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https://www.rdocumentation.org/packages/spatstat/versions/1.25-1/topics/vcov.ppm	# vcov.ppm 0th Percentile ##### Variance-Covariance Matrix for a Fitted Point Process Model Returns the variance-covariance matrix of the estimates of the parameters of a fitted point process model. Keywords models, methods, spatial ##### Usage ## S3 method for class 'ppm': vcov(object, \dots, what = "vcov", verbose = TRUE, gam.action="warn", matrix.action="warn", hessian=FALSE) ##### Arguments object A fitted point process model (an object of class "ppm".) ... Ignored. what Character string (partially-matched) that specifies what matrix is returned. Options are "vcov" for the variance-covariance matrix, "corr" for the correlation matrix, and "fisher" or "Fisher" verbose Logical. If TRUE, a message will be printed if various minor problems are encountered. gam.action String indicating what to do if object was fitted by gam. Options are "fatal", "warn" and "silent". matrix.action String indicating what to do if the matrix is ill-conditioned (so that its inverse cannot be calculated). Options are "fatal", "warn" and "silent". hessian Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of the Fisher information. ##### Details This function computes the asymptotic variance-covariance matrix of the estimates of the canonical parameters in the point process model object. It is a method for the generic function vcov. object should be an object of class "ppm", typically produced by ppm. The canonical parameters of the fitted model object are the quantities returned by coef.ppm(object). The function vcov calculates the variance-covariance matrix for these parameters. The argument what provides three options: [object Object],[object Object],[object Object] In all three cases, the result is a square matrix. The rows and columns of the matrix correspond to the canonical parameters given by coef.ppm(object). The row and column names of the matrix are also identical to the names in coef.ppm(object). For models fitted by maximum pseudolikelihood (which is the default in ppm), the current implementation only works for Poisson point processes. The calculations are based on standard asymptotic theory for the maximum likelihood estimator. The observed Fisher information matrix of the fitted model object is first computed, by summing over the Berman-Turner quadrature points in the fitted model. The asymptotic variance-covariance matrix is calculated as the inverse of the observed Fisher information. The correlation matrix is then obtained by normalising. For models fitted by the Huang-Ogata method (method="ho" in the call to ppm), the implementation works for all models. A Monte Carlo estimate of the Fisher information matrix is calculated using the results of the original fit. The argument verbose makes it possible to suppress some diagnostic messages. The asymptotic theory is not correct if the model was fitted using gam (by calling ppm with use.gam=TRUE). The argument gamaction determines what to do in this case. If gamaction="fatal", an error is generated. If gamaction="warn", a warning is issued and the calculation proceeds using the incorrect theory for the parametric case, which is probably a reasonable approximation in many applications. If gamaction="silent", the calculation proceeds without a warning. If hessian=TRUE then the negative Hessian (second derivative) matrix of the log pseudolikelihood, and its inverse, will be computed. For non-Poisson models, this is not a valid estimate of variance, but is useful for other calculations. ##### Value • A square matrix. ##### Error messages An error message that reports system is computationally singular indicates that the determinant of the Fisher information matrix was either too large or too small for reliable numerical calculation. This can occur either because of numerical overflow or because of collinearity in the covariates. Most commonly it occurs because of numerical overflow: to check this, rescale the coordinates of the data points and refit the model. See the Examples. • vcov.ppm ##### Examples X <- rpoispp(42) fit <- ppm(X, ~ x + y) vcov(fit) vcov(fit, what="Fish") # example of singular system data(demopat) m <- ppm(demopat, ~polynom(x,y,2)) try(v <- vcov(m)) # rescale x, y coordinates to range [0,1] x [0,1] approximately demopat <- rescale(demopat, 10000) m <- ppm(demopat, ~polynom(x,y,2)) v <- vcov(m) Documentation reproduced from package spatstat, version 1.25-1, License: GPL (>= 2) ### Community examples Looks like there are no examples yet.	{"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": 0.5837872624397278, "perplexity": 1738.2216669585719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540529006.88/warc/CC-MAIN-20191210205200-20191210233200-00132.warc.gz"}
http://math.stackexchange.com/users/21019/mcarthey?tab=summary	McArthey Reputation Top tag Next privilege 125 Rep. Vote down 3 Impact ~147 people reached ### Question (1) 2 Is it possible to define a directed graph “facing” direction? ### Reputation (113) +10 Is it possible to define a directed graph “facing” direction? This user has not answered any questions ### Accounts (34) Stack Overflow 707 rep 1134 Spanish Language 398 rep 18 Home Improvement 331 rep 12 English Language & Usage 279 rep 26 Arqade 224 rep 239	{"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": 0.8677253723144531, "perplexity": 9128.78264956408}, "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/1429246639057.4/warc/CC-MAIN-20150417045719-00134-ip-10-235-10-82.ec2.internal.warc.gz"}
http://pps2017.soic.indiana.edu/author/mislove/	## Probabilistic Programming and a Domain Theoretic Approach to Skorohod’s Theorem Skorohod’s Theorem is a fundamental result from stochastic process theory. It makes two assertions: 1. Any Borel probability measure $\mu$ on a Polish space $P$ can be realized as $\mu = X_\,\lambda$ for some random variable $X\colon [0,1]\to P$, where $\lambda$ denotes Lebesgue measure and $X_\, \lambda$ is the “law of $X$,” the push forward of $\lambda$ under $X$. 2. Moreover, if $\mu_n \to_w \mu$ is a weakly convergent sequence of probability measures on $P$, then the associated random variables $X_n\to_{a.s} X$ converge almost surely wrt $\lambda$. Recently, the author devised a domain-theoretic proof of Skorohod’s Theorem. Since domain theory is a mainstay of denotational semantics, and since Probabilistic Programming concerns modeling probability measures and stochastic processes, it’s reasonable to look for connections between the domain-theoretic proof of Skorohod’s Theorem and the semantics of Probabilistic Programming languages. But there is a difference in how computability is realized in the domain-theoretic approach used to prove Skorohod’s Theorem, versus the standard approach to computable metric spaces and their probability measures: • The domain-theoretic model used in the proof of Skorohod’s Theorem is built from topological subsets of the Polish space $P$. For example, in the case of locally compact spaces, the domain is the family of compact subsets ordered by reverse inclusion. Computability is “implicit” in the domain model, but can be instantiated by an enumeration of a countable basis of the domain, i.e., of a countable neighborhood basis of $P$ that forms a basis for the domain model. • In the standard approach, a computable metric space $(P,{\mathcal S},d)$ consists of a Polish space $(P,d)$, a countable subset ${\mathcal S}\subseteq P$, with the assumption that $d\colon {\mathcal S}\times {\mathcal S}\to {\mathbb R}_+$ is computable. The computable structure is then lifted to $\textsf{Prob}(P)$ via simple measures with rational coefficients that are concentrated on finite subsets of ${\mathcal S}$. This difference can be seen in how computable simple measures are represented. In contrast to the standard approach to computable metric spaces where computable simple measures are concentrated on finite subsets of ${\mathcal S}$, in the domain-theroetic model, each point mass arising in a computable simple measure has for its mass a subset of the Polish space. Devising an application of the domain-theoretic proof of Skorohod’s Theorem to Probabilistic Programming first requires an understanding of how the computable structure on $P$ and one on the domain model built from appropriate topological susbsets of $P$ are related. ### Extended Abstract: Probabilistic Programming and a Domain Theoretic Approach to Skorohod’s Theorem	{"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": 24, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9792745113372803, "perplexity": 269.92139032290413}, "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/1500549423839.97/warc/CC-MAIN-20170722002507-20170722022507-00637.warc.gz"}
https://math.stackexchange.com/questions/2856225/reference-request-extension-of-homomorphism-to-multiplier-algebra/2856262	# Reference request: extension of $$-homomorphism to multiplier algebra Let $A$ be a $C^$-algebra and $f:A\rightarrow\mathbb{C}$ a $^$-homomorphism. Does $f$ always extend to a $^$-homomorphism $\tilde{f}:M(A)\rightarrow\mathbb{C}$, where $M(A)$ is the multiplier algebra of $A$? I suspect the answer is no, but I'm not sure how to show this. In case the answer is yes, a proof or reference would be greatly appreciated. Thanks! The answer is yes. This depends on two facts: • $A$ is an ideal in $M(A)$. • if $J\subset B$ is an ideal, any non-degenerate representation $J\to B(H)$ can be extended to a representation $B\to B(H)$. This is for instance Lemma I.9.14 in Davidson's C$^$-Algebras By Example. Here we have $J=A$, $B=M(A)$, $H=\mathbb C$. The argument works for any $$-homomorphism $A\to B(H)$. • Can this somehow be used to show that any $$-homomorphism of $C^$-algebras $A\rightarrow B$ extends to a $$-homomorphism $M(A)\rightarrow M(B)$? (I'm slightly worried about the non-degeneracy requirement in Davidson.) – ougoah Nov 2 '18 at 2:34 • As far as I can tell, you usually require the $$-homomorphism to be onto for that. In that case, it looks like the above argument would indeed work. – Martin Argerami Nov 2 '18 at 3:51 • If the representation $\pi:A→B(H)$ is degenerate,can we also get the unique extension $\tilde{\pi}:M(A)→B(H)$ by defining $\tilde{\pi}(\tilde a)=lim\pi(a)e_n$,where $\{e_n\}$ is a.u for A. – math112358 Apr 27 at 22:23 • You will never get uniqueness of an extension of a degenerate representation; that's precisely the problem. As for the definition, I see no reason for your limit to converge. – Martin Argerami Apr 27 at 22:34 • math.stackexchange.com/questions/2940320/… answered the question before.In Blackdar's book,there also states the non-degeneracy of the representation. – math112358 Apr 27 at 22:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9633263945579529, "perplexity": 247.53983262570316}, "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/1560627999210.22/warc/CC-MAIN-20190620105329-20190620131329-00555.warc.gz"}
https://api.philpapers.org/s/Nye%20Howard	Results for 'Nye Howard' 1000+ found Order: 1. Non-Consequentialism Demystified.Howard Nye, David Plunkett & John Ku - 2015 - Philosophers' Imprint 15 (4):1-28. Morality seems important, in the sense that there are practical reasons — at least for most of us, most of the time — to be moral. A central theoretical motivation for consequentialism is that it appears clear that there are practical reasons to promote good outcomes, but mysterious why we should care about non-consequentialist moral considerations or how they could be genuine reasons to act. In this paper we argue that this theoretical motivation is mistaken, and that because many arguments (...) Export citation Bookmark 9 citations 2. In this article I attempt to show conclusively that the apparent intrinsic difference between causing collateral damage and directly attacking innocents is an illusion. I show how eleven morally irrelevant alterations can transform an apparently permissible case of harming as a side-effect into an apparently impermissible case of harming as a means. The alterations are as obviously irrelevant as the victims’ skin colour, and consistently treating them as relevant would have unacceptable implications for choices between more and less harmful ways (...) Export citation Bookmark 4 citations 3. Directly Plausible Principles.Howard Nye - 2015 - In Christopher Daly (ed.), The Palgrave Handbook of Philosophical Methods. Palgrave MacMillan. pp. 610-636. In this chapter I defend a methodological view about how we should conduct substantive ethical inquiries in the fields of normative and practical ethics. I maintain that the direct plausibility and implausibility of general ethical principles – once fully clarified and understood – should be foundational in our substantive ethical reasoning. I argue that, in order to expose our ethical intuitions about particular cases to maximal critical scrutiny, we must determine whether they can be justified by directly plausible principles. To (...) Export citation Bookmark 2 citations 4. Why Should We Try to Be Sustainable? Expected Consequences and the Ethics of Making an Indeterminate Difference.Howard Nye - 2021 - In Chelsea Miya, Oliver Rossier & Geoffrey Rockwell (eds.), Right Research: Modelling Sustainable Research Practices in the Anthropocene. Open Book Publishers. pp. 3-35. Why should we refrain from doing things that, taken collectively, are environmentally destructive, if our individual acts seem almost certain to make no difference? According to the expected consequences approach, we should refrain from doing these things because our individual acts have small risks of causing great harm, which outweigh the expected benefits of performing them. Several authors have argued convincingly that this provides a plausible account of our moral reasons to do things like vote for policies that will reduce (...) Export citation Bookmark 5. Chaos and Constraints.Howard Nye - 2014 - In David Boersema (ed.), Dimensions of Moral Agency. Cambridge Scholars Press. pp. 14-29. Agent-centered constraints on harming hold that some harmful upshots of our conduct cannot be justified by its generating equal or somewhat greater benefits. In this paper I argue that all plausible theories of agent-centered constraints on harming are undermined by the likelihood that our actions will have butterfly effects, or cause cascades of changes that make the world dramatically different than it would have been. Theories that impose constraints against only intended harming or proximally caused harm have unacceptable implications for (...) Export citation Bookmark 2 citations 6. Artificial Moral Patients: Mentality, Intentionality, and Systematicity.Howard Nye & Tugba Yoldas - 2021 - International Review of Information Ethics 29:1-10. In this paper, we defend three claims about what it will take for an AI system to be a basic moral patient to whom we can owe duties of non-maleficence not to harm her and duties of beneficence to benefit her: (1) Moral patients are mental patients; (2) Mental patients are true intentional systems; and (3) True intentional systems are systematically flexible. We suggest that we should be particularly alert to the possibility of such systematically flexible true intentional systems developing (...) Export citation Bookmark 7. Objective Double Effect and the Avoidance of Narcissism.Howard Nye - 2013 - In Mark Timmons (ed.), Oxford Studies in Normative Ethics, Volume 3. Oxford University Press. pp. 260-286. The Doctrine of Double Effect [DDE] states roughly that it is harder to justify causing or allowing harm as a means to an end than it is to justify conduct that results in harm as a side effect. This chapter argues that a theory of deontological constraints on harming needs something like the DDE in order to avoid the charge that it reflects a narcissistic obsession with the cleanliness of our own hands. Unfortunately, the DDE is often interpreted as maintaining (...) Export citation Bookmark 1 citation 8. Ethics, Fitting Attitudes, and Practical Reason: A Theory of Normative Facts.Howard Nye - 2009 - Dissertation, University of Michigan I present and defend (1) an account of ethical judgments as judgments about our reasons to feel specific motivationally laden attitudes, (2) an account of what an agent should do in terms of what would achieve ends that she has reason to be motivated to pursue, and (3) an account of an agent’s reasons for motivation (and thus action) in terms of the prescriptions of the most fundamental principles that guide her deliberations. Using these accounts, I explain the connection between (...) Export citation Bookmark 2 citations 9. Expressivism, Pragmatism, and Representationalism, by Huw Price: Cambridge: Cambridge University Press, 2013, Pp. X–Xii + 204, US$29.99.Howard Nye - 2016 - Australasian Journal of Philosophy 94 (1):196-199. Direct download (6 more) Export citation Bookmark 10. Technological Displacement and the Duty to Increase Living Standards: From Left to Right.Howard Nye - 2020 - International Review of Information Ethics 28:1-16. Many economists have argued convincingly that automated systems employing present-day artificial intelligence have already caused massive technological displacement, which has led to stagnant real wages, fewer middle- income jobs, and increased economic inequality in developed countries like Canada and the United States. To address this problem various individuals have proposed measures to increase workers’ living standards, including the adoption of a universal basic income, increased public investment in education, increased minimum wages, increased worker control of firms, and investment in a (...) Direct download (3 more) Export citation Bookmark 11. Well‐Being, Self‐Regarding Reasons, and Morality.Howard L. M. Nye - 2014 - Thought: A Journal of Philosophy 3 (4):332-341. It seems that we should want to avoid becoming intellectually disabled. It is common for philosophers to infer from this that those of us without intellectual disabilities are intrinsically better off than individuals with intellectual disabilities, and that there are consequently stronger moral reasons for others to preserve our lives than to preserve the lives of intellectually disabled individuals. In this article, I argue against this inference from what states we should prefer for ourselves to how much moral reason others (...) Direct download Export citation Bookmark 12. Expressivism, Pragmatism, and Representationalism, by Huw Price: Cambridge: Cambridge University Press, 2013, Pp. X–Xii + 204, US$29.99. [REVIEW]Howard Nye - 2016 - Australasian Journal of Philosophy 94 (1):196-199. No categories Export citation Bookmark 13. The Wrong Kind of Reasons.Nye Howard - 2017 - In Tristram McPherson & David Plunkett (eds.), The Routledge Handbook of Metaethics. New York: Routledge. pp. 340-354. Export citation Bookmark 14. In the World Library of Educationalists series, international experts themselves compile career-long collections of what they judge to be their finest pieces--extracts from books, key articles, salient research findings, major theoretical and/practical contributions--so the work can read them in a single manageable volume. Readers will be able to follow the themes and strands of their work and see their contribution to the development of a field. A developmental psychologist by training, Howard Gardner has spent the last 30 years researching, (...) Export citation Bookmark 4 citations 15. Michael Polanyi and His Generation: Origins of the Social Construction of Science.Mary Jo Nye - 2013 - University of Chicago Press. In _Michael Polanyi and His Generation_, Mary Jo Nye investigates the role that Michael Polanyi and several of his contemporaries played in the emergence of the social turn in the philosophy of science. This turn involved seeing science as a socially based enterprise that does not rely on empiricism and reason alone but on social communities, behavioral norms, and personal commitments. Nye argues that the roots of the social turn are to be found in the scientific culture and political events (...) Export citation Bookmark 14 citations 16. Howard Schuman.Howard Schuman - 1994 - Social Epistemology 8 (1):27-33. Translate Export citation Bookmark 17. Reviews : Mickael W. Howard -- From Commodity Fetishism to Market Socialism: Critical Notes on Stanley Moore.Michael W. Howard - 1980 - Philosophy and Social Criticism 7 (2):184-214. Export citation Bookmark 18. Michael W. Howard -- Utopianism and Nuclear Deterrence.Michael W. Howard - 1984 - Philosophy and Social Criticism 10 (3-4):53-65. Export citation Bookmark 19. Laws of Nature: Necessary and Contingent.Samuel Kimpton-Nye - forthcoming - The Philosophical Quarterly. This paper shows how a niche account of the metaphysics of laws of nature and physical properties—the Powers-BSA—can underpin both a sense in which the laws are metaphysically necessary and a sense in which it is true that the laws could have been different. The ability to reconcile entrenched disagreement should count in favour of a philosophical theory, so this paper constitutes a novel argument for the Powers-BSA by showing how it can reconcile disagreement about the laws’ modal status. This (...) Export citation Bookmark 20. Aristotle and the Virtues.Howard J. Curzer - 2012 - Oxford University Press. Howard J. Curzer presents a fresh new reading of Aristotle's Nicomachean Ethics, which brings each of the virtues alive. He argues that justice and friendship are symbiotic in Aristotle's view; reveals how virtue ethics is not only about being good, but about becoming good; and describes Aristotle's ultimate quest to determine happiness. Export citation Bookmark 42 citations 21. Real Politics and Metaethical Baggage.Sebastian Nye - 2015 - Ethical Theory and Moral Practice 18 (5):1083-1100. So-called 'realists' have argued that political philosophers should engage with real politics, but that mainstream 'non-realist' political philosophers fail to do so. Perhaps surprisingly, many of the discussions between realists and their critics have not drawn much on debates in metaethics. In this paper, I argue that this is an oversight. There are important connections between the realism/non-realism debate and certain controversies in metaethics. Both realism and non-realism come with metaethical baggage. By considering several arguments that could be made for (...) Export citation Bookmark 5 citations 22. The first full-scale history of cognitive science, this work addresses a central issue: What is the nature of knowledge? Export citation Bookmark 204 citations 23. The New Mr. Coffee: Howard Schultz.Howard Schultz & Mary Scott - 1995 - Business Ethics: The Magazine of Corporate Responsibility 9 (6):26-29. Export citation Bookmark 24. .Howard Caygill - 2016 No categories Export citation Bookmark 17 citations 25. The Significance of Religious Experience.Howard Wettstein - 2012 - Oxford University Press USA. In this volume of essays, Howard Wettstein explores the foundations of religious commitment. His orientation is broadly naturalistic, but not in the mode of reductionism or eliminativism. This collection explores questions of broad religious interest, but does so through a focus on the author's religious tradition, Judaism. Among the issues explored are the nature and role of awe, ritual, doctrine, religious experience; the distinction between belief and faith; problems of evil and suffering with special attention to the Book of (...) Export citation Bookmark 12 citations 26. Aristotle's Painful Path to Virtue.Howard J. Curzer - 2002 - Journal of the History of Philosophy 40 (2):141-162. Howard J. Curzer - Aristotle's Painful Path to Virtue - Journal of the History of Philosophy 40:2 Journal of the History of Philosophy 40.2 141-162 Aristotle's Painful Path to Virtue Howard J. Curzer [P]unishment . . . is a kind of cure . . . . We think young people should be prone to shame . . . . 1. Two Questions FOR ARISTOTLE, THE GOAL OF MORAL development is, of course, to become virtuous. Aristotle provides a partial (...) Export citation Bookmark 22 citations 27. From the Knowledge Argument to Mental Substance: Resurrecting the Mind.Howard Robinson - 2016 - Cambridge University Press. This book presents a strong case for substance dualism and offers a comprehensive defense of the knowledge argument, showing that materialism cannot accommodate or explain the 'hard problem' of consciousness. Bringing together the discussion of reductionism and semantic vagueness in an original and illuminating way, Howard Robinson argues that non-fundamental levels of ontology are best treated by a conceptualist account, rather than a realist one. In addition to discussing the standard versions of physicalism, he examines physicalist theories such as (...) Export citation Bookmark 6 citations 28. Howard Mumford Jones: O Strange New WorldO Strange New World.Henry Nash Smith & Howard Mumford Jones - 1965 - Journal of the History of Ideas 26 (3):435. Export citation Bookmark 29. Book Review: Journalism and Justice: An Essay Review by Howard Ziff. [REVIEW]Howard M. Ziff - 1990 - Journal of Mass Media Ethics 5 (3):203 – 211. Export citation Bookmark Export citation Bookmark 28 citations 31. Reconsidering the Dispositional Essentialist Canon.Samuel Kimpton-Nye - 2021 - Philosophical Studies 178 (10):3421-3441. Dispositional Essentialism is a unified anti-Humean account of the metaphysics of low-level physical properties and laws of nature. In this paper, I articulate the view that I label Canonical Dispositional Essentialism, which comprises a structuralist metaphysics of properties and an account of laws as relations in the property structure. I then present an alternative anti-Humean account of properties and laws. This account rejects CDE’s structuralist metaphysics of properties in favour of a view of properties as qualitative grounds of dispositions and (...) Translate Export citation Bookmark 5 citations 32. Selfishness, Altruism, and Rationality: A Theory of Social Choice.Howard Margolis - 1982 - Cambridge University Press. No categories Export citation Bookmark 24 citations 33. Games and Decisions: Introduction and Critical Survey.Robert Duncan Luce & Howard Raiffa - 1957 - New York: Wiley. "The best book available for non-mathematicians." — Contemporary Psychology. Superb nontechnical introduction to game theory and related disciplines, primarily as applied to the social sciences. Clear, comprehensive coverage of utility theory, 2-person zero-sum games, 2-person non-zero-sum games, n-person games, individual and group decision-making, much more. Appendixes. Bibliography. Graphs and figures. Export citation Bookmark 197 citations 34. The Two Pragmatisms: From Peirce to Rorty.Howard Mounce - 1997 - Routledge. _The Two Pragmatisms - From Peirce to Rorty_ maps the main movements within the pragmatist tradition. Two distinct forms of pragmatism are identified, that of Peirce and that of the `second' pragmatism stemming from James' interpretation of Peirce and seen in the work of Dewey and above all Rorty. Both the influential work of Rorty and the way in which he has transformed contemporary philosophy's understanding of pragmatism are clearly explained. _The Two Pragmatisms - From Peirce to Rorty_ is essential (...) Export citation Bookmark 11 citations 35. Matter and Sense: A Critique of Contemporary Materialism.Howard Robinson - 1984 - Oxford University Press. Published in 1982 by CUP (pb. 2009) it discusses the forms of materialism then current, including Davidson, early Rorty, but concentrating on Smart and Armstrong, and arguing that central state materialism fails to give a better 'occurrent' account of conscious states than does behaviourism/functionalism, as Armstrong claims. The book starts with a version of the 'knowledge argument' and ends with a chapter claiming that our conception of matter/the physical is more problematic than our conception of mind. No categories Export citation Bookmark 23 citations 36. Can Hardcore Actualism Validate S5?Samuel Kimpton-Nye - 2021 - Philosophy and Phenomenological Research 102 (2):342-358. Hardcore actualism (HA) grounds all modal truths in the concrete constituents of the actual world (see, e.g., Borghini and Williams (2008), Jacobs (2010), Vetter (2015)). I bolster HA, and elucidate the very nature of possibility (and necessity) according to HA, by considering if it can validate S5 modal logic. Interestingly, different considerations pull in different directions on this issue. To resolve the tension, we are forced to think hard about the nature of the hardcore actualist's modal reality and how radically (...) Export citation Bookmark 5 citations 37. Hooked: Ethics, the Medical Profession, and the Pharmaceutical Industry.Howard Brody - 2007 - Rowman & Littlefield Publishers. This book explores the controversial relationship between physicians and the pharmaceutical industry, identifies the ethical tensions and controversies, and proposes numerous reforms both for medicine's own professional integrity and for effective public regulation of the industry. Export citation Bookmark 40 citations 38. The Scoundrel and the Visionary: On Reasonable Hope and the Possibility of a Just Future.Dana S. Howard - 2019 - Journal of Political Philosophy 27 (3):294-317. John Rawls, among others, has argued that one of aims political philosophy is that it provides reasonable hope in the possibility of justice in the future. But what makes hope reasonable? What sorts of theories of justice are best suited to foster reasonable hope in us? To answer these questions, this paper investigates Rawls’s conception of reasonable hope and the kinds of unreasonableness Rawls sought to guard against with his account of a realistic utopia. Rawls's idea of reasonable hope goes (...) Export citation Bookmark 5 citations 39. The Healer's Power.Howard Brody - 1992 - Yale University Press. Although the physician’s use and misuse of power have been discussed in the social sciences and in literature, they have never been explored in medical ethics until now. In this book, Dr. Howard Brody argues that the central task is not to reduce the physician’s power, as others have suggested, but to develop guidelines for its use, so that the doctor shares with the patient both information and the responsibility for deciding on appropriate treatment. Dr. Brody first reviews literary (...) No categories Export citation Bookmark 20 citations 40. Export citation Bookmark 41. Export citation Bookmark 42. Feminist Theory and the Philosophies of Man.Andrea Nye - 1988 - Routledge. First published in 1990. Routledge is an imprint of Taylor & Francis, an informa company. No categories Export citation Bookmark 3 citations 43. A Kant Dictionary.Howard Caygill - 1996 - Journal of Nietzsche Studies 11:64-66. Export citation Bookmark 44 citations 44. Narrative Ethics:A Narrative.Howard Brody & Mark Clark - 2014 - Hastings Center Report 44 (s1):S7-S11. Export citation Bookmark 14 citations 45. Humean Laws in an unHumean World.Samuel Kimpton-nye - 2017 - Journal of the American Philosophical Association 3 (2):129-147. I argue that an unHumean ontology of irreducibly dispositional properties might be fruitfully combined with what has typically been thought of as a Humean account of laws, namely, the best-system account, made popular by David Lewis (e.g., 1983, 1986, 1994). In this paper I provide the details of what I argue is the most defensible account of Humean laws in an unHumean world. This package of views has the benefits of upholding scientific realism while doing without any suspect metaphysical entities (...) Export citation Bookmark 6 citations 46. Hardcore Actualism and Possible Non‐Existence.Samuel Kimpton‐Nye - 2018 - Thought: A Journal of Philosophy 7 (2):122-131. According to hardcore actualism (HA), all modal truths are grounded in the concrete constituents of the actual world. In this paper, I discuss some problems faced by HA when it comes to accounting for certain alleged possibilities of non‐existence. I focus particular attention on Leech (2017)'s dilemma for HA, according to which HA must either sacrifice extensional correctness or admit mere possibilia. I propose a solution to Leech's dilemma, which relies on a distinction between weak and strong possibility. It remains (...) Export citation Bookmark 4 citations 47. Common Ground for Laws and Metaphysical Modality.Samuel Kimpton-Nye - 2018 - Dissertation, King's College London Export citation Bookmark 3 citations 48. Necessary Laws and the Problem of Counterlegals.Samuel Kimpton-Nye - 2020 - Philosophy of Science 87 (3):518-535. Substantive counterlegal discourse poses a problem for those according to whom the laws of nature are metaphysically necessary. I discern two types of necessitarianism about laws: dispositional essentialism and modal necessitarianism. I argue that Toby Handfield’s response to the problem of counterlegals cannot help the modal necessitarian, according to whom all possible worlds are identical with respect to the laws. I thus propose a fictionalist treatment of counterlegals. Fictions are not limited by metaphysical possibility; hence, fictionalism affords the modal necessitarian (...) Export citation Bookmark 2 citations 49. Taking Chances: Essays on Rational Choice.Jordan Howard Sobel - 1994 - Cambridge, England: Cambridge University Press. J. Howard Sobel has long been recognized as an important figure in philosophical discussions of rational decision. He has done much to help formulate the concept of causal decision theory. In this volume of essays Sobel explores the Bayesian idea that rational actions maximize expected values, where an action's expected value is a weighted average of its agent's values for its possible total outcomes. Newcomb's Problem and The Prisoner's Dilemma are discussed, and Allais-type puzzles are viewed from the perspective (...)	{"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": 0.2986873686313629, "perplexity": 6725.9825042294515}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662517485.8/warc/CC-MAIN-20220517130706-20220517160706-00731.warc.gz"}
https://worldwidescience.org/topicpages/w/wavelength+small+amplitude.html	#### Sample records for wavelength small amplitude 1. Wavelength selection in injection-driven Hele-Shaw flows: A maximum amplitude criterion Science.gov (United States) Dias, Eduardo; Miranda, Jose 2013-11-01 As in most interfacial flow problems, the standard theoretical procedure to establish wavelength selection in the viscous fingering instability is to maximize the linear growth rate. However, there are important discrepancies between previous theoretical predictions and existing experimental data. In this work we perform a linear stability analysis of the radial Hele-Shaw flow system that takes into account the combined action of viscous normal stresses and wetting effects. Most importantly, we introduce an alternative selection criterion for which the selected wavelength is determined by the maximum of the interfacial perturbation amplitude. The effectiveness of such a criterion is substantiated by the significantly improved agreement between theory and experiments. We thank CNPq (Brazilian Sponsor) for financial support. 2. Modeling Small-Amplitude Perturbations in Inertial Confinement Fusion Pellets Science.gov (United States) Zalesak, Steven; Metzler, N.; Velikovich, A. L.; Gardner, J. H.; Manheimer, W. 2005-10-01 Recent advances in inertial confinement fusion (ICF) technology serve to ensure that imploding laser-driven ICF pellets will spend a significantly larger portion of their time in what is regarded as the linear'' portion of their perturbation evolution, i.e., in the presence of small-amplitude but nonetheless evolving perturbations. Since the evolution of these linear perturbations collectively form the initial conditions for the subsequent nonlinear evolution of the pellet, which in turn determines the energy yield of the pellet, the accurate numerical modeling of these small-amplitude perturbations has taken on an increased importance. This modeling is difficult despite the expected linear evolution of the perturbations themselves, because these perturbations are embedded in a highly nonlinear, strongly-shocked, and highly complex flow field which in and of itself stresses numerical computation capabilities, and whose simulation often employs numerical techniques which were not designed with the proper treatment of small-amplitude perturbations in mind. In this paper we will review some of the techniques that we have recently found to be of use toward this end. 3. Nuclear-Mechanical Coupling: Small Amplitude Mechanical Vibrations and High Amplitude Power Oscillations in Nuclear Reactors International Nuclear Information System (INIS) Suarez Antola, R. 2008-11-01 The cores of nuclear reactors, including its structural parts and cooling fluids, are complex mechanical systems able to vibrate in a set of normal modes and frequencies, if suitable perturbed. The cyclic variations in the strain state of the core materials may produce changes in density. Changes in density modify the reactivity. Changes in reactivity modify thermal power. Modifications in thermal power produce variations in temperature fields. Variations in temperature produce variations in strain due to thermal-elastic effects. If the variation of the temperature field is fast enough and if the Doppler Effect and other stabilizing prompt effects in the fuel are weak enough, a fast oscillatory instability could be produced, coupled with mechanical vibrations of small amplitude. A recently constructed, simple mathematical model of nuclear reactor kinetics, that improves the one due to A.S. Thompson, is reviewed. It was constructed in order to study, in a first approximation, the stability of the reactor: a nonlinear nuclear-thermal oscillator (that corresponds to reactor point kinetics with thermal-elastic feedback and with frozen delayed neutron effects) is coupled nonlinearly with a linear mechanical-thermal oscillator (that corresponds to the first normal mode of mechanical vibrations excited by thermo-elastic effects). This mathematical model is studied here from the standpoint of mechanical vibrations. It is shown how, under certain conditions, a suitable mechanical perturbation could elicit fast and growing oscillatory instabilities in the reactor power. Applying the asymptotic method due to Krylov, Bogoliubov and Mitropolsky, analytical formulae that may be used in the calculation of the time varying amplitude and phase of the mechanical oscillations are given, as functions of the mechanical, thermal and nuclear parameters of the reactor. The consequences for the mechanical integrity of the reactor are assessed. Some conditions, mainly, but not exclusively 4. Elimination of residual amplitude modulation in tunable diode laser wavelength modulation spectroscopy using an optical fiber delay line. Science.gov (United States) Chakraborty, Arup Lal; Ruxton, Keith; Johnstone, Walter; Lengden, Michael; Duffin, Kevin 2009-06-08 A new fiber-optic technique to eliminate residual amplitude modulation in tunable diode laser wavelength modulation spectroscopy is presented. The modulated laser output is split to pass in parallel through the gas measurement cell and an optical fiber delay line, with the modulation frequency / delay chosen to introduce a relative phase shift of pi between them. The two signals are balanced using a variable attenuator and recombined through a fiber coupler. In the absence of gas, the direct laser intensity modulation cancels, thereby eliminating the high background. The presence of gas induces a concentration-dependent imbalance at the coupler's output from which the absolute absorption profile is directly recovered with high accuracy using 1f detection. 5. Calculation of the real part of the nuclear amplitude at high s and small t from the Coulomb amplitude Energy Technology Data Exchange (ETDEWEB) Gauron, P.; Nicolescu, B. [Universite Pierre et Marie Curie, Theory Group, Lab. de Physique Nucleaire et des Hautes Energies (LPNHE), CNRS 75 - Paris (France) 2005-07-01 A new method for the determination of the real part of the elastic scattering amplitude is examined for high energy proton-proton at small momentum transfer. This method allows us to decrease the number of model assumptions, to obtain the real part in a narrow region of momentum transfer and to test different models. The possible non-exponential behavior of the real part was found on the base of the analysis of the ISR experimental data. (authors) 6. Disappearance of Anisotropic Intermittency in Large-amplitude MHD Turbulence and Its Comparison with Small-amplitude MHD Turbulence Science.gov (United States) Yang, Liping; Zhang, Lei; He, Jiansen; Tu, Chuanyi; Li, Shengtai; Wang, Xin; Wang, Linghua 2018-03-01 Multi-order structure functions in the solar wind are reported to display a monofractal scaling when sampled parallel to the local magnetic field and a multifractal scaling when measured perpendicularly. Whether and to what extent will the scaling anisotropy be weakened by the enhancement of turbulence amplitude relative to the background magnetic strength? In this study, based on two runs of the magnetohydrodynamic (MHD) turbulence simulation with different relative levels of turbulence amplitude, we investigate and compare the scaling of multi-order magnetic structure functions and magnetic probability distribution functions (PDFs) as well as their dependence on the direction of the local field. The numerical results show that for the case of large-amplitude MHD turbulence, the multi-order structure functions display a multifractal scaling at all angles to the local magnetic field, with PDFs deviating significantly from the Gaussian distribution and a flatness larger than 3 at all angles. In contrast, for the case of small-amplitude MHD turbulence, the multi-order structure functions and PDFs have different features in the quasi-parallel and quasi-perpendicular directions: a monofractal scaling and Gaussian-like distribution in the former, and a conversion of a monofractal scaling and Gaussian-like distribution into a multifractal scaling and non-Gaussian tail distribution in the latter. These results hint that when intermittencies are abundant and intense, the multifractal scaling in the structure functions can appear even if it is in the quasi-parallel direction; otherwise, the monofractal scaling in the structure functions remains even if it is in the quasi-perpendicular direction. 7. Fault-tolerant linear optical quantum computing with small-amplitude coherent States. Science.gov (United States) Lund, A P; Ralph, T C; Haselgrove, H L 2008-01-25 Quantum computing using two coherent states as a qubit basis is a proposed alternative architecture with lower overheads but has been questioned as a practical way of performing quantum computing due to the fragility of diagonal states with large coherent amplitudes. We show that using error correction only small amplitudes (alpha>1.2) are required for fault-tolerant quantum computing. We study fault tolerance under the effects of small amplitudes and loss using a Monte Carlo simulation. The first encoding level resources are orders of magnitude lower than the best single photon scheme. 8. Numerical modeling study of the momentum deposition of small amplitude gravity waves in the thermosphere Energy Technology Data Exchange (ETDEWEB) Liu, X. [Chinese Academy of Sciences, Beijing (China). State Key Lab. of Space Weather; Henan Normal Univ., Xinxiang (China). College of Mathematics and Information Science; Xu, J. [Chinese Academy of Sciences, Beijing (China). State Key Lab. of Space Weather; Yue, J. [National Center for Atmospheric Research, Boulder, CO (United States). High Altitude Observatory; Hampton Univ., VA (United States). Atmospheric and Planetary Sciences; Vadas, S.L. [North West Research Associates, Inc., Boulder, CO (United States) 2013-03-01 We study the momentum deposition in the thermosphere from the dissipation of small amplitude gravity waves (GWs) within a wave packet using a fully nonlinear two-dimensional compressible numerical model. The model solves the nonlinear propagation and dissipation of a GW packet from the stratosphere into the thermosphere with realistic molecular viscosity and thermal diffusivity for various Prandtl numbers. The numerical simulations are performed for GW packets with initial vertical wavelengths ({lambda}{sub z}) ranging from 5 to 50 km. We show that {lambda}{sub z} decreases in time as a GW packet dissipates in the thermosphere, in agreement with the ray trace results of Vadas and Fritts (2005) (VF05). We also find good agreement for the peak height of the momentum flux (z{sub diss}) between our simulations and VF05 for GWs with initial {lambda}{sub z} {<=} 2{pi}H in an isothermal, windless background, where H is the density scale height.We also confirm that z{sub diss} increases with increasing Prandtl number. We include eddy diffusion in the model, and find that the momentum deposition occurs at lower altitudes and has two separate peaks for GW packets with small initial {lambda}{sub z}. We also simulate GW packets in a non-isothermal atmosphere. The net {lambda}{sub z} profile is a competition between its decrease from viscosity and its increase from the increasing background temperature. We find that the wave packet disperses more in the non-isothermal atmosphere, and causes changes to the momentum flux and {lambda}{sub z} spectra at both early and late times for GW packets with initial {lambda}{sub z} {>=} 10 km. These effects are caused by the increase in T in the thermosphere, and the decrease in T near the mesopause. (orig.) 9. Analytical solutions for the surface response to small amplitude perturbations in boundary data in the shallow-ice-stream approximation Directory of Open Access Journals (Sweden) G. H. Gudmundsson 2008-07-01 Full Text Available New analytical solutions describing the effects of small-amplitude perturbations in boundary data on flow in the shallow-ice-stream approximation are presented. These solutions are valid for a non-linear Weertman-type sliding law and for Newtonian ice rheology. Comparison is made with corresponding solutions of the shallow-ice-sheet approximation, and with solutions of the full Stokes equations. The shallow-ice-stream approximation is commonly used to describe large-scale ice stream flow over a weak bed, while the shallow-ice-sheet approximation forms the basis of most current large-scale ice sheet models. It is found that the shallow-ice-stream approximation overestimates the effects of bed topography perturbations on surface profile for wavelengths less than about 5 to 10 ice thicknesses, the exact number depending on values of surface slope and slip ratio. For high slip ratios, the shallow-ice-stream approximation gives a very simple description of the relationship between bed and surface topography, with the corresponding transfer amplitudes being close to unity for any given wavelength. The shallow-ice-stream estimates for the timescales that govern the transient response of ice streams to external perturbations are considerably more accurate than those based on the shallow-ice-sheet approximation. In particular, in contrast to the shallow-ice-sheet approximation, the shallow-ice-stream approximation correctly reproduces the short-wavelength limit of the kinematic phase speed given by solving a linearised version of the full Stokes system. In accordance with the full Stokes solutions, the shallow-ice-sheet approximation predicts surface fields to react weakly to spatial variations in basal slipperiness with wavelengths less than about 10 to 20 ice thicknesses. 10. Sexual dimorphism of short-wavelength photoreceptors in the small white butterfly, Pieris rapae crucivora NARCIS (Netherlands) Arikawa, K; Wakakuwa, M; Qiu, XD; Kurasawa, M; Stavenga, DG; Qiu, Xudong 2005-01-01 The eyes of the female small white butterfly, Pieris rapae crucivora, are furnished with three classes of short-wavelength photoreceptors, with sensitivity peaks in the ultraviolet (UV) (lambda(max) = 360 nm), violet (V) (lambda max = 425 nm), and blue (B) (lambda(max) = 453 nm) wavelength range. 11. Interferometry on small quantum systems at short wavelength International Nuclear Information System (INIS) Usenko, Sergey 2017-01-01 The present work concentrates on prototypical studies of light-induced correlated many-body dynamics in complex systems. In its course a reflective split-and-delay unit (SDU) for phase-resolved one-color pump-probe experiments with gas phase samples using VUV-XUV laser pulses was built. The collinear propagation of pump and probe pulses is ensured by the special geometry of the SDU and allows to perform phase-resolved (coherent) autocorrelation measurements. The control of the pump-probe delay with attosecond precision is established by a specially developed diagnostic tool based on an in-vacuum white light interferometer that allows to monitor the relative displacement of the SDU reflectors with nanometer resolution. Phase-resolved (interferometric) pump-probe experiments with developed SDU require spatially-resolved imaging of the ionization volume. For this an electron-ion coincidence spectrometer was built. The spectrometer enables coincident detection of photoionization products using velocity map imaging (VMI) technique for electrons and VMI or spatial imaging for ions. In first experiments using the developed SDU and the spectrometer in the ion spatial-imaging mode linear field autocorrelation of free-electron laser pulses at the central wavelength of 38 nm was recorded. A further focus of the work were energy- and time-resolved resonant two-photon ionization experiments using short tunable UV laser pulses on C_6_0 fullerene. The experiments demonstrated that dipole-selective excitation on a timescale faster than the characteristic intramolecular energy dissipation limits the number of accessible excitation pathways and thus results in a narrow resonance. Time-dependent one-color pump-probe study showed that nonadiabatic (vibron) coupling is the dominant energy dissipation mechanism for high-lying electronic excited states in C_6_0. 12. Interferometry on small quantum systems at short wavelength Energy Technology Data Exchange (ETDEWEB) Usenko, Sergey 2017-01-15 The present work concentrates on prototypical studies of light-induced correlated many-body dynamics in complex systems. In its course a reflective split-and-delay unit (SDU) for phase-resolved one-color pump-probe experiments with gas phase samples using VUV-XUV laser pulses was built. The collinear propagation of pump and probe pulses is ensured by the special geometry of the SDU and allows to perform phase-resolved (coherent) autocorrelation measurements. The control of the pump-probe delay with attosecond precision is established by a specially developed diagnostic tool based on an in-vacuum white light interferometer that allows to monitor the relative displacement of the SDU reflectors with nanometer resolution. Phase-resolved (interferometric) pump-probe experiments with developed SDU require spatially-resolved imaging of the ionization volume. For this an electron-ion coincidence spectrometer was built. The spectrometer enables coincident detection of photoionization products using velocity map imaging (VMI) technique for electrons and VMI or spatial imaging for ions. In first experiments using the developed SDU and the spectrometer in the ion spatial-imaging mode linear field autocorrelation of free-electron laser pulses at the central wavelength of 38 nm was recorded. A further focus of the work were energy- and time-resolved resonant two-photon ionization experiments using short tunable UV laser pulses on C{sub 60} fullerene. The experiments demonstrated that dipole-selective excitation on a timescale faster than the characteristic intramolecular energy dissipation limits the number of accessible excitation pathways and thus results in a narrow resonance. Time-dependent one-color pump-probe study showed that nonadiabatic (vibron) coupling is the dominant energy dissipation mechanism for high-lying electronic excited states in C{sub 60}. 13. Small amplitude variable charge dust Bernstein-Greene-Kruskal double layers Energy Technology Data Exchange (ETDEWEB) Amour, Rabia [Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Sciences - Physics, U.S.T.H.B, Bab-Ezzouar, B.P. 32, El Alia, Algiers 16111 (Algeria); Tribeche, Mouloud [Plasma Physics Group, Theoretical Physics Laboratory, Faculty of Sciences - Physics, U.S.T.H.B, Bab-Ezzouar, B.P. 32, El Alia, Algiers 16111 (Algeria)], E-mail: mouloud-tribeche@lycos.com 2009-05-11 A first theoretical attempt is made to investigate small amplitude, variable charge dust Bernstein-Greene-Kruskal (BGK) double layers (DLs). The nature of the dust BGK-DLs (compressive or rarefactive), their strength and thickness depend sensitively on the net negative charge residing on the grain surface, the dust grain dynamics and, more interestingly, on the ion-to-electron temperatures ratio. 14. Small amplitude variable charge dust Bernstein-Greene-Kruskal double layers International Nuclear Information System (INIS) Amour, Rabia; Tribeche, Mouloud 2009-01-01 A first theoretical attempt is made to investigate small amplitude, variable charge dust Bernstein-Greene-Kruskal (BGK) double layers (DLs). The nature of the dust BGK-DLs (compressive or rarefactive), their strength and thickness depend sensitively on the net negative charge residing on the grain surface, the dust grain dynamics and, more interestingly, on the ion-to-electron temperatures ratio. 15. On the Numerical Modeling of Fluid Instabilities in the Small-Amplitude Limit Science.gov (United States) Zalesak, Steven; Schmitt, A. J.; Velikovich, A. L. 2008-11-01 The problem we wish to address is that of accurately modeling the evolution of small-amplitude perturbations to a time- dependent flow, where the unperturbed flow itself exhibits large-amplitude temporal and spatial variations. In particular, we wish to accurately model the evolution of small-amplitude perturbations to an imploding ICF pellet, which is subject to both Richtmyer-Meshkov and Rayleigh-Taylor instabilities. This modeling is difficult despite the expected linear evolution of the perturbations themselves, because these perturbations are embedded in a highly nonlinear, strongly- shocked, and highly complex flow field which in and of itself stresses numerical computation capabilities, and whose simulation often employs numerical techniques which were not designed with the proper treatment of small-amplitude perturbations in mind. We will review some of the techniques that we have found to be of use toward this end, including the imposition of a differentiability condition'' on the component numerical algorithms of the codes which implement such modeling, the appropriate representation of interfaces in an Eulerian hydrodynamics context, and the role of exact energy conservation. 16. Small-amplitude limit of the spectral transform for the periodic Korteweg-de Vries equation Energy Technology Data Exchange (ETDEWEB) Osborne, A R; Bergamasco, L 1985-02-01 The inverse spectral transform for the periodic Korteweg-de Vries equation is investigated in the limit for small-amplitude waves and the inverse Fourier transform is recovered. In the limiting process we find that the widths of the forbidden bands approach the amplitudes of the Fourier spectrum. The number of spectral bands is estimated from Fourier theory and depends explicitly on the assumed spatial discretization in the wave amplitude function (potential). This allows one to estimate the number of degrees of freedom in a discrete (and, therefore, finite-banded) potential. An essential feature of the calculations is that all results for the periodic problem are cast in terms of the infinite-line reflection and transmission coefficients b(k), a(k). Thus the connection between the whole-line and periodic problems is clear at every stage of the computations. 17. Small--radiation-amplitude dynamical voltage model of an irradiated, externally unbiased Josephson tunnel junction International Nuclear Information System (INIS) 1988-01-01 A theory is presented for the nonequilibrium voltage states of an irradiated Josephson junction shunted by an external resistor but with no external current or voltage biasing. This device, referred to as a free-running Josephson junction, is modeled in a small--radiation-amplitude, deterministic regime extending the previous work of Shenoy and Agarwal. The time-averaged induced voltage is treated as a dynamical variable, the external radiation is modeled as a current source, and the induced junction-radiation vector potential, with and without a mode structure, is treated to first order in the driving currents. A dynamical equation for the time-averaged induced voltage yields a (nonequilibrium) steady-state relation between the time-averaged induced voltage and the incident radiation amplitude valid for a wide range of voltages, including zero. Regions of bistability occur in the voltage--versus--incident-amplitude curves, some of which are dependent on the external resistor. The zero-voltage state breaks down, as the external radiation amplitude is increased, at a critical value of the incident-radiation amplitude inversely proportional to the external resistance 18. Dynamics of the nuclear one-body density: small amplitude regime International Nuclear Information System (INIS) Nemes, M.C.; Toledo Piza, A.F.R. de. 1984-01-01 A microscopic treatment for the small amplitude limite of the equations of motion for the nuclear one-body density is presented. These were derived previously by means of projection techniques, and allow for the explicit separation of mean-field and collision effects which result from the dynamics of many-body correlations. The form of the nuclear response in the presence of collision effects is derived. An illustrative application to a soluble model is discussed. (Author) [pt 19. Dual Shapiro-Virasoro amplitudes in the dipole picture of QCD at small x Science.gov (United States) Peschanski, R. 1997-02-01 Using the dipole picture describing the 1/(NC) limit of QCD at small x and the conformal invariance properties of the BFKL kernel in transverse coordinate space, we show that the 1 ==> p dipole densities can be expressed in terms of dual Shapiro-Virasoro amplitudes B2p+2 and their generalization including non-zero conformal spins. We discuss the possibility of an effective closed string theory of interacting QCD dipoles. 20. Small and large amplitude movement of the unstable interface between two immiscible fluids Energy Technology Data Exchange (ETDEWEB) Aribert, J M; Thirriot, C 1970-01-01 The flow of immiscible fluids in a confined flow channel is accompanied by a deformation of the surface of separation when the stability conditions are not fulfilled. A simplified schematic for the problem is given, and the characteristic surface perturbation is calculated analytically and numerically. The perturbation is characterized by a wavelength, an amplitude, and the shape of the perturbation at a sufficient distance from the front. Two asymptotic cases are fully discussed: the creation of a wave in the surface, and the shape of a fully developed perturbation. Experimental results from 2 Hele-Shaw models are in satisfactory agreement with the theoretical predictions. Further studies will be concerned with variable rate flow, geometrically divergent flow, layered flow with variable viscosity between layers, and non-Newtonian flow. 1. Effect of Pressure Anisotropy on the m = 1 Small Wavelength Modes in Z-Pinches Science.gov (United States) Faghihi, M. 1987-05-01 A generalization of Freidberg's perpendicular MHD model is used to investigate the effect of pressure anisotropy on the small wavelength internal kink (m = 1) mode instability in a Z-Pinch. A normal mode analysis of perturbed motion of an incompressible, collisionless and cylindrical plasma is performed. The stability criterion is (rΣB2)' = 0. 2. Stability of small axial wavelength internal kink modes of an anisotropic plasma International Nuclear Information System (INIS) Faghihi, M.; Scheffel, J. 1987-03-01 The double adiabatic equations are used to study the stability of a cylindrical Z-pinch with respect to small axial wavelength, internal kink (m>=1)modes. It is found that marginally (ideally) unstable, isotropic equilibria are stabilized. Also constant current density equilibria can be stabilized for P per >P par and large β per . (authors) 3. High-Amplitude Atlantic Hurricanes Produce Disparate Mortality in Small, Low-Income Countries. Science.gov (United States) Dresser, Caleb; Allison, Jeroan; Broach, John; Smith, Mary-Elise; Milsten, Andrew 2016-12-01 Hurricanes cause substantial mortality, especially in developing nations, and climate science predicts that powerful hurricanes will increase in frequency during the coming decades. This study examined the association of wind speed and national economic conditions with mortality in a large sample of hurricane events in small countries. Economic, meteorological, and fatality data for 149 hurricane events in 16 nations between 1958 and 2011 were analyzed. Mortality rate was modeled with negative binomial regression implemented by generalized estimating equations to account for variable population exposure, sequence of storm events, exposure of multiple islands to the same storm, and nonlinear associations. Low-amplitude storms caused little mortality regardless of economic status. Among high-amplitude storms (Saffir-Simpson category 4 or 5), expected mortality rate was 0.72 deaths per 100,000 people (95% confidence interval [CI]: 0.16-1.28) for nations in the highest tertile of per capita gross domestic product (GDP) compared with 25.93 deaths per 100,000 people (95% CI: 13.30-38.55) for nations with low per capita GDP. Lower per capita GDP and higher wind speeds were associated with greater mortality rates in small countries. Excessive fatalities occurred when powerful storms struck resource-poor nations. Predictions of increasing storm amplitude over time suggest increasing disparity between death rates unless steps are taken to modify the risk profiles of poor nations. (Disaster Med Public Health Preparedness. 2016;10:832-837). 4. Observation of small sub-pulses out of the delayed-interference signal-wavelength converter DEFF Research Database (Denmark) Sakaguchi, J.; Nielsen, Mads Lønstrup; Ohira, T. 2005-01-01 The generation of small sub-pulses in the delayed-interference signal-wavelength converter (DISC), which has been studied for use in future 160-Gb/s optical time division multiplexing-wavelength division multiplexing (OTDM-WDM) communication systems, was recently predicted as a potential problem....... In this work, we have experimentally verified the generation of such pulses and its mechanism. In the experiments we used 3.8-ps-long 1.56-mu m input pulses with repetition frequencies from 12.5 to 25.0 GHz and a cross-correlation monitoring system with a time resolution of approximately 2 ps.... 5. The response of a turbulent boundary layer to a small-amplitude traveling wave International Nuclear Information System (INIS) Howes, F.A. 1986-01-01 We study the response of a turbulent boundary layer to an outer-flow disturbance in the form of a small-amplitude wave travelling along the bottom of a smooth channel. In a previous paper we proposed a model for the viscous attenuation of a wave propagating along the interface between two superposed fluids inside a laminar boundary layer attached to the bottom wall. We obtained precise estimates on the amount of attenuation suffered by the oscillatory component of the motion as a result of viscous dissipation. This was accomplished by means of a representation of the solution as the asymptotic sum of a Blasius boundary layer profile and a modified Stokes layer profile. The present paper contains a similar asymptotic decomposition of the solution of the appropriate turbulent Prandtl equations when the outer flow is a small-amplitude travelling wave, and so it may be considered an extension of our previous work to the more realistic case of turbulent flow. 4 refs 6. Interactions between finite amplitude small and medium-scale waves in the MLT region. Science.gov (United States) Heale, C. J.; Snively, J. B. 2016-12-01 Small-scale gravity waves can propagate high into the thermosphere and deposit significant momentum and energy into the background flow [e.g., Yamada et al., 2001, Fritts et al., 2014]. However, their propagation, dissipation, and spectral evolution can be significantly altered by other waves and dynamics and the nature of these complex interactions are not yet well understood. While many ray-tracing and time-dependent modeling studies have been performed to investigate interactions between waves of varying scales [e.g., Eckermann and Marks .1996, Sartelet. 