from circumstellar shells to extragalactic sources. The presentupdate provides an increase of almost a factor of two over the previousedition. Additionally, it includes several corrections and improvements,as well as a cross-check with the valuable public release observationsof the ESO Very Large Telescope Interferometer (VLTI). A total of 8231entries for 3238 unique sources are now present in CHARM2. Thisrepresents an increase of a factor of 3.4 and 2.0, respectively, overthe contents of the previous version of CHARM.The catalog is only available in electronic form at the CDS viaanonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/431/773 Local kinematics of K and M giants from CORAVEL/Hipparcos/Tycho-2 data. Revisiting the concept of superclustersThe availability of the Hipparcos Catalogue has triggered many kinematicand dynamical studies of the solar neighbourhood. Nevertheless, thosestudies generally lacked the third component of the space velocities,i.e., the radial velocities. This work presents the kinematic analysisof 5952 K and 739 M giants in the solar neighbourhood which includes forthe first time radial velocity data from a large survey performed withthe CORAVEL spectrovelocimeter. It also uses proper motions from theTycho-2 catalogue, which are expected to be more accurate than theHipparcos ones. An important by-product of this study is the observedfraction of only 5.7% of spectroscopic binaries among M giants ascompared to 13.7% for K giants. After excluding the binaries for whichno center-of-mass velocity could be estimated, 5311 K and 719 M giantsremain in the final sample. The UV-plane constructed from these datafor the stars with precise parallaxes (σπ/π≤20%) reveals a rich small-scale structure, with several clumpscorresponding to the Hercules stream, the Sirius moving group, and theHyades and Pleiades superclusters. A maximum-likelihood method, based ona Bayesian approach, has been applied to the data, in order to make fulluse of all the available stars (not only those with precise parallaxes)and to derive the kinematic properties of these subgroups. Isochrones inthe Hertzsprung-Russell diagram reveal a very wide range of ages forstars belonging to these groups. These groups are most probably relatedto the dynamical perturbation by transient spiral waves (as recentlymodelled by De Simone et al. \cite{Simone2004}) rather than to clusterremnants. A possible explanation for the presence of younggroup/clusters in the same area of the UV-plane is that they have beenput there by the spiral wave associated with their formation, while thekinematics of the older stars of our sample has also been disturbed bythe same wave. The emerging picture is thus one of dynamical streamspervading the solar neighbourhood and travelling in the Galaxy withsimilar space velocities. The term dynamical stream is more appropriatethan the traditional term supercluster since it involves stars ofdifferent ages, not born at the same place nor at the same time. Theposition of those streams in the UV-plane is responsible for the vertexdeviation of 16.2o ± 5.6o for the wholesample. Our study suggests that the vertex deviation for youngerpopulations could have the same dynamical origin. The underlyingvelocity ellipsoid, extracted by the maximum-likelihood method afterremoval of the streams, is not centered on the value commonly acceptedfor the radial antisolar motion: it is centered on < U > =-2.78±1.07 km s-1. However, the full data set(including the various streams) does yield the usual value for theradial solar motion, when properly accounting for the biases inherent tothis kind of analysis (namely, < U > = -10.25±0.15 kms-1). This discrepancy clearly raises the essential questionof how to derive the solar motion in the presence of dynamicalperturbations altering the kinematics of the solar neighbourhood: doesthere exist in the solar neighbourhood a subset of stars having no netradial motion which can be used as a reference against which to measurethe solar motion?Based on observations performed at the Swiss 1m-telescope at OHP,France, and on data from the ESA Hipparcos astrometry satellite.Full Table \ref{taba1} is only available in electronic form at the CDSvia anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/430/165} Ca II K Emission-Line Asymmetries Among Red GiantsMeasurements of the asymmetry of the K2 emission line of CaII have been made for a sample of bright field giants with B-V>1.15observed with the Cassegrain echelle spectrometer on the McDonaldObservatory 2.1 m telescope. The asymmetry of the Ca II K2line is quantified through measurement of a parameter V/R, which isdefined as the ratio between the maximum counts recorded in the violetand red components of the double-peaked emission profile. Red-maximumasymmetry (V/R<1.0) is found in our sample of 35 giants only amongstars with B-V>1.35, a trend that is still maintained (with oneexception) with the inclusion of an additional sample of giantspreviously observed by us with the same spectrograph. Althoughexceptional stars can be found in the literature, the data support anearlier finding by R. Stencel that among luminosity class III fieldgiants the occurrence of V/R<1.0 is generally restricted to effectivetemperatures cooler than 4320 K. This limit may coincide with the onsetof pulsation. A Low-Mass Central Black Hole in the Bulgeless Seyfert 1 Galaxy NGC 4395NGC 4395 is one of the least luminous and nearest known type 1 Seyfertgalaxies, and it also lacks a bulge. We present a Hubble Space Telescope(HST) I-band image of its nuclear region and Keck high-resolution (~8 kms-1) echelle spectra containing the Ca II near-infraredtriplet. In addition to the unresolved point source, there is a nuclearstar cluster of size r~3.9 pc; the upper limit on its velocitydispersion is only 30 km s-1. We thus derive an upper limitof ~6.2×106 Msolar for the mass of thecompact nucleus. Based on the amount of spatially resolved light in theHST image, a sizable fraction of this is likely to reside in stars.Hence, this estimate sets a stringent upper limit on the mass of thecentral black hole. We argue, from other lines of evidence, that thetrue mass of the black hole is likely to be~104-105 Msolar. Although the blackhole is much less massive than those thought to exist in classicalactive galactic nuclei (AGNs), its accretion rate ofLbol/LEdd~2×10-2 to2×10-3 is consistent with the mass-luminosity relationobeyed by classical AGNs. This may explain why NGC 4395 has ahigh-excitation (Seyfert) emission-line spectrum; active galaxies havinglow-ionization spectra seem to accrete at significantly lower rates. NGC4395, a pure disk galaxy, demonstrates that supermassive black holes arenot associated exclusively with bulges. A catalogue of calibrator stars for long baseline stellar interferometryLong baseline stellar interferometry shares with other techniques theneed for calibrator stars in order to correct for instrumental andatmospheric effects. We present a catalogue of 374 stars carefullyselected to be used for that purpose in the near infrared. Owing toseveral convergent criteria with the work of Cohen et al.(\cite{cohen99}), this catalogue is in essence a subset of theirself-consistent all-sky network of spectro-photometric calibrator stars.For every star, we provide the angular limb-darkened diameter, uniformdisc angular diameters in the J, H and K bands, the Johnson photometryand other useful parameters. Most stars are type III giants withspectral types K or M0, magnitudes V=3-7 and K=0-3. Their angularlimb-darkened diameters range from 1 to 3 mas with a median uncertaintyas low as 1.2%. The median distance from a given point on the sky to theclosest reference is 5.2degr , whereas this distance never exceeds16.4degr for any celestial location. The catalogue is only available inelectronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr(130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/393/183 CHARM: A Catalog of High Angular Resolution MeasurementsThe Catalog of High Angular Resolution Measurements (CHARM) includesmost of the measurements obtained by the techniques of lunaroccultations and long-baseline interferometry at visual and infraredwavelengths, which have appeared in the literature or have otherwisebeen made public until mid-2001. A total of 2432 measurements of 1625sources are included, along with extensive auxiliary information. Inparticular, visual and infrared photometry is included for almost allthe sources. This has been partly extracted from currently availablecatalogs, and partly obtained specifically for CHARM. The main aim is toprovide a compilation of sources which could be used as calibrators orfor science verification purposes by the new generation of largeground-based facilities such as the ESO Very Large Interferometer andthe Keck Interferometer. The Catalog is available in electronic form atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/386/492, and from theauthors on CD-Rom. Long-Term VRI Photometry of Small-Amplitude Red Variables. I. Light Curves and PeriodsWe report up to 5000 days of VRI photometry, from a robotic photometrictelescope, of 34 pulsating red giants, namely, TV Psc, EG And, Z Psc, RZAnd, 4 Ori, RX Lep, UW Lyn, η Gem, μ Gem, ψ1 Aur,V523 Mon, V614 Mon, HD 52690, Y Lyn, BC CMi, X Cnc, UX Lyn, RS Cnc, VYUMa, ST UMa, TU CVn, FS Com, SW Vir, 30 Her, α1 Her,V642 Her, R Lyr, V450 Aql, V1293 Aql, δ Sge, EU Del, V1070 Cyg, WCyg, and μ Cep, as well as a few variable comparison stars. V, R, andI variations are generally in phase. The length and density of the dataenable us to look for variations on timescales ranging from days toyears. We use both power-spectrum (Fourier) analysis and autocorrelationanalysis, as well as light-curve analysis; these three approaches arecomplementary. The variations range from regular to irregular, but inmost of the stars, we find a period in the range of 20-200 days, whichis probably due to low-order radial pulsation. In many of the stars, wealso find a period which is an order of magnitude longer. It may be dueto rotation, or it may be due to a new kind of convectively inducedoscillatory thermal mode, recently proposed by P. Wood. Revision and Calibration of MK Luminosity Classes for Cool Giants by HIPPARCOS ParallaxesThe Hipparcos parallaxes of cool giants are utilized in two ways in thispaper. First, a plot of reduced parallaxes of stars brighter than 6.5,as a function of spectral type, for the first time separates members ofthe clump from stars in the main giant ridge. A slight modification ofthe MK luminosity standards has been made so that luminosity class IIIbdefines members of the clump, and nearly all of the class III stars fallwithin the main giant ridge. Second, a new calibration of MK luminosityclasses III and IIIb in terms of visual absolute magnitudes has beenmade. Spectral Irradiance Calibration in the Infrared. X. A Self-Consistent Radiometric All-Sky Network of Absolutely Calibrated Stellar SpectraWe start from our six absolutely calibrated continuous stellar spectrafrom 1.2 to 35 μm for K0, K1.5, K3, K5, and M0 giants. These wereconstructed as far as possible from actual observed spectral fragmentstaken from the ground, the Kuiper Airborne Observatory, and the IRAS LowResolution Spectrometer, and all have a common calibration pedigree.From these we spawn 422 calibrated spectral templates'' for stars withspectral types in the ranges G9.5-K3.5 III and K4.5-M0.5 III. Wenormalize each template by photometry for the individual stars usingpublished and/or newly secured near- and mid-infrared photometryobtained through fully characterized, absolutely calibrated,combinations of filter passband, detector radiance response, and meanterrestrial atmospheric transmission. These templates continue ourongoing effort to provide an all-sky network of absolutely calibrated,spectrally continuous, stellar standards for general infrared usage, allwith a common, traceable calibration heritage. The wavelength coverageis ideal for calibration of many existing and proposed ground-based,airborne, and satellite sensors, particularly low- tomoderate-resolution spectrometers. We analyze the statistics of probableuncertainties, in the normalization of these templates to actualphotometry, that quantify the confidence with which we can assert thatthese templates truly represent the individual stars. Each calibratedtemplate provides an angular diameter for that star. These radiometricangular diameters compare very favorably with those directly observedacross the range from 1.6 to 21 mas. Catalogs of temperatures and [Fe/H] averages for evolved G and K starsA catalog of mean values of [Fe/H] for evolved G and K stars isdescribed. The zero point for the catalog entries has been establishedby using differential analyses. Literature sources for those entries areincluded in the catalog. The mean values are given with rms errors andnumbers of degrees of freedom, and a simple example of the use of thesestatistical data is given. For a number of the stars with entries in thecatalog, temperatures have been determined. A separate catalogcontaining those data is briefly described. Catalog only available atthe CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/Abstract.html Vitesses radiales. Catalogue WEB: Wilson Evans Batten. Subtittle: Radial velocities: The Wilson-Evans-Batten catalogue.We give a common version of the two catalogues of Mean Radial Velocitiesby Wilson (1963) and Evans (1978) to which we have added the catalogueof spectroscopic binary systems (Batten et al. 1989). For each star,when possible, we give: 1) an acronym to enter SIMBAD (Set ofIdentifications Measurements and Bibliography for Astronomical Data) ofthe CDS (Centre de Donnees Astronomiques de Strasbourg). 2) the numberHIC of the HIPPARCOS catalogue (Turon 1992). 3) the CCDM number(Catalogue des Composantes des etoiles Doubles et Multiples) byDommanget & Nys (1994). For the cluster stars, a precise study hasbeen done, on the identificator numbers. Numerous remarks point out theproblems we have had to deal with. Spectral classifications in the near infrared of stars with composite spectra. I. The study of MK standards.Up to now the spectral classifications of the cool components ofcomposite spectra obtained in the 3800-4800A wavelength region have beenvery disparate. These disparities are due to the fact that the spectraof the evolved cool component are strongly veiled by that of the hotterdwarf component, which makes a classification very difficult. We proposeto study these systems in the near infrared (8380-8780A). In thisspectral domain the magnitude difference between the spectra of thecomponents is in general sufficiently large so that one observespractically only the spectrum of the cool component. In this first paperwe provide, for a sample of MK standards, the relations between theequivalent width (Wlambda_ ) of certain lines and thespectral classifications. For the cool G, K and M type stars, the linesconsidered are those of the calcium triplet (Ca II 8498, 8542 and 8662),of iron (Fe I 8621 and 8688), of titanium (Ti I 8426 and 8435) and ofthe blend λ8468. The use of certain line intensity ratiospermits, after eliminating partially the luminosity effects, a firstapproach to the spectral type. For the hotter stars of types O, B, A andF we study the behavior of the hydrogen lines (P12 and P14), the calciumlines (Ca II 8498 and 8542) as well as those of the oxygen (O I 8446).The latter line presents a very characteristic profile for stars of lowrotation and therefore in Am stars, which are frequently found among thecomposite spectrum binaries. Among the cooler stars of our sample, only6% present real anomalies with respect to the MK classifications. Thisresult is very encouraging for undertaking the classification of asample of composite spectra. The spectra were taken at the Observatoirede Haute-Provence (OHP) with the CARELEC spectrograph at the 193 cmtelescope, with a dispersion of 33 A/mm. A critical appraisal of published values of (Fe/H) for K II-IV stars'Primary' (Fe/H) averages are presented for 373 evolved K stars ofluminosity classes II-IV and (Fe/H) values beween -0.9 and +0.21 dex.The data define a 'consensus' zero point with a precision of + or -0.018 dex and have rms errors per datum which are typically 0.08-0.16dex. The primary data base makes recalibration possible for the large(Fe/H) catalogs of Hansen and Kjaergaard (1971) and Brown et al. (1989).A set of (Fe/H) standard stars and a new DDO calibration are given whichhave rms of 0.07 dex or less for the standard star data. For normal Kgiants, CN-based values of (Fe/H) turn out to be more precise than manyhigh-dispersion results. Some zero-point errors in the latter are alsofound and new examples of continuum-placement problems appear. Thushigh-dispersion results are not invariably superior to photometricmetallicities. A review of high-dispersion and related work onsupermetallicity in K III-IV star is also given. Third preliminary catalogue of stars observed with the photoelectric astrolabe of the Beijing Astronomical Observatory.Not Available Large and kinematically unbiased samples of G- and K-type stars. IV - Evolved stars of the old disk populationModified Stromgren and (R,I) photometry, along with DDO and Genevaphotometry, are presented for a complete sample of evolved old-disk Gand K giants in the Bright Star Catalogue. Stars with ages of between1.5 x 10 to the 9th and 10 to the 10th yr are found to have anear-normal distribution of heavy element abundances, centered on anFe/H abundance ratio of -0.1 dex. The old disk clusters NGC 3680 and IC4651 contain red-straggler young-disk giants that are probablycontemporaries of the blue stragglers in the clusters. Large and kinematically unbiased samples of G- and K-type stars. II - Observations of evolved stars in the Bright Star sample. III - Evolved young disk stars in the Bright Star sampleFour color and RI observations were obtained for a large sample ofG-type and K-type stars in the Bright Star Catalogue. Data are firstpresented for 110 evolved stars. Photometry of evolved young diskpopulation stars have then been calibrated for luminosity, reddening,and metallicity on the basis of results for members of the Hyades andSirius superclusters. New DDO results are given for 120 stars. A list of MK standard starsNot Available A search for lithium-rich giant starsLithium abundances or upper limits have been determined for 644 brightG-K giant stars selected from the DDO photometric catalog. Two of thesegiants possess surface lithium abundances approaching the 'cosmic' valueof the interstellar medium and young main-sequence stars, and eight moregiants have Li contents far in excess of standard predictions. At leastsome of these Li-rich giants are shown to be evolved to the stage ofhaving convectively mixed envelopes, either from the direct evidence oflow surface carbon isotope ratios, or from the indirect evidence oftheir H-R diagram positions. Suggestions are given for the uniqueconditions that might have allowed these stars to produce or accrete newlithium for their surface layers, or simply to preserve from destructiontheir initial lithium contents. The lithium abundance of the remainingstars demonstrates that giants only very rarely meet the expectations ofstandard first dredge-up theories; the average extra Li destructionrequired is about 1.5 dex. The evolutionary states of these giants andtheir average masses are discussed briefly, and the Li distribution ofthe giants is compared to predictions of Galactic chemical evolution. The Perkins catalog of revised MK types for the cooler starsA catalog is presented listing the spectral types of the G, K, M, and Sstars that have been classified at the Perkins Observatory in therevised MK system. Extensive comparisons have been made to ensureconsistency between the MK spectral types of stars in the Northern andSouthern Hemispheres. Different classification spectrograms have beengradually improved in spite of some inherent limitations. In thecatalog, the full subclasses used are the following: G0, G5, G8, K0, K1,K2, K3, K4, K5, M0, M1, M2, M3, M4, M5, M6, M7, and M8. Theirregularities are the price paid for keeping the general scheme of theoriginal Henry Draper classification. 1988 Revised MK Spectral Standards for Stars GO and LaterNot Available Narrow-band photometry of late-type stars. IIThis paper presents extensive narrow-band photometry in the Uppsalasystem supplementing earlier published mesurements so that data now areavailable for all late-type stars brighter than V = 6.05 and a number ofgalactic cluster members. Numerous UBV and BV measurements are alsopublished. The data are used to determine relations for the predictionof UBV intrinsic colors for late-type stars from the narrow-bandmeasurements. The main purpose of the data is to constitute the basisfor the determination of solar-neighborhood space densities of late-typestars, mainly giants of different kinds; these space densities will becombined with narrow-band data for fainter stars in the north Galacticpole region to yield the decrease of space density with distance fromthe galactic plane for many kinds of late-type stars. Radial velocities of standard starsRadial-velocity observations obtained over a five-year period with theMcDonald Observatory photoelectric radial-velocity spectrometer arereported for those stars not known to be velocity variable. There are259 stars included in the 1650 observations of the spectral type rangeF0 to M0 and brighter than a V of 6.5 mag. For the best-observed 134stars, the standard error of the mean velocity is typically better than+ or - 0.9 km/s. Sixteen stars are shown to be constant to a sufficientlevel to warrant standard-star status. Six possible spectroscopicbinaries are found. 1985 revised MK spectral standards : stars GO and laterNot Available Revised MK Spectral Standard Stars Later than G0Not Available The absolute magnitudes of G5-M3 stars near the giant branchThe absolute magnitudes of stars on the red giant branch (G-K-M) havebeen determined using both trigonometric and statistical parallaxes,from a sample of 212 stars classified in the Revised MK System (Keenanand Pitts, 1980). The results of both methods are summarized in a table.A good agreement is found and the difference between trigonometric andstatistical parallaxes is found not to be greater than + or - 0.002. Thecomputed absolute magnitudes and space motions are tabulated. UBV Photoelectric Photometry of 259 PZT StarsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1980PASP...92..215G&db_key=AST Revised MK spectral types for G, K, and M starsA catalog of spectral types of 552 G, K, and M stars is presented, whichis classified on the revised MK system. Stellar representatives of thehalo, disk, and arm populations in all parts of the sky are included.Photoelectric V magnitudes are given, as are intensity estimates of anyfeatures which make the spectrum appear peculiar as compared to thespectrum of a similar normal star. Abundance indices are also providedin the following lines or bands: CN, barium, Fe, calcium, and CH. Catalog of Indidual Radial Velocities, 0h-12h, Measured by Astronomers of the Mount Wilson ObservatoryAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1970ApJS...19..387A&db_key=AST - and Broad-Band Photometry of Red Stars. Northern GiantsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1967ApJS...14..307E&db_key=AST The Application of an Oscilloscopic Microphotometer to the Spectral Classification of Late-Type Stars.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1954ApJ...119..613H&db_key=AST
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### 下列团体成员
#### 观测天体数据
星座: 雙魚座 右阿森松: 01h10m19.40s 赤纬: +25°27'28.0" 视星: 5.8 距离: 110.375 天文距离 右阿森松适当运动: 2.8 赤纬适当运动: -114.9 B-T magnitude: 7.751 V-T magnitude: 5.972 | 2019-11-18 14:37:31 | {"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": 0.6376286745071411, "perplexity": 6537.302142132072}, "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-2019-47/segments/1573496669795.59/warc/CC-MAIN-20191118131311-20191118155311-00182.warc.gz"} |
http://onlineprediction.net/?n=Main.KernelMethodsIntroduction | # Kernel Methods Introduction
Kernel methods are methods of machine learning based on the so-called "kernel trick" of solving non-linear problems by the means of linear methods. A very detailed article about kernel methods, their motivation, theory, and RKHS can be found in Kernel Methods.
Kernel trick transforms observations into some higher-dimensional space (sometimes called feature space), where the linear problem is solved. The kernel trick can be applied to an algorithm which uses the observations only in the form of scalar products . Instead of the scalar products, this algorithm is forced to use the value of some continuous, symmetric, non-negative definite kernel function . For some learning algorithm, like Support Vector Machine, the use of kernel tricks allows to find a non-linear class separation. The features of this non-linearity are defined by the means of the chosen kernel. Kernel trick can be applied in Conformal Prediction, as one of the examples.
In Competitive on-line prediction the kernel trick is often applied to the algorithms for online linear regression in order to compete with wider prediction rules. Gammerman et al.(2004) apply the Strong Aggregating Algorithm to compete with all functions from the separable RKHS (Reproducing Kernel Hilbert space), and Vovk (2006) shows that this approach gives the regret term of order .
In practice, the popular kernels are Gaussian (or RBF - Radial Basis Function) kernel: , polynomial . More complicated ANNOVA kernel is . If is set to 1 in the ANOVA kernel, the outcome becomes the sum of all individual sub-kernel results. Alternatively, if is set to equal the size of vector , the outcome becomes the product of all individual sub-kernel results.
Each kernel corresponds to some unique Reproducing Kernel Hilbert space. It is a non trivial problem to find an appropriate norm in this space for the given kernel, and backwards to find a kernel by a given norm. Some set of kernels and norms are described in Wahba (1990).
To generalize the kernel trick to Banach spaces is an open problem: Banach learning.
### Bibliography
• Alex Gammerman, Yuri Kalnishkan, and Vladimir Vovk. On-line Prediction with Kernels and the Complexity Approximation Principle. In: Uncertainty in Artificial Intelligence, Proceedings of the Twentieth Conference, pages 170--176. AUAI Press, 2004.
• Bernhard Scholkopf B. and Alexander J. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002.
• Vladimir Vovk. On-line regression competitive with reproducing kernel Hilbert spaces (extended abstract). In: Theory and Applications of Models of Computation. Proceedings of the Third Annual Conference on Computation and Logic (ed. by J.-Y. Cai, S. B. Cooper and A. Li), Lecture Notes in Computer Science, vol. 3959, pp. 452 – 463. Berlin: Springer, 2006
• Grace Wahba. Spline Models for Observational Data, volume 59 of CBMSNSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA, 1990.
• Nachman Aronszajn, La théorie des noyaux reproduisants et ses applications, Proceedings of the Cambridge Philosophical Society, vol. 39 (1943), pp. 133—153 (in French).
• Nachman Aronszajn, Theory of Reproducing Kernels, Transactions of the American Mathematical Society, volume 68, number 3, pages 337-404, 1950.
• Saburou Saitoh, Theory of reproducing kernels and its applications, Longman Scientific and Technical, 1988.
• Saburou Saitoh, Integral Transforms, Reproducing Kernels and Their Applications, Addison Wesley Longman Limited, 1997. | 2019-07-20 16:12:04 | {"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": 0.8132832050323486, "perplexity": 994.5782788897804}, "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-2019-30/segments/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00395.warc.gz"} |
https://www.khanacademy.org/math/pre-algebra/pre-algebra-equations-expressions/pre-algebra-evaluating-expressions-word-problems/e/evaluating-expressions-3 | # Evaluating expressions with variables word problems
Practice plugging in values to evaluate real-world algebraic expressions. These are introductory problems, so the expressions aren't too complicated.
You might need: Calculator
### Problem
The expression 1, point, 08, s, plus, 1, point, 02, b predicts the end-of-year value of a financial portfolio where s is the value of stocks and b is the value of bonds in the portfolio at the beginning of the year.
What is the predicted end-of-year value of a portfolio that begins the year with dollar sign, 200 in stocks and dollar sign, 100 in bonds?
dollar sign | 2016-09-28 15:36:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 13, "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": 0.3828108012676239, "perplexity": 2489.361495270118}, "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-2016-40/segments/1474738661555.40/warc/CC-MAIN-20160924173741-00295-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://paultopia.github.io/posts-output/ng2/ | ## Lecture 3 – locally weighted regression
Nonparametric algorithms reduce "the need to choose features very carefully" (I guess that makes sense if you think of features as mathematical transformations on stuff observed rather than stuff observed in general... a nonparemetric algorithm surely can't avoid the fact that you left something off, though I guess it can help avoid the fact that you threw a bunch of extra stuff in...)
Formal definition of a nonparametric algorithm is an algorithm where the number of parameters grows with m. Which also means it needs to hold onto the entire training set even after training. As a student said in questions, "it's like you're not even really building a model at all." You just fit for every training example. (This seems really expensive!!)
### Locally weighted regression (loess/lowss)
Concept: consider a value for x, the vector of features, of a single observation. To make a prediction with OLS, we'd find the vector of weights (parameters) $\theta$ s.t. they minimize the cost function, then return $\theta^tx$ as prediction.
For loess, we'd take a region around x, and work on the subset of data around there. So, geometrically, rather than predicting y based a line fitted to the entire dataset, predicts y based a line fitted to a subset of the dataset around x.
formally, in loess we fit $\theta$ to minimize a weighted version of the same loss function we use in OLS, where the weights are chosen such that we give more weight to training examples closer to what we're trying to predict.
i.e., OLS minimizes:
$$\sum_i(y^{(i)}-\theta^Tx^{(i)})^2$$
while loess minimizes:
$$\sum_iw^{(i)}(y^{(i)}-\theta^Tx^{(i)})^2$$
the trick is in the definition of the weight function. As I understand it it the fit is made at the time of prediction, so x without a subscript in the below is the value of the feature for which you're trying to predict the output (and that's why you have to keep your training data around even after training, as Ng noted earlier. Do you even train at all in advance? Maybe there's some optimization trick that allows you to pre-train something? Kinda doubting it from the "not even a model" chat above.). So with that preamble, the weight function is
$$w^{(i)}=e^{(-\frac{(x^{(i)}-x)^2}{2})}$$
Actually, he said there are lots of possible weight functions, but the point is to have something that gets close to zero when $x^{(i)}$ is far from x and close to 1 when they're close together. Which, obviously, this satisfies.
A more common form of the weight is
$$w^{(i)}=e^{(-\frac{(x^{(i)}-x)^2}{2\tau^2})}$$
where tau is a "bandwidth parameter" that controls the rate at which the weighting function falls off with distance from x.
Another student question: this is indeed very costly, "every time you make a prediction you need to fit theta to your entire training set again." However, "turns out there are ways to make this much more efficient." He referred to Andrew Moore's kd-trees as this method.
### Probabilistic interpretation of linear regression
Why are we minimizing the sum of squared error as opposed to the absolute value or something? Assumptions that make this work. (Oh boy, are we going to do BLUE again? Might skim past this.)
First we "endow the least squares model with probabilistic semantics."
Yeah, this is the same stuff. Assume y is a function of the model plus error, assume error is IID and distributed normally with mean zero, all the good social science stats stuff I already know. Then we do the standard probability and algebra and get the maximum likelihood estimator, which turns out to be the OLS cost function. And that was like a whole class in grad school. The central limit theorem came up, as it does. All the good stuff. For his derivation, see pages 11-13 of lecture notes 1.
There is one useful notation note though. Semicolon indicates not a random variable but as something we're trying to estimate in the world, i.e. this:
$$P(y^{(i)}|x^{(i)};\theta)$$
indicates "the probability of $y^{(i)}$ conditioned on $x^{(i)}$ as parameterized by $\theta$ " while this:
$$P(y^{(i)}|x^{(i)},\theta)$$
means "the probability of $y^{(i)}$ conditioned on $x^{(i)}$ and $\theta$ " which is wrong, because theta isn't a random variable, it's a property of the world we're trying to estimate (in frequentist terms).
The conditional probability (the correct one) = the likelihood function, only y gets an arrow (to indicate it's observed?). So maximum likelihood.
### Classification
Started off with standard stuff, instead of choosing a linear function, we choose a nonlinear function. And for logistic regression, that's the sigmoid function, a.k.a. the logistic function:
$$h_\theta(x) = \frac{1}{1 + e^{-\theta^Tx}}$$
One important point is that for logistic gradient descent we're actually ascending, that is, we're trying to maximize not minimize, so we add the gradient rather than subtract it.
Interestingly, it comes out to the same update rule with the sign swapped when the dust settles. It's not the same math because the function that generates the hypothesis is obviously different (the linear function vs the logistic function), but it has the same functional form. Logistic gradient ascent update rule:
$$\theta_j : = \theta_j + \alpha (y^{(i)} - h_{\theta}(x^{(i)})) \cdot x_j^{(i)}$$
Why does maximizing this work exactly? It's just maximum likelihood again. It turns out that for OLS, maximizing the likelihood function, after the math dust settles, is the same as minimizing the least squares cost function. (See notes pg. 13) But for logistic regression, when we maximize the log likelihood of the parameters, the gradient ascent that we use to directly maximize the likelihood just simplifies to the same form.
(An explanation of why turned up elsewhere: logistic regression likelihood is concave. page 11 of this)
### digression on perceptrons
a perceptron is just the same update rule but with this threshold function that maps positive values to 1 and negative values to 0 rather than logistic function's mapping of everything to the space from 0-1.
Hard to interpret perceptrons probabilistically though.
Tags: math meta test | 2018-06-18 15:21:22 | {"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": 0.9183850884437561, "perplexity": 640.4148574599537}, "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/1529267860570.57/warc/CC-MAIN-20180618144750-20180618164750-00563.warc.gz"} |
https://cseducators.stackexchange.com/tags/python/hot | # Tag Info
Accepted
### Interesting Programming Exercises to Teach Inheritance?
I'm not as familiar with Python as I am with other languages, but I'm sure your students have played Minecraft. If you haven't, I suggest taking a few minutes to find some introductory "Lets Play" ...
Accepted
### What to consider when choosing version of Python?
A few years ago, the answer to this would have been "stick with Python 2; the libraries aren't ready for Python 3 yet". In many cases, that would have been a deal-breaker, because many of the older ...
• 3,485
Accepted
### IDE vs Editor and terminal for CS1
TL;DR Those two aren't your only options. The main concern is cognitive load: learning to program is difficult enough without adding incidental complexity. We've seen an explosion of hybrids in ...
• 799
### Interesting Programming Exercises to Teach Inheritance?
Too many examples that you find are (IMO) fatally flawed. The Animal->Dog is especially flawed, though widely used. The problem is that these sorts of examples almost require that the superclass has a ...
• 34.9k
### IDE vs Editor and terminal for CS1
Here are my thoughts on this. Editor and Terminal This is most likely the more lightweight solution. Editor and terminal often don't consume much space (or, at least, come bundled with the operating ...
• 961
Accepted
### How do I approach teaching Python to 12-year-olds as a first-time teacher?
Check out the Python books written by Al Sweigart. His homepage Invent with Python includes some great, free resources that are geared to the age range of your students. In particular he focuses on ...
• 9,032
### IDE vs Editor and terminal for CS1
Of course it depends on your overall goals. For me, however, the answer is clear: Use the most powerful IDE that I can find (Eclipse or NetBeans fit my def). I started programming on primitive ...
• 34.9k
### Any simple Python GUI projects for beginner/novice programming students?
In my experience, good introductory programming courses meet three overarching goals: Empower students to create simple programs outside of the scope of the class by giving them the technical skills ...
Accepted
### First programming Language : C or Python?
The question is actually more complex than it might appear, and really the answer can depend on the context. For example, at what age are the students when they are first taught to program? Is this ...
### IDE vs Editor and terminal for CS1
The real question is this: do you want to teach your students what is actually going on, or teach them which magic buttons to press in an IDE? Of course for professional programming work nobody would ...
• 613
### Starting open source
Since you specifically mention high-schoolers getting started with open source, i have to recommend the Google Code-In. I participated all 4 years as a high-schooler and really learned a lot about ...
• 758
### Interesting Programming Exercises to Teach Inheritance?
I've got one that might help, modified/simplified from an actual problem I had to solve at my current job. Imagine you're writing a Content Management system - this system will store four types of ...
• 290
Accepted
### Will I Regret Using Python As A Teaching Language When I Later Need to Teach Static Typing?
If you teach Python's typing system correctly, you should have no problem later. The rule in Python is that names don't have a type associated with them, but all values do. It isn't that "things" ...
• 34.9k
Accepted
### Any simple Python GUI projects for beginner/novice programming students?
Unfortunately, GUI programming is sufficiently different from algorithmic programming that if you start with it students can get the wrong idea about what a program should look like. For example, ...
• 34.9k
### Pedagogical issues with Stack Implementation
Actually, the code is terrible, but I don't think its purpose is to illustrate a stack so much as to illustrate in a very rudimentary way how heap allocation works. (Worse than "terrible", it isn't "...
• 34.9k
### Interesting Programming Exercises to Teach Inheritance?
My coding school gave one particular (weeks-long) project that I felt nailed the concept of inheritance, and why it could be useful: Simulating a circuit board with logic gates. The framework of the ...
### Explaining why arrays are important for statisticians
If you are tutoring her, it is wonderful that you are trying to motivate the material in a practical way, but don't beat yourself up too much if you aren't that successful at persuading her. Some ...
• 31.5k
### What is the best way to learn an object oriented programming language with framework, data structure and alogrithms?
You certainly don't need a list longer than this one. If you do even half of this you will have learned enough to know pretty much what should be next. Having a complete list now gives you very little....
• 34.9k
### How do I approach teaching Python to 12-year-olds as a first-time teacher?
I agree with Peter. It needs to be fun, and games really help. It doesn't take much to exercise basic programming concepts such as variables, loops, selection, input, and output. I would start with ...
• 1,044
### Explaining why arrays are important for statisticians
Have you tried a simple statistics formula such as $$mean = \left(\sum_{i=1}^{n} x_i\right) / n$$ This maps exactly to array notation. Explain that the array x ...
• 1,113
### Starting open source
Before contributing to an open source project it can be useful to become familiar with the tools and concepts involved in version control systems. Write some code that is unfinished with some ...
• 806
### Do Python and Java lead students to construct different mental models of memory?
I think that this is an interesting question, and one the textbooks often ignore or treat as unimportant. I've noticed it because I end up teaching Python, Java and C++ and have seen how my student's ...
### Do Python and Java lead students to construct different mental models of memory?
I am not aware that cognitive modeling of memory structures in early programming education has been directly studied, so anything that I say here is entirely speculative. However, I suspect that the ...
• 31.5k
### IDE vs Editor and terminal for CS1
I like to ease people in. You have mentioned that this is for first year. So, yeah, easing in would really have a positive impact on the overall learning experience. I would like to draw from my own ...
• 1,874
### Will I Regret Using Python As A Teaching Language When I Later Need to Teach Static Typing?
Maybe it is a bit overkill, but I'm teaching python and C in parallel to CS beginners. The languages are syntactically similar enough to lower the burden of learning two languages, but the students ...
• 594
### Explaining why arrays are important for statisticians
I had a student some time back who also really struggled with concepts like this, but who was interested in research in psychology which for her was largely about statistics. Statistical data is ...
• 101
### What is the best way to learn an object oriented programming language with framework, data structure and alogrithms?
Your list of items is very complete and would fill a good part of a Bachelor's degree curriculum. One thing to not overlook is that some of the aspects you highlight make little sense in the context ...
• 727 | 2022-07-02 02:09:26 | {"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": 0.28545546531677246, "perplexity": 1632.3046744154294}, "config": {"markdown_headings": true, "markdown_code": false, "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/1656103983398.56/warc/CC-MAIN-20220702010252-20220702040252-00545.warc.gz"} |
https://meta.discourse.org/t/code-blocks-with-incorrect-language-string-cause-problems/54345 | # Code blocks with incorrect language string cause problems
(Sebastian Brandt) #1
It seems that there is a problem with too many code blocks. Sorry for the long post but it seems the post have to be this long.
The Example:
Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.
Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text:
``````Some random sql
``````
Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text. Some text.
``````some random sql
``````
## A heading which is not working anymore
``````some random sql
``````
(Gerhard Schlager) #2
It’s the `SQL` in ````SQL` that’s causing this. Replace it with ````sql` (lowercase) and everything works fine.
(Sebastian Brandt) #3
So that is not a bug?
(Mittineague) #4
No, it’s a “I can break Markdown if I do this”
(Darix) #5
This can also be broken without specifying any languages in the code blocks. not just “wrong” language specifiers.
any suggestions how to fix it in that case?
(Darix) #6
also forcing all code blocks into e.q language “sql” as shown in the first comment does not fix it. | 2018-06-22 07:32:54 | {"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": 0.9406681656837463, "perplexity": 12633.62817657891}, "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-2018-26/segments/1529267864364.38/warc/CC-MAIN-20180622065204-20180622085204-00521.warc.gz"} |
http://artent.net/category/multi-armed-bandit-problem/page/2/ | Multi-Armed Bandit Problem
You are currently browsing the archive for the Multi-Armed Bandit Problem category.
“Analysis of Thompson Sampling for the multi-armed bandit problem”
In “Analysis of Thompson Sampling for the multi-armed bandit problem“, Agrawal and Goyal show that Thompson Sampling (“The basic idea is to choose an arm to play according to its probability of being the best arm.”) has logarithmic regret. More specifically, if there are $n$ bandits and the regret $R$ at time $T$ is defined by
$$R(T) := \sum_{t=1}^T (\mu^* – \mu_{i(t)})$$
where $\mu_i$ is the expected return of the $i$th arm and $\mu^* = \max_{i = 1, \ldots, n} \mu_i$, then
$$E[R(T)] \in O\left(\left(\sum_{i\neq i^*} 1/\Delta_i^2\right)^2 \log T \right)$$
where $\mu(i^*) = \mu^*$ and $\Delta_i = \mu^* – \mu_i$.
“BOA: The Bayesian Optimization Algorithm”
In “BOA: The Bayesian Optimization Algorithm“, Pelikan, Goldberg, and Cant´u-Paz introduce an adaptive improvement over genetic optimization algorithms (See also [1]). They write, “In this paper, an algorithm based on the concepts of genetic algorithms that uses an estimation of a probability distribution of promising solutions in order to generate new candidate solutions is proposed. To estimate the distribution, techniques for modeling multivariate data by Bayesian networks are used.” and
“The algorithm proposed in this paper is also capable of covering higher order interactions. It uses techniques from the field of modeling data by Bayesian networks in order to estimate the joint distribution of promising solutions. The class of distributions that are considered is identical to the class of conditional distributions used in the FDA. Therefore, the theory of the FDA can be used in order to demonstrate the power of the proposed algorithm to solve decomposable problems. However, unlike the FDA, our algorithm does not require any prior information about the problem. It discovers the structure of a problem on the fly.”
where FDA refers to the Factorized Distribution Algorithm (Mühlenbein et al., 1998). The algorithm consists of the following steps
The Bayesian Optimization Algorithm (BOA)
(1) set t ← 0 randomly generate initial population P(0)
(2) select a set of promising strings S(t) from P(t)
(3) construct the network B using a chosen metric and constraints
(4) generate a set of new strings O(t) according to the joint distribution encoded by B
(5) create a new population P(t+1) by replacing some strings from P(t) with O(t) set t ← t + 1
(6) if the termination criteria are not met, go to (2)
where B is a Bayesian network.
Check out the NIPS 2011 workshop.
The 20 most striking papers, workshops, and presentations from NIPS 2012
NIPS was pretty fantastic this year. There were a number of breakthroughs in the areas that interest me most: Markov Decision Processes, Game Theory, Multi-Armed Bandits, and Deep Belief Networks. Here is the list of papers, workshops, and presentations I found the most interesting or potentially useful:
Unfortunately, when you have 30 full day workshops in a two day period, you miss most of them. I could only attend the three listed above. There were many other great ones.
Product of Experts
“A Product of Experts model (PoE) (Hinton 2002) combines a number of individual component models (the experts) by taking their product and normalizing the result.” Here’s the Scholarpedia page.
“Algorithms for Infinitely Many-Armed Bandits”
In “Algorithms for Infinitely Many-Armed Bandits”, Wang, Audibert, and Munos (2008) describe some algorithms for the multi-armed bandit problem when a large number or infinitely many arms are available. Their algorithms are designed for the situation where all rewards are contained in $[0,1]$ and “the probability that a new arm is $\epsilon$-optimal is of order $\epsilon^\beta$”. More precisely, there exist real numbers $c, \mu^*,$ and $\beta$ such that the expected value of an unexplored arm $\mu$ obeys
$$P(\mu^* – \mu < \epsilon) < c \epsilon^\beta.$$
They prove that the total regret is at most of order $n^{\beta/(\beta+1)}\log^2(n)$ if $\beta > 1$ and $\log^2(n)\sqrt{n}$ otherwise. Additionally, they prove a lower bound of order $n^{\beta / (\beta + 1)}$ for any algorithm. Their algorithm applies UCB to the first $n^{\beta/(\beta+1)}$ arms. (The case where $\beta = 1$ was explored in “Bandit problems with infinitely many arms” by Berry, Chen, Zame, Heath, and Shepp (1997).)
“Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness”
In “Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness“, Remi Munos (2011) develops an optimization algorithm similar to hierarchical optimistic optimization, Lipschitz optimization, the Zooming algorithm (Klienberg, Slivkins, & Upfal 2008), branch and bound, and the upper confidence bound algorithm. The algorithm does not minimize regret, rather it attempts to maximize $\max_n f(x_n)$ over all the samples $x_n$. Munos’s first algorithm, Deterministic Optimistic Optimization, requires that the function be smooth with respect to a known semi-metric. His second algorithm, Simultaneous Optimistic Optimization, does not require knowledge of the smoothness semi-metric. He proves performance bounds, gives examples for both algorithms, and finishes the paper by comparing the algorithm to the well known DIRECT (DIviding RECTangles) algorithm (Jones, Perttunen, Stuckman 1993).
“From Bandits to Experts: On the Value of Side-Observations”
I was reading the Machine Learning‘s article “Coactive Learning” and they referred to that paper “From Bandits to Experts: On the Value of Side-Observations” by Mannor and Shamir (2011). This paper develops algorithms for the situation where the learner gets information about neighboring bandits after it chooses which bandit arm to pull. Recall that in the mixture of experts situation, the leaner gets to see the results of all the experts (bandits) after choosing which arm to pull.
ICML Tutorial on bandits
This 2011 tutorial (Parts 1 and 2) is the best introduction to stochastic and adversarial bandits and details recent advances in research that I have seen. Many of the bandit papers that I have linked to in previous posts are quoted. Some experimental results and suggested parameter values are given. Part one mentions that regret may be much larger than the expected regret as per the paper “Exploration-exploitation tradeoff using variance estimates in multi-armed bandits” by Audibert, Munos, Szepesvári (2009). It describes the UCB-Horizon algorithm and Hoeffding-based GCL∗ policy which alleviate this problem. Part two is titled “Bandits with large sets of actions” and mostly deals with the case of infinitely many bandits. Many algorithms are described including Hierarchical Optimistic Optimization and a huge bibliography is provided at the end of the tutorial.
“Finite-time Analysis of the Multiarmed Bandit Problem”
In the paper “Finite-time Analysis of the Multiarmed Bandit Problem“, Auer, Cesa-Bianchi, and Fisher (2002) prove that for the fixed reward distributions, “the optimal logarithmic regret is also achievable uniformly over time, with simple and efficient policies, and for all reward distributions with bounded support.”
The paper discusses the following three ideas:
1) The expected regret for the upper confidence bound algorithm UCB1 is bounded above by
$$\left[ 8 \sum_{i:\mu_i < \mu^*} \left({\log n \over {\mu^* - \mu_i}}\right)\right] + \left( 1 + {\pi^2\over 3}\right)\left(\sum_{j=1}^K (\mu^*-\mu_j) \right)$$
where $\mu_i$ is the mean of the $i$th distribution, $n$ is the number of pulls, and $K$ is the number of arms if the rewards are between 0 and 1.
2) They improve on the coefficient of $\log n$ with a slightly more complex algorithm UCB2.
3) The $\epsilon_n = 1/n$ greedy algorithm has $O(\log n)$ regret.
