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http://dustintran.com/blog/on-asymptotic-convergence-of-averaged-sgd | Stochastic gradient descent (SGD) has seen wide application for learning problems on large scale data, whether this be for generalized linear models [6], SVMs [1], or neural networks [2] alike. We review a result by Polyak and Juditsky [3] on the theory of performing SGDs with averaged estimates, which shows that it is asymptotically as good as a second-order method.
Recall that in the typical formulation, the problem is to minimize some loss function:
$\theta^* = {\mathrm{arg\, min}}\, Q(\theta) = \sum_{n=1}^N Q_n(\theta),$
where $Q_n(\theta)$ is the only part of the function associated with the $n^{th}$ data point. In the statistical formulation, we can formalize this by assuming all $N$ observations are i.i.d., and the loss function corresponds to the negative log-likelihood for some choice of model and its set of parameters $\theta=(\theta_1,\ldots,\theta_p)$.
Let the learning rate $\{\alpha_n\}$ be a specified decreasing sequence. Stochastic gradient descent uses the update
where at each iteration one chooses a randomly sampled data point $i_n\in \{1,\ldots,N\}$. This is a stochastic approximation procedure [5], and it is proven to converge to a point where the expected loss $\mathbb{E}[\nabla Q(\theta^*)]=0$; for convex loss, $\theta^*$ is the global minimum.
Remark. It was well-known heuristically that one should sample without replacement and make enough passes of the data for SGD, but not well-understood why this is actally better than sampling with replacement. Recht and Re [4] have an interesting proof for this using noncommutative arithmetic-geometric means.
Here is a table which summarizes the most common optimization methods.
method Running time complexity Convergence rate
Newton $$\mathcal{O}(Np^{2+\epsilon})$$ quadratic
Quasi-Newton (e.g. BFGS, DFP) $$\mathcal{O}(Np^{1+\epsilon})$$ superlinear
Gradient descent $$\mathcal{O}(Np^{1-\epsilon})$$ linear
Stochastic gradient descent $$\mathcal{O}(p)$$ sublinear
It’s a clear win for SGD in terms of running time. Newton’s method requires inverting a Hessian $[\nabla^2 Q(\theta_n)]^{-1}$ and computing the full gradient $\nabla Q(\theta_n)$ at each step; Quasi-Newton only slightly relaxes this by using a low rank approximation of the Hessian; and gradient descent doesn’t use it at all but still requires computing $\nabla Q(\theta_n)$.
However, this comes at the cost of convergence rates. Naturally, SGD can only obtain a sublinear convergence rate $\mathcal{O}(1/n)$ for convex functions, whereas the above can achieve better convergence by using more information per iteration.
While there will always be a point in which SGD surpasses any of the above methods as the size of the data becomes larger (thus causing running time complexity to be the limiting factor), we can still do better than this.
The simplest variant which improves SGD use Polyak-Ruppert averaging:
That is, take the running average of the estimates during each iteration. We define averaged stochastic gradient descent (ASGD) as applying SGD with Polyak-Ruppert averaging.
Proposition. (Polyak and Judisty [3]) Asymptotically, ASGD is as fast as a second-order SGD, i.e.,
While the convergence rate of a second-order SGD is still sublinear, the constants are much improved. This is reflected in practice [7], and it comes at very little cost due to the simplicity of the small change in implementation.
As I’m currently reviewing the literature on stochastic gradient methods, I’ll post the proof on another page as I think it has a nice derivation.
## References
[1] L. Bottou. Online Algorithms and Stochastic Approximations, Online Learning and Neural Networks, Edited by David Saad, Cambridge University Press, 1998.
[2] L. Bottou. Stochastic Gradient Learning in Neural Networks, Proceedings of Neuro-Nîmes 91, EC2, 1991.
[3] B. Polyak and A. Juditsky. Acceleration of stochastic approximation by averaging. Automation and Remote Control, 30(4):838–855, 1992.
[4] B. Recht and C. Re. Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences. preprint arXiv:1202.4184, 2012.
[5] H. Robbins and S. Monro. A stochastic approximation method. The Annals of Mathematical Statistics, pages 400–407, 1951.
[6] P. Toulis, E. Airoldi, and J. Rennie. Statistical analysis of stochastic gradient methods for generalized linear models. JMLR W&CP, 32(1):667–675, 2014.
[7] W. Xu. Towards optimal one pass large scale learning with averaged stochastic gradient descent. preprint arXiv:1107.2490, 2011. | {"extraction_info": {"found_math": true, "script_math_tex": 13, "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": 0.938345730304718, "perplexity": 740.5831650597357}, "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/1508187824733.32/warc/CC-MAIN-20171021095939-20171021115939-00114.warc.gz"} |
https://www.physicsforums.com/threads/equivalence-relations.462767/ | # Equivalence Relations!
• Start date
• #1
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1. Let R be a relation on X that satisfies
a) for all a in X, (a,a) is in R
b) for a,b,c in X, if (a,b) and (b,c) in R, then (c,a) in R.
Show that R is an equivalence relation.
2. In order for R to be an equivalence relation, the following must be true:
1) for all a in X, (a,a) is in R
2) for a,b in X, if (a,b) is in R, then (b,a) is also in R
3) for a,b,c in X, if (a,b) and (b,c) is in R, then (a,c) is in R.
3. The first part is given by the definition of an equivalence relation. I'm stuck on proving part b. Help please!
## The Attempt at a Solution
Related Calculus and Beyond Homework Help News on Phys.org
• #2
lanedance
Homework Helper
3,304
2
how about considering (a,b) and (b,b) for the second part?
• #3
115
1
how about considering (a,b) and (b,b) for the second part?
Proof by contradiction, using lanedance's example.
Or consider (b,c) and (c,c).
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https://www.saralstudy.com/study-eschool-ncertsolution/12th/mathematics/relations-and-functions | Home NCERT Solutions NCERT Exemplar CBSE Sample Papers NCERT Books Class 12 Class 10
Relations and Functions
Fundamentals of relations and functions are already known to us such as their definitions, types, graphs, domain and range, ete. In this chapter, we will study these topics in detail. Once we come and over these topics further chapters of our curriculum will be easily understood. Topics of this chapter are - types of relations : reflexive, symmetric, transitive and equivalence relations, one to one and onto functions, inverse of a function, composite functions, Binary operations.
Download pdf of NCERT Solutions for Class mathematics Chapter 1 Relations and Functions
Download pdf of NCERT Examplar with Solutions for Class mathematics Chapter 1 Relations and Functions
Exercise 1
• Q1 Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, 3,13, 14} defined as R = {(x, y): 3x − y = 0} (ii) Relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y): y is divisible by x} (iv) Relation R in the set Z of all integers defined as R = {(x, y): x − y is as integer} (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {(x, y): x and y work at the same place} (b) R = {(x, y): x and y live in the same locality} (c) R = {(x, y): x is exactly 7 cm taller than y} (d) R = {(x, y): x is wife of y} (e) R = {(x, y): x is father of y} Ans: (i) A = {1, 2, 3 … 13, 14} R = {(x, y): 3x − y = 0} ∴ R = {(1, 3), (2, 6), (3, 9), (4, 12)} R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R. Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0] Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R. [3(1) − 9 ≠ 0] Hence, R is neither reflexive, nor symmetric, nor transitive. (ii) R = {(x, y): y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)} It is seen that (1, 1) ∉ R. ∴ R is not reflexive. (1, 6) ∈R But, (1, 6) ∉ R. ∴ R is not symmetric. Now, since there is no pair in R such that (x, y) and (y, z) ∈R, then (x, z) cannot belong to R. ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. (iii) A = {1, 2, 3, 4, 5, 6} R = {(x, y): y is divisible by x} We know that any number (x) is divisible by itself. (x, x) ∈R ∴ R is reflexive. Now, (2, 4) ∈R [as 4 is divisible by 2] But, (4, 2) ∉ R. [as 2 is not divisible by 4] ∴ R is not symmetric. Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y. ∴ z is divisible by x. ⇒ (x, z) ∈R ∴ R is transitive. Hence, R is reflexive and transitive but not symmetric. (iv) R = {(x, y): x − y is an integer} Now, for every x ∈ Z, (x, x) ∈R as x − x = 0 is an integer. ∴ R is reflexive. Now, for every x, y ∈ Z if (x, y) ∈ R, then x − y is an integer. ⇒ −(x − y) is also an integer. ⇒ (y − x) is an integer. ∴ (y, x) ∈ R ∴ R is symmetric. Now, Let (x, y) and (y, z) ∈R, where x, y, z ∈ Z. ⇒ (x − y) and (y − z) are integers. ⇒ x − z = (x − y) + (y − z) is an integer. ∴ (x, z) ∈R ∴ R is transitive. Hence, R is reflexive, symmetric, and transitive. (v) (a) R = {(x, y): x and y work at the same place} (x, x) ∈ R ∴ R is reflexive. If (x, y) ∈ R, then x and y work at the same place. ⇒ y and x work at the same place. ⇒ (y, x) ∈ R. ∴ R is symmetric. Now, let (x, y), (y, z) ∈ R ⇒ x and y work at the same place and y and z work at the same place. ⇒ x and z work at the same place. ⇒ (x, z) ∈R ∴ R is transitive. Hence, R is reflexive, symmetric, and transitive. (b) R = {(x, y): x and y live in the same locality} Clearly (x, x) ∈ R as x and x is the same human being. ∴ R is reflexive. If (x, y) ∈R, then x and y live in the same locality. ⇒ y and x live in the same locality. ⇒ (y, x) ∈ R ∴ R is symmetric. Now, let (x, y) ∈ R and (y, z) ∈ R. ⇒ x and y live in the same locality and y and z live in the same locality. ⇒ x and z live in the same locality. ⇒ (x, z) ∈ R ∴ R is transitive. Hence, R is reflexive, symmetric, and transitive. (c) R = {(x, y): x is exactly 7 cm taller than y} Now, (x, x) ∉ R Since human being x cannot be taller than himself. ∴ R is not reflexive. Now, let (x, y) ∈R. ⇒ x is exactly 7 cm taller than y. Then, y is not taller than x. ∴ (y, x) ∉R Indeed if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x. ∴R is not symmetric. Now, Let (x, y), (y, z) ∈ R. ⇒ x is exactly 7 cm taller than y and y is exactly 7 cm taller than z. ⇒ x is exactly 14 cm taller than z . ∴ (x, z) ∉R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. (d) R = {(x, y): x is the wife of y} Now, (x, x) ∉ R Since x cannot be the wife of herself. ∴R is not reflexive. Now, let (x, y) ∈ R ⇒ x is the wife of y. Clearly y is not the wife of x. ∴ (y, x) ∉ R Indeed if x is the wife of y, then y is the husband of x. ∴ R is not transitive. Let (x, y), (y, z) ∈ R ⇒ x is the wife of y and y is the wife of z. This case is not possible. Also, this does not imply that x is the wife of z. ∴ (x, z) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. (e) R = {(x, y): x is the father of y} Now (x, x) ∉ R As x cannot be the father of himself. ∴R is not reflexive. Now, let (x, y) ∈R. ⇒ x is the father of y. ⇒ y cannot be the father of y. Indeed, y is the son or the daughter of y. ∴ (y, x) ∉ R ∴ R is not symmetric. Now, let (x, y) ∈ R and (y, z) ∈ R. ⇒ x is the father of y and y is the father of z. ⇒ x is not the father of z. Indeed x is the grandfather of z. ∴ (x, z) ∉ R ∴R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. Q2 Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b2} is neither reflexive nor symmetric nor transitive. Ans: R = {(a, b): a ≤ b2} It can be observed that \begin{align} \left(\frac{1}{2},\frac{1}{2}\right) ∉ R , since \frac{1}{2}>\left(\frac{1}{2}\right)^2 = \frac{1}{4}\end{align} ∴R is not reflexive. Now, (1, 4) ∈ R as 1 < 42 But, 4 is not less than 12. ∴(4, 1) ∉ R ∴R is not symmetric. Now, (3, 2), (2, 1.5) ∈ R (as 3 < 22 = 4 and 2 < (1.5)2 = 2.25) But, 3 > (1.5)2 = 2.25 ∴(3, 1.5) ∉ R ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. Q3 Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. Ans: Let A = {1, 2, 3, 4, 5, 6}. A relation R is defined on set A as: R = {(a, b): b = a + 1} ∴ R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} We can find (a, a) ∉ R, where a ∈ A. For instance, (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R ∴ R is not reflexive. It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R. ∴ R is not symmetric. Now, (1, 2), (2, 3) ∈ R But, (1, 3) ∉ R ∴R is not transitive Hence, R is neither reflexive, nor symmetric, nor transitive. Q4 Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric. Ans: R = {(a, b); a ≤ b} Clearly (a, a) ∈ R as a = a. ∴R is reflexive. Now, (2, 4) ∈ R (as 2 < 4) But, (4, 2) ∉ R as 4 is greater than 2. ∴ R is not symmetric. Now, let (a, b), (b, c) ∈ R. Then, a ≤ b and b ≤ c ⇒ a ≤ c ⇒ (a, c) ∈ R ∴R is transitive. Hence,R is reflexive and transitive but not symmetric. Q5 Check whether the relation R in R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive. Ans: R = {(a, b): a ≤ b3} It is observed that \begin{align} \left(\frac{1}{2},\frac{1}{2}\right) ∉ R , as \frac{1}{2}>\left(\frac{1}{2}\right)^3 = \frac{1}{8}\end{align} ∴ R is not reflexive. Now, (1, 2) ∈ R (as 1 < 23 = 8) But, (2, 1) ∉ R (as 23 > 1) ∴ R is not symmetric. We have \begin{align} \left(3,\frac{3}{2}\right),\left(\frac{3}{2},\frac{6}{5}\right) ∉ R , as 3>\left(\frac{3}{2}\right)^3 and \frac{3}{2}<\left(\frac{6}{5}\right)^3 \end{align} But \begin{align} \left(3,\frac{6}{5}\right) ∉ R , as 3>\left(\frac{6}{5}\right)^3 \end{align} ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. Q6 Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. Ans: Let A = {1, 2, 3}. A relation R on A is defined as R = {(1, 2), (2, 1)}. It is seen that (1, 1), (2, 2), (3, 3) ∉R. ∴ R is not reflexive. Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric. Now, (1, 2) and (2, 1) ∈ R However, (1, 1) ∉ R ∴ R is not transitive. Hence, R is symmetric but neither reflexive nor transitive. Q7 Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation. Ans: Set A is the set of all books in the library of a college. R = {x, y): x and y have the same number of pages} Now, R is reflexive since (x, x) ∈ R as x and x has the same number of pages. Let (x, y) ∈ R ⇒ x and y have the same number of pages. ⇒ y and x have the same number of pages. ⇒ (y, x) ∈ R ∴R is symmetric. Now, let (x, y) ∈R and (y, z) ∈ R. ⇒ x and y and have the same number of pages and y and z have the same number of pages. ⇒ x and z have the same number of pages. ⇒ (x, z) ∈ R ∴R is transitive. Hence, R is an equivalence relation. Q8 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}. Ans: A = {1, 2, 3, 4, 5} R = { (a,b) ; |a – b| is even} It is clear that for any element a ∈A, we have |a -a| = 0(which is even). ∴R is reflexive. Let (a, b) ∈ R. => |a –b| is even. => |- (a –b)| = |b - a| is also even. => (b, a) ∈ R is even. A = {1, 2, 3, 4, 5} R = { (a, b) : | a – b| is even} It is clear that for any element a ∈A, we have |a - a | = 0 (which is even). ∴R is reflexive. Let (a, b) ∈ R. ⇒ |a –b| is even. ⇒ |- (a –b)| = |b - a| is also even. ⇒ (b, a) ∈ R is even. ∴R is symmetric. Now, let (a, b) ∈ R and (b, c) ∈ R. ⇒ |a –b| is even and |(b –c)| is even. ⇒ (a – b) is even and (b –c ) is even. ⇒ (a –c ) = (a – b) + (b – c ) is even. [ Sum of two even integers is even] ⇒ |a – c | is even. ⇒ (a, c) ∈ R ∴R is transitive. Hence, R is an equivalence relation. Now, all elements of the set {1, 3, 5} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even. Similarly, all elements of the set {2, 4} are related to each other as all the elements of this subset are even. Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even. ∴R is symmetric. Now, let (a, b) ∈ R and (b, c) ∈ R. ⇒ |a –b| is even and |(b –c)| is even. ⇒ (a – b) is even and (b –c ) is even. ⇒ (a –c ) = (a – b) + (b – c ) is even. [ Sum of two even integers is even] ⇒ |a – c | is even. ⇒ (a, c) ∈ R ∴R is transitive. Hence, R is an equivalence relation. Now, all elements of the set {1, 3, 5} are related to each other as all the elements of this subset are odd. Thus, the modulus of the difference between any two elements will be even. Similarly, all elements of the set {2, 4} are related to each other as all the elements of this subset are even. Also, no element of the subset {1, 3, 5} can be related to any element of {2, 4} as all elements of {1, 3, 5} are odd and all elements of {2, 4} are even. Thus, the modulus of the difference between the two elements (from each of these two subsets) will not be even. Q9 Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by (i) R = { (a,b) : |a - b| is a multiple of 4} (ii) R = {(a,b):a = b} is an equivalence relation. Find the set of all elements related to 1 in each case. Ans: A = { x ∈ Z : 0 ≤ x ≤ 12} = {0,1,2,3,4,5,6,7,8,9,10,11,12} (i) R = { (a,b) : |a - b| is a multiple of 4} For any element a ∈A, we have (a, a) ∈ R as |a - a = 0|is a multiple of 4. ∴R is reflexive. Now, let (a, b) ∈ R ⇒ |a - b| is a multiple of 4. ⇒ |-(a - b)| = ⇒ |b - a| is a multiple of 4. ⇒ (b, a) ∈ R ∴R is symmetric. Now, let (a, b), (b, c) ∈ R. ⇒ |(a - b)| is a multiple of 4 and |(b - c)| is a multiple of 4. ⇒ (a - b) is a multiple of 4 and (b - c) is a multiple of 4. ⇒ (a - c) = (a – b) + (b – c) is a multiple of 4. ⇒ |a - c| is a multiple of 4. ⇒ (a, c) ∈R ∴ R is transitive. Hence, R is an equivalence relation. The set of elements related to 1 is {1, 5, 9} since |1 - 1| = 0 is a multiple of 4, |5 - 1| = 4 is a multiple of 4, and |9 - 1| = 8 is a multiple of 4. (ii) R = {(a, b): a = b} For any element a ∈A, we have (a, a) ∈ R, since a = a. ∴R is reflexive. Now, let (a, b) ∈ R. ⇒ a = b ⇒ b = a ⇒ (b, a) ∈ R ∴R is symmetric. Now, let (a, b) ∈ R and (b, c) ∈ R. ⇒ a = b and b = c ⇒ a = c ⇒ (a, c) ∈ R ∴ R is transitive. Hence, R is an equivalence relation. The elements in R that are related to 1 will be those elements from set A which are equal to 1. Hence, the set of elements related to 1 is {1}. Q10 Given an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symmetric and transitive but not reflexive. Ans: (i) Let A = {5, 6, 7}. Define a relation R on A as R = {(5, 6), (6, 5)}. Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R. Now, as (5, 6) ∈ R and also (6, 5) ∈ R, R is symmetric. => (5, 6), (6, 5) ∈ R, but (5, 5) ∉ R ∴R is not transitive. Hence, relation R is symmetric but not reflexive or transitive. (ii) Consider a relation R in R defined as: R = {(a, b): a < b} For any a ∈ R, we have (a, a) ∉ R since a cannot be strictly less than a itself. In fact, a = a. ∴ R is not reflexive. Now, (1, 2) ∈ R (as 1 < 2) But, 2 is not less than 1. ∴ (2, 1) ∉ R ∴ R is not symmetric. Now, let (a, b), (b, c) ∈ R. ⇒ a < b and b < c ⇒ a < c ⇒ (a, c) ∈ R ∴ R is transitive. Hence, relation R is transitive but not reflexive and symmetric. (iii) Let A = {4, 6, 8}. Define a relation R on A as: A = {(4, 4), (6, 6), (8, 8), (4, 6), (6, 4), (6, 8), (8, 6)} Relation R is reflexive since for every a ∈ A, (a, a) ∈R i.e., (4, 4), (6, 6), (8, 8)} ∈ R. Relation R is symmetric since (a, b) ∈ R ⇒ (b, a) ∈ R for all a, b ∈ R. Relation R is not transitive since (4, 6), (6, 8) ∈ R, but (4, 8) ∉ R. Hence, relation R is reflexive and symmetric but not transitive. (iv) Define a relation R in R as: R = {a, b): a3 ≥ b3} Clearly (a, a) ∈ R as a3 = a3. ∴ R is reflexive. Now, (2, 1) ∈ R (as 23 ≥ 13) But, (1, 2) ∉ R (as 13 < 23) ∴ R is not symmetric. Now, Let (a, b), (b, c) ∈ R. ⇒ a3 ≥ b3 and b3 ≥ c3 ⇒ a3 ≥ c3 ⇒ (a, c) ∈ R ∴ R is transitive. Hence, relation R is reflexive and transitive but not symmetric. (v) Let A = {−5, −6}. Define a relation R on A as: R = {(−5, −6), (−6, −5), (−5, −5)} Relation R is not reflexive as (−6, −6) ∉ R. Relation R is symmetric as (−5, −6) ∈ R and (−6, −5}∈R. It is seen that (−5, −6), (−6, −5) ∈ R. Also, (−5, −5) ∈ R. ∴ The relation R is transitive. Hence, relation R is symmetric and transitive but not reflexive. Q11 Show that the relation R in the set A of points in a plane given by R = {(P, Q): distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all point related to a point P ≠ (0, 0) is the circle passing through P with origin as centre. Ans: R = {(P, Q): distance of point P from the origin is the same as the distance of point Q from the origin} Clearly, (P, P) ∈ R since the distance of point P from the origin is always the same as the distance of the same point P from the origin. ∴R is reflexive. Now, Let (P, Q) ∈ R. ⇒ The distance of point P from the origin is the same as the distance of point Q from the origin. ⇒ The distance of point Q from the origin is the same as the distance of point P from the origin. ⇒ (Q, P) ∈ R ∴R is symmetric. Now, Let (P, Q), (Q, S) ∈ R. ⇒ The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same. ⇒ The distance of points P and S from the origin is the same. ⇒ (P, S) ∈ R ∴R is transitive. Therefore, R is an equivalence relation. The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin. In other words, if O (0, 0) is the origin and OP = k, then the set of all points related to P is at a distance of k from the origin. Hence, this set of points forms a circle with the centre as the origin and this circle passes through point P. Q12 Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related? Ans: R = {(T1, T2): T1 is similar to T2} R is reflexive since every triangle is similar to itself. Further, if (T1, T2) ∈ R, then T1 is similar to T2. ⇒ T2 is similar to T1. ⇒ (T2, T1) ∈R ∴R is symmetric. Now, Let (T1, T2), (T2, T3) ∈ R. ⇒ T1 is similar to T2 and T2 is similar to T3. ⇒ T1 is similar to T3. ⇒ (T1, T3) ∈ R ∴ R is transitive. Thus, R is an equivalence relation. Now, we can observe that: \begin{align} \frac {3}{6}=\frac {4}{8}=\frac {5}{10} = \left(\frac {1}{2}\right) \end{align} ∴The corresponding sides of triangles T1 and T3 are in the same ratio. Then, triangle T1 is similar to triangle T3. Hence, T1 is related to T3. Q13 Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5? Ans: R = {(P1, P2): P1 and P2 have same the number of sides} R is reflexive since (P1, P1) ∈ R as the same polygon has the same number of sides with itself. Let (P1, P2) ∈ R. ⇒ P1 and P2 have the same number of sides. ⇒ P2 and P1 have the same number of sides. ⇒ (P2, P1) ∈ R ∴R is symmetric. Now, Let (P1, P2), (P2, P3) ∈ R. ⇒ P1 and P2 have the same number of sides. Also, P2 and P3 have the same number of sides. ⇒ P1 and P3 have the same number of sides. ⇒ (P1, P3) ∈ R ∴R is transitive. Hence, R is an equivalence relation. The elements in A related to the right-angled triangle (T) with sides 3, 4, and 5 are those polygons which have 3 sides (since T is a polygon with 3 sides). Hence, the set of all elements in A related to triangle T is the set of all triangles. Q14 Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4. Ans: R = {(L1, L2): L1 is parallel to L2} R is reflexive as any line L1 is parallel to itself i.e., (L1, L1) ∈ R. Now, Let (L1, L2) ∈ R. ⇒ L1 is parallel to L2. ⇒ L2 is parallel to L1. ⇒ (L2, L1) ∈ R ∴ R is symmetric. Now, Let (L1, L2), (L2, L3) ∈R. ⇒ L1 is parallel to L2. Also, L2 is parallel to L3. ⇒ L1 is parallel to L3. ∴R is transitive. Hence, R is an equivalence relation. The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4. Slope of line y = 2x + 4 is m = 2 It is known that parallel lines have the same slopes. The line parallel to the given line is of the form y = 2x + c, where c ∈R. Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R. Q15 Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer. (A) R is reflexive and symmetric but not transitive. (B) R is reflexive and transitive but not symmetric. (C) R is symmetric and transitive but not reflexive. (D) R is an equivalence relation. Ans: R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} It is seen that (a, a) ∈ R, for every a ∈{1, 2, 3, 4}. ∴ R is reflexive. It is seen that (1, 2) ∈ R, but (2, 1) ∉ R. ∴R is not symmetric. Also, it is observed that (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R for all a, b, c ∈ {1, 2, 3, 4}. ∴ R is transitive. Hence, R is reflexive and transitive but not symmetric. The correct answer is B. Q16 Let R be the relation in the set N given by R = {(a, b): a = b − 2, b > 6}. Choose the correct answer. (A) (2, 4) ∈ R (B) (3, 8) ∈R (C) (6, 8) ∈R (D) (8, 7) ∈ R Ans: R = {(a, b): a = b − 2, b > 6} Now, since b > 6, (2, 4) ∉ R Also, as 3 ≠ 8 − 2, (3, 8) ∉ R And, as 8 ≠ 7 − 2 ∴(8, 7) ∉ R Now, consider (6, 8). We have 8 > 6 and also, 6 = 8 − 2. ∴(6, 8) ∈ R The correct answer is C.
Exercise 2
• Q1 Show that the function f : R* → R* defined by f(x) = 1/x is one-one and onto,where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R* ? Ans: It is given that f: R* → R* is defined by One-one: ∴f is one-one. Onto: It is clear that for y∈R*, there exists such that ∴f is onto. Thus, the given function (f) is one-one and onto. Now, consider function g: N → R* defined by We have, ∴g is one-one. Further, it is clear that g is not onto as for 1.2 ∈R* there does not exit any x in N such that g(x) =. Hence, function g is one-one but not onto. Q2 Check the injectivity and surjectivity of the following functions: (i) f : N → N given by f(x) = x2 (ii) f : Z → Z given by f(x) = x2 (iii) f : R → R given by f(x) = x2 (iv) f : N → N given by f(x) = x3 (v) f : Z → Z given by f(x) = x3 Ans: (i) f: N → N is given by, f(x) = x2 It is seen that for x, y ∈N, f(x) = f(y) ⇒ x2 = y2 ⇒ x = y. ∴f is injective. Now, 2 ∈ N. But, there does not exist any x in N such that f(x) = x2 = 2. ∴ f is not surjective. Hence, function f is injective but not surjective. (ii) f: Z → Z is given by, f(x) = x2 It is seen that f(-1) = f(1) = 1, but -1 ≠ 1. ∴ f is not injective. Now,-2 ∈ Z. But, there does not exist any element x ∈Z such that f(x) = x2 = -2. ∴ f is not surjective. Hence, function f is neither injective nor surjective. (iii) f: R → R is given by, f(x) = x2 It is seen that f(-1) = f(1) = 1, but -1 ≠ 1. ∴ f is not injective. Now,-2 ∈ R. But, there does not exist any element x ∈ R such that f(x) = x2 = -2. ∴ f is not surjective. Hence, function f is neither injective nor surjective. (iv) f: N → N given by, f(x) = x3 It is seen that for x, y ∈N, f(x) = f(y) ⇒ x3 = y3 ⇒ x = y. ∴f is injective. Now, 2 ∈ N. But, there does not exist any element x in domain N such that f(x) = x3 = 2. ∴ f is not surjective Hence, function f is injective but not surjective. (v) f: Z → Z is given by, f(x) = x3 It is seen that for x, y ∈ Z, f(x) = f(y) ⇒ x3 = y3 ⇒ x = y. ∴ f is injective. Now, 2 ∈ Z. But, there does not exist any element x in domain Z such that f(x) = x3 = 2. ∴ f is not surjective. Hence, function f is injective but not surjective. Q3 Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. Ans: f: R → R is given by, f(x) = [x] It is seen that f(1.2) = [1.2] = 1, f(1.9) = [1.9] = 1. ∴ f(1.2) = f(1.9), but 1.2 ≠ 1.9. ∴ f is not one-one. Now, consider 0.7 ∈ R. It is known that f(x) = [x] is always an integer. Thus, there does not exist any element x ∈ R such that f(x) = 0.7. ∴ f is not onto. Hence, the greatest integer function is neither one-one nor onto. Q4 Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. Ans: f : R → R is given by, It is seen that. ∴f( - 1) = f(1), but - 1 ≠ 1. ∴ f is not one-one. Now, consider - 1 ∈ R. It is known that f(x) = |x| is always non-negative. Thus, there does not exist any element x in domain R such that f(x) = |x| = - 1. ∴ f is not onto. Hence, the modulus function is neither one-one nor onto. Q5 Show that the Signum Function f : R → R, given by is neither one-one nor onto. Ans: f : R → R, given by It is seen that f(1) = f(2) = 1, but 1 ≠ 2. ∴f is not one-one. Now, as f(x) takes only 3 values (1, 0, or - 1) for the element - 2 in co-domain R, there does not exist any x in domain R such that f(x) = - 2. ∴ f is not onto. Hence, the signum function is neither one-one nor onto. Q6 Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one. Ans: It is given that A = {1, 2, 3}, B = {4, 5, 6, 7}. f: A → B is defined as f = {(1, 4), (2, 5), (3, 6)}. ∴ f (1) = 4, f (2) = 5, f (3) = 6 It is seen that the images of distinct elements of A under f are distinct. Hence, function f is one-one. Q7 In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) f : R → R defined by f(x) = 3 – 4x (ii) f : R → R defined by f(x) = 1 + x2 Ans: (i) f: R → R is defined as f(x) = 3 - 4x. . ∴ f is one-one. For any real number (y) in R, there existsin R such that ∴f is onto. Hence, f is bijective. (ii) f: R → R is defined as . . ∴does not imply that x1 = x2 For instance, ∴ f is not one-one. Consider an element - 2 in co-domain R. It is seen thatis positive for all x ∈ R. Thus, there does not exist any x in domain R such that f(x) = - 2. ∴ f is not onto. Hence, f is neither one-one nor onto. Q8 Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function. Ans: f: A × B → B × A is defined as f(a, b) = (b, a). . ∴ f is one-one. Now, let (b, a) ∈ B × A be any element. Then, there exists (a, b) ∈A × B such that f(a, b) = (b, a). [By definition of f] ∴ f is onto. Hence,f is bijective. Q9 Let f : N → N be defined by State whether the function f is bijective. Justify your answer. Ans: f: N → N is defined as It can be observed that: ∴ f is not one-one. Consider a natural number (n) in co-domain N. Case I: n is odd ∴n = 2r + 1 for some r ∈ N. Then, there exists 4r + 1∈N such that . Case II: n is even ∴n = 2r for some r ∈ N. Then,there exists 4r ∈N such that. ∴ f is onto. Hence, f is not a bijective function Q10 Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by . Is f one-one and onto? Justify your answer. Ans: A = R - {3}, B = R - {1} f: A → B is defined as. . ∴ f is one-one. Let y ∈B = R - {1}. Then, y ≠ 1. The function f is onto if there exists x ∈A such that f(x) = y. Now, Thus, for any y ∈ B, there existssuch that Hence, function f is one-one and onto. Q11 Let f : R → R be defined as f(x) = x4. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto. Ans: f: R → R is defined as f(x) = x4 Let x, y ∈ R such that f(x) = f(y). ∴does not imply that x1=x2. For instance, f(1) = f(-1) = 1 ∴ f is not one-one. Consider an element 2 in co-domain R. It is clear that there does not exist any x in domain R such that f(x) = 2. ∴ f is not onto. Hence, function f is neither one-one nor onto. The correct answer is D. Q12 Let f : R → R be defined as f(x) = 3x. Choose the correct answer. (A) f is one-one onto (B) f is many-one onto (C) f is one-one but not onto (D) f is neither one-one nor onto. Ans: f: R → R defined as f(x) = 3x. Let x, y ∈ R such that f(x) = f(y). ⇒ 3x = 3y ⇒ x = y ∴f is one-one. Also, for any real number (y) in co-domain R, there exists in R such that. ∴f is onto. Hence, function f is one-one and onto. The correct answer is A.
Exercise 3
• Q1 Let f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. Ans: The functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} are defined as f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Q2 Let f, g and h be functions from R to R. Show that (f + g)oh = foh + goh (f . g)oh = (foh) . (goh) Ans: To prove: (f + g)oh = foh + goh To Prove: (f . g)oh = (foh) . (goh) Hence, (f . g)oh = (foh) . (goh) Q3 Find gof and fog, if (i) f(x) = | x | and g(x) = | 5x – 2 | (ii) f(x) = 8x3 and g(x) = x1/3 . Ans: (i) f(x) = | x | and g(x) = | 5x – 2 | (ii) f(x) = 8x3 and g(x) = x1/3 Q4 If f(x) = , show that fof(x) = x, for all x ≠ 2/3. What is the inverse of f ? Ans: It is given that. Hence, the given function f is invertible and the inverse of f is f itself. Q5 State with reason whether following functions have inverse (i) f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)} (ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)} (iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)} Ans: (i) f: {1, 2, 3, 4} → {10} defined as: f = {(1, 10), (2, 10), (3, 10), (4, 10)} From the given definition of f, we can see that f is a many one function as: f(1) = f(2) = f(3) = f(4) = 10 ∴f is not one-one. Hence, function f does not have an inverse. (ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} defined as: g = {(5, 4), (6, 3), (7, 4), (8, 2)} From the given definition of g, it is seen that g is a many one function as: g(5) = g(7) = 4. ∴g is not one-one, Hence, function g does not have an inverse. (iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} defined as: h = {(2, 7), (3, 9), (4, 11), (5, 13)} It is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under h. ∴Function h is one-one. Also, h is onto since for every element y of the set {7, 9, 11, 13}, there exists an element x in the set {2, 3, 4, 5}such that h(x) = y. Thus, h is a one-one and onto function. Hence, h has an inverse. Q6 Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f. (Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,) Ans: f: [ - 1, 1] → R is given as Let f(x) = f(y). ∴ f is a one-one function. It is clear that f: [ - 1, 1] → Range f is onto. ∴ f: [ - 1, 1]→ Range f is one-one and onto and therefore, the inverse of the function: f: [ - 1, 1] → Range f exists. Let g: Range f → [ - 1, 1] be the inverse of f. Let y be an arbitrary element of range f. Since f: [ - 1, 1] → Range f is onto, we have: Now, let us define g: Range f → [ - 1, 1] as ∴gof =I[-1, 1]and fog = IRange f ∴ f - 1 = g ⇒ Q7 Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f. Ans: f: R → R is given by, f(x) = 4x + 3 One-one: Let f(x) = f(y). ∴ f is a one-one function. Onto: For y ∈ R, let y = 4x + 3. Therefore, for any y ∈ R, such that ∴ f is onto. Thus, f is one-one and onto and therefore, f - 1 exists. Let us define g: R→ R by. ∴ Hence, f is invertible and the inverse of f is given by Q8 Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by , where R+ is the set of all non-negative real numbers. Ans: f : R+ → [4, ∞) is given as f(x) = x2 + 4. One-one: Let f(x) = f(y). ∴ f is a one-one function. Onto: For y ∈ [4, ∞), let y = x2 + 4. Therefore, for any y ∈ R, there exists such that . ∴ f is onto. Thus, f is one-one and onto and therefore, f - 1 exists. Let us define g: [4, ∞) → R+ by, ∴ Hence, f is invertible and the inverse of f is given by Q9 Consider f : R+ → [– 5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible with . Ans: f: R+ → [ - 5, ∞) is given as f(x) = 9x2 + 6x - 5. Let y be an arbitrary element of [ - 5, ∞). Let y = 9x2 + 6x - 5. ∴f is onto, thereby range f = [ - 5, ∞). Let us define g: [ - 5, ∞) → R+ as We now have: ∴ gof = IR+ and Hence, f is invertible and the inverse of f is given by Q10 Let f : X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f). Ans: Let f: X → Y be an invertible function. Also, suppose f has two inverses (say g1 and g2). Then, for all y ∈Y, we have: Hence, f has a unique inverse. Q11 Consider f : {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f –1 and show that (f –1)–1 = f. Ans: Function f: {1, 2, 3} → {a, b, c} is given by, f(1) = a, f(2) = b, and f(3) = c If we define g: {a, b, c} → {1, 2, 3} as g(a) = 1, g(b) = 2, g(c) = 3, then we have: gof = Ix and fog = Iy where X = {1, 2, 3} and Y= {a, b, c}. Thus, the inverse of f exists and f - 1 = g. ∴f - 1: {a, b, c} → {1, 2, 3} is given by, f - 1(a) = 1, f - 1(b) = 2, f-1(c) = 3 Let us now find the inverse of f - 1 i.e., find the inverse of g. If we define h: {1, 2, 3} → {a, b, c} as h(1) = a, h(2) = b, h(3) = c, then we have: ∴, where X = {1, 2, 3} and Y = {a, b, c}. Thus, the inverse of g exists and g - 1 = h ⇒ (f - 1) - 1 = h. It can be noted that h = f. Hence, (f - 1) - 1 = f. Q12 Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f. Ans: Let f: X → Y be an invertible function. Then, there exists a function g: Y → X such that gof = Ix and fog = Iy Here, f-1 = g. Now, gof = Ix and fog = Iy ⇒ f-1of = Ix and fof-1= Iy Hence, f-1: Y → X is invertible and f is the inverse of f-1 i.e., (f-1)-1 = f. Q13 If f: R → R be given by f(x) = , then fof(x) is (A) (B) x3 (C) x (D) (3 – x3). Ans: : R → R is given as. The correct answer is C. Q14 Letbe a function defined as. The inverse of f is map g: Range (A) (B) (C) (D) Ans: It is given that is defined as Let y be an arbitrary element of Range f. Then, there exists x ∈such that y = f(x) Let us define g: Rangeas Now, ∴ Thus, g is the inverse of f i.e., f - 1 = g. Hence, the inverse of f is the map g: Range, which is given by . The correct answer is B.
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https://www.physicsforums.com/threads/is-the-natural-behavior-of-matter-to-lose-energy.795535/ | Is the natural behavior of matter to lose energy?
Tags:
1. Feb 2, 2015
Goran
Hello everyone this is my first post here. I'm sorry if my spelling is terrible.
I am no physicists so I would like to ask if my thinking was going in the right direction.
When I say energy I really mean electromagnetic radiation. The earth receives it constantly from the sun but there is no constant increase of the earth's temperature. Presumably because it's radiated in to the cosmos. I also know that all matter radiates depending on it's temperature.
Know that is leading me to think that in a closed system with no input of radiated energy all the matter would drop down to the lowest energy state?
Also I am interested to now if that is true then what implications does it make on Newtons first law of motion (also I have no idea if that law holds in general relativity)? Is it possible that a body flying trough proposed closed system will not change it's speed as a whole but the individual atoms would drop to 0 K. Sounds logical but then again not really, just more questions.
2. Feb 2, 2015
Staff: Mentor
If the system is closed, then no energy can escape. Over time, everything would reach the same temperature. If you have a lot of empty space in your system, this temperature can be very low, but it won't be zero. If you put an object into space far away from stars, it will reach a temperature of about 3 K, then it is in equilibrium between its own outgoing radiation and the incoming radiation.
None. Temperature is related to unordered motion within the object, the net motion of the object as a whole has nothing to do with temperature.
3. Feb 2, 2015
Goran
Thank you for replaying (and moving the thread, that was my original intention :) ).
So I'm correct in assuming that matter will lose energy if there is no incoming radiation ?
I get that energy can't escape a closed system I just didn't considered that some of that radiation might come back. But that is a consequence of the system being finite?
If it was infinite original matters temperature should converge to 0 K ?
4. Feb 2, 2015
Staff: Mentor
If it has a temperature above 0 K, yes.
Right.
Not if you have a non-zero energy density everywhere, like we have in space today (mainly from the cosmic microwave background).
5. Feb 8, 2015
ezfzx
Read up a little bit on the Law of Entropy ... 2nd Law of Thermodynamics. :)
6. Feb 8, 2015
haruspex
Not so fast. You appear to be thinking of black body radiation. That's an idealised circumstance.
In practice, emission / absorption of electromagnetic radiation requires the acceleration of a charge. Diatomic gases in the atmosphere do not readily do either at their ambient temperatures.
Each molecule is electrically neutral, so there's no net acceleration of charge during molecular collisions.
The atoms within the molecule are also electrically neutral, so there's no net charge acceleration in the spinning of the molecule or in the internal vibrations.
To get these molecules involved in EMR, you need to go to ionising energy levels. These are higher than visible light energies. This explains why the atmosphere is clear and, in turn, forms part of the explanation of why our eyes are sensitive to the frequency ranges that they are. (No point in being sensitive to frequencies that don't penetrate the atmosphere.)
Triatomic gases, such as H2O and CO2, are another matter. They have vibrational modes which do involve charge acceleration. This is why they can absorb and emit certain bands within the IR range.
Metals have free electrons. These can easily undergo accelerations which interact with any frequency. Of course, the interaction often consists of reflection, though.
The surface of a planet is generally such a mishmash of complex molecules that it can be largely treated as a 'black body', absorbing anything, and correspondingly emitting according to whatever thermal energy it has.
To add a little to mfb's answer, emission and absorption are highly symmetric. The facility with which a given substance at a given temperature emits EMR of a given frequency equals that with which it would absorb the same frequency. 'Emissivity' covers both. That is why a steady state can be reached in which the bodies emit EMR in exactly the same power distribution as they absorb it from the ambient space. (But note that if they are distinctly not black bodies, the bodies could theoretically settle at different temperatures. You could have one subset only emitting and absorbing in one narrow band, with another only doing so in a separate narrow band.)
7. Feb 8, 2015
Staff: Mentor
Then you would need exactly zero emissivity over some large range - you won't get that. Not even the diatomic or even monoatomic gases have that. They still absorb and emit a tiny bit of radiation based on collisions.
8. Feb 8, 2015
haruspex
Yes, that's true... the rate of interchange could be very low, but it won't vanish.
9. Feb 10, 2015
Jewish_Vulcan
Does entropy state that the amount of unusable energy increases naturally in the Universe? My physics book did a bad job explaining it for me.
Last edited: Feb 10, 2015
10. Feb 10, 2015
Staff: Mentor
unsable is one letter away from both unusable and usable.
The amount of unusable energy increases, neglecting some technical details.
11. Feb 10, 2015
Jewish_Vulcan
Right, thx.
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https://link.springer.com/article/10.1007/s10287-020-00368-3 | ## Introduction
The standard unit commitment (UC) problem for power systems operations involves determining which thermal generators should be scheduled to meet projected demand for power over a given time horizon, while ensuring physical and operational constraints are satisfied. The time horizon is typically one or two days at an hourly resolution. Feasible operating constraints for thermal generators include limits on ramping, minimum up and down times, startup and shutdown ramp rates, and minimum and maximum production levels. The UC problem is extensively studied in the optimization and power systems literature (Anjos and Conejo 2017; Damcı-Kurt et al. 2016; Ostrowski et al. 2015; Padhy 2004; Queyranne and Wolsey 2017; Silbernagl 2016). It can be formulated as a mixed-integer linear program (MILP) and solved with commercial branch-and-cut software packages (Knueven et al. 2020, 2018b; Morales-España et al. 2013; O’Neill 2017). Prior to recent advances in MILP solver technology and UC formulations, even modest instances were computationally challenging to solve in time limits required by operations. Consequently, techniques such as Lagrangian decomposition (Borghetti et al. 2003), Benders decomposition (Wu and Shahidehpour 2010), and metaheuristics (Kazarlis et al. 1996) were previously employed. Presently, MILP solvers are regularly employed by Independent System Operators (ISOs) and Vertically Integrated Utilities to solve UC instances for real time operations; see, e.g. O’Neill (2017) and Ott (2010).
The standard UC model is implicitly a deterministic MILP, with inherently uncertain parameters such as demand being populated with their forecasted “point” quantities, with potential deviations from forecasted quantities being addressed by reserve margins. With the growing deployment of renewable energy sources, particularly wind and solar generators, there is increasing interest in explicitly treating uncertain aspects of the UC problem—yielding stochastic UC models; see, e.g., van Ackooij et al. (2018), Wang et al. (2012) and Zhao et al. (2013). Takriti et al. (2000) present a stochastic single-UC formulation with uncertain electricity prices, but from the perspective of an individual generator participating in a market. Uncertain demand is also a factor driving interest in stochastic UC formulations; see, e.g., Ozturk et al. (2004).
Stochastic UC is commonly formulated as an extensive stochastic program; see, e.g., Birge and Louveaux (2011) for an introduction to this decision formalism. Standard stochastic programming models assume that uncertainty is captured via a finite set of discrete scenarios, and that any solution to the model ensures feasibility of all constraints in all scenarios. A less stringent approach is to assume that some constraints can be violated with a small probability, resulting in chance constrained stochastic programming models. Due to their flexibility and explicit acknowledgment that certain constraints may be violated while ensuring reliability, chance constrained models are frequently found in the stochastic UC literature; see, e.g., Pozo and Contreras (2013), Singh et al. (2018), Wang et al. (2012) and Wu et al. (2014). Chance constrained models are seldom solved using a naive extensive formulation; see (Ozturk et al. 2004) for an approximation using iteratively updated union bounds, Pozo and Contreras (2013) for an approximation using conditional value-at-risk, Zhao et al. (2014) for a Monte Carlo sample average approximation, and Watson et al. (2010) for decomposition methods.
Operational limits on thermal generators are key inputs to both deterministic and stochastic UC models. Often, these limits are based on engineering judgment or economic considerations, rather than physical limitations. For example, minimum power outputs are commonly set to levels such that lower production levels are not profitable for operators, while maximum power outputs can be exceeded in practice for short periods of time to ensure reliable system operation. Consequently, system operators can, and do, run thermal generators beyond these limits (Kraemer 2013). We further note that such limits are also explicitly recognized by operators as “soft” constraints (Blumsack 2018), reinforcing the notion that they can be occasionally violated in practice. ISOs within the US, such as MISO (2018) and System Operations Division, PJM (2019), have mechanisms in place to allow for infrequent exceedance of normal production (i.e., dispatch) limits. Such mechanisms are different from spinning reserves and other ancillary service products in that they are typically held back for when net-load is significantly different from the forecasted quantity, whereas ancillary services are commonly dispatched to address contingencies (generator and/or transmission) and more typical deviations from net-load forecasts.
In principle, a stochastic programming model for UC can address anticipated variations in net-load (Wang and Hobbs 2014). For reserve products, market operators typically procure sufficient generation to cover a $$(1-\varepsilon )$$ proportion of possible net-load realizations (Cornelius 2014). Compared to holding energy in reserve, stochastic programming can yield overly conservative schedules by scheduling sufficient generation to cover all—or a very large range of—possible load realizations. Our proposed stochastic UC model yields a method for incorporating the practical consideration of procuring sufficient generation to address a large proportion of possible net-load realizations into stochastic UC, while allowing for a currently-used recourse action (emergency capacity) if a low probability net-load realization is not covered under normal operations.
To the best of our knowledge, previous work has not analyzed the effect of periodically violating these soft operational constraints associated with thermal generators, in support of either cost reductions or system reliability. We acknowledge that a thermal generator should not be operated outside its prescribed engineering and/or economic limits for significant amounts of time, as this would likely incur increased maintenance costs and lifetime reductions (Dahal and Chakpitak 2007). Further, running thermal generators beyond their prescribed ratings is not fuel efficient (Knudsen et al. 2017). However, occasional violations may not significantly inflate maintenance costs or reduce lifetime, and may yield significant overall operational cost savings. Because of the scales involved, even a 1% savings in energy production costs is worth more than $10 billion per year globally, and more than$1 billion per year in the U.S. alone (O’Neill 2007). Thus, it is important to understand the trade-offs between the frequency and magnitude of these violations and any potential cost savings.
Here, we explicitly allow thermal generators to occasionally produce beyond their prescribed minimum and maximum ratings, while simultaneously limiting the extent of violations. We informally refer to the operation of a thermal generator in this mode as an “non-nominality”. We limit the number of these non-nominalities to a small quantity using a chance constraint, in the broader context of a stochastic UC formulation. Unlike traditional chance-constrained stochastic programming models where violations are generally unbounded, thermal generators are still restricted by their absolute minimum and maximum ratings. As these absolute maximum ratings are typically proprietary information and consequently unknown to system operators, we take the absolute ratings to be a few percent greater than the prescribed ratings. Thus, unlike a traditional chance constraint where the magnitude of violations could be unrestricted, in our work the non-nominalities qualify as (tightly) bounded violations.
• We present a mathematical formulation for a chance-constrained stochastic UC model where thermal generators are allowed to produce modestly beyond their technical minimum and maximum ratings with a small probability, in order to address discrepancies between forecasted and actual net-load.
• We show that an extensive form of the resulting chance-constrained stochastic UC model can be solved directly by commercial MILP branch-and-cut solvers for modestly sized problem instances.
• We demonstrate significant operational cost savings (on the order of $$\approx$$ 1%) can be obtained via infrequent and modest exceedance of nominal thermal generator production limits.
• We analyze the structure of optimal solutions to our chance-constrained UC model and assess the generalizability of our results to distinct test cases and operational restrictions.
The remainder of this paper is organized as follows. In Sect. 2, we present our chance-constrained stochastic UC model. In Sect. 3, we describe the data and sources associated with two different UC test cases that serve as the basis for our computational experiments. Our experimental results are detailed in Sect. 4, and include a sensitivity analysis to the parameters associated with the limit violation’s frequency and magnitude. We conclude in Sect. 5 with a summary of our contributions and plans for future work.
## Chance-constrained stochastic unit commitment formulation
We now present our chance-constrained stochastic unit commitment formulation. We begin in Sect. 2.1 by introducing the notation, and then formally describe the mathematical programming model in Sect. 2.2.
### Notation
Indices and Sets
$$g \in \mathcal {G}$$ :
Thermal generators.
$$t \in \mathcal {T}$$ :
Hourly time steps: $$1, \ldots , T$$; i.e., $$[a, b) \in \mathcal {T}\times \mathcal {T}$$ such that $$b \ge a + {UT}^g$$.
$$l \in \mathcal {L}^g$$ :
Piecewise production cost intervals for generator g: $$1, \ldots , {L}_g$$.
$$s \in \mathcal {S}^g$$ :
Start-up categories for generator g, from hottest (1) to coldest ($$S_g$$).
$$\omega \in \varOmega$$ :
Scenarios: $$\omega _1, \ldots , \omega _N$$.
Parameters: First Stage
$$C^{l,g}$$ :
Marginal cost for piecewise segment l for generator g ($/MWh). $${\overline{C}}^g$$ : Marginal cost for production above $${\overline{P}}^g$$ ($/MWh).
$${\underline{C}}^g$$ :
Marginal cost for production below $${\underline{P}}^g$$ ($/MWh). $$C^{R,g}$$ : Cost of generator g running and operating at minimum production $${\underline{P}}_g$$ ($/h).
$$C^{s,g}$$ :
Start-up cost of category s for generator g ($). $${DT}^g$$ : Minimum down time for generator g (h). $${\overline{P}}^g$$ : Maximum power output for generator g under normal operations (MW). $${\overline{\overline{P}}}^g$$ : Maximum power output for generator g under non-nominal operations (MW). $${\underline{P}}^g$$ : Minimum power output for generator g under normal operations (MW). $${\underline{\underline{P}}}^g$$ : Minimum power output for generator g under non-nominal operations (MW). $${\overline{P}}^{l,g}$$ : Maximum power available for piecewise segment l for generator g (MW) (with $${\overline{P}}^{0,g} = {\underline{P}}^g$$). $${RD}^g$$ : Ramp-down rate for generator g (MW/h). $${RU}^g$$ : Ramp-up rate for generator g (MW/h). $${SD}^g$$ : Shutdown ramp rate for generator g (MW/h). $${SU}^g$$ : Start-up ramp rate for generator g (MW/h). $${TC}^g$$ : Time down after which generator g goes cold (h). $${\underline{T}}^{s,g}$$ : Time offline after which the start-up category s is available (h) (with $${\underline{T}}^{1,g} = {DT}^g$$, $${\underline{T}}^{S^g,g} = {TC}^g$$) $${UT}^g$$ : Minimum up time for generator g (h). Parameters: Second Stage $${D}_t^{\omega }$$ : Load (demand) at time t in scenario $$\omega$$ (MW). $${\overline{W}}_t^{\omega }$$ : Maximum power from renewables at time t in scenario $$\omega$$ (MW). $${\underline{W}}_t^{\omega }$$ : Minimum power from renewables at time t in scenario $$\omega$$ (MW). Variables: First Stage $$c^{SU,g}_t$$ : Start-up cost of generator g at time t ($).
$$u^g_t$$ :
Commitment status of generator g at time t, $$\in \{0,1\}$$.
$$v^g_t$$ :
Start-up status of generator g at time t, $$\in \{0,1\}$$.
$$w^g_t$$ :
Shutdown status of generator g at time t, $$\in \{0,1\}$$.
$$x^g_{[t,t')}$$ :
Indicator arc for shutdown at time t, start-up at time $$t'$$, uncommitted for $$i \in [t, t')$$, for generator g, $$\in \{0,1\}$$, $$[t,t')$$ such that $$t + {DT}^g \le t' \le t+{TC}^g -1$$.
Variables: Second Stage
$$p_t^{g,\omega }$$ :
Power above minimum from generator g at time t in scenario $$\omega$$ (MW).
$${\overline{p}}_t^{g,\omega }$$ :
Power above maximum from generator g at time t in scenario $$\omega$$ (MW).
$${\underline{p}}_t^{g,\omega }$$ :
Power below minimum from generator g at time t in scenario $$\omega$$ (MW).
$$p^{l,g,\omega }_t$$ :
Power from piecewise interval l for generator g at time t in scenario $$\omega$$ (MW).
$$r^{n,\omega }_t$$ :
Power from renewables at time t in scenario $$\omega$$ (MW).
$$y^{g,\omega }_t$$ :
Non-nominal operation status of generator g at time t in scenario (MW).
### Mathematical programming model
We assume that the generator operating cost is increasing, piecewise linear and convex in $$p_t^{g,\omega }$$ under normal operating conditions. Further, let $$\mathcal {G}^{>1} = \{ g \in \mathcal {G}\mid {UT}^g > 1 \}$$ and $$\mathcal {G}^1 = \{g \in \mathcal {G}\mid {UT}^g = 1 \}$$; i.e., $$\mathcal {G}^{>1}$$ and $$\mathcal {G}^1$$ denote the set of generators with an uptime greater than one and equal to one, respectively. We use the so-called “3-bin” formulation from (Knueven et al. 2020; Morales-España et al. 2013), with the ramping constraints from (Damcı-Kurt et al. 2016).
\begin{aligned} \text {min} \sum _{g \in \mathcal {G}} \sum _{t \in \mathcal {T}} \left( \sum _{l \in \mathcal {L}^g} \mathbb {E}[C^{l,g} \, p^{l,g,\omega }_t + {\overline{C}}^{g} \, {\overline{p}}^{g,\omega }_t + {\underline{C}}^g \, {\underline{p}}^{g,\omega }] \> + C^{R,g} \, u^g_t + c^{SU,g}_t \right) \end{aligned}
(1)
subject to:
\begin{aligned} u^g_t - u^g_{t-1}&= v^g_t - w^g_t \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G} \end{aligned}
(2a)
\begin{aligned} \sum _{i= t-{UT}^g + 1}^t v^g_i&\le u^g_t \qquad \qquad \forall t \in [{UT}^g, T], \, \forall g \in \mathcal {G} \end{aligned}
(2b)
\begin{aligned} \sum _{i= t-{DT}^g + 1}^t w^g_i&\le 1 - u^g_t \qquad \qquad \forall t \in [{DT}^g, T], \> \, \forall g \in \mathcal {G} \end{aligned}
(2c)
\begin{aligned} \sum _{t'= t-{TC}^g+1}^{t-{DT}^g} x^g_{[t',t)}&\le v^g_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G} \end{aligned}
(2d)
\begin{aligned} \sum _{t' = t+{DT}^g}^{t+{TC}^g-1} x^g_{[t,t')}&\le w^g_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}\end{aligned}
(2e)
\begin{aligned} c^{SU, g}_t&= C^{S,g} v^g_t + \sum _{s=1}^{S^g-1} (C^{s,g} - C^{S,g}) \left( \sum _{ t' = t-{\underline{T}}^{s+1,g}+1 }^{t-{\underline{T}}^{s,g}} x^g_{[t',t)} \right) \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G} \end{aligned}
(2f)
\begin{aligned} p_t^{g,\omega }&\le ({\overline{P}}^g-{\underline{P}}^g) u^g_t - ({\overline{P}}^g - {SU}^g) v^g_t - ({\overline{P}}^g - {SD}^g) w^g_{t+1} \qquad \qquad \forall t \in \mathcal {T}, \forall g \in \mathcal {G}^{>1}, \forall \omega \in \varOmega \end{aligned}
(3a)
\begin{aligned} p_t^{g,\omega }&\le ({\overline{P}}^g-{\underline{P}}^g) u^g_t - ({\overline{P}}^g - {SU}^g) v^g_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}^1, \, \forall \omega \in \varOmega \end{aligned}
(3b)
\begin{aligned} p_t^{g,\omega }&\le ({\overline{P}}^g-{\underline{P}}^g) u^g_t - ({\overline{P}}^g - {SD}^g) w^g_{t+1} \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}^1, \, \forall \omega \in \varOmega \end{aligned}
(3c)
\begin{aligned} p_t^{g,\omega } - p^{g, \omega }_{t-1}&\le ({SU}^g - {RU}^g - {\underline{P}}^g) v^g_t + {RU}^g u^g_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(3d)
\begin{aligned} p^{g, \omega }_{t-1} - p_t^{g,\omega }&\le ({SD}^g - {RD}^g - {\underline{P}}^g) w^g_t + {RD}^g u^g_{t-1} \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(3e)
\begin{aligned} p_t^{g,\omega }&= \sum _{l \in \mathcal {L}^g} p^{l,g,\omega }_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(3f)
\begin{aligned} p^{l,g,\omega }_t&\le ({\overline{P}}^{l,g}- {\overline{P}}^{l-1,g})u^g_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall l \in \mathcal {L}^g, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(3g)
\begin{aligned} y^{g,\omega }_t&\le u^g_t - v^g_t - w^g_{t+1} \qquad \qquad \forall t \in \mathcal {T}, \forall g \in \mathcal {G}^{>1}, \forall \omega \in \varOmega \end{aligned}
(4a)
\begin{aligned} y^{g,\omega }_t&\le u^g_t - v^g_t \qquad \qquad \forall t \in \mathcal {T}, \forall g \in \mathcal {G}^1, \forall \omega \in \varOmega \end{aligned}
(4b)
\begin{aligned} y^{g,\omega }_t&\le u^g_t - w^g_{t+1} \qquad \qquad \forall t \in \mathcal {T}, \forall g \in \mathcal {G}^1, \forall \omega \in \varOmega \end{aligned}
(4c)
\begin{aligned} {\overline{p}}^{g, \omega }_t&\le \left( {\overline{\overline{P}}}- {\overline{P}}\right) y^{g,\omega }_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(4d)
\begin{aligned} {\underline{p}}^{g,\omega }_t&\le \left( {\underline{P}}- {\underline{\underline{P}}}\right) y^{g, \omega }_t \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(4e)
\begin{aligned}&\sum _{g \in \mathcal {G}} \left( p_t^{g,\omega } + {\overline{p}}^{g,\omega }_t -{\underline{p}}^{g,\omega }_t + {\underline{P}}^g u^g_t \right) + r_t^{\omega } = {D}_t^{\omega } \quad \forall t \in \mathcal {T}, \, \forall \omega \in \varOmega \end{aligned}
(5)
\begin{aligned}&\frac{1}{|\mathcal {G}||\mathcal {T}||\varOmega |} \sum _{\omega \in \varOmega } \sum _{g \in \mathcal {G}} \sum _{t \in T} y_t^{g,\omega } \le \varepsilon \end{aligned}
(6)
\begin{aligned}&p^{l,g,\omega }_t \in \mathbb {R}_+ \qquad \qquad \forall t \in \mathcal {T}, \, \forall l \in \mathcal {L}^g, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(7a)
\begin{aligned}&p_t^{g,\omega }, {\overline{p}}_t^{g,\omega }, {\underline{p}}_t^{g,\omega } \in \mathbb {R}_+ \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G}, \, \forall \omega \in \varOmega \end{aligned}
(7b)
\begin{aligned}&r_t^{n,\omega } \in \left[ {\underline{W}}^{n,\omega }_t, {\overline{W}}^{n,\omega }_t \right] \qquad \qquad \forall t \in \mathcal {T}, \forall n \in \mathcal {N}, \forall \omega \in \varOmega \end{aligned}
(7c)
\begin{aligned}&u^g_t, v^g_t, w^g_t \in \{0, 1\} \qquad \qquad \forall t \in \mathcal {T}, \, \forall g \in \mathcal {G} \end{aligned}
(7d)
\begin{aligned}&x^g_{[t,t')} \in \{0, 1\} \qquad \qquad \forall [t,t') \in \mathcal {X}^g, \, \forall g \in \mathcal {G} \end{aligned}
(7e)
\begin{aligned}&y^{g,\omega }_t \in \{0,1\} \qquad \qquad \forall t \in \mathcal {T}, \forall g \in \mathcal {G}, \forall \omega \in \varOmega . \end{aligned}
(7f)
We introduce this model only briefly here, details are available in Morales-España et al. (2013). Constraints (2) describe the use of the generator with its on and off variables and encode the start-up costs (which are directly substituted into the objective function via equation (2f). Constraints (3) represent the generator’s start-up and shutdown requirements, ramping requirements, and piecewise power production. Constraint (5) ensures we meet the uncertain demand exactly with the thermal and renewable generators. Variable $$y_t^{g,\omega }$$ is one when the generator g is operating in a non-nominal mode at hour t in scenario $$\omega$$, and zero under normal operating conditions. Turning to constraints (4), first notice by constraints (4a)–(4c) that a generator can only be in non-nomimal mode at time t if it is on ($$u_t=1$$), has been on ($$v_t=0$$), and will be on for at least one hour ($$w_{t+1}=0$$). Constraints (4d) and (4e) enforce that in a non-nominal mode the generator can produce power up to $${\overline{\overline{P}}}^g$$ or down to $${\underline{\underline{P}}}^g$$. Notice that the total output of the generator at t in scenario $$\omega$$ is represented by the quantity $$\left( p_t^{g,\omega } + {\overline{p}}^{g,\omega }_t -{\underline{p}}^{g,\omega }_t + {\underline{P}}^g u^g_t \right)$$. Finally, constraint (6) restricts the proportion of non-nominalities across all generators, times and scenarios to be no more than $$\varepsilon$$. Here, $$\varepsilon$$ is a small number less than one, such as 0.01 or 0.05. We explain the particular choice of this chance constraint in Section 2.3. The remaining constraints ensure the non-negativity and binary restrictions on the relevant decision variables and the stochastic bounds on the renewable power.
Because of the additional variables used to represent dispatch over $${\overline{P}}^g$$ and under $${\underline{P}}^g$$, this formulation implicitly relaxes the ramping requirements. We believe this to be reasonable as $${\overline{\overline{P}}}^g$$ ($${\underline{\underline{P}}}^g$$) will be not too different from $${\overline{P}}^g$$ ($${\underline{P}}^g$$), so the relaxation of the ramping requirements will be of the same scale as the relaxation of the operating limits.
In the two-stage stochastic program represented by model (1)–(7), we decide each generator’s operation status in the first stage. Then, we observe the uncertainty in the demand and wind power. After this, we decide the amount of power to employ from each generator that was declared to be “on” in the first stage. Under normal operating conditions, generator g can use power up to $${\overline{P}}^g$$. This limit is increased to $${\overline{\overline{P}}}^g$$ under non-nominal operations. Similarly, if there is the potential for an over-generation event, the lower limit can be reduced from $${\underline{P}}^g$$ to $${\underline{\underline{P}}}^g$$. Also notice that power below minimum $${\underline{p}}^{g,\omega }_t$$ is penalized with a positive term in the objective function, and similarly for $${\overline{p}}^{g,\omega }_t$$. When the provided cost curve $$C^{l,g}$$ is convex, the overall production costs with non-nominalities will be convex when $${\overline{C}}^g \ge C^{L_g,g}$$ and $${\underline{C}}^g \ge -C^{1,g}$$. The first condition says the marginal cost of providing non-nominal power by dispatching above $${\overline{P}}^g$$is at least that of the generator operating at $${\overline{P}}^g$$, and the second says the marginal cost of dispatching below $${\underline{P}}^g$$ is at least that of being dispatched at $${\underline{P}}^g$$. In practice we will consider $${\underline{C}}^g > 0$$ so as to compensate generators for dispatch below $${\underline{P}}^g$$.
During a non-nominality, we allow generator g to produce power up to $${\overline{\overline{P}}}^g > {\overline{P}}^g$$, at a cost greater than that during normal operations. Similarly, we allow generator g to produce power down to $${\underline{\underline{P}}}^g < {\underline{P}}^g$$, at cost $${\underline{C}}^g$$. Given the lack of data, we selected parameters $$\beta , \gamma > 0$$ for use in our study:
\begin{aligned} {\overline{\overline{P}}}^g&= (1+ \beta ) {\overline{P}}^g \\ {\underline{\underline{P}}}^g&= (1 - \beta ) {\underline{P}}^g \\ {\overline{C}}^g&= (1+ \gamma ) C^{L^g,g} \\ {\underline{C}}^g&= (1+ \gamma ) C^{L^g, g}. \end{aligned}
### Choice of the chance constraint
We begin with a few definitions, motivated by Prékopa (1988); Singh and Watson (2019). Define the event $$E_t^g = \{\omega : y_t^{g,\omega } = 0 \}$$. Then, the complement of $$E_t^g$$ is $$\overline{E}_t^g = \{\omega : y_t^{g,\omega } = 1 \}$$. Define the event $$F =\{ \omega : \cap _{t \in \mathcal {T}, g \in \mathcal {G}} E_t^g \}$$. Then, the complement of F is $$\overline{F}= \{ \omega : \cup _{t \in \mathcal {T}, g \in \mathcal {G}} \overline{E}_t^g \}$$. Consider the following probabilistic constraints:
\begin{aligned} \mathbb {P}(\overline{F})\le & {} \varepsilon . \end{aligned}
(8a)
\begin{aligned} \sum _{g \in \mathcal {G}} \sum _{t \in T} \mathbb {P}\left( \overline{E}_t^g \right)\le & {} \varepsilon . \end{aligned}
(8b)
### Lemma 1
The constraint (8b) implies the constraint (8a).
### Proof
The proof follows from the definitions of $$\overline{F}$$ and $$\overline{E}_t^g$$, and the classical probability inequality: $$\mathbb {P}\left( \cup _{t \in \mathcal {T}, g \in \mathcal {G}} \overline{E}_t^g \right) \le \sum _{g \in \mathcal {G}} \sum _{t \in T} \mathbb {P}\left( \overline{E}_t^g \right)$$. $$\square$$
Next, we construct a sample average approximation (SAA) of the above probabilistic constraints with $$|\varOmega |$$ samples. To this end, we define a new binary variable: $$z^\omega =0$$ if $$y_t^{g,\omega } = 0, \forall t \in \mathcal {T}, g \in \mathcal {G}$$, and $$z^\omega =1$$ otherwise. We can relate the z and y variables as follows:
\begin{aligned} z^\omega\le & {} \sum _{g \in \mathcal {G}} \sum _{t \in T} y_t^{g,\omega }, \forall \omega \in \varOmega \end{aligned}
(9a)
\begin{aligned} z^\omega\ge & {} y_t^{g,\omega }, \forall t \in \mathcal {T}, \forall g \in \mathcal {G}, \forall \omega \in \varOmega . \end{aligned}
(9b)
The SAA of the probabilistic constraints (8a) and (8b) are respectively:
\begin{aligned} \frac{1}{|\varOmega |} \sum _{\omega \in \varOmega } z^\omega\le & {} \varepsilon . \end{aligned}
(10a)
\begin{aligned} \frac{1}{|\varOmega |} \sum _{g \in \mathcal {G}} \sum _{t \in T} \sum _{\omega \in \varOmega } y_t^{g, \omega }\le & {} \varepsilon . \end{aligned}
(10b)
Equations (9b) and (10a) imply:
\begin{aligned} \frac{1}{|\varOmega ||\mathcal {T}||\mathcal {G}|} \sum _{g \in \mathcal {G}} \sum _{t \in T} \sum _{\omega \in \varOmega } y_t^{g, \omega } \le \varepsilon , \end{aligned}
(11)
which is weaker than constraint (10b). The above discussion can be summarized in the following lemma.
### Lemma 2
Under Eq. (9), the constraint (10b) implies the constraint (10a) which further implies the constraint (11).
In model (1)–(7), we choose the weakest of the above three probabilistic constraints. If the SAA is constructed using only a few scenarios, as is often the case with stochastic UC, the constraints (10b) and constraint (10a) might be too stringent; e.g., a model with 99 scenarios and $$\varepsilon =0.01$$ would have $$z^\omega = 0, \forall \omega \in \varOmega$$. To summarize the above discussion, we define a non-nominality as an operation outside of the prescribed technical ratings in a {generator, time, scenario} triplet. And, we restrict the proportion of these non-nominalities to a small positive quantity, namely $$\varepsilon$$.
In Sect. 4 we analyze the sensitivity of the stochastic solution to $$\beta$$, $$\gamma$$, and $$\varepsilon$$. Larger values of $$\beta$$ relax the nominal operating constraints further, and larger values for $$\gamma$$ penalize deviation from the nominal operating constraints through the cost via the objective function.
## Case study
To analyze the impact of considering bounded exceedances of generator ratings directly in a stochastic UC model, we consider two case studies. The first case study considers the WECC240++ system (Rachunok et al. 2018). WECC240++ UC instances have 85 thermal generating units. Demand profiles are based on real-world data from 2004. Wind generation scenarios are based on data from 2013, scaled to achieve approximately 30% penetration levels. We selected one day in the 2013 simulation set, 11 May 2013, and consider 50 probabilistic scenarios and a time horizon of 48 hours at hourly resolution. Scenarios were constructed using the Markov-Chain Monte-Carlo procedure to model wind production; demand is treated as deterministic. Wind generation is fully curtailable. For additional details, the reader is referred to Rachunok et al. (2018).
The second case study is based on the recently introduced RTS-GMLC system (RTS-GMLC 2018; Barrows et al. 2020). The RTS-GMLC system has 73 thermal generators (oil, coal, gas, and nuclear) and 81 renewable generators (wind, hydro, utility-scale photo-voltaic, and rooftop photo-voltaic). Simulated load and renewable generation is provided on a 5-minute basis for the year 2020. We consider the date 10 July 2020 with a 48 hour planning horizon, at hourly resolution, and use 16 probabilistic wind scenarios obtained from Staid et al. (2017). All other data is considered deterministic for this study. We additionally assume hydro units are self-scheduling and the rooftop photo-voltaic is must-take. However, the wind and utility-scale photo-voltaic are fully curtailable. Hence, the aggregate renewables at each bus in the system is a mix of must-take and curtailable resources.
## Computational experiments
### Computational setup
We encoded all models using the Pyomo 5.5 (Hart et al. 2017) algebraic modeling language. All models are solved using the commercial Gurobi 8.0.1 (Gurobi Optimization 2018) MILP solver, on a laptop comprising of a 2.8 GHz Intel Core i7 processor and 16 GB of RAM. We consider $$\varepsilon = 0.01$$ and 0.05, corresponding respectively to use of nominal generator operations at least 99% and 95%, across all scenarios, generators, and time periods. We attempt to solve all problems to a MILP optimality gap of 0.001 (i.e., 0.1%) within a time limit of 1800 seconds. Additionally, we set the Gurobi parameter Method=1, which dictates that dual simplex is to be used to solve the root LP relaxations. All other Gurobi solver parameter settings were preserved at their defaults. We report the MILP optimality gap at termination when a problem could not be solved within the 1800 second time limit. The expected cost at $$\varepsilon =0$$ is the baseline for operations when no non-nominal states are permitted; i.e., standard stochastic UC. In this case, the values of $$\beta$$ and $$\gamma$$ are irrelevant. Setting $$\varepsilon > 0$$ allows for varying degrees of non-nominal operations. In addition to absolute cost, we report the percentage of savings relative to the base case ($$\varepsilon =0$$). In general, this percentage represents the minimum percentage cost saved, as some problems could not be solved within the time limit, and in any case all models are only solved to a termination criteria of a 0.1% MILP gap.
### Comparison of the two test systems
In Table 1 we present the results for the WECC-240++ case. All parameterizations of this case study were solved to 0.1% optimality gap within the time limit; this is reflected using “-” in the MILP gap column. In Table 2, we present analogous results for a single representative day for the RTS-GMLC case. Here, we do observe non-zero MILP gaps in some parameterizations of our model.
Both of our baseline case studies consider data from late spring and summer months in the US. Winter and fall months necessarily exhibit different load profiles. Next, we consider RTS-GMLC case results using data from the winter (for 10 January 2020). The results, shown in Table 3, indicate qualitatively identical behavior for our model.
Despite having a similar number of generators and fewer scenarios, the RTS-GMLC system is generally more computationally demanding than the WECC-240++ system. However, despite this difference, the qualitative performance of our model with respect to parameterizations of $$\varepsilon$$, $$\beta$$, and $$\gamma$$ across the two cases is identical. We observe that higher values of $$\varepsilon$$ results in lower operation costs, but at the expense of a larger computation time. This increase is conceptually consistent with the increase in combinatorics (in terms of the number of possible combinations of scenarios, generators, and time periods) allowed by a larger $$\varepsilon$$. Second, increasing the value of $$\beta$$ increases the percentage of costs savings. This is expected as larger values of $$\beta$$ allow the emergency limits to be larger. However, the effect of changes in $$\beta$$ on computational difficulty is not consistent. Third, increasing $$\gamma$$ decreases the percentage of costs savings. This is again expected as the piecewise cost for operation during an emergency is larger for a larger $$\gamma$$. Again, the effect of $$\gamma$$ on computational times in not consistent. For a system operator, the results with $$\varepsilon =0.01$$ are likely more valuable and relevant, as a 5% exceedance regime may be too disruptive. For the WECC240++ case, cost savings are between 0.3% and 1.6%. For the RTS-GMLC case, the savings are larger and lie between 1.2% and 2.1%. Although the two systems we consider are test systems, we note that a 1% savings can result in system operators saving several billions of dollars saved per year (O’Neill 2007). Finally, in comparison to some other chance-constrained variants of UC (Kargarian et al. 2016; Singh et al. 2018; Wang et al. 2012), our solution results in no loss of load in any scenario as we require demand to be satisfied when possible.
### Sensitivity analysis of optimal solution
Next, we analyze the structure of an optimal solution (subject to the MILP optimality gap) to the RTS-GMLC system with $$\varepsilon =0.01$$, $$\beta =0.1$$, and $$\gamma =0.1$$, in order to analyze differences relative to an optimal solution to the baseline stochastic UC model. Figure 1a presents the aggregate number of non-nominalities per scenario (i.e., $$\sum _{g\in \mathcal {G}}\sum _{t\in \mathcal {T}}y_t^{g,\omega })$$. The total number of non-nominalities is 560, which is 1% of $$|\mathcal {G}||\mathcal {T}||\varOmega |$$. Clearly, the model incentivizes choosing a lower cost generator whenever possible. As a result, the sole nuclear generator in the RTS-GMLC system—which is the cheapest marginal unit—enters a non-nominal mode in all of the 16 scenarios for at least one hour in the operating horizon. We further observe (not shown) that this non-nominality is largely consistent in timing across scenarios; i.e., the nuclear generator is in a non-nominal operations mode for almost all of the same hours across all the 16 scenarios. While nuclear generators tend to have low marginal costs, in practice, they are operated in narrow windows and are subject to rigorous oversight. Thus, we conducted another analysis to see the impacts if the nuclear generator were not allowed to run at all in a non-nominal mode. Figure 1b presents the aggregated non-nominalities per scenario for this case. The variability in the non-nominalities per scenario is notably larger in Fig. 1b than Fig. 1a. Specifically, the standard deviation in the number of non-nominalities across the 16 scenarios is 20.2 for Fig. 1b and 14.8 for Fig. 1a. While the cost savings relative to the stochastic solution for Fig. 1a is 1.51%, the cost savings for the solution represented in Fig. 1b is only 1.15%, indicating that shifting of non-nominal operations to more expensive units does decrease the overall cost benefit. This is, again, expected as the latter solution is suboptimal.
To examine the sensitivity of cost savings relative to any one generator, we now consider the instance analyzed in Table 2 under another restrictive operational regime; the results are shown in Table 4. The first considers only allowing units to activate non-nominal operations once per day; i.e., adds the following constraints
\begin{aligned}&\sum _{t = 1}^{24} y_t^{g,\omega } \le 1 \qquad \qquad \forall g \in \mathcal {G}, \ \forall \omega \in \varOmega \\&\sum _{t = 25}^{48} y_t^{g,\omega } \le 1 \qquad \qquad \forall g \in \mathcal {G}, \ \forall \omega \in \varOmega \end{aligned}
to the model (1)–(7). Results for this regime are reported in the column labeled “Limited”. The second, mentioned above, disables non-nominal operation for the nuclear unit in this system, and is reported in the column labeled “No nuclear”. The “Optimal” column reports the savings achieved using the non-modified model (same as Table 2). We note that we can still observe cost savings on the order of 1% in both cases, which is particularly surprising for the much more conservative Limited case. Hence, despite placing these additional restrictions on when non-nominal generation can be dispatched, significant cost savings can still be realized.
To analyze the generalizability of our observed cost savings, we next solve our model for another six days of the RTS-GMLC system. Table 5 reports these results, for $$\varepsilon =0.01, \beta =0.05, 0.1, \gamma =0.1$$. Of the 12 instances considered, only one could not be solved to within the specified MILP optimality gap in the 1800 second time limit. The percentage cost savings do not differ qualitatively from those in Table 2, which were generated under the same parameterization. The average and standard deviation across the six days in terms of percentage cost savings for the $$\beta =0.05$$$$(\beta =0.1)$$ case are 1.29% (2.34%) and 0.56% (0.86%).
We next comment on the economic implications for our study. While saving operational costs is clearly desirable, our proposed non-nominality UC would also incentivize less expensive generators to relax their operating constraints, which would shift revenue from more expensive peaking units. However, this effect may not be so determinative if a more restrictive policy such as Limited is considered. Future extensions could consider a fixed cost for operating a generator in non-nominal mode or a maximum up-time for non-nominal mode, so as to fully capture the additional costs or restrictions for operating outside of non-nominal mode. In the context of stochastic UC, wide variation in renewables generation and/or load across scenarios may lead to solutions that are overly conservative. That is, there may be generators that are dispatched to provide power only for an event with low probability; see, e.g., Rachunok et al. (2018). Although we did not examine this for this study, the use of non-nominal modes could serve as an inexpensive way to add additional peaking capacity to a system so as to enable higher renewables penetration.
We observe that all of our computational experiments were conducted on a modest laptop. The problems are generally tractable even without any custom algorithms (e.g., decomposition) for the number of scenarios considered above. An ISO or a utility would likely run the optimizations on larger machines and with a greater number of scenarios. In this context, we next test our stochastic UC model by using (i) a small batch of 10 scenarios and (ii) a large batch of 100 scenarios, both for the WECC-240++ case for 11 May 2013. Tables 6a and 6b report results analogous to Table 1, respectively, for the 10 and 100 scenario variants. These computational results reinforce two intuitive observations. First, as expected, the respective cost values do not differ significantly between the three instances, i.e., with 10, 50, and 100 scenarios. Second, the computational effort required increases with an increase in the number of scenarios. However, only one of the instances in Table 6b could not be solved within the 1800 second time limit. The 100 scenario instances have approximately half a million binary variables after Gurobi’s presolve. Future research could examine specialized algorithms that would further assist in tractability, as well as examining larger test cases with hundreds or thousands of generators.
## Conclusion
We presented a chance-constrained unit commitment formulation to incorporate small violations of the technical ratings of a generator. Our motivation comes from the fact that system operators occasionally run generators in non-nominal conditions; this can create significant economical benefits, especially in the context of stochastic unit commitment with a wide diversity in scenarios. The model can also be useful when there is an outage, either unplanned or planned, since system operators often analyze contingencies on a case-by-case basis (Power Water 2017). In our study, we demonstrated a small percentage savings in the costs which could translate to a significant amount of dollars saved over the year; see, also (O’Neill 2007). We analyzed the sensitivity of the optimal solution under restricted regimes as well. The models we presented are generally tractable, however future work could examine tailored algorithms to achieve even faster solutions. | {"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": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9931807518005371, "perplexity": 1742.4849380448054}, "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/1669446710462.59/warc/CC-MAIN-20221128002256-20221128032256-00319.warc.gz"} |
http://mathhelpforum.com/calculus/132438-integral-print.html | integral
• Mar 7th 2010, 03:11 AM
dapore
1 Attachment(s)
integral
find
• Mar 7th 2010, 09:57 AM
shawsend
Write it as:
$\int x^{-3/2}-x^{-3/2} e^{-x} dx$
then use parts on the second one to get the expression $\frac{e^{-x}}{\sqrt{x}}$, then use the substitution $w=\sqrt{x}$ on that expression to get it in terms of $\int_0^{\infty}e^{-w^2}dw=\frac{\sqrt{\pi}}{2}$. May have to take limits since it's indeterminate at zero. Mathematica returns $2\sqrt{\pi}$ but I haven't worked out all the steps so fix it if I left some kinks ok.
• Mar 11th 2010, 02:24 PM
dapore
Quote:
Originally Posted by shawsend
Write it as:
$\int x^{-3/2}-x^{-3/2} e^{-x} dx$
then use parts on the second one to get the expression $\frac{e^{-x}}{\sqrt{x}}$, then use the substitution $w=\sqrt{x}$ on that expression to get it in terms of $\int_0^{\infty}e^{-w^2}dw=\frac{\sqrt{\pi}}{2}$. May have to take limits since it's indeterminate at zero. Mathematica returns $2\sqrt{\pi}$ but I haven't worked out all the steps so fix it if I left some kinks ok.
Thank you, but my teacher says this method is not working, and says that the correct way is to insert (x)in the bow and the integration is a two parts, one directly and the other gamma function
• Mar 11th 2010, 03:27 PM
chisigma
Integrating by parts we obtain...
$\int_{0}^{\infty} x^{-\frac{3}{2}}\cdot (1-e^{-x})\cdot dx = |-2\cdot x^{-\frac{1}{2}}\cdot (1-e^{-x})|_{0}^{\infty} +2\cdot \int_{0}^{\infty} x^{-\frac{1}{2}}\cdot e^{-x}\cdot dx = 2\cdot \Gamma(\frac{1}{2}) = 2\cdot \sqrt{\pi}$ (1)
Kind regards
$\chi$ $\sigma$
• Mar 11th 2010, 03:40 PM
dapore
Quote:
Originally Posted by chisigma
Integrating by parts we obtain...
$\int_{0}^{\infty} x^{-\frac{3}{2}}\cdot (1-e^{-x})\cdot dx = |-2\cdot x^{-\frac{1}{2}}\cdot (1-e^{-x})|_{0}^{\infty} +2\cdot \int_{0}^{\infty} x^{-\frac{1}{2}}\cdot e^{-x}\cdot dx = 2\cdot \Gamma(\frac{1}{2}) = 2\cdot \sqrt{\pi}$ (1)
Kind regards
$\chi$ $\sigma$
Thank you, my dear, indeed this is the answer | {"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": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9604713916778564, "perplexity": 1011.369702175454}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816647.80/warc/CC-MAIN-20180225150214-20180225170214-00281.warc.gz"} |
https://www.science.gov/topicpages/p/path+integrals+physically.html | #### Sample records for path integrals physically
1. Path-integral approach to 't Hooft's derivation of quantum physics from classical physics
SciTech Connect
Blasone, Massimo; Jizba, Petr; Kleinert, Hagen
2005-05-15
We present a path-integral formulation of 't Hooft's derivation of quantum physics from classical physics. The crucial ingredient of this formulation is Gozzi et al.'s supersymmetric path integral of classical mechanics. We quantize explicitly two simple classical systems: the planar mathematical pendulum and the Roessler dynamical system.
2. The universal path integral
Lloyd, Seth; Dreyer, Olaf
2016-02-01
Path integrals calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration. This paper defines a universal path integral, which sums over all computable structures. This path integral contains as sub-integrals all possible computable path integrals, including those of field theory, the standard model of elementary particles, discrete models of quantum gravity, string theory, etc. The universal path integral possesses a well-defined measure that guarantees its finiteness. The probabilities for events corresponding to sub-integrals can be calculated using the method of decoherent histories. The universal path integral supports a quantum theory of the universe in which the world that we see around us arises out of the interference between all computable structures.
3. Canonical path-integral measures for Holst and Plebanski gravity: II. Gauge invariance and physical inner product
Han, Muxin
2010-12-01
This paper serves as a continuation for the discussion in Engle et al (2010, Class. Quantum Grav. 27 245014). We analyze the invariance properties of the gravity path-integral measure derived from canonical framework and discuss which path-integral formula may be employed in the concrete computation e.g. constructing a spin-foam model, so that the final model can be interpreted as a physical inner product in the canonical theory. This paper is divided into two parts, the first part is concerning the gauge invariance of the canonical path-integral measure for gravity from the reduced phase space quantization. We show that the path-integral measure is invariant under all the gauge transformations generated by all the constraints. These gauge transformations are the local symmetries of the gravity action, which is implemented without anomaly at the quantum level by the invariant path-integral measure. However, these gauge transformations coincide with the spacetime diffeomorphisms only when the equations of motion are imposed. But the path-integral measure is not invariant under spacetime diffeomorphisms, i.e. the local symmetry of spacetime diffeomorphisms become anomalous in the reduced phase space path-integral quantization. In the second part, we present a path-integral formula, which formally solves all the quantum constraint equations of gravity, and further results in a rigging map in the sense of refined algebraic quantization (RAQ). Then we give a formal path-integral expression of the physical inner product in loop quantum gravity (LQG). This path-integral expression is simpler than the one from reduced phase space quantization, since all the gauge-fixing conditions are removed except the time gauge. The resulting path-integral measure is different from the product Lebesgue measure up to a local measure factor containing both the spacetime volume element and the spatial volume element. This formal path-integral expression of the physical inner product can be
4. Path Integrals and Hamiltonians
Baaquie, Belal E.
2014-03-01
1. Synopsis; Part I. Fundamental Principles: 2. The mathematical structure of quantum mechanics; 3. Operators; 4. The Feynman path integral; 5. Hamiltonian mechanics; 6. Path integral quantization; Part II. Stochastic Processes: 7. Stochastic systems; Part III. Discrete Degrees of Freedom: 8. Ising model; 9. Ising model: magnetic field; 10. Fermions; Part IV. Quadratic Path Integrals: 11. Simple harmonic oscillators; 12. Gaussian path integrals; Part V. Action with Acceleration: 13. Acceleration Lagrangian; 14. Pseudo-Hermitian Euclidean Hamiltonian; 15. Non-Hermitian Hamiltonian: Jordan blocks; 16. The quartic potential: instantons; 17. Compact degrees of freedom; Index.
5. Path Integrals and Supersolids
Ceperley, D. M.
2008-11-01
Recent experiments by Kim and Chan on solid 4He have been interpreted as discovery of a supersolid phase of matter. Arguments based on wavefunctions have shown that such a phase exists, but do not necessarily apply to solid 4He. Imaginary time path integrals, implemented using Monte Carlo methods, provide a definitive answer; a clean system of solid 4He should be a normal quantum solid, not one with superfluid properties. The Kim-Chan phenomena must be due to defects introduced when the solid is formed.
6. Path Integral Approach to Atomic Collisions
Harris, Allison
2016-09-01
The Path Integral technique is an alternative formulation of quantum mechanics that is based on a Lagrangian approach. In its exact form, it is completely equivalent to the Hamiltonian-based Schrödinger equation approach. Developed by Feynman in the 1940's, following inspiration from Dirac, the path integral approach has been widely used in high energy physics, quantum field theory, and statistical mechanics. However, only in limited cases has the path integral approach been applied to quantum mechanical few-body scattering. We present a theoretical and computational development of the path integral method for use in the study of atomic collisions. Preliminary results are presented for some simple systems. Ultimately, this approach will be applied to few-body ion-atom collisions. Work supported by NSF grant PHY-1505217.
7. Scattering theory with path integrals
SciTech Connect
Rosenfelder, R.
2014-03-15
Starting from well-known expressions for the T-matrix and its derivative in standard nonrelativistic potential scattering, I rederive recent path-integral formulations due to Efimov and Barbashov et al. Some new relations follow immediately.
8. Fermionic path integrals and local anomalies
Roepstorff, G.
2003-05-01
No doubt, the subject of path integrals proved to be an immensely fruitful human, i.e. Feynman's idea. No wonder it is more timely than ever. Some even claim that it is the most daring, innovative and revolutionary idea since the days of Heisenberg and Bohr. It is thus likely to generate enthusiasm, if not addiction among physicists who seek simplicity together with perfection. Professor Devreese's long-lasting interest in, if not passion on the subject stems from his firm conviction that, beyond being the tool of choice, path integration provides the key to all quantum phenomena, be it in solid state, atomic, molecular or particle physics as evidenced by the impressive list of publications at the address http://lib.ua.ac.be/AB/a867.html. In this note, I review a pitfall of fermionic path integrals and a way to get around it in situations relevant to the Standard Model of particle physics.
9. Bergman Kernel from Path Integral
Douglas, Michael R.; Klevtsov, Semyon
2010-01-01
We rederive the expansion of the Bergman kernel on Kähler manifolds developed by Tian, Yau, Zelditch, Lu and Catlin, using path integral and perturbation theory, and generalize it to supersymmetric quantum mechanics. One physics interpretation of this result is as an expansion of the projector of wave functions on the lowest Landau level, in the special case that the magnetic field is proportional to the Kähler form. This is relevant for the quantum Hall effect in curved space, and for its higher dimensional generalizations. Other applications include the theory of coherent states, the study of balanced metrics, noncommutative field theory, and a conjecture on metrics in black hole backgrounds discussed in [24]. We give a short overview of these various topics. From a conceptual point of view, this expansion is noteworthy as it is a geometric expansion, somewhat similar to the DeWitt-Seeley-Gilkey et al short time expansion for the heat kernel, but in this case describing the long time limit, without depending on supersymmetry.
10. Squeezed states and path integrals
NASA Technical Reports Server (NTRS)
Daubechies, Ingrid; Klauder, John R.
1992-01-01
The continuous-time regularization scheme for defining phase-space path integrals is briefly reviewed as a method to define a quantization procedure that is completely covariant under all smooth canonical coordinate transformations. As an illustration of this method, a limited set of transformations is discussed that have an image in the set of the usual squeezed states. It is noteworthy that even this limited set of transformations offers new possibilities for stationary phase approximations to quantum mechanical propagators.
11. Polymer quantum mechanics some examples using path integrals
SciTech Connect
Parra, Lorena; Vergara, J. David
2014-01-14
In this work we analyze several physical systems in the context of polymer quantum mechanics using path integrals. First we introduce the group averaging method to quantize constrained systems with path integrals and later we use this procedure to compute the effective actions for the polymer non-relativistic particle and the polymer harmonic oscillator. We analyze the measure of the path integral and we describe the semiclassical dynamics of the systems.
12. Perturbative Methods in Path Integration
Johnson-Freyd, Theodore Paul
This dissertation addresses a number of related questions concerning perturbative "path" integrals. Perturbative methods are one of the few successful ways physicists have worked with (or even defined) these infinite-dimensional integrals, and it is important as mathematicians to check that they are correct. Chapter 0 provides a detailed introduction. We take a classical approach to path integrals in Chapter 1. Following standard arguments, we posit a Feynman-diagrammatic description of the asymptotics of the time-evolution operator for the quantum mechanics of a charged particle moving nonrelativistically through a curved manifold under the influence of an external electromagnetic field. We check that our sum of Feynman diagrams has all desired properties: it is coordinate-independent and well-defined without ultraviolet divergences, it satisfies the correct composition law, and it satisfies Schrodinger's equation thought of as a boundary-value problem in PDE. Path integrals in quantum mechanics and elsewhere in quantum field theory are almost always of the shape ∫ f es for some functions f (the "observable") and s (the "action"). In Chapter 2 we step back to analyze integrals of this type more generally. Integration by parts provides algebraic relations between the values of ∫ (-) es for different inputs, which can be packaged into a Batalin--Vilkovisky-type chain complex. Using some simple homological perturbation theory, we study the version of this complex that arises when f and s are taken to be polynomial functions, and power series are banished. We find that in such cases, the entire scheme-theoretic critical locus (complex points included) of s plays an important role, and that one can uniformly (but noncanonically) integrate out in a purely algebraic way the contributions to the integral from all "higher modes," reducing ∫ f es to an integral over the critical locus. This may help explain the presence of analytic continuation in questions like the
13. Integrated assignment and path planning
Murphey, Robert A.
2005-11-01
A surge of interest in unmanned systems has exposed many new and challenging research problems across many fields of engineering and mathematics. These systems have the potential of transforming our society by replacing dangerous and dirty jobs with networks of moving machines. This vision is fundamentally separate from the modern view of robotics in that sophisticated behavior is realizable not by increasing individual vehicle complexity, but instead through collaborative teaming that relies on collective perception, abstraction, decision making, and manipulation. Obvious examples where collective robotics will make an impact include planetary exploration, space structure assembly, remote and undersea mining, hazardous material handling and clean-up, and search and rescue. Nonetheless, the phenomenon driving this technology trend is the increasing reliance of the US military on unmanned vehicles, specifically, aircraft. Only a few years ago, following years of resistance to the use of unmanned systems, the military and civilian leadership in the United States reversed itself and have recently demonstrated surprisingly broad acceptance of increasingly pervasive use of unmanned platforms in defense surveillance, and even attack. However, as rapidly as unmanned systems have gained acceptance, the defense research community has discovered the technical pitfalls that lie ahead, especially for operating collective groups of unmanned platforms. A great deal of talent and energy has been devoted to solving these technical problems, which tend to fall into two categories: resource allocation of vehicles to objectives, and path planning of vehicle trajectories. An extensive amount of research has been conducted in each direction, yet, surprisingly, very little work has considered the integrated problem of assignment and path planning. This dissertation presents a framework for studying integrated assignment and path planning and then moves on to suggest an exact
14. Test on the effectiveness of the sum over paths approach in favoring the construction of an integrated knowledge of quantum physics in high school
Malgieri, Massimiliano; Onorato, Pasquale; De Ambrosis, Anna
2017-06-01
In this paper we present the results of a research-based teaching-learning sequence on introductory quantum physics based on Feynman's sum over paths approach in the Italian high school. Our study focuses on students' understanding of two founding ideas of quantum physics, wave particle duality and the uncertainty principle. In view of recent research reporting the fragmentation of students' mental models of quantum concepts after initial instruction, we collected and analyzed data using the assessment tools provided by knowledge integration theory. Our results on the group of n =14 students who performed the final test indicate that the functional explanation of wave particle duality provided by the sum over paths approach may be effective in leading students to build consistent mental models of quantum objects, and in providing them with a unified perspective on both the photon and the electron. Results on the uncertainty principle are less clear cut, as the improvements over traditional instruction appear less significant. Given the low number of students in the sample, this work should be interpreted as a case study, and we do not attempt to draw definitive conclusions. However, our study suggests that (i) the sum over paths approach may deserve more attention from researchers and educators as a possible route to introduce basic concepts of quantum physics in high school, and (ii) more research should be focused not only on the correctness of students' mental models on individual concepts, but also on the ability of students to connect different ideas and experiments related to quantum theory in an organized whole.
15. Two-path plasmonic interferometer with integrated detector
DOEpatents
Dyer, Gregory Conrad; Shaner, Eric A.; Aizin, Gregory
2016-03-29
An electrically tunable terahertz two-path plasmonic interferometer with an integrated detection element can down convert a terahertz field to a rectified DC signal. The integrated detector utilizes a resonant plasmonic homodyne mixing mechanism that measures the component of the plasma waves in-phase with an excitation field that functions as the local oscillator in the mixer. The plasmonic interferometer comprises two independently tuned electrical paths. The plasmonic interferometer enables a spectrometer-on-a-chip where the tuning of electrical path length plays an analogous role to that of physical path length in macroscopic Fourier transform interferometers.
16. White Noise Path Integrals in Stochastic Neurodynamics
Carpio-Bernido, M. Victoria; Bernido, Christopher C.
2008-06-01
The white noise path integral approach is used in stochastic modeling of neural activity, where the primary dynamical variables are the relative membrane potentials, while information on transmembrane ionic currents is contained in the drift coefficient. The white noise path integral allows a natural framework and can be evaluated explicitly to yield a closed form for the conditional probability density.
17. Simulating biochemical physics with computers: 1. Enzyme catalysis by phosphotriesterase and phosphodiesterase; 2. Integration-free path-integral method for quantum-statistical calculations
Wong, Kin-Yiu
We have simulated two enzymatic reactions with molecular dynamics (MD) and combined quantum mechanical/molecular mechanical (QM/MM) techniques. One reaction is the hydrolysis of the insecticide paraoxon catalyzed by phosphotriesterase (PTE). PTE is a bioremediation candidate for environments contaminated by toxic nerve gases (e.g., sarin) or pesticides. Based on the potential of mean force (PMF) and the structural changes of the active site during the catalysis, we propose a revised reaction mechanism for PTE. Another reaction is the hydrolysis of the second-messenger cyclic adenosine 3'-5'-monophosphate (cAMP) catalyzed by phosphodiesterase (PDE). Cyclicnucleotide PDE is a vital protein in signal-transduction pathways and thus a popular target for inhibition by drugs (e.g., ViagraRTM). A two-dimensional (2-D) free-energy profile has been generated showing that the catalysis by PDE proceeds in a two-step SN2-type mechanism. Furthermore, to characterize a chemical reaction mechanism in experiment, a direct probe is measuring kinetic isotope effects (KIEs). KIEs primarily arise from internuclear quantum-statistical effects, e.g., quantum tunneling and quantization of vibration. To systematically incorporate the quantum-statistical effects during MD simulations, we have developed an automated integration-free path-integral (AIF-PI) method based on Kleinert's variational perturbation theory for the centroid density of Feynman's path integral. Using this analytic method, we have performed ab initio pathintegral calculations to study the origin of KIEs on several series of proton-transfer reactions from carboxylic acids to aryl substituted alpha-methoxystyrenes in water. In addition, we also demonstrate that the AIF-PI method can be used to systematically compute the exact value of zero-point energy (beyond the harmonic approximation) by simply minimizing the centroid effective potential.
18. Selecta from a Life-Long Obsession with Path Integrals
SciTech Connect
Klauder, John R.
2008-06-18
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.
19. Test on the Effectiveness of the Sum over Paths Approach in Favoring the Construction of an Integrated Knowledge of Quantum Physics in High School
ERIC Educational Resources Information Center
Malgieri, Massimiliano; Onorato, Pasquale; De Ambrosis, Anna
2017-01-01
In this paper we present the results of a research-based teaching-learning sequence on introductory quantum physics based on Feynman's sum over paths approach in the Italian high school. Our study focuses on students' understanding of two founding ideas of quantum physics, wave particle duality and the uncertainty principle. In view of recent…
20. Test on the Effectiveness of the Sum over Paths Approach in Favoring the Construction of an Integrated Knowledge of Quantum Physics in High School
ERIC Educational Resources Information Center
Malgieri, Massimiliano; Onorato, Pasquale; De Ambrosis, Anna
2017-01-01
In this paper we present the results of a research-based teaching-learning sequence on introductory quantum physics based on Feynman's sum over paths approach in the Italian high school. Our study focuses on students' understanding of two founding ideas of quantum physics, wave particle duality and the uncertainty principle. In view of recent…
1. Characterizing regulatory path motifs in integrated networks using perturbational data
PubMed Central
2010-01-01
We introduce Pathicular http://bioinformatics.psb.ugent.be/software/details/Pathicular, a Cytoscape plugin for studying the cellular response to perturbations of transcription factors by integrating perturbational expression data with transcriptional, protein-protein and phosphorylation networks. Pathicular searches for 'regulatory path motifs', short paths in the integrated physical networks which occur significantly more often than expected between transcription factors and their targets in the perturbational data. A case study in Saccharomyces cerevisiae identifies eight regulatory path motifs and demonstrates their biological significance. PMID:20230615
2. Dressed coordinates: The path-integral approach
Casana, R.; Flores-Hidalgo, G.; Pimentel, B. M.