2003, Liu et al. 2008, Vanderhoff et al., 2008, Senf and Achatz., 2011, Heale et al., 2015], the majority of these have considered waves of larger (tidal) scales, or have simplified one of the waves to be an imposed "background" and discount (or limit) the nonlinear feedback mechanisms between the two waves. In reality, both waves will influence each other, especially at finite amplitudes when nonlinear effects become important or dominant. We present a study of fully nonlinear interactions between small-scale 10s km, 10 min period) and medium-scale wave packets at finite amplitudes, which include feedback between the two waves and the ambient atmosphere. Time-dependence of the larger-scale wave has been identified as an important factor in reducing reflection [Heale et al., 2015] and critical level effects [Sartelet, 2003, Senf and Achatz, 2011], we choose medium-scale waves of different periods, and thus vertical scales, to investigate how this influences the propagation, filtering, and momentum and energy deposition of the small-scale waves, and in turn how these impacts affect the medium-scale waves. We also consider the observable features of these interactions in the mesosphere and lower thermosphere. 7. A velocity-amplified electromagnetic energy harvester for small amplitude vibration Science.gov (United States) Klein, J.; Zuo, L. 2017-09-01 Dedicated, self-powered wireless sensors are widely being studied for use throughout many industries to monitor everyday operations, maintain safety, and report performance characteristics. To enable sensors to power themselves, harvesting energy from machine vibration has been studied, however, its overall effectiveness can be hampered due to small vibration amplitudes and thus limited harvestable energy density. This paper addresses the issue by proposing a novel vibration energy harvester architecture in which a compliant mechanism and proof mass system is used to amplify the vibrational velocity of machine vibration for a linear electromagnetic generator. A prototype has been fabricated and experimentally characterized to verify its effectiveness. When operating at its natural frequency in a low base amplitude, 0.001 inch (25.4 μm) at 19.4 Hz, during lab tests, the harvester has been shown to produce up to 0.91 V AC open voltage, and a maximum power of 2 mW, amplifying the relative proof mass velocity by approximately 5.4 times. This method of locally increasing the machine vibrational velocity has been shown to be a viable option for increasing the potential power output of an energy harvester. In addition, a mathematical model is created based on pseudo-rigid-body dynamics and the analysis matches closely with experiments. 8. Small-amplitude vibrations at a finite temperature in the liquid drop model International Nuclear Information System (INIS) Providencia, J. da Jr. 1991-01-01 The ground state of a hot nucleus is studied in the classical limit. The equations of motion and boundary conditions of the liquid drop model are derived from the variational principle. The effect of the surface tension is taken into account. The temperature dependence of small-amplitude vibrations in the liquid drop model is investigated. It is shown that the breathing mode suffers a 6.3% decrease in energy when the temperature increases from 0 to 5 MeV. The present model allows for a description of surface modes with an A -1/2 dependence of the energy. It is also found that the surface modes will show an appreciable temperature dependence if a reasonable temperature dependence of the surface tension is postulated. It is shown that the model satisfies the energy-weighted sum rule and the inverse energy-weighted sum rule. (orig.)D 9. Effect of pressure anisotropy on the m=1 small wavelength modes in Z-pinches Energy Technology Data Exchange (ETDEWEB) Faghihi, M. (Royal Inst. of Tech., Stockholm, Sweden. Dept. of Plasma Physics and Fusion Research) 1987-05-01 A generalization of Freidberg's perpendicular MHD model is used to investigate the effect of pressure anisotropy on the small wavelength internal kink (m=1) mode instability in a Z-Pinch. A normal mode analysis of perturbed motion of an incompressible, collisionless and cylindrical plasma is performed. The stability criterion is (r{SIGMA}B{sup 2})' {le} 0, where {SIGMA} = 1 - (P{sub parallel} - P{sub perpendicular} {sub to})/B/sup 2/. It cannot be fulfilled without violation of the fire hose stability condition {SIGMA} {ge} 0. 10. Effect of pressure anisotropy on the m=1 small wavelength modes in Z-pinches International Nuclear Information System (INIS) Faghihi, M. 1987-01-01 A generalization of Freidberg's perpendicular MHD model is used to investigate the effect of pressure anisotropy on the small wavelength internal kink (m=1) mode instability in a Z-Pinch. A normal mode analysis of perturbed motion of an incompressible, collisionless and cylindrical plasma is performed. The stability criterion is (rΣB 2 )' ≤ 0, where Σ = 1 - (P parallel - P perpendicular to )/B 2 . It cannot be fulfilled without violation of the fire hose stability condition Σ ≥ 0. (orig.) 11. Interpretation of the shape of electron diffraction spots from small polyhedral crystals by means of the crystal shape amplitude International Nuclear Information System (INIS) Neumann, W.; Hofmeister, H.; Heydenreich, J.; Komrska, J. 1988-01-01 The influence of the crystal shape on the fine structure of transmission electron diffraction (TED) patterns described by the crystal shape amplitude is discussed. A general algebraic expression for the crystal shape amplitude of any crystal polyhedron is used for computing the intensity distribution of TED reflections. The computer simulation method is applied to the analysis of the fine structure of TED patterns of small gold and palladium crystals having octahedral and tetrahedral habits. (orig.) 12. Finite Amplitude Ocean Waves Indian Academy of Sciences (India) wavelength, they are called shallow water waves. In the ... Deep and intermediate water waves are dispersive as the velocity of these depends on wavelength. This is not the ..... generation processes, the finite amplitude wave theories are very ... 13. Linking optical and electrical small amplitude perturbation techniques for dynamic performance characterization of dye solar cells. Science.gov (United States) Halme, Janne 2011-07-21 This paper unifies the analytical models used widely but thus far mostly separately for electrical and optical small amplitude perturbation measurements of nanostructured electrochemical dye solar cells (DSC): electrochemical impedance spectroscopy (EIS), intensity-modulated photocurrent spectroscopy (IMPS) and intensity-modulated photovoltage spectroscopy (IMVS). The models are linked by expressing the kinetic boundary condition used for solving the time-dependent continuity equation of electrons in IMPS and IMVS analysis in terms of the series and parallel impedance components found in the complete equivalent circuit impedance model of DSC. As a result, analytical expressions are derived for potentiostatic IMPS and galvanostatic IMVS transfer functions of complete DSCs that are applicable at any operating point along the solar cell current-voltage (IV) curve. In agreement with the theory, impedance spectrum calculated as a ratio of IMVS and IMPS transfer functions measured near the maximum power point matches exactly with the impedance spectrum measured directly with EIS. Consequently, both IMPS-IMVS and EIS yield equal estimates for the electron diffusion length. The role of the chemical capacitance of the nanostructured semiconductor photoelectrode in the interpretation of the so-called RC attenuation of the IMPS response is clarified, as well as the capacitive frequency dispersion in IMPS and IMVS. This journal is © the Owner Societies 2011 14. Small amplitude two dimensional electrostatic excitations in a magnetized dusty plasma with q-distributed electrons Science.gov (United States) 2016-07-01 The propagation of linear and nonlinear electrostatic waves is investigated in magnetized dusty plasma with stationary negatively or positively charged dust, cold mobile ions and non-extensive electrons. Two normal modes are predicted in the linear regime, whose characteristics are investigated parametrically, focusing on the effect of electrons non-extensivity, dust charge polarity, concentration of dust and magnetic field strength. Using the reductive perturbation technique, a Zakharov-Kuznetsov (ZK) type equation is derived which governs the dynamics of small-amplitude solitary waves in magnetized dusty plasma. The properties of the solitary wave structures are analyzed numerically with the system parameters i.e. electrons non-extensivity, concentration of dust, polarity of dust and magnetic field strength. Following Allen and Rowlands (J. Plasma Phys. 53:63, 1995), we have shown that the pulse soliton solution of the ZK equation is unstable, and have analytically traced the dependence of the instability growth rate on the nonextensive parameter q for electrons, dust charge polarity and magnetic field strength. The results should be useful for understanding the nonlinear propagation of DIA solitary waves in laboratory and space plasmas. 15. Small amplitude waves and linear firehose and mirror instabilities in rotating polytropic quantum plasma Science.gov (United States) Bhakta, S.; Prajapati, R. P.; Dolai, B. 2017-08-01 The small amplitude quantum magnetohydrodynamic (QMHD) waves and linear firehose and mirror instabilities in uniformly rotating dense quantum plasma have been investigated using generalized polytropic pressure laws. The QMHD model and Chew-Goldberger-Low (CGL) set of equations are used to formulate the basic equations of the problem. The general dispersion relation is derived using normal mode analysis which is discussed in parallel, transverse, and oblique wave propagations. The fast, slow, and intermediate QMHD wave modes and linear firehose and mirror instabilities are analyzed for isotropic MHD and CGL quantum fluid plasmas. The firehose instability remains unaffected while the mirror instability is modified by polytropic exponents and quantum diffraction parameter. The graphical illustrations show that quantum corrections have a stabilizing influence on the mirror instability. The presence of uniform rotation stabilizes while quantum corrections destabilize the growth rate of the system. It is also observed that the growth rate stabilizes much faster in parallel wave propagation in comparison to the transverse mode of propagation. The quantum corrections and polytropic exponents also modify the pseudo-MHD and reverse-MHD modes in dense quantum plasma. The phase speed (Friedrichs) diagrams of slow, fast, and intermediate wave modes are illustrated for isotropic MHD and double adiabatic MHD or CGL quantum plasmas, where the significant role of magnetic field and quantum diffraction parameters on the phase speed is observed. 16. Effect of gyroviscosity on the small axial wavelength internal kink instability in the Z-pinch International Nuclear Information System (INIS) Scheffel, J.; Faghihi, M. 1987-03-01 The effect of ion gyroviscosity on the stability of m=1 small axial wavelength kinks in a z-pinch with purely poloidal magnetic field is investigated by normal mode methods. We use the Incompressible FLR MHD model; a collisionless fluid model introduced to include Finite Larmor Radius effects. For vanishing Larmor radius, a stability criterion which coincides with that of ideal MHD is found; the Kadomtsev criterion 2rdp/dr+m/sp2/B/sp2///my//sb0/> or = 0. This criterion predicts instability unless the current density becomes singular at the centre of the pinch. When the Larmor radius terms of the ion pressure tensor are included, we find that marginally unstable (ideal) modes are stabilized. (authors) 17. Assessment of UVA-Riboflavin Corneal Cross-Linking Using Small Amplitude Oscillatory Shear Measurements. Science.gov (United States) Aslanides, Ioannis M; Dessi, Claudia; Georgoudis, Panagiotis; Charalambidis, Georgios; Vlassopoulos, Dimitris; Coutsolelos, Athanassios G; Kymionis, George; Mukherjee, Achyut; Kitsopoulos, Theofanis N 2016-04-01 The effect of ultraviolet (UV)-riboflavin cross-linking (CXL) has been measured primarily using the strip extensometry technique. We propose a simple and reliable methodology for the assessment of CXL treatment by using an established rheologic protocol based on small amplitude oscillatory shear (SAOS) measurements. It provides information on the average cross-link density and the elastic modulus of treated cornea samples. Three fresh postmortem porcine corneas were used to study the feasibility of the technique, one serving as control and two receiving corneal collagen cross-linking treatment. Subsequently, five pairs of fresh postmortem porcine corneas received corneal collagen cross-linking treatment with riboflavin and UVA-irradiation (370 nm; irradiance of 3 mW/cm2) for 30 minutes (Dresden protocol); the contralateral porcine corneas were used as control samples. After the treatment, the linear viscoelastic moduli of the corneal samples were measured using SAOS measurements and the average cross-linking densities extracted. For all cases investigated, the dynamic moduli of the cross-linked corneas were higher compared to those of the corresponding control samples. The increase of the elastic modulus of the treated samples was between 122% and 1750%. The difference was statistically significant for all tested samples (P = 0.018, 2-tailed t-test). We report a simple and accurate methodology for quantifying the effects of cross-linking on porcine corneas treated with the Dresden protocol by means of SAOS measurements in the linear regime. The measured dynamic moduli, elastic and viscous modulus, represent the energy storage and energy dissipation, respectively. Hence, they provide a means to assess the changing physical properties of the cross-linked collagen networks after CXL treatment. 18. Dissipation and oscillatory solvation forces in confined liquids studied by small amplitude atomic force spectroscopy NARCIS (Netherlands) de Beer, Sissi; van den Ende, Henricus T.M.; Mugele, Friedrich 2010-01-01 We determine conservative and dissipative tip–sample interaction forces from the amplitude and phase response of acoustically driven atomic force microscope (AFM) cantilevers using a non-polar model fluid (octamethylcyclotetrasiloxane, which displays strong molecular layering) and atomically flat 19. Investigation of Planets and Small Bodies Using Decameter Wavelength Radar Sounders Science.gov (United States) Safaeinili, A. 2003-12-01 Decameter wavelength radar sounders provide a unique capability for the exploration of subsurface of planets and internal structure of small bodies. Recently, a number of experimental radar sounding instruments have been proposed and/or are planned to become operational in the near future. The first of these radar sounders is MARSIS (Picardi et al.) that is about to arrive at Mars on ESA's Mars Express for a two-year mission. The second radar sounder, termed SHARAD (Seu et. al), will fly on NASA's Mars Reconnaissance orbiter in 2005. MARSIS and SHARAD have complementary science objectives in that MARSIS (0.1-5.5 MHz) is designed to explore the deep subsurface with a depth resolution of ˜100 m while SHARAD (15-25 MHz) focuses its investigation to near-surface (generation of radar sounders will benefit from high power and high data rate capability that is made available through the use of Nuclear Electric generators. An example of such high-capability mission is the Jovian Icy Moons Orbiter (JIMO) where, for example, the radar sounder can be used to explore beneath the icy surfaces of Europa in search of the ice/ocean interface. The decameter wave radar sounder is probably the only instrument that has the potential of providing an accurate estimate for the ocean depth. Another exciting and rewarding area of application for planetary radar sounding is the investigation of the deep interior of small bodies (asteroids and comets). The small size of asteroids and comets provides the opportunity to collect data in a manner that enables Radio Reflection Tomographic (RRT) reconstruction of the body in the same manner that a medical ultrasound probe can image the interior of our body. This paper provides an overview of current technical capabilities and challenges and the potential of radio sounders in the investigation of planets and small bodies. 20. The small amplitude of density turbulence in the inner solar wind Directory of Open Access Journals (Sweden) S. R. Spangler 2003-01-01 Full Text Available Very Long Baseline Interferometer (VLBI observations were made of radio sources close to the Sun, whose lines of sight pass through the inner solar wind (impact parameters 16-26 RE. Power spectra were analyzed of the interferometer phase fluctuations due to the solar wind plasma. These power spectra provide information on the level of plasma density fluctuations on spatial scales of roughly one hundred to several thousand kilometers. By specifying an outer scale to the turbulence spectrum, we can estimate the root-mean-square (rms amplitude of the density fluctuations. The data indicate that the rms fluctuation in density is only about 10% of the mean density. This value is low, and consistent with extrapolated estimates from more distant parts of the solar wind. Physical speculations based on this result are presented. 1. Effect of electron temperature on small-amplitude electron acoustic solitary waves in non-planar geometry Science.gov (United States) Bansal, Sona; Aggarwal, Munish; Gill, Tarsem Singh 2018-04-01 Effects of electron temperature on the propagation of electron acoustic solitary waves in plasma with stationary ions, cold and superthermal hot electrons is investigated in non-planar geometry employing reductive perturbation method. Modified Korteweg-de Vries equation is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdV equation reveal that the phase velocity of the electron acoustic waves increases as one goes from planar to non planar geometry. It is shown that the electron temperature ratio changes the width and amplitude of the solitary waves and when electron temperature is not taken into account,our results completely agree with the results of Javidan & Pakzad (2012). It is found that at small values of τ , solitary wave structures behave differently in cylindrical ( {m} = 1), spherical ( {m} = 2) and planar geometry ( {m} = 0) but looks similar at large values of τ . These results may be useful to understand the solitary wave characteristics in laboratory and space environments where the plasma have multiple temperature electrons. 2. Numerical models of salt diapir formation by down-building : the role of sedimentation rate, viscosity contrast, initial amplitude and wavelength OpenAIRE Fuchs, Lukas; Schmeling, H.; Koyi, Hemin 2011-01-01 Formation of salt diapirs has been described to be due to upbuilding (i. e. Rayleigh-Taylor like instability of salt diapirs piercing through a denser sedimentary overburden) or syndepositional down-building process (i. e. the top of the salt diapir remains at the surface all the time). Here we systematically analyse this second end-member mechanism by numerical modelling. Four parameters are varied: sedimentation rate nu(sed), salt viscosity eta(salt), amplitude delta of the initial perturba... 3. Oscillations in the hadron scattering amplitude at high energy and small momentum transfer Energy Technology Data Exchange (ETDEWEB) Nicolescu, B. [Institut de Physique Nucleaire, 91 - Orsay (France). Div. de Physique Theorique 1997-12-31 It is shown that the high precision dN/dt UA4/2 data at {radical}s = 541 GeV are compatible with the presence of Auberson - Kinoshita - Martin (AKM) type of oscillations at very small momentum transfers. These oscillations seem to be periodic in {radical}\|t\|, the corresponding period being {approx_equal} 2 x 10{sup -2} GeV. The existence of such visible oscillations suggests a general mechanism of saturation of axiomatic bounds. As an illustration the consequences for extracting the parameter {rho} = ReF/ImF from dN/dt data are also discussed. The necessity of specific future experiments in the crucially interesting TeV region of energy - at Tevatron, RHIC and LHC - is underlined. (author) 8 refs. 4. Dependence of the frequency spectrum of small amplitude vibrations superimposed on finite deformations of a nonlinear, cylindrical elastic body on residual stress KAUST Repository Gorb, Yuliya; Walton, Jay R. 2010-01-01 We model and analyze the response of nonlinear, residually stressed elastic bodies subjected to small amplitude vibrations superimposed upon large deformations. The problem derives from modeling the use of intravascular ultrasound (IVUS) imaging 5. Stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks Science.gov (United States) Ferruzzo Correa, Diego P.; Bueno, Átila M.; Castilho Piqueira, José R. 2017-04-01 In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results. 6. Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks Science.gov (United States) Pogan, Alin; Zumbrun, Kevin 2018-06-01 We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman-Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation. 7. Simulation of channeling and radiation of 855 MeV electrons and positrons in a small-amplitude short-period bent crystal Energy Technology Data Exchange (ETDEWEB) Korol, Andrei V., E-mail: korol@mbnexplorer.com [MBN Research Center, Altenhöferallee 3, 60438 Frankfurt am Main (Germany); Bezchastnov, Victor G. [A.F. Ioffe Physical-Technical Institute, Politechnicheskaya Str. 26, 194021 St. Petersburg (Russian Federation); Peter the Great St. Petersburg Polytechnic University, Politechnicheskaya 29, 195251 St. Petersburg (Russian Federation); Sushko, Gennady B.; Solov’yov, Andrey V. [MBN Research Center, Altenhöferallee 3, 60438 Frankfurt am Main (Germany) 2016-11-15 Channeling and radiation are studied for the relativistic electrons and positrons passing through a Si crystal periodically bent with a small amplitude and a short period. Comprehensive analysis of the channeling process for various bending amplitudes is presented on the grounds of numerical simulations. The features of the channeling are highlighted and elucidated within an analytically developed continuous potential approximation. The radiation spectra are computed and discussed. 8. Multi-fluid Approach to High-frequency Waves in Plasmas. II. Small-amplitude Regime in Partially Ionized Media Energy Technology Data Exchange (ETDEWEB) Martínez-Gómez, David; Soler, Roberto; Terradas, Jaume, E-mail: david.martinez@uib.es [Departament de Física, Universitat de les Illes Balears, E-07122, Palma de Mallorca (Spain) 2017-03-01 The presence of neutral species in a plasma has been shown to greatly affect the properties of magnetohydrodynamic waves. For instance, the interaction between ions and neutrals through momentum transfer collisions causes the damping of Alfvén waves and alters their oscillation frequency and phase speed. When the collision frequencies are larger than the frequency of the waves, single-fluid magnetohydrodynamic approximations can accurately describe the effects of partial ionization, since there is a strong coupling between the various species. However, at higher frequencies, the single-fluid models are not applicable and more complex approaches are required. Here, we use a five-fluid model with three ionized and two neutral components, which takes into consideration Hall’s current and Ohm’s diffusion in addition to the friction due to collisions between different species. We apply our model to plasmas composed of hydrogen and helium, and allow the ionization degree to be arbitrary. By analyzing the corresponding dispersion relation and numerical simulations, we study the properties of small-amplitude perturbations. We discuss the effect of momentum transfer collisions on the ion-cyclotron resonances and compare the importance of magnetic resistivity, and ion–neutral and ion–ion collisions on the wave damping at various frequency ranges. Applications to partially ionized plasmas of the solar atmosphere are performed. 9. MULTI-FLUID APPROACH TO HIGH-FREQUENCY WAVES IN PLASMAS. I. SMALL-AMPLITUDE REGIME IN FULLY IONIZED MEDIUM Energy Technology Data Exchange (ETDEWEB) Martínez-Gómez, David; Soler, Roberto; Terradas, Jaume, E-mail: david.martinez@uib.es [Departament de Física, Universitat de les Illes Balears, E-07122, Palma de Mallorca (Spain) 2016-12-01 Ideal magnetohydrodynamics (MHD) provides an accurate description of low-frequency Alfvén waves in fully ionized plasmas. However, higher-frequency waves in many plasmas of the solar atmosphere cannot be correctly described by ideal MHD and a more accurate model is required. Here, we study the properties of small-amplitude incompressible perturbations in both the low- and the high-frequency ranges in plasmas composed of several ionized species. We use a multi-fluid approach and take into account the effects of collisions between ions and the inclusion of Hall’s term in the induction equation. Through the analysis of the corresponding dispersion relations and numerical simulations, we check that at high frequencies ions of different species are not as strongly coupled as in the low-frequency limit. Hence, they cannot be treated as a single fluid. In addition, elastic collisions between the distinct ionized species are not negligible for high-frequency waves, since an appreciable damping is obtained. Furthermore, Coulomb collisions between ions remove the cyclotron resonances and the strict cutoff regions, which are present when collisions are not taken into account. The implications of these results for the modeling of high-frequency waves in solar plasmas are discussed. 10. Hall effect upon small wavelength kink instabilities near an elliptic magnetic stagnation line International Nuclear Information System (INIS) Spies, G.O.; Faghihi, M. 1985-12-01 To explore the question whether the Hall effect removes the unphysical feature of ideal magnetohydrodynamics of predicting unstable small wavelenght kinks near any elliptic magnetic stagnation line, a spectral analysis is performed of the motion of an incompressible plasma about cylindrical Z-pinch equilibria with circular sections. The eigenvalue loci in the complex frequency plane are derived for the equilibrium with constant current density. These show that every particular mode becomes stable as the Hall parameter exceeds a critical value. However, this critical value is a decreasing function of the ideal growth rate and has a pole at the origin, implying that there always remains an infinite reservoir of slowly growing instabilities. Correspondingly, for equilibiria with arbitrary current distributions, the stability criterion is unaffected by the Hall term. (author) 11. A new method for the determination of the real part of the hadron elastic scattering amplitude at small angles and high energies Energy Technology Data Exchange (ETDEWEB) Gauron, P. [Theory Group, Laboratoire de Physique Nucleaire et des Hautes Energies (LPNHE), CNRS, and Universite Pierre et Marie Curie, Paris (France)]. E-mail: gauron@in2p3.fr; Nicolescu, B. [Theory Group, Laboratoire de Physique Nucleaire et des Hautes Energies (LPNHE), CNRS, and Universite Pierre et Marie Curie, Paris (France)]. E-mail: nicolesc@lpnhep.in2p3.fr; Selyugin, O.V. [BLTP, JINR, Dubna, Moscow region (Russian Federation)]. E-mail: selugin@thsun1.jinr.ru 2005-11-24 A new method for the determination of the real part of the elastic scattering amplitude is examined for high energy proton-proton at small momentum transfer. This method allows us to decrease the number of model assumptions, to obtain the real part in a narrow region of momentum transfer and to test different models. The real part is computed at a given point t{sub min} near t=0 from the known Coulomb amplitude. Hence one obtains an important constraint on the real part of the forward scattering amplitude and therefore on the {rho}-parameter (measuring the ratio of the real to imaginary part of the scattering amplitude at t=0), which can be tested at LHC. 12. High-energy hadron spin-flip amplitude at small momentum transfer and new AN data from RHIC International Nuclear Information System (INIS) Cudell, J.-R.; Selyugin, O.V.; Predazzi, E. 2004-01-01 In the case of elastic high-energy hadron-hadron scattering, the impact of the large-distance contributions on the behaviour of the slopes of the spin-non-flip and of the spin-flip amplitudes is analysed. It is shown that the long tail of the hadronic potential in impact parameter space leads to a value of the slope of the reduced spin-flip amplitude larger than that of the spin-non-flip amplitude. This effect is taken into account in the calculation of the analysing power in proton-nucleus reactions at high energies. It is shown that the preliminary measurement of A N for p 12 C obtained by the E950 Collaboration indeed favours a spin-flip amplitude with a large slope. Predictions for A N at p L =250/ GeV/c are given. (orig.) 13. An analysis of the small amplitude variations in the light curves of the R Coronae Borealis variables, S Apodis and UW Centauri International Nuclear Information System (INIS) Kilkenny, D.; Flanagan, C. 1983-01-01 The small amplitude variations in the light curves of the R CrB stars, S Aps and UW Cen, are examined. The periodicity of these variations appears to be changing rapidly in S Aps and slowly, if at all, in UW Cen. The evolutionary consequences of these effects are discussed. (author) 14. Photogrammetric measurement of two-dimensional small-amplitude waves; Hakuso suiryu no nijigen bisho shinpukuha no shashin sokuryoho ni yoru sokutei Energy Technology Data Exchange (ETDEWEB) Yoshino, F. [Tottori University, Tottori (Japan). Faculty of Engineering; Urata, K. [Hitachi Zosen Corp., Osaka (Japan); Kishi, H. 1996-01-25 A photogrammetric measurement method for two-dimensional small-amplitude waves were proposed where a diffuse reflection spot is used as an index point. An equation used to obtain the still water depth was introduced. This equation was confirmed experimentally by using a laser displacement sensor which is equivalent to a camera-index-point system in principle. To confirm the applicability of this method to waves form measurement, numerical simulations of measurement by this method were carried out for sinusoidal waves and a composed wave. The results of these simulations show that the small-amplitude waves can be measured with sufficient accuracy when the water surface inclination is small. 4 refs., 14 figs., 1 tab. 15. Dependence of the frequency spectrum of small amplitude vibrations superimposed on finite deformations of a nonlinear, cylindrical elastic body on residual stress KAUST Repository Gorb, Yuliya 2010-11-01 We model and analyze the response of nonlinear, residually stressed elastic bodies subjected to small amplitude vibrations superimposed upon large deformations. The problem derives from modeling the use of intravascular ultrasound (IVUS) imaging to interrogate atherosclerotic plaques in vivo in large arteries. The goal of this investigation is twofold: (i) introduce a modeling framework for residual stress that unlike traditional Fung type classical opening angle models may be used for a diseased artery, and (ii) investigate the sensitivity of the spectra of small amplitude high frequency time harmonic vibrations superimposed on a large deformation to the details of the residual stress stored in arteries through a numerical simulation using physiologic parameter values under both low and high blood pressure loadings. The modeling framework also points the way towards an inverse problem using IVUS techniques to estimate residual stress in healthy and diseased arteries. © 2010 Elsevier Ltd. All rights reserved. 16. Structural changes of small amplitude kinetic Alfvén solitary waves due to second-order corrections International Nuclear Information System (INIS) Choi, Cheong R. 2015-01-01 The structural changes of kinetic Alfvén solitary waves (KASWs) due to higher-order terms are investigated. While the first-order differential equation for KASWs provides the dispersion relation for kinetic Alfvén waves, the second-order differential equation describes the structural changes of the solitary waves due to higher-order nonlinearity. The reductive perturbation method is used to obtain the second-order and third-order partial differential equations; then, Kodama and Taniuti's technique [J. Phys. Soc. Jpn. 45, 298 (1978)] is applied in order to remove the secularities in the third-order differential equations and derive a linear second-order inhomogeneous differential equation. The solution to this new second-order equation indicates that, as the amplitude increases, the hump-type Korteweg-de Vries solution is concentrated more around the center position of the soliton and that dip-type structures form near the two edges of the soliton. This result has a close relationship with the interpretation of the complex KASW structures observed in space with satellites 17. Detection of small-amplitude periodic surface pressure fluctuation by pressure-sensitive paint measurements using frequency-domain methods Science.gov (United States) Noda, Takahiro; Nakakita, Kazuyki; Wakahara, Masaki; Kameda, Masaharu 2018-06-01 Image measurement using pressure-sensitive paint (PSP) is an effective tool for analyzing the unsteady pressure field on the surface of a body in a low-speed air flow, which is associated with wind noise. In this study, the surface pressure fluctuation due to the tonal trailing edge (TE) noise for a two-dimensional NACA 0012 airfoil was quantitatively detected using a porous anodized aluminum PSP (AA-PSP). The emission from the PSP upon illumination by a blue laser diode was captured using a 12-bit high-speed complementary metal-oxide-semiconductor (CMOS) camera. The intensities of the captured images were converted to pressures using a standard intensity-based method. Three image-processing methods based on the fast Fourier transform (FFT) were tested to determine their efficiency in improving the signal-to-noise ratio (SNR) of the unsteady PSP data. In addition to two fundamental FFT techniques (the full data and ensemble averaging FFTs), a technique using the coherent output power (COP), which involves the cross correlation between the PSP data and the signal measured using a pointwise sound-level meter, was tested. Preliminary tests indicated that random photon shot noise dominates the intensity fluctuations in the captured PSP emissions above 200 Hz. Pressure fluctuations associated with the TE noise, whose dominant frequency is approximately 940 Hz, were successfully measured by analyzing 40,960 sequential PSP images recorded at 10 kfps. Quantitative validation using the power spectrum indicates that the COP technique is the most effective method of identification of the pressure fluctuation directly related to TE noise. It is possible to distinguish power differences with a resolution of 10 Pa^2 (4 Pa in amplitude) when the COP was employed without use of another wind-off data. This resolution cannot be achieved by the ensemble averaging FFT because of an insufficient elimination of the background noise. 18. The AKM theorem and oscillations in the hadron scattering amplitude at high energy and small momentum transfer Energy Technology Data Exchange (ETDEWEB) Gauron, P.; Nicolescu, B. [Paris-11 Univ., 91 - Orsay (France). Inst. de Physique Nucleaire; Selyugin, O.V. [Joint Inst. for Nuclear Research, Dubna (Russian Federation). Bogoliubov Lab. of Theoretical Physics 1996-10-01 It is shown that the high precision UA4/2 data for differential cross sections p-barp scattering are compatible with the presence of Auberson -Kinoshita - Martin (AKM) type of oscillations at very small momentum transfers. These oscillations seem to be periodic in {radical}\|t\|. The existence of such visible oscillations suggests a general mechanism of saturation of axiomatic bounds. As an illustration the consequences for extracting the parameter {rho} = ReF/ImF from dN/dt data are also discussed. (K.A.). 19 refs. 19. A note on conservative transport in anisotropic, heterogeneous porous media in the presence of small-amplitude transients Science.gov (United States) Naff, R.L. 1998-01-01 The late-time macrodispersion coefficients are obtained for the case of flow in the presence of a small-scale deterministic transient in a three-dimensional anisotropic, heterogeneous medium. The transient is assumed to affect only the velocity component transverse to the mean flow direction and to take the form of a periodic function. For the case of a highly stratified medium, these late-time macrodispersion coefficients behave largely as the standard coefficients used in the transport equation. Only in the event that the medium is isotropic is it probable that significant deviations from the standard coefficients would occur. 20. Increase in the Amplitude of Line-of-sight Velocities of the Small-scale Motions in a Solar Filament before Eruption Energy Technology Data Exchange (ETDEWEB) Seki, Daikichi; Isobe, Hiroaki [Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University, Sakyo, Kyoto 606-8306 (Japan); Otsuji, Kenichi; Ishii, Takako T.; Sakaue, Takahito; Hirose, Kumi, E-mail: seki@kwasan.kyoto-u.ac.jp [Kwasan and Hida Observatories, Kyoto University, Yamashina, Kyoto 607-8471 (Japan) 2017-07-10 We present a study on the evolution of the small-scale velocity field in a solar filament as it approaches the eruption. The observation was carried out by the Solar Dynamics Doppler Imager (SDDI) that was newly installed on the Solar Magnetic Activity Research Telescope at Hida Observatory. The SDDI obtains a narrowband full-disk image of the Sun at 73 channels from H α − 9.0 Å to H α + 9.0 Å, allowing us to study the line-of-sight (LOS) velocity of the filament before and during the eruption. The observed filament is a quiescent filament that erupted on 2016 November 5. We derived the LOS velocity at each pixel in the filament using the Becker’s cloud model, and made the histograms of the LOS velocity at each time. The standard deviation of the LOS velocity distribution can be regarded as a measure for the amplitude of the small-scale motion in the filament. We found that the standard deviation on the previous day of the eruption was mostly constant around 2–3 km s{sup −1}, and it slightly increased to 3–4 km s{sup −1} on the day of the eruption. It shows a further increase, with a rate of 1.1 m s{sup −2}, about three hours before eruption, and another increase, with a rate of 2.8 m s{sup −2}, about an hour before eruption. From this result we suggest that the increase in the amplitude of the small-scale motions in a filament can be regarded as a precursor of the eruption. 1. Small-pixel long wavelength infrared focal plane arrays based on InAs/GaSb Type-II superlattice Science.gov (United States) Han, Xi; Jiang, Dongwei; Wang, Guowei; Hao, Hongyue; Sun, Yaoyao; Jiang, Zhi; Lv, Yuexi; Guo, Chunyan; Xu, Yingqiang; Niu, Zhichuan 2018-03-01 The paper reports a 640 × 512 long wavelength infrared focal plane arrays (FPAs) with 15 × 15 μm2 pixels pitch based on the type II InAs/GaSb superlattice. Material grown on a 3 in. GaSb substrate exhibits a 50% cutoff wavelength of 10.2 μm across the entire wafer. The peak quantum efficiency of the detector reaches 28% at 9.1 μm without anti-reflecting coating. Maximal resistance-area products of 8.95 Ω·cm2 at 77 K and 24.4 Ω·cm2 at 45 K are achieved in a single element device indicating that the generation-recombination and tunneling mechanisms dominate the device dark current, respectively. The peak Johnson Detectivity reaches 9.66 × 1011 cm Hz1/2/W at 9.1 μm with the bias voltage of 80 mV. In the whole zone, the operability and non-uniformity for the responsivity are 97.74% and 6.41% respectively. The average noise equivalent temperature difference of 31.9 mK at 77 K is achieved with an integration time of 0.5 ms, a 300 K background and f/2 optics. 2. Exploiting Performance of Different Low-Cost Sensors for Small Amplitude Oscillatory Motion Monitoring: Preliminary Comparisons in View of Possible Integration Directory of Open Access Journals (Sweden) Elisa Benedetti 2016-01-01 Full Text Available We address the problem of low amplitude oscillatory motion detection through different low-cost sensors: a LIS3LV02DQ MEMS accelerometer, a Microsoft Kinect v2 range camera, and a uBlox 6 GPS receiver. Several tests were performed using a one-direction vibrating table with different oscillation frequencies (in the range 1.5–3 Hz and small challenging amplitudes (0.02 m and 0.03 m. A Mikrotron EoSens high-resolution camera was used to give reference data. A dedicated software tool was developed to retrieve Kinect v2 results. The capabilities of the VADASE algorithm were employed to process uBlox 6 GPS receiver observations. In the investigated time interval (in the order of tens of seconds the results obtained indicate that displacements were detected with the resolution of fractions of millimeters with MEMS accelerometer and Kinect v2 and few millimeters with uBlox 6. MEMS accelerometer displays the lowest noise but a significant bias, whereas Kinect v2 and uBlox 6 appear more stable. The results suggest the possibility of sensor integration both for indoor (MEMS accelerometer + Kinect v2 and for outdoor (MEMS accelerometer + uBlox 6 applications and seem promising for structural monitoring applications. 3. Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation. Science.gov (United States) Van Gorder, Robert A 2013-04-01 We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely. 4. Small amplitude atomic force spectroscopy NARCIS (Netherlands) de Beer, Sissi; van den Ende, Henricus T.M.; Ebeling, Daniel; Mugele, Friedrich Gunther; Bhushan, Bharat 2011-01-01 Over the years atomic force microscopy has developed from a pure imaging technique to a tool that can be employed for measuring quantitative tip–sample interaction forces. In this chapter we provide an overview of various techniques to extract quantitative tip–sample forces focusing on both 5. Mean flow produced by small-amplitude vibrations of a liquid bridge with its free surface covered with an insoluble surfactant Science.gov (United States) Carrión, Luis M.; Herrada, Miguel A.; Montanero, José M.; Vega, José M. 2017-09-01 As is well known, confined fluid systems subject to forced vibrations produce mean flows, called in this context streaming flows. These mean flows promote an overall mass transport in the fluid that has consequences in the transport of passive scalars and surfactants, when these are present in a fluid interface. Such transport causes surfactant concentration inhomogeneities that are to be counterbalanced by Marangoni elasticity. Therefore, the interaction of streaming flows and Marangoni convection is expected to produce new flow structures that are different from those resulting when only one of these effects is present. The present paper focuses on this interaction using the liquid bridge geometry as a paradigmatic system for the analysis. Such analysis is based on an appropriate post-processing of the results obtained via direct numerical simulation of the system for moderately small viscosity, a condition consistent with typical experiments of vibrated millimetric liquid bridges. It is seen that the flow patterns show a nonmonotone behavior as the Marangoni number is increased. In addition, the strength of the mean flow at the free surface exhibits two well-defined regimes as the forcing amplitude increases. These regimes show fairly universal power-law behaviors. 6. Oscillations in the hadron scattering amplitude at high energy and small momentum transfer; Oscillations dans lamplitude de diffusion hadronique a haute energie et petites moments de transfer Energy Technology Data Exchange (ETDEWEB) Gauron, Pierre; Basarab Nicolescu [Theoretical Physics Division, Inst. de Physique Nucleaire, Paris-11 Univ., 91 - Orsay (France); Selyugin, O.V. [Lab. of Theoretical Physics, Joint Inst. for Nuclear Research, Dubna (Russian Federation) 1999-10-01 We show that the high precision dN/dt UA4/2 data at {radical} = 541 GeV are compatible with the presence of Auberson-Kinoshita-Martin (AKM) type of oscillations at very small momentum transfer. These oscillations seem to be periodic in {radical}\|t\|, the corresponding period being {approx_equal} 2 {center_dot}10{sup -2} GeV. The existence of such visible oscillations suggests a general mechanism of saturation of axiomatic bounds. As an illustration the consequences for extracting the parameter {rho} = ReF/ImF from dN/dt data are also discussed. (authors) 1 ref., 2 figs. 7. Multi-wavelength high-resolution observations of a small-scale emerging magnetic flux event and the chromospheric and coronal response Energy Technology Data Exchange (ETDEWEB) Vargas Domínguez, Santiago; Kosovichev, Alexander; Yurchyshyn, Vasyl, E-mail: svargas@bbso.njit.edu [Big Bear Solar Observatory, NJIT, 40386 North Shore Lane, Big Bear City, CA 92314-9672 (United States) 2014-10-20 State-of-the-art solar instrumentation is now revealing magnetic activity of the Sun with unprecedented temporal and spatial resolutions. Observations with the 1.6 m aperture New Solar Telescope (NST) of the Big Bear Solar Observatory are making next steps in our understanding of the solar surface structure. Granular-scale magnetic flux emergence and the response of the solar atmosphere are among the key research topics of high-resolution solar physics. As part of a joint observing program with NASA's Interface Region Imaging Spectrograph (IRIS) mission on 2013 August 7, the NST observed active region NOAA 11,810 in the photospheric TiO 7057 Å band with a resolution of pixel size of 0.''034 and chromospheric He I 10830 Å and Hα 6563 Å wavelengths. Complementary data are provided by the Solar Dynamics Observatory (SDO) and Hinode space-based telescopes. The region displayed a group of solar pores, in the vicinity of which we detect a small-scale buoyant horizontal magnetic flux tube causing granular alignments and interacting with the preexisting ambient field in the upper atmospheric layers. Following the expansion of distorted granules at the emergence site, we observed a sudden appearance of an extended surge in the He I 10830 Å data (bandpass of 0.05 Å). The IRIS transition region imaging caught ejection of a hot plasma jet associated with the He I surge. The SDO/HMI data used to study the evolution of the magnetic and Doppler velocity fields reveal emerging magnetic loop-like structures. Hinode/Ca II H and IRIS filtergrams detail the connectivities of the newly emerged magnetic field in the lower solar chromosphere. From these data, we find that the orientation of the emerging magnetic field lines from a twisted flux tube formed an angle of ∼45° with the overlying ambient field. Nevertheless, the interaction of emerging magnetic field lines with the pre-existing overlying field generates high-temperature emission regions and boosts the 8. Topology optimised wavelength dependent splitters DEFF Research Database (Denmark) Hede, K. K.; Burgos Leon, J.; Frandsen, Lars Hagedorn A photonic crystal wavelength dependent splitter has been constructed by utilising topology optimisation1. The splitter has been fabricated in a silicon-on-insulator material (Fig. 1). The topology optimised wavelength dependent splitter demonstrates promising 3D FDTD simulation results....... This complex photonic crystal structure is very sensitive against small fabrication variations from the expected topology optimised design. A wavelength dependent splitter is an important basic building block for high-performance nanophotonic circuits. 1J. S. Jensen and O. Sigmund, App. Phys. Lett. 84, 2022... 9. Observation of Rayleigh - Taylor growth to short wavelengths on Nike International Nuclear Information System (INIS) Pawley, C.J.; Bodner, S.E.; Dahlburg, J.P.; Obenschain, S.P.; Schmitt, A.J.; Sethian, J.D.; Sullivan, C.A.; Gardner, J.H.; Aglitskiy, Y.; Chan, Y.; Lehecka, T. 1999-01-01 The uniform and smooth focal profile of the Nike KrF laser [S. Obenschain et al., Phys. Plasmas 3, 2098 (1996)] was used to ablatively accelerate 40 μm thick polystyrene planar targets with pulse shaping to minimize shock heating of the compressed material. The foils had imposed small-amplitude sinusoidal wave perturbations of 60, 30, 20, and 12.5 μm wavelength. The shortest wavelength is near the ablative stabilization cutoff for Rayleigh - Taylor growth. Modification of the saturated wave structure due to random laser imprint was observed. Excellent agreement was found between the two-dimensional simulations and experimental data for most cases where the laser imprint was not dominant. copyright 1999 American Institute of Physics 10. Making Displaced Holograms At Two Wavelengths Science.gov (United States) Witherow, William K.; Ecker, Andreas 1989-01-01 Two-wavelength holographic system augmented with pair of prisms to introduce small separation between holograms formed simultaneously at two wavelengths on holographic plate. Principal use in study of flows. Gradients in index of refraction of fluid caused by variations in temperature, concentration, or both. Holography at one wavelength cannot be used to distinguish between two types of variations. Difference between spacings of fringes in photographs reconstructed from holograms taken simultaneously at two different wavelengths manipulated mathematically to determine type of variation. 11. Nonlinear Saturation Amplitude in Classical Planar Richtmyer–Meshkov Instability International Nuclear Information System (INIS) Liu Wan-Hai; Jiang Hong-Bin; Ma Wen-Fang; Wang Xiang 2016-01-01 The classical planar Richtmyer–Meshkov instability (RMI) at a fluid interface supported by a constant pressure is investigated by a formal perturbation expansion up to the third order, and then according to definition of nonlinear saturation amplitude (NSA) in Rayleigh–Taylor instability (RTI), the NSA in planar RMI is obtained explicitly. It is found that the NSA in planar RMI is affected by the initial perturbation wavelength and the initial amplitude of the interface, while the effect of the initial amplitude of the interface on the NSA is less than that of the initial perturbation wavelength. Without marginal influence of the initial amplitude, the NSA increases linearly with wavelength. The NSA normalized by the wavelength in planar RMI is about 0.11, larger than that corresponding to RTI. (paper) 12. Wavelength converter technology DEFF Research Database (Denmark) Kloch, Allan; Hansen, Peter Bukhave; Poulsen, Henrik Nørskov 1999-01-01 Wavelength conversion is important since it ensures full flexibility of the WDM network layer. Progress in optical wavelength converter technology is reviewed with emphasis on all-optical wavelength converter types based on semiconductor optical amplifiers.......Wavelength conversion is important since it ensures full flexibility of the WDM network layer. Progress in optical wavelength converter technology is reviewed with emphasis on all-optical wavelength converter types based on semiconductor optical amplifiers.... 13. Diphoton generalized distribution amplitudes International Nuclear Information System (INIS) El Beiyad, M.; Pire, B.; Szymanowski, L.; Wallon, S. 2008-01-01 We calculate the leading order diphoton generalized distribution amplitudes by calculating the amplitude of the process γγ→γγ in the low energy and high photon virtuality region at the Born order and in the leading logarithmic approximation. As in the case of the anomalous photon structure functions, the γγ generalized distribution amplitudes exhibit a characteristic lnQ 2 behavior and obey inhomogeneous QCD evolution equations. 14. Two Photon Distribution Amplitudes International Nuclear Information System (INIS) El Beiyad, M.; Pire, B.; Szymanowski, L.; Wallon, S. 2008-01-01 The factorization of the amplitude of the process γγ→γγ in the low energy and high photon virtuality region is demonstrated at the Born order and in the leading logarithmic approximation. The leading order two photon (generalized) distribution amplitudes exhibit a characteristic ln Q 2 behaviour and obey new inhomogeneous evolution equations 15. An Empirical Study of Atmospheric Correction Procedures for Regional Infrasound Amplitudes with Ground Truth. Science.gov (United States) Howard, J. E. 2014-12-01 This study focusses on improving methods of accounting for atmospheric effects on infrasound amplitudes observed on arrays at regional distances in the southwestern United States. Recordings at ranges of 150 to nearly 300 km from a repeating ground truth source of small HE explosions are used. The explosions range in actual weight from approximately 2000-4000 lbs. and are detonated year-round which provides signals for a wide range of atmospheric conditions. Three methods of correcting the observed amplitudes for atmospheric effects are investigated with the data set. The first corrects amplitudes for upper stratospheric wind as developed by Mutschlecner and Whitaker (1999) and uses the average wind speed between 45-55 km altitudes in the direction of propagation to derive an empirical correction formula. This approach was developed using large chemical and nuclear explosions and is tested with the smaller explosions for which shorter wavelengths cause the energy to be scattered by the smaller scale structure of the atmosphere. The second approach isa semi-empirical method using ray tracing to determine wind speed at ray turning heights where the wind estimates replace the wind values in the existing formula. Finally, parabolic equation (PE) modeling is used to predict the amplitudes at the arrays at 1 Hz. The PE amplitudes are compared to the observed amplitudes with a narrow band filter centered at 1 Hz. An analysis is performed of the conditions under which the empirical and semi-empirical methods fail and full wave methods must be used. 16. Amplitudes, acquisition and imaging Energy Technology Data Exchange (ETDEWEB) Bloor, Robert 1998-12-31 Accurate seismic amplitude information is important for the successful evaluation of many prospects and the importance of such amplitude information is increasing with the advent of time lapse seismic techniques. It is now widely accepted that the proper treatment of amplitudes requires seismic imaging in the form of either time or depth migration. A key factor in seismic imaging is the spatial sampling of the data and its relationship to the imaging algorithms. This presentation demonstrates that acquisition caused spatial sampling irregularity can affect the seismic imaging and perturb amplitudes. Equalization helps to balance the amplitudes, and the dealing strategy improves the imaging further when there are azimuth variations. Equalization and dealiasing can also help with the acquisition irregularities caused by shot and receiver dislocation or missing traces. 2 refs., 2 figs. 17. Color guided amplitudes Energy Technology Data Exchange (ETDEWEB) Broedel, Johannes [Stanford Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA (United States); Dixon, Lance J. [SLAC National Accelerator Laboratory, Stanford University, Stanford, CA (United States) 2012-07-01 Amplitudes in gauge thoeries obtain contributions from color and kinematics. While these two parts of the amplitude seem to exhibit different symmetry structures, it turns out that they can be reorganized in a way to behave equally, which leads to the so-called color-kinematic dual representations of amplitudes. Astonishingly, the existence of those representations allows squaring to related gravitational theories right away. Contrary to the Kawaii-Levellen-Tye relations, which have been used to relate gauge theories and gravity previously, this method is applicable not only to tree amplitudes but also at loop level. In this talk, the basic technique is introduced followed by a discussion of the existence of color-kinematic dual representations for amplitudes derived from gauge theory actions which are deformed by higher-operator insertions. In addition, it is commented on the implications for deformed gravitational theories. 18. Real topological string amplitudes Energy Technology Data Exchange (ETDEWEB) Narain, K.S. [The Abdus Salam International Centre for Theoretical Physics (ICTP),Strada Costiera 11, Trieste, 34151 (Italy); Piazzalunga, N. [Simons Center for Geometry and Physics, State University of New York,Stony Brook, NY, 11794-3636 (United States); International School for Advanced Studies (SISSA) and INFN, Sez. di Trieste,via Bonomea 265, Trieste, 34136 (Italy); Tanzini, A. [International School for Advanced Studies (SISSA) and INFN, Sez. di Trieste,via Bonomea 265, Trieste, 34136 (Italy) 2017-03-15 We discuss the physical superstring correlation functions in type I theory (or equivalently type II with orientifold) that compute real topological string amplitudes. We consider the correlator corresponding to holomorphic derivative of the real topological amplitude G{sub χ}, at fixed worldsheet Euler characteristic χ. This corresponds in the low-energy effective action to N=2 Weyl multiplet, appropriately reduced to the orientifold invariant part, and raised to the power g{sup ′}=−χ+1. We show that the physical string correlator gives precisely the holomorphic derivative of topological amplitude. Finally, we apply this method to the standard closed oriented case as well, and prove a similar statement for the topological amplitude F{sub g}. 19. The Small Breathing Amplitude at the Upper Lobes Favors the Attraction of Polymorphonuclear Neutrophils to Mycobacterium tuberculosis Lesions and Helps to Understand the Evolution toward Active Disease in An Individual-Based Model. Science.gov (United States) Cardona, Pere-Joan; Prats, Clara 2016-01-01 Infection with Mycobacterium tuberculosis (Mtb) can induce two kinds of lesions, namely proliferative and exudative. The former are based on the presence of macrophages with controlled induction of intragranulomatous necrosis, and are even able to stop its physical progression, thus avoiding the induction of active tuberculosis (TB). In contrast, the most significant characteristic of exudative lesions is their massive infiltration with polymorphonuclear neutrophils (PMNs), which favor enlargement of the lesions and extracellular growth of the bacilli. We have built an individual-based model (IBM) (known as "TBPATCH") using the NetLogo interface to better understand the progression from Mtb infection to TB. We have tested four main factors previously identified as being able to favor the infiltration of Mtb-infected lesions with PMNs, namely the tolerability of infected macrophages to the bacillary load; the capacity to modulate the Th17 response; the breathing amplitude (BAM) (large or small in the lower and upper lobes respectively), which influences bacillary drainage at the alveoli; and the encapsulation of Mtb-infected lesions by the interlobular septae that structure the pulmonary parenchyma into secondary lobes. Overall, although all the factors analyzed play some role, the small BAM is the major factor determining whether Mtb-infected lesions become exudative, and thus induce TB, thereby helping to understand why this usually takes place in the upper lobes. This information will be very useful for the design of future prophylactic and therapeutic approaches against TB. 20. Detection of a periodic structure hidden in random background: the role of signal amplitude in the matched filter detection method International Nuclear Information System (INIS) Vani, V C; Chatterjee, S 2010-01-01 The matched filter method for detecting a periodic structure on a surface hidden behind randomness is known to detect up to (r 0 /Λ)≥0.11, where r 0 is the coherence length of light on scattering from the rough part and Λ is the wavelength of the periodic part of the surface-the above limit being much lower than what is allowed by conventional detection methods. The primary goal of this technique is the detection and characterization of the periodic structure hidden behind randomness without the use of any complicated experimental or computational procedures. This paper examines this detection procedure for various values of the amplitude a of the periodic part beginning from a=0 to small finite values of a. We thus address the importance of the following quantities: '(a/λ)', which scales the amplitude of the periodic part with the wavelength of light, and (r 0 /Λ), in determining the detectability of the intensity peaks. 1. Wavelength selection in traveling-wave convection in a fluid mixture International Nuclear Information System (INIS) Surko, C.M.; Eaton, K.D.; Baxter, G.W.; Iwata, K. 1993-01-01 The mechanisms by which a one-dimensional pattern of traveling waves changes wavelength (i.e. the Eckhaus instability) is studied in a binary fluid mixture. Propagating wavelength modulations develop when the Rayleigh number of the system is decreased below a wavelength-dependent threshold, commonly referred to as the Eckhaus boundary. These wavelength modulations increase in amplitude and narrow in spatial extent until they trigger the creation or annihilation of convection roll pairs and thereby change the average wavelength of the system. The authors find qualitatively different dynamics for wavelength-increasing events and wavelength-decreasing events; these differences are due to the strong wavelength dependence of the group velocity 2. Amplitude and Ascoli analysis International Nuclear Information System (INIS) Hansen, J.D. 1976-01-01 This article discusses the partial wave analysis of two, three and four meson systems. The difference between the two approaches, referred to as amplitude and Ascoli analysis is discussed. Some of the results obtained with these methods are shown. (B.R.H.) 3. Reinforcing Saccadic Amplitude Variability Science.gov (United States) Paeye, Celine; Madelain, Laurent 2011-01-01 Saccadic endpoint variability is often viewed as the outcome of neural noise occurring during sensorimotor processing. However, part of this variability might result from operant learning. We tested this hypothesis by reinforcing dispersions of saccadic amplitude distributions, while maintaining constant their medians. In a first experiment we… 4. Wavelength dependence of superhumps in VW Hyi International Nuclear Information System (INIS) Amerongen, S. van; Bovenschen, H.; Paradijs, J. van 1987-01-01 Results are presented of five-colour photometric observations of the SU UMa system VW Hyi, made on six nights during the November 1984 superoutburst. The light curve is dominated by superhump variations, whose amplitude in all passbands decreases with time (in the V-band from 0.16 mag about 4.5 day after the superoutburst reached maximum brightness, to 0.10 mag about 5 day later). The superhump light curve depends strongly on wavelength. In particular it appears that the light curves in different passbands are mutually shifted: the larger the wavelength, the more the light curve is delayed. (author) 5. Finite Amplitude Electron Plasma Waves in a Cylindrical Waveguide DEFF Research Database (Denmark) Juul Rasmussen, Jens 1978-01-01 The nonlinear behaviour of the electron plasma wave propagating in a cylindrical plasma waveguide immersed in an infinite axial magnetic field is investigated using the Krylov-Bogoliubov-Mitropolsky perturbation method, by means of which is deduced the nonlinear Schrodinger equation governing...... the long-time slow modulation of the wave amplitude. From this equation the amplitude-dependent frequency and wavenumber shifts are calculated, and it is found that the electron waves with short wavelengths are modulationally unstable with respect to long-wavelength, low-frequency perturbations... 6. Wavelength conversion devices DEFF Research Database (Denmark) Mikkelsen, Benny; Durhuus, Terji; Jørgensen, Carsten 1996-01-01 system requirements. The ideal wavelength converter should be transparent to the bit rate and signal format and provide an unchirped output signal with both a high extinction ratio and a large signal-to-noise ratio. It should allow conversion to both shorter and longer wavelengths with equal performance... 7. Light Meson Distribution Amplitudes CERN Document Server Arthur, R.; Brommel, D.; Donnellan, M.A.; Flynn, J.M.; Juttner, A.; de Lima, H.Pedroso; Rae, T.D.; Sachrajda, C.T.; Samways, B. 2010-01-01 We calculated the first two moments of the light-cone distribution amplitudes for the pseudoscalar mesons ($\\pi$ and $K$) and the longitudinally polarised vector mesons ($\\rho$, $K^$ and $\\phi$) as part of the UKQCD and RBC collaborations' $N_f=2+1$ domain-wall fermion phenomenology programme. These quantities were obtained with a good precision and, in particular, the expected effects of $SU(3)$-flavour symmetry breaking were observed. Operators were renormalised non-perturbatively and extrapolations to the physical point were made, guided by leading order chiral perturbation theory. The main results presented are for two volumes, $16^3\\times 32$ and $24^3\\times 64$, with a common lattice spacing. Preliminary results for a lattice with a finer lattice spacing, $32^3\\times64$, are discussed and a first look is taken at the use of twisted boundary conditions to extract distribution amplitudes. 8. Amplitude-modulated fiber-ring laser DEFF Research Database (Denmark) Caputo, J. G.; Clausen, Carl A. Balslev; Sørensen, Mads Peter 2000-01-01 Soliton pulses generated by a fiber-ring laser are investigated by numerical simulation and perturbation methods. The mathematical modeling is based on the nonlinear Schrödinger equation with perturbative terms. We show that active mode locking with an amplitude modulator leads to a self......