They conclude the paper with numerical results.
Sleeping Experts
In “Regret Bounds for Sleeping Experts and Bandits” Kleinberg, Niculescu-Mizil, and Sharma (2008) propose an algorithm for selecting among a list of experts some of whom may be sleeping. At each time step $t$, the algorithm ranks the experts in order of preference with a permutation vector $\sigma(t)$. The first analyzed algorithm is Follow The Awake Leader (FTAL) for the case where all of the expert predictions and rewards are observed. Next they explore a natural extension of upper confidence bound algorithm for the multi-armed bandit setting. In the adversarial setting, they prove that the regret is at least $O(\sqrt{t\, K \log K})$ for $K$ bandits at time $t$. | 2017-10-24 02:18:00 | {"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": 0.7793279886245728, "perplexity": 1345.005009223449}, "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-2017-43/segments/1508187827853.86/warc/CC-MAIN-20171024014937-20171024034937-00684.warc.gz"} |
https://www.flyingcoloursmaths.co.uk/ask-uncle-colin-a-parametric-integration/ | Dear Uncle Colin,
I have the parametric equations $x = (t+1)^2$ and $y = \frac{1}{2}t^3 + 3$ and the lines $y = 16 - x$ and $x=1$. I need to find the area enclosed by the curve, these two lines and the $x$-axis, but my answer doesn’t agree with the book’s! Can you help?
- Frankly I’m Surprised How Errors Result
Hi, FISHER, and thanks for your message!
Before we begin, let’s sketch the diagram:
There are clearly two bits to integrate: a wiggly bit on the left and a triangle on the right. The triangle is probably easiest to begin with.
The triangle
We need to find the intersection point of the curve and the line, so that $(t+1)^2 = 13 - \frac{1}{2}t^3$.
Doubling and expanding: $2t^2 + 4t + 2 = 26 - t^3$, or $t^3 + 2t^2 + 4t - 24 = 0$.
This factorises as $(t-2)(t^2 + 4t + 12)$, and the only real solution to that is at $t=2$.
That corresponds to the point $(9,7)$; we can immediately see that the triangle has height of 7 and base of 7, and therefore an area of $\frac{49}{2}$.
The wiggle
I can see two reasonable ways to solve this, so let’s try both. The first, and I think easiest, is parametric integration; the second is to make it cartesian and integrate directly.
Parametric
When $x=1$, $t=0$, so we would need to calculate $\int_0^2 y \diff{x}{t} \dt$.
This works out to be $\int_0^2 (t^3 + 6)(t+1) \dt$, or $\int_0^2 t^4 + t^3 + 6t + 6 \dt$.
That’s $\left[ \frac{1}{5}t^5 + \frac{1}{4}t^4 + 3t^2 + 6t\right]_0^2$, which evaluates to $\frac{32}{5} + 4 + 12 + 12$; I get $\frac{172}{5}$.
Cartesian
Alternatively, we can rearrange the first equation to find that $t = x^{\frac{1}{2}}-1$.
In this case, $y = \frac{1}{2} \left( x^{\frac{1}{2}}-1\right)+3$, which we can expand: $y = \frac{1}{2} \left( x^{\frac{3}{2}} - 3x + 3x^{\frac{1}{2}} + 5\right)$, bringing the 3 inside the bracket.
We need $\int_1^9 y \dx$, which is $\frac{1}{2}\left[ \frac{2}{5}x^{\frac{5}{2} } - \frac{3}{2}x^2 + 2x^\frac{3}{2} + 5x\right]_1^9$.
Evaluating this, I get $\frac{1}{2}\left[ \frac{2}{5}\times 242 - \frac{3}{2}\times 80 + 2 \times 26 + 5 \times 8 \right]$.
This gives $\frac{242}{5} - 60 + 26 + 20$, or $\frac{172}{5}$ again. Phew.
Altogether
Finally, we need to add $\frac{172}{5}$ to $\frac{49}{2}$, which gives $\frac{344 + 245}{10} = 58.9$ square units.
Hope that helps!
- Uncle Colin | 2021-11-30 06:58:45 | {"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": 0.9850675463676453, "perplexity": 341.8245919494745}, "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-2021-49/segments/1637964358953.29/warc/CC-MAIN-20211130050047-20211130080047-00251.warc.gz"} |
https://docs.itascacg.com/itasca900/3dec/block/doc/manual/block_manual/block_commands/block/cmd_block.densify.html | block densify command
Syntax
block densify keyword ... <range>
Primary keywords:
Increase block density. Within the given range, subdivide blocks to increase density of blocking. Blocks are subdivided by cutting joints parallel to the global $$x$$, $$y$$, $$z$$ axes unless the tetrahedra keyword is given.
Blocks chosen for “densification” will be checked to attempt to ensure that adjacent blocks differ by, at most, one level of densification. This may mean more blocks will be densified than actually fall into the range.
hexahedra
Use this keyword if blocks in the range are hexahedra (8-sided). Certain efficiencies are achieved when densifying hexahedral blocks and the command will be executed significantly faster. If blocks in the range are not hexahedra, the keyword is ignored and the default densification scheme is used.
joint-set i
Specify joint ID i for faces and joints created with this command.
join
Join blocks across the joints created with this command.
maximum-length f1 <f2 <f3 >>
This specifies the maximum edge lengths to be densified, in order. If f2 and f3 are not specified, they will be set to the same number as f1.
repeat <i >
Repeat the densification i times, causing a recursive application of the range and densification settings. This can, for example, be used to create an octree grid. Note that if no number is given, then 3DEC will repeat until the maximum-length length is reached, applying the division specified in segments each time.
segments i1 <i2 <i3 >>
This specifies the number of subdivisions in the $$x$$-, $$y$$- and $$z$$-directions. If i2 and i3 are not specified, they will be set to the same number as i1.
tetrahedra
Tetrahedral blocks will be split by first making cuts through the midpoint of each edge. This results in four new tetrahedra and an octahedron in the center. The octahedron is then split into four tetrahedra with two further cuts along the diagonals (see the figure below). Note that if the tet keyword is given, the $$segments$$ value is ignored for tetrahedral blocks. Blocks that are not tetrahedral are split normally along the global x,y and z axes. | 2023-03-27 20:16:52 | {"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": 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": 0.42534399032592773, "perplexity": 1691.4235183344847}, "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-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00644.warc.gz"} |
https://www.physicsforums.com/threads/speed-of-time.903245/ | # I Speed of time
Tags:
1. Feb 9, 2017
### Jens K Munk
Dear Forum,
The title "Speed of time" is the best I can come up with. We know that time isn't constant, rather it depends on the speed at which we move as well as the gravity we are subjected to (see https://en.wikipedia.org/wiki/Gravitational_time_dilation for a reference to the latter). As a consequence of the latter, and as stated on the Wikipiedia page, Earth is stated as being 2.5 years younger at its core (due to higher gravity in there).
Thoughts on this:
1. Isn't gravity zero at Earth's core (disregarding the gravity of the Sun)? I would argue that at the core, you have an equal amount of Earth at each side of you, so they cancel out.
2. Instead of "gravitational time dilation", do we experience "density-dependent time dilation"? Higher density (such as that in Earth's core) slows down time. This would explain why clocks in space record time going faster than at Earth's surface. It may also explain why extreme density such as that in black holes causes light to reach zero speed (relative to observer's perspective).
Let's stop here. Please comment. Thanks.
2. Feb 9, 2017
### Orodruin
Staff Emeritus
Gravitational time dilation depends on the gravitational potential, not the gravitational field.
No. Also, there is no such thing as light reaching zero speed and the Schwarzschild black hole is a vacuum solution to the Einstein field equations, i.e., the density is zero at the event horizon.
Also note that you should not be using the A-level tag unless you have knowledge in the subject equivalent to that of a graduate student or higher. It is clear from your post that you do not. The level system is in place for you to tell us your level of expertise so that the answers can be aimed at that level of understanding. <Moderator's note: level changed>
Last edited by a moderator: Feb 9, 2017
3. Feb 23, 2017
### Albrecht
I do not understand the comment to the time dilation and the speed of light in a gravitational field.
The Schwarzschild metrics say about the proper time in a gravitational field
d(tau) = (1 - r0/r)1/2dt where r0is the Schwarzschild radius.
This means that the flow of time stops completely at r = r0 where is the event horizon. Correspondingly the speed of light reaches zero at that range.
4. Feb 23, 2017
### Orodruin
Staff Emeritus
No it doesn't. It is just a matter of the coordinate t not being a good coordinate at the Schwarzschild radius. The "speed of light" that you are referring to is not the invariant speed of light - it is a coordinate velocity.
5. Feb 24, 2017
### Albrecht
So, what means: d(tau) = 0 ?
Doesn't it mean that the local, proper time stops at that point? Whatever the general meaning of t might be? - I have listened to a talk of nobel price winner Gerard t'Hoft about cosmology. And he just stated this. Was he wrong?
With respect to the speed of light I of course meant the coordiante velocity. The one which was for instance measured in the Shapiro experiment. That should go to zero. The invariant speed of light is the one measured by a local observer. It is easy to see that this observer will always measure the nominal speed of light as his tools (clocks and rulers) change accordingly in the gravitational field. - This is at least my understanding.
6. Feb 24, 2017
### Orodruin
Staff Emeritus
You do not generally get $d\tau = 0$ at $r = r_S$ - you only get this for null world lines. It is not any stranger than having a general null world line. The only locally peculiar thing is that you have chosen to use coordinates such that the coordinate lines for the $t$ coordinate are null world lines.
7. Feb 25, 2017
### Albrecht
So I understand that all points on the event horizon are world null lines. Which consequences?
An observer outside the black hole and outside the event horizon will see any motion stopping at the event horizon. For him (and so for us) there is c = 0 and as well any other motion and also - as a general conclusion - dτ = 0. From which the question follows how a black hole can collect material.
To say it again: We are the observers looking from outside and we have the expectation that material moves into the black hole. But how?
8. Feb 25, 2017
### Orodruin
Staff Emeritus
There is no local Lorentz frame where the event horizon is stationary. No observer can be stationary at the event horizon.
This is an empty statement. The observer is not located at the event horizon and does not measure the coordinate speed of light there. The symbol $c$ is the invariant speed of light in vacuum, not the coordinate speed of light.
9. Feb 27, 2017
### Albrecht
I think that it is just the other way around. If an observer would be located at the event horizon then for him anything would look normal. But seen from our view, so from our Lorentz frame, any speed at the event horizon goes to zero. That is what also Gerard t'Hoft has said in his talk which I have mentioned earlier.
If we transport a clock into a gravitational field and take it back later then this clock will have a delay compared to our clocks. That is according to General Relativity and also the result of measurements. If we would transport a clock close to the event horizon and would later be able to take it back that this clock would have a considerable delay. This is in my understanding a clear result of the Schwarzschild metric. What else could happen?
10. Feb 27, 2017
### Orodruin
Staff Emeritus
You do realise that this is a GR equivalent of referring to an observer travelling at the speed of light in SR? It simply does not make sense. The most probable is that t'Hooft was making some sort of popularisation or that you misunderstood him (or you took the popularisation too far). Without seeing the talk, there is no way we can tell.
11. Feb 27, 2017
### Albrecht
It is a clear result of the Schwarzschild metric that
dτ = dt*sqrt(1-rs/r) and c = c0*sqrt(1-rs/r) (for tangential motion)
It has in fact a singularity for r = rs.
One can of course avoid the singularity by introducing a coordinate system which adapts gradually with the approximation to the horizon so as to avoid a singularity (like Eddington-Finkelstein coordinates). But if we discuss the development of the universe we use our coordinates, for instance to say that its age since the Big Bang is 13 billion years. Such an adapted coordinate system will tell us something different. But what would be the use of that for our understanding?
12. Feb 27, 2017
### Orodruin
Staff Emeritus
No it won't. The physical predictions of GR are coordinate independent. This is the entire point.
13. Feb 27, 2017 | 2018-02-22 03:23:56 | {"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": 0.66301429271698, "perplexity": 538.4341058218156}, "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/1518891813883.34/warc/CC-MAIN-20180222022059-20180222042059-00439.warc.gz"} |
https://nforum.ncatlab.org/discussion/9339/ | # Start a new discussion
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## Site Tag Cloud
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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 3rd 2018
stub entry, for the moment just so as to satisfy links
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 24th 2020
added two sentences to the Idea-section, to provide more of an explanation:
If the gauge group is SU(N) and fermions are in the fundamental representation $\mathbf{N}$ of dimension $N$, so that $\mathbf{N}$ has a linear basis of $N$ elements, one says that these are $N$ colors of the given fermion particle.
Hence, more generally one could reasonably say that the “number of colors” of a given fundamental particle in gauge theory is physics synonym for the choice of linear representation of the gauge group that defines this particle. | 2022-01-17 01:42:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "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": 0.7185676097869873, "perplexity": 1439.878432889862}, "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-2022-05/segments/1642320300253.51/warc/CC-MAIN-20220117000754-20220117030754-00535.warc.gz"} |
https://math.stackexchange.com/questions/3049692/reduced-homology-of-a-point-is-trivial-from-axioms | # Reduced homology of a point is trivial from axioms
In the section Axioms for homology from Hatcher's Algebraic Topology (page 161) he says:
Note that $$\tilde{h}_n(x_0) = 0$$ for all $$n$$, as can be seen by looking at the long exact sequence of reduced homology groups of the pair $$(x_0,x_0)$$.
I guess that he means the exact sequence of the second axiom, which gives the boundary map $$\partial :\tilde{h}_n(X/A)\to\tilde{h}_{n-1}(A)$$.
From this it is clear that $$\tilde{h}_n(x_0)=\tilde{h}_{n-1}(x_0)$$ for all $$n$$. But Hatcher is not assuming the dimension axiom, so I cannot say that $$\tilde{h}_n(x_0)=0$$ for $$n\neq 0$$, so how could I conclude that $$\tilde{h}_n(x_0) = 0$$ for all $$n$$?
I've thought that this could be shown using the first axiom if I could show that constant maps induce trivial maps in homology, but I don't see how to deduce this fact from the axioms because the proof I know for singular, cellular or simplicial homology uses that the homology of a point is trivial.
• I'm not sure how you concluded that $\tilde{h}_n(x_0)=\tilde{h}_{n-1}(x_0)$. There is a boundary map between them, but why would that map be an isomorphism? (In fact it is since both groups are trivial, but I don't know what logic you're using.) Dec 22 '18 at 18:47
• @EricWofsey in this case, $X=A=x_0$, so $X/A=A$, and the boundary map is an isomorphism.
– Javi
Dec 22 '18 at 18:49
• But...why is it an isomorphism? Dec 22 '18 at 18:50
• @EricWofsey I was going to write my argument but I found the mistake, thanks.
– Javi
Dec 22 '18 at 18:54
We have an exact sequence $$\tilde{h}_n(A)\stackrel{i_*}\to\tilde{h}_n(X)\stackrel{q_*}\to\tilde{h}_n(X/A)$$ where the first map is induced by the inclusion $$i:A\to X$$ and the second map is induced by the quotient map $$q:X\to X/A$$. But if $$X$$ and $$A$$ are both just a point, then $$i$$ and $$q$$ are both homeomorphisms, so $$i_*$$ and $$q_*$$ are both isomorphisms. But the image of $$i_*$$ is the kernel of $$q_*$$, so this implies $$q_*=0$$, and so $$\tilde{h}_n(X)=0$$ since $$q_*$$ is an isomorphism.
Hatcher's definition of a reduced homology theory is limited to CW-pairs. A more general definition for general pairs is this: Omit axiom 3 and replace the long exact sequence in axiom 2 by $$\dots \stackrel{\partial}{\rightarrow} \tilde{h}_n(A) \stackrel{i_*}{\rightarrow} \tilde{h}_n(X) \stackrel{j_*}{\rightarrow} \tilde{h}_n(X \cup CA) \stackrel{\partial}{\rightarrow} \dots$$ Here, $$CA$$ is the unreduced cone on $$A$$ and $$X \cup CA$$ is the adjunction space in which the equivalence class $$[a,0] \in CA$$ in the base of $$CA$$ is identified with $$a \in A \subset X$$. For CW-pairs this is equivalent to Hatcher's long exact sequence. In fact, if $$(X,A)$$ is CW-pair, then $$(X \cup CA, CA)$$ is also a CW-pair, in a particular $$CA \hookrightarrow X \cup CA$$ is cofibration. Since $$CA$$ is contractible, the quotient map $$p : X \cup CA \to (X \cup CA)/CA = X/A$$ is a homotopy equivalence.
Now consider $$(X,A) = (x_0,x_0)$$. Then $$X \cup CA$$ is homeomorphic to the unit interval whence $$j_*$$ is an isomorphism. Since $$i_*$$ trivially is an isomorphism, we conclude that $$\tilde{h}_n(x_0) = 0$$ for all $$n$$.
Hatcher's approach to axiomatic homology is not standard. In fact, he gives an axiomatic definition of a reduced single space homology theory for unbased spaces. This means that no relative groups $$\tilde{h}_n(X,A)$$ are used.
Such theories for unbased spaces are considered only rarely in the literature (but is a common to work with reduced single space homology theory for based spaces, although their definition is different from Hatchers's).
If you consider singular homology defined for all pairs of toplogical spaces, then it is well known that reduced singular homology groups can be defined via the augmented chain complex.
A more general approach is this: If we have a generalized homology theory $$(H_n,\partial)$$ satisfying the Eilenberg-Steenrod axioms with the possible exception of the dimension axiom, we can define reduced groups by $$\tilde{H}_n(X) = \ker (p_* : H_n(X) \to H_n(*)) \subset H_n(X)$$ where $$p : X \to *$$ is the unique map to a one-point space $$*$$. It is then an easy exercise to show that we get a long exact sequence $$\dots \stackrel{\partial}{\rightarrow} \tilde{H}_n(A) \stackrel{i_*}{\rightarrow} \tilde{H}_n(X) \stackrel{j_*}{\rightarrow} H_n(X,A) \stackrel{\partial}{\rightarrow} \tilde{H}_{n-1} (A)\stackrel{i_*}{\rightarrow} \dots$$ The excision axiom shows that $$H_n(X,A) \approx H_n(X \cup CA,CA)$$ and the latter is easily seen to be isomorphic to $$\tilde{H}_n(X \cup CA)$$. This yields the exact sequence from above. Note that it does not agree with Hatcher's exact sequence, except if restrict to CW-pairs.
In the case of reduced singular homology the definition via the augmented chain complex and our general definition agree and we have $$\tilde{H}_n(X) = H_n(X)$$ for $$n > 0$$. But reduced singular homology does not satisfy Hatchers's axiom 2 (long exact sequence with quotients $$X/A$$). Consider for example the pair $$(T,J)$$ where $$T = \{ x,\sin(1/x)) \mid x \in (0,1] \} \cup \{ 0 \} \times [-1,1]$$ is the closed topologist's sine curve and $$J = \{ 0 \} \times [-1,1]$$. The space $$T$$ has two path components which are contractible and $$T/J$$ is homeomorphic to the interval $$[0,1]$$, i.e. contractible. Hence $$\tilde{H}_0(J) = 0, \tilde{H}_0(T) = \mathbb {Z}, \tilde{H}_0(T/J) = 0$$ which shows that $$\tilde{H}_0(J) \stackrel{i_*}{\rightarrow} \tilde{H}_0(T) \stackrel{p_*}{\rightarrow} H_0(T/J)$$ cannot be exact. | 2022-01-28 09:56:51 | {"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": 61, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9641242027282715, "perplexity": 80.05570322730787}, "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-2022-05/segments/1642320305423.58/warc/CC-MAIN-20220128074016-20220128104016-00505.warc.gz"} |
http://archived.moe/a/thread/13391651 | 235KiB, 765x1024, gantz2771.jpg
## GANTZ 277 scans
!!gh3hyLhv/6n No.13391651
well....fuck
I'm posting 2 of these, the 3rd one is the last page in the chapter, so I will spoiler tag it | 2016-10-28 03:05:49 | {"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": 0.8473291993141174, "perplexity": 7110.306195171963}, "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/1476988721555.36/warc/CC-MAIN-20161020183841-00167-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://email.esm.psu.edu/pipermail/macosx-emacs/2009-January/001225.html | [OS X Emacs] latex on aquamacs
Peter Dyballa Peter_Dyballa at Web.DE
Wed Jan 21 18:57:59 EST 2009
Am 21.01.2009 um 16:24 schrieb Jan Rosinski:
> 1. I am rather an advanced user of latex but a novice on aquamacs.
> Is there a good source to learn aquamacs specifics for latex?
> (Sample macros, key-bindings for latex? A screencast?)
I don't think Aquamacs Emacs has "its own" TeX mode, it's either
that, which comes with regular Emacsen or it's AUCTeX – this has a
>
>
> 2. I have a problem using preview in line in book style from within
> included chapters.
> I opened my book's master file, typesetted it and and from sync in
> Skim I opened
> one of the included chapters. I changed one formula in the chapter
> and wanted to create
> at point preview.
> I got error messages (no \begin{document}, etc.) suggesting that I
> file to run preview for whole book. This is very inconvenient.
> So, the question is how to run preview in at point, section, etc.
> from within included chapters?
I think Preview tries to motivate you to think before doing anything.
By removing the previews from the whole document you will be able to
create previews for particular regions individually.
--
Greetings
Pete
Experience is what you get when you don't get what you want. | 2020-10-31 11:20:33 | {"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": 0.8649680018424988, "perplexity": 8449.193521235105}, "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-2020-45/segments/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00542.warc.gz"} |
https://leanprover-community.github.io/archive/stream/217875-Is-there-code-for-X%3F/topic/(commutative)-semirng.html | ## Stream: Is there code for X?
### Topic: (commutative)-semirng
#### Damiano Testa (Dec 13 2020 at 16:32):
Is there already a definition of a type with two operations, addition and multiplication, that is just like a semiring, except that it does not need to have a unit for multiplication. I do not mind if multiplication is commutative, although it is in my intended application. I looked for semirng (as well as semring and semrng), but did not find anything.
#### Damiano Testa (Dec 13 2020 at 16:41):
Nevermind, I think that add_comm_monoid is what I want.
#### Adam Topaz (Dec 13 2020 at 16:41):
This doesn't have a multipication
#### Damiano Testa (Dec 13 2020 at 16:42):
Ah, so I am misunderstanding this definition:
/-- An additive commutative monoid is an additive monoid with commutative (+). -/
I thought that the add_monoid and add_comm_semigroup implied different operations. Is this not the case?
#### Damiano Testa (Dec 13 2020 at 16:43):
(The doc_string seems contradicting what I think)
#### Damiano Testa (Dec 13 2020 at 16:45):
So, I guess that I would like two add_comm structures, one of which is a monoid, and a distributive law.
#### Adam Topaz (Dec 13 2020 at 16:45):
the to_additive thing is a metaprogramming trick that takes something like the class comm_monoid and automatically makes add_comm_monoid.
#### Adam Topaz (Dec 13 2020 at 16:45):
But a priori there is no relationship between comm_monoid and add_comm_monoid.
#### Damiano Testa (Dec 13 2020 at 16:46):
ah, so I could build one like what i want by piling up two add structures and if I do not put the to_additive they would be a + and a *?
#### Adam Topaz (Dec 13 2020 at 16:46):
E.g. look at the source for src#comm_monoid
#### Adam Topaz (Dec 13 2020 at 16:47):
I think what you want is a class that extends add_comm_monoid and the (nonexistent, as far as I know) semigroup_with_zero and the class that ensures distributivity of + and * that I can't remember the name of right now...
#### Damiano Testa (Dec 13 2020 at 16:48):
distrib should be the distributive property:
/-- A typeclass stating that multiplication is left and right distributive
class distrib (R : Type*) extends has_mul R, has_add R :=
(left_distrib : ∀ a b c : R, a * (b + c) = (a * b) + (a * c))
(right_distrib : ∀ a b c : R, (a + b) * c = (a * c) + (b * c))
#### Damiano Testa (Dec 13 2020 at 16:50):
I am happy to assume comm_semigroup_with_zero, but I really do not have a unit for multiplication: I want to put this structure on finsets.
#### Adam Topaz (Dec 13 2020 at 16:50):
Oh, BTW my explanation of to_additive is probably not 100% correct. It looks like add_comm_monoid is actually declared explicitly with a to_additive declaration right after to connect it to comm_monoid. I guess to_additive makes some of the API for you.
#### Adam Topaz (Dec 13 2020 at 16:50):
semigroups don't have units IIRC
#### Damiano Testa (Dec 13 2020 at 16:51):
Ok, thanks for the correction: I know so little of this, at the moment, that I had filed it in my head as "there is a system to keep track of whether you call it + or *"!
#### Reid Barton (Dec 13 2020 at 16:51):
That's not to_additive, that's this add_ stuff
#### Reid Barton (Dec 13 2020 at 16:52):
The classes with add_ in their name are built on top of +, the ones without are built on *
#### Damiano Testa (Dec 13 2020 at 16:52):
ok, so by default, an operation is *, unless is it declared with a add_ and then it is a +, right?
#### Adam Topaz (Dec 13 2020 at 16:53):
I think to_additive takes lemmas of the form mul_foo and automatically makes lemmas of the form add_foo, if I understand what Reid is saying here...
#### Damiano Testa (Dec 13 2020 at 16:53):
I was typing my comment, while you were explaining yours, thanks!)
#### Reid Barton (Dec 13 2020 at 16:53):
right, to_additive is some automation which means we don't have to duplicate everything between the additive and multiplicative versions by hand
#### Adam Topaz (Dec 13 2020 at 16:54):
so you have to tell lean that it should generate the add_foo lemmas for add_comm_monoid using the mul_foo lemmas for comm_monoid, and that's the to_additive declaration that we noticed above.
#### Reid Barton (Dec 13 2020 at 16:54):
So, I think you want to extend add_comm_group, semigroup and distrib... that will imply 0 * x = x * 0 = 0 already, right?
#### Reid Barton (Dec 13 2020 at 16:55):
I guess you can also add monoid_with_zero just in case
#### Adam Topaz (Dec 13 2020 at 16:55):
Yes. but he doesn't have negation and that's not true for additive monoids
#### Reid Barton (Dec 13 2020 at 16:55):
oh I missed "semi"
#### Damiano Testa (Dec 13 2020 at 16:55):
Yes, negation I also want to avoid.
#### Reid Barton (Dec 13 2020 at 16:55):
oh then there might not be any combination which does exactly what you want
#### Reid Barton (Dec 13 2020 at 16:56):
You can't actually use monoid_with_zero because it implies monoid
#### Adam Topaz (Dec 13 2020 at 16:56):
Yeah, this is why I was suggesting the semigroup_with_zero class :rofl:
#### Reid Barton (Dec 13 2020 at 16:56):
semigroup is such a hard name for category theorists :head_bandage:
#### Damiano Testa (Dec 13 2020 at 16:57):
well, if it is only the with_zero part that is missing, I can add the two axioms zero_mul and mul_zero, right?
#### Reid Barton (Dec 13 2020 at 16:58):
What goofy names... mul_zero_class and distrib
#### Damiano Testa (Dec 13 2020 at 16:58):
so, extends add_comm_monoid, semigroup, distrib, mul_zero_class?
#### Damiano Testa (Dec 13 2020 at 16:58):
I will try it and see if I can pull it off!
#### Damiano Testa (Dec 13 2020 at 17:03):
Just to avoid being stuck for silly reasons: proving the instance should simply be a matter of following my nose, right?
import algebra.algebra.basic
set_option old_structure_cmd true
variables {α : Type*}
class semirng (α : Type*) extends add_comm_monoid α, semigroup α, distrib α, mul_zero_class α
instance : semirng (finset α) :=
#### Kevin Buzzard (Dec 13 2020 at 17:05):
If all the other structures are on finset then it might be a matter of by apply_instance. If it isn't then you can make the other structures first. But this might be one of those places where some people don't want those instances, so maybe it should be a def instead.
#### Adam Topaz (Dec 13 2020 at 17:05):
I assume this is the semiring structure given by the lattice structure?
#### Damiano Testa (Dec 13 2020 at 17:05):
Yes, union=+ and intersection=*.
#### Adam Topaz (Dec 13 2020 at 17:06):
Umm... addition is symmetric difference no?
#### Damiano Testa (Dec 13 2020 at 17:06):
Ah, I had union in mind...
#### Damiano Testa (Dec 13 2020 at 17:06):
It is not the boolean-ish algebra that I want
#### Kevin Buzzard (Dec 13 2020 at 17:07):
If there are two natural choices for addition then probably you don't want anything to be an instance, because then everyone who wants the other one is stuck with it
#### Adam Topaz (Dec 13 2020 at 17:07):
This is essentially a direct sum of $\mathbf{Z}/2$ indexed by the type
#### Adam Topaz (Dec 13 2020 at 17:07):
where you identify a finite set with its indicator function.
#### Damiano Testa (Dec 13 2020 at 17:08):
In what I want, once an element is in, it will never go out
#### Damiano Testa (Dec 13 2020 at 17:08):
(with addition, you can remove it with intersections, of course)
#### Adam Topaz (Dec 13 2020 at 17:08):
Is that distributive?
#### Damiano Testa (Dec 13 2020 at 17:09):
I thought that it was distributive "both ways"
#### Adam Topaz (Dec 13 2020 at 17:09):
Yeah you're probably right
#### Reid Barton (Dec 13 2020 at 17:09):
This already basically exists as distrib_lattice
#### Damiano Testa (Dec 13 2020 at 17:09):
it is according to wikipedia:
https://en.wikipedia.org/wiki/Algebra_of_sets
#### Damiano Testa (Dec 13 2020 at 17:10):
Ah, I will look at distributive lattice then!
#### Damiano Testa (Dec 13 2020 at 17:10):
(the name sounds good)
#### Adam Topaz (Dec 13 2020 at 17:11):
Yeah it's clearly distribute. But I agree with what Kevin said above, because as far as I know the "standard" way to define a (semi)ring structure from a boolean algebra is by letting addition be the symmetric difference.
#### Damiano Testa (Dec 13 2020 at 17:13):
Ok, it is simply that for doing "induction", unions is what I need, rather than symmetric differences
#### Damiano Testa (Dec 13 2020 at 17:13):
I will not make an instance, and I will check out distrib_lattice
#### Kevin Buzzard (Dec 13 2020 at 17:18):
@Damiano Testa $\cup$ and $\cap$ are "builtin" notation for mathematicians when using sets, but lattice theory is a generalisation of this stuff and the notation used is these square $\sqcup$ and $\sqcap$ stuff. It is actually a pain that set theory in Lean sticks to this set union and intersection notation. When ideals were defined and given a lattice structure, the lattice notation was used, and initially I was shocked that I was supposed to write $I\leq J$ instead of $I\subseteq J$, but now actually I see the benefits of this: we are using lots of different notations for lattice notation in various situations when we're dealing with lattices which we don't perceive as lattices, e.g. the subgroup generated by $H$ and $K$ is just $H\sqcup K$ in Lean. Sticking to lattice notation everywhere is a good convention, I think.
#### Damiano Testa (Dec 13 2020 at 17:19):
Good, I will follow the convention! How do I type the square union and intersection?
#### Damiano Testa (Dec 13 2020 at 17:19):
(hovering in VSCode answers my question \lub)
#### Damiano Testa (Dec 13 2020 at 17:21):
So, I should define inclusion of sets, rather than union. Have I understood correctly? I am slightly confused.
#### Adam Topaz (Dec 13 2020 at 17:24):
I agree with what Kevin said about the union intersection notation, but just one small comment. I would argue that addition being defined as symmetric difference is the only reasonable choice. For example if you work in a finite type, then the structure you get is a ring if you use symmetric difference, but not if you use union.
#### Damiano Testa (Dec 13 2020 at 17:27):
I thought that this would have worked, but Lean does not want me to use neither ⊔ nor ∪. I suspect that you already told me what I should do, but I did not understand it...
import algebra.algebra.basic
def finset_lattice (X : Type*) : distrib_lattice (finset X) :=
begin
refine ⟨_, _, _, _, _, _, _, _, _, _, _, _, _, _, _⟩,
use (λ a b, a ⊔ b), -- failed to instantiate goal with fun (a : 4._.40) (b : 4._.41), ((frozen_name has_sup.sup) a b)
end
#### Mario Carneiro (Dec 13 2020 at 17:27):
ooh, what a fun error message
#### Damiano Testa (Dec 13 2020 at 17:28):
I agree with what Kevin said about the union intersection notation, but just one small comment. I would argue that addition being defined as symmetric difference is the only reasonable choice. For example if you work in a finite type, then the structure you get is a ring if you use symmetric difference, but not if you use union.
Ok, I will use this "simple-minded union" structure sparingly!
#### Adam Topaz (Dec 13 2020 at 17:31):
Don't we have this? docs#finset.distrib_lattice
#### Mario Carneiro (Dec 13 2020 at 17:31):
note the decidable_eq requirement
#### Damiano Testa (Dec 13 2020 at 17:32):
Thanks! I opened the classical locale and now Lean is having more fun: thanks!
#### Damiano Testa (Dec 13 2020 at 17:33):
Don't we have this? docs#finset.distrib_lattice
I will try to do it by hand first, and then compare with what is already in mathlib, just to get some practice. Thanks for the pointer, though!
#### Adam Topaz (Dec 13 2020 at 17:47):
My only issue with lattice notation is that $\sqcup$ is disjoint union in my mind. But I got used to $\oplus$
#### Damiano Testa (Dec 13 2020 at 18:47):
Is it possible to see the name of each field that I am trying to prove? For instance, the first one asks to produce something with type finset X \to finset X \to finset X. I "guessed" that it was the union. Others later on are similarly ambiguous. Is there a way to know what name each property has?
#### Damiano Testa (Dec 13 2020 at 18:47):
(instead of type-checking myself and second guessing what I see that I will need to prove later on!)
#### Sebastien Gouezel (Dec 13 2020 at 18:49):
If you type {! } where you are supposed to construct a structure, a little light bulb will show on. If you click on the light bulb and select generate a skeleton for the structure under construction, it will give you the name of the different fields.
#### Damiano Testa (Dec 13 2020 at 18:52):
Wow: this is great!!! Thank you @Sebastien Gouezel !
#### Mario Carneiro (Dec 13 2020 at 18:58):
Actually you can just use _
#### Damiano Testa (Dec 13 2020 at 19:04):
I had never thought of clickling the yellow light-bulbs
:light_bulb:
#### Damiano Testa (Dec 14 2020 at 08:13):
To get some practice, I thought that I would define a new structure sus on subsets of a set, just like topological_space, except that I drop the requirement of arbitrary unions and only allow finite unions. (The definition only involves pairwise unions, although, as a mathematician, I still feel a bit funny having to specify this! As a consequence, for those who, like me, think often about the empty set, this implies that I need to add the hypothesis that ∅ is in my collection of sets, since I can no longer take the empty union.)
Thus, sus with + = ∪ and * = ∩ is a comm_semiring, which is where I am headed. This comm_semiring structure is also the structure that I want to put on "finsets with univ".
I copied the beginning of the topology/basic file and started changing as needed. When I got to
@[ext]
lemma topological_space_eq : ∀ {f g : topological_space α}, f.is_open = g.is_open → f = g
| ⟨a, _, _, _⟩ ⟨b, _, _, _⟩ rfl := rfl
I got completely stuck. I can understand the statement: if two instances of topological space on the same type have the same open sets, then they agree. However, the proof in indecipherable to me. Can anyone give me some guidance or even simply a proof? Below is the code that I have so far. Feel free to give more advice!
import order.filter.ultrafilter
import order.filter.partial
open set filter classical
open_locale classical filter
/-- A sus on α. -/
@[protect_proj] structure sus (α : Type*) :=
(is_su : set α → Prop)
(is_su_univ : is_su univ)
(is_su_empty : is_su ∅)
(is_su_inter : ∀s t, is_su s → is_su t → is_su (s ∩ t))
(is_su_union : ∀s t, is_su s → is_su t → is_su (s ∪ t))
attribute [class] sus --I do not know what this does, but it was there
/-- A constructor for sus using complements of the given sus structure. -/
-- this definition is not relevant to what I care about: I simply say that if I have a sus, then the collection of complements also forms a sus
def sus.comp {α : Type*} (f : sus α) : sus α :=
{ is_su := λ X, f.is_su Xᶜ,
is_su_univ := by simp [sus.is_su_empty],
is_su_empty := by simp [sus.is_su_univ],
is_su_inter := λ s t hs ht, by { rw compl_inter, exact sus.is_su_union f sᶜ tᶜ hs ht },
is_su_union := λ s t hs ht, by { rw compl_union, exact sus.is_su_inter f sᶜ tᶜ hs ht },
}
section sus
variables {α : Type*} {β : Type*} {ι : Sort*} {a : α} {s s₁ s₂ : set α} {p p₁ p₂ : α → Prop}
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
-- this was the proof for topological spaces
-- | ⟨a, _, _, _, _⟩ ⟨b, _, _, _, _⟩ rfl := rfl
begin
sorry
end
#### Alex J. Best (Dec 14 2020 at 08:22):
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
-- this was the proof for topological spaces
-- | ⟨a, _, _, _, _⟩ ⟨b, _, _, _, _⟩ rfl := rfl
begin
rintro ⟨f_is_su, f_is_su_univ, f_is_su_empty, f_is_su_inter, f_is_su_union⟩ ⟨g_is_su,
g_is_su_univ,
g_is_su_empty,
g_is_su_inter,
g_is_su_union⟩ ⟨⟩,refl,
end
#### Alex J. Best (Dec 14 2020 at 08:22):
The first part was generated by rintros?
#### Damiano Testa (Dec 14 2020 at 08:23):
Thanks! I will try it!
#### Damiano Testa (Dec 14 2020 at 08:25):
I am not sure if this makes it clearer or opaquer, but also what is below works.
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
begin
rintro ⟨_⟩ ⟨_⟩ ⟨⟩,
refl,
end
#### Alex J. Best (Dec 14 2020 at 08:28):
If you do
@[protect_proj, ext] structure sus (α : Type*) :=
lean will generate a lemma sus.ext and sus.ext_iff for you
#### Kevin Buzzard (Dec 14 2020 at 08:30):
To prove two instances of a structure are equal, do cases on them. Look at how I prove the ext lemma for complex numbers in the complex number game. The equation compiler proof is just doing cases.
#### Damiano Testa (Dec 14 2020 at 08:33):
Thank you both!
Indeed, with
@[protect_proj, ext] structure sus (α : Type*) :=
the proof is simply sus.ext.
Otherwise, cases all the way reduces the proof to refl:
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
begin
intros f g h,
cases f,
cases g,
cases h,
refl,
end
#### Damiano Testa (Dec 14 2020 at 08:34):
Kevin Buzzard said:
To prove two instances of a structure are equal, do cases on them. Look at how I prove the ext lemma for complex numbers in the complex number game. The equation compiler proof is just doing cases.
What does "equation compiler" mean? Is it what you see when you use show_term?
#### Alex J. Best (Dec 14 2020 at 08:40):
The equation compiler proof is the first one you mentioned:
@[ext]
lemma topological_space_eq : ∀ {f g : topological_space α}, f.is_open = g.is_open → f = g
| ⟨a, _, _, _⟩ ⟨b, _, _, _⟩ rfl := rfl
#### Damiano Testa (Dec 14 2020 at 08:48):
Ah, thanks! I am also seeing through this proof more. However, now I do not understand why is what is below not a proof?
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
| ⟨_, _, _, _, _⟩ ⟨_, _, _, _, _⟩ ⟨⟩ := rfl -- invalid expression\\ command expected
While this one is?
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
begin
rintros ⟨_⟩ ⟨_⟩ ⟨⟩,
refl,
end
#### Mario Carneiro (Dec 14 2020 at 08:52):
you probably need rfl instead of ⟨⟩ in the first version
#### Mario Carneiro (Dec 14 2020 at 08:54):
you can also use ⟨_⟩
#### Mario Carneiro (Dec 14 2020 at 08:54):
because the constructor of eq is eq.refl which has one explicit argument
#### Mario Carneiro (Dec 14 2020 at 08:55):
rintro doesn't really care if you don't give it enough arguments:
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
by rintro ⟨⟩ ⟨⟩ ⟨⟩; refl
#### Damiano Testa (Dec 14 2020 at 09:00):
I tried with
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g :=
| ⟨_, _, _, _, _⟩ ⟨_, _, _, _, _⟩ rfl := rfl -- invalid expression \\ command expected
and it does not work. In any case, I have enough proofs of a completely trivial result that the computer can even figure out on its own! I was just curious about the proof style with | that I had never seen before, hence my curiousity!
#### Mario Carneiro (Dec 14 2020 at 09:02):
The following compiles for me with no errors:
import order.filter.ultrafilter
import order.filter.partial
open set filter classical
open_locale classical filter
/-- A sus on α. -/
@[protect_proj] structure sus (α : Type*) :=
(is_su : set α → Prop)
(is_su_univ : is_su univ)
(is_su_empty : is_su ∅)
(is_su_inter : ∀s t, is_su s → is_su t → is_su (s ∩ t))
(is_su_union : ∀s t, is_su s → is_su t → is_su (s ∪ t))
attribute [class] sus --I do not know what this does, but it was there
/-- A constructor for sus using complements of the given sus structure. -/
-- this definition is not relevant to what I care about: I simply say that if I have a sus, then the collection of complements also forms a sus
def sus.comp {α : Type*} (f : sus α) : sus α :=
{ is_su := λ X, f.is_su Xᶜ,
is_su_univ := by simp [sus.is_su_empty],
is_su_empty := by simp [sus.is_su_univ],
is_su_inter := λ s t hs ht, by { rw compl_inter, exact sus.is_su_union f sᶜ tᶜ hs ht },
is_su_union := λ s t hs ht, by { rw compl_union, exact sus.is_su_inter f sᶜ tᶜ hs ht },
}
section sus
variables {α : Type*} {β : Type*} {ι : Sort*} {a : α} {s s₁ s₂ : set α} {p p₁ p₂ : α → Prop}
@[ext]
lemma sus_eq : ∀ {f g : sus α}, f.is_su = g.is_su → f = g
| ⟨a, _, _, _, _⟩ ⟨b, _, _, _, _⟩ rfl := rfl
#### Mario Carneiro (Dec 14 2020 at 09:02):
Oh, you still have the := on the first line
#### Mario Carneiro (Dec 14 2020 at 09:03):
when you use an equation compiler proof you have to take that off
#### Damiano Testa (Dec 14 2020 at 09:03):
Ah, I was puzzled by the fact that yours compiled, while mine did not. I had taken no notice of the :=
#### Damiano Testa (Dec 14 2020 at 09:04):
so then, why the a and b?
#### Mario Carneiro (Dec 14 2020 at 09:04):
it makes the intermediate state display a little nicer
I see
#### Mario Carneiro (Dec 14 2020 at 09:05):
it's totally optional here
#### Damiano Testa (Dec 14 2020 at 09:07):
Ok, I am understanding this proof a little better now. At first, it looked like non-sense, now it suggests that the statement is entirely trivial!
#### Damiano Testa (Dec 14 2020 at 09:38):
Kevin Buzzard said:
To prove two instances of a structure are equal, do cases on them. Look at how I prove the ext lemma for complex numbers in the complex number game. The equation compiler proof is just doing cases.
I am doing the Complex Numbers Game. If you are interested, I found a couple of typos in Level_00_basic.lean.
#### Kevin Buzzard (Dec 14 2020 at 09:43):
Open an issue on github and I'll deal with it one day.
Last updated: May 17 2021 at 16:26 UTC | 2021-05-17 16:37:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 12, "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": 0.5480043292045593, "perplexity": 2583.6865026020196}, "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/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00408.warc.gz"} |
https://wiki.onion.io/Tutorials/Connecting-Omega-to-Wifi-Hotspot-that-Requres-Login | # Connecting the Omega to Wi-Fi Hotspot that Requires Login
Last edited by Boken Lin, 2015-11-28 00:08:19
In order to connect to some Wi-Fi hostspots in certain hotels or restaurants, you will need to first connect to the Wi-Fi and login via your browser before you can access the Internet. Since the Omega does not have a browser, you will need to use your computer or smartphone to login for the Omega. To connect the Omega to one of these Wi-Fi hotspots, following the following instruction:
## 1. Enable the Access Point (AP) on the Omega
Access Point on the Omega should be enabled by default. If it isn't already enabled, you will need to connect to the Omega via the serial terminal. Then, to enable the Access Point, you will need to edit /etc/config/wireless. Uncomment or add the following lines to the file:
config wifi-iface
option network 'wlan'
option mode 'ap'
option ssid 'Omega AP'
option encryption 'psk2'
option key 'onioneer'
Then, to restart the Wi-Fi service with the new configuration:
wifi
Your Access Point should be turned on at this point.
