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http://swmath.org/software/1448 | # dsoa
dsoa: the implementation of a dynamic system optimization algorithm This paper describes the ANSI C/C++ computer program dsoa, which implements an algorithm for the approximate solution of dynamics system optimization problems. The algorithm is a direct method that can be applied to the optimization of dynamic systems described by index-1 differential-algebraic equations (DAEs). The types of problems considered include optimal control problems and parameter identification problems. The numerical techniques are employed to transform the dynamic system optimization problem into a parameter optimization problem by: (i) parameterizing the control input as piecewise constant on a fixed mesh, and (ii) approximating the DAEs using a linearly implicit Runge-Kutta method. The resultant nonlinear programming (NLP) problem is solved via a sequential quadratic programming technique. The program dsoa is evaluated using 83 nontrivial optimal control problems that have appeared in the literature. Here we compare the performance of the algorithm using two different NLP problem solvers, and two techniques for computing the derivatives of the functions that define the problem.
## Keywords for this software
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## References in zbMATH (referenced in 6 articles )
Showing results 1 to 6 of 6.
Sorted by year (citations) | {"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": 0.8183316588401794, "perplexity": 492.69531222329573}, "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-2018-43/segments/1539583512836.0/warc/CC-MAIN-20181020142647-20181020164147-00243.warc.gz"} |
https://physics.stackexchange.com/questions/229108/what-is-the-metric-tensor-for/514110 | # What is the metric tensor for?
I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their applications are well defined.
In that book, is explained how to get the metric tensor for coordinate system transformation, such as from spherical coordinates to ordinary Cartesian coordinates or even from cylindrical coordinates to Cartesian coordinates; which are easy to obtain, given enough practice. But what do such metric tensors mean (in practice), how does one use such a tensor in an actual math/physics problem?
• A metric is usually associated with a particular coordinate system. Can you elaborate on how you are using one in a transformation? Usually that would be done by the Jacobian. – user10851 Jan 13 '16 at 0:09
• For example the trasformation from polar coordinates (r,θ)= (x^1,x^2) to Cartesian coordinates (x,y) goes as follows: x=rcos(θ) and y = rsin(θ) and the components of the metric are g ij = [(dx/x^i)(dx/d^j)+(dy/dx^i)(dy/d^j)] where i and j take on values [1 or 2]. This yeilds a 2 x 2 matrix in this case. – Investor Jan 13 '16 at 0:17
• @ChrisWhite - The metric is most certainly not associated with any coordinate system (no tensor is!). – Prahar Mitra Jan 13 '16 at 0:25
• @Prahar The components (you know what I mean). Certainly a metric is never associated with a coordinate transformation. – user10851 Jan 13 '16 at 0:26
• @ChrisWhite - Agreed. – Prahar Mitra Jan 13 '16 at 0:26
The metric measures lengths in various directions, and also angles between various directions. For example if $\vec{e}_{(1)}$ is the basis vector in the $x^1$-direction, it will have length (squared) given by $$\lVert \vec{e}_{(1)} \rVert^2 = g(\vec{e}_{(1)}, \vec{e}_{(1)}) = g_{11}.$$ If we also have the basis vector $\vec{e}_{(2)}$ in the $x^2$-direction, then the angle $\theta$ between these vectors obeys $$\lVert \vec{e}_{(1)} \rVert \cdot \lVert \vec{e}_{(2)} \rVert \cos\theta = \vec{e}_{(1)} \cdot \vec{e}_{(2)} = g(\vec{e}_{(1)}, \vec{e}_{(2)}) = g_{12}.$$
So far we haven't made any mention of transforming coordinates. Now coordinate transformations are something we'd like to be able to do, and the rule for the metric (or indeed any rank-(0,2) tensor) is $$g_{ij} = \sum_{\hat{\imath},\hat{\jmath}} \frac{\partial x^\hat{\imath}}{\partial x^i} \frac{\partial x^\hat{\jmath}}{\partial x^j} g_{\hat{\imath}\hat{\jmath}}. \qquad \text{(all coordinate transformations)}$$ If the hatted coordinate system is normal Euclidean space with normal Cartesian coordinates, $g_{\hat{\imath}\hat{\jmath}} = \delta_{\hat{\imath}\hat{\jmath}}$ and we are left with $$g_{ij} = \sum_{\hat{\imath}} \frac{\partial x^\hat{\imath}}{\partial x^i} \frac{\partial x^\hat{\imath}}{\partial x^j}. \qquad \text{(Cartesian hatted coordinates only)}$$ But this is just a rule for transforming the metric from one coordinate system to another. The real use of the metric is to calculate lengths and angles in a particular coordinate system (as above), or to describe the local geometry of space(time) in a concise, abstract way (in which case you don't even find its components in any particular coordinate system).
The metric is an important concept in general relativity.
In GR, vectors correspond to weighted directions in spacetime (by "weighted", I mean any scalar multiple of a vector corresponds to the same direction, but weighted differently). The metric tensor can then tell us about the angle between two directions or the magnitude of a given vector, which gives us a notion of length in spacetime.
The metric also appears in Einstein's equtaions, relating the distribution of energy and momentum through spacetime to the curvature--which involves the metric and its derivatives. That is, curvature--and therefore the metric--of spacetime are determined by the distribution of energy and momentum.
• But how do you use me metric in transformation from spherical coordinates to Cartesian coordinates, as taught in the book I read? – Investor Jan 13 '16 at 0:06
• Is that your question--whether the metric plays a role in the transformation from one set of coordinates to another? If so, then I would ask what you're trying to compare against. Do you want to look at the expressions for the spherical basis vectors in terms of the Cartesian ones? Do you want to look at the new metric compared to the old one? – Muphrid Jan 13 '16 at 0:16
• I added a comment above, if it can help. – Investor Jan 13 '16 at 0:19
• Sure, that computes the new metric (in polar coordinates) when the old metric (in Cartesian coordinates) is the identity matrix. That formula would not be correct if the metric in Cartesian coordinates were some other symmetric matrix (positive definite as well, for Euclidean spaces). – Muphrid Jan 13 '16 at 0:22
• It may mean curved space; it is not simple to tell if space is curved or not just by looking at the metric. The metric may be flat and still not identity (for instance, a multiple of the identity is still flat); conversely, even in polar coordinates the space is still flat. But in general, the space isn't necessarily flat. Consider, for example, the metric for a Schwarzschild black hole. Even if you considered the coordinates there to be related to Cartesian ones by the usual transformation, the resulting metric for those Cartesian coordinates would not be the identity. – Muphrid Jan 13 '16 at 0:30
The metric is a rank two tensor that defines several features of a differential manifold. It defines how to relate changes in distance to changes in coordinates, how to take take the inner product of two vectors. Given an inner product we have a way of measuring angles.
More indirectly, the metric describes geodesics in a manifold, the shortest distance between two points. Just as the shortest distance between two points on the surface of a sphere is not a straight line, this is also the case in a general non-flat space. Speaking of, the metric contains information about the curvature of a manifold as well as how coordinate basis vectors change within the manifold.
Given its relation with the inner product, the metric provides a means to generalize Gauss' and Stoke's Laws to higher dimensions.
This is an excellent introduction to these topics:A First Course in General Relativity by Schultz | {"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": 0.9435696005821228, "perplexity": 236.00702629259737}, "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-2021-17/segments/1618038098638.52/warc/CC-MAIN-20210417011815-20210417041815-00176.warc.gz"} |
https://physics.stackexchange.com/questions/386701/mn-dim-hilbert-space-vs-product-of-m-and-n-hilbert-spaces | # $mn$-dim Hilbert space vs product of $m$ and $n$ Hilbert spaces
Assume one has a Hilbert space $\mathcal{H}_K$ of dimension $K=mn$. Associated to it, there exists a set of all possible quantum transformations $\mathcal{E}$. Of those, the most important are, of course, the unitary transformations $\mathcal{U}$ responsible for all the possible ways of evolution, and the set of measurements $\mathcal{M}$.
We also state that $\widetilde{\mathcal{H}}_K$ is a tensor product of two Hilbert spaces $\mathcal{H}_m$ and $\mathcal{H}_n$.
When learning about Hilbert spaces obtained by means of the tensor product, like $\widetilde{\mathcal{H}}_K$, we are always taught about the fundamental difference between the states which are represented by elementary tensors, ${|u\rangle_1\otimes|v\rangle_2}$, and the entangled ones, having more than one term in the Schmidt decomposition. But what is the reflection of this difference in terms of vectors from $\mathcal{H}_K$? In other words, how do we describe the entanglement in terms of a single system $\mathcal{H}_K$?
Do all operations from $\widetilde{\mathcal{E}}$ have their counterparts in $\mathcal{E}$, and vice versa? How to describe measurements within $\mathcal{H}_m$ (but not $\mathcal{H}_n$) in the formalism of $\mathcal{H}_K$? What will be the state of the system after such a measurement in the $\mathcal{H}_K$ language?
Numerous questions arise...
## 2 Answers
Two finite-dimensional Hilbert spaces of the same dimension are isomorphic, and that's it. Sure anything in one has a counterpart in the other. What you are facing here is that some object in $\tilde{\mathcal{H}}_K$ are just more special than their counterparts in $\mathcal{H}_K$ would be.
Take, for example, the state space of two qubits versus a generic 4-dimensional space (so $m = n = 2$), and their bases $\{|0\rangle, |1\rangle\}^{\otimes 2}$ versus $\{|a\rangle,|b\rangle,|c\rangle,|d\rangle\}$. You wouldn't consider $b+d$ any more special than $b+c$, for example, while the difference between $|01\rangle + |11\rangle$ and $|01\rangle + |10\rangle$ is that one we call separated and the other entangled. This semantic difference is lost when we erase the difference between the two spaces. The same goes for quantum operations.
Perhaps the most illustrative example is a change of basis. In the tensor product space, you would likely primarily consider new bases made by tensor multiplying two bases of $\mathbb{C}^2$, to maintain the separation. But you might also just as well use the basis of Bell states, $$\{(|00\rangle + |11\rangle)/\sqrt2, (|00\rangle - |11\rangle)/\sqrt2, (|01\rangle + |10\rangle)/\sqrt2, (|01\rangle - |10\rangle)/\sqrt2\}.$$ Some things are clearer in it, but not e.g. telling which states were separable and which not – that suddenly requires more calculation, basically reducing to transforming back and then doing what you would have done in the original basis. This is because the $\tilde{\mathcal{H}}_4$ was treated like $\mathcal{H}_4$. No information has been lost, it's only a matter of your choice of mathematical description of the same thing.
• I see. What would you say about the measurements? For example, what are the counterparts of measuring the second bit in (b+d) and (b+c) states? (in one case it tells us nothing about the first bit, in another case it tells everything) – mavzolej Feb 16 '18 at 14:16
• Measurements of the first qubit of a two-qubit system are given by (a projector) ⊗ (identity), in the tensor decomposition. These operator are used on the "whole" state space to determine the probabilities and post-measurement states as usual, and give the results you write. But they are just special cases of 4×4 matrices (projectors, Hermitian matrices, measurement operators – depending on the preferred description of quantum measurement) and could be substituted by more general forms. And just like the states they can also easily lose their tensor product form in result of a basis change. – The Vee Feb 16 '18 at 15:37
Assume one has a Hilbert space of dimension ${K= mn}$. It is possible to describe it in (at least) two ways — simply as a $K$-dimensional space, or as a product of $m$- and $n$-dimensional spaces. In the former case we shall call it $\mathcal{H}$, in the latter — $\widetilde{\mathcal{H}}$. All the states in $\mathcal{H}$ and ${\widetilde{\mathcal{H}}=\widetilde{\mathcal{H}}_1\otimes \widetilde{\mathcal{H}}_2}$, as well as the quantum operations, should be in one-to-one correspondence with each other. So, mathematically these two descriptions are equivalent. Let us now explore whether anything is changing from the physical point of view.
For example, in the case of $\widetilde{\mathcal{H}}$ we can measure only a part' of the state corresponding to one of the Hilbert spaces $\widetilde{\mathcal{H}}_1$ and $\widetilde{\mathcal{H}}_2$. This will cause the partial collapse of the wave function. By performing measurements in $\widetilde{\mathcal{H}}_1$, one gets (here and in what follows for simplicity we assume ${\dim \widetilde{\mathcal{H}}_1 = \dim \widetilde{\mathcal{H}}_2 = 2}$, and also that the measurements in $\widetilde{\mathcal{H}}_{1,2}$ are performed in the computational basis):
\begin{alignat}{9} \label{pr1} \dfrac{1}{2}(|0\rangle+|1\rangle) \otimes (|0\rangle+|1\rangle) &\longrightarrow \left[\begin{alignedat}{9} &|0\rangle\otimes \dfrac{1}{\sqrt{2}} (|0\rangle+|1\rangle)\\ &|1\rangle\otimes \dfrac{1}{\sqrt{2}} (|0\rangle+|1\rangle) \end{alignedat}\right.\quad&&,\qquad\qquad(1)\\ \label{pr2} \dfrac{1}{\sqrt{2}} (|0\rangle\otimes|0\rangle+|1\rangle\otimes|1\rangle) &\longrightarrow \left[\begin{alignedat}{9} &|0\rangle\otimes |0\rangle \\ &|1\rangle\otimes |1\rangle \end{alignedat}\right.\quad&&.\qquad\qquad(2) \end{alignat} In the former case, the measurement in $\widetilde{\mathcal{H}}_1$ does not affect the part of the state belonging to $\widetilde{\mathcal{H}}_2$. In the latter case the situation is opposite — due to entanglement, the outcome of the $\widetilde{\mathcal{H}}_1$ measurement ensues the collapse of $\widetilde{\mathcal{H}}_2$. It natural to ask what are the reflections of these effects in the case of $\mathcal{H}$.
Denoting the four states of $\mathcal{H}$ with letters, we assign the states as: \begin{alignat}{99} |00\rangle & = |a\rangle \quad&&,\qquad |01\rangle & = |b\rangle \quad&&,\qquad |10\rangle & = |c\rangle \quad&&,\qquad |11\rangle & = |d\rangle \quad&&.\qquad\qquad(3) \end{alignat}
We now want to recast (1) and (2) in terms of $\mathcal{H}$: \begin{alignat}{9} \label{pr3} \dfrac{1}{2}(|a\rangle+|b\rangle+|c\rangle+|d\rangle) &\longrightarrow \left[\begin{alignedat}{9} &\dfrac{1}{\sqrt{2}} (|a\rangle+|b\rangle) \\ &\dfrac{1}{\sqrt{2}} (|c\rangle+|d\rangle) \end{alignedat}\right.\quad&&,\qquad\qquad(4)\\ \label{pr4} \dfrac{1}{\sqrt{2}} (|a\rangle+|d\rangle) &\longrightarrow \left[\begin{alignedat}{9} &|a\rangle \\ &|d\rangle \end{alignedat}\right.\quad&&.\qquad\qquad(5) \end{alignat}
It's tempting to call the operations above projecting onto the states on the RHS of (1)'. To convince ourselves, let us take a look at the quantum channels corresponding to these measurements. The $\widetilde{\mathcal{H}}_1$ measurement operators of $\widetilde{\mathcal{H}}$ are: $$\widetilde{M}_1 = \dfrac{1}{\sqrt{2}} (|0\rangle\langle0|) \otimes {\rm 1\hspace{-0.4ex}\rule{0.1ex}{1.52ex}\hspace{0.2ex}} \qquad,\quad \widetilde{M}_2 = \dfrac{1}{\sqrt{2}} (|1\rangle\langle1|) \otimes {\rm 1\hspace{-0.4ex}\rule{0.1ex}{1.52ex}\hspace{0.2ex}} \quad.\qquad\qquad(6)$$ They represent a complete set of measurement operators in $\widetilde{\mathcal{H}}_1$, and, of course, an incomplete set in $\widetilde{\mathcal{H}}$. Rewriting those in the basis of $\mathcal{H}$ renders: \begin{alignedat}{9} M_1 &= \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \otimes {\rm 1\hspace{-0.4ex}\rule{0.1ex}{1.52ex}\hspace{0.2ex}} = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\\end{pmatrix} = |a\rangle\langle a| + |b\rangle\langle b| \quad&&,\qquad\qquad(7)\\ M_2 &= \dfrac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \otimes {\rm 1\hspace{-0.4ex}\rule{0.1ex}{1.52ex}\hspace{0.2ex}} = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1\\\end{pmatrix} = |c\rangle\langle c| + |d\rangle\langle d| \quad&&.\qquad\qquad(8) \end{alignedat} As expected, these are the projectors onto $\operatorname{span}\{|a\rangle,|b\rangle\}$ and $\operatorname{span}\{|c\rangle,|d\rangle\}$.
Now, speaking of physics. What would one say about the state ${\dfrac{1}{2}(|0\rangle+|1\rangle) \otimes (|0\rangle+|1\rangle)}$ when measuring in the computational basis? — 'Not sure about either of two qubits'. And about the post-measurement state ${|0\rangle\otimes \dfrac{1}{\sqrt{2}} (|0\rangle+|1\rangle)}$? — 'Not sure about the second qubit but the first one is definitely $|0\rangle$'. Clearly, (4) permits for the similar interpretation.
The celebrated feature of the maximally entangled Bell state is the observer's ability to completely identify the post-measurement state of a composite system by performing a measurement on its part. By looking at (5), we see how this mechanism works in the language of $\mathcal{H}$: the identification of a general post-measurement state after projecting the input on a two-dimensional subspace is, of course, impossible. However, for certain special states such a `soft' measurement may give the full information about the post-measurement state. | {"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": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9698377251625061, "perplexity": 302.2786393172814}, "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-2021-17/segments/1618038089289.45/warc/CC-MAIN-20210416191341-20210416221341-00094.warc.gz"} |
https://www.physicsforums.com/threads/a-homogeneous-linear-system-w-constant-coefficients.394009/ | # A Homogeneous Linear System w/ Constant Coefficients
1. Apr 10, 2010
### Jamin2112
I'll make this post short.
The problem just asks me to something in the form x'=Ax (A is a 2x2 of constants) and then describe the behavior of the solution as t approaches infinity.
My solution is x=C1e-2t(2/3 1)T + C2e-t(1 1)T.
Since both vectors are multiplied by 1/et, my solution just goes to (0 0)T as t->∞, right?
2. Apr 10, 2010
### tiny-tim
Hi Jamin2112!
Yes …
but from which direction?
(and if your solution is for x, not xT, why is it a transpose? )
3. Apr 10, 2010
### Jamin2112
Should indeed be xT
From the sketch I drew, it approaches the origin along y=x in quadrant 1 and goes away from the origin along y=x in quadrant 3. Correct?
4. Apr 10, 2010
### tiny-tim
Yes, it approaches the origin along y=x.
But what do you mean by "goes away from the origin" …
isn't there only one way of getting to t = ∞ ?
(and what if C2 = 0 ? )
5. Apr 10, 2010
### Jamin2112
The equation was x'=[1 -2; 3 -4]x (wrote the matrix MATLAB-style)
So at x=(1 1)T, x'=(-1 -1)T, and so on ...
And at x=(-1 -1)T, x'=(1 1)T, and so on ...
See what I mean?
6. Apr 10, 2010
### Jamin2112
In general, I'm confused about these questions about what x(t) approaches as t approaches infinity.
For a different problem on my homework, I got a solution x= 2e3t(1 5)T -e-t(1 1)T. Obviously, the first part, 2e3t(1 5)T, will dominate as t --> ∞. Does that mean the x approaches (∞ 5∞)T?
7. Apr 10, 2010
### tiny-tim
What happens as t approaches ∞ is the same for matrices as it is for ordinary numbers …
so yes in your example, 2e3t(1 5)T, will dominate, and that approaches infinity along the (1 5)T direction.
8. Apr 10, 2010
### Jamin2112
I see. Can I ask one more question?
9. Apr 10, 2010
### tiny-tim
uhhh? depends what it is!!
10. Apr 10, 2010
### Jamin2112
Consider the vectors x(1)(t)=(t 1)T and x(2)(t)=(t2 2t)T.
(a) Compute the Wronskian x(1) and x(2).
(b) In what intervals are x(1) and x(2) linearly independent?
(c) What conclusion can be drawn about the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2)?
(d) Find this system of equations and verify the conclusions of part (c).
------------------------*snip*----------------------
Wronskian ≠ 0 <==> system has a solution
W|(t 1)T, (t2 2t)T|= t2, which only equals zero when t=0.
Making x(1), x(2) into a system of homogenous differential equations would look like
C1(t 1)T + C2(t2 2t)T=(0 0)T.
I don't understand what the point of (c) is. Where are they going with this? Please explain in as much or as little detail as possible parts (c) and (d). I'm ready to learn.
11. Apr 11, 2010
### tiny-tim
Hi Jamin2112!
(just got up :zzz: …)
Sorry, Wronskians are not my field
you'd better start another thread on this one.
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https://www.physicsforums.com/threads/a-equation-to-prove.203964/ | # A equation to prove
1. Dec 11, 2007
### inferi
hi,
I have this equation that i need to prove i used numbers and worked but you have to prove it without substitution, here is the question.
let a1,a2,..........,an be all real number with the property that:
a0+(a1/2)+(a2/3)+.........(an/n+1)=0
prove that:
a0+a1x+a2x^2+........+anx^n=0
this question is consider a challenging problem.
2. Dec 11, 2007
### SiddharthM
from rudin, this is actually one of the easier problems. But you didn't really state the problem fully: it's asking you to show there exists a x in (0,1) so that
a0+a1x+a2x^2+........+anx^n=0
You will need to use the Mean Value Theorem. If f'(x)=a0+a1x+a2x^2+........+anx^n, what is f(x)?
3. Dec 11, 2007
### inferi
sorry i did not put the full question it is really the interval (0,1) but there is no f(x) in the question only equations are:
1- a0+(a1/2)+(a2/3)+.........(an/n+1)=0
2- a0+a1x+a2x^2+........+anx^n=0
that's it and if you use the mean value theorem you are going only to get a0 when x=0 and the same second equation when x=1
so howto do it? hank for your help
4. Dec 11, 2007
### SiddharthM
IF you put
f'(x)=a0+a1x+a2x^2+........+anx^n
WHAT MUST f(x) be?
Use the Mean value theorem on f(x).
I think you need to review the statement of the mean value theorem and the theory in general because it your last post indicates you don't understand it well. The MVT works for FUNCTIONS not simply equations.
The problem REQUIRES you to CONSTRUCT a polynomial function based on the constants a_0,..,a_n - it is here that you are required to be somewhat clever (if you've ever differentiated or integrated before the proper function you need to create it obvious) And from there use the two equations given to show that the function vanishes at 1 and at 0, and the rest is an application of the mean value theorem.
5. Dec 14, 2007
### therector24
i started it three times with the mean value theorem (roll's, general mean value theorem,& integral mean value theorem), how ever, it didn't work so if you were in my case what will be your way of solving this question!!!
6. Dec 15, 2007
### inferi
ok we have to integral
f'(x)=a0+a1x+a2x^2+........+anx^n
so we can apply the mean value theorem, and the integral of this equation is a0+(a1/2)+(a2/3)+.........(an/n+1)
is that right????
7. Dec 15, 2007
### HallsofIvy
Staff Emeritus
No, it's not. The integral is the FUNCTION
$$c+ a_0 x+ \frac{a_1}{2} x^2+ \frac{a_2}{3} x^3+ \cdot\cdot\cdot + \frac{a_n}{n+1} x^{n+1}$$
where c can be any constant.
Apply the mean value theorem to THAT function. | {"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": 0.9451919794082642, "perplexity": 716.8838336136504}, "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-2017-09/segments/1487501171900.13/warc/CC-MAIN-20170219104611-00162-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=3354860&postcount=1 | View Single Post
P: 45 1. The problem statement, all variables and given/known data The charge is distributed with uniform surface density σ on the disk of radius R. Find the potential at the axis of the disk. 2. Relevant equations Coulomb's law and the definition of a electric potential at point x 3. The attempt at a solution I have a solution in front of me but can't understand some step inside it: The potential can be defined now phi(x)= (1/4pi*epsilon0)Integral[(sigma(x')/|x-x'|)dS'] and the solution for the integral from 0 to R is: (sigma/2*epsilon0)(sqrt(x^2+R^2)-z) Now, the electric field at this point is: E(z)=(sigma/2*epsilon0)(1-(z/sqrt(z^2+R^2)) I can clearly follow until now, but then the book says that for z>>R we get E=Q/(4pi*epsilon0*z) where Q is the total charge of the disc. How can it be proportional to 1/z?? when I take z>>R - I get E(z)=0... | {"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": 0.984152615070343, "perplexity": 326.28608128267285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829839.93/warc/CC-MAIN-20140820021349-00284-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://clay6.com/qa/50326/choose-the-correct-statement- | Browse Questions
# Choose the correct statement:
(A) Phosphates have no biological significance in humans (B) Nitrates are less abundant in the earth's crust compared to phosphates (C) Phosphates are less abundant in the earth's crust compared to nitrates (D) Oxidation of nitrates is possible in soil
Answer: (B) Nitrates are less abundant in the earth's crust compared to phosphates
As Nitrates are more soluble in water, they are less abundant in the earth's crust compared to phosphates. | {"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": 0.8383071422576904, "perplexity": 4958.562617804103}, "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-2017-13/segments/1490218188213.41/warc/CC-MAIN-20170322212948-00193-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://papers.nips.cc/paper/2018/hash/ea8fcd92d59581717e06eb187f10666d-Abstract.html | #### Authors
Jaeho Lee, Maxim Raginsky
#### Abstract
<p>As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wasserstein distance between the source and target domain distributions can be reliably estimated from unlabeled samples.</p> | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9244478940963745, "perplexity": 491.45528658667405}, "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/1614178359082.48/warc/CC-MAIN-20210227174711-20210227204711-00530.warc.gz"} |
https://michael.cai-schmidt.org/talk/201406-flasy/ | Origin of Neutrino Mass and the LHC
Date
2014-06-18 00:00
Event
FLASY 2014
Location
Sussex, UK
Dr Michael A Schmidt
Senior Lecturer of Theoretical Particle Physics
Michael’s research interests include neutrino physics, dark matter, flavour physics and in general physics beyond the Standard Model. | {"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": 0.9839025735855103, "perplexity": 4109.175673679608}, "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-2021-10/segments/1614178368431.60/warc/CC-MAIN-20210304021339-20210304051339-00447.warc.gz"} |
https://dandelife.com/what-are-the-things-you-should-know-about-complex-numbers/ | # What Are the Things You Should Know About Complex Numbers?
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In recent times, it has become quite difficult for students to study strategically and earn their desired goals. Not having appropriate guidance is one of the key factors responsible for students lagging in their curriculum. One of the subjects which kids, in general, find difficult and boring is Maths. The concept of Complex numbers is one of the sub-parts of maths that can be quite easy if you know the rules of solving complex numbers. Various online platforms provide guidance and clarity to solve such problems.
Cuemath is an online learning platform that provides classes and worksheets for all your maths queries. It is quite practical and simple to use this website. All the worksheets are separated based on grades and topics, which makes it effortless and a time-saver to find what you are looking for.
## What is a complex number?
A complex number is represented as a sum of a real number and an imaginary number. It is represented in the form of ‘a+ib’, where i is the imaginary part, ‘a’ is the real part and is written as Re (z) and, ‘b’ is the imaginary part of the number that is written as Img (z). This number as a whole is represented as ‘z’.
i’is called iota and its value is-1. It is valuable in finding the roots of negative numbers. The value of i2= -1 which is also known as the fundamental aspect of a complex number. If you wish to learn more Cuemath online classes are the best and easiest way to do so under the guidance of expert teachers.
## Modulus of a complex number:
The modulus of a complex number is the square root of the sum of the square of the real and imaginary part of the complex number. It is represented as
z= a2+b2 . In other terms, it is represented as the distance of a complex number represented as a point in the argand plane i.e. the distance from the point (0,0) to the point (a, ib).
## The argument of a complex number:
The argument of a complex number is defined as the angle formed by the line joining the origin with the point (a, ib) and the positive x-axis. It is known as Θ that is the inverse of tan of the imaginary part divided by the real part. It is represented asArgz= ba .
## Polar representation of a complex number:
With the modulus and argument of a complex number, it can be represented in the polar formz=rθ +isin ) , which is also written as rcisΘ, where r is the modulus of a complex number and Θ is the argument of the complex number that is represented as=ba
The polar coordinates of a complex number are (r, Θ) and (r, Θ+2KΠ), K being an integer. Therefore, the polar representation of a complex number is not unique.
## Operations on a complex number:
• The addition of a complex number is quite simple and similar to the addition of integers.
• While adding two complex numbers i.e. Z1+Z2, the real parts are added together and the imaginary parts are added together.
• Considering,
z1=(a+ib), z2=c+id, z1+z2=a+c+i(b+d)
### #2. Subtraction:
• The subtraction of a complex number is performed in the same way as the addition.
• When subtracting two complex numbers, the real parts are subtracted together and the imaginary parts are subtracted together.
• Considering,
z1=a+ib, z2=c+idz1z2=a-c+i(b-d)
### #3. Multiplication:
• It is different from the multiplication of integers. Since it involves the term iota, therefore, the multiplication of two complex numbers gives us the term i2= -1.
• Considering,
• Multiplication in polar form is different from the above method.
• If z1=r1(cos 1+i sin 1), z2=r2cos 2 +isin θ 2,
z1z2=r1r2(cos 1+2)+i sin 1+2))
### #4. Division:
• Division of a complex number is performed with the property of reciprocal of a complex number.
• Considering,
z1=a+ib, z2=c+id, z1z2=a+ib*(c-id)(c2+d2)
The above mentioned are the most basic and important things to know about complex numbers.
Anna Hoffman is a part-time blogger who blog about Business Technology, Digital Marketing, Real Estate, Digital Currencies, and Educational topics. | {"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": 0.9019128084182739, "perplexity": 517.2719240889468}, "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/1656104683020.92/warc/CC-MAIN-20220707002618-20220707032618-00288.warc.gz"} |
http://math.stackexchange.com/questions/240678/how-many-2x2-matrices-are-invertible-in-mod-p?answertab=oldest | # how many 2x2 matrices are invertible in mod p
I am trying to solve this problem for homework but unable to get anything. The question is to find the number of invertible 2x2 matrices in mod p? Each entery can bee from the set {0,1,2....(p-1)} where p is prime.
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How many choices do you have for your first column? Then what about the second column once you've chosen a first? – Derek Allums Nov 19 '12 at 16:32
Sorry, in a hurry I left some information – Aman Deep Gautam Nov 19 '12 at 16:36
Right, I was asking those as hints to you mainly. It's just a counting problem once you interpret invertibility the right way. The answer below is precisely was I was getting at. – Derek Allums Nov 19 '12 at 16:47
The answer is $(p^2-1)(p^2-p)$, since there are $(p^2-1)$ ways to choose the first column such that it is non-zero, then the second column can be chosen in any way out of the $p$ multiples of the first column.
- | {"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": 0.9473460912704468, "perplexity": 366.5107636546334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861623301.66/warc/CC-MAIN-20160428164023-00148-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/69702/mechanics-of-a-rolling-drum | # Mechanics of a rolling drum
I have no clue on how to approach this. The professor only discussed centripetal acceleration and angular velocity (As in $2πr\over T$ $= ωr$). Does the acceleration along the axis of the drum act in the same way that centripetal acceleration does? I understand that the angular velocity in this case is $ωR$, and the derivative of that is the angular acceleration, but how to I find $ω$? How does $a$ affect the period, and how does its position (In this case along the axis) affect that?
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You have been given a translational acceleration of $a$. You just need to relate this to the angular version $\alpha$. How can we relate the two? – Will Jul 1 at 15:10
Here's a hint: what is the relationship between $v$ and $\omega$? Think about where this comes from. Sure it acts on the entire drum, but the drum has symmetry so we can consider the gravitational acceleration through the center of mass. Anyway that's not the problem. You have been given an acceleration $a$, yes you know it's due to gravity but it doesn't matter where is comes from, you know its value. – Will Jul 1 at 15:20
Would you know what to do if you were given the acceleration of the edge of the drum with respect to the center? – Will Jul 1 at 15:37
Great, hopefully you understand where that relationship is coming from? So we want to know the acceleration of the edge. Now, if the cylinder's center has acceleration $a$ with respect to the slope, how is the bottom edge accelerating with respect to the center of the cylinder? – Will Jul 1 at 17:00
Right, so the edge is accelerating with magnitude $a$ with respect to the center and you can now find $\alpha$. – Will Jul 1 at 18:10
There is no oscillation or period on this problem. All you know is the slip condition which means that the center of the drum moves by $v = \omega R$.
Now differentiate this relationship to find $a$ as a function of $\alpha = \dot{\omega}$. | {"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": 0.9661489725112915, "perplexity": 225.46169158595424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345762590/warc/CC-MAIN-20131218054922-00090-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/47963/is-resistance-to-motion-directly-proportional-to-the-speed-of-a-moving-object | # Is resistance to motion directly proportional to the speed of a moving object?
Power is known to be equal to the force x velocity (P=FV).
Im solving a question that states the following :
Car with engine working at 32 kW, mass of 1 tonne, travels at a constant speed of 40m/s along a level road.
Given that the resistance to motion is directly proportional to the speed at which the car is travelling, find the resistance experienced when the car is travelling at 30m/s
How is the text in Bold correct ?? P=FV .. Which means F=P/V, Velocity(speed) of car is indirectly proportional to the Force !! Can someone please explain this ?
- | {"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": 0.9662984609603882, "perplexity": 492.123416182526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609532480.36/warc/CC-MAIN-20140416005212-00530-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://yuvalfilmus.cs.technion.ac.il/publications/manuscripts/?id=817 | # PublicationsManuscripts
## A positive proof of Dehn's theorem
Yuval Filmus
Dehn proved that if a rectangle can be tiled by rectangles whose sides are commensurable, then the tiled rectangle is also commensurable. His proof, as described in Proofs from the book, applies a homomorphism which results in possibly negative side lengths. We modify his proof so that all side lengths are positive.
The crucial ingredient is the following lemma: for each finite set of positive reals there is a basis (over the rationals) of positive reals such that every element in the set is a non-negative integral combination of base elements. We provide two proofs of this lemma, one due to us and one due to Avinoam Braverman. | {"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": 0.9779108166694641, "perplexity": 309.4358294798495}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056890.28/warc/CC-MAIN-20210919125659-20210919155659-00392.warc.gz"} |
https://www.snapxam.com/problems/46540937/derivdefx | Try NerdPal! Our new app on iOS and Android
# Find the derivative of $x$ using the definition
## Step-by-step Solution
Go!
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### Videos
$1$
Got another answer? Verify it here!
## Step-by-step Solution
Problem to solve:
$derivdef\left(x\right)$
Specify the solving method
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Find the derivative of $x$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
$\lim_{h\to0}\left(\frac{x+h-x}{h}\right)$
2
Cancel like terms $x$ and $-x$
$\lim_{h\to0}\left(\frac{h}{h}\right)$
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Simplify the fraction $\frac{h}{h}$ by $h$
$\lim_{h\to0}\left(1\right)$
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The limit of a constant is just the constant
$1$
$1$
SnapXam A2
### beta Got another answer? Verify it!
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https://readtiger.com/wkp/en/Spin%E2%80%93orbit_interaction | # Spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.
For atoms, energy level split produced by the spin-orbit interaction is usually of the same order in size to the relativistic corrections to the kinetic energy and the zitterbewegung effect. The addition of these three corrections is known as the fine structure. The interaction between the magnetic field created by the electron and the magnetic moment of the nucleus is a slighter correction to the energy levels known as the hyperfine structure.
## In atomic energy levels
This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to a hydrogen-like atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations.
A rigorous calculation of the same result would use relativistic quantum mechanics, using Dirac equation, and would include many-body interactions. Achieving an even more precise result would involve calculating small corrections from quantum electrodynamics.
### Energy of a magnetic moment
The energy of a magnetic moment in a magnetic field is given by
${\displaystyle \Delta H=-{\boldsymbol {\mu }}\cdot \mathbf {B} ,}$
where μ is the magnetic moment of the particle, and B is the magnetic field it experiences.
### Magnetic field
We shall deal with the magnetic field first. Although in the rest frame of the nucleus, there is no magnetic field acting on the electron, there is one in the rest frame of the electron (see classical electromagnetism and special relativity). Ignoring for now that this frame is not inertial, in SI units we end up with the equation
${\displaystyle \mathbf {B} =-{\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}},}$
where v is the velocity of the electron, and E is the electric field it travels through. Here, in the non-relativistic limit, we assume that the Lorentz factor ${\displaystyle \gamma \backsimeq 1}$. Now we know that E is radial, so we can rewrite ${\displaystyle \mathbf {E} =|E/r|\mathbf {r} }$. Also we know that the momentum of the electron ${\displaystyle \mathbf {p} =m_{\text{e}}\mathbf {v} }$. Substituting this in and changing the order of the cross product gives
${\displaystyle \mathbf {B} ={\frac {\mathbf {r} \times \mathbf {p} }{m_{\text{e}}c^{2}}}\left|{\frac {E}{r}}\right|.}$
Next, we express the electric field as the gradient of the electric potential ${\displaystyle \mathbf {E} =-\nabla V}$. Here we make the central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen and hydrogen-like systems. Now we can say that
${\displaystyle |E|={\frac {\partial V}{\partial r}}={\frac {1}{e}}{\frac {\partial U(r)}{\partial r}},}$
where ${\displaystyle U=eV}$ is the potential energy of the electron in the central field, and e is the elementary charge. Now we remember from classical mechanics that the angular momentum of a particle ${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$. Putting it all together, we get
${\displaystyle \mathbf {B} ={\frac {1}{m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} .}$
It is important to note at this point that B is a positive number multiplied by L, meaning that the magnetic field is parallel to the orbital angular momentum of the particle, which is itself perpendicular to the particle's velocity.
### Magnetic moment of the electron
${\displaystyle {\boldsymbol {\mu }}_{S}=-g_{\text{s}}\mu _{\text{B}}{\frac {\mathbf {S} }{\hbar }},}$
where ${\displaystyle \mathbf {S} }$ is the spin angular-momentum vector, ${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton, and ${\displaystyle g_{\text{s}}\approx 2}$ is the electron-spin g-factor. Here ${\displaystyle {\boldsymbol {\mu }}}$ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.
The spin–orbit potential consists of two parts. The Larmor part is connected to the interaction of the magnetic moment of the electron with the magnetic field of the nucleus in the co-moving frame of the electron. The second contribution is related to Thomas precession.
### Larmor interaction energy
The Larmor interaction energy is
${\displaystyle \Delta H_{\text{L}}=-{\boldsymbol {\mu }}\cdot \mathbf {B} .}$
Substituting in this equation expressions for the magnetic moment and the magnetic field, one gets
${\displaystyle \Delta H_{\text{L}}={\frac {2\mu _{\text{B}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} \cdot \mathbf {S} .}$
Now we have to take into account Thomas precession correction for the electron's curved trajectory.
### Thomas interaction energy
In 1926 Llewellyn Thomas relativistically recomputed the doublet separation in the fine structure of the atom.[1] Thomas precession rate ${\displaystyle {\boldsymbol {\Omega }}_{\text{T}}}$ is related to the angular frequency of the orbital motion ${\displaystyle {\boldsymbol {\omega }}}$ of a spinning particle as follows:[2][3]
${\displaystyle {\boldsymbol {\Omega }}_{\text{T}}={\boldsymbol {\omega }}(\gamma -1),}$
where ${\displaystyle \gamma }$ is the Lorentz factor of the moving particle. The Hamiltonian producing the spin precession ${\displaystyle {\boldsymbol {\Omega }}_{\text{T}}}$ is given by
${\displaystyle \Delta H_{\text{T}}={\boldsymbol {\Omega }}_{\text{T}}\cdot \mathbf {S} .}$
To the first order in ${\displaystyle (v/c)^{2}}$, we obtain
${\displaystyle \Delta H_{\text{T}}=-{\frac {\mu _{\text{B}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} \cdot \mathbf {S} .}$
### Total interaction energy
The total spin–orbit potential in an external electrostatic potential takes the form
${\displaystyle \Delta H\equiv \Delta H_{\text{L}}+\Delta H_{\text{T}}={\frac {\mu _{\text{B}}}{\hbar m_{\text{e}}ec^{2}}}{\frac {1}{r}}{\frac {\partial U(r)}{\partial r}}\mathbf {L} \cdot \mathbf {S} .}$
The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor 1/2, which came to be known as the Thomas half.
### Evaluating the energy shift
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. Note that Lz and Sz are no longer conserved quantities. In particular, we wish to find a new basis that diagonalizes both H0 (the non-perturbed Hamiltonian) and ΔH. To find out what basis this is, we first define the total angular momentum operator
${\displaystyle \mathbf {J} =\mathbf {L} +\mathbf {S} .}$
Taking the dot product of this with itself, we get
${\displaystyle \mathbf {J} ^{2}=\mathbf {L} ^{2}+\mathbf {S} ^{2}+2\,\mathbf {L} \cdot \mathbf {S} }$
(since L and S commute), and therefore
${\displaystyle \mathbf {L} \cdot \mathbf {S} ={\frac {1}{2}}(\mathbf {J} ^{2}-\mathbf {L} ^{2}-\mathbf {S} ^{2})}$
It can be shown that the five operators H0, J2, L2, S2, and Jz all commute with each other and with ΔH. Therefore, the basis we were looking for is the simultaneous eigenbasis of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five quantum numbers: ${\displaystyle n}$ (the "principal quantum number"), ${\displaystyle j}$ (the "total angular momentum quantum number"), ${\displaystyle \ell }$ (the "orbital angular momentum quantum number"), ${\displaystyle s}$ (the "spin quantum number"), and ${\displaystyle j_{z}}$ (the "z component of total angular momentum").
To evaluate the energies, we note that
${\displaystyle \left\langle {\frac {1}{r^{3}}}\right\rangle ={\frac {2}{a^{3}n^{3}\;\ell (\ell +1)(2\ell +1)}}}$
for hydrogenic wavefunctions (here ${\displaystyle a=\hbar /(Z\alpha m_{\text{e}}c)}$ is the Bohr radius divided by the nuclear charge Z); and
${\displaystyle \left\langle \mathbf {L} \cdot \mathbf {S} \right\rangle ={\frac {1}{2}}{\big (}\langle \mathbf {J} ^{2}\rangle -\langle \mathbf {L} ^{2}\rangle -\langle \mathbf {S} ^{2}\rangle {\big )}={\frac {\hbar ^{2}}{2}}{\big (}j(j+1)-\ell (\ell +1)-s(s+1){\big )}.}$
### Final energy shift
We can now say that
${\displaystyle \Delta E={\frac {\beta }{2}}{\big (}j(j+1)-\ell (\ell +1)-s(s+1){\big )},}$
where
${\displaystyle \beta =\beta (n,l)=Z^{4}{\frac {\mu _{0}}{4\pi }}g_{\text{s}}\mu _{\text{B}}^{2}{\frac {1}{n^{3}a_{0}^{3}\;\ell (\ell +1/2)(\ell +1)}}.}$
For the exact relativistic result, see the solutions to the Dirac equation for a hydrogen-like atom.
## In solids
A crystalline solid (semiconductor, metal etc.) is characterized by its band structure. While on the overall scale (including the core levels) the spin–orbit interaction is still a small perturbation, it may play a relatively more important role if we zoom in to bands close to the Fermi level (${\displaystyle E_{\text{F}}}$). The atomic ${\displaystyle \mathbf {L} \cdot \mathbf {S} }$ (spin-orbit) interaction, for example, splits bands that would be otherwise degenerate, and the particular form of this spin–orbit splitting (typically of the order of few to few hundred millielectronvolts) depends on the particular system. The bands of interest can be then described by various effective models, usually based on some perturbative approach. An example of how the atomic spin–orbit interaction influences the band structure of a crystal is explained in the article about Rashba and Dresselhaus interactions.
In crystalline solid contained paramagnetic ions, e.g. ions with unclosed d or f atomic subshell, localized electronic states exist.[4][5] In this case, atomic-like electronic levels structure is shaped by intrinsic magnetic spin–orbit interactions and interactions with crystalline electric fields.[6] Such structure is named the fine electronic structure. For rare-earth ions the spin–orbit interactions are much stronger than the crystal electric field (CEF) interactions.[7] The strong spin–orbit coupling makes J a relatively good quantum number, because the first excited multiplet is at least ~130 meV (1500 K) above the primary multiplet. The result is that filling it at room temperature (300 K) is negligibly small. In this case, a (2J + 1)-fold degenerated primary multiplet split by an external CEF can be treated as the basic contribution to the analysis of such systems' properties. In the case of approximate calculations for basis ${\displaystyle |J,J_{z}\rangle }$, to determine which is the primary multiplet, the Hund principles, known from atomic physics, are applied:
• The ground state of the terms' structure has the maximal value S allowed by the Pauli exclusion principle.
• The ground state has a maximal allowed L value, with maximal S.
• The primary multiplet has a corresponding J = |LS| when the shell is less than half full, and J = L + S, where the fill is greater.
The S, L and J of the ground multiplet are determined by Hund's rules. The ground multiplet is 2J + 1 degenerated – its degeneracy is removed by CEF interactions and magnetic interactions. CEF interactions and magnetic interactions resemble, somehow, Stark and Zeeman effect known from atomic physics. The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2J + 1)-dimensional matrix. The fine electronic structure can be directly detected by many different spectroscopic methods, including the inelastic neutron scattering (INS) experiments. The case of strong cubic CEF[8][9] (for 3d transition-metal ions) interactions form group of levels (e.g. T2g, A2g), which are partially split by spin–orbit interactions and (if occur) lower-symmetry CEF interactions. The energies and eigenfunctions of the discrete fine electronic structure (for the lowest term) are obtained by diagonalization of the (2L + 1)(2S + 1)-dimensional matrix. At zero temperature (T = 0 K) only the lowest state is occupied. The magnetic moment at T = 0 K is equal to the moment of the ground state. It allows the evaluation of the total, spin and orbital moments. The eigenstates and corresponding eigenfunctions ${\displaystyle |\Gamma _{n}\rangle }$ can be found from direct diagonalization of Hamiltonian matrix containing crystal field and spin–orbit interactions. Taking into consideration the thermal population of states, the thermal evolution of the single-ion properties of the compound is established. This technique is based on the equivalent operator theory[10] defined as the CEF widened by thermodynamic and analytical calculations defined as the supplement of the CEF theory by including thermodynamic and analytical calculations.
### Examples of effective Hamiltonians
Hole bands of a bulk (3D) zinc-blende semiconductor will be split by ${\displaystyle \Delta _{0}}$ into heavy and light holes (which form a ${\displaystyle \Gamma _{8}}$ quadruplet in the ${\displaystyle \Gamma }$-point of the Brillouin zone) and a split-off band (${\displaystyle \Gamma _{7}}$ doublet). Including two conduction bands (${\displaystyle \Gamma _{6}}$ doublet in the ${\displaystyle \Gamma }$-point), the system is described by the effective eight-band model of Kohn and Luttinger. If only top of the valence band is of interest (for example when ${\displaystyle E_{\text{F}}\ll \Delta _{0}}$, Fermi level measured from the top of the valence band), the proper four-band effective model is
${\displaystyle H_{\text{KL}}(k_{\text{x}},k_{\text{y}},k_{\text{z}})=-{\frac {\hbar ^{2}}{2m}}\left[(\gamma _{1}+{\textstyle {\frac {5}{2}}\gamma _{2}})k^{2}-2\gamma _{2}(J_{\text{x}}^{2}k_{\text{x}}^{2}+J_{\text{y}}^{2}k_{\text{y}}^{2}+J_{\text{z}}^{2}k_{\text{z}}^{2})-2\gamma _{3}\sum _{m\neq n}J_{m}J_{n}k_{m}k_{n}\right]}$
where ${\displaystyle \gamma _{1,2,3}}$ are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and ${\displaystyle J_{{\text{x}},{\text{y}},{\text{z}}}}$ are angular momentum 3/2 matrices (${\displaystyle m}$ is the free electron mass). In combination with magnetization, this type of spin–orbit interaction will distort the electronic bands depending on the magnetization direction, thereby causing magnetocrystalline anisotropy (a special type of magnetic anisotropy). If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting. Within the four bands (light and heavy holes), the dominant term is
${\displaystyle H_{{\text{D}}3}=b_{41}^{8{\text{v}}8{\text{v}}}[(k_{\text{x}}k_{\text{y}}^{2}-k_{\text{x}}k_{\text{z}}^{2})J_{\text{x}}+(k_{\text{y}}k_{\text{z}}^{2}-k_{\text{y}}k_{\text{x}}^{2})J_{\text{y}}+(k_{\text{z}}k_{\text{x}}^{2}-k_{\text{z}}k_{\text{y}}^{2})J_{\text{z}}]}$
where the material parameter ${\displaystyle b_{41}^{8{\text{v}}8{\text{v}}}=-81.93\,{\text{meV}}\cdot {\text{nm}}^{3}}$ for GaAs (see pp. 72 in Winkler's book, according to more recent data the Dresselhaus constant in GaAs is 9 eVÅ3;[11] the total Hamiltonian will be ${\displaystyle H_{\text{KL}}+H_{{\text{D}}3}}$). Two-dimensional electron gas in an asymmetric quantum well (or heterostructure) will feel the Rashba interaction. The appropriate two-band effective Hamiltonian is
${\displaystyle H_{0}+H_{\text{R}}={\frac {\hbar ^{2}k^{2}}{2m^{*}}}\sigma _{0}+\alpha (k_{\text{y}}\sigma _{\text{x}}-k_{\text{x}}\sigma _{\text{y}})}$
where ${\displaystyle \sigma _{0}}$ is the 2 × 2 identity matrix, ${\displaystyle \sigma _{{\text{x}},{\text{y}}}}$ the Pauli matrices and ${\displaystyle m^{*}}$ the electron effective mass. The spin–orbit part of the Hamiltonian, ${\displaystyle H_{\text{R}}}$ is parametrized by ${\displaystyle \alpha }$, sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.
Above expressions for spin–orbit interaction couple spin matrices ${\displaystyle {\boldsymbol {J}}}$ and ${\displaystyle {\boldsymbol {\sigma }}}$ to the quasi-momentum ${\displaystyle {\boldsymbol {k}}}$, and to the vector potential ${\displaystyle {\boldsymbol {A}}}$ of an AC electric field through the Peierls substitution ${\displaystyle {\boldsymbol {k}}=-i\nabla -({\frac {e}{\hbar c}}){\boldsymbol {A}}}$. They are lower order terms of the Luttinger–Kohn k·p perturbation theory in powers of ${\displaystyle k}$. Next terms of this expansion also produce terms that couple spin operators of the electron coordinate ${\displaystyle {\boldsymbol {r}}}$. Indeed, a cross product ${\displaystyle ({\boldsymbol {\sigma }}\times {\boldsymbol {k}})}$ is invariant with respect to time inversion. In cubic crystals, it has a symmetry of a vector and acquires a meaning of a spin–orbit contribution ${\displaystyle {\boldsymbol {r}}_{\text{SO}}}$ to the operator of coordinate. For electrons in semiconductors with a narrow gap ${\displaystyle E_{G}}$ between the conduction and heavy hole bands, Yafet derived the equation[12][13]
${\displaystyle {\boldsymbol {r}}_{\text{SO}}={\frac {\hbar ^{2}g}{4m_{0}}}\left({\frac {1}{E_{G}}}+{\frac {1}{E_{G}+\Delta _{0}}}\right)({\boldsymbol {\sigma }}\times {\boldsymbol {k}})}$
where ${\displaystyle m_{0}}$ is a free electron mass, and ${\displaystyle g}$ is a ${\displaystyle g}$-factor properly renormalized for spin–orbit interaction. This operator couples electron spin ${\displaystyle {\boldsymbol {S}}={\tfrac {1}{2}}{\boldsymbol {\sigma }}}$ directly to the electric field ${\displaystyle {\boldsymbol {E}}}$ through the interaction energy ${\displaystyle -e({\boldsymbol {r}}_{\text{SO}}\cdot {\boldsymbol {E}})}$.
### Oscillating electromagnetic field
Electric dipole spin resonance (EDSR) is the coupling of the electron spin with an oscillating electric field. Similar to the electron spin resonance (ESR) in which electrons can be excited with an electromagnetic wave with the energy given by the Zeeman effect, in EDSR the resonance can be achieved if the frequency is related to the energy band split given by the spin-orbit coupling in solids. While in ESR the coupling is obtained via the magnetic part of the EM wave with the electron magnetic moment, the ESDR is the coupling of the electric part with the spin and motion of the electrons. This mechanism has been proposed for controlling the spin of electrons in quantum dots and other mesoscopic systems.[14]
## References
1. ^ Thomas, Llewellyn H. (1926). "The Motion of the Spinning Electron". Nature. 117 (2945): 514. doi:10.1038/117514a0. ISSN 0028-0836.
2. ^ L. Föppl and P. J. Daniell, Zur Kinematik des Born'schen starren Körpers, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 519 (1913).
3. ^ C. Møller, The Theory of Relativity, (Oxford at the Claredon Press, London, 1952).
4. ^ A. Abragam & B. Bleaney (1970). Electron Paramagnetic Resonance of Transition Ions. Clarendon Press, Oxford.
5. ^ J. S. Griffith (1970). The Theory of Transition Metal Ions. The Theory of Transition Metal Ions, Cambridge University Press.
6. ^ J. Mulak, Z. Gajek (2000). The effective crystal field potential. Elsevier Science Ltd, Kidlington, Oxford, UK.
7. ^ Fulde. Handbook on the Physics and Chemistry Rare Earth Vol. 2. North-Holland. Inc. (1979).
8. ^ R. J. Radwanski, R. Michalski, Z. Ropka, A. Błaut (1 July 2002). "Crystal-field interactions and magnetism in rare-earth transition-metal intermetallic compounds". Physica B. 319 (1–4): 78–89. doi:10.1016/S0921-4526(02)01110-9.CS1 maint: Multiple names: authors list (link)
9. ^ Radwanski, R. J.; Michalski, R.; Ropka, Z.; Błaut, A. (2002). "Crystal-field interactions and magnetism in rare-earth transition-metal intermetallic compounds". Physica B: Condensed Matter. 319 (1–4): 78–89. doi:10.1016/s0921-4526(02)01110-9. ISSN 0921-4526.
10. ^ Watanabe, Hiroshi (1966). Operator methods in ligand field theory. Prentice-Hall.
11. ^ Krich, Jacob J.; Halperin, Bertrand I. (2007). "Cubic Dresselhaus Spin-Orbit Coupling in 2D Electron Quantum Dots". Physical Review Letters. 98 (22): 226802. arXiv:cond-mat/0702667. doi:10.1103/PhysRevLett.98.226802. PMID 17677870.
12. ^ Yafet, Y. (1963), g Factors and Spin-Lattice Relaxation of Conduction Electrons, Solid State Physics, 14, Elsevier, pp. 1–98, doi:10.1016/s0081-1947(08)60259-3, ISBN 9780126077148, retrieved 2018-06-27
13. ^ E. I. Rashba and V. I. Sheka, Electric-Dipole Spin-Resonances, in: Landau Level Spectroscopy, (North Holland, Amsterdam) 1991, p. 131; [arxiv.org]
14. ^ Rashba, Emmanuel I. (2005). "Spin Dynamics and Spin Transport". Journal of Superconductivity. 18 (2): 137–144. arXiv:cond-mat/0408119. doi:10.1007/s10948-005-3349-8. ISSN 0896-1107.
## Textbooks
• Condon, Edward U. & Shortley, G. H. (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 978-0-521-09209-8.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall.
• Landau, Lev; Lifshitz, Evgeny. "${\displaystyle \S }$72. Fine structure of atomic levels". Quantum Mechanics: Non-Relativistic Theory, Volume 3.
• Yu, Peter Y.; Cardona, Manuel (1995). Fundamentals of Semiconductors. Springer.
• Winkler, Roland (2003). Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems. Springer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 83, "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": 0.9326755404472351, "perplexity": 907.9315732302206}, "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/1566027314732.59/warc/CC-MAIN-20190819114330-20190819140330-00139.warc.gz"} |
https://www.mdrc.org/publication/quantifying-cross-site-impact-variation-some-important-lessons | Methodology
# Quantifying Cross-Site Impact Variation: Some Important Lessons
04/2018
This post is one in a series highlighting MDRC’s methodological work. Contributors discuss the refinement and practical use of research methods being employed across our organization.
In a recently published three-article set, MDRC researchers and colleagues discuss quantifying cross-site impact variation using data from multisite randomized trials. The papers show that the extent to which the effects of an intervention vary across settings has important implications for policy, practice, science, and research design. This post distills some key considerations for research design and for reporting and interpreting cross-site impact variation.
The three articles, published in the Journal of Research on Educational Effectiveness, are based on research funded by the Spencer Foundation, the William T. Grant Foundation, and the Institute for Education Sciences. The first paper (Bloom et al., 2017) considers how to estimate, report, and interpret cross-site impact variation.[1] The second paper (Bloom and Spybrook, 2017) considers how to design multisite trials with adequate precision in the presence of this variation. The third paper (Weiss et al., 2017) applies methods from the first two papers to data from 16 multisite trials in education and training research to quantify the cross-site impact variation they reflect.
The benefits of understanding variation apply on multiple levels. Local policymakers and practitioners need to know both the average impact of an intervention and its variation across settings to properly assess its likely benefits and risks for their jurisdictions. For social scientists, cross-site impact variation offers opportunities to learn about mechanisms or mediators through which interventions produce their impacts and characteristics of settings and sample members that influence or moderate these impacts. And for researchers designing studies, cross-site impact variation can markedly affect the statistical precision of effect estimates and hence influence the sample size requirements for these estimates.
One important lesson illustrated by our papers involves reporting cross-site impact variation and reflects the difference between variation in impact estimates and variation in true impacts. It is a simple matter to produce internally valid estimates of the mean impact of an intervention for each site in a multisite randomized trial. However, reporting cross-site variation in these estimates through a frequency distribution or a standard deviation can greatly overstate the amount of true variation that exists. This can occur because differences between site-specific impact estimates have two sources: (1) true cross-site impact differences and (2) differences in random, site-specific estimation error. For studies without very large site samples, most of the variation in site-specific impact estimates reflects random estimation error. Thus, it is essential to use a rigorous method for inferring the magnitude of true cross-site impact variation.
Another lesson involves interpreting cross-site impact variation and reflects the fact that the impact of an intervention is by necessity defined with respect to a specific counterfactual alternative or set of alternatives. As Holland (1986, p. 950) aptly notes, “the effect of a cause is always relative to another cause.” For example, the impact of lottery-based assignment to a charter school is defined as the difference between the mean outcome for students assigned to the charter school and the mean outcome for comparable students not assigned to it and thus attending other schools. Consequently, cross-site impact variation may reflect the fact that both (1) charter schools can vary in their ability to produce educational gains for students with a given educational background and potential and (2) alternatives to charter schools can also vary in this regard. Indeed it is possible, in principle, to have no such variation in the effectiveness of specific charter schools but considerable variation in the corresponding effectiveness of their counterfactual schools, and thus to have considerable variation in charter school impacts. This very real fact of evaluation life illustrates the importance of focusing not just on the treatment being studied but also on the treatment contrast.
Interpretation of cross-site impact variation is also complicated by the possibility that some observed variation in impacts between sites is due to differences in the composition of their sample members; likewise, some observed variation in impacts between subgroups of individuals may be due to differences in their distribution across sites. Thus when studying these sources of impact variation it is essential to account for their potential conflation.
Once you acknowledge the possibility of impact variation across sites or individuals, it is essential to clearly specify your target population. Researchers must decide whether to limit their estimates of a mean intervention impact to the sites in their sample (a fixed-effect inference) or to project those estimates to a super-population of sites represented by their sample (a random-effects inference).[2] In addition, researchers must decide whether to infer study findings to a population of sites or a population of individuals. Specifying these aspects of your target population has important implications both conceptually (it helps to define your parameters of interest) and for estimation (it determines how sites and sample members should be weighted).
Yet another lesson from our research is the importance of specifying — and where possible assessing — the full range of assumptions that underlie the model used to estimate cross-site impact variation. In this regard, seemingly little things (like specifying the individual-level error distribution) can make a big difference in the magnitude and statistical significance of estimates of cross-site impact variation.
Finally, it is important to note that there may be no clear relationship between cross-site impact variation and the magnitude of overall mean impact. The present empirical research identified interventions with (1) near-zero mean impact and substantial cross-site impact variation (charter schools in multiple states); (2) substantial mean impact and substantial impact variation (New York City’s small high schools of choice); (3) near-zero mean impact and near-zero impact variation (after-school reading programs in multiple states); and (4) substantial mean impact and near-zero impact variation (high-school career academies in multiple states). These diverse findings illustrate the need for future empirical research on the patterns of impact variation that exist and factors that predict these patterns.
The presence of impact variation across sites and individuals offers a rich opportunity to learn how interventions work, if at all, and for whom. Careful attention to the complications involved is crucial to ensure the validity of the analysis.
[1]This paper received the 2017 Outstanding Article award from the Journal of Research on Educational Effectiveness.
[2]This broader inference can be justified by the fact that even for a convenience sample of sites (the basis for most past multisite trials) the ultimate target of interest is typically not just sites in a study’s sample but rather some population of sites represented by the sample where the intervention being tested might be implemented. | {"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": 0.8036519885063171, "perplexity": 2180.553299924864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662593428.63/warc/CC-MAIN-20220525182604-20220525212604-00464.warc.gz"} |
http://www.jiskha.com/display.cgi?id=1335400383 | Friday
October 24, 2014
# Homework Help: ALGEBRA
Posted by Randy on Wednesday, April 25, 2012 at 8:33pm.
Use the definition of logarithm to simplify each expression.
text((a) )log_(3b) \(3b\)
text((b) )log_(4b) \((4b)^6\)
text((c) )log_(7b) \((7b)^(-11)\)
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Members | {"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": 0.8647099733352661, "perplexity": 2902.37963148895}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119646484.16/warc/CC-MAIN-20141024030046-00293-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://lindeinstitute.caltech.edu/events/50764 | Search
Market Dynamics of Best-Response with Lookahead
Friday, June 3, 2016
12:00 PM - 1:00 PM
Location: Baxter 125
Krishnamurthy (Dj) Dvijotham, Postdoctoral Scholar in Computing and Mathematical Science, Caltech
Abstract:
In both general equilibrium theory and game theory, the dominant mathematical models rest on a fully rational solution concept in which every player's action is a best-response to the actions of the other players. In both theories there is less agreement on suitable out-of-equilibrium modeling, but one attractive approach is the level k model in which a level 0 player adopts a very simple response to current conditions, a level 1 player best-responds to a model in which others take level 0 actions, and so forth. (This is analogous to k-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address this question in the setting of Fisher Markets with constant elasticities of substitution (CES) utilities, in the weak gross substitutes (WGS) regime. We show that despite the inconsistency of the mental models, and even if players' models change arbitrarily from round to round, the market converges to its unique equilibrium. (We show this for both synchronous and asynchronous discrete-time updates.) Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. Even for the simple case of (level 0) best-response dynamics, where we observe that convergence at some rate can be derived from recent results in convex optimization, our result is the first to demonstrate a linear rate of convergence.
Joint work with Leonard Schulman and Yuval Rabani
Series: Linde Institute/Social and Information Sciences Laboratory Seminar Series (SISL) | {"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": 0.8301456570625305, "perplexity": 898.1497160766644}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647556.43/warc/CC-MAIN-20180321004405-20180321024405-00161.warc.gz"} |
http://link.springer.com/article/10.1007%2FBF01298184 | manuscripta mathematica
, Volume 32, Issue 1, pp 91–100
Puiseux expansion for space curves
• Joseph Maurer
Article
DOI: 10.1007/BF01298184
Maurer, J. Manuscripta Math (1980) 32: 91. doi:10.1007/BF01298184
Abstract
For any ideal I in a convergent power series ring ℌ {X1,..,Xn} (n≥2) with one dimensional zero set X ⊂ (ℌn, 0) we give a method of computing a parametrization of each irreducible component of the reduction of X. This generalizes the well-known method of the Newton polygon or the so called Puiseux expansion for plane curves (see [N], [P], and [B]). The slope of a side of the Newton polygon is generalized to what we calltropism of the ideal. It may be visualized as the direction of a hyperplane touching the Newton polyhedron of every element of the ideal at least along an edge. | {"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": 0.9300893545150757, "perplexity": 748.3041872284633}, "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-2016-44/segments/1476988718987.23/warc/CC-MAIN-20161020183838-00311-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://mishabucko.wordpress.com/2012/09/29/my-task/ | Adding next integers vs adding next primes, and the AIR project
$\lim\limits_{n \rightarrow \infty}{{(1+{{1}\over{n}})(2+{{1}\over{n}})(3+{{1}\over{n}})..(n+{{1}\over{n}})}\over{n!}}$
Solutions:
dr Sonnhard Graubner
$\frac{\prod_{i=1}^n\left(i+\frac{1}{n}\right)}{n!}=\frac{\left(\frac{n^2+1}{n}\right)!}{n!\left(\frac{1}{n}!\right)}$. The searched limit is .
Look at the identity:
$\Gamma(z) = \frac{\Gamma(z+n)}{z(z+1)(z+2)\cdot\cdot\cdot (z+n-1)}$, remembering that $\Gamma(n+1)=n!$. Let $z=1+\frac{1}{n}$
JoeBlow
It’s equal to:
$\prod_{k=1}^n \left(1+\frac{1}{kn}\right)=1+\frac{1}{n}\sum_{k=1}^n \frac{1}{k}+O(n^{-1})=1+\frac{\log n}{n}+O(n^{-1})$.
WWW
Another approach: Apply $\ln$ to get $\sum_{k=1}^{n}\ln(k+1/n)-\sum_{k=1}^{n}\ln k =\sum_{k=1}^{n}\ln(1+1/(kn))$. Now $0<\ln (1+h) < h$ for $h>0$, so the last sum lies between $0$ and $\frac {\sum_{k=1}^{n}1/k}{n}.$ This $\to 0$ (Cesaro-Stolz or the well known fact that $\sum_{k=1}^{n}1/k \sim \ln n$). Thus the original limit is $1.$
Indeed,from the very definition, as JoeBlow mentioned. What about the following?
R-limit = $\lim\limits_{n \rightarrow \infty}{{(p_1+{{1}\over{n}})(p_2+{{1}\over{n}})(p_3+{{1}\over{n}})..(p_n+{{1}\over{n}})}\over{n!}}$, where $p_i$ is prime for $i \in N$.
I already received one answer, but haven’t analyzed it for correctness yet. Shown below.
Xantos C. Guin
Let $A_n = \dfrac{1}{n!}\prod_{k=1}^{n}(p_k + \tfrac{1}{n})$ and $B_n = \dfrac{1}{n!}\prod_{k=1}^{n}p_k$.
Clearly, $0 < B_n < A_n$ for all $n \in \mathbb{N}$. Also, $\dfrac{B_{n}}{B_{n-1}} = \dfrac{p_n}{n} \ge 2$ for all integers $n \ge 5$.
Hence, $\lim_{n \to \infty} B_n = \infty$, and thus, $\lim_{n \to \infty} A_n = \infty$ as well. | {"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": 26, "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": 0.9690526723861694, "perplexity": 1030.7166151273407}, "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/1510934807044.45/warc/CC-MAIN-20171123233821-20171124013821-00510.warc.gz"} |
http://www.ams.org/joursearch/servlet/DoSearch?f1=msc&v1=44A20 | AMS eContent Search Results
Matches for: msc=(44A20) AND publication=(all) Sort order: Date Format: Standard display
Results: 1 to 9 of 9 found Go to page: 1
[1] Jack Buttcane. The Spectral Kuznetsov Formula on $SL(3)$. Trans. Amer. Math. Soc. 368 (2016) 6683-6714. Abstract, references, and article information View Article: PDF [2] Ana F. Loureiro, P. Maroni and S. Yakubovich. On a polynomial sequence associated with the Bessel operator. Proc. Amer. Math. Soc. 142 (2014) 467-482. Abstract, references, and article information View Article: PDF [3] Sarah Chisholm, Alyson Deines and Holly Swisher. Recent Advances for Ramanujan Type Supercongruences. Contemporary Mathematics 606 (2013) 189-206. Book volume table of contents View Article: PDF [4] Luís Daniel Abreu and Fethi Bouzeffour. A Paley-Wiener theorem for the Askey-Wilson function transform. Proc. Amer. Math. Soc. 138 (2010) 2853-2862. MR 2644898. Abstract, references, and article information View Article: PDF This article is available free of charge [5] Tibor K. Pogány and Endre Süli. Integral representation for Neumann series of Bessel functions. Proc. Amer. Math. Soc. 137 (2009) 2363-2368. MR 2495270. Abstract, references, and article information View Article: PDF This article is available free of charge [6] K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song and I. Yoo. Sequential Fourier-Feynman transform, convolution and first variation. Trans. Amer. Math. Soc. 360 (2008) 1819-1838. MR 2366964. Abstract, references, and article information View Article: PDF This article is available free of charge [7] Óscar Ciaurri and Krzysztof Stempak. Transplantation and multiplier theorems for Fourier-Bessel expansions. Trans. Amer. Math. Soc. 358 (2006) 4441-4465. MR 2231384. Abstract, references, and article information View Article: PDF This article is available free of charge [8] Ahmed I. Zayed. On the singularities of the continuous Jacobi transform when $\alpha+\beta=0$ . Proc. Amer. Math. Soc. 101 (1987) 67-75. MR 897072. Abstract, references, and article information View Article: PDF This article is available free of charge [9] A. M. Cormack. The Radon transform on a family of curves in the plane. II . Proc. Amer. Math. Soc. 86 (1982) 293-298. MR 667292. Abstract, references, and article information View Article: PDF This article is available free of charge
Results: 1 to 9 of 9 found Go to page: 1 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.939236044883728, "perplexity": 3216.5785787569785}, "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/1510934804976.22/warc/CC-MAIN-20171118151819-20171118171819-00449.warc.gz"} |
https://kluedo.ub.uni-kl.de/frontdoor/index/index/year/2008/docId/1991 | ## Determination of interaction between MCT1 and CAII via a mathematical and physiological approach
• The enzyme carbonic anhydrase isoform II (CAII), catalysing the hydration and dehydration of CO2, enhances transport activity of the monocarboxylate transporter isoform I (MCT1, SLC16A1) expressed in Xenopus oocytes by a mechanism that does not require CAII catalytic activity (Becker et al. (2005) J. Biol. Chem., 280). In the present study, we have investigated the mechanism of the CAII induced increase in transport activity by using electrophysiological techniques and a mathematical model of the MCT1 transport cycle. The model consists of six states arranged in cyclic fashion and features an ordered, mirror-symmetric, binding mechanism were binding and unbinding of the proton to the transport protein is considered to be the rate limiting step under physiological conditions. An explicit rate expression for the substrate °ux is derived using model reduction techniques. By treating the pools of intra- and extracellular MCT1 substrates as dynamic states, the time dependent kinetics are obtained by integration using the derived expression for the substrate °ux. The simulations were compared with experimental data obtained from MCT1-expressing oocytes injected with di®erent amounts of CAII. The model suggests that CAII increases the e®ective rate constants of the proton reactions, possibly by working as a proton antenna.
$Rev: 13581$ | {"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": 0.9216337203979492, "perplexity": 4620.300772218881}, "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-2016-07/segments/1454701165378.58/warc/CC-MAIN-20160205193925-00326-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3933751/prove-sum-b-n-converges-if-sum-a-n-converges-and-lim-fraca-nb-n-6 | Prove $\sum b_n$ converges if $\sum a_n$ converges and $\lim \frac{a_n}{b_n}=6$
I recently received this problem on a real analysis exam and I'm still having a hard time understanding it.
"Let $$(a_n)$$ and $$(b_n)$$ be sequences of $$\mathbb{R}^+$$. Suppose $$\lim \frac{a_n}{b_n}=6.$$ Prove that if the infinite series $$\sum a_n$$ converges, then $$\sum b_n$$ converges."
Here's what I understand so far: we can use the Algebraic Limit Theorem to simplify and get $$\lim (a_n) = 6*\lim(b_n)$$. Also, since we know $$\lim \frac{a_n}{b_n}=6$$, we know that division by $$\lim(b_n)$$ is valid, so $$\lim(b_n) \ne 0$$.
What's confusing me is that there is a theorem that says if an infinite series $$\sum s_n$$ converges, then $$\lim(s_n) =0$$. If we attempt the problem and suppose that $$\sum a_n$$ converges, wouldn't that mean that $$\lim(a_n) =0$$, so $$\lim \frac{a_n}{b_n}=6$$ can't occur? What am I missing?
Edit: I understand now that the Algebraic Limit Thm may not come into play with this problem. However, I'm still confused how to use the given information to prove that $$\sum b_n$$ converges.
• "...we know that division by $lim(b_n)$ is valid, so $lim(b_n) \ne 0$". No. Surely $a_n$ and $b_n$ can both tend to $0$, e.g. $a_n = 6/n^2$ and $b_n = 1/n^2$.. Dec 4, 2020 at 1:29
• I'm not sure what the Algebraic Limit Theorem is. Please can you state clearly what it is (preferably, write it in your question). Dec 4, 2020 at 1:35
• You cannot quite use the algebraic limit theorem that way, it supposes that $\lim(b_n)$ exists, which is what you're trying to prove. Dec 4, 2020 at 1:35
• @AdamRubinson thank you for the clarification! That helps with the first have of my confusion, however I'm still confused about the second part. By the Algebraic Limit Thm we have $lim \frac{a_n}{b_n} = 6 \rightarrow \frac{lim(a_n)}{lim(b_n)}$. But by the theorem mentioned, if we know $\sum a_n$ converges then $lim (a_n)$ must equal 0, correct? Dec 4, 2020 at 1:37
• @Oreomair you're right, thank you for the clarification! Dec 4, 2020 at 1:42
Hint: Let $$\epsilon>0$$. Since $$\frac{a_n}{b_n} \to 6,$$ there exists $$N$$ such that $$\frac{a_n}{b_n}>6-\epsilon$$ for all $$n\geq N$$. Show that $$\sum b_n$$ is bounded above. It is obviously monotone.
To prove it, I would do the following:
$$\lim_{n \to \infty} \left(\frac{a_n}{b_n}\right) = 6 \implies \exists N \in \mathbb{N}$$ such that $$\frac{a_n}{7} < b_n < \frac{a_n}{5} \quad \forall n \geq N.$$
Can you finish this now?
Formal proof of the above statement, as requested by OP:
By definition of the limit of a sequence, given $$\varepsilon>0, \ \exists N \in \mathbb{N}$$ such that $$\left|\frac{a_n}{b_n} - 6\right| < \varepsilon \quad \forall \ n \geq N.$$
Let $$\varepsilon = \frac12.$$
Then $$\exists N$$ such that:
$$\left|\frac{a_n \ - \ 6b_n}{b_n}\right| < \frac12 \quad \forall n \geq N,\$$ which implies that
$$\left|a_n-6b_n\right| < \frac12 b_n \quad \forall \ n \geq N \quad (*),$$
where $$|b_n| = b_n$$ is justified because $$b_n \in \mathbb{R}^+$$ was assumed in the question.
If $$a_n > 6b_n,\$$ then $$\left|a_n-6b_n\right| = a_n - 6b_n$$, so $$(*) \implies a_n < 6 \frac12 b_n\ < 7b_n \quad \forall \ n \geq N.$$
Else if $$a_n \leq 6b_n\ ,$$ then $$\left|a_n-6b_n\right| = -a_n + 6b_n$$, so $$(*) \implies a_n > 5 \frac12b_n\ > 5b_n \quad \forall \ n \geq N.$$
So $$5b_n < a_n < 7b_n \quad \forall \ n \geq N$$, i.e.:
$$\frac{a_n}{7} < b_n < \frac{a_n}{5} \quad \forall n \geq N.$$
• Can you describe how you got there? Using the formal definition of a limit, I understand that $\forall \ \epsilon > 0 \ \exists \ N \in \mathbb{N}$ such that $|\frac{a_n}{b_n} -6| < \epsilon$ Dec 4, 2020 at 1:51
• It's kind of "obviously true" if you're familiar with limits. However, I will modify my answer shortly to give a formal proof of my statement. Dec 4, 2020 at 2:05
• Thank you! My professor is quite the stickler for proofs, so I like to make sure I fully understand the exercises without taking any shortcuts, however obvious they may be. Dec 4, 2020 at 2:08 | {"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": 41, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874224662780762, "perplexity": 184.94173928998305}, "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-2022-21/segments/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00032.warc.gz"} |
https://causal-fermion-system.com/intro-phys/?_ga=2.262692684.171960012.1622538747-1640362779.1622538747 | # The Theory of Causal Fermion Systems
## Introduction for Physicists
### Introduction for physicists
The Theory of Causal Fermion Systems is a new approach to unify Quantum Theory and Gravitation. The starting point is the concept that every spacetime point corresponds to an operator on a Hilbert space. Setting up an action principle which relates the spacetime points to each other, one can derive the classical field equations of the Standard Model of particle physics as well the Einstein equations. Moreover, one gets close connections to Quantum Field Theory.
Below we give a sketch of the theory aimed at readers who first want to get a general impression before delving into the more detailed presentation of the theory.
If you prefer a more substantiated discussion from the get go, you find an
overview of the detailed introductions for physicists here
. If you want to dive right in, you can start straight away with the abstract definition of a causal fermion system.
## “We can’t solve problems by using the same kind of thinking we used when we created them.”
—Albert Einstein
We begin our introduction to the physical aspects of the Theory of Causal Fermion Systems with this well-known quote for a very good reason: the core of what we present on this website is not only a new model for our physical reality, but a completely new mathematical framework within which this novel theory can be formulated.
We first give a broad sketch of the mathematical framework before we explain how one can construct a particular physical model based on this framework. Learning a new mathematical framework takes time, there is no way around that, but here we try to give you the gist of it, so you can get a rough intuition for the physical implications.
(An introduction focusing on the mathematical ideas can be found here).
The Mathematical Framework
When we set out to look for a good mathematical structure, we first have to remind ourselves what is supposed to be achieved. So what are the central elements of the theories we try to unify? Two of the core features of General Relativity are that it is background independent and diffeomorphism invariant. Hence, spacetime, which in mathematical terms is a Lorentzian manifold, does not exist inside any external entity, it is everything there is, and any two observers inside a spacetime agree on the physical laws of the universe independent of their state of motion and independent of what coordinate system they choose to describe the world around them.
Quantum Theories, on the other hand, usually start with a set of selfadjoint linear operators on a Hilbert Space, where the Hilbert space is usually taken as a suitable space of functions defined in a given background spacetime.
The mathematical framework of causal fermion systems merges these concepts in a novel manner in the concept of an operator manifold with an intrinsic causal structure. We will now give a rough idea of how this is achieved.
The setup of the mathematical framework starts with an abstract Hilbert space as a background structure. It turns out that a suitable subset of the linear operators acting on this Hilbert space has in fact a manifold structure. We will refer to this operator manifold as the space of all possible events for a physical system. We are then able to formulate causal relations between events that tells us whether they are timelike, spacelike or lightlike separated. These relations are defined via the structure of the eigenvalues of operator products to ensure basis independence, which takes the place of observer independence in General Relativity.
Finally, we can formulate an action principle over the space of possible events. The causal relations between possible events enter the action principle in a crucial manner, and spacelike separated events do not contribute to the action. Physical systems are then realized as minimizers of this so-called causal action principle. It selects which events are actually realized in a particular physical system, as the system will in general not be supported in the entire operator manifold. One intuitive way of looking at it the causal action principle is that a minimizer tries to maximize the number of pairs of events that are spacelike separated.
Thus the causal action principle selects a set of events based on their causal relations. This set is then referred to as spacetime. One can then define a rich set of emergent objects with respect to this selected set of events.
Last but not least, every vector in the abstract Hilbert Space can be represented by a wave function in the emergent spacetime. This representation eventually gives a usual “Hilbert space formed of wave functions in spacetime,” giving a connection to the standard formalism of quantum theory. Moreover, we can derive the geometric structure from the net of causal relations between all events which are realized in a particular physical system.
How the Theory gives rise to a Model
The crucial step for a physical theory is that it must be able to model our reality. The Standard Model of particle physics is one particular Quantum Field Theory, and the concordance model of cosmology is one particular model in Einstein’s General Relativity. We now explain how to construct a particular causal fermion system to model the world we live in. Only when we succeed in this step, the mathematical framework becomes a candidate for a physical theory.
In physics, one usually begins with an action, derives the Euler Lagrange equations, then solves these equations, and finally studies the solutions to obtain predictions for physical systems.
In the CFS framework, one cannot proceed directly in this way. The reason is that the causal action principle allows for many different minimizers, and at present little is known on the set of all minimizing measures. Instead, we must begin with a known physical model and look for a way to represent this model as a minimizing causal fermion system. Once we succeed, we can then study general (finite) perturbations of that minimizer to explore its neighborhood in the space of minimizers.
This program has been carried out in detail for the simplest spacetime geometry, namely Minkowski space. In order to obtain a causal fermion system, we first have to choose a Hilbert space. The space of negative-energy solutions of the Dirac equation (i.e. the Dirac sea) turns out to be a good choice. In the next step, we must find a way to assign an operator to every point in Minkowski space. Here the local correlation operator does the job. For every pair of vectors in the Hilbert space it returns their correlation at a particular point in spacetime. However, for this construction to be well-defined, we need to introduce a regularization on a length scale $\varepsilon$. Then the local correlation operator associates to every point of Minkowski space a symmetric linear operator on a Hilbert space. By varying the spacetime point, we obtain the local correlation map, which maps Minkowski space to the linear operators and thereby defines a Causal Fermion System. In this way, one can describe Minkowski space by a causal fermion system. Finally, in order to recover the structures of Minkowski space from the structures of the causal fermion system, we have to consider the limiting case when the regularization parameter $\varepsilon$ tends to zero. This is the so called continuum limit for causal fermion systems. Taking this limit is in fact highly non-trivial and involves a lot of subtle analysis. However, currently the study of the continuum limit is our best tool to obtain statements about the real world if modelled by a causal fermion system.
As a side remark, it is worth noting that the Dirac sea vacuum is to be seen as an effective model describing a particular minimizing causal fermion system. It is one particular physical system that we can describe as a minimizing causal fermion system. But we should really only think of it as an effective description, in the sense that it describes only the macroscopic structure of spacetime, whereas its microscopic structure on the Planck scale is essentially unknown. This is an absolutely crucial point, as in the following we will always start from the “wrong” direction, with a local correlation map of an effective description, to crank out the physical properties of that system in the continuum limit. The idea of the Dirac Sea did, however, play an important role in the conception of the causal fermion systems framework, and most of the existing literature is written with that point of view in mind. A more detailed motivation for why it is a natural starting point can be found here.
The World as we know it
When setting up a new fundamental theory, the first question we have to answer is: how do the familiar theories and valuable concepts from these theories emerge from the new fundamental description. At this point, it is worth noting that the Theory of Causal Fermion Systems fits into the “emergent gravity” corner among the various approaches to unification. Hence in the strict sense, the theory does not give us a quantization of gravity. In fact, it turns out that all the classical fields are emergent as an effective description. But let us better proceed step by step.
On a heuristic level, the physical world as we experience it emerges from perturbations of the local correlation map, which we can think of as describing perturbations of the Dirac sea. To give you an idea for how the bosonic fields emerge, we start again from the “wrong” direction: Suppose we look at the Dirac sea in Minkowski space and turn on a small electromagnetic field. Since the Dirac equation couples to the electromagnetic field, this leads to a “deformation” of all the states in the Dirac Sea. This again leads to a modification of the causal correlations and hence of the causal action.
Now in the Theory of Causal Fermion Systems, the logic is essentially reversed (here you find a detailed discussion of the logic underlying the emergence of the classical field equations in the Continuum limit). Minimizing the causal action determines the causal correlations of the states in the Dirac Sea. The bosonic fields then emerge as an effective description of the collective “deformation” of the states in the Dirac Sea. The field equations for the bosonic fields then emerge from the Euler Lagrange equations of the causal action principle in the limiting regime when the regularization length tends to zero. Namely, we demand for the singularities in the continuum limit to be well behaved. This concept is the work horse behind all the results discussed below. One of the interesting observations from the analysis of the electromagnetic field is that, in order to remove the singularity related to the vacuum polarization, we need to assume that the Hilbert space involves the negative-energy states of at least three generations of fermions.
A Sketch of the Standard Model and Gravity
In this final section we provide a brief and heavily condensed overview of physically relevant results obtained in the framework of causal fermion systems. Note that most of these results have so far only been obtained by analyzing causal fermion systems in a neighborhood of Minkowski space (this is explained in more detail here).
The fermionic one-particle state and gravity emerge in a similar fashion as the electromagnetic field when considering different perturbations of the Dirac sea. In order to obtain the Standard Model of particle physics, we put in the number of fermionic sectors (eight) and the fact that the chiral symmetry is broken in the neutrino sector. The requirement that the Euler-Lagrange equations are well-behaved and satisfied in the continuum limit requires for the coupling between the eight fermionic sectors to obey the gauge group of the standard model. This leads to a so-called spontaneous block formation where two sectors pair up, one for each isospin, and the remaining sectors encode the charged leptons and neutrinos. At this point we have to mention that the continuum limit for the Higgs field has not been worked out yet. It is clear from the considerations in the book [cfs16] that for the Higgs field to emerge it takes at least two fermionic sectors. Under the assumption that the Higgs field can be derived, we thus get the full standard model including a good theoretical motivation for the fact that we observe three generations of fermions in each sector.
In this last paragraphs we discuss a couple of assorted facts that seem relevant with respect to various open questions in theoretical physics.
It is interesting to observe that the coupling constants for the gauge fields of the standard model are of order one in the continuum limit. Gravitation, on the other hand, enters the scheme as a next to next to leading order effect and hence the coupling constant is of order epsilon squared. (For a simpler example of next-to-leading order effects in classical theories see eg. the gravitational Spin Hall effect)
This is interesting with respect to the hierarchy problem. We collected a number of quantum gravity wish lists from the literature and comment on those points where the Theory of Causal Fermion Systems might provide an answer, or at least novel insights. It can also be shown that the spacetime emerging in the continuum limit has exactly one time dimension. Furthermore, the maximal dimension of the Clifford Algebra is five-dimensional. If we reserve one dimension for chirality, this tells us that the emerging spacetime has at most three spatial dimensions.
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https://math.stackexchange.com/questions/1448684/ways-to-arrange-the-letters-in-bookkeeper-if-vowels-must-appear-in-alphabeti | # Ways to arrange the letters in **BOOKKEEPER** if vowels must appear in alphabetical order?
Question
How many different ways can I arrange the letters in BOOKKEEPER if vowels must appear in alphabetical order?
How many different ways can I arrange the vowels? That would be 1 way. No I glue the term $eeeoo$ together and treat it as a letter of its own. That would leave me with $6$ letters left. Different ways to arrange them? $${6! \over 2!}$$ since there are $2 \;k's$. Did I do it right? Thank you in advance for your time.
• Presumably you meant to write $6!/2!$, not $6/2!$. That answer is correct if you interpret the instructions as keeping the vowels together in alphabetical order. But if you allow arrangements such as KEBEPEKORO, you get a larger number. – Barry Cipra Sep 23 '15 at 21:51
• I completely didn't account for that, how do I go about it then... – ponderingdev Sep 23 '15 at 22:02
Method 1: We use symmetry.
For the moment, let's consider the number of distinguishable ways we can permute the ten letters in BOOKKEEPER. We can place the three E's in three of the ten locations in $\binom{10}{3}$ ways. We can place the two O's in two of the remaining seven places in $\binom{7}{2}$ ways. We can place the two K's in two of the remaining five places in $\binom{5}{2}$ ways. There are then $3!$ ways of arranging the B, P, and R in the three remaining places. Hence, the number of distinguishable arrangements of BOOKKEEPER is $$\binom{10}{3}\binom{7}{2}\binom{5}{2} \cdot 3! = \frac{10!}{7!3!} \cdot \frac{7!}{5!2!} \cdot \frac{5!}{3!2!} \cdot 3! = \frac{10!}{3!2!2!}$$
Now, let's restrict our attention to arrangements of the five vowels in BOOKKEEPER. Since there are three E's and two O's, a given permutation of EEEOO is determined by in which three of the five positions the E's are placed. There are $\binom{5}{3} = 10$ ways to do this, of which just one is in alphabetical order.
Hence, the number of permutations of the letters of BOOKKEEPER in which the vowels appear in alphabetical order is $$\frac{1}{10} \cdot \frac{10!}{3!2!2!} = \frac{9!}{3!2!2!}$$
Method 2: We place the consonants first.
There are $\binom{10}{2}$ ways of choosing the positions of the two K's, eight ways to place the B, seven ways to place the P, and six ways to place the R. Once the consonants have been placed, there is only way to fill the five remaining positions with the vowels in alphabetical order. Hence, the number of distinguishable arrangements of the letters of BOOKKEEPER in which the vowels appear in alphabetical order is $$\binom{10}{2} \cdot 8 \cdot 7 \cdot 6$$
Method 3: We place the vowels first.
There are five vowels in BOOKKEEPER, which has ten letters. We can select positions for the five vowels in $\binom{10}{5}$ ways. There is only one way to arrange the vowels in those positions in alphabetical order. There are $\binom{5}{2}$ ways to place the K's in two of the remaining five positions. There are $3!$ ways to arrange the B, P, and R in the remaining three positions. Hence, the number of distinguishable arrangements of BOOKKEEPER in which the vowels appear in alphabetical order is $$\binom{10}{5}\binom{5}{2} \cdot 3!$$
• (+1) There are some differences in our answers, but if you think they are too close, I will remove mine since yours was here first. – robjohn Sep 23 '15 at 22:46
• @robjohn (+1) for your answer since you provided alternate ways of looking at both parts of my answer. – N. F. Taussig Sep 23 '15 at 22:49
• I don't understand why you divide by 10 though..are you treating it like a probability problem? sorry if i just don't see it – ponderingdev Sep 23 '15 at 23:03
• For a given permutation of the consonants, there are ten ways of arranging the vowels, of which just one is in alphabetical order. Thus, one tenth of the total number of permutations is in alphabetical order. Yes, you could view this argument as the probability that the vowels are in alphabetical order is $1/10$. – N. F. Taussig Sep 23 '15 at 23:09
• the alternate method makes more sense to me in an instant! – ponderingdev Sep 23 '15 at 23:44
One Approach
Without the alphabetical vowel constraint, there are $$\frac{10!}{\underbrace{1!}_{\text{B}}\underbrace{2!}_{\text{O}}\underbrace{2!}_{\text{K}}\underbrace{3!}_{\text{E}}\underbrace{1!}_{\text{P}}\underbrace{1!}_{\text{R}}}$$ Arrangements of the letters. For each arrangement with the vowels in alphabetical order, there are $\binom{5}{2}$ unconstrained arrangements of the vowels (just count the arrangements of "EEEOO", of which there is only one that is in alphabetical order).
Thus, there are $$\frac1{\binom{5}{2}}\frac{10!}{1!2!2!3!1!1!}=15120$$ arrangements of the letters where the vowels are in alphabetical order.
Another Approach
Consider the vowels as one letter. This works since once we have selected where the vowels go, we know the order of the individual vowels within the group of vowels. Thus, there are $$\frac{10!}{\underbrace{1!}_{\text{B}}\underbrace{5!}_{\begin{array}{c}\text{v}\\[-4pt]\text{o}\\[-4pt]\text{w}\\[-4pt]\text{e}\\[-2pt]\text{l}\\[-4pt]\text{s}\end{array}}\underbrace{2!}_{\text{K}}\underbrace{1!}_{\text{P}}\underbrace{1!}_{\text{R}}}=15120$$ arrangements of the letters where the vowels are in alphabetical order.
Imagine the letters as Scrabble tiles. Lay down the tiles EEEOO in the required alphabetical order. Now take the first consonant and place it in any of the $6$ positions before, between, or after what's there. Then take the first K and place it in any of the now $7$ positions before, between, or after what's on the table. Repeating this with the second K in any of the $8$ possible positions, the P in any of the $9$, and the R in any of the $10$ positions, we see there are $6\cdot7\cdot8\cdot9\cdot10$ ways of creating the arrangements. Finally, dividing by $2$ (or $2!$, if you like) to account for the two K's, we have the answer
$${6\cdot7\cdot8\cdot9\cdot10\over2}=15{,}120$$
Another method is to count the number of ways of rearranging the symbols $vvvvvccccc$ where $v$ means vowel and $c$ means consonant. That's $10 \choose 5$. Then for each such arrangement count the number of ways of rearranging the vowels. That's $1$. Then count the number of weighs to rearrange the consonants. That's $5! \over 1!2!1!1!$. Finally multiply: ${10 \choose 5} \times 1 \times {5! \over 1!2!1!1!} = 15120$ | {"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": 0.8339532017707825, "perplexity": 179.30350913437803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991921.61/warc/CC-MAIN-20210516232554-20210517022554-00334.warc.gz"} |
https://uh-ir.tdl.org/uh-ir/handle/10657/1/browse?type=subject&value=variational+multi-scale+formalism | Now showing items 1-1 of 1
• #### Modification to Darcy Model for High Pressure and High Velocity Applications and Associated Mixed Finite Element Formulations
(2013-05)
The Darcy model is based on a plethora of assumptions. One of the most important assumptions is that the Darcy model assumes the drag coefficient to be constant. However, there is irrefutable experimental evidence that ... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8761010766029358, "perplexity": 535.0574570269957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2018-51/segments/1544376830305.92/warc/CC-MAIN-20181219005231-20181219031221-00039.warc.gz"} |
http://math.stackexchange.com/questions/29047/vector-space-of-dimension-2n | # Vector space of dimension $2^n$
Let $V$ be a finite dimensional vector space over $\mathbb{R}$ of dimension $n$.
The vector space $\underbrace{V \oplus V \oplus \cdots \oplus V}_k$ has dimension $kn$, and the vector space $\underbrace{V \otimes V \otimes \cdots \otimes V}_k$ has dimension $n^k$.
Is there a similar construction (that does not depend on a basis of $V$), that gives a vector space of dimension $2^n$?
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Is there any particular reason (apart from curiousity) why you ask this question? (as far as I know the answer to your question is no) – Fabian Mar 25 '11 at 18:26
Similar construction to what? The one for $k=2$ where your vectors are $(\vec{a},\vec{b})$, addition is componentwise and scalar multiplication is $c(\vec{a},\vec{b})=(c\vec{a},c\vec{b})$? – Ross Millikan Mar 25 '11 at 18:35
You might be looking for the exterior algebra of $V$. Its definition does not depend on choosing a basis, and its dimension is $2^n$.
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Which is a special case of the Clifford algebra . – Myself Mar 25 '11 at 18:34
This is exactly what I am looking for. I can't believe I forgot about it in the first place. – NymSudo Mar 25 '11 at 18:46
@Jonas: Doesn't the exterior algebra of an $n$-dimensional space have dimension $\binom{n}{2}$? – Arturo Magidin Mar 25 '11 at 18:59
the full algebra has dimension $\sum{n\choose k}=2^n$ – yoyo Mar 25 '11 at 19:01
@Arturo: What yoyo said; $\binom{n}{2}$ is the dimension of the exterior square. – Jonas Meyer Mar 25 '11 at 19:15 | {"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": 0.9476258158683777, "perplexity": 278.8976358448277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701159031.19/warc/CC-MAIN-20160205193919-00136-ip-10-236-182-209.ec2.internal.warc.gz"} |
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## New answers tagged signal-processing
0
You can use a reverse saturable absorber in such a way that the threshold of the RSA must be such that the RSA transmits only the AC part and acts as an opaque object for the DC signal.
1
The propagation speed and the symbol rate are completely different concepts. The symbol rate is the rate at which you can transmit information (hence 'symbols'), and it is closely related to the bandwidth, which is the span of frequencies which can propagate well through the wire. The relationship between the symbol rate and the bandwidth was recently ...
Top 50 recent answers are included | {"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": 0.8271880149841309, "perplexity": 528.5668309921853}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535919886.18/warc/CC-MAIN-20140909051552-00268-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://link.springer.com/article/10.1007/s100520100627?error=cookies_not_supported&code=5951915c-e3a4-4073-a3dc-08fa1d47b494 | # Precise determination of the Z resonance parameters at LEP: “Zedometry”
## Abstract.
This final analysis of hadronic and leptonic cross-sections and of leptonic forward-backward asymmetries in $${\mathrme}^+{\mathrme}^-$$ collisions with the OPAL detector makes use of the full LEP 1 data sample comprising $$161 \mathrm{pb}^{-1}$$ of integrated luminosity and $$4.5\times10^6$$ selected Z decays. An interpretation of the data in terms of contributions from pure Z exchange and from $$\gamma/$$Z interference allows the parameters of the Z resonance to be determined in a model-independent way. Our results are in good agreement with lepton universality and consistent with the vector and axial-vector couplings predicted in the Standard Model. A fit to the complete dataset yields the fundamental Z resonance parameters: $$m_{\mathrm{Z}} = (91.1852 \pm 0.0030)$$ GeV, $$\Gamma_{\mathrm{Z}} = (2.4948 \pm 0.0041)$$ GeV, $$\sigma^0_{\mathrm{h}} = (41.501 \pm 0.055)$$ nb, $$R_{\ell} = 20.823 \pm 0.044$$, and $$A_{\mathrm{FB}}^{0,\ell} = 0.0145 \pm 0.0017$$. Transforming these parameters gives a measurement of the ratio between the decay width into invisible particles and the width to a single species of charged lepton, $$\Gamma_{\mathrm{inv}}/\Gamma_{\ell\ell} = 5.942 \pm 0.027$$. Attributing the entire invisible width to neutrino decays and assuming the Standard Model couplings for neutrinos, this translates into a measurement of the effective number of light neutrino species, $$N_{\nu} = 2.984 \pm 0.013$$. Interpreting the data within the context of the Standard Model allows the mass of the top quark, $$m_{\mathrm{t}} = (162 ^{+29}_{-16})$$ GeV, to be determined through its influence on radiative corrections. Alternatively, utilising the direct external measurement of $$m_{\mathrm{t}}$$ as an additional constraint leads to a measurement of the strong coupling constant and the mass of the Higgs boson: $$\alpha_{\mathrm{s}}(m_{\mathrm{Z}}) = 0.127 \pm 0.005$$ and $$m_{\mathrm{H}} = (390^{+750}_{-280})$$ GeV.
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Received: 1 December 2000 / Published online: 6 April 2001
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The OPAL Collaboration., Abbiendi et al., G. Precise determination of the Z resonance parameters at LEP: “Zedometry”. Eur. Phys. J. C 19, 587–651 (2001). https://doi.org/10.1007/s100520100627 | {"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": 0.9102746248245239, "perplexity": 1298.471359262906}, "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/1634323585696.21/warc/CC-MAIN-20211023130922-20211023160922-00184.warc.gz"} |
http://aa.quae.nl/en/reken/saros.html | $$\DeclareMathOperator{\trunc}{trunc} \def\floorratio#1#2{\left\lfloor \frac{#1}{#2} \right\rfloor} \def\ceilratio#1#2{\left\lceil \frac{#1}{#2} \right\rceil}$$
## 1. Introduction
At least 2500 years ago astronomers in Babylon discovered that solar eclipses and lunar eclipses often recur under similar circumstances after 223 synodical months (counted according to the phases of the Moon, not according to the western calendar) or about 6585⅓ days. This period is nowadays called the saros. We call a series of eclipses that each occur one saros after the previous one a saros series.
To get a solar eclipse or lunar eclipse, two independents series of periodic phenomena must come together, namely the proper phase of the Moon (New Moon for a solar eclipse, or Full Moon for a lunar eclipse) and passage of the Moon through a node of the orbit of the Moon around the Earth.
## 2. Strictly Periodic Phenomena
We study the coming together of two different periodic phenomena A and B. When they coincide then we have the combination phenomenon which we'll call Z. We assume that phenomenon A has a period $$P_A$$, phenomenon B a period $$P_B$$, and that the ratio of those periods is a rational number:
$$\frac{P_A}{P_B} = γ' = \frac{a}{b}$$
with $$a$$ and $$b$$ whole numbers. That means that $$b$$ periods $$P_A$$ are exactly as long as $$a$$ periods $$P_B$$. That long after a previous Z there will be a next Z. We call the period $$y = b P_A$$ the prediction period. We assume that $$a$$ and $$b$$ have no factors in common. If they do, then you can just divide them out.
For eclipses Z, A is reaching the proper lunar phase (Full Moon for a lunar eclipse, New Moon for a solar eclipse) and B is reaching a node in the orbit of the Moon. For the saros, $$a$$ is equal to 484, $$b$$ is equal to 223, and $$y$$ is just over 18 years.
If an A and a B happen at the same time then we get the combination phenomenon Z. Often there is a little leeway so that A and B may have a small time difference and yet give a Z. We can find two boundaries $$d_1$$ and $$d_2$$ so that a Z certainly occurs if the time difference between an A and the closest B is less than $$d_1 P_B$$, and a Z certainly does not occur if the time difference is more than $$d_2 P_B$$.
In our example, Z is a solar eclipse or lunar eclipse. For such eclipses, $$d_1$$ = 0.090 and $$d_2$$ = 0.117.
If you go $$y$$ forward or backward in time from a Z then you find another Z. All of those phenomena Z form a collection that we call a prediction series. We call the collection of phenomena A that is associated with such a prediction series a prediction series (of A) as well, and likewise for B. In a prediction series of A or B or Z, the different phenomena of the same kind are all $$y$$ apart.
We'll ignore the prediction series of B below, but everything works the same way for Bs as for As, except that you have to use some special values that hold for Bs instead of the corresponding values that hold for As.
We can associate each A with a prediction series. There are then $$b$$ different prediction series. The phenomena Z won't happen in all of those prediction series.
We first assign an ordinal number to each A that is 1 greater for each than the number of the previous A. We use $$k$$ to denote the ordinal number of A. A suitable formula for calculating the number of the prediction series is then
$$s ≡ a k + s_0 \pmod{b} \label{eq:sarosnummer}$$
where the $$\bmod b$$ means "except for multiples of $$b$$", and the $$s_0$$ is the number of the series that $$k$$ = 0 belongs to. If you want to go the other way and find which $$k$$ belong to the series with number $$s$$, then you can use
$$k = l (s - s_0) + n b \label{eq:ks}$$
where $$l$$ is a number such that $$a l ≡ 1 \pmod{b}$$ and $$n$$ is an ordinal number that counts phenomena A in a prediction series. In general,
$$l ≡ a^{φ(b) - 1} \pmod{b}$$
with $$φ(b)$$ the number of non-negative whole numbers less than $$b$$ that are prime relative to $$b$$.
For example, the solar eclipse of 5 February 2000 belongs, according to a popular scheme, to saros number 150. If we assign the number $$k$$ = 2 to that New Moon (so that the first New Moon of the year 2000 has the number 1), then $$s_0$$ = 74 and the formula to calculate the saros number of a New Moon is
$$s = 484 k + 74 = 38 k + 74 \pmod{223}$$
The other way around, all $$k$$ that belong to a particular saros number $$s$$ can be found from
$$k = 135 (s - 74) = 135 s + 45 \pmod{223}$$
There are saros numbers also for Full Moons and lunar eclipses. The lunar eclipse of 21 January 2000 belongs, according to a popular scheme, to (lunar) saros 124. If we assign $$k$$ = 1 to the first Full Moon of the year 2000, then $$s_0$$ = 86 and the formulas to go from $$k$$ to $$s$$ or the other way around are, for Full Moons,
\begin{align} s & = 484 k + 86 = 38 k + 86 \pmod{223} \\ k & = 135 (s - 86) = 135 s + 209 \pmod{223} \end{align}
The condition of phenomena B at the moment of an A is
$$S'_k = S_0 + γ' k \label{eq:b-toestand}$$
If $$S'_k$$ is a whole number, then we are exactly in the middle of a B, and thus also exactly in the middle of a Z. $$S_0$$ is the condition of B at the moment that corresponds to $$k$$ = 0. With the definition
$$s'_* = a k + s_0 - b S'_k \label{eq:bestenummer}$$
we find that
$$s'_* = s_0 - b S_0$$
which is a constant. The whole number closest to $$s'_*$$ is the number (except for multiples of $$b$$) of the series for which A and B are closest together. We refer to that series as the best series.
For solar eclipses, $$S_0$$ at the beginning of 2000 was equal to −0.2775775 and $$s'_*$$ was then equal to 135.8998. For lunar eclipse, $$S_0$$ was then equal to −0.1923817 and $$s'_*$$ was then 128.9011.
If we insert equation \ref{eq:ks} into equation \ref{eq:b-toestand} then we find
$$S'_k = S_0 + \frac{a}{b} l (s - s_0) + a n = S_0 - \frac{a}{b} s_0 + \frac{a l}{b} s + a n$$
If we modify $$n$$ by adding $$∆n$$ to it, or $$s$$ by adding $$∆s$$ to it, then $$S'_k$$ gets $$∆S'_k$$ added to it. Only the difference of $$S'_k$$ from the nearest whole number is important for determining whether a Z will happen, so we can calculate $$\bmod 1$$. Then we find
$$∆S'_k = \frac{a l}{b} ∆s + a ∆n = \frac{1}{b} ∆s \pmod{1} \label{eq:delta-s}$$
The term $$a ∆n$$ disappears because it is always a whole number, and $$\frac{a l}{b} ≡ \frac{1}{b} \pmod{1}$$ because $$a l ≡ 1 \pmod{b}$$ by definition.
If the $$S'_k$$ of a prediction series differs less than $$d_1$$ from a whole number, then there is certainly a Z in that series. If the difference lies between $$d_1$$ and $$d_2$$, then there is sometimes a Z, and if the difference is greater than $$d_2$$, then there is certainly no Z. The best prediction series has $$s = [s'_*]$$. For that series, $$S'_k$$ is closest to a whole number. It follows from equation \ref{eq:delta-s} that there is a Z for every A in prediction series for which $$s$$ lies between $$s'_* - d_1 b$$ and $$s'_* + d_1 b$$ (inclusive). For prediction series between $$s'_* - d_2 b$$ and $$s'_* - d_1 b - 1$$ or between $$s'_* + d_1 b + 1$$ and $$s'_* + d_2 b$$ there is a Z for some but not all A. If $$s$$ is outside of those ranges, then there is no Z in that series.
For solar eclipses, at the beginning of the year 2000, saros series number 136 was the best one. At that time, eclipses occurred in saros series between 117 and 156. For lunar eclipses, saros series 129 was then the best one, and eclipses occurred in series 109 through 150.
## 3. Almost Periodic Phenomena With Fixed Periods
Up till now we've assumed that the ratio of the periods of A and B was a rational number (a division involving only whole numbers) which we knew exactly. If one or more of these assumptions is not met, then we can still approximate the ratio with a rational number, but then our approximation won't quite fit the reality. We call the real ratio of the periods $$γ$$ and continue to use $$γ'$$ for our approximation of the ratio:
\begin{align} γ & = \frac{P_A}{P_B} \\ γ' & = \frac{a}{b} \end{align}
We call the difference between the approximation and the real ratio $$δ$$:
$$δ = γ' - γ = \frac{a}{b} - γ$$
For the Moon, at the beginning of the year 2000, we had $$P_A$$ = 29.530588853 days and $$P_B$$ = 13.606110408 days, so $$γ$$ = 2.170391681. If we use the saros as an approximation (so $$a$$ = 484 and $$b$$ = 223) then $$δ$$ was equal to 0.000011906 at the beginning of the year 2000.
We can continue to use equation \ref{eq:sarosnummer} to calculate the number of the prediction series. For the condition of B at the time of an A we must adjust equation \ref{eq:b-toestand} to
$$S_k = S_0 + γ k \label{eq:b-toestand-2}$$
and we adjust equation \ref{eq:bestenummer} for the number of the best prediction series to
$$s_* = a k + s_0 - b S_k \label{eq:bestenummer-2}$$
from which
$$s_* = b δ k + s_0 - b S_0 = b δ k + s'_*$$
Now $$s_*$$ is no longer a constant, so the range of prediction series in which phenomena Z occur shifts with time at a rate of 1 per period $$\frac{P_A}{b δ}$$.
That period is now about 30 years for solar eclipses and lunar eclipses.
The prediction series that yield phenomena Z around a given date won't do that indefinitely. If they yield phenomena Z then we call them active and otherwise inactive. Each prediction series will switch between active and inactive and back again in a grand period which we'll indicate as $$c$$ and which is equal to
$$c = \frac{P_A}{δ}$$
With this, equation \ref{eq:delta-s} becomes
$$∆S_k = l \left( \frac{a}{b} - δ \right) ∆s + (a - b δ) ∆n = \left( \frac{1}{b} - l δ \right) ∆s - b δ ∆n \pmod{1} \label{eq:sk-bijna}$$
If $$b δ ∆n$$ is less than $$d_1$$, then there is certainly a Z, and if it is greater than $$d_2$$ then there is certainly no Z. It follows that an active period of a prediction series contains between $$\frac{2 d_1}{b δ}$$ and $$\frac{2 d_2}{b δ}$$ phenomena Z.
If you go to a different $$s$$ then you go not just to a different prediction series but also (with equation \ref{eq:ks}) to a different $$k$$, hence to a different time. This was not important in the previous section, because there the prediction series did not depend on time (because the prediction period was exactly right), but here this is important. We want to see what happens to $$S_k$$ if you go to a different prediction series without going to a different date. We can do this, because equation \ref{eq:ks} shows that $$∆k = l ∆s + b ∆n$$ so $$∆k = 0$$ if $$∆n = -\frac{l}{b} ∆s$$. If we insert that into equation \eqref{eq:sk-bijna} then we find, like for strictly periodic phenomena,
$$∆S_k = \frac{1}{b} ∆s \pmod{1}$$
This means that between $$2 d_1 b$$ and $$2 d_2 b$$ prediction series are active at any one time.
The grand period of the saros is 6790 years (about 377 saros periods) and each active period of a saros series contains between 70 and 86 eclipses. Between 40 and 52 saros series are active at any given time for solar and lunar eclipses separately.
Because prediction series now switch between active and inactive, it is interesting to give each active period its own prediction series number. We can do that, because equation \ref{eq:sarosnummer} has $$\bmod b$$ in it, so we can choose to given each active period from a given prediction series a number that is $$b$$ greater than that of the previous active period. For example, like this:
$$s = \left( \left( a k + s_0 - \left\lfloor s_* - \frac{b}{2} \right\rfloor \right) \bmod b \right) + \left\lfloor s_* - \frac{b}{2} \right\rfloor \label{eq:sarosnummer-2}$$
Because $$(x×y) \bmod b = ((x \bmod b)×(y \bmod b)) \bmod b$$, you may replace $$a k$$ from equation \ref{eq:sarosnummer-2} with $$(a \bmod b)×(k \bmod b)$$, which may be useful to keep the calculation numbers as small as possible when $$k$$ becomes very large.
For example, saros 114 was active between the years 651 and 1931, and will be active again roughly between the years 7400 and 8700. The New Moon with $$k = 75422$$ belongs to saros number 114, because
$ak + s_0 = 484×75422 + 74 = 36504322 = 114 \pmod{223}$
but that $$k$$ falls in the year 8097, long past the end of the most recently active period of saros 114 (which ended in 1931). We'd like to assign saros number 114 + 223 = 337 to that $$k$$ to show that it belongs to the same series as 114 but is one great period later.
Then
$s_* = bδk + s'_* = 223×0.000011906×75422 + 135.8998 = 336.1481$
and earlier we saw that $$s = 114 \bmod 223$$. With equation \ref{eq:sarosnummer-2} we find
\begin{align*} \left\lfloor s_* - \frac{b}{2} \right\rfloor & = \left\lfloor 336.1481 - \frac{223}{2} \right\rfloor \\ & = ⌊225.1481⌋ = 225 \\ s & = ((484×75422 + 74 - 225) \bmod 223) + 225 \\ & = (36504097 \bmod 223) + 225 = 112 + 225 = 337 \end{align*}
The intermediate result 36504097 is large, but we may apply the $$\bmod b$$ also separately to the factors within the formula that is itself $$\bmod b$$, so
\begin{align*} s & = (((484 \bmod 223)×(75422 \bmod 223) + 74 - 225) \bmod 223 + 225 \\ & = (38×48 + 74 - 225) \bmod 223 + 225 \\ & = 1673 \bmod 223 + 225 = 112 + 225 = 337 \end{align*}
If $$s_*$$ is equal to a whole number, then A and B exactly coincide, so then we've found the middle of the active period. $$s_*$$ is equal to a particular $$s$$ (a whole number) if $$k$$ is equal to
$$k_s = \frac{s - s_0 + b S_0}{b δ} = \frac{s - s'_*}{b δ}$$
For solar eclipses and the saros we find
$$k_s = (s - 135.8998)×376.6284$$
and for lunar eclipses
$$k_s = (s - 128.9011)×376.6284$$
This $$k_s$$ need not be a whole number and need not even fit in prediction series $$s$$, but the $$k$$ from series $$s$$ that is closest to $$k_s$$ indicates the middle of that series. With this, we can assign ordinal numbers
$$m = \left[ \frac{k - k_s}{b} \right] \label{eq:m}$$
to the different Z from a prediction series. In that formula, $$k$$ must fit the prediction series $$s$$ (e.g., using equation \ref{eq:ks}). If $$m$$ is equal to 0, then we've found the middle of the active period. If $$m$$ is greater than 0, then we are past the middle, and if $$m$$ is less than 0, then we are still before the middle.
For the solar eclipse of 5 February 2000 ($$s$$ = 150, $$k$$ = 2) we then find $$k_s$$ = 5310.537 and $$m$$ = −24. This solar eclipse is therefore a very early one in its series, because there are still 23 eclipses to follow in that series before the middle of the series is reached.
For the lunar eclipse of 21 January 2000 ($$s$$ = 124, $$k$$ = 1) we find $$k_s$$ = −1845.891 and $$m$$ = +8. So, that lunar eclipse is past the middle of its saros series.
Equation \ref{eq:m} depends in two different ways on $$k$$: once because it contains $$k$$ explicitly, and once because $$k_s$$ depends on $$k$$ via $$s$$. Using equation \ref{eq:sarosnummer} we can say a bit more about how $$m$$ depends on $$k$$.
\begin{align} m' & ≡ \frac{k - k_s}{b} \\ m & = [m'] \end{align}
$$\begin{split} m' & = \frac{k - k_s}{b} \\ & = \frac{k - \frac{s - s'_*}{bδ}}{b} \\ & = \frac{k - \frac{ak + s_0 \bmod b - s'_*}{bδ}}{b} \\ & = \frac{k - \frac{ak + s_0 - s'_*}{bδ}}{b} \bmod \frac{1}{bδ} \\ & = \left( 1 - \frac{a}{bδ} \right) \frac{k}{b} + \frac{s'_* - s_0}{b^2δ} \bmod \frac{1}{bδ} \\ & = -\frac{γk}{bδ} + \frac{s'_* - s_0}{b^2δ} \bmod \frac{1}{bδ} \\ & = \frac{-γk + \frac{s'_* - s_0}{b} \bmod 1}{bδ} \end{split} \label{eq:m'}$$
For example, for the saros of solar eclipses we have $$a = 484$$, $$b = 223$$, $$γ = 2.170391681$$, $$δ = 0.000011906$$, $$s'_* = 135.8998$$, so
\begin{align*} m' & = \frac{−2.170391681k + 0.27758 \bmod 1}{223×0.000011906} \\ & = (−0.170391681k + 0.27758 \bmod 1)×376.6284 \\ & = −64.1743454k + 104.54 \bmod 376.6284 \end{align*}
The equation for lunar eclipses is the same except that 104.54 should be replaced with 72.46.