2007-02-01
The recently introduced dressed coordinates are studied in the path-integral approach. These coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path-integral functional measure. The analysis also generalizes the sum rules established in a previous work.
3. Asymptotic properties of path integral ideals
SciTech Connect
Bogojevic, A.; Balaz, A.; Belic, A.
2005-09-01
We introduce and analyze an interesting quantity, the path integral ideal, governing the flow of generic discrete theories to the continuum limit and greatly increasing their convergence. The said flow is classified according to the degree of divergence of the potential at spatial infinity. Studying the asymptotic behavior of path integral ideals we isolate the dominant terms in the effective potential that determine the behavior of a generic theory for large discrete time steps.
4. Local-time representation of path integrals.
PubMed
Jizba, Petr; Zatloukal, Václav
2015-12-01
We derive a local-time path-integral representation for a generic one-dimensional time-independent system. In particular, we show how to rephrase the matrix elements of the Bloch density matrix as a path integral over x-dependent local-time profiles. The latter quantify the time that the sample paths x(t) in the Feynman path integral spend in the vicinity of an arbitrary point x. Generalization of the local-time representation that includes arbitrary functionals of the local time is also provided. We argue that the results obtained represent a powerful alternative to the traditional Feynman-Kac formula, particularly in the high- and low-temperature regimes. To illustrate this point, we apply our local-time representation to analyze the asymptotic behavior of the Bloch density matrix at low temperatures. Further salient issues, such as connections with the Sturm-Liouville theory and the Rayleigh-Ritz variational principle, are also discussed.
5. On the path integral of constrained systems
SciTech Connect
Muslih, Sami I.
2004-10-04
Constrained Hamiltonian systems are investigated by using Gueler's method. Integration of a set of equations of motion and the action function is discussed. It is shown that the canonical path integral quantization is obtained directly as an integration over the canonical phase-space coordinates without any need to enlarge the initial phase-space by introducing extra- unphysical variables as in the Batalin-Fradkin-Tyutin (BFT) method. The abelian Proca model is analyzed by the two methods.
6. The path integral for dendritic trees.
PubMed
Abbott, L F; Farhi, E; Gutmann, S
1991-01-01
We construct the path integral for determining the potential on any dendritic tree described by a linear cable equation. This is done by generalizing Brownian motion from a line to a tree. We also construct the path integral for dendritic structures with spatially-varying and/or time-dependent membrane conductivities due, for example, to synaptic inputs. The path integral allows novel computational techniques to be applied to cable problems. Our analysis leads ultimately to an exact expression for the Green's function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules providing a fast and efficient method for solving complex cable problems.
7. Path Integrals on Manifolds with Boundary
Ludewig, Matthias
2017-09-01
We give time-slicing path integral formulas for solutions to the heat equation corresponding to a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold with boundary. More specifically, we show that such a solution can be approximated by integrals over finite-dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval. We consider a subclass of mixed boundary conditions which includes standard Dirichlet and Neumann boundary conditions.
8. Optical tomography with discretized path integral
PubMed Central
Yuan, Bingzhi; Tamaki, Toru; Kushida, Takahiro; Mukaigawa, Yasuhiro; Kubo, Hiroyuki; Raytchev, Bisser; Kaneda, Kazufumi
2015-01-01
Abstract. We present a framework for optical tomography based on a path integral. Instead of directly solving the radiative transport equations, which have been widely used in optical tomography, we use a path integral that has been developed for rendering participating media based on the volume rendering equation in computer graphics. For a discretized two-dimensional layered grid, we develop an algorithm to estimate the extinction coefficients of each voxel with an interior point method. Numerical simulation results are shown to demonstrate that the proposed method works well. PMID:26839903
9. Optical tomography with discretized path integral.
PubMed
Yuan, Bingzhi; Tamaki, Toru; Kushida, Takahiro; Mukaigawa, Yasuhiro; Kubo, Hiroyuki; Raytchev, Bisser; Kaneda, Kazufumi
2015-07-01
We present a framework for optical tomography based on a path integral. Instead of directly solving the radiative transport equations, which have been widely used in optical tomography, we use a path integral that has been developed for rendering participating media based on the volume rendering equation in computer graphics. For a discretized two-dimensional layered grid, we develop an algorithm to estimate the extinction coefficients of each voxel with an interior point method. Numerical simulation results are shown to demonstrate that the proposed method works well.
10. The formal path integral and quantum mechanics
SciTech Connect
Johnson-Freyd, Theo
2010-11-15
Given an arbitrary Lagrangian function on R{sup d} and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by 'Feynman diagrams', although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a 'Fubini theorem' expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by 'cutting and pasting' and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic 'formal path integral' for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.
11. The Use of Path Integral Ideals: Deriving the Euler Summation Formula for Path Integrals
SciTech Connect
Bogojevic, Aleksandar; Balaz, Antun; Belic, Aleksandar
2006-03-29
We present and comment on a new quantity that we have recently introduced: the path integral ideal. The new quantity governs the flow of a discrete quantum theory to its continuum limit. Path integral ideals satisfy a unique integral equation - the distinction between different quantum theories being in the boundary conditions. An asymptotic expansion of this equation has led to the derivation of a generalization of Euler's summation formula for path integrals. The new analytical method has brought about a systematic improvement of the convergence of path integrals. Applied to numerical procedures, the new analytical input has resulted in the speedup of numerical simulations by many orders of magnitude. On the analytical side, the integral equation for ideals may turn out to be a useful setting for extending the obtained results to a wider setting - e.g. to p-adic valued theories and theories on non-commuting space-times.
12. Bead-Fourier path integral molecular dynamics.
PubMed
Ivanov, Sergei D; Lyubartsev, Alexander P; Laaksonen, Aatto
2003-06-01
Molecular dynamics formulation of Bead-Fourier path integral method for simulation of quantum systems at finite temperatures is presented. Within this scheme, both the bead coordinates and Fourier coefficients, defining the path representing the quantum particle, are treated as generalized coordinates with corresponding generalized momenta and masses. Introduction of the Fourier harmonics together with the center-of-mass thermostating scheme is shown to remove the ergodicity problem, known to pose serious difficulties in standard path integral molecular dynamics simulations. The method is tested for quantum harmonic oscillator and hydrogen atom (Coulombic potential). The simulation results are compared with the exact analytical solutions available for both these systems. Convergence of the results with respect to the number of beads and Fourier harmonics is analyzed. It was shown that addition of a few Fourier harmonics already improves the simulation results substantially, even for a relatively small number of beads. The proposed Bead-Fourier path integral molecular dynamics is a reliable and efficient alternative to simulations of quantum systems.
13. Path integration in tactile perception of shapes.
PubMed
Moscatelli, Alessandro; Naceri, Abdeldjallil; Ernst, Marc O
2014-11-01
Whenever we move the hand across a surface, tactile signals provide information about the relative velocity between the skin and the surface. If the system were able to integrate the tactile velocity information over time, cutaneous touch may provide an estimate of the relative displacement between the hand and the surface. Here, we asked whether humans are able to form a reliable representation of the motion path from tactile cues only, integrating motion information over time. In order to address this issue, we conducted three experiments using tactile motion and asked participants (1) to estimate the length of a simulated triangle, (2) to reproduce the shape of a simulated triangular path, and (3) to estimate the angle between two-line segments. Participants were able to accurately indicate the length of the path, whereas the perceived direction was affected by a direction bias (inward bias). The response pattern was thus qualitatively similar to the ones reported in classical path integration studies involving locomotion. However, we explain the directional biases as the result of a tactile motion aftereffect. Copyright © 2014 Elsevier B.V. All rights reserved.
14. Bead-Fourier path integral molecular dynamics
Ivanov, Sergei D.; Lyubartsev, Alexander P.; Laaksonen, Aatto
2003-06-01
Molecular dynamics formulation of Bead-Fourier path integral method for simulation of quantum systems at finite temperatures is presented. Within this scheme, both the bead coordinates and Fourier coefficients, defining the path representing the quantum particle, are treated as generalized coordinates with corresponding generalized momenta and masses. Introduction of the Fourier harmonics together with the center-of-mass thermostating scheme is shown to remove the ergodicity problem, known to pose serious difficulties in standard path integral molecular dynamics simulations. The method is tested for quantum harmonic oscillator and hydrogen atom (Coulombic potential). The simulation results are compared with the exact analytical solutions available for both these systems. Convergence of the results with respect to the number of beads and Fourier harmonics is analyzed. It was shown that addition of a few Fourier harmonics already improves the simulation results substantially, even for a relatively small number of beads. The proposed Bead-Fourier path integral molecular dynamics is a reliable and efficient alternative to simulations of quantum systems.
15. A taxonomy of integral reaction path analysis
SciTech Connect
Grcar, Joseph F.; Day, Marcus S.; Bell, John B.
2004-12-23
W. C. Gardiner observed that achieving understanding through combustion modeling is limited by the ability to recognize the implications of what has been computed and to draw conclusions about the elementary steps underlying the reaction mechanism. This difficulty can be overcome in part by making better use of reaction path analysis in the context of multidimensional flame simulations. Following a survey of current practice, an integral reaction flux is formulated in terms of conserved scalars that can be calculated in a fully automated way. Conditional analyses are then introduced, and a taxonomy for bidirectional path analysis is explored. Many examples illustrate the resulting path analysis and uncover some new results about nonpremixed methane-air laminar jets.
16. An alternative path integral for quantum gravity
Krishnan, Chethan; Kumar, K. V. Pavan; Raju, Avinash
2016-10-01
We define a (semi-classical) path integral for gravity with Neumann boundary conditions in D dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.
17. Path-integral approach to lattice polarons
Kornilovitch, P. E.
2007-06-01
The basic principles behind a path integral approach to the lattice polaron are reviewed. Analytical integration of phonons reduces the problem to one self-interacting imaginary-time path, which is then simulated by Metropolis Monte Carlo. Projection operators separate states of different symmetry, which provides access to various excited states. Shifted boundary conditions in imaginary time enable calculation of the polaron mass, spectrum and density of states. Other polaron characteristics accessible by the method include the polaron energy, number of excited phonons and isotope exponent on mass. Monte Carlo updates are formulated in continuous imaginary time on infinite lattices and as such provide statistically unbiased results without finite-size and finite time-step errors. Numerical data are presented for models with short-range and long-range electron-phonon interactions.
18. Path integration: effect of curved path complexity and sensory system on blindfolded walking.
PubMed
Koutakis, Panagiotis; Mukherjee, Mukul; Vallabhajosula, Srikant; Blanke, Daniel J; Stergiou, Nicholas
2013-02-01
Path integration refers to the ability to integrate continuous information of the direction and distance traveled by the system relative to the origin. Previous studies have investigated path integration through blindfolded walking along simple paths such as straight line and triangles. However, limited knowledge exists regarding the role of path complexity in path integration. Moreover, little is known about how information from different sensory input systems (like vision and proprioception) contributes to accurate path integration. The purpose of the current study was to investigate how sensory information and curved path complexity affect path integration. Forty blindfolded participants had to accurately reproduce a curved path and return to the origin. They were divided into four groups that differed in the curved path, circle (simple) or figure-eight (complex), and received either visual (previously seen) or proprioceptive (previously guided) information about the path before they reproduced it. The dependent variables used were average trajectory error, walking speed, and distance traveled. The results indicated that (a) both groups that walked on a circular path and both groups that received visual information produced greater accuracy in reproducing the path. Moreover, the performance of the group that received proprioceptive information and later walked on a figure-eight path was less accurate than their corresponding circular group. The groups that had the visual information also walked faster compared to the group that had proprioceptive information. Results of the current study highlight the roles of different sensory inputs while performing blindfolded walking for path integration.
19. Quantitative molecular thermochemistry based on path integrals.
PubMed
Glaesemann, Kurt R; Fried, Laurence E
2005-07-15
The calculation of thermochemical data requires accurate molecular energies and heat capacities. Traditional methods rely upon the standard harmonic normal-mode analysis to calculate the vibrational and rotational contributions. We utilize path-integral Monte Carlo for going beyond the harmonic analysis and to calculate the vibrational and rotational contributions to ab initio energies. This is an application and an extension of a method previously developed in our group [J. Chem. Phys. 118, 1596 (2003)].
20. Quantitative Molecular Thermochemistry Based on Path Integrals
SciTech Connect
Glaesemann, K R; Fried, L E
2005-03-14
The calculation of thermochemical data requires accurate molecular energies and heat capacities. Traditional methods rely upon the standard harmonic normal mode analysis to calculate the vibrational and rotational contributions. We utilize path integral Monte Carlo (PIMC) for going beyond the harmonic analysis, to calculate the vibrational and rotational contributions to ab initio energies. This is an application and extension of a method previously developed in our group.
1. Systematically Accelerated Convergence of Path Integrals
SciTech Connect
Bogojevic, A.; Balaz, A.; Belic, A.
2005-05-13
We present a new analytical method that systematically improves the convergence of path integrals of a generic N-fold discretized theory. Using it we calculate the effective actions S{sup (p)} for p{<=}9, which lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit as 1/N{sup p}. We checked this derived speedup in convergence by performing Monte Carlo simulations on several different models.
2. Fractional Levy motion through path integrals
SciTech Connect
Calvo, Ivan; Sanchez, Raul; Carreras, Benjamin A
2009-01-01
Fractional Levy motion (fLm) is the natural generalization of fractional Brownian motion in the context of self-similar stochastic processes and stable probability distributions. In this paper we give an explicit derivation of the propagator of fLm by using path integral methods. The propagators of Brownian motion and fractional Brownian motion are recovered as particular cases. The fractional diffusion equation corresponding to fLm is also obtained.
3. Path integration in desert ants, Cataglyphis fortis
PubMed Central
Müller, Martin; Wehner, Rüdiger
1988-01-01
Foraging desert ants, Cataglyphis fortis, continually keep track of their own posotions relative to home— i.e., integrate their tortuous outbound routes and return home along straight (inbound) routes. By experimentally manipulating the ants' outbound trajectories we show that the ants solve this path integration problem not by performing a true vector summation (as a human navigator does) but by employing a computationally simple approximation. This approximation is characterized by small, but systematic, navigational errors that helped us elucidate the ant's way of computing its mean home vector. PMID:16593958
4. BOOK REVIEW: Path Integrals in Field Theory: An Introduction
Ryder, Lewis
2004-06-01
In the 1960s Feynman was known to particle physicists as one of the people who solved the major problems of quantum electrodynamics, his contribution famously introducing what are now called Feynman diagrams. To other physicists he gained a reputation as the author of the Feynman Lectures on Physics; in addition some people were aware of his work on the path integral formulation of quantum theory, and a very few knew about his work on gravitation and Yang--Mills theories, which made use of path integral methods. Forty years later the scene is rather different. Many of the problems of high energy physics are solved; and the standard model incorporates Feynman's path integral method as a way of proving the renormalisability of the gauge (Yang--Mills) theories involved. Gravitation is proving a much harder nut to crack, but here also questions of renormalisability are couched in path-integral language. What is more, theoretical studies of condensed matter physics now also appeal to this technique for quantisation, so the path integral method is becoming part of the standard apparatus of theoretical physics. Chapters on it appear in a number of recent books, and a few books have appeared devoted to this topic alone; the book under review is a very recent one. Path integral techniques have the advantage of enormous conceptual appeal and the great disadvantage of mathematical complexity, this being partly the result of messy integrals but more fundamentally due to the notions of functional differentiation and integration which are involved in the method. All in all this subject is not such an easy ride. Mosel's book, described as an introduction, is aimed at graduate students and research workers in particle physics. It assumes a background knowledge of quantum mechanics, both non-relativistic and relativistic. After three chapters on the path integral formulation of non-relativistic quantum mechanics there are eight chapters on scalar and spinor field theory, followed
5. Master equations and the theory of stochastic path integrals.
PubMed
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
6. Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
7. Breakdown of the Coherent State Path Integral: Two Simple Examples
SciTech Connect
Wilson, Justin H.; Galitski, Victor
2011-03-18
We show how the time-continuous coherent state path integral breaks down for both the single-site Bose-Hubbard model and the spin-path integral. Specifically, when the Hamiltonian is quadratic in a generator of the algebra used to construct coherent states, the path integral fails to produce correct results following from an operator approach. As suggested by previous authors, we note that the problems do not arise in the time-discretized version of the path integral.
8. Path integral evaluation of equilibrium isotope effects
Zimmermann, Tomáš; Vaníček, Jiří
2009-07-01
A general and rigorous methodology to compute the quantum equilibrium isotope effect is described. Unlike standard approaches, ours does not assume separability of rotational and vibrational motions and does not make the harmonic approximation for vibrations or rigid rotor approximation for the rotations. In particular, zero point energy and anharmonicity effects are described correctly quantum mechanically. The approach is based on the thermodynamic integration with respect to the mass of isotopes and on the Feynman path integral representation of the partition function. An efficient estimator for the derivative of free energy is used whose statistical error is independent of the number of imaginary time slices in the path integral, speeding up calculations by a factor of ˜60 at 500 K and more at room temperature. We describe the implementation of the methodology in the molecular dynamics package AMBER 10. The method is tested on three [1,5] sigmatropic hydrogen shift reactions. Because of the computational expense, we use ab initio potentials to evaluate the equilibrium isotope effects within the harmonic approximation and then the path integral method together with semiempirical potentials to evaluate the anharmonicity corrections. Our calculations show that the anharmonicity effects amount up to 30% of the symmetry reduced reaction free energy. The numerical results are compared with recent experiments of Doering et al., [J. Am. Chem. Soc. 128, 9080 (2006); J. Am. Chem. Soc.129, 2488 (2007)] confirming the accuracy of the most recent measurement on 2,4,6,7,9-pentamethyl-5-(5,5-H22)methylene-11,11a-dihydro-12H-naphthacene as well as concerns about compromised accuracy, due to side reactions, of another measurement on 2-methyl-10-(10,10-H22)methylenebicyclo[4.4.0]dec-1-ene.
9. Path integral quantization of generalized quantum electrodynamics
SciTech Connect
Bufalo, R.; Pimentel, B. M.; Zambrano, G. E. R.
2011-02-15
In this paper, a complete covariant quantization of generalized electrodynamics is shown through the path integral approach. To this goal, we first studied the Hamiltonian structure of the system following Dirac's methodology and, then, we followed the Faddeev-Senjanovic procedure to obtain the transition amplitude. The complete propagators (Schwinger-Dyson-Fradkin equations) of the correct gauge fixation and the generalized Ward-Fradkin-Takahashi identities are also obtained. Afterwards, an explicit calculation of one-loop approximations of all Green's functions and a discussion about the obtained results are presented.
10. Path integral quantization of generalized quantum electrodynamics
Bufalo, R.; Pimentel, B. M.; Zambrano, G. E. R.
2011-02-01
In this paper, a complete covariant quantization of generalized electrodynamics is shown through the path integral approach. To this goal, we first studied the Hamiltonian structure of the system following Dirac’s methodology and, then, we followed the Faddeev-Senjanovic procedure to obtain the transition amplitude. The complete propagators (Schwinger-Dyson-Fradkin equations) of the correct gauge fixation and the generalized Ward-Fradkin-Takahashi identities are also obtained. Afterwards, an explicit calculation of one-loop approximations of all Green’s functions and a discussion about the obtained results are presented.
11. Efficient stochastic thermostatting of path integral molecular dynamics
Ceriotti, Michele; Parrinello, Michele; Markland, Thomas E.; Manolopoulos, David E.
2010-09-01
The path integral molecular dynamics (PIMD) method provides a convenient way to compute the quantum mechanical structural and thermodynamic properties of condensed phase systems at the expense of introducing an additional set of high frequency normal modes on top of the physical vibrations of the system. Efficiently sampling such a wide range of frequencies provides a considerable thermostatting challenge. Here we introduce a simple stochastic path integral Langevin equation (PILE) thermostat which exploits an analytic knowledge of the free path integral normal mode frequencies. We also apply a recently developed colored noise thermostat based on a generalized Langevin equation (GLE), which automatically achieves a similar, frequency-optimized sampling. The sampling efficiencies of these thermostats are compared with that of the more conventional Nosé-Hoover chain (NHC) thermostat for a number of physically relevant properties of the liquid water and hydrogen-in-palladium systems. In nearly every case, the new PILE thermostat is found to perform just as well as the NHC thermostat while allowing for a computationally more efficient implementation. The GLE thermostat also proves to be very robust delivering a near-optimum sampling efficiency in all of the cases considered. We suspect that these simple stochastic thermostats will therefore find useful application in many future PIMD simulations.
12. On the Path Integral in Non-Commutative (nc) Qft
Dehne, Christoph
2008-09-01
As is generally known, different quantization schemes applied to field theory on NC spacetime lead to Feynman rules with different physical properties, if time does not commute with space. In particular, the Feynman rules that are derived from the path integral corresponding to the T*-product (the so-called naïve Feynman rules) violate the causal time ordering property. Within the Hamiltonian approach to quantum field theory, we show that we can (formally) modify the time ordering encoded in the above path integral. The resulting Feynman rules are identical to those obtained in the canonical approach via the Gell-Mann-Low formula (with T-ordering). They preserve thus unitarity and causal time ordering.
13. High order path integrals made easy
Kapil, Venkat; Behler, Jörg; Ceriotti, Michele
2016-12-01
The precise description of quantum nuclear fluctuations in atomistic modelling is possible by employing path integral techniques, which involve a considerable computational overhead due to the need of simulating multiple replicas of the system. Many approaches have been suggested to reduce the required number of replicas. Among these, high-order factorizations of the Boltzmann operator are particularly attractive for high-precision and low-temperature scenarios. Unfortunately, to date, several technical challenges have prevented a widespread use of these approaches to study the nuclear quantum effects in condensed-phase systems. Here we introduce an inexpensive molecular dynamics scheme that overcomes these limitations, thus making it possible to exploit the improved convergence of high-order path integrals without having to sacrifice the stability, convenience, and flexibility of conventional second-order techniques. The capabilities of the method are demonstrated by simulations of liquid water and ice, as described by a neural-network potential fitted to the dispersion-corrected hybrid density functional theory calculations.
14. Building a cognitive map by assembling multiple path integration systems.
PubMed
Wang, Ranxiao Frances
2016-06-01
Path integration and cognitive mapping are two of the most important mechanisms for navigation. Path integration is a primitive navigation system which computes a homing vector based on an animal's self-motion estimation, while cognitive map is an advanced spatial representation containing richer spatial information about the environment that is persistent and can be used to guide flexible navigation to multiple locations. Most theories of navigation conceptualize them as two distinctive, independent mechanisms, although the path integration system may provide useful information for the integration of cognitive maps. This paper demonstrates a fundamentally different scenario, where a cognitive map is constructed in three simple steps by assembling multiple path integrators and extending their basic features. The fact that a collection of path integration systems can be turned into a cognitive map suggests the possibility that cognitive maps may have evolved directly from the path integration system.
15. Path Integral Monte Carlo Methods for Fermions
Ethan, Ethan; Dubois, Jonathan; Ceperley, David
2014-03-01
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems. Support from DOE DE-FG52-09NA29456, DE-AC52-07NA27344, LLNL LDRD 10- ERD-058, and the Lawrence Scholar program.
16. Path Integral Simulations of Solid ^4He
Ceperley, David
2006-03-01
Kim and Chan have found indications that solid ^4He is a supersolid by measuring the period of a torsional oscillator. To understand the state of solid ^4He at low temperature, we [1] have calculated tunnelling frequencies for ring exchanges in bulk solid helium with Path Integral Monte Carlo by finding the free energy of a path that begins with the atoms in one configuration and ends with a permutation of those positions. The exchange frequencies are found to be described by a lattice model which does not show superfluid behavior. However, simulations [2] of ^4He absorbed in Vycor find that ^4He forms a layered structure with the first layer solid-like and highly localized, the second layer disordered with some atoms delocalized and possibly superfluid. This persistent liquid layer mechanism can only be relevant for bulk ^4He in a very disordered crystal. New calculations [3] of the single particle density matrix, (the fourier transform of the momentum distribution) to measure ODLRA will also be discussed. [1] D. M. Ceperley and B. Bernu, Phys. Rev. Letts. 93, 155303(2004).[2] S. A. Khairallah and D. M. Ceperley, Phys. Rev. Letts. 95, 185301 (2005). [3] B. Clark and D. M. Ceperley, unpublished.
17. The path integral picture of quantum systems
Ceperley, David
2011-03-01
The imaginary time path integral formalism'' was introduced in 1953 by Feynman to understand the superfluid transition in liquid helium. The equilibrium properties of quantum many body systems is isomorphic to the classical statistical mechanics of cross-linking polymer-like objects. With the Markov Chain Monte Carlo method, invented by Metropolis et al., also in 1953, a potential way of calculating properties of correlated quantum systems was in place. But calculations for many-body quantum systems did not become routine until computers and algorithms had become sufficiently powerful three decades later. Once such simulations could happen, it was realized that simulations provided a deeper insight into boson superfluids, in particular the relation of bose condensation to the polymer end-to-end distance, and the superfluid density to the polymer winding number.'' Some recent developments and applications to supersolids, and helium droplets will be given. Finally, limitations of the methodology e.g. to fermion systems are discussed.
18. An automated integration-free path-integral method based on Kleinert's variational perturbation theory
Wong, Kin-Yiu; Gao, Jiali
2007-12-01
Based on Kleinert's variational perturbation (KP) theory [Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. (World Scientific, Singapore, 2004)], we present an analytic path-integral approach for computing the effective centroid potential. The approach enables the KP theory to be applied to any realistic systems beyond the first-order perturbation (i.e., the original Feynman-Kleinert [Phys. Rev. A 34, 5080 (1986)] variational method). Accurate values are obtained for several systems in which exact quantum results are known. Furthermore, the computed kinetic isotope effects for a series of proton transfer reactions, in which the potential energy surfaces are evaluated by density-functional theory, are in good accordance with experiments. We hope that our method could be used by non-path-integral experts or experimentalists as a "black box" for any given system.
19. Mixed time slicing in path integral simulations
Steele, Ryan P.; Zwickl, Jill; Shushkov, Philip; Tully, John C.
2011-02-01
A simple and efficient scheme is presented for using different time slices for different degrees of freedom in path integral calculations. This method bridges the gap between full quantization and the standard mixed quantum-classical (MQC) scheme and, therefore, still provides quantum mechanical effects in the less-quantized variables. Underlying the algorithm is the notion that time slices (beads) may be "collapsed" in a manner that preserves quantization in the less quantum mechanical degrees of freedom. The method is shown to be analogous to multiple-time step integration techniques in classical molecular dynamics. The algorithm and its associated error are demonstrated on model systems containing coupled high- and low-frequency modes; results indicate that convergence of quantum mechanical observables can be achieved with disparate bead numbers in the different modes. Cost estimates indicate that this procedure, much like the MQC method, is most efficient for only a relatively few quantum mechanical degrees of freedom, such as proton transfer. In this regime, however, the cost of a fully quantum mechanical simulation is determined by the quantization of the least quantum mechanical degrees of freedom.
20. Optimal combination of environmental cues and path integration during navigation.
PubMed
Sjolund, Lori A; Kelly, Jonathan W; McNamara, Timothy P
2017-08-21
Navigation is influenced by body-based self-motion cues that are integrated over time, in a process known as path integration, as well as by environmental cues such as landmarks and room shape. In two experiments we explored whether humans combine path integration and environmental cues (Exp. 1: room shape; Exp. 2: room shape, single landmark, and multiple landmarks) to reduce response variability when returning to a previously visited location. Participants walked an outbound path in an immersive virtual environment before attempting to return to the path origin. Path integration and an environmental cue were both available during the outbound path, but experimental manipulations created single- and dual-cue conditions during the return path. The response variance when returning to the path origin was reduced when both cues were available, consistent with optimal integration predicted on the basis of Bayesian principles. The findings indicate that humans optimally integrate multiple spatial cues during navigation. Additionally, a large (but not a small) cue conflict caused participants to assign a higher weight to path integration than to environmental cues, despite the relatively greater precision afforded by the environmental cues.
1. Path-Integral Derivation of Lifshitz Tails
Sa-Yakanit, V.
2008-11-01
The behavior of the electron density of states (DOS) for the Lifshitz tail states is studied in the limit of low energy using the Feynman path-integral method. This method was used to study the heavily doped semiconductors for the case of a Gaussian random potential. The main results obtained are that the tail states behave as DOS exp (-B(E)), with B(E) = En, n = {1 / 2} for short-range interaction and n = 2 for long-range interaction. In this study it is shown that without the Gaussian approximation, the behavior of the Lifshitz tails for the Poisson distribution is obtained as DOS exp (-B(E)) with B(E) = En, n = {3 / 2} . As in the case of heavily doped semiconductor, the method can be easily generalized to long-range interactions. A comparison with the method developed by Friedberg and Luttinger based on the reformulation of the problem in terms of Brownian motion is given.
2. Differential neural network configuration during human path integration
PubMed Central
Arnold, Aiden E. G. F; Burles, Ford; Bray, Signe; Levy, Richard M.; Iaria, Giuseppe
2014-01-01
Path integration is a fundamental skill for navigation in both humans and animals. Despite recent advances in unraveling the neural basis of path integration in animal models, relatively little is known about how path integration operates at a neural level in humans. Previous attempts to characterize the neural mechanisms used by humans to visually path integrate have suggested a central role of the hippocampus in allowing accurate performance, broadly resembling results from animal data. However, in recent years both the central role of the hippocampus and the perspective that animals and humans share similar neural mechanisms for path integration has come into question. The present study uses a data driven analysis to investigate the neural systems engaged during visual path integration in humans, allowing for an unbiased estimate of neural activity across the entire brain. Our results suggest that humans employ common task control, attention and spatial working memory systems across a frontoparietal network during path integration. However, individuals differed in how these systems are configured into functional networks. High performing individuals were found to more broadly express spatial working memory systems in prefrontal cortex, while low performing individuals engaged an allocentric memory system based primarily in the medial occipito-temporal region. These findings suggest that visual path integration in humans over short distances can operate through a spatial working memory system engaging primarily the prefrontal cortex and that the differential configuration of memory systems recruited by task control networks may help explain individual biases in spatial learning strategies. PMID:24808849
3. Differential neural network configuration during human path integration.
PubMed
Arnold, Aiden E G F; Burles, Ford; Bray, Signe; Levy, Richard M; Iaria, Giuseppe
2014-01-01
Path integration is a fundamental skill for navigation in both humans and animals. Despite recent advances in unraveling the neural basis of path integration in animal models, relatively little is known about how path integration operates at a neural level in humans. Previous attempts to characterize the neural mechanisms used by humans to visually path integrate have suggested a central role of the hippocampus in allowing accurate performance, broadly resembling results from animal data. However, in recent years both the central role of the hippocampus and the perspective that animals and humans share similar neural mechanisms for path integration has come into question. The present study uses a data driven analysis to investigate the neural systems engaged during visual path integration in humans, allowing for an unbiased estimate of neural activity across the entire brain. Our results suggest that humans employ common task control, attention and spatial working memory systems across a frontoparietal network during path integration. However, individuals differed in how these systems are configured into functional networks. High performing individuals were found to more broadly express spatial working memory systems in prefrontal cortex, while low performing individuals engaged an allocentric memory system based primarily in the medial occipito-temporal region. These findings suggest that visual path integration in humans over short distances can operate through a spatial working memory system engaging primarily the prefrontal cortex and that the differential configuration of memory systems recruited by task control networks may help explain individual biases in spatial learning strategies.
4. Path integrals and the WKB approximation in loop quantum cosmology
Ashtekar, Abhay; Campiglia, Miguel; Henderson, Adam
2010-12-01
We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.
5. A note on the path integral representation for Majorana fermions
Greco, Andrés
2016-04-01
Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.
6. The path integral formulation of climate dynamics.
PubMed
Navarra, Antonio; Tribbia, Joe; Conti, Giovanni
2013-01-01
The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent) attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model.
7. The Path Integral Formulation of Climate Dynamics
PubMed Central
Navarra, Antonio; Tribbia, Joe; Conti, Giovanni
2013-01-01
The chaotic nature of the atmospheric dynamics has stimulated the applications of methods and ideas derived from statistical dynamics. For instance, ensemble systems are used to make weather predictions recently extensive, which are designed to sample the phase space around the initial condition. Such an approach has been shown to improve substantially the usefulness of the forecasts since it allows forecasters to issue probabilistic forecasts. These works have modified the dominant paradigm of the interpretation of the evolution of atmospheric flows (and oceanic motions to some extent) attributing more importance to the probability distribution of the variables of interest rather than to a single representation. The ensemble experiments can be considered as crude attempts to estimate the evolution of the probability distribution of the climate variables, which turn out to be the only physical quantity relevant to practice. However, little work has been done on a direct modeling of the probability evolution itself. In this paper it is shown that it is possible to write the evolution of the probability distribution as a functional integral of the same kind introduced by Feynman in quantum mechanics, using some of the methods and results developed in statistical physics. The approach allows obtaining a formal solution to the Fokker-Planck equation corresponding to the Langevin-like equation of motion with noise. The method is very general and provides a framework generalizable to red noise, as well as to delaying differential equations, and even field equations, i.e., partial differential equations with noise, for example, general circulation models with noise. These concepts will be applied to an example taken from a simple ENSO model. PMID:23840577
8. A Note on Feynman Path Integral for Electromagnetic External Fields
Botelho, Luiz C. L.
2017-08-01
We propose a Fresnel stochastic white noise framework to analyze the nature of the Feynman paths entering on the Feynman Path Integral expression for the Feynman Propagator of a particle quantum mechanically moving under an external electromagnetic time-independent potential.
9. The application of path integral for log return probability calculation
Palupi, D. S.; Hermanto, A.; Tenderlilin, E.; Rosyid, M. F.
2014-10-01
Log return probability has been calculated using path integral method. The stock price is assumed obeying the stochastic differential equation of a geometric Brownian motion and the volatility is assumed following Ornstein Uhlenbeck process. The stochastic differential equation of stock price and volatility lead to Fokker-Plank equation. The Fokker-Plank equation is solved using path integral method. Distribution of log return can be used to take the valuation ln return stock.
10. Variational path integral molecular dynamics study of a water molecule
Miura, Shinichi
2013-08-01
In the present study, a variational path integral molecular dynamics method developed by the author [Chem. Phys. Lett. 482, 165 (2009)] is applied to a water molecule on the adiabatic potential energy surface. The method numerically generates an exact wavefunction using a trial wavefunction of the target system. It has been shown that even if a poor trial wavefunction is employed, the exact quantum distribution is numerically extracted, demonstrating the robustness of the variational path integral method.
11. Sensory feedback in a bump attractor model of path integration.
PubMed
Poll, Daniel B; Nguyen, Khanh; Kilpatrick, Zachary P
2016-04-01
Mammalian spatial navigation systems utilize several different sensory information channels. This information is converted into a neural code that represents the animal's current position in space by engaging place cell, grid cell, and head direction cell networks. In particular, sensory landmark (allothetic) cues can be utilized in concert with an animal's knowledge of its own velocity (idiothetic) cues to generate a more accurate representation of position than path integration provides on its own (Battaglia et al. The Journal of Neuroscience 24(19):4541-4550 (2004)). We develop a computational model that merges path integration with feedback from external sensory cues that provide a reliable representation of spatial position along an annular track. Starting with a continuous bump attractor model, we explore the impact of synaptic spatial asymmetry and heterogeneity, which disrupt the position code of the path integration process. We use asymptotic analysis to reduce the bump attractor model to a single scalar equation whose potential represents the impact of asymmetry and heterogeneity. Such imperfections cause errors to build up when the network performs path integration, but these errors can be corrected by an external control signal representing the effects of sensory cues. We demonstrate that there is an optimal strength and decay rate of the control signal when cues appear either periodically or randomly. A similar analysis is performed when errors in path integration arise from dynamic noise fluctuations. Again, there is an optimal strength and decay of discrete control that minimizes the path integration error.
12. Path Integrals and Exotic Options:. Methods and Numerical Results
Bormetti, G.; Montagna, G.; Moreni, N.; Nicrosini, O.
2005-09-01
In the framework of Black-Scholes-Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price path dependent options on multidimensional and correlated underlying assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the case of Asian call options. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at the money (ATM) and out of the money (OTM) options, path integral exhibits competitive performances.
13. Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes
Pérez, Alejandro; Tuckerman, Mark E.
2011-08-01
Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.
14. Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes.
PubMed
Pérez, Alejandro; Tuckerman, Mark E
2011-08-14
Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.
15. An Access Path Model for Physical Database Design.
DTIC Science & Technology
1979-12-28
target system. 4.1 Algebraic Structure for Physical Design For the purposes of implementation-oriented design, we shall use the logical access paths...subsection, we present an algorithm for gen- erating a maximal labelling that specifies superior support for the access paths most heavily travelled. Assume...A.C.M. SIGMOD Conf., (May 79). [CARD731 Cardenas , A. F., "Evaluation and Selection of File Organization - A Model and a System," Comm. A.C.M., V 16, N
16. Alternative paths to observing Efimov physics
Esry, B. D.; Wang, Yujun; D'Incao, J. P.
2008-12-01
We present some of our work on the ultracold behavior of three-body collisions and their relation to the recent experiment (Kraemer [1]). In particular, we discuss the role of Efimov physics in this experiment and other ultracold three-body collisions. We also suggest one way to make observation of the key feature of the Efimov effect - geometrical scaling - more experimentally feasible.
17. A subterranean mammal uses the magnetic compass for path integration.
PubMed
Kimchi, Tali; Etienne, Ariane S; Terkel, Joseph
2004-01-27
Path integration allows animals to navigate without landmarks by continuously processing signals generated through locomotion. Insects such as bees and ants have evolved an accurate path integration system, assessing and coding rotations with the help of a general directional reference, the sun azimuth. In mammals, by contrast, this process can take place through purely idiothetic (mainly proprioceptive and vestibular) signals. However, without any stable external reference for measuring direction, path integration is highly affected by cumulative errors and thus has been considered so far as valid only for short-distance navigation. Here we show through two path integration experiments (homing and shortcut finding) that the blind mole rat assesses direction both through internal signals and by estimating its heading in relation to the earth's magnetic field. Further, it is shown that the greater the circumvolution and length of the traveled path, the more the animal relies on the geomagnetic field. This path integration system strongly reduces the accumulation of errors due to inaccuracies in the estimation of rotations and thus allows the mole rat to navigate efficiently in darkness, without the help of any landmark, over both short and long distances.
18. The perturbative approach to path integrals: A succinct mathematical treatment
Nguyen, Timothy
2016-09-01
We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal parameter irrespective of convergence properties. We establish invariance properties of such a Wick expansion under coordinate changes and the action of a Lie group of symmetries, and we use this to study essential features of path integral manipulations, including coordinate changes, Ward identities, Schwinger-Dyson equations, Faddeev-Popov gauge-fixing, and eliminating fields by their equation of motion. We also discuss the asymptotic nature of the Wick expansion and the implications this has for defining path integrals perturbatively and nonperturbatively.
19. Medial Temporal Lobe Roles in Human Path Integration
PubMed Central
Yamamoto, Naohide; Philbeck, John W.; Woods, Adam J.; Gajewski, Daniel A.; Arthur, Joeanna C.; Potolicchio, Samuel J.; Levy, Lucien; Caputy, Anthony J.
2014-01-01
Path integration is a process in which observers derive their location by integrating self-motion signals along their locomotion trajectory. Although the medial temporal lobe (MTL) is thought to take part in path integration, the scope of its role for path integration remains unclear. To address this issue, we administered a variety of tasks involving path integration and other related processes to a group of neurosurgical patients whose MTL was unilaterally resected as therapy for epilepsy. These patients were unimpaired relative to neurologically intact controls in many tasks that required integration of various kinds of sensory self-motion information. However, the same patients (especially those who had lesions in the right hemisphere) walked farther than the controls when attempting to walk without vision to a previewed target. Importantly, this task was unique in our test battery in that it allowed participants to form a mental representation of the target location and anticipate their upcoming walking trajectory before they began moving. Thus, these results put forth a new idea that the role of MTL structures for human path integration may stem from their participation in predicting the consequences of one's locomotor actions. The strengths of this new theoretical viewpoint are discussed. PMID:24802000
20. Medial temporal lobe roles in human path integration.
PubMed
Yamamoto, Naohide; Philbeck, John W; Woods, Adam J; Gajewski, Daniel A; Arthur, Joeanna C; Potolicchio, Samuel J; Levy, Lucien; Caputy, Anthony J
2014-01-01
Path integration is a process in which observers derive their location by integrating self-motion signals along their locomotion trajectory. Although the medial temporal lobe (MTL) is thought to take part in path integration, the scope of its role for path integration remains unclear. To address this issue, we administered a variety of tasks involving path integration and other related processes to a group of neurosurgical patients whose MTL was unilaterally resected as therapy for epilepsy. These patients were unimpaired relative to neurologically intact controls in many tasks that required integration of various kinds of sensory self-motion information. However, the same patients (especially those who had lesions in the right hemisphere) walked farther than the controls when attempting to walk without vision to a previewed target. Importantly, this task was unique in our test battery in that it allowed participants to form a mental representation of the target location and anticipate their upcoming walking trajectory before they began moving. Thus, these results put forth a new idea that the role of MTL structures for human path integration may stem from their participation in predicting the consequences of one's locomotor actions. The strengths of this new theoretical viewpoint are discussed.
1. Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations
SciTech Connect
Weinstein, Marvin; /SLAC
2009-02-12
It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will
2. Low level constraints on dynamic contour path integration.
PubMed
Hall, Sophie; Bourke, Patrick; Guo, Kun
2014-01-01
Contour integration is a fundamental visual process. The constraints on integrating discrete contour elements and the associated neural mechanisms have typically been investigated using static contour paths. However, in our dynamic natural environment objects and scenes vary over space and time. With the aim of investigating the parameters affecting spatiotemporal contour path integration, we measured human contrast detection performance of a briefly presented foveal target embedded in dynamic collinear stimulus sequences (comprising five short 'predictor' bars appearing consecutively towards the fovea, followed by the 'target' bar) in four experiments. The data showed that participants' target detection performance was relatively unchanged when individual contour elements were separated by up to 2° spatial gap or 200 ms temporal gap. Randomising the luminance contrast or colour of the predictors, on the other hand, had similar detrimental effect on grouping dynamic contour path and subsequent target detection performance. Randomising the orientation of the predictors reduced target detection performance greater than introducing misalignment relative to the contour path. The results suggest that the visual system integrates dynamic path elements to bias target detection even when the continuity of path is disrupted in terms of spatial (2°), temporal (200 ms), colour (over 10 colours) and luminance (-25% to 25%) information. We discuss how the findings can be largely reconciled within the functioning of V1 horizontal connections.
3. A whole-path importance-sampling scheme for Feynman path integral calculations of absolute partition functions and free energies.
PubMed
Mielke, Steven L; Truhlar, Donald G
2016-01-21
Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function.
4. Canonical formulation and path integral for local vacuum energy sequestering
Bufalo, R.; KlusoÅ, J.; Oksanen, M.
2016-08-01
We establish the Hamiltonian analysis and the canonical path integral for a local formulation of vacuum energy sequestering. In particular, by considering the state of the Universe as a superposition of vacuum states corresponding to different values of the cosmological and gravitational constants, the path integral is extended to include integrations over the cosmological and gravitational constants. The result is an extension of the Ng-van Dam form of the path integral of unimodular gravity. It is argued to imply a relation between the fraction of the most likely values of the gravitational and cosmological constants and the average values of the energy density and pressure of matter over spacetime. Finally, we construct and analyze a Becchi-Rouet-Stora-Tyutin-exact formulation of the theory, which can be considered as a topological field theory.
5. Linearized path integral approach for calculating nonadiabatic time correlation functions.
PubMed
Bonella, Sara; Montemayor, Daniel; Coker, David F
2005-05-10
We show that quantum time correlation functions including electronically nonadiabatic effects can be computed by using an approach in which their path integral expression is linearized in the difference between forward and backward nuclear paths while the electronic component of the amplitude, represented in the mapping formulation, can be computed exactly, leading to classical-like equations of motion for all degrees of freedom. The efficiency of this approach is demonstrated in some simple model applications.
6. Integrated Schools: Finding a New Path
ERIC Educational Resources Information Center
Orfield, Gary; Frankenberg, Erica; Siegel-Hawley, Genevieve
2010-01-01
Research shows that schools remain a powerful tool for shoring up individual opportunity and for attaining a thriving, multiracial democratic society. The authors point to social science evidence that demonstrates how segregated schooling limits the prospects of both minority and majority students and how integrated education can close the…
7. Path integral representation for polymer quantized scalar fields
Kajuri, Nirmalya
2015-12-01
According to loop quantum gravity, matter fields must be quantized in a background-independent manner. For scalar fields, such a background-independent quantization is called polymer quantization and is inequivalent to the standard Schrödinger quantization. It is therefore important to obtain predictions from the polymer quantized scalar field theory and compare with the standard results. As a step towards this, we develop a path integral representation for the polymer quantized scalar field. We notice several crucial differences from the path integral for the Schrödinger quantized scalar field. One important difference is the appearance of an extra summation at each point in the path integral for the polymer quantized theory. A second crucial difference is the loss of manifest Lorentz symmetry for a polymer quantized theory on Minkowski space.
8. Faddeev-Jackiw quantization and the path integral
Toms, David J.
2015-11-01
The method for quantization of constrained theories that was suggested originally by Faddeev and Jackiw along with later modifications is discussed. The particular emphasis of this paper is to show how it is simple to implement their method within the path integral framework using the natural geometric structure that their method utilizes. The procedure is exemplified with the analysis of two models: a quantum mechanical particle constrained to a surface (of which the hypersphere is a special case), and a quantized Schrödinger field interacting with a quantized vector field for both the massive and the massless cases. The results are shown to agree with what is found using the Dirac method for constrained path integrals. We comment on a previous path integral analysis of the Faddeev-Jackiw method. We also discuss why a previous criticism of the Faddeev-Jackiw method is unfounded and why suggested modifications of their method are unnecessary.
9. Predator-prey quasicycles from a path-integral formalism.
PubMed
Butler, Thomas; Reynolds, David
2009-03-01
The existence of beyond mean-field quasicycle oscillations in a simple spatial model of predator-prey interactions is derived from a path-integral formalism. The results agree substantially with those obtained from analysis of similar models using system size expansions of the master equation. In all of these analyses, the discrete nature of predator-prey populations and finite-size effects lead to persistent oscillations in time, but spatial patterns fail to form. The path-integral formalism goes beyond mean-field theory and provides a focus on individual realizations of the stochastic time evolution of population not captured in the standard master-equation approach.
10. Action with Acceleration i: Euclidean Hamiltonian and Path Integral
Baaquie, Belal E.
2013-10-01
An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.
11. Quantum-classical path integral with self-consistent solvent-driven reference propagators.
PubMed
Banerjee, Tuseeta; Makri, Nancy
2013-10-24
Efficient procedures for evaluating the quantum-classical path integral (QCPI) [J. Chem. Phys. 2013, 137, 22A552] are described. The main idea is to identify a trajectory-specific reference Hamiltonian that captures the dominant effects of the classical "solvent" degrees of freedom on the dynamics of the quantum "system". This time-dependent reference is used to construct a system propagator that is valid for large time increments. Residual "quantum memory" interactions are included via the path integral representation of the density matrix, which converges with large time steps. Two physically motivated reference schemes are considered. The first involves the dynamics of the solvent unperturbed by the system, which forms the basis for the "classical path" approximation. The second is based on solvent trajectories determined self-consistently with the evolution of the system, according to the time-dependent self-consistent field or Ehrenfest model. Application to dissipative two-level systems indicates that both reference schemes allow a substantial increase of the path integral time step, leading to rapid convergence of the path sum. In addition, the time-dependent reference propagators automatically weigh state-to-state coupling against solvent reorganization in the determination of transition probabilities, further enhancing the convergence of the path integral.
12. Looping probabilities of elastic chains: a path integral approach.
PubMed
2010-11-01
We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end
13. Linear Path Integration Deficits in Patients with Abnormal Vestibular Afference
PubMed Central
Arthur, Joeanna C.; Kortte, Kathleen B.; Shelhamer, Mark; Schubert, Michael C.
2014-01-01
Effective navigation requires the ability to keep track of one’s location and maintain orientation during linear and angular displacements. Path integration is the process of updating the representation of body position by integrating internally-generated self-motion signals over time (e.g., walking in the dark). One major source of input to path integration is vestibular afference. We tested patients with reduced vestibular function (unilateral vestibular hypofunction, UVH), patients with aberrant vestibular function (benign paroxysmal positional vertigo, BPPV), and healthy participants (controls) on two linear path integration tasks: experimenter-guided walking and target-directed walking. The experimenter-guided walking task revealed a systematic underestimation of self-motion signals in UVH patients compared to the other groups. However, we did not find any difference in the distance walked between the UVH group and the control group for the target-directed walking task. Results from neuropsychological testing and clinical balance measures suggest that the errors in experimenter-guided walking were not attributable to cognitive and/or balance impairments. We conclude that impairment in linear path integration in UVH patients stem from deficits in self-motion perception. Importantly, our results also suggest that patients with a UVH deficit do not lose their ability to walk accurately without vision to a memorized target location. PMID:22726251
14. Path Integrals and the Statistical Thermodynamics of Black Holes.
Martinez, Erik Andres
The path integral is an important element in modern approaches to the quantization of the gravitational field. Path integral representations of partition functions for static and stationary black hole systems as well as path integrals for minisuperspace models of cosmology are presented. The functional integral is defined throughout as a sum over Lorentzian histories. A consistent formulation of Feynman's prescription to construct partition functions in terms of path integrals for general gravitational systems is presented and contrasted with other "Euclideanization" prescriptions. It is shown that the central object in the description of black hole systems is the gravitational action. In particular, the additivity of the entropies of matter and black holes in thermal equilibrium is a consequence of the additivity of their corresponding actions, and thermodynamic potentials like the energy or the pressure are not in general addivite when gravity plays an important role. Partition functions as stationary phase approximations of functional integrals for all the thermodynamic ensembles are then constructed by including gravitation as a part of the thermodynamical system. We show that a complex geometry is required to derive the thermodynamic properties of stationary geometries from the sum over histories. The corresponding real "thermodynamical" action is calculated explicitly and the thermodynamical data that imply thermal equilibrium in the presence of a rotating black hole in interaction with matter fields are presented and related to geometrical data. Some of the consequences for Kerr-Newman black hole systems are also discussed. For minisuperspace cosmologies the Lorentzian path integral is a Green function for the Wheeler-DeWitt operator, and its real part is a solution to the Wheeler -DeWitt equation. It is computed explicitly for the de Sitter minisuperspace model. The resulting Green function is then related to both the Hartle-Hawking and tunneling wave
15. Transport path optimization algorithm based on fuzzy integrated weights
Hou, Yuan-Da; Xu, Xiao-Hao
2014-11-01
Natural disasters cause significant damage to roads, making route selection a complicated logistical problem. To overcome this complexity, we present a method of using a trapezoidal fuzzy number to select the optimal transport path. Using the given trapezoidal fuzzy edge coefficients, we calculate a fuzzy integrated matrix, and incorporate the fuzzy multi-weights into fuzzy integrated weights. The optimal path is determined by taking two sets of vertices and transforming undiscovered vertices into discoverable ones. Our experimental results show that the model is highly accurate, and requires only a few measurement data to confirm the optimal path. The model provides an effective, feasible, and convenient method to obtain weights for different road sections, and can be applied to road planning in intelligent transportation systems.
16. Sex-Integrated Physical Education
ERIC Educational Resources Information Center
Kneer, Marian E.
1978-01-01
Title IX requires that the practice of segregating physical education classes on the basis of sex must cease. This article offers suggestions on how to ease the transition to sex-integrated physical education programs. (Author/DS)
17. Physics thematic paths: laboratorial activities and historical scientific instruments
Pantano, O.; Talas, S.
2010-03-01
The Physics Department of Padua University keeps an important collection of historical physics instruments which alludes to the fruitful scientific activity of Padua through the centuries. This heritage led to the suggestion of setting up laboratory activities connected to the Museum collection for secondary school students. This article shows how different thematic paths have been developed, reflecting on the importance of historical perspectives in science teaching. We also show how a scientific historical museum can play a central role in improving the learning of physics concepts.
18. Path Integral for Dirac oscillator with generalized uncertainty principle
SciTech Connect
Benzair, H.; Boudjedaa, T.; Merad, M.
2012-12-15
The propagator for Dirac oscillator in (1+1) dimension, with deformed commutation relation of the Heisenberg principle, is calculated using path integral in quadri-momentum representation. As the mass is related to momentum, we then adapt the space-time transformation method to evaluate quantum corrections and this latter is dependent from the point discretization interval.
19. Piloting Systems Reset Path Integration Systems during Position Estimation
ERIC Educational Resources Information Center
Zhang, Lei; Mou, Weimin
2017-01-01
During locomotion, individuals can determine their positions with either idiothetic cues from movement (path integration systems) or visual landmarks (piloting systems). This project investigated how these 2 systems interact in determining humans' positions. In 2 experiments, participants studied the locations of 5 target objects and 1 single…
20. The functional measure for the in-in path integral
Kaya, Ali
2015-05-01
The in-in path integral of a scalar field propagating in a fixed background is formulated in a suitable function space. The free kinetic operator, whose inverse gives the propagators of the in-in perturbation theory, becomes essentially self adjoint after imposing appropriate boundary conditions. An explicit spectral representation is given for the scalar in the flat space, and the standard propagators are rederived using this representation. In this way the subtle boundary path integral over the field configurations at the return time is handled straightforwardly. It turns out that not only the values of the forward (+) and the backward (-) evolving fields but also their time derivatives must be matched at the return time, which is mainly overlooked in the literature. This formulation also determines the field configurations that are included in the path integral uniquely. We show that some of the recently suggested instanton-like solutions corresponding to the stationary phases of the cosmological in-in path integrals can be rigorously identified as limits of sequences in the function space.
1. Piloting and Path Integration within and across Boundaries
ERIC Educational Resources Information Center
Mou, Weimin; Wang, Lin
2015-01-01
Three experiments investigated whether navigation is less efficient across boundaries than within boundaries. In an immersive virtual environment, participants learned objects' locations in a large room or a small room. Participants then pointed to the objects' original locations after physically walking a circuitous path without vision.…
2. Piloting and Path Integration within and across Boundaries
ERIC Educational Resources Information Center
Mou, Weimin; Wang, Lin
2015-01-01
Three experiments investigated whether navigation is less efficient across boundaries than within boundaries. In an immersive virtual environment, participants learned objects' locations in a large room or a small room. Participants then pointed to the objects' original locations after physically walking a circuitous path without vision.…
3. SU(2) Coherent State Path Integrals Labeled by a Full Set of Euler Angles: Basic Formulation
Matsumoto, Masao
2012-11-01
We develop a basic formulation of the spin (SU(2)) coherent state path integrals based not on the conventional highest or lowest weight vectors but on arbitrary fiducial vectors. The coherent states, being defined on a 3-sphere, are specified by a full set of Euler angles. They are generally considered as states without classical analogues. The overcompleteness relation holds for the states, by which we obtain the time evolution of general systems in terms of the path integral representation; the resultant Lagrangian in the action has a monopole-type term à la Balachandran et al. as well as some additional terms, both of which depend on fiducial vectors in a simple way. The process of the discrete path integrals to the continuous ones is clarified. Complex variable forms of the states and path integrals are also obtained. During the course of all steps, we emphasize the analogies and correspondences to the general canonical coherent states and path integrals that we proposed some time ago. In this paper we concentrate on the basic formulation. The physical applications as well as criteria in choosing fiducial vectors for real Lagrangians, in relation to fictitious monopoles and geometric phases, will be treated in subsequent papers separately.
4. Integrating Physical Education, Math, and Physics
ERIC Educational Resources Information Center
Hatch, Greg M.; Smith, Darla R.
2004-01-01
The primary goal of education is to help students gain the knowledge and skills necessary to solve the problems that occur in real-life situations. One of the most touted methods for achieving this goal is through curriculum integration. Physical education students can benefit from an integrated curriculum. NASPE's (National Association tar Sport…
5. Quantum tunneling splittings from path-integral molecular dynamics
Mátyus, Edit; Wales, David J.; Althorpe, Stuart C.
2016-03-01
We illustrate how path-integral molecular dynamics can be used to calculate ground-state tunnelling splittings in molecules or clusters. The method obtains the splittings from ratios of density matrix elements between the degenerate wells connected by the tunnelling. We propose a simple thermodynamic integration scheme for evaluating these elements. Numerical tests on fully dimensional malonaldehyde yield tunnelling splittings in good overall agreement with the results of diffusion Monte Carlo calculations.
6. What is the right time for path integrals?
Jones, Eric; Bach, Roger; Batelaan, Herman
2013-05-01
The Feynman path integral formulation of quantum mechanics has proven to be a powerful tool for calculations in matter optics. It is natural to introduce the path integral in the context of Young's double slit experiment for matter waves as Feynman did, perhaps after discussing the analogous situation for optics. While intuitive, this approach can lead to a pedagogical misrepresentation of the theory, namely in the phase accumulated along single free-particle trajectories. How is the use of the accumulated phase, 2 πL /λdB , along a path of length L justified? The free-particle action gives a phase that differs by a factor of two. The guiding principle that interference occurs only for two paths that are indistinguishable from one another provides a correct solution: interfering paths must originate and terminate at equal times. We will present several simple thought experiments to illustrate incorrect and correct methods for determining phase shifts. We gratefully acknowledge support from NSF and the DoE GAANN. We thank Ron Cappelletti for interesting discussions.
7. Path integrals and large deviations in stochastic hybrid systems
Bressloff, Paul C.; Newby, Jay M.
2014-04-01
We construct a path-integral representation of solutions to a stochastic hybrid system, consisting of one or more continuous variables evolving according to a piecewise-deterministic dynamics. The differential equations for the continuous variables are coupled to a set of discrete variables that satisfy a continuous-time Markov process, which means that the differential equations are only valid between jumps in the discrete variables. Examples of stochastic hybrid systems arise in biophysical models of stochastic ion channels, motor-driven intracellular transport, gene networks, and stochastic neural networks. We use the path-integral representation to derive a large deviation action principle for a stochastic hybrid system. Minimizing the associated action functional with respect to the set of all trajectories emanating from a metastable state (assuming that such a minimization scheme exists) then determines the most probable paths of escape. Moreover, evaluating the action functional along a most probable path generates the so-called quasipotential used in the calculation of mean first passage times. We illustrate the theory by considering the optimal paths of escape from a metastable state in a bistable neural network.
8. Integration across Time Determines Path Deviation Discrimination for Moving Objects.
PubMed
Whitaker, David; Levi, Dennis M; Kennedy, Graeme J
2008-04-16
Human vision is vital in determining our interaction with the outside world. In this study we characterize our ability to judge changes in the direction of motion of objects-a common task which can allow us either to intercept moving objects, or else avoid them if they pose a threat. Observers were presented with objects which moved across a computer monitor on a linear path until the midline, at which point they changed their direction of motion, and observers were required to judge the direction of change. In keeping with the variety of objects we encounter in the real world, we varied characteristics of the moving stimuli such as velocity, extent of motion path and the object size. Furthermore, we compared performance for moving objects with the ability of observers to detect a deviation in a line which formed the static trace of the motion path, since it has been suggested that a form of static memory trace may form the basis for these types of judgment. The static line judgments were well described by a 'scale invariant' model in which any two stimuli which possess the same two-dimensional geometry (length/width) result in the same level of performance. Performance for the moving objects was entirely different. Irrespective of the path length, object size or velocity of motion, path deviation thresholds depended simply upon the duration of the motion path in seconds. Human vision has long been known to integrate information across space in order to solve spatial tasks such as judgment of orientation or position. Here we demonstrate an intriguing mechanism which integrates direction information across time in order to optimize the judgment of path deviation for moving objects.
9. Amplitudes for spacetime regions and the quantum Zeno effect: pitfalls of standard path integral constructions
Halliwell, J. J.; Yearsley, J. M.
2013-06-01
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect - the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale epsilon, describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as epsilon >> 1/E, where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to
10. Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals
Liko, Tomáš
2012-05-01
The Barbero-Immirzi parameter γ appears in the real connection formulation of gravity in terms of the Ashtekar variables, and gives rise to a one-parameter quantization ambiguity in loop quantum gravity. In this paper, we investigate the conditions under which γ will have physical effects in Euclidean quantum gravity. This is done by constructing a well-defined Euclidean path integral for the Holst action with a non-zero cosmological constant on a manifold with a boundary. We find that two general conditions must be satisfied by the spacetime manifold in order for the Holst action and its surface integral to be non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryagin number of the manifold has to be non-zero. The latter is a strong topological condition and rules out many of the known solutions to the Einstein field equations. This result leads us to evaluate the on-shell first-order Holst action and corresponding Euclidean partition function on the Taub-NUT-ADS solution. We find that γ shows up as a finite rotation of the on-shell partition function which corresponds to shifts in the energy and entropy of the NUT charge. In an appendix, we also evaluate the Holst action on the Taub-NUT and Taub-bolt solutions in flat spacetime and find that in that case as well γ shows up in the energy and entropy of the NUT and bolt charges. We also present an example whereby the Euler characteristic of the manifold has a non-trivial effect on black hole mergers. Communicated by PRLV Moniz
11. Computational Physics as a Path for Physics Education
Landau, Rubin H.
2008-04-01
Evidence and arguments will be presented that modifications in the undergraduate physics curriculum are necessary to maintain the long-term relevance of physics. Suggested will a balance of analytic, experimental, computational, and communication skills, that in many cases will require an increased inclusion of computation and its associated skill set into the undergraduate physics curriculum. The general arguments will be followed by a detailed enumeration of suggested subjects and student learning outcomes, many of which have already been adopted or advocated by the computational science community, and which permit high performance computing and communication. Several alternative models for how these computational topics can be incorporated into the undergraduate curriculum will be discussed. This includes enhanced topics in the standard existing courses, as well as stand-alone courses. Applications and demonstrations will be presented throughout the talk, as well as prototype video-based materials and electronic books.
12. Free energies from integral equation theories: enforcing path independence.
PubMed
Kast, Stefan M
2003-04-01
A variational formalism is constructed for deriving the chemical potential and the Helmholtz free energy in various statistical-mechanical integral equation theories of fluids. Nonzero bridge functions extending the scope of the theories beyond the hypernetted chain approximation can be classified as to whether or not they imply path dependence of the free energy. Classes of bridge functions free of the path dependence problem are derived, based on which a route is devised toward direct computation of free energies from the simulation of a single state.
13. Tackling higher derivative ghosts with the Euclidean path integral
SciTech Connect
Fontanini, Michele; Trodden, Mark
2011-05-15
An alternative to the effective field theory approach to treat ghosts in higher derivative theories is to attempt to integrate them out via the Euclidean path integral formalism. It has been suggested that this method could provide a consistent framework within which we might tolerate the ghost degrees of freedom that plague, among other theories, the higher derivative gravity models that have been proposed to explain cosmic acceleration. We consider the extension of this idea to treating a class of terms with order six derivatives, and find that for a general term the Euclidean path integral approach works in the most trivial background, Minkowski. Moreover we see that even in de Sitter background, despite some difficulties, it is possible to define a probability distribution for tensorial perturbations of the metric.
14. Reparametrization invariance of perturbatively defined path integrals. II. integrating products of distributions
Kleinert, H.; Chervyakov, A.
2000-03-01
We show how to perform integrals over products of distributions in coordinate space such as to reproduce the results of momentum space Feynman integrals in dimensional regularization. This ensures the invariance of path integrals under coordinate transformations. The integrals are performed by expressing the propagators in /1-ɛ dimensions in terms of modified Bessel functions.
15. Beam spread functions calculated using Feynman path integrals
Kilgo, Paul; Tessendorf, Jerry
2017-07-01
A method of solving the radiative transfer equation using Feynman path integrals (FPIs) is discussed. The FPI approach is a mathematical framework for computing multiple scattering in participating media. Its numerical behavior is not well known, and techniques are being developed to solve the FPI approach numerically. A missing numerical technique is detailed and used to calculate beam spread functions (BSFs), a commonly studied experimental property of many types of media. The calculations are compared against measured BSFs of sea ice. Analysis shows differently-shaped BSFs, and suggests the width parameter of the calculated BSF's Gaussian fit approaches a value in the limit of the number of path segments. A projection is attempted, but suggests a larger number of path segments would not increase the width of the calculated BSF. The trial suggests the approach is numerically stable, but requires further testing to ensure scientific accuracy.
16. Path integration and perturbation theory with complex Euclidean actions
SciTech Connect
Alexanian, Garnik; MacKenzie, R.; Paranjape, M. B.; Ruel, Jonathan
2008-05-15
The Euclidean path integral quite often involves an action that is not completely real, i.e. a complex action. This occurs when the Minkowski action contains t-odd CP-violating terms. This usually consists of topological terms, such as the Chern-Simons term in odd dimensions, the Wess-Zumino term, the {theta} term or Chern character in 4-dimensional gauge theories, or other topological densities. Analytic continuation to Euclidean time yields an imaginary term in the Euclidean action. It also occurs when the action contains fermions, the fermion path integral being in general a sum over positive and negative real numbers. Negative numbers correspond to the exponential of i{pi} and hence indicate the presence of an imaginary term in the action. In the presence of imaginary terms in the Euclidean action, the usual method of perturbative quantization can fail. Here the action is expanded about its critical points, the quadratic part serving to define the Gaussian free theory and the higher order terms defining the perturbative interactions. For a complex action, the critical points are generically obtained at complex field configurations. Hence the contour of path integration does not pass through the critical points and the perturbative paradigm cannot be directly implemented. The contour of path integration has to be deformed to pass through the complex critical point using a generalized method of steepest descent, in order to do so. Typically, this procedure is not followed. Rather, only the real part of the Euclidean action is considered, and its critical points are used to define the perturbation theory, a procedure that can lead to incorrect results. In this article we present a simple example to illustrate this point. The example consists of N scalar fields in 0+1 dimensions interacting with a U(1) gauge field in the presence of a Chern-Simons term. In this example the path integral can be done exactly, the procedure of deformation of the contour of path
17. Three dimensional path integration in the house mouse (Mus domestica).
PubMed
Bardunias, P M; Jander, R
2000-12-01
Previous studies have explored two-dimensional path integration in rodents by recording responses to passive rotation on a horizontal plane. This study adds the element of passive rotation in a vertical plane, necessitating the mouse to integrate positional information from three dimensions. Mice were trained to climb a wire mesh joining two horizontal planes. The whole arena was rotated 90 degrees while the mouse was vertically oriented as it moved between planes. Rotation was conducted both clockwise and counter-clockwise, controls being provided by rotation of the arena while the mouse was in its nest-box. All 16 mice tested altered their direction of travel subsequent to rotation in the vertical plane, compensating with a change in their path on the following horizontal plane.
18. A path integral approach to age dependent branching processes
Greenman, Chris D.
2017-03-01
Age dependent population dynamics are frequently modeled with generalizations of the classic McKendrick–von Foerster equation. These are deterministic systems, and a stochastic generalization was recently reported in Greenman and Chou (2016 Phys. Rev. E 93 012112, 2016 J. Stat. Phys. 16449). Here we develop a fully stochastic theory for age-structured populations via quantum field theoretical Doi–Peliti techniques. This results in a path integral formulation where birth and death events correspond to cubic and quadratic interaction terms. This formalism allows us to efficiently recapitulate the results in Greenman and Chou (2016 Phys. Rev. E 93 012112, 2016 J. Stat. Phys. 16449), exemplifying the utility of Doi–Peliti methods. Furthermore, we find that the path integral formulation for age-structured moments has an exact perturbative expansion that explicitly relates to the hereditary structure between correlated individuals. These methods are then generalized with a binary fission model of cell division.
19. Three dimensional path integration in the house mouse (Mus domestica)
Bardunias, Paul M.; Jander, R.
Previous studies have explored two-dimensional path integration in rodents by recording responses to passive rotation on a horizontal plane. This study adds the element of passive rotation in a vertical plane, necessitating the mouse to integrate positional information from three dimensions. Mice were trained to climb a wire mesh joining two horizontal planes. The whole arena was rotated 90° while the mouse was vertically oriented as it moved between planes. Rotation was conducted both clockwise and counter-clockwise, controls being provided by rotation of the arena while the mouse was in its nest-box. All 16 mice tested altered their direction of travel subsequent to rotation in the vertical plane, compensating with a change in their path on the following horizontal plane.
20. Path integral Liouville dynamics for thermal equilibrium systems.
PubMed
Liu, Jian
2014-06-14
We show a new imaginary time path integral based method--path integral Liouville dynamics (PILD), which can be derived from the equilibrium Liouville dynamics [J. Liu and W. H. Miller, J. Chem. Phys. 134, 104101 (2011)] in the Wigner phase space. Numerical tests of PILD with the simple (white noise) Langevin thermostat have been made for two strongly anharmonic model problems. Since implementation of PILD does not request any specific form of the potential energy surface, the results suggest that PILD offers a potentially useful approach for general condensed phase molecular systems to have the two important properties: conserves the quantum canonical distribution and recovers exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits.
1. Path integral Liouville dynamics for thermal equilibrium systems
SciTech Connect
Liu, Jian
2014-06-14
We show a new imaginary time path integral based method—path integral Liouville dynamics (PILD), which can be derived from the equilibrium Liouville dynamics [J. Liu and W. H. Miller, J. Chem. Phys. 134, 104101 (2011)] in the Wigner phase space. Numerical tests of PILD with the simple (white noise) Langevin thermostat have been made for two strongly anharmonic model problems. Since implementation of PILD does not request any specific form of the potential energy surface, the results suggest that PILD offers a potentially useful approach for general condensed phase molecular systems to have the two important properties: conserves the quantum canonical distribution and recovers exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits.
2. Path integral for the relativistic particle in curved space
Ferraro, Rafael
1992-02-01
The propagator for a single relativistic particle in a (D+1)-dimensional curved background is obtained by evaluating the canonical path integral in the true 2D-dimensional phase space. Since only paths moving forward in time are integrated, the resulting propagator depends on how the time is chosen; i.e., it depends on the reference system. In order for the propagator to satisfy the properties of a unitary theory, the time must be attached to a Killing vector. Although the measure is unique (it is the Liouville measure), the skeletonization of the phase-space functional action is ambiguous. One such ambiguity is exploited to obtain different propagators obeying the Klein-Gordon equation with different couplings to quantities related to the shape of the reference system (spatial curvature, etc.).
3. Spin foam models for quantum gravity from lattice path integrals
SciTech Connect
Bonzom, Valentin
2009-09-15
Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations that satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models correspond to a special choice of gluing. In this case, the equations of motion describe genuine geometries, where the constraints of area-angle Regge calculus are satisfied. Furthermore, the Immirzi parameter drops out of the on-shell action, and stationarity with respect to area variations requires spacetime geometry to be flat.
4. Asymptotics of Selberg-like integrals by lattice path counting
SciTech Connect
Novaes, Marcel
2011-04-15
We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations.
5. A path to integration in an academic health science center.
PubMed
Panko, W B; Wilson, W
1992-01-01
This article describes a networking and integration strategy in use at the University of Michigan Medical Center. This strategy builds upon the existing technology base and is designed to provide a roadmap that will direct short-term development along a productive, long-term path. It offers a way to permit the short-term development of incremental solutions to current problems while at the same time maximizing the likelihood that these incremental efforts can be recycled into a more comprehensive approach.
6. Path integral analysis of Jarzynski's equality: analytical results.
PubMed
Minh, David D L; Adib, Artur B
2009-02-01
We apply path integrals to study nonequilibrium work theorems in the context of Brownian dynamics, deriving in particular the equations of motion governing the most typical and most dominant trajectories. For the analytically soluble cases of a moving harmonic potential and a harmonic oscillator with a time-dependent natural frequency, we find such trajectories, evaluate the work-weighted propagators, and validate Jarzynski's equality.
7. Integration of Hierarchical Goal Network Planning and Autonomous Path Planning
DTIC Science & Technology
2016-03-01
world robotic systems. This report documents work to integrate a hierarchical goal network planning algorithm with low-level path planning. The system...goals state. The domain is a robot operating in a known office environment with labeled rooms and doors that can be manipulated. The report goes on to...discuss future improvement of the system with the goal of creating a robust system that can operate on a robotic platform in a dynamic environment
8. Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?
Penney, Mark D.; Enshan Koh, Dax; Spekkens, Robert W.
2017-07-01
It is straightforward to compute the transition amplitudes of a quantum circuit using the sum-over-paths methodology when the gates in the circuit are balanced, where a balanced gate is one for which all non-zero transition amplitudes are of equal magnitude. Here we consider the question of whether, for such circuits, the relative phases of different discrete-time paths through the configuration space can be defined in terms of a classical action, as they are for continuous-time paths. We show how to do so for certain kinds of quantum circuits, namely, Clifford circuits where the elementary systems are continuous-variable systems or discrete systems of odd-prime dimension. These types of circuit are distinguished by having phase-space representations that serve to define their classical counterparts. For discrete systems, the phase-space coordinates are also discrete variables. We show that for each gate in the generating set, one can associate a symplectomorphism on the phase-space and to each of these one can associate a generating function, defined on two copies of the configuration space. For discrete systems, the latter association is achieved using tools from algebraic geometry. Finally, we show that if the action functional for a discrete-time path through a sequence of gates is defined using the sum of the corresponding generating functions, then it yields the correct relative phases for the path-sum expression. These results are likely to be relevant for quantizing physical theories where time is fundamentally discrete, characterizing the classical limit of discrete-time quantum dynamics, and proving complexity results for quantum circuits.
9. Path integral approach to the quantum fidelity amplitude
PubMed Central
2016-01-01
The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and in its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet non-trivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called ‘dephasing representation,’ circumventing the use of a semiclassical propagator as in the original derivation. We also obtain an approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact. PMID:27140973
10. Path integral approach to the quantum fidelity amplitude.
PubMed
Vaníček, Jiří; Cohen, Doron
2016-06-13
The Loschmidt echo is a measure of quantum irreversibility and is determined by the fidelity amplitude of an imperfect time-reversal protocol. Fidelity amplitude plays an important role both in the foundations of quantum mechanics and in its applications, such as time-resolved electronic spectroscopy. We derive an exact path integral formula for the fidelity amplitude and use it to obtain a series of increasingly accurate semiclassical approximations by truncating an exact expansion of the path integral exponent. While the zeroth-order expansion results in a remarkably simple, yet non-trivial approximation for the fidelity amplitude, the first-order expansion yields an alternative derivation of the so-called 'dephasing representation,' circumventing the use of a semiclassical propagator as in the original derivation. We also obtain an approximate expression for fidelity based on the second-order expansion, which resolves several shortcomings of the dephasing representation. The rigorous derivation from the path integral permits the identification of sufficient conditions under which various approximations obtained become exact.
11. Electronic Stroke CarePath: Integrated Approach to Stroke Care.
PubMed
Katzan, Irene L; Fan, Youran; Speck, Micheal; Morton, Johanna; Fromwiller, Lauren; Urchek, John; Uchino, Ken; Griffith, Sandra D; Modic, Michael
2015-10-01
We describe the development, implementation, and outcomes of the first 2 years of the Electronic Stroke CarePath, an initiative developed for management of ischemic stroke patients in an effort to improve efficiency and quality of care for patients. The CarePath consists of care pathways for ischemic stroke that are integrated within the electronic health record. Patient-reported outcomes are collected using an external software platform. Documentation tools, order sets, and clinical decision support were designed to improve efficiency, optimize process measure adherence, and produce clinical data as a byproduct of care that are available for future analyses. Inpatient mortality and length of stay were compared before and after CarePath implementation in ischemic stroke patients after adjustment for case-mix. Postdischarge functional outcomes of patients with ischemic stroke were compared between the first 3 months of rollout and remainder of the study period. From January 2011 to December 2012, there were 1106 patients with ischemic stroke on the CarePath. There was a decline in inpatient mortality in patients with ischemic stroke, but not in control patients with intracerebral or subarachnoid hemorrhage. Completion rate of patient-reported questionnaires at postdischarge stroke follow-up was 72.9%. There was a trend toward improved functional outcomes at follow-up with CarePath implementation. Implementation of the Electronic Stroke CarePath is feasible and may be associated with a benefit in multiple different outcomes after ischemic stroke. This approach may be an important strategy for optimizing stroke care in the future.
12. A review of path-independent integrals in elastic-plastic fracture mechanics
NASA Technical Reports Server (NTRS)
Kim, Kwang S.; Orange, Thomas W.
1988-01-01
The objective of this paper is to review the path-independent (P-I) integrals in elastic plastic fracture mechanics which have been proposed in recent years to overcome the limitations imposed on the J-integral. The P-I integrals considered are the J-integral by Rice (1968), the thermoelastic P-I integrals by Wilson and Yu (1979) and Gurtin (1979), the J-integral by Blackburn (1972), the J(theta)-integral by Ainsworth et al. (1978), the J-integral by Kishimoto et al. (1980), and the Delta-T(p) and Delta T(p)-asterisk integrals by Alturi et al. (1982). The theoretical foundation of the P-I integrals is examined with an emphasis on whether or not the path independence is maintained in the presence of nonproportional loading and unloading in the plastic regime, thermal gradient, and material inhomogeneities. The simularities, difference, salient features, and limitations of the P-I integrals are discussed. Comments are also made with regard to the physical meaning, the possibility of experimental measurement, and computational aspects.
13. A review of path-independent integrals in elastic-plastic fracture mechanics, task 4
NASA Technical Reports Server (NTRS)
Kim, K. S.
1985-01-01
The path independent (P-I) integrals in elastic plastic fracture mechanics which have been proposed in recent years to overcome the limitations imposed on the J integral are reviewed. The P-I integrals considered herein are the J integral by Rice, the thermoelastic P-I integrals by Wilson and Yu and by Gurtin, the J* integral by Blackburn, the J sub theta integral by Ainsworth et al., the J integral by Kishimoto et al., and the delta T sub p and delta T* sub p integrals by Atluri et al. The theoretical foundation of these P-I integrals is examined with emphasis on whether or not path independence is maintained in the presence of nonproportional loading and unloading in the plastic regime, thermal gradients, and material inhomogeneities. The similarities, differences, salient features, and limitations of these P-I integrals are discussed. Comments are also made with regard to the physical meaning, the possibility of experimental measurement, and computational aspects.
14. A review of path-independent integrals in elastic-plastic fracture mechanics
NASA Technical Reports Server (NTRS)
Kim, Kwang S.; Orange, Thomas W.
1988-01-01
The objective of this paper is to review the path-independent (P-I) integrals in elastic plastic fracture mechanics which have been proposed in recent years to overcome the limitations imposed on the J-integral. The P-I integrals considered are the J-integral by Rice (1968), the thermoelastic P-I integrals by Wilson and Yu (1979) and Gurtin (1979), the J-integral by Blackburn (1972), the J(theta)-integral by Ainsworth et al. (1978), the J-integral by Kishimoto et al. (1980), and the Delta-T(p) and Delta T(p)-asterisk integrals by Alturi et al. (1982). The theoretical foundation of the P-I integrals is examined with an emphasis on whether or not the path independence is maintained in the presence of nonproportional loading and unloading in the plastic regime, thermal gradient, and material inhomogeneities. The simularities, difference, salient features, and limitations of the P-I integrals are discussed. Comments are also made with regard to the physical meaning, the possibility of experimental measurement, and computational aspects.
15. Physical limits for scaling of integrated circuits
Nawrocki, Waldemar
2010-11-01
In this paper we discuss some physical limits for scaling of devices and conducting paths inside of semiconductor integrated circuits (ICs). Since 40 years only a semiconductor technology, mostly the CMOS and the TTL technologies, are used for fabrication of integrated circuits in the industrial scale. Miniaturization of electronic devices in integrated circuits has technological limits and physical limits as well. In 2010 best parameters of commercial ICs shown the dual-core Intel Core i5-670 processor manufactured in the technology of 32 nm. Its clock frequency in turbo mode is 3.73 GHz. A forecast of the development of the semiconductor industry (ITRS 2009) predicts that sizes of electronic devices in ICs circuits will be smaller than 10 nm in the next 10 years. The physical gate length in a MOSFET will even amount 7 nm in the year 2024. At least 5 physical effects should be taken into account if we discuss limits of scaling of integrated circuits.
16. Integrating Physically Handicapped Pupils.
ERIC Educational Resources Information Center
Hodgson, Ann
1984-01-01
Intended for regular class teachers working with mainstreamed physically handicapped pupils, the article offers guidelines regarding school organization, conditions directly affecting the pupil, and classroom considerations. A brief review of the major conditions (such as allergies, cerebral palsy, and muscular dystrophy) is presented. (CL)
17. Integrating Physically Handicapped Pupils.
ERIC Educational Resources Information Center
Hodgson, Ann
1984-01-01
Intended for regular class teachers working with mainstreamed physically handicapped pupils, the article offers guidelines regarding school organization, conditions directly affecting the pupil, and classroom considerations. A brief review of the major conditions (such as allergies, cerebral palsy, and muscular dystrophy) is presented. (CL)
18. 77 FR 74027 - Certain Integrated Circuit Packages Provided with Multiple Heat-Conducting Paths and Products...
Federal Register 2010, 2011, 2012, 2013, 2014
2012-12-12
... From the Federal Register Online via the Government Publishing Office INTERNATIONAL TRADE COMMISSION Certain Integrated Circuit Packages Provided with Multiple Heat- Conducting Paths and Products... integrated circuit packages provided with multiple heat-conducting paths and products containing same...
19. Self-organizing path integration using a linked continuous attractor and competitive network: path integration of head direction.
PubMed
Stringer, Simon M; Rolls, Edmund T
2006-12-01
A key issue is how networks in the brain learn to perform path integration, that is update a represented position using a velocity signal. Using head direction cells as an example, we show that a competitive network could self-organize to learn to respond to combinations of head direction and angular head rotation velocity. These combination cells can then be used to drive a continuous attractor network to the next head direction based on the incoming rotation signal. An associative synaptic modification rule with a short term memory trace enables preceding combination cell activity during training to be associated with the next position in the continuous attractor network. The network accounts for the presence of neurons found in the brain that respond to combinations of head direction and angular head rotation velocity. Analogous networks in the hippocampal system could self-organize to perform path integration of place and spatial view representations.
20. Physical Layer Security Using Two-Path Successive Relaying.
PubMed
Liau, Qian Yu; Leow, Chee Yen; Ding, Zhiguo
2016-06-09
Relaying is one of the useful techniques to enhance wireless physical-layer security. Existing literature shows that employing full-duplex relay instead of conventional half-duplex relay improves secrecy capacity and secrecy outage probability, but this is at the price of sophisticated implementation. As an alternative, two-path successive relaying has been proposed to emulate operation of full-duplex relay by scheduling a pair of half-duplex relays to assist the source transmission alternately. However, the performance of two-path successive relaying in secrecy communication remains unexplored. This paper proposes a secrecy two-path successive relaying protocol for a scenario with one source, one destination and two half-duplex relays. The relays operate alternately in a time division mode to forward messages continuously from source to destination in the presence of an eavesdropper. Analytical results reveal that the use of two half-duplex relays in the proposed scheme contributes towards a quadratically lower probability of interception compared to full-duplex relaying. Numerical simulations show that the proposed protocol achieves the ergodic achievable secrecy rate of full-duplex relaying while delivering the lowest probability of interception and secrecy outage probability compared to the existing half duplex relaying, full duplex relaying and full duplex jamming schemes.
1. Physical Layer Security Using Two-Path Successive Relaying
PubMed Central
Liau, Qian Yu; Leow, Chee Yen; Ding, Zhiguo
2016-01-01
Relaying is one of the useful techniques to enhance wireless physical-layer security. Existing literature shows that employing full-duplex relay instead of conventional half-duplex relay improves secrecy capacity and secrecy outage probability, but this is at the price of sophisticated implementation. As an alternative, two-path successive relaying has been proposed to emulate operation of full-duplex relay by scheduling a pair of half-duplex relays to assist the source transmission alternately. However, the performance of two-path successive relaying in secrecy communication remains unexplored. This paper proposes a secrecy two-path successive relaying protocol for a scenario with one source, one destination and two half-duplex relays. The relays operate alternately in a time division mode to forward messages continuously from source to destination in the presence of an eavesdropper. Analytical results reveal that the use of two half-duplex relays in the proposed scheme contributes towards a quadratically lower probability of interception compared to full-duplex relaying. Numerical simulations show that the proposed protocol achieves the ergodic achievable secrecy rate of full-duplex relaying while delivering the lowest probability of interception and secrecy outage probability compared to the existing half duplex relaying, full duplex relaying and full duplex jamming schemes. PMID:27294924
2. From conformal blocks to path integrals in the Vaidya geometry
Anous, Tarek; Hartman, Thomas; Rovai, Antonin; Sonner, Julian
2017-09-01
Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must be included in order to correctly reproduce complex saddles in the Lorentzian theory. During thermalization, a complex saddle dominates under certain circumstances; in this case, the CFT correlator is not given by the Virasoro identity block in any channel, but can be recovered by summing heavy operators. This effectively converts the conformal block expansion in CFT from a sum over intermediate states to a sum over channels that mimics the bulk path integral.
3. Path integral based calculations of symmetrized time correlation functions. II.
PubMed
Bonella, S; Monteferrante, M; Pierleoni, C; Ciccotti, G
2010-10-28
Schofield's form of quantum time correlation functions is used as the starting point to derive a computable expression for these quantities. The time composition property of the propagators in complex time is exploited to approximate Schofield's function in terms of a sequence of short time classical propagations interspersed with path integrals that, combined, represent the thermal density of the system. The approximation amounts to linearization of the real time propagators and it becomes exact with increasing number of propagation legs. Within this scheme, the correlation function is interpreted as an expectation value over a probability density defined on the thermal and real path space and calculated by a Monte Carlo algorithm. The performance of the algorithm is tested on a set of benchmark problems. Although the numerical effort required is considerable, we show that the algorithm converges systematically to the exact answer with increasing number of iterations and that it is stable for times longer than those accessible via a brute force, path integral based, calculation of the correlation function. Scaling of the algorithm with dimensionality is also examined and, when the method is combined with commonly used filtering schemes, found to be comparable to that of alternative semiclassical methods.
4. Effective descriptions of complex quantum systems: path integrals and operator ordering problems
Eckern, U.; Gruber, M. J.; Schwab, P.
2005-09-01
[Dedicated to Bernhard Mühlschlegel on the occasion ofhis 80th birthday]We study certain aspects of the effective, occasionally called collective, description of complex quantum systems within the framework of the path integral formalism, in which the environment is integrated out. Generalising the standard Feynman-Vernon Caldeira-Leggett model to include a non-linear coupling between particle and environment, and considering a particular spectral density of the coupling, a coordinate-dependent mass (or velocity-dependent potential) is obtained. The related effective quantum theory, which depends on the proper discretisation of the path integral, is derived and discussed. As a result, we find that in general a simple effective low-energy Hamiltonian, in which only the coordinate-dependent mass enters, cannot be formulated. The quantum theory of weakly coupled superconductors and the quantum dynamics of vortices in Josephson junction arrays are physical examples where these considerations, in principle, are of relevance.
5. Physics Integration KErnels (PIKE)
SciTech Connect
Pawlowski, Roger
2014-07-31
Pike is a software library for coupling and solving multiphysics applications. It provides basic interfaces and utilities for performing code-to-code coupling. It provides simple black-box Picard iteration methods for solving the coupled system of equations including Jacobi and Gauss-Seidel solvers. Pike was developed originally to couple neutronics and thermal fluids codes to simulate a light water nuclear reactor for the Consortium for Simulation of Light-water Reactors (CASL) DOE Energy Innovation Hub. The Pike library contains no physics and just provides interfaces and utilities for coupling codes. It will be released open source under a BSD license as part of the Trilinos solver framework (trilinos.org) which is also BSD. This code provides capabilities similar to other open source multiphysics coupling libraries such as LIME, AMP, and MOOSE.
6. Path Integrals for Electronic Densities, Reactivity Indices, and Localization Functions in Quantum Systems
PubMed Central
Putz, Mihai V.
2009-01-01
The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr’s quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions – all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems. PMID:20087467
7. Path integrals for electronic densities, reactivity indices, and localization functions in quantum systems.
PubMed
Putz, Mihai V
2009-11-10
The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr's quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions - all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems.
8. Path integral molecular dynamics at zero thermal temperature
Willow, Soohaeng Yoo
2017-04-01
Path integral molecular dynamics (PIMD) simulations at the zero thermal temperature still remain inconceivable. Herein, the quantum-mechanical partition function is revised in conjunction with the time-independent Schrödinger equation. The imaginary temperature for the quantum-mechanical partition function is introduced as an independent variable and defined under the guidance of the virial theorem. In the end, computational evidences are provided showing that this revised PIMD simulation at the zero thermal temperature reproduces both the zero-point energy and the probability density obtained from the Schrödinger equation for the harmonic oscillator.
9. Quantum corrections from a path integral over reparametrizations
SciTech Connect
Makeenko, Yuri; Olesen, Poul
2010-08-15
We study the path integral over reparametrizations that has been proposed as an ansatz for the Wilson loops in the large-N QCD and reproduces the area law in the classical limit of large loops. We show that a semiclassical expansion for a rectangular loop captures the Luescher term associated with d=26 dimensions and propose a modification of the ansatz that reproduces the Luescher term in other dimensions, which is observed in lattice QCD. We repeat the calculation for an outstretched ellipse advocating the emergence of an analog of the Luescher term and verify this result by a direct computation of the determinant of the Laplace operator and the conformal anomaly.
10. Path-integral formula for local thermal equilibrium
Hongo, Masaru
2017-08-01
We develop a complete path-integral formulation of relativistic quantum fields in local thermal equilibrium, which brings about the emergence of thermally induced curved spacetime. The resulting action is shown to have full diffeomorphism invariance and gauge invariance in thermal spacetime with imaginary-time independent backgrounds. This leads to the notable symmetry properties of emergent thermal spacetime: Kaluza-Klein gauge symmetry, spatial diffeomorphism symmetry, and gauge symmetry. A thermodynamic potential in local thermal equilibrium, or the so-called Massieu-Planck functional, is identified as a generating functional for conserved currents such as the energy-momentum tensor and the electric current.
11. Path-integral approach to the Wigner-Kirkwood expansion.
PubMed
Jizba, Petr; Zatloukal, Václav
2014-01-01
We study the high-temperature behavior of quantum-mechanical path integrals. Starting from the Feynman-Kac formula, we derive a functional representation of the Wigner-Kirkwood perturbation expansion for quantum Boltzmann densities. As shown by its applications to different potentials, the presented expansion turns out to be quite efficient in generating analytic form of the higher-order expansion coefficients. To put some flesh on the bare bones, we apply the expansion to obtain basic thermodynamic functions of the one-dimensional anharmonic oscillator. Further salient issues, such as generalization to the Bloch density matrix and comparison with the more customary world-line formulation, are discussed.
12. A path to integration in an academic health science center.
PubMed Central
Panko, W. B.; Wilson, W.
1992-01-01
This article describes a networking and integration strategy in use at the University of Michigan Medical Center. This strategy builds upon the existing technology base and is designed to provide a roadmap that will direct short-term development along a productive, long-term path. It offers a way to permit the short-term development of incremental solutions to current problems while at the same time maximizing the likelihood that these incremental efforts can be recycled into a more comprehensive approach. PMID:1336413
13. Exact path integral treatment of a diatomic molecule potential
SciTech Connect
Benamira, F.; Guechi, L.; Mameri, S.; Sadoun, M. A.
2007-03-15
A rigorous evaluation of the path integral for Green's function associated with a four-parameter potential for a diatomic molecule is presented. A closed form of Green's function is obtained for different shapes of this potential. When the deformation parameter {lambda} is {lambda}<0 or 0<{lambda}<1, it is found that the quantization conditions are transcendental equations that require a numerical solution. For {lambda}{>=}1 and r set-membership sign](1/{eta})ln {lambda},{infinity}[, the energy spectrum and the normalized wave functions of the bound states are derived. Particular cases of this potential which appear in the literature are also briefly discussed.
14. Remarks on the Origin of Path Integration:. Einstein and Feynman
Sauer, T.
2008-11-01
I offer some historical comments about the origins of Feynman's path-integral approach, as an alternative approach to standard quantum mechanics. Looking at the interaction between Einstein and Feynman, which was mediated by Feynman's thesis supervisor John Wheeler, it is argued that, contrary to what one might expect, the significance of the interaction between Einstein and Feynman pertained to a critique of classical field theory, rather than to a direct critique of quantum mechanics itself. Nevertheless, the critical perspective on classical field theory became a motivation and point of departure for Feynman's space-time approach to non-relativistic quantum mechanics.
15. Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation.
PubMed
Liang, L Z J; Lemmens, D; Tempere, J
2011-05-01
In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen atom with path integrals is performed to price timer options under stochastic volatility models. We present general pricing formulas for both the perpetual timer call options and the finite time-horizon timer call options. These general results allow us to find closed-form pricing formulas for both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility model as well as under the Heston stochastic volatility model. For the treatment of timer options under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston model we will rely on the Kratzer potential. © 2011 American Physical Society
16. Path integral methods for the dynamics of stochastic and disordered systems
Hertz, John A.; Roudi, Yasser; Sollich, Peter
2017-01-01
We review some of the techniques used to study the dynamics of disordered systems subject to both quenched and fast (thermal) noise. Starting from the Martin-Siggia-Rose/Janssen-De Dominicis-Peliti path integral formalism for a single variable stochastic dynamics, we provide a pedagogical survey of the perturbative, i.e. diagrammatic, approach to dynamics and how this formalism can be used for studying soft spin models. We review the supersymmetric formulation of the Langevin dynamics of these models and discuss the physical implications of the supersymmetry. We also describe the key steps involved in studying the disorder-averaged dynamics. Finally, we discuss the path integral approach for the case of hard Ising spins and review some recent developments in the dynamics of such kinetic Ising models.
17. A unified scheme for ab initio molecular orbital theory and path integral molecular dynamics
Shiga, Motoyuki; Tachikawa, Masanori; Miura, Shinichi
2001-11-01
We present a general approach for accurate calculation of chemical substances which treats both nuclei and electrons quantum mechanically, adopting ab initio molecular orbital theory for the electronic structure and path integral molecular dynamics for the nuclei. The present approach enables the evaluation of physical quantities dependent on the nuclear configuration as well as the electronic structure, within the framework of Born-Oppenheimer adiabatic approximation. As an application, we give the path integral formulation of electric response properties—dipole moment and polarizability, which characterize the changes both in electronic structure and nuclear configuration at a given temperature when uniform electrostatic field is present. We also demonstrate the calculation of a water molecule using the present approach and the result of temperature and isotope effects is discussed.
18. Idiothetic Path Integration in the Fruit Fly Drosophila melanogaster.
PubMed
Kim, Irene S; Dickinson, Michael H
2017-08-07
After discovering a small drop of food, hungry flies exhibit a peculiar behavior in which they repeatedly stray from, but then return to, the newly discovered resource. To study this behavior in more detail, we tracked hungry Drosophila as they explored a large arena, focusing on the question of how flies remain near the food. To determine whether flies use external stimuli, we individually eliminated visual, olfactory, and pheromonal cues. In all cases, flies still exhibited a centralized search behavior, suggesting that none of these cues are absolutely required for navigation back to the food. To simultaneously eliminate visual and olfactory cues associated with the position of the food, we constructed an apparatus in which the food could be rapidly translated from the center of the arena. Flies continued to search around the original location, even after the food was moved to a new position. A random search model based on measured locomotor statistics could not reproduce the centered nature of the animal's trajectory. We conclude that this behavior is best explained by a form of path integration in which the flies use idiothetic cues to search near the location of the food. We argue that the use of path integration to perform a centered local search is not a specialization of Drosophila but rather represents an ancient behavioral mode that is homologous to the more elaborate foraging strategies of central place foragers such as ants. Copyright © 2017 Elsevier Ltd. All rights reserved.
19. Path integral polymer propagator of relativistic and nonrelativistic particles
Morales-Técotl, Hugo A.; Rastgoo, Saeed; Ruelas, Juan C.
2017-03-01
A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so-called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer propagators previously obtained by spectral methods for a nonrelativistic polymer particle, both free and in a box, are regained with this method and as a new result we obtain the polymer propagator of the relativistic particle. All of them reduce to their standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Our results are robust thanks to their analytic and exact character which in turn come from the fact that presented models are solvable. They lend support to the vertex expansion scheme of the polymer path integral explored before in a formal way for cosmological models. Some possible future developments are commented upon in the discussion.
20. Path integration mediated systematic search: a Bayesian model.
PubMed
Vickerstaff, Robert J; Merkle, Tobias
2012-08-21
The systematic search behaviour is a backup system that increases the chances of desert ants finding their nest entrance after foraging when the path integrator has failed to guide them home accurately enough. Here we present a mathematical model of the systematic search that is based on extensive behavioural studies in North African desert ants Cataglyphis fortis. First, a simple search heuristic utilising Bayesian inference and a probability density function is developed. This model, which optimises the short-term nest detection probability, is then compared to three simpler search heuristics and to recorded search patterns of Cataglyphis ants. To compare the different searches a method to quantify search efficiency is established as well as an estimate of the error rate in the ants' path integrator. We demonstrate that the Bayesian search heuristic is able to automatically adapt to increasing levels of positional uncertainty to produce broader search patterns, just as desert ants do, and that it outperforms the three other search heuristics tested. The searches produced by it are also arguably the most similar in appearance to the ant's searches. Copyright © 2012 Elsevier Ltd. All rights reserved.
1. Path integral Monte Carlo on a lattice. II. Bound states
O'Callaghan, Mark; Miller, Bruce N.
2016-07-01
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and the atoms are restricted to sites on a one-dimensional lattice. A path integral formalism developed within the context of the canonical ensemble is utilized, where the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp in the context of a square well potential, forcing the qp to occupy bound states. We consider a one-dimensional square well potential where all atoms on the lattice are occupied with one atom with an on-site potential except for a contiguous set of sites of various lengths centered at the middle of the lattice. Comparison of the potential energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the numerical calculations.
2. Quantum-classical path integral. II. Numerical methodology
Lambert, Roberto; Makri, Nancy
2012-12-01
We present a quantum-classical methodology for propagating the density matrix of a system coupled to a polyatomic (large molecular or solvent) environment. The system is treated via a full path integral, while the dynamics of the environment is approximated in terms of classical trajectories. We obtain quantum-classical path integral (QCPI) expressions in which the trajectories can undergo transitions to other quantum states at regular time intervals, but the cumulative probability of these transitions is governed by the local strength of the state-to-state coupling as well as the magnitude of the solvent reorganization energy. If quantum effects in the coordinates of the environment are relatively weak, an inexpensive random hop approximation leads to accurate descriptions of the dynamics. We describe a systematic iterative scheme for including quantum mechanical corrections for the solvent by gradually accounting for nonlocal "quantum memory" effects. As the length of the included memory approaches the decoherence time of the environment, the iterative QCPI procedure converges to the full QCPI result. The methodology is illustrated with application to dissipative symmetric and asymmetric two-level systems.
3. Path-integral molecular dynamics simulation of diamond
Ramírez, Rafael; Herrero, Carlos P.; Hernández, Eduardo R.
2006-06-01
Diamond is studied by path-integral molecular dynamics simulations of the atomic nuclei in combination with a tight-binding Hamiltonian to describe its electronic structure and total energy. This approach allows us to quantify the influence of quantum zero-point vibrations and finite temperatures on both the electronic and vibrational properties of diamond. The electron-phonon coupling mediated by the zero-point vibration reduces the direct electronic gap of diamond by 10%. The calculated decrease of the direct gap with temperature shows good agreement with the experimental data available up to 700K . Anharmonic vibrational frequencies of the crystal have been obtained from a linear-response approach based on the path integral formalism. In particular, the temperature dependence of the zone-center optical phonon has been derived from the simulations. The anharmonicity of the interatomic potential produces a red shift of this phonon frequency. At temperatures above 500K , this shift is overestimated in comparison to available experimental data. The predicted temperature shift of the elastic constant c44 displays reasonable agreement with the available experimental results.