-starting of stable solitonic pulses from small random noise, provided the modulation depth is small. The perturbative analysis leads to a nonlinear coupled return map for the amplitude, phase, and position of the soliton pulses circulating in the fiber-ring laser. We established the validity of this approach... 9. Long wavelength irregularities in the equatorial electrojet OpenAIRE Kudeki, E.; Farley, D. T.; Fejer, Bela G. 1982-01-01 We have used the radar interferometer technique at Jicamarca to study in detail irregularities with wavelengths of a few kilometers generated in the unstable equatorial electrojet plasma during strong type 1 conditions. In-situ rocket observations of the same instability process are discussed in a companion paper. These large scale primary waves travel essentially horizontally and have large amplitudes. The vertical electron drift velocities driven by the horizontal wave electric fields reach... 10. Effective string theory and QCD scattering amplitudes International Nuclear Information System (INIS) Makeenko, Yuri 2011-01-01 QCD string is formed at distances larger than the confinement scale and can be described by the Polchinski-Strominger effective string theory with a nonpolynomial action, which has nevertheless a well-defined semiclassical expansion around a long-string ground state. We utilize modern ideas about the Wilson-loop/scattering-amplitude duality to calculate scattering amplitudes and show that the expansion parameter in the effective string theory is small in the Regge kinematical regime. For the amplitudes we obtain the Regge behavior with a linear trajectory of the intercept (d-2)/24 in d dimensions, which is computed semiclassically as a momentum-space Luescher term, and discuss an application to meson scattering amplitudes in QCD. 11. Short wavelength FELS International Nuclear Information System (INIS) Sheffield, R.L. 1991-01-01 The generation of coherent ultraviolet and shorter wavelength light is presently limited to synchrotron sources. The recent progress in the development of brighter electron beams enables the use of much lower energy electron rf linacs to reach short-wavelengths than previously considered possible. This paper will summarize the present results obtained with synchrotron sources, review proposed short- wavelength FEL designs and then present a new design which is capable of over an order of magnitude higher power to the extreme ultraviolet. 17 refs., 10 figs 12. Short wavelength FELS Energy Technology Data Exchange (ETDEWEB) Sheffield, R.L. 1991-01-01 The generation of coherent ultraviolet and shorter wavelength light is presently limited to synchrotron sources. The recent progress in the development of brighter electron beams enables the use of much lower energy electron rf linacs to reach short-wavelengths than previously considered possible. This paper will summarize the present results obtained with synchrotron sources, review proposed short- wavelength FEL designs and then present a new design which is capable of over an order of magnitude higher power to the extreme ultraviolet. 17 refs., 10 figs. 13. Performance Analysis of Wavelength Multiplexed Sac Ocdma Codes in Beat Noise Mitigation in Sac Ocdma Systems Science.gov (United States) Alhassan, A. M.; Badruddin, N.; Saad, N. M.; Aljunid, S. A. 2013-07-01 In this paper we investigate the use of wavelength multiplexed spectral amplitude coding (WM SAC) codes in beat noise mitigation in coherent source SAC OCDMA systems. A WM SAC code is a low weight SAC code, where the whole code structure is repeated diagonally (once or more) in the wavelength domain to achieve the same cardinality as a higher weight SAC code. Results show that for highly populated networks, the WM SAC codes provide better performance than SAC codes. However, for small number of active users the situation is reversed. Apart from their promising improvement in performance, these codes are more flexible and impose less complexity on the system design than their SAC counterparts. 14. Sub-wavelength metamaterial cylinders with multiple dipole resonances DEFF Research Database (Denmark) Arslanagic, Samel; Breinbjerg, Olav 2009-01-01 It has been shown that the sub-wavelength resonances of the individual MTM cylinders also occur for electrically small configurations combining 2 or 4 cylinders. For the 2-and 4-cylinder configurations the overall size is 1/20 and 1/12.5 of the smallest wavelength, respectively. These MTM...... configuration thus offer the possibility for multi-resonant electrically small configurations.... 15. Wavelength conversion technology DEFF Research Database (Denmark) Stubkjær, Kristian 1998-01-01 Optical wavelength conversion is currently attracting much interest. This is because it enables full flexibility and eases management of WDM fibre networks. The tutorial will review existing and potential application areas. Examples of node architectures and network demonstrators that use wavelen... 16. Unifying relations for scattering amplitudes Science.gov (United States) Cheung, Clifford; Shen, Chia-Hsien; Wen, Congkao 2018-02-01 We derive new amplitudes relations revealing a hidden unity among a wideranging variety of theories in arbitrary spacetime dimensions. Our results rely on a set of Lorentz invariant differential operators which transmute physical tree-level scattering amplitudes into new ones. By transmuting the amplitudes of gravity coupled to a dilaton and two-form, we generate all the amplitudes of Einstein-Yang-Mills theory, Dirac-Born-Infield theory, special Galileon, nonlinear sigma model, and biadjoint scalar theory. Transmutation also relates amplitudes in string theory and its variants. As a corollary, celebrated aspects of gluon and graviton scattering like color-kinematics duality, the KLT relations, and the CHY construction are inherited traits of the transmuted amplitudes. Transmutation recasts the Adler zero as a trivial consequence of the Weinberg soft theorem and implies new subleading soft theorems for certain scalar theories. 17. Long wavelength irregularities in the equatorial electrojet International Nuclear Information System (INIS) Kudeki, E.; Farley, D.T.; Fejer, B.G. 1982-01-01 We have used the radar interferometer technique at Jicamarca to study in detail irregularities with wavelengths of a few kilometers generated in the unstable equatorial electrojet plasma during strong type 1 conditions. In-situ rocket observations of the same instability process are discussed in a companion paper. These large scale primary waves travel essentially horizontally and have large amplitudes. The vertical electron drift velocities driven by the horizontal wave electric fields reach or exceed the ion-acoustic velocity even though the horizontal phase velocity of the wave is considerably smaller. A straightforward extension to the long wavelength regime of the usual linear theory of the electrojet instability explains this and several other observed features of these dominant primary waves 18. Hidden beauty in multiloop amplitudes International Nuclear Information System (INIS) Cachazo, Freddy; Spradlin, Marcus; Volovich, Anastasia 2006-01-01 Planar L-loop maximally helicity violating amplitudes in N = 4 supersymmetric Yang-Mills theory are believed to possess the remarkable property of satisfying iteration relations in L. We propose a simple new method for studying iteration relations for four-particle amplitudes which involves the use of certain linear differential operators and eliminates the need to fully evaluate any loop integrals. We carry out this procedure in explicit detail for the two-loop amplitude and prove that this method can be applied to any multiloop integral, allowing a conjectured iteration relation for any given amplitude to be tested up to polynomials in logarithms 19. Two-Loop Splitting Amplitudes International Nuclear Information System (INIS) Bern, Z. 2004-01-01 Splitting amplitudes govern the behavior of scattering amplitudes at the momenta of external legs become collinear. In this talk we outline the calculation of two-loop splitting amplitudes via the unitarity sewing method. This method retains the simple factorization properties of light-cone gauge, but avoids the need for prescriptions such as the principal value or Mandelstam-Leibbrandt ones. The encountered loop momentum integrals are then evaluated using integration-by-parts and Lorentz invariance identities. We outline a variety of applications for these splitting amplitudes 20. Two-loop splitting amplitudes International Nuclear Information System (INIS) Bern, Z.; Dixon, L.J.; Kosower, D.A. 2004-01-01 Splitting amplitudes govern the behavior of scattering amplitudes at the momenta of external legs become collinear. In this talk we outline the calculation of two-loop splitting amplitudes via the unitarity sewing method. This method retains the simple factorization properties of light-cone gauge, but avoids the need for prescriptions such as the principal value or Mandelstam-Leibbrandt ones. The encountered loop momentum integrals are then evaluated using integration-by-parts and Lorentz invariance identities. We outline a variety of applications for these splitting amplitudes 1. Wavelength sweepable laser source DEFF Research Database (Denmark) 2014-01-01 Wavelength sweepable laser source is disclosed, wherein the laser source is a semiconductor laser source adapted for generating laser light at a lasing wavelength. The laser source comprises a substrate, a first reflector, and a second reflector. The first and second reflector together defines...... and having a rest position, the second reflector and suspension together defining a microelectromechanical MEMS oscillator. The MEMS oscillator has a resonance frequency and is adapted for oscillating the second reflector on either side of the rest position.; The laser source further comprises electrical...... connections adapted for applying an electric field to the MEMS oscillator. Furthermore, a laser source system and a method of use of the laser source are disclosed.... 2. Multichannel conformal blocks for scattering amplitudes Science.gov (United States) Belitsky, A. V. 2018-05-01 By performing resummation of small fermion-antifermion pairs within the pentagon form factor program to scattering amplitudes in planar N = 4 superYang-Mills theory, we construct multichannel conformal blocks within the flux-tube picture for N-sided NMHV polygons. This procedure is equivalent to summation of descendants of conformal primaries in the OPE framework. The resulting conformal partial waves are determined by multivariable hypergeometric series of Lauricella-Saran type. 3. MIT wavelength tables. Volume 2. Wavelengths by element International Nuclear Information System (INIS) Phelps, F.M. III. 1982-01-01 This volume is the first stage of a project to expand and update the MIT wavelength tables first compiled in the 1930's. For 109,325 atomic emission lines, arranged by element, it presents wavelength in air, wavelength in vacuum, wave number and intensity. All data are stored on computer-readable magnetic tape 4. Jump phenomena. [large amplitude responses of nonlinear systems Science.gov (United States) Reiss, E. L. 1980-01-01 The paper considers jump phenomena composed of large amplitude responses of nonlinear systems caused by small amplitude disturbances. Physical problems where large jumps in the solution amplitude are important features of the response are described, including snap buckling of elastic shells, chemical reactions leading to combustion and explosion, and long-term climatic changes of the earth's atmosphere. A new method of rational functions was then developed which consists of representing the solutions of the jump problems as rational functions of the small disturbance parameter; this method can solve jump problems explicitly. 5. Construction of two-dimensional Schrodinger operator with given scattering amplitude at fixed energy International Nuclear Information System (INIS) Novikov, R.G. 1986-01-01 The classical necessary properties of the scattering amplitude (reciprocity and unitarity) are, provided its L 2 norm is small, sufficient for the existence of a two-dimensional Schrodinger operator with the given scattering amplitude at fixed energy 6. Off-shell CHY amplitudes Energy Technology Data Exchange (ETDEWEB) Lam, C.S., E-mail: Lam@physics.mcgill.ca [Department of Physics, McGill University, Montreal, Q.C., H3A 2T8 (Canada); Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1 (Canada); Yao, York-Peng, E-mail: yyao@umich.edu [Department of Physics, The University of Michigan Ann Arbor, MI 48109 (United States) 2016-06-15 The Cachazo–He–Yuan (CHY) formula for on-shell scattering amplitudes is extended off-shell. The off-shell amplitudes (amputated Green's functions) are Möbius invariant, and have the same momentum poles as the on-shell amplitudes. The working principles which drive the modifications to the scattering equations are mainly Möbius covariance and energy momentum conservation in off-shell kinematics. The same technique is also used to obtain off-shell massive scalars. A simple off-shell extension of the CHY gauge formula which is Möbius invariant is proposed, but its true nature awaits further study. 7. Multiscalar production amplitudes beyond threshold CERN Document Server Argyres, E N; Kleiss, R H 1993-01-01 We present exact tree-order amplitudes for $H^ \\to n~H$, for final states containing one or two particles with non-zero three-momentum, for various interaction potentials. We show that there are potentials leading to tree amplitudes that satisfy unitarity, not only at threshold but also in the above kinematical configurations and probably beyond. As a by-product, we also calculate $2\\to n$ tree amplitudes at threshold and show that for the unbroken $\\phi^4$ theory they vanish for $n>4~$, for the Standard Model Higgs they vanish for $n\\ge 3~$ and for a model potential, respecting tree-order unitarity, for $n$ even and $n>4~$. Finally, we calculate the imaginary part of the one-loop $1\\to n$ amplitude in both symmetric and spontaneously broken $\\phi^4$ theory. 8. Scattering amplitudes in gauge theories CERN Document Server Henn, Johannes M 2014-01-01 At the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has been made in our understanding of and computational abilities with regard to scattering amplitudes in gauge theories, going beyond the traditional textbook approach. These advances build upon on-shell methods that focus on the analytic structure of the amplitudes, as well as on their recently discovered hidden symmetries. In fact, when expressed in suitable variables the amplitudes are much simpler than anticipated and hidden patterns emerge. These modern methods are of increasing importance in phenomenological applications arising from the need for high-precision predictions for the experiments carried out at the Large Hadron Collider, as well as in foundational mathematical physics studies on the S-matrix in quantum ... 9. Amplitude damping of vortex modes CSIR Research Space (South Africa) Dudley, Angela L 2010-09-01 Full Text Available An interferometer, mimicking an amplitude damping channel for vortex modes, is presented. Experimentally the action of the channel is in good agreement with that predicted theoretically. Since we can characterize the action of the channel on orbital... 10. Motivic amplitudes and cluster coordinates International Nuclear Information System (INIS) Golden, J.K.; Goncharov, A.B.; Spradlin, M.; Vergu, C.; Volovich, A. 2014-01-01 In this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf n (ℙ 3 ) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A 7,2 M and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf n (ℙ 3 ). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to A 7,2 M being expressible in terms of classical polylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster X-coordinates of a single algebra. In this respect it is similar to Abel’s 5-term dilogarithm identity 11. Nonsinglet pentagons and NMHV amplitudes Directory of Open Access Journals (Sweden) A.V. Belitsky 2015-07-01 Full Text Available Scattering amplitudes in maximally supersymmetric gauge theory receive a dual description in terms of the expectation value of the super Wilson loop stretched on a null polygonal contour. This makes the analysis amenable to nonperturbative techniques. Presently, we elaborate on a refined form of the operator product expansion in terms of pentagon transitions to compute twist-two contributions to NMHV amplitudes. To start with, we provide a novel derivation of scattering matrices starting from Baxter equations for flux-tube excitations propagating on magnon background. We propose bootstrap equations obeyed by pentagon form factors with nonsinglet quantum numbers with respect to the R-symmetry group and provide solutions to them to all orders in 't Hooft coupling. These are then successfully confronted against available perturbative calculations for NMHV amplitudes to four-loop order. 12. Nonsinglet pentagons and NMHV amplitudes Energy Technology Data Exchange (ETDEWEB) Belitsky, A.V., E-mail: andrei.belitsky@asu.edu 2015-07-15 Scattering amplitudes in maximally supersymmetric gauge theory receive a dual description in terms of the expectation value of the super Wilson loop stretched on a null polygonal contour. This makes the analysis amenable to nonperturbative techniques. Presently, we elaborate on a refined form of the operator product expansion in terms of pentagon transitions to compute twist-two contributions to NMHV amplitudes. To start with, we provide a novel derivation of scattering matrices starting from Baxter equations for flux-tube excitations propagating on magnon background. We propose bootstrap equations obeyed by pentagon form factors with nonsinglet quantum numbers with respect to the R-symmetry group and provide solutions to them to all orders in 't Hooft coupling. These are then successfully confronted against available perturbative calculations for NMHV amplitudes to four-loop order. 13. Cluster polylogarithms for scattering amplitudes International Nuclear Information System (INIS) Golden, John; Paulos, Miguel F; Spradlin, Marcus; Volovich, Anastasia 2014-01-01 Motivated by the cluster structure of two-loop scattering amplitudes in N=4 Yang-Mills theory we define cluster polylogarithm functions. We find that all such functions of weight four are made up of a single simple building block associated with the A 2 cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A 2 building blocks arrange themselves to form a unique function associated with the A 3 cluster algebra. This A 3 function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n = 7 amplitude as an example. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’. (paper) 14. Topological amplitudes in string theory International Nuclear Information System (INIS) Antoniadis, I.; Taylor, T.R. 1993-07-01 We show that certain type II string amplitudes at genus g are given by the topological partition F g discussed recently by Bershadsky, Cecotti, Ooguri and Vafa. These amplitudes give rise to a term in the four-dimensional effective action of the form Σ g F g W 2g , where W is the chiral superfield of N = 2 supergravitational multiplet. The holomorphic anomaly of F g is related to non-localities of the effective action due to the propagation of massless states. This result generalizes the holomorphic anomaly of the one loop case which is known to lead to non-harmonic gravitational couplings. (author). 22 refs, 2 figs 15. N-loop string amplitude International Nuclear Information System (INIS) Mandelstam, S. 1986-06-01 Work on the derivation of an explicit perturbation series for string and superstring amplitudes is reviewed. The light-cone approach is emphasized, but some work on the Polyakov approach is also mentioned, and the two methods are compared. The calculation of the measure factor is outlined in the interacting-string picture 16. Scattering Amplitudes from Intersection Theory. Science.gov (United States) Mizera, Sebastian 2018-04-06 We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation. 17. Positivity of spin foam amplitudes International Nuclear Information System (INIS) Baez, John C; Christensen, J Daniel 2002-01-01 The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e iS ) rather than imaginary-time e -S path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett-Crane model 18. Employing Helicity Amplitudes for Resummation NARCIS (Netherlands) Moult, I.; Stewart, I.W.; Tackmann, F.J.; Waalewijn, W.J. 2015-01-01 Many state-of-the-art QCD calculations for multileg processes use helicity amplitudes as their fundamental ingredients. We construct a simple and easy-to-use helicity operator basis in soft-collinear effective theory (SCET), for which the hard Wilson coefficients from matching QCD onto SCET are 19. Scattering amplitudes in gauge theories Energy Technology Data Exchange (ETDEWEB) Henn, Johannes M. [Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences; Plefka, Jan C. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik 2014-03-01 First monographical text on this fundamental topic. Course-tested, pedagogical and self-contained exposition. Includes exercises and solutions. At the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has been made in our understanding of and computational abilities with regard to scattering amplitudes in gauge theories, going beyond the traditional textbook approach. These advances build upon on-shell methods that focus on the analytic structure of the amplitudes, as well as on their recently discovered hidden symmetries. In fact, when expressed in suitable variables the amplitudes are much simpler than anticipated and hidden patterns emerge. These modern methods are of increasing importance in phenomenological applications arising from the need for high-precision predictions for the experiments carried out at the Large Hadron Collider, as well as in foundational mathematical physics studies on the S-matrix in quantum field theory. Bridging the gap between introductory courses on quantum field theory and state-of-the-art research, these concise yet self-contained and course-tested lecture notes are well-suited for a one-semester graduate level course or as a self-study guide for anyone interested in fundamental aspects of quantum field theory and its applications. The numerous exercises and solutions included will help readers to embrace and apply the material presented in the main text. 20. Scattering amplitudes in gauge theories International Nuclear Information System (INIS) Henn, Johannes M.; Plefka, Jan C. 2014-01-01 First monographical text on this fundamental topic. Course-tested, pedagogical and self-contained exposition. Includes exercises and solutions. At the fundamental level, the interactions of elementary particles are described by quantum gauge field theory. The quantitative implications of these interactions are captured by scattering amplitudes, traditionally computed using Feynman diagrams. In the past decade tremendous progress has been made in our understanding of and computational abilities with regard to scattering amplitudes in gauge theories, going beyond the traditional textbook approach. These advances build upon on-shell methods that focus on the analytic structure of the amplitudes, as well as on their recently discovered hidden symmetries. In fact, when expressed in suitable variables the amplitudes are much simpler than anticipated and hidden patterns emerge. These modern methods are of increasing importance in phenomenological applications arising from the need for high-precision predictions for the experiments carried out at the Large Hadron Collider, as well as in foundational mathematical physics studies on the S-matrix in quantum field theory. Bridging the gap between introductory courses on quantum field theory and state-of-the-art research, these concise yet self-contained and course-tested lecture notes are well-suited for a one-semester graduate level course or as a self-study guide for anyone interested in fundamental aspects of quantum field theory and its applications. The numerous exercises and solutions included will help readers to embrace and apply the material presented in the main text. 1. Employing helicity amplitudes for resummation International Nuclear Information System (INIS) Moult, Ian; Stewart, Iain W.; Tackmann, Frank J.; Waalewijn, Wouter J.; Amsterdam Univ. 2015-08-01 Many state-of-the-art QCD calculations for multileg processes use helicity amplitudes as their fundamental ingredients. We construct a simple and easy-to-use helicity operator basis in soft-collinear effective theory (SCET), for which the hard Wilson coefficients from matching QCD onto SCET are directly given in terms of color-ordered helicity amplitudes. Using this basis allows one to seamlessly combine fixed-order helicity amplitudes at any order they are known with a resummation of higher-order logarithmic corrections. In particular, the virtual loop amplitudes can be employed in factorization theorems to make predictions for exclusive jet cross sections without the use of numerical subtraction schemes to handle real-virtual infrared cancellations. We also discuss matching onto SCET in renormalization schemes with helicities in 4- and d-dimensions. To demonstrate that our helicity operator basis is easy to use, we provide an explicit construction of the operator basis, as well as results for the hard matching coefficients, for pp → H+0,1,2 jets, pp → W/Z/γ+0,1,2 jets, and pp → 2,3 jets. These operator bases are completely crossing symmetric, so the results can easily be applied to processes with e + e - and e - p collisions. 2. Discontinuity formulas for multiparticle amplitudes International Nuclear Information System (INIS) Stapp, H.P. 1976-03-01 It is shown how discontinuity formulas for multiparticle scattering amplitudes are derived from unitarity and analyticity. The assumed analyticity property is the normal analytic structure, which was shown to be equivalent to the space-time macrocausality condition. The discontinuity formulas to be derived are the basis of multi-particle fixed-t dispersion relations 3. Distribution amplitudes of vector mesons Energy Technology Data Exchange (ETDEWEB) Braun, V.M. [Regensburg Univ. (Germany). Inst. fuer Theoretische Physik; Broemmel, D. [Deutsches Elektronen-Synchrotron, Hamburg (Germany); Goeckeler, M. [Regensburg Univ. (DE). Inst. fuer Theoretische Physik] (and others) 2007-11-15 Results are presented for the lowest moment of the distribution amplitude for the K{sup } vector meson. Both longitudinal and transverse moments are investigated. We use two flavours of O(a) improved Wilson fermions, together with a non-perturbative renormalisation of the matrix element. (orig.) 4. Spin and wavelength multiplexed nonlinear metasurface holography Science.gov (United States) Ye, Weimin; Zeuner, Franziska; Li, Xin; Reineke, Bernhard; He, Shan; Qiu, Cheng-Wei; Liu, Juan; Wang, Yongtian; Zhang, Shuang; Zentgraf, Thomas 2016-06-01 Metasurfaces, as the ultrathin version of metamaterials, have caught growing attention due to their superior capability in controlling the phase, amplitude and polarization states of light. Among various types of metasurfaces, geometric metasurface that encodes a geometric or Pancharatnam-Berry phase into the orientation angle of the constituent meta-atoms has shown great potential in controlling light in both linear and nonlinear optical regimes. The robust and dispersionless nature of the geometric phase simplifies the wave manipulation tremendously. Benefitting from the continuous phase control, metasurface holography has exhibited advantages over conventional depth controlled holography with discretized phase levels. Here we report on spin and wavelength multiplexed nonlinear metasurface holography, which allows construction of multiple target holographic images carried independently by the fundamental and harmonic generation waves of different spins. The nonlinear holograms provide independent, nondispersive and crosstalk-free post-selective channels for holographic multiplexing and multidimensional optical data storages, anti-counterfeiting, and optical encryption. 5. On expansion of scattering amplitude at large momentum transfers International Nuclear Information System (INIS) Edneral, V.F.; Troshin, S.M.; Tyurin, N.E. 1979-01-01 The aim of the paper is to construct an iterative approximation for hadronic scattering amplitude and to search for the related small parameters. The expansion of the amplitude is obtained. A series is derived where the role of the small parameter is played by the quantity dependent on the momentum transfer. The appearance of the small parameter is directly related to the growth of total cross section. For the case g 2 not equal to 0 in the framework of the strong interaction theory model, based on the solution of three-domensional dynamical equation an expression is obtained for scattering amplitude in the form of a series over the quantity decreasing with the growth of momentum transfer 6. Scruncher phase and amplitude control International Nuclear Information System (INIS) DeHaven, R.A.; Morris, C.L.; Johnson, R.; Davis, J.; O'Donnell, J.M. 1992-01-01 The analog controller for phase and amplitude control of a 402.5 MHz super conducting cavity is described in this paper. The cavity is a single cell with niobium explosively bonded to a copper cavity. It is used as an energy compressor for pions at the Clinton P. Anderson Meson Physics Facility (LAMPF). The controller maintains cavity frequency to within 4 degrees in phase of the LAMPF beam frequency. Field amplitude is maintained to within 2 percent. This control is accomplished at critical coupling (Q load of 1 x 10 9 ) with the use of only a 30 watt rf amplifier for accelerating fields of 6 MV/m. The design includes the use of piezoelectric crystals for fast resonance control. Three types of control, self excited, VCO, and a reference frequency driven, were tried on this cavity and we present a comparison of their performance. (Author) 4 figs., ref 7. SCRUNCHER phase and amplitude control International Nuclear Information System (INIS) DeHaven, R.A.; Morris, C.L.; Johnson, R.; Davis, J.; O'Donnell, J.M. 1992-01-01 The analog controller for phase and amplitude control of a 402.5 MHz super conducting cavity is described in this paper. The cavity is a single cell with niobium explosively bonded to a copper cavity. It is used as an energy compressor for pions at the Clinton P. Anderson Meson Physics Facility (LAMPF). The controller maintains cavity frequency to within 4 degrees in phase of the LAMPF beam frequency. Field amplitude is maintained to within 2 percent. This control is accomplished at critical coupling (Q loaded of 1 x 10 9 ) with the use of only a 30 watt rf amplifier for accelerating fields of 6 MV/m. The design includes the use of piezoelectric crystals for fast resonance control. Three types of control, self excited VCO, and a reference frequency driven, were tried on this cavity and we present a comparison of their performance 8. Periodic instantons and scattering amplitudes International Nuclear Information System (INIS) Khlebnikov, S.Yu.; Rubakov, V.A.; Tinyakov, P.G. 1991-04-01 We discuss the role of periodic euclidean solutions with two turning points and zero winding number (periodic instantons) in instanton induced processes below the sphaleron energy E sph . We find that the periodic instantons describe certain multiparticle scattering events leading to the transitions between topologically distinct vacua. Both the semiclassical amplitudes and inital and final states of these transitions are determined by the periodic instantons. Furthermore, the corresponding probabilities are maximal among all states of given energy. We show that at E ≤ E sph , the periodic instantons can be approximated by infinite chains of ordinary instantons and anti-instantons, and they naturally emerge as deformations of the zero energy instanton. In the framework of 2d abelian Higgs model and 4d electroweak theory we show, however, that there is not obvious relation between periodic instantons and two-particle scattering amplitudes. (orig.) 9. Determination of the scattering amplitude International Nuclear Information System (INIS) Gangal, A.D.; Kupsch, J. 1984-01-01 The problem to determine the elastic scattering amplitude from the differential cross-section by the unitarity equation is reexamined. We prove that the solution is unique and can be determined by a convergent iteration if the parameter lambda=sin μ of Newton and Martin is bounded by lambda 2 approx.=0.86. The method is based on a fixed point theorem for holomorphic mappings in a complex Banach space. (orig.) 10. Pulse amplitude modulated chlorophyll fluorometer Science.gov (United States) Greenbaum, Elias; Wu, Jie 2015-12-29 Chlorophyll fluorometry may be used for detecting toxins in a sample because of changes in micro algae. A portable lab on a chip ("LOAC") based chlorophyll fluorometer may be used for toxin detection and environmental monitoring. In particular, the system may include a microfluidic pulse amplitude modulated ("PAM") chlorophyll fluorometer. The LOAC PAM chlorophyll fluorometer may analyze microalgae and cyanobacteria that grow naturally in source drinking water. 11. Semiclassical approach to fidelity amplitude International Nuclear Information System (INIS) García-Mata, Ignacio; Vallejos, Raúl O; Wisniacki, Diego A 2011-01-01 The fidelity amplitude (FA) is a quantity of paramount importance in echo-type experiments. We use semiclassical theory to study the average FA for quantum chaotic systems under external perturbation. We explain analytically two extreme cases: the random dynamics limit - attained approximately by strongly chaotic systems - and the random perturbation limit, which shows a Lyapunov decay. Numerical simulations help us to bridge the gap between both the extreme cases. (paper) 12. Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography International Nuclear Information System (INIS) Nissilae, Ilkka; Noponen, Tommi; Kotilahti, Kalle; Katila, Toivo; Lipiaeinen, Lauri; Tarvainen, Tanja; Schweiger, Martin; Arridge, Simon 2005-01-01 In this article, we describe the multichannel implementation of an intensity modulated optical tomography system developed at Helsinki University of Technology. The system has two time-multiplexed wavelengths, 16 time-multiplexed source fibers and 16 parallel detection channels. The gain of the photomultiplier tubes (PMTs) is individually adjusted during the measurement sequence to increase the dynamic range of the system by 10 4 . The PMT used has a high quantum efficiency in the near infrared (8% at 800 nm), a fast settling time, and low hysteresis. The gain of the PMT is set so that the dc anode current is below 80 nA, which allows the measurement of phase independently of the intensity. The system allows measurements of amplitude at detected intensities down to 1 fW, which is sufficient for transmittance measurements of the female breast, the forearm, and the brain of early pre-term infants. The mean repeatability of phase and the logarithm of amplitude (ln A) at 100 MHz were found to be 0.08 deg. and 0.004, respectively, in a measurement of a 7 cm phantom with an imaging time of 5 s per source and source optical power of 8 mW. We describe a three-step method of calibrating the phase and amplitude measurements so that the absolute absorption and scatter in tissue may be measured. A phantom with two small cylindrical targets and a second phantom with three rods are measured and reconstructions made from the calibrated data are shown and compared with reconstructions from simulated data 13. AWG Filter for Wavelength Interrogator Science.gov (United States) Black, Richard J. (Inventor); Costa, Joannes M. (Inventor); Faridian, Fereydoun (Inventor); Moslehi, Behzad (Inventor); Sotoudeh, Vahid (Inventor) 2015-01-01 A wavelength interrogator is coupled to a circulator which couples optical energy from a broadband source to an optical fiber having a plurality of sensors, each sensor reflecting optical energy at a unique wavelength and directing the reflected optical energy to an AWG. The AWG has a detector coupled to each output, and the reflected optical energy from each grating is coupled to the skirt edge response of the AWG such that the adjacent channel responses form a complementary pair response. The complementary pair response is used to convert an AWG skirt response to a wavelength. 14. Time-amplitude converter; Convertisseur temps-amplitude Energy Technology Data Exchange (ETDEWEB) Banner, M [Commissariat a l' Energie Atomique, Saclay (France).Centre d' Etudes Nucleaires 1961-07-01 It is normal in high energy physics to measure the time of flight of a particle in order to determine its mass. This can be done by the method which consists in transforming the time measurement into an analysis of amplitude, which is easier; a time-amplitude converter has therefore been built for this purpose. The apparatus here described uses a double grid control tube 6 BN 6 whose resolution time, as measured with a pulse generator, is 5 x 10{sup -11} s. The analysis of the response of a particle counter, made up of a scintillator and a photomultiplier, indicates that a time of resolution of 5 x 10{sup -10} s. can be obtained. A time of this order of magnitude is obtained experimentally with the converter. This converter has been used in the study of the time of flight of particles in a secondary beam of the accelerator Saturne. It has thus been possible to measure the energy spectrum of {pi}-mesons, of protons, and of deutons emitted from a polyethylene target bombarded by 1,4 and 2 GeV protons. (author) [French] Pour determiner la masse d'une particule, il est courant, en physique des hautes energies, de mesurer le temps de vol de cette particule. Cela peut etre fait par la methode qui consiste a transformer la mesure d'un temps en une analyse d'amplitude, plus aisee; aussi a-t-on, a cet effet, cree un convertisseur temps-amplitude. L'appareillage decrit dans cet article utilise un tube a double grille de commande 6 BN 6 dont le temps de resolution mesure avec un generateur d'impulsion est de 5.10{sup -11} s. L'analyse de la reponse d'un compteur de particules, constitue par un scintillateur et un photomultiplicateur, indique qu'un temps de resolution de 5.10{sup -10} s peut etre obtenu. Un temps de cet ordre est atteint experimentalement avec le convertisseur. Ce convertisseur a servi a l'etude du temps de vol des particules dans un faisceau secondaire de l'accelerateur Saturne. On a mesure ainsi le spectre d'energie des mesons {pi}, des protons, des deutons 15. Constraining lowermost mantle structure with PcP/P amplitude ratios from large aperture arrays Science.gov (United States) Ventosa, S.; Romanowicz, B. A. 2015-12-01 Observations of weak short-period teleseismic body waves help to resolve lowermost mantle structure at short wavelengths, which is essential for understanding mantle dynamics and the interactions between the mantle and core. Their limited amount and uneven distribution are however major obstacles to solve for volumetric structure of the D" region, topography of the core-mantle boundary (CMB) and D" discontinuity, and the trade-offs among them. While PcP-P differential travel times provide important information, there are trade-offs between velocity structure and core-mantle boundary topography, which PcP/P amplitude ratios can help resolve, as long as lateral variations in attenuation and biases due to focusing are small or can be corrected for. Dense broadband seismic networks help to improve signal-to-noise ratio (SNR) of the target phases and signal-to-interference ratio (SIR) of other mantle phases when the slowness difference is large enough. To improve SIR and SNR of teleseismic PcP data, we have introduced the slant-stacklet transform to define coherent-guided filters able to separate and enhance signals according to their slowness, time of arrival and frequency content. We thus obtain optimal PcP/P amplitude ratios in the least-square sense using two short sliding windows to match the P signal with a candidate PcP signal. This method allows us to dramatically increase the amount of high-quality observations of short-period PcP/P amplitude ratios by allowing for smaller events and wider epicentral distance and depth ranges.We present the results of measurement of PcP/P amplitude ratios, sampling regions around the Pacific using dense arrays in North America and Japan. We observe that short-period P waves traveling through slabs are strongly affected by focusing, in agreement with the bias we have observed and corrected for due to mantle heterogeneities on PcP-P travel time differences. In Central America, this bias is by far the stronger anomaly we observe 16. Towards short wavelengths FELs workshop International Nuclear Information System (INIS) Ben-Zvi, I.; Winick, H. 1993-01-01 This workshop was caged because of the growing perception in the FEL source community that recent advances have made it possible to extend FEL operation to wavelengths about two orders of magnitude shorter than the 240 nm that has been achieved to date. In addition short wavelength FELs offer the possibilities of extremely high peak power (several gigawatts) and very short pulses (of the order of 100 fs). Several groups in the USA are developing plans for such short wavelength FEL facilities. However, reviewers of these plans have pointed out that it would be highly desirable to first carry out proof-of-principle experiments at longer wavelengths to increase confidence that the shorter wavelength devices will indeed perform as calculated. The need for such experiments has now been broadly accepted by the FEL community. Such experiments were the main focus of this workshop as described in the following objectives distributed to attendees: (1) Define measurements needed to gain confidence that short wavelength FELs will perform as calculated. (2) List possible hardware that could be used to carry out these measurements in the near term. (3) Define a prioritized FEL physics experimental program and suggested timetable. (4) Form collaborative teams to carry out this program 17. Towards short wavelengths FELs workshop Science.gov (United States) Ben-Zvi, I.; Winick, H. 1993-11-01 This workshop was caged because of the growing perception in the FEL source community that recent advances have made it possible to extend FEL operation to wavelengths about two orders of magnitude shorter than the 240 nm that has been achieved to date. In addition short wavelength FEL's offer the possibilities of extremely high peak power (several gigawatts) and very short pulses (of the order of 100 fs). Several groups in the USA are developing plans for such short wavelength FEL facilities. However, reviewers of these plans have pointed out that it would be highly desirable to first carry out proof-of-principle experiments at longer wavelengths to increase confidence that the shorter wavelength devices will indeed perform as calculated. The need for such experiments has now been broadly accepted by the FEL community. Such experiments were the main focus of this workshop as described in the following objectives distributed to attendees: (1) Define measurements needed to gain confidence that short wavelength FEL's will perform as calculated. (2) List possible hardware that could be used to carry out these measurements in the near term. (3) Define a prioritized FEL physics experimental program and suggested timetable. (4) Form collaborative teams to carry out this program. 18. Amplitude modulation reflectometer for FTU International Nuclear Information System (INIS) Zerbini, M.; Buratti, P.; Centioli, C.; Amadeo, P. 1995-06-01 Amplitude modulation (AM) reflectometry is a modification of the classical frequency sweep technique which allows to perform unambiguous phase delay measurements. An eight-channel AM reflectometer has been realized for the measurement of density profiles on the FTU tokamak in the range. The characteristics of the instrument have been determined in extensive laboratory tests; particular attention has been devoted to the effect of interference with parasitic reflections. The reflectometer is now operating on FTU. Some examples of the first experimental data are discussed 19. Superstring amplitudes and contact interactions International Nuclear Information System (INIS) Greensite, J. 1987-08-01 We show that scattering amplitudes computed from light-cone superstring field theory are divergent at tree level. The divergences can be eliminated, and supersymmetry restored, by the addition of certain counter terms to the light-cone Hamiltonian. These counter terms have the form of local contact interactions, whose existence we had previously deduced on grounds of vacuum stability, and closure of the super-Poincare algebra. The quartic contact interactions required in Type I and Type IIB superstring theories are constructed in detail. (orig.) 20. Forward amplitude in pion deuteron International Nuclear Information System (INIS) Ferreira, E.M.; Munguia, G.A.P.; Rosa, L.P.; Thome, Z.D. 1979-06-01 The data on total cross section for πd scattering is analysed in terms of a single scattering calculation with Fermi motion dependence, in order to obtain a criterion to fix the value of the energy entering the two body meson nucleon amplitude. It is found that the prescription derived from the non-relativistic three body kinematics gives reasonable results. The introduction of a shift in the energy value, possibly representing nuclear binding effects, leads to a very good fitting of the data. The results are compared with those obtained in direct calculations of Faddeev equations and with the Brueckner model of fixed scatterers. (Author) [pt 1. Modeling solar flare induced lower ionosphere changes using VLF/LF transmitter amplitude and phase observations at a midlatitude site Science.gov (United States) Schmitter, E. D. 2013-04-01 Remote sensing of the ionosphere bottom using long wave radio signal propagation is a still going strong and inexpensive method for continuous monitoring purposes. We present a propagation model describing the time development of solar flare effects. Based on monitored amplitude and phase data from VLF/LF transmitters gained at a mid-latitude site during the currently increasing solar cycle no. 24 a parameterized electron density profile is calculated as a function of time and fed into propagation calculations using the LWPC (Long Wave Propagation Capability). The model allows to include lower ionosphere recombination and attachment coefficients, as well as to identify the relevant forcing X-ray wavelength band, and is intended to be a small step forward to a better understanding of the solar-lower ionosphere interaction mechanisms within a consistent framework. 2. Control of surface ripple amplitude in ion beam sputtered polycrystalline cobalt films Energy Technology Data Exchange (ETDEWEB) Colino, Jose M., E-mail: josemiguel.colino@uclm.es [Institute of Nanoscience, Nanotechnology and Molecular Materials, University of Castilla-La Mancha, Campus de la Fabrica de Armas, Toledo 45071 (Spain); Arranz, Miguel A. [Facultad de Ciencias Quimicas, University of Castilla-La Mancha, Ciudad Real 13071 (Spain) 2011-02-15 We have grown both polycrystalline and partially textured cobalt films by magnetron sputter deposition in the range of thickness (50-200 nm). Kinetic roughening of the growing film leads to a controlled rms surface roughness values (1-6 nm) increasing with the as-grown film thickness. Ion erosion of a low energy 1 keV Ar+ beam at glancing incidence (80{sup o}) on the cobalt film changes the surface morphology to a ripple pattern of nanometric wavelength. The wavelength evolution at relatively low fluency is strongly dependent on the initial surface topography (a wavelength selection mechanism hereby confirmed in polycrystalline rough surfaces and based on the shadowing instability). At sufficiently large fluency, the ripple wavelength steadily increases on a coarsening regime and does not recall the virgin surface morphology. Remarkably, the use of a rough virgin surface makes the ripple amplitude in the final pattern can be controllably increased without affecting the ripple wavelength. 3. Frequency domain phase retrieval of simultaneous multi-wavelength phase-shifting interferometry International Nuclear Information System (INIS) Yin, Zhenxing; Zhong, Liyun; Xu, Xiaofei; Zhang, Wangping; Lu, Xiaoxu; Tian, Jindong 2016-01-01 In simultaneous multi-wavelength phase-shifting interferometry, we propose a novel frequency domain phase retrieval (FDPR) algorithm. First, using only a one-time phase-shifting operation, a sequence of simultaneous multi-wavelength phase-shifting interferograms (SPSMWIs) are captured by a monochrome charge-coupled device. Second, by performing a Fourier transform for each pixel of SPSMWIs, the wrapped phases of each wavelength can be retrieved from the complex amplitude located in the spectral peak of each wavelength. Finally, the phase of the synthetic wavelength can be obtained by the subtraction between the wrapped phases of a single wavelength. In this study, the principle and the application condition of the proposed approach are discussed. Both the simulation and the experimental result demonstrate the simple and convenient performance of the proposed FDPR approach. (paper) 4. A pulse amplitude dividing circuit for nuclear applications International Nuclear Information System (INIS) Ediss, C.; McQuarrie, S.A. 1981-01-01 A pulse dividing circuit has been developed to provide analogue and digital outputs proportional to the ratio of the amplitudes of two nuclear pulses. Input pulses ranging from 200 mV to 10 V may be processed by the device. The pulse dividing circuit has been successfully incorporated as part of a small gamma camera. (orig.) 5. Nonlinear Amplitude Evolution During Spontaneous Patterning of Ion-Bombarded Si(001) International Nuclear Information System (INIS) Chason, Eric; Erlebacher, Jonah; Aziz, Michael J.; Floro, Jerold A.; Sinclair, Michael B. 1999-01-01 The time evolution of the amplitude of periodic nanoscale ripple patterns formed on Ar+ sputtered Si(OOl ) surfaces was examined using a recently developed in situ spectroscopic technique. At sufficiently long times, we find that the amplitude does not continue to grow exponentially as predicted by the standard Bradley-Harper sputter rippling model. In accounting for this discrepancy, we rule out effects related to the concentration of mobile species, high surface curvature, surface energy anisotropy, and ion-surface interactions. We observe that for all wavelengths the amplitude ceases to grow when the width of the topmost terrace of the ripples is reduced to approximately 25 nm. This observation suggests that a short circuit relaxation mechanism limits amplitude . growth. A strategy for influencing the ultimate ripple amplitude is discussed 6. Wavelength dependence of interstellar polarization International Nuclear Information System (INIS) Mavko, G.E. 1974-01-01 The wavelength dependence of interstellar polarization was measured for twelve stars in three regions of the Milky Way. A 120A bandpass was used to measure the polarization at a maximum of sixteen wavelengths evenly spaced between 2.78μ -1 (3600A) and 1.28μ -1 (7800A). For such a wide wavelength range, the wavelength resolution is superior to that of any previously reported polarization measurements. The new scanning polarimeter built by W. A. Hiltner of the University of Michigan was used for the observations. Very broad structure was found in the wavelength dependence of the polarization. Extensive investigations were carried out to show that the structure was not caused by instrumental effects. The broad structure observed is shown to be in agreement with concurrent extinction measurements for the same stars. Also, the observed structure is of the type predicted when a homogeneous silicate grain model is fitted to the observed extinction. The results are in agreement with the hypothesis that the very broad band structure seen in the extinction is produced by the grains. (Diss. Abstr. Int., B) 7. Covariant amplitudes in Polyakov string theory International Nuclear Information System (INIS) Aoyama, H.; Dhar, A.; Namazie, M.A. 1986-01-01 A manifestly Lorentz-covariant and reparametrization-invariant procedure for computing string amplitudes using Polyakov's formulation is described. Both bosonic and superstring theories are dealt with. The computation of string amplitudes is greatly facilitated by this formalism. (orig.) 8. Examining the time dependence of DAMA's modulation amplitude Science.gov (United States) Kelso, Chris; Savage, Christopher; Sandick, Pearl; Freese, Katherine; Gondolo, Paolo 2018-03-01 If dark matter is composed of weakly interacting particles, Earth's orbital motion may induce a small annual variation in the rate at which these particles interact in a terrestrial detector. The DAMA collaboration has identified at a 9.3σ confidence level such an annual modulation in their event rate over two detector iterations, DAMA/NaI and DAMA/LIBRA, each with ˜ 7 years of observations. This data is well fit by a constant modulation amplitude for the two iterations of the experiment. We statistically examine the time dependence of the modulation amplitudes, which "by eye" appear to be decreasing with time in certain energy ranges. We perform a chi-squared goodness of fit test of the average modulation amplitudes measured by the two detector iterations which rejects the hypothesis of a consistent modulation amplitude at greater than 80, 96, and 99.6% for the 2-4, 2-5 and 2-6 keVee energy ranges, respectively. We also find that among the 14 annual cycles there are three ≳ 3σ departures from the average in our estimated data in the 5-6 keVee energy range. In addition, we examined several phenomenological models for the time dependence of the modulation amplitude. Using a maximum likelihood test, we find that descriptions of the modulation amplitude as decreasing with time are preferred over a constant modulation amplitude at anywhere between 1σ and 3σ , depending on the phenomenological model for the time dependence and the signal energy range considered. A time dependent modulation amplitude is not expected for a dark matter signal, at least for dark matter halo morphologies consistent with the DAMA signal. New data from DAMA/LIBRA-phase2 will certainly aid in determining whether any apparent time dependence is a real effect or a statistical fluctuation. 9. Grassmannian geometry of scattering amplitudes CERN Document Server Arkani-Hamed, Nima; Cachazo, Freddy; Goncharov, Alexander; Postnikov, Alexander; Trnka, Jaroslav 2016-01-01 Outlining a revolutionary reformulation of the foundations of perturbative quantum field theory, this book is a self-contained and authoritative analysis of the application of this new formulation to the case of planar, maximally supersymmetric Yang–Mills theory. The book begins by deriving connections between scattering amplitudes and Grassmannian geometry from first principles before introducing novel physical and mathematical ideas in a systematic manner accessible to both physicists and mathematicians. The principle players in this process are on-shell functions which are closely related to certain sub-strata of Grassmannian manifolds called positroids - in terms of which the classification of on-shell functions and their relations becomes combinatorially manifest. This is an essential introduction to the geometry and combinatorics of the positroid stratification of the Grassmannian and an ideal text for advanced students and researchers working in the areas of field theory, high energy physics, and the... 10. Determination of backward pion nucleon scattering amplitudes International Nuclear Information System (INIS) Pietarinen, E. 1978-04-01 Backward C(sup(+-))πN amplitudes are determined from πN→Nπ and NantiN→2π differential cross sections in such a way that they are consistent with the analyticity properties and information of the unphysical ππ→NantiN amplitudes. Combining the result with forward C(sup(+-)) amplitudes positive and negative parity resonances are extracted. An error analysis of the amplitudes is performed. (author) 11. Wavelength conversion of QAM signals in a low loss CMOS compatible spiral waveguide DEFF Research Database (Denmark) Da Ros, Francesco; Porto da Silva, Edson; Zibar, Darko 2017-01-01 We demonstrate wavelength conversion of quadrature amplitude modulation (QAM) signals, including 32-GBd quadrature phase-shift keying and 10-GBd 16-QAM, in a 50-cm long high index doped glass spiral waveguide. The quality of the generated idlers for up to 20 nm of wavelength shift is sufficient...... to achieve a BER performance below the hard decision forward error correction threshold BER performance (... 12. Wavelength conversion techniques and devices DEFF Research Database (Denmark) Danielsen, Søren Lykke; Mikkelsen, Benny; Hansen, Peter Bukhave 1997-01-01 Taking into account the requirements to the converters e.g., bit rate transparency (at least up to 10 Gbit/s), polarisation independence, wavelength independence, moderate input power levels, high signal-to-noise ratio and high extinction ratio interferometric wavelength convertors are very...... interesting for use in WDM optical fibre networks. However, the perfect converter has probably not yet been fabricated and new techniques such as conversion relying on cross-absorption modulation in electro-absorption modulators might also be considered in pursue of effective conversion devices... 13. Sub-wavelength plasmon laser Science.gov (United States) Bora, Mihail; Bond, Tiziana C. 2016-04-19 A plasmonic laser device has resonant nanocavities filled with a gain medium containing an organic dye. The resonant plasmon frequencies of the nanocavities are tuned to align with both the absorption and emission spectra of the dye. Variables in the system include the nature of the dye and the wavelength of its absorption and emission, the wavelength of the pumping radiation, and the resonance frequencies of the nanocavities. In addition the pumping frequency of the dye is selected to be close to the absorption maximum. 14. Wavelength standards in the infrared CERN Document Server Rao, KN 2012-01-01 Wavelength Standards in the Infrared is a compilation of wavelength standards suitable for use with high-resolution infrared spectrographs, including both emission and absorption standards. The book presents atomic line emission standards of argon, krypton, neon, and xenon. These atomic line emission standards are from the deliberations of Commission 14 of the International Astronomical Union, which is the recognized authority for such standards. The text also explains the techniques employed in determining spectral positions in the infrared. One of the techniques used includes the grating con 15. Light-cone distribution amplitudes of the baryon octet Energy Technology Data Exchange (ETDEWEB) Bali, Gunnar S. [Institut für Theoretische Physik, Universität Regensburg,Universitätsstraße 31, D-93040 Regensburg (Germany); Department of Theoretical Physics, Tata Institute of Fundamental Research,Homi Bhabha Road, Mumbai 400005 (India); Braun, Vladimir M.; Göckeler, Meinulf; Gruber, Michael; Hutzler, Fabian; Schäfer, Andreas; Schiel, Rainer W.; Simeth, Jakob; Söldner, Wolfgang [Institut für Theoretische Physik, Universität Regensburg,Universitätsstraße 31, D-93040 Regensburg (Germany); Sternbeck, Andre [Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena,Max-Wien-Platz 1, D-07743 Jena (Germany); Wein, Philipp [Institut für Theoretische Physik, Universität Regensburg,Universitätsstraße 31, D-93040 Regensburg (Germany) 2016-02-10 We present results of the first ab initio lattice QCD calculation of the normalization constants and first moments of the leading twist distribution amplitudes of the full baryon octet, corresponding to the small transverse distance limit of the associated S-wave light-cone wave functions. The P-wave (higher twist) normalization constants are evaluated as well. The calculation is done using N{sub f}=2+1 flavors of dynamical (clover) fermions on lattices of different volumes and pion masses down to 222 MeV. Significant SU(3) flavor symmetry violation effects in the shape of the distribution amplitudes are observed. 16. Light-cone distribution amplitudes of the baryon octet International Nuclear Information System (INIS) Bali, Gunnar S.; Braun, Vladimir M.; Göckeler, Meinulf; Gruber, Michael; Hutzler, Fabian; Schäfer, Andreas; Schiel, Rainer W.; Simeth, Jakob; Söldner, Wolfgang; Sternbeck, Andre; Wein, Philipp 2016-01-01 We present results of the first ab initio lattice QCD calculation of the normalization constants and first moments of the leading twist distribution amplitudes of the full baryon octet, corresponding to the small transverse distance limit of the associated S-wave light-cone wave functions. The P-wave (higher twist) normalization constants are evaluated as well. The calculation is done using N_f=2+1 flavors of dynamical (clover) fermions on lattices of different volumes and pion masses down to 222 MeV. Significant SU(3) flavor symmetry violation effects in the shape of the distribution amplitudes are observed. 17. Finite amplitude effects on drop levitation for material properties measurement Science.gov (United States) Ansari Hosseinzadeh, Vahideh; Holt, R. Glynn 2017-05-01 The method of exciting shape oscillation of drops to extract material properties has a long history, which is most often coupled with the technique of acoustic levitation to achieve non-contact manipulation of the drop sample. We revisit this method with application to the inference of bulk shear viscosity and surface tension. The literature is replete with references to a "10% oscillation amplitude" as a sufficient condition for the application of Lamb's analytical expressions for the shape oscillations of viscous liquids. Our results show that even a 10% oscillation amplitude leads to dynamic effects which render Lamb's results inapplicable. By comparison with samples of known viscosity and surface tension, we illustrate the complicating finite-amplitude effects (mode-splitting and excess dissipation associated with vorticity) that can occur and then show that sufficiently small oscillations allow us to recover the correct material properties using Lamb's formula. 18. PASTA - An RF Phase and Amplitude Scan and Tuning Application CERN Document Server Galambos, J; Deibele, C; Henderson, S 2005-01-01 To assist the beam commissioning in the Spallation Neutron Source (SNS) linac, a general purpose RF tuning application has been written to help set RF phase and amplitude. It follows the signature matching procedure described in Ref. The method involves varying an upstream Rf cavity amplitude and phase settings and comparing the measured downstream beam phase responses to model predictions. The model input for cavity phase and amplitude calibration and for the beam energy are varied to best match observations. This scheme has advantages over other RF tuning techniques of not requiring intercepting devices (e.g. Faraday Cups), and not being restricted to a small linear response regime near the design values. The application developed here is general and can be applied to different RF structure types in the SNS linac. Example applications in the SNS Drift Tube Linac (DTL) and Coupled Cavity Linac (CCL) structures will be shown. 19. Metasurface axicon lens design at visible wavelengths Science.gov (United States) Alyammahi, Saleimah; Zhan, Qiwen 2017-08-01 The emerging field of metasurfaces is promising to realize novel optical devices with miniaturized flat format and added functionalities. Metasurfaces have been demonstrated to exhibit full control of amplitude, phase and polarization of electromagnetic waves. Using the metasurface, the wavefront of light can be manipulated permitting new functionalities such as focusing and steering of the beams and imaging. One optical component which can be designed using metasurfaces is the axicon. Axicons are conical lenses used to convert Gaussian beams into nondiffraction Bessel beams. These unique devices are utilized in different applications ranging from optical trapping and manipulation, medical imaging, and surgery. In this work, we study axicon lens design comprising of planar metasurfaces which generate non-diffracting Bessel beams at visible wavelengths. Dielectric metasurfaces have been used to achieve high efficiency and low optical loss. We measured the spot size of the resulted beams at different planes to demonstrate the non-diffraction properties of the resulted beams. We also investigated how the spot size is influenced by the axicon aperture. Furthermore, we examined the achromatic properties of the designed axicon. Comparing with the conventional lens, the metasurface axicon lens design enables the creation of flat optical device with wide range of depth of focus along its optical axis. 20. Relative amplitude preservation processing utilizing surface consistent amplitude correction. Part 3; Surface consistent amplitude correction wo mochiita sotai shinpuku hozon shori. 