For the solar eclipse of 5 February 2000 ($$k$$ = 2) we then find
$m' = −64.1743454×2 + 104.54 \bmod 376.6284 = −23.81 \bmod 376.6284$
so $$m = [m'] = −24$$. For the lunar eclipse of 21 January 2000 ($$k$$ = 1) we then find
$m' = −64.1743454×1 + 72.45 \bmod 376.6284 = 8.28 \bmod 376.6284$
so $$m = [m'] =+8$$.
## 4. Almost Periodic Phenomena With Varying Periods
Up till now we've assumed that the ratio of periods is constant, but in practice the ratio usually slowly varies with time. If the variation is slow, then we can still use reasonably simple formulas. We assume that the ratio $$γ$$ is not a constant but varies according to
$$γ_k = γ_0 + γ_1 k$$
The only important adjustments relative to the case of constant periods are to formulas that feature $$δ$$ or $$S_k$$. We find
\begin{align} δ & = \frac{a}{b} - γ_0 - γ_1 k = δ_0 - γ_1 k \\ S_k & = S_0 + γ_0 k + \frac{1}{2} γ_1 k^2 \\ s_* & = a k + s_0 - S_k = b δ_0 k - \frac{1}{2} b γ_1 k^2 + s'_* \end{align}
To find the best Z of prediction series $$s$$ we must now solve a quadratic equation:
$$-\frac{1}{2} b γ_1 k_s^2 + b δ_0 k_s + s'_* - s = 0$$
The solution is
$$k_s = \frac{δ_0}{γ_1} \left( 1 - \sqrt{1 - \frac{2γ_1}{bδ_0^2}(s - s'_*)} \right)$$
If $$|s - s'_*|$$ is much less than $$\left| \frac{bδ_0^2}{2γ_1} \right||$$, then the solution is approximately equal to
\begin{align} k_s & = \frac{s - s'_*}{bδ_0} + \frac{γ_1 (s - s'_*)^2}{2b^2δ_0^3} \\ k_s & ≡ k_{s0} + k_{s1} \\ k_{s0} & ≡ \frac{s - s'_*}{bδ_0} \\ k_{s1} & ≡ \frac{γ_1 (s - s'_*)^2}{2b^2δ_0^3} \end{align}
The calculation of $$m'$$ then goes in a couple of steps:
\begin{align} m' & = m_0' + m_1' \\ m_0' & = \frac{k - k_{s0}}{b} \notag \\ & = -\frac{γ_0k}{bδ_0} + \frac{s'_* - s_0}{b^2δ_0} \bmod \frac{1}{bδ_0} \notag \\ & = \frac{-γ_0k + \frac{s'_* - s_0}{b} \bmod 1}{bδ_0} \\ m_1' & = -\frac{k_{s1}}{b} = -\frac{γ_1(k - bm_0')^2}{2bδ_0} \end{align}
So first calculate $$m'$$ from equation \ref{eq:m'} as if $$γ = γ_0$$ and call it $$m_0'$$, and then calculate $$m_1'$$ as a correction to $$m_0'$$.
For the saros $$γ_1$$ is approximately equal to $$−1.2×10^{−11}$$ so $$\frac{bδ_0^2}{2γ_1}$$ is equal to about −1300. For saros numbers much less than 1300 we can use the previous equations, which then become (for solar eclipses):
\begin{align*} k_s & = (s - 135.8998) × 376.6284 - 0.071482 (s - 135.8998)^2 \\ m_0' & = −64.1743454k + 104.54 \bmod 376.6284 \\ m_1' & = 2.3×10^{−9} (k - 223m_0')^2 \\ m' & = m_0' + m_1' \end{align*}
For the solar eclipse of 5 February 2000 we then find $$k_s$$ = 5296.324 but still $$m$$ = −24. For lunar eclipses the formulas are the same as for lunar eclipses, except that you should use 128.9011 rather than 135.8998, and 72.45 instead of 104.54.
## 5. Related Phenomena
Suppose you can find for each phenomenon A a related phenomenon A′ which happens $$ε P_A$$ later than A, with $$ε a$$ a whole number. Then you'll find the same kind of behavior for the combination phenomena Z′ of A′ and B as for the combination phenomena Z of A and B, and you can use the same formulas for A′ and Z′ as for A and Z, except that some constants are different. You could, for example, insert $$k + ε$$ in place of plain $$k$$ into the formulas for A and Z. The formula to calculate the number of the prediction series then becomes
$$s′ = a (k + ε) + s_0 = a k + (s_0 + a ε) \bmod b$$
You then find that the best prediction series have the exact same number ($$s'_*$$) for strictly periodic Z and Z′, so (almost) the same range of prediction series numbers is active for Z and for Z′.
For lunar eclipses as related phenomena to solar eclipses (with $$ε = \frac{1}{2}$$) you'd then find $$s_{0\text{moon}} = s_{0\text{sun}} + 484×\frac{1}{2} = s_{0\text{sun}} + 19 = 93 \bmod 223$$, but as we saw earlier $$s_{0\text{moon}}$$ = 86 according to the oft-used scheme, so that scheme runs 93 - 86 = 7 saros series behind, and that exactly fits the difference between the numbers of the best saros series for solar eclipses and for lunar eclipses, which we earlier found to be 136 and 129 in the year 2000.
## 6. The Next Events
Suppose that event A or Z occurs for a particular $$k$$ corresponding to a particular $$s$$ and $$m$$. Then when are the next interesting events A or A' or Z or Z'?
• The next A (for $$k + 1$$) then belongs to saros $$s + a \bmod b$$ and has approximately $$m + \frac{γb}{δ}$$.
• The next A that belongs to the same saros $$s$$ then has number $$k + b$$ and has $$m + 1$$.
• The next A that belongs to the next saros $$s + 1$$ then has number $$k + 1$$ and has approximately $$m + \frac{γl}{bδ} \bmod \frac{1}{bδ}$$.
• The next A that belongs to the previous saros $$s - 1$$ then has number $$k + b - l$$ and has approximately $$m + \frac{γ×(b - l)}{bδ} \bmod \frac{1}{bδ}$$, which is approximately $$±1$$
• .
• The A' that is related to A (and thus occurs a half $$P_A$$ later for the same $$k$$) belongs to A-saros $$s + aε \bmod b$$.
• Suppose that a Z occurred at this A. When is the next Z? Look at the first couple of multiples of $$a$$, modulus $$b$$. If those are close to 0 (or to $$b$$) then there is a chance of another Z. If $$a \bmod b ≤ 2d_2b$$, then the very next A might be a Z again (if the previous Z was sufficiently early in its series, i.e., if $$m$$ is quite negative). After that, $$∆k = \frac{b}{a \bmod b}$$ is a good candidate.
These rules are summarized in Table 1.
Table 1: Next Events
$${∆s}$$$${∆k}$$$${∆m}$$
A $${a \bmod b}$$1 $${≈ γ/bδ}$$
A 0 $${b}$$+1
A +1 $${l}$$$${≈ γl/bδ}$$
A −1 $${b - l}$$$${≈ γ(b - l)/bδ}$$
A' $${aε \bmod b}$$0
Z $${a \bmod b}$$1? $${≈ γ/bδ?}$$
Z $${ab/(a \bmod b)}$$$${b/(a \bmod b)?}$$$${≈ γ/(a \bmod b)δ?}$$
For solar and lunar eclipses and the Saros, we find
$${∆k}$$ 0 1 2 3 4 5 6 7 8 $${∆s}$$ 0 38 76 −109 −71 −33 5 43 81 $${∆m}$$ 0 −64 −128 +184 120 56 −8 −73 −137
and $$|2d_2b| = 52$$, so if $$|∆s| ≤ 52$$ then $$∆k$$ months after a solar or lunar eclipse there is a chance of another solar or lunar eclipse.
$${∆s}$$$${∆k}$$$${∆m}$$
A 38 1 ≈ −64
A 0 223 1
A +1 135 ≈ −1
A −1 88 ≈ 2
A' 19 0 ≈ −89
Z 38 1? ≈ −64?
Z 5 6? ≈ −8?
## 7. Comparison with the Eclipse Calculations by Fred Espenak
The "Catalog of Solar Eclipses" of November 1999 by Fred Espenak lists all 11897 solar eclipses during the years from −1999 to +3000, including their dates and saros numbers. The "Catalog of Lunar Eclipses" of November 1999 by Fred Espenak lists all 12186 lunar eclipses in the same period, including their dates and saros numbers.
(For the solar eclipses I noticed that the saros numbers of 4 of them in the catalog are wrong: the saros numbers of the 512th, 649th, 3627th, and 7553rd solar eclipse from the catalog are equal to 0 but should have been −9, −4, 33, and 91. The solar eclipses are sorted by date in the catalog, but the lunar eclipse aren't ― at least not perfectly.)
I calculated $$k$$ (from the date) and also $$m$$ and $$s$$ for all of those eclipses, using the formulas given above. "My" saros number $$s$$ is equal to that from the catalogs, except for the four solar eclipses that I mentioned. For succesive solar eclipses and successive lunar eclipses from the catalogs I find only the following combinations of $$∆s$$, $$∆k$$, and $$∆m$$:
$${∆s}$$$${∆k}$$$${∆m}$$
38 1 −68…−63
−33 5 55…59
5 6 −9…−8
After every solar eclipse there is a lunar eclipse ½ or 5½ months later. For the first lunar eclipse after each solar eclipse I find (with each $$∆x = x_\text{Moon} - x_\text{Sun}$$)
$${∆s}$$$${∆k}$$$${∆m}$$
12 0 −34…−31
−21 5 23…25
Each lunar eclipse is followed by a solar eclipse ½ or 5½ months later. For the first solar eclipse after each lunar eclipse I find (with each $$∆x = x_\text{Moon} - x_\text{Sun}$$)
$${∆s}$$$${∆k}$$$${∆m}$$
26 1 −34…−31
−7 6 23…25
languages: [en] [nl]
//aa.quae.nl/en/reken/saros.html;
Last updated: 2017-09-19 | {"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": 6, "equation": 28, "x-ck12": 0, "texerror": 0, "math_score": 0.982468843460083, "perplexity": 664.4518428668014}, "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/1510934806353.62/warc/CC-MAIN-20171121113222-20171121133222-00349.warc.gz"} |
http://oftankonyv.reak.bme.hu/tiki-index.php?page=Statistical+image+reconstruction+strategies&structure=Book+for+Physicists | # Statistical image reconstruction strategies
Let us add to our earlier equation the assumption that the linear system of equations are only valid as an expected value of some statistical quantities:
thus the incoming information, the measured data is not "carved into stone", exact values but -even when no noise is present- it can take "legally" multiple values while the x unknowns do not change. The linearity of the operator projecting the unknowns to the space of the measure data can be also interpreted as statistical independence, e.g. in emission tomography the radioactive decays happening in one voxel are independent of the decays happening in the adjacent voxel, while for transmission tomography linearity only holds if we disregard spectral changes.
Let us generalize our model where the acquired data and the unknowns are connected by a probability density function (pdf) describing the probability of obtaining y data given the unknowns take a value x; let us choose the notation for this pdf as . When also x is regarded as a random variable, we should include that in the notation:
### Maximum-Likelihood estimation
A basic statistical estimation theory tool is the Maximum-Likelihood estimation, when we look for the x parameters where a given measured data would occur with the maximum likelihood, i.e. our estimation could be formalized as
The majority of the measurable physical quantities can be modeled using the exponential distribution family therefore the maximization criteria is set to the logarithm of the likelihood function .
The Maximum Likelihood estimation is in theory and practice both a very basic mathematical tool that we are not going to dwell on but only show an example for those wishing to brush up their memories.
Example for teh Maximum Likelihood estimator can be found on a separate page.
### Bayes estimation and the Maximum Aposteriori criteria
Now let us look at the x model parameters as random variables, and look at the probability that if we acquired y measured data what would the pdf of the parameters be:
The Bayes-estimation maximizes this probability, that is, how the reply is given to the question of what would be the most probable parameter set given the measured data?. This probability can be considered as aposteriori, i.e. post-measurement pdf, hence is the name Maximum Aposteriori (MAP) technique.
In mathematical symbols using the Bayes-theorem:
The first maximization term is already known to us from the Maximum-Likelihood estimation, the second depends on the pdf of the parameters this one contains our apriori knowledge, and is called the prior. Its interpretation is that the parameters cannot take any arbitrary value since the activity concentration value for example can never exceed the activity concentration of the radiopharmacon administered.
Our pre-examination knowledge may be very hard to mathematically formulate. In practice the priors are very simple conditions like the continuity or smoothness of the spatial distribution of x. As the ln functionis often applied the priors are also given often in an exponential form, called Gibbs-priors:
For example if our apriori knowledge is that if a voxel has a certain reconstructed value than its neighbors should not be much different, the U(x) potential a can be proportional with the distance of the voxel from the voxel in question.
### Maximum-Likelihood Expectation Maximisation (ML-EM ) algorithm
An extension of the Maximum Likelihood method is the iterative Maximum-Likelihood Expectation Maximisation method with involving virtual random variables in the modelling that increases modeling accuracy while simplifies the maximisation procedure. The ML-EM method can also involve priors, then it is called the MAP-EM method.
As the ML-EM method is a fundamental tool in medical imaging we will go into details in the next section. | {"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": 0.9277796149253845, "perplexity": 667.1664750515555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146809.98/warc/CC-MAIN-20200227191150-20200227221150-00531.warc.gz"} |
https://www.physicsforums.com/threads/hypothetical-qm-question.105216/ | # Hypothetical QM Question
1. Dec 27, 2005
Suppose an experiment was published next week in Science that conclusively falsified quantum mechanics, how would the science of physics change ? Here I assume that QM is a true scientific theory and via Popper must be open to falsification (that is, if QM cannot ever be falsified it is not science). Perhaps a bit too much toward philosophy, so move it to that area of the forum if need be.
2. Dec 27, 2005
Staff Emeritus
It would depend on just what feature of QM was falsified. For example if the experiement showed that realism was true - that particles have properties independent of observation, the results would be different than if it were shown that evolution is something nonlinear instead of unitary.
What no new experiment could disprove is all the prior experiments. Any new theory would have to account for them as well as for the ground breaking new one.
3. Dec 27, 2005
Thank you "selfAdjoint". So, it would appear that this Science paper could in theory take three separate forms, and that any of these would falsify quantum mechanic theory:
1. Show that non-realism is false.
2. Show that evolution is nonlinear
3. Show both 1 & 2 above
4. Dec 28, 2005
### kleinwolf
Non-realism is "particle don't have properties (because in superposition ?)before a measurement (?)"...and the measurement disturbs the system collapsing into a measured state...but what if the system itself distubrbs the observer or measurement apparatus...is this non-realism (hwo do you read the result of observation if the measurement apparatus does not change it's state during the measurement process ?)
5. Dec 28, 2005
Staff Emeritus
There is at least one other property, localism (I am following Dr. Chinese here). So you have three independent (well....) properties, call them R, U, and L, and QM is ~R/U/L, and your conjectured experiment could show that reality is R, ~U, or ~L, in any combination.
6. Dec 28, 2005
Excellent, so many ways to falsify QM using the scientific method--so, when can I realistically expect to see this Science paper in print ? Next I would like to have a discussion on the following, which would also help with the question by "kleinwolf":
1. Type of experimental design required to show that reality is R, which would then falsify QM
2. Type of experimental design required to show that reality is ~U....
3. Type of experimental design required to show that reality is ~L....
That is, I would like to better understand the format required for the experimental designs that would conclusively falsify QM based on your ~ R/U/L criteria.
7. Dec 28, 2005
Staff Emeritus
Well I don't have any idea how such an experiment would work. And I more than half suspect no such experiment is possible; that is I believe that quantum mechanics captures some essential features of reality. It doesn't have to be a perfect theory to do that.
8. Dec 28, 2005
Well then, all that I can conclude up to this point in the thread discussion is that this hypothetical paper in Science will never appear because QM theory is not a scientific theory that can be falsified via experimentation. Is this the majority view of this physics forum, that QM is not a theory that can be falsified via experimentation ? If not, would someone please provide me the required experiments that would conclusively falsify QM as identified by "selfAdjoint" and given in post #6.
9. Dec 29, 2005
### Staff: Mentor
On the face of it, this seems a silly statement. QM has made countless specific, surprising (compared to classical physics) predictions which--so far--have all been verified. Had these predictions failed to be verified, then QM would easily be falsified. Just because a theory is successful (so far) hardly makes it unscientific or unfalsifiable! A theory that makes specific, unexpected predictions--like QM--is eminently falsifiable.
And when future experiments push the envelope so far that even QM fails, then it too will need to be replaced by a better theory.
It seems to me that what you are really asking is "What one experiment will, now and forever, conclusively show that QM is correct?" Such an experiment is impossible. But any of the many experiments done to date could have shown QM to be wrong.
10. Dec 29, 2005
Staff Emeritus
And in addition to what Doc Al said, which is absolutely true, the fact that there is no experiment we can imagine now doesn't mean there can never be one. QM is a powerful theory that makes new predictions as we investigate it more. It is logically possible that some future experiment dealing with some future prediction will falsify it.
I will note that QM workers nearly all believe that the Aspect and other experiments testing the Bell inequalities have confirmed ~R, so based on what we know today it's only U and L that are up for grabs. And indeed you see papers suggesting non-linear replacements for QM and nonlocal replacements for relativity - all conformed to agree with the theories they replace at the scales we have been able to investigate so far.
11. Dec 30, 2005
[To Doc Al and selfAdjoint] I appreciate your comments, but can you see how they may lead to confusion ? First, selfAdjoint indicates that "no experiment can be imagined that would falsify QM", then Doc Al informs that in fact many such experiments have already been published and QM has never been falsified. Then selfAdjoint states that R (Realism as defined above) most likely has been falsified via experiments, but were not these experiments designed to falsify R and not QM ? Finally, selfAdjoint informs that U and L are still fair game, that is, someday an experimental design may be thought of that will falsify the predictions of QM concerning U and L--thus my quantum mental state.
And, to Doc Al, no, I really am not looking for a way to show that QM is correct, such is not the way of science, we do not set out to prove a theory true, we design an experiment to falsify the theory (more often a hypothesis in the null form). And thus the reason for my thread, I would like to understand the details of the experimental designs that would be required to "falsify" QM as it predicts ~ R, ~ U, ~ L as presented above by selfAdjoint.
Finally, I still have no answer to my post # 1 question, e.g., how would science of physics change if next week the paper in Science appeared that "falsified" QM ?---it is a hypothetical question, thus hypothetical answers are allowed I would think.
12. Dec 31, 2005
### inha
Physicists are mostly very practical people so we would keep on using QM within it's domain of appliacability and work on it's successor. There really isn't anything more to it.
Last edited: Dec 31, 2005
13. Dec 31, 2005
### cartuz
14. Dec 31, 2005
### inha
There is no article. This was a hypothetical question.
15. Jan 1, 2006
### Sherlock
What is unique to QM? What is QM's fundamental hypothesis? That there is a fundamental quantum of action?
Last edited: Jan 1, 2006
16. Jan 1, 2006
### DrChinese
QM is falsifiable per Popper. However, it was not falsified, it was supported. So it is no longer possible to publish such a paper as you describe. The only thing that could be published is a paper that is in contradiction with some secondary area of QM - i.e. a prediction which is a consequence of QM. (This would not actually be a falsification of QM, it would merely be setting limits on its domain of applicability.)
HOWEVER... none of that really matters. Every theory has a domain over which it may be useful. Outside of such domain, it ceases to be useful. That does not change it from being a "true" theory to a "false" theory. Popper's ideas about theory evaluation are themselves useful tools, but they are not absolute in any sense.
Theories compete - may the best theory win! Perhaps someday, someone will publish a BETTER theory than QM.
17. Jan 11, 2006
### RandallB
I think I can give an answer to your question, as long as you allow for some range of results based on “how wrong” QM is found to be.
Example: Ptolemy and Tyco theories are considered completely wrong in their geocentric orbital theories. BUT if the deferent and epicycle paths are made ecliptics the results would not just be close but identical to Copernicus with Kepler.
But Newton’s Theories are considered ‘wrong’ or incomplete in fine detail and stand as corrected by relativity. But still considered correct with appropriate limitations. EX: Newton could never explain the source for the additional energy required to account for the Precession of Mercury, where GR accounts for the precession without any added energy though curved space-time. But short of things like that Newton’s is considered OK.
So what will it mean to physics if QM is Proven to be wrong. Well it depends on how wrong, it will not be able to be shown as in accurate, that much seems to be sure.
So worst case; It would be an accurate and highly successful, useful, and productive analogy. Much as you have to admit both Ptolemy and Tyco theories could have been if they’d cracked the ellipse problem first.
Best case; It in could be ‘wrong’ but able to be ‘completed’ (like Newton’s) with the addition of something currently unknown.
I cannot imagine that the formulas that have been generated by QM based on the real success they have given in practical science would not survive. Either in a new and replacing theory, or as part of the ‘completed’ from of some combined theory(ies).
Now as to what hypothetical proof could show QM wrong, “or falsified”.
I think Dr C would agree, it would take something like Einstein’s “unknown variables” being discovered in a manner that allows them to SUCCESSFULLY make some sort of prediction, that QM is unable to predict or predicts incorrectly based on experimental results. ( I consider Dr C the expert here on the various arguments between entanglement and local variables. Plus any replacing theory would certainly IMO need to explain entanglement.)
Is this a little more on target to your OP hypothetical?
18. Jan 15, 2006
Yes, thank you for taking the time. You seem to take the position that you believe it unlikely that QM, if ever falsified, will not be completely replaced by a deeper theory. I have seen nothing added to this thread to indicate that you are incorrect.
I also found the comments of "selfAdjoint" very useful for they showed exactly the type of experiments that would need to be conducted, which I repeat below:
1. Experimental design to show that reality of wavefunction is R, which would then falsify QM
2. Experimental design to show that reality of wavefunction is ~U, which would falsify QM
3. Experimental design to show that reality of wavefunction is ~L, which would falsify QM
where, R = real, ~U = not unitary (linear), ~L = not local
So, clearly, your comments are well taken, the "answer" to my OP must depend on which of the above experiments first falsify QM, and to what degree and under what conditions (context dependent) they falsify QM. Now, numerous comments have been given that the experiment # 1 above will "never" be found, valid experiments have been conducted, it can be falsified, but never has been falsified. Thus, it would appear that the hypothetical paper in Science will have to address experiments # 2 & 3, or ideally both at the same time.
19. Jan 16, 2006
### reilly
Rade -- Perhaps i'm a bit naive, but I do not understand what the "reality" means in the phrase "reality of the wave function". And, I don't have a clue about how to design the experiments you discuss.
There's virtually no chance that QM will be falsified in the realms in which the theory has been spectacularly successful. But, it's not out of the question that QM could have problems at very, very high energy; there are certainly issues surrounding QM and gravity, Generally, the new stuff tends to come from folks who push the boundaries, go over the top into new areas, and so forth.
Regards,
Reilly Atkinson
20. Jan 16, 2006
### JesseM
What do you mean by "completely replaced" though? Would you say that General Relativity completely replaces Newtonian Gravity, despite the fact that its predictions can be shown to come arbitrarily close to Newtonian predictions in certain well-defined limits? As long as you assume that all the existing experiments which have confirmed QM's predictions will continue to do so when repeatedly performed in the future, and that the only way QM could be shown wrong would be in making wrong predictions about new experiments that no one has performed yet, it seems that any new theory that replaces QM will have to reproduce quite a lot of the predictions of QM in a variety of circumstances.
If R is just supposed to represent "realism", the idea that particles have well-defined properties at all times, doesn't Bohm's interpretation show that it's possible to have a realistic nonlocal theory that reproduces all the same empirical predictions as ordinary QM? Maybe what you mean is that existing evidence shows it's impossible for R to be true and for L to be true.
Last edited: Jan 16, 2006
Similar Discussions: Hypothetical QM Question | {"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": 0.8027818202972412, "perplexity": 1199.3814651399894}, "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-2017-22/segments/1495463607849.21/warc/CC-MAIN-20170524173007-20170524193007-00497.warc.gz"} |
https://indico.cern.ch/event/331032/contributions/1720190/ | Please note that conversion to PDF is currently not available. More details here
# SUSY 2015, 23rd International Conference on Supersymmetry and Unification of Fundamental Interactions
23-29 August 2015
Lake Tahoe
US/Pacific timezone
## Precise knowledge of the Higgs-boson mass in the SM and in the MSSM
25 Aug 2015, 09:30
30m
Mountain Ballroom ()
Plenary Talks
### Speaker
Pietro Slavich (LPTHE Paris)
### Description
The measurement of the Higgs-boson mass at the LHC completes our knowledge of the fundamental parameters of the SM, and imposes important constraints on the parameter space of its SUSY extensions. To properly exploit this long-awaited experimental information, high-precision calculations of the predictions of the SM and its extensions are necessary. In this talk I review recent advances in the precise determination of the Higgs sector in both the SM and the MSSM, and discuss their implications for the energy scale at which new physics should be expected, and for the prospects of discovering SUSY particles at the LHC.
### Primary author
Pietro Slavich (LPTHE Paris) | {"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": 0.8339980840682983, "perplexity": 2417.5589635321426}, "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-2019-47/segments/1573496670389.25/warc/CC-MAIN-20191120010059-20191120034059-00177.warc.gz"} |
http://tex.stackexchange.com/questions/13842/the-graphicx-package | # The graphicx package [closed]
The following may interest the LaTeX3 Team, but I would first like to hear the views of other people (especially that I haven't followed every line of implementation of graphicx).
The following function in the graphicx package has two points of worry, in my experience and view, anyway:
1. The local group found in it is perhaps meant to localize the contents of \toks0, but this can slow down \setkeys if xkeyval's \setkeys is used, which is often the case these days.
2. The contents of \toks0 may change (if by a novice programmer) from the time it is built in this macro and the time it is invoked later (after \setkeys). Another token register (or macro) would have been preferred. Moreover, the conditional in this macro is left open for too long, in my view, and TeX has to keep track of it. Because of pointers and possible re-entrance, xkeyval's \setkeys can be quite expensive.
\def\Gin@ii[#1]#2{%
\def\@tempa{[}\def\@tempb{#2}%
\ifx\@tempa\@tempb
\def\@tempa{\Gin@iii[#1][}%
\expandafter\@tempa
\else
\begingroup
\@tempswafalse
\toks@{\Ginclude@graphics{#2}}%
\setkeys{Gin}{#1}%
\Gin@esetsize
\the\toks@
\endgroup
\fi
}
-
I'm not sure I understand the concern. Why is \setkeys enough slower inside one group more to worry about? Where do you see \toks@ getting used incorrectly? Are you thinking that it could be misused in \setkeys somewhere? Can you give an example of the issue? Which conditional are you seeing here? The only thing related I see is \@tempswafalse which should just \let\if@tempswa\iffalse inside the group. – TH. Mar 18 '11 at 21:36
Um, where is the question here? – Joseph Wright Mar 18 '11 at 22:13
This code was written many years ago, and is unlikely to alter unless there is a clear bug. A bug is not likely to include 'a third-party package might behave badly and mess things up'. Getting any change to the core LaTeX2e stuff is very hard. – Joseph Wright Mar 18 '11 at 22:35
@Ahmed: Why don't you enlighten me. For example, explain what problems your version avoids. The only thing I see is what I mentioned before about \setkeys and then likely only if someone defines a new key via \define@key that uses \toks@ not in a group. – TH. Mar 19 '11 at 1:59
Please direct discussion of this kind to the latex-l mailing list. I don't think this site is the best place for it. – Will Robertson Mar 19 '11 at 4:07
## closed as not a real question by Will Robertson, lockstep, Matthew Leingang, Hendrik Vogt, CaramdirMar 19 '11 at 17:56
It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.
I had many gripes with xkeyval (see CTT among others). In the end I have reverted back to the original definition.
\def\origsetkeys#1#2{% Original def
\def\KV@prefix{KV@#1@}%
\let\@tempc\relax
\KV@do#2,\relax,}
\def\Gin@ii[#1]#2{%
\def\@tempa{[}\def\@tempb{#2}%
\ifx\@tempa\@tempb
\def\@tempa{\Gin@iii[#1][}%
\expandafter\@tempa
\else
\begingroup
\@tempswafalse
\toks@{\Ginclude@graphics{#2}}%
\origsetkeys{Gin}{#1}%<----------
\Gin@esetsize
\the\toks@
\endgroup
\fi}
-
Xkeyval is better than it used to be in this respect -- I remember talking about this with you years ago. I finally convinced Morten to write a patch that allowed this behaviour. – Will Robertson Mar 19 '11 at 4:06 @Will: Thanks, I did not know about that patch. I will revisit xkeyval again, because it is a very usefull package. – Danie Els Mar 19 '11 at 4:18
I have been using the following patch:
\newcommand*\@pdf@strcmp[2]{%
\@nameuse{@\ifnum\pdfstrcmp{\detokenize{#1}}%
{\detokenize{#2}}=\z@ first\else second\fi oftwo}%
}
\def\Gin@ii[#1]#2{%
\@pdf@strcmp[{#2}{%
\Gin@iii[#1][%
}{%
\begingroup
\@tempswafalse
\toks@{\Ginclude@graphics{#2}}%
\setkeys{Gin}{#1}%
\Gin@esetsize
\the\toks@
\endgroup
}%
}
-
This needs e-TeX - no good for graphicx, which only requires TeX82 (as for most of the LaTeX required and tools stuff). – Joseph Wright Mar 18 '11 at 22:33
Actually, it requires pdfTeX of a sufficiently new version: this won't work as-written with either XeTeX or LuaTeX, and will break with relatively-recent XeTeX versions (which don't have \strcmp). – Joseph Wright Mar 18 '11 at 22:47
@Joseph: As you know very well, we can avoid the use of pdfTeX here without difficulty but to the same effect. My concern comes from a personal experience. That was what took me into the code for graphicx. Eg, \@tempswa and \toks0 are used often by many codes and packages without being restricted to local groups. Relying on their previous states after setting keys is not, in my opinion, very safe. However, as suggested earlier, this is perhaps not the right place for this discussion. – Ahmed Musa Mar 21 '11 at 21:42 | {"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": 0.805915117263794, "perplexity": 1786.401839577355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368696383077/warc/CC-MAIN-20130516092623-00051-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/determine-the-horizontal-force-p-that-must-be-applied-perpendicular-to-the-handle-of-the-l-q3228792 | ## Static - Friction
Determine the horizontal force P that must be applied perpendicular to the handle of the lever at A in order to develop a compressive force of 12 kN on the material. Each single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction at all contacting surfaces of the wedges is 0.2, and the coefficient of static friction at the screw is 0.15. Diagram is 8-80 and is located at: http://www.flickr.com/photos/upload/
• Anonymous commented
so is that the final ans ? | {"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": 0.915805995464325, "perplexity": 606.6414983354084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368698289937/warc/CC-MAIN-20130516095809-00048-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://moniker.name/worldmaking/?p=744 | # Hyperbolic Translations
One of the crucial issues in the construction of the implementation of the transversion automaton is how it interpolate its geometric relationships (transversions and rigid body motions) through space to generate a surface in 3D. The input data is pentagonal tiling of the surface unfolded into a (5, 4) hyperbolic tiling with the state of the transversion automaton defined at the vertices of the tiling.
Interpolation the state of the automaton across the tiling is subject to the constraint that any given arc through the tiling represents a complete cycle around some part of the surface and must therefore be a harmonic function of the transversion state.
Ideally, the interpolation scheme chosen will evenly sample the space over which the interpolation is performed in order describe the most surface for the least number of interpolation points. In the case of the transversion automaton, the interpolation space is hyperbolic. As a result, our familiar Euclidean intuition about what constitutes evenly spaced samples doesn’t apply. The most straightforward way to interpolate hyperbolically is to work with a mapping of hyperbolic space into Euclidean space and develop the geometry of interpolation through that mapping. In this case, the mapping is the Poincaré disc model of the hyperbolic plane.
## The Poincaré Disc in Geometric Algebra
In the conformal model of Geometric Algebra (GA), the usual Euclidean axes are augmented with two additional axes representing the point at infinity and the point at the origin. In other words, the point at infinity is explicitly represented in the model of Euclidean space. As a result, new symmetries appear enable exotic transformations that smoothly interpolate between circles and lines or spheres and planes since lines are essentially circles of infinite radius and planes are spheres of infinite radius. Furthermore, translations and rotations collapse into a single representation since translations are really just rotations about the infinite point.
In GA, the particular representation of infinity is fungible and depending on which representation is chosen, a particular geometry will arise. For Euclidean geometry, infinity is taken as a point. For spherical and hyperbolic geometry, infinity is typically taken as the unit sphere where the only difference is that spherical geometry is given by the imaginary dual unit sphere and hyperbolic geometry by the real dual unit sphere. Imaginary spheres have a negative radius while real spheres have a positive radius. In the Poincaré disc model of hyperbolic space, infinity is also represented by a unit sphere in 3D and a 2D sphere (aka a circle) in 2D space. The beauty of all this is that all of our previously acquired knowledge about working with spatial transformations in the conformal model of Euclidean space still apply to hyperbolic space. All that needs to be done is to swap the usual $\infty$ with $o – \frac{\infty}{2}$.
For example, given two points $p$ and $q$, the line through them is given by $L = p\wedge{}q\wedge{}\infty$. The hyperbolic line through them is given by $L = p\wedge{}q\wedge{}(o – \frac{\infty}{2})$. Of course, the hyperbolic line is not a line in the Euclidean sense but instead a circular arc.
The reason for this difference is that hyperbolic space expands as you move away from the center, so the mapping to Euclidean space squishes it toward the unit circle, bending the line into a circle.
## Translations
Translations in hyperbolic space also occur along circular arcs and from a Euclidean point of view move quickly through the center of the disc and slowly toward the perimeter. In Euclidean space, translations along lines are described by $exp(-\infty{}\rfloor{}\frac{L}{2})$ with the equivalent hyperbolic expression being $exp(-e\rfloor{}\frac{L}{2})$ where $e = o – \frac{\infty}{2}$.
The real trick with hyperbolic translations is how to get them into a form that is useful for interpolation purposes. Essentially, we want to calculate a hyperbolic translation that will move between two points in a fixed number of steps. The form of the translation given above won’t do this because the line is not normalized and thus scales the distance covered by translation by a factor. To calculate a hyperbolic translation that moves between two points in N steps, the term $-e\rfloor{}L$ has to be normalized and then scaled by half the hyperbolic distance between the points.
Hyperbolic distance can be calculated in GA by first normalizing the given points in terms of hyperbolic space. The condition for normalization is $-e\dot{}p = -1$. The distance between two points is then $d(p, q) = acosh(1 – \frac{-p}{e\dot{}p}\dot{}\frac{-q}{e\dot{}q})$. The algorithm for computing the interpolating translation is:
1. Construct a translation bivector between $p$ and $q$ as $T = e\rfloor{}(p\wedge{}q\wedge{}e)$.
2. Normalize it $T_n = \frac{T}{\sqrt{T*T}}$.
3. Calculate the distance between $p$ and $q$ using $d(p, q) = acosh(1 – \frac{-p}{e\dot{}p}\dot{}\frac{-q}{e\dot{}q})$.
4. Scale the normalized bivector and exponentiate it $V = exp(T_n\frac{d}{2N})$ where N is the number of steps to take between the points.
V is then the versor used to translate $p$ into $q$ over N steps.
In the image above, the edges of the pentagon are drawn by interpolating between the vertices 20 steps. The pink points are midpoints while the blue divide the edges into quarters. The pink and blue points were calculated with a hyperbolic lerp constructed from the hyperbolic translation algorithm.
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https://hakavuori.fi/publication/web/gradings/section-applications.html | Section4Applications
In this subsection we show how maximal gradings may be used to find some structural information of Lie algebras. We start by studying how maximal gradings reveal the structure of a direct product. A similar result can be found in 1.6.5. of [15].
Consider the Lie algebra $L_{6,22}(1)$ in [2] with basis $\{X_1,\dots,X_6 \}\text{,}$ where the only non-zero bracket relations are
\begin{equation*} [X_1,X_2]=X_5,\;\; [X_1,X_3]=X_6,\;\; [X_2,X_4]=X_6,\;\; [X_3,X_4]=X_5\text{.} \end{equation*}
In a basis $\{Y_1,\dots,Y_6 \}$ adapted to the maximal grading, the bracket relations are
\begin{equation*} [Y_1,Y_2]=Y_3,\;\;[Y_4,Y_5]=Y_6\text{.} \end{equation*}
From these bracket relations one sees more easily that the Lie algebra $L_{6,22}(1)$ is isomorphic to $L_{3,2} \times L_{3,2}\text{,}$ where $L_{3,2}$ is the first Heisenberg Lie algebra.
We say that a split torus $\mathfrak{t}\subset\der(\mathfrak{g})$ is non-degenerate if the intersection of the kernels of the maps $D \in \mathfrak{t}$ is trivial. That is, a split torus is non-degenerate if and only if the $\mathfrak{t}^*$-grading it induces does not have zero as a weight.
We expect that the following result is known even without the non-degeneracy assumption, however we have been unable to locate a reference. We will therefore give a direct proof of the simpler claim.
Denoting $\mathfrak{t} = \mathfrak{t}_1 \times \mathfrak{t}_2\text{,}$ let $D \in C(\mathfrak{t})$ be a diagonalizable derivation in the centralizer $C(\mathfrak{t})\text{.}$ To show the maximality of $\mathfrak{t}\text{,}$ it suffices to show that $D \in \mathfrak{t}\text{.}$ In a basis adapted to the product we may represent
\begin{equation*} D = \begin{bmatrix} E_1 \amp F_1 \\ F_2 \amp E_2 \end{bmatrix}\text{,} \end{equation*}
where $E_1\in \der(\mathfrak{g}_1)\text{,}$ $E_2\in \der(\mathfrak{g}_2)\text{,}$ and $F_1\colon \mathfrak{g}_2\to\mathfrak{g}_1$ and $F_2\colon \mathfrak{g}_1\to \mathfrak{g}_2$ are some linear maps. We are going to demonstrate that $E_1\in \mathfrak t_1\text{,}$ $E_2\in \mathfrak t_2$ and $F_1=F_2=0\text{,}$ which would prove that $D = E_1\times E_2 \in \mathfrak t\text{.}$
Let $D_1 \in \mathfrak t_1\text{.}$ By assumption $D$ commutes with $D_1\times 0 \in \mathfrak t\text{,}$ so a simple computation shows that $E_1$ commutes with $D_1$ and $D_1 F_1 = 0\text{.}$ Since $D_1$ is arbitrary, we obtain $E_1 \in C(\mathfrak t_1) \text{.}$ From the fact that $D_1F_1 = 0$ for every $D_1 \in \mathfrak t_1$ we get
\begin{equation*} \operatorname{Im}(F_1) \subset \bigcap_{D_1\in \mathfrak t_1} \ker(D_1) = \{0\}, \end{equation*}
where the last equality follows from the non-degeneracy of $\mathfrak t_1\text{.}$ Consequently, $F_1 = 0\text{.}$
A similar argument shows that $E_2 \in C(\mathfrak t_2)$ and $F_2=0\text{.}$ Since $D$ is assumed diagonalizable, it follows that $E_1$ and $E_2$ are diagonalizable. Then by maximality of $\mathfrak t_1$ and $\mathfrak t_2$ we have $E_1\in \mathfrak t_1$ and $E_2\in \mathfrak t_2\text{,}$ which shows that $D = E_1\times E_2 \in \mathfrak t\text{.}$
For gradings, the above lemma implies the following. Suppose $\mathcal V : \mathfrak g_1 = \bigoplus_{\alpha \in A } V_\alpha$ and $\mathcal W : \mathfrak g_2 = \bigoplus_{\beta \in B } W_\beta$ are maximal gradings of Lie algebras $\mathfrak g_1$ and $\mathfrak g_2\text{,}$ and suppose zero is not a weight for either $\mathcal{V}$ or $\mathcal{W}\text{.}$ Then
$$\mathcal V \times \mathcal W : \Bigl(\bigoplus_{(\alpha,0) \in A \times B} V_\alpha \times \{0\} \Bigr) \oplus \Bigl( \bigoplus_{(0,\beta) \in A \times B} \{0\} \times W_\beta \Bigr)\label{eq-product-grading}\tag{4.1}$$
is a maximal grading of $\mathfrak g = \mathfrak g_1 \times \mathfrak g_2 \text{.}$ Indeed, the gradings $\mathcal{V}$ and $\mathcal{W}$ are the universal realizations of gradings induced by the respective maximal split tori $\mathfrak{t}_1$ and $\mathfrak{t}_2$ of the Lie algebras $\mathfrak{g}_1$ and $\mathfrak{g}_2\text{.}$ By Lemma 4.2, the product torus $\mathfrak t_1 \times \mathfrak t_2$ is maximal. The universal realization of the grading induced by $\mathfrak t_1 \times \mathfrak t_2$ is equivalent to the product grading (4.1).
For a grading $\mathfrak{g} = \bigoplus_{\alpha \in \Omega} V_\alpha \text{,}$ consider the graph with vertices $\Omega$ defined as follows: Whenever $0 \neq [V_\alpha, V_\beta] \subset V_\gamma \text{,}$ we define edges between all the three vertices $\alpha,\beta,\gamma \in \Omega\text{.}$ If the graph $\Omega$ admits a partition $\Omega = \Omega_1 \sqcup \Omega_2$ such that no edges exist between $\Omega_1$ and $\Omega_2\text{,}$ then the Lie algebra $\mathfrak{g}$ is a direct product of the ideals $\mathfrak{g}_1 = \bigoplus_{\alpha \in \Omega_1} V_\alpha$ and $\mathfrak{g}_2 = \bigoplus_{\beta \in \Omega_2} V_\beta\text{.}$ In this situation we say the grading $\mathcal{V}$ detects the product structure $\mathfrak{g}_1 \times \mathfrak{g}_2$ of the Lie algebra $\mathfrak{g}\text{.}$ We gather the observations made above into the following proposition.
We remark that while maximal gradings are able to detect product structures as indicated above, they are not able to detect some other algebraic properties. The Lie algebra $L_{6,24}(1)$ in [2] provides examples of two such phenomena. First, the layers of its maximal grading are not contained in the terms of its lower central series (this behavior can be also achieved by examples where the maximal grading is very coarse). Secondly, this Lie algebra has a “nice” basis (see [10] for the precise definition and its motivation), but it can be shown that no basis adapted to a maximal grading is nice.
Despite these negative results, maximal gradings have another structural application in simplifying the problem of deciding whether two Lie algebras are isomorphic or not.
Remark4.4.
If two Lie algebras $\mathfrak g_1$ and $\mathfrak g_2$ are isomorphic, then any isomorphism maps the maximal grading of $\mathfrak{g}_1$ to a maximal grading of $\mathfrak{g}_2\text{.}$ Therefore, if the maximal gradings of $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are given, then deciding if $\mathfrak{g}_1$ and $\mathfrak{g}_2$ are isomorphic reduces to determining the existence of an isomorphism between the maximal gradings. In many cases this is significantly easier than naively solving the original isomorphism problem. For example, in low dimensions, the majority of the layers of the maximal grading are one-dimensional, in which case searching for possible isomorphisms becomes a combinatorial problem.
Subsection4.2Classification of gradings in low dimension
Following the strategy outlined in Section 3.7 of [22], we classify torsion-free gradings, i.e., gradings that admit a torsion-free realization, in nilpotent Lie algebras of dimension up to 6 over $\mathbb{C}\text{.}$ The main part of the classification is the construction of a maximal grading (Algorithm 3.12) and the enumeration of torsion-free gradings (Proposition 2.25). Here we will give a brief overview of the gradings of each Lie algebra.
We give a complete listing of the universal realizations for the 669 equivalence classes of gradings for the 46 complex Lie algebras of dimension up to 6 in [19]. We also include a similar listing for an extensive (but incomplete) family of 7 dimensional Lie algebras over $\mathbb{C}\text{.}$ The listing in dimension 7 is incomplete because there are a few uncountable families of 7 dimensional complex Lie algebras that depend on a complex parameter $\lambda\text{.}$ In these cases, following the study carried out in [24], we focus on those singular values of $\lambda$ for which either the Lie algebra cohomology or the adjoint cohomology have different dimensions compared to the rest of the Lie algebras in the same family. We will also include a few examples corresponding to non-singular values.
As a starting point we used the classifications of nilpotent Lie algebras given in [11] for dimensions less than 6, [2] for dimension 6, and [17] for dimension 7. The classification up to dimension 6 has a pre-existing computer implementation in the GAP package [3]. However these Lie algebras are not always given in a basis adapted to any maximal grading, so we first compute the maximal grading using the methods described in Subsection 3.3 and switch to a basis adapted to the resulting grading.
The presentations we use for the nilpotent Lie algebras up to dimension 6 are listed in Table 4.5. The Lie brackets $[Y_a,Y_b]=Y_c$ are listed in the condensed form $ab=c\text{.}$ Lie algebras $\mathfrak{g}\times\mathbb{R}^d$ with abelian factors have identical structure coefficients with the nonabelian factor $\mathfrak{g}$ and are omitted from the list. For example $L_{4,2}=L_{3,2}\times\mathbb{R}$ has the basis $Y_1,\ldots,Y_4$ with the bracket relation $[Y_1,Y_2]=Y_3$ from $L_{3,2}\text{.}$
With all the maximal gradings computed, we enumerate all torsion-free gradings as in Proposition 2.25. For the classification up to equivalence, we first introduce some easy-to-check invariants for gradings. Recall that by Lemma 2.14, the grading groups of the obtained gradings are isomorphic to some groups $\mathbb{Z}^k\text{.}$ The dimension $k$ is called the rank of the grading. We recall also an invariant from Section 3.2 of [22]: The type of a grading is the tuple $(n_1,n_2,\ldots,n_k)\text{,}$ where $k$ is the dimension of the largest layer, and each $n_i$ is the number of $i$-dimensional layers.
From the full list of torsion-free gradings, we initially collect together gradings using the following criteria:
1. The ranks of the gradings are equal.
2. The types of the gradings are equal.
3. There exists a homomorphism between the grading groups of the universal realizations mapping layers to layers of equal dimensions.
In this way we get for each Lie algebra families $I_1,I_2,\ldots,I_k$ of gradings such that the gradings of $I_i$ and $I_j$ are not equivalent for $i\neq j\text{.}$
To compute the precise equivalence classes, we naively check if the gradings within each family $I_i$ are equivalent. For each pair of an $A$-grading $\mathfrak{g}=\bigoplus_{\alpha\in A} V_\alpha$ and a $B$-grading $\mathfrak{g}=\bigoplus_{\beta\in B} W_\beta\text{,}$ there are usually only a few homomorphisms $f\colon A\to B$ with $\dim V_\alpha=\dim W_{f(\beta)}\text{.}$ For each such homomorphism $f\text{,}$ we need to check whether there exists an automorphism $\Phi\in\Aut(\mathfrak{g})$ such that $\Phi(V_\alpha)=W_{f(\beta)}$ for all weights $\alpha\text{.}$ These identities define a system of quadratic equations. Since we are working over an algebraically closed field, the system has no solution if and only if 1 is contained in the ideal defined by the polynomial equations. The dimensions of the layers are generally quite small in the cases we need to check, so Gröbner basis methods work well.
For nilpotent Lie algebras of dimension up to 6, an overview of our classification of gradings is compiled in Table 4.6. For each Lie algebra, we list its label in the classification of [2], the rank of its maximal grading ($k$), whether it is stratifiable or not (s?), the number of gradings (#), and the number of gradings with a positive realization ($\#\mathbb{Z}_+$).
We present our method of classifying gradings explicitly in the simple case of the Lie algebra $L_{4,2}=L_{3,2}\times\mathbb{R}$ given in the basis $Y_1,\ldots,Y_4$ with the only bracket $[Y_1,Y_2]=Y_3\text{.}$ The maximal grading is over $\mathbb{Z}^3$ with the layers
\begin{align*} V_{(1,0,0)}\amp=\langle Y_1\rangle,\amp V_{(0,1,0)}\amp=\langle Y_2\rangle,\amp V_{(1,1,0)}\amp=\langle Y_3\rangle,\amp V_{(0,0,1)}\amp=\langle Y_4\rangle\text{.} \end{align*}
Ignoring scalar multiples, the difference set $\Omega-\Omega$ of weights consists of the 6 elements $e_1\text{,}$ $e_2\text{,}$ $e_1-e_2\text{,}$ $e_1-e_3\text{,}$ $e_2-e_3\text{,}$ and $e_1+e_2-e_3\text{,}$ where $e_1,e_2,e_3$ are the standard basis elements of $\mathbb{Z}^3\text{.}$ Subsets of these points span the trivial subspace, 6 one-dimensional subspaces, 7 two-dimensional subspaces ${\langle e_1,e_2\rangle}\text{,}$ ${\langle e_1,e_3 \rangle}\text{,}$ ${\langle e_2,e_3 \rangle}\text{,}$ ${\langle e_1 - e_3,e_2 \rangle}\text{,}$ ${\langle e_1,e_2 - e_3 \rangle}\text{,}$ ${\langle e_1 - e_3,e_2 - e_3 \rangle}\text{,}$ ${\langle 2e_1 - e_3,2e_2 - e_3 \rangle}\text{,}$ and the full space $\mathbb{Z}^3\text{.}$
In this case, each of these 15 subspaces $S$ defines a torsion-free quotient $\mathbb{Z}^3/S\text{.}$ For instance parametrizing the quotient $\pi\colon\mathbb{Z}^3\to\mathbb{Z}^3/\langle e_1 - e_3,e_2-e_3 \rangle$ as $\mathbb{Z}$ using the complementary line $\mathbb{Z}e_3$ gives the weights
\begin{equation*} \pi(e_1) = \pi(e_2) = \pi(e_3) = 1,\quad \pi(e_1+e_2) = 2\text{,} \end{equation*}
so a push-forward grading for the quotient $\mathbb{Z}^3/\langle e_1 - e_3,e_2-e_3 \rangle$ is the $\mathbb{Z}$-grading
\begin{equation*} V_1 = \langle Y_1, Y_2, Y_4\rangle, \quad V_2=\langle Y_3\rangle\text{.} \end{equation*}
To determine the distinct equivalence classes out of the 15 gradings, we first consider the simple criteria listed earlier. The trivial grading and the maximal grading are distinguished by the rank. The six $\mathbb{Z}^2$-gradings all have 2 one-dimensional layers and 1 two-dimensional layer. There exists a homomorphism that preserves the dimensions of the layers for two pairs of the gradings: one between the quotients by $\langle e_1 \rangle$ and $\langle e_2\rangle\text{,}$ and one between the quotients by $\langle e_1-e_3\rangle$ and $\langle e_2-e_3\rangle\text{.}$
Out of the seven $\mathbb{Z}$-gradings, the four quotients by
\begin{equation*} {\langle e_1,e_2\rangle}, {\langle e_1 - e_3,e_2 \rangle}, {\langle e_1,e_2 - e_3 \rangle}, {\langle e_1 - e_3,e_2 - e_3 \rangle} \end{equation*}
define gradings with 1 one-dimensional layer and 1 three-dimensional layer, and the three quotients by
\begin{equation*} {\langle e_1,e_3 \rangle}, {\langle e_2,e_3 \rangle}, {\langle 2e_1 - e_3,2e_2 - e_3 \rangle} \end{equation*}
define gradings with 2 two-dimensional layers. In both families there is exactly one pair of gradings admitting a homomorphism: the pair $\langle e_1 - e_3,e_2 \rangle$ and $\langle e_1,e_2 - e_3 \rangle\text{,}$ and the pair $\langle e_1,e_3 \rangle$ and $\langle e_2,e_3 \rangle\text{.}$
In all of these cases, the homomorphism between the quotients is induced by the isomorphism $f\colon\mathbb{Z}^3\to\mathbb{Z}^3$ swapping $e_1$ and $e_2\text{.}$ All of the mentioned pairs of $\mathbb{Z}^2$- and $\mathbb{Z}$-gradings are in fact equivalent, since there is a corresponding Lie algebra automorphism swapping the basis elements $Y_1$ and $Y_2$ that preserves the subspaces $\langle Y_3\rangle$ and $\langle Y_4\rangle\text{.}$ This reduces the list of 15 gradings down to 11 distinct equivalence classes. Universal realizations for each equivalence class of torsion-free gradings are listed in Table 4.8.
Subsection4.3Enumerating Heintze groups
In this section, we present how our complete list of gradings for a given nilpotent Lie algebra $\mathfrak{g}$ can be used to determine a list of Heintze groups over $\mathfrak{g}\text{.}$
Definition4.9.
A Heintze group is a simply connected Lie group over $\mathbb{R}$ whose Lie algebra is a semidirect product of a nilpotent Lie algebra $\mathfrak{g}$ and $\mathbb{R}$ via a derivation $\alpha \in \der(\mathfrak{g})$ whose eigenvalues have strictly positive real parts.
Positive gradings for a given Lie algebra are naturally identified with diagonalizable derivations with strictly positive eigenvalues, see Subsection 2.3. Hence, to any positively graded Lie algebra $\mathfrak{g}$ we may associate a Heintze group over $\mathfrak{g}\text{.}$ We shall call these groups diagonal Heintze groups.
The quasi-isometric classification of Heintze groups reduces to the study of so called purely real Heintze groups, for which the associated derivation has real eigenvalues. Note that diagonal Heintze groups form a subset of purely real Heintze groups. By [9] the quasi-isometric classification problem of diagonal Heintze groups reduces to an algebraic problem of finding all the possible derivations defining non-isomorphic Heintze groups. Proposition 4.10 is a tool for tackling this algebraic problem using positive gradings. We will prove this result later in this section after discussing its role in the enumeration of Heintze groups.
The enumeration of positive gradings we have established immediately gives the corresponding enumeration of diagonal Heintze groups over $\mathfrak{g}\text{.}$ The enumeration of positive gradings can be understood in two different ways, see Remark 3.11. The corresponding enumeration of Heintze groups has similar character: it is either a parametrization via the projections or a finite list that does not contain all the isomorphism classes of Heintze groups but a representative for each family in terms of the layers. If one is able to eliminate equivalent gradings from the enumeration of positive gradings, then by Proposition 4.10 the corresponding list of Heintze groups does not contain isomorphic Heintze groups.
Remark4.11.
The enumeration of Heintze groups has a few caveats:
1. Already over $\mathfrak{g}=\mathbb{R}^2$ there are uncountably many isomorphism classes of Heintze groups due to the multitude of possible projections $\mathbb{Z}^2 \to \mathbb{R}\text{.}$ The same is true more generally whenever the maximal torus has dimension at least 2.
2. Our methods are in general able to find maximal gradings only for Lie algebras over algebraically closed fields. On the contrary, the base field of Heintze groups is $\mathbb{R}\text{.}$
Before proving Proposition 4.10, we need the following lemmas.
Let $Y_1,\ldots,Y_n$ be a basis of $\mathfrak{g}$ that diagonalizes $\delta\text{.}$ Fix some $Y=Y_i$ and let $w_Y$ and $w_X$ be the eigenvalues of the eigenvectors $Y$ and $X\text{.}$ Since $\delta$ is a derivation and $X$ and $Y$ are eigenvectors, the vectors $\ad_X^kY$ are also eigenvectors, and have the eigenvalues $kw_X+w_Y\text{.}$ Using this fact, and recalling that $\Ad_{\exp(X)} = e^{\ad_X}\text{,}$ see Proposition 1.91 of [21], we compute
By cancellation of $\Ad_{\exp(X)}$ and $\Ad_{\exp(-X)}\text{,}$ the claimed formula follows.
Since the map $\Ad_{\exp(X)}$ is a Lie algebra homomorphism and is the inverse of $\Ad_{\exp(-X)}\text{,}$ we have
for every $Z\in\mathfrak{g}\text{.}$
For a vector $X\in\mathfrak{g}\text{,}$ denote by $C_X\colon\der(\mathfrak{g})\to\der(\mathfrak{g})$ the conjugation map
Let $X_1,\ldots,X_n$ be a basis of $\mathfrak{g}$ that diagonalizes $\delta\text{.}$ Consider the map
\begin{equation*} \Phi\colon \mathbb{R}^n\to \der(\mathfrak{g}),\quad \Phi(x_1,\ldots,x_n)=C_{x_nX_n}\circ\cdots\circ C_{x_1X_1}(\delta)\text{.} \end{equation*}
By repeated application of Lemma 4.12 and Lemma 4.13, it follows that $\Phi(x) = \delta - \ad_{\phi(x)}\text{,}$ where $\phi\colon \mathbb{R}^n\to\mathfrak{g}$ is the map
Since the composition of conjugations is a conjugation, it suffices to prove that the map $\phi$ is surjective.
Let $w_1,\ldots,w_n\gt 0$ be the eigenvalues of the vectors $X_1,\ldots,X_n$ for the derivation $\delta\text{.}$ Since the maps $x_i\mapsto \operatorname{sign}(x_i)\abs{x_i}^{w_i}$ are all invertible, the map $\phi\colon \mathbb{R}^n\to\mathfrak{g}$ is surjective if and only if the map $\tilde{\phi}\colon \mathbb{R}^n\to\mathfrak{g}$ defined by
$$\tilde{\phi}(x_1,\ldots,x_n)=\phi(\operatorname{sign}(x_1)\abs{x_1}^{w_1},\ldots,\operatorname{sign}(x_n)\abs{x_n}^{w_n})\label{eq-rescaled-adjoint-sum}\tag{4.3}$$
is surjective.
Let $D_\lambda\in\Aut(\mathfrak{g})\text{,}$ $\lambda\gt 0\text{,}$ be the one-parameter family of dilations defined by the derivation $\delta\text{.}$ Then for each $i=1,\ldots,n$ the dilation is given by $D_\lambda(X_i)=\lambda^{w_i}X_i$ and we have the dilation equivariance
Applying the above equivariance to the definition (4.3) we find that the map $\tilde{\phi}$ is $D_\lambda$-homogeneous, i.e., $\tilde{\phi}(\lambda x) = D_\lambda(\tilde{\phi}(x))$ for all $x\in\mathbb{R}^n$ and $\lambda\gt 0\text{.}$ Consequently the map $\tilde{\phi}$ is surjective if and only if it is open at zero. Since the change of parameters in (4.3) is a homeomorphism, the same is true also for the map $\phi\text{.}$
By the definition (4.2), the map $\phi$ is smooth. The derivative of each summand $\Ad_{\exp(x_nX_n)}\cdots\Ad_{\exp(x_{i+1}X_{i+1})}\delta(x_iX_i)$ at zero is the map $x\mapsto \delta(x_iX_i)\text{,}$ so the derivative $D_0\phi$ of the map $\phi$ at zero is
\begin{equation*} D_0\phi(x_1,\ldots,x_n) = \delta(x_1X_1+\cdots+x_nX_n)\text{.} \end{equation*}
By the strictly positive eigenvalue assumption, the map $\delta$ is invertible. Since $X_1,\ldots,X_n$ is a basis of $\mathfrak{g}\text{,}$ it follows that the map $\phi$ is open at zero, concluding the proof.
Rescaling the derivations by a scalar, we may assume the smallest of the eigenvalues for both the derivations to be 1. Since the Heintze groups are assumed to be isomorphic, it is straightforward to see that there is a vector $X \in \mathfrak{g}$ so that the derivation $\alpha$ is conjugate by a Lie algebra automorphism of $\mathfrak{g}$ to the derivation $\beta + \ad_X\text{.}$ We use Lemma 4.14 to find that actually $\alpha$ and $\beta$ are conjugate. Applying Lemma 2.19i to the split tori spanned by $\alpha$ and $\beta$ gives the desired result.
The following example shows that Lemma 4.14 is false for some derivations with eigenvalues of different signs.
Consider the Engel Lie algebra, i.e., the 4-dimensional Lie algebra given by the basis $X_1,X_2,X_3,X_4$ satisfying the bracket relations
\begin{equation*} [X_1,X_2]=X_3\;,\;[X_1,X_3]=X_4\text{.} \end{equation*}
Consider in this basis the derivations $\alpha = \operatorname{diag}(-3,1,-2,-1)$ and $\beta = \alpha + \ad_{X_1}\text{.}$ Then $\beta$ is similar to $\alpha\text{,}$ but the similarity is not an automorphism.
Subsection4.4Bounds for non-vanishing $\ell^{q,p}$ cohomology
Knowing all the possible positive gradings of a nilpotent Lie algebra $\mathfrak{g}$ has one further application in the realm of quasi-isometric classifications. Different positive gradings can be used to obtain better estimates in the computation of the $\ell^{q,p}$ cohomology of a nilpotent Lie group, which is a well-known quasi-isometry invariant.
By definition, the $\ell^{q,p}$ cohomology of a Riemannian manifold with bounded geometry is the $\ell^{q,p}$ cohomology of every bounded geometry simplicial complex quasi-isometric to it. A crucial result of [25] shows that in the case of contractible Lie groups, the $\ell^{q,p}$ cohomology of the manifold is isomorphic to its $L^{q,p}$ cohomology.
Definition4.16.
The $L^{q,p}$ cohomology of a nilpotent Lie group $G$ is defined as
\begin{equation*} L^{q,p}H^\bullet(G)=\frac{\lbrace \text{closed forms in }L^p\rbrace}{d\big(\lbrace \text{forms in }L^q\rbrace\big)\cap L^p}\text{.} \end{equation*}
In Theorem 1.1 of [25] it is shown that the Rumin complex constructed on a Carnot group allows for sharper computations regarding $L^{q,p}H^\bullet(G)$ when compared to the usual de Rham complex. Defining and reviewing the properties of the Rumin complex $(E_0^\bullet,d_c)$ goes beyond the scope of this paper. For the following discussion, it is sufficient to know that the space of Rumin $h$-forms $E_0^h$ is a subspace of the space of smooth differential $h$-forms of the underlying nilpotent Lie group $G\text{.}$
Definition4.17.
Let us consider a positive grading $\mathcal{V} : \mathfrak{g}=\bigoplus_{\alpha\in\mathbb{R}}V_\alpha\text{.}$ Then a left-invariant 1-form $\theta$ has weight $\alpha\text{,}$ that is $w(\theta)=\alpha\text{,}$ if $\theta=X^\ast$ for $X \in V_\alpha\text{.}$ In other words, $\theta$ is the dual of a vector field belonging to the subspace $V_\alpha$ at the identity of the group. In general, given a left-invariant $h$-form, we will say that it has weight $p$ if it can be expressed as a linear combination of left-invariant $h$-forms $\theta_{i_1,\ldots,i_h} = \theta_{i_1}\wedge\cdots\wedge\theta_{i_h}$ such that $w(\theta_{i_1})+\cdots+w(\theta_{i_h})=p\text{.}$
Given a positive grading $\mathcal V :\mathfrak{g} = \bigoplus_{\alpha \in \mathbb{R}} V_\alpha\text{,}$ we call the quantity
\begin{equation*} Q = \sum_{\alpha \in \mathbb{R}_+}\alpha \dim V_\alpha \end{equation*}
the homogeneous dimension of $\mathcal V\text{.}$ We also define for each degree $h$ the number
\begin{gather*} \delta N_{\min}(h)= \min_{\theta\in E_0^h} w(\theta)-\max_{\tilde{\theta}\in E_0^{h-1}}w(\tilde{\theta})\,. \end{gather*}
The following is Theorem 1.1(ii) of [25].
Moreover, in Theorem 9.2 of the same paper it is shown how the non-vanishing statement has a wider scope, as it can be applied to Carnot groups equipped with a homogeneous structure that comes from a positive grading. This result has been further extended in [30] to arbitrary positively graded nilpotent Lie groups.
A natural question that stems from these considerations is whether it is possible to identify which choice of positive grading will yield the best interval for non-vanishing cohomology. This problem can be easily presented in terms of maximising the value of the fraction $\delta N_{\min}(h)/Q$ among all the possible positive gradings for a given Lie group $G\text{.}$
Let us describe the maximization procedure in more detail. Let $\mathcal W :\mathfrak{g}=\bigoplus_{n\in \mathbb{Z}^k}W_n$ be a maximal grading of $\mathfrak g$ and let $\Omega$ be the set of weights of $\mathcal{W}\text{.}$ For $\mathbf{a}=(a_1,\ldots,a_k)\in\mathbb{R}^k\text{,}$ let $\pi^{\mathbf a} \colon \mathbb{Z}^k \to \mathbb{R}$ be the projection given by $\pi^{\mathbf a}(e_i)=a_i\text{.}$ Let
$$\positiveset = \{\mathbf{a}\in\mathbb{R}^k: \pi^{\mathbf a}(n)\gt 0\,\forall n\in\Omega \}\text{.}\label{eq-positive-set}\tag{4.4}$$
The push-forward $\pi^{\mathbf a}_*(\mathcal W)$ is a positive grading if and only if $\mathbf{a}\in\positiveset\text{.}$
In the sequel we shall identify any positive grading of $\mathfrak{g}$ with the corresponding vector $\mathbf{a}\in\positiveset\text{.}$ In particular, if $\theta$ is the dual of $X\in W_n\text{,}$ then the weight of $\theta$ with respect to the grading $\pi^{\mathbf a}_*(\mathcal W)$ is $w(\theta)_{\mathbf{a}} = \pi^\mathbf{a}(n)\text{.}$ Then we want to find the value of the following expression for each degree $h\text{:}$
\begin{gather*} \max_{\mathbf{a}\in\positiveset}\bigg\{\frac{\min_{\theta\in E_0^h}w(\theta)_\mathbf{a}-\max_{\tilde{\theta}\in E_0^{h-1}}w(\tilde{\theta})_{\mathbf{a}}}{Q_{\mathbf{a}}}\bigg\}\text{,} \end{gather*}
where $Q_{\mathbf{a}}$ is the homogeneous dimension of $\pi^{\mathbf a}_*(\mathcal W)\text{.}$
A problem of this form can be converted into a linear optimization problem as follows:
1. replace $\min_{\theta\in E_0^h} w(\theta)_\mathbf{a}$ with a new variable $x\text{,}$ and add the constraint $x\le w(\theta)_\mathbf{a}$ for each $\theta\in E_0^h\text{;}$
2. replace $\max_{\tilde{\theta}\in E_0^{h-1}}w(\tilde{\theta})$ with a new variable $y\text{,}$ and add the constraint $y\ge w(\tilde{\theta})_\mathbf{a}$ for each $\tilde{\theta}\in E_0^{h-1}\text{;}$
3. normalize the expression by imposing $Q_\mathbf{a}=1.$
We are then left with the following expression for our original maximization problem
which can easily be solved by a computer, yielding the optimal bound for non-vanishing cohomology using the method of Theorem 4.18.
Let us consider the non-stratifiable Lie group $G$ of dimension 6, whose Lie algebra is denoted as $L_{6,10}$ in [11], with the non-trivial brackets
\begin{equation*} [X_1,X_2]=X_3\;,\;[X_1,X_3]=[X_5,X_6]=X_4\text{.} \end{equation*}
The space of Rumin forms in $G$ is
\begin{align*} E_0^1\amp=\langle\theta_1,\theta_2,\theta_5,\theta_6\rangle;\\ E_0^2\amp=\langle\theta_{5,6}-\theta_{1,3},\theta_{1,5},\theta_{1,6},\theta_{2,3},\theta_{2,5},\theta_{2,6}\rangle;\\ E_0^3\amp=\langle\theta_{2,5,6}+\theta_{1,2,3},\theta_{2,3,5},\theta_{2,3,6},\theta_{1,3,4}-\theta_{4,5,6},\theta_{1,4,5},\theta_{1,4,6}\rangle. \end{align*}
For the Lie algebra $L_{6,10}\text{,}$ the maximal grading is over $\mathbb{Z}^3$ with the layers
\begin{align*} V_{(0,1,0)}\amp=\langle X_1\rangle,\amp V_{(0,0,1)}\amp=\langle X_2\rangle,\amp V_{(0,1,1)}\amp=\langle X_3\rangle\\ V_{(0,2,1)}\amp=\langle X_4\rangle,\amp V_{(1,0,0)}\amp=\langle X_5\rangle,\amp V_{(-1,2,1)}\amp=\langle X_6\rangle\text{.} \end{align*}
The family of projections $\pi^{\mathbf{a}} \colon \mathbb{Z}^3 \to \mathbb{R}$ giving positive gradings is parametrized by $(a_1,a_2,a_3)=\mathbf{a}\in\positiveset$ as in (4.4). The weights of left-invariant 1-forms are
\begin{align*} w(\theta_1)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(0,1,0)=a_2;\\ w(\theta_2)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(0,0,1)=a_3;\\ w(\theta_3)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(0,1,1)=a_2+a_3;\\ w(\theta_4)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(0,2,1)=2a_2+a_3;\\ w(\theta_5)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(1,0,0)=a_1;\\ w(\theta_6)_{\mathbf{a}}\amp=\pi^{\mathbf{a}}(-1,2,1)=2a_2+a_3-a_1\text{.} \end{align*}
From this computation we get the explicit expression
\begin{equation*} \positiveset = \{\mathbf{a}\in\mathbb{R}^3: a_1\gt 0,\,a_2\gt 0,\,a_3\gt 0,\,-a_1+2a_2+a_3\gt 0 \} \end{equation*}
and the homogeneous dimension $Q_{\mathbf{a}}=6a_2+4a_3\text{.}$
Let us first consider the bound for non-vanishing cohomology in degree 1. We express
\begin{gather*} \max_{\mathbf{a}\in\positiveset}\bigg\{\frac{\delta N_{\min}(1)}{Q_{\mathbf{a}}}\bigg\}=\max_{\mathbf{a}\in\positiveset}\bigg\{\frac{\min\{a_1,a_2,a_3,2a_2+a_3-a_1\}}{6a_2+4a_3}\bigg\}\text{.} \end{gather*}
as the linear optimization problem
\begin{align*} \text{Maximize}\quad \amp x\\ \text{subject to}\quad x\amp\le a_1,\; x\leq a_2,\; x\leq a_3,\\ x\amp\leq 2a_2+a_3-a_1,\\ 1\amp=6a_2+4a_3,\\ a_1\amp,a_2,a_3\gt 0,\; 2a_2+a_3-a_1\gt 0\text{.} \end{align*}
A solver finds the solution $\frac{1}{10}\text{,}$ which is obtained by choosing $a_1=a_2=a_3=\frac{1}{10}\text{.}$ Since the quantity $\frac{\delta N_{\min}(1)}{Q_{\mathbf{a}}}$ is scaling invariant, we find that the grading defined by $a_1=a_2=a_3=1$ gives $\ell^{q,p}H^1(G)\neq 0$ with the optimal bound $\frac{1}{p}-\frac{1}{q}\lt \frac{1}{10}\text{.}$
Similarly, once we re-express
\begin{gather*} \max_{\mathbf{a}\in\positiveset}\bigg\{\frac{\delta N_{\min}(2)}{Q_{\mathbf{a}}}\bigg\} \text{.} \end{gather*}
as a linear optimization problem and feed it into a solver, we get the result $\frac{1}{10}\text{,}$ obtained (up to rescaling) by taking $a_2=a_3=2$ and $a_1=3\text{.}$ Therefore $\ell^{q,p}H^2(G)\neq 0$ for $\frac{1}{p}-\frac{1}{q}\lt\frac{1}{10}\text{.}$
Likewise, we obtain the optimal bound $\frac{1}{p}-\frac{1}{q}\lt\frac{1}{10}$ for $\ell^{q,p}H^3(G)\neq 0$ by taking $a_1=a_2=a_3=1\text{.}$
Finally, by Hodge duality, see Theorem 7.3 of [30], we obtain the optimal bounds for $\ell^{q,p}$ cohomology in complementary degree, that is $\ell^{q,p}H^4(G)\neq 0\text{,}$ $\ell^{q,p}H^5(G)\neq 0\text{,}$ and $\ell^{q,p}H^6(G)\neq 0\text{,}$ for $\frac{1}{p}-\frac{1}{q}\lt\frac{1}{10}\text{.}$
Remark4.20.
Example 9.5 of [25] describes an explicit positive grading in the Engel group that gives an improved bound for the non-vanishing of the $L^{q,p}$ cohomology in degree 2. By a similar computation as the one shown in Example 4.19, one can verify that the value given in Example 9.5 of [25] is indeed the optimal bound. | {"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": 1, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9528873562812805, "perplexity": 257.4606951361705}, "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-2021-04/segments/1610704835901.90/warc/CC-MAIN-20210128040619-20210128070619-00354.warc.gz"} |
http://math.stackexchange.com/questions/188877/name-of-this-division-property | # Name of this division property
Let us take two integers, $a$ and $b$. Let us then take $\lfloor a / b\rfloor = c$ and $a \bmod b = d$. Obviously, it follows that $a = bc + d$. Our professor claimed that this was called the "Fundamental Theorem of Arithmetic" at his high school. Clearly, this isn't the case anymore, if it ever was.
What is this property actually called?
-
For the benefit of other readers, $a/b=c$ is apparently referring to "integer division". Apparently programmers know about this convention but I don't think many mathematicians do. Personally I think this made-up ad-hoc pseudo-operation was invented to confuse and annoy mathematicians. (This is all written jocularly, of course!) – rschwieb Aug 30 '12 at 16:48
In some sense, the "division algorithm" theorem is the definition of the integer operations $a/b$ and $a%b$. – Thomas Andrews Aug 30 '12 at 16:52
Where are you located? The FTA has been pretty well-established as unique factorization for a long time. I knew a prof who said FTA should really be "If $a|bc$ and $a$ and $b$ are relative prime, then $a|c$." But he was an outlier. – Thomas Andrews Aug 30 '12 at 16:55
The result says that for any integers $a$ and $b$, with $b\gt 0$, there exist unique integers $q$ and $r$ such that $0\le r\lt b$ and $a=bq+r$. | {"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": 0.8556188344955444, "perplexity": 540.7549751910211}, "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-2016-18/segments/1461860111592.84/warc/CC-MAIN-20160428161511-00166-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://scicomp.stackexchange.com/questions/8376/computing-sin-pi-2-numerically | # Computing $\sin(\pi/2)$ numerically
I'm trying to understand the types of numerical errors, to do this I want to calculate $\sin(\pi/2)=1$ numerically. To do this I use the Taylor series of $\sin(x)$ in 0: $$\sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+ \ldots$$
To make the calcultaions I did the following Fortran program:
program main
implicit none
real*8 pi, x, res, error
pi=3.14159265
x = pi/2
res = x - x**3/6 + x**5/120 - x**7/5040
error = 1-res
print*, "Result: ",res
print*, "Error: ", error
end
And I get the following results depending of the degreee of the Taylor series and in the decimals of pi.
If $\pi=3.14$
• Taylor 3rd Degree: 0.92501782114084341
• Taylor 5th Degree: 1.0045086615502121
• Taylor 7th Degree: 0.99984349522599403
• Taylor 9th Degree: 1.0000032059098956
• Taylor 11thDegree: 0.99999962708361323
If $\pi=3.14159265$
• Taylor 3rd Degree: 0.92483221907327295
• Taylor 5th Degree: 1.0045248564076874
• Taylor 7th Degree: 0.99984310136039689
• Taylor 9th Degree: 1.0000035425853664
• Taylor 11thDegree: 0.99999994374102952
And I observe that the error gets bigger if I put more decimals in my $\pi$ approximation. Why is this happening? Edit: Adding more terms to the Taylor expansion (11th degree) makes adding more $\pi$ digits better.
Also, How can I make a better approximation? What method will be better to a problem like this one?
How can I predict the size of these errors?
• What happens if you start with an approximation to $\pi$ that is larger than the actual value, such as $3.15$? Sep 1 '13 at 1:18
• @Brian If I start with an approximation of 3.15 and a 3rd degree polynomial, the answer is: 0.92383592604100528 which is worse than with the 3.14 approximation. Sep 1 '13 at 10:45
• Taylor series converge very slowly. What happens if you go to higher orders? Sep 1 '13 at 15:57
• Since this is most likely a homework exercise, I'll simply suggest that you consider two different sources of error in this computation: 1. you've truncated the Taylor series and 2. you've approximated $\pi$. For each of these errors you'll need to find a way to bound the error and then compare the sizes of the two errors (hint: use Taylor's theorem with a remainder term.) Sep 1 '13 at 16:10
• @Guido When I added the 11th degree I get better results with the longer $\pi$ approximation, so that is interesting. Sep 1 '13 at 18:01
The first is that you are looking at finite approximating Taylor polynomials $p_3(x), p_5(x), p_7(x)$ evaluated at $x=\pi/2$. Taylor's theorem says that for analytic functions (which the sine is), $p_N(\pi/2)\rightarrow 1$ as $N\to \infty$. Your results appear to confirm this, at least for the small set of examples you consider.
On the other hand, you are not considering $p_N(\pi/2)$ but $p_N(x)$ for some $x\approx \pi/2$. Now, you expect that $|p_N(x)-1|$ becomes smaller as $x\to \pi/2$, but there is really no reason why that should be so. All you know is that $p_N(x)\to p_N(\pi/2)$ as $x\to \pi/2$. So what to you looks like the error getting larger is because you think that the error is $|p_N(x)-1|$, but that's not correct. The error is in fact $|p_N(x)-p_N(\pi/2)|$ for which you have not shown results.
• That clarifies a lot. Thank you! But how I estimate the error in $\|p_N(x)-p_N(\pi/2)\|$ depending on mi $\pi$ approximation? Sep 1 '13 at 18:58
• @Msegade: You can estimate the error by using that $|p_N(x)-p_N(y)|\approx |p'_N(x)|\;|x-y|$ whenever $x$ and $y$ are close together. If you can evaluate $p_N(x)$, then you can also evaluate $p'_N(x)$. Sep 2 '13 at 3:40 | {"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": 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": 0.9148532152175903, "perplexity": 662.4057542305426}, "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/1634323585280.84/warc/CC-MAIN-20211019171139-20211019201139-00464.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/5957-span-set.html | # Thread: Span of a Set
1. ## Span of a Set
The question is.......
Let u and v be any vectors in R^n. Prove that the spans of {u,v} and {u+v, u-v} are equal.
I'm thinking if they u and v were inverses of each other then this would work, but i'm not sure.
2. Originally Posted by TreeMoney
The question is.......
Let u and v be any vectors in R^n. Prove that the spans of {u,v} and {u+v, u-v} are equal.
I'm thinking if they u and v were inverses of each other then this would work, but i'm not sure.
Let x in Span(u,v), then there exists a, b in R, such that:
x=au+bv.
Now put a'=(a+b)/2 and b'=(a-b)/2, then x=a'(u+v) +b'(u-v) so x is in
Span(u+v, u-v).
A similar argument (left to the reader) will show that if y in Span(u+v, u-v)
then y is also in Span(u,v).
RonL
3. Originally Posted by TreeMoney
The question is.......
Let u and v be any vectors in R^n. Prove that the spans of {u,v} and {u+v, u-v} are equal.
I'm thinking if they u and v were inverses of each other then this would work, but i'm not sure.
We need to show that the sets,
S={au+bv|a,b in R and u,v in R^n}
T={a(u+v)+b(u-v)|a,b in R}
Are equal.
Equivalently, S is subset of T and T is subset of S.
In set T if a=1/2 and b=1/2 then, u is in S
In set T if a=1/2 and b=-1/2 then v is in S.
Thus, their linear combination is in T.
Thus, T contains S.
In set S if a=1 and b=1 then u+v is in T.
In set S if a=1 and b=-1 then u-v is in T.
Thus, their linear combination is in T.
Thus, S contains T.
Thus, T=S | {"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": 0.9061970710754395, "perplexity": 2513.3346097337767}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947957.81/warc/CC-MAIN-20180425193720-20180425213720-00400.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/understand-units-inductance-potential-energy-stored-inductor-consequences-inductance-circu-q3357144 | To understand the units of inductance, the potential energy stored in an inductor, and some of the consequences of having inductance in a circuit. After batteries, resistors, and capacitors, the most common elements in circuits are inductors. Inductors usually look like tightly wound coils of fine wire. Unlike capacitors, which produce a physical break in the circuit between the capacitor plates, the wire of an inductor provides an unbroken continuous path in which current can flow. When the current in a circuit is constant, an inductor acts essentially like a short circuit (i.e., a zero-resistance path). In reality, there is always at least a small amount of resistance in the windings of an inductor, a fact that is usually neglected in introductory discussions. Recall that current flowing through a wire generates a magnetic field in the vicinity of the wire. If the wire is coiled , such as in a solenoid or an inductor, the magnetic field is strongest within the coil parallel to its axis. The magnetic field associated with current flowing through an inductor takes time to create, and time to eliminate when the current is turned off. When the current changes, an EMF is generated in the inductor, according to Faraday's law, that opposes the change in current flow. Thus inductors provide electrical inertia to a circuit by reducing the rapidity of change in the current flow. Inductance is usually denoted by L and is measured in SI units of henries (also written henrys, and abbreviated \rm{H}), named after Joseph Henry, a contemporary of Michael Faraday. The EMF EMF produced in a coil with inductance L is, according to Faraday's law, given by {\cal{E}} = - {L} \frac {\Delta I} {\Delta t}. Here \Delta I/ \Delta t characterizes the rate at which the current I through the inductor is changing with time t. Part A Based on the equation given in the introduction, what are the units of inductance L in terms of the units of EMF, t, and I (respectively volts \rm{V}, seconds \rm{s}, and amperes \rm{A}? | {"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": 0.8977287411689758, "perplexity": 483.4683575103677}, "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-2014-42/segments/1414119646849.7/warc/CC-MAIN-20141024030046-00294-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://rkganti.wordpress.com/2012/01/ | You are currently browsing the monthly archive for January 2012.
We will look a nice application of Carathéodory’s theorem in information theory. Let ${e_i =[0,0,...0,1,0,...,0]}$ essentially a vector with ${1}$ int he ${i}$-th position and zero elsewhere. Before we go to the main example, we require the following lemma.
Lemma 1 Let ${\mathop{\mathbb P}_n}$ denote the “vertices” (extreme points) of the ${\Delta_{n-1}}$ simplex, \ie ${\mathop{\mathbb P}_n =\{e_1,e_2,...,e_n\}}$ are the standard basis of ${{\mathbb R}^n}$. For ${i=1,....k}$, let ${f_i(x):\mathop{\mathbb P} \rightarrow {\mathbb R}}$ be ${k}$ real valued functions on ${\mathop{\mathbb P}_n}$. Also, let ${\beta_i \geq 0, i=1,...,n}$ be such that ${\sum_{i=1}^n \beta_i=1}$. Then there exists ${\alpha_i\geq 0}$, ${i=1,...,k+1}$, ${\sum_{i=1}^{k+1}\alpha_i=1}$ and ${\{\hat{e_1}, ....,\hat{e_{k+1}}\}\subset \mathop{\mathbb P}_n}$ such that
$\displaystyle \sum_{i=1}^n \beta_i f_m(e_i ) = \sum_{i=1}^{k+1}\alpha_i f_m(\hat{e_i}), \ \text{ for } m=1,...,k$
Proof: We have
$\displaystyle \sum_{i=1}^n \beta_i\left[ \begin{array}{c} f_1(e_i)\\ f_2(e_i)\\ \vdots\\ f_k(e_i)\\ \end{array} \right] \in {conv}\left(\left[ \begin{array}{c} f_1(e_1)\\ f_2(e_1)\\ \vdots\\ f_k(e_1)\\ \end{array} \right], \left[ \begin{array}{c} f_1(e_2)\\ f_2(e_2)\\ \vdots\\ f_k(e_2)\\ \end{array} \right] \hdots \left[ \begin{array}{c} f_1(e_n)\\ f_2(e_n)\\ \vdots\\ f_k(e_n)\\ \end{array} \right] \right)$
Hence by Caratheodory’s theorem, there exists scalars ${\alpha_1,...\alpha_{k+1}}$ and ${k+1}$ vectors such that
$\displaystyle \sum_{i=1}^n \beta_i\left[ \begin{array}{c} f_1(e_i)\\ f_2(e_i)\\ \vdots\\ f_k(e_i)\\ \end{array} \right] = \sum_{i=1}^{k+1} \alpha_i\left[ \begin{array}{c} f_1(\hat{e_i})\\ f_2(\hat{e_i})\\ \vdots\\ f_k(\hat{e_i})\\ \end{array} \right],$
proving the theorem. $\Box$
For a more general version of the Lemma look at Lemma 3 in “Source coding with side information and a converse for degraded broadcast channels”
Bounds on the cardinality of channel inputs using Caratheodory’s theorem
We will now look at the capacity theorem for channel capacity of Discrete memoryless channels (DMC). Please see Wikipedia and related links for more introduction on the topic of channel capacity. Refer to the the technical report by M SALEHI for more advanced techniques to bound the cardinality of random variables in Information theory.
Consider a channel with input from the alphabet ${\mathcal{X}}$ and output alphabet ${\mathcal{Y}}$. A channel is specified by a conditional probability distribution for each input, ${P(y|x)}$, \ie the probability of obtaining ${y}$ when ${x}$ is transmitted. For the discussion below, let us assume that the input and the output alphabet are discrete. Shannon proved that the channel capacity of a DMC is
$\displaystyle C= \max_{P_x }I(X; \ Y),$
where the maximization is over all the probability distributions on the input alphabet. Let for notational simplicity that the input alphabet cardinality ${|\mathcal{X}| =n}$ and ${|\mathcal{Y}| =m}$. Observe that in the above optimization problem, the search domain is the set of all discrete probability distrbutions with mass at ${n}$-points, \ie, the search space is the ${n-1}$-simplex. We will now use Caratheodeory theorem to show that this search space can be reduced drastically if the cardinality of the output alphabet ${m}$ is small.
Let ${X}$ and ${Y}$ be two discrete random variables with PMF ${p(x), x\in \mathcal{X}}$ and conditional PMF ${p(y| x), x\in \mathcal{X}, y \in \mathcal{Y}}$. Once the channel ${P(y|x)}$ is specified, fixing the input distribution ${p(x)}$ fixes the output distribution as ${p(y) =\sum_{x\in \mathcal{X}}p(x)p(y|x)}$. So the only flexibility we have is the distribution of the input.
The mutual information between the random variables ${X}$ and ${Y}$ is denoted by ${I(X; Y)}$ and equals
$\displaystyle I(X;Y) = H(Y) - H(Y|X),$
where ${H(Y)}$ is the entropy of the random variable ${Y}$ defined as
$\displaystyle H(Y) =\sum_{y\in \mathcal{Y}}p(y)\log\left(\frac{1}{p(y)}\right),$
and the conditional entropy is defined as
$\displaystyle H(Y|X) =\sum_{x\in \mathcal{X}}p(x)\sum_{y\in \mathcal{Y}}p(y|x)\log\left(\frac{1}{p(y|x)}\right),$
Observe that the entropies and mutual information depend only on the distribution of the random variables. Let ${p^*(x)}$ be the optimal distribution on the input alphabet that achieves the capacity ${C^*}$. As mentioned earlier, this optimal distribution induces a distribution and conditional distribution ${p^*(y)}$ and ${p^*(y|x)}$.
Now we will show that there exists one more probability distribution on ${\mathcal{X}}$ ${P^{**}}$ such that the same mutual information ${C^*}$. However the new distribution will have a non-zero probability only on ${m+1}$ terms of ${\mathcal{X}}$. Observe that if ${m+1 then the search space will reduce.
Since ${p^*(x)}$ is a ${n}$ dimensional (term) probability distribution, it belongs to the ${n-1}$ simplex and hence can be represented as a convex combination of the vertex points, \ie,
$\displaystyle P^*_X = \sum_{i=1}^n \beta_i e_i,$
for some ${\beta_i\geq 0}$ and ${\sum_{i=1}^n \beta_i=1}$ and ${e_i}$ are the standard ${n}$-dimensional basis defined in the previous lemma. Here we have used the notation for the optimal distribution
$\displaystyle P^*_X=\left[ \begin{array}{c} p^*(x_1)\\ p^*(x_2)\\ \vdots\\ p^*(x_n)\\ \end{array} \right]$
where ${x_1,...x_n}$ are the alphabet of ${\mathcal{X}}$. We now consider the following functions on ${\mathop{\mathbb P}_n}$ (see the previous lemma for the definition of ${\mathop{\mathbb P}_n}$)and use the previous lemma to reduce the cardinality. Let ${B(a)=[p(a|x_1),p(a|x_2),....,p(a|x_n)]^T}$. Let ${y_1,y_2,...y_m}$ be the alphabet of the output Define For any vector ${P\in \mathop{\mathbb P}_n}$,
$\displaystyle \begin{array}{rcl} f_{1}(P)&=& P^T B(y_1).\\ f_{2}(P)&=& P^T B(y_2).\\ \vdots &=&\vdots\\ f_{{m-1}}(P)&=& P^T B(y_{m-1}).\\ f_{m}(P)&=& P^T\left [\sum_{y}p(y|x_1)\log\left(\frac{1}{p(y|x_1)}\right),.,\sum_{y}p(y|x_n)\log\left(\frac{1}{p(y|x_n)}\right)\right]^T.\\ \end{array}$
We make the following observation for ${1\leq k\leq m-1}$
$\displaystyle \begin{array}{rcl} \sum_{i=1}^n \beta_i f_{k}(e_i)&=&\sum_{i=1}^n \beta_i( e_i^T B(y_i))\\ &=&\left(\sum_{i=1}^n \beta_i e_i^T\right) B(y_i)\\ &=&P_X^*B(y_i)\\ &=&\sum_{x\in \mathcal{X}}p^*(x) p(y_i|x)\\ & =& p^*(y_i), \end{array}$
where ${ p^*(y_i)}$ is the probability of ${y_i}$ in the optimal distribution of the channel output. Similarly it follows that
$\displaystyle \sum_{i=1}^n \beta_i f_{m}(e_i) = H(Y^*|X^*).$
So we have
$\displaystyle \left[ \begin{array}{c} p^*(y_1)\\ p^*(y_2)\\ \vdots\\ p^*(y_{m-1})\\ H(Y^*|X^*) \end{array} \right]=\left[ \begin{array}{c} \sum_{i=1}^n \beta_i f_{1}(e_i)\\ \sum_{i=1}^n \beta_i f_{2}(e_i)\\ \vdots\\ \sum_{i=1}^n \beta_i f_{m-1}(e_i)\\ \sum_{i=1}^n \beta_i f_{m}(e_i)\\ \end{array} \right]$
So the right hand side corresponds to the optimal distribution ${P^*_Y}$ and the optimal entropy distribution ${H(Y^*|X^*)}$. But by previous lemma, there exists constants ${\alpha_1,...\alpha_m \geq 0}$ and ${\sum_{i=1}^{m+1}\alpha_i =1}$ and ${\tilde{e_1},...,\tilde{e_{m+1}}}$such that
$\displaystyle \left[ \begin{array}{c} \sum_{i=1}^n \beta_i f_{1}(e_i)\\ \sum_{i=1}^n \beta_i f_{2}(e_i)\\ \vdots\\ \sum_{i=1}^n \beta_i f_{m-1}(e_i)\\ \sum_{i=1}^n \beta_i f_{m}(e_i)\\ \end{array} \right]= \left[ \begin{array}{c} \sum_{i=1}^{m+1} \alpha_i f_{1}(\tilde{e_i})\\ \sum_{i=1}^{m+1} \alpha_i f_{2}(\tilde{e_i})\\ \vdots\\ \sum_{i=1}^{m+1} \alpha_i f_{m-1}(\tilde{e_i})\\ \sum_{i=1}^{m+1} \alpha_i f_{m}(\tilde{e_i})\\ \end{array} \right]$
But from the definitions of ${f_i}$, it follows that
$\displaystyle \left[ \begin{array}{c} \sum_{i=1}^{m+1} \alpha_i f_{1}(\tilde{e_i})\\ \sum_{i=1}^{m+1} \alpha_i f_{2}(\tilde{e_i})\\ \vdots\\ \sum_{i=1}^{m+1} \alpha_i f_{m-1}(\tilde{e_i})\\ \sum_{i=1}^{m+1} \alpha_i f_{m}(\tilde{e_i})\\ \end{array} \right]= \left[ \begin{array}{c} (\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T )B(y_1)\\ (\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T )B(y_2)\\ \vdots\\ (\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T )B(y_{m-1})\\ (\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T )D\\ \end{array} \right],$
where ${D=\left [\sum_{y\in \mathcal{Y}}p(y|x_1)\log\left(\frac{1}{p(y|x_1)}\right),.....,\sum_{y\in \mathcal{Y}}p(y|x_n)\log\left(\frac{1}{p(y|x_n)}\right)\right]}$ Hence it follows that choosing the distribution ${(\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T )}$ on ${\mathcal{X}}$ would lead to the same optimal distribution on ${Y}$ as ${Y^*}$ and the same conditional entropy ${H(Y^*|X^*)}$. But observe that the new optimal distribution.
$\displaystyle P^{**}=\sum_{i=1}^{m+1} \alpha_i \tilde{e_i}^T ,$
has positive mass only on ${m+1}$ terms of the PMF. This is because ${e_i}$ has one only one coordinate, the ${i}$-th coordinate. This proves the claim. Hence it is sufficient to search over all the ${n=|\mathcal{X}|}$ probability distributions with positive mass on ${\min\{|\mathcal{X}|, |\mathcal{Y}|+1\}}$ coordinates.
Definition 1 (Convex hull) The set of all convex combinations of points from a set ${A}$ is called the convex hull of ${A}$. It is denoted by ${\text{conv}(A)}$.
Similar to a closure of a set, the convex hull of a set is the smallest convex set containing ${A}$. We will now see some examples.
1. Some basic geometric examples.
2. Birkhoff – Von Neumann Theorem: Consider the set of permutation matrices ${\{X_\sigma, \sigma \in P_n\}}$. Then the convex hull of this set is the set of all doubly stochastic matrices.
Hw 1 Prove that the convex hull of a set is convex.
We will now see an interesting application of convex hull.
Theorem 2 (Gauss-Lucas Theorem) Let ${f(z)}$ be a ${n}$-th degree polynomial with real coefficients. By fundamental theorem of algebra there are ${n}$-roots of the polynomial (they might be complex). Call the roots ${\{z_1 = x_1+i y_1, z_2=x_2+iy_2,...,z_n=x_n+iy_n\}}$ be the set of all the roots. Observe that each root can be represented as a point in the plane ${{\mathbb R}^2}$. Then the roots of the ${n-1}$ degree polynomial ${f'(z) = \frac{d f(z)}{dz}}$ lies in the convex hull ${{conv}(z_1,z_2,...,z_n)}$.
Proof: Without loss of generality the polynomial can be represented as
$\displaystyle f(z)= (z-z_1)(z-z_2)....(z-z_n).$
So we have
$\displaystyle f'(z)= \sum_{k=1}^n \prod_{i\neq k}(z-z_i).$
Let ${w}$ be a root of ${f'(z)}$. Then we have ${f'(w)=0}$ and hence
$\displaystyle f'(w)= \sum_{k=1}^n \prod_{i\neq k}(w-z_i) =0$
Multiplying both sides with ${\prod_{i\neq k}(\bar w-\bar z_i)}$, we obatin
$\displaystyle \sum_{k=1}^n \frac{1}{( w - z_i)} \prod_{i=1}^n |w-z_i|^2 =0$
which implies
$\displaystyle \prod_{i=1}^n |w-z_i|^2\sum_{k=1}^n \frac{1}{( w - z_i)} =0.$
If ${\prod_{i=1}^n |w-z_i|^2 =0}$ then ${w = z_i}$ for some ${i}$ and hence the theorem is proved. Else if
$\displaystyle \sum_{k=1}^n \frac{1}{(w - z_k)} =0.$
which equals
$\displaystyle \sum_{k=1}^n \frac{\bar{w}-\bar{z_k}}{|w - z_k|^2} =0,$
which implies
$\displaystyle \bar{w} = \sum_{k=1}^n\frac{\frac{\bar{z_k}}{|w - z_k|^2}}{\sum_{i=1}^n\frac{\bar{z_k}}{|w - z_i|^2}}.$
Taking the conjugate we get the result ${w =\sum_{i=1}^n z_i \alpha_i}$ where ${\alpha_i =\frac{\frac{1}{|w - z_k|^2}}{\sum_{i=1}^n\frac{\bar{z_k}}{|w - z_i|^2}}}$. $\Box$
We will now look at one more theorem that looks at convex combinations.
Theorem 3 (Caratheodory’s theorem ) Let ${S \subset {\mathbb R}^d}$. Then any point ${x \in {conv}(S)}$ can be represented as a convex combination of ${d+1}$ points of ${S}$
Some observations before the proof:
1. The ${(d+1)}$ points depend on ${x}$ in consideration.
2. ${d+1}$ points are necessary. Let ${S}$ be three non-collinear points. Let ${x}$ be the the centroid of the triangle formed by these three points. Observe that the convex combination of the three points is necessary to obtain ${x}$.
3. Observe that the ${d+1}$ points are from ${S}$ and not ${{conv}(S)}$.
Proof: Any point ${y\in {conv}(S)}$ can be represented as a convex combination of some ${m}$ points in ${S}$. So
$\displaystyle y=\sum_{i=1}^m \alpha_i x_i,$
where ${x_i \in S}$ and ${\sum_{i=1}^m \alpha_i=1}$. If ${m \leq d+1}$ we are done. Hence we will only consider the case of ${m > d+1}$. Let ${x_i =(x_i^1,x_i^2,....,x_i^{d})}$. Now consider the set of equations ${\sum_{i=1}^m\gamma_i x_i=0}$ and ${\sum_{i=1}^m \gamma_i=0}$. This system of equations can be put in matrix form as
$\displaystyle \left[ \begin{array}{cccc} x_1^1 & x_2^1 &...&x_m^d \\ x_1^2 & x_2^2 &...&x_m^d \\ \vdots&\vdots&\vdots&\vdots\\ x_1^d & x_2^d &...&x_m^d \\ 1&1&...&1 \end{array} \right]\left[ \begin{array}{c} \gamma_1\\ \gamma_2\\ \vdots\\ \gamma_m \\ \end{array} \right] =o_{d+1 \times 1}$
Since the matrix is of size ${(d+1)\times m}$, and ${d+1 < m}$, there are non trivial solutions of this equation (look at rank-nullity theorem to observe that ${\dim(\mathcal{N}(A))\geq m-(d+1)>0}$). From now on Let ${\gamma_1, ....,\gamma_m}$ be one such solution. Now let ${\tau >0}$. Since ${\sum_{i=1}^m \gamma_i x_i =0}$, the point ${y}$ can be represented as
$\displaystyle y = \sum_{i=1}^m \alpha_i x_i -\tau \sum_{i=1}^m \gamma_i x_i = \sum_{i=1}^m (\alpha_i -\tau \gamma_i)x_i$
Also observe that ${\sum_{i=1}^m (\alpha_i -\tau \gamma_i) =1}$ since ${\sum_{i=1}^m \gamma_i =0}$. Now we will choose ${\tau}$ such that ${\alpha_i -\tau\gamma_i \geq 0}$ and equals ${0}$ for one term so that ${\sum_{i=1}^m (\alpha_i -\tau \gamma_i)x_i}$ becomes a convex combination. Since ${\sum \gamma_i =0}$ implies some ${\gamma_i}$‘s are positive and others are negative. If for some ${i}$, ${\gamma_i <0}$, then ${\alpha-\tau\gamma_i \geq0}$. So we need to consider only the terms with ${\gamma_i >0}$. For all ${i}$ such that ${\gamma_i >0}$, we want
$\displaystyle \alpha_i-\tau\gamma_i \geq 0,$
which implies
$\displaystyle \frac{\alpha_i}{\gamma_i}-\tau \geq 0.$
Hence if we choose ${\tau =\min\{\frac{\alpha_i}{\gamma_i}, \gamma_i >0\}}$, then one term ${\alpha_i -\tau\gamma_i =0}$ and every other term will be positive. So we are able to express
$\displaystyle y = \sum_{i=1}^m (\alpha_i -\tau \gamma_i)x_i,$
where one of the coefficients is zero, \ie, as a convex combination of ${m-1}$ points. We can continue this procedure till ${m\leq d+1}$ $\Box$
We will now see a simple application of Caratheodory’s theorem.
Lemma 4 If a set ${S\subset {\mathbb R}^d}$ is compact, then its convex hull ${{conv}(S)}$ is also compact.
Proof: Consider a sequence ${\{x_n\} \in {conv}(S)}$. If we show that there is a convergent subsequence which converges to a point in ${{conv}(S)}$ then we are done.
By Caratheodory’s theorem, each ${x_i \in {conv}(S)}$ can be represented as a convex sum of ${d+1}$ points in ${S}$. So
$\displaystyle x_i =\sum_{k=1}^{d+1}{\alpha_k^i}y_k^i, \quad \sum_{k=1}^{d+1}\alpha_k^i =1 \text{ and } y_k^i \in S$
Now consider the following matrix of dimensions ${\infty \times 2(d+1)}$
$\displaystyle \left[ \begin{array}{ccccccc} y_1^1 & \alpha_1^1& y_2^1 &\alpha_2^1&...&x_{d+1}^1&\alpha_{d+1}^1 \\ y_1^2& \alpha_1^2& y_2^2 &\alpha_2^2&...&x_{d+1}^2&\alpha_{d+1}^2 \\ y_1^3 & \alpha_1^3& y_2^3 &\alpha_2^3&...&x_{d+1}^3&\alpha_{d+1}^3\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ y_1^n & \alpha_1^n& y_2^n &\alpha_2^n&...&x_{d+1}^n&\alpha_{d+1}^n\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \end{array} \right]$
Now consider the first column ${ \{y_k^1\}}$. Since this sequence lies in ${S}$ and ${S}$ is compact, there is a sub-sequence in that that converges to a point ${y^1 \in S}$. The indices of this subsequence are denoted by ${I_1 \subset \{1,2,3,...\}}$. Now consider the second column, but consider only rows corresponding to the index set ${I_1}$. Since ${\alpha_1^{k} \in [0,1]}$ a compact set, there exists a subsequene of ${\{\alpha_1^{k}\}_{k\in I_1}}$ that converges to some point ${\alpha_1}$ in ${[0,1]}$. Denote the indices corresponding to this subsequence by the set ${I_2 \subset I_1}$. Now consider the third column corresponding to ${y_2^{k}}$ but only corresponding to the rows ${I_2}$. Since ${\{y_2^{k}\} \subset S}$ and ${S}$ is compact, we can find a subsequence which converges to ${y_2 \in S}$. Denote the set of the indices corresponding to this sub sequence by ${I_3 \subset I_2}$. Continue this till all the ${2(d+1)}$ are exhausted. So we will finally have an index set ${I_{2(d+1)}}$ corresponding to the last column.
Now consider the original sequence ${\{x_n\}}$. It is easy to see that, the subsequence corresponding to the index set ${I_{2(d+1)}}$ converges to a point
$\displaystyle \sum_{i=1}^{d+1}\alpha_i y_i.$
which belongs to the set ${{conv}(S)}$ proving the result. $\Box$
1. Convex Sets
Definition 1 (Line) Let ${x_1}$ and ${x_2}$ be two points. Then the line passing through ${x_1}$ and ${x_2}$ can be represented as
$\displaystyle y=\theta x_1 + (1-\theta) x_2,\ \theta \in {\mathbb R}.$
The value ${\theta =1}$ corresponds to the point ${x_1}$ and ${\theta=0}$ is the point ${x_2}$. Also observe that for a line ${\theta}$ can any real value. When ${\theta}$ is restricted to ${[0,1]}$, \ie, ${\theta \in[0,1]}$, a line segment connecting ${x_1}$ and ${x_2}$ is obtained. We denote the line segment between ${x_1}$ and ${x_2}$ as ${[x_1,x_2]}$. Let us see some examples:
1. The concept of line and line segment is very clear if ${x}$ and ${y}$ belong to ${{\mathbb R}^d}$.
2. Let ${X \in S}$ and ${Y \in S}$. Then it is very clear that ${\theta X \in S}$ for all ${t\in {\mathbb R}}$. Also it is very clear that ${\theta X +(1-\theta)Y \in S}$. Hence the concept of line is well defined on the set ${S}$.
3. Let ${X \in S_{++}}$ and ${Y \in S_{++}}$. Then it is very clear that ${\theta X \in S_{++}}$ for ${\theta \in [0,1]}$. From the properties of SD matrices, it follows that
$\displaystyle \theta X +(1-\theta)Y \in S_{++},\ \ \theta \in [0,1].$
So the notion of line segment is well defined on ${S_{++}}$.
A set ${C}$ is convex if the line segment joining any two points in ${C}$ lies in ${C}$. Mathematically the set ${C}$ is convex if and only if for any ${x, y\in C}$ implies ${[x,y]\subseteq C}$. First observe that convexity is not a topological property, that is it does not depend on the norm. Let us see some examples:
1. the empty set ${\emptyset}$ is always convex.
2. Provide other geometrical examples.
3. The set
$\displaystyle \Delta=\{(x_1,x_2,...,x_{d+1}): x_1+x_2+...+x_{d+1}=1, x_i\geq 0 \}$
is a convex set. This is also called as the ${d}$-dimensional simplex. Observe that this set corresponds to all ${d+1}$ valued discrete random variables.
4. Any ball ${B(o,R)}$ is simplex. Let ${x}$ and${y}$ belong to the ball ${B(o,R)}$. Then by the property of norm
$\displaystyle \begin{array}{rcl} \|\theta x+ (1-\theta)y\| &\leq \|\theta x\| + \|(1-t)y\|\\ &=\theta\| x\| + (1-t)\|y\|\\ &\leq \theta R +(1-t)R =R \end{array}$
This implies ${\theta x+ (1-\theta)y \in B(o,R)}$ which implies ${B(o,R)}$ is convex. In general, ${B(x,R)}$ is a convex set for any ${x}$ and ${R}$ (Home work). Observe that the ball ${B(x,R)}$ can be represented as (prove it)
$\displaystyle B(y,R) = \{y+Rx:\ \|x\|\leq 1\}.$
5. Ellipsoid: Prove that the set ${\{y+Ax:\ \|x\|\leq 1\}}$, where ${A \succ 0 }$ is a convex set.
6. Hyperplane and halfspace: Let ${a\in {\mathbb R}^d}$ and ${b\in {\mathbb R}}$, then the set
$\displaystyle \{x: a^tx=b\}$
is called a hyperplane. The set
$\displaystyle \{x: a^tx\leq b\}$
is the halfspace.
7. Systems of linear equations or Polyhedron: Let ${c_i}$ be vectors in ${{\mathbb R}^d}$ and ${\beta_i}$ be some scalars. Then the set
$\displaystyle P=\{x\in {\mathbb R}^d: c_i^T x \leq \beta_i,\ i=1,2,...,n\},$
is called a polyhedron. Succinctly, this can be represented as ${Cx\preceq \beta}$.
8. The set of PSD matrices ${S_+}$ and ${S_{++}}$ is convex. Proof: Let ${Z}$ and ${Y}$ belong to ${S_{++}}$.
$\displaystyle x^t(\theta Z+(1-\theta)Y )x = \theta x^t Zx + (1-\theta) x^t Y x \succ 0.$
9. Voronoi tessellation: Let ${x_1, x_2,\hdots, x_n}$ denote ${n}$ points in ${{\mathbb R}^d}$. For any point ${x_i}$ let ${V(x_i)}$ denote the Voronoi region of the point ${x_i}$, i.e.m
$\displaystyle V(x_i) = \{x: \|x-x_i| \leq \|x-x_j\|, \forall j=1,...,n\}.$
Prove that ${V(x_i)}$ is a Voronoi set.
Definition 2 (Convex combination) Let ${\{x_1,x_2,\hdots, x_n\}\subset {\mathbb R}^d}$ denote a set of ${n}$ points. Let ${\beta_i >0}$ and ${\sum_{i=1}^n \beta_i =1}$. Then
$\displaystyle x=\sum_{i=1}^n \beta_i x_i,$
is called a convex combination of the points ${x_1,\hdots,x_n}$.
Theorem 3 A set ${A}$ is convex if and only if it contains all convex combinations of its points.
Proof: It is easy to see that if a set contains all convex combinations of its points, then the set is convex.
Now to prove the other way. We prove it by induction on the number of points. By the definition, it is is true for ${n=2}$. Let us assume it is true for any ${n}$ points, \ie, any convex combination of ${n}$-points lies in the set. Now consider ${n+1}$ points ${\{x_1,x_2,...,x_{n+1}\}}$. Consider
$\displaystyle x=\sum_{i=1}^{n+1} a_i x_i,$
such that ${\sum_{i=1}^{n+1}a_i=1}$. Alternatively,
$\displaystyle x=\sum_{i=1}^{n+1} a_i x_i= a_{n+1}x_{n+1}+(1-a_{n+1})\left(\sum_{i=1}^n \frac{a_i}{1-a_{n+1}} x_i \right).$
Observe by induction assumption, ${y=\sum_{i=1}^n \frac{a_i}{1-a_{n+1}} x_i \in A}$, since ${\sum_{i=1}^n\frac{a_i}{1-a_{n+1}} = \frac{1-a_i}{1-a_i}=1}$. So,
$\displaystyle x=a_{n+1}x_{n+1}+(1-a_{n+1})y,$
is a two points convex combination of ${x_{n+1}}$ and the points ${y}$ and hence ${x}$ belongs to the set ${A}$.
$\Box$
We will now define a process of convexifying any set.
Definition 4 (Convex hull) The set of all convex combinations of points from a set ${A}$ is called the convex hull of ${A}$. It is denoted by ${\text{conv}(A)}$.
Similar to a closure of a set, the convex hull of a set is the smallest convex set containing ${A}$. We will now see some examples.
1. Some basic geometric examples.
2. Birkhoff – Von Neumann Theorem: Consider the set of permutation matrices ${\{X_\sigma, \sigma \in P_n\}}$. Then the convex hull of this set is the set of all doubly stochastic matrices.
Hw 1 Prove that the convex hull of a set is convex.
1. Linear Algebra
Let ${A\in {\mathbb R}^{m\times n}}$ be a real valued matrix. The matrix ${A}$ can be considered as a linear operator from ${{\mathbb R}^n}$ to ${{\mathbb R}^m}$. Let us see what a linear operator is: A function ${f(x):{\mathbb R}^n \rightarrow {\mathbb R}^m}$ is called a linear operator (functional) if for any ${x,y \in {\mathbb R}^n}$, and ${a, b \in {\mathbb R}}$,
$\displaystyle f(ax+by)=af(x)+bf(y).$
It turns out that on ${{\mathbb R}^m}$, the only linear functional’s are matrix operators, \ie, the linear functionals are of the form,
$\displaystyle f(x) = Ax, \quad A\in {\mathbb R}^{m\times n}, x\in {\mathbb R}^n.$
For every linear functional, there are a couple of important associated spaces.
1. Range space: The range space of the matrix is a subspace of ${{\mathbb R}^m}$ and is defined as
$\displaystyle \mathcal{R}(A)=\{y:\ y \in {\mathbb R}^m, y = Ax \text{ for some } x\in {\mathbb R}^n\}.$
A space is a subspace if for scalars ${a, b}$ and ${x, y}$ in the space, ${ax+by }$ belongs to the space.
Let ${a_1, a_2, ..., a_n}$ be the columns of ${A}$, \ie, ${A =[a_1, a_2,\hdots, a_n]}$. Let ${x={\mathbb R}^n}$, ${x=(x_1,x_2,...,x_n)}$
$\displaystyle Ax = a_1 x_1 + a_2 x_2+\hdots+a_n x_n.$
This means that ${Ax}$ is a linear combination of the columns of ${A}$. Alternatively ${\mathcal{R}(A)}$ is the space spanned by the columns of ${A}$. Hence the Range space of ${A}$ is also the column space of ${A}$.
2. Null space or Kernel space: The kernel space of a matrix is a subspace of ${{\mathbb R}^n}$ of ${A}$ is
$\displaystyle \mathcal{N}(A)=\mathcal{K}(A)= \{x :\ Ax =o, x\in {\mathbb R}^n\}.$
Essentially, null space is the set of solution to the homogeneous equation ${Ax=o}$. Let ${\hat{a_1}, \hat{a_2},\hdots, \hat{a_n}}$ denote the row of ${A}$. Then
$\displaystyle Ax=\left[ \begin{array}{c} a_1 x\\ a_2 x\\ \vdots\\ a_n x \end{array} \right].$
So ${Ax=0}$ requires that ${a_1x=0}$, ${a_2x=0}$, ${\hdots}$, ${a_nx=0}$. This implies that ${x \perp a_1}$, ${x \perp a_2}$, ${\hdots}$, ${x \perp a_n}$. So for any scalars ${c_1, c_2, \hdots, c_n \in {\mathbb R}}$ and the linearity of the inner product,
$\displaystyle (c_1 a_1+c_2a_2+\hdots+c_na_n)x =0$
So ${x \in \mathcal{N}(A)}$ if and only if ${x}$ is orthogonal to the subspace spanned by the rows of the matrix ${A}$. From the previous argument observe that the set of ${x \in {\mathbb R}^m }$ such that ${A^tx=0}$ equals the orthogonal subspace to the row space of ${A^t}$ or the column space of ${A}$, \ie, ${\mathcal{R}(A)}$. So
$\displaystyle \mathcal{R}(A)\perp \mathcal{N}(A^t)$
The following is a fundamental theorem in linear algebra
Theorem 1 Let ${A \in {\mathbb R}^{m\times n}}$. Then
$\displaystyle \mathcal{R}(A)\stackrel{\perp}{\bigoplus} \mathcal{N}(A^t) = {\mathbb R}^m$
This theorem means that any vector of ${{\mathbb R}^n}$ belongs to ${\mathcal{R}(A)}$ or ${\mathcal{N}(A^t)}$ or can be expressed as a sum of two vectors one belonging to ${ \mathcal{R}(A)}$ and the other to ${\mathcal{N}(A^t)}$.
Lemma 2 (Fredholm Alternative) ${Ax=b}$ has a solution if and only if for any ${y}$ such that ${A^Ty=0}$, \ie, ${y\in \mathcal{N}(A^t)}$, implies ${y^T b=0}$, \ie, ${b\in \mathcal{R}(A)=\perp\mathcal{N}(A^t)}$.
Hw 1 Let ${A\in {\mathbb R}^{m\times n}}$. Show that ${\mathcal{R}(A)}$ and ${\mathcal{N}(A)}$ are subspaces.
Lemma 3 Consider the solution of ${Ax=b}$. Let ${b \in {\mathbb R}^m}$ and the range space of ${A}$. Let ${x'}$ be a solution to ${Ax=b}$. Then any solution to the equation ${Ax =b}$ belongs to the set
$\displaystyle \{x'+y: y\in \mathcal{N}(A)\}.$
Proof:We have ${Ax'=b}$. So the equation to be solved can be reformulated to be
$\displaystyle Ax = Ax' \iff A(x-x')=o.$
So any solution of the original equation is of the form ${x-x'=y}$, where ${y}$ belongs to the kernel of ${A}$. Hence ${x=x'+y}$. $\Box$
Some observations:
1. So ${Ax=b}$ has a unique solution only when the kernel of ${A}$ has only one element, \ie the zero vector. This means that the dimension of ${\mathcal{K}(A)}$ is zero.