4. Potential theory, path integrals and the Laplacian of the indicator
Lange, Rutger-Jan
2012-11-01
This paper links the field of potential theory — i.e. the Dirichlet and Neumann problems for the heat and Laplace equation — to that of the Feynman path integral, by postulating the following seemingly ill-defined potential: V(x):=∓ {{σ^2}}/2nabla_x^2{1_{{xin D}}} where the volatility is the reciprocal of the mass (i.e. m = 1/ σ 2) and ħ = 1. The Laplacian of the indicator can be interpreted using the theory of distributions: it is the d-dimensional analogue of the Dirac δ'-function, which can formally be defined as partial_x^2{1_{x>0 }} . We show, first, that the path integral's perturbation series (or Born series) matches the classical single and double boundary layer series of potential theory, thereby connecting two hitherto unrelated fields. Second, we show that the perturbation series is valid for all domains D that allow Green's theorem (i.e. with a finite number of corners, edges and cusps), thereby expanding the classical applicability of boundary layers. Third, we show that the minus (plus) in the potential holds for the Dirichlet (Neumann) boundary condition; showing for the first time a particularly close connection between these two classical problems. Fourth, we demonstrate that the perturbation series of the path integral converges as follows:
mode of convergence absorbed propagator reflected propagator convex domain alternating monotone
5. PhytoPath: an integrative resource for plant pathogen genomics
PubMed Central
Pedro, Helder; Maheswari, Uma; Urban, Martin; Irvine, Alistair George; Cuzick, Alayne; McDowall, Mark D.; Staines, Daniel M.; Kulesha, Eugene; Hammond-Kosack, Kim Elizabeth; Kersey, Paul Julian
2016-01-01
PhytoPath (www.phytopathdb.org) is a resource for genomic and phenotypic data from plant pathogen species, that integrates phenotypic data for genes from PHI-base, an expertly curated catalog of genes with experimentally verified pathogenicity, with the Ensembl tools for data visualization and analysis. The resource is focused on fungi, protists (oomycetes) and bacterial plant pathogens that have genomes that have been sequenced and annotated. Genes with associated PHI-base data can be easily identified across all plant pathogen species using a BioMart-based query tool and visualized in their genomic context on the Ensembl genome browser. The PhytoPath resource contains data for 135 genomic sequences from 87 plant pathogen species, and 1364 genes curated for their role in pathogenicity and as targets for chemical intervention. Support for community annotation of gene models is provided using the WebApollo online gene editor, and we are working with interested communities to improve reference annotation for selected species. PMID:26476449
6. 77 FR 33486 - Certain Integrated Circuit Packages Provided With Multiple Heat-Conducting Paths and Products...
Federal Register 2010, 2011, 2012, 2013, 2014
2012-06-06
... COMMISSION Certain Integrated Circuit Packages Provided With Multiple Heat- Conducting Paths and Products.... International Trade Commission has received a complaint entitled Certain Integrated Circuit Packages Provided... sale within the United States after importation of certain integrated circuit packages provided...
7. Coupled exciton-photon Bose condensate in path integral formalism
Elistratov, A. A.; Lozovik, Yu. E.
2016-03-01
We study the behavior of exciton polaritons in an optical microcavity with an embedded semiconductor quantum well. We use a two-component exciton-photon approach formulated in terms of path integral formalism. In order to describe spatial distributions of the exciton and photon condensate densities, the two coupled equations of the Gross-Pitaevskii type are derived. For a homogeneous system, we find the noncondensate photon and exciton spectra, calculate the coefficients of transformation from the exciton-photon basis to the lower-upper polariton basis, and obtain the exciton and photon occupation numbers of the lower and upper polariton branches for nonzero temperatures. For an inhomogeneous system, the set of coupled equations of the Bogoliubov-de Gennes type is derived. The equations govern the spectra and spatial distributions of noncondensate photons and excitons.
8. Path integral quantization corresponding to the deformed Heisenberg algebra
SciTech Connect
Pramanik, Souvik; Moussa, Mohamed; Faizal, Mir; Ali, Ahmed Farag
2015-11-15
In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.
9. Path integral treatment of a noncentral electric potential
Ghoumaid, Ali; Benamira, Farid; Guechi, Larbi; Khiat, Zohra
2013-01-01
We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential V(r,θ ) = V(r) + tfrac{1} {{r^2 }}V(θ ) . It is shown that the Green's function can be calculated in spherical coordinate system for V(θ ) = frac{{hbar ^2 }} {{2μ }}frac{{γ + β sin ^2 θ + α sin ^4 θ }} {{sin ^2 θ \\cos ^2 θ }} . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.
10. Hippocampal “Time Cells”: Time versus Path Integration
PubMed Central
Kraus, Benjamin J.; Robinson, Robert J.; White, John A.; Eichenbaum, Howard; Hasselmo, Michael E.
2014-01-01
SUMMARY Recent studies have reported the existence of hippocampal “time cells,” neurons that fire at particular moments during periods when behavior and location are relatively constant. However, an alternative explanation of apparent time coding is that hippocampal neurons “path integrate” to encode the distance an animal has traveled. Here, we examined hippocampal neuronal firing patterns as rats ran in place on a treadmill, thus “clamping” behavior and location, while we varied the treadmill speed to distinguish time elapsed from distance traveled. Hippocampal neurons were strongly influenced by time and distance, and less so by minor variations in location. Furthermore, the activity of different neurons reflected integration over time and distance to varying extents, with most neurons strongly influenced by both factors and some significantly influenced by only time or distance. Thus, hippocampal neuronal networks captured both the organization of time and distance in a situation where these dimensions dominated an ongoing experience. PMID:23707613
11. Thermal Momentum Distribution from Path Integrals with Shifted Boundary Conditions
Giusti, Leonardo; Meyer, Harvey B.
2011-04-01
For a thermal field theory formulated in the grand canonical ensemble, the distribution of the total momentum is an observable characterizing the thermal state. We show that its cumulants are related to thermodynamic potentials. In a relativistic system, for instance, the thermal variance of the total momentum is a direct measure of the enthalpy. We relate the generating function of the cumulants to the ratio of (a) a partition function expressed as a Matsubara path integral with shifted boundary conditions in the compact direction and (b) the ordinary partition function. In this form the generating function is well suited for Monte Carlo evaluation, and the cumulants can be extracted straightforwardly. We test the method in the SU(3) Yang-Mills theory and obtain the entropy density at three different temperatures.
12. Path integral approach to electron scattering in classical electromagnetic potential
Chuang, Xu; Feng, Feng; Ying-Jun, Li
2016-05-01
As is known to all, the electron scattering in classical electromagnetic potential is one of the most widespread applications of quantum theory. Nevertheless, many discussions about electron scattering are based upon single-particle Schrodinger equation or Dirac equation in quantum mechanics rather than the method of quantum field theory. In this paper, by using the path integral approach of quantum field theory, we perturbatively evaluate the scattering amplitude up to the second order for the electron scattering by the classical electromagnetic potential. The results we derive are convenient to apply to all sorts of potential forms. Furthermore, by means of the obtained results, we give explicit calculations for the one-dimensional electric potential. Project supported by the National Natural Science Foundation of China (Grant Nos. 11374360, 11405266, and 11505285) and the National Basic Research Program of China (Grant No. 2013CBA01504).
13. Path To Ignition: US Indirect Target Physics (LIRPP Vol. 12)
Cray, M.; Campbell, E. M.
2016-10-01
The United States ICF Program has been pursuing an aggressive research program in preparation for an ignition demonstration on the National Ignition Facility. Los Alamos and Livermore laboratories have collaborated on resolving indirect drive target physics issues on the Nova laser at Livermore National Laboratory. This combined with detailed modeling of laser heated indirectly driven targets likely to achieve ignition, has provided the basis for planning for the NIF. A detailed understanding of target physics, laser performance, and target fabrication is required for developing robust ignition targets. We have developed large-scale computational models to simulate complex physics which occurs in an indirectly driven target. For ignition, detailed understanding of hohlraum and implosion physics is required in order to control competing processes at the few percent level. From crucial experiments performed by Los Alamos and Livermore on the Nova laser, a comprehensive indirect drive database has been assembled. Time integrated and time dependent measurements of radiation drive and symmetry coupled with a detailed set of plasma instability measurements have confirmed our ability to predict hohlraum energetics. Implosion physics campaigns are focused on underdstanding detailed capsule hydrodynamics and instability growth. Target fabrication technology is also an active area of research at Los Alamos, Livermore, and General Atomics for NIF. NIF targets require developing technology in cryogenics and manufacturing in such areas as beryllium shell manufacture. Descriptions of our NIF target designs, experimental results, and fabrication technology supporting NIF target performance predictions will be given.
14. Conservation laws and path-independent integrals in mechanical-diffusion-electrochemical reaction coupling system
Yu, Pengfei; Wang, Hailong; Chen, Jianyong; Shen, Shengping
2017-07-01
In this study, the conservation laws οf dissipative mechanical-diffusion-electrochemical reaction system are systematically obtained based on Noether's theorem. According to linear, irreversible thermodynamics, dissipative phenomena can be described by an irreversible force and an irreversible flow. Additionally, the Lagrange function, L and the generalized Hamilton least-action principle are proposed to be used to obtain the conservation integrals. A group of these integrals, including the J-, M-, and L-integrals, can be then obtained using the classical Noether approach for dissipative processes. The relation between the J-integral and the energy release rate is illustrated. The path-independence of the J-integral is then proven. The J-integral, derived based on Noether's theorem, is a line integral, contrary to the propositions of existing published works that describe it both as a line and an area integral. Herein, we prove that the outcomes are identical, and identify the physical meaning of the area integral, a concept that was not explained previously. To show that the J-integral can dominate the distribution of the corresponding field quantities, an example of a partial, stress-diffusion coupling process is disscussed.
15. An adiabatic linearized path integral approach for quantum time correlation functions: electronic transport in metal-molten salt solutions.
PubMed
Causo, Maria Serena; Ciccotti, Giovanni; Montemayor, Daniel; Bonella, Sara; Coker, David F
2005-04-14
We generalize the linearized path integral approach to evaluate quantum time correlation functions for systems best described by a set of nuclear and electronic degrees of freedom, restricting ourselves to the adiabatic approximation. If the operators in the correlation function are nondiagonal in the electronic states, then this adiabatic linearized path integral approximation for the thermal averaged quantum dynamics presents interesting and distinctive features, which we derive and explore in this paper. The capability of these approximations to accurately reproduce the behavior of physical systems is demonstrated by calculating the diffusion constant for an excess electron in a metal-molten salt solution.
16. An introduction to stochastic control theory, path integrals and reinforcement learning
Kappen, Hilbert J.
2007-02-01
Control theory is a mathematical description of how to act optimally to gain future rewards. In this paper I give an introduction to deterministic and stochastic control theory and I give an overview of the possible application of control theory to the modeling of animal behavior and learning. I discuss a class of non-linear stochastic control problems that can be efficiently solved using a path integral or by MC sampling. In this control formalism the central concept of cost-to-go becomes a free energy and methods and concepts from statistical physics can be readily applied.
17. Seismic Imaging, One-Way Wave Equations, Pseudodifferential Operators, Path Integrals, and all that Jazz
Artoun, Ojenie; David-Rus, Diana; Emmett, Matthew; Fishman, Lou; Fital, Sandra; Hogan, Chad; Lim, Jisun; Lushi, Enkeleida; Marinov, Vesselin
2006-05-01
In this report we summarize an extension of Fourier analysis for the solution of the wave equation with a non-constant coefficient corresponding to an inhomogeneous medium. The underlying physics of the problem is exploited to link pseudodifferential operators and phase space path integrals to obtain a marching algorithm that incorporates the backward scattering into the evolution of the wave. This allows us to successfully apply single-sweep, one-way marching methods in inherently two-way environments, which was not achieved before through other methods for this problem.
18. Age differences in virtual environment and real world path integration
PubMed Central
Adamo, Diane E.; Briceño, Emily M.; Sindone, Joseph A.; Alexander, Neil B.; Moffat, Scott D.
2012-01-01
Accurate path integration (PI) requires the integration of visual, proprioceptive, and vestibular self-motion cues and age effects associated with alterations in processing information from these systems may contribute to declines in PI abilities. The present study investigated age-related differences in PI in conditions that varied as a function of available sources of sensory information. Twenty-two healthy, young (23.8 ± 3.0 years) and 16 older (70.1 ± 6.4 years) adults participated in distance reproduction and triangle completion tasks (TCTs) performed in a virtual environment (VE) and two “real world” conditions: guided walking and wheelchair propulsion. For walking and wheelchair propulsion conditions, participants wore a blindfold and wore noise-blocking headphones and were guided through the workspace by the experimenter. For the VE condition, participants viewed self-motion information on a computer monitor and used a joystick to navigate through the environment. For TCTs, older compared to younger individuals showed greater errors in rotation estimations performed in the wheelchair condition, and for rotation and distance estimations in the VE condition. Distance reproduction tasks (DRTs), in contrast, did not show any age effects. These findings demonstrate that age differences in PI vary as a function of the available sources of information and by the complexity of outbound pathway. PMID:23055969
19. Path integral in area tensor Regge calculus and complex connections
Khatsymovsky, V. M.
2006-06-01
Euclidean quantum measure in Regge calculus with independent area tensors is considered using example of the Regge manifold of a simple structure. We go over to integrations along certain contours in the hyperplane of complex connection variables. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. We argue that upon integrating out the latter the resulting measure can be well-defined on physical hypersurface (for the area tensors corresponding to certain edge vectors, i.e. to certain metric) as positive and having exponential cutoff at large areas on condition that we confine ourselves to configurations which do not pass through degenerate metrics.
20. Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure
LaChapelle, J.
1996-09-01
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce quantum corrections'' into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.
1. A comparison experiment between open and closed path eddy covariance devices using an integrated open path sensor
Hogan, Patrick; Parajka, Juraj; Blöschl, Günter
2014-05-01
The eddy covariance method has become one of the most common methods for measuring evaporation and carbon dioxide fluxes as it makes direct measurements and can be used at different spatial scales. Eddy covariance measurement devices are divided into two different designs, designated open path and closed path depending on where the gas of interest is measured. There is currently no preferred eddy covariance design, with the decision on which design to use usually based on the local precipitation conditions and power availability. A recent long term field comparison by Haslwanter et al. (2009) found differences in the measured and corrected evaporation between the different designs, with the largest differences in the latent heat flux occurring during periods of above average meteorological conditions. All previous comparison studies have been performed using the LI-7500 OP analyser which must be placed a distance away from the closed path intake and the path of the sonic anemometer. This must be accounted for by including corrections for high frequency filtering and sensor heating. The objective of this study is to use the IRGASON open-path design from Campbell Scientific where the gas analyser and sonic anemometer will be directly aligned with the intake to a closed path sensor to compare the different sensor designs. The measurements will be performed at the HOAL catchment at Petzenkirchen, Austria, which is equipped with a weather and energy balance station as well as an extensive soil moisture network to measure evaporation. This project will perform a comparison between open path and closed path eddy covariance systems using a new integrated open path design. The measurements will then be used to study the differences between the corrections required for the different designs and the effects of meteorological variables on the measured latent heat fluxes to address the issue of open and closed path gas analyser comparisons.
2. The retrosplenial cortex is necessary for path integration in the dark.
PubMed
Elduayen, Coralie; Save, Etienne
2014-10-01
An increasing amount of data indicates that the retrosplenial cortex (RSC) plays a role in navigation and spatial memory. Moreover, it has been suggested that the RSC integrates mnemonic spatial features of the environment with self-motion information therefore enabling accurate path integration in darkness. This hypothesis rests on data obtained from animals trained in spatial memory tasks involving the conjoint use of allothetic and idiothetic information [8]. We examined the contribution of the RSC when animals are submitted to a path integration task with minimal memory requirement in light and dark. We found that RSC-lesioned rats exhibited a path integration deficit in the dark but not in the light. This suggests that the RSC is important for path integration and incorporates visuospatial information to maintain path integration accuracy.
3. Keeping track of the distance from home by leaky integration along veering paths.
PubMed
Lappe, Markus; Stiels, Maren; Frenz, Harald; Loomis, Jack M
2011-07-01
When humans use vision to gauge the travel distance of an extended forward movement, they often underestimate the movement's extent. This underestimation can be explained by leaky path integration, an integration of the movement to obtain distance. Distance underestimation occurs because this integration is imperfect and contains a leak that increases with distance traveled. We asked human observers to estimate the distance from a starting location for visually simulated movements in a virtual environment. The movements occurred along curved paths that veered left and right around a central forward direction. In this case, the distance that has to be integrated (i.e., the beeline distance between origin and endpoint) and the distance that is traversed (the path length along the curve) are distinct. We then tested whether the leak accumulated with distance from the origin or with traversed distance along the curved path. Leaky integration along the path makes the seemingly counterintuitive prediction that the estimated origin-to-endpoint distance should decrease with increasing veering, because the length of the path over which the integration occurs increases, leading to a larger leak effect. The results matched the prediction: movements of identical origin-to-endpoint distance were judged as shorter when the path became longer. We conclude that leaky path integration from visual motion is performed along the traversed path even when a straight beeline distance is calculated.
4. Quantum effects in graphene monolayers: Path-integral simulations.
PubMed
Herrero, Carlos P; Ramírez, Rafael
2016-12-14
Path-integral molecular dynamics (PIMD) simulations have been carried out to study the influence of quantum dynamics of carbon atoms on the properties of a single graphene layer. Finite-temperature properties were analyzed in the range from 12 to 2000 K, by using the LCBOPII effective potential. To assess the magnitude of quantum effects in structural and thermodynamic properties of graphene, classical molecular dynamics simulations have been also performed. Particular emphasis has been laid on the atomic vibrations along the out-of-plane direction. Even though quantum effects are present in these vibrational modes, we show that at any finite temperature classical-like motion dominates over quantum delocalization, provided that the system size is large enough. Vibrational modes display an appreciable anharmonicity, as derived from a comparison between kinetic and potential energies of the carbon atoms. Nuclear quantum effects are found to be appreciable in the interatomic distance and layer area at finite temperatures. The thermal expansion coefficient resulting from PIMD simulations vanishes in the zero-temperature limit, in agreement with the third law of thermodynamics.
5. Computational Convergence of the Path Integral for Real Dendritic Morphologies
PubMed Central
2012-01-01
Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of sub-threshold dendritic currents. The Green’s function obtained for a given dendritic geometry provides this functional relationship for passive or quasi-active dendrites and can be constructed by a sum-over-trips approach based on a path integral formalism. In this paper, we introduce a number of efficient algorithms for realisation of the sum-over-trips framework and investigate the convergence of these algorithms on different dendritic geometries. We demonstrate that the convergence of the trip sampling methods strongly depends on dendritic morphology as well as the biophysical properties of the cell membrane. For real morphologies, the number of trips to guarantee a small convergence error might become very large and strongly affect computational efficiency. As an alternative, we introduce a highly-efficient matrix method which can be applied to arbitrary branching structures. PMID:23174188
6. Path integral regularization of pure Yang-Mills theory
SciTech Connect
Jacquot, J. L.
2009-07-15
In enlarging the field content of pure Yang-Mills theory to a cutoff dependent matrix valued complex scalar field, we construct a vectorial operator, which is by definition invariant with respect to the gauge transformation of the Yang-Mills field and with respect to a Stueckelberg type gauge transformation of the scalar field. This invariant operator converges to the original Yang-Mills field as the cutoff goes to infinity. With the help of cutoff functions, we construct with this invariant a regularized action for the pure Yang-Mills theory. In order to be able to define both the gauge and scalar fields kinetic terms, other invariant terms are added to the action. Since the scalar fields flat measure is invariant under the Stueckelberg type gauge transformation, we obtain a regularized gauge-invariant path integral for pure Yang-Mills theory that is mathematically well defined. Moreover, the regularized Ward-Takahashi identities describing the dynamics of the gauge fields are exactly the same as the formal Ward-Takahashi identities of the unregularized theory.
7. Quantum effects in graphene monolayers: Path-integral simulations
Herrero, Carlos P.; Ramírez, Rafael
2016-12-01
Path-integral molecular dynamics (PIMD) simulations have been carried out to study the influence of quantum dynamics of carbon atoms on the properties of a single graphene layer. Finite-temperature properties were analyzed in the range from 12 to 2000 K, by using the LCBOPII effective potential. To assess the magnitude of quantum effects in structural and thermodynamic properties of graphene, classical molecular dynamics simulations have been also performed. Particular emphasis has been laid on the atomic vibrations along the out-of-plane direction. Even though quantum effects are present in these vibrational modes, we show that at any finite temperature classical-like motion dominates over quantum delocalization, provided that the system size is large enough. Vibrational modes display an appreciable anharmonicity, as derived from a comparison between kinetic and potential energies of the carbon atoms. Nuclear quantum effects are found to be appreciable in the interatomic distance and layer area at finite temperatures. The thermal expansion coefficient resulting from PIMD simulations vanishes in the zero-temperature limit, in agreement with the third law of thermodynamics.
8. Nearest neighbor interaction in the Path Integral Renormalization Group method
de Silva, Wasanthi; Clay, R. Torsten
2014-03-01
The Path Integral Renormalization Group (PIRG) method is an efficient numerical algorithm for studying ground state properties of strongly correlated electron systems. The many-body ground state wave function is approximated by an optimized linear combination of Slater determinants which satisfies the variational principle. A major advantage of PIRG is that is does not suffer the Fermion sign problem of quantum Monte Carlo. Results are exact in the noninteracting limit and can be enhanced using space and spin symmetries. Many observables can be calculated using Wick's theorem. PIRG has been used predominantly for the Hubbard model with a single on-site Coulomb interaction U. We describe an extension of PIRG to the extended Hubbard model (EHM) including U and a nearest-neighbor interaction V. The EHM is particularly important in models of charge-transfer solids (organic superconductors) and at 1/4-filling drives a charge-ordered state. The presence of lattice frustration also makes studying these systems difficult. We test the method with comparisons to small clusters and long one dimensional chains, and show preliminary results for a coupled-chain model for the (TMTTF)2X materials. This work was supported by DOE grant DE-FG02-06ER46315.
9. Isotope effects in ice Ih: a path-integral simulation.
PubMed
Herrero, Carlos P; Ramírez, Rafael
2011-03-07
Ice Ih has been studied by path-integral molecular dynamics simulations, using the effective q-TIP4P/F potential model for flexible water. This has allowed us to analyze finite-temperature quantum effects in this solid phase from 25 to 300 K at ambient pressure. Among these effects we find a negative thermal expansion of ice at low temperatures, which does not appear in classical molecular dynamics simulations. The compressibility derived from volume fluctuations gives results in line with experimental data. We have analyzed isotope effects in ice Ih by considering normal, heavy, and tritiated water. In particular, we studied the effect of changing the isotopic mass of hydrogen on the kinetic energy and atomic delocalization in the crystal as well as on structural properties such as interatomic distances and molar volume. For D(2)O ice Ih at 100 K we obtained a decrease in molar volume and intramolecular O-H distance of 0.6% and 0.4%, respectively, as compared to H(2)O ice.
10. Path integral duality modified propagators in spacetimes with constant curvature
Kothawala, Dawood; Sriramkumar, L.; Shankaranarayanan, S.; Padmanabhan, T.
2009-08-01
The hypothesis of path integral duality provides a prescription to evaluate the propagator of a free, quantum scalar field in a given classical background, taking into account the existence of a fundamental length, say, the Planck length LP in a locally Lorentz invariant manner. We use this prescription to evaluate the duality modified propagators in spacetimes with constant curvature (exactly in the case of one spacetime, and in the Gaussian approximation for another two), and show that (i) the modified propagators are ultraviolet finite, (ii) the modifications are nonperturbative in LP, and (iii) LP seems to behave like a “zero point length” of spacetime intervals such that ⟨σ2(x,x')⟩=[σ2(x,x')+O(1)LP2], where σ(x,x') is the geodesic distance between the two spacetime points x and x', and the angular brackets denote (a suitable) average over the quantum gravitational fluctuations. We briefly discuss the implications of our results.
11. Factors Affecting Counselor Educators' Integration of Educational Technology: A Path Analysis
ERIC Educational Resources Information Center
Kennedy, John F.
2011-01-01
This study used path analysis to explore the effects of individual and institutional-level factors on counselor educators' integration of technology in counselor education. The study fills a gap in the literature by providing a research-based path model describing counselor educators' integration of technology in counselor education. Counselor…
12. Factors Affecting Counselor Educators' Integration of Educational Technology: A Path Analysis
ERIC Educational Resources Information Center
Kennedy, John F.
2011-01-01
This study used path analysis to explore the effects of individual and institutional-level factors on counselor educators' integration of technology in counselor education. The study fills a gap in the literature by providing a research-based path model describing counselor educators' integration of technology in counselor education. Counselor…
13. A reductionist perspective on quantum statistical mechanics: Coarse-graining of path integrals.
PubMed
Sinitskiy, Anton V; Voth, Gregory A
2015-09-07
Computational modeling of the condensed phase based on classical statistical mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature quantum systems at equilibrium has been based on Feynman's imaginary time path integral approach and the ensuing quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of quantum mechanical operators. The theory offers a simplified perspective on quantum statistical mechanics, revealing its most reductionist connection to classical statistical physics. By doing so, it can facilitate a simpler representation of certain quantum effects in complex molecular environments.
14. A reductionist perspective on quantum statistical mechanics: Coarse-graining of path integrals
SciTech Connect
Sinitskiy, Anton V.; Voth, Gregory A.
2015-09-07
Computational modeling of the condensed phase based on classical statistical mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature quantum systems at equilibrium has been based on Feynman’s imaginary time path integral approach and the ensuing quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of quantum mechanical operators. The theory offers a simplified perspective on quantum statistical mechanics, revealing its most reductionist connection to classical statistical physics. By doing so, it can facilitate a simpler representation of certain quantum effects in complex molecular environments.
15. Color path-integral Monte Carlo simulations of quark-gluon plasma
Filinov, V. S.; Ivanov, Yu. B.; Bonitz, M.; Fortov, V. E.; Levashov, P. R.
2012-02-01
Thermodynamic properties of a strongly coupled quark-gluon plasma (QGP) of constituent quasiparticles are studied by a color path-integral Monte Carlo simulations (CPIMC). For our simulations we have presented QGP partition function in the form of color path integral with new relativistic measure instead of Gaussian one used in Feynman and Wiener path integrals. For integration over color variable we have also developed procedure of sampling color variables according to the group SU(3) Haar measure. It is shown that this method is able to reproduce the available quantum lattice chromodynamics (QCD) data.
16. A path-independent integral for fracture of solids under combined electrochemical and mechanical loadings
2014-11-01
In this study, we first demonstrate that the J-integral in classical linear elasticity becomes path-dependent when the solid is subjected to combined electrical, chemical and mechanical loadings. We then construct an electro-chemo-mechanical J-integral that is path-independent under such combined multiple driving forces. Further, we show that this electro-chemo-mechanical J-integral represents the rate at which the grand potential releases per unit crack growth. As an example, the path-independent nature of the electro-chemo-mechanical J-integral is demonstrated by solving the problem of a thin elastic film delaminated from a thick elastic substrate.
17. Canonical quantization, path integral representations, and pseudoclassical description of massive Weyl neutrinos in external backgrounds
Dvornikov, Maxim; Gitman, D. M.
2012-11-01
We study massive 1/2-spin particles in various external backgrounds keeping in mind applications to neutrino physics. We are mainly interested in massive Majorana (Weyl) fields. However, massive neutral Dirac particles are also considered. We formulate classical Lagrangian theory of the massive Weyl field in terms of Grassmann-odd two-component spinors. Then we construct the Hamiltonian formulation of such a theory, which turns out to be a theory with second-class constraints. Using this formulation we canonically quantize the massive free Weyl field. We derive propagators of the Weyl field and relate them to the propagator of a massive Dirac particle. We also study the massive Weyl particles propagating in the background mater. We find the path integral representation for the propagator of such a field, as well as the corresponding pseudoclassical particle action. The massless limit of the Weyl field interacting with the matter is considered and compared with results of other works. Finally, the path integral representation for the propagator of the neutral massive Dirac particle with an anomalous magnetic moment moving in the background matter and external electromagnetic field, as well as the corresponding pseudoclassical particle action are constructed.
18. Square-root actions, metric signature, and the path integral of quantum gravity
Carlini, A.; Greensite, J.
1995-12-01
We consider quantization of the Baierlein-Sharp-Wheeler form of the gravitational action, in which the lapse function is determined from the Hamiltonian constraint. This action has a square root form, analogous to the actions of the relativistic particle and Nambu string. We argue that path-integral quantization of the gravitational action should be based on a path integrand exp[ √i S] rather than the familiar Feynman expression exp[iS], and that unitarity requires integration over manifolds of both Euclidean and Lorentzian signature. We discuss the relation of this path integral to our previous considerations regarding the problem of time, and extend our approach to include fermions.
19. Physics as an Integrative Theme in Oceanography.
ERIC Educational Resources Information Center
Myers, Richard L.
1990-01-01
The teaching of physics as an integral part of an undergraduate oceanography course is described. A general outline of oceanography and the corresponding physics topics is given. The objectives, organization, and difficulties of such a course are discussed. (CW)
20. The path to active living: physical activity through community design in Somerville, Massachusetts.
PubMed
Burke, Noreen M; Chomitz, Virginia R; Rioles, Nicole A; Winslow, Stephen P; Brukilacchio, Lisa B; Baker, Jessie C
2009-12-01
Somerville, Massachusetts, an ethnically diverse, urban community northwest of Boston, presents opportunities and challenges for active living. With a dense street grid, well-maintained sidewalks, neighborhood parks, and existing Community Path, Somerville is very walkable. However, two major surface arteries traverse and bisect neighborhoods, creating pedestrian safety and environmental justice issues. Major goals included promoting increased collaboration and communication among existing active-living efforts; managing the Community Path extension project; encouraging Portuguese-speaking adults to incorporate daily physical activity; leveraging existing urban planning work to establish secure, attractive walking/biking corridors; and embedding active-living messages in everyday life. The Somerville Active Living by Design Partnership (ALbD) successfully created a robust task force that was integrated with citywide active-living efforts, secured resources to increase infrastructure and support for active living, including city-level coordinator positions, and changed decision-making practices that led to incorporation of pedestrian and bicycle transportation priorities into city planning and that influenced the extension of the Community Path. Partnerships must employ sustainability planning early on, utilize skilled facilitative leaders to manage leadership transitions, and engage new partners. Identifying, cultivating, and celebrating champions, especially those with political power, are critical. Working closely with research partners leads to rich data sources for planning and evaluation. Changing the built environment is difficult; working toward smaller wins is realistic and achievable. The synergy of ALbD and other community interventions created a foundation for short-term successes and accelerated political-cultural changes already underway with respect to active living.
1. PathCase-SB: integrating data sources and providing tools for systems biology research
PubMed Central
2012-01-01
Background Integration of metabolic pathways resources and metabolic network models, and deploying new tools on the integrated platform can help perform more effective and more efficient systems biology research on understanding the regulation of metabolic networks. Therefore, the tasks of (a) integrating under a single database environment regulatory metabolic networks and existing models, and (b) building tools to help with modeling and analysis are desirable and intellectually challenging computational tasks. Results PathCase Systems Biology (PathCase-SB) is built and released. This paper describes PathCase-SB user interfaces developed to date. The current PathCase-SB system provides a database-enabled framework and web-based computational tools towards facilitating the development of kinetic models for biological systems. PathCase-SB aims to integrate systems biology models data and metabolic network data of selected biological data sources on the web (currently, BioModels Database and KEGG, respectively), and to provide more powerful and/or new capabilities via the new web-based integrative framework. Conclusions Each of the current four PathCase-SB interfaces, namely, Browser, Visualization, Querying, and Simulation interfaces, have expanded and new capabilities as compared with the original data sources. PathCase-SB is already available on the web and being used by researchers across the globe. PMID:22697505
2. Path integrals, supersymmetric quantum mechanics, and the Atiyah-Singer index theorem for twisted Dirac
Fine, Dana S.; Sawin, Stephen
2017-01-01
Feynman's time-slicing construction approximates the path integral by a product, determined by a partition of a finite time interval, of approximate propagators. This paper formulates general conditions to impose on a short-time approximation to the propagator in a general class of imaginary-time quantum mechanics on a Riemannian manifold which ensure that these products converge. The limit defines a path integral which agrees pointwise with the heat kernel for a generalized Laplacian. The result is a rigorous construction of the propagator for supersymmetric quantum mechanics, with potential, as a path integral. Further, the class of Laplacians includes the square of the twisted Dirac operator, which corresponds to an extension of N = 1/2 supersymmetric quantum mechanics. General results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the Atiyah-Singer index theorem.
3. A brief view of known landmarks reorientates path integration in hamsters
Etienne, A. S.; Boulens, V.; Maurer, R.; Rowe, T.; Siegrist, C.
In darkness, hamsters commute between their nest and a feeding site through path integration only, and therefore show cumulative errors in the return direction to the nest. We examined whether a brief presentation of familiar room cues could reset the path integrator. The hamsters could see the room cues either during, or at the end of, the outward journey to the food place, in a conflict situation where motion cues and visual information were set at variance. In both conditions, the animals used mainly visual information to return home. Thus, hamsters can determine their azimuth, and possibly their location, through a visual fix, and can reset their path integrator through the fix. This allows them to update their position during further locomotion in the dark and thus to compute a correct homing vector with respect to a visually induced reference frame. Taking episodic positional fixes may greatly enhance the functional value of path integration.
4. A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat
Liu, Jian; Li, Dezhang; Liu, Xinzijian
2016-07-01
We introduce a novel simple algorithm for thermostatting path integral molecular dynamics (PIMD) with the Langevin equation. The staging transformation of path integral beads is employed for demonstration. The optimum friction coefficients for the staging modes in the free particle limit are used for all systems. In comparison to the path integral Langevin equation thermostat, the new algorithm exploits a different order of splitting for the phase space propagator associated to the Langevin equation. While the error analysis is made for both algorithms, they are also employed in the PIMD simulations of three realistic systems (the H2O molecule, liquid para-hydrogen, and liquid water) for comparison. It is shown that the new thermostat increases the time interval of PIMD by a factor of 4-6 or more for achieving the same accuracy. In addition, the supplementary material shows the error analysis made for the algorithms when the normal-mode transformation of path integral beads is used.
5. A simple and accurate algorithm for path integral molecular dynamics with the Langevin thermostat.
PubMed
Liu, Jian; Li, Dezhang; Liu, Xinzijian
2016-07-14
We introduce a novel simple algorithm for thermostatting path integral molecular dynamics (PIMD) with the Langevin equation. The staging transformation of path integral beads is employed for demonstration. The optimum friction coefficients for the staging modes in the free particle limit are used for all systems. In comparison to the path integral Langevin equation thermostat, the new algorithm exploits a different order of splitting for the phase space propagator associated to the Langevin equation. While the error analysis is made for both algorithms, they are also employed in the PIMD simulations of three realistic systems (the H2O molecule, liquid para-hydrogen, and liquid water) for comparison. It is shown that the new thermostat increases the time interval of PIMD by a factor of 4-6 or more for achieving the same accuracy. In addition, the supplementary material shows the error analysis made for the algorithms when the normal-mode transformation of path integral beads is used.
6. Rotational error in path integration: encoding and execution errors in angle reproduction.
PubMed
Chrastil, Elizabeth R; Warren, William H
2017-03-16
Path integration is fundamental to human navigation. When a navigator leaves home on a complex outbound path, they are able to keep track of their approximate position and orientation and return to their starting location on a direct homebound path. However, there are several sources of error during path integration. Previous research has focused almost exclusively on encoding error-the error in registering the outbound path in memory. Here, we also consider execution error-the error in the response, such as turning and walking a homebound trajectory. In two experiments conducted in ambulatory virtual environments, we examined the contribution of execution error to the rotational component of path integration using angle reproduction tasks. In the reproduction tasks, participants rotated once and then rotated again to face the original direction, either reproducing the initial turn or turning through the supplementary angle. One outstanding difficulty in disentangling encoding and execution error during a typical angle reproduction task is that as the encoding angle increases, so does the required response angle. In Experiment 1, we dissociated these two variables by asking participants to report each encoding angle using two different responses: by turning to walk on a path parallel to the initial facing direction in the same (reproduction) or opposite (supplementary angle) direction. In Experiment 2, participants reported the encoding angle by turning both rightward and leftward onto a path parallel to the initial facing direction, over a larger range of angles. The results suggest that execution error, not encoding error, is the predominant source of error in angular path integration. These findings also imply that the path integrator uses an intrinsic (action-scaled) rather than an extrinsic (objective) metric.
7. A Path Integral Approach to Option Pricing with Stochastic Volatility: Some Exact Results
Baaquie, Belal E.
1997-12-01
The Black-Scholes formula for pricing options on stocks and other securities has been generalized by Merton and Garman to the case when stock volatility is stochastic. The derivation of the price of a security derivative with stochastic volatility is reviewed starting from the first principles of finance. The equation of Merton and Garman is then recast using the path integration technique of theoretical physics. The price of the stock option is shown to be the analogue of the Schrödinger wavefunction of quantum mechanics and the exact Hamiltonian and Lagrangian of the system is obtained. The results of Hull and White are generalized to the case when stock price and volatility have non-zero correlation. Some exact results for pricing stock options for the general correlated case are derived.
8. Momentum distribution functions of strongly correlated systems of particles: Wigner approach and path integrals
Larkin, A. S.; Filinov, V. S.; Fortov, V. E.
2016-11-01
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of the strongly interacting systems of particles has been developed for extreme conditions, when there are no small physical parameters and analytical approximations used in different kind of perturbation theories can not be applied. The new path integral representation of the quantum Wigner function in the phase space has been developed for canonical ensemble. Explicit analytical expression of the Wigner function has been obtained in linear and harmonic approximations. The new quantum Monte-Carlo method for calculations of average values of arbitrary quantum operators has been proposed. Preliminary calculations of the momentum distribution function of the Coulomb systems of particles have been carried out. Comparison with classical Maxwell-Boltzmann distribution shows the significant influence of quantum effects on the high energy asymptotics (“tails’) of the calculated momentum distribution functions, which resulted in appearance of sharp oscillations.
9. Ultra-light and strong: The massless harmonic oscillator and its singular path integral
Modanese, Giovanni
2017-09-01
In classical mechanics, a light particle bound by a strong elastic force just oscillates at high frequency in the region allowed by its initial position and velocity. In quantum mechanics, instead, the ground state of the particle becomes completely de-localized in the limit m → 0. The harmonic oscillator thus ceases to be a useful microscopic physical model in the limit m → 0, but its Feynman path integral has interesting singularities which make it a prototype of other systems exhibiting a “quantum runaway” from the classical configurations near the minimum of the action. The probability density of the coherent runaway modes can be obtained as the solution of a Fokker-Planck equation associated to the condition S = Smin. This technique can be applied also to other systems, notably to a dimensional reduction of the Einstein-Hilbert action.
10. Continuum tensor network field states, path integral representations and spatial symmetries
Jennings, David; Brockt, Christoph; Haegeman, Jutho; Osborne, Tobias J.; Verstraete, Frank
2015-06-01
A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled-pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class’ definition must be Lorentz-invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.
11. 77 FR 39735 - Certain Integrated Circuit Packages Provided With Multiple Heat-Conducting Paths and Products...
Federal Register 2010, 2011, 2012, 2013, 2014
2012-07-05
... Integrated Circuit Packages Provided With Multiple Heat- Conducting Paths and Products Containing Same... within the United States after importation of certain integrated circuit packages provided with multiple... importation, or the sale within the United States after importation of certain integrated circuit...
12. Physics Thematic Paths: Laboratorial Activities and Historical Scientific Instruments
ERIC Educational Resources Information Center
Pantano, O.; Talas, S.
2010-01-01
The Physics Department of Padua University keeps an important collection of historical physics instruments which alludes to the fruitful scientific activity of Padua through the centuries. This heritage led to the suggestion of setting up laboratory activities connected to the Museum collection for secondary school students. This article shows how…
13. Which way and how far? Tracking of translation and rotation information for human path integration.
PubMed
Chrastil, Elizabeth R; Sherrill, Katherine R; Hasselmo, Michael E; Stern, Chantal E
2016-10-01
Path integration, the constant updating of the navigator's knowledge of position and orientation during movement, requires both visuospatial knowledge and memory. This study aimed to develop a systems-level understanding of human path integration by examining the basic building blocks of path integration in humans. To achieve this goal, we used functional imaging to examine the neural mechanisms that support the tracking and memory of translational and rotational components of human path integration. Critically, and in contrast to previous studies, we examined movement in translation and rotation tasks with no defined end-point or goal. Navigators accumulated translational and rotational information during virtual self-motion. Activity in hippocampus, retrosplenial cortex (RSC), and parahippocampal cortex (PHC) increased during both translation and rotation encoding, suggesting that these regions track self-motion information during path integration. These results address current questions regarding distance coding in the human brain. By implementing a modified delayed match to sample paradigm, we also examined the encoding and maintenance of path integration signals in working memory. Hippocampus, PHC, and RSC were recruited during successful encoding and maintenance of path integration information, with RSC selective for tasks that required processing heading rotation changes. These data indicate distinct working memory mechanisms for translation and rotation, which are essential for updating neural representations of current location. The results provide evidence that hippocampus, PHC, and RSC flexibly track task-relevant translation and rotation signals for path integration and could form the hub of a more distributed network supporting spatial navigation. Hum Brain Mapp 37:3636-3655, 2016. © 2016 Wiley Periodicals, Inc.
14. A variational path integral molecular dynamics study of a solid helium-4
Miura, Shinichi
2011-01-01
In the present study, a variational path integral molecular dynamics method developed by the author [Chem. Phys. Lett. 482 (2009) 165] is applied to a solid helium-4 in the ground state. The method is a molecular dynamics algorithm for a variational path integral method which can be used to generate the exact ground state numerically. The solid state is shown to successfully be realized by the method, although a poor trial wavefunction that cannot describe the solid state is used.
15. Path Integral Computation of Quantum Free Energy Differences Due to Alchemical Transformations Involving Mass and Potential.
PubMed
Pérez, Alejandro; von Lilienfeld, O Anatole
2011-08-09
Thermodynamic integration, perturbation theory, and λ-dynamics methods were applied to path integral molecular dynamics calculations to investigate free energy differences due to "alchemical" transformations. Several estimators were formulated to compute free energy differences in solvable model systems undergoing changes in mass and/or potential. Linear and nonlinear alchemical interpolations were used for the thermodynamic integration. We find improved convergence for the virial estimators, as well as for the thermodynamic integration over nonlinear interpolation paths. Numerical results for the perturbative treatment of changes in mass and electric field strength in model systems are presented. We used thermodynamic integration in ab initio path integral molecular dynamics to compute the quantum free energy difference of the isotope transformation in the Zundel cation. The performance of different free energy methods is discussed.
16. An Anatomically Constrained Model for Path Integration in the Bee Brain.
PubMed
Stone, Thomas; Webb, Barbara; Adden, Andrea; Weddig, Nicolai Ben; Honkanen, Anna; Templin, Rachel; Wcislo, William; Scimeca, Luca; Warrant, Eric; Heinze, Stanley
2017-10-04
17. Phase-space path-integral calculation of the Wigner function
Samson, J. H.
2003-10-01
The Wigner function W(q, p) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the midpoint of their ends; short paths where the midpoint is close to (q, p) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a sequence of density matrices interpolating between a Poissonian number distribution and a number state, each member of which can be represented exactly by a discretized path integral with a finite number of vertices. Saddle-point evaluation of these integrals recovers (up to a constant factor) the WKB approximation to the Wigner function of a number state.
18. The path dependency theory: analytical framework to study institutional integration. The case of France
PubMed Central
Trouvé, Hélène; Couturier, Yves; Etheridge, Francis; Saint-Jean, Olivier; Somme, Dominique
2010-01-01
Background The literature on integration indicates the need for an enhanced theorization of institutional integration. This article proposes path dependence as an analytical framework to study the systems in which integration takes place. Purpose PRISMA proposes a model for integrating health and social care services for older adults. This model was initially tested in Quebec. The PRISMA France study gave us an opportunity to analyze institutional integration in France. Methods A qualitative approach was used. Analyses were based on semi-structured interviews with actors of all levels of decision-making, observations of advisory board meetings, and administrative documents. Results Our analyses revealed the complexity and fragmentation of institutional integration. The path dependency theory, which analyzes the change capacity of institutions by taking into account their historic structures, allows analysis of this situation. The path dependency to the Bismarckian system and the incomplete reforms of gerontological policies generate the coexistence and juxtaposition of institutional systems. In such a context, no institution has sufficient ability to determine gerontology policy and build institutional integration by itself. Conclusion Using path dependence as an analytical framework helps to understand the reasons why institutional integration is critical to organizational and clinical integration, and the complex construction of institutional integration in France. PMID:20689740
19. Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes.
PubMed
Wang, Qian; Hammes-Schiffer, Sharon
2006-11-14
A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.
20. The development of path integration: combining estimations of distance and heading.
PubMed
Smith, Alastair D; McKeith, Laura; Howard, Christina J
2013-12-01
1. In Franklin's Path: Establishing Physics at the University of Pennsylvania
Halpern, Paul
2008-04-01
In 1751 Benjamin Franklin established the Academy of Philadelphia, the precursor of the University of Pennsylvania. Among its curricular mandates he envisioned included Natural and Mechanic History,'' using a popular text he suggested by No"el Antoine Pluche that encompassed optics and celestial dynamics among its subjects. This talk will trace the history of physics research and education at Penn from its establishment, to the appointment of the first designated physics professor, George Frederic Barker, in 1873, to the opening of the Randall Morgan Laboratory in 1901 under the directorship of Arthur Goodspeed, and finally to the inauguration of the David Rittenhouse Laboratory in 1954 under the university leadership of Gaylord Harnwell.
2. FUNdamental Integrative Training (FIT) for Physical Education
ERIC Educational Resources Information Center
Bukowsky, Michael; Faigenbaum, Avery D.; Myer, Gregory D.
2014-01-01
There is a growing need for physical education teachers to integrate different types of fitness activities into their lessons in order to provide opportunities for all students to learn and practice a variety of movement skills that will enhance their physical fitness and support free-time physical activity. An increased focus on age-appropriate…
3. FUNdamental Integrative Training (FIT) for Physical Education
ERIC Educational Resources Information Center
Bukowsky, Michael; Faigenbaum, Avery D.; Myer, Gregory D.
2014-01-01
There is a growing need for physical education teachers to integrate different types of fitness activities into their lessons in order to provide opportunities for all students to learn and practice a variety of movement skills that will enhance their physical fitness and support free-time physical activity. An increased focus on age-appropriate…
4. Quantum-classical path integral. I. Classical memory and weak quantum nonlocality.
PubMed
Lambert, Roberto; Makri, Nancy
2012-12-14
We consider rigorous path integral descriptions of the dynamics of a quantum system coupled to a polyatomic environment, assuming that the latter is well approximated by classical trajectories. Earlier work has derived semiclassical or purely classical expressions for the influence functional from the environment, which should be sufficiently accurate for many situations, but the evaluation of quantum-(semi)classical path integral (QCPI) expressions has not been practical for large-scale simulation because the interaction with the environment introduces couplings nonlocal in time. In this work, we analyze the nature of the effects on a system from its environment in light of the observation [N. Makri, J. Chem. Phys. 109, 2994 (1998)] that true nonlocality in the path integral is a strictly quantum mechanical phenomenon. If the environment is classical, the path integral becomes local and can be evaluated in a stepwise fashion along classical trajectories of the free solvent. This simple "classical path" limit of QCPI captures fully the decoherence of the system via a classical mechanism. Small corrections to the classical path QCPI approximation may be obtained via an inexpensive random hop QCPI model, which accounts for some "back reaction" effects. Exploiting the finite length of nonlocality, we argue that further inclusion of quantum decoherence is possible via an iterative evaluation of the path integral. Finally, we show that the sum of the quantum amplitude factors with respect to the system paths leads to a smooth integrand as a function of trajectory initial conditions, allowing the use of Monte Carlo methods for the multidimensional phase space integral.
5. How Do They Get Here?: Paths into Physics Education Research
ERIC Educational Resources Information Center
Barthelemy, Ramon S.; Henderson, Charles; Grunert, Megan L.
2013-01-01
Physics education research (PER) is a relatively new and rapidly growing area of Ph.D. specialization. To sustain the field of PER, a steady pipeline of talented scholars needs to be developed and supported. One aspect of building this pipeline is understanding how students come to graduate and postdoctoral work in PER and what their career goals…
6. Variational path integral molecular dynamics and hybrid Monte Carlo algorithms using a fourth order propagator with applications to molecular systems
Kamibayashi, Yuki; Miura, Shinichi
2016-08-01
In the present study, variational path integral molecular dynamics and associated hybrid Monte Carlo (HMC) methods have been developed on the basis of a fourth order approximation of a density operator. To reveal various parameter dependence of physical quantities, we analytically solve one dimensional harmonic oscillators by the variational path integral; as a byproduct, we obtain the analytical expression of the discretized density matrix using the fourth order approximation for the oscillators. Then, we apply our methods to realistic systems like a water molecule and a para-hydrogen cluster. In the HMC, we adopt two level description to avoid the time consuming Hessian evaluation. For the systems examined in this paper, the HMC method is found to be about three times more efficient than the molecular dynamics method if appropriate HMC parameters are adopted; the advantage of the HMC method is suggested to be more evident for systems described by many body interaction.
7. Non-uniqueness of quantum transition state theory and general dividing surfaces in the path integral space.
PubMed
Jang, Seogjoo; Voth, Gregory A
2017-05-07
Despite the fact that quantum mechanical principles do not allow the establishment of an exact quantum analogue of the classical transition state theory (TST), the development of a quantum TST (QTST) with a proper dynamical justification, while recovering the TST in the classical limit, has been a long standing theoretical challenge in chemical physics. One of the most recent efforts of this kind was put forth by Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)], which can be specified for any cyclically invariant dividing surface defined in the space of the imaginary time path integral. The present work revisits the issue of the non-uniqueness of QTST and provides a detailed theoretical analysis of HA-QTST for a general class of such path integral dividing surfaces. While we confirm that HA-QTST reproduces the result based on the ring polymer molecular dynamics (RPMD) rate theory for dividing surfaces containing only a quadratic form of low frequency Fourier modes, we find that it produces different results for those containing higher frequency imaginary time paths which accommodate greater quantum fluctuations. This result confirms the assessment made in our previous work [Jang and Voth, J. Chem. Phys. 144, 084110 (2016)] that HA-QTST does not provide a derivation of RPMD-TST in general and points to a new ambiguity of HA-QTST with respect to its justification for general cyclically invariant dividing surfaces defined in the space of imaginary time path integrals. Our analysis also offers new insights into similar path integral based QTST approaches.
8. Path-Integration Computation of the Transport Properties of Polymers Nanoparticles and Complex Biological Structures
Douglas, Jack
2014-03-01
One of the things that puzzled me when I was a PhD student working under Karl Freed was the curious unity between the theoretical descriptions of excluded volume interactions in polymers, the hydrodynamic properties of polymers in solution, and the critical properties of fluid mixtures, gases and diverse other materials (magnets, superfluids,etc.) when these problems were formally expressed in terms of Wiener path integration and the interactions treated through a combination of epsilon expansion and renormalization group (RG) theory. It seemed that only the interaction labels changed from one problem to the other. What do these problems have in common? Essential clues to these interrelations became apparent when Karl Freed, myself and Shi-Qing Wang together began to study polymers interacting with hyper-surfaces of continuously variable dimension where the Feynman perturbation expansions could be performed through infinite order so that we could really understand what the RG theory was doing. It is evidently simply a particular method for resuming perturbation theory, and former ambiguities no longer existed. An integral equation extension of this type of exact calculation to surfaces'' of arbitrary fixed shape finally revealed the central mathematical object that links these diverse physical models- the capacity of polymer chains, whose value vanishes at the critical dimension of 4 and whose magnitude is linked to the friction coefficient of polymer chains, the virial coefficient of polymers and the 4-point function of the phi-4 field theory,...Once this central object was recognized, it then became possible solve diverse problems in material science through the calculation of capacity, and related virials'' properties, through Monte Carlo sampling of random walk paths. The essential ideas of this computational method are discussed and some applications given to non-trivial problems: nanotubes treated as either rigid rods or ensembles worm-like chains having
9. Mnemonic Discrimination Relates to Perforant Path Integrity: An Ultra-High Resolution Diffusion Tensor Imaging Study
PubMed Central
Bennett, Ilana J.; Stark, Craig E.L.
2015-01-01
Pattern separation describes the orthogonalization of similar inputs into unique, non-overlapping representations. This computational process is thought to serve memory by reducing interference and to be mediated by the dentate gyrus of the hippocampus. Using ultra-high in-plane resolution diffusion tensor imaging (hrDTI) in older adults, we previously demonstrated that integrity of the perforant path, which provides input to the dentate gyrus from entorhinal cortex, was associated with mnemonic discrimination, a behavioral outcome designed to load on pattern separation. The current hrDTI study assessed the specificity of this perforant path integrity-mnemonic discrimination relationship relative to other cognitive constructs (identified using a factor analysis) and white matter tracts (hippocampal cingulum, fornix, corpus callosum) in 112 healthy adults (20–87 years). Results revealed age-related declines in integrity of the perforant path and other medial temporal lobe (MTL) tracts (hippocampal cingulum, fornix). Controlling for global effects of brain aging, perforant path integrity related only to the factor that captured mnemonic discrimination performance. Comparable integrity-mnemonic discrimination relationships were also observed for the hippocampal cingulum and fornix. Thus, whereas perforant path integrity specifically relates to mnemonic discrimination, mnemonic discrimination may be mediated by a broader MTL network. PMID:26149893
10. Mnemonic discrimination relates to perforant path integrity: An ultra-high resolution diffusion tensor imaging study.
PubMed
Bennett, Ilana J; Stark, Craig E L
2016-03-01
Pattern separation describes the orthogonalization of similar inputs into unique, non-overlapping representations. This computational process is thought to serve memory by reducing interference and to be mediated by the dentate gyrus of the hippocampus. Using ultra-high in-plane resolution diffusion tensor imaging (hrDTI) in older adults, we previously demonstrated that integrity of the perforant path, which provides input to the dentate gyrus from entorhinal cortex, was associated with mnemonic discrimination, a behavioral outcome designed to load on pattern separation. The current hrDTI study assessed the specificity of this perforant path integrity-mnemonic discrimination relationship relative to other cognitive constructs (identified using a factor analysis) and white matter tracts (hippocampal cingulum, fornix, corpus callosum) in 112 healthy adults (20-87 years). Results revealed age-related declines in integrity of the perforant path and other medial temporal lobe (MTL) tracts (hippocampal cingulum, fornix). Controlling for global effects of brain aging, perforant path integrity related only to the factor that captured mnemonic discrimination performance. Comparable integrity-mnemonic discrimination relationships were also observed for the hippocampal cingulum and fornix. Thus, whereas perforant path integrity specifically relates to mnemonic discrimination, mnemonic discrimination may be mediated by a broader MTL network.
11. Feynman-Kac Path Integral Calculation of the Ground State Energy of Atoms
Orr, David Edwin
In a paper written in 1950, the mathematician, Marc Kac, established a rigorous basis for the Feynman path-integral formulation of quantum mechanics. The original Feynman path integral lacks mathematical rigor in the definition of "summing over all paths", which are infinite in number (a theorem by Cameron states that a finite, real or complex, Lebesgue measure of the path defined by Feynman does not exist). The difficulty of using Feynman's method in a computation is supported by the observation that an accurate path integral solution of the hydrogen groundstate was only recently computed (in 1984). Since its introduction in 1950, the Feynman-Kac path integral (FKPI) has received limited attention despite its simplicity and power in solving quantum many-body problems. This work demonstrates that the FKPI method can be used to find the groundstate and excited states of small atomic systems to within experimental accuracy, and is ideally suited for the new massively parallel computer architectures, such as the Thinking Machines CM-5, the INTEL Paragon, et al., or, can be effectively used in a cluster of loosely -coupled workstations. It also demonstrates a simple procedure for incorporating into the FKPI computational method restrictions on the many-body wavefunction imposed by permutation symmetries of identical particles.
12. Physically Important Integrals Without Calculus
ERIC Educational Resources Information Center
Reif, F.; Larkin, Jill H.
1976-01-01
Develops a geometric solution to be used with spatial integrals needed for finding electric and magnetic fields. The method is used in the calculation of the electric fields due to a uniformly charged plane, line and sphere. (CP)
13. Application of path-independent integrals to elevated temperature crack growth
NASA Technical Reports Server (NTRS)
Kim, K. S.; Van Stone, R. H.
1990-01-01
The applicability of the J-integral in elasto-plastic fracture mechanics is limited to isothermal, monotonic loading conditions from the theoretical viewpoint, while in many applications, for instance gas turbine engines, crack growth occurs in the presence of cyclic inelastic loading, thermomechanical loading and temperature gradients. A number of path-independent (P-I) integrals have been proposed which do not have the restrictions of the J-integral. A review indicates that four of these integrals, although they are not the classical conservation integrals, are path-independent under these complex loading conditions. This paper describes a combined analytical and experimental effort to evaluate the ability of these four P-I integrals to correlate the crack growth data of Alloy 718 at elevated temperatures. Results for uniform temperature, 538 C, cases indicate that all these integrals are capable of correlating the crack growth data over a wide range of cyclic plasticity.
14. Factor Ordering and Path Integral Measure for Quantum Gravity in (1+1) Dimensions
Haga, John; Maitra, Rachel Lash
2017-06-01
We develop a mathematically rigorous path integral representation of the time evolution operator for a model of (1+1) quantum gravity that incorporates factor ordering ambiguity. In obtaining a suitable integral kernel for the time-evolution operator, one requires that the corresponding Hamiltonian is self-adjoint; this issue is subtle for a particular category of factor orderings. We identify and parametrize a complete set of self-adjoint extensions and provide a canonical description of these extensions in terms of boundary conditions. Moreover, we use Trotter-type product formulae to construct path-integral representations of time evolution.
15. Automatic Tool Path Generation for Robot Integrated Surface Sculpturing System
Zhu, Jiang; Suzuki, Ryo; Tanaka, Tomohisa; Saito, Yoshio
In this paper, a surface sculpturing system based on 8-axis robot is proposed, the CAD/CAM software and tool path generation algorithm for this sculpturing system are presented. The 8-axis robot is composed of a 6-axis manipulator and a 2-axis worktable, it carves block of polystyrene foams by heated cutting tools. Multi-DOF (Degree of Freedom) robot benefits from the faster fashion than traditional RP (Rapid Prototyping) methods and more flexibility than CNC machining. With its flexibility driven from an 8-axis configuration, as well as efficient custom-developed software for rough cutting and finish cutting, this surface sculpturing system can carve sculptured surface accurately and efficiently.
16. How do they get here?: Paths into physics education research
Barthelemy, Ramón S.; Henderson, Charles; Grunert, Megan L.
2013-12-01
17. Feynman path integral application on deriving black-scholes diffusion equation for european option pricing
Utama, Briandhika; Purqon, Acep
2016-08-01
Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.
18. A path-integral Langevin equation treatment of low-temperature doped helium clusters
Ing, Christopher; Hinsen, Konrad; Yang, Jing; Zeng, Toby; Li, Hui; Roy, Pierre-Nicholas
2012-06-01
We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)], 10.1063/1.3489925 sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of HeN-CO2 clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], 10.1002/(SICI)1096-987X(20000130)21:2<79::AID-JCC1>3.0.CO;2-B, an open-source molecular simulation package.
19. A path-integral Langevin equation treatment of low-temperature doped helium clusters.
PubMed
Ing, Christopher; Hinsen, Konrad; Yang, Jing; Zeng, Toby; Li, Hui; Roy, Pierre-Nicholas
2012-06-14
We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)] sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of He(N)-CO(2) clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], an open-source molecular simulation package.
20. Phase space path integral approach to harmonic oscillator with a time-dependent force constant
Janakiraman, Deepika; Sebastian, K. L.
2015-09-01
The quantum statistical mechanical propagator for a harmonic oscillator with a time-dependent force constant, mω2(t) , has been investigated in the past and was found to have only a formal solution in terms of the solutions of certain ordinary differential equations. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical expressions for such path integrals is of great interest. In a previous work, we had obtained the exact propagator for motion in an arbitrary time-dependent harmonic potential in the overdamped limit of friction using phase space path integrals in the context of Lévy flights - a result that can be easily extended to Brownian motion. In this paper, we make a connection between the overdamped Brownian motion and the imaginary time propagator of quantum mechanics and thereby get yet another way to evaluate the latter exactly. We find that explicit analytic solution for the quantum statistical mechanical propagator can be written when the time-dependent force constant has the form ω2(t) =λ2(t) - dλ(t)/dt, where λ(t) is any arbitrary function of t and use it to evaluate path integrals which have not been evaluated previously. We also employ this method to arrive at a formal solution of the propagator for both Lévy flights and Brownian subjected to a time-dependent harmonic potential in the underdamped limit of friction.
1. Social pain and physical pain: shared paths to resilience
PubMed Central
Sturgeon, John A; Zautra, Alex J
2016-01-01
Although clinical models have traditionally defined pain by its consequences for the behavior and internal states of the sufferer, recent evidence has highlighted the importance of examining pain in the context of the broader social environment. Neuroscience research has highlighted commonalities of neural pathways connecting the experience of physical and social pain, suggesting a substantial overlap between these phenomena. Further, interpersonal ties, support and aspects of the social environment can impair or promote effective adaptation to chronic pain through changes in pain perception, coping and emotional states. The current paper reviews the role of social factors in extant psychological interventions for chronic pain, and discusses how greater attention to these factors may inform future research and clinical care. PMID:26678402
2. Travel distance estimation from visual motion by leaky path integration.
PubMed
Lappe, Markus; Jenkin, Michael; Harris, Laurence R
2007-06-01
Visual motion can be a cue to travel distance when the motion signals are integrated. Distance estimates from visually simulated self-motion are imprecise, however. Previous work in our labs has given conflicting results on the imprecision: experiments by Frenz and Lappe had suggested a general underestimation of travel distance, while results from Redlick, Jenkin and Harris had shown an overestimation of travel distance. Here we describe a collaborative study that resolves the conflict by tracing it to differences in the tasks given to the subjects. With an identical set of subjects and identical visual motion simulation we show that underestimation of travel distance occurs when the task involves a judgment of distance from the starting position, and that overestimation of travel distance occurs when the task requires a judgment of the remaining distance to a particular target position. We present a leaky integrator model that explains both effects with a single mechanism. In this leaky integrator model we introduce the idea that, depending on the task, either the distance from start, or the distance to target is used as a state variable. The state variable is updated during the movement by integration over the space covered by the movement, rather than over time. In this model, travel distance mis-estimation occurs because the integration leaks and because the transformation of visual motion to travel distance involves a gain factor. Mis-estimates in both tasks can be explained with the same leak rate and gain in both conditions. Our results thus suggest that observers do not simply integrate traveled distance and then relate it to the task. Instead, the internally represented variable is either distance from the origin or distance to the goal, whichever is relevant.
3. PathPPI: an integrated dataset of human pathways and protein-protein interactions.
PubMed
Tang, HaiLin; Zhong, Fan; Liu, Wei; He, FuChu; Xie, HongWei
2015-06-01
Integration of pathway and protein-protein interaction (PPI) data can provide more information that could lead to new biological insights. PPIs are usually represented by a simple binary model, whereas pathways are represented by more complicated models. We developed a series of rules for transforming protein interactions from pathway to binary model, and the protein interactions from seven pathway databases, including PID, BioCarta, Reactome, NetPath, INOH, SPIKE and KEGG, were transformed based on these rules. These pathway-derived binary protein interactions were integrated with PPIs from other five PPI databases including HPRD, IntAct, BioGRID, MINT and DIP, to develop integrated dataset (named PathPPI). More detailed interaction type and modification information on protein interactions can be preserved in PathPPI than other existing datasets. Comparison analysis results indicate that most of the interaction overlaps values (O AB) among these pathway databases were less than 5%, and these databases must be used conjunctively. The PathPPI data was provided at http://proteomeview.hupo.org.cn/PathPPI/PathPPI.html.
4. Computational Acoustics: Computational PDEs, Pseudodifferential Equations, Path Integrals, and All That Jazz
Fishman, Louis
2000-11-01
The role of mathematical modeling in the physical sciences will be briefly addressed. Examples will focus on computational acoustics, with applications to underwater sound propagation, electromagnetic modeling, optics, and seismic inversion. Direct and inverse wave propagation problems in both the time and frequency domains will be considered. Focusing on fixed-frequency (elliptic) wave propagation problems, the usual, two-way, partial differential equation formulation will be exactly reformulated, in a well-posed manner, as a one-way (marching) problem. This is advantageous for both direct and inverse considerations, as well as stochastic modeling problems. The reformulation will require the introduction of pseudodifferential operators and their accompanying phase space analysis (calculus), in addition to path integral representations for the fundamental solutions and their subsequent computational algorithms. Unlike the more traditional, purely numerical applications of, for example, finite-difference and finite-element methods, this approach, in effect, writes the exact, or, more generally, the asymptotically correct, answer as a functional integral and, subsequently, computes it directly. The overall computational philosophy is to combine analysis, asymptotics, and numerical methods to attack complicated, real-world problems. Exact and asymptotic analysis will stress the complementary nature of the direct and inverse formulations, as well as indicating the explicit structural connections between the time- and frequency-domain solutions.
5. MPI CyberMotion Simulator: implementation of a novel motion simulator to investigate multisensory path integration in three dimensions.
PubMed
Barnett-Cowan, Michael; Meilinger, Tobias; Vidal, Manuel; Teufel, Harald; Bülthoff, Heinrich H
2012-05-10
Path integration is a process in which self-motion is integrated over time to obtain an estimate of one's current position relative to a starting point (1). Humans can do path integration based exclusively on visual (2-3), auditory (4), or inertial cues (5). However, with multiple cues present, inertial cues - particularly kinaesthetic - seem to dominate (6-7). In the absence of vision, humans tend to overestimate short distances (<5 m) and turning angles (<30°), but underestimate longer ones (5). Movement through physical space therefore does not seem to be accurately represented by the brain. Extensive work has been done on evaluating path integration in the horizontal plane, but little is known about vertical movement (see (3) for virtual movement from vision alone). One reason for this is that traditional motion simulators have a small range of motion restricted mainly to the horizontal plane. Here we take advantage of a motion simulator (8-9) with a large range of motion to assess whether path integration is similar between horizontal and vertical planes. The relative contributions of inertial and visual cues for path navigation were also assessed. 16 observers sat upright in a seat mounted to the flange of a modified KUKA anthropomorphic robot arm. Sensory information was manipulated by providing visual (optic flow, limited lifetime star field), vestibular-kinaesthetic (passive self motion with eyes closed), or visual and vestibular-kinaesthetic motion cues. Movement trajectories in the horizontal, sagittal and frontal planes consisted of two segment lengths (1st: 0.4 m, 2nd: 1 m; ±0.24 m/s(2) peak acceleration). The angle of the two segments was either 45° or 90°. Observers pointed back to their origin by moving an arrow that was superimposed on an avatar presented on the screen. Observers were more likely to underestimate angle size for movement in the horizontal plane compared to the vertical planes. In the frontal plane observers were more likely to
PubMed
Miller, Andrew; Rudland, Joy; Hurrell, Michael; Ali, Anthony
2009-01-01
The anatomical pathology tutorials for the fourth year of our undergraduate medical course at the University of Otago, Christchurch, New Zealand, were re-developed with the aim of promoting more active participation and application of learning by the students in their small group work. Radiology was integrated into half of the pathology tutorials with the objective of enhancing learning of both disciplines. The tutorials were designed to be easy to run for the tutors, who are mostly hospital-based staff. A set of 24 tutor-directed undergraduate anatomical pathology tutorials was re-developed into a case-based, student-centred format. Radiology learning was integrated into 12 tutorials. Student assessment was by way of three short answer examinations spaced throughout the course. Students and tutors completed evaluations of the tutorials. Student evaluations were positive, indicating the tutorials were considered valuable, relevant and a stimulus for thinking. Students and tutors endorsed the benefits of integrated pathology and radiology learning. Assessment results have demonstrated effective learning of pathology and radiology in the tutorials. The tutors found that the tutorials stimulated student engagement and were easy to run. The re-developed anatomical pathology tutorials with the integration of radiology appear to offer a number of benefits for learning of pathology and radiology.
7. Existence of short-time approximations of any polynomial order for the computation of density matrices by path integral methods
Predescu, Cristian
2004-05-01
In this paper I provide significant mathematical evidence in support of the existence of direct short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and bounded from below potentials. While for Theorem 2, which is “experimental,” I only provide a “physicist’s” proof, I believe the present development is mathematically sound. As a verification, I explicitly construct two short-time approximations to the density matrix having convergence orders 3 and 4, respectively. Furthermore, in Appendix B, I derive the convergence constant for the trapezoidal Trotter path integral technique. The convergence orders and constants are then verified by numerical simulations. While the two short-time approximations constructed are of sure interest to physicists and chemists involved in Monte Carlo path integral simulations, the present paper is also aimed at the mathematical community, who might find the results interesting and worth exploring. I conclude the paper by discussing the implications of the present findings with respect to the solvability of the dynamical sign problem appearing in real-time Feynman path integral simulations.
8. Path integral approach to eikonal and next-to-eikonal exponentiation
Laenen, Eric; Stavenga, Gerben; White, Chris D.
2009-03-01
We approach the issue of exponentiation of soft gauge boson corrections to scattering amplitudes from a path integral point of view. We show that if one represents the amplitude as a first quantized path integral in a mixed coordinate-momentum space representation, a charged particle interacting with a soft gauge field is represented as a Wilson line for a semi-infinite line segment, together with calculable fluctuations. Combining such line segments, we show that exponentiation in an abelian field theory follows immediately from standard path-integral combinatorics. In the non-abelian case, we consider color singlet hard interactions with two outgoing external lines, and obtain a new viewpoint for exponentiation in terms of webs'', with a closed form solution for their corresponding color factors. We investigate and clarify the structure of next-to-eikonal corrections.
9. On the relation between operator constraint, master constraint, reduced phase space and path integral quantization
Han, Muxin; Thiemann, T.
2010-11-01
Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space formulation. In this paper we review this rather involved procedure in full generality. Moreover, we demonstrate that the reduced phase space path integral formulation formally agrees with the Dirac's operator constraint quantization and, more specifically, with the master constraint quantization for first-class constraints. For first-class constraints with nontrivial structure functions the equivalence can only be established by passing to Abelian(ized) constraints which is always possible locally in phase space. Generically, the correct configuration space path integral measure deviates from the exponential of the Lagrangian action. The corrections are especially severe if the theory suffers from second-class secondary constraints. In a companion paper we compute these corrections for the Holst and Plebanski formulations of GR on which current spin foam models are based.
10. Integrated Learning with Physical Education and Music
ERIC Educational Resources Information Center
Humphries, Charlotte A.; Bidner, Sara; Edwards, Cheryl
2011-01-01
Integrated learning is well established in education, primarily in the classroom subjects. This article describes settings and ways for extending integrated instruction to physical education and music. Benefits of these connections include reinforcing content and better meeting the needs of students whose intelligences include the bodily…
11. Integrated Learning with Physical Education and Music
ERIC Educational Resources Information Center
Humphries, Charlotte A.; Bidner, Sara; Edwards, Cheryl
2011-01-01
Integrated learning is well established in education, primarily in the classroom subjects. This article describes settings and ways for extending integrated instruction to physical education and music. Benefits of these connections include reinforcing content and better meeting the needs of students whose intelligences include the bodily…
12. Robust path integration in the entorhinal grid cell system with hippocampal feed-back.
PubMed
Samu, Dávid; Eros, Péter; Ujfalussy, Balázs; Kiss, Tamás
2009-07-01
Animals are able to update their knowledge about their current position solely by integrating the speed and the direction of their movement, which is known as path integration. Recent discoveries suggest that grid cells in the medial entorhinal cortex might perform some of the essential underlying computations of path integration. However, a major concern over path integration is that as the measurement of speed and direction is inaccurate, the representation of the position will become increasingly unreliable. In this paper, we study how allothetic inputs can be used to continually correct the accumulating error in the path integrator system. We set up the model of a mobile agent equipped with the entorhinal representation of idiothetic (grid cell) and allothetic (visual cells) information and simulated its place learning in a virtual environment. Due to competitive learning, a robust hippocampal place code emerges rapidly in the model. At the same time, the hippocampo-entorhinal feed-back connections are modified via Hebbian learning in order to allow hippocampal place cells to influence the attractor dynamics in the entorhinal cortex. We show that the continuous feed-back from the integrated hippocampal place representation is able to stabilize the grid cell code.
13. Physics Research Integrated Development Environment (PRIDE)
SciTech Connect
Burton, J.; Cormell, L.
1993-12-01
Past efforts to implement a Software Engineering approach to High Energy Physics computing have been met with significant resistance and have been, in many cases, only marginally successful. At least a portion of the problem has been the Lick of an integrated development environment, tailored to High Energy Physics and incorporating a suite of Computer Aided Software Engineering tools. The Superconducting Super Collider Physics Research Division Computing Department is implementing pilot projects to develop just such an environment.
14. PathJam: a new service for integrating biological pathway information.
PubMed
Glez-Peña, Daniel; Reboiro-Jato, Miguel; Domínguez, Rubén; Gómez-López, Gonzalo; Pisano, David G; Fdez-Riverola, Florentino
2010-10-28
Biological pathways are crucial to much of the scientific research today including the study of specific biological processes related with human diseases. PathJam is a new comprehensive and freely accessible web-server application integrating scattered human pathway annotation from several public sources. The tool has been designed for both (i) being intuitive for wet-lab users providing statistical enrichment analysis of pathway annotations and (ii) giving support to the development of new integrative pathway applications. PathJam’s unique features and advantages include interactive graphs linking pathways and genes of interest, downloadable results in fully compatible formats, GSEA compatible output files and a standardized RESTful API.
15. Path integral measure, constraints and ghosts for massive gravitons with a cosmological constant
SciTech Connect
Metaxas, Dimitrios
2009-12-15
For massive gravity in a de Sitter background one encounters problems of stability when the curvature is larger than the graviton mass. I analyze this situation from the path integral point of view and show that it is related to the conformal factor problem of Euclidean quantum (massless) gravity. When a constraint for massive gravity is incorporated and the proper treatment of the path integral measure is taken into account one finds that, for particular choices of the DeWitt metric on the space of metrics (in fact, the same choices as in the massless case), one obtains the opposite bound on the graviton mass.
16. Ab initio path integral ring polymer molecular dynamics: Vibrational spectra of molecules
Shiga, Motoyuki; Nakayama, Akira
2008-01-01
The path integral ring polymer molecular dynamics method is combined with 'on-the-fly' ab initio electronic structure calculations and applied to vibrational spectra of small molecules, LiH and H 2O, at the room temperature. The results are compared with those of the numerically exact solution and ab initio path integral centroid molecular dynamics calculation. The peak positions in the calculated spectra are found to be reasonable, showing the red-shift due to potential anharmonicity. This unification enables the investigation of real-time quantum dynamics of chemically complex molecular systems on the ab initio Born-Oppenheimer potential energy surface.
17. A path integral approach to asset-liability management
Decamps, Marc; De Schepper, Ann; Goovaerts, Marc
2006-05-01
Functional integrals constitute a powerful tool in the investigation of financial models. In the recent econophysics literature, this technique was successfully used for the pricing of a number of derivative securities. In the present contribution, we introduce this approach to the field of asset-liability management. We work with a representation of cash flows by means of a two-dimensional delta-function perturbation, in the case of a Brownian model and a geometric Brownian model. We derive closed-form solutions for a finite horizon ALM policy. The results are numerically and graphically illustrated.
18. Enzymatic Kinetic Isotope Effects from Path-Integral Free Energy Perturbation Theory.
PubMed
Gao, J
2016-01-01
Path-integral free energy perturbation (PI-FEP) theory is presented to directly determine the ratio of quantum mechanical partition functions of different isotopologs in a single simulation. Furthermore, a double averaging strategy is used to carry out the practical simulation, separating the quantum mechanical path integral exactly into two separate calculations, one corresponding to a classical molecular dynamics simulation of the centroid coordinates, and another involving free-particle path-integral sampling over the classical, centroid positions. An integrated centroid path-integral free energy perturbation and umbrella sampling (PI-FEP/UM, or simply, PI-FEP) method along with bisection sampling was summarized, which provides an accurate and fast convergent method for computing kinetic isotope effects for chemical reactions in solution and in enzymes. The PI-FEP method is illustrated by a number of applications, to highlight the computational precision and accuracy, the rule of geometrical mean in kinetic isotope effects, enhanced nuclear quantum effects in enzyme catalysis, and protein dynamics on temperature dependence of kinetic isotope effects. © 2016 Elsevier Inc. All rights reserved.
19. Cyber/Physical Security Vulnerability Assessment Integration
SciTech Connect
MacDonald, Douglas G.; Clements, Samuel L.; Patrick, Scott W.; Perkins, Casey J.; Muller, George; Lancaster, Mary J.; Hutton, William J.
2013-02-28
Securing high value and critical assets is one of the biggest challenges facing this nation and others around the world. In modern integrated systems, there are four potential modes of attack available to an adversary: • physical only attack, • cyber only attack, • physical-enabled cyber attack, • cyber-enabled physical attack. Blended attacks involve an adversary working in one domain to reduce system effectiveness in another domain. This enables the attacker to penetrate further into the overall layered defenses. Existing vulnerability assessment (VA) processes and software tools which predict facility vulnerabilities typically evaluate the physical and cyber domains separately. Vulnerabilities which result from the integration of cyber-physical control systems are not well characterized and are often overlooked by existing assessment approaches. In this paper, we modified modification of the timely detection methodology, used for decades in physical security VAs, to include cyber components. The Physical and Cyber Risk Analysis Tool (PACRAT) prototype illustrates an integrated vulnerability assessment that includes cyber-physical interdependencies. Information about facility layout, network topology, and emplaced safeguards is used to evaluate how well suited a facility is to detect, delay, and respond to attacks, to identify the pathways most vulnerable to attack, and to evaluate how often safeguards are compromised for a given threat or adversary type. We have tested the PACRAT prototype on critical infrastructure facilities and the results are promising. Future work includes extending the model to prescribe the recommended security improvements via an automated cost-benefit analysis.
20. An integration of integrated information theory with fundamental physics
PubMed Central
2014-01-01
To truly eliminate Cartesian ghosts from the science of consciousness, we must describe consciousness as an aspect of the physical. Integrated Information Theory states that consciousness arises from intrinsic information generated by dynamical systems; however existing formulations of this theory are not applicable to standard models of fundamental physical entities. Modern physics has shown that fields are fundamental entities, and in particular that the electromagnetic field is fundamental. Here I hypothesize that consciousness arises from information intrinsic to fundamental fields. This hypothesis unites fundamental physics with what we know empirically about the neuroscience underlying consciousness, and it bypasses the need to consider quantum effects. PMID:24550877
1. An integration of integrated information theory with fundamental physics.
PubMed
2014-01-01
To truly eliminate Cartesian ghosts from the science of consciousness, we must describe consciousness as an aspect of the physical. Integrated Information Theory states that consciousness arises from intrinsic information generated by dynamical systems; however existing formulations of this theory are not applicable to standard models of fundamental physical entities. Modern physics has shown that fields are fundamental entities, and in particular that the electromagnetic field is fundamental. Here I hypothesize that consciousness arises from information intrinsic to fundamental fields. This hypothesis unites fundamental physics with what we know empirically about the neuroscience underlying consciousness, and it bypasses the need to consider quantum effects.
2. Managing the physics of the economics of integrated health care.
PubMed
Zismer, Daniel K; Werner, Mark J
2012-01-01
The physics metaphor, as applied to the economics (and financial performance) of the integrated health system, seems appropriate when considered together with the nine principles of management framework provided. The nature of the integrated design enhances leaders' management potential as they consider organizational operations and strategy in the markets ahead. One question begged by this argument for the integrated design is the durability, efficiency and ultimate long-term survivability of the more "traditional" community health care delivery models, which, by design, are fragmented, internally competitive and less capital efficient. They also cannot exploit the leverage of teams, optimal access management or the pursuit of revenues made available in many forms. For those who wish to move from the traditional to the more integrated community health system designs (especially those who have not yet started the journey), the path requires: * Sufficient balance sheet capacity to fund the integration process-especially as the model requires physician practice acquisitions and electronic health record implementations * A well-prepared board13, 14 * A functional, durable and sustainable physician services enterprise design * A redesigned organizational and governance structure * Favorable internal financial incentives alignment design * Effective accountable physician leadership * Awareness that the system is not solely a funding strategy for acquired physicians, rather a fully -.. committed clinical and business model, one in which patient-centered integrated care is the core service (and not acute care hospital-based services) A willingness to create and exploit the implied and inherent potential of an integrated design and unified brand Last, it's important to remember that an integrated health system is a tool that creates a "new potential" (a physics metaphor reference, one last time). The design doesn't operate itself. Application of the management principles
3. Integrative Families and Systems Treatment: A Middle Path toward Integrating Common and Specific Factors in Evidence-Based Family Therapy
ERIC Educational Resources Information Center
Fraser, J. Scott; Solovey, Andrew D.; Grove, David; Lee, Mo Yee; Greene, Gilbert J.
2012-01-01
A moderate common factors approach is proposed as a synthesis or middle path to integrate common and specific factors in evidence-based approaches to high-risk youth and families. The debate in family therapy between common and specific factors camps is reviewed and followed by suggestions from the literature for synthesis and creative flexibility…
4. Integrative Families and Systems Treatment: A Middle Path toward Integrating Common and Specific Factors in Evidence-Based Family Therapy
ERIC Educational Resources Information Center
Fraser, J. Scott; Solovey, Andrew D.; Grove, David; Lee, Mo Yee; Greene, Gilbert J.
2012-01-01
A moderate common factors approach is proposed as a synthesis or middle path to integrate common and specific factors in evidence-based approaches to high-risk youth and families. The debate in family therapy between common and specific factors camps is reviewed and followed by suggestions from the literature for synthesis and creative flexibility…
5. Singular path-independent energy integrals for elastic bodies with thin elastic inclusions
Shcherbakov, V. V.
2016-06-01
An equilibrium problem for a two-dimensional homogeneous linear elastic body containing a thin elastic inclusion and an interfacial crack is considered. The thin inclusion is modeled within the framework of Euler-Bernoulli beam theory. An explicit formula for the first derivative of the energy functional with respect to the crack perturbation along the interface is presented. It is shown that the formulas for the derivative associated with translation and self-similar expansion of the crack are represented as path-independent integrals along smooth contour surrounding one or both crack tips. These path-independent integrals consist of regular and singular terms and are analogs of the well-known Eshelby-Cherepanov-Rice J-integral and Knowles-Sternberg M-integral.
6. Integrated flight path planning system and flight control system for unmanned helicopters.
PubMed
Jan, Shau Shiun; Lin, Yu Hsiang
2011-01-01
This paper focuses on the design of an integrated navigation and guidance system for unmanned helicopters. The integrated navigation system comprises two systems: the Flight Path Planning System (FPPS) and the Flight Control System (FCS). The FPPS finds the shortest flight path by the A-Star (A*) algorithm in an adaptive manner for different flight conditions, and the FPPS can add a forbidden zone to stop the unmanned helicopter from crossing over into dangerous areas. In this paper, the FPPS computation time is reduced by the multi-resolution scheme, and the flight path quality is improved by the path smoothing methods. Meanwhile, the FCS includes the fuzzy inference systems (FISs) based on the fuzzy logic. By using expert knowledge and experience to train the FIS, the controller can operate the unmanned helicopter without dynamic models. The integrated system of the FPPS and the FCS is aimed at providing navigation and guidance to the mission destination and it is implemented by coupling the flight simulation software, X-Plane, and the computing software, MATLAB. Simulations are performed and shown in real time three-dimensional animations. Finally, the integrated system is demonstrated to work successfully in controlling the unmanned helicopter to operate in various terrains of a digital elevation model (DEM).
7. Quantum mechanical path integrals in curved spaces and the type-A trace anomaly
Bastianelli, Fiorenzo; Corradini, Olindo; Vassura, Edoardo
2017-04-01
Path integrals for particles in curved spaces can be used to compute trace anomalies in quantum field theories, and more generally to study properties of quantum fields coupled to gravity in first quantization. While their construction in arbitrary coordinates is well understood, and known to require the use of a regularization scheme, in this article we take up an old proposal of constructing the path integral by using Riemann normal coordinates. The method assumes that curvature effects are taken care of by a scalar effective potential, so that the particle lagrangian is reduced to that of a linear sigma model interacting with the effective potential. After fixing the correct effective potential, we test the construction on spaces of maximal symmetry and use it to compute heat kernel coefficients and type-A trace anomalies for a scalar field in arbitrary dimensions up to d = 12. The results agree with expected ones, which are reproduced with great efficiency and extended to higher orders. We prove explicitly the validity of the simplified path integral on maximally symmetric spaces. This simplified path integral might be of further use in worldline applications, though its application on spaces of arbitrary geometry remains unclear.
8. Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman Path Integrals Method
ERIC Educational Resources Information Center
Fanaro, Maria de los Angeles; Otero, Maria Rita; Arlego, Marcelo
2012-01-01
This paper discusses the teaching of basic quantum mechanics in high school. Rather than following the usual formalism, our approach is based on Feynman's path integral method. Our presentation makes use of simulation software and avoids sophisticated mathematical formalism. (Contains 3 figures.)
9. Teaching Basic Quantum Mechanics in Secondary School Using Concepts of Feynman Path Integrals Method
ERIC Educational Resources Information Center
Fanaro, Maria de los Angeles; Otero, Maria Rita; Arlego, Marcelo
2012-01-01
This paper discusses the teaching of basic quantum mechanics in high school. Rather than following the usual formalism, our approach is based on Feynman's path integral method. Our presentation makes use of simulation software and avoids sophisticated mathematical formalism. (Contains 3 figures.)
10. Coherent-state path integrals in the continuum: The SU(2) case
Kordas, G.; Kalantzis, D.; Karanikas, A. I.
2016-09-01
We define the time-continuous spin coherent-state path integral in a way that is free from inconsistencies. The proposed definition is used to reproduce known exact results. Such a formalism opens new possibilities for applying approximations with improved accuracy and can be proven useful in a great variety of problems where spin Hamiltonians are used.
11. Path-dependent J-integral evaluations around an elliptical hole for large deformation theory
Unger, David J.
2016-08-01
An exact expression is obtained for a path-dependent J-integral for finite strains of an elliptical hole subject to remote tensile tractions under the Tresca deformation theory for a thin plate composed of non-work hardening material. Possible applications include an analytical resistance curve for the initial stage of crack propagation due to crack tip blunting.
12. SU(1,1) coherent-state dynamics: A path-integral approach
Gerry, Christopher C.
1989-02-01
We use SU(1,1) coherent-state path integrals to describe the time evolution of systems driven by coherence-preserving Hamiltonians. Such Hamiltonians are composed of linear combinations of the group generators. The method described here is an alternative to the well-known Wei-Norman procedures. Application is made to the degenerate parametric amplifier of quantum optics.
13. Quantum-Classical Path Integral Simulation of Ferrocene-Ferrocenium Charge Transfer in Liquid Hexane.
PubMed
Walters, Peter L; Makri, Nancy
2015-12-17
We employ the quantum-classical path integral methodology to simulate the outer sphere charge-transfer process of the ferrocene-ferrocenium pair in liquid hexane with unprecedented accuracy. Comparison of the simulation results to those obtained by mapping the solvent on an effective harmonic bath demonstrates the accuracy of linear response theory in this system.
14. A combined explicit-implicit method for high accuracy reaction path integration.
PubMed
Burger, Steven K; Yang, Weitao
2006-06-14
We present the use of an optimal combined explicit-implicit method for following the reaction path to high accuracy. This is in contrast to most purely implicit reaction path integration algorithms, which are only efficient on stiff ordinary differential equations. The defining equation for the reaction path is considered to be stiff, however, we show here that the reaction path is not uniformly stiff and instead is only stiff near stationary points. The optimal algorithm developed in this work is a combination of explicit and implicit methods with a simple criterion to switch between the two. Using three different chemical reactions, we combine and compare three different integration methods: the implicit trapezoidal method, an explicit stabilized third order algorithm implemented in the code DUMKA3 and the traditional explicit fourth order Runge-Kutta method written in the code RKSUITE. The results for high accuracy show that when the implicit trapezoidal method is combined with either explicit method the number of energy and gradient calculations can potentially be reduced by almost a half compared with integrating either method alone. Finally, to explain the improvements of the combined method we expand on the concepts of stability and stiffness and relate them to the efficiency of integration methods.
15. Integrated Flight Path Planning System and Flight Control System for Unmanned Helicopters
PubMed Central
Jan, Shau Shiun; Lin, Yu Hsiang
2011-01-01
This paper focuses on the design of an integrated navigation and guidance system for unmanned helicopters. The integrated navigation system comprises two systems: the Flight Path Planning System (FPPS) and the Flight Control System (FCS). The FPPS finds the shortest flight path by the A-Star (A*) algorithm in an adaptive manner for different flight conditions, and the FPPS can add a forbidden zone to stop the unmanned helicopter from crossing over into dangerous areas. In this paper, the FPPS computation time is reduced by the multi-resolution scheme, and the flight path quality is improved by the path smoothing methods. Meanwhile, the FCS includes the fuzzy inference systems (FISs) based on the fuzzy logic. By using expert knowledge and experience to train the FIS, the controller can operate the unmanned helicopter without dynamic models. The integrated system of the FPPS and the FCS is aimed at providing navigation and guidance to the mission destination and it is implemented by coupling the flight simulation software, X-Plane, and the computing software, MATLAB. Simulations are performed and shown in real time three-dimensional animations. Finally, the integrated system is demonstrated to work successfully in controlling the unmanned helicopter to operate in various terrains of a digital elevation model (DEM). PMID:22164029
16. Factors Affecting Technology Integration in K-12 Classrooms: A Path Model
ERIC Educational Resources Information Center
Inan, Fethi A.; Lowther, Deborah L.
2010-01-01
The purpose of this study was to examine the direct and indirect effects of teachers' individual characteristics and perceptions of environmental factors that influence their technology integration in the classroom. A research-based path model was developed to explain causal relationships between these factors and was tested based on data gathered…
17. Accelerated path integral methods for atomistic simulations at ultra-low temperatures
Uhl, Felix; Marx, Dominik; Ceriotti, Michele
2016-08-01
Path integral methods provide a rigorous and systematically convergent framework to include the quantum mechanical nature of atomic nuclei in the evaluation of the equilibrium properties of molecules, liquids, or solids at finite temperature. Such nuclear quantum effects are often significant for light nuclei already at room temperature, but become crucial at cryogenic temperatures such as those provided by superfluid helium as a solvent. Unfortunately, the cost of converged path integral simulations increases significantly upon lowering the temperature so that the computational burden of simulating matter at the typical superfluid helium temperatures becomes prohibitive. Here we investigate how accelerated path integral techniques based on colored noise generalized Langevin equations, in particular the so-called path integral generalized Langevin equation thermostat (PIGLET) variant, perform in this extreme quantum regime using as an example the quasi-rigid methane molecule and its highly fluxional protonated cousin, CH5+. We show that the PIGLET technique gives a speedup of two orders of magnitude in the evaluation of structural observables and quantum kinetic energy at ultralow temperatures. Moreover, we computed the spatial spread of the quantum nuclei in CH4 to illustrate the limits of using such colored noise thermostats close to the many body quantum ground state.
18. Applications of Path Integral Langevin Dynamics to Weakly Bound Clusters and Biological Molecules
Ing, Christopher; Hinsen, Conrad; Yang, Jing; Roy, Pierre-Nicholas
2011-06-01
We present the use of path integral molecular dynamics (PIMD) in conjunction with the path integral Langevin equation thermostat for sampling systems that exhibit nuclear quantum effects, notably those at low temperatures or those consisting mainly of hydrogen or helium. To test this approach, the internal energy of doped helium clusters are compared with white-noise Langevin thermostatting and high precision path integral monte carlo (PIMC) simulations. We comment on the structural evolution of these clusters in the absence of rotation and exchange as a function of cluster size. To quantify the importance of both rotation and exchange in our PIMD simulation, we compute band origin shifts for (He)_N-CO_2 as a function of cluster size and compare to previously published experimental and theoretical shifts. A convergence study is presented to confirm the systematic error reduction introduced by increasing path integral beads for our implementation in the Molecular Modelling Toolkit (MMTK) software package. Applications to carbohydrates are explored at biological temperatures by calculating both equilibrium and dynamical properties using the methods presented. M. Ceriotti, M. Parrinello, and D. E. Manolopoulos, J Chem Phys 133, 124104. H. Li, N. Blinov, P.-N. Roy, and R. J. L. Roy, J Chem Phys 130, 144305.
19. Accelerated path integral methods for atomistic simulations at ultra-low temperatures.
PubMed
Uhl, Felix; Marx, Dominik; Ceriotti, Michele
2016-08-07
Path integral methods provide a rigorous and systematically convergent framework to include the quantum mechanical nature of atomic nuclei in the evaluation of the equilibrium properties of molecules, liquids, or solids at finite temperature. Such nuclear quantum effects are often significant for light nuclei already at room temperature, but become crucial at cryogenic temperatures such as those provided by superfluid helium as a solvent. Unfortunately, the cost of converged path integral simulations increases significantly upon lowering the temperature so that the computational burden of simulating matter at the typical superfluid helium temperatures becomes prohibitive. Here we investigate how accelerated path integral techniques based on colored noise generalized Langevin equations, in particular the so-called path integral generalized Langevin equation thermostat (PIGLET) variant, perform in this extreme quantum regime using as an example the quasi-rigid methane molecule and its highly fluxional protonated cousin, CH5 (+). We show that the PIGLET technique gives a speedup of two orders of magnitude in the evaluation of structural observables and quantum kinetic energy at ultralow temperatures. Moreover, we computed the spatial spread of the quantum nuclei in CH4 to illustrate the limits of using such colored noise thermostats close to the many body quantum ground state.
20. Path Integral Monte Carlo Simulations of Warm Dense Plasmas with mid-Z Elements
Driver, Kevin; Soubiran, Francois; Zhang, Shuai; Militzer, Burkhard
2016-10-01
Theoretical studies of warm dense plasmas are crucial for improving our knowledge of giant planets, astrophysics, shock physics, and new plasma energy technologies, such as inertial confined fusion. Path integral Monte Carlo (PIMC) and density functional theory molecular dynamics (DFT-MD) provide consistent, first-principles descriptions of warm, dense matter over a wide range of density and temperature conditions. Here, we report simulation results for a variety of first- and second-row elements. DFT-MD algorithms are well-suited for low temperatures, while PIMC has been restricted to relatively high temperatures due to the free-particle approximation of the nodal surface. For heavier, second-row elements, we have developed a new, localized nodal surface, which allows us to treat bound states within the PIMC formalism. By combining PIMC and DFT-MD pressures and internal energies, we produce a coherent, first-principles equation of state, bridging the entire warm dense matter regime. Pair-correlation functions and the density of electronic states reveal an evolving plasma structure. The degree of ionization is affected by both temperature and density. Finally, shock Hugoniot curves show an increase in compression as the first and second shells are ionized. Funding provided by the DOE (DE-SC0010517). Computational resources provided by the NCAR/CISL, NERSC, and NASA.
1. Singular Function Integration in Computational Physics
Hasbun, Javier
2009-03-01
In teaching computational methods in the undergraduate physics curriculum, standard integration approaches taught include the rectangular, trapezoidal, Simpson, Romberg, and others. Over time, these techniques have proven to be invaluable and students are encouraged to employ the most efficient method that is expected to perform best when applied to a given problem. However, some physics research applications require techniques that can handle singularities. While decreasing the step size in traditional approaches is an alternative, this may not always work and repetitive processes make this route even more inefficient. Here, I present two existing integration rules designed to handle singular integrals. I compare them to traditional rules as well as to the exact analytic results. I suggest that it is perhaps time to include such approaches in the undergraduate computational physics course.
2. Iterative blip-summed path integral for quantum dynamics in strongly dissipative environments.
PubMed
Makri, Nancy
2017-04-07
The iterative decomposition of the blip-summed path integral [N. Makri, J. Chem. Phys. 141, 134117 (2014)] is described. The starting point is the expression of the reduced density matrix for a quantum system interacting with a harmonic dissipative bath in the form of a forward-backward path sum, where the effects of the bath enter through the Feynman-Vernon influence functional. The path sum is evaluated iteratively in time by propagating an array that stores blip configurations within the memory interval. Convergence with respect to the number of blips and the memory length yields numerically exact results which are free of statistical error. In situations of strongly dissipative, sluggish baths, the algorithm leads to a dramatic reduction of computational effort in comparison with iterative path integral methods that do not implement the blip decomposition. This gain in efficiency arises from (i) the rapid convergence of the blip series and (ii) circumventing the explicit enumeration of between-blip path segments, whose number grows exponentially with the memory length. Application to an asymmetric dissipative two-level system illustrates the rapid convergence of the algorithm even when the bath memory is extremely long.
3. Iterative blip-summed path integral for quantum dynamics in strongly dissipative environments
Makri, Nancy
2017-04-01
The iterative decomposition of the blip-summed path integral [N. Makri, J. Chem. Phys. 141, 134117 (2014)] is described. The starting point is the expression of the reduced density matrix for a quantum system interacting with a harmonic dissipative bath in the form of a forward-backward path sum, where the effects of the bath enter through the Feynman-Vernon influence functional. The path sum is evaluated iteratively in time by propagating an array that stores blip configurations within the memory interval. Convergence with respect to the number of blips and the memory length yields numerically exact results which are free of statistical error. In situations of strongly dissipative, sluggish baths, the algorithm leads to a dramatic reduction of computational effort in comparison with iterative path integral methods that do not implement the blip decomposition. This gain in efficiency arises from (i) the rapid convergence of the blip series and (ii) circumventing the explicit enumeration of between-blip path segments, whose number grows exponentially with the memory length. Application to an asymmetric dissipative two-level system illustrates the rapid convergence of the algorithm even when the bath memory is extremely long.
4. Teaching quantum physics by the sum over paths approach and GeoGebra simulations
Malgieri, M.; Onorato, P.; De Ambrosis, A.
2014-09-01
We present a research-based teaching sequence in introductory quantum physics using the Feynman sum over paths approach. Our reconstruction avoids the historical pathway, and starts by reconsidering optics from the standpoint of the quantum nature of light, analysing both traditional and modern experiments. The core of our educational path lies in the treatment of conceptual and epistemological themes, peculiar of quantum theory, based on evidence from quantum optics, such as the single photon Mach-Zehnder and Zhou-Wang-Mandel experiments. The sequence is supported by a collection of interactive simulations, realized in the open source GeoGebra environment, which we used to assist students in learning the basics of the method, and help them explore the proposed experimental situations as modeled in the sum over paths perspective. We tested our approach in the context of a post-graduate training course for pre-service physics teachers; according to the data we collected, student teachers displayed a greatly improved understanding of conceptual issues, and acquired significant abilities in using the sum over path method for problem solving.
5. Draw Your Physics Homework? Art as a Path to Understanding in Physics Teaching
ERIC Educational Resources Information Center
van der Veen, Jatila
2012-01-01
The persistent fear of physics by learners motivated the author to take action to increase all students' interest in the subject via a new curriculum for introductory college physics that applies Greene's model of Aesthetic Education to the study of contemporary physics, utilizing symmetry as the mathematical foundation of physics as well as the…
6. Draw Your Physics Homework? Art as a Path to Understanding in Physics Teaching
ERIC Educational Resources Information Center
van der Veen, Jatila
2012-01-01
The persistent fear of physics by learners motivated the author to take action to increase all students' interest in the subject via a new curriculum for introductory college physics that applies Greene's model of Aesthetic Education to the study of contemporary physics, utilizing symmetry as the mathematical foundation of physics as well as the…
7. PRELIMINARY PROJECT PLAN FOR LANSCE INTEGRATED FLIGHT PATHS 11A, 11B, 12, and 13
SciTech Connect
D. H. BULTMAN; D. WEINACHT - AIRES CORP.
2000-08-01
This Preliminary Project Plan Summarizes the Technical, Cost, and Schedule baselines for an integrated approach to developing several flight paths at the Manual Lujan Jr. Neutron Scattering Center at the Los Alamos Neutron Science Center. For example, the cost estimate is intended to serve only as a rough order of magnitude assessment of the cost that might be incurred as the flight paths are developed. Further refinement of the requirements and interfaces for each beamline will permit additional refinement and confidence in the accuracy of all three baselines (Technical, Cost, Schedule).
8. Path integral pricing of Wasabi option in the Black-Scholes model
Cassagnes, Aurelien; Chen, Yu; Ohashi, Hirotada
2014-11-01
In this paper, using path integral techniques, we derive a formula for a propagator arising in the study of occupation time derivatives. Using this result we derive a fair price for the case of the cumulative Parisian option. After confirming the validity of the derived result using Monte Carlo simulation, a new type of heavily path dependent derivative product is investigated. We derive an approximation for our so-called Wasabi option fair price and check the accuracy of our result with a Monte Carlo simulation.