## 2. Connect the Omega to the Wifi router
Next, you will need to connect the Omega to the Wi-Fi hotspot of the hotel/restaurants you are in. To do this, you will use the wifisetup command:
root@Omega-0104:/# wifisetup
Onion Omega Wifi Setup
Select from the following:
1) Scan for Wifi networks
2) Type network info
q) Exit
Selection:
Just follow the instruction to scan the Wi-Fi network and connect to it. | 2017-11-23 01:54:14 | {"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": 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": 0.17473170161247253, "perplexity": 3213.1426020050953}, "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/1510934806715.73/warc/CC-MAIN-20171123012207-20171123032207-00116.warc.gz"} |
http://math.stackexchange.com/questions/181760/use-the-laws-of-algebra-of-sets-to-show-a-cup-b-cap-c-cap-a-cup-c | # Use the laws of algebra of sets to show $(A \cup ( B \cap C')) \cap ( A \cup C ) = A$
Use the laws of algebra of sets to show that:
$$(A \cup ( B \cap C')) \cap ( A \cup C ) = A$$
Can someone please tell me how to work out such questions and what are the rules that can be used when using laws to prove such quesion.
-
First using distributivity $$(A∪(B∩C'))∩(A∪C) = ((A∪B)∩ (A∪C'))∩(A∪C)$$ then associativity $$((A∪B)∩ (A∪C'))∩(A∪C) = \color{teal}{(A∪B)}∩ \color{blue}{(A∪C') ∩ (A∪C)}$$ Can you take it from here?
To show that $\color{teal}{(A∪B)}∩ \color{blue}{(A∪C') ∩ (A∪C)} =A$ we will first show that $$\color{blue}{(A∪C')∩(A∪C)} = \color{blue}{A} \tag{1}$$ Then we'll use the absorption law to finish with $$\text{LHS} = \color{teal}{(A∪B)} ∩ \color{blue}{A} = A = \text{RHS}.$$
To prove $(1)$ we use distributivity to get: $$\color{red}{[A ∩ (A∪C)]} ∪ [C' ∩ (A∪C)]$$ But by absorption law we have $$\color{red}{A} ∪ [C' ∩ (A∪C)]$$ But $C' ∩ (A∪C) = (C' ∩ A) ∪ (C' ∩ C) = (C' ∩ A) ∪ \phi = (C' ∩ A) .$ So by absorption again we have $$\color{red}{A} ∪ [C' ∩ (A∪C)] = \color{red}{A} ∪ (C' ∩ A) = A.$$ QED.
-
Strictly speaking, your second righthand side ought to be $$(A\cup B)\cap\Big((A\cup C')\cap(A\cup C)\Big)\;.$$ It works: you then use distributivity in the big parenthesis, do a couple of obvious things, and finish with absorption. – Brian M. Scott Aug 12 '12 at 17:11
I just don't understand how to get rid of the C′ ? – Zee123 Aug 17 '12 at 0:14
@Zee123 can you edit your question & add your steps? In which step did you get stuck? – user2468 Aug 17 '12 at 0:44
First using distributivity (A∪(B∩C′))∩(A∪C)=((A∪B)∩(A∪C′))∩(A∪C) then associativity ((A∪B)∩(A∪C′))∩(A∪C)=(A∪B)∩(A∪C′)∩(A∪C) then used complement law to get ((A∪B)∩(A∪C′))∩(A∪C)=(A∪B)∩ ∅ .....but I'm not sure if that's what the next step is or if that's how you get rid of the C′ using this law or if there are other laws that can be used? – Zee123 Aug 17 '12 at 1:02
@Zee123 see my update. I posted the full solution. Let me know if you have more questions. If this is a satisfactory answer, then please consider the accept button to the right below the upvote button. – user2468 Aug 17 '12 at 3:06
By the laws of algebra, I assume you mean the Boolean Algebra axiom for the Boolean Algebra of Sets. Converting into Boolean notation, you would have that $+$ is $\cup$, $\cdot$ is $\cap$, $0 = \emptyset$, and $1 = X$ where $X$ is the universe of the particular boolean algebra
I will use this frequently, the property of absorption is
$u \cap (u \cup x) = u \text{ and } u \cup (u \cap x) = u$
for all $u$ and $x$. Jech's Set Theory includes it among his axioms, but derives the identity axiom. With the identity axiom (like the definition from wikipedia), you can derive absorption. Depending on what exactly are your boolean algebra axioms, you may or may not need to prove absorption or identity.
By Distributivity,
$$(A \cap (A \cup C)) \cup ((B \cap C') \cap (A \cup C))$$
By absorption,
$A \cap (A \cup C) = A$
So you have
$A \cup ((B \cap C') \cap (A \cup C))$
By associativity of $\cap$
$A \cup (B \cap (C' \cap (A \cup C)))$
By Distributivity
$A \cup (B \cap ((C' \cap A) \cup (C' \cap C)))$
By complementation, $C' \cap C = \emptyset$ so
$A \cup (B \cap (C' \cap A) \cup \emptyset))$
By the identity axiom, $u \cup \emptyset = u$ for any $u$. So
$A \cup (B \cap (C' \cap A))$
By Distibutivity
$A \cup ((B \cap C') \cap A)$
By absorption
$A$
The proof is complete.
- | 2016-07-25 12:26:28 | {"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": 0.8871198296546936, "perplexity": 362.06168489201}, "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/1469257824226.79/warc/CC-MAIN-20160723071024-00052-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://docs.sui.io/devnet/build/programming-with-objects/ch1-object-basics | Chapter 1 - Object Basics
Define Sui Object
In Move, besides primitive data types, we can define organized data structures using struct. For example:
struct Color {
red: u8,
green: u8,
blue: u8,
}
The above struct defines a data structure that can represent RGB color. structs like this can be used to organize data with complicated semantics. However, instances of structs like Color are not Sui objects yet. To define a struct that represents a Sui object type, we must add a key capability to the definition, and the first field of the struct must be the id of the object with type UID from the object module:
use sui::object::UID;
struct ColorObject has key {
id: UID,
red: u8,
green: u8,
blue: u8,
}
Now ColorObject represents a Sui object type and can be used to create Sui objects that can be eventually stored on the Sui chain.
📚In both core Move and Sui Move, the key ability denotes a type that can appear as a key in global storage. However, the structure of global storage is a bit different: core Move uses a (type, address)-indexed map, whereas Sui Move uses a map keyed by object IDs.
💡The UID type is internal to Sui, and you most likely won't need to deal with it directly. For curious readers, it contains the "unique ID" that defines an object. It is unique in the sense that no two values of type UID will ever have the same underlying set of bytes.
Create Sui object
Now that we have learned how to define a Sui object type, how do we create/instantiate a Sui object? In order to create a new Sui object from its type, we must assign an initial value to each of the fields, including id. The only way to create a new UID for a Sui object is to call object::new. The new function takes the current transaction context as an argument to generate unique IDs. The transaction context is of type &mut TxContext and should be passed down from an entry function (a function that can be called directly from a transaction). Let's look at how we may define a constructor for ColorObject:
// object creates an alias to the object module, which allows us call
// functions in the module, such as the new function, without fully
// qualifying, e.g. sui::object::new.
use sui::object;
// tx_context::TxContext creates an alias to the the TxContext struct in tx_context module.
use sui::tx_context::TxContext;
fun new(red: u8, green: u8, blue: u8, ctx: &mut TxContext): ColorObject {
ColorObject {
id: object::new(ctx),
red,
green,
blue,
}
}
💡Move supports field punning, which allows us to skip the field values if the field name happens to be the same as the name of the value variable it is bound to. The code above leverages this to write "red," as shorthand for "red: red,".
Store Sui object
We have defined a constructor for the ColorObject. Calling this constructor will put the value in a local variable where it can be returned from the current function, passed to other functions, or stored inside another struct. And of course, the object can be placed in persistent global storage so it can be read by the outside world and accessed in subsequent transactions.
All of the APIs for adding objects to persistent storage live in the transfer module. One key API is:
public fun transfer<T: key>(obj: T, recipient: address)
This places obj in global storage along with metadata that records recipient as the owner of the object. In Sui, every object must have an owner, which can be either an address, another object, or "shared"--see object ownership for more details.
💡In core Move, we would call move_to<T>(a: address, t: T) to add the entry (a, T) -> t to the global storage. But because (as explained above) the schema of Sui Move's global storage is different, we use the Transfer APIs instead of move_to or the other global storage operators in core Move. These operators cannot be used in Sui Move.
A common use of this API is to transfer the object to the sender/signer of the current transaction (e.g., mint an NFT owned by you). The only way to obtain the sender of the current transaction is to rely on the transaction context passed in from an entry function. The last argument to an entry function must be the current transaction context, defined as ctx: &mut TxContext. To obtain the current signer's address, one can call tx_context::sender(ctx).
Below is the code that creates a new ColorObject and makes it owned by the sender of the transaction:
use sui::transfer;
// This is an entry function that can be called directly by a Transaction.
public entry fun create(red: u8, green: u8, blue: u8, ctx: &mut TxContext) {
let color_object = new(red, green, blue, ctx);
transfer::transfer(color_object, tx_context::sender(ctx))
}
💡Naming convention: Constructors are typically named new, which returns an instance of the struct type. The create function is typically defined as an entry function that constructs the struct and transfers it to the desired owner (most commonly the sender).
We can also add a getter to ColorObject that returns the color values so that modules outside of ColorObject are able to read their values:
public fun get_color(self: &ColorObject): (u8, u8, u8) {
(self.red, self.green, self.blue)
}
Find the full code online in color_object.move.
To compile the code, make sure you have installed Sui so that sui is in PATH. In the code root directory (where Move.toml is), run:
sui move build
Writing unit tests
After defining the create function, we want to test this function in Move using unit tests, without having to go all the way through sending Sui transactions. Since Sui manages global storage separately outside of Move, there is no direct way to retrieve objects from global storage within Move. This poses a question: after calling the create function, how do we check that the object is properly transferred?
To assist easy testing in Move, we provide a comprehensive testing framework in the test_scenario module that allows us to interact with objects put into the global storage. This allows us to test the behavior of any function directly in Move unit tests. A lot of this is also covered in our Move testing doc.
The idea of test_scenario is to emulate a series of Sui transactions, each sent from a particular address. A developer writing a test starts the first transaction using the test_scenario::begin function that takes the address of the user sending this transaction as an argument and returns an instance of the Scenario struct representing a test scenario.
An instance of the Scenario struct contains a per-address object pool emulating Sui's object storage, with helper functions provided to manipulate objects in the pool. Once the first transaction is finished, subsequent transactions can be started using the test_scenario::next_tx function that takes an instance of the Scenario struct representing the current scenario and an address of a (new) user as arguments.
Now let's try to write a test for the create function. Tests that need to use test_scenario must be in a separate module, either under a tests directory, or in the same file but in a module annotated with #[test_only]. This is because test_scenario itself is a test-only module, and can be used only by test-only modules.
First of all, we begin the test with a hardcoded test address, which will also give us a transaction context as if we are sending the transaction started with test_scenario::begin from this address. We then call the create function, which should create a ColorObject and transfer it to the test address:
let owner = @0x1;
// Create a ColorObject and transfer it to @owner.
let scenario = &mut test_scenario::begin(&owner);
{
let ctx = test_scenario::ctx(scenario);
color_object::create(255, 0, 255, ctx);
};
📚Note there is a ";" after "}". ; is required to sequence a series of expressions, and even the block { ... } is an expression! Refer to the Move book for a detailed explanation.
Now, after the first transaction completes (and only after the first transaction completes), address @0x1 should own the object. Let's first make sure it's not owned by anyone else:
let not_owner = @0x2;
// Check that not_owner does not own the just-created ColorObject.
test_scenario::next_tx(scenario, ¬_owner);
{
assert!(!test_scenario::can_take_owned<ColorObject>(scenario), 0);
};
test_scenario::next_tx switches the transaction sender to @0x2, which is a new address different from the previous one. test_scenario::can_take_owned checks whether an object with the given type actually exists in the global storage owned by the current sender of the transaction. In this code, we assert that we should not be able to remove such an object, because @0x2 does not own any object.
💡The second parameter of assert! is the error code. In non-test code, we usually define a list of dedicated error code constants for each type of error that could happen in production. For unit tests though, it's usually unnecessary because there will be way too many assetions and the stacktrace upon error is sufficient to tell where the error happened. Hence we recommend just putting 0 there in unit tests for assertions.
Finally we check that @0x1 owns the object and the object value is consistent:
test_scenario::next_tx(scenario, &owner);
{
let object = test_scenario::take_owned<ColorObject>(scenario);
let (red, green, blue) = color_object::get_color(&object);
assert!(red == 255 && green == 0 && blue == 255, 0);
test_scenario::return_owned(scenario, object);
};
test_scenario::take_owned removes the object of given type from global storage that's owned by the current transaction sender (it also implicitly checks can_take_owned). If this line of code succeeds, it means that owner indeed owns an object of type ColorObject. We also check that the field values of the object match with what we set in creation. At the end, we must return the object back to the global storage by calling test_scenario::return_owned so that it's back to the global storage. This also ensures that if any mutations happened to the object during the test, the global storage is aware of the changes.
Again, you can find the full code in color_object.move.
To run the test, simply run the following in the code root directory:
sui move test
On-chain Interactions
Now let's try to call create in actual transactions and see what happens. To do this, we need to start Sui and the CLI client. Follow the Sui CLI client guide to start the Sui network and set up the client.
Before starting, let's take a look at the default client address (this is the address that will eventually own the object later):
$sui client active-address This will tell you the current client address. First, we need to publish the code on-chain. Assuming the path to the root of the repository containing Sui source code is$ROOT:
$sui client publish --path$ROOT/sui_programmability/examples/objects_tutorial --gas-budget 10000
You can find the published package object ID in the Publish Results output:
----- Publish Results ----
The newly published package object: (0x57258f32746fd1443f2a077c0c6ec03282087c19, SequenceNumber(1), o#b3a8e284dea7482891768e166e4cd16f9749e0fa90eeb0834189016c42327401)
Note that the exact data you see will be different. The first hex string in that triple is the package object ID (0x57258f32746fd1443f2a077c0c6ec03282087c19 in this case). For convenience, let's save it to an environment variable:
$export PACKAGE=0x57258f32746fd1443f2a077c0c6ec03282087c19 Next we can call the function to create a color object: $ sui client call --gas-budget 1000 --package $PACKAGE --module "color_object" --function "create" --args 0 255 0 In the Transaction Effects portion of the output, you will see an object showing up in the list of Created Objects, like this: Created Objects: 0x5eb2c3e55693282faa7f5b07ce1c4803e6fdc1bb SequenceNumber(1) o#691b417670979c6c192bdfd643630a125961c71c841a6c7d973cf9429c792efa Again, for convenience, let's save the object ID: $ export OBJECT=0x5eb2c3e55693282faa7f5b07ce1c4803e6fdc1bb
We can inspect this object and see what kind of object it is:
$sui client object --id$OBJECT
This will show you the metadata of the object with its type:
Owner: AddressOwner(k#5db53ebb05fd3ea5f1d163d9d487ee8cd7b591ee)
Version: 1
ID: 0x5eb2c3e55693282faa7f5b07ce1c4803e6fdc1bb
Type: 0x57258f32746fd1443f2a077c0c6ec03282087c19::color_object::ColorObject
As we can see, it's owned by the current default client address that we saw earlier. And the type of this object is ColorObject!
You can also look at the data content of the object by adding the --json parameter:
$sui client object --id$OBJECT --json
This will print the values of all the fields in the Move object, such as the values of red, green, and blue.
Congratulations! You have learned how to define, create, and transfer objects. You should also know how to write unit tests to mock transactions and interact with the objects. In the next chapter, we will learn how to use the objects that we own.
Last update 8/18/2022, 4:04:52 PM | 2022-09-30 16:40:25 | {"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": 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": 0.17205937206745148, "perplexity": 2321.4200627151736}, "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-2022-40/segments/1664030335491.4/warc/CC-MAIN-20220930145518-20220930175518-00557.warc.gz"} |
http://www.mathynomial.com/problem/1324 | # Problem #1324
1324 Point $P$ is inside equilateral $\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$? $\mathrm {(A)} 4\qquad \mathrm {(B)} 3\sqrt{3}\qquad \mathrm {(C)} 6\qquad \mathrm {(D)} 4\sqrt{3}\qquad \mathrm {(E)} 9$ This problem is copyrighted by the American Mathematics Competitions.
Note: you aren't logged in. If you log in, we'll keep a record of which problems you've solved. | 2018-02-18 06:51:57 | {"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": 14, "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": 0.785031259059906, "perplexity": 260.0448263275571}, "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/1518891811794.67/warc/CC-MAIN-20180218062032-20180218082032-00511.warc.gz"} |
https://www.hackmath.net/en/math-problem/533?tag_id=50 | # Motion
If you go at speed 3.7 km/h, you come to the station 42 minutes after leaving the train. If you go by bike to the station at speed 27 km/h, you come to the station 56 minutes before its departure.
How far is the train station?
Result
s = 7.003 km
#### Solution:
$v_{1}=3.7 \ \text{km/h} \ \\ v_{2}=27 \ \text{km/h} \ \\ t_{1}=(x + 42)/60 \ \\ t_{2}=(x - 56)/60 \ \\ s=v_{1} \cdot \ t_{1}=v_{2} \cdot \ t_{2} \ \\ v_{1}( x+ 42 )=v_{2}( x - 56 ) \ \\ x=(v_{1} \cdot \ 42+v_{2} \cdot \ 56)/(v_{2}-v_{1})=(3.7 \cdot \ 42+27 \cdot \ 56)/(27-3.7) \doteq 71.5622 \ \text{min} \ \\ t_{1}=(x + 42)/60=(71.5622 + 42)/60 \doteq \dfrac{ 441 }{ 233 } \doteq 1.8927 \ \text{h} \ \\ t_{2}=(x - 56)/60=(71.5622 - 56)/60 \doteq 0.2594 \ \text{h} \ \\ \ \\ s=s_{1}=s_{2} \ \\ s_{1}=v_{1} \cdot \ t_{1}=3.7 \cdot \ 1.8927 \doteq 7.003 \ \text{km} \ \\ s=v_{2} \cdot \ t_{2}=27 \cdot \ 0.2594 \doteq 7.003 \doteq 7.003 \ \text{km}$
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15. The copper wire
The copper wire bundle with a diameter of 2.8mm has a weight of 5kg. How many meters of wire is bundled if 1m3 of copper weighs 8930kg? | 2020-05-26 00:26:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.5530619025230408, "perplexity": 800.2448092832051}, "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-2020-24/segments/1590347390437.8/warc/CC-MAIN-20200525223929-20200526013929-00319.warc.gz"} |
https://betterexplained.com/articles/math-implied-subject/ | # Learning Math: Find The Implied Subject
In English, we often drop the subject of a phrase:
Who are these signs written for? It's really You, stop or You, yield or You, be alert for bears (I'm not doing it).
After internalizing a language, we can take hints without explicit instructions. But, to put it politely, math isn't usually well-internalized.
Let's get clear about who the "math signs" are referring to.
## Imaginary Numbers
Imaginary numbers are often defined as $i^2 = -1$, and written this way they're utterly baffling.
A better restatement is:
$\displaystyle{i * i = -1}$
$\displaystyle{1 * i * i = -1}$
It's getting clearer. The instructions are: "Start with 1, multiply by i, multiply by i again, and (somehow) end up at -1".
What could do that? Realizing we start at 1 and end up at -1 helps us visualize something like this:
Aha! $i$ is a change (visualized as a rotation) that moves us from positive to negative in two steps (More about imaginary numbers.)
Writing $i^2 = -1$ without a clear subject is confusing. (Don't get me started with $i = \sqrt{-1}$)
Missing the implied subject of "1" in $1* i * i$ caused me years of confusion. I wish this sign was hanging on the classroom wall:
## Exponents
Why is $x^0 = 1$ for any value of x? How do we ask for 0 of something and get 1 back out?
Again, let's break it down with a simple example. Here's a typical exponent:
$\displaystyle{x * x = x^2}$
But it's missing a subject. It's written better as:
$\displaystyle{1 (* x) (* x) = x^2}$
We start at 1 (our default multiplicative scaling factor), scale by x, then scale by x again, ending up at $x^2$. The size of the exponent (2) tells us the number of times to use our "times x" scaling machine.
Stepping back, multiplication is about scaling: 3 is really "1 * 3", or the unit quantity enlarged 3 times.
If we want to scale by x (just once), we write:
$\displaystyle{1 (* x) = x}$
What if we don't plan on using our scaling machine at all?
$\displaystyle{1(* x^0) = 1 () = 1}$
The notation is a bit weird, but I'm using empty parenthesis to indicate a lack of action. See, the zero in $x^0$ is that of indifference -- taking no action -- and not obliteration. "Using" $x^0$ means we haven't scaled our original quantity at all.
Subtle, right?
We can take this "growth machine" idea further with the Expand-o-tron 3000.
## Imaginary Exponents
Let's combine insights. What does a strange exponent like $e^{i \pi}$ represent?
With our new "implicitly start at 1" perspective, it's really:
$\displaystyle{1 \cdot e^{i\pi}}$
Start at 1 and then apply the growth engine. Here, growth is aimed sideways ($i$) with enough fuel to last for half a circle ($\pi$).
The essence of Euler's Identity is that we are starting at 1 and transforming it with a spin. We aren't creating a negative number out of seemingly positive exponents directly. (See article and video.)
## Calculus
Calculus has numerous notational shortcuts. When we write:
$\displaystyle{\int x^2}$
it really means:
$\displaystyle{\int x^2 \ dx}$
which really means:
$\displaystyle{\int x^2 \ dx \ (\text{at each position}) }$
Here's the tricky part. There isn't a single "dx", there's a whole chain of them along the number line. The sentence is something like:
"Hey everyone on the number line! You're all spaced "dx" apart. Take your current position and square it. Then I'll come by and add you all up."
The integral addresses not a single "you" like 1.0, but "them", the countless positions on the number line.
Find the implied subject in an equation, then work to shorten it (Bears!).
Happy math. | 2022-10-03 02:46:20 | {"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": 10, "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": 0.7473952174186707, "perplexity": 2414.900799236928}, "config": {"markdown_headings": true, "markdown_code": false, "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-2022-40/segments/1664030337371.9/warc/CC-MAIN-20221003003804-20221003033804-00513.warc.gz"} |
https://meetonfriday.com/posts/8cad39eb/ | # [論文速速讀]Hierarchical Attention Networks for Document Classification
Posted by John on 2020-05-19
Words 1.2k and Reading Time 5 Minutes
Viewed Times
〖想觀看更多中文論文導讀，至[論文速速讀]系列文章介紹可以看到目前已發布的所有文章！〗
## Abstract
We propose a hierarchical attention network for document classification. Our model has two distinctive characteristics:
(i) it has a hierarchical structure that mirrors the hierarchical structure of documents;
(ii) it has two levels of attention mechanisms applied at the word and sentence-level, enabling it to attend differentially to more and less important content when constructing the document representation.
1. 使用了可以反映文章結構的的階層結構
2. 在word level和sentence level上都使用了attention mechanism，使得在建構representation時能夠注意到比較重要的內容
## Introduction
Although neural-network–based approaches to text classification have been quite effective (Kim, 2014; Zhang et al., 2015; Johnson and Zhang, 2014; Tang et al., 2015), in this paper we test the hypothesis that better representations can be obtained by incorporating knowledge of document structure in the model architecture. The intuition underlying our model is that not all parts of a document are equally relevant for answering a query and that determining the relevant sections involves modeling the interactions of the words, not just their presence in isolation
First, since documents have a hierarchical structure (words form sentences, sentences form a document), we likewise construct a document representation by first building representations of sentences and then aggregating those into a document representation. Second, it is observed that different words and sentences in a documents are differentially informative. Moreover, the importance of words and sentences are highly context dependent.
1. 不同的word跟sentence在文章中具有不同的資訊量
2. word和sentence的重要性很大程度上取決於上下文
## Hierarchical Attention Networks
HAN包含了以下幾個部分:
• word sequence encoder
• word-level attention layer
• sentence encoder
• sentence-level attention layer
GRU的new state:
### Hierarchical Attention
#### Word Attention
$h_{it}=[\overrightarrow{h_{it}}, \overleftarrow{h_{it}}]$進行一次transformation，然後透過一個content vector $u_w$來進行attention，這裡的$u_w$可以被視為a high level representation of a fixed query “what is the imformative word” over the words like that used in menory networks
Jumping
## Experiments
### Datasets
• 80% training, 10% validation, 10% testing
1. Yelp reviews(有三年，一年一個)
2. IMDB reviews
4. Amazon reviews
### Model configuration and training
• 使用Stanford’s CoreNLP (Manning et al., 2014)切sentence和word
• word2vec做word embedding, dimension=200
• frequency小於5的換成
• GRU dimension=50
• 所以biGRU就會是100
• word/sentence context vector dimension=100
• batch size=64
• optimizer use SGD
### Results and analysis
The experimental results on all data sets are shown in Table 2. We refer to our models as HN-{AVE, MAX, ATT}. Here HN stands for Hierarchical Network, AVE indicates averaging, MAX indicates max-pooling, and ATT indicates our proposed hierarchical attention model. Results show that HNATT gives the best performance across all data sets
HAN棒棒噠
> | 2021-03-08 21:26:02 | {"extraction_info": {"found_math": true, "script_math_tex": 2, "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": 0.6506341695785522, "perplexity": 7560.452227449417}, "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-2021-10/segments/1614178385529.97/warc/CC-MAIN-20210308205020-20210308235020-00155.warc.gz"} |
http://en.wikipedia.org/wiki/Canonicalization | Canonicalization
Not to be confused with Canonization.
In computer science, canonicalization (sometimes standardization or normalization) is a process for converting data that has more than one possible representation into a "standard", "normal", or canonical form. This can be done to compare different representations for equivalence, to count the number of distinct data structures, to improve the efficiency of various algorithms by eliminating repeated calculations, or to make it possible to impose a meaningful sorting order.
The term canonicalization is sometimes abbreviated c14n, where 14 represents the number of letters between the C and the N.
Usage cases
Web servers
Canonicalization of filenames is important for computer security. For example, a web server may have a security rule stating "only execute files under the cgi directory (C:\inetpub\wwwroot\cgi-bin)". The rule is enforced by checking that the path starts with "C:\inetpub\wwwroot\cgi-bin\", and if it does, the file is executed.
Should file "C:\inetpub\wwwroot\cgi-bin\..\..\..\Windows\System32\cmd.exe" be executed? No, because this trick path goes back up the directory hierarchy (through use of the '..' path specifier), not staying within cgi-bin. Accepting it at face value would be an error due to failure to canonicalize the filename to the unique (simplest) representation, namely: "C:\Windows\System32\cmd.exe", before doing the path check. This type of fault is called a directory traversal vulnerability.
Unicode
Variable-length encodings in the Unicode standard, in particular UTF-8, have more than one possible encoding for most common characters.[1] This makes string validation more complicated, since every possible encoding of each string character must be considered. A software implementation which does not consider all character encodings runs the risk of accepting strings considered invalid in the application design, which could cause bugs or allow attacks. The solution is to allow a single encoding for each character. Canonicalization is then the process of translating every string character to its single allowed encoding. An alternative is for software to determine whether a string is canonicalized, and then reject it if it is not. In this case, in a client/server context, the canonicalization would be the responsibility of the client.
Search engines and SEO
In web search and search engine optimization (SEO), URL canonicalization deals with web content that has more than one possible URL. Having multiple URLs for the same web content can cause problems for search engines - specifically in determining which URL should be shown in search results.[2]
Example:
All of these URLs point to the homepage of Wikipedia, but a search engine will only consider one of them to be the canonical form of the URL.
XML
A Canonical XML document is by definition an XML document that is in XML Canonical form, defined by The Canonical XML specification. Briefly, canonicalization removes whitespace within tags, uses particular character encodings, sorts namespace references and eliminates redundant ones, removes XML and DOCTYPE declarations, and transforms relative URIs into absolute URIs.
Simple example: Given two versions of the same XML:
• "<node1>Data</node1> <node2>Data</node2>"
• "<node1>Data</node1> <node2>Data</node2>"
Note the extra spaces in the samples, the canonicalized version of these two might be:
• "<node1>Data</node1><node2>Data</node2>"
Note that the spaces are removed — this is one thing a canonicalizer does. A real canonicalizer may make other changes as well.
A full summary of canonicalization changes is listed below:
• The document is encoded in UTF-8
• Line breaks normalized to #xA on input, before parsing
• Attribute values are normalized, as if by a validating processor
• Character and parsed entity references are replaced
• CDATA sections are replaced with their character content
• The XML declaration and document type declaration are removed
• Empty elements are converted to start-end tag pairs
• Whitespace outside of the document element and within start and end tags is normalized
• All whitespace in character content is retained (excluding characters removed during line feed normalization)
• Attribute value delimiters are set to quotation marks (double quotes)
• Special characters in attribute values and character content are replaced by character references
• Superfluous namespace declarations are removed from each element
• Default attributes are added to each element
• Fixup of xml:base attributes is performed
• Lexicographic order is imposed on the namespace declarations and attributes of each element | 2014-09-21 11:27:10 | {"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": 0.4040094017982483, "perplexity": 2790.9086174947724}, "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-2014-41/segments/1410657135396.68/warc/CC-MAIN-20140914011215-00325-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://unionems.com/12z9wv/acd55c-the-nature-of-electromagnetic-wave-is | photoelectric effect. Ask Question Asked 5 years, 8 months ago. Any two orthogonal basis vectors could be used to describe the polarization, and in some cases the right- and left-handed circular ba-sis is used. Nature of Electromagnetic waves. Viewed 2k times 4. The wave nature of light, which is electromagnetic radiation, was discovered in 1801 by Thomas Young in England. Switch; Flag; Bookmark; Suppose that the electric field amplitude of an electromagnetic wave is E 0 = 120 N/C and that its frequency is v = 50.0 MHz. A wave-type experiment shows the wave nature, and a particle-type experiment shows particle nature. The plane waves may be viewed as the limiting case of spherical waves at a very large (ideally infinite) distance from the source. (a) Determine, B 0, ω, k, and λ. We can get a good understanding of electromagnetic waves (EM) by considering how they are produced. PLAY. 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A particle of light is called a photon. what is a electromagnetic wave Preview this quiz on Quizizz. 8th … Wave nature of electromagnetic radiation is characterized by its following three properties – Wavelength Frequency Velocity Wavelength – The distance of one full cycle of the oscillation is called wavelength or the distance between two adjacent crests or troughs of a wave is called the wavelength. This means wave propagation ---à x-axis , electric field -----> y-axis, magnetic field --à z-axis. smanya9503_84270. Since the oscillation is with respect to time, the wave moves along the axis of velocity in the … Start studying Chapter 17 the nature of electromagnetic waves. One cannot test the wave and the particle nature at the same time. It refers to waves of any one of these various frequencies of oscillations of electric and magnetic fields. A photon wave function and a time-dependent electromagnetic field are completely different things. Explanation: No explanation available. Another name for light is electromagnetic radiation. Draw a diagram showing the propagation of an electromagnetic. Water waves transmit energy through space by the periodic oscillation of matter (the water). In some cases, this is easy to answer. The Nature of Electromagnetic Waves (pages 70–73) What Is an Electromagnetic Wave? Play this game to review Science. what is a electromagnetic wave. 100% average accuracy. electromagnetic radiation. In the 1860's, James Clerk Maxwell, a Scottish mathematician and physicist, determined light was electromagnetic radiation. These waves propagate with a constant velocity of 3*10 8 m/s and do not need any specific medium to propagate.. EM waves are said to be of transverse nature hence is … Electric field of EM wave is represented as: E y = E 0 sin(kx–ωt) Where Ey= electric field along y-axis and x=direction of propagation of wave. Such waves are called "mechanical waves", which require a material medium to travel in in order to exist. electromagnetic wave. Played 2 times. NATURE OF AN ELECTROMAGNETIC WAVE* We now know something quite unusual about an electromagnetic wave and about a photon. STUDY. 2 times. Here we find that a photon is a single oscillation of the E-field vector and the H-field vector as shown in the diagram. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in Figure $$\PageIndex{1}$$. Reason: The refractive index of the ionosphere becomes very high for frequencies higher than the critical frequency. From what I understand, the 50 Hz power frequency wave in our 230 V supply at home (60Hz and a lesser voltage in countries other than India) is also an electromagnetic wave. In this context, if I want to talk about light whose frequency falls in the range to which our eyes are sensitive, I refer to it as visible light. Edit. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Two main classes of solutions are known, namely plane waves and spherical waves. Science. Transverse nature of electromagnetic waves means the electric and magnetic fields in an electromagnetic wave are perpendicular to each other and to the direction of propagation. Electromagnetic Radiation. If both assertion and reason are true and reason is the correct explanation of assertion. The Nature of Electromagnetic Waves | Physics| Lesson VideoElectromagnetic waves are produced by a charge that changes its direction or speed. Active 5 years, 8 months ago. It is represented by $\lambda$. Also, the electromagnetic field and photons have to be handled with relativistic quantum … So, it explains the transverse nature of E M waves. The energy is called electromagnetic radiation. Correct Answer: Polarisation phenomena. • An electromagnetic wave is a transverse wave that carries electrical and magnetic energy. Electromagnetic waves include, radio waves, microwaves, infrared, light, ultraviolet, x-rays, and gamma rays. 0. asked Jan 21, 2019 in Physics by kajalk (77.7k points) electromagnetic waves… Figure 4a shows how a photon may be represented. What is meant by the transverse nature of electromagnetic waves ? Radio wave, x-ray, infrared, gamma, light, microwaves are known as electromagnetic waves. the nature of electromagnetic waves DRAFT. All EM waves are made up of photons that travel through space until they interact with matter; some waves are absorbed and others are reflected. Assertion: Electromagnetic waves with frequencies smaller than the critical frequencies of ionosphere cannot be used for communication using sky wave propagation. Start studying The Nature of Waves:. the light that passes through a polarized filter. Electromagnetic waves are transverse in nature as is evident by polarization interference and diffraction explain the wave nature of light or E M waves polarization is the phenomenon by which we restrict the vibration of wave in a particular direction perpendicular to direction of W are propagation. Waves classed by the type of disturbance; name disturbance is… examples; transverse waves: perpendicular to propagation: light and all electromagnetic waves, gravitational waves, matter waves, nerve impulses, peristalsis, secondary seismic waves (S waves a.k.a. (Figure $$\PageIndex{3}$$). Class 12: Physics: Electromagnetic Waves: Nature of Electromagnetic Waves. I've been trying to work out what the physical nature of electromagnetic waves is, since I reasoned that given electromagnetic waves have wavelengths that are given in distance units, rather than units of energy or some other more abstract/non-physical unit, then electromagnetic waves must have a physical description.. 0. 6 months ago . 6 months ago. This raises the question "How can the disturbance itself move?" the nature of electromagnetic waves DRAFT. 3 \$\begingroup\$ We know that light is an electromagnetic wave. Students can understand this chapter with the help of notes, equations, tips and list of best books provided by the experts at learn,careers360.com What are Electromagnetic Waves - Wave is nothing but a pattern of disturbance which propagates and carry energy with it. Dual Nature of Electromagnetic Radiations(Part-5)- Black Body Radiation- Structure of Atom #11 - … The entire wave shown in Figure 21.1 may be thought of as moving from left to right. Wave … 1 answer. Save. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Electromagnetic waves in free space must be solutions of Maxwell's electromagnetic wave equation. They were predicted by J. C. Maxwell in 1864 and verified experimentally by H. Hertz in … 26 NATURE AND PROPERTIES OF ELECTROMAGNETIC WAVES c02.qxd 1/20/2006 12:57 PM Page 26. polarization and the direction of propagation, and corresponds to the case in which the electric vector is in the plane of incidence. Electromagnetic Waves: Electromagnetic waves are ever-present in the world around us. These forces are thought to be derived from one ultimate force, which is encapsulated in many theories concerning cosmology and the theory of everything. Electromagnetic waves are transverse waves. by smanya9503_84270. The nature of a wave . 14. Because of this EM waves are transverse waves in nature. Gradually other electromagnetic waves were found The wave nature of light causes different colors to be reflected differently by a surface ruled in fine parallel scratches--which is why a compact laser disk (for music or computer use) shimmers in all colors of the rainbow. Assertion : The electromagnetic waves are transverse in nature. Options (a) Interference phenomena (b) Diffraction phenomena (c) Dispersion phenomena (d) Polarisation phenomena. asked Sep 28, 2019 in Physics by Deepak01 (58.6k points) electromagnetic waves; class-12; 0 votes. Lots of waves travel through a material, in which case it is the material of the medium that is being disturbed. In contrast, energy that is transmitted, or radiated, through space in the form of periodic oscillations of electric and magnetic fields is known as electromagnetic radiation. Electromagnetic waves are dealt with more fully in another section of this book. 878 Views. Photon. Electromagnetic waves travel through a vacuum at the speed of light, c = 2.9979 × 10 8 m s –1. Electromagnetic wave theory seems to indicate that there are intrinsic similarities between gravitational waves, light, and the space-time forces behind all of the universes main atomic forces. An electromagnetic wave is a concentration of energy and other properties in space and time; and it's also spread out over a larger region of space and time. The transverse nature of electromagnetic waves is proved by which of the following? So is a radio frequency wave. Powered by Create your own unique website with customizable templates. Be the first to write the explanation for this question by commenting below. Their vibrations or oscillations are changes in electrical and magnetic fields at right angles to the direction of wave travel. Related Questions: When 1 kg of ice … A photon is a quantum (energy packet) of light. The term electric wave, or hertzian wave, is often applied to electromagnetic waves in the radar and radio range. Nature of Electromagnetic Waves. the energy transferred through space by electromagnetic waves . shear waves), locomotion in snakes and eels, stringed instruments, drums: longitudinal waves: parallel to propagation Thus at position P, where the electric field had maximum amplitude at the instant the figure was drawn there is a progressive decrease in amplitude with time. 8th grade. Edit. The electromagnetic (EM) spectrum encompasses all wave frequencies, including radio, visible light and X-rays. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Gravitational waves. Electromagnetic waves may be confined in tubes, such as wave guides, or guided by transmission lines. (pages 71–72) Key Concept: An electromagnetic wave consists of vibrating electric and magnetic fields that move through space at the speed of light. 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Healing Cap Implant, Marks And Spencer Garden Furniture Ireland, White Ledger Stone Backsplash, Glencoe Massacre Map, Vocabulary For Achievement Second Course Pdf, Wool Couture Company Reviews, | 2021-06-20 15:02:48 | {"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": 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": 0.5656354427337646, "perplexity": 875.1402411076592}, "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-2021-25/segments/1623488249738.50/warc/CC-MAIN-20210620144819-20210620174819-00585.warc.gz"} |
https://physics.stackexchange.com/questions/437104/what-exactly-is-voltage-drop-across-a-circuit-element | # What exactly is “voltage drop” across a circuit element?
what exactly is "voltage drop" across a circuit element? I don't quite understand what this means.
• If you found an answer helpful, consider accepting it. This marks the question as cleared up. – ahemmetter Oct 29 '18 at 11:49
If you have a simple circuit with resistors, the voltage at the source decreases throughout the circuit. At the positive terminal of a battery for example, you might have a voltage of 9 V, while the negative terminal can be taken as the ground potential 0V.
The voltage in between those terminals (that means through the circuit) decreases from the positive voltage 9V to the ground potential 0 V.
How does it decrease? We assume that we have perfect conductors as wires, which means that they have the same potential at every point, i.e. at the beginning and the end of the wire, the voltage is equal. The only points in a circuit where the voltage can decrease are the resistors in that case. In front of the resistor is a higher voltage, and behind it a lower voltage. The difference is the voltage drop. It corresponds to the amount of power dissipated in the resistor and is proportional to the resistance.
Here is the schematic of a simple circuit. As you can see, there are three resistors in series and a voltage source of 9 V.
A current of $$0.5 mA$$ flows through the circuit, and it flows through all circuit elements. We can calculate this current by finding the equivalent resistance of the circuit. For a series connection of three resistors, we just add their resistances: $$R = R_1 + R_2 + R_3 = 3 k\Omega + 10 k\Omega + 5 k\Omega = 18 k\Omega$$
From $$U = R I$$ we can find the current flowing through every element: $$I = \frac{U}{R} = \frac{9 V}{18 k\Omega} = 0.5 mA$$
Now we can find how much voltage drops at every element. It is proportional to its resistance. Since we now know the current and the resistance at each element, we can calculate the voltage drop.
For the voltage that falls off at $$R_1$$, we write: $$\Delta U_1 = R_1 I = 3 k\Omega \cdot 0.5 mA = 1.5 V$$
Similarly for the other resistors we get $$5 V$$ for $$R_2$$ and $$2.5 V$$ for $$R_3$$.
In the circuit above we can see how this changes the voltage levels at each part of the circuit. Coming from the positive terminal of the voltage source, the voltage is 9 V. It stays this until it reaches the first resistor. At the front of the resistors (in direction of the current, clockwise), the voltage is 9 V (the red part), but the voltage behind the resistor is lower: it is now only $$9 V - 1.5 V = 7.5 V$$ (the yellow part). The voltage dropped over the resistor. If you'd probe the voltage behind the resistor, you should measure 7.5 V.
Similarly this works for all subsequent resistors: the 10k resistor will drop 5 V, which means that in front of it we have a level of 7.5 V (yellow), and behind it $$7.5 V - 5 V = 2.5 V$$ (green). Finally, the last resistor drops 2.5 V, bringing the voltage at the end down to 0 V (blue).
when electrical current flows through a circuit component (easy example: a resistor), the resistor dissipates power; as it does, the voltage decreases steadily from the high-voltage side to the low-voltage side. this decrease in voltage is called the voltage drop.
In the case of a resistor of resistance R with a voltage difference across it of V volts, the power dissipated is equal to (V^2)/R.
What exactly it means is, how much the opposition the circuit element is doing against the current flow..!!
If the circuit element has no resistance, meaning it doesn't oppose the current flow at all, the voltage drop across that element will be absolute zero. But that's true only for superconductor.
In all normal case all circuit elements will have some resistance and they oppose current flow and thus drops some voltage across them by obeying the Ohm's Law.
There is nothing special about the term "voltage drop" - it is just a voltage across a particular component or any part of a circuit.
It is typically used in the context of a circuit with a known total voltage, in reference to voltages across specific parts of the circuit, especially, when we want to know how the total voltage is distributed between various parts of the circuit.
Let's say, we have a battery that measures exactly $$9$$V. We connect it to a circuit consisting of two components connected in series, say, a resistor and an LED.
A typical question here could be: "What is the voltage drop on the resistor?" or "What is the voltage drop on the diode?", which is the same as asking "What is the voltage across the resistor?" or "What is the voltage across the diode?".
Let's say, we measured those voltages and came up with $$7$$V on the resistor and $$1.97$$V on the LED. We re-measure the voltage on the battery and it is $$8.98$$V. So, we say that there is $$0.02$$V voltage drop on the internal resistance of the battery.
But that leaves $$0.01$$V unaccounted for. So, we say that there must be another $$0.01$$V voltage drop somewhere in the circuit, perhaps, on the long wires or on the contacts.
Voltage is a term which is sometimes confusing because people use it different ways in different contexts. Technically, voltage is the difference in electrical potential between two points. Electric potential is a physics concept which describes how much potential energy will be added or removed from a system if a unit charge is added or removed.
For example, if the electric potential (I like to use the symbol $$\phi$$ for electric potential) at a point is 93 V, then adding a 1 coulomb charge at that point will add 93 J of potential energy to the system. Next, if we move that charge to a point where the potential is 100 V, we have effectively added 100 J but removed 93 J, for a net change of 7 J. The voltage, potential difference, ($$V=\Delta \phi$$) between those points is +7 V. On the other hand, if we started with the charge at $$\phi =$$100 V potential point and moved it to the $$\phi =$$93 V potential point, we would have a voltage, $$V=\Delta\phi=\phi_{final}-\phi_{initial}=-7\text{ V}$$ and a potential energy decrease of 7 J. That potential energy is converted into some other form of energy, depending on the system. This decrease of potential, or negative voltage, is commonly called a voltage drop.
I personally don't like adding the word "drop" to voltage. I simply say voltage, and I always emphasize which end of a circuit element is higher, because if the voltage "drops" going left to right, then it "rises" going right to left. The word voltage by itself is enough to convey a difference of potential. | 2019-07-18 08:52:42 | {"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": 24, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5882914662361145, "perplexity": 273.40358842388406}, "config": {"markdown_headings": true, "markdown_code": false, "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-2019-30/segments/1563195525587.2/warc/CC-MAIN-20190718083839-20190718105839-00044.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-01444560v2 | Abstract : The p-adic Kummer--Leopoldt constant kappa_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n >> 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This constant has been computed by Assim & Nguyen Quang Do using Iwasawa's techniques, after intricate studies and calculations by many authors. We give an elementary p-adic proof and an improvement of these results, then a class field theory interpretation of kappa_K. We give some applications (including generalizations of Kummer's lemma on regular pth cyclotomic fields) and a natural definition of the normalized p-adic regulator for any K and any p≥2. This is done without analytical computations, using only class field theory and especially the properties of the so-called p-torsion group T_K of Abelian p-ramification theory over K.