2. Also observe that ${Ax}$ is one to one only if ${\mathcal{N}(A)=\{o\}}$.
We will now see when the linear map ${Ax:{\mathbb R}^n \rightarrow {\mathbb R}^m}$ is continuous.
Theorem 4 The map ${Ax:{\mathbb R}^n \rightarrow {\mathbb R}^m}$ is a continuous function if any matrix norm of ${A}$ is finite.
Proof:Let ${\{x_n \rightarrow {\mathbb R}^n\}}$ and ${\|\|_b}$ a induced matrix norm. So we have to show that ${Ax_n \rightarrow Ax}$ in ${{\mathbb R}^m}$. Let ${\epsilon >0}$. Then
$\displaystyle \|Ax_n -Ax\| = \|A(x_n-x)\|\stackrel{(a)}{\leq}\|A\|\|x_n-x\| .$
Since ${\|x_n -x\| \rightarrow 0}$ and ${\|A\|<\infty}$, ${\|Ax_n -Ax\| \rightarrow 0}$ which proves the convergence and hence continuity.
Also since all norms are equivalent, it is sufficient that ${\|A\|}$ be finite with respect to one norm. $\Box$
We will now describe the rank of a matrix. The rank of a matrix is the dimension of the range space of ${A}$. The following is a fundamental theorem of linear algebra
Theorem 5 The dimension of the range (or the column space) of a matrix ${A}$ equals the dimension of the range of ${A^T}$ (or the row space) and equals the rank of the matrix. Or equivalently, The dimension of the row space equals that of the column space
Proof: Follows from rank decomposition ${A=CF}$ where the matrix ${C}$ is the matrix of independent vectors of column space. …… $\Box$
Theorem 6 Rank-nullity theorem: Let ${A \in {\mathbb R}^{m\times n}}$, then
$\displaystyle \dim(N(A))+\dim(\mathcal{R}(A))=n.$
Proof:Let ${k}$ denote the dimension of the null space or ${k=\dim(N(A))}$. Hence there exists ${\{v_1,v_2, \hdots v_k\} \subset {\mathbb R}^n}$ a basis of ${N(A)}$. We can add vectors ${\{v_{k+1},v_{k+2},...,v_{n}\}}$, so that ${\{v_{1},\hdots,v_n\}}$ form a basis of ${{\mathbb R}^n}$. From the definition of the range space,
$\displaystyle \mathcal{R}(A)= A* \text{LinearComb}(\{v_{1},\hdots,v_n\})$
which by the linearity equals,
$\displaystyle \mathcal{R}(A)= \text{LinearComb}(\{Av_{1},Av_2,...,Av_k,Av_{k+1},\hdots,v_n\})$
$\displaystyle \mathcal{R}(A)= \text{LinearComb}(\{o,o,...,o,Av_{k+1},\hdots,v_n\})$
$\displaystyle \mathcal{R}(A)= \text{LinearComb}(\{Av_{k+1},\hdots,v_n\})$
Observe that the dimension of the RHS is ${n-k}$ if all the vectors are linearly independent. Suppose there exists some scalars ${c_{k+1},c_{k+2},...c_n}$ so that
$\displaystyle c_{k+1}Av_{k+1}+c_{k+2}Av_{k+2}+...+c_nAv_n=o$
Which implies
$\displaystyle A(c_{k+1}v_{k+1}+c_{k+2}v_{k+2}+...+c_nv_n)=o,$
\ie, ${(c_{k+1}v_{k+1}+c_{k+2}v_{k+2}+...+c_nv_n) \in \mathcal{N}(A)}$. Hence there exists ${c_1, ....c_k}$ such that
$\displaystyle c_{k+1}v_{k+1}+c_{k+2}v_{k+2}+...+c_nv_n= c_1v_1+....+c_kv_k,$
which implies
$\displaystyle -c_1v_1-....-c_kv_kc_{k+1}v_{k+1}+c_{k+2}v_{k+2}+...+c_nv_n=o,$
which is a contradiction since we assumed ${v_1,....v_n}$ are a basis of ${{\mathbb R}^n}$ unless ${c_1=c_2=...=c_n=0}$ which proves our theorem. $\Box$
Lemma 7 So the linear map ${Ax :{\mathbb R}^n \rightarrow {\mathbb R}^n}$ is invertible if and only if either of the conditions hold true
• ${\mathcal{N}(A)=\{o\}}$, \ie, ${A}$ is one-one.
• ${A}$ is onto, \ie, ${\dim\mathcal{R}(A) =n}$
Proof: Follows from the rank nullity theorem. $\Box$
So the homogeneous set of equations ${Ax=0}$ has a non trivial solution only when ${rank(A) .
1.1. Determinant
The determinant of a matrix ${A}$ is a function from the space of square matrices to real numbers. One definition of determinant is that it is the volume of the parallelepiped formed by the row or column vectors.
(Leibniz formula)
$\displaystyle \det(A)=\sum_{\sigma \in \Pi_n}\text{sgn}(\sigma)\prod_{i=1}^nA_{i\sigma(i)}$
Look at the following Wikipedia page for more details about the permutation group $\Pi_n$ and the sign of permutation $\text{sgn}(\sigma)$. From this definition it follows that the determinant is a continuous function of the matrices. Very formally, the determinant if the unique function on ${n\times n}$ matrices that is ${n}$-linear alternating in the columns and takes a unit value on the identity matrix. Some properties of the determinants:
1. ${\det(AB)=\det(A)\det(B)}$.
2. ${\det(I+AB) =\det(I+BA)}$
3. ${\det(A^{-1}) = \frac{1}{\det(A)}}$
1.2. Eigenvalues
A vector ${u \neq o}$ is an eigenvector of an ${n \times n}$ matrix if there exists a scalar ${\lambda}$ such that
$\displaystyle Au=\lambda u.$
The scalar is called the eigenvalue of the matrix ${A}$. The eigenvalues can alternatively be characterized as the roots of the polynomial equation
$\displaystyle \det(A-\lambda I)=0.$
The set of eigenvectors form a subspace (Why?) and is called the eigenspace. If there are ${n}$ independent eigenvectors then the matrix ${A}$ can be decomposed
$\displaystyle A= U\Lambda U^{-1},$
where the ${n}$-eigenvectors form the columns of ${U}$. Some facts about eigenvalues:
1. ${\det(A)=\prod_{i=1}^n \lambda_i}$.
2. ${\text{tr}(A) =\sum_{i=1}^n \lambda_i}$
1.3. Symmetric matrices
A matrix is symmetric if ${A^T =A}$. The space of Symmetric ${n\times n}$ matrices is denoted by ${S}$. We have the following properties of a symmetric matrix
1. The dimension of a symmetric matrix is ${n(n+1)/2}$.
2. The eigenvalues of a symmetric matrix are real. (The eigenvectors are real and non trivial if restricted to ${{\mathbb R}^n}$)
3. Eigenvectors corresponding to different eigenvalues are orthogonal.
4. Indeed ${A}$ can be decomposed as ${U^T\Lambda U}$ for a real orthogonal matrix ${U^TU=I}$.
5. If ${u_i}$ denote the moralized eigenvectors of ${A}$, then ${A= \lambda_1 u_1u_1^T + \lambda_2 u_2u_2^T +...+ \lambda_n u_nu_n^T }$
1.4. Rayleigh quotient
Let ${A}$ be a symmetric real matrix. Then the Rayleigh quotient is defined as
$\displaystyle R(x) = \frac{x^TAx}{x^Tx}.$
It turns out that Rayleigh quotient is a very important quantity in the analysis of eigenvalues. For example the following lemma characterizes the maximum eigenvalue of a symmetric matrix.
Lemma 8 Let ${\lambda_{\max}}$ denote the maximum eigenvalue of the symmetric matrix ${A}$. Then
$\displaystyle \lambda_{\max}(A) = \sup\{R(x): x\in {\mathbb R}^n\},$
$\displaystyle \lambda_{\min}(A) = \inf\{R(x): x\in {\mathbb R}^n\}.$
Proof:Let ${U ^T \Lambda U}$ be the eigen decomposition of ${A}$. Then
$\displaystyle R(x) = \frac{x ^TU^ T\Lambda U x}{x^Tx}.$
Since ${U^TU=I}$, ${x^Tx = X^TU^TUx}$. So
$\displaystyle R(x) = \frac{x ^TU^ T\Lambda U x}{X^TU^TUx}= \frac{y^T\lambda y}{y^T y},$
where ${Y=Ux}$ some other vector. Observe that ${U}$ is invertible and hence onto. So for ${x}$ spanning the whole of ${{\mathbb R}^n}$, ${y}$ also spans the whole of ${{\mathbb R}^n}$. Hence
$\displaystyle \sup _{x\in {\mathbb R}^n} R(x)=\sup _{y\in {\mathbb R}^n} \frac{y^T\lambda y}{y^T y}$
This equals
$\displaystyle \sup _{x\in {\mathbb R}^n} R(x)= \sup\frac{\sum_{i=1}^n\lambda_i y_i^2}{\sum_{i=1}^n y_i^2}=\sup_{p_i\geq0, \sum_{i=1}^n p_i=1}\sum_{i=1}^n \lambda_i p_i =\max{\lambda_i}.$
Similarly we have the following result for the other eigenvalue. $\Box$
Hw 2 How do you characterize the other eigenvalues?
\subsubsection{Positive definite matrices} A matrix ${A\in S}$ is positive definite if for any ${x\neq o}$ and ${x\in {\mathbb R}^n}$,
$\displaystyle x^tAx >0.$
This is denoted by ${A \succ 0}$. The set of positive definite matrices is denoted by ${S_{++}}$.
Small digression as to why we consider only the subsets of ${S}$:Technically, a positive definite matrix need not be symmetric. Every matrix can be represented as
$\displaystyle A= B+C=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T).$
Observe that ${C=(A-A^T)/2}$ is skew symmetric, \ie, ${C=-C^T}$. Also ${B=B^T}$. Hence
$\displaystyle =+.$
We have
$\displaystyle ==-.$
This implies ${=0}$. So, ${=}$. Hence to understand the positive definite property of a matrix, it suffices to consider only the symmetric part.)
Some important properties:
1. A matrix ${A}$ is positive definite if and only if all its eigenvalues are positive.
2. If ${A \in S_{++}}$ then ${\det(A)>0}$ and hence invertible.
3. Let ${A_S}$ be the matrix obtained by removing the rows ans the columns indexed by ${S}$. Then ${A_S \succ 0}$. This implies all the diagonal elements should be greater than ${0}$.
4. There exists a unique matrix called the square root such that ${A= BB^T}$. It is easy to see that ${B= U^T \sqrt{\Lambda}U}$.
5. If ${A,B \in S_{++}}$ then ${A+B\in S_{++}}$
6. If ${A \in S_{++}}$, ${A^m \in S_{++}}$.
7. ${A \in S_{++}}$ if and only if for any invertible matrix ${M}$, ${MAM^T \in S_{++}}$
(Positive seimidefinite matrices) A matrix ${A\in S^n}$ is positive definite if for any ${x\neq o}$ and ${x\in {\mathbb R}^n}$,
$\displaystyle x^tAx \geq0.$
This is denoted by ${A \succeq 0}$. The set of positive definite matrices is denoted by ${S_+}$.
1. A matrix ${A}$ is positive semidefinite if and only if all its eigenvalues are non-negative.
2. Let ${A_D}$ be the matrix obtained by removing the rows ans the columns indexed by ${D}$. Then ${A_D \succeq 0}$. This implies all the diagonal elements should be non-negative
3. There exists a unique matrix called the square root such that ${A= BB^T}$. It is easy to see that ${B= U^T \sqrt{\Lambda}U}$.
1.5. Schur Complement
Let ${X \in S }$. If ${X}$ is partitioned as
$\displaystyle \left( \begin{array}{ccc} A &B\\ B^T & C \\ \end{array}\right)$
Then if ${\det(C) >0}$ then the Schur complement of ${C}$ in ${X}$ is defined as
$\displaystyle S_c=A-B^TC^{-1}B.$
We have the following lemma
Lemma 9 For any symmetric matrix ${X}$, that can be partitioned as
$\displaystyle \left( \begin{array}{ccc} A &B\\ B^T & C \\ \end{array}\right),$
Then,
• The matrix ${X\succ 0}$ if and only if ${C\succ0}$ and ${A-B^TC^{-1}B \succ 0}$.
• If ${C \succ 0}$, then ${X \succeq 0}$ if and only if ${A-B^TC^{-1}B \succeq 0}$
Proof:
$\displaystyle X= \left( \begin{array}{ccc} I &BC^{-1}\\ 0 & I \\ \end{array}\right)\left( \begin{array}{ccc} A-B^TC^{-1}B &0\\ 0 & D \\ \end{array}\right)\left( \begin{array}{ccc} I &BC^{-1}\\ 0 & I \\ \end{array}\right)^T$
$\Box$
Also we have
$\displaystyle \left( \begin{array}{ccc} I &BC^{-1}\\ 0 & I \\ \end{array}\right)^{-1} = \left( \begin{array}{ccc} I &-BC^{-1}\\ 0 & I \\ \end{array}\right).$
So
$\displaystyle \left( \begin{array}{ccc} I &-BC^{-1}\\ 0 & I \\ \end{array}\right) X \left( \begin{array}{ccc} I &-BC^{-1}\\ 0 & I \\ \end{array}\right)^T =\left( \begin{array}{ccc} A-B^TC^{-1}B &0\\ 0 & D \\ \end{array}\right)$
Since for any matrix ${D \in S_{++}}$ if and only if for any invertible matrix ${M}$, ${MDM^T \in S_{++}}$ This implies ${A-B^{T}C^{-1}B \succ 0}$
1. Compact sets
We will now move to an important class of sets. These sets are desirable (most analysts) since they are very nice and easy to work with. There are many definitions of compact sets. Since we are in ${{\mathbb R}^d}$, we will use a sequence definition). There are alternative ways to define compact sets, however we will not concentrate on them (See the appendix for an alternative definition).
We first begin with a notion for a subsequence. Given a sequence ${\{x_n\}}$ a subsequence is denoted by ${\{x_{n_k}\}}$ such that ${n_1\leq n_2 \leq n_3 \leq ...}$.
A set ${A\subset {\mathbb R}^d}$ is compact if and only if for any sequence ${\{x_n\} \subset A}$, there is a subsequence that converges to a point in ${A}$.
The above definition does not provide us with a clear picture of how compact sets look. The next theorem provides a better characterization of compact sets in ${{\mathbb R}^d}$.
Theorem 1 (Bolzano–Weierstrass theorem) A set ${A\subset {\mathbb R}^d}$ is compact if and only if it is closed and bounded in ${{\mathbb R}^d}$
Before we go into the proof let us see some examples of compact sets.
1. The set ${[0,2]}$ is compact since it is closed and bounded.
2. ${(0,2]}$ is not compact since it is not closed.
3. ${[0,\infty)}$ is not compact since it is not bounded.
4. The ball ${B(0,R)}$ in ${{\mathbb R}^d}$ is not compact since it is an open set. However the closure of the ball
$\displaystyle \overline{B(o,R)} = \{x: \|x\|\leq R , \ x\in {\mathbb R}^d\}$
is compact.
Proof: I will prove the theorem in ${{\mathbb R}^2}$. A similar proof with more notation will hold true for general ${{\mathbb R}^d}$. We will first prove that a closed and bounded set ${A\subset {\mathbb R}^2}$ is compact. So we should prove that given any sequence in ${A}$, there exists a convergence subsequence with limit in ${A}$.
Let ${\{x_n\} \subset A}$ be a sequence in ${A}$. Since ${A}$ is bounded, it can be fitted in a big square say ${S_1=[-a, a]^2}$, that is ${A\subset [-a,a]^2}$. Now divide the square ${S_1}$ into ${4}$ equal parts. So now each square is of side ${-a/2}$. Since the sequence ${x_n}$ has infinite number of points, at least one of the square should contain infinite number of points (If not we are done). Call that square ${S_2}$. So ${S_2}$ is some square of side length ${a/2}$ . Now divide the square ${S_2}$ into ${4}$ equal squares of side length ${a/4}$ . Again, at least one of these squares will contain infinite number of points. Pick one such square and call it ${S_3}$. Now again divide the ${S_3}$ square into ${4}$ equal squares and continue this procedure.
Observe that the side length of ${S_n}$ is ${1/2^{n-1}}$. So will pick the subsequence this way: Pick any point of the original sequence from square ${S_1}$ and denote it by ${x_{n_1}}$. Pick the second point from the original sequence with ${n_2 > n_1}$ and lying in square ${S_2}$. Call it ${x_{n_2}}$. Continue this way to obtain a subsequence ${\{x_{n_k}\}}$. You should always be able to do this since there are infinite number of points in any square we picked. Also observe the following: The sets ${S_n}$ are monotocnically decreasing, i.e., ${S_{n+1}\subset S_n}$ and they are closed. Since the arbitrary intersection of closed sets is closed, ${\cap_{n=1}^\infty S_n }$ is closed. It turns out that the intersection of these sets will contain only one point which we denote by , i.e., ${\cap_{n=1}^\infty S_n =\{x\}}$ (It is interesting to prove this and will leave it to you).
The claim is that the subsequence ${\{x_{n_k}\}}$ converges to the point ${x}$ in ${A}$. It is sufficient to prove that the subsequence converses. The limit will automatically lie in the set ${A}$ since the set ${A}$ is closed. So let us now prove that the subsequence converges. Fix an ${\epsilon >0}$. It is easy to observe that for all ${m>k}$, ${x_{n_m}}$ will lie in ${S_k}$ and also ${x}$ lies in ${S_k}$. Since the width of ${S_k}$ is ${1/2^{k-1}}$ it follows that
$\displaystyle \|x_{n_m}-x\|<\frac{1}{2^{k-1}}$
We can choose a large ${k}$ such that ${\frac{1}{2^{k-1}} <\epsilon}$ (for example ${k=\log(2/\epsilon)}$) and hence ${\|x_{n_m}-x\|<\epsilon}$, which implies ${x_{n_m}\rightarrow x}$ as ${m\rightarrow \infty}$. This proves the existence of a subsequence of any sequence converging to a point in ${A}$.
The converse (showing that compact implies closed and bounded) is left as HW. $\Box$
In some sense compact sets are the best kind of sets to deal with. Since they are closed all the limit points exist inside the set and the set itself is bounded.
Hw 1 Are the following sets compact: (prove or disprove)
1. ${A_1= [5,10]\cup\{0\}}$
2. ${A_2 = \{0,1/2,1/3,1/4,/1/5,....\}}$ as a subset of ${{\mathbb R}}$
3. ${A_3 = \{x; x \in {\mathbb R}^d, \|x\|_\alpha \leq 1\}}$
4. ${A_4 = \{x; x \in {\mathbb R}^d, Ax \geq b\}}$, ${A}$ is invertible ${d \times d}$ matrix and ${b\neq 0}$
5. ${A_5 = A_4 \cap B(0,10)}$
6. ${A_6 = (0,1)}$ as a subset of ${{\mathbb R}^2}$.
7. ${A_7 ={\mathbb R}}$. Try to prove or disprove using the sequential definition of compactness.
Hw 2 Prove that the union of two compact sets is compact.
Hw 3 Let ${S_i}$ be the squares defined in the previous theorem. Prove that ${\cap_{i=1}^n S_i =\{x\}}$. (Hint: Look at the coordinates of the corners of the square. See if they form a monotonic bounded sequence. )
Hw 4 If ${A}$ is a compact set and ${B}$ is a closed subset of ${A}$, i.e., ${B\subset A}$ then ${B}$ is compact.
If ${A\subset {\mathbb R}}$ is a compact set, since ${A}$ is bounded, ${\sup(A)}$ and ${\inf(A)}$ exist and since it is closed the ${\sup}$ and ${\inf}$ belong to the set ${A}$. We now come to the most important theorem from an optimization perspective.
Lemma 2 Let ${U\in {\mathbb R}^d}$ be a compact set and ${f(x)}$ be a continuous function. Then the image of the set ${U}$, \ie ${f(U)}$ is compact.
Proof: Let ${x_n}$ be a sequence in the set ${f(U)}$. Then the pre image of the sequence ${y_n =f^{-1}(x_n)}$ is in the compact set ${U}$. By the definition of compactness, there is some subsequence ${y_{n_k}}$ that converges to a point ${y}$ in ${U}$. Since the function ${f}$ is continuous, by the definition, ${f(y_{n_k})}$ converges to ${f(y)}$. This means ${f(f^{-1}(x_{n_k})) = x_{n_k}\rightarrow f(y)}$ where ${f(y)}$ is some point in ${f(U)}$. Hence for every sequence ${x_n}$ there exists a subsequence ${x_{n_k}}$ which converges to a point in ${f(U)}$. Hence the set ${f(U)}$ is compact. $\Box$
We will see some examples soon, but first an important corollary.
Lemma 3 If ${f(x):A\subset {\mathbb R}^d \rightarrow R}$ is a continuous function. Then ${f(x)}$ is bounded on the set ${A}$ and attains its maximum or minimum in the set ${A}$.
Proof: Since ${A}$ is a compact set and ${f(x)}$ is continuous the image ${f(A)\subset {\mathbb R}}$ is a compact set. Since ${f(A)}$ is compact it is closed and bounded. Hence the supremum and the infimum of the set ${f(A)}$ exist and they belong to the set. $\Box$
Lemma 4 Any two norms ${\|.\|_a}$ and ${\|.\|_b}$ on the Euclidean space ${{\mathbb R}^d}$ are equivalent.
Proof: Let ${S}$ be the unit sphere in ${{\mathbb R}^d}$ with respect to the standard Euclidean norm ${\|.\|_2}$, i.e., ${S=\{ x: \ \|x\|_2 =1\}}$.
First claim: It suffices to prove that ${\|.\|_a}$ and ${\|.\|_b}$ are equivalent on ${S}$.
Reason: Suppose you are able to prove that ${\|.\|_a}$ and ${\|.\|_b}$ are equivalent on ${S}$, i.e.,
$\displaystyle C_1\|x\|_a \leq \|x\|_b \leq C_2\|x\|_a, \ \forall \ x \text{ such that } \|x\|_2 =1. \ \ \ \ \ (1)$
Let ${y}$ be an arbitrary point in ${{\mathbb R}^d}$. Observe that ${\|\frac{y}{\|y\|_2}\|_2 = \frac{\|y\|_2}{\|y\|_2} =1}$, i.e., the point ${ \frac{y}{\|y\|_2} \in S}$. Hence
$\displaystyle C_1\|\frac{y}{\|y\|_2}\|_a \leq \|\frac{y}{\|y\|_2}\|_b \leq C_2\|\frac{y}{\|y\|_2}\|_a.$
By the property of the norms,
$\displaystyle \frac{C_1}{\|y\|_2}\|y\|_a \leq \frac{1}{\|y\|_2}\|y\|_b \leq \frac{C_2}{\|y\|_2}\|y\|_a,$
which proves that
$\displaystyle C_1\|y\|_a \leq \frac{1}{\|y\|_2}\|y\|_b \leq C_2\|y\|_a,\quad \forall y \in {\mathbb R}^d.$
We will now prove (1). Consider the function ${f(x)= \frac{\|x\|_b}{\|x\|_a}}$ on ${S}$. Since ${x\neq 0}$, and ${\|x\|_a}$ and ${\|x\|_b}$ are continuous functions with respect to the standard Euclidean norm, their quotient is also continuous with respect to the Euclidean norm. Also observe that ${S}$ is a closed and bounded set with respect to Euclidean norm and hence compact. By the previous theorem the image ${f(S)}$ is compact. Hence the supremum and infimum of the set ${f(S)}$ are attained. Hence for any ${x \in S}$
$\displaystyle C_1\leq f(x)\leq C_2,$
which proves the result. $\Box$
We will state the following theorem without proof:
Theorem 5 (Tychonoff’s theorem) If ${A}$ and ${B}$ are compact sets, then their Cartesian product ${A\times B}$ is also compact. (This holds true for product of an arbitrary number of sets)
2. Appendix (optional)
The following characterization of compact sets is fundamental compared to the sequential definition as it depends only on the underlying topology (open sets)
2.1. An open cover description of compact sets
An open cover of a set ${A}$ is a collection of sets ${Y_a, a \in I }$ such that ${A\subset \cup_{a\in I}Y_a}$. In plain English, an open cover of ${A}$ is a collection of open sets that cover the set ${A}$. Observe that the set ${I}$ need not be finite. Let’s see some examples.
• Let ${A=[0,1]}$. Then an open cover of ${A}$ is the set of sets ${\{(x-1,x+1); x\in [0,1]\}}$. This is because ${A\subset \cup_{x \in [0,1]}(x-1,x+1)}$. Observe that the cardinality of the cover is infinite.
• Let ${A=[0,1]}$. Another open cover of ${A}$ is ${\{(-0.5, 0.6), (0.5,1.5) \}}$ since ${[0,1] \subset (-0.5, 0.6)\cup(0.5,1.5)}$. Here the cardinality of the cover is finite.
• Let ${A=B(0,1)\subset {\mathbb R}^d}$. Then the set ${\{B(x,\epsilon): x \in B(o,1)\}}$ is an open cover since ${A\subset \cup_{x\in B(o,1)}B(a,\epsilon)}$. Here again the cardinality of the over cover is not finite.
A set ${A}$ is compact if any open cover of ${A}$ has finite sub cover. This means that given any cover of compact set ${A}$, you will be able to pick a finite number of subsets of the open cover to cover ${A}$. The emphasis is on the word any. You should be able to pick a finite sub cover for any cover. Lets look at some examples:
1. Consider the set ${A=[0,1]}$. Then one open cover of ${A}$ is the set of sets ${\{(x-1,x+1); x\in [0,1]\}}$. Let’s see if we can find a finite subcover. Observe that ${(-1,1)}$ and ${(-1,2)}$ belong to the previous cover and ${[0,1] \subset (-1,1)\cup (-1,2)}$. It turns out (we will see later) that ${[0,1]}$ is compact.
2. Consider the set ${A=(0,1)}$. Then one open cover of ${A}$ is the set of sets ${\{(x-1,x+1); x\in (0,1])\}}$. Let’s see if we can find a finite subcover. Observe that ${(-1,1)}$ and ${(-1,2)}$ belong to the previous cover and ${(0,1) \subset (-1,1)\cup (-1,2)}$. Let see if we find an open cover of ${(0,1)}$ which has no finite sub cover.
Consider the open cover ${\{(1/n , 1), (1/(n+1), 2/n), (1(n+2), 2/(n+1)),....\}}$. This is an open cover of ${(0,1)}$. You can check, how much ever you try you cannot pick a finite number of these sets to cover ${(0,1)}$. ${(0,1)}$ is not a compact set. As this example shows, the finite subcover property should hold for any open cover.
The following theorem shows that the earlier definition of compact sets based on sequences and this definition based on open sets are both equivalent on ${{\mathbb R}^d}$.
Theorem 6 (Heine–Borel theorem) A set ${S\subset {\mathbb R}^d}$ is compact if and only if ${S}$ is bounded and closed.
1. Continuous functions
A function ${f(x)}$ is continuous at ${x}$ if for any sequence ${x_n \rightarrow x}$, ${f(x_n) \rightarrow f(x)}$. A function is continuous if it is continuous on every point of its domain. This is a very intuitive definition. Observe that the definition does not restrict the function to be in real spaces. Let us know see some examples. Let us see an example of a discontinuous function for the definition to be clear.
$\displaystyle S(x)= -1, x<0; \ 0, x=0; 1, x>0;$
It is obvious that this function is discontinuous. However, we will use the definition to make this point. Consider the sequence ${x_n =1/n}$. We have that ${\lim_{n\rightarrow \infty} 1/n =0}$. We have
$\displaystyle S(x_n)=S(1/n)=1,$
since ${1/n >0}$. However we have
$\displaystyle S(\lim_{n\rightarrow \infty}1/n) =S(0) =0.$
So,
$\displaystyle \lim_{n\rightarrow\infty}S(x_n)\neq S(\lim_{n\rightarrow \infty}1/n).$
Since we are able to find a sequence ${x_n \rightarrow 0}$ such that ${\lim_{n\rightarrow \infty}S(x_n)\neq S(0)}$ which implies the function ${S(x)}$ is not continuous at ${x=0}$. One important observation is the following which I state as a lemma.
Lemma 1 For any sequence ${x_n}$ and a function ${f(x)}$
$\displaystyle \lim_{n \rightarrow \infty}f(x_n) = f(\lim_{n \rightarrow \infty}x_n),$
if and only if ${f(x)}$ is continuous.
So before taking the limit operation inside a function, it is necessary to check if the function is continuous. Alternatively, a function is continuous at ${x}$ if for any ${\epsilon >0}$, there exits a ${\delta>0}$ such that ${\|f(x)-f(x')\|<\epsilon}$ whenever ${\|x-x'\|<\delta}$. This is just a reformulation of the continuity. Observe that the notion of continuity depends on the notion of Balls and norm. (It turns out that in ${{\mathbb R}^n}$ norm doesn’t matter since all norms are equivalent).
Let ${\|x\|_a}$ be some norm in ${{\mathbb R}^n}$. Consider the function ${f(x)=\|x\|_a}$. We will now show that the function ${f(x)}$ is continuous with respect to the topology generated by the the Euclidean norm ${\|.\|_2}$.
Lemma 2 Let ${\|.\|_a}$ denote some norm on ${R^n}$. Then the function ${f(x)=\|x\|:{\mathbb R}^n \rightarrow [0,\infty)}$ is a continuous function with respect to the Euclidean norm.
Proof: Let ${x_n}$ be some sequence in ${{\mathbb R}^n}$ and let ${x_n}$ converge to ${x}$ in the ${\|.\|_2}$ norm. This means that ${\|x_k-x\|_2\rightarrow 0}$ as ${k}$ becomes large. To prove continuity, we will have to show that ${f(x_k)\rightarrow f(x)}$ as ${k\rightarrow \infty}$. From triangle inequality,
$\displaystyle f(x_k)=\|x_k\|_a = \|x_k -y +y\|_a \leq \|x_k-y\|_a+\|y\|_a.$
Hence rearranging and taking absolute value
$\displaystyle \|x_k\|_a-\|y\|_a| \leq \|x_k -y\|_a$
Let ${e_i, e_2, ..., e_n}$ be the standard basis of ${{\mathbb R}^n}$, i.e., ${e_i =(0,0,..0,1,0,..0,0)}$ a vector with ${1}$ in the ${i}$-th position. Then we have
$\displaystyle \|x_k -y\|_a=\|\sum_{i=1}^n (x_k^{(i)} - y_k^{(i)} )e_i\|_a,$
where ${x_k^{(i)}}$ is the ${i}$-th coordinate of ${x_k}$ (recall that ${x_k \in {\mathbb R}^n)}$. Again using triangle inequality (or the property of the norm),
$\displaystyle \|x_k -y\|_a\leq\sum_{i=1}^n \|(x_k^{(i)} - y_k^{(i)} )e_i\|_a.$
Observe that ${(x_k^{(i)} - y_k^{(i)} )}$ is a real number. Using the scaling property of a norm,
$\displaystyle \|x_k -y\|_a\leq\sum_{i=1}^n \|(x_k^{(i)} - y_k^{(i)} )e_i\|_a=\sum_{i=1}^n |x_k^{(i)} - y_k^{(i)} |\|e_i\|_a.$
Now using Cauchy-Schwartz inequality,
$\displaystyle \sum_{i=1}^n |x_k^{(i)} - y_k^{(i)} |\|e_i\|_a\leq \sqrt{\sum_{i=1}^n |x_k^{(i)} - y_k^{(i)} |^2}\sqrt{\sum_{i=1}^n \|e_i\|_a^2}$
Hence
$\displaystyle \|x_k -y\|_a\leq \|x_k -x\|_2 C,$
where ${C=\sqrt{\sum_{i=1}^n \|e_i\|_a^2}}$ is some finite constant. So,
$\displaystyle \begin{array}{rcl} \|x_k\|_a-\|y\|_a|& \leq& \|x_k -y\|_a\\ &\leq&\|x_k -x\|_2 C \rightarrow 0 \end{array}$
as ${k\rightarrow \infty}$ . Hence ${\|x_k\|}$ converges to ${\|x\|}$ as ${k \rightarrow \infty}$. This implies that the function ${f(x)}$ is continuous.
But we have for all ${n>k}$, ${\|x_n-y\|<\epsilon}$. So
$\displaystyle |\|x_n\|-\|y\|| \leq \epsilon$
Hence ${\|x_n\|}$ converges to ${\|y\|}$. This implies ${\|.\|}$ is continuous. $\Box$
While the definition is analytically good, its not “easy” to check if a function is continuous at every point. We now provide a stronger version of continuity that is often useful and desired. A function ${f(x)}$ is Lipschitz continuous, if there exists a constant ${K>0}$, such that for any ${x,y}$ in the domain of the function
$\displaystyle \|f(x)-f(y)\| \leq K\|x-y\|.$
${K}$ is called the Lipschitz constant. It is easy to see that every Lipschitz continuous function is continuous. Let ${x_n \rightarrow x}$. Hence
$\displaystyle \lim_{n \rightarrow \infty}\|f(x_n)-f(x)\| \leq K\lim_{n \rightarrow \infty}\|x_n-x\|.$
Since ${\|.\|}$ is continuous we can move the limit inside the norm on the RHS.
$\displaystyle \lim_{n \rightarrow \infty}\|f(x_n)-f(x)\| \leq K\lim_{n \rightarrow \infty}\|x_n-x\| = 0.$
This implies ${f(x_n)\rightarrow f(x)}$ which implies the continuity of ${f}$. Indeed we can show that every Lipschitz function is also differentiable and its derivative bounded. Examples of Lipschitz functions:
1. ${\exp(-x)}$ is Lipschitz function with constant ${1}$.
2. ${\sin(x)}$ and ${\cos(x)}$ are Lipschitz function with constant ${1}$.
3. ${\exp(-x^2)}$ is Lipschitz function with constant ${1/e}$.
We will now look at an alternative definition of continuity that uses open sets.
Lemma 3 A function ${f(x):{\mathbb R}^n \rightarrow {\mathbb R}^m}$ is continuous if and only if the inverse image of any open set is an open set. More precisely, ${f^{-1}(A)}$ is open for any open set ${A\subset {\mathbb R}^m}$.
Proof: We will prove that for a continuous function, the inverse image of an open set ${A}$ is open.
1. Let ${x \in f^{-1}(A)}$. This implies ${y=f(x) \in A}$.
2. Since ${A}$ is an open set, there exists an ${\epsilon}$ such that ${B(y,\epsilon)\subset A}$.
3. Since ${f(x)}$ is continuous, there exits a ${\delta}$ such that if ${z \in B(x,\delta)}$, then ${f(z) \in B(y,\epsilon)}$. This implies that ${f(B(x,\delta))\subset B(f(x),\epsilon)\subset A}$
4. This implies that ${B(x,\delta) \subset f^{-1}(A)}$.
5. Hence ${f^{-1}(A)}$ is an open set.
The converse can be proved similarly (HW). $\Box$
Observe that the ambient spaces (${R^n}$ and ${ R^m}$) are important. A function can be continuous in one space and not in another. Look at ${1/x}$ on ${ (0,1)}$ and ${[0.1)}$.
Let us check for the constant function, ${f(x)=c}$. Its range is ${\{c\}}$ which is a closed set. Now take any open set ${A}$ in ${{\mathbb R}}$. Then ${f^{-1}(A) = \emptyset}$ which is open. Hence the function is continuous. Let us look at the step function ${S(x)}$ that I have defined earlier. Consider the open set ${(1/2 , 2)}$. Its inverse image is ${S^{-1}(1/2,2)}$ and equals ${\{x: x>0\} \cup \{0\}}$. Observe that${\{x: x>0\} \cup \{0\}}$ is not an open set and hence the function is not continuous.
Hw 1 Use this definition to check the continuity of ${\exp(-x)}$.
Hw 2 Is the function ${1/x}$ continuous on ${[0,1]}$?
1. Open sets
On ${{\mathbb R}^d}$ the standard topology can be described using the balls generated by the Euclidean norm. We begin with the central object of topology, an open set.
Definition 1 A set ${A \subset {\mathbb R}^d}$ is open if for every ${x \in A}$, there exists an ${\epsilon}$ such that ${B(x,\epsilon)\subset A}$.
Simply put, a set is open if for every point in the set we can find a radius ${\epsilon}$ such that a ball of radius ${\epsilon}$ centered around the point fits entirely in the set. Observe that the ${\epsilon}$ can be arbitrary small and also can depend on the position ${x}$. We now provide a few examples.
1. By convention, empty set ${\emptyset}$ is an open set.
2. Let ${d=1}$, \ie, the real line. The interval ${I=(3,4)}$ is an open set. It is obvious that if ${x=2.99}$, we can choose ${\epsilon =0.00001}$ and ${B(2.99, 0.00001)}$ will fit entirely in ${(3,4)}$. For any ${x\in (3,4)}$, choosing ${\epsilon =0.5\min\{x-3, 4-x\}}$ would suffice. Also observe that we do not have to worry about the end points ${3}$ and ${4}$ since they do not belong to ${I}$.
3. Observe that if ${I_1}$ and ${I_2}$ are open intervals then ${I_1 \cup I_2}$ is open. This proof is obvious.
4. If ${I_1}$ and ${I_2}$ are open sets in ${{\mathbb R}^d}$, show that ${I_1\cap I_2}$ are open sets.
5. ${(0,\infty)}$ is open in ${{\mathbb R}}$.
6. ${[0,1)}$ is not an open set. Reason out why? (Check what happens when you try to center a ball around ${0}$).
7. A ${d}$-dimensional ball ${B(x,r)}$ is an open set in ${{\mathbb R}^d}$. (Prove it)
Hw 1 Check if the following sets are open subsets of ${{\mathbb R}^2}$
$\displaystyle A_1=\{(x,y): x>0, y >0, 3x+2y < 1 \}.$
$\displaystyle A_2=\{(x,y): x>0, y >0, 3x+2y > 1 \}.$
$\displaystyle A_3=\{(x,y): x>0, y >0, 3x+2y \leq 1 \}.$
$\displaystyle A_4=\{(x,y): 0 < x< \pi, \sin(x)<1/2\}.$
please provide reasons.
Theorem 2 Let ${A_1}$ and ${A_2}$ be two open sets. Then the intersection of the sets ${A_1 \cap A_2}$ is open.
Proof: Let ${x \in A_1 \cap A_2}$. This implies that ${x}$ belongs to both ${A_1}$ and ${A_2}$. Since ${A_1}$ is an open set, there exists an ${\epsilon_1}$ such that ${B(x,\epsilon_1)\subset A_1}$ and an ${\epsilon_2}$ such that ${B(x,\epsilon_2)\subset A_2}$. It is easy to see that, ${B(x,\min\{\epsilon_1,\epsilon_2\}) \subset A_1\cap A_2}$, which implies ${A_1\cap A_2}$ is open. $\Box$
The previous two results can be extended to any finite collection of sets by induction. More precisely, if ${A_1, A_2, \hdots A_n}$ are ${n}$ open sets. Then ${A_1\cap A_2\cap ... \cap A_n}$ is open. Let us see an example. If ${A_1=(-1,2)}$ and ${A_2=(0,1)}$, then their intersection is ${(0,1)}$ which is an open set. Let us also see an example which highlights the importance of finite number of sets. Let
$\displaystyle A_n= (1-1/n, 2+ 1/n).$
Observe that ${A_n}$ is an open set. It is also easy to observe that the intersection of ${A_1, A_2}$ or for that case any finite number of ${A_i}$ is open. However,
$\displaystyle \cap_{i=1}^\infty A_i = [1,2],$
which is closed. Hence one cannot claim that the intersection of an arbitrary number of open sets is open.
It turns out that the union of open sets is always open regardless of the number of sets.
Theorem 3 Let ${I}$ be any set. Then if ${A_\alpha}$, is an open set for every ${\alpha \in I}$, then their union
$\displaystyle \cup_{\alpha \in I} A_\alpha$
is always open.
Proof: HW $\Box$
The set ${I}$ in the above theorem is called an indexing set. For example ${I=\{1,2\}}$ or ${I=\{1,2,3,4,...\}}$ or ${I=(0,1)}$. So the number of sets in consideration depends on the cardinality of the index set. The above theorem indicates that for any cardinality of the index sets , the union is still open.
Hw 2 This problem emphasizes the fact that the notion of a set being open (or closed) depends on the ambient space in consideration. As a subset of ${{\mathbb R}}$ check if ${(0,1)}$ is an open set. Is ${(0,1)}$ an open set of ${{\mathbb R}^2}$? (Use the definition of open set)?
Hw 3 Let ${A= (0,10) \setminus \{1, 2, 3 ,4 ,5\}}$. Is the set ${A}$ an open subset of ${{\mathbb R}}$? “${\setminus}$” denotes set minus.
Hw 4 Let ${Q}$ denote the set of rational numbers in ${(0,1)}$. That is
$\displaystyle Q=\left\{\frac{m}{n} : m, n \in \{1,2,3,4,....\} \text{ and } \frac{m}{n}<1\right\} .$
Is the set ${Q}$ an open set of ${{\mathbb R}}$?
Hw 5 Let ${A}$ be a ${n\times n}$ invertible matrix and ${b\in{\mathbb R}^n}$ belong to the range space of ${A}$. Show that the set
$\displaystyle \{x: x\in {\mathbb R}^n, Ax < b\},$
is an open subset of ${{\mathbb R}^n}$. What happens if ${A}$ is not invertible?
Hw 6 The following exercise shows the properties of sequences in an open set. Let ${I=(0,1)}$ and ${x_n = 1/n,\ n>2}$. Verify the following:
• Does ${x_n \in I}$ ?
• What is the limit ${ \lim_{n\rightarrow \infty}x_n}$? Does it belong to the set ${I}$?
• Let ${y_n = \frac{1}{2} +\frac{1}{n}}$. Does the limit of the sequence ${y_n}$ belong to ${I}$?
This shows that even though a sequence belongs to an open set, the limit of the sequence (if it exists) may not belong to the set.
Hw 7 Observe that the definition of an open set depends on the notion of a Ball which in turn depends on the norm used. Suppose ${A\subset {\mathbb R}^d}$ is an open set with respect to the norm ${\|.\|_2}$ (the standard Euclidean norm). Is the set ${A}$ an open set in ${{\mathbb R}^d}$, when we use a different norm, for example the ${\|.\|_1}$ norm? Hint: Use the equivalence of norms in ${{\mathbb R}^d}$ and the definition of open sets.
2. Closed sets
A set ${A \subset {\mathbb R}^n}$ is a closed set if its complement ${{\mathbb R}^n \setminus\ A}$ is open.
So ${(-\infty,3]\cup[4,\infty)}$ is a closed set in ${{\mathbb R}}$ since its complement ${(3,4)}$ is an open set. Any set with finite cardinality (for example ${\{3\}}$ or ${\{0,4\}}$) is a closed set. Also observe that the entire set ${{\mathbb R}^d}$ is both a closed and open set with respect to ${{\mathbb R}^d}$. So a closed or an open set need not be bounded. Also by convention, the empty set ${\emptyset}$ is a closed set.
Hw 8 Check if the following sets are closed subsets of ${{\mathbb R}^2}$
$\displaystyle A_1=\{(x,y): x>0, y >0, 3x+2y \leq 1 \}.$
$\displaystyle A_2=\{(x,y): x>0, y >0, 3x+2y > 1 \}.$
$\displaystyle A_4=\{(x,y): 0 < x< \pi, \sin(x)\leq1/2\}.$
Explain why.
From the definition of closed sets, we have the following important property which we state as a theorem.
Theorem 4 Let ${A}$ be an open set in ${{\mathbb R}^n}$. Then its complement ${{\mathbb R}^n\setminus A}$ denoted by ${A^c}$ is closed.
We will now use the definition of closed sets and De-Morgan laws to characterize the union and intersection of closed sets.
Theorem 5 Let ${B_1}$ and ${B_2}$ be two closed sets. Then the union of the sets ${B_1 \cap B_2}$ is closed.
Proof: We have ${(B_1\cup B_2)^c = B_1^c \cap B_2^c}$ Since ${B_1^c}$ and ${B_2^c}$ are open sets (complements of closed set), it follows from the open sets intersection theorem that ${B_1^c \cap B_2^c}$ is open. Hence ${(B_1\cup B_2)^c}$ is open implying its complement ${(B_1\cap B_2)}$ is closed. $\Box$
Observe that as in the open set case, the above theorem can be extended to any finite collection of closed sets. It turns out that the intersection of closed sets is always closed regardless of the number of sets.
Theorem 6 Let ${I}$ be any set. Then if ${B_\alpha}$, is a closed set for every ${\alpha \in I}$, then their intersection
$\displaystyle \cap_{\alpha \in I} B_\alpha$
is always closed.
Proof: HW $\Box$
If ${B_1, B_2, \hdots B_n}$ are ${n}$ closed sets. Then ${B_1\cap B_2\cap ... \cap B_n}$ is closed and follows from the previous result by induction.
While the above definition characterizes a closed set, we now look at an alternate definition of closed sets that is more intuitive. For that we require the notion of a limit point.
Definition 7 ${x\in {\mathbb R}^n}$ is a limit point of a set ${S\subset {\mathbb R}^n}$ if for every ${\epsilon>0}$, ${B(x,\epsilon)\cap S\setminus\{x\}}$ is non-empty.
So a point is a limit point of a set when every ball centered around the point has a non-empty intersection with the set. For example if ${S=(0,2)}$ then every point of ${S}$ is a limit point of the set. In addition, the points ${0}$ and ${2}$ are limit points of the set ${S}$ (check why?). However ${2.00000001}$ is not a limit point of the set ${S}$ (Why?). So one can think of a limit point as a point in the set, or a point which is arbitrary close to some point in the set.
Hw 9 Find the limit points of the following sets.
1. ${A_1 = \{1,2,3,4\}}$.
2. ${A_2 =[0,1]}$
3. ${A_3= (0,1)}$
4. ${A_4=\{\|x\|<1: x\in {\mathbb R}^2\}}$
5. ${Q}$ the set of rational numbers.
Theorem 8 A set ${B\subset {\mathbb R}^n}$ is closed if and only if it contains all of its limit points.
Proof: We we prove by contradiction. Let ${x}$ be a limit point of the closed set ${B}$ and let ${x \notin B}$. We will provide a contradiction.
Since ${B}$ is closed, ${B^c}$ is open. Also since ${x\notin B}$ implies ${x \in B^c}$. Since ${B^c}$ is open, by the definition of an open set, there exists an ${\epsilon}$ such that ${B(x,\epsilon) \subset B^c}$. This implies that for this ${\epsilon}$ ${B(x,\epsilon)\cap B =\empty}$. Hence ${x}$ is not a limit point of ${B}$. This contradicts our assumption. $\Box$
Hw 10 Prove that the supremum and the inifimum of a closed set ${S\subset {\mathbb R}}$ belong to the set.
The closure of a set ${S}$ denoted by ${\bar{S}}$ equals the union of the set ${S}$ and all its limit points, \ie,
$\displaystyle \bar{S} = S\cup \{\text{Limit points of } S\}$
So closure of ${S=(0,2)}$ equals ${[0,2]}$.
Hw 11 Find the closures of the following sets
1. ${A_1 = \{1,2,3,4\}}$.
2. ${A_2 =[0,1]}$
3. ${A_3= (0,1)}$
4. ${A_4=\{\|x\|<1: x\in {\mathbb R}^2\}}$
5. ${ A_5=(0,1)\cup\{2\}}$
6. ${ A_6=\{x: Ax< b, x\in {\mathbb R}^n, A \in {\mathbb R}^{n\times n}, b\in {\mathbb R}^n\}}$
7. ${A_7=Q}$ the set of rational numbers.
We will now define the interior and the boundary of a set. Before that we require the notion of an interior point. Let ${x\in S}$ be a point such that there exists a ball of some radius ${\epsilon>0}$ such that the ball fits entirely in ${S}$. The set of all such points of ${S}$ is called the interior of ${S}$ and is denoted by ${S^o}$
$\displaystyle S^o=\{x; x\in S, B(x,\epsilon)\subset S, \text{ for some } \epsilon >0\}.$
The definition of the interior is similar to that of an open set. In fact the interior of a set ${S}$ is the largest open set contained in ${S}$. The interior of the set ${(0,1)}$ is the set ${(0,1)}$ itself. The interior of the set ${[0,1]}$ is the set ${(0,1)}$. The interior of the set ${\{(x,y); x^2+y^2\leq 1\}}$ is the set ${\{(x,y); x^2+y^2< 1\}}$. Also observe that the set might not have any interior.
Hw 12 Find the interior of the sets ${A_1, A_2, A_3, A_4,A_,A_6,A_7}$ in HW problem 11
Hw 13 What is the interior of the set ${ x^2+y^2\leq 1}$ as a subset of ${{\mathbb R}^3}$ (with respect to the metric on ${{\mathbb R}^3}$). What is its interior as a subset of ${{\mathbb R}^2}$ (with respect to the metric on ${{\mathbb R}^2}$).
The above exercise shows that the ambient space is important to the definition of the interior of a set.
Hw 14 Prove that a set ${S}$ is open if and only if all of its elements are inter points, \ie ${S=S^o}$. Hint: Look at both the definitions.
The boundary of a set ${S}$ denoted by ${\partial{S}}$ equals the set difference between the closure of the set and the interior of the set, \ie,
$\displaystyle \partial S = \bar{S}\setminus S^o.$
For example the boundary of the set ${(0,1)}$ are the points ${\{0,1\}}$. The boundary of the set ${\{(x,y); x^2+y^2\leq 1\}}$ is the set ${\{(x,y); x^2+y^2=1\}}$.
Hw 15 What is the boundary of the set ${ x^2+y^2\leq 1}$ as a subset of ${{\mathbb R}^3}$ (with respect to the metric on ${{\mathbb R}^3}$). What is its boundary of the above set as a subset of ${{\mathbb R}^2}$ (with respect to the metric on ${{\mathbb R}^2}$).
Hw 16 Find the boundary of the sets ${A_1, A_2, A_3, A_4,A_,A_6,A_7}$ in Exercise 11
Hw 17 Prove that the boundary of a set is a closed set.
A set ${S\subset {\mathbb R}^d}$ is bounded if there exists a ball of radius ${R<\infty}$ centered around the origin ${B(o,R)}$ that contains the set. The square ${[0, 2]^2}$ is bounded while the set ${\{(x,y); x>0, y>0\}}$ is not bounded.
This lecture notes will cover the basics of Analysis and Linear Algebra that is required for the optimization theory course EE5121. This is by no stretch a comprehensive treatment of either Real Analysis or Linear Algebra. There are many excellent books that deal with both these subjects. Here are a few references:
• Andrei Nikolaevich Kolmogorov, Sergei Vasilevich Fomin, “Introductory real analysis”.
• H. L. Royden, “Real Analysis”.
• Walter Rudin, “Principles of mathematical analysis”.
• Gilbert Strang, “Introduction to linear algebra”.
• Roger A. Horn, Charles R. Johnson, “Matrix analysis”.
1. Real numbers
The set of real number is denoted by ${{\mathbb R}}$. We begin with an important property (axiom?) of the real number system. The real number system can be ordered. In plain English this means any two real number can be compared. We take this property for granted. However, it turns out to be a very important property that in general does not hold for lots of other fields.
Since any two numbers can be compared we can provide an upper bound to a set of real numbers. For example, consider the set of numbers ${A=\{0, -3, -\pi, \pi, 2.718, 10\}}$. There are lot of real numbers that upper bound this set. Any number in the set ${\{11, 12, 14, 16\}}$, upper bound the numbers in ${A}$ while any number in the set ${\{-100, -200,-300\}}$, lower bound the real numbers in ${A}$. If you consider the interval ${(-10,10)}$, then any number in the set ${[10, \infty)}$ upper bounds ${A}$ while ${(-\infty,-10]}$ lower bounds ${A}$. It turns out, whenever a set has some upper bound and some lower bound, we can always pick the best upper bound and the best lower bound.
Don’t try to prove this as this is a distinguishing axiom of real numbers: “Every non empty set of real numbers that has an upper bound has a least upper bound. Similarly, Every non empty set of real numbers that has a lower bound has a greatest lower bound.” The least upper bound of a set is often denoted by ${\sup(S)}$ and the greatest lower bound is denoted by ${\inf(S)}$. Also observe that the ${\sup(S)}$ and ${\inf(S)}$ need not belong to the set ${S}$.
If we consider the set ${A=(-10,10)}$, we can see that ${\sup(A)=10}$ and ${\inf(A)=-10}$. For the set ${B=\{1,2,3,4 ,...\}}$, there is no supremum as it is not upper bounded. Its infimum ${\inf(B) =1}$. However, if we extended the real number system by including ${-\infty}$ and ${\infty}$ infimum and supremum exist for any set (they might just be ${\infty}$ and ${-\infty}$).
Hw 1 Find the ${\sup}$ and the ${\inf}$ of the following sets:
1. ${A_1=\{\frac{1}{n};\ n=1,2,3,\hdots\}}$
2. ${A_2=\{2^{1/n},n=1,2,\hdots\}}$
3. ${A_3=\{x: \arcsin(x) < 1/2\}}$
4. ${A_4 =(-3,4)}$
5. ${A_5 =[-3,4)}$
6. ${A_6 =[-3,4]}$
Hw 2 For any ${A\subset {\mathbb R}}$, can ${\sup(A)=\inf(A)}$? Can you tell anything about the set ${A}$?
Hw 3 Is it necessary that the supremeum ${\sup(A)}$ belongs to the set ${A}$. Similar question for the infimum. Look at the sets ${A_4}$, ${A_5}$ and ${A_6}$.
If the supremeum of a set ${A}$ belongs to the set, ie, ${\sup(A)\in A}$, then the supremum is usually replaced with the maximum ${\max(A)}$. So ${\max([-1,0]) }$ makes sense while ${\max((-1,0))}$ doesn’t make sense. So if you are unsure, always use ${\sup}$ and ${\inf}$ instead of ${\max}$ and ${\min}$.
2. Basic Topology
2.1. Norm and Inner Product
One of the nice properties of real spaces is that we can define a norm and a notion of distance in a rather concrete manner. Most of the course deals with real functions and subsets of ${n}$-dimensional real space. So unless otherwise stated the discussion will be about real spaces and functions. We first begin with the definition of a norm. A norm on ${{\mathbb R}^n}$ is a function ${f(x)}$ that maps every ${x \in {\mathbb R}^d}$ to a non-negative real number with the following properties:
1. Let ${a \in {\mathbb R}}$ and ${x \in {\mathbb R}^d}$, then ${f(ax)=|a|f(x)}$.
2. ${f(x+y)\leq f(x)+f(y)}$.
3. If ${f(x)=0}$ then ${x=0}$
The most popular norm on ${{\mathbb R}^d}$ is the Euclidean norm. For any point ${x =(x_1,\hdots,x_d) \in {\mathbb R}^d}$, its Euclidean norm is defined as
$\displaystyle \|x\|_2 = \sqrt{\sum_{i=1}^d x_i^2} =\sqrt{x^Tx}$
Other examples of norms are
1. ${\|x\|_1=\sum_{i=1}^d |x_i|}$
2. ${\|x\|_\infty=\max_{i=1,...d}|x_i|}$
3. ${\|x\|_p =(\sum_{i=1}^d |x_i|^p)^{1/p}}$
Hw 4 Prove that ${\|x\|_1}$,${\|x\|_2}$, ${\|x\|_\infty}$ satisfy the properties of a norm.
Two norms ${\| . \|}$ and ${\| . \|'}$ are equivalent, if there exists two positive constants ${c_1}$ and ${c_2}$ such that for any ${x \in {\mathbb R}^d}$
$\displaystyle c_2 \|x\|' \leq \| x\|\leq c_1 \|x\|'.$
Observe that ${c_1}$ and ${c_2}$ do not depend on ${x}$. Let us now show that ${\|.\|_2}$ and ${\|.\|_\infty}$ are equivalent. Let ${x = (x_1,\hdots, x_n)}$ be some element in ${{\mathbb R}^n}$. Then
$\displaystyle \begin{array}{rcl} \|x\|_2^2 &=& \sum_{i=1}^n x_i^2,\\ &\stackrel{(a)}{\leq}&\sum_{i=1}^n \max (x_i^2),\\ &=&\max (x_i^2)\sum_{i=1}^n 1= \|x\|_{\infty}^2 n. \end{array}$
Where ${(a)}$ follows since any ${i}$, ${x_i \leq \max(x_i)}$. Taking square root on both sides we have, ${\|x\|_2\leq \sqrt{n}\|x\|_{\infty}}$. We also have
$\displaystyle \begin{array}{rcl} \|x\|_2^2 &=& \sum_{i=1}^n x_i^2,\\ &\geq &\max({x_i}^2),\\ &=&\|x\|_{\infty}^2. \end{array}$
Taking square root we obtain ${\|x\|_2\geq \|x\|_{\infty}}$. Hence we have
$\displaystyle \|x\|_{\infty} \leq \|x\|_2\leq \sqrt{n}\|x\|_{\infty},$
which show that the norms ${\|x\|_\infty}$ and ${\|x\|_2}$ are equivalent. The following theorem shows that all norms in ${{\mathbb R}^d}$ are equivalent.
Theorem 1 In ${{\mathbb R}^d}$ (or on any finite dimensional vector space), all the norms are equivalent.
Proof: The proof of this follows from a simple optimization problem. We will prove this later in the course. $\Box$
Hw 5 Show that ${\|x\|_2}$, ${\|x\|_1}$ and ${\|x\|_\infty}$ norms are equivalent.
From now on we assume that we are provided with some norm on ${{\mathbb R}^n}$ which we denote by ${\|.\|}$. A norm can be used to define a distance metric as
$\displaystyle d(x,y) = \|x-y\|.$
It is easy to prove that ${d(x,y)}$ is a metric. The only thing to prove is the triangle inequality which follows in a straight forward manner from the norm definition.
One of the most fundamental geometric object is a ball. A ball centered at ${x}$ and radius ${r}$ is denoted by ${B(x,r)}$ and is defined as
$\displaystyle B(x,r)= \{y : y \in {\mathbb R}^2, \|y-x\|
Observe the strict inequality. This is called as an open ball.
Hw 6
1. Plot ${B(o,1)}$ when the norm is given by ${\|.\|_1}$ in ${{\mathbb R}^2}$.
2. Plot ${B(o,1)}$ when the norm is given by ${\|.\|_2}$ in ${{\mathbb R}^2}$.
3. Plot ${B(o,1)}$ when the norm is given by ${\|.\|_\infty}$ in ${{\mathbb R}^2}$.
Explain what equivalence of these three norms in terms of the plots.
Hw 7 Let ${A}$ be a ${n\times n}$ real matrix. For any ${x \in {\mathbb R}^n}$ define
$\displaystyle \|x\|_A =\|Ax\|_2.$
Is ${\|x\|_A }$ a norm in ${{\mathbb R}^n}$ a norm for any matrix ${A}$?. What are the properties of ${A}$ that will make ${\|.\|_A }$ a norm?
Hw 8 Let ${A}$ be a ${n\times n}$ real matrix. For any ${x \in {\mathbb R}^n}$ define
$\displaystyle \|x\|_A =\sqrt{x^T A x}.$
Is ${\|x\|_A }$ a norm in ${{\mathbb R}^n}$ a norm for any matrix ${A}$?. What are the properties of ${A}$ that will make ${\|.\|_A }$ a norm?
Hw 9 Let ${S}$ denote a symmetric ${n\times n}$ matrix. A norm on the set of symmetric matrices can be defined as
$\displaystyle \|S\| =\sqrt{ Tr(SS^T)}=\sqrt{\sum_{i=1}^n\sum_{j=1}^n s_{ij}^2}.$
Show that ${\|S\|}$ is a norm on symmetric real matrices.
\subsubsection{Dual norm} For any norm ${\|\|}$, its dual is defined as
$\displaystyle \|x\|_* = \sup\{x^Ty \ |\quad \|y\|\leq 1\}$
We will first show that ${\|x\|_* }$ is a norm. We will show all the properties
1. Homogeneity: First assume ${a>0}$,
$\displaystyle \begin{array}{rcl} \|ax\|_* &=& \sup\{ax^Ty \ |\quad \|y\|\leq 1\}\\ &=&a\sup\{x^Ty \ |\quad \|y\|\leq 1\}\\ &=&a\|x\|_* \end{array}$
Now if ${a<0}$, then
$\displaystyle \begin{array}{rcl} \|ax\|_* &=&\sup\{ax^Ty \ |\quad \|y\|\leq 1\}\\ &=&|a|\sup\{-x^Ty \ |\quad \|y\|\leq 1\}\\ &=&|a|\sup\{x^T(-y) \ |\quad \|y\|\leq 1\}\\ &=&|a|\sup\{x^Ty \ |\quad \|y\|\leq 1\}\\ &=&|a|\|x\|_* \end{array}$
2. Suppose ${\|x\|_*=0}$. This implies ${\sup\{x^Ty \ |\quad \|y\|\leq 1\}=0}$. However if we choose ${y = x/\|x\|}$, then ${x^Ty =1}$ which is a contradiction. Hence ${x=o}$.
3. Triangle inequality:
$\displaystyle \begin{array}{rcl} \|x+z\|_* &=& \sup\{(x+y)^Ty \ |\quad \|y\|\leq 1\}\\ &=& \sup\{x^Ty +z^Ty \ |\quad \|y\|\leq 1\}\\ &\leq&\sup\{x^Ty \ |\quad \|y\|\leq 1\}+\sup\{z^Ty \ |\quad \|y\|\leq 1\}\\ &=&\|x\|_*+\|z\|_* \end{array}$
Also observe that from the proof we observe that ${\|x\|_* \geq 1}$ when ${x\neq o}$. The dual norm of ${\| \|_2}$ is again the ${\| \|_2}$ norm, while for ${\| \|_p}$ is the conjugate norm of ${\| \|_q}$, where ${p^{-1}+q^{-1}=1}$. We have the following important lemma.
Lemma 2
$\displaystyle x^Tz\leq \|x\|_*\|z\| .$
Proof: From the definition we have
$\displaystyle \|x\|_* = \sup\{x^Ty \ |\quad \|y\|\leq 1\}.$
Since ${\|x\|_*}$ is the supremum of ${x^Ty }$ over all norm ${1}$ ${y}$ vectors, picking any particular one will only give a lower bound on ${\|x\|_*}$. Pick ${y= z/\|z\|}$. Observe that ${\|y\|=1}$. Hence
$\displaystyle \|x\|_* \geq \frac{ x^T z}{\|z\|},$
which proves the result.
$\Box$
2.2. Sequences
A sequence ${X_n = \{x_1,x_2,x_3,x_4,\dots\}}$ in ${{\mathbb R}}$ (or any space), is a map from the set of integers to ${{\mathbb R}}$ (or the space in consideration). We will now look at some important properties regarding the convergence of sequences. We say that a sequence ${x_n}$ converges to ${x}$ if for every ${\epsilon >0}$, there exists some ${k}$ such that the distance between ${x_n}$ and ${x}$ for all ${n>k}$ is less than ${\epsilon}$, \ie,
$\displaystyle d(x_n,x)=\|x_n-x\|< \epsilon, \quad \forall n> k.$
Let us think about a game: Suppose you want to argue (prove) that a sequence ${x_n}$ converges to some limit ${x}$ to your friend who is not convinced and challenges you. Your friend gives you an ${\epsilon}$, say ${\epsilon =0.01}$. Then you should show that you can find some ${k}$ such that the distance between the terms of the sequence after ${x_k}$ and ${x}$, \ie, ${\|x_n-x\|, n>k}$ is smaller than the ${\epsilon}$ provided by your friend. Observe that you can choose whatever ${k}$ you want. So ${\epsilon}$ is provided by someone (your friend) and you will have to find the required ${k}$. Now let us see some examples.
1. Let ${x_n = 1}$ for all ${n}$. It is obvious that the limit of the sequence is ${1}$. However, let us check it more formally. Suppose your friend gives you ${\epsilon =0.1}$. You can observe that you can choose ${k=1}$, since ${\|x_n-1\|=0<0.1}$. Even if he had given you ${\epsilon =0.000001}$, ${k=1}$ would be valid.
2. Let ${x_n =1/n}$. Of course the limit is ${x=0}$. Suppose your friend gives ${\epsilon =1}$. Let us see how to choose the correct ${k}$. So we want
$\displaystyle \|x_n-0\|= \|1/n\|<1,$
which is always true. So you can pick ${k=1}$. Now suppose your friend challenges you for ${\epsilon =0.01}$, then you would want
$\displaystyle \|1/n\| = 1/n < 0.01.$
This is possible if ${n \geq 101}$. So you can answer your friend that the ${k}$ corresponding to ${\epsilon =0.01}$ is ${101}$.
3. Let ${x_n=1/n}$ except that we replace some values as follows: ${x_{102}= 1}$, ${x_{200}=10}$, ${x_{500}=-10}$ and ${x_{1000}=100}$. You claim that the sequence still converges to ${x=0}$. Then your friend challenges to ${\epsilon =0.01}$. So what do you pick ${k}$ as. As last time picking ${k=101}$ will not work since ${\|x_{102}-0\|= |1-0|> 0.01}$. But you can see that choosing ${k= 1001}$ will work, since ${\|x_{n}-0\|<0.01}$ for all ${n>1001}$. So only the values of ${x_n}$ for large ${n}$ will matter and fluctuations for a finite set of values wont matter.
Hw 10 Formally prove the following limits.
1. ${\lim_{n \rightarrow \infty}\frac{1}{1+n}= 0}$
2. ${\lim_{n\rightarrow\infty}(-1)^n \exp(-n) =0}$
Let ${x_n =1+1/n}$. Observe that ${x_n \in (0,1)}$ for all ${n}$. However, the limit ${0}$ does not belong to the set ${(0,1)}$. In the coming sections, we will characterize sets so that when the sequence belong to the set, the limit also belongs to the set. | {"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": 1392, "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": 0.9955594539642334, "perplexity": 253.20174176361147}, "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-2020-34/segments/1596439735916.91/warc/CC-MAIN-20200805065524-20200805095524-00367.warc.gz"} |
https://asu.pure.elsevier.com/en/publications/parameterized-maximum-and-average-degree-approximation-in-topic-b-3 | # Parameterized maximum and average degree approximation in topic-based publish-subscribe overlay network design
Melih Onus, Andrea Richa
Research output: Contribution to journalArticlepeer-review
11 Scopus citations
## Abstract
Publish/subscribe communication systems where nodes subscribe to many different topics of interest are becoming increasingly more common. Designing overlay networks that connect the nodes subscribed to each distinct topic is hence a fundamental problem in these systems. For scalability and efficiency, it is important to keep the degree of the nodes in the publish/subscribe system low. Ideally one would like to be able not only to keep the average degree of the nodes low, but also to ensure that all nodes have equally the same degree, giving rise to the following problem: Given a collection of nodes and their topic subscriptions, connect the nodes into a graph with low average and maximum degree such that for each topic t, the graph induced by the nodes interested in t is connected. We present the first polynomial time parameterized sublinear approximation algorithm for this problem. We also propose a heuristic for constructing topic-connected networks with low average degree and diameter 2 and validate our results through simulations.
Original language English (US) 307-317 11 Computer Networks 94 https://doi.org/10.1016/j.comnet.2015.10.023 Published - Jan 15 2016
## Keywords
• multicast
• optimization problems
• overlay networks
• peer to peer
• publish/subscribe
## ASJC Scopus subject areas
• Computer Networks and Communications
## Fingerprint
Dive into the research topics of 'Parameterized maximum and average degree approximation in topic-based publish-subscribe overlay network design'. Together they form a unique fingerprint. | {"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": 0.8946542739868164, "perplexity": 1735.6454408433724}, "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/1634323587659.72/warc/CC-MAIN-20211025092203-20211025122203-00712.warc.gz"} |
https://nips.cc/Conferences/2015/ScheduleMultitrack?event=5766 | `
Timezone: »
Poster
Lifted Inference Rules With Constraints
Happy Mittal · Anuj Mahajan · Vibhav Gogate · Parag Singla
Wed Dec 09 04:00 PM -- 08:59 PM (PST) @ 210 C #56 #None
Lifted inference rules exploit symmetries for fast reasoning in statistical rela-tional models. Computational complexity of these rules is highly dependent onthe choice of the constraint language they operate on and therefore coming upwith the right kind of representation is critical to the success of lifted inference.In this paper, we propose a new constraint language, called setineq, which allowssubset, equality and inequality constraints, to represent substitutions over the vari-ables in the theory. Our constraint formulation is strictly more expressive thanexisting representations, yet easy to operate on. We reformulate the three mainlifting rules: decomposer, generalized binomial and the recently proposed singleoccurrence for MAP inference, to work with our constraint representation. Exper-iments on benchmark MLNs for exact and sampling based inference demonstratethe effectiveness of our approach over several other existing techniques. | {"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": 0.8453201651573181, "perplexity": 4040.653153234034}, "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-2022-27/segments/1656103984681.57/warc/CC-MAIN-20220702040603-20220702070603-00565.warc.gz"} |
https://www.gradesaver.com/beowulf/q-and-a/what-is-the-probable-effect-of-the-lines-above-on-beowulf-190-232-260072 | # What is the probable effect of the lines above on Beowulf? (190-232)
What is the probable effect of the lines above on Beowulf? (190-232) | {"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": 0.9578354954719543, "perplexity": 4418.926314231688}, "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-2018-43/segments/1539583517628.91/warc/CC-MAIN-20181024001232-20181024022732-00461.warc.gz"} |
http://math.stackexchange.com/questions/147701/all-groups-of-order-175-are-abelian | # All groups of order 175 are abelian?
Question. All groups of order 175 are abelian?
I can show that there exists only one Sylow 5-subgroup of order 25, call it $H$, and one Sylow 7-subgroup of order 7, denote $K$.
I know that $K$ is cyclic, and thus abelian. I know that $|H| = p^2$, where $p=5$ is prime, and so $H$ is abelian too. I also know the $|G| = |H| \cdot |K|$.
Further I know that G happens to be the direct product of these two groups as they intersect trivially, and this completes the proof.
1. Why the group is the direct product, is this always so if the groups intersect trivially, and the product of the orders of subgroups matches the group order?
2. Why the direct product is abelian. Is this always the case if the subgroups $H$ and $K$ intersect trivialy, or is it because they are both abelian too?
3. Anything else I should know?
Thanks
-
Ad 2) The direct product of abelian groups is abelian. As the product is direct by (1), $G \cong H \times K$ is abelian. – martini May 21 '12 at 11:07
Only because they intersect trivially? – rk101 May 21 '12 at 11:10
For (1) , you need normality too. As in, you need $H$ and $K$ to be both normal (i.e. there is precisely one Sylow $p$-group and one Sylow $q$ groups where $|G|=p^aq^b$). This works, as for $G$ to be a direct product you need to find two normal subgroups $H$ and $K$ which intersect trivially (which you have) and such that $G=HK$ (which you also have, but proving this is slightly more subtle - but I am sure this is not outwith your abilities. For (3), there are many things you should know. See, for example, bbc.co.uk/news and tearfund.org) – user1729 May 21 '12 at 11:12
(Also, for (1), let $G$ be a group where $G=HK$ and $H\cap K=1$. If both of $H$ and $K$ are normal $G$ is called a direct product of $H$ and $K$, $G=H\times K$, if one one is normal ($H$ say) then $G$ is called a semidirect direct product of $H$ and $K$, $H\rtimes K$, while if neither is normal then $G$ is called a Zappa-Szep product of $H$ and $K$, $G=H\bowtie K$ (see my answer here math.stackexchange.com/questions/107781/…) – user1729 May 21 '12 at 11:16
@rk101, see also math.stackexchange.com/questions/67407/… – Nicky Hekster May 21 '12 at 22:28 | {"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": 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": 0.9462315440177917, "perplexity": 224.80634149836314}, "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-2014-23/segments/1404776428349.3/warc/CC-MAIN-20140707234028-00000-ip-10-180-212-248.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/68083/better-looking-nested-subscripts-and-superscripts | # Better looking nested subscripts and superscripts
I have a situation where I need to subscript a superscript in math mode (and actually, subscript that subscript in some cases). (Please don't waste time telling me to change my notation; this is the notation used historically and I can't change it). I've noticed that LaTeX doesn't handle this very well, and it is very difficult to tell that the subscripted superscript is subordinate, particularly in the case of numbers like 1, 2, and 3. Is there a way to adjust this? I've tried
A^{a_{{}_1}}
which looks okay, but is far from ideal. Any suggestions?
-
see also tex.stackexchange.com/q/38100/5001 for a discussion of how to typeset subscripts of subscripts – Mico Aug 21 '12 at 21:26
You can change more or less anything without changing your markup, the font size and position of superscripts is all customisable with care.
You didn't say too much about what you didn't like about the default setting. This shows some possibilities using different font sizes or setting the fontdimen parameters to affect the script positioning.
\documentclass{article}
\textwidth3cm
\begin{document}
1$A^{a_{{}_1}}$
2$A^{a_1}$
3$A^{\textstyle a_1}$
4$A^{\textstyle a_{\scriptscriptstyle 1}}$
\fontdimen16\the\scriptfont2=5pt
5$A^{a_1}$
\end{document}
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https://math.msu.edu/seminars/TalkView.aspx?talk=837 | ## Colloquium
• Sean Li, University of Chicago
• Beyond Euclidean rectifiability
• 01/23/2017
• 4:10 PM - 5:00 PM
• C304 Wells Hall
Rectifiable spaces are a class of metric measure spaces that are Lipschitz analogues of differentiable manifolds (for example, they admit a parameterization by Lipschitz charts) and arise naturally in many areas of analysis and geometry. Due to the important works of Federer, Mattila, Preiss, and many others, we now have a good understanding of the geometric properties of rectifiability in Euclidean spaces. In this talk, we examine some generalizations of rectifiability to the setting of non-Euclidean spaces and discuss the similarities and differences between rectifiability in the Euclidean setting and these generalizations.
## Contact
Department of Mathematics
Michigan State University | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8990663886070251, "perplexity": 1164.720051714148}, "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/1544376826306.47/warc/CC-MAIN-20181214184754-20181214210754-00466.warc.gz"} |
https://www.physicsforums.com/threads/fourier-series-matlab-syntax.313078/ | Fourier Series/ MatLab syntax
1. May 10, 2009
cambone69
1. The problem statement, all variables and given/known data
(a) For each function determine the first 10 non zero elements in the Fourier series representation. Use Matlab to compute the coefficients, and graph the result.
2. Relevant equations
N/A
3. The attempt at a solution
I noted the even symmetry of the function, and thus I do not have to compute the "bn" (sine) coefficients. Also the fact that I can compute the cosine coefficients by integrating over half of the period, ie 0--->T/2. I know how to compute the coefficients by hand, however I am required to use Matlab.
In MatLab how would I define the function f (which appears to be a unit step function on half the period) in MatLab, with which I could multiply by cosine and then integrate appropriately?
This is a simple problem, however Matlab has become the biggest roadblock here.
Any help would be greatly appreciated.
Cam
Can you offer guidance or do you also need help?
Draft saved Draft deleted
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https://fabrizziosoares.netlify.app/publication/2018/franca2018/ | # Scientific computing in Statistical Mechanics: time decay of orientation order in 2D hard disk system
### Abstract
In this work we successfully applied diverse computational techniques to calculate important quantities in one problem in Statistical Mechanics: the hard disk system in two dimensions. We calculated the global and local orientation order and the time decay constant of the global orientation correlation function. We also computed the Voronoi construction to visualize the spatial distribution of the local orientation order.
Type
Publication
In Proceedings of XI Academic Meeting of Computational Modeling (XI ENCONTRO ACADÊMICO MODELAGEM COMPUTACIONAL)
Date | {"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": 0.9174817204475403, "perplexity": 1385.9428879502132}, "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/1596439735867.94/warc/CC-MAIN-20200804131928-20200804161928-00554.warc.gz"} |
https://brilliant.org/problems/a-problem-by-david-mattingly/ | # A problem by David Mattingly
Level 2
If light travels infinitely fast then Finn's clock will not tick between the sending and receiving of the light. Therefore $$F_1=A_r=F_f$$. This means that every observer has the exact same tick value on their synchronized clocks - i.e. there is a global time function $$t$$ defined everywhere. No matter where various observers are, on the earth, moon, Jupiter, etc. they can all set their clocks to tick together. We define time as the single, unique global value of the ticking clocks. (Remember, this is in Newtonian mechanics).
This physical assumption about how light travels is reflected in the basic structure of Newtonian mechanics, although you may never have realized it. Consider the Galilean transformations that relate the $$x,t$$ coordinates of one observer (Finn) with the $$x',t'$$ coordinates of another observer (Aaron) moving at a relative velocity. Do the Galilean transformations change the time used by the two observers? i.e. is $$t'=t$$ or not?
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https://www.science.gov/topicpages/0-9/14-mev+incident+neutrons.html | #### Sample records for 14-mev incident neutrons
1. LOW VOLTAGE 14 Mev NEUTRON SOURCE
DOEpatents
Little, R.N. Jr.; Graves, E.R.
1959-09-29
An apparatus yielding high-energy neutrons at the rate of 10/sup 8/ or more per second by the D,T or D,D reactions is described. The deuterium gas filling is ionized by electrons emitted from a filament, and the resulting ions are focused into a beam and accelerated against a fixed target. The apparatus is built in accordance with the relationship V/sub s/ = A--B log pd, where V/sub s/ is the sparking voltage, p the gas pressure, and d the gap length between the high voltage electrodes. Typical parameters to obtain the high neutron yields are 55 to 80 kv, 0.5 to 7.0 ma beam current, 5 to 12 microns D/sub 2/, and a gap length of 1 centimeter.
2. Material Classification by Analysis of Prompt Photon Spectra Induced by 14-Mev Neutrons
Barzilov, Alexander; Novikov, Ivan
Neutron based technologies are widely used in the field of bulk material analysis. These methods employ characteristic prompt gamma rays induced by a neutron probe for classification of the interrogated object using the elemental parameters extracted from the spectral data. Automatic data analysis and material classification algorithms are required for applications where access to nuclear spectroscopy expertise is limited and/or the autonomous robotic operation is necessary. Data obtained with neutron based systems differ from elemental composition evaluations based on chemical formulae due to statistical nature of nuclear reactions, presence of shielding and cladding, and other environmental conditions. Experimental data that are produced by the spectral decomposition can be expressed graphically as sets of overlapping classes in a multidimensional space of measured elemental intensities. To discriminate between classes of various materials, decision-tree and pattern recognition algorithms were studied. Results of application of these methods to data sets obtained for a pulsed 14-MeV neutron generator based active interrogation system are discussed.
3. FEASIBILITY OF MEASURING IRON IN VIVO USING FAST 14 MEV NEUTRONS.
SciTech Connect
WIELOPOLSKI, L.
2005-05-01
In this short report, I reassess the feasibility of measuring iron in vivo in the liver and heart of thalassemia patients undergoing chelation therapy. Despite the multiplicity of analytical methods for analyzing iron, only two, magnetic resonance imaging, and magnetic susceptibility, are suitable for in vivo applications, and these are limited to the liver because of the heart's beat. Previously, a nuclear method, gamma-resonance scattering, offered a quantitative measure of iron in these organs; however, it was abandoned because it necessitated a nuclear reactor to produce the radioactive source. I reviewed and reassessed the status of two alternative nuclear methods, based on iron spectroscopy of gamma rays induced by fast neutron inelastic scattering and delayed activation in iron. Both are quantitative methods with high specificity for iron and adequate penetrating power to measure it in organs sited deep within the human body. My experiments demonstrated that both modalities met the stated qualitative objectives to measure iron. However, neutron dosimetry revealed that the intensity of the neutron radiation field was too weak to reliably assess the minimum detection limits, and to allow quantitative extrapolations to measurements in people. A review of the literature, included in this report, showed that these findings agree qualitatively with the published results, although the doses reported were about three orders-of-magnitude higher than those I used. Reviewing the limitations of the present work, steps were outlined for overcoming some of the shortcomings. Due to a dearth of valid quantitative alternatives for determining iron in vivo, I conclude that nuclear methods remain the only viable option. However, from the lessons learned, further systematic work is required before embarking on clinical studies.
4. Grazing Incidence Neutron Optics
NASA Technical Reports Server (NTRS)
Gubarev, Mikhail V. (Inventor); Ramsey, Brian D. (Inventor); Engelhaupt, Darell E. (Inventor)
2013-01-01
Neutron optics based on the two-reflection geometries are capable of controlling beams of long wavelength neutrons with low angular divergence. The preferred mirror fabrication technique is a replication process with electroform nickel replication process being preferable. In the preliminary demonstration test an electroform nickel optics gave the neutron current density gain at the focal spot of the mirror at least 8 for neutron wavelengths in the range from 6 to 20.ANG.. The replication techniques can be also be used to fabricate neutron beam controlling guides.
5. Grazing incidence neutron optics
NASA Technical Reports Server (NTRS)
Gubarev, Mikhail V. (Inventor); Ramsey, Brian D. (Inventor); Engelhaupt, Darell E. (Inventor)
2012-01-01
Neutron optics based on the two-reflection geometries are capable of controlling beams of long wavelength neutrons with low angular divergence. The preferred mirror fabrication technique is a replication process with electroform nickel replication process being preferable. In the preliminary demonstration test an electroform nickel optics gave the neutron current density gain at the focal spot of the mirror at least 8 for neutron wavelengths in the range from 6 to 20 .ANG.. The replication techniques can be also be used to fabricate neutron beam controlling guides.
6. Grazing-Incidence Neutron Optics based on Wolter Geometries
NASA Technical Reports Server (NTRS)
Gubarev, M. V.; Ramsey, B. D.; Mildner, D. F. R.
2008-01-01
The feasibility of grazing-incidence neutron imaging optics based on the Wolter geometries have been successfully demonstrated. Biological microscopy, neutron radiography, medical imaging, neutron crystallography and boron neutron capture therapy would benefit from high resolution focusing neutron optics. Two bounce optics can also be used to focus neutrons in SANS experiments. Here, the use of the optics would result in lower values of obtainable scattering angles. The high efficiency of the optics permits a decrease in the minimum scattering vector without lowering the neutron intensity on sample. In this application, a significant advantage of the reflective optics over refractive optics is that the focus is independent of wavelength, so that the technique can be applied to polychromatic beams at pulsed neutron sources.
7. Development of Grazing Incidence Optics for Neutron Imaging and Scattering
NASA Technical Reports Server (NTRS)
Gubarev, M. V.; Khaykovich, B.; Liu, D.; Ramsey, B. D.; Zavlin, V. E.; Kilaru, K.; Romaine, S.; Rosati, R. E.; Bruni, R.; Moncton, D. E.
2012-01-01
Because of their wave nature, thermal and cold neutrons can be reflected from smooth surfaces at grazing incidence angles, be reflected by multilayer coatings or be refracted at boundaries of different materials. The optical properties of materials are characterized by their refractive indices which are slightly less than unity for most elements and their isotopes in the case of cold and thermal neutrons as well as for x-rays. The motivation for the optics use for neutrons as well as for x-rays is to increase the signal rate and, by virtue of the optic's angular resolution, to improve the signal-to-noise level by reducing the background so the efficiency of the existing neutron sources use can be significantly enhanced. Both refractive and reflective optical techniques developed for x-ray applications can be applied to focus neutron beams. Typically neutron sources have lower brilliance compared to conventional x-ray sources so in order to increase the beam throughput the neutron optics has to be capable of capturing large solid angles. Because of this, the replicated optics techniques developed for x-ray astronomy applications would be a perfect match for neutron applications, so the electroformed nickel optics under development at the Marshall Space Flight Center (MSFC) can be applied to focus neutron beams. In this technique, nickel mirror shells are electroformed onto a figured and superpolished nickel-plated aluminum cylindrical mandrel from which they are later released by differential thermal contraction. Cylindrical mirrors with different diameters, but the same focal length, can be nested together to increase the system throughput. The throughput can be increased further with the use of the multilayer coatings deposited on the reflectivr surface of the mirror shells. While the electroformed nickel replication technique needs to be adopted for neutron focusing, the technology to coat the inside of cylindrical mirrors with neutron multilayers has to be
8. An Evaluation of Grazing-Incidence Optics for Neutron Imaging
NASA Technical Reports Server (NTRS)
Gubarev, M. V.
2007-01-01
The refractive index for most materials is slightly less than unity, which opens an opportunity to develop the grazing incidence neutron imaging optics. The ideal material for the optics would be natural nickel and its isotopes. Marshall Space Flight Center (MSFC) has active development program on the nickel replicated optics for use in x-ray astronomy. Brief status report on the program is presented. The results of the neutron focusing optic test carried by the MSFC team at National Institute of Standards and Technology (NIST) are also presented. Possible applications of the optics are briefly discussed.
9. Experimental verification of improved depth-dose distribution using hyper-thermal neutron incidence in neutron capture therapy.
PubMed
Sakurai, Y; Kobayashi, T
2001-01-01
We have proposed the utilization of 'hyper-thermal neutrons' for neutron capture therapy (NCT) from the viewpoint of the improvement in the dose distribution in a human body. In order to verify the improved depth-dose distribution due to hyper-thermal neutron incidence, two experiments were carried out using a test-type hyper-thermal neutron generator at a thermal neutron irradiation field in Kyoto University Reactor (KUR), which is actually utilized for NCT clinical irradiation. From the free-in-air experiment for the spectrum-shift characteristics, it was confirmed that the hyper-thermal neutrons of approximately 860 K at maximum could be obtained by the generator. From the phantom experiment, the improvement effect and the controllability for the depth-dose distribution were confirmed. For example, it was found that the relative neutron depth-dose distribution was about 1 cm improved with the 860 K hyper-thermal neutron incidence, compared to the normal thermal neutron incidence.
10. Neutron-induced fission: properties of prompt neutron and γ rays as a function of incident energy
Stetcu, I.; Talou, P.; Kawano, T.
2016-06-01
We have applied the Hauser-Feshbach statistical theory, in a Monte-Carlo implementation, to the de-excitation of fission fragments, obtaining a reasonable description of the characteristics of neutrons and gamma rays emitted before beta decays toward stability. Originally implemented for the spontaneous fission of 252Cf and the neutroninduced fission of 235U and 239Pu at thermal neutron energy, in this contribution we discuss the extension of the formalism to incident neutron energies up to 20 MeV. For the emission of pre-fission neutrons, at incident energies beyond second-chance fission, we take into account both the pre-equilibrium and statistical pre-fission components. Phenomenological parameterizations of mass, charge and TKE yields are used to obtain the initial conditions for the fission fragments that subsequently decay via neutron and emissions. We illustrate this approach for 239Pu(n,f).
11. An Evaluation of Grazing-Incidence Optics for Neutron Imaging
NASA Technical Reports Server (NTRS)
Gubarev, M. V.; Ramsey, B. D.; Engelhaupt, D. E.; Burgess, J.; Mildner, D. F. R.
2007-01-01
The focusing capabilities of neutron imaging optic based on the Wolter-1 geometry have been successfully demonstrated with a beam of long wavelength neutrons with low angular divergence.. A test mirror was fabricated using an electroformed nickel replication process at Marshall Space Flight Center. The neutron current density gain at the focal spot of the mirror is found to be at least 8 for neutron wavelengths in the range from 6 to 20 A. Possible applications of the optics are briefly discussed.
12. Analysis of incident-energy dependence of delayed neutron yields in actinides
SciTech Connect
Nasir, Mohamad Nasrun bin Mohd Metorima, Kouhei Ohsawa, Takaaki Hashimoto, Kengo
2015-04-29
The changes of delayed neutron yields (ν{sub d}) of Actinides have been analyzed for incident energy up to 20MeV using realized data of precursor after prompt neutron emission, from semi-empirical model, and delayed neutron emission probability data (P{sub n}) to carry out a summation method. The evaluated nuclear data of the delayed neutron yields of actinide nuclides are still uncertain at the present and the cause of the energy dependence has not been fully understood. In this study, the fission yields of precursor were calculated considering the change of the fission fragment mass yield based on the superposition of fives Gaussian distribution; and the change of the prompt neutrons number associated with the incident energy dependence. Thus, the incident energy dependent behavior of delayed neutron was analyzed.The total number of delayed neutron is expressed as ν{sub d}=∑Y{sub i} • P{sub ni} in the summation method, where Y{sub i} is the mass yields of precursor i and P{sub ni} is the delayed neutron emission probability of precursor i. The value of Y{sub i} is derived from calculation of post neutron emission mass distribution using 5 Gaussian equations with the consideration of large distribution of the fission fragments. The prompt neutron emission ν{sub p} increases at higher incident-energy but there are two different models; one model says that the fission fragment mass dependence that prompt neutron emission increases uniformly regardless of the fission fragments mass; and the other says that the major increases occur at heavy fission fragments area. In this study, the changes of delayed neutron yields by the two models have been investigated.
13. Neutron scattering in concrete and wood: Part II--Oblique incidence.
PubMed
Facure, A; Silva, A X; Rivera, J C; Falcão, R C
2008-01-01
The knowledge of neutron reflection coefficients is of practical interest when projecting the shielding of radiotherapy rooms, since it is known that about 75% of the neutrons at the maze entrance of these rooms are scattered neutrons. In a previous paper, the energy spectra of photoneutrons were calculated, when reflected by ordinary, high-density concrete and wood barriers, using the MCNP5 code, considering normal incidence and neutron incident energies varying between 0.1 and 10 MeV. It was found that the mean energy of the reflected neutrons does not depend on the reflection angle and that these mean energies are lower in wood and barytes concrete, compared with ordinary concrete. In the present work, the simulation of neutron reflection coefficients were completed, considering the case when these particles do not collide frontally with the barriers, which constitute the radiotherapy room walls. Some simulations were also made to evaluate how neutron equivalent doses at the position of the room door is affected when the maze walls are lined with neutron absorbing materials, such as wood itself or borated polyethylene. Finally, capture gamma rays dose at the entrance of rooms with different maze lengths were also simulated. The results were discussed in the light of the albedo concepts presented in the literature and some of these results were confronted with others, finding good agreement between them.
14. Grazing incidence neutron spin echo spectroscopy: instrumentation aspects and scientific opportunities
Holderer, O.; Frielinghaus, H.; Wellert, S.; Lipfert, F.; Monkenbusch, M.; von Klitzing, R.; Richter, D.
2014-07-01
Grazing Incidence Neutron Spin Echo Spectroscopy (GINSES) opens new possibilities for observing the thermally driven dynamics of macromolecules close to a rigid interface. The information about the dynamics can be retrieved as a function of scattering depth of the evanescent neutron wave, on the length scale in the range of some 10-100 nm. Using a classical neutron spin echo spectrometer with a laterally collimated beam, dynamics can be measured in grazing incidence geometry. We show examples of how the interface modifies the dynamics of microemulsions, membranes and microgels. Instrumental details and possible improvements for this technique will be presented. The key issue is the low intensity for dynamics measurements with an evanescent neutron wave. Conceptual questions how a specialised instrument could improve the experimental technique will be discussed.
15. Detailed characterisation of the incident neutron beam on the TOSCA spectrometer
Pinna, Roberto S.; Rudić, Svemir; Capstick, Matthew J.; McPhail, David J.; Pooley, Daniel E.; Howells, Gareth D.; Gorini, Giuseppe; Fernandez-Alonso, Felix
2017-10-01
We report a detailed characterisation of the incident neutron beam on the TOSCA spectrometer. A bespoke time-of-flight neutron monitor has been designed, constructed and used to perform extensive spatially resolved measurements of the absolute neutron flux and its underlying time structure at the instrument sample position. The obtained data give a quantitative understanding of the current instrument beyond neutronic simulations and provide a baseline in order to assess the performance of the upgraded instrument. At an average proton current-on-target of 153 μA (ISIS Target Station 1; at the time of measurements) we have found that the wavelength-integrated neutron flux (from 0.28 Å to 4.65 Å) at the position of the TOSCA instrument sample (spatially averaged across the 3 × 3cm2 surface centred around (0,0) position) is approximately 1 . 2 × 106 neutrons cm-2s-1, while the whole beam has a homogeneous distribution across the 3 . 0 × 3 . 5cm2 sample surface. The spectra reproduced the well-known shape of the neutrons moderated by the room temperature water moderator and exhibit a neutron flux of 7 . 3 × 105 neutrons cm-2s-1Å-1 at 1 Å.
16. Neutron lifetime measurement with pulsed beam at J-PARC:Incident Beam Flux
Sakakibara, Risa; Shimizu, Hirohiko M.; Kitaguchi, Masaaki; Hirota, Katsuya; Sugino, Tomoaki; Yamashita, Satoru; Katayama, Ryo; Yamada, Takahito; Higashi, Nao; Yokoyama, Harumichi; Sumino, Hirochika; Yoshioka, Tamaki; Otono, Hidetoshi; Tanaka, Genki; Sumi, Naoyuki; Iwashita, Yoshihisa; Kitahara, Ryunosuke; Oide, Hideyuki; Shima, Tatsushi; Ino, Takashi; Mishima, Kenji; Taketani, Kaoru; Seki, Yoshichika; NOP Collaboration
2014-09-01
The neutron lifetime is one of the important parameters in the estimation of the abundance of the light elements in the early universe through the Big Bang Nucleosynthesis (BBN). The accuracy of 0.1% is desired in the neutron lifetime to quantitatively discuss the BBN in combination with the observation of the anisotropy of the cosmic microwave. We have started a lifetime measurement with pulsed neutrons at J-PARC/BL05. To measure the lifetime, we detect the decay electrons from the bunched neutrons and the incident neutron flux in the TPC at the same time. By diluting a small amount of 3He gas into the TPC, the incident flux is estimated by counting protons via 3He(n,p)3H reactions. The accuracy of the selection of 3He(n,p)3H events and the influence of the contamination of nitrogen gas are the major systematic errors. In this paper, the estimation of the systematic error in the incident flux is reported.
17. New opportunities for quasielastic and inelastic neutron scattering at steady-state sources using mechanical selection of the incident and final neutron energy
SciTech Connect
Mamantov, Eugene
2015-06-12
We propose a modification of the neutron wide-angle velocity selector (WAVES) device that enables inelastic (in particular, quasielastic) scattering measurements not relying on the neutron time-of-flight. The proposed device is highly suitable for a steady-state neutron source, somewhat similar to a triple-axis spectrometer, but with simultaneous selection of the incident and final neutron energy over a broad range of scattering momentum transfer. Both the incident and final neutron velocities are defined by the WAVES geometry and rotation frequency. The variable energy transfer is achieved through the natural variation of the velocity of the transmitted neutrons as a function of the scattering angle component out of the equatorial plane.
18. New opportunities for quasielastic and inelastic neutron scattering at steady-state sources using mechanical selection of the incident and final neutron energy
DOE PAGES
Mamantov, Eugene
2015-06-12
We propose a modification of the neutron wide-angle velocity selector (WAVES) device that enables inelastic (in particular, quasielastic) scattering measurements not relying on the neutron time-of-flight. The proposed device is highly suitable for a steady-state neutron source, somewhat similar to a triple-axis spectrometer, but with simultaneous selection of the incident and final neutron energy over a broad range of scattering momentum transfer. Both the incident and final neutron velocities are defined by the WAVES geometry and rotation frequency. The variable energy transfer is achieved through the natural variation of the velocity of the transmitted neutrons as a function of themore » scattering angle component out of the equatorial plane.« less
19. Energy measurement of prompt fission neutrons in 239Pu(n,f) for incident neutron energies from 1 to 200 MeV
SciTech Connect
Haight, Robert C; Devlin, Matthew J; Nelson, Ronald O; O' Donnell, John M; Chatillon, Audrey; Granier, Thierry; Taieb, Julien; Belier, Gilbert; Laurent, Benoit; Noda, Shusaku
2010-01-01
An experimental campaign was started in 2002 in the framework of a collaboration belween CEA-DAM and the Los Alamos National Laboratory to measure the prompt fission neutron spectra (PFNS) for incident neutron energies from 1 to 200 MeV with consistent error uncertainties over the whole energy range. The prompt neutron spectra in {sup 235,238}U(n,f) and {sup 237}Np(n,f) have been already studied successfully. A first attempt to characterize the prompt neutrons emitted during the fission of the {sup 239}Pu was done in 2007. This contribution will focus on the results obtained during the final experiment to measure the PFNS in {sup 239}Pu(n,f) performed in 2008. Prompt fission neutron spectra in the neutron-induced fission of {sup 239}Pu have been measured for incident neutron energies from 1 to 200 MeV at the Los Alamos Neutron Science Center. Mean energies obtained from the spectra are discussed and compared to theoretical model calculation.
20. Neutron equivalent doses and associated lifetime cancer incidence risks for head & neck and spinal proton therapy
Athar, Basit S.; Paganetti, Harald
2009-08-01
In this work we have simulated the absorbed equivalent doses to various organs distant to the field edge assuming proton therapy treatments of brain or spine lesions. We have used computational whole-body (gender-specific and age-dependent) voxel phantoms and considered six treatment fields with varying treatment volumes and depths. The maximum neutron equivalent dose to organs near the field edge was found to be approximately 8 mSv Gy-1. We were able to clearly demonstrate that organ-specific neutron equivalent doses are age (stature) dependent. For example, assuming an 8-year-old patient, the dose to brain from the spinal fields ranged from 0.04 to 0.10 mSv Gy-1, whereas the dose to the brain assuming a 9-month-old patient ranged from 0.5 to 1.0 mSv Gy-1. Further, as the field aperture opening increases, the secondary neutron equivalent dose caused by the treatment head decreases, while the secondary neutron equivalent dose caused by the patient itself increases. To interpret the dosimetric data, we analyzed second cancer incidence risks for various organs as a function of patient age and field size based on two risk models. The results show that, for example, in an 8-year-old female patient treated with a spinal proton therapy field, breasts, lungs and rectum have the highest radiation-induced lifetime cancer incidence risks. These are estimated to be 0.71%, 1.05% and 0.60%, respectively. For an 11-year-old male patient treated with a spinal field, bronchi and rectum show the highest risks of 0.32% and 0.43%, respectively. Risks for male and female patients increase as their age at treatment time decreases.
1. Grazing-incidence small-angle neutron scattering from structures below an interface
PubMed Central
Nouhi, Shirin; Hellsing, Maja S.; Kapaklis, Vassilios; Rennie, Adrian R.
2017-01-01
Changes of scattering are observed as the grazing angle of incidence of an incoming beam increases and probes different depths in samples. A model has been developed to describe the observed intensity in grazing-incidence small-angle neutron scattering (GISANS) experiments. This includes the significant effects of instrument resolution, the sample transmission, which depends on both absorption and scattering, and the sample structure. The calculations are tested with self-organized structures of two colloidal samples with different size particles that were measured on two different instruments. The model allows calculations for various instruments with defined resolution and can be used to design future improved experiments. The possibilities and limits of GISANS for different studies are discussed using the model calculations. PMID:28808432
2. Grazing-incidence small-angle neutron scattering from structures below an interface.
PubMed
Nouhi, Shirin; Hellsing, Maja S; Kapaklis, Vassilios; Rennie, Adrian R
2017-08-01
Changes of scattering are observed as the grazing angle of incidence of an incoming beam increases and probes different depths in samples. A model has been developed to describe the observed intensity in grazing-incidence small-angle neutron scattering (GISANS) experiments. This includes the significant effects of instrument resolution, the sample transmission, which depends on both absorption and scattering, and the sample structure. The calculations are tested with self-organized structures of two colloidal samples with different size particles that were measured on two different instruments. The model allows calculations for various instruments with defined resolution and can be used to design future improved experiments. The possibilities and limits of GISANS for different studies are discussed using the model calculations.
3. Inspection of 56Fe γ-Ray angular distributions as a function of incident neutron energy using optical model approaches
Vanhoy, J. R.; Ramirez, A. P.; Alcorn-Dominguez, D. K.; Hicks, S. F.; Peters, E. E.; McEllistrem, M. T.; Mukhopadhyay, S.; Yates, S. W.
2017-09-01
Neutron inelastic scattering cross sections measured directly through (n,n) or deduced from γ-ray production cross sections following inelastic neutron scattering (n,n'γ) are a focus of basic and applied research at the University of Kentucky Accelerator Laboratory (www.pa.uky.edu/accelerator). For nuclear data applications, angle-integrated cross sections are desired over a wide range of fast neutron energies. Several days of experimental beam time are required for a data set at each incident neutron energy, which limits the number of angular distributions that can be measured in a reasonable amount of time. Approximations can be employed to generate cross sections with a higher energy resolution, since at 125o, the a2P2 term of the Legendre expansion is identically zero and the a4P4 is assumed to be very small. Provided this assumption is true, a single measurement at 125o would produce the γ-ray production cross section. This project tests these assumptions and energy dependences using the codes CINDY/SCAT and TALYS/ECIS06/SCAT. It is found that care must be taken when interpreting γ-ray excitation functions as cross sections when the incident neutron energy is < 1000 keV above threshold or before the onset of feeding.
4. Point-by-point model calculation of the prompt neutron multiplicity distribution ν(A) in the incident neutron energy range of multi-chance fission
Tudora, Anabella; Hambsch, Franz-Josef; Tobosaru, Viorel
2017-09-01
Prompt neutron multiplicity distributions ν(A) are required for prompt emission correction of double energy (2E) measurements of fission fragments to determine pre-neutron fragment properties. The lack of experimental ν(A) data especially at incident neutron energies (En) where the multi-chance fission occurs impose the use of ν(A) predicted by models. The Point-by-Point model of prompt emission is able to provide the individual ν(A) of the compound nuclei of the main and secondary nucleus chains undergoing fission at a given En. The total ν(A) is obtained by averaging these individual ν(A) over the probabilities of fission chances (expressed as total and partial fission cross-section ratios). An indirect validation of the total ν(A) results is proposed. At high En, above 70 MeV, the PbP results of individual ν(A) of the first few nuclei of the main and secondary nucleus chains exhibit an almost linear increase. This shape is explained by the damping of shell effects entering the super-fluid expression of the level density parameters. They tend to approach the asymptotic values for most of the fragments. This fact leads to a smooth and almost linear increase of fragment excitation energy with the mass number that is reflected in a smooth and almost linear behaviour of ν(A).
5. Nuclear Poincaré cycle synchronizes with the incident de Broglie wave to predict regularity in neutron resonance energies
Ohkubo, Makio
2016-06-01
In observed neutron resonances, long believed to be a form of quantum chaos, regular family structures are found in the s-wave resonances of many even-even nuclei in the tens keV to MeV region [M.Ohkubo, Phys. Rev. C 87, 014608(2013)]. Resonance reactions take place when the incident de Broglie wave synchronizes with the Poincaré cycle of the compound nucleus, which is composed of several normal modes with periods that are time quantized by inverse Fermi energy. Based on the breathing model of the compound nucleus, neutron resonance energies in family structures are written by simple arithmetic expressions using Sn and small integers. Family structures in observed resonances of 40Ca+n and 37Cl+n are described as simple cases. A model for time quantization is discussed.
6. Predicted risks of second malignant neoplasm incidence and mortality due to secondary neutrons in a girl and boy receiving proton craniospinal irradiation
PubMed Central
Taddei, Phillip J; Mahajan, Anita; Mirkovic, Dragan; Zhang, Rui; Giebeler, Annelise; Kornguth, David; Harvey, Mark; Woo, Shiao; Newhauser, Wayne D
2010-01-01
The purpose of this study was to compare the predicted risks of second malignant neoplasm (SMN) incidence and mortality from secondary neutrons for a 9-year-old girl and a 10-year-old boy who received proton craniospinal irradiation (CSI). SMN incidence and mortality from neutrons were predicted from equivalent doses to radiosensitive organs for cranial, spinal and intracranial boost fields. Therapeutic proton absorbed dose and equivalent dose from neutrons were calculated using Monte Carlo simulations. Risks of SMN incidence and mortality in most organs and tissues were predicted by applying risks models from the National Research Council of the National Academies to the equivalent dose from neutrons; for non-melanoma skin cancer, risk models from the International Commission on Radiological Protection were applied. The lifetime absolute risks of SMN incidence due to neutrons were 14.8% and 8.5%, for the girl and boy, respectively. The risks of a fatal SMN were 5.3% and 3.4% for the girl and boy, respectively. The girl had a greater risk for any SMN except colon and liver cancers, indicating that the girl’s higher risks were not attributable solely to greater susceptibility to breast cancer. Lung cancer predominated the risk of SMN mortality for both patients. This study suggests that the risks of SMN incidence and mortality from neutrons may be greater for girls than for boys treated with proton CSI. PMID:21076189
7. Predicted risks of second malignant neoplasm incidence and mortality due to secondary neutrons in a girl and boy receiving proton craniospinal irradiation.
PubMed
Taddei, Phillip J; Mahajan, Anita; Mirkovic, Dragan; Zhang, Rui; Giebeler, Annelise; Kornguth, David; Harvey, Mark; Woo, Shiao; Newhauser, Wayne D
2010-12-07
The purpose of this study was to compare the predicted risks of second malignant neoplasm (SMN) incidence and mortality from secondary neutrons for a 9-year-old girl and a 10-year-old boy who received proton craniospinal irradiation (CSI). SMN incidence and mortality from neutrons were predicted from equivalent doses to radiosensitive organs for cranial, spinal and intracranial boost fields. Therapeutic proton absorbed dose and equivalent dose from neutrons were calculated using Monte Carlo simulations. Risks of SMN incidence and mortality in most organs and tissues were predicted by applying risks models from the National Research Council of the National Academies to the equivalent dose from neutrons; for non-melanoma skin cancer, risk models from the International Commission on Radiological Protection were applied. The lifetime absolute risks of SMN incidence due to neutrons were 14.8% and 8.5%, for the girl and boy, respectively. The risks of a fatal SMN were 5.3% and 3.4% for the girl and boy, respectively. The girl had a greater risk for any SMN except colon and liver cancers, indicating that the girl's higher risks were not attributable solely to greater susceptibility to breast cancer. Lung cancer predominated the risk of SMN mortality for both patients. This study suggests that the risks of SMN incidence and mortality from neutrons may be greater for girls than for boys treated with proton CSI.
8. Predicted risks of second malignant neoplasm incidence and mortality due to secondary neutrons in a girl and boy receiving proton craniospinal irradiation
Taddei, Phillip J.; Mahajan, Anita; Mirkovic, Dragan; Zhang, Rui; Giebeler, Annelise; Kornguth, David; Harvey, Mark; Woo, Shiao; Newhauser, Wayne D.
2010-12-01
The purpose of this study was to compare the predicted risks of second malignant neoplasm (SMN) incidence and mortality from secondary neutrons for a 9-year-old girl and a 10-year-old boy who received proton craniospinal irradiation (CSI). SMN incidence and mortality from neutrons were predicted from equivalent doses to radiosensitive organs for cranial, spinal and intracranial boost fields. Therapeutic proton absorbed dose and equivalent dose from neutrons were calculated using Monte Carlo simulations. Risks of SMN incidence and mortality in most organs and tissues were predicted by applying risks models from the National Research Council of the National Academies to the equivalent dose from neutrons; for non-melanoma skin cancer, risk models from the International Commission on Radiological Protection were applied. The lifetime absolute risks of SMN incidence due to neutrons were 14.8% and 8.5%, for the girl and boy, respectively. The risks of a fatal SMN were 5.3% and 3.4% for the girl and boy, respectively. The girl had a greater risk for any SMN except colon and liver cancers, indicating that the girl's higher risks were not attributable solely to greater susceptibility to breast cancer. Lung cancer predominated the risk of SMN mortality for both patients. This study suggests that the risks of SMN incidence and mortality from neutrons may be greater for girls than for boys treated with proton CSI.
9. Influence of beam incidence and irradiation parameters on stray neutron doses to healthy organs of pediatric patients treated for an intracranial tumor with passive scattering proton therapy.
PubMed
Bonfrate, A; Farah, J; De Marzi, L; Delacroix, S; Hérault, J; Sayah, R; Lee, C; Bolch, W E; Clairand, I
2016-04-01
10. Study of the in-plane magnetic structure of a layered system using polarized neutron scattering under grazing incidence geometry
Maruyama, R.; Bigault, T.; Wildes, A. R.; Dewhurst, C. D.; Soyama, K.; Courtois, P.
2016-05-01
The in-plane magnetic structure of a layered system with a polycrystalline grain size less than the ferromagnetic exchange length was investigated using polarized neutron off-specular scattering and grazing incidence small angle scattering measurements to gain insight into the mechanism that controls the magnetic properties which are different from the bulk. These complementary measurements with different length scales and the data analysis based on the distorted wave Born approximation revealed the lateral correlation on a length scale of sub- μm due to the fluctuating orientation of the magnetization in the layer. The obtained in-plane magnetic structure is consistent with the random anisotropy model, i.e. competition between the exchange interactions between neighboring spins and the local magnetocrystalline anisotropy.
11. Measurements of the neutron yields from 7Li(p,n)7Be reaction (thick target) with incident energies from 1.885 to 2.0 MeV.
PubMed
Yu, W; Yue, G; Han, X; Chen, J; Tian, B
1998-07-01
Accelerator-based neutron source have been considered to be practical for boron neutron capture therapy (BNCT). Based on experience with a parameters of the Brookhaven National Laboratory BMRR reactor neutron source, which has been used in treatment experiments, the future accelerator-based neutron source for BNCT should have the properties of low energy distribution (< 100 keV) and high flux (about 10(9) neutrons per second per square centimeter) in the patient zone. Using protons to bombard thick 7Li targets, generating neutrons via the 7Li(p,n)7Be reaction, is one of the optimal choices for this kind of neutron source. Neutron yield data versus incident energy are necessary in order to select the proper incident energy and for estimating how high the incident proton current should be. The required proton beam current intensity is one of the key parameters for an accelerator useful for BNCT. In the present work, neutron yields of the 7Li(p,n)7Be reaction with a thick lithium target and incident energies of 1.885 and 1.9 MeV were measured at 0 degree with respect to the incident beam direction. The results are (3.08 +/- 0.17) x 10(12) and (5.71 +/- 0.32) x 10(12) neutrons/C sr, respectively. Neutron yield angular distribution measurements at 2 MeV incident energy were also performed. The proton beams were generated by the Peking University 4.5 MV electrostatic accelerator. The emitted neutrons from these reactions have the advantages of low energy distribution and forward angular distribution, which are requirements for a BNCT neutron source. The data obtained in this work can be used as a reference to study the accelerator-based neutron sources for BNCT.
12. Fission fragment mass and energy distributions as a function of incident neutron energy measured in a lead slowing-down spectrometer
SciTech Connect
Romano, C.; Danon, Y.; Block, R.; Thompson, J.; Blain, E.; Bond, E.
2010-01-15
A new method of measuring fission fragment mass and energy distributions as a function of incident neutron energy in the range from below 0.1 eV to 1 keV has been developed. The method involves placing a double-sided Frisch-gridded fission chamber in Rensselaer Polytechnic Institute's lead slowing-down spectrometer (LSDS). The high neutron flux of the LSDS allows for the measurement of the energy-dependent, neutron-induced fission cross sections simultaneously with the mass and kinetic energy of the fission fragments of various small samples. The samples may be isotopes that are not available in large quantities (submicrograms) or with small fission cross sections (microbarns). The fission chamber consists of two anodes shielded by Frisch grids on either side of a single cathode. The sample is located in the center of the cathode and is made by depositing small amounts of actinides on very thin films. The chamber was successfully tested and calibrated using 0.41+-0.04 ng of {sup 252}Cf and the resulting mass distributions were compared to those of previous work. As a proof of concept, the chamber was placed in the LSDS to measure the neutron-induced fission cross section and fragment mass and energy distributions of 25.3+-0.5 mug of {sup 235}U. Changes in the mass distributions as a function of incident neutron energy are evident and are examined using the multimodal fission mode model.
13. Neutron reflecting supermirror structure
DOEpatents
Wood, James L.
1992-01-01
An improved neutron reflecting supermirror structure comprising a plurality of stacked sets of bilayers of neutron reflecting materials. The improved neutron reflecting supermirror structure is adapted to provide extremely good performance at high incidence angles, i.e. up to four time the critical angle of standard neutron mirror structures. The reflection of neutrons striking the supermirror structure at a high critical angle provides enhanced neutron throughput, and hence more efficient and economical use of neutron sources.
14. Neutron reflecting supermirror structure
DOEpatents
Wood, J.L.
1992-12-01
An improved neutron reflecting supermirror structure comprising a plurality of stacked sets of bilayers of neutron reflecting materials. The improved neutron reflecting supermirror structure is adapted to provide extremely good performance at high incidence angles, i.e. up to four time the critical angle of standard neutron mirror structures. The reflection of neutrons striking the supermirror structure at a high critical angle provides enhanced neutron throughput, and hence more efficient and economical use of neutron sources. 2 figs.
15. Angular distribution measurements of photo-neutron yields produced by 2.0 GeV electrons incident on thick targets.
PubMed
Lee, Hee-Seock; Ban, Syuichi; Sanami, Toshiya; Takahashi, Kazutoshi; Sato, Tatsuhiko; Shin, Kazuo; Chung, Chinwha
2005-01-01
A study of differential photo-neutron yields by irradiation with 2 GeV electrons has been carried out. In this extension of a previous study in which measurements were made at an angle of 90 degrees relative to incident electrons, the differential photo-neutron yield was obtained at two other angles, 48 degrees and 140 degrees, to study its angular characteristics. Photo-neutron spectra were measured using a pulsed beam time-of-flight method and a BC418 plastic scintillator. The reliable range of neutron energy measurement was 8-250 MeV. The neutron spectra were measured for 10 Xo-thick Cu, Sn, W and Pb targets. The angular distribution characteristics, together with the previous results for 90 degrees, are presented in the study. The experimental results are compared with Monte Carlo calculation results. The yields predicted by MCNPX 2.5 tend to underestimate the measured ones. The same trend holds for the comparison results using the EGS4 and PICA3 codes.
16. Neutron tubes
DOEpatents
Leung, Ka-Ngo; Lou, Tak Pui; Reijonen, Jani
2008-03-11
A neutron tube or generator is based on a RF driven plasma ion source having a quartz or other chamber surrounded by an external RF antenna. A deuterium or mixed deuterium/tritium (or even just a tritium) plasma is generated in the chamber and D or D/T (or T) ions are extracted from the plasma. A neutron generating target is positioned so that the ion beam is incident thereon and loads the target. Incident ions cause D-D or D-T (or T-T) reactions which generate neutrons. Various embodiments differ primarily in size of the chamber and position and shape of the neutron generating target. Some neutron generators are small enough for implantation in the body. The target may be at the end of a catheter-like drift tube. The target may have a tapered or conical surface to increase target surface area.
17. Accessible length scale of the in-plane structure in polarized neutron off-specular and grazing-incidence small-angle scattering measurements
Maruyama, R.; Bigault, T.; Wildes, A. R.; Dewhurst, C. D.; Saerbeck, T.; Honecker, D.; Yamazaki, D.; Soyama, K.; Courtois, P.