9. Quantum Mechanics, Path Integrals and Option Pricing:. Reducing the Complexity of Finance
Baaquie, Belal E.; Corianò, Claudio; Srikant, Marakani
2003-04-01
Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connection between these fields, we illustrate some of the methods of lattice simulations of path integrals for the pricing of options. The ideas are sketched out for simple models, such as the Black-Scholes model, where analytical and numerical results are compared. Application of the method to nonlinear systems is also briefly overviewed. More general models, for exotic or path-dependent options are discussed.
10. Visual influence on path integration in darkness indicates a multimodal representation of large-scale space.
PubMed
Tcheang, Lili; Bülthoff, Heinrich H; Burgess, Neil
2011-01-18
Our ability to return to the start of a route recently performed in darkness is thought to reflect path integration of motion-related information. Here we provide evidence that motion-related interoceptive representations (proprioceptive, vestibular, and motor efference copy) combine with visual representations to form a single multimodal representation guiding navigation. We used immersive virtual reality to decouple visual input from motion-related interoception by manipulating the rotation or translation gain of the visual projection. First, participants walked an outbound path with both visual and interoceptive input, and returned to the start in darkness, demonstrating the influences of both visual and interoceptive information in a virtual reality environment. Next, participants adapted to visual rotation gains in the virtual environment, and then performed the path integration task entirely in darkness. Our findings were accurately predicted by a quantitative model in which visual and interoceptive inputs combine into a single multimodal representation guiding navigation, and are incompatible with a model of separate visual and interoceptive influences on action (in which path integration in darkness must rely solely on interoceptive representations). Overall, our findings suggest that a combined multimodal representation guides large-scale navigation, consistent with a role for visual imagery or a cognitive map.
11. Visual influence on path integration in darkness indicates a multimodal representation of large-scale space
PubMed Central
Tcheang, Lili; Bülthoff, Heinrich H.; Burgess, Neil
2011-01-01
Our ability to return to the start of a route recently performed in darkness is thought to reflect path integration of motion-related information. Here we provide evidence that motion-related interoceptive representations (proprioceptive, vestibular, and motor efference copy) combine with visual representations to form a single multimodal representation guiding navigation. We used immersive virtual reality to decouple visual input from motion-related interoception by manipulating the rotation or translation gain of the visual projection. First, participants walked an outbound path with both visual and interoceptive input, and returned to the start in darkness, demonstrating the influences of both visual and interoceptive information in a virtual reality environment. Next, participants adapted to visual rotation gains in the virtual environment, and then performed the path integration task entirely in darkness. Our findings were accurately predicted by a quantitative model in which visual and interoceptive inputs combine into a single multimodal representation guiding navigation, and are incompatible with a model of separate visual and interoceptive influences on action (in which path integration in darkness must rely solely on interoceptive representations). Overall, our findings suggest that a combined multimodal representation guides large-scale navigation, consistent with a role for visual imagery or a cognitive map. PMID:21199934
12. Path integration of head direction: updating a packet of neural activity at the correct speed using axonal conduction delays.
PubMed
Walters, Daniel; Stringer, Simon; Rolls, Edmund
2013-01-01
The head direction cell system is capable of accurately updating its current representation of head direction in the absence of visual input. This is known as the path integration of head direction. An important question is how the head direction cell system learns to perform accurate path integration of head direction. In this paper we propose a model of velocity path integration of head direction in which the natural time delay of axonal transmission between a linked continuous attractor network and competitive network acts as a timing mechanism to facilitate the correct speed of path integration. The model effectively learns a "look-up" table for the correct speed of path integration. In simulation, we show that the model is able to successfully learn two different speeds of path integration across two different axonal conduction delays, and without the need to alter any other model parameters. An implication of this model is that, by learning look-up tables for each speed of path integration, the model should exhibit a degree of robustness to damage. In simulations, we show that the speed of path integration is not significantly affected by degrading the network through removing a proportion of the cells that signal rotational velocity.
13. Path Integration of Head Direction: Updating a Packet of Neural Activity at the Correct Speed Using Axonal Conduction Delays
PubMed Central
Walters, Daniel; Stringer, Simon; Rolls, Edmund
2013-01-01
The head direction cell system is capable of accurately updating its current representation of head direction in the absence of visual input. This is known as the path integration of head direction. An important question is how the head direction cell system learns to perform accurate path integration of head direction. In this paper we propose a model of velocity path integration of head direction in which the natural time delay of axonal transmission between a linked continuous attractor network and competitive network acts as a timing mechanism to facilitate the correct speed of path integration. The model effectively learns a “look-up” table for the correct speed of path integration. In simulation, we show that the model is able to successfully learn two different speeds of path integration across two different axonal conduction delays, and without the need to alter any other model parameters. An implication of this model is that, by learning look-up tables for each speed of path integration, the model should exhibit a degree of robustness to damage. In simulations, we show that the speed of path integration is not significantly affected by degrading the network through removing a proportion of the cells that signal rotational velocity. PMID:23526976
14. Spatial representations of place cells in darkness are supported by path integration and border information.
PubMed
Zhang, Sijie; Schönfeld, Fabian; Wiskott, Laurenz; Manahan-Vaughan, Denise
2014-01-01
Effective spatial navigation is enabled by reliable reference cues that derive from sensory information from the external environment, as well as from internal sources such as the vestibular system. The integration of information from these sources enables dead reckoning in the form of path integration. Navigation in the dark is associated with the accumulation of errors in terms of perception of allocentric position and this may relate to error accumulation in path integration. We assessed this by recording from place cells in the dark under circumstances where spatial sensory cues were suppressed. Spatial information content, spatial coherence, place field size, and peak and infield firing rates decreased whereas sparsity increased following exploration in the dark compared to the light. Nonetheless it was observed that place field stability in darkness was sustained by border information in a subset of place cells. To examine the impact of encountering the environment's border on navigation, we analyzed the trajectory and spiking data gathered during navigation in the dark. Our data suggest that although error accumulation in path integration drives place field drift in darkness, under circumstances where border contact is possible, this information is integrated to enable retention of spatial representations.
15. Spatial representations of place cells in darkness are supported by path integration and border information
PubMed Central
Zhang, Sijie; Schönfeld, Fabian; Wiskott, Laurenz; Manahan-Vaughan, Denise
2014-01-01
Effective spatial navigation is enabled by reliable reference cues that derive from sensory information from the external environment, as well as from internal sources such as the vestibular system. The integration of information from these sources enables dead reckoning in the form of path integration. Navigation in the dark is associated with the accumulation of errors in terms of perception of allocentric position and this may relate to error accumulation in path integration. We assessed this by recording from place cells in the dark under circumstances where spatial sensory cues were suppressed. Spatial information content, spatial coherence, place field size, and peak and infield firing rates decreased whereas sparsity increased following exploration in the dark compared to the light. Nonetheless it was observed that place field stability in darkness was sustained by border information in a subset of place cells. To examine the impact of encountering the environment’s border on navigation, we analyzed the trajectory and spiking data gathered during navigation in the dark. Our data suggest that although error accumulation in path integration drives place field drift in darkness, under circumstances where border contact is possible, this information is integrated to enable retention of spatial representations. PMID:25009477
16. Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.
PubMed
Bressloff, Paul C
2015-01-01
We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous
17. IntPath--an integrated pathway gene relationship database for model organisms and important pathogens
PubMed Central
2012-01-01
Background Pathway data are important for understanding the relationship between genes, proteins and many other molecules in living organisms. Pathway gene relationships are crucial information for guidance, prediction, reference and assessment in biochemistry, computational biology, and medicine. Many well-established databases--e.g., KEGG, WikiPathways, and BioCyc--are dedicated to collecting pathway data for public access. However, the effectiveness of these databases is hindered by issues such as incompatible data formats, inconsistent molecular representations, inconsistent molecular relationship representations, inconsistent referrals to pathway names, and incomprehensive data from different databases. Results In this paper, we overcome these issues through extraction, normalization and integration of pathway data from several major public databases (KEGG, WikiPathways, BioCyc, etc). We build a database that not only hosts our integrated pathway gene relationship data for public access but also maintains the necessary updates in the long run. This public repository is named IntPath (Integrated Pathway gene relationship database for model organisms and important pathogens). Four organisms--S. cerevisiae, M. tuberculosis H37Rv, H. Sapiens and M. musculus--are included in this version (V2.0) of IntPath. IntPath uses the "full unification" approach to ensure no deletion and no introduced noise in this process. Therefore, IntPath contains much richer pathway-gene and pathway-gene pair relationships and much larger number of non-redundant genes and gene pairs than any of the single-source databases. The gene relationships of each gene (measured by average node degree) per pathway are significantly richer. The gene relationships in each pathway (measured by average number of gene pairs per pathway) are also considerably richer in the integrated pathways. Moderate manual curation are involved to get rid of errors and noises from source data (e.g., the gene ID errors
18. Functional integration of vertical flight path and speed control using energy principles
NASA Technical Reports Server (NTRS)
Lambregts, A. A.
1984-01-01
A generalized automatic flight control system was developed which integrates all longitudinal flight path and speed control functions previously provided by a pitch autopilot and autothrottle. In this design, a net thrust command is computed based on total energy demand arising from both flight path and speed targets. The elevator command is computed based on the energy distribution error between flight path and speed. The engine control is configured to produce the commanded net thrust. The design incorporates control strategies and hierarchy to deal systematically and effectively with all aircraft operational requirements, control nonlinearities, and performance limits. Consistent decoupled maneuver control is achieved for all modes and flight conditions without outer loop gain schedules, control law submodes, or control function duplication.
19. Moving to a Soft Path for Water: Integrated Research and Management Needs
Gleick, P. H.
2011-12-01
Water on Earth in its three fundamental phases is integral to the functioning, dynamics, and variability of the global climatological and biological support systems. From a purely scientific point of view, understanding the complexity of the hydrological cycle is of paramount interest and central to our understanding of other planetary geological, atmospheric, chemical, and physical processes. But water is more than that: water is key to economic, social, and political issues as well, including some of the core challenges of our time such central to issues of poverty, health, environmental sustainability, conflict, and economic prosperity. The more society seeks to solve these challenges, the more obvious it becomes that we must improve more than just our understanding of the fundamental science of the hydrological cycle and its links with related global processes; we must also improve our understanding of the complex social, economic, and structural challenges facing water managers and users. We must move to a different paradigm where water is managed in a far more integrated way - what I call the "soft path for water." Central to our basic science needs are (1) an expansion of the frequency and nature of the data we collect, (2) the development of systems for managing, sharing, and analyzing those data, and (3) improvements in our ability to model and forecast the hydrological cycle together with other climatological, geophysical, and biochemical systems. These improvements would lead to a far better understanding of the local, regional, and global details of the water balance on timescales from minutes to millennia. These needs are increasingly well understood in the research community and extensive efforts in these areas are underway under the auspices of national research centers, universities, and international scientific collaborations. But it is also becoming increasingly apparent that many of the current water challenges facing society are not going to be
20. Classical to path-integral adaptive resolution in molecular simulation: towards a smooth quantum-classical coupling.
PubMed
Poma, A B; Delle Site, L
2010-06-25
Simulations that couple different molecular models in an adaptive way by changing resolution on the fly allow us to identify the relevant degrees of freedom of a system. This, in turn, leads to a detailed understanding of the essential physics which characterizes a system. While the delicate process of transition from one model to another is well understood for the adaptivity between classical molecular models the same cannot be said for the quantum-classical adaptivity. The main reason for this is the difficulty in describing a continuous transition between two different kinds of physical principles: probabilistic for the quantum and deterministic for the classical. Here we report the basic principles of an algorithm that allows for a continuous and smooth transition by employing the path integral description of atoms.
1. The role of spatial memory and frames of reference in the precision of angular path integration.
PubMed
Arthur, Joeanna C; Philbeck, John W; Kleene, Nicholas J; Chichka, David
2012-09-01
Angular path integration refers to the ability to maintain an estimate of self-location after a rotational displacement by integrating internally-generated (idiothetic) self-motion signals over time. Previous work has found that non-sensory inputs, namely spatial memory, can play a powerful role in angular path integration (Arthur et al., 2007, 2009). Here we investigated the conditions under which spatial memory facilitates angular path integration. We hypothesized that the benefit of spatial memory is particularly likely in spatial updating tasks in which one's self-location estimate is referenced to external space. To test this idea, we administered passive, non-visual body rotations (ranging 40°-140°) about the yaw axis and asked participants to use verbal reports or open-loop manual pointing to indicate the magnitude of the rotation. Prior to some trials, previews of the surrounding environment were given. We found that when participants adopted an egocentric frame of reference, the previously-observed benefit of previews on within-subject response precision was not manifested, regardless of whether remembered spatial frameworks were derived from vision or spatial language. We conclude that the powerful effect of spatial memory is dependent on one's frame of reference during self-motion updating.
2. Automatic integration of the reaction path using diagonally implicit Runge-Kutta methods.
PubMed
Burger, Steven K; Yang, Weitao
2006-12-28
The diagonally implicit Runge-Kutta framework is shown to be a general form for constructing stable, efficient steepest descent reaction path integrators, of any order. With this framework tolerance driven, adaptive step-size methods can be constructed by embedding methods to obtain error estimates of each step without additional computational cost. There are many embedded and nonembedded, diagonally implicit Runge-Kutta methods available from the numerical analysis literature and these are reviewed for orders two, three, and four. New embedded methods are also developed which are tailored to the application of reaction path following. All integrators are summarized and compared for three systems: the Muller-Brown [Theor. Chem. Acta 53, 75 (1979)] potential and two gas phase chemical reactions. The results show that many of the methods are capable of integrating efficiently while reliably keeping the error bound within the desired tolerance. This allows the reaction path to be determined through automatic integration by only specifying the desired accuracy and transition state.
3. Color path integral equation of state of the quark-gluon plasma at nonzero chemical potential
Filinov, V. S.; Bonitz, M.; Ivanov, Yu B.; Ilgenfritz, E.-M.; Fortov, V. E.
2015-04-01
Based on the constituent quasiparticle model of the quark-gluon plasma (QGP), a color quantum path-integral Monte-Carlo (PIMC) method for calculation of the thermodynamic properties of the QGP is developed. We show that the PIMC method can be used for calculations of the equation of state at zero and non-zero baryon chemical potential not only above but also below the QCD critical temperature. Our results agree with lattice QCD calculations based on a Taylor expansion around zero baryon chemical potential. In our approach the QGP partition function is presented in the form of a color path integral with a relativistic measure replacing the Gaussian one traditionally used in the Feynman-Wiener path integrals. A procedure of sampling color variables according to the SU(3) group Haar measure is used for integration over the color variables. We expect that this approach will be useful to predict additional properties of the QGP that are still unaccesible in lattice QCD.
4. The Role of Spatial Memory and Frames of Reference in the Precision of Angular Path Integration
PubMed Central
Arthur, Joeanna C.; Philbeck, John W.; Kleene, Nicholas J.; Chichka, David
2012-01-01
Angular path integration refers to the ability to maintain an estimate of self-location after a rotational displacement by integrating internally-generated (idiothetic) self-motion signals over time. Previous work has found that non-sensory inputs, namely spatial memory, can play a powerful role in angular path integration (Arthur et al., 2007, 2009). Here we investigated the conditions under which spatial memory facilitates angular path integration. We hypothesized that the benefit of spatial memory is particularly likely in spatial updating tasks in which one’s self-location estimate is referenced to external space. To test this idea, we administered passive, nonvisual body rotations (ranging 40° – 140°) about the yaw axis and asked participants to use verbal reports or open-loop manual pointing to indicate the magnitude of the rotation. Prior to some trials, previews of the surrounding environment were given. We found that when participants adopted an egocentric frame of reference, the previously-observed benefit of previews on within-subject response precision was not manifested, regardless of whether remembered spatial frameworks were derived from vision or spatial language. We conclude that the powerful effect of spatial memory is dependent on one’s frame of reference during self-motion updating. PMID:22885073
5. Blip-summed quantum-classical path integral with cumulative quantum memory.
PubMed
Makri, Nancy
2016-12-22
The quantum-classical path integral (QCPI) offers a rigorous methodology for simulating quantum mechanical processes in condensed-phase environments treated in full atomistic detail. This paper describes the implementation of QCPI on system-bath models, which are frequently employed in studying the dynamics of reactive processes. The QCPI methodology incorporates all effects associated with stimulated phonon absorption and emission as its crudest limit, thus can (in some regimes) converge faster than influence functional-based path integral methods specifically designed for system-bath Hamiltonians. It is shown that the QCPI phase arising from a harmonic bath can be summed analytically with respect to the discrete bath degrees of freedom and expressed in terms of precomputed influence functional coefficients, avoiding the explicit enumeration of forced oscillator trajectories, whose number grows exponentially with the length of quantum memory. Further, adoption of the blip decomposition (which classifies the system paths based on the time length over which their forward and backward components are not identical) and a cumulative treatment of the QCPI phase between blips allows elimination of the majority of system paths, leading to a dramatic increase in efficiency. The generalization of these acceleration techniques to anharmonic environments is discussed.
6. Reducing Uncertainty in GMPE's Through Physical Explanations of the Path Term
Sahakian, V. J.; Baltay, A.; Hanks, T. C.; Buehler, J. S.; Vernon, F.; Kilb, D. L.
2016-12-01
The residuals, or differences between recorded ground motions and median predictions made by Ground Motion Prediction Equations (GMPEs) can be considerable and largely scattered, leading to large standard deviations. These large standard deviations may result in large ground motion amplitudes at low probabilities of exceedance. As such, reducing uncertainty in GMPE's is integral in decreasing the variability in predicted ground motions at such low probabilities. Our approach for reducing uncertainty in the predictions is to understand the separate effects of source-, path-, and site-specific information. The knowable, repeatable parts of these effects can be identified and removed from population of residuals, which results in location-specific GMPEs. This study focuses on correlating the path term to material properties along each recording's raypath. To do this, we employ a large database of events recorded in Southern California on the Anza, San Jacinto Fault Zone, Plate Boundary Observatory, Southern California Seismic Network, and UC Santa Barbara seismic networks. The Anza network in particular has been in operation since 1981, resulting in redundancy in source-to-station paths; the majority of the events in database have magnitudes in the range 1 < M < 3, with just a few hundred events larger than M 3. We invert the recordings to form a regional GMPE, and perform the residual decomposition into source, site and path terms using this GMPE. Using the regional tomographic model of Fang et al. (2016), we compute the raypaths of each recording, as well as an index representing the path integral of material properties along the raypath. Such properties include seismic velocity, as well as the gradient of velocity, and we correlate these indices with the path term. A number of parameters may affect these correlations, such as velocity grid spacing, temporal variability, frequency-dependence, and the dataset bias towards smaller magnitudes. We discuss analyses that
7. The path-independent M Integral around Röthlisberger channels
Meyer, C. R.; Rice, J. R.
2015-12-01
Röthlisberger channels are essential components of subglacial hydrologic systems. Deviations from the Nye creep closure of the ice around a Röthlisberger channel have been long recognized and enhancement factors or a more complex rheology for ice have been suggested as ameliorations to account for channels closing faster than predicted. Here we use the MM integral, a path-independent integral of the equations of continuum mechanics, with a Glen power-law rheology to unify descriptions of creep closure under a variety of stress states surrounding the Röthlisberger channel. The advantage of this approach is that the MM integral around the Röthlisberger channel is equivalent to the integral around the far field. In this way, the creep closure on the channel wall can be determined as a function of the far-field loading, e.g. antiplane shear as well as overburden pressure. We start by analyzing the case of axisymmetric creep closure and we see that the Nye solution is implied by the path-independence of MM integral. We then examine the effects of antiplane shear in several geometries and derive scalings for the creep closure rate based on the MM integral. The results are compared to observations for tunnel closure measurements in a variety of stress states and it is shown that the additional stress components can account for the deviations from the Nye solution. Furthermore, creep closure can be succinctly written in terms of the path-independent MM integral and the variation with applied shear can be found via scalings, which is useful for subglacial hydrology models.
8. Path-integral and Ornstein-Zernike study of quantum fluid structures on the crystallization line
Sesé, Luis M.
2016-03-01
Liquid neon, liquid para-hydrogen, and the quantum hard-sphere fluid are studied with path integral Monte Carlo simulations and the Ornstein-Zernike pair equation on their respective crystallization lines. The results cover the whole sets of structures in the r-space and the k-space and, for completeness, the internal energies, pressures and isothermal compressibilities. Comparison with experiment is made wherever possible, and the possibilities of establishing k-space criteria for quantum crystallization based on the path-integral centroids are discussed. In this regard, the results show that the centroid structure factor contains two significant parameters related to its main peak features (amplitude and shape) that can be useful to characterize freezing.
9. Two-scale large deviations for chemical reaction kinetics through second quantization path integral
Li, Tiejun; Lin, Feng
2016-04-01
Motivated by the study of rare events for a typical genetic switching model in systems biology, in this paper we aim to establish the general two-scale large deviations for chemical reaction systems. We build a formal approach to explicitly obtain the large deviation rate functionals for the considered two-scale processes based upon the second quantization path integral technique. We get three important types of large deviation results when the underlying two timescales are in three different regimes. This is realized by singular perturbation analysis to the rate functionals obtained by the path integral. We find that the three regimes possess the same deterministic mean-field limit but completely different chemical Langevin approximations. The obtained results are natural extensions of the classical large volume limit for chemical reactions. We also discuss its implication on the single-molecule Michaelis-Menten kinetics. Our framework and results can be applied to understand general multi-scale systems including diffusion processes.
10. Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
Caputa, Pawel; Kundu, Nilay; Miyaji, Masamichi; Takayanagi, Tadashi; Watanabe, Kento
2017-08-01
We introduce a new optimization procedure for Euclidean path integrals, which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently, this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space, and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model, and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.
11. Proton momentum distribution in water: an open path integral molecular dynamics study.
PubMed
Morrone, Joseph A; Srinivasan, Varadharajan; Sebastiani, Daniel; Car, Roberto
2007-06-21
Recent neutron Compton scattering experiments have detected the proton momentum distribution in water. The theoretical calculation of this property can be carried out via "open" path integral expressions. In this work, present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize a flexible, single point charge empirical force field to model the system's interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.
12. Quantum free-energy differences from nonequilibrium path integrals. II. Convergence properties for the harmonic oscillator.
PubMed
van Zon, Ramses; Hernández de la Peña, Lisandro; Peslherbe, Gilles H; Schofield, Jeremy
2008-10-01
Nonequilibrium path-integral methods for computing quantum free-energy differences are applied to a quantum particle trapped in a harmonic well of uniformly changing strength with the purpose of establishing the convergence properties of the work distribution and free energy as the number of degrees of freedom M in the regularized path integrals goes to infinity. The work distribution is found to converge when M tends to infinity regardless of the switching speed, leading to finite results for the free-energy difference when the Jarzynski nonequilibrium work relation or the Crooks fluctuation relation are used. The nature of the convergence depends on the regularization method. For the Fourier method, the convergence of the free-energy difference and work distribution go as 1/M , while both quantities converge as 1/M(2) when the bead regularization procedure is used. The implications of these results to more general systems are discussed.
13. A path-independent integral for the characterization of solute concentration and flux at biofilm detachments
USGS Publications Warehouse
Moran, B.; Kulkarni, S.S.; Reeves, H.W.
2007-01-01
A path-independent (conservation) integral is developed for the characterization of solute concentration and flux in a biofilm in the vicinity of a detachment or other flux limiting boundary condition. Steady state conditions of solute diffusion are considered and biofilm kinetics are described by an uptake term which can be expressed in terms of a potential (Michaelis-Menten kinetics). An asymptotic solution for solute concentration at the tip of the detachment is obtained and shown to be analogous to that of antiplane crack problems in linear elasticity. It is shown that the amplitude of the asymptotic solution can be calculated by evaluating a path-independent integral. The special case of a semi-infinite detachment in an infinite strip is considered and the amplitude of the asymptotic field is related to the boundary conditions and problem parameters in closed form for zeroth and first order kinetics and numerically for Michaelis-Menten kinetics. ?? Springer Science+Business Media, Inc. 2007.
14. Path-integral and Ornstein-Zernike study of quantum fluid structures on the crystallization line.
PubMed
Sesé, Luis M
2016-03-07
Liquid neon, liquid para-hydrogen, and the quantum hard-sphere fluid are studied with path integral Monte Carlo simulations and the Ornstein-Zernike pair equation on their respective crystallization lines. The results cover the whole sets of structures in the r-space and the k-space and, for completeness, the internal energies, pressures and isothermal compressibilities. Comparison with experiment is made wherever possible, and the possibilities of establishing k-space criteria for quantum crystallization based on the path-integral centroids are discussed. In this regard, the results show that the centroid structure factor contains two significant parameters related to its main peak features (amplitude and shape) that can be useful to characterize freezing.
15. From path integrals to tensor networks for the AdS /CFT correspondence
Miyaji, Masamichi; Takayanagi, Tadashi; Watanabe, Kento
2017-03-01
In this paper, we discuss tensor network descriptions of AdS /CFT from two different viewpoints. First, we start with a Euclidean path-integral computation of ground state wave functions with a UV cutoff. We consider its efficient optimization by making its UV cutoff position dependent and define a quantum state at each length scale. We conjecture that this path integral corresponds to a time slice of anti-de Sitter (AdS) spacetime. Next, we derive a flow of quantum states by rewriting the action of Killing vectors of AdS3 in terms of the dual two-dimensional conformal field theory (CFT). Both approaches support a correspondence between the hyperbolic time slice H2 in AdS3 and a version of continuous multiscale entanglement renormalization ansatz. We also give a heuristic argument about why we can expect a sub-AdS scale bulk locality for holographic CFTs.
16. Path-integral action of a particle in the noncommutative phase-space
2017-01-01
In this paper we construct a path integral formulation of quantum mechanics on noncommutative phase-space. We first map the system to an equivalent system on the noncommutative plane. Then by applying the formalism of representing a quantum system in the space of Hilbert-Schmidt operators acting on noncommutative configuration space, the path integral action of a particle is derived. It is observed that the action has a similar form to that of a particle in a magnetic field in the noncommutative plane. From this action the energy spectrum is obtained for the free particle and the harmonic-oscillator potential. We also show that the nonlocal nature (in time) of the action yields a second-class constrained system from which the noncommutative Heisenberg algebra can be recovered.
17. Efficient algorithms for semiclassical instanton calculations based on discretized path integrals
SciTech Connect
Kawatsu, Tsutomu E-mail: [email protected]; Miura, Shinichi E-mail: [email protected]
2014-07-14
Path integral instanton method is a promising way to calculate the tunneling splitting of energies for degenerated two state systems. In order to calculate the tunneling splitting, we need to take the zero temperature limit, or the limit of infinite imaginary time duration. In the method developed by Richardson and Althorpe [J. Chem. Phys. 134, 054109 (2011)], the limit is simply replaced by the sufficiently long imaginary time. In the present study, we have developed a new formula of the tunneling splitting based on the discretized path integrals to take the limit analytically. We have applied our new formula to model systems, and found that this approach can significantly reduce the computational cost and gain the numerical accuracy. We then developed the method combined with the electronic structure calculations to obtain the accurate interatomic potential on the fly. We present an application of our ab initio instanton method to the ammonia umbrella flip motion.
18. Path integral investigation of the electronic spectra of He-tetracene clusters
Whitley, Heather D.; Whaley, K. Birgitta
2008-03-01
Planar aromatic molecules (PAMs) are nanoscale precursors to bulk graphite. Their electronic spectra have been extensively studied in ^4He nanodroplets and show a number of unusual spectroscopic features. We have conducted many-body quantum simulations of tetracene in He nanodroplets to probe the 1.1 cm-1 spectral splitting of the electronic origin seen for this PAM. We calculate spectral shifts and He density profiles via path integral quantum Monte Carlo simulations. The spectral splitting is examined using a path integral correlation function approach to determine the lowest-lying vibrational excitation frequencies for small HeN-tetracene clusters. Simulations in the S1 state of tetracene utilize a semi-empirical perturbative interaction potential for a He atom with a PAM. Results for the splitting of the electronic origin and the spectral shifts are in good agreement with experiment. Prepared by LLNL under Contract DE-AC52-07NA27344.
19. Proton momentum distributions in water: A path integral molecular dynamics study
Srinivasan, Varadharajan; Morrone, Joseph A.; Sebastiani, Daniel; Car, Roberto
2007-03-01
Recent neutron Compton scattering experiments have detected the proton momentum distributions of water. This density in momentum space is a quantum mechanical property of the proton, due to the confining anharmonic potential from covalent and hydrogen bonds. The theoretical calculation of this property can be carried out via open'' path integral expressions. In this work, we present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize the SPC/F2 empirical force field to model the system's interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.
20. Proton momentum distribution in water: an open path integral molecular dynamics study
Morrone, Joseph A.; Srinivasan, Varadharajan; Sebastiani, Daniel; Car, Roberto
2007-06-01
Recent neutron Compton scattering experiments have detected the proton momentum distribution in water. The theoretical calculation of this property can be carried out via "open" path integral expressions. In this work, present an extension of the staging path integral molecular dynamics method, which is then employed to calculate the proton momentum distributions of water in the solid, liquid, and supercritical phases. We utilize a flexible, single point charge empirical force field to model the system's interactions. The calculated momentum distributions depict both agreement and discrepancies with experiment. The differences may be explained by the deviation of the force field from the true interactions. These distributions provide an abundance of information about the environment and interactions surrounding the proton.
1. Simulations of one- and two-electron systems by Bead-Fourier path integral molecular dynamics
Ivanov, Sergei D.; Lyubartsev, Alexander P.
2005-07-01
The Bead-Fourier path integral molecular dynamics technique introduced earlier [S. D. Ivanov, A. P. Lyubartsev, and A. Laaksonen, Phys. Rev. E 67 066710 (2003)] is applied for simulation of electrons in the simplest molecules: molecular hydrogen, helium atom, and their ions. Special attention is paid to the correct description of electrons in the core region of a nucleus. In an attempt to smooth the Coulomb potential at small distances, a recipe is suggested. The simulation results are in excellent agreement with the analytical solution for the "harmonic helium atom", as well as with the vibrational potential of the H2 molecule and He ionization energies. It is demonstrated, that the Bead-Fourier path integral molecular dynamics technique is able to provide the accuracy required for the description of electron structure and chemical bonds in cases when electron exchange effects need not be taken into account.
2. Ab Initio Path Integral Molecular Dynamics Simulation of Hydrogen in Silicon
Probert, M. I. J.; Glover, M. J.
2006-05-01
We report results of a first-principles theoretical study of an isolated neutral hydrogen atom in crystalline silicon. Spin-polarised density functional theory is used to treat the electrons, and the path-integral molecular dynamics method is used to describe the quantum properties of the nucleus at finite temperature. This is necessary as the hydrogen atom has sufficiently low mass that it exhibits significant nuclear quantum delocalisation and zero-point motion even at room temperature. Unlike post-hoc treatments, such as calculating a static potential energy surface, the path-integral treatment enables such effects to be included "on-the-fly". This is found to be significant, as a coupling is found between the structure of the host silicon lattice and the quantum delocalisation of the hydrogen defect.
3. A 2-Micron Pulsed Integrated Path Differential Absorption Lidar Development For Atmospheric CO2 Concentration Measurements
NASA Technical Reports Server (NTRS)
Yu, Jirong; Petros, Mulugeta; Reithmaier, Karl; Bai, Yingxin; Trieu, Bo C.; Refaat, Tamer F.; Kavaya, Michael J.; Singh, Upendra N.
2012-01-01
A 2-micron pulsed, Integrated Path Differential Absorption (IPDA) lidar instrument for ground and airborne atmospheric CO2 concentration measurements via direct detection method is being developed at NASA Langley Research Center. This instrument will provide an alternate approach to measure atmospheric CO2 concentrations with significant advantages. A high energy pulsed approach provides high-precision measurement capability by having high signal-to-noise level and unambiguously eliminates the contamination from aerosols and clouds that can bias the IPDA measurement.
4. Path integral solution for a deformed radial Rosen-Morse potential
Kadja, A.; Benamira, F.; Guechi, L.
2017-03-01
An exact path integral treatment of a particle in a deformed radial Rosen-Morse potential is presented. For this problem with the Dirichlet boundary conditions, the Green's function is constructed in a closed form by adding to Vq(r) a δ-function perturbation and making its strength infinitely repulsive. A transcendental equation for the energy levels E_{nr} and the wave functions of the bound states can then be deduced.
5. Path integral formalism for the spectral line shape in plasmas: Lyman-{alpha} with fine structure
SciTech Connect
Bedida, N.; Meftah, M. T.; Boland, D.; Stamm, R.
2008-10-22
We examine in this work the expression of the dipolar autocorrelation function for an emitter in the plasma using the path integrals formalism. The results for Lyman alpha lines with fine structure are retrieved in a compact formula. The expression of the dipolar autocorrelation function takes into account the ions dynamics and the fine structure effects. The electron's effect is represented by the impact operator {phi}{sub e} in the final formula.
6. An efficient algorithm for integrated task sequencing and path planning for robotic remote laser welding
2017-07-01
Different planning problems for robotic remote laser welding are of considerable interest. In this paper, we consider the problem of integrated task sequencing and path planning for robotic remote laser welding. We propose an efficient approach to solve the problem. In particular, we consider an explicit reduction from the decision version of the problem to the satisfiability problem. We present the results of computational experiments for different satisfiability algorithms.
7. Path-integral simulations of zero-point effects for implanted muons in benzene
Valladares, R. M.; Fisher, A. J.; Hayes, W.
1995-08-01
We describe a simulation method which is capable of treating the quantum fluctuations of an implanted muon and the electronic structure of the system simultaneously. The partition function for the muon is evaluated using a discretized imaginary-time path-integral technique, using electronic energies and forces evaluated from a semi-empirical quantum chemical treatment of the electronic structure. An application to the cyclohexadienyl radical (C 6H 7) and its muonated analogue (C 6H 6Mu) is presented.
8. Equation of State of Strongly Coupled Quark-Gluon Plasma - Path Integral Monte Carlo Results
Filinov, V. S.; Bonitz, M.; Ivanov, Y. B.; Skokov, V. V.; Levashov, P. R.; Fortov, V. E.
2009-09-01
A strongly coupled plasma of quark and gluon quasiparticles at temperatures from $1.1 T_c$ to $3 T_c$ is studied by path integral Monte Carlo simulations. This method extends previous classical nonrelativistic simulations based on a color Coulomb interaction to the quantum regime. We present the equation of state and find good agreement with lattice results. Further, pair distribution functions and color correlation functions are computed indicating strong correlations and liquid-like behavior.
9. Path integrals, differential renormalization-group, and stochastic systems near criticality
NASA Technical Reports Server (NTRS)
Chang, Tom
1992-01-01
It is demonstrated, using the techniques of path integrals and renormalization-group, that nonlinear stochastic systems near criticality (including self-organized criticality) generally exhibit low-dimensional behavior. The symmetry which characterizes a particular criticality can be broken by the appearance of relevant scaling fields. A connection is made between the fractal dimensions of finite-dimensional chaotic systems and the anomalous dimensions in stochastic systems near criticality. The effect of additional random noise on stochastic systems is also delineated.
10. Permutation blocking path integral Monte Carlo approach to the uniform electron gas at finite temperature.
PubMed
Dornheim, Tobias; Schoof, Tim; Groth, Simon; Filinov, Alexey; Bonitz, Michael
2015-11-28
The uniform electron gas (UEG) at finite temperature is of high current interest due to its key relevance for many applications including dense plasmas and laser excited solids. In particular, density functional theory heavily relies on accurate thermodynamic data for the UEG. Until recently, the only existing first-principle results had been obtained for N = 33 electrons with restricted path integral Monte Carlo (RPIMC), for low to moderate density, rs=r¯/aB≳1. These data have been complemented by configuration path integral Monte Carlo (CPIMC) simulations for rs ≤ 1 that substantially deviate from RPIMC towards smaller rs and low temperature. In this work, we present results from an independent third method-the recently developed permutation blocking path integral Monte Carlo (PB-PIMC) approach [T. Dornheim et al., New J. Phys. 17, 073017 (2015)] which we extend to the UEG. Interestingly, PB-PIMC allows us to perform simulations over the entire density range down to half the Fermi temperature (θ = kBT/EF = 0.5) and, therefore, to compare our results to both aforementioned methods. While we find excellent agreement with CPIMC, where results are available, we observe deviations from RPIMC that are beyond the statistical errors and increase with density.
11. Phase space path-integral formulation of the above-threshold ionization
Milošević, D. B.
2013-04-01
Atoms and molecules submitted to a strong laser field can emit electrons of high energies in the above-threshold ionization (ATI) process. This process finds a highly intuitive and also quantitative explanation in terms of Feynman's path integral and the concept of quantum orbits [P. Salières et al., Science 292, 902 (2001)], 10.1126/science.108836. However, the connection with the Feynman path-integral formalism is explained only by intuition and analogy and within the so-called strong-field approximation (SFA). Using the phase space path-integral formalism we have obtained an exact result for the momentum-space matrix element of the total time-evolution operator. Applying this result to the ATI we show that the SFA and the so-called improved SFA are, respectively, the zeroth- and the first-order terms of the expansion in powers of the laser-free effective interaction of the electron with the rest of the atom (molecule). We have also presented the second-order term of this expansion which is responsible for the ATI with double scattering of the ionized electron.
12. Integrating cell on chip—Novel waveguide platform employing ultra-long optical paths
Fohrmann, Lena Simone; Sommer, Gerrit; Pitruzzello, Giampaolo; Krauss, Thomas F.; Petrov, Alexander Yu.; Eich, Manfred
2017-09-01
Optical waveguides are the most fundamental building blocks of integrated optical circuits. They are extremely well understood, yet there is still room for surprises. Here, we introduce a novel 2D waveguide platform which affords a strong interaction of the evanescent tail of a guided optical wave with an external medium while only employing a very small geometrical footprint. The key feature of the platform is its ability to integrate the ultra-long path lengths by combining low propagation losses in a silicon slab with multiple reflections of the guided wave from photonic crystal (PhC) mirrors. With a reflectivity of 99.1% of our tailored PhC-mirrors, we achieve interaction paths of 25 cm within an area of less than 10 mm2. This corresponds to 0.17 dB/cm effective propagation which is much lower than the state-of-the-art loss of approximately 1 dB/cm of single mode silicon channel waveguides. In contrast to conventional waveguides, our 2D-approach leads to a decay of the guided wave power only inversely proportional to the optical path length. This entirely different characteristic is the major advantage of the 2D integrating cell waveguide platform over the conventional channel waveguide concepts that obey the Beer-Lambert law.
13. Calculation of heat capacities of light and heavy water by path-integral molecular dynamics
Shiga, Motoyuki; Shinoda, Wataru
2005-10-01
As an application of atomistic simulation methods to heat capacities, path-integral molecular dynamics has been used to calculate the constant-volume heat capacities of light and heavy water in the gas, liquid, and solid phases. While the classical simulation based on conventional molecular dynamics has estimated the heat capacities too high, the quantum simulation based on path-integral molecular dynamics has given reasonable results based on the simple point-charge/flexible potential model. The calculated heat capacities (divided by the Boltzmann constant) in the quantum simulation are 3.1 in the vapor H2O at 300 K, 6.9 in the liquid H2O at 300 K, and 4.1 in the ice IhH2O at 250 K, respectively, which are comparable to the experimental data of 3.04, 8.9, and 4.1, respectively. The quantum simulation also reproduces the isotope effect. The heat capacity in the liquid D2O has been calculated to be 10% higher than that of H2O, while it is 13% higher in the experiment. The results demonstrate that the path-integral simulation is a promising approach to quantitatively evaluate the heat capacities for molecular systems, taking account of quantum-mechanical vibrations as well as strongly anharmonic motions.
14. i-PI: A Python interface for ab initio path integral molecular dynamics simulations
Ceriotti, Michele; More, Joshua; Manolopoulos, David E.
2014-03-01
Recent developments in path integral methodology have significantly reduced the computational expense of including quantum mechanical effects in the nuclear motion in ab initio molecular dynamics simulations. However, the implementation of these developments requires a considerable programming effort, which has hindered their adoption. Here we describe i-PI, an interface written in Python that has been designed to minimise the effort required to bring state-of-the-art path integral techniques to an electronic structure program. While it is best suited to first principles calculations and path integral molecular dynamics, i-PI can also be used to perform classical molecular dynamics simulations, and can just as easily be interfaced with an empirical forcefield code. To give just one example of the many potential applications of the interface, we use it in conjunction with the CP2K electronic structure package to showcase the importance of nuclear quantum effects in high-pressure water. Catalogue identifier: AERN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AERN_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 138626 No. of bytes in distributed program, including test data, etc.: 3128618 Distribution format: tar.gz Programming language: Python. Computer: Multiple architectures. Operating system: Linux, Mac OSX, Windows. RAM: Less than 256 Mb Classification: 7.7. External routines: NumPy Nature of problem: Bringing the latest developments in the modelling of nuclear quantum effects with path integral molecular dynamics to ab initio electronic structure programs with minimal implementational effort. Solution method: State-of-the-art path integral molecular dynamics techniques are implemented in a Python interface. Any electronic structure code can be patched to receive the atomic
15. Transport coefficients of normal liquid helium-4 calculated by path integral centroid molecular dynamics simulation
Imaoka, Haruna; Kinugawa, Kenichi
2017-03-01
Thermal conductivity, shear viscosity, and bulk viscosity of normal liquid 4He at 1.7-4.0 K are calculated using path integral centroid molecular dynamics (CMD) simulations. The calculated thermal conductivity and shear viscosity above lambda transition temperature are on the same order of magnitude as experimental values, while the agreement of shear viscosity is better. Above 2.3 K the CMD well reproduces the temperature dependences of isochoric shear viscosity and of the time integral of the energy current and off-diagonal stress tensor correlation functions. The calculated bulk viscosity, not known in experiments, is several times larger than shear viscosity.
16. Treatment of the hydrogen atom in an electric field by the path-integral formalism
SciTech Connect
Chetouani, L.; Hammann, T.F.
1986-12-01
The Feynman path-integral method is applied to solve the problem of a H atom in an electric field. In accordance with the midpoint philosophy, the propagators are symmetrized in every time interval and, via several transformations, one of which is the Langer modification, the Green's function is calculated in parabolic coordinates, decomposed into partial propagators, and expressed in terms of two one-dimensional Green's functions. The perturbation method being no more valid for high excited levels, the spectrum is given, according to the WKB method, as a solution of a system of two elliptic integral equations. The exact spectrum of the H atom is obtained for a zero electric field.
17. Accurate path integration in continuous attractor network models of grid cells.
PubMed
Burak, Yoram; Fiete, Ila R
2009-02-01
Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat's velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of approximately 10-100 meters and approximately 1-10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.
18. Accurate Path Integration in Continuous Attractor Network Models of Grid Cells
PubMed Central
Burak, Yoram; Fiete, Ila R.
2009-01-01
Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animal's position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rat's velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other. PMID:19229307
19. CYBER/PHYSICAL SECURITY VULNERABILITY ASSESSMENT INTEGRATION
SciTech Connect
MacDonald, Douglas G.; Key, Brad; Clements, Samuel L.; Hutton, William J.; Craig, Philip A.; Patrick, Scott W.; Crawford, Cary E.
2011-07-17
This internally funded Laboratory-Directed R&D project by the Pacific Northwest National Laboratory, in conjunction with QinetiQ North America, is intended to identify and properly assess areas of overlap (and interaction) in the vulnerability assessment process between cyber security and physical protection. Existing vulnerability analysis (VA) processes and software tools exist, and these are heavily utilized in the determination of predicted vulnerability within the physical and cyber security domains. These determinations are normally performed independently of one another, and only interact on a superficial level. Both physical and cyber security subject matter experts have come to realize that though the various interactive elements exist, they are not currently quantified in most periodic security assessments. This endeavor aims to evaluate both physical and cyber VA techniques and provide a strategic approach to integrate the interdependent relationships of each into a single VA capability. This effort will also transform the existing suite of software currently utilized in the physical protection world to more accurately quantify the risk associated with a blended attack scenario. Performance databases will be created to support the characterization of the cyber security elements, and roll them into prototype software tools. This new methodology and software capability will enable analysts to better identify and assess the overall risk during a vulnerability analysis.
20. Path-integral virial estimator for reaction rate calculation basedon the quantum instanton aproximation
SciTech Connect
Yang, Sandy; Yamamoto, Takeshi; Miller, William H.
2005-11-28
The quantum instanton approximation is a type of quantum transition state theory that calculates the chemical reaction rate using the reactive flux correlation function and its low order derivatives at time zero. Here we present several path-integral estimators for the latter quantities, which characterize the initial decay profile of the flux correlation function. As with the internal energy or heat capacity calculation, different estimators yield different variances (and therefore different convergence properties) in a Monte Carlo calculation. Here we obtain a virial(-type) estimator by using a coordinate scaling procedure rather than integration by parts, which allows more computational benefits. We also consider two different methods for treating the flux operator, i.e., local-path and global-path approaches, in which the latter achieves a smaller variance at the cost of using second-order potential derivatives. Numerical tests are performed for a one-dimensional Eckart barrier and a model proton transfer reaction in a polar solvent, which illustrates the reduced variance of the virial estimator over the corresponding thermodynamic estimator.
1. Spatial memory and path integration studied by self-driven passive linear displacement. I. Basic properties.
PubMed
Israël, I; Grasso, R; Georges-Francois, P; Tsuzuku, T; Berthoz, A
1997-06-01
According to path integration, the brain is able to compute the distance of a traveled path. In this research we applied our previously reported method for studying memory of linear distance, a crucial mechanism in path integration; our method is based on the overt reconstruction of a passive transport. Passive transport is a special case of navigation in which no active control is performed. Blindfolded subjects were first asked to travel 2 m forward, in darkness, by driving with a joystick the robot on which they were seated. The results show that all subjects but two undershot this distance, i.e., overestimated their own displacement. Then, subjects were submitted to a passive linear forward displacement along 2, 4, 6, 8, or 10 m, and had to reproduce the same distance, still blindfolded. The results show that the distance of the stimulus was accurately reproduced, as well as stimulus duration, peak velocity, and velocity profile. In this first condition, the imposed velocity profile was triangular and therefore stimulus distance and duration were correlated. In a second condition, it was shown that distance was correctly reproduced also when the information about stimulus duration was kept constant. Here, different velocity profiles were used as stimuli, and most subjects also reproduced the velocity profile. Statistical analyses indicated that distance was not reproduced as a consequence of duration, peak velocity, or velocity profile reproduction, but was uniquely correlated to stimulus distance. The previous hypothesis of a double integration of the otolith signal to provide a distance estimate can explain our results. There was a large discrepancy between the accuracy with which the subjects matched the velocity profiles and that of distance reproduction. It follows that, whereas the dynamics of passive motion are stored and available to further use, distance is independently estimated. It is concluded that vestibular and somatosensory signals excited by
2. Integrating physical stress, growth, and development.
PubMed
Uyttewaal, Magalie; Traas, Jan; Hamant, Olivier
2010-02-01
Linking the gene regulatory network to morphogenesis is a central question in developmental biology. Shape relies on the combined actions of biochemistry and biophysics, two parameters that are under local genetic control. The blooming of molecular biology since the 1970s has promoted a biochemical view of development, leaving behind the contribution of physical forces. Recently, the development of new techniques, such as live imaging, micromechanical approaches, and computer modeling, has revitalized the biomechanics field. In this review, we use shoot apical meristem development to illustrate how biochemistry and biomechanics cooperate to integrate the local cellular gene input into global growth patterns. Copyright 2009 Elsevier Ltd. All rights reserved.
3. Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent
Yamamoto, Takeshi; Miller, William H.
2005-01-01
The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of ˜2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.
4. Dynamic response characteristics of dual flow-path integrally bladed rotors
Beck, Joseph A.; Brown, Jeffrey M.; Scott-Emuakpor, Onome E.; Cross, Charles J.; Slater, Joseph C.
2015-02-01
5. Accelerating ab initio path integral molecular dynamics with multilevel sampling of potential surface
SciTech Connect
Geng, Hua Y.
2015-02-15
A multilevel approach to sample the potential energy surface in a path integral formalism is proposed. The purpose is to reduce the required number of ab initio evaluations of energy and forces in ab initio path integral molecular dynamics (AI-PIMD) simulation, without compromising the overall accuracy. To validate the method, the internal energy and free energy of an Einstein crystal are calculated and compared with the analytical solutions. As a preliminary application, we assess the performance of the method in a realistic model—the FCC phase of dense atomic hydrogen, in which the calculated result shows that the acceleration rate is about 3 to 4-fold for a two-level implementation, and can be increased up to 10 times if extrapolation is used. With only 16 beads used for the ab initio potential sampling, this method gives a well converged internal energy. The residual error in pressure is just about 3 GPa, whereas it is about 20 GPa for a plain AI-PIMD calculation with the same number of beads. The vibrational free energy of the FCC phase of dense hydrogen at 300 K is also calculated with an AI-PIMD thermodynamic integration method, which gives a result of about 0.51 eV/proton at a density of r{sub s}=0.912.
6. Algorithms and novel applications based on the isokinetic ensemble. I. Biophysical and path integral molecular dynamics
Minary, Peter; Martyna, Glenn J.; Tuckerman, Mark E.
2003-02-01
In this paper (Paper I) and a companion paper (Paper II), novel new algorithms and applications of the isokinetic ensemble as generated by Gauss' principle of least constraint, pioneered for use with molecular dynamics 20 years ago, are presented for biophysical, path integral, and Car-Parrinello based ab initio molecular dynamics. In Paper I, a new "extended system" version of the isokinetic equations of motion that overcomes the ergodicity problems inherent in the standard approach, is developed using a new theory of non-Hamiltonian phase space analysis [M. E. Tuckerman et al., Europhys. Lett. 45, 149 (1999); J. Chem. Phys. 115, 1678 (2001)]. Reversible multiple time step integrations schemes for the isokinetic methods, first presented by Zhang [J. Chem. Phys. 106, 6102 (1997)] are reviewed. Next, holonomic constraints are incorporated into the isokinetic methodology for use in fast efficient biomolecular simulation studies. Model and realistic examples are presented in order to evaluate, critically, the performance of the new isokinetic molecular dynamic schemes. Comparisons are made to the, now standard, canonical dynamics method, Nosé-Hoover chain dynamics [G. J. Martyna et al., J. Chem. Phys. 97, 2635 (1992)]. The new isokinetic techniques are found to yield more efficient sampling than the Nosé-Hoover chain method in both path integral molecular dynamics and biophysical molecular dynamics calculations. In Paper II, the use of isokinetic methods in Car-Parrinello based ab initio molecular dynamics calculations is presented.
7. Accelerating ab initio path integral molecular dynamics with multilevel sampling of potential surface
Geng, Hua Y.
2015-02-01
A multilevel approach to sample the potential energy surface in a path integral formalism is proposed. The purpose is to reduce the required number of ab initio evaluations of energy and forces in ab initio path integral molecular dynamics (AI-PIMD) simulation, without compromising the overall accuracy. To validate the method, the internal energy and free energy of an Einstein crystal are calculated and compared with the analytical solutions. As a preliminary application, we assess the performance of the method in a realistic model-the FCC phase of dense atomic hydrogen, in which the calculated result shows that the acceleration rate is about 3 to 4-fold for a two-level implementation, and can be increased up to 10 times if extrapolation is used. With only 16 beads used for the ab initio potential sampling, this method gives a well converged internal energy. The residual error in pressure is just about 3 GPa, whereas it is about 20 GPa for a plain AI-PIMD calculation with the same number of beads. The vibrational free energy of the FCC phase of dense hydrogen at 300 K is also calculated with an AI-PIMD thermodynamic integration method, which gives a result of about 0.51 eV/proton at a density of rs = 0.912.
8. Hippocampus and Retrosplenial Cortex Combine Path Integration Signals for Successful Navigation
PubMed Central
Erdem, Uğur M.; Ross, Robert S.; Brown, Thackery I.; Hasselmo, Michael E.; Stern, Chantal E.
2013-01-01
9. Path-integral calculation of the second virial coefficient including intramolecular flexibility effects
SciTech Connect
Garberoglio, Giovanni; Jankowski, Piotr; Szalewicz, Krzysztof; Harvey, Allan H.
2014-07-28
We present a path-integral Monte Carlo procedure for the fully quantum calculation of the second molecular virial coefficient accounting for intramolecular flexibility. This method is applied to molecular hydrogen (H{sub 2}) and deuterium (D{sub 2}) in the temperature range 15–2000 K, showing that the effect of molecular flexibility is not negligible. Our results are in good agreement with experimental data, as well as with virials given by recent empirical equations of state, although some discrepancies are observed for H{sub 2} between 100 and 200 K.
10. Error Reduction Methods for Integrated-path Differential-absorption Lidar Measurements
NASA Technical Reports Server (NTRS)
Chen, Jeffrey R.; Numata, Kenji; Wu, Stewart T.
2012-01-01
We report new modeling and error reduction methods for differential-absorption optical-depth (DAOD) measurements of atmospheric constituents using direct-detection integrated-path differential-absorption lidars. Errors from laser frequency noise are quantified in terms of the line center fluctuation and spectral line shape of the laser pulses, revealing relationships verified experimentally. A significant DAOD bias is removed by introducing a correction factor. Errors from surface height and reflectance variations can be reduced to tolerable levels by incorporating altimetry knowledge and "log after averaging", or by pointing the laser and receiver to a fixed surface spot during each wavelength cycle to shorten the time of "averaging before log".
11. All-Electron Path Integral Simulations of Warm, Dense Matter: Application to Water and Carbon
Driver, Kevin; Militzer, Burkhard
2012-02-01
We develop an all-electron path integral Monte Carlo (PIMC) method for warm dense matter and apply it to study water and carbon. PIMC pressures, internal energies, and pair-correlation functions compare well with density functional theory molecular dynamics (DFT-MD) at lower temperatures and enable the construction of a coherent equation of state over a density-temperature range of 3--12 g/cm^3 and 10^2--10^9 K. PIMC results converge to the Debye-Huckel limiting law at high-temperatures and illuminate the breakdown of DFT pseudopotentials due to core excitations.
12. Path-integral calculation of the third virial coefficient of quantum gases at low temperatures
SciTech Connect
Garberoglio, Giovanni; Harvey, Allan H.
2011-04-07
We derive path-integral expressions for the second and third virial coefficients of monatomic quantum gases. Unlike previous work that considered only Boltzmann statistics, we include exchange effects (Bose-Einstein or Fermi-Dirac statistics). We use state-of-the-art pair and three-body potentials to calculate the third virial coefficient of {sup 3}He and {sup 4}He in the temperature range 2.6-24.5561 K. We obtain uncertainties smaller than those of the limited experimental data. Inclusion of exchange effects is necessary to obtain accurate results below about 7 K.
13. Path-integral calculation of the third virial coefficient of quantum gases at low temperatures.
PubMed
Garberoglio, Giovanni; Harvey, Allan H
2011-04-07
We derive path-integral expressions for the second and third virial coefficients of monatomic quantum gases. Unlike previous work that considered only Boltzmann statistics, we include exchange effects (Bose-Einstein or Fermi-Dirac statistics). We use state-of-the-art pair and three-body potentials to calculate the third virial coefficient of (3)He and (4)He in the temperature range 2.6-24.5561 K. We obtain uncertainties smaller than those of the limited experimental data. Inclusion of exchange effects is necessary to obtain accurate results below about 7 K.
14. Path-integral calculation of the second virial coefficient including intramolecular flexibility effects
Garberoglio, Giovanni; Jankowski, Piotr; Szalewicz, Krzysztof; Harvey, Allan H.
2014-07-01
We present a path-integral Monte Carlo procedure for the fully quantum calculation of the second molecular virial coefficient accounting for intramolecular flexibility. This method is applied to molecular hydrogen (H2) and deuterium (D2) in the temperature range 15-2000 K, showing that the effect of molecular flexibility is not negligible. Our results are in good agreement with experimental data, as well as with virials given by recent empirical equations of state, although some discrepancies are observed for H2 between 100 and 200 K.
15. Path-integral calculation of the second virial coefficient including intramolecular flexibility effects.
PubMed
Garberoglio, Giovanni; Jankowski, Piotr; Szalewicz, Krzysztof; Harvey, Allan H
2014-07-28
We present a path-integral Monte Carlo procedure for the fully quantum calculation of the second molecular virial coefficient accounting for intramolecular flexibility. This method is applied to molecular hydrogen (H2) and deuterium (D2) in the temperature range 15-2000 K, showing that the effect of molecular flexibility is not negligible. Our results are in good agreement with experimental data, as well as with virials given by recent empirical equations of state, although some discrepancies are observed for H2 between 100 and 200 K.
16. Path integral representation of spin foam models of 4D gravity
2008-12-01
We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter γ, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= γ < 1, the states can be restricted to SU(2) spin networks.
17. Option pricing formulas and nonlinear filtering: a Feynman path integral perspective
Balaji, Bhashyam
2013-05-01
Many areas of engineering and applied science require the solution of certain parabolic partial differential equa tions, such as the Fokker-Planck and Kolmogorov equations. The fundamental solution, or the Green's function, for such PDEs can be written in terms of the Feynman path integral (FPI). The partial differential equation arising in the valuing of options is the Kolmogorov backward equation that is referred to as the Black-Scholes equation. The utility of this is demonstrated and numerical examples that illustrate the high accuracy of option price calculation even when using a fairly coarse grid.
18. Path integral Liouville dynamics: Applications to infrared spectra of OH, water, ammonia, and methane
SciTech Connect
Liu, Jian; Zhang, Zhijun
2016-01-21
Path integral Liouville dynamics (PILD) is applied to vibrational dynamics of several simple but representative realistic molecular systems (OH, water, ammonia, and methane). The dipole-derivative autocorrelation function is employed to obtain the infrared spectrum as a function of temperature and isotopic substitution. Comparison to the exact vibrational frequency shows that PILD produces a reasonably accurate peak position with a relatively small full width at half maximum. PILD offers a potentially useful trajectory-based quantum dynamics approach to compute vibrational spectra of molecular systems.
19. Temperature-dependent isovector pairing gap equations using a path integral approach
SciTech Connect
Fellah, M.; Allal, N. H.; Belabbas, M.; Oudih, M. R.; Benhamouda, N.
2007-10-15
Temperature-dependent isovector neutron-proton (np) pairing gap equations have been established by means of the path integral approach. These equations generalize the BCS ones for the pairing between like particles at finite temperature. The method has been numerically tested using the one-level model. It has been shown that the gap parameter {delta}{sub np} has a behavior analogous to that of {delta}{sub nn} and {delta}{sub pp} as a function of the temperature: one notes the presence of a critical temperature. Moreover, it has been shown that the isovector pairing effects remain beyond the critical temperature that corresponds to the pairing between like particles.
20. Path integral Liouville dynamics: Applications to infrared spectra of OH, water, ammonia, and methane.
PubMed
Liu, Jian; Zhang, Zhijun
2016-01-21
Path integral Liouville dynamics (PILD) is applied to vibrational dynamics of several simple but representative realistic molecular systems (OH, water, ammonia, and methane). The dipole-derivative autocorrelation function is employed to obtain the infrared spectrum as a function of temperature and isotopic substitution. Comparison to the exact vibrational frequency shows that PILD produces a reasonably accurate peak position with a relatively small full width at half maximum. PILD offers a potentially useful trajectory-based quantum dynamics approach to compute vibrational spectra of molecular systems.
1. Path integral action and exact renormalization group dualities for quantum systems in noncommutative plane
2015-06-01
We employ the path integral approach developed in Gangopadhyay S. and Scholtz F. E., Phys. Rev. Lett., 102 (2009) 241602 to discuss the (generalized) harmonic oscillator in a noncommutative plane. The action for this system is derived in the coherent state basis with additional degrees of freedom. From this the action in the coherent state basis without any additional degrees of freedom is obtained. This gives the ground-state spectrum of the system. We then employ the exact renormalization group approach to show that an equivalence can be constructed between this (noncommutative) system and a commutative system.
2. The chemical shift of deprotonated water dimer: Ab initio path integral simulation
Shiga, Motoyuki; Suzuki, Kimichi; Tachikawa, Masanori
2010-03-01
The H1 NMR chemical shift in deprotonated water dimer H3O2- has been studied by ab initio path integral simulation. The simulation predicts that the isotropic shielding of hydrogen-bonded proton increases as a function of temperature by about 0.003 ppm/K. This change is about an order of magnitude larger than that of the nonhydrogen-bonded proton. It is concluded that this is caused by the significant difference in the quantum distribution of proton at high and low temperatures in the low barrier hydrogen bond.
3. Local time of Lévy random walks: A path integral approach
Zatloukal, Václav
2017-05-01
The local time of a stochastic process quantifies the amount of time that sample trajectories x (τ ) spend in the vicinity of an arbitrary point x . For a generic Hamiltonian, we employ the phase-space path-integral representation of random walk transition probabilities in order to quantify the properties of the local time. For time-independent systems, the resolvent of the Hamiltonian operator proves to be a central tool for this purpose. In particular, we focus on the local times of Lévy random walks (Lévy flights), which correspond to fractional diffusion equations.
4. Ab initio Path Integral Molecular Dynamics Based on Fragment Molecular Orbital Method
Fujita, Takatoshi; Watanabe, Hirofumi; Tanaka, Shigenori
2009-10-01
We have developed an ab initio path integral molecular dynamics method based on the fragment molecular orbital method. This “FMO-PIMD” method can treat both nuclei and electrons quantum mechanically, and is useful to simulate large hydrogen-bonded systems with high accuracy. After a benchmark calculation for water monomer, water trimer and glycine pentamer have been studied using the FMO-PIMD method to investigate nuclear quantum effects on structure and molecular interactions. The applicability of the present approach is demonstrated through a number of test calculations.
5. Formation of bound states in expanded metal studied via path integral molecular dynamics
Deymier, P. A.; Oh, Ki-Dong
2004-03-01
The usefulness of the restricted path integral molecular dynamics method for the study of strongly correlated electrons is demonstrated by studying the formation of bound electronic states in a half-filled expanded three-dimensional hydrogenoid body-centred cubic lattice at finite temperature. Starting from a metallic state with one-component plasma character, we find that bound electrons form upon expansion of the lattice. The bound electrons are spatially localized with their centre for the motion of gyration located at ionic positions. The number of bound electrons increases monotonically with decreasing density.
6. Many-body quantum dynamics by adiabatic path-integral molecular dynamics: Disordered Frenkel Kontorova models
Krajewski, Florian R.; Müser, Martin H.
2005-07-01
The spectral density of quantum mechanical Frenkel Kontorova chains moving in disordered, external potentials is investigated by means of path-integral molecular dynamics. If the second moment of the embedding potential is well defined (roughness exponent H=0), there is one regime in which the chain is pinned (large masses m of chain particles) and one in which it is unpinned (small m). If the embedding potential can be classified as a random walk on large length scales ( H=1/2), then the chain is always pinned irrespective of the value of m. For H=1/2, two phonon-like branches appear in the spectra.
7. Path-integral molecular dynamics simulations for water anion clusters (HO)5- and (DO)5-
Takayanagi, Toshiyuki; Yoshikawa, Takehiro; Motegi, Haruki; Shiga, Motoyuki
2009-11-01
Quantum path-integral molecular dynamics simulations have been performed for the (HO)5- and (DO)5- anion clusters on the basis of a semiempirical one-electron pseudopotential-polarization model. Due to larger zero-point vibrational amplitudes for H atoms than that of D atoms, hydrogen-bond lengths in the (HO)5- cluster are slightly larger than those in (DO)5-. The distribution of the vertical detachment energies for (HO)5- also show a broader feature than that for (DO)5-. The present PIMD simulations thus demonstrate the importance of nuclear quantum effects in water anion clusters.
8. Restricted Path-Integral Molecular Dynamics for Simulating the Correlated Electron Plasma in Warm Dense Matter
Kapila, Vivek; Deymier, Pierre; Runge, Keith
2011-10-01
Several areas of study including heavy ion beam, large scale laser, and high pressure or Thomson scattering studies necessitate a fundamental understanding of warm dense matter (WDM) i.e. matter at high temperature and high density. The WDM regime, however, lacks any adequate highly developed class of simulation methods. Recent progress to address this deficit has been the development of orbital-free Density Functional Theory (ofDFT). However, scant benchmark information is available on temperature and pressure dependence of simple but realistic models in WDM regime. The present work aims to fill this critical gap using the restricted path-integral molecular dynamics (rPIMD) method. Within the discrete path integral representation, electrons are described as harmonic necklaces. Quantum exchange takes the form of cross linking between electron necklaces. The fermion sign problem is addressed by restricting the density matrix to positive values. The molecular dynamics algorithm is employed to sample phase space. Here, we focus on the behavior of strongly correlated electron plasmas under WDM conditions. We compute the kinetic and potential energies and compare them to those obtained with the ofDFT method. Several areas of study including heavy ion beam, large scale laser, and high pressure or Thomson scattering studies necessitate a fundamental understanding of warm dense matter (WDM) i.e. matter at high temperature and high density. The WDM regime, however, lacks any adequate highly developed class of simulation methods. Recent progress to address this deficit has been the development of orbital-free Density Functional Theory (ofDFT). However, scant benchmark information is available on temperature and pressure dependence of simple but realistic models in WDM regime. The present work aims to fill this critical gap using the restricted path-integral molecular dynamics (rPIMD) method. Within the discrete path integral representation, electrons are described as
9. Path Integral Molecular Dynamics for Hydrogen with Orbital-Free Density Functional Theory
Runge, Keith; Karasiev, Valentin; Deymier, Pierre
2014-03-01
The computational bottleneck for performing path-integral molecular dynamics (PIMD) for nuclei on a first principles electronic potential energy surface has been the speed with which forces from the electrons can be generated. Recent advances in orbital-free density functional theory (OF-DFT) not only allow for faster generation of first principles forces but also include the effects of temperature on the electron density. We will present results of calculations on hydrogen in warm dense matter conditions where the protons are described by PIMD and the electrons by OF-DFT. Work supported by U.S. Dept. of Energy, grant DE-SC0002139.
10. Unified path integral approach to theories of diffusion-influenced reactions
Prüstel, Thorsten; Meier-Schellersheim, Martin
2017-08-01
Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we develop a general approach for computational modeling of diffusion-influenced reactions that is capable of capturing not only the classical Smoluchowski picture but also alternative theories, as is here exemplified by a volume reactivity model. In particular, we prove the path decomposition expansion of various Green's functions describing the irreversible and reversible reaction of an isolated pair of molecules. To this end, we exploit a connection between boundary value and interaction potential problems with δ - and δ'-function perturbation. We employ a known path-integral-based summation of a perturbation series to derive a number of exact identities relating propagators and survival probabilities satisfying different boundary conditions in a unified and systematic manner. Furthermore, we show how the path decomposition expansion represents the propagator as a product of three factors in the Laplace domain that correspond to quantities figuring prominently in stochastic spatially resolved simulation algorithms. This analysis will thus be useful for the interpretation of current and the design of future algorithms. Finally, we discuss the relation between the general approach and the theory of Brownian functionals and calculate the mean residence time for the case of irreversible and reversible reactions.
11. Development of a Pulsed 2-Micron Integrated Path Differential Absorption Lidar for CO2 Measurement
NASA Technical Reports Server (NTRS)
Singh, Upendra N.; Yu, Jirong; Petros, Mulugeta; Refaat, Tamer; Refaat, Tamer
2013-01-01
Atmospheric carbon dioxide (CO2) is an important greenhouse gas that significantly contributes to the carbon cycle and global radiation budget on Earth. Active remote sensing of CO2 is important to address several limitations that contend with passive sensors. A 2-micron double-pulsed, Integrated Path Differential Absorption (IPDA) lidar instrument for ground and airborne atmospheric CO2 concentration measurements via direct detection method is being developed at NASA Langley Research Center. This active remote sensing instrument will provide an alternate approach of measuring atmospheric CO2 concentrations with significant advantages. A high energy pulsed approach provides high-precision measurement capability by having high signal-to-noise ratio level and unambiguously eliminates the contamination from aerosols and clouds that can bias the IPDA measurement. Commercial, on the shelf, components are implemented for the detection system. Instrument integration will be presented in this paper as well as a background for CO2 measurement at NASA Langley research Center
12. High-order sampling schemes for path integrals and Gaussian chain simulations of polymers
SciTech Connect
Müser, Martin H.; Müller, Marcus
2015-05-07
In this work, we demonstrate that path-integral schemes, derived in the context of many-body quantum systems, benefit the simulation of Gaussian chains representing polymers. Specifically, we show how to decrease discretization corrections with little extra computation from the usual O(1/P{sup 2}) to O(1/P{sup 4}), where P is the number of beads representing the chains. As a consequence, high-order integrators necessitate much smaller P than those commonly used. Particular emphasis is placed on the questions of how to maintain this rate of convergence for open polymers and for polymers confined by a hard wall as well as how to ensure efficient sampling. The advantages of the high-order sampling schemes are illustrated by studying the surface tension of a polymer melt and the interface tension in a binary homopolymers blend.
13. Development of a pulsed 2-micron integrated path differential absorption lidar for CO2 measurement
Singh, Upendra N.; Yu, Jirong; Petros, Mulugeta; Refaat, Tamer; Reithmaier, Karl
2013-09-01
Atmospheric carbon dioxide (CO2) is an important greenhouse gas that significantly contributes to the carbon cycle and global radiation budget on Earth. Active remote sensing of CO2 is important to address several limitations that contend with passive sensors. A 2-micron double-pulsed, Integrated Path Differential Absorption (IPDA) lidar instrument for ground and airborne atmospheric CO2 concentration measurements via direct detection method is being developed at NASA Langley Research Center. This active remote sensing instrument will provide an alternate approach of measuring atmospheric CO2 concentrations with significant advantages. A high energy pulsed approach provides high-precision measurement capability by having high signal-to-noise ratio level and unambiguously eliminates the contamination from aerosols and clouds that can bias the IPDA measurement. Commercial, on the shelf, components are implemented for the detection system. Instrument integration will be presented in this paper as well as a background for CO2 measurement at NASA Langley research Center.
14. Path integral representation of Lorentzian spinfoam model, asymptotics and simplicial geometries
Han, Muxin; Krajewski, Thomas
2014-01-01
A new path integral representation of Lorentzian Engle-Pereira-Rovelli-Livine spinfoam model is derived by employing the theory of unitary representation of {SL}(2, {C}). The path integral representation is taken as a starting point of semiclassical analysis. The relation between the spinfoam model and classical simplicial geometry is studied via the large-spin asymptotic expansion of the spinfoam amplitude with all spins uniformly large. More precisely, in the large-spin regime, there is an equivalence between the spinfoam critical configuration (with certain nondegeneracy assumption) and a classical Lorentzian simplicial geometry. Such an equivalence relation allows us to classify the spinfoam critical configurations by their geometrical interpretations, via two types of solution-generating maps. The equivalence between spinfoam critical configuration and simplical geometry also allows us to define the notion of globally oriented and time-oriented spinfoam critical configuration. It is shown that only at the globally oriented and time-oriented spinfoam critical configuration, the leading-order contribution of spinfoam large-spin asymptotics gives precisely an exponential of Lorentzian Regge action of General Relativity. At all other (unphysical) critical configurations, spinfoam large-spin asymptotics modifies the Regge action at the leading-order approximation.
15. Spectroscopic fingerprints of toroidal nuclear quantum delocalization via ab initio path integral simulations.
PubMed
Schütt, Ole; Sebastiani, Daniel
2013-04-05
We investigate the quantum-mechanical delocalization of hydrogen in rotational symmetric molecular systems. To this purpose, we perform ab initio path integral molecular dynamics simulations of a methanol molecule to characterize the quantum properties of hydrogen atoms in a representative system by means of their real-space and momentum-space densities. In particular, we compute the spherically averaged momentum distribution n(k) and the pseudoangular momentum distribution n(kθ). We interpret our results by comparing them to path integral samplings of a bare proton in an ideal torus potential. We find that the hydroxyl hydrogen exhibits a toroidal delocalization, which leads to characteristic fingerprints in the line shapes of the momentum distributions. We can describe these specific spectroscopic patterns quantitatively and compute their onset as a function of temperature and potential energy landscape. The delocalization patterns in the projected momentum distribution provide a promising computational tool to address the intriguing phenomenon of quantum delocalization in condensed matter and its spectroscopic characterization. As the momentum distribution n(k) is also accessible through Nuclear Compton Scattering experiments, our results will help to interpret and understand future measurements more thoroughly.
16. Interactions of the polarization and the sun compass in path integration of desert ants.
PubMed
Lebhardt, Fleur; Ronacher, Bernhard
2014-08-01
Desert ants, Cataglyphis fortis, perform large-scale foraging trips in their featureless habitat using path integration as their main navigation tool. To determine their walking direction they use primarily celestial cues, the sky's polarization pattern and the sun position. To examine the relative importance of these two celestial cues, we performed cue conflict experiments. We manipulated the polarization pattern experienced by the ants during their outbound foraging excursions, reducing it to a single electric field (e-)vector direction with a linear polarization filter. The simultaneous view of the sun created situations in which the directional information of the sun and the polarization compass disagreed. The heading directions of the homebound runs recorded on a test field with full view of the natural sky demonstrate that none of both compasses completely dominated over the other. Rather the ants seemed to compute an intermediate homing direction to which both compass systems contributed roughly equally. Direct sunlight and polarized light are detected in different regions of the ant's compound eye, suggesting two separate pathways for obtaining directional information. In the experimental paradigm applied here, these two pathways seem to feed into the path integrator with similar weights.
17. Accelerating Ab Initio Path Integral Simulations via Imaginary Multiple-Timestepping.
PubMed
Cheng, Xiaolu; Herr, Jonathan D; Steele, Ryan P
2016-04-12
This work investigates the use of multiple-timestep schemes in imaginary time for computationally efficient ab initio equilibrium path integral simulations of quantum molecular motion. In the simplest formulation, only every n(th) path integral replica is computed at the target level of electronic structure theory, whereas the remaining low-level replicas still account for nuclear motion quantum effects with a more computationally economical theory. Motivated by recent developments for multiple-timestep techniques in real-time classical molecular dynamics, both 1-electron (atomic-orbital basis set) and 2-electron (electron correlation) truncations are shown to be effective. Structural distributions and thermodynamic averages are tested for representative analytic potentials and ab initio molecular examples. Target quantum chemistry methods include density functional theory and second-order Møller-Plesset perturbation theory, although any level of theory is formally amenable to this framework. For a standard two-level splitting, computational speedups of 1.6-4.0x are observed when using a 4-fold reduction in time slices; an 8-fold reduction is feasible in some cases. Multitiered options further reduce computational requirements and suggest that quantum mechanical motion could potentially be obtained at a cost not significantly different from the cost of classical simulations.
18. Quantum free-energy differences from nonequilibrium path integrals. I. Methods and numerical application.
PubMed
van Zon, Ramses; Hernández de la Peña, Lisandro; Peslherbe, Gilles H; Schofield, Jeremy
2008-10-01
In this paper, the imaginary-time path-integral representation of the canonical partition function of a quantum system and nonequilibrium work fluctuation relations are combined to yield methods for computing free-energy differences in quantum systems using nonequilibrium processes. The path-integral representation is isomorphic to the configurational partition function of a classical field theory, to which a natural but fictitious Hamiltonian dynamics is associated. It is shown that if this system is prepared in an equilibrium state, after which a control parameter in the fictitious Hamiltonian is changed in a finite time, then formally the Jarzynski nonequilibrium work relation and the Crooks fluctuation relation hold, where work is defined as the change in the energy as given by the fictitious Hamiltonian. Since the energy diverges for the classical field theory in canonical equilibrium, two regularization methods are introduced which limit the number of degrees of freedom to be finite. The numerical applicability of the methods is demonstrated for a quartic double-well potential with varying asymmetry. A general parameter-free smoothing procedure for the work distribution functions is useful in this context.
19. Contact Potential Instability in the Path-Integral Description of Itinerant Ferromagnetism
Vermeyen, E.; Tempere, J.
2015-05-01
It has long been predicted that a two-component non-localized Fermi gas will exhibit spontaneous polarization for sufficiently strong repulsive interactions, a phenomenon which is called itinerant ferromagnetism. Recent experiments with ultracold atomic gases have reached the interaction strength for which theoretical models have predicted the occurrence of the normal-to-itinerant-ferromagnetic phase transition, but so far this transition has not been observed. The instability of the repulsive branch of the Feshbach resonance prevents the formation of the itinerant ferromagnetic state, but it is not clear whether this is the only instability impeding its experimental realization. In this article, we use the path-integral formalism with density fields in the Hubbard-Stratonovich transformation to study the stability of a homogeneous two-component Fermi gas with contact interactions. Within the saddle-point approximation we show that none of the extrema of the action are minima, meaning all extrema are unstable to small density fluctuations. This implies a more general mechanical instability of the polarized (itinerant ferromagnetic) and normal states of the system in the path-integral formalism. We find that it is important to consider the stability of the system when studying itinerant ferromagnetism. Since (mechanical) stability may be influenced by the details of the interaction potential, we suggest the use of a more realistic potential than the contact potential in future theoretical descriptions.
20. Path integral method for predicting relative binding affinities of protein-ligand complexes
PubMed Central
Mulakala, Chandrika; Kaznessis, Yiannis N.
2009-01-01
We present a novel approach for computing biomolecular interaction binding affinities based on a simple path integral solution of the Fokker-Planck equation. Computing the free energy of protein-ligand interactions can expedite structure-based drug design. Traditionally, the problem is seen through the lens of statistical thermodynamics. The computations can become, however, prohibitively long for the change in the free energy upon binding to be determined accurately. In this work we present a different approach based on a stochastic kinetic formalism. Inspired by Feynman's path integral formulation, we extend the theory to classical interacting systems. The ligand is modeled as a Brownian particle subjected to the effective non-bonding interaction potential of the receptor. This allows the calculation of the relative binding affinities of interacting biomolecules in water to be computed as a function of the ligand's diffusivity and the curvature of the potential surface in the vicinity of the binding minimum. The calculation is thus exceedingly rapid. In test cases, the correlation coefficient between actual and computed free energies is >0.93 for accurate data-sets. PMID:19275144
1. Path-integral description of combined Hamiltonian and non-Hamiltonian dynamics in quantum dissipative systems
Barth, A. M.; Vagov, A.; Axt, V. M.
2016-09-01
We present a numerical path-integral iteration scheme for the low-dimensional reduced density matrix of a time-dependent quantum dissipative system. Our approach simultaneously accounts for the combined action of a microscopically modeled pure-dephasing-type coupling to a continuum of harmonic oscillators representing, e.g., phonons, and further environmental interactions inducing non-Hamiltonian dynamics in the inner system represented, e.g., by Lindblad-type dissipation or relaxation. Our formulation of the path-integral method allows for a numerically exact treatment of the coupling to the oscillator modes and moreover is general enough to provide a natural way to include Markovian processes that are sufficiently described by rate equations. We apply this new formalism to a model of a single semiconductor quantum dot which includes the coupling to longitudinal acoustic phonons for two cases: (a) external laser excitation taking into account a phenomenological radiative decay of the excited dot state and (b) a coupling of the quantum dot to a single mode of an optical cavity taking into account cavity photon losses.
2. Transport properties of liquid para-hydrogen: The path integral centroid molecular dynamics approach
Yonetani, Yoshiteru; Kinugawa, Kenichi
2003-11-01
Several fundamental transport properties of a quantum liquid para-hydrogen (p-H2) at 17 K have been numerically evaluated by means of the quantum dynamics simulation called the path integral centroid molecular dynamics (CMD). For comparison, classical molecular dynamics (MD) simulations have also been performed under the same condition. In accordance with the previous path integral simulations, the calculated static properties of the liquid agree well with the experimental results. For the diffusion coefficient, thermal conductivity, and shear viscosity, the CMD predicts the values closer to the experimental ones though the classical MD results are far from the reality. The agreement of the CMD result with the experimental one is especially good for the shear viscosity with the difference less than 5%. The calculated diffusion coefficient and the thermal conductivity agree with the experimental values at least in the same order. We predict that the ratio of bulk viscosity to shear viscosity for liquid p-H2 is much larger than classical van der Waals simple liquids such as rare gas liquids.
3. Testing an Integrated Self-Determined Work Motivation Model for People With Disabilities: A Path Analysis.
PubMed
Tansey, Timothy N; Iwanaga, Kanako; Bezyak, Jill; Ditchman, Nicole
2017-05-04
Individuals with disabilities are more likely to live in poverty, have more health issues, and be less likely to be employed than their same-aged peers. Although these issues may be attenuated by vocational rehabilitation services, amotivation and ambivalence to employment can limit the readiness of persons with disabilities to engage in these services. Drawing on self-efficacy, self-determination, and stages of change theories, the purpose of this study was to develop and test an integrated self-determined work motivation model for people with disabilities. Participants included 277 people with disabilities recruited through vocational rehabilitation agencies across 8 states. Path analysis was used to evaluate the contribution of functional disability, self-determination, and social efficacy variables in a hypothesized integrated self-determined work motivation model. Model estimations used maximum likelihood estimation and model-data fit was examined using several goodness-of-fit indices. The initial path analysis indicated a less than optimal fit between the model and the observed data. Post hoc model modifications were conducted based on examination of the critical ratios and modification indices and theoretical consideration. The respecified integrated self-determined work motivation model fit the data very well, χ2/df = 1.88, CFI = .99, and RMSEA = 0.056. The R2 for the endogenous variables in the model ranged from .19 to .54. Findings from this study support the integrated self-determined work motivation model in vocational rehabilitation as a useful framework for understanding the relationship among functioning levels, self-determination and self-efficacy factors, vocational rehabilitation engagement, and readiness for employment. (PsycINFO Database Record (c) 2017 APA, all rights reserved).
4. An integrated radiation physics computer code system.
NASA Technical Reports Server (NTRS)
Steyn, J. J.; Harris, D. W.
1972-01-01
An integrated computer code system for the semi-automatic and rapid analysis of experimental and analytic problems in gamma photon and fast neutron radiation physics is presented. Such problems as the design of optimum radiation shields and radioisotope power source configurations may be studied. The system codes allow for the unfolding of complex neutron and gamma photon experimental spectra. Monte Carlo and analytic techniques are used for the theoretical prediction of radiation transport. The system includes a multichannel pulse-height analyzer scintillation and semiconductor spectrometer coupled to an on-line digital computer with appropriate peripheral equipment. The system is geometry generalized as well as self-contained with respect to material nuclear cross sections and the determination of the spectrometer response functions. Input data may be either analytic or experimental.
5. An integrated radiation physics computer code system.
NASA Technical Reports Server (NTRS)
Steyn, J. J.; Harris, D. W.
1972-01-01
An integrated computer code system for the semi-automatic and rapid analysis of experimental and analytic problems in gamma photon and fast neutron radiation physics is presented. Such problems as the design of optimum radiation shields and radioisotope power source configurations may be studied. The system codes allow for the unfolding of complex neutron and gamma photon experimental spectra. Monte Carlo and analytic techniques are used for the theoretical prediction of radiation transport. The system includes a multichannel pulse-height analyzer scintillation and semiconductor spectrometer coupled to an on-line digital computer with appropriate peripheral equipment. The system is geometry generalized as well as self-contained with respect to material nuclear cross sections and the determination of the spectrometer response functions. Input data may be either analytic or experimental.
6. a Latent Variable Path Analysis Model of Secondary Physics Enrollments in New York State.
Sobolewski, Stanley John
The Percentage of Enrollment in Physics (PEP) at the secondary level nationally has been approximately 20% for the past few decades. For a more scientifically literate citizenry as well as specialists to continue scientific research and development, it is desirable that more students enroll in physics. Some of the predictor variables for physics enrollment and physics achievement that have been identified previously includes a community's socioeconomic status, the availability of physics, the sex of the student, the curriculum, as well as teacher and student data. This study isolated and identified predictor variables for PEP of secondary schools in New York. Data gathered by the State Education Department for the 1990-1991 school year was used. The source of this data included surveys completed by teachers and administrators on student characteristics and school facilities. A data analysis similar to that done by Bryant (1974) was conducted to determine if the relationships between a set of predictor variables related to physics enrollment had changed in the past 20 years. Variables which were isolated included: community, facilities, teacher experience, number of type of science courses, school size and school science facilities. When these variables were isolated, latent variable path diagrams were proposed and verified by the Linear Structural Relations computer modeling program (LISREL). These diagrams differed from those developed by Bryant in that there were more manifest variables used which included achievement scores in the form of Regents exam results. Two criterion variables were used, percentage of students enrolled in physics (PEP) and percent of students enrolled passing the Regents physics exam (PPP). The first model treated school and community level variables as exogenous while the second model treated only the community level variables as exogenous. The goodness of fit indices for the models was 0.77 for the first model and 0.83 for the second
7. Coarse-grained representation of the quasi adiabatic propagator path integral for the treatment of non-Markovian long-time bath memory
Richter, Martin; Fingerhut, Benjamin P.
2017-06-01
The description of non-Markovian effects imposed by low frequency bath modes poses a persistent challenge for path integral based approaches like the iterative quasi-adiabatic propagator path integral (iQUAPI) method. We present a novel approximate method, termed mask assisted coarse graining of influence coefficients (MACGIC)-iQUAPI, that offers appealing computational savings due to substantial reduction of considered path segments for propagation. The method relies on an efficient path segment merging procedure via an intermediate coarse grained representation of Feynman-Vernon influence coefficients that exploits physical properties of system decoherence. The MACGIC-iQUAPI method allows us to access the regime of biological significant long-time bath memory on the order of hundred propagation time steps while retaining convergence to iQUAPI results. Numerical performance is demonstrated for a set of benchmark problems that cover bath assisted long range electron transfer, the transition from coherent to incoherent dynamics in a prototypical molecular dimer and excitation energy transfer in a 24-state model of the Fenna-Matthews-Olson trimer complex where in all cases excellent agreement with numerically exact reference data is obtained.
8. Differential recruitment of the hippocampus, medial prefrontal cortex, and the human motion complex during path integration in humans.
PubMed
Wolbers, Thomas; Wiener, Jan M; Mallot, Hanspeter A; Büchel, Christian
2007-08-29
Path integration, the ability to sense self-motion for keeping track of changes in orientation and position, constitutes a fundamental mechanism of spatial navigation and a keystone for the development of cognitive maps. Whereas animal path integration is predominantly supported by the head-direction, grid, and place cell systems, the neural foundations are not well understood in humans. Here we used functional magnetic resonance imaging and a virtual rendition of a triangle completion paradigm to test whether human path integration recruits a cortical system similar to that of rodents and nonhuman primates. Participants traveled along two legs of a triangle before pointing toward the starting location. In accordance with animal models, stronger right hippocampal activation predicted more accurate updating of the starting location on a trial-by-trial basis. Moreover, between-subjects fluctuations in response consistency were negatively correlated with bilateral hippocampal and medial prefrontal activation, and bilateral recruitment of the human motion complex (hMT+) covaried with individual path integration capability. Given that these effects were absent in a perceptual control task, the present study provides the first evidence that visual path integration is related to the dynamic interplay of self-motion processing in hMT+, higher-level spatial processes in the hippocampus, and spatial working memory in medial prefrontal cortex.
9. Real-time Feynman path integral with Picard–Lefschetz theory and its applications to quantum tunneling
SciTech Connect
Tanizaki, Yuya; Koike, Takayuki
2014-12-15
Picard–Lefschetz theory is applied to path integrals of quantum mechanics, in order to compute real-time dynamics directly. After discussing basic properties of real-time path integrals on Lefschetz thimbles, we demonstrate its computational method in a concrete way by solving three simple examples of quantum mechanics. It is applied to quantum mechanics of a double-well potential, and quantum tunneling is discussed. We identify all of the complex saddle points of the classical action, and their properties are discussed in detail. However a big theoretical difficulty turns out to appear in rewriting the original path integral into a sum of path integrals on Lefschetz thimbles. We discuss generality of that problem and mention its importance. Real-time tunneling processes are shown to be described by those complex saddle points, and thus semi-classical description of real-time quantum tunneling becomes possible on solid ground if we could solve that problem. - Highlights: • Real-time path integral is studied based on Picard–Lefschetz theory. • Lucid demonstration is given through simple examples of quantum mechanics. • This technique is applied to quantum mechanics of the double-well potential. • Difficulty for practical applications is revealed, and we discuss its generality. • Quantum tunneling is shown to be closely related to complex classical solutions.
10. Combining Meteosat-10 satellite image data with GPS tropospheric path delays to estimate regional integrated water vapor (IWV) distribution
Leontiev, Anton; Reuveni, Yuval
2017-02-01
Using GPS satellites signals, we can study different processes and coupling mechanisms that can help us understand the physical conditions in the lower atmosphere, which might lead or act as proxies for severe weather events such as extreme storms and flooding. GPS signals received by ground stations are multi-purpose and can also provide estimates of tropospheric zenith delays, which can be converted into accurate integrated water vapor (IWV) observations using collocated pressure and temperature measurements on the ground. Here, we present for the first time the use of Israel's dense regional GPS network for extracting tropospheric zenith path delays combined with near-real-time Meteosat-10 water vapor (WV) and surface temperature pixel intensity values (7.3 and 10.8 µm channels, respectively) in order to assess whether it is possible to obtain absolute IWV (kg m-2) distribution. The results show good agreement between the absolute values obtained from our triangulation strategy based solely on GPS zenith total delays (ZTD) and Meteosat-10 surface temperature data compared with available radiosonde IWV absolute values. The presented strategy can provide high temporal and special IWV resolution, which is needed as part of the accurate and comprehensive observation data integrated in modern data assimilation systems and is required for increasing the accuracy of regional numerical weather prediction systems forecast.
11. Adaptive integral LOS path following for an unmanned airship with uncertainties based on robust RBFNN backstepping.
PubMed
Zheng, Zewei; Zou, Yao
2016-11-01
This paper investigates the path following control problem for an unmanned airship in the presence of unknown wind and uncertainties. The backstepping technique augmented by a robust adaptive radial basis function neural network (RBFNN) is employed as the main control framework. Based on the horizontal dynamic model of the airship, an improved adaptive integral line-of-sight (LOS) guidance law is first proposed, which suits any parametric paths. The guidance law calculates the desired yaw angle and estimates the wind. Then the controller is extended to cope with the airship yaw tracking and velocity control by resorting to the augmented backstepping technique. The uncertainties of the dynamics are compensated by using the robust RBFNNs. Each robust RBFNN utilizes an nth-order smooth switching function to combine a conventional RBFNN with a robust control. The conventional RBFNN dominates in the neural active region, while the robust control retrieves the transient outside the active region, so that the stability range can be widened. Stability analysis shows that the controlled closed-loop system is globally uniformly ultimately bounded. Simulations are provided to validate the effectiveness of the proposed control approach. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.
12. Path integral centroid variables and the formulation of their exact real time dynamics
Jang, Seogjoo; Voth, Gregory A.
1999-08-01
A formalism is presented in this paper which, for the first time, establishes the theoretical basis for the quantum time evolution of path integral centroid variables and also provides clear motivation for using these variables to study condensed phase quantum dynamics. The equilibrium centroid distribution is first shown to be a well-defined distribution function which is specific to the canonical density operator. A quantum mechanical quasi-density operator (QDO) is associated with each value of the distribution so that, upon application of the standard quantum mechanical formalism, the QDO can be used to provide a rigorous definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables are derived, including the perspective that the centroid constraint on the imaginary time paths introduces a nonstationarity in the equilibrium ensemble which, in turn, can be shown to yield information on the correlations of spontaneous fluctuations. The analytic solution for the harmonic oscillator and a numerical solution for a double well system are provided which illustrate the various aspects of the theory. The theory contained herein provides the basis for a derivation of Centroid Molecular Dynamics, as well as the systematic improvements of that theory.
13. Information Entropy Exchange in the Path Integral Formulation of Transition Amplitudes
Deeter, Daniel; Petridis, Athanasios
2016-09-01
The quantum mechanical transition amplitude for a free particle is calculated using the path integral formalism. This amplitude is the kernel of the Schrödinger equation. A Wick rotation of the time increment transforms the kernel into a partition function that depends on the space and time intervals of the transition, with the temperature being proportional to the inverse of the time increment. The information entropy exchange between the system and the observer during the transition is calculated from the partition function. The requirement that this be real-valued leads to uncertainty-type relations. Furthermore, the transition exhibits positive information entropy exchange for small time intervals and negative entropy for large ones. The related statistical weight is inversely proportional to the square root of the time interval. Calculations for interacting systems are in progress.
14. Path integral study of the correlated electronic states of Na4-Na6
Hall, Randall W.
1990-12-01
Feynman's path integral formulation of quantum mechanics is used to study the correlated electronic states of Na4-Na6. Two types of simulations are performed: in the first, the nuclei are allowed to move at finite temperature in order to find the most stable geometries. In agreement with previous calculations, we find that planar structures are the most stable and that there is significant vibrational amplitude at finite temperatures, indicating that the Born-Oppenheimer surface is relatively flat. In the second type of simulation, the nuclei are held fixed at symmetric and asymmetric geometries and the correlated electron density is found. Our results show that the electrons are localized, rather than delocalized as previous workers have concluded from examination of the single-particle orbitals. We find that the best picture of these clusters is that they contain three-center, two-electron bonds.
15. All-Electron Path Integral Simulations of Warm, Dense Matter: Application to Water and Carbon
Driver, K. P.; Militzer, B.
2012-12-01
We develop an all-electron path integral Monte Carlo (PIMC) method for warm dense matter and apply it to study water and carbon. PIMC pressures, internal energies, and pair-correlation functions compare well with density functional theory molecular dynamics (DFT-MD) at lower temperatures and enable the construction of a coherent equation of state over a density-temperature range of 3--12 g/cm3 and 102--109 K. PIMC results converge to the Debye-Huckel limiting law at high-temperatures and illuminate the breakdown of DFT pseudopotentials due to core excitations. Funding provided by the NSF (DMS-1025370). Computational resources provided by the National Center for Atmospheric Research and Lawrence Berkeley National Laboratory.
16. Torsional path integral Monte Carlo method for the quantum simulation of large molecules
Miller, Thomas F.; Clary, David C.
2002-05-01
A molecular application is introduced for calculating quantum statistical mechanical expectation values of large molecules at nonzero temperatures. The Torsional Path Integral Monte Carlo (TPIMC) technique applies an uncoupled winding number formalism to the torsional degrees of freedom in molecular systems. The internal energy of the molecules ethane, n-butane, n-octane, and enkephalin are calculated at standard temperature using the TPIMC technique and compared to the expectation values obtained using the harmonic oscillator approximation and a variational technique. All studied molecules exhibited significant quantum mechanical contributions to their internal energy expectation values according to the TPIMC technique. The harmonic oscillator approximation approach to calculating the internal energy performs well for the molecules presented in this study but is limited by its neglect of both anharmonicity effects and the potential coupling of intramolecular torsions.
17. Quantum Mechanical Single Molecule Partition Function from PathIntegral Monte Carlo Simulations
SciTech Connect
Chempath, Shaji; Bell, Alexis T.; Predescu, Cristian
2006-10-01
An algorithm for calculating the partition function of a molecule with the path integral Monte Carlo method is presented. Staged thermodynamic perturbation with respect to a reference harmonic potential is utilized to evaluate the ratio of partition functions. Parallel tempering and a new Monte Carlo estimator for the ratio of partition functions are implemented here to achieve well converged simulations that give an accuracy of 0.04 kcal/mol in the reported free energies. The method is applied to various test systems, including a catalytic system composed of 18 atoms. Absolute free energies calculated by this method lead to corrections as large as 2.6 kcal/mol at 300 K for some of the examples presented.
18. A PATH INTEGRAL FORMULATION OF THE WRIGHT-FISHER PROCESS WITH GENIC SELECTION
PubMed Central
SCHRAIBER, JOSHUA G.
2014-01-01
The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient. PMID:24269333
19. Euclidean path integral formalism in deformed space with minimum measurable length
Bernardo, Reginald Christian S.; Esguerra, Jose Perico H.
2017-04-01
We study time-evolution at the quantum level by developing the Euclidean path-integral approach for the general case where there exists a minimum measurable length. We derive an expression for the momentum-space propagator which turns out to be consistent with recently developed β-canonical transformation. We also construct the propagator for maximal localization which corresponds to the amplitude that a state which is maximally localized at location ξ' propagates to a state which is maximally localized at location ξ″ in a given time. Our expression for the momentum-space propagator and the propagator for maximal localization is valid for any form of time-independent Hamiltonian. The nonrelativistic free particle, particle in a linear potential, and the harmonic oscillator are discussed as examples.
20. Path integral centroid molecular dynamics simulation of para-hydrogen sandwiched by graphene sheets
Minamino, Yuki; Kinugawa, Kenichi
2016-11-01
The carbon-hydrogen composite systems of para-hydrogen (p-H2) sandwiched by a couple of graphene sheets have been investigated by means of path integral centroid molecular dynamics simulations at 17 K. It has been shown that sandwiched hydrogen is liquid-like but p-H2 molecules are preferably adsorbed onto the graphene sheets because of attractive graphene-hydrogen interaction. The diffusion coefficient of p-H2 molecules in the direction parallel to the graphene sheets is comparable to that in pure liquid p-H2. There exists a characteristic mode of 140 cm-1 of the p-H2 molecules, attributed to adsorption-binding motion perpendicular to the graphene sheets.
1. Path-integral Monte Carlo simulation of the warm dense homogeneous electron gas.
PubMed
Brown, Ethan W; Clark, Bryan K; DuBois, Jonathan L; Ceperley, David M
2013-04-05
We perform calculations of the 3D finite-temperature homogeneous electron gas in the warm-dense regime (r(s) ≡ (3/4πn)(1/3)a(0)(-1) = 1.0-40.0 and Θ ≡ T/T(F) = 0.0625-8.0) using restricted path-integral Monte Carlo simulations. Precise energies, pair correlation functions, and structure factors are obtained. For all densities, we find a significant discrepancy between the ground state parametrized local density approximation and our results around T(F). These results can be used as a benchmark for developing finite-temperature density functionals, as well as input for orbital-free density function theory formulations.
2. Excitonic effects in two-dimensional semiconductors: Path integral Monte Carlo approach
DOE PAGES
2015-11-01
The most striking features of novel two-dimensional semiconductors (e.g., transition metal dichalcogenide monolayers or phosphorene) is a strong Coulomb interaction between charge carriers resulting in large excitonic effects. In particular, this leads to the formation of multicarrier bound states upon photoexcitation (e.g., excitons, trions, and biexcitons), which could remain stable at near-room temperatures and contribute significantly to the optical properties of such materials. In our work we have used the path integral Monte Carlo methodology to numerically study properties of multicarrier bound states in two-dimensional semiconductors. Specifically, we have accurately investigated and tabulated the dependence of single-exciton, trion, and biexcitonmore » binding energies on the strength of dielectric screening, including the limiting cases of very strong and very weak screening. Our results of this work are potentially useful in the analysis of experimental data and benchmarking of theoretical and computational models.« less
3. Excitonic effects in two-dimensional semiconductors: Path integral Monte Carlo approach
SciTech Connect
2015-11-01
The most striking features of novel two-dimensional semiconductors (e.g., transition metal dichalcogenide monolayers or phosphorene) is a strong Coulomb interaction between charge carriers resulting in large excitonic effects. In particular, this leads to the formation of multicarrier bound states upon photoexcitation (e.g., excitons, trions, and biexcitons), which could remain stable at near-room temperatures and contribute significantly to the optical properties of such materials. In our work we have used the path integral Monte Carlo methodology to numerically study properties of multicarrier bound states in two-dimensional semiconductors. Specifically, we have accurately investigated and tabulated the dependence of single-exciton, trion, and biexciton binding energies on the strength of dielectric screening, including the limiting cases of very strong and very weak screening. Our results of this work are potentially useful in the analysis of experimental data and benchmarking of theoretical and computational models.
4. LIGHT SCATTERING: Fast path-integration technique in simulation of light propagation through highly scattering objects
Voronov, Aleksandr V.; Tret'yakov, Evgeniy V.; Shuvalov, Vladimir V.
2004-06-01
Based on the path-integration technique and the Metropolis method, the original calculation scheme is developed for solving the problem of light propagation through highly scattering objects. The elimination of calculations of 'unnecessary' realisations and the phenomenological description of processes of multiple small-angle scattering provided a drastic increase (by nine and more orders of magnitude) in the calculation rate, retaining the specific features of the problem (consideration of spatial inhomogeneities, boundary conditions, etc.). The scheme allows one to verify other fast calculation algorithms and to obtain information required to reconstruct the internal structure of highly scattering objects (of size ~1000 scattered lengths and more) by the method of diffusion optical tomography.
5. Path integration: how the head direction signal maintains and corrects spatial orientation
PubMed Central
Valerio, Stephane; Taube, Jeffrey S.