Keywords :
Type de document :
Pré-publication, Document de travail
To appear in International Journal of Number Theory'' (2018). 2017
Littérature citée [18 références]
https://hal.archives-ouvertes.fr/hal-01444560
Contributeur : Georges Gras <>
Soumis le : vendredi 31 mars 2017 - 14:07:48
Dernière modification le : mardi 4 avril 2017 - 01:02:34
Document(s) archivé(s) le : samedi 1 juillet 2017 - 13:28:23
### Fichiers
Kummer-Leopoldt.HAL.pdf
Fichiers produits par l'(les) auteur(s)
### Identifiants
• HAL Id : hal-01444560, version 2
• ARXIV : 1701.06857
### Citation
Georges Gras. The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator. To appear in International Journal of Number Theory'' (2018). 2017. 〈hal-01444560v2〉
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https://developers.google.com/earth-engine/guides/featureview_styling | FeatureView Styling
Stay organized with collections Save and categorize content based on your preferences.
The style of features in a FeatureView asset are specified using rules defined in a JavaScript object. Style can be set during initial definition of a FeatureViewLayer or anytime after. The styling system allows you to set broad style rules that apply to large groups of features, as well as more specific rules for particular features. Feature style can be defined by constant values or data driven, based on feature characteristics.
Style object
The basic structure of a style object is shown below. There are two types of rules: broad rules and specific rules. Broad rules affect all features in a FeatureView asset, while specific rules affect a subset of features.
{
// Broad style rules.
opacity: …,
polygonFillColor: …,
// Specific style rules.
rules: [
{ … },
{ … }
]
};
To apply style properties to all features (or those of a specific geometry type), specify the style properties at the top level in the style object.
{
opacity: 0.5,
pointShape: 'triangle',
lineWidth: 10,
polygonFillColor: 'green'
};
Specific rules
To apply style properties to a subset of features, use the rules field. The rules field accepts a list of JavaScript objects, each with a filter that selects features according to conditions defined by an ee.Filter object, followed by a series of style properties. In the example below, there is a rule that sets polygonStrokeWidth and polygonFillColor only if the "REP_AREA" property is less than 100. Specific rules override the style properties of broad rules, and rules near the end of the rules list override those near the beginning (specific rules are evaluated first to last).
{
rules: [
{
filter: ee.Filter.lt('REP_AREA', 100),
polygonStrokeWidth: 0.5,
polygonFillColor: 'blue'
},
{ … } // Optionally include additional rules.
]
};
Setting style
Feature style can be set when a FeatureViewLayer is declared or anytime after.
FeatureViewLayer declaration
To set the visualization parameters when declaring a FeatureViewLayer, use the visParams parameter.
var visParams = {
opacity: 0.5,
lineWidth: 10,
polygonFillColor: 'purple'
};
var layer = ui.Map.FeatureViewLayer({
assetId: 'WCMC/WDPA/current/polygons_FeatureView',
visParams: visParams
});
Existing FeatureViewLayer
To set the visualization parameters for an existing FeatureViewLayer, use the setVisParams function. It replaces any previously specified style rules; unspecified properties are set to their default.
var layer = ui.Map.FeatureViewLayer('WCMC/WDPA/current/polygons_FeatureView');
layer.setVisParams({
opacity: 0.5,
lineWidth: 10,
polygonFillColor: 'purple'
});
Symbology
For each style property, you can specify either a constant style rule or a data-driven style rule. The data-driven option uses feature property values to determine symbolization, which can either be categorical or interpolated. For a full list of the style properties, see the style properties table.
Constant
A constant style rule consists of a style property to set and its value. The following example sets the polygon fill color to blue.
var visParams = {
polygonFillColor: 'blue'
};
Categorical
A categorical style rule consists of a style property to set and a JavaScript object with three properties:
• property — a feature property name whose value will affect the style.
• categories — a list of lists that map feature property values to style property symbologies. Each category includes a property value followed by a symbology value to apply. The property value that defines a category must be a string.
• defaultValue — a default symbology to apply to features whose property value is not defined in categories. If it is null/undefined, default style settings will be applied.
For example, the following object sets the color style property based on the "MARINE" feature property. Features in "MARINE" category "0" are set as purple, "1" as green, "2" as blue, and any other category as white.
var visParams = {
color: {
property: 'MARINE',
categories: [
['0', 'purple'],
['1', 'green'],
['2', 'blue']
],
defaultValue: 'white'
}
};
Interpolated
An interpolated style rule consists of a style property to set and a JavaScript object with up to five properties:
• property — a feature property name whose value will affect the style.
• mode — the interpolation mode, either 'linear' or 'interval'.
• palette — a list of colors, opacities, or widths to interpolate input property values between. The format depends on the mode, see the Linear and Interval sections for more details.
Applies only to 'linear' mode
• min — the property value to map to the first element in the palette list.
• max — the property value to map to the last element in the palette list.
Linear
Linear interpolation mode sets a feature style property by mapping input values in the range min to max linearly between a list of symbology values defined in the palette property. The input values are clamped to the range set by min and max.
For example, the following object sets the color style property based on the "REP_AREA" feature property. The palette property is a list of colors indicating that input values between min and max should grade linearly from yellow to red to blue. A value between 1 and 500 is interpolated between yellow and red, and a value between 500 and 1000 is interpolated between red and blue.
var visParams = {
color: {
property: 'REP_AREA',
mode: 'linear',
palette: ['yellow', 'red', 'blue'],
min: 1,
max: 1000
}
};
Interval
Interval interpolation mode sets a feature style property by mapping input values to class breaks and then applying a class-specific symbology. Input values from the selected feature property are assigned to the nearest class break value by rounding down. The palette property is formatted as a list of lists, where each inner list contains a class break value followed by a style property value. Features whose property value are less than the minimum class break value maintain their default style property setting.
In the following example, feature fill opacity is set according to graduated classes of the "REP_AREA" property. Class definition and style symbology are provided in the palette property as a list of lists. It indicates that there should be 4 classes with breaks at value 0, 80, 2000, and 5000, with respective feature opacities of 0.5, 0.35, 0.22, and 0.15. In other words, features with "REP_AREA" values in the interval $0 \le x < 80$ will have a fill opacity of 0.5, values in the interval $80 \le x < 2000$ will have fill opacity of 0.35, and so on.
var visParams = {
fillOpacity: {
property: 'REP_AREA',
mode: 'interval',
palette: [
[0, 0.5],
[80, 0.35],
[2000, 0.22],
[5000, 0.15]
]
}
};
All style properties
Below are all of the style properties you can specify in the style object. Setting style properties for specific geometry types overrides the corresponding style properties set for "All geometries" (for example, setting polygonFillColor overrides the value set in fillColor).
Property Type Description Supports Interpolated Rule
All geometries
isVisible Boolean Sets whether the feature is visible. No
color String Sets fill/stroke color for all geometry types. Must be a hex value or a CSS3 color. Yes
opacity Double Sets fill/stroke opacity for all geometry types. Must be a double between 0 and 1. Yes
width Double Sets stroke width for all geometry types. Yes
fillColor String Sets fill color for all geometry types. Must be a hex value or a CSS3 color. Yes
Point geometries
pointShape String Sets the shape of point geometries. Supports the same shapes as ee.FeatureCollection.style (circle, square, diamond, cross, plus, pentagram, hexagram, triangle, triangle_up, triangle_down, triangle_left, triangle_right, pentagon, hexagon, star5, star6). No
pointSize Double Sets the width of point geometries (in px). Yes
pointFillColor String Sets fill color for point geometries. Must be a hex value or a CSS3 color. Yes
pointFillOpacity Double Sets fill opacity for point geometries. Must be a double between 0 and 1. Yes
Line geometries
lineType String Sets the line type. Supports the same types as ee.FeatureCollection.style (solid, dashed, dotted). No
lineWidth Double Sets the line width (in px). Yes
lineColor String Sets color for line geometries. Must be a hex value or a CSS3 color. Yes
lineOpacity Double Sets opacity for line geometries. Must be a double between 0 and 1. Yes
Polygon geometries
polygonStrokeWidth Double Sets the stroke width of polygons (in px). Yes
polygonStrokeType String Sets the line type for polygons. Supports the same types as ee.FeatureCollection.style (solid, dashed, dotted). No
polygonStrokeColor String Sets stroke color for polygon geometries. Must be a hex value or a CSS3 color. Yes
polygonStrokeOpacity Double Sets stroke opacity for polygon geometries. Must be a double between 0 and 1. Yes
polygonFillColor String Sets fill color for polygon geometries. Must be a hex value or a CSS3 color. Yes
polygonFillOpacity Double Sets fill opacity for polygon geometries. Must be a double between 0 and 1. Yes
[{ "type": "thumb-down", "id": "missingTheInformationINeed", "label":"Missing the information I need" },{ "type": "thumb-down", "id": "tooComplicatedTooManySteps", "label":"Too complicated / too many steps" },{ "type": "thumb-down", "id": "outOfDate", "label":"Out of date" },{ "type": "thumb-down", "id": "samplesCodeIssue", "label":"Samples / code issue" },{ "type": "thumb-down", "id": "otherDown", "label":"Other" }]
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https://www.safaribooksonline.com/library/view/discrete-structures-logic/9781284070408/ch11_answers.html | ## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.
No credit card required
### Chapter 11
#### Section 11.1
1. a. {a, b}.
c. {a, Λ, b, bb, . . ., bn, . . . }.
e. {a, b, ab, bc, abb, bcc, ..., abn, bcn, . . . }.
2. a. a + b + c. c. ab* + ba*. e. Λ + a(bb)*. g. Λ + c*a + bc*. i. a*bc*.
3. 0 + 1(0 + 1)*.
4. a. (aa + ab + ba + bb)*. c. (a + b)*aba(a + b)*.
5. a. (ab)*. c. a (a + b)*.
6. a.
$\begin{array}{ll}b+a{b}^{*}+a{a}^{*}b+a{a}^{*}a{b}^{*}& =b+a{b}^{*}+a{a}^{*}\left(b+a{b}^{*}\right)\\ =\left(\mathrm{\Lambda }+a{a}^{*}\right)\left(b+a{b}^{*}\right)\\ ={a}^{*}\left(b+a{b}^{*}\right)\phantom{0000000000}\left(\mathrm{by}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(11.1.1\mathrm{e}\right)\end{array}$
c. By using property 7 of (11.1.1g) the subexpression (a + bb*a)* of the left side am be written (a*bb*a)*a*. So the left expression has the following form:
ab*a(a + bb*a)*b = ab*a(a*bb*a)*a*b.
Similarly, the subexpression (b + aa*b)* of the right side of the original equation can be written as b*(aa*bb*)*. So the right expression has the following form:
a(b + ...
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No credit card required | 2018-08-16 01:25:27 | {"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": 0.24587446451187134, "perplexity": 8820.689345187864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "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-2018-34/segments/1534221210387.7/warc/CC-MAIN-20180815235729-20180816015729-00191.warc.gz"} |
https://www.researcher-app.com/paper/148514 | 3 years ago
# Excited states of two-dimensional solitons supported by the spin-orbit coupling and field-induced dipole-dipole repulsion.
Shimei Liu, Chunqing Huang, Wei Pang, Yongyao Li, B. A. Malomed, Hexiang He, Yuebo Ye
It was recently found that excited states of semi-vortex and mixed-mode solitons (ESVs and EMMs) are unstable in spin-orbit-coupled Bose-Einstein condensates (BECs) with contact interactions. We demonstrate a possibility to stabilize such excited states in a setting based on repulsive dipole-dipole interactions induced by a polarizing field, oriented perpendicular to the plane in which the dipolar BEC is trapped. The strength of the field is assumed to grow in the radial direction $\sim$ $r^{4}$. ESVs have vorticities $S$ and $S+1$ in their two components, each being an eigenstate of the angular momentum. The ESVs are fully stable up to $S=5$. EMMs feature interweaving necklace structures with opposite fractional values of the angular momentum in the two components. They are stable if they are built of dominant angular harmonics $\pm S$, with $S\leq 4$. Characteristics and stability of these two types of previously unknown higher-order solitons are systematically analyzed.
Publisher URL: http://arxiv.org/abs/1711.02874
DOI: arXiv:1711.02874v1
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Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.
Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article. | 2022-09-25 10:16:40 | {"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": 0.3486133813858032, "perplexity": 2820.5243564095463}, "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-2022-40/segments/1664030334528.24/warc/CC-MAIN-20220925101046-20220925131046-00415.warc.gz"} |
http://hal.in2p3.fr/in2p3-00382021 | # Growth of Long Range Forward-Backward Multiplicity Correlations with Centrality in Au+Au Collisions at $\sqrt{s_{NN}}$ = 200 GeV
Abstract : Forward-backward multiplicity correlation strengths have been measured for the first time with the STAR detector for Au+Au and $\textit{p+p}$ collisions at $\sqrt{s_{NN}}$ = 200 GeV. Strong short and long range correlations are seen in central (0-10%) Au+Au collisions. The magnitude of these correlations decrease with decreasing centrality until only short range correlations are observed in 40-50% Au+Au collisions. The results are in agreement with predictions from the Dual Parton and Color Glass Condensate models.
Document type :
Journal articles
http://hal.in2p3.fr/in2p3-00382021
Contributor : Dominique Girod <>
Submitted on : Thursday, May 7, 2009 - 10:11:30 AM
Last modification on : Sunday, January 3, 2021 - 8:58:01 AM
### Citation
B.I. Abelev, M. M. Aggarwal, Z. Ahammed, B. D. Anderson, D. Arkhipkin, et al.. Growth of Long Range Forward-Backward Multiplicity Correlations with Centrality in Au+Au Collisions at $\sqrt{s_{NN}}$ = 200 GeV. Physical Review Letters, American Physical Society, 2009, 103, pp.172301-6. ⟨10.1103/PhysRevLett.103.172301⟩. ⟨in2p3-00382021⟩
Record views | 2021-01-15 15:43:44 | {"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": 0.6704813241958618, "perplexity": 7226.007338986516}, "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-2021-04/segments/1610703495901.0/warc/CC-MAIN-20210115134101-20210115164101-00616.warc.gz"} |
https://de.maplesoft.com/support/help/maple/view.aspx?path=Finance%2FGetSize | Finance - Maple Programming Help
Home : Support : Online Help : Mathematics : Finance : Lattice Methods : Finance/GetSize
Finance
GetSize
get the size of a binomial/trinomial tree at the given level
Calling Sequence GetSize(T, i)
Parameters
T - binomial or trinomial tree data structure i - positive integer; level
Description
• The GetSize command returns the number of nodes at level i of the tree T. See GetDescendants for more details about node indexing.
Examples
> $\mathrm{with}\left(\mathrm{Finance}\right):$
> $M≔\mathrm{VasicekModel}\left(0.05,0.03,0.5,0.03\right)$
${M}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (1)
> $T≔\mathrm{ShortRateTree}\left(M,3,20\right)$
${T}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (2)
> $\mathrm{TreePlot}\left(T,\mathrm{thickness}=2,\mathrm{axes}=\mathrm{BOXED},\mathrm{gridlines}=\mathrm{true}\right)$
> $\mathrm{GetSize}\left(T,1\right)$
${1}$ (3)
> $\mathrm{GetSize}\left(T,2\right)$
${3}$ (4)
> $\mathrm{GetSize}\left(T,3\right)$
${5}$ (5)
> $\mathrm{GetSize}\left(T,10\right)$
${15}$ (6)
> $\mathrm{GetSize}\left(T,11\right)$
${15}$ (7)
Compatibility
• The Finance[GetSize] command was introduced in Maple 15. | 2021-01-21 05:48:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 16, "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": 0.9407531023025513, "perplexity": 2961.7613082910552}, "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-2021-04/segments/1610703522242.73/warc/CC-MAIN-20210121035242-20210121065242-00618.warc.gz"} |
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Book%3A_Semiconductor_Devices_-_Theory_and_Application_(Fiore)/09%3A_BJT_Class_B_Power_Amplifiers/9.5%3A_Exercises | 9.5: Exercises
9.5.1: Analysis Problems
1. For the circuit of Figure $$\PageIndex{1}$$, determine compliance, $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$. $$V_{CC}$$ = 15 V, $$V_{EE}$$ = −15 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 680 $$\Omega$$, $$R_2$$ = 680 $$\Omega$$.
2. For the circuit of Figure $$\PageIndex{1}$$, determine $$Z_{in}$$. $$V_{CC}$$ = 15 V, $$V_{EE}$$ = −15 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 680 $$\Omega$$, $$R_2$$ = 680 $$\Omega$$.
3. For the circuit of Figure $$\PageIndex{1}$$, determine $$Z_{in}$$. $$V_{CC}$$ = 25 V, $$V_{EE}$$ = −25 V, $$\beta$$ = 70, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 560 $$\Omega$$, $$R_2$$ = 560 $$\Omega$$.
Figure $$\PageIndex{1}$$
4. For the circuit of Figure $$\PageIndex{1}$$, determine compliance $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$. $$V_{CC}$$ = 25 V, $$V_{EE}$$ = −25 V, $$\beta$$ = 70, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 560 $$\Omega$$, $$R_2$$ = 560 $$\Omega$$.
5. For the circuit of Figure $$\PageIndex{2}$$, determine $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$. $$V_{CC}$$ = 15 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 630 $$\Omega$$, $$R_2$$ = 630 $$\Omega$$.
Figure $$\PageIndex{2}$$
6. For the circuit of Figure $$\PageIndex{2}$$, determine $$Z_{in}$$. $$V_{CC}$$ = 15 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 630 $$\Omega$$, $$R_2$$ = 630 $$\Omega$$.
7. For the circuit of Figure $$\PageIndex{2}$$, determine $$Z_{in}$$. $$V_{CC}$$ = 25 V, $$\beta$$ = 70, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 560 $$\Omega$$, $$R_2$$ = 560 $$\Omega$$.
8. For the circuit of Figure $$\PageIndex{2}$$, determine $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$. $$V_{CC}$$ = 25 V, $$\beta$$ = 70, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 510 $$\Omega$$, $$R_2$$ = 510 $$\Omega$$.
9. For the circuit of Figure $$\PageIndex{3}$$, determine $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$ for the output transistors. $$V_{CC}$$ = 24 V, $$V_{EE}$$ = −24 V, $$\beta$$ = 75, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 2.5 k$$\Omega$$, $$R_2$$ = 300 $$\Omega$$, $$R_3$$ = 330 $$\Omega$$, $$R_4$$ = 63 $$\Omega$$.
Figure $$\PageIndex{3}$$
10. For the circuit of Figure $$\PageIndex{3}$$, determine $$A_v$$ and $$Z_{in}$$. $$V_{CC}$$ = 24 V, $$V_{EE}$$ = −24 V, $$\beta$$ = 75, $$R_L$$ = 8 $$\Omega$$, $$R_1$$ = 2.5 k$$\Omega$$, $$R_2$$ = 300 $$\Omega$$, $$R_3$$ = 330 $$\Omega$$, $$R_4$$ = 63 $$\Omega$$.
Figure $$\PageIndex{4}$$
11. For the circuit of Figure $$\PageIndex{4}$$, determine $$P_{load(max)}$$, $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$ for the output transistors. $$V_{CC}$$ = 24 V, $$V_{EE}$$ = −24 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 600 $$\Omega$$, $$R_2$$ = 5 k$$\Omega$$, $$R_3$$ = 63 $$\Omega$$, $$R_4$$ = 330 $$\Omega$$.
12. For the circuit of Figure $$\PageIndex{4}$$, determine $$A_v$$ and $$Z_{in}$$. $$V_{CC}$$ = 24 V, $$V_{EE}$$ = −24 V, $$\beta$$ = 75, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 600 $$\Omega$$, $$R_2$$ = 5 k$$\Omega$$, $$R_3$$ = 63 $$\Omega$$, $$R_4$$ = 330 $$\Omega$$.
13. Determine the limit current for the circuit of Figure $$\PageIndex{5}$$ if $$R_E$$ = 0.2 $$\Omega$$.
Figure $$\PageIndex{5}$$
14. Determine $$P_{load(max)}$$, and $$P_{D(max)}$$, $$BV_{CEO}$$ and $$I_{C(max)}$$ for the output and driver transistors of Figure $$\PageIndex{6}$$. $$V_{CC}$$ = 50 V, $$V_{EE}$$ = −50 V, $$\beta$$ = 75, $$R_L$$ = 8 $$\Omega$$, $$R_5$$ through $$R_8$$ = 0.05 $$\Omega$$. Assume all other components produce proper bias.
9.5.2: Design Problems
15. For the circuit of Figure $$\PageIndex{3}$$, determine values for $$R_1$$ and $$R_2$$ for proper bias. $$V_{CC}$$ = 32 V, $$V_{EE}$$ = −32 V, $$\beta$$ = 75, $$R_L$$ = 8 $$\Omega$$, $$R_3$$ = 330 $$\Omega$$, $$R_4$$ = 63 $$\Omega$$.
16. Determine a value for $$R_E$$ to set the limit current for the circuit of Figure $$\PageIndex{5}$$ to 2 A.
9.5.3: Challenge Problems
17. For the circuit of Figure $$\PageIndex{6}$$, determine values for $$R_1$$ and $$R_2$$ for proper bias. $$V_{CC}$$ = 50 V, $$V_{EE}$$ = −50 V, $$\beta$$ = 85, $$R_L$$ = 8 $$\Omega$$, $$R_5$$ through $$R_8$$ = 0.05 $$\Omega$$, $$R_3$$ = 2.2 k$$\Omega$$, $$R_4$$ = 330 $$\Omega$$.
18. For the circuit of Figure $$\PageIndex{7}$$, determine a value for $$R_5$$ for proper bias. $$V_{CC}$$ = 30 V, $$V_{EE}$$ = −30 V, $$\beta$$ = 100, $$R_L$$ = 16 $$\Omega$$, $$R_1$$ = 2.2 k$$\Omega$$, $$R_2$$ = 8.2 k$$\Omega$$, $$R_3$$ = 1.2 k$$\Omega$$, $$R_4$$ = 47 $$\Omega$$, $$R_5$$ = 330 $$\Omega$$, $$R_6$$ = 470 $$\Omega$$, $$R_7$$ = 68 $$\Omega$$.
Figure $$\PageIndex{6}$$
9.5.4: Computer Simulation Problems
19. Perform a transient analysis on the circuit of Problem 1 to verify the compliance.
20. Perform a transient analysis on the circuit of Problem 4 to verify the compliance.
21. Perform a DC analysis on the design from Problem 15 to verify the results.
22. Perform a DC analysis on the design from Problem 17 to verify the results.
Figure $$\PageIndex{7}$$
This page titled 9.5: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James M. Fiore via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. | 2023-02-06 12:37:53 | {"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": 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": 0.9094340205192566, "perplexity": 468.3652616049413}, "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-2023-06/segments/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00023.warc.gz"} |
https://docs.infrascale.com/dr/cfa/management-console/settings/tools/appliance-network-share | › Backup & Disaster Recovery › Appliance Management Console › Settings tab › Appliance Network Share
# Appliance Network Share
## Overview
Appliance network share is a local folder on the Cloud Failover Appliance (CFA), which allows you to keep files and folders restored from backups. Restoring to this folder is much faster compared with restoring to other destinations.
Size of the appliance network share is not predefined, but varies depending on the free space on the CFA.
You can see the free space in the lower-left part of the Management Console, on the RAID bar.
Here, you can find the list of folders stored in the appliance network share, how much space they take, and when they were created, that is restored. Also, you can remove folders completely to free up space on the CFA.
For security reasons, the appliance network share does not allow accessing it anonymously. Instead, the system provides the pre-generated credentials (username and password). Also, you can generate new credentials, if necessary.
The Appliance Network Share shows information in the table format with the following columns:
Column Description
Folder Name Name of the restored folder
Network Share Path Network path to the restored folder
## Access appliance network share
You can use the tools built in the operating system or the appropriate third-party SMB client to browse and manage the restored files and folders.
You may have to provide user credentials to access the appliance network share. These are the same used to log in to the CFA Management Console.
For example, in Windows, open your Internet browser or File Explorer, enter \\ip_address\restored_jobs, where ip_address is the IP address of the CFA, and enter the network share credentials when prompted.
In macOS, open Finder, click GoConnect to Server, enter smb://ip_address/restored_jobs, where ip_address is the IP address of the CFA, and then enter the network share credentials when prompted.
## Restore to the appliance network share
When browsing and restoring files and folders from a client or from a backup job, select Appliance Network Share as the destination point. | 2022-10-07 16:32:42 | {"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": 0.39627158641815186, "perplexity": 5196.2358175247555}, "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-2022-40/segments/1664030338213.55/warc/CC-MAIN-20221007143842-20221007173842-00623.warc.gz"} |
https://texblog.net/latex-archive/uncategorized/changing-margins-paragraph/ | Changing margins for just one paragraph
If you want to indent a paragraph just by a certain length but the standard LaTeX environments don’t meet the requirements you could use the TeX primitive \leftskip. To limit its effect it can be enclosed in \begingroup … \endgroup. The equivalent for the right margin is \rightskip. Here’s an example:
\documentclass[a4paper,10pt]{article} \usepackage[english]{babel} \usepackage{blindtext} \parindent0em \parskip\baselineskip \begin{document} \blindtext \par \begingroup \leftskip4em \rightskip\leftskip \blindtext \par \endgroup \blindtext \end{document}
A downscaled screenshot of the output:
This topic was discussed on Matheplanet and ChemieOnline.
02. May 2008 by stefan
Categories: Uncategorized | 3 comments
1. Thanks dear. It has made life certainly easy.
2. Thx a lot Stefan Kottwitz……
3. try this | 2020-05-30 01:11:35 | {"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": 0.9949254989624023, "perplexity": 13798.95587664793}, "config": {"markdown_headings": false, "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-2020-24/segments/1590347407001.36/warc/CC-MAIN-20200530005804-20200530035804-00229.warc.gz"} |
http://codeforces.com/blog/entry/107461 | ### awoo's blog
By awoo, history, 2 months ago, translation,
1739A - Immobile Knight
Idea: BledDest
Tutorial
Solution 1 (awoo)
Solution 2 (awoo)
1739B - Array Recovery
Idea: BledDest
Tutorial
Solution (Neon)
1739C - Card Game
Idea: BledDest
Tutorial
Solution 1 (BledDest)
Solution 2 (BledDest)
1739D - Reset K Edges
Idea: BledDest
Tutorial
Solution (awoo)
1739E - Cleaning Robot
Idea: BledDest
Tutorial
Solution (awoo)
1739F - Keyboard Design
Idea: BledDest
Tutorial
Solution (BledDest)
• +59
» 2 months ago, # | 0 can anyone help me in D 174113824
• » » 2 months ago, # ^ | 0 I need help too; (●'◡'-)
• » » 2 months ago, # ^ | 0 Take a look at Ticket 16232 from CF Stress for a counter example.
» 2 months ago, # | +2 Can someone suggest problems similar to Div2 C: Card Game, which requires DP, Combinatorics and some thinking. Thanks!
• » » 4 weeks ago, # ^ | 0 sum of product 2 ....codechef
• » » » 4 weeks ago, # ^ | 0 Thanks will look into it
• » » » » 4 weeks ago, # ^ | 0 btw this question can be done in O(n) , with some combinatorics and some thinking..wont require dp..
» 2 months ago, # | ← Rev. 4 → +10 For those who just want to see the recurrence used in solution for C:$dp(n, p) = {n-1 \choose \frac{n}{2}}+ dp(n-2, p^{-1})$ Let $p$ represent some player, and $p^{-1}$ represent the opponent of that player.
• » » 2 months ago, # ^ | 0 What is the analytical base case of this recurrence ?
• » » » 2 months ago, # ^ | 0 If a Alice has got the biggest card, he will win. Otherwise If the opponent will lose in the round with $n$ cards, the player will win in the round with $n + 2$ cards. Otherwise : he will lose.
• » » 3 days ago, # ^ | 0 W Observation. I did this
» 2 months ago, # | +28 When does the robot break? Let the robot currently be in the cell (j,i) (0-indexed) and the next column with a dirty cell be nxti (possibly, nxti=i). The robot breaks only if both (1−j,nxti) and (j,nxti) are dirty.I think there is a typo. break condition should be (1-j,nxti) and (j,nxti + 1)
» 2 months ago, # | ← Rev. 3 → 0 My solution to Prblem C in Div 2.This is very interesting.We can find that: if Alice has the biggest number, he will win. Because he can use this card first. Otherwise Boris has the biggest number, it's obvious that Boris will be "stronger" in the first turn, and he get first in the secound turn. So the number of ways that Alice wins is equal to the number of ways that Boris wins with $n - 2$ cards. So we have $fst_i = \frac{C_{i}^{i/2}}{2} + lst_{i - 2}$.There is another key to the task : the number of way to make a draw is equal to $1$. So $lst_i = C_{i}^{i/2} - fst_i - 1$.My submission : 174191151
• » » 2 months ago, # ^ | ← Rev. 2 → 0 _ So the number of ways that Alice wins is equal to the number of ways that Boris wins with n−2 cards_ i dont understand this part, can you explain how you get to that observaton ?
• » » » 2 months ago, # ^ | 0 Because in the first round, Boris is Stronger and plays cards first in the second round. (This is because Boris has got the biggest card). If Boris loses the round with $n - 2$ card, then Alise will win in the round with $n$ cards. For example.Alice : $2, 3$Boris : $1, 4$Alice uses card $2$ and Boris uses card $4$, then Alice has card $3$ and Boris has card $1$. in this case ($2$ cards in total) Alice will win in the game ($4$ cards in total).
• » » » » 2 months ago, # ^ | ← Rev. 3 → 0 I think i slightly get your idea, but waittt isnt the example you gave should be a draw
» 2 months ago, # | 0 In D, why does this greedy solution starting from the root and cutting whenever the depth exceeds the candidate does not work? TIA
• » » 2 months ago, # ^ | +1 Consider the following inputInput: 1 7 1 1 1 3 2 5 5 Output: 2 If you cut the tree whenever depth exceeds 2, you will require 2 operations, while the min. operations to make depth 2 is actually 1. Just make the tree and see why this happening :)
• » » » 2 months ago, # ^ | 0 Thank you! I should have observed that removing a complete sub tree at (h-1) is better than removing each of its children at (h).
» 2 months ago, # | 0 Why are the constraints so low for C?I wrote a $O(n)$ solution which should easily work for $\Sigma n \geq 10^6$ The solution
» 2 months ago, # | 0 Can anyone tell why this is giving WA? https://codeforces.com/contest/1739/submission/174471060
• » » 2 months ago, # ^ | 0 Take a look at Ticket 16237 from CF Stress for a counter example.
» 2 months ago, # | 0 Someone please help with my solution to problem E:https://codeforces.com/contest/1739/submission/174503555idea: dp[i][j][k] ; 0<=i<=1, 0<=j<=n-1, 0<=k<=2dp[i][j][0] = num cells to clean manually in [0-1][j..n-1] rectangle if we are currently at clean (i,j)dp[i][j][1] = num cells to clean manually in [0-1][j..n-1] rectangle if we are currently at clean (i,j), and cell (1-i,j) is already cleandp[i][j][2] = num cells to clean manually in [0-1][j..n-1] rectangle if we are currently at clean (i,j), and cells (1-i,j) and (1-i,j+1) are already cleanI store the index of next clean cell in row i in k00,k01,k02; and of row 1-i in k10,k11,k12
• » » 2 months ago, # ^ | +3 Take a look at Ticket 16235 from CF Stress for a counter example.
• » » » 2 months ago, # ^ | 0 Thank you so much! Can you please clarify on the point below as well though?
• » » » » 2 months ago, # ^ | +3 Your statement looks reasonable to me. Probably a typo in the editorial.
» 2 months ago, # | 0 For problem E:" When does the robot break? Let the robot currently be in the cell $(j,i)$ (0-indexed) and the next column with a dirty cell be $nxt_i$ (possibly, $nxt_i=i$). The robot breaks only if both $(1−j,nxt_i)$ and $(j,nxt_i)$ are dirty "Shouldn't it break when $(1-j,nxt_i)$ and $(j,nxt_i+1 )$ are both dirty, and $( j,nxt_i )$ is clean ?
• » » 2 months ago, # ^ | 0 yep, that's a mistake in the editorial
» 2 months ago, # | 0 BledDest!Aho-Corasick is new algo for me and and read about this in cp-algorithms. They said if you need to find all strings from a given set in text, you need to use "exit links" to make code faster: storing the nearest leaf vertex that is reachable using suffix links (this is sometimes called the exit link).Link As I see, you don't use this links or ncost array does this job?
• » » 2 months ago, # ^ | +3 Yeah, ncost does the trick. Usually the exit links are helpful to traverse the whole path to the root along the suffix links without actually visiting non-terminal states of the Aho-Corasick (for example, this allows to find all occurrences of all strings we have to search for in $O(Ans)$); but in this problem, we are not interested in the actual occurrences themselves, only in their total cost, so precalculating it for each state is both easier and faster.
» 2 months ago, # | 0 Can somebody explain problem E why it is not always optimal when the robot is at j-th row i-th column and $(1 - j, i)$ is dirty, $(j, i + 1)$ is clean, and we just always go to clean $(1 - j, i)$? I have been stuck by this for a long time but could not find any counter case :/
• » » 2 months ago, # ^ | +13 See this case: 8 00100110 10001101 You can see that if you clean the cell in (1, 3) (1-indexed, (row, col)) you will avoid doing some forced cleaning twice near the end.If you are doing DP, probably you just need to add one more transition to take care of this.
• » » » 2 months ago, # ^ | +3 Thanks you! It gets AC after I add this transition!
» 2 months ago, # | 0 can anyone tell what is the turn variable used for in Div2 C: Card Game solution 1
» 7 weeks ago, # | 0 Can't figure out my mistake in problem D.https://codeforces.com/contest/1739/submission/176283387
• » » 7 weeks ago, # ^ | +3 Take a look at Ticket 16293 from CF Stress for a counter example.
• » » » 7 weeks ago, # ^ | 0 Thanks!
» 6 weeks ago, # | 0 why my code for problem D always gives WA ? any help please .
• » » 6 weeks ago, # ^ | +3 Take a look at Ticket 16324 from CF Stress for a counter example.
• » » » 6 weeks ago, # ^ | 0 Thanks a lot. it helps me to get my code accepted now. the problem is to clear visit and level arrays after each step from the binary search.
» 2 weeks ago, # | ← Rev. 2 → 0 BledDest Problem C-Card Game (Did you converted combination recurrence relation of solution 2 to bottom up dp in solution 1. How did you exactly come up with Solution 1.)'we can use dynamic programming of the form dp x,y,t, where x is the number of characters A we used, y is the number of characters B we used'Please could you give some recurrence relation to understand DPone more thing to notice it is mentioned t=0 means draw but t=2 is draw in Solution1.t is 0, 1 or 2 depending on whether our string coincides with the draw string (t=0),
» 13 days ago, # | 0 Reverse everything and make a problem and throw at our face. | 2022-12-05 20:15:48 | {"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": 0.28635066747665405, "perplexity": 1926.4784846658085}, "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/1669446711045.18/warc/CC-MAIN-20221205200634-20221205230634-00619.warc.gz"} |
http://docmadhattan.fieldofscience.com/2017/11/ | ### Paolo Nespoli reads Emily Dickinson
#1695
There is a solitude of space
A solitude of sea
A solitude of death, but these
Society shall be
Compared with that profounder site
That polar privacy
Finite Infinity.
### Sputnik-2 or: Laika, Our Hero
Laika was a Soviet space dog who became one of the first animals in space, and the first animal to orbit the Earth. Laika, a stray dog from the streets of Moscow, was selected to be the occupant of the Soviet spacecraft Sputnik 2 that was launched into outer space on November 3, 1957.
Little was known about the impact of spaceflight on living creatures at the time of Laika's mission, and the technology to de-orbit had not yet been developed, so Laika's survival was never expected. Some scientists believed humans would be unable to survive the launch or the conditions of outer space, so engineers viewed flights by animals as a necessary precursor to human missions. The experiment aimed to prove that a living passenger could survive being launched into orbit and endure a Micro-g environment, paving the way for human spaceflight and providing scientists with some of the first data on how living organisms react to spaceflight environments.
Laika died within hours from overheating, possibly caused by a failure of the central R-7 sustainer to separate from the payload. The true cause and time of her death were not made public until 2002; instead, it was widely reported that she died when her oxygen ran out on day six or, as the Soviet government initially claimed, she was euthanised prior to oxygen depletion.
On April 11, 2008, Russian officials unveiled a monument to Laika. A small monument in her honour was built near the military research facility in Moscow that prepared Laika's flight to space. It features a dog standing on top of a rocket. She also appears on the Monument to the Conquerors of Space in Moscow.
video via Popular Science
### Abstract: The Universe at Lattice-Fields
Guido, G. and Filippelli, G. (2017) The Universe at Lattice-Fields. Journal of High Energy Physics, Gravitation and Cosmology, 3, 828-860. doi:10.4236/jhepgc.2017.34060.
We formulate the idea of a Universe crossing different evolving phases $U_k^*$ where in each phase one can define a basic field at lattice structure $U_k$ increasing in mass (Universe-lattice). The mass creation in $U_k$ has a double consequence for the equivalence "mass-space": Increasing gravity (with varying metric) and increasing space (expansion). We demonstrate that each phase is at variable metric beginning by open metric and to follow a flat metric and after closed. Then we define the lattice-field of intersection between two lattice fields of base into universe and we analyse the universe in the Nucleo-synthesis phase and in the that of recombination. We show that the phase is built on the intersection of the lattices of the proton and electron. We show $U_H$ [the intersection between proton's anch electron's lattices] to be at variable metric (open in the past, flat in the present and closed in the future). Then, we explain some fundamental aspects of this universe $U_H$: Hubble's law by creating the mass-space in it, its age (13.82 million of Years) as time for reaching the flat metric phase and the value of critic density. In last we talk about dark universe lattice, having hadronic nature, and calculating its spatial step and its density in present phase of [the universe].
For some personal problems, I cannot add the LaTeX figures, so I uploaded them on researchgate. | 2022-07-04 06:03:30 | {"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": 0.2611657679080963, "perplexity": 2894.2046570252005}, "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/1656104354651.73/warc/CC-MAIN-20220704050055-20220704080055-00462.warc.gz"} |
http://blog.wolfram.com/2012/10/23/calculating-the-energy-between-two-cubes/ | # Calculating the Energy between Two Cubes
October 23, 2012 — Michael Trott, Chief Scientist
In my last blog post, we discussed 3D charge configurations that have sharp edges. Reader Rich Heart commented on it and asked whether Mathematica can calculate the force between two charged cubes, as done by Bengt Fornberg and Nick Hale and in the appendix of Lloyd N. Trefethen’s book chapter.
The answer to the question from the post is: Yes, we can; I mean, yes, Mathematica can.
Actually, it is quite straightforward to treat a more general problem than two just-touching cubes of equal size:
• We can deal with two cubes of different edge lengths L1 L2
• We can calculate the force for any separation X, where X is the distance between the two cube centers (including the case of penetrating cubes; think plasma)
• We will use a method that can be generalized to higher-dimensional cubes without having to do more nested integrals
Instead of calculating the force between the two cubes, we will calculate the total electrostatic energy of the system of the two cubes. The force is then simply the negative gradient of the total energy with respect to X. The electrostatic energy (in appropriate units) is given by:
(In the following calculations, we will skip the constant [with respect to X] prefactors Q1 L1-3 Q2 L2-3 or Q1 Q2 if not needed.)
Approaching this integral head-on doing one integral after another is possible, but a very tedious and time-consuming operation. Instead, to avoid having to carry out a nested six-dimensional integral, we remember the Laplace transform of 1 / √s.
Meaning that, modulo a numeric factor, the inverse of the square root function is self-reciprocal with respect to the Laplace transform.
Using this integral transform for the term ((x1 – (Xx2))2 + (y1y2)2 + (z1z2)2)½ changes the nested six-dimensional integral into three factors of double integrals. (Only one of the double integrals depends on X; this means the generalization to two four-dimensional cubes is straightforward at this point.) We calculate the function Vs that has to be integrated over s to obtain the full interaction energy. The three double integrals can all be carried out in closed form, and only a one-dimensional integral over s remains to be carried out.
Now we have to carry out the remaining integral over s. The forms of the last expressions first suggest a change of variable sq = s½ to get a nicer-looking integrand.
The factor q-5 in the last expression suggests we carry out partial integrations, so we define a rule to carry out the partial integrations and apply this rule recursively to remove all higher negative powers of q. (We skip the fully integrated parts of the partial integration because they all vanish.)
And we define some functions to apply this rule to sums and products with numeric factors.
The resulting expression now has more terms, but a simpler structure. A single term in Vq will result in many more terms after partial integration. This process removes all terms containing q-5, q-4, q-3, and q-2 and only keeps terms containing q-1. Here is an example of partially integrating out one summand out of the 264 summands.
Carrying out the partial integrations increases the size of our q-integrands from 264 to 1,196.
Here are three of them randomly selected.
A quick inspection of the terms shows that there are only four different types with respect to their q-dependence, with at most a q-1 factor. As our goal is to carry out the remaining integral over q, we define a quick pattern-matched integration function for the four types of integrands that occur in Vq2List (this will be much faster than using Integrate). One is just a Gaussian integral, and the other three integrals contain error functions multiplied by a Gaussian.
Now we have a closed form for the potential. We carry out a quick numerical check of the integration by comparing the integration with a numerically carried out integral over q:
Now we can easily calculate the resulting force. The result is relatively large. Inspecting the explicit form shows terms containing arcsinh and terms containing arctan. We simplify these terms individually:
The resulting expression is now smaller by more than a factor of 10.
It is still large, but manageable. Here are some of the 68 summands of V2 shown.
Here are two plots of the total interaction energy and the force between the two cubes as a function of the edge length L2 of the second cube and the separation distance X. (We assume the first cube has unit edge length.)
The last graphic of the force shows the intuitive, to-be-expected behavior: For large separation distances X, the force decreases; for zero separation (concentric cubes), the force vanishes because of symmetry. And the sharp peak at X = 1/2 for small L2 is the maximum of the field strength at the center of a face of the unit cube.
Let’s have a quick look at the behavior of two cubes separated by a large distance. The term proportional X5 gives the first correction to the Coulomb law.
As a quick check, we compare these leading terms with an integration over the series of the integrand.
Now, for the rest, let’s specialize to two cubes of equal edge length. Because of the presence of removable singularities, we cannot just substitute L2L1, but have to be slightly more careful. Making series expansions, it turns out that all indeterminate terms vanish in the limit L2L1.
This gives the following final result for the interaction potential of two identical cubes separated by a distance X. After some more simplifications, we get a result that fits on a single page. (We now include the prefactors [L-3]2.)
The last expression has the form Vc = L-1 f(x) where the dimensionless x is defined as x = X/L. Here is a quick look at the function f(x) and its first two derivatives. The second derivative is no longer a smooth function at x = 1.
And these are the asymptotic behaviors of f(x) at x = 0 and x = .
In the series expansion around x = 1, we see the cusp of the second derivative reflected in the different values of the coefficient of (x – 1)3.
The difference of the coefficients is just 2π/3.
We obtain the force between the two cubes by differentiating VC with respect to X. (Analogous to the well-known Newtonian force law between two homogeneous spherical objects, a natural name for this distance-dependent force law would be the Cubonian force law.)
And here is the exact expression for the force between two cubes touching each other along a face (meaning X = L). We use Series instead of substitution in FC due to indeterminate expressions arising in the substitution process.
And here is the numerical value we were looking for with 100 digits.
There are various ways to represent this expression; here are some of them.
Here are some more root reduced versions that better show the algebraic and transcendental part of the result.
Having the force between the two cubes as a function of the distance, a natural question to ask is: What is the force between two penetrable cubes, and how does it compare to the force between two spheres? The force between two penetrable, homogeneously charged spheres was calculated by Kermode, Mustafa, and Rowley. (We again ignore prefactors).
We will use a sphere with the same volume as a unit cube.
Here are plots of the total interaction energy and the force between the two spheres. The blue curves are for the two cubes and the red curves for the two spheres. (The two vertical lines indicate the sphere radius and the half edge lengths.)
We see that the maximum force between two cubes is slightly larger than the maximum force between two spheres of equal volume. The maximal force between two equal spheres occurs when the two sphere centers are separated by a distance equal to the sphere’s radius.
The maximum force between two cubes occurs when the two cube centers are separated by about 68% of the cube edge length, and the maximal force between two cubes in this case is about 3.5% smaller than the maximal force between two spheres of equal volume.
So, now that we have the distance-dependent Cubonian force between two cubes, what can we calculate with it? For instance, we could build a soft-shell Newtonian cradle using charged, penetrable cubes.
We assume the strings to be very long so that the cubes all move in 1D and model the gravitational force as a linear restoring force to the cube’s resting positions.
We assume unit cubes positioned initially at positions xn] = n Δ for some given initial spacing Δ.
Here is an example (download the CDF at the end of the post to interact with it):
Because the cubes are penetrable, the initially elongated cube carries out about 50 oscillations through the other cubes before the neighboring tubes start to move with a larger amplitude.