2017-06-01
Polarized neutron off-specular and grazing-incidence small-angle scattering measurements are useful methods to investigate the in-plane structure and its correlation of layered systems. Although these measurements give information on complementary and overlapping length scale, the different characteristics between them need to be taken into account when performed. In this study, the difference in the accessible length scale of the in-plane structure, which is one of the most important characteristics, was discussed using an Fe/Si multilayer together with simulations based on the distorted wave Born approximation.
18. Point-by-Point model calculation of the prompt neutron multiplicity distribution ν (A ) for 238U(n ,f ) at incident neutron energies ranging from 1 MeV to 80 MeV
Tudora, A.; Hambsch, F.-J.; Tobosaru, V.
2016-10-01
Prompt neutron multiplicity distributions ν(A ) are generally required for prompt emission correction of double energy (2 E ) measurements of fission fragments in order to determine pre-neutron fragment properties. The lack of experimental ν(A ) data especially at higher incident neutron energy imposes the use of prompt emission models to predict ν(A ). At incident neutron energies (En ) where multiple fission chances are involved, the Point-by-Point (PbP) model of prompt emission is able to provide the individual ν(A ) of the compound nuclei of the main and secondary nucleus chains that are undergoing fission at a given En . The total ν(A ) are obtained by averaging these individual ν(A ) over the fission chance probabilities (expressed as total and partial fission cross-section ratios). An indirect validation of the total ν(A ) results is proposed. At high En (above 70 MeV) the PbP results of individual ν(A ) of the first few nuclei of the main and secondary nucleus chains exhibit an almost linear increase. This shape is explained by the damping of shell effects entering the superfluid expression of the level-density parameters. They tend to approach the asymptotic values for a great part of the fragments. This fact leads to a smooth and almost linear increase of fragment excitation energy with the fragment mass number that is reflected in a smooth and almost linear behavior of individual ν(A ). The comparison of the present results with those of the GEF code reveals different shapes of ν(A ) as well as different total average neutron multiplicity as a function of the En . At high En the PbP calculations definitely reflect the influence of the almost linear shape of individual ν(A ) of the first few nuclei of the U and Pa chains. The differences between the total ν(A ) obtained by averaging the PbP results of individual ν(A ) over fission cross-section ratios of different evaluations are insignificant.
19. Estimation of prompt neutron and residual gamma dose rates and induced activity from 0.1-1.0 GeV protons incident on lead-bismuth target
Sunil, C.; Biju, K.; Sarkar, P. K.
2013-08-01
The important parameters from the radiological safety points of view in the case of a lead bismuth target irradiated by protons of energy 100 MeV to 1 GeV are presented here. The prompt neutron yield and the neutron dose rates, time evolution of the induced activity and the resulting gamma dose rates from a lead-bismuth target irradiated by protons of 1 mA beam current for a period of one month are calculated using the FLUKA Monte Carlo code. Some of the results are compared with the values obtained by empirical expressions available in the literature for a medium Z target. The existing empirical relations are found to under predict the prompt neutron yield and the ambient dose equivalent values. A new set of empirical equations are proposed to calculate these parameters more accurately for 100 MeV-1 GeV protons incident on lead-bismuth target. 195Au, 201Tl, 203Pb, 205Bi, 206Bi, 207Bi, 208Po, 210Po and 3H are found to be the important radionuclides from the target handling perspective. The residual gamma dose rates obtained by the empirical relations appear to reproduce the results obtained in this study during the initial days of cooling.
20. Design of axisymmetric multi-mirror grazing incidence system to increase the numerical aperture of neutron and X-ray microscopes
2016-04-01
An axisymmetric multi-mirror system for neutron and X-ray microscopes is proposed to increase their numerical aperture and collection efficiency. A Wolter type-I mirror is used as the basis of the multi-mirror system at grazing incidence. The addition of an even number of hyperboloid mirrors to the Wolter type-I mirror can satisfy both an equal optical path length and Abbe's sine condition. The numerical aperture increases in proportion to the number of mirrors. The optical parameters of the system with four tandem mirrors are calculated for neutrons and X-rays with a wavelength of 0.4 nm by assuming that the average grazing angle of incidence is 5.4 mrad and the magnification is 10. The inner diameters of the mirrors are limited to <10 mm considering the total length of the optical system. Tolerance of off-axis distance was calculated using a ray-tracing computer simulation. Ray tracing shows that a blur size <14 nm will be possible at an off-axis displacement of 10 μm.
1. Neutron reflecting supermirror structure
DOEpatents
Wood, James L.
1992-01-01
An improved neutron reflecting supermirror structure comprising a plurality of stacked sets of bilayers of neutron reflecting materials. The improved neutron reflecting supermirror structure is adapted to provide extremely good performance at high incidence angles, i.e. up to four time the critical angle of standard neutron mirror structures. The reflection of neutrons striking the supermirror structure at a high critical angle provides enhanced neutron throughput, and hence more efficient and economical use of neutron sources. One layer of each set of bilayers consist of titanium, and the second layer of each set of bilayers consist of an alloy of nickel with carbon interstitially present in the nickel alloy.
2. Differential cross section of γn→K+Σ- on bound neutrons with incident photons from 1.1 to 3.6 GeV
DOE PAGES
Pereira, S. Anefalos; Mirazita, M.; Rossi, P.; ...
2010-05-01
Differential cross sections of the reaction γd → K+Σ–(p) have been measured with the CLAS detector at Jefferson Lab using incident photons with energies between 1.1 and 3.6 GeV. This is the first complete set of strangeness photoproduction data on the neutron covering a broad angular range. At energies close to threshold and up to Eγ ~ 1.8 GeV, the shape of the angular distribution is suggestive of the presence of s -channel production mechanisms. For Eγ > 1.8 GeV, a clear forward peak appears and becomes more prominent as the photon energy increases, suggesting contributions from t-channel production mechanisms.more » Furthermore, these data can be used to constrain future analysis of this reaction.« less
3. 10B(n,α)7Li and 10B(n,α1γ)7Li cross section data up to 3 MeV incident neutron energy
Bevilacqua, Riccardo; Hambsch, Franz-Josef; Vidali, Marzio; Ruskov, Ivan; Lamia, Livio
2017-09-01
The 10B(n,α) reaction cross-section is a well-established neutron cross-section standard for incident neutron energies up to 1 MeV. However, above this energy limit there are only scarce direct (n,α) measurements available and these few experimental data are showing large inconsistencies with each other. These discrepancies are reflected in the evaluated data libraries: ENDF/B-VII.1, JEFF-3.1.2 and JENDL-4.0 are in excellent agreement up to 100 keV incident neutrons, whereas the 10B(n,α) data in the different libraries show large differences in the MeV region. To address these inconsistencies, we have measured the cross section of the two branches of the 10B(n,α) reaction for incident neutron energies up to 3 MeV. We present here the 10B(n,α) and the 10B(n,α1γ) reactions cross section data, their branching ratio and the total 10B(n,α) reaction cross section. The measurements were conducted with a dedicated Frisch-grid ionization chamber installed at the GELINA pulsed neutron source of the EC-JRC. We compare our results with existing experimental data and evaluations.
4. Neutron-emission measurements at a white neutron source
SciTech Connect
Haight, Robert C
2010-01-01
Data on the spectrum of neutrons emittcd from neutron-induced reactions are important in basic nuclear physics and in applications. Our program studies neutron emission from inelastic scattering as well as fission neutron spectra. A ''white'' neutron source (continuous in energy) allows measurements over a wide range of neutron energies all in one experiment. We use the tast neutron source at the Los Alamos Neutron Science Center for incident neutron energies from 0.5 MeV to 200 MeV These experiments are based on double time-of-flight techniques to determine the energies of the incident and emitted neutrons. For the fission neutron measurements, parallel-plate ionization or avalanche detectors identify fission in actinide samples and give the required fast timing pulse. For inelastic scattering, gamma-ray detectors provide the timing and energy spectroscopy. A large neutron-detector array detects the emitted neutrons. Time-of-flight techniques are used to measure the energies of both the incident and emitted neutrons. Design considerations for the array include neutron-gamma discrimination, neutron energy resolution, angular coverage, segmentation, detector efficiency calibration and data acquisition. We have made preliminary measurements of the fission neutron spectra from {sup 235}U, {sup 238}U, {sup 237}Np and {sup 239}Pu. Neutron emission spectra from inelastic scattering on iron and nickel have also been investigated. The results obtained will be compared with evaluated data.
5. Neutron reflectivity
Cousin, Fabrice; Menelle, Alain
2015-10-01
The specular neutron reflectivity is a technique enabling the measurement of neutron scattering length density profile perpendicular to the plane of a surface or an interface, and thereby the profile of chemical composition. The characteristic sizes that are probed range from around 5 Å up 5000 Å. It is a scattering technique that averages information on the entire surface and it is therefore not possible to obtain information within the plane of the interface. The specific properties of neutrons (possibility of tuning the contrast by isotopic substitution, sensitivity to magnetism, negligible absorption, low energy of the incident neutrons) makes it particularly interesting in the fields of soft matter, biophysics and magnetic thin films. This course is a basic introduction to the technique and does not address the magnetic reflectivity. It is composed of three parts describing respectively its principle and its formalism, the experimental aspects of the method (spectrometers, samples) and two examples related to the materials for energy.
6. Neutron measurements
SciTech Connect
McCall, R.C.
1981-01-01
Methods of neutron detection and measurement are discussed. Topics include sources of neutrons, neutrons in medicine, interactions of neutrons with matter, neutron shielding, neutron measurement units, measurement methods, and neutron spectroscopy. (ACR)
7. Neutron and X-ray Scattering From Single Supported Lipid Bilayers: Reflectometry, Grazing Incidence In-Plane Diffraction and Off-Specular Scattering
Majewski, Jaroslaw
2010-03-01
Biological membranes mediate transport and communication between the cell and its surroundings. They defend the cell against invasive agents, and most present day drugs interact with membrane components. Complexity of the cell membranes renders many of their characteristics impenetrable to fundamental physical studies. As a result, a significant emphasis has been placed on developing model lipid membranes that facilitate the physical and chemical characterization of particular membrane features. X-ray (XR) and neutron reflectivity (NR) and grazing incidence X-ray diffraction (GIXD) techniques can be utilized to measure the structure of single, supported lipid bilayers in bulk water. GIXD studies demonstrated that bilayers formed by vesicle fusion have more disorder in the inner leaflet compared to structures prepared using the Langmuir-Blodgett/Schaeffer (LB/S) technique. In both cases, only a modest water cushion was detected between the bilayer and substrate. Diffraction from in-plane ordered domains was observed from bilayers prepared by either technique. In the case of 1,2-Dipalmitoyl-sn-Glycero-3-Phosphocholine (DPPC) bilayers, the ordered domains were coupled across both leaflets, scattering as one entity. Contrastingly, the ordered domains were uncoupled in 1,2-Dipalmitoyl-sn-Glycero-3-Phosphoethanolamine (DPPE) bilayers. NR can be effectively used to study polymer-supported single lipid bilayers in bulk water. Using NR and fluorescence microscopy, we demonstrated that a hydrated, surface-tethered polymer network capable of five-fold change in thickness over a 25-37 C temperature range can be a novel support for single DPPC bilayers in a liquid environment. Moderate temperature change swells the polymer, lifting the membrane from the substrate, creating a nearly aqueous cushion. Additionally, as the polymer swells, it promotes both in- and out-of-plane undulations in the supported membrane. Off-specular neutron scattering was used to deduce the in
8. Neutron range spectrometer
DOEpatents
Manglos, Stephen H.
1989-06-06
A neutron range spectrometer and method for determining the neutron energy spectrum of a neutron emitting source are disclosed. Neutrons from the source are collimnated along a collimation axis and a position sensitive neutron counter is disposed in the path of the collimated neutron beam. The counter determines positions along the collimation axis of interactions between the neutrons in the neutron beam and a neutron-absorbing material in the counter. From the interaction positions, a computer analyzes the data and determines the neutron energy spectrum of the neutron beam. The counter is preferably shielded and a suitable neutron-absorbing material is He-3. The computer solves the following equation in the analysis: ##EQU1## where: N(x).DELTA.x=the number of neutron interactions measured between a position x and x+.DELTA.x, A.sub.i (E.sub.i).DELTA.E.sub.i =the number of incident neutrons with energy between E.sub.i and E.sub.i +.DELTA.E.sub.i, and C=C(E.sub.i)=N .sigma.(E.sub.i) where N=the number density of absorbing atoms in the position sensitive counter means and .sigma. (E.sub.i)=the average cross section of the absorbing interaction between E.sub.i and E.sub.i +.DELTA.E.sub.i.
9. Neutron cross sections
SciTech Connect
Not Available
1988-01-01
This handbook displays curves of neutron cross sections in the energy range of 0.01 eV to 200 MeV (and associated information) as a function of incident neutron energy. Tables include reference to all data. Information on isomeric state production is also included. This book represents the fourth edition of what was previously known as BNL-325, Neutron Cross Sections, Volume 2, the third edition of which was published in 1976.
10. Laser generated neutron source for neutron resonance spectroscopy
SciTech Connect
Higginson, D. P.; Bartal, T.; McNaney, J. M.; Swift, D. C.; Hey, D. S.; Le Pape, S.; Mackinnon, A.; Kodama, R.; Tanaka, K. A.; Mariscal, D.; Beg, F. N.; Nakamura, H.; Nakanii, N.
2010-10-15
A neutron source for neutron resonance spectroscopy has been developed using high-intensity, short-pulse lasers. This technique will allow robust measurement of interior ion temperature of laser-shocked materials and provide insight into material equation of state. The neutron generation technique uses laser-accelerated protons to create neutrons in LiF through (p,n) reactions. The incident proton beam has been diagnosed using radiochromic film. This distribution is used as the input for a (p,n) neutron prediction code which is validated with experimentally measured neutron yields. The calculation infers a total fluence of 1.8x10{sup 9} neutrons, which are expected to be sufficient for neutron resonance spectroscopy temperature measurements.
11. Neutron Emission Spectra of 104,105,106,108,110Pd Isotopes for (p,xn) Reactions at 21.6 MeV Proton Incident Energy
Büyükuslu, H.; Kaplan, A.; Tel, E.; Aydin, A.; Yıldırım, G.
2010-02-01
Palladium, which is a rare and lustrous silvery-white color from precious metals, plays important role in fusion-fission reactions and different fields of nuclear technology. In addition, it is used for not only cold fusion experiments but also separation of hydrogen isotopes researches for fusion reactors. In this study, neutron-emission spectra produced by (p,xn) reactions for structural fusion material 104,105,106,108,110Pd isotopes have been investigated by a proton beam at 21.6 MeV. Moreover, multiple pre-equilibrium mean free paths constant from internal transition, and the pre-equilibrium and equilibrium level density parameters have been analyzed for some (p,xn) neutron-emission spectra calculated. New evaluated hybrid model and geometry dependent hybrid model, full exciton model and cascade exciton model were used to calculate the pre-equilibrium neutron-emission spectra. For the reaction equilibrium component, Weisskopf-Ewing model calculations were preferred. The obtained results have been discussed and compared with the available experimental data and found agreement with each other.
12. Energy dependence of fission product yields from 235U, 238U and 239Pu for incident neutron energies between 0.5 and 14.8 MeV
DOE PAGES
Gooden, M. E.; Arnold, C. W.; Becker, J. A.; ...
2016-01-06
-fission chamber and gamma-ray counted using shielded HPGe detectors for a period of 1-2 months to determine the yield of various fission products. To the extent possible all irradiation and counting procedures were kept the same to minimize sources of systematic errors. FPY have been determined at incident neutron energies of 0.6, 1.4, 2.4, 3.5, 4.6, 5.5, 8.9 and 14.8 MeV.« less
13. Energy dependence of fission product yields from 235U, 238U and 239Pu for incident neutron energies between 0.5 and 14.8 MeV
SciTech Connect
Gooden, M. E.; Arnold, C. W.; Becker, J. A.; Bhatia, C.; Bhike, M.; Bond, E. M.; Bredeweg, T. A.; Fallin, B.; Fowler, M. M.; Howell, C. R.; Kelley, J. H.; Krishichayan, .; Macri, R.; Rusev, G.; Ryan, C.; Sheets, S. A.; Stoyer, M. A.; Tonchev, A. P.; Tornow, W.; Vieira, D. J.; Wilhelmy, J. B.
2016-01-06
from the dual-fission chamber and gamma-ray counted using shielded HPGe detectors for a period of 1-2 months to determine the yield of various fission products. To the extent possible all irradiation and counting procedures were kept the same to minimize sources of systematic errors. FPY have been determined at incident neutron energies of 0.6, 1.4, 2.4, 3.5, 4.6, 5.5, 8.9 and 14.8 MeV.
14. Energy Dependence of Fission Product Yields from {sup 235}U, {sup 238}U and {sup 239}Pu for Incident Neutron Energies Between 0.5 and 14.8 MeV
SciTech Connect
Gooden, M.E.; Arnold, C.W.; Becker, J.A.; Bhatia, C.; Bhike, M.; Bond, E.M.; Bredeweg, T.A.; Fallin, B.; Fowler, M.M.; Howell, C.R.; Kelley, J.H.; Krishichayan; Macri, R.; Rusev, G.; Ryan, C.; Sheets, S.A.; Stoyer, M.A.; Tonchev, A.P.; Tornow, W.; and others
2016-01-15
and gamma-ray counted using shielded HPGe detectors for a period of 1-2 months to determine the yield of various fission products. To the extent possible all irradiation and counting procedures were kept the same to minimize sources of systematic errors. FPY have been determined at incident neutron energies of 0.6, 1.4, 2.4, 3.5, 4.6, 5.5, 8.9 and 14.8 MeV.
15. Energy Dependence of Fission Product Yields from 235U, 238U and 239Pu for Incident Neutron Energies Between 0.5 and 14.8 MeV
Gooden, M. E.; Arnold, C. W.; Becker, J. A.; Bhatia, C.; Bhike, M.; Bond, E. M.; Bredeweg, T. A.; Fallin, B.; Fowler, M. M.; Howell, C. R.; Kelley, J. H.; Krishichayan; Macri, R.; Rusev, G.; Ryan, C.; Sheets, S. A.; Stoyer, M. A.; Tonchev, A. P.; Tornow, W.; Vieira, D. J.; Wilhelmy, J. B.
2016-01-01
-ray counted using shielded HPGe detectors for a period of 1-2 months to determine the yield of various fission products. To the extent possible all irradiation and counting procedures were kept the same to minimize sources of systematic errors. FPY have been determined at incident neutron energies of 0.6, 1.4, 2.4, 3.5, 4.6, 5.5, 8.9 and 14.8 MeV.
16. Energy Dependence of Fission Product Yields from 235U, 238U and 239Pu for Incident Neutron Energies Between 0.5 and 14.8 MeV
SciTech Connect
Gooden, M. E.; Arnold, C. W.; Becker, J. A.; Bhatia, C.; Bhike, M.; Bond, E. M.; Bredeweg, T. A.; Fallin, B.; Fowler, M. M.; Howell, C. R.; Kelley, J. H.; Krishichayan,; Macri, R.; Rusev, G.; Ryan, C.; Sheets, S. A.; Stoyer, M. A.; Tonchev, A. P.; Tornow, W.; Vieira, D. J.; Wilhelmy, J. B.
2016-01-01
from the dual-fission chamber and gamma-ray counted using shielded HPGe detectors for a period of 1-2 months to determine the yield of various fission products. To the extent possible all irradiation and counting procedures were kept the same to minimize sources of systematic errors. FPY have been determined at incident neutron energies of 0.6, 1.4, 2.4, 3.5, 4.6, 5.5, 8.9 and 14.8 MeV.
17. Microscopic cluster model for the description of new experimental results on the 13C(18O,16O) 15C two-neutron transfer at 84 MeV incident energy
Carbone, D.; Ferreira, J. L.; Cappuzzello, F.; Lubian, J.; Agodi, C.; Cavallaro, M.; Foti, A.; Gargano, A.; Lenzi, S. M.; Linares, R.; Santagati, G.
2017-03-01
The 13C(18O,16O) 15C reaction is studied at 84 MeV incident energy. Excitation energy spectra and absolute cross-section angular distributions for the strongest transitions are measured with good energy and angular resolutions. Strong selectivity for two-neutron configurations in the states of the residual nucleus is found. The measured cross-section angular distributions are analyzed by exact finite-range coupled reaction channel calculations. The two-particle wave functions are extracted using the extreme cluster and the independent coordinate scheme with shell-model derived coupling strengths. A new approach also is introduced, the microscopic cluster, in which the spectroscopic amplitudes in the center-of-mass reference frame are derived from shell-model calculations using the Moshinsky transformation brackets. This new model is able to describe well the experimental cross section and to highlight cluster configurations in the involved wave functions.
18. Energy Dependence of Fission Product Yields from 235U, 238U and 239Pu for Incident Neutron Energies Between 0.5 and 14.8 MeV
Gooden, M.; Arnold, C.; Bredeweg, T.; Vieira, D.; Wilhelmy, J.; Tonchev, A.; Stoyer, M.; Bhike, M.; Krishichayan, F.; Tornow, W.; Fowler, M.
2015-10-01
Under a joint collaboration between TUNL-LANL-LLNL, a set of absolute fission product yield measurements has been performed. The energy dependence of a number of cumulative fission product yields (FPY) have been measured using quasi-monoenergetic neutron beams for three actinide targets, 235U, 238U and 239Pu, between 0.5 and 14.8 MeV. The FPYs were measured by a combination of fission counting using specially designed dual-fission chambers and ?-ray counting. Each dual-fission chamber is a back-to-back ionization chamber encasing an activation target in the center with thin deposits of the same target isotope in each chamber. This method allows for the direct measurement of the total number of fissions in the activation target with no reference to the fission cross-section, thus reducing uncertainties. ?-ray counting of the activation target was performed on well-shielded HPGe detectors over a period of 2 months post irradiation to properly identify fission products. Reported are absolute cumulative fission product yields for incident neutron energies of 0.5, 1.37, 2.4, 3.6, 4.6, 5.5, 7.5, 8.9 and 14.8 MeV. These results are compared to previous measurements and theoretical estimates. This work was performed under the auspices of the USDoE by Los Alamos National Security, LLC under Contract DE-AC52-06NA25396.
19. Apparatus for measuring a flux of neutrons
DOEpatents
Stringer, James L.
1977-01-01
A flux of neutrons is measured by disposing a detector in the flux and applying electronic correlation techniques to discriminate between the electrical signals generated by the neutron detector and the unwanted interfering electrical signals generated by the incidence of a neutron flux upon the cables connecting the detector to the electronic measuring equipment at a remote location.
20. Portable neutron spectrometer and dosimeter
DOEpatents
Waechter, David A.; Erkkila, Bruce H.; Vasilik, Dennis G.
1985-01-01
The disclosure relates to a battery operated neutron spectrometer/dosimeter utilizing a microprocessor, a built-in tissue equivalent LET neutron detector, and a 128-channel pulse height analyzer with integral liquid crystal display. The apparatus calculates doses and dose rates from neutrons incident on the detector and displays a spectrum of rad or rem as a function of keV per micron of equivalent tissue and also calculates and displays accumulated dose in millirads and millirem as well as neutron dose rates in millirads per hour and millirem per hour.
1. Portable neutron spectrometer and dosimeter
DOEpatents
Waechter, D.A.; Erkkila, B.H.; Vasilik, D.G.
The disclosure relates to a battery operated neutron spectrometer/dosimeter utilizing a microprocessor, a built-in tissue equivalent LET neutron detector, and a 128-channel pulse height analyzer with integral liquid crystal display. The apparatus calculates doses and dose rates from neutrons incident on the detector and displays a spectrum of rad or rem as a function of keV per micron of equivalent tissue and also calculates and displays accumulated dose in millirads and millirem as well as neutron dose rates in millirads per hour and millirem per hour.
2. TART calculations of neutron attenuation and neutron-induced photons on 5% and 20% borated polyethylene slabs
SciTech Connect
Wuest, C.R.
1993-04-20
The coupled neutron/photon transport code TART has been used to calculate the attenuation of neutrons and the production of induced photons for neutrons incidents on 5% and 20% borated polyethylene slabs. The neutron attenuation lengths are found to be 2.4 cm and 2.9 cm for 5% and 20% borated polyethylene, respectively.
3. Characterizing a Neutron Energy Spectrum Using a "Forward Edge" Neutron Time-of-Flight Spectroscopy Technique
DTIC Science & Technology
2005-03-01
energy spectrum of monoenergetic neutron source [9:562]..............................................27 16. Example data set...is valid only when the absorber is composed of one material and the incident neutrons are monoenergetic . In the case where the absorber consists of...source due to its creation from monoenergetic neutrons . Figure 15: Proton recoil energy spectrum of monoenergetic neutron source [9:562
4. Experiments with neutron Schroedinger waves
1986-09-01
Four neutron Schroedinger wave interference experiments are described: (1) null Fizeau effect for thermal neutrons in thermal matter; (2) precision measurements of the energy dependent scattering length of U-235; (3) gravitationally induced quantum interference; and (4) investigation of incident beam coherence questions in dynamical diffraction experiments. In the Fizeau effect experiment it was verified that a neutron de Broglie wave is not affected by the state of motion of matter within fixed boundaries through which the neutron is propagating. Upper limits for the energy dependence of the neutron optical potential and the wavelength dependence of s-wave neutron scattering lengths were deduced. Using the neutron interferometric technique the energy dependent bound coherent scattering lengths of U-235 were directly measured in the energy range of 30 to 92 meV with an error of 0.3%. In the gravity induced quantum interference experiments (COW), the results showed that the observed loss of contrast of the interference pattern is caused in part by the bending of the perfect crystal neutron interferometer. Lastly, mathematical and numerical calculation techniques are presented to investigate the effect of wavefront modulation and transverse coherence of an incident neutron beam in dynamical diffraction experiments. Experimental and theoretical pendelloesung interference patterns in perfect silicon crystal were obtained and compared.
5. Wolter Optics for Neutron Focusing
NASA Technical Reports Server (NTRS)
Mildner, D. F. R.; Gubarev, M. V.
2010-01-01
Focusing optics based on Wolter optical geometries developed for x-ray grazing incidence beams can be designed for neutron beams. Wolter optics are formed by grazing incidence reflections from two concentric conic sections (for example, a paraboloid and a hyperboloid). This has transformed observational X-ray astronomy by increasing the sensitivity by many orders of magnitude for research in astrophysics and cosmology. To increase the collection area, many reflecting mirrors of different diameters are nested with a common focal plane. These mirrors are fabricated using nickel-electroformed replication techniques. We apply these ideas to neutron focusing using nickel mirrors. We show an initial test of a conical mirror using a beam of cold neutrons. key words: electroformed nickel replication, focusing optics, grazing angle incidence, mirror reflection, neutron focusing, Wolter optics
6. A fundamental study on hyper-thermal neutrons for neutron capture therapy.
PubMed
Sakurai, Y; Kobayashi, T; Kanda, K
1994-12-01
The utilization of hyper-thermal neutrons, which have an energy spectrum with a Maxwellian distribution at a higher temperature than room temperature (300 K), was studied in order to improve the thermal neutron flux distribution at depth in a living body for neutron capture therapy. Simulation calculations were carried out using a Monte Carlo code 'MCNP-V3' in order to investigate the characteristics of hyper-thermal neutrons, i.e. (i) depth dependence of the neutron energy spectrum, and (ii) depth distribution of the reaction rate in a water phantom for materials with 1/v neutron absorption. It is confirmed that hyper-thermal neutron irradiation can improve the thermal neutron flux distribution in the deeper areas in a living body compared with thermal neutron irradiation. When hyper-thermal neutrons with a 3000 K Maxwellian distribution are incident on a body, the reaction rates of 1/v materials such as 14N, 10B etc are about twice that observed for incident thermal neutrons at 300 K, at a depth of 5 cm. The limit of the treatable depth for tumours having 30 ppm 10B is expected to be about 1.5 cm greater by utilizing hyper-thermal neutrons at 3000 K compared with the incidence of thermal neutrons at 300 K.
7. Nested Focusing Optics for Compact Neutron Sources
NASA Technical Reports Server (NTRS)
Nabors, Sammy A.
2015-01-01
NASA's Marshall Space Flight Center, the Massachusetts Institute of Technology (MIT), and the University of Alabama Huntsville (UAH) have developed novel neutron grazing incidence optics for use with small-scale portable neutron generators. The technology was developed to enable the use of commercially available neutron generators for applications requiring high flux densities, including high performance imaging and analysis. Nested grazing incidence mirror optics, with high collection efficiency, are used to produce divergent, parallel, or convergent neutron beams. Ray tracing simulations of the system (with source-object separation of 10m for 5 meV neutrons) show nearly an order of magnitude neutron flux increase on a 1-mm diameter object. The technology is a result of joint development efforts between NASA and MIT researchers seeking to maximize neutron flux from diffuse sources for imaging and testing applications.
8. Neutron skins and neutron stars
SciTech Connect
Piekarewicz, J.
2013-11-07
The neutron-skin thickness of heavy nuclei provides a fundamental link to the equation of state of neutron-rich matter, and hence to the properties of neutron stars. The Lead Radius Experiment ('PREX') at Jefferson Laboratory has recently provided the first model-independence evidence on the existence of a neutron-rich skin in {sup 208}Pb. In this contribution we examine how the increased accuracy in the determination of neutron skins expected from the commissioning of intense polarized electron beams may impact the physics of neutron stars.
9. Neutron Imaging Camera
NASA Technical Reports Server (NTRS)
Hunter, Stanley; deNolfo, G. A.; Barbier, L. M.; Link, J. T.; Son, S.; Floyd, S. R.; Guardala, N.; Skopec, M.; Stark, B.
2008-01-01
The Neutron Imaging Camera (NIC) is based on the Three-dimensional Track Imager (3DTI) technology developed at GSFC for gamma-ray astrophysics applications. The 3-DTI, a large volume time-projection chamber, provides accurate, approximately 0.4 mm resolution, 3-D tracking of charged particles. The incident direction of fast neutrons, En > 0.5 MeV, are reconstructed from the momenta and energies of the proton and triton fragments resulting from (sup 3)He(n,p) (sup 3)H interactions in the 3-DTI volume. The performance of the NIC from laboratory and accelerator tests is presented.
10. Atmospheric neutrons
NASA Technical Reports Server (NTRS)
Korff, S. A.; Mendell, R. B.; Merker, M.; Light, E. S.; Verschell, H. J.; Sandie, W. S.
1979-01-01
Contributions to fast neutron measurements in the atmosphere are outlined. The results of a calculation to determine the production, distribution and final disappearance of atmospheric neutrons over the entire spectrum are presented. An attempt is made to answer questions that relate to processes such as neutron escape from the atmosphere and C-14 production. In addition, since variations of secondary neutrons can be related to variations in the primary radiation, comment on the modulation of both radiation components is made.
11. Neutron dosimetry
DOEpatents
Quinby, Thomas C.
1976-07-27
A method of measuring neutron radiation within a nuclear reactor is provided. A sintered oxide wire is disposed within the reactor and exposed to neutron radiation. The induced radioactivity is measured to provide an indication of the neutron energy and flux within the reactor.
12. Neutron guide
DOEpatents
Greene, Geoffrey L.
1999-01-01
A neutron guide in which lengths of cylindrical glass tubing have rectangular glass plates properly dimensioned to allow insertion into the cylindrical glass tubing so that a sealed geometrically precise polygonal cross-section is formed in the cylindrical glass tubing. The neutron guide provides easier alignment between adjacent sections than do the neutron guides of the prior art.
13. From X-Ray Telescopes to Neutron Focusing
NASA Technical Reports Server (NTRS)
Gubarev, M. V.; Khaykovich, B.; Ramsey, B.; Moncton, D. E.
2011-01-01
In the case of neutrons the refractive index is slightly less than unity for most elements and their isotopes. Consequently, thermal and cold neutrons can be reflected from smooth surfaces at grazing-incidence angles. Hence, the optical technologies developed for x-ray astronomy can be applied for neutron focusing. The focusing capabilities of grazing incidence neutron imaging optics have been successfully demonstrated using nickel mirrors. The mirrors were fabricated using an electroformed nickel replication process at Marshall Space Flight Center. Results of the neutron optics experiments will be presented. Challenges of the neutron imaging optics as well as possible applications of the optics will be discussed.
14. Fast Neutron Sensitivity with HPGe
SciTech Connect
Seifert, Allen; Hensley, Walter K.; Siciliano, Edward R.; Pitts, W. K.
2008-01-22
In addition to being excellent gamma-ray detectors, germanium detectors are also sensitive to fast neutrons. Incident neutrons undergo inelastic scattering {Ge(n,n')Ge*} off germanium nuclei and the resulting excited states emit gamma rays or conversion electrons. The response of a standard 140% high-purity germanium (HPGe) detector with a bismuth germanate (BGO) anti-coincidence shield was measured for several neutron sources to characterize the ability of the HPGe detector to detect fast neutrons. For a sensitivity calculation performed using the characteristic fast neutron response peak that occurs at 692 keV, the 140% germanium detector system exhibited a sensitivity of ~175 counts / kg of WGPumetal in 1000 seconds at a source-detector distance of 1 meter with 4 in. of lead shielding between source and detector. Theoretical work also indicates that it might be possible to use the shape of the fast-neutron inelastic scattering signatures (specifically, the end-point energy of the long high energy tail of the resulting asymmetric peak) to gain additional information about the energy distribution of the incident neutron spectrum. However, the experimentally observed end-point energies appear to be almost identical for each of the fast neutron sources counted. Detailed MCNP calculations show that the neutron energy distributions impingent on the detector for these sources are very similar in this experimental configuration, due to neutron scattering in a lead shield (placed between the neutron source and HPGe detector to reduce the gamma ray flux), the BGO anti-coincidence detector, and the concrete floor.
15. Direction sensitive neutron detector
DOEpatents
Ahlen, Steven; Fisher, Peter; Dujmic, Denis; Wellenstein, Hermann F.; Inglis, Andrew
2017-01-31
A neutron detector includes a pressure vessel, an electrically conductive field cage assembly within the pressure vessel and an imaging subsystem. A pressurized gas mixture of CF.sub.4, .sup.3He and .sup.4He at respective partial pressures is used. The field cage establishes a relatively large drift region of low field strength, in which ionization electrons generated by neutron-He interactions are directed toward a substantially smaller amplification region of substantially higher field strength in which the ionization electrons undergo avalanche multiplication resulting in scintillation of the CF.sub.4 along scintillation tracks. The imaging system generates two-dimensional images of the scintillation patterns and employs track-finding to identify tracks and deduce the rate and direction of incident neutrons. One or more photo-multiplier tubes record the time-profile of the scintillation tracks permitting the determination of the third coordinate.
16. Measurement of neutron scattering lengths using neutron interferometry
Shahi, Chandra B.
This thesis describes the details on building a new Neutron Interferometry and Optics Facility (NIOFa), the measurement of the incoherent neutron scattering length bi of 3He, and the measurement of the coherent neutron scattering length bc of 4He at National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR). A new monochromatic beamline and facility has been installed at the NCNR devoted to neutron interferometry in the research areas of spin control, spin manipulation, quantum mechanics, quantum information science, spintronics, and material science. This facility is possible in part because of advances in decoherence free subspace interferometer designs that have demonstrated consistent contrast in the presence of vibrational noise; a major environmental constraint that has prevented neutron interferometry from being applied at other neutron facilities. This new facility, NIOFa, is located in the guide hall of the NCNR upstream of the existing Neutron Interferometry and Optics Facility (NIOF) and has several advantages over the NIOF including higher incident flux, better neutron polarization, and increased accessibility. The measurement of the incoherent neutron scattering length bi of 3He was done using a (220) single silicon crystal skew symmetric interferometer. This experiment requires both a polarized beam and a polarized target. We report bi = -2.35 +/- 0.014 (stat.) +/- 0.014 (syst.). This experiment is a revision of the previous experiment which was done in 2008, and partially explains the non-zero phase shift seen in 2008 experiment even if target cell was completely unpolarized. The measurement of the coherent neutron scattering length b c of the 4He was done using a (111) single silicon crystal interferometer. The neutron interferometry and optics facility at NIST had been used previously to determine the coherent scattering lengths for n- 1H, n-2H, and n-3He to less than 1% relative uncertainty. We report bc of the 4He
17. Properties and uses of cold neutron beams
SciTech Connect
Clark, David D.
1992-07-01
Cold neutrons are conventionally defined as those with energy below 0.005 eV; the corresponding velocity and wavelength arc 980 m/s and 4 angstroms. The first extensive use of cold neutrons was in the 1960's by condensed matter physicists for investigations of spatial structure and internal dynamics of solids and liquids. Different experiments place different requirements on neutron beams, but it is usually advantageous to eliminate the faster neutrons and the gamma rays that are present in normal reactor beams. Several types of filters that pass only the low-energy portion of an incident Maxwellian spectrum have been developed and will be discussed. Examples include single crystal quartz or bismuth (room temperature or cooled), polycrystalline beryllium, and neutron guides. For any of these shifting the incident neutrons to a lower energy spectrum by use of a cold moderator leads to large increases in the intensity of cold neutrons. The properties of the beams resulting from the particular combination of a cold moderator and a neutron guide will be discussed. These include the changes in beam intensity and spectral shape as warm neutrons in a typical reactor spectrum first interact with a cold moderator and then pass through a straight or curved neutron guide. The spatial and angular distribution of the neutrons at the exit of the guide will be described. One further important effect for cold neutron beam experiments involving nuclear reactions is the increase in reaction rates because of the usual 1/v dependence of reaction cross sections and another is the considerable simplification with cold neutrons in the problems of collimating, shielding, and stopping the beam. The resulting benefits for studies of nuclear energy levels by neutron capture gamma-ray and conversion electron experiments and for the analysis of materials by PGNAA will be discussed. Neutron depth profiling is also improved with cold neutrons. (author)
18. Wolter Mirrors for Neutron Imaging
Wu, Huarui; Khaykovich, Boris; Wang, Xuewu; Hussey, Daniel S.
Conventional radiography based on attenuation of a well-collimated beam remains the mainstay of neutron imaging. The spatial resolution attained with this pinhole-camera method depends on the beam collimation; therefore, achieving the spatial resolution of a few microns is practically difficult, since collimating the neutron beam results in a low flux. The use of focusing devices allows maintaining sufficient spatial resolution without collimating the beam. Therefore, axisymmetric grazing-incidence focusing mirrors (Wolter mirrors) have begun to be introduced to neutron imaging. In this paper, a design of a neutron microscope for NIST Center for Neutron Research (NCNR) is presented. We evaluate the spatial resolution and study field curvature aberrations of Wolter mirrors through ray-tracing simulations. A general formula is found describing the field curvature, and ways to counter these aberrations are discussed.
Heller, A. K.; Brenizer, J. S.
Neutron radiography and its related two-dimensional (2D) neutron imaging techniques have been established as invaluable nondestructive inspection methods and quantitative measurement tools. They have been used in a wide variety of applications ranging from inspection of aircraft engine turbine blades to study of two-phase fluid flow in operating proton exchange membrane fuel cells. Neutron radiography is similar to X-ray radiography in that the method produces a 2D attenuation map of neutron radiation that has penetrated the object being examined. However, the images produced differ and are often complementary due to the differences between X-ray and neutron interaction mechanisms. The uses and types of 2D neutron imaging have expanded over the past 15 years as a result of advances in imaging technology and improvements in neutron generators/sources and computers. Still, high-intensity sources such as those from reactors and spallation neutron sources, together with conventional film radiography, remain the mainstay of high-resolution, large field-of-view neutron imaging. This chapter presents a summary of the history, methods, and related variations of neutron radiography techniques.
20. Hybrid superconducting neutron detectors
SciTech Connect
Merlo, V.; Lucci, M.; Ottaviani, I.; Salvato, M.; Cirillo, M.; Scherillo, A.; Celentano, G.; Pietropaolo, A.
2015-03-16
A neutron detection concept is presented that is based on superconductive niobium (Nb) strips coated by a boron (B) layer. The working principle of the detector relies on the nuclear reaction, {sup 10}B + n → α + {sup 7}Li, with α and Li ions generating a hot spot on the current-biased Nb strip which in turn induces a superconducting-normal state transition. The latter is recognized as a voltage signal which is the evidence of the incident neutron. The above described detection principle has been experimentally assessed and verified by irradiating the samples with a pulsed neutron beam at the ISIS spallation neutron source (UK). It is found that the boron coated superconducting strips, kept at a temperature T = 8 K and current-biased below the critical current I{sub c}, are driven into the normal state upon thermal neutron irradiation. As a result of the transition, voltage pulses in excess of 40 mV are measured while the bias current can be properly modulated to bring the strip back to the superconducting state, thus resetting the detector. Measurements on the counting rate of the device are presented and the basic physical features of the detector are discussed.
1. Hybrid superconducting neutron detectors
Merlo, V.; Salvato, M.; Cirillo, M.; Lucci, M.; Ottaviani, I.; Scherillo, A.; Celentano, G.; Pietropaolo, A.
2015-03-01
A neutron detection concept is presented that is based on superconductive niobium (Nb) strips coated by a boron (B) layer. The working principle of the detector relies on the nuclear reaction, 10B + n → α + 7Li, with α and Li ions generating a hot spot on the current-biased Nb strip which in turn induces a superconducting-normal state transition. The latter is recognized as a voltage signal which is the evidence of the incident neutron. The above described detection principle has been experimentally assessed and verified by irradiating the samples with a pulsed neutron beam at the ISIS spallation neutron source (UK). It is found that the boron coated superconducting strips, kept at a temperature T = 8 K and current-biased below the critical current Ic, are driven into the normal state upon thermal neutron irradiation. As a result of the transition, voltage pulses in excess of 40 mV are measured while the bias current can be properly modulated to bring the strip back to the superconducting state, thus resetting the detector. Measurements on the counting rate of the device are presented and the basic physical features of the detector are discussed.
2. Superconducting thermal neutron detectors
Merlo, V.; Pietropaolo, A.; Celentano, G.; Cirillo, M.; Lucci, M.; Ottaviani, I.; Salvato, M.; Scherillo, A.; Schooneveld, E. M.; Vannozzi, A.
2016-09-01
A neutron detection concept is presented that is based on superconductive niobium nitride (NbN) strips coated by a boron (B) layer. The working principle is well described by a hot spot mechanism: upon the occurrence of the nuclear reactions n + 10B → α + 7Li + 2.8 MeV, the energy released by the secondary particles into the strip induces a superconducting-normal state transition. The latter is recognized as a voltage signal which is the evidence of the incident neutron. The above described detection principle has been experimentally assessed and verified by irradiating the samples with a pulsed neutron beam at the ISIS spallation neutron source (UK). It is found that the boron coated superconducting strips, kept at a temperature T below 11K and current-biased below the critical current IC, are driven into the normal state upon thermal neutron irradiation. Measurements on the counting rate of the device are presented and the basic physical features of the detector are discussed and compared to those of a borated Nb superconducting strip.
3. Development of An Epi-thermal Neutron Field for Fundamental Researches for BNCT with A DT Neutron Source
Osawa, Yuta; Imoto, Shoichi; Kusaka, Sachie; Sato, Fuminobu; Tanoshita, Masahiro; Murata, Isao
2017-09-01
Boron Neutron Capture Therapy (BNCT) is known to be a new promising cancer therapy suppressing influence against normal cells. In Japan, Accelerator Based Neutron Sources (ABNS) are being developed for BNCT. For the spread of ABNS based BNCT, we should characterize the neutron field beforehand. For this purpose, we have been developing a low-energy neutron spectrometer based on 3He position sensitive proportional counter. In this study, a new intense epi-thermal neutron field was developed with a DT neutron source for verification of validity of the spectrometer. After the development, the neutron field characteristics were experimentally evaluated by using activation foils. As a result, we confirmed that an epi-thermal neutron field was successfully developed suppressing fast neutrons substantially. Thereafter, the neutron spectrometer was verified experimentally. In the verification, although a measured detection depth distribution agreed well with the calculated distribution by MCNP, the unfolded spectrum was significantly different from the calculated neutron spectrum due to contribution of the side neutron incidence. Therefore, we designed a new neutron collimator consisting of a polyethylene pre-collimator and boron carbide neutron absorber and confirmed numerically that it could suppress the side incident neutrons and shape the neutron flux to be like a pencil beam.
4. Neutron detector
DOEpatents
Stephan, Andrew C.; Jardret; Vincent D.
2011-04-05
A neutron detector has a volume of neutron moderating material and a plurality of individual neutron sensing elements dispersed at selected locations throughout the moderator, and particularly arranged so that some of the detecting elements are closer to the surface of the moderator assembly and others are more deeply embedded. The arrangement captures some thermalized neutrons that might otherwise be scattered away from a single, centrally located detector element. Different geometrical arrangements may be used while preserving its fundamental characteristics. Different types of neutron sensing elements may be used, which may operate on any of a number of physical principles to perform the function of sensing a neutron, either by a capture or a scattering reaction, and converting that reaction to a detectable signal. High detection efficiency, an ability to acquire spectral information, and directional sensitivity may be obtained.
5. Analysis of the neutron generation from a D-Li neutron source
SciTech Connect
Gomes, I.
1994-02-01
The study of the neutron generation from the D-Li reaction is an important issue to define the optimum combination of the intervening parameters during the design phase of a D-Li neutron source irradiation facility. The major players in defining the neutron yield from the D-Li reaction are the deuteron incident energy and the beam current, provided that the lithium target is thick enough to stop all incident deuterons. The incident deuteron energy also plays a role on the angular distribution of the generated neutrons, on the energy distribution of the generated neutrons, and on the maximum possible energy of the neutrons. The D-Li reaction produces neutrons with energies ranging from eVs to several MeVs. The angular distribution of these neutrons is dependent on the energy of both, incident deuterons and generated neutrons. The deuterons lose energy interacting with the lithium target material in such a way that the energy of the deuterons inside the lithium target varies from the incident deuteron energy to essentially zero. The first part of this study focuses in analyzing the neutron generation rate from the D-Li reaction as a function of the intervening parameters, in defining the source term, in terms of the energy and angular distributions of the generated neutrons, and finally in providing some insights of the impact of varying input parameters on the generation rate and correlated distributions. In the second part an analytical description of the Monte Carlo sampling procedure of the neutron from the D-Li reaction is provided with the aim at further Monte Carlo transport of the D-Li neutrons.
6. Atmospheric neutrons
NASA Technical Reports Server (NTRS)
Preszler, A. M.; Moon, S.; White, R. S.
1976-01-01
Additional calibrations of the University of California double-scatter neutron detector and additional analysis corrections lead to slightly changed neutron fluxes. The theoretical angular distributions of Merker (1975) are in general agreement with the reported experimental fluxes but do not give the peaks for vertical upward and downward moving neutrons. The theoretical neutron escape current is in agreement with the experimental values from 10 to 100 MeV. The experimental fluxes obtained agree with those of Kanbach et al. (1974) in the overlap region from 70 to 100 MeV.
7. Neutron Stars
NASA Technical Reports Server (NTRS)
Cottam, J.
2007-01-01
Neutron stars were discovered almost 40 years ago, and yet many of their most fundamental properties remain mysteries. There have been many attempts to measure the mass and radius of a neutron star and thereby constrain the equation of state of the dense nuclear matter at their cores. These have been complicated by unknown parameters such as the source distance and burning fractions. A clean, straightforward way to access the neutron star parameters is with high-resolution spectroscopy. I will present the results of searches for gravitationally red-shifted absorption lines from the neutron star atmosphere using XMM-Newton and Chandra.
SciTech Connect
Berger, H.; Iddings, F.
1998-08-01
9. Fast neutrons incident on rotors : - tantalum.
SciTech Connect
Smith, A. B.
2005-03-21
Reports in the Argonne National Laboratory Nuclear Data and Measurement Series present results of studies in the field of microscopic nuclear data. The primary objective of the series is the dissemination of information in the comprehensive form required for nuclear technology applications. This series is devoted to: (a) measured microscopic nuclear parameters, (b) experimental techniques and facilities employed in measurements, (c) the analysis, correlation and interpretation of nuclear data, and (d) the compilation and evaluation of nuclear data.
10. NEUTRONIC REACTOR
DOEpatents
Fermi, E.; Zinn, W.H.; Anderson, H.L.
1958-09-16
Means are presenied for increasing the reproduction ratio of a gaphite- moderated neutronic reactor by diminishing the neutron loss due to absorption or capture by gaseous impurities within the reactor. This means comprised of a fluid-tight casing or envelope completely enclosing the reactor and provided with a valve through which the casing, and thereby the reactor, may be evacuated of atmospheric air.
11. Neutronic reactor
DOEpatents
Wende, Charles W. J.
1976-08-17
A safety rod for a nuclear reactor has an inner end portion having a gamma absorption coefficient and neutron capture cross section approximately equal to those of the adjacent shield, a central portion containing materials of high neutron capture cross section and an outer end portion having a gamma absorption coefficient at least equal to that of the adjacent shield.
12. Neutron spin echo scattering angle measurement (SESAME)
SciTech Connect
Pynn, R.; Fitzsimmons, M.R.; Fritzsche, H.; Gierlings, M.; Major, J.; Jason, A.
2005-05-15
We describe experiments in which the neutron spin echo technique is used to measure neutron scattering angles. We have implemented the technique, dubbed spin echo scattering angle measurement (SESAME), using thin films of Permalloy electrodeposited on silicon wafers as sources of the magnetic fields within which neutron spins precess. With 30-{mu}m-thick films we resolve neutron scattering angles to about 0.02 deg. with neutrons of 4.66 A wavelength. This allows us to probe correlation lengths up to 200 nm in an application to small angle neutron scattering. We also demonstrate that SESAME can be used to separate specular and diffuse neutron reflection from surfaces at grazing incidence. In both of these cases, SESAME can make measurements at higher neutron intensity than is available with conventional methods because the angular resolution achieved is independent of the divergence of the neutron beam. Finally, we discuss the conditions under which SESAME might be used to probe in-plane structure in thin films and show that the method has advantages for incident neutron angles close to the critical angle because multiple scattering is automatically accounted for.
13. Neutron source
DOEpatents
Cason, J.L. Jr.; Shaw, C.B.
1975-10-21
A neutron source which is particularly useful for neutron radiography consists of a vessel containing a moderating media of relatively low moderating ratio, a flux trap including a moderating media of relatively high moderating ratio at the center of the vessel, a shell of depleted uranium dioxide surrounding the moderating media of relatively high moderating ratio, a plurality of guide tubes each containing a movable source of neutrons surrounding the flux trap, a neutron shield surrounding one part of each guide tube, and at least one collimator extending from the flux trap to the exterior of the neutron source. The shell of depleted uranium dioxide has a window provided with depleted uranium dioxide shutters for each collimator. Reflectors are provided above and below the flux trap and on the guide tubes away from the flux trap.
14. The DAMPE Neutron Detector
Yan, Zhang; Tao, Ma; Yongyi, Huang
2016-07-01
The first Chinese space observatory DAMPE (DArk Matter Particle Explorer) was successfully launched on Dec. 17th, 2015. One major scientific object of DAMPE is to measure electrons between 5GeV to 10TeV with excellent energy resolution (1.5% at 800GeV) to search for possible dark matter signatures. The detector consists of four subsystems: a plastic scintillator detector (PSD), a silicon-tungsten tracker (STK), a BGO calorimeter (BGO), and a neutron detector (NUD). The NUD on board DAMPE is designed to detect moderated neutrons via the boron capture of thermal neutrons in boron-doped plastics. Given the fact that hadron showers initiated in the BGO calorimeter by incident nuclei tend to be followed by significantly more neutron activities comparing to electromagnetic cascades triggered by electrons, the NUD provides an additional order of magnitude hadron rejection capability to improve the overall e/p discrimination of DAMPE up to 10 ^{5}. Preliminary analysis of the in-orbit data is given, together with comparisons to the results obtained by a detailed GEANT4 simulation of the NUD instrument.
15. Thermal neutron detection system
DOEpatents
Peurrung, Anthony J.; Stromswold, David C.
2000-01-01
According to the present invention, a system for measuring a thermal neutron emission from a neutron source, has a reflector/moderator proximate the neutron source that reflects and moderates neutrons from the neutron source. The reflector/moderator further directs thermal neutrons toward an unmoderated thermal neutron detector.
16. Neutron decay of the Giant Pairing Vibration in 15C
Cavallaro, M.; Agodi, C.; Assié, M.; Azaiez, F.; Cappuzzello, F.; Carbone, D.; de Séréville, N.; Foti, A.; Pandola, L.; Scarpaci, J. A.; Sgouros, O.; Soukeras, V.
2016-06-01
The neutron decay of the resonant states of light neutron-rich nuclei is an important and poorly explored property, useful to extract valuable nuclear structure information. The neutron decay of the 15C resonances populated via the two-neutron transfer reaction 13C(18O,16O n) at 84 MeV incident energy is studied using an innovative technique which couples the MAGNEX magnetic spectrometer and the EDEN neutron detector array. The data show that the recently observed 15C Giant Pairing Vibration at 13.7 MeV mainly decays via two-neutron emission.
17. NEUTRON SOURCE
DOEpatents
Bernander, N.K. et al.
1960-10-18
An apparatus is described for producing neutrons through target bombardment with deuterons. Deuterium gas is ionized by electron bombardment and the deuteron ions are accelerated through a magnetic field to collimate them into a continuous high intensity beam. The ion beam is directed against a deuteron pervious metal target of substantially the same nnaterial throughout to embed the deuterous therein and react them to produce neutrons. A large quantity of neutrons is produced in this manner due to the increased energy and quantity of ions bombarding the target.
18. Neutron polycapillary optics: back to future
Ioffe, Alexander
2005-08-01
At the sunrise of the Kumakhov optics, the latter was enormously attractive for neutron scientists eager to take advantage from focused neutron beams. However, in spite of this initial enthusiasm, rear neutron applications of polycapillary optics are mostly limited either to feasibility studies or to non-scattering techniques. The reason for such a drawback is a serious degradation in the momentum resolution of neutron scattering methods caused by increased divergence of focused beams. However, limited brightness of present day neutron sources (both existing and coming) requires rethinking the current situation. Neutron lenses are successfully tested for low Q-resolution applications providing an enormous increase of neutron flux at small samples. Moreover, some recent developments in the field of neutron instrumentation allow us to overcome the resolution problem by decoupling the angular resolution and the incident beam divergence thus opening the opportunity for the use of focusing neutron optics in some high Q-resolution application. Further advances in technology required to improve the performance of neutron polycapillary optics, also in combination with modem methods of polarization of neutron beams, are discussed.
19. Compact neutron imaging system using axisymmetric mirrors
SciTech Connect
Khaykovich, Boris; Moncton, David E; Gubarev, Mikhail V; Ramsey, Brian D; Engelhaupt, Darell E
2014-05-27
A dispersed release of neutrons is generated from a source. A portion of this dispersed neutron release is reflected by surfaces of a plurality of nested, axisymmetric mirrors in at least an inner mirror layer and an outer mirror layer, wherein the neutrons reflected by the inner mirror layer are incident on at least one mirror surface of the inner mirror layer N times, wherein N is an integer, and wherein neutrons reflected by the outer mirror are incident on a plurality of mirror surfaces of the outer layer N+i times, where i is a positive integer, to redirect the neutrons toward a target. The mirrors can be formed by a periodically reversed pulsed-plating process.
20. A NEW SINGLE-CRYSTAL FILTERED THERMAL NEUTRON SOURCE FOR NEUTRON CAPTURE THERAPY RESEARCH AT THE UNIVERSITY OF MISSOURI
SciTech Connect
John D. Brockman; David W. Nigg; M. Frederick Hawthorne
2008-09-01
Parameter studies, design calculations and initial neutronic performance measurements have been completed for a new thermal neutron beamline to be used for neutron capture therapy cell and small-animal radiobiology studies at the University of Missouri Research Reactor. The beamline features the use of single-crystal silicon and bismuth sections for neutron filtering and for reduction of incident gamma radiation. The calculated and measured thermal neutron flux produced at the irradiation location is on the order of 9.5x108 neutrons/cm2-s, with a measured cadmium ratio (Au foils) of 105, indicating a well-thermalized spectrum.
1. NEUTRONIC REACTORS
DOEpatents
Wigner, E.P.
1960-11-22
A nuclear reactor is described wherein horizontal rods of thermal- neutron-fissionable material are disposed in a body of heavy water and extend through and are supported by spaced parallel walls of graphite.
2. NEUTRONIC REACTOR
DOEpatents
1958-09-16
This patent relates to a reflector means for a neutronic reactor. A reflector comprised of a plurality of vertically movable beryllium control members is provided surrounding the sides of the reactor core. An absorber of fast neutrons comprised of natural uramum surrounds the reflector. An absorber of slow neutrons surrounds the absorber of fast neutrons and is formed of a plurality of beryllium blocks having natural uranium members distributcd therethrough. in addition, a movable body is positioned directly below the core and is comprised of a beryllium reflector and an absorbing member attached to the botiom thereof, the absorbing member containing a substance selected from the goup consisting of natural urantum and Th/sup 232/.
3. Neutron Imaging Camera
NASA Technical Reports Server (NTRS)
Hunter, Stanley D.; DeNolfo, Georgia; Floyd, Sam; Krizmanic, John; Link, Jason; Son, Seunghee; Guardala, Noel; Skopec, Marlene; Stark, Robert
2008-01-01
We describe the Neutron Imaging Camera (NIC) being developed for DTRA applications by NASA/GSFC and NSWC/Carderock. The NIC is based on the Three-dimensional Track Imager (3-DTI) technology developed at GSFC for gamma-ray astrophysics applications. The 3-DTI, a large volume time-projection chamber, provides accurate, approximately 0.4 mm resolution. 3-D tracking of charged particles. The incident direction of fast neutrons, E(sub N) > 0.5 MeV. arc reconstructed from the momenta and energies of the proton and triton fragments resulting from 3He(n,p)3H interactions in the 3-DTI volume. We present angular and energy resolution performance of the NIC derived from accelerator tests.
4. Neutron Imaging Camera
NASA Technical Reports Server (NTRS)
Hunter, Stanley D.; DeNolfo, Georgia; Floyd, Sam; Krizmanic, John; Link, Jason; Son, Seunghee; Guardala, Noel; Skopec, Marlene; Stark, Robert
2008-01-01
We describe the Neutron Imaging Camera (NIC) being developed for DTRA applications by NASA/GSFC and NSWC/Carderock. The NIC is based on the Three-dimensional Track Imager (3-DTI) technology developed at GSFC for gamma-ray astrophysics applications. The 3-DTI, a large volume time-projection chamber, provides accurate, approximately 0.4 mm resolution. 3-D tracking of charged particles. The incident direction of fast neutrons, E(sub N) > 0.5 MeV. arc reconstructed from the momenta and energies of the proton and triton fragments resulting from 3He(n,p)3H interactions in the 3-DTI volume. We present angular and energy resolution performance of the NIC derived from accelerator tests.
5. NEUTRON SOURCES
DOEpatents
Richmond, J.L.; Wells, C.E.
1963-01-15
A neutron source is obtained without employing any separate beryllia receptacle, as was formerly required. The new method is safer and faster, and affords a source with both improved yield and symmetry of neutron emission. A Be container is used to hold and react with Pu. This container has a thin isolating layer that does not obstruct the desired Pu--Be reaction and obviates procedures previously employed to disassemble and remove a beryllia receptacle. (AEC)
6. NEUTRONIC REACTOR
DOEpatents
Fraas, A.P.; Mills, C.B.
1961-11-21
A neutronic reactor in which neutron moderation is achieved primarily in its reflector is described. The reactor structure consists of a cylindrical central "island" of moderator and a spherical moderating reflector spaced therefrom, thereby providing an annular space. An essentially unmoderated liquid fuel is continuously passed through the annular space and undergoes fission while contained therein. The reactor, because of its small size, is particularly adapted for propulsion uses, including the propulsion of aircraft. (AEC)
7. NEUTRONIC REACTOR
DOEpatents
Wigner, E.P.
1958-04-22
A nuclear reactor for isotope production is described. This reactor is designed to provide a maximum thermal neutron flux in a region adjacent to the periphery of the reactor rather than in the center of the reactor. The core of the reactor is generally centrally located with respect tn a surrounding first reflector, constructed of beryllium. The beryllium reflector is surrounded by a second reflector, constructed of graphite, which, in tune, is surrounded by a conventional thermal shield. Water is circulated through the core and the reflector and functions both as a moderator and a coolant. In order to produce a greatsr maximum thermal neutron flux adjacent to the periphery of the reactor rather than in the core, the reactor is designed so tbat the ratio of neutron scattering cross section to neutron absorption cross section averaged over all of the materials in the reflector is approximately twice the ratio of neutron scattering cross section to neutron absorption cross section averaged over all of the material of the core of the reactor.
8. Neutron-induced background by an α-beam incident on a deuterium gas target and its implications for the study of the 2H(α,γ)6Li reaction at LUNA
Anders, M.; Trezzi, D.; Bellini, A.; Aliotta, M.; Bemmerer, D.; Broggini, C.; Caciolli, A.; Costantini, H.; Corvisiero, P.; Davinson, T.; Elekes, Z.; Erhard, M.; Formicola, A.; Fülöp, Zs.; Gervino, G.; Guglielmetti, A.; Gustavino, C.; Gyürky, Gy.; Junker, M.; Lemut, A.; Marta, M.; Mazzocchi, C.; Menegazzo, R.; Prati, P.; Rossi Alvarez, C.; Scott, D.; Somorjai, E.; Straniero, O.; Szücs, T.
2013-02-01
The production of the stable isotope 6Li in standard Big Bang nucleosynthesis has recently attracted much interest. Recent observations in metal-poor stars suggest that a cosmological 6Li plateau may exist. If true, this plateau would come in addition to the well-known Spite plateau of 7Li abundances and would point to a predominantly primordial origin of 6Li , contrary to the results of standard Big Bang nucleosynthesis calculations. Therefore, the nuclear physics underlying Big Bang 6Li production must be revisited. The main production channel for 6Li in the Big Bang is the 2H(α,γ)6Li reaction. The present work reports on neutron-induced effects in a high-purity germanium detector that were encountered in a new study of this reaction. In the experiment, an α-beam from the underground accelerator LUNA in Gran Sasso, Italy, and a windowless deuterium gas target are used. A low neutron flux is induced by energetic deuterons from elastic scattering and, subsequently, the 2H(d,n)3He reaction. Due to the ultra-low laboratory neutron background at LUNA, the effect of this weak flux of 2-3MeV neutrons on well-shielded high-purity germanium detectors has been studied in detail. Data have been taken at 280 and 400keV α-beam energy and for comparison also using an americium-beryllium neutron source.
9. Neutron spectra and angular distributions of concrete-moderated neutron calibration fields at JAERI.
PubMed
Yoshizawa, M; Tanimura, Y; Saegusa, J; Nemoto, H; Yoshida, M
2004-01-01
The Facility of Radiation Standards of Japan Atomic Energy Research Institute has been equipped with concrete-moderated neutron calibration fields as simulated workplace neutron fields. The fields use an 241Am-Be neutron source placed in the narrow space surrounded by concrete bricks, walls and floor. The neutron spectra and the neutron fluence rates of the fields were measured with the Bonner multi-sphere spectrometer system (BMS), spherical recoil-proton proportional counters, and a liquid scintillation counter (NE-213). The results were compared with each other. The reference values of H*(10) were determined from the results of BMS. The angular distributions of neutron fluence were calculated using MCNP-4B2 to obtain the reference values of Hp(10). The calculated results show that the scattered neutrons have a wide range of incident angles. The reference Hp(10) values considered the angular distribution were found to be 10-18% smaller than those without consideration.
10. FOREWORD: Neutron metrology Neutron metrology
Thomas, David J.; Nolte, Ralf; Gressier, Vincent
2011-12-01
11. Computational characterization and experimental validation of the thermal neutron source for neutron capture therapy research at the University of Missouri
SciTech Connect
Broekman, J. D.; Nigg, D. W.; Hawthorne, M. F.
2013-07-01
Parameter studies, design calculations and neutronic performance measurements have been completed for a new thermal neutron beamline constructed for neutron capture therapy cell and small-animal radiobiology studies at the University of Missouri Research Reactor. The beamline features the use of single-crystal silicon and bismuth sections for neutron filtering and for reduction of incident gamma radiation. The computational models used for the final beam design and performance evaluation are based on coupled discrete-ordinates and Monte Carlo techniques that permit detailed modeling of the neutron transmission properties of the filtering crystals with very few approximations. Validation protocols based on neutron activation spectrometry measurements and rigorous least-square adjustment techniques show that the beam produces a neutron spectrum that has the anticipated level of thermal neutron flux and a somewhat higher than expected, but radio-biologically insignificant, epithermal neutron flux component. (authors)
12. Neutron Lifetime Measurement Using Magnetically Trapped Ultracold Neutrons
Huffer, Craig; Huffman, P. R.; Schelhammer, K. W.; Dewey, M. S.; Huber, M. G.; Hughes, P. P.; Mumm, H. P.; Thompson, A. K.; Coakley, K.; Yue, A. T.; O'Shaughnessy, C. M.
2017-01-01
The neutron beta-decay lifetime is important in both nuclear astrophysics and in understanding weak interactions in the framework of the Standard Model. An experiment based at the NIST Center for Neutron Research was designed to address statistical and systematic limitations of former measurements. In our approach, a beam of 0.89 nm neutrons is incident on a superfluid 4He target within the minimum field region of an Ioffe-type magnetic trap. Some of the neutrons are subsequently downscattered by single phonons in the helium to low energies (< 100 neV) and those in the appropriate spin state become trapped. The inverse process, upscattering of UCN, is suppressed by the low phonon density in the < 300 mK helium, allowing the neutron to travel undisturbed through the helium. When the neutron decays the energetic electron produces a scintillation signal in the helium that is detected in real time using photomultiplier tubes. The current measurement is limited by larger than expected systematic corrections. We will discuss the result of the latest dataset and comment on the potential of future measurements.
13. A neutron sensor based on synthetic single crystal diamond
SciTech Connect
Schmid, G J; Koch, J A; Lerche, R A; Moran, M J
2003-10-17
We report the first neutron data for a single crystal Chemical Vapor Deposition (CVD) diamond sensor. Results are presented for 2.5, 14.1, and 14.9 MeV incident neutrons. We show that the energy resolution for 14.1 MeV neutrons is at least 2.9% (as limited by the energy spread of the incident neutrons), and perhaps as good as 0.4% (as extrapolated from high resolution {alpha} particle data). This result could be relevant to fusion neutron spectroscopy at machines like the International Thermonuclear Experimental Reactor (ITER). We also show that our sensor has a high neutron linear attenuation coefficient, due to the high atomic density of diamond, and this could lead to applications in fission neutron detection.
14. Neutron therapy of cancer
NASA Technical Reports Server (NTRS)
Frigerio, N. A.; Nellans, H. N.; Shaw, M. J.
1969-01-01
Reports relate applications of neutrons to the problem of cancer therapy. The biochemical and biophysical aspects of fast-neutron therapy, neutron-capture and neutron-conversion therapy with intermediate-range neutrons are presented. Also included is a computer program for neutron-gamma radiobiology.
15. Neutron-proton bremsstrahlung experiments
SciTech Connect
Koster, J.E.; Nelson, R.O.; Schillaci, M.E.; Wender, S.A.; Mayo, D.; Brady, F.P.; Romero, J.; Krofcheck, D.; Blann, M.; Anthony, P.; Brown, V.R.; Hansen, L.; Pohl, B.; Sangster, T.C.; Nifenecker, H.; Pinston, J.A.
1992-12-01
It is well known that charged particles emit bremsstrahlung radiation when they are accelerated. Classical electron bremsstrahlung occurs when a photon is emitted by an electron accelerated in the field of a nucleus. The bremsstrahlung process also occurs in the scattering of nucleons, for which it is the lowest energy inelastic process that can occur. Like electron bremsstrahlung, nucleon-nucleon bremsstrahlung also requires the exchange of a virtual particle to conserve energy and momentum. In electron bremsstrahlung a virtual photon is exchanged but with two nucleons a meson can be exchanged. Unlike electron bremsstrahlung, in nucleon-nucleon bremsstrahlung the photon can originate from the exchanged meson. This exchange contribution has been shown in calculations to be a significant fraction of bremsstrahlung events. Thus bremsstrahlung serves as a probe of exchange currents in the nucleon-nucleon interaction. Because of a lack of a free neutron target or an intense neutron beam, few measurements of neutron-proton bremsstrahlung exist, each having poor statistical accuracy and poor energy resolution. The white neutron source at the Weapons Neutron Research (WNR) target area at the Los Alamos Meson Physics Facility (LAMPF) produces neutrons with energies from below 50 to above 400 MeV. Using time-of-flight techniques and a liquid hydrogen target, we are measuring the outgoing photons of energies up to 250 MeV at gamma ray angles of around 90{degree} relative to the incident beam. Protons scattered at very forward angles are also detected in coincidence with the gamma rays.
16. NEUTRONIC REACTOR
DOEpatents
Hurwitz, H. Jr.; Brooks, H.; Mannal, C.; Payne, J.H.; Luebke, E.A.
1959-03-24
A reactor of the heterogeneous, liquid cooled type is described. This reactor is comprised of a central region of a plurality of vertically disposed elongated tubes surrounded by a region of moderator material. The central region is comprised of a central core surrounded by a reflector region which is surrounded by a fast neutron absorber region, which in turn is surrounded by a slow neutron absorber region. Liquid sodium is used as the primary coolant and circulates through the core which contains the fuel elements. Control of the reactor is accomplished by varying the ability of the reflector region to reflect neutrons back into the core of the reactor. For this purpose the reflector is comprised of moderator and control elements having varying effects on reactivity, the control elements being arranged and actuated by groups to give regulation, shim, and safety control.
17. NEUTRON SOURCE
DOEpatents
Foster, J.S. Jr.
1960-04-19
A compact electronic device capable of providing short time high density outputs of neutrons is described. The device of the invention includes an evacuated vacuum housing adapted to be supplied with a deuterium, tritium, or other atmosphere and means for establishing an electrical discharge along a path through the gas. An energized solenoid is arranged to constrain the ionized gas (plasma) along the path. An anode bearing adsorbed or adherent target material is arranged to enclose the constrained plasma. To produce neutrons a high voltage is applied from appropriate supply means between the plasma and anode to accelerate ions from the plasma to impinge upcn the target material, e.g., comprising deuterium.
18. NEUTRONIC REACTOR
DOEpatents
Fermi, E.; Szilard, L.
1957-09-24
Reactors of the type employing plates of natural uranium in a moderator are discussed wherein the plates are um-formly disposed in parallel relationship to each other thereby separating the moderator material into distinct and individual layers. Each plate has an uninterrupted sunface area substantially equal to the cross-sectional area of the active portion of the reactor, the particular size of the plates and the volume ratio of moderator to uranium required to sustain a chain reaction being determinable from the known purity of these materials and other characteristics such as the predictable neutron losses due to the formation of radioactive elements of extremely high neutron capture cross section.
19. NEUTRONIC REACTOR
DOEpatents
Wigner, E.P.; Weinberg, A.W.; Young, G.J.
1958-04-15
A nuclear reactor which uses uranium in the form of elongated tubes as fuel elements and liquid as a coolant is described. Elongated tubular uranium bodies are vertically disposed in an efficient neutron slowing agent, such as graphite, for example, to form a lattice structure which is disposed between upper and lower coolant tanks. Fluid coolant tubes extend through the uranium bodies and communicate with the upper and lower tanks and serve to convey the coolant through the uranium body. The reactor is also provided with means for circulating the cooling fluid through the coolant tanks and coolant tubes, suitable neutron and gnmma ray shields, and control means.
20. Prompt fission neutron spectra in fast-neutron-induced fission of 238U
Desai, V. V.; Nayak, B. K.; Saxena, A.; Suryanarayana, S. V.; Capote, R.
2015-07-01
Prompt fission neutron spectrum (PFNS) measurements for the neutron-induced fission of 238U are carried out at incident neutron energies of 2.0, 2.5, and 3.0 MeV, respectively. The time-of-flight technique is employed to determine the energy of fission neutrons. The prompt fission neutron energy spectra so obtained are analyzed using Watt parametrization to derive the neutron multiplicity and average prompt fission neutron energy. The present experimental PFNS data are compared with the evaluated spectra taken from the ENDF/B-VII.1 library and the predictive calculations carried out using the empire-3.2 (Malta) code with built-in Los Alamos (LA) and Kornilov PFNS models. The sensitivity of the empire-3.2 LA model-calculated PFNS to the nuclear level density parameter of the average fission fragment and to the total kinetic energy is investigated. empire-3.2 LA model PFNS calculations that use Madland 2006-recommended values [D. G. Madland, Nucl. Phys. A 772, 113 (2006), 10.1016/j.nuclphysa.2006.03.013] of the total kinetic energy and the level density parameter a =A /(10 ±0.5 ) compare very well to measured data at all incident neutron incident energies.
1. The measurement of gamma ray induced heating in a mixed neutron and gamma ray environment
SciTech Connect
Chiu, H.K.
1991-10-01
The problem of measuring the gamma heating in a mixed DT neutron and gamma ray environment was explored. A new detector technique was developed to make this measurement. Gamma heating measurements were made in a low-Z assembly irradiated with 14-Mev neutrons and (n, n{prime}) gammas produced by a Texas Nuclear Model 9400 neutron generator. Heating measurements were made in the mid-line of the lattice using a proportional counter operating in the Continuously-varied Bias-voltage Acquisition mode. The neutron-induced signal was separated from the gamma-induced signal by exploiting the signal rise-time differences inherent to radiations of different linear energy transfer coefficient, which are observable in a proportional counter. The operating limits of this measurement technique were explored by varying the counter position in the low-Z lattice, hence changing the irradiation spectrum observed. The experiment was modelled numerically to help interpret the measured results. The transport of neutrons and gamma rays in the assembly was modelled using the one- dimensional radiation transport code ANISN/PC. The cross-section set used for these calculations was derived from the ENDF/B-V library using the code MC{sup 2}-2 for the case of DT neutrons slowing down in a low-Z material. The calculated neutron and gamma spectra in the slab and the relevant mass-stopping powers were used to construct weighting factors which relate the energy deposition in the counter fill-gas to that in the counter wall and in the surrounding material. The gamma energy deposition at various positions in the lattice is estimated by applying these weighting factors to the measured gamma energy deposition in the counter at those locations.
2. Neutronic reactor
DOEpatents
Carleton, John T.
1977-01-25
A graphite-moderated nuclear reactor includes channels between blocks of graphite and also includes spacer blocks between adjacent channeled blocks with an axis of extension normal to that of the axis of elongation of the channeled blocks to minimize changes in the physical properties of the graphite as a result of prolonged neutron bombardment.
3. NEUTRONIC REACTORS
DOEpatents
Anderson, H.L.
1958-10-01
The design of control rods for nuclear reactors are described. In this design the control rod consists essentially of an elongated member constructed in part of a neutron absorbing material and having tube means extending therethrough for conducting a liquid to cool the rod when in use.
4. Detecting energy dependent neutron capture distributions in a liquid scintillator
Balmer, Matthew J. I.; Gamage, Kelum A. A.; Taylor, Graeme C.
2015-03-01
A novel technique is being developed to estimate the effective dose of a neutron field based on the distribution of neutron captures in a scintillator. Using Monte Carlo techniques, a number of monoenergetic neutron source energies and locations were modelled and their neutron capture response was recorded. Using back propagation Artificial Neural Networks (ANN) the energy and incident direction of the neutron field was predicted from the distribution of neutron captures within a 6Li-loaded liquid scintillator. Using this proposed technique, the effective dose of 252Cf, 241AmBe and 241AmLi neutron fields was estimated to within 30% for four perpendicular angles in the horizontal plane. Initial theoretical investigations show that this technique holds some promise for real-time estimation of the effective dose of a neutron field.
5. Methods for absorbing neutrons
DOEpatents
Guillen, Donna P [Idaho Falls, ID; Longhurst, Glen R [Idaho Falls, ID; Porter, Douglas L [Idaho Falls, ID; Parry, James R [Idaho Falls, ID
2012-07-24
A conduction cooled neutron absorber may include a metal matrix composite that comprises a metal having a thermal neutron cross-section of at least about 50 barns and a metal having a thermal conductivity of at least about 1 W/cmK. Apparatus for providing a neutron flux having a high fast-to-thermal neutron ratio may include a source of neutrons that produces fast neutrons and thermal neutrons. A neutron absorber positioned adjacent the neutron source absorbs at least some of the thermal neutrons so that a region adjacent the neutron absorber has a fast-to-thermal neutron ratio of at least about 15. A coolant in thermal contact with the neutron absorber removes heat from the neutron absorber.
6. Recent Advances in Neutron Physics
ERIC Educational Resources Information Center
Feshbach, Herman; Sheldon, Eric
1977-01-01
Discusses new studies in neutron physics within the last decade, such as ultracold neutrons, neutron bottles, resonance behavior, subthreshold fission, doubly radiative capture, and neutron stars. (MLH)
7. Recent Advances in Neutron Physics
ERIC Educational Resources Information Center
Feshbach, Herman; Sheldon, Eric
1977-01-01
Discusses new studies in neutron physics within the last decade, such as ultracold neutrons, neutron bottles, resonance behavior, subthreshold fission, doubly radiative capture, and neutron stars. (MLH)
8. 237Np absolute delayed neutron yield measurements
Doré, D.; Ledoux, X.; Nolte, R.; Gagnon-Moisan, F.; Thulliez, L.; Litaize, O.; Roettger, S.; Serot, O.
2017-09-01
237Np absolute delayed neutron yields have been measured at different incident neutron energies from 1.5 to 16 MeV. The experiment was performed at the Physikalisch-Technische Bundesanstalt (PTB) facility where the Van de Graaff accelerator and the cyclotron CV28 delivered 9 different neutron energy beams using p+T, d+D and d+T reactions. The detection system is made up of twelve 3He tubes inserted into a polyethylene cylinder. In this paper, the experimental setup and the data analysis method are described. The evolution of the absolute DN yields as a function of the neutron incident beam energies are presented and compared to experimental data found in the literature and data from the libraries.
9. A feasibility study on the use of phantoms with statistical lung masses for determining the uncertainty in the dose absorbed by the lung from broad beams of incident photons and neutrons
PubMed Central
Khankook, Atiyeh Ebrahimi; Hakimabad, Hashem Miri
2017-01-01
Abstract Computational models of the human body have gradually become crucial in the evaluation of doses absorbed by organs. However, individuals may differ considerably in terms of organ size and shape. In this study, the authors sought to determine the energy-dependent standard deviations due to lung size of the dose absorbed by the lung during external photon and neutron beam exposures. One hundred lungs with different masses were prepared and located in an adult male International Commission on Radiological Protection (ICRP) reference phantom. Calculations were performed using the Monte Carlo N-particle code version 5 (MCNP5). Variation in the lung mass caused great uncertainty: ~90% for low-energy broad parallel photon beams. However, for high-energy photons, the lung-absorbed dose dependency on the anatomical variation was reduced to <1%. In addition, the results obtained indicated that the discrepancy in the lung-absorbed dose varied from 0.6% to 8% for neutron beam exposure. Consequently, the relationship between absorbed dose and organ volume was found to be significant for low-energy photon sources, whereas for higher energy photon sources the organ-absorbed dose was independent of the organ volume. In the case of neutron beam exposure, the maximum discrepancy (of 8%) occurred in the energy range between 0.1 and 5 MeV. PMID:28077627
10. Neutron scattering and models: Titanium
SciTech Connect
Smith, A.B.
1997-07-01
Differential neutron elastic-scattering cross sections of elemental titanium were measured from 4.5 {r_arrow} 10.0 MeV in incident energy increments of {approx} 0.5 MeV. At each energy the measurements were made at forty or more scattering angles distributed between {approx} 17 and 160{degree}. Concurrently, differential neutron inelastic-scattering cross sections were measured for observed excitations of 0.975 {+-} 0.034, 1.497 {+-} 0.033, 2.322 {+-} 0.058, 3.252 {+-} 0.043, 3.700 {+-} 0.093, 4.317 {+-} 0.075 and 4.795 {+-} 0.100 MeV. All of the observed inelastically-scattered neutron groups were composites of contributions from several isotopes and/or levels. The experimental results were used to develop energy-average optical, statistical and coupled-channels models.
11. Epithermal Neutron Source for Neutron Resonance Spectroscopy (NRS) using High Intensity, Short Pulse Lasers
SciTech Connect
Higginson, D P; McNaney, J M; Swift, D C; Bartal, T; Hey, D S; Pape, S L; Mackinnon, A; Mariscal, D; Nakamura, H; Nakanii, N; Beg, F N
2010-04-22
A neutron source for neutron resonance spectroscopy (NRS) has been developed using high intensity, short pulse lasers. This measurement technique will allow for robust measurements of interior ion temperature of laser-shocked materials and provide insight into equation of state (EOS) measurements. The neutron generation technique uses protons accelerated by lasers off of Cu foils to create neutrons in LiF, through (p,n) reactions with {sup 7}Li and {sup 19}F. The distribution of the incident proton beam has been diagnosed using radiochromic film (RCF). This distribution is used as the input for a (p,n) neturon prediction code which is compared to experimentally measured neutron yields. From this calculation, a total fluence of 1.8 x 10{sup 9} neutrons is infered, which is shown to be a reasonable amount for NRS temperature measurement.
12. Neutrons and Granite: Transport and Activation
SciTech Connect
Bedrossian, P J
2004-04-13
In typical ground materials, both energy deposition and radionuclide production by energetic neutrons vary with the incident particle energy in a non-monotonic way. We describe the overall balance of nuclear reactions involving neutrons impinging on granite to demonstrate these energy-dependencies. While granite is a useful surrogate for a broad range of soil and rock types, the incorporation of small amounts of water (hydrogen) does alter the balance of nuclear reactions.
13. Neutron cross sections: Book of curves
SciTech Connect
McLane, V.; Dunford, C.L.; Rose, P.F.
1988-01-01
Neuton Cross Sections: Book of Curves represents the fourth edition of what was previously known as BNL-325, Neutron Cross Sections, Volume 2, CURVES. Data is presented only for (i.e., intergrated) reaction cross sections (and related fission parameters) as a function of incident-neutron energy for the energy range 0.01 eV to 200 MeV. For the first time, isometric state production cross sections have been included. 11 refs., 4 figs.
14. Algebraic method for calculating a neutron albedo
Ignatovich, V. K.; Shabalin, E. P.
2007-02-01
A neutron albedo from arbitrary homogeneous and finely grained substances is examined on the basis of a new, algebraic, method. In the approximation of an isotropic distribution of incident and reflected neutrons, it is shown that, in the case of thermal neutrons, coherent scattering on individual particles of finely grained media increases only slightly the transport cross section, but, at a given wall thickness, it reduces the albedo because of a decrease in the density of the substance. A significant increase in the albedo is possible only for neutrons of wavelength on the order of dimensions of a powder grain. The angular distribution of reflected neutrons is discussed, and it is proven that a deviation of this distribution from an isotropic one does not lead to a change in the magnitude of the albedo.
15. Method and apparatus for detecting neutrons
DOEpatents
Perkins, Richard W.; Reeder, Paul L.; Wogman, Ned A.; Warner, Ray A.; Brite, Daniel W.; Richey, Wayne C.; Goldman, Don S.
1997-01-01
The instant invention is a method for making and using an apparatus for detecting neutrons. Scintillating optical fibers are fabricated by melting SiO.sub.2 with a thermal neutron capturing substance and a scintillating material in a reducing atmosphere. The melt is then drawn into fibers in an anoxic atmosphere. The fibers may then be coated and used directly in a neutron detection apparatus, or assembled into a geometrical array in a second, hydrogen-rich, scintillating material such as a polymer. Photons generated by interaction with thermal neutrons are trapped within the coated fibers and are directed to photoelectric converters. A measurable electronic signal is generated for each thermal neutron interaction within the fiber. These electronic signals are then manipulated, stored, and interpreted by normal methods to infer the quality and quantity of incident radiation. When the fibers are arranged in an array within a second scintillating material, photons generated by kinetic neutrons interacting with the second scintillating material and photons generated by thermal neutron capture within the fiber can both be directed to photoelectric converters. These electronic signals are then manipulated, stored, and interpreted by normal methods to infer the quality and quantity of incident radiation.
16. Method and apparatus for detecting neutrons
DOEpatents
Perkins, R.W.; Reeder, P.L.; Wogman, N.A.; Warner, R.A.; Brite, D.W.; Richey, W.C.; Goldman, D.S.
1997-10-21
The instant invention is a method for making and using an apparatus for detecting neutrons. Scintillating optical fibers are fabricated by melting SiO{sub 2} with a thermal neutron capturing substance and a scintillating material in a reducing atmosphere. The melt is then drawn into fibers in an anoxic atmosphere. The fibers may then be coated and used directly in a neutron detection apparatus, or assembled into a geometrical array in a second, hydrogen-rich, scintillating material such as a polymer. Photons generated by interaction with thermal neutrons are trapped within the coated fibers and are directed to photoelectric converters. A measurable electronic signal is generated for each thermal neutron interaction within the fiber. These electronic signals are then manipulated, stored, and interpreted by normal methods to infer the quality and quantity of incident radiation. When the fibers are arranged in an array within a second scintillating material, photons generated by kinetic neutrons interacting with the second scintillating material and photons generated by thermal neutron capture within the fiber can both be directed to photoelectric converters. These electronic signals are then manipulated, stored, and interpreted by normal methods to infer the quality and quantity of incident radiation. 5 figs.
17. Applicability of convex hull in multiple detector response space for neutron dose measurements.
PubMed
Hashimoto, Makoto; Iimoto, Takeshi; Kosako, Toshiso
2009-08-01
A novel neutron dose measurement method that flexibly responds to variations in the neutron field is being developed by Japan Atomic Energy Agency. This is an implementation of the multi-detector method (first introduced in 1960s) for neutron dose evaluation using a convex hull in the response space defined for multiple detectors. The convex hull provides a range of possible neutron dose corresponding to the incident neutron spectrum. Feasibility of the method was studied using a simulated response of mixed gas proportional counter. Monochromatic neutrons are shown to be fundamentally suitable for mapping the convex. The convex hull can be further reduced taking into consideration a priori information about physically possible incident neutron spectra, for example, theoretically derived moderated neutron spectra originated from a fission neutron source.
18. NEUTRONIC REACTORS
DOEpatents
Vernon, H.C.
1959-01-13
A neutronic reactor of the heterogeneous, fluid cooled tvpe is described. The reactor is comprised of a pressure vessel containing the moderator and a plurality of vertically disposed channels extending in spaced relationship through the moderator. Fissionable fuel material is placed within the channels in spaced relationship thereto to permit circulation of the coolant fluid. Separate means are provided for cooling the moderator and for circulating a fluid coolant thru the channel elements to cool the fuel material.
19. Development of First Responders Equipment at RN Incident Sites
Tsuchiya, K.; Kuroki, K.; Kurosawa, K.; Akiba, N.
2015-10-01
On site categorization and collection of radioactive and nuclear materials are required at radiological and nuclear incident site. We are developing portable equipment and radiation protection for radiological emergency response team to carry out emergency missions safely at the incident sites. In this report, we review radiation monitoring system including wireless dosimeter system and neutron shield with water developed in our institute. Also the development of fast-neutron directional detector with a micro pattern gas detector is described.
20. Fission neutron spectra measurements at LANSCE - status and plans
SciTech Connect
Haight, Robert C; Noda, Shusaku; Nelson, Ronald O; O' Donnell, John M; Devlin, Matt; Chatillon, Audrey; Granier, Thierry; Taieb, Julien; Laurent, Benoit; Belier, Gilbert; Becker, John A; Wu, Ching - Yen
2009-01-01
A program to measure fission neutron spectra from neutron-induced fission of actinides is underway at the Los Alamos Neutron Science Center (LANSCE) in a collaboration among the CEA laboratory at Bruyeres-le-Chatel, Lawrence Livermore National Laboratory and Los Alamos National Laboratory. The spallation source of fast neutrons at LANSCE is used to provide incident neutron energies from less than 1 MeV to 100 MeV or higher. The fission events take place in a gas-ionization fission chamber, and the time of flight from the neutron source to that chamber gives the energy of the incident neutron. Outgoing neutrons are detected by an array of organic liquid scintillator neutron detectors, and their energies are deduced from the time of flight from the fission chamber to the neutron detector. Measurements have been made of the fission neutrons from fission of {sup 235}U, {sup 238}U, {sup 237}Np and {sup 239}Pu. The range of outgoing energies measured so far is from 1 MeV to approximately 8 MeV. These partial spectra and average fission neutron energies are compared with evaluated data and with models of fission neutron emission. Results to date will be presented and a discussion of uncertainties will be given in this presentation. Future plans are to make significant improvements in the fission chambers, neutron detectors, signal processing, data acquisition and the experimental environment to provide high fidelity data including mea urements of fission neutrons below 1 MeV and improvements in the data above 8 MeV.
SciTech Connect
Gavron, A.; Morley, K.; Morris, C.; Seestrom, S.; Ullmann, J.; Yates, G.; Zumbro, J.
1996-06-01
High-energy spallation neutron sources are now being considered in the US and elsewhere as a replacement for neutron beams produced by reactors. High-energy and high intensity neutron beams, produced by unmoderated spallation sources, open potential new vistas of neutron radiography. The authors discuss the basic advantages and disadvantages of high-energy neutron radiography, and consider some experimental results obtained at the Weapons Neutron Research (WNR) facility at Los Alamos.
2. Measurement of neutron capture on 136Xe
Albert, J. B.; Daugherty, S. J.; Johnson, T. N.; O'Conner, T.; Kaufman, L. J.; Couture, A.; Ullmann, J. L.; Krtička, M.
2016-09-01
136Xe is a 0 ν β β decay candidate isotope, and is used in multiple experiments searching for this hypothetical decay mode. These experiments require precise information about neutron capture for their background characterization and minimization. Thermal and resonant neutron capture on 136Xe have been measured at the Detector for Advanced Neutron Capture Experiments (DANCE) at the Los Alamos Neutron Science Center. A neutron beam ranging from thermal energy to greater than 100 keV was incident on a gas cell filled with isotopically pure 136Xe. The relative neutron capture cross sections for neutrons at thermal energies and the first resonance at 2.154 keV have been measured, yielding a new absolute measurement of 0.238 ±0.019 b for the thermal neutron capture cross section. Additionally, the γ cascades for captures at both energies have been measured, and cascade models have been developed which may be used by 0 ν β β experiments using 136Xe.
3. Variable control of neutron albedo in toroidal fusion devices
DOEpatents
Jassby, D.L.; Micklich, B.J.
1983-06-01
This invention pertains to methods of controlling in the steady state, neutron albedo in toroidal fusion devices, and in particular, to methods of controlling the flux and energy distribution of collided neutrons which are incident on an outboard wall of a toroidal fusion device.
4. Neutron-neutron and neutron-photon correlations with FREYA
Vogt, R.; Randrup, J.
2017-09-01
For many years, the state of the art for modeling fission in radiation transport codes has involved sampling from average distributions. However, in a true fission event, the energies, momenta and multiplicities of emitted particles are correlated. The FREYA (Fission Reaction Event Yield Algorithm) code generates complete fission events. Event-by-event techniques such as those of FREYA are particularly useful because it is possible to obtain complete kinematic information on the prompt neutrons and photons emitted during the fission process. It is therefore possible to extract any desired correlation observables. We describe FREYA and compare our results with neutron-neutron, neutron-light fragment and neutron-photon correlation data.
5. NEUTRON COUNTER
DOEpatents
Curtis, C.D.; Carlson, R.L.; Tubinis, M.P.
1958-07-29
An ionization chamber instrument is described for cylindrical electrodes with an ionizing gag filling the channber. The inner electrode is held in place by a hermetic insulating seal at one end of the outer electrode, the other end of the outer electrode being closed by a gas filling tube. The outer surface of the inner electrode is coated with an active material which is responsive to neutron bombardment, such as uranium235 or boron-10, to produce ionizing radiations in the gas. The transverse cross sectional area of the inner electrode is small in relation to that of the channber whereby substantially all of the radiations are directed toward the outer electrode.
6. NEUTRON SOURCE
DOEpatents
Reardon, W.A.; Lennox, D.H.; Nobles, R.G.
1959-01-13
A neutron source of the antimony--beryllium type is presented. The source is comprised of a solid mass of beryllium having a cylindrical recess extending therein and a cylinder containing antimony-124 slidably disposed within the cylindrical recess. The antimony cylinder is encased in aluminum. A berylliunn plug is removably inserted in the open end of the cylindrical recess to completely enclose the antimony cylinder in bsryllium. The plug and antimony cylinder are each provided with a stud on their upper ends to facilitate handling remotely.
7. Neutron Zeeman beam-splitting for the investigation of magnetic nanostructures
Kozhevnikov, S. V.; Ott, F.; Semenova, E.
2017-03-01
Zeeman spatial splitting of a neutron beam takes place during a neutron spin-flip in magnetically non-collinear systems at grazing incidence geometry. We apply the neutron beam-splitting method for the investigation of magnetically non-collinear clusters of submicron size in a thin film. The experimental results are compared with ones obtained by other methods.
8. Small angle neutron scattering
Cousin, Fabrice
2015-10-01
Small Angle Neutron Scattering (SANS) is a technique that enables to probe the 3-D structure of materials on a typical size range lying from ˜ 1 nm up to ˜ a few 100 nm, the obtained information being statistically averaged on a sample whose volume is ˜ 1 cm3. This very rich technique enables to make a full structural characterization of a given object of nanometric dimensions (radius of gyration, shape, volume or mass, fractal dimension, specific area…) through the determination of the form factor as well as the determination of the way objects are organized within in a continuous media, and therefore to describe interactions between them, through the determination of the structure factor. The specific properties of neutrons (possibility of tuning the scattering intensity by using the isotopic substitution, sensitivity to magnetism, negligible absorption, low energy of the incident neutrons) make it particularly interesting in the fields of soft matter, biophysics, magnetic materials and metallurgy. In particular, the contrast variation methods allow to extract some informations that cannot be obtained by any other experimental techniques. This course is divided in two parts. The first one is devoted to the description of the principle of SANS: basics (formalism, coherent scattering/incoherent scattering, notion of elementary scatterer), form factor analysis (I(q→0), Guinier regime, intermediate regime, Porod regime, polydisperse system), structure factor analysis (2nd Virial coefficient, integral equations, characterization of aggregates), and contrast variation methods (how to create contrast in an homogeneous system, matching in ternary systems, extrapolation to zero concentration, Zero Averaged Contrast). It is illustrated by some representative examples. The second one describes the experimental aspects of SANS to guide user in its future experiments: description of SANS spectrometer, resolution of the spectrometer, optimization of spectrometer
9. A preliminary area survey of neutron radiation levels associated with the NASA variable energy cyclotron horizontal neutron delivery system
NASA Technical Reports Server (NTRS)
Roberts, W. K.; Leonard, R. F.
1976-01-01
The 25 MeV deuteron beam from the NASA variable energy cyclotron incident on a thick beryllium target will deliver a tissue neutron dose rate of 2.14 rad micron A-min at a source to skin distance of 125 cm. A neutron survey of the existing hallways with various shielding configurations made during operating of the horizontal neutron delivery system indicates that minimal amounts of additional neutron shielding material are required to provide a low level radiation environment within a self-contained neutron therapy control station. Measurements also indicate that the primary neutron distribution delivered by a planned vertical delivery system will be minimally perturbed by neutrons backscattered from the floor.
10. Measurement of the Response Function of a BC501A Neutron Detector
Miller, J.; Alexander, D.; Daniel, A.; Hungerford, E. V.; Ahmed, M. W.; Sikora, M.
2015-10-01
A 5'' X 2'' BC501A neutron detector was used to measure proton recoil spectra at a number of mono-energetic incident neutron energies between 2 and 6 MeV at the Triangle Universities Nuclear Laboratory. The goal of the experiment was to characterize the response function for a variety of known neutron energies so that an unknown neutron spectrum can be obtained by unfolding the detector response to the incident spectrum. We discuss calibration, optimization of the neutron-gamma discrimination, and the progress of the analysis. Supported in part by the U.S. Department of Energy.
11. Neutron Stars
van den Heuvel, Ed
Radio pulsars are unique laboratories for a wide range of physics and astrophysics. Understanding how they are created, how they evolve and where we find them in the Galaxy, with or without binary companions, is highly constraining of theories of stellar and binary evolution. Pulsars' relationship with a recently discovered variety of apparently different classes of neutron stars is an interesting modern astrophysical puzzle which we consider in Part I of this review. Radio pulsars are also famous for allowing us to probe the laws of nature at a fundamental level. They act as precise cosmic clocks and, when in a binary system with a companion star, provide indispensable venues for precision tests of gravity. The different applications of radio pulsars for fundamental physics will be discussed in Part II. We finish by making mention of the newly discovered class of astrophysical objects, the Fast Radio Bursts, which may or may not be related to radio pulsars or neutron stars, but which were discovered in observations of the latter.
12. A Method for Neutron Scattering Quantification and Correction Applied to Neutron Imaging
Raventos, Marc; Harti, Ralph P.; Lehmann, Eberhard; Grünzweig, Christian
Either the composition, the thickness or the density of a sample can be derived, given the other two are known, by measuring the attenuation of a given incident radiation intensity through the sample. However, in the case of neutron imaging, the separation of the transmitted intensity from the scattered intensity using scintillator-based detection systems is yet to be solved. Several methods have been proposed for the correction of disturbing neutron scattering, but they are only applicable to specific materials or require some a-priori knowledge of the sample. Here we present a method for white beam neutron imaging which compares transmitted neutron images at different distances from the scintillator to improve the quantification capabilities of neutron imaging.
13. Validation of neutron emission profiles in MAST with a collimated neutron monitora)
Sangaroon, S.; Cecconello, M.; Conroy, S.; Weiszflog, M.; Turnyanskiy, M.; Wodniak, I.; Ericsson, G.
2012-10-01
A neutron camera with liquid scintillator detectors is used in MAST to measure the neutron emissivity from D(d,n)3He reactions along collimated lines of sight. In this work, the measured recoil proton pulse height spectra generated in the detectors by the incident neutrons is modelled taking into account the energy spectrum of the generated neutrons, their spatial distribution and transport to the detectors as well as the detector's response function. The contribution of scattered neutrons to the pulse height spectrum is also modelled. Good agreement is found between the experimental data and the simulations. Examples are given showing the sensitivity of the recoil proton pulse height spectra to different observation angles with respect the neutral beam injection and the plasma rotation direction.
14. Neutron matter, symmetry energy and neutron stars
SciTech Connect
Stefano, Gandolfi; Steiner, Andrew W
2016-01-01
Recent progress in quantum Monte Carlo with modern nucleon-nucleon interactions have enabled the successful description of properties of light nuclei and neutron-rich matter. Of particular interest is the nuclear symmetry energy, the energy cost of creating an isospin asymmetry, and its connection to the structure of neutron stars. Combining these advances with recent observations of neutron star masses and radii gives insight into the equation of state of neutron-rich matter near and above the saturation density. In particular, neutron star radius measurements constrain the derivative of the symmetry energy.
15. Neutron matter, symmetry energy and neutron stars
Gandolfi, S.; Steiner, A. W.
2016-01-01
Recent progress in quantum Monte Carlo with modern nucleon-nucleon interactions have enabled the successful description of properties of light nuclei and neutron- rich matter. Of particular interest is the nuclear symmetry energy, the energy cost of creating an isospin asymmetry, and its connection to the structure of neutron stars. Combining these advances with recent observations of neutron star masses and radii gives insight into the equation of state of neutron-rich matter near and above the saturation density. In particular, neutron star radius measurements constrain the derivative of the symmetry energy.
16. Neutron standard data
SciTech Connect
Peelle, R.; Conde, H.
1988-01-01
The neutron standards are reviewed with emphasis on the evaluation for ENDFB-VI. Also discussed are the neutron spectrum of /sup 252/Cf spontaneous fission, activation cross sections for neutron flux measurement, and standards for neutron energies greater than 20 MeV. Recommendations are made for future work. 21 refs., 6 figs., 3 tabs.
17. Borner Ball Neutron Detector
NASA Technical Reports Server (NTRS)
2002-01-01
The Bonner Ball Neutron Detector measures neutron radiation. Neutrons are uncharged atomic particles that have the ability to penetrate living tissues, harming human beings in space. The Bonner Ball Neutron Detector is one of three radiation experiments during Expedition Two. The others are the Phantom Torso and Dosimetric Mapping.
18. Neutron and proton activation measurements from Skylab
NASA Technical Reports Server (NTRS)
Fishman, G. J.
1974-01-01
Radioactivity induced by high-energy protons and secondary neutrons (from nuclear interactions) in various samples returned from different locations in Skylab was measured directly by gamma-ray spectroscopy measurements of decay gamma rays from the samples. Incident fluxes were derived from the activation measurements, using known nuclear cross-section. Neutron and proton flux values were found to range from 0.2 to 5 particles/sq cm-sec, depending on the energy range and location in Skylab. The thermal neutron flux was less than 0.07 neutrons/sq cm-sec. The results are useful for data analysis and planning of future high-energy astronomy experiments.
19. Neutrons in cancer therapy
Allen, Barry J.
1995-03-01
The role of neutrons in the management of cancer has a long history. However, it is only in recent years that neutrons are beginning to find an accepted place as an efficacious radiation modality. Fast neutron therapy is already well established for the treatment of certain cancers, and clinical trials are ongoing. Californium neutron sources are being used in brachytherapy. Boron neutron capture therapy has been well tested with thermal neutrons and epithermal neutron dose escalation studies are about to commence in the USA and Europe. Possibilities of neutron induced auger electron therapy are also discussed. With respect to chemotherapy, prompt neutron capture analysis is being used to study the dose optimization of chemotherapy in the management of breast cancer. The rationales behind these applications of neutrons in the management of cancer are examined.
20. Nuclear reactor neutron shielding
DOEpatents
Speaker, Daniel P; Neeley, Gary W; Inman, James B
2017-09-12
A nuclear reactor includes a reactor pressure vessel and a nuclear reactor core comprising fissile material disposed in a lower portion of the reactor pressure vessel. The lower portion of the reactor pressure vessel is disposed in a reactor cavity. An annular neutron stop is located at an elevation above the uppermost elevation of the nuclear reactor core. The annular neutron stop comprises neutron absorbing material filling an annular gap between the reactor pressure vessel and the wall of the reactor cavity. The annular neutron stop may comprise an outer neutron stop ring attached to the wall of the reactor cavity, and an inner neutron stop ring attached to the reactor pressure vessel. An excore instrument guide tube penetrates through the annular neutron stop, and a neutron plug comprising neutron absorbing material is disposed in the tube at the penetration through the neutron stop.
1. NEUTRONIC REACTOR
DOEpatents
Stewart, H.B.
1958-12-23
A nuclear reactor of the type speclfically designed for the irradiation of materials is discussed. In this design a central cyllndrical core of moderating material ls surrounded by an active portlon comprlsed of an annular tank contalning fissionable material immersed ln a liquid moderator. The active portion ls ln turn surrounded by a reflector, and a well ls provided in the center of the core to accommodate the materlals to be irradiated. The over-all dimensions of the core ln at least one plane are equal to or greater than twice the effective slowing down length and equal to or less than twlce the effective diffuslon length for neutrons in the core materials.
2. NEUTRONIC REACTORS
DOEpatents
Wigner, E.P.; Young, G.J.
1958-10-14
3. Neutron production from 200-500 MeV proton interaction with spacecraft materials.
PubMed
Maurer, Richard H; Kinnison, James D; Roth, David R
2005-01-01
We report on detailed energy spectra of neutron production > 14 MeV from collisions of 200-500 MeV protons with combinations of aluminium, graphite and polyethylene. Comparisons of normalised neutron spectra are made with respect to incident proton energy, angle of neutron production and material. In general, carbon (graphite) or polyethylene (by itself or in combination with aluminium) reduce secondary neutron production > 14 MeV relative to the production from interactions in aluminium.
4. A new polarized neutron interferometry facility at the NCNR
Shahi, C. B.; Arif, M.; Cory, D. G.; Mineeva, T.; Nsofini, J.; Sarenac, D.; Williams, C. J.; Huber, M. G.; Pushin, D. A.
2016-03-01
A new monochromatic beamline and facility has been installed at the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR) devoted to neutron interferometry in the research areas of spin control, spin manipulation, quantum mechanics, quantum information science, spintronics, and material science. This facility is possible in part because of advances in decoherence free subspace interferometer designs that have demonstrated consistent contrast in the presence of vibrational noise; a major environmental constraint that has prevented neutron interferometry from being applied at other neutron facilities. Neutron interferometry measures the phase difference between a neutron wave function propagating along two spatially separated paths. It is a practical example of self interference and due to its modest path separation of a few centimeters allows the insertion of samples and macroscopic neutron spin rotators. Phase shifts can be caused by gravitational, magnetic and nuclear interactions as well as purely quantum mechanical effects making interferometer a robust tool in neutron research. This new facility is located in the guide hall of the NCNR upstream of the existing Neutron Interferometry and Optics Facility (NIOF) and has several advantages over the NIOF including higher incident flux, better neutron polarization, and increased accessibility. The long term goal for the new facility is to be a user supported beamline and makes neutron interferometer more generally available to the scientific community. This paper addresses both the capabilities and characteristics of the new facility.
5. Design of a transportable high efficiency fast neutron spectrometer
DOE PAGES
Roecker, C.; Bernstein, A.; Bowden, N. S.; ...
2016-04-12
A transportable fast neutron detection system has been designed and constructed for measuring neutron energy spectra and flux ranging from tens to hundreds of MeV. The transportability of the spectrometer reduces the detector-related systematic bias between different neutron spectra and flux measurements, which allows for the comparison of measurements above or below ground. The spectrometer will measure neutron fluxes that are of prohibitively low intensity compared to the site-specific background rates targeted by other transportable fast neutron detection systems. To measure low intensity high-energy neutron fluxes, a conventional capture-gating technique is used for measuring neutron energies above 20 MeV andmore » a novel multiplicity technique is used for measuring neutron energies above 100 MeV. The spectrometer is composed of two Gd containing plastic scintillator detectors arranged around a lead spallation target. To calibrate and characterize the position dependent response of the spectrometer, a Monte Carlo model was developed and used in conjunction with experimental data from gamma ray sources. Multiplicity event identification algorithms were developed and used with a Cf-252 neutron multiplicity source to validate the Monte Carlo model Gd concentration and secondary neutron capture efficiency. The validated Monte Carlo model was used to predict an effective area for the multiplicity and capture gating analyses. For incident neutron energies between 100 MeV and 1000 MeV with an isotropic angular distribution, the multiplicity analysis predicted an effective area of 500 cm2 rising to 5000 cm2. For neutron energies above 20 MeV, the capture-gating analysis predicted an effective area between 1800 cm2 and 2500 cm2. As a result, the multiplicity mode was found to be sensitive to the incident neutron angular distribution.« less
6. Design of a transportable high efficiency fast neutron spectrometer
Roecker, C.; Bernstein, A.; Bowden, N. S.; Cabrera-Palmer, B.; Dazeley, S.; Gerling, M.; Marleau, P.; Sweany, M. D.; Vetter, K.
2016-08-01
A transportable fast neutron detection system has been designed and constructed for measuring neutron energy spectra and flux ranging from tens to hundreds of MeV. The transportability of the spectrometer reduces the detector-related systematic bias between different neutron spectra and flux measurements, which allows for the comparison of measurements above or below ground. The spectrometer will measure neutron fluxes that are of prohibitively low intensity compared to the site-specific background rates targeted by other transportable fast neutron detection systems. To measure low intensity high-energy neutron fluxes, a conventional capture-gating technique is used for measuring neutron energies above 20 MeV and a novel multiplicity technique is used for measuring neutron energies above 100 MeV. The spectrometer is composed of two Gd containing plastic scintillator detectors arranged around a lead spallation target. To calibrate and characterize the position dependent response of the spectrometer, a Monte Carlo model was developed and used in conjunction with experimental data from gamma ray sources. Multiplicity event identification algorithms were developed and used with a Cf-252 neutron multiplicity source to validate the Monte Carlo model Gd concentration and secondary neutron capture efficiency. The validated Monte Carlo model was used to predict an effective area for the multiplicity and capture gating analyses. For incident neutron energies between 100 MeV and 1000 MeV with an isotropic angular distribution, the multiplicity analysis predicted an effective area of 500 cm2 rising to 5000 cm2. For neutron energies above 20 MeV, the capture-gating analysis predicted an effective area between 1800 cm2 and 2500 cm2. The multiplicity mode was found to be sensitive to the incident neutron angular distribution.
7. Design of a transportable high efficiency fast neutron spectrometer
SciTech Connect
Roecker, C.; Bernstein, A.; Bowden, N. S.; Cabrera-Palmer, B.; Dazeley, S.; Gerling, M.; Marleau, P.; Sweany, M. D.; Vetter, K.
2016-04-12
A transportable fast neutron detection system has been designed and constructed for measuring neutron energy spectra and flux ranging from tens to hundreds of MeV. The transportability of the spectrometer reduces the detector-related systematic bias between different neutron spectra and flux measurements, which allows for the comparison of measurements above or below ground. The spectrometer will measure neutron fluxes that are of prohibitively low intensity compared to the site-specific background rates targeted by other transportable fast neutron detection systems. To measure low intensity high-energy neutron fluxes, a conventional capture-gating technique is used for measuring neutron energies above 20 MeV and a novel multiplicity technique is used for measuring neutron energies above 100 MeV. The spectrometer is composed of two Gd containing plastic scintillator detectors arranged around a lead spallation target. To calibrate and characterize the position dependent response of the spectrometer, a Monte Carlo model was developed and used in conjunction with experimental data from gamma ray sources. Multiplicity event identification algorithms were developed and used with a Cf-252 neutron multiplicity source to validate the Monte Carlo model Gd concentration and secondary neutron capture efficiency. The validated Monte Carlo model was used to predict an effective area for the multiplicity and capture gating analyses. For incident neutron energies between 100 MeV and 1000 MeV with an isotropic angular distribution, the multiplicity analysis predicted an effective area of 500 cm2 rising to 5000 cm2. For neutron energies above 20 MeV, the capture-gating analysis predicted an effective area between 1800 cm2 and 2500 cm2. As a result, the multiplicity mode was found to be sensitive to the incident neutron angular distribution.
8. Edge localized modes of cold neutrons in periodic condensed media
Belyakov, V. A.
2017-06-01
It is found that for certain energies of discreet cold neutrons, quasi-stationary eigen solutions of the corresponding Schrodinger equation, which are localized in the layer of a periodic medium, exist. The localization time of these solutions is strongly dependent on the layer thickness, being finite for a finite layer thickness and increasing indefinitely upon a infinite growth of the layer thickness as the third power of the layer thickness. The problem has been solved in the two-wave approximation of the dynamic diffraction theory for the neutron propagation direction coinciding with the periodicity axes (normal incidence of the neutron beam on the layer). The expressions for neutron eigenwave functions in a periodic medium, the reflection and transmission coefficients, and the neutron wavefunction in the layer as a function of the neutron energy incident on the layer have been determined. It turns out that for the certain discrete neutron energies, the amplitudes of the neutron wavefunction in the layer reach sharp maxima. The corresponding energies are just outside of the neutron stop band (energies forbidden for neutron propagation in the layer) and determine the energies of neutron edge modes (NEMs) localized in the layer, which are direct analogs of the optical edge modes for photonic crystals. The dispersion equation for the localized neutron edge modes has been obtained and analytically solved for the case of thick layers. A rough estimate for the localization length L is L ( db N)-1, where b is the neutron scattering length, d is the crystal period, and N is the density of nuclei in the crystal. The estimates of the localized thermal neutron lifetime show that acheaving of a lifetime close to the free neutron lifetime seems nonrealistic due to absorption of thermal neutrons and requires a perfect large size crystal. Nevertheless, acheaving the localized neutron lifetime exceeding by 104 times the neutron time of flight through the layer appears as
9. Measurement of neutron fluence spectra up to 150 MeV using a stacked scintillator neutron spectrometer.
PubMed
Brooks, F D; Allie, M S; Buffler, A; Dangendorf, V; Herbert, M S; Makupula, S A; Nolte, R; Smit, F D
2004-01-01
A stacked scintillator neutron spectrometer (S3N) consisting of three slabs of liquid organic scintillator is described. A pulsed beam providing a broad spectrum of neutron energies is used to determine the detection efficiency of the spectrometer as a function of incident neutron energy and to measure the pulse height response matrix of the system. Neutron spectra can then be determined for beams with any kind of time structure by unfolding pulse height spectra measured by the S3N. Examples of fluence spectrum measurements in the energy range 20-150 MeV are presented.
10. Neutron field for boron neutron capture therapy
SciTech Connect
Kanda, K.; Kobayashi, T.
1986-01-01
Recently, the development of an epithermal neutron source has been required by medical doctors for deeper neutron penetrations, which is to be used for deep tumor treatment and diagnosis of metastasis. Several attempts have already been made to realize an epithermal neutron field, such as the undermoderated neutron beam, the filtered neutron beam, and the use of a fission plate. At present, these facilities can not be used for actual therapy. For the treatment of deep tumor, another method has been also proposed in normal water in the body is replaced by heavy water to attain a deeper neutron penetration. At Kyoto University's Research Reactor Institute, almost all physics problems have been settled relative to thermal neutron capture therapy that has been used for treating brain tumors and for biological experiments on malignant melanoma. Very recently feasibility studies to use heavy water have been started both theoretically and experimentally. The calculation shows the deeper penetration of neutrons as expected. Two kinds of experiments were done by using the KUR guide tube: 1. Thermal neutron penetration measurement. 2. Heavy water uptake in vitro sample. In addition to the above experiment using heavy water, the development of a new epithermal neutron source using a large fission plate is in progress, which is part of a mockup experiment of an atomic bomb field newly estimated.
11. The Prompt Fission Neutron Spectrum (PFNS) Measurement Program at LANSCE
Haight, R. C.; Lee, H. Y.; Taddeucci, T. N.; O'Donnell, J. M.; Perdue, B. A.; Fotiades, N.; Devlin, M.; Ullmann, J. L.; Laptev, A.; Bredeweg, T.; Jandel, M.; Nelson, R. O.; Wender, S. A.; White, M. C.; Wu, C. Y.; Kwan, E.; Chyzh, A.; Henderson, R.; Gostic, J.
2014-05-01
The prompt neutron spectrum from neutron-induced fission needs to be known in designing new fast reactors, predicting criticality for safety analyses, and developing techniques for global security application. A program to measure this spectrum for neutron-induced fission of 239Pu is underway at the Los Alamos Neutron Science Center. The goal is to obtain data on the shape of the spectrum with a small uncertainty over the emitted neutron energy range of 100 keV to 12 MeV with additional data below and above this range. The incident neutron energy range will be from 0.5 to 30 MeV. The status of this program including results of initial experimental measurements is described here.
12. Radiative neutron capture cross sections on 176Lu at DANCE
Roig, O.; Jandel, M.; Méot, V.; Bond, E. M.; Bredeweg, T. A.; Couture, A. J.; Haight, R. C.; Keksis, A. L.; Rundberg, R. S.; Ullmann, J. L.; Vieira, D. J.