3 Energy Technology Data Exchange (ETDEWEB) Saeki, T [Japan National Oil Corporation, Tokyo (Japan). Technology Research Center 1996-10-01 For the seismic reflection method conducted on the ground surface, generator and geophone are set on the surface. The observed waveforms are affected by the ground surface and surface layer. Therefore, it is required for discussing physical properties of the deep underground to remove the influence of surface layer, preliminarily. For the surface consistent amplitude correction, properties of the generator and geophone were removed by assuming that the observed waveforms can be expressed by equations of convolution. This is a correction method to obtain records without affected by the surface conditions. In response to analysis and correction of waveforms, wavelet conversion was examined. Using the amplitude patterns after correction, the significant signal region, noise dominant region, and surface wave dominant region would be separated each other. Since the amplitude values after correction of values in the significant signal region have only small variation, a representative value can be given. This can be used for analyzing the surface consistent amplitude correction. Efficiency of the process can be enhanced by considering the change of frequency. 3 refs., 5 figs. 1. Stable Dual-Wavelength Fibre Laser with Bragg Gratings Fabricated in a Polarization-Maintaining Erbium-Doped Fibre International Nuclear Information System (INIS) Lin, Wang; Feng-Ping, Yan; Xiang-Qiao, Mao; Shui-Sheng, Jian 2008-01-01 A new polarization-independent dual-wavelength fibre laser by fabricating a uniform FBG and a chirped FBG in a polarization-maintaining erbium-doped fibre (PM-EDF) is proposed and demonstrated. The wavelength spacing is 0.18nm and the optical signal-to-noise ratio is greater than 50dB with pump power of 246mW. Chirped FBG is used to make the reflectivity wavelengths of two PM-FBGs match easier. Since both EDF and FBGs are polarization-maintaining without splices and the two wavelengths are polarization-independent, the maximum amplitude variation and wavelength shifts for both lasing wavelength with 3-min intervals over a period of six hours are less than 0.2 dB and 0.005 nm, respectively, which shows stable dual-wavelength output 2. Review of short wavelength lasers International Nuclear Information System (INIS) Hagelstein, P.L. 1985-01-01 There has recently been a substantial amount of research devoted to the development of short wavelength amplifiers and lasers. A number of experimental results have been published wherein the observation of significant gain has been claimed on transitions in the EUV and soft x-ray regimes. The present review is intended to discuss the main approaches to the creation of population inversions and laser media in the short wavelength regime, and hopefully aid workers in the field by helping to provide access to a growing literature. The approaches to pumping EUV and soft x-ray lasers are discussed according to inversion mechanism. The approaches may be divided into roughly seven categories, including collisional excitation pumping, recombination pumping, direct photoionization and photoexcitation pumping, metastable state storage plus optical pumping, charge exchange pumping, and finally, the extension of free electron laser techniques into the EUV and soft x-ray regimes. 250 references 3. Review of short wavelength lasers Energy Technology Data Exchange (ETDEWEB) Hagelstein, P.L. 1985-03-18 There has recently been a substantial amount of research devoted to the development of short wavelength amplifiers and lasers. A number of experimental results have been published wherein the observation of significant gain has been claimed on transitions in the EUV and soft x-ray regimes. The present review is intended to discuss the main approaches to the creation of population inversions and laser media in the short wavelength regime, and hopefully aid workers in the field by helping to provide access to a growing literature. The approaches to pumping EUV and soft x-ray lasers are discussed according to inversion mechanism. The approaches may be divided into roughly seven categories, including collisional excitation pumping, recombination pumping, direct photoionization and photoexcitation pumping, metastable state storage plus optical pumping, charge exchange pumping, and finally, the extension of free electron laser techniques into the EUV and soft x-ray regimes. 250 references. 4. Dual-wavelength phase-shifting digital holography selectively extracting wavelength information from wavelength-multiplexed holograms. Science.gov (United States) Tahara, Tatsuki; Mori, Ryota; Kikunaga, Shuhei; Arai, Yasuhiko; Takaki, Yasuhiro 2015-06-15 Dual-wavelength phase-shifting digital holography that selectively extracts wavelength information from five wavelength-multiplexed holograms is presented. Specific phase shifts for respective wavelengths are introduced to remove the crosstalk components and extract only the object wave at the desired wavelength from the holograms. Object waves in multiple wavelengths are selectively extracted by utilizing 2π ambiguity and the subtraction procedures based on phase-shifting interferometry. Numerical results show the validity of the proposed technique. The proposed technique is also experimentally demonstrated. 5. On the singularities of massive superstring amplitudes International Nuclear Information System (INIS) Foda, O. 1987-01-01 Superstring one-loop amplitudes with massive external states are shown to be in general ill-defined due to internal on-shell propagators. However, we argue that since any massive string state (in the uncompactified theory) has a finite lifetime to decay into massless particles, such amplitudes are not terms in the perturbative expansion of physical S-matrix elements: These can be defined only with massless external states. Consistent massive amplitudes repuire an off-shell formalism. (orig.) 6. On the singularities of massive superstring amplitudes Energy Technology Data Exchange (ETDEWEB) Foda, O. 1987-06-04 Superstring one-loop amplitudes with massive external states are shown to be in general ill-defined due to internal on-shell propagators. However, we argue that since any massive string state (in the uncompactified theory) has a finite lifetime to decay into massless particles, such amplitudes are not terms in the perturbative expansion of physical S-matrix elements: These can be defined only with massless external states. Consistent massive amplitudes repuire an off-shell formalism. 7. New relations for graviton-matter amplitudes CERN Multimedia CERN. Geneva 2018-01-01 I report on recent progress in finding compact expressions for scattering amplitudes involving gravitons and gluons as well as massive scalar and fermionic matter particles. At tree level the single graviton emission amplitudes may be expressed as linear combination of purely non-gravitational ones. At the one-loop level recent results on all four point Einstein-Yang-Mills amplitudes with at most one opposite helicity state using unitarity methods are reported. 8. Analytical properties of multiple production amplitudes Energy Technology Data Exchange (ETDEWEB) Medvedev, B V; Pavlov, V P; Polivanov, M K; Sukhanov, A D [Gosudarstvennyj Komitet po Ispol' zovaniyu Atomnoj Ehnergii SSSR, Moscow. Inst. Teoreticheskoj i Ehksperimental' noj Fiziki; AN SSSR, Moscow. Matematicheskij Inst.) 1984-05-01 Local analytical properties of amplitudes 2..-->..3 and 2..-->..4 are studied. The amplitudes are shown to be analytical functions of total and partial energies at fixed momentum transfers in the neighbourhood of any physical point on the energy shell 14 (for the 2..-->..3 case) and 242 (for the 2..-->..4 case) boundary values are expressed through the amplitudes of real processes. 9. Coherence techniques at extreme ultraviolet wavelengths Energy Technology Data Exchange (ETDEWEB) Chang, Chang [Univ. of California, Berkeley, CA (United States) 2002-01-01 The renaissance of Extreme Ultraviolet (EUV) and soft x-ray (SXR) optics in recent years is mainly driven by the desire of printing and observing ever smaller features, as in lithography and microscopy. This attribute is complemented by the unique opportunity for element specific identification presented by the large number of atomic resonances, essentially for all materials in this range of photon energies. Together, these have driven the need for new short-wavelength radiation sources (e.g. third generation synchrotron radiation facilities), and novel optical components, that in turn permit new research in areas that have not yet been fully explored. This dissertation is directed towards advancing this new field by contributing to the characterization of spatial coherence properties of undulator radiation and, for the first time, introducing Fourier optical elements to this short-wavelength spectral region. The first experiment in this dissertation uses the Thompson-Wolf two-pinhole method to characterize the spatial coherence properties of the undulator radiation at Beamline 12 of the Advanced Light Source. High spatial coherence EUV radiation is demonstrated with appropriate spatial filtering. The effects of small vertical source size and beamline apertures are observed. The difference in the measured horizontal and vertical coherence profile evokes further theoretical studies on coherence propagation of an EUV undulator beamline. A numerical simulation based on the Huygens-Fresnel principle is performed. 10. DVCS amplitude with kinematical twist-3 terms International Nuclear Information System (INIS) Radyushkin, A.V.; Weiss, C. 2000-01-01 The authors compute the amplitude of deeply virtual Compton scattering (DVCS) using the calculus of QCD string operators in coordinate representation. To restore the electromagnetic gauge invariance (transversality) of the twist-2 amplitude they include the operators of twist-3 which appear as total derivatives of twist-2 operators. The results are equivalent to a Wandzura-Wilczek approximation for twist-3 skewed parton distributions. They find that this approximation gives a finite result for the amplitude of a longitudinally polarized virtual photon, while the amplitude for transverse polarization is divergent, i.e., factorization breaks down in this term 11. Amplitude structure of off-shell processes International Nuclear Information System (INIS) Fearing, H.W.; Goldstein, G.R.; Moravcsik, M.J. 1984-01-01 The structure of M matrices, or scattering amplitudes, and of potentials for off-shell processes is discussed with the objective of determining how one can obtain information on off-shell amplitudes of a process in terms of the physical observables of a larger process in which the first process is embedded. The procedure found is inevitably model dependent, but within a particular model for embedding, a determination of the physically measurable amplitudes of the larger process is able to yield a determination of the off-shell amplitudes of the embedded process 12. Covariant N-string amplitude International Nuclear Information System (INIS) Di Vecchia, P.; Sciuto, S.; Nakayama, R.; Petersen, J.L.; Sidenius, J.R. 1986-11-01 The BRST-invariant N-Reggeon vertex (for the bosonic string) previously given by us in the operator formulation is considered in more detail. In particular we present a direct derivation from the string path integral. Several crucial symmetry properties found a posteriori before, become a priori clearer in this formulation. A number of delicate points related to zero modes, cut off procedures and normal ordering prescriptions are treated in some detail. The old technique of letting the string field acquire a small dimension ε/2 → 0 + is found especially elegant. (orig.) 13. Wavelength stabilized multi-kW diode laser systems Science.gov (United States) Köhler, Bernd; Unger, Andreas; Kindervater, Tobias; Drovs, Simon; Wolf, Paul; Hubrich, Ralf; Beczkowiak, Anna; Auch, Stefan; Müntz, Holger; Biesenbach, Jens 2015-03-01 We report on wavelength stabilized high-power diode laser systems with enhanced spectral brightness by means of Volume Holographic Gratings. High-power diode laser modules typically have a relatively broad spectral width of about 3 to 6 nm. In addition the center wavelength shifts by changing the temperature and the driving current, which is obstructive for pumping applications with small absorption bandwidths. Wavelength stabilization of high-power diode laser systems is an important method to increase the efficiency of diode pumped solid-state lasers. It also enables power scaling by dense wavelength multiplexing. To ensure a wide locking range and efficient wavelength stabilization the parameters of the Volume Holographic Grating and the parameters of the diode laser bar have to be adapted carefully. Important parameters are the reflectivity of the Volume Holographic Grating, the reflectivity of the diode laser bar as well as its angular and spectral emission characteristics. In this paper we present detailed data on wavelength stabilized diode laser systems with and without fiber coupling in the spectral range from 634 nm up to 1533 nm. The maximum output power of 2.7 kW was measured for a fiber coupled system (1000 μm, NA 0.22), which was stabilized at a wavelength of 969 nm with a spectral width of only 0.6 nm (90% value). Another example is a narrow line-width diode laser stack, which was stabilized at a wavelength of 1533 nm with a spectral bandwidth below 1 nm and an output power of 835 W. 14. Wavelength selection in the crown splash Science.gov (United States) Zhang, Li V.; Brunet, Philippe; Eggers, Jens; Deegan, Robert D. 2010-12-01 The impact of a drop onto a liquid layer produces a splash that results from the ejection and dissolution of one or more liquid sheets, which expand radially from the point of impact. In the crown splash parameter regime, secondary droplets appear at fairly regularly spaced intervals along the rim of the sheet. By performing many experiments for the same parameter values, we measure the spectrum of small-amplitude perturbations growing on the rim. We show that for a range of parameters in the crown splash regime, the generation of secondary droplets results from a Rayleigh-Plateau instability of the rim, whose shape is almost cylindrical. In our theoretical calculation, we include the time dependence of the base state. The remaining irregularity of the pattern is explained by the finite width of the Rayleigh-Plateau dispersion relation. Alternative mechanisms, such as the Rayleigh-Taylor instability, can be excluded for the experimental parameters of our study. 15. Automation of loop amplitudes in numerical approach International Nuclear Information System (INIS) Fujimoto, J.; Ishikawa, T.; Shimizu, Y.; Kato, K.; Nakazawa, N.; Kaneko, T. 1997-01-01 An automatic calculating system GRACE-L1 of one-loop Feynman amplitude is reviewed. This system can be applied to 2 to 2-body one-loop processes. A sample calculation of 2 to 3-body one-loop amplitudes is also presented. (orig.) 16. On the singularities of massive superstring amplitudes NARCIS (Netherlands) Foda, O. 1987-01-01 Superstring one-loop amplitudes with massive external states are shown to be in general ill-defined due to internal on-shell propagators. However, we argue that since any massive string state (in the uncompactified theory) has a finite lifetime to decay into massless particles, such amplitudes are 17. Scattering Amplitudes via Algebraic Geometry Methods DEFF Research Database (Denmark) Feynman diagrams. The study of multiloop scattering amplitudes is crucial for the new era of precision phenomenology at the Large Hadron Collider (LHC) at CERN. Loop-level scattering amplitudes can be reduced to a basis of linearly independent integrals whose coefficients are extracted from generalized... 18. Full amplitude models of 15 day Cepheids International Nuclear Information System (INIS) Cogan, B.C.; Cox, A.N.; King, D.S. 1980-01-01 Numerical models of Cepheids have been computed with a range of effective temperatures and compositions. The amplitudes increase if the helium abundance increases or if the effective temperature decreases. The latter effect is contrary to observational data. The models also exhibit velocity amplitudes which are much lower than those observed 19. Helicity amplitudes for matter-coupled gravity International Nuclear Information System (INIS) Aldrovandi, R.; Novaes, S.F.; Spehler, D. 1992-07-01 The Weyl-van der Waerden spinor formalism is applied to the evaluation of helicity invariant amplitudes in the framework of linearized gravitation. The graviton couplings to spin-0, 1 - 2 , 1, and 3 - 2 particles are given, and, to exhibit the reach of this method, the helicity amplitudes for the process electron + positron → photon + graviton are obtained. (author) 20. New relations for gauge-theory amplitudes International Nuclear Information System (INIS) Bern, Z.; Carrasco, J. J. M.; Johansson, H. 2008-01-01 We present an identity satisfied by the kinematic factors of diagrams describing the tree amplitudes of massless gauge theories. This identity is a kinematic analog of the Jacobi identity for color factors. Using this we find new relations between color-ordered partial amplitudes. We discuss applications to multiloop calculations via the unitarity method. In particular, we illustrate the relations between different contributions to a two-loop four-point QCD amplitude. We also use this identity to reorganize gravity tree amplitudes diagram by diagram, offering new insight into the structure of the Kawai-Lewellen-Tye (KLT) relations between gauge and gravity tree amplitudes. This insight leads to similar but novel relations. We expect this to be helpful in higher-loop studies of the ultraviolet properties of gravity theories. 1. Spectral Amplitude Coding (SAC)-OCDMA Network with 8DPSK Science.gov (United States) Aldhaibani, A. O.; Aljunid, S. A.; Fadhil, Hilal A.; Anuar, M. S. 2013-09-01 Optical code division multiple access (OCDMA) technique is required to meet the increased demand for high speed, large capacity communications in optical networks. In this paper, the transmission performance of a spectral amplitude coding (SAC)-OCDMA network is investigated when a conventional single-mode fiber (SMF) is used as the transmission link using 8DPSK modulation. The DW has a fixed weight of two. Simulation results reveal that the transmission distance is limited mainly by the fiber dispersion when high coding chip rate is used. For a two-user SAC-OCDMA network operating with 2 Gbit/s data rate and two wavelengths for each user, the maximum allowable transmission distance is about 15 km. 2. Injection-locked single-mode VCSEL for orthogonal multiplexing and amplitude noise suppression DEFF Research Database (Denmark) 2017-01-01 It has been shown earlier, that the injection locked semiconductor lasers enable effective amplitude noise suppression [1] and makes possible an extra level of signal multiplexing-orthogonal modulation [2], where DPSK and ASK NRZ channels propagate at the same wavelength [3]. In our work we use...... an injection-locked 1550 nm VCSEL as a slave laser providing separation of amplitude and phase modulations, carrying independent information flows. To validate the possibility of phase modulation extraction by an injection-locked VCSEL, an experimental setup shown in Fig. 1 has been built.... 3. Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators Science.gov (United States) Premalatha, K.; Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M. 2018-03-01 We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera, and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes, while the desynchronized group of oscillators oscillates with small amplitudes, and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time. Then, we analyze the stability of the amplitude chimera states under various circumstances, including variations in system parameters and coupling strength, and perturbations in the initial states of the oscillators. For an increase in the value of the system parameter, namely, the nonisochronicity parameter, the transient chimera state becomes a stable chimera state for a sufficiently large value of coupling strength. In addition, we also analyze the stability of these states by perturbing the initial states of the oscillators. We find that while a small perturbation allows one to perturb a large number of oscillators resulting in a stable amplitude chimera state, a large perturbation allows one to perturb a small number of oscillators to get a stable amplitude chimera state. We also find the stability of the transient and stable amplitude chimera states and traveling wave states for an appropriate number of oscillators using Floquet theory. In addition, we also find the stability of the incoherent oscillation death states. 4. Gravothermal catastrophe of finite amplitude Energy Technology Data Exchange (ETDEWEB) Hachisu, I; Sugimoto, D [Tokyo Univ. (Japan). Coll. of General Education; Nakada, Y; Nomoto, K 1978-08-01 Development of the gravothermal catastrophe is followed numerically for self-gravitating gas system enclosed by an adiabatic wall, which is isothermal in the initial state. It is found that the final fate of the catastrophe is in two ways depending on the initial perturbations. When the initial perturbation produces a temperature distribution decreasing outward, the contraction proceeds in the central region and the central density increases unlimitedly, as the heat flows outward. When the initial temperature distribution is increasing outward, on the other hand, the central region expands as the heat flows into the central region. Then the density contrast is reduced and finally the system reaches another isothermal configuration with the same energy but with a lower density contrast and a higher entropy. This final configuration is gravothermally stable and may be called a thermal system. In the former case of the unlimited contraction, the final density profile is determined essentially by the density and temperature dependence of the heat conductivity. In the case of a system under the force of the inverse square law, the final density distribution is well approximated by a power law so that the mass contained in the condensed core is relatively small. A possibility of formation of a black hole in stellar systems is also discussed. 5. Gravothermal catastrophe of finite amplitude International Nuclear Information System (INIS) Hachisu, Izumi; Sugimoto, Daiichiro; Nakada, Yoshikazu; Nomoto, Ken-ichi. 1978-01-01 Development of the gravothermal catastrophe is followed numerically for self-gravitating gas system enclosed by an adiabatic wall, which is isothermal in the initial state. It is found that the final fate of the catastrophe is in two ways depending on the initial perturbations. When the initial perturbation produces a temperature distribution decreasing outward, the contraction proceeds in the central region and the central density increases unlimitedly, as the heat flows outward. When the initial temperature distribution is increasing outward, on the other hand, the central region expands as the heat flows into the central region. Then the density contrast is reduced and finally the system reaches another isothermal configuration with the same energy but with a lower density contrast and a higher entropy. This final configuration is gravothermally stable and may be called a thermal system. In the former case of the unlimited contraction, the final density profile is determined essentially by the density and temperature dependence of the heat conductivity. In the case of a system under the force of the inverse square law, the final density distribution is well approximated by a power law so that the mass contained in the condensed core is relatively small. A possibility of formation of a black hole in stellar systems is also discussed. (author) 6. Scattering of light from small nematic spheres with radial dielectric anisotropy International Nuclear Information System (INIS) Karacali, H.; Risser, S.M.; Ferris, K.F. 1997-01-01 We have calculated the scattering cross sections of small anisotropic nematic droplets embedded in a polymer matrix as a function of the dielectric constants of the nematic and the polymer. We have derived the general form for the Helmholtz wave equation for a droplet which has spatially varying radial anisotropy, and have explicitly solved this equation for three distinct models of the dielectric anisotropy, including one model where the anisotropy increases linearly with droplet radius. Numerical calculations of the scattering amplitudes for droplets much smaller than the wavelength of the incident radiation show that droplets with continual variation in the dielectric anisotropy have much larger scattering amplitude than droplets with fixed anisotropy. The scattering from droplets with linearly varying anisotropy exhibits a scattering minimum for much smaller polymer dielectric constants than the other models. These results show that the scattering from small anisotropic droplets is sensitive to details of the internal structure and anisotropy of the droplet. copyright 1997 The American Physical Society 7. The Dynamics of Large-Amplitude Motion in Energized Molecules Energy Technology Data Exchange (ETDEWEB) Perry, David S. [Univ. of Akron, OH (United States). Dept. of Chemistry 2016-05-27 Chemical reactions involve large-amplitude nuclear motion along the reaction coordinate that serves to distinguish reactants from products. Some reactions, such as roaming reactions and reactions proceeding through a loose transition state, involve more than one large-amplitude degree of freedom. Because of the limitation of exact quantum nuclear dynamics to small systems, one must, in general, define the active degrees of freedom and separate them in some way from the other degrees of freedom. In this project, we use large-amplitude motion in bound model systems to investigate the coupling of large-amplitude degrees of freedom to other nuclear degrees of freedom. This approach allows us to use the precision and power of high-resolution molecular spectroscopy to probe the specific coupling mechanisms involved, and to apply the associated theoretical tools. In addition to slit-jet spectra at the University of Akron, the current project period has involved collaboration with Michel Herman and Nathalie Vaeck of the Université Libre de Bruxelles, and with Brant Billinghurst at the Canadian Light Source (CLS). 8. Effects of strength training on mechanomyographic amplitude International Nuclear Information System (INIS) DeFreitas, Jason M; Beck, Travis W; Stock, Matt S 2012-01-01 The aim of the present study was to determine if the patterns of mechanomyographic (MMG) amplitude across force would change with strength training. Twenty-two healthy men completed an 8-week strength training program. During three separate testing visits (pre-test, week 4, and week 8), the MMG signal was detected from the vastus lateralis as the subjects performed isometric step muscle actions of the leg extensors from 10–100% of maximal voluntary contraction (MVC). During pre-testing, the MMG amplitude increased linearly with force to 66% MVC and then plateaued. Conversely, weeks 4 and 8 demonstrated an increase in MMG amplitude up to ∼85% of the subject's original MVC before plateauing. Furthermore, seven of the ten force levels (30–60% and 80–100%) showed a significant decrease in mean MMG amplitude values after training, which consequently led to a decrease in the slope of the MMG amplitude/force relationship. The decreases in MMG amplitude at lower force levels are indicative of hypertrophy, since fewer motor units would be required to produce the same absolute force if the motor units increased in size. However, despite the clear changes in the mean values, analyses of individual subjects revealed that only 55% of the subjects demonstrated a significant decrease in the slope of the MMG amplitude/force relationship. (paper) 9. Joint Inversion of Phase and Amplitude Data of Surface Waves for North American Upper Mantle Science.gov (United States) Hamada, K.; Yoshizawa, K. 2015-12-01 For the reconstruction of the laterally heterogeneous upper-mantle structure using surface waves, we generally use phase delay information of seismograms, which represents the average phase velocity perturbation along a ray path, while the amplitude information has been rarely used in the velocity mapping. Amplitude anomalies of surface waves contain a variety of information such as anelastic attenuation, elastic focusing/defocusing, geometrical spreading, and receiver effects. The effects of elastic focusing/defocusing are dependent on the second derivative of phase velocity across the ray path, and thus, are sensitive to shorter-wavelength structure than the conventional phase data. Therefore, suitably-corrected amplitude data of surface waves can be useful for improving the lateral resolution of phase velocity models. In this study, we collect a large-number of inter-station phase velocity and amplitude ratio data for fundamental-mode surface waves with a non-linear waveform fitting between two stations of USArray. The measured inter-station phase velocity and amplitude ratios are then inverted simultaneously for phase velocity maps and local amplification factor at receiver locations in North America. The synthetic experiments suggest that, while the phase velocity maps derived from phase data only reflect large-scale tectonic features, those from phase and amplitude data tend to exhibit better recovery of the strength of velocity perturbations, which emphasizes local-scale tectonic features with larger lateral velocity gradients; e.g., slow anomalies in Snake River Plain and Rio Grande Rift, where significant local amplification due to elastic focusing are observed. Also, the spatial distribution of receiver amplification factor shows a clear correlation with the velocity structure. Our results indicate that inter-station amplitude-ratio data can be of help in reconstructing shorter-wavelength structures of the upper mantle. 10. The wavelength dependence of polarization in NGC 2023 International Nuclear Information System (INIS) Rolph, C. D.; Scarrott, S. M. 1989-01-01 NGC 2023 is a bright reflection nebula illuminated by the central star HD37903. At 2 microns the nebula is seen solely by reflected light from the central star but in the NIR there is excess radiation that is supposed to arise from thermal emission from a population of small grains (Sellgren, 1984). The unexpectedly high surface brightness at R and I wavelengths has led to the suggestion that even at these wavelengths there is a significant contribution from this thermal emission process (Witt, Schild, and Kraiman, 1984). If the nebula is seen by reflected starlight then this radiation will be linearly polarized. The level of polarization depends on the scattering geometry, grain size distribution, etc., and is typically 20 to 40 percent for nebulae such as NGC 1999 which is morphologically similar to NGC 2023. If, in any waveband, there is a contribution of radiation from emission processes this radiation will be unpolarized and will serve to dilute the scattered radiation to give a lower level of observed polarization. A study of the wavelength dependence of polarization in nebulae in which there may be thermal emission from grains will indicate the contribution from this process to the total luminosity. Polarization maps were produced in BVRI wavebands for the NGC 2023 nebulosity which confirm that at all wavelengths it is a reflection nebula illuminated by a central star. The wavelength dependence of polarization at representative points in the nebula and in a scatter plot of polarization in V and I wavebands at all points at which measurements are given. Results indicate that throughout the nebula there is a general trend for the level of polarization to increase with wavelength and that maximum levels of polarization occur at the longest wavelengths. No evidence is seen in the data for any significant contribution from the thermal emission from grains in the BVRI luminosity of NGC 2023 11. Holographic corrections to meson scattering amplitudes Energy Technology Data Exchange (ETDEWEB) Armoni, Adi; Ireson, Edwin, E-mail: 746616@swansea.ac.uk 2017-06-15 We compute meson scattering amplitudes using the holographic duality between confining gauge theories and string theory, in order to consider holographic corrections to the Veneziano amplitude and associated higher-point functions. The generic nature of such computations is explained, thanks to the well-understood nature of confining string backgrounds, and two different examples of the calculation in given backgrounds are used to illustrate the details. The effect we discover, whilst only qualitative, is re-obtainable in many such examples, in four-point but also higher point amplitudes. 12. Amplitude-Mode Dynamics of Polariton Condensates International Nuclear Information System (INIS) Brierley, R. T.; Littlewood, P. B.; Eastham, P. R. 2011-01-01 We study the stability of collective amplitude excitations in nonequilibrium polariton condensates. These excitations correspond to renormalized upper polaritons and to the collective amplitude modes of atomic gases and superconductors. They would be present following a quantum quench or could be created directly by resonant excitation. We show that uniform amplitude excitations are unstable to the production of excitations at finite wave vectors, leading to the formation of density-modulated phases. The physical processes causing the instabilities can be understood by analogy to optical parametric oscillators and the atomic Bose supernova. 13. Hidden simplicity of gauge theory amplitudes Energy Technology Data Exchange (ETDEWEB) Drummond, J M, E-mail: drummond@lapp.in2p3.f [LAPTH, Universite de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, Cedex (France) 2010-11-07 These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the Britto, Cachzo, Feng and Witten (BCFW) recursion relations we solve the tree-level S-matrix in N=4 super Yang-Mills theory and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries. 14. Linear response to long wavelength fluctuations using curvature simulations Energy Technology Data Exchange (ETDEWEB) Baldauf, Tobias; Zaldarriaga, Matias [School of Natural Sciences, Institute for Advanced Study, Princeton, NJ (United States); Seljak, Uroš [Physics Department, Astronomy Department and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA (United States); Senatore, Leonardo, E-mail: baldauf@ias.edu, E-mail: useljak@berkeley.edu, E-mail: senatore@stanford.edu, E-mail: matiasz@ias.edu [Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA (United States) 2016-09-01 We study the local response to long wavelength fluctuations in cosmological N -body simulations, focusing on the matter and halo power spectra, halo abundance and non-linear transformations of the density field. The long wavelength mode is implemented using an effective curved cosmology and a mapping of time and distances. The method provides an alternative, more direct, way to measure the isotropic halo biases. Limiting ourselves to the linear case, we find generally good agreement between the biases obtained from the curvature method and the traditional power spectrum method at the level of a few percent. We also study the response of halo counts to changes in the variance of the field and find that the slope of the relation between the responses to density and variance differs from the naïve derivation assuming a universal mass function by approximately 8–20%. This has implications for measurements of the amplitude of local non-Gaussianity using scale dependent bias. We also analyze the halo power spectrum and halo-dark matter cross-spectrum response to long wavelength fluctuations and derive second order halo bias from it, as well as the super-sample variance contribution to the galaxy power spectrum covariance matrix. 15. Amplitude growth due to random, correlated kicks International Nuclear Information System (INIS) Michelotti, L.; Mills, F. 1989-03-01 Historically, stochastic processes, such as gas scattering or stochastic cooling, have been treated by the Fokker-Planck equation. In this approach, usually considered for one dimension only, the equation can be considered as a continuity equation for a variable which would be a constant of the motion in the absence of the stochastic process, for example, the action variable, I = ε/2π for betatron oscillations, where ε is the area of the Courant-Snyder ellipse, or energy in the case of unbunched beams, or the action variable for phase oscillations in case the beam is bunched. A flux, /Phi/, including diffusive terms can be defined, usually to second order. /Phi/ = M 1 F(I) + M 2 ∂F/∂I + /hor ellipsis/. M 1 and M 2 are the expectation values of δI and (δI) 2 due to the individual stochastic kicks over some period of time, long enough that the variance of these quantities is sufficiently small. Then the Fokker-Planck equation is just ∂F/∂I + ∂/Phi//∂I = 0. In many cases those where the beam distribution has already achieved its final shape, it is sufficient to find the rate of increase of by taking simple averages over the Fokker-Planck equation. At the time this work was begun, there was good knowledge of the second moment for general stochastic processes due to stochastic cooling theory, but the form of the first moment was known only for extremely wideband processes. The purposes of this note are to derive an expression relating the expected single particle amplitude growth to the noise autocorrelation function and to obtain, thereby, the form of M 1 for narrow band processes. 4 refs 16. Numerical study of turbulent channel flow perturbed by spanwise topographic heterogeneity: Amplitude and frequency modulation within low- and high-momentum pathways Science.gov (United States) Awasthi, Ankit; Anderson, William 2018-04-01 We have studied the effects of topographically driven secondary flows on inner-outer interaction in turbulent channel flow. Recent studies have revealed that large-scale motions in the logarithmic region impose an amplitude and frequency modulation on the dynamics of small-scale structures near the wall. This led to development of a predictive model for near-wall dynamics, which has practical relevance for large-eddy simulations. Existing work on amplitude modulation has focused on smooth-wall flows; however, Anderson [J. Fluid Mech. 789, 567 (2016), 10.1017/jfm.2015.744] addressed the problem of rough-wall turbulent channel flow in which the correlation profiles for amplitude modulation showed trends similar to those reported by Mathis et al. [Phys. Fluids 21, 111703 (2009), 10.1063/1.3267726]. For the present study, we considered flow over surfaces with a prominent spanwise heterogeneity, such that domain-scale turbulent secondary flows in the form of counter-rotating vortices are sustained within the flow. (We also show results for flow over a homogeneous roughness, which serves as a benchmark against the spanwise-perturbed cases.) The vortices are anchored to the topography such that prominent upwelling and downwelling occur above the low and high roughness, respectively. We have quantified the extent to which such secondary flows disrupt the distribution of spectral density across constituent wavelengths throughout the depth of the flow, which has direct implications for the existence of amplitude and frequency modulation. We find that the distinct outer peak associated with large-scale motions—the "modulators"—is preserved within the upwelling zone but vanishes in the downwelling zone. Within the downwelling zones, structures are steeper and shorter. Single- and two-point correlations for inner-outer amplitude and frequency modulation demonstrate insensitivity to resolution across cases. We also show a pronounced crossover between the single- and two 17. Wavelength modulation spectroscopy of semiconductors Energy Technology Data Exchange (ETDEWEB) Kohn, S.E. 1977-10-01 The use of modulation spectroscopy to study the electronic properties of solids has been very productive. The construction of a wide range Wavelength Modulation Spectrometer to study the optical properties of solids is described in detail. Extensions of the working range of the spectrometer into the vacuum ultraviolet are discussed. Measurements of the reflectivity and derivative reflectivity spectra of the lead chalcogenides, the chalcopyrite ZnGeP/sub 2/, the layer compounds GaSe and GaS and their alloys, the ferroelectric SbSI, layer compounds SnS/sub 2/ and SnSe/sub 2/, and HfS/sub 2/ were made. The results of these measurements are presented along with their interpretation in terms of band structure calculations. 18. Analytic continuation of dual Feynman amplitudes International Nuclear Information System (INIS) Bleher, P.M. 1981-01-01 A notion of dual Feynman amplitude is introduced and a theorem on the existence of analytic continuation of this amplitude from the convergence domain to the whole complex is proved. The case under consideration corresponds to massless power propagators and the analytic continuation is constructed on the propagators powers. Analytic continuation poles and singular set of external impulses are found explicitly. The proof of the theorem on the existence of analytic continuation is based on the introduction of α-representation for dual Feynman amplitudes. In proving, the so-called ''trees formula'' and ''trees-with-cycles formula'' are established that are dual by formulation to the trees and 2-trees formulae for usual Feynman amplitudes. (Auth.) 19. Amplitude-Integrated EEG in the Newborn Directory of Open Access Journals (Sweden) J Gordon Millichap 2008-11-01 Full Text Available Th value of amplitude-integrated electroencephalography (aEEG in the newborn is explored by researchers at Washington University, St Louis; Wilhelmina Children’s Hospital, Utrecht, Netherlands; and Uppsala University Hospital, Sweden. 20. Scattering Amplitudes via Algebraic Geometry Methods CERN Document Server Søgaard, Mads; Damgaard, Poul Henrik This thesis describes recent progress in the understanding of the mathematical structure of scattering amplitudes in quantum field theory. The primary purpose is to develop an enhanced analytic framework for computing multiloop scattering amplitudes in generic gauge theories including QCD without Feynman diagrams. The study of multiloop scattering amplitudes is crucial for the new era of precision phenomenology at the Large Hadron Collider (LHC) at CERN. Loop-level scattering amplitudes can be reduced to a basis of linearly independent integrals whose coefficients are extracted from generalized unitarity cuts. We take advantage of principles from algebraic geometry in order to extend the notion of maximal cuts to a large class of two- and three-loop integrals. This allows us to derive unique and surprisingly compact formulae for the coefficients of the basis integrals. Our results are expressed in terms of certain linear combinations of multivariate residues and elliptic integrals computed from products of ... 1. Effective gluon interactions from superstring disk amplitudes Energy Technology Data Exchange (ETDEWEB) Oprisa, D. 2006-05-15 In this thesis an efficient method for the calculation of the N-point tree-level string amplitudes is presented. Furthermore it is shown that the six-gluon open-superstring disk amplitude can be expressed by a basis of six triple hypergeometric functions, which encode the full {alpha}' dependence. In this connection material for obtaining the {alpha}' expansion of these functions is derived. Hereby many Euler-Zagier sums are calculated including multiple harmonic series. (HSI) 2. Effective gluon interactions from superstring disk amplitudes International Nuclear Information System (INIS) Oprisa, D. 2006-05-01 In this thesis an efficient method for the calculation of the N-point tree-level string amplitudes is presented. Furthermore it is shown that the six-gluon open-superstring disk amplitude can be expressed by a basis of six triple hypergeometric functions, which encode the full α' dependence. In this connection material for obtaining the α' expansion of these functions is derived. Hereby many Euler-Zagier sums are calculated including multiple harmonic series. (HSI) 3. Employing helicity amplitudes for resummation in SCET International Nuclear Information System (INIS) Moult, Ian; Stewart, Iain W.; Tackmann, Frank J.; Waalewijn, Wouter J.; Nikhef, Amsterdam 2016-05-01 Helicity amplitudes are the fundamental ingredients of many QCD calculations for multi-leg processes. We describe how these can seamlessly be combined with resummation in Soft-Collinear Effective Theory (SCET), by constructing a helicity operator basis for which the Wilson coefficients are directly given in terms of color-ordered helicity amplitudes. This basis is crossing symmetric and has simple transformation properties under discrete symmetries. 4. The amplitude of quantum field theory International Nuclear Information System (INIS) Medvedev, B.V.; Pavlov, V.P.; Polivanov, M.K.; Sukhanov, A.D. 1989-01-01 General properties of the transition amplitude in axiomatic quantum field theory are discussed. Bogolyubov's axiomatic method is chosen as the variant of the theory. The axioms of this method are analyzed. In particular, the significance of the off-shell extension and of the various forms of the causality condition are examined. A complete proof is given of the existence of a single analytic function whose boundary values are the amplitudes of all channels of a process with given particle number 5. The Cepheid bump progression and amplitude equations International Nuclear Information System (INIS) Kovacs, G.; Buchler, J.R. 1989-01-01 It is shown that the characteristic and systematic behavior of the low-order Fourier amplitudes and phases of hydrodynamically generated radial velocity and light curves of Cepheid model sequences is very well captured not only qualitatively but also quantitatively by the amplitude equation formalism. The 2:1 resonance between the fundamental and the second overtone plays an essential role in the behavior of the models 8 refs 6. FY 1998 results of the regional consortium R and D project/the venture promotion type regional consortium R and D (small-/medium-size venture creation type). Development of a new X-ray wavelength-modulated diffraction system; 1998 nendo hacho henco kaisetsuho wo mochiita shin x sen kaisetsu hochi no kaihatsu seika hokokusho Energy Technology Data Exchange (ETDEWEB) NONE 2000-03-01 In the wavelength-modulated X-ray diffraction method, noting that synchrotron radiation (SR) is a high luminance white light with the continuous wavelength distribution, the Blagg refection from crystals is recorded by continuously changing the wavelength. The paper described the results of the development. The camera distance/diameter were made variable in connection with the wavelength modulation. By expanding the area (enlargement of screen) of imaging plate (IP) by Weissenberg camera method, the number of diffraction dots increased, and the data on wavelength gradient were made exact. The camera was completed as designed by introducing X-ray beams, adjusting collimator, specimens, and position of IP. The concentrating mirror device is integrated into the beam line of the SR center, and mirror effects were confirmed through the actual synchrotron radiation. The function of wavelength modulation of double crystal spectrometer was also confirmed. Two kinds of ferrocene derivatives were synthesized of which the wavelength modulation effect is expected as specimen for functional test. Monocrystals were successfully raised, and measured by wavelength-modulated X-ray diffraction device. The validity of phase determination function of the developmental device was confirmed with the results of the existing structure analysis as reference. (NEDO) 7. Effect of Nonlinearity by the Amplitude Variation in coherent transmission in Laser Heterodyne Interferometric International Nuclear Information System (INIS) Chen, H F; Ding, X M; Zhong, Z; Xie, Z L; Yue, H 2006-01-01 To reduce the nonlinearity of nanometer measurement in laser heterodyne interferometric, the influence mechanics of the amplitude variation in coherent transmission upon nonlinearity must be confirmed. Based on the mechanics of nonlinearity, the models about how first-harmonic and second-harmonic nonlinearity caused by the amplitude variation in coherent transmission are proposed. The emulation result shows that different amplitude between measurement arm and reference arm increases the first-harmonic nonlinearity when laser beams nonorthogonality errors exist, but it doesn't change the relationship between nonlinearity and half wavelength. When the rotation angle error β of polarizing beam splitter (PBS) exists, amplitude variation only affects the first-harmonic nonlinearity. With a constant rotation angle of PBS β = 4 0 , when the amplitude factor of measurement arm reduces from 1 to 0.6, the nonlinearity increases from 0.25 nm to 3.81 nm, and the nonlinearity is simple superposition of first-harmonic and second-harmonic. Theoretic analysis and emulation show that the reduction of amplitude variation in coherent transmission can reduce influence on nonlinearity 8. Technique for rapid at-wavelength inspection of extreme ultraviolet mask blanks International Nuclear Information System (INIS) Spector, S. J.; White, D. L.; Tennant, D. M.; Ocola, L. E.; Novembre, A. E.; Peabody, M. L.; Wood, O. R. II 1999-01-01 We have developed two new methods for at-wavelength inspection of mask blanks for extreme-ultraviolet (EUV) lithography. In one method an EUV photoresist is applied directly to a mask blank which is then flood exposed with EUV light and partially developed. In the second method, the photoresist is applied to an EUV transparent membrane that is placed in close proximity to the mask and then exposed and developed. Both reflectivity defects and phase defects alter the exposure of the resist, resulting in mounds of resist at defect sites that can then be located by visual inspection. In the direct application method, a higher contrast resist was shown to increase the height of the mounds, thereby improving the sensitivity of the technique. In the membrane method, a holographic technique was used to reconstruct an image of the mask, revealing the presence of very small defects, approximately 0.2 μm in size. The demonstrated clean transfer of phase and amplitude defects to resist features on a membrane will be important when flagging defects in an automatic inspection tool. (c) 1999 American Vacuum Society 9. Wavelength-Modulated Differential Photoacoustic (WM-DPA) imaging: a high dynamic range modality towards noninvasive diagnosis of cancer Science.gov (United States) Dovlo, Edem; Lashkari, Bahman; Choi, Sung soo Sean; Mandelis, Andreas 2016-03-01 This study explores wavelength-modulated differential photo-acoustic (WM-DPA) imaging for non-invasive early cancer detection via sensitive characterization of functional information such as hemoglobin oxygenation (sO2) levels. Well-known benchmarks of tumor formation such as angiogenesis and hypoxia can be addressed this way. While most conventional photo-acoustic imaging has almost entirely employed high-power pulsed lasers, frequency-domain photo-acoustic radar (FD-PAR) has seen significant development as an alternative technique. It employs a continuous wave laser source intensity-modulated and driven by frequency-swept waveforms. WM-DPA imaging utilizes chirp modulated laser beams at two distinct wavelengths for which absorption differences between oxy- and deoxygenated hemoglobin are minimum (isosbestic point, 805 nm) and maximum (680 nm) to simultaneously generate two signals detected using a standard commercial array transducer as well as a single-element transducer that scans the sample. Signal processing is performed using Lab View and Matlab software developed in-house. Minute changes in total hemoglobin concentration (tHb) and oxygenation levels are detectable using this method since background absorption is suppressed due to the out-of-phase modulation of the laser sources while the difference between the two signals is amplified, thus allowing pre-malignant tumors to become identifiable. By regulating the signal amplitude ratio and phase shift the system can be tuned to applications like cancer screening, sO2 quantification and hypoxia monitoring in stroke patients. Experimental results presented demonstrate WM-DPA imaging of sheep blood phantoms in comparison to single-wavelength FD-PAR imaging. Future work includes the functional PA imaging of small animals in vivo. 10. Dependence of wavelength of Xe ion-induced rippled structures on the fluence in the medium ion energy range Energy Technology Data Exchange (ETDEWEB) Hanisch, Antje; Grenzer, Joerg [Institute of Ion Beam Physics and Materials Research, Dresden (Germany); Biermanns, Andreas; Pietsch, Ullrich [Institute of Physics, University of Siegen (Germany) 2010-07-01 Ion-beam eroded self-organized nanostructures on semiconductors offer new ways for the fabrication of high density memory and optoelectronic devices. It is known that wavelength and amplitude of noble gas ion-induced rippled structures tune with the ion energy and the fluence depending on the energy range, ion type and substrate. The linear theory by Makeev predicts a linear dependence of the ion energy on the wavelength for low temperatures. For Ar{sup +} and O{sub 2}{sup +} it was observed by different groups that the wavelength grows with increasing fluence after being constant up to an onset fluence and before saturation. In this coarsening regime power-law or exponential behavior of the wavelength with the fluence was monitored. So far, investigations for Xe ions on silicon surfaces mainly concentrated on energies below 1 keV. We found a linear dependence of both the ion energy and the fluence on the wavelength and amplitude of rippled structures over a wide range of the Xe{sup +} ion energy between 5 and 70 keV. Moreover, we estimated the ratio of wavelength to amplitude to be constant meaning a shape stability when a threshold fluence of 2.10{sup 17} cm{sup -2} was exceeded. 11. Solar Observations at Submillimeter Wavelengths Science.gov (United States) Kaufmann, P. We review earlier to recent observational evidences and theoretical motivations leading to a renewed interest to observe flares in the submillimeter (submm) - infrared (IR) range of wavelengths. We describe the new solar dedicated submillimeter wave telescope which began operations at El Leoncito in the Argentina Andes: the SST project. It consists of focal plane arrays of two 405 GHz and four 212 GHz radiometers placed in a 1.5-m radome-enclosed Cassegrain antenna, operating simultaneously with one millisecond time resolution. The first solar events analyzed exhibited the onset of rapid submm-wave spikes (100-300 ms), well associated to other flare manifestations, especially at X-rays. The spikes positions were found scattered over the flaring source by tens of arcseconds. For one event an excellent association was found between the gamma-ray emission time profile and the rate of occurrence of submm-wave rapid spikes. The preliminary results favour the idea that bulk burst emissions are a response to numerous fast energetic injections, discrete in time, produced at different spatial positions over the flaring region. Coronal mass ejections were associated to the events studied. Their trajectories extrapolated to the solar surface appear to correspond to the onset time of the submm-wave spikes, which might represent an early signature of the CME's initial acceleration process. 12. Scattering amplitudes in open superstring theory Energy Technology Data Exchange (ETDEWEB) Schlotterer, Oliver 2011-07-15 The present thesis deals with the theme field of the scattering amplitudes in theories of open superstrings. Especially two different formalisms for the handling of superstrings are introduced and applied for the calaculation of tree-level amplitudes - the Ramond- Neveu-Schwarz (RNS) and the Pure-Spinor (PS) formalism. The RNS approach is proved as flexible in order to describe compactification of the initially ten flat space-time dimensions to four dimensions. We solve the technical problems, which result from the interacting basing world-sheet theory with conformal symmetry. This is used to calculate phenomenologically relevant scattering amplitudes of gluons and quarks as well as production rates of massive harmonic vibrations, which were already identified as virtual exchange particles on the massless level. In the case of a low string mass scale in the range of some Tev the string-specific signatures in parton collisions can be observed in the near future in the LHC experiment at CERN and indicated as first experimental proof of the string theory. THose string effects occur universally for a wide class of string ground states respectively internal geometries and represent an elegant way to avoid the so-called landscape problem of the string theory. A further theme complex in this thesis is based on the PS formalism, which allows a manifestly supersymmetric treatment of scattering amplitudes in ten space-time dimension with sixteen supercharges. We introduce a family of superfields, which occur in massless amplitudes of the open string and can be naturally identified with diagrams of three-valued knots. Thereby we reach not only a compact superspace representation of the n-point field-theory amplitude but can also write the complete superstring n-point amplitude as minimal linear combination of partial amplitudes of the field theory as well as hypergeometric functions. The latter carry the string effects and are analyzed from different perspectives, above all 13. Scattering amplitudes in open superstring theory International Nuclear Information System (INIS) Schlotterer, Oliver 2011-01-01 The present thesis deals with the theme field of the scattering amplitudes in theories of open superstrings. Especially two different formalisms for the handling of superstrings are introduced and applied for the calaculation of tree-level amplitudes - the Ramond- Neveu-Schwarz (RNS) and the Pure-Spinor (PS) formalism. The RNS approach is proved as flexible in order to describe compactification of the initially ten flat space-time dimensions to four dimensions. We solve the technical problems, which result from the interacting basing world-sheet theory with conformal symmetry. This is used to calculate phenomenologically relevant scattering amplitudes of gluons and quarks as well as production rates of massive harmonic vibrations, which were already identified as virtual exchange particles on the massless level. In the case of a low string mass scale in the range of some Tev the string-specific signatures in parton collisions can be observed in the near future in the LHC experiment at CERN and indicated as first experimental proof of the string theory. THose string effects occur universally for a wide class of string ground states respectively internal geometries and represent an elegant way to avoid the so-called landscape problem of the string theory. A further theme complex in this thesis is based on the PS formalism, which allows a manifestly supersymmetric treatment of scattering amplitudes in ten space-time dimension with sixteen supercharges. We introduce a family of superfields, which occur in massless amplitudes of the open string and can be naturally identified with diagrams of three-valued knots. Thereby we reach not only a compact superspace representation of the n-point field-theory amplitude but can also write the complete superstring n-point amplitude as minimal linear combination of partial amplitudes of the field theory as well as hypergeometric functions. The latter carry the string effects and are analyzed from different perspectives, above all 14. Direct amplitude detuning measurement with ac dipole Directory of Open Access Journals (Sweden) S. White 2013-07-01 Full Text Available In circular machines, nonlinear dynamics can impact parameters such as beam lifetime and could result in limitations on the performance reach of the accelerator. Assessing and understanding these effects in experiments is essential to confirm the accuracy of the magnetic model and improve the machine performance. A direct measurement of the machine nonlinearities can be obtained by characterizing the dependency of the tune as a function of the amplitude of oscillations (usually defined as amplitude detuning. The conventional technique is to excite the beam to large amplitudes with a single kick and derive the tune from turn-by-turn data acquired with beam position monitors. Although this provides a very precise tune measurement it has the significant disadvantage of being destructive. An alternative, nondestructive way of exciting large amplitude oscillations is to use an ac dipole. The perturbation Hamiltonian in the presence of an ac dipole excitation shows a distinct behavior compared to the free oscillations which should be correctly taken into account in the interpretation of experimental data. The use of an ac dipole for direct amplitude detuning measurement requires careful data processing allowing one to observe the natural tune of the machine; the feasibility of such a measurement is demonstrated using experimental data from the Large Hadron Collider. An experimental proof of the theoretical derivations based on measurements performed at injection energy is provided as well as an application of this technique at top energy using a large number of excitations on the same beam. 15. Color-Kinematics Duality for QCD Amplitudes CERN Document Server Johansson, Henrik 2016-01-01 We show that color-kinematics duality is present in tree-level amplitudes of quantum chromodynamics with massive flavored quarks. Starting with the color structure of QCD, we work out a new color decomposition for n-point tree amplitudes in a reduced basis of primitive amplitudes. These primitives, with k quark-antiquark pairs and (n-2k) gluons, are taken in the (n-2)!/k! Melia basis, and are independent under the color-algebra Kleiss-Kuijf relations. This generalizes the color decomposition of Del Duca, Dixon, and Maltoni to an arbitrary number of quarks. The color coefficients in the new decomposition are given by compact expressions valid for arbitrary gauge group and representation. Considering the kinematic structure, we show through explicit calculations that color-kinematics duality holds for amplitudes with general configurations of gluons and massive quarks. The new (massive) amplitude relations that follow from the duality can be mapped to a well-defined subset of the familiar BCJ relations for gluo... 16. Effective wavelength calibration for moire fringe projection International Nuclear Information System (INIS) Purcell, Daryl; Davies, Angela; Farahi, Faramarz 2006-01-01 The fringe patterns seen when using moire instruments are similar to the patterns seen in traditional interferometry but differ in the spacing between consecutive fringes. In traditional interferometry, the spacing is constant and related to the wavelength of the source. In moire fringe projection, the spacing (the effective wavelength) may not be constant over the field of view and the spacing depends on the system geometry. In these cases, using a constant effective wavelength over the field of view causes inaccurate surface height measurements. We examine the calibration process of the moirefringe projection measurement, which takes this varying wavelength into account to produce a pixel-by-pixel wavelength map. The wavelength calibration procedure is to move the object in the out-of-plane direction a known distance until every pixel intensity value goes through at least one cycle. A sinusoidal function is then fit to the data to extract the effective wavelength pixel by pixel, yielding an effective wavelength map. A calibrated step height was used to validate the effective wavelength map with results within 1% of the nominal value of the step height. The error sources that contributed to the uncertainty in determining the height of the artifact are also investigated 17. High energy hadron spin-flip amplitude International Nuclear Information System (INIS) Selyugin, O.V. 2016-01-01 The high-energy part of the hadron spin-flip amplitude is examined in the framework of the new high-energy general structure (HEGS) model of the elastic hadron scattering at high energies. The different forms of the hadron spin-flip amplitude are compared in the impact parameter representation. It is shown that the existing experimental data of the proton-proton and proton-antiproton elastic scattering at high energy in the region of the diffraction minimum and at large momentum transfer give support in the presence of the energy-independent part of the hadron spin-flip amplitude with the momentum dependence proposed in the works by Galynskii-Kuraev. [ru 18. Scaling of saturation amplitudes in baroclinic instability International Nuclear Information System (INIS) Shepherd, T.G. 1994-01-01 By using finite-amplitude conservation laws for pseudomomentum and pseudoenergy, rigorous upper bounds have been derived on the saturation amplitudes in baroclinic instability for layered and continuously-stratified quasi-geostrophic models. Bounds have been obtained for both the eddy energy and the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity. This approach provides an efficient way of extracting an analytical estimate of the dynamical scalings of the saturation amplitudes in terms of crucial non-dimensional parameters. A possible use is in constructing eddy parameterization schemes for zonally-averaged climate models. The scaling dependences are summarized, and compared with those derived from weakly-nonlinear theory and from baroclinic-adjustment estimates 19. Analytic representations of Yang–Mills amplitudes Energy Technology Data Exchange (ETDEWEB) Bjerrum-Bohr, N.E.J. [Niels Bohr International Academy and Discovery Center, The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø (Denmark); Bourjaily, Jacob L., E-mail: bourjaily@nbi.ku.dk [Niels Bohr International Academy and Discovery Center, The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø (Denmark); Damgaard, Poul H. [Niels Bohr International Academy and Discovery Center, The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen Ø (Denmark); Feng, Bo [Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou City, 310027 (China) 2016-12-15 Scattering amplitudes in Yang–Mills theory can be represented in the formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary projective space—fully localized on the support of the scattering equations. Because solving the scattering equations is difficult and summing over the solutions algebraically complex, a method of directly integrating the terms that appear in this representation has long been sought. We solve this important open problem by first rewriting the terms in a manifestly Möbius-invariant form and then using monodromy relations (inspired by analogy to string theory) to decompose terms into those for which combinatorial rules of integration are known. The result is the foundations of a systematic procedure to obtain analytic, covariant forms of Yang–Mills tree-amplitudes for any number of external legs and in any number of dimensions. As examples, we provide compact analytic expressions for amplitudes involving up to six gluons of arbitrary helicities. 