2012-01-01
Head direction (HD) cells have frequently been regarded as an internal “compass” that can be used for navigation, although there is little evidence showing a link between their activity and spatial behaviour. In a navigational task requiring the use of internal cues to return to a home base location without vision (path integration), we found a robust correlation between HD cell activity and the rat's heading error in their homing behaviour. Furthermore, we observed two different correction processes that animals used to improve performance after an error. The more frequent one consists of resetting' the cell whenever the animal returns to the home location. However, we found that when large errors occur the HD system has the ability to remap' and set a new reference frame, which is then used in subsequent trials. We also offer some insight into how these two correction processes operate when animals make an error. PMID:22983210
6. Time travel paradoxes, path integrals, and the many worlds interpretation of quantum mechanics
Everett, Allen
2004-06-01
We consider two approaches to evading paradoxes in quantum mechanics with closed timelike curves. In a model similar to Politzer’s, assuming pure states and using path integrals, we show that the problems of paradoxes and of unitarity violation are related; preserving unitarity avoids paradoxes by modifying the time evolution so that improbable events become certain. Deutsch has argued, using the density matrix, that paradoxes do not occur in the “many worlds interpretation.” We find that in this approach account must be taken of the resolution time of the device that detects objects emerging from a wormhole or other time machine. When this is done one finds that this approach is viable only if macroscopic objects traversing a wormhole interact with it so strongly that they are broken into microscopic fragments.
7. Semiclassical Path Integral Dynamics: Photosynthetic Energy Transfer with Realistic Environment Interactions
Lee, Mi Kyung; Huo, Pengfei; Coker, David F.
2016-05-01
This article reviews recent progress in the theoretical modeling of excitation energy transfer (EET) processes in natural light harvesting complexes. The iterative partial linearized density matrix path-integral propagation approach, which involves both forward and backward propagation of electronic degrees of freedom together with a linearized, short-time approximation for the nuclear degrees of freedom, provides an accurate and efficient way to model the nonadiabatic quantum dynamics at the heart of these EET processes. Combined with a recently developed chromophore-protein interaction model that incorporates both accurate ab initio descriptions of intracomplex vibrations and chromophore-protein interactions treated with atomistic detail, these simulation tools are beginning to unravel the detailed EET pathways and relaxation dynamics in light harvesting complexes.
8. WORM ALGORITHM PATH INTEGRAL MONTE CARLO APPLIED TO THE 3He-4He II SANDWICH SYSTEM
Al-Oqali, Amer; Sakhel, Asaad R.; Ghassib, Humam B.; Sakhel, Roger R.
2012-12-01
We present a numerical investigation of the thermal and structural properties of the 3He-4He sandwich system adsorbed on a graphite substrate using the worm algorithm path integral Monte Carlo (WAPIMC) method [M. Boninsegni, N. Prokof'ev and B. Svistunov, Phys. Rev. E74, 036701 (2006)]. For this purpose, we have modified a previously written WAPIMC code originally adapted for 4He on graphite, by including the second 3He-component. To describe the fermions, a temperature-dependent statistical potential has been used. This has proven very effective. The WAPIMC calculations have been conducted in the millikelvin temperature regime. However, because of the heavy computations involved, only 30, 40 and 50 mK have been considered for the time being. The pair correlations, Matsubara Green's function, structure factor, and density profiles have been explored at these temperatures.
9. Approximate path integral solution for a Dirac particle in a deformed Hulthén potential
Kadja, A.; Benamira, F.; Guechi, L.
2017-05-01
The problem of a Dirac particle moving in a deformed Hulthén potential is solved in the framework of the path integral formalism. With the help of the Biedenharn transformation, the construction of a closed form for the Green's function of the second-order Dirac equation is done by using a proper approximation to the centrifugal term and the Green's function of the linear Dirac equation is calculated. The energy spectrum for the bound states is obtained from the poles of the Green's function. A Dirac particle in the standard Hulthén potential ( q = 1) and a Dirac hydrogen-like ion ( q = 1 and a → ∞) are considered as particular cases.
10. Gaussian white noise analysis and its application to Feynman path integral
Suryawan, Herry Pribawanto
2016-02-01
In applied science, Gaussian white noise (the time derivative of Brownian motion) is often chosen as a mathematical idealization of phenomena involving sudden and extremely large fluctuations. It is also possible to define and study Gaussian white noise in a mathematically rigorous framework. In this survey paper we review the Gaussian white noise as an object in an infinite dimensional topological vector space. A brief construction of Gaussian white noise space and Gaussian white noise distributions will be presented. Gaussian white noise analysis provides a framework which offers various generalization of concept known from finite dimensional analysis to the infinite dimensional case, among them are differential operators, Fourier transform, and distribution theory. We will also present some recent developments and results on the application of Gaussian white noise theory to Feynman's path integral approach for quantum mechanics.
11. Path integration: how the head direction signal maintains and corrects spatial orientation.
PubMed
Valerio, Stephane; Taube, Jeffrey S
2012-10-01
Head-direction cells have frequently been regarded as an internal 'compass' that can be used for navigation, although there is little evidence showing a link between their activity and spatial behavior. In a navigational task requiring the use of internal cues to return to a home location without vision (path integration), we found a robust correlation between head-direction cell activity and the rat's heading error in the rat's homing behavior. We observed two different correction processes that rats used to improve performance after an error. The more frequent one consists of 'resetting' the cell whenever the rat returns to the home location. However, we found that when large errors occur, the head-direction system has the ability to 'remap' and set a new reference frame, which is then used in subsequent trials. We also offer some insight into how these two correction processes operate when rats make an error.
12. Proton transfer dynamics in the propionic acid dimer from path integral molecular dynamics calculations.
PubMed
Durlak, Piotr; Latajka, Zdzisław
2011-09-01
The double proton transfer process in the cyclic dimer of propionic acid in the gas phase was studied using a path integral molecular dynamics method. Structures, energies and proton trajectories were determined. Very large amplitude motions of the skeleton of a propionic acid molecule were observed during the simulations, and almost free rotation of the C(2)H(5) group around the C(α)-C bond. A double-well symmetric potential with a very small energy barrier was determined from the free energy profile for the proton motions. Infrared spectra for different isotopomers were calculated, and comparative vibrational analysis was performed. The vibrational results from CPMD appear to be in qualitative agreement with the experimental ones.
13. Systematic speedup of path integrals of a generic N-fold discretized theory
SciTech Connect
Bogojevic, A.; Balaz, A.; Belic, A.
2005-08-01
We present and discuss a detailed derivation of an analytical method that systematically improves the convergence of path integrals of a generic N-fold discretized theory. We develop an explicit procedure for calculating a set of effective actions S{sup (p)}, for p=1,2,3,... which have the property that they lead to the same continuum amplitudes as the starting action, but converge to that continuum limit ever faster. Discretized amplitudes calculated using the p-level effective action differ from the continuum limit by a term of order 1/N{sup p}. We obtain explicit expressions for the effective actions for levels p{<=}9. We end by analyzing the speedup of Monte Carlo simulations of two different models: an anharmonic oscillator with quartic coupling and a particle in a modified Poeschl-Teller potential.
14. A New Perspective on Path Integral Quantum Mechanics in Curved Space-Time
2013-09-01
Abstract. A new approach to path integral quantum mechanics in curved space-time is presented for scalar particle propagation, expressed in terms of Lie transport and Fermi or Riemann normal co-ordinates to describe local curvature. While the presence of local curvature results in a strictly non-unitary representation of local time translation, the formalism nevertheless correctly recovers the free-particle Lagrangian in curved space-time, along with new terms that predict a simultaneous breakdown of time-reversal symmetry and a quantum violation of the weak equivalence principle at the particle's Compton wavelength scale. Furthermore, the formalism reveals the prediction of a gauge-invariant phase factor interpreted as the gravitational Aharonov-Bohm effect and Berry's phase.
15. Path-integral action of a particle in the noncommutative plane.
PubMed
2009-06-19
Noncommutative quantum mechanics can be viewed as a quantum system represented in the space of Hilbert-Schmidt operators acting on noncommutative configuration space. Taking this as a departure point, we formulate a coherent state approach to the path-integral representation of the transition amplitude. From this we derive an action for a particle moving in the noncommutative plane and in the presence of an arbitrary potential. We find that this action is nonlocal in time. However, this nonlocality can be removed by introducing an auxilary field, which leads to a second class constrained system that yields the noncommutative Heisenberg algebra upon quantization. Using this action, the propagator of the free particle and harmonic oscillator are computed explicitly.
16. Unified path integral treatment for generalized Hulthen and Woods-Saxon potentials
SciTech Connect
Benamira, F.; Guechi, L. . E-mail: [email protected]; Mameri, S.; Sadoun, M.A.
2007-09-15
A rigorous path integral discussion of the s states for a diatomic molecule potential with varying shape, which generalizes the Hulthen and the Woods-Saxon potentials, is presented. A closed form of the Green's function is obtained for different shapes of this potential. For {lambda}>=1 and (1/{eta})ln{lambda}
17. Topics in mode conversion theory and the group theoretical foundations of path integrals
Richardson, Andrew Stephen
discrete Beisenberg-Wey1 group to construct the symbol of a matrix. We then go on to show how the path integral arises when calculating the symbol of a function of an operator. We also show how the phase space and configuration space path integrals arise when considering reductions of the regular representation of the Heisenberg-Wey1 group to the primary representations and irreducible representations, respectively. We also show how the path integral can be interpreted as a Fourier transform on the space of measures, opening up the possibility of using tools from statistical mechanics (such as maximum entropy techniques) to analyze the path integral. We conclude with a survey of ideas for future research and describe several potential applications of this group theoretical perspective to problems in mode conversion.
18. Quantum tautomerization in porphycene and its isotopomers: Path-integral molecular dynamics simulations
Yoshikawa, Takehiro; Sugawara, Shuichi; Takayanagi, Toshiyuki; Shiga, Motoyuki; Tachikawa, Masanori
2012-02-01
Path-integral molecular dynamics simulations have been performed for porphycene and its isotopic variants in order to understand the effect of isotopic substitution of inner protons on the double proton transfer mechanism. We have used an on-the-fly direct dynamics technique at the semiempirical PM6 level combined with specific reaction parameterization. Our quantum simulations show that double proton transfer of the unsubstituted porphycene at T = 300 K mainly occurs via a so-called concerted mechanism through the D2h second-order saddle point. In addition, we found that both isotopic substitution and temperature significantly affect the double proton transfer mechanism. For example, the contribution of the stepwise mechanism increases with a temperature increase. We have also carried out hypothetical simulations with the porphycene configurations being completely planar. It has been found that out-of-plane vibrational motions significantly decrease the contribution of the concerted proton transfer mechanism.
19. On the applicability of centroid and ring polymer path integral molecular dynamics for vibrational spectroscopy
Witt, Alexander; Ivanov, Sergei D.; Shiga, Motoyuki; Forbert, Harald; Marx, Dominik
2009-05-01
Centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) are two conceptually distinct extensions of path integral molecular dynamics that are able to generate approximate quantum dynamics of complex molecular systems. Both methods can be used to compute quasiclassical time correlation functions which have direct application in molecular spectroscopy; in particular, to infrared spectroscopy via dipole autocorrelation functions. The performance of both methods for computing vibrational spectra of several simple but representative molecular model systems is investigated systematically as a function of temperature and isotopic substitution. In this context both CMD and RPMD feature intrinsic problems which are quantified and investigated in detail. Based on the obtained results guidelines for using CMD and RPMD to compute infrared spectra of molecular systems are provided.
20. Geometric isotope effects on small chloride ion water clusters with path integral molecular dynamics simulations
Wang, Qi; Suzuki, Kimichi; Nagashima, Umpei; Tachikawa, Masanori; Yan, Shiwei
2013-11-01
The geometric isotope effects on the structures of hydrated chloride ionic hydrogen bonded clusters are explored by carrying out path integral molecular dynamics simulations. First, an outer shell coordinate is selected to display the rearrangement of single and multi hydration shell cluster structures. Next, to show the competition of intramolecular and intermolecular nuclear quantum effects, the intramolecular OH∗ stretching and intermolecular ion-water wagging motions are studied for single and multi shell structures, respectively. The results indicate that the intermolecular nuclear quantum effects stabilize the ionic hydrogen bonds in single shell structures, while they are destabilized through the competition with intramolecular nuclear quantum effects in multi shell structures. In addition, the correlations between ion-water stretching motion and other cluster vibrational coordinates are discussed. The results indicate that the intermolecular nuclear quantum effects on the cluster structures are strongly related to the cooperation of the water-water hydrogen bond interactions.
1. Determination of the experimental equilibrium structure of solid nitromethane using path-integral molecular dynamics simulations
Reilly, Anthony M.; Habershon, Scott; Morrison, Carole A.; Rankin, David W. H.
2010-03-01
Path-integral molecular dynamics (PIMD) simulations with an empirical interaction potential have been used to determine the experimental equilibrium structure of solid nitromethane at 4.2 and 15 K. By comparing the time-averaged molecular structure determined in a PIMD simulation to the calculated minimum-energy (zero-temperature) molecular structure, we have derived structural corrections that describe the effects of thermal motion. These corrections were subsequently used to determine the equilibrium structure of nitromethane from the experimental time-averaged structure. We find that the corrections to the intramolecular and intermolecular bond distances, as well as to the torsion angles, are quite significant, particularly for those atoms participating in the anharmonic motion of the methyl group. Our results demonstrate that simple harmonic models of thermal motion may not be sufficiently accurate, even at low temperatures, while molecular simulations employing more realistic potential-energy surfaces can provide important insight into the role and magnitude of anharmonic atomic motions.
2. Mapping variable ring polymer molecular dynamics: A path-integral based method for nonadiabatic processes
Ananth, Nandini
2013-09-01
We introduce mapping-variable ring polymer molecular dynamics (MV-RPMD), a model dynamics for the direct simulation of multi-electron processes. An extension of the RPMD idea, this method is based on an exact, imaginary time path-integral representation of the quantum Boltzmann operator using continuous Cartesian variables for both electronic states and nuclear degrees of freedom. We demonstrate the accuracy of the MV-RPMD approach in calculations of real-time, thermal correlation functions for a range of two-state single-mode model systems with different coupling strengths and asymmetries. Further, we show that the ensemble of classical trajectories employed in these simulations preserves the Boltzmann distribution and provides a direct probe into real-time coupling between electronic state transitions and nuclear dynamics.
3. Low-temperature metallic liquid hydrogen: an ab-initio path-integral molecular dynamics perspective
Chen, Ji; Li, Xin-Zheng; Zhang, Qianfan; Probert, Matthew; Pickard, Chris; Needs, Richard; Michaelides, Angelos; Wang, Enge
2013-03-01
Experiments and computer simulations have shown that the melting temperature of solid hydrogen drops with pressure above about 65 GPa, suggesting that a low temperature liquid state might exist. It has also been suggested that this liquid state might be non-molecular and metallic, although evidence for such behaviour is lacking. Using a combination of ab initio path-integral molecular dynamics and the two-phase methods, we have simulated the melting of solid hydrogen under finite temperatures. We found an atomic solid phase from 500 to 800 GPa which melts at < 200 K. Beyond this and up to pressures of 1,200 GPa a metallic atomic liquid is stable at temperatures as low as 50 K. The quantum motion of the protons is critical to the low melting temperature in this system as ab initio simulations with classical nuclei lead to a considerably higher melting temperature of ~300 K across the entire pressure range considered.
4. Hydrogen and muonium in diamond: A path-integral molecular dynamics simulation
Herrero, Carlos P.; Ramírez, Rafael; Hernández, Eduardo R.
2006-06-01
Isolated hydrogen, deuterium, and muonium in diamond have been studied by path-integral molecular dynamics simulations in the canonical ensemble. Finite-temperature properties of these point defects were analyzed in the range from 100 to 800K . Interatomic interactions were modeled by a tight-binding potential fitted to density-functional calculations. The most stable position for these hydrogenic impurities is found at the C-C bond center. Vibrational frequencies have been obtained from a linear-response approach, based on correlations of atom displacements at finite temperatures. The results show a large anharmonic effect in impurity vibrations at the bond center site, which hardens the vibrational modes with respect to a harmonic approximation. Zero-point motion causes an appreciable shift of the defect level in the electronic gap, as a consequence of electron-phonon interaction. This defect level goes down by 70meV when replacing hydrogen by muonium.
5. Low-temperature anharmonicity of barium titanate: A path-integral molecular-dynamics study
Geneste, Grégory; Dammak, Hichem; Hayoun, Marc; Thiercelin, Mickael
2013-01-01
We investigate the influence of quantum effects on the dielectric and piezoelectric properties of barium titanate in its (low-temperature) rhombohedral phase, and show the strongly anharmonic character of this system even at low temperature. For this purpose, we perform path-integral molecular-dynamics simulations under fixed pressure and fixed temperature, using an efficient Langevin thermostat-barostat, and an effective Hamiltonian derived from first-principles calculations. The quantum fluctuations are shown to significantly enhance the static dielectric susceptibility (≈ by a factor of 2) and the piezoelectric constants, reflecting the strong anharmonicity of this ferroelectric system even at very low temperature. The slow temperature-evolution of the dielectric properties observed below ≈100 K is attributed (i) to zero-point energy contributions and (ii) to harmonic behavior if the quantum effects are turned off.
6. Isotope effects in water as investigated by neutron diffraction and path integral molecular dynamics
Zeidler, Anita; Salmon, Philip S.; Fischer, Henry E.; Neuefeind, Jörg C.; Simonson, J. Mike; Markland, Thomas E.
2012-07-01
The structures of heavy and light water at 300 K were investigated by using a joint approach in which the method of neutron diffraction with oxygen isotope substitution was complemented by path integral molecular dynamics simulations. The diffraction results, which give intra-molecular O-D and O-H bond distances of 0.985(5) and 0.990(5) Å, were found to be in best agreement with those obtained by using the flexible anharmonic TTM3-F water model. Both techniques show a difference of ≃ 0.5% between the O-D and O-H intra-molecular bond lengths, and the results support a competing quantum effects model for water in which its structural and dynamical properties are governed by an offset between intra-molecular and inter-molecular quantum contributions. Further consideration of the O-O correlations is needed in order to improve agreement with experiment.
7. Path Integral Monte Carlo and Density Functional Molecular Dynamics Simulations of Warm Dense Matter
Militzer, Burkhard; Driver, Kevin
2011-10-01
We analyze the applicability of two first-principles simulation techniques, path integral Monte Carlo (PIMC) and density functional molecular dynamics (DFT-MD), to study the regime of warm dense matter. We discuss the advantages as well as the limitations of each method and propose directions for future development. Results for dense, liquid helium, where both methods have been applied, demonstrate the range of each method's applicability. Comparison of the equations of state from simulations with analytical theories and free energy models show that DFT is useful for temperatures below 100000 K and then PIMC provides accurate results for all higher temperatures. We characterize the structure of the liquid in terms of pair correlation functions and study the closure of the band gap with increasing density and temperature. Finally, we discuss simulations of heavier elements and demonstrate the reliability are both methods in such cases with preliminary results.
8. Airborne 2-Micron Double-Pulsed Integrated Path Differential Absorption Lidar for Column CO2 Measurement
NASA Technical Reports Server (NTRS)
Singh, Upendra N.; Yu, Jirong; Petros, Mulugeta; Refaat, Tamer F.; Remus, Ruben G.; Fay, James J.; Reithmaier, Karl
2014-01-01
Double-pulse 2-micron lasers have been demonstrated with energy as high as 600 millijouls and up to 10 Hz repetition rate. The two laser pulses are separated by 200 microseconds and can be tuned and locked separately. Applying double-pulse laser in DIAL system enhances the CO2 measurement capability by increasing the overlap of the sampled volume between the on-line and off-line. To avoid detection complicity, integrated path differential absorption (IPDA) lidar provides higher signal-to-noise ratio measurement compared to conventional range-resolved DIAL. Rather than weak atmospheric scattering returns, IPDA rely on the much stronger hard target returns that is best suited for airborne platforms. In addition, the IPDA technique measures the total integrated column content from the instrument to the hard target but with weighting that can be tuned by the transmitter. Therefore, the transmitter could be tuned to weight the column measurement to the surface for optimum CO2 interaction studies or up to the free troposphere for optimum transport studies. Currently, NASA LaRC is developing and integrating a double-Pulsed 2-micron direct detection IPDA lidar for CO2 column measurement from an airborne platform. The presentation will describe the development of the 2-micron IPDA lidar system and present the airborne measurement of column CO2 and will compare to in-situ measurement for various ground target of different reflectivity.
9. Airborne 2-micron double-pulsed integrated path differential absorption lidar for column CO2 measurement
Singh, Upendra N.; Yu, Jirong; Petros, Mulugeta; Refaat, Tamer F.; Remus, Ruben G.; Fay, James J.; Reithmaier, Karl
2014-10-01
Double-pulse 2-micron lasers have been demonstrated with energy as high as 600 mJ and up to 10 Hz repetition rate. The two laser pulses are separated by 200 µs and can be tuned and locked separately. Applying double-pulse laser in DIAL system enhances the CO2 measurement capability by increasing the overlap of the sampled volume between the on-line and off-line. To avoid detection complicity, integrated path differential absorption (IPDA) lidar provides higher signal-to-noise ratio measurement compared to conventional range-resolved DIAL. Rather than weak atmospheric scattering returns, IPDA rely on the much stronger hard target returns that is best suited for airborne platforms. In addition, the IPDA technique measures the total integrated column content from the instrument to the hard target but with weighting that can be tuned by the transmitter. Therefore, the transmitter could be tuned to weight the column measurement to the surface for optimum CO2 interaction studies or up to the free troposphere for optimum transport studies. Currently, NASA LaRC is developing and integrating a double-Pulsed 2-µm direct detection IPDA lidar for CO2 column measurement from an airborne platform. The presentation will describe the development of the 2-μm IPDA lidar system and present the airborne measurement of column CO2 and will compare to in-situ measurement for various ground target of different reflectivity.
10. Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature.
PubMed
Filinov, V S; Fortov, V E; Bonitz, M; Moldabekov, Zh
2015-03-01
The uniform electron gas (UEG) at finite temperature has recently attracted substantial interest due to the experimental progress in the field of warm dense matter. To explain the experimental data, accurate theoretical models for high-density plasmas are needed that depend crucially on the quality of the thermodynamic properties of the quantum degenerate nonideal electrons and of the treatment of their interaction with the positive background. Recent fixed-node path-integral Monte Carlo (RPIMC) data are believed to be the most accurate for the UEG at finite temperature, but they become questionable at high degeneracy when the Brueckner parameter rs=a/aB--the ratio of the mean interparticle distance to the Bohr radius--approaches 1. The validity range of these simulations and their predictive capabilities for the UEG are presently unknown. This is due to the unknown quality of the used fixed nodes and of the finite-size scaling from N=33 simulated particles (per spin projection) to the macroscopic limit. To analyze these questions, we present alternative direct fermionic path integral Monte Carlo (DPIMC) simulations that are independent from RPIMC. Our simulations take into account quantum effects not only in the electron system but also in their interaction with the uniform positive background. Also, we use substantially larger particle numbers (up to three times more) and perform an extrapolation to the macroscopic limit. We observe very good agreement with RPIMC, for the polarized electron gas, up to moderate densities around rs=4, and larger deviations for the unpolarized case, for low temperatures. For higher densities (high electron degeneracy), rs≲1.5, both RPIMC and DPIMC are problematic due to the increased fermion sign problem.
11. Maintaining a Cognitive Map in Darkness: The Need to Fuse Boundary Knowledge with Path Integration
PubMed Central
Cheung, Allen; Ball, David; Milford, Michael; Wyeth, Gordon; Wiles, Janet
2012-01-01
Spatial navigation requires the processing of complex, disparate and often ambiguous sensory data. The neurocomputations underpinning this vital ability remain poorly understood. Controversy remains as to whether multimodal sensory information must be combined into a unified representation, consistent with Tolman's “cognitive map”, or whether differential activation of independent navigation modules suffice to explain observed navigation behaviour. Here we demonstrate that key neural correlates of spatial navigation in darkness cannot be explained if the path integration system acted independently of boundary (landmark) information. In vivo recordings demonstrate that the rodent head direction (HD) system becomes unstable within three minutes without vision. In contrast, rodents maintain stable place fields and grid fields for over half an hour without vision. Using a simple HD error model, we show analytically that idiothetic path integration (iPI) alone cannot be used to maintain any stable place representation beyond two to three minutes. We then use a measure of place stability based on information theoretic principles to prove that featureless boundaries alone cannot be used to improve localization above chance level. Having shown that neither iPI nor boundaries alone are sufficient, we then address the question of whether their combination is sufficient and – we conjecture – necessary to maintain place stability for prolonged periods without vision. We addressed this question in simulations and robot experiments using a navigation model comprising of a particle filter and boundary map. The model replicates published experimental results on place field and grid field stability without vision, and makes testable predictions including place field splitting and grid field rescaling if the true arena geometry differs from the acquired boundary map. We discuss our findings in light of current theories of animal navigation and neuronal computation, and
12. A transformed path integral approach for solution of the Fokker-Planck equation
Subramaniam, Gnana M.; Vedula, Prakash
2017-10-01
A novel path integral (PI) based method for solution of the Fokker-Planck equation is presented. The proposed method, termed the transformed path integral (TPI) method, utilizes a new formulation for the underlying short-time propagator to perform the evolution of the probability density function (PDF) in a transformed computational domain where a more accurate representation of the PDF can be ensured. The new formulation, based on a dynamic transformation of the original state space with the statistics of the PDF as parameters, preserves the non-negativity of the PDF and incorporates short-time properties of the underlying stochastic process. New update equations for the state PDF in a transformed space and the parameters of the transformation (including mean and covariance) that better accommodate nonlinearities in drift and non-Gaussian behavior in distributions are proposed (based on properties of the SDE). Owing to the choice of transformation considered, the proposed method maps a fixed grid in transformed space to a dynamically adaptive grid in the original state space. The TPI method, in contrast to conventional methods such as Monte Carlo simulations and fixed grid approaches, is able to better represent the distributions (especially the tail information) and better address challenges in processes with large diffusion, large drift and large concentration of PDF. Additionally, in the proposed TPI method, error bounds on the probability in the computational domain can be obtained using the Chebyshev's inequality. The benefits of the TPI method over conventional methods are illustrated through simulations of linear and nonlinear drift processes in one-dimensional and multidimensional state spaces. The effects of spatial and temporal grid resolutions as well as that of the diffusion coefficient on the error in the PDF are also characterized.
13. Maintaining a cognitive map in darkness: the need to fuse boundary knowledge with path integration.
PubMed
Cheung, Allen; Ball, David; Milford, Michael; Wyeth, Gordon; Wiles, Janet
2012-01-01
Spatial navigation requires the processing of complex, disparate and often ambiguous sensory data. The neurocomputations underpinning this vital ability remain poorly understood. Controversy remains as to whether multimodal sensory information must be combined into a unified representation, consistent with Tolman's "cognitive map", or whether differential activation of independent navigation modules suffice to explain observed navigation behaviour. Here we demonstrate that key neural correlates of spatial navigation in darkness cannot be explained if the path integration system acted independently of boundary (landmark) information. In vivo recordings demonstrate that the rodent head direction (HD) system becomes unstable within three minutes without vision. In contrast, rodents maintain stable place fields and grid fields for over half an hour without vision. Using a simple HD error model, we show analytically that idiothetic path integration (iPI) alone cannot be used to maintain any stable place representation beyond two to three minutes. We then use a measure of place stability based on information theoretic principles to prove that featureless boundaries alone cannot be used to improve localization above chance level. Having shown that neither iPI nor boundaries alone are sufficient, we then address the question of whether their combination is sufficient and--we conjecture--necessary to maintain place stability for prolonged periods without vision. We addressed this question in simulations and robot experiments using a navigation model comprising of a particle filter and boundary map. The model replicates published experimental results on place field and grid field stability without vision, and makes testable predictions including place field splitting and grid field rescaling if the true arena geometry differs from the acquired boundary map. We discuss our findings in light of current theories of animal navigation and neuronal computation, and elaborate on
14. Effect of (+)-Methamphetamine on Path Integration Learning, Novel Object Recognition, and Neurotoxicity in Rats
PubMed Central
Herring, Nicole R.; Schaefer, Tori L.; Gudelsky, Gary A.; Vorhees, Charles V.; Williams, Michael T.
2008-01-01
Rationale Methamphetamine (MA) has been implicated in cognitive deficits in humans after chronic use. Animal models of neurotoxic MA exposure reveal persistent damage to monoaminergic systems, but few associated cognitive effects. Objectives Since, questions have been raised about the typical neurotoxic dosing regimen used in animals and whether it adequately models human cumulative drug exposure, these experiments examined two different dosing regimens. Methods Rats were treated with one of two regimens, one the typical neurotoxic regimen (4 × 10 mg/kg every 2 h) and one based on pharmacokinetic modeling (Cho et al. 2001) designed to better represent accumulating plasma concentrations of MA as seen in human users (24 ×1.67 mg/kg once every 15 min); matched for total daily dose. In two separate experiments, dosing regimens were compared for their effects on markers of neurotoxicity or on behavior. Results On markers of neurotoxicity, MA showed decreased DA and 5-HT, and increased glial fibrillary acidic protein and increased corticosterone levels regardless of dosing regimen 3 days post-treatment. Behaviorally, MA-treated groups, regardless of dosing regimen, showed hypoactivity, increased initial hyperactivity to a subsequent MA challenge, impaired novel object recognition, impaired learning in a multiple-T water maze test of path integration, and no differences on spatial navigation or reference memory in the Morris water maze. After behavioral testing, reductions of DA and 5-HT remained. Conclusions MA treatment induces an effect on path integration learning not previously reported. Dosing regimen had no differential effects on behavior or neurotoxicity. PMID:18509623
15. Short-time asymptotics of a rigorous path integral for N = 1 supersymmetric quantum mechanics on a Riemannian manifold
SciTech Connect
Fine, Dana S.; Sawin, Stephen
2014-06-15
Following Feynman's prescription for constructing a path integral representation of the propagator of a quantum theory, a short-time approximation to the propagator for imaginary-time, N = 1 supersymmetric quantum mechanics on a compact, even-dimensional Riemannian manifold is constructed. The path integral is interpreted as the limit of products, determined by a partition of a finite time interval, of this approximate propagator. The limit under refinements of the partition is shown to converge uniformly to the heat kernel for the Laplace-de Rham operator on forms. A version of the steepest descent approximation to the path integral is obtained, and shown to give the expected short-time behavior of the supertrace of the heat kernel.
16. TOWARD EFFICIENT RIPARIAN RESTORATION: INTEGRATING ECONOMIC, PHYSICAL, AND BIOLOGICAL MODELS
EPA Science Inventory
This paper integrates economic, biological, and physical models to determine the efficient combination and spatial allocation of conservation efforts for water quality protection and salmonid habitat enhancement in the Grande Ronde basin, Oregon. The integrated modeling system co...
17. TOWARD EFFICIENT RIPARIAN RESTORATION: INTEGRATING ECONOMIC, PHYSICAL, AND BIOLOGICAL MODELS
EPA Science Inventory
This paper integrates economic, biological, and physical models to determine the efficient combination and spatial allocation of conservation efforts for water quality protection and salmonid habitat enhancement in the Grande Ronde basin, Oregon. The integrated modeling system co...
18. Movement and Learning: Integrating Physical Activity into the Classroom
ERIC Educational Resources Information Center
Reeves, Emily; Miller, Stacia; Chavez, Crystal
2016-01-01
We know the benefits of physical activity, and yet recess and physical education classes are being cut or scaled back to make room for meeting academic standards. Is cutting recess and physical education really benefiting academics? A look at some recent studies suggests that it is not. Integrating physical activity into the classroom may increase…
19. Movement and Learning: Integrating Physical Activity into the Classroom
ERIC Educational Resources Information Center
Reeves, Emily; Miller, Stacia; Chavez, Crystal
2016-01-01
We know the benefits of physical activity, and yet recess and physical education classes are being cut or scaled back to make room for meeting academic standards. Is cutting recess and physical education really benefiting academics? A look at some recent studies suggests that it is not. Integrating physical activity into the classroom may increase…
20. Dogs, Cats, and Kids: Integrating Yoga into Elementary Physical Education
ERIC Educational Resources Information Center
Toscano, Lisa; Clemente, Fran
2008-01-01
This article describes the benefits of integrating yoga into elementary physical education classes. Taught as warm-up exercises or as an entire class, yoga offers children of any age and physical ability the opportunity to experience success in physical activity. Children need to experience joy while participating in physical activity in order to…
1. Dogs, Cats, and Kids: Integrating Yoga into Elementary Physical Education
ERIC Educational Resources Information Center
Toscano, Lisa; Clemente, Fran
2008-01-01
This article describes the benefits of integrating yoga into elementary physical education classes. Taught as warm-up exercises or as an entire class, yoga offers children of any age and physical ability the opportunity to experience success in physical activity. Children need to experience joy while participating in physical activity in order to…
2. A new approach to calculate charge carrier transport mobility in organic molecular crystals from imaginary time path integral simulations
SciTech Connect
Song, Linze; Shi, Qiang
2015-05-07
We present a new non-perturbative method to calculate the charge carrier mobility using the imaginary time path integral approach, which is based on the Kubo formula for the conductivity, and a saddle point approximation to perform the analytic continuation. The new method is first tested using a benchmark calculation from the numerical exact hierarchical equations of motion method. Imaginary time path integral Monte Carlo simulations are then performed to explore the temperature dependence of charge carrier delocalization and mobility in organic molecular crystals (OMCs) within the Holstein and Holstein-Peierls models. The effects of nonlocal electron-phonon interaction on mobility in different charge transport regimes are also investigated.
3. A new approach to calculate charge carrier transport mobility in organic molecular crystals from imaginary time path integral simulations.
PubMed
Song, Linze; Shi, Qiang
2015-05-07
We present a new non-perturbative method to calculate the charge carrier mobility using the imaginary time path integral approach, which is based on the Kubo formula for the conductivity, and a saddle point approximation to perform the analytic continuation. The new method is first tested using a benchmark calculation from the numerical exact hierarchical equations of motion method. Imaginary time path integral Monte Carlo simulations are then performed to explore the temperature dependence of charge carrier delocalization and mobility in organic molecular crystals (OMCs) within the Holstein and Holstein-Peierls models. The effects of nonlocal electron-phonon interaction on mobility in different charge transport regimes are also investigated.
4. Path integration of head direction: updating a packet of neural activity at the correct speed using neuronal time constants.
PubMed
Walters, D M; Stringer, S M
2010-07-01
A key question in understanding the neural basis of path integration is how individual, spatially responsive, neurons may self-organize into networks that can, through learning, integrate velocity signals to update a continuous representation of location within an environment. It is of vital importance that this internal representation of position is updated at the correct speed, and in real time, to accurately reflect the motion of the animal. In this article, we present a biologically plausible model of velocity path integration of head direction that can solve this problem using neuronal time constants to effect natural time delays, over which associations can be learned through associative Hebbian learning rules. The model comprises a linked continuous attractor network and competitive network. In simulation, we show that the same model is able to learn two different speeds of rotation when implemented with two different values for the time constant, and without the need to alter any other model parameters. The proposed model could be extended to path integration of place in the environment, and path integration of spatial view.
5. Triple-Pulsed Two-Micron Integrated Path Differential Absorption Lidar: A New Active Remote Sensing Capability with Path to Space
NASA Technical Reports Server (NTRS)
Singh, Upendra N.; Refaat, Tamer F.; Petros, Mulugeta; Yu, Jirong
2015-01-01
The two-micron wavelength is suitable for monitoring atmospheric water vapor and carbon dioxide, the two most dominant greenhouse gases. Recent advances in 2-micron laser technology paved the way for constructing state-of-the-art lidar transmitters for active remote sensing applications. In this paper, a new triple-pulsed 2-micron integrated path differential absorption lidar is presented. This lidar is capable of measuring either two species or single specie with two different weighting functions, simultaneously and independently. Development of this instrument is conducted at NASA Langley Research Center. Instrument scaling for projected future space missions will be discussed.
6. Effect of surface corrugation on low temperature phases of adsorbed (p-H2)7: A quantum path integral Monte Carlo study
Cruz, Anthony; López, Gustavo E.
2014-04-01
By using path integral Monte Carlo simulations coupled to Replica Exchange algorithms, various phases of (p-H2)7 physically adsorbed on a model graphite surface were identified at low temperatures. At T=0.5 K, the expected superfluid phase was observed for flat and slightly corrugated surfaces. At intermediate and high corrugations, a "supersolid" phase in C7/16 registry and a solid phase in C1/3 registry were observed, respectively. At higher temperatures, the superfluid is converted to a fluid and the "supersolid" to a solid.
7. PathText: a text mining integrator for biological pathway visualizations
PubMed Central
Kemper, Brian; Matsuzaki, Takuya; Matsuoka, Yukiko; Tsuruoka, Yoshimasa; Kitano, Hiroaki; Ananiadou, Sophia; Tsujii, Jun'ichi
2010-01-01
Motivation: Metabolic and signaling pathways are an increasingly important part of organizing knowledge in systems biology. They serve to integrate collective interpretations of facts scattered throughout literature. Biologists construct a pathway by reading a large number of articles and interpreting them as a consistent network, but most of the models constructed currently lack direct links to those articles. Biologists who want to check the original articles have to spend substantial amounts of time to collect relevant articles and identify the sections relevant to the pathway. Furthermore, with the scientific literature expanding by several thousand papers per week, keeping a model relevant requires a continuous curation effort. In this article, we present a system designed to integrate a pathway visualizer, text mining systems and annotation tools into a seamless environment. This will enable biologists to freely move between parts of a pathway and relevant sections of articles, as well as identify relevant papers from large text bases. The system, PathText, is developed by Systems Biology Institute, Okinawa Institute of Science and Technology, National Centre for Text Mining (University of Manchester) and the University of Tokyo, and is being used by groups of biologists from these locations. Contact: [email protected]. PMID:20529930
8. Ab initio path-integral molecular dynamics and the quantum nature of hydrogen bonds
Yexin, Feng; Ji, Chen; Xin-Zheng, Li; Enge, Wang
2016-01-01
The hydrogen bond (HB) is an important type of intermolecular interaction, which is generally weak, ubiquitous, and essential to life on earth. The small mass of hydrogen means that many properties of HBs are quantum mechanical in nature. In recent years, because of the development of computer simulation methods and computational power, the influence of nuclear quantum effects (NQEs) on the structural and energetic properties of some hydrogen bonded systems has been intensively studied. Here, we present a review of these studies by focussing on the explanation of the principles underlying the simulation methods, i.e., the ab initio path-integral molecular dynamics. Its extension in combination with the thermodynamic integration method for the calculation of free energies will also be introduced. We use two examples to show how this influence of NQEs in realistic systems is simulated in practice. Project supported by the National Natural Science Foundation of China (Grant Nos. 11275008, 91021007, and 10974012) and the China Postdoctoral Science Foundation (Grant No. 2014M550005).
9. PathText: a text mining integrator for biological pathway visualizations.
PubMed
Kemper, Brian; Matsuzaki, Takuya; Matsuoka, Yukiko; Tsuruoka, Yoshimasa; Kitano, Hiroaki; Ananiadou, Sophia; Tsujii, Jun'ichi
2010-06-15
Metabolic and signaling pathways are an increasingly important part of organizing knowledge in systems biology. They serve to integrate collective interpretations of facts scattered throughout literature. Biologists construct a pathway by reading a large number of articles and interpreting them as a consistent network, but most of the models constructed currently lack direct links to those articles. Biologists who want to check the original articles have to spend substantial amounts of time to collect relevant articles and identify the sections relevant to the pathway. Furthermore, with the scientific literature expanding by several thousand papers per week, keeping a model relevant requires a continuous curation effort. In this article, we present a system designed to integrate a pathway visualizer, text mining systems and annotation tools into a seamless environment. This will enable biologists to freely move between parts of a pathway and relevant sections of articles, as well as identify relevant papers from large text bases. The system, PathText, is developed by Systems Biology Institute, Okinawa Institute of Science and Technology, National Centre for Text Mining (University of Manchester) and the University of Tokyo, and is being used by groups of biologists from these locations.
10. A reliable acoustic path: Physical properties and a source localization method
Duan, Rui; Yang, Kun-De; Ma, Yuan-Liang; Lei, Bo
2012-12-01
The physical properties of a reliable acoustic path (RAP) are analysed and subsequently a weighted-subspace-fitting matched field (WSF-MF) method for passive localization is presented by exploiting the properties of the RAP environment. The RAP is an important acoustic duct in the deep ocean, which occurs when the receiver is placed near the bottom where the sound velocity exceeds the maximum sound velocity in the vicinity of the surface. It is found that in the RAP environment the transmission loss is rather low and no blind zone of surveillance exists in a medium range. The ray theory is used to explain these phenomena. Furthermore, the analysis of the arrival structures shows that the source localization method based on arrival angle is feasible in this environment. However, the conventional methods suffer from the complicated and inaccurate estimation of the arrival angle. In this paper, a straightforward WSF-MF method is derived to exploit the information about the arrival angles indirectly. The method is to minimize the distance between the signal subspace and the spanned space by the array manifold in a finite range-depth space rather than the arrival-angle space. Simulations are performed to demonstrate the features of the method, and the results are explained by the arrival structures in the RAP environment.
11. The Sensitivity of Sediment Path-Lengths to Channel Morphology: Results from Physical Models of Braided Rivers
Kasprak, A.; Ashmore, P.; Peirce, S.; Wheaton, J. M.
2013-12-01
The travel distances of coarse sediment in gravel-bed rivers during floods, or their path-lengths, exhibit strong dependencies on the arrangement of geomorphic units they are sourced from, routed through or around, and finally deposited on. Despite previous research on both braiding and single-thread meandering streams, a coherent rule set which relates particle path lengths to morphology remains elusive; such a rule set has the potential to vastly simplify models which seek to predict sediment transport or morphodynamics in these channels. Here we seek to understand the sensitivity of particle path-length distributions to morphology via a physical model of a braided stream, by using fluorescent tracer particles to track sediment path-lengths. These flume experiments provide a rich dataset composed of before-and-after bed photogrammetry, along with high-resolution photography and high-speed video documentation of particle deposition and mobility. Initial analyses indicate a strong coupling between particle path-length and the spatial arrangement of in-channel geomorphic units, with bar heads and point bars frequently acting as sink locations for tracer particles. This is interesting as it suggests that morphodynamics are in part contingent on morphology with strong positive feedbacks on hydraulics, deposition and negative feedbacks on path-length. Several mechanisms of braiding were captured in the simulations as erosional source processes for the tracers, including chute cutoff of point bars, bank erosion, channel incision, and bar edge trimming. These results may help inform the development of morphodynamic models for braided rivers which rely on particle path-lengths to simplify sediment transport algorithms, and such relations between channel morphology and path-length hold the potential to benefit numerous other modeling efforts, as well as provide a heuristic framework with which to understand fluvial morphodynamics.
12. Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space
Gomes, Henrique
2017-09-01
In an accompanying paper Gomes (arXiv:1504.02818, 2015), we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space Q of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, M-O, if the projection of extremal curves on Q onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to e^{i\\hat{H}t}circ prO= prOcirc e^{i\\hat{H}t}, where prO:Q→ Q_O^{partial O} is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.
13. Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space
Gomes, Henrique
2017-06-01
In an accompanying paper Gomes (arXiv:1504.02818, 2015), we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe M, existing on timeless configuration space Q of some field over M. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, M-O , if the projection of extremal curves on Q onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to e^{i\\hat{H}t}° prO= prO° e^{i\\hat{H}t} , where prO:Q→ Q_O^{partial O} is a linear projection. This criterion for locality can be made approximate—an impossible feat had it been already postulated—and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.
14. Review of computer simulations of isotope effects on biochemical reactions: From the Bigeleisen equation to Feynman's path integral.
PubMed
Wong, Kin-Yiu; Xu, Yuqing; Xu, Liang
2015-11-01
Enzymatic reactions are integral components in many biological functions and malfunctions. The iconic structure of each reaction path for elucidating the reaction mechanism in details is the molecular structure of the rate-limiting transition state (RLTS). But RLTS is very hard to get caught or to get visualized by experimentalists. In spite of the lack of explicit molecular structure of the RLTS in experiment, we still can trace out the RLTS unique "fingerprints" by measuring the isotope effects on the reaction rate. This set of "fingerprints" is considered as a most direct probe of RLTS. By contrast, for computer simulations, oftentimes molecular structures of a number of TS can be precisely visualized on computer screen, however, theoreticians are not sure which TS is the actual rate-limiting one. As a result, this is an excellent stage setting for a perfect "marriage" between experiment and theory for determining the structure of RLTS, along with the reaction mechanism, i.e., experimentalists are responsible for "fingerprinting", whereas theoreticians are responsible for providing candidates that match the "fingerprints". In this Review, the origin of isotope effects on a chemical reaction is discussed from the perspectives of classical and quantum worlds, respectively (e.g., the origins of the inverse kinetic isotope effects and all the equilibrium isotope effects are purely from quantum). The conventional Bigeleisen equation for isotope effect calculations, as well as its refined version in the framework of Feynman's path integral and Kleinert's variational perturbation (KP) theory for systematically incorporating anharmonicity and (non-parabolic) quantum tunneling, are also presented. In addition, the outstanding interplay between theory and experiment for successfully deducing the RLTS structures and the reaction mechanisms is demonstrated by applications on biochemical reactions, namely models of bacterial squalene-to-hopene polycyclization and RNA 2'-O
15. Derivation of the Schrodinger Equation from the Hamilton-Jacobi Equation in Feynman's Path Integral Formulation of Quantum Mechanics
ERIC Educational Resources Information Center
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
16. White Noise Path Integral Treatment of a Two-dimensional Dirac Oscillator in a Uniform Magnetic Field
SciTech Connect
Bastatas, Lyndon D.; Bornales, Jinky B.
2008-06-18
White noise path integral prescription is applied to solve the Dirac equation for a two-dimensional Dirac oscillator in a uniform magnetic field. The energy spectrum obtained agrees with the result obtained by Villalba and Maggiolo using the differential approach.
17. Derivation of the Schrodinger Equation from the Hamilton-Jacobi Equation in Feynman's Path Integral Formulation of Quantum Mechanics
ERIC Educational Resources Information Center
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
18. PLANE-INTEGRATED OPEN-PATH FOURIER TRANSFORM INFRARED SPECTROMETRY METHODOLOGY FOR ANAEROBIC SWINE LAGOON EMISSION MEASUREMENTS
EPA Science Inventory
Emissions of ammonia and methane from an anaerobic lagoon at a swine animal feeding operation were evaluated five times over a period of two years. The plane-integrated (PI) open-path Fourier transform infrared spectrometry (OP-FTIR) methodology was used to transect the plume at ...
19. Creativity, Spirituality, and Transcendence: Paths to Integrity and Wisdom in the Mature Self. Publications in Creativity Research.
ERIC Educational Resources Information Center
Miller, Melvin E., Ed.; Cook-Greuter, Susanne R., Ed.
This book contains 11 papers on creativity, spirituality, and transcendence as paths to integrity and wisdom in the mature self. The book begins with the paper "Introduction--Creativity in Adulthood: Personal Maturity and Openness to Extraordinary Sources of Inspiration" (Susanne R. Cook-Greuter, Melvin E. Miller). The next four papers,…
20. Architectural constraints are a major factor reducing path integration accuracy in the rat head direction cell system.
PubMed
Page, Hector J I; Walters, Daniel; Stringer, Simon M
2015-01-01
Head direction cells fire to signal the direction in which an animal's head is pointing. They are able to track head direction using only internally-derived information (path integration)In this simulation study we investigate the factors that affect path integration accuracy. Specifically, two major limiting factors are identified: rise time, the time after stimulation it takes for a neuron to start firing, and the presence of symmetric non-offset within-layer recurrent collateral connectivity. On the basis of the latter, the important prediction is made that head direction cell regions directly involved in path integration will not contain this type of connectivity; giving a theoretical explanation for architectural observations. Increased neuronal rise time is found to slow path integration, and the slowing effect for a given rise time is found to be more severe in the context of short conduction delays. Further work is suggested on the basis of our findings, which represent a valuable contribution to understanding of the head direction cell system.
1. Architectural constraints are a major factor reducing path integration accuracy in the rat head direction cell system
PubMed Central
Page, Hector J. I.; Walters, Daniel; Stringer, Simon M.
2015-01-01
Head direction cells fire to signal the direction in which an animal's head is pointing. They are able to track head direction using only internally-derived information (path integration)In this simulation study we investigate the factors that affect path integration accuracy. Specifically, two major limiting factors are identified: rise time, the time after stimulation it takes for a neuron to start firing, and the presence of symmetric non-offset within-layer recurrent collateral connectivity. On the basis of the latter, the important prediction is made that head direction cell regions directly involved in path integration will not contain this type of connectivity; giving a theoretical explanation for architectural observations. Increased neuronal rise time is found to slow path integration, and the slowing effect for a given rise time is found to be more severe in the context of short conduction delays. Further work is suggested on the basis of our findings, which represent a valuable contribution to understanding of the head direction cell system. PMID:25705190
2. Atomic orbitals of the nonrelativistic hydrogen atom in a four-dimensional Riemann space through the path integral formalism
SciTech Connect
Grinberg, H.; Maranon, J.; Vucetich, H.
1983-01-15
The Kustaanheimo--Stiefel transformation together with the well-known expansion of the kernel of an isotropic harmonic oscillator is used to generate the atomic orbitals of the nonrelativistic hydrogen atom in a four-dimensional Riemann space through the path integral formalism. Group theoretical implications of the present problem are briefly discussed.
3. PLANE-INTEGRATED OPEN-PATH FOURIER TRANSFORM INFRARED SPECTROMETRY METHODOLOGY FOR ANAEROBIC SWINE LAGOON EMISSION MEASUREMENTS
EPA Science Inventory
Emissions of ammonia and methane from an anaerobic lagoon at a swine animal feeding operation were evaluated five times over a period of two years. The plane-integrated (PI) open-path Fourier transform infrared spectrometry (OP-FTIR) methodology was used to transect the plume at ...
4. Path-integral solution for a two-dimensional model with axial-vector-current--pseudoscalar derivative interaction
SciTech Connect
Botelho, L.C.L.
1985-03-15
We study a two-dimensional quantum field model with axial-vector-current--pseudoscalar derivative interaction using path-integral methods. We construct an effective Lagrangian by performing a chiral change in the fermionic variables leading to an exact solution of the model.
5. Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations.
PubMed
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
6. Creativity, Spirituality, and Transcendence: Paths to Integrity and Wisdom in the Mature Self. Publications in Creativity Research.
ERIC Educational Resources Information Center
Miller, Melvin E., Ed.; Cook-Greuter, Susanne R., Ed.
This book contains 11 papers on creativity, spirituality, and transcendence as paths to integrity and wisdom in the mature self. The book begins with the paper "Introduction--Creativity in Adulthood: Personal Maturity and Openness to Extraordinary Sources of Inspiration" (Susanne R. Cook-Greuter, Melvin E. Miller). The next four papers,…
7. Ab initio molecular orbital calculation considering the quantum mechanical effect of nuclei by path integral molecular dynamics
Shiga, Motoyuki; Tachikawa, Masanori; Miura, Shinichi
2000-12-01
We present an accurate calculational scheme for many-body systems composed of electrons and nuclei, by path integral molecular dynamics technique combined with the ab initio molecular orbital theory. Based upon the scheme, the simulation of a water molecule at room temperature is demonstrated, applying all-electron calculation at the Hartree-Fock level of theory.
8. Coherent-state path integral versus coarse-grained effective stochastic equation of motion: From reaction diffusion to stochastic sandpiles.
PubMed
Wiese, Kay Jörg
2016-04-01
We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.
9. Vector Schwinger Model with a Photon Mass Term:. Gauge-Invariant Reformulation, Operator Solutions and Hamiltonian and Path Integral Formulations
Kulshreshtha, Usha
We consider the vector Schwinger model (VSM) describing two-dimensional electrodynamics with massless fermions, where the left-handed and right-handed fermions are coupled to the electromagnetic field with equal couplings, with a mass term for the U(1) gauge field and then study its operator solutions and the Hamiltonian and path integral formulations. We emphasize here that although the VSM has been studied in the literature rather widely but only without a photon mass term (which was a consequence of demanding the regularization for the VSM to be gauge-invariant (GI)). The VSM with a photon mass term is seen to be a gauge-noninvariant (GNI) theory. Using the standard Stueckelberg formalism we then construct a GI theory corresponding to the proposed GNI model. From this reformulated GI theory, we further recover the physical contents of the proposed GNI theory under a very special gauge choice. The theory proposed and studied here presents a new class of models in the two-dimensional quantum electrodynamics with massless fermions but with a photon mass term.
10. Path ANalysis
SciTech Connect
Snell, Mark K.
2007-07-14
The PANL software determines path through an Adversary Sequence Diagram (ASD) with minimum Probability of Interruption, P(I), given the ASD information and data about site detection, delay, and response force times. To accomplish this, the software generates each path through the ASD, then applies the Estimate of Adversary Sequence Interruption (EASI) methodology for calculating P(I) to each path, and keeps track of the path with the lowest P(I). Primary use is for training purposes during courses on physical security design. During such courses PANL will be used to demonstrate to students how more complex software codes are used by the US Department of Energy to determine the most-vulnerable paths and, where security needs improvement, how such codes can help determine physical security upgrades.
11. Computer Integrated Manufacturing: Physical Modelling Systems Design. A Personal View.
ERIC Educational Resources Information Center
Baker, Richard
A computer-integrated manufacturing (CIM) Physical Modeling Systems Design project was undertaken in a time of rapid change in the industrial, business, technological, training, and educational areas in Australia. A specification of a manufacturing physical modeling system was drawn up. Physical modeling provides a flexibility and configurability…
12. On the calculation of the static structure factor of path-integral quantum simple fluids far from exchange
Sesé, Luis M.
This paper addresses several points of interest concerning the computation of the static structure factor of path-integral monatomic quantum fluids. First of all, the connection between the structure factor and the path-integral linear response pair radial correlation function is shown as its defining quantity by assuming a generalized Fermi's potential for the neutron- nuclei interactions, which is to be included in the general expression of the dynamic structure factor. Second, the possibilities of finding Ornstein-Zernike equations for full path-integral fluids, and also for the effective potential models of fluids derived from the path-integral formalism, are explored by working in the grand canonical ensemble. By so doing, the success and features for improvement of the weak-field approach used previously in this context of determining quantum static structure factors [SESE,L.M.,1996, Molec. Phys., 89, 1783; SESE, L.M., and LEDESMA,R., 1997, J. chem. Phys., 106, 1134] can be understood. New numerical applications are performed within this weak-field approach taking as probes the quantum hard-sphere fluid and dense fluid helium-4, the latter being described through LennardJones and Aziz-Slaman underlying interactions. The results show that the structure factors associated with the linear response and instantaneous path-integral pair radial correlation functions differ noticeably from each other with increasing quantum effects. In particular, the linear response description leads to more compressible fluids than the instantaneous one. Besides, the equality between the isothermal compressibilities fixed via the linear response and the quantum particle centre-of-gravity pair radial correlation functions does not hold beyond the situations that can be treated with the Gaussian Feynman-Hibbs effective potential picture. Comparison with experiment in the case of helium-4 (T = 4.2 K) reveals clearly that, under strong quantum conditions, an operative framework more
13. Discovering Integration through a Physical Phenomenon
ERIC Educational Resources Information Center
Burt, Derek; Magnes, Jenny; Schwarz, Gregory; Hartke, John
2008-01-01
This article outlines a method of conducting a laboratory designed to discover mathematical integration with students. The results are produced and verified in the laboratory by students. Understanding that an integral is defined by the area bounded by a function of x and the x-axis from a point a to a point b is challenging. Students often have…
14. Integrating Physical Activity into Academic Pursuits
ERIC Educational Resources Information Center
Gaus, Mark D.; Simpson, Cynthia G.
2009-01-01
Children of today may be the first generation in the United States in more than 200 years to have a life expectancy shorter than their parents. Low levels of fitness caused by physical inactivity and poor nutritional habits of many of today's youth may be a contributing factor. Combating low fitness levels with physical activity is of utmost…
15. Integrating Physical Activity into Academic Pursuits
ERIC Educational Resources Information Center
Gaus, Mark D.; Simpson, Cynthia G.
2009-01-01
Children of today may be the first generation in the United States in more than 200 years to have a life expectancy shorter than their parents. Low levels of fitness caused by physical inactivity and poor nutritional habits of many of today's youth may be a contributing factor. Combating low fitness levels with physical activity is of utmost…
16. Integrating Condensed Matter Physics into a Liberal Arts Physics Curriculum
Collett, Jeffrey
2008-03-01
The emergence of nanoscale science into the popular consciousness presents an opportunity to attract and retain future condensed matter scientists. We inject nanoscale physics into recruiting activities and into the introductory and the core portions of the curriculum. Laboratory involvement and research opportunity play important roles in maintaining student engagement. We use inexpensive scanning tunneling (STM) and atomic force (AFM) microscopes to introduce students to nanoscale structure early in their college careers. Although the physics of tip-surface interactions is sophisticated, the resulting images can be interpreted intuitively. We use the STM in introductory modern physics to explore quantum tunneling and the properties of electrons at surfaces. An interdisciplinary course in nanoscience and nanotechnology course team-taught with chemists looks at nanoscale phenomena in physics, chemistry, and biology. Core quantum and statistical physics courses look at effects of quantum mechanics and quantum statistics in degenerate systems. An upper level solid-state physics course takes up traditional condensed matter topics from a structural perspective by beginning with a study of both elastic and inelastic scattering of x-rays from crystalline solids and liquid crystals. Students encounter reciprocal space concepts through the analysis of laboratory scattering data and by the development of the scattering theory. The course then examines the importance of scattering processes in band structure and in electrical and thermal conduction. A segment of the course is devoted to surface physics and nanostructures where we explore the effects of restricting particles to two-dimensional surfaces, one-dimensional wires, and zero-dimensional quantum dots.
17. Integrated Health and Physical Education Program to Reduce Media Use and Increase Physical Activity in Youth
ERIC Educational Resources Information Center
Clocksin, Brian D.; Wattson, Doris L.; Williams, Daniel P.; Randsell, Lynda
2009-01-01
The purpose of this project was to compare an integrated health and physical education curriculum, focused on reducing media use and on increasing physical activity in middle school adolescents, to traditional and nonintegrated health and physical education curricula. Two middle schools' health and physical education classes were assigned to an…
18. Psychosocial determinants of physical activity in children attending afterschool programs: a path analysis.
PubMed
Ling, Jiying; Robbins, Lorraine B; McCarthy, Valerie Lander; Speck, Barbara J
2015-01-01
Physical activity (PA) is important for controlling childhood obesity, but a comprehensive PA model for school-aged children is lacking. Guided by the youth PA promotion (YPAP) model, this study estimated the direct and indirect effects of self-efficacy, enjoyment, parental influence, and environment on self-reported PA and pedometer steps. A secondary purpose was to explore the association between self-reported PA and pedometer steps. An observational and prospective study was conducted among 133 children, aged 8-11 years old, from 10 elementary schools with afterschool programs in a Midwestern U.S. school district from August through October 2013. PA was assessed by a 7-day recall scale and pedometers. Other variables were assessed by validated questionnaires. Approximately 65 (49%) children were overweight or obese-only 17 (13%) met national PA recommendations-and body mass index z-score was negatively correlated with pedometer steps (r = -.18 p = .04). A path analysis showed that self-efficacy had a direct effect on self-reported PA and pedometer steps, enjoyment had only a direct effect on self-reported PA, and parental influence had a direct effect on pedometer steps and an indirect effect on self-reported PA through self-efficacy and enjoyment. The association between self-reported PA and pedometer steps was not significant. Because this study only partially supports the YPAP model, studies with a larger sample size and longitudinal design are essential to further examine this model. The nonsignificant relationship of self-reported PA with pedometer steps may be due to the systematic error resulting from a common method artifact of self-report. Given the importance of parental influence, enjoyment, and self-efficacy, targeting these three determinants in future interventions to increase PA among children is recommended.
19. A path-integral Monte Carlo study of a small cluster: The Ar trimer
Pérez de Tudela, R.; Márquez-Mijares, M.; González-Lezana, T.; Roncero, O.; Miret-Artés, S.; Delgado-Barrio, G.; Villarreal, P.
2010-06-01
The Ar3 system has been studied between T =0 K and T =40 K by means of a path-integral Monte Carlo (PIMC) method. The behavior of the average energy in terms of the temperature has been explained by comparison with results obtained with the thermal averaged rovibrational spectra estimated via: (i) a quantum mechanical method based on distributed Gaussian functions for the interparticle distances and (ii) an analytical model which precisely accounts for the participation of the dissociative continua Ar2+Ar and Ar+Ar+Ar. Beyond T ˜20 K, the system explores floppier configurations than the rigid equilateral geometry, as linear and Ar-Ar2-like arrangements, and fragmentates around T ˜40 K. A careful investigation of the specific heat in terms of a confining radius in the PIMC calculation seems to discard a proper phase transition as in larger clusters, in apparent contradiction with previous reports of precise values for a liquid-gas transition. The onset of this noticeable change in the dynamics of the trimer occurs, however, at a remarkably low value of the temperature in comparison with Arn systems formed with more Ar atoms. Quantum mechanical effects are found of relevance at T ≤15 K, with both energies and radial distributions obtained with a quantum PIMC deviating from the corresponding classical results, thus precluding exclusively classical approaches for a precise description of the system at this low temperature range.
20. Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis
PubMed Central
Chang, Joshua C.; Fok, Pak-Wing; Chou, Tom
2015-01-01
Dynamic single-molecule force spectroscopy is often used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, are used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complicated bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule. To address these challenges, we develop a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experimental and statistical parameters in our method are estimated directly from the data. Upon testing our method on simulated data, our regularized approach requires less data and allows simultaneous inference of both complex bond potentials and diffusivity profiles. Crucially, we show that the accuracy of the reconstructed bond potential is sensitive to the spatially varying diffusivity and accurate reconstruction can be expected only when both are simultaneously inferred. Moreover, after providing a means for self-consistently choosing regularization parameters from data, we derive posterior probability distributions, allowing for uncertainty quantification. PMID:26331254
1. On the Intramolecular Hydrogen Bond in Solution: Car-Parrinello and Path Integral Molecular Dynamics Perspective.
PubMed
Dopieralski, Przemyslaw; Perrin, Charles L; Latajka, Zdzislaw
2011-11-08
The issue of the symmetry of short, low-barrier hydrogen bonds in solution is addressed here with advanced ab initio simulations of a hydrogen maleate anion in different environments, starting with the isolated anion, going through two crystal structures (sodium and potassium salts), then to an aqueous solution, and finally in the presence of counterions. By Car-Parrinello and path integral molecular dynamics simulations, it is demonstrated that the position of the proton in the intramolecular hydrogen bond of an aqueous hydrogen maleate anion is entirely related to the solvation pattern around the oxygen atoms of the intramolecular hydrogen bond. In particular, this anion has an asymmetric hydrogen bond, with the proton always located on the oxygen atom that is less solvated, owing to the instantaneous solvation environment. Simulations of water solutions of hydrogen maleate ion with two different counterions, K(+) and Na(+), surprisingly show that the intramolecular hydrogen-bond potential in the case of the Na(+) salt is always asymmetric, regardless of the hydrogen bonds to water, whereas for the K(+) salt, the potential for H motion depends on the location of the K(+). It is proposed that repulsion by the larger and more hydrated K(+) is weaker than that by Na(+) and competitive with solvation by water.
2. Ground return signal simulation and retrieval algorithm of spaceborne integrated path DIAL for CO2 measurements
Liu, Bing-Yi; Wang, Jun-Yang; Liu, Zhi-Shen
2014-11-01
Spaceborne integrated path differential absorption (IPDA) lidar is an active-detection system which is able to perform global CO2 measurement with high accuracy of 1ppmv at day and night over ground and clouds. To evaluate the detection performance of the system, simulation of the ground return signal and retrieval algorithm for CO2 concentration are presented in this paper. Ground return signals of spaceborne IPDA lidar under various ground surface reflectivity and atmospheric aerosol optical depths are simulated using given system parameters, standard atmosphere profiles and HITRAN database, which can be used as reference for determining system parameters. The simulated signals are further applied to the research on retrieval algorithm for CO2 concentration. The column-weighted dry air mixing ratio of CO2 denoted by XCO2 is obtained. As the deviations of XCO2 between the initial values for simulation and the results from retrieval algorithm are within the expected error ranges, it is proved that the simulation and retrieval algorithm are reliable.
3. Path-integral formalism for stochastic resetting: Exactly solved examples and shortcuts to confinement
Roldán, Édgar; Gupta, Shamik
2017-08-01
We study the dynamics of overdamped Brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space-dependent rate of resetting. We present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as (i) the propagator prior to first reset, (ii) the distribution of the first-reset time, and (iii) the spatial distribution of the particle at long times. We apply our approach to several representative and hitherto unexplored examples of resetting dynamics. A particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a Brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle. We find that using energy-dependent resetting processes is more effective in achieving spatial confinement of Brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential.
4. Oscillator-interference models of path integration do not require theta oscillations.
PubMed
Orchard, Jeff
2015-03-01
Navigation and path integration in rodents seems to involve place cells, grid cells, and theta oscillations (4-12 Hz) in the local field potential. Two main theories have been proposed to explain the neurological underpinnings of how these phenomena relate to navigation and to each other. Attractor network (AN) models revolve around the idea that local excitation and long-range inhibition connectivity can spontaneously generate grid-cell-like activity patterns. Oscillator interference (OI) models propose that spatial patterns of activity are caused by the interference patterns between neural oscillators. In rats, these oscillators have a frequency close to the theta frequency. Recent studies have shown that bats do not exhibit a theta cycle when they crawl, and yet they still have grid cells. This has been interpreted as a criticism of OI models. However, OI models do not require theta oscillations. We explain why the absence of theta oscillations does not contradict OI models and discuss how the two families of models might be distinguished experimentally.
5. A new approach to gravitational clustering: A path-integral formalism and large-N expansions
Valageas, P.
2004-07-01
We show that the formation of large-scale structures through gravitational instability in the expanding universe can be fully described through a path-integral formalism. We derive the action S[f] which gives the statistical weight associated with any phase-space distribution function f({x},{p},t). This action S describes both the average over the Gaussian initial conditions and the Vlasov-Poisson dynamics. Next, applying a standard method borrowed from field theory we generalize our problem to an N-field system and we look for an expansion over powers of 1/N. We describe three such methods and we derive the corresponding equations of motion at the lowest non-trivial order for the case of gravitational clustering. This yields a set of non-linear equations for the mean /line{f} and the two-point correlation G of the phase-space distribution f, as well as for the response function R. These systematic schemes match the usual perturbative expansion on quasi-linear scales but should also be able to treat the non-linear regime. Our approach can also be extended to non-Gaussian initial conditions and may serve as a basis for other tools borrowed from field theory. Appendix A is only available in electronic form at http://www.edpsciences.org
6. Path-integral simulation of ice VII: Pressure and temperature effects
Herrero, Carlos P.; Ramírez, Rafael
2015-11-01
The effects of pressure and temperature on structural and thermodynamic properties of ice VII have been studied by using path-integral molecular dynamics (PIMD) simulations. Temperatures between 25 and 450 K, as well as pressures up to 12 GPa were considered. Interatomic interactions were modeled by using the effective q-TIP4P/F potential for flexible water. We analyze the pressure dependence of the molar volume, bulk modulus, interatomic distances, kinetic energy, and atomic delocalization at various temperatures. Results of PIMD simulations are compared with those derived from a quasi-harmonic approximation (QHA) of vibrational modes, which helps to assess the importance of anharmonic effects, as well as the influence of the different modes on the properties of ice VII. The accuracy of the QHA for describing this high-pressure phase decreases for rising temperature, but this approximation becomes more reliable as pressure grows, since anharmonicity becomes less relevant. Comparisons with low-pressure cubic ice are presented.
7. Theoretical study on isotope and temperature effect in hydronium ion using ab initio path integral simulation
Tachikawa, Masanori; Shiga, Motoyuki
2004-09-01
We have applied the ab initio path integral molecular dynamics simulation to study hydronium ion and its isotopes, which are the simplest systems for hydrated proton and deuteron. In this simulation, all the rotational and vibrational degrees of freedom are treated fully quantum mechanically, while the potential energies of the respective atomic configurations are calculated "on the fly" using ab initio quantum chemical approach. With the careful treatment of the ab initio electronic structure calculation by relevant choices in electron correlation level and basis set, this scheme is theoretically quite rigorous except for Born-Oppenheimer approximation. This accurate calculation allows a close insight into the structural shifts for the isotopes of hydronium ion by taking account of both quantum mechanical and thermal effects. In fact, the calculation is shown to be successful to quantitatively extract the geometrical isotope effect with respect to the Walden inversion. It is also shown that this leads to the isotope effect on the electronic structure as well as the thermochemical properties.
8. Exact path-integral evaluation of the heat distribution function of a trapped Brownian oscillator
Chatterjee, Debarati; Cherayil, Binny J.
2010-11-01
Using path integrals, we derive an exact expression—valid at all times t —for the distribution P(Q,t) of the heat fluctuations Q of a Brownian particle trapped in a stationary harmonic well. We find that P(Q,t) can be expressed in terms of a modified Bessel function of zeroth order that in the limit t→∞ exactly recovers the heat distribution function obtained recently by Imparato [Phys. Rev. E 76, 050101(R) (2007)]10.1103/PhysRevE.76.050101 from the approximate solution to a Fokker-Planck equation. This long-time result is in very good agreement with experimental measurements carried out by the same group on the heat effects produced by single micron-sized polystyrene beads in a stationary optical trap. An earlier exact calculation of the heat distribution function of a trapped particle moving at a constant speed v was carried out by van Zon and Cohen [Phys. Rev. E 69, 056121 (2004)]10.1103/PhysRevE.69.056121; however, this calculation does not provide an expression for P(Q,t) itself, but only its Fourier transform (which cannot be analytically inverted), nor can it be used to obtain P(Q,t) for the case v=0 .
9. Iterative quantum-classical path integral with dynamically consistent state hopping
SciTech Connect
Walters, Peter L.; Makri, Nancy
2016-01-28
We investigate the convergence of iterative quantum-classical path integral calculations in sluggish environments strongly coupled to a quantum system. The number of classical trajectories, thus the computational cost, grows rapidly (exponentially, unless filtering techniques are employed) with the memory length included in the calculation. We argue that the choice of the (single) trajectory branch during the time preceding the memory interval can significantly affect the memory length required for convergence. At short times, the trajectory branch associated with the reactant state improves convergence by eliminating spurious memory. We also introduce an instantaneous population-based probabilistic scheme which introduces state-to-state hops in the retained pre-memory trajectory branch, and which is designed to choose primarily the trajectory branch associated with the reactant at early times, but to favor the product state more as the reaction progresses to completion. Test calculations show that the dynamically consistent state hopping scheme leads to accelerated convergence and a dramatic reduction of computational effort.
10. Neural Network-Based Solutions for Stochastic Optimal Control Using Path Integrals.
PubMed
Rajagopal, Karthikeyan; Balakrishnan, Sivasubramanya Nadar; Busemeyer, Jerome R
2017-03-01
In this paper, an offline approximate dynamic programming approach using neural networks is proposed for solving a class of finite horizon stochastic optimal control problems. There are two approaches available in the literature, one based on stochastic maximum principle (SMP) formalism and the other based on solving the stochastic Hamilton-Jacobi-Bellman (HJB) equation. However, in the presence of noise, the SMP formalism becomes complex and results in having to solve a couple of backward stochastic differential equations. Hence, current solution methodologies typically ignore the noise effect. On the other hand, the inclusion of noise in the HJB framework is very straightforward. Furthermore, the stochastic HJB equation of a control-affine nonlinear stochastic system with a quadratic control cost function and an arbitrary state cost function can be formulated as a path integral (PI) problem. However, due to curse of dimensionality, it might not be possible to utilize the PI formulation for obtaining comprehensive solutions over the entire operating domain. A neural network structure called the adaptive critic design paradigm is used to effectively handle this difficulty. In this paper, a novel adaptive critic approach using the PI formulation is proposed for solving stochastic optimal control problems. The potential of the algorithm is demonstrated through simulation results from a couple of benchmark problems.
11. Efficient real-time path integrals for non-Markovian spin-boson models
Strathearn, A.; Lovett, B. W.; Kirton, P.
2017-09-01
Strong coupling between a system and its environment leads to the emergence of non-Markovian dynamics, which cannot be described by a time-local master equation. One way to capture such dynamics is to use numerical real-time path integrals, where assuming a finite bath memory time enables manageable simulation scaling. However, by comparing to the exactly soluble independent boson model, we show that the presence of transient negative decay rates in the exact dynamics can result in simulations with unphysical exponential growth of density matrix elements when the finite memory approximation is used. We therefore reformulate this approximation in such a way that the exact dynamics are reproduced identically and then apply our new method to the spin-boson model with superohmic environmental coupling, commonly used to model phonon environments, but which cannot be solved exactly. Our new method allows us to easily access parameter regimes where we find revivals in population dynamics which are due to non-Markovian backflow of information from the bath to the system.
12. Path-integral and Ornstein-Zernike computations of quantum fluid structures under strong fluctuations
Sesé, Luis M.
2017-02-01
This work deals with the computation of the structure factors of quantum fluids under complex conditions involving substantial density fluctuations and/or large particle delocalization effects. The method is based on the combination of path-integral Monte Carlo (PIMC) simulations and the pair Ornstein-Zernike framework (OZ2). PIMC provides the radial correlation functions (centroid, instantaneous, and thermalized-continuous total linear response), which are used as data input to the OZ2 calculations that lead to their associated structure factors. To undertake this project normal liquid 4He and supercritical 3He are selected, studying conditions in the range (T = 4.2 K; 0.01886 <ρN/Å-3 < 0.02687). Full inter-comparison between the structure factors determined via both OZ2 and direct PIMC calculations is made. In addition, comparison with experimental data, including thermodynamic properties, is made wherever possible. The results establish that, even under severe thermodynamic and/or quantum fluctuation conditions, OZ2 remains in the quantum domain as a highly reliable and cost-effective framework to determine accurate structure factors, also allowing one to understand the related isotopic shifts in fluid He.
13. Path Integral Monte Carlo finite-temperature electronic structure of quantum dots
Leino, Markku; Rantala, Tapio T.
2003-03-01
Quantum Monte Carlo methods allow a straightforward procedure for evaluation of electronic structures with a proper treatment of electronic correlations. This can be done even at finite temperatures [1]. We apply the Path Integral Monte Carlo (PIMC) simulation method [2] for one and two electrons in a single and double quantum dots. With this approach we evaluate the electronic distributions and correlations, and finite temperature effects on those. Temperature increase broadens the one-electron distribution as expected. This effect is smaller for correlated electrons than for single ones. The simulated one and two electron distributions of a single and two coupled quantum dots are also compared to those from experiments and other theoretical (0 K) methods [3]. Computational capacity is found to become the limiting factor in simulations with increasing accuracy. This and other essential aspects of PIMC and its capability in this type of calculations are also discussed. [1] R.P. Feynman: Statistical Mechanics, Addison Wesley, 1972. [2] D.M. Ceperley, Rev.Mod.Phys. 67, 279 (1995). [3] M. Pi, A. Emperador and M. Barranco, Phys.Rev.B 63, 115316 (2001).
14. The quantum nature of the hydrogen bond: insight from path-integral molecular dynamics
Walker, Brent; Li, Xin-Zheng; Michaelides, Angelos
2011-03-01
Hydrogen (H) bonds are weak, generally intermolecular bonds, that hold together much of soft matter, the condensed phases of water, network liquids, and many ferroelectric crystals. The small mass of H means H-bonds are inherently quantum mechanical; effects such as zero point motion and tunneling should be considered, although often are not. In particular, a consistent picture of quantum nuclear effects on the strength of H-bonds and consequently the structure of H-bonded systems is still absent. Here, we report ab initio path-integral molecular dynamics studies on the quantum nature of the H-bond. Systematic examination of a range of H-bonded systems shows that quantum nuclei weaken weak H-bonds but strengthen relatively strong ones. This correlation arises from a competition between anharmonic intermolecular bond bending and intramolecular bond stretching. A simple rule of thumb enables predictions to be made for H-bonded bonded materials in general with merely classical knowledge (e.g. H-bond strength or H-bond length). Our work rationalizes the contrasting influence of quantum nuclear dynamics on a wide variety of materials, including liquid water and HF, and highlights the need for flexible molecules in force-field based studies of quantum nuclear dynamics.
15. Path-integral Monte Carlo simulations for electronic dynamics on molecular chains. II. Transport across impurities
Mühlbacher, Lothar; Ankerhold, Joachim
2005-05-01
Electron transfer (ET) across molecular chains including an impurity is studied based on a recently improved real-time path-integral Monte Carlo (PIMC) approach [L. Mühlbacher, J. Ankerhold, and C. Escher, J. Chem. Phys. 121 12696 (2004)]. The reduced electronic dynamics is studied for various bridge lengths and defect site energies. By determining intersite hopping rates from PIMC simulations up to moderate times, the relaxation process in the extreme long-time limit is captured within a sequential transfer model. The total transfer rate is extracted and shown to be enhanced for certain defect site energies. Superexchange turns out to be relevant for extreme gap energies only and then gives rise to different dynamical signatures for high- and low-lying defects. Further, it is revealed that the entire bridge compound approaches a steady state on a much shorter time scale than that related to the total transfer. This allows for a simplified description of ET along donor-bridge-acceptor systems in the long-time range.
16. Path integral Monte Carlo and density functional molecular dynamics simulations of hot, dense helium
Militzer, B.
2009-04-01
Two first-principles simulation techniques, path integral Monte Carlo (PIMC) and density functional molecular dynamics (DFT-MD), are applied to study hot, dense helium in the density-temperature range of 0.387-5.35gcm-3 and 500K-1.28×108K . One coherent equation of state is derived by combining DFT-MD data at lower temperatures with PIMC results at higher temperatures. Good agreement between both techniques is found in an intermediate-temperature range. For the highest temperatures, the PIMC results converge to the Debye-Hückel limiting law. In order to derive the entropy, a thermodynamically consistent free-energy fit is used that reproduces the internal energies and pressure derived from the first-principles simulations. The equation of state is presented in the form of a table as well as a fit and is compared with different free-energy models. Pair-correlation functions and the electronic density of states are discussed. Shock Hugoniot curves are compared with recent laser shock-wave experiments.
17. Path-integral molecular dynamics simulation of 3C-SiC
Ramírez, Rafael; Herrero, Carlos P.; Hernández, Eduardo R.; Cardona, Manuel
2008-01-01
Molecular dynamics simulations of 3C-SiC have been performed as a function of pressure and temperature. These simulations treat both electrons and atomic nuclei by quantum mechanical methods. While the electronic structure of the solid is described by an efficient tight-binding Hamiltonian, the nuclei dynamics is treated by the path-integral formulation of statistical mechanics. To assess the relevance of nuclear quantum effects, the results of quantum simulations are compared to others where either the Si nuclei, the C nuclei, or both atomic nuclei are treated as classical particles. We find that the experimental thermal expansion of 3C-SiC is realistically reproduced by our simulations. The calculated bulk modulus of 3C-SiC and its pressure derivative at room temperature show also good agreement with the available experimental data. The effect of the electron-phonon interaction on the direct electronic gap of 3C-SiC has been calculated as a function of temperature and related to results obtained for bulk diamond and Si. Comparison to available experimental data shows satisfactory agreement, although we observe that the employed tight-binding model tends to overestimate the magnitude of the electron-phonon interaction. The effect of treating the atomic nuclei as classical particles on the direct gap of 3C-SiC has been assessed. We find that nonlinear quantum effects related to the atomic masses are particularly relevant at temperatures below 250K .
18. Path integration and cognitive mapping in a continuous attractor neural network model.
PubMed
Samsonovich, A; McNaughton, B L
1997-08-01
A minimal synaptic architecture is proposed for how the brain might perform path integration by computing the next internal representation of self-location from the current representation and from the perceived velocity of motion. In the model, a place-cell assembly called a "chart" contains a two-dimensional attractor set called an "attractor map" that can be used to represent coordinates in any arbitrary environment, once associative binding has occurred between chart locations and sensory inputs. In hippocampus, there are different spatial relations among place fields in different environments and behavioral contexts. Thus, the same units may participate in many charts, and it is shown that the number of uncorrelated charts that can be encoded in the same recurrent network is potentially quite large. According to this theory, the firing of a given place cell is primarily a cooperative effect of the activity of its neighbors on the currently active chart. Therefore, it is not particularly useful to think of place cells as encoding any particular external object or event. Because of its recurrent connections, hippocampal field CA3 is proposed as a possible location for this "multichart" architecture; however, other implementations in anatomy would not invalidate the main concepts. The model is implemented numerically both as a network of integrate-and-fire units and as a "macroscopic" (with respect to the space of states) description of the system, based on a continuous approximation defined by a system of stochastic differential equations. It provides an explanation for a number of hitherto perplexing observations on hippocampal place fields, including doubling, vanishing, reshaping in distorted environments, acquiring directionality in a two-goal shuttling task, rapid formation in a novel environment, and slow rotation after disorientation. The model makes several new predictions about the expected properties of hippocampal place cells and other cells of the
19. Equilibrium fractionation of H and O isotopes in water from path integral molecular dynamics
Pinilla, Carlos; Blanchard, Marc; Balan, Etienne; Ferlat, Guillaume; Vuilleumier, Rodolphe; Mauri, Francesco
2014-06-01
The equilibrium fractionation factor between two phases is of importance for the understanding of many planetary and environmental processes. Although thermodynamic equilibrium can be achieved between minerals at high temperature, many natural processes involve reactions between liquids or aqueous solutions and solids. For crystals, the fractionation factor α can be theoretically determined using a statistical thermodynamic approach based on the vibrational properties of the phases. These calculations are mostly performed in the harmonic approximation, using empirical or ab-initio force fields. In the case of aperiodic and dynamic systems such as liquids or solutions, similar calculations can be done using finite-size molecular clusters or snapshots obtained from molecular dynamics (MD) runs. It is however difficult to assess the effect of these approximate models on the isotopic fractionation properties. In this work we present a systematic study of the calculation of the D/H and 18O/16O equilibrium fractionation factors in water for the liquid/vapour and ice/vapour phases using several levels of theory within the simulations. Namely, we use a thermodynamic integration approach based on Path Integral MD calculations (PIMD) and an empirical potential model of water. Compared with standard MD, PIMD takes into account quantum effects in the thermodynamic modeling of systems and the exact fractionation factor for a given potential can be obtained. We compare these exact results with those of modeling strategies usually used, which involve the mapping of the quantum system on its harmonic counterpart. The results show the importance of including configurational disorder for the estimation of isotope fractionation in liquid phases. In addition, the convergence of the fractionation factor as a function of parameters such as the size of the simulated system and multiple isotope substitution is analyzed, showing that isotope fractionation is essentially a local effect in
20. Integrating two-dimensional paths: do desert ants process distance information in the absence of celestial compass cues?
PubMed
Ronacher, B; Westwig, E; Wehner, R
2006-09-01
When performing foraging trips desert ants of the genus Cataglyphis continuously process and update a ;home vector' that enables them to return to their nest on the shortest route. This capacity of path integration requires two types of information: (i) information about the travelling directions, and (ii) odometric information about the distances travelled in a particular direction. We have investigated how these two necessary pieces of information interact within the path integration processor. The specific question is: how do the ants process distance information if there is no simultaneous input from the sky compass available. Ants were trained to forage in a ;Z'-shaped channel system, the three segments of which joined at right angles. Individual animals were transferred from the feeder to a test field where their homing paths could be observed. In the crucial tests the middle segment of the maze was covered by orange Perspex that did not transmit the UV part of the spectrum, and thus precluded the perception of polarization patterns. Changes of the ant's processing of odometric information within this channel segment directly translate into a change in homing direction on the test field. The results indicate that the odometric information about travelling distance is largely ignored for path integration if there is no simultaneous input from the sky-view-based compass. They further show that idiothetic information cannot adequately substitute for the polarization compass to infer travelling directions. | {"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": 0.890559732913971, "perplexity": 1485.6547467836724}, "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-2018-05/segments/1516084887621.26/warc/CC-MAIN-20180118210638-20180118230638-00210.warc.gz"} |
https://stacks.math.columbia.edu/tag/0A22 | ## 32.42 Curves
In the Stacks project we will use the following as our definition of a curve.
Two standard examples of curves over $k$ are the affine line $\mathbf{A}^1_ k$ and the projective line $\mathbf{P}^1_ k$. The scheme $X = \mathop{\mathrm{Spec}}(k[x, y]/(f))$ is a curve if and only if $f \in k[x, y]$ is irreducible.
Our definition of a curve has the same problems as our definition of a variety, see the discussion following Definition 32.3.1. Moreover, it means that every curve comes with a specified field of definition. For example $X = \mathop{\mathrm{Spec}}(\mathbf{C}[x])$ is a curve over $\mathbf{C}$ but we can also view it as a curve over $\mathbf{R}$. The scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$ isn't a curve, even though the schemes $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and $\mathbf{A}^1_{\mathbf{F}_ p}$ behave similarly in many respects.
Lemma 32.42.2. Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$. Let $x \in X$ be a closed point. The open subscheme $X \setminus \{ x\}$ is not proper over $k$.
Proof. Namely, choose a specialization $x' \leadsto x$ with $x' \not= x$ (for example take $x'$ to be the generic point). By Schemes, Lemma 25.20.4 there exists a morphism $a : \mathop{\mathrm{Spec}}(A) \to X$ where $A$ is a valuation ring with fraction field $K$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps to $x'$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$. The morphism $\mathop{\mathrm{Spec}}(K) \to X \setminus \{ x\}$ does not extend to a morphism $b : \mathop{\mathrm{Spec}}(A) \to X \setminus \{ x\}$ since by the uniqueness in Schemes, Lemma 25.22.1 we would have $a = b$ as morphisms into $X$ which is absurd. Hence the valuative criterion (Schemes, Proposition 25.20.6) shows that $X \to \mathop{\mathrm{Spec}}(k)$ is not universally closed, hence not proper. $\square$
Lemma 32.42.3. Let $X$ be a separated finite type scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-quasi-projective over $k$.
Proof. By Proposition 32.37.12 the scheme $X$ has an ample invertible sheaf $\mathcal{L}$. By Morphisms, Lemma 28.37.3 we see that $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^ n_ k$ over $\mathop{\mathrm{Spec}}(k)$. This is the definition of being H-quasi-projective over $k$, see Morphisms, Definition 28.38.1. $\square$
Lemma 32.42.4. Let $X$ be a proper scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-projective over $k$.
Proof. By Lemma 32.42.3 we see that $X$ is a locally closed subscheme of $\mathbf{P}^ n_ k$ for some field $k$. Since $X$ is proper over $k$ it follows that $X$ is a closed subscheme of $\mathbf{P}^ n_ k$ (Morphisms, Lemma 28.39.7). $\square$
Lemma 32.42.5. Let $X$ be a separated scheme of finite type over $k$. If $\dim (X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ with the following properties
1. $\overline{X}$ is H-projective over $k$, i.e., $\overline{X}$ is a closed subscheme of $\mathbf{P}^ d_ k$ for some $d$,
2. $j(X) \subset \overline{X}$ is dense and scheme theoretically dense,
3. $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\}$ for some closed points $x_ i \in \overline{X}$.
Proof. By Lemma 32.42.3 we may assume $X$ is a locally closed subscheme of $\mathbf{P}^ d_ k$ for some $d$. Let $\overline{X} \subset \mathbf{P}^ d_ k$ be the scheme theoretic image of $X \to \mathbf{P}^ d_ k$, see Morphisms, Definition 28.6.2. The description in Morphisms, Lemma 28.7.7 gives properties (1) and (2). Then $\dim (X) = 1 \Rightarrow \dim (\overline{X}) = 1$ for example by looking at generic points, see Lemma 32.20.3. As $\overline{X}$ is Noetherian, it then follows that $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\}$ is a finite set of closed points. $\square$
Lemma 32.42.6. Let $X$ be a separated scheme of finite type over $k$. If $X$ is reduced and $\dim (X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ such that
1. $\overline{X}$ is H-projective over $k$, i.e., $\overline{X}$ is a closed subscheme of $\mathbf{P}^ d_ k$ for some $d$,
2. $j(X) \subset \overline{X}$ is dense and scheme theoretically dense,
3. $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\}$ for some closed points $x_ i \in \overline{X}$,
4. the local rings $\mathcal{O}_{\overline{X}, x_ i}$ are discrete valuation rings for $i = 1, \ldots , n$.
Proof. Let $j : X \to \overline{X}$ be as in Lemma 32.42.5. Consider the normalization $X'$ of $\overline{X}$ in $X$. By Lemma 32.27.2 the morphism $X' \to \overline{X}$ is finite. By Morphisms, Lemma 28.42.16 $X' \to \overline{X}$ is projective. By Morphisms, Lemma 28.41.16 we see that $X' \to \overline{X}$ is H-projective. By Morphisms, Lemma 28.41.7 we see that $X' \to \mathop{\mathrm{Spec}}(k)$ is H-projective. Let $\{ x'_1, \ldots , x'_ m\} \subset X'$ be the inverse image of $\{ x_1, \ldots , x_ n\} = \overline{X} \setminus X$. Then $\dim (\mathcal{O}_{X', x'_ i}) = 1$ for all $1 \leq i \leq m$. Hence the local rings $\mathcal{O}_{X', x'}$ are discrete valuation rings by Morphisms, Lemma 28.51.16. Then $X \to X'$ and $\{ x'_1, \ldots , x'_ m\}$ is as desired. $\square$
Observe that if an affine scheme $X$ over $k$ is proper over $k$ then $X$ is finite over $k$ (Morphisms, Lemma 28.42.11) and hence has dimension $0$ (Algebra, Lemma 10.52.2 and Proposition 10.59.6). Hence a scheme of dimension $> 0$ over $k$ cannot be both affine and proper over $k$. Thus the possibilities in the following lemma are mutually exclusive.
Lemma 32.42.7. Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$.
Proof. Choose $X \to \overline{X}$ as in Lemma 32.42.5. By Lemma 32.37.4 we can find a globally generated invertible sheaf $\mathcal{L}$ on $\overline{X}$ and a section $s \in \Gamma (\overline{X}, \mathcal{L})$ such that $X = \overline{X}_ s$. Choose a basis $s = s_0, s_1, \ldots , s_ m$ of the finite dimensional $k$-vector space $\Gamma (\overline{X}, \mathcal{L})$ (Cohomology of Schemes, Lemma 29.19.2). We obtain a corresponding morphism
$f : \overline{X} \longrightarrow \mathbf{P}^ m_ k$
such that the inverse image of $D_{+}(T_0)$ is $X$, see Constructions, Lemma 26.13.1. In particular, $f$ is non-constant, i.e., $\mathop{\mathrm{Im}}(f)$ has more than one point. A topological argument shows that $f$ maps the generic point $\eta$ of $\overline{X}$ to a nonclosed point of $\mathbf{P}^ n_ k$. Hence if $y \in \mathbf{P}^ n_ k$ is a closed point, then $f^{-1}(\{ y\} )$ is a closed set of $\overline{X}$ not containing $\eta$, hence finite. By Cohomology of Schemes, Lemma 29.21.21 we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$ is affine. $\square$
The following lemma combined with Lemma 32.42.2 tells us that given a separated scheme $X$ of dimension $1$ and of finite type over $k$, then $X \setminus Z$ is affine, whenever the closed subset $Z$ meets every irreducible component of $X$.
Lemma 32.42.8. Let $X$ be a separated scheme of finite type over $k$. If $\dim (X) \leq 1$ and no irreducible component of $X$ is proper of dimension $1$, then $X$ is affine.
Proof. Let $X = \bigcup X_ i$ be the decomposition of $X$ into irreducible components. We think of $X_ i$ as an integral scheme (using the reduced induced scheme structure, see Schemes, Definition 25.12.5). In particular $X_ i$ is a singleton (hence affine) or a curve hence affine by Lemma 32.42.7. Then $\coprod X_ i \to X$ is finite surjective and $\coprod X_ i$ is affine. Thus we see that $X$ is affine by Cohomology of Schemes, Lemma 29.13.3. $\square$
[1] One can avoid using this lemma which relies on the theorem of formal functions. Namely, $\overline{X}$ is projective hence it suffices to show a proper morphism $f : X \to Y$ with finite fibres between quasi-projective schemes over $k$ is finite. To do this, one chooses an affine open of $X$ containing the fibre of $f$ over a point $y$ using that any finite set of points of a quasi-projective scheme over $k$ is contained in an affine. Shrinking $Y$ to a small affine neighbourhood of $y$ one reduces to the case of a proper morphism between affines. Such a morphism is finite by Morphisms, Lemma 28.42.7.
Comment #1425 by Erik Visse on
In the paragraph following Lemma 32.28.4 there is a reference to Lemma 28.44.7. In section 28.44 there is another lemma (28.44.10) whose proof is basically Lemma 28.44.7 and some easy facts that states exactly the (first) result in the above mentioned paragraph. Maybe the reference could be changed to 28.44.10 for convenience.
Comment #4588 by Fred Vu on
In Lemma 32.42.2 (tag 0A24), the last sentence of the proof should read $X \setminus \\{ x \\} \to \mathop{\mathrm{Spec}}(k)$ instead of $X \to \mathop{\mathrm{Spec}}(k)$.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9981867074966431, "perplexity": 127.83380361192246}, "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/1573496664437.49/warc/CC-MAIN-20191111191704-20191111215704-00088.warc.gz"} |
http://mathhelpforum.com/calculus/90421-volume-ice-cream-cone-print.html | Volume of ice cream cone
• May 25th 2009, 08:26 AM
Frostking
Volume of ice cream cone
I need to find the volume of an ice cream cone bound by the hemisphere z = (8 = x^2 - y^2)^1/2 and the cone z = (x^2 + y^2)^1/2
I know that is is a double integral but I can't even get started from there. Would someone please push or point me in the right direction? Thanks much for your effort. Frostking
• May 25th 2009, 08:44 AM
Spec
$\sqrt{x^2+y^2}\leq z \leq \sqrt{8 - x^2 - y^2}$
The projection on the xy-plane, $E$, is where the two surfaces meet: $\sqrt{x^2+y^2}=\sqrt{8 - x^2 - y^2} \Longleftrightarrow x^2+y^2=4$
So you can calculate it like this: $\iint_E \left( \sqrt{8 - x^2 - y^2}-\sqrt{x^2+y^2}\right)dxdy$
• May 25th 2009, 08:52 AM
Spec
To understand why we are doing this, consider this: $\int_{\sqrt{x^2+y^2}}^{\sqrt{8 - x^2 - y^2}}\left(\iint_E dxdy\right)dz=\iint_E \left( \sqrt{8 - x^2 - y^2}-\sqrt{x^2+y^2}\right)dxdy$
• May 25th 2009, 08:59 AM
Frostking
So, if x^2 + y^2 is four then my bounds should be from 0 to 2 for both x and y is this correct? Frostking
• May 25th 2009, 09:07 AM
Spec
$x^2+y^2=4$ is a circle with the radius 2. So both x and y go from -2 to 2. I always switch to polar coordinates for problems like these though.
Either go with $-\sqrt{4-x^2}\leq y \leq\sqrt{4-x^2},\ -2\leq x\leq 2$ or the other way around $-\sqrt{4-y^2}\leq x \leq\sqrt{4-y^2},\ -2\leq y\leq 2$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9029474258422852, "perplexity": 442.90598874407675}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982958896.58/warc/CC-MAIN-20160823200918-00217-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://d2mvzyuse3lwjc.cloudfront.net/pdfs/NAG26/Manual/html/f08/f08jyc.html | # NAG Library Function Document
## 1Purpose
nag_zstegr (f08jyc) computes selected eigenvalues and, optionally, the corresponding eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix.
## 2Specification
#include #include
void nag_zstegr (Nag_OrderType order, Nag_JobType job, Nag_RangeType range, Integer n, double d[], double e[], double vl, double vu, Integer il, Integer iu, Integer *m, double w[], Complex z[], Integer pdz, Integer isuppz[], NagError *fail)
## 3Description
nag_zstegr (f08jyc) computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix $T$. That is, the function computes the (partial) spectral factorization of $T$ given by
$ZΛZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the selected eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the corresponding eigenvectors, ${z}_{i}$, of $T$. Thus
$Tzi= λi zi , i = 1,2,…,m$
where $m$ is the number of selected eigenvectors computed.
The function stores the real orthogonal matrix $Z$ in a complex array, so that it may also be used to compute selected eigenvalues and the corresponding eigenvectors of a complex Hermitian matrix $A$ which has been reduced to tridiagonal form $T$:
$QZΛQZH, where Q is unitary.$
In this case, the matrix $Q$ must be explicitly applied to the output matrix $Z$. The functions which must be called to perform the reduction to tridiagonal form and apply $Q$ are:
full matrix nag_zhetrd (f08fsc) and nag_zunmtr (f08fuc) full matrix, packed storage nag_zhptrd (f08gsc) and nag_zupmtr (f08guc) band matrix nag_zhbtrd (f08hsc) with ${\mathbf{vect}}=\mathrm{Nag_FormQ}$ and nag_zgemm (f16zac).
This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. nag_zstegr (f08jyc) can usually compute all the eigenvalues and eigenvectors in $O\left({n}^{2}\right)$ floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the $QR$ algorithm.
## 4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151
## 5Arguments
1: $\mathbf{order}$Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{job}$Nag_JobTypeInput
On entry: indicates whether eigenvectors are computed.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Only eigenvalues are computed.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$.
3: $\mathbf{range}$Nag_RangeTypeInput
On entry: indicates which eigenvalues should be returned.
${\mathbf{range}}=\mathrm{Nag_AllValues}$
All eigenvalues will be found.
${\mathbf{range}}=\mathrm{Nag_Interval}$
All eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\mathrm{Nag_Indices}$
The ilth through iuth eigenvectors will be found.
Constraint: ${\mathbf{range}}=\mathrm{Nag_AllValues}$, $\mathrm{Nag_Interval}$ or $\mathrm{Nag_Indices}$.
4: $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
5: $\mathbf{d}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ diagonal elements of the tridiagonal matrix $T$.
On exit: d is overwritten.
6: $\mathbf{e}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: ${\mathbf{e}}\left[0\right]$ to ${\mathbf{e}}\left[{\mathbf{n}}-2\right]$ are the subdiagonal elements of the tridiagonal matrix $T$. ${\mathbf{e}}\left[{\mathbf{n}}-1\right]$ need not be set.
On exit: e is overwritten.
7: $\mathbf{vl}$doubleInput
8: $\mathbf{vu}$doubleInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, vl and vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Indices}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
9: $\mathbf{il}$IntegerInput
10: $\mathbf{iu}$IntegerInput
On entry: if ${\mathbf{range}}=\mathrm{Nag_Indices}$, il and iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$ or $\mathrm{Nag_Interval}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
• if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.
11: $\mathbf{m}$Integer *Output
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_AllValues}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\mathrm{Nag_Indices}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
12: $\mathbf{w}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: the eigenvalues in ascending order.
13: $\mathbf{z}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array z must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdz}}×{\mathbf{n}}\right)$ when ${\mathbf{job}}=\mathrm{Nag_DoBoth}$;
• $1$ otherwise.
The $\left(i,j\right)$th element of the matrix $Z$ is stored in
• ${\mathbf{z}}\left[\left(j-1\right)×{\mathbf{pdz}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{z}}\left[\left(i-1\right)×{\mathbf{pdz}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, then if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the columns of z contain the orthonormal eigenvectors of the matrix $T$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left[i-1\right]$.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, z is not referenced.
14: $\mathbf{pdz}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
Constraints:
• if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdz}}\ge 1$.
15: $\mathbf{isuppz}\left[\mathit{dim}\right]$IntegerOutput
Note: the dimension, dim, of the array isuppz must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$.
On exit: the support of the eigenvectors in $Z$, i.e., the indices indicating the nonzero elements in $Z$. The $i$th eigenvector is nonzero only in elements ${\mathbf{isuppz}}\left[2×i-2\right]$ through ${\mathbf{isuppz}}\left[2×i-1\right]$.
16: $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).
## 6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_CONVERGENCE
Inverse iteration failed to converge.
The $\mathrm{dqds}$ algorithm failed to converge.
NE_ENUM_INT_2
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_DoBoth}$, ${\mathbf{pdz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdz}}\ge 1$.
NE_ENUM_INT_3
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{il}}=〈\mathit{\text{value}}〉$, ${\mathbf{iu}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
if ${\mathbf{range}}=\mathrm{Nag_Indices}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.
NE_ENUM_REAL_2
On entry, ${\mathbf{range}}=〈\mathit{\text{value}}〉$, ${\mathbf{vl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{vu}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{range}}=\mathrm{Nag_Interval}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdz}}>0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
## 7Accuracy
See Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.
## 8Parallelism and Performance
nag_zstegr (f08jyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zstegr (f08jyc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to ${n}^{2}$.
The real analogue of this function is nag_dstegr (f08jlc).