2016-03-01
The cross section of the neutron capture reaction 176Lu(n ,γ ) has been measured for a wide incident neutron energy range with the Detector for Advanced Neutron Capture Experiments at the Los Alamos Neutron Science Center. The thermal neutron capture cross section was determined to be (1912 ±132 ) b for one of the Lu natural isotopes, 176Lu. The resonance part was measured and compared to the Mughabghab's atlas using the R -matrix code, sammy. At higher neutron energies the measured cross sections are compared to ENDF/B-VII.1, JEFF-3.2, and BRC evaluated nuclear data. The Maxwellian averaged cross sections in a stellar plasma for thermal energies between 5 keV and 100 keV were extracted using these data.
13. High-energy neutron spectroscopy with thick silicon detectors
NASA Technical Reports Server (NTRS)
Kinnison, James D.; Maurer, Richard H.; Roth, David R.; Haight, Robert C.
2003-01-01
The high-energy neutron component of the space radiation environment in thick structures such as the International Space Station contributes to the total radiation dose received by an astronaut. Detector design constraints such as size and mass have limited the energy range of neutron spectrum measurements in orbit to about 12 MeV in Space Shuttle studies. We present a new method for high-energy neutron spectroscopy using small silicon detectors that can extend these measurements to more than 500 MeV. The methodology is based on measurement of the detector response function for high-energy neutrons and inversion of this response function with measured deposition data to deduce neutron energy spectra. We also present the results of an initial shielding study performed with the thick silicon detector system for high-energy neutrons incident on polyethylene.
Ovechkina, Lena; Riley, Kent; Miller, Stuart; Bell, Zane; Nagarkar, Vivek
2009-08-01
Gadolinium has the highest thermal neutron absorption cross section of any naturally occurring element, and emits conversion electrons as well as atomic X-rays in over 50% of its neutron captures, which makes it a useful dopant in scintillators for detecting thermal neutrons. Gadolinium isopropoxide was studied as a possible dopant for styrene-based plastic scintillators as a convenient and inexpensive method to produce high-efficiency thermal neutron detectors. Plastic scintillators with gadolinium weight concentrations of up to 3% were transparent, uniform and defect-free and were characterized with spectral measurements performed under x-ray and neutron irradiation. The new material has the same characteristic emission of styrene with a maximum at approximately 425 nm, and a light output of 76% relative to the undoped plastic. A 13 mm thick sample containing 0.5% gadolinium by weight detected 46% of incident thermal neutrons, which makes this an attractive material for a variety of applications.
15. Spin distribution in neutron induced preequilibrium reactions
SciTech Connect
Dashdorj, D; Kawano, T; Chadwick, M; Devlin, M; Fotiades, N; Nelson, R O; Mitchell, G E; Garrett, P E; Agvaanluvsan, U; Becker, J A; Bernstein, L A; Macri, R; Younes, W
2005-10-04
The preequilibrium reaction mechanism makes an important contribution to neutron-induced reactions above E{sub n} {approx} 10 MeV. The preequilibrium process has been studied exclusively via the characteristic high energy neutrons produced at bombarding energies greater than 10 MeV. They are expanding the study of the preequilibrium reaction mechanism through {gamma}-ray spectroscopy. Cross-section measurements were made of prompt {gamma}-ray production as a function of incident neutron energy (E{sub n} = 1 to 250 MeV) on a {sup 48}Ti sample. Energetic neutrons were delivered by the Los Alamos National Laboratory spallation neutron source located at the Los Alamos Neutron Science Center facility. The prompt-reaction {gamma} rays were detected with the large-scale Compton-suppressed Germanium Array for Neutron Induced Excitations (GEANIE). Neutron energies were determined by the time-of-flight technique. The {gamma}-ray excitation functions were converted to partial {gamma}-ray cross sections taking into account the dead-time correction, target thickness, detector efficiency and neutron flux (monitored with an in-line fission chamber). Residual state population was predicted using the GNASH reaction code, enhanced for preequilibrium. The preequilibrium reaction spin distribution was calculated using the quantum mechanical theory of Feshback, Kerman, and Koonin (FKK). The multistep direct part of the FKK theory was calculated for a one-step process. The FKK preequilibrium spin distribution was incorporated into the GNASH calculations and the {gamma}-ray production cross sections were calculated and compared with experimental data. The difference in the partial {gamma}-ray cross sections using spin distributions with and without preequilibrium effects is significant.
16. NEUTRONIC REACTOR
DOEpatents
Ohlinger, L.A.; Wigner, E.P.; Weinberg, A.M.; Young, G.J.
1958-09-01
17. Neutron streak camera
DOEpatents
Wang, Ching L.
1983-09-13
Apparatus for improved sensitivity and time resolution of a neutron measurement. The detector is provided with an electrode assembly having a neutron sensitive cathode which emits relatively low energy secondary electrons. The neutron sensitive cathode has a large surface area which provides increased sensitivity by intercepting a greater number of neutrons. The cathode is also curved to compensate for differences in transit time of the neutrons emanating from the point source. The slower speeds of the secondary electrons emitted from a certain portion of the cathode are matched to the transit times of the neutrons impinging thereupon.
18. Neutron streak camera
DOEpatents
Wang, C.L.
1981-05-14
Apparatus for improved sensitivity and time resolution of a neutron measurement. The detector is provided with an electrode assembly having a neutron sensitive cathode which emits relatively low energy secondary electrons. The neutron sensitive cathode has a large surface area which provides increased sensitivity by intercepting a greater number of neutrons. The cathode is also curved to compensate for differences in transit time of the neutrons emanating from the point source. The slower speeds of the secondary electrons emitted from a certain portion of the cathode are matched to the transit times of the neutrons impinging thereupon.
19. Neutron streak camera
DOEpatents
Wang, C.L.
1983-09-13
Disclosed is an apparatus for improved sensitivity and time resolution of a neutron measurement. The detector is provided with an electrode assembly having a neutron sensitive cathode which emits relatively low energy secondary electrons. The neutron sensitive cathode has a large surface area which provides increased sensitivity by intercepting a greater number of neutrons. The cathode is also curved to compensate for differences in transit time of the neutrons emanating from the point source. The slower speeds of the secondary electrons emitted from a certain portion of the cathode are matched to the transit times of the neutrons impinging thereupon. 4 figs.
20. Organic metal neutron detector
DOEpatents
Butler, M.A.; Ginley, D.S.
1984-11-21
A device for detection of neutrons comprises: as an active neutron sensing element, a conductive organic polymer having an electrical conductivity and a cross-section for said neutrons whereby a detectable change in said conductivity is caused by impingement of said neutrons on the conductive organic polymer which is responsive to a property of said polymer which is altered by impingement of said neutrons on the polymer; and means for associating a change in said alterable property with the presence of neutrons at the location of said device.
1. Layered semiconductor neutron detectors
DOEpatents
Mao, Samuel S; Perry, Dale L
2013-12-10
Room temperature operating solid state hand held neutron detectors integrate one or more relatively thin layers of a high neutron interaction cross-section element or materials with semiconductor detectors. The high neutron interaction cross-section element (e.g., Gd, B or Li) or materials comprising at least one high neutron interaction cross-section element can be in the form of unstructured layers or micro- or nano-structured arrays. Such architecture provides high efficiency neutron detector devices by capturing substantially more carriers produced from high energy .alpha.-particles or .gamma.-photons generated by neutron interaction.
2. Characterization of Neutron Backgrounds for Direct Dark Matter Searches
Sweany, Melinda Dominique
Direct dark matter experiments generally cannot distinguish between nuclear recoils caused by Weakly Interacting Massive Particles (WIMPs) and those caused by neutron backgrounds. It is therefore crucial that all sources of neutron background are well understood and accounted for when claiming a discovery or reporting limits on the WIMP-nucleon cross section. One source of neutrons that is not well understood results from cosmogenic muon interactions in the material surrounding a detector. The Neutron Multiplicity Meter in the Soudan cavern is a gadolinium-doped water Cherenkov detector capable of detecting high multiplicity neutron showers resulting from fast neutrons incident on a lead target. This measurement is the first such measurement obtained without a liquid scintillator detector medium; muon and neutron spallation is media-dependent, and because neutron shield technology for dark matter detectors is moving towards water, this is an important measurement. The integrated fast neutron flux in the Soudan cavern is reported as a linear function of the power, alpha, of the neutron angular distribution with the zenith angle: F = 4.8x10-9 +/- 3.5x10-10 + (5.4x10-10 +/- 1.5x10-10)alpha. Technological studies of neutron detection with gadolinium-doped water are also reported here. The neutron detection efficiency of a cylindrical 3.5 kL detector is measured at 70% for neutrons in the center of the detector. In addition, other improvements to water Cherenkov technology are explored, namely the addition of water-soluble wavelength-shifting chemicals. The wavelength-shifting chemical 4-Methylumbelliferone has been shown here to increase the measured light output of Cherenkov radiation resulting from neutron capture showers by a factor of 1.7.
3. Neutronic Reactor Design to Reduce Neutron Loss
DOEpatents
Miles, F. T.
1961-05-01
A nuclear reactor construction is described in which an unmoderated layer of the fissionable material is inserted between the moderated portion of the reactor core and the core container steel wall. The wall is surrounded by successive layers of pure fertile material and moderator containing fertile material. The unmoderated layer of the fissionable material will insure that a greater portion of fast neutrons will pass through the steel wall than would thermal neutrons. Since the steel has a smaller capture cross section for the fast neutrons, greater nunnbers of neutrons will pass into the blanket, thereby increasing the over-all efficiency of the reactor. (AEC)
4. NEUTRONIC REACTOR DESIGN TO REDUCE NEUTRON LOSS
DOEpatents
Mills, F.T.
1961-05-01
A nuclear reactor construction is described in which an unmoderated layer of the fissionable material is inserted between the moderated portion of the reactor core and the core container steel wall which is surrounded by successive layers of pure fertile material and fertile material having moderator. The unmoderated layer of the fissionable material will insure that a greater portion of fast neutrons will pass through the steel wall than would thermal neutrons. As the steel has a smaller capture cross-section for the fast neutrons, then greater numbers of the neutrons will pass into the blanket thereby increasing the over-all efficiency of the reactor.
5. C7LYC Scintillators and Fast Neutron Spectroscopy
Chowdhury, P.; Brown, T.; Doucet, E.; Lister, C. J.; Wilson, G. L.; D'Olympia, N.; Devlin, M.; Mosby, S.
2016-09-01
Cs2 LiYCl6 (CLYC) scintillators detect both gammas and neutrons with excellent pulse shape discrimination. At UML, fast neutron measurements with a 16-element 1''x1'' CLYC array show promise for low energy nuclear science. CLYC detects fast neutrons via the 35Cl (n,p) reaction (resolution < 10 % at < 8 MeV). In our 7Li-enriched C7LYC, the thermal neutron response from the 6Li(n, α)t reaction is virtually eliminated. The low intrinsic efficiency of CLYC for fast neutrons (< 1 %) is offset by increased solid angle with the array placed near the target, since TOF is not needed for energy resolution. The array was tested at LANL for measuring elastic and inelastic neutron scattering on 56Fe. The incident energy from the white neutron source was measured via TOF, and the scattered neutron energy via the pulse height in CLYC. The array was also tested at CARIBU for measuring beta-delayed neutrons. Larger CLYC crystals are now a reality. Measurements with the first 3'' x 3'' C7LYC crystal are in progress at UML. Results will be discussed in the context of constructing a C7LYC array at FRIB for reaction and decay spectroscopy of neutron-rich fragments. Supported by the NNSA Stewardship Science Academic Alliance Program under Grant DE-NA00013008.
6. Neutron polycapillary optics: present status and future trends
Ioffe, A.
2005-07-01
Up to present, neutron applications of polycapillary optics have been rather rear and mostly limited either to feasibility studies or to non-scattering techniques. The practical use of polycapillary focusing neutron optics for purposes of neutron scattering have been hindered by a serious degradation of momentum resolution caused by increased divergence of focused beams. However, limited size of samples available for research as well as limited brightness of present day neutron sources (both reactors and spallation sources) requires to rethink the current situation. Neutron lenses are successfully tested for low Q-resolution applications providing an enormous increase of neutron flux at small samples. Moreover, some recent developments in the field of neutron instrumentation allow us to overcome the resolution problem by decoupling the angular resolution and the incident beam divergence thus opening the opportunity for the use of focusing neutron optics in some high Q-resolution application. Further advances in technology required to improve the performance of neutron polycapillary optics, also in combination with modem methods of polarization of neutron beams, are discussed.
7. Isotope identification capabilities using time resolved prompt gamma emission from epithermal neutrons
Festa, G.; Arcidiacono, L.; Pappalardo, A.; Minniti, T.; Cazzaniga, C.; Scherillo, A.; Andreani, C.; Senesi, R.
2016-03-01
We present a concept of integrated measurements for isotope identification which takes advantage of the time structure of spallation neutron sources for time resolved γ spectroscopy. Time resolved Prompt Gamma Activation Analysis (T-PGAA) consists in the measurement of gamma energy spectrum induced by the radioactive capture as a function of incident neutron Time Of Flight (TOF), directly related with the energy of incident neutrons. The potential of the proposed concept was explored on INES (Italian Neutron Experimental Station) at the ISIS spallation neutron source (U.K.). Through this new technique we show an increase in the sensitivity to specific elements of archaeometric relevance, through incident neutron energy selection in prompt γ spectra for multicomponent samples. Results on a standard bronze sample are presented.
8. ATW neutron spectrum measurements at LAMPF
SciTech Connect
Butler, G.W.; Littleton, P.E.; Morgan, G.L.
1995-10-01
Accelerator transmutation of waste (ATW) is a proposal to use a high flux of accelerator-produced thermalized neutrons to transmute both fission product and higher actinide commercial nuclear waste into stable or short-lived radioactive species in order to avoid long-term storage of nuclear waste. At LAMPF the authors recently performed experiments that were designed to measure the spectrum of neutrons produced per incident proton for full-scale proposed ATW targets of lead and lithium. The neutrons produced in such targets have a spectrum of energies that extends up to the energy of the incident proton beam, but the distribution peaks between 1 and 5 MeV. Transmutation reactions and fission of actinides are most efficient when the neutron energy is below a few eV, so the target must be surrounded by a non-absorbing material (blanket) to produce additional neutrons and reduce the energy of high energy neutrons without loss. The experiments with the lead target, 25 cm diameter by 40 cm long, were conducted with 800 MeV protons, while those with the lithium target, 25 cm diameter by 175 cm long, were conducted with 400 MeV protons. The blanket in both sets of experiments was a 60 cm diameter by 200 cm long annulus of lead that surrounded the target. Surrounding the blanket was a steel water tank with dimensions of 250 cm diameter by 300 cm long that simulated the transmutation region. A small sample pipe penetrated the length of the lead blanket and other sample pipes penetrated the length of the water tank at different radii from the beam axis so that the neutron spectra at different locations could be measured by foil activation. After irradiation the activated foil sets were extracted and counted with calibrated high resolution germanium gamma ray detectors at the Los Alamos nuclear chemistry counting facility.
9. Neutron anatomy
SciTech Connect
Bacon, G.E.
1994-12-31
The familiar extremes of crystalline material are single-crystals and random powders. In between these two extremes are polycrystalline aggregates, not randomly arranged but possessing some preferred orientation and this is the form taken by constructional materials, be they steel girders or the bones of a human or animal skeleton. The details of the preferred orientation determine the ability of the material to withstand stress in any direction. In the case of bone the crucial factor is the orientation of the c-axes of the mineral content - the crystals of the hexagonal hydroxyapatite - and this can readily be determined by neutron diffraction. In particular it can be measured over the volume of a piece of bone, utilizing distances ranging from 1mm to 10mm. The major practical problem is to avoid the intense incoherent scattering from the hydrogen in the accompanying collagen; this can best be achieved by heat-treatment and it is demonstrated that this does not affect the underlying apatite. These studies of bone give leading anatomical information on the life and activities of humans and animals - including, for example, the life history of the human femur, the locomotion of sheep, the fracture of the legs of racehorses and the life-styles of Neolithic tribes. We conclude that the material is placed economically in the bone to withstand the expected stresses of life and the environment. The experimental results are presented in terms of the magnitude of the 0002 apatite reflection. It so happens that for a random powder the 0002, 1121 reflections, which are neighboring lines in the powder pattern, are approximately equal in intensity. The latter reflection, being of manifold multiplicity, is scarcely affected by preferred orientation so that the numerical value of the 0002/1121 ratio serves quite accurately as a quantitative measure of the degree of orientation of the c-axes in any chosen direction for a sample of bone.
10. Pulsed-neutron monochromator
DOEpatents
Mook, Jr., Herbert A.
1985-01-01
In one aspect, the invention is an improved pulsed-neutron monochromator of the vibrated-crystal type. The monochromator is designed to provide neutron pulses which are characterized both by short duration and high density. A row of neutron-reflecting crystals is disposed in a neutron beam to reflect neutrons onto a common target. The crystals in the row define progressively larger neutron-scattering angles and are vibrated sequentially in descending order with respect to the size of their scattering angles, thus generating neutron pulses which arrive simultaneously at the target. Transducers are coupled to one end of the crystals to vibrate them in an essentially non-resonant mode. The transducers propagate transverse waves in the crystal which progress longitudinally therein. The wave are absorbed at the undriven ends of the crystals by damping material mounted thereon. In another aspect, the invention is a method for generating neutron pulses characterized by high intensity and short duration.
11. Pulsed-neutron monochromator
DOEpatents
Mook, H.A. Jr.
1984-01-01
In one aspect, the invention is an improved pulsed-neutron monochromator of the vibrated-crystal type. The monochromator is designed to provide neutron pulses which are characterized both by short duration and high density. A row of neutron-reflecting crystals is disposed in a neutron beam to reflect neutrons onto a common target. The crystals in the row define progressively larger neutron-scattering angles and are vibrated sequentially in descending order with respect to the size of their scattering angles, thus generating neutron pulses which arrive simultaneously at the target. Transducers are coupled to one end of the crystals to vibrate them in an essentially non-resonant mode. The transducers propagate transverse waves in the crystal which progress longitudinally therein. The waves are absorbed at the undriven ends of the crystals by damping material mounted thereon. In another aspect, the invention is a method for generating neutron pulses characterized by high intensity and short duration.
12. Dose equivalent neutron dosimeter
DOEpatents
Griffith, Richard V.; Hankins, Dale E.; Tomasino, Luigi; Gomaa, Mohamed A. M.
1983-01-01
A neutron dosimeter is disclosed which provides a single measurements indicating the amount of potential biological damage resulting from the neutron exposure of the wearer, for a wide range of neutron energies. The dosimeter includes a detecting sheet of track etch detecting material such as a carbonate plastic, for detecting higher energy neutrons, and a radiator layer containing conversion material such as .sup.6 Li and .sup.10 B lying adjacent to the detecting sheet for converting moderate energy neutrons to alpha particles that produce tracks in the adjacent detecting sheet. The density of conversion material in the radiator layer is of an amount which is chosen so that the density of tracks produced in the detecting sheet is proportional to the biological damage done by neutrons, regardless of whether the tracks are produced as the result of moderate energy neutrons striking the radiator layer or as the result of higher energy neutrons striking the sheet of track etch material.
13. On neutron surface waves
SciTech Connect
Ignatovich, V. K.
2009-01-15
It is shown that neutron surface waves do not exist. The difference between the neutron wave mechanics and the wave physics of electromagnetic and acoustic processes, which allows the existence of surface waves, is analyzed.
14. Neutron dose equivalent meter
DOEpatents
Olsher, Richard H.; Hsu, Hsiao-Hua; Casson, William H.; Vasilik, Dennis G.; Kleck, Jeffrey H.; Beverding, Anthony
1996-01-01
A neutron dose equivalent detector for measuring neutron dose capable of accurately responding to neutron energies according to published fluence to dose curves. The neutron dose equivalent meter has an inner sphere of polyethylene, with a middle shell overlying the inner sphere, the middle shell comprising RTV.RTM. silicone (organosiloxane) loaded with boron. An outer shell overlies the middle shell and comprises polyethylene loaded with tungsten. The neutron dose equivalent meter defines a channel through the outer shell, the middle shell, and the inner sphere for accepting a neutron counter tube. The outer shell is loaded with tungsten to provide neutron generation, increasing the neutron dose equivalent meter's response sensitivity above 8 MeV.
Nico, J. S.
2006-11-01
Precision measurements of neutron beta decay address basic questions in nuclear and particle physics, astrophysics, and cosmology. As the simplest semileptonic decay system, the free neutron plays an important role in understanding the physics of the weak interaction, and improving the precision of the neutron lifetime is fundamental to testing the validity of the theory. The neutron lifetime also directly affects the relative abundance of primordial helium in big bang nucleosynthesis. There are two distinct strategies for measuring the lifetime. Experiments using cold neutrons measure the absolute specific activity of a beam of neutrons by counting decay protons; experiments using confined, ultracold neutrons determine the lifetime by counting neutrons that remain after some elapsed time. The status of the recent lifetime measurements using both of these techniques is discussed.
SciTech Connect
Nico, J. S.
2006-11-17
Precision measurements of neutron beta decay address basic questions in nuclear and particle physics, astrophysics, and cosmology. As the simplest semileptonic decay system, the free neutron plays an important role in understanding the physics of the weak interaction, and improving the precision of the neutron lifetime is fundamental to testing the validity of the theory. The neutron lifetime also directly affects the relative abundance of primordial helium in big bang nucleosynthesis. There are two distinct strategies for measuring the lifetime. Experiments using cold neutrons measure the absolute specific activity of a beam of neutrons by counting decay protons; experiments using confined, ultracold neutrons determine the lifetime by counting neutrons that remain after some elapsed time. The status of the recent lifetime measurements using both of these techniques is discussed.
17. ULTRASONIC NEUTRON DOSIMETER
DOEpatents
Truell, R.; de Klerk, J.; Levy, P.W.
1960-02-23
A neutron dosimeter is described which utilizes ultrasonic waves in the megacycle region for determination of the extent of neutron damage in a borosilicate glass through ultrasonic wave velocity and attenuation measurements before and after damage.
18. Ultrafast neutron detector
DOEpatents
Wang, C.L.
1985-06-19
A neutron detector of very high temporal resolution is described. It may be used to measure distributions of neutrons produced by fusion reactions that persist for times as short as about 50 picoseconds.
19. Free fall onto magnetized neutron stars
Salpeter, E. E.
Some compact X-ray sources show evidence of cyclotron line radiation from excited electron Landau orbits, powered by hydrogen and helium falling onto a neutron star atmosphere along the magnetic field. The slowing of the incident matter is discussed, including the spread in energy loss due to Coulomb scattering and direct nuclear reactions for disintegrating the α particles. The α disintegrations, followed by neutron capture, lead to nuclear γ rays; the γ-ray intensity is (indirectly) coupled to the Coulomb energy loss and the cyclotron line emission.
20. Neutron capture cross section of Am241
Jandel, M.; Bredeweg, T. A.; Bond, E. M.; Chadwick, M. B.; Clement, R. R.; Couture, A.; O'Donnell, J. M.; Haight, R. C.; Kawano, T.; Reifarth, R.; Rundberg, R. S.; Ullmann, J. L.; Vieira, D. J.; Wilhelmy, J. B.; Wouters, J. M.; Agvaanluvsan, U.; Parker, W. E.; Wu, C. Y.; Becker, J. A.
2008-09-01
The neutron capture cross section of Am241 for incident neutrons from 0.02 eV to 320 keV has been measured with the detector for advanced neutron capture experiments (DANCE) at the Los Alamos Neutron Science Center. The thermal neutron capture cross section was determined to be 665±33 b. Our result is in good agreement with other recent measurements. Resonance parameters for En<12 eV were obtained using an R-matrix fit to the measured cross section. The results are compared with values from the ENDF/B-VII.0, Mughabghab, JENDL-3.3, and JEFF-3.1 evaluations. Γn neutron widths for the first three resonances are systematically larger by 5-15% than the ENDF/B-VII.0 values. The resonance integral above 0.5 eV was determined to be 1553±7 b. Cross sections in the resolved and unresolved energy regions above 12 eV were calculated using the Hauser-Feshbach theory incorporating the width-fluctuation correction of Moldauer. The calculated results agree well with the measured data, and the extracted averaged resonance parameters in the unresolved resonance region are consistent with those for the resolved resonances.
1. Broad energy range neutron spectroscopy using a liquid scintillator and a proportional counter: Application to a neutron spectrum similar to that from an improvised nuclear device
Xu, Yanping; Randers-Pehrson, Gerhard; Marino, Stephen A.; Garty, Guy; Harken, Andrew; Brenner, David J.
2015-09-01
A novel neutron irradiation facility at the Radiological Research Accelerator Facility (RARAF) has been developed to mimic the neutron radiation from an Improvised Nuclear Device (IND) at relevant distances (e.g. 1.5 km) from the epicenter. The neutron spectrum of this IND-like neutron irradiator was designed according to estimations of the Hiroshima neutron spectrum at 1.5 km. It is significantly different from a standard reactor fission spectrum, because the spectrum changes as the neutrons are transported through air, and it is dominated by neutron energies from 100 keV up to 9 MeV. To verify such wide energy range neutron spectrum, detailed here is the development of a combined spectroscopy system. Both a liquid scintillator detector and a gas proportional counter were used for the recoil spectra measurements, with the individual response functions estimated from a series of Monte Carlo simulations. These normalized individual response functions were formed into a single response matrix for the unfolding process. Several accelerator-based quasi-monoenergetic neutron source spectra were measured and unfolded to test this spectroscopy system. These reference neutrons were produced from two reactions: T(p,n)3He and D(d,n)3He, generating neutron energies in the range between 0.2 and 8 MeV. The unfolded quasi-monoenergetic neutron spectra indicated that the detection system can provide good neutron spectroscopy results in this energy range. A broad-energy neutron spectrum from the 9Be(d,n) reaction using a 5 MeV deuteron beam, measured at 60 degrees to the incident beam was measured and unfolded with the evaluated response matrix. The unfolded broad neutron spectrum is comparable with published time-of-flight results. Finally, the pair of detectors were used to measure the neutron spectrum generated at the RARAF IND-like neutron facility and a comparison is made to the neutron spectrum of Hiroshima.
2. Broad Energy Range Neutron Spectroscopy using a Liquid Scintillator and a Proportional Counter: Application to a Neutron Spectrum Similar to that from an Improvised Nuclear Device
PubMed Central
Randers-Pehrson, Gerhard; Marino, Stephen A.; Garty, Guy; Harken, Andrew; Brenner, David J.
2015-01-01
A novel neutron irradiation facility at the Radiological Research Accelerator Facility (RARAF) has been developed to mimic the neutron radiation from an Improvised Nuclear Device (IND) at relevant distances (e.g. 1.5 km) from the epicenter. The neutron spectrum of this IND-like neutron irradiator was designed according to estimations of the Hiroshima neutron spectrum at 1.5 km. It is significantly different from a standard reactor fission spectrum, because the spectrum changes as the neutrons are transported through air, and it is dominated by neutron energies from 100 keV up to 9 MeV. To verify such wide energy range neutron spectrum, detailed here is the development of a combined spectroscopy system. Both a liquid scintillator detector and a gas proportional counter were used for the recoil spectra measurements, with the individual response functions estimated from a series of Monte Carlo simulations. These normalized individual response functions were formed into a single response matrix for the unfolding process. Several accelerator-based quasi-monoenergetic neutron source spectra were measured and unfolded to test this spectroscopy system. These reference neutrons were produced from two reactions: T(p,n)3He and D(d,n)3He, generating neutron energies in the range between 0.2 and 8 MeV. The unfolded quasi-monoenergetic neutron spectra indicated that the detection system can provide good neutron spectroscopy results in this energy range. A broad-energy neutron spectrum from the 9Be(d,n) reaction using a 5 MeV deuteron beam, measured at 60 degrees to the incident beam was measured and unfolded with the evaluated response matrix. The unfolded broad neutron spectrum is comparable with published time-of-flight results. Finally, the pair of detectors were used to measure the neutron spectrum generated at the RARAF IND-like neutron facility and a comparison is made to the neutron spectrum of Hiroshima. PMID:26273118
3. Broad Energy Range Neutron Spectroscopy using a Liquid Scintillator and a Proportional Counter: Application to a Neutron Spectrum Similar to that from an Improvised Nuclear Device.
PubMed
Xu, Yanping; Randers-Pehrson, Gerhard; Marino, Stephen A; Garty, Guy; Harken, Andrew; Brenner, David J
2015-09-11
A novel neutron irradiation facility at the Radiological Research Accelerator Facility (RARAF) has been developed to mimic the neutron radiation from an Improvised Nuclear Device (IND) at relevant distances (e.g. 1.5 km) from the epicenter. The neutron spectrum of this IND-like neutron irradiator was designed according to estimations of the Hiroshima neutron spectrum at 1.5 km. It is significantly different from a standard reactor fission spectrum, because the spectrum changes as the neutrons are transported through air, and it is dominated by neutron energies from 100 keV up to 9 MeV. To verify such wide energy range neutron spectrum, detailed here is the development of a combined spectroscopy system. Both a liquid scintillator detector and a gas proportional counter were used for the recoil spectra measurements, with the individual response functions estimated from a series of Monte Carlo simulations. These normalized individual response functions were formed into a single response matrix for the unfolding process. Several accelerator-based quasi-monoenergetic neutron source spectra were measured and unfolded to test this spectroscopy system. These reference neutrons were produced from two reactions: T(p,n)(3)He and D(d,n)(3)He, generating neutron energies in the range between 0.2 and 8 MeV. The unfolded quasi-monoenergetic neutron spectra indicated that the detection system can provide good neutron spectroscopy results in this energy range. A broad-energy neutron spectrum from the (9)Be(d,n) reaction using a 5 MeV deuteron beam, measured at 60 degrees to the incident beam was measured and unfolded with the evaluated response matrix. The unfolded broad neutron spectrum is comparable with published time-of-flight results. Finally, the pair of detectors were used to measure the neutron spectrum generated at the RARAF IND-like neutron facility and a comparison is made to the neutron spectrum of Hiroshima.
4. Dibaryons in neutron stars
NASA Technical Reports Server (NTRS)
Olinto, Angela V.; Haensel, Pawel; Frieman, Joshua A.
1991-01-01
The effects are studied of H-dibaryons on the structure of neutron stars. It was found that H particles could be present in neutron stars for a wide range of dibaryon masses. The appearance of dibaryons softens the equations of state, lowers the maximum neutron star mass, and affects the transport properties of dense matter. The parameter space is constrained for dibaryons by requiring that a 1.44 solar mass neutron star be gravitationally stable.
5. Intense fusion neutron sources
Kuteev, B. V.; Goncharov, P. R.; Sergeev, V. Yu.; Khripunov, V. I.
2010-04-01
The review describes physical principles underlying efficient production of free neutrons, up-to-date possibilities and prospects of creating fission and fusion neutron sources with intensities of 1015-1021 neutrons/s, and schemes of production and application of neutrons in fusion-fission hybrid systems. The physical processes and parameters of high-temperature plasmas are considered at which optimal conditions for producing the largest number of fusion neutrons in systems with magnetic and inertial plasma confinement are achieved. The proposed plasma methods for neutron production are compared with other methods based on fusion reactions in nonplasma media, fission reactions, spallation, and muon catalysis. At present, intense neutron fluxes are mainly used in nanotechnology, biotechnology, material science, and military and fundamental research. In the near future (10-20 years), it will be possible to apply high-power neutron sources in fusion-fission hybrid systems for producing hydrogen, electric power, and technological heat, as well as for manufacturing synthetic nuclear fuel and closing the nuclear fuel cycle. Neutron sources with intensities approaching 1020 neutrons/s may radically change the structure of power industry and considerably influence the fundamental and applied science and innovation technologies. Along with utilizing the energy produced in fusion reactions, the achievement of such high neutron intensities may stimulate wide application of subcritical fast nuclear reactors controlled by neutron sources. Superpower neutron sources will allow one to solve many problems of neutron diagnostics, monitor nano-and biological objects, and carry out radiation testing and modification of volumetric properties of materials at the industrial level. Such sources will considerably (up to 100 times) improve the accuracy of neutron physics experiments and will provide a better understanding of the structure of matter, including that of the neutron itself.
6. A High Count Rate Neutron Beam Monitor for Neutron Scattering Facilities
SciTech Connect
Barnett, Amanda; Crow, Lowell; Diawara, Yacouba; Hayward, J P; Hayward, Jason P; Menhard, Kocsis; Sedov, Vladislav N; Funk, Loren L
2013-01-01
Abstract Beam monitors are an important diagnostic tool in neutron science facilities. Present beam monitors use either ionization chambers in integration mode, which are slow and have no timing information, or pulse counters which can easily be saturated by high beam intensities. At high flux neutron scattering facilities, neutron beam monitors with very low intrinsic efficiency (10-5) are presently selected to keep the counting rate within a feasible range, even when a higher efficiency would improve the counting statistics and yield a better measurement of the incident beam. In this work, we report on a high count rate neutron beam monitor. This beam monitor offers good timing with an intrinsic efficiency of 10-3 and a counting rate capability of over 1,000,000 cps without saturation.
7. Neutron removal cross section as a measure of neutron skin
SciTech Connect
Fang, D. Q.; Ma, Y. G.; Cai, X. Z.; Tian, W. D.; Wang, H. W.
2010-04-15
We study the relation between neutron removal cross section (sigma{sub -N}) and neutron skin thickness for finite neutron-rich nuclei using the statistical abrasion ablation model. Different sizes of neutron skin are obtained by adjusting the diffuseness parameter of neutrons in the Fermi distribution. It is demonstrated that there is a good linear correlation between sigma{sub -N} and the neutron skin thickness for neutron-rich nuclei. Further analysis suggests that the relative increase of neutron removal cross section could be used as a quantitative measure for neutron skin thickness in neutron-rich nuclei.
8. LGB neutron detector
Quist, Nicole
2012-10-01
The double pulse signature of the Gadolinium Lithium Borate Cerium doped plastic detector suggests its effectiveness for analyzing neutrons while providing gamma ray insensitivity. To better understand this detector, a californium gamma/neutron time of flight facility was constructed in our lab. Reported here are efforts to understand the properties and applications of the LGB detector with regards to neutron spectroscopy.
9. Perforated diode neutron sensors
McNeil, Walter J.
A novel design of neutron sensor was investigated and developed. The perforated, or micro-structured, diode neutron sensor is a concept that has the potential to enhance neutron sensitivity of a common solid-state sensor configuration. The common thin-film coated diode neutron sensor is the only semiconductor-based neutron sensor that has proven feasible for commercial use. However, the thin-film coating restricts neutron counting efficiency and severely limits the usefulness of the sensor. This research has shown that the perforated design, when properly implemented, can increase the neutron counting efficiency by greater than a factor of 4. Methods developed in this work enable detectors to be fabricated to meet needs such as miniaturization, portability, ruggedness, and adaptability. The new detectors may be used for unique applications such as neutron imaging or the search for special nuclear materials. The research and developments described in the work include the successful fabrication of variant perforated diode neutron detector designs, general explanations of fundamental radiation detector design (with added focus on neutron detection and compactness), as well as descriptive theory and sensor design modeling useful in predicting performance of these unique solid-state radiation sensors. Several aspects in design, fabrication, and operational performance have been considered and tested including neutron counting efficiency, gamma-ray response, perforation shapes and depths, and silicon processing variations. Finally, the successfully proven technology was applied to a 1-dimensional neutron sensor array system.
DOEpatents
Branagan, Daniel J.; Smolik, Galen R.
2000-01-01
A neutron absorbing material and method utilizing rare earth elements such as gadolinium, europium and samarium to form metallic glasses and/or noble base nano/microcrystalline materials, the neutron absorbing material having a combination of superior neutron capture cross sections coupled with enhanced resistance to corrosion, oxidation and leaching.
11. Arsenic activation neutron detector
DOEpatents
Jacobs, Eddy L.
1981-01-01
A detector of bursts of neutrons from a deuterium-deuteron reaction includes a quantity of arsenic adjacent a gamma detector such as a scintillator and photomultiplier tube. The arsenic is activated by the 2.5 Mev neutrons to release gamma radiation which is detected to give a quantitative representation of detected neutrons.
12. Arsenic activation neutron detector
DOEpatents
Jacobs, E.L.
1980-01-28
A detector of bursts of neutrons from a deuterium-deuteron reaction includes a quantity of arsenic adjacent a gamma detector such as a scintillator and photomultiplier tube. The arsenic is activated by the 2.5-MeV neutrons to release gamma radiation which is detected to give a quantitative representation of detected neutrons.
13. Use of Neutron Transfer Reactions to Indirectly Determine Neutron Capture Cross Sections on Neutron-Rich Nuclei
SciTech Connect
McCleskey, M.; Mukhamedzhanov, A. M.; Tribble, R. E.; Simmons, E.; Spiridon, A.; Banu, A.; Roeder, B.; Goldberg, V.; Trache, L.; Chen, X. F.; Lui, Y.-W.
2010-03-01
{sup 14}C(n,gamma){sup 15}C is being used as a test case in the development of an indirect method to determine neutron capture cross sections on neutron-rich unstable nuclei at astrophysical energies. Our approach makes use of two reactions: one peripheral used to find the asymptotic normalization coefficient (ANC) and a second non-peripheral reaction to determine the spectroscopic factor. The ANC for {sup 15}C has been determined using a HI neutron transfer reaction with a 12 MeV/nucleon {sup 14}C beam on a {sup 13}C target. The spectroscopic factor will be determined using {sup 14}C(d,p) in forward kinematics with an incident deuteron energy of 60 MeV. Both experiments were performed using the MDM high-resolution spectrometer at Texas A and M University.
14. Neutron computed tomography.
PubMed
Koeppe, R A; Brugger, R M; Schlapper, G A; Larsen, G N; Jost, R J
1981-02-01
A neutron-transmission computed tomography scanning system has been built for scanning biological materials. An oxygen filtered beam of 2.35 MeV neutrons was used for the measurements. The studies to date show that the interactions of these energy neutrons with samples simulating biological materials are more sensitive than X-rays to variations in the content of the material, thus providing the ability to produce high quality images. The neutron scans suggest that neutrons can be an effective radiation for the imaging of biological materials.
15. PERSONNEL NEUTRON DOSIMETER
DOEpatents
Fitzgerald, J.J.; Detwiler, C.G. Jr.
1960-05-24
A description is given of a personnel neutron dosimeter capable of indicating the complete spectrum of the neutron dose received as well as the dose for each neutron energy range therein. The device consists of three sets of indium foils supported in an aluminum case. The first set consists of three foils of indium, the second set consists of a similar set of indium foils sandwiched between layers of cadmium, whereas the third set is similar to the second set but is sandwiched between layers of polyethylene. By analysis of all the foils the neutron spectrum and the total dose from neutrons of all energy levels can be ascertained.
16. Organic metal neutron detector
DOEpatents
Butler, Michael A.; Ginley, David S.
1987-01-01
A device for detecting neutrons comprises a layer of conductive polymer sandwiched between electrodes, which may be covered on each face with a neutron transmissive insulating material layer. Conventional electrodes are used for a non-imaging integrating total neutron fluence-measuring embodiment, while wire grids are used in an imaging version of the device. The change in conductivity of the polymer after exposure to a neutron flux is determined in either case to provide the desired data. Alternatively, the exposed conductive polymer layer may be treated with a chemical reagent which selectively binds to the sites altered by neutrons to produce an image of the flux detected.
17. Neutron activation analysis system
DOEpatents
Taylor, M.C.; Rhodes, J.R.
1973-12-25
A neutron activation analysis system for monitoring a generally fluid media, such as slurries, solutions, and fluidized powders, including two separate conduit loops for circulating fluid samples within the range of radiation sources and detectors is described. Associated with the first loop is a neutron source that emits s high flux of slow and thermal neutrons. The second loop employs a fast neutron source, the flux from which is substantially free of thermal neutrons. Adjacent to both loops are gamma counters for spectrographic determination of the fluid constituents. Other gsmma sources and detectors are arranged across a portion of each loop for deterMining the fluid density. (Official Gazette)
18. High energy neutron dosimeter
DOEpatents
Sun, Rai Ko S.F.
1994-01-01
A device for measuring dose equivalents in neutron radiation fields. The device includes nested symmetrical hemispheres (forming spheres) of different neutron moderating materials that allow the measurement of dose equivalents from 0.025 eV to past 1 GeV. The layers of moderating material surround a spherical neutron counter. The neutron counter is connected by an electrical cable to an electrical sensing means which interprets the signal from the neutron counter in the center of the moderating spheres. The spherical shape of the device allows for accurate measurement of dose equivalents regardless of its positioning.
19. Neutron scatter camera
DOEpatents
Mascarenhas, Nicholas; Marleau, Peter; Brennan, James S.; Krenz, Kevin D.
2010-06-22
An instrument that will directly image the fast fission neutrons from a special nuclear material source has been described. This instrument can improve the signal to background compared to non imaging neutron detection techniques by a factor given by ratio of the angular resolution window to 4.pi.. In addition to being a neutron imager, this instrument will also be an excellent neutron spectrometer, and will be able to differentiate between different types of neutron sources (e.g. fission, alpha-n, cosmic ray, and D-D or D-T fusion). Moreover, the instrument is able to pinpoint the source location.
20. Semiconductor neutron detector
DOEpatents
Ianakiev, Kiril D.; Littlewood, Peter B.; Blagoev, Krastan B.; Swinhoe, Martyn T.; Smith, James L.; Sullivan, Clair J.; Alexandrov, Boian S.; Lashley, Jason Charles
2011-03-08
A neutron detector has a compound of lithium in a single crystal form as a neutron sensor element. The lithium compound, containing improved charge transport properties, is either lithium niobate or lithium tantalate. The sensor element is in direct contact with a monitor that detects an electric current. A signal proportional to the electric current is produced and is calibrated to indicate the neutrons sensed. The neutron detector is particularly useful for detecting neutrons in a radiation environment. Such radiation environment may, e.g. include gamma radiation and noise.
1. Development of an Array of Liquid Scintillators to Measure the Prompt Fission Neutron Spectrum at LANSCE
Perdue, B. A.; Taddeucci, T. N.; Haight, R. C.; Bredeweg, T.; Devlin, M.; Fotiades, N.; Jandel, M.; Laptev, A.; Lee, H. Y.; Nelson, R. O.; O'Donnell, J. M.; Ullmann, J. L.; Wender, S. A.; Wu, C. Y.; Kwan, E.; Chyzh, A.; Henderson, R. A.; Gostic, J. M.
2014-05-01
Higher quality measurements of outgoing prompt neutron spectra from neutron-induced fission as a function of the incoming neutron energy are needed. These data can be used in designing new fast reactors, predicting criticality for safety analyses, and developing techniques for global security applications. As part of the program to measure the prompt fission neutron spectra (PFNS) from the fission of 239Pu at the Los Alamos Neutron Science Center, we are developing a new array of liquid-scintillator detectors. This array will be used to measure the PFNS over a range of outgoing neutron energies from approximately 600 keV to 12 MeV and incident neutron energies from 0.5 to 30 MeV. A complete characterization of the detectors and the array as a whole will be carried out, targeted at understanding the light-output curves, efficiencies, and the neutron multiple-scattering backgrounds.
2. Radiative neutron capture cross section from 236U
Baramsai, B.; Jandel, M.; Bredeweg, T. A.; Bond, E. M.; Roman, A. R.; Rusev, G.; Walker, C. L.; Couture, A.; Mosby, S.; O'Donnell, J. M.; Ullmann, J. L.; Kawano, T.
2017-08-01
The 236U(n ,γ ) reaction cross section has been measured for the incident neutron energy range from 10 eV to 800 keV by using the Detector for Advanced Neutron Capture Experiments (DANCE) γ -ray calorimeter at the Los Alamos Neutron Science Center. The cross section was determined with the ratio method, which is a technique that uses the 235U(n ,f ) reaction as a reference. The results of the experiment are reported in the resolved and unresolved resonance energy regions. Individual neutron resonance parameters were obtained below 1 keV incident energy by using the R -matrix code sammy. The cross section in the unresolved resonance region is determined with improved experimental uncertainty. It agrees with both ENDF/B-VII.1 and JEFF-3.2 nuclear data libraries. The results above 10 keV agree better with the JEFF-3.2 library.
3. The Neutron Star Interior Composition Explorer (NICER)
NASA Technical Reports Server (NTRS)
Wilson-Hodge, Colleen A.; Gendreau, K.; Arzoumanian, Z.
2014-01-01
The Neutron Star Interior Composition Explorer (NICER) is an approved NASA Explorer Mission of Opportunity dedicated to the study of the extraordinary gravitational, electromagnetic, and nuclear-physics environments embodied by neutron stars. Scheduled to be launched in 2016 as an International Space Station payload, NICER will explore the exotic states of matter, using rotation-resolved spectroscopy of the thermal and non-thermal emissions of neutron stars in the soft (0.2-12 keV) X-ray band. Grazing-incidence "concentrator" optics coupled with silicon drift detectors, actively pointed for a full hemisphere of sky coverage, will provide photon-counting spectroscopy and timing registered to GPS time and position, with high throughput and relatively low background. The NICER project plans to implement a Guest Observer Program, which includes competitively selected user targets after the first year of flight operations. I will describe NICER and discuss ideas for potential Be/X-ray binary science.
4. Neoplasia in fast neutron-irradiated beagles
SciTech Connect
Bradley, E.W.; Zook, B.C.; Casarett, G.W.; Deye, J.A.; Adoff, L.M.; Rogers, C.C.
1981-09-01
One hundred fifty-one beagle dogs were irradiated with either photons or fast neutrons (15 MeV) to one of three dose-limiting normal tissues--spinal cord, lung, or brain. The radiation was given in four fractions per week for 5 weeks (spinal cord), 6 weeks (lung), or 7 weeks (brain) to total doses encompassing those given clinically for cancer management. To date, no nonirradiated dogs or photon-irradiated dogs have developed any neoplasms. Seven dogs receiving fast neutrons have developed 9 neoplasms within the irradiated field. Of the neutron-irradiated dogs at risk, the incidence of neoplasia was 15%. The latent period for radiation-induced cancers has varied from 1 to 4 1/2 years at this time in the study.
5. Use of ultracold neutrons for condensed-matter studies
SciTech Connect
Michaudon, A.
1997-05-01
Ultracold neutrons have such low velocities that they are reflected by most materials at all incident angles and can be stored in material bottles for long periods of time during which their intrinsic properties can be studied in great detail. These features have been mainly used for fundamental-physics studies including the detection of a possible neutron electric dipole moment and the precise determination of neutron-decay properties. Ultracold neutrons can also play a role in condensed-matter studies with the help of high-resolution spectrometers that use gravity as a strongly dispersive medium for low-velocity neutrons. Such studies have so far been limited by the low intensity of existing ultracold-neutron sources but could be reconsidered with more intense sources, which are now envisaged. This report provides a broad survey of the properties of ultracold neutrons (including their reflectivity by different types of samples), of ultracold-neutron spectrometers that are compared with other high-resolution instruments, of results obtained in the field of condensed matter with these instruments, and of neutron microscopes. All these subjects are illustrated by numerous examples.
6. Upgrades to the Polarized Neutron Reflectometer Asterix at LANSCE
SciTech Connect
Pynn, Roger
2015-03-16
We have upgraded the polarized neutron reflectometer, Asterix, at the Lujan Neutron Scattering Center at Los Alamos for the benefit of the research communities that study magnetic and complex-fluid films, both of which play important roles in support of the DOE’s energy mission. The upgrades to the instrument include: • A secondary spectrometer that was integrated with a Huber sample goniometer purchased with other funds just prior to the start of our project. The secondary spectrometer provides a flexible length for the scattered flight path, includes a mechanism to select among 3 alternative polarization analyzers as well as a support for new neutron detectors. Also included is an optic rail for reproducible positioning of components for Spin Echo Scattering Angle Measurement (SESAME). The entire secondary spectrometer is now non-magnetic, as required for neutron Larmor labeling. • A broad-band neutron polarizer for the incident neutron beam based on the V geometry. • A wide-angle neutron polarization analyzer • A 2d position-sensitive neutron detector • Electromagnetic coils (Wollaston prisms) for SESAME plus the associated power supplies, cooling, safety systems and integration into the data acquisition system. The upgrades allowed a nearly effortless transition between configurations required to serve the polarized neutron reflectometry community, users of the 11 T cryomagnet and users of SESAME.
7. GEM-based detectors for thermal and fast neutrons
Croci, G.; Claps, G.; Cazzaniga, C.; Foggetta, L.; Muraro, A.; Valente, P.
2015-06-01
Lately the problem of 3He replacement for neutron detection stimulated an intense activity research on alternative technologies based on alternative neutron converters. This paper presents briefly the results obtained with new GEM detectors optimized for fast and thermal neutrons. For thermal neutrons, we realized a side-on GEM detector based on a series of boron-coated alumina sheets placed perpendicularly to the incident neutron beam direction. This prototype has been tested at n@BTF photo-production neutron facilty in order to test its effectiveness under a very high flux gamma background. For fast neutrons, we developed new GEM detectors (called nGEM) for the CNESM diagnostic system of the SPIDER NBI prototype for ITER (RFX-Consortium, Italy) and as beam monitor for fast neutrons lines at spallation sources. The nGEM is a Triple GEM gaseous detector equipped with a polyethylene layer used to convert fast neutrons into recoil protons through the elastic scattering process. This paper describes the results obtained by testing a medium size (30 × 25 cm2 active area) nGEM detector at the ISIS spallation source on the VESUVIO beam line.
8. High-Energy Neutron Spectra and Flux Measurements Below Ground
Roecker, Caleb; Bernstein, Adam; Marleau, Peter; Vetter, Kai
2016-03-01
High-energy neutrons are a ubiquitous and often poorly measured background. Below ground, these neutrons could potentially interfere with antineutrino based reactor monitoring experiments as well as other rare-event neutral particle detectors. We have designed and constructed a transportable fast neutron detection system for measuring neutron energy spectra and flux ranging from tens to hundreds of MeV. The spectrometer uses a multiplicity technique in order to have a higher effective area than traditional transportable high-energy neutron spectrometers. Transportability ensures a common detector-related systematic bias for future measurements. The spectrometer is composed of two Gd containing plastic scintillator detectors arranged around a lead spallation target. A high-energy neutron may interact in the lead producing many secondary neutrons. The detector records the correlated secondary neutron multiplicity. Over many events, the response can be used to infer the incident neutron energy spectrum and flux. As a validation of the detector response, surface measurements have been performed; results confirm agreement with previous experiments. Below ground measurements have been performed at 3 depths (380, 600, and 1450 m.w.e.); results from these measurements will be presented.
9. NEUTRON DENSITY CONTROL IN A NEUTRONIC REACTOR
DOEpatents
Young, G.J.
1959-06-30
The method and means for controlling the neutron density in a nuclear reactor is described. It describes the method and means for flattening the neutron density distribution curve across the reactor by spacing the absorbing control members to varying depths in the central region closer to the center than to the periphery of the active portion of the reactor to provide a smaller neutron reproduction ratio in the region wherein the members are inserted, than in the remainder of the reactor thereby increasing the over-all potential power output.
10. Ultracold neutron detector for neutron lifetime measurements
Andreev, V.; Vassiljev, A.; Ivanov, E.; Ilyin, D.; Krivshich, A.; Serebrov, A.
2017-02-01
The gas-filled detector of ultracold neutrons has been designed and constructed for the spectrometer of the neutron lifetime measurements at the ILL, Grenoble, France. The detector has been successfully tested and is currently being used at this spectrometer. We could show that minimization of the ;wall; effect is a key factor to ensure efficient background suppression and to maximize the detection efficiency. This effect is primarily related to the composition of the gas mixture, which crucially depends on the neutron velocity spectrum.
11. Evaluating the 239Pu prompt fission neutron spectrum induced by thermal to 30 MeV neutrons
SciTech Connect
Neudecker, Denise; Talou, Patrick; Kawano, Toshihiko; Kahler, Albert Comstock; Rising, Michael Evan; White, Morgan Curtis
2016-03-15
We present a new evaluation of the 239Pu prompt fission neutron spectrum (PFNS) induced by thermal to 30 MeV neutrons. Compared to the ENDF/B-VII.1 evaluation, this one includes recently published experimental data as well as an improved and extended model description to predict PFNS. For instance, the pre-equilibrium neutron emission component to the PFNS is considered and the incident energy dependence of model parameters is parametrized more realistically. Experimental and model parameter uncertainties and covariances are estimated in detail. Also, evaluated covariances are provided between all PFNS at different incident neutron energies. In conclusion, selected evaluation results and first benchmark calculations using this evaluation are briefly discussed.
12. Evaluating the 239Pu prompt fission neutron spectrum induced by thermal to 30 MeV neutrons
DOE PAGES
Neudecker, Denise; Talou, Patrick; Kawano, Toshihiko; ...
2016-03-15
We present a new evaluation of the 239Pu prompt fission neutron spectrum (PFNS) induced by thermal to 30 MeV neutrons. Compared to the ENDF/B-VII.1 evaluation, this one includes recently published experimental data as well as an improved and extended model description to predict PFNS. For instance, the pre-equilibrium neutron emission component to the PFNS is considered and the incident energy dependence of model parameters is parametrized more realistically. Experimental and model parameter uncertainties and covariances are estimated in detail. Also, evaluated covariances are provided between all PFNS at different incident neutron energies. In conclusion, selected evaluation results and first benchmarkmore » calculations using this evaluation are briefly discussed.« less
13. Neutron beam design, development, and performance for neutron capture therapy
SciTech Connect
Harling, O.K.; Bernard, J.A. ); Zamenhof, R.G. )
1990-01-01
The report presents topics presented at a workshop on neutron beams and neutron capture therapy. Topics include: neutron beam design; reactor-based neutron beams; accelerator-based neutron beams; and dosimetry and treatment planning. Individual projects are processed separately for the databases. (CBS)
14. Replica neutron guides for experiments with ultracold neutrons
Serebrov, A. P.; Vasil'ev, A. V.; Lasakov, M. S.; Siber, E. V.; Murashkin, A. N.; Egorov, A. I.; Fomin, A. K.; Sbitnev, S. V.; Geltenbort, P.; Zimmer, O.
2017-01-01
The method for producing neutron guides for ultracold neutrons based on the replica method has been described. A comparative analysis of the quality of replica neutron guides, neutron guides made from polished anode-mechanical steel tubes, and neutron guides from electropolished tubes has been given.
15. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, Michiko; Slatkin, Daniel N.
1995-10-03
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na.sub.4 B.sub.12 I.sub.11 SSB.sub.12 I.sub.11, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy.
16. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, Michiko; Slatkin, Daniel N.
1997-03-18
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na.sub.4 B.sub.12 I.sub.11 SSB.sub.12 I.sub.11, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy.
17. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, Michiko; Slatkin, Daniel N.
1997-08-05
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized. by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na.sub.4 B.sub.12 I.sub.11 SSB.sub.12 I.sub.11, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy.
18. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, M.; Slatkin, D.N.
1997-08-05
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na{sub 4}B{sub 12}I{sub 11}SSB{sub 12}I{sub 11}, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy. 1 fig.
19. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, M.; Slatkin, D.N.
1995-10-03
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na{sub 4}B{sub 12}I{sub 11}SSB{sub 12}I{sub 11}, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy. 1 fig.
20. Neutron spectrometry using CR-39 track etch detectors.
PubMed
Phillips, Gary W; Spann, Jerrette E; Bogard, James S; VoDinh, Tuan; Emfietzoglou, Dimitris; Devine, Robert T; Moscovitch, Marko
2006-01-01
Track-size distributions were measured for chemically etched CR-39 foils exposed to monoenergetic neutrons with energies ranging from 0.144 to 19 MeV and to various broad-spectrum neutron sources including spontaneous fission neutrons from (238)Pu. These tracks are due to energetic charged particles resulting from interactions of the neutrons with the CR-39. The tracks are visible with an optical microscope after chemical etching and vary in size and configuration depending on the particle, energy and angle of incidence. The foils were analysed using an automatic analysis system that scans the foils, identifies valid tracks and records the track-size parameters. The track-size distributions vary with neutron energy for the monoenergetic sources and with the hardness of the broad-spectrum sources. The distribution from the (238)Pu fission source is readily distinguishable from the other sources measured and from distributions owing to the background.
1. Halogenated sulfidohydroboranes for nuclear medicine and boron neutron capture therapy
DOEpatents
Miura, M.; Slatkin, D.N.
1997-03-18
A method for performing boron neutron capture therapy for the treatment of tumors is disclosed. The method includes administering to a patient an iodinated sulfidohydroborane, a boron-10-containing compound. The site of the tumor is localized by visualizing the increased concentration of the iodine labelled compound at the tumor. The targeted tumor is then irradiated with a beam of neutrons having an energy distribution effective for neutron capture. Destruction of the tumor occurs due to high LET particle irradiation of the tissue secondary to the incident neutrons being captured by the boron-10 nuclei. Iodinated sulfidohydroboranes are disclosed which are especially suitable for the method of the invention. In a preferred embodiment, a compound having the formula Na{sub 4}B{sub 12}I{sub 11}SSB{sub 12}I{sub 11}, or another pharmaceutically acceptable salt of the compound, may be administered to a cancer patient for boron neutron capture therapy. 1 fig.
2. Introduction to Pesticide Incidents
EPA Pesticide Factsheets
Pesticides incidents must be reported by pesticide registrants. Others, such as members of the public and environmental professionals, would like to report pesticide incidents. This website will explain and facilitate such incident reporting.
SciTech Connect
Berger, H.
1993-12-31
Neutron radiography is a recognized method for nondestructive testing (NDT). It is one of eight established NDT methods offered for certification by ASNT. There are ASTM standards describing selected characteristics associated with neutron radiography, as discussed later in this session. Neutron radiography standards are proceeding in the international community (International Organization for Standardization, ISO). A primary advantage of using neutrons for radiologic inspection follows from the fact that the attenuation of thermal neutrons is very different from that of X-rays. A comparison of the attenuation of the elements for thermal neutrons (small dots) and 125 kV X-rays (solid line) is shown. As opposed to the increasing attenuation with increasing atomic number (Z) for X-rays, the neutron attenuation pattern is scattered. If there is a pattern for the neutrons, it tends to be the reverse of the X-ray case, namely, high neutron attenuation for light materials, such as hydrogen, lithium, and boron (H, Li, and B) and low attenuation for the heavy materials, such as lead, bismuth, and uranium (Pb, Bi, and U). The capability of thermal neutrons to image low-Z material, particularly hydrogen, in metal assemblies is in sharp contrast to that of X-rays. This opens up many practical inspection applications involving, for example, explosives, adhesives, corrosion, water intrusion, and hydriding of metals.
4. Neutron interaction and their transport with bulk materials
SciTech Connect
2015-05-15
In the current paper an attempt was made to study and provide fundamental information about neutron interactions that are important to nuclear material measurements. The application of this study is explained about macroscopic interactions with bulk compound materials through a program in DEV C++ language which is done by enabling interaction of neutrons in nature. The output of the entire process depends upon the random number (i.e., incident neutron number), thickness of the material and mean free path as input parameters. Further the current study emphasizes on the usage of materials in shielding.
5. Actinide neutron-induced fission cross section measurements at LANSCE
SciTech Connect
Tovesson, Fredrik K; Laptev, Alexander B; Hill, Tony S
2010-01-01
Fission cross sections of a range of actinides have been measured at the Los Alamos Neutron Science Center (LANSCE) in support of nuclear energy applications in a wide energy range from sub-thermal energies up to 200 MeV. A parallel-plate ionization chamber are used to measure fission cross sections ratios relative to the {sup 235}U standard while incident neutron energies are determined using the time-of-flight method. Recent measurements include the {sup 233,238}U, {sup 239-242}Pu and {sup 243}Am neutron-induced fission cross sections. Obtained data are presented in comparison with ex isting evaluations and previous data.
6. IMPROVED COMPUTATIONAL CHARACTERIZATION OF THE THERMAL NEUTRON SOURCE FOR NEUTRON CAPTURE THERAPY RESEARCH AT THE UNIVERSITY OF MISSOURI
SciTech Connect
Stuart R. Slattery; David W. Nigg; John D. Brockman; M. Frederick Hawthorne
2010-05-01
Parameter studies, design calculations and initial neutronic performance measurements have been completed for a new thermal neutron beamline to be used for neutron capture therapy cell and small-animal radiobiology studies at the University of Missouri Research Reactor. The beamline features the use of single-crystal silicon and bismuth sections for neutron filtering and for reduction of incident gamma radiation. The computational models used for the final beam design and performance evaluation are based on coupled discrete-ordinates and Monte Carlo techniques that permit detailed modeling of the neutron transmission properties of the filtering crystals with very few approximations. This is essential for detailed dosimetric studies required for the anticipated research program.
7. Neutron metrology laboratory facility simulation.
PubMed
Pereira, Mariana; Salgado, Ana P; Filho, Aidano S; Pereira, Walsan W; Patrão, Karla C S; Fonseca, Evaldo S
2014-10-01
The Neutron Low Scattering Laboratory in Brazil has been completely rebuilt. Evaluation of air attenuation parameters and neutron component scattering in the room was done using Monte Carlo simulation code. Neutron fields produced by referenced neutron source were used to calculate neutron scattering and air attenuation.
8. Neutron chopper development at LANSCE
SciTech Connect
Nutter, M.; Lewis, L.; Tepper, S.; Silver, R.N.; Heffner, R.H.
1985-01-01
Progress is reported on neutron chopper systems for the Los Alamos Neutron Scattering Center pulsed spallation neutron source. This includes the development of 600+ Hz active magnetic bearing neutron chopper and a high speed control system designed to operate with the Proton Storage Ring to phase the chopper to the neutron source. 5 refs., 3 figs.
9. Studies of 54,56Fe Neutron Scattering Cross Sections
Hicks, S. F.; Vanhoy, J. R.; French, A. J.; Henderson, S. L.; Howard, T. J.; Pecha, R. L.; Santonil, Z. C.; Crider, B. P.; Liu, S.; McEllistrem, M. T.; Peters, E. E.; Prados-Estévez, F. M.; Ross, T. J.; Yates, S. W.
2015-05-01
Elastic and inelastic neutron scattering differential cross sections and γ-ray production cross sections have been measured on 54,56Fe at several incident energies in the fast neutron region between 1.5 and 4.7 MeV. All measurements were completed at the University of Kentucky Accelerator Laboratory (UKAL) using a 7-MV Model CN Van de Graaff accelerator, along with the neutron production and neutron and γ-ray detection systems located there. The facilities at UKAL allow the investigation of both elastic and inelastic scattering with nearly mono-energetic incident neutrons. Time-of-flight techniques were used to detect the scattered neutrons for the differential cross section measurements. The measured cross sections are important for fission reactor applications and also for testing global model calculations such as those found at ENDF, since describing both the elastic and inelastic scattering is important for determining the direct and compound components of the scattering mechanism. The γ-ray production cross sections are used to determine cross sections to unresolved levels in the neutron scattering experiments. Results from our measurements and comparisons to model calculations are presented.
10. Boron neutron capture therapy (BNCT): implications of neutron beam and boron compound characteristics.
PubMed
Wheeler, F J; Nigg, D W; Capala, J; Watkins, P R; Vroegindeweij, C; Auterinen, I; Seppälä, T; Bleuel, D
1999-07-01
The potential efficacy of boron neutron capture therapy (BNCT) for malignant glioma is a significant function of epithermal-neutron beam biophysical characteristics as well as boron compound biodistribution characteristics. Monte Carlo analyses were performed to evaluate the relative significance of these factors on theoretical tumor control using a standard model. The existing, well-characterized epithermal-neutron sources at the Brookhaven Medical Research Reactor (BMRR), the Petten High Flux Reactor (HFR), and the Finnish Research Reactor (FiR-1) were compared. Results for a realistic accelerator design by the E. O. Lawrence Berkeley National Laboratory (LBL) are also compared. Also the characteristics of the compound p-Boronophenylaline Fructose (BPA-F) and a hypothetical next-generation compound were used in a comparison of the BMRR and a hypothetical improved reactor. All components of dose induced by an external epithermal-neutron beam fall off quite rapidly with depth in tissue. Delivery of dose to greater depths is limited by the healthy-tissue tolerance and a reduction in the hydrogen-recoil and incident gamma dose allow for longer irradiation and greater dose at a depth. Dose at depth can also be increased with a beam that has higher neutron energy (without too high a recoil dose) and a more forward peaked angular distribution. Of the existing facilities, the FiR-1 beam has the better quality (lower hydrogen-recoil and incident gamma dose) and a penetrating neutron spectrum and was found to deliver a higher value of Tumor Control Probability (TCP) than other existing beams at shallow depth. The greater forwardness and penetration of the HFR the FiR-1 at greater depths. The hypothetical reactor and accelerator beams outperform at both shallow and greater depths. In all cases, the hypothetical compound provides a significant improvement in efficacy but it is shown that the full benefit of improved compound is not realized until the neutron beam is fully
11. Neutron radiography using neutron imaging plate.
PubMed
Chankow, Nares; Punnachaiya, Suvit; Wonglee, Sarinrat
2010-01-01
The aims of this research are to study properties of a neutron imaging plate (NIP) and to test it for use in nondestructive testing (NDT) of materials. The experiments were carried out by using a BAS-ND 2040 Fuji NIP and a neutron beam from the Thai Research Reactor TRR-1/M1. The neutron intensity and Cd ratio at the specimen position were approximately 9x10(5) ns/cm(2) s and 100 respectively. It was found that the photostimulated luminescence (PSL) readout of the imaging plate was directly proportional to the exposure time and approximately 40 times faster than the conventional NR using Gd converter screen/X-ray film technique. The sensitivities of the imaging plate to slow neutron and to Ir-192 gamma-rays were found to be approximately 4.2x10(-3) PSL/mm(2) per neutron and 6.7x10(-5) PSL/mm(2) per gamma-ray photon respectively. Finally, some specimens containing light elements were selected to be radiographed with neutrons using the NIP and the Gd converter screen/X-ray film technique. The image quality obtained from the two recording media was found to be comparable. Crown Copyright 2009. Published by Elsevier Ltd. All rights reserved.
12. Average neutronic properties of prompt fission products
SciTech Connect
Foster, D.G. Jr.; Arthur, E.D.
1982-02-01
Calculations of the average neutronic properties of the ensemble of fission products producted by fast-neutron fission of /sup 235/U and /sup 239/Pu, where the properties are determined before the first beta decay of any of the fragments, are described. For each case we approximate the ensemble by a weighted average over 10 selected nuclides, whose properties we calculate using nuclear-model parameters deduced from the systematic properties of other isotopes of the same elements as the fission fragments. The calculations were performed primarily with the COMNUC and GNASH statistical-model codes. The results, available in ENDF/B format, include cross sections, angular distributions of neutrons, and spectra of neutrons and photons, for incident-neutron energies between 10/sup -5/ eV and 20 MeV. Over most of this energy range, we find that the capture cross section of /sup 239/Pu fission fragments is systematically a factor of two to five greater than for /sup 235/U fission fragments.
13. Neutron scattering studies of 54,56Fe with monoenergetic neutrons
Hicks, S. F.; Combs, B. M.; Henderson, S. L.; Sidwell, L. C.; Vanhoy, J. R.; Garza, E.; Steves, J.; Chakraborty, A.; Crider, B. P.; Prados-Estevez, F. M.; Kumar, A.; McEllistrem, M. T.; Peters, E. E.; Ross, T. J.; Yates, S. W.
2013-10-01
Neutron scattering data for Fe are important for the development of next generation fission reactors, since Fe is an important structural material in all proposed reactor designs, as well as in existing reactors. How neutrons interact with Fe has an important impact on fuel performance during irradiations and the overall efficiency of fission reactors. While differential scattering cross sections have been previously measured at several incident neutron energies in the fast neutron region, questions remain regarding the uncertainties for existing cross sections and for neutron inelastic scattering. Elastic and inelastic differential scattering cross sections have been measured on 54,56Fe at the University of Kentucky Accelerator Laboratory in the fast neutron energy region between 1.7 and 4 MeV. Results from our measurements and comparisons to model calculations will be presented. This material is based on work supported by the Department of Energy under grant NEUP: NU-12-KY-UK-0201-05 and by the Cowan Physics Fund at the Univ. of Dallas.
14. Neutron sources and applications
SciTech Connect
Price, D.L.; Rush, J.J.
1994-01-01
Review of Neutron Sources and Applications was held at Oak Brook, Illinois, during September 8--10, 1992. This review involved some 70 national and international experts in different areas of neutron research, sources, and applications. Separate working groups were asked to (1) review the current status of advanced research reactors and spallation sources; and (2) provide an update on scientific, technological, and medical applications, including neutron scattering research in a number of disciplines, isotope production, materials irradiation, and other important uses of neutron sources such as materials analysis and fundamental neutron physics. This report summarizes the findings and conclusions of the different working groups involved in the review, and contains some of the best current expertise on neutron sources and applications.
15. Italian neutron sources
Prata, M.; Alloni, D.; De Felice, P.; Palomba, M.; Pietropaolo, A.; Pillon, M.; Quintieri, L.; Santagata, A.; Valente, P.
2014-11-01
Many research activities, instrumental analysis, studies of radiation damage, etc., require neutron sources. The main neutron sources present in Italy are described in three different sections: nuclear research reactors, accelerator driven, and metrology stations. The nuclear research reactors of LENA (University of Pavia) and ENEA Casaccia are described in terms of irradiation facilities available, neutron flux for each of them and the main activities carried out by each research centre. In the second section, the Frascati Neutron Generator (FNG), the Frascati Beam-Test Facility (BTF) and their main features are reported. In the last section there is a detailed description of the institutional role and the main activities carried out in the field of neutron metrology by the National Institute for Metrology of Ionizing Radiation (INMRI) with a brief description of neutron sources of which the institute is endowed.
16. Prototype Stilbene Neutron Collar
SciTech Connect
Prasad, M. K.; Shumaker, D.; Snyderman, N.; Verbeke, J.; Wong, J.
2016-10-26
A neutron collar using stilbene organic scintillator cells for fast neutron counting is described for the assay of fresh low enriched uranium (LEU) fuel assemblies. The prototype stilbene collar has a form factor similar to standard He-3 based collars and uses an AmLi interrogation neutron source. This report describes the simulation of list mode neutron correlation data on various fuel assemblies including some with neutron absorbers (burnable Gd poisons). Calibration curves (doubles vs 235U linear mass density) are presented for both thermal and fast (with Cd lining) modes of operation. It is shown that the stilbene collar meets or exceeds the current capabilities of He-3 based neutron collars. A self-consistent assay methodology, uniquely suited to the stilbene collar, using triples is described which complements traditional assay based on doubles calibration curves.
SciTech Connect
Raman, S.; Hayter, J.B.
1990-01-01
The Advanced Neutron Source (ANS) is a new user experimental facility planned to be operational at Oak Ridge in the late 1990's. The centerpiece of the ANS will be a steady-state research reactor of unprecedented thermal neutron flux ({phi}{sub th} {approx} 8 {times} 10{sup 19} m{sup {minus}2} {center dot}s{sup {minus}1}) accompanied by extensive and comprehensive equipment and facilities for neutron-based research.
SciTech Connect
Hayter, J.B.
1989-01-01
The Advanced Neutron Source (ANS) is a new user experimental facility planned to be operational at Oak Ridge in the late 1990's. The centerpiece of the ANS will be a steady-state research reactor of unprecedented thermal neutron flux ({phi}{sub th} {approx} 9{center dot}10{sup 19} m{sup -2}{center dot}s{sup -1}) accompanied by extensive and comprehensive equipment and facilities for neutron-based research. 5 refs., 5 figs.
19. The DIORAMA Neutron Emitter
SciTech Connect
Terry, James Russell
2016-05-05
Emission of neutrons in a given event is modeled by the DioramaEmitterNeutron object, a subclass of the abstract DioramaEmitterModule object. The GenerateEmission method of this object is the entry point for generation of a neutron population for a given event. Shown in table 1, this method requires a number of parameters to be defined in the event definition.
20. Portable Neutron Source
DTIC Science & Technology
2008-05-30
distribution were studied as a function of peak laser intensity, laser pulse duration and primary target thickness. The proposed scheme for neutron ...Fig. 2. The next step is to calculate the neutron yield from the ion beam-target deposition model. The secondary target is a thick (~ 1mm) slab of...specific directions of observations. Figure 4 displays a typical angular distribution function of neutrons from a CD2 target. In Fig. 4 we plot the
NASA Technical Reports Server (NTRS)
1972-01-01
The design, development and application of a neutron radiographic viewing system for use in nondestructive testing applications is considered. The system consists of a SEC vidicon camera, neutron image intensifier system, disc recorder, and TV readout. Neutron bombardment of the subject is recorded by an image converter and passed through an optical system into the SEC vidicon. The vidicon output may be stored, or processed for visual readout.
2. Neutron absorptiometric titration.
PubMed
Tölgyessy, J; Varga, S; Dillinger, P
1967-03-01
A method is outlined for detection of two-phase titration end-points by means of an abrupt change in the neutron-absorption characteristics of one of the phases. One of the components of the precipitate must have a large neutron absorption cross-section, and the disappearance or appearance of neutron absorption by the supernatant liquid from the precipitation reaction then marks the completion of precipitation.
3. ATR neutron spectral characterization
SciTech Connect
Rogers, J.W.; Anderl, R.A.
1995-11-01
4. On accelerator-based neutron sources and neutron field characterization with low energy neutron spectrometer based on position sensitive 3He counter.
PubMed
Murata, I; Miyamaru, H; Kato, I; Mori, Y
2009-07-01
The development of new neutron sources for BNCT applications, based on particle accelerators is currently underway all over the world. Though nuclear reactors were used for a long time as the only neutron source available having the requested flux levels, the accelerator-based ones have recently been investigated on the other hand due to its easy-to-use and acceptable performances. However, when using an accelerator, various secondary particles would be emitted which forms a troublesome background. Moreover, the neutrons produced have usually an energy spectrum somewhat different from the requested one and thus should be largely moderated. An additional issue to be taken into account is the patient positioning, which should be close to the neutron source, in order to take advantage of a neutron flux level high enough to limit the BNCT treatment time within 1h. This implies that, inside a relatively narrow space, neutrons should be moderated, while unnecessary secondary particles should be shielded. Considering that a background-free neutron field from an accelerator-driven neutron source dedicated to BNCT application is generally difficult to be provided, the characterization of such a neutron field will have to be clearly assessed. In the present study, a low energy neutron spectrometer has been thus designed and is now being developed to measure the accelerator-based neutron source performance. The presently proposed spectrometer is based on a (3)He proportional counter, which is 50 cm long and 5 cm in diameter, with a gas pressure of 0.5 MPa. It is quite unique that the spectrometer is set up in parallel with the incident neutron beam and a reaction depth distribution is measured by it as a position sensitive detector. Recently, a prototype detector has been developed and the signal test is now underway. In this paper, the feature of the accelerator-based neutron sources is outlined and importance of neutron field characterization is discussed. And the developed
5. Introduction to neutron stars
SciTech Connect
Lattimer, James M.
2015-02-24
Neutron stars contain the densest form of matter in the present universe. General relativity and causality set important constraints to their compactness. In addition, analytic GR solutions are useful in understanding the relationships that exist among the maximum mass, radii, moments of inertia, and tidal Love numbers of neutron stars, all of which are accessible to observation. Some of these relations are independent of the underlying dense matter equation of state, while others are very sensitive to the equation of state. Recent observations of neutron stars from pulsar timing, quiescent X-ray emission from binaries, and Type I X-ray bursts can set important constraints on the structure of neutron stars and the underlying equation of state. In addition, measurements of thermal radiation from neutron stars has uncovered the possible existence of neutron and proton superfluidity/superconductivity in the core of a neutron star, as well as offering powerful evidence that typical neutron stars have significant crusts. These observations impose constraints on the existence of strange quark matter stars, and limit the possibility that abundant deconfined quark matter or hyperons exist in the cores of neutron stars.
6. Investigating Time and Spectral Dependence in Neutron Radiation Environments for Semiconductor Damage Studies
DTIC Science & Technology
2014-09-18
Charged particle accelerators bombard a Device under Test (DuT) with ions and equate the resultant damage to a fluence of neutrons via a transfer function ... transport in electronics; ionizing effects can be tran- sient, as well as accumulative, and neutron effects are generally permanent. The test...Φ(E) = incident neutron spectrum, energy dependent FD,mat(E) = displacement damage function for material being irradiated, also energy dependent FD
7. Three-port beam splitter for slow neutrons using holographic nanoparticle-polymer composite diffraction gratings
SciTech Connect
Klepp, J.; Fally, M.; Tomita, Y.; Pruner, C.; Kohlbrecher, J.
2012-10-08
Diffraction of slow neutrons by nanoparticle-polymer composite gratings has been observed. By carefully choosing grating parameters such as grating thickness and spacing, a three-port beam splitter operation for slow neutrons - splitting the incident neutron intensity equally into the {+-}1st and the 0th diffraction orders - has been realized. As a possible application, a Zernike three-path interferometer is briefly discussed.
8. Cross correlation calculations and neutron scattering analysis for a portable solid state neutron detection system
Saltos, Andrea
In efforts to perform accurate dosimetry, Oakes et al. [Nucl. Intrum. Mehods. (2013)] introduced a new portable solid state neutron rem meter based on an adaptation of the Bonner sphere and the position sensitive long counter. The system utilizes high thermal efficiency neutron detectors to generate a linear combination of measurement signals that are used to estimate the incident neutron spectra. The inversion problem associated to deduce dose from the counts in individual detector elements is addressed by applying a cross-correlation method which allows estimation of dose with average errors less than 15%. In this work, an evaluation of the performance of this system was extended to take into account new correlation techniques and neutron scattering contribution. To test the effectiveness of correlations, the Distance correlation, Pearson Product-Moment correlation, and their weighted versions were performed between measured spatial detector responses obtained from nine different test spectra, and the spatial response of Library functions generated by MCNPX. Results indicate that there is no advantage of using the Distance Correlation over the Pearson Correlation, and that weighted versions of these correlations do not increase their performance in evaluating dose. Both correlations were proven to work well even at low integrated doses measured for short periods of time. To evaluate the contribution produced by room-return neutrons on the dosimeter response, MCNPX was used to simulate dosimeter responses for five isotropic neutron sources placed inside different sizes of rectangular concrete rooms. Results show that the contribution of scattered neutrons to the response of the dosimeter can be significant, so that for most cases the dose is over predicted with errors as large as 500%. A possible method to correct for the contribution of room-return neutrons is also assessed and can be used as a good initial estimate on how to approach the problem.
9. Predicted performance of neutron spectrometers using scintillating fibers
SciTech Connect
RA Craig; M Bliss
2000-02-14
A variety of needs exists for knowing the energy spectral content of a neutron flux. Among these needs are arms-control and national-security applications, which arise because different neutron sources produce different neutron energy spectra. This work is primarily directed at these applications. The concept described herein is a spectrometer in the same sense as a Bonner sphere. The instrument response reflects a statistical average of the energy spectrum. The Bonner sphere is an early rendition of this class. In this, a neutron detector is placed at the center of a moderating (and absorbing) sphere (of varying thickness and composition). Spectral unfolding is required, and the resolution and efficiency are, typically, poor, although the potential bandwidth is very large. A recent variation on the Bonner-sphere approach uses {sup 3}He gas proportional counters with resistive wires to locate the position of the event (Toyokawa et al 1996). The spectrometer concept investigated here has the potential for better resolution and much improved neutron efficiency compared to Bonner spheres and similar devices. These improvements are possible because of the development of neutron-sensitive, scintillating-glass fibers. These fibers can be precisely located in space, which allows a corresponding precision in energy resolution. Also, they can be fabricated into arrays that intercept a large fraction of incident thermal neutrons, providing the improvement in neutron economy.
10. α and 2 p 2 n emission in fast neutron-induced reactions on 60Ni
Fotiades, N.; Devlin, M.; Haight, R. C.; Nelson, R. O.; Kunieda, S.; Kawano, T.
2015-06-01
Background: The cross sections for populating the residual nucleus in the reaction ZAX(n,x) Z -2 A -4Y exhibit peaks as a function of incident neutron energy corresponding to the (n ,n'α ) reaction and, at higher energy, to the (n ,2 p 3 n ) reaction. The relative magnitudes of these peaks vary with the Z of the target nucleus. Purpose: Study fast neutron-induced reactions on 60Ni. Locate experimentally the nuclear charge region along the line of stability where the cross sections for α emission and for 2 p 2 n emission in fast neutron-induced reactions are comparable as a further test of reaction models. Methods: Data were taken by using the Germanium Array for Neutron-Induced Excitations. The broad-spectrum pulsed neutron beam of the Los Alamos Neutron Science Center's Weapons Neutron Research facility provided neutrons in the energy range from 1 to 250 MeV. The time-of-flight technique was used to determine the incident-neutron energies. Results: Absolute partial cross sections for production of seven discrete Fe γ rays populated in 60Ni (n ,α /2 p x n γ ) reactions with 2 ≤x ≤5 were measured for neutron energies 1 MeVneutron energies while discrepancies appear at higher neutron energies. The cross section for producing an isotope in fast neutron-induced reactions on stable targets via α emission at the peak of the (n ,α ) and (n ,n'α ) reactions is comparable to that for 2 p 2 n and 2 p 3 n emission at higher incident energies in the nuclear charge region around Fe.
11. Accelerator-based neutron source using a cold deuterium target with degenerate electrons
SciTech Connect
Phillips, R. E.; Ordonez, C. A.
2013-07-15
A neutron generator is considered in which a beam of tritons is incident on a hypothetical cold deuterium target with degenerate electrons. The energy efficiency of neutron generation is found to increase substantially with electron density. Recent reports of potential targets are discussed.
12. Development of the FNIT detector for 2-20 MeV solar neutrons
Bravar, Ulisse; Bruillard, Paul J.; Flueckiger, Erwin O.; MacKinnon, Alec L.; Macri, John R.; McConnell, Mark L.; Moser, Michael R.; Ryan, James M.
2006-04-01
The Fast Neutron Imaging Telescope (FNIT) is a newly developed neutron detector with imaging and energy measurement capabilities, sensitive to neutrons in the 2-20 MeV energy range. FNIT was conceived as a candidate instrument for the Solar Sentinels program. Its design is optimized to measure neutrons produced in solar flares from the inner heliosphere. The detection principle is based on multiple elastic neutron-proton scatterings in plastic scintillators. By measuring the scattering coordinates and determining the energy of recoil protons and time of flight of scattered neutrons, the energy spectrum and incident direction of primary neutrons can be reconstructed. We present the results of recent laboratory efforts and describe the performance of the FNIT prototype.
13. Neutron decay of 15C resonances by measurements of neutron time-of-flight
Cavallaro, M.; Agodi, C.; Assié, M.; Azaiez, F.; Cappuzzello, F.; Carbone, D.; de Séréville, N.; Foti, A.; Pandola, L.; Scarpaci, J. A.; Sgouros, O.; Soukeras, V.; Tropea, S.
2016-06-01
The neutron decay of the resonant states of light neutron-rich nuclei is an important and poorly explored property, useful to extract valuable nuclear structure information. In the present paper the neutron decay of the 15C resonances populated via the two-neutron transfer reaction 13C(18O,16On ) at 84-MeV incident energy is reported for the first time using an innovative technique which couples the MAGNEX magnetic spectrometer and the EDEN neutron detector array. Experimental data show that the resonances below the one-neutron emission threshold decay to the 14C ground state via one-neutron emission with an almost 100 % total branching ratio, whereas the recently observed 15C giant pairing vibration at 13.7 MeV mainly decays via two-neutron emission.
14. Fission signal detection using helium-4 gas fast neutron scintillation detectors
SciTech Connect
Lewis, J. M. Kelley, R. P.; Jordan, K. A.; Murer, D.
2014-07-07
We demonstrate the unambiguous detection of the fission neutron signal produced in natural uranium during active neutron interrogation using a deuterium-deuterium fusion neutron generator and a high pressure {sup 4}He gas fast neutron scintillation detector. The energy deposition by individual neutrons is quantified, and energy discrimination is used to differentiate the induced fission neutrons from the mono-energetic interrogation neutrons. The detector can discriminate between different incident neutron energies using pulse height discrimination of the slow scintillation component of the elastic scattering interaction between a neutron and the {sup 4}He atom. Energy histograms resulting from this data show the buildup of a detected fission neutron signal at higher energies. The detector is shown here to detect a unique fission neutron signal from a natural uranium sample during active interrogation with a (d, d) neutron generator. This signal path has a direct application to the detection of shielded nuclear material in cargo and air containers. It allows for continuous interrogation and detection while greatly minimizing the potential for false alarms.
15. Polarized neutron reflectometry of magnetic nanostructures
Toperverg, B. P.
2015-12-01
Among a number of methods employed to characterize various types of magnetic nano-structures Polarized Neutron Reflectometry (PNR) is shown to be a unique tool providing a scope of quantitative information on magnetization arrangement over relevant scales. Deeply penetrating into materials neutron spins are able to resolve vectorial profile of magnetic induction with accuracy of a fraction of Oersted over a fraction of nano-meters. This property is exploited in measurements of specular PNR which hence constitutes the method of depth resolved vector magnetometry widely used to examine magnetic states in exchange coupled magnetic superlattices, exchange bias systems, spin valves, exchange springs, superconducting/ferromagnetic heterostructure, etc. Off-specular polarized neutron scattering (OS-PNS) measures the in-plane magnetization distribution over scales from hundreds of nanoto hundreds of micrometers providing, in combination with specular PNR, access to lateral long range fluctuations of the magnetization vector and magnetic domains in these systems. OSPNS is especially useful in studies of co-operative magnetization reversal processes in various films and multilayers laterally patterned into periodic arrays of stripes, or islands of various dimentions, shapes, internal structures, etc., representing an interest for e.g. spintronics. Smaller sizes of 10?100 nm are accessed with the method of Polarized Neutrons Grazing Incidence Small Angle Scattering (PN-GISAS), which in a combination with specular PNR and OS-PNS is used to study self-assembling of magnetic nano-particles on flat surfaces, while Polarized Neutron Grazing Incidence Diffraction (PN-GID) complete the scope of magnetic information over wide range of scales in 3D space. The review of recent results obtained employing the methods listed above is preceded by the detailed theoretical consideration and exemplified by new developments addressing with PNR fast magnetic kinetics in nano-systems.
16. Studies on depth-dose-distribution controls by deuteration and void formation in boron neutron capture therapy.
PubMed
Sakurai, Yoshinori
2004-08-07
Physical studies on (i) replacement of heavy water for body water (deuteration), and (ii) formation of a void in human body (void formation) were performed as control techniques for dose distribution in a human head under neutron capture therapy. Simulation calculations were performed for a human-head-size cylindrical phantom using a two-dimensional transport calculation code for mono-energetic incidences of higher-energy epi-thermal neutrons (1.2-10 keV), lower-energy epi-thermal neutrons (3.1-23 eV) and thermal neutrons (1 meV to 0.5 eV). The deuteration was confirmed to be effective both in thermal neutron incidence and in epi-thermal neutron incidence from the viewpoints of improvement of the thermal neutron flux distribution and elimination of the secondary gamma rays. For the void formation, a void was assumed to be 4 cm in diameter and 3 cm in depth at the surface part in this study. It was confirmed that the treatable depth was improved almost 2 cm for any incident neutron energy in the case of the 10 cm irradiation field diameter. It was made clear that the improvement effect was larger in isotropic incidence than in parallel incidence, in the case that an irradiation field size was delimited fitting into a void diameter.
NASA Technical Reports Server (NTRS)
Leysath, W.; Brown, R. L.
1972-01-01
Neutron radiographic viewing system consisting of camera head and control processor is developed for use in nondestructive testing applications. Camera head consists of neutron-sensitive image intensifier system, power supply, and SEC vidicon camera head. Both systems, with their optics, are housed on test mount.
18. Neutron capture therapies
SciTech Connect
Yanch, Jacquelyn C.; Shefer, Ruth E.; Klinkowstein, Robert E.
1999-01-01
In one embodiment there is provided an application of the .sup.10 B(n,.alpha.).sup.7 Li nuclear reaction or other neutron capture reactions for the treatment of rheumatoid arthritis. This application, called Boron Neutron Capture Synovectomy (BNCS), requires substantially altered demands on neutron beam design than for instance treatment of deep seated tumors. Considerations for neutron beam design for the treatment of arthritic joints via BNCS are provided for, and comparisons with the design requirements for Boron Neutron Capture Therapy (BNCT) of tumors are made. In addition, exemplary moderator/reflector assemblies are provided which produce intense, high-quality neutron beams based on (p,n) accelerator-based reactions. In another embodiment there is provided the use of deuteron-based charged particle reactions to be used as sources for epithermal or thermal neutron beams for neutron capture therapies. Many d,n reactions (e.g. using deuterium, tritium or beryllium targets) are very prolific at relatively low deuteron energies.
19. Neutron capture therapies
SciTech Connect
Yanch, J.C.; Shefer, R.E.; Klinkowstein, R.E.
1999-11-02
In one embodiment there is provided an application of the {sup 10}B(n,{alpha}){sup 7}Li nuclear reaction or other neutron capture reactions for the treatment of rheumatoid arthritis. This application, called Boron Neutron Capture Synovectomy (BNCS), requires substantially altered demands on neutron beam design than for instance treatment of deep seated tumors. Considerations for neutron beam design for the treatment of arthritic joints via BNCS are provided for, and comparisons with the design requirements for Boron Neutron Capture Therapy (BNCT) of tumors are made. In addition, exemplary moderator/reflector assemblies are provided which produce intense, high-quality neutron beams based on (p,n) accelerator-based reactions. In another embodiment there is provided the use of deuteron-based charged particle reactions to be used as sources for epithermal or thermal neutron beams for neutron capture therapies. Many d,n reactions (e.g. using deuterium, tritium or beryllium targets) are very prolific at relatively low deuteron energies.
20. Compact neutron generator
DOEpatents
Leung, Ka-Ngo; Lou, Tak Pui
2005-03-22
A compact neutron generator has at its outer circumference a toroidal shaped plasma chamber in which a tritium (or other) plasma is generated. A RF antenna is wrapped around the plasma chamber. A plurality of tritium ion beamlets are extracted through spaced extraction apertures of a plasma electrode on the inner surface of the toroidal plasma chamber and directed inwardly toward the center of neutron generator. The beamlets pass through spaced acceleration and focusing electrodes to a neutron generating target at the center of neutron generator. The target is typically made of titanium tubing. Water is flowed through the tubing for cooling. The beam can be pulsed rapidly to achieve ultrashort neutron bursts. The target may be moved rapidly up and down so that the average power deposited on the surface of the target may be kept at a reasonable level. The neutron generator can produce fast neutrons from a T-T reaction which can be used for luggage and cargo interrogation applications. A luggage or cargo inspection system has a pulsed T-T neutron generator or source at the center, surrounded by associated gamma detectors and other components for identifying explosives or other contraband.
1. Hyperons in neutron stars
SciTech Connect
Glendenning, N.K.
1986-04-01
Generalized beta equilibrium involving nucleons, hyperons, and isobars is examined for neutron star matter. The hyperons produce a considerable softening of the equation of state. It is shown that the observed masses of neutron stars can be used to settle a recent controversy concerning the nuclear compressibility. Compressibilities less than 200 MeV are incompatible with observed masses. 7 refs., 9 figs.
2. Shielding for thermal neutrons.
PubMed
McCall, R C
1997-01-01
The problem of calculating the neutron capture gamma-ray dose rate due to thermal neutron capture in a boron or cadmium rectangular shield is considered. An example is given for shielding for a door at the exit of medical accelerator room maze in order to determine the optimum location of lead relative to the borated polyethylene.
3. NEUTRONIC REACTOR CONTROL
DOEpatents
Untermyer, S.; Hutter, E.
1959-08-01
This patent relates to "shadow" control of a nuclear reactor. The control means comprises a plurality ot elongated rods disposed adjacent and parallel to each other, The morphology and effects of gases generated within sections of neutron absorbing materials and equal length sections of neutron permeable materials together with means for longitudinally pcsitioning the rcds relative to each other.
4. Neutron filters for producing monoenergetic neutron beams
SciTech Connect
Harvey, J.A.; Hill, N.W.; Harvey, J.R.
1982-01-01
Neutron transmission measurements have been made on high-purity, highly-enriched samples of /sup 58/Ni (99.9%), /sup 60/Ni (99.7%), /sup 64/Zn (97.9%) and /sup 184/W (94.5%) to measure their neutron windows and to assess their potential usefulness for producing monoenergetic beams of intermediate energies from a reactor. Transmission measurements on the Los Alamos Sc filter (44.26 cm Sc and 1.0 cm Ti) have been made to determine the characteristics of the transmitted neutron beam and to measure the total cross section of Sc at the 2.0 keV minimum. When corrected for the Ti and impurities, a value of 0.35 +- 0.03 b was obtained for this minimum.
5. Prompt Neutron Multiplicity Measurements with Portable Detectors
SciTech Connect
S. Mukhopadhyay, R. Wolff, R. Maurer, S. Mitchell, E. X. Smith, P. Guss, J. L. Lacy, L. Sun, A. Athanasiades
2011-09-01
proportional counter gas (a mixture of 90% Ar and 10% CO2). The tubes operate in proportional counter mode and attract mobile charged particles (alpha-particles) created in the nuclear interaction 10B(n, a)7Li. Several MCNPX calculations covering the substantial design parameter space of neutron multiplicity detectors have been performed and are presented. Our detector’s thermal neutron detection efficiency is compared to two Ortec commercial products, the Fission Meter and Detective-EX. Pulse height spectra originating from the charged particles created in the nuclear reaction 10B(n, a) 7Li* + 2.310 MeV (94%) – excited state (1) and 10B(n, a) 7Li + 2.792 MeV (6%) – ground state (2) are examined, and the response to incident gamma rays are demonstrated.
6. Neutron scattering and models : molybdenum.
SciTech Connect
Smith, A.B.
1999-05-26
A comprehensive interpretation of the fast-neutron interaction with elemental and isotopic molybdenum at energies of {le} 30 MeV is given. New experimental elemental-scattering information over the incident energy range 4.5 {r_arrow} 10 MeV is presented. Spherical, vibrational and dispersive models are deduced and discussed, including isospin, energy-dependent and mass effects. The vibrational models are consistent with the ''Lane potential''. The importance of dispersion effects is noted. Dichotomies that exist in the literature are removed. The models are vehicles for fundamental physical investigations and for the provision of data for applied purposes. A ''regional'' molybdenum model is proposed. Finally, recommendations for future work are made.
7. NEUTRON SHIELDING STRUCTURE
DOEpatents
Mattingly, J.T.
1962-09-25
A lightweight neutron shielding structure comprises a honeycomb core which is filled with a neutron absorbing powder. The honeycomb core is faced with parallel planar facing sheets to form a lightweight rigid unit. Suitable absorber powders are selected from among the following: B, B/sub 4/C, B/sub 2/O/ sub 3/, CaB/sub 6/, Li/sub 2/CO3, LiOH, LiBO/sub 2/, Li/s ub 2/O. The facing sheets are constructed of a neutron moderating material, so that fast neutrons will be moderated while traversing the facing sheets, and ultimately be absorbed by the absorber powder in the honeycomb. Beryllium is a preferred moderator material for use in the facing sheets. The advantage of the structure is that it combines the rigidity and light weight of a honeycomb construction with the neutron absorption properties of boron and lithium. (AEC)
8. Neutrons against cancer
Dovbnya, A. N.; Kuplennikov, E. L.; Kandybey, S. S.; Krasiljnikov, V. V.
2014-09-01
The review is devoted to the analysis and generalization of the research carried out during recent years in industrially advanced countries on the use of fast, epithermal, and thermal neutrons for therapy of malignant tumors. Basic facilities for neutron production used for cancer treatment are presented. Optimal parameters of therapeutic beams are described. Techniques using neutrons of different energy regions are discussed. Results and medical treatment efficiency are given. Comparison of the current state of neutron therapy of tumors and alternative treatments with beams of protons and carbon ions has been conducted. Main attention is given to the possibility of the practical use of accumulated experience of application of neutron beams for cancer therapy.
9. Pulsed neutron detector
DOEpatents
Robertson, deceased, J. Craig; Rowland, Mark S.
1989-03-21
A pulsed neutron detector and system for detecting low intensity fast neutron pulses has a body of beryllium adjacent a body of hydrogenous material the latter of which acts as a beta particle detector, scintillator, and moderator. The fast neutrons (defined as having En>1.5 MeV) react in the beryllium and the hydrogenous material to produce larger numbers of slow neutrons than would be generated in the beryllium itself and which in the beryllium generate hellium-6 which decays and yields beta particles. The beta particles reach the hydrogenous material which scintillates to yield light of intensity related to the number of fast neutrons. A photomultiplier adjacent the hydrogenous material (scintillator) senses the light emission from the scintillator. Utilization means, such as a summing device, sums the pulses from the photo-multiplier for monitoring or other purposes.
10. THERMAL NEUTRON BACKSCATTER IMAGING.
SciTech Connect
VANIER,P.; FORMAN,L.; HUNTER,S.; HARRIS,E.; SMITH,G.
2004-10-16
Objects of various shapes, with some appreciable hydrogen content, were exposed to fast neutrons from a pulsed D-T generator, resulting in a partially-moderated spectrum of backscattered neutrons. The thermal component of the backscatter was used to form images of the objects by means of a coded aperture thermal neutron imaging system. Timing signals from the neutron generator were used to gate the detection system so as to record only events consistent with thermal neutrons traveling the distance between the target and the detector. It was shown that this time-of-flight method provided a significant improvement in image contrast compared to counting all events detected by the position-sensitive {sup 3}He proportional chamber used in the imager. The technique may have application in the detection and shape-determination of land mines, particularly non-metallic types.
11. Neutron stars - General review
NASA Technical Reports Server (NTRS)
Cameron, A. G. W.; Canuto, V.
1974-01-01
A review is presented of those properties of neutron stars upon which there is general agreement and of those areas which currently remain in doubt. Developments in theoretical physics of neutron star interiors are summarized with particular attention devoted to hyperon interactions and the structure of interior layers. Determination of energy states and the composition of matter is described for successive layers, beginning with the surface and proceeding through the central region into the core. Problems encountered in determining the behavior of matter in the ultra-high density regime are discussed, and the effects of the magnetic field of a neutron star are evaluated along with the behavior of atomic structures in the field. The evolution of a neutron star is outlined with discussion centering on carbon detonation, cooling, vibrational damping, rotation, and pulsar glitches. The role of neutron stars in cosmic-ray propagation is considered.
12. Pocked surface neutron detector
DOEpatents
McGregor, Douglas; Klann, Raymond
2003-04-08
The detection efficiency, or sensitivity, of a neutron detector material such as of Si, SiC, amorphous Si, GaAs, or diamond is substantially increased by forming one or more cavities, or holes, in its surface. A neutron reactive material such as of elemental, or any compound of, .sup.10 B, .sup.6 Li, .sup.6 LiF, U, or Gd is deposited on the surface of the detector material so as to be disposed within the cavities therein. The portions of the neutron reactive material extending into the detector material substantially increase the probability of an energetic neutron reaction product in the form of a charged particle being directed into and detected by the neutron detector material.
13. The Neutron Structure Function
Holt, Roy
2013-10-01
Knowledge of the neutron structure function is important for testing models of the nucleon, for a complete understanding of deep inelastic scattering (DIS) from nuclei, and for high energy experiments. As there exist no free neutron targets, neutron structure functions have been determined from deep inelastic scattering from the deuteron. Unfortunately, the short-range part of the deuteron wave function becomes important in extracting the neutron structure function at very high Bjorken x. New methods have been devised for Jefferson Lab experiments to mitigate this problem. The BONUS experiment involves tagging spectator neutrons in the deuteron, while the MARATHON experiment minimizes nuclear structure effects by a comparison of DIS from 3H and 3He. A summary of the status and future plans will be presented. This work supported by the U. S. Department of Energy, Office of Nuclear Physics, under contract DE-AC02-06CH11357.
14. SINGLE CRYSTAL NEUTRON DIFFRACTION.
SciTech Connect
KOETZLE,T.F.
2001-03-13
Single-crystal neutron diffraction measures the elastic Bragg reflection intensities from crystals of a material, the structure of which is the subject of investigation. A single crystal is placed in a beam of neutrons produced at a nuclear reactor or at a proton accelerator-based spallation source. Single-crystal diffraction measurements are commonly made at thermal neutron beam energies, which correspond to neutron wavelengths in the neighborhood of 1 Angstrom. For high-resolution studies requiring shorter wavelengths (ca. 0.3-0.8 Angstroms), a pulsed spallation source or a high-temperature moderator (a ''hot source'') at a reactor may be used. When complex structures with large unit-cell repeats are under investigation, as is the case in structural biology, a cryogenic-temperature moderator (a ''cold source'') may be employed to obtain longer neutron wavelengths (ca. 4-10 Angstroms). A single-crystal neutron diffraction analysis will determine the crystal structure of the material, typically including its unit cell and space group, the positions of the atomic nuclei and their mean-square displacements, and relevant site occupancies. Because the neutron possesses a magnetic moment, the magnetic structure of the material can be determined as well, from the magnetic contribution to the Bragg intensities. This latter aspect falls beyond the scope of the present unit; for information on magnetic scattering of neutrons see Unit 14.3. Instruments for single-crystal diffraction (single-crystal diffractometers or SCDs) are generally available at the major neutron scattering center facilities. Beam time on many of these instruments is available through a proposal mechanism. A listing of neutron SCD instruments and their corresponding facility contacts is included in an appendix accompanying this unit.
15. SINGLE CRYSTAL NEUTRON DIFFRACTION.
SciTech Connect
KOETZLE,T.F.
2001-03-13
Single-crystal neutron diffraction measures the elastic Bragg reflection intensities from crystals of a material, the structure of which is the subject of investigation. A single crystal is placed in a beam of neutrons produced at a nuclear reactor or at a proton accelerator-based spallation source. Single-crystal diffraction measurements are commonly made at thermal neutron beam energies, which correspond to neutron wavelengths in the neighborhood of 1 Angstrom. For high-resolution studies requiring shorter wavelengths (ca. 0.3-0.8 Angstroms), a pulsed spallation source or a high-temperature moderator (a ''hot source'') at a reactor may be used. When complex structures with large unit-cell repeats are under investigation, as is the case in structural biology, a cryogenic-temperature moderator (a ''cold source'') may be employed to obtain longer neutron wavelengths (ca. 4-10 Angstroms). A single-crystal neutron diffraction analysis will determine the crystal structure of the material, typically including its unit cell and space group, the positions of the atomic nuclei and their mean-square displacements, and relevant site occupancies. Because the neutron possesses a magnetic moment, the magnetic structure of the material can be determined as well, from the magnetic contribution to the Bragg intensities. This latter aspect falls beyond the scope of the present unit; for information on magnetic scattering of neutrons see Unit 14.3. Instruments for single-crystal diffraction (single-crystal diffractometers or SCDs) are generally available at the major neutron scattering center facilities. Beam time on many of these instruments is available through a proposal mechanism. A listing of neutron SCD instruments and their corresponding facility contacts is included in an appendix accompanying this unit.
16. Optical polarizing neutron devices designed for pulsed neutron sources
SciTech Connect
Takeda, M.; Kurahashi, K.; Endoh, Y.; Itoh, S.
1997-09-01
We have designed two polarizing neutron devices for pulsed cold neutrons. The devices have been tested at the pulsed neutron source at the Booster Synchrotron Utilization Facility of the National Laboratory for High Energy Physics. These two devices proved to have a practical use for experiments to investigate condensed matter physics using pulsed cold polarized neutrons.
17. Gadolinium-doped water cerenkov-based neutron and high energy gamma-ray detector and radiation portal monitoring system
DOEpatents
Dazeley, Steven A; Svoboda, Robert C; Bernstein, Adam; Bowden, Nathaniel
2013-02-12
A water Cerenkov-based neutron and high energy gamma ray detector and radiation portal monitoring system using water doped with a Gadolinium (Gd)-based compound as the Cerenkov radiator. An optically opaque enclosure is provided surrounding a detection chamber filled with the Cerenkov radiator, and photomultipliers are optically connected to the detect Cerenkov radiation generated by the Cerenkov radiator from incident high energy gamma rays or gamma rays induced by neutron capture on the Gd of incident neutrons from a fission source. The PMT signals are then used to determine time correlations indicative of neutron multiplicity events characteristic of a fission source.
18. A novel laser-collider used to produce monoenergetic 13.3 MeV 7Li (d, n) neutrons
PubMed Central
Zhao, J. R.; Zhang, X. P.; Yuan, D. W.; Li, Y. T.; Li, D. Z.; Rhee, Y. J.; Zhang, Z.; Li, F.; Zhu, B. J.; Li, Yan F.; Han, B.; Liu, C.; Ma, Y.; Li, Yi F.; Tao, M. Z.; Li, M. H.; Guo, X.; Huang, X. G.; Fu, S. Z.; Zhu, J. Q.; Zhao, G.; Chen, L. M.; Fu, C. B.; Zhang, J.
2016-01-01
Neutron energy is directly correlated with the energy of the incident ions in experiments involving laser-driven nuclear reactions. Using high-energy incident ions reduces the energy concentration of the generated neutrons. A novel “laser-collider” method was used at the Shenguang II laser facility to produce monoenergetic neutrons via 7Li (d, n) nuclear reactions. The specially designed K-shaped target significantly increased the numbers of incident d and Li ions at the keV level. Ultimately, 13.3 MeV neutrons were obtained. Considering the time resolution of the neutron detector, we demonstrated that the produced neutrons were monoenergetic. Interferometry and a Multi hydro-dynamics simulation confirmed the monoenergetic nature of these neutrons. PMID:27250660
19. A novel laser-collider used to produce monoenergetic 13.3 MeV (7)Li (d, n) neutrons.
PubMed
Zhao, J R; Zhang, X P; Yuan, D W; Li, Y T; Li, D Z; Rhee, Y J; Zhang, Z; Li, F; Zhu, B J; Li, Yan F; Han, B; Liu, C; Ma, Y; Li, Yi F; Tao, M Z; Li, M H; Guo, X; Huang, X G; Fu, S Z; Zhu, J Q; Zhao, G; Chen, L M; Fu, C B; Zhang, J
2016-06-02
Neutron energy is directly correlated with the energy of the incident ions in experiments involving laser-driven nuclear reactions. Using high-energy incident ions reduces the energy concentration of the generated neutrons. A novel "laser-collider" method was used at the Shenguang II laser facility to produce monoenergetic neutrons via (7)Li (d, n) nuclear reactions. The specially designed K-shaped target significantly increased the numbers of incident d and Li ions at the keV level. Ultimately, 13.3 MeV neutrons were obtained. Considering the time resolution of the neutron detector, we demonstrated that the produced neutrons were monoenergetic. Interferometry and a Multi hydro-dynamics simulation confirmed the monoenergetic nature of these neutrons.
20. A novel laser-collider used to produce monoenergetic 13.3 MeV 7Li (d, n) neutrons
Zhao, J. R.; Zhang, X. P.; Yuan, D. W.; Li, Y. T.; Li, D. Z.; Rhee, Y. J.; Zhang, Z.; Li, F.; Zhu, B. J.; Li, Yan F.; Han, B.; Liu, C.; Ma, Y.; Li, Yi F.; Tao, M. Z.; Li, M. H.; Guo, X.; Huang, X. G.; Fu, S. Z.; Zhu, J. Q.; Zhao, G.; Chen, L. M.; Fu, C. B.; Zhang, J.
2016-06-01
Neutron energy is directly correlated with the energy of the incident ions in experiments involving laser-driven nuclear reactions. Using high-energy incident ions reduces the energy concentration of the generated neutrons. A novel “laser-collider” method was used at the Shenguang II laser facility to produce monoenergetic neutrons via 7Li (d, n) nuclear reactions. The specially designed K-shaped target significantly increased the numbers of incident d and Li ions at the keV level. Ultimately, 13.3 MeV neutrons were obtained. Considering the time resolution of the neutron detector, we demonstrated that the produced neutrons were monoenergetic. Interferometry and a Multi hydro-dynamics simulation confirmed the monoenergetic nature of these neutrons.
1. DETECTORS AND EXPERIMENTAL METHODS: Study on spatial resolution of micromegas as a neutron detector under condition of high neutron flux and γ ray background
Wang, Wen-Xin; Zhang, Yi; Wang, Ji-Jin; Hu, Bi-Tao
2009-02-01
In this paper Micromegas has been designed to detect neutrons. The simulation of the spatial resolution of Micromegas as neutron detector is carried out by GEANT4 toolkit. The neutron track reconstruction method based on the time coincidence technology is employed in the present work. The influence of the flux of incident 14 MeV neutron and high gamma background on the spatial resolution is carefully studied. Our results show that the spatial resolution of the detector is sensitive to the neutron flux, but insensitive to the intensity of γ background if the neutron track reconstruction method proposed by our group is used. The γ insensitivity makes it possible for us to use the Micromegas detector under condition which has high γ-rays background.
2. Rhodium self-powered neutron detector as a suitable on-line thermal neutron flux monitor in BNCT treatments
SciTech Connect
Miller, Marcelo E.; Sztejnberg, Manuel L.; Gonzalez, Sara J.; Thorp, Silvia I.; Longhino, Juan M.; Estryk, Guillermo
2011-12-15
Purpose: A rhodium self-powered neutron detector (Rh SPND) has been specifically developed by the Comision Nacional de Energia Atomica (CNEA) of Argentina to measure locally and in real time thermal neutron fluxes in patients treated with boron neutron capture therapy (BNCT). In this work, the thermal and epithermal neutron response of the Rh SPND was evaluated by studying the detector response to two different reactor spectra. In addition, during clinical trials of the BNCT Project of the CNEA, on-line neutron flux measurements using the specially designed detector were assessed. Methods: The first calibration of the detector was done with the well-thermalized neutron spectrum of the CNEA RA-3 reactor thermal column. For this purpose, the reactor spectrum was approximated by a Maxwell-Boltzmann distribution in the thermal energy range. The second calibration was done at different positions along the central axis of a water-filled cylindrical phantom, placed in the mixed thermal-epithermal neutron beam of CNEA RA-6 reactor. In this latter case, the RA-6 neutron spectrum had been well characterized by both calculation and measurement, and it presented some marked differences with the ideal spectrum considered for SPND calibrations at RA-3. In addition, the RA-6 neutron spectrum varied with depth in the water phantom and thus the percentage of the epithermal contribution to the total neutron flux changed at each measurement location. Local (one point-position) and global (several points-positions) and thermal and mixed-field thermal neutron sensitivities were determined from these measurements. Thermal neutron flux was also measured during BNCT clinical trials within the irradiation fields incident on the patients. In order to achieve this, the detector was placed on patient's skin at dosimetric reference points for each one of the fields. System stability was adequate for this kind of measurement. Results: Local mixed-field thermal neutron sensitivities and global
3. Rhodium self-powered neutron detector as a suitable on-line thermal neutron flux monitor in BNCT treatments.
PubMed
Miller, Marcelo E; Sztejnberg, Manuel L; González, Sara J; Thorp, Silvia I; Longhino, Juan M; Estryk, Guillermo
2011-12-01
A rhodium self-powered neutron detector (Rh SPND) has been specifically developed by the Comisión Nacional de Energía Atómica (CNEA) of Argentina to measure locally and in real time thermal neutron fluxes in patients treated with boron neutron capture therapy (BNCT). In this work, the thermal and epithermal neutron response of the Rh SPND was evaluated by studying the detector response to two different reactor spectra. In addition, during clinical trials of the BNCT Project of the CNEA, on-line neutron flux measurements using the specially designed detector were assessed. The first calibration of the detector was done with the well-thermalized neutron spectrum of the CNEA RA-3 reactor thermal column. For this purpose, the reactor spectrum was approximated by a Maxwell-Boltzmann distribution in the thermal energy range. The second calibration was done at different positions along the central axis of a water-filled cylindrical phantom, placed in the mixed thermal-epithermal neutron beam of CNEA RA-6 reactor. In this latter case, the RA-6 neutron spectrum had been well characterized by both calculation and measurement, and it presented some marked differences with the ideal spectrum considered for SPND calibrations at RA-3. In addition, the RA-6 neutron spectrum varied with depth in the water phantom and thus the percentage of the epithermal contribution to the total neutron flux changed at each measurement location. Local (one point-position) and global (several points-positions) and thermal and mixed-field thermal neutron sensitivities were determined from these measurements. Thermal neutron flux was also measured during BNCT clinical trials within the irradiation fields incident on the patients. In order to achieve this, the detector was placed on patient's skin at dosimetric reference points for each one of the fields. System stability was adequate for this kind of measurement. Local mixed-field thermal neutron sensitivities and global thermal and mixed
4. Effect of wall thickness on measurement of dose for high energy neutrons.
PubMed
Perez-Nunez, Delia; Braby, Leslie A
2010-01-01
Neutrons produced from the interaction between galactic cosmic rays and spacecraft materials are responsible for a very important portion of the dose received by astronauts. The neutron energy spectrum depends on the incident charged particle spectrum and the scattering environment but generally extends to beyond 100 MeV. Tissue-equivalent proportional counters (TEPC) are used to measure the dose during the space mission, but their weight and size are very important factors for their design and construction. To achieve ideal neutron dosimetry, the wall thickness should be at least the range of a proton having the maximum energy of the neutrons to be monitored. This proton range is 0.1 cm for 10 MeV neutrons and 7.6 cm for 100 MeV neutrons. A 7.6 cm wall thickness TEPC would provide charged particle equilibrium (CPE) for neutrons up to 100 MeV, but for space applications it would not be reasonable in terms of weight and size. In order to estimate the errors in measured dose due to absence of CPE, MCNPX simulations of energy deposited by 10 MeV and 100 MeV neutrons in sites with wall thickness between 0.1 cm and 8.5 cm were performed. The results for 100 MeV neutrons show that energy deposition per incident neutron approaches a plateau as the wall thickness approaches 7.6 cm. For the 10 MeV neutrons, energy deposition per incident neutron decreases as the wall thickness increases above 0.1 cm due to attenuation.
5. Comparison of DD, DT and Cf-252 neutron excitation of light and medium mass nuclei for field PGNAA applications
Seabury, E. H.; Blackburn, B. W.; Chichester, D. L.; Wharton, C. J.; Caffrey, A. J.
2007-08-01
Prompt Gamma Ray Neutron activation analysis can offer significant cost and safety advantages in the identification of explosives and toxic chemicals. As an example, the US military examined over a thousand suspect chemical munitions with Idaho National Laboratory's PINS Chemical Assay System last year. PGNAA requires, of course, a neutron source to excite the atomic nuclei of the item under test via neutron capture and inelastic neutron scattering reactions and the choice of neutron source can drastically affect PGNAA system performance. We have carried out Monte Carlo and laboratory experiments comparing DD, DT and Cf-252 neutrons incident on light and medium mass chemical elements, toward optimizing the design of future neutron-generator-based PGNAA systems for field use. We report the excitation of (n, γ) and (n, n‧) gamma rays from these elements by each type of neutron source.