Posted in: Mathematics
### One Comment
hello That’s a nice post.Thank you for sharing. Posted by nike dunk June 3, 2014 at 5:08 am | 2018-12-18 19:35:56 | {"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": 0.8262162804603577, "perplexity": 491.6534448841338}, "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-2018-51/segments/1544376829568.86/warc/CC-MAIN-20181218184418-20181218210418-00083.warc.gz"} |
https://mdsoar.org/handle/11603/57 | ### Browse by
The mission of the Department of Physics at UMBC is based on three pillars: education, research and service to society. We strive to provide outstanding educational opportunities, through instruction and research, for undergraduate physics majors seeking preparation for graduate school or entry into the workforce. We also aim to train graduate students to be leaders in their field of research. Our research goal is to grow and sustain internationally recognized research groups in atmospheric physics, astrophysics, condensed matter physics and quantum optics and information science. Through our teaching of physics to non-science and non-physics majors and through our Physics for Secondary Education Teachers program, we provide professional service to the university community and the State.
### Recent Submissions
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(APS, 2022-07-19)
Geometric quantum gates are often expected to be more resilient than dynamical gates against certain types of error, which would make them ideal for robust quantum computing. However, this is still up for debate due to ...
• #### BASS. XXVIII. Near-infrared Data Release 2: High-ionization and Broad Lines in Active Galactic Nuclei*
(AAS, 2022-07-15)
We present the BAT AGN Spectroscopic Survey (BASS) Near-infrared Data Release 2 (DR2), a study of 168 nearby ($\bar{z}=0.04$, z < 0.6) active galactic nuclei (AGN) from the all-sky Swift Burst Array Telescope X-ray survey ...
• #### Where does the dust deposited over the Sierra Nevada snow come from?
(EGU, 2022-07-06)
Mineral dust contributes up to one-half of surface aerosol loading in spring over the southwestern U.S., posing an environmental challenge that threatens human health and the ecosystem. Using the self-organizing map (SOM) ...
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(2022-07-11)
We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. ...
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In semiconductor spin qubits which typically interact through short-range exchange coupling, shuttling of spin is a practical way to generate quantum operations between distant qubits. Although the exchange is often tunable ...
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We make the first observation-based calculation of the energy that goes into cosmic ray protons versus cosmic ray electrons in shock acceleration during structure formation. We find a ratio of energy in cosmic ray protons ...
• #### Shortcuts to thermodynamic quasistaticity
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The operation of near-term quantum technologies requires the development of feasible, implementable, and robust strategies of controlling complex many body systems. However, currently existing shortcuts to adiabaticity'' ...
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(2022-06-27)
Quantum mechanics is an inherently linear theory. However, collective effects in many body quantum systems can give rise to effectively nonlinear dynamics. In the present work, we analyze whether and to what extent such ...
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(2022-06-14)
Super-massive black holes residing at the centres of galaxies can launch powerful radio-emitting plasma jets which reach scales of hundreds of thousands of light-years, well beyond their host galaxies.1 The advent of ...
• #### Low-Frequency Oscillations in Optical Measurements of Metal-Nanoparticle Vibrations
(ACS, 2022-06-14)
Time-resolved optical measurements of vibrating metal nanoparticles have been used extensively to probe the ultrafast mechanical properties of the nanoparticles and of the surrounding liquid, but nearly all investigations ...
• #### Galactic Extinction: How Many Novae Does It Hide and How Does It Affect the Galactic Nova Rate?
(AAS, 2021-11-16)
There is a long-standing discrepancy between the observed Galactic classical nova rate of ∼10 yr−1 and the predicted rate from Galactic models of ∼30–50 yr−1. One explanation for this discrepancy is that many novae are ...
• #### Size-Resolved Dust Direct Radiative Effect Efficiency Derived from Satellite Observations
(EGU, 2022-06)
The role of mineral dust aerosol in global radiative energy budget is often quantified by the dust direct radiative effect (DRE). The dust DRE strongly depends on dust aerosol optical depth (DAOD), therefore, DRE efficiency ...
• #### Controlled etching and tapering of Au nanorods using cysteamine
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While gold nanorods (AuNRs) have found many applications due to their unique optical properties, a few challenges persist in their synthesis. Namely, it is often difficult to reproducibly synthesize AuNRs with specific and ...
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Dyads containing two near-infrared absorbing and emitting bacteriochlorins with distinct spectral properties have been prepared and characterized by absorption, emission, and transient-absorption spectroscopies. The dyads ...
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(2022-05-30)
FUV spectra of Eta Car, recorded across two decades with HST/STIS, document multiple changes in resonant lines caused by dissipating extinction in our line of sight. The FUV flux has increased nearly ten-fold which has led ...
• #### Optimal finite-time processes in weakly driven overdamped Brownian motion
(2022-06-08)
The complete physical understanding of the optimization of the thermodynamic work still is an important open problem in stochastic thermodynamics. We address this issue using the Hamiltonian approach of linear response ...
• #### Strong coupling of emitters to single plasmonic nanoparticles: Exciton-induced transparency and Rabi splitting
(Royal Society of Chemistry, 2019-07-22)
Strong coupling between plasmons in metal nanoparticles and single excitons in molecules or semiconductor nanomaterials has recently attracted considerable experimental effort for potential applications in quantum-mechanical ...
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(AIP, 2020-08-17)
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(AAS, 2021-11-10)
Massive black holes at the centers of galaxies can launch powerful wide-angle winds that, if sustained over time, can unbind the gas from the stellar bulges of galaxies. These winds may be responsible for the observed ... | 2022-08-09 19:27:38 | {"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": 0.36248353123664856, "perplexity": 6605.935161598923}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2022-33/segments/1659882571086.77/warc/CC-MAIN-20220809185452-20220809215452-00570.warc.gz"} |
https://www.doubtnut.com/question-answer/negative-rational-number-a-rational-number-is-said-to-be-negative-if-its-numerator-and-denominator-a-1527862 | Home
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# Negative rational number a rational number is said to be negative if its numerator and denominator are such that one of them is positive integer and another one is a negative integer.
Updated On: 27-06-2022 | 2022-12-07 07:53:19 | {"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": 0.8336508870124817, "perplexity": 734.6396302576132}, "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/1669446711150.61/warc/CC-MAIN-20221207053157-20221207083157-00104.warc.gz"} |
http://clay6.com/qa/47812/a-survey-shows-that-63-of-the-americans-like-cheese-whereas-76-like-apples- | Browse Questions
Home >> AIMS >> Class11 >> Math >> Sets
# A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, find the value of x.
39 $\leq$ x $\leq$ 63 | 2016-12-11 14:00:09 | {"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": 0.48718616366386414, "perplexity": 533.1491130006974}, "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/1480698544679.86/warc/CC-MAIN-20161202170904-00340-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://codereview.stackexchange.com/questions/131428/custom-hash-tablek-t-that-take-responsibility-to-assign-hashcode-index | # custom hash table<K,T> that take responsibility to assign hashcode/index
## Introduction
I try to create a game engine with component-based architecture.
There are many type game components and different game systems.
For example, the components can be
• Com_HP (health), Com_Physics , Com_Graphics, ...
The systems can be
• Physics_Manager, Graphics_Manager, Binder_Manager, ...
Most systems want to cache relation (map{Key->Value}) that has game component as Key, but the Value can be any type.
For example,
System_Binder : Hashtable<Com_Graphics*,Com_Physics*>.
System_PhysicFactory : Hashtable<Com_Physics*,someInternalData*>.
The components that are cached are only subset of all instance of components of that type.
In other words, not all Com_Graphics instances in the game engine are in the key of the System_Binder's Hashtable.
This problem can be solved easily with some standard library.
However, the retrieve, insert, erase performance is very crucial.
- After test, I found that standard hashmap/hashtable's performance is not good enough.
Therefore, I developed a custom hash table, let's name it AMap.
## The custom HashTable (hashmap)
AMap works almost like general hash table, but it take responsibility to create hashCode (instead of the Key's responsibility), and assign hashCode to a special field of Key.
The special field of Key have to be fed to the AMap, let's call this field as dedicated###.
Here is pseudo code of AMap :-
template<class Key,class Value, int Key::*dedicatedField>class AMap{
Array<pair<Key,Value>> chunk; //like a pool
// Key & Value are usually pointers.
Queue<int> recycleNotUsed; //contain index of "chunk" that is not used
int indexRunner=0; //count from 0 and so on ...
int index; // replace traditional hashCode
if( recycleNotUsed.isEmpty() ){
index = indexRunner++; //count from 0 and so on ...
}else{
index = recycleNotUsed.popFirstElement(); //use recycled key
}
key.dedicatedField = index ; //### assign to the dedicated field
chunk[index ] = {key,value};
//... house keeping about flag that [index] is valid ....
}
void remove(Key key){
recycleNotUsed.pushLastElement(key.dedicatedField ); //###
//... house keeping about flag that the [key.dedicatedField] is invalid ....
}
Value get(Key key){
chunk.get(key.dedicatedField).second; //### so fast
}
//.... other function like iterator, get, set, ....
}
The ### line mark dedicated field. I will show how to feed it to a certain AMap.
When I want to create any AMap, I have to create a dedicated field in the Key class.
// System_Binder.h : This is a field that I want.
AMap<Com_Graphics*,Com_Physics*> graphic_to_physics;
// Com_Graphics.h : I have to add a dedicated field to Com_Graphics.h
int dedicated_System_Binder = -1; //###
// System_Binder.h : This is what I have to declare
AMap<Com_Graphics*,
Com_Physics*,
&Com_Graphics::dedicated_System_Binder> //###
graphic_to_physics;
After another test, AMap, compared to std::unorder_map, is astonishing fast (30-40x). Memory is also packed.
## Disadvantage of my custom HashMap
• I have to create 1 dedicated field for any Com_XXX per 1 hash map that use that Com_XXX as a key, as in the example.
• For example, if a system has 2 maps, both use Com_HP as key, I have to create 2 dedicated fields within Com_HP.
• Lower maintainability, e.g.,
if I want to remove the field ...
// System_Binder.h :
AMap<Com_Graphics*,Com_Physics*, ...dedicate...> graphic_to_physics;
... I should also remove this line
// Com_Graphics.h :
int dedicated_System_Binder = -1;
• A limited number of map have to be defined beforehand, because each one has to have a corresponding dedicated variable in the Key class. For example, if there are 2 instances of the same system (e.g. 2 Physic_Manager) , the game will potentially crash.
## Question :
• Is the code good in term of design?
• Are there any better alternatives that provide similar/better performance, but higher maintainability, flexibility and usability? | 2019-08-19 08:57:55 | {"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": 0.194292813539505, "perplexity": 7590.882538690995}, "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/1566027314696.33/warc/CC-MAIN-20190819073232-20190819095232-00077.warc.gz"} |
https://studies.quantimo.do/tag/against-the-grain-pesto-pizza-consumption/ | # Against the Grain Pesto Pizza Consumption Studies
### Higher Against The Grain Pesto Pizza Consumption Predicts Very Slightly Lower Guiltiness
This individual’s Guiltiness is generally lowest after a daily total of 32 serving of Against The Grain Pesto Pizza consumption over the previous 7 days. | 2019-09-21 00:51:48 | {"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": 0.8435288667678833, "perplexity": 12974.227191336242}, "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/1568514574159.19/warc/CC-MAIN-20190921001810-20190921023810-00017.warc.gz"} |
https://brilliant.org/discussions/thread/computer-science-section-in-brilliant/ | # Computer Science section in Brilliant
Hello Brilliantians!
I saw that the computer science section was removed from Brilliant.org. My first question is when will a new one be back?
If it does come back, I have a few requests that I'm very sure many of you out there will have:
1. Return of the Data Structures & Algorithms section!
2. Focus not only on Python but other popular ones like C++/C, Java, etc also so that non-python users will also be able to participate and solve problems.
3. Strong and difficult theoretical questions which may be associated with real world problems and not necessarily require a computer and compiler to solve.
That's all for the present. I you guys have something in mind, please tell Brilliant! I'd grateful to the Brilliant team if they bring back programming.
Thanks
Note by Kou\$htav Chakrabarty
4 years, 8 months ago
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I also think that c++ should also be included.
- 3 years, 11 months ago
Admin should focus on the computer science section to publish more set of problems as they will bring attention of few Indian coders towards it.So have an attention towards Computer science section once.
- 4 years, 1 month ago | 2018-07-22 11:03:57 | {"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": 0.9738667607307434, "perplexity": 5541.13332058476}, "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/1531676593208.44/warc/CC-MAIN-20180722100513-20180722120513-00108.warc.gz"} |
https://scirate.com/arxiv/cs.MA?range=31 | # Multiagent Systems (cs.MA)
• We present an approach for implementing a specific form of collaborative industrial practices-called Industrial Symbiotic Networks (ISNs)-as MC-Net cooperative games and address the so called ISN implementation problem. This is, the characteristics of ISNs may lead to inapplicability of fair and stable benefit allocation methods even if the collaboration is a collectively desired one. Inspired by realistic ISN scenarios and the literature on normative multi-agent systems, we consider regulations and normative socioeconomic policies as two elements that in combination with ISN games resolve the situation and result in the concept of coordinated ISNs.
• Apr 20 2018 cs.MA cs.AI arXiv:1804.07178v1
Interest in emergent communication has recently surged in Machine Learning. The focus of this interest has largely been either on investigating the properties of the learned protocol or on utilizing emergent communication to better solve problems that already have a viable solution. Here, we consider self-driving cars coordinating with each other and focus on how communication influences the agents' collective behavior. Our main result is that communication helps (most) with adverse conditions.
• Apr 18 2018 cs.MA arXiv:1804.06011v1
Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in $1 + \frac{2\pi}{3} + \sqrt{3} \approx 4.8264$ time units and this is optimal~[Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most $4.81854$. Furthermore, we show that any strategy for saving the queen requires time at least $3 + \pi/6 + \sqrt{3}/2 \approx 4.3896$ in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to $3.8327$ for two servants, and $3.3738$ for three servants. Finally we show lower bounds for these cases of $3.6307$ (two servants) and $3.2017$ (three servants). The case of $n\geq 4$ is the subject of an independent study by Queen Daniela's Royal Scientific Team.
• In the parable of Simon's Ant, an ant follows a complex path along a beach on to reach its goal. The story shows how the interaction of simple rules and a complex environment result in complex behavior. But this relationship can be looked at in another way - given path and rules, we can infer the environment. With a large population of agents - human or animal - it should be possible to build a detailed map of a population's social and physical environment. In this abstract, we describe the development of a framework to create such maps of human belief space. These maps are built from the combined trajectories of a large number of agents. Currently, these maps are built using multidimensional agent-based simulation, but the framework is designed to work using data from computer-mediated human communication. Maps incorporating human data should support visualization and navigation of the "plains of research", "fashionable foothills" and "conspiracy cliffs" of human belief spaces.
• Distributed controllers are often necessary for a multi-agent system to satisfy safety properties such as collision avoidance. Communication and coordination are key requirements in the implementation of a distributed control protocol, but maintaining an all-to-all communication topology is unreasonable and not always necessary. Given a safety objective and a controller implementation, we consider the problem of identifying when agents need to communicate with one another and coordinate their actions to satisfy the safety constraint. We define a coordination-free controllable predecessor operator that is used to derive a subset of the state space that allows agents to act independently, without consulting other agents to double check that the action is safe. Applications are shown for identifying an upper bound on connection delays and a self-triggered coordination scheme. Examples are provided which showcase the potential for designers to visually interpret a system's ability to tolerate delays when initializing a network connection.
• In this work, we present a programming paradigm allowing the control of swarms with a minimum communication bandwidth in a simple manner, yet allowing the emergence of diverse complex behaviors and autonomy of the swarm. Communication in the proposed paradigm is based on single bit "ping"-signals propagating as information-waves throughout the swarm. We show that even this minimum bandwidth communication between agents suffices for the design of a substantial set of behaviors in the domain of essential behaviors of a collective, including locomotion and self awareness of the swarm.
• The ability of algorithms to evolve or learn (compositional) communication protocols has traditionally been studied in the language evolution literature through the use of emergent communication tasks. Here we scale up this research by using contemporary deep learning methods and by training reinforcement-learning neural network agents on referential communication games. We extend previous work, in which agents were trained in symbolic environments, by developing agents which are able to learn from raw pixel data, a more challenging and realistic input representation. We find that the degree of structure found in the input data affects the nature of the emerged protocols, and thereby corroborate the hypothesis that structured compositional language is most likely to emerge when agents perceive the world as being structured.
• Multi-agent reinforcement learning offers a way to study how communication could emerge in communities of agents needing to solve specific problems. In this paper, we study the emergence of communication in the negotiation environment, a semi-cooperative model of agent interaction. We introduce two communication protocols -- one grounded in the semantics of the game, and one which is \textita priori ungrounded and is a form of cheap talk. We show that self-interested agents can use the pre-grounded communication channel to negotiate fairly, but are unable to effectively use the ungrounded channel. However, prosocial agents do learn to use cheap talk to find an optimal negotiating strategy, suggesting that cooperation is necessary for language to emerge. We also study communication behaviour in a setting where one agent interacts with agents in a community with different levels of prosociality and show how agent identifiability can aid negotiation.
• We present a novel algorithm for computing collision-free navigation for heterogeneous road-agents such as cars, tricycles, bicycles, and pedestrians in dense traffic. Our approach currently assumes the positions, shapes, and velocities of all vehicles and pedestrians are known and computes smooth trajectories for each agent by taking into account the dynamic constraints. We describe an efficient optimization-based algorithm for each road-agent based on reciprocal velocity obstacles that takes into account kinematic and dynamic constraints. Our algorithm uses tight fitting shape representations based on medial axis to compute collision-free trajectories in dense traffic situations. We evaluate the performance of our algorithm in real-world dense traffic scenarios and highlight the benefits over prior reciprocal collision avoidance schemes.
• Decentralized (PO)MDPs provide an expressive framework for sequential decision making in a multiagent system. Given their computational complexity, recent research has focused on tractable yet practical subclasses of Dec-POMDPs. We address such a subclass called CDEC-POMDP where the collective behavior of a population of agents affects the joint-reward and environment dynamics. Our main contribution is an actor-critic (AC) reinforcement learning method for optimizing CDEC-POMDP policies. Vanilla AC has slow convergence for larger problems. To address this, we show how a particular decomposition of the approximate action-value function over agents leads to effective updates, and also derive a new way to train the critic based on local reward signals. Comparisons on a synthetic benchmark and a real-world taxi fleet optimization problem show that our new AC approach provides better quality solutions than previous best approaches.
• The social community in open source software developers has a complex network structure. The network structure represents the relations between the project and the engineer in the software developer's community. A project forms some teams which consist of engineers categorized into some task group. Source Forge is well known to be one of open source websites. The node and arc in the network structure means the engineer and their connection among engineers in the Source Forge. In the previous study, we found the growing process of project becomes strong according to the number of developers joining into the project. In the growing phase, we found some characteristic patterns between the number of agents and the produced projects. By such observations, we developed a simulation model of performing the growing process of project. In this paper, we introduced the altruism behavior as shown in the Army Ant model into the software developer's simulation model. The efficiency of the software developing process was investigated by some experimental simulation results.
• We present a novel algorithm for reciprocal collision avoidance between heterogeneous agents of different shapes and sizes. We present a novel CTMAT representation based on medial axis transform to compute a tight fitting bounding shape for each agent. Each CTMAT is represented using tuples, which are composed of circular arcs and line segments. Based on the reciprocal velocity obstacle formulation, we reduce the problem to solving a low-dimensional linear programming between each pair of tuples belonging to adjacent agents. We precompute the Minkowski Sums of tuples to accelerate the runtime performance. Finally, we provide an efficient method to update the orientation of each agent in a local manner. We have implemented the algorithm and highlight its performance on benchmarks corresponding to road traffic scenarios and different vehicles. The overall runtime performance is comparable to prior multi-agent collision avoidance algorithms that use circular or elliptical agents. Our approach is less conservative and results in fewer false collisions.
• This paper describes an agent based simulation used to model human actions in belief space, a high-dimensional subset of information space associated with opinions. Using insights from animal collective behavior, we are able to simulate and identify behavior patterns that are similar to nomadic, flocking and stampeding patterns of animal groups. These behaviors have analogous manifestations in human interaction, emerging as solitary explorers, the fashion-conscious, and members of polarized echo chambers. We demonstrate that a small portion of nomadic agents that widely traverse belief space can disrupt a larger population of stampeding agents. Extending the model, we introduce the concept of Adversarial Herding, where bad actors can exploit properties of technologically mediated communication to artificially create self sustaining runaway polarization. We call this condition the Pishkin Effect as it recalls the large scale buffalo stampedes that could be created by native Americans hunters. We then discuss opportunities for system design that could leverage the ability to recognize these negative patterns, and discuss affordances that may disrupt the formation of natural and deliberate echo chambers.
• Self-organization has been an important concept within a number of disciplines, which Artificial Life (ALife) also has heavily utilized since its inception. The term and its implications, however, are often confusing or misinterpreted. In this work, we provide a mini-review of self-organization and its relationship with ALife, aiming at initiating discussions on this important topic with the interested audience. We first articulate some fundamental aspects of self-organization, outline its usage, and review its applications to ALife within its soft, hard, and wet domains. We also provide perspectives for further research.
• Real-time strategy games have been an important field of game artificial intelligence in recent years. This paper presents a reinforcement learning and curriculum transfer learning method to control multiple units in StarCraft micromanagement. We define an efficient state representation, which breaks down the complexity caused by the large state space in the game environment. Then a parameter sharing multi-agent gradientdescent Sarsa(\lambda) (PS-MAGDS) algorithm is proposed to train the units. The learning policy is shared among our units to encourage cooperative behaviors. We use a neural network as a function approximator to estimate the action-value function, and propose a reward function to help units balance their move and attack. In addition, a transfer learning method is used to extend our model to more difficult scenarios, which accelerates the training process and improves the learning performance. In small scale scenarios, our units successfully learn to combat and defeat the built-in AI with 100% win rates. In large scale scenarios, curriculum transfer learning method is used to progressively train a group of units, and shows superior performance over some baseline methods in target scenarios. With reinforcement learning and curriculum transfer learning, our units are able to learn appropriate strategies in StarCraft micromanagement scenarios.
• The authors present an overview of a hierarchical framework for coordinating task- and motion-level operations in multirobot systems. Their framework is based on the idea of using simple temporal networks to simultaneously reason about precedence/causal constraints required for task-level coordination and simple temporal constraints required to take some kinematic constraints of robots into account. In the plan-generation phase, the framework provides a computationally scalable method for generating plans that achieve high-level tasks for groups of robots and take some of their kinematic constraints into account. In the plan-execution phase, the framework provides a method for absorbing an imperfect plan execution to avoid time-consuming re-planning in many cases. The authors use the multirobot path-planning problem as a case study to present the key ideas behind their framework for the long-term autonomy of multirobot systems.
• In many real-world settings, a team of agents must coordinate their behaviour while acting in a decentralised way. At the same time, it is often possible to train the agents in a centralised fashion in a simulated or laboratory setting, where global state information is available and communication constraints are lifted. Learning joint action-values conditioned on extra state information is an attractive way to exploit centralised learning, but the best strategy for then extracting decentralised policies is unclear. Our solution is QMIX, a novel value-based method that can train decentralised policies in a centralised end-to-end fashion. QMIX employs a network that estimates joint action-values as a complex non-linear combination of per-agent values that condition only on local observations. We structurally enforce that the joint-action value is monotonic in the per-agent values, which allows tractable maximisation of the joint action-value in off-policy learning, and guarantees consistency between the centralised and decentralised policies. We evaluate QMIX on a challenging set of StarCraft II micromanagement tasks, and show that QMIX significantly outperforms existing value-based multi-agent reinforcement learning methods.
• Apr 02 2018 cs.MA cs.CY arXiv:1803.11457v1
The emerging field of morphogenetic engineering proposes to design complex heterogeneous system focused on the paradigm of emergence. Necessarily at the interface of disciplines, its concepts can be defined through multiple viewpoints. This contribution aims at linking a co-evolutionary perspective on such systems with morphogenesis, and therein at bringing a novel conceptual approach to the bottom-up design of complex systems which allows to fully consider co-evolutive processes. We first situate systems of interest at the interface between biological and social systems, and introduce a multidisciplinary perspective on co-evolution. Building on Holland's signals and boundaries theory of complex adaptive systems, we finally suggest that morphogenetic systems are equivalent to combinations of co-evolutionary niches. This introduces an entry to morphogenetic engineering focused on co-evolution between components of a system. Applications can be found in a broad range of subjects, which we illustrate with the example of planning in territorial systems, suggesting an extended scope for the relevance of morphogenetic engineering concepts.
• This paper develops an optimal relative output-feedback based solution to the containment control problem of linear heterogeneous multi-agent systems. A distributed optimal control protocol is presented for the followers to not only assure that their outputs fall into the convex hull of the leaders' output (i.e., the desired or safe region), but also optimizes their transient performance. The proposed optimal control solution is composed of a feedback part, depending of the followers' state, and a feed-forward part, depending on the convex hull of the leaders' state. To comply with most real-world applications, the feedback and feed-forward states are assumed to be unavailable and are estimated using two distributed observers. That is, since the followers cannot directly sense their absolute states, a distributed observer is designed that uses only relative output measurements with respect to their neighbors (measured for example by using range sensors in robotic) and the information which is broadcasted by their neighbors to estimate their states. Moreover, another adaptive distributed observer is designed that uses exchange of information between followers over a communication network to estimate the convex hull of the leaders' state. The proposed observer relaxes the restrictive requirement of knowing the complete knowledge of the leaders' dynamics by all followers. An off-policy reinforcement learning algorithm on an actor-critic structure is next developed to solve the optimal containment control problem online, using relative output measurements and without requirement of knowing the leaders' dynamics by all followers. Finally, the theoretical results are verified by numerical simulations.
• The amount of personal data collected in our everyday interactions with connected devices offers great opportunities for innovative services fueled by machine learning, as well as raises serious concerns for the privacy of individuals. In this paper, we propose a massively distributed protocol for a large set of users to privately compute averages over their joint data, which can then be used to learn predictive models. Our protocol can find a solution of arbitrary accuracy, does not rely on a third party and preserves the privacy of users throughout the execution in both the honest-but-curious and malicious adversary models. Specifically, we prove that the information observed by the adversary (the set of maliciours users) does not significantly reduce the uncertainty in its prediction of private values compared to its prior belief. The level of privacy protection depends on a quantity related to the Laplacian matrix of the network graph and generally improves with the size of the graph. Furthermore, we design a verification procedure which offers protection against malicious users joining the service with the goal of manipulating the outcome of the algorithm.
• Applications in robotics, such as multi-robot target tracking, involve the execution of information acquisition tasks by teams of mobile robots. However, in failure-prone or adversarial environments, robots get attacked, their communication channels get jammed, and their sensors fail, resulting in the withdrawal of robots from the collective task, and, subsequently, the inability of the remaining active robots to coordinate with each other. As a result, traditional design paradigms become insufficient and, in contrast, resilient designs against system-wide failures and attacks become important. In general, resilient design problems are hard, and even though they often involve objective functions that are monotone and (possibly) submodular, scalable approximation algorithms for their solution have been hitherto unknown. In this paper, we provide the first algorithm, enabling the following capabilities: minimal communication, i.e., the algorithm is executed by the robots based only on minimal communication between them, system-wide resiliency, i.e., the algorithm is valid for any number of denial-of-service attacks and failures, and provable approximation performance, i.e., the algorithm ensures for all monotone and (possibly) submodular objective functions a solution that is finitely close to the optimal. We support our theoretical analyses with simulated and real-world experiments, by considering an active information acquisition application scenario, namely, multi-robot target tracking.
• Mar 28 2018 cs.MA cs.SY math.OC arXiv:1803.08950v1
We consider a multi-agent framework for distributed optimization where each agent in the network has access to a local convex function and the collective goal is to achieve consensus on the parameters that minimize the sum of the agents' local functions. We propose an algorithm wherein each agent operates asynchronously and independently of the other agents in the network. When the local functions are strongly-convex with Lipschitz-continuous gradients, we show that a subsequence of the iterates at each agent converges to a neighbourhood of the global minimum, where the size of the neighbourhood depends on the degree of asynchrony in the multi-agent network. When the agents work at the same rate, convergence to the global minimizer is achieved. Numerical experiments demonstrate that Asynchronous Subgradient-Push can minimize the global objective faster than state-of-the-art synchronous first-order methods, is more robust to failing or stalling agents, and scales better with the network size.
• In this paper, we study the problem of distributed multi-agent optimization over a network, where each agent possesses a local cost function that is smooth and strongly convex. The global objective is to find a common solution that minimizes the average of all cost functions. Assuming agents only have access to unbiased estimates of the gradients of their local cost functions, we consider a distributed stochastic gradient tracking method. We show that, in expectation, the iterates generated by each agent are attracted to a neighborhood of the optimal solution, where they accumulate exponentially fast (under a constant step size choice). More importantly, the limiting (expected) error bounds on the distance of the iterates from the optimal solution decrease with the network size, which is a comparable performance to a centralized stochastic gradient algorithm. Numerical examples further demonstrate the effectiveness of the method.
• When scheduling public works or events in a shared facility one needs to accommodate preferences of a population. We formalize this problem by introducing the notion of a collective schedule. We show how to extend fundamental tools from social choice theory---positional scoring rules, the Kemeny rule and the Condorcet principle---to collective scheduling. We study the computational complexity of finding collective schedules. We also experimentally demonstrate that optimal collective schedules can be found for instances with realistic sizes.
• The spread of autonomous systems into safety-critical areas has increased the demand for their formal verification, not only due to stronger certification requirements but also to public uncertainty over these new technologies. However, the complex nature of such systems, for example, the intricate combination of discrete and continuous aspects, ensures that whole system verification is often infeasible. This motivates the need for novel analysis approaches that modularise the problem, allowing us to restrict our analysis to one particular aspect of the system while abstracting away from others. For instance, while verifying the real-time properties of an autonomous system we might hide the details of the internal decision-making components. In this paper we describe verification of a range of properties across distinct dimesnions on a practical hybrid agent architecture. This allows us to verify the autonomous decision-making, real-time aspects, and spatial aspects of an autonomous vehicle platooning system. This modular approach also illustrates how both algorithmic and deductive verification techniques can be applied for the analysis of different system subcomponents.
• Making decisions is a great challenge in distributed autonomous environments due to enormous state spaces and uncertainty. Many online planning algorithms rely on statistical sampling to avoid searching the whole state space, while still being able to make acceptable decisions. However, planning often has to be performed under strict computational constraints making online planning in multi-agent systems highly limited, which could lead to poor system performance, especially in stochastic domains. In this paper, we propose Emergent Value function Approximation for Distributed Environments (EVADE), an approach to integrate global experience into multi-agent online planning in stochastic domains to consider global effects during local planning. For this purpose, a value function is approximated online based on the emergent system behaviour by using methods of reinforcement learning. We empirically evaluated EVADE with two statistical multi-agent online planning algorithms in a highly complex and stochastic smart factory environment, where multiple agents need to process various items at a shared set of machines. Our experiments show that EVADE can effectively improve the performance of multi-agent online planning while offering efficiency w.r.t. the breadth and depth of the planning process.
• This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics. In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions. In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focusing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance. In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.
• With the emergence of autonomous vehicles, it is important to understand their impact on the transportation system. However, conventional traffic simulations are time-consuming. In this paper, we introduce an analytical traffic model for unmanaged intersections accounting for microscopic vehicle interactions. The macroscopic property, i.e., delay at the intersection, is modeled as an event-driven stochastic dynamic process, whose dynamics encode the microscopic vehicle behaviors. The distribution of macroscopic properties can be obtained through either direct analysis or event-driven simulation. They are more efficient than conventional (time-driven) traffic simulation, and capture more microscopic details compared to conventional macroscopic flow models. We illustrate the efficiency of this method by delay analyses under two different policies at a two-lane intersection. The proposed model allows for 1) efficient and effective comparison among different policies, 2) policy optimization, 3) traffic prediction, and 4) system optimization (e.g., infrastructure and protocol).
• We studied the long-term dynamics of evolutionary Swarm Chemistry by extending the simulation length ten-fold compared to earlier work and by developing and using a new automated object harvesting method. Both macroscopic dynamics and microscopic object features were characterized and tracked using several measures. Results showed that the evolutionary dynamics tended to settle down into a stable state after the initial transient period, and that the extent of environmental perturbations also affected the evolutionary trends substantially. In the meantime, the automated harvesting method successfully produced a huge collection of spontaneously evolved objects, revealing the system's autonomous creativity at an unprecedented scale.
• Hierarchical Modular Reinforcement Learning (HMRL), consists of 2 layered learning where Profit Sharing works to plan a prey position in the higher layer and Q-learning method trains the state-actions to the target in the lower layer. In this paper, we expanded HMRL to multi-target problem to take the distance between targets to the consideration. The function, called `AT field', can estimate the interests for an agent according to the distance between 2 agents and the advantage/disadvantage of the other agent. Moreover, the knowledge related to state-action rules is extracted by C4.5. The action under the situation is decided by using the acquired knowledge. To verify the effectiveness of proposed method, some experimental results are reported.
• The concept of truth, as a public good is the production of a collective understanding, which emerges from a complex network of social interactions. The recent impact of social networks on shaping the perception of truth in political arena shows how such perception is corroborated and established by the online users, collectively. However, investigative journalism for discovering truth is a costly option, given the vast spectrum of online information. In some cases, both journalist and online users choose not to investigate the authenticity of the news they receive, because they assume other actors of the network had carried the cost of validation. Therefore, the new phenomenon of "fake news" has emerged within the context of social networks. The online social networks, similarly to System of Systems, cause emergent properties, which makes authentication processes difficult, given availability of multiple sources. In this study, we show how this conflict can be modeled as a volunteer's dilemma. We also show how the public contribution through news subscription (shared rewards) can impact the dominance of truth over fake news in the network.
• In this letter we discuss cost optimization of sensor networks monitoring structurally full-rank systems under distributed observability constraint. Using structured systems theory, the problem is relaxed into two subproblems: (i) sensing cost optimization and (ii) networking cost optimization. Both problems are reformulated as combinatorial optimization problems. The sensing cost optimization is shown to have a polynomial order solution. The networking cost optimization is shown to be NP-hard in general, but has a polynomial order solution under specific conditions. A 2-approximation polynomial order relaxation is provided for general networking cost optimization, which is applicable in large-scale system monitoring.
• The behavior of heterogeneous multi-agent systems is studied when the coupling matrices are possibly all different and/or singular (that is, its rank is less than the system dimension). Rank-deficient coupling allows exchange of limited state information, which is suitable for study of output coupling in multi-agent systems. We present a coordinate change that transforms the heterogeneous multi-agent system into a singularly perturbed form. The slow dynamics is still a reduced-order multi-agent system consisting of a weighted average of the vector fields of all agents, and some sub-dynamics of agents. The weighted average is an emergent dynamics, which we call a blended dynamics. By analyzing or synthesizing the blended dynamics, one can predict or design the behavior of heterogeneous multi-agent system when the coupling gain is sufficiently large. For this result, stability of the blended dynamics is required. Since stability of individual agent is not asked, stability of the blended dynamics is the outcome of trading stability among the agents. It can be seen that, under stability of the blended dynamics, the initial conditions of individual agents are forgotten as time goes on, and thus, the behavior of the synthesized multi-agent system are initialization-free and suitable for plug-and-play operation. As a showcase, we apply the proposed tool to two application problems; distributed state estimation for linear systems, and practical synchronization of heterogeneous Van der Pol oscillators (for which phase cohesiveness is achieved). We also present underlying intuition for two more applications; estimation of the number of nodes in a network, and a problem of distributed optimization.
• We consider a scenario consisting of a set of heterogeneous mobile agents located at a depot, and a set of tasks dispersed over a geographic area. The agents are partitioned into different types. The tasks are partitioned into specialized tasks that can only be done by agents of a certain type, and generic tasks that can be done by any agent. The distances between each pair of tasks are specified, and satisfy the triangle inequality. Given this scenario, we address the problem of allocating these tasks among the available agents (subject to type compatibility constraints) while minimizing the maximum cost to tour the allocation by any agent and return to the depot. This problem is NP-hard, and we give a three phase algorithm to solve this problem that provides 5-factor approximation, regardless of the total number of agents and the number of agents of each type. We also show that in the special case where there is only one agent of each type, the algorithm has an approximation factor of 4.
• Groups of humans are often able to find ways to cooperate with one another in complex, temporally extended social dilemmas. Models based on behavioral economics are only able to explain this phenomenon for unrealistic stateless matrix games. Recently, multi-agent reinforcement learning has been applied to generalize social dilemma problems to temporally and spatially extended Markov games. However, this has not yet generated an agent that learns to cooperate in social dilemmas as humans do. A key insight is that many, but not all, human individuals have inequity averse social preferences. This promotes a particular resolution of the matrix game social dilemma wherein inequity-averse individuals are personally pro-social and punish defectors. Here we extend this idea to Markov games and show that it promotes cooperation in several types of sequential social dilemma, via a profitable interaction with policy learnability. In particular, we find that inequity aversion improves temporal credit assignment for the important class of intertemporal social dilemmas. These results help explain how large-scale cooperation may emerge and persist.
• Chu Spaces and Channel Theory are well established areas of investigation in the general context of category theory. We review a range of examples and applications of these methods in logic and computer science, including Formal Concept Analysis, distributed systems and ontology development. We then employ these methods to describe human object perception, beginning with the construction of uncategorized object files and proceeding through categorization, individual object identification and the tracking of object identity through time. We investigate the relationship between abstraction and mereological categorization, particularly as these affect object identity tracking. This we accomplish in terms of information flow that is semantically structured in terms of local logics, while at the same time this framework also provides an inferential mechanism towards identification and perception. We show how a mereotopology naturally emerges from the representation of classifications by simplicial complexes, and briefly explore the emergence of geometric relations and interactions between objects.
• Demand Responsive Shared Transport DRST services take advantage of Information and Communication Technologies ICT, to provide on demand transport services booking in real time a ride on a shared vehicle. In this paper, an agent-based model ABM is presented to test different the feasibility of different service configurations in a real context. First results show the impact of route choice strategy on the system performance.
• This paper presents a distributed position synchronization strategy that also preserves the initial communication links for single-integrator multi-agent systems with time-varying delays. The strategy employs a coordinating proportional control derived from a specific type of potential energy, augmented with damping injected through a dynamic filter. The injected damping maintains all agents within the communication distances of their neighbours, and asymptotically stabilizes the multi-agent system, in the presence of time delays. Regarding the closed-loop single-integrator multi-agent system as a double-integrator system suggests an extension of the proposed strategy to connectivity-preserving coordination of Euler-Lagrange networks with time-varying delays. Lyapunov stability analysis and simulation results validate the two designs.
• Distributed model predictive control (MPC) has been proven a successful method in regulating the operation of large-scale networks of constrained dynamical systems. This paper is concerned with cooperative distributed MPC in which the decision actions of the systems are usually derived by the solution of a system-wide optimization problem. However, formulating and solving such large-scale optimization problems is often a hard task which requires extensive information communication among the individual systems and fails to address privacy concerns in the network. Hence, the main challenge is to design decision policies with a prescribed structure so that the resulting system-wide optimization problem to admit a loosely coupled structure and be amendable to distributed computation algorithms. In this paper, we propose a decentralized problem synthesis scheme which only requires each system to communicate sets which bound its states evolution to neighboring systems. The proposed method alleviates concerns on privacy since this limited communication scheme does not reveal the exact characteristics of the dynamics within each system. In addition, it enables a distributed computation of the solution, making our method highly scalable. We demonstrate in a number of numerical studies, inspired by engineering and finance, the efficacy of the proposed approach which leads to solutions that closely approximate those obtained by the centralized formulation only at a fraction of the computational effort.
• In this paper, we propose a distributed model predictive control (DMPC) scheme for linear time-invariant constrained systems which admit a separable structure. To exploit the merits of distributed computation algorithms, the stabilizing terminal controller, value function and invariant terminal set of the DMPC optimization problem need to respect the loosely coupled structure of the system. Although existing methods in the literature address this task, they typically decouple the synthesis of terminal controllers and value functions from the one of terminal sets. In addition, these approaches do not explicitly consider the effect of the current state of the system in the synthesis process. These limitations can lead the resulting DMPC scheme to poor performance since it may admit small or even empty terminal sets. Unlike other approaches, this paper presents a unified framework to encapsulate the synthesis of both the stabilizing terminal controller and invariant terminal set into the DMPC formulation. Conditions for Lyapunov stability and invariance are imposed in the synthesis problem in a way that allows the value function and invariant terminal set to admit the desired distributed structure. We illustrate the effectiveness of the proposed method on several examples including a benchmark spring-mass-damper problem.
• This paper is about a new model of opinion dynamics with opinion-dependent connectivity. We assume that agents update their opinions asynchronously and that each agent's new opinion depends on the opinions of the $k$ agents that are closest to it. We show that the resulting dynamics is substantially different from comparable models in the literature, such as bounded-confidence models. We study the equilibria of the dynamics, observing that they are robust to perturbations caused by the introduction of new agents. We also prove that if the number of agents $n$ is smaller than $2k$, the dynamics converge to consensus. This condition is only sufficient. | 2018-04-22 16:12:05 | {"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": 0.46990829706192017, "perplexity": 816.7642690253791}, "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-2018-17/segments/1524125945624.76/warc/CC-MAIN-20180422154522-20180422174522-00441.warc.gz"} |
https://www.sarthaks.com/919269/show-that-circumference-orbit-hydrogen-integral-multiple-broglie-wavelength-associated | # Show that the circumference of the Bohr orbit for the hydrogen atom is an integral multiple of the de Broglie wavelength associated
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Show that the circumference of the Bohr orbit for the hydrogen atom is an integral multiple of the de Broglie wavelength associated with the electron revolving around the orbit.
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According to Bohr’s theory,
According to de Brogue equation,
Thus, the circumference (2πr) of the Bohr orbit for hydrogen atom is an into the de Broglie wave length. | 2022-11-30 03:06:10 | {"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": 0.9564711451530457, "perplexity": 635.0892792389604}, "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/1669446710719.4/warc/CC-MAIN-20221130024541-20221130054541-00508.warc.gz"} |
https://www.physicsoverflow.org/32543/unitarity-constraint-perturbation-superchiral-perturbation | # What is the analogue of the unitarity constraint of chiral perturbation theory in superchiral perturbation theory?
+ 3 like - 0 dislike
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I am trying to reconstruct something and I would appreciate someone helping me filling the gaps. To motivate my question let's first consider chiral perturbation theory with up and down quarks. This is the lowest order Lagrangian
$$\begin{equation*} \mathcal{L}_2=\frac{f^2}{4}\langle{}\partial_{\mu}U^{\dagger}\partial^{\mu}U\rangle \end{equation*}$$
where $f$ is some parameter with mass dimensions and $U$ is defined via
$$U=e^{it^a\phi_a/f}$$
where $t^a$ are the Pauli matrices and $\phi_a$ are complex scalar fields. For some reason I don't know, people take this matrix to be unitary. So my first question is, why does $U$ have to be unitary?
In any case, if we demand $U$ to be unitary
$$UU^{\dagger}=1$$
$$e^{it^a\phi_a/f}e^{-it^a\phi_a^*/f}=1$$
and introducing $e^{-it^a\phi_a/f}$ on both sides
$$e^{-it^a\phi_a^*/f}=e^{-it^a\phi_a/f}$$
$$t^a\phi_a^*=t^a\phi_a$$
$$\begin{pmatrix} \pi_3&\pi_1-i\pi_2\\ \pi_1+i\pi_2&-\pi_3 \end{pmatrix}= \begin{pmatrix} \pi_3^*&\pi_1^*-i\pi_2^*\\ \pi_1^*+i\pi_2^*&-\pi_3^* \end{pmatrix}$$
now, redefining $\pi_3\equiv\pi_0$, $\pi_1+i\pi_2\equiv\sqrt{2}\pi_-$ and $\pi_1-i\pi_2\equiv\sqrt{2}\pi_+$ we see that the unitarity of $U$ imposes that $\pi_0$ is a real field after all and $\pi_-^*=\pi_+$
This up till now is nothing but the usual pion theory. I want to supersymmetrize this theory and this is where most of my doubts arise.
I have been told that I can embed the Lagrangian considered so far in
$$\mathcal{L}=f^2\int{}d^4\theta\langle\mathcal{U}^{\dagger}\mathcal{U}\rangle$$
where
$$\mathcal{U}=e^{it^a\Phi_a/f}$$
where $t^a$ are still the Pauli matrices and $\Phi_a$ are chiral superfields. Now my second question. In the nonsupersymmetric theory we imposed a unitarity constraint in the matrix $U$. I have been told that there is an analogous constraint on $t^a\Phi_a$ but I don't know which it is, let alone where it comes from. So which constraint must I apply and why? My third question would be about how I can relate the pion fields of the nonsupersymetric theory with the superfields I have just introduced, but of course I cannot properly state this question without first applying the constraint on $t^a\Phi_a$ (If the post gets anwsered this will go in a follow up question).
asked Jul 13, 2015
## 1 Answer
+ 3 like - 0 dislike
Since no answers appeared yet, I will try to make a contribution. My answer concerns only the first of your questions, i.e. why is $U$ unitary.