20. Nonlinear (super)symmetries and amplitudes Energy Technology Data Exchange (ETDEWEB) Kallosh, Renata [Physics Department, Stanford University,382 Via Pueblo Mall, Stanford, CA 94305-4060 (United States) 2017-03-07 There is an increasing interest in nonlinear supersymmetries in cosmological model building. Independently, elegant expressions for the all-tree amplitudes in models with nonlinear symmetries, like D3 brane Dirac-Born-Infeld-Volkov-Akulov theory, were recently discovered. Using the generalized background field method we show how, in general, nonlinear symmetries of the action, bosonic and fermionic, constrain amplitudes beyond soft limits. The same identities control, for example, bosonic E{sub 7(7)} scalar sector symmetries as well as the fermionic goldstino symmetries. We present a universal derivation of the vanishing amplitudes in the single (bosonic or fermionic) soft limit. We explain why, universally, the double-soft limit probes the coset space algebra. We also provide identities describing the multiple-soft limit. We discuss loop corrections to N≥5 supergravity, to the D3 brane, and the UV completion of constrained multiplets in string theory. 1. Scattering Amplitudes and Worldsheet Models of QFTs CERN Multimedia CERN. Geneva 2016-01-01 I will describe recent progress on the study of scattering amplitudes via ambitwistor strings and the scattering equations. Ambitwistor strings are worldsheet models of quantum field theories, inspired by string theory. They naturally lead to a representation of amplitudes based on the scattering equations. While worldsheet models and related ideas have had a wide-ranging impact on the modern study of amplitudes, their direct application at loop level is a very recent success. I will show how a major difficulty in the loop-level story, the technicalities of higher-genus Riemann surfaces, can be avoided by turning the higher-genus surface into a nodal Riemann sphere, with the nodes representing the loop momenta. I will present new formulas for the one-loop integrands of gauge theory and gravity, with or without supersymmetry, and also some two-loop results. 2. Relativistic amplitudes in terms of wave functions International Nuclear Information System (INIS) Karmanov, V.A. 1978-01-01 In the framework of the invariant diagram technique which arises at the formulation of the fueld theory on the light front the question about conditions at which the relativistic amplitudes may be expressed through the wave functions is investigated. The amplitudes obtained depend on four-vector ω, determining the light front surface. The way is shown to find such values of the four-vector ω, at which the contribution of diagrams not expressed through wave functions is minimal. The investigation carried out is equivalent to the study of the dependence of amplitudes of the old-fashioned perturbation theory in the in the infinite momentum frame on direction of the infinite momentum 3. Multiphoton amplitude in a constant background field Science.gov (United States) Ahmad, Aftab; Ahmadiniaz, Naser; Corradini, Olindo; Kim, Sang Pyo; Schubert, Christian 2018-01-01 In this contribution, we present our recent compact master formulas for the multiphoton amplitudes of a scalar propagator in a constant background field using the worldline fomulation of quantum field theory. The constant field has been included nonperturbatively, which is crucial for strong external fields. A possible application is the scattering of photons by electrons in a strong magnetic field, a process that has been a subject of great interest since the discovery of astrophysical objects like radio pulsars, which provide evidence that magnetic fields of the order of 1012G are present in nature. The presence of a strong external field leads to a strong deviation from the classical scattering amplitudes. We explicitly work out the Compton scattering amplitude in a magnetic field, which is a process of potential relevance for astrophysics. Our final result is compact and suitable for numerical integration. 4. Born amplitudes and seagull term in meson-soliton scattering International Nuclear Information System (INIS) Liang, Y.G.; Li, B.A.; Liu, K.F.; Su, R.K. 1990-01-01 The meson-soliton scattering for the φ 4 theory in 1+1 dimensions is calculated. We show that when the seagull term from the equal time commutator is included in addition to the Born amplitudes, the t-matrix from the reduction formula approach is identical to that of the potential scattering with small quantum fluctuations to leading order in weak coupling. The seagull term is equal to the Born term in the potential scattering. This confirms the speculation that the leading order Yukawa coupling is derivable from the classical soliton. (orig.) 5. Measurements of acoustic pressure at high amplitudes and intensities International Nuclear Information System (INIS) Crum, L A; Bailey, M R; Kaczkowski, P; McAteer, J A; Pishchalnikov, Y A; Sapozhnikov, O A 2004-01-01 In our research group, we desire measurements of the large pressure amplitudes generated by the shock waves used in shock wave lithotripsy (SWL) and the large acoustic intensities used in High Intensity Focused Ultrasound (HIFU). Conventional piezoelectric or PVDF hydrophones can not be used for such measurements as they are damaged either by cavitation, in SWL applications, or heat, in HIFU applications. In order to circumvent these difficulties, we have utilized optical fiber hydrophones in SWL that do not cavitate, and small glass probes and a scattering technique for measurements of large HIFU intensities. Descriptions of these techniques will be given as well as some typical data 6. Interferometric Imaging Directly with Closure Phases and Closure Amplitudes Science.gov (United States) Chael, Andrew A.; Johnson, Michael D.; Bouman, Katherine L.; Blackburn, Lindy L.; Akiyama, Kazunori; Narayan, Ramesh 2018-04-01 Interferometric imaging now achieves angular resolutions as fine as ∼10 μas, probing scales that are inaccessible to single telescopes. Traditional synthesis imaging methods require calibrated visibilities; however, interferometric calibration is challenging, especially at high frequencies. Nevertheless, most studies present only a single image of their data after a process of “self-calibration,” an iterative procedure where the initial image and calibration assumptions can significantly influence the final image. We present a method for efficient interferometric imaging directly using only closure amplitudes and closure phases, which are immune to station-based calibration errors. Closure-only imaging provides results that are as noncommittal as possible and allows for reconstructing an image independently from separate amplitude and phase self-calibration. While closure-only imaging eliminates some image information (e.g., the total image flux density and the image centroid), this information can be recovered through a small number of additional constraints. We demonstrate that closure-only imaging can produce high-fidelity results, even for sparse arrays such as the Event Horizon Telescope, and that the resulting images are independent of the level of systematic amplitude error. We apply closure imaging to VLBA and ALMA data and show that it is capable of matching or exceeding the performance of traditional self-calibration and CLEAN for these data sets. 7. Amplitude Models for Discrimination and Yield Estimation Energy Technology Data Exchange (ETDEWEB) Phillips, William Scott [Los Alamos National Lab. (LANL), Los Alamos, NM (United States) 2016-09-01 This seminar presentation describes amplitude models and yield estimations that look at the data in order to inform legislation. The following points were brought forth in the summary: global models that will predict three-component amplitudes (R-T-Z) were produced; Q models match regional geology; corrected source spectra can be used for discrimination and yield estimation; three-component data increase coverage and reduce scatter in source spectral estimates; three-component efforts must include distance-dependent effects; a community effort on instrument calibration is needed. 8. Singularity Structure of Maximally Supersymmetric Scattering Amplitudes DEFF Research Database (Denmark) Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy 2014-01-01 We present evidence that loop amplitudes in maximally supersymmetric (N=4) Yang-Mills theory (SYM) beyond the planar limit share some of the remarkable structures of the planar theory. In particular, we show that through two loops, the four-particle amplitude in full N=4 SYM has only logarithmic ...... singularities and is free of any poles at infinity—properties closely related to uniform transcendentality and the UV finiteness of the theory. We also briefly comment on implications for maximal (N=8) supergravity theory (SUGRA).... 9. Amplitude modulation detection with concurrent frequency modulation. Science.gov (United States) Nagaraj, Naveen K 2016-09-01 Human speech consists of concomitant temporal modulations in amplitude and frequency that are crucial for speech perception. In this study, amplitude modulation (AM) detection thresholds were measured for 550 and 5000 Hz carriers with and without concurrent frequency modulation (FM), at AM rates crucial for speech perception. Results indicate that adding 40 Hz FM interferes with AM detection, more so for 5000 Hz carrier and for frequency deviations exceeding the critical bandwidth of the carrier frequency. These findings suggest that future cochlear implant processors, encoding speech fine-structures may consider limiting the FM to narrow bandwidth and to low frequencies. 10. Gluon amplitudes as 2 d conformal correlators Science.gov (United States) Pasterski, Sabrina; Shao, Shu-Heng; Strominger, Andrew 2017-10-01 Recently, spin-one wave functions in four dimensions that are conformal primaries of the Lorentz group S L (2 ,C ) were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wave functions. The answers have the same conformal covariance as correlators of spin-one primaries in a 2 d CFT. The Britto-Cachazo-Feng-Witten (BCFW) recursion relation between three- and four-point gluon amplitudes is recast into this conformal basis. 11. Precise generator of stability amplitude pulses International Nuclear Information System (INIS) Zhuk, N.A.; Zdesenko, Yu.G.; Kuts, V.N. 1989-01-01 A generator of stability amplitude pulses, designed for stabilization of a low-noise semiconducting spectrometer, used in investigations of 76 Ge2β-decay, is described. The generator contains a permanent-voltage source, a storage element and a switch based on a Hg relay. A thermostatic source provides a relative voltage instability less than ±5x10 -6 per 80h (standard deviation). The Hg relay is placed into a separate thermostat. The relative instability of output generator pulse amplitude does not exceed ±1.5x10 -5 per 24h 12. High energy multi-gluon exchange amplitudes International Nuclear Information System (INIS) Jaroszewicz, T. 1980-11-01 We examine perturbative high energy n-gluon exchange amplitudes calculated in the Coulomb gauge. If n exceeds the minimum required by the t-channel quantum numbers, such amplitudes are non-leading in lns. We derive a closed system of coupled integral equations for the corresponding two-particle n-gluon vertices, obtained by summing the leading powers of ln(N μ psup(μ)), where psup(μ) is the incident momentum and Nsup(μ) the gauge-defining vector. Our equations are infra-red finite, provided the external particles are colour singlets. (author) 13. Study on modulation amplitude stabilization method for PEM based on FPGA in atomic magnetometer Science.gov (United States) Wang, Qinghua; Quan, Wei; Duan, Lihong 2017-10-01 Atomic magnetometer which uses atoms as sensitive elements have ultra-high precision and has wide applications in scientific researches. The photoelastic modulation method based on photoelastic modulator (PEM) is used in the atomic magnetometer to detect the small optical rotation angle of a linearly polarized light. However, the modulation amplitude of the PEM will drift due to the environmental factors, which reduces the precision and long-term stability of the atomic magnetometer. Consequently, stabilizing the PEM's modulation amplitude is essential to precision measurement. In this paper, a modulation amplitude stabilization method for PEM based on Field Programmable Gate Array (FPGA) is proposed. The designed control system contains an optical setup and an electrical part. The optical setup is used to measure the PEM's modulation amplitude. The FPGA chip, with the PID control algorithm implemented in it, is used as the electrical part's micro controller. The closed loop control method based on the photoelastic modulation detection system can directly measure the PEM's modulation amplitude in real time, without increasing the additional optical devices. In addition, the operating speed of the modulation amplitude stabilization control system can be greatly improved because of the FPGA's parallel computing feature, and the PID control algorithm ensures flexibility to meet different needs of the PEM's modulation amplitude set values. The Modelsim simulation results show the correctness of the PID control algorithm, and the long-term stability of the PEM's modulation amplitude reaches 0.35% in a 3-hour continuous measurement. 14. WEATHER ON OTHER WORLDS. III. A SURVEY FOR T DWARFS WITH HIGH-AMPLITUDE OPTICAL VARIABILITY International Nuclear Information System (INIS) Heinze, Aren N.; Metchev, Stanimir; Kellogg, Kendra 2015-01-01 We have monitored 12 T dwarfs with the Kitt Peak 2.1 m telescope using an F814W filter (0.7-0.95 μm) to place in context the remarkable 10%-20% variability exhibited by the nearby T dwarf Luhman 16B in this wavelength regime. The motivation was the poorly known red optical behavior of T dwarfs, which have been monitored almost exclusively at infrared wavelengths, where variability amplitudes greater than 10% have been found to be very rare. We detect highly significant variability in two T dwarfs. The T2.5 dwarf 2MASS 13243559+6358284 shows consistent ∼17% variability on two consecutive nights. The T2 dwarf 2MASS J16291840+0335371 exhibits ∼10% variability that may evolve from night to night, similarly to Luhman 16B. Both objects were previously known to be variable in the infrared, but with considerably lower amplitudes. We also find evidence for variability in the T6 dwarf J162414.37+002915.6, but since it has lower significance, we conservatively refrain from claiming this object as a variable. We explore and rule out various telluric effects, demonstrating that the variations we detect are astrophysically real. We suggest that high-amplitude photometric variability for T dwarfs is likely more common in the red optical than at longer wavelengths. The two new members of the growing class of high-amplitude variable T dwarfs offer excellent prospects for further study of cloud structures and their evolution 15. WEATHER ON OTHER WORLDS. III. A SURVEY FOR T DWARFS WITH HIGH-AMPLITUDE OPTICAL VARIABILITY Energy Technology Data Exchange (ETDEWEB) Heinze, Aren N.; Metchev, Stanimir [Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800 (United States); Kellogg, Kendra, E-mail: aren.heinze@stonybrook.edu, E-mail: smetchev@uwo.ca [Department of Physics and Astronomy, The University of Western Ontario, 1151 Richmond St, London, ON N6A 3K7 (Canada) 2015-03-10 We have monitored 12 T dwarfs with the Kitt Peak 2.1 m telescope using an F814W filter (0.7-0.95 μm) to place in context the remarkable 10%-20% variability exhibited by the nearby T dwarf Luhman 16B in this wavelength regime. The motivation was the poorly known red optical behavior of T dwarfs, which have been monitored almost exclusively at infrared wavelengths, where variability amplitudes greater than 10% have been found to be very rare. We detect highly significant variability in two T dwarfs. The T2.5 dwarf 2MASS 13243559+6358284 shows consistent ∼17% variability on two consecutive nights. The T2 dwarf 2MASS J16291840+0335371 exhibits ∼10% variability that may evolve from night to night, similarly to Luhman 16B. Both objects were previously known to be variable in the infrared, but with considerably lower amplitudes. We also find evidence for variability in the T6 dwarf J162414.37+002915.6, but since it has lower significance, we conservatively refrain from claiming this object as a variable. We explore and rule out various telluric effects, demonstrating that the variations we detect are astrophysically real. We suggest that high-amplitude photometric variability for T dwarfs is likely more common in the red optical than at longer wavelengths. The two new members of the growing class of high-amplitude variable T dwarfs offer excellent prospects for further study of cloud structures and their evolution. 16. Amplitude ratios in ρ0 leptoproductions and GPDs Directory of Open Access Journals (Sweden) Goloskokov S.V. 2017-01-01 Using the model results we calculate the ratio of different helicity amplitudes for a transversely polarized proton target to the leading twist longitudinal amplitude. Our results are close to the amplitude ratios measured by HERMES. 17. Coupling dynamics in speech gestures: amplitude and rate influences. Science.gov (United States) van Lieshout, Pascal H H M 2017-08-01 Speech is a complex oral motor function that involves multiple articulators that need to be coordinated in space and time at relatively high movement speeds. How this is accomplished remains an important and largely unresolved empirical question. From a coordination dynamics perspective, coordination involves the assembly of coordinative units that are characterized by inherently stable coupling patterns that act as attractor states for task-specific actions. In the motor control literature, one particular model formulated by Haken et al. (Biol Cybern 51(5):347-356, 1985) or HKB has received considerable attention in the way it can account for changes in the nature and stability of specific coordination patterns between limbs or between limbs and external stimuli. In this model (and related versions), movement amplitude is considered a critical factor in the formation of these patterns. Several studies have demonstrated its role for bimanual coordination and similar types of tasks, but for speech motor control such studies are lacking. The current study describes a systematic approach to evaluate the impact of movement amplitude and movement duration on coordination stability in the production of bilabial and tongue body gestures for specific vowel-consonant-vowel strings. The vowel combinations that were used induced a natural contrast in movement amplitude at three speaking rate conditions (slow, habitual, fast). Data were collected on ten young adults using electromagnetic articulography, recording movement data from lips and tongue with high temporal and spatial precision. The results showed that with small movement amplitudes there is a decrease in coordination stability, independent from movement duration. These findings were found to be robust across all individuals and are interpreted as further evidence that principles of coupling dynamics operate in the oral motor control system similar to other motor systems and can be explained in terms of coupling 18. Wavelength scaling of laser plasma coupling International Nuclear Information System (INIS) Kruer, W.L. 1983-01-01 The use of shorter wavelength laser light both enhances collisional absorption and reduces deleterious collective plasma effects. Coupling processes which can be important in reactor-size targets are briefly reviewed. Simple estimates are presented for the intensity-wavelength regime in which collisional absorption is high and collective effects are minimized 19. Azimuthal Seismic Amplitude Variation with Offset and Azimuth Inversion in Weakly Anisotropic Media with Orthorhombic Symmetry Science.gov (United States) Pan, Xinpeng; Zhang, Guangzhi; Yin, Xingyao 2018-01-01 Seismic amplitude variation with offset and azimuth (AVOaz) inversion is well known as a popular and pragmatic tool utilized to estimate fracture parameters. A single set of vertical fractures aligned along a preferred horizontal direction embedded in a horizontally layered medium can be considered as an effective long-wavelength orthorhombic medium. Estimation of Thomsen's weak-anisotropy (WA) parameters and fracture weaknesses plays an important role in characterizing the orthorhombic anisotropy in a weakly anisotropic medium. Our goal is to demonstrate an orthorhombic anisotropic AVOaz inversion approach to describe the orthorhombic anisotropy utilizing the observable wide-azimuth seismic reflection data in a fractured reservoir with the assumption of orthorhombic symmetry. Combining Thomsen's WA theory and linear-slip model, we first derive a perturbation in stiffness matrix of a weakly anisotropic medium with orthorhombic symmetry under the assumption of small WA parameters and fracture weaknesses. Using the perturbation matrix and scattering function, we then derive an expression for linearized PP-wave reflection coefficient in terms of P- and S-wave moduli, density, Thomsen's WA parameters, and fracture weaknesses in such an orthorhombic medium, which avoids the complicated nonlinear relationship between the orthorhombic anisotropy and azimuthal seismic reflection data. Incorporating azimuthal seismic data and Bayesian inversion theory, the maximum a posteriori solutions of Thomsen's WA parameters and fracture weaknesses in a weakly anisotropic medium with orthorhombic symmetry are reasonably estimated with the constraints of Cauchy a priori probability distribution and smooth initial models of model parameters to enhance the inversion resolution and the nonlinear iteratively reweighted least squares strategy. The synthetic examples containing a moderate noise demonstrate the feasibility of the derived orthorhombic anisotropic AVOaz inversion method, and the 20. Stora's fine notion of divergent amplitudes Directory of Open Access Journals (Sweden) Joseph C. Várilly 2016-11-01 Full Text Available Stora and coworkers refined the notion of divergent quantum amplitude, somewhat upsetting the standard power-counting recipe. This unexpectedly clears the way to new prototypes for free and interacting field theories of bosons of any mass and spin. 1. Connected formulas for amplitudes in standard model Energy Technology Data Exchange (ETDEWEB) He, Song [CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics, Chinese Academy of Sciences,Beijing 100190 (China); School of Physical Sciences, University of Chinese Academy of Sciences,No. 19A Yuquan Road, Beijing 100049 (China); Zhang, Yong [Department of Physics, Beijing Normal University,Beijing 100875 (China); CAS Key Laboratory of Theoretical Physics,Institute of Theoretical Physics, Chinese Academy of Sciences,Beijing 100190 (China) 2017-03-17 Witten’s twistor string theory has led to new representations of S-matrix in massless QFT as a single object, including Cachazo-He-Yuan formulas in general and connected formulas in four dimensions. As a first step towards more realistic processes of the standard model, we extend the construction to QCD tree amplitudes with massless quarks and those with a Higgs boson. For both cases, we find connected formulas in four dimensions for all multiplicities which are very similar to the one for Yang-Mills amplitudes. The formula for quark-gluon color-ordered amplitudes differs from the pure-gluon case only by a Jacobian factor that depends on flavors and orderings of the quarks. In the formula for Higgs plus multi-parton amplitudes, the massive Higgs boson is effectively described by two additional massless legs which do not appear in the Parke-Taylor factor. The latter also represents the first twistor-string/connected formula for form factors. 2. Fatigue Reliability under Multiple-Amplitude Loads DEFF Research Database (Denmark) Talreja, R. 1979-01-01 for the initial tensile strength and the fatigue life, the probability distributions for the residual tensile strength in both the crack initiation and the crack propagation stages of fatigue are determined. The method is illustrated for two-amplitude loads by means of experimental results obtained by testing... 3. Ward identities for amplitudes with reggeized gluons International Nuclear Information System (INIS) Bartles, J.; Vacca, G.P. 2012-05-01 Starting from the effective action of high energy QCD we derive Ward identities for Green's functions of reggeized gluons. They follow from the gauge invariance of the effective action, and allow to derive new representations of amplitudes containing physical particles as well as reggeized gluons. We explicitly demonstrate their validity for the BFKL kernel, and we present a new derivation of the kernel. 4. Scattering amplitudes in super-renormalizable gravity International Nuclear Information System (INIS) Donà, Pietro; Giaccari, Stefano; Modesto, Leonardo; Rachwał, Lesław; Zhu, Yiwei 2015-01-01 We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d’Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell n−point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action. 5. Particle Distribution Modification by Low Amplitude Modes International Nuclear Information System (INIS) White, R.B.; Gorelenkov, N.; Heidbrink, W.W.; Van Zeeland, M.A. 2009-01-01 Modification of a high energy particle distribution by a spectrum of low amplitude modes is investigated using a guiding center code. Only through resonance are modes effective in modifying the distribution. Diagnostics are used to illustrate the mode-particle interaction and to find which effects are relevant in producing significant resonance, including kinetic Poincare plots and plots showing those orbits with time averaged mode-particle energy transfer. Effects of pitch angle scattering and drag are studied, as well as plasma rotation and time dependence of the equilibrium and mode frequencies. A specific example of changes observed in a DIII-D deuterium beam distribution in the presence of low amplitude experimentally validated Toroidal Alfven (TAE) eigenmodes and Reversed Shear Alfven (RSAE) eigenmodes is examined in detail. Comparison with experimental data shows that multiple low amplitude modes can account for significant modification of high energy beam particle distributions. It is found that there is a stochastic threshold for beam profile modification, and that the experimental amplitudes are only slightly above this threshold. 6. Kaon decay amplitudes using staggered fermions International Nuclear Information System (INIS) Sharpe, S.R. 1986-12-01 A status report is given of an attempt, using staggered fermions to calculate the real and imaginary parts of the amplitudes for K → ππ,. Semi-quantitative results are found for the imaginary parts, and these suggest that ε' might be smaller than previously expected in the standard model 7. Constraints on low energy Compton scattering amplitudes International Nuclear Information System (INIS) Raszillier, I. 1979-04-01 We derive the constraints and correlations of fairly general type for Compton scattering amplitudes at energies below photoproduction threshold and fixed momentum transfer, following from (an upper bound on) the corresponding differential cross section above photoproduction threshold. The derivation involves the solution of an extremal problem in a certain space of vector - valued analytic functions. (author) 8. Analytic properties of many-particle amplitudes Energy Technology Data Exchange (ETDEWEB) Medvedev, B V; Pavlov, V P; Polivanov, M K; Sukhanov, A D [Gosudarstvennyj Komitet po Ispol' zovaniyu Atomnoj Ehnergii SSSR, Moscow. Inst. Teoreticheskoj i Ehksperimental' noj Fiziki; AN SSSR, Moscow. Matematicheskij Inst.) 1982-08-01 In the framework of N. N. Bogolyubov axiomatic approach the complete proof of the existence of an analytic function the boundary values of which are the amplitudes of any channel of n-particle process is given. The one-particle structure of this function is described. 9. Low-Profile, Dual-Wavelength, Dual-Polarized Antenna Science.gov (United States) Carswell, James R. 2010-01-01 A single-aperture, low-profile antenna design has been developed that supports dual-polarization and simultaneous operation at two wavelengths. It realizes multiple beams in the elevation plane, and supports radiometric, radar, and conical scanning applications. This antenna consists of multiple azimuth sticks, with each stick being a multilayer, hybrid design. Each stick forms the h-plane pattern of the C and Ku-band vertically and horizontally polarized antenna beams. By combining several azimuth sticks together, the elevation beam is formed. With a separate transceiver for each stick, the transmit phase and amplitude of each stick can be controlled to synthesize a beam at a specific incidence angle and to realize a particular side-lobe pattern. By changing the transmit phase distribution through the transceivers, the transmit antenna beam can be steered to different incidence angles. By controlling the amplitude distribution, different side lobe patterns and efficiencies can be realized. The receive beams are formed using digital beam synthesis techniques, resulting in very little loss in the receive path, thus enabling a very-low loss receive antenna to support passive measurements. 10. Short wavelength FELs using the SLAC linac International Nuclear Information System (INIS) Winick, H.; Bane, K.; Boyce, R. 1993-08-01 Recent technological developments have opened the possibility to construct a device which we call a Linac Coherent Light Source (LCLS); a fourth generation light source, with brightness, coherence, and peak power far exceeding other sources. Operating on the principle of the free electron laser (FEL), the LCLS would extend the range of FEL operation to much aborter wavelength than the 240 mn that has so far been reached. We report the results of studies of the use of the SLAC linac to drive an LCLS at wavelengths from about 3-100 nm initially and possibly even shorter wavelengths in the future. Lasing would be achieved in a single pass of a low emittance, high peak current, high energy electron beam through a long undulator. Most present FELs use an optical cavity to build up the intensity of the light to achieve lasing action in a low gain oscillator configuration. By eliminating the optical cavity, which is difficult to make at short wavelengths, laser action can be extended to shorter wavelengths by Self-Amplified-Spontaneous-Emission (SASE), or by harmonic generation from a longer wavelength seed laser. Short wavelength, single pass lasers have been extensively studied at several laboratories and at recent workshops 11. Propagation of nonlinear ion acoustic wave with generation of long-wavelength waves International Nuclear Information System (INIS) Ohsawa, Yukiharu; Kamimura, Tetsuo 1978-01-01 The nonlinear propagation of the wave packet of an ion acoustic wave with wavenumber k 0 asymptotically equals k sub(De) (the electron Debye wavenumber) is investigated by computer simulations. From the wave packet of the ion acoustic wave, waves with long wavelengths are observed to be produced within a few periods for the amplitude oscillation of the original wave packet. These waves are generated in the region where the original wave packet exists. Their characteristic wavelength is of the order of the length of the wave packet, and their propagation velocity is almost equal to the ion acoustic speed. The long-wavelength waves thus produced strongly affect the nonlinear evolution of the original wave packet. (auth.) 12. Microscopic single-crystal refractometry as a function of wavelength International Nuclear Information System (INIS) DeLoach, L.D. 1994-01-01 The refractive indices of crystal fragments 50--200 μm in size can be measured for light wavelengths between 365 and 1100 nm with a spindle-stage refractometer. Established methods from optical crystallograpy are used to orient a crystal on the microscope spindle stage and then to match its refractive index to an immersion fluid. The refractive index of the fluid for the wavelength of light and matching temperature is determined by comparison of a reference crystal on a second spindle axis with the fluid under the match conditions. Investigations of new nonlinear-optical crystals admirably demonstrate the advantages of measuring the refractive index to ± 0.0004 in small single crystals 13. A readout system for the wavelength-shifting optical module Energy Technology Data Exchange (ETDEWEB) Foesig, Carl-Christian; Boeser, Sebastian [Johannes Gutenberg-Universitaet, Mainz (Germany); Collaboration: IceCube-Collaboration 2016-07-01 The success of IceCube and the plans for an IceCube-Gen2 stimulate the development of new photo sensors. The approach of the Wavelength-shifting Optical Module is to provide a device which has a low dark noise rate combined with a high detection efficiency. A small PMT is used to detect red shifted photons guided in a coated PMMA tube, originally emitted by a wavelength shifting coating that absorbs photons in the UV Region. We have studied several PMTs for their usability with the IceCube-Gen2 readout system. Relevant parameters are the pulse widths in relation to the bandwidth of the IceCube-Gen2 readout electronics and the dark noise rates. 14. System and Method for Multi-Wavelength Optical Signal Detection Science.gov (United States) McGlone, Thomas D. (Inventor) 2017-01-01 The system and method for multi-wavelength optical signal detection enables the detection of optical signal levels significantly below those processed at the discrete circuit level by the use of mixed-signal processing methods implemented with integrated circuit technologies. The present invention is configured to detect and process small signals, which enables the reduction of the optical power required to stimulate detection networks, and lowers the required laser power to make specific measurements. The present invention provides an adaptation of active pixel networks combined with mixed-signal processing methods to provide an integer representation of the received signal as an output. The present invention also provides multi-wavelength laser detection circuits for use in various systems, such as a differential absorption light detection and ranging system. 15. Photon-counting chirped amplitude modulation lidar system using superconducting nanowire single-photon detector at 1550-nm wavelength Science.gov (United States) Zhou, Hui; He, Yu-Hao; Lü, Chao-Lin; You, Li-Xing; Li, Zhao-Hui; Wu, Guang; Zhang, Wei-Jun; Zhang, Lu; Liu, Xiao-Yu; Yang, Xiao-Yan; Wang, Zhen 2018-01-01 Not Available Project supported by National Key R&D Program of China (Grant No. 2017YFA0304000), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61501442 and 61671438), and the Joint Research Fund in Astronomy (U1631240) under Cooperative Agreement between the NSFC and Chinese Academy of Sciences (CAS). 16. Cascaded Amplitude Modulations in Sound Texture Perception DEFF Research Database (Denmark) McWalter, Richard Ian; Dau, Torsten 2017-01-01 . In this study, we investigated the perception of sound textures that contain rhythmic structure, specifically second-order amplitude modulations that arise from the interaction of different modulation rates, previously described as "beating" in the envelope-frequency domain. We developed an auditory texture...... model that utilizes a cascade of modulation filterbanks that capture the structure of simple rhythmic patterns. The model was examined in a series of psychophysical listening experiments using synthetic sound textures-stimuli generated using time-averaged statistics measured from real-world textures....... In a texture identification task, our results indicated that second-order amplitude modulation sensitivity enhanced recognition. Next, we examined the contribution of the second-order modulation analysis in a preference task, where the proposed auditory texture model was preferred over a range of model... 17. Source amplitudes for active exterior cloaking International Nuclear Information System (INIS) Norris, Andrew N; Amirkulova, Feruza A; Parnell, William J 2012-01-01 The active cloak comprises a discrete set of multipole sources that destructively interfere with an incident time harmonic scalar wave to produce zero total field over a finite spatial region. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. The field generated by the active sources vanishes in the infinite region exterior to a set of circles defined by the relative positions of the sources. The results provide a direct solution to the inverse problem of determining the source amplitudes. They also define a broad class of non-radiating discrete sources. (paper) 18. Constructing QCD one-loop amplitudes International Nuclear Information System (INIS) Forde, D 2008-01-01 In the context of constructing one-loop amplitudes using a unitarity bootstrap approach we discuss a general systematic procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted after examining the behavior of the cut integrand as the unconstrained parameters of a specifically chosen parameterization of the cut loop momentum approach infinity. Measurements of new physics at the forthcoming experimental program at CERN's Large Hadron Collider (LHC) will require a precise understanding of processes at next-to-leading order (NLO). This places increased demands for the computation of new one-loop amplitudes. This in turn has spurred recent developments towards improved calculational techniques. Direct calculations using Feynman diagrams are in general inefficient. Developments of more efficient techniques have usually centered around unitarity techniques [1], where tree amplitudes are effectively 'glued' together to form loops. The most straightforward application of this method, in which the cut loop momentum is in D = 4, allows for the computation of 'cut-constructible' terms only, i.e. (poly)logarithmic containing terms and any related constants. QCD amplitudes contain, in addition to such terms, rational pieces which cannot be derived using such cuts. These 'missing' rational parts can be extracted using cut loop momenta in D = 4-2 (var e psilon). The greater difficulty of such calculations has restricted the application of this approach, although recent developments [3, 4] have provided new promise for this technique. Recently the application of on-shell recursion relations [5] to obtaining the 'missing' rational parts of one-loop processes [6] has provided an alternative very promising solution to this problem. In combination with unitarity methods an 'on-shell bootstrap' approach provides an efficient technique for computing complete one-loop QCD amplitudes [7]. Additionally 19. Differential equations, associators, and recurrences for amplitudes Directory of Open Access Journals (Sweden) Georg Puhlfürst 2016-01-01 Full Text Available We provide new methods to straightforwardly obtain compact and analytic expressions for ϵ-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for recurrence relations connecting different ϵ-orders of a power series solution in ϵ of a differential equation. This strategy generalizes the usual iteration by Picard's method. Our tools are demonstrated for generalized hypergeometric functions. Furthermore, we match the ϵ-expansion of specific generalized hypergeometric functions with the underlying Drinfeld associator with proper Lie algebra and monodromy representations. We also apply our tools for computing ϵ-expansions for solutions to generic first-order Fuchsian equations (Schlesinger system. Finally, we set up our methods to systematically get compact and explicit α′-expansions of tree-level superstring amplitudes to any order in α′. 20. Unitarity and amplitudes for high energies International Nuclear Information System (INIS) Efimov, G.V. 1997-01-01 It is shown that in the quantum field theory of scalar particles with mass m the following inequalities for the upper bound for the amplitude of elastic scattering Μ(s,t) \|Μ(s,t)\| 0 )s, (\|t\|≥\|t 0 \|>0) and for the total cross section of scalar particles σ tot (s)≤C\|d/dt ln Im Μ(s,t)\| t=0 , (s → ∞) are valid. This result is based on the unitarity of the S-matrix on the mass shell and on a natural assumption that the imaginary part of the elastic scattering Im Μ(s,t) is a differentiable and convex down function in some vicinity of t=0. The locality of the theory and the analyticity of the elastic amplitude in the Martin-Lehmann ellipse are not used in proving these inequalities 1. Large amplitude waves and fields in plasmas International Nuclear Information System (INIS) Angelis, U. de; Naples Univ. 1990-02-01 In this review, based mostly on the results of the recent workshop on ''Large Amplitude Waves and Fields in Plasmas'' held at ICTP (Trieste, Italy) in May 1989 during the Spring College on Plasma Physics, I will mostly concentrate on underdense, cold, homogeneous plasmas, discussing some of the alternative (to fusion) uses of laser-plasma interaction. In Part I an outline of some basic non-linear processes is given, together with some recent experimental results. The processes are chosen because of their relevance to the applications or because new interesting developments have been reported at the ICTP workshop (or both). In Part II the excitation mechanisms and uses of large amplitude plasma waves are presented: these include phase-conjugation in plasmas, plasma based accelerators (beat-wave, plasma wake-field and laser wake-field), plasma lenses and plasma wigglers for Free Electron Lasers. (author) 2. Integrable spin chains and scattering amplitudes Energy Technology Data Exchange (ETDEWEB) Bartels, J.; Prygarin, A. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Lipatov, L.N. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Petersburg Nuclear Physics Institute (Russian Federation); Sankt-Peterburgskij Univ., St. Petersburg (Russian Federation) 2011-04-15 In this review we show that the multi-particle scattering amplitudes in N=4 SYM at large N{sub c} and in the multi-Regge kinematics for some physical regions have the high energy behavior appearing from the contribution of the Mandelstam cuts in the complex angular momentum plane of the corresponding t-channel partial waves. These Mandelstam cuts or Regge cuts are resulting from gluon composite states in the adjoint representation of the gauge group SU(N{sub c}). In the leading logarithmic approximation (LLA) their contribution to the six point amplitude is in full agreement with the known two-loop result. The Hamiltonian for the Mandelstam states constructed from n gluons in LLA coincides with the local Hamiltonian of an integrable open spin chain. We construct the corresponding wave functions using the integrals of motion and the Baxter-Sklyanin approach. (orig.) 3. Understanding the amplitudes of noise correlation measurements Science.gov (United States) Tsai, Victor C. 2011-01-01 Cross correlation of ambient seismic noise is known to result in time series from which station-station travel-time measurements can be made. Part of the reason that these cross-correlation travel-time measurements are reliable is that there exists a theoretical framework that quantifies how these travel times depend on the features of the ambient noise. However, corresponding theoretical results do not currently exist to describe how the amplitudes of the cross correlation depend on such features. For example, currently it is not possible to take a given distribution of noise sources and calculate the cross correlation amplitudes one would expect from such a distribution. Here, we provide a ray-theoretical framework for calculating cross correlations. This framework differs from previous work in that it explicitly accounts for attenuation as well as the spatial distribution of sources and therefore can address the issue of quantifying amplitudes in noise correlation measurements. After introducing the general framework, we apply it to two specific problems. First, we show that we can quantify the amplitudes of coherency measurements, and find that the decay of coherency with station-station spacing depends crucially on the distribution of noise sources. We suggest that researchers interested in performing attenuation measurements from noise coherency should first determine how the dominant sources of noise are distributed. Second, we show that we can quantify the signal-to-noise ratio of noise correlations more precisely than previous work, and that these signal-to-noise ratios can be estimated for given situations prior to the deployment of seismometers. It is expected that there are applications of the theoretical framework beyond the two specific cases considered, but these applications await future work. 4. On the infinities of closed superstring amplitudes International Nuclear Information System (INIS) Restuccia, A.; Taylor, J.G. 1988-01-01 The authors present an analysis of possible infinities that may be present in uncompactified multi-loop heterotic and type II superstring amplitudes constructed, without use of the short-string limit, in the light-cone gauge, and with use of a closed [10]-SUSY field theory algebra. Various types of degenerations of the integrand are discussed on the string worldsheet. No infinities are found, modulo (for type II) a particular identity for Green's functions 5. Deep Inelastic Scattering at the Amplitude Level International Nuclear Information System (INIS) Brodsky, Stanley J. 2005-01-01 The deep inelastic lepton scattering and deeply virtual Compton scattering cross sections can be interpreted in terms of the fundamental wavefunctions defined by the light-front Fock expansion, thus allowing tests of QCD at the amplitude level. The AdS/CFT correspondence between gauge theory and string theory provides remarkable new insights into QCD, including a model for hadronic wavefunctions which display conformal scaling at short distances and color confinement at large distances 6. How to calculate the Coulomb scattering amplitude International Nuclear Information System (INIS) Grosse, H.; Narnhofer, H.; Thirring, W. 1974-01-01 The derivation of scattering amplitudes for Coulomb scattering is discussed. A derivation of the S-matrix elements for a dense set of states in momentum space is given in the framework of time dependent scattering theory. The convergence of the S-matrix is studied. A purely algebraic derivation of the S-matrix elements and phase shifts is also presented. (HFdV) 7. Seismic amplitude measurements suggest foreshocks have different focal mechanisms than aftershocks Science.gov (United States) Lindh, A.; Fuis, G.; Mantis, C. 1978-01-01 The ratio of the amplitudes of P and S waves from the foreshocks and aftershocks to three recent California earthquakes show a characteristic change at the time of the main events. As this ratio is extremely sensitive to small changes in the orientation of the fault plane, a small systematic change in stress or fault configuration in the source region may be inferred. These results suggest an approach to the recognition of foreshocks based on simple measurements of the amplitudes of seismic waves. Copyright ?? 1978 AAAS. 8. Accommodative Amplitude in School-Age Children Directory of Open Access Journals (Sweden) Ikaunieks Gatis 2017-10-01 Full Text Available In children, intensive near-work affects the accommodation system of the eye. Younger children, due to anatomical parameters, read at smaller distance than older children and we can expect that the accommodation system of younger can be affected more than that of older children. We wanted to test this hypothesis. Some authors showed that the norms of amplitude of accommodation (AA developed by Hofstetter (1950 not always could be applied for children. We also wanted to verify these results. A total of 106 (age 7-15 children participated in the study. Distance visual acuity was measured for all children and only data of children with good visual acuity 1.0 or more (dec. units were analysed (73 children. Accommodative amplitude was measured before and after lessons using subjective push-up technique (with RAF Near Point Ruler. The results showed that the amplitude of accommodation reduced significantly (p < 0.05 during the day and decrease of AA was similar in different age groups (about ~0.70 D. Additional measurements are needed to verify that the observed changes in AA were associated with fatigue effect. The results showed lower accommodation values compared to average values calculated according to the Hofstetter equation (p < 0.05. 9. Scattering amplitudes from multivariate polynomial division Energy Technology Data Exchange (ETDEWEB) Mastrolia, Pierpaolo, E-mail: pierpaolo.mastrolia@cern.ch [Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Muenchen (Germany); Dipartimento di Fisica e Astronomia, Universita di Padova, Padova (Italy); INFN Sezione di Padova, via Marzolo 8, 35131 Padova (Italy); Mirabella, Edoardo, E-mail: mirabell@mppmu.mpg.de [Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Muenchen (Germany); Ossola, Giovanni, E-mail: GOssola@citytech.cuny.edu [New York City College of Technology, City University of New York, 300 Jay Street, Brooklyn, NY 11201 (United States); Graduate School and University Center, City University of New York, 365 Fifth Avenue, New York, NY 10016 (United States); Peraro, Tiziano, E-mail: peraro@mppmu.mpg.de [Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Muenchen (Germany) 2012-11-15 We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the weak Nullstellensatz theorem and on the division modulo the Groebner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself. 10. Cascaded Amplitude Modulations in Sound Texture Perception Directory of Open Access Journals (Sweden) Richard McWalter 2017-09-01 Full Text Available Sound textures, such as crackling fire or chirping crickets, represent a broad class of sounds defined by their homogeneous temporal structure. It has been suggested that the perception of texture is mediated by time-averaged summary statistics measured from early auditory representations. In this study, we investigated the perception of sound textures that contain rhythmic structure, specifically second-order amplitude modulations that arise from the interaction of different modulation rates, previously described as “beating” in the envelope-frequency domain. We developed an auditory texture model that utilizes a cascade of modulation filterbanks that capture the structure of simple rhythmic patterns. The model was examined in a series of psychophysical listening experiments using synthetic sound textures—stimuli generated using time-averaged statistics measured from real-world textures. In a texture identification task, our results indicated that second-order amplitude modulation sensitivity enhanced recognition. Next, we examined the contribution of the second-order modulation analysis in a preference task, where the proposed auditory texture model was preferred over a range of model deviants that lacked second-order modulation rate sensitivity. Lastly, the discriminability of textures that included second-order amplitude modulations appeared to be perceived using a time-averaging process. Overall, our results demonstrate that the inclusion of second-order modulation analysis generates improvements in the perceived quality of synthetic textures compared to the first-order modulation analysis considered in previous approaches. 11. Transversity Amplitudes in Hypercharge Exchange Processes International Nuclear Information System (INIS) Aguilar Benitez de Lugo, M. 1979-01-01 ' In this work we present several techniques developed for the extraction of the. Transversity amplitudes governing quasi two-body meson baryon reactions with hypercharge exchange. We review the methods used in processes having a pure spin configuration, as well as the more relevant results obtained with data from K p and Tp interactions at intermediate energies. The predictions of the additive quark model and the ones following from exchange degeneracy and etoxicity are discussed. We present a formalism for amplitude analysis developed for reactions with mixed spin configurations and discuss the methods of parametric estimation of the moduli and phases of the amplitudes, as well as the various tests employed to check the goodness of the fits. The calculation of the generalized joint density matrices is given and we propose a method based on the generalization of the idea of multipole moments, which allows to investigate the structure of the decay angular correlations and establishes the quality of the fits and the validity of the simplifying assumptions currently used in this type of studies. (Author) 43 refs 12. Multi-wavelength lasers using AWGs NARCIS (Netherlands) Besten, den J.H. 2003-01-01 Multiwavelength lasers using AWGs can be used as digitally tunable lasers with simple channel selection, and for generating multiple wavelengths simultanously. In this paper a number of different configurations is reviewed. 13. Wavelength mismatch effect in electromagnetically induced absorption Energy Technology Data Exchange (ETDEWEB) Bharti, Vineet [Department of Physics, Indian Institute of Science, Bangalore 560012 (India); Wasan, Ajay [Department of Physics, Indian Institute of Technology, Roorkee 247667 (India); Natarajan, Vasant [Department of Physics, Indian Institute of Science, Bangalore 560012 (India) 2016-07-15 We present a theoretical investigation of the phenomenon of electromagnetically induced absorption (EIA) in a 4-level system consisting of vee and ladder subsystems. The four levels are coupled using one weak probe field, and two strong control fields. We consider an experimental realization using energy levels of Rb. This necessitates dealing with different conditions of wavelength mismatch—near-perfect match where all three wavelengths are approximately equal; partial mismatch where the wavelength of one control field is less than the other fields; and complete mismatch where all three wavelengths are unequal. We present probe absorption profiles with Doppler averaging at room temperature to account for experiments in a room temperature Rb vapor cell. Our analysis shows that EIA resonances can be studied using Rydberg states excited with diode lasers. - Highlights: • Wavelength mismatch effect is investigated in electromagnetically induced absorption (EIA). • An experimental realization of 4-level vee + ladder system using energy levels of rubidium atom is presented. • EIA resonances are studied under different conditions of wavelength mismatch. • Possibility of observation of EIA using Rydberg states excited with diode lasers. 14. Wavelength mismatch effect in electromagnetically induced absorption International Nuclear Information System (INIS) Bharti, Vineet; Wasan, Ajay; Natarajan, Vasant 2016-01-01 We present a theoretical investigation of the phenomenon of electromagnetically induced absorption (EIA) in a 4-level system consisting of vee and ladder subsystems. The four levels are coupled using one weak probe field, and two strong control fields. We consider an experimental realization using energy levels of Rb. This necessitates dealing with different conditions of wavelength mismatch—near-perfect match where all three wavelengths are approximately equal; partial mismatch where the wavelength of one control field is less than the other fields; and complete mismatch where all three wavelengths are unequal. We present probe absorption profiles with Doppler averaging at room temperature to account for experiments in a room temperature Rb vapor cell. Our analysis shows that EIA resonances can be studied using Rydberg states excited with diode lasers. - Highlights: • Wavelength mismatch effect is investigated in electromagnetically induced absorption (EIA). • An experimental realization of 4-level vee + ladder system using energy levels of rubidium atom is presented. • EIA resonances are studied under different conditions of wavelength mismatch. • Possibility of observation of EIA using Rydberg states excited with diode lasers. 15. Multi-wavelength study of PPDs using an OPO tunable pulse laser microscope system International Nuclear Information System (INIS) Yoshimura, Koji; Nakamura, Isamu 2012-01-01 We have developed a new pulsed laser microscope system whose wavelength is continuously tunable from 410 nm to 2200 nm by using an optical parametric oscillator (OPO) laser system. The laser spot can be focused to ∼2μm diameter, small enough to measure pixel-by-pixel performance of PPDs (pixelated photon detectors). Using multi-wavelength laser light, we plan to probe PPDs at various depths, thanks to their different penetration lengths in the silicon layer. In this paper, details of the commissioning of the laser microscope system and pilot measurements on a PPD at several wavelengths will be presented. 16. Multi-wavelength study of PPDs using an OPO tunable pulse laser microscope system Energy Technology Data Exchange (ETDEWEB) Yoshimura, Koji, E-mail: koji.yoshimura@kek.jp [High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801 (Japan); Nakamura, Isamu [High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba, Ibaraki 305-0801 (Japan) 2012-12-11 We have developed a new pulsed laser microscope system whose wavelength is continuously tunable from 410 nm to 2200 nm by using an optical parametric oscillator (OPO) laser system. The laser spot can be focused to {approx}2{mu}m diameter, small enough to measure pixel-by-pixel performance of PPDs (pixelated photon detectors). Using multi-wavelength laser light, we plan to probe PPDs at various depths, thanks to their different penetration lengths in the silicon layer. In this paper, details of the commissioning of the laser microscope system and pilot measurements on a PPD at several wavelengths will be presented. 17. Investigating the amplitude of interactive footstep sounds and soundscape reproduction DEFF Research Database (Denmark) Turchet, Luca; Serafin, Stefania 2013-01-01 In this paper, we study the perception of amplitude of soundscapes and interactively generated footstep sounds provided both through headphones and a surround sound system. In particular, we investigate whether there exists a value for the amplitude of soundscapes and footstep sounds which...... of soundscapes does not significantly affect the selected amplitude of footstep sounds. Similarly, the perception of the soundscapes amplitude is not significantly affected by the selected amplitude of footstep sounds.... 18. One-loop triple collinear splitting amplitudes in QCD Energy Technology Data Exchange (ETDEWEB) Badger, Simon; Buciuni, Francesco; Peraro, Tiziano [Higgs Centre for Theoretical Physics, School of Physics and Astronomy, University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom) 2015-09-28 We study the factorisation properties of one-loop scattering amplitudes in the triple collinear limit and extract the universal splitting amplitudes for processes initiated by a gluon. The splitting amplitudes are derived from the analytic Higgs plus four partons amplitudes. We present compact results for primitive helicity splitting amplitudes making use of super-symmetric decompositions. The universality of the collinear factorisation is checked numerically against the full colour six parton squared matrix elements. 19. Classical analogues of a quantum system in spatial and temporal domains: A probability amplitude approach Directory of Open Access Journals (Sweden) 2016-12-01 Full Text Available We have simulated the similar features of the well-known classical phenomena in quantum domain under the formalism of probability amplitude method. The identical pattern of interference fringes of a Fabry–Perot interferometer (especially on reflection mode is obtained through the power-broadened spectral line shape of the population distribution in the excited state with careful delineation of a coherently driven two-level atomic model. In a unit wavelength domain, such pattern can be substantially modified by controlling typical spatial field arrangement in one and two dimensions, which is found complementary to the findings of recent research on atom localization in sub-wavelength domain. The spatial dependence of temporal dynamics has also been studied at a particular condition, which is equivalent to that could be obtained under Raman–Nath diffraction controlled by spatial phase. 20. A 12 GHz wavelength spacing multi-wavelength laser source for wireless communication systems Science.gov (United States) Peng, P. C.; Shiu, R. K.; Bitew, M. A.; Chang, T. L.; Lai, C. H.; Junior, J. I. 2017-08-01 This paper presents a multi-wavelength laser source with 12 GHz wavelength spacing based on a single distributed feedback laser. A light wave generated from the distributed feedback laser is fed into a frequency shifter loop consisting of 50:50 coupler, dual-parallel Mach-Zehnder modulator, optical amplifier, optical filter, and polarization controller. The frequency of the input wavelength is shifted and then re-injected into the frequency shifter loop. By re-injecting the shifted wavelengths multiple times, we have generated 84 optical carriers with 12 GHz wavelength spacing and stable output power. For each channel, two wavelengths are modulated by a wireless data using the phase modulator and transmitted through a 25 km single mode fiber. In contrast to previously developed schemes, the proposed laser source does not incur DC bias drift problem. Moreover, it is a good candidate for radio-over-fiber systems to support multiple users using a single distributed feedback laser. 