## 10Example
This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
$T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .$
### 10.1Program Text
Program Text (f08jyce.c)
### 10.2Program Data
Program Data (f08jyce.d)
### 10.3Program Results
Program Results (f08jyce.r)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 140, "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": 0.9841901063919067, "perplexity": 2330.205087322679}, "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/1642320303717.35/warc/CC-MAIN-20220121222643-20220122012643-00022.warc.gz"} |
https://math.stackexchange.com/questions/2990370/proof-that-uparrow-is-the-unique-solution-of-tinyg-g/2992628 | # Proof that “$\uparrow$ is the unique solution of $tiny(G) = G$”
Tiny & miny games can be defined as: $$tiny(G) = \{0||0|-G\}$$ $$miny(G) = -tiny(G) = \{G|0||0\}$$
Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "$$\uparrow$$ is the unique solution of $$tiny(G) = G$$"
$$\uparrow$$ is generally defined as $$\{0|*\}$$. Interestingly, $$tiny(0)=\{0||0|0\}=\{0|*\}=\uparrow$$ & $$miny(0)=\{0|0||0\}=\{*|0\}=\downarrow$$.
How is $$\uparrow$$ the unique solution to $$tiny(G) = G$$?
Additionally, if not apparent, what makes this amusing?
I will use $$\pmb{+}_G$$ for $$\text{tiny-}G$$ throughout, but the unicode symbol ⧾ is arguably more correct for that plus sign.
## Why Amusing?
This is a typo in the wikipedia page at the time of writing. On page 215 of On Numbers and Games, it actually says:
...it is amusing to verify that for any game $$G$$, we have $$\pmb{+}_{\pmb{+}_{\pmb{+}_G}}=\uparrow$$, so that in particular, $$\uparrow$$ is the unique solution of $$G=\pmb{+}_G$$.
Without asking, I can't be certain why John H. Conway found it amusing, but I personally find it amusing to work through why something complicated and with an arbitrary game parameter like $$\pmb{+}_{\pmb{+}_{\pmb{+}_G}}\cong\{ 0\Vert0|-\{ 0\Vert0|-\{ 0\Vert0|-G\} \} \}$$ simplifies to something as simple as $$\uparrow\cong\{ 0\Vert0|0\}$$.
## Why $$\uparrow$$?
Verifying the claims in the quote above is a problem in Chapter 5 of Lessons in Play: An Introduction to Combinatorial Game Theory (tinies and minies are introduced in section 5.4). It's an amusing exercise, so I don't want to spoil the whole thing, but I can clarify the second part a bit.
Once we have $$\pmb{+}_{\pmb{+}_{\pmb{+}_G}}=\uparrow$$, then proving that $$\uparrow$$ is the unique solution (up to equality) to $$\pmb{+}_G=G$$ does not require any game theory. It's a general fact that if we have a function $$f:X\to X$$ and a particular $$y\in X$$ such that $$\forall x\in X,f(f(f(x)))=y$$, then $$y$$ is the only solution to $$f(x)=x$$. Can you see why?
• Thank you! This is incredibly helpful. I will take a crack at proving $\pmb{+}_{\pmb{+}_{\pmb{+}_G}}=\uparrow$ for myself. Also, I am not sure what exactly is meant by $*\cong\{ 0\Vert0|0\}$ It seems this would imply $*\cong\{0|*\}$ and thus $*\cong\uparrow$. What does $\cong$ mean in this case? Is confused with? – meowzz Nov 10 '18 at 16:46
• That was a typo of mine. I meant $\uparrow$ there, not $*$. $\cong$ can mean "isomorphic to"/"identical to" since in CGT $=$ is merely an equivalence relation. – Mark S. Nov 10 '18 at 17:11
• Got it! What symbol in CGT is best for fuzzy /“is confused with” ($\cong$, $=$, ?)? Additionally, does fuzzy (equal?) isomorphic? Or? – meowzz Nov 10 '18 at 17:28
• @meowzz That's a little off topic, but I've just added the standard symbols for "confused with" to my answer distinguishing "confused with" from "fuzzy". "Isomorphic" is a general/common term across mathematics. In the context of combinatorial games, Lessons in Play says "two games are isomorphic if they have identical game trees". This has nothing to do with the relation of being confused; two isomorphic games are necessarily equal, so they can't possibly be confused with each other. – Mark S. Nov 10 '18 at 17:46
• @meowzz, Do you already know theorems about (in)equality of combinatorial games? You'll need at least a couple to do this exercise from the middle of a textbook. If you don't have the background already, I'd really recommend tracking down a copy of Lessons in Play. And if you do, but are stuck in the calculations, you could post a new question and show all your work/where you're getting stuck. – Mark S. Nov 11 '18 at 3:24 | {"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": 27, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8401667475700378, "perplexity": 292.5909407881124}, "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-2019-22/segments/1558232256215.47/warc/CC-MAIN-20190521022141-20190521044141-00077.warc.gz"} |
http://en.dsplib.org/content/resampling_spline.html | Farrow Filter of Signal Digital Resampling on the Basis of Spline Interpolation
Contents
Introduction
In the previous section we considered digital resamplers on the basis of Lagrange piecewise and polynomial interpolation. The equation for polynomial coefficients was received in the form of:
\begin{equation*} \begin{array} & \mathbf{a} = \mathbf{M^{-1}} \cdot \mathbf{s} = \left[ \begin{array}{cccc} 0 & 0 & 1 & 0\\ \frac{1}{6} & -1 & \frac{1}{2} & \frac{1}{3}\\ 0 & \frac{1}{2} & -1 & \frac{1}{2}\\ -\frac{1}{6} & \frac{1}{2} & -\frac{1}{2}& \frac{1}{6}\\ \end{array} \right] \cdot \left[ \begin{array}{c} s(n-3) \\ s(n-2) \\ s(n-1) \\ s(n)\\ \end{array} \right], \end{array} \end{equation*}
(1)
where $\mathbf{a}$ is a vector of polynomial coefficients, $\mathbf{M^{-1}}$is an inverse system matrix, $\mathbf{s}$ is a vector of delayed input signal samples.
As a result of multiplier minimization equations for polynomial coefficients were received:
$$\begin{cases} a_0 = s(n-1),\\ a_3= \frac{1}{6} \cdot \left( s(n) - s(n-3)\right) + \frac{1}{2} \cdot \left( s(n-2) - s(n-1)\right), \\ a_1 = \frac{1}{2} \cdot \left( s(n) - s(n-2) \right)-a_3,\\ a_2 = s(n) - s(n-1) - a_1 - a_3.\\ \end{cases}$$
(2)
The functional chart of the optimized signal digital resampling filter on the basis of Lagrange piecewise and polynomial interpolation corresponding to (2) is given in Figure 1.
Figure 1. Functional Chart of the Optimized Digital Resampling Filter
Thus, we received the filter structure which calculates coefficients of Lagrange polynomial interpolation using only one multiplier by $\frac{1}{6}$ and two trivial multipliers by $\frac{1}{2}$.
We also considered in details examples of using Farrow filter for problems of signal digital resampling including digital compensation of fractional delay and signal interpolation.
Upon considering examples we mentioned that the pulse characteristic $h(n)$ of Farrow filter under signal digital interpolation has no continuous derivative in interpolation knots. As a result, the rejection level within the scope of the received filter blocking is only 28 dB as it is shown in Figure 2.
Figure 2. Pulse Characteristic and AFR of Digital Interpolation Farrow Filter
Besides Lagrange interpolation there are also other methods of piecewise and polynomial interpolation, for example, spline interpolation [1] which provides continuous derivatives in interpolation knots in contrast to Lagrange polynomial interpolation.
In this section we will consider creating Farrow filter on the basis of Hermite splines [1].
Creating Cubic Hermite Spline
Upon creating cubic Lagrange polynomial four signal samples $s(n) \ldots s(n-3)$ are used and the received polynomial interpolation $p(t)$ passes through these knots as it is shown in Figure 3a.
Figure 3. Creating Polynomial Interpolation: a - Cubic Lagrange Polynomial b - Cubic Hermite Spline
However such creating polynomial does not impose restrictions upon values of derivatives in extreme points $t = 1$ and $t = -2$. As a result there is discontinuity of the pulse characteristic derivative of the interpolator filter as it is shown in Figure 2.
To provide derivative continuity upon “conjugating” cubic polynomial we will use cubic Hermite spline which is created in the interval $t = [-1 \ldots 0]$ as it is shown in Figure 3b.
We will calculate Hermite spline coefficients by solving the linear equations set. Two set equations result from Figure 3b:
\begin{eqnarray*} { \begin{cases} p(0)= s(n-1), \\ p(-1) = s(n-2); \end{cases} } & \Longrightarrow & { \begin{cases} a_0 + a_1 \cdot 0 + a_2 \cdot 0^2 + a_3 \cdot 0^3 = s(n-1), \\ a_0 + a_1 \cdot (-1) + a_2 \cdot (-1)^2 + a_3 \cdot (-1)^3 = s(n-2). \end{cases} } \end{eqnarray*}
(3)
It is necessary to add two more equations into the set (3). For this purpose we demand for Hermite spline derivatives $p^\prime(t)$ in the knots $t = 0$ and $t = -1$ to be equal to derivatives of the input signal $s^\prime(n)$ in these points, i.e.
\begin{eqnarray*} { \begin{cases} p^\prime(0)= s^\prime(n-1), \\ p^\prime(-1) = s^\prime(n-2); \end{cases} } & \Longrightarrow & { \begin{cases} a_1 + 2 \cdot a_2 \cdot 0 + 3 \cdot a_3 \cdot 0^2 = s^\prime(n-1), \\ a_1 + 2 \cdot a_2 \cdot (-1) + 3\cdot a_3 \cdot (-1)^2 = s^\prime(n-2). \end{cases} } \end{eqnarray*}
(4)
Uniting Equations (3) and (4) in the unified equations set to calculate cubic Hermite spline coefficients, we get:
\begin{equation*} \begin{cases} a_0 &+ & a_1 \cdot 0 + a_2 \cdot 0^2 + a_3 \cdot 0^3 & = s(n-1), \\ a_0 & + & a_1 \cdot (-1) + a_2 \cdot (-1)^2 + a_3 \cdot (-1)^3 & = s(n-2), \\ & & a_1 + 2 \cdot a_2 \cdot 0 + 3 \cdot a_3 \cdot 0^2 & = s^\prime(n-1), \\ & & a_1 + 2 \cdot a_2 \cdot (-1) + 3\cdot a_3 \cdot (-1)^2 & = s^\prime(n-2), \end{cases} \end{equation*}
(5)
Or in the matrix form:
\begin{equation*} \begin{array} & \mathbf{M \cdot a = s} & \Longrightarrow & \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 1 & -1 & 1 & -1 \\ 0 & 1 & 0 & 0\\ 0 & 1 & -2 & 3 \end{array} \right] & \cdot & \left[ \begin{array}{cccc} a_0 \\ a_1 \\ a_2 \\ a_3 \end{array} \right] & = & \left[ \begin{array}{cccc} s(n-1) \\ s(n-2) \\s^\prime(n-1) \\ s^\prime(n-2) \end{array} \right]. \end{array} \end{equation*}
(6)
Then the system solution (6) can be as follows:
\begin{equation*} \begin{array} & \mathbf{ a = M^{-1} \cdot s} & \Longrightarrow & \left[ \begin{array}{cccc} a_0 \\ a_1 \\ a_2 \\ a_3 \end{array} \right] & = & \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -3 & 3 & 2 & 1\\ -2 & 2 & 1 & 1 \end{array} \right] & \cdot & \left[ \begin{array}{cccc} s(n-1) \\ s(n-2) \\s^\prime(n-1) \\ s^\prime(n-2) \end{array} \right]. \end{array} \end{equation*}
(7)
After multiplying the return matrix $\mathbf{M^{-1}}$ by the vector $\mathbf{s}$ we get the equations for cubic Hermite spline coefficients in the form:
\begin{equation*} \begin{cases} a_0 = s(n-1), \\ a_1 = s^\prime(n-1), \\ a_2 = -3 \cdot s(n-1) + 3 \cdot s(n-2) + 2 \cdot s^\prime(n-1) + s^\prime(n-2), \\ a_3 = -2 \cdot s(n-1) + 2 \cdot s(n-2) + s^\prime(n-1) + s^\prime(n-2). \end{cases} \end{equation*}
(8)
Thus, we received the equations for cubic Hermite spline coefficients. At the same time the coefficients depend on values of signal derivatives $s^\prime(n-1)$ and $s^\prime(n-2)$ which we have to value taking into account input signal samples.
Values of a Discrete Signal Derivative
The numerical differentiation problem of discretely set signals is solved when using approximation of a signal derivative by means of final differences. The simplest derivative approximation is the finite difference of the form:
$$s^\prime(n-1) \approx s(n-1) - s(n-2), \\ s^\prime(n-2) \approx s(n-2) - s(n-3).$$
(9)
Values of derivatives (9) are realized upon using the input signal linear interpolation. This method requires only one subtraction, however, it has the greatest inaccuracy as the residual member in case of derivative value (9) is equal to $O(h)$ [2], i.e. it decreases linearly with reducing the sampling step.
The more exact derivative value method is the central difference method:
$$s^\prime(n-1) \approx \frac{s(n) - s(n-2)}{2}, \\ s^\prime(n-2) \approx \frac{s(n-1) - s(n-3)}{2}.$$
(10)
The central difference results from the derivative value with using the input signal parabolic interpolation, and the residual member in case of the derivative value (10) is equal to $O(h^2)$ [2], i.e. it decreases quadratically in case of reducing the sampling step $h$.
At the same time the central difference (10) requires additional multiplication by $\frac{1}{2}$ which can be realized in integer arithmetics as the single position shifting to the right.
Thus, we can use values of derivative (10) in Equation (8) to calculate Hermite spline coefficients and provide derivative continuity at the digital resampling filter output.
Optimized Structure of Farrow Filter on the Basis of Hermite Splines
Plug (10) into (8) and get equations for Hermite spline coefficients in the form:
\begin{equation*} \begin{cases} a_0 = s(n-1), \\ a_1 = \frac{1}{2}\cdot (s(n) - s(n-2)), \\ a_2 = -3 \cdot s(n-1) + 3 \cdot s(n-2) + s(n) - s(n-2) + \frac{1}{2}\cdot (s(n-1) - s(n-3)), \\ a_3 = -2 \cdot s(n-1) + 2 \cdot s(n-2) + \frac{1}{2}\cdot (s(n) - s(n-2)) + \frac{1}{2}\cdot (s(n-1) - s(n-3)). \end{cases} \end{equation*}
(11)
Equation (11) allows to note that:
$$a_2 - a_3 = -s(n-1) + s(n-2) + \underbrace{\frac{1}{2}\cdot (s(n) - s(n-2))}_{a_1},$$
(12)
then
$$a_2 = s(n-2)-s(n-1) + a_3 + a_1,$$
(13)
and finally it is possible to write down:
$$\begin{cases} a_0 = s(n-1), \\ a_1 = \frac{1}{2}\cdot \Big(s(n) - s(n-2)\Big), \\ a_3 = 2 \cdot \Big(s(n-2) - s(n-1)\Big) + a_1 + \frac{1}{2}\cdot \Big(s(n-1) - s(n-3)\Big), \\ a_2 = s(n-2)-s(n-1) + a_3 + a_1. \end{cases}$$
(14)
Calculation of Hermite spline coefficients requires only multiplication by $2$ and $\frac{1}{2}$ which can be considered as trivial. Besides we can get rid of multiplication by $\frac{1}{2}$ when calculating $a_3$ if we consider that $\frac{1}{2}\cdot \Big(s(n-1) - s(n-3)\Big)$ is nothing but the delayed single position value $a_1 = \frac{1}{2}\cdot \Big(s(n) - s(n-2)\Big)$.
The functional chart of the optimized digital resampling filter on the basis of cubic Hermite splines is shown in Figure 4.
Figure 4. Optimized Digital Resampling Filter on the Basis of Cubic Hermite Splines
Comparing Figures 1 and 4, it is possible to note that the optimized digital resampling filter on the basis of cubic Hermite splines requires only one trivial multiplication by $\frac{1}{2}$ (multiplication by 2 can be replaced with one adder) while the filter on the basis of Lagrange polynomial interpolation requires two trivial multiplication by $\frac{1}{2}$ and one multiplier by $\frac{1}{6}$. The total quantity of adders required to calculate coefficients is also less when using cubic Hermite splines.
Conclusions
In this section the Farrow filter structure of signal digital resampling on the basis of cubic Hermite splines is considered. The question of input signal derivative values to provide derivative continuity when using cubic Hermite splines is also considered.
The received digital resampling filter requires only one multiplication by $\frac{1}{2}$ which can be considered as trivial. The total quantity of adders required to calculate polynomial coefficients is also less when using cubic Hermite splines in comparison with Lagrange polynomial interpolation.
In the following section we will consider examples of using the Farrow filter of signal digital resampling on the basis of cubic Hermite splines and compare results of this using with the signal resampling filter on the basis of Lagrange polynomial interpolation.
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Reference
[1] Kahaner D., Moler C., Nash S. Numerical Methods and Software. Prentice Hall, 1988.
[2] Farrow C.W. A Continuously Variable Digital Delay Element. Circuits and Systems, IEEE International Symposium. 1988, p. 2641–2645. vol. 3
[3] Gardner Floyd M. Interpolation in Digital Modems-Part I: Fundamentals: IEEE Transactions on Communications, Vol. 41, No. 3, March 1993, P. 501-507.
[4] Erup L., Gardner Floyd M., Harris Robert A. Interpolation in Digital Modems-Part II: Implementation and Performance: IEEE Transactions on Communications, Vol. 41, No. 6, June 1993, p.998-1008.
[5] Franck A. Efficient Algorithms for Arbitrary Sample Rate Conversion with Application to Wave Field Synthesis. PhD thesis. Universitätsverlag Ilmenau, Ilmenau, 2012. [PDF]
[6] McConnell J. Analysis of Algorithms: An Active Learning Approach. Jones and Bartlett Publishers, 2001.
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https://mathinsight.org/assess/math2241/solving_2x2_linear_systems/overview | # Math Insight
### Overview of: Solving 2x2 linear systems
Converting between systems of linear equations and matrix equations. Using the method of elimination to solve the systems.
#### Points and due date summary
Total points: 1
Assigned: Jan. 26, 2022, 11:15 a.m.
Due: Feb. 2, 2022, 11:59 p.m. | {"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": 0.9247517585754395, "perplexity": 4531.094908934294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00272.warc.gz"} |
http://math.stackexchange.com/users/54388/piotr-semenov?tab=activity&sort=comments | # Piotr Semenov
less info
reputation
8
bio website linkedin.com/in/piotrsemenov location Russian Federation, Saint-Petersburg age 28 member for 1 year, 7 months seen May 12 at 20:44 profile views 33
Researcher and software engineer.
Feb7 comment How many ways to distribute $k$ indistinguishable balls over $m$ of $n$ distinguishable bins of finite capacity $l$? sure, I specified the upper limit just to see the finite summation. Thank you for your answer! Feb6 comment How many ways to distribute $k$ indistinguishable balls over $m$ of $n$ distinguishable bins of finite capacity $l$? Just to improve your answer, the upper limit of sum is $\lfloor \frac{k}{l} \rfloor-1$. Feb6 comment How many ways to distribute $k$ indistinguishable balls over $m$ of $n$ distinguishable bins of finite capacity $l$? Just a small typo in your answer. Add $l$ balls to first bin, one have to distribute $k-l$ balls (instead of "$m-l$ units"). Feb6 comment How many ways to distribute $k$ indistinguishable balls over $m$ of $n$ distinguishable bins of finite capacity $l$? Can you kindly explain, why $\binom{k-jl-1}{m-1}$ and not $\binom{k-jl-1}{m-j-1}$? Feb3 comment Number of ways to put $n$ unlabeled balls in $k$ bins with a max of $m$ balls in each bin @Brian What are the sum limits in your formula? The expression from book Enumerative Combinatorics (p. $360$) is $L(n,k,m)=\sum_{j=0}^k (-1)^j C_k^j C_{k+n-j(m+1)-1}^{k-1}$ according to your notation, where $C_k^j$ means the binomial coefficient. Unfortunately, $L(n,k,m)=0$ for any $n \leq k \cdot m$ (Mathematica' showed me that). Somewhere the mistake lives. Can you kindly help me with this problem? Oct21 comment When is a cyclotomic polynomial over a finite field a minimal polynomial? Thanks! So I did a mistake while suggesting that cyclotomic polynomials are also irreducible in case of finite fields. Oct21 comment When is a cyclotomic polynomial over a finite field a minimal polynomial? Please explain it for the following example. Given finite field $\mathrm{F}_q$, I have a decomposition $x^n-1=\prod_{i=0}^{m-1}f_i(x)$ over $\mathrm{F}_q$, where all $f_i(x)$ are cyclotomic. I am not sure that all $f_i(x)$ are minimal polynomials for some elements from extension $[\mathrm{F}_q^r : \mathrm{F}_q]$, where $r$ is a smallest number that $n$ divides $q^r - 1$. What is wrong in that? Jun26 comment Finding Eigenvalues and Eigenvectors weird equations I got my answers just be definition of eigenvector: for matrix $A$ the eigenvector $v$ corresponding to eigenvalue $\lambda$ is the solution of system of linear equations $(A - \lambda \cdot I) \cdot v = 0$. In another words, dot product of matrix $(A - \lambda \cdot I)$ rows and vector $v$ vanishes. So I have got the equation $2x - y - 2z = 0$ for $[x, y, z]$ being the eigenvector we are looking for. Setting $x$ to $1$ you can got the eigenvectors $[1, 0, 1]$ and $[1, 2, 0]$ correspondingly. Jun26 comment Finding Eigenvalues and Eigenvectors weird equations Just consider equations $2 \cdot x - y - 2 \cdot z = 0$. These are for example, $[x,y,z] = [1, 0, 1]$ and $[x,y,z] = [1,2,0]$. These ones are the basis of eigenspace corresponding to eigenvalue $1$. May23 comment Multivariate normal distribution density function @Babla Sorry for delay. Let $Y_1, Y_2$ have the covariance matrices $A, B$ correspondingly. Let $Cov(Y_1,Y_2)=0$. So the covariance matrix of $2n$-dimensional variable $(Y_1,Y_2)$ is the just $\Sigma=\left(\begin{array} AA & 0 \\ 0 & B \end{array}\right)$. Note that $|\Sigma|=|A||B|$ and $\Sigma^{-1}=\left(\begin{array} AA^{-1} & 0 \\ 0 & B^{-1} \end{array}\right)$. Put this in density function $f_{(Y_1,Y_2)}(x)$ and it decomposes to $f_{Y_1}(x) \cdot f_{Y_2}(x)$. This is the independence of Gaussian variables $Y_1,Y_2$. Hope, it helps. Unfortunately, I cannot update my answer with this text. May18 comment Multivariate normal distribution density function @Babla $Y_1$ and $Y_2$ are uncorrelated if and only if $Cov(Y_1, Y_2) = 0$. This means that $Y_1$ and $Y_2$ variates in ways those do not share any statistical data between. From $Cov(Y_1, Y_2) = 0$ you can not reason anything about independence of $Y_1$ and $Y_2$. It must be only the a-priori assumption. May18 comment $0$-th moment of product of gaussian and sinc function @JFNJr See my answer. Unfortunately, Mathematica was able to evaluate only the first integral. May18 comment Multivariate normal distribution density function @Babla I see. I have rewritten my answer according to your comment. May18 comment Multivariate normal distribution density function @Babla $Cov(\mathbf{c} + B \cdot \mathbf{X}, \mathbf{c} + B \cdot \mathbf{X}) = B \cdot \Sigma \cdot B^\top$, where $\Sigma$ is covariance matrix of random Gaussian vector $\Sigma$. May be I do not understand your problem... May18 comment $0$-th moment of product of gaussian and sinc function If you need only the answer, I can calculate it with help of Wolfram Mathematica. | {"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": 0.9719989895820618, "perplexity": 325.1547084101113}, "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/1406510267075.55/warc/CC-MAIN-20140728011747-00156-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/144656-parametric-equation.html | 1. ## parametric equation
Show by eliminating the parameter $\theta$ that the following parametric equations represent a hyperbola:
$
x = a\tan\theta
$
$
y = b\sec\theta
$
Not really sure how to even start this problem. Any help would be appreciated!
2. Originally Posted by kevin11
Show by eliminating the parameter $\theta$ that the following parametric equations represent a hyperbola:
$
x = a\tan\theta
$
$
y = b\sec\theta
$
Not really sure how to even start this problem. Any help would be appreciated!
$x^2 = a^2\tan^2\theta$
$y^2 = b^2\sec^2\theta$
Use the identity $\sec^2\theta - tan^2\theta = 1$
Solve the problem. | {"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": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9823066592216492, "perplexity": 340.0257027911413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662336.11/warc/CC-MAIN-20160924173742-00293-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://brilliant.org/problems/shells-losing-energy/ | # Shells Losing Energy !
An arrangement of two concentric conducting spherical shells charged initially such that the inner shell (radius = $a$) carries a net charge of $+Q$ while outer shell (radius $b$) carries net charge $0$. If the switch "$S$" is closed, how much electrostatic potential energy would the system lose?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 10, "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": 0.9702472686767578, "perplexity": 904.130341527989}, "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/1566027316783.70/warc/CC-MAIN-20190822042502-20190822064502-00541.warc.gz"} |
http://electricalacademia.com/dc-machines/dc-generator-operation/ | Home / DC Machines / DC Generator Operation
DC Generator Operation
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When operated as a generator, the armature of the DC machine is driven by a prime mover. As the armature coils move through the flux created by the stator field, a voltage is induced in them. To understand the operation of the DC generator we need a specific relationship for the generated voltage in terms of the flux, the physical design of the machine, and its speed.
Generated voltage
$\begin{matrix} {{E}_{a}}=N\frac{d{{\phi }_{p}}}{dt}\text{ }or\text{ }{{E}_{a}}=N\frac{\Delta {{\phi }_{p}}}{\Delta t} & {} & \left( 1 \right) \\\end{matrix}$
Where N is the number of series conductors.
Rather than use the derivative form of equation 1, we can use the different form. During each revolution, a conductor cuts the flux of P poles, where P is the number of poles in the machine. Thus, the total amount of flux cut in one revolution is
$\begin{matrix} \Delta \phi =P{{\phi }_{p}} & {} & \left( 2 \right) \\\end{matrix}$
Where ϕp is the flux per pole.
If the speed of the machine in RPM is represented by n, then the time to complete one revolution is
$\begin{matrix} \Delta t=\frac{1}{{}^{n}/{}_{60}}=\frac{60}{n} & {} & \left( 3 \right) \\\end{matrix}$
Substituting equations 2 and 3 into 1 yields
$\begin{matrix} {{E}_{a}}=NP{{\phi }_{p}}\frac{n}{60} & {} & \left( 4 \right) \\\end{matrix}$
To find the number of series conductors in the armature winding, let Za be the total number of armature conductors and a be the number of parallel paths.
We have seen that the number of parallel paths in the armature winding is equal to the number of poles for a lap winding and is 2 for a wave winding.
The number of series conductors is
$\begin{matrix} N=\frac{{{Z}_{a}}}{a} & {} & \left( 5 \right) \\\end{matrix}$
Equation 5 can be substituted into equation 4 to yield an expression for the generated voltage of a DC generator:
$\begin{matrix} {{E}_{a}}=\left( \frac{P{{Z}_{a}}}{60a} \right)n{{\phi }_{p}}={{K}_{g}}n{{\phi }_{p}} & {} & \left( 6 \right) \\\end{matrix}$
Where
$\begin{matrix} {{K}_{g}}=\frac{P{{Z}_{a}}}{60a} & {} & \left( 7 \right) \\\end{matrix}$
Equation 6 is called the generated voltage equation for a DC generator and is an extremely important equation for the DC generator.
Kg is called the generator constant for a DC machine and is solely a function of the design of the machine— specifically, the number of poles and the type of winding. We can note several things about the voltage equation:
• Because of saturation effects in the iron of the machine, the flux per pole is a nonlinear function of the field current.
• Ea is a linear function of the flux per pole, so it is also nonlinear with respect to the field current.
• Ea is linear with respect to the rotational speed of the machine—to in radians/s or n in RPM.
DC Generator Characteristics Curves
Figure 1 shows relationships between generated voltage, field current, ϕ, and speed. The relationship between the flux per pole and the field current is shown in Figure 1 (a). Because the stator structure is ferromagnetic, it will exhibit the effects of both hysteresis and saturation. Thus, the flux shows both of these effects.
As the field current is increased, the flux increases until saturation of the core cause the flux to roll off. When the field current is then decreased, the domains within the steel are already aligned, so there is more flux when the field current is decreased than when it is increased.
Because the voltage is directly proportional to the flux, as shown by equation 5, the voltage will also exhibit hysteresis and saturation as the field current is changed with the machine running at constant speed. The magnetization curve is described in more detail in the next section.
FIGURE 1: DC Generator Characteristics Curves
Figure 1(b) shows the voltage as a function of field current; however, the hysteresis effect has been omitted. Three voltage curves are shown for three machine operating speeds, n 1 < n 2 < n3. Finally, Figure 1(c) shows the voltage as a function of speed for three values of field current.
The relationship between generated voltage and speed is linear if the field current (and thus the flux) is held constant. Because the generated voltage is directly proportionate to the speed of the machine, the voltage-speed curve passes through the origin.
DC Generator Magnetization Curve
Figure 2 showed the equivalent circuit for a separately excited DC generator. The field is supplied from a separate DC source that is applied to the shunt field of the machine.
FIGURE 2: Complete equivalent circuit of a separately excited DC generator.
The variation of the generated voltage, Ea, with the field current is shown by the magnetization curve or open-circuit characteristic, as shown in Figure 1(b).
The magnetization curve is obtained by driving the machine at a constant speed and varying the field current.
Assuming the field has been previously energized, the steel of the stator pole-faces will have some residual magnetism. This provides some flux, so the machine will have an output of a few volts even with the field circuit disconnected.
As the field current is increased, the flux is almost directly proportionate to the field current, so the voltage increases linearly with the field current. As the steel in the machine begins to saturate, however, the curve will bend over.
After some maximum value of field current is reached, reducing the field current will cause hysteresis to be observed—the voltage output of the machine will be higher when the current is decreased than it was when the current was increased. This difference may be only a few volts but is an easily observed demonstration of hysteresis.
The magnetization curve only has to be done at one speed because we can obtain one at any other speed from the equation for the generated voltage. Consider if we have two voltages at two different speeds, with the flux per pole held constant:
$\begin{matrix} {{E}_{a1}}={{K}_{g}}{{\phi }_{p}}{{n}_{1}} & {} & \left( 8 \right) \\\end{matrix}$
$\begin{matrix} {{E}_{a2}}={{K}_{g}}{{\phi }_{p}}{{n}_{2}} & {} & \left( 9 \right) \\\end{matrix}$
Dividing (8) by (9) and rearranging yields
$\begin{matrix} {{E}_{a2}}={{E}_{a1}}\frac{{{n}_{2}}}{{{n}_{1}}} & {} & \left( 10 \right) \\\end{matrix}$
Equation 10 allows us to easily calculate the generated voltage at any speed if we know it for one speed, assuming the field current is held constant.
Voltage Regulation
The terminal voltage for the separately excited DC generator shown in Figure 2 can be found by writing a voltage loop equation:
$\begin{matrix} {{V}_{t}}={{E}_{a}}-{{I}_{a}}{{R}_{a}} & {} & \left( 11 \right) \\\end{matrix}$
If the field voltage is kept constant, the field current and the flux per pole, ϕp, will also be constant.
If the generator is driven at constant speed, the internal voltage, Ea, will be constant, since all the terms on the right-hand side of equation 6 are constant. Under these conditions, equation 11 represents a linear relationship between the terminal voltage and the armature current, with a negative slope. Thus, as the armature load current increases, the terminal voltage will decrease linearly, as shown by the solid line in Figure 3.
FIGURE 3: Variation of terminal voltage with load current for a separately excited DC generator.
Generators are designed to produce rated voltage when delivering rated load current. For the separately excited generator, the terminal voltage is above the rated voltage at less than full load. To quantify the variation of terminal voltage with output, we define the voltage regulation, VR, in the same manner as for the transformer:
$\begin{matrix} VR=\frac{{{V}_{nl}}-{{V}_{fl}}}{{{V}_{fl}}}\times 100 & {} & \left( 12 \right) \\\end{matrix}$
Did you find apk for android? You can find new Free Android Games and apps. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.905386745929718, "perplexity": 696.1193967380065}, "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/1542039743732.41/warc/CC-MAIN-20181117185331-20181117211331-00253.warc.gz"} |
https://www.lessonplanet.com/teachers/global-winds | # Global Winds
Middle schoolers extend their understanding of convection to consider global winds and the effect of the earth's rotation on the creation of patterns of prevailing wind direction. | {"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": 0.9542814493179321, "perplexity": 1395.5364679486888}, "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-2018-09/segments/1518891812880.33/warc/CC-MAIN-20180220050606-20180220070606-00186.warc.gz"} |
https://planetmath.org/phasorsdemystified1 | # Phasors Demystified
Phasors Demystified Swapnil Sunil Jain Aug 7, 2006
Phasors Demystified
Suppose the following integro-differential equation is given in the time-domain11This is not the most general integro-differential equation but it has all the basic elements required for this discussion and hence the reader can easily extend this discussion for the more generalized case.:
$\displaystyle C_{1}\frac{d}{dt}y(t)+C_{2}\int_{-\infty}^{t}y(t)dt+C_{3}y(t)=x(t)$ (1)
where $y(t)$ and $x(t)$ are sinusoidal waveforms of the same frequency. Now, since $y(t)$ is a sinusoidal function it can be represented as $A_{y}\cos(\omega t+\phi_{y})$ and, similarly, x(t) can be represented as $A_{x}\cos(\omega t+\phi_{x})$. Furthermore, using the properties of complex numbers we can write
$\displaystyle y(t)=A_{y}\cos(\omega t+\phi_{y})=\Re(A_{y}e^{j\phi_{y}}e^{j% \omega t})$ $\displaystyle x(t)=A_{x}\cos(\omega t+\phi_{x})=\Re(A_{x}e^{j\phi_{x}}e^{j% \omega t})$
Now if we define the quantities $\tilde{Y}$ as $A_{y}e^{j\phi_{y}}$ and $\tilde{X}$ as $A_{x}e^{j\phi_{x}}$ (where $\tilde{Y}$ and $\tilde{X}$ are called phasors), then we can write the above expression in a more compact form as
$\displaystyle y(t)=\Re(\tilde{Y}e^{j\omega t})$ $\displaystyle x(t)=\Re(\tilde{X}e^{j\omega t})$
Now, using the above expression for $y(t)$ and $x(t)$ we can rewrite our original integro-differential equation as
$\displaystyle C_{1}\frac{d}{dt}\Re[\tilde{Y}e^{j\omega t}]+C_{2}\int_{-\infty}% ^{t}\Re[\tilde{Y}e^{j\omega t}]dt+C_{3}\Re[\tilde{Y}e^{j\omega t}]=\Re[\tilde{% X}e^{j\omega t}]$
Moving the derivative and the integral inside the $\Re$ operator we get
$\displaystyle C_{1}\Re\Big{[}\frac{d}{dt}\tilde{Y}e^{j\omega t}\Big{]}+C_{2}% \Re\Bigg{[}\int_{-\infty}^{t}\tilde{Y}e^{j\omega t}dt\Bigg{]}+C_{3}\Re[\tilde{% Y}e^{j\omega t}]=\Re[\tilde{X}e^{j\omega t}]$ $\displaystyle\Rightarrow C_{1}\Re[\tilde{Y}j\omega e^{j\omega t}]+C_{2}\Re[% \tilde{Y}\frac{e^{j\omega t}}{j\omega}]+C_{3}\Re[\tilde{Y}e^{j\omega t}]=\Re[% \tilde{X}e^{j\omega t}]$ $\displaystyle\Rightarrow\Re[\tilde{Y}j\omega C_{1}e^{j\omega t}]+\Re[\frac{% \tilde{Y}C_{2}}{j\omega}e^{j\omega t}]+\Re[\tilde{Y}C_{3}e^{j\omega t}]-\Re[% \tilde{X}e^{j\omega t}]=0$
$\displaystyle\Rightarrow\Re\Big{[}\tilde{Y}j\omega C_{1}e^{j\omega t}+\frac{% \tilde{Y}C_{2}}{j\omega}e^{j\omega t}+\tilde{Y}C_{3}e^{j\omega t}-\tilde{X}e^{% j\omega t}\Big{]}=0$ $\displaystyle\Rightarrow\Re\Big{[}e^{j\omega t}\Big{(}\tilde{Y}j\omega C_{1}+% \frac{\tilde{Y}C_{2}}{j\omega}+\tilde{Y}C_{3}-\tilde{X}\Big{)}\Big{]}=\Re[0]+j% \Im[0]$
Equating the real parts above we get,
$\displaystyle\Rightarrow e^{j\omega t}\Big{(}\tilde{Y}j\omega C_{1}+\frac{% \tilde{Y}C_{2}}{j\omega}+\tilde{Y}C_{3}-\tilde{X}\Big{)}=0$
$\displaystyle\Rightarrow\tilde{Y}j\omega C_{1}+\frac{\tilde{Y}C_{2}}{j\omega}+% \tilde{Y}C_{3}-\tilde{X}=0\quad\mbox{(for t\neq-\infty)}$ (2)
Hence, we have now arrive at the phasor domain expression for (1). You can see from the analysis above that we aren’t adding or losing any information when we transform equation (1) into the ”phasor domain” and arrive at equation (2). One can easily get to (2) by using simple algebraic properties of real and complex numbers. Furthermore, since (2) can be derived readily from (1), in practice we don’t even bother to do all the intermediate steps and just skip straight to (2) calling this ”skipping of steps” as ”transforming the equation into the phasor domain.”
We can now continue the analysis even further and solve for $y(t)$ which is the whole motivation behind the use of phasors. Solving for $\tilde{Y}$ in (2) we get
$\displaystyle\Rightarrow\tilde{Y}=\frac{\tilde{X}}{\Big{(}j\omega C_{1}+\frac{% C_{2}}{j\omega}+C_{3}\Big{)}}$
Now, since
$\displaystyle y(t)=\Re[\tilde{Y}e^{j\omega t}]$
we have
$\displaystyle y(t)=\Re\Bigg{[}\frac{\tilde{X}e^{j\omega t}}{\Big{(}j\omega C_{% 1}+\frac{C_{2}}{j\omega}+C_{3}\Big{)}}\Bigg{]}$ $\displaystyle\Rightarrow y(t)=\Re\Bigg{[}\frac{A_{x}e^{j\phi_{x}}e^{j\omega t}% }{\Big{(}j\omega C_{1}+\frac{C_{2}}{j\omega}+C_{3}\Big{)}}\Bigg{]}$
The above equation makes sense because you can see that the output y(t) is given completely in terms of the variables $A_{x}$ and $\phi_{x}$ (which depend only on the input sinusoid $x(t)$) and the constants $C_{1}$, $C_{2}$ and $C_{3}$—as we expected! So by converting the integro-differential equation (1) into the phasor domain (2), all the complicated integration and differentiation operations become simple manipulation of complex variables—which is why phasors are so useful!
Title Phasors Demystified PhasorsDemystified1 2013-03-11 19:26:47 2013-03-11 19:26:47 swapnizzle (13346) (0) 1 swapnizzle (0) Definition | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 39, "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": 0.9828711748123169, "perplexity": 269.3417096809912}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742117.38/warc/CC-MAIN-20181114150002-20181114172002-00129.warc.gz"} |
https://neurips.cc/Conferences/2020/ScheduleMultitrack?event=18027 | Timezone: »
Poster
Geometric Exploration for Online Control
Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1557
We study the control of an \emph{unknown} linear dynamical system under general convex costs. The objective is minimizing regret vs the class of strongly-stable linear policies. In this work, we first consider the case of known cost functions, for which we design the first polynomial-time algorithm with $n^3\sqrt{T}$-regret, where $n$ is the dimension of the state plus the dimension of control input. The $\sqrt{T}$-horizon dependence is optimal, and improves upon the previous best known bound of $T^{2/3}$. The main component of our algorithm is a novel geometric exploration strategy: we adaptively construct a sequence of barycentric spanners in an over-parameterized policy space. Second, we consider the case of bandit feedback, for which we give the first polynomial-time algorithm with $poly(n)\sqrt{T}$-regret, building on Stochastic Bandit Convex Optimization. | {"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": 0.9691268801689148, "perplexity": 786.9726550747116}, "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/1674764499934.48/warc/CC-MAIN-20230201112816-20230201142816-00119.warc.gz"} |
https://physics.stackexchange.com/questions/222631/would-quantum-fluctuations-cause-problems-for-scalar-field-inflation | Would quantum fluctuations cause problems for scalar-field inflation?
Wheeler once said that spacetime would be highly curved at very small scales because of the uncertainty principle for energy-momentum. In which case the spacetime becomes very bumpy and not smooth anymore, which Wheeler called spacetime foam. It seems that such a picture doesn't bother us because in most cases we are dealing with physics of larger scales and the spacetime becomes smooth again at an averaged level over the large scale.
But when we extend the picture to cosmology, problems appear even at a semiclassical level. Now let's consider the $\phi^4$ scalar theory with $$V(\phi)=\frac{\lambda}{4}\phi^4.$$ For the vacuum, because of the uncertainty principle, $\phi$ cannot stay at 0 every where and all the time. If that the field would have definite configuration and definite velocity (field momentum) which violates the uncertainty principle. At the Planck scale $l_p$, the energy should have an uncertainty of $M_p$. Thus for every Plack volume, $\phi$ may take values between $-M_p/(\lambda)^{1/4}$ and $M_p/(\lambda)^{1/4}$ so that $V(\phi)=\frac{\lambda}{4}\phi^{4}\sim M_p^4$.
However in such case, this small patch of space will be driven to inflation. $$a(t)=a_0\exp(Ht),$$ where $$H=\left(\frac{8\pi}{3}V(\phi)/M^{2}_p\right)^{1/2}.$$ Note that the analysis here applies not only to the early universe but also the current universe. But such inflation at small scales will surely cause problems such as the inhomogeneity in our universe which is of course not the case of our observed universe.
So what's the problem with the analysis given above?
The picture in this post is basically wrong. The uncertainty is used for the length scale of the whole system. For example, if we consider a particle moving in a box of volume $L^3$, we would say that the particle has momentum uncertainty ~$\hbar/L$ and therefore energy uncertainty $\hbar^2/(2m L^2)$. Hence for a large box, the energy uncertainty is nearly zero. But we can not say in this case that, we can look at a smaller region, and in that region, there is a large energy uncertainty which may lead to a large total energy uncertainty if we add them up. That is, we should distinguish the scale of the whole system and the distance from point to point which can be arbitrary small. Likewise, the uncertainty principle does not force us away from the spacetime concept between two arbitrary close points. But it tell us that, a quantum system with a very small scale is meaningless since a black hole will be formed from the large energy fluctuations. | {"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": 0.9789738655090332, "perplexity": 207.19446508836543}, "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-2019-26/segments/1560627999163.73/warc/CC-MAIN-20190620065141-20190620091141-00450.warc.gz"} |
http://export.arxiv.org/abs/1506.08247 | math.OC
(what is this?)
# Title: First order constrained optimization algorithms with feasibility updates
Abstract: We propose first order algorithms for convex optimization problems where the feasible set is described by a large number of convex inequalities that is to be explored by subgradient projections. The first algorithm is an adaptation of a subgradient algorithm, and has convergence rate $1/\sqrt{k}$. The second algorithm has convergence rate 1/k when (1) one has linear metric inequality in the feasible set, (2) the objective function is strongly convex, differentiable and has Lipschitz gradient, and (3) it is easy to optimize the objective function on the intersection of two halfspaces. This second algorithm generalizes Haugazeau's algorithm. The third algorithm adapts the second algorithm when condition (3) is dropped. We give examples to show that the second algorithm performs poorly when the objective function is not strongly convex, or when the linear metric inequality is absent.
Comments: 26 pages, 2 figures Subjects: Optimization and Control (math.OC) MSC classes: 90C25, 68Q25, 47J25 Cite as: arXiv:1506.08247 [math.OC] (or arXiv:1506.08247v1 [math.OC] for this version)
## Submission history
From: Chin How Jeffrey Pang [view email]
[v1] Sat, 27 Jun 2015 01:09:08 GMT (29kb)
Link back to: arXiv, form interface, contact. | {"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": 0.8671632409095764, "perplexity": 1210.23584408603}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038862159.64/warc/CC-MAIN-20210418224306-20210419014306-00628.warc.gz"} |
https://www.physicsforums.com/threads/doppler-effect-of-a-whistle.222324/ | # Doppler Effect of a whistle
1. Mar 16, 2008
### danago
A whistle of frequency 577 Hz moves in a circle of radius 73.2 cm at an angular speed of 16.1 rad/s. What are (a) the lowest and (b) the highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle? (Take the speed of sound in air to be 343 m/s.)
The linear velocity of the whistle is given by:
$$v = r\omega = (0.732)(16.1) = 11.7852ms^{ - 1}$$
The component of the velocity in the direction of the listener is at a maximum/minimum when it moves directly towards/away from the listener, with the velocity in this direction being 11.7852m/s.
The greatest frequency will be heard when the velocity of the whistle towards the listener is greatest, thus the effective frequency will be:
$$f' = 577 \times \frac{{343}}{{343 - 11.7852}} = 597.53Hz$$
The smallest frequency will be heard when the velocity of the whistle towards the listener is smallest, or when the whistle moves away from the listener with greatest velocity, thus the effective frequency will be:
$$f' = 577 \times \frac{{343}}{{343 + 11.7852}} = 557.833Hz$$
According to the solutions, those answers are incorrect. Anyone able to shed some light on where my reasoning is flawed?
Dan.
2. Mar 16, 2008
### danago
Ah nevermind, i just realised that i was submitting the solutions the wrong way round, giving the highest one instead of the lowest one. Guess thats i sign i should head off to bed
Similar Discussions: Doppler Effect of a whistle | {"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": 0.8983166217803955, "perplexity": 693.7787194005111}, "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/1518891816370.72/warc/CC-MAIN-20180225110552-20180225130552-00539.warc.gz"} |
https://thiscondensedlife.wordpress.com/2020/06/04/meissner-effect-as-amplified-atomic-diamagnetism/ | Meissner effect as amplified atomic diamagnetism
As you can probably tell from my previous post, I have found the recent activism inspiring and genuinely hopeful of it translating into some form of justice and meaningful action. At the end of this post I share a few videos that I found particularly poignant.
It’s hard to imagine the history of condensed matter physics without both the discovery and theory of superconductivity. Superconductivity has played and continues to play an outsized role in our field, and it is quite easy to understand why this is the case. All one has to do is to imagine what our world would look like with room temperature superconductivity. Besides the potential technological implications, it has also garnered attention because of the wealth of stunning effects associated with it. A few examples include the Josephson effect, flux quantization, persistent superconducting currents, vortex lattices and the Meissner effect.
Now, these effects occur for various reasons, but there are a couple of them that can be viewed to some extent as a microscopic effect on a macroscopic scale. To show what I mean by that, I am going to focus on the Meissner effect and talk about how we can view it as an amplification of atomic diamagnetism. One could also extend the this microscopic to macroscopic amplification picture to the relationship between a Josephson junction in a superconducting ring and the Aharonov-Bohm effect, but I’ll leave that discussion to another day.
To understand what I mean by amplification, let’s first look at atomic diamagnetism. Here we can use a similar logic that led to the Bohr model of the atom. Two conditions are important here — (i) the de Broglie relation $\lambda = h/p$ and (ii) the Bohr quantization condition $n\lambda = 2\pi r$ which states that only integer wavelengths are allowed in a closed loop (such as an atomic orbit). See the image below for a simple picture (click the image for the source).
We can use the classical relation for the momentum $p=mv$ in addition to equations (i) and (ii) above to get $mvr = n\hbar$, which is what Bohr got in his atomic model. It’s worth noting here that when the atom is in its ground state (i.e. $n=0$), there is no “atomic current”, meaning that $j = ev = 0$. Without this current, however, it is not possible to have a diamagnetic response.
So how do we understand atomic diamagnetism? To do so, we need to incorporate the applied field into the deBroglie relation by using the canonical momentum. By making the “Peierls substitution”, we can write that $p = mv+eA$. Using the same logic as above, our quantization condition is now $mvr = n\hbar - eAr$. Now, however, something has changed; we do get a non-zero current in the ground state (i.e. $j = ev = -e^2A/m$ for $n=0$). Qualitatively, this current circulates to screen out the field that is trying to “mess up” the integer-number-of-wavelengths-around-the-loop condition. Note also that we have a response that is strictly quantum mechanical in nature; the current is responding to the vector potential. (I realize that the relation is not strictly gauge invariant, but it makes sense in the “Coulomb gauge”, i.e. when $\nabla\cdot A=0$ or when the vector potential is strictly transverse). In some sense, we already knew that our answer must look obviously quantum mechanical because of the Bohr-van Leeuwen theorem.
If we examine the equation for the electromagnetic response to a superconductor, i.e. the London equation, we obtain a similar equation $j = n_sev = -n_se^2A/m$, where $n_s$ is the superfluid density. The resemblance between the two equations is far from superficial. It is this London equation which allows us to understand the origin of the Meissner effect and the associated spectacular diamagnetism. Roughly speaking then, we can understand the Meissner effect as an amplification of an atomic effect that results in a non-zero ground state “screening” current.
I would also like to add that the Meissner effect is also visible in a multiply connected geometry (see below). This time, the magnetic field (for sufficiently small magnetic fields) is forbidden from going through the center of the ring.
What is particularly illuminating about this ring geometry is that you don’t have to have a magnetic field like in the image above. In fact, it is totally possible to have a superconducting ring under so-called Aharonov-Bohm conditions, where a solenoid passes through the center but the ring never sees the magnetic field. Instead, the superconducting ring “feels the vector potential”. In some sense, this latter experiment emphasizes the equation above where the current really responds (in a gauge-invariant way) to a vector potential and not just the magnetic field.
Understanding the Meissner effect in this way helps us divorce the Meissner effect from the at-first-sight similar effect of persistent currents in a superconducting ring. In the Meissner effect, as soon as the magnetic field is turned off, the current dies and goes back to zero. This is because through this entire process, the superconductor remains in its ground state. Only when the superconductor is excited to higher states (i.e. $n=1,2,3$…) does the current persist in a metastable fashion for a quasi-infinitely long time.
To me, understanding the Meissner effect in this way, which exposes the connection of the microscopic to the macroscopic, harks back to an old post I made about Frank Wilczek’s concept of upward inheritence. The Meissner effect somehow seems clearer through his lens.
Now as promised, here are the couple videos (if the videos don’t play, click on the panel to take you to the twitter website because these videos are worth watching!): | {"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": 14, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8250148296356201, "perplexity": 327.6439814042662}, "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-2021-31/segments/1627046154471.78/warc/CC-MAIN-20210803191307-20210803221307-00643.warc.gz"} |
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Results: 1 to 30 of 73 found Go to page: 1 2 3
[1] Alina Marian, Dragos Oprea and Rahul Pandharipande. The First Chern Class of the Verlinde Bundles. Proceedings of Symposia in Pure Mathematics 90 87-111. Book volume table of contents View Article: PDF [2] A. B. Zheglov. On rings of commuting partial differential operators. St. Petersburg Math. J. 25 (2014) 775-814. Abstract, references, and article information View Article: PDF [3] Arend Bayer and Emanuele Macrì. Projectivity and birational geometry of Bridgeland moduli spaces. J. Amer. Math. Soc. 27 (2014) 707-752. Abstract, references, and article information View Article: PDF [4] Han-Bom Moon. Log canonical models for the moduli space of stable pointed rational curves. Proc. Amer. Math. Soc. 141 (2013) 3771-3785. Abstract, references, and article information View Article: PDF [5] Thorsten Weist. Localization in quiver moduli spaces. Represent. Theory 17 (2013) 382-425. Abstract, references, and article information View Article: PDF [6] Bumsig Kim, Andrew Kresch and Yong-Geun Oh. A compactification of the space of maps from curves. Trans. Amer. Math. Soc. 366 (2014) 51-74. Abstract, references, and article information View Article: PDF [7] Jason Lo. Moduli of PT-semistable objects II. Trans. Amer. Math. Soc. 365 (2013) 4539-4573. Abstract, references, and article information View Article: PDF [8] Mark Gross and Bernd Siebert. Logarithmic Gromov-Witten invariants. J. Amer. Math. Soc. 26 (2013) 451-510. Abstract, references, and article information View Article: PDF [9] Adam S. Sikora. Character varieties. Trans. Amer. Math. Soc. 364 (2012) 5173-5208. Abstract, references, and article information View Article: PDF [10] C. G. Madonna. On some moduli spaces of bundles on $K3$ surfaces, II. Proc. Amer. Math. Soc. 140 (2012) 3397-3408. Abstract, references, and article information View Article: PDF [11] Najmuddin Fakhruddin. Chern classes of conformal blocks. Contemporary Mathematics 564 (2012) 145-176. Book volume table of contents View Article: PDF [12] Indranil Biswas, Norbert Hoffmann, Amit Hogadi and Alexander H. W. Schmitt. The Brauer group of moduli spaces of vector bundles over a real curve. Proc. Amer. Math. Soc. 139 (2011) 4173-4179. MR 2823062. Abstract, references, and article information View Article: PDF [13] Wei-Ping Li and Zhenbo Qin. $1$-point Gromov-Witten invariants of the moduli spaces of sheaves over the projective plane. Trans. Amer. Math. Soc. 363 (2011) 2551-2569. MR 2763726. Abstract, references, and article information View Article: PDF [14] Yukinobu Toda. Curve counting theories via stable objects I. DT/PT correspondence. J. Amer. Math. Soc. 23 (2010) 1119-1157. MR 2669709. Abstract, references, and article information View Article: PDF This article is available free of charge [15] Sukmoon Huh. Dominance of a rational map to the Coble quartic. Proc. Amer. Math. Soc. 138 (2010) 777-786. MR 2566543. Abstract, references, and article information View Article: PDF This article is available free of charge [16] Vladimir Baranovsky. Algebraization of bundles on non-proper schemes. Trans. Amer. Math. Soc. 362 (2010) 427-439. MR 2550158. Abstract, references, and article information View Article: PDF This article is available free of charge [17] Takeshi Abe. Projective normality of the moduli space of rank $2$ vector bundles on a generic curve. Trans. Amer. Math. Soc. 362 (2010) 477-490. MR 2550160. Abstract, references, and article information View Article: PDF This article is available free of charge [18] Sean Lawton. Poisson geometry of $\mathrm {SL}(3,\mathbb {C})$-character varieties relative to a surface with boundary. Trans. Amer. Math. Soc. 361 (2009) 2397-2429. MR 2471924. Abstract, references, and article information View Article: PDF This article is available free of charge [19] Jaya NN. Iyer. Murre's conjectures and explicit Chow-Künneth projectors for varieties with a nef tangent bundle. Trans. Amer. Math. Soc. 361 (2009) 1667-1681. MR 2457413. Abstract, references, and article information View Article: PDF This article is available free of charge [20] William M. Goldman and Eugene Z. Xia. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces. Memoirs of the AMS 193 (2008) MR 2400111. Book volume table of contents [21] Prakash Belkale. Quantum generalization of the Horn conjecture. J. Amer. Math. Soc. 21 (2008) 365-408. MR 2373354. Abstract, references, and article information View Article: PDF This article is available free of charge [22] Ravi Vakil and Aleksey Zinger. A natural smooth compactification of the space of elliptic curves in projective space. Electron. Res. Announc. Amer. Math. Soc. 13 (2007) 53-59. MR 2320682. Abstract, references, and article information View Article: PDF [23] Christian Pauly. A smooth counterexample to Nori's conjecture on the fundamental group scheme. Proc. Amer. Math. Soc. 135 (2007) 2707-2711. MR 2317943. Abstract, references, and article information View Article: PDF This article is available free of charge [24] Prakash Belkale. The strange duality conjecture for generic curves. J. Amer. Math. Soc. 21 (2008) 235-258. MR 2350055. Abstract, references, and article information View Article: PDF This article is available free of charge [25] Birge Huisgen-Zimmermann. Classifying representations by way of Grassmannians. Trans. Amer. Math. Soc. 359 (2007) 2687-2719. MR 2286052. Abstract, references, and article information View Article: PDF This article is available free of charge [26] O. García-Prada, P. B. Gothen and V. Muñoz. Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. Memoirs of the AMS 187 (2007) MR 2308696. Book volume table of contents [27] Javier Fernández de Bobadilla. Moduli spaces of polynomials in two variables. Memoirs of the AMS 173 (2005) MR 2110626. Book volume table of contents [28] Tawanda Gwena. Degenerations of cubic threefolds and matroids. Proc. Amer. Math. Soc. 133 (2005) 1317-1323. MR 2111937. Abstract, references, and article information View Article: PDF This article is available free of charge [29] Megumi Harada and Nicholas Proudfoot. Hyperpolygon spaces and their cores. Trans. Amer. Math. Soc. 357 (2005) 1445-1467. MR 2115372. Abstract, references, and article information View Article: PDF This article is available free of charge [30] Donghoon Hyeon and David Murphy. Note on the stability of principal bundles. Proc. Amer. Math. Soc. 132 (2004) 2205-2213. MR 2052395. Abstract, references, and article information View Article: PDF This article is available free of charge
Results: 1 to 30 of 73 found Go to page: 1 2 3 | {"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": 0.9493603706359863, "perplexity": 2033.7591377197193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982294097.9/warc/CC-MAIN-20160823195814-00168-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/55136/general-questions-about-level-sets-of-polynomials-of-two-variables | # General questions about level sets of polynomials of two variables
Sorry if I'm being too general here, but here it goes. I'm trying to find out more about levels sets of polynomials of two variables of degree $d$
$$C = \{ (x,y) \ : \sum_{1 \leq i + j \leq d} c_{i,j} x^i y^j = 1 \}$$
In particular, I want a feeling of what these curves "look like" for low $d$ ($d = 3,4,5...$), and also what kind of things can be computed "exactly" (probably need special functions), for instance:
1. When is $C$ made up of closed curve(s)?
2. When is $C$ connected?
3. What is the arc length of $C$?
4. What is the curvature of $C$?
Any tips, or paper references would be helpful too, thanks!!
-
heres one result: en.wikipedia.org/wiki/Harnack%27s_curve_theorem – yoyo Aug 2 '11 at 15:15 | {"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": 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": 0.8625268936157227, "perplexity": 515.1936559453528}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657139669.58/warc/CC-MAIN-20140914011219-00079-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/445558/do-there-exist-normal-multi-valued-interpretations-for-the-equivlaential-calculu | # Do There Exist Normal Multi-Valued Interpretations for the Equivlaential Calculus?
Suppose we define a propositional calculus, just by its (object language) theorem set and its rules of inference. For example, suppose we define the C-N propositional calculus by the set of theorems deducible from
1. CCpCqrCCpqCpr (self-distribution)
2. CpCqp (simplifcation)
3. CCNpNqCqp
under the rules of inference of C-detachment "From $\vdash$C$\alpha$$\beta, as well as \vdash$$\alpha$, we may infer $\vdash$$\beta," and uniform substitution. If we do this, at least some logical systems can admit of different matrices, since the above C-N propositional calculus satisfies both the two-valued matrix: C| 1 0| N ------------ 1| 1 0| 0 0| 1 1| 1 As well as Slupecki's three-valued matrix (and other multi-valued matrices too): C| 1 .5 0| N ------------------ 1| 1 .5 0| 0 .5| 1 1 1| 1 0| 1 1 1| 1 Slupecki's matrix qualifies as a normal 3-valued matrix in the sense that if the atomic formulas take on the values {0, 1}, then the valuation of Cpq, denoted v(Cpq) \epsilon {0, 1} and v(Np) \epsilon {0, 1}. What I've read indicates that the equivalential calculus can get axiomizated by these two axioms 1. EEpqEqp "commutation" 2. EEEpqrEpEqr "association" with rules of inference of uniform substitution, and E-detachment "From \vdashE\alpha$$\beta$, as well as $\vdash$$\alpha, we may infer \vdash$$\beta$." But, I've only see authors refer to a two-valued matrix such as:
E| 0 1
--------
0| 1 0
1| 0 1
when talking about the equivalential calculus.
I feel inclined to believe that we can't have a normal 3-valued matrix which satisfies these two axioms of the equivalential calculus and still has E-detachment as a valid rule of inference, nor will any odd-valued matrix work. But, could we have a 4, 6, or an n-valued (normal) matrix where n does not equal 2? Could we have an odd-valued matrix which satisfies those axioms? If not, how do we disprove it?
As I understand things, the equivalential calculus has what gets called the two-property, which means that a formula F (formulas only involving E) qualifies as a theorem iff every lower case letter appears in F an even number of times.
-
Yes, Bochvar's 3-valued logic works out this way, since it has the same set of tautologies as two-valued logic does, and it contains an equivalence, E-connective. Consequently, due to truth tables, we can in principle show that the set of tautologies for the fragment of Bochvar's 3-valued logic which has just the E-connective, has the same set of tautologies as 2-valued logic with just the E-connective.
- | {"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": 0.9621548652648926, "perplexity": 674.3892862024377}, "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-2015-35/segments/1440644064865.49/warc/CC-MAIN-20150827025424-00337-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://brilliant.org/problems/do-you-know-frogs-can-fly/ | # Did you know Frogs can fly?
A frog can levitate if it has a strong magnetic field below it.
This is possible because the body of the frog behaves as
× | {"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": 0.8667004108428955, "perplexity": 1747.1273648377573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267161638.66/warc/CC-MAIN-20180925123211-20180925143611-00274.warc.gz"} |
https://www.physicsforums.com/threads/abelian-group-what-to-do-if-the-set-is-g-r-1-3.215302/ | # Abelian Group; what to do if the set is G=R-{1/3}?
1. Feb 13, 2008
### lostinmath08
1. The problem statement, all variables and given/known data
On the set G=R-{1/3} the following operation is defined:
*G: GxG arrow G
(x,y) arrow x*y=x+y-3xy
Show that (G,*) is an abelian group.
2. Relevant equations
To proove something is an abelian group:
The Associative Law need to hold true x*(y*x)=(x*y)*x
Neutral Element needs to be true e*x=x*e=x for all or any x E G
Inverse Elements x*x'=x'*x=e where x' is called the inverse element of x
Commutativity x*y=y*x
3. The attempt at a solution
I don't know how to solve this, especially with a set like this. Any Hints or Advice would be greatly appreciated
2. Feb 13, 2008
### Dick
You posted a problem like this before. Commutativity should be obvious. Try solving for e. Just get started with any property you want to start out with.
3. Feb 13, 2008
### lostinmath08
I have already solved for the properties, but I thought the set matters.
4. Feb 13, 2008
### Dick
The set does matter. Because once you have figured out that e=0, then 1/3 is the only element of R that doesn't have an inverse under the operation '*'. So if you include 1/3, it's not a group. Just like R-{0} is a multiplicative group and R isn't.
Last edited: Feb 13, 2008 | {"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": 0.9039555191993713, "perplexity": 1397.1814617943992}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698540563.83/warc/CC-MAIN-20161202170900-00208-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://chemistry.stackexchange.com/questions/28221/difference-between-intermediates-and-transition-states/28224#28224 | # Difference between intermediates and transition states
Can someone please explain what exactly the difference between an intermediate and a transition state is?
I understand that they are formed as part of the process in converting the reactants to products in a chemical reaction. Other than that, all I know is that usually a transition state exists very briefly (if at all) whereas an intermediate can be isolated. But why?
An intermediate is a short-lived unstable molecule in a reaction which is formed inbetween the reaction when reactants change into products.
Whereas, transition state is just the state before formation of new molecule(involves breaking of bonds of reactants and formation of new ones)
An intermediate differs from a transition state in that the intermediate has a discrete lifetime (be it a few nanoseconds or many days), whereas a transition state lasts for just one bond vibration cycle. Intermediates may be unstable molecules (in which case they are called reactive intermediates) or highly stable molecules. The difference between them can be better described through the energy profile diagram.
Transition states are local energy maximums and have partial bonds. This might be one of the reasons why they cant be isolated as intermediates.
Throughout, I will be using definitions from a Journal of Chemical Education article$^1$ by Keith Laidler.
I'll start with describing an intermediate, as I think it is an easier concept to grasp. An intermediate, according to Laidler, is a short-lived compound formed during a composite reaction. This intermediate is a product of one elementary reaction and a reactant for another. An intermediate must be a minimum on the potential energy surface; this means it will have some finite lifetime (dependent on how deep this minimum is) and could potentially be isolated under the right conditions.
A transition state is a saddle point on the potential energy surface, meaning it is a maximum along the reaction coordinate and a minimum along all other coordinates (this is a slight simplification as it is possible, though rare, to have higher order saddle points where it is a maximum along more than one coordinate). Note that this definition only applies to an elementary reaction; a composite reaction doesn't have one overarching transition state. Laidler defines the broader category of transition species to refer to any structure that occurs while moving along the reaction coordinate between reactants and products. This is an important distinction; note that while a transition state is impossible to detect (it exists only for a very small range of atomic configurations and thus its concentration is always vanishingly small) transition species as whole can be detected as they exist, however fleetingly, in much higher concentration due to being defined for a wider range of atomic configurations. Laidler gives as an example the reaction $$\ce{F + Na2 ->F...Na...Na->NaF +Na^*}$$ where certain emissions surrounding the sodium D-line have been attributed to these transition species.
As some what of an aside, it is worth mention how potential energy extrema and saddle points are determined from a computational perspective. First, we find extrema and saddle points by determining regions where the force vector (first derivative of the energy with respect to all coordinates) is zero. Given the Hessian matrix for a molecule, which gives the second derivative of the energy with respect to all pairs of the 3N coordinates of the molecule (these might be normal modes or cartesian coordinates), one can determine whether the compound is an min/saddle point by examining the eigenvalues of the matrix. If all the eigenvalues are positive, this means the compound is a minimum; if exactly one eigenvalue is negative, it is a saddle point; if $n$ eigenvalues are negative, it is an $n^{th}$ order saddle point; if all eigenvalues are negative, it is a maximum (this is hardly ever the case for chemical systems).
1. Laidler, K. J. Just what is a transition state? J. Chem. Educ. 1988, 65 (6), 540. DOI: 10.1021/ed065p540.
A transition state cannot be isolated while an intermediate can be isolated. A transition state is a chemical species which has only fleeting existence and represents an energy maxima on reaction coordination diagram . While an intermediate lies in depression on potential energy curve . Therefore actual lifetime of an intermediate depends on the depth of the depression. A shallow depression implies a low activation energy for subsequent steps and therefore short lifetime. While deeper the depression longer is the lifetime of intermediate. | {"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": 0.8887457847595215, "perplexity": 564.9041124108459}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662594414.79/warc/CC-MAIN-20220525213545-20220526003545-00567.warc.gz"} |
http://www.gradesaver.com/textbooks/science/physics/fundamentals-of-physics-extended-10th-edition/chapter-2-motion-along-a-straight-line-problems-page-33/18c | ## Fundamentals of Physics Extended (10th Edition)
$-12 m/s^2$
The position is given by $x = 12t^2 - 2t^3$ Since the velocity equation is the derivative of the position equation we need to find $\frac{d}{dt} (12t^2 - 2t^3)$ Since $\frac{d}{dt} t^n = n*t^(n-1)$ then $\frac{d}{dt} (12t^2 - 2t^3) = 12(2)t - 2(3)t^2 = 24t - 6t^2$ Therefore $v = 24t - 6t^2$ The acceleration is given by the derivative of the velocity equation This is $\frac{d}{dt} (24t - 6t^2) = 24 - 6(2)t = 24 - 12t$ Therefore, to find acceleration at t =3, we plug in t = 3 into the equation This gives us $24 -12(3) = 24-36 = -12$ Therefore the acceleration is $-12m/s^2$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9773697257041931, "perplexity": 125.09154565649618}, "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/1492917118310.2/warc/CC-MAIN-20170423031158-00517-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://hal.archives-ouvertes.fr/hal-02300800 | # Finite-density QCD transition in magnetic field background
Abstract : Using numerical simulations of lattice QCD with physical quark masses, we reveal the influence of magnetic-field background on chiral and deconfinement crossovers in finite-temperature QCD at low baryonic density. In the absence of thermodynamic singularity, we identify these transitions with inflection points of the approximate order parameters: normalized light-quark condensate and renormalized Polyakov loop, respectively. We show that the quadratic curvature of the chiral transition temperature in the temperature--chemical potential'' plane depends rather weakly on the strength of the background magnetic field. At weak magnetic fields, the thermal width of the chiral crossover gets narrower as the density of the baryon matter increases, possibly indicating a proximity to a real thermodynamic phase transition. Remarkably, the curvature of the chiral thermal width flips its sign at $eB_{\mathrm{fl}} \simeq 0.6\,\mathrm{GeV}^2$, so that above the flipping point $B > B_{\mathrm{fl}}$, the chiral width gets wider as the baryon density increases. Approximately at the same strength of magnetic field, the chiral and deconfining crossovers merge together at $T \approx 140\,\mathrm{MeV}$. The phase diagram in the parameter space temperature-chemical potential-magnetic field'' is outlined, and single-quark entropy and single-quark magnetization are explored. The curvature of the chiral thermal width allows us to estimate an approximate position of the chiral critical endpoint at zero magnetic field: $(T_c^{\text{CEP}}, \mu_B^{\text{CEP}})= (100(25)\, \text{MeV},\ 800(140)\,\text{MeV})$.
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https://hal.archives-ouvertes.fr/hal-02300800
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### Citation
V. V. Braguta, M. N. Chernodub, A. Yu. Kotov, A. V. Molochkov, A. A. Nikolaev. Finite-density QCD transition in magnetic field background. Physical Review D, American Physical Society, 2019, 100, pp.114503. ⟨10.1103/PhysRevD.100.114503⟩. ⟨hal-02300800⟩
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https://cs.stackexchange.com/questions/21367/a-quine-in-pure-lambda-calculus/21368 | # A quine in pure lambda calculus
I would like an example of a quine in pure lambda calculus. I was quite surprised that I couldn't find one by googling. The quine page lists quines for many "real" languages, but not for lambda calculus.
Of course, this means defining what I mean by a quine in the lambda calculus, which I do below. (I'm asking for something quite specific.)
In a few places, e.g. Larkin and Stocks (2004), I see the following quoted as a "self-replicating" expression: $(\lambda x.x \; x)\;(\lambda x.x \; x)$. This reduces to itself after a single beta-reduction step, giving it a somehow quine-like feel. However, it's un-quine-like in that it doesn't terminate: further beta-reductions will keep producing the same expression, so it will never reduce to normal form. To me a quine is a program that terminates and outputs itself, and so I would like a lambda expression with that property.
Of course, any expression that contains no redexes is already in normal form, and will therefore terminate and output itself. But that's too trivial. So I propose the following definition in the hope that it will admit a non-trivial solution:
definition (tentative): A quine in lambda calculus is an expression of the form $$(\lambda x . A)$$ (where $A$ stands for some specific lambda calculus expression) such that $((\lambda x . A)\,\, y)$ becomes $(\lambda x . A)$, or something equivalent to it under changes of variable names, when reduced to normal form, for any input $y$.
Given that the lambda calculus is as Turing equivalent as any other language, it seems as if this should be possible, but my lambda calculus is rusty, so I can't think of an example.
Reference
James Larkin and Phil Stocks. (2004) "Self-replicating expressions in the Lambda Calculus" Conferences in Research and Practice in Information Technology, 26 (1), 167-173. http://epublications.bond.edu.au/infotech_pubs/158
• Not an answer to my question, but for my own future reference (and for future visitors) it will be useful to have a link to wiki.haskell.org/Combinatory_logic , in which someone has much deeper thoughts about quines than I did. – Nathaniel May 15 '16 at 10:04
• Note that a quine needs to produce its own source code. Producing the function it represents is not sufficient. – PyRulez Dec 14 '18 at 20:49
• @PyRulez what is the source code for a lambda expression? If it's a sequence of characters then it's impossible for a lambda expression to output it, and consequently we can define the word "quine" to mean something slightly different for lambda expressions without fear of ambiguity. On the other hand, if you think of the source code as being the lambda expesssion itself then "the source code" and "the function it represents" are the same thing. So I think I'm ok here. – Nathaniel Dec 15 '18 at 2:42
• there is a church encoding for strings. A lambda calculus quine should output the church encoding of the string of characters representing it. – PyRulez Dec 15 '18 at 4:20
• Sure, that is not hard to do, if you define it that way. This question was about a different thing. – Nathaniel Dec 15 '18 at 9:04
You want a term $Q$ such that $\forall M \in \Lambda$:
$$QM \rhd_\beta Q$$
I will specify no further restrictions on $Q$ (e.g. regarding its form and whether it is normalising) and I will show you that it definitely must be non-normalising.
1. Assume $Q$ is in normal form. Choose $M \equiv x$ (we can do so because the theorem needs to hold for all $M$). Then there are three cases.
• $Q$ is some atom $a$. Then $QM \equiv ax$. This is not reducible to $a$.
• $Q$ is some application $(RS)$. Then $QM \equiv (RS)x$. $(RS)$ is a normal form by hypothesis, so $(RS)x$ is also in normal form and not reducible to $(RS)$.
• $Q$ is some abstraction $(\lambda x.A)$ (if $x$ is supposed to be free in $A$, then for simplicity we can just choose $M$ equivalent to whatever variable $\lambda$ abstracts over). Then $QM \equiv (\lambda x.A)x \rhd_\beta A[x/x] \equiv A$. Since $(\lambda x.A)$ is in normal form, so is $A$. Consequently we cannot reduce $A$ to $(\lambda x.A)$.
So if such a $Q$ exists, it cannot be in normal form.
2. For completeness, suppose $Q$ has a normal form, but is not in normal form (perhaps it is weakly normalising), i.e. $\exists N \in \beta\text{-nf}$ with $N \not\equiv Q$ such that $\forall M \in \Lambda$: $$QM \rhd_\beta Q \rhd_\beta N$$
Then with $M \equiv x$ there must also be exist a reduction sequence $Qx \rhd_\beta Nx \rhd_\beta N$, because:
• $Qx \rhd_\beta Nx$ is possible by the fact that $Q \rhd_\beta N$.
• $Nx$ must normalise since $N$ is a $\beta$-nf and $x$ is just an atom.
• If $Nx$ were to normalise to anything other than $N$, then $Qx$ has two $\beta$-nfs, which is not possible by a corollary to the Church-Rosser theorem. (The Church-Rosser theorem essentially states that reductions are confluent, as you probably already know.)
But note that $Nx \rhd_\beta N$ is not possible by argument (1) above, so our assumption that $Q$ has a normal form is not tenable.
3. If we permit such a $Q$, then, we are certain that it must be non-normalising. In that case we can simply use a combinator that eliminates any argument it receives. Denis's suggestion works just fine: $$Q \equiv (\lambda z.(λx.λz.(x x)) (λx.λz.(x x)))$$ Then in only two $\beta$-reductions: \begin{align} QM &\equiv (\lambda z.(λx.λz.(x x)) (λx.λz.(x x))) M \\ & \rhd_{1\beta} (λx.λz.(x x)) (λx.λz.(x x)) \\ & \rhd_{1\beta} (λz.((λx.λz.(x x))(λx.λz.(x x))) \\ & \equiv Q \end{align}
This result is not very surprising, since you are essentially asking for a term that eliminates any argument it receives, and this is something I often see mentioned as a direct application of the fixed-point theorem.
• If I could accept Denis' answer as well then I would, but (after I'd learned a bit more and was able to fully understand it) it was this answer that really convinced me that this "quine combinator" cannot be implemented by a lambda expression in normal form. – Nathaniel May 15 '16 at 5:43
One one hand this is impossible, because a quine is supposed to output its own code, and the pure lambda calculus has no means for performing output.
On the other hand, if you assume that the resulting term is the output, then every normal form is a quine.
For example, lambda term $(\lambda x. x)$ is already a normal form, then assuming that its output is the resulting normal form, the output is $(\lambda x. x)$. Thus $(\lambda x. x)$ is a quine.
• That's an interesting point. In the question I tried to give a definition of what might count as a non-trivial quine in lambda calculus: a function that, when applied to any input, beta-reduces to itself (up to variable name substitutions). It might be that this is impossible, but it's not obvious, at least to me. – Nathaniel Feb 6 '14 at 10:58
Here is a proposition:
We choose $A$ to be a fixpoint of the function $f=\lambda t. (\lambda z.t)$.
This can be done by using the fixpoint combinator $Y=\lambda g.((\lambda x.g\ (x\ x))\ (\lambda x.g\ (x\ x)))$, and setting $A=Y f=(\lambda x.\lambda z.(x\ x))\ (\lambda x.\lambda z.(x\ x))$.
Now we show that $A$ is a quine. Indeed $A$ reduces to $\lambda z.A$, so it means that for any $y$, $(\lambda z.A)y \to_\beta A \to_\beta (\lambda z.A)$.
• This is pretty neat, and answers the question as I asked it, so I feel bad for not accepting it --- but unfortunately I made a slight mistake in specifying what I want. I actually want $(\lambda z.A)\;y$ to become $(\lambda z.A)$ when reduced to normal form, not just after a beta reduction step. (See the updated question for why.) This means that $A$ can't contain any redexes, because if it does then the reduction will not terminate. – Nathaniel Feb 7 '14 at 5:03
• Ah in this case i'm pretty sure it is impossible, because of the following intuition (not a proof but almost): you want $y$ to play no role since it has to work for every $y$, so $y$ should not be free in $A$. Then $(\lambda z.A)y$ just reduces to $A$. Now you want $A$ to reduce to $\lambda z.A$. This last expression cannot be a normal form, since the $A$ inside can again be reduced... – Denis Feb 7 '14 at 13:28
• This behaviour is not very surprising, because thince the "printing" of $\lambda-calculus$ are again instructions, a quine printing its own code is always executable. What you are asking is similar to asking for a quine such that if you execute the output, it prints nothing (which is impossible by definition). – Denis Feb 7 '14 at 13:32
• Ahh, you're right of course. I should have seen that. I'm not sure whether to accept your answer or edit the question to ask for a better definition. I'll give it a bit of thought. (It still seems to me that it should be possible to give a non-trivial definition where you're asking for something that will terminate, but I'm not sure how.) – Nathaniel Feb 9 '14 at 5:42
• Though having said that, is it really true that $z$ (I assume you mean $z$) has to not be free in $A$? E.g. $A$ could be something along the lines of if z==p then return q, otherwise return q. (Pseudocode because I'm not sure if it's even possible to define the equality operator for arbitrary expressions in lambda calculus, but I think you see what I mean.) – Nathaniel Feb 9 '14 at 5:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9459348320960999, "perplexity": 341.15553474103353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250605075.24/warc/CC-MAIN-20200121192553-20200121221553-00466.warc.gz"} |
https://www.physicsforums.com/threads/divergence-of-the-navier-stokes-equation.869977/ | # Divergence of the Navier-Stokes Equation
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The Navier-Stokes equation is:
(DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p)
where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is non-solenoidal (∇⋅U ≠ 0).
How, then, can we take the divergence of this equation and get the following result?:
I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.
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The Navier-Stokes equation is:
(DUj/Dt) = v [(∂2Ui/∂xj∂xi) + (∂2Uj/∂xi∂xi)] – 1/ρ (∇p)
where D/Dt is the material (substantial) derivative, v is the kinematic viscosity and ∇p is the modified pressure gradient (taking into account gravity and pressure). Note that the velocity field is non-solenoidal (∇⋅U ≠ 0).
How, then, can we take the divergence of this equation and get the following result?:
View attachment 100112
I can follow all of the terms other than the one underlined in blue. I know that it comes from the blue section of the Navier-Stokes written above, since I can easily get all the other terms.
The terms in blue can not come from the blue section of the NS. They are non-linear in velocity, and blue section of the NS equation is linear in velocity. The blue terms in your relationship must come from the left side of the NS equation.
My advice to you is to write out the NS equation for each of the three components. Then take the partial of the x component with respect to x, the partial of the y component with respect to y, and the partial of the z component with respect to z. Then add the resulting 3 equations. This will be bulletproof.
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2K | {"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": 0.8832619190216064, "perplexity": 986.949823514471}, "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/1614178355937.26/warc/CC-MAIN-20210225211435-20210226001435-00064.warc.gz"} |
http://www.astro.physik.uni-potsdam.de/~www/research/abstracts/Shenar2016_SMC.html | # Wolf-Rayet stars in the Small Magellanic Cloud: II. Analysis of the binaries
### T. Shenar, R. Hainich, H. Todt, A. Sander, W.-R. Hamann, A. F. J. Moffat, J. J. Eldridge, H. Pablo, L. M. Oskinova, N. D. Richardson
Massive WR stars are evolved massive stars characterized by strong mass-loss. Hypothetically, they can form either as single stars or as mass donors in close binaries. About 40% of the known WR stars are confirmed binaries, raising the question as to the impact of binarity on the WR population. By performing a spectral analysis of all multiple WR systems in the SMC, we obtain the full set of stellar parameters for each individual component. Mass-luminosity relations are tested, and the importance of the binary evolution channel is assessed. The spectral analysis is performed with the PoWR model atmosphere code by superimposing model spectra that correspond to each component. Evolutionary channels are constrained using the BPASS evolution tool. Significant Hydrogen mass fractions (0.1 - 0.4) are detected in all WN components. A comparison with mass-luminosity relations and evolutionary tracks implies that the majority of the WR stars in our sample are not chemically homogeneous. The WR component in the binary AB 6 is found to be very luminous (Log L ~ 6.3 [Lsun]) given its orbital mass (~10 Msun), presumably because of observational contamination by a third component. Evolutionary paths derived for our objects suggest that Roche lobe overflow had occurred in most systems, affecting their evolution. However, the implied initial masses are large enough for the primaries to have entered the WR phase, regardless of binary interaction. Together with the results for the putatively single SMC WR stars, our study suggests that the binary evolution channel does not dominate the formation of WR stars at SMC metallicity. | {"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": 0.926555335521698, "perplexity": 3694.318735010105}, "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/1544376823614.22/warc/CC-MAIN-20181211083052-20181211104552-00239.warc.gz"} |
https://www.mendeley.com/research-papers/relationship-between-leaky-lamb-modes-reflection-coefficient-zeroes-fluidcoupled-elastic-layer/ | ## Relationship between leaky Lamb modes and reflection coefficient zeroes for a fluid-coupled elastic layer
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#### Abstract
It is shown in this article how the zeroes of the plane‐wave reflection coefficient for a fluid‐coupled elastic layer are modified under fluid loading, throughout the entire range of fluid density. Transformation and exchange behavior of branches of the reflection coefficient zero spectrum with increasing fluid density has been found and discussed. This study extends an earlier investigation of the complex mode spectrum(reflection coefficient poles). The behavior of the zeroes of the reflection coefficient is compared with that of the poles as a function of the fluid density. In addition, we derive an analytical expression for the intersection of the fluid‐modified fundamental zero branches and the Rayleigh wave speed. This number can be useful as a measure of the sensitivity of a solid material to fluid loading. More significantly, the results of the current study have immediate relevance for an important class of advanced materials: polymer‐matrix composites. It is demonstrated that the spectral modifications similar to those induced in the reflection coefficient of aluminum at high fluid density occur below the density of water, if the solid plate is a polymer‐matrix composite.
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https://physics.stackexchange.com/questions/362145/what-are-the-gravitational-binding-energies-of-giant-planets | # What are the gravitational binding energies of giant planets?
What are the gravitational binding energies of the planets in our solar system? In particular, interested in the giant planets: Jupiter, Saturn, Uranus, and Neptune. Ideally the information would be from a paper or other peer-reviewed source, but I'll take what I can get. What I'd like to avoid is the approximation $$U_\circ=\frac{3G}{5}\cdot\frac{m^2}{r}=\frac{16\pi^2G}{15}r^5\rho^2$$ which comes from assuming that the planet is a sphere of uniform density. This isn't all that realistic for rocky planets, but it's far worse for ice and gas giants.
I know that there's been a lot of revision recently on the composition and layers of Jupiter and Saturn, and a lot is based on our ability to model metallic hydrogen about which we know very little. So recent sources would be better than old sources (though I'll take what I can get).
## Attempt
Arturo don Juan and A. C. A. C. asked how much this effect matters, so here are some calculations. I'll use Jupiter as my example. Its mass is 1.8986e27 kg and its mean radius is 69911 km, giving it $U_\circ = 2.065 \cdot 10^{36}\text{ J}.$ Using the crude approximation $$\rho_x=\begin{cases} 25 & \text{ if }x \le .14\\ 6.0533 - 6.3166x & \text{ if }.14<x<.8\\ \frac52(1-x) & \text{ if }x \ge .8 \end{cases}$$ (where 0 is the core and 1 is the edge of the clouds and the density is in g/cm^3) based on the diagram here I compute $$U \approx 1.775 \cdot 10^{36}\text{ J}.$$
## Background
I recently answered a question about the Death Star's destructive ability in Star Wars. My answer relied heavily on the uniform density approximation for scaling, and I'm not sure how much that affected it. I hoped that starting from reasonable baselines would help, but I wasn't really able to do that because I couldn't find any information on the GBE for planets other than Earth. (I could find plenty of exercises asking students to apply the above approximation of course.)
I have to imagine there's something out there, maybe just giving the best-guess density $\rho_x$ at a given depth $x$, from which one can integrate $$U = \frac{16\pi^2G}{3}\int_0^r x^4\rho_x^2 dx$$ to get the GBE.
• I think a good start would be to first show that assuming a slightly more realistic mass distribution actually does give a (wildly) different binding energy, i.e. that the answer truly is sensitive to which distribution you choose. – Arturo don Juan Oct 10 '17 at 20:02
• @ArturodonJuan If you have a slightly more realistic mass distribution, I'd be happy to check. Of course then you'd actually have an answer to my question, so you could post it just as easily. – Charles Oct 10 '17 at 20:08
• I don't know the density at different depth to have any idea of this but you can try doing the exercise of integrating that with simple function to see how much things change. – A. C. A. C. Oct 10 '17 at 20:56
• @Charles Sorry I had that in mind but was in a bit of a rush so I decided to plant the seed of my idea in a comment and see if it grew. While I look into it, a possible parametric distribution could be of the kind $\rho (r) \propto r^{\alpha}(R-r)^{\beta}$ (beta distributed), where you could play around with the parameters $\alpha$ and $\beta$. – Arturo don Juan Oct 10 '17 at 21:41
• @ArturodonJuan I did a quick piecewise linear model to show sensitivity. The model was designed to have the same mean density (if I did my calculations correctly) and turns out to be ~15% smaller. Actually I expected the opposite, maybe I made a mistake...? – Charles Oct 11 '17 at 13:39
This is an answer to the point I brought up in the comments.
To begin, I'd like to point out that the quoted formula for the gravitational binding energy of a spherically symmetric body isn't quite right. You can see in the wiki article that the factor of $3$ in the denominator comes from assuming a uniform mass distribution. Just to have it on record, here's a proper derivation.
To send a layer of mass off the surface of a sphere - its shell - to infinity takes $dU$ energy:
$$dU(r)=\frac{GM_{\text{inside}}(r)\,m_{\text{shell}}(r)}{r}=\frac{GM_{\text{inside}}(r)}{r}(4\pi r^2 \rho(r)\,dr)$$
The mass inside this shell is
$$M_{\text{inside}}(r)=\int_0^r\rho(r') \,(4\pi r'^2\, dr')$$
To get the total binding energy, we send all the shells off to infinity, i.e. we integrate from the core ($r=0$) to the outer radius ($r=R$):
$$U_{\text{binding}}=\int dU = 16\pi^2 G \int_0^R dr_1 \rho(r_1) r_1 \int_0^{r_1} dr_2\, \rho(r_2) r_2^2$$
Maybe with some trick you could reduce that double integral to a single integral, but I can't think of how to do it off the top of my head.
If we assume the distribution goes like, say, $\rho(r)=\rho_0 e^{-\alpha r}$, where $\rho_0$ is fixed by the condition that the total mass is $M$, you can carry out the lengthy calculations and find the binding energy as a function of the parameter $\alpha$. Setting $R=M=G=1$, my calculations gave me:
$$U_{\text{binding}}(\alpha)=\frac{\alpha}{4}\frac{\frac{5}{8}-2e^{-\alpha}(\alpha+1)+\frac{1}{8}(2\alpha(\alpha(2\alpha+7)+11)+11)e^{-2\alpha}}{\left(1-(1+\alpha+\frac{1}{2}\alpha^2)e^{-\alpha}\right)^2}$$
Here's a plot of it (courtesy of wolframalpha):
First, as a sanity check, notice that at $\alpha=0$ we recover the uniform-density answer,
$$U_{\text{binding}}(\alpha=0)=\frac{3 M^2 G}{5R}$$
What this plot tells us is that if we have a mass distribution which decays like $e^{-4r}$, the binding energy will change by $\approx (0.8-0.6)/0.6\times 100\%=33\%$. I have specifically just chosen $\alpha=4$ because that gives approximately the mass-density of Jupiter in the link you provided (ignoring the gaseous hydrogen part, whose inclusion would only make the difference more profound).
However, for a rocky planet like earth, the density near the surface is approximately a factor $3/13\approx e^{-1.5}$ less than the density at the core, then $U_{\text{binding}}(\alpha=1.5)=0.649$, which is only $8\%$ different than the uniform-density approximation.
Conclusion: The exact mass distribution for a gas giant does matter, because the calculated binding energy of a planet semi-realistic distribution (exponential decay) differs from the uber-naive distribution (uniform) by a little more than $33\%$. However, for a rocky planet whose distribution tapers off much more slowly, the semi-realistic distribution only changes the naive answer about $8\%$.
So, approximating the Earth by a sphere of uniform mass-density isn't so bad, unless you want the answer to better than $8\%$.
Next Step: A possible next step would be to do a piecewise distribution, using a different distribution for each "layer". I don't think this would change the answer too much though, but it's a possible direction.
• This is terrific -- probably the best I can hope for, in the absence of anything published. I appreciate your effort. – Charles Oct 11 '17 at 18:34
The gravitational potential energy of a polytropic sphere of gas (i..e governed by a polytropic equation of state), with a polytropic index $n$ is given by $$\Omega = -\frac{3}{5-n}\ \frac{GM^2}{R}$$ See for example here.
A constant density sphere would have to have to be incompressible. Since a polytrope has a pressure $p \propto \rho^{1 +1/n}$, this corresponds to $n=0$ and gives you your result for a uniform sphere.
Higher values of $n$ have more centrally condensed profiles and larger binding energies.
The appropriate value of the polytropic index for brown dwarfs and gas giants, where energy transport is convective or where the gas is (non-relativistically) degenerate is $n=3/2$. In which case, your leading coefficient increases from 3/5 to 6/7. i.e. No big deal for an approximate calculation. | {"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": 0.8888591527938843, "perplexity": 355.8080386956131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998369.29/warc/CC-MAIN-20190617022938-20190617044938-00392.warc.gz"} |
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http://jaac.ijournal.cn/ch/reader/view_abstract.aspx?file_no=JAAC-2016-0120 | For REFEREES
Volume 8, Number 6, 2018, Pages 1645-1663 Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation Yali Duan,Linghua Kong,Xianjin Chen,Min Guo Keywords:Lattice Boltzmann method, Sine-Gordon equation, Chapman-Enskog expansion, soliton. Abstract: The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation. PDF Download reader | {"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": 0.8932161331176758, "perplexity": 603.04242013334}, "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-2018-51/segments/1544376823817.62/warc/CC-MAIN-20181212091014-20181212112514-00631.warc.gz"} |
http://mathhelpforum.com/algebra/157244-polynomial-function-imaginary-root.html | # Thread: Polynomial function with imaginary root
1. ## Polynomial function with imaginary root
The polynomial 3x^3-13x^2+ax+b=0, a and b real numbers, has 2-i as one of its roots. Where does the graph of y=3x^3-13x^2+ax+b cross the x-axis?
The solution is 1/3. Through what method(s) could i find that answer?
2. Originally Posted by ceinstein1
has 2-i as one of its roots.
One of the other roots is its complex conjugate 2+i, then use these with the fact that if P(a) = 0 then (x-a) is a factor.
Also if you want to graph this you with have to use a complex plane with an imaginary and real axis.
3. If $2-i$ is a root of $f(x) = 3x^3-13x^2+ax+b$, then so is $2+i$.
$(x-[2-i])(x-[2+i]) = x^2-4x+5$, so we can write
$3x^3-13x^2+ax+b = (x^2-4x+5)(cx+d)$ for some $c, d \in\mathbb{R}$.
By expanding the RHS, or otherwise, we see from the coefficient of $x^3$ that $c = 3$,
and from that of $x^2$ we have $d-4c = -13$, and as $c = 3$, we have $d = -1$.
Thus it crosses the x-axis at [LaTeX ERROR: Convert failed] | {"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": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9032647609710693, "perplexity": 515.490662882525}, "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/1487501170940.45/warc/CC-MAIN-20170219104610-00285-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.esaral.com/q/find-the-value-of-x-y-and-z-from-the-following-equation-56628 | # Find the value of $x, y$, and $z$ from the following equation:
Question:
Find the value of $x, y$, and $z$ from the following equation:
(i) $\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]$
(ii) $\left[\begin{array}{ll}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$
(iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$
Solution:
(i) $\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]$
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x = 1, y = 4, and z = 3
(ii) $\left[\begin{array}{ll}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{ll}6 & 2 \\ 5 & 8\end{array}\right]$
As the given matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
x + y = 6, xy = 8, 5 + = 5
Now, 5 + z = 5 z = 0
We know that:
$(x-y)^{2}=(x+y)^{2}-4 x y$
$\Rightarrow(x-y)^{2}=36-32=4$
$\Rightarrow x-y=\pm 2$
Now, when x − y = 2 and x + y = 6, we get x = 4 and y = 2
When x − = − 2 and x + y = 6, we get x = 2 and = 4
x = 4, y = 2, and z = 0 or x = 2, y = 4, and z = 0
(iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$
As the two matrices are equal, their corresponding elements are also equal.
Comparing the corresponding elements, we get:
$x+y+z=9 \ldots(1)$
$x+z=5 \ldots(2)$
$y+z=7 \ldots$ (3)
From (1) and (2), we have:
$y+5=9$
$\Rightarrow y=4$
Then, from (3), we have:
$4+z=7$
$\Rightarrow z=3$
$\therefore x+z=5$
$\Rightarrow x=2$
$\therefore x=2, y=4$, and $z=3$ | {"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": 0.9705731272697449, "perplexity": 722.9899010106515}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499857.57/warc/CC-MAIN-20230131091122-20230131121122-00048.warc.gz"} |
http://math.stackexchange.com/questions/104901/what-is-v-s-to-v | # What is $V : S \to V$?
I came across a problem where I don't know what field to consult. The problem is simple to explain:
I have a geometric object $S$, and a finite set of maps $V$ from the points in this object to $V$.
Intuitively, any $v \in V$ partitions $S$, and chosing one of these partitions yields a new such partitioning.
What kind of thing is this and/or what could I google to get more information on it or similar structures? (I am particularly interested in iterated applications, given an initial element of $V$ (partitioning), to a given sequence of points.)
-
where did you come across this problem? – bgins Feb 2 '12 at 8:59
I'm working in logic and this seemed a succinct way of describing an aspect of the semantics I'm investigating (tentatively). – tap Feb 2 '12 at 9:05
This seems like a fairly general concept. For one, it seems to be a particular generalization of finite symmetry groups. Below is some food for thought.
Note that $V^{(k)}=\{v^k|v\in V\}\subset V$ for $k\in\mathbb{Z}^+$.
Note that we could also view each $s\in S$ as defining an action $\sigma(s)$ on $V$ by $v\rightarrow v(s)$. In other words, $(\sigma(s))(v)=v(s)$. For each $s$, let us call $\sigma(s):V\rightarrow V$ the induced action of $s$ on $V$ and let us call $\sigma:S\rightarrow V^V$ the "complementary" map.
Since composition of maps is associative, $V$ is a semigroup under map composition. If there is a unique element $e \in V$ so that $e(S)=\{e\}$ (i.e., a unique simultaneous fixed point of each "complementary" map $s:V\rightarrow V$), then $V$ is a monoid. Note, however, that composition of maps is not commutative. For example, we have no guarantee that right-inverses and left-inverses of each $v \in V$ exist (in $V$) or that they are the same. Of course, if this is the case, then $V$ is a group as well. But in general, $V$ will be a semigroup and we need to be aware of concepts like pseudoinverse, idempotents, Green's relations and absorbing elements before we try to think of $V$ as the more familiar group. Each $V^{(k)}$ above will be a sub-object of the respective type. In case $V$ is in fact a group (which is not true in general), it would help to have some background on the symmetric group of permutations.
If there exists an $n>1$ so that $v^n=v$ for all $v$ or, more generally, so that $V^n=V$ these would be interesting cases to study. The latter case (when the map $v\rightarrow v^n$ is onto or surjective) might imply that $V$ is a group with unique identity, particularly for $n=|V|$, but you might also need to assume a limit on the number of idempotents. Whether left or right inverses are always well-defined seems to also hinge on the number of distinct idempotents.
If left or right inverses are well-defined and there is at most one idempotent, I think we can extend $V$ to include an identity and left or right inverses of each element. Each stepwise extension would still be finite. Whether the recursive extension (like a closure) is finite is another question.
$V=S$ would be another interesting case to study, but might reduce or specialize to the study of symmetry groups, the symmetric group, and group actions. If $V$ were allowed to be infinite, elliptic curves would be another interesting example.
Intuitively, I'm thinking of the (finite sets) $V$ as "views" or perspectives or projections of the (possibly infinite sets) $S$, and as the "complementary" map $\sigma$ as an important key to decomposing $V$. If $S$ is itself already a group, then this "complementary" map could define a group action of $S$ on $V$, providing a rich source of examples.
Perhaps there are also fruitful connections with category theory.
-
Thank you for your elaborate answer. I will certainly look into the relfections you've offered. – tap Feb 2 '12 at 13:28 | {"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": 0.9038525819778442, "perplexity": 218.0754732358596}, "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/1393999664754/warc/CC-MAIN-20140305060744-00097-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=2366949&postcount=1 | View Single Post
P: 12 1. The problem statement, all variables and given/known data A triangle with x as the angle, 'a' as the adjacent, 8 as the opposite side and 'h' as the hypotenuse. An intermediate question was to show that sin(x)=cot(x). Now i have to write 'sin(x)=cot(x)' in terms of cos(x) only and solve the quadratic, stating correct value for cos(x) 2. Relevant equations 2. "sin(x)=cot(x)" in terms of cos(x). a*h=8^2 3. The attempt at a solution 3. I have used some trig identities and have got cos^2(x)+cos(x)-1=0. Is this correct? To solve i solved x^2+x-1 using quadratic formula and got 2 answers, one negative. So cos(x) is the positive value..? | {"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": 0.8773965239524841, "perplexity": 1073.6032540016824}, "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/1393999654293/warc/CC-MAIN-20140305060734-00044-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://labs.tib.eu/arxiv/?author=G.%20Przeau | • This paper presents a study of the ISW effect from the Planck 2015 temperature and polarization data release. The CMB is cross-correlated with different LSS tracers: the NVSS, SDSS and WISE catalogues, and the Planck 2015 lensing map. This cross-correlation yields a detection at $4\,\sigma$, where most of the signal-to-noise is due to the Planck lensing and NVSS. In fact, the ISW effect is detected only from the Planck data (through the ISW-lensing bispectrum) at $\approx 3\,\sigma$, which is similar to the detection level achieved by combining the cross-correlation signal coming from all the catalogues. The ISW signal allow us to detect $\Omega_\Lambda$ at more than $3\,\sigma$. This cross-correlation analysis is performed only with the Planck temperature data, since the polarization scales available in the 2015 release do not permit significant improvement of the CMB-LSS cross-correlation detectability. Nevertheless, polarization data is used to study the anomalously large ISW signal previously reported through the stacking of CMB features at the locations of known superstructures. We find that the current Planck polarization data do not exclude that this signal could be caused by the ISW effect. In addition, the stacking of the Planck lensing map on the locations of superstructures exhibits a positive cross-correlation with these large-scale structures. Finally, we have improved our previous reconstruction of the ISW temperature fluctuations by combining the information encoded in all the previously mentioned LSS tracers. In particular, we construct a map of the ISW secondary anisotropies and the corresponding uncertainties map, obtained from simulations. We also explore the reconstruction of the ISW anisotropies caused by the LSS traced by the 2MPZ survey by directly inverting the density field into the gravitational potential field. | {"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": 0.8842895030975342, "perplexity": 1342.5414760718108}, "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-2021-10/segments/1614178364932.30/warc/CC-MAIN-20210302221633-20210303011633-00617.warc.gz"} |
http://mathhelpforum.com/calculus/49871-limit-proof-print.html | # limit proof
• September 20th 2008, 12:14 PM
PvtBillPilgrim
limit proof
I have that the limit of f(x) as x goes to c exists and that the limit of g(x) as x goes to c does not exist.