To my understanding, the unitarity of $U$ is a consequence of the nature of the chiral symmetry breaking. The assumption is, that it is the bilinear combination of quarks that form a condensate breaking the symmetry
$$(\bar{q}_{R})_j(q_L)^i=v^3 \delta^{i}_j$$
Here $i,j$ are flavour indices while spinor and color indices are summed over to form a Lorentz and a color scalar. Let for the sake of definiteness only consider $u$ and $d$ quarks. Under the $SU(2)_L\times SU(2)_R$ the quarks transform as follows
$$q_L^i\to L^i_{i'}q_L^{i'},\quad q_R^i\to R^j_{j'}q_R^{j'}$$
The vacuum condensate then transforms as
$$(\bar{q}_{R})_j(q_L)^i\to (R^\dagger)^{j'}_{j} v^3\delta^{i'}_{j'}L^{i}_{i'}=v^3(LR^\dagger)^i_j$$
For the vectorial subgroup of $SU(2)_L\times SU(2)_R$ characterized by $L=R$ the vacuum condensate stays invariant. On the other hand, for $L\neq R^\dagger$ the vaccum condensate changes by a unitary matrix $\mathcal{U}=LR^\dagger$. Value $\mathcal{U}=1$ corresponds to the absence of the Goldstone bosons (pions). On general grounds we know that a chiral transformation with $L\neq R^\dagger$ will produce pions (non-linear realization of symmetry). Hence, the quark condensate with some unitary matrix $\mathcal{U}$ in place of the unit matrix describes a state with pions rather then the vacuum. By letting $\mathcal{U}$ vary with the space-time point $\mathcal{U}=\mathcal{U}(x)$ we account for a dynamical pion field. Thus we identify $\mathcal{U}(x)=U(x)$, the same $U(x)$ that appears in your Lagrangian. It is unitary by construction and under the chiral transformations it changes as
$$U(x)\to LU(x)R^\dagger$$
This explanation is not crystal-clear to me. Some conclusions do not seem to follow inevitably. Moreover, it can be plain wrong. Hopefully, someone more qualified will correct me.
answered Jul 15, 2015 by (220 points)
## Your answer
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http://math.stackexchange.com/questions/235350/what-is-the-difference-between-kernel-and-null-space/235353 | What is the difference between kernel and null space?
What is the difference, if any, between kernel and null space?
I previously understood the kernel to be of a linear map and the null space to be of a matrix: i.e., for any linear map $f : V \to W$,
$$\ker(f) \cong \operatorname{null}(A),$$
where
• $\cong$ represents isomorphism with respect to $+$ and $\cdot$, and
• $A$ is the matrix of $f$ with respect to some source and target bases.
However, I took a class with a professor last year who used $\ker$ on matrices. Was that just an abuse of notation or have I had things mixed up all along?
-
"Was that just an abuse of notation or have I had things mixed up all along?" Neither. Different courses/books will maintain/not maintain such a distinction. If a matrix represents some underlying linear transformation of a vector space, then the kernel of the matrix might mean the set of vectors sent to 0 by that transformation, or the set of lists of numbers (interpreted as vectors in $\mathbb{R}^n$ representing those vectors in a given basis, etc. | 2015-07-03 22:22:40 | {"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": 0.8531627655029297, "perplexity": 265.3600723778699}, "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-2015-27/segments/1435375096287.97/warc/CC-MAIN-20150627031816-00044-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://cpaadsacademy.com/sodium-carbonate-dwvvu/e5e8c9-floor-and-ceiling-functions | # floor and ceiling functions
⌊ for ceiling and <. This function is also declared in “cmath” header file in C++ language. ] ⌋ Topics similar to or like Floor and ceiling functions. One of the requirements can then be formulated asf−1(y)f^{-1}(y)f−1(y) must be integer fo… Flooring and Ceiling Functions. ⌊ x The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to .The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. {\displaystyle \lceil x\rceil } ) Floor And Ceiling Functions In Js. double floor ( double x ); Floor function and its antiderivatives.svg 720 × 540; 32 KB. But the online help provided in 2010 does not reflect this behavior. This identity doesn't in any way help understanding what the floor function is. + Ceiling function.svg 1,000 × 1,000; 16 KB. − 4 ( ⌋ The ceiling function is implemented in the Wolfram Language as Ceiling[z], where it is generalized to complex values of as illustrated above.. Log InorSign Up. The CEILING function. to the nearest integer with tie breaking towards positive infinity is given by ⌉ Choose the greatest one (which is 2 in this case), The greatest integer that is less than (or equal to) 2.31 is 2, Floor Function: the greatest integer that is less than or equal to x, Ceiling Function: the least integer that is greater than or equal to x. Properties of the Floor and Ceiling Functions. ⌊ a [27], The floor function appears in several formulas characterizing prime numbers. This function returns the rounded up number which is nearest to the specified multiple of significance. {\displaystyle {\text{rpi}}(x)} ⌈ Syntax Syntax: veil(x) Where x is a numeric value Example of ceil() Which leads to our definition: Floor Function: the greatest integer that is less than or equal to x. We can round a number upwards to the nearest integer (with a ceiling function), or down with a floor function. {\displaystyle \operatorname {ceil} (2.4)=\lceil 2.4\rceil =3} ) ⌊ or x 2 ⌈ ( ⌉ Figure 2. 1 There are lots of integers less than 2.31. Although the details differ between programs, most implementations support a second parameter—a multiple of which the given number is to be rounded to. An example could be f(x)=xf(x) = \sqrt{x}f(x)=x. The floor function is similar to the ceiling function, which rounds up. is the same as The ceil() function will return the mathematical ceiling value i.e. Round ceil and floor matlab datenumbers file exchange rounding mode ceiling matlab simulink شرح خصائص الـ round fix ceil floor الخاصه ببرنامج الماتلاب 2 4 one sided limits. ( x ⌈ The truncation of a negative number is given by Ceil and floor functions are different in many respects. [ {\displaystyle \lceil x\rceil } The reason for this is historical, as the first machines used ones' complement and truncation was simpler to implement (floor is simpler in two's complement). ) n ⌉ x 3 ⌋ , and 1994).. The Floor of 2.31 is 2 Rounding and truncating numbers in javascript pawelgrzybek com php ceil function w3resource postgresql ceiling function w3resource how to use the excel ceiling function exceljet . for floor. ( floor() floor() method in Python returns floor of x i.e., the largest integer not greater than x. Syntax: import math math.floor(x) Parameter: x-numeric expression.Returns: largest integer not greater than x. ] [ x None of the functions discussed in this article are continuous, but all are piecewise linear: the functions ⌊ The function will return a number that is rounded up to a supplied number that is away from zero to the nearest multiple of a given number. x ⌋ ⌊ It takes single value whoes floor value is to be calculated. {\displaystyle [x]} The fractional part function also shows up in integral representations of the Riemann zeta function. = ceiling(x) Where x = input vector or a value. This definition can be extended to real x and y, y ≠ 0, by the formula. The Ceiling of 2.31 is 3. The symbols for floor and ceiling are like the square brackets [ ] with the top or bottom part missing: But I prefer to use the word form: floor(x) and ceil(x). 1.3, p. 46. These characters are provided in Unicode: In the LaTeX typesetting system, these symbols can be specified with the \lfloor, \rfloor, \lceil and \rceil commands in math mode. n ⌋ {\displaystyle x} Both of these functions take a numerical value as an argument. x ⌈ floor 2 x 1. ⌊ Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. the largest integer value which is not greater than the numerical value passed. is The number $$n$$ is assumed to be an integer. Define bxcto be the integer n such that n x < n +1: Definition (The Ceiling Function) Let x 2R. 0 ≤ r < 1. x ⌊ n 2 The notation for the ceiling function is: ceil (x) = ⌈x⌉. 2 Mahler[37] has proved there can only be a finite number of such k; none are known. Jump to navigation Jump to search ← Archive 1 | Archive 2 | Archive 3 Formula disrupts article flow. ) ⌈ Note: Both floor() and ceiling() values will round of the given input values. x FORTRAN was defined to require this behavior and thus almost all processors implement conversion this way. Floor and ceiling in R is demonstrated with examples in this chapter. Floor and Ceiling question. ⌋ x . ⌊ Ceil (short for ceiling) and floor function are both mathematical functions. CEILING(value, [factor]) Unlike MROUND and FLOOR, this time I’m taking you directly to the examples. n Free Floor/Ceiling Equation Calculator - calculate equations containing floor/ceil values and expressions step by step This website uses cookies to ensure you get the best experience. The floor function is a type of step function where the function is constant between any two integers. Floor and Ceiling Functions - Problem Solving. Floor Function. By using this website, you agree to our Cookie Policy. Similarly, the ceiling function maps ] . | [citation needed]. n is any function with a continuous derivative in the closed interval [a, b], Letting masuzi 8 hours ago Uncategorized Leave a comment 0 Views. or ]x[ for ceiling. At points of continuity the series converges to the true value. CEILING and FLOOR functions. x x . − floor() function takes the vector or column of the dataframe in R and rounds down those values. ⌊ ⌋ ⌉ The floor Function. s If m and n are coprime integers, then ∑ 1≤i≤n-1 floor(im/n) = (m-1)(n-1)/2. [7][8] Sometimes The integral part or integer part of a number (partie entière in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. ⌈ for floor and >. where By using this website, you agree to our Cookie Policy. 2.4 Assuming such shifts are "premature optimization" and replacing them with division can break software. ⌊ ⌋ 6 {\displaystyle [\![x]\!]} {\displaystyle \left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor } Figure 1. For other uses, see Floor (disambiguation) and Ceiling (disambiguation). ⌈ ⌊ ⌈ ⌊ x These characters are provided in Unicode: U+2308 ⌈ LEFT CEILING (HTML ⌈⧼dot-separator⧽ ⌈) ] ( x ) {\displaystyle \lfloor x\rfloor } The "Int" function (short for "integer") is like the "Floor" function, BUT some calculators and computer programs show different results when given negative numbers: With the Floor Function, we "throw away" the fractional part. In mathematics and computer science, the floor function is the function that takes as input a real number $${\displaystyle x}$$, and gives as output the greatest integer less than or equal to $${\displaystyle x}$$, denoted $${\displaystyle \operatorname {floor} (x)}$$ or $${\displaystyle \lfloor x\rfloor }$$. The floor and ceiling functions give us the nearest integer up or down. Ceilfloor nt.png 360 × 252; 1 KB. ⌉ Excel CEILING and FLOOR Functions allow you to round values up or down to the nearest value divisible by a specified number. , and rounding towards even can be expressed with the more cumbersome x 0. ⌈ 3 − [ It is a straightforward deduction from Wilson's theorem that[31], None of the formulas in this section are of any practical use. x 4 Share. is given by a version of Legendre's formula[22]. 1 The Floor and Ceiling Functions 2 Theorems 3 Applications 4 Assignment Robb T. Koether (Hampden-Sydney College) Direct Proof – Floor and Ceiling Wed, Feb 13, 2013 2 / 21 . ri Oh no! This identity doesn't in any way help understanding what the floor function is. 1994, p. 67). Un article de Wikipédia, l'encyclopédie libre. These characters are provided in Unicode: Ceil vs Floor Functions. How do the FLOOR and CEILING Functions Work? Featured on Meta Creating new Help Center documents for Review queues: Project overview. But floor function will round off the nearest values which should also be less than the input value.In the case of the ceiling function, it rounds off the nearest value which should also be greater than the input value.. x ⌈ smallest integer value … 2 ⌊ Rounding And Truncating Numbers In Javascript Pawelgrzybek Com Php Ceil Function W3resource Postgresql Ceiling … [50] This has followed through to the Office Open XML file format. Figure 1. + [} 2 x for floor and Z ] The study of Waring's problem has led to an unsolved problem: Are there any positive integers k ≥ 6 such that[36]. + − ϕ = sgn The value of 21 on applying floor() function is: 21 The value of -23.6 on applying floor() function is: -24 The value of 14.2 on applying floor() function is: 14 ceil() It accepts a number with decimal as parameter and returns the integer which is greater than the number itself. Floor (0) = ⌊0⌋ = 0. It would use the same arithmetic sign (positive or negative) as per the provided number argument. For an arbitrary real number This module includes two object type functions, math.floor() and math.ceil(). {\displaystyle x} Microsoft Excel used almost exactly the opposite of standard notation, with INT for floor, and FLOOR meaning round-toward-zero, and CEILING meaning round-away-from-zero. Since none of the functions discussed in this article are continuous, none of them have a power series expansion. For any ADC the mapping from input voltage to digital output value is not exactly a floor or ceiling function as it should be. rpi 2 Floor And Ceiling Functions In Javascript. ⌋ Floor (2.1) = ⌊2.1⌋ = 2. Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Proof involving Big O and floor. , denoted is itself 2 1 x n or . ⌈ 6 rpi Related. x n Graham, Knuth, & Patashnik, p. 85 and Ex. Commonalities in both these functions. ⌊ x k We invoke Math.Ceiling and Floor, often with Doubles with fractional parts.Double. In some sources, boldface or double brackets The table below shows values for the function from -5 to 5, along with the corresponding graph: Learn how and when to remove this template message, J.E.blazek, Combinatoire de N-modules de Catalan, https://en.wikipedia.org/w/index.php?title=Floor_and_ceiling_functions&oldid=992707368, Short description is different from Wikidata, Articles with unsourced statements from November 2020, Articles lacking reliable references from July 2019, Articles with unsourced statements from November 2018, Articles with unsourced statements from March 2019, Articles needing additional references from August 2008, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 6 December 2020, at 18:10. floor(x) function in R rounds to the nearest integer that’s smaller than x. {\displaystyle \operatorname {floor} (2.4)=\lfloor 2.4\rfloor =2} BUT many calculators and computer programs use frac(x) = x − int(x), and so their result depends on how they calculate int(x): So be careful using this function with negative values. That's easy: no change! • ⌈ x ⌉ = the smallest integer greater than or equal to x. The floor and ceiling functions are usually typeset with left and right square brackets, where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing (⌊ ⌋ for floor and ⌈ ⌉ for ceiling). {\displaystyle \lfloor x\rceil =\left\lfloor x+{\tfrac {1}{2}}\right\rfloor +\left\lceil {\tfrac {2x-1}{4}}\right\rceil -\left\lfloor {\tfrac {2x-1}{4}}\right\rfloor -1} [ floor() and ceil() function Python; Floor and Ceil from a BST in C++; Find floor and ceil in an unsorted array using C++. Topic. Floor (3) = ⌊3⌋ = 3. ( 1. x =CEILING(number, significance) The function uses the following arguments: 1. ( } The floor() function will return the mathematical floor value of that numerical value passed as argument i.e. Free Floor/Ceiling Equation Calculator - calculate equations containing floor/ceil values and expressions step by step This website uses cookies to ensure you get the best experience. e.g. Mathematical functions taking a real input and rounding it down or up, respectively. Analog-to-digital converter-Wikipedia. n + Floor and ceiling functions. In words, this is the integer that has the largest absolute value less than or equal to the absolute value of x. The infinite upper limit of the sum can be replaced with, Ribenboim, p.180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations ... ", Hardy & Wright, pp.344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α ... could be expressed independently of the primes. The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, follows the mathematical definition of ceiling for its ceiling function, with an optional parameter for Excel compatibility. = x 0. , rounding = 1 Obviously the truncation of Proving Floor and Ceiling of a Rational Number . , The floor()function will return the mathematical floor value of that numerical value passed as argument i.e. ( The math module which comes pre-installed with Python. The greatest integer that is less than (or equal to) 2.31 is 2. ceil . I've also tried the one below but same error: Similarly, the ceiling function maps $${\displaystyle x}$$ to the least integer greater than or equal to $${\displaystyle x}$$, denoted $${\displaystyle \operatorname {ceil} (x)}$$ or $${\displaystyle \lceil x\rceil }$$. x ( {\displaystyle \operatorname {floor} (x)} 1 ⌋ 2 floor and ceiling functions ... Media in category "Floor and ceiling" The following 12 files are in this category, out of 12 total. = The floor and ceiling functions look like a staircase and have a jump discontinuity at each integer point. The number of digits in base b of a positive integer k is, Let n be a positive integer and p a positive prime number. = ⌋ So if you want more details (not necessary for learning the Ceiling function), please refer my tutorial on the use of FLOOR function. {\displaystyle {\text{rpi}}(x)=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =\left\lceil {\tfrac {\lfloor 2x\rfloor }{2}}\right\rceil } , where sgn is the sign function. There are many interesting and useful properties involving the floor and ceiling functions, some of which are listed below. ) The J Programming Language , a follow on to APL that is designed to use standard keyboard symbols, uses >. Won't mind having to use awk i fneed be, but not sure how to call the function. [48] The language APL uses ⌊x for floor. 2 Note that being continuous and monotonically increasing ensures a well-defined inverse f−1f^{-1}f−1. ⌊ ) [ The truncation of any real number can be given by: Excel MROUND, CEILING and FLOOR function examples Excel How Tos, Shortcuts, Tutorial, Tips and Tricks on Excel Office. x x [citation needed], The fractional part is the sawtooth function, denoted by floor 2 1 Floor and ceiling in R is demonstrated with examples in this chapter. , and gives as output the greatest integer less than or equal to ϕ Carl Friedrich Gauss introduced the square bracket notation [35], (i) ⌈ 2 . The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to .The name and symbol for the floor function were coined by K. E. Iverson (Graham et al. 3 Commenting is not possible for this post, but feel free to leave a question, correction or any comment by using the contact page 3.15, Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function, These formulas are from the Wikipedia article, Crandall & Pomerance, Ex. {\displaystyle \{x\}} 2. [ The floor and ceiling function are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, and, e.g., in the LaTeX typesetting system these symbols can be specified with the \lfloor, \rfloor, \lceil and \rceil commands in … 2.4 {\displaystyle \lfloor \,\rfloor } ⌊ { + title Definition (The Floor Function) Let x 2R. [citation needed], A bit-wise right-shift of a signed integer ⌉ x x and ⌉ { n [33][34], Ramanujan submitted these problems to the Journal of the Indian Mathematical Society. ALGOL usesentier for floor. {\displaystyle 0} , floor and ceiling may be defined by the equations, Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation. + Number (required argument) – This is the value that we wish to round off. + ... Hello, My round and floor functions in C program behaves weird. These formulas can be used to simplify expressions involving floors and ceilings.[10]. The floor corner brackets ⌊ and ⌋, the ceiling corner brackets ⌈ and ⌉ are used to denote the integer floor and ceiling functions. The ceiling function is usually denoted by ceil(x) or less commonly ceiling(x) in non-APL computer languages that have a notation for this function. ⌊ 4 ⌉ ⌋ ( and n {\displaystyle x} = 0\le r <1. Nor is it somthing special: there are probably dozens of identities involving the floor function. In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. ] {\displaystyle \operatorname {sgn}(x)\lfloor |x|\rfloor } floor(x) function in R rounds to the nearest integer that’s smaller than x. x {\displaystyle \lfloor x\rfloor } − The Floor of 5 is 5 The Ceiling of 5 is 5… The above arguments in the syntax are the same in FLOOR function. The datatype of variable should be double/ float/ long double only. The floor function returns the largest possible integer value which is equal to the value or smaller than that. ⌋ {\displaystyle \phi (x)} + is equal to 1 if m divides n, and to 0 otherwise, it follows that a positive integer n is a prime if and only if[28], One may also give formulas for producing the prime numbers. 2 The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. The ceiling function is implemented in the Wolfram Language as Ceiling[z], where it is generalized to complex values of as illustrated above.. {\displaystyle x} = The Floor Function and the Ceiling Function Main Concept The floor of a real number x , denoted by , is defined to be the largest integer no larger than x . Cassels, Hardy & Wright, and Ribenboim use Gauss's notation, Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's. = Likewise for Ceiling: Ceiling Function: the least integer that is greater than or equal to x. | ⌊ may also be taken as the definition of floor and ceiling. Floor Function. ⌊ Browse other questions tagged functions ceiling-and-floor-functions or ask your own question. The exponent of the highest power of p that divides n! The floor and ceiling functions are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing. ⌉ {\displaystyle \lfloor 2\rfloor =\lceil 2\rceil =2} 0 There seems no likelihood of this, but it cannot be ruled out as entirely impossible.". Both these function can take negative and positive numbers. ⌊ for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives, This formula is valid for all s with real part greater than −1, (except s = 1, where there is a pole) and combined with the Fourier expansion for {x} can be used to extend the zeta function to the entire complex plane and to prove its functional equation.[26]. The fractional part function has Fourier series expansion[18]. x Given real numbers x and y, integers k, m, n and the set of integers In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals. The ceil function returns the smallest value, whereas the floor function returns the largest value for the specified number. The Wikipedia page Floor and ceiling functions furthermore lists a lot of properties (very few proofs or derivations, though). k 1 | 2 In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. (e.g., ⌊3.7⌋ = 3.) For example, since ∑ 4 ⌋ What if we want the floor or ceiling of a number that is already an integer? = Definition (The Floor Function) Let x 2R. ⌋ ⌉ = {\displaystyle \{x\}} 1 ⌊ minus an integrality indicator for The floor and ceiling functions give you the nearest integer up or down. The input to the ceiling function is any real number x and its output is the smallest integer greater than or equal to x. m Whats people lookup in this blog: Matlab Floor And Ceiling Functions ] Figure 2. 2 ⌊ x m {\displaystyle \left\lfloor {\frac {x}{2^{n}}}\right\rfloor } {\displaystyle {\tfrac {2x-1}{4}}} in his third proof of quadratic reciprocity (1808). + Define dxeto be the integer n such that n 1 < x n: Robb T. Koether (Hampden-Sydney College) Direct Proof – Floor and Ceiling Wed, Feb 13, 2013 3 / 21 2 Suppose the floor and ceiling of 4 are 4 for both of them. | 2021-05-07 12:11:05 | {"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": 2, "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": 0.6457322239875793, "perplexity": 1110.7632483075063}, "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-2021-21/segments/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00621.warc.gz"} |
https://saravananthirumuruganathan.wordpress.com/2010/02/17/introduction-to-bayesian-decision-theory-part-1/ | Feeds:
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## Introduction to Bayesian Decision Theory – Part 1
In this series of articles , I intend to discuss Bayesian Decision Theory and its most important basic ideas. The articles are mostly based on the classic book "Pattern Classification" by Duda,Hart and Stork. If you want the ideas in all its glory, go get the book !
As I was reading the book, I realized that in its heart , this field is a set of mostly common sense ideas validated by rigorous mathematics. So I will try to discuss the basic ideas in plain english without much mathematical rigor. Since I am still learning how to best explain these complex ideas, any comments on how to improve will be welcome !
You may ask what happened to PCA . Well, I still intend to write more on it but have not found enough time to sit and write it all. I have written a draft version of it but felt it was too technical and not very intuitive.So I am hoping to rewrite it again.
### Background
You would need to know basics of probability to understand the following. In particular the ideas of Prior , Posterior and the idea of Likelihood . Of course if you know all these , you will know the big idea of Bayes Theorem . I will try to explain them lightly but if you have any doubts check out Wikipedia or some old books.
We will take the example used in Duda et al. There are two types of fish : Sea Bass and Salmon. We catch a lot of fish of these two types but which are mixed together and our aim is to separate them by automation. We have a conveyor belt in which the fishes come one by one and we need to decide if the current fish is sea bass or salmon. Of course, we want to make it as accurate as possible but also don’t want to spend lot of money on this project. This is in its heart a classification project. We will be given a few examples of both sea bass and salmon and based on it we need to infer the general characteristics using which we can distinguish them.
### Basic Probability Ideas
Now let us be slightly more formal. We say that there are two "classes" of fish – sea bass and salmon. According to our system, there is no other type of fish. If we treat it as a state machine , then our system has two states. The book uses a notation of $\omega_1 \; and \; \omega_2$ to represent them. We will use the names seabass and salmon as it is more intuitive.
The first basic idea is that of prior probability . This is represented as $P(seabass) \; and \; P(salmon)$ which basically give the probability that the next fish in the conveyor is seabass or salmon. Of course, both of them have to sum to one. From the Bayesian perspective, this probability is usually obtained from prior (domain) knowledge. We will not talk about Frequentist interpretation as we will focus on Bayesian decision theory.
Let us assume that we use length of the fish to differentiate the fishes. So whenever the fish comes to the conveyor belt, we calculate its length (how, we don’t really care here ) . So we have transformed the fish into a simple representation using a single number, its length. So the length is a feature that we use to classify and the step of converting the fish into length is called feature extraction .
In a real life scenario, we will have multiple features and the input will converted to a vector. For eg we may use length , lightness of skin , fin length etc as feature. In this case , the fish will be transformed into a triplet. Converting the input to a feature vector makes further processing easy and more robust. We will usually use the letter x to represent the feature. So you can consider $P(x)$ is the probability of evidence. Eg lets say we got a fish (we dont know what it is yet) of length 5 inches. Now $P(x)$ gives the probability that some fish (either seabass or salmon) has the length 5 inches.
The next idea is that of likelihood . It is also called class conditional probability. It is represented as either $P(x|seabass) \; or \; P(x|salmon)$ . The interpretation is simple. This answers the question that if the fish is seabass what is the probability that it will have length $x$ inches (ditto salmon). Alternatively , what is the probability that a 5 inch seabass exists and so on. Or even how "likely" is a 5 inch seabass ?
The posterior probability is the other side of the story. This is represented by $P(seabass|x) \; or \; P(salmon|x)$ . Intuitively, given that we have a fish of length $x$ inches , what is the probability that it is a seabass (or salmon). The interesting thing is that knowing prior probability and likelihood we can calculate posterior probability using the famous "Bayes Theorem". We can represent it in words as ,
$posterior = \frac{likelihood \times prior}{evidence}$
This gives another rationale for the word "likelihood". Among all other things being equal , the item with higher likelihood is more "likely" to final result. For eg if the likelihood of a 10 inch seabass is more than that of salmon then when we observe an unknown fish of length 10 inches , it is most likely a seabass.
PS : There is an excellent (but long) tutorial on Bayes Theorem at "An Intuitive Explanation of Bayes’ Theorem" . True to its title, it does try to explain the bizarre (atleast initially) result of Bayes Theorem using multiple examples. I highly recommend reading it.
### Bayesian Decision Theory
Let us enter into the decision theory at last. In a very high level definition, you can consider decision theory as a field which studies about "decisions" (to classify as seabass or not to be) – more exactly, it considers these decisions in terms of cost or loss functions. (More on that later). In essence , you can think of decision theory as providing a decision rule which tells us what action to taken when we make a particular observation. Decision theory can be thought of as all about evaluating decision rules. (Of course, I am grossly simplifying things, but I think I have conveyed the essence).
### Informal Discussion of Decision Theory for Two Class System with Single Feature
Let us take a look at the simplest application of decision theory to our problem. We have a two class system (seabass,salmon) and we are using a single feature (length) to make a decision. Be aware that length is not an ideal feature because many a time you will be having both seabass and salmon of same length (say 5 inches). So when we come across a fish with length 5 inches, we are stuck. We don’t know what decision to take as we know both seabass and salmon can be 5 inches. Decision theory to the rescue !
Instead of providing the theoretical ideas, I will discuss various scenarios and what is the best decision theoretic action to do. In all the scenarios let us assume that we want to be as accurate as possible.
### Case I : We don’t know anything and we are not allowed to see the fish
This is the worst case to be in. We have no idea about seabass and salmon (a vegetarian , perhaps ?🙂 ). You are also not allowed to see the fish. But you are asked is the next fish in conveyor a seabass or salmon ? All is not lost – The best thing to do is to randomize. So the decision rule is with probability 50% say the next fish is seabass and with probability 50% say it is salmon.
Convince yourself that this is the best thing to do – Not only when the seabass and salmon are in 50:50 , even when they are in 90:10 ratio.
### Case II : You know the prior probability but still you don’t see the fish
We are in a slightly better position here. We don’t get to see the fish yet , but we know the prior probability that the next fish is a seabass or salmon. ie We are give $P(seabass) \; and \; P(salmon)$ Remember, we want to be as accurate as possible and we want to as reliable about accuracy rate as possible.
A common mistake to do is to randomize again. ie with $P(seabass)$ say that the next fish is seabass and salmon otherwise. For eg let us say, $P(seabass) = .70 \; and \; P(salmon) = 0.3$ . Let me attempt an informal reasoning – In the (sample) worst case, you will get first 40 as seabass, next 30 as salmon and next 30 as seabass. But you say first 30 as salmon and next 70 as seabass. In this hypothetical example you are only at the most 40% accurate even though you can theoretically do better.
What does decision theory say here ? If $P(seabass) > P(salmon)$ then ALWAYS say seabass. Else ALWAYS say salmon. In this case the accuracy rate is $max(P(seabass),P(salmon))$ . Conversely, the error rate is the minimum of both the prior probabilities. Convince yourself that this is the best you can do . It sure is counterintuitive to always say seabass when you know you will get salmon too. But we can easily prove that this is the best you can do "reliably".
Mathematically, decision rule is decide seabass if $P(seabass) > P(salmon)$ else decide salmon .
Error is $min(P(seabass),P(salmon)$ .
### Case III : You know the likelihood function and the length but not the prior probability
This case is really hypothetical. ie we can see the fish and hence find its length. Let say x inches. We have $P(x inches|seabass) \; and \; P(x inches | salmon)$ but we don’t know the prior probability. The decision rule here is : For each fish , find the appropriate likelihood values. If the likelihood of seabass is higher than that of salmon , say the fish is seabass and salmon otherwise.
Note that we are making a decision based on our "observation" in contrast to previous cases. Unless, you are really unlucky and the prior probabilities are really skewed you can do well with this decision rule.
Mathematically, decision rule is decide seabass if $P(x|seabass) > P(x|salmon)$ else decide salmon .
### Case IV : You know the length, prior probability and the likelihood function
This is the scenario we are mostly in. We know the length of the fish (say 5 inches). We know the prior probability (say 60% salmon and 40% seabass). We also know the likelihood of them. (say P(5 inches|seabass) is 60% and P(5 inches|salmon) is 10% )
Now we can apply our favorite Bayes rule to get posterior probability. If the length of the fish is 5 inches then what is the probability that it a seabass ? A salmon ? Once you can calculate the posterior , the decision rule becomes simpler. If the posterior probability of seabass is higher than say the fish is seabass else say it is salmon.
Mathematically, decision rule is decide seabass if $P(seabass|x) > P(salmon|x)$ else decide salmon . This rule is very important and is called as Bayes Decision Rule.
For this decision rule, the error is $min(P(seabass|x),P(salmon|x)$
We can expand Bayes decision rule using Bayes theorem.
Decide seabass if $p(x|seabass)P(seabass) > p(x|salmon)P(salmon)$ else decide salmon.
There are two special cases.
1. If likelihood are equal then our decision depends on prior probabilities.
2. If prior probabilities are equal then our decisions depend on likelihoods.
We have only scratched the surface of decision theory. In particular we did not focus much on bounding the error today. Also we did not discuss the cases where there are multiple classes or features. Hopefully, I will discuss them in a future post.
### Reference
Pattern Classification by Duda,Hart and Stork. Chapter 2. | 2016-12-03 06:42:29 | {"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": 21, "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": 0.7266072034835815, "perplexity": 605.0396517099508}, "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/1480698540909.75/warc/CC-MAIN-20161202170900-00332-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://cforall.uwaterloo.ca/trac/changeset/31127335c8c2e22e0ffdd7913bd847d003707d55/ | # Changeset 3112733
Ignore:
Timestamp:
Apr 12, 2022, 1:52:32 PM (2 months ago)
Branches:
enum, master
Children:
6b06abe
Parents:
e88c2fb
Message:
Filled in all of Chapter 4.
It's not great but it's worth discussing
File:
1 edited
### Legend:
Unmodified
re88c2fb Finally, the last important part of the \io subsystem is it's interface. There are multiple approaches that can be offered to programmers, each with advantages and disadvantages. The new \io subsystem can replace the C runtime's API or extend it. And in the later case the interface can go from very similar to vastly different. The following sections discuss some useful options using @read@ as an example. The standard Linux interface for C is : @ssize_t read(int fd, void *buf, size_t count);@. @ssize_t read(int fd, void *buf, size_t count);@ \subsection{Replacement} Replacing the C \glsxtrshort{api} Replacing the C \glsxtrshort{api} is the more intrusive and draconian approach. The goal is to convince the compiler and linker to replace any calls to @read@ to direct them to the \CFA implementation instead of glibc's. This has the advantage of potentially working transparently and supporting existing binaries without needing recompilation. It also offers a, presumably, well known and familiar API that C programmers can simply continue to work with. However, this approach also entails a plethora of subtle technical challenges which generally boils down to making a perfect replacement. If the \CFA interface replaces only \emph{some} of the calls to glibc, then this can easily lead to esoteric concurrency bugs. Since the gcc ecosystems does not offer a scheme for such perfect replacement, this approach was rejected as being laudable but infeasible. \subsection{Synchronous Extension} An other interface option is to simply offer an interface that is different in name only. For example: @ssize_t cfa_read(int fd, void *buf, size_t count);@ \noindent This is much more feasible but still familiar to C programmers. It comes with the caveat that any code attempting to use it must be recompiled, which can be a big problem considering the amount of existing legacy C binaries. However, it has the advantage of implementation simplicity. \subsection{Asynchronous Extension} It is important to mention that there is a certain irony to using only synchronous, therefore blocking, interfaces for a feature often referred to as non-blocking'' \io. A fairly traditional way of doing this is using futures\cit{wikipedia futures}. As simple way of doing so is as follows: @future(ssize_t) read(int fd, void *buf, size_t count);@ \noindent Note that this approach is not necessarily the most idiomatic usage of futures. The definition of read above returns'' the read content through an output parameter which cannot be synchronized on. A more classical asynchronous API could look more like: @future([ssize_t, void *]) read(int fd, size_t count);@ \noindent However, this interface immediately introduces memory lifetime challenges since the call must effectively allocate a buffer to be returned. Because of the performance implications of this, the first approach is considered preferable as it is more familiar to C programmers. \subsection{Interface directly to \lstinline{io_uring}} Finally, an other interface that can be relevant is to simply expose directly the underlying \texttt{io\_uring} interface. For example: @array(SQE, want) cfa_io_allocate(int want);@ @void cfa_io_submit( const array(SQE, have) & );@ \noindent This offers more flexibility to users wanting to fully use all of the \texttt{io\_uring} features. However, it is not the most user-friendly option. It obviously imposes a strong dependency between user code and \texttt{io\_uring} but at the same time restricting users to usages that are compatible with how \CFA internally uses \texttt{io\_uring}. | 2022-06-26 11:31:47 | {"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": 0.24534249305725098, "perplexity": 3563.246584392811}, "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/1656103205617.12/warc/CC-MAIN-20220626101442-20220626131442-00498.warc.gz"} |
https://stringi.gagolewski.com/rapi/stri_startsendswith.html | # stri_startsendswith: Determine if the Start or End of a String Matches a Pattern
## Description
These functions check if a string starts or ends with a match to a given pattern. Also, it is possible to check if there is a match at a specific position.
## Usage
stri_startswith(str, ..., fixed, coll, charclass)
stri_endswith(str, ..., fixed, coll, charclass)
stri_startswith_fixed(
str,
pattern,
from = 1L,
negate = FALSE,
...,
opts_fixed = NULL
)
stri_endswith_fixed(
str,
pattern,
to = -1L,
negate = FALSE,
...,
opts_fixed = NULL
)
stri_startswith_charclass(str, pattern, from = 1L, negate = FALSE)
stri_endswith_charclass(str, pattern, to = -1L, negate = FALSE)
stri_startswith_coll(
str,
pattern,
from = 1L,
negate = FALSE,
...,
opts_collator = NULL
)
stri_endswith_coll(
str,
pattern,
to = -1L,
negate = FALSE,
...,
opts_collator = NULL
)
## Arguments
str
character vector
...
supplementary arguments passed to the underlying functions, including additional settings for opts_collator, opts_fixed, and so on.
pattern, fixed, coll, charclass
character vector defining search patterns; for more details refer to stringi-search
from
integer vector
negate
single logical value; whether a no-match to a pattern is rather of interest
to
integer vector
opts_collator, opts_fixed
a named list used to tune up the search engine’s settings; see stri_opts_collator and stri_opts_fixed, respectively; NULL for the defaults
## Details
Vectorized over str, pattern, and from or to (with recycling of the elements in the shorter vector if necessary).
If pattern is empty, then the result is NA and a warning is generated.
Argument start controls the start position in str where there is a match to a pattern. to gives the end position.
Indexes given by from or to are of course 1-based, i.e., an index 1 denotes the first character in a string. This gives a typical R look-and-feel.
For negative indexes in from or to, counting starts at the end of the string. For instance, index -1 denotes the last code point in the string.
If you wish to test for a pattern match at an arbitrary position in str, use stri_detect.
stri_startswith and stri_endswith are convenience functions. They call either stri_*_fixed, stri_*_coll, or stri_*_charclass, depending on the argument used. Relying on these underlying functions directly will make your code run slightly faster.
Note that testing for a pattern match at the start or end of a string has not been implemented separately for regex patterns. For that you may use the ‘^’ and ‘\$’ meta-characters, see stringi-search-regex.
## Value
Each function returns a logical vector.
## Author(s)
Marek Gagolewski and other contributors
The official online manual of stringi at https://stringi.gagolewski.com/
Gagolewski M., stringi: Fast and portable character string processing in R, Journal of Statistical Software 103(2), 2022, 1-59, doi: 10.18637/jss.v103.i02
Other search_detect: about_search, stri_detect()
## Examples
stri_startswith_charclass(' trim me! ', '\\p{WSpace}')
## [1] TRUE
stri_startswith_fixed(c('a1', 'a2', 'b3', 'a4', 'c5'), 'a')
## [1] TRUE TRUE FALSE TRUE FALSE
stri_detect_regex(c('a1', 'a2', 'b3', 'a4', 'c5'), '^a')
## [1] TRUE TRUE FALSE TRUE FALSE
stri_startswith_fixed('ababa', 'ba')
## [1] FALSE
stri_startswith_fixed('ababa', 'ba', from=2)
## [1] TRUE
stri_startswith_coll(c('a1', 'A2', 'b3', 'A4', 'C5'), 'a', strength=1)
## [1] TRUE TRUE FALSE TRUE FALSE
pat <- stri_paste('\u0635\u0644\u0649 \u0627\u0644\u0644\u0647 ',
'\u0639\u0644\u064a\u0647 \u0648\u0633\u0644\u0645XYZ')
stri_endswith_coll('\ufdfa\ufdfa\ufdfaXYZ', pat, strength=1)
## [1] TRUE | 2022-10-06 20:24:56 | {"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": 0.2655527889728546, "perplexity": 8733.719934229865}, "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-2022-40/segments/1664030337855.83/warc/CC-MAIN-20221006191305-20221006221305-00777.warc.gz"} |
http://www.lastfm.es/user/bedontlikeyou/library/music/The+Strokes/_/The+End+Has+No+End?setlang=es | # Colección
Música » The Strokes »
## The End Has No End
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The End Has No End 3:05 23 Ago 2011, 21:00 | 2014-03-13 19:26:03 | {"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": 0.9644241333007812, "perplexity": 14610.634650799502}, "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/1394678680766/warc/CC-MAIN-20140313024440-00035-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.tutorialspoint.com/How-does-del-operator-work-on-list-in-Python | # How does del operator work on list in Python?
PythonServer Side ProgrammingProgramming
The del operator removes a specific index from given list. For example, if you want to remove the element on index 1 from list a, you'd use:
## Example
a = [3, "Hello", 2, 1]
del a[1]
print(a)
## Output
This will give the output −
[3, 2, 1]
Note that del removes the elements in place, ie, it doesn't create a new list.
Updated on 12-Jun-2020 06:20:34 | 2022-08-13 00:49:43 | {"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": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.1731211096048355, "perplexity": 5166.606571714753}, "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-2022-33/segments/1659882571847.45/warc/CC-MAIN-20220812230927-20220813020927-00433.warc.gz"} |
https://codereview.stackexchange.com/questions/245751/a-sieve-of-eratosthenes | # A Sieve of Eratosthenes
I have written this Sieve of Eratosthenes. This is a bit of software that will find all of the prime numbers between 2 and n, where n is a user-defined value.
The code does seem quite bloated and any help at making it more elegant would be great. I know that I am not very good at list comprehensions and if it is possible to utilise them more in this situation I would love to know how.
def sieve_of_eratosthenes(n):
"""
Create a Sieve of Eratosthenes to find all of the prime
numbers between 2 and n
"""
n += 1
num_list = []
[num_list.append(i) for i in range(2, n)]
test = 0
while test < len(num_list):
for i, x in enumerate(num_list):
if num_list[i] % num_list[test] == 0:
if num_list[i] != num_list[test]:
del num_list[i]
test += 1
return num_list
• That is not the Sieve of Eratosthenes, see e.g. my remarks here: codereview.stackexchange.com/a/194762. The Sieve of Eratosthenes algorithms computes multiples, not remainders. Jul 20 '20 at 13:50
Replace the while loop, making it clearer and more pythonic: for test in num_list: | 2021-10-16 22:11:44 | {"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": 0.5102439522743225, "perplexity": 1136.8110637426385}, "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-2021-43/segments/1634323585025.23/warc/CC-MAIN-20211016200444-20211016230444-00071.warc.gz"} |
http://www.aorinevo.com/bcc/mat-180-precalculus/3-exponential-and-logarithmic-functions/3-1-exponential-functions-and-their-graphs/ | # 3.1 | Exponential Functions and Their Graphs
The exponential function $$f$$ with base $$a$$ is denoted by
$$f(x) = a^x$$
where $$a>0, a\not = 1,$$ and $$x$$ is any real number.
In general, to graph
$$f(x) = a \cdot b^{kx-c} + d$$
you can rely on three defining characteristics of the graph of $$f$$:
• The horizontal asymptote: $$y = d$$
• The $$y$$-intercept (one always exists)
Plot the horizontal asymptote using a dashed line, plot the $$y$$-intercept and an additional point. Use a smooth curve to connect the two points and extend the graph so that the curve becomes asymptotic to the line $$y = d$$. You should also extend the curve in the other direction and so that the curve moves away from the line $$y = d$$. Essentially you want to preserve the general shape of the basic exponential function, when graphing its variations.
After $$t$$ years, the balance $$A$$ in an account with principal $$P$$ and annual interest rate $$r$$ (in decimal form) is given by the following formulas.
1. For $$n$$ compoundings per year: $$\displaystyle A = P \left(1 + \frac{r}{n} \right)^{nt}$$
2. For continuous compounding: $$A = Pe^{rt}$$
Graph
$$f(x) = -2^{x-1} +3.$$ | 2017-11-19 21:12:57 | {"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": 2, "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": 0.8683719038963318, "perplexity": 414.72325887095644}, "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-2017-47/segments/1510934805809.59/warc/CC-MAIN-20171119210640-20171119230640-00600.warc.gz"} |
https://stackabuse.com/python-for-nlp-working-with-text-and-pdf-files/ | ## Python for NLP: Working with Text and PDF Files
This is the first article in my series of articles on Python for Natural Language Processing (NLP). In this article, we will start with the basics of Python for NLP. We will see how we can work with simple text files and PDF files using Python.
### Working with Text Files
Text files are probably the most basic types of files that you are going to encounter in your NLP endeavors. In this section, we will see how to read from a text file in Python, create a text file, and write data to the text file.
Create a text file with the following text and save it in your local directory with a ".txt" extension.
Welcome to Natural Language Processing
It is one of the most exciting research areas as of today
We will see how Python can be used to work with text files.
In my case, I stored the file named "myfile.txt" in my root "D:" directory.
Now let's see how we can read the whole contents of the file. The first step is to specify the path of the file, as shown below:
myfile = open("D:\myfile.txt")
To open the file, you can use Python's built-in open function. If you execute the above piece of code and do not see an error, that means your file was successfully opened. Make sure to change the file path to the location in which you saved your text file.
Let's now see what is stored in the myfile variable:
print(myfile)
The output looks like this:
<_io.TextIOWrapper name='D:\\myfile.txt' mode='r' encoding='cp1252'>
The output reads that myfile variable is a wrapper to the myfile.txt file and opens the file in read-only mode.
If you specify the wrong file path, you are likely to get the following error:
myfile222 = open("D:\myfile222.txt")
print(myfile222)
FileNotFoundError: [Errno 2] No such file or directory: 'D:\\myfile222.txt'
Whenever you get Errno 2, there can be two reasons. Either your file doesn't exist or you provided the wrong file path to the open function.
Now, let's read the contents of the file. To do so, you need to call the read() function on the myfile variable, as shown below:
myfile = open("D:\myfile.txt")
In the output, you should see the text of the file, as shown below:
Welcome to Natural Language Processing
It is one of the most exciting research areas as of today
We will see how Python can be used to work with text files.
Now if you try to call the read method again, you will see that nothing will be printed on the console:
print(myfile.read())
This is because once you call the read method, the cursor is moved to the end of the text. Therefore, when you call read again, nothing is displayed since there is no more text to print.
A solution to this problem is that after calling the read() method, call the seek() method and pass 0 as the argument. This will move the cursor back to the start of the text file. Look at the following script to see how this works:
myfile = open("D:\myfile.txt")
myfile.seek(0)
In the output, you will see the contents of the text file printed twice.