6. Spectral Performance of a Composite Single-Crystal Filtered Thermal Neutron Beam for BNCT Research at the University of Missouri
SciTech Connect
J. Brockman; D. W. Nigg; M. F. Hawthorne; C. McKibben
2009-07-01
Parameter studies, design calculations and initial neutronic performance measurements have been completed for a new thermal neutron beamline to be used for neutron capture therapy cell and small-animal radiobiology studies at the University of Missouri Research Reactor. The beamline features the use of single-crystal silicon and bismuth sections for neutron filtering and for reduction of incident gamma radiation. The calculated and measured thermal neutron fluxes produced at the irradiation location are 9.6x108 and 8.8x108 neutrons/cm2-s, respectively. Calculated and measured cadmium ratios (Au foils) are 217 and 132. These results indicate a well-thermalized neutron spectrum with sufficient thermal neutron flux for a variety of small animal BNCT studies.
7. Critical incident reporting systems.
PubMed
Ahluwalia, Jag; Marriott, Lin
2005-02-01
Approximately 10% of all hospital admissions are complicated by critical incidents in which harm is caused to the patient - this amounts to more than 850,000 incidents annually. Critical incident reporting (CIR) systems refer to the structured reporting, collation and analysis of such incidents. This article describes the attributes required for an effective CIR system. Example neonatal trigger events and a management pathway for handling a critical incident report are described. The benefits and limitations of CIR systems, reactive and prospective approaches to the analysis of actual or potential critical incidents and the assessment of risk are also reviewed. Individual human error is but one contributor in the majority of critical incidents. Recognition of this and the fostering of an organisational culture that views critical incident reports as an opportunity to learn and to improve future patient care is vital if CIR systems are to be effective.
8. Neutron energy-dependent initial DNA damage and chromosomal exchange.
PubMed
Tanaka, K; Gajendiran, N; Endo, S; Komatsu, K; Hoshi, M; Kamada, N
1999-12-01
This study was undertaken to investigate the biological effect of monoenergetic neutrons on human lymphocyte DNA and chromosomes. Monoenergetic neutrons of 2.3, 1.0, 0.79, 0.57, 0.37 and 0.186 MeV were generated, and 252Cf neutrons and 60Co gamma-rays were also used for comparison. Biological effect was evaluated two ways. The RBE values with the comet assay were estimated as 6.3 and 5.4 at 0.37 MeV and 0.57 MeV relative to that of 60Co gamma-rays, and chromosome aberration rates were also observed in these different levels of monoenergetic neutrons. The yield of chromosome aberrations per unit dose was high at lower neutron energies with a gradual decline with 0.186 MeV neutron energy. The RBE was increased to 10.7 at 0.57 MeV from 3.9 at 252Cf neutrons and reached 16.4 as the highest RBE at 0.37 MeV, but the value decreased to 11.2 at 0.186 MeV. The response patterns of initial DNA damage and chromosome exchange were quite similar to that of LET. These results show that the intensity of DNA damage and chromosomal exchange is LET dependent. RBE of low energy neutrons is higher than that of fission neutrons. Low energy neutrons containing Hiroshima atomic bomb radiation may have created a significantly higher incidence of biological effect in atomic bomb survivors.
9. Neutron-neutron correlations in ^6He
Atramentov, Oleksiy; Vary, James; Navrátil, Petr
2004-05-01
We evaluate 2-neutron correlations in ^6He within the ab initio no-core nuclear shell model using a realistic effective 2-body Hamiltonian. The distribution of relative separation between neutrons, including spin dependence, is obtained for the ground, and lowest 2^+ and 1^- states. We compare angular distributions and relative kinetic energy distributions of neutron-pairs in these ^6He states directly with 2-neutron correlation measurements from dissociation experiments (Aumann, Phys. Rev. C 59, 1252 (1999); Wang, Phys. Rev. C 65,034306 (2002)). The reasonable agreement between these theoretical 1^- and experimental distributions supports an interpretation of the experiment as a simple process: Coulomb excitation of the 1^- followed by dissociation without significant final state interactions. We argue that such a simple picture may be reasonable for the breakup of this halo nucleus. We present predictions for additional spin-dependent correlation experiments that will sensitively test this simple picture. Work supported in part by USDOE grant DE-FG02-87ER40371 and was performed, in part, under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.
SciTech Connect
Wietfeldt, Fred E.; Greene, Geoffrey L.
2011-10-01
The decay of the free neutron into a proton, electron, and antineutrino is the prototype semileptonic weak decay and is the simplest example of nuclear beta decay. It played a key role in the early Universe as it determined the ratio of neutrons to protons during the era of primordial light element nucleosynthesis. Neutron decay is physically related to important processes in solar physics and neutrino detection. The mean neutron lifetime has been the subject of more than 20 major experiments done, using a variety of methods, between 1950 and the present. The most precise recent measurements have stated accuracies approaching 0.1%, but are not in good agreement as they differ by as much as 5{sigma} using quoted uncertainties. The history of neutron lifetime measurements is reviewed and the different methods used are described, giving important examples of each. The discrepancies and some systematic issues in the experiments that may be responsible are discussed, and it is shown by means of global averages that the neutron lifetime is likely to lie in the range of 880-884 s. Plans and prospects for future experiments are considered that will address these systematic issues and improve our knowledge of the neutron lifetime.
SciTech Connect
Greene, Geoffrey L; Wietfeldt, F
2011-01-01
The decay of the free neutron into a proton, electron, and antineutrino is the prototype semileptonic weak decay and is the simplest example of nuclear beta decay. It played a key role in the early Universe as it determined the ratio of neutrons to protons during the era of primordial light element nucleosynthesis. Neutron decay is physically related to important processes in solar physics and neutrino detection. The mean neutron lifetime has been the subject of more than 20 major experiments done, using a variety of methods, between 1950 and the present. The most precise recent measurements have stated accuracies approaching 0.1%, but are not in good agreement as they differ by as much as 5 sigma using quoted uncertainties. The history of neutron lifetime measurements is reviewed and the different methods used are described, giving important examples of each. The discrepancies and some systematic issues in the experiments that may be responsible are discussed, and it is shown by means of global averages that the neutron lifetime is likely to lie in the range of 880 884 s. Plans and prospects for future experiments are considered that will address these systematic issues and improve our knowledge of the neutron lifetime.
12. Multigroup neutron dose calculations for proton therapy
SciTech Connect
Kelsey Iv, Charles T; Prinja, Anil K
2009-01-01
We have developed tools for the preparation of coupled multigroup proton/neutron cross section libraries. Our method is to use NJOY to process evaluated nuclear data files for incident particles below 150 MeV and MCNPX to produce data for higher energies. We modified the XSEX3 program of the MCNPX code system to produce Legendre expansions of scattering matrices generated by sampling the physics models that are comparable to the output of the GROUPR routine of NJOY. Our code combines the low and high energy scattering data with user input stopping powers and energy deposition cross sections that we also calculated using MCNPX. Our code also calculates momentum transfer coefficients for the library and optionally applies an energy straggling model to the scattering cross sections and stopping powers. The motivation was initially for deterministic solution of space radiation shielding calculations using Attila, but noting that proton therapy treatment planning may neglect secondary neutron dose assessments because of difficulty and expense, we have also investigated the feasibility of multi group methods for this application. We have shown that multigroup MCNPX solutions for secondary neutron dose compare well with continuous energy solutions and are obtainable with less than half computational cost. This efficiency comparison neglects the cost of preparing the library data, but this becomes negligible when distributed over many multi group calculations. Our deterministic calculations illustrate recognized obstacles that may have to be overcome before discrete ordinates methods can be efficient alternatives for proton therapy neutron dose calculations.
13. The neutron channeling phenomenon.
PubMed
Khanouchi, A; Sabir, A; Boulkheir, M; Ichaoui, R; Ghassoun, J; Jehouani, A
1997-01-01
Shields, used for protection against radiation, are often pierced with vacuum channels for passing cables and other instruments for measurements. The neutron transmission through these shields is an unavoidable phenomenon. In this work we study and discuss the effect of channels on neutron transmission through shields. We consider an infinite homogeneous slab, with a fixed thickness (20 lambda, with lambda the mean free path of the neutron in the slab), which contains a vacuum channel. This slab is irradiated with an infinite source of neutrons on the left side and on the other side (right side) many detectors with windows equal to 2 lambda are placed in order to evaluate the neutron transmission probabilities (Khanouchi, A., Aboubekr, A., Ghassoun, J. and Jehouani, A. (1994) Rencontre Nationale des Jeunes Chercheurs en Physique. Casa Blanca Maroc; Khanouchi, A., Sabir, A., Ghassoun, J. and Jehouani, A. (1995) Premier Congré International des Intéractions Rayonnements Matière. Eljadida Maroc). The neutron history within the slab is simulated by the Monte Carlo method (Booth, T. E. and Hendricks, J. S. (1994) Nuclear Technology 5) and using the exponential biasing technique in order to improve the Monte Carlo calculation (Levitt, L. B. (1968) Nuclear Science and Engineering 31, 500-504; Jehouani, A., Ghassoun, J. and Aboubker, A. (1994) In Proceedings of the 6th International Symposium on Radiation Physics, Rabat, Morocco). Then different geometries of the vacuum channel have been studied. For each geometry we have determined the detector response and calculated the neutron transmission probability for different detector positions. This neutron transmission probability presents a peak for the detectors placed in front of the vacuum channel. This study allowed us to clearly identify the neutron channeling phenomenon. One application of our study is to detect vacuum defects in materials.
14. Incidents of Security Concern
SciTech Connect
Atencio, Julian J.
2014-05-01
This presentation addresses incidents of security concern and an incident program for addressing them. It addresses the phases of an inquiry, and it divides incidents into categories based on severity and interest types based on whether security, management, or procedural interests are involved. A few scenarios are then analyzed according to these breakdowns.
15. Control of neutron albedo in toroidal fusion reactors
SciTech Connect
Micklich, B.J.; Jassby, D.L.
1983-07-01
The MCNP and ANISN codes have been used to obtain basic neutron albedo data for materials of interest for fusion applications. Simple physical models are presented which explain albedo dependence on pre- and post-reflection variables. The angular distribution of reflected neutrons. The energy spectra of reflected neutrons are presented, and it is shown that substantial variations in the total neutron current at the outboard wall of a torus can be effected by changing materials behind the inboard wall. Analyses show that a maximum of four isolated incident-current environments may be established simultaneously on the outboard side of a torus. With suitable inboard reflectors, global tritium breeding ratios significantly larger than unity can be produced in limited-coverage breeding blankets when the effects of outboard penetrations are included.
16. Neutron response function characterization of 4He scintillation detectors
DOE PAGES
Kelley, Ryan P.; Rolison, Lucas M.; Lewis, Jason M.; ...
2015-04-15
Time-of-flight measurements were conducted to characterize the neutron energy response of pressurized 4He fast neutron scintillation detectors for the first time, using the Van de Graaff generator at Ohio University. The time-of-flight spectra and pulse height distributions were measured. This data was used to determine the light output response function, which was found to be linear at energies below 3.5 MeV. The intrinsic efficiency of the detector as a function of incident energy was also calculated: the average efficiency up to 10 MeV was 3.1%, with a maximum efficiency of 6.6% at 1.05 MeV. Furthermore, these results will enable developmentmore » of neutron spectrum unfolding algorithms for neutron spectroscopy applications with these detectors.« less
17. Neutron diffraction optics of films with noncollinear magnetic depth structures
Korneev, Daniel A.; Chernenko, L. P.
1992-11-01
A model of a thin film with depth inhomogeneous or noncollinear magnetization is used to describe its neutron-optical, diffraction, and polarization properties. A quantum mechanical method of calculation is developed based on a numerical solution of the Pauli equation for a neutron in inhomogeneous matter with boundary conditions. This solution permits the uniform calculation of intensities and the polarization vector of the scattered beam over a wide range of conditions from specular reflection to neutron diffraction. The method is applied to certain model structures. Examples of its application to model structures (ferromagnetic spirals, long- period soliton) are given and the calculation can be programmed for a personal computer, taking a few seconds for each value of the incident neutron wave vector.
18. NEUTRONIC REACTOR CONTROL
DOEpatents
Metcalf, H.E.
1958-10-14
Methods of controlling reactors are presented. Specifically, a plurality of neutron absorber members are adjustably disposed in the reactor core at different distances from the center thereof. The absorber members extend into the core from opposite faces thereof and are operated by motive means coupled in a manner to simultaneously withdraw at least one of the absorber members while inserting one of the other absorber members. This feature effects fine control of the neutron reproduction ratio by varying the total volume of the reactor effective in developing the neutronic reaction.
19. METHOD OF PRODUCING NEUTRONS
DOEpatents
Imhoff, D.H.; Harker, W.H.
1964-01-14
This patent relates to a method of producing neutrons in which there is produced a heated plasma containing heavy hydrogen isotope ions wherein heated ions are injected and confined in an elongated axially symmetric magnetic field having at least one magnetic field gradient region. In accordance with the method herein, the amplitude of the field and gradients are varied at an oscillatory periodic frequency to effect confinement by providing proper ratios of rotational to axial velocity components in the motion of said particles. The energetic neutrons may then be used as in a blanket zone containing a moderator and a source fissionable material to produce heat and thermal neutron fissionable materials. (AEC)
20. FABRICATION OF NEUTRON SOURCES
DOEpatents
Birden, J.H.
1959-04-21
A method is presented for preparing a neutron source from polonium-210 and substances, such as beryllium and boron, characterized by emission of neutrons upon exposure to alpha particles from the polonium. According to the invention, a source is prepared by placing powdered beryllium and a platinum foil electroplated with polonium-2;.0 in a beryllium container. The container is sealed and then heated by induction to a temperature of 450 to 1100 deg C to volatilize the polonium off the foil into the powder. The heating step is terminated upon detection of a maximum in the neutron flux level.
1. Fruits of neutron research
SciTech Connect
Krause, C.
1994-12-31
Car windshields that dont break during accidents and jets that fly longer without making a refueling stop. Compact discs, credit cards, and pocket calculators. Refrigerator magnets and automatic car window openers. Beach shoes, food packaging, and bulletproof vests made of tough plastics. The quality and range of consumer products have improved steadily since the 1970s. One of the reasons: neutron research. Industries, employing neutron scattering techniques, to study materials properties, to act as diagnostics in tracing system performance, or as sources for radioactive isotopes used in medical fields for diagnostics or treatment, have all benefited from the fruits of advanced work with neutron sources.
NASA Technical Reports Server (NTRS)
Delorenzo, J. T.; Kennedy, E. J.; Blalock, T. V.; Rochelle, J. M.; Chiles, M. M.; Valentine, K. H.
1981-01-01
An experimental wireless, in-vessel neutron monitor was developed to measure the reactivity of an advanced breeder reactor as the core is loaded for the first time to preclude an accidental critically incident. The environment is liquid sodium at a temperature of approx. 220 C, with negligible gamma or neutron radiation. With ultrasonic transmission of neutron data, no fundamental limitation was observed after tests at 230 C for 2000 h. The neutron sensitivity was approx. 1 count/s-nv, and the potential data transmission rate was approx. 10,000 counts/s.
3. Progress in development of neutron energy spectrometer for deuterium plasma operation in KSTAR
SciTech Connect
Tomita, H. Yamashita, F.; Nakayama, Y.; Morishima, K.; Yamamoto, Y.; Sakai, Y.; Hayashi, S.; Kawarabayashi, J.; Iguchi, T.; Cheon, M. S.; Isobe, M.; Ogawa, K.
2014-11-15
Two types of DD neutron energy spectrometer (NES) are under development for deuterium plasma operation in KSTAR to understand behavior of beam ions in the plasma. One is based on the state-of-the-art nuclear emulsion technique. The other is based on a coincidence detection of a recoiled proton and a scattered neutron caused by an elastic scattering of an incident DD neutron, which is called an associated particle coincidence counting-NES. The prototype NES systems were installed at J-port in KSTAR in 2012. During the 2012 and 2013 experimental campaigns, multiple shots-integrated neutron spectra were preliminarily obtained by the nuclear emulsion-based NES system.
4. A neutron diagnostic for high current deuterium beams
SciTech Connect
Rebai, M.; Perelli Cippo, E.; Cavenago, M.; Dalla Palma, M.; Pasqualotto, R.; Tollin, M.; Croci, G.; Gervasini, G.; Ghezzi, F.; Grosso, G.; Tardocchi, M.; Murtas, F.; Gorini, G.
2012-02-15
A neutron diagnostic for high current deuterium beams is proposed for installation on the spectral shear interferometry for direct electric field reconstruction (SPIDER, Source for Production of Ion of Deuterium Extracted from RF plasma) test beam facility. The proposed detection system is called Close-contact Neutron Emission Surface Mapping (CNESM). The diagnostic aims at providing the map of the neutron emission on the beam dump surface by placing a detector in close contact, right behind the dump. CNESM uses gas electron multiplier detectors equipped with a cathode that also serves as neutron-proton converter foil. The cathode is made of a thin polythene film and an aluminium film; it is designed for detection of neutrons of energy >2.2 MeV with an incidence angle < 45 deg. CNESM was designed on the basis of simulations of the different steps from the deuteron beam interaction with the beam dump to the neutron detection in the nGEM. Neutron scattering was simulated with the MCNPX code. CNESM on SPIDER is a first step towards the application of this diagnostic technique to the MITICA beam test facility, where it will be used to resolve the horizontal profile of the beam intensity.
5. Gamma discrimination in pillar structured thermal neutron detectors
SciTech Connect
Shao, Q; Radev, R P; Conway, A M; Voss, L F; Wang, T F; Nikolic, R J; Deo, N; Cheung, C L
2012-03-26
Solid-state thermal neutron detectors are desired to replace {sup 3}He tube based technology for the detection of special nuclear materials. {sup 3}He tubes have some issues with stability, sensitivity to microphonics and very recently, a shortage of {sup 3}He. There are numerous solid-state approaches being investigated that utilize various architectures and material combinations. By using the combination of high-aspect-ratio silicon PIN pillars, which are 2 {micro}m wide with a 2 {micro}m separation, arranged in a square matrix, and surrounded by {sup 10}B, the neutron converter material, a high efficiency thermal neutron detector is possible. Besides intrinsic neutron detection efficiency, neutron to gamma discrimination is an important figure of merit for unambiguous signal identification. In this work, theoretical calculations and experimental measurements are conducted to determine the effect of structure design of pillar structured thermal neutron detectors including: intrinsic layer thickness, pillar height, substrate doping and incident gamma energy on neutron to gamma discrimination.
6. The plain truth about forming a plane wave of neutrons
Wagh, Apoorva G.; Abbas, Sohrab; Treimer, Wolfgang
2011-04-01
We have attained the first sub-arcsecond collimation of a monochromatic neutron beam by diffracting neutrons from a Bragg prism, viz. a single crystal prism operating in the vicinity of Bragg incidence. Analytical as well numerical computations based on the dynamical diffraction theory, led to the optimised collimator configuration of a silicon {1 1 1} Bragg prism for 5.26 Å neutrons. We fabricated a Bragg prism to these specifications, tested and operated it at the double diffractometer setup in Helmholtz Zentrum Berlin to produce a 0.58 arcsec wide monochromatic neutron beam. With a similarly optimised Bragg prism analyser of opposite asymmetry, we recorded a 0.62 arcsec wide virgin rocking curve for this ultra-parallel beam. With this nearly plane-wave neutron beam, we have recorded the first ever USANS spectrum in Q˜10-6 Å-1 range with a hydroxyapatite casein protein sample and demonstrated the instrument capability to characterise agglomerates up to 150 μm in size. The super-collimated monochromatic beam has also enabled us to record the first neutron diffraction pattern from a macroscopic grating of 200 μm period. The transverse coherence length of 175 μm (FWHM) of the ultra-parallel beam derived from the analysis of this pattern, is the greatest achieved to date for Å wavelength neutrons.
7. Improved fission neutron energy discrimination with 4He detectors through pulse filtering
Zhu, Ting; Liang, Yinong; Rolison, Lucas; Barker, Cathleen; Lewis, Jason; Gokhale, Sasmit; Chandra, Rico; Kiff, Scott; Chung, Heejun; Ray, Heather; Baciak, James E.; Enqvist, Andreas; Jordan, Kelly A.
2017-03-01
This paper presents experimental and computational techniques implemented for 4He gas scintillation detectors for induced fission neutron detection. Fission neutrons are produced when natural uranium samples are actively interrogated by 2.45 MeV deuterium-deuterium fusion reaction neutrons. Fission neutrons of energies greater than 2.45 MeV can be distinguished by their different scintillation pulse height spectra since 4He detectors retain incident fast neutron energy information. To enable the preferential detection of fast neutrons up to 10 MeV and suppress low-energy event counts, the detector photomultiplier gain is lowered and trigger threshold is increased. Pile-up and other unreliable events due to the interrogating neutron flux and background radiation are filtered out prior to the evaluation of pulse height spectra. With these problem-specific calibrations and data processing, the 4He detector's accuracy at discriminating fission neutrons up to 10 MeV is improved and verified with 252Cf spontaneous fission neutrons. Given the 4He detector's ability to differentiate fast neutron sources, this proof-of-concept active-interrogation measurement demonstrates the potential of special nuclear materials detection using a 4He fast neutron detection system.
8. Switchable radioactive neutron source device
DOEpatents
Stanford, G.S.; Rhodes, E.A.; Devolpi, A.; Boyar, R.E.
1987-11-06
This invention is a switchable neutron generating apparatus comprised of a pair of plates, the first plate having an alpha emitter section on it and the second plate having a target material portion on it which generates neutrons when its nuclei absorb an alpha particle. In operation, the alpha portion of the first plate is aligned with the neutron portion of the second plate to produce neutrons and brought out of alignment to cease production of neutrons. 3 figs.
9. Switchable radioactive neutron source device
DOEpatents
Boyar, Robert E.; DeVolpi, Alexander; Stanford, George S.; Rhodes, Edgar A.
1989-01-01
This invention is a switchable neutron generating apparatus comprised of a pair of plates, the first plate having an alpha emitter section on it and the second plate having a target material portion on it which generates neutrons when its nuclei absorb an alpha particle. In operation, the alpha portion of the first plate is aligned with the neutron portion of the second plate to produce neutrons and brought out of alignment to cease production of neutrons.
10. Isotope-Identifying neutron reflectometry
SciTech Connect
Nikitenko, Yu. V. Petrenko, A. V.; Gundorin, N. A.; Gledenov, Yu. M.; Aksenov, V. L.
2015-07-15
The possibilities of an isotope-indentifying study of layered structures in different regimes of a neutron wave field are considered. The detection of specularly reflected neutrons and secondary radiation (caused by neutron capture) in the form of charged particles, γ quanta, and nuclear fission fragments, as well as neutrons spin-flipped in a noncollinear magnetic field and on nuclei of elements with spin, makes it possible to implement isotope-indentifying neutron reflectometry.
11. CORRECTIONS ASSOCIATED WITH ON-PHANTOM CALIBRATIONS OF NEUTRON PERSONAL DOSEMETERS.
PubMed
Hawkes, N P; Thomas, D J; Taylor, G C
2016-09-01
The response of neutron personal dosemeters as a function of neutron energy and angle of incidence is typically measured by mounting the dosemeters on a slab phantom and exposing them to neutrons from an accelerator-based or radionuclide source. The phantom is placed close to the source (75 cm) so that the effect of scattered neutrons is negligible. It is usual to mount several dosemeters on the phantom together. Because the source is close, the source distance and the neutron incidence angle vary significantly over the phantom face, and each dosemeter may receive a different dose equivalent. This is particularly important when the phantom is angled away from normal incidence. With accelerator-produced neutrons, the neutron energy and fluence vary with emission angle relative to the charged particle beam that produces the neutrons, contributing further to differences in dose equivalent, particularly when the phantom is located at other than the straight-ahead position (0° to the beam). Corrections for these effects are quantified and discussed in this article. © Crown copyright 2015.
12. Determination of the response function for two personal neutron dosemeter designs based on PADC.
PubMed
Mayer, S; Assenmacher, F; Boschung, M
2014-10-01
Since 1998 neutron dosimetry based on PADC (poly allyl diglycol carbonate) is done with a so-called original Paul Scherrer Institute (PSI) design at PSI. The original design (i.e. holder) was later changed. Both designs are optimised for use in workplaces around high-energy accelerators, where the neutron energy spectra are dominated by fast neutrons ranging up to some 100 MeV. In addition to the change of the dosemeter design a new evaluation method based on a microscope scanning technique has been introduced and the etching conditions have been optimised. In the present work, the responses obtained with the original and the new dosemeter designs are compared for fields of radionuclide sources and monoenergetic reference fields using the new evaluation method. The response curves in terms of the personal dose equivalent for normally incident neutrons were built as functions of the incident neutron energy.
13. Prediction of In-Phantom Dose Distribution Using In-Air Neutron Beam Characteristics for Boron Neutron Capture Synovectomy
SciTech Connect
Verbeke, Jerome M.; Chen, Allen S.; Vujic, Jasmina L.; Leung, Ka-Ngo
2000-08-15
A monoenergetic neutron beam simulation study was carried out to determine the optimal neutron energy range for treatment of rheumatoid arthritis using radiation synovectomy. The goal of the treatment is the ablation of diseased synovial membranes in joints such as knees and fingers. This study focuses on human knee joints. Two figures of merit are used to measure the neutron beam quality, the ratio of the synovium-absorbed dose to the skin-absorbed dose, and the ratio of the synovium-absorbed dose to the bone-absorbed dose. It was found that (a) thermal neutron beams are optimal for treatment and that (b) similar absorbed dose rates and therapeutic ratios are obtained with monodirectional and isotropic neutron beams. Computation of the dose distribution in a human knee requires the simulation of particle transport from the neutron source to the knee phantom through the moderator. A method was developed to predict the dose distribution in a knee phantom from any neutron and photon beam spectra incident on the knee. This method was revealed to be reasonably accurate and enabled one to reduce the particle transport simulation time by a factor of 10 by modeling the moderator only.
14. Neutron focusing system for the Texas Cold Neutron Source
Wehring, Bernard W.; Kim, Jong-Youl; Ünlü, Kenan
1994-12-01
A "converging neutron guide" focusing system located at the end of the Texas Cold Neutron Source (TCNS) "curved neutron guide" would increase the neutron flux for neutron capture experiments. Our design for a converging guide is based on using several rectangular truncated cone sections. Each rectangular truncated cone consists of four 20-cm long Si plates coated with NiC-Ti supermirrors. Dimensions of each section were determined by a three-dimensional Monte Carlo optimization calculation. The two slant angles of the truncated cones were varied to optimize the neutron flux at the focal area of the focusing system. Different multielement converging guides were designed and their performance analyzed. From the performance results and financial considerations, we selected a four-section 80-cm long converging guide focusing system for construction and use with the TCNS. The focused cold neutron beam will be used for neutron capture experiment, e.g., prompt gamma activation analysis and neutron depth profiling.
15. Cylindrical neutron generator
DOEpatents
Leung, Ka-Ngo
2005-06-14
A cylindrical neutron generator is formed with a coaxial RF-driven plasma ion source and target. A deuterium (or deuterium and tritium) plasma is produced by RF excitation in a cylindrical plasma ion generator using an RF antenna. A cylindrical neutron generating target is coaxial with the ion generator, separated by plasma and extraction electrodes which contain many slots. The plasma generator emanates ions radially over 360.degree. and the cylindrical target is thus irradiated by ions over its entire circumference. The plasma generator and target may be as long as desired. The plasma generator may be in the center and the neutron target on the outside, or the plasma generator may be on the outside and the target on the inside. In a nested configuration, several concentric targets and plasma generating regions are nested to increase the neutron flux.
16. Cylindrical neutron generator
DOEpatents
Leung, Ka-Ngo [Hercules, CA
2008-04-22
A cylindrical neutron generator is formed with a coaxial RF-driven plasma ion source and target. A deuterium (or deuterium and tritium) plasma is produced by RF excitation in a cylindrical plasma ion generator using an RF antenna. A cylindrical neutron generating target is coaxial with the ion generator, separated by plasma and extraction electrodes which contain many slots. The plasma generator emanates ions radially over 360.degree. and the cylindrical target is thus irradiated by ions over its entire circumference. The plasma generator and target may be as long as desired. The plasma generator may be in the center and the neutron target on the outside, or the plasma generator may be on the outside and the target on the inside. In a nested configuration, several concentric targets and plasma generating regions are nested to increase the neutron flux.
17. Cylindrical neutron generator
DOEpatents
Leung, Ka-Ngo
2009-12-29
A cylindrical neutron generator is formed with a coaxial RF-driven plasma ion source and target. A deuterium (or deuterium and tritium) plasma is produced by RF excitation in a cylindrical plasma ion generator using an RF antenna. A cylindrical neutron generating target is coaxial with the ion generator, separated by plasma and extraction electrodes which contain many slots. The plasma generator emanates ions radially over 360.degree. and the cylindrical target is thus irradiated by ions over its entire circumference. The plasma generator and target may be as long as desired. The plasma generator may be in the center and the neutron target on the outside, or the plasma generator may be on the outside and the target on the inside. In a nested configuration, several concentric targets and plasma generating regions are nested to increase the neutron flux.
18. Cooling of neutron stars
NASA Technical Reports Server (NTRS)
Pethick, C. J.
1992-01-01
It is at present impossible to predict the interior constitution of neutron stars based on theory and results from laboratory studies. It has been proposed that it is possible to obtain information on neutron star interiors by studying thermal radiation from their surfaces, because neutrino emission rates, and hence the temperature of the central part of a neutron star, depend on the properties of dense matter. The theory predicts that neutron stars cool relatively slowly if their cores are made up of nucleons, and cool faster if the matter is in an exotic state, such as a pion condensate, a kaon condensate, or quark matter. This view has recently been questioned by the discovery of a number of other processes that could lead to copious neutrino emission and rapid cooling.
19. Global Neutron View
NASA Image and Video Library
2002-03-01
In this image taken by NASA Mars Odyssey spacecraft during its first week of mapping, soil enriched in hydrogen is indicated by the deep blue colors, which show a low intensity of epithermal neutrons.
20. Shifting scintillator neutron detector
DOEpatents
Clonts, Lloyd G; Cooper, Ronald G; Crow, Jr., Morris Lowell; Hannah, Bruce W; Hodges, Jason P; Richards, John D; Riedel, Richard A
2014-03-04
Provided are sensors and methods for detecting thermal neutrons. Provided is an apparatus having a scintillator for absorbing a neutron, the scintillator having a back side for discharging a scintillation light of a first wavelength in response to the absorbed neutron, an array of wavelength-shifting fibers proximate to the back side of the scintillator for shifting the scintillation light of the first wavelength to light of a second wavelength, the wavelength-shifting fibers being disposed in a two-dimensional pattern and defining a plurality of scattering plane pixels where the wavelength-shifting fibers overlap, a plurality of photomultiplier tubes, in coded optical communication with the wavelength-shifting fibers, for converting the light of the second wavelength to an electronic signal, and a processor for processing the electronic signal to identify one of the plurality of scattering plane pixels as indicative of a position within the scintillator where the neutron was absorbed.
1. Cooling of neutron stars
NASA Technical Reports Server (NTRS)
Pethick, C. J.
1992-01-01
It is at present impossible to predict the interior constitution of neutron stars based on theory and results from laboratory studies. It has been proposed that it is possible to obtain information on neutron star interiors by studying thermal radiation from their surfaces, because neutrino emission rates, and hence the temperature of the central part of a neutron star, depend on the properties of dense matter. The theory predicts that neutron stars cool relatively slowly if their cores are made up of nucleons, and cool faster if the matter is in an exotic state, such as a pion condensate, a kaon condensate, or quark matter. This view has recently been questioned by the discovery of a number of other processes that could lead to copious neutrino emission and rapid cooling.
2. Effective neutron targets
SciTech Connect
Gao, H.
1997-07-01
Because of the lack of a free neutron target, deuterium targets have been used extensively in studying the neutron structure. The unique spin structure of the {sup 3}He ground state wave function and the recent developments in laser technologies made polarized {sup 3}He targets widely used in many experiments from neutron electromagnetic form factor studies to nucleon spin structure function measurements at all major electron accelerator facilities. In this talk, the current status of the polarized {sup 3}He targets will be reviewed. The author will focus on neutron electromagnetic form factor studies using polarized {sup 3}He targets. The polarized nucleon spin structure function measurements using polarized {sup 3}He targets will also be discussed.
3. NEUTRONIC REACTOR FUEL COMPOSITION
DOEpatents
Thurber, W.C.
1961-01-10
Uranium-aluminum alloys in which boron is homogeneously dispersed by adding it as a nickel boride are described. These compositions have particular utility as fuels for neutronic reactors, boron being present as a burnable poison.
4. Neutron resonance averaging
SciTech Connect
Chrien, R.E.
1986-10-01
The principles of resonance averaging as applied to neutron capture reactions are described. Several illustrations of resonance averaging to problems of nuclear structure and the distribution of radiative strength in nuclei are provided. 30 refs., 12 figs.
5. Neutron personnel dosimetry
SciTech Connect
Griffith, R.V.
1981-06-16
The current state-of-the-art in neutron personnel dosimetry is reviewed. Topics covered include dosimetry needs and alternatives, current dosimetry approaches, personnel monitoring devices, calibration strategies, and future developments. (ACR)
6. Neoplasia in fast neutron-irradiated beagles
SciTech Connect
Bradley, E.W.; Zook; B.C.; Casarett, G.W.
1981-09-01
One hundred fifty-one beagle dogs were irradiated with either photons or fast neutrons (15 MeV) to one of three dose-limiting normal tissues - spinal cord, lung, or brain. The radiation was given in four fractions per week for 5 weeks (spinal cord), 6 weeks (lung), 7 weeks (brain) to total doses encompassing those given clinically for cancer management. To date, no nonirradiated dogs or photon-irradiated dogs have developed neoplasms within the irradiated field. Of the neutron-irradiated dogs at risk, the incidence of neoplasia was 15%. The latent period for radiation-induced cancers has varied from 1 to 4 1/2 years at this time in the study.
7. Plastic neutron detectors.
SciTech Connect
Wilson, Tiffany M.S; King, Michael J.; Doty, F. Patrick
2008-12-01
This work demonstrated the feasibility and limitations of semiconducting {pi}-conjugated organic polymers for fast neutron detection via n-p elastic scattering. Charge collection in conjugated polymers in the family of substituted poly(p-phenylene vinylene)s (PPV) was evaluated using band-edge laser and proton beam ionization. These semiconducting materials can have high H/C ratio, wide bandgap, high resistivity and high dielectric strength, allowing high field operation with low leakage current and capacitance noise. The materials can also be solution cast, allowing possible low-cost radiation detector fabrication and scale-up. However, improvements in charge collection efficiency are necessary in order to achieve single particle detection with a reasonable sensitivity. The work examined processing variables, additives and environmental effects. Proton beam exposure was used to verify particle sensitivity and radiation hardness to a total exposure of approximately 1 MRAD. Conductivity exhibited sensitivity to temperature and humidity. The effects of molecular ordering were investigated in stretched films, and FTIR was used to quantify the order in films using the Hermans orientation function. The photoconductive response approximately doubled for stretch-aligned films with the stretch direction parallel to the electric field direction, when compared to as-cast films. The response was decreased when the stretch direction was orthogonal to the electric field. Stretch-aligned films also exhibited a significant sensitivity to the polarization of the laser excitation, whereas drop-cast films showed none, indicating improved mobility along the backbone, but poor {pi}-overlap in the orthogonal direction. Drop-cast composites of PPV with substituted fullerenes showed approximately a two order of magnitude increase in photoresponse, nearly independent of nanoparticle concentration. Interestingly, stretch-aligned composite films showed a substantial decrease in
8. Pulsed spallation neutron sources
SciTech Connect
Carpenter, J.M.
1996-05-01
This paper reviews the early history of pulsed spallation neutron source development ar Argonne and provides an overview of existing sources world wide. A number of proposals for machines more powerful than currently exist are under development, which are briefly described. The author reviews the status of the Intense Pulsed Neutron Source, its instrumentation, and its user program, and provide a few examples of applications in fundamental condensed matter physics, materials science and technology.
9. Pulsed spallation Neutron Sources
SciTech Connect
Carpenter, J.M.
1994-12-31
This paper reviews the early history of pulsed spallation neutron source development at Argonne and provides an overview of existing sources world wide. A number of proposals for machines more powerful than currently exist are under development, which are briefly described. The author reviews the status of the Intense Pulsed Neutron Source, its instrumentation, and its user program, and provides a few examples of applications in fundamental condensed matter physics, materials science and technology.
10. NEUTRON FLUX INTENSITY DETECTION
DOEpatents
Russell, J.T.
1964-04-21
A method of measuring the instantaneous intensity of neutron flux in the core of a nuclear reactor is described. A target gas capable of being transmuted by neutron bombardment to a product having a resonance absorption line nt a particular microwave frequency is passed through the core of the reactor. Frequency-modulated microwave energy is passed through the target gas and the attenuation of the energy due to the formation of the transmuted product is measured. (AEC)
11. FABRICATION OF NEUTRON SOURCES
DOEpatents
Birden, J.H.
1959-01-20
A method is presented for preparing a more efficient neutron source comprising inserting in a container a quantity of Po-210, inserting B powder coated with either Ag, Pt, or Ni. The container is sealed and then slowly heated to about 450 C to volatilize the Po and effect combination of the coated powder with the Po. The neutron flux emitted by the unit is moritored and the heating step is terminated when the flux reaches a maximum or selected level.
12. Matter accreting neutron stars
NASA Technical Reports Server (NTRS)
Meszaros, P.
1981-01-01
Some of the fundamental neutron star parameters, such as the mass and the magnetic field strength, were experimentally determined in accreting neutron star systems. Some of the relevant data and the models used to derive useful information from them, are reviewed concentrating mainly on X-ray pulsars. The latest advances in our understanding of the radiation mechanisms and the transfer in the strongly magnetized polar cap regions are discussed.
13. Coded source neutron imaging
SciTech Connect
Bingham, Philip R; Santos-Villalobos, Hector J
2011-01-01
Coded aperture techniques have been applied to neutron radiography to address limitations in neutron flux and resolution of neutron detectors in a system labeled coded source imaging (CSI). By coding the neutron source, a magnified imaging system is designed with small spot size aperture holes (10 and 100 m) for improved resolution beyond the detector limits and with many holes in the aperture (50% open) to account for flux losses due to the small pinhole size. An introduction to neutron radiography and coded aperture imaging is presented. A system design is developed for a CSI system with a development of equations for limitations on the system based on the coded image requirements and the neutron source characteristics of size and divergence. Simulation has been applied to the design using McStas to provide qualitative measures of performance with simulations of pinhole array objects followed by a quantitative measure through simulation of a tilted edge and calculation of the modulation transfer function (MTF) from the line spread function. MTF results for both 100um and 10um aperture hole diameters show resolutions matching the hole diameters.
14. Neutron Nucleic Acid Crystallography.
PubMed
Chatake, Toshiyuki
2016-01-01
The hydration shells surrounding nucleic acids and hydrogen-bonding networks involving water molecules and nucleic acids are essential interactions for the structural stability and function of nucleic acids. Water molecules in the hydration shells influence various conformations of DNA and RNA by specific hydrogen-bonding networks, which often contribute to the chemical reactivity and molecular recognition of nucleic acids. However, X-ray crystallography could not provide a complete description of structural information with respect to hydrogen bonds. Indeed, X-ray crystallography is a powerful tool for determining the locations of water molecules, i.e., the location of the oxygen atom of H2O; however, it is very difficult to determine the orientation of the water molecules, i.e., the orientation of the two hydrogen atoms of H2O, because X-ray scattering from the hydrogen atom is very small.Neutron crystallography is a specialized tool for determining the positions of hydrogen atoms. Neutrons are not diffracted by electrons, but are diffracted by atomic nuclei; accordingly, neutron scattering lengths of hydrogen and its isotopes are comparable to those of non-hydrogen atoms. Therefore, neutron crystallography can determine both of the locations and orientations of water molecules. This chapter describes the current status of neutron nucleic acid crystallographic research as well as the basic principles of neutron diffraction experiments performed on nucleic acid crystals: materials, crystallization, diffraction experiments, and structure determination.
15. Neutron scattering in Australia
SciTech Connect
Knott, R.B.
1994-12-31
Neutron scattering techniques have been part of the Australian scientific research community for the past three decades. The High Flux Australian Reactor (HIFAR) is a multi-use facility of modest performance that provides the only neutron source in the country suitable for neutron scattering. The limitations of HIFAR have been recognized and recently a Government initiated inquiry sought to evaluate the future needs of a neutron source. In essence, the inquiry suggested that a delay of several years would enable a number of key issues to be resolved, and therefore a more appropriate decision made. In the meantime, use of the present source is being optimized, and where necessary research is being undertaken at major overseas neutron facilities either on a formal or informal basis. Australia has, at present, a formal agreement with the Rutherford Appleton Laboratory (UK) for access to the spallation source ISIS. Various aspects of neutron scattering have been implemented on HIFAR, including investigations of the structure of biological relevant molecules. One aspect of these investigations will be presented. Preliminary results from a study of the interaction of the immunosuppressant drug, cyclosporin-A, with reconstituted membranes suggest that the hydrophobic drug interdigitated with lipid chains.
16. Neutron Velocity Selector
Fermi, Enrico
This Patent presents a detailed description of the construction and operation of a velocity selector for neutrons with velocities up to 6000÷7000 m/s. This apparatus employs a rotating shutter designed in such a way that neutrons are passed during a portion of each rotation of the shutter, the shutter blocking all neutron radiation at other times. The selector is built up with alternate laminations of a material with high neutron capture cross section (such as, for example, cadmium, boron or gadolinium), and parallel laminations of a material with low capture probability (such as, for example, aluminium, magnesium or beryllium). This is required in order to provide a path through the shutter to the neutrons, which then pass into a ionization chamber. The timing mechanism, adopted to activate or deactivate the neutron detection, and measuring means at given times following each opening or closing of the shutter, is electronic (not mechanic), controlled by a photocell unit. The reference published article for the main topic of the present Patent is [Fermi (1947)].
17. Neutron Speed Echo Spectroscopy
Ioffe, A.
Neutron speed echo (NSPE) technique is in a way a generalization of the neutron spin echo (NSE) technique. Similar to NSE spectrometers, the resolution of such NSPE spectrometer is extremely high and is not connected with the monochromatization of the incoming beam. However, in contrast to NSE spectrometers, the operation of proposed spectrometer does not necessarily require a polarized neutron beam. Such decoupling the polarization and the resolution is in clear contrast to NSE technique. Because the resolution of a NSPE spectrometer can be a few orders higher than the resolution of NSE spectrometers, one can achieve the energy resolution of about 10-14 eV by the use of ultra cold neutrons; a fact that can be used in some fundamental physics experiments. Though the scattering on the sample impose limitations on the resolution of a NSPE spectrometer, the use of the proposed technique in a low-resolution mode can be useful in the combination with triple-axis spectrometers and allow for the significant improvement of their energy resolution, however, without the use of polarized neutrons. This fact opens new possibilities for the study of magnetic phenomena in solids, where the NSE method is principally not applicable because of the neutron precession in the sample, especially by combining polarization analysis with high-resolution spectroscopy. The proposed technique also allows for an easy implementation of the principle of the NSE focusing, when the resolution ellipse is aligned along a dispersion curve.
18. Polarized neutrons in RHIC
SciTech Connect
Courant, E.D.
1998-04-20
There does not appear to be any obvious way to accelerate neutrons, polarized or otherwise, to high energies by themselves. To investigate the behavior of polarized neutrons the authors therefore have to obtain them by accelerating them as components of heavier nuclei, and then sorting out the contribution of the neutrons in the analysis of the reactions produced by the heavy ion beams. The best neutron carriers for this purpose are probably {sup 3}He nuclei and deuterons. A polarized deuteron is primarily a combination of a proton and a neutron with their spins pointing in the same direction; in the {sup 3}He nucleus the spins of the two protons are opposite and the net spin (and magnetic moment) is almost the same as that of a free neutron. Polarized ions other than protons may be accelerated, stored and collided in a ring such as RHIC provided the techniques proposed for polarized proton operation can be adapted (or replaced by other strategies) for these ions. This paper discusses techniques for accelerating polarized {sup 3}He nuclei and deuterons.
19. Coded source neutron imaging
Bingham, Philip; Santos-Villalobos, Hector; Tobin, Ken
2011-03-01
Coded aperture techniques have been applied to neutron radiography to address limitations in neutron flux and resolution of neutron detectors in a system labeled coded source imaging (CSI). By coding the neutron source, a magnified imaging system is designed with small spot size aperture holes (10 and 100μm) for improved resolution beyond the detector limits and with many holes in the aperture (50% open) to account for flux losses due to the small pinhole size. An introduction to neutron radiography and coded aperture imaging is presented. A system design is developed for a CSI system with a development of equations for limitations on the system based on the coded image requirements and the neutron source characteristics of size and divergence. Simulation has been applied to the design using McStas to provide qualitative measures of performance with simulations of pinhole array objects followed by a quantitative measure through simulation of a tilted edge and calculation of the modulation transfer function (MTF) from the line spread function. MTF results for both 100μm and 10μm aperture hole diameters show resolutions matching the hole diameters.
20. MAGNETIC NEUTRON SCATTERING
SciTech Connect
ZALIZNYAK,I.A.; LEE,S.H.
2004-07-30
Much of our understanding of the atomic-scale magnetic structure and the dynamical properties of solids and liquids was gained from neutron-scattering studies. Elastic and inelastic neutron spectroscopy provided physicists with an unprecedented, detailed access to spin structures, magnetic-excitation spectra, soft-modes and critical dynamics at magnetic-phase transitions, which is unrivaled by other experimental techniques. Because the neutron has no electric charge, it is an ideal weakly interacting and highly penetrating probe of matter's inner structure and dynamics. Unlike techniques using photon electric fields or charged particles (e.g., electrons, muons) that significantly modify the local electronic environment, neutron spectroscopy allows determination of a material's intrinsic, unperturbed physical properties. The method is not sensitive to extraneous charges, electric fields, and the imperfection of surface layers. Because the neutron is a highly penetrating and non-destructive probe, neutron spectroscopy can probe the microscopic properties of bulk materials (not just their surface layers) and study samples embedded in complex environments, such as cryostats, magnets, and pressure cells, which are essential for understanding the physical origins of magnetic phenomena. Neutron scattering is arguably the most powerful and versatile experimental tool for studying the microscopic properties of the magnetic materials. The magnitude of the cross-section of the neutron magnetic scattering is similar to the cross-section of nuclear scattering by short-range nuclear forces, and is large enough to provide measurable scattering by the ordered magnetic structures and electron spin fluctuations. In the half-a-century or so that has passed since neutron beams with sufficient intensity for scattering applications became available with the advent of the nuclear reactors, they have became indispensable tools for studying a variety of important areas of modern science
1. Fundamental neutron physics at LANSCE
SciTech Connect
Greene, G.
1995-10-01
Modern neutron sources and science share a common origin in mid-20th-century scientific investigations concerned with the study of the fundamental interactions between elementary particles. Since the time of that common origin, neutron science and the study of elementary particles have evolved into quite disparate disciplines. The neutron became recognized as a powerful tool for studying condensed matter with modern neutron sources being primarily used (and justified) as tools for neutron scattering and materials science research. The study of elementary particles has, of course, led to the development of rather different tools and is now dominated by activities performed at extremely high energies. Notwithstanding this trend, the study of fundamental interactions using neutrons has continued and remains a vigorous activity at many contemporary neutron sources. This research, like neutron scattering research, has benefited enormously by the development of modern high-flux neutron facilities. Future sources, particularly high-power spallation sources, offer exciting possibilities for continuing this research.
2. Neutron dosimetry in boron neutron capture therapy
SciTech Connect
Fairchild, R.G.; Miola, U.J.; Ettinger, K.V.
1981-01-01
The recent development of various borated compounds and the utilization of one of these (Na/sub 2/B/sub 12/H/sub 11/SH) to treat brain tumors in clinical studies in Japan has renewed interest in neutron capture therapy. In these procedures thermal neutrons interact with /sup 10/B in boron containing cells through the /sup 10/B(n,..cap alpha..)/sup 7/Li reaction producing charged particles with a maximum range of approx. 10..mu..m in tissue. Borated analogs of chlorpromazine, porphyrin, thiouracil and deoxyuridine promise improved tumor uptake and blood clearance. The therapy beam from the Medical Research Reactor in Brookhaven contains neutrons from a modified and filtered fission spectrum and dosimetric consequences of the use of the above mentioned compounds in conjunction with thermal and epithermal fluxes are discussed in the paper. One of the important problems of radiation dosimetry in capture therapy is determination of the flux profile and, hence, the dose profile in the brain. This has been achieved by constructing a brain phantom made of TE plastic. The lyoluminescence technique provides a convenient way of monitoring the neutron flux distributions; the detectors for this purpose utilize /sup 6/Li and /sup 10/B compounds. Such compounds have been synthesized specially for the purpose of dosimetry of thermal and epithermal beams. In addition, standard lyoluminescent phosphors, like glutamine, could be used to determine the collisional component of the dose as well as the contribution of the /sup 14/N(n,p)/sup 14/C reaction. Measurements of thermal flux were compared with calculations and with measurements done with activation foils.
3. The p + 9Be(thin target) reaction as a source of quasi-monoenergetic neutrons
Novák, J.; Bém, P.; Majerle, M.; Mrázek, J.; Šimečková, E.; Štefánik, M.; Yasin, Z.
2017-09-01
The cyclotron driven fast quasi-monoenergetic neutron source based on the p+ 9Be (thin target) reaction was studied at the proton energy around 30 MeV Due to higher melting point of Be, the p+ 9Be(thin), reaction could be considered as an alternative to the most used p+Li(thin) neutron source, providing a similar quasi-monoenergetic neutron spectrum at significantly higher neutron output owing to advance in higher incident proton beam current. The neutron spectrum measured by the time-of-flight method agrees with other experimental data and indicates dominant contribution of ground and first excited states leading to only two peaks, separated by some 2-3 MeV, in the p+thin 9Be neutron spectrum.
4. Feasibility of a boron loaded scintillation detector for dose measurements related to boron neutron capture therapy
Kim, Don-Soo; Egan, James J.; Kegel, Gunter H. R.; Desimone, David
2002-04-01
The feasibility of the use of a boron loaded scintillation detector in a head phantom for boron neutron capture therapy dose estimates was evaluated. Several monoenergetic neutron groups were produced via the ^7Li(p,n)^7Be reaction in a metallic lithium target using the Van de Graaff accelerator at University of Massachusetts Lowell. The pulse-height spectra were taken from a natural boron loaded (10205-, 304-, 407-, 507-, 570-, 702-, and 780-keV incident neutrons. The results shows that a boron loaded scintillator could be used to distinguish the doses from different radiation sources in boron neutron capture therapy. This detector may be used in the estimation of doses due to fast neutrons, alpha particles and recoil lithium from ^10B(n,α)^7Li, and photons at the same time during neutron irradiation procedures.
5. Design of the cold neutron triple-axis spectrometer at the China Advanced Research Reactor
Cheng, P.; Zhang, Hongxia; Bao, W.; Schneidewind, A.; Link, P.; Grünwald, A. T. D.; Georgii, R.; Hao, L. J.; Liu, Y. T.
2016-06-01
The design of the first cold neutron triple-axis spectrometer at the China Advanced Research Reactor is presented. Based on the Monte Carlo simulations using neutron ray-tracing program McStas, the parameters of major neutron optics in this instrument are optimized. The neutron flux at sample position is estimated to be 5.6 ×107 n/cm2/s at neutron incident energy Ei=5 meV when the reactor operates normally at the designed 60 MW power. The performances of several neutron supermirror polarizing devices are compared and their critical parameters are optimized for this spectrometer. The polarization analysis will be realized with a flexible switch from the unpolarized experimental mode.
6. Thermal neutron response of a boron-coated GEM detector via GEANT4 Monte Carlo code.
PubMed
Jamil, M; Rhee, J T; Kim, H G; Ahmad, Farzana; Jeon, Y J
2014-10-22
In this work, we report the design configuration and the performance of the hybrid Gas Electron Multiplier (GEM) detector. In order to make the detector sensitive to thermal neutrons, the forward electrode of the GEM has been coated with the enriched boron-10 material, which works as a neutron converter. A total of 5×5cm(2) configuration of GEM has been used for thermal neutron studies. The response of the detector has been estimated via using GEANT4 MC code with two different physics lists. Using the QGSP_BIC_HP physics list, the neutron detection efficiency was determined to be about 3%, while with QGSP_BERT_HP physics list the efficiency was around 2.5%, at the incident thermal neutron energies of 25meV. The higher response of the detector proves that GEM-coated with boron converter improves the efficiency for thermal neutrons detection. Copyright © 2014 Elsevier Ltd. All rights reserved.
7. Neutron Capture Reactions for Stockpile Stewardship and Basic Science
SciTech Connect
Parker, W; Agvaanluvsan, U; Becker, J; Wilk, P; Wu, C; Bredeweg, T; Couture, A; Haight, R; Jandel, M; O'Donnell, J; Reifarth, R; Rundberg, R; Ullmann, J; Vieira, D; Wouters, J; Sheets, S; Mitchell, G; Becvar, F; Krticka, M
2007-08-04
The capture process is a nuclear reaction in which a target atom captures an incident projectile, e.g. a neutron. The excited-state compound nucleus de-excites by emitting photons. This process creates an atom that has one more neutron than the target atom, so it is a different isotope of the same element. With low energy (slow) neutron projectiles, capture is the dominant reaction, other than elastic scattering. However, with very heavy nuclei, fission competes with capture as a method of de-excitation of the compound nucleus. With higher energy (faster) incident neutrons, additional reactions are also possible, such as emission of protons or emission of multiple neutrons. The probability of a particular reaction occurring (such as capture) is referred to as the cross section for that reaction. Cross sections are very dependent on the incoming neutron's energy. Capture reactions can be studied either using monoenergetic neutron sources or 'white' neutron sources. A 'white' neutron source has a wide range of neutron energies in one neutron beam. The advantage to the white neutron source is that it allows the study of cross sections as they depend on neutron energies. The Los Alamos Neutron Science Center, located at Los Alamos National Laboratory, provides an intense white neutron source. Neutrons there are created by a high-energy proton beam from a linear accelerator striking a heavy metal (tungsten) target. The neutrons range in energy from subthermal up to very fast - over 100 MeV in energy. Low-energy neutron reaction cross sections fluctuate dramatically from one target to another, and they are very difficult to predict by theoretical modeling. The cross sections for particular capture reactions are important for defense sciences, advanced reactor concepts, transmutation of radioactive wastes and nuclear astrophysics. We now have a strong collaboration between Lawrence Livermore National Laboratory, Los Alamos National Laboratory, North Carolina State
8. Neutron Capture Gamma-Ray Libraries for Nuclear Applications
SciTech Connect
Sleaford, B W; Firestone, R B; Summers, N; Escher, J; Hurst, A; Krticka, M; Basunia, S; Molnar, G; Belgya, T; Revay, Z; Choi, H D
2010-11-04
The neutron capture reaction is useful in identifying and analyzing the gamma-ray spectrum from an unknown assembly as it gives unambiguous information on its composition. this can be done passively or actively where an external neutron source is used to probe an unknown assembly. There are known capture gamma-ray data gaps in the ENDF libraries used by transport codes for various nuclear applications. The Evaluated Gamma-ray Activation file (EGAF) is a new thermal neutron capture database of discrete line spectra and cross sections for over 260 isotopes that was developed as part of an IAEA Coordinated Research project. EGAF is being used to improve the capture gamma production in ENDF libraries. For medium to heavy nuclei the quasi continuum contribution to the gamma cascades is not experimentally resolved. The continuum contains up to 90% of all the decay energy and is modeled here with the statistical nuclear structure code DICEBOX. This code also provides a consistency check of the level scheme nuclear structure evaluation. The calculated continuum is of sufficient accuracy to include in the ENDF libraries. This analysis also determines new total thermal capture cross sections and provides an improved RIPL database. For higher energy neutron capture there is less experimental data available making benchmarking of the modeling codes more difficult. They are investigating the capture spectra from higher energy neutrons experimentally using surrogate reactions and modeling this with Hauser-Feshbach codes. This can then be used to benchmark CASINO, a version of DICEBOX modified for neutron capture at higher energy. This can be used to simulate spectra from neutron capture at incident neutron energies up to 20 MeV to improve the gamma-ray spectrum in neutron data libraries used for transport modeling of unknown assemblies.
9. Neutron-Induced Failures in Semiconductor Devices
SciTech Connect
Wender, Stephen Arthur
2016-04-06
This slide presentation explores single event effect, environmental neutron flux, system response, the Los Alamos Neutron Science Center (LANSCE) neutron testing facility, examples of SEE measurements, and recent interest in thermal neutrons.
10. GEANT4 Simulation of Neutron Detector for DAMPE
He, M.; Ma, T.; Chang, J.; Zhang, Y.; Huang, Y. Y.; Zang, J. J.; Wu, J.; Dong, T. K.
2016-01-01
During recent tens of years dark matter has gradually become a hot topic in astronomical research field, and related theory researches and experiment projects change with each passing day. The Dark Matter Particle Explorer (DAMPE) of our country is proposed under this background. As the probing object involves high energy electrons, appropriate methods must be taken to distinguish them from protons in order to reduce the event probability of other charged particles (e.g. a proton) being mistaken as electrons. The experiments show that, the hadronic shower of high energy proton in BGO electromagnetic calorimeter, which is usually accompanied by the emitting of large number of secondary neutrons, is significantly different from the electromagnetic shower of high energy electron. Through the detection of secondary neutron signal emitting from the bottom of BGO electromagnetic calorimeter and the shower shape of incident particles in BGO electromagnetic calorimeter, we can effectively distinguish whether the incident particles are high energy protons or electrons. This paper introduces the structure and detecting principle of DAMPE neutron detector. We use Monte-Carlo method with GEANT4 software to simulate the signal emitting from protons and electrons at characteristic energy in the neutron detector, and finally summarize the neutron detector's ability to distinguish protons and electrons under different electron acception efficiencies.
11. GEANT4 Simulation of Neutron Detector for DAMPE
Ming, He; Tao, Ma; Jin, Chang; Yan, Zhang; Yong-yi, Huang; Jing-jing, Zang; Jian, Wu; Tie-kuang, Dong
2016-10-01
In recent decades, dark matter has gradually become a hot topic in astronomical research, and the related theoretical research and experimental project are updated with each passing day. The Dark Matter Particle Explorer (DAMPE) of our country was proposed under this background. As the detected object involves high-energy electrons, appropriate methods must be taken to distinguish them from protons, in order to reduce the event probability of other charged particles (for example protons) being mistaken as electrons. The experiments show that the hadron shower of high-energy proton in BGO (Bismuth Germanium Oxide) calorimeter, which is usually accompanied with the emitting of a large number of secondary neutrons, is significantly different from the electromagnetic shower of high-energy electron. Through the detection of secondary neutron signals emerging from the bottom of BGO calorimeter, and the shower shape of incident particles in the BGO calorimeter, we can effectively distinguish whether the incident particles are high-energy protons or electrons. This paper introduces the structure and detection principle of the DAMPE neutron detector. We use the Monte-Carlo method and the GEANT4 software to simulate the signals produced by protons and electrons at the characteristic energy in the neutron detector, and finally summarize the neutron detector's ability to distinguish protons and electrons under different electron acceptabilities.
12. Proposed neutron activation analysis facilities in the Advanced Neutron Source
SciTech Connect
Robinson, L.; Dyer, F.F.; Emery, J.F.
1990-01-01
A number of analytical chemistry experimental facilities are being proposed for the Advanced Neutron Source. Experimental capabilities will include gamma-ray analysis and neutron depth profiling. This paper describes the various systems proposed and some of their important characteristics.
13. Neutron lifetime measurements using gravitationally trapped ultracold neutrons
SciTech Connect
Serebrov, A. P.; Varlamov, V. E.; Kharitonov, A. G.; Fomin, A. K.; Krasnoschekova, I. A.; Lasakov, M. S.; Taldaev, R. R.; Vassiljev, A. V.; Zherebtsov, O. M.; Pokotilovski, Yu. N.; Geltenbort, P.
2008-09-15
Our experiment using gravitationally trapped ultracold neutrons (UCN) to measure the neutron lifetime is reviewed. Ultracold neutrons were trapped in a material bottle covered with perfluoropolyether. The neutron lifetime was deduced from comparison of UCN losses in the traps with different surface-to-volume ratios. The precise value of the neutron lifetime is of fundamental importance to particle physics and cosmology. In this experiment, the UCN storage time is brought closer to the neutron lifetime than in any experiments before: the probability of UCN losses from the trap was only 1% of that for neutron {beta} decay. The neutron lifetime obtained, 878.5{+-}0.7{sub stat}{+-}0.3{sub sys} s, is the most accurate experimental measurement to date.
14. A multitask neutron beam line for spallation neutron sources
Pietropaolo, A.; Festa, G.; Grazzi, F.; Barzagli, E.; Scherillo, A.; Schooneveld, E. M.; Civita, F.
2011-08-01
Here we present a new concept for a time-of-flight neutron scattering instrument allowing for simultaneous application of three different techniques: time-of-flight neutron diffraction, neutron resonance capture analysis and Bragg edge transmission analysis. The instrument can provide average resolution neutron radiography too. The potential of the proposed concept was explored by implementing the necessary equipment on INES (Italian Neutron Experimental Station) at the ISIS spallation neutron source (UK). The results obtained show the effectiveness of the proposed instrument to acquire relevant quantitative information in a non-invasive way on a historical metallurgical sample, namely a Japanese hand guard (tsuba). The aforementioned neutron techniques simultaneously exploited the extended neutron energy range available from 10 meV to 1 keV. This allowed a fully satisfactory characterization of the sample in terms of metal components and their combination in different phases, and forging and assembling methods.
15. The Fundamental Neutron Physics Beamline at the Spallation Neutron Source
PubMed Central
Greene, Geoffrey; Cianciolo, Vince; Koehler, Paul; Allen, Richard; Snow, William Michael; Huffman, Paul; Gould, Chris; Bowman, David; Cooper, Martin; Doyle, John
2005-01-01
The Spallation Neutron Source (SNS), currently under construction at Oak Ridge National Laboratory with an anticipated start-up in early 2006, will provide the most intense pulsed beams of cold neutrons in the world. At a projected power of 1.4 MW, the time averaged fluxes and fluences of the SNS will approach those of high flux reactors. One of the flight paths on the cold, coupled moderator will be devoted to fundamental neutron physics. The fundamental neutron physics beamline is anticipated to include two beam-lines; a broad band cold beam, and a monochromatic beam of 0.89 nm neutrons for ultracold neutron (UCN) experiments. The fundamental neutron physics beamline will be operated as a user facility with experiment selection based on a peer reviewed proposal process. An initial program of five experiments in neutron decay, hadronic weak interaction and time reversal symmetry violation have been proposed. PMID:27308112
16. High Energy Neutron Induced Gamma Production
SciTech Connect
Brown, D A; Johnson, M; Navratil, P
2007-09-28
N Division has an interest in improving the physics and accuracy of the gamma data it provides to its customers. It was asked to look into major gamma producing reactions for 14 MeV incident neutrons for several low-Z materials and determine whether LLNL's processed data files faithfully represent the current state of experimental and theoretical knowledge for these reactions. To address this, we surveyed the evaluations of the requested materials, made recommendations for the next ENDL release and noted isotopes that will require further experimental study. This process uncovered several major problems in our translation and processing of the ENDF formatted evaluations, most of which have been resolved.
17. Structure in the Proton and the Neutron
DOE R&D Accomplishments Database
1958-06-01
between the limits 0.61 x 10{sup -13} cm and 0.80 x 10{sup -13} cm. The first value (0.61 x 10{sup -13} cm) is determined by examining the total deuteron electro-disintegration cross section at a given angle and incident energy and comparing this cross section with that of the free proton under the same conditions. The second value (0.80 x 10{sup -13} cm) is found by examining the peak of the deuteron electro-disintegration cross section. Because of possible contributions to the total cross section by mesonic exchange effects, the second method is believed to be slightly more accurate. The neutron size is, therefore, approximately (0.70 plus or minus 0.10) x 10{sup -13} cm and probably the larger size 0.90 x 10{sup -13} cm is correct. Thus the magnetic clouds of the neutron and proton are closely the same. The bearing of these results on the validity of electrodynamics is discussed. Because of the small radius implied by the neutron-electron experiments, there is an anomaly between the neutron and the proton. This is represented by the small charge radius for the neutron and the much larger radius of the proton. Additional information of the structure of the deuteron and on the production of pions by electrons is also furnished by the same experiments and will be discussed at the meeting. (auth)
18. Characterization of a Tunable Quasi-Monoenergetic Neutron Beamfrom Deuteron Breakup
SciTech Connect
Bleuel, D.L.; McMahan, M.A.; Ahle, L.; Barquest, B.R.; Cerny, J.; Heilbronn, L.H.; Jewett, C.C.
2006-12-14
A neutron irradiation facility is being developed at the88-Inch Cyclotron at Lawrence Berkeley National Laboratory for thepurposes of measuring neutron reaction cross sections on radioactivetargets and for radiation effects testing. Applications are of benefit tostockpile stewardship, nuclear astrophysics, next generation advancedfuel reactors, and cosmic radiation biology and electronics in space. Thefacility will supply a tunable, quasi-monoenergetic neutron beam in therange of 10-30 MeV or a white neutron source, produced by deuteronbreakup reactions on thin and thick targets, respectively. Because thedeuteron breakup reaction has not been well studied at intermediateincident deuteron energies, above the target Coulomb barrier and below 56MeV, a detailed characterization was necessary of the neutron spectraproduced by thin targets.Neutron time of flight (TOF) methods have beenused to measure the neutron spectra produced on thin targets of low-Z(titanium) and high-Z (tantalum) materials at incident deuteron energiesof 20 MeV and 29 MeV at 0 deg. Breakup neutrons at both energies fromlow-Z targets appear to peak at roughly half of the available kineticenergy, while neutrons from high-Z interactions peak somewhat lower inenergy, owing to the increased proton energy due to breakup within theCoulomb field. Furthermore, neutron spectra appear narrower for high-Ztargets. These centroids are consistent with recent preliminary protonenergy measurements using silicon telescope detectors conducted at LBNL,though there is a notable discrepancy with spectral widths.
19. Use of Apollo 17 Epoch Neutron Spectrum as a Benchmark in Testing LEND Collimated Sensor
NASA Technical Reports Server (NTRS)
Chin, Gordon; Sagdeev, R.; Milikh, G.
2011-01-01
The Apollo 17 neutron experiment LPNE provided a unique set of data on production of neutrons in the Lunar soil bombarded by Galactic Cosmic Rays (GCR). It serves as valuable "ground-truth" in the age of orbital remote sensing. We used the neutron data attributed to Apollo 17 epoch as a benchmark for testing the LEND's collimated sensor, as introduced by the geometry of collimator and efficiency of He3 counters. The latter is defined by the size of gas counter and pressure inside it. The intensity and energy spectrum of neutrons escaping the lunar surface are dependent on incident flux of Galactic Cosmic Rays (GCR) whose variability is associated with Solar Cycle and its peculiarities. We obtain first the share of neutrons entering through the field of view of collimator as a fraction of the total neutron flux by using the angular distribution of neutron exiting the Moon described by our Monte Carlo code. We computed next the count rate of the 3He sensor by using the neutron energy spectrum from McKinney et al. [JGR, 2006] and by consider geometry and gas pressure of the LEND sensor. Finally the neutron count rate obtained for the Apollo 17 epoch characterized by intermediate solar activity was adjusted to the LRO epoch characterized by low solar activity. It has been done by taking into account solar modulation potential, which affects the GCR flux, and in turn changes the neutron albedo flux.
20. Neutron production for 250 MeV protons bombarding on thick grain-made tungsten target
Zhang, Xueying; Zhang, Yanbin; Ma, Fei; Ju, Yongqin; Chen, Liang; Zhang, Hongbin; Li, Yanyan; Wan, Bo; Wang, Jianguo; Ge, Honglin
2015-08-01
Neutron yield for 250 MeV protons incident on a tungsten target has been measured using the water bath method. The target was made of many randomly placed tungsten grains. Through analyzing the activity of Au foils, the neutron flux distribution in water was obtained. The neutrons slowing down process shows that the neutrons from tungsten have an average energy lower than neutrons from the lead target. The neutron yield was experimentally determined to be 2.02 ± 0.15 neutron/proton. Detailed simulation was also performed with the Geant4 toolkit. Comparison has been made with the experimentally derived neutron yield. It was found that, around 250 MeV, experimental results were described satisfactorily with a combination of high-energy spallation, low-energy neutron reaction and scattering. It was shown that the grain-packed target does not affect much the main neutronic properties, which are of crucial importance for the design of the spallation target.
1. The neutron dose conversion coefficients calculation in human tooth enamel in an anthropomorphic phantom.
PubMed
Khailov, A M; Ivannikov, A I; Skvortsov, V G; Stepanenko, V F; Tsyb, A F; Trompier, F; Hoshi, M
2010-02-01
In the present study, MCNP4B simulation code is used to simulate neutron and photon transport. It gives the conversion coefficients that relate neutron fluence to the dose in tooth enamel (molars and pre-molars only) for 20 energy groups of monoenergetic neutrons with energies from 10-9 to 20 MeV for five different irradiation geometries. The data presented are intended to provide the basis for connection between EPR dose values and standard protection quantities defined in ICRP Publication 74. The results of the calculations for critical organs were found to be consistent with ICRP data, with discrepancies generally less than 10% for the fast neutrons. The absorbed dose in enamel was found to depend strongly on the incident neutron energy for neutrons over 10 keV. The dependence of the data on the irradiation geometry is also shown. Lower bound estimates of enamel radiation sensitivity to neutrons were made using obtained coefficients for the secondary photons. Depending on neutron energy, tooth enamel was shown to register 10-120% of the total neutron dose in the human body in the case of pure neutron exposure and AP irradiation geometry.
2. Influence of the neutron transport tube on neutron resonance densitometry
Kitatani, Fumito; Tsuchiya, Harufumi; Koizumi, Mitsuo; Takamine, Jun; Hori, Junichi; Sano, Tadafumi
2017-09-01
Neutron Resonance Densitometry (NRD) is a non-destructive assay technique of nuclear materials in particle-like debris that contains various materials. An aim of NRD is to quantify nuclear materials in a melting fuel of Fukusima Daiichi plant, spent nuclear fuel and annihilation disposal fuel etc. NRD consists of two techniques of Neutron Resonance Transmission Analysis (NRTA) and Neutron Resonance Capture Analysis (NRCA) or Prompt Gamma-ray Analysis (PGA). A density of nuclear material isotopes is decided with NRTA. The materials absorbing a neutron in a wide energy range such as boron in a sample are identified by NRCA/PGA. The information of NRCA/PGA is used in NRTA analysis to quantify nuclear material isotopes. A neutron time of flight (TOF) method is used in NRD measurements. A facility, consisting of a neutron source, a neutron flight path, and a detector is required. A short flight path and a strong neutron source are needed to downsize such a facility and put NRD into practical use. A neutron transport tube covers a flight path to prevent noises. In order to investigate the effect of neutron transport tube and pulse width of a neutron source, we carried out NRTA experiments with a 2-m short neutron transport tube constructed at Kyoto University Research Reactor Institute - Linear Accelerator (KURRI-LINAC), and impacts of shield of neutron transport tube and influence of pulse width of a neutron source were examined. A shield of the neutron transport tube reduced a background and had a good influence on the measurement. The resonance dips of 183W at 27 eV was successfully observed with a pulse width of a neutron source less than 2 μs.
3. Single event upsets calculated from new ENDF/B-VI proton and neutron data up to 150 MeV
SciTech Connect
1999-06-01
Single-event upsets (SEU) in microelectronics are calculated from newly-developed silicon nuclear reaction recoil data that extend up to 150 MeV, for incident protons and neutrons. Calculated SEU cross sections are compared with measured data.
4. DETECTORS AND EXPERIMENTAL METHODS: Monte Carlo studies of micromegas as a neutron detector and its track reconstruction
Zhang, Yi; Zhang, Xiao-Dong; Wang, Wen-Xin; Yang, He-Run; Yang, Zheng-Cai; Hu, Bi-Tao
2009-01-01
In this paper a two dimensional readout micromegas detector with a polyethylene foil as converter was simulated on GEANT4 toolkit and GARFIELD for fast neutron detection. A new track reconstruction method based on time coincidence technology was developed in the simulation to obtain the incident neutron position. The results showed that with this reconstruction method higher spatial resolution was achieved.
5. Apollo 16 neutron stratigraphy.
NASA Technical Reports Server (NTRS)
Russ, G. P., III
1973-01-01
The Apollo 16 soils have the largest low-energy neutron fluences yet observed in lunar samples. Variations in the isotopic ratios Gd-158/Gd-157 and Sm-150/Sm-149 (up to 1.9 and 2.0%, respectively) indicate that the low-energy neutron fluence in the Apollo 16 drill stem increases with depth throughout the section sampled. Such a variation implies that accretion has been the dominant regolith 'gardening' process at this location. The data may be fit by a model of continuous accretion of pre-irradiated material or by models involving as few as two slabs of material in which the first slab could have been deposited as long as 1 b.y. ago. The ratio of the number of neutrons captured per atom by Sm to the number captured per atom by Gd is lower than in previously measured lunar samples, which implies a lower energy neutron spectrum at this site. The variation of this ratio with chemical composition is qualitatively similar to that predicted by Lingenfelter et al. (1972). Variations are observed in the ratio Gd-152/Gd-160 which are fluence-correlated and probably result from neutron capture by Eu-151.
6. Twisting Neutron Waves
Pushin, Dmitry
Most waves encountered in nature can be given a twist'', so that their phase winds around an axis parallel to the direction of wave propagation. Such waves are said to possess orbital angular momentum (OAM). For quantum particles such as photons, atoms, and electrons, this corresponds to the particle wavefunction having angular momentum of Lℏ along its propagation axis. Controlled generation and detection of OAM states of photons began in the 1990s, sparking considerable interest in applications of OAM in light and matter waves. OAM states of photons have found diverse applications such as broadband data multiplexing, massive quantum entanglement, optical trapping, microscopy, quantum state determination and teleportation, and interferometry. OAM states of electron beams have been used to rotate nanoparticles, determine the chirality of crystals and for magnetic microscopy. Here I discuss the first demonstration of OAM control of neutrons. Using neutron interferometry with a spatially incoherent input beam, we show the addition and conservation of quantum angular momenta, entanglement between quantum path and OAM degrees of freedom. Neutron-based quantum information science heretofore limited to spin, path, and energy degrees of freedom, now has access to another quantized variable, and OAM modalities of light, x-ray, and electron beams are extended to a massive, penetrating neutral particle. The methods of neutron phase imprinting demonstrated here expand the toolbox available for development of phase-sensitive techniques of neutron imaging. Financial support provided by the NSERC Create and Discovery programs, CERC and the NIST Quantum Information Program is acknowledged.
7. Apollo 16 neutron stratigraphy.
NASA Technical Reports Server (NTRS)
Russ, G. P., III
1973-01-01
The Apollo 16 soils have the largest low-energy neutron fluences yet observed in lunar samples. Variations in the isotopic ratios Gd-158/Gd-157 and Sm-150/Sm-149 (up to 1.9 and 2.0%, respectively) indicate that the low-energy neutron fluence in the Apollo 16 drill stem increases with depth throughout the section sampled. Such a variation implies that accretion has been the dominant regolith 'gardening' process at this location. The data may be fit by a model of continuous accretion of pre-irradiated material or by models involving as few as two slabs of material in which the first slab could have been deposited as long as 1 b.y. ago. The ratio of the number of neutrons captured per atom by Sm to the number captured per atom by Gd is lower than in previously measured lunar samples, which implies a lower energy neutron spectrum at this site. The variation of this ratio with chemical composition is qualitatively similar to that predicted by Lingenfelter et al. (1972). Variations are observed in the ratio Gd-152/Gd-160 which are fluence-correlated and probably result from neutron capture by Eu-151.
8. Origin of Neutron Stars
Brecher, K.
1999-12-01
The origin of the concept of neutron stars can be traced to two brief, incredibly insightful publications. Work on the earlier paper by Lev Landau (Phys. Z. Sowjetunion, 1, 285, 1932) actually predated the discovery of neutrons. Nonetheless, Landau arrived at the notion of a collapsed star with the density of a nucleus (really a "nucleus star") and demonstrated (at about the same time as, and independent of, Chandrasekhar) that there is an upper mass limit for dense stellar objects of about 1.5 solar masses. Perhaps even more remarkable is the abstract of a talk presented at the December 1933 meeting of the American Physical Society published by Walter Baade and Fritz Zwicky in 1934 (Phys. Rev. 45, 138). It followed the discovery of the neutron by just over a year. Their report, which was about the same length as the present abstract: (1) invented the concept and word supernova; (2) suggested that cosmic rays are produced by supernovae; and (3) in the authors own words, proposed "with all reserve ... the view that supernovae represent the transitions from ordinary stars to neutron stars (italics), which in their final stages consist of extremely closely packed neutrons." The abstract by Baade and Zwicky probably contains the highest density of new, important (and correct) ideas in high energy astrophysics ever published in a single paper. In this talk, we will discuss some of the facts and myths surrounding these two publications.
9. Neutron counting with cameras
SciTech Connect
Van Esch, Patrick; Crisanti, Marta; Mutti, Paolo
2015-07-01
A research project is presented in which we aim at counting individual neutrons with CCD-like cameras. We explore theoretically a technique that allows us to use imaging detectors as counting detectors at lower counting rates, and transits smoothly to continuous imaging at higher counting rates. As such, the hope is to combine the good background rejection properties of standard neutron counting detectors with the absence of dead time of integrating neutron imaging cameras as well as their very good spatial resolution. Compared to Xray detection, the essence of thermal neutron detection is the nuclear conversion reaction. The released energies involved are of the order of a few MeV, while X-ray detection releases energies of the order of the photon energy, which is in the 10 KeV range. Thanks to advances in camera technology which have resulted in increased quantum efficiency, lower noise, as well as increased frame rate up to 100 fps for CMOS-type cameras, this more than 100-fold higher available detection energy implies that the individual neutron detection light signal can be significantly above the noise level, as such allowing for discrimination and individual counting, which is hard to achieve with X-rays. The time scale of CMOS-type cameras doesn't allow one to consider time-of-flight measurements, but kinetic experiments in the 10 ms range are possible. The theory is next confronted to the first experimental results. (authors)
10. Reflections on Critical Incidents.
ERIC Educational Resources Information Center
Thiel, Teresa
1999-01-01
Focuses on the reporting of critical incidents as a tool for self-reflection in a program to train English-as-a-Second-language (ESL) secondary school teachers in Malaysia. Case studies of critical incidents highlight the potential value of encouraging ESL trainee teachers to record, analyze, and reflect on significant personal events in their…
11. Incident analysis report
SciTech Connect
Gregg, D.W.; Buerer, A.; Leeds, S.
1996-02-20
This document presents information about a fire that occurred in January 1996 at Lawrence Livermore National Laboratory. This fire was caused by the spontaneous combustion of 100% fuming nitric acid. Topics discussed include: Summary of the incident; technical background; procedural background; supervision; previous incidents with 100% fuming nitric acid; and judgment of potential hazards.
12. Measurement of High-Energy Neutron Flux Above Ground Utilizing a Spallation Based Multiplicity Technique
Roecker, Caleb; Bernstein, Adam; Marleau, Peter; Vetter, Kai
2016-12-01
Cosmogenic high-energy neutrons are a ubiquitous, difficult to shield, poorly measured background. Above ground the high-energy neutron energy-dependent flux has been measured, with significantly varying results. Below ground, high-energy neutron fluxes are largely unmeasured. Here we present a reconstruction algorithm to unfold the incident neutron energy-dependent flux measured using the Multiplicity and Recoil Spectrometer (MARS), simulated test cases to verify the algorithm, and provide a new measurement of the above ground high-energy neutron energy-dependent flux with a detailed systematic uncertainty analysis. Uncertainty estimates are provided based upon the measurement statistics, the incident angular distribution, the surrounding environment of the Monte Carlo model, and the MARS triggering efficiency. Quantified systematic uncertainty is dominated by the assumed incident neutron angular distribution and surrounding environment of the Monte Carlo model. The energy-dependent neutron flux between 90 MeV and 400 MeV is reported. Between 90 MeV and 250 MeV the MARS results are comparable to previous Bonner sphere measurements. Over the total energy regime measured, the MARS result are located within the span of previous measurements. These results demonstrate the feasibility of future below ground measurements with MARS.
13. Measurement of high-energy neutron flux above ground utilizing a spallation based multiplicity technique
DOE PAGES
Roecker, Caleb; Bernstein, Adam; Marleau, Peter; ...
2016-11-14
Cosmogenic high-energy neutrons are a ubiquitous, difficult to shield, poorly measured background. Above ground the high-energy neutron energy-dependent flux has been measured, with significantly varying results. Below ground, high-energy neutron fluxes are largely unmeasured. Here we present a reconstruction algorithm to unfold the incident neutron energy-dependent flux measured using the Multiplicity and Recoil Spectrometer (MARS), simulated test cases to verify the algorithm, and provide a new measurement of the above ground high-energy neutron energy-dependent flux with a detailed systematic uncertainty analysis. Uncertainty estimates are provided based upon the measurement statistics, the incident angular distribution, the surrounding environment of the Montemore » Carlo model, and the MARS triggering efficiency. Quantified systematic uncertainty is dominated by the assumed incident neutron angular distribution and surrounding environment of the Monte Carlo model. The energy-dependent neutron flux between 90 MeV and 400 MeV is reported. Between 90 MeV and 250 MeV the MARS results are comparable to previous Bonner sphere measurements. Over the total energy regime measured, the MARS result are located within the span of previous measurements. Lastly, these results demonstrate the feasibility of future below ground measurements with MARS.« less
14. Measurement of high-energy neutron flux above ground utilizing a spallation based multiplicity technique
SciTech Connect
Roecker, Caleb; Bernstein, Adam; Marleau, Peter; Vetter, Kai
2016-11-14
Cosmogenic high-energy neutrons are a ubiquitous, difficult to shield, poorly measured background. Above ground the high-energy neutron energy-dependent flux has been measured, with significantly varying results. Below ground, high-energy neutron fluxes are largely unmeasured. Here we present a reconstruction algorithm to unfold the incident neutron energy-dependent flux measured using the Multiplicity and Recoil Spectrometer (MARS), simulated test cases to verify the algorithm, and provide a new measurement of the above ground high-energy neutron energy-dependent flux with a detailed systematic uncertainty analysis. Uncertainty estimates are provided based upon the measurement statistics, the incident angular distribution, the surrounding environment of the Monte Carlo model, and the MARS triggering efficiency. Quantified systematic uncertainty is dominated by the assumed incident neutron angular distribution and surrounding environment of the Monte Carlo model. The energy-dependent neutron flux between 90 MeV and 400 MeV is reported. Between 90 MeV and 250 MeV the MARS results are comparable to previous Bonner sphere measurements. Over the total energy regime measured, the MARS result are located within the span of previous measurements. Lastly, these results demonstrate the feasibility of future below ground measurements with MARS.
15. Global Map of Thermal Neutrons
NASA Image and Video Library
2002-05-28
Observations by NASA Mars Odyssey spacecraft show a global view of Mars in low energy, or thermal, neutrons. Thermal neutrons are sensitive to the presence of hydrogen and the presence of carbon dioxide, in this case dry ice frost.
16. Neutron Star Compared to Manhattan
NASA Image and Video Library
A pulsar is a neutron star, the crushed core of a star that has exploded. Neutron stars crush half a million times more mass than Earth into a sphere no larger than Manhattan, as animated in this s...
17. Development of a gaseous recoil-proton detector for neutron flux measurements between 0.2 and 2 MeV neutron energy
Marini, P.; Mathieu, L.; Aiche, M.; Cheron, T.; Hellmuth, P.; Pedroza, J. L.; Czajkowski, S.; Jurado, B.; Tsekhanovich, I.
2017-09-01
Absolute measurements of neutron fluence are an essential prerequisite of neutron-induced cross section measurements, neutron beam lines characterisation and dosimetric investigations. Precise neutron flux measurements can be performed with respect to the H(n,p) elastic cross section. The use of this technique, with silicon proton recoil detectors, is not straightforward below incident neutron energy of 1 MeV, due to a high background in the detected proton spectrum. Experiments carried out at the AIFIRA facility identified its origin. Based on these investigations, a gaseous recoil-proton detector has been designed, with a reduced low energy background. Preliminary results of the first tests of the developed detector are discussed here.
18. Quasifree photoproduction of eta mesons off the neutron.
PubMed
Jaegle, I; Mertens, T; Anisovich, A V; Bacelar, J C S; Bantes, B; Bartholomy, O; Bayadilov, D; Beck, R; Beloglazov, Y A; Castelijns, R; Crede, V; Dutz, H; Ehmanns, A; Elsner, D; Essig, K; Ewald, R; Fabry, I; Fuchs, M; Funke, Ch; Gothe, R; Gregor, R; Gridnev, A B; Gutz, E; Höffgen, S; Hoffmeister, P; Horn, I; Junkersfeld, J; Kalinowsky, H; Kammer, S; Kleber, V; Klein, Frank; Klein, Friedrich; Klempt, E; Konrad, M; Kotulla, M; Krusche, B; Lang, M; Langheinrich, J; Löhner, H; Lopatin, I V; Lotz, J; Lugert, S; Menze, D; Messchendorp, J G; Metag, V; Morales, C; Nanova, M; Nikonov, V A; Novinski, D; Novotny, R; Ostrick, M; Pant, L M; van Pee, H; Pfeiffer, M; Radkov, A; Roy, A; Sarantsev, A V; Schadmand, S; Schmidt, C; Schmieden, H; Schoch, B; Shende, S V; Sokhoyan, V; Süle, A; Sumachev, V V; Szczepanek, T; Thoma, U; Trnka, D; Varma, R; Walther, D; Weinheimer, Ch; Wendel, Ch
2008-06-27
Quasifree photoproduction of eta mesons off nucleons bound in the deuteron has been measured with the CBELSA/TAPS detector for incident photon energies up to 2.5 GeV at the Bonn ELSA accelerator. The eta mesons have been detected in coincidence with recoil protons and recoil neutrons, which allows a detailed comparison of the quasifree n(gamma,eta)n and p(gamma,eta)p reactions. The excitation function for eta production off the neutron shows a pronounced bumplike structure at W=1.68 GeV (E{gamma} approximately 1 GeV), which is absent for the proton.
19. Simulating Makrofol as a detector for neutron-induced recoils.
PubMed
Zhang, G; Becker, F; Urban, M; Xuan, Y
2011-03-01
The response of solid-state nuclear track detector is extremely dependent on incident angles of neutrons, which determine the angular distribution of secondary particles. In this paper, the authors present a method to investigate the angular response of Makrofol detectors. Using the C++-based Monte-Carlo tool-kit Geant4 in combination with SRIM and our MATLAB codes, we simulated the angular response of Makrofol. The simulations were based on the restricted energy loss model, and the concept of energy threshold and critical angle. Experiments were carried out with (252)Cf neutrons to verify the simulation results.
20. Quasifree Photoproduction of η Mesons off the Neutron
Jaegle, I.; Mertens, T.; Anisovich, A. V.; Bacelar, J. C. S.; Bantes, B.; Bartholomy, O.; Bayadilov, D.; Beck, R.; Beloglazov, Y. A.; Castelijns, R.; Crede, V.; Dutz, H.; Ehmanns, A.; Elsner, D.; Essig, K.; Ewald, R.; Fabry, I.; Fuchs, M.; Funke, Ch.; Gothe, R.; Gregor, R.; Gridnev, A. B.; Gutz, E.; Höffgen, S.; Hoffmeister, P.; Horn, I.; Junkersfeld, J.; Kalinowsky, H.; Kammer, S.; Kleber, V.; Klein, Frank; Klein, Friedrich; Klempt, E.; Konrad, M.; Kotulla, M.; Krusche, B.; Lang, M.; Langheinrich, J.; Löhner, H.; Lopatin, I. V.; Lotz, J.; Lugert, S.; Menze, D.; Messchendorp, J. G.; Metag, V.; Morales, C.; Nanova, M.; Nikonov, V. A.; Novinski, D.; Novotny, R.; Ostrick, M.; Pant, L. M.; van Pee, H.; Pfeiffer, M.; Radkov, A.; Roy, A.; Sarantsev, A. V.; Schadmand, S.; Schmidt, C.; Schmieden, H.; Schoch, B.; Shende, S. V.; Sokhoyan, V.; Süle, A.; Sumachev, V. V.; Szczepanek, T.; Thoma, U.; Trnka, D.; Varma, R.; Walther, D.; Weinheimer, Ch.; Wendel, Ch.
2008-06-01
Quasifree photoproduction of η mesons off nucleons bound in the deuteron has been measured with the CBELSA/TAPS detector for incident photon energies up to 2.5 GeV at the Bonn ELSA accelerator. The η mesons have been detected in coincidence with recoil protons and recoil neutrons, which allows a detailed comparison of the quasifree n(γ,η)n and p(γ,η)p reactions. The excitation function for η production off the neutron shows a pronounced bumplike structure at W=1.68GeV (Eγ≈1GeV), which is absent for the proton.
1. Gravitational radiation from dual neutron star elliptical binaries
NASA Technical Reports Server (NTRS)
Hils, Dieter
1991-01-01
General expressions are derived for the gravitational radiation incident on earth due to elliptical binary systems in the Galaxy. These results are applied to dual neutron star elliptical binaries. Calculations show that eccentric dual neutron star binaries lead to a moderate increase in gravitational flux density compared with circular systems for frequencies above approximately 0.0001 Hz. Tables of various quantities such as average gravitational luminosity, number of sources per unit bandwidth, energy spectral flux density, and gravitational wave strain density are given.
2. Neutron capture cross section of {sup 241}Am
SciTech Connect
Jandel, M.; Bredeweg, T. A.; Bond, E. M.; Chadwick, M. B.; Clement, R. R.; Couture, A.; O'Donnell, J. M.; Haight, R. C.; Kawano, T.; Reifarth, R.; Rundberg, R. S.; Ullmann, J. L.; Vieira, D. J.; Wilhelmy, J. B.; Wouters, J. M.; Agvaanluvsan, U.; Parker, W. E.; Wu, C. Y.; Becker, J. A.
2008-09-15
The neutron capture cross section of {sup 241}Am for incident neutrons from 0.02 eV to 320 keV has been measured with the detector for advanced neutron capture experiments (DANCE) at the Los Alamos Neutron Science Center. The thermal neutron capture cross section was determined to be 665{+-}33 b. Our result is in good agreement with other recent measurements. Resonance parameters for E{sub n}<12 eV were obtained using an R-matrix fit to the measured cross section. The results are compared with values from the ENDF/B-VII.0, Mughabghab, JENDL-3.3, and JEFF-3.1 evaluations. {gamma}{sub n} neutron widths for the first three resonances are systematically larger by 5-15% than the ENDF/B-VII.0 values. The resonance integral above 0.5 eV was determined to be 1553{+-}7 b. Cross sections in the resolved and unresolved energy regions above 12 eV were calculated using the Hauser-Feshbach theory incorporating the width-fluctuation correction of Moldauer. The calculated results agree well with the measured data, and the extracted averaged resonance parameters in the unresolved resonance region are consistent with those for the resolved resonances.
3. Neutron-multiplicity experiments for enhanced fission modelling
Al-Adili, Ali; Tarrío, Diego; Hambsch, Franz-Josef; Göök, Alf; Jansson, Kaj; Solders, Andreas; Rakapoulos, Vasileios; Gustavsson, Cecilia; Lantz, Mattias; Mattera, Andrea; Oberstedt, Stephan; Prokofiev, Alexander V.; Sundén, Erik A.; Vidali, Marzio; Österlund, Michael; Pomp, Stephan
2017-09-01
The nuclear de-excitation process of fission fragments (FF) provides fundamental information for the understanding of nuclear fission and nuclear structure in neutron-rich isotopes. The variation of the prompt-neutron multiplicity, ν(A), as a function of the incident neutron energy (En) is one of many open questions. It leads to significantly different treatments in various fission models and implies that experimental data are analyzed based on contradicting assumptions. One critical question is whether the additional excitation energy (Eexc) is manifested through an increase of ν(A) for all fragments or for the heavy ones only. A systematic investigation of ν(A) as a function of En has been initiated. Correlations between prompt-fission neutrons and fission fragments are obtained by using liquid scintillators in conjunction with a Frisch-grid ionization chamber. The proof-of-principle has been achieved on the reaction 235U(nth,f) at the Van De Graff (VdG) accelerator of the JRC-Geel using a fully digital data acquisition system. Neutrons from 252Cf(sf) were measured separately to quantify the neutron-scattering component due to surrounding shielding material and to determine the intrinsic detector efficiency. Prelimenary results on ν(A) and spectrum in correlation with FF properties are presented.
4. High frame-rate neutron radiography of dynamic events
Bossi, R. H.; Robinson, A. H.; Barton, J. P.
1981-11-01
A system was developed to perform neutron radiographic analysis of dynamic events having a duration of several milliseconds. The system is operated in the range of 2000 to 10,000 frames/second. Synchronization provided high speed motion neutron radiographs for evaluation of the firing cycle of 7.62 mm munition rounds within a steel rifle barrel. The system was also used to demonstrate the ability to produce neutron radiographic movies of two phase flow. The equipment uses the Oregon State University TRIGA reactor capable of pulsing to 3000 MW peak power, a neutron beam collimator, a scintillator neutron conversion screen coupled to an image intensifier, and a 16 mm high speed movie camera. The peak neutron flux incident at the object position is approximately 4 x 10 to the 11th power n/sq cm with a pulse full width at half maximum, of 9 ms. The scintillator conversion screens and on the effects of statistical limitations on the image quality were studied and modulation transfer function analysis was used to assist in the evaluation of the system performance.
5. Event-by-event evaluation of the prompt fission neutron spectrum from 239Pu(n,f)
SciTech Connect
Vogt, R; Randrup, J; Brown, D A; Descalle, M A; Ormand, W E
2011-11-28
light and heavy fragments, governing the number and energy of neutrons emitted from each fragment. The latter two parameters are assumed to be independent of the incident neutron energy while the first varies with incident energy. We describe our method and the subsequent spectral evaluation and present the results of several standard validation calculations that test our new evaluation. These benchmarks include critical assemblies, sensitive to criticality in fast systems; pulsed sphere measurements testing the spectra at incident neutron energies of 14 MeV; and other tests.
6. Incidence of Menkes disease.
PubMed
Tønnesen, T; Kleijer, W J; Horn, N
1991-02-01
We have calculated the incidence of Menkes disease for Denmark, France, The Netherlands, the United Kingdom and West Germany, based on known Menkes patients born during the time period 1976-87. Considering live-born Menkes patients, the combined incidence for these five countries is 1 Menkes patient per 298,000 live-born babies. If the number of affected aborted fetuses are taken into account, the incidence is 1 Menkes per 254,000 live-born babies. This incidence, which is 2-4 times lower than earlier published incidence figures, places Menkes disease as an extremely rare disease. The mutation rate for Menkes disease is estimated to be 1.96 x 10(-6), based on the number of isolated Menkes cases born during the time period 1976-87 and the total number of newborn males during this time.
7. GINA-A polarized neutron reflectometer at the Budapest Neutron Centre
SciTech Connect
Bottyan, L.; Merkel, D. G.; Nagy, B.; Sajti, Sz.; Deak, L.; Endroczi, G.; Fuezi, J.; Petrenko, A. V.; Major, J.
2013-01-15
The setup, capabilities, and operation parameters of the neutron reflectometer GINA, the recently installed 'Grazing Incidence Neutron Apparatus' at the Budapest Neutron Centre, are introduced. GINA, a dance-floor-type, constant-energy, angle-dispersive reflectometer is equipped with a 2D position-sensitive detector to study specular and off-specular scattering. Wavelength options between 3.2 and 5.7 A are available for unpolarized and polarized neutrons. Spin polarization and analysis are achieved by magnetized transmission supermirrors and radio-frequency adiabatic spin flippers. As a result of vertical focusing by a five-element pyrolytic graphite monochromator, the reflected intensity from a 20 Multiplication-Sign 20 mm{sup 2} sample has been doubled. GINA is dedicated to studies of magnetic films and heterostructures, but unpolarized options for non-magnetic films, membranes, and other surfaces are also provided. Shortly after its startup, reflectivity values as low as 3 Multiplication-Sign 10{sup -5} have been measured by the instrument. The instrument capabilities are demonstrated by a non-polarized and a polarized reflectivity experiment on a Si wafer and on a magnetic film of [{sup 62}Ni/{sup nat}Ni]{sub 5} isotope-periodic layer composition. The facility is now open for the international user community. Its further development is underway establishing new sample environment options and spin analysis of off-specularly scattered radiation as well as further decreasing the background.
8. GINA--A polarized neutron reflectometer at the Budapest Neutron Centre
Bottyán, L.; Merkel, D. G.; Nagy, B.; Füzi, J.; Sajti, Sz.; Deák, L.; Endrőczi, G.; Petrenko, A. V.; Major, J.
2013-01-01
The setup, capabilities, and operation parameters of the neutron reflectometer GINA, the recently installed "Grazing Incidence Neutron Apparatus" at the Budapest Neutron Centre, are introduced. GINA, a dance-floor-type, constant-energy, angle-dispersive reflectometer is equipped with a 2D position-sensitive detector to study specular and off-specular scattering. Wavelength options between 3.2 and 5.7 Å are available for unpolarized and polarized neutrons. Spin polarization and analysis are achieved by magnetized transmission supermirrors and radio-frequency adiabatic spin flippers. As a result of vertical focusing by a five-element pyrolytic graphite monochromator, the reflected intensity from a 20 × 20 mm2 sample has been doubled. GINA is dedicated to studies of magnetic films and heterostructures, but unpolarized options for non-magnetic films, membranes, and other surfaces are also provided. Shortly after its startup, reflectivity values as low as 3 × 10-5 have been measured by the instrument. The instrument capabilities are demonstrated by a non-polarized and a polarized reflectivity experiment on a Si wafer and on a magnetic film of [62Ni/natNi]5 isotope-periodic layer composition. The facility is now open for the international user community. Its further development is underway establishing new sample environment options and spin analysis of off-specularly scattered radiation as well as further decreasing the background.
9. Hyperons and neutron stars
Vidaña, Isaac
2015-02-01
In this lecture I will briefly review some of the effects of hyperons on the properties of neutron and proto-neutron stars. In particular, I will revise the problem of the strong softening of the EoS, and the consequent reduction of the maximum mass, induced by the presence of hyperons, a puzzle which has become more intringuing and difficult to solve due the recent measurements of the unusually high masses of the millisecond pulsars PSR J1903+0327 (1.667±0.021M⊙), PSR J1614-2230 (1.97±0.04M⊙), and PSR J0348+0432 (2.01±0.04M⊙). Finally, I will also examine the role of hyperons on the cooling properties of newly born neutron stars and on the so-called r-mode instability.
10. Hyperons in Neutron Stars
Vidaña, Isaac
2016-01-01
In this work I briefly review some of the effects of hyperons on the properties of neutron and proto-neutron stars. In particular, I revise the problem of the strong softening of the EoS, and the consequent reduction of the maximum mass, induced by the presence of hyperons, a puzzle which has become more intringuing and difficult to solve because of the recent measurements of the unusually high masses of the millisecond pulsars PSR J1903+0327 (1.667 ± 0.021M⊙), PSR J1614-2230 (1.97 ± 0.04M⊙), and PSR J0348+0432 (2.01 ± 0.04M⊙). Some of the solutions proposed to tackle this problem are discussed. Finally, I re-examine also the role of hyperons on the cooling properties of newly born neutron stars and on the so-called r-mode instability.
11. Personnel electronic neutron dosimeter
DOEpatents
Falk, R.B.; Tyree, W.H.
1982-03-03
A personnel electronic dosimeter includes a neutron-proton and neutron-alpha converter for providing an electrical signal having a magnitude proportional to the energy of a detected proton or alpha particle produced from the converter, a pulse generator circuit for generating a pulse having a duration controlled by the weighed effect of the amplitude of the electrical signal, an oscillator enabled by the pulse for generating a train of clock pulses for a time dependent upon the pulse length, a counter for counting the clock pulses, and an indicator for providing a direct reading and aural alarm when the count indicates that the wearer has been exposed to a selected level of neutron dose equivalent.
12. Personnel electronic neutron dosimeter
DOEpatents
Falk, Roger B.; Tyree, William H.
1984-12-18
A personnel electronic dosimeter includes a neutron-proton and neutron-alpha converter for providing an electrical signal having a magnitude proportional to the energy of a detected proton or alpha particle produced from the converter, a pulse generator circuit for generating a pulse having a duration controlled by the weighed effect of the amplitude of the electrical signal, an oscillator enabled by the pulse for generating a train of clock pulses for a time dependent upon the pulse length, a counter for counting the clock pulses, and an indicator for providing a direct reading and aural alarm when the count indicates that the wearer has been exposed to a selected level of neutron dose equivalent.
13. Spherical neutron generator
DOEpatents
Leung, Ka-Ngo
2006-11-21
A spherical neutron generator is formed with a small spherical target and a spherical shell RF-driven plasma ion source surrounding the target. A deuterium (or deuterium and tritium) ion plasma is produced by RF excitation in the plasma ion source using an RF antenna. The plasma generation region is a spherical shell between an outer chamber and an inner extraction electrode. A spherical neutron generating target is at the center of the chamber and is biased negatively with respect to the extraction electrode which contains many holes. Ions passing through the holes in the extraction electrode are focused onto the target which produces neutrons by D-D or D-T reactions.
14. Ultrafast neutron detector
DOEpatents
Wang, Ching L.
1987-01-01
The invention comprises a neutron detector (50) of very high temporal resolution that is particularly well suited for measuring the fusion reaction neutrons produced by laser-driven inertial confinement fusion targets. The detector comprises a biased two-conductor traveling-wave transmission line (54, 56, 58, 68) having a uranium cathode (60) and a phosphor anode (62) as respective parts of the two conductors. A charge line and Auston switch assembly (70, 72, 74) launch an electric field pulse along the transmission line. Neutrons striking the uranium cathode at a location where the field pulse is passing, are enabled to strike the phosphor anode and produce light that is recorded on photographic film (64). The transmission line may be variously configured to achieve specific experimental goals.
15. METHOD OF PRODUCING NEUTRONS
DOEpatents
Imhoff, D.H.; Harker, W.H.
1964-02-01
A method for producing neutrons is described in which there is employed a confinement zone defined between longitudinally spaced localized gradient regions of an elongated magnetic field. Changed particles and neutralizing electrons, more specifically deuterons and tritons and neutralizng electrons, are injected into the confinement field from ion sources located outside the field. The rotational energy of the parrticles is increased at the gradients by imposing an oscillating transverse electrical field thereacross. The imposition of such oscillating transverse electrical fields improves the reflection capability of such gradient fielda so that the reactive particles are retained more effectively within the zone. With the attainment of appropriate densities of plasma particles and provided that such particles are at a sufficiently high temperature, neutron-producing reactions ensue and large quantities of neutrons emerge from the containment zone. (AEC)
16. Neutronic reactor construction
DOEpatents
Huston, Norman E.
1976-07-06
1. A neutronic reactor comprising a moderator including horizontal layers formed of horizontal rows of graphite blocks, alternate layers of blocks having the rows extending in one direction, the remaining alternate layers having the rows extending transversely to the said one direction, alternate rows of blocks in one set of alternate layers having longitudinal ducts, the moderator further including slotted graphite tubes positioned in the ducts, the reactor further comprising an aluminum coolant tube positioned within the slotted tube in spaced relation thereto, bodies of thermal-neutron-fissionable material, and jackets enclosing the bodies and being formed of a corrosion-resistant material having a low neutron-capture cross section, the bodies and jackets being positioned within the coolant tube so that the jackets are spaced from the coolant tube.
17. NEUTRONIC REACTOR CORE INSTRUMENT
DOEpatents
Mims, L.S.
1961-08-22
A multi-purpose instrument for measuring neutron flux, coolant flow rate, and coolant temperature in a nuclear reactor is described. The device consists essentially of a hollow thimble containing a heat conducting element protruding from the inner wall, the element containing on its innermost end an amount of fissionsble materinl to function as a heat source when subjected to neutron flux irradiation. Thermocouple type temperature sensing means are placed on the heat conducting element adjacent the fissionable material and at a point spaced therefrom, and at a point on the thimble which is in contact with the coolant fluid. The temperature differentials measured between the thermocouples are determinative of the neutron flux, coolant flow, and temperature being measured. The device may be utilized as a probe or may be incorporated in a reactor core. (AE C)
18. Hyperons and neutron stars
SciTech Connect
Vidaña, Isaac
2015-02-24
In this lecture I will briefly review some of the effects of hyperons on the properties of neutron and proto-neutron stars. In particular, I will revise the problem of the strong softening of the EoS, and the consequent reduction of the maximum mass, induced by the presence of hyperons, a puzzle which has become more intringuing and difficult to solve due the recent measurements of the unusually high masses of the millisecond pulsars PSR J1903+0327 (1.667±0.021M{sub ⊙}), PSR J1614–2230 (1.97±0.04M{sub ⊙}), and PSR J0348+0432 (2.01±0.04M{sub ⊙}). Finally, I will also examine the role of hyperons on the cooling properties of newly born neutron stars and on the so-called r-mode instability.
19. Neutron star crusts
NASA Technical Reports Server (NTRS)
Lorenz, C. P.; Ravenhall, D. G.; Pethick, C. J.
1993-01-01
We calculate properties of neutron star matter at subnuclear densities using an improved nuclear Hamiltonian. Nuclei disappear and the matter becomes uniform at a density of about 0.6n(s), where n(s) of about 0.16/cu fm is the saturation density of nuclear matter. As a consequence, the mass of matter in the crusts of neutron stars is only about half as large as previously estimated. In about half of that crustal mass, nuclear matter occurs in shapes very different from the roughly spherical nuclei familiar at lower densities. The thinner crust and the unusual nuclear shape have important consequences for theories of the rotational and thermal evolution of neutron stars, especialy theories of glitches.
20. Dutch perfusion incident survey.
PubMed
Groenenberg, Ingrid; Weerwind, Patrick W; Everts, Peter A M; Maessen, Jos G
2010-09-01
Cardiopulmonary bypass procedures remain complex, involving many potential risks. Therefore, a nationwide retrospective study was conducted to gain insight into the number of incidents and accidents in Dutch adult perfusion practice. An anonymous postal survey (85 questions about hardware, disposables, fluids and medication, air emboli, anticoagulation, practice, and safety measures) was sent to all Dutch perfusionists involved in adult cardiovascular perfusion during 2006 and 2007. To guarantee complete anonymity, respondents were asked to return the survey to a notary who discarded personal information. The net response rate was 72% and covered 23,500 perfusions. Individual respondents performed 240 ± 103 perfusions during the 2-year study period and had 13.8 ± 8.7 years of practical experience. The incident rate was 1 per 15.6 perfusions and the adverse event rate was 1 per 1,236 perfusions. The three most reported incidents were: (1) persistent inability to raise the activated coagulation time above 400s during perfusion (184 incidents); (2) an allergic or anaphylactic reaction to drugs, fluids, or blood products (114 incidents); and (3) clotting formation in the extracorporeal circuit (74 incidents). Furthermore, pre-bypass safety measures showed no statistically significant association with the reported incidents. In comparison with data from the recent literature, the reported number of incidents is high. Nevertheless, the adverse outcome rate is well matched to other published surveys. The relatively high response rate conveys the impression that the Dutch perfusionist is vigilant and willing to report incidents. Hence, a web-based Dutch perfusion incident registration system is recommended.
1. Corrosion resistant neutron absorbing coatings
DOEpatents
Choi, Jor-Shan [El Cerrito, CA; Farmer, Joseph C [Tracy, CA; Lee, Chuck K [Hayward, CA; Walker, Jeffrey [Gaithersburg, MD; Russell, Paige [Las Vegas, NV; Kirkwood, Jon [Saint Leonard, MD; Yang, Nancy [Lafayette, CA; Champagne, Victor [Oxford, PA
2012-05-29
A method of forming a corrosion resistant neutron absorbing coating comprising the steps of spray or deposition or sputtering or welding processing to form a composite material made of a spray or deposition or sputtering or welding material, and a neutron absorbing material. Also a corrosion resistant neutron absorbing coating comprising a composite material made of a spray or deposition or sputtering or welding material, and a neutron absorbing material.
2. Corrosion resistant neutron absorbing coatings
DOEpatents
Choi, Jor-Shan; Farmer, Joseph C; Lee, Chuck K; Walker, Jeffrey; Russell, Paige; Kirkwood, Jon; Yang, Nancy; Champagne, Victor
2013-11-12
A method of forming a corrosion resistant neutron absorbing coating comprising the steps of spray or deposition or sputtering or welding processing to form a composite material made of a spray or deposition or sputtering or welding material, and a neutron absorbing material. Also a corrosion resistant neutron absorbing coating comprising a composite material made of a spray or deposition or sputtering or welding material, and a neutron absorbing material.
3. Pixelated neutron image plates
Schlapp, M.; Conrad, H.; von Seggern, H.
2004-09-01
Neutron image plates (NIPs) have found widespread application as neutron detectors for single-crystal and powder diffraction, small-angle scattering and tomography. After neutron exposure, the image plate can be read out by scanning with a laser. Commercially available NIPs consist of a powder mixture of BaFBr : Eu2+ and Gd2O3 dispersed in a polymer matrix and supported by a flexible polymer sheet. Since BaFBr : Eu2+ is an excellent x-ray storage phosphor, these NIPs are particularly sensitive to ggr-radiation, which is always present as a background radiation in neutron experiments. In this work we present results on NIPs consisting of KCl : Eu2+ and LiF that were fabricated into ceramic image plates in which the alkali halides act as a self-supporting matrix without the necessity for using a polymeric binder. An advantage of this type of NIP is the significantly reduced ggr-sensitivity. However, the much lower neutron absorption cross section of LiF compared with Gd2O3 demands a thicker image plate for obtaining comparable neutron absorption. The greater thickness of the NIP inevitably leads to a loss in spatial resolution of the image plate. However, this reduction in resolution can be restricted by a novel image plate concept in which a ceramic structure with square cells (referred to as a 'honeycomb') is embedded in the NIP, resulting in a pixelated image plate. In such a NIP the read-out light is confined to the particular illuminated pixel, decoupling the spatial resolution from the optical properties of the image plate material and morphology. In this work, a comparison of experimentally determined and simulated spatial resolutions of pixelated and unstructured image plates for a fixed read-out laser intensity is presented, as well as simulations of the properties of these NIPs at higher laser powers.
4. POLARIZED NEUTRONS IN RHIC
SciTech Connect
COURANT,E.D.
1998-04-27
There does not appear to be any obvious way to accelerate neutrons, polarized or otherwise, to high energies by themselves. To investigate the behavior of polarized neutrons the authors therefore have to obtain them by accelerating them as components of heavier nuclei, and then sorting out the contribution of the neutrons in the analysis of the reactions produced by the heavy ion beams. The best neutron carriers for this purpose are probably {sup 3}He nuclei and deuterons. A polarized deuteron is primarily a combination of a proton and a neutron with their spins pointing in the same direction; in the {sup 3}He nucleus the spins of the two protons are opposite and the net spin (and magnetic moment) is almost the same as that of a free neutron. Polarized ions other than protons may be accelerated, stored and collided in a ring such as RHIC provided the techniques proposed for polarized proton operation can be adapted (or replaced by other strategies) for these ions. To accelerate polarized particles in a ring, one must make provisions for overcoming the depolarizing resonances that occur at certain energies. These resonances arise when the spin tune (ratio of spin precession frequency to orbit frequency) resonates with a component present in the horizontal field. The horizontal field oscillates with the vertical motion of the particles (due to vertical focusing); its frequency spectrum is dominated by the vertical oscillation frequency and its modulation by the periodic structure of the accelerator ring. In addition, the magnet imperfections that distort the closed orbit vertically contain all integral Fourier harmonics of the orbit frequency.
5. Improvement of dose distribution by central beam shielding in boron neutron capture therapy.
PubMed
Sakurai, Yoshinori; Ono, Koji
2007-12-21
Since boron neutron capture therapy (BNCT) with epithermal neutron beams started at the Kyoto University Reactor (KUR) in June 2002, nearly 200 BNCT treatments have been carried out. The epithermal neutron irradiation significantly improves the dose distribution, compared with the previous irradiation mainly using thermal neutrons. However, the treatable depth limit still remains. One effective technique to improve the limit is the central shield method. Simulations were performed for the incident neutron energies and the annular components of the neutron source. It was clear that thermal neutron flux distribution could be improved by decreasing the lower energy neutron component and the inner annular component of the incident beam. It was found that a central shield of 4-6 cm diameter and 10 mm thickness is effective for the 12 cm diameter irradiation field. In BNCT at KUR, the depth dose distribution can be much improved by the central shield method, resulting in a relative increase of the dose at 8 cm depth by about 30%. In addition to the depth dose distribution, the depth dose profile is also improved. As the dose rate in the central area is reduced by the additional shielding, the necessary irradiation time, however, increases by about 30% compared to normal treatment.
SciTech Connect
Shaikh, A. M.; Shylaja, D.
2011-07-15
In the present paper we give a new methodology named, 'neutron induced beta radiography-NIBR' which makes use of neutron activated Dy or In foils as source of (3-radiation. Radiographs are obtained with an aluminium cassette containing image plate, a sample under inspection and the activated Dy or In foil kept in tight contact. The sensitivity of the technique to thickness was evaluated for different materials in the form of step wedges. Some radiographs are presented to demonstrate potential of method to inspect thin samples.
7. Semiconductor neutron detectors
Gueorguiev, Andrey; Hong, Huicong; Tower, Joshua; Kim, Hadong; Cirignano, Leonard; Burger, Arnold; Shah, Kanai
2016-09-01
Lithium Indium Selenide (LiInSe2) has been under development in RMD Inc. and Fisk University for room temperature thermal neutron detection due to a number of promising properties. The recent advances of the crystal growth, material processing, and detector fabrication technologies allowed us to fabricate large detectors with 100 mm2 active area. The thermal neutron detection sensitivity and gamma rejection ratio (GRR) were comparable to 3He tube with 10 atm gas pressure at comparable dimensions. The synthesis, crystal growth, detector fabrication, and characterization are reported in this paper.
8. Simplified fast neutron dosimeter
DOEpatents
Sohrabi, Mehdi
1979-01-01
Direct fast-neutron-induced recoil and alpha particle tracks in polycarbonate films may be enlarged for direct visual observation and automated counting procedures employing electrochemical etching techniques. Electrochemical etching is, for example, carried out in a 28% KOH solution at room temperature by applying a 2000 V peak-to-peak voltage at 1 kHz frequency. Such recoil particle amplification can be used for the detection of wide neutron dose ranges from 1 mrad. to 1000 rads. or higher, if desired.
9. Fast neutron dosimetry
SciTech Connect
DeLuca, P.M. Jr.; Pearson, D.W.
1992-01-01
This progress report concentrates on two major areas of dosimetry research: measurement of fast neutron kerma factors for several elements for monochromatic and white spectrum neutron fields and determination of the response of thermoluminescent phosphors to various ultra-soft X-ray energies and beta-rays. Dr. Zhixin Zhou from the Shanghai Institute of Radiation Medicine, People's Republic of China brought with him special expertise in the fabrication and use of ultra-thin TLD materials. Such materials are not available in the USA. The rather unique properties of these materials were investigated during this grant period.