How do I prove that the limit of h(x) as x goes to c does not exist (where h = f + g)?
What can you say about the limit of g using epsilon notation to reach a contradiction?
• September 20th 2008, 12:24 PM
hwhelper
lim g(x) = lim h(x) - lim f(x)
because the "limit" operation distributes over addition/subtraction. <-- Are you allowed to assume this in your problem or do you essentially have to prove this?
If RHS is finite then LHS is finite. QED.
• September 20th 2008, 12:32 PM
PvtBillPilgrim
I guess that does work, but I don't want to use that.
abs(g(x) - L) >= epsilon I think, but g doesn't technically have a limit, so what do you say?
• September 20th 2008, 12:57 PM
Jhevon
Quote:
Originally Posted by PvtBillPilgrim
I guess that does work, but I don't want to use that.
abs(g(x) - L) >= epsilon I think, but g doesn't technically have a limit, so what do you say?
i guess that could work.
you have for all $\epsilon > 0$ and all those $\delta$'s and stuff :p that
$|f(x) - F| < \frac {\epsilon}2$ but $|g(x) - G| \ge \frac {\epsilon}2 = \frac {\epsilon}2 + \zeta$, where $\zeta > 0$
where $F$ and $G$ are the limits of $f(x)$ and $g(x)$ respectfully.
had $\lim_{x \to c}h(x)$ existed, we would have $|h(x) - H| < \epsilon$, where $H = F + G$ is the limit of $h(x)$
but, $|h(x) - H| = |[f(x) + g(x)] - [F + G]| = |[f(x) - F] + [g(x) - G]|$ $\le |f(x) - F| + |g(x) - G| = \frac {\epsilon}2 + \bigg( \frac {\epsilon}2 + \zeta \bigg) > \epsilon$
i don't really like this though. hwhelper's way is perhaps better. maybe you can fix it up
• September 20th 2008, 01:09 PM
PvtBillPilgrim
it's weird because at the end you get that
abs(f - F) < epsilon
and abs(g- G) >= epsilon
does this imply that abs(h - (F+G)) >= epsilon?
• September 20th 2008, 01:25 PM
Jhevon
Quote:
Originally Posted by PvtBillPilgrim
it's weird because at the end you get that
abs(f - F) < epsilon
and abs(g- G) >= epsilon
does this imply that abs(h - (F+G)) >= epsilon?
yeah, the inequalities turned in opposing directions does make things unclear. that's why i liked hwhelper's solution better. it is more straight forward. an epsilon proof is more subtle, i'll have to think about it more
• September 20th 2008, 05:22 PM
ThePerfectHacker
Quote:
Originally Posted by PvtBillPilgrim
I have that the limit of f(x) as x goes to c exists and that the limit of g(x) as x goes to c does not exist.
How do I prove that the limit of h(x) as x goes to c does not exist (where h = f + g)?
What can you say about the limit of g using epsilon notation to reach a contradiction?
If the limit of (f+g) existed then the limit of [(f+g)-f] = g would have existed. | {"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": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9360914826393127, "perplexity": 925.3162392394845}, "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-2015-27/segments/1435375098464.55/warc/CC-MAIN-20150627031818-00049-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2930863/what-is-substitution-simultaneously | # What is substitution simultaneously?
I have been reading Kleene "Introduction to Metamathematics" and found out that, even though he has been using a notion of "substituting simultaneously", he has never defined it.
On page 78 he says "The substitution of term $$t$$ for a variable $$x$$ in a term or formula $$A$$ shall consist in replacing simultaneously each free occurrence of $$x$$ in $$A$$ by an occurrence of $$t$$."
On page 79 he says "Similarly, we define substitution performed simultaneously for a number of distinct variables."
So, what does "substitution performed simultaneously" actually means?
My thoughts:
if we are given formula $$A(x)$$ and we want to replace $$t$$ for a variable $$x$$ in it then, first of all, we find all places in $$A(x)$$ (which is a finite string of symbols) where $$x$$ is. Then, we go from left to right through the string $$A(x)$$ and write $$t$$ in each place where we identified the $$x$$ is.
One then might define substitution in $$A(x_1, ... , x_n)$$ for $$t_1, ..., t_n$$ recursively but then I guess one runs into problem that $$t_2$$ for example might be $$x_3$$. Then, the recursive mechanism is not the same as one would intuitively think as "simultaneous substitution", at least I think so.
So, probably, one could define simultaneous substitution in the case of many variables as, first of all, because $$A(x_1, ..., x_n)$$ is still a finite string, we find all occurences of $$x_1, ... x_n$$. Then, we go from the left to right in a given string and whenever we have occurence of some $$x_i$$ we replace it by $$t_i$$. Then, we continue until the string ends. Is this how it is supposed to be defined?
As a concrete example, take the formula $$A(x,y)$$ to be $$x=y$$.
Substituting $$x+y$$ for $$x$$ and $$x-y$$ for $$y$$ simultaneously in $$A$$ yields $$x+y = x-y$$ Substituting $$x+y$$ for $$x$$ and then substituting $$x-y$$ for $$y$$ in $$A$$ yields $$x+(x-y)=x-y$$ These are evidently not the same formula! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 36, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8973663449287415, "perplexity": 115.3294573413238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250604397.40/warc/CC-MAIN-20200121132900-20200121161900-00471.warc.gz"} |
https://wikidev.seg.org/index.php?title=Exploring_nonlinear_inversions:_A_1D_magnetotelluric_example&printable=yes | # Exploring nonlinear inversions: A 1D magnetotelluric example
At some point in many geophysical workflows, an inversion is a necessary step for answering the geoscientific question at hand, whether it is recovering a reflectivity series from a seismic trace in a deconvolution problem, finding a susceptibility model from magnetic data, or recovering conductivity from an electromagnetic survey. This is particularly true when working with data sets where it may not even be clear how to plot the data: 3D direct current resistivity and induced polarization surveys (it is not necessarily clear how to organize data into a pseudosection) or multicomponent data, such as electromagnetic data (we can measure three spatial components of electric and/or magnetic fields through time over a range of frequencies). Inversion is a tool for translating these data into a model we can interpret. The goal of the inversion is to find a “model” — some description of the earth's physical properties — that is consistent with both the data and geologic knowledge.
In a general inverse problem, we start from a forward problem, of the form ${\displaystyle {\mathcal {F}}[\mathbf {m} ]=\mathbf {d} }$, where ${\displaystyle {\mathcal {F}}}$ is the forward operator (the mathematical description of the physics/problem), d is our data, and m is our earth model (an array of numbers that describes the physical properties of the earth). Matt Hall kicked off the discussion of inversions in The Leading Edge in his Linear Inversion tutorial.[1] He walked through how to solve the classic linear inverse problem in which the forward simulation takes the form ${\displaystyle {\mathcal {F}}[\mathbf {m} ]=\mathbf {G} \mathbf {m} =\mathbf {d} }$. The example he demonstrated is a deconvolution problem; in that case, G is a convolution matrix, m is the reflectivity series, and d is a seismic trace. He introduced the concepts of an underdetermined problem, motivated the need for regularization, formulated the inversion in terms of an optimization problem, and solved the linear inverse problem (in true polyglot fashion, using Python, Lua, Julia, and R). In this tutorial, we will pick up from there and explore a nonlinear forward problem, of the form ${\displaystyle {\mathcal {F}}[\mathbf {m} ]=\mathbf {d} }$; in this case, our forward operator is a function of the model. In the accompanying notebooks (https://github.com/seg), we use SimPEG (http://simpeg.xyz) for the implementation in Python of the physics simulations, optimization, and structure necessary to perform an inversion.[2]
## Magnetotellurics
We will explore the 1D magnetotelluric (MT) survey technique, which is a natural source electromagnetic method. In MT, plane-wave source fields are generated by solar wind (giving us low-frequency signals <1 Hz) and lightning strikes worldwide (giving us higher frequency signals >1 Hz). In MT, the model m is a description of the earth's electrical conductivity ${\displaystyle \sigma }$, and ${\displaystyle {\mathcal {F}}[\mathbf {m} ]}$ solves Maxwell's equations — giving the electric field ${\displaystyle {\vec {E}}}$ and the magnetic field ${\displaystyle {\vec {H}}}$ — for a plane-wave source. In the continuous world, Maxwell's equations are:
${\displaystyle \nabla \times {\vec {E}}+i\omega \mu {\vec {H}}=0}$ ${\displaystyle \nabla \times {\vec {H}}+\sigma {\vec {E}}=0}$ ${\displaystyle {\text{with boundary conditions}}}$ (1)
where ${\displaystyle \mu }$ is magnetic permeability, generally taken to be the permeability of free space ${\displaystyle \mu _{0}}$, and ${\displaystyle \omega =2\pi f}$ is the angular frequency (${\displaystyle f}$ is frequency in Hz). When we discretize Maxwell's equations so they can be solved numerically (see the first notebook), we obtain a matrix system of the form
${\displaystyle \mathbf {A} (\mathbf {m} )\mathbf {u} =\mathbf {b} }$ (2)
where A(m) is a matrix capturing the physics, u is a vector of the electric and magnetic fields everywhere in our simulation domain, and b includes the boundary conditions that describe the plane wave source. This problem is nonlinear because this matrix A depends on the conductivity model. When we solve equation 2 for u, we obtain the electric and magnetic fields everywhere on our mesh. Our data consist of samples of the electric and magnetic fields where we have receivers. For the MT problem, the measured data, d, are impedances; for the 1D problem, the impedance ${\displaystyle Z_{xy}}$ at a single frequency is given by
${\displaystyle Z_{xy}=-{\frac {E_{x}}{H_{y}}}.}$ (3)
Note that the impedance is a complex number, consisting of real and imaginary parts. Impedance is a nonintuitive quantity; often, we instead consider apparent resistivity (Note that resistivity is the inverse of conductivity ${\displaystyle \rho =1/\sigma }$) ${\displaystyle \rho _{\mathrm {a} }}$ and phase ${\displaystyle \psi }$, given by
${\displaystyle \rho _{\mathrm {a} }={\frac {1}{\mu _{0}\omega }}{\big |}Z_{xy}{\big |}^{2},\quad \psi =\tan ^{-1}\left({\frac {{\text{Im}}(Z_{xy})}{{\text{Re}}(Z_{xy})}}\right).}$ (4)
For an earth that is a half-space, the apparent resistivity equals the true resistivity, and the phase is 45°. When we implement the computation of the data, we define a method that: (1) selects the values of the electric and magnetic fields at the surface of the earth from the fields that we computed everywhere in our domain u and (2) computes their ratio to provide us with impedance data, that is: P(u) = d.
In summary, to implement the forward simulation for the MT problem (${\displaystyle ({\mathcal {F}}[\mathbf {m} ]=\mathbf {d} }$), we break it into two steps:
1. solve A(m)u = b; and
2. compute the impedance data d = P(u).
In the first notebook, we provide details on how each step is performed using a finite difference approach. If you are looking for more on numerical discretization, we wrote a tutorial on finite volume methods.[3]
The inversion aims to solve ${\displaystyle {\mathcal {F}}^{-1}[\mathbf {d} ]}$ for a model. Just as in the linear problem, we require regularization to select a model from the infinitely many that can fit the data. Before we tackle this ill-posed inverse problem, let's explore an example of nonuniqueness: how can different models give us the same data?
## Go forwards
A classic example that demonstrates the nonuniqueness of MT data is the equivalence of the conductivity-thickness product (conductance) of a thin layer. If we start with a layer that has a conductivity of ${\displaystyle \sigma }$, halve its thickness, and double its conductivity, the resulting data will be similar. In Figure 1, we show apparent resistivity and phase data for five models, each of which has the same conductance. In all of the simulations, the data show a decrease in apparent resistivity and an increase in phase starting at ∼10 Hz. Thus, in all of the data we have evidence of a conductive layer, and the frequency range at which it appears is an indicator of the depth of the layer (you can explore by changing the depth variable in the model setup of the second notebook). However, all scenarios produce similar data. Even with a small amount of noise, we cannot expect an inversion code to separate the conductivity and thickness of a conductive unit without incorporating additional information. When setting up the inverse problem and defining regularization (next up), it is important to realize that the choices we make there will influence the character of the model we recover, as the data alone do not provide us with a unique model.
Figure 1. MT responses from five models, each having an equivalent conductivity-thickness product for the conductive layer. (a) Conductivity models, (b) apparent resistivity (${\displaystyle \rho _{a}}$), and (c) phase (${\displaystyle \phi }$).
## Go backwards
There are many models that can fit the data, so we need a means of regularizing our inversion so that we can select a single, reasonable model from the (infinitely) many that agree with the data. We will come back to how the regularization ${\displaystyle \phi _{\text{m}}}$ is defined, but for now, consider it as a measure of how “reasonable” the model is based on our prior knowledge about the earth. Formally, we pose the inversion as an optimization problem:
${\displaystyle {\underset {\mathbf {m} }{\text{minimize}}}\quad \phi _{\text{m}}(\mathbf {m} )}$ ${\displaystyle {\text{subject to }}\quad {\mathcal {F}}[\mathbf {m} ]=\mathbf {d} _{\text{obs}}}$ (5)
Essentially we are saying “find the model m that best fits the assumptions we are making in the regularization ${\displaystyle \phi _{\text{m}}(\mathbf {m} )}$, and that agrees with the observed data ${\displaystyle \mathbf {d} _{\text{obs}}}$.” In practice, our data are noisy, so there is no sense in fitting them exactly. Rather, we pose the optimization problem as a trade-off between fitting the data and fitting the regularization, so our inverse problem can be stated as:
${\displaystyle {\underset {\mathbf {m} }{\text{minimize}}}\quad \phi (\mathbf {m} )=\phi _{\mathrm {d} }(\mathbf {m} )+\beta \phi _{\mathrm {m} }(\mathbf {m} )}$ (6)
where ${\displaystyle \phi _{d}(\mathbf {m} )}$ is the data misfit, a measure of how far our simulated data are from the observed data; ${\displaystyle \phi _{m}(\mathbf {m} )}$ is the regularization; and ${\displaystyle \beta }$ is a trade-off parameter that weights the relative importance of the data misfit and regularization in the optimization. A larger ${\displaystyle \beta }$ says that we want our model to do a good job minimizing the regularization, while a smaller ${\displaystyle \beta }$ turns down the importance of the regularization and says that fitting the data is more important in the inversion.
The data misfit is often taken to be a weighted ${\displaystyle \ell _{2}}$ norm:
${\displaystyle \phi _{d}(\mathbf {m} )={\frac {1}{2}}\|\mathbf {W_{d}} ({\mathcal {F}}(\mathbf {m} )-\mathbf {d} _{\text{obs}})\|_{2}^{2}}$ (7)
where ${\displaystyle \mathbf {W_{d}} }$ captures the noise model (typically it is a diagonal matrix containing the standard deviation of each datum).
The regularization is one place where a priori information about the geologic setting can be brought in. There are a variety of regularization functionals that can be chosen, but one of the most widely used is Tikhonov regularization, which again uses ${\displaystyle \ell _{2}}$ norms:
${\displaystyle \phi _{m}(\mathbf {m} )={\frac {1}{2}}{\big (}\alpha _{\mathrm {s} }\|\mathbf {W_{\mathrm {s} }} (\mathbf {m} -\mathbf {m} _{\text{ref}})\|_{2}^{2}+\alpha _{\mathrm {z} }\|\mathbf {W_{\mathrm {z} }} (\mathbf {m} )\|_{2}^{2}{\big )}}$ (8)
The first term is often referred to as the "smallness" as it measures the "size" of the model (in the ${\displaystyle \ell _{2}}$ sense). The matrix ${\displaystyle \mathbf {W_{\mathrm {s} }} }$ is generally taken to be a diagonal matrix that may contain information about the length scales of the model or be used to weight the relative importance of various parameters in the model. The scalar ${\displaystyle \alpha _{s}}$ weights the relative importance of this term in the regularization. Notice that we include a reference model, ${\displaystyle \mathbf {m} _{\text{ref}}}$. Often this is defined as a constant value, but if more information is known about the background, that can be used to construct a more intricate reference model. Here, we will not delve too far into how the reference model impacts the recovered results, but you are encouraged to change mref in the notebooks and investigate its impact.
The second term is often referred to as the "smoothness". The matrix ${\displaystyle \mathbf {W_{\mathrm {z} }} }$ approximates the derivative of the model with respect to depth, and is hence a measure of how "smooth" the model is. The term ${\displaystyle \alpha _{\mathrm {z} }}$ weights the relative importance of smoothness in the regularization.
From this setup, we see that there are quite a number of choices to make: defining uncertainties on the data (${\displaystyle \mathbf {W_{\mathrm {d} }} }$), selecting a reference model (${\displaystyle \mathbf {m} _{\text{ref}}}$), choosing the importance of smallness and smoothness (${\displaystyle \alpha _{\mathrm {s} }}$ and ${\displaystyle \alpha _{\mathrm {z} }}$), and selecting a trade-off parameter (${\displaystyle \beta }$). Let’s start by assuming a known noise model, fix ${\displaystyle \alpha _{\mathrm {s} }}$ and ${\displaystyle \alpha _{\mathrm {z} }}$, and explore the impact of the trade-off parameter ${\displaystyle \beta }$. Our forward problem depends upon the electrical conductivity. For the inverse problem, however, we are free to use any function of the conductivity as a parameter. The electrical conductivity of earth materials varies by many orders of magnitude and is strictly positive. Thus it is advantageous to use ${\displaystyle \mathbf {\log(\sigma )} }$ as the model in the inverse problem. For a nonlinear problem, we also have the additional choice of the initial model ${\displaystyle \mathbf {m} _{0}}$ at which to start the inversion. Although we will not discuss the choice of ${\displaystyle \mathbf {m} _{0}}$, you are encouraged to change the initial model in the notebooks and examine the impact it makes because it can be significant!
### The ${\displaystyle \beta }$ knob
If the noise is Gaussian, then the sum of squares (our data misfit) is a Chi-squared distribution, which has an expected value of ${\displaystyle N_{\text{data}}}$ (in our case, we divide this by two to match our definition of ${\displaystyle \phi _{\mathrm {d} }}$). Thus, the ideal choice of ${\displaystyle \beta }$ is one that gives us ${\displaystyle \phi _{\mathrm {d} }^{*}\approx {\frac {1}{2}}N_{\text{data}}}$. To demonstrate the effect of ${\displaystyle \beta }$, we consider a five-layer model, originally shown in Whitall and Oldenburg (1992), and will demonstrate inversions when we achieve the target misfit, underfit the data, and overfit the data.[4] The conductivity model used is the solid black line in Figure 2a. For these inversions we fix the regularization parameters to ${\displaystyle \alpha _{\mathrm {s} }=10^{-2}}$, ${\displaystyle \alpha _{\mathrm {z} }=1}$ and set ${\displaystyle \mathbf {m} _{\text{ref}}=10^{-2}S/m}$, and the initial model, ${\displaystyle \mathbf {m} _{0}=\mathbf {m} _{\text{ref}}}$ (feel free to change them in the notebook!). We start the inversion with a large ${\displaystyle \beta }$ and decrease its value to plot the trade-off or Tikhonov curve (Figure 2b). In blue, we show the inversion that is stopped when the data misfit approximately equals the target misfit (the star in Figure 2b). Figures 2c and 2d show the data as apparent resistivity and phase, which is a visualization of our complex-valued impedance data. The blue line in Figure 2a shows the recovered model, which identifies the general structure and conductivity values of the five layers. In this case, we are employing a smooth regularization, thus we expect to recover smoothly varying structures.
Figure 2. Inversions that fit the data (blue), underfit the data (orange), and overfit the data (green). (a) True (black) and recovered electrical conductivity models. (b) Tikhonov curve showing the trade-off between the misfit and regularization, target misfit (red star), and achieved misfit corresponding to each of the inversion results shown (c) observed (black) and predicted apparent resistivity data, (d) observed (black) and predicted phase data.
If we instead choose a larger ${\displaystyle \beta }$, reducing the contribution of the data misfit to the objective function, we underfit the data, as is shown in orange in Figure 2. Although we still see evidence of two conductive structures, we do not recover their amplitudes and do a poor job resolving the location and widths of the conductive layers. (If you had to pick the top of the first layer, where should it be?) Examining the plots in Figures 2c and 2d, there is more insight about the subsurface conductivity that can be learned by pushing the inversion to extract more from the data.
On the other extreme, we can choose a very small ${\displaystyle \beta }$ and try to fit all of the details in the data. Doing this, we obtain the results shown in green in Figure 2. When we push the inversion to fit the (noisy) data very closely, we end up fitting the noise. To do this, conductivity contrasts are exaggerated and oscillatory and erroneous conductivity structures are introduced in the inversion.
### The ${\displaystyle \alpha }$ knobs
For the inversions shown in Figure 2, we prescribed the values ${\displaystyle \alpha _{s}}$, ${\displaystyle \alpha _{z}}$. What impact do they have on the character of the model we recover?
In Figure 3, we compare two inversions with different regularization parameters: (1) a smooth inversion (blue line), with ${\displaystyle \alpha _{s}=10^{-5}}$ and ${\displaystyle \alpha _{z}=1}$, and (2) a small inversion (brown line): with ${\displaystyle \alpha _{s}=1}$ and ${\displaystyle \alpha _{z}=10^{-5}}$. In both, ${\displaystyle \beta }$ was chosen so that a desired target misfit was achieved. The smooth inversion penalizes large gradients; the resulting model has two smooth peaks. Note that we smooth over the resistive third layer, overestimating its conductivity. The small inversion instead favors models that are close to the reference model; this model has more structure. The resistivity of the first layer matches well, and the conductivity of the third layer is closer to its true value, but additional oscillatory structures are introduced at depth. In the third notebook, you can explore the impact of these parameters yourself.
Figure 3. Comparing the use of smooth regularization versus small regularization in the inversion.
In practice, these parameters are often determined by experimentation; strategies such as examining length scales are often successfully adopted (see page 38 in Oldenburg and Li, 2005)[5]. Changing the relative values of ${\displaystyle \alpha _{s}}$ and ${\displaystyle \alpha _{z}}$ is one way to bring in a priori information. If we know very little, often starting with a smooth inversion is a good option; this penalizes structure (high gradients) while showing general trends. If more structure is expected, or a reliable reference model can be built from additional data such as physical property measurements, well logs, or additional geophysical/geologic data, then the influence of the smallness term may be increased. There are a few other ways to bring in additional a priori information. If we are expecting a more “blocky” model, we can choose a different norm (such as an ${\displaystyle \ell _{1}}$ norm), or if we have structural constraints, we can introduce other weighting structures (e.g., on the smoothness); these are knobs for another tutorial, and there is discussion in Oldenburg and Li (2005).[5]
## Summary
In this tutorial, we have introduced the forward simulation for MT and explored a few aspects of the inverse problem. Prior to jumping into an inversion, it is important to know the limitations of the survey and data, and what you can and cannot resolve, even if there is no noise. Forward modeling is a powerful tool for setting realistic expectations of an inversion.
To set up and solve the inverse problem, we posed the inversion as an optimization problem that searches for a model of the earth that minimizes an objective function consisting of a data misfit and a regularization term. There are many choices to be made in defining the various elements of the inverse problem, including how to assign uncertainties, selecting a trade-off parameter, defining the regularization function, and choosing initial and reference models. In this tutorial we explored two of the knobs: (1) the trade-off parameter and (2) the relative importance of smallness and smoothness contributions in Tikhonov regularization. The interactive notebooks that are provided allow you to change parameters and experiment with their impact.
## References
1. Hall, M., 2016, Linear inversion: The Leading Edge, 35, no. 12, 1085–1087, http://dx.doi.org/10.1190/tle35121085.1
2. Cockett, R., S. Kang, L. J. Heagy, A. Pidlisecky, and D. W. Oldenburg, 2015, SimPEG: An open source framework for simulation and gradient based parameter estimation in geophysical applications: Computers & Geosciences, 85, 142–154, http://dx.doi.org/10.1016/j.cageo.2015.09.015
3. Cockett, R., L. J. Heagy, and D. W. Oldenburg, 2016, Pixels and their neighbors: Finite volume: The Leading Edge, 35, no. 8, 703–706, http://dx.doi.org/10.1190/tle35080703.1.
4. Whittall, K., and D. Oldenburg, 1992, Inversion of magnetotelluric data for a one-dimensional conductivity: Society of Exploration Geophysicists. http://dx.doi.org/10.1190/1.9781560802419
5. Oldenburg, D. W., and Y. Li, 2005, Inversion for applied geophysics: A tutorial, in D. K. Butler, ed., Investigations in Geophysics: Near-Surface Geophysics, 89–150, http://dx.doi.org/10.1190/1.9781560801719.ch5
## Corresponding author
• Corresponding author: Seogi Kang, University of British Columbia Geophysical Inversion Facility, skangeos.ubc.ca | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 86, "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": 0.9044074416160583, "perplexity": 539.4139167916974}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201996.61/warc/CC-MAIN-20190319143502-20190319165502-00075.warc.gz"} |
https://www.physicsforums.com/threads/lorentz-invariance-and-general-invariance.445237/ | # Lorentz invariance and General invariance
• Start date
#### mtak0114
47
0
Hi
I am confused about these two related but different terms
Lorentz invariance/covariance and General invariance/covariance
As I understand it a Lorentz invariant is a scalar which is the same in all inertial reference frames i.e. it acts trivially under a Lorentz transformation
an example would be rest mass $$p^\mu p_\mu = m^2$$, all observers would agree on the value of the mass. But is this true for all scalars say for example the inner product between two arbitrary 4-vectors $$V^\mu W_\mu = C$$ would all inertial observers agree on the value of C? A good example may be the inner product between the 4-velocity and the 4-acceleration $$u^\mu a_\mu = 0$$.
How does this change for general invariance
$$p^\mu p_\mu = m$$ it is true that all observers would agree on the mass but how about for:
$$V^\mu W_\mu = C$$
is this still a constant C or is it a spacetime dependant quantity $$C(x^\mu)$$?
What about $$u^\mu a_\mu = 0$$
I understand that this is still true in general relativity but is this a special scalar?
still very confused hope you can help
thanks
Mark
Related Special and General Relativity News on Phys.org
#### atyy
13,627
1,684
A scalar g(u,v) is a generally invariant quantity, and hence a Lorentz invariant quantity. However, it can be a spacetime dependent quantity - ie. has a different value at different points in spacetime (spacetime dependence), but at any particular point in spacetime it has the same value under arbitrary coordinate changes (generally invariant).
However, a different use of Lorentz invariant is the "form" of the laws of physics. In special relativity, Lorentz covariance means the laws are supposed to maintain the "same form" under a Lorentz transformation, where the "same form" means that we are NOT allowed to use Christoffel symbols (covariant derivatives disallowed) to write the laws of physics.
If we are allowed to use Christoffel symbols (covariant derivatives allowed), then any law is generally covariant, and hence also Lorentz covariant.
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https://arxiv.org/list/astro-ph.EP/new | # Earth and Planetary Astrophysics
## New submissions
[ total of 12 entries: 1-12 ]
[ showing up to 2000 entries per page: fewer | more ]
### New submissions for Thu, 24 May 18
[1]
Title: Phantom Inflated Planets in Occurrence Rate Based Samples
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
The recently launched Transiting Exoplanet Survey Satellite (TESS) is expected to produce many new exoplanet discoveries which will be especially amenable to follow-up study. Assessments of the planet discovery yield of TESS, such as Sullivan et al. (2015) and Barclay et al. (2018), will be important for planning follow-up work. Analyzing these predicted planet samples, however, we find that giant planet radii derived from the current bulk transiting planet sample have been used at all potential orbits without accounting for the temperature dependence of radius inflation. These phantom inflated planets (PIPs) make up just over 1.1% of the Sullivan et al. (2015) predicted population and about 8% of the Barclay et al. (2018) sample. Similar population predictions for direct imaging studies should likewise take care not to include inflated planets at large separations as such planets will appear larger and, depending on assumed albedo, brighter and more easily detectable than physically possible. Despite their high false positive rate, giant planets are the testbeds for atmospheric characterization techniques and care should be taken to understand and account for potential contaminating factors in this population.
[2]
Title: KMT-2016-BLG-1107: A New Hollywood-Planet Close/Wide Degeneracy
Comments: 22 pages, 7 figures, submitted to AAS
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
We show that KMT-2016-BLG-1107 displays a new type of degeneracy between wide-binary and close-binary Hollywood events in which a giant-star source envelops the planetary caustic. The planetary anomaly takes the form of a smooth, two-day "bump" far out on the falling wing of the light curve, which can be interpreted either as the source completely enveloping a minor-image caustic due to a close companion with mass ratio $q=0.036$, or partially enveloping a major-image caustic due to a wide companion with $q=0.004$. The best estimates of the companion masses are both in the planetary regime ($3.3^{+3.5}_{-1.8}\,M_{\rm jup}$ and $0.090^{+0.096}_{-0.037}\,M_{\rm jup}$) but differ by an even larger factor than the mass ratios due to different inferred host masses. We show that the two solutions can be distinguished by high-resolution imaging at first light on next-generation ("30m") telescopes. We provide analytic guidance to understand the conditions under which this new type of degeneracy can appear.
[3]
Title: Rings in the Solar System: a short review
Comments: Accepted for the Handbook of Exoplanets
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
Rings are ubiquitous around giant planets in our Solar System. They evolve jointly with the nearby satellite system. They could form either during the giant planet formation process or much later, as a result of large scale dynamical instabilities either in the local satellite system, or at the planetary scale. We review here the main characteristics of rings in our solar system, and discuss their main evolution processes and possible origin. We also discuss the recent discovery of rings around small bodies.
[4]
Title: Bubbles in Titan's seas: nucleation, growth and RADAR signature
Comments: Published in The Astrophysical Journal
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
In the polar regions of Titan, the main satellite of Saturn, hydrocarbon seas have been discovered by the Cassini-Huygens mission. RADAR observations have revealed surprising and transient bright areas over Ligeia Mare surface. As suggested by recent research, bubbles could explain these strange features. However, the nucleation and growth of such bubbles, together with their RADAR reflectivity, have never been investigated. All of these aspects are critical to an actual observation. We have thus applied the classical nucleation theory to our context, and we developed a specific radiative transfer model that is appropriate for bubbles streams in cryogenic liquids. According to our results, the sea bed appears to be the most plausible place for the generation of bubbles, leading to a signal comparable to observations. This conclusion is supported by thermodynamic arguments and by RADAR properties of a bubbly column. The latter are also valid in the case of bubble plumes, due to gas leaking from the sea floor.
[5]
Title: An interstellar origin for Jupiter's retrograde co-orbital asteroid
Comments: Published in Monthly Notices of the Royal Astronomical Society: Letters
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
Asteroid (514107) 2015 BZ509 was discovered recently in Jupiter's co-orbital region with a retrograde motion around the Sun. The known chaotic dynamics of the outer Solar System have so far precluded the identification of its origin. Here, we perform a high-resolution statistical search for stable orbits and show that asteroid (514107) 2015 BZ509 has been in its current orbital state since the formation of the Solar System. This result indicates that (514107) 2015 BZ509 was captured from the interstellar medium 4.5 billion years in the past as planet formation models cannot produce such a primordial large-inclination orbit with the planets on nearly-coplanar orbits interacting with a coplanar debris disk that must produce the low-inclination small-body reservoirs of the Solar System such as the asteroid and Kuiper belts. This result also implies that more extrasolar asteroids are currently present in the Solar System on nearly-polar orbits.
[6]
Title: Radar-based Re-Entry Predictions with very limited tracking capabilities: the GOCE case study
Comments: 41 pages, 30 figures, presented at the 4th International Workshop on Space Debris Re-entry, 28Feb-01Mar 2018, ESOC, Darmstadt
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
The problem of the re-entry predictions of GOCE has been deeply investigated in the literature, due to the large amount of data, mainly radar and GPS, available until re-entry. The accurate GPS and attitude measurements are used to compute a precise reference orbit for the three weeks of decay, and to extrapolate the ballistic coefficient evolution of the object. In previous works, the capabilities of radar-based solutions for the re-entry predictions of GOCE and of similar objects were investigated, focusing on the german TIRA radar. In this work we have performed additional analysis, focusing on the northern european radar EISCAT-UHF, located in Troms\o, Norway. This sensor, conceived for atmospheric studies, has recently been considered for space debris applications. Due to its limited tracking capabilities, we are interested in testing its effectiveness in supporting re-entry predictions. We have simulated reliable re-entry prediction scenarios, with different availability of data from EISCAT and TIRA. The main conclusion is that, provided a suitable amount of observations, EISCAT-based re-entry predictions are comparable to TIRA-based, but also to GPS and TLE-based, corresponding ones. In general, EISCAT is not able to determine an orbit with the same accuracy of TIRA, but the results are equivalent in terms of re-entry predictions, if we consider the relevant parameters involved and their effects on the re-entry time. What is very important is the difficulty in predicting atmospheric and attitude significant variations in between the current epoch and the actual re-entry. Thus, it is not easy to keep the accuracy of predictions much lower than 10% of the residual lifetime, apart from cases with constant area to mass ratio, and low atmospheric variations with respect to the models. An experiment with real data is presented for the object 2012-006K, with consistent results.
[7]
Title: Primordial N2 provides a cosmochemical explanation for the existence of Sputnik Planitia, Pluto
Comments: 37 pages, 7 figures, 2 tables
Subjects: Earth and Planetary Astrophysics (astro-ph.EP)
The presence of N2 in the surface environment of Pluto is critical in creating Pluto's richness of features and processes. Here, we propose that the nitrogen atoms in the N2 observed on Pluto were accreted in that chemical form during the formation of Pluto. We use New Horizons data and models to estimate the amounts of N2 in the following exterior reservoirs: atmosphere, escape, photochemistry, and surface. The total exterior inventory is deduced to be dominated by a glacial sheet of N2-rich ices at Sputnik Planitia, or by atmospheric escape if past rates of escape were much faster than at present. Pluto's atmosphere is a negligible reservoir of N2, and photochemical destruction of N2 may also be of little consequence. Estimates are made of the amount of N2 accreted by Pluto based on cometary and solar compositions. It is found that the cometary model can account for the amount of N2 in Sputnik Planitia, while the solar model can provide a large initial inventory of N2 that would make prodigious atmospheric escape possible. These consistencies can be considered preliminary evidence in support of a primordial origin of Pluto's N2. However, both models predict accreted ratios of CO/N2 that are much higher than that in Pluto's atmosphere. Possible processes to explain "missing CO" that are given quantitative support here are fractional crystallization from the atmosphere resulting in CO burial at the surface, and aqueous destruction reactions of CO subject to metastable thermodynamic equilibrium in the subsurface. The plausibility of primordial N2 as the primary source of Pluto's nitrogen (vs. NH3 or organic N) can be tested more rigorously using future constraints on the 14N/15N ratio in N2 and the 36Ar/N2 ratio.
### Cross-lists for Thu, 24 May 18
[8] arXiv:1805.08820 (cross-list from astro-ph.SR) [pdf, other]
Title: Kepler-503b: An Object at the Hydrogen Burning Mass Limit Orbiting a Subgiant Star
Subjects: Solar and Stellar Astrophysics (astro-ph.SR); Earth and Planetary Astrophysics (astro-ph.EP)
Using spectroscopic radial velocities with the APOGEE instrument and \emph{Gaia} distance estimates, we demonstrate that Kepler-503b, currently considered a validated {\it Kepler} planet, is in fact a brown-dwarf/low-mass star in a nearly circular 7.2-day orbit around a subgiant star. Using a mass estimate for the primary star derived from stellar models, we derive a companion mass and radius of $0.075\pm0.003 \ M_{\odot}$ ($78.6\pm3.1 \ M_{Jup}$) and $0.099^{+0.006}_{-0.004}\ R_{\odot}$ ($0.96^{+0.06}_{-0.04}\ R_{Jup}$), respectively. Assuming the system is coeval, the evolutionary state of the primary indicates the age is $\sim6.7$ Gyr. Kepler-503b sits right at the hydrogen burning mass limit, straddling the boundary between brown dwarfs and very low-mass stars. More precise radial velocities and secondary eclipse spectroscopy with James Webb Space Telescope will provide improved measurements of the physical parameters and age of this important system to better constrain and understand the physics of these objects and their spectra. This system emphasizes the value of radial velocity observations to distinguish a genuine planet from astrophysical false positives, and is the first result from the SDSS-IV monitoring of Kepler planet candidates with the multi-object APOGEE instrument.
[9] arXiv:1805.08844 (cross-list from astro-ph.SR) [pdf, other]
Title: Stellar Companions of Exoplanet Host Stars in K2
Comments: Accepted for publication in The Astronomical Journal. 18 pages, 9 figures, 3 tables
Subjects: Solar and Stellar Astrophysics (astro-ph.SR); Earth and Planetary Astrophysics (astro-ph.EP)
It is well established that roughly half of all nearby solar-type stars have at least one companion. Stellar companions can have significant implications for the detection and characterization of exoplanets, including triggering false positives and masking the true radii of planets. Determining the fraction of exoplanet host stars that are also binaries allows us to better determine planetary characteristics as well as establish the relationship between binarity and planet formation. Using high angular resolution speckle imaging, we detect stellar companions within ~1 arcsec of K2 planet candidate host stars. Comparing our detected companion rate to TRILEGAL star count simulations and known detection limits of speckle imaging we estimate the binary fraction of K2 planet host stars to be 40 - 50%, similar to that of Kepler exoplanet hosts and field stars.
[10] arXiv:1805.09276 (cross-list from astro-ph.IM) [pdf, other]
Title: MAROON-X: A Radial Velocity Spectrograph for the Gemini Observatory
Comments: 10 pages, 9 figures, submitted for SPIE Astronomical Telescopes + Instrumentation 2018
Subjects: Instrumentation and Methods for Astrophysics (astro-ph.IM); Earth and Planetary Astrophysics (astro-ph.EP)
MAROON-X is a red-optical, high precision radial velocity spectrograph currently nearing completion and undergoing extensive performance testing at the University of Chicago. The instrument is scheduled to be installed at Gemini North in the first quarter of 2019. MAROON-X will be the only RV spectrograph on a large telescope with full access by the entire US community. In these proceedings we discuss the latest addition of the red wavelength arm and the two science grade detector systems, as well as the design and construction of the telescope front end. We also present results from ongoing RV stability tests in the lab. First results indicate that MAROON-X can be calibrated at the sub-m/s level, and perhaps even much better than that using a simultaneous reference approach.
### Replacements for Thu, 24 May 18
[11] arXiv:1803.02008 (replaced) [pdf]
Title: Planetary Spectrum Generator: an accurate online radiative transfer suite for atmospheres, comets, small bodies and exoplanets | {"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": 0.8247899413108826, "perplexity": 2696.625575995147}, "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-2018-22/segments/1526794866894.26/warc/CC-MAIN-20180524224941-20180525004941-00616.warc.gz"} |
https://people.maths.bris.ac.uk/~matyd/GroupNames/97/C5xD12.html | Copied to
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## G = C5×D12order 120 = 23·3·5
### Direct product of C5 and D12
Aliases: C5×D12, C156D4, C604C2, C203S3, C121C10, D61C10, C10.15D6, C30.20C22, C4⋊(C5×S3), C31(C5×D4), (S3×C10)⋊4C2, C2.4(S3×C10), C6.3(C2×C10), SmallGroup(120,23)
Series: Derived Chief Lower central Upper central
Derived series C1 — C6 — C5×D12
Chief series C1 — C3 — C6 — C30 — S3×C10 — C5×D12
Lower central C3 — C6 — C5×D12
Upper central C1 — C10 — C20
Generators and relations for C5×D12
G = < a,b,c | a5=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D12
On 60 points
Generators in S60
(1 39 60 33 14)(2 40 49 34 15)(3 41 50 35 16)(4 42 51 36 17)(5 43 52 25 18)(6 44 53 26 19)(7 45 54 27 20)(8 46 55 28 21)(9 47 56 29 22)(10 48 57 30 23)(11 37 58 31 24)(12 38 59 32 13)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 14)(15 24)(16 23)(17 22)(18 21)(19 20)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 58)(50 57)(51 56)(52 55)(53 54)(59 60)
G:=sub<Sym(60)| (1,39,60,33,14)(2,40,49,34,15)(3,41,50,35,16)(4,42,51,36,17)(5,43,52,25,18)(6,44,53,26,19)(7,45,54,27,20)(8,46,55,28,21)(9,47,56,29,22)(10,48,57,30,23)(11,37,58,31,24)(12,38,59,32,13), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60)>;
G:=Group( (1,39,60,33,14)(2,40,49,34,15)(3,41,50,35,16)(4,42,51,36,17)(5,43,52,25,18)(6,44,53,26,19)(7,45,54,27,20)(8,46,55,28,21)(9,47,56,29,22)(10,48,57,30,23)(11,37,58,31,24)(12,38,59,32,13), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,14)(15,24)(16,23)(17,22)(18,21)(19,20)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,58)(50,57)(51,56)(52,55)(53,54)(59,60) );
G=PermutationGroup([(1,39,60,33,14),(2,40,49,34,15),(3,41,50,35,16),(4,42,51,36,17),(5,43,52,25,18),(6,44,53,26,19),(7,45,54,27,20),(8,46,55,28,21),(9,47,56,29,22),(10,48,57,30,23),(11,37,58,31,24),(12,38,59,32,13)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14),(15,24),(16,23),(17,22),(18,21),(19,20),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,58),(50,57),(51,56),(52,55),(53,54),(59,60)])
C5×D12 is a maximal subgroup of
C15⋊D8 C5⋊D24 C20.D6 D12.D5 D12⋊D5 D125D5 C20⋊D6 C5×S3×D4
45 conjugacy classes
class 1 2A 2B 2C 3 4 5A 5B 5C 5D 6 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A 15B 15C 15D 20A 20B 20C 20D 30A 30B 30C 30D 60A ··· 60H order 1 2 2 2 3 4 5 5 5 5 6 10 10 10 10 10 ··· 10 12 12 15 15 15 15 20 20 20 20 30 30 30 30 60 ··· 60 size 1 1 6 6 2 2 1 1 1 1 2 1 1 1 1 6 ··· 6 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2
45 irreducible representations
dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 S3 D4 D6 D12 C5×S3 C5×D4 S3×C10 C5×D12 kernel C5×D12 C60 S3×C10 D12 C12 D6 C20 C15 C10 C5 C4 C3 C2 C1 # reps 1 1 2 4 4 8 1 1 1 2 4 4 4 8
Matrix representation of C5×D12 in GL2(𝔽11) generated by
4 0 0 4
,
6 2 5 0
,
0 2 6 0
G:=sub<GL(2,GF(11))| [4,0,0,4],[6,5,2,0],[0,6,2,0] >;
C5×D12 in GAP, Magma, Sage, TeX
C_5\times D_{12}
% in TeX
G:=Group("C5xD12");
// GroupNames label
G:=SmallGroup(120,23);
// by ID
G=gap.SmallGroup(120,23);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-3,221,106,2004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export
×
𝔽 | {"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": 0.9458785057067871, "perplexity": 2538.287147995876}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251684146.65/warc/CC-MAIN-20200126013015-20200126043015-00092.warc.gz"} |
https://experts.arizona.edu/en/publications/influence-of-a-mortality-trade-off-on-the-spreading-rate-of-cane- | Emeric Bouin, Matthew H. Chan, Christopher Henderson, Peter S. Kim
Research output: Contribution to journalArticlepeer-review
3 Scopus citations
## Abstract
We study the influence of a mortality trade-off in a nonlocal reaction-diffusion-mutation equation that we introduce to model the invasion of cane toads in Australia. This model is built off of one that has attracted attention recently, in which the population of toads is structured by a phenotypical trait that governs the spatial diffusion. We are concerned with the case when the diffusivity can take unbounded values and the mortality trade-off depends only on the trait variable. Depending on the rate of increase of the penalization term, we obtain the rate of spreading of the population. We identify two regimes, an acceleration regime when the penalization is weak and a linear spreading regime when the penalization is strong. While the development of the model comes from biological principles, the bulk of the article is dedicated to the mathematical analysis of the model, which is very technical. The upper and lower bounds are proved via the Li-Yau estimates of the fundamental solution of the heat equation with potential on Riemannian manifolds and a moving ball technique, respectively, and the traveling waves by a Leray-Schauder fixed point argument. We also present a simple method for a priori L bounds.
Original language English (US) 1627-1671 45 Communications in Partial Differential Equations 43 11 https://doi.org/10.1080/03605302.2018.1523190 Published - Nov 2 2018 Yes
## Keywords
• 35C07
• 35Q92
• 45K05
• Front acceleration
• reaction-diffusion equations
• structured populations
## ASJC Scopus subject areas
• Analysis
• Applied Mathematics
## Fingerprint
Dive into the research topics of 'Influence of a mortality trade-off on the spreading rate of cane toads fronts'. 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": 0.8462252616882324, "perplexity": 1573.6458028883328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570879.37/warc/CC-MAIN-20220809003642-20220809033642-00552.warc.gz"} |
https://en.wikipedia.org/wiki/Half-logistic_distribution | # Half-logistic distribution
Support Probability density function Cumulative distribution function ${\displaystyle k\in [0;\infty )\!}$ ${\displaystyle {\frac {2e^{-k}}{(1+e^{-k})^{2}}}\!}$ ${\displaystyle {\frac {1-e^{-k}}{1+e^{-k}}}\!}$ ${\displaystyle \log _{e}(4)=1.386\ldots }$ ${\displaystyle \log _{e}(3)=1.0986\ldots }$ 0 ${\displaystyle \pi ^{2}/3-(\log _{e}(4))^{2}=1.368\ldots }$
In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for
${\displaystyle X=|Y|\!}$
where Y is a logistic random variable, X is a half-logistic random variable.
## Specification
### Cumulative distribution function
The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F(k) is the cdf for the logistic distribution, then G(k) = 2F(k) − 1 is the cdf of a half-logistic distribution. Specifically,
${\displaystyle G(k)={\frac {1-e^{-k}}{1+e^{-k}}}{\mbox{ for }}k\geq 0.\!}$
### Probability density function
Similarly, the probability density function (pdf) of the half-logistic distribution is g(k) = 2f(k) if f(k) is the pdf of the logistic distribution. Explicitly,
${\displaystyle g(k)={\frac {2e^{-k}}{(1+e^{-k})^{2}}}{\mbox{ for }}k\geq 0.\!}$
## References
• Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "23.11". Continuous univariate distributions. 2 (2nd ed.). New York: Wiley. p. 150.
• George, Olusengun; Meenakshi Devidas (1992). "Some Related Distributions". In N. Balakrishnan. Handbook of the Logistic Distribution. New York: Marcel Dekker, Inc. pp. 232–234. ISBN 0-8247-8587-8.
• Olapade, A.K. (2003), "On characterizations of the half-logistic distribution" (PDF), InterStat, 2003 (February): 2, ISSN 1941-689X | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 9, "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": 0.9940783977508545, "perplexity": 2158.7220444636764}, "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/1492917122174.32/warc/CC-MAIN-20170423031202-00283-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://forum.allaboutcircuits.com/threads/rc-low-pass-filter-with-sine-wave-input.179473/ | # RC Low Pass Filter with Sine Wave Input
#### jaydnul
Joined Apr 2, 2015
155
Hey Everyone,
I am trying to gain a level of fundamental understanding of an RC circuit sine wave response through the mathematics and was wondering if someone could help me work it out.
Fundamentally a sine wave is represented by the equation y=-ky'' . When a sine wave is used as the input of an RC low pass filter, you get a sine wave out (just phase and amplitude shifted). So I know that when the input waveform has the form Vin=-kVin'' then somehow the output also retains that relationship, Vout=-kVout''. I am having trouble proving this with the equation that describes an RC circuit response: Vout' = 1/RC*(Vin-Vout).
Any help would be great, thanks!
#### MrChips
Joined Oct 2, 2009
23,536
Is this a school related question?
#### Papabravo
Joined Feb 24, 2006
16,160
I'd be surprised if you could prove it. The simple answer is that you cannot make a 1st order system behave like a 2nd order system. To to that you need to add an inductor to the RC circuit and then you will have a second order system
#### jaydnul
Joined Apr 2, 2015
155
Just for an intuitive visualization I am more looking for an equation in the form of a differential equation (instead of the solutions to the differential equation), just to show that the relationship between acceleration and voltage remains the same without converting to sines or exp functions. I know that getting a sine wave at the output when there is sine wave at the input must mean the the output voltage retains this voltage/acceleration information.
I understand the phasor/exponential/sinusoidal explanations, but is there a way to prove this without exp or sine functions, just keeping it in the form of the differential equations?
My attempt here:
$$(1) Vin = -k\frac {d^2 Vin} {dt^2}$$
Plugged into the RC circuit equation
$$(2) \frac {d Vout} {dt} = \frac {Vin - Vout}{RC}$$
$$(3) \frac {d Vout} {dt} = \frac {-k\frac {d^2 Vin} {dt^2} - Vout}{RC}$$
Differentiating equation (2)
$$(4) \frac {d^2 Vout} {dt^2} = \frac {\frac {d Vin} {dt} - \frac {d Vout} {dt}} {RC}$$
Then plugging equation (3) into equation (4)
$$(5) \frac {d^2 Vout} {dt^2} = \frac {\frac {d Vin} {dt} - \frac {-k\frac {d^2 Vin} {dt^2} - Vout}{RC}} {RC}$$
Not sure where to go from there, or if im going in completely the wrong direction...
#### Papabravo
Joined Feb 24, 2006
16,160
That's all fine and dandy, but the RC circuit does not behave according to 2nd Order ODE (Ordinary Differential Equation).
It behaves according to a 1st Order ODE. so all your fine machinations mean nothing when it comes to understanding what is going on if you start with the wrong premise.
What is will do for a given frequency is provide attenuation in the magnitude and a shift in the phase.
If you calculate and plot the magnitude and phase you will see immediately the low pass characteristic in the magnitude, and the shift in phase at the corner frequency given by the values of R and C.
Here is a simulation of a typical RC circuit. The dotted blue lone is the phase of the output with respect to the input and it starts a 0° for low frequency AC inputs. As you approach the corner frequency the phase begins to lag and at high frequency it reach 90°. In order for a sign change to happen, as is the case in a 2nd Order system the phase needs to shift by 180°,
In case you are interested the corner frequency is:
$$f_c\;=\;\cfrac{1}{2\pi RC}\;\approx\;159.15\;\text {Hertz}$$
Edit: One more thing the homogeneous ODE you should be looking at is:
$$R\cfrac{di}{dt}\;+\;\cfrac{1}{C}i\;=\;0$$
You can transform this to a differential equation with voltage as the dependent variable because the current is the same in every part of a series circuit. Can you do that?
#### Attachments
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Last edited:
#### bogosort
Joined Sep 24, 2011
674
Fundamentally a sine wave is represented by the equation y=-ky'' .
Note that this isn't strictly true. A cosine is also a solution to that DE, and so the general solution is a sum of sine and cosine.
#### Papabravo
Joined Feb 24, 2006
16,160
@jaydnul
There are potentially solutions in the complex domain as well. The differential equation represents a much larger class of functions than you think. The solution to a differential equation consists of two parts:
1. The first part is the solution to the homogeneous equation which you keep trying to misuse.
2. The second part is the particular solution, given a forcing function which you have neglected.
You have also neglected to specify any initial conditions which are required for the general solution to any ODE. Your attempted definition of a sine function as the solution to a homogeneous ODE is doomed. It won't work because it is not unique. AFAIK, to be useful a definition must be unique.
So riddle me this:
The sine function is a odd function. That is $$\;\;sin(-x)\;=\;-sin(x)$$
The cosine function is an even function. That is $$\;\;cos(-x)\;=\;cos(x)$$
Your definition is not able to create a single function that is both odd and even.
Last edited:
#### bogosort
Joined Sep 24, 2011
674
There are potentially solutions in the complex domain as well. The differential equation represents a much larger class of functions than you think. The solution to a differential equation consists of two parts:
1. The first part is the solution to the homogeneous equation which you keep trying to misuse.
2. The second part is the particular solution, given a forcing function which you have neglected.
You have also neglected to specify any initial conditions which are required for the general solution to any ODE. Your attempted definition of a sine function as the solution to a homogeneous ODE is doomed. It won't work because it is not unique.
Why are you telling me this, I've done no such things.
#### Papabravo
Joined Feb 24, 2006
16,160
Why are you telling me this, I've done no such things.
Because my reply was for the TS, not for you.
I fixed it. Are we happier now?
#### bogosort
Joined Sep 24, 2011
674
Are we happier now?
Good ol' Grumpy Papabravo. Yes, I'm happier now, thank you.
#### Papabravo
Joined Feb 24, 2006
16,160
Good ol' Grumpy Papabravo. Yes, I'm happier now, thank you.
At least I know you well enough not to insult you intentionally. I'll try and see that it doesn't happen again.
#### Tesla23
Joined May 10, 2009
445
Hey Everyone,
I am trying to gain a level of fundamental understanding of an RC circuit sine wave response through the mathematics and was wondering if someone could help me work it out.
Fundamentally a sine wave is represented by the equation y=-ky'' . When a sine wave is used as the input of an RC low pass filter, you get a sine wave out (just phase and amplitude shifted). So I know that when the input waveform has the form Vin=-kVin'' then somehow the output also retains that relationship, Vout=-kVout''. I am having trouble proving this with the equation that describes an RC circuit response: Vout' = 1/RC*(Vin-Vout).
Any help would be great, thanks!
I think you will have trouble as I don't think it is true. Whilst it is true that the response of the RC network to a sinusoid is a sinusoid (shifted in amplitude and phase), your assumption that "Fundamentally a sine wave is represented by the equation y=-ky" needs the added note that k > 0. Without this you admit all the exponentials with real exponents.
Restricting yourself to steady-state sinusoidal excitation, you can analyze the circuit in the Fourier transform domain - and here the relationship is clear - so it should be possible to prove it in the time domain from differential equations.
Without the requirement k>0, you can no longer rely on a fourier representation and you are trying to prove it in the Laplace domain, and at first glance it appears that the circuit introduces extra exponentials in response to non-steady state excitation, | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9205416440963745, "perplexity": 467.07252521107625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487649731.59/warc/CC-MAIN-20210619203250-20210619233250-00498.warc.gz"} |
http://mathhelpforum.com/differential-equations/87624-annual-proportionate-growth-rate.html | ## annual proportionate growth rate
A herd of goats was introduced onto an island on 1 march 1970 and the size of the population was estimated on 1 march in subsequent year.
The size of the initial population was 50, and it had grown to approx 110 after one year. After another year the size of the population was approx 200 and following year it was approx 260. Assume that the behavior of the population satisfies the logistic model.
A) show that the annual proportionate growth rate for the population of size 50 was approx 1.2 and that the annual proportionate growth rate for the population of size 200 was approx 0.3.
B) find the corresponding values of the annual proportionate growth rate for low population level R and the equilibrium population size E. | {"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": 0.9906696677207947, "perplexity": 702.9402780858637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930109.71/warc/CC-MAIN-20150521113210-00135-ip-10-180-206-219.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/difference-between-volume-displaced-fluid-and-volume-of-the-object.576056/ | # Difference between volume displaced fluid and volume of the object
1. Feb 9, 2012
### paulh428
What is the difference between the volume displaced and the object of the volume, according to Archimedes' Principle? Because I'm trying to find logs floating on water and my book gives an equation for buoyant force. Here it is Fb = mg => rho-fluid * V-displaced * g = rho-object * V-object * g. Hope that equation helps. Essentially, my question leads to: shouldn't the V-displaced and V-object be the same thing?
Hope this makes sense. Please ask if you want me to try and make some more sense. :D
2. Feb 9, 2012
### James Leighe
Only if the object is fully submerged.
If the object floats, it will only displace water equal to the volume of the submerged portion.
3. Feb 9, 2012
### HallsofIvy
Staff Emeritus
If the log has volume v and it floats so that exactly half of it is submerged and half is above the water it would displace v/2 volume of water. And that would mean, of course, that the density of the log is 1/2 the density of water. | {"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": 0.8139421343803406, "perplexity": 1150.5043616341823}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806258.88/warc/CC-MAIN-20171120223020-20171121003020-00779.warc.gz"} |
https://zulfahmed.wordpress.com/2014/05/06/dissenting-view-on-quantum-gravity/ | Feeds:
Posts
## Dissenting view on quantum gravity
If our goal in physics is to produce a coherent mathematical model of nature then we should take a step back from the currently accepted theories and examine the phenomena of nature to be independent of these theories. The great movements of the twentieth century were general relativity and quantum field theory. General relativity describes gravity via a metric on a Lorentzian manifold. The metric arises as a critical point of the Einstein-Hilbert action and relates matter to a stress-energy tensor. Quantum field theory arises from an a numerically successful effort to understand subatomic physics. These are theories that have a monumental amount of thought and effort behind them.
I have been engaged in attempting to employ parsimony as a guiding principle to a different sort of unity between gravity and quantum phenomena. I have produced empirical evidence that the universe is a scaled 4-sphere. With this new fact in hand, we can very simply explain quantization of energy directly while at the same time make a very small technical change to general relativity and gain conflictless unity of gravity and quantum phenomena. We can identify possible physical universes with hypersurfaces of a 4-sphere and define the Einstein-Hilbert action over hypersurfaces. This small change does not affect the empirical basis of general relativity because the Euler-Lagrange equations for the action is not different from the Einstein field equations but the possible metrics are all hypersurface induced metrics. There is a fixed cosmological constant term that comes from the curvature of the ambient space, which is $h^2$ where $1/h$ is the radius of the universe which must be set with $h$ the Planck constant in order to ensure that spectrum of the Dirac operator is $\pm(2+k)h$ with $k=0,1,2,...$ with spacing consistent with energy quantization empirically observed.
We can formulate the Dirac equation for fermions naturally using classical spinor fields on $S^4(1/h) \times \mathbf{R}$ with a metric totally orthogonal in space and time components. The latter because it is parsimonious to use absolute time which is sensible. At the moment, our SI unit of second cannot be defined without assuming a universality of the behavior of cesium-133 atoms around the universe which is compact. The situation here is much simpler than most of the theories of either general relativity or quantum field theory for curved spacetimes allows. The important issue is not the level of generality and complexity but the match to observed nature. The picture here contains a fundamental link between macroscopic and microscopic phenomena via setting the radius of the universe to $1/h$.
Of course there is a question about whether it is sensible to accept a higher macroscopic space dimension. There is crystallographic evidence for four macroscopic space dimensions. There is of course the question of the discrepancy between our apparent experience of three space dimensions but it is best to ask what ‘space dimensions’ will mean and what will the relation between space dimensions be to our actual experience. One could easily understand ‘experienced space dimensions’ through the result of choosing a hypersurface of the 4-sphere as the physical universe. Some thought will convince the reader that if the physical universe is determined by a gravitational field equation, then it will vary smoothly in a 4-sphere and if observers stay on the hypersurface then they might experience three space dimensions on the hypersurface differently from the ‘fourth space dimension’.
The picture presented above have some qualitiative implications. Since quantization of energy is determined by the 4-sphere geometry, we are better off seeking a simplification of quantum field theory rather than trying to ‘quantize’ gravity. | {"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": 7, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9059424996376038, "perplexity": 409.1477912996079}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423320.19/warc/CC-MAIN-20170720181829-20170720201829-00526.warc.gz"} |
http://www.newton.ac.uk/programmes/DIS/seminars/2009051416309.html | DIS
Seminar
Admissibility of solutions of discrete dynamical systems
Halburd, R (UCL)
Thursday 14 May 2009, 16:30-17:30
Satellite
Abstract
For discrete equations on a number field, the rate of growth of the heights of iterates is a good detector of integrability. In the case of a rational number, the height is just the maximum of the absolute value of the denominator and numerator. A solution is called admissible if its height grows much faster than the heights of the coefficients in the equation. For certain classes of equations it is shown that the existence of a single slow-growing admissible solution is enough to guarantee that the equation is a discrete Painleve equation. Inadmissible solutions are also explored. These solutions correspond to pre-periodic orbits for classical (autonomous) dynamical systems. The classical theory is extended to better understand these solutions in the non-autonomous setting.
Video
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If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible. | {"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": 0.9075927138328552, "perplexity": 362.83543020922536}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999676834/warc/CC-MAIN-20140305060756-00021-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://edurev.in/course/quiz/attempt/1734_Test-Continuity/cf6c3f8a-42d1-4f72-9f0c-3b70263d1530 | Courses
# Test: Continuity
## 10 Questions MCQ Test Mathematics (Maths) Class 12 | Test: Continuity
Description
This mock test of Test: Continuity for JEE helps you for every JEE entrance exam. This contains 10 Multiple Choice Questions for JEE Test: Continuity (mcq) to study with solutions a complete question bank. The solved questions answers in this Test: Continuity quiz give you a good mix of easy questions and tough questions. JEE students definitely take this Test: Continuity exercise for a better result in the exam. You can find other Test: Continuity extra questions, long questions & short questions for JEE on EduRev as well by searching above.
QUESTION: 1
### Which of the following functions are not continuous.
Solution:
Here, f(x)=[X] could be expressed graphically as:
From the graph it is clear that the function is discontinuous.
QUESTION: 2
### If f (x) = [xsinπx] { where [x] denotes greatest integer function}, then f (x) is
Solution:
► By the definition of [x], we have f(x)=[xsinπx]=0 for −1≤x≤1, because 0 ≤ xsin(πx) ≤1
Also, f(x)=[xsinπx]=−1, for 1<x<1+h for some small appropriate h> 0, because sinπx is negative and ≥−1 for 1<x<1+h.
Thus f(x) is constant and equal to 0 in the interval [−1;1] and so it is continuous and differentiable in (−1,1).
In particular, f(x) is continuous at x=0
QUESTION: 3
### Examine the continuity of the function
Solution:
f’(x) = (x2-4)/(x-2)
= [(x-2)(x+2)]/(x-2)
= (x+2) which is continuous everywhere.
QUESTION: 4
For what values of a and b, f is a continuous function.
Solution:
QUESTION: 5
Discuss the continuity of function f(x) = |x-1| + |x+1|, x R
Solution:
f(x) = |x-1| + |x+1| , x € R
► For x ≥ 1,
f(x) = x-1 + x+1
f(x) = 2x
► For -1 ≤ x ≤ 1,
f(x) = -x + 1 + x+1
f(x) = 2
► For x ≤ -1,
f(x) = -x + 1 - x-1
f(x) = -2x
As the graph of f(x) shows, the function is continuous throughout its domain.
QUESTION: 6
Examine the continuity of function f(x) = (x-1)(x-2)
Solution:
Lim f (x) = lim (x-1)(x-2) at x tend to k
► So it get k2-3k+2
► Now f (k) = k2 -3k+2
► So f (x) =f (k) so continous at everywhere
QUESTION: 7
What is the point of discontinuity for signum function?
Solution:
Sgn(x)=|x|/x
► for x>0, sgn(x)=1
► for x=0, sgn(x)=0
► for x<0, sgn(x)="">
► Now see graphically or theoretically sgn(x) is discontinuous at x=0
QUESTION: 8
Function f(x) = log x + is continuous at
Solution:
[-1,1] cannot be continuous interval because log is not defined at 0
The value of x cannot be greater than 1 because then the function will become complex
(0,1) will not be considered because its continuous at 1 as well. Hence D is the correct option
QUESTION: 9
Solution:
Function is not continuous at x=0
QUESTION: 10
A real function f is said to be continuous if it is continuous at every point in …… .
Solution: | {"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": 0.9164478182792664, "perplexity": 3005.7137936014315}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703587074.70/warc/CC-MAIN-20210125154534-20210125184534-00487.warc.gz"} |
https://www.cfd-online.com/Forums/fluent/75287-mass-flow-rate-calculation-v-s-computation.html | # Mass flow rate: calculation v/s computation
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April 21, 2010, 07:33 Mass flow rate: calculation v/s computation #1 New Member Annon Join Date: Apr 2010 Posts: 16 Rep Power: 9 Hi all, In my simulations density (Rho) is constant. I have computed mass flow rate (M) through an interior face using standard mass flow rate calculation option from fluent. Then I calculated mass averaged velocity (Vm), area weighted averaged velocity (Va) and facet averaged velocity (Vf). To recheck, I calculated the mass flow rate as M' = Rho*A*V. By using all the three above mentioned velocities in the above formula I am not able to get both mass flow rates the same (i.e. M' and M are not equal). Which is the apropriate value of velocity that one should consider to re-calculate accurately the mass flow rate? I guess, the mass flow rate is computed in fluent as summation of (Rho*A*V) values at all the facets. Could any body throw some light on this issue? Thanks. -aditya
March 23, 2013, 11:31 #2 New Member Andre Joubert Join Date: Mar 2013 Posts: 4 Rep Power: 6 Hi beguxa, I believe the velocity you want to use is actually the axial velocity. When running my current simulation of flow rate through a nozzle I used surface integrals in the report menu to get an accurate reading but when calculating the flow through use of a custom field function the velocity which returned the same value for flow rate was the axial velocity.
March 24, 2013, 04:44 #3 Member Thiagu Join Date: Oct 2012 Location: India Posts: 59 Rep Power: 6 you can use standard massflow option in fluent. but to check this use velocity normal to the surface in your calculation. I think both should agree.
March 28, 2013, 12:04 #4 Senior Member OJ Join Date: Apr 2012 Location: United Kindom Posts: 475 Rep Power: 13 beguxa, There are subtle differences in all four values. For correct mass flow rate, we want addition of all the mass flow rates going into the every surface element of your surface. Let's compare all four approaches with this bold statement. Area averaged velocity is calculated as . This is a good representation of average velocity over the surface. But depending on velocity distribution and mesh, smaller mesh elements may have higher velocities and vice versa, so this doesn't equate to bold sentence at the beginning. Facet average is calculated as , where n is no. of facets in the surface. If your mesh is not uniform and if there are more mesh elements in regions of high velocity, your facet average velocity will be unnecessarily higher than the average velocity. You can't multiply this with area and still get the definition of bold statement at the beginning. Mass weighted average is . We are slowly getting closer now, since there is a slight reference of actual mass going into every element - - in the denominator. But again since it is "weighted" it bears same limitations as above. Mass flow rate is calculated as: . This definition is closest to the bold statement at the beginning. And hence is recommended. Although, if mesh is of high quality, structured and if solution is fully converged, the differences between these should become smaller. But in real world, with often-used unstructured meshes, this is not always possible. OJ delaneyluke and Behnam Ghadimi like this.
June 8, 2013, 16:17 #5 New Member behnam Join Date: Sep 2011 Location: Iran Posts: 20 Rep Power: 7 Dear beguxa If you define a variable in the Define> Custom Field Functions... as Density*Velocity, and then integrated this variable in the report>surface integral using the "integral" in the "Report type" you can observe that this value is equal to the mass flow rate in your desired surface, which is reported by FLUENT post processing from Report> FLuxes
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All times are GMT -4. The time now is 05:55. | {"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": 0.8662174940109253, "perplexity": 1983.275595293981}, "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/1492917123484.45/warc/CC-MAIN-20170423031203-00261-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/16221/covington-gloss-in-parbox-makebox-minipage?answertab=oldest | # Covington Gloss in Parbox/Makebox/Minipage
I am using LyX and I'm trying to place a Convington Gloss (Covington usage) inside a \parbox, \makebox, or minipage but I get the error:
http://pastie.org/1814468
Here is the file:
http://pastebin.tlhiv.org/iHOSWyqg
-
In general, it looks like I'm having difficulty with the Gloss. For example, I cannot place it in a table. pastebin.tlhiv.org/z_uSCKGJ – Gus Apr 20 '11 at 9:18
In general, covington should be avoided for both examples and glosses. Instead use either gb4e or linguex (which both use the cgloss4e package for glosses.) Independent of this, what effect are you trying to achieve overall? There may be a better way to do this. – Alan Munn Apr 20 '11 at 18:19
You don't need to use minipage at all. It's sufficient to put it in a \vtop and then box that. Note that I subtracted the space added and the width of the rules.
\documentclass{article}
\usepackage{covington}
\begin{document}
\setbox0\vtop{
\hsize\dimexpr\columnwidth - 2\fboxsep - 2\fboxrule\relax
\gll dat Jan het boek lest
@Gus: I can't help you with LyX and I've found over the years that minipage is almost never needed. (Okay, clearly it is never needed, but it is almost never useful.) – TH. Apr 20 '11 at 7:44
@Gus: In this case, my guess is it's related to the fact that \fbox and thus \framebox tokenizes its argument needlessly, but I didn't actually check how covington works. – TH. Apr 20 '11 at 7:46
@Gus: Replace \columnwidth with .5\textwidth and it should work just fine. – TH. Apr 20 '11 at 8:37 | {"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": 0.9056729674339294, "perplexity": 2107.4157567592615}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398464396.48/warc/CC-MAIN-20151124205424-00207-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://math.libretexts.org/Courses/Long_Beach_City_College/Book%3A_Intermediate_Algebra/Text/01%3A_Foundations/1.03%3A_Use_the_Language_of_Algebra | $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
# 1.3: Use the Language of Algebra
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
Learning Objectives
By the end of this section, you will be able to:
• Use variables and algebraic symbols
• Simplify expressions using the order of operations
• Evaluate an expression
• Identify and combine like terms
• Translate an English phrase to an algebraic expression
## Use Variables and Algebraic Symbols
Suppose this year Greg is $$20$$ years old and Alex is $$23$$. You know that Alex is $$3$$ years older than Greg. When Greg was $$12$$, Alex was $$15$$. When Greg is $$35$$, Alex will be $$38$$. No matter what Greg’s age is, Alex’s age will always be 3 years more, right? In the language of algebra, we say that Greg’s age and Alex’s age are variables and the $$3$$ is a constant. The ages change (“vary”) but the $$3$$ years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by $$3$$ years, $$3$$ is the constant. In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age $$g$$, then we could use $$g + 3g + 3$$ to represent Alex’s age. See Table $$\PageIndex{1}$$.
Table $$\PageIndex{1}$$
Greg's age Alex's age
$$12$$ $$15$$
$$20$$ $$23$$
$$35$$ $$38$$
$$g$$ $$g+3$$
The letters used to represent these changing ages are called variables. The letters most commonly used for variables are $$x, y, a, b,$$ and $$c$$.
Definition: VARIABLE
A variable is a letter that represents a number whose value may change.
Definition: CONSTANT
A constant is a number whose value always stays the same.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using.
There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations below (Table $$\PageIndex{2}$$). You’ll probably recognize some of them. $$\require{enclose}$$
Table $$\PageIndex{2}$$
Operation Notation Say: The result is...
Addition $$a+b$$ $$a$$ plus $$b$$ the sum of $$a$$ and $$b$$
Subtraction $$a−b$$ $$a$$ minus $$b$$ the difference of $$a$$ and $$b$$
Multiplication $$a·b,ab,(a)(b),(a)b,a(b)$$ $$a$$ times $$b$$ the product of $$a$$ and $$b$$
Division $$a\div{b}, a/b,\dfrac{a}{b}, b \enclose{longdiv}{a}$$ $$a$$ divided by $$b$$ the quotient of $$a$$ and $$b$$, $$a$$ is called the dividend, and $$b$$ is called the divisor
We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.”
• The difference of $$9$$ and $$2$$ means subtract $$9$$ and $$2$$ , in other words, $$9$$ minus $$2$$ , which we write symbolically as $$9−2$$.
• The product of $$4$$ and $$8$$ means multiply $$4$$ and $$8$$, in other words $$4$$ times $$8$$, which we write symbolically as $$4\cdot 8$$.
In algebra, the cross symbol, $$\times$$, is not used to show multiplication because that symbol may cause confusion. Does $$3xy$$ mean $$3\times y$$ (‘three times $$y$$’) or $$3\cdot x \cdot y$$ (three times $$x$$ times $$y$$)? To make it clear, use $$\cdot$$ or parentheses for multiplication.
When two quantities have the same value, we say they are equal and connect them with an equal sign.
EQUALITY SYMBOL
$$a = b$$ is read “$$a$$ is equal to $$b$$”
The symbol $$“=”$$ is called the equal sign.
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols $$“<”$$ and $$“>"$$.
INEQUALITY
$$a<b$$ is read “$$a$$ is less than $$b$$”
$$a$$ is to the left of $$b$$ on the number line
$$a>b$$ is read "$$a$$ is greater than $$b$$”
$$a$$ is to the right of $$b$$ on the number line
The expressions $$a < b$$ or $$a > b$$ can be read from left to right or right to left, though in English we usually read from left to right Table $$\PageIndex{3}$$. In general, $$a < b$$ is equivalent to $$b > a$$. For example $$7 < 11$$ is equivalent to $$11 > 7$$. And $$a > b$$ is equivalent to $$b < a$$. For example $$17 > 4$$ is equivalent to $$4 < 17$$.
Table $$\PageIndex{3}$$
Inequality Symbols Words
$$a \neq b$$ $$a$$ is not equal to $$b$$
$$a < b$$ $$a$$ is less than $$b$$
$$a \leq b$$ $$a$$ is less than or equal to $$b$$
$$a > b$$ $$a$$ is greater than $$b$$
$$a \geq b$$ $$a$$ is greater than or not equal to $$b$$
Exercise $$\PageIndex{1}$$
Translate from algebra into English:
1. $$17 \leq 26$$
2. $$8 \neq 17 - 3$$
3. $$12 > 27 \div 3$$
4. $$y + 7 < 19$$
1. $$17 \leq 26$$, $$17$$ is less than or equal to $$26$$
2. $$8 \neq 17 - 3$$, $$8$$ is not equal to $$17$$ minus $$3$$
3. $$12 > 27 \div 3$$, $$12$$ is greater than $$27$$ divided by $$3$$
4. $$y + 7 < 19$$, $$y$$ plus $$7$$ is less than $$19$$
Exercise $$\PageIndex{2}$$
Translate from algebra into English:
1. $$14 \leq 27$$
2. $$19 - 2 \neq 8$$
3. $$12 > 4 \div 2$$
4. $$x - 7 < 1$$
1. $$14$$ is less than or equal to $$27$$
2. $$19$$ minus $$2$$ is not equal to $$8$$
3. $$12$$ is greater than $$4$$ divided by $$2$$
4. $$x$$ minus $$7$$ is less than $$1$$
Exercise $$\PageIndex{3}$$
Translate from algebra into English:
1. $$19 \leq 15$$
2. $$7 = 12 - 5$$
3. $$15 \div 3 < 8$$
4. $$y + 3 < 6$$
1. $$19$$ is greater then or equal to $$15$$
2. $$7$$ is equal to $$12$$ minus $$5$$
3. $$15$$ divided by $$3$$ is less than $$8$$
4. $$y$$ plus $$3$$ is greater than $$6$$
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help to make clear which expressions are to be kept together and separate from other expressions. We will introduce three types now.
GROUPING SYMBOLS
\begin{align*} & \text{Parentheses} & & ( ) \\ & \text{Brackets} & & [ ] \\ & \text{Braces} & & \{ \} \end{align*}
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
$8(14−8) \qquad 21−3[2 + 4(9−8)] \qquad 24\div \{ 13−2[1(6−5)+4] \nonumber\}$
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and equations.
EXPRESSION
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
An expression is like an English phrase. Here are some examples of expressions:
Table $$\PageIndex{4}$$
Expression Words English Phrase
$$3 + 5$$ $$3$$ plus $$5$$ the sum of three and five
$$n − 1$$ $$n$$ minus one the difference of $$n$$ and one
$$6\cdot 7$$ $$6$$ times $$7$$ the product of six and seven
$$\dfrac{x}{y}$$ $$x$$ divided by $$y$$ the quotient of $$x$$ and $$y$$
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
Definition: EQUATION
An equation is two expressions connected by an equal sign.
Here are some examples of equations.
Table $$\PageIndex{5}$$
Equation English Sentence
$$3+5=8$$ the sum of three and five is equal to eight
$$n−1=14$$ $$n$$ minus one equals fourteen
$$6 \cdot 7=42$$ The product of six ans seven is equal to forty-two
$$x=53$$ $$x$$ is equal to fifty-three
$$y+9=2y−3$$ $$y$$ plus nine is equal to two $$y$$ minus three
Exercise $$\PageIndex{4}$$
Determine if each is an expression or an equation:
1. $$2(x + 3) = 10$$
2. $$4(y - 1) + 1$$
3. $$x \div 25$$
4. $$y + 8 = 40$$
1. $$2(x + 3) = 10$$. This is an equation — two expressions are connected with an equal sign.
2. $$4(y - 1) + 1$$. This is an expression — no equal sign.
3. $$x \div 25$$. This is an expression — no equal sign.
4. $$y + 8 = 40$$. This is an equation — two expressions are connected with an equal sign.
Exercise $$\PageIndex{5}$$
Determine if each is an expression or an equation:
1. $$3(x - 7) = 27$$
2. $$5(4y - 2) - 7$$
1. equation
2. expression
Exercise $$\PageIndex{6}$$
Determine if each is an expression or an equation:
1. $$y^{3} \div 14$$
2. $$4x - 6 = 22$$
1. expression
2. equation
Suppose we need to multiply nine factors of $$2$$. We could write this as $$2\cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$$. This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write $$2\cdot 2 \cdot 2$$ as $$2^{3}$$ and $$2\cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$$ as $$2^{9}$$. In expressions such as $$2^{3}$$, the $$2$$ is called the base and the $$3$$ is called the exponent. The exponent tells us how many times we need to multiply the base.
We read $$2^{3}$$ as “two to the third power” or “two cubed.”
We say $$2^{3}$$ is in exponential notation and $$2\cdot 2 \cdot 2$$ is in expanded notation.
EXPONENTIAL NOTATION
$$a^{n}$$ means the product of $$n$$ factors of $$a$$.
The expression $$a^{n}$$ is read $$a$$ to the $$n^{th}$$ power.
While we read $$a^{n}$$ as “$$a$$ to the $$n^{th}$$ power,” we usually read:
• $$a^{2}$$ “a squared”
• $$a^{3}$$ “a cubed”
We’ll see later why $$a^{2}$$ and $$a^{3}$$ have special names.
Table $$\PageIndex{6}$$ shows how we read some expressions with exponents.
Table $$\PageIndex{6}$$
Expression In Words
$$7^{2}$$ $$7$$ to the second power or $$7$$ squared
$$5^{3}$$ $$5$$ to the third power or $$5$$ cubed
$$9^{4}$$ $$9$$ to the fourth power
$$12^{5}$$ $$12$$ to the fifth power
Exercise $$\PageIndex{7}$$
Simplify: $$3^{4}$$
$\quad 3^{4}\nonumber$
\begin{align*} & Expand the expression & & 3 \cdot 3 \cdot 3 \cdot 3 \\[5pt] & \text{Multiply left to right} & & 9 \cdot 3 \cdot 3 \\[5pt] & \text{Multiply} & & 27 \cdot 3 \\[5pt] & \text{Multiply} & & 81 \end{align*}
Exercise $$\PageIndex{8}$$
Simplify:
1. $$5^{3}$$
2. $$1^{7}$$
1. $$125$$
2. $$1$$
Exercise $$\PageIndex{9}$$
1. $$7^{2}$$
2. $$0^{5}$$
1. $$49$$
2. $$0$$
## Simplify Expressions Using the Order of Operations
To simplify an expression means to do all the math possible. For example, to simplify $$4\cdot 2 + 1$$ we’d first multiply $$4\cdot 2$$ to get $$8$$ and then add the $$1$$ to get $$9$$. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
$4\cdot 2 + 1\nonumber$
$8 + 1\nonumber$
$9\nonumber$
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
SIMPLIFY AN EXPRESSION
To simplify an expression, do all operations in the expression.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression:
$4 + 3\cdot 7\nonumber$
If you simplify this expression, what do you get?
Some students say $$49$$,
$4 + 3\cdot 7\nonumber$
Since $$4+3$$ gives $$7$$.
$7 \cdot 7\nonumber$
And $$7\cdot 7$$ is $$49$$ $49\nonumber$
Others say $$25$$,
$4 + 3\cdot 7\nonumber$
Since $$3\cdot 7$$ is $$21$$.
$4 + 21\nonumber$
And $$21 + 4$$ makes $$25$$.
$25\nonumber$
Imagine the confusion in our banking system if every problem had several different correct answers!
The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations.
PERFORM THE ORDER OF OPERATIONS.
1. Parentheses and Other Grouping Symbols
• Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents
• Simplify all expressions with exponents.
3. Multiplication and Division
• Perform all multiplication and division in order from left to right. These operations have equal priority.
• Perform all addition and subtraction in order from left to right. These operations have equal priority.
Note
Doing the Manipulative Mathematics activity “Game of 24” will give you practice using the order of operations.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”
\begin{align*} &\textbf{P}\text{arentheses} & & \textbf{P}\text{lease} \\[5pt] &\textbf{E}\text{xponents} & & \textbf{E}\text{xcuse} \\[5pt] &\textbf{M}\text{ultiplication}\space\textbf{D}\text{ivision} & & \textbf{M}\text{y}\space\textbf{D}\text{ear} \\[5pt] &\textbf{A}\text{ddition}\space\textbf{S}\text{ubtraction} & & \textbf{A}\text{unt}\space\textbf{S}\text{ally} \end{align*}
It’s good that “$$\textbf{M}\text{y}\space\textbf{D}\text{ear}$$” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “$$\textbf{A}\text{unt}\space\textbf{S}\text{ally}$$” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
Let’s try an example.
Exercise $$\PageIndex{10}$$
Simplify:
1. $$4 + 3\cdot 7$$
2. $$(4 + 3)\cdot 7$$
1.
$$4 + 3 \cdot 7$$ Are there any parentheses? No. Are there any exponents? No. Is there any multiplication or division? Yes. Multiply first. $$4 + {\color{red}{3 \cdot 7}}$$ Add. $$4+21$$ $$25$$
2.
$$(4 + 3)\cdot 7$$ Are there any parentheses? Yes. $${\color{red}{(4 + 3)}}\cdot 7$$ Simplify inside the parentheses. $$({\color{red}{7}})7$$ Are there any exponents? No. Is there any multiplication or division? Yes. Multiply. $$49$$
Exercise $$\PageIndex{11}$$
Simplify:
1. $$12 - 5\cdot 2$$
2. $$(12 - 5)\cdot 2$$
1. $$2$$
2. $$14$$
Exercise $$\PageIndex{12}$$
Simplify:
1. $$8 + 3\cdot 9$$
2. $$(8 + 3)\cdot 9$$
1. $$35$$
2. $$99$$
Exercise $$\PageIndex{13}$$
Simplify: $$18\div 6 + 4(5 - 2)$$
Parentheses? Yes, subtract first. $$18\div 6 + 4(5 - 2)$$ $$18\div 6 + 4(3)$$ Exponents? No. Multiplication or division? Yes. $${\color{red}{18\div 6}} + {\color{red}{4(3)}}$$ Divide first because we multiply and divide left to right. $$3+{\color{red}{4(3)}}$$ Any other multiplication or division? Yes. Multiply. $$3 + 12$$ Any other multiplication or division? No. Any addition or subtraction? Yes. $$15$$
Exercise $$\PageIndex{14}$$
Simplify: $$30\div 5 + 10(3 - 2)$$
$$16$$
Exercise $$\PageIndex{15}$$
Simplify: $$70\div 10 + 4(6 - 2)$$
$$23$$
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
Exercise $$\PageIndex{16}$$
Simplify: $$5 + 2^{3} + 3[6 - 3(4 - 2)]$$.
$$5 + 2^{3} + 3[6 - 3(4 - 2)]$$ Are there any parentheses (or other grouping symbol)? Yes. Focus on the parentheses that are inside the brackets. $$5 + 2^{3} + 3[6 - 3{\color{red}{(4 - 2)}}]$$ Subtract. $$5 + 2^{3} + 3[6 - {\color{red}{3(2)}}]$$ Continue inside the brackets and multiply. $$5 + 2^{3} + 3[{\color{red}{6 - 6}}]$$ Continue inside the brackets and subtract. $$5 + 2^{3} + 3[{\color{red}{0}}]$$ The expression inside the brackets requires no further simplification. Are there any exponents? Yes. $$5 + {\color{red}{2^{3}}}+ 3[0]$$ Simplify exponents. $$5 + 8 + {\color{red}{3[0]}}$$ Is there any multiplication or division? Yes. Multiply. $${\color{red}{5 + 8}}+0$$ Is there any addition or subtraction? Yes. Add. $${\color{red}{13 + 0}}$$ Add. $$13$$
Exercise $$\PageIndex{17}$$
Simplify: $$9 + 5^{3} - [4(9 + 3)]$$.
$$86$$
Exercise $$\PageIndex{18}$$
Simplify: $$7^{2} - 2[4(5 + 1)]$$.
$$1$$
## Evaluate an Expression
In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
EVALUATE AN EXPRESSION
To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.
Exercise $$\PageIndex{19}$$
Evaluate $$7x - 4$$, when
1. $$x = 5$$
2. $$x = 1$$
1.
when $$x = {\color{red}{5}}$$ $$7x - 4$$ $$7({\color{red}{5}}) - 4$$ Multiply. $$35 - 4$$ Subtract. $$31$$
2.
when $$x = {\color{red}{1}}$$ $$7x - 4$$ $$7({\color{red}{1}}) - 4$$ Multiply. $$7 - 4$$ Subtract. $$3$$
Exercise $$\PageIndex{20}$$
Evaluate $$8x - 3$$, when
1. $$x = 2$$
2. $$x = 1$$
1. $$13$$
2. $$5$$
Exercise $$\PageIndex{21}$$
Evaluate $$4y - 4$$, when
1. $$y = 3$$
2. $$y = 5$$
1. $$8$$
2. $$16$$
Exercise $$\PageIndex{22}$$
Evaluate $$x = 4$$, when
1. $$x^{2}$$
2. $$3^{x}$$
1.
$$x^{2}$$ Replace $$x$$ with $${\color{red}{4}}$$. $$({\color{red}{4}})^{2}$$ Use definition of exponent. $$4\cdot 4$$ Simplify. $$16$$
2.
$$3^{x}$$ Replace $$x$$ with $${\color{red}{4}}$$. $$3^ParseError: invalid DekiScript (click for details)Callstack: at (Bookshelves/Algebra/Book:_Elementary_Algebra_(OpenStax)/01:_Foundations/1.03:_Use_the_Language_of_Algebra), /content/body/div[4]/div[5]/dl/dd/table[2]/tbody/tr[2]/td[2]/span, line 1, column 1 at wiki.page() at (Courses/Long_Beach_City_College/Book:_Intermediate_Algebra/Text/01:_Foundations/1.03:_Use_the_Language_of_Algebra), /content/body/div[1]/pre, line 2, column 10 $$ Use definition of exponent. $$3\cdot3\cdot3\cdot3$$ Simplify. $$81$$
Exercise $$\PageIndex{23}$$
Evaluate $$x = 3$$, when
1. $$x^{2}$$
2. $$4^{x}$$
1. $$9$$
2. $$64$$
Exercise $$\PageIndex{24}$$
Evaluate $$x = 6$$, when
1. $$x^{3}$$
2. $$2^{x}$$
1. $$216$$
2. $$64$$
Exercise $$\PageIndex{25}$$
Evaluate $$2x^{2} + 3x + 8$$ when $$x = 4$$.
$$2x^{2} + 3x + 8$$ Substitute $$x = {\color{red}{4}}$$. $$\small{2x^{2} + 3x + 8}$$ $$2({\color{red}{4}})^{2} + 3({\color{red}{4}}) + 8$$ Follow the order of operations. $$2(16)+3(4)+8$$ $$32+12+8$$ $$52$$
Exercise $$\PageIndex{26}$$
Evaluate $$3x^{2} + 4x + 1$$ when $$x = 3$$.
$$40$$
Exercise $$\PageIndex{27}$$
Evaluate $$6x^{2} - 4x - 7$$ when $$x = 2$$.
$$9$$
## Identify and Combine Like Terms
Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
TERM
A term is a constant, or the product of a constant and one or more variables.
Examples of terms are $$7, y, 5x^{2}, 9a$$, and $$b^{5}$$.
The constant that multiplies the variable is called the coefficient.
COEFFICIENT
The coefficient of a term is the constant that multiplies the variable in a term.
Think of the coefficient as the number in front of the variable. The coefficient of the term $$3x$$ is $$3$$. When we write $$x$$, the coefficient is $$1$$, since $$x=1\cdot x$$.
Exercise $$\PageIndex{28}$$
Identify the coefficient of each term:
1. $$14y$$
2. $$15x^{2}$$
3. $$a$$
1. The coefficient of $$14y$$ is $$14$$
2. The coefficient of $$15x^{2}$$ is $$15$$
3. The coefficient of $$a$$ is $$1$$ since $$a=1a$$.
Exercise $$\PageIndex{29}$$
Identify the coefficient of each term:
1. $$17x$$
2. $$41b^{2}$$
3. $$z$$
1. $$14$$
2. $$41$$
3. $$1$$
Exercise $$\PageIndex{30}$$
Identify the coefficient of each term:
1. $$9p$$
2. $$13a^{2}$$
3. $$y^{3}$$
1. $$9$$
2. $$13$$
3. $$1$$
Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common?
$5x \qquad 7 \qquad n^{2} \qquad 4 \qquad 3x \qquad 9n^{2}\nonumber$
The $$7$$ and the $$4$$ are both constant terms.
The $$5x$$ and the $$3x$$ are both terms with $$x$$.
The $$n^{2}$$ and the $$9n^{2}$$ are both terms with $$n^{2}$$.
When two terms are constants or have the same variable and exponent, we say they are like terms.
• $$7$$ and $$4$$ are like terms.
• $$5x$$ and $$3x$$ are like terms.
• $$x^{2}$$ and $$9x^{2}$$ are like terms.
LIKE TERMS
Terms that are either constants or have the same variables raised to the same powers are called like terms.
Exercise $$\PageIndex{31}$$
Identify the like terms: $$y^{3},7x^{2}, 14, 23, 4y^{3}, 9x, 5x^{2}$$.
$$y^{3}$$ and $$4y^{3}$$ are like terms because both have $$y^{3}$$; the variable and the exponent match.
$$7x^{2}$$ and $$5x^{2}$$ are like terms because both have $$x^{2}$$; the variable and the exponent match.
$$14$$ and $$23$$ are like terms because both are constants.
There is no other term like $$9x$$.
Exercise $$\PageIndex{32}$$
Identify the like terms: $$9, 2x^{3},y^{2}, 8x^{3}, 15, 9y, 11y^{2}$$.
$$9$$ and $$15$$, $$y^{2}$$ and $$11y^{2}$$, $$2x^{3}$$ and $$8x^{3}$$
Exercise $$\PageIndex{33}$$
Identify the like terms: $$4x^{3},8x^{2}, 19, 3x^{3}, 24, 6x^{3}$$.
$$19$$ and $$24$$, $$8x^{2}$$ and $$3x^{2}$$, $$4x^{3}$$ and $$6x^{3}$$
Adding or subtracting terms forms an expression. In the expression $$2x^{2} + 3x + 8$$, from Example, the three terms are $$2x^{2}$$, $$3x$$, and $$8$$.
Exercise $$\PageIndex{34}$$
Identify the terms in each expression.
1. $$9x^{2}+7x+12$$
2. $$8x+3y$$
1. The terms of $$9x^{2}+7x+12$$ are $$9x^{2}, 7x$$, and $$12$$.
2. The terms of $$8x+3y$$ are $$8x$$ and $$3y$$.
Exercise $$\PageIndex{35}$$
Identify the terms in the expression $$4x^{2}+5x+17$$.
$$4x^{2}, 5x, 17$$
Exercise $$\PageIndex{36}$$
Identify the terms in the expression $$5x+2y$$.
$$5x, 2y$$
If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think $$4x+7x+x$$ would simplify to? If you thought $$12x$$, you would be right!
$\begin{array} { c } { 4 x + 7 x + x } \\ { x + x + x + x \quad + x + x + x + x + x + x + x \quad+ x } \\ { 12 x } \end{array}$
Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.
Simplify: $$4x+7x+x$$
Add the coefficients. $$12x$$
Exercise $$\PageIndex{37}$$: How To Combine Like Terms
Simplify: $$2x^{2} + 3x + 7 + x^{2} + 4x + 5$$
Exercise $$\PageIndex{38}$$
Simplify: $$3x^{2} + 7x + 9 + 7x^{2} + 9x + 8$$.
$$10x^{2}+16x+17$$
Exercise $$\PageIndex{39}$$
Simplify: $$4y^{2} + 5y + 2 + 8y^{2} + 4y + 5$$.
$$12y^{2}+9y+7$$
COMBINE LIKE TERMS.
1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add or subtract the coefficients and keep the same variable for each group of like terms.
## Translate an English Phrase to an Algebraic Expression
In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table $$\PageIndex{7}$$ summarizes them.
Operation Phrase Expression
Addition $$a$$ plus $$b$$
the sum of $$a$$ and $$b$$
$$a$$ increased by $$b$$
$$b$$ more than $$a$$
the total of $$a$$ and $$b$$
$$b$$ added to $$a$$
$a+b$
Subtraction $$a$$ minus $$b$$
the difference of $$a$$ and $$b$$
$$a$$ decreased by $$b$$
$$b$$ less than $$a$$
$$b$$ subtracted from $$a$$
$a−b$
Multiplication $$a$$ times $$b$$
the product of $$a$$ and $$b$$
twice $$a$$
$a\cdot b, ab, a(b), (a)(b)$
$2a$
Division $$a$$ divided by $$b$$
the quotient of $$a$$ and $$b$$
the ratio of $$a$$ and $$b$$
$$b$$ divided into $$a$$
$a\div b, a/b, \frac{a}{b}, b \enclose{longdiv}{a}$
Table $$\PageIndex{7}$$
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
Exercise $$\PageIndex{40}$$
Translate each English phrase into an algebraic expression:
1. the difference of $$17x$$ and $$5$$
2. the quotient of $$10x^{2}$$ and $$7$$.
1. The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.
2. The key word is “quotient,” which tells us the operation is division.
This can also be written $$10x^{2}/7$$ or $$\dfrac{10x^{2}}{7}$$.
Exercise $$\PageIndex{41}$$
Translate each English phrase into an algebraic expression:
1. the difference of $$14x^{2}$$ and $$13$$
2. the quotient of $$12x$$ and $$2$$.
1. $$14x^{2} - 13$$
2. $$12x \div 2$$
Exercise $$\PageIndex{42}$$
Translate each English phrase into an algebraic expression:
1. the sum of $$17y^{2}$$ and $$19$$
2. the product of $$7$$ and $$y$$.
1. $$17y^{2} + 19$$
2. $$7y$$
How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.
Exercise $$\PageIndex{43}$$
Translate the English phrase into an algebraic expression:
1. Seventeen more than $$y$$
2. Nine less than $$9x^{2}$$.
1. The key words are more than. They tell us the operation is addition. More than means “added to.”
$$\begin{array} { c } { \text { Seventeen more than } y } \\ { \text { Seventeen added to } y } \\ { y + 17 } \end{array}$$
2. The key words are less than. They tell us to subtract. Less than means “subtracted from.”
$$\begin{array} { c } { \text { Nine less than } 9 x ^ { 2 } } \\ { \text { Nine subtracted from } 9 x ^ { 2 } } \\ { 9 x ^ { 2 } - 9 } \end{array}$$
Exercise $$\PageIndex{44}$$
Translate the English phrase into an algebraic expression:
1. Eleven more than x
2. Fourteen less than $$11a$$.
1. $$x+11$$
2. $$11a−14$$
Exercise $$\PageIndex{45}$$
Translate the English phrase into an algebraic expression:
1. $$13$$ more than $$z$$
2. $$18$$ less than $$8x$$.
1. $$z+13$$
2. $$8x−18$$
Exercise $$\PageIndex{46}$$
Translate the English phrase into an algebraic expression:
1. five times the sum of $$m$$ and $$n$$
2. the sum of five times $$m$$ and $$n$$.
There are two operation words—times tells us to multiply and sum tells us to add.
1. Because we are multiplying $$5$$ times the sum we need parentheses around the sum of $$m$$ and $$n$$, $$(m+n)$$. This forces us to determine the sum first. (Remember the order of operations.)
$\begin{array} { c } { \text { five times the sum of } m \text { and } n } \\ { 5 ( m + n ) } \end{array}$
2. To take a sum, we look for the words “of” and “and” to see what is being added. Here we are taking the sum of five times $$m$$ and $$n\.) $\begin{array} { c } { \text { the sum of five times } m \text { and } n } \\ { 5 m + n } \end{array}$ Exercise \(\PageIndex{47}$$
Translate the English phrase into an algebraic expression:
1. four times the sum of $$p$$ and $$q$$
2. the sum of four times $$p$$ and $$q$$.
1. $$4(p+q)$$
2. $$4p+q$$
Exercise $$\PageIndex{48}$$
Translate the English phrase into an algebraic expression:
1. the difference of two times x and $$8$$,
2. two times the difference of x and $$8$$.
1. $$2x−8$$
2. $$2(x−8)$$
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
Exercise $$\PageIndex{49}$$
The length of a rectangle is $$6$$ less than the width. Let $$w$$ represent the width of the rectangle. Write an expression for the length of the rectangle.
$\begin{array} { l l } { \text { Write a phrase about the length of the rectangle. } } &{ 6 \text { less than the width } } \\ { \text { Substitute } w \text { for "the width." } } &{\text{6 less then w}} \\ { \text { Rewrite "less than" as "subtracted from." } } &{\text{6 subtracted from w}} \\ { \text { Translate the phrase into algebra. } } &{w - 6} \end{array}$
Exercise $$\PageIndex{50}$$
The length of a rectangle is $$7$$ less than the width. Let $$w$$ represent the width of the rectangle. Write an expression for the length of the rectangle.
$$w - 7$$
Exercise $$\PageIndex{51}$$
The width of a rectangle is $$6$$ less than the length. Let $$l$$ represent the length of the rectangle. Write an expression for the width of the rectangle.
$$l - 6$$
Exercise $$\PageIndex{52}$$
June has dimes and quarters in her purse. The number of dimes is three less than four times the number of quarters. Let $$q$$ represent the number of quarters. Write an expression for the number of dimes.
$\begin{array} { ll } { \text { Write the phrase about the number of dimes. } } &{\text{three less than four times the number of quarters}} \\ { \text { Substitute } q \text { for the number of quarters. } } &{\text{3 less than 4 times q}} \\ { \text { Translate "4 times } q \text { ." } } &{\text{3 less than 4q}} \\ { \text { Translate the phrase into algebra. } } &{\text{4q - 3}} \end{array}$
Exercise $$\PageIndex{53}$$
Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let $$q$$ represent the number of quarters. Write an expression for the number of dimes.
$$4q - 8$$
Exercise $$\PageIndex{54}$$
Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let $$n$$ represent the number of nickels. Write an expression for the number of dimes.
$$7n + 3$$
## Key Concepts
• Notation The result is…
$$\begin{array} { l l } {\bullet \space a + b } &{ \text { the sum of } a \text { and } b } \\ { \bullet \space a - b } &{ \text { the difference of } a \text { and } b } \\ {\bullet\space a \cdot b , a b , ( a ) ( b ) ( a ) b , a ( b ) } &{ \text { the product of } a \text { and } b } \\ {\bullet\space a \div b , a / b , \frac { a } { b } , b ) \overline{a} } &{ \text { the quotient of } a \text { and } b } \end{array}$$
• Inequality
$$\begin{array} { l l } { \bullet \space a < b \text { is read "a is less than } b ^ { \prime \prime } } &{a \text { is to the left of } b \text { on the number line } } \\ { \bullet \space a > b \text { is read "a is greater than } b ^ { \prime \prime } } & { a \text { is to the right of } b \text { on the number line } } \end{array}$$
• Inequality Symbols Words
$$\begin{array} {ll} { \bullet a \neq b } &{ a \text { is not equal to } b } \\ { \bullet a < b } &{ a \text { is less than } b } \\ { \bullet a \leq b } &{ a \text { is less than or equal to } b } \\ { \bullet a > b } & { a \text { is greater than } b } \\ { \bullet a \geq b } & { a \text { is greater than or equal to } b } \end{array}$$
• Grouping Symbols
• Parentheses ()
• Brackets []
• Braces {}
• Exponential Notation
• $$a^{n}$$ means the product of $$n$$ factors of $$a$$. The expression $$a^{n}$$ is read $$a$$ to the $$n^{th}$$ power.
• Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
1. Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
2. Exponents: Simplify all expressions with exponents.
3. Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
4. Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
• Combine Like Terms
1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add or subtract the coefficients and keep the same variable for each group of like terms.