1. Wavelength converter placement for different RWA algorithms in wavelength-routed all-optical networks Science.gov (United States) Chu, Xiaowen; Li, Bo; Chlamtac, Imrich 2002-07-01 Sparse wavelength conversion and appropriate routing and wavelength assignment (RWA) algorithms are the two key factors in improving the blocking performance in wavelength-routed all-optical networks. It has been shown that the optimal placement of a limited number of wavelength converters in an arbitrary mesh network is an NP complete problem. There have been various heuristic algorithms proposed in the literature, in which most of them assume that a static routing and random wavelength assignment RWA algorithm is employed. However, the existing work shows that fixed-alternate routing and dynamic routing RWA algorithms can achieve much better blocking performance. Our study in this paper further demonstrates that the wavelength converter placement and RWA algorithms are closely related in the sense that a well designed wavelength converter placement mechanism for a particular RWA algorithm might not work well with a different RWA algorithm. Therefore, the wavelength converter placement and the RWA have to be considered jointly. The objective of this paper is to investigate the wavelength converter placement problem under fixed-alternate routing algorithm and least-loaded routing algorithm. Under the fixed-alternate routing algorithm, we propose a heuristic algorithm called Minimum Blocking Probability First (MBPF) algorithm for wavelength converter placement. Under the least-loaded routing algorithm, we propose a heuristic converter placement algorithm called Weighted Maximum Segment Length (WMSL) algorithm. The objective of the converter placement algorithm is to minimize the overall blocking probability. Extensive simulation studies have been carried out over three typical mesh networks, including the 14-node NSFNET, 19-node EON and 38-node CTNET. We observe that the proposed algorithms not only outperform existing wavelength converter placement algorithms by a large margin, but they also can achieve almost the same performance comparing with full wavelength 2. Wavelength variation of a standing wave along a vertical spring Science.gov (United States) Welsch, Dylan; Baker, Blane 2018-03-01 Hand-driven resonance can be observed readily in a number of mechanical systems including thin boards, rods, strings, and springs. In order to show such behavior in the vertical spring pictured in Fig. 1, a section of spring is grasped at a location about one meter from its free end and driven by small, circular motions of the hand. At driving frequencies of a few hertz, a dramatic standing wave is generated. One of the fascinating features of this particular standing wave is that its wavelength varies along the length of the spring. 3. Effect of initial perturbation amplitude on Richtmyer-Meshkov flows induced by strong shocks Energy Technology Data Exchange (ETDEWEB) Dell, Z.; Abarzhi, S. I., E-mail: snezhana.abarzhi@gmail.com, E-mail: sabarji@andrew.cmu.edu [Mellon College of Science and Carnegie Mellon University – Qatar, Carnegie Mellon University, Pittsburgh, Pennsylvania 15231 (United States); Stellingwerf, R. F. [Stellingwerf Consulting, Huntsville, Alabama 35803 (United States) 2015-09-15 We systematically study the effect of the initial perturbation on Richtmyer-Meshkov (RM) flows induced by strong shocks in fluids with contrasting densities. Smooth Particle Hydrodynamics simulations are employed. A broad range of shock strengths and density ratios is considered. The amplitude of the initial single mode sinusoidal perturbation of the interface varies from 0% to 100% of its wavelength. The simulations results are compared, wherever possible, with four rigorous theories, and with other experiments and simulations, achieving good quantitative and qualitative agreement. Our study is focused on early time dynamics of the Richtmyer-Meshkov instability (RMI). We analyze the initial growth-rate of RMI immediately after the shock passage, when the perturbation amplitude increases linearly with time. For the first time, to the authors' knowledge, we find that the initial growth-rate of RMI is a non-monotone function of the initial perturbation amplitude, thus restraining the amount of energy that can be deposited by the shock at the interface. The maximum value of the initial growth-rate depends on the shock strength and the density ratio, whereas the corresponding value of the initial perturbation amplitude depends only slightly on the shock strength and density ratio. 4. Q factor and resonance amplitude of Josephson tunnel junctions International Nuclear Information System (INIS) Broom, R.F.; Wolf, P. 1977-01-01 The surface impedance of the superconducting films comprising the electrodes of Josephson tunnel junctions has been derived from the BCS theory in the extreme London limit. Expressions have been obtained for (i) the dependence of the penetration depth lambda on frequency and temperature, and (ii) the quality factor Q of the junction cavity, attributable to surface absorption in the electrodes. The effect of thin electrodes (t 9 or approx. = lambda) is also included in the calculations. Comparison of the calculated frequency dependence of lambda with resonance measurements on Pb-alloy and all-Nb tunnel junctions yields quite good agreement, indicating that the assumptions made in the theory are reasonable. Measurements of the (current) amplitude of the resonance peaks of the junctions have been compared with the values obtained from inclusion of the calculated Q in the theory by Kulik. In common with observations on microwave cavities by other workers, we find that a small residual conductivity must be added to the real part of the BCS value. With its inclusion, good agreement is found between calculation and experiment, within the range determined by the simplifying assumptions of Kulik's theory. From the results, we believe the calculation of Q to be reasonably accurate for the materials investigated. It is shown that the resonance amplitude of Josephson junctions can be calculated directly from the material constants and a knowledge of the residual conductivity 5. Research proposal on : amplitude modulated reflectometry system for JET divertor International Nuclear Information System (INIS) Sanchez, J.; Branas, T.; Estrada, T.; Luna, E. de la. 1992-01-01 Amplitude Modulated reflectometry is presented here as a tool for density profile measurements in the JET divertor plasmas. One of the main problems which has been presented in most reflectometers during the last years is the need for a coherent tracking of the phase delay: fast density fluctuations and strong modulation on the amplitude of the reflected signal usually bring to fringe jumps' in the phase signal, which are a big problem when the phase values are much larger than 2 pi. The conditions in the JET divertor plasmas: plasma geometry, access and long oversized broad-band waveguide paths makes very difficult the phase measurements at the millimeter wave range. AM reflectometry is to some extension an intermediate solution between the classical phase delay reflectometry, so far applied to small distances, and the time domain reflectometry, used for ionospheric studies and recently also proposed for fusion plasma. the main advantage is to allow the use of millimeter wave reflectometry with moderate phase shifts (approx 2 pi). (author) 6. Ethnic differences in electrocardiographic amplitude measurements International Nuclear Information System (INIS) Mansi, Ishak A.; Nash, Ira S. 2004-01-01 There is a controversy regarding ethnic differences in electrocardiographic (ECG) patterns because of the potentially confounding socioeconomic, nutritional, environmental and occupational factors. We reviewed the first 1000 medical files of a multiethnic community, where all individuals shared similar living conditions. Only healthy adults age 15 to 60 years were included. Wave amplitudes were measured manually from the standard 12lead ECG. Minnesota coding was used. ECG from 597 subjects were included in the study: 350 Saudi Arabians, 95 Indians, 17 Sri-Lankans, 39 Filipinos, and 57 Caucasians; 349 were men. the mean +-SD of Sokolow-Lyon voltage (SLV) in men was signifcantly different among ethnic groups (2.9+-0.86, 2.64+-0.79, 2.73+-0.72, 3.23+-0.61, 2.94+-0.6, 2.58+-0.79 mV, P=0.0006, for Saudi's, Indians, Jordanians, Filipinos, Sri-Lankans, and Caucasians, respectively). SLV was similar among ethnic groups in women. The prevalence of early transition pattern was also different among ethnic groups in men but not women (15.8%, 34.6%, 17.9%, 21.7%, 35.3%, 26.8% in Suadi, Indian, Jordanian, Filipino, Sri-Lankan, and Caucasian, respectively, P=0.037). T wave amplitude was significantly different among ethnic groups in selected lead. ECG wave amplitude differs with ethnic region even when other factors are similar. Using SLV of 3.5 mV as a criterion may overestimate the incidence of left ventricular hypertrophy in some ethnic groups. The pattern of high R wave in lead V1is common in healthy adults in certain ethnic groups. T wave height differs with ethnic origin and sex. (author) 7. Loop amplitudes in an extended gravity theory Science.gov (United States) Dunbar, David C.; Godwin, John H.; Jehu, Guy R.; Perkins, Warren B. 2018-05-01 We extend the S-matrix of gravity by the addition of the minimal three-point amplitude or equivalently adding R3 terms to the Lagrangian. We demonstrate how Unitarity can be used to simply examine the renormalisability of this theory and determine the R4 counter-terms that arise at one-loop. We find that the combination of R4 terms that arise in the extended theory is complementary to the R4 counter-term associated with supersymmetric Lagrangians. 8. Loop-quantum-gravity vertex amplitude. Science.gov (United States) Engle, Jonathan; Pereira, Roberto; Rovelli, Carlo 2007-10-19 Spin foam models are hoped to provide the dynamics of loop-quantum gravity. However, the most popular of these, the Barrett-Crane model, does not have the good boundary state space and there are indications that it fails to yield good low-energy n-point functions. We present an alternative dynamics that can be derived as a quantization of a Regge discretization of Euclidean general relativity, where second class constraints are imposed weakly. Its state space matches the SO(3) loop gravity one and it yields an SO(4)-covariant vertex amplitude for Euclidean loop gravity. 9. Speech production in amplitude-modulated noise DEFF Research Database (Denmark) Macdonald, Ewen N; Raufer, Stefan 2013-01-01 The Lombard effect refers to the phenomenon where talkers automatically increase their level of speech in a noisy environment. While many studies have characterized how the Lombard effect influences different measures of speech production (e.g., F0, spectral tilt, etc.), few have investigated...... the consequences of temporally fluctuating noise. In the present study, 20 talkers produced speech in a variety of noise conditions, including both steady-state and amplitude-modulated white noise. While listening to noise over headphones, talkers produced randomly generated five word sentences. Similar...... of noisy environments and will alter their speech accordingly.... 10. Optical twists in phase and amplitude DEFF Research Database (Denmark) Daria, Vincent R.; Palima, Darwin; Glückstad, Jesper 2011-01-01 where both phase and amplitude express a helical profile as the beam propagates in free space. Such a beam can be accurately referred to as an optical twister. We characterize optical twisters and demonstrate their capacity to induce spiral motion on particles trapped along the twisters’ path. Unlike LG...... beams, the far field projection of the twisted optical beam maintains a high photon concentration even at higher values of topological charge. Optical twisters have therefore profound applications to fundamental studies of light and atoms such as in quantum entanglement of the OAM, toroidal traps... 11. First order correction to quasiclassical scattering amplitude International Nuclear Information System (INIS) Kuz'menko, A.V. 1978-01-01 First order (with respect to h) correction to quasiclassical with the aid of scattering amplitude in nonrelativistic quantum mechanics is considered. This correction is represented by two-loop diagrams and includes the double integrals. With the aid of classical equations of motion, the sum of the contributions of the two-loop diagrams is transformed into the expression which includes one-dimensional integrals only. The specific property of the expression obtained is that the integrand does not possess any singularities in the focal points of the classical trajectory. The general formula takes much simpler form in the case of one-dimensional systems 12. Tomography for amplitudes of hard exclusive processes International Nuclear Information System (INIS) Polyakov, M.V. 2008-01-01 We discuss which part of information about hadron structure encoded in the Generalized Parton Distributions (GPDs) [part of total GPD image] can be restored from the known amplitude of a hard exclusive process. The physics content of this partial image is analyzed. Among other things, we show that this partial image contains direct information about how the target hadron responses to the (string) quark-antiquark operator of arbitrary spin J. Explicit equations relating physics content of the partial image of GPDs directly to the data are derived. Also some new results concerning the dual parametrization of GPDs are presented 13. Inverse amplitude method and Adler zeros International Nuclear Information System (INIS) Gomez Nicola, A.; Pelaez, J. R.; Rios, G. 2008-01-01 The inverse amplitude method is a powerful unitarization technique to enlarge the energy applicability region of effective Lagrangians. It has been widely used to describe resonances in hadronic physics, combined with chiral perturbation theory, as well as in the strongly interacting symmetry breaking sector. In this work we show how it can be slightly modified to also account for the subthreshold region, incorporating correctly the Adler zeros required by chiral symmetry and eliminating spurious poles. These improvements produce negligible effects on the physical region. 14. Survey of vibration amplitudes throughout the linac International Nuclear Information System (INIS) Werner, K.L. 1984-01-01 The magnitude of vibrations of the Linac structure due to on site disturbances, such as cooling towers, pumps, generators, Highway 280 overpass traffic, is of interest. CN-263, for example, discusses tolerances of random (i.e., uncorrelated) quad jitter and suggests that amplitudes should not exceed 0.7 microns rms. This note describes the results of a series of measurements carried out in the summer of 1983. In general, the tolerance is not exceeded, but there appears not to be a good safety factor at low frequencies 15. Conformist-contrarian interactions and amplitude dependence in the Kuramoto model Science.gov (United States) Lohe, M. A. 2014-11-01 We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist-contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties. 16. Conformist–contrarian interactions and amplitude dependence in the Kuramoto model International Nuclear Information System (INIS) Lohe, M A 2014-01-01 We derive exact formulas for the frequency of synchronized oscillations in Kuramoto models with conformist–contrarian interactions, and determine necessary conditions for synchronization to occur. Numerical computations show that for certain parameters repulsive nodes behave as conformists, and that in other cases attractive nodes can display frustration, being neither conformist nor contrarian. The signs of repulsive couplings can be placed equivalently outside the sum, as proposed in Hong and Strogatz (2011 Phys. Rev. Lett. 106 054102), or inside the sum as in Hong and Strogatz (2012 Phys. Rev. E 85 056210), but the two models have different characteristics for small magnitudes of the coupling constants. In the latter case we show that the distributed coupling constants can be viewed as oscillator amplitudes which are constant in time, with the property that oscillators of small amplitude couple only weakly to connected nodes. Such models provide a means of investigating the effect of amplitude variations on synchronization properties. (paper) 17. A tunable and switchable single-longitudinal-mode dual-wavelength fiber laser with a simple linear cavity. Science.gov (United States) He, Xiaoying; Fang, Xia; Liao, Changrui; Wang, D N; Sun, Junqiang 2009-11-23 A simple linear cavity erbium-doped fiber laser based on a Fabry-Perot filter which consists of a pair of fiber Bragg gratings is proposed for tunable and switchable single-longitudinal-mode dual-wavelength operation. The single-longitudinal-mode is obtained by the saturable absorption of an unpumed erbium-doped fiber together with a narrow-band fiber Bragg grating. Under the high pump power (>166 mW) condition, the stable dual-wavelength oscillation with uniform amplitude can be realized by carefully adjusting the polarization controller in the cavity. Wavelength selection and switching are achieved by tuning the narrow-band fiber Bragg grating in the system. The spacing of the dual-wavelength can be selected at 0.20 nm (approximately 25.62 GHz), 0.22 nm (approximately 28.19 GHz) and 0.54 nm (approximately 69.19 GHz). 18. Local Analysis Approach for Short Wavelength Geopotential Variations Science.gov (United States) Bender, P. L. 2009-12-01 results from different arcs crossing the region can be seen most easily for an orbit with moderate inclination, such as 50 to 65 deg., so that the crossing angle between south-to-north (S-N) and N-S passes is fairly large over most regions well away from the poles. In that case, after filtering to pass the shorter wavelengths, the results for a given time interval can be combined to give the short wavelength W-E variations in the geopotential efficiently. For continents with extensive meteorological measurements available, like Europe and North America, a very rough guess at the surface mass density variation uncertainties is about 3 kg/m^2. This is based on the apparent accuracy of carefully calibrated surface pressure measurements. If a substantial part of the resulting uncertainties in the geopotential height at satellite altitude are at wavelengths less than about 1,500 km, they will dominate the measurement uncertainty at short spatial wavelengths for a GRACE-type mission with laser interferometry. This would be the case, even if the uncertainty in the atmospheric and oceanic mass distribution at large distances has a fairly small effect. However, the geopotential accuracy would still be substantially better than for the results achievable with a microwave ranging system. 19. On the influence of crystal size and wavelength on native SAD phasing. Science.gov (United States) Liebschner, Dorothee; Yamada, Yusuke; Matsugaki, Naohiro; Senda, Miki; Senda, Toshiya 2016-06-01 Native SAD is an emerging phasing technique that uses the anomalous signal of native heavy atoms to obtain crystallographic phases. The method does not require specific sample preparation to add anomalous scatterers, as the light atoms contained in the native sample are used as marker atoms. The most abundant anomalous scatterer used for native SAD, which is present in almost all proteins, is sulfur. However, the absorption edge of sulfur is at low energy (2.472 keV = 5.016 Å), which makes it challenging to carry out native SAD phasing experiments as most synchrotron beamlines are optimized for shorter wavelength ranges where the anomalous signal of sulfur is weak; for longer wavelengths, which produce larger anomalous differences, the absorption of X-rays by the sample, solvent, loop and surrounding medium (e.g. air) increases tremendously. Therefore, a compromise has to be found between measuring strong anomalous signal and minimizing absorption. It was thus hypothesized that shorter wavelengths should be used for large crystals and longer wavelengths for small crystals, but no thorough experimental analyses have been reported to date. To study the influence of crystal size and wavelength, native SAD experiments were carried out at different wavelengths (1.9 and 2.7 Å with a helium cone; 3.0 and 3.3 Å with a helium chamber) using lysozyme and ferredoxin reductase crystals of various sizes. For the tested crystals, the results suggest that larger sample sizes do not have a detrimental effect on native SAD data and that long wavelengths give a clear advantage with small samples compared with short wavelengths. The resolution dependency of substructure determination was analyzed and showed that high-symmetry crystals with small unit cells require higher resolution for the successful placement of heavy atoms. 20. Multichannel amplitude analyser for nuclear spectrometry International Nuclear Information System (INIS) Jankovic, S.; Milovanovic, B. 2003-01-01 A multichannel amplitude analyser with 4096 channels was designed. It is based on a fast 12-bit analog-to-digital converter. The intended purpose of the instrument is recording nuclear spectra by means of scintillation detectors. The computer link is established through an opto-isolated serial connection cable, thus reducing instrument sensitivity to disturbances originating from digital circuitry. Refreshing of the data displayed on the screen occurs on every 2.5 seconds. The impulse peak detection is implemented through the differentiation of the amplified input signal, while the synchronization with the data coming from the converter output is established by taking advantage of the internal 'pipeline' structure of the converter itself. The mode of operation of the built-in microcontroller provides that there are no missed impulses, and the simple logic network prevents the initiation of the amplitude reading sequence for the next impulse in case it appears shortly after its precedent. The solution proposed here demonstrated a good performance at a comparatively low manufacturing cost, and is thus suitable for educational purposes (author) 1. Getting superstring amplitudes by degenerating Riemann surfaces International Nuclear Information System (INIS) Matone, Marco; Volpato, Roberto 2010-01-01 We explicitly show how the chiral superstring amplitudes can be obtained through factorisation of the higher genus chiral measure induced by suitable degenerations of Riemann surfaces. This powerful tool also allows to derive, at any genera, consistency relations involving the amplitudes and the measure. A key point concerns the choice of the local coordinate at the node on degenerate Riemann surfaces that greatly simplifies the computations. As a first application, starting from recent ansaetze for the chiral measure up to genus five, we compute the chiral two-point function for massless Neveu-Schwarz states at genus two, three and four. For genus higher than three, these computations include some new corrections to the conjectural formulae appeared so far in the literature. After GSO projection, the two-point function vanishes at genus two and three, as expected from space-time supersymmetry arguments, but not at genus four. This suggests that the ansatz for the superstring measure should be corrected for genus higher than four. 2. The Construction of Spin Foam Vertex Amplitudes Directory of Open Access Journals (Sweden) Eugenio Bianchi 2013-01-01 Full Text Available Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine. 3. Subducted slabs and lateral viscosity variations: effects on the long-wavelength geoid Science.gov (United States) Tosi, Nicola; Čadek, Ondřej; Martinec, Zdeněk 2009-11-01 The characteristic broad local maxima exhibited by the long-wavelength geoid over subduction zones are investigated with a numerical model of mantle flow. In a spherical axisymmetric geometry, a synthetic model of buoyancy driven subduction is used to test the effects on the geoid caused by the depth of penetration of the lithosphere into the mantle, by the viscosity stratification and by lateral viscosity variations (LVV) in the lithosphere, upper and lower mantle. The presence of anomalous slab density in the lower mantle guarantees geoid amplitudes comparable with the observations, favouring the picture of slabs that penetrate the transition zone and sink into the deep mantle. The viscosity of the lower mantle controls the long-wavelength geoid to the first order, ensuring a clear positive signal when it is at least 30-times greater than the upper-mantle viscosity. The presence of LVV in the lithosphere, in the form of weak plate margins, helps to increase the contribution of the surface topography, causing a pronounced reduction of the geoid. Localized LVV associated with the cold slab play a secondary role if they are in the upper mantle. On the other hand, highly viscous slabs in the lower mantle exert a large influence on the geoid. They cause its amplitude to increase dramatically, way beyond the values typically observed over subduction zones. Long-wavelength flow becomes less vigorous as the slab viscosity increases. Deformation in the upper mantle becomes more localized and power is transferred to short wavelengths, causing the long-wavelength surface topography to diminish and the total geoid to increase. Slabs may be then weakened in the lower mantle or retain their high viscosity while other mechanisms act to lower the geoid. It is shown that a phase change from perovskite to post-perovskite above the core-mantle boundary can cause the geoid to reduce significantly, thereby helping to reconcile models and observations. 4. Fatigue life prediction of rotor blade composites: Validation of constant amplitude formulations with variable amplitude experiments International Nuclear Information System (INIS) Westphal, T; Nijssen, R P L 2014-01-01 The effect of Constant Life Diagram (CLD) formulation on the fatigue life prediction under variable amplitude (VA) loading was investigated based on variable amplitude tests using three different load spectra representative for wind turbine loading. Next to the Wisper and WisperX spectra, the recently developed NewWisper2 spectrum was used. Based on these variable amplitude fatigue results the prediction accuracy of 4 CLD formulations is investigated. In the study a piecewise linear CLD based on the S-N curves for 9 load ratios compares favourably in terms of prediction accuracy and conservativeness. For the specific laminate used in this study Boerstra's Multislope model provides a good alternative at reduced test effort 5. Fatigue life prediction of rotor blade composites: Validation of constant amplitude formulations with variable amplitude experiments Science.gov (United States) Westphal, T.; Nijssen, R. P. L. 2014-12-01 The effect of Constant Life Diagram (CLD) formulation on the fatigue life prediction under variable amplitude (VA) loading was investigated based on variable amplitude tests using three different load spectra representative for wind turbine loading. Next to the Wisper and WisperX spectra, the recently developed NewWisper2 spectrum was used. Based on these variable amplitude fatigue results the prediction accuracy of 4 CLD formulations is investigated. In the study a piecewise linear CLD based on the S-N curves for 9 load ratios compares favourably in terms of prediction accuracy and conservativeness. For the specific laminate used in this study Boerstra's Multislope model provides a good alternative at reduced test effort. 6. Propagation of Finite Amplitude Sound in Multiple Waveguide Modes. Science.gov (United States) van Doren, Thomas Walter 1993-01-01 This dissertation describes a theoretical and experimental investigation of the propagation of finite amplitude sound in multiple waveguide modes. Quasilinear analytical solutions of the full second order nonlinear wave equation, the Westervelt equation, and the KZK parabolic wave equation are obtained for the fundamental and second harmonic sound fields in a rectangular rigid-wall waveguide. It is shown that the Westervelt equation is an acceptable approximation of the full nonlinear wave equation for describing guided sound waves of finite amplitude. A system of first order equations based on both a modal and harmonic expansion of the Westervelt equation is developed for waveguides with locally reactive wall impedances. Fully nonlinear numerical solutions of the system of coupled equations are presented for waveguides formed by two parallel planes which are either both rigid, or one rigid and one pressure release. These numerical solutions are compared to finite -difference solutions of the KZK equation, and it is shown that solutions of the KZK equation are valid only at frequencies which are high compared to the cutoff frequencies of the most important modes of propagation (i.e., for which sound propagates at small grazing angles). Numerical solutions of both the Westervelt and KZK equations are compared to experiments performed in an air-filled, rigid-wall, rectangular waveguide. Solutions of the Westervelt equation are in good agreement with experiment for low source frequencies, at which sound propagates at large grazing angles, whereas solutions of the KZK equation are not valid for these cases. At higher frequencies, at which sound propagates at small grazing angles, agreement between numerical solutions of the Westervelt and KZK equations and experiment is only fair, because of problems in specifying the experimental source condition with sufficient accuracy. 7. Large amplitude solitary waves in a multicomponent plasma with negative ions International Nuclear Information System (INIS) Nakamura, Y.; Tsukabayashi, I.; Ludwig, G.O.; Ferreira, J.L. 1987-09-01 When the concentration of negative ions is larger than a critical value, a small compressive pulse evolves into a subsonic wave train and a large pulse develops into a solitary wave. The threshold amplitude and velocity of the solitary waves are measured and compared with predictions using the pseudopotential method. (author) [pt 8. Wavelength division multiplexing a practical engineering guide CERN Document Server Grobe, Klaus 2013-01-01 In this book, Optical Wavelength Division Multiplexing (WDM) is approached from a strictly practical and application-oriented point of view. Based on the characteristics and constraints of modern fiber-optic components, transport systems and fibers, the text provides relevant rules of thumb and practical hints for technology selection, WDM system and link dimensioning, and also for network-related aspects such as wavelength assignment and resilience mechanisms. Actual 10/40 Gb/s WDM systems are considered, and a preview of the upcoming 100 Gb/s systems and technologies for even higher bit rate 9. Short wavelength sources and atoms and ions International Nuclear Information System (INIS) Kennedy, E.T. 2008-01-01 The interaction of ionizing radiation with atoms and ions is a key fundamental process. Experimental progress has depended in particular on the development of short wavelength light sources. Laser-plasma and synchrotron sources have been exploited for several decades and most recently the development of short wavelength Free Electron Laser (FEL) sources is revolutionizing the field. This paper introduces laser plasma and synchrotron sources through examples of their use in studies of the interaction of ionizing radiation with atoms and ions, ranging from few-electron atomic and ionic systems to the many-electron high atomic number actinides. The new FEL source (FLASH) at DESY is introduced. (author) 10. Large amplitude ion-acoustic solitary waves and double layers in multicomponent plasma with positrons International Nuclear Information System (INIS) Sabry, R. 2009-01-01 A finite amplitude theory for ion-acoustic solitary waves and double layers in multicomponent plasma consisting of hot positrons, cold ions, and electrons with two-electron temperature distributions is presented. Conditions are obtained under which large amplitude stationary ion-acoustic solitary waves and double layers can exist. For the physical parameters of interest, the ion-acoustic solitary wave (double layers) profiles and the relationship between the maximum soliton (double layers) amplitude and the Mach number are found. Also, we have presented the region of existence of the large amplitude ion-acoustic waves by analyzing the structure of the pseudopotential. For the selected range of parameters, it is found that only positive solitary waves and double layers can exist. An analysis for the small amplitude limit through the Sagdeev pseudopotential analysis and the reductive perturbation theory shows the existence of positive and negative ion-acoustic solitary waves and double layers. The effects of positron concentration and temperature ratio on the characteristics of the solitary ion-acoustic waves and double layers (namely, the amplitude and width) are discussed in detail. The relevance of this investigation to space and laboratory plasmas is pointed out. 11. Design of integrated optics all-optical label swappers for spectral amplitude code label swapping optical packet networks on active/passive InP technology NARCIS (Netherlands) Habib, C.; Munoz, P.; Leijtens, X.J.M.; Chen, Lawrence; Smit, M.K.; Capmany, J. 2009-01-01 In this paper the designs of optical label swapper devices, for spectral amplitude coded labels, monolithically integrated on InP active/passive technology are pre sented. The devices are based on cross-gain modulation in a semiconductor optical amplifier. Multi-wavelength operation is enabled by 12. Frequency-domain inversion using the amplitude of the derivative wavefield with respect to the angular frequency KAUST Repository Choi, Yun Seok 2012-01-01 The instantaneous traveltime based inversion was developed to solve the phase wrapping problem, thus generating long-wavelength structures even for a high single-frequency. However, it required aggressive damping to insure proper convergence. A reason for that is the potential for unstable division in the calculation of the instantaneous traveltime for low damping factors. Thus, we propose an inversion algorithm using the amplitude of the derivative wavefield to avoid the unstable division process. Since the amplitude of the derivative wavefield contains the unwrapped-phase information, its inversion has the potential to provide robust inversion results. On the other hand, the damping term rapidly diminishes the amplitude of the derivative wavefield at far source-receiver offsets. As an alternative, we suggest using the logarithmic amplitude of the derivative wavefield. The gradient of this inversion algorithm is obtained by the back-propagation approach, based on the adjoint-state technique. Numerical examples show that the logarithmic-amplitude approach yields better convergent results than the instantaneous traveltime inversion, whereas the pure-amplitude approach does not show much convergence. 13. Wavelength dispersive μPIXE setup for the ion microprobe Energy Technology Data Exchange (ETDEWEB) Fazinić, S., E-mail: stjepko.fazinic@irb.hr [Laboratory for Ion Beam Interactions, Division of Experimental Physics, Rudjer Bošković Institute, Bijenicka cesta 54, 10000 Zagreb (Croatia); Božičević Mihalić, I.; Tadić, T.; Cosic, D.; Jakšić, M. [Laboratory for Ion Beam Interactions, Division of Experimental Physics, Rudjer Bošković Institute, Bijenicka cesta 54, 10000 Zagreb (Croatia); Mudronja, D. [Croatian Conservation Institute, Grškovićeva 23, 10000 Zagreb (Croatia) 2015-11-15 We have developed a small wavelength dispersive X-ray spectrometer to explore the possibility of performing chemical speciation on microscopic samples utilizing focused ion beams available at the Rudjer Boskovic Institute ion microprobe. Although PIXE spectra are in principle chemically invariant, small influence of chemical effects could be observed even with Si(Li) or SDD detectors. Such chemical effects can be clearly seen with high resolution crystal X-ray spectrometers having energy resolution of several eV. A dedicated vacuum chamber, housing the diffraction crystal, sample holder and CCD X-ray detector, was constructed and positioned behind the main ion microprobe vacuum chamber. Here we will briefly describe the spectrometer, and illustrate its capabilities on measured K X-ray spectra of selected sulfur compounds. We will also demonstrate its abilities to resolve K and M X-ray lines irresolvable by solid state ED detectors usually used in PIXE. 14. Thrombelastography Early Amplitudes in bleeding and coagulopathic trauma patients DEFF Research Database (Denmark) Laursen, Thomas Holst; Meyer, Martin A S; Meyer, Anna Sina P 2018-01-01 BACKGROUND: Early amplitudes in the viscoelastic hemostatic assays Thrombelastography (TEG) and Rotation Thromboelastometry (ROTEM) provide fast results, which is critical in resuscitation of bleeding patients. This study investigated associations between TEG early amplitudes and standard TEG var... 15. Variational principles for the projected breakup amplitude International Nuclear Information System (INIS) Hahn, Y. 1976-01-01 Two alternate forms of variational principles for the breakup amplitude describing the two- to three-cluster transition are derived such that all the integrals involved in the intermediate stages are well defined. The first form contains a trial Green's function with which both the initial and final state trial wave functions are constructed. The earlier form of the Kohn-type variational principle derived by Lieber, Rosenberg, and Spruch is recovered, however, when this connection between the trial functions is removed. The second form of the variational principle is derived by projecting out from the trial functions all the open channel components which correspond to the two-cluster structures including the rearrangement channels. The remaining part of the wave functions describes the channels with three-cluster structures, and the integrals involving this part are then mathematically well defined 16. Single isospin decay amplitude and CP violation Energy Technology Data Exchange (ETDEWEB) Deshpande, N.G. [Oregon Univ., Eugene, OR (United States). Inst. of Theoretical Science; He, Xiaogang [Melbourne Univ., Parkville, VIC (Australia). School of Physics; Pakvasa, S. [Hawaii Univ., Honolulu, HI (United States). Dept. of Physics and Astronomy 1996-06-01 While for K meson or hyperon decays, the partial rate asymmetries are always zero if the final states are single isospin states, in B decays the situation is dramatically different and partial rate asymmetries can be non-zero if the final states are single isospin states. Partial rate asymmetries were calculated for several B decays with single isospin amplitude in the finale states using factorization approximation. It was found that more intermediate on-shell states with different Cabbibbo-Kobayashi-Maskawa factors are allowed in B decay and CP violating partial rate asymmetries need not to be zero even if the final state contains only a single isospin state. 17 refs., 4 figs. 17. Large amplitude parallel propagating electromagnetic oscillitons International Nuclear Information System (INIS) Cattaert, Tom; Verheest, Frank 2005-01-01 Earlier systematic nonlinear treatments of parallel propagating electromagnetic waves have been given within a fluid dynamic approach, in a frame where the nonlinear structures are stationary and various constraining first integrals can be obtained. This has lead to the concept of oscillitons that has found application in various space plasmas. The present paper differs in three main aspects from the previous studies: first, the invariants are derived in the plasma frame, as customary in the Sagdeev method, thus retaining in Maxwell's equations all possible effects. Second, a single differential equation is obtained for the parallel fluid velocity, in a form reminiscent of the Sagdeev integrals, hence allowing a fully nonlinear discussion of the oscilliton properties, at such amplitudes as the underlying Mach number restrictions allow. Third, the transition to weakly nonlinear whistler oscillitons is done in an analytical rather than a numerical fashion 18. Amplitude and phase modulation with waveguide optics International Nuclear Information System (INIS) Burkhart, S.C.; Wilcox, R.B.; Browning, D.; Penko, F.A. 1996-01-01 We have developed amplitude and phase modulation systems for glass lasers using integrated electro-optic modulators and solid state high-speed electronics. The present and future generation of lasers for Inertial Confinement Fusion require laser beams with complex temporal and phase shaping to compensate for laser gain saturation, mitigate parametric processes such as transverse stimulated Brillouin scattering in optics, and to provide specialized drive to the fusion targets. These functions can be performed using bulk optoelectronic modulators, however using high-speed electronics to drive low voltage integrated optical modulators has many practical advantages. In particular, we utilize microwave GaAs transistors to perform precision, 250 ps resolution temporal shaping. Optical bandwidth is generated using a microwave oscillator at 3 GHz amplified by a solid state amplifier. This drives an integrated electrooptic modulator to achieve laser bandwidths exceeding 30 GHz 19. Topological amplitudes in heterotic superstring theory International Nuclear Information System (INIS) Antoniadis, I.; Taylor, T.R. 1996-06-01 We show that certain heterotic string amplitudes are given in terms of correlators of the twisted topological (2,0) SCFT, corresponding to the internal sector of the N = 1 spacetime supersymmetric background. The genus g topological partition function F g corresponds to a term in the effective action of the form W 2g , where W is the gauge or gravitational superfield. We study also recursion relations related to holomorphic anomalies, showing that, contrary to the type II case, they involve correlators of anti-chiral superfields. The corresponding terms in the effective action are of the form W 2g II n , where II is a chiral superfield obtained by chiral projection of a general superfield. We observe that the structure of the recursion relations is that of N = 1 spacetime supersymmetry Ward identity. We give also a solution of the tree level recursion relations and discuss orbifold examples. (author). 23 refs, 2 figs 20. Polynomial structures in one-loop amplitudes International Nuclear Information System (INIS) Britto, Ruth; Feng Bo; Yang Gang 2008-01-01 A general one-loop scattering amplitude may be expanded in terms of master integrals. The coefficients of the master integrals can be obtained from tree-level input in a two-step process. First, use known formulas to write the coefficients of (4-2ε)-dimensional master integrals; these formulas depend on an additional variable, u, which encodes the dimensional shift. Second, convert the u-dependent coefficients of (4-2ε)-dimensional master integrals to explicit coefficients of dimensionally shifted master integrals. This procedure requires the initial formulas for coefficients to have polynomial dependence on u. Here, we give a proof of this property in the case of massless propagators. The proof is constructive. Thus, as a byproduct, we produce different algebraic expressions for the scalar integral coefficients, in which the polynomial property is apparent. In these formulas, the box and pentagon contributions are separated explicitly. 1. Amplitude correlations for inelastic proton scattering from 48Ti International Nuclear Information System (INIS) Chou, B.H.; Mitchell, G.E.; Bilpuch, E.G.; Westerfeldt, C.R. 1981-01-01 The magnitudes and relative signs of inelastic proton channel amplitudes were determined for three decay channels for 45 5/2 + resonances in 49 V. The reduced widths in each channel follow a Porter-Thomas distribution, but extremely large amplitude correlations are observed - for one pair of channel amplitudes the relative sign is positive for 43 of 45 resonances. These results provide the first direct test of the Krieger-Porter reduced width amplitude distribution. (orig.) 2. Tree-level gluon amplitudes on the celestial sphere Science.gov (United States) Schreiber, Anders Ø.; Volovich, Anastasia; Zlotnikov, Michael 2018-06-01 Pasterski, Shao and Strominger have recently proposed that massless scattering amplitudes can be mapped to correlators on the celestial sphere at infinity via a Mellin transform. We apply this prescription to arbitrary n-point tree-level gluon amplitudes. The Mellin transforms of MHV amplitudes are given by generalized hypergeometric functions on the Grassmannian Gr (4 , n), while generic non-MHV amplitudes are given by more complicated Gelfand A-hypergeometric functions. 3. Correlations for reduced-width amplitudes in 49V International Nuclear Information System (INIS) Chou, B.H.; Mitchell, G.E.; Bilpuch, E.G.; Westerfeldt, C.R. 1980-01-01 Measurement of the relative sign of inelastic proton-channel amplitudes permits the determination of amplitude correlations. Data were obtained for 45 5/2 + resonances in 49 V. Although the reduced widths in each channel followed a Porter-Thomas distribution, large amplitude correlations were observed. The results are compared with the reduced-width--amplitude distribution of Krieger and Porter. This is the first direct test of the Krieger-Porter distribution 4. Wavelength interrogation of fiber Bragg grating sensors using tapered hollow Bragg waveguides. Science.gov (United States) Potts, C; Allen, T W; Azar, A; Melnyk, A; Dennison, C R; DeCorby, R G 2014-10-15 We describe an integrated system for wavelength interrogation, which uses tapered hollow Bragg waveguides coupled to an image sensor. Spectral shifts are extracted from the wavelength dependence of the light radiated at mode cutoff. Wavelength shifts as small as ~10 pm were resolved by employing a simple peak detection algorithm. Si/SiO₂-based cladding mirrors enable a potential operational range of several hundred nanometers in the 1550 nm wavelength region for a taper length of ~1 mm. Interrogation of a strain-tuned grating was accomplished using a broadband amplified spontaneous emission (ASE) source, and potential for single-chip interrogation of multiplexed sensor arrays is demonstrated. 5. Topology Optimization of Sub-Wavelength Antennas DEFF Research Database (Denmark) Erentok, Aycan; Sigmund, Ole 2011-01-01 We propose a topology optimization strategy for the systematic design of a three-dimensional (3D), conductor-based sub-wavelength antenna. The post-processed finite-element (FE) models of the optimized structure are shown to be self-resonant, efficient and exhibit distorted omnidirectional... 6. Characterization of ethanol concentrations at ultraviolet wavelength ... African Journals Online (AJOL) This paper presents the measurement of optical absorption spectrum for different concentrations of ethanol at ultraviolet wavelength. Ethanol absorption spectrum was measured using portable spectroscopy setup from Avantes. It consists of Balanced Deuterium Halogen light source and spectrometer. The light source can ... 7. Smart wavelength meter for integrated photonics NARCIS (Netherlands) Benelajla, Meryem; Taballione, Caterina; Boller, Klaus J. 2017-01-01 Thermally tunable SiN waveguide microring resonators in connection with neural network readout algorithms appear promising for use as integrated optical wavelength meters. So far, we have observed long-term reliability and a temperature immunity of the readout across several degrees of ambient 8. An automated wavelength selection for flame spectroscopy International Nuclear Information System (INIS) Hurteau, M.; Mislan, J.P.; Ashley, R.W. 1976-01-01 A simple electro-mechanical programming system is described for use with a flame spectrophotometer. Its application for automated sequential multi-element analysis is illustrated. Reproducibility of wavelength settings are within +-0.5 A. Precision and sensitivities are at least as good as those obtained for single element determinations. (author) 9. Alien wavelength modeling tool and field trial DEFF Research Database (Denmark) Sambo, N.; Sgambelluri, A.; Secondini, M. 2015-01-01 A modeling tool is presented for pre-FEC BER estimation of PM-QPSK alien wavelength signals. A field trial is demonstrated and used as validation of the tool's correctness. A very close correspondence between the performance of the field trial and the one predicted by the modeling tool has been... 10. Two-wavelength spatial-heterodyne holography Science.gov (United States) Hanson, Gregory R.; Bingham, Philip R.; Simpson, John T.; Karnowski, Thomas P.; Voelkl, Edgar 2007-12-25 Systems and methods are described for obtaining two-wavelength differential-phase holograms. A method includes determining a difference between a filtered analyzed recorded first spatially heterodyne hologram phase and a filtered analyzed recorded second spatially-heterodyned hologram phase. 11. Corrections to the box diagram amplitude due to kaon mass International Nuclear Information System (INIS) Datta, A.; Kumbhakar, D. 1985-08-01 The K 0 -anti-K 0 mixing amplitude is calculated without using the standard zero external momentum approximation. The resulting corrections are numerically significant for the real part of the amplitude. In the imaginary part of the amplitude the effects of similar corrections are less important. Implications for Δmsub(k) and epsilon are discussed. (author) 12. MHV Vertices And Tree Amplitudes In Gauge Theory International Nuclear Information System (INIS) Cachazo, Freddy; Svrcek, Peter; Witten, Edward 2004-01-01 As an alternative to the usual Feynman graphs, tree amplitudes in Yang-Mills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off shell in a particular fashion. The formalism leads to new and relatively simple formulas for many amplitudes, and can be heuristically derived from twistor space. (author) 13. Modulating the amplitude and phase of the complex spectral degree of coherence with plasmonic interferometry Science.gov (United States) Li, Dongfang; Pacifici, Domenico The spectral degree of coherence describes the correlation of electromagnetic fields, which plays a key role in many applications, including free-space optical communications and speckle-free bioimaging. Recently, plasmonic interferometry, i.e. optical interferometry that employs surface plasmon polaritons (SPPs), has enabled enhanced light transmission and high-sensitivity biosensing, among other applications. It offers new ways to characterize and engineer electromagnetic fields using nano-structured thin metal films. Here, we employ plasmonic interferometry to demonstrate full control of spatial coherence at length scales comparable to the wavelength of the incident light. Specifically, by measuring the diffraction pattern of several double-slit plasmonic structures etched on a metal film, the amplitude and phase of the degree of spatial coherence is determined as a function of slit-slit separation distance and incident wavelength. When the SPP contribution is turned on (i.e., by changing the polarization of the incident light from TE to TM illumination mode), strong modulation of both amplitude and phase of the spatial coherence is observed. These findings may help design compact modulators of optical spatial coherence and other optical elements to shape the light intensity in the far-field. 14. Calibration-free wavelength-modulation spectroscopy based on a swiftly determined wavelength-modulation frequency response function of a DFB laser. Science.gov (United States) Zhao, Gang; Tan, Wei; Hou, Jiajia; Qiu, Xiaodong; Ma, Weiguang; Li, Zhixin; Dong, Lei; Zhang, Lei; Yin, Wangbao; Xiao, Liantuan; Axner, Ove; Jia, Suotang 2016-01-25 A methodology for calibration-free wavelength modulation spectroscopy (CF-WMS) that is based upon an extensive empirical description of the wavelength-modulation frequency response (WMFR) of DFB laser is presented. An assessment of the WMFR of a DFB laser by the use of an etalon confirms that it consists of two parts: a 1st harmonic component with an amplitude that is linear with the sweep and a nonlinear 2nd harmonic component with a constant amplitude. Simulations show that, among the various factors that affect the line shape of a background-subtracted peak-normalized 2f signal, such as concentration, phase shifts between intensity modulation and frequency modulation, and WMFR, only the last factor has a decisive impact. Based on this and to avoid the impractical use of an etalon, a novel method to pre-determine the parameters of the WMFR by fitting to a background-subtracted peak-normalized 2f signal has been developed. The accuracy of the new scheme to determine the WMFR is demonstrated and compared with that of conventional methods in CF-WMS by detection of trace acetylene. The results show that the new method provides a four times smaller fitting error than the conventional methods and retrieves concentration more accurately. 15. Investigation of boiling water reactor stability and limit-cycle amplitude International Nuclear Information System (INIS) Damiano, B.; March-Leuba, J.A.; Euler, J.A. 1991-01-01 Galerkin's method has been applied to a boiling water reactor (BWR) dynamics model consisting of the point kinetics equations, which describe the neutronics, and a feedback transfer function, which describes the thermal hydraulics. The result is a low-order approximate solution describing BWR behavior during small-amplitude limit-cycle oscillations. The approximate solution has been used to obtain a stability condition, show that the average reactor power must increase during limit-cycle oscillations, and qualitatively determine how changes in transfer function values affect the limit-cycle amplitude. 6 refs., 2 figs., 2 tabs 16. Report from LHC MD 2171: Amplitude dependent closest tune approach from normal and skew octupoles CERN Document Server Maclean, Ewen Hamish; Persson, Tobias Hakan Bjorn; Carlier, Felix Simon; CERN. Geneva. ATS Department 2018-01-01 Simulation-based studies predict signiﬁcant amplitude-dependent closest tune approach can be generated by skew octupole sources in conjunction with their normal octupolar counterparts. This has the potential to signiﬁcantly inﬂuence Landau damping at small β∗, where skew octupole errors in the experimental IRs, together with b4 introduced by the Landau octupoles, is predicted to cause large distortion of the tune footprint. This MD aimed to perform a ﬁrst exploration of these predictions with beam, by enhancing skew octupole sources in the IRs at injection and measuring amplitude detuning with free kicks in the plane approaching the coupling resonance. 17. Heterodyne technique for measuring the amplitude and phase transfer functions of an optical modulator DEFF Research Database (Denmark) Romstad, Francis Pascal; Birkedal, Dan; Mørk, Jesper 2002-01-01 In this letter, we propose a technique based on heterodyne detection for accurately and simultaneously measuring the amplitude and phase transfer functions of an optical modulator. The technique is used to characterize an InGaAsp multiple quantum-well electroabsorption modulator. From the measure...... the measurements we derive the small-signal alpha-parameter and the time-dependent chirp for different operation conditions.......In this letter, we propose a technique based on heterodyne detection for accurately and simultaneously measuring the amplitude and phase transfer functions of an optical modulator. The technique is used to characterize an InGaAsp multiple quantum-well electroabsorption modulator. From... 18. Measurement of the amplitude and phase transfer functions of an optical modulator using a heterodyne technique DEFF Research Database (Denmark) Romstad, Francis Pascal; Birkedal, Dan; Mørk, Jesper 2001-01-01 We present a new technique that measures the full amplitude and phase transfer curves of the modulator as a function of the applied bias, from which the small signal α-parameter can be calculated. The technique measures the amplitude and phase transfer functions simultaneously and directly......, compared to techniques where a time-consuming data analysis is necessary to calculate the a-parameter and an additional measurement is necessary to estimate the phase. Additionally, the chirp profile for all operation points can be calculated.... 19. Non-supersymmetric loop amplitudes and MHV vertices International Nuclear Information System (INIS) Bedford, James; Brandhuber, Andreas; Spence, Bill; Travaglini, Gabriele 2005-01-01 We show how the MHV diagram description of Yang-Mills theories can be used to study non-supersymmetric loop amplitudes. In particular, we derive a compact expression for the cut-constructible part of the general one-loop MHV multi-gluon scattering amplitude in pure Yang-Mills theory. We show that in special cases this expression reduces to known amplitudes-the amplitude with adjacent negative-helicity gluons, and the five gluon non-adjacent amplitude. Finally, we briefly discuss the twistor space interpretation of our result 20. Phase and amplitude detection system for the Stanford Linear Accelerator International Nuclear Information System (INIS) Fox, J.D.; Schwarz, H.D. 1983-01-01 A computer controlled phase and amplitude detection system to measure and stabilize the rf power sources in the Stanford Linear Accelerator is described. This system measures the instantaneous phase and amplitude of a 1 microsecond 2856 MHz rf pulse and will be used for phase feedback control and for amplitude and phase jitter detection. This paper discusses the measurement system performance requirements for the operation of the Stanford Linear Collider, and the design and implementation of the phase and amplitude detection system. The fundamental software algorithms used in the measurement are described, as is the performance of the prototype phase and amplitude detector system 1. Modeling of large amplitude plasma blobs in three-dimensions Energy Technology Data Exchange (ETDEWEB) Angus, Justin R. [Naval Research Laboratory, 4555 Overlook Avenue, Washington, DC 20375 (United States); Umansky, Maxim V. [Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550 (United States) 2014-01-15 Fluctuations in fusion boundary and similar plasmas often have the form of filamentary structures, or blobs, that convectively propagate radially. This may lead to the degradation of plasma facing components as well as plasma confinement. Theoretical analysis of plasma blobs usually takes advantage of the so-called Boussinesq approximation of the potential vorticity equation, which greatly simplifies the treatment analytically and numerically. This approximation is only strictly justified when the blob density amplitude is small with respect to that of the background plasma. However, this is not the case for typical plasma blobs in the far scrape-off layer region, where the background density is small compared to that of the blob, and results obtained based on the Boussinesq approximation are questionable. In this report, the solution of the full vorticity equation, without the usual Boussinesq approximation, is proposed via a novel numerical approach. The method is used to solve for the evolution of 2D and 3D plasma blobs in a regime where the Boussinesq approximation is not valid. The Boussinesq solution under predicts the cross field transport in 2D. However, in 3D, for parameters typical of current tokamaks, the disparity between the radial cross field transport from the Boussinesq approximation and full solution is virtually non-existent due to the effects of the drift wave instability. 2. Modeling of large amplitude plasma blobs in three-dimensions International Nuclear Information System (INIS) Angus, Justin R.; Umansky, Maxim V. 2014-01-01 Fluctuations in fusion boundary and similar plasmas often have the form of filamentary structures, or blobs, that convectively propagate radially. This may lead to the degradation of plasma facing components as well as plasma confinement. Theoretical analysis of plasma blobs usually takes advantage of the so-called Boussinesq approximation of the potential vorticity equation, which greatly simplifies the treatment analytically and numerically. This approximation is only strictly justified when the blob density amplitude is small with respect to that of the background plasma. However, this is not the case for typical plasma blobs in the far scrape-off layer region, where the background density is small compared to that of the blob, and results obtained based on the Boussinesq approximation are questionable. In this report, the solution of the full vorticity equation, without the usual Boussinesq approximation, is proposed via a novel numerical approach. The method is used to solve for the evolution of 2D and 3D plasma blobs in a regime where the Boussinesq approximation is not valid. The Boussinesq solution under predicts the cross field transport in 2D. However, in 3D, for parameters typical of current tokamaks, the disparity between the radial cross field transport from the Boussinesq approximation and full solution is virtually non-existent due to the effects of the drift wave instability 3. Leading Wave Amplitude of a Tsunami Science.gov (United States) Kanoglu, U. 2015-12-01 Okal and Synolakis (EGU General Assembly 2015, Geophysical Research Abstracts-Vol. 17-7622) recently discussed that why the maximum amplitude of a tsunami might not occur for the first wave. Okal and Synolakis list observations from 2011 Japan tsunami, which reached to Papeete, Tahiti with a fourth wave being largest and 72 min later after the first wave; 1960 Chilean tsunami reached Hilo, Hawaii with a maximum wave arriving 1 hour later with a height of 5m, first wave being only 1.2m. Largest later waves is a problem not only for local authorities both in terms of warning to the public and rescue efforts but also mislead the public thinking that it is safe to return shoreline or evacuated site after arrival of the first wave. Okal and Synolakis considered Hammack's (1972, Ph.D. Dissertation, Calif. Inst. Tech., 261 pp., Pasadena) linear dispersive analytical solution with a tsunami generation through an uplifting of a circular plug on the ocean floor. They performed parametric study for the radius of the plug and the depth of the ocean since these are the independent scaling lengths in the problem. They identified transition distance, as the second wave being larger, regarding the parameters of the problem. Here, we extend their analysis to an initial wave field with a finite crest length and, in addition, to a most common tsunami initial wave form of N-wave as presented by Tadepalli and Synolakis (1994, Proc. R. Soc. A: Math. Phys. Eng. Sci., 445, 99-112). We compare our results with non-dispersive linear shallow water wave results as presented by Kanoglu et al. (2013, Proc. R. Soc. A: Math. Phys. Eng. Sci., 469, 20130015), investigating focusing feature. We discuss the results both in terms of leading wave amplitude and tsunami focusing. Acknowledgment: The research leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under grant agreement no 603839 (Project ASTARTE - Assessment, Strategy and Risk 4. Amplitude modulation of atomic wave functions. Final report Energy Technology Data Exchange (ETDEWEB) NONE 1998-11-01 The major theoretical advance has been to show that one can modulate Rydberg wave functions using either of two methods: (1) the amplitude modulation technique which depends on autoionization to deplete part of the wave function, or (2) a phase modulation method, which uses a change in the core potential to create a localized phase shift in the wave function. Essentially, these two methods can both be seen as using the core potential to change the Rydberg wave function, using the imaginary part of the potential to do amplitude modulation, or using the real part of the potential to do phase modulation. This work will be published as the authors acquire experimental results which show the differences between the two methods. One of the results of this theoretical study is that the initial proposal to study Barium 6snd states had a significant flaw. Neither the autoionization time, nor the quantum defect shifts are very large in these cases. This means that the modulation is relatively small. This shows itself primarily in the difficulty of seeing significant population redistribution into different 6snd states. The authors intend to correct this in the next funding cycle either: (a) by using the more quickly decaying Ba 6pnf states to modulate 6snd states, or (b) by using Sr 5 snd states, as outlined in this report. Their first, low power experiments are complete. These experiments have used two pulses to do a temporal version of the Ramsey separated oscillatory fields excitation. The two pulses are generated by passing the single pulse through a Michelson-Morley interferometer, which is computer controlled to sweep one arm through 2.5 {micro}m in steps of 10 nm. The second pulses excitation interferes with that of the first pulse, and so the total excitation has a sinusoidal variation (with a time period equal to the optical period) on top of a constant background. The amplitude of the total variation should decay at half of the rate decay rate of the autoionizing 5. Effects of amplitude modulation on perception of wind turbine noise Energy Technology Data Exchange (ETDEWEB) Yoon, Ki Seop; Lee, Soo Gab; Gwak, Doo Young [Dept. of Mechanical and Aerospace Engineering, Seoul National University, Seoul (Korea, Republic of); Seong, Yeol Wan [Ammunition Engineering Team, Defense Agency for Technology and Quality, Daejeon (Korea, Republic of); Lee, Seung Hoon [Aerodynamics Research Team, Korea Aerospace Research Institute, Daejeon (Korea, Republic of); Hong, Ji Young [Transportation Environmental Research Team, Green Transport and Logistics Institute, Korea Railroad Research Institute, Uiwang (Korea, Republic of) 2016-10-15 Wind turbine noise is considered to be easily detectable and highly annoying at relatively lower sound levels than other noise sources. Many previous studies attributed this characteristic to amplitude modulation. However, it is unclear whether amplitude modulation is the main cause of these properties of wind turbine noise. Therefore, the aim of the current study is to identify the relationship between amplitude modulation and these two properties of wind turbine noise. For this investigation, two experiments were conducted. In the first experiment, 12 participants determined the detection thresholds of six target sounds in the presence of background noise. In the second experiment, 12 participants matched the loudness of modified sounds without amplitude modulation to that of target sounds with amplitude modulation. The results showed that the detection threshold was lowered as the modulation depth increased; additionally, sounds with amplitude modulation had higher subjective loudness than those without amplitude modulation. 