10. FAST NEUTRONIC REACTOR
DOEpatents
Snell, A.H.
1957-12-01
This patent relates to a reactor and process for carrying out a controlled fast neutron chain reaction. A cubical reactive mass, weighing at least 920 metric tons, of uranium metal containing predominantly U/sup 238/ and having a U/sup 235/ content of at least 7.63% is assembled and the maximum neutron reproduction ratio is limited to not substantially over 1.01 by insertion and removal of a varying amount of boron, the reactive mass being substantially freed of moderator.
11. Neutrinos from neutron stars
NASA Technical Reports Server (NTRS)
Helfand, D. J.
1979-01-01
A calculation of the flux of ultra-high energy neutrinos from galactic neutron stars is presented. The calculation is used to determine the number of point sources detectable at the sensitivity threshold of a proposed deep underwater muon and neutrino detector array. The detector array would have a point source detection threshold of about 100 eV/sq cm-sec. Analysis of neutrino luminosities and the number of detectable sources suggests that the deep underwater detector may make a few discoveries. In particular, a suspected neutron star in the Cyg X-3 source seems a promising target for the deep underwater array.
12. Lunar neutron stratigraphy.
NASA Technical Reports Server (NTRS)
Russ, G. P., III; Burnett, D. S.; Wasserburg, G. J.
1972-01-01
Study of the isotopic composition of gadolinium and samarium in four soil and seven drill stem samples returned by the Apollo 15 mission. The results show the possibility to date sedimentary processes on the lunar surface for time scales of around 100 million years because of the particular dependence of neutron capture reactions on depth. The neutron flux has a distinct peak as a function of depth. This peak appears to lie below the level of shallow cratering for time scales of less than one billion years and consequently forms a readily identified marker layer of both depth and time.
13. Lunar neutron stratigraphy.
NASA Technical Reports Server (NTRS)
Russ, G. P., III; Burnett, D. S.; Wasserburg, G. J.
1972-01-01
Study of the isotopic composition of gadolinium and samarium in four soil and seven drill stem samples returned by the Apollo 15 mission. The results show the possibility to date sedimentary processes on the lunar surface for time scales of around 100 million years because of the particular dependence of neutron capture reactions on depth. The neutron flux has a distinct peak as a function of depth. This peak appears to lie below the level of shallow cratering for time scales of less than one billion years and consequently forms a readily identified marker layer of both depth and time.
14. GUIDE FOR POLARIZED NEUTRONS
DOEpatents
Sailor, V.L.; Aichroth, R.W.
1962-12-01
The plane of polarization of a beam of polarized neutrons is changed by this invention, and the plane can be flipped back and forth quicitly in two directions in a trouble-free manner. The invention comprises a guide having a plurality of oppositely directed magnets forming a gap for the neutron beam and the gaps are spaced longitudinally in a spiral along the beam at small stepped angles. When it is desired to flip the plane of polarization the magnets are suitably rotated to change the direction of the spiral of the gaps. (AEC)
15. The neutron star zoo
Harding, Alice K.
2013-12-01
Neutron stars are a very diverse population, both in their observational and their physical properties. They prefer to radiate most of their energy at X-ray and gamma-ray wavelengths. But whether their emission is powered by rotation, accretion, heat, magnetic fields or nuclear reactions, they are all different species of the same animal whose magnetic field evolution and interior composition remain a mystery. This article will broadly review the properties of inhabitants of the neutron star zoo, with emphasis on their high-energy emission.
16. NEUTRONIC REACTOR CONTROL ELEMENT
DOEpatents
Newson, H.W.
1960-09-13
A novel composite neutronic reactor control element is offered. The element comprises a multiplicity of sections arranged in end-to-end relationship, each of the sections having a markedly different neutron-reactive characteristic. For example, a three-section control element could contain absorber, moderator, and fuel sections. By moving such an element longitudinally through a reactor core, reactivity is decreased by the absorber, increased slightly by the moderator, or increased substantially by the fuel. Thus, control over a wide reactivity range is provided.
17. Neutron activated switch
DOEpatents
Barton, David M.
1991-01-01
A switch for reacting quickly to a neutron emission. A rod consisting of fissionable material is located inside a vacuum tight body. An adjustable contact is located coaxially at an adjustable distance from one end of the rod. Electrical leads are connected to the rod and to the adjustable contact. With a vacuum drawn inside the body, a neutron bombardment striking the rod causes it to heat and expand longitudinally until it comes into contact with the adjustable contact. This circuit closing occurs within a period of a few microseconds.
18. New compact neutron polarizer
Krist, Th; Kennedy, S. J.; Hicks, T. J.; Mezei, F.
A new type of a neutron polarizing bender was developed in co-operation with BENSC and ANSTO. It is based upon bent thin silicon wafers coated on one side with SiFeCo polarizing supermirrors and on the other side with Gd. Initial tests at BENSC in a 300 Oe magnetic field yielded a transmission of spin-up neutrons of about 55% over an angle range of 0.75° and flipping ratios > 30. Subsequent tests at ANSTO at 1200 Oe yielded a transmission of 48% with a flipping ratio > 45.
19. Helium 3 neutron precision polarimetry
Menard, Christopher
2009-10-01
Measuring neutron polarization to a high degree of precision is critical for the next generation of neutron decay correlation experiments. Polarized neutrons are also used in experiments to probe the hadronic weak interaction which contributes a small portion (˜10-7) of the force between nucleons. Using a beam of cold neutrons at Los Alamos Neutron Science Center (LANSCE), we polarized neutrons and measured their absolute polarization to ˜0.1%. Neutrons were polarized by passing them through a ^3He spin filter, relying on the maximally spin dependent 3He neutron absorption cross section. The neutron polarization can be determined by measuring the wavelength-dependent neutron transmission through the ^3He cell. An independent measurement of the neutron polarization was also obtained by passing the polarized beam through an RF spin flipper and a second polarized ^3He cell, used as an analyzer. To measure the efficiency of the spin flipper, the same measurements were made after reversing the ^3He polarization in the polarizer by using NMR techniques (adiabatic fast passage). We will show the consistency of these two measurements and the resulting precision of neutron polarimetry using these techniques.
20. Characterisation of the incident beam and current diffraction capabilities on the VESUVIO spectrometer
Romanelli, G.; Krzystyniak, M.; Senesi, R.; Raspino, D.; Boxall, J.; Pooley, D.; Moorby, S.; Schooneveld, E.; Rhodes, N. J.; Andreani, C.; Fernandez-Alonso, F.
2017-09-01
The VESUVIO spectrometer at the ISIS pulsed neutron and muon source is a unique instrument amongst those available at neutron facilities. This is the only inverted-geometry neutron spectrometer accessing values of energy and wavevector transfer above tens of eV and {\\mathringA}-1 , respectively, and where deep inelastic neutron scattering experiments are routinely performed. As such, the procedure at the base of the technique has been previously described in an article published by this journal (Mayers and Reiter 2012 Meas. Sci. Technol. 23 045902). The instrument has recently witnessed an upsurge of interest due to a new trend to accommodate, within a single experiment, neutron diffraction and transmission measurements in addition to deep inelastic neutron scattering. This work presents a broader description of the instrument following these recent developments. In particular, we assess the absolute intensity and two-dimensional profile of the incident neutron beam and the capabilities of the backscattering diffraction banks. All results are discussed in the light of recent changes to the moderator viewed by the instrument. We find that VESUVIO has to be considered a high-resolution diffractometer as much as other diffractometers at ISIS, with a resolution as high as 2× 10-3 in backscattering. Also, we describe the extension of the wavelength range of the instrument to include lower neutron energies for diffraction measurements, an upgrade that could be readily applied to other neutron instruments as well.
1. Neutron-Mirror-Neutron Oscillations: How Fast Might They Be?
SciTech Connect
Berezhiani, Zurab; Bento, Luis
2006-03-03
We discuss the phenomenological implications of the neutron (n) oscillation into the mirror neutron (n{sup '}), a hypothetical particle exactly degenerate in mass with the neutron but sterile to normal matter. We show that the present experimental data allow a maximal n-n{sup '} oscillation in vacuum with a characteristic time {tau} much shorter than the neutron lifetime, in fact as small as 1 sec. This phenomenon may manifest in neutron disappearance and regeneration experiments perfectly accessible to present experimental capabilities and may also have interesting astrophysical consequences, in particular, for the propagation of ultra high energy cosmic rays.
2. SPECTER: neutron damage calculations for materials irradiations
SciTech Connect
Greenwood, L.R.; Smither, R.K.
1985-01-01
Neutron displacement damage-energy cross sections have been calculated for 41 isotopes in the energy range from 10/sup -10/ to 20 MeV. Calculations were performed on a 100-point energy grid using nuclear cross sections from ENDF/B-V and the DISCS computer code. Elastic scattering is treated exactly including angular distributions from ENDF/B-V. Inelastic scattering calculations consider both discrete and continuous nuclear level distributions. Multiple (n,xn) reactions use a Monte Carlo technique to derive the recoil distributions. The (n,d) and (n,t) reactions are treated as (n,p) and (n,/sup 3/He) as (n,/sup 4/He). The (n,..gamma..) reaction and subsequent ..beta..-decay are also included, using a new treatment of ..gamma..-..gamma.. coincidences, angular correlations, ..beta..-neutrino correlations, and the incident neutron energy. The Lindhard model was used to compute the energy available for nuclear displacement at each recoil energy. The SPECTER computer code has been developed to simplify damage calculations. The user need only specify a neutron energy spectrum. SPECTER will then calculate spectral-averaged displacements, recoil spectra, gas production, and total damage energy (Kerma). The SPECTER computer code package is readily accessible to the fusion community via the National Magnetic Fusion Energy Computer Center (NMFECC) at Lawrence Livermore National laboratory.
3. Fast neutron measurements with 7Li and 6Li enriched CLYC scintillators
Giaz, A.; Blasi, N.; Boiano, C.; Brambilla, S.; Camera, F.; Cattadori, C.; Ceruti, S.; Gramegna, F.; Marchi, T.; Mattei, I.; Mentana, A.; Million, B.; Pellegri, L.; Rebai, M.; Riboldi, S.; Salamida, F.; Tardocchi, M.
2016-07-01
The recently developed Cs2LiYCl6:Ce (CLYC) crystals are interesting scintillation detectors not only for their gamma energy resolution (<5% at 662 keV) but also for their capability to identify and measure the energy of both gamma rays and fast/thermal neutrons. The thermal neutrons were detected by the 6Li(n,α)t reaction while for the fast neutrons the 35Cl(n,p)35S and 35Cl(n,α)32P neutron-capture reactions were exploited. The energy of the outgoing proton or α particle scales linearly with the incident neutron energy. The kinetic energy of the fast neutrons can be measured using both the Time Of Flight (TOF) technique and using the CLYC energy signal. In this work, the response to monochromatic fast neutrons (1.9-3.8 MeV) of two CLYC 1″×1″ crystals was measured using both the TOF and the energy signal. The observables were combined to identify fast neutrons, to subtract the thermal neutron background and to identify different fast neutron-capture reactions on 35Cl, in other words to understand if the detected particle is an α or a proton. We performed a dedicated measurement at the CN accelerator facility of the INFN Legnaro National Laboratories (Italy), where the fast neutrons were produced by impinging a proton beam (4.5, 5.0 and 5.5 MeV) on a 7LiF target. We tested a CLYC detector 6Li-enriched at about 95%, which is ideal for thermal neutron measurements, in parallel with another CLYC detector 7Li-enriched at more than 99%, which is suitable for fast neutron measurements.
4. Smaller, Lower-Power Fast-Neutron Scintillation Detectors
NASA Technical Reports Server (NTRS)
Patel, Jagdish; Blaes, Brent
2008-01-01
Scintillation-based fast-neutron detectors that are smaller and less power-hungry than mainstream scintillation-based fast-neutron detectors are undergoing development. There are numerous applications for such detectors in monitoring fast-neutron fluxes from nuclear reactors, nuclear materials, and natural sources, both on Earth and in outer space. A particularly important terrestrial application for small, low-power, portable fast-neutron detectors lies in the requirement to scan for nuclear materials in cargo and baggage arriving at international transportation facilities. The present development of miniature, low-power scintillation-based fast-neutron detectors exploits recent advances in the fabrication of avalanche photodiodes (APDs). Basically, such a detector includes a plastic scintillator, typically between 300 and 400 m thick with very thin silver mirror coating on all its faces except the one bonded to an APD. All photons generated from scintillation are thus internally reflected and eventually directed to the APD. This design affords not only compactness but also tight optical coupling for utilization of a relatively large proportion of the scintillation light. The combination of this tight coupling and the avalanche-multiplication gain (typically between 750 and 1,000) of the APD is expected to have enough sensitivity to enable monitoring of a fast-neutron flux as small as 1,000 cm(exp -2)s(exp -1). Moreover, pulse-height analysis can be expected to provide information on the kinetic energies of incident neutrons. It has been estimated that a complete, fully developed fast-neutron detector of this type, would be characterized by linear dimensions of the order of 10 cm or less, a mass of no more than about 0.5 kg, and a power demand of no more than a few watts.
5. Dose-equivalent neutron dosimeter
DOEpatents
Griffith, R.V.; Hankins, D.E.; Tomasino, L.; Gomaa, M.A.M.
1981-01-07
A neutron dosimeter is disclosed which provides a single measurement indicating the amount of potential biological damage resulting from the neutron exposure of the wearer, for a wide range of neutron energies. The dosimeter includes a detecting sheet of track etch detecting material such as a carbonate plastic, for detecting higher energy neutrons, and a radiator layer contaning conversion material such as /sup 6/Li and /sup 10/B lying adjacent to the detecting sheet for converting moderate energy neutrons to alpha particles that produce tracks in the adjacent detecting sheet.
6. Neutron proton crystallography station (PCS)
SciTech Connect
Fisher, Zoe; Kovalevsky, Andrey; Johnson, Hannah; Mustyakimov, Marat
2009-01-01
The PCS (Protein Crystallography Station) at Los Alamos Neutron Science Center (LANSCE) is a unique facility in the USA that is designed and optimized for detecting and collecting neutron diffraction data from macromolecular crystals. PCS utilizes the 20 Hz spallation neutron source at LANSCE to enable time-of-flight measurements using 0.6-7.0 {angstrom} neutrons. This increases the neutron flux on the sample by using a wavelength range that is optimal for studying macromolecular crystal structures. The diagram below show a schematic of PCS and photos of the detector and instrument cave.
7. Solid state neutron detector array
DOEpatents
Seidel, John G.; Ruddy, Frank H.; Brandt, Charles D.; Dulloo, Abdul R.; Lott, Randy G.; Sirianni, Ernest; Wilson, Randall O.
1999-01-01
A neutron detector array is capable of measuring a wide range of neutron fluxes. The array includes multiple semiconductor neutron detectors. Each detector has a semiconductor active region that is resistant to radiation damage. In one embodiment, the array preferably has a relatively small size, making it possible to place the array in confined locations. The ability of the array to detect a wide range of neutron fluxes is highly advantageous for many applications such as detecting neutron flux during start up, ramp up and full power of nuclear reactors.
8. Solid state neutron detector array
DOEpatents
Seidel, J.G.; Ruddy, F.H.; Brandt, C.D.; Dulloo, A.R.; Lott, R.G.; Sirianni, E.; Wilson, R.O.
1999-08-17
A neutron detector array is capable of measuring a wide range of neutron fluxes. The array includes multiple semiconductor neutron detectors. Each detector has a semiconductor active region that is resistant to radiation damage. In one embodiment, the array preferably has a relatively small size, making it possible to place the array in confined locations. The ability of the array to detect a wide range of neutron fluxes is highly advantageous for many applications such as detecting neutron flux during start up, ramp up and full power of nuclear reactors. 7 figs.
9. Accelerator based epithermal neutron source
2015-11-01
We review the current status of the development of accelerator sources of epithermal neutrons for boron neutron capture therapy (BNCT), a promising method of malignant tumor treatment. Particular attention is given to the source of epithermal neutrons on the basis of a new type of charged particle accelerator: tandem accelerator with vacuum insulation and lithium neutron-producing target. It is also shown that the accelerator with specialized targets makes it possible to generate fast and monoenergetic neutrons, resonance and monoenergetic gamma-rays, alpha-particles, and positrons.
10. Solid state neutron detector array
SciTech Connect
Seidel, J.G.; Ruddy, F.H.; Brandt, C.D.; Dulloo, A.R.; Lott, R.G.; Sirianni, E.; Wilson, R.O.
1999-08-17
A neutron detector array is capable of measuring a wide range of neutron fluxes. The array includes multiple semiconductor neutron detectors. Each detector has a semiconductor active region that is resistant to radiation damage. In one embodiment, the array preferably has a relatively small size, making it possible to place the array in confined locations. The ability of the array to detect a wide range of neutron fluxes is highly advantageous for many applications such as detecting neutron flux during start up, ramp up and full power of nuclear reactors. 7 figs.
11. Neutron-image intensifier
NASA Technical Reports Server (NTRS)
Berger, H.
1970-01-01
Electronic intensifier tube with a demagnification ratio of 9-1 enhances the usefulness of neutron-radiographic techniques. A television signal can be obtained by optical coupling of a small-output phosphor-light image to a television camera.
12. NEUTRONIC REACTOR SHIELDING
DOEpatents
Borst, L.B.
1961-07-11
A special hydrogenous concrete shielding for reactors is described. In addition to Portland cement and water, the concrete essentially comprises 30 to 60% by weight barytes aggregate for enhanced attenuation of fast neutrons. The biological shields of AEC's Oak Ridge Graphite Reactor and Materials Testing Reactor are particular embodiments.
13. Neutronic reactor thermal shield
DOEpatents
Wende, Charles W. J.
1976-06-15
1. The method of operating a water-cooled neutronic reactor having a graphite moderator which comprises flowing a gaseous mixture of carbon dioxide and helium, in which the helium comprises 40-60 volume percent of the mixture, in contact with the graphite moderator.
14. Neutron Star Phenomena
NASA Technical Reports Server (NTRS)
Ruderman, Malvin
1998-01-01
Various phenomena involving neutron stars are addressed. Electron-positron production in the near magnetosphere of gamma-ray pulsars is discussed along with magnetic field evolution in spun-up and spinning-down pulsars. Glitches and gamma-ray central engines are also discussed.
15. Neutron star evolutionary sequences
NASA Technical Reports Server (NTRS)
Richardson, M. B.; Van Horn, H. M.; Ratcliff, K. F.; Malone, R. C.
1982-01-01
Detailed numerical calculations which are solutions of the full set of general relativistic equations describing the evolution of a spherical star are presented, for the case of the evolution of neutron stars that are cooling over the central temperatures range of 10 to the 10th to 10 to the 7th K. The effects of nucleon superfluidity in the inner crust and core are included, and models are constructed with and without a pion condensate at high densities. It is found that the localized neutrino cooling which dominates the early evolution of neutron stars is so rapid that heat transport within the star cannot keep pace, and temperature distribution is not isothermal. The residual contraction of the neutron star during the early cooling phase contributes little to the heat budget of the star, and most of the gravitational energy released raises the Fermi energy of the degenerate nucleons. It is concluded that since calculations with and without pion condensate are consistent with the upper limits of current observations, these are not sufficient in distinguishing between the various models of neutron star cooling.
16. NEUTRONIC REACTOR CONTROL ELEMENT
DOEpatents
Beaver, R.J.; Leitten, C.F. Jr.
1962-04-17
A boron-10 containing reactor control element wherein the boron-10 is dispersed in a matrix material is describeri. The concentration of boron-10 in the matrix varies transversely across the element from a minimum at the surface to a maximum at the center of the element, prior to exposure to neutrons. (AEC)
17. Neutron metrology for SBSS
SciTech Connect
Morris, C.L.; Armijo, V.; Gomez, J.J.; Hart, G.W.; Lee, D.M.; Zumbro, J.D.; Espinosa, C.; Anaya, J.M.; Bowles, T.J.; Hill, R.E.; Morley, K.B.; Seestrom, S.J.; Ullmann, J.; Taddeucci, T.N.; Teasdale, W.A.; Hahn, S.F.
1998-01-01
This is the final report of a two-year, Laboratory Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). The goal of this work is to develop new detector technologies for Science-Based Stockpile Stewardship (SBSS) at the Los Alamos Neutron Scattering Center (LANSCE) using existing expertise and infrastructure from the nuclear and particle physics programs at LANL.
18. Neutron metrology for SBSS
SciTech Connect
Morris, C.L.; Anaya, J.M.; Armijo, V.
1998-12-31
This is the final report of a two-year, Laboratory Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). The goal of this work is to develop new detector technologies for Science-Based Stockpile Stewardship (SBSS) at the Los Alamos Neutron Scattering Center (LANSCE) using existing expertise and infrastructure from the nuclear and particle physics programs at LANL.
19. Neutronic Reactor Structure
DOEpatents
Vernon, H. C.; Weinberg, A. M.
1961-05-30
The neutronic reactor is comprised of a core consisting of natural uranium and heavy water with a K-factor greater than unity. The core is surrounded by a reflector consisting of natural uranium and ordinary water with a Kfactor less than unity. (AEC)
20. Neutron Absorbing Alloys
SciTech Connect
Mizia, Ronald E.; Shaber, Eric L.; DuPont, John N.; Robino, Charles V.; Williams, David B.
2004-05-04
The present invention is drawn to new classes of advanced neutron absorbing structural materials for use in spent nuclear fuel applications requiring structural strength, weldability, and long term corrosion resistance. Particularly, an austenitic stainless steel alloy containing gadolinium and less than 5% of a ferrite content is disclosed. Additionally, a nickel-based alloy containing gadolinium and greater than 50% nickel is also disclosed.
1. Surface Mounted Neutron Generators
Elizondo-Decanini, Juan M.
2012-10-01
A deuterium-tritium (DT) base reaction pulsed neutron generator packaged in a flat computer chip shape of 1.54 cm (0.600 in) wide by 3.175 cm (1.25 in) length and 0.3 cm (0.120 in) thick has been successfully demonstrated to produce 14 MeV neutrons at a rate of 10^9 neutrons per second. The neutron generator is based on a deuterium ion beam accelerated to impact a tritium loaded target. The accelerating voltage is in the 15 to 20 kV in a 3 mm (0.120 in) gap, the ion beam is shaped by using a lens design to produce a flat ion beam that conforms to the flat rectangular target. The ion source is a simple surface mounted deuterium filled titanium film with a fused gap that operates at a current-voltage design to release the deuterium during a pulse length of about 1 μs. We present the general description of the working prototypes, which we have labeled the NEUTRISTOR.''[4pt] Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration. Work funded by the LDRD office.
2. Small Angle Neutron Scattering
SciTech Connect
Urban, Volker S
2012-01-01
Small Angle Neutron Scattering (SANS) probes structural details at the nanometer scale in a non-destructive way. This article gives an introduction to scientists who have no prior small-angle scattering knowledge, but who seek a technique that allows elucidating structural information in challenging situations that thwart approaches by other methods. SANS is applicable to a wide variety of materials including metals and alloys, ceramics, concrete, glasses, polymers, composites and biological materials. Isotope and magnetic interactions provide unique methods for labeling and contrast variation to highlight specific structural features of interest. In situ studies of a material s responses to temperature, pressure, shear, magnetic and electric fields, etc., are feasible as a result of the high penetrating power of neutrons. SANS provides statistical information on significant structural features averaged over the probed sample volume, and one can use SANS to quantify with high precision the structural details that are observed, for example, in electron microscopy. Neutron scattering is non-destructive; there is no need to cut specimens into thin sections, and neutrons penetrate deeply, providing information on the bulk material, free from surface effects. The basic principles of a SANS experiment are fairly simple, but the measurement, analysis and interpretation of small angle scattering data involves theoretical concepts that are unique to the technique and that are not widely known. This article includes a concise description of the basics, as well as practical know-how that is essential for a successful SANS experiment.
3. NEUTRONIC REACTOR STRUCTURE
DOEpatents
Weinberg, A.M.; Vernon, H.C.
1961-05-30
A neutronic reactor is described. It has a core consisting of natural uranium and heavy water and having a K-factor greater than unity which is surrounded by a reflector consisting of natural uranium and ordinary water having a Kfactor less than unity.
4. Neutron multiplicity analysis tool
SciTech Connect
Stewart, Scott L
2010-01-01
I describe the capabilities of the EXCOM (EXcel based COincidence and Multiplicity) calculation tool which is used to analyze experimental data or simulated neutron multiplicity data. The input to the program is the count-rate data (including the multiplicity distribution) for a measurement, the isotopic composition of the sample and relevant dates. The program carries out deadtime correction and background subtraction and then performs a number of analyses. These are: passive calibration curve, known alpha and multiplicity analysis. The latter is done with both the point model and with the weighted point model. In the current application EXCOM carries out the rapid analysis of Monte Carlo calculated quantities and allows the user to determine the magnitude of sample perturbations that lead to systematic errors. Neutron multiplicity counting is an assay method used in the analysis of plutonium for safeguards applications. It is widely used in nuclear material accountancy by international (IAEA) and national inspectors. The method uses the measurement of the correlations in a pulse train to extract information on the spontaneous fission rate in the presence of neutrons from ({alpha},n) reactions and induced fission. The measurement is relatively simple to perform and gives results very quickly ({le} 1 hour). By contrast, destructive analysis techniques are extremely costly and time consuming (several days). By improving the achievable accuracy of neutron multiplicity counting, a nondestructive analysis technique, it could be possible to reduce the use of destructive analysis measurements required in safeguards applications. The accuracy of a neutron multiplicity measurement can be affected by a number of variables such as density, isotopic composition, chemical composition and moisture in the material. In order to determine the magnitude of these effects on the measured plutonium mass a calculational tool, EXCOM, has been produced using VBA within Excel. This
5. 2011 Japanese Nuclear Incident
EPA Pesticide Factsheets
EPA’s RadNet system monitored the environmental radiation levels in the United States and parts of the Pacific following the Japanese Nuclear Incident. Learn about EPA’s response and view historical laboratory data and news releases.
6. Coated Fiber Neutron Detector Test
SciTech Connect
Lintereur, Azaree T.; Ely, James H.; Kouzes, Richard T.; Stromswold, David C.
2009-10-23
Radiation portal monitors used for interdiction of illicit materials at borders include highly sensitive neutron detection systems. The main reason for having neutron detection capability is to detect fission neutrons from plutonium. The currently deployed radiation portal monitors (RPMs) from Ludlum and Science Applications International Corporation (SAIC) use neutron detectors based upon 3He-filled gas proportional counters, which are the most common large neutron detector. There is a declining supply of 3He in the world, and thus, methods to reduce the use of this gas in RPMs with minimal changes to the current system designs and sensitivity to cargo-borne neutrons are being investigated. Reported here are the results of tests of the 6Li/ZnS(Ag)-coated non-scintillating plastic fibers option. This testing measured the required performance for neutron detection efficiency and gamma ray rejection capabilities of a system manufactured by Innovative American Technology (IAT).
7. Nanostructure Neutron Converter Layer Development
NASA Technical Reports Server (NTRS)
Park, Cheol (Inventor); Sauti, Godfrey (Inventor); Kang, Jin Ho (Inventor); Lowther, Sharon E. (Inventor); Thibeault, Sheila A. (Inventor); Bryant, Robert G. (Inventor)
2016-01-01
Methods for making a neutron converter layer are provided. The various embodiment methods enable the formation of a single layer neutron converter material. The single layer neutron converter material formed according to the various embodiments may have a high neutron absorption cross section, tailored resistivity providing a good electric field penetration with submicron particles, and a high secondary electron emission coefficient. In an embodiment method a neutron converter layer may be formed by sequential supercritical fluid metallization of a porous nanostructure aerogel or polyimide film. In another embodiment method a neutron converter layer may be formed by simultaneous supercritical fluid metallization of a porous nanostructure aerogel or polyimide film. In a further embodiment method a neutron converter layer may be formed by in-situ metalized aerogel nanostructure development.
8. INCIDENCE OF ENDOMETRIAL HYPERPLASIA
PubMed Central
REED, Susan D.; NEWTON, Katherine M.; CLINTON, Walter L.; EPPLEIN, Meira; GARCIA, Rochelle; ALLISON, Kimberly; VOIGT, Lynda F.; Weiss, Noel S.
2009-01-01
Objective Estimate age-specific incidence of endometrial hyperplasia: simple, complex, and atypical, in order of increasing likelihood of progression to carcinoma. Study design Women ages 18–90 years with endometrial pathology specimens (1985–2003) at a large integrated health plan were identified using automated data. Incidence rates were obtained by dividing the number of cases by the estimated number of female health plan enrollees who retained a uterus. Results Endometrial hyperplasia peak incidence was: simple-142/100,000 woman-years, complex-213/100,000 woman-years, both in the early 50s; and atypical-56/100,000 woman-years in the early 60s. Age-adjusted incidence decreased over the study period, especially for atypical hyperplasia. Conclusions Endometrial hyperplasia incidence without and with atypia peaks in the early postmenopausal years and in the early 60s, respectively. Given that some cases of endometrial hyperplasia likely go undiagnosed, the figures provided should be viewed as minimum estimates of the true incidence. PMID:19393600
9. Gamma-ray bursts and neutron star field decay
NASA Technical Reports Server (NTRS)
Hartmann, Dieter; Blumenthal, George; Chuang, Kuan-Wen; Hurley, Kevin; Kargatis, Vincent; Liang, Edison; Linder, Eric
1992-01-01
Assuming a Galactic origin of gamma-ray bursts, we use pulsar data to calculate the spatial distribution of neutron stars and determine the sampling depths of current detectors. Based on these distance limits, we calculate the corresponding age distribution of Galactic neutron stars and apply an exponential field decay model to test whether the observed high incidence rate of cyclotron lines is consistent with suggested field decay time scales of order 10 exp 7 years. We find that the properties of the observed population of gamma-ray bursts are inconsistent with the idea that bursts originate at arbitrary times on neutron stars whose fields decay on time scales shorter than about 10 exp 9 years. Possible interpretations of this inconsistency are discussed.
10. A new electronic neutron dosimeter (END) for reliable personal dosimetry
Ing, H.; Cousins, T.; Andrews, H. R.; Machrafi, R.; Voevodskiy, A.; Kovaltchouk, V.; Clifford, E. T. H.; Robins, M.; Larsson, C.; Hugron, R.; Brown, J.
2008-04-01
Tests of existing electronic neutron dosimeters by military and civilian groups have revealed significant performance limitations. To meet the operational requirements of emergency response personnel to a radiological/nuclear incident as well as those in the nuclear industry, a new END has been developed. It is patterned after a unique commercial neutron spectral dosemeter known as the N-probe. It uses a pair of small special scintillators on tiny photomultiplier tubes. Special electronics were designed to minimize power consumption to allow for weeks of operation on a single charge. The size, performance, and data analysis for the END have been designed to meet/exceed international standards for electronic neutron dosimeters. Results obtained with the END prototype are presented.
11. Determining fragmentation dynamics through a study of neutron multiplicity at the NSCL
Stephenson, Sharon; Christ, Peter; Mazza, Maria; MoNA Collaboration
2017-01-01
In nuclear fragmentation reactions the number of neutrons and the excitation energy of the final fragment are related to the excitation energies of prefragments, which are produced in the reaction target but not directly observed. The MoNA Collaboration designed and performed an experiment to measure the number of neutrons in coincidence with charged projectile fragments to determine the excitation mechanisms of specific prefragments. All prior MoNA experimental campaigns concentrated on neutrons emitted from discrete levels in near dripline nuclei and treated any evaporation neutrons as an underlying background. This experiment capitalizes on those evaporation neutrons, focusing on sodium, neon and fluorine reaction products. For the experiment a 32Mg secondary beam with energy 86 MeV/u was incident on a Be reaction target. This target is upstream from the Sweeper, a superconducting dipole steering magnet with a bending angle of 43° and a vertical gap that permits forward-focused neutrons to get to the MoNA, the Modular Neutron Array. The rigidity of the Sweeper was varied during this experiment to increase the detection range. Analysis of the neutron-neutron hit distribution in coincidence with each sodium, neon, or fluorine charged fragment will be presented. This work was supported in part by the National Science Foundation Award 1613429 and the Howard Hughes Medical Institute Award 52007540.
12. Prompt fission neutron spectra of n+235U above the (n,nf) fission threshold
Shu, Neng-Chuan; Jia, Min; Chen, Yong-Jing; Liu, Ting-Jin
2015-05-01
Calculations of prompt fission neutron spectra (PFNS) from the 235U(n, f) reaction were performed with a semi-empirical method for En = 7.0 and 14.7 MeV neutron energies. The total PFNS were obtained as a superposition of (n,xnf) pre-fission neutron spectra and post-fission spectra of neutrons which were evaporated from fission fragments, and these two kinds of spectra were taken as an expression of the evaporation spectrum. The contributions of (n,xnf) fission neutron spectra on the calculated PFNS were discussed. The results show that emission of one or two neutrons in the (n,nf) or (n,2nf) reactions influences the PFNS shape, and the neutron spectra of the (n,xnf) fission-channel are soft compared with the neutron spectra of the (n,f) fission channel. In addition, analysis of the multiple-chance fission component showed that second-chance fission dominates the PFNS with an incident neutron energy of 14.7 MeV whereas first-chance fission dominates the 7 MeV case. Supported by National Natural Science Foundation of China (11205246, 91126010, U1230127, 91226102), IAEA CRP (15905), and Defense Industrial Technology Development Program (B0120110034)
13. Fast Neutron Inelastic Scattering Cross Sections in THORIUM-232.
Ciarcia, Christopher Albert
Fast neutron inelastic scattering cross sections for levels between 700-1550-keV excitation energy in the actinide nucleus, ('232)Th, have been measured using the (n,n') time-of-flight technique. Two series of measurements were undertaken using neutrons with a typical energy spread of 8-10 keV, generated by the ('7)Li(p,n)('7)Be reaction. These measurments for 125(DEGREES)-differential scattering cross sections were performed over the incident neutron energy regions of (i) 0.950-1.550 MeV, in 50-keV intervals with the time-of-flight spectrometer optimized to detect 0.200 -0.400-MeV scattered neutrons and (ii) 1.200-2.000 MeV, in 100-keV intervals with the time-of-flight spectrometer optimized to detect 0.400-0.800-MeV scattered neutrons. Over these scattered energy regions, an overall energy resolution of less than 15 keV was maintained. The relative neutron fluence was determined for each individual measurement, by positioning the main detector at 0(DEGREES) to view the primary neutron flux. Relative normalization was achieved by measuring the direct neutron flux from the lithium target with a fixed overhead monitor detector in both measurements. Main detector response was determined by comparison with a ('235)U fission chamber of known efficiency. Techniques for unfolding the complicated spectra obtained from these (n,n') studies were developed, employing user interactive computer codes to (i) generate simulated scattered neutron group response functions, (ii) subtract background effects from the measured spectra, (iii) approximate the background subtracted spectra in a weighted least-squares fashion by a superposition of response functions and (iv) make corrections for neutron absorption, finite scatterer size effects and multiple neutron scattering. Support codes consisting of graphics interaction packages, data file manipulation and transfer utility routines were created to assist in the spectral analysis procedure. Excitation function and angular distribution
14. Neutron scattering measurements in {sup 197}Au from 850 keV to 2.0 MeV
SciTech Connect
OConnor, M.; Chen, J.; Egan, J.J.
1995-10-01
Differential elastic and inelastic neutron scattering cross-sections for low lying levels in {sup 197}Au have been measured for incident neutron energies of 1.0 MeV, 1.5 MeV and 2.0 MeV. In addition, the total neutron cross sections in {sup 197}Au was measured from 850 keV to 1.5 MeV. For both experiments the UML 5.5 MV Van-de-Graaff accelerator with a Mobley post acceleration compression system, produced subnanosecond proton pulses which generated neutrons via the {sup 7}Li(p,n) {sup 7}Be reaction.
15. Superconducting Gamma/Neutron Spectrometer Task 1 Completion Report Evaluation of Candidate Neutron-Sensitive Materials
SciTech Connect
Bell, Z.W.; Lamberti, V.E.
2002-06-20
A review of the scientific literature regarding boron- and lithium-containing compounds was completed. Information such as Debye temperature, heat capacity, superconductivity properties, physical and chemical characteristics, commercial availability, and recipes for synthesis was accumulated and evaluated to develop a list of neutron-sensitive materials likely to perform properly in the spectrometer. The best candidate borides appear to be MgB{sub 2} (a superconductor with T{sub c} = 39 K), B{sub 6}Si, B{sub 4}C, and elemental boron; all are commercially available. Among the lithium compounds are LiH, LiAl, Li{sub 12}Si{sub 7}, and Li{sub 7}Sn{sub 2}. These materials have or are expected to have high Debye temperatures and sufficiently low heat capacities at 100 mK to produce a useful signal. The responses of {sup 10}B and {sup 6}Li to a fission neutron spectrum were also estimated. These demonstrated that the contribution of scattering events is no more than 3% in a boron-based system and 1.5% in a lithium-based system. This project is concerned with the development of materials for use in a cryogenic neutron spectrometer and is complementary to work in progress by Labov at LLNL to develop a cryogenic gamma ray spectrometer. Refrigeration to 100 mK lowers the heat capacity of these materials to the point that the energy of absorbed gamma and x rays, nuclei scattered by fast neutrons, and ions from (n, {alpha}) reactions produce a measurable heat pulse, from which the energy of the incident radiation may be deduced. The objective of this project is the discovery, fabrication, and testing of candidate materials with which a cryogenic neutron spectrometer may be realized.
16. Improvements in neutron beam applications by using capillary neutron optics
Downing, Robert G.; Xiao, Qi-Fan; Sharov, V. A.; Ponomarev, Igor Y.; Ullrich, Johannes B.; Gibson, David M.; Chen-Mayer, Huaiyu H.; Mildner, David F. R.; Lamaze, G. P.
1997-02-01
Capillary neutron optics improve the capabilities of neutron beam techniques such as neutron depth profiling and prompt gamma activation analysis. Millions of glass capillaries are configured to capture and guide low-energy neutrons by grazing total reflection from the smooth inner surface of the hollow channels. By precise orientation of the capillaries, beams of neutrons are readily collimated with good angular control or can be finely focused - as required by the application. In addition, the optics can improve the signal-to-noise ratio by diverting a neutron beam to a convenient off-axis direction, thereby circumventing interferences from gamma rays and fast neutrons characteristic of simple aperture collimation. The focused intensity of neutrons obtained in an area of 0.03 mm2 may be increased up to a hundred times over that previously available for NDP or PGAA techniques. Furthermore, the spatial resolution can be improved by up to 100 times. Consequently, small samples, or small volumes within larger samples, may be better and more rapidly investigated with neutron probe techniques. We report on developments in the application of capillary neutron optics.
17. Observation of Neutron Skyshine from an Accelerator Based Neutron Source
Franklyn, C. B.
2011-12-01
A key feature of neutron based interrogation systems is the need for adequate provision of shielding around the facility. Accelerator facilities adapted for fast neutron generation are not necessarily suitably equipped to ensure complete containment of the vast quantity of neutrons generated, typically >1011 nṡs-1. Simulating the neutron leakage from a facility is not a simple exercise since the energy and directional distribution can only be approximated. Although adequate horizontal, planar shielding provision is made for a neutron generator facility, it is sometimes the case that vertical shielding is minimized, due to structural and economic constraints. It is further justified by assuming the atmosphere above a facility functions as an adequate radiation shield. It has become apparent that multiple neutron scattering within the atmosphere can result in a measurable dose of neutrons reaching ground level some distance from a facility, an effect commonly known as skyshine. This paper describes a neutron detection system developed to monitor neutrons detected several hundred metres from a neutron source due to the effect of skyshine.
18. Neutron logging tool readings and neutron parameters of formations
Czubek, Jan A.
1995-03-01
A case history of the calibration of neutron porosity tools is given in the paper. The calibration of neutron porosity tools is one of the most difficult, complicated, and time consuming tasks in the well logging operations in geophysics. A semi empirical approach to this problem is given in the paper. It is based on the correlation of the tool readings observed in known environments with the apparent neutron parameters sensed by the tools. The apparent neutron parameters are functions of the true neutron parameters of geological formations and of the borehole material, borehole diameter, and the tool position inside the borehole. The true integral neutron transport parameters are obtained by the multigroup diffusion approximation for slowing down of neutrons and by one thermal neutron group for the diffusion. In the latter, the effective neutron temperature is taken into account. The problem of the thermal neutron absorption cross section of rocks is discussed in detail from the point of view of its importance for the well logging results and for the experimental techniques being used.
19. Photon and neutron dose contributions and mean quality factors phantoms of different size irradiated by monoenergetic neutrons.
PubMed
Dietze, G; Siebert, B R
1994-10-01
The International Commission on Radiological Protection (ICRP) in its Publication 60 introduced important changes in the concept of risk-related quantities. For external neutron radiation in particular the introduction of the equivalent dose with the radiation weighting factor wR instead of the dose equivalent concept with the quality factor Q(L) has many consequences. The value of wR is defined by the external neutron radiation field, while the radiation quality in the phantom depends on the radiation field at the position of interest and hence on the size of and the position in the phantom. It has been investigated to what extent the size of the phantom influences the mean radiation quality in the phantoms. For incident monoenergetic neutrons, mean photon dose contributions and mean quality factors have been calculated. Results are presented for various phantoms which characterize the conditions for a mouse, a rat, the ICRU sphere and a human body.
20. Investigation of the Statistical Properties of Stable Eu Nuclei using Neutron-Capture Reactions
SciTech Connect
Agvaanluvsan, U; Alpizar-Vicente, A; Becker, J A; Becvar, F; Bredeweg, T A; Clement, R; Esch, E; Folden, III, C M; Hatarik, R; Haight, R C; Hoffman, D C; Krticka, M; Macri, R A; Mitchell, G E; Nitsche, H; O'Donnell, J M; Parker, W; Reifarth, R; Rundberg, R S; Schwantes, J M; Sheets, S A; Ullmann, J L; Vieira, D J; Wilhelmy, J B; Wilk, P; Wouters, J M; Wu, C Y
2005-10-04
Neutron capture for incident neutron energies <1eV up to 100 keV has been measured for {sup 151,153}Eu targets. The highly efficient DANCE (Detector for Advanced Neutron Capture Experiments) array coupled with the intense neutron beam at Los Alamos Neutron Science Center is used for the experiment. Stable Eu isotopes mass separated and electroplated on Be backings were used. Properties of well-resolved, strong resonances in two Eu nuclei are examined. The parameters for most of these resonances are known. Detailed multiplicity information for each resonance is obtained employing the high granularity of the DANCE array. The radiative decay cascades corresponding to each resonance are obtained in the experiment. The measurements are compared to simulation of these cascades which calculated with various models for the radiative strength function. Comparison between the experimental data and simulation provides an opportunity to investigate the average quantities.
1. Observation of the one- to six-neutron transfer reactions at sub-barrier energies
SciTech Connect
Jiang, C.L.; Rehm, K.E.; Gehring, J.
1995-08-01
It was suggested many years ago that when two heavy nuclei are in contact during a grazing collision, the transfer of several correlated neutron-pairs could occur. Despite considerable experimental effort, however, so far only cross sections for up to four-neutron transfers have been uniquely identified. The main difficulties in the study of multi-neutron transfer reactions are the small cross sections encountered at incident energies close to the barrier, and various experimental uncertainties which can complicate the analysis of these reactions. We have for the first time found evidence for multi-neutron transfer reactions covering the full sequence from one- to six-neutron transfer reactions at sub-barrier energies in the system {sup 58}Ni + {sup 100}Mo.
2. Gravity-Induced Neutron Interference Phases Using a Two-Wavelength Difference Technique
Werner, S. A.
1997-04-01
We have used a new technique to measure the quantum phase shift of neutron deBroglie waves due to the Earth's gravitational field (COW effect). The total measured phase shift is composed of two parts: that due to gravity which is proportional to the neutron wavelength, and a part due to the bending of the interferometer crystal as it is tilted about the incident beam, which is inversely proportional to the neutron wavelength. Previous experiments have used x-rays to separately measure the phase shift due to bending. In the present experiment, the phase shifts due to gravity and bending are measured using nearly harmonic pairs of neutron wavelengths. A discrepancy of order 1 percent between theory and experiment is again observed in the data sets obtained in experiments on two different interferometers of quite different geometries. These results will be described in the historical context of previous versions of the COW neutron experiments.
3. Neutron yield enhancement in laser-induced deuterium-deuterium fusion using a novel shaped target
SciTech Connect
Zhao, J. R.; Chen, L. M. Li, Y. T.; Li, F.; Zhu, B. J.; Li, Yan. F.; Liao, G. Q.; Huang, K.; Ma, Y.; Li, Yi. F.; Zhang, X. P.; Fu, C. B.; Yuan, D. W.; Zhang, K.; Han, B.; Zhao, G.; Rhee, Y. J.; Liu, C.; Xiong, J.; Huang, X. G.; and others
2015-06-15
Neutron yields have direct correlation with the energy of incident deuterons in experiments of laser deuterated target interaction [Roth et al., Phys. Rev. Lett. 110, 044802 (2013) and Higginson et al., Phys. Plasmas 18, 100703 (2011)], while deuterated plasma density is also an important parameter. Experiments at the Shenguang II laser facility have produced neutrons with energy of 2.45 MeV using d (d, n) He reaction. Deuterated foil target and K-shaped target were employed to study the influence of plasma density on neutron yields. Neutron yield generated by K-shaped target (nearly 10{sup 6}) was two times higher than by foil target because the K-shaped target results in higher density plasma. Interferometry and multi hydro-dynamics simulation confirmed the importance of plasma density for enhancement of neutron yields.
4. Thermal neutron imaging through XRQA2 GAFCHROMIC films coupled with a cadmium radiator
Sacco, D.; Bedogni, R.; Bortot, D.; Palomba, M.; Pola, A.; Introini, M. V.; Lorenzoli, M.; Gentile, A.; Strigari, L.; Pressello, C.; Soriani, A.; Gómez-Ros, J. M.
2015-10-01
A simple and inexpensive method to perform passive thermal neutron imaging on large areas was developed on the basis of XRQA2 GAFCHROMIC films, commonly employed for quality assurance in radiology. To enhance their thermal neutron response, the sensitive face of film was coupled with a 1 mm thick cadmium radiator, forming a sandwich. By exchanging the order of Cd filter and sensitive film with respect to the incident neutron beam direction, two different configurations (beam-Cd-film and beam-film-Cd) were identified. These configurations were tested at thermal neutrons fluence values in the range 109-1010 cm-2, using the ex-core radial thermal neutron column of the ENEA Casaccia - TRIGA reactor. The results are presented in this work.
5. Invited Article: Polarization Down Under'': The polarized time-of-flight neutron reflectometer PLATYPUS
Saerbeck, T.; Klose, F.; Le Brun, A. P.; Füzi, J.; Brule, A.; Nelson, A.; Holt, S. A.; James, M.
2012-08-01
This review presents the implementation and full characterization of the polarization equipment of the time-of-flight neutron reflectometer PLATYPUS at the Australian Nuclear Science and Technology Organisation (ANSTO). The functionality and efficiency of individual components are evaluated and found to maintain a high neutron beam polarization with a maximum of 99.3% through polarizing Fe/Si supermirrors. Neutron spin-flippers with efficiencies of 99.7% give full control over the incident and scattered neutron spin direction over the whole wavelength spectrum available in the instrument. The first scientific experiments illustrate data correction mechanisms for finite polarizations and reveal an extraordinarily high reproducibility for measuring magnetic thin film samples. The setup is now fully commissioned and available for users through the neutron beam proposal system of the Bragg Institute at ANSTO.
6. Invited article: polarization "down under": the polarized time-of-flight neutron reflectometer PLATYPUS.
PubMed
Saerbeck, T; Klose, F; Le Brun, A P; Füzi, J; Brule, A; Nelson, A; Holt, S A; James, M
2012-08-01
This review presents the implementation and full characterization of the polarization equipment of the time-of-flight neutron reflectometer PLATYPUS at the Australian Nuclear Science and Technology Organisation (ANSTO). The functionality and efficiency of individual components are evaluated and found to maintain a high neutron beam polarization with a maximum of 99.3% through polarizing Fe/Si supermirrors. Neutron spin-flippers with efficiencies of 99.7% give full control over the incident and scattered neutron spin direction over the whole wavelength spectrum available in the instrument. The first scientific experiments illustrate data correction mechanisms for finite polarizations and reveal an extraordinarily high reproducibility for measuring magnetic thin film samples. The setup is now fully commissioned and available for users through the neutron beam proposal system of the Bragg Institute at ANSTO.
7. Focused neutron beam dose deposition profiles in tissue equivalent materials: a pilot study for BNCT
Mayer, Rulon R.; Welsh, James; Chen-Mayer, Huaiyu H.
1997-02-01
Boron Neutron Capture Therapy (BNCT) has been limited by the inability to direct neutrons toward the therapeutic target and away from sensitive normal tissues. The recently developed Kumakhov lens has focused a broad incident low energy neutron beam in air to a sub-mm spot. This study examines the radiation does distribution of a converging beam passing through tissue equivalent materials. A neutron beam exiting a focusing lens is directed toward a stack of thin radiochromic media sandwiched between plastic sheets. The depth dose and beam profile within the tissue equivalent materials are determined by optical scanning and image processing of the individual radiochromic media sheets, a polymer based dosimetry medium which darkens upon exposure to ionizing radiation. The alpha particle emission from boron is examined by substituting a plastic sheet with a 6Li enriched lithium carbonate sheet positioned at the focal plane. The information will help determine the feasibility of applying the focused neutron beam to BNCT for therapy.
8. Neutron yield enhancement in laser-induced deuterium-deuterium fusion using a novel shaped target
Zhao, J. R.; Zhang, X. P.; Yuan, D. W.; Chen, L. M.; Li, Y. T.; Fu, C. B.; Rhee, Y. J.; Li, F.; Zhu, B. J.; Li, Yan. F.; Liao, G. Q.; Zhang, K.; Han, B.; Liu, C.; Huang, K.; Ma, Y.; Li, Yi. F.; Xiong, J.; Huang, X. G.; Fu, S. Z.; Zhu, J. Q.; Zhao, G.; Zhang, J.
2015-06-01
Neutron yields have direct correlation with the energy of incident deuterons in experiments of laser deuterated target interaction [Roth et al., Phys. Rev. Lett. 110, 044802 (2013) and Higginson et al., Phys. Plasmas 18, 100703 (2011)], while deuterated plasma density is also an important parameter. Experiments at the Shenguang II laser facility have produced neutrons with energy of 2.45 MeV using d (d, n) He reaction. Deuterated foil target and K-shaped target were employed to study the influence of plasma density on neutron yields. Neutron yield generated by K-shaped target (nearly 106) was two times higher than by foil target because the K-shaped target results in higher density plasma. Interferometry and multi hydro-dynamics simulation confirmed the importance of plasma density for enhancement of neutron yields.
9. Investigating Prompt Fission Neutron Emission from 235U(n,f) in the Resolved Resonance Region
Göök, Alf; Hambsch, Franz-Josef; Oberstedt, Stephan
2016-03-01
Investigations of prompt emission in fission is of importance in understanding the fission process in general and the sharing of excitation energy among the fission fragments in particular. Experimental activities at IRMM on prompt neutron emission from fission in response to OECD/NEA nuclear data requests is presented in this contribution. Main focus lies on currently on-going investigations of prompt neutron emission from the reaction 235U(n,f) in the region of the resolved resonances. For this reaction strong fluctuations of fission fragment mass distributions and mean total kinetic energy have been observed [Nucl. Phys. A 491, 56 (1989)] as a function of incident neutron energy in the resonance region. In addition fluctuations of prompt neutron multiplicities were also observed [Phys. Rev. C 13, 195 (1976)]. The goal of the present study is to verify the current knowledge of prompt neutron multiplicity fluctuations and to study correlations with fission fragment properties.
10. One directional polarized neutron reflectometry with optimized reference layer method
SciTech Connect
Masoudi, S. Farhad; Jahromi, Saeed S.
2012-09-01
In the past decade, several neutron reflectometry methods for determining the modulus and phase of the complex reflection coefficient of an unknown multilayer thin film have been worked out among which the method of variation of surroundings and reference layers are of highest interest. These methods were later modified for measurement of the polarization of the reflected beam instead of the measurement of the intensities. In their new architecture, these methods not only suffered from the necessity of change of experimental setup but also another difficulty was added to their experimental implementations. This deficiency was related to the limitations of the technology of the neutron reflectometers that could only measure the polarization of the reflected neutrons in the same direction as the polarization of the incident beam. As the instruments are limited, the theory has to be optimized so that the experiment could be performed. In a recent work, we developed the method of variation of surroundings for one directional polarization analysis. In this new work, the method of reference layer with polarization analysis has been optimized to determine the phase and modulus of the unknown film with measurement of the polarization of the reflected neutrons in the same direction as the polarization of the incident beam.
11. Materials and neutronic research at the Low Energy Neutron Source
Baxter, David V.
2016-04-01
In the decade since the Low Energy Neutron Source (LENS) at Indiana University Center for Exploration of Energy and Matter (CEEM) produced its first neutrons, the facility has made important contributions to the international neutron scattering community. LENS employs a 13MeV proton beam at up to 4kW beam power onto one of two Be targets to produce neutrons for research in fields ranging from radiation effects in electronics to studies of the structure of fluids confined in nanoporous materials. The neutron source design at the heart of LENS facilitates relatively rapid hands-on access to most of its components which provides a foundation for a research program in experimental neutronics and affords numerous opportunities for novel educational experiences. We describe in some detail a number of the unique capabilities of this facility.
12. SELF-REACTIVATING NEUTRON SOURCE FOR A NEUTRONIC REACTOR
DOEpatents
Newson, H.W.
1959-02-01
Reactors of the type employing beryllium in a reflector region around the active portion and to a neutron source for use therewith are discussed. The neutron source is comprised or a quantity of antimony permanently incorporated in, and as an integral part of, the reactor in or near the beryllium reflector region. During operation of the reactor the natural occurring antimony isotope of atomic weight 123 absorbs neutrons and is thereby transformed to the antimony isotope of atomic weight 124, which is radioactive and emits gamma rays. The gamma rays react with the beryllium to produce neutrons. The beryllium and antimony thus cooperate to produce a built in neutron source which is automatically reactivated by the operation of the reactor itself and which is of sufficient strength to maintain the slow neutron flux at a sufficiently high level to be reliably measured during periods when the reactor is shut down.
13. Neutron-$$\\gamma$$ competition for β-delayed neutron emission
DOE PAGES
Mumpower, Matthew Ryan; Kawano, Toshihiko; Moller, Peter
2016-12-19
Here we present a coupled quasiparticle random phase approximation and Hauser-Feshbach (QRPA+HF) model for calculating delayed particle emission. This approach uses microscopic nuclear structure information, which starts with Gamow-Teller strength distributions in the daughter nucleus and then follows the statistical decay until the initial available excitation energy is exhausted. Explicitly included at each particle emission stage is γ-ray competition. We explore this model in the context of neutron emission of neutron-rich nuclei and find that neutron-γ competition can lead to both increases and decreases in neutron emission probabilities, depending on the system considered. Finally, a second consequence of this formalismmore » is a prediction of more neutrons on average being emitted after β decay for nuclei near the neutron drip line compared to models that do not consider the statistical decay.« less
14. Neutron-$\\gamma$ competition for β-delayed neutron emission
SciTech Connect
Mumpower, Matthew Ryan; Kawano, Toshihiko; Moller, Peter
2016-12-19
Here we present a coupled quasiparticle random phase approximation and Hauser-Feshbach (QRPA+HF) model for calculating delayed particle emission. This approach uses microscopic nuclear structure information, which starts with Gamow-Teller strength distributions in the daughter nucleus and then follows the statistical decay until the initial available excitation energy is exhausted. Explicitly included at each particle emission stage is γ-ray competition. We explore this model in the context of neutron emission of neutron-rich nuclei and find that neutron-γ competition can lead to both increases and decreases in neutron emission probabilities, depending on the system considered. Finally, a second consequence of this formalism is a prediction of more neutrons on average being emitted after β decay for nuclei near the neutron drip line compared to models that do not consider the statistical decay.
15. High intensity, pulsed thermal neutron source
DOEpatents
Carpenter, J.M.
1973-12-11
This invention relates to a high intensity, pulsed thermal neutron source comprising a neutron-producing source which emits pulses of fast neutrons, a moderator block adjacent to the last neutron source, a reflector block which encases the fast neutron source and the moderator block and has a thermal neutron exit port extending therethrough from the moderator block, and a neutron energy- dependent decoupling reflector liner covering the interior surfaces of the thermal neutron exit port and surrounding all surfaces of the moderator block except the surface viewed by the thermal neutron exit port. (Official Gazette)
16. Anatomy of an incident
SciTech Connect
Cournoyer, Michael E.; Trujillo, Stanley; Lawton, Cindy M.; Land, Whitney M.; Schreiber, Stephen B.
2016-03-23
A traditional view of incidents is that they are caused by shortcomings in human competence, attention, or attitude. It may be under the label of “loss of situational awareness,” procedure “violation,” or “poor” management. A different view is that human error is not the cause of failure, but a symptom of failure – trouble deeper inside the system. In this perspective, human error is not the conclusion, but rather the starting point of investigations. During an investigation, three types of information are gathered: physical, documentary, and human (recall/experience). Through the causal analysis process, apparent cause or apparent causes are identified as the most probable cause or causes of an incident or condition that management has the control to fix and for which effective recommendations for corrective actions can be generated. A causal analysis identifies relevant human performance factors. In the following presentation, the anatomy of a radiological incident is discussed, and one case study is presented. We analyzed the contributing factors that caused a radiological incident. When underlying conditions, decisions, actions, and inactions that contribute to the incident are identified. This includes weaknesses that may warrant improvements that tolerate error. Measures that reduce consequences or likelihood of recurrence are discussed.
17. Anatomy of an incident
DOE PAGES
Cournoyer, Michael E.; Trujillo, Stanley; Lawton, Cindy M.; ...
2016-03-23
A traditional view of incidents is that they are caused by shortcomings in human competence, attention, or attitude. It may be under the label of “loss of situational awareness,” procedure “violation,” or “poor” management. A different view is that human error is not the cause of failure, but a symptom of failure – trouble deeper inside the system. In this perspective, human error is not the conclusion, but rather the starting point of investigations. During an investigation, three types of information are gathered: physical, documentary, and human (recall/experience). Through the causal analysis process, apparent cause or apparent causes are identifiedmore » as the most probable cause or causes of an incident or condition that management has the control to fix and for which effective recommendations for corrective actions can be generated. A causal analysis identifies relevant human performance factors. In the following presentation, the anatomy of a radiological incident is discussed, and one case study is presented. We analyzed the contributing factors that caused a radiological incident. When underlying conditions, decisions, actions, and inactions that contribute to the incident are identified. This includes weaknesses that may warrant improvements that tolerate error. Measures that reduce consequences or likelihood of recurrence are discussed.« less
18. Skylab neutron environment experiment (Science Demonstration SD-34 (TV108)). Description and preliminary results
NASA Technical Reports Server (NTRS)
Fishman, G. J.
1974-01-01
Neutron and proton induced radioactivity at various locations within Skylab were measured. Samples of five metals were formed into activation packets and deployed at the following locations on the Skylab 4 mission: orbital workshop film vault, water storage tank, and two opposing orbital workshop internal locations. Radioactive nuclides were produced in the packets by nuclear interactions of high-energy protons and secondary neutrons within Skylab. Low-level gamma ray spectroscopy measurements were made on the returned packets to determine the incident neutron and proton fluxes and spectra and their variations with mass distribution.
19. Radiation damage in silicon due to albedo neutrons emitted from hadronic beam dumps (Fe and U)
SciTech Connect
Gabriel, T.A.; Bishop, B.L.
1987-01-01
Calculations have been carried out to determine the level of radiation damage that can be expected from albedo neutrons when 1- and 5-GeV negative pions are incident on iron and uranium beam dumps. The calculated damage data are presented in several ways including neutron fluence above 0.111 MeV, 1 MeV equivalent neutron fluence, damage energy deposition, and DPA or displacements per atom. Details are presented as to the method of calculation. 14 refs., 1 fig., 1 tab.
20. Flux and dose transmission through concrete of neutrons from proton induced reactions on various target elements | {"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": 0.8514183759689331, "perplexity": 3565.623199836408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084891750.87/warc/CC-MAIN-20180123052242-20180123072242-00772.warc.gz"} |
https://www.physicsforums.com/threads/work-of-an-object-moving-down-inclined-plane.652363/ | 1. PF Insights is off to a great start! Fresh and interesting articles on all things science and math. Here: PF Insights
Work of an Object moving down Inclined Plane
1. Astrum
274
1. The problem statement, all variables and given/known data
Experiment set up.
A 208g weight is attached via a pulley system to a block on an inclined plane. What is the work done by the suspended mass as the car is lowering at a constant velocity? And work done by gravity?
Distance weight moves down - 24.7cm
Incline is 30 degrees
2. Relevant equations
$F=ma$
$w=Fdcos(\theta)$
3. The attempt at a solution
So, in this case, work would be negative, right? Because the direction of the force from the suspended mass is going UP the incline, and the direction of moment is DOWN the incline?
The block is 474g
I'm not sure where to start. If I use the basic work formula and do
$\(-208*980)*24.7$?
2. Simon Bridge
14,950
The thing losing energy is doing the work.
The description of the experiment is incomplete - we are told about the suspended mass and then some car is introduced out of nowhere... but it looks like you are expected to use conservation of energy. Your descriptions of what you have tried are also incomplete so it is not clear what you have done.
Try expressing your working symbolically - do all the algebra before you put numbers in.
3. Astrum
274
Sorry, I wrote it really quickly.
The setup was an inclined plane with a pulley system which was set off the leg of the triangle. The pulley system was attached to the car. We had to set a weight on the pulley so that the car would go down the plane with a constant velocity.
I'm really not sure what to do. The car is moving down the car, therefore losing potential energy. If work can be defined as the difference in U.
So, if then ΔU= (474*980*12.5)-(474-980*0)= 5,806,500 erg? | {"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": 0.8069053292274475, "perplexity": 575.0372956808089}, "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-2015-18/segments/1430453938418.92/warc/CC-MAIN-20150501041858-00013-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://digest.kndise.gov.ua/en/modern-understanding-of-installation-in-forensic-technical-examination-of-documents/ | Міжвідомчий Науково-методичний збірник
"Криміналістика і судова експертиза"
ISSN: 0130-2655
PDF Kulykovska Завантажень: 76, розмiр: 321.0 KB
DOI: https://doi.org/10.33994/kndise.2021.66.59
V. Kulykovska
The article presents an attempt to provide a general methodological concept of document mounting and describe its main methods and features.
The article considers practical examples in which a scheme-algorithm for conducting research is described and a logical chain of established facts is built, which helped in each case to come to a reasoned conclusion.
The article also draws attention to the fact that today the schemes for solving problems to establish the execution of documents using installation are the least developed both theoretically and scientifically-methodically – there is no single terminological approach that allows you to unambiguously interpret and evaluate the results obtained.
Key words: falsified documents by editing, establishing the fact of editing, technical editing of the document, computer-technical editing of the document | {"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": 0.8084067106246948, "perplexity": 4117.638898797774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00272.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/135433-rings-id-fields.html | # Math Help - Rings, ID, and Fields
1. ## Rings, ID, and Fields
Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then r' is commutative.
2. Let $\phi : R \rightarrow R'$
If R is commumative then
$\forall a,b \in R: ab = ba$
$\Rightarrow \phi(ab) = \phi(ba)$
Also $\phi(ab) = \phi(a)\phi(b)$ and $\phi(ba) = \phi(b)\phi(a)$
and $\phi$ is bijective.
This should be enough for you to see why that R' is communative
3. Originally Posted by bookie88
Assume that the ring R is isomorphic to the ring R'. Prove that if R is commutative, then r' is commutative.
Let $\phi:R\to R'$ denote an isomorphism, and $x,y\in R'$. Then there are $a,b\in R$ with $\phi(a)=x$ and $\phi(b)=y$. Furthermore, $xy=\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a) =yx$. So $R'$ is commutative.
EDIT: Looks like someone beat me to it. Oh well. | {"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": 13, "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": 0.9311702847480774, "perplexity": 596.1075138148574}, "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-2016-07/segments/1454701158609.98/warc/CC-MAIN-20160205193918-00232-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://discretemath.org/ads/s-what-is-a-tree.html | $\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$
## Section10.1What Is a Tree?
### Subsection10.1.1Definition
What distinguishes trees from other types of graphs is the absence of certain paths called cycles. Recall that a path is a sequence of consecutive edges in a graph, and a circuit is a path that begins and ends at the same vertex.
###### Definition10.1.1Cycle
A cycle is a circuit whose edge list contains no duplicates. It is customary to use $C_n$ to denote a cycle with $n$ edges.
The simplest example of a cycle in an undirected graph is a pair of vertices with two edges connecting them. Since trees are cycle-free, we can rule out all multigraphs from consideration as trees.
Trees can either be undirected or directed graphs. We will concentrate on the undirected variety in this chapter.
###### Definition10.1.2Tree
An undirected graph is a tree if it is connected and contains no cycles or self-loops.
1. Graphs i, ii and iii in Figure 10.1.4 are all trees, while graphs iv, v, and vi are not trees.
2. A $K_2$ is a tree. However, if $n\geq 3$, a $K_n$ is not a tree.
3. In a loose sense, a botanical tree is a mathematical tree. There are usually no cycles in the branch structure of a botanical tree.
4. The structures of some chemical compounds are modeled by a tree. For example, butane Figure 10.1.5 consists of four carbon atoms and ten hydrogen atoms, where an edge between two atoms represents a bond between them. A bond is a force that keeps two atoms together. The same set of atoms can be linked together in a different tree structure to give us the compound isobutane Figure 10.1.6. There are some compounds whose graphs are not trees. One example is benzene Figure 10.1.7.
One type of graph that is not a tree, but is closely related, is a forest.
###### Definition10.1.8Forest
A forest is an undirected graph whose components are all trees.
The top half of Figure 10.1.4 can be viewed as a forest of three trees. Graph (vi) in this figure is also a forest.
### Subsection10.1.2Conditions for a graph to be a tree
We will now examine several conditions that are equivalent to the one that defines a tree. The following theorem will be used as a tool in proving that the conditions are equivalent.
Let $p_1= \left(e_1, e_2, \ldots , e_m \right)$ and $p_2=\left(f_1,f_2,\ldots , f_n\right)$ be two different simple paths from $v_a$ to $v_b$. The first step we will take is to delete from $p_1$ and $p_2$ the initial edges that are identical. That is, if $e_1= f_1$, $e_2= f_2, \dots$, $e_{j}= f_j$, and $e_{j+1}\neq f_{j+1}$ delete the first $j$ edges of both paths. Once this is done, both paths start at the same vertex, call it $v_c$, and both still end at $v_b$. Now we construct a cycle by starting at $v_c$ and following what is left of $p_1$ until we first meet what is left of $p_2$. If this first meeting occurs at vertex $v_d$, then the remainder of the cycle is completed by following the portion of the reverse of $p_2$ that starts at $v_d$ and ends at $v_c$.
Proof Strategy. Most of this theorem can be proven by proving the following chain of implications: $(1) \Rightarrow (2)$, $(2) \Rightarrow (3)$, $(3)\Rightarrow (4)$, and $(4) \Rightarrow (1)$. Once these implications have been demonstrated, the transitive closure of $\Rightarrow$ on ${1, 2, 3, 4}$ establishes the equivalence of the first four conditions. The proof that Statement 5 is equivalent to the first four can be done by induction, which we will leave to the reader.
$(1) \Rightarrow (2)$ (Indirect). Assume that $G$ is a tree and that there exists a pair of vertices between which there is either no path or there are at least two distinct paths. Both of these possibilities contradict the premise that $G$ is a tree. If no path exists, $G$ is disconnected, and if two paths exist, a cycle can be obtained by Theorem 10.1.11.
$(2) \Rightarrow (3)$. We now use Statement 2 as a premise. Since each pair of vertices in $V$ are connected by exactly one path, $G$ is connected. Now if we select any edge $e$ in $E\text{,}$ it connects two vertices, $v_1$ and $v_2$. By (2), there is no simple path connecting $v_1$ to $v_2$ other than $e$. Therefore, no path at all can exist between $v_1$ and $v_2$ in $(V, E - \{e\})$. Hence $(V, E - \{e\})$ is disconnected.
$(3)\Rightarrow (4)$. Now we will assume that Statement 3 is true. We must show that $G$ has no cycles and that adding an edge to $G$ creates a cycle. We will use an indirect proof for this part. Since (4) is a conjunction, by DeMorgan's Law its negation is a disjunction and we must consider two cases. First, suppose that $G$ has a cycle. Then the deletion of any edge in the cycle keeps the graph connected, which contradicts (3). The second case is that the addition of an edge to $G$ does not create a cycle. Then there are two distinct paths between the vertices that the new edge connects. By Lemma 10.1.10, a cycle can then be created, which is a contradiction.
$(4) \Rightarrow (1)$ Assume that $G$ contains no cycles and that the addition of an edge creates a cycle. All that we need to prove to verify that $G$ is a tree is that $G$ is connected. If it is not connected, then select any two vertices that are not connected. If we add an edge to connect them, the fact that a cycle is created implies that a second path between the two vertices can be found which is in the original graph, which is a contradiction.
The usual definition of a directed tree is based on whether the associated undirected graph, which is created by “erasing” its directional arrows, is a tree. In Section 10.3 we will introduce the rooted tree, which is a special type of directed tree.
### SubsectionExercises for Section 10.1
###### 1
Given the following vertex sets, draw all possible undirected trees that connect them.
1. $V_a= \{\text{right},\text{left}\}$
2. $V_b = \{+,-,0\}$
3. $V_c = \{\text{north}, \text{south}, \text{east}, \text{west}\}$.
The number of trees are: (a) 1, (b) 3, and (c) 16. The trees that connect $V_c$ are:
###### 2
Are all trees planar? If they are, can you explain why? If they are not, you should be able to find a nonplanar tree.
###### 3
Prove that if $G$ is a simple undirected graph with no self-loops, then $G$ is a tree if and only if $G$ is connected and $\lvert E \rvert = \lvert V \rvert - 1$.
Hint
Use induction on $\lvert E\rvert$.
###### 4
1. Prove that if $G = (V, E)$ is a tree and $e \in E$, then $(V, E - \{e\})$ is a forest of two trees.
2. Prove that if $\left(V_1,E_1\right.$) and $\left(V_2,E_2\right)$ are disjoint trees and $e$ is an edge that connects a vertex in $V_1$ to a vertex in $V_2$, then $\left(V_1\cup V_2, E_1\cup E_2\cup \{e\}\right)$ is a tree.
###### 5
1. Prove that any tree with at least two vertices has at least two vertices of degree 1.
2. Prove that if a tree has $n$ vertices, $n \geq 4$, and is not a path graph, $P_n$, then it has at least three vertices of degree 1.
1. Assume that $(V,E)$ is a tree with $\left| V\right| \geq 2$, and all but possibly one vertex in $V$ has degree two or more.
2. The proof of this part is similar to part a in that we can infer $2\lvert E\rvert \geq 2 \lvert V\rvert -1$, using the fact that a non-chain tree has at least one vertex of degree three or more. | {"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": 0.8331575989723206, "perplexity": 139.49288151645763}, "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/1531676594790.48/warc/CC-MAIN-20180723012644-20180723032644-00181.warc.gz"} |
https://www.physicsforums.com/threads/non-equilibrium-applications-of-newtons-laws.341987/ | # Non-Equilibrium Applications of Newtons Laws
1. Oct 1, 2009
### IAmSparticus
1. In the drawing, the weight of the block on the table is 388 N and that of the hanging block is 175 N. Ignore all frictional effects, and assuming the pulley and the cord to be massless. Find the acceleration of the two blocks as well as the tension in the cord
2. Fnet = mass of the object * acceleration
3. The force acting on the hanging block is gravity which has a magnitude of 9.8 m/s/s, and the mass is 17.86 kg. So the net force, which is just equal to the gravitational force, would be 175 N, which is incorrect. What am I doing wrong?
#### Attached Files:
• ###### Pulley.gif
File size:
4.4 KB
Views:
104
Last edited: Oct 1, 2009
2. Oct 2, 2009
### Redbelly98
Staff Emeritus
There is another force acting on the hanging block. Do you see what it is, from looking at the diagram?
Hint: what stops the hanging block from falling as if it were dropped?
Similar Discussions: Non-Equilibrium Applications of Newtons Laws | {"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": 0.9078155755996704, "perplexity": 994.4624397295215}, "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/1518891813431.5/warc/CC-MAIN-20180221044156-20180221064156-00428.warc.gz"} |
http://philosophersanon.blogspot.com/2012/05/department-meeting-rule-7.html?showComment=1336527580085 | ## Tuesday, May 8, 2012
### Department Meeting Rule #7
7.1 That something was done at your PhD-granting institution at the time you were a graduate student is no argument for doing it now in your present department.
7.2 If you earned your PhD more than 15 years ago, that something was done at your PhD-granting institution at the time you were a graduate student is an argument for not doing it now in your present department.
PhD 1992 said...
Those weren't the rules in my PhD granting institution when I was a grad student!
Anonymous said...
Amen to these rules, for reals.
Anonymous said...
But don't you understand WHERE I got my PhD?
Anonymous said...
But don't you understand WHERE I got my PhD?
Anonymous said...
Do I need to say it again?
Anonymous said...
Those WERE the rules in the dpeartment where I got my PhD.
Ho hum.
Anonymous said...
Department Meeting Rule #7 is a rule in my PhD-granting institution more than 15 years ago.
Anonymous said...
@9:39 (and @9:39, but I repeat myself):
Don't you understand where I got MY PhD? | {"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": 0.893389105796814, "perplexity": 3142.5679710593618}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463104.7/warc/CC-MAIN-20150226074103-00108-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://docs.scipy.org/doc/scipy/reference/tutorial/stats/continuous_geninvgauss.html | # Generalized Inverse Gaussian Distribution¶
The probability density function is given by:
\begin{eqnarray*} f(x; p, b) = x^{p-1} \exp(-b(x + 1/x)/2) / (2 K_p(b)), \end{eqnarray*}
where $$x > 0$$ is a real number and the parameters $$p, b$$ satisfy $$b > 0$$. $$K_v$$ is the modified Bessel function of second kind of order $$v$$ (scipy.special.kv).
If X is geninvgauss(p, b), then the distribution of 1/X is geninvgauss(-p, b). The inverse Gaussian distribution (scipy.stats.invgauss) is a special case with p=-1/2.
Implementation: scipy.stats.geninvgauss
#### Previous topic
Generalized Half-Logistic Distribution
#### Next topic
Generalized Normal Distribution | {"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": 0.9936301112174988, "perplexity": 2117.2910156251096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141202590.44/warc/CC-MAIN-20201129184455-20201129214455-00672.warc.gz"} |
http://link.springer.com/article/10.1007/s11098-011-9790-6 | Philosophical Studies
, Volume 163, Issue 1, pp 133–149
# How many thoughts are there? Or why we likely have no Tegmark duplicates$$10^{{10^{115} }}$$ m away
Article
DOI: 10.1007/s11098-011-9790-6
Porpora, D.V. Philos Stud (2013) 163: 133. doi:10.1007/s11098-011-9790-6
## Abstract
Physicist Max Tegmark argues that if there are infinite universes or sub-universes, we will encounter our exact duplicates infinite times, the nearest within $$10^{{10^{115} }}$$ m. Tegmark assumes Humean supervenience and a finite number of possible combinations of elementary quantum states. This paper argues on the contrary that Tegmark’s argument fails to hold if possible thoughts, persons, and life histories are all infinite in number. Are there infinite thoughts we could possibly think? This paper will show that there are. If so, then it is not only Tegmark’s specific claim about our duplication that is called into question. We additionally acquire another strong argument against Humean supervenience.
### Keywords
Supervenience Multiverse Personal identity Mind–body Number of mental states | {"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": 0.8204564452171326, "perplexity": 2429.032072997003}, "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-2017-17/segments/1492917118707.23/warc/CC-MAIN-20170423031158-00465-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/129660/ | # Measurement of forward $W→eν$ production in pp collisions at $\sqrt{s} = 8$ TeV
LHCb Collaboration; Bernet, R; Müller, K; Steinkamp, O; Straumann, U; et al (2016). Measurement of forward $W→eν$ production in pp collisions at $\sqrt{s} = 8$ TeV. Journal of High Energy Physics, 2016(10):30.
## Abstract
A measurement of the cross-section for $W → eν$ production in pp collisions is presented using data corresponding to an integrated luminosity of 2 $fb^{−1}$ collected by the LHCb experiment at a centre-of-mass energy of $\sqrt{s} = 8$ TeV. The electrons are required to have more than 20 GeV of transverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. The inclusive W production cross-sections, where the W decays to eν, are measured to be
$σ_{W^+}→_{{e^+}{ν_e}}=1124.4±2.1±21.5±11.2±13.0pb$,
$σ_{W^−}→_{{e^+}{\overlineν_e}}=809.0±1.9±18.1±7.0±9.4pb$,
where the first uncertainties are statistical, the second are systematic, the third are due to the knowledge of the LHC beam energy and the fourth are due to the luminosity determination.
Differential cross-sections as a function of the electron pseudorapidity are measured. The $W^+/W^−$ cross-section ratio and production charge asymmetry are also reported. Results are compared with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. Finally, in a precise test of lepton universality, the ratio of W boson branching fractions is determined to be
$\mathcal{B}(W→eν)/\mathcal{B}(W→μν)=1.020±0.002±0.019$,
where the first uncertainty is statistical and the second is systematic.
## Abstract
A measurement of the cross-section for $W → eν$ production in pp collisions is presented using data corresponding to an integrated luminosity of 2 $fb^{−1}$ collected by the LHCb experiment at a centre-of-mass energy of $\sqrt{s} = 8$ TeV. The electrons are required to have more than 20 GeV of transverse momentum and to lie between 2.00 and 4.25 in pseudorapidity. The inclusive W production cross-sections, where the W decays to eν, are measured to be
$σ_{W^+}→_{{e^+}{ν_e}}=1124.4±2.1±21.5±11.2±13.0pb$,
$σ_{W^−}→_{{e^+}{\overlineν_e}}=809.0±1.9±18.1±7.0±9.4pb$,
where the first uncertainties are statistical, the second are systematic, the third are due to the knowledge of the LHC beam energy and the fourth are due to the luminosity determination.
Differential cross-sections as a function of the electron pseudorapidity are measured. The $W^+/W^−$ cross-section ratio and production charge asymmetry are also reported. Results are compared with theoretical predictions at next-to-next-to-leading order in perturbative quantum chromodynamics. Finally, in a precise test of lepton universality, the ratio of W boson branching fractions is determined to be
$\mathcal{B}(W→eν)/\mathcal{B}(W→μν)=1.020±0.002±0.019$,
where the first uncertainty is statistical and the second is systematic.
## Statistics
### Citations
Dimensions.ai Metrics
22 citations in Web of Science®
25 citations in Scopus®
### Altmetrics
Detailed statistics | {"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": 0.9980090260505676, "perplexity": 1298.1697257297988}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304798.1/warc/CC-MAIN-20220125070039-20220125100039-00520.warc.gz"} |
https://sites.psu.edu/johnroe/2013/09/12/a-geometric-version-of-the-analytic-surgery-sequence/ | # A geometric version of the analytic surgery sequence?
In our Mapping surgery to analysis papers, Nigel and I proposed an analytic counterpart of the surgery exact sequence which summarizes the main results of the (Browder, Novikov, Sullivan, Wall) theory of high-dimensional manifolds. This exact sequence identifies the set of manifold structures within a given homotopy type $$X$$ (the structure set) as the fiber of an assembly map
$H_*(X; {\mathbb L}(e)) \to L_*({\mathbb Z}\pi_1(X))$
which abstracts the ides of obtaining “signature obstructions” from a “surgery problem”.
Analogously, we constructed an analytic structure set (actually the K-theory of a certain C*-algebra) as the fiber of a Baum-Connes type assembly map, and showed that index theory provides a natural transformation from the topological surgery exact sequence to our analytic surgery exact sequence.
Our structure set is defined in purely analytic terms. However, in a subsequent paper where we related our exact sequence to the theory of $$\eta$$ invariants, it became useful to have a more geometrical approach to the structure set also. (The relation between the “more geometrical” and “more analytical” approaches is roughly the same as that between the Baum-Douglas and Kasparov models of K-homology.) Our paper didn’t give a geometric definition of the structure set – just a geometric approach to certain elements.
A recent arXiv paper by Deeley and Goffeng proposes to take this idea to its logical conclusion by constructing a Baum-Douglas type model for the whole analytic structure set. The basic idea is this: An element of the structure set should be “an elliptic operator together with a reason that its index vanishes”. The cobordism invariance of the index shows that one example of such a “reason” is that our elliptic operator is actually defined on the boundary of some manifold (and that our operator is a boundary operator). Therefore a first approximation to a Baum-Douglas model of the structure set should have as cycles spin-c manifolds with boundary $$(M,\partial M)$$ together with maps $$\partial M \to X$$.
But of course this (cobordism) is not the only known reason for the vanishing of an index (e.g., as I understand it, the fundamental question about positive scalar curvature metrics is whether positive scalar curvature implies some bordism condition). So suppose you have an elliptic operator whose index vanishes for some “positive scalar curvature type” reason. How are you to build a structure class? It seems to me that Deeley-Goffeng deal with this by incorporating quite a lot of analysis into their geometric cycles – as well as the bordism that I have described, there are also projective module bundles over the group algebra, etc… this makes the desired exactness true, but perhaps at the cost of making the groups less geometrical; they are a “geometry-analysis hybrid”. And that is inevitable in this problem.
I should mention that several other applications of the analytic surgery sequence depend on constructing an appropriate ncie model for the structure set: e.g. Siegel, Xie-Yu (see below). I’m not sure whether our original model is “nice” for anybody!
#### References
Deeley, Robin, and Magnus Goffeng. Realizing the Analytic Surgery Group of Higson and Roe Geometrically, Part I: The Geometric Model. ArXiv e-print, August 27, 2013. http://arxiv.org/abs/1308.5990.
Higson, Nigel, and John Roe. “Mapping Surgery to Analysis. I. Analytic Signatures.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 277–299. doi:10.1007/s10977-005-1561-8.
———. “Mapping Surgery to Analysis. II. Geometric Signatures.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 301–324. doi:10.1007/s10977-005-1559-2.
———. “Mapping Surgery to Analysis. III. Exact Sequences.” K-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of K-Theory in the Mathematical Sciences 33, no. 4 (2005): 325–346. doi:10.1007/s10977-005-1554-7.
Higson, Nigel, and John Roe. “$$K$$-homology, Assembly and Rigidity Theorems for Relative Eta Invariants.” Pure and Applied Mathematics Quarterly 6, no. 2, Special Issue: In honor of Michael Atiyah and Isadore Singer (2010): 555–601.
Siegel, Paul. “The Mayer-Vietoris Sequence for the Analytic Structure Group.” arXiv:1212.0241 (December 2, 2012). http://arxiv.org/abs/1212.0241.
Siegel, Paul. “Homological Calculations with the Analytic Structure Group.” PhD Thesis, Penn State, 2012. https://etda.libraries.psu.edu/paper/16113/.
Xie, Zhizhang, and Guoliang Yu. “A Relative Higher Index Theorem, Diffeomorphisms and Positive Scalar Curvature.” arXiv:1204.3664 (April 16, 2012). http://arxiv.org/abs/1204.3664.
Xie, Zhizhang, and Guoliang Yu. “Positive Scalar Curvature, Higher Rho Invariants and Localization Algebras.” arXiv:1302.4418 (February 18, 2013). http://arxiv.org/abs/1302.4418. | {"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": 0.900704562664032, "perplexity": 1364.3955865035662}, "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-2017-51/segments/1512948587577.92/warc/CC-MAIN-20171216104016-20171216130016-00349.warc.gz"} |
https://www.physicsforums.com/threads/hamiltonian-and-symmetries.408075/ | # Homework Help: Hamiltonian and symmetries
1. Jun 5, 2010
### eoghan
[QM] Hamiltonian and symmetries
1. The problem statement, all variables and given/known data
Let there be the hamiltonian:
$$H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2(x^2+y^2+z^2)+kxyz+\frac{k^2}{\hbar \omega}x^2y^2z^2$$
Find the expectation value of the three components of $$\vec r$$ in the ground state using ONLY the symmetry properties of the hamiltonian.
2. Relevant equations
3. The attempt at a solution
I define this parity:
$$\Pi_{xy}: x\rightarrow-x\ \ \ \ y\rightarrow-y$$
Then the hamiltonian commutes with this parity: $$[H, \Pi_{xy}]=0$$
The ground state is not degenerate, so it has a definite parity with respect to $$\Pi_{xy}$$:
$$<gs|x|gs>=<gs|\Pi_{xy}\Pi_{xy}x\Pi_{xy}\Pi_{xy}|gs>=-<gs|\Pi_{xy}x\Pi_{xy}|gs>=-<gs|x|gs>$$
So <gs|x|gs>=0;
Is it right?
Last edited: Jun 5, 2010
2. Jun 5, 2010
### kuruman
Re: [QM] Hamiltonian and symmetries
That is not the definition of the parity operator I am familiar with. What happened to
$$z\rightarrow-z?$$
Only because you defined "this parity" so that it commutes.
Last edited: Jun 5, 2010
3. Jun 6, 2010
### eoghan
Yes, I know this is not the usual parity... but I don't really know any other way to solve this problem
4. Jun 6, 2010
### kuruman
As I mentioned earlier, your Hamiltonian is invariant under cyclic permutations. In other words, the system cannot distinguish x from y from z (it doesn't know the alphabet). What do you think that implies about the expectation values <x>, <y> and <z>?
5. Jun 7, 2010
### eoghan
That they are all the same?
6. Jun 7, 2010
### kuruman
Correct. Since they are all the same, calculating one of them will give you the others. So what do you think that value can be and why? | {"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": 0.9482713341712952, "perplexity": 1295.7480861348702}, "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/1544376826354.54/warc/CC-MAIN-20181214210553-20181214232553-00290.warc.gz"} |
http://mathhelpforum.com/latex-help/135403-how-place-subtitle-paper.html | ## How to place subtitle in paper
If I have the title as "This is my Title", then I'd code:
\title{This is my Title}
and in \documentclass{amsart}, that is all in capital letters.
How can I then specify "This is my subtitle" and have it appear below the title but in lower case with capital first letters or whatever is standard for a subtitle? I see there is no \subtitle command and when I try to specify two lines in the title command, all are capitalized.
Thanks guys, | {"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": 0.9707292318344116, "perplexity": 2607.6157026677824}, "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/1510934808254.76/warc/CC-MAIN-20171124142303-20171124162303-00212.warc.gz"} |
http://www.sawaal.com/statement-and-conclusions-questions-and-answers/statements-in-a-recent-survey-report-it-has-been-stated-that-those-who-undertake-physical-exercise-f_11192 | 2
Q:
# Statements : In a recent survey report, it has been stated that those who undertake physical exercise for at least half an hour a day are less prone to have any heart ailments.Conclusions :a) Moderate level of physical exercise is necessary for leading a healthy life.b) All people who do desk-bound jobs definitely suffer from heart ailments.
A) Only a follows B) Both a & b follows C) Only b follows D) Neither a nor b follows
Explanation:
The statement mentions that chances of heart ailments are greatly reduced by a regular half-hour exercise. So, a follows. However, it talks of only reducing the probability which does not mean that persons involved in sedentary jobs shall definitely suffer from heart ailments. So, b does not follow.
Hence, concusion a only follows the given statement.
Q:
In each question given below four conclusions are followed by statements. You have to take the three given statements to be true (even if they seem to be at variance from the commonly known facts). Read the conclusions and decide which of the following set of statement indicate that conclusion is logically follows.
Statements :
1. All chairs are tents.
2. No chairs is jugs.
3. No jugs is glasses.
4. No glasses is pots.
Conclusions :
I. All pots are tents.
II. All glasses are chair.
III. Some jugs are tents.
A) Only conclusion I follows B) Both conclusion II & III follows C) None follows D) Only conclusion II follows
Explanation:
3 154
Q:
Find which of the two conclusions I and II given below is/are definitely true from the the given statements.
Statements: I > W > T > N, I = F = G = C
Conclusions: I. W > I
II. C > N
A) Only Conclusion I is true B) Only Conclusion II is true C) Both conclusions I & II are true D) Neither conclusion I nor II is true
Explanation:
From the given Statements :
1. I > W > T > N
2. F = G = C
Conclusions are :
I. W > I (False)
II. C > N (True)
7 172
Q:
Study the question and all the three statements given and decide whether any information provided in the statements are redundand and can be dispensed with while answering the questions.
Question :
What is the rate of interest per year ?
Statements :
1. The amount invested is Rs. 6000.
2. The amount becomes Rs. 6741.60 in 2 years at compound interest.
3. The difference between compound interest and simple interest in 2 years is Rs. 21.60.
A) Only 3 B) Both 2 & 3 C) Any two of (1, 2 & 3) D) None
Answer & Explanation Answer: C) Any two of (1, 2 & 3)
Explanation:
Using Both 1 & 2 statements we get the rate of interest as in (1) we have given principle amount and in (2) compound interest for 2 years. By this data we get the rate%.
Using Both 1 & 3 statements we can get the rate% as we have principle amount & difference between compound interest and simple interest in 2 years.
Using Both 2 & 3 statements we get compound interest & simple interest by which we get principle amount. So that we can calculate %rate.
Hence by using any two of the three statements(1,2&3) we get rate of interest.
5 248
Q:
Statements :-
All kids are God.
No God is a Human.
Some Human are Son.
All Son are Men.
Conclusions :-
(a) Some Son are not God.
(b) Some Men are Human.
A) Only A follows B) Only B follows C) Both (A) and (B) follows D) Neither (A) nor (B) follows
Explanation:
7 373
Q:
In the following question, two equations numbered 1 and 2 are given. You have to solve both the equations and determine the relation between a and b.
1. $5{a}^{2}-18a+9=0$
2. $3{b}^{2}+5b-2=0$
A) a >= b B) a <= b C) a < b D) a > b
Explanation:
From solving 1 and 2 we get,
1.
5a(a-3)-3(a-3) = 0
(5a-3)(a-3) = 0
a = 3 or 3/5
2.
3b(b+2)-1(b+2) = 0
b = -2 or b = 1/3
Here when a = 3, a > b for b = -2 and b = 1/3
when a = 3/5. a > b for b = -2 and b = 1/3.
Hence, it is clear that a > b.
3 335
Q:
Assuming the given statements to be true, find which of the two conclusions A and B given below is/are definitely true.
Statements: C < T < L= P > Q, N > C > Y
Conclusions: A. P > Y B. Y < P
A) only conclusion B is true. B) only conclusion A is true. C) neither conclusion I nor II is true. D) either conclusion I or II is true.
Explanation:
From given statements, we can conclude that
N > C < T < L= P > Q .....(1)
Here given that C > Y but in eq(1) we got that C < T < T <= P => Y is definitely less than P.
So only conclusion B is True.
3 293
Q:
Q : Which train did Harish catch to go to office ?
Statements :
A. Harish missed his usual train of 4.15 p.m. A train comes in every 15 minutes.
B. Harish did not catch the 4.45 p.m. train or any train after that time.
A) If statement B alone is sufficient but statement A alone is not sufficient. B) If statement A alone is sufficient but statement B alone is not sufficient. C) If both statement together are sufficient, but neither statement alone is sufficient. D) If statement A and B together are not sufficient.
Answer & Explanation Answer: D) If statement A and B together are not sufficient.
Explanation:
From both statements we cannot conclude the train catched by Harish
Since he missed at 4.15 and train coes at 4.30, 4.45, 5.00,...
But in B given that he didn't catch the train at 4.45 and after that.
So both statements A & B together are not sufficient to answer the question.
3 325
Q:
What is the value of KL ?
Statement A: ${K}^{2}$= 4.
Statement B: L = 0.
A) Only A is sufficient B) Only B is sufficient C) Both (A) and (B) are sufficient D) None of the above | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "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": 0.8203544020652771, "perplexity": 1691.3793702740795}, "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/1521257650262.65/warc/CC-MAIN-20180324112821-20180324132821-00254.warc.gz"} |
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Mathematics , 2014, Abstract: We prove uniqueness for Calder\'on's problem with Lipschitz conductivities in higher dimensions. Combined with the recent work of Haberman, who treated the three and four dimensional cases, this confirms a conjecture of Uhlmann. Our proof builds on the work of Sylvester and Uhlmann, Brown, and Haberman and Tataru who proved uniqueness for $C^1$ conductivities and Lipschitz conductivities sufficiently close to the identity.
Boaz Haberman Mathematics , 2014, DOI: 10.1007/s00220-015-2460-3 Abstract: We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(\Omega)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.
Guo Zhang Mathematics , 2012, DOI: 10.1088/0266-5611/28/10/105008 Abstract: In this paper we study the inverse conductivity problem with partial data. Moreover, we show that, in dimension $n\geq 3$ the uniqueness of the Calder\'{o}n problem holds for the $C^{1}\bigcap H^{3/2, 2}$ conductivities.
Mathematics , 2014, DOI: 10.1088/0266-5611/31/1/015008 Abstract: We consider the stability issue of the inverse conductivity problem for a conformal class of anisotropic conductivities in terms of the local Dirichlet-to-Neumann map. We extend here the stability result obtained by Alessandrini and Vessella in Advances in Applied Mathematics 35:207-241, where the authors considered the piecewise constant isotropic case.
Mathematics , 2015, Abstract: We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $\Omega\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.
Mathematics , 2014, Abstract: We consider the inverse problem of determining the Lam\'{e} parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lam\'{e} parameters and the density are assumed to be piecewise constant on a given domain partition.
Mathematics , 2012, Abstract: In these notes we prove log-type stability for the Calder\'on problem with conductivities in $C^{1,\varepsilon}(\bar{\Omega})$. We follow the lines of a recent work by Haberman and Tataru in which they prove uniqueness for $C^1(\bar{\Omega})$.
Mathematics , 2013, Abstract: In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lam\'{e} parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.
Mathematics , 2015, Abstract: We extend a global uniqueness result for the Calder\'on problem with partial data, due to Kenig-Sj\"ostrand-Uhlmann, to the case of less regular conductivities. Specifically, we show that in dimensions $n\ge 3$, the knowledge of the Diricihlet-to-Neumann map, measured on possibly very small subsets of the boundary, determines uniquely a conductivity having essentially $3/2$ derivatives in an $L^2$ sense.
P. A. Krutitskii International Journal of Mathematics and Mathematical Sciences , 2013, DOI: 10.1155/2013/302628 Abstract: We study the Dirichlet problem for the equation in the exterior of nonclosed Lipschitz surfaces in . The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution of the problem are proved. The integral representation for a solution is obtained in the form of single-layer potential. The density in the potential is defined as a solution of the operator (integral) equation, which is uniquely solvable. Weak solvability of elliptic boundary value problems with Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary conditions in Lipschitz domains has been studied in [1–6]. It is pointed out in the book [1, page 91] that domains with cracks (cuts) are not Lipschitz domains. So, solvability of elliptic boundary value problems in domains with cracks does not follow from general results on solvability of elliptic boundary value problems in Lipschitz domains. In the present paper, the weak solvability of the Dirichlet problem for the equation in the exterior of nonclosed Lipschitz surfaces (cracks) in is studied. The Dirichlet problem for the Laplace equation is a particular case of our problem. Theorems on existence and uniqueness of a weak solution are proved, integral representation for a solution in the form of single-layer potential is obtained, the problem is reduced to the uniquely solvable operator equation. The weak solvability of the Neumann problem for the Laplace equation in the exterior of several smooth nonclosed surfaces in has been studied in [7]. Boundary value problems for the Helmholtz equation in the exterior of smooth nonclosed screens have been studied in [8, 9]. In Cartesian coordinates in consider bounded Lipschitz domain with the boundary , that is, is closed Lipschitz surface. Let be nonempty subset of the boundary and . Assume that is a nonclosed Lipschitz surface with Lipschitz boundary in the space , and assume that includes its limiting points, or, alternatively, assume that is a union of finite number of such nonclosed surfaces, which do not have common points, in particular, they do not have common boundary points. In the latter case, is not a connected set. Notice that is a closed set. Let us introduce Sobolev spaces on as follows: Spaces and are dual spaces in the sense of scalar product in [1, pages 91-92]. Furthermore, one can set for (see [1, page 79]), and (see [1, pages 77, 99]). Spaces and on closed Lipschitz surface and their norms are defined, for example, in [1, page 98]. Let be Laplacian in , then for the equation consider the single-layer
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https://topfuturepoint.com/35-percent-of-20/ | # 35 percent of 20
Let us know about 35 percent of 20.
35 percent of 20 is 7 .
Similarly, which number is 40 percent of 20? Answer: 40% of 20 is 8 .
Let’s find 40% of 20.
How do you find 60 percent of a number? To convert the fraction 60/100 to a percentage, you must first divide the fraction 60 by the denominator 100 to convert 60/100 to a decimal. This means that 60/100 = 0.6 . Then, multiply 0.6 by 100 = 60%.
What is 60 out of 25? So 60 out of 25 is up to 41.67% .
Second 90 out of what percentage is 300? Nearby results
## Which number is 50 percent of 20?
Answer: 50 percent of 20 is 10 .
Let’s find 50 percent of 20.
So how do I figure out the percentage? The percentage can be calculated by dividing the value by the total value and then multiplying the result by 100. The formula used to calculate the percentage is: (Price/Total Value)×100% .
How do you find 25% of a number? To calculate 25 percent of a number, simply multiply it by 4. divide by . For example, 12 percent of 25 is 12 divided by 4, or 3.
### What is 60 percent in a grade?
60 out of 40 is what percent? Now we can see that our fraction is 66.666666666667/100, which means that 40/60 as a percentage is up to 66.6667% .
What is 60 out of 30?
Take help of what is x percentile of y calculator, an online math tool that easily calculates 60% of 30 along with step by step solutions that explain how the result came to 18.
Nearby Results.
75 out of 60 is what percent? Now we can see that our fraction is 80/100, which means that 60/75 as a percentage is up to 80% .
#### What percent is 60 out of 3?
Now we can see that our fraction is 5/100, which means that 3/60 as a percentage is 5% . And there you have it!
How do you write 2/10 as a percentage?
Now we can see that our fraction is 20/100, which means that 2/10 as a percentage is up to 20% .
75 out of 30 is what percent? Now we can see that our fraction is 40/100, which means that 30/75 as a percentage is up to 40% .
Which number is 90% of 20? Answer: 90% of 20 is 18 .
Let’s find 90% of 20.
#### What number is 45% of 20?
Answer: 45% of 20 is 9 .
Let us find 45% of 20.
Which number is 80 percent of 20? Answer: 80% of 20 is 16 .
Let’s find 80% of 20.
How do you figure out percentages without a calculator?
How do you calculate percentage on calculator online? To calculate the percentage of a percentage, convert both percentages to 100 or decimal fractions and multiply them by . For example, 50% of 40% is calculated; (50/100) x (40/100) = 0.50 x 0.40 = 0.20 = 20/100 = 20%.
### How do you find the percentage of a whole number?
find percentage
For this type of problem, you can simply divide the number you want to convert to a percentage by the whole . So, using this example, you would divide 2 by 5. This equation will give you 0.4. You would then multiply 0.4 by 100 to get 40, or 40%.
How do you find 76 percent of a number?
What percentage is 80 out of 12?
Now we can see that our fraction is 15/100, which means that 12/80 as a percentage is up to 15% .
What is 45 135 as a percentage? Solution and How to Convert 45/135 to Percentage
0.33 times 100 = 33.33 . that’s all there is to it! | {"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": 0.8266540765762329, "perplexity": 633.5145675774543}, "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/1669446710417.25/warc/CC-MAIN-20221127173917-20221127203917-00458.warc.gz"} |
https://planetmath.org/osculatingcurve | osculating curve
Definition. From a family of plane curves, the osculating curve of the curve $y=f(x)$ in a certain point is the curve of the family which has the highest order contact with the curve $y=f(x)$ in that point.
Example 1. From the family of the graphs of the polynomial functions
$P_{n}(x):=c_{0}+c_{1}(x-x_{0})+\ldots+c_{n}(x-x_{0})^{n},$
the osculating curve of $y=f(x)$ in $(x_{0},\,f(x_{0}))$ is the Taylor polynomial of degree $n$ of the function $f$.
Example 2. Determine the osculating hyperbola with axes parallel to the coordinate axes for the curve $y=\cos{x}$ in the point $(0,\,1)$. What is the order of contact?
We may seek the osculating hyperbola from the three-parametric family
$\displaystyle\frac{x^{2}}{a^{2}}-\frac{(y-y_{0})^{2}}{b^{2}}=-1.$ (1)
Removing the denominators and differentiating six times successively yield the equations
$\displaystyle\begin{cases}b^{2}x^{2}-a^{2}(y-y_{0})^{2}+a^{2}b^{2}=0,\\ b^{2}x-a^{2}(y-y_{0})y^{\prime}=0,\\ b^{2}-a^{2}y^{\prime 2}-a^{2}(y-y_{0})y^{\prime\prime}=0,\\ 3y^{\prime}y^{\prime\prime}+(y-y_{0})y^{\prime\prime\prime}=0,\\ 3y^{\prime\prime 2}+4y^{\prime}y^{\prime\prime\prime}+(y-y_{0})y^{\prime\prime% \prime\prime}=0,\\ 10y^{\prime\prime}y^{\prime\prime\prime}+5y^{\prime}y^{\prime\prime\prime% \prime}+(y-y_{0})y^{(5)}=0,\\ 10y^{\prime\prime\prime 2}+15y^{\prime\prime}y^{\prime\prime\prime\prime}+6y^{% \prime}y^{(5)}+(y-y_{0})y^{(6)}=0.\end{cases}$ (2)
Into these equations we can substitute the coordinates $x=0,\,y=1$ of the contact point and the values of the derivatives
$y^{\prime}=-\sin{x},\;y^{\prime\prime}=-\cos{x},\;y^{\prime\prime\prime}=\sin{% x},\;y^{\prime\prime\prime\prime}=\cos{x},\;y^{(5)}=-\sin{x},\;y^{(6)}=-\cos{x}$
of cosine in that point; the values are $0,\;-1,\;0,\;1,\,0,\;-1$. The first, third and fifth of the equations (2) give the result $y_{0}=4,\;\,b^{2}=9,\;\,a^{2}=3$, whence the osculating hyperbola is
$\frac{x^{2}}{3}-\frac{(y-4)^{2}}{9}=-1.$
When we substitute the pertinent values of the cosine derivatives into the two last equations (2), we see that only the former of them is satisfied. It means that the order of contact between the cosine curve and the hyperbola is 5.
Example 3. The osculating parabola of the exponential curve (http://planetmath.org/ExponentialFunction) $y=e^{x}$ in the point $(0,\,1)$ is
$4x^{2}+y^{2}+4xy+14x-20y+19=0.$
The order of contact is only 3.
Title osculating curve OsculatingCurve 2013-03-22 17:57:17 2013-03-22 17:57:17 pahio (2872) pahio (2872) 7 pahio (2872) Definition msc 51N05 msc 53A04 OrderOfVanishing CircleOfCurvature Cosine QuadraticCurves | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 19, "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": 0.8107964992523193, "perplexity": 1648.049994788797}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743248.7/warc/CC-MAIN-20181117020225-20181117042225-00349.warc.gz"} |
https://chemistry.stackexchange.com/questions/16003/how-much-energy-does-it-cost-to-have-electron-configurations-that-are-not-in-acc | # How much energy does it cost to have electron configurations that are not in accordance with Hund's rules?
What is a ballpark figure for the difference in energy for an atom that follows Hund's rule vs one that has two electrons with opposite spins? I'd be interested to know carbon and nitrogen. Is there a bigger difference when there are two electrons with one spin and one electron with the opposite spin vs two electrons with opposite spins?
• May I ask: What electron configurations were the ones you actually asked for? – Philipp Sep 9 '14 at 2:38
In the following pictures the notation $m_{S} = + \frac{1}{2} \overset{\scriptsize{\text{def}}}{=} \, \uparrow$ and $m_{S} = - \frac{1}{2} \overset{\scriptsize{\text{def}}}{=} \, \downarrow$ will be used for the spin quantum number $m_{S}$.
The data for carbon can be found here. The ground state for carbon is ${}^{3}\mathrm{P}_{0}$ - it is assigned the energy $0.00 \, \mathrm{cm}^{-1}$. From your question it's not 100 % clear what exact electron configuration you want. I assume it is either ${}^{3}\mathrm{P}_{2}$ or ${}^{1}\mathrm{S}_{0}$ therefore I will give the values for both of them (for comparison: the thermal energy at room temperature is about $200 \, \mathrm{cm}^{-1}$).
The data for nitrogen can be found here. The ground state for nitrogen is ${}^{4}\mathrm{S}_{3/2}$ - it is assigned the energy $0.00 \, \mathrm{cm}^{-1}$. From your question it's not 100 % clear what exact electron configuration you want. I assume it is either ${}^{2}\mathrm{D}_{5/2}$ or ${}^{2}\mathrm{P}_{3/2}$ therefore I will give the values for both of them.
• @LordStryker The $^3P_0$ and $^3P_2$ differs in J, i.e. the mutual orientation of L and S. This energy difference is coming from the spin-orbit coupling (which is per definition proportional with $LS$). Since spin-orbit coupling is small for light elements that is why the energy gap is so small. – Greg Sep 9 '14 at 13:04 | {"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": 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": 0.8296761512756348, "perplexity": 293.80466830995704}, "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/1568514573176.51/warc/CC-MAIN-20190918024332-20190918050332-00072.warc.gz"} |
http://www.mathjournals.org/jot/2012-067-002/2012-067-002-009.html | Previous issue · Next issue · Most recent issue · All issues
# Journal of Operator Theory
Volume 67, Issue 2, Spring 2012 pp. 437-467.
Proper actions of groupoids on $C^*$-algebras
Authors Jonathan Henry Brown
Author institution: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand
Summary: In 1990, Rieffel defined a notion of proper action of a group $H$ on a $C^*$-algebra $A$. He then defined a generalized fixed point algebra $A^{\alpha}$ for this action and showed that $A^{\alpha}$ is Morita equivalent to an ideal of the reduced crossed product. We generalize Rieffel's notion to define proper groupoid dynamical systems and show that the generalized fixed point algebra for proper groupoid actions is Morita equivalent to a subalgebra of the reduced crossed product. We give some nontrivial examples of proper groupoid dynamical systems and show that if $(\A, G, \alpha)$ is a groupoid dynamical system such that $G$ is principal and proper, then the action of $G$ on $\A$ is saturated, that is the generalized fixed point algebra is Morita equivalent to the reduced crossed product.
Contents Full-Text PDF | {"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": 0.9787947535514832, "perplexity": 317.24264861064626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571234.82/warc/CC-MAIN-20220811042804-20220811072804-00555.warc.gz"} |
http://mathhelpforum.com/calculus/32280-simple-cal-2-integral.html | # Thread: Simple Cal 2 integral
1. ## Simple Cal 2 integral
integral of $\int \, 2x/x-4
$
got a test tomorrow and I have a feeling this may be on the test.
2. Originally Posted by p00ndawg
integral of $2x/x-4
$
got a test tomorrow and I have a feeling this may be on the test.
rewrite as follows or use long division
$2\int \frac{x}{x-4}dx=2\int \frac{(x-4)+4}{x-4}dx=$
$2\int dx + 2\int \frac{4}{x-4}dx=2x+8 \ln|x-4|+c$
3. Originally Posted by TheEmptySet
rewrite as follows or use long division
$2\int \frac{x}{x-4}dx=2\int \frac{(x-4)+4}{x-4}dx=$
$2\int dx + 2\int \frac{4}{x-4}=2x+8 \ln(x-4)+c$
Brilliant.
4. This is in essence exactly the same procedure that the others have used, but with slightly different symbols (my old calc teacher used to call it "tricky" substitution)...
choose u=x-4. This means that du=dx, but it also means that u+4=x. As a result we have the following integral:
$\int \frac{2(u+4)}{u} du$
$
=\int \frac{2u}{u} +\frac{8}{u} du$
$
=2\int du + 8\int \frac{1}{u} du$
$=2u+8\ln{u}+c$
$=2(x-4)+8\ln{|x-4|}+c$
$=2x+8\ln{|x-4|}+C$
5. Originally Posted by teuthid
This is in essence exactly the same procedure that the others have used, but with slightly different symbols (my old calc teacher used to call it "tricky" substitution)...
choose u=x-4. This means that du=dx, but it also means that u+4=x. As a result we have the following integral:
$\int \frac{2(u+4)}{u} du$
$
=\int \frac{2u}{u} +\frac{8}{u} du$
$
=2\int du + 8\int \frac{1}{u} du$
$=2u+8\ln{u}+c$
$=2(x+4)+8\ln{(x-4)}+c$
$=2x+8\ln{(x-4)}+C$
what happens to the 8 when you distribute 2 to x + 4 and why is it x + 4, shouldnt it be x - 4?
6. TheEmptySet & teuthid are missing something important here: it's actually $\ln|x-4|.$
7. your right, it should be x-4. sorry about the typo. The 8 got dumped into the arbitrary constant that is added to the end. Any time you have an antiderivative with spare constants lying around you can do that. For example if your antiderivative was $F(x)=x^2+1+\sin(x)+2+e^x+5$, then the indefinite integral could be written as $x^2+\sin(x)+e^x+C$. You'll notice that I switched from c to C to show this "dumping". | {"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": 21, "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": 0.8754021525382996, "perplexity": 729.8492426414008}, "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/1502886109157.57/warc/CC-MAIN-20170821152953-20170821172953-00458.warc.gz"} |
https://www.math.uci.edu/category/event-category/special-colloquium | # Statistical estimation under algebraic constraints: total positivity
Elina Robeva
MIT
## Time:
Thursday, January 10, 2019 - 4:00pm to 5:00pm
## Location:
RH 306
My research focuses on studying models that depict complex dependencies between random variables. Such models include directed graphical models with hidden variables, discrete mixture models (which give rise to low rank tensors), and models that impose strong positive dependence called total positivity. In this talk I will give a brief overview of my work in these areas, and will particularly focus on the problem of density estimation under total positivity.
Nonparametric density estimation is a challenging statistical problem -- in general the maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints such as total positivity allows a path forward. Though they possess very special structure, totally positive random variables are quite common in real world data and exhibit appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the MLE exists with probability one if there are at least 3 samples. We characterize the domain of the MLE, and give algorithms to compute it. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a piecewise linear function and can be computed via a certain convex program. Finally, I will discuss statistical guarantees for the convergence of the MLE, and will conclude with a variety of further research directions.
# When combinatorics meets Littlewood-Offord theory
Asaf Ferber
MIT
## Time:
Friday, January 11, 2019 - 4:00pm to 5:00pm
## Location:
RH 306
Given an integer vector $a = (a_1,\dots,a_n)$, let $\rho(a)$ be the number of solutions to $a\cdot x=0$, with $x\in \{\pm 1\}^n$. In 1945, Erdos gave a beautiful combinatorial solution to the following problem that was posed by Littlewood and Offord: how large can $\rho(a)$ be if all the entries of $a$ are non-zero?
Following his breakthrough result, several extensions of this problem have been intensively studied by various researchers. In classical works, Erdos-Moser, Sarkozy-Szemeredi, and Halasz obtained better bounds on $\rho(a)$ under additional assumptions on $a$, while Kleitman, Frankl-Furedi, Esseen, Halasz and many others studied generalizations to higher dimensions. In recent years, motivated by deep Freiman-type inverse theorems from additive combinatorics, Tao and Vu brought a new view to this problem by asking for the underlying structural reason for $\rho(a)$ to be large --this is known as the Inverse Littlewood-Offord problem, which is a cornerstone of modern random matrix theory.
In this talk, we will discuss further extensions and improvements for both forward and inverse Littlewood-Offord problems where combinatorial tools and insights have proved to be especially powerful. We also present several applications in (discrete) matrix theory such as: a resilience' version of it, a non-trivial upper bound on the number of Hadamard matrices, an upper bound on the number of $\pm 1$ normal matrices, and a unified approach for counting the number of singular $\pm1$ matrices from various popular models.
# Large deviations for sparse random graphs
Nick Cook
UCLA
## Time:
Monday, January 14, 2019 - 3:00pm to 4:00pm
## Location:
RH 306
Let $G=G(N,p)$ be an Erd\H{o}s--R\'enyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles (triples of vertices with all three pairs connected) as we would expect? I will discuss recent progress on this infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.
# Graphs in algebraic and arithmetic geometry
Farbod Shokrieh
## Institution:
University of Copenhagen
## Time:
Tuesday, January 22, 2019 - 4:00pm to 5:00pm
## Location:
RH 306
Graphs can be viewed as (non-archimedean) analogues of Riemann surfaces. For example, there is a notion of Jacobians for graphs. More classically, graphs can be viewed as electrical networks. I will explain the interplay between these points of view, as well as some recent application in arithmetic geometry.
# Modern Non''-Problems in Optimization: Applications to Statistics and Machine Learning
Ying Cui
## Institution:
University of Southern California
## Time:
Wednesday, January 23, 2019 - 4:00pm to 5:00pm
## Location:
RH 306
Classical continuous optimization starts with linear and nonlinear programming. In the past two decades, convex optimization (e.g., sparse linear regressions with convex regularizers) has been a very effective computational tool in applications to statistical estimations and machine learning. However, many modern data-science problems involve some basic `non’’-properties that are ignored by the convex approach for the sake of the computation convenience. These non-properties include the coupling of the non-convexity, non-differentiability and non-(Clarke) regularity. In this talk, we present a rigorous computational treatment for solving two non-problems: the piecewise affine regression and the feed-forward deep neural network. The algorithmic framework is an integration of the first order non-convex majorization-minimization method and the second order non-smooth Newton methods. Numerical experiments demonstrate the effectiveness of our proposed approach. Contrary to existing methods for solving non-problems which provide at best very weak guarantees on the computed solutions obtained in practical implementation, our rigorous mathematical treatment aims to understand properties of these computed solutions with reference to both the empirical and the population risk minimizations. This is based on joint work with Jong-Shi Pang, Bodhisattva Sen and Ziyu He.
# Universal arithmetical hierarchy of eigenfunctions for supercritical almost Mathieu operators
Wencai Liu
## Institution:
University of California, Irvine
## Time:
Tuesday, January 8, 2019 - 4:00pm to 5:00pm
## Location:
RH 306
The Harper's model is a tight-binding description of Bloch electrons on $\mathbb{Z}^2$ under a constant transverse magnetic field. In 1964, Mark Azbel predicted that both spectra and eigenfunctions of this model have self-similar hierarchical structure driven by the continued fraction expansion of the irrational magnetic flux. In 1976, the hierarchical structure of spectra was discovered numerically by Douglas Hofstadter, and was later observed in various experiments. The mathematical study of Harper's model led to the development of spectral theory of the almost Mathieu operator, with the solution of the Ten Martini Problem partially confirming the fractal structure of the spectrum.