## Glossary
coefficient
The coefficient of a term is the constant that multiplies the variable in a term.
constant
A constant is a number whose value always stays the same.
equality symbol
The symbol “$$=$$” is called the equal sign. We read $$a=b$$ as “$$a$$ is equal to $$b$$.”
equation
An equation is two expressions connected by an equal sign.
evaluate an expression
To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
expression
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
like terms
Terms that are either constants or have the same variables raised to the same powers are called like terms.
simplify an expression
To simplify an expression, do all operations in the expression.
term
A term is a constant or the product of a constant and one or more variables.
variable
A variable is a letter that represents a number whose value may change. | {"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": 0.887201189994812, "perplexity": 690.3610060812128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886802.13/warc/CC-MAIN-20200704232817-20200705022817-00206.warc.gz"} |
http://www.ebooks.com/334651/pocket-book-of-integrals-and-mathematical-formulas/tallarida-ronald-j/ | for Kindle Fire, Apple, Android, Nook, Kobo, PC, Mac, BlackBerry ...
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# Pocket Book of Integrals and Mathematical Formulas, 4th Edition
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Convenient Organization of Essential Material so You Can Look up Formulas Fast
Containing a careful selection of standard and timely topics, the Pocket Book of Integrals and Mathematical Formulas, Fourth Edition presents many numerical and statistical tables, scores of worked examples, and the most useful mathematical formulas for engineering and scientific applications. This fourth edition of a bestseller provides even more comprehensive coverage with the inclusion of several additional topics, all while maintaining its accessible, clear style and handy size.
New to the Fourth Edition
• An expanded chapter on series that covers many fascinating properties of the natural numbers that follow from number theory
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• Revised format of the table of integrals for easier use of the forms and functions
Easy to Use on the Go
The book addresses a range of areas, from elementary algebra, geometry, matrices, and trigonometry to calculus, vector analysis, differential equations, and statistics. Featuring a convenient, portable size, it is sure to remain in the pockets or on the desks of all who use mathematical formulas and tables of integrals and derivatives.
Taylor and Francis; April 2008
288 pages; ISBN 9781420063073 | {"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": 0.883095383644104, "perplexity": 2237.217179428254}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422122102237.39/warc/CC-MAIN-20150124175502-00032-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/gravitational-field-inside-the-earth.113389/ | # Gravitational field inside the earth
1. Mar 7, 2006
### jackiefrost
Hi all - I broke my foot and had surgury so I'm bored (for a couple more weeks) which explains my recent posts.
[This isn't "Homework" like some class assignment - it's just something I've been playing with since reading some thought experiments that John Wheeler expressed - and some Feynman stuff on the E field inside a charged sphere]
I'm kind of interested in the nature of fields inside of variously shaped "sources" and would be interested to here any comments about how to best think about any of the following problems so they can be simply expressed mathematically. I could just go look up the appropriate equations but what fun is that?
1. If the earth (mass M) were a perfect sphere with a perfectly homogenous rigid mass distribution, what would be the vector field equation for the earth's gravitational field at any location on the inside region of the sphere? Disregard all external forces and interactions. Any coordinate system is OK but let the origin be at the center of the sphere.
2. Assume a test mass m is suspended in a cylindrical tube of negligible total volume, bored perfectly along a diameter, from one side of the planet to the other through its center. The tube contains a perfect vacuum except for the presence of the test mass, and its walls exert no influence on the test mass. The test mass is initially suspended at the surface level of the earth and is motionless relative to it. What would be the vector equation of motion for the body after it's released to free fall?
3. Disregarding any relativistic effects, would the earth's rotation matter at all to any of the above?
4. Would the equations remain unchanged for any diameterical path we could choose for the tube; e.g. pole to pole vs two diametrical points on the equator.
5. What happens to G (the gravitational constant) inside the sphere?
Any little feedback would be welcome - except maybe "hey jackiemoron - go kill yourself... slowly!!!" or "get well VERY, VERY, VERY, soon, we're all beggin' ya..." or something in that vain
jf
Last edited: Mar 7, 2006
2. Mar 7, 2006
### Staff: Mentor
Just about all of these questions can be answered with a bit of thought, assuming you know the basics of Newtonian gravity. Since you have time on your hands , why don't you tell us what you think the answers would be and why?
3. Mar 7, 2006
### ZapperZ
Staff Emeritus
If you know Gauss's law in electrostatics, there is a gauss's law equivalent for gravitational field. That would answer a few of your questions.
http://scienceworld.wolfram.com/physics/GausssLaw.html
Regardless on whether this is a "homework" question, these still qualify as homework/schoolwork-TYPE questions, and belong in that forum.
Zz.
4. Mar 7, 2006
### qtp
the gravitational field inside of the earth depends only on the mass contained inside of the radius. b/c of the geometry of a sphere the mass outside of a given radius has all of its gravitational effects cancel. also any mass dropped down the cylinder will oscillate back and forth with simple harmonic motion in a $\frac{d^2r}{dt^2} = -\frac{k}{m}r$ fashion
5. Mar 8, 2006
### jackiefrost
Sorry for posting to wrong forum.
Crap!!! Disregard all this. #1 is NO GOOD! Back to the drawing board :-(
1. I may have messed up? Anyway, for deriving a vector field eq for inside the earth F(r) where r is position vector relative to center of the earth and |r| < re (radius of earth), I took the vector eq for the field at the surface of the earth and multiplied it by a scalar function called called "ratio" that varies as a function of r. That scaler ratio is that of the mass inside the Gaussian surface at r to that of the mass of the shell outside that surface. I did it that way since only the mass inside the surface at r contributes to the field. Since the density is constant throught the earth, I calculated the ratio in terms of volumes instead of masses, getting rid of any need for a mass density constant.
I can't seem to get LaTeX of eqs. to preview??? Not sure if it supposed to preview OK or not? Here's a brief text version for now:
The desired vector equation for inside the earth:
F(r) = F_surface(re) * ratio(r)
where F_surface(re) is the vector field eq. at the surface of the earth and re is any position vector <x,y,z> located on the surface of the earth and |re| = radius of the earth. Vector r is any position inside the sphere.
F_surface(re) = -(mMG/|re|^3)re
ratio(r) = (Vol inside Gaussian surface of radius |r|)/(Vol of Earth - Vol inside Gaussian surface of radius |r|)
2. I haven't messed with this yet.
For the other questions:
3 & 4: I can't see any way that rotation of the earth or the choice of diametrical path for the tube through the earth can have any influence on the motion of the test mass.
5. G is G :uhh: I couldn't tell ya what I was "thinking" there...
Does thie reasoning in (1) look OK?
Last edited: Mar 8, 2006
6. Mar 8, 2006
### pervect
Staff Emeritus
Here are some hints: the gravity inside a uniform spherical shell of matter is zero.
The gravity of a sphere acts as if all the mass were concentrated into a point.
Now, divide up the Earth into two parts - a hollow sphere for r>R, and a sphere r<=R, where R is the location of the particle.
1) What is the net gravitational force due to the hollow sphere (see hint #1)?
2) What is the net gravitational force due to the sphere (see hint #2).
7. Mar 9, 2006
### jackiefrost
OK - let's try again :surprised
Let:
F = gravitational vector field function inside of the earth
r = position vector, <x,y,z>, with origin at center of the earth
me = mass of earth
m = mass of test particle
If I want to express the grav field vector inside the earth as a function of a position vector r (where |r| < radius of earth), the only mass that contributes to F(r) is inside a Gaussian spherical surface of radius |r|. The external "shell" of mass contributes nothing to the field according to Gauss's Law. Therefore, that quantity of mass is also a function of r, m(r). That mass m(r) is the volume of the sphere of radius |r| times the mass density. Mass density is just the volume the earth divide by the mass of the earth = ((4/3)*Pi*re^3)/me
m(r) = ((4/3)*Pi*|r|^3) * ((4/3)*Pi*re^3)/me
Then
F(r) = -((G * m * m(r))/|r|^3) * r
F(r) = -G*m*((16 * Pi^2 * re^3*)/9 * me) * r
Is that correct? I wish I could do it in tex - it would be more readable.
jf
Last edited: Mar 9, 2006
8. Mar 9, 2006
### Staff: Mentor
You have that inverted:
$$\rho = m_e / (4/3 \pi r_e^3)$$
Latex works. It just doesn't display in preview mode (a pain in the butt, for sure).
9. Mar 9, 2006
### jackiefrost
Crap!!! How stupid :yuck: That's what I get for hurrying and not double checking the details.
Oh well - I just gotta work harder at not being so sloppy (in my thinking). Anyone have a spare brain I can use for a while - like 30 or 40 years?
Ohh!! That's good, though. I wasn't sure. OK - at least can go back and Edit.
BTW - I never really understood why that mass-shell, external to the surface, canceled EVERYWHERE in the interior until studying this little problem. I like working thru problems - even if I do screw up the first five attempts at most.
Thanks,
jackie
10. Mar 9, 2006
### Staff: Mentor
That's the only way to learn. If you aren't making mistakes, the problems are too easy.
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https://www.physicsforums.com/threads/van-der-waals-equation-question.386029/ | # Van der waals equation question
• Start date
• #1
1
0
Hi everyone,
Struggling with the following question - maths and physics never been my strongest subjects. Thanks in advance
1. An equation of state that has been used to model the behaviour of a fixed amount of a real gas is:
(P+a/v2)(v-b)=ZT
In the equation Z, a, b are constants, and P represents pressure, V = volume, T = temp. The constant Z depends on the amount of gas.
Consider what the dimensions of a and b are in terms of the fundamental quantities of mass (M), length (l), and time (t).
2. Q1: The dimensions of b are:
a) L3
b) L6
c) M.L-1.T-2
d) M-1.L.T2
Q2: The dimensions of a are:
a) L6
b) M.L5.T-2
c) M.L-1.T-2
d) M.L-5.T-1
Q3: The value of the constant Z could be expressed in terms of the unit:
a) W
b) W.N.s-1
c) N.J.K-1
d) J.K-1
3. According to std van der Waal's equantion Z is equivalent of R, therefore Q3 = D, not sure about the calculations for Q1 or Q2
Any help would be greatly appreciated.
Thanks
## The Attempt at a Solution
• #2
chemisttree
Homework Helper
Gold Member
3,588
622
Hi everyone,
Struggling with the following question - maths and physics never been my strongest subjects. Thanks in advance
1. An equation of state that has been used to model the behaviour of a fixed amount of a real gas is:
(P+a/v2)(v-b)=ZT
In the equation Z, a, b are constants, and P represents pressure, V = volume, T = temp. The constant Z depends on the amount of gas.
Consider what the dimensions of a and b are in terms of the fundamental quantities of mass (M), length (l), and time (t).
2. Q1: The dimensions of b are:
a) L3
b) L6
c) M.L-1.T-2
d) M-1.L.T2
I'll help you with this one since it is the easiest. Without just giving you the answer I will explain by analogy. The portion of the equation that contains b is (v-b), right? You are subtracting b from the volume.
Now here is the hint/analogy... Can you subtract gallons from miles? Gallons from degrees C? No. What must the units of b... err be?
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11K | {"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": 0.9454735517501831, "perplexity": 1750.0328523008795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988882.7/warc/CC-MAIN-20210508121446-20210508151446-00061.warc.gz"} |
https://www.physicsforums.com/threads/a-perfectly-inelastic-collision.241576/ | # A perfectly inelastic collision?
1. Jun 22, 2008
### 00PS
1. The problem statement, all variables and given/known data
A 5.00 g bullet moving with an initial speed of 410 m/s is fired into and passes through a 1.00 kg block. The block, initially at rest on a frictionless, horizontal surface, is connected to a spring of force constant 965 N/m.
1. If the block moves 5.20 cm to the right after impact, find the speed at which the bullet emerges from the block.
wrong check mark
2. Relevant equations
p=mv (momentum)
(1/2)mv^2 (kinetic energy)
(1/2kx^2 (elastic potential energy of a spring, where x is the distance the spring is compressed)
-Energy is always conserved in a isolated system
-Momentum is always conserved in a closed isolated system
3. The attempt at a solution
I am having a hard time grappeling what type of problem this is. A perfectly inelastic collision? But the bullet then exits the block...so a little tricky. Here is my attempt:
(1/2)mv^2(initial)=(1/2)mv^2 (final) - (1/2)kx^2
solving for v (final) i got 28.4 m/s
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
#### Attached Files:
• ###### p9-71.gif
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2. Jun 22, 2008
### EngageEngage
you are on the right track; this can definitely be solved with Energy considerations. First, you should check the sign on your 'spring energy' expression. What you are implying with that statement is that as the bullet compresses the spring, your bullet obtains more energy, when it is the opposite.
3. Jun 23, 2008
### tiny-tim
Welcome to PF!
Hi 00PS! Welcome to PF!
No, energy is not always conserved in a isolated system … in fact energy is never conserved unless the question tells you it is!
There is no way that a bullet tearing a hole in something conserves energy!
In this case, you have to work backwards …
you know how far the spring compresses, so you know what energy it got.
That energy came from the block, not the bullet, so you can work out the KE of the block after the bullet left it.
Then use conservation of momentum to find the speed of the bullet.
4. Jun 23, 2008
### 00PS
oops...already making use of my user name.
I simply forgot the word 'mechanichal'. According to my textbook, mechanical energy of a isolated system is conserved. Nonetheless, I asked my professor about the problem this morning in class, and he explained it exactly as you did, thanks! I knew conservation of momentum was involved somehow, but could not put my finger on how to apply it.
btw, thanks for the welcome! I found this site sort of as a joke using google, and to my surpise a phyics homework help does exist!
EDIT: Do you know how you could find how much mechanichal energy was transferred to internal energy?
Last edited: Jun 23, 2008
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http://math.stackexchange.com/questions/869150/a-function-fx-that-increases-from-0-to-1-when-x-increases-from-0-to-infinity | # A function f(x) that increases from 0 to 1 when x increases from 0 to infinity.
I am looking for a function f(x) with a value range of [0,1].
f(x) should increase from 0 to 1 while its parameter x increases from 0 to +infinity.
f(x) increases very fast when x is small, and then very slow and eventually approach 1 when x is infinity.
Here is a figure. The green curve is what I am looking for:
Thanks.
It would be great if I can adjust the slope of the increase. Although this is not a compulsory requirement.
-
Do you want it to actually take on the values 0 and 1? – Mike Miller Jul 16 '14 at 17:20
@MikeMiller value 0 yes. value 1 no. f(x)=0 when x=0, =1 when x=infinity, and = a value between 0 and 1 otherwise. – Leo Jul 16 '14 at 17:22
Consider $f(x)=1-\frac1{x+1}$ – abiessu Jul 16 '14 at 17:22
@Leo do you want this $f$ to be $\mathcal{C}^0$? or something more strict perhaps? – DanZimm Jul 16 '14 at 19:20
It is always going to be relative fast when $x$ is small, a simple consequence of the fact that you require it to be bounded and monotone. – Gina Jul 16 '14 at 19:21
I think this should work well for your purposes: $$f(x) = \frac{x}{x + a}$$ Where $a$ can be any number bigger than $0$. The smaller $a$ is, the sharper the increase will be.
ADDENDUM: if you want to extend this to an odd (and continuously differentiable) function, simply take $$f(x) = \frac{x}{|x| + a}$$
-
I love this function so much. So simple, elegant, and nice.... – Leo Jul 16 '14 at 17:30
And in particular, the slope at $x=0$ is $\dfrac{1}{a}$. – Théophile Jul 16 '14 at 17:35
I like this, but it would be great to be able to parameterise the rate of convergence to the asymptote as well as the slope at the origin. – Keith Jul 17 '14 at 3:15
@Keith in what sense do you mean? Would something like $f(x) = \frac{x^n}{x^n + a}$ be sufficient for those purposes? – Omnomnomnom Jul 17 '14 at 10:56
@Omnomnomnom That $f(x)$ does not have the right gradient at $x=0$. $$g(x) = f(x)^{1+bf(x)}$$ would do, but seems a bit ugly. – Keith Jul 17 '14 at 23:31
Here are two simple functions with slope $k$ at $x=0$, for some $k>0$:
$$f(x) = 1-e^{-kx}$$
and
$$g(x) = \frac{2}{\pi}\arctan(\frac{\pi}{2}kx)$$
The first of these approaches $1$ more quickly than the second.
-
One nice thing about the arctan version: it also approaches -1 as x approaches -infinity. – Brilliand Jul 16 '14 at 21:00
One I use very often is : $$\tanh x = \frac{\sinh x}{\cosh x} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = \frac{1 - e^{-2x}} {1 + e^{-2x}}$$
The increase at the begining around 0 is "only" linear but can do the work. and you can choose the slope at $x \mapsto 0$
-
Use $f(x) = \tanh mx$ if you want slope $m$ at $x = 0$. – David K Jul 17 '14 at 20:18
I'm surprised no one has mentioned erf(x) aka the "Error Function", defined roughly as the normalized area under the bell curve as a function of the upper limit of integration. For x > 1, it satisfies your requirement and also has a slope that is easily controlled (and made arbitrarily large at 0) by the width of your Gaussian.
-
Why is this answer downvoted so much? The error function is in fact a sigmoid curve with limit $1$ as $x\to\infty$ and is different from the other suggestions. Perhaps the fact that it isn't an elementary function makes it less appealing, but it certainly meets the criteria in the OP. – Mario Carneiro Jul 17 '14 at 3:27
$f(x) = 1 - \exp(-x/\epsilon)$ for $\epsilon > 0$ small will do.
-
Some time ago I had an interest in such a function, but with the added requirement that it should map negative values of $x$ onto the range $(-1,0)$, asymptotically approaching $-1$ as $x$ goes to $-\infty$. In fact, I wanted $f(-x) = -f(x)$.
I came up with something like this:
$$f(x) = \frac{x}{\sqrt{a^2 + x^2}} .$$
This has slope $\frac{1}{|a|\,}$ at $x = 0$.
-
Simple version: $$f(x)=1-\mathrm e^{-a\sqrt{x}}\qquad (a\gt0)$$
Slightly more elaborate version:$$f(x)=1-\mathrm e^{-a\sqrt{x}-bx}\qquad (a\gt0,\ b\geqslant0)$$ Every such function fits every requisite in the question, including the infinite slope at $0$. The parameter $a$ can help to tune the increase near $0$. The parameter $b$ can help to tune the increase near $+\infty$. To get even quicker convergence to $1$ when $x\to+\infty$, one can replace $-bx$ in the exponent by $-bx^n$ for some $b\gt0$ and $n\gt1$.
Once one understands the principle, a host of other solutions springs to mind.
-
Try $f(x) = 1-e^{-x^2}$ , which will be 0 for x = 0 and approach 1 rapidly for big x.
-
But doesn't increase very fast when $x$ is small. – Henning Makholm Jul 16 '14 at 17:27
Judging from the scale of his sketch, i thought it would be very sufficiently fast. – ChocolateBar Jul 16 '14 at 17:28
Look at the derivative of the function at $x = 0$. It seems the OP wants something with initially a large positive slope that gradually flattens out, not a slope that starts flat, increases, then decreases. – David K Jul 16 '14 at 17:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9197863936424255, "perplexity": 399.5037244940514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246643283.35/warc/CC-MAIN-20150417045723-00183-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=3969 | 2019
Том 71
№ 7
# Ciarlettci M.
Articles: 1
Article (English)
### On the Spatial and Temporal Behavior in Dynamics of Porous Elastic Mixtures
Ukr. Mat. Zh. - 2002. - 54, № 4. - pp. 527-544
In this paper, we study the spatial and temporal behavior of dynamic processes in porous elastic mixtures. For the spatial behavior, we use the time-weighted surface power function method in order to obtain a more precise determination of the domain of influence and establish spatial-decay estimates of the Saint-Venant type with respect to time-independent decay rate for the inside of the domain of influence. For the asymptotic temporal behavior, we use the Cesáro means associated with the kinetic and strain energies and establish the asymptotic equipartition of the total energy. A uniqueness theorem is proved for finite and infinite bodies, and we note that it is free of any kind of a priori assumptions on the solutions at infinity. | {"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": 0.8315683007240295, "perplexity": 448.5481541103754}, "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/1566027315865.44/warc/CC-MAIN-20190821085942-20190821111942-00469.warc.gz"} |
http://mathhelpforum.com/algebra/75240-series-problem-print.html | # series problem .
• February 22nd 2009, 10:55 PM
thereddevils
series problem .
Find the sum to the n terms of the series :
$\frac{1}{(2r-1)(2r+1)(2r+3)}$
• February 23rd 2009, 05:47 AM
Sum of series
Hello thereddevils
Quote:
Originally Posted by thereddevils
Find the sum to the n terms of the series :
$\frac{1}{(2r-1)(2r+1)(2r+3)}$
Using Partial Fractions, you find that
$\frac{1}{(2r-1)(2r+1)(2r+3)} = \frac{1}{8}\left(\frac{1}{2r-1}-\frac{2}{2r+1}+\frac{1}{2r+3}\right)$
So if the sum to n terms is $S$:
$8S = \sum_{r=1}^n\left(\frac{1}{2r-1}-\frac{2}{2r+1}+\frac{1}{2r+3}\right)$
$= \sum_{r=1}^n\frac{1}{2r-1} -\sum_{r=2}^{n+1}\frac{2}{2r-1}+\sum_{r=3}^{n+2}\frac{1}{2r-1}$
$= 1 + \frac{1}{3}+\sum_{r=3}^n\frac{1}{2r-1}-(\frac{2}{3}+2\sum_{r=3}^n\frac{1}{2r-1} + \frac{2}{2n+1})+\sum_{r=3}^n\frac{1}{2r-1}+\frac{1}{2n+1}+\frac{1}{2n+3}$
$=\frac{2}{3}-\frac{2}{2n+1}+\frac{1}{2n+1}+\frac{1}{2n+3}$
$\Rightarrow S=\frac{1}{8}\left(\frac{2}{3}-\frac{1}{2n+1}+\frac{1}{2n+3}\right)$ | {"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": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420454502105713, "perplexity": 3750.8475823466583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657114105.77/warc/CC-MAIN-20140914011154-00218-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://math.stackexchange.com/questions/280986/type-homotopy-of-stiefel-variety | # Type homotopy of Stiefel variety
Why Stiefel variety $V_{n}(\mathbb{C}^{k})$ of $n$-frame in $\mathbb{C}^{k}$ is $2k-2n$ connected? I know that all homotopy groups of infinite Stiefel variety $V_{n}(\mathbb{C}^{\infty})$ vanish and Stiefel variety has a CW structure. Then how can I use Whitehead theorem to prove that it is contractible?
-
For the first question, there are two manifolds commonly called the Stiefel manifold. The first is the collection of $n$-tuples of linearly independent vectors in $\mathbb{C}^k$, the second is the collection of $n$-tuples of orthonormal vectors in $\mathbb{C}^k$.
The Gram-Schmidt process gives a deformation retract from the first manifold to the second. Thus, we may as well focus on the second description for answering your question.
The group $U(k)$ acts transitively on the set of all $n$-tuples of orthonormal vectors. For a proof, given any orthonormal $n$-tuple $\{v_1,...,v_n\}$, extend it to an orthonormal basis of $\mathbb{C}^k$. The matrix which has these vectors as columns will be in $U(k)$ and $U(k)\cdot \{e_1,....,e_n\} = \{v_1,...,v_n\}$ (where $e_i$ is the standard orthonormal basis of $\mathbb{C}^k.$)
The stabilizer of this action, at the point $\{e_1,...,e_n\}$ is given by all matrices in $U(k)$ for which the initial $n\times n$ block is the identity. It's easy to see that that such a matrix must be block diagonal, with the second block consisting of an arbitrary element of $U(k-n)$.
This proves $V_n(\mathbb{C^k})$ is diffeomorphic to $U(k)/U(k-n)$. It remains to compute the topology of $U(k)/U(k-n)$.
Since $U(k-n+1)/U(k-n) \cong S^{2(k-n+1)-1} = S^{2k-2n+1}$ is $2k-2n$ connected, this implies the inclusion map $$U(k-n)\rightarrow U(k-n+1)$$ is $2k-2n$ connected as well. A similar argument shows the inclusion $$U(k-n+1)\rightarrow U(k-n+2)$$ is even more connected. So the inclusion map $$U(k-n)\rightarrow U(k-n+2)$$ is $2k-2n$ connected. Continuing, by induction, we see the inclusion $$U(k-n)\rightarrow U(k)$$ is also $2k-2n$ connected, so the quotient $U(k)/U(k-n)$ is at least $2k-2n$ connected as claimed.
For you second question, try applying Whitehead's theorem to the inclusion of a point into $V_n(\mathbb{C}^\infty)$.
-
How do you define a connected map? What do you want to say writing that the inclusion map is $2k-2n$-connected? – ArthurStuart Jan 24 '13 at 10:17
en.wikipedia.org/wiki/N-connected#n-connected_map. In short, a continuous function is called $n-$ connected if the induced map on $\pi_k$ is an isomorphism for $k < n$ and surjective for $k = n$. As far as the inclusion map part is concerned, any subset $A\subseteq B$ gives rise to a function $i:A\rightarrow B$ given by $i(a) = a$. This is a continuous function, so it makes sense to talk about how connected it is. – Jason DeVito Jan 24 '13 at 14:16 | {"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": 0.9566634297370911, "perplexity": 76.81178341647339}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500829393.78/warc/CC-MAIN-20140820021349-00058-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://blog.shuningbian.net/2012/04/dft-and-window-spectrum.html | ## April 18, 2012
### DFT and the Window Spectrum
In case I forget again why the window spectrum is computed when doing DFT, the answer is that discretisation leads to aliasing artefacts.
Consider the DFT of a x(t)=1. Its DFT is calculated by calculating the sum sin(2 pi f t)*x(t) and cos(2 pi f t)*x(t) individually, then squaring the resulting sums, add them together, then take the square root. Since x(t) = 1, for any frequency you are essentially summing cos(t) and sin(t), which you expect to be zero, and thus the power spectrum should be 0 for any frequency.
If you actually do this with, you will find that unless the points in time where sampling occurs covers an integer number of wavelengths, and is coherent in phase, i.e. sampling occurs in the same phase of each cycle, you will find that you don't get 0. You might get 0.1, 0.15, 0.09, 0.01, etc, but not 0. This is made more obvious by non-uniformely sampled data, such as those you might get in the real world.
This present a problem: the residue can produces peaks where there would otherwise be none, and this is more pronounced for lower frequency, because their wavelengths are longer and thus less likely to fit perfectly inside the window.
One way of rectifying this issue is to compute a correction coefficient c(f)=1-w(f) where f is the frequency, w(f) the window spectrum. The "corrected" power spectrum can then be calculated as correctp(f) = p(f) * c(f), where p(f) is the power spectrum.
I should add that the above method is not at all mathematical. It is chosen because any correction strategy that involves dividing by w(f) is unstable as sometimes w(f) gets very close to 0. Simply subtracting w(f) from p(f) won't work either because p(f)'s magnitude clearly depends on x(t) whereas w(f) doesn't.
An obvious failure of the above correction is that it doesn't make the residue disappear - it only reduces it. A superior method would correct the DFT for a DC signal so it is 0 everywhere.
Update
The obvious answer is to simply normalise x(t) so the maximum amplitude is 1.0. Then the window power can simply be subtracted. Duh.
Cheers,
Steve | {"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": 0.9013497233390808, "perplexity": 1086.4066908692378}, "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/1642320304261.85/warc/CC-MAIN-20220123111431-20220123141431-00539.warc.gz"} |
https://www.physicsforums.com/threads/general-fbd-issue.651131/ | # General FBD issue
1. Nov 10, 2012
### Kugan
Ok so here is the issue I'm having:
For problems with inclined planes I have always broken the Fg force down into its components:
Fgy= mgcos(theta)
Fgx = mgsin(theta)
Usually the Fnety= Fgy+ Fn(there is no motion in the Y direction for lets say a car going around a banked curve) Fnety=0 so we cant just say -Fgy= Fn, so Fn = -m(-g)cos(theta) finally arriving at Fn= mgcos(theta)
If however I were to break the normal force vector into components and not the Gravitational force I get: Fn= mg/Cos(theta)
Fny= Fncos(theta)
Fnx = Fnsin(theta)
Fnety = Fny+ mg;
0 = Fny+ mg;
Fny= mg;
Fncos(theta)= mg;
Fn = mg/ cos(theta)
Can someone clarify why, I get two different normal force vectors depending on whether I use the Fg force or the Fn force to orient the axis and break the vector into the components ?
Thank you.
2. Nov 10, 2012
### Simon Bridge
Here, let me tidy up:
If you put your axis so +y is normal to the slope, +x points down the slope, and the slope has angle $\theta$ to the horizontal (as the usual setup) then you will have, in the absence of any other, applied, force:
* a normal force $\vec{N}=N\hat{\jmath}$,
* a weight (gravitational force) $\vec{w}=mg\sin(\theta)\hat{\imath}-mg\cos(\theta)\hat{\jmath}$; and
* a friction force $\vec{f}=-\mu N \hat{\imath}$
(note - I find it is useful to avoid subscripts where I can.)
For a car going around a banked curve, there is a net unbalanced force acting radially inwards... so it is not correct to write that $F_{net}{}_y=0$
In this situation it may be more useful to adopt a different coordinate system. i.e.
If you orient your axis so +y = $-\vec{w}$ (i.e. "upwards") and +x pointing at the center of the turn (i.e not into the ground): then the three forces resolve into:
* a normal force $\vec{N}=N\sin(\theta)\hat{\imath}+N\cos(\theta) \hat{\jmath}$,
* a weight (gravitational force) $\vec{w}=-mg\hat{\jmath}$; and
* a friction force $\vec{f}=-\mu N\cos(\theta)\hat{\imath}+\mu N\sin(\theta)\hat{\jmath}$
Thus, $F_{net}{}_y = N_y-mg+f_y=0 \Rightarrow N_y=mg-f_y$
For the sake of an argument - say that $f_y=0$, then we recover your:
$N_y=N\cos(\theta)=mg \Rightarrow N_y=mg/\cos(\theta)$
So lets see what the issue is:
Well, clearly, you shouldn't. The trouble is that you are comparing situations that are not equivalent. I suspect you are treating the normal force as an fixed applied force rather than a dynamical force.
Lets take the case of a block accelerating down a frictionless incline ... in that case, the forces normal to the slope should cancel.
We would expect to find that $N=mg\cos(\theta)$ using either coordinate system.
Using gravity to set the coordinates - it is not correct to write $N\cos(\theta)=mg$ because some of the gravity force must be unbalanced to give the acceleration down the slope. In fact $\vec{N}$ will always work so that $\vec{N}+\vec{w}$ will point down the slope.
Now take the case of a block sitting stationary on a slope with friction.
In this case normal components will cancel and so will perpendicular components.
We'd expect to find that $N=mg\cos{\theta}$ and $\mu N=mg\sin(\theta)$ no matter what coordinates we use.
I think you should be able to do it from there.
Similar Discussions: General FBD issue | {"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": 0.8910987973213196, "perplexity": 1037.7601057038694}, "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-2017-39/segments/1505818687428.60/warc/CC-MAIN-20170920175850-20170920195850-00019.warc.gz"} |
http://popflock.com/learn?s=Gravitational_field | Gravitational Field
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Gravitational Field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body.[1] Thus, a gravitational field is used to explain gravitational phenomena, and is measured in newtons per kilogram (N/kg). In its original concept, gravity was a force between point masses. Following Isaac Newton, Pierre-Simon Laplace attempted to model gravity as some kind of radiation field or fluid, and since the 19th century, explanations for gravity have usually been taught in terms of a field model, rather than a point attraction.
In a field model, rather than two particles attracting each other, the particles distort spacetime via their mass, and this distortion is what is perceived and measured as a "force".[] In such a model one states that matter moves in certain ways in response to the curvature of spacetime,[2] and that there is either no gravitational force,[3] or that gravity is a fictitious force.[4]
Gravity is distinguished from other forces by its obedience to the equivalence principle.
## Classical mechanics
In classical mechanics, a gravitational field is a physical quantity.[5] A gravitational field can be defined using Newton's law of universal gravitation. Determined in this way, the gravitational field g around a single particle of mass M is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the universal law, and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass, ?, at each point in space associated with the force fields; this is called gravitational potential.[6] The gravitational field equation is[7]
${\displaystyle \mathbf {g} ={\frac {\mathbf {F} }{m}}={\frac {\mathrm {d} ^{2}\mathbf {R} }{\mathrm {d} t^{2}}}=-GM{\frac {\mathbf {\hat {R}} }{\left|\mathbf {R} \right|^{2}}}=-\nabla \Phi }$
where F is the gravitational force, m is the mass of the test particle, R is the position of the test particle (or for Newton's second law of motion which is a time dependent function, a set of positions of test particles each occupying a particular point in space for the start of testing), R? is a unit vector in the radial direction of R, t is time, G is the gravitational constant, and ? is the del operator.
This includes Newton's law of universal gravitation, and the relation between gravitational potential and field acceleration. Note that and are both equal to the gravitational acceleration g (equivalent to the inertial acceleration, so same mathematical form, but also defined as gravitational force per unit mass[8]). The negative signs are inserted since the force acts antiparallel to the displacement. The equivalent field equation in terms of mass density ? of the attracting mass is:
${\displaystyle \nabla \cdot \mathbf {g} =-\nabla ^{2}\Phi =-4\pi G\rho }$
which contains Gauss's law for gravity, and Poisson's equation for gravity. Newton's and Gauss's law are mathematically equivalent, and are related by the divergence theorem.
These classical equations are differential equations of motion for a test particle in the presence of a gravitational field, i.e. setting up and solving these equations allows the motion of a test mass to be determined and described.
The field around multiple particles is simply the vector sum of the fields around each individual particle. An object in such a field will experience a force that equals the vector sum of the forces it would experience in these individual fields. This is mathematically[9]
${\displaystyle \mathbf {g} _{j}^{\text{(net)}}=\sum _{i\neq j}\mathbf {g} _{i}={\frac {1}{m_{j}}}\sum _{i\neq j}\mathbf {F} _{i}=-G\sum _{i\neq j}m_{i}{\frac {\mathbf {\hat {R}} _{ij}}{\left|\mathbf {R} _{i}-\mathbf {R} _{j}\right|^{2}}}=-\sum _{i\neq j}\nabla \Phi _{i}}$
i.e. the gravitational field on mass mj is the sum of all gravitational fields due to all other masses mi, except the mass mj itself. The unit vector R?ij is in the direction of Ri - Rj.
## General relativity
In general relativity, the Christoffel symbols play the role of the gravitational force field and the metric tensor plays the role of the gravitational potential.
In general relativity, the gravitational field is determined by solving the Einstein field equations[10]
${\displaystyle \mathbf {G} =\kappa \mathbf {T} ,}$
where T is the stress-energy tensor, G is the Einstein tensor, and ? is the Einstein gravitational constant. The latter is defined as ? = 8?G/c4, where G is the Newtonian constant of gravitation and c is the speed of light.
These equations are dependent on the distribution of matter and energy in a region of space, unlike Newtonian gravity, which is dependent only on the distribution of matter. The fields themselves in general relativity represent the curvature of spacetime. General relativity states that being in a region of curved space is equivalent to accelerating up the gradient of the field. By Newton's second law, this will cause an object to experience a fictitious force if it is held still with respect to the field. This is why a person will feel himself pulled down by the force of gravity while standing still on the Earth's surface. In general the gravitational fields predicted by general relativity differ in their effects only slightly from those predicted by classical mechanics, but there are a number of easily verifiable differences, one of the most well known being the deflection of light in such fields.
## Notes
1. ^ Feynman, Richard (1970). The Feynman Lectures on Physics. I. Addison Wesley Longman. ISBN 978-0-201-02115-8.
2. ^ Geroch, Robert (1981). General Relativity from A to B. University of Chicago Press. p. 181. ISBN 978-0-226-28864-2.
3. ^ Grøn, Øyvind; Hervik, Sigbjørn (2007). Einstein's General Theory of Relativity: with Modern Applications in Cosmology. Springer Japan. p. 256. ISBN 978-0-387-69199-2.
4. ^ Foster, J.; Nightingale, J. D. (2006). A Short Course in General Relativity (3 ed.). Springer Science & Business. p. 55. ISBN 978-0-387-26078-5.
5. ^ Feynman, Richard (1970). The Feynman Lectures on Physics. II. Addison Wesley Longman. ISBN 978-0-201-02115-8. A "field" is any physical quantity which takes on different values at different points in space.
6. ^ Forshaw, J. R.; Smith, A. G. (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.[page needed]
7. ^ Lerner, R. G.; Trigg, G. L., eds. (1991). Encyclopaedia of Physics (2nd ed.). Wiley-VCH. ISBN 978-0-89573-752-6.[page needed]
8. ^ Whelan, P. M.; Hodgeson, M. J. (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 978-0-7195-3382-2.[page needed]
9. ^ Kibble, T. W. B. (1973). Classical Mechanics. European Physics Series (2nd ed.). UK: McGraw Hill. ISBN 978-0-07-084018-8.[page needed]
10. ^ Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973). Gravitation. W. H. Freeman & Co. ISBN 978-0-7167-0344-0.[page needed] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.946637749671936, "perplexity": 473.78164095533435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704795033.65/warc/CC-MAIN-20210126011645-20210126041645-00704.warc.gz"} |
https://blog.assafrinot.com/?cat=91 | # Category Archives: Open Problems
## 6th European Set Theory Conference, July 2017
I gave a 3-lectures tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading
Posted in Invited Talks, Open Problems | | 4 Comments
## Prikry forcing may add a Souslin tree
A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading
Posted in Blog, Open Problems | | 9 Comments
## Partitioning the club guessing
In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading
## Syndetic colorings with applications to S and L
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
Posted in Blog, Expository, Open Problems | | 1 Comment
## The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems | | 4 Comments
## Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
## An inconsistent form of club guessing
In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading
Posted in Blog, Open Problems | Tagged | 3 Comments
## Dushnik-Miller for regular cardinals (part 1)
This is the first out of a series of posts on the following theorem. Theorem (Erdos-Dushnik-Miller, 1941). For every infinite cardinal $\lambda$, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Namely, for any coloring $c:[\lambda]^2\rightarrow\{0,1\}$ there exists either a subset $A\subseteq \lambda$ of order-type $\lambda$ with … Continue reading
Posted in Blog, Expository, Open Problems | | 14 Comments
## The order-type of clubs in a square sequence
Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, … Continue reading
Posted in Blog, Open Problems | Tagged | 12 Comments
## Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading | {"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": 0.8353211879730225, "perplexity": 1344.210121745822}, "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/1502886106996.2/warc/CC-MAIN-20170820223702-20170821003702-00580.warc.gz"} |
http://mathhelpforum.com/calculus/153718-unusual-use-integral-explanation-why.html | # Math Help - Unusual use for integral, explanation why?
1. ## Unusual use for integral, explanation why?
Hi,
So I have the following problem:
A vessel is in the form of an inverted circular cone with a semi-vertical angle of 30 degrees. Water is poured in at 5cm^3/min and leaks out from a hole at the bottom at a rate of $\frac{\sqrt{h}}{30}$cm^3/min. Set up the integral that gives the time for the depth to change from 0.5cm to 5cm.
I have got the (correct) integral:
$\frac{dt}{dh} = \frac{10*\pi*h^2}{150-\sqrt{h}}$
Then the answers say I should take the integral of the above equation, with respect to h, from h=0.5 to h=5.
I'm wondering, why? Doesn't taking that integral just find the area of the curve? How does it relate to the question? Or does doing that subtract the time at t=5 from the time at t=0.5?
Just confused about the why. Thanks!
2. You are right! It does find the area UNDER the curve. However, it is not just any ordinary area. The x-axis (or h-axis) measures distance (that is length). The y-axix measures dt/dh which is the rate of change of time with respect to distance (h). That is the y axis measures time/length. If you have a square with corners (0,0),(a,0),(0,b),(a,b) in the dt/dh versus h plane, it's area is just a*b and the units are length*time/length = time. So the area actually signifies time!
3. Ah, much clearer now. Thanks heaps! | {"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": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9588125944137573, "perplexity": 864.0483254566867}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936464193.61/warc/CC-MAIN-20150226074104-00295-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/equilibrium-problem.210082/ | # Equilibrium Problem
1. Jan 21, 2008
### yasar1967
1. Four people carry a uniform beam of length L (with mass M) holding it horizontal. Two men hold it at the ends while the remaining two are inward but nevertheless all men are equally apart from each other i.e. if the beam is 4m long the distance from 1st to 2nd is 1m; 2nd to 3rd is 2m and 3rd to 4th is 1m.
Calculate the normal forces acting upon the men.
2. System is in equilibrium, static. general force and torque formulas apply.
3. I assumed due to symmetry, outward Fn forces and inward Fn forces must be equal. Therefore Mg=2xFno+2xFni
The problem is the SAME equation stems from torque formula as well and as I have two identical formulas with two unknowns, I get nowhere.
Where did I go wrong?
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Jan 21, 2008
### Staff: Mentor
You didn't go wrong. The problem is statically indeterminate--you just don't have enough information. For example: The two end people can lift 90% of the weight if they want and the two guys in the middle could just pretend to help. That and many other combinations of forces are consistent with the given information.
Of course, you could just assume that the load is evenly distributed.
(Also: If they are evenly spaced and the board is 4m long, they are 4/3 m apart.)
3. Jan 27, 2008
thank you | {"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": 0.9627202153205872, "perplexity": 1025.6541598759745}, "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/1521257647660.83/warc/CC-MAIN-20180321141313-20180321161313-00542.warc.gz"} |
https://proofwiki.org/wiki/Definition:Pseudometrizable_Topology | # Definition:Pseudometrizable Topology
## Definition
Let $\left({S, d}\right)$ be a pseudometric space.
Let $\left({S, \tau_d}\right)$ be the topological space induced by $d$.
Then for any topological space which is homeomorphic to such a $\left({S, \tau_d}\right)$, it and its topology are defined as pseudometrizable.
## Linguistic Note
The UK English spelling of this is pseudometrisable, but it is rarely found. | {"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": 0.8368067145347595, "perplexity": 820.9857750820818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897168.4/warc/CC-MAIN-20200714145953-20200714175953-00026.warc.gz"} |
https://arxiv.org/abs/1612.09296 | cs.LG
(what is this?)
# Title: Symmetry, Saddle Points, and Global Geometry of Nonconvex Matrix Factorization
Abstract: We propose a general theory for studying the geometry of nonconvex objective functions with underlying symmetric structures. In specific, we characterize the locations of stationary points and the null space of the associated Hessian matrices via the lens of invariant groups. As a major motivating example, we apply the proposed general theory to characterize the global geometry of the low-rank matrix factorization problem. In particular, we illustrate how the rotational symmetry group gives rise to infinitely many non-isolated strict saddle points and equivalent global minima of the objective function. By explicitly identifying all stationary points, we divide the entire parameter space into three regions: ($\cR_1$) the region containing the neighborhoods of all strict saddle points, where the objective has negative curvatures; ($\cR_2$) the region containing neighborhoods of all global minima, where the objective enjoys strong convexity along certain directions; and ($\cR_3$) the complement of the above regions, where the gradient has sufficiently large magnitudes. We further extend our result to the matrix sensing problem. This allows us to establish strong global convergence guarantees for popular iterative algorithms with arbitrary initial solutions.
Subjects: Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML) Cite as: arXiv:1612.09296 [cs.LG] (or arXiv:1612.09296v2 [cs.LG] for this version)
## Submission history
From: Tuo Zhao [view email]
[v1] Thu, 29 Dec 2016 20:57:19 GMT (2880kb,D)
[v2] Sun, 1 Jan 2017 18:14:33 GMT (2882kb,D) | {"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": 0.8408738970756531, "perplexity": 826.2691349017144}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320679.64/warc/CC-MAIN-20170626050425-20170626070425-00336.warc.gz"} |
https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | ## Theorem 36: the Cantor set is an uncountable set with zero measure
This week’s post is by Laura Irvine, who is just about to start her second year reading Mathematics at Murray Edwards College, Cambridge. Thanks very much, Laura!
First of all, what is the Cantor set?
To form the Cantor set, start with the closed interval $[0,1]$ (this means $0$ and $1$ are included in the interval) and remove the middle open third of the interval (i.e. remove ($\frac{1}{3}$, $\frac{2}{3}$) where the curved brackets mean the interval is open, so $\frac{1}{3}$ and $\frac{2}{3}$ are not themselves in the interval). You should be left with two disjoint closed intervals: $[0, \frac{1}{3}]$ and $[\frac{2}{3}, 1]$. I’m going to call this step the first iteration.
Then do the same thing to each of these intervals: remove the middle third of each to get the new intervals as $[0, \frac{1}{9}]$, $[\frac{2}{9}, \frac{1}{3}]$, $[\frac{2}{3}, \frac{7}{9}]$, $[\frac{8}{9}, 1]$. Then remove the middle third of each of these intervals. Keep repeating this process and the Cantor set is the set of all the points in the interval $[0,1]$ that are never removed. So the first few steps of the process look like this:
Do you understand how this set is formed? Can you think of some points that are in the Cantor set?
Well, $0$ will never be removed: the first closed interval after the $n^{\textrm{th}}$ iteration would be $[0, \frac{1}{3^n}]$ so $0$ will be in the infinite intersection of the first interval of each step. The other interval endpoints are points in the set too. It turns out that there are also points that are in the set that aren’t interval endpoints.
An interesting and, in my opinion, rather surprising property of the Cantor set is that it has measure $0$, despite being an uncountable set! The fact it is uncountable means there is no way of writing all the numbers in the Cantor set in a list.
So, intuitively, this is saying that if all the points in the Cantor set were lined up next to each other, the line would have length $0$ and yet there are infinitely many points in the set. How weird is that?!
How can we show the Cantor set is uncountable? Well, it would be good if we could show that there’s a surjection from the set to another set that we already know is uncountable. This would show that there are at least as many points in the Cantor set as there are in this other uncountable set. We know that there are uncountably many real numbers so how about trying to construct a surjection from the Cantor set to the real numbers?
First, we need to work in a ternary system. This is a similar idea to the binary system (which is where numbers are written using just the digits $0$ and $1$) or the decimal system (where you use the digits $0$ to $9$). However, in a ternary system there are three digits that can be used: $0$, $1$ and $2$.
To write a number in the decimal system, you might imagine column headings of units, tens, hundreds etc. (i.e. $1$, $10$, $10^2$, $10^3$, …). In binary, you would imagine column headings of $1$, $2$, $4$, $8$, $16$, etc. (i.e. $1$, $2$, $2^2$, $2^3$, …).
In ternary, the column headings would be (from right to left) $1$, $3$, $9$, $27$, ….
So, to write a given number, say $66$, in ternary, think what the biggest power of $3$ is that is still less than, in this case, $66$. So that will be $27$$27$ goes into $66$ twice so in the $27$ column’, write $2$.
$66-2\times 27 = 12$ so we now see what the largest power of three that is less than $12$ is. It’s $9$ and $9$ only goes into $12$ once so write $1$ in the $9$ column’.
Carry on in this way.
$12 - 9 = 3$ so in the `$3$ column’ write $1$$3 -3 = 0$ so put $0$ in the final column.
So, in ternary, $66 = 2110$. Does that make sense? What would $32$ be in ternary?
For numbers between $0$ and $1$, the column headings will be $\frac{1}{3}$, $\frac{1}{9}$, $\frac{1}{27}$, etc. So $\frac{1}{3}$ would be $0.1$ and $\frac{2}{3}$ would be $0.2$. Just as another example, $\frac{1}{2} = 0.111111111...$ in ternary. Can you see how I calculated that?
In the first iteration when forming the Cantor set, all numbers which don’t have $0.0$ or $0.2$ as their beginning in ternary are removed. You might think: but what about $0.1$? But that’s alright, because in ternary, $0.1$ is the same as $0.0222222...$. Then, in the second iteration, get rid of all the numbers that don’t have $0$ or $2$ in the second position after the decimal point. And so on, so we can see that in the Cantor set, every number is made up of $0$s and $2$s.
Now, what we have to do is to map every $2$ in any number in the Cantor set to a $1$. So, for example, $0.20002$ would become $0.10001$. This will give the full set of numbers in the interval $[0,1]$ in binary. This means there is a mapping which has its image as the whole of the interval $[0,1]$. That is, there is a surjection from the Cantor set to all the real numbers in the interval $[0,1]$. Since the real numbers are uncountable, so the Cantor set must be!
Here comes the cool part!
Theorem The Cantor set has measure $0$.
OK, so how can we prove that? Well, how about calculating how much of the interval $[0,1]$ is removed when forming the Cantor set. Then we could subtract this number from $1$:
Measure of Cantor set = (Measure of the interval $[0,1]$) – (All the stuff removed from $[0,1]$ to get the Cantor set).
So, to form the Cantor set, we started with an interval of measure $1$ and subtracted $\frac{1}{3}$. Then of each of the remaining thirds, we took the middle thirds of each away (so we subtracted $\frac{2}{9}$). Then there were $4$ intervals, each of which had its middle third removed so $\frac{4}{27}$ was subtracted. Then $\frac{8}{81}$ were removed, then $\frac{16}{243}$ and so on.
Can you see what the difference between the measure of the $n^{\textrm{th}}$ and the $(n-1)^{\textrm{th}}$ iteration would be?
Well, the size of each interval being removed is a third smaller than in the previous step so the denominator will be $3^n$. The numerator, on the other hand, doubles with each step as each interval splits into two new intervals. So it is $2^{n-1}$.
Therefore, all these intervals that we’re removing add up to $\sum_{n=1}^{\infty} \frac{2^{n-1}}{3^n} = \frac{1}{2} \sum_{n=1}^{\infty} (\frac{2}{3})^n$.
Now $\sum_{n=1}^{\infty} (\frac{2}{3})^n$ is a geometric progression with starting value $a = \frac{2}{3}$ and where each term is $\frac{2}{3}$ of the previous term so the common ratio is $r = \frac{2}{3}$. The formula for the sum to infinity of a geometric progression is $\frac{a}{1-r} = \frac{2/3}{1/3} = 2$ so $\frac{2^{n-1}}{3^n} = \frac{1}{2} \times 2 = 1$.
$1 - 1 = 0$ so the measure of the Cantor set is $0$. So the theorem is proved!
### 10 Responses to “Theorem 36: the Cantor set is an uncountable set with zero measure”
1. theoremoftheweek Says:
If you haven’t come across these ideas before, in addition to reading Laura’s very nice post you might also like to have a look at some resources on NRICH:
The Cantor Set (which includes the opportunity to listen to the Cantor set!)
How Long is the Cantor Set?
and
How Many Elements Are There in the Cantor Set? (which also includes an interactivity).
2. Xiaoning Wu Says:
Hi Laura! It’s nice to see your work – very interesting post! I’m working on an environmental project in Plymouth at the moment and would like to just say Hi! Drop me an email if you come across this comment somehow. Enjoy your Maths at Cambridge and hoping to see you sometime! x
3. Troels Says:
very nice work but it does not conflict with the idea of a measure.
Borel:
Every set whose measure is not zero is uncountable.
But yes maybe it can seem suprising in the old days before Borel.
\Troels
4. Franciscus Rebro Says:
It doesn’t conflict with Borel’s result (no one said it does), but it is probably the easiest to understand example of an uncountable set with measure 0. Matter of fact, I don’t know of a different example…
5. ehnoj_21 Says:
Reblogged this on The Coolest Anti-Stereotypic, Nerdy, Geeky Statistics Guy and commented:
And because of the take home portion of an exam… I luckily found this nerdy but awesome blog
6. Daniel Says:
gr8t job laura keep it up
7. Abdul Haq Khan Says:
very nice way to explain that how the cantor set is uncountable. other things are also well. so keep its up
8. Sani Kassim Says:
Your article is very helpfull but I still not much clear about comparing the closed interval from 0 to 1 and the set of real numbers as you mentioned!
9. mike Says:
what is the measure of a cantor middle fifth 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": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 117, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9047250151634216, "perplexity": 337.1311456126292}, "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/1544376825728.30/warc/CC-MAIN-20181214114739-20181214140239-00319.warc.gz"} |
http://www.math.kit.edu/ianm3/edu/numahighlyoscil2018s/ | Research Group 3: Scientific Computing
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# Numerical analysis of highly-oscillatory problems (Summer Semester 2018)
Lecturer: JProf. Dr. Katharina Schratz Lecture (0161300), Problem class (0161310) 2+2
In this lecture, we construct and analyze efficient numerical integration schemes for highly oscillatory Klein-Gordon type and related equations. The understanding of the theoretical basis of these methods from the lecture shall be deepened in the accompanying problem class by theoretical exercises and as well practical implementation of simple examples with the software MATLAB (or Python).
Please register here for the participation in the problem class. Additional material to the lecture/problem class will be provided there.
As a student of KIT you can get a free license key for MATLAB at the KIT software shop. Then you can download the software from mathworks.com after registration.
Python is a free software with a syntax similar to the one of MATLAB.
Lecture: Problem class: Thursday 9:45-11:15 SR 3.69 Friday 11:30-13:00 SR 2.58
Lecturer JProf. Dr. Katharina Schratz Office hours: By Appointment Room Kollegiengebäude Mathematik (20.30) Email: Dr. Patrick Krämer Office hours: by appointment Room 3.025 Kollegiengebäude Mathematik (20.30) Email: [email protected]
# Exercises
Please register here for the participation in the problem class. Additional material to the lecture/problem class will be provided there.
Exercise Sheet 1 Program Files uploaded to the internal area
Exercise Sheet 2 Program Files uploaded to the internal area
Exercise Sheet 3 Program Files (and auxiliary material) uploaded to the internal area
Exercise Sheet 4 Auxiliary material uploaded to the internal area
Exercise Sheet 5
# Motivation
The nature of highly oscillatory oscillatory problems can be explained with a simple example, the harmonic oscillator. Initially, a mass attached to a spring with stiffness is deflected units away from its equilibrium. The moment we release the mass, the system starts to oscillate. The rapidness of the oscillations is thereby strictly determined by the value of , i.e. the larger the faster are the oscillations. The position of the mass (with respect to the equilibrium) at time is then determined via the solution of the second order ODE (here with an initial deflection of 1 unit in downward direction and with initial velocity 0)
Harmonic oscillator with
• a nonstiff spring with (green, left)
• an intermediate spring with (yellow, middle) and
• a stiff spring with (red, right).
# Content of the Lecture
The Klein-Gordon equation is a fundamental equation in physics describing the motion of a spin-less particle and is the model problem in our lecture. Its efficient and robust numerical time integration in the nonrelativistic limit regime, i.e. when the speed of light formally tends to infinity, has been subject to current research in the recent years. In this regime, the highly oscillatory nature of the solution is numerically very delicate since standard time integration schemes require severe restrictions on the time step depending on the small parameter which leads to high computational costs.
In order to overcome this numerical challenge, we construct and analyze two types of numerical integrators to efficiently integrate the Klein-Gordon equation in the highly oscillatory
nonrelativistic limit regime as well as in slowly oscillatory relativistic regime:
• The first type of method relies on the asymptotic behaviour of the solution as which allows us to break down the numerical task to solving a non-oscillatory Schrödinger limit system with a standard Strang splitting scheme. We thus obtain a scheme which satifies error bounds of order without any time step restriction. Despite that this scheme provides brilliant results at very low cost in the highly oscillatory regime , the approximation to the exact solution in the intermediate and slowly oscillatory relativist regimes is limited through the error term of order . This motivates us to construct a second type of integrator which allows uniformly accurate in error bounds.
• The second type of method is based on the idea of exponential integrators applied to so-called "twisted variables". These "twisted variables" satisfy a first order in time differential equation with a bounded (with respect to ) right hand side. Thereby, starting from Duhamel's formula (variation of constants formula) for the latter equation, we integrate the linear terms exactly and approximate the "nice" nonlinearity via a truncated Taylor series expansion. The resulting scheme allows first order in time error bounds in independent of the parameter . We thus call this scheme uniformly accurate in , first order in time. | {"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": 0.8180856108665466, "perplexity": 1527.9881002235475}, "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/1581875145941.55/warc/CC-MAIN-20200224102135-20200224132135-00350.warc.gz"} |
https://www.revyuh.com/science-and-research/scientific-research/ripples-ancient-star-helped-limit-change-gravitational-constant/ | # Ripples of an ancient star helped limit the change in the gravitational constant
Astronomers managed to impose restrictions on the change in the gravitational constant over billions of years of the existence of the universe. To do this, they evaluated the effect of this change on the evolution and vibrational properties of one of the oldest known stars in the Universe – KIC 7970740, whose age is approximately 11 billion years. It was not possible to identify a significant change, scientists write.
It is generally believed that the so-called fundamental physical constants are the same at all points in space and at any given time. However, scientists are building theories in which a gradual change in these quantities is allowed. Sometimes this change is very small, which does not allow us to fix it directly, and in other models, these changes are responsible for some observed effects, which are usually interpreted differently.
Such fundamental constants can include the speed of light in vacuum, the elementary charge, the mass of the electron and proton, the Planck constant, and the gravitational constant. It is difficult to directly measure some quantities with high accuracy, therefore, the drift of their combinations is sought, for example, a fine structure constant that connects the electron charge, the electric constant, the speed of light and the Planck constant. Relations between spectral lines, which can be measured with high accuracy, depending on this value.
The modern theory of gravity – the general theory of relativity by Albert Einstein – is based on the principle of equivalence, which indicates the independence of the outcome of a local experiment in a freely falling reference frame from a position in space-time. In other words, in the framework of GR, the gravitational constant G is assumed to be unchanged by definition. At the same time, from the point of view of more complex concepts, such as string theory and some types of modified gravitations, this quantity is not a fundamental constant and can change over time.
Drift G is experimentally sought in several ways. The most accurate method at the moment is the laser location of the moon, which allows you to very accurately calculate the distance to the nearest cosmic body thanks to the reflectors placed there. The current restriction on the variation of this constant, defined as the ratio of its time derivative to the value Ġ / G itself, is (7.1 ± 7.6) × 10 −14 year −1. Within the error limits, this value does not differ from zero, therefore, it is said that there is no significant trend.
Danish astrophysicists led by Earl Patrick Bellinger of Aarhus University have presented a new estimate of the temporal drift G based on observation and modeling of the ancient low-metallic star KIC 7970740. The authors’ idea is to simulate the pulsations of the star based on data obtained from three-year observations using the Kepler Space Telescope. The properties of such oscillations are associated with the parameters of the star and allow us to calculate its global characteristics. If we also assume that the existing theory of stellar evolution is sufficiently accurate, then we can establish restrictions on the age and history of the star by comparing model results and observations.
The authors obtained an estimate of Ġ / G = (2.1 ± 2.9) × 10 −12 year −1, which is also compatible with the absence of change. This estimate is somewhat worse than the most accurate ones already existing, but differs from them in an important aspect: it limits the change in the constant over billions of years, while the laser location of the moon is carried out only for several decades.
The gravitational constant affects several parameters of the star. In particular, the total luminosity is proportional to G to the 7th degree, therefore, a constant change in time directly affects the rate of evolution of the star. Also, changes in gravity will cause displacements of the frequencies of stellar pulsations and their properties.
Astronomers write that this method can be developed by applying it to ensembles of stars for which accurate observational data are available. This will make it possible to establish even stronger restrictions on the drift of G. Also, in a similar way, you can try to find a change in other fundamental constants that affect the properties of the stars.
The most accurate value of the gravitational constant was obtained by physicists last year, and the fine structure constant and the mass ratio of the proton and electron, according to experimental data, have not changed over the past three billion years. There were attempts to explain the possible stability of beryllium-8 in the early Universe with the help of constant drift, but they were unsuccessful.
Vie | arXiv
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Jiya Saini is a Journalist and Writer at Revyuh.com. She has been working with us since January 2018. After studying at Jamia Millia University, she is fascinated by smart lifestyle and smart living. She covers technology, games, sports and smart living, as well as good experience in press relations. She is also a freelance trainer for macOS and iOS, and In the past, she has worked with various online news magazines in India and Singapore. Email: jiya (at) revyuh (dot) com | {"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": 0.8615471720695496, "perplexity": 363.88279136325946}, "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-2019-51/segments/1575540544696.93/warc/CC-MAIN-20191212153724-20191212181724-00530.warc.gz"} |
https://chemistry.stackexchange.com/questions/73760/how-to-derive-the-gibbs-helmholtz-equation | # How to derive the Gibbs-Helmholtz equation? [closed]
Can someone please explain to me how to derive the Gibbs-Helmholtz relationship from $G = H - TS$?
• en.wikipedia.org/wiki/… May 2, 2017 at 10:37
• @orthocresol can you please outline the chain rule step in the derivation? This is the part I don't understand: May 2, 2017 at 10:42
• It's more of the product rule rather than chain rule. It may be more familiar to you to look at it this way: you should know that $\mathrm{d}(fg)/\mathrm{d}x = f(\mathrm{d}g/\mathrm{d}x) + g(\mathrm{d}f/\mathrm{d}x)$. Now let $x = T$, $f = G$, $g = 1/T$. May 2, 2017 at 10:43
You need to know that
$$\require{begingroup} \begingroup \newcommand{\d}[0]{\mathrm{d}} \d G = V\,\d p - S\,\d T$$
from which you can determine that
$$\newcommand{\pdiff}[3]{\left(\frac{\partial #1}{\partial #2}\right)_{\!#3}} \pdiff{G}{p}{T} = V \qquad \pdiff{G}{T}{p} = -S$$
Therefore (by the quotient rule)
\begin{align}\pdiff{}{T}{p}\left(\frac{G}{T}\right) &= \frac{T(\partial G/\partial T)_p - G(\partial T/\partial T)_p} {T^2} \\[8pt] &= \frac{T(-S) - G(1)}{T^2} \\[8pt] &= \frac{-TS-G}{T^2} \\[8pt] &= -\frac{H}{T^2} \end{align}
as desired (since $G = H - TS$).
• Why it also holds that $$\Delta \left(\frac{\partial{G/T}}{\partial{T}}\right)_p=-\frac{\Delta H}{T^2}$$ if initial and final temperatures are not the same? Jun 17, 2020 at 17:38
$$dG=dH-TdS-SdT=dU+VdP+PdV-TdS-SdT$$ But, $$dU=TdS-PdV$$ Adding the above two equations together, we get $$dG=VdP-SdT$$ At constant pressure, we have $$dG=-SdT$$But, from the definition of G, $$-S=\frac{G-H}{T}$$Substituting for -S yields: $$\frac{dG}{dT}=\frac{G-H}{T}\tag{constant P}$$ The rest is strightforward math. | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.996443510055542, "perplexity": 1009.2445925118093}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337625.5/warc/CC-MAIN-20221005105356-20221005135356-00604.warc.gz"} |
http://mathhelpforum.com/pre-calculus/90502-converting-rectangular-equations-into-polar-form.html | # Math Help - converting rectangular equations into polar form
1. ## converting rectangular equations into polar form
rectangular equation : x^2 - y^2 = 4
use the double angle forumla
i have no idea what to do?
2. ## Converting to a polar equation
Hello tomatoes
Originally Posted by tomatoes
rectangular equation : x^2 - y^2 = 4
use the double angle forumla
i have no idea what to do?
Using $x = r\cos\theta, y = r\sin\theta$, we get:
$r^2\cos^2\theta - r^2\sin^2\theta = 4$
$\Rightarrow r^2(\cos^2\theta - \sin^2\theta) = 4$
But, using the double-angle formula, $\cos^2\theta - \sin^2\theta = \cos2\theta$, this becomes
$r^2\cos2\theta = 4$
or $r^2 = 4\sec2\theta$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9845278859138489, "perplexity": 3485.6297750094104}, "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/1405997893859.88/warc/CC-MAIN-20140722025813-00061-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://calciii.wordpress.com/2009/05/14/make-it-new/ | ## Make It New
Last year’s posts about Arc Length On The Cardiod, Vector Products, and integral-secant-cubed are relevant again. The cardiod was a recent in-class example; the “vector” note corrects a mistake I didn’t make until yesterday; the “surprisingly useful integral” is one I had students to look up in a table on the recent exam (and work with… nobody had this one all the way right but me though in principle it’s an “easy” problem when the table is available…).
What the heck. Here’s the problem: The polar equation $r = \theta$, for $\theta \in [0, \infty)$, determines an “Archimedean Spiral”, S. Determine the arc length along S for $\theta$ between 0 and $\pi\over6$ (exactly; a messy “formula”… check the work numerically). And the answer: ${{\pi\sqrt{36+ \pi^2}}\over{72}} + {1\over2} ln({{\pi +\sqrt{36+\pi^2}}\over6})$. This evaluates to about .5466… as does $\int_0^{\pi\over6} \sqrt{\theta^2 + 1}d\theta$… so we can be reasonably sure this is right.
In my own work messy expressions like this are almost never right the first time… on the copies of the handwritten solution I distributed yesterday, quite a bit of erased work is clearly visible. Rooting out every last little mistake until things are just right is of course a vital part of the process for a lot of problems. One student was real close. | {"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": 6, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9134814143180847, "perplexity": 1340.2412906856725}, "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-2017-26/segments/1498128321961.50/warc/CC-MAIN-20170627235941-20170628015941-00415.warc.gz"} |
https://arxiv.org/abs/1702.01722 | astro-ph.CO
(what is this?)
Title: The Halo Boundary of Galaxy Clusters in the SDSS
Abstract: Mass around dark matter halos can be divided into "infalling" material and "collapsed" material that has passed through at least one pericenter. Analytical models and simulations predict a rapid drop in the halo density profile associated with the transition between these two regimes. Using data from SDSS, we explore the evidence for such a feature in the density profiles of galaxy clusters and investigate the connection between this feature and a possible phase space boundary. We first estimate the steepening of the outer galaxy density profile around clusters: the profiles show an abrupt steepening, providing evidence for truncation of the halo profile. Next, we measure the galaxy density profile around clusters using two sets of galaxies selected based on color. We find evidence of an abrupt change in the galaxy colors that coincides with the location of the steepening of the density profile. Since galaxies are likely to be quenched of star formation and turn red inside of clusters, this change in the galaxy color distribution can be interpreted as the transition from an infalling regime to a collapsed regime. We also measure this transition using a model comparison approach which has been used recently in studies of the "splashback" phenomenon, but find that this approach is not a robust way to quantify the significance of detecting a splashback-like feature. Finally, we perform measurements using an independent cluster catalog to test for potential systematic errors associated with cluster selection. We identify several avenues for future work: improved understanding of the small-scale galaxy profile, lensing measurements, identification of proxies for the halo accretion rate, and other tests. With upcoming data from the DES, KiDS and HSC surveys, we can expect significant improvements in the study of halo boundaries.
Comments: 17 pages, 8 figures Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); Astrophysics of Galaxies (astro-ph.GA) Cite as: arXiv:1702.01722 [astro-ph.CO] (or arXiv:1702.01722v2 [astro-ph.CO] for this version)
Submission history
From: Eric Baxter [view email]
[v1] Mon, 6 Feb 2017 17:48:39 GMT (1953kb,D)
[v2] Tue, 7 Feb 2017 16:53:30 GMT (1954kb,D) | {"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": 0.8441733121871948, "perplexity": 1637.7393611895964}, "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-2017-17/segments/1492917119120.22/warc/CC-MAIN-20170423031159-00380-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2451978/find-sum-of-lengths-of-intervals-satisfying-cot2x-tan2x-ge-frac2 | # Find sum of lengths of intervals satisfying $|\cot(2x) - \tan(2x)| \ge \frac{2}{3}\sqrt{3}$ for $0 \le x \le 2\pi$.
Find sum of lengths of intervals satisfying $|\cot(2x) - \tan(2x)| \ge \frac{2}{3}\sqrt{3}$ for $0 \le x \le 2\pi$. WolframAlpha shows that the total length of intervals satisfying this equation is $\frac{4}{3} \pi$. I've tried to solve it using $\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$, but it lead me to solving complicated rational functions inequalities which would be quite exhausting to solve. Do you know any way to solve this problem faster?
To solve the equality case, don't use the double angle formula. Instead, solve for $2x$ directly (or, if you prefer, substitute $y=2x$ and solve for $y$, and use that to solve for $x$.)
$$\cot y-\tan y=2\cot2y=\dfrac2{\tan2y}$$
So, we need $-\sqrt3\le\tan4x\le\sqrt3$
$\implies m\pi-\dfrac\pi3\le4x\le m\pi+\dfrac\pi3$ where $m$ is any integer | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785780310630798, "perplexity": 115.42535207451348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704843561.95/warc/CC-MAIN-20210128102756-20210128132756-00751.warc.gz"} |
http://mathoverflow.net/questions/98638/expanding-the-sqare-of-sum | # expanding the sqare of sum
If there any way to expand the following?
$$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}}$$
and more generally, a way to expand
$$\left(\sum_{i=1}^nx_i\right)^{\frac{p}{q}}$$
where $gcd(p,q) = 1$
More closed to my original problem, is there any formula for: $$\left(\sum_{i=1}^nx_i\right)^{\frac{1}{2}} - \sum_{i=1}^nx_i^{\frac{1}{2}}$$
-
What do you mean by "formula"? – Igor Rivin Jun 2 '12 at 19:30
This really does not see to be of the appropriate level. Voting to close. I am sure stackexchange would be more receptive. – Igor Rivin Jun 2 '12 at 19:46
I think, you can use binomial theorem for rational exponents inductively for expanding $(\sum_{i=1}^{n}x_i)^\frac{1}{2}$. I am not very sure whether one can use multinomial theorem for rational exponents.
When you expand $(1+x)^{\frac{1}{2}}$, It has infinitely many terms, so to your other question, It is very less likely to have any formula, as one of the term in the expansion has infinitely many terms while $\sum_{i=1}^{n}x_i^{\frac{1}{2}}$ has only finitely many.
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http://www.zora.uzh.ch/122044/ | # Production and characterization of $^{228}$Th calibration sources with low neutron emission for GERDA
Baudis, L; Benato, G; Carconi, P; Cattadori, C; De Felice, P; Eberhardt, K; Eichler, R; Petrucci, A; Tarka, M; Walter, M (2015). Production and characterization of $^{228}$Th calibration sources with low neutron emission for GERDA. Journal of Instrumentation, 10(12):P12005.
## Abstract
The GERDA experiment at the Laboratori Nazionali del Gran Sasso (LNGS) searches for the neutrinoless double beta decay of (76)Ge. In view of the GERDA Phase II data collection, four new (228)Th radioactive sources for the calibration of the germanium detectors enriched in (76)Ge have been produced with a new technique, leading to a reduced neutron emission rate from (α, n) reactions. The gamma activities of the sources were determined with a total uncertainty of ~4% using an ultra-low background HPGe detector operated underground at LNGS. The neutron emission rate was determined using a low background LiI(Eu) detector and a (3)He counter at LNGS. In both cases, the measured neutron activity is ~10(−)(6) n/(s⋅Bq), with a reduction of about one order of magnitude with respect to commercially available (228)Th sources. Additionally, a specific leak test with a sensitivity to leaks down to ~10 mBq was developed to investigate the tightness of the stainless steel capsules housing the sources after their use in cryogenic environment.
## Abstract
The GERDA experiment at the Laboratori Nazionali del Gran Sasso (LNGS) searches for the neutrinoless double beta decay of (76)Ge. In view of the GERDA Phase II data collection, four new (228)Th radioactive sources for the calibration of the germanium detectors enriched in (76)Ge have been produced with a new technique, leading to a reduced neutron emission rate from (α, n) reactions. The gamma activities of the sources were determined with a total uncertainty of ~4% using an ultra-low background HPGe detector operated underground at LNGS. The neutron emission rate was determined using a low background LiI(Eu) detector and a (3)He counter at LNGS. In both cases, the measured neutron activity is ~10(−)(6) n/(s⋅Bq), with a reduction of about one order of magnitude with respect to commercially available (228)Th sources. Additionally, a specific leak test with a sensitivity to leaks down to ~10 mBq was developed to investigate the tightness of the stainless steel capsules housing the sources after their use in cryogenic environment.
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Detailed statistics
Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English 2015 18 Feb 2016 10:05 10 Dec 2016 01:00 IOP Publishing 1748-0221 https://doi.org/10.1088/1748-0221/10/12/P12005 arXiv:1508.05731v1 | {"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": 0.9363649487495422, "perplexity": 4828.508524889172}, "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/1487501171608.86/warc/CC-MAIN-20170219104611-00055-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.math-only-math.com/percentage-into-decimal.html | # Percentage into Decimal
We know that percentage represents parts per hundred. So, any number written as percent is a fraction with 100 as its denominator. We write the number as the numerator of a fraction and 100 in the denominator. The symbol % is not written in the fraction form. So, 75% is written as $$\frac{75}{100}$$ = 0.75.
How to convert a given percentage into decimal?
We will follow the following steps for converting a percentage into a decimal:
Step I: Obtain the percentage which is to be converted into decimal
Step II: Remove the percentage sign (%) and divide it by 100.
Step III: Express the fraction in the decimal form
Remember: Remove % sign and move the decimal two places to the left.
1. Express each of the following percentage as decimals:
(i) 23 % = 23/100 = 0.23
(ii) 16 % = 16/100 = 0.16
(iii) 47 % = 47/100 = 0.47
(iv) 1 % = 1/100 = 0.01
(v) 25 % = 25/100 = 0.25
Note: Decimal shifts by 2 places to the left.
2. Express each of the following decimal percentage as decimals:
(i) 1.75 % = 1.75/100 = (1.75 × 100)/(100 × 100) = 175/10000 = 0.0175
(ii) 7.5 % = 7.5/100 = (7.5 × 10)/(100 × 10) = 75/1000 = 0.075
(iii) 31.5 % = 31.5/100 = (31.5 × 10)/(100 × 10) = 315/1000 = 0.315
(iv) 0.18 % = 0.18/100 = (0.18 × 100)/(100 × 100) = 18/10000 = 0.0018
(v) 231.2 % = 231.2/100 = (231.2 × 10)/(100 × 10) = 2312/1000 = 2.312
3. Express the percentage as fraction and in decimal form.
4. Express each of the following fraction percentage as decimals:
(i) 4/5 % = 0.8 % = 0.8/100 = (0.8 × 10)/(100 × 10) = 8/1000 = 0.008
(ii) 9/20 % = 0.45 % = 0.45/100 = (0.45 × 100)/(100 × 100) = 45/10000 = 0.0045
(iii) 1/80 % = 0.0125 % = 0.0125/100 = (0.0125 × 10000)/(100 × 10000) = 125/1000000 = 0.000125
(iv) 13/50 % = 0.26 % = 0.26/100 = (0.26 × 100)/(100 × 100) = 26/10000 = 0.0026
(v) 44/5 % = 8.8 % = 8.8/100 = (8.8 × 10)/(100 × 10) = 88/1000 = 0.088
Questions and Answers on Percentage into Decimal:
I. Convert the following percentages to decimals:
(i) 5%
(ii) 13%
(iii) 20%
(iv) 75%
(v) 44%
(vi) 96%
(i) 0.05
(ii) 0.13
(iii) 0.2
(iv) 0.75
(v) 0.44
(vi) 0.96
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Application of Percentage | {"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": 0.851111114025116, "perplexity": 2695.5136195631144}, "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-2021-31/segments/1627046153897.89/warc/CC-MAIN-20210729203133-20210729233133-00227.warc.gz"} |
https://www.physicsforums.com/threads/population-growth-model.619397/ | # Population Growth Model
1. Jul 8, 2012
### chaoseverlasting
1. The problem statement, all variables and given/known data
I'm trying to model how a population would grow wrt the following constraints:
1. Number of people within the population
2. How many people each person of the population informs
3. Probability of each person of the population informing a member outside the population.
4. There is an upper limit to the population growth, but this is not the most important constraint as of now.
It's almost like a virus growth model and I have some idea of where to begin, but it's been a while since I've solved these problems.
2. Relevant equations
Because of the first constraint, this looks like an exponential growth model to begin with. Thus,
$$\frac{dN}{dT}=kN$$
is the basic equation to be used.
3. The attempt at a solution
From the second constraint onwards, I'm having a problem forming the DE.
The growth is dependent on the probability of each person within the population telling others about it and the number of people each person will tell.
Thus, if $$\alpha (t)$$ is the probability of a person within the population telling others
and $$m(t)$$ are the number of people each person tells, then I think the model would look like this:
$$\frac{dN}{dT}=\alpha (t)m(t)N$$
Is this right? Again, it's been a while since I've done this stuff, so I'm not sure, but I think I'm on the right track.
For the final constraint, if I have to put an upper limit on the model, I think I can do a (k-N) growth model where k is the upper limit.
2. Jul 8, 2012
### chaoseverlasting
I believe that the model I'm trying to derive is akin to the Logistic Growth Model. However, I cannot understand the derivation of the same. Could somebody guide me through the process?
From what I understand, as there is an upper limit on the population here (k), the rate of growth is directly proportional to (k-N). However, the model is also proportional to the current population, N.
I do not understand why we multiply the two and not add them. Thus, why is the growth proportional to N(k-N) ?
Furthermore, the logistic growth model gives the following equation:
$$\frac{dN}{dt}=\frac{r}{k} N(k-N)$$
Why do we divide throughout by k? Also, what is r?
3. Jul 12, 2012
### chaoseverlasting
Can someone help me out with this? This is what I have so far:
Since the rate of growth is directly proportional to the current population, N and there is a cap on the total possible population K.
As each person reaches out to m other persons, the probability of telling someone who is not a part of the current population is given by $$\alpha k C m(1-\frac{N}{K})$$
Thus, if the acceptance rate is R, then the rate of growth should be:
$$\frac{dN}{dt}=R\alpha kCm(1-\frac{N}{k})N$$
Is this right?
Similar Discussions: Population Growth Model | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.981687605381012, "perplexity": 295.4686841276624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812579.21/warc/CC-MAIN-20180219091902-20180219111902-00193.warc.gz"} |
https://www.physicsforums.com/threads/proving-that-a-linear-space-is-infinite-dimensional.465432/ | # Proving that a linear space is infinite-dimensional
• Thread starter scotto3394
• Start date
• #1
6
0
## Homework Statement
Let $$V$$ denote the linear space of all real functions continuous on the interval $$[-\pi,\pi]$$. Let $$S$$ be that subset of $$V$$ consisting of all $$f$$ satisfying the three equations $$$\int_{-\pi}^{\pi} f(t) dt = 0, ~~~\int_{-\pi}^{\pi} f(t) cos(t) dt = 0, ~~~ \int_{-\pi}^{\pi} f(t)sin(t) dt = 0$$$
(c) Prove that $$S$$ is infinite-dimensional.
## Homework Equations
From parts (a) and (b) of the question we know that $$S$$ is a subspace of $$V$$ and that $$S$$ contains the functions $$f(x) = cos (nx), ~~ f(x) = sin(nx)$$ for $$n=2,3,...$$. Also I'm not sure whether it's relevant, but the sections before this group of exercises covered null spaces, rank, and linear transformations. So if it is relevant I would imagine that dim N + dim T = dim V would be useful where N is the null space, T is the range, V is the domain (which is a linear space), and dim is dimension.
## The Attempt at a Solution
My idea is that you would try a proof by contradiction where you would assume that $$S$$ is finite-dimensional. From there I had a few random ideas, such as trying to show that the null space is infinite-dimensional and then use the formula above to show that $$S$$ would somehow be infinite-dimensional. Also I was thinking of assuming that there are $$n$$ independent elements and somehow showing that they are not actually independent using part b or by applying the fact that they are in $$S$$, though I'm not really sure how that would work.
Really I'm not looking for an answer, so much as helpful hints. I'm actually working through this book on my own so I don't really have a teacher to ask questions to. Also, I hope that I provided enough relevant data. Thank you for your time.
## Answers and Replies
• #2
CompuChip
Homework Helper
4,302
47
You already mention infinitely many functions that you know are in S.
So, correct me if I'm wrong, but if you show directly that they are all independent then you are done, right?
• #3
6
0
If I could show that it would work, but the functions that I know are there (sine and cosine) aren't actually independent I think. Let's say within the elements I chose $$sin(2x)$$ and $$sin(4x)$$. $$sin(4x) = (2cos(2x))sin(2x)$$ which is a multiple of $$sin(2x)$$ making it dependent. Then again, I might be wrong...
I'll definitely give it a try though.
• #4
CompuChip
Homework Helper
4,302
47
My definition of independent is that
$$\langle f, g \rangle = 0$$
where the inner product for function spaces is usually
$$\langle f, g \rangle := \int_{-\pi}^\pi f(x) g(x) \, dx$$
But whichever definition you use, I am pretty sure that, sin(px) and sin(qx) are independent in either of them, for p and q (distinct) primes.
• #5
Dick
Homework Helper
26,260
619
My definition of independent is that
$$\langle f, g \rangle = 0$$
where the inner product for function spaces is usually
$$\langle f, g \rangle := \int_{-\pi}^\pi f(x) g(x) \, dx$$
But whichever definition you use, I am pretty sure that, sin(px) and sin(qx) are independent in either of them, for p and q (distinct) primes.
That's not a definition of independent. It's a definition of orthogonal. Independent means c1*v1+...+cn*vn=0 implies c1=...=cn=0. You shouldn't have any trouble showing that a mutually orthogonal set of vectors is also independent. sin(px) and sin(qx) being orthogonal doesn't have anything to do with p and q being prime. p and q just need to be distinct positive integers. Show that.
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https://www.physicsforums.com/threads/tension-problem-finding-acceleration-with-kinetic-friction.786679/ | # Tension problem, finding acceleration with kinetic friction
Tags:
1. Dec 9, 2014
### p0ps1c1e
Three blocks are connected as shown in the uploaded picture. The strings and friction-less pulleys have negligible masses, and the coefficient of kinetic friction between the 2.0 kg block and the table is 0.17. What is the acceleration of the 2.0 kg block?
2. Relevant equations
∑F = ma
3. The attempt at a solution
So i wrote 3 newton equations:
k-T1+T2 = m2a
T1-m1g = m1a
T2-m3g = m3a
Putting them together I got:
km2g - m1g + m3g = a(m2 + m1 - m3)
and then tried to solve for acceleration. Problem is every time I do it I end up with a 0 in the denominator so I don't know if I'm doing something wrong conceptually or if I am just messing up in my algebra. I'm also not sure if all three of the blocks have the same acceleration. I would think so since they are all on a string.
#### Attached Files:
• ###### PhysicsDrawing1.png
File size:
23.3 KB
Views:
152
Last edited: Dec 9, 2014
2. Dec 9, 2014
### Simon Bridge
Your equation do not match your drawing... but I think I can figure it out.
m1=1kg, m2=2kg and m3=3kg right?
The first equation is incorrect - what is the equation governing friction?
The last equation is inconsistent with the first two - check the direction for positive acceleration.
3. Dec 9, 2014
### p0ps1c1e
Oh derp.... haha. Thanks.
I'm looking through my old exams and I'm finding so many little mistakes like that.
Draft saved Draft deleted
Similar Discussions: Tension problem, finding acceleration with kinetic friction | {"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": 0.8792548775672913, "perplexity": 1438.0957016221755}, "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/1508187825227.80/warc/CC-MAIN-20171022113105-20171022133105-00571.warc.gz"} |
https://brilliant.org/problems/all-russian-olympiad-problem-5/ | # All-Russian Olympiad Problem 5
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).
###### This problem is part of this set.
×
Problem Loading...
Note Loading...
Set Loading... | {"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": 0.9881408214569092, "perplexity": 1657.5699242895143}, "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-2018-05/segments/1516084887414.4/warc/CC-MAIN-20180118131245-20180118151245-00793.warc.gz"} |
http://mathhelpforum.com/geometry/15767-geometry-marathon-2.html | # Math Help - Geometry Marathon
1. If your proof doesn't use trig. then it's okay.
2. Originally Posted by blurgh
i've figured out the solution but the proof doesn't use similar triangles.
3. ## my proof
correction, my proof does use similar triangles. i found a mistake in my proof and had to fix it. (hehe, oops)
um, i haven't got the hang of Latex, so i hope you don't mind if i give you a link to a pdf file instead. (you'll have to download it, sorry!)
although i think and am more fluent in English, i always solve math problems in Vietnamese, so my proof is basically a literal translation of what i would've written in Vietnamese, so I'm sorry if it doesn't make sense in some parts. All symbols are what the Vietnamese use in our math, if there is anything different from what most people would write please tell me.
my proof
4. impressive proof!
5. Nice proof.
You can shorten it using similar triangles.
In your sketch we have that $\overline{AD}=\overline{CD}=\overline{ND}$, let's denote this segments "k". On the other hand we have that $\measuredangle~CBN=\measuredangle~BCN=45^\circ$.
Nevertheless, $\triangle{ACB}\sim\triangle{BCD}\implies\overline{ BC}=k\sqrt2$. Finally, $\overline{BN}=\overline{CN}=k\implies\triangle{CDN }~\text{is equilateral}$, then $\measuredangle~NDC=2x=60^\circ~\therefore~x=30^\ci rc$ and the proof is completed.
6. sweet! i like your approach!
ok here's one of my favourite geometry problems. well actually i've changed it a bit, now it's a bit harder! if someone wants a hint i'll give them the original version... maybe.
Let there be a rectangle that has a length that is twice the size of it's width, as shown in the diagram below.
Prove x=60 degrees.
7. Has anyone tried to solve this?
I have tried many ideas, but failed.
8. Prove that $\overline{AB}=\overline{AC}$
9. Originally Posted by Krizalid
Prove that $\overline{AB}=\overline{AC}$
I had good idea, and tried many variants of it, but like I said I failed.
If anyone wants I will post it, maybe someone will get new idea.
10. I found another proof:
• Take a point $F$ on $\overline{CD}$ such that $\measuredangle~CBF=60^\circ\implies\overline{BF}=2 m~\therefore~\overline{AB}=\overline{BF}~\blacktri angleleft$
• From $\blacktriangleleft$ we have that $\measuredangle~FAB=75^\circ\implies\triangle{ADF}\ cong\triangle{BCE}~\therefore~\overline{AF}=\overl ine{BE}$
• Nevertheless, $\overline{AB}\parallel\overline{EF}\implies\text{q uadrilateral }ABEF~\text{isosceles trapezium}$ (and cyclic).
• Finally $x=\measuredangle~EAD=\measuredangle~FAD+\measureda ngle~EAF=15^\circ+45^\circ=60^\circ$ and the proof is completed $\blacksquare$
11. Nice proof.
12. very nice proof! i like it alot! my proof involved creating an equadrilateral triangle with one side being BE and overlaps triangle ABE. once again, very sweet approach.
well, knock us out with another question. *shudders* i'm scared of what your hard questions are like... scared and intrigued.
13. Originally Posted by blurgh
correction, my proof does use similar triangles. i found a mistake in my proof and had to fix it. (hehe, oops)
um, i haven't got the hang of Latex, so i hope you don't mind if i give you a link to a pdf file instead. (you'll have to download it, sorry!)
although i think and am more fluent in English, i always solve math problems in Vietnamese, so my proof is basically a literal translation of what i would've written in Vietnamese, so I'm sorry if it doesn't make sense in some parts. All symbols are what the Vietnamese use in our math, if there is anything different from what most people would write please tell me.
my proof
Wow, how were you able to make it so neat? A special program?
14. Originally Posted by blurgh
very nice proof! i like it alot! my proof involved creating an equadrilateral triangle with one side being BE and overlaps triangle ABE. once again, very sweet approach.
well, knock us out with another question. *shudders* i'm scared of what your hard questions are like... scared and intrigued. | {"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": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8312320709228516, "perplexity": 811.1777451371139}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042981856.5/warc/CC-MAIN-20150728002301-00198-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/pram/A_S_Joshipura | • A S Joshipura
Articles written in Pramana – Journal of Physics
• Exact potential minimization for a supergravity model
The low energy effective scalar potential arising from the supergravity model proposed by Nilles, Srednicki and Wyler is minimized exactly. Bounds are derived for the parameters of the theory from the requirement that SU(2) × U(1) be broken at the tree level. These results support earlier approximate results.
• Interpretation of the recent Kolar events
We give plausible interpretations of the unusual events seen in the proton decay detector at Kolar Gold Fields indicating the existence of a massive (≳2GeV) long lived (10−8−10−9s) particle. We show that it is possible to accommodate the particle in the standard model as a fourth generation neutrino, or inE6 grand unified theory as a neutral fermion occurring in27 representation or in supersymmetric theory as a scalar neutrino. However, there is a difficulty in explaining the large production rate for the particle.
• # Pramana – Journal of Physics
Volume 94, 2020
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | {"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": 0.9386330246925354, "perplexity": 2277.363407552438}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251802249.87/warc/CC-MAIN-20200129194333-20200129223333-00434.warc.gz"} |
http://mathhelpforum.com/algebra/14911-fractions.html | # Math Help - Fractions
1. ## Fractions
Yea umm a little help on adding and subtracting rational expressions.
I think I did this right . .
2x-4/x2+8x-20
2(x-2)/(x-2)(x+10)
2/(x-10)
Is this right and If so I kinda need help on this one too
x2+6x/10*4/x2-36
2. ## Re:
Originally Posted by thebest
Yea umm a little help on adding and subtracting rational expressions.
I think I did this right . .
2x-4/x2+8x-20
2(x-2)/(x-2)(x+10)
2/(x-10)
RE:
Attached Thumbnails
3. Originally Posted by thebest
x2+6x/10*4/x2-36
Is this a typo?
4. Originally Posted by thebest
...
Is this right and If so I kinda need help on this one too
x2+6x/10*4/x2-36
Hello,
if I understand the problem correctly it reads:
Code:
x² + 6x 4
-------- * ------- =
10 x² -36
x(x + 6) 4 2x
-------- * ---------- = -------
10 (x+6)(x-6) 5(x-6) | {"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": 0.9580180048942566, "perplexity": 4075.7064275124476}, "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-2016-30/segments/1469257828009.82/warc/CC-MAIN-20160723071028-00017-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://quantiki.org/wiki/entanglement-formation | Entanglement of formation
The entanglement of formation is an entanglement measure for bipartite quantum states.
It is defined as
Ef(ρ) = minEpiEE(∣ψi⟩) where the minimization is over all ensembles of pure states E = {(pi, ∣ψi⟩)} that realizes the given state, ρ = ∑ipiψi⟩⟨ψi∣, and EE(∣ψ⟩) is the entropy of entanglement which is defined for pure states. This kind of extension of a quantity defined on pure states to mixed states is called a convex roof construction.
Entanglement of formation quantifies how many bell states are needed per copy of to prepare many copies of ρ using the following specific LOCC procedure:
• For each copy, select which pure state ∣ϕi⟩ to prepare from a probability distribution qi.
• For each of the different ∣ϕi⟩, prepare the required number of copies from bell states.
• Discard the information about which copy is in which pure state.
Until recently, it was not known if entanglement of formation was equal to the entanglement cost. It was shown in 2008 that the quantities are not equal, and furthermore that entanglement of formation is not additive (see section XIV D). However, the entanglement cost is equal to the regularization of the entanglement of formation,
$$E_c(\rho) = \lim_{n \to \infty} \frac{1}{n} E_f(\rho^{\otimes n}).$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9641013741493225, "perplexity": 595.4645774459235}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526818.17/warc/CC-MAIN-20190721020230-20190721042230-00244.warc.gz"} |
https://research-information.bris.ac.uk/en/publications/strict-inequalities-for-minimal-degrees-of-direct-products | Strict inequalities for minimal degrees of direct products
Neil J Saunders
Research output: Contribution to journalArticle (Academic Journal)
4 Citations (Scopus)
Abstract
The minimal faithful permutation degree μ(G) of a finite group G is the least non-negative integer n such that G embeds in the symmetric group Sym(n). Work of Johnson and Wright in the 1970s established conditions for when μ(H×K)=μ(H)+μ(K), for finite groups H and K. Wright asked whether this is true for all finite groups. A counter-example of degree 15 was provided by the referee and was added as an addendum in Wright’s paper. Here we provide two counter-examples; one of degree 12 and the other of degree 10.
Original language English 23-30 8 Bulletin of the Australian Mathematical Society 79 1 https://doi.org/10.1017/S0004972708000956 Published - Feb 2009
Keywords
• faithful permutation representations; complex reflection groups; monomial reflection groups | {"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": 0.8308616280555725, "perplexity": 1250.0783604508786}, "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-2020-40/segments/1600400210996.32/warc/CC-MAIN-20200923113029-20200923143029-00211.warc.gz"} |
https://www.physicsforums.com/threads/precal-inequalities-involving-absolute-value.509570/ | # Precal, Inequalities involving absolute value
1. Jun 25, 2011
1. The problem statement, all variables and given/known data
DIRECTIONS: Express the intercal in terms of an inequality involving absolute value.
PROBLEM: (-4, 4)
MY STEPS:
1: (-4, 4)
2: -4<x<4
Is that correct? Is step 3 correct? The only reason that I included that part is becuase it says "Involving absolute value," but I am not sure if that is what the question is really asking. Please advise if you can. Thank you in advance.
2. Jun 25, 2011
### tiny-tim
yes, that's fine
(and step 3 is obvious from step 1, you don't need step 2)
3. Jun 25, 2011
lol. thanks tiny tim!
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http://mathhelpforum.com/algebra/100630-radical-equation.html | $n \in {\rm N}$
Solve in R the equation :
$\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {4^n x + 3} } } } - \sqrt x = 1$
2. Originally Posted by dhiab
$n \in {\rm N}$
Solve in R the equation :
$\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt {4^n x + 3} } } } - \sqrt x = 1$
Interesting problem. Assume the problem to be:
$\sqrt {x + \sqrt {4x + \sqrt {16x + \sqrt{ 64x+... +\sqrt {4^n x + 3} } } } } - \sqrt x = 1$
- Put $\sqrt x$ on the RHS, square both sides, cancel out the terms $4^k x, k=0, 1, 2, ..., n$, would finally get $x=4^{-n}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9874210953712463, "perplexity": 1250.4975372733275}, "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/1492917121267.21/warc/CC-MAIN-20170423031201-00212-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://engcourses-uofa.ca/books/vibrations-and-sound/forced-vibrations-of-damped-single-degree-of-freedom-systems/base-excitation-in-a-damped-system/ | ## Forced Vibrations of Damped Single Degree of Freedom Systems: Base Excitation in a Damped System
We can also consider the response of a damped spring–mass system subjected to base excitation. Among other applications, these results are useful when considering the operation of transducers that measure vibrations.
Figure 5.11: Damped spring–mass system subjected to harmonic forcing function
Figure 5.11(a) illustrates such a situation schematically while Figure 5.11(b) shows the corresponding FBD/MAD diagram. Applying Newton’s Laws we find
(5.23)
or since
(5.24)
This can also be expressed as
where
[\
\ensuremath{\mathbb{R}} = a \sqrt{k^2 + \bigl(c \omega\bigr)^2}, \qquad
\alpha = \tan^{-1}{\biggl[\frac{-c \omega}{k}\biggr]}.
\] As a result, we can see that the base excitation is equivalent to applying a harmonic force of magnitude \ensuremath{\mathbb{R}} to a mass with a fixed base. We have previously found the steady state solution for such a case (equations (5.7) and (5.9)) which here becomes
where
Therefore we could write the response as
where
and
However, this in often not the most convenient form for the phase angle. It can be shown (after some algebra) that
which can also be expressed in non-dimensional form as
To summarize, the response of the damped spring–mass system to harmonic base excitation is given by
(5.25)
where
(5.26)
and
(5.27)
In dimensionless form these become
(5.28)
and
(5.27)
Figure 5.12: Response amplitude of damped spring–mass system subjected to harmonic base excitation
The term can be thought of as the dynamic magnification factor in this situation as it represents the ratio of the amplitude of the motion of the mass to the amplitude of motion of the base.
Figure 5.12 illustrates this relationship. We can see that this factor is exactly the same as
the transmissibility for the forced damped situation considered previously (equation (5.19) and
Figure 5.7). Therefore the problem of isolating a mass from the motion of the base is essentially
the same problem as preventing a disturbing force acting on the mass from being transmitted to
the supporting structure (the base).
A precision grinding machine is supported on an isolator that has a stiffness of 1 MN/m and a viscous damping constant of 1 kN·s/m. The floor on which the machine is mounted is subjected to a harmonic disturbing force due to the operation of a unbalanced engine in the vicinity of the grinding machine operating at a speed of 6000 RPM. Determine the maximum acceptable amplitude of the floor motion () if the resulting amplitude of vibration of the grinding wheel is restricted to be 10 −6 m. Assume that the grinding machine and the wheel are a rigid body of weight 5000 N.
### Relative Motion Between the Mass and the Base
In the previous section we considered the response of a damped spring–mass system subjected to harmonic base excitation and found the \emph{absolute} response of the mass. Those results are not always the most convenient to use. For example, in many vibration measurement instruments we usually directly measure only the \emph{relative} motion between the mass and the base and then try to use this measurement to infer the underlying motion of the base (which is typically what we would be trying to measure).
Considering Figure 5.11(a) once again, the relative motion z between the mass and the base
is given by
However, as we have seen previously there is a phase difference between and , given by equation (5.29), which must be taken into account. For example we could take our previous solutions and expand them using trigonometric identities to account for this phase difference. An alternative (and often easier) approach is to consider the relative motion directly.
Since
we can write
The original equation of motion in (5.23)
can then be written as
However, since , so the equation of motion becomes
(5.30)
We note that this equation is identical to that obtained for the forced vibration situation (and particularly the case of a rotating imbalance) which we have already considered. The solution to this equation is therefore
(5.31)
where
(5.32)
and
(5.33)
Equation (5.32) can be rearranged as
(5.34)
which determines the ratio of the amplitude of the \emph{relative motion} to that of the base motion itself. This relationship is shown in Figure 5.13. (This is the same relationship found for the ratio in Equation (5.22) and Figure 5.10). Note again that as this gives us the same results as in the undamped case.
We can apply these results to the use of a variety of vibration measurement instruments.
Figure 5.13: Amplitude of the relative motion for a damped spring–mass system subjected to harmonic base excitation
##### Vibrometer or Seismometer
These instruments tend to be large and have soft springs so that their natural frequencies are small. As a result they are usually used in the range where \ensuremath{\frac{\omega}{\ensuremath{p}}} is large (). In this range, we can see from Figure 5.13 that as ,
so that the amplitude of the \emph{relative} motion \mathbb{Z} between the mass and the base is the same as the amplitude of the \emph{actual} motion of the base. In operation, the mass remains approximately stationary while the base moves with the supporting structure (whose vibrations are being measured).\par
These devices tend to have little damping, but as can be seen from Figure 5.13, even moderate
amounts of damping have only a small effect in this region.
##### Accelerometer
Accelerometers are widely used to measure acceleration levels. They tend to be small and stiff and as a result usually have very high natural frequencies. They are useful for measuring vibrations at frequencies that are much smaller than their own natural frequency ().
We can see from equation 5.34 that as \ensuremath{\frac{\omega}{\ensuremath{p}}} approaches zero,
where is the magnitude of the acceleration of the base.
Therefore, in this region the amplitude of the \emph{relative} motion is approximately \emph{proportional} to the magnitude of the \emph{acceleration} of the base.
To understand the response better, note that equation (5.34) could be rearranged as
where is a constant for the accelerometer that is usually determined by calibration.
As can be seen, the value of the factor
approaches 1 as but for this factor is different from one and this difference contributes to the error in the measurement.
To see this effect, figure 5.14 illustrates this region of the response for an undamped accelerometer. For a frequency ratio , the value of the factor is approximately 1.01, which corresponds to approximately a 1\% error. At a frequency ratio , the value of the factor increases to approximately 1.04, which corresponds to a 4\% error. Further increases in lead to rapidly increasing errors.
Figure 5.14: Accelerometer response for small ratios
Figure 5.15: Accelerometer response for small ratios
We can take advantage of damping to improve this behaviour. Figure 5.15 shows the effect of damping on the response. As shown in this figure, for damping ratios between about 0.65 and 0.7 this factor remains much closer to one over a wider range of frequency ratios. For example, with , the error introduced by assuming this factor to be unity is less than 1\% over a frequency range of , and is less than 0.01\% for frequencies up to 20\% of the natural frequency of the unit.
Many accelerometers, typically piezoelectric, have almost no damping so the usable frequency range is much smaller, up to about 10\% of the natural frequency for a 1\% error. Note however that these accelerometers tend to have quite large natural frequencies to begin with so the usable range is still quite appreciable.
### Force Transmitted in Damped Base Excitation
During base excitation of a damped spring–mass system, the force transmitted between the support
and the mass depends on the relative motion between the two and is transmitted through both the
spring and the damper. Referring to Figure 5.11(b), the total force transmitted can be expressed
as
(5.35)
or, in terms of the relative motion,
We could use either of these expressions since we have already found solutions for and previously for this situation. However, it is much easier to proceed by comparing equations (5.35) and (5.23) to note that
We have already shown (equation (5.28)) that the response of the mass is given by
so that the maximum acceleration is . The maximum force transmitted is then
which can be rearranged as
(5.36)
Equation 5.36 represents the transmissibility in the case of base excitation of a damped spring–mass system. As with the undamped situation discussed previously, the maximum disturbing force is again interpreted here to be . Figure 5.16 represents this relationship.
Figure 5.16: Transmissibility for base excitation of a damped system | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9633240699768066, "perplexity": 577.2885245540912}, "config": {"markdown_headings": true, "markdown_code": false, "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-2020-50/segments/1606141184870.26/warc/CC-MAIN-20201125213038-20201126003038-00581.warc.gz"} |
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