Once you are done working with a file, it is important to close the file so that other applications can access the file. To do so, you need to call the close() method.
myfile.close()
##### Reading a File Line by Line
Instead of reading all the contents of the file at once, we can also read the file contents line by line. To do so, we need to execute the readlines() method, which returns each line in the text file as list item.
myfile = open("D:\myfile.txt")
In the output, you will see each line in the text file as a list item:
['Welcome to Natural Language Processing\n', 'It is one of the most exciting research areas as of today\n', 'We will see how Python can be used to work with text files.']
In many cases this makes the text easier to work with. For example, we can now easily iterate through each line and print the first word in the line.
myfile = open("D:\myfile.txt")
for lines in myfile:
print(lines.split()[0])
The output looks like this:
Welcome
It
We
#### Writing to a Text File
To write to a text file, you simply have to open a file with mode set to w or w+. The former opens a file in the write mode, while the latter opens the file in both read and write mode. If the file doesn't exist, it will be created. It is important to mention that if you open a file that already contains some text with w or w+ mode, all the existing file contents will be removed, as shown below:
myfile = open("D:\myfile.txt", 'w+')
In the output, you will see nothing printed on the screen since the file is opened using the w+ mode, all the contents of the file have been removed. If you want to avoid this then you'll want to append text instead, which I cover below as well.
Now, let's write some content in the file using the write() method.
myfile = open("D:\myfile.txt", 'w+')
myfile.write("The file has been rewritten")
myfile.seek(0)
In the script above, we write text to the file and then call the seek() method to shift the cursor back to the start and then call the read method to read the contents of the file. In the output, you will see the newly added content as shown below:
The file has been rewritten
Often times, you dont simply need to wipe out the existing contents of the file. Rather, you may need to add the contents at the end of the file.
To do so, you need to open the file with a+ mode which refers to append plus read.
Again create a file with the following contents and save it as "myfile.txt" in the "D" directory:
Welcome to Natural Language Processing
It is one of the most exciting research areas as of today
We will see how Python can be used to work with text files.
Execute the following script to open the file with the append mode:
myfile = open("D:\myfile.txt", 'a+')
myfile.seek(0)
In the output, you will see the contents of the file.
Next, let's append some text to the file.
myfile.write("\nThis is a new line")
Let's now again read the file contents:
myfile.seek(0)
In the output, you will see the newly appended line at the end of the text as shown below:
Welcome to Natural Language Processing
It is one of the most exciting research areas as of today
We will see how Python can be used to work with text files.
This is a new line
Finally, before moving on to the next section, let's see how context manager can be used to automatically close the file after performing the desired operations.
with open("D:\myfile.txt") as myfile:
Using the with keyword, as shown above, you don't need to explicitly close the file. Rather, the above script opens the file, reads its contents, and then closes it automatically.
### Working with PDF Files
In addition to text files, we often need to work with PDF files to perform different natural language processing tasks. By default, Python doesn't come with any built-in library that can be used to read or write PDF files. Rather, we can use the PyPDF2 library.
Before we can use the PyPDF2 library, we need to install it. If you are using pip installer, you can use the following command to install PyPDF2 library:
$pip install PyPDF2 Alternatively, if you are using Python from Anaconda environment, you can execute the following command at the conda command prompt: $ conda install -c conda-forge pypdf2
Note: It is important to mention here that a PDF document can be created from different sources like word processing documents, images, etc. In this article, we will only be dealing with the PDF documents created using word processors. For the PDF documents created using images, there are other specialized libraries that I will explain in a later article. For now, we will only work with the PDF documents generated using word processors.
To read a PDF document, we first have to open it like any ordinary file. Look at the following script:
import PyPDF2
mypdf = open('D:\Lorem-Ipsum.pdf', mode='rb')
It is important to mention that while opening a PDF file, the mode must be set to rb, which stands for "read binary" since most of the PDF files are in binary format.
Once the file is opened, we will need to call the PdfFileReader() function of the PyPDF2 library, as shown below.
pdf_document = PyPDF2.PdfFileReader(mypdf)
Now using the pdf_document variable, we can perform a variety of read functions. For instance, to get the total number of pages in the PDF document, we can use the numPages attribute:
pdf_document.numPages
Since we only have one 1 page, in our PDF document, you will see 1 in the output.
Finally, to extract the text from the PDF document, you first need to get the page of the PDF document using the getPage() function.
Next, you can call the extractText() function to extract the text from that particular page.
The following script extracts the text from the first page of the PDF and then prints it on the console.
first_page = pdf_document.getPage(0)
print(first_page.extractText())
In the output, you should see the text from the first page of the PDF.
#### Writing to a PDF Document
It is not possible to directly write Python strings to PDF document using the PyPDF2 library due to fonts and other constraints. However, for the sake of demonstration, we will read contents from our PDF document and then will write that content to another PDF file that we will create.
Let's first read the contents of the first page of our PDF document.
import PyPDF2
mypdf = open('D:\Lorem-Ipsum.pdf', mode='rb')
pdf_document.numPages
page_one = pdf_document.getPage(0)
The above script reads the first page of our PDF document. Now we can write the contents from the first page to a new PDF document using the following script:
pdf_document_writer = PyPDF2.PdfFileWriter()
The script above creates an object that can be used to write content to a PDF file. First, we will add a page to this object and pass it the page that we retrieved from the other PDF.
pdf_document_writer.addPage(page_one)
Next, we need to open a new file with wb (write binary) permissions. Opening a file with such permissions creates a new file if one doesn't exist.
pdf_output_file = open('new_pdf_file.pdf', 'wb')
Finally, we need to call the write() method on the PDF writer object and pass it the newly created file.
pdf_document_writer.write(pdf_output_file)
Close both the mypdf and pdf_output_file files and go to the program's working directory. You should see a new file new_pdf_file.pdf in your editor. Open the file and you should see that it contains the contents from the first page from our original PDF.
Let's try to read the contents of our newly created PDF document:
import PyPDF2
pdf_document.numPages
page_one = pdf_document.getPage(0)
print(page_one.extractText())
http://ctan.math.utah.edu/ctan/tex-archive/macros/latex/contrib/lipsum/lipsum.pdf
Execute the following script to see the number of pages in the file:
import PyPDF2
mypdf = open(r'D:\lipsum.pdf', mode='rb')
pdf_document.numPages
In the output, you will see 87 printed out since there are 87 pages in the PDF. Let's print all the pages in the document on the console:
import PyPDF2
mypdf = open(r'D:\lipsum.pdf', mode='rb') | 2020-04-08 11:33:39 | {"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": 0.5969080328941345, "perplexity": 954.8789654609157}, "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/1585371813538.73/warc/CC-MAIN-20200408104113-20200408134613-00272.warc.gz"} |
https://www.theresearchkitchen.com/archives/category/r/page/16 | Categories
## Project Euler Problem #2 (R)
Here is a solution for Project Euler’s Problem #2 in R, which is stated as:
Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Find the sum of all the even-valued terms in the sequence which do not exceed four million.
Firstly ,we will define a function to calculate the Fibonacci numbers below N, where N in this case is 4*10^6:
[code lang=”R”]
fib < – function(n) {
x <- c(length=10)
x[1] <- 1; x[2] <- 2
for (i in 3:n) {
y <- x[i-1]+x[i-2];
if (y > n){ break }
else { x[i] < – y }
}
x
}
[/code]
Next, we just calculate the sum of the even-valued problems, in a similar manner to the way we solved problem #1:
[code lang=”R”]
f < – fib(4E6)
sum(f[f %% 2 ==0])
[/code]
Categories
## Project Euler Problem #1 (R)
Here is a solution for Project Euler’s problem #1 in R. The problem is expressed as:
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
As usual with Project Euler questions, there is an obvious way, and a less obvious, but much more efficient way. In this case, the obvious way is:
[code lang=”R”]
x < – seq(1,999)
sum(x[x %% 3 ==0 | x %% 5 == 0])
[/code]
Which very concisely returns the correct answer, 233168. However, if we use the following intuition:
$S_N = S_3 + S_5 – (S_{3,5})$
i.e. the sum of all numbers divisible by 3 or 5 is the sum of all numbers divisible by 3, plus the sum of all numbers divisible by 5, minus the sum of all numbers divisible by 3 and 5 (as we have double counted them), then we get the correct answer.
Since $S_n = \frac{n(a_1 + a_n)}{2}$, where $$n = \lfloor \frac{N}{n} \rfloor$$, and $$a_n = a_1n$$, the last piece of the puzzle is what to use for $$S_{3,5}$$. This is straightforward, we just use the lowest common multiple of 3 and 5, which in this case is 15. Hence, the R representation of this is:
[code lang=”R”]sum(333*((3+333*3)/2),199*((5+199*5)/2)-66*((15+66*15)/2))[/code]
Categories
## Black-Scholes in R
Here is a simple implementation of the Black-Scholes pricing formula in R. This will return a two-element vector containing the calculated call and put price, respectively.
[source lang=”r”]
# Black-Scholes Option Value
# Call value is returned in values[1], put in values[2]
blackscholes <- function(S, X, rf, T, sigma) {
values <- c(2)
d1 <- (log(S/X)+(rf+sigma^2/2)*T)/(sigma*sqrt(T))
d2 <- d1 – sigma * sqrt(T)
values[1] <- S*pnorm(d1) – X*exp(-rf*T)*pnorm(d2)
values[2] <- X*exp(-rf*T) * pnorm(-d2) – S*pnorm(-d1)
values
}
[/source]
Example use:
> blackscholes(100,110,.05,1,.2) [1] 6.040088 10.675325 | 2021-11-28 07:57:28 | {"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": 0.7619252800941467, "perplexity": 621.1685856047559}, "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-2021-49/segments/1637964358480.10/warc/CC-MAIN-20211128073830-20211128103830-00056.warc.gz"} |
http://stackoverflow.com/questions/18722471/when-to-use-double-star-in-glob-syntax-within-java/18722567 | # When to use ** (double star) in glob syntax within JAVA
Directly from this Java Oracle tutorial:
Two asterisks, **, works like * but crosses directory boundaries. This syntax is generally used for matching complete paths.
Could anybody do a real example out of it? What do they mean with "crosses directory boundary"? Crossing the directory boundary, I imagine something like checking the file from root to getNameCount()-1. Again a real example explaining the difference between * and ** in practice would be great.
-
It means it will recursively go through all sub-directories, where * will only grab files from the current directory, ignoring sub-directories – StormeHawke Sep 10 '13 at 15:06
The javadoc for FileSystem#getPathMatcher() has some pretty good examples and explanations
*.java Matches a path that represents a file name ending in .java
*.* Matches file names containing a dot
*.{java,class}} Matches file names ending with .java or .class
foo.? Matches file names starting with foo. and a single character extension
/home/*/* Matches /home/gus/data on UNIX platforms
/home/** Matches /home/gus and /home/gus/data on UNIX platforms
C:\\* Matches C:\foo and C:\bar on the Windows platform (note that the backslash is escaped; as a string literal in the Java Language the pattern would be "C:\\\\*")
So /home/** would match /home/gus/data, but /home/* wouldn't.
/home/* is saying every file directly in the /home directory.
/home/** is saying every file in any directory inside /home.
-
That API seems to cover glob syntax quite well. Thanks, I was looking for some glob specification and not finding them. Thanks again – Rollerball Sep 10 '13 at 15:14
@Rollerball You're welcome. The PathMatcher class seems to use glob syntax a lot, so you might have more luck looking at the related methods and classes. – Sotirios Delimanolis Sep 10 '13 at 15:15 | 2014-03-17 13:40:48 | {"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": 0.4850555956363678, "perplexity": 6139.755821846757}, "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/1394678705611/warc/CC-MAIN-20140313024505-00042-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://blender.stackexchange.com/questions/26133/how-do-i-randomise-a-glossy-nodes-roughness-within-a-specific-range | # How do I randomise a glossy node's roughness within a specific range?
I have a glossy node with the roughness set to 0.05. I am using this material on many objects, and want to randomize the roughness by a small bit on each model (to be specific, between 0.02 and 0.05). I know that the way to randomize is to add an Object Info node, but what should I put between the Object Info node and the Glossy node to put the values 0.02 and 0.05 and plug into the roughness?
With an input of 1: 1 * .03 = .03, .03 + .02 = .05
With an input of 0: 0 * .03 = 0, 0 + .02 = .02 | 2019-11-21 15:55:12 | {"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": 0.4704131484031677, "perplexity": 645.7954928936716}, "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-2019-47/segments/1573496670921.20/warc/CC-MAIN-20191121153204-20191121181204-00558.warc.gz"} |
https://cms.math.ca/cjm/msc/20G25?fromjnl=cjm&jnl=CJM | Canadian Mathematical Society www.cms.math.ca
location: Publications → journals
Search results
Search: MSC category 20G25 ( Linear algebraic groups over local fields and their integers )
Expand all Collapse all Results 1 - 5 of 5
1. CJM 2016 (vol 69 pp. 107)
Kamgarpour, Masoud
On the Notion of Conductor in the Local Geometric Langlands Correspondence Under the local Langlands correspondence, the conductor of an irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection. Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representationsCategories:17B67, 17B69, 22E50, 20G25
2. CJM 2014 (vol 67 pp. 184)
McReynolds, D. B.
Geometric Spectra and Commensurability The work of Reid, Chinburg-Hamilton-Long-Reid, Prasad-Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form $(\operatorname{SL}(d,\mathbf{R}))^r \times (\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\operatorname{SL}_d$ and the failure of determining the form via certain classes of algebraic subgroups. Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifoldsCategory:20G25
3. CJM 2010 (vol 62 pp. 1310)
Lee, Kyu-Hwan
Iwahori--Hecke Algebras of $SL_2$ over $2$-Dimensional Local Fields In this paper we construct an analogue of Iwahori--Hecke algebras of $\operatorname{SL}_2$ over $2$-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on $\operatorname{SL}_2$, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori--Matsumoto type relations. Categories:22E50, 20G25
4. CJM 2009 (vol 62 pp. 34)
Campbell, Peter S.; Nevins, Monica
Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples. Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groupsCategories:20G25, 20G05
5. CJM 2005 (vol 57 pp. 648)
Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25
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© Canadian Mathematical Society, 2017 : https://cms.math.ca/ | 2017-11-22 21:58:03 | {"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": 0.7958840131759644, "perplexity": 667.6672819009561}, "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/1510934806676.57/warc/CC-MAIN-20171122213945-20171122233945-00420.warc.gz"} |
https://scioly.org/wiki/index.php?title=Wind_Power&curid=235&diff=100284&oldid=92941 | # Difference between revisions of "Wind Power"
Wind Power
Type Physics
Category Lab
Event Information
Latest Appearance 2017
Wind Power, previously known as Physical Science Lab and Physics Lab, was an event for the 2017 and 2016 seasons which involves the construction of a device that can turn wind into energy and the answering of questions relating to alternative energy.
## The Basics of the Event
Half of the scoring for Wind Power comes from the building portion. Competitors must build a wind turbine mounted to an unmodified 12cm diameter CD. During competition, the judges will attach the device to a DC motor. They will then place a fan in front of the device and turn it on. The blades will spin, thereby creating voltage.
The other portion of the event is a written test on wind power, power generation, and alternative energy. One 3-ring binder full of notes from any source is allowed for each team, as long as the material in the binder is attached securely. Teams may bring multiple calculators of any type.
Category B safety spectacles are required for the blade testing portion of this event.
## The Building Section
When planning to build a Wind Power device, there are many factors to consider. Some of them include:
• Weight
• Wind Turbine Diameter
• Curve of the Blade
• Number of Blades
In 2017, the event changed compared to the previous year. Instead of trying to get the blades to generate as much power as possible with 5-ohm resistance, there were 5 to 25 ohms of resistance the blades had to 'push' against, which were wired into the setup. Resistors can be bought in varying sizes or wired in parallel or in series to generate the correct resistance. For example, two 14 ohm resistors in parallel = 7 ohms of resistance.
Examples: Blades for 2017
### Factors
#### Weight
Lighter blades will spin faster- however, since the build will be scored for power production due to the resistors, one may use a slightly heavier blade in order to produce higher momentum. There are 2.5 minutes of preparation in order to get the blades up to their highest attainable speed, so they can be somewhat heavy.
Various items can be used to construct the blades, but metal is not allowed in any capacity. Balsa wood or cardboard are often used because they are lighter materials. Everyday items can often be constructed into blades. For example, a plastic cup can be cut into fourths. Experimenting with different materials is key, as sometimes the best blades are found simply through testing.
#### Diameter
According to the New York Coaches Conference, doubling the diameter of the circle made by the blades produces a 4-fold increase in power. However, keep in mind that the radius of the circle created by the blades can be no more than 20cm (Div B) and 14cm (Div C).
Many teams have found that since real life wind turbines can create more power (watts, not voltage) by increasing the diameter of the blades, it is advantageous to do so.
It is advised that the blade contains some curve, although it is not necessary in order to generate spin. A V-shape might also work. Materials may be morphed into the desired shape, whether it be by cutting or molding. With some materials, such as wood, a certain shape can be achieved by soaking and bending the material in water. Not much curve is necessary. It is recommended that the blades have a curve to simulate an airfoil, while still keeping the blades aerodynamic. Keep in mind that more force is generated when the wind from the fan is blowing on the outside of the blade curve. However, many competitors curve the blade so that the wind hits the inside of the curve. This provides quick acceleration (which isn't totally necessary, as long as it accelerates to top speed within 2.5 minutes), but also results in a top speed that is lower than if the wind was on the outside of the blade.
Blade pitch refers to the angle of the blades relative to the CD. Generally, a lower pitch results in a higher speed and thus a higher power (it will have a much slower acceleration but a higher top speed). One may want a higher pitch the closer the blade gets to the center of the turbine (eg. 20 degrees towards the center with the front edge's angle gradually getting lower until it reaches around 5-10 degrees or so at the outside). Make sure your blade pitch is not 0 degrees, as the blades will not generate lift.
#### Number of Blades
The number of blades is another adjustable attribute of the turbine. 2-3 blades is the most popular amount. Remember that more weight will gain more momentum, but at a point, too much weight will lower the high speed of the turbine. 4 or more blades can often be too heavy if the material is not light enough. However, if the blades are light enough, 4 may work better than 2 or 3. If there are more than four or so, the added weight might, literally, outweigh the benefits.
#### Balance
Balance is also an important factor. If a turbine is terribly off balance, it may wobble on the mount, and possibly even break. Even a slight imbalance can create instability in the device. As such, a well-balanced turbine will be able to spin faster since there will be less friction against the mount and less energy will be lost to vibration. It is best if a turbine's center of gravity is at or near the center. This can be accomplished by taking some mass off of the turbine on the heavy side, or by adding mass on the lighter side. A way to help make sure that blades are built as balanced as possible is making a device with a thin, 1-foot squaer piece of plywood with a metal or wood rod sticking straight out, roughly the size of the hole in the CD (it could be slightly smaller, but not larger). Lines can be drawn on the sheet of plywood going away from the rod at equally angled increments on which you can line up your blades as you glue them to the CD. When gluing, it is best to use some drawn guidelines or template either around or on the CD. This will allow for more accurate spacing to be implemented in the construction. This helps to ensure a more well-balanced turbine without having to add or subtract weight in ways that might affect the aerodynamics of the device .
Although it may seem to be a good idea to cut holes in the CD to reduce its weight, the rules say that "modification of the CD is not allowed (except to affix the blades via tape, glue, etc.)".
Make sure that the turbine is clearly within all specifications and labeled with team name and number.
## The Written Section
The other half of the score is the written portion. These rules have varied over the years for Wind Power. In 2017, the written test focused on rotor/fan blade design, power generators for different sources of power, power storage, power transmission, and historical wind power designs.
### History
#### People
• Lucien Gaulard and John Dixon Gibbs: built what they called the secondary generator (an early transformer provided with 1:1 turn ratio and open magnetic circuit) in 1881, allowing for transmission of electric power with alternating current (AC)
• James Blyth: Scottish academic, installed the first electricity-generating wind turbine (a battery charging machine) in 1887 to light his holiday home in Marykirk, Scotland.
• Charles F. Brush: American inventor, built the first automatically operated wind turbine in 1888 in Cleveland, Ohio.
• Poul la Cour: Danish scientist, constructed a wind turbine to generate electricity in 1891, which was used to produce hydrogen by electrolysis. With his Askov mill, he made windmills more efficient.
• Albert Betz: German physicist, discovered the theory of wind energy in 1919.
#### Events
• The first use of windmills was recorded in Europe during the Middle Ages. They were primarily used to make flour by turning grinding mills and transport water out from the lowlands by rotating water screws.
### Laws
Newton's Three Laws of motion describe the properties of forces. Forces cause objects to move or otherwise do something; it is the ability to do work.
• The first law states that an object at rest tends to remain at rest, while an object moving at constant velocity tends to remain at constant velocity unless acted on by an outside force. This law describes inertia. Inertia is not a force; in fact, it is the absence of force. It describes an object's resistance to change in motion.
• The second law is the basis for everything in forces. It is simply $F=ma$. Sum of all forces on an object equals object mass times object acceleration.
• The third law states "For every action there is an equal yet opposite reaction." But if this is true, how could anything happen at all; if the reaction is equal and opposite, why should anything happen. It is because the action and reaction occurs on different objects. For example, take two blocks. When one is pushed into the other, the two blocks move to the right. Why is this possible? Because the reaction occurs on a different object than the action.
### Power Generation
#### Alternative Energy
Other forms of power generation using alternative energy exist. These are provided below.
• Solar power converts sunlight into energy via photovoltaic panels or concentration. In the long-term it is relatively constant; however, days are longer in the summer and shorter in the winter, resulting in less solar energy generated in the winter. Also, solar power is only available during the day and is less powerful when there are many clouds.
• Hydroelectric power converts the movement of water into energy, generally with turbines as well. The flow of water is abundant, constant, and hydroelectric plants can store energy for times when it is more necessary, but there are concerns in respect to the reservoirs created by dams.
• Tidal power is a major subset of hydroelectric power. It uses the movement of water created by tides to get energy. Tides are regular and predictable, but less power can be generated this way than some other ways.
• Ocean Thermal Energy Conversion (OTEC) utilizes the difference in temperature from shallow water to deeper water. When heat goes from the warmer water to cooler water, an engine converts the flow to energy. This is constant, but not very cost-effective and finding locations where this technology can be used is difficult.
• Geothermal power converts underground heat into energy. This method is reliable, cost-effective, and environmentally friendly, but is mostly limited to areas with a high tectonic activity.
To conserve energy, the adage "reduce, reuse, recycle" can be followed. These three concepts can be applied to many techniques used to conserve energy.
### Energy Transmission
Electric power generation is defined as the movement of a large amount of electrical energy from a generation site to an electrical substation.
The method of three-phase electric power is commonly used for alternating current electric power transmission.
#### History
• The first long-distance AC line was built for the 1884 International Exhibition of Turin, Italy and extended a total of 34 kilometers (21 miles).
• Through the invention of Lucien Gaulard's and John Dixon Gibbs' secondary generator in 1881, transmission of electric power using an alternating current was made possible.
### Energy Storage
Energy stockpiling is a vital part of balancing the supply and demand of power into the grid.
Originally, non-renewable energy sources were burned with the need for power. However, there was a need for a sustainable power source in electricity generation that would appease the concerns of pollution. But, renewable energy outlets were inconsistent; winds were uncontrolled and sunlight based power fluctuates because of cloud coverage and must be collected during the day. Now, most technology has the ability to rapidly and efficiently discharge power into the grid at peak demand (usually in the evening), allowing for grid stability.
The U.S. has around 23 gigawatts (GW) of storage capacity, with pumped hydroelectric storage accounting for 96%. Pumped hydroelectric capacity allows for energy storage at the grid’s transmission stage, by accumulating any excess generation for later use. Flywheels can reserve energy by conserving angular momentum in a spinning mass which, in turn, enables them to benefit the grid at both either transmission or distribution level. Nuclear power plants, on the other hand, are not designed to increase or decrease and therefore have a steady generation for the duration of the day.
### Energy (Thermodynamics)
Note: The following spoiler box contains information that is not relevant to the current rules of Wind Power. For more complete information about Thermodynamics, please see Thermodynamics.
Energy (Thermodynamics)
Energy is the capacity to perform work. Work, in turn, is when a force is exerted on an object and the object moves parallel to the direction of the force. Work, in joules (J), can be calculated by multiplying the force applied in Newtons (N) to the distance traveled in meters (m). Power is the rate at which this work is performed, and is found by dividing the work by the time. It is measured in watts.
There are two major types of energy:
• Kinetic energy is the energy contained by an object in motion
• Potential energy is how much work an outside force like gravity can do on an object depending on its position.
The Conservation of Energy principle states that energy cannot be created nor destroyed, and that it can only be transformed from one form to another.
#### Heat
Heat is the transfer of energy from a high-temperature object to a low-temperature object. Temperature is the average energy contained in the particles of an object. It is produced by the thermal energy in free particles. Many scales can measure temperature.
Heat Transfer
There are three main vehicles for transferring heat:
• Conduction is the transfer of heat by direct contact. The heat transfer for convection can be calculated by the following formula:
$Q=\frac {kA\left (T_{hot}-T_{cold}\right )}{d}$
Where
Q is the heat transferred in $Watts$
k is the barrier's thermal conductivity in $k=\frac {Watts}{m K}$
A is the cross-sectional area in $m^2$
T is the temperature in °C ($T_{hot}$ represents the warmer temperature and $T_{cold}$ represents the cooler temperature)
d is the barrier's length in $m$
Conduction
• Convection is the transfer of heat from a solid or liquid to another fluid. Forced convection is a fluid flowing over a surface. Natural convection is when a fluid is heated and rises due to buoyancy such as when hot air rises and cooler air sinks, creating circular currents. The heat transfer for convection can be calculated by the following formula:
$Q=hA(T_{hot}-T_{cold})$
Where
Q is the heat transferred in $Watts$
h is the film coefficient in $h=\frac{watts}{m^2 K}$
A is the surface area in contact with the fluid in $m^2$
T is the temperature in °C ($T_{hot}$ represents the warmer temperature and $T_{cold}$ represents the cooler temperature)
Convection
• Radiation is where the heat is carried by electromagnetic waves. The Stefan-Boltzmann Law can help calculate this transfer:
$P=e\sigma A \left (T^4-{T_C}^4\right )$
Where
P is the radiated power in $Watts$
e is the object's emissivity
σ is Stefan's constant (equal to $5.6703 \cdot10^{-8} \frac {W}{m^2K^4}$)
A is the radiating area in $m^2$
T is the temperature of the radiating source in K and $T_C$ is the surrounding temperature.
Specific Heat
This is the amount of heat per unit mass needed to raise a substance's temperature by 1 °C. It is represented by the following formula:
$Q=mC\Delta T$
Where Q is the heat needed, C is the specific heat, m is the mass, and ΔT is the change in temperature. Every substance has its own specific heat. For example, water's specific heat is 4.184 J/g*K
#### Laws
Betz’ law/limit
Only a certain amount of energy can be harnessed from the wind; 59.3%; the theoretical maximum coefficient of power for any wind turbine.
Joule's Law
Energy losses are directly proportional to the square of the current. Thus, reducing the current by a factor of two will lower the energy lost to conductor resistance by a factor of four for any given size of conductor.
Kelvin's Law
The optimum size of a conductor for a given voltage and current can be estimated by Kelvin's law for conductor size. Kelvin's Law states that the size is at its optimum when the annual cost of energy wasted in the resistance is equal to the annual capital charges of providing the conductor. At times of lower interest rates, Kelvin's law indicates that thicker wires are optimal; while, when metals are expensive, thinner conductors are indicated: however, power lines are designed for long-term use, so Kelvin's law has to be used in conjunction with long-term estimates of the price of copper and aluminum as well as interest rates for capital.
Laws of Thermodynamics
Zeroth Law of Thermodynamics: If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.
First Law of Thermodynamics (also known as Law of Conservation of Energy): Energy cannot be created or destroyed in an isolated system.
Second Law of Thermodynamics: The entropy of any isolated system always increases.
Third Law of Thermodynamics: The entropy of a system approaches a constant value as the temperature approaches absolute zero.
## The Competition
For the building set-up, there will be 2 stations: high speed and low speed. These may be at the same fan, but if they are at different fans, all teams will test at high speed at one and at low speed at the other. The event supervisors will provide the testing mount for your blades (don't bring your stand, ONLY the cd with blades attached). They will also provide the fan, motor/generator, load resistor, and device to measure voltage. There is a 3-minute time limit for each station. You will get a warning at 2 minutes. In the first 2.5 minutes, you may adjust, modify, and start and stop your blades. Within 2.5 minutes of the start of the testing period, you must tell the ES to begin measuring your voltage for 30 seconds. The ES must record the peak voltage that is measured during that period.
The written test will take place after impound. If a lengthier test is used for the event, teams will be called up separately to stations and resume the written portion after testing their build.
### Scoring
As of 2017, the scoring for the building section is calculated as follows:
$25*\frac{ \text{Low speed voltage} }{ \text{Highest low speed voltage out of all teams} } + 25*\frac{ \text{High speed voltage} }{ \text{Highest high speed voltage out of all teams} }$
The maximum possible score on the building section is 50.
As of 2017, the scoring for the test section is calculated as follows:
$50*\frac{ \text{Test score} }{ \text{Highest test score out of all teams} }$
The maximum possible score on the test section is 50.
The test score and the building score are added together to determine a team's score, with the highest score winning. | 2020-08-08 04:04:52 | {"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": 0.5364291071891785, "perplexity": 854.6536488245401}, "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-2020-34/segments/1596439737238.53/warc/CC-MAIN-20200808021257-20200808051257-00350.warc.gz"} |
https://brilliant.org/discussions/thread/doubts-help-needed/ | ×
# Doubts ! Help Needed
Please give a solution to the below questions.
Note by Rajdeep Dhingra
10 months ago
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Congratulations for being selected to represent India in ijso. Also what about JOMPC? · 10 months ago
Thanks. It will be out day after tomorrow. I have marked all the solutions I'll compile and post it on monday. · 10 months ago
Which book is it? · 1 month, 2 weeks ago
I answered it before also , an old JEE subjective book. Just Mechanics for JEE is written on the book , nothing else. · 1 month, 2 weeks ago
who's the author of the book? · 1 month, 2 weeks ago
?? · 4 weeks ago
Bro is your second doubt cleared ? i am getting it's answer as root 5g/7(R-r) tell if not cleared i will give the solution ( if my answer is correct.) . · 7 months, 1 week ago
Cleared. Thanks anyways. P.S : your answer is correct. · 7 months, 1 week ago
Hey @Rajdeep Dhingra why not coming to fiitjee ? PLease tell the truth !! · 7 months, 3 weeks ago
Which book is it ? · 10 months ago
I don't know. Some old book for JEE subjective. · 10 months ago
Hy rajdeep, can u give some tips for prepartion of nsejs , as my brother will be giving this year, presently he is in class 9. · 10 months ago
Read 9th and 10th class NCERT books thoroughly , if possible read 11th and 12th class NCERT also.(only relevant topics). For NSEJS most important is your speed of solving questions. · 10 months ago
If he starts from June 10 , is he will be able to clear nsejs ? He also goes to coaching ... · 10 months ago
I think so. · 10 months ago
Solution for Question 1:
· 10 months ago
Thanks ! · 10 months ago
You are welcome · 10 months ago
Question 2:
Let $$f$$denote the friction force acting on the body. Now, ablut the center of mass writing the torque equation,
$$f \cdot r = \dfrac{MR^{2}\alpha}{2}$$
$$f = \dfrac{MR\alpha}{2}$$
But since the cylinder doesn't slip $$r \alpha = a_{cm}$$
$$f = \dfrac{ma_{cm}}{2}$$ Now,
$$mg\sin(\theta) - f = ma_{cm}$$
$$mg\sin(\theta) = \dfrac{3ma_{cm}}{2}$$
$$a_{cm} = \dfrac{2g\sin(\theta)}{3}$$
$$\alpha = \dfrac{2g \sin(\theta)}{3r}$$
For small values of $$\theta$$, $$\sin(\theta) \approx \theta$$
$$\alpha = - \dfrac{2g \theta}{3r}$$
The negative sign just denotes that direction of alpha is opposite to displacement.
This is the rotational analogue of $$a = -\omega^{2}x$$
$$\therefore \omega = \sqrt{\dfrac{2g}{3r}}$$
Frequency $$\dfrac{\omega}{2\pi} = \dfrac{1}{2\pi} \sqrt{\dfrac{2g}{3r}}$$ · 10 months ago
It is wrong bro. $$\theta$$ is the angle you moved the center of mass not the angle by which you rotated the sphere. Hence you can't equate $${\omega}^2$$ to $$\dfrac{2g}{3r}$$. · 8 months, 4 weeks ago
it would be 5g/7r instead ! · 4 months ago
Thanks · 10 months ago | 2017-03-28 21:58:05 | {"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": 0.9059062004089355, "perplexity": 4929.313575741038}, "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-2017-13/segments/1490218189903.83/warc/CC-MAIN-20170322212949-00611-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://flaviocopes.com/how-to-cut-string-words-javascript/ | Use the split() method of a string instance. It accepts an argument we can use to cut the string when we have a space:
const text = "Hello World! Hey, hello!"
text.split(" ")
The result is an array. In this case, an array with 4 items:
[ 'Hello', 'World!', 'Hey,', 'hello!' ] | 2020-10-28 11:51:50 | {"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": 0.3788727819919586, "perplexity": 5332.746794647309}, "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-2020-45/segments/1603107898499.49/warc/CC-MAIN-20201028103215-20201028133215-00583.warc.gz"} |
https://academic.oup.com/aje/article-lookup/doi/10.1093/aje/kwf185 | ## Abstract
When conducting epidemiologic case-control studies, some investigators include only controls who can be interviewed within a certain time after contact and/or do not recontact potential participants who initially refuse, whereas others expend considerable effort to recruit reluctant respondents. This additional effort is only worthwhile if it results in a sample that is more representative of the target population. In this study, the authors used data collected from in-person interviews of 5,616 female controls to compare characteristics of willing, accessible respondents with those of their less accessible or less willing counterparts to determine whether or not the two groups differed with respect to lifestyle, socioeconomic status, health history, and demographic characteristics. Late responders were younger, were more likely to be non-White, were less likely to have attended college, and were more likely to be current smokers than early responders. Initial refusers were similar to late responders with respect to education and race. Initial refusers were also older, were less likely to be currently married, were less likely to have a managerial occupation, had fewer lifetime sexual partners, and were more likely to have a history of diabetes than early responders. These findings suggest that additional effort expended in recruiting reluctant respondents may often result in more accurate estimates of population characteristics that are of interest in epidemiologic research.
Received for publication January 28, 2002; accepted for publication August 13, 2002.
In most epidemiologic studies, investigators expend considerable effort to recruit difficult-to-reach respondents so that participants will be representative of the population of interest. The validity of a study comes into question if response is poor (13). Response proportions generally improve as effort to recruit reluctant respondents increases. However, the pursuit of reluctant respondents is expensive and time-consuming. It is unclear whether the additional expense of attempting to convert persons who refuse to participate is worthwhile. Furthermore, some investigators restrict the amount of time between contact and interview to minimize recall bias or to avoid delaying the completion of the study.
Several studies have compared respondents who refuse to participate, or those who initially refuse and later agree, with respondents who readily agree to participate. Most have found that reluctant or difficult-to-reach respondents are older and less educated than respondents who readily agree (49). Differences with respect to income, occupation, race, and marital status have been inconsistent (4, 5, 7, 912). Three studies have compared reproductive or lifestyle characteristics of initial refusers with those of initial responders (3, 7, 11). Three studies have found that while characteristics of initially refusing respondents differed from those of other respondents, the inclusion of those persons’ data had little overall effect on the results (4, 7, 11). In addition, one study found that data from initial refusers were less useful because these persons refused to answer more questions than other respondents (9). Since recruiting reluctant respondents is expensive and lengthens the time required to complete interviews, it is important to determine whether or not this additional expense is warranted.
We undertook this study to further address the question of differences in demographic, reproductive, health, and lifestyle characteristics between control respondents according to the level of effort required to recruit them.
## MATERIALS AND METHODS
We combined interview data obtained from 5,616 female controls selected by random digit dialing (RDD) for six population-based case-control studies of anogenital cancers, breast cancer, endometrial cancer, thyroid cancer, oral cancer, and rheumatoid arthritis (1318). All six studies were conducted in western Washington State. All of the studies were approved by the institutional review boards of the participating institutions, and all respondents gave written informed consent. The controls were not screened for the presence of any of these diseases prior to interview. Table 1 shows the response proportions and characteristics of each study. Details on these studies are presented elsewhere (1318). All controls were selected using either unrestricted RDD or the Waksberg-Mitofsky (19) modification of RDD. Phone numbers for the unrestricted RDD were created by selecting an area code and telephone prefix at random from all such combinations in the geographic area of interest and adding a random four-digit number to create a telephone number. Studies that used the Waksberg-Mitofsky modification added a second stage of number selection. The area code, prefix, and next two digits of all telephone numbers that were found to belong to residential households in the primary stage of calling became the “primary sampling unit” for the secondary stage. Two random numbers were added to the sampling unit to create an additional phone number. This phone number was called and screened according to the study protocol. Additional phone numbers were generated and resolved until a total of “k” residential numbers were achieved for each sampling unit. This clustering factor, “k,” was 2 for some studies and 5 for others.
The RDD screening protocol was similar for all of the studies. Each number was called at least nine times at different times and on different days of the week over a 3-week period before it was abandoned. Residential status could not be determined for 4.3 percent of the 96,806 phone numbers dialed (3.7 percent of calls were never answered, 0.6 percent always resulted in a fast-busy tone, and 0.3 percent always resulted in a slow-busy tone). When a residential household was reached, the RDD interviewer screened the household for multiple telephone numbers, county of residence, and the ages and genders of household members. If there was more than one phone number in the household and the phone was not dedicated to a computer or fax line, the household was considered ineligible if the last digit of the phone number was odd and eligible if this digit was even.