In this talk, we will present necessary background and discuss the main ideas behind our confirmation (joint with S. Jitomirskaya) of Azbel's second prediction of the structure of the eigenfunctions. More precisely, we show that the eigenfunctions of the almost Mathieu operators in the localization regime, feature self-similarity governed by the continued fraction expansion of the frequency. These results also lead to the proof of sharp arithmetic transitions between pure point and singular continuous spectra, both in the frequency and the phase, as conjectured in 1994. | {"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": 0.856057345867157, "perplexity": 657.6755335878668}, "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-2019-04/segments/1547583657097.39/warc/CC-MAIN-20190116073323-20190116095323-00230.warc.gz"} |
https://briankoberlein.com/post/gravity-check/ | Gravity Check
3 September 2013
Yesterday I wrote about how we test whether unitless constants such as alpha (α) change over the history of the universe. You might also have noticed that I said if such constants did change, then it would mean either fundamental physical constants change or there is some exotic physics going on. We looked at the physical constants yesterday, so today let’s look for exotic physics.
As I mentioned yesterday, the value of alpha can affect the line spectra of atoms and molecules, which makes it easy to observe astronomically. In testing for changes in alpha over space and time, astronomers looked at distant molecular clouds. These are low-gravity environments, so they are similar to the type of environment we have here on Earth. But if you want to look for exotic physics, you probably want to look for situations that are either very high energy, or very high gravity.
Since alpha depends on three constants (electron charge, light speed, and Planck’s constant), it shouldn’t be affected by strong gravitational fields such as the ones near a white dwarf or neutron star. But there are theoretical models that connect Planck’s constant to the universal gravitational constant (G). These models attempt to develop a theory of quantum gravity which would connect quantum theory to general relativity. Some of these models introduce things that would allow the gravitational field to change the value of alpha. Of course this change would only be seen with very strong gravity cases.
Recently, a paper was published in Physical Review Letters that looked at line spectra from a white dwarf known as G191-B2B.1 Since the surface gravity of a white dwarf is about 100,000 times that of Earth’s, this provided a good test of whether alpha is affected by gravity. Comparing the line spectra with those observed on Earth, they found no discernable difference, which means any variation in alpha can be no larger than one part in 10,000.
These results put a damper on any quantum gravity models where alpha could vary significantly (such as those with strong interactions between scalar and electromagnetic fields). However 1 in 10,000 is not too strong a constraint, so the results still leave the door open on more subtle variations that might give us hints about quantum gravity.
For now, however, alpha remains constant.
1. Berengut, J. C., et al. “Limits on the dependence of the fine-structure constant on gravitational potential from white-dwarf spectra.” Physical review letters 111.1 (2013): 010801. ↩︎ | {"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": 0.9013700485229492, "perplexity": 509.80564676997943}, "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-2021-04/segments/1610703565376.63/warc/CC-MAIN-20210125061144-20210125091144-00588.warc.gz"} |
http://vesnik.math.rs/cap/browse.php?p=mv133 | Volume 65 , issue 3 ( 2013 ) back
Convergence theorems of two-step implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings 285$-$298 Gurucharan Singh Saluja
Abstract
The objective of this paper is to study the weak and strong convergence of two-step implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), Zhou and Chang (2002) and Gu and Lu (2006).
Keywords: Asymptotically nonexpansive mapping; implicit iteration process with errors; common fixed point; strong convergence; weak convergence.
MSC: 47H09, 47H10
On certain univalent class associated with functions of non-Bazilevič type 299$-$305 Rabha W. Ibrahim
Abstract
In this work, we study certain differential inequalities and first order differential subordinations. As their applications, we obtain some sufficient conditions for univalence, which generalize and refine some previous results.
Keywords: Univalent functions; starlike functions; convex functions; close-to-convex functions; differential subordination; subordination; superordination; unit disk; $\Phi$-like functions; non-Bazilevič type; Dziok-Srivastava linear operator; sandwich theorem.
MSC: 30C45
Compact-like properties in hyperspaces 306$-$318 J. Angoa, Y. F. Ortiz-Castillo and Á. Tamariz-Mascar\'ua
Abstract
$\mathcal{CL}(X)$ and $\mathcal{K}(X)$ denote the hyperspaces of non-empty closed and non-empty compact subsets of $X$, respectively, with the Vietoris topology. For an infinite cardinal number $\alpha$, a space $X$ is $\alpha$-hyperbounded if for every family $\{S_{\xi}:\xi<\alpha\}$ of non-empty compact subsets of $X$, $Cl_X(\bigcup_{\xi<\alpha}S_{\xi})$ is a compact set, and a space $X$ is pseudo-$\omega$-bounded if for each countable family $\mathcal{U}$ of non-empty open subsets of $X$, there exists a compact set $K\subseteq X$ such that each element in $\mathcal{U}$ has a non-empty intersection with $K$. We prove that $X$ is $\alpha$-hyperbounded if and only if $\mathcal{K}(X)$ is $\alpha$-hyperbounded, if and only if $\mathcal{K}(X)$ is initially $\alpha$-compact. Moreover, $\mathcal{K}(X)$ is pseudocompact if and only if $X$ is pseudo-$\omega$-bounded. Also, we show than if $\mathcal{K}(X)$ is normal and $C^{*}$-embbeded in $\mathcal{CL}(X)$, then $X$ is $\omega$-hyperbounded, and $X$ is $\alpha$-bounded if and only if $X$ is $\alpha$-hyperbounded, for every infinite cardinal number $\alpha$.
Keywords: Hyperspaces; Vietoris topology; $\alpha$-hyperbounded spaces; pseudo-$\omega$-bounded spaces; normal and $C^*$-embedded spaces.
MSC: 54B20, 54D99, 54D15, 54C45
Mapping properties of some classes of analytic functions under a general integral operator defined by the Hadamard product 319$-$325 Serap Bulut and Pranay Goswami
Abstract
In this paper, we consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under a general integral operator defined by the Hadamard product.
Keywords: Bounded boundary and bounded radius rotations; integral operator; Hadamard product (convolution).
MSC: 30C45
On the rings on torsion-free groups 326$-$333 F. Karimi and H. Mohtadifar
Abstract
The typeset of a torsion-free group is one of the important concepts in the theory of abelian groups. We use the typeset of an abelian group to study the rings that exist over such groups. Moreover, we consider the types of rational groups belonging to an independent set of a group and obtain some results about their relation with the rings over the group.
Keywords: Typeset; ring on a group; nil group.
MSC: 20K15
On some new characterizations of near paracompactness and associated results 334$-$345 M. N. Mukherjee and Dhananjoy Mandal
Abstract
Near paracompactness is a concept, in Set Topology, which is weaker than paracompactness; in this paper, several characterizations of this concept have been enunciated and proved. In the process, several tools have been utilized. The main theorem uses the selection theory of Michael.
Keywords: Near paracompactness; almost regularity; regular open set; semiregularization; strongly lower semicontinuous carrier.
MSC: 54D20, 54C65, 54D99
Some results on trans-Sasakian manifolds 346$-$352 Rajendra Prasad and Vibha Srivastava
Abstract
The object of the present paper is to study $\phi$-conformally (resp. conharmonically, projectively) flat trans-Sasakian manifolds.
Keywords: $\phi$-conformally flat; $\phi$-conharmonically flat; $\phi$-projectively flat and trans-Sasakian manifold
MSC: 53C50, 53C15
Weighted Hankel operators and matrices 353$-$363 Gopal Datt and Deepak Kumar Porwal
Abstract
In this paper, the notions of weighted Hankel matrix along with weighted Hankel operator $S_{\phi}^{\beta}$, with $\phi ın L^{ınfty}({\beta})$ on the space $L^2(\beta)$, $\beta=\{\beta_n\}_{nın \Bbb{Z}}$ being a sequence of positive numbers with $\beta_0=1$, are introduced. It is proved that an operator on $L^2(\beta)$ is a weighted Hankel operator on $L^2(\beta)$ if and only if its matrix is a weighted Hankel matrix. Various properties of the weighted Hankel operators $S_{\phi}^{\beta}$ on $L^2(\beta)$ are also discussed.
Keywords: Weighted Hankel matrix; weighted Hankel operator.
MSC: 47B35, 47B20
Relative EXT groups of Abelian categories 364$-$372 Chaoling Huang and Kai Deng
Abstract
In this article, we characterize the relative Ext groups of abelian categories relative to a fixed precovering class $\Cal{F}$ and give some examples.
Keywords: Relative Ext group; precovering; $\mathcal{F}$-resolution.
MSC: 18G15, 18G25
Certain sufficient conditions for a subclass of analytic functions associated with Hohlov operator 373$-$382 S. Sivasubramanian, Thomas Rosy and K. Muthunagai
Abstract
Making use of the Hohlov operator, we obtain inclusion relations between the classes of certain normalized analytic functions. Relevant connections of our work with the earlier works are pointed out.
Keywords: Starlike function; convolution, negative coefficients; coefficient inequalities; growth and distortion theorems.
MSC: 33C45, 33A30, 30C45
Decomposition of an integer as a sum of two cubes to a fixed modulus 383$-$386 David Tsirekidze and Ala Avoyan
Abstract
The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime power there is given an inductive way to find its remainders that can be represented as the sum of two cubes to a fixed modulus $N$. Moreover, it is possible to find the components of this representation.
Keywords: Sum of two cubes; Diophantine equation.
MSC: 11A07, 11B50, 11D25
Signed total distance $k$-domatic numbers of graphs 387$-$393 S. M. Sheikholeslami and L. Volkmann
Abstract
In this paper we initiate the study of signed total distance $k$-domatic numbers in graphs and we present its sharp upper bounds.
Keywords: Signed total distance $k$-domatic number; signed total distance $k$-dominating function; signed total distance $k$-domination number.
MSC: 05C69
Univalence conditions of general integral operator 394$-$402 B. A. Frasin and D. Breaz
Abstract
In this paper, we obtain new univalence conditions for the integral operator $$I_{\xi}^{\alpha_{i},\beta_{i}}(f_{1},\dots,f_{n})(z)=\left[\xiınt_{0}^{z}t^{\xi-1}(f_{1}^{\prime}(t))^{\alpha_{1}} (\frac{f_{1}(t)}{t})^{\beta_{1}}\cdots(f_{n}^{\prime}(t))^{\alpha_{n}}(\frac{f_{n}(t)}{t})^{\beta_{n}}\,dt\right]^{\frac{1}{\xi}}$$ of analytic functions defined in the open unit disc.
Keywords: Analytic function; univalent function; integral operator.
MSC: 30C45
Majorization problem for a subclass of $p$-valently analytic functions defined by the Wright generalized hypergeometric function 403$-$411 S. P. Goyal and Sanjay Kumar Bansal
Abstract
In this paper we investigate the majorization problem for a subclass of $p$-valently analytic functions involving the Wright generalized hypergeometric function. Some useful consequences of the main result are mentioned and relevance with some of the earlier results are also pointed out.
Keywords: Analytic, $p$-valent, majorization, Wright generalized hypergeometric function.
MSC: 30C45
Ore type condition and Hamiltonian graphs 412$-$418 Kewen Zhao
Abstract
In 1960, Ore proved that if $G$ is a graph of order $n\geq3$ such that $d(x)+d(y)\geq n$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian. In 1985, Ainouche and Christofides proved that if $G$ is a 2-connected graph of order $n\geq 3$ such that $d(x)+d(y)\geq n-1$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian or $G$ belongs to two classes of exceptional graphs. In this paper, we prove that if $G$ is a connected graph of order $n\geq 3$ such that $d(x)+d(y)\geq n-2$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian or $G$ belongs to one of several classes of well-structured graphs.
Keywords: Ore type condition; Hamiltonian graphs.
MSC: 05C38, 05C45
Totally bounded endomorphisms on a topological ring 419$-$424 Madjid Mirzavaziri and Omid Zabeti
Abstract
Let $X$ be a topological ring. In this paper, we consider the three classes ($btb$-bounded, $tbtb$-bounded, and $tbb$-bounded) of endomorphisms defined on $X$ and denote these classes by $B_{btb}(X), B_{tbtb}(X)$, and $B_{tbb}(X)$, respectively. We equip them with an appropriate topology and we find some sufficient conditions under which, each class of these endomorphisms is complete.
Keywords: Totally bounded endomorphism; completeness; topological ring.
MSC: 54H13, 16W80 | {"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": 0.9650974869728088, "perplexity": 917.6950831163598}, "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-2018-51/segments/1544376829812.88/warc/CC-MAIN-20181218204638-20181218230638-00454.warc.gz"} |
https://www.techyv.com/questions/where-i-can-find-vmware-network-traffic-monitoring-application/ | ## Where I can find vmware network traffic monitoring application
Asked By 110 points N/A Posted on -
Hi I want to trace and monitor my network in vmware. Where I can download such application or software vmware network traffic monitoring that I can capture’s to those of passing through to my network. So that I can minimize and trace where is the part of the network has low part. I hope you can help me to find a software. Thank you.
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Answered By 0 points N/A #199331
## Where I can find vmware network traffic monitoring application
Try using the LANGuardian to monitor the VMWare physical and virtual networks. It allows you to monitor the virtual network traffic just like the way you monitor it on the physical network. It brings combined monitoring for the physical and virtual network with one user interface.
It seizes and analyzes data packets running on the core switch of the network and then save the details on the database and then submit the data on the browser. | {"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": 0.8676897287368774, "perplexity": 1604.9550161779437}, "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-2018-39/segments/1537267158045.57/warc/CC-MAIN-20180922044853-20180922065253-00261.warc.gz"} |
http://mathhelpforum.com/calculus/161217-double-integrals-problem-print.html | # double integrals problem
• Oct 27th 2010, 04:36 PM
Evan.Kimia
double integrals problem
Im doing this problem below:
http://i51.tinypic.com/k96gi8.png
here is the solution and i understand everything except how they integrated the r(sqrt(1+r^2)). could anyone point out how they accomplished that? Thank you.
solution:
http://i56.tinypic.com/2vlqihz.jpg
• Oct 27th 2010, 04:51 PM
harish21
$\displaystyle{\int 2r - r \sqrt{1+r^2} dr = \int 2r dr - \int r \sqrt{1+r^2} dr}$
$\displaystyle{\int 2r dr} = 2 \times \dfrac{r^2}{2} = r^2$
for $\displaystyle{ \int r \sqrt{1+r^2} dr}$ use substitution rule by supposing $u = 1+r^2$
So, $du = 2r \mbox{dr} \implies dr = \dfrac{du}{2}$
• Oct 27th 2010, 05:58 PM
Evan.Kimia
ah, i should of know that.
Thanks a bunch. | {"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": 5, "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": 0.9734726548194885, "perplexity": 4846.95115806965}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187824775.99/warc/CC-MAIN-20171021114851-20171021134851-00583.warc.gz"} |
https://www.physicsforums.com/threads/help-with-a-matrix.763478/ | # Help with a matrix
1. Jul 26, 2014
### skujesco2014
Hi, all. I'm in desperate need of assistance with a matrix I can't get my head around of. I want to solve a system of equations of the type $Ax=b$, where
$$A= \begin{pmatrix} 2 & 5 & -3 \\ 1 & -2 & 1 \\ 7 & 4 & -3 \end{pmatrix}$$
and where
$$b= \begin{pmatrix} -2 \\ -1 \\ -7 \end{pmatrix}$$
that is, $b$ is the negative of the first column. Written as it is above, $A$ has zero determinant and the determinant formed when the $k^{th}$ column of $A$ is substituted by the vector $b$ is clearly zero as well. A theorem says that in this case the system has infinite solutions. If one reduces the system to reduced row-echelon form the solutions can be parameterized as, for example, $x_3=t, x_2=5t/9, x_1=t-1$. An immediate solution by inspection is $x=(-1,0,0)^T$ which one obtains letting $t=0$.
But let's give another value of $t$, for example, $t=1$ which gives $x=(0,5/9,1)^T$. This is one of the parameterized solutions and yet it does not satisfy the original system. It does, however, satisfy the reduced system obtained from the original by gaussian elimination and should be equivalent, i.e.,
$$\begin{cases} x_1 - x_3 & = -1\\ 9x_2 - 5x_3 & = 0 \end{cases}$$
But shouldn't my parameterized solution satisfy both original and reduced systems, no matter what? Yet, the only satisfying solution for the original system seems to be $x=(-1,0,0)^T$. What am I not seeing here?
2. Jul 26, 2014
### skujesco2014
I've discovered my mistake! I've sinned against the elementary row operations on matrix $A$! The reduced system actually is
$$\begin{cases} x_1 -x_3/9& = -1\\ x_2-5x_3/9 & = 0 \end{cases}$$
The displayed erroneous reduced system above was obtained after mistakenly adding a multiple of a row to a multiple of another row. This isn't an allowed elementary operation. It was very intriguing that this little mistake lead to the puzzle that inspired this post.
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http://mlg.eng.cam.ac.uk/yarin/blog_2248.html | ## Uncertainty in Deep Learning (PhD Thesis)
October 13th, 2016 (Updated: June 4th, 2017)
##### Function draws from a dropout neural network. This new visualisation technique depicts the distribution over functions rather than the predictive distribution (see demo below).
So I finally submitted my PhD thesis (given below). In it I organised the already published results on how to obtain uncertainty in deep learning, and collected lots of bits and pieces of new research I had lying around (which I hadn't had the time to publish yet). The questions I got about the work over the past year were a great help in guiding my writing, with the greatest influence on my writing, I reckon, being the work of Professor Sir David MacKay (and his thesis specifically). Weirdly enough, I would consider David's writing style to be the equivalent of modern blogging, and would highly recommend reading his thesis. I attempted to follow David's writing style in my own writing, explaining topics through examples and remarks, resulting in what almost looks like a 120 pages long blog post. So hopefully it can now be seen as a more complete body of work, accessible to as large an audience as possible, and also acting as an introduction to the field of what people refer to today as Bayesian Deep Learning. One of the interesting results which I will demonstrate below touches on uncertainty visualisation in Bayesian neural networks. It's something that almost looks trivial, yet it has gone unnoticed for quite some time! But before that, I'll review quickly some of the new bits and pieces in the thesis for people already familiar with the work. For others I would suggest starting with the introduction: The Importance of Knowing What We Don't Know.
### Thesis: Uncertainty in Deep Learning
Some of the work in the thesis was previously presented in [Gal, 2015; Gal and Ghahramani, 2015a,b,c,d; Gal et al., 2016], but the thesis contains many new pieces of work as well. The most notable of these are
1. some discussions: a discussion of AI safety and model uncertainty (§1.3), a historical survey of Bayesian neural networks (§2.2),
2. some theoretical analysis: a theoretical analysis of the variance of the re-parametrisation trick and other Monte Carlo estimators used in variational inference (the re-parametrisation trick is not a universal variance reduction technique! §3.1.1§3.1.2), a survey of measures of uncertainty in classification tasks (§3.3.1),
3. some empirical results: an empirical analysis of different Bayesian neural network priors (§4.1) and posteriors with various approximating distributions (§4.2), new quantitative results comparing dropout to existing techniques (§4.3), tools for heteroscedastic model uncertainty in Bayesian neural networks (§4.6),
4. some applications: a survey of recent applications in language, biology, medicine, and computer vision making use of the tools presented in this thesis (§5.1), new applications in active learning with image data (§5.2),
5. and more theoretical results: a discussion of what determines what our model uncertainty looks like (§6.1§6.2), an analytical analysis of the dropout approximating distribution in Bayesian linear regression (§6.3), an analysis of ELBO-test log likelihood correlation (§6.4), discrete prior models (§6.5), an interpretation of dropout as a proxy posterior in spike and slab prior models (§6.6, relating dropout to works by MacKay, Nowlan, and Hinton from 1992), as well as a procedure to optimise the dropout probabilities based on the variational interpretation to separate the different sources of uncertainty (§6.7).
The thesis can be obtained as a Single PDF (9.1M), or as individual chapters (since the single file is fairly large):
• Contents (PDF, 36K)
• Chapter 1: The Importance of Knowing What We Don't Know (PDF, 393K)
• Chapter 2: The Language of Uncertainty (PDF, 136K)
• Chapter 3: Bayesian Deep Learning (PDF, 302K)
• Chapter 4: Uncertainty Quality (PDF, 2.9M)
• Chapter 5: Applications (PDF, 648K)
• Chapter 6: Deep Insights (PDF, 939K)
• Chapter 7: Future Research (PDF, 28K)
• Bibliography (PDF, 72K)
• Appendix A: KL condition (PDF, 71K)
• Appendix B: Figures (PDF, 2M)
• Appendix C: Spike and slab prior KL (PDF, 28K)
I would appreciate it if you could cite this thesis (BiBTex) if you intend to use any of the new results.
One of the nice practical new results in section §4.1 for example affects function visualisation. It's a minor change that has gone unnoticed until now, but which is significant in understanding our functions.
### Function visualisation
There are two factors at play when visualising uncertainty in dropout Bayesian neural networks: the dropout masks and the dropout probability of the first layer. Uncertainty depictions in my previous blog posts drew new dropout masks for each test point—which is equivalent to drawing a new prediction from the predictive distribution for each test point $-2 \leq \x \leq 2$. More specifically, for each test point $\x_i$ we drew a set of network parameters from the dropout approximate posterior $\boh_{i} \sim q_\theta(\bo)$, and conditioned on these parameters we drew a prediction from the likelihood $\y_i \sim p(\y | \x_i, \boh_{i})$. Since the predictive distribution has \begin{align*} p&(\y_i | \x_i, \X_\train, \Y_\train) \\ &= \int p(\y_i | \x_i, \bo) p(\bo | \X_\train, \Y_\train) \text{d} \bo \\ &\approx \int p(\y_i | \x_i, \bo) q_\theta(\bo) \text{d} \bo \\ &=: q_\theta(\y_i | \x_i) \end{align*} we have that $\y_i$ is a draw from an approximation to the predictive distribution.
##### Figure A: In black is a draw from the predictive distribution of a dropout neural network $\yh \sim q_\theta(\y | \x)$ for each test point $-2 \leq \x \leq 2$, compared to the function draws in figure B.
This process is equivalent to drawing a new function for each test point, which results in extremely erratic depictions that have peaks at different locations (seen in figure A taken from the previous blog post). Drawing a new function for each test point makes no difference if all we care about is obtaining the predictive mean and predictive variance (actually, for these two quantities this process is preferable to the one I will describe below), but this process does not result in draws from the induced distribution over functions. This is because different network parameters correspond to different functions, and a distribution over the network parameters therefore induces a distribution over functions. Under a Bayesian interpretation, we identify a draw $\boh$ from the posterior over network parameters $q_\theta(\bo)$ with a single function draw. To get a draw from our induced posterior over functions, we would need to sample a single network for all test points then, rather than sample a particular prediction for each test point.
##### Figure B: Each solid black line is a function drawn from a dropout neural network posterior over functions, induced by a draw from the approximate posterior over the weights $\boh \sim q_\theta(\bo)$.
To visualise our predictive distribution in a more appealing way we could draw a single network for the entire test set. With dropout, this can be done by drawing a single set of masks to be used with all test points. Our induced functions look very different now (seen in figure B, and with a demo below). In the new visualisation the functions are smooth, even though they are drawn from a dropout approximating distribution (which randomly sets whole rows of the weight matrix to zero). Note that to calculate predictive mean and predictive variance, using different masks is actually preferable since it results in lower variance estimators.
Another important factor affecting visualisation is the dropout probability of the first layer. In the previous posts we depicted scalar functions and set all dropout probabilities to $0.1$. As a result, with probability $0.1$, the sampled functions from the posterior would be identically zero. This is because a zero draw from the Bernoulli distribution in the first layer together with a scalar input leads the model to completely drop its input (explaining the points where the function touches the $x$-axis in figure A). This is a behaviour we might not believe the posterior should exhibit (especially when a single set of masks is drawn for the entire test set), and could change this by setting a different probability for the first layer. Setting $p_1 = 0$ for example is identical to placing a delta approximating distribution over the first weight layer.
In the demo below we use $p_1 = 0$, and depict draws from the approximate predictive distribution evaluated on the entire test set $q_{\theta_i}(\Y | \X, \boh_i)$ ($\boh_i \sim q_{\theta_i}(\bo)$, function draws), as the variational parameters $\theta_i$ change and adapt to minimise the divergence to the true posterior (with old samples disappearing after 20 optimisation steps). You can change the function the data is drawn from (with two functions, one from the last blog post and one from the appendix in this paper), and the model used (a homoscedastic model or a heteroscedastic model, see section §4.6 in the thesis for example or this blog post).
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## Acknowledgements
To finish this blog post I would like to thank the people that helped through comments and discussions during the writing of the various papers composing the thesis above. I would like to thank (in alphabetical order) Christof Angermueller, Yoshua Bengio, Phil Blunsom, Yutian Chen, Roger Frigola, Shane Gu, Alex Kendall, Yingzhen Li, Rowan McAllister, Carl Rasmussen, Ilya Sutskever, Gabriel Synnaeve, Nilesh Tripuraneni, Richard Turner, Oriol Vinyals, Adrian Weller, Mark van der Wilk, Yan Wu, and many other reviewers for their helpful comments and discussions. I would further like to thank my collaborators Rowan McAllister, Carl Rasmussen, Richard Turner, Mark van der Wilk, and my supervisor Zoubin Ghahramani.
Lastly, I would like to thank Google for supporting three years of my PhD with the Google European Doctoral Fellowship in Machine Learning, and Qualcomm for supporting my fourth year with the Qualcomm Innovation Fellowship.
PS. there might be some easter eggs hidden in the introduction :)
## Contact me
#### Email
yg279 -at- cam.ac.uk
#### Post
Cambridge University
Engineering Department
Cambridge, CB2 1PZ
United Kingdom | {"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": 0.84390789270401, "perplexity": 1194.732666225906}, "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/1620243991801.49/warc/CC-MAIN-20210515100825-20210515130825-00523.warc.gz"} |
https://infoscience.epfl.ch/record/117537 | Infoscience
Journal article
Structural and genetic heterogeneity of the receptor mediating translocation of immunoglobulin A dimer antibodies across epithelia in the rabbit
Secretory component (SC), synthesized as a transmembrane protein, acts as the receptor that binds IgA dimers and mediates their transepithelial transport. Cleavage of the receptor (membrane SC) apparently occurs during transport and a fragment, the secreted form, is generated, which remains tightly bound to the IgA dimer. In the rabbit, variation in the size of membrane SC is observed with both a high and a low molecular weight family, each composed either of two or of four distinct polypeptides depending on the individual rabbit. The same degree of size heterogeneity is observed for secreted SC. Part of this size heterogeneity is related to genetic polymorphism. The milk of individual rabbits typed with anti-SC-allotype sera reveals three different banding patterns. The simplest pattern, found in t61/t61 and t62/t62 homozygotes, consists of an upper and a lower doublet. Since each band of these two doublets in the t62 allotype migrates slightly faster than its counterpart in the t61 allotype, a composite pattern is observed in the heterozygotes (t61/t62). Within a given allotypic group, all SC polypeptide chains expressed the identical allotypic specificity. The 2000 to 4000 difference in molecular weight between the two forms of a doublet probably reflects differences in the number of glycosylated asparagine residues, since individual bands of a doublet show identical peptide maps. High and low molecular weight families are also structurally related to each other as shown by one-dimensional peptide maps and identical NH2- and COOH-terminal amino acid sequences. These results indicate that the 25-kDa size difference between SC from the high and low molecular weight families reflects an intramolecular deletion.
Reference
• GR-KUHN-ARTICLE-1983-002
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https://www.microwave-link.com/page/2/ | WR5 | WG30 | R1800 Waveguide
WR5 | WG30 | R1800 Waveguide Size
• EIA Standard:WR5
• RSCS Standard (British Military):WG30
• IEC Standard:R1800
WR5 Specifications
• Recommended Frequency Band:140 to 220 GHz
• Cutoff Frequency of Lowest Order Mode:115.714 GHz
• Cutoff Frequency of Upper Mode:231.429 GHz
• Dimension:0.051 Inches [1.2954 mm] x 0.0255 Inches [0.6477 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR6 | WG29 | R1400 Waveguide
WR6 | WG29 | R1400 Waveguide Size
• EIA Standard:WR6
• RSCS Standard (British Military):WG29
• IEC Standard:R1400
WR6 Specifications
• Recommended Frequency Band:110 to 170 GHz
• Cutoff Frequency of Lowest Order Mode:90.791 GHz
• Cutoff Frequency of Upper Mode:181.583 GHz
• Dimension:0.065 Inches [1.651 mm] x 0.0325 Inches [0.8255 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR8 | WG28 | R1200 Waveguide
WR8 | WG28 | R1200 Waveguide Size
• EIA Standard (British Military):WR8
• RSCS Standard:WG28
• IEC Standard:R1200
WR8 Specifications
• Recommended Frequency Band:90 to 140 GHz
• Cutoff Frequency of Lowest Order Mode:73.768 GHz
• Cutoff Frequency of Upper Mode:147.536 GHz
• Dimension:0.08 Inches [2.032 mm] x 0.04 Inches [1.016 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR10 | WG27 | R900 Waveguide
WR10 | WG27 | R900 Waveguide Size
• EIA Standard:WR10
• RSCS Standard (British Military):WG27
• IEC Standard:R900
WR10 Specifications
• Recommended Frequency Band:75 to 110 GHz
• Cutoff Frequency of Lowest Order Mode:59.015 GHz
• Cutoff Frequency of Upper Mode:118.03 GHz
• Dimension:0.1 Inches [2.54 mm] x 0.05 Inches [1.27 mm]
Commonly Used for Millimeter Wave (MMW) Link Bands
• E-Band (71-76GHz and 81-86GHz): 70GHz, 80GHz
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR12 | WG26 | R740 Waveguide
WR12 | WG26 | R740 Waveguide Size
• EIA Standard:WR12
• RSCS Standard (British Military):WG26
• IEC Standard:R740
WR12 Specifications
• Recommended Frequency Band:60 to 90 GHz
• Cutoff Frequency of Lowest Order Mode:48.373 GHz
• Cutoff Frequency of Upper Mode:96.746 GHz
• Dimension:0.122 Inches [3.0988 mm] x 0.061 Inches [1.5494 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR15 | WG25 | R620 Waveguide
WR15 | WG25 | R620 Waveguide Size
• EIA Standard:WR15
• RSCS Standard (British Military):WG25
• IEC Standard:R620
WR15 Specifications
• Recommended Frequency Band:50.00 to 75 GHz
• Cutoff Frequency of Lowest Order Mode:39.875 GHz
• Cutoff Frequency of Upper Mode:79.75 GHz
• Dimension:0.148 Inches [3.7592 mm] x 0.074 Inches [1.8796 mm]
Commonly Used for Millimeter Wave (MMW) Link Bands
• V-Band (58GHz to 64GHz): 58GHz, 60GHz
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR19 | WG24 | R500 Waveguide
WR19 | WG24 | R500 Waveguide Size
• EIA Standard:WR19
• RSCS Standard (British Military):WG24
• IEC Standard:R500
WR19 Specifications
• Recommended Frequency Band:40.00 to 60 GHz
• Cutoff Frequency of Lowest Order Mode:31.391 GHz
• Cutoff Frequency of Upper Mode:62.782 GHz
• Dimension:0.188 Inches [4.7752 mm] x 0.094 Inches [2.3876 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR22 | WG23 | R400 Waveguide
WR22 | WG23 | R400 Waveguide Size
• EIA Standard:WR2300
• RSCS Standard (British Military):WG0.0
• IEC Standard:R3
WR2300 Specifications
• Recommended Frequency Band:33.00-50.00 GHz
• Cutoff Frequency of Lowest Order Mode:26.346 GHz
• Cutoff Frequency of Upper Mode:52.692 GHz
• Dimension:0.224 Inches [5.6896 mm] x 0.112 Inches [2.8448 mm]
• 42GHz
Other Names:
UBR400, PBR400
These are different flange types. U = Unpressurised, P = Pressurised
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR28 | WG22 | R320 Waveguide
WR28 | WG22 | R320 Waveguide Size
• EIA Standard:WR28
• RSCS Standard (British Military):WG22
• IEC Standard:R320
WR28 Specifications
• Recommended Frequency Band:26.50 to 40 GHz
• Cutoff Frequency of Lowest Order Mode:21.077 GHz
• Cutoff Frequency of Upper Mode:42.154 GHz
• Dimension:0.28 Inches [7.112 mm] x 0.14 Inches [3.556 mm]
• 28GHz
• 32GHz
• 38GHz
Other Names:
UBR320, PBR320
These are different flange types. U = Unpressurised, P = Pressurised
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal.
WR34 | WG21 | R260 Waveguide
WR34 | WG21 | R260 Waveguide Size
• EIA Standard:WR34
• RSCS Standard (British Military):WG21
• IEC Standard:R260
WR34 Specifications
• Recommended Frequency Band:22.00 to 33 GHz
• Cutoff Frequency of Lowest Order Mode:17.357 GHz
• Cutoff Frequency of Upper Mode:34.715 GHz
• Dimension:0.34 Inches [8.636 mm] x 0.17 Inches [4.318 mm]
What is a Waveguide?
A waveguide is an electromagnetic feed line that is used for high frequency signals. Waveguides conduct microwave energy at lower loss than coaxial cables and are used in microwave communications, radars and other high frequency applications.
The waveguide must have a certain minimum cross section, relative to the wavelength of the signal to function properly. If wavelength of the signal is too long (Frequency is too low) when compared to the cross section of the waveguide, the electromagnetic fields cannot propagate. The lowest frequency range at which a waveguide will operate is where the cross section is large enough to fit one complete wavelength of the signal. | {"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": 0.9388150572776794, "perplexity": 4765.799602359716}, "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-2019-30/segments/1563195525634.13/warc/CC-MAIN-20190718125048-20190718151048-00530.warc.gz"} |
http://stacks.math.columbia.edu/tag/0544 | # The Stacks Project
## Tag 0544
Lemma 10.31.2. Let $R \to R'$ be a ring map and let $I \subset R$ be a locally nilpotent ideal. Then $IR'$ is a locally nilpotent ideal of $R'$.
Proof. This follows from the fact that if $x, y \in R'$ are nilpotent, then $x + y$ is nilpotent too. Namely, if $x^n = 0$ and $y^m = 0$, then $(x + y)^{n + m - 1} = 0$. $\square$
The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 5910–5914 (see updates for more information).
\begin{lemma}
\label{lemma-locally-nilpotent}
Let $R \to R'$ be a ring map and let $I \subset R$ be a locally nilpotent
ideal. Then $IR'$ is a locally nilpotent ideal of $R'$.
\end{lemma}
\begin{proof}
This follows from the fact that if $x, y \in R'$ are nilpotent, then
$x + y$ is nilpotent too. Namely, if $x^n = 0$ and $y^m = 0$, then
$(x + y)^{n + m - 1} = 0$.
\end{proof}
There are no comments yet for this tag.
## Add a comment on tag 0544
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner). | {"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": 0.973462700843811, "perplexity": 318.1837929062594}, "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-2017-17/segments/1492917118477.15/warc/CC-MAIN-20170423031158-00237-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.mathdoubts.com/base/ | # Base of Exponential form
A quantity that splits another quantity as its factors is called the base.
## Introduction
A quantity can be expressed in product form on the basis of another quantity and the product form expression can be written simply in exponential notation as per exponentiation. The quantity that is considered to split a quantity as a product of its terms is called the base of exponential form.
### Example
$81$ is a number and express it in product form on the basis of another number $3$ by factorization.
$81 \,=\, 3 \times 3 \times 3 \times 3$
According to exponentiation, write the product form expression in exponential notation.
$81 \,=\, 3^4$
On the basis of number $3$, the number $81$ is expressed as factors of $3$. Hence, the number $3$ is called the base of the exponential notation for the number $81$.
#### More Examples
Observe the following examples to understand what exactly the base is in exponential notation.
$(1)\,\,\,\,\,\,\,$ $8 = 2 \times 2 \times 2 = 2^3$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $2$ is the base of the exponential term.
$(2)\,\,\,\,\,\,\,$ $16 = 4 \times 4 = 4^2$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $4$ is the base of the exponential term.
$(3)\,\,\,\,\,\,\,$ $3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $5$ is the base of the exponential term.
$(4)\,\,\,\,\,\,\,$ $343 = 7 \times 7 \times 7 = 7^3$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $7$ is the base of the exponential term.
$(5)\,\,\,\,\,\,\,$ $28561 = 13 \times 13 \times 13 \times 13 = 13^4$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ In this example, $13$ is the base of the exponential term.
#### Algebraic form
$m$ is literal and it represents a quantity. Take, the quantity $m$ is divided as factors on the basis of the quantity $b$. The total number of factors of $b$ is $n$ by the factorisation.
$m$ $\,=\,$ $\underbrace{b \times b \times b \times … \times b}_{\displaystyle n factors}$ $\,=\,$ $b^n$
Therefore, the number $b$ is called the base of the exponential notation $b^n$.
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###### Math Problems
Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.
Learn more | {"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": 0.8962085247039795, "perplexity": 379.2974165733267}, "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-2020-45/segments/1603107911027.72/warc/CC-MAIN-20201030153002-20201030183002-00584.warc.gz"} |
http://mathhelpforum.com/statistics/125948-2-questions-i-need-clarification.html | # Math Help - 2 Questions----I need clarification.
1. ## 2 Questions----I need clarification.
The following are 2 practice questions for my math class. There will be similar questions on my test so I need to know how to do them correctly. Any help would be great, thanks!
1) Following are 3 boxes containing letters: [tex] [AND] [HISTORY].
a) From box 1, three letters are drawn one-by-one without replacement and recorded in that order. What is the probability that the outcome is HAT?
-I did (1/4)(1/3)(1/2) = 1/24 Is this correct?
b) If a box is chosen at random and then a letter is drawn at random from the box, what is the probability that the outcome is A?
-I'm thinking along the lines of (2/3)(1/4)(1/3), but am unsure...
2) Box 1: [2 black balls & 3 white balls] Box 2: [4 black balls & 2 white balls] A ball is drawn at random from box 1, then at random from box 2. The colors of the balls are recorded in order. The boxes look like this:
1:[BBWWW] 2:[BBBBWW] Find:
a) The probability of 2 whites.
-I did (3/5)(2/6) = 1/5 Is this correct?
b) The probability of at least one black. (I'm stumped)
c) The probability of at most one black. (I'm stumped again)
d) The probability of a black followed by a white, or a white followed by a black.
-I did (2/5)(2/6) = 2/15 or (3/5)(4/6) = 2/5 Is this correct?
2. Hello, mdenham2!
1) There are 3 boxes containing letters: . $[M,A,T,H]\;\; [A,N,D]\;\; [H,I,S,T,O,R,Y].$
a) From box 1, three letters are drawn one-by-one without replacement.
What is the probability that the outcome is $HAT$?
I did: . $\frac{1}{4}\cdot\frac{1}{3}\cdot\frac{1}{2} \,=\,\frac{1}{24}$ . . . Is this correct? . . Yes!
b) If a box is chosen at random and then a letter is drawn at random from the box,
what is the probability that the outcome is $A$?
The box is chosen at random: . $P(\text{box 1}) \,=\,P(\text{box 2}) \,=\,P(\text{box 3}) \,=\,\frac{1}{3}$
From box 1, $P(A) = \frac{1}{4}$
. . Hence: . $P(\text{box 1 }\wedge \text{ A}) \:=\:\frac{1}{3}\cdot\frac{1}{4} \:=\:\frac{1}{12}$
From box 2, $P(A) = \frac{1}{3}$
.Hence: . $P(\text{box 2 }\wedge\text{ A}) \:=\:\frac{1}{3}\cdot\frac{1}{3} \:=\:\frac{1}{9}$
From box 3, $P(A) = 0$
. . Hence: . $P(\text{box 3 }\wedge\text{ A}) \:=\:\frac{1}{3}\cdot0 \:=\:0$
Therefore: . $P(A) \;=\;\frac{1}{12} + \frac{1}{9} + 0 \;=\;\frac{7}{36}$
2) Box 1: [2 black balls & 3 white balls]
. . Box 2: [4 black balls & 2 white balls]
A ball is drawn at random from box 1, then at random from box 2.
a) The probability of 2 whites.
I did: . $\frac{3}{5}\cdot\frac{2}{6} \,=\, \frac{1}{5}$ . . . Is this correct? . Yes!
b) The probability of at least one black.
The opposite of "at least one black" is "no blacks" (both white).
You found: $P(\text{2 whites}) = \frac{1}{5}$ .in part (a).
Therefore: . $P(\text{at least one black}) \;=\;1-\frac{1}{5} \;=\;\frac{4}{5}$
c) The probability of at most one black.
The opposite of "at most one black" is "both black".
$P(\text{both black}) \:=\:\frac{2}{5}\cdot\frac{4}{6} \:=\:\frac{4}{15}$
Therefore: . $P(\text{at most one black}) \;=\;1-\frac{4}{15} \;=\;\frac{11}{15}$
d) The probability of a black followed by a white, or a white followed by a black.
I did : . $\frac{2}{5}\cdot\frac{2}{6} \,=\,\frac{2}{15}\:\text{ or }\:\frac{3}{5}\cdot\frac{4}{6} \,=\,\frac{2}{5}$ . . . Is this correct? . Um, not quite ...
With an "or" statement, we add the probabilties.
Therefore: . $P(\text{BW }\vee\text{ WB}) \;=\;\frac{2}{15} + \frac{2}{5} \;=\;\frac{8}{15}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Here's an alternate approach to part (d).
We want: . $P(\text{opposite colors})$
In part (a), we found that: . $P(\text{2 whites}) \,=\,\frac{1}{5}$
In part (c), we found that: . $P(\text{2 blacks}) \,=\,\frac{4}{15}$
. . Hence: . $P(\text{same color}) \:=\:\frac{1}{5} + \frac{4}{15} \:=\:\frac{7}{15}$
Therefore: . $P(\text{opposite colors}) \;=\;1 - \frac{7}{15} \;=\;\frac{8}{15}$
3. ## Thank you!
Thank you so much, I couldn't have asked for better explanations!
4. Thanks!!! | {"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": 24, "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": 0.9318273067474365, "perplexity": 763.8307025707571}, "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/1394678694885/warc/CC-MAIN-20140313024454-00088-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.imperial.ac.uk/people/j.gauntlett/publications.html | # ProfessorJeromeGauntlett
Faculty of Natural SciencesDepartment of Physics
Chair in Theoretical Physics
//
### Contact
+44 (0)20 7594 1275j.gauntlett
//
### Assistant
Mrs Graziela De Nadai-Sowrey +44 (0)20 7594 7843
//
### Location
602BHuxley BuildingSouth Kensington Campus
//
## Publications
Publication Type
Year
to
159 results found
Cheung KCM, Gauntlett JP, Rosen C, Consistent KK truncations for M5-branes wrapped on Riemann surfaces, Classical and Quantum Gravity, ISSN: 0264-9381
We construct a consistent Kaluza-Klein reduction of D = 11 supergravityon Σ2 × S4, where Σ2 = S2, R2or H2, or a quotient thereof, at the levelof the bosonic fields. The result is a gauged N = 4, D = 5 supergravitytheory coupled to three vector multiplets, with the gauging lying in anSO(2) × SE(3) ⊂ SO(5, 3) subgroup of the SO(1, 1) × SO(5, 3) globalsymmetry group of the ungauged theory. For Σ2 = H2, the D = 5 theoryhas a maximally supersymmetric AdS5 vacuum which uplifts to the knownsolution of D = 11 supergravity corresponding to M5-branes wrapping aRiemann surface with genus greater than one and dual to an N = 2 SCFTin d = 4. For Σ2 = S2, we find two AdS5solutions, one of which is new,and both of which are unstable. There is an additional subtruncationto an N = 2 gauged supergravity coupled to two vector multiplets, withvery special real manifold SO(1, 1)×SO(1, 1), and a single hypermultiplet,with quaternionic K¨ahler manifold SU(2, 1)/S[U(2) × U(1)] and gaugingassociated with an SO(2) × R ⊂ SU(2, 1) subgroup.
Journal article
Gauntlett JP, Martelli D, Sparks J, 2019, Toric geometry and the dual of I-extremization, Publisher: arXiv
We consider d=3, N=2 gauge theories arising on membranes sitting at the apex of an arbitrary toric Calabi-Yau 4-fold cone singularity that are then further compactified on a Riemann surface, Σg, with a topological twist that preserves two supersymmetries. If the theories flow to a superconformal quantum mechanics in the infrared, then they have a D=11 supergravity dual of the form AdS2×Y9, with electric four-form flux and where Y9 is topologically a fibration of a Sasakian Y7 over Σg. These D=11 solutions are also expected to arise as the near horizon limit of magnetically charged black holes in AdS4×Y7, with a Sasaki-Einstein metric on Y7. We show that an off-shell entropy function for the dual AdS2 solutions may be computed using the toric data and Kähler class parameters of the Calabi-Yau 4-fold, that are encoded in a master volume, as well as a set of integers that determine the fibration of Y7 over Σg and a Kähler class parameter for Σg. We also discuss the class of supersymmetric AdS3×Y7 solutions of type IIB supergravity with five-form flux only in the case that Y7 is toric, and show how the off-shell central charge of the dual field theory can be obtained from the toric data. We illustrate with several examples, finding agreement both with explicit supergravity solutions as well as with some known field theory results concerning I-extremization.
Working paper
Arav I, Gauntlett JP, Roberts M, Rosen Cet al., 2019, Spatially modulated and supersymmetric deformations of ABJM theory, Journal of High Energy Physics, Vol: 2019, ISSN: 1029-8479
We construct supersymmetric solutions of $D=11$ supergravity, preserving 1/4of the supersymmetry, that are holographically dual to ABJM theory which hasbeen deformed by spatially varying mass terms depending on one of the twospatial directions. We show that the BPS equations reduce to the Helmholtzequation on the complex plane leading to rich classes of new solutions. Inparticular, the construction gives rise to infinite classes of supersymmetricboomerang RG flows, as well as generalising a known Janus solution.
Journal article
Couzens C, Gauntlett JP, Martelli D, Sparks Jet al., 2019, A geometric dual of c-extremization, Journal of High Energy Physics, Vol: 2019, Pages: 1-55, ISSN: 1029-8479
We consider supersymmetric AdS3 × Y7 and AdS2 × Y9 solutions of type IIBand D = 11 supergravity, respectively, that are holographically dual to SCFTs with (0, 2)supersymmetry in two dimensions and N = 2 supersymmetry in one dimension. Thegeometry of Y2n+1, which can be defined for n ≥ 3, shares many similarities with SasakiEinstein geometry, including the existence of a canonical R-symmetry Killing vector, butthere are also some crucial differences. We show that the R-symmetry Killing vector maybe determined by extremizing a function that depends only on certain global, topologicaldata. In particular, assuming it exists, for n = 3 one can compute the central charge ofan AdS3 × Y7 solution without knowing its explicit form. We interpret this as a geometricdual of c-extremization in (0, 2) SCFTs. For the case of AdS2 × Y9 solutions we show thatthe extremal problem can be used to obtain properties of the dual quantum mechanics,including obtaining the entropy of a class of supersymmetric black holes in AdS4. We alsostudy many specific examples of the type AdS3×T2×Y5, including a new family of explicitsupergravity solutions. In addition we discuss the possibility that the (0, 2) SCFTs dualto these solutions can arise from the compactification on T2 of certain d = 4 quiver gaugetheories associated with five-dimensional Sasaki-Einstein metrics and, surprisingly, cometo a negative conclusion.
Journal article
Gauntlett JP, Martelli D, Sparks J, 2019, Toric geometry and the dual of c-extremization, Journal of High Energy Physics, Vol: 2019, ISSN: 1029-8479
We consider D3-brane gauge theories at an arbitrary toric Calabi-Yau 3-fold cone singularity that are then further compactified on a Riemann surface Σg, with an arbitrary partial topological twist for the global U(1) symmetries. This constitutes a rich, infinite class of two-dimensional (0, 2) theories. Under the assumption that such a theory flows to a SCFT, we show that the supergravity formulas for the central charge and R-charges of BPS baryonic operators of the dual AdS3 solution may be computed using only the toric data of the Calabi-Yau 3-fold and the topological twist parameters. We exemplify the procedure for both the Yp,q and Xp,q 3-fold singularities, along with their associated dual quiver gauge theories, showing that the new supergravity results perfectly match the field theory results obtained using c-extremization, for arbitrary twist over Σg. We furthermore conjecture that the trial central charge Open image in new window , which we define in gravity, matches the field theory trial c-function off-shell, and show this holds in non-trivial examples. Finally, we check our general geometric formulae against a number of explicitly known supergravity solutions.
Journal article
Gauntlett JP, Rosen C, 2018, Susy Q and spatially modulated deformations of ABJM theory, Journal of High Energy Physics, Vol: 2018, ISSN: 1029-8479
Within a holographic framework we construct supersymmetric Q-lattice (‘Susy Q’) solutions that describe RG flows driven by supersymmetric and spatially modulated deformations of the dual CFTs. We focus on a specific D = 4 supergravity model which arises as a consistent KK truncation of D = 11 supergravity on the seven sphere that preserves SO(4) × SO(4) symmetry. The Susy Q solutions are dual to boomerang RG flows, flowing from ABJM theory in the UV, deformed by spatially modulated mass terms depending on one of the spatial directions, back to the ABJM vacuum in the far IR. For large enough deformations the boomerang flows approach the well known Poincaré invariant RG dielectric flow. The spatially averaged energy density vanishes for the Susy Q solutions.
Journal article
Donos A, Gauntlett JP, Griffin T, Ziogas Vet al., 2018, Incoherent transport for phases that spontaneously break translations, JOURNAL OF HIGH ENERGY PHYSICS, Vol: 2018, ISSN: 1029-8479
We consider phases of matter at finite charge density which spontaneously break spatial translations. Without taking a hydrodynamic limit we identify a boost invariant incoherent current operator. We also derive expressions for the small frequency behaviour of the thermoelectric conductivities generalising those that have been derived in a translationally invariant context. Within holographic constructions we show that the DC conductivity for the incoherent current can be obtained from a solution to a Stokes flow for an auxiliary fluid on the black hole horizon combined with specific thermodynamic quantities associated with the equilibrium black hole solutions.
Journal article
Donos A, Gauntlett JP, Rosen C, Sosa-Rodriguez Oet al., 2018, Boomerang RG flows with intermediate conformal invariance, JOURNAL OF HIGH ENERGY PHYSICS, Vol: 2018, ISSN: 1029-8479
For a class of D = 5 holographic models we construct boomerang RG flow solutions that start in the UV at an AdS5 vacuum and end up at the same vacuum in the IR. The RG flows are driven by deformations by relevant operators that explicitly break translation invariance. For specific models, such that they admit another AdS5 solution, AdS 5 c , we show that for large enough deformations the RG flows approach an intermediate scaling regime with approximate conformal invariance governed by AdS 5 c . For these flows we calculate the holographic entanglement entropy and the entropic c-function for the RG flows. The latter is not monotonic, but it does encapsulate the degrees of freedom in each scaling region. For a different set of models, we find boomerang RG flows with intermediate scaling governed by an AdS2 × ℝ3 solution which breaks translation invariance. Furthermore, for large enough deformations these models have interesting and novel thermal insulating ground states for which the entropy vanishes as the temperature goes to zero, but not as a power-law. Remarkably, the thermal diffusivity and the butterfly velocity for these new insulating ground states are related via D = Ev B 2 /(2πT), with E(T) → 0.5 as T → 0.
Journal article
Donos A, Gauntlett JP, Ziogas V, 2018, Diffusion for holographic lattices, JOURNAL OF HIGH ENERGY PHYSICS, ISSN: 1029-8479
Journal article
Donos A, Gauntlett JP, Ziogas V, 2017, Diffusion in inhomogeneous media, Physical Review D - Particles, Fields, Gravitation and Cosmology, Vol: 96, ISSN: 1550-2368
We consider the transport of conserved charges in spatially inhomogeneous quantum systems with a discrete lattice symmetry. We analyze the retarded two-point functions involving the charges and the associated currents at long wavelengths, compared to the scale of the lattice, and, when the dc conductivities are finite, extract the hydrodynamic modes associated with diffusion of the charges. We show that the dispersion relations of these modes are related to the eigenvalues of a specific matrix constructed from the dc conductivities and certain thermodynamic susceptibilities, thus obtaining generalized Einstein relations. We illustrate these general results in the specific context of relativistic hydrodynamics where translation invariance is broken using spatially inhomogeneous and periodic deformations of the stress tensor and the conserved U(1) currents. Equivalently, this corresponds to considering hydrodynamics on a curved manifold, with a spatially periodic metric and chemical potential, and we obtain the dispersion relations for the heat and charge diffusive modes.
Journal article
Donos A, Gauntlett JP, Rosen C, Sosa-Rodriguez Oet al., 2017, Boomerang RG flows in M-theory with intermediate scaling, JOURNAL OF HIGH ENERGY PHYSICS, Vol: 2017, ISSN: 1029-8479
We construct novel RG flows of D=11 supergravity that asymptotically approach AdS4 × S7 in the UV with deformations that break spatial translations in the dual field theory. In the IR the solutions return to exactly the same AdS4 × S7 vacuum, with a renormalisation of relative length scales, and hence we refer to the flows as ‘boomerang RG flows’. For sufficiently large deformations, on the way to the IR the solutions also approach two distinct intermediate scaling regimes, each with hyperscaling violation. The first regime is Lorentz invariant with dynamical exponent z = 1 while the second has z = 5/2. Neither ofthe two intermediatescaling regimesare associatedwith exact hyperscaling violation solutions of D = 11 supergravity. The RG flow solutions are constructed using the four dimensional N = 2 STU gauged supergravity theory with vanishing gauge fields, but non-vanishing scalar and pseudoscalar fields. In the ABJM dual field theory the flows are driven by spatially modulated deformation parameters for scalar and fermion bilinear operators.
Journal article
Donos A, Gauntlett JP, Griffin T, Lohitsiri N, Melgar Let al., 2017, Holographic DC conductivity and Onsager relations, Journal of High Energy Physics, Vol: 2017, ISSN: 1029-8479
Within holography the DC conductivity can be obtained by solving a system of Stokes equations for an auxiliary fluid living on the black hole horizon. We show that these equations can be derived from a novel variational principle involving a functional that depends on the fluid variables of interest as well as the time reversed quantities. This leads to simple derivation of the Onsager relations for the conductivity. We also obtain the relevant Stokes equations for bulk theories of gravity in four dimensions including a ϑF ∧ F term in the Lagrangian, where ϑ is a function of dynamical scalar fields. We discuss various realisations of the anomalous Hall conductivity that this term induces and also solve the Stokes equations for holographic lattices which break translations in one spatial dimension.
Journal article
Donos A, Gauntlett JP, Griffin T, Melgar Let al., 2017, DC conductivity and higher derivative gravity, Classical and Quantum Gravity, Vol: 34, ISSN: 0264-9381
For Gauss–Bonnet gravity and in the context of holography we show how the thermal DC conductivity can be obtained by solving a generalised system of Stokes equations for an auxiliary fluid on a curved black hole horizon. For more general higher derivative theories of gravity coupled to gauge-fields, we also analyse the linearised thermal and electric currents that are produced by DC thermal and electric sources. We show how suitably defined DC transport current fluxes of the dual field theory are given by current fluxes defined at the black horizon.
Journal article
Banks E, Donos A, Gauntlett JP, Griffin T, Melgar Let al., 2017, Thermal backflow in CFTs, Physical Review D, Vol: 95, ISSN: 1550-7998
We study the thermal transport properties of general conformal field theories (CFTs) on curved spacetimes in the leading order viscous hydrodynamic limit. At the level of linear response, we show that the thermal transport is governed by a system of forced linearized Navier-Stokes equations on a curved space. Our setup includes CFTs in flat spacetime that have been deformed by spatially dependent and periodic local temperature variations or strains that have been applied to the CFT, and hence is relevant to CFTs arising in condensed matter systems at zero charge density. We provide specific examples of deformations which lead to thermal backflow driven by a dc source: that is, the thermal currents locally flow in the opposite direction to the applied dc thermal source. We also consider thermal transport for relativistic quantum field theories that are not conformally invariant.
Journal article
Banks E, Donos A, Gauntlett JP, Griffin T, Melgar Let al., 2017, Thermal backflow in CFTs, PHYSICAL REVIEW D, Vol: 95, ISSN: 2470-0010
Journal article
Banks E, Donos A, Gauntlett JP, Griffin T, Melgar Let al., 2017, Holographic thermal DC response in the hydrodynamic limit, Classical and Quantum Gravity, Vol: 34, ISSN: 1361-6382
We consider black hole solutions of Einstein gravity that describe deformations of CFTs at finite temperature in which spatial translations have been broken explicitly. We focus on deformations that are periodic in the non-compact spatial directions, which effectively corresponds to considering the CFT on a spatial torus with a non-trivial metric. We apply a DC thermal gradient and show that in a hydrodynamic limit the linearised, local thermal currents can be determined by solving linearised, forced Navier–Stokes equations for an incompressible fluid on the torus. We also show how sub-leading corrections to the thermal current can be calculated as well as showing how the full stress tensor response that is generated by the DC source can be obtained. We also compare our results with the fluid-gravity approach.
Journal article
Donos A, Gauntlett JP, Sosa-Rodriguez O, 2016, Anisotropic plasmas from axion and dilaton deformations, The Journal of High Energy Physics, Vol: 2016, ISSN: 1029-8479
We construct black hole solutions of type IIB supergravity that are holographically dual to anisotropic plasmas arising from deformations of an infinite class of four-dimensional CFTs. The CFTs are dual to AdS5 × X5, where X5 is an Einstein manifold, and the deformations involve the type IIB axion and dilaton, with non-trivial periodic dependence on one of the spatial directions of the CFT. At low temperatures the solutions approach smooth domain wall solutions with the same AdS5 × X5 solution appearing in the far IR. For sufficiently large deformations an intermediate scaling regime appears which is governed by a Lifshitz-like scaling solution. We calculate the DC thermal conductivity and some components of the shear viscosity tensor.
Journal article
Gauntlett J, 2016, Thomas Kibble (1932-2016) Theoretical physicist and Higgs-boson pioneer. OBITUARY, Nature, Vol: 534, Pages: 622-622, ISSN: 0028-0836
Journal article
Donos A, Gauntlett JP, 2016, Minimally packed phases in holography, Journal of High Energy Physics, Vol: 2016, ISSN: 1126-6708
We numerically construct asymptotically AdS black brane solutions of $D=4$Einstein-Maxwell theory coupled to a pseudoscalar. The solutions areholographically dual to $d=3$ CFTs at finite chemical potential and in aconstant magnetic field, which spontaneously break translation invarianceleading to the spontaneous formation of abelian and momentum magnetisationcurrents flowing around the plaquettes of a periodic Bravais lattice. Weanalyse the three-dimensional moduli space of lattice solutions, which aregenerically oblique, and show, for a specific value of the magnetic field, thatthe free energy is minimised by the triangular lattice, associated with minimalpacking of circles in the plane. We show that the average stress tensor for thethermodynamically preferred phase is that of a perfect fluid and that thisresult applies more generally to spontaneously generated periodic phases. Thetriangular structure persists at low temperatures indicating the existence ofnovel crystalline ground states.
Journal article
Donos A, Gauntlett JP, Griffin T, Melgar Let al., 2016, DC Conductivity of Magnetised Holographic Matter, Journal of High Energy Physics, Vol: 2016, ISSN: 1126-6708
We consider general black hole solutions of Einstein-Maxwell-scalar theorythat are holographically dual to conformal field theories at finite chargedensity with non-vanishing magnetic fields and local magnetisation currents,which generically break translation invariance explicitly. We show that thethermoelectric DC conductivity of the field theory can be obtained by solving asystem of generalised Stokes equations on the black hole horizon. For variousexamples, including Q-lattices and one-dimensional lattices, we solve theStokes equations explicitly and obtain expressions for the DC conductivity interms of the solution at the black hole horizon.
Journal article
Donos A, Gauntlett JP, 2015, Navier-Stokes equations on black hole horizons and DC thermoelectric conductivity, Physical Review D, Vol: 92, ISSN: 1550-7998
Within the context of the AdS/CFT correspondence, we show that the DC thermoelectric conductivity can be obtained by solving the linearized, time-independent, and forced Navier-Stokes equations on the black hole horizon for an incompressible and charged fluid.
Journal article
Banks E, Donos A, Gauntlett JP, 2015, Thermoelectric DC conductivities and Stokes flows on black hole horizons, Journal of High Energy Physics, Vol: 2015, ISSN: 1126-6708
We consider a general class of electrically charged black holes of EinsteinMaxwell-scalartheory that are holographically dual to conformal field theories at finitecharge density which break translation invariance explicitly. We examine the linearisedperturbations about the solutions that are associated with the thermoelectric DC conductivity.We show that there is a decoupled sector at the black hole horizon which must solvegeneralised Stokes equations for a charged fluid. By solving these equations we can obtainthe DC conductivity of the dual field theory. For Q-lattices and one-dimensional latticeswe solve the fluid equations to obtain closed form expressions for the DC conductivity interms of the solution at the black hole horizon. We also determine the leading order DCconductivity for lattices that can be expanded as a perturbative series about translationallyinvariant solutions.
Journal article
Banks E, Gauntlett JP, 2015, A new phase for the anisotropic N=4 super Yang-Mills plasma, JOURNAL OF HIGH ENERGY PHYSICS, Vol: 2015, ISSN: 1029-8479
Black hole solutions of type IIB supergravity have been previously constructedthat describe the N=4 supersymmetric Yang-Mills plasma with an anisotropic spatial deformation.The zero temperature limit of these black holes approach a Lifshitz-like scalingsolution in the infrared. We show that these black holes become unstable at low temperatureand we construct a new class of black hole solutions which are thermodynamicallypreferred. The phase transition is third order and incorporates a spontaneous breaking ofthe SO(6) global symmetry down to SO(4) × SO(2). The critical exponents for the phasetransition are given by (α, β, γ, δ) = (−1, 1, 1, 2) which differ from the standard mean-fieldexponents usually seen in holography. At low temperatures the black holes approach anovel kind of scaling behaviour in the far IR with spatial anisotropy and hyperscaling violation.We show that the new ground states are thermal insulators in the direction of theanisotropy.
Journal article
Donos A, Gauntlett JP, Pantelidou C, 2015, Conformal field theories in d=4 with a helical twist, PHYSICAL REVIEW D, Vol: 91, ISSN: 1550-7998
Journal article
Donos A, Gauntlett JP, 2015, The thermoelectric properties of inhomogeneous holographic lattices, Journal of High Energy Physics, Vol: 2015, ISSN: 1126-6708
We consider inhomogeneous, periodic, holographic lattices of D=4Einstein-Maxwell theory. We show that the DC thermoelectric conductivity matrixcan be expressed analytically in terms of the horizon data of the correspondingblack hole solution. We numerically construct such black hole solutions forlattices consisting of one, two and ten wave-numbers. We numerically determinethe AC electric conductivity which reveals Drude physics as well as resonancesassociated with sound modes. No evidence for an intermediate frequency scalingregime is found. All of the monochromatic lattice black holes that we haveconstructed exhibit scaling behaviour at low temperatures which is consistentwith the appearance of $AdS_2\times\mathbb{R}^2$ in the far IR at T=0.
Journal article
Donos A, Gauntlett JP, 2014, Thermoelectric DC conductivities from black hole horizons, Journal of High Energy Physics, Vol: 2014, ISSN: 1029-8479
An analytic expression for the DC electrical conductivity in terms of black hole horizon data was recently obtained for a class of holographic black holes exhibiting momentum dissipation. We generalise this result to obtain analogous expressions for the DC thermoelectric and thermal conductivities. We illustrate our results using some holographic Q-lattice black holes as well as for some black holes with linear massless axions, in both D = 4 and D = 5 bulk spacetime dimensions, which include both spatially isotropic and anisotropic examples. We show that some recently constructed ground states of holographic Q-lattices, which can be either electrically insulating or metallic, are all thermal insulators.
Journal article
Donos A, Gauntlett JP, 2014, Flowing from AdS(5) to AdS(3) with T-1,T- 1, JOURNAL OF HIGH ENERGY PHYSICS, ISSN: 1029-8479
Journal article
Donos A, Gauntlett JP, 2014, Novel metals and insulators from holography, JOURNAL OF HIGH ENERGY PHYSICS, ISSN: 1029-8479
Journal article
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Donos A, Gauntlett JP, Pantelidou C, 2014, Competing p-wave orders, CLASSICAL AND QUANTUM GRAVITY, Vol: 31, ISSN: 0264-9381
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# c-decay
c-decay theory[note 1] is a pseudoscientific creationist cosmology put forward by cdesign proponentsists. It attempts to solve the starlight problem by claiming that the speed of light in a vacuum was faster in the past and has since decayed to the value we observe it to be today.[1]
Since the development of electronic digital counters and pulsed lasers, it has been possible to measure the speed of light in the laboratory with extraordinary precision. Even if the speed of light reached the proximity of its final value decades ago, there would be enough residual decay as the value reached its limit asymptotically for our modern apparatus to detect. There is none, forcing the proponents of the theory to toss out the exponential decay which governs nearly every phenomenon in the universe in favor of wild trigonometric functions they found by brute force curve-fitting, without an underlying explanation of "why". Ultimately, even many creationists have abandoned c-decay.[2]
Keep in mind that, in order to be useful for validating an age of the universe less than 10,000 years rather than more than 10 billion years, the speed of light needs to be more than a million times faster, a difference which would be difficult to miss. We're not talking about a difference of a fraction of a percent.
Creationists would have us believe that the speed of light is arbitrary and somehow separated from the rest of reality; however, c is not just "the speed of light." It is a universal constant which is observed as unchanging no matter who is observing it. It can be thought of as the speed which all objects fly through the four dimensions of spacetime — if you move along "space" you have less of the speed left to go through "time" and you experience the effect of time dilation. "c" is also used in many equations related to electromagnetic phenomenon such as Maxwell's equations; and it is the fact that these equations mention c without asking what the speed is relative to as should happen in a relativistic universe that lead to the notion that c is constant for all observers. It is also a key component in Einstein's famous E = mc2 equation; in this case, if c was larger in the past, then matter would have had more unit of energy per unit of mass in the past. Spontaneously losing this energy would at least violate the law of conservation of energy in some way.
The speed of light may be intimately related to two other physical constants: the vacuum permittivity ε0 and the vacuum permeability μ0:
$c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$
Therefore, a variation on the value of c implies variation either on ε0 or μ0 or both. These properties of space are measured in experiments that don't even involve light, such as experiments with capacitors and magnets.
c-decay, therefore, does not simply mean that "light travels a bit faster;" it means that the very fabric of reality would be subject to change in the temporal dimension. So believe us, anyone who postulates that "c may decay" needs to understand that it has very serious and very far-reaching consequences. One of the most far-reaching being that the speed of light changing with time implies that energy is not conserved.
## c-decay and the fine-tuned universe
c-decay may be in conflict with the argument that the fundamental constants of the universe are, and must be, fine-tuned for life as we know it. Indeed, the speed of light is closely related to the fine-structure constant α, which may be defined as
$\alpha =\ \frac{e^2}{(4 \pi \varepsilon_0)\hbar c}\ =\ \frac{e^2 c \mu_0}{2 h} = \frac{k_\mathrm{e} e^2}{\hbar c},$
where:
• e is the elementary charge;
• ħ = h/2π is the reduced Planck constant;
• c is the speed of light in vacuum;
• ε0 is the electric constant or permittivity of free space;
• or µ0 is the magnetic constant or permeability of free space;
• or ke is the Coulomb constant.
J.D Barrow published in 2001 that stable matter, and therefore life and intelligent beings, could not exist if the value of α were much different. Barrow wrote that if α to increase by 4%, "there would be a disaster... Stars would rapidly... collapse to degenerate states or black holes"; while if α were 10% smaller, the "pathways to the buildup of biological elements would be blocked."[3]
## Definition of the meter
Currently, the second is defined in terms of the frequency of the hyperfine transition in the cesium atom, and the meter is defined in terms of how far light travels in a vacuum in one 1/299,792,458 of a second. In other words, the unit of distance depends on the speed of light and a measure of time, not on the length of some specific physical object (although it used to between 1889 and 1960). If an experiment were to indicate that it takes more cesium transitions for light to travel to a previously measured distance, then based on the definition of our measurement units, we would actually be forced to conclude that the measured distance has increased, i.e., that everything is getting larger, not that the speed of light has decreased.
## Not to be confused with
• Carbon dating, a scientific method for dating organic matter, based on the radioactive decay of 14C. | {"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": 2, "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": 0.8635544776916504, "perplexity": 569.2762227574958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823997.21/warc/CC-MAIN-20171020082720-20171020102720-00352.warc.gz"} |
http://physics.stackexchange.com/questions/97713/schr%c3%b6dingers-cat-why-was-it-necessary | # Schrödinger's cat; why was it necessary?
Could someone please explain to me the idea that Schrödinger was trying to illustrate by the cat in his box? I understand that he was trying to introduce the notion of the cat being both alive and dead at the same time. But why was it necessary to introduce this thought experiment and what did it achieve?
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The original paper is relatively short and very readable. – MBN Feb 10 at 11:08
## 4 Answers
First, a historical subtlety: Schrödinger has actually stolen the idea of the cat from Einstein.
Second, both men – Einstein and Schrödinger – used the thought experiment to "explain" a point that was wrong. They thought it was absurd for quantum mechanics to say that the state $a|{\rm alive}\rangle+b|{\rm dead}\rangle$ was possible in Nature (it was claimed to be possible in quantum mechanics) because it allowed the both "incompatible" types of the cat to exist simultaneously.
Third, they were wrong because quantum mechanics does imply that such superpositions are totally allowed and must be allowed and this fact can be experimentally verified – not really with cats but with objects of a characteristic size that has been increasing. Macroscopic objects have already been put to similar "general superposition states".
The men introduced it to fight against the conventional, Copenhagen-like interpretations of quantum mechanics, and that's how most people are using the meme today, too. But the men were wrong, so from a scientifically valid viewpoint, the thought experiment shows that superpositions are indeed always allowed – it is a postulate of quantum mechanics – even if such states are counterintuitive. Similar superpositions of common-sense states are measured so that only $|a|^2$ and $|b|^2$ from the coefficients matter and may be interpreted as (more or less classical) probabilities. Due to decoherence, the relative phase is virtually unmeasurable for large, chaotic systems like cats, but in principle, even the relative phase matters.
Quite generally, the people who are wrong – who have a problem with quantum mechanics – like to say that the superposition means that the cat is alive "and" dead. But the right, quantum answer is that the addition in the wave function doesn't mean "and". Instead, it means a sort of "or", so the superposition simply says that the cat is dead or alive, with the appropriate probabilities (quantum mechanics determines not only the probabilities but also their complex phases, and those may matter for other questions).
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Reading up on Wikipedia, I presume that the theft of the idea of the cat was actually the theft of an idea that lacked a cat. – Glen The Udderboat Feb 4 at 17:28
The thought experiment aimed at illustrating one concept, and questioning the validity of such concept: 1/ The concept: in quantum mechanics, before an observation is made (note the cat is in a closed box, and nobody can see what is going on in the box), a system is not in a defined state, but only has a certain probability to be in any state - here the system is a cat, and the states are "dead" and "alive", and it is indeed a striking picture. With this, the goal of making eyebrows raise, and listeners question realize the significance of the concept is indeed reached.
2/ The questioning: by applying the concept of superposition to life and death issues, and also to a macroscopic object, the aim was to highlight the really revolutionary nature of the concept itself - in particular to highlight our complete ignorance of the reason why this quantum behaviour is ubiquitous at the atomic scale, but more rarely observed at the macroscopic scale (even though some manifestations do exist: superconductivity, BEC...)
So my answer is: it was not necessary, but it was a striking way to convey the concept of quantum superposition, and ask what is the defining limit between the microscopic and macroscopic laws of nature, if there is any. This question is still in debate, and is one of the goals of unification theories, with the major question of a quantum theory of gravity still unsolved.
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Schrödinger's cat is a thought experiment, sometimes described as a paradox. The scenario presents a cat that may be both alive and dead, depending on an earlier random event. Although the original "experiment" was imaginary, similar principles have been researched and used in practical applications.
The thought experiment is also often featured in theoretical discussions of the interpretations of quantum mechanics. Schrödinger wrote: One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid.
If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.
It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself, it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.
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This is why I think Schrodinger introduced this thought experiment. You can explain why Schrodinger developed this thought experiment by first considering the double slit experiment. You fire a photon at a double slit, and the photon goes through the double slit device and then strikes a screen. If you don’t try to determine which slit the photon has gone through, then the photon will (almost certainly) strike a spot on a screen that corresponds to the position of an interference antinode (you won’t generate an interference pattern of antinodes with just one photon, but one photon will land in one of the places that would be an antinode if you had used more light (and had sent many more photons the double slit). This tells you that the photon could not have gone through one slit or the other. If it had gone through one slit or the other, it would have traveled along one of the two possible classical paths. The photon would have traveled along a straight line from the light source to the slit that it went through and continued traveling in a straight line until it hit the screen at a point that is in-line with the slit and source. But it didn’t do this – it stuck the screen at an antinode position as if the photon were a wave and traveled through both slits at the same time. So the outcome of this experiment requires that you regard the traveling photon to exist along all possible paths (i.e. it takes all possible paths) between the source and the point where the photon eventually strikes the screen.
The lesson? When you don’t interfere with a quantum entity it evolves according to a wavefunction and is in a superposition of many states until it is observed.
Now for Schrodinger’s cat. If an atom with an unstable nucleus is not being observed, it evolves according to a wavefunction and is simultaneously in an undecayed state and a decayed state (where the value of the decayed state coefficient in the wavefunction increases as time elapses, and the coefficient of the undecayed coefficient decreases as time elapses). While you can regard the wavefunction as a means of predicting the probability that the photon strikes the screen at a particular point, you cannot regard it to describe the probability that the photon has gone through one slit or the other, because that would not be consistent with the observation that the photon struck the screen at an antinode position rather than strike a point that is in-line with a particular slit and the light source. Classically, the atom must exist in one state or the other but not both simultaneously; the classical states are mutually exclusive. But in quantum mechanics, the atom can exist in both states at the same time when it isn’t observed. Schrodinger’s thought experiment shows how ludicrous quantum reality is (but doesn’t refute it). You can imagine an experiment where the superposition of atom- states requires that a macroscopic object (the cat) is also in a superposition of mutually exclusive states (alive and dead), rather than existing in one classical state or the other.
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## protected by Qmechanic♦Jul 13 at 18:34
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Would you like to answer one of these unanswered questions instead? | {"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": 0.8271428346633911, "perplexity": 477.1199934551697}, "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-2014-42/segments/1413558067467.26/warc/CC-MAIN-20141017150107-00318-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://xianblog.wordpress.com/tag/tpa/ | ## Another harmonic mean approximation
Posted in R, Statistics with tags , , , , , , on June 27, 2010 by xi'an
Martin Weinberg posted on arXiv a revision of his paper, Computing the Bayesian Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution, that is submitted to Bayesian Analysis. I have already mentioned this paper in a previous post, but I remain unconvinced of the appeal of the paper method, given that it recovers the harmonic mean approximation to the marginal likelihood… The method is very close to John Skilling’s nested sampling, except that the simulation is run from the posterior rather than from the prior, hence the averaging on the inverse likelihoods and hence the harmonic mean connection. The difficulty with the original (Michael Newton and Adrian Raftery’s) harmonic mean estimator is attributed to “a few outlying terms with abnormally small values of” the likelihood, while, as clearly spelled out by Radford Neal, the poor behaviour of the harmonic mean estimator has nothing abnormal and is on the opposite easily explainable.
I must admit I found the paper difficult to read, partly because of the use of poor and ever-changing notations and partly because of the lack of mathematical rigour (see, e.g., eqn (11)). (And maybe also because of the current heat wave.) In addition to the switch from prior to posterior in the representation of the evidence, a novel perspective set in the paper seems to be an extension of the standard harmonic mean identity that relates to the general expression of Gelfand and Dey (1994, Journal of the Royal Statistical Society B) when using an indicator function as an instrumental function. There is therefore a connection with our proposal (made with Jean-Michel Marin) of considering an HPD region for excluding the tails of the likelihood, even though the set of integration is defined as “eliminating the divergent samples with $L_i \ll 1$“. This is essentially the numerical Lebesgue algorithm advanced as one of two innovative algorithms by Martin Weinberg. I wonder how closely related the second (volume tesselation) algorithm is to Huber and Schott’s TPA algorithm, in the sense that TPA also requires a “smaller” integral…. | {"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": 1, "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": 0.952885091304779, "perplexity": 644.9925096661733}, "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/1656104248623.69/warc/CC-MAIN-20220703164826-20220703194826-00640.warc.gz"} |
http://www.map.mpim-bonn.mpg.de/index.php?title=Talk:Structure_set_(Ex)&oldid=11122 | # Talk:Structure set (Ex)
Solution:
We begin with the map from $\mathcal{S}^s(M) \to \mathcal{M}(M)$$; '''Solution''': We begin with the map from \mathcal{S}^s(M) \to \mathcal{M}(M), from the simple structure set of manifolds simply homotopy equivalent to M and then we show two things: * If manifolds N and N' simply homotopy equivalent to M are diffeomorphic then their images by the map belong to the same orbit of \mathcal{E}^s(M)-action on \mathcal{S}^s(M). * If two elements of \mathcal{S}^s(M) belong to the same orbit of \mathcal{E}^s(M)-action, then we will show that they are diffeomorphic. Let N be a smooth manifold and f\colon N\to M a simple homotopy equivalence. Consider a map which takes N to [(N,f)]\in \mathcal{S}^c(M). Suppose now that N' is a manifold diffeomorphic to N, and f'\colon N'\to M a simple homotopy equivalence (possibly N'=N and f\nsim f'). Then there exists h\colon M\to M such that the following diagram commutes. \xymatrix{ N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\ N'\ar[r]^{f'}& M } Map h is given by composition f'\circ d\circ f^{-1} (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in \mathcal{S}^s(M) we have the following equalities. h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')], where \cdot denotes the \mathcal{E}^s(M)-action. Therefore [(N,f)] and [(N',f')] belong to the same orbit. Suppose now, that [(N,f)],[(N',f')]\in \mathcal{S}^s(M) belong to the same orbit of \mathcal{E}^s(M)-action. It means, that there exist a simple homotopy equivalence h\colon M\to M such that h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')]. But equality in the simple structure set \mathcal{S}^s(M) is just the existence of a diffeomorphism d\colon N\to N' making the following diagram commute. \xymatrix{ N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\ N'\ar[r]^{f'}& M } . \mathcal{S}^s(M) \to \mathcal{M}(M)$, from the simple structure set of manifolds simply homotopy equivalent to $M$$M$ and then we show two things:
• If manifolds $N$$N$ and $N'$$N'$ simply homotopy equivalent to $M$$M$ are diffeomorphic then their images by the map belong to the same orbit of $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$-action on $\mathcal{S}^s(M)$$\mathcal{S}^s(M)$.
• If two elements of $\mathcal{S}^s(M)$$\mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$-action, then we will show that they are diffeomorphic.
Let $N$$N$ be a smooth manifold and $f\colon N\to M$$f\colon N\to M$ a simple homotopy equivalence. Consider a map which takes $N$$N$ to $[(N,f)]\in \mathcal{S}^c(M)$$[(N,f)]\in \mathcal{S}^c(M)$. Suppose now that $N'$$N'$ is a manifold diffeomorphic to $N$$N$, and $f'\colon N'\to M$$f'\colon N'\to M$ a simple homotopy equivalence (possibly $N'=N$$N'=N$ and $f\nsim f'$$f\nsim f'$). Then there exists $h\colon M\to M$$h\colon M\to M$ such that the following diagram commutes.
$\displaystyle \xymatrix{ N \ar[r]^{f} \ar[d]^d_{\cong} & M \ar@{.>}[d]^{h}\\ N'\ar[r]^{f'}& M }$
Map $h$$h$ is given by composition $f'\circ d\circ f^{-1}$$f'\circ d\circ f^{-1}$ (the homotopy inverse) and hence is a simple homotopy equivalence. The commutativity of the diagram tells us that in $\mathcal{S}^s(M)$$\mathcal{S}^s(M)$ we have the following equalities.
$\displaystyle h\cdot [(N,f)] = [(N,h\circ f)]=[(N,f'\circ d\circ f^{-1}\circ f\simeq f'\circ d)]=[(N',f')],$
where $\cdot$$\cdot$ denotes the $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$-action. Therefore $[(N,f)]$$[(N,f)]$ and $[(N',f')]$$[(N',f')]$ belong to the same orbit.
Suppose now, that $[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$$[(N,f)],[(N',f')]\in \mathcal{S}^s(M)$ belong to the same orbit of $\mathcal{E}^s(M)$$\mathcal{E}^s(M)$-action. It means, that there exist a simple homotopy equivalence $h\colon M\to M$$h\colon M\to M$ such that
$\displaystyle h\cdot[(N,f)]=[(N,h\circ f)]=[(N',f')].$
But equality in the simple structure set $\mathcal{S}^s(M)$$\mathcal{S}^s(M)$ is just the existence of a diffeomorphism $d\colon N\to N'$$d\colon N\to N'$ making the following diagram commute.
$\displaystyle \xymatrix{ N \ar[r]^{f} \ar@{.>}[d]^d & M \ar[d]^{h}\\ N'\ar[r]^{f'}& M } .$ | {"extraction_info": {"found_math": true, "script_math_tex": 31, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 35, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9921615123748779, "perplexity": 4458.434306370862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446708046.99/warc/CC-MAIN-20221126180719-20221126210719-00462.warc.gz"} |
http://math.stackexchange.com/questions/87417/transfinite-recursion-theorem | # Transfinite Recursion Theorem
I have the following homework assignment:
Let $\textbf{V} \models ZFC$ and let $\mathbb{P} = (P, \leq)$ be a forcing partial order in $\textbf{V}$. Define a class function $F: \textbf{V} \to \textbf{V}$ such that using the transfinite recursion theorem there exists a function $G: \Omega \to \textbf{V}$ such that for all $\alpha \in \Omega$: $$G(\alpha) = V_\alpha^{\mathbb{P}}$$
What follows is my answer, can you please tell me if I have it right:
(i) For $\alpha = \emptyset$ we want $G(\alpha) = V_\emptyset^\mathbb{P} = 0 =: F(0)$
(ii) For $\alpha = 1$ we want $G(1) = V_1^\mathbb{P} = \mathcal{P}( V_\emptyset^\mathbb{P}\times \mathbb{P}) = \mathcal{P}( \emptyset \times \mathbb{P}) \cong \mathcal{P}(\emptyset) = \{\emptyset\} = 1 =: F(G|_1)$
For $\alpha = 2$ we want $G(2) = V_2^\mathbb{P} = \mathcal{P}(V_1^\mathbb{P} \times \mathbb{P}) = \mathcal{P}(\{\emptyset\} \times \mathbb{P}) =: F(G|_2)$
So for a successor ordinal $\alpha \in \Omega$ we define $F(G|_{\alpha + 1}) := \mathcal{P}(F(G|_\alpha) \times \mathbb{P})$
(iii) For a limit ordinal $\alpha \in \Omega$, we have $V_\alpha^\mathbb{P} = \cup_{\beta \in \alpha} V_\beta^\mathbb{P}$ and so we want to define $G(\alpha) = \cup_{\beta \in \alpha} V_\beta^\mathbb{P} = \cup_{\beta \in \alpha} F(G|_\beta) =: F(G|_\alpha)$
Is this all I need to do? Or do I need to show that $F$ is well-defined or anything else? Many thanks for your help!
-
Note that $V^\mathbb P_0 = \varnothing$, so $V^\mathbb P_1 = \{\varnothing\}$. Only from the $2$ you get actual content. – Asaf Karagila Dec 1 '11 at 16:06
@AsafKaragila: How so? The definition in the notes says $$V_0^\mathbb{P}=\emptyset$$ $$V_\alpha^\mathbb{P} = \cup_{\beta \in \alpha}V_\beta^\mathbb{P}$$ for a limit ordinal $\alpha$ and $$V_{\alpha + 1}^\mathbb{P} = \mathcal{P}(V_\alpha^\mathbb{P} \times \mathbb{P})$$ So with $\alpha = 1 = \{ \emptyset \}$ I should get $\mathcal{P}(V_{\emptyset}^\mathbb{P} \times \mathbb{P})$ – Matt N. Dec 1 '11 at 16:31
Which is $\mathcal P(\varnothing\times\mathbb P) = \mathcal P(\varnothing)$. – Asaf Karagila Dec 1 '11 at 16:37
It depends on the precise statement of the Rec.Thrm. It appears that you are following something close to Kunen, wherein it is is stated as follows. If $F:V\to V$, then there is a unique $G:\mathrm{On}\to V$ such that $(\forall\alpha\in\mathrm{On})(G(\alpha)=F(G\restriction\alpha))$. So first of all, you want $F$ to be defined on all sets (though you can make trivial choices in cases where the value at an argument is not important). You should be aware that the interesting case is when the argument to $F$ is an $\alpha$-sequence (since that is what $G\restriction\alpha$ is). – Arthur Fischer Dec 1 '11 at 18:43
@ArthurFischer: Thanks! Do you mean I just need to define $F(X) := \emptyset$ for any $X$ not an ordinal? I don't understand what you mean by $\alpha$-sequence. As for the theorem: you guessed right, we are using what you stated above. – Matt N. Dec 1 '11 at 21:26
To answer a small question, by an $\alpha$-sequence we simply mean any function with domain $\alpha$ (where $\alpha$ is an ordinal). And we will follow standard notation that by $\langle x_\xi \rangle_{\xi<\alpha}$ we denote the $\alpha$-sequence $g$ where $g (\xi) = x_\xi$ for all $\xi < \alpha$.
I'm not certain how pedantic you are supposed to be in your own solution, but since they are even asking you to do this, you might have to be fairly thorough, so let's look at the Recursion Theorem closely. It says that given $F : V \to V$ there is a (unique) $G : \mathrm{On} \to V$ such that for all $\alpha$ $$G ( \alpha ) = F ( G \restriction \alpha ).$$ Looking at the right-hand-side, we have "the value of $F$ at the argument $G \restriction \alpha$." Clearly, $G \restriction \alpha$ is an $\alpha$-sequence, so we should focus our main attention to defining $F$ on $\alpha$-sequences. As we intend to define the "$\mathrm{On}$-sequence" $G = \langle V_\alpha^P \rangle_{\alpha \in \mathrm{On}}$, we note for $\alpha \in \mathrm{On}$ that $G \restriction \alpha = \langle V_\xi^P \rangle_{\xi < \alpha}$. So the case that we are really, really interested in is when we have an $\alpha$-sequence of the form $\langle V_\xi^P \rangle_{\xi < \alpha}$. Now how to you move from this to $V_\alpha^P$? Clearly this will depend on whether $\alpha$ is a successor or a limit ordinal, but I am sure that by looking at these cases closely for a few seconds you will see how to do this -- and we have thus begun our definition of $F$. Whatever you do will likely be easily extended to arbitrary $\alpha$-sequences, where the values of the sequence are not necessarily of the form desired, so you might as well define $F$ in this similar way (it doesn't hurt), and then take $F (x) = \emptyset$ whenever $x$ is not an $\alpha$-sequence for any $\alpha$. In this way we will get $F : V \to V$. | {"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": 0.971836507320404, "perplexity": 136.14495579888555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368708783242/warc/CC-MAIN-20130516125303-00067-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/1794822/is-the-riemann-integral-of-a-strictly-smaller-function-strictly-smaller | # Is the Riemann integral of a strictly smaller function strictly smaller?
We all know that if $f\leq{}g$ in $[a,b]$ then $$\int_a^bf\,dx\leq\int_a^bg\,dx$$ now, imagine that we have $f<g$, is it true that
$$\int_a^bf\,dx<\int_a^bg\,dx$$
• Hint: make it one integral in your inequality. – Steve D May 22 '16 at 1:38
• @SteveD's hint : this reduces to the question of whether or not $h >0 \implies \int h > 0$ – Chill2Macht May 22 '16 at 1:39
• This is indeed true, but I am not sure if we have any proof that avoids ideas from measure theory. – Sangchul Lee May 22 '16 at 1:46
• @SangchulLee: huh? $(g-f)$ has a positive minimum ($[a,b]$ is compact)... – Steve D May 22 '16 at 2:10
• @SteveD If it's not continuous, the infimums may be $0$. – MathematicsStudent1122 May 22 '16 at 2:11
Here is an argument which borrowed ideas from measure theory, but does not assume any direct knowledge on it.
Proof. It suffices to prove the following claim:
Claim. If $h \in \mathscr{R}([a, b])$ satisfies $h \geq 0$ and $\int_{a}^{b} h \, \mathrm{d}x = 0$, then $h(c) = 0$ for some $c \in [a, b]$.
Step 1. To this end, assume that $h$ satisfies the assumptions of the claim. Then we have the following observation:
Observation. For any $\epsilon > 0$ and $\delta > 0$, there exists a relatively open subset $U \subseteq [a, b]$ such that
1. $U$ is the union of finitely many relatively open subintervals of $[a, b]$,
2. the lengths of $U$ is less than $\delta$, and
3. $\{ x \in [a, b] : h(x) > \epsilon \} \subseteq U$.
We first check that this indeed implies the claim. For each $n \geq 1$, choose $U_n$ as in Observation with $\epsilon = 1/n$ and $\delta = 3^{-n}(b-a)$, so that
1. the length of $U_n$ is less than $3^{-n}(b-a)$, and
2. $\{ x \in [a, b] : h(x) > 1/n \} \subseteq U_n$.
Then we find that
$$\{ x \in [a, b] : h(x) > 0 \} = \bigcup_{n=1}^{\infty} \{ x \in [a, b] : h(x) > 1/n \} \subseteq \bigcup_{n=1}^{\infty} U_n.$$
Now assume otherwise that $h > 0$ on all of $[a, b]$. Then it follows that $\bigcap_{n=1}^{\infty} U_n = [a, b]$ and thus $\{ U_n : n \geq 1 \}$ is an open cover of $[a, b]$. So we can pick a finite subcover, say $\{ U_{n_1}, \dots, U_{n_K} \}$. This implies that
$$[a, b] = U_{n_1} \cup \cdots \cup U_{n_K}.$$
This is a contradiction since the right-hand side has length at most
$$\sum_{n=1}^{\infty} 3^{-n}(b-a) < b-a.$$
Step 2. It now remains to prove the observation. (The proof is essentially a variant of the Markov's inequality.)
Choose a partition $P$ such that $U(P, h) < \delta \epsilon$. Write $P = \{a = x_0 < \cdots < x_N = b\}$ an define $M_j = \sup_{[x_{j-1}, x_j]} h$ and $\Delta x_j = x_j - x_{j-1}$. Then we know that $U(P, h) = \sum_{j=1}^{N} M_j \Delta x_j < \delta \epsilon$. On the other hand, let $J$ be the set of indices $j$ for which $M_j > \epsilon$. Then
$$\sum_{j \in J} \Delta x_j \leq \frac{1}{\epsilon} \sum_{j \in J} M_j \Delta x_j \leq \frac{1}{\epsilon} U(P, h) < \delta$$
and that $\cup_{j \notin J} [x_{j-1}, x_j]$ is a finite union of closed intervals on which $h \leq \max_{j \notin J} M_j \leq \epsilon$ holds. Therefore the observation follows by taking $U$ as the complement of $\cup_{j \notin J} [x_{j-1}, x_j]$.
Suppose on the contrary that $\int f=\int g$, we can assume $f,g$ vanishes outside the interval, then they are both $\mathscr L^1$, then we have $$\int_\Bbb R(f-g)dx=0.$$ since $f- g\ge 0$, indeed $f=g$ a.e. on $\Bbb R$, contradiction.
I'd appreciate an elementary proof without any appeal to Lebesgue theory.
• It suffices to show that there is at least one point where $f-g$ is continuous. This answer gives a proof which does not formally require measure theory, although it has a measure-theoretic flavor. It would be nice to see a more elementary proof, but i haven't found one. – Bungo May 22 '16 at 6:04
• @Bungo fair observation. Reduces the problem to a considerably more trivial one. – Vim May 22 '16 at 6:25
Assuming $$f+h=g$$
and $$h>0$$
$$\int_a^b f dx+\int_a^b h dx=\int_a^b g dx$$
since
$$\int_a^b h dx>(b-a)\times\min(h(x))>0$$ We can write
$$\int_a^b f dx < \int_a^b g dx$$
but what if $h$ has no minimum? it is enough to find any piece where $h$ has a minimum to prove $\int_a^b h dx >0$. Unless $h$ has no minimum at any neighborhood. But if $f$ and $g$ are integrable then $h$ must be integrable too.
• Might be quite tricky to show that in at least one neighbourhood $h$ attains a minimum. The idea is inspiring, though. – Vim May 22 '16 at 3:25
• @Vim Not that tricky. Riemann integrable functions are continuous almost everywhere. – MathematicsStudent1122 May 22 '16 at 3:25
• @MathematicsStudent1122 that's an appeal to measure theory, which I think Arashium wants to avoid using here. – Vim May 22 '16 at 3:28
• @Vim, Arashium is an engineer and not a mathematician. I have no idea about measure theory :) – Arashium May 22 '16 at 3:31
• @Vim Can we prove the weaker statement $\text{discontinuous everywhere} \Longrightarrow \text{not Riemann integrable}$ without measure theory? If so, we're done. Intuitively, it seems clear; in such a situation, regardless of how fine one makes a partition, the $\sup$ and $\inf$ of $f$ in every subinterval will still be "far away" from each other. – MathematicsStudent1122 May 22 '16 at 5:38 | {"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": 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": 0.9818047285079956, "perplexity": 191.91289370436095}, "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-2019-22/segments/1558232262311.80/warc/CC-MAIN-20190527085702-20190527111702-00076.warc.gz"} |
https://math.stackexchange.com/questions/3010766/how-can-prove-this-inequality-tau2-gamma-leq-c-5-gamma-frac1-tau | How can prove this inequality $|\tau|^{2\gamma} \leq c_5(\gamma) \frac{1+|\tau|}{1+|\tau|^{1-2\gamma}}$?
I'm reading a demonstration that uses this following inequality. For a fixed $$\gamma<1/4$$, exists a $$c_5(\gamma)$$, such that
$$|\tau|^{2\gamma} \leq c_5(\gamma) \frac{1+|\tau|}{1+|\tau|^{1-2\gamma}}, \forall \tau \in \mathbb{R}$$.
I tried to deduce that using the fact that $$0<\frac{1}{1+|\tau|}\leq1$$, but i couldn't get in anywhere.
Reorganizing, this is equivalent to proving that the function $$f\colon\mathbb{R}\to\mathbb{R}$$ given by $$f(\tau) = \frac{|\tau|+|\tau|^{2\gamma}} {|\tau|+1}$$ is bounded. Note that it is continuous, and $$\lim_{+\infty} f = \lim_{-\infty} f = 1$$ (using the fact that $$2\gamma <1$$); therefore, it is bounded.
• (also, as shown in this proof, the result holds for $\gamma < 1/2$, not only $\gamma<1/4$.) – Clement C. Nov 23 '18 at 19:59
• Actually it will not hold for $\gamma < 1/2$, the reason why it holds is because $2\gamma<1-2\gamma$ when $\gamma < 1/4$. I figured this, but it wasn't suffice to prove that. But thank you so much for help me. – João Paulo Andrade Nov 23 '18 at 22:38
• @JoãoPauloAndrade Glad it helped, but -- it does hold for all $\gamma \leq 1/2$ (the case $1/2$ is proven similarly, but the limit of $f$ at $\infty$ is $2$, not 1). – Clement C. Nov 23 '18 at 22:40
• For instance, for $\gamma=1/2$, take $c_5(1/2)=2$. You have $$|\tau| \leq 2\cdot \frac{1+|\tau|}{2}$$ for all $\tau$. – Clement C. Nov 23 '18 at 22:46
Let $$|\tau| = x \geq 0$$ and Consider: $$f(x) = \dfrac{x^{2\gamma}+x}{1+x}$$ and prove this has a maximum by taking derivative. | {"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": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9623385071754456, "perplexity": 219.000434359213}, "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-2019-51/segments/1575540497022.38/warc/CC-MAIN-20191207082632-20191207110632-00559.warc.gz"} |
http://physics.stackexchange.com/questions/103053/can-a-human-body-supercavitate-to-survive-water-impact | # Can a human body supercavitate to survive water impact?
Inspired by this analysis of a human (OK, Captain America) hitting water feet first at terminal velocity, I'm wondering if supercavitation would be possible and whether it would improve your chances of survival?
Would a human body reach a high enough speed to supercavitate on impact? Would this reduce the g-force to a survivable level (eg by allowing deeper penetration into the water)? And crucially, when the supercavitation collapses do you still have to deal with an unsurvivable acceleration?
-
The problem with supercavitation is that it occurs at a pressure drop, not an increase. The thing that kills you on impact is the pressure in front of you - the water that cannot get out of the way fast enough. Skin drag on the body happens later - when more of the body is already submerged. Most likely you are dead by then...
Let me try (with my rusty hydrodynamics) to estimate a few things.
For a sphere moving through water, the apparent inertia experienced in laminar flow (because when you move the sphere, you move water) is the mass of the sphere plus half the mass of the displaced liquid (this is called the added mass). Of course a human is not even close to a spherical shape, but this gives us an order of magnitude of the forces we are looking for.
For a human with a density approximately that of water, the effect of the added mass can be thought of as hitting another human half their weight (before accounting for skin drag) when they hit the water. You don't hit that human all at once, of course, since your body isn't hitting the water all at once - but at a first approximation, the time it takes to hit that body is your velocity divided by your length, and by conservation of momentum your velocity must decrease by 1/3 in that time. If the density of the water is lower, this ratio will become smaller, and in general is
\begin{align}ratio &= 1-\frac{\rho_1}{\rho_1 + 0.5 * \rho_2}\\ &= \frac{\rho_2}{2\rho_1+\rho_2}\end{align}
Where $\rho_1$ is the density of the body, and $\rho_2$ is the density of the water. From this you can see that if $\rho_2$ becomes $<<\rho_1$, the ratio becomes very small - and there is less initial deceleration. But let's keep going with regular water first.
If you have length $l$ and velocity $v$, the time during which you decelerate will be approximately $l/v$ and in that time your velocity decreases by $v/3$ and momentum decreases by $mv/3$ for a mass $m$. This means an average force
\begin{align}F &= \frac{mv}{3\frac{l}{v}}\\ &= \frac{mv^2}{3l}\end{align}
Putting terminal velocity at 56 m/s (from Wolfram Alpha), length of 2 m (with arms extended above head), and a mass of 70 kg, you get an average force of about 36 kN for the 1/20th of a second that it takes to submerge - that's like an elephant sitting on you for a single frame of a video. Enough to do some real damage to your internal organs (losing 1/3 of your velocity in that time is roughly equivalent to 40 g of deceleration).
You need to lower the density of the water before you hit it. I suppose if you had a machine gun on you and shot a rapid burst at the water just before impact, two good things would happen: you would slow yourself down (because of the recoil of the gun - bullets accelerate down = you accelerate up), and you might be able to create cavitation in the water where you land. This lowers the effective density of the water, meaning that you will decelerate more slowly and have a slightly better chance of surviving that initial hit.
As an aside - the current record for a free fall into water stands at 54 m, which puts the velocity at about 30 m/s (assuming some drag) and the initial deceleration at 20 g. That's about double what fighter pilots get, but it's just about believable that a stunt man can survive that on a good day.
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https://curriculum.illustrativemathematics.org/MS/teachers/3/8/3/preparation.html | Lesson 3
Rational and Irrational Numbers
Lesson Narrative
In previous lessons, students learned that square root notation is used to write the side length of a square given the area of the square. For example, a square whose area is 17 square units has a side length of $$\sqrt{17}$$ units.
In this lesson, students build on their work with square roots to learn about a new mathematical idea, irrational numbers. Students recall the definition of rational numbers (MP6) and use this definition to search for a rational number $$x$$ such that $$x^2 = 2$$. Students should not be left with the impression that looking for and failing to find a rational number whose square is 2 is a proof that $$\sqrt{2}$$ is irrational; this exercise is simply meant to reinforce what it means to be irrational and to provide some plausibility for the claim. Students are not expected to prove that $$\sqrt{2}$$ is irrational in grade 8, and so ultimately must just accept it as a fact for now.
In the next lesson, students will learn strategies for finding the approximate location of an irrational number on a number line.
Learning Goals
Teacher Facing
• Comprehend the term “irrational number” (in spoken language) to mean a number that is not rational and that $\sqrt{2}$ is an example of an irrational number.
• Comprehend the term “rational number” (in written and spoken language) to mean a fraction or its opposite.
• Determine whether a given rational number is a solution to the equation $x^2=2$ and explain (orally) the reasoning.
Required Preparation
It would be useful throughout this unit to have a list of perfect squares for easy reference. Consider hanging up a poster that shows the 20 perfect squares from 1 to 400. It is particularly handy in this lesson.
Student Facing
• I know what an irrational number is and can give an example.
• I know what a rational number is and can give an example.
Building On
Building Towards
Glossary Entries
• irrational number
An irrational number is a number that is not a fraction or the opposite of a fraction.
Pi ($$\pi$$) and $$\sqrt2$$ are examples of irrational numbers.
• rational number
A rational number is a fraction or the opposite of a fraction.
Some examples of rational numbers are: $$\frac74,0,\frac63,0.2,\text-\frac13,\text-5,\sqrt9$$
• square root
The square root of a positive number $$n$$ is the positive number whose square is $$n$$. It is also the the side length of a square whose area is $$n$$. We write the square root of $$n$$ as $$\sqrt{n}$$.
For example, the square root of 16, written as $$\sqrt{16}$$, is 4 because $$4^2$$ is 16.
$$\sqrt{16}$$ is also the side length of a square that has an area of 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": 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": 0.8885753154754639, "perplexity": 323.2558222670776}, "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-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00556.warc.gz"} |
https://www.physicsforums.com/threads/center-of-mass.100397/ | # Center of mass
1. Nov 17, 2005
### jacy
hi,
I am finding the center of mass in this problem. I have it as an attachment. Please take a look, thanks.
#### Attached Files:
• ###### center_of_mass.doc
File size:
19.5 KB
Views:
38
2. Nov 18, 2005
### Staff: Mentor
Your attachment shows three masses, but I can't tell their positions from what's written. (It's just not clear to me.)
3. Nov 18, 2005
### jacy
Here is that file again. Thanks for your help.
#### Attached Files:
• ###### center_of_mass.doc
File size:
20.5 KB
Views:
31
4. Nov 18, 2005
### Staff: Mentor
OK, now it's a bit clearer. The first thing to realize is that the masses are not point masses, but have length. Assuming that they are uniform, each mass has its own center of mass, right at its center. When calculating the center of mass of the system, you need to measure the distance of the center of each mass from your reference point.
Give it another shot.
5. Nov 18, 2005
### jacy
Thanks again, so the distance for mass 1 will be 1m from my reference point, for mass 2 it will be 6m, for mass 3 it will be 8m. Am i correct.
center of mass = (20(1) + 30(6) + 40(8))/ 90
= 5.78 m
6. Nov 18, 2005
### neutrino
Recheck the distance to the cm of the third mass.
7. Nov 18, 2005
### Staff: Mentor
Measured from the left edge of mass 1, I'd say that your distances are correct for masses 1 and 2, but not for mass 3.
8. Nov 18, 2005
### jacy
Thanks, the distance for mass 3 will be 10m from the left edge of mass 1, correct.
center of mass = (20(1) + 30(6) + 40(10))/90
= 6.67 m
This will be the answer, thanks for ur help.
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https://www.physicsforums.com/threads/homework-help.189620/ | # Homework Help: Homework Help
1. Oct 7, 2007
1. The problem statement, all variables and given/known data
A large package weighing 517 N is accelerated across a gym floor at 3 m/sec^2 because it is being pushed at a force of 285 N at 50 degrees to the floor. What is the normal force of the package? What is the coefficient of sliding friction between the package and the floor? Approximately what percentage of the 285 N force is being wasted due to the excessive angle produced by the pusher?
2. Relevant equations
F=MA
HORIZONTALLY
FORCE APPLIED, X= (285)(cos 50)= 183.194
VERTICALLY
FORCE OF Y=0 (not rising off the ground)
FORCE APPLIED, Y= (285)(sin 50)= 218.323
3. The attempt at a solution
Thanks
1. The problem statement, all variables and given/known data
In order to slide a 400 N box up a 25 degree ramp at a constant speed, a force of 250 N parallel to to the surface is required
2. Relevant equations
3. The attempt at a solution
Last edited: Oct 7, 2007
2. Oct 7, 2007
### bob1182006
for the first problem:
what is the normal force of the box?
how would you setup Newton's Second Law in order to find the coefficient of friction?
and to find the % being wasted take (Force Applied - Total Force being Exerted)/Total Force being Exerted * 100%
And what is the question for your second problem?
3. Oct 7, 2007
Wouldn't you use F=MA to find normal force? | {"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": 0.9224899411201477, "perplexity": 1581.0256459174486}, "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-2018-43/segments/1539583512499.22/warc/CC-MAIN-20181019232929-20181020014429-00237.warc.gz"} |
https://stats.stackexchange.com/questions/370241/how-to-minimize-chi-square-using-the-cdf-instead-of-the-pdf | # How to minimize Chi-Square using the CDF instead of the PDF?
Suppose one has data that is suspected to obey a normal distribution. One computes a histogram of the data, and performs Pearson's Chi-Squared Test. To perform this test, one must compare the observed counts from each bin in the histogram against the expected counts in the corresponding bins by the formula below:
To compute the expected counts, one integrates the probability density function of the normal distribution from the left bin-edge to the right bin-edge for all bins using fixed parameters mu and sigma. The smallest Chi-Squared statistic (assuming the associated pvalue is not too high or too low) should correspond to the fixed parameters that best fit the data.
(If anything is incorrect, please correct me).
I have read that a better optimization technique is to use the cdf instead of the pdf. If one still wanted to use the cdf (instead of the pdf) in conjunction with Pearson's Chi-Squared Test to optimize the distribution fit, how would one go about this?
My initial thought is that the histogram of observations should be converted into a cumulative histogram, such that the expected counts can be obtained from the integration of the cdf instead of the pdf. Is this wrong?
(I am asking because I have a lognormal dataset; the Chi-Square fit of looks great but the pvalue is small enough to be approximated as zero. Any advice or corrections regarding misunderstood information would be appreciated).
• Chi-square testing goes back to 1900 and wasn't the method of choice even then for assessing fit to a normal distribution, as normal quantile plots (modern terminology) were already in use. Either way, binning discards information -- and in an arbitrary way. Jan 9 at 18:25
## 1 Answer
CDF is the integral of PDF from negative infinity. The CDF values don't tell you the expected counts in each bin because as mentioned, it starts the integration from negative infinity.
You will need to apply integration for each bin endpoint.
• The integration starts at negative infinity, but I think the CDF will be zero from -inf until the first bin. Are you saying that to apply this Chi-Squared Test with the CDF, I would integrate the first bin from -inf to the first right-side bin-edge, would integrate the second bin from -inf to the second right-side bin-edge, etc. ?
– user146123
Oct 5, 2018 at 5:44 | {"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": 0.9247338175773621, "perplexity": 535.8627938896358}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662594414.79/warc/CC-MAIN-20220525213545-20220526003545-00395.warc.gz"} |
https://www.physicsforums.com/threads/how-does-nmr-work.723996/ | # How does NMR work
1. Nov 20, 2013
### CrimpJiggler
I've read two different explanations so I'm confused. The way I thought NMR works is that the protons spins align with the external magnetic field, then the RF radiation applied causes the spins to flip from -1/2 to 1/2, if the frequency of the radio waves are equal to the energy difference between these two spin states.
Now I read a different explanation which has me confused. Here:
http://www.simplyphysics.com/page2_4.html
it says that the protons actually wobble around the axis of the external magnetic field, and when you apply radio waves with energy equal to the frequency of this wobbling (Larmor frequency), then the protons absorb it. Which is it? Is it the energy difference between the two spin states, or is it the energy of the Larmor frequency? According to this animation:
the absorbed RF energy causes the magnetic vector axis to drop to the XY plane, so that it is now tangent to the axis of the external magnetic field. This clearly isn't the same thing as a spin flip, so I'm confused.
2. Nov 20, 2013
### DrDu
Both descriptions are correct but it is not so easy to explain why without a basic understanding of quantum mechanics. The point is that as long as you don't explicitly analyze whether a given spin point up or down, it can also be in a superposition which corresponds to the spin pointing at least partially somewhere into the x-y plane. This is not an energy eigenstate, so it will evolve in time. Actually this turns out to a revolution of the axis of the spin in the x-y plane, i.e. the wobbling with the Lamor frequency. Now if the spin interacts with radiation you change that superposition and rotate the spin into or out of the x-y plane.
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https://www.physicsforums.com/threads/repeated-eigenvalue-of-a-n-3-system-of-differential-equations.689379/ | Repeated Eigenvalue of a n=3 system of differential equations
1. May 2, 2013
tehdiddulator
1. The problem statement, all variables and given/known data
x' = \begin{pmatrix}0&1&3\\2&-1&2\\-1&0&-2\end{pmatrix}*x
3. The attempt at a solution
I've found the repeated eigenvalues to be λ$_{1,2,3}$=-1
I've also found the first (and only non zero eigenvector) to be \begin{pmatrix}1&2&-1\end{pmatrix}, but I'm not entirely sure where to go from here. Everything I've found talks about 2x2 matrices. Even looking around on MIT opencoursewares site, they say that its possible but is beyond the scope of that class...I did find some reference to using Jourdan Canonical form may possibly be used, but I don't remember him talking too much about that.
2. May 2, 2013
Staff: Mentor
That's Jordan.. Here's a wiki article about Jordan Normal Form. The section on generalized eigenvectors that might be of help.
3. May 3, 2013
Ray Vickson
For a matrix A having a single eigenvalue $r$ of multiplicity 3 the analytic matrix function $$f(A) = c_0 I + c_1 A + c_2 A^2 + c_3 A^3 + \cdots$$ (corresponding to the ordinary analytic function $f(x) = c_0 + c_1 x + c_2 x^2 + \cdots$) is of the form
$$f(A) = E_1 f(r) + E_2 f'(r) + E_3 f''(r)$$ for some fixed matrices $E_1, E_2, E_3$ which are the same for all functions f. We can determine the $E_1$ by looking at special cases of f: for $f(x) = x^0 = 1$ we have $f'x) = f''(x) = 0$, so $I = A^0 = 1 E_1 + 0 E_2 + 0 E_3 = E_1$. For$f(x) = x$ we have $f'(x) = 1, f''(x) = 0$ and so $A = r E_1 + 1 E_2$. For $f(x) = x^2$ we have $f'(x) = 2x, f''(x) = 2$ and so $A^2 = r^2 E_1 + 2r E_2 + 2 E_3.$ Altogether, we have the three equations
$$E_1 = I\\ r E_1 + E_2 = A\\ r^2 E_1 + 2r E_2 + 2E_3 = A^2.$$
Once we have the $E_i$ we can solve the differential equation system using the matrix exponential $M(t) = e^{At}$. This will have the form
$$M(t) = e^{rt} E_1 + t e^{rt} E_2 + t^2 e^{rt} E_3,$$
obtained from the function $f(x) = e^{xt}.$
This all follows by looking at the Jordan Canonical Form. Note, however, that the discussion above applies equally whether the matrix has a diagonal Jordan form, or one block of size 1 and one of size 2, or a single Jordan block of size 3; those separate cases will just correspond to situations in which some $E_i$ are zero. For example, if A were diagonalizable but with a single eigenvalue r of multiplicity 3, we would just have $E_2 = E_3 = 0$.
4. May 3, 2013
Ray Vickson
For a matrix A having a single eigenvalue $r$ of multiplicity 3 the analytic matrix function $$f(A) = c_0 I + c_1 A + c_2 A^2 + c_3 A^3 + \cdots$$ (corresponding to the ordinary analytic function $f(x) = c_0 + c_1 x + c_2 x^2 + \cdots$) is of the form
$$f(A) = E_1 f(r) + E_2 f'(r) + E_3 f''(r)$$ for some fixed matrices $E_1, E_2, E_3$ which a re the same for all functions f. We can determine the $E_1$ by looking at special cases of f: for $f(x) = x^0 = 1$ we have $f'x) = f''(x) = 0$, so $I = A^0 = 1 E_1 + 0 E_2 + 0 E_3 = E_1$. For$f(x) = x$ we have $f'(x) = 1, f''(x) = 0$ and so $A = r E_1 + 1 E_2$. For $f(x) = x^2$ we have $f'(x) = 2x, f''(x) = 2$ and so $A^2 = r^2 E_1 + 2r E_2 + 2 E_3.$ Altogether, we have the three equations
$$E_1 = I\\ r E_1 + E_2 = A\\ r^2 E_1 + 2r E_2 + 2E_3 = A^2.$$
Once we have the $E_i$ we can solve the differential equation system using the matrix exponential $M(t) = e^{At}$. This will have the form
$$M(t) = e^{rt} E_1 + t e^{rt} E_2 + t^2 e^{rt} E_3,$$
obtained from the function $f(x) = e^{xt}.$
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http://mathhelpforum.com/algebra/58913-i-m-not-sure-about.html | # Math Help - I'm not sure about this...
1. ## I'm not sure about this...
Can someone check this for me, please.
Attached Thumbnails
2. Originally Posted by Alienis Back
Can someone check this for me, please.
Stop at the middle line. The last line is wrong (as you would readily see if you expanded)
3. Why? Isn't the second line a perfect square trinomial?
4. Originally Posted by Alienis Back
Why? Isn't the second line a perfect square trinomial?
I hope that when you expanded $(3a - 3b)^2$ (as I suggested) you got $9a^2 - 18ab + 9b^2$ ....
This is obviously not the same as $3a^2 - 4ab + 3b^2$.
5. Originally Posted by Alienis Back
Why? Isn't the second line a perfect square trinomial?
===============================================
Alienis Back,
2nd line is not correct.
see my attachment.
6. Ok. Last line is certainly wrong. No doubt. So, It shouldn't be further factor-able, should it? And where is your attachement, 2976math?
7. simplfy.pdf
Here we go.
8. 2976math, look for the clip.
Attached Thumbnails
9. Originally Posted by 2976math
simplfy.pdf
Here we go.
Your answer is useful in that it clearly shows why the trinomial in brackets cannot be factorised (because it's a sum of two squares). But it's not what I'd give as an answer to the original question (I'd stop at the middle line).
10. Originally Posted by Alienis Back
2976math, look for the clip.
Can you see attachment?
11. simplfy.pdf
I did in hurry.
18(a^2)(b^2) should be always there from 3rd line to 5th line.
12. Yep!
2976math, That was quite useful, but is a lot more that I expected. Quite useful but as I see it, it doesn't mean my aproach is wrong. I'll keep on studying yours anyway.
Thanks a lot guys. | {"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": 3, "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": 0.849961519241333, "perplexity": 2133.1408812962095}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "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-2014-10/segments/1394010980041/warc/CC-MAIN-20140305091620-00041-ip-10-183-142-35.ec2.internal.warc.gz"} |
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