6. Effects of amplitude modulation on perception of wind turbine noise International Nuclear Information System (INIS) Yoon, Ki Seop; Lee, Soo Gab; Gwak, Doo Young; Seong, Yeol Wan; Lee, Seung Hoon; Hong, Ji Young 2016-01-01 Wind turbine noise is considered to be easily detectable and highly annoying at relatively lower sound levels than other noise sources. Many previous studies attributed this characteristic to amplitude modulation. However, it is unclear whether amplitude modulation is the main cause of these properties of wind turbine noise. Therefore, the aim of the current study is to identify the relationship between amplitude modulation and these two properties of wind turbine noise. For this investigation, two experiments were conducted. In the first experiment, 12 participants determined the detection thresholds of six target sounds in the presence of background noise. In the second experiment, 12 participants matched the loudness of modified sounds without amplitude modulation to that of target sounds with amplitude modulation. The results showed that the detection threshold was lowered as the modulation depth increased; additionally, sounds with amplitude modulation had higher subjective loudness than those without amplitude modulation 7. Single-shot dual-wavelength in-line and off-axis hybrid digital holography Science.gov (United States) Wang, Fengpeng; Wang, Dayong; Rong, Lu; Wang, Yunxin; Zhao, Jie 2018-02-01 We propose an in-line and off-axis hybrid holographic real-time imaging technique. The in-line and off-axis digital holograms are generated simultaneously by two lasers with different wavelengths, and they are recorded using a color camera with a single shot. The reconstruction is carried using an iterative algorithm in which the initial input is designed to include the intensity of the in-line hologram and the approximate phase distributions obtained from the off-axis hologram. In this way, the complex field in the object plane and the output by the iterative procedure can produce higher quality amplitude and phase images compared to traditional iterative phase retrieval. The performance of the technique has been demonstrated by acquiring the amplitude and phase images of a green lacewing's wing and a living moon jellyfish. 8. First-Light Galaxies or Intrahalo Stars: Multi-Wavelength Measurements of the Infrared Background Anisotropies Science.gov (United States) Cooray, Asantha The research program described in this proposal can be broadly described as data analysis, measurement, and interpretation of the spatial fluctuations of the unresolved cosmic IR background. We will focus primarily on the background at optical and near-IR wavelengths as probed by Hubble and Spitzer. As absolute background intensity measurements are challenging, the focus is on the spatial fluctuations similar to the anisotropiesof the cosmic microwave background (CMB). Measurements of the unresolved Spitzer fluctuations by two independent teams on multiple fields agree within the measurement errors. However, there are now two interpretations on the origin of the unresolved IRAC fluctuations. One involves a population of faint sources at very high redshifts (z > 6) during the epoch of reionization. The second interpretation involves the integrated emission from intrahalo light associated with diffuse stars in the outskirts of z of 1 to 3 dark matter halos of galaxies. We now propose to further test these two interpretations with a new set of measurements at shorter IR and optical wavelengths with HST/ACS and WFC3 overlapping with deep IRAC surveys. A multi-wavelength study from 0.5 to 4.5 micron will allow us to independently determine the relative contribution of intrahalo light and z > 8 faint galaxies to the unresolved IR fluctuations. We will also place strong limits on the surface density of faint sources at z > 8. Such a limit will be useful for planning deep surveys with JWST. Moving to the recent wide IRAC fields with the warm mission, we propose to study fluctuations at tens of degree angular scales. At such large angular scales IRAC fluctuations should trace diffuse Galactic light (DGL), ISM dust-scattered starlight in our Galaxy. We will measure the amplitude and slope of the DGL power spectrum and compare them to measurements of the Galactic dust power spectrum from IRAS and Planck and study if the large degree-scale fluctuations seen in CIBER can be 9. The pulsed amplitude unit for the SLC International Nuclear Information System (INIS) Rolfe, J.; Browne, M.J.; Jobe, R.K. 1987-02-01 There is a recurring requirement in the SLC for the control of devices such as magnets, phase shifters, and attenuators on a beam-by-beam basis. The Pulsed Amplitude Unit (PAU) is a single width CAMAC module developed for this purpose. It provides digitally programmed analog output voltages on a beam-by-beam basis. Up to 32 preprogrammed values of output voltage are available from the single analog output of the module, and any of these values can be associated with any of the 256 possible SLC beam definitions. A 12-bit Analog-to-Digital Converter (ADC) digitizes an analog input signal at the appropriate beam time and stores it in a buffer memory. This feature is normally used to monitor the response of the device being controlled by the PAU at each beam time. Initial application of the PAU is a part of the system that controls the output of Klystrons in the SLC. The PAU combines several different functions in a single module. In order to accommodate these functions in a single width CAMAC module, field programmed logic is used extensively. Field Programmable Logic Arrays, Programmed Array Logic, and a Field Programmable Logic Sequencer are employed 10. Gearbox Vibration Signal Amplitude and Frequency Modulation Directory of Open Access Journals (Sweden) Fakher Chaari 2012-01-01 Full Text Available Gearboxes usually run under fluctuating load conditions during service, however most of papers available in the literature describe models of gearboxes under stationary load conditions. Main task of published papers is fault modeling for their detection. Considering real situation from industry, the assumption of stationarity of load conditions cannot be longer kept. Vibration signals issued from monitoring in maintenance operations differ from mentioned models (due to load non-stationarity and may be difficult to analyze which lead to erroneous diagnosis of the system. The objective of this paper is to study the influence of time varying load conditions on a gearbox dynamic behavior. To investigate this, a simple spur gear system without defects is modeled. It is subjected to a time varying load. The speed-torque characteristic of the driving motor is considered. The load variation induces speed variation, which causes a variation in the gearmesh stiffness period. Computer simulation shows deep amplitude modulations with sidebands that don't differ from those obtained when there is a defective tooth. In order to put in evidence the time varying load effects, Short Time Fourier Transform and then Smoothed Wigner-Ville distribution are used. Results show that the last one is well suited for the studied case. 11. Electroweak amplitudes in chiral quark models International Nuclear Information System (INIS) Fiolhais, Manuel 2004-01-01 After referring to some basic features of chiral models for baryons, with quarks and mesons, we describe how to construct model states representing physical baryons. We consider soliton models such as the Linear Sigma Model or the Chromodielectric Model, and bag models such as the Cloudy Bag Model. These models are solved approximately using variational approaches whose starting point is a mean-field description. We go beyond the mean-field description by introducing quantum fluctuations in the mesonic degrees of freedom. This is achieved, in a first step, by using a quantum state to represent meson clouds and, secondly, by performing an angular momentum and isospin projection from the mean-field state (actually a coherent state). Model states for baryons (nucleon, Delta, Roper) constructed in this way are used to determine several physical properties. I this seminar we paid a particular attention to the nucleon-delta electromagnetic and weak transition, presenting the model predictions for the electromagnetic and axial amplitudes 12. Nonlinear amplitude dynamics in flagellar beating. Science.gov (United States) 2017-03-01 The physical basis of flagellar and ciliary beating is a major problem in biology which is still far from completely understood. The fundamental cytoskeleton structure of cilia and flagella is the axoneme, a cylindrical array of microtubule doublets connected by passive cross-linkers and dynein motor proteins. The complex interplay of these elements leads to the generation of self-organized bending waves. Although many mathematical models have been proposed to understand this process, few attempts have been made to assess the role of dyneins on the nonlinear nature of the axoneme. Here, we investigate the nonlinear dynamics of flagella by considering an axonemal sliding control mechanism for dynein activity. This approach unveils the nonlinear selection of the oscillation amplitudes, which are typically either missed or prescribed in mathematical models. The explicit set of nonlinear equations are derived and solved numerically. Our analysis reveals the spatio-temporal dynamics of dynein populations and flagellum shape for different regimes of motor activity, medium viscosity and flagellum elasticity. Unstable modes saturate via the coupling of dynein kinetics and flagellum shape without the need of invoking a nonlinear axonemal response. Hence, our work reveals a novel mechanism for the saturation of unstable modes in axonemal beating. 13. An amplitude modulated radio frequency plasma generator Science.gov (United States) Lei, Fan; Li, Xiaoping; Liu, Yanming; Liu, Donglin; Yang, Min; Xie, Kai; Yao, Bo 2017-04-01 A glow discharge plasma generator and diagnostic system has been developed to study the effects of rapidly variable plasmas on electromagnetic wave propagation, mimicking the plasma sheath conditions encountered in space vehicle reentry. The plasma chamber is 400 mm in diameter and 240 mm in length, with a 300-mm-diameter unobstructed clear aperture. Electron densities produced are in the mid 1010 electrons/cm3. An 800 W radio frequency (RF) generator is capacitively coupled through an RF matcher to an internally cooled stainless steel electrode to form the plasma. The RF power is amplitude modulated by a waveform generator that operates at different frequencies. The resulting plasma contains electron density modulations caused by the varying power levels. A 10 GHz microwave horn antenna pair situated on opposite sides of the chamber serves as the source and detector of probe radiation. The microwave power feed to the source horn is split and one portion is sent directly to a high-speed recording oscilloscope. On mixing this with the signal from the pickup horn antenna, the plasma-induced phase shift between the two signals gives the path-integrated electron density with its complete time dependent variation. Care is taken to avoid microwave reflections and extensive shielding is in place to minimize electronic pickup. Data clearly show the low frequency modulation of the electron density as well as higher harmonics and plasma fluctuations. 14. Open string topological amplitudes and gaugino masses International Nuclear Information System (INIS) Antoniadis, I.; Narain, K.S.; Taylor, T.R. 2005-09-01 We discuss the moduli-dependent couplings of the higher derivative F-terms (TrW 2 ) h-1 , where W is the gauge N =1 chiral superfield. They are determined by the genus zero topological partition function F (0,h) , on a world-sheet with h boundaries. By string duality, these terms are also related to heterotic topological amplitudes studied in the past, with the topological twist applied only in the left-moving supersymmetric sector of the internal N =(2,0) superconformal field theory. The holomorphic anomaly of these couplings relates them to terms of the form Π n (TrW 2 ) h-2 , where Π's represent chiral projections of non-holomorphic functions of chiral superfields. An important property of these couplings is that they violate R-symmetry for h ≥ 3. As a result, once supersymmetry is broken by D-term expectation values, (TrW 2 ) 2 generates gaugino masses that can be hierarchically smaller than the scalar masses, behaving as m 1/2 ∼ m 0 4 in string units. Similarly, ΠTrW 2 generates Dirac masses for non-chiral brane fermions, of the same order of magnitude. This mechanism can be used for instance to obtain fermion masses at the TeV scale for scalar masses as high as m 0 ∼ O (10 13 ) GeV. We present explicit examples in toroidal string compactifications with intersecting D-branes. (author) 15. Casimir amplitudes in topological quantum phase transitions. Science.gov (United States) Griffith, M A; Continentino, M A 2018-01-01 Topological phase transitions constitute a new class of quantum critical phenomena. They cannot be described within the usual framework of the Landau theory since, in general, the different phases cannot be distinguished by an order parameter, neither can they be related to different symmetries. In most cases, however, one can identify a diverging length at these topological transitions. This allows us to describe them using a scaling approach and to introduce a set of critical exponents that characterize their universality class. Here we consider some relevant models of quantum topological transitions associated with well-defined critical exponents that are related by a quantum hyperscaling relation. We extend to these models a finite-size scaling approach based on techniques for calculating the Casimir force in electromagnetism. This procedure allows us to obtain universal Casimir amplitudes at their quantum critical points. Our results verify the validity of finite-size scaling in these systems and confirm the values of the critical exponents obtained previously. 16. Effective anisotropy through traveltime and amplitude matching KAUST Repository Wang, Hui 2014-08-05 Introducing anisotropy to seismic wave propagation reveals more realistic physics of our Earth\\'s subsurface as compared to the isotropic assumption. However wavefield modeling, the engine of seismic inverse problems, in anisotropic media still suffers from computational burdens, in particular with complex anisotropy such as transversely isotropic (TI) and Orthorhombic anisotropy. We develop effective isotropic velocity and density models to package the effects of anisotropy such that the wave propagation behavior using these effective models approximate those of the original anisotropic model. We build these effective models through the high frequency asymptotic approximation based on the eikonal and transport equations. We match the geometrical behavior of the wave-fields, given by traveltimes, from the anisotropic and isotropic eikonal equations. This matching yields the effective isotropic velocity that approximates the kinematics of the anisotropic wavefield. Equivalently, we calculate the effective densities by equating the anisotropic and isotropic transport equations. The effective velocities and densities are then fed into the isotropic acoustic variable density wave equation to obtain cheaper anisotropic wavefields. We justify our approach by testing it on an elliptical anisotropic model. The numerical results demonstrate a good matching of both traveltime and amplitude between anisotropic and effective isotropic wavefields. 17. The pulsed amplitude unit for the SLC International Nuclear Information System (INIS) Rolfe, J.; Browne, M.J.; Jobe, R.K. 1987-01-01 There is a recurring requirement in the SLC for the control of devices such as magnets, phase shifters, and attenuators on a beam-by-beam basis. The Pulsed Amplitude Unit (PAU) is a single width CAMAC module developed for this purpose. It provides digitally programmed analog output voltages on a beam-by-beam basis. Up to 32 preprogrammed values of output voltage are available from the single analog output of the module, and any of these values can be associated with any of the 256 possible SLC beam definitions. A 12-bit Analog-to-Digital converter (ADC) digitizes an analog input signal at the appropriate beam time and stores it in a buffer memory. This feature is normally used to monitor the response of the device being controlled by the PAU at each beam time. Initial application of the PAU at is as part of the system that controls the output of Klystorns in the SLC. The PAU combines several different functions in a single module. In order to accommodate these functions in a single width CAMAC module, field programmed logic is used extensively. Field Programmable Logic Arrays, Programmed Array Logic, and a Field Programmable Logic Sequencer are employed 18. Beyond Panglossian Optimism: Larger N2 Amplitudes Probably Signal a Bilingual Disadvantage in Conflict Monitoring Directory of Open Access Journals (Sweden) Kenneth R. Paap 2015-01-01 Full Text Available In this special issue on the brain mechanisms that lead to cognitive benefits of bilingualism we discussed six reasons why it will be very difficult to discover those mechanisms. Many of these problems apply to the article by Fernandez, Acosta, Douglass, Doshi, and Tartar that also appears in the special issue. These concerns include the following: 1 an overly optimistic assessment of the replicability of bilingual advantages in behavioral studies, 2 reliance on risky small samples sizes, 3 failures to match the samples on demographic characteristics such as immigrant status, and 4 language group differences that occur in neural measures (i.e., N2 amplitude, but not in the behavioral data. Furthermore the N2 amplitude measure in general suffers from valence ambiguity: larger N2 amplitudes reported for bilinguals are more likely to reflect poorer conflict resolution rather than enhanced inhibitory control. 19. Large amplitude ion-acoustic waves in a plasma with an electron beam International Nuclear Information System (INIS) Nejoh, Y.; Sanuki, H. 1995-01-01 The nonlinear wave structures of large amplitude ion-acoustic waves are studied in a plasma with an electron beam, by the pseudopotential method. The region of the existence of large amplitude ion-acoustic waves is examined, showing that the condition of the existence sensitively depends on the parameters such as the electron beam temperature, the ion temperature, the electrostatic potential, and the concentration of the electron beam density. It turns out that the region of the existence spreads as the beam temperature increases but the effect of the electron beam velocity is relatively small. New findings of large amplitude ion-acoustic waves in a plasma with an electron beam are predicted. copyright 1995 American Institute of Physics 20. Finite amplitude, horizontal motion of a load symmetrically supported between isotropic hyperelastic springs. Science.gov (United States) Beatty, Millard F; Young, Todd R 2012-03-01 The undamped, finite amplitude horizontal motion of a load supported symmetrically between identical incompressible, isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is formulated in general terms. The small amplitude motion of the load about the deformed static state is discussed; and the periodicity of the arbitrary finite amplitude motion is established for all such elastic materials for which certain conditions on the engineering stress and the strain energy function hold. The exact solution for the finite vibration of the load is then derived for the classical neo-Hookean model. The vibrational period is obtained in terms of the complete Heuman lambda-function whose properties are well-known. Dependence of the period and hence the frequency on the physical parameters of the system is investigated and the results are displayed graphically. 1. Large-Amplitude Long-Wave Instability of a Supersonic Shear Layer Science.gov (United States) Messiter, A. F. 1995-01-01 For sufficiently high Mach numbers, small disturbances on a supersonic vortex sheet are known to grow in amplitude because of slow nonlinear wave steepening. Under the same external conditions, linear theory predicts slow growth of long-wave disturbances to a thin supersonic shear layer. An asymptotic formulation is given here which adds nonzero shear-layer thickness to the weakly nonlinear formulation for a vortex sheet. Spatial evolution is considered, for a spatially periodic disturbance having amplitude of the same order, in Reynolds number, as the shear-layer thickness. A quasi-equilibrium inviscid nonlinear critical layer is found, with effects of diffusion and slow growth appearing through nonsecularity condition. Other limiting cases are also considered, in an attempt to determine a relationship between the vortex-sheet limit and the long-wave limit for a thin shear layer; there appear to be three special limits, corresponding to disturbances of different amplitudes at different locations along the shear layer. 2. Complex amplitude reconstruction by iterative amplitude-phase retrieval algorithm with reference Science.gov (United States) Shen, Cheng; Guo, Cheng; Tan, Jiubin; Liu, Shutian; Liu, Zhengjun 2018-06-01 Multi-image iterative phase retrieval methods have been successfully applied in plenty of research fields due to their simple but efficient implementation. However, there is a mismatch between the measurement of the first long imaging distance and the sequential interval. In this paper, an amplitude-phase retrieval algorithm with reference is put forward without additional measurements or priori knowledge. It gets rid of measuring the first imaging distance. With a designed update formula, it significantly raises the convergence speed and the reconstruction fidelity, especially in phase retrieval. Its superiority over the original amplitude-phase retrieval (APR) method is validated by numerical analysis and experiments. Furthermore, it provides a conceptual design of a compact holographic image sensor, which can achieve numerical refocusing easily. 3. Amplitude various angles (AVA) phenomena in thin layer reservoir: Case study of various reservoirs Energy Technology Data Exchange (ETDEWEB) Nurhandoko, Bagus Endar B., E-mail: bagusnur@bdg.centrin.net.id, E-mail: bagusnur@rock-fluid.com [Wave Inversion and Subsurface Fluid Imaging Research Laboratory (WISFIR), Basic Science Center A 4" t" hfloor, Physics Dept., FMIPA, Institut Teknologi Bandung (Indonesia); Rock Fluid Imaging Lab., Bandung (Indonesia); Susilowati, E-mail: bagusnur@bdg.centrin.net.id, E-mail: bagusnur@rock-fluid.com [Rock Fluid Imaging Lab., Bandung (Indonesia) 2015-04-16 Amplitude various offset is widely used in petroleum exploration as well as in petroleum development field. Generally, phenomenon of amplitude in various angles assumes reservoir’s layer is quite thick. It also means that the wave is assumed as a very high frequency. But, in natural condition, the seismic wave is band limited and has quite low frequency. Therefore, topic about amplitude various angles in thin layer reservoir as well as low frequency assumption is important to be considered. Thin layer reservoir means the thickness of reservoir is about or less than quarter of wavelength. In this paper, I studied about the reflection phenomena in elastic wave which considering interference from thin layer reservoir and transmission wave. I applied Zoeppritz equation for modeling reflected wave of top reservoir, reflected wave of bottom reservoir, and also transmission elastic wave of reservoir. Results show that the phenomena of AVA in thin layer reservoir are frequency dependent. Thin layer reservoir causes interference between reflected wave of top reservoir and reflected wave of bottom reservoir. These phenomena are frequently neglected, however, in real practices. Even though, the impact of inattention in interference phenomena caused by thin layer in AVA may cause inaccurate reservoir characterization. The relation between classes of AVA reservoir and reservoir’s character are different when effect of ones in thin reservoir and ones in thick reservoir are compared. In this paper, I present some AVA phenomena including its cross plot in various thin reservoir types based on some rock physics data of Indonesia. 4. Amplitude various angles (AVA) phenomena in thin layer reservoir: Case study of various reservoirs International Nuclear Information System (INIS) thfloor, Physics Dept., FMIPA, Institut Teknologi Bandung (Indonesia); Rock Fluid Imaging Lab., Bandung (Indonesia))" data-affiliation=" (Wave Inversion and Subsurface Fluid Imaging Research Laboratory (WISFIR), Basic Science Center A 4thfloor, Physics Dept., FMIPA, Institut Teknologi Bandung (Indonesia); Rock Fluid Imaging Lab., Bandung (Indonesia))" >Nurhandoko, Bagus Endar B.; Susilowati 2015-01-01 Amplitude various offset is widely used in petroleum exploration as well as in petroleum development field. Generally, phenomenon of amplitude in various angles assumes reservoir’s layer is quite thick. It also means that the wave is assumed as a very high frequency. But, in natural condition, the seismic wave is band limited and has quite low frequency. Therefore, topic about amplitude various angles in thin layer reservoir as well as low frequency assumption is important to be considered. Thin layer reservoir means the thickness of reservoir is about or less than quarter of wavelength. In this paper, I studied about the reflection phenomena in elastic wave which considering interference from thin layer reservoir and transmission wave. I applied Zoeppritz equation for modeling reflected wave of top reservoir, reflected wave of bottom reservoir, and also transmission elastic wave of reservoir. Results show that the phenomena of AVA in thin layer reservoir are frequency dependent. Thin layer reservoir causes interference between reflected wave of top reservoir and reflected wave of bottom reservoir. These phenomena are frequently neglected, however, in real practices. Even though, the impact of inattention in interference phenomena caused by thin layer in AVA may cause inaccurate reservoir characterization. The relation between classes of AVA reservoir and reservoir’s character are different when effect of ones in thin reservoir and ones in thick reservoir are compared. In this paper, I present some AVA phenomena including its cross plot in various thin reservoir types based on some rock physics data of Indonesia 5. Design alternatives for wavelength routing networks Science.gov (United States) Miliotis, K.; Papadimitriou, G. I.; Pomportsis, A. S. 2003-03-01 This paper attempts to provide a high level overview of many of the technologies employed in optical networks with a focus on wavelength-routing networks. Optical networks involve a number of technologies from the physics of light through protocols and networks architectures. In fact there is so much technology and know-how that most people involved with optical networks only have a full understanding of the narrow area they deal with. We start first examining the principles that govern light and its use as a wave guide, and then turn our focus to the various components that constitute an optical network and conclude with the description of all optical networks and wavelength-routed networks in greater detail. 6. Immersion lithography defectivity analysis at DUV inspection wavelength Science.gov (United States) Golan, E.; Meshulach, D.; Raccah, N.; Yeo, J. Ho.; Dassa, O.; Brandl, S.; Schwarz, C.; Pierson, B.; Montgomery, W. 2007-03-01 Significant effort has been directed in recent years towards the realization of immersion lithography at 193nm wavelength. Immersion lithography is likely a key enabling technology for the production of critical layers for 45nm and 32nm design rule (DR) devices. In spite of the significant progress in immersion lithography technology, there remain several key technology issues, with a critical issue of immersion lithography process induced defects. The benefits of the optical resolution and depth of focus, made possible by immersion lithography, are well understood. Yet, these benefits cannot come at the expense of increased defect counts and decreased production yield. Understanding the impact of the immersion lithography process parameters on wafer defects formation and defect counts, together with the ability to monitor, control and minimize the defect counts down to acceptable levels is imperative for successful introduction of immersion lithography for production of advanced DR's. In this report, we present experimental results of immersion lithography defectivity analysis focused on topcoat layer thickness parameters and resist bake temperatures. Wafers were exposed on the 1150i-α-immersion scanner and 1200B Scanner (ASML), defect inspection was performed using a DUV inspection tool (UVision TM, Applied Materials). Higher sensitivity was demonstrated at DUV through detection of small defects not detected at the visible wavelength, indicating on the potential high sensitivity benefits of DUV inspection for this layer. The analysis indicates that certain types of defects are associated with different immersion process parameters. This type of analysis at DUV wavelengths would enable the optimization of immersion lithography processes, thus enabling the qualification of immersion processes for volume production. 7. Optical Detection in Ultrafast Short Wavelength Science International Nuclear Information System (INIS) Fullagar, Wilfred K.; Hall, Chris J. 2010-01-01 A new approach to coherent detection of ionising radiation is briefly motivated and recounted. The approach involves optical scattering of coherent light fields by colour centres in transparent solids. It has significant potential for diffractive imaging applications that require high detection dynamic range from pulsed high brilliance short wavelength sources. It also motivates new incarnations of Bragg's X-ray microscope for pump-probe studies of ultrafast molecular structure-dynamics. 8. Short wavelength striations on expanding plasma clouds International Nuclear Information System (INIS) Winske, D.; Gary, S.P. 1989-01-01 The growth and evolution of short wavelength (< ion gyroradius) flute modes on a plasma expanding across an ambient magnetic field have been actively studied in recent years, both by means of experiments in the laboratory as well as in space and through numerical simulations. We review the relevant observations and simulations results, discuss the instability mechanism and related linear theory, and describe recent work to bring experiments and theory into better agreement. 30 refs., 6 figs 9. Transversity Amplitudes in Hypercharge Exchange Processes; Amplitudes de transversidad en procesos de intercambio de hipercarga Energy Technology Data Exchange (ETDEWEB) Aguilar Benitez de Lugo, M. 1979-07-01 In this work we present several techniques developed for the extraction of the. Transversity amplitudes governing quasi two-body meson baryon reactions with hypercharge exchange. We review the methods used In processes having a pure spin configuration, as well as the more relevant results obtained with data from K{sup p} and Tp interactions at intermediate energies. The predictions of the additive quark model and the ones following from exchange degeneracy and etoxicity are discussed. We present a formalism for amplitude analysis developed for reactions with mixed spin configurations and discuss the methods of parametric estimation of the moduli and phases of.the amplitudes, as well as the various tests employed to check the goodness of the fits. The calculation of the generalized joint density matrices is given and we propose a method based on the generalization of the idea of multipole moments, which allows to investigate the structure of the decay angular correlations and establishes the quality of the fits and the validity of the simplifying assumptions currently used in this type of studies. (Author) 43 refs. 10. Optically coupled cavities for wavelength switching Energy Technology Data Exchange (ETDEWEB) Costazo-Caso, Pablo A; Granieri, Sergio; Siahmakoun, Azad, E-mail: pcostanzo@ing.unlp.edu.ar, E-mail: granieri@rose-hulman.edu, E-mail: siahmako@rose-hulman.edu [Department of Physics and Optical Engineering, Rose-Hulman Institute of Technology, 5500 Wabash Avenue, Terre Haute, IN 47803 (United States) 2011-01-01 An optical bistable device which presents hysteresis behavior is proposed and experimentally demonstrated. The system finds applications in wavelength switching, pulse reshaping and optical bistability. It is based on two optically coupled cavities named master and slave. Each cavity includes a semiconductor optical amplifier (SOA), acting as the gain medium of the laser, and two pair of fiber Bragg gratings (FBG) which define the lasing wavelength (being different in each cavity). Finally, a variable optical coupler (VOC) is employed to couple both cavities. Experimental characterization of the system performance is made analyzing the effects of the coupling coefficient between the two cavities and the driving current in each SOA. The properties of the hysteretic bistable curve and switching can be controlled by adjusting these parameters and the loss in the cavities. By selecting the output wavelength ({lambda}{sub 1} or {lambda}{sub 2}) with an external filter it is possible to choose either the invert or non-invert switched signal. Experiments were developed employing both optical discrete components and a photonic integrated circuit. They show that for 8 m-long cavities the maximum switching frequency is about 500 KHz, and for 4 m-long cavities a minimum rise-time about 21 ns was measured. The switching time can be reduced by shortening the cavity lengths and using photonic integrated circuits. 11. Wavelength switching in an optical klystron International Nuclear Information System (INIS) Berryman, K.W.; Smith, T.I. 1995-01-01 A symmetric optical klystron consists of two identical undulator sections separated a dispersive section. For a device of a given length, an optical klystron is capable of producing much more bunching, and therefore more gain, than a traditional undulator. Another consequence of introducing dispersion between two undulator sections is that the overall spontaneous radiation pattern results from the interference between the two undulator sections, and as such resembles a standard undulator radiation pattern modulated by a sinusoidal interference term. The presence of several wavelength peaks in the spontaneous lineshape implies an equal number of peaks in the gain spectrum. If the strength of the dispersion section is adjusted to provide nearly equal gain on the two largest of these peaks, then they will compete, and the FEL may switch wavelengths based on noise, cavity length, or other perturbations. We provide the first observations of this behavior, using the FIREFLY system at the Stanford Picosecond FEL Center. In FIREFLY, relative wavelength switching by more than 3%--more than twice the laser linewidth-has been observed by varying dispersion section strength, while at intermediate points stable switching has also been observed as a function of cavity length 12. Multiple wavelength X-ray monochromators International Nuclear Information System (INIS) Steinmeyer, P.A. 1992-01-01 An improved apparatus and method is provided for separating input x-ray radiation containing first and second x-ray wavelengths into spatially separate first and second output radiation which contain the first and second x-ray wavelengths, respectively. The apparatus includes a crystalline diffractor which includes a first set of parallel crystal planes, where each of the planes is spaced a predetermined first distance from one another. The crystalline diffractor also includes a second set of parallel crystal planes inclined at an angle with respect to the first set of crystal planes where each of the planes of the second set of parallel crystal planes is spaced a predetermined second distance from one another. In one embodiment, the crystalline diffractor is comprised of a single crystal. In a second embodiment, the crystalline diffractor is comprised of a stack of two crystals. In a third embodiment, the crystalline diffractor includes a single crystal that is bent for focusing the separate first and second output x-ray radiation wavelengths into separate focal points. 3 figs 13. Stability of short-axial-wavelength internal kink modes of an anisotropic plasma Science.gov (United States) Faghihi, M.; Scheffel, J. 1987-12-01 The double adiabatic equations are used to study the stability of a cylindrical Z-pinch with respect to small axial wavelength, internal kink (m ≥ 1) modes. It is found that marginally (ideally) unstable, isotropic equilibria are stabilized. Also, constant-current-density equilibria can be stabilized for P > P and large β 14. Stability of short-axial-wavelength internal kink modes of an anisotropic plasma International Nuclear Information System (INIS) Faghihi, M.; Schefffel, J. 1987-01-01 The double adiabatic equations are used to study the stability of a cylindrical Z-pinch with respect to small axial wavelength, internal kink (m ≥ 1) modes. It is found that marginally (ideally) unstable, isotropic equilibria are stabilized. Also, constant-current-density equilibria can be stabilized for Psub(perpendicular) > Psub(parallel) and large βsub(perpendicular). (author) 15. Modal effects on amplitude perturbations on subionospheric signals (trimpis) deduced from two-frequency measurements International Nuclear Information System (INIS) Dowden, R.L.; Adams, C.D.D. 1989-01-01 Interference between the first two modes of Earth-ionosphere waveguide propagation at the high end of the VLF band (> 18 kHz) increases with distance from the transmitter out to very large distances and can add amplitude perturbations to the phase perturbations (trimpis) produced by lightning-induced electron precipitation (LEP) on the great circle path. Since the two modes have slightly different phase velocities, an interference pattern or standing wave is formed which is shifted slightly along the propagation path by the LEP-induced change in differential phase velocity. The model effect at the receiver depends on the local gradient (along the great circle path) of amplitude with respect to the differential phase. Since this differential or mode beat phase varies with frequency, measurement of the resultant amplitude at two close frequencies enables an estimation of the modal effects. In this study, measurements were made at Dunedin at the two MSK frequencies, 22,250 Hz and 22,350 Hz, of the transmitter NWC, during a night of frequent one-dimensional trimpis (i.e., those produced by large-area LEP occurring close to the great circle path) and of strong and varying modal interference. Modal generation or modification of trimpi amplitude was related to the local gradient of amplitude as expected. From these results it was deduced that modal modification of echo trimpis (those produced by small area LEP occurring well off the great circle path), even under extreme conditions, is insignificant 16. Dynamic force microscopy with quartz tuning forks at high oscillation amplitudes International Nuclear Information System (INIS) Labardi, M 2007-01-01 Dynamic force microscopy (DFM) with the self-oscillator (SO) method allows reasonably high scanning rates even with high Q-factors of the resonant force sensor, typical of cantilevers in ultra-high vacuum and of quartz tuning forks. However, due to simpler interpretation of force spectroscopy measurements, small oscillation amplitudes (sub-nm level) are generally preferred. In applications like 'apertureless' scanning near-field optical microscopy (SNOM), oscillation amplitudes of the order of 5-10 nm are needed to increase optical sensitivity and to apply standard optical artefact suppression methods. This motivates the study of the behaviour of tuning forks driven at such high amplitudes, as compared to usual air-operated cantilevers. Both constant-excitation-amplitude (CE) and constant-oscillation-amplitude (CA) modes of SO-DFM are analysed, since the CA mode is more convenient for SNOM applications, denoting remarkable differences. In particular, possible instability effects, previously found in CE mode, are not anticipated for CA mode. It is shown how resonance and approach ('isophase') curves in both modes can be conveniently described in terms of the usual 'normalized frequency shift' γ and of a 'normalized gain' η, defined as a measurement of surface dissipation 17. Scattering amplitudes in four- and six-dimensional gauge theories International Nuclear Information System (INIS) Schuster, Theodor 2014-01-01 We study scattering amplitudes in quantum chromodynamics (QCD), N=4 super Yang-Mills (SYM) theory and the six-dimensional N=(1,1) SYM theory, focusing on the symmetries of and relations between the tree-level scattering amplitudes in these three gauge theories. We derive the tree level and one-loop color decomposition of an arbitrary QCD amplitude into primitive amplitudes. Furthermore, we derive identities spanning the null space among the primitive amplitudes. We prove that every color ordered tree amplitude of massless QCD can be obtained from gluon-gluino amplitudes of N=4 SYM theory. Furthermore, we derive analytical formulae for all gluon-gluino amplitudes relevant for QCD. We compare the numerical efficiency and accuracy of evaluating these closed analytic formulae for color ordered QCD tree amplitudes to a numerically efficient implementation of the Berends-Giele recursion. We derive the symmetries of massive tree amplitudes on the coulomb branch of N=4 SYM theory, which in turn can be obtained from N=(1,1) SYM theory by dimensional reduction. Furthermore, we investigate the tree amplitudes of N=(1, 1) SYM theory and explain how analytical formulae can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation and investigate a potential uplift of the massless tree amplitudes of N=4 SYM theory. Finally we study an alternative to dimensional regularization of N=4 SYM theory. The infrared divergences are regulated by masses obtained from a Higgs mechanism. The corresponding string theory set-up suggests that the amplitudes have an exact dual conformal symmetry. We confirm this expectation and illustrate the calculational advantages of the massive regulator by explicit calculations. 18. CBET Experiments with Wavelength Shifting at the Nike Laser Science.gov (United States) Weaver, James; McKenty, P.; Bates, J.; Myatt, J.; Shaw, J.; Obenschain, K.; Oh, J.; Kehne, D.; Obenschain, S.; Lehmberg, R. H.; Tsung, F.; Schmitt, A. J.; Serlin, V. 2016-10-01 Studies conducted at NRL during 2015 searched for cross-beam energy transport (CBET) in small-scale plastic targets with strong gradients in planar, cylindrical, and spherical geometries. The targets were irradiated by two widely separated beam arrays in a geometry similar to polar direct drive. Data from these shots will be presented that show a lack of a clear CBET signature even with wavelength shifting of one set of beams. This poster will discuss the next campaign being planned, in part, with modelling codes developed at LLE. The next experiments will use a target configuration optimized to create stronger SBS growth. The primary path under consideration is to increase scale lengths 5-10x over the previous study by using exploding foils or low density foams. In addition to simulations, the presentation will also discuss improvements to the diagnostic suite and laser operations; for example, a new set of etalons will be available for the next campaign that should double the range of wavelength shifting between the two beam arrays. Work supported by DoE/NNSA. 19. Relative amplitude preservation processing utilizing surface consistent amplitude correction. Part 4; Surface consistent amplitude correction wo mochiita sotai shinpuku hozon shori. 4 Energy Technology Data Exchange (ETDEWEB) Saeki, T [Japan National Oil Corp., Tokyo (Japan). Technology Research Center 1997-10-22 Discussions were given on seismic exploration from the ground surface using the reflection method, for surface consistent amplitude correction from among effects imposed from the ground surface and a surface layer. Amplitude distribution on the reflection wave zone is complex. Therefore, items to be considered in making an analysis are multiple, such as estimation of spherical surface divergence effect and exponential attenuation effect, not only amplitude change through the surface layer. If all of these items are taken into consideration, burden of the work becomes excessive. As a method to solve this problem, utilization of amplitude in initial movement of a diffraction wave may be conceived. Distribution of the amplitude in initial movement of the diffraction wave shows a value relatively close to distribution of the vibration transmitting and receiving points. The reason for this is thought because characteristics of the vibration transmitting and receiving points related with waveline paths in the vicinity of the ground surface have no great difference both on the diffraction waves and on the reflection waves. The lecture described in this paper introduces an attempt of improving the efficiency of the surface consistent amplitude correction by utilizing the analysis of amplitude in initial movement of the diffraction wave. 4 refs., 2 figs. 20. Analyticity properties of two-body helicity amplitudes; Proprietes d'analyticite des amplitudes d'helicite a deux corps Energy Technology Data Exchange (ETDEWEB) Navelet-Noualhier, H [Commissariat a l' Energie Atomique, Saclay (France). Centre d' Etudes Nucleaires 1967-06-15 Helicity amplitudes are expressed via the spinor amplitudes in terms of the Joos invariant which have been shown by Williams to be free from kinematical singularities. This procedure allows to analyze the kinematical singularities of helicity amplitudes and separate them out, which results into the definition of regularized helicity amplitudes. A crossing matrix for helicity amplitudes, is written down, corresponding to the continuation path used to cross spinor amplitudes. We verify explicitly that the corresponding crossing matrix for regularized helicity amplitudes is uniform as it should be. Kinematical constraints which generalize, to the case of arbitrary spins and masses, relations which must hold between helicity amplitudes at some values of the energy variable in {pi}N {yields} {pi}N, {pi}{pi} {yields} NN-bar and NN-bar {yields} NN-bar reactions, appear as a consequence of the existence of poles in the crossing matrix between regularized helicity amplitudes. An english version of this work has been written with G. Cohen-Tannoudji and A. Morel and submitted for publication to Annals of Physics. (author) [French] Les amplitudes d'helicite pour une reaction a deux corps sont exprimees, par l'intermediaire des amplitudes spinorielles, en fonction d'amplitudes invariantes de Joos qui sont, comme l'a montre Williams, sans singularites cinematiques. Ce procede nous permet d'analyser puis d'eliminer les singularites cinematiques des amplitudes d'helicite. Ceci nous conduit a la definition d'amplitudes d'helicite 'regularisees'. Une relation de 'croisement' entre amplitudes d'helicite est ecrite; elle realise leur prolongement analytique le long du chemin utilise pour 'croiser' les amplitudes spinorielles. Nous verifions que les elements de la matrice de croisement entre amplitudes d'helicite 'regularisees' sont bien uniformes. Les contraintes cinematiques qui generalisent, au cas de masses et de spins arbitraires, les relations obtenues dans les reactions {pi 1. Eikonal representation of N-body Coulomb scattering amplitudes International Nuclear Information System (INIS) Fried, H.M.; Kang, K.; McKellar, B.H.J. 1983-01-01 A new technique for the construction of N-body Coulomb scattering amplitudes is proposed, suggested by the simplest case of N = 2: Calculate the scattering amplitude in eikonal approximation, discard the infinite phase factors which appear upon taking the limit of a Coulomb potential, and treat the remainder as an amplitude whose absolute value squared produces the exact, Coulomb differential cross section. The method easily generalizes to the N-body Coulomb problem for elastic scattering, and for inelastic rearrangement scattering of Coulomb bound states. We give explicit results for N = 3 and 4; in the N = 3 case we extract amplitudes for the processes (12)+3->1+2+3 (breakup), (12)+3->1+(23) (rearrangement), and (12)+3→(12)'+3 (inelastic scattering) as residues at the appropriate poles in the free-free amplitude. The method produces scattering amplitudes f/sub N/ given in terms of explicit quadratures over (N-2) 2 distinct integrands 2. Expansion of all multitrace tree level EYM amplitudes Science.gov (United States) Du, Yi-Jian; Feng, Bo; Teng, Fei 2017-12-01 In this paper, we investigate the expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes. First, we propose two types of recursive expansions of tree level EYM amplitudes with an arbitrary number of gluons, gravitons and traces by those amplitudes with fewer traces or/and gravitons. Then we give many support evidence, including proofs using the Cachazo-He-Yuan (CHY) formula and Britto-Cachazo-Feng-Witten (BCFW) recursive relation. As a byproduct, two types of generalized BCJ relations for multitrace EYM are further proposed, which will be useful in the BCFW proof. After one applies the recursive expansions repeatedly, any multitrace EYM amplitudes can be given in the Kleiss-Kuijf (KK) basis of tree level color ordered Yang-Mills (YM) amplitudes. Thus the Bern-Carrasco-Johansson (BCJ) numerators, as the expansion coefficients, for all multitrace EYM amplitudes are naturally constructed. 3. All-fiber femtosecond Cherenkov laser at visible wavelengths DEFF Research Database (Denmark) Liu, Xiaomin; Lægsgaard, Jesper; Møller, Uffe Visbech 2013-01-01 -matching condition [1]. The resonant ultrafast wave conversion via the fiber-optic CR mechanism is instrumental for applications in biophotonics such as bio-imaging and microscopy [2]. In this work, we demonstrate a highly-stable all-fiber, fully monolithic CR system based on an Yb-fiber femtosecond laser, producing...... to be as low as -103 dBc/Hz. This is 2 orders of magnitudes lower noise as compared to spectrally-sliced supercontinuum, which is the current standard of ultrafast fiber-optic generation at visible wavelength. The layout of the laser system is shown in Fig. 1(a). The system consists of two parts: an all-fiber......Fiber-optic Cherenkov radiation (CR), also known as dispersive wave generation or non-solitonic radiation, is produced in small-core photonic crystal fibers (PCF) when a soliton perturbed by fiber higher-order dispersion co-propagates with a dispersive wave fulfilling a certain phase... 4. Tensor exchange amplitudes in K +- N charge exchange reactions International Nuclear Information System (INIS) Svec, M. 1979-01-01 Tensor (A 2 ) exchange amplitudes in K +- N charge exchange (CEX) are constructed from the K +- N CEX data supplemented by information on the vector (rho) exchange amplitudes from πN sca tering. We observed new features in the t-structure of A 2 exchange amplitudes which contradict the t-de pendence anticipated by most of the Regge models. The results also provide evidence for violation of weak exchange degeneracy 5. Loop Amplitudes in Pure Yang-Mills from Generalised Unitarity OpenAIRE Brandhuber, Andreas; McNamara, Simon; Spence, Bill; Travaglini, Gabriele 2005-01-01 We show how generalised unitarity cuts in D = 4 - 2 epsilon dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions. 6. Loop amplitudes in pure Yang-Mills from generalised unitarity International Nuclear Information System (INIS) Brandhuber, Andreas; McNamara, Simon; Spence, Bill; Travaglini, Gabriele 2005-01-01 We show how generalised unitarity cuts in D = 4-2ε dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions 7. Loop amplitudes in pure Yang-Mills from generalised unitarity Energy Technology Data Exchange (ETDEWEB) Brandhuber, Andreas [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); McNamara, Simon [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); Spence, Bill [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom); Travaglini, Gabriele [Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS (United Kingdom) 2005-10-15 We show how generalised unitarity cuts in D = 4-2{epsilon} dimensions can be used to calculate efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We test the validity of our method by re-deriving the one-loop ++++, -+++, --++, -+-+ and +++++ gluon scattering amplitudes using generalised quadruple cuts and triple cuts in D dimensions. 8. Calculation and modular properties of multiloop superstring amplitudes International Nuclear Information System (INIS) Danilov, G. S. 2013-01-01 Multiloop superstring amplitudes are calculated within an extensively used gauge where the two-dimensional gravitino field carries Grassmann moduli. In general, the amplitudes possess, instead of modular symmetry, symmetry with respect to modular transformation supplemented with appropriate transformations of two-dimensional local supersymmetry. If the number of loops is larger than three, the integrationmeasures are notmodular forms, while the expression for the amplitude contains integrals along the boundary of the fundamental region of the modular group. 9. Efficient analytic computation of higher-order QCD amplitudes International Nuclear Information System (INIS) Bern, Z.; Chalmers, G.; Dunbar, D.C.; Kosower, D.A. 1995-01-01 The authors review techniques simplifying the analytic calculation of one-loop QCD amplitudes with many external legs, for use in next-to-leading-order corrections to multi-jet processes. Particularly useful are the constraints imposed by perturbative unitarity, collinear singularities and a supersymmetry-inspired organization of helicity amplitudes. Certain sequences of one-loop helicity amplitudes with an arbitrary number of external gluons have been obtained using these constraints 10. Improved pion pion scattering amplitude from dispersion relation formalism International Nuclear Information System (INIS) Cavalcante, I.P.; Coutinho, Y.A.; Borges, J. Sa 2005-01-01 Pion-pion scattering amplitude is obtained from Chiral Perturbation Theory at one- and two-loop approximations. Dispersion relation formalism provides a more economic method, which was proved to reproduce the analytical structure of that amplitude at both approximation levels. This work extends the use of the formalism in order to compute further unitarity corrections to partial waves, including the D-wave amplitude. (author) 11. Ambitwistor strings and reggeon amplitudes in N=4 SYM Directory of Open Access Journals (Sweden) L.V. Bork 2017-11-01 Full Text Available We consider the description of reggeon amplitudes (Wilson lines form factors in N=4 SYM within the framework of four dimensional ambitwistor string theory. The latter is used to derive scattering equations representation for reggeon amplitudes with multiple reggeized gluons present. It is shown, that corresponding tree-level string correlation function correctly reproduces previously obtained Grassmannian integral representation of reggeon amplitudes in N=4 SYM. 12. The five-gluon amplitude and one-loop integrals International Nuclear Information System (INIS) Bern, Z.; Dixon, L.; Kosower, D.A. 1992-12-01 We review the conventional field theory description of the string motivated technique. This technique is applied to the one-loop five-gluon amplitude. To evaluate the amplitude a general method for computing dimensionally regulated one-loop integrals is outlined including results for one-loop integrals required for the pentagon diagram and beyond. Finally, two five-gluon helicity amplitudes are given 13. Renormalization in the complete Mellin representation of Feynman amplitudes International Nuclear Information System (INIS) Calan, C. de; David, F.; Rivasseau, V. 1981-01-01 The Feynmann amplitudes are renormalized in the formalism of the CM representation. This Mellin-Barnes type integral representation, previously introduced for the study of asymptotic behaviours, is shown to have the following interesting property: in contrast with the usual subtraction procedures, the renormalization leaves the CM intergrand unchanged, and only results into translations of the integration path. The explicit CM representation of the renormalized amplitudes is given. In addition, the dimensional regularization and the extension to spinor amplitudes are sketched. (orig.) 14. Algebraic evaluation of rational polynomials in one-loop amplitudes International Nuclear Information System (INIS) Binoth, Thomas; Guillet, Jean-Philippe; Heinrich, Gudrun 2007-01-01 One-loop amplitudes are to a large extent determined by their unitarity cuts in four dimensions. We show that the remaining rational terms can be obtained from the ultraviolet behaviour of the amplitude, and determine universal form factors for these rational parts by applying reduction techniques to the Feynman diagrammatic representation of the amplitude. The method is valid for massless and massive internal particles. We illustrate this method by evaluating the rational terms of the one-loop amplitudes for gg→H, γγ→γγ, gg→gg,γγ→ggg and γγ→γγγγ 15. Phase and amplitude control system for Stanford Linear Accelerator International Nuclear Information System (INIS) Yoo, S.J. 1983-01-01 The computer controlled phase and amplitude detection system measures the instantaneous phase and amplitude of a 1 micro-second 2856 MHz rf pulse at a 180 Hz rate. This will be used for phase feedback control, and also for phase and amplitude jitter measurement. The program, which was originally written by John Fox and Keith Jobe, has been modified to improve the function of the system. The software algorithms used in the measurement are described, as is the performance of the prototype phase and amplitude detector system 16. New rigorous asymptotic theorems for inverse scattering amplitudes International Nuclear Information System (INIS) 1984-01-01 The rigorous asymptotic theorems both of integral and local types obtained earlier and establishing logarithmic and in some cases even power correlations aetdeen the real and imaginary parts of scattering amplitudes Fsub(+-) are extended to the inverse amplitudes 1/Fsub(+-). One also succeeds in establishing power correlations of a new type between the real and imaginary parts, both for the amplitudes themselves and for the inverse ones. All the obtained assertions are convenient to be tested in high energy experiments when the amplitudes show asymptotic behaviour 17. Proof of the fundamental BCJ relations for QCD amplitudes International Nuclear Information System (INIS) Cruz, Leonardo de la; Kniss, Alexander; Weinzierl, Stefan 2015-01-01 The fundamental BCJ-relation is a linear relation between primitive tree amplitudes with different cyclic orderings. The cyclic orderings differ by the insertion place of one gluon. The coefficients of the fundamental BCJ-relation are linear in the Lorentz invariants 2p_ip_j. The BCJ-relations are well established for pure gluonic amplitudes as well as for amplitudes in N=4 super-Yang-Mills theory. Recently, it has been conjectured that the BCJ-relations hold also for QCD amplitudes. In this paper we give a proof of this conjecture. The proof is valid for massless and massive quarks. 18. Amplitude dependent damping in single crystalline high purity molybdenum International Nuclear Information System (INIS) Zelada-Lambri, G.I; Lambri, O.A; Garcia, J.A; Lomer, J.N 2004-01-01 Amplitude dependent damping measurements were performed on high purity single crystalline molybdenum at several different constant temperatures between room temperature and 1273K. The employed samples were single crystals with the orientation, having a residual resistivity ratio of about 8000. Previously to the amplitude dependent damping tests, the samples were subjected to different thermomechanical histories. Amplitude dependent damping effects appear only during the first heating run in temperature where the samples have the thermomechanical state of the deformation process at room temperature. In the subsequent run-ups in temperature, i.e, after subsequent annealings, amplitude dependent damping effects were not detected (au) 19. Analytical structure of the 3. -->. 3 forward scattering amplitude Energy Technology Data Exchange (ETDEWEB) Logunov, A A; Medvedev, B V; Muzafarov, L M; Pavlov, V P; Polivanov, M K; Sukhanov, A D [AN SSSR, Moscow. Matematicheskij Inst. 1979-08-01 Analytical properties of the amplitude of 3..-->..3 forward scattering established in the framework of the Bogolyubov axiomatic approach are described. The amplitudes of the different channels of the process are boundary values of a unique analytical function of invariant variables. Crossing-symmetry property of the amplitude is proved. Analysis of the absorptive part of the amplitude is performed and the generalized optical theorem is proved which connects one of the contributions into the absorptive part with the distribution function of the inclusive process. 20. Generalized unitarity for N=4 super-amplitudes Energy Technology Data Exchange (ETDEWEB) Drummond, J.M.; Henn, J. [LAPTH, Université de Savoie, CNRS B.P. 110, F-74941 Annecy-le-Vieux Cedex (France); Korchemsky, G.P., E-mail: Gregory.Korchemsky@cea.fr [Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette Cedex (France); Sokatchev, E. [LAPTH, Université de Savoie, CNRS B.P. 110, F-74941 Annecy-le-Vieux Cedex (France) 2013-04-21 We develop a manifestly supersymmetric version of the generalized unitarity cut method for calculating scattering amplitudes in N=4 SYM theory. We illustrate the power of this method by computing the one-loop n-point NMHV super-amplitudes. The result confirms two conjectures which we made in Drummond, et al., [1]. Firstly, we derive the compact, manifestly dual superconformally covariant form of the NMHV tree amplitudes for arbitrary number and types of external particles. Secondly, we show that the ratio of the one-loop NMHV to the MHV amplitude is dual conformal invariant. 1. Tree-level amplitudes and dual superconformal symmetry Energy Technology Data Exchange (ETDEWEB) Drummond, J M, E-mail: drummond@lapp.in2p3.fr [PH-TH Division, CERN, CH-1211, Geneva 23 (Switzerland); LAPTH, Universite de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux Cedex (France) 2011-11-11 We review the structure of gauge theory scattering amplitudes at tree level and describe how a compact expression can be found which encodes all the tree-level amplitudes in the maximally supersymmetric N=4 theory. The expressions for the amplitudes reveal a dual superconformal symmetry. We describe how these ideas can be extended to leading singularities and the loop integrand in the planar theory and discuss the appearance of dual conformal symmetry in higher-dimensional gauge theories. This paper is an invited review for a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Scattering amplitudes in gauge theories'. (review) 2. New relations for Einstein–Yang–Mills amplitudes International Nuclear Information System (INIS) Stieberger, Stephan; Taylor, Tomasz R. 2016-01-01 We obtain new relations between Einstein–Yang–Mills (EYM) amplitudes involving N gauge bosons plus a single graviton and pure Yang–Mills amplitudes involving N gauge bosons plus one additional vector boson inserted in a way typical for a gauge boson of a “spectator” group commuting with the group associated to original N gauge bosons. We show that such EYM amplitudes satisfy U(1) decoupling relations similar to Kleiss–Kuijf relations for Yang–Mills amplitudes. We consider a D-brane embedding of EYM amplitudes in the framework of disk amplitudes involving open and closed strings. A new set of monodromy relations is derived for mixed open–closed amplitudes with one closed string inserted on the disk world-sheet and a number of open strings at the boundary. These relations allow expressing the latter in terms of pure open string amplitudes and, in the field-theory limit, they yield the U(1) decoupling relations for EYM amplitudes. 