Controls were selected using a one-step stratification design (20). Controls were selected so that each 5-year age group would have approximately the same number of controls as did the cases for that particular study. Eligible controls were asked whether they would be willing to receive a letter describing the study. If they were willing, a letter describing the study was mailed within 2 weeks of initial contact. A trained interviewer contacted the potential control after the letter was received to answer questions and to schedule an in-person interview. All phone numbers at which the respondent refused to answer the screening questions for determination of eligibility or, if screened and eligible, refused to receive a letter about the study were recontacted by a different RDD interviewer 3–6 months after the original phone call. In addition, most phone numbers at which all nine calls were answered by a machine were recontacted at a later date. The recontact included a question as to whether the respondent had had that phone number at the time of the original phone call. Approximately half of the respondents who originally either refused screening or (if eligible after screening) refused to receive a letter agreed on the recontact call. Screening was completed for almost half of the households in which all nine original phone calls had been answered by machine. Interviews were conducted in person by trained interviewers at a place and time most convenient to the respondent. The interviews lasted about an hour. Respondents for the anogenital and oral cancer studies were paid $10 for participation. Respondents for the other studies were not paid. All respondents were assigned a “reference” year that approximated the distribution of diagnosis years in the corresponding case group. Each control was also assigned a randomly selected “reference” month. Only events that occurred prior to this reference date were recorded during the interview. We categorized the respondents recruited by RDD into early responders (those who were interviewed within 1 month of the date the letter describing the study was mailed (n = 2,747)), intermediate responders (those who were interviewed 2–6 months after initial letter contact (n = 837)), late responders (those who were interviewed more than 6 months after initial letter contact (n = 444)), and initial refusers (those who initially refused to participate but agreed after recontact (n = 376)). All respondents who initially refused were grouped together regardless of how much time had elapsed between initial letter and interview. We excluded those who were interviewed 1–2 months after initial letter contact (n = 1,212), because some of those interviews might have been delayed as a result of interviewer schedules rather than respondent delay. Our categorization was motivated by a common assumption, described by Lin and Schaeffer (21), that the characteristics of people who are difficult to interview become more like those of refusers as the difficulty of interviewing them increases. Table 2 shows the distribution of responses by study. We selected questions from each interview that related to health history and to demographic, reproductive, and lifestyle characteristics. Body mass index was computed as weight in kilograms divided by height in meters squared. Data that were not collected in a similar manner across studies were excluded from analysis. The footnotes in table 3 identify variables for which data were not available for all studies. Some data were missing because of respondent refusal or because the respondent answered “don’t know.” The case-control study of rheumatoid arthritis included a second control group selected from the enrollment files of the Group Health Cooperative, a health maintenance organization serving western Washington State (18). Seventy-eight percent of the selected controls completed the interview. The Group Health Cooperative research department abstracted medical record data without identifiers for all 59 of the selected controls who refused to be interviewed and a random sample (n = 49) of the controls who were interviewed. The abstractors recorded only events that occurred prior to the assigned reference date. Data were recorded from the medical visits that occurred closest in time to the reference date. The following data were abstracted from the medical records: age at reference date, number of pregnancies, ever having a livebirth, ever using hormones, smoking history, height, body weight, and date on which weight was recorded. All proportions for the intermediate responders, late responders, and initial refusers were standardized to the age distribution of the early responders so that the proportions of respondents with each characteristic could be compared with each other without distortion from the different age structures of the four groups. All variables were included simultaneously in polytomous logistic regression models with the early responders designated the reference group, using Stata statistical software (22). After adjustment for all other variables analyzed, p values were computed for individual variables using Wald’s test. Differences were considered statistically significant if the two-sided p value was less than 0.05. Results of likelihood ratio tests for inclusion of each variable in models adjusted only for age were similar to the Wald’s test statistics. Additional significance tests were performed using Bonferroni’s correction to account for multiple comparisons (23). Results of both tests are presented in the tables. ## RESULTS ### Comparison of intermediate and late responders with early responders The intermediate- and late-responding women were slightly younger, were more likely to be non-White, and were less likely to have ever attended college than early responders (table 3). Early responders were more likely to have a family history of cancer in a first- or second-degree female relative than the intermediate- or late-responding women. Current marital status and occupation were similar for all three groups. Women who were intermediate or late responders were similar to early responders with respect to number of pregnancies and oral contraceptive use but were less likely to have ever been tested for infertility. Late responders were less likely to have ever had an induced abortion and to have used noncontraceptive hormones than early responders, but the differences were not statistically significant. Intermediate- and late-responding women had similar histories of hypertension, diabetes, and cancer and similar body mass indexes. Late-responding women were more likely than early responders to be current smokers, whereas the smoking history of intermediate responders was similar to that of early responders. Ever use of alcoholic beverages and recent exercise were similar in all three groups. ### Comparison of initial refusers with early responders Initial refusers were older and were more likely to be non-White than early responders. They were less likely to be currently married, to have attended college, and to be employed in a managerial or professional job. They were also less likely than early responders to have a family history of cancer and to use noncontraceptive hormones, but these differences were not statistically significant. Early responders and initial refusers were similar with respect to all reproductive characteristics except lifetime number of sexual partners. Initial refusers had fewer lifetime sexual partners than early responders. Initial refusers and early responders had similar health histories, with the exception of diabetes mellitus. Initial refusers were more likely to report a history of diabetes. Intermediate responders, late responders, and initial refusers were similar to each other and different from early responders with respect to education, non-White race, and history of infertility testing. In addition, late responders were similar to initial refusers with respect to marital status, number of sexual partners, and use of noncontraceptive hormones. ### Medical record data There were no significant differences in medical record data between women who were interviewed and those who refused to be interviewed in the Group Health Cooperative population. Refusers had a higher body mass index than those who were interviewed, but the difference was not statistically significant (table 4). There were very few questions that the respondents refused to answer or for which the response was “don’t know.” The question that was most often refused was the question on income. Of the early-responding, intermediate-responding, late-responding, and initially-refusing female respondents, 76 (2.8 percent), 21 (2.5 percent), 9 (2.0 percent), and 16 (4.3 percent), respectively, refused to identify which category of income corresponded to their annual household income or responded that they did not know their household income. The item nonresponse for all other questions was less than 1 percent. ## DISCUSSION Our ability to find differences between the response groups was limited by the relatively small number of respondents who completed the interview more than 6 months after contact and who initially refused to be interviewed but were subsequently recruited. We found only one other study that compared respondents who agreed to be interviewed but delayed the interview with respondents who were interviewed soon after contact (24). Robins (24) conducted in-person interviews of former clients of a child guidance clinic and found that the only difference between early and late responders was proximity to the study site. In contrast, we found several differences between late and early responders and/or intermediate and early responders. Late responders may be as willing to be interviewed as early responders but be less accessible than early respondents because of lifestyle factors and younger age. Our finding that initial refusers were older, less educated, and less likely to be employed in professional or managerial occupations than early responders is in general agreement with most (4, 7, 11, 12, 24, 25) but not all (25) prior studies. One other study concurred with our results that respondents who initially refused were more likely to be non-White than early responders (9), whereas three others did not (4, 5, 12). Our finding that respondents who initially refused were less likely to be currently married than early responders is in contrast to the studies of both Fitzgerald and Fuller (5) and Kristal et al. (7). The only other similar study that compared reproductive characteristics differed from our study in that only women of reproductive age who were at risk of pregnancy were included (11). Our findings are consistent with this study with respect to lifetime number of sexual partners but not gravidity or history of induced abortion. Our results did not change when we examined reproductive characteristics only among women of reproductive age. Our finding that histories of hypertension and cancer were similar for early responders and initial refusers but that initial refusers were more likely to report a history of diabetes is consistent with another comparison of female participants and refusers (3). Kristal et al. (7) found that initial female refusers were less likely to use alcohol and to smoke cigarettes than those who initially agreed, whereas Criqui et al. (3) found that female refusers were more likely to be current smokers than study participants in a study of heart disease. Those results are in contrast with our findings of no differences with respect to alcohol and cigarette use. Since all of the studies except one were related to cancer, it is not surprising that early responders were much more likely to have a first- or second-degree relative who had been diagnosed with cancer than either intermediate responders, late responders, or initial refusers. This supports Massey et al.’s (26) observation that interest in the study topic improves response. A similar result was found in a study of risk factors for cardiovascular disease (3). In that study, female participants were more likely to have a family history of cardiovascular disease than nonparticipants (3). Although a less-biased group of participants may be recruited if potential respondents are not aware of the specific disease under study, response may be slower. Although our comparison of medical record data between members of a health maintenance organization who agreed to be interviewed and members who refused to be interviewed revealed no significant differences, the small number of respondents in this group limited our ability to find differences. Some of the differences between our study and other studies that have compared respondents who initially agreed to participate with those who initially refused may be related to the type of interview administered. Most other studies used telephone interviews or mailed questionnaires, whereas ours utilized in-person interviews. Another reason may be geographic variation or differences over time. Only three of the prior studies were conducted during the 1990s (4, 7, 9); the remainder were conducted prior to 1990. Two of the prior studies that compared respondents who initially agreed to participate with those who initially refused were carried out in the same general geographic area as our study (7, 11). Another difference between our study and prior studies is that we separated respondents who initially agreed into early and late responders, whereas most other studies combined these respondents and compared the combined group with persons who had initially refused. We found several significant differences when intermediate responders, late responders, and initial refusers were compared with early responders. However, no clear pattern of characteristics that could be useful in generalizing these differences emerged. The only variables that showed a consistent gradient according to difficulty of recruitment were education and race. Age differences between late and early responders and initial refusers and early responders pointed in opposite directions. These findings argue against the common assumption that the characteristics of respondents become more like those of refusers as the difficulty of recruiting them increases. Lin and Schaeffer (21) and Fitzgerald and Fuller (5) reached similar conclusions. To reduce recall bias, epidemiologic case-control studies which examine events that occurred before the diagnosis date in cases or a similar “reference” date in controls often exclude respondents who are not interviewed within a specified time period after the diagnosis/reference date. In the current study, 24.4 percent of the initial female respondents completed the interview 2 or more months after the letter describing the study was mailed (which was generally within 2 weeks of the RDD contact), and 8.5 percent completed the interview more than 6 months after original contact. Since the initial refusers were recontacted 3–6 months after the original RDD contact, all of them were interviewed 3 or more months after original contact. The reduction of recall bias must be weighed against the increase in response bias. Weighting factors that are thought to be associated with nonresponse would not adequately compensate for the loss of late responders or initial refusers, since the characteristics of these two groups appear to be different. Initial refusers comprised only 6.7 percent of our total sample, so excluding them from any of the studies would generally have little effect on the overall distribution of the control group by the characteristics examined. Three studies have shown this to be the case (4, 7, 11). However, analyses by subgroups could be distorted, particularly if the analyses involved two characteristics that were differentially distributed by response. This potential for bias will increase if study participants become more difficult to recruit in the future and an effort is not made to include reluctant respondents (1, 3). Characteristics of reluctant respondents may well vary over time and by geographic area, so it is difficult to quantify the potential bias resulting from exclusion of inaccessible or less-willing participants. Expending the additional effort required to convert refusers into participants and to delay interviews rather than accept refusal remains the best defense against the creation of unrepresentative or biased samples. ## ACKNOWLEDGMENTS This study was funded in part by grants RO1-CA47749, RO1-CA41410, RO1-CA52656, RO1-CA48996, and PO1-CA42792 from the National Cancer Institute and by contract N01-HD-62914 with the National Institute of Child Health and Human Development. Correspondence to Dr. Lynda Voigt, Fred Hutchinson Cancer Research Center, P.O. Box 19024, Seattle, WA 98109-1024 (e-mail: [email protected]). TABLE 1. Response proportions and characteristics of random digit dialed female controls from six studies conducted in western Washington State, 1987–1998 Study Interview dates Random digit dialing screening proportion* Interview response agreement† Geographic area Breast cancer, endometrial cancer, and rheumatoid arthritis‡ 1987–1994 0.96 0.80 Metropolitan counties§ Female anogenital cancer¶ 1987–1998 0.92 0.69 Metropolitan counties§ plus 10 surrounding counties Thyroid cancer 1992–1996 0.95 0.78 Metropolitan counties§ Oral cancer# 1991–1996 0.96 0.69 Metropolitan counties§ Study Interview dates Random digit dialing screening proportion* Interview response agreement† Geographic area Breast cancer, endometrial cancer, and rheumatoid arthritis‡ 1987–1994 0.96 0.80 Metropolitan counties§ Female anogenital cancer¶ 1987–1998 0.92 0.69 Metropolitan counties§ plus 10 surrounding counties Thyroid cancer 1992–1996 0.95 0.78 Metropolitan counties§ Oral cancer# 1991–1996 0.96 0.69 Metropolitan counties§ * Number of women who answered the screening questions divided by the total number of residential phone numbers. † Number of women who completed the interview divided by the total number eligible. ‡ These three studies recruited controls jointly. § King, Pierce, and Snohomish counties of Washington State. ¶ Vulvar, vaginal, cervical, and anal cancer. # Excludes controls recruited jointly with those of the anogenital cancer study. TABLE 2. Distribution of responses (%) of random digit dialed female controls from six studies conducted in western Washington State, by study and response category, 1987–1998 Months between letter and interview date Total no. ≤1 1–2 2.1–5.9 ≥6 Initial refusers Breast cancer, endometrial cancer, and rheumatoid arthritis 47.2 20.8 17.8 8.1 6.0 2,781 Anogenital cancer 50.2 22.7 11.9 7.8 7.4 2,176 Thyroid cancer 51.6 21.6 12.5 7.3 7.0 574 Oral cancer 52.9 20.0 10.6 5.9 10.6 85 Total 48.9 21.6 14.9 7.9 6.7 5,616 Months between letter and interview date Total no. ≤1 1–2 2.1–5.9 ≥6 Initial refusers Breast cancer, endometrial cancer, and rheumatoid arthritis 47.2 20.8 17.8 8.1 6.0 2,781 Anogenital cancer 50.2 22.7 11.9 7.8 7.4 2,176 Thyroid cancer 51.6 21.6 12.5 7.3 7.0 574 Oral cancer 52.9 20.0 10.6 5.9 10.6 85 Total 48.9 21.6 14.9 7.9 6.7 5,616 TABLE 3. Characteristics (%) of early responders, intermediate responders, late responders, and initial refusers among random digit dialed female controls from six studies conducted in western Washington State, 1987–1998 Early responders(n = 2,747) Intermediate responders†(n = 837) Late responders†(n = 444) Initial refusers†(n = 376) Demographic characteristics Age (years) at reference date‡ ≤30 7.6 11.5 9.2 7.7 30–39 25.1 26.2 23.0 22.3 40–49 19.6 21.3 21.8 20.7 50–59 20.7 19.7 22.5 20.2 ≥60 27.1 21.4 23.4 29.0 Mean 48.6 46.2 47.6 49.7 p value <0.001* 0.056 0.54 Ever attending college‡ 58.0 54.7 48.0 51.3 p value 0.133 <0.001* 0.003 Marital status‡ Not married 27.9 27.4 29.2 31.7 Currently married or living as married 72.1 72.6 70.8 68.3 p value 0.96 0.11 0.01 Race‡ Non-White 5.6 7.6 9.5 9.4 p value 0.03 0.008 0.007 Annual household income‡ <$45,000 70.1 68.5 66.9 69.7 ≥$45,000 29.9 31.5 33.1 30.3 p value 0.33 0.01 0.28 Occupation at reference date§,¶,# Managerial or professional 22.2 23.1 19.2 12.9 Other occupation 36.1 38.4 39.0 49.3 p value 0.49 0.97 0.002 Retired 9.4 9.5 6.2 9.6 p value 0.25 0.07 0.31 Housewife or student 32.3 29.1 35.6 28.2 p value 0.18 0.84 0.05 Any first- or second-degree female relative with cancer§,¶,# 59.9 54.8 56.5 51.0 p value 0.04 0.41 0.11 Reproductive characteristics No. of pregnancies#,** 0 13.0 12.6 12.8 15.0 1 or 2 33.7 34.7 32.0 36.1 p value 0.50 0.94 0.77 >2 53.2 52.8 55.3 48.9 p value 0.66 0.48 0.13 Ever having an induced abortion#,** among women who were ever pregnant 16.7 14.9 12.9 15.6 p value 0.14 0.09 0.88 Ever use of oral contraceptives‡ 62.3 61.3 59.8 61.6 p value 0.28 0.52 0.89 Ever having a test for infertility#,**,†† Yes 11.7 9.3 7.6 9.7 p value 0.03 0.01 0.25 Lifetime no. of sexual partners‡ 0 6.4 6.6 6.8 9.2 1 38.9 38.5 41.0 38.5 p value 0.58 0.58 0.49 2–4 32.5 32.1 33.1 33.6 p value 0.50 0.40 0.046 ≥5 22.2 22.9 19.1 18.7 p value 0.42 0.095 0.04 Health history Ever being diagnosed with hypertension#,** 23.1 23.4 23.8 22.4 p value 0.92 0.49 0.46 Ever having cancer#,** 8.6 8.9 9.3 8.1 p value 0.49 0.50 0.93 Ever being diagnosed with diabetes‡ 4.0 3.7 4.1 7.9 p value 0.31 0.55 0.02 Ever use of noncontraceptive hormones among women aged ≥50 years§,¶,# 60.1 61.0 52.6 51.9 p value 0.13 0.55 0.51 Body mass index‡‡ at reference date‡ <25 66.5 65.7 66.3 64.0 25–29.9 20.9 20.9 22.0 23.7 p value 0.72 0.94 0.65 ≥30 12.6 13.4 11.7 12.3 p value 0.45 0.59 0.53 Mean 24.3 24.2 24.2 24.5 Lifestyle factors Smoking history‡ Never smoker 50.7 48.2 46.9 48.1 Former smoker 26.4 25.7 23.2 29.1 p value 0.93 0.53 0.65 Current smoker 22.9 26.1 29.8 22.7 p value 0.07 0.001* 0.53 Ever drinking alcohol‡ 79.9 82.7 75.8 80.6 p value 0.14 0.19 0.49 Regular exercise within 2 years of reference date§,¶,# 51.2 51.2 42.6 46.1 p value 0.87 0.10 0.26 Early responders(n = 2,747) Intermediate responders†(n = 837) Late responders†(n = 444) Initial refusers†(n = 376) Demographic characteristics Age (years) at reference date‡ ≤30 7.6 11.5 9.2 7.7 30–39 25.1 26.2 23.0 22.3 40–49 19.6 21.3 21.8 20.7 50–59 20.7 19.7 22.5 20.2 ≥60 27.1 21.4 23.4 29.0 Mean 48.6 46.2 47.6 49.7 p value <0.001* 0.056 0.54 Ever attending college‡ 58.0 54.7 48.0 51.3 p value 0.133 <0.001* 0.003 Marital status‡ Not married 27.9 27.4 29.2 31.7 Currently married or living as married 72.1 72.6 70.8 68.3 p value 0.96 0.11 0.01 Race‡ Non-White 5.6 7.6 9.5 9.4 p value 0.03 0.008 0.007 Annual household income‡ <$45,000 70.1 68.5 66.9 69.7 ≥\$45,000 29.9 31.5 33.1 30.3 p value 0.33 0.01 0.28 Occupation at reference date§,¶,# Managerial or professional 22.2 23.1 19.2 12.9 Other occupation 36.1 38.4 39.0 49.3 p value 0.49 0.97 0.002 Retired 9.4 9.5 6.2 9.6 p value 0.25 0.07 0.31 Housewife or student 32.3 29.1 35.6 28.2 p value 0.18 0.84 0.05 Any first- or second-degree female relative with cancer§,¶,# 59.9 54.8 56.5 51.0 p value 0.04 0.41 0.11 Reproductive characteristics No. of pregnancies#,** 0 13.0 12.6 12.8 15.0 1 or 2 33.7 34.7 32.0 36.1 p value 0.50 0.94 0.77 >2 53.2 52.8 55.3 48.9 p value 0.66 0.48 0.13 Ever having an induced abortion#,** among women who were ever pregnant 16.7 14.9 12.9 15.6 p value 0.14 0.09 0.88 Ever use of oral contraceptives‡ 62.3 61.3 59.8 61.6 p value 0.28 0.52 0.89 Ever having a test for infertility#,**,†† Yes 11.7 9.3 7.6 9.7 p value 0.03 0.01 0.25 Lifetime no. of sexual partners‡ 0 6.4 6.6 6.8 9.2 1 38.9 38.5 41.0 38.5 p value 0.58 0.58 0.49 2–4 32.5 32.1 33.1 33.6 p value 0.50 0.40 0.046 ≥5 22.2 22.9 19.1 18.7 p value 0.42 0.095 0.04 Health history Ever being diagnosed with hypertension#,** 23.1 23.4 23.8 22.4 p value 0.92 0.49 0.46 Ever having cancer#,** 8.6 8.9 9.3 8.1 p value 0.49 0.50 0.93 Ever being diagnosed with diabetes‡ 4.0 3.7 4.1 7.9 p value 0.31 0.55 0.02 Ever use of noncontraceptive hormones among women aged ≥50 years§,¶,# 60.1 61.0 52.6 51.9 p value 0.13 0.55 0.51 Body mass index‡‡ at reference date‡ <25 66.5 65.7 66.3 64.0 25–29.9 20.9 20.9 22.0 23.7 p value 0.72 0.94 0.65 ≥30 12.6 13.4 11.7 12.3 p value 0.45 0.59 0.53 Mean 24.3 24.2 24.2 24.5 Lifestyle factors Smoking history‡ Never smoker 50.7 48.2 46.9 48.1 Former smoker 26.4 25.7 23.2 29.1 p value 0.93 0.53 0.65 Current smoker 22.9 26.1 29.8 22.7 p value 0.07 0.001* 0.53 Ever drinking alcohol‡ 79.9 82.7 75.8 80.6 p value 0.14 0.19 0.49 Regular exercise within 2 years of reference date§,¶,# 51.2 51.2 42.6 46.1 p value 0.87 0.10 0.26
* p < 0.05 compared with early responders. Data were adjusted for other variables and were corrected for multiple comparisons using Bonferroni’s correction.
† Proportions for all variables except age were adjusted to the age distribution of early responders.
‡ The p value was adjusted for all other variables in the table except family history of cancer, number of pregnancies, induced abortion, infertility testing, hypertension, history of cancer, use of noncontraceptive hormones, exercise, and occupation.
§ The p value was adjusted for all other variables in the table.
¶ This question was not asked in the anogenital cancer study.
# This question was not asked in the oral cancer study.
** The p value was adjusted for all other variables in the table except occupation, family history of cancer, infertility testing, use of noncontraceptive hormones, and exercise.
†† This question was added to the anogenital cancer interview after the study began.
‡‡ Weight (kg)/height (m)2.
TABLE 4.
Comparison (%) between interviewed female respondents and refusing respondents with regard to Group Health Cooperative medical record data, western Washington State, 1987–1991
Interviewed(n = 49) Refused(n = 59) Age (years) 15–24 10.2 15.2 25–34 8.2 11.9 35–44 24.5 15.3 45–54 16.3 18.6 55–64 40.8 39.0 Mean 46.6 45.7 p value 0.70 No. of pregnancies 0 15.6 16.1 1 42.2 39.3 ≥2 42.2 44.6 p value 0.96 Ever having a livebirth 77.8 78.6 p value 0.92 Ever use of hormones 20.0 17.5 p value 0.75 Smoking history Never smoker 63.6 60.0 Former smoker 11.4 12.7 Current smoker 25.0 27.3 p value 0.93 Body mass index* within 2 years of reference date† <30 82.1 70.7 ≥30 17.9 29.3 Mean 25.4 26.5 p value 0.18
Interviewed(n = 49) Refused(n = 59) Age (years) 15–24 10.2 15.2 25–34 8.2 11.9 35–44 24.5 15.3 45–54 16.3 18.6 55–64 40.8 39.0 Mean 46.6 45.7 p value 0.70 No. of pregnancies 0 15.6 16.1 1 42.2 39.3 ≥2 42.2 44.6 p value 0.96 Ever having a livebirth 77.8 78.6 p value 0.92 Ever use of hormones 20.0 17.5 p value 0.75 Smoking history Never smoker 63.6 60.0 Former smoker 11.4 12.7 Current smoker 25.0 27.3 p value 0.93 Body mass index* within 2 years of reference date† <30 82.1 70.7 ≥30 17.9 29.3 Mean 25.4 26.5 p value 0.18
* Weight (kg)/height (m)2.
† Data for the 2-year period prior to the reference date were not available for 21 of the women interviewed and 18 of those who refused.
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https://www.darwinproject.ac.uk/letter/?docId=letters/DCP-LETT-7069.xml;query=Gray,%20Asa;brand=default;hit.rank=2 | skip to content
# From William Winwood Reade [c. 8 or 9 April 1870]1
Swanzy’s Factory | Accra | West Coast of Africa
My dear Sir
I read your last letter with very great pleasure. I should consider a letter from Darwin a treat anywhere—how much more so out here!2
Alex Agassiz has the reputation in the U.S. of being a patient & earnest investigator.3 I knew he differed from his father who has a perfect right to believe in the immutability of species but who has no right (scientifically speaking) to say “we are the children of God not the children of monkeys”, & other such catch-penny or catch-parson statements.4 I am pleased & surprised he has written something contrary to his previously expressed opinions.5 To tell the truth I feared he was not honest enough, & brave enough to change his assertions even if he changed his opinions. What is a naturalist if he is not sincere? Sincerity is to him what faith hope & charity are to the religieux; if he shows a deficiency in that the most splendid talents can scarcely save him from contempt.
I thought you wd like the Jollof instance—6 I need scarcely say that anything I write to you is fully at your disposal. My only fear is that I cannot send you anything worth having—
The hand to mouth expression of astonishment I am told is common on the coast. I made a statement about a girl in her presence wh. I knew wd. excite her astonishment & watched her face. She protruded her lips (much in that curious manner in wh. the chimpanzee does) as if in the act of blowing.7
I discovered a small fly in the Niger country (ie near Kankan)8 which makes honey in holes of trees. It does not sting. I found a nest & identified the species, specimens of which I sent to Mr. Swanzy of 122 Cannon St. E.C.9 I also sent an insect which carries a load of rubbish on its back perhaps to imitate dead leaves. Perhaps Bates can tell you something about these insects if they are worth inquiring about, for he knows Swanzy & I have asked him to look at them.10 I am ignorant of things entomological.
I am going tomorrow to a German mission station 1500 ft above sea-level. There I shall stay for some weeks & recruit.11 I shall moreover be among Europeans who really live among the natives, speak their language & know something of their inner life. In this country the traveller gets little profit from the experience of residents. Most men come out for a few years; & those who do spend their lives here are usually traders who take no interest in anything but trade. As regards natural history there is not to my knowledge an observer on the coast, except perhaps the present Governor of the Gold Coast who has little time for science—if indeed he can boast of anything beyond a taste for it.12 Since the days of Adanson naturalists (especially in his branch) have come out from time to time—13 I see you gleaned one fact of importance from Mann’s energetic researches.14 Do you know if Rohlfs15 made any discovery of a scientific nature apart from geography?
I hope to have the pleasure of being personally questioned by you when I return wh. will be probably next autumn.16 In the mean-time any hint as to ethnological inquiry will be welcome to me. I forget whether I told you that I had seen a blue-eyed negress (not an albino nor cd. she have had European blood) in the interior.17 This instance I believe stands alone. I hope you will not delay your work too long.18 The best way & the surest to get information upon the points which are detaining you is I imagine to publish; facts will then stream in you can then add to subsequent editions— I conjecture it is painful to you to put forward anything in an incomplete form; there is a finish about your work in a literary sense which cd hardly I think have been achieved without much pains. But consider your disciples. Your new work will doubtless prove a revelation to many; and will certainly suggest to ethnologists fresh methods of investigation. I know for my own part that it will be a great loss to me not to have seen your book till after I have left my present field to which I shall never return— I shall have spent probably 3$\frac{1}{2}$ or 4 years in Africa; to spend more would be to enslave myself to one idea. I am beginning to understand this race; that is up to a certain point. The complex man of civilization is quite undecipherable: women & savages are a little easier; & there is less variety among them. But still the laws of human nature are difficult to get at, even in their simplest form
Hoping to hear from you soon | I remain | Yours truly | Winwood Reade
The above address will find me.
## CD annotations
1.1 I read … having— 3.3] crossed blue crayon
4.1 The hand … blowing. 4.4] enclosed in square brackets pencil; scored pencil
5.1 I … form 7.18] crossed blue crayon
Top of letter: ‘Africa’ blue crayon
## Footnotes
The date range is established by the relationship between this letter and the letter from Reade of 24 April 1870, in which he says that he has been at the Akropong mission station for a fortnight. Akropong is about thirty miles from Accra.
CD’s letter to Reade has not been found. Reade’s career as an explorer and his relationship with CD are discussed in Driver 2001.
Alexander Agassiz had visited CD in late November or early December 1869 (see Correspondence vol. 17, letter to Fritz Müller, 1 December [1869]).
Louis Agassiz had been a vigorous opponent of CD’s theory of natural selection (see Lurie 1960, pp. 252–350).
CD may have reported to Reade a message received from Louis Agassiz via Elizabeth Agassiz and Asa Gray (see letter from Asa Gray, 27 February and 1 March 1870 and n. 4, and letter to Asa Gray, 15 March [1870]).
See Correspondence vol. 17, letter from W. W. Reade, 26 December 1869. CD quoted Reade’s account of the ‘Jollofs’ in Descent 2: 357. The Wolof people (also spelled Ouolof) now live primarily in Senegal, Gambia, and Mauritania (Appiah and Gates eds. 2005, 5: 430).
The city of Kankan is now in Guinea.
Andrew Swanzy, who traded with the Gold Coast, had sponsored Reade’s explorations (Reade 1873, 2: 352–3). The fly has not been identified.
Reade refers to Henry Walter Bates. The insect has not been identified.
The mission station was at Akropong (see letter from W. W. Reade, 24 April 1870). Akropong is now in Ghana. The mission station took in invalids; Reade stayed there for two months (see Reade 1873, 2: 125). Recruit: reinvigorate (Chambers).
The Gold Coast colony had not had a governor since 1866, since when it had been governed from Sierra Leone. The administrator of the government was Herbert Taylor Ussher. (Colonial Office list 1870.)
Michel Adanson studied the natural history of Senegal from 1748 to 1754 (DBF).
Reade refers to Gustav Mann, and to Origin 4th ed., p. 450: ‘So again, on the island of Fernando Po, in the Gulf of Guinea, Mr. Mann found temperate European forms first beginning to appear at the height of about five thousand feet.’
Gerhard Friedrich Rohlfs.
Reade arrived back in London in August 1870 (letter from W. W. Reade, 3 September 1870).
Reade refers to CD’s work on Descent and Expression.
## Bibliography
Chambers: The Chambers dictionary. Edinburgh: Chambers Harrap Publishers. 1998.
Colonial Office list: The Colonial Office list … or, general register of the colonial dependencies of Great Britain. London: Edward Stanford; Harrison & Sons. 1862–99.
Correspondence: The correspondence of Charles Darwin. Edited by Frederick Burkhardt et al. 27 vols to date. Cambridge: Cambridge University Press. 1985–.
DBF: Dictionnaire de biographie Française. Under the direction of J. Balteau et al. 21 vols. (A–Le Nain) to date. Paris: Librairie Letouzey & Ané. 1933–.
Descent: The descent of man, and selection in relation to sex. By Charles Darwin. 2 vols. London: John Murray. 1871.
Driver, Felix. 2001. Geography militant: cultures of exploration and empire. Oxford: Blackwell.
Lurie, Edward. 1960. Louis Agassiz: a life in science. Chicago: University of Chicago Press.
Origin 4th ed.: On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. 4th edition, with additions and corrections. By Charles Darwin. London: John Murray. 1866.
Reade, William Winwood. 1873. The African sketch-book. 2 vols. London: Smith, Elder, and Co.
## Summary
Brief observations on expression in Africa.
Alexander Agassiz is a good investigator, who differs with his father on evolution.
The behaviour of women and savages is a little easier to understand than that of civilised men.
## Letter details
Letter no.
DCP-LETT-7069
From
William Winwood Reade
To
Charles Robert Darwin
Sent from
Accra
Source of text
DAR 176: 36
Physical description
4pp †
## Please cite as
Darwin Correspondence Project, “Letter no. 7069,” accessed on 13 May 2021, https://www.darwinproject.ac.uk/letter/DCP-LETT-7069.xml
Also published in The Correspondence of Charles Darwin, vol. 18
letter | 2021-05-13 10:29:15 | {"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": 0.4679704010486603, "perplexity": 6248.5542271163295}, "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-2021-21/segments/1620243990584.33/warc/CC-MAIN-20210513080742-20210513110742-00503.warc.gz"} |
https://academic.oup.com/nar/article-lookup/doi/10.1093/nar/gkl114 | ## Abstract
TarFisDock is a web-based tool for automating the procedure of searching for small molecule–protein interactions over a large repertoire of protein structures. It offers PDTD (potential drug target database), a target database containing 698 protein structures covering 15 therapeutic areas and a reverse ligand–protein docking program. In contrast to conventional ligand–protein docking, reverse ligand–protein docking aims to seek potential protein targets by screening an appropriate protein database. The input file of this web server is the small molecule to be tested, in standard mol2 format; TarFisDock then searches for possible binding proteins for the given small molecule by use of a docking approach. The ligand–protein interaction energy terms of the program DOCK are adopted for ranking the proteins. To test the reliability of the TarFisDock server, we searched the PDTD for putative binding proteins for vitamin E and 4H-tamoxifen. The top 2 and 10% candidates of vitamin E binding proteins identified by TarFisDock respectively cover 30 and 50% of reported targets verified or implicated by experiments; and 30 and 50% of experimentally confirmed targets for 4H-tamoxifen appear amongst the top 2 and 5% of the TarFisDock predicted candidates, respectively. Therefore, TarFisDock may be a useful tool for target identification, mechanism study of old drugs and probes discovered from natural products. TarFisDock and PDTD are available at http://www.dddc.ac.cn/tarfisdock/ .
## INTRODUCTION
Recent advances in the development of tools for docking small molecules to proteins, i.e. virtual screening, has demonstrated the efficiency of this approach for the discovery of potential lead compounds for drug development in the postgenomic era ( 13 ). Numerous docking programs ( 410 ) have been used to seek ligands which recognize the 3D structure of a given target obtained by X-ray crystallography, NMR spectroscopy or even by homology modeling [for a review comparing and evaluating docking tools see ref. ( 11 )]. However, identification and validation of druggable targets from amongst thousands of candidate macromolecules is still a challenging task ( 12 , 13 ). A proteomic approach for identification of binding proteins for a given small molecule involves comparison of the protein expression profiles for a given cell or tissue in the presence or absence of the given molecule. This method has not proved very successful in target discovery because it is laborious and time-consuming ( 14 ). Thus an efficient computational method for identifying the targets of a small molecule which had been demonstrated experimentally to have an important biological activity would provide a tool of great potential value. An alternative approach that has shown promise in recent years is to use computational methods to find putative binding proteins for a given compound from either genomic or protein databases, and subsequently use experimental procedures to validate the computational result ( 1518 ). One such computational approach, which is the reverse of docking a set of ligands into a given target, is to dock a compound with a known biological activity into the binding sites of all the 3D structures in a given protein database. Protein ‘hits’ so identified can then serve as potential candidates for experimental validation. Accordingly, this approach is referred to as reverse docking.
Herein, we present a web-based tool Target Fishing Dock (TarFisDock) for seeking potential binding proteins for a given ligand. It makes use of a ligand–protein reverse docking strategy to search out all possible binding proteins for a small molecule from the potential drug target database (PDTD). The small molecule might be a biologically active compound detected in a cell- or animal-based bioassay screen, a natural product or an existing drug whose molecular target(s) is (are) unknown. Thus, TarFisDock may serve as a valuable tool for identifying targets for a novel synthetic compound or for a newly isolated natural product, for a compound with known biological activity, or for an existing drug whose mechanism of action is unknown.
## METHODS
### Construction of the potential drug target database
TarFisDock requires a sufficient number of known protein structures covering a diverse range of drug targets. The target proteins collected in PDTD were selected from the literature ( 1922 ), and from several online databases, such as DrugBank ( http://redpoll.pharmacy.ualberta.ca/drugbank/ ) ( 23 ), and TTD ( http://bidd.nus.edu.sg/group/cjttd/ ) ( 24 ). Only proteins with known 3D structures were deposited in PDTD, the Protein Data Bank (PDB) ( 25 ) being the major source of their coordinates. PDTD currently consists of 698 entries covering 371 drug targets. These drug targets may be categorized into 15 types, according to their therapeutic areas ( 20 , 22 ), as shown in Table 1 . Because TarFisDock does not take into account protein flexibility, PDTD includes redundant entries for proteins known to be flexible. Thus, for example, there are seven entries for HIV-1 ( Figure 1 ).
Water molecules and complexed ligands were removed from the protein structures, after which hydrogen atoms were added, and KOLLMAN charges ( 26 ), with the protonation state of the individual residues being taken into account during charge assignment. A mo12 file (Mol2 file (.mol2) developed by SYBYL, Tripos Inc., St Louis, USA ( http://www.tripos.com/ ) is a complete, portable representation of a SYBYL molecule. It is an ASCII file which contains all the information needed to reconstruct a SYBYL molecule.) was then constructed for each protein. The active site of each protein was defined as all residues within 6.5 Å of the ligand bound, and a sphere file for the active site was generated using the SPHGEN program ( 27 ). The PDB, mol2 and sphere files for each protein were stored in PDTD.
### Reverse docking procedure using TarFisDock
TarFisDock consists of two parts, a front-end web interface written in both PHP and HTML, with MySQL as database system, and a back-end tool for reverse docking. TarFisDock was developed on the basis of the widely used docking program, DOCK (version 4.0) ( 5 , 27 ). The reverse docking procedure is as follows: (i) TarFisDock either generates a protein target list according to the user's preference (see INPUT) or selects all the protein entries in the PDTD if the user intends to find a new target or targets for an active compound; (ii) TarFisDock docks a given small molecule into the possible binding sites of proteins in the target list, and the interaction energies between the small molecule and the proteins are calculated and recorded; (iii) TarFisDock analyzes the reverse docking result. In general, TarFisDock may output the top 2, 5 or 10% of the ranking list, from which the user may select protein candidates for further biological study. So far, TarFisDock has taken into account the flexibility of the small molecules, but has not yet taken into account protein flexibility. Putative binding proteins are selected by ranking the values of the interaction energy ( Einter ), which is composed of van der Waals and electrostatic interaction terms ( Equation 1 ),
${E}_{\hbox{ inter }}={\displaystyle \sum _{i=1}^{\mathit{lig}}}{\displaystyle \sum _{j=1}^{\mathit{rec}}}\left(\frac{{A}_{\mathit{ij}}}{{r}_{\mathit{ij}}^{a}}-\frac{{B}_{\mathit{ij}}}{{r}_{\mathit{ij}}^{b}}+332.0\frac{{q}_{i}{q}_{j}}{{\mathit{Dr}}_{\mathit{ij}}}\right),$
where each term is a double sum over ligand atoms i and receptor atoms j ; r ij is the distance between atom i in the ligand and atom j in the putative receptor protein; A ij and B ij are van der Waals repulsion and attraction parameters, respectively; a and b are the van der Waals repulsion and attraction exponents, respectively; q i and q j are point charges on atoms i and j ; D is dielectric function; and 332.0 is the factor that converts the electrostatic energy into kcal/mol. The Amber force field ( 26 ) was used for the energy calculation.
## INPUT, OUTPUT AND OPTIONS
The input file consists of only the test small molecule in standard mol2 format. The 2D structure of a small molecule can be either sketched using ISIS/Draw (ISIS/Draw, MDL Informations Systems, Inc., San Leandro, CA 945577) or ChemDraw (ChemDraw, CambridgeSoft Corporation, 875 Massachusetts Avenue, Cambridge, MA 02139, USA) or taken from such chemical databases as CCD ( http://www.chemnetbase.com/ ), ACD ( http://www.mdli.com/ ) and SPECS ( http://www.specs.net/ ). The user can convert the small molecule from its 2D structures to the 3D structures by using CORINA ( 28 ) ( http://www2.chemie.uni-erlangen.de/software/corina/free_struct.html ) or other modeling software. The structures can be minimized by means of molecular mechanics, and Gasteiger charges ( 29 ) should be assigned to them. Finally, the 3D structure of the small molecule is saved in a mol2 file.
Users can register free of charge for using the TarFisDock server, including access to PDTD. The user must provide his/her email address and username so as to receive the result. After registration, the user can login to the server to upload the mol2 file of the test molecule, customize a target list from PDTD, and submit a job ( Figure 2 ). A job identity number, the ‘job_id’, is assigned to each job by the web server, and the number is appended to a job queue in the back-end server. The user may use the job_id to check the status of his/her job.
The output is delivered in ascending order of energy score (interaction energy). The archive file contains a list of the scores, together with binding models (in mol2 format) of the small molecule tested within the binding sites of the candidate targets. The user can also browse the ‘Categories’ dropdown menu of PDTD to obtain detailed information for the potential target proteins identified by TarFisDock: the ‘PDB_ID’ field contains a hyperlink to the PDB website; the ‘TARGET NAME’ field also contains a hyperlink to the DrugBank website ( Figures 1 and 2 ), and any information linking targets to diseases is contained in the ‘RELATED DISEASE’ field taken from TTD.
## TEST CASES
To test the reliability of the TarFisDock server, we searched for the candidate binding proteins for vitamin E and for 4H-tamoxifen. The results and their comparison with the published experimental data are described below.
### Potential binding proteins for vitamin E
Vitamin E is an antioxidant which is widely used as a dietary supplement ( 30 ). It has also been shown to be of therapeutic value in the treatment of a number of diseases, such as cardiovascular disease and some forms of cancer, and to enhance the immune response ( 31 ). It is thus likely that vitamin E may interact with multiple target proteins. Indeed, 12 targets for vitamin E have already been reported ( 16 ) (Supplementary Table S1). Candidate vitamin E-binding proteins identified using TarFisDock are listed in Supplementary Table S2. The top 2% candidates identified by TarFisDock, ranked by interaction energies, included 4 out of the 12 targets identified experimentally. Three more of these experimentally identified targets were in the top 10% of the proteins ranked by interaction energy. The top 2 and 10% candidates of vitamin E-binding proteins identified by TarFisDock cover 30 and 50%, respectively, of reported targets verified or implicated by experiments. Other targets, such as glutathione S -transferase, glutathione synthetase, D-amino acid oxidase, and guanylyl cyclase (it is not available in PDTD), were not identified by TarFisDock ( Table 2 ). The main reason may be that TarFisDock does not take into account protein flexibility. It is of interest that many of the top 10% candidate vitamin E-binding proteins are associated with cancer, cardiovascular diseases, immune function and dementia (Supplementary Table S2).
### Potential binding proteins for 4H-tamoxifen
4H-tamoxifen is used as an adjuvant therapy in the treatment of breast cancer ( 32 ). Like vitamin E, it is a multiple target drug. So far, 10 proteins have been identified as interaction targets for 4H-tamoxifen or for its metabolite, tamoxifen ( 16 ) (Supplementary Table S1). To test the reliability of our TarFisDock server, we used it to search for candidate binding proteins for 4H-tamoxifen in the PDTD. The target candidates so thus identified are listed in Supplementary Table S3, and those which correspond to proteins identified experimentally are shown in Table 3 . Three amongst the top 2% of the candidates are known targets of 4H-tamoxifen, namely dihydrofolate reductase, immunoglobulin and glutathione transferase. The top 5% of the candidates include two additional targets identified experimentally, i.e. human fibroblast collagenase and 17β-hydroxysteroid dehydrogenase. Of experimentally confirmed targets for 4H-tamoxifen 30 and 50% appear amongst the top 2 and 5% of the TarFisDock predicted candidates, respectively, indicating the reliability of this server tool again.
TarFisDock has been in use for about 9 months, and over 1000 small molecules, including synthetic compounds, existing drugs and natural products, have been screened. Five groups outside the authors' labs have become involved in screening. Experimental evidence has been obtained to confirm that binding proteins identified by TarFisDock for several compounds indeed display binding activity. In one case, that of a binding protein for a natural product, not only was binding verified experimentally, but a complex was obtained whose 3D crystal structure was solved by X-ray crystallography (data not shown). The computing time required depends on the flexibility of the given compound. Thus, TarFisDock may finish the PDTD search within 5–20 h using one CPU of the SGI Origin3800 superserver.
## SUMMARY
In bringing together the target database PDTD and the reverse docking program, TarFisDock server is a convenient tool for identification of potential binding proteins for small molecules such as drugs, lead compounds and natural products. Totally, this web server has already been tested for over 1000 small molecules, the binding proteins for several molecules have been verified by bioassay including crystal structure determination (data not shown). This web server can also be used in mapping the regulation genomic network for an existing drug or a drug candidate. In general, one drug molecule may interact with several targets including targets associated with side effect (toxicity). As illustrated by the examples for identifying potential binding proteins of vitamin E and 4H-tamxifen, TarFisDock provides multiple options for selecting protein targets. These are useful clues for further experimental test in evaluating the efficacy and toxicity of the drug. On the other hand, the targets information produced by TarFisDock is also significant for functional genomic study with the chemical biology paradigm ( 33 ). In general, TarFisDock web sever is a convenient tool for ‘fishing’ the target proteins of small molecules, the user just inputs the structure of querying compound and customizes a target list from PDTD (a list of all the targets is recommended).
However, TarFisDock still has certain limitations. The major one is that the protein entries are not enough for covering all the protein information of disease related genomes. The second one is that TarFisDock has not considered the flexibility of proteins during docking simulation. These two aspects will produce negative false. Another limitation is that the scoring function for reverse docking is not accurate enough, which will produce positive false. To overcome these shortages, we are (i) collecting proteins structures (experimental and modeling structures) as more as possible for enlarging PDTD, (ii) developing new docking program including protein flexibility, and (iii) establishing accurate scoring function. TarFisDock and PDTD are available at http://www.dddc.ac.cn/tarfisdock/ .
## SUPPLEMENTARY DATA
Supplementary Data are available at NAR online.
Figure 1
An example of PDTD querying and finding out 22 targets records of ‘[HIV] DISEASE’.
Figure 1
An example of PDTD querying and finding out 22 targets records of ‘[HIV] DISEASE’.
Table 1
Diseases categories of drug targets in PDTD
(1) Synaptic And Neuroeffector Junctional Sites And Central Nervous System (2) Inflammation (3) Renal And Cardiovascular Functions (4) Gastrointestinal Functions (5) Uterine Motility (6) Bacterial Infections (7) Fungal Infections (8) Viral Infections (9) Parasitic Infectious Diseases (10) Immunomodulation (11) Blood And Blood-Forming Organs (12) Neoplastic Diseases (13) Hormones And Hormone Antagonists (14) The Vitamins (15) Undefined
(1) Synaptic And Neuroeffector Junctional Sites And Central Nervous System (2) Inflammation (3) Renal And Cardiovascular Functions (4) Gastrointestinal Functions (5) Uterine Motility (6) Bacterial Infections (7) Fungal Infections (8) Viral Infections (9) Parasitic Infectious Diseases (10) Immunomodulation (11) Blood And Blood-Forming Organs (12) Neoplastic Diseases (13) Hormones And Hormone Antagonists (14) The Vitamins (15) Undefined
Figure 2
An example of the input and output of TarFisDock.
Figure 2
An example of the input and output of TarFisDock.
Table 2
The protein target candidates of vitamin E identified by TarFisDock
Rank PDB_ID Energy score Target name
1VXR −32.61 Acetylcholinesterase
1DHT −32.49 Estrogenic 17β-hydroxysteroid dehydrogenase
2NSE −32.07 Nitric oxide synthase
2ACS −30.49 Aldose reductase
20 1M9M −29.28 Nitric oxide synthase
28 1GPN −28.56 Acetylcholinesterase
49 1DBJ −27.4 Fab' fragment of monoclonal antibody Db3
53 4FAB −27.37 4-4-20 (IgG2A) Fab fragment
62 5P2P −27.14 Phospholipase A2
Rank PDB_ID Energy score Target name
1VXR −32.61 Acetylcholinesterase
1DHT −32.49 Estrogenic 17β-hydroxysteroid dehydrogenase
2NSE −32.07 Nitric oxide synthase
2ACS −30.49 Aldose reductase
20 1M9M −29.28 Nitric oxide synthase
28 1GPN −28.56 Acetylcholinesterase
49 1DBJ −27.4 Fab' fragment of monoclonal antibody Db3
53 4FAB −27.37 4-4-20 (IgG2A) Fab fragment
62 5P2P −27.14 Phospholipase A2
Table 3
The protein target candidates of 4H-tamoxifen identified by TarFisDock
Rank PDB_ID Energy score Target name
1DHF −36.8 Dihydrofolate reductase
10 1MCR −35.4 Immunoglobulin λ light chain dimer
12 1K3Y −34.66 Glutathione transferase
13 1DBM −34.51 Fab' fragment of monoclonal antibody Db3
17 4DFR −34.07 Dihydrofolate reductase
19 2CGR −33.68 Igg2B (κ) Fab fragment
21 4AYK −33.53 Human fibroblast collagenase
22 4FAB −33.49 4-4-20 (IgG2A) Fab fragment
25 1DBJ −33.26 Fab' fragment of monoclonal antibody Db3
27 1DHT −33.12 17β-Hydroxysteroid dehydrogenase
31 1AYK −32.9 Human fibroblast collagenase
46 1RA2 −31.75 Dihydrofolate reductase
48 1DIH −31.71 Microbial dihydrofolate reductase
50 1MCB −31.66 Immunoglobulin λ light chain dimer
70 1BHS −30.81 17β-Hydroxysteroid dehydrogenase
Rank PDB_ID Energy score Target name
1DHF −36.8 Dihydrofolate reductase
10 1MCR −35.4 Immunoglobulin λ light chain dimer
12 1K3Y −34.66 Glutathione transferase
13 1DBM −34.51 Fab' fragment of monoclonal antibody Db3
17 4DFR −34.07 Dihydrofolate reductase
19 2CGR −33.68 Igg2B (κ) Fab fragment
21 4AYK −33.53 Human fibroblast collagenase
22 4FAB −33.49 4-4-20 (IgG2A) Fab fragment
25 1DBJ −33.26 Fab' fragment of monoclonal antibody Db3
27 1DHT −33.12 17β-Hydroxysteroid dehydrogenase
31 1AYK −32.9 Human fibroblast collagenase
46 1RA2 −31.75 Dihydrofolate reductase
48 1DIH −31.71 Microbial dihydrofolate reductase
50 1MCB −31.66 Immunoglobulin λ light chain dimer
70 1BHS −30.81 17β-Hydroxysteroid dehydrogenase
Correspondence may also be addressed to Xicheng Wang. Tel: +86 411 84706223; Fax: +86 411 84709390; Email: [email protected]
The authors wish it to be known that, in their opinion, the first three authors should be regarded as joint First Authors
The authors thank Prof. Irwin D. Kuntz for providing the source code of DOCK4.0. The Shanghai Supercomputing Center and Computer Network Information Center are acknowledged for allocation of computing time. The authors thank Prof. Israel Silman at Weizmann Institute of Science and the reviewers for critical reading and helpful comments on the manuscript. This study was supported by the Special Fund for the Major State Basic Research Project of China (grants 2002CB512802 and 2002CB512807) from Ministry of Science and Technology of China and the National Natural Science Foundation of China (grant 10572033).
Conflict of interest statement . None declared.
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http://registry.gimp.org/node/27107 | # Preview in Windows Explorer
Windows displays small to xlarge thumbnails/previews of jpg/gif/png/etc files as you browse directories in explorer and file open dialogs. Is there a way to get this to happen with gimp xcf files? Thanks for reading!
### You'll have to make sure that
You'll have to make sure that entries are created in the Windows thumbnail caches (see http://en.wikipedia.org/wiki/Windows_thumbnail_cache). This could probably be done instead of or alongside the current thumbnail creation (which is using the Thumbnail Managing Standard, see http://specifications.freedesktop.org/thumbnail-spec/thumbnail-spec-late...)
This will require change to the GIMP source code. | 2015-05-27 21:51:34 | {"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": 0.8189151883125305, "perplexity": 5341.640993800693}, "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-2015-22/segments/1432207929171.55/warc/CC-MAIN-20150521113209-00225-ip-10-180-206-219.ec2.internal.warc.gz"} |
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