3. Wavelength tuning of porous silicon microcavities International Nuclear Information System (INIS) Mulders, J.; Reece, P.; Zheng, W.H.; Lerondel, G.; Sun, B.; Gal, M. 2002-01-01 Full text: In the last decade much attention has been given to porous silicon (PS) for optoelectronic applications, which include efficient room temperature light emission as well as microcavity formation. Due to the large specific surface area, the use of porous silicon microcavities (PSMs) has been proposed for chemical sensing. Large wavelength shifts have indicated that the optical properties of PSMs are indeed strongly dependent on the environment. In this paper, we report the shifting of the resonance frequency of high quality PSMs, with the aim of tuning a future PS device to a certain required wavelength. The PSM samples were prepared by anodically etching p + -doped (5mΩcm) bulk silicon wafer in a solution (25%) of aqueous HF and ethanol. The device structure consisted of a PS layer sandwiched between 2 stacks of thin PS layers with alternating high and low effective refractive indices (RI), i.e. distributed Bragg mirrors (DBM). The layer thickness depends on the etch time while the porosity and hence refractive index is determined by the current density as the Si is etched. The position and the width of the stop-band can be fully controlled by the design of the DBMs, with the microcavity resonance mode sitting within the stop-band. We achieved tuning of the microcavity resonance by a number of methods, including temperature dependent tuning. The temperature induced wavelength shift was found to be of the order of 10 -15 nm. Computer modeling of these changes in the reflectivity spectra allowed us to quantify the changes of the effective refractive index and the respective layer thicknesses 4. Large-amplitude mesospheric response to an orographic wave generated over the Southern Ocean Auckland Islands (50.7°S) during the DEEPWAVE project Science.gov (United States) Pautet, P.-D.; Taylor, M. J.; Fritts, D. C.; Bossert, K.; Williams, B. P.; Broutman, D.; Ma, J.; Eckermann, S. D.; Doyle, J. D. 2016-02-01 The Deep Propagating Gravity Wave Experiment (DEEPWAVE) project was conducted over New Zealand and the surrounding regions during June and July 2014, to more fully understand the generation, propagation, and effects of atmospheric gravity waves. A large suite of instruments collected data from the ground to the upper atmosphere (~100 km), with several new remote-sensing instruments operating on board the NSF Gulfstream V (GV) research aircraft, which was the central measurement platform of the project. On 14 July, during one of the research flights (research flight 23), a spectacular event was observed as the GV flew in the lee of the sub-Antarctic Auckland Islands (50.7°S). An apparent "ship wave" pattern was imaged in the OH layer (at ~83.5 km) by the Utah State University Advanced Mesospheric Temperature Mapper and evolved significantly over four successive passes spanning more than 4 h. The waves were associated with orographic forcing generated by relatively strong (15-20 m/s) near-surface wind flowing over the rugged island topography. The mountain wave had an amplitude T' ~ 10 K, a dominant horizontal wavelength ~40 km, achieved a momentum flux exceeding 300 m2 s-2, and eventually exhibited instability and breaking at the OH altitude. This case of deep mountain wave propagation demonstrates the potential for strong responses in the mesosphere arising from a small source under suitable propagation conditions and suggests that such cases may be more common than previously believed. 5. A radiation research apparatus sensitive to wavelength International Nuclear Information System (INIS) 1980-01-01 The apparatus described is equipped with a radiation source with a tuning device for the generation of X radiation of at least two different wavelength spectra. The detector with ionisation chamber is able to discriminate between these spectra. This is done with the aid of an auxillary electrode between the entrance window and a high voltage electrode. With a lower source of voltage this electrode has a potential equal to the high voltage electrode potential and with a higher voltage source it has a potential equal to the signal electrode potential. (Th.P.) 6. Wavelength-agnostic WDM-PON System DEFF Research Database (Denmark) Wagner, Christoph; Eiselt, Michael; Zou, S. 2016-01-01 on the standardization status of this lowcost system in the new ITU-T G.metro draft recommendation, in the context of autonomous tuning. We also discuss some low-effort implementations of the pilot-tone labels and investigate the impact of these labels on the transmission channels.......Next-generation WDM-PON solutions for metro and access systems will take advantage of remotely controlled wavelength-tunable ONUs to keep system costs as low as possible. For such a purpose, each ONU signal can be labeled by a pilot tone modulated onto the optical data stream. We report... 7. Sub-wavelength imaging at radio frequency International Nuclear Information System (INIS) Wiltshire, M C K; Pendry, J B; Hajnal, J V 2006-01-01 A slab of material with a negative permeability can act as a super-lens for magnetic fields and generate images with a sub-wavelength resolution. We have constructed an effective medium using a metamaterial with negative permeability in the region of 24 MHz, and used this to form images in free space of radio frequency magnetic sources. Measurements of these images show that a resolution of approximately λ/64 has been achieved, consistent with both analytical and numerical predictions. (letter to the editor) 8. Gold Photoluminescence: Wavelength and Polarization Engineering DEFF Research Database (Denmark) Andersen, Sebastian Kim Hjælm; Pors, Anders Lambertus; Bozhevolnyi, Sergey I. 2015-01-01 We demonstrate engineering of the spectral content and polarization of photoluminescence (PL) from arrayed gold nanoparticles atop a subwavelength-thin dielectric spacer and optically-thick gold film, a configuration that supports gap-surface plasmon resonances (GSPRs). Choice of shapes...... and dimensions of gold nanoparticles influences the GSPR wavelength and polarization characteristics, thereby allowing us to enhance and spectrally mold the plasmon-assisted PL while simultaneously controlling its polarization. In order to understand the underlying physics behind the plasmon-enhanced PL, we... 9. Dephasing due to Nuclear Spins in Large-Amplitude Electric Dipole Spin Resonance. Science.gov (United States) Chesi, Stefano; Yang, Li-Ping; Loss, Daniel 2016-02-12 We analyze effects of the hyperfine interaction on electric dipole spin resonance when the amplitude of the quantum-dot motion becomes comparable or larger than the quantum dot's size. Away from the well-known small-drive regime, the important role played by transverse nuclear fluctuations leads to a Gaussian decay with characteristic dependence on drive strength and detuning. A characterization of spin-flip gate fidelity, in the presence of such additional drive-dependent dephasing, shows that vanishingly small errors can still be achieved at sufficiently large amplitudes. Based on our theory, we analyze recent electric dipole spin resonance experiments relying on spin-orbit interactions or the slanting field of a micromagnet. We find that such experiments are already in a regime with significant effects of transverse nuclear fluctuations and the form of decay of the Rabi oscillations can be reproduced well by our theory. 10. Momentum space dipole amplitude for DIS and inclusive hadron production International Nuclear Information System (INIS) Basso, E. A.; Gay Ducati, M. B.; De Oliveira, E. G. 2013-01-01 We show how the AGBS model, originally developed for deep inelastic scattering applied to HERA data on the proton structure function, can also describe the RHIC data on single inclusive hadron yield for d+Au and p+p collisions through a new simultaneous fit. The single inclusive hadron production is modeled through the color glass condensate, which uses the quark(and gluon) condensate amplitudes in momentum space. The AGBS model is also a momentum space model based on the asymptotic solutions of the BK equation, although a different definition of the Fourier transform is used. This description entirely in transverse momentum of both processes arises for the first time. The small difference between the simultaneous fit and the one for HERA data alone suggests that the AGBS model describes very well both kind of processes and thus emerges as a good tool to investigate the inclusive hadron production data. We use this model for predictions at LHC energies, which agree quite well with available experimental data. 11. Momentum space dipole amplitude for DIS and inclusive hadron production Energy Technology Data Exchange (ETDEWEB) Basso, E. A.; Gay Ducati, M. B. [Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 - Porto Alegre, RS (Brazil); De Oliveira, E. G. [Instituto de Fisica, Universidade de Sao Paulo, Caixa Postal 66318, 05314-970 Sao Paulo, SP (Brazil) 2013-03-25 We show how the AGBS model, originally developed for deep inelastic scattering applied to HERA data on the proton structure function, can also describe the RHIC data on single inclusive hadron yield for d+Au and p+p collisions through a new simultaneous fit. The single inclusive hadron production is modeled through the color glass condensate, which uses the quark(and gluon) condensate amplitudes in momentum space. The AGBS model is also a momentum space model based on the asymptotic solutions of the BK equation, although a different definition of the Fourier transform is used. This description entirely in transverse momentum of both processes arises for the first time. The small difference between the simultaneous fit and the one for HERA data alone suggests that the AGBS model describes very well both kind of processes and thus emerges as a good tool to investigate the inclusive hadron production data. We use this model for predictions at LHC energies, which agree quite well with available experimental data. 12. Streaming vorticity flux from oscillating walls with finite amplitude Science.gov (United States) Wu, J. Z.; Wu, X. H.; Wu, J. M. 1993-01-01 How to describe vorticity creation from a moving wall is a long standing problem. This paper discusses relevant issues at the fundamental level. First, it is shown that the concept of 'vorticity flux due to wall acceleration' can be best understood by following fluid particles on the wall rather than observing the flow at fixed spatial points. This is of crucial importance when the time-averaged flux is to be considered. The averaged flux has to be estimated in a wall-fixed frame of reference (in which there is no flux due to wall acceleration at all); or, if an inertial frame of reference is used, the generalized Lagrangian mean (GLM) also gives the same result. Then, for some simple but typical configurations, the time-averaged vorticity flux from a harmonically oscillating wall with finite amplitude is analyzed, without appealing to small perturbation. The main conclusion is that the wall oscillation will produce an additional mean vorticity flux (a fully nonlinear streaming effect), which is partially responsible for the mechanism of vortex flow control by waves. The results provide qualitative explanation for some experimentally and/or computationally observed phenomena. 13. Wavelength converter placement in optical networks with dynamic traffic DEFF Research Database (Denmark) Buron, Jakob Due; Ruepp, Sarah Renée; Wessing, Henrik 2008-01-01 We evaluate the connection provisioning performance of GMPLS-controlled wavelength routed networks under dynamic traffic load and using three different wavelength converter placement heuristics. Results show that a simple uniform placement heuristic matches the performance of complex heuristics... 14. Smooth Pursuit Saccade Amplitude Modulation During Exposure to Microgravity Science.gov (United States) Reschke, M. F.; Kozlovskaya, I. B.; Sayenko, D. G.; Sayenko, I.; Somers, J. T.; Paloski, W. H. 2002-01-01 main sequence . This difference in the regression slopes between flight phase, head/eye condition (EO or EH), and pursuit target frequency was observed across all subjects (statistically significant at the ptendency (not significant) for both saccade amplitude and peak velocity to increase during the postflight testing. This tendency had vanished by R+ 1. Of particular interest was the redistribution of saccades during the latter stages of the flight and immediately postflight in the EO condition. At the 1.0 Hz frequency the saccades tended to be clustered near the lowest target velocity. It was also interesting to note that gaze performance (eye in skull + head in space) was consistently better during the EH condition; a finding also observed by our Russian colleagues. As the results of the long duration flight become available we expect that they will not only show that postflight effects will be similar to those observed during the short duration flights, but will also last for a greater period of time following flight. It is not clear what mechanism is responsible for the decreased peak saccadic velocity during flight unless the change is related to the control of retinal slip. For example, it is possible that saccades will tend to initially undershoot their targets by a small percentage and these saccades are then followed, if vision is available, by a small augmenting corrective saccade. It has been postulated that the functional significance of this undershooting tendency is to maintain the spatial representation of the target on the same side of the fovea (as opposedo racing across the fovea) and hence in the same cerebral hemisphere that initiated the primary saccade thus minimizing delays caused by an intra-hemispheric transfer of information . One could also speculate that with saccade velocities greater than normal, additional corrective saccades would be required to bring the target back on the fovea. A less plausible explanation of our findings could be 15. Ultrasonic Measurement of Interfacial Layer Thickness of Sub-Quarter-Wavelength Energy Technology Data Exchange (ETDEWEB) Kim, No Hyu; Lee, Sang Soon [Korea University of Technology and Education, Cheonan (Korea, Republic of) 2003-12-15 This paper describes a new technique for thickness measurement of a very thin layer less than one-quarter of the wavelength of ultrasonic wave used in the ultrasonic pulse-echo measurements. The technique determines the thickness of a thin layer in a tapered medium from constructive interference of multiple reflection waves. The interference characteristics are derived and investigated in theoretical and experimental approaches. Modified total reflection wave g(t) defined as difference between total and first reflection waves increases in amplitude as the interfacial layer thickness decreases down to zero. A layer thickness less than one-tenth of the ultrasonic wavelength is measured using the maximum amplitude of g(t) with a good accuracy and sensitivity. The method also requires no inversion process to extract the thickness information from the waveforms of reflected waves, so that it makes possible to have the on-line thickness measurement of a thin layer such as a lubricating oil film in thrust bearings and journal bearings during manufacturing process 16. Ultrasonic Measurement of Interfacial Layer Thickness of Sub-Quarter-Wavelength International Nuclear Information System (INIS) Kim, No Hyu; Lee, Sang Soon 2003-01-01 This paper describes a new technique for thickness measurement of a very thin layer less than one-quarter of the wavelength of ultrasonic wave used in the ultrasonic pulse-echo measurements. The technique determines the thickness of a thin layer in a tapered medium from constructive interference of multiple reflection waves. The interference characteristics are derived and investigated in theoretical and experimental approaches. Modified total reflection wave g(t) defined as difference between total and first reflection waves increases in amplitude as the interfacial layer thickness decreases down to zero. A layer thickness less than one-tenth of the ultrasonic wavelength is measured using the maximum amplitude of g(t) with a good accuracy and sensitivity. The method also requires no inversion process to extract the thickness information from the waveforms of reflected waves, so that it makes possible to have the on-line thickness measurement of a thin layer such as a lubricating oil film in thrust bearings and journal bearings during manufacturing process 17. Source-Space Cross-Frequency Amplitude-Amplitude Coupling in Tinnitus Directory of Open Access Journals (Sweden) Oliver Zobay 2015-01-01 Full Text Available The thalamocortical dysrhythmia (TCD model has been influential in the development of theoretical explanations for the neurological mechanisms of tinnitus. It asserts that thalamocortical oscillations lock a region in the auditory cortex into an ectopic slow-wave theta rhythm (4–8 Hz. The cortical area surrounding this region is hypothesized to generate abnormal gamma (>30 Hz oscillations (“edge effect” giving rise to the tinnitus percept. Consequently, the model predicts enhanced cross-frequency coherence in a broad range between theta and gamma. In this magnetoencephalography study involving tinnitus and control cohorts, we investigated this prediction. Using beamforming, cross-frequency amplitude-amplitude coupling (AAC was computed within the auditory cortices for frequencies (f1,f2 between 2 and 80 Hz. We find the AAC signal to decompose into two distinct components at low (f1,f230 Hz frequencies, respectively. Studying the correlation of AAC with several key covariates (age, hearing level (HL, tinnitus handicap and duration, and HL at tinnitus frequency, we observe a statistically significant association between age and low-frequency AAC. Contrary to the TCD predictions, however, we do not find any indication of statistical differences in AAC between tinnitus and controls and thus no evidence for the predicted enhancement of cross-frequency coupling in tinnitus. 18. Passively synchronized dual-wavelength Q-switched lasers DEFF Research Database (Denmark) Janousek, Jiri; Tidemand-Lichtenberg, Peter; Mortensen, Jesper Liltorp We present a simple and efficient way of generating synchronized Q-switched pulses at wavelengths hundreds of nanometers apart. This principle can result in new pulsed all-solid-state light sources at new wavelengths based on SFG.......We present a simple and efficient way of generating synchronized Q-switched pulses at wavelengths hundreds of nanometers apart. This principle can result in new pulsed all-solid-state light sources at new wavelengths based on SFG.... 19. True amplitude wave equation migration arising from true amplitude one-way wave equations Science.gov (United States) Zhang, Yu; Zhang, Guanquan; Bleistein, Norman 2003-10-01 One-way wave operators are powerful tools for use in forward modelling and inversion. Their implementation, however, involves introduction of the square root of an operator as a pseudo-differential operator. Furthermore, a simple factoring of the wave operator produces one-way wave equations that yield the same travel times as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for almost all points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the travel time satisfy the same differential equations as the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse spatial variables. Here, we address the fully heterogeneous case. Singling out depth as the preferred direction of propagation, we introduce a representation of the square-root operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows us to carry out explicit asymptotic analysis of the resulting one-way wave equations. To do this, we introduce an auxiliary function that satisfies a lower dimensional wave equation in transverse spatial variables only. We prove that ray theory for these one-way wave equations leads to one-way eikonal equations and the correct leading order transport equation for the full wave equation. We then introduce appropriate boundary conditions at z = 0 to generate waves at depth whose quotient leads to a reflector map and an estimate of the ray theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a 'true amplitude wave equation migration' (WEM) method. In fact, we prove that applying the WEM imaging condition 20. Meter-wavelength VLBI. III. Pulsars International Nuclear Information System (INIS) Vandenberg, N.R.; Clark, T.A.; Clark, W.C.; Erickson, W.C.; Resch, G.M.; Broderick, J.J. 1976-01-01 The results and analysis of observations of pulsars, especially the Crab Nebula pulsar, taken during a series of meter-wavelength very long baseline interferometry (VLBI) experiments are discussed. Based on a crude 144 MHz visibility curve which is consistent with a Gaussian brightness distribution, the measured visibilities at 196, 111, and 74 MHz were interpreted to yield apparent angular diameters (at half-power) of 0 .03 +- 0 .01, 0 .07 +- 0 .01, and 0 .18 +- 0 .01, respectively. These sizes scale approximately as wavelength-squared, and the 74 MHz size agrees with recent observations using interplanetary scintillation techniques.The VLBI-measured total flux densities lie on the extrapolation from higher frequencies of the pulsing flux densities. Variations in the total flux density up to 25 percent were observed. A lack of fine structure other than the pulsar in the nebula is indicated by our simple visibility curves. The pulse shapes observed with the interferometer are similar to single-dish measurements at 196 MHz but reveal a steady, nonpulsing component at 111 MHz. The ratio of pulsing to total power was approximately equal to one-half but varied with time. No pulsing power was detected at 74 MHz. It was found that four strong, low-dispersion pulsars were only slightly resolved 1. Long-wavelength microinstabilities in toroidal plasmas International Nuclear Information System (INIS) Tang, W.W.; Rewoldt, G. 1993-01-01 Realistic kinetic toroidal eigenmode calculations have been carried out to support a proper assessment of the influence of long-wavelength microturbulence on transport in tokamak plasmas. In order to efficiently evaluate large-scale kinetic behavior extending over many rational surfaces, significant improvements have been made to a toroidal finite element code used to analyze the fully two-dimensional (r,θ) mode structures of trapped-ion and toroidal ion temperature gradient (ITG) instabilities. It is found that even at very long wavelengths, these eigenmodes exhibit a strong ballooning character with the associated radial structure relatively insensitive to ion Landau damping at the rational surfaces. In contrast to the long-accepted picture that the radial extent of trapped-ion instabilities is characterized by the ion-gyroradius-scale associated with strong localization between adjacent rational surfaces, present results demonstrate that under realistic conditions, the actual scale is governed by the large-scale variations in the equilibrium gradients. Applications to recent measurements of fluctuation properties in TFTR L-mode plasmas indicate that the theoretical trends appear consistent with spectral characteristics as well as rough heuristic estimates of the transport level. Benchmarking calculations in support of the development of a three-dimensional toroidal gyrokinetic code indicate reasonable agreement with respect to both the properties of the eigenfunctions and the magnitude of the eigenvalues during the linear phase of the simulations of toroidal ITG instabilities 2. Dye mixtures for ultrafast wavelength shifters Energy Technology Data Exchange (ETDEWEB) Gangopadhyay, S.; Liu, L.; Palsule, C.; Borst, W.; Wigmans, R. [Texas Tech Univ., Lubbock, TX (United States). Dept. of Physics; Barashkov, N. [Karpov Inst. of Physical Chemistry, Moscow (Russian Federation) 1994-12-31 Particle detectors based on scintillation processes have been used since the discovery of radium about 100 years ago. The fast signals that can be obtained with these detectors, although often considered a nice asset, were rarely essential for the success of experiments. However, the new generation of high energy particle accelerators require particle detectors with fast response time. The authors have produced fast wavelength shifters using mixtures of various Coumarin dyes with DCM in epoxy-polymers (DGEBA+HHPA) and measured the properties of these wavelength shifters. The particular mixtures were chosen because there is a substantial overlap between the emission spectrum of Coumarin and the absorption spectrum of DCM. The continuous wave and time-resolved fluorescence spectra have been studied as a function of component concentration to optimize the decay times, emission peaks and quantum yields. The mean decay times of these mixtures are in the range of 2.5--4.5 ns. The mean decay time increases with an increase in Coumarin concentration at a fixed DCM concentration or with a decrease in DCM concentration at a fixed Coumarin concentration. This indicates that the energy transfer is radiative at lower relative DCM concentrations and becomes non-radiative at higher DCM concentrations. 3. Dye mixtures for ultrafast wavelength shifters International Nuclear Information System (INIS) Gangopadhyay, S.; Liu, L.; Palsule, C.; Borst, W.; Wigmans, R. 1994-01-01 Particle detectors based on scintillation processes have been used since the discovery of radium about 100 years ago. The fast signals that can be obtained with these detectors, although often considered a nice asset, were rarely essential for the success of experiments. However, the new generation of high energy particle accelerators require particle detectors with fast response time. The authors have produced fast wavelength shifters using mixtures of various Coumarin dyes with DCM in epoxy-polymers (DGEBA+HHPA) and measured the properties of these wavelength shifters. The particular mixtures were chosen because there is a substantial overlap between the emission spectrum of Coumarin and the absorption spectrum of DCM. The continuous wave and time-resolved fluorescence spectra have been studied as a function of component concentration to optimize the decay times, emission peaks and quantum yields. The mean decay times of these mixtures are in the range of 2.5--4.5 ns. The mean decay time increases with an increase in Coumarin concentration at a fixed DCM concentration or with a decrease in DCM concentration at a fixed Coumarin concentration. This indicates that the energy transfer is radiative at lower relative DCM concentrations and becomes non-radiative at higher DCM concentrations 4. WDM cross-connect cascade based on all-optical wavelength converters for routing and wavelength slot interchanging using a reduced number of internal wavelengths DEFF Research Database (Denmark) Pedersen, Rune Johan Skullerud; Mikkelsen, Benny; Jørgensen, Bo Foged 1998-01-01 interchanging can be used to create a robust and nonblocking OXC. However, for an OXC with n fiber inlets each carrying m wavelengths the OXC requires n×m internal wavelengths, which constrains the size of the cross-connect. In this paper we therefore propose and demonstrate an architecture that uses a reduced......Optical transport layers need rearrangeable wavelength-division multiplexing optical cross-connects (OXCs) to increase the capacity and flexibility of the network. It has previously been shown that a cross-connect based on all-optical wavelength converters for routing as well as wavelength slot...... set of internal wavelengths without sacrificing cross-connecting capabilities. By inserting a partly equipped OXC with the new architecture in a 10-Gbit/s re-circulating loop setup we demonstrate the possibility of cascading up to ten OXCs. Furthermore, we investigate the regenerating effect... 5. Behavior of the hadron potential at large distances and properties of the hadron spin-flip amplitude International Nuclear Information System (INIS) Predazzi, E.; Selyugin, O.V. 2002-01-01 The impact of the form of the hadron potential at large distances on the behavior of the hadron spin-flip amplitude at small angles is examined. The t-dependence of the spin-flip amplitude of high-energy hadron elastic scattering is analyzed under different assumptions on the hadron interaction. It is shown that the long tail of the nonGaussian form of the hadron potential of the hadron interaction in the impact parameter representation leads to a large value of the slope of the spin-flip amplitude (without the kinematical factor √(vertical stroke t vertical stroke)) as compared with the slope of the spin-nonflip amplitude. This effect can explain the form of the differential cross-section and the analyzing power at small transfer momenta. The methods for the definition of the spin-dependent part of the hadron scattering amplitude are presented. A possibility to investigate the structure of the hadron spin-flip amplitude from the accurate measure of the differential cross-section and the spin correlation parameters is shown. (orig.) 6. Upper Extremity Freezing and Dyscoordination in Parkinson’s Disease: Effects of Amplitude and Cadence Manipulations Directory of Open Access Journals (Sweden) April J. Williams 2013-01-01 Full Text Available Purpose. Motor freezing, the inability to produce effective movement, is associated with decreasing amplitude, hastening of movement, and poor coordination. We investigated how manipulations of movement amplitude and cadence affect upper extremity (UE coordination as measured by the phase coordination index (PCI—only previously measured in gait—and freezing of the upper extremity (FO-UE in people with Parkinson's disease (PD who experience freezing of gait (PD + FOG, do not experience FOG (PD-FOG, and healthy controls. Methods. Twenty-seven participants with PD and 18 healthy older adults made alternating bimanual movements between targets under four conditions: Baseline; Fast; Small; SmallFast. Kinematic data were recorded and analyzed for PCI and FO-UE events. PCI and FO-UE were compared across groups and conditions. Correlations between UE PCI, gait PCI, FO-UE, and Freezing of Gait Questionnaire (FOG-Q were determined. Results. PD + FOG had poorer coordination than healthy old during SmallFast. UE coordination correlated with number of FO-UE episodes in two conditions and FOG-Q score in one. No differences existed between PD−/+FOG in coordination or number of FO-UE episodes. Conclusions. Dyscoordination and FO-UE can be elicited by manipulating cadence and amplitude of an alternating bimanual task. It remains unclear whether FO-UE and FOG share common mechanisms. 7. Double logarithmic asymptotics of quark amplitudes with flavour exchange International Nuclear Information System (INIS) Kirschner, R. 1982-01-01 Results on the quark scattering and annihilation amplitudes in the Regge region are presented. The perturbative contribution to those amplitudes in the double logarithmic approximation are calculated. In the calculations a method based on dispersion relations and gauge invariance is used. (M.F.W.) 8. Multi-loop string amplitudes and Riemann surfaces International Nuclear Information System (INIS) Taylor, J.G. 1986-01-01 The paper was presented at the workshop on 'Supersymmetry and its applications', Cambridge, United Kingdom, 1985. Super-string theory is discussed under the following topic headings: the functional approach to the string amplitude, Rieman surfaces, the determinants Δsub(epsilon)(1) and Δsub(epsilon)(2), Green's functions, total amplitude, and divergence analysis. (U.K.) 9. Damping and Frequency Shift of Large Amplitude Electron Plasma Waves DEFF Research Database (Denmark) Thomsen, Kenneth; Juul Rasmussen, Jens 1983-01-01 The initial evolution of large-amplitude one-dimensional electron waves is investigated by applying a numerical simulation. The initial wave damping is found to be strongly enhanced relative to the linear damping and it increases with increasing amplitude. The temporal evolution of the nonlinear... 10. Singularities of elastic scattering amplitude by long-range potentials International Nuclear Information System (INIS) Kvitsinsky, A.A.; Komarov, I.V.; Merkuriev, S.P. 1982-01-01 The angular peculiarities and the zero energy singularities of the elastic scattering amplitude by a long-range potential are described. The singularities of the elastic (2 → 2) scattering amplitude for a system of three Coulomb particles are considered [ru 11. Properties of the scattering amplitude for electron-atom collisions International Nuclear Information System (INIS) Combes, J.M.; Tip, A. 1983-02-01 For the scattering of an electron by an atom finiteness of the amplitude at non threshold energies is proved in the framework of the N-body Schroedinger equation. It is also shown that both the direct and exchange amplitudes have analytic continuations for complex values of incident momentum, with pole or cut singularities on the imaginary axis 12. Miracles in Scattering Amplitudes: from QCD to Gravity Energy Technology Data Exchange (ETDEWEB) Volovich, Anastasia [Brown Univ., Providence, RI (United States) 2016-10-09 The goal of my research project "Miracles in Scattering Amplitudes: from QCD to Gravity" involves deepening our understanding of gauge and gravity theories by exploring hidden structures in scattering amplitudes and using these rich structures as much as possible to aid practical calculations. 13. Weak decay amplitudes in large N/sub c/ QCD International Nuclear Information System (INIS) Bardeen, W.A. 1988-10-01 A systematic analysis of nonleptonic decay amplitudes is presented using the large N/sub c/ expansion of quantum chromodynamics. In the K-meson system, this analysis is applied to the calculation of the weak decay amplitudes, weak mixing and CP violation. 10 refs., 5 figs., 2 tabs 14. Abnormal Selective Attention Normalizes P3 Amplitudes in PDD Science.gov (United States) Hoeksma, Marco R.; Kemner, Chantal; Kenemans, J. Leon; van Engeland, Herman 2006-01-01 This paper studied whether abnormal P3 amplitudes in PDD are a corollary of abnormalities in ERP components related to selective attention in visual and auditory tasks. Furthermore, this study sought to clarify possible age differences in such abnormalities. Children with PDD showed smaller P3 amplitudes than controls, but no abnormalities in… 15. Multiphoton states and amplitude k-th power squeezing International Nuclear Information System (INIS) Buzek, V.; Jex, I. 1991-01-01 On the basis of the work of d'Ariano and coworkers a new type of multiphoton states is introduced. Amplitude k-th power squeezing of the multiphoton states are analysed. In particular, it is shown that even if the multiphoton states do not exhibit ordinary squeezing they can be amplitude k-th power squeezed 16. Automated force controller for amplitude modulation atomic force microscopy Energy Technology Data Exchange (ETDEWEB) Miyagi, Atsushi, E-mail: atsushi.miyagi@inserm.fr, E-mail: simon.scheuring@inserm.fr; Scheuring, Simon, E-mail: atsushi.miyagi@inserm.fr, E-mail: simon.scheuring@inserm.fr [U1006 INSERM, Université Aix-Marseille, Parc Scientifique et Technologique de Luminy, 163 Avenue de Luminy, 13009 Marseille (France) 2016-05-15 Atomic Force Microscopy (AFM) is widely used in physics, chemistry, and biology to analyze the topography of a sample at nanometer resolution. Controlling precisely the force applied by the AFM tip to the sample is a prerequisite for faithful and reproducible imaging. In amplitude modulation (oscillating) mode AFM, the applied force depends on the free and the setpoint amplitudes of the cantilever oscillation. Therefore, for keeping the applied force constant, not only the setpoint amplitude but also the free amplitude must be kept constant. While the AFM user defines the setpoint amplitude, the free amplitude is typically subject to uncontrollable drift, and hence, unfortunately, the real applied force is permanently drifting during an experiment. This is particularly harmful in biological sciences where increased force destroys the soft biological matter. Here, we have developed a strategy and an electronic circuit that analyzes permanently the free amplitude of oscillation and readjusts the excitation to maintain the free amplitude constant. As a consequence, the real applied force is permanently and automatically controlled with picoNewton precision. With this circuit associated to a high-speed AFM, we illustrate the power of the development through imaging over long-duration and at various forces. The development is applicable for all AFMs and will widen the applicability of AFM to a larger range of samples and to a larger range of (non-specialist) users. Furthermore, from controlled force imaging experiments, the interaction strength between biomolecules can be analyzed. 17. Numerical evaluation of one-loop QCD amplitudes DEFF Research Database (Denmark) Badger, Simon David; Biedermann, Benedikt; Uwer, Peter 2012-01-01 We present the publicly available program NGluon allowing the numerical evaluation of primitive amplitudes at one-loop order in massless QCD. The program allows the computation of one-loop amplitudes for an arbitrary number of gluons. The focus of the present article is the extension to one-loop ... 18. Conformal higher spin scattering amplitudes from twistor space Energy Technology Data Exchange (ETDEWEB) Adamo, Tim [Blackett Laboratory, Imperial College, London, SW7 2AZ (United Kingdom); Hähnel, Philipp; McLoughlin, Tristan [School of Mathematics, Trinity College Dublin, College Green, Dublin 2 (Ireland) 2017-04-04 We use the formulation of conformal higher spin (CHS) theories in twistor space to study their tree-level scattering amplitudes, finding expressions for all three-point (MHV)-bar amplitudes and all MHV amplitudes involving positive helicity conformal gravity particles and two negative helicity higher spins. This provides the on-shell analogue for the covariant coupling of CHS fields to a conformal gravity background. We discuss the restriction of the theory to a ghost-free unitary subsector, analogous to restricting conformal gravity to general relativity with a cosmological constant. We study the flat-space limit and show that the restricted amplitudes vanish, supporting the conjecture that in the unitary sector the S-matrix of CHS theories is trivial. However, by appropriately rescaling the amplitudes we find non-vanishing results which we compare with chiral flat-space higher spin theories. 19. Simplicity in the structure of QED and gravity amplitudes Energy Technology Data Exchange (ETDEWEB) Badger, Simon [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Bjerrum-Bohr, N.E.J. [Institute for Advanced Study, Princeton, NJ (United States). School of Natural Sciences; Vanhove, Pierre [Institut des Hautes Etudes Scientifiques IHES, Bures sur Yvette (France); CEA, IPhT, CNRS, URA, Gif-sur-Yvette, (France). Inst. de Physique Theorique 2008-11-15 We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED 'no-triangle' property is true for all helicity configurations and has similarities to the 'notriangle' property found in the case of maximal N=8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude. (orig.) 20. Simplicity in the structure of QED and gravity amplitudes International Nuclear Information System (INIS) Badger, Simon; Bjerrum-Bohr, N.E.J.; Vanhove, Pierre; CEA, IPhT, CNRS, URA, Gif-sur-Yvette, 2008-11-01 We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED 'no-triangle' property is true for all helicity configurations and has similarities to the 'notriangle' property found in the case of maximal N=8 supergravity. Results are derived both via a world-line formalism as well as using on-shell unitarity methods. We show that the simple structure of the loop amplitude originates from the extremely good BCFW scaling behaviour of the QED tree-amplitude. (orig.) 1. High Frequency Amplitude Detector for GMI Magnetic Sensors Directory of Open Access Journals (Sweden) Aktham Asfour 2014-12-01 Full Text Available A new concept of a high-frequency amplitude detector and demodulator for Giant-Magneto-Impedance (GMI sensors is presented. This concept combines a half wave rectifier, with outstanding capabilities and high speed, and a feedback approach that ensures the amplitude detection with easily adjustable gain. The developed detector is capable of measuring high-frequency and very low amplitude signals without the use of diode-based active rectifiers or analog multipliers. The performances of this detector are addressed throughout the paper. The full circuitry of the design is given, together with a comprehensive theoretical study of the concept and experimental validation. The detector has been used for the amplitude measurement of both single frequency and pulsed signals and for the demodulation of amplitude-modulated signals. It has also been successfully integrated in a GMI sensor prototype. Magnetic field and electrical current measurements in open- and closed-loop of this sensor have also been conducted. 2. Conformal higher spin scattering amplitudes from twistor space International Nuclear Information System (INIS) Adamo, Tim; Hähnel, Philipp; McLoughlin, Tristan 2017-01-01 We use the formulation of conformal higher spin (CHS) theories in twistor space to study their tree-level scattering amplitudes, finding expressions for all three-point (MHV)-bar amplitudes and all MHV amplitudes involving positive helicity conformal gravity particles and two negative helicity higher spins. This provides the on-shell analogue for the covariant coupling of CHS fields to a conformal gravity background. We discuss the restriction of the theory to a ghost-free unitary subsector, analogous to restricting conformal gravity to general relativity with a cosmological constant. We study the flat-space limit and show that the restricted amplitudes vanish, supporting the conjecture that in the unitary sector the S-matrix of CHS theories is trivial. However, by appropriately rescaling the amplitudes we find non-vanishing results which we compare with chiral flat-space higher spin theories. 3. Development of a two-wavelength IR laser absorption diagnostic for propene and ethylene Science.gov (United States) Parise, T. C.; Davidson, D. F.; Hanson, R. K. 2018-05-01 A two-wavelength infrared laser absorption diagnostic for non-intrusive, simultaneous quantitative measurement of propene and ethylene was developed. To this end, measurements of absorption cross sections of propene and potential interfering species at 10.958 µm were acquired at high-temperatures. When used in conjunction with existing absorption cross-section measurements of ethylene and other species at 10.532 µm, a two-wavelength diagnostic was developed to simultaneously measure propene and ethylene, the two small alkenes found to generally dominate the final decomposition products of many fuel hydrocarbon pyrolysis systems. Measurements of these two species is demonstrated using this two-wavelength diagnostic scheme for propene decomposition between 1360 and 1710 K. 4. Multiple scattering wavelength dependent backscattering of kaolin dust in the IR: Measurements and theory Science.gov (United States) Ben-David, Avishai 1992-01-01 Knowing the optical properties of aerosol dust is important for designing electro-optical systems and for modeling the effect on propagation of light in the atmosphere. As CO2 lidar technology becomes more advanced and is used for multiwavelength measurements, information on the wavelength dependent backscattering of aerosol dust particles is required. The volume backscattering coefficient of aerosols in the IR is relatively small. Thus, only a few field measurements of backscattering, usually at only a few wavelengths, are reported in the literature. We present spectral field measurements of backscattering of kaolin dust in the 9-11 micron wavelength range. As the quantity of dust increases, multiple scattering contributes more to the measured backscattered signal. The measurements show the effect of the dust quantity of the spectral backscatter measurements. A simple analytical two stream radiative transfer model is applied to confirm the measurements and to give insight to the multiple scattering spectra of backscattering. 5. Parallel implementation of the PHOENIX generalized stellar atmosphere program. II. Wavelength parallelization International Nuclear Information System (INIS) Baron, E.; Hauschildt, Peter H. 1998-01-01 We describe an important addition to the parallel implementation of our generalized nonlocal thermodynamic equilibrium (NLTE) stellar atmosphere and radiative transfer computer program PHOENIX. In a previous paper in this series we described data and task parallel algorithms we have developed for radiative transfer, spectral line opacity, and NLTE opacity and rate calculations. These algorithms divided the work spatially or by spectral lines, that is, distributing the radial zones, individual spectral lines, or characteristic rays among different processors and employ, in addition, task parallelism for logically independent functions (such as atomic and molecular line opacities). For finite, monotonic velocity fields, the radiative transfer equation is an initial value problem in wavelength, and hence each wavelength point depends upon the previous one. However, for sophisticated NLTE models of both static and moving atmospheres needed to accurately describe, e.g., novae and supernovae, the number of wavelength points is very large (200,000 - 300,000) and hence parallelization over wavelength can lead both to considerable speedup in calculation time and the ability to make use of the aggregate memory available on massively parallel supercomputers. Here, we describe an implementation of a pipelined design for the wavelength parallelization of PHOENIX, where the necessary data from the processor working on a previous wavelength point is sent to the processor working on the succeeding wavelength point as soon as it is known. Our implementation uses a MIMD design based on a relatively small number of standard message passing interface (MPI) library calls and is fully portable between serial and parallel computers. copyright 1998 The American Astronomical Society 6. Dynamic Sensor Interrogation Using Wavelength-Swept Laser with a Polygon-Scanner-Based Wavelength Filter Science.gov (United States) Kwon, Yong Seok; Ko, Myeong Ock; Jung, Mi Sun; Park, Ik Gon; Kim, Namje; Han, Sang-Pil; Ryu, Han-Cheol; Park, Kyung Hyun; Jeon, Min Yong 2013-01-01 We report a high-speed (∼2 kHz) dynamic multiplexed fiber Bragg grating (FBG) sensor interrogation using a wavelength-swept laser (WSL) with a polygon-scanner-based wavelength filter. The scanning frequency of the WSL is 18 kHz, and the 10 dB scanning bandwidth is more than 90 nm around a center wavelength of 1,540 nm. The output from the WSL is coupled into the multiplexed FBG array, which consists of five FBGs. The reflected Bragg wavelengths of the FBGs are 1,532.02 nm, 1,537.84 nm, 1,543.48 nm, 1,547.98 nm, and 1,553.06 nm, respectively. A dynamic periodic strain ranging from 500 Hz to 2 kHz is applied to one of the multiplexed FBGs, which is fixed on the stage of the piezoelectric transducer stack. Good dynamic performance of the FBGs and recording of their fast Fourier transform spectra have been successfully achieved with a measuring speed of 18 kHz. The signal-to-noise ratio and the bandwidth over the whole frequency span are determined to be more than 30 dB and around 10 Hz, respectively. We successfully obtained a real-time measurement of the abrupt change of the periodic strain. The dynamic FBG sensor interrogation system can be read out with a WSL for high-speed and high-sensitivity real-time measurement. PMID:23899934 7. Temperature Characteristics of Monolithically Integrated Wavelength-Selectable Light Sources International Nuclear Information System (INIS) Han Liang-Shun; Zhu Hong-Liang; Zhang Can; Ma Li; Liang Song; Wang Wei 2013-01-01 The temperature characteristics of monolithically integrated wavelength-selectable light sources are experimentally investigated. The wavelength-selectable light sources consist of four distributed feedback (DFB) lasers, a multimode interferometer coupler, and a semiconductor optical amplifier. The oscillating wavelength of the DFB laser could be modulated by adjusting the device operating temperature. A wavelength range covering over 8.0nm is obtained with stable single-mode operation by selecting the appropriate laser and chip temperature. The thermal crosstalk caused by the lateral heat spreading between lasers operating simultaneously is evaluated by oscillating-wavelength shift. The thermal crosstalk approximately decreases exponentially as the increasing distance between lasers 8. Analysis of subsystems in wavelength-division-multiplexing networks DEFF Research Database (Denmark) Liu, Fenghai 2001-01-01 Wavelength division multiplexing (WDM) technology together with optical amplification has created a new era for optical communication. Transmission capacity is greatly increased by adding more and more wavelength channels into a single fiber, as well as by increasing the line rate of each channel...... in semiconductor optical amplifiers (SOAs), and dispersion managed fiber sections. New subsystems are also proposed in the thesis: a modular 2×2 multiwavelength cross-connect using wavelength switching blocks, a wavelength converter based on cross phase modulation in a semiconductor modulator, a wavelength... 9. Multi-wavelength time-coincident optical communications system and methods thereof Science.gov (United States) Lekki, John (Inventor); Nguyen, Quang-Viet (Inventor) 2009-01-01 An optical communications transmitter includes a oscillator source, producing a clock signal, a data source, producing a data signal, a modulating circuit for modulating the clock signal using the data signal to produce modulating signals, optical drivers, receiving the modulating signals and producing optical driving signals based on the modulating signals and optical emitters, producing small numbers of photons based on the optical driving signals. The small numbers of photons are time-correlated between at least two separate optical transmission wavelengths and quantum states and the small number of photons can be detected by a receiver to reform the data signal. 10. Underdense radiation sources: Moving towards longer wavelengths Energy Technology Data Exchange (ETDEWEB) Back, C.A.; Kilkenny, J.D. [General Atomics, San Diego, California (United States); Seely, J.F.; Weaver, J.L. [Naval Research Laboratory, Washington, DC (United States); Feldman, U. [Artep Inc., Ellicott City, MD (United States); Tommasini, R.; Glendinning, S.G.; Chung, H.K.; Rosen, M.; Lee, R.W.; Scott, H.A. [Lawrence Livermore National Laboratory, California (United States); Tillack, M. [U. C. San Diego, La Jolla, CA (United States) 2006-06-15 Underdense radiation sources have been developed to provide efficient laboratory multi-keV radiation sources for radiography and radiation hardening studies. In these plasmas laser absorption by inverse Bremsstrahlung leads to high x-ray conversion efficiency because of efficient ionization of the low density aerogel or gas targets. Now we performing experiments in the soft x-ray energy regime where the atomic physics models are much more complicated. In recent experiments at the NIKE laser, we have irradiated a Ti-doped SiO{sub 2} aerogel with up to 1650 J of 248 nm wavelength light. The absolute Ti L-shell emission in the 200-800 eV range is measured with a diagnostic that uses a transmission grating coupled to Si photodiodes. We will give an overview of the temporally-resolved absolutely calibrated spectra obtained over a range of conditions. (authors) 11. Underdense radiation sources: Moving towards longer wavelengths International Nuclear Information System (INIS) Back, C.A.; Kilkenny, J.D.; Seely, J.F.; Weaver, J.L.; Feldman, U.; Tommasini, R.; Glendinning, S.G.; Chung, H.K.; Rosen, M.; Lee, R.W.; Scott, H.A.; Tillack, M. 2006-01-01 Underdense radiation sources have been developed to provide efficient laboratory multi-keV radiation sources for radiography and radiation hardening studies. In these plasmas laser absorption by inverse Bremsstrahlung leads to high x-ray conversion efficiency because of efficient ionization of the low density aerogel or gas targets. Now we performing experiments in the soft x-ray energy regime where the atomic physics models are much more complicated. In recent experiments at the NIKE laser, we have irradiated a Ti-doped SiO 2 aerogel with up to 1650 J of 248 nm wavelength light. The absolute Ti L-shell emission in the 200-800 eV range is measured with a diagnostic that uses a transmission grating coupled to Si photodiodes. We will give an overview of the temporally-resolved absolutely calibrated spectra obtained over a range of conditions. (authors) 12. Aligning of single and multiple wavelength chromatographic DEFF Research Database (Denmark) Nielsen, Niels-Peter Vest; Carstensen, Jens Michael; Smedsgaard, Jørn 1998-01-01 optimised warping (COW) using two input parameters which can be estimated from the observed peak width. COW is demonstrated on constructed single trace chromatograms and on single and multiple wavelength chromatograms obtained from HPLC diode detection analyses of fungal extractsA copy of the C program......The use of chemometric data processing is becoming an important part of modern chromatography. Most chemometric analyses are performed on reduced data sets using areas of selected peaks detected in the chromatograms, which means a loss of data and introduces the problem of extracting peak data from...... to utilise the entire data matrix or rely on peak detection, thus having the same limitations as the commonly used chemometric procedures. The method presented uses the entire chromatographic data matrices and does not require any preprocessing e.g., peak detection. It relies on piecewise linear correlation... 13. Single wavelength standard wiggler for PEP International Nuclear Information System (INIS) Brunk, W.; Fischer, G.; Spencer, J. 1979-03-01 A 1lambda planar wiggler has been designed that will be used for the initial operation of the 4 to 18 GeV storage ring PEP. Three of these wigglers will be installed symmetrically around the ring at 120 0 intervals in three of six available 5 m straight sections with the purpose of providing: (1) beam size control to obtain better luminosities below 15 GeV, and (2) decreased damping times to obtain better injection rates at lower energies. Design goals are discussed and a description of the final system including cost estimates is given. Expected results and usage in PEP are discussed. Some possibilities for production of synchrotron radiation and beam monitoring with shorter wavelength, multiple-period wigglers at PEP energies are also discussed. Comparison to a wiggler now operating in SPEAR is given 14. Dual-wavelength external cavity laser device for fluorescence suppression in Raman spectroscopy Science.gov (United States) Zhang, Xuting; Cai, Zhijian; Wu, Jianhong 2017-10-01 Raman spectroscopy has been widely used in the detection of drugs, pesticides, explosives, food additives and environmental pollutants, for its characteristics of fast measurement, easy sample preparation, and molecular structure analyzing capability. However, fluorescence disturbance brings a big trouble to these applications, with strong fluorescence background covering up the weak Raman signals. Recently shifted excitation Raman difference spectroscopy (SERDS) not only can completely remove the fluorescence background, but also can be easily integrated into portable Raman spectrometers. Usually, SERDS uses two lasers with small wavelength gap to excite the sample, then acquires two spectra, and subtracts one to the other to get the difference spectrum, where the fluorescence background will be rejected. So, one key aspects of successfully applying SERDS method is to obtain a dual-wavelength laser source. In this paper, a dual-wavelength laser device design based on the principles of external cavity diode laser (ECDL) is proposed, which is low-cost and compact. In addition, it has good mechanical stability because of no moving parts. These features make it an ideal laser source for SERDS technique. The experiment results showed that the device can emit narrow-spectral-width lasers of two wavelengths, with the gap smaller than 2 nanometers. The laser power corresponding to each wavelength can be up to 100mW. 15. Dielectric Covered Planar Antennas at Submillimeter Wavelengths for Terahertz Imaging Science.gov (United States) Chattopadhyay, Goutam; Gill, John J.; Skalare, Anders; Lee, Choonsup; Llombart, Nuria; Siegel, Peter H. 2011-01-01 Most optical systems require antennas with directive patterns. This means that the physical area of the antenna will be large in terms of the wavelength. When non-cooled systems are used, the losses of microstrip or coplanar waveguide lines impede the use of standard patch or slot antennas for a large number of elements in a phased array format. Traditionally, this problem has been solved by using silicon lenses. However, if an array of such highly directive antennas is to be used for imaging applications, the fabrication of many closely spaced lenses becomes a problem. Moreover, planar antennas are usually fed by microstrip or coplanar waveguides while the mixer or the detector elements (usually Schottky diodes) are coupled in a waveguide environment. The coupling between the antenna and the detector/ mixer can be a fabrication challenge in an imaging array at submillimeter wavelengths. Antennas excited by a waveguide (TE10) mode makes use of dielectric superlayers to increase the directivity. These antennas create a kind of Fabry- Perot cavity between the ground plane and the first layer of dielectric. In reality, the antenna operates as a leaky wave mode where a leaky wave pole propagates along the cavity while it radiates. Thanks to this pole, the directivity of a small antenna is considerably enhanced. The antenna consists of a waveguide feed, which can be coupled to a mixer or detector such as a Schottky diode via a standard probe design. The waveguide is loaded with a double-slot iris to perform an impedance match and to suppress undesired modes that can propagate on the cavity. On top of the slot there is an air cavity and on top, a small portion of a hemispherical lens. The fractional bandwidth of such antennas is around 10 percent, which is good enough for heterodyne imaging applications.The new geometry makes use of a silicon lens instead of dielectric quarter wavelength substrates. This design presents several advantages when used in the submillimeter 16. Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement Science.gov (United States) Shiba, Hayato; Keim, Peter; Kawasaki, Takeshi 2018-03-01 It has recently been revealed that long-wavelength fluctuation exists in two-dimensional (2D) glassy systems, having the same origin as that given by the Mermin-Wagner theorem for 2D crystalline solids. In this paper, we discuss how to characterise quantitatively the long-wavelength fluctuation in a molecular dynamics simulation of a lightly supercooled liquid. We employ the cage-relative mean-square displacement (MSD), defined on relative displacement to its cage, to quantitatively separate the long-wavelength fluctuation from the original MSD. For increasing system size the amplitude of acoustic long wavelength fluctuations not only increases but shifts to later times causing a crossover with structural relaxation of caging particles. We further analyse the dynamic correlation length using the cage-relative quantities. It grows as the structural relaxation becomes slower with decreasing temperature, uncovering an overestimation by the four-point correlation function due to the long-wavelength fluctuation. These findings motivate the usage of cage-relative MSD as a starting point for analysis of 2D glassy dynamics. 17. Long-wavelength microinstabilities in toroidal plasmas International Nuclear Information System (INIS) Tang, W.M.; Rewoldt, G. 1993-01-01 Realistic kinetic toroidal eigenmode calculations have been carried out to support a proper assessment of the influence of long-wavelength microturbulence on transport in tokamak plasmas. In order to efficiently evaluate large-scale kinetic behavior extending over many rational surfaces, significant improvements have been made to a toroidal finite element code used to analyze the fully two-dimensional (r,θ) mode structures of trapped-ion and toroidal ion temperature gradient (ITG) instabilities. It is found that even at very long wavelengths, these eigenmodes exhibit a strong ballooning character with the associated radial structure relatively insensitive to ion Landau damping at the rational surfaces. In contrast to the long-accepted picture that the radial extent of trapped-ion instabilities is characterized by the ion-gyroradius-scale associated with strong localization between adjacent rational surfaces, present results demonstrate that under realistic conditions, the actual scale is governed by the large-scale variations in the equilibrium gradients. Applications to recent measurements of fluctuation properties in Tokamak Fusion Test Reactor (TFTR) [Plasma Phys. Controlled Nucl. Fusion Res. (International Atomic Energy Agency, Vienna, 1985), Vol. 1, p. 29] L-mode plasmas indicate that the theoretical trends appear consistent with spectral characteristics as well as rough heuristic estimates of the transport level. Benchmarking calculations in support of the development of a three-dimensional toroidal gyrokinetic code indicate reasonable agreement with respect to both the properties of the eigenfunctions and the magnitude of the eigenvalues during the linear phase of the simulations of toroidal ITG instabilities 18. Amplitude death and spatiotemporal bifurcations in nonlocally delay-coupled oscillators International Nuclear Information System (INIS) Guo, Yuxiao; Niu, Ben 2015-01-01 Amplitude death and spatiotemporal oscillations are remarkable patterns in coupled systems. We consider a ring of n identical oscillators with distance-dependent couplings and time delay. The amplitude death region is the intersection of three stable regions. Employing the method of multiple scales and normal form theory, the stability and criticality of spatiotemporal oscillations are determined. Around the amplitude death boundary there exist one branch of synchronized oscillations, n − 3 branches of co-existing phase-locked oscillations, n branches of mirror-reflecting oscillations, n branches of standing-wave oscillations, one branch of quasiperiodic oscillations and two branches of co-existing synchronized oscillations. It is proved that amplitude death is robust to small inhomogeneity of couplings, and the stability of synchronized or phase-locked oscillations inherits that of the individual decoupled oscillator. For the arbitrary form of coupling functions, some general results are also obtained for the thermodynamic limit. Finally, two examples are given to support the main results. (paper) 19. Differential Service in a Bidirectional Radio-over-Fiber System over a Spectral-Amplitude-Coding OCDMA Network Directory of Open Access Journals (Sweden) Chao-Chin Yang 2016-10-01 Full Text Available A new scheme of radio-over-fiber (RoF network based on spectral-amplitude-coding (SAC optical code division multiple access (OCDMA is herein proposed. Differential service is provided by a power control scheme that classifies users into several classes and assigns each of them with a specific power level. Additionally, the wavelength reuse technique is adapted to support bidirectional transmission and reduce base station (BS cost. Both simulation and numerical results show that significantly differential quality-of-service (QoS in bit-error rate (BER is achieved in both downlink and uplink transmissions. 20. Large amplitude ion-acoustic solitons in dusty plasmas International Nuclear Information System (INIS) Tiwari, R. S.; Jain, S. L.; Mishra, M. K. 2011-01-01 Characteristics of ion-acoustic soliton in dusty plasma, including the dynamics of heavily charged massive dust grains, are investigated following the Sagdeev Potential formalism. Retaining fourth order nonlinearities of electric potential in the expansion of the Sagdeev Potential in the energy equation for a pseudo particle and integrating the resulting energy equation, large amplitude soliton solution is determined. Variation of amplitude (A), half width (W) at half maxima and the product P = AW 2 of the Korteweg-deVries (KdV), dressed and large amplitude soliton as a function of wide range of dust concentration are numerically studied for recently observed parameters of dusty plasmas. We have also presented the region of existence of large amplitude ion-acoustic soliton in the dusty plasma by analyzing the structure of the pseudo potential. It is found that in the presence of positively charged dust grains, system supports only compressive solitons, on the other hand, in the presence of negatively charged dust grains, the system supports compressive solitons up to certain critical concentration of dust grains and above this critical concentration, the system can support rarefactive solitons also. The effects of dust concentration, charge, and mass of the dust grains, on the characteristics of KdV, dressed and large amplitude the soliton, i.e., amplitude (A), half width at half maxima (W), and product of amplitude (A) and half width at half maxima (P = AW 2 ), are discussed in detail 1. Fringe image analysis based on the amplitude modulation method. Science.gov (United States) Gai, Shaoyan; Da, Feipeng 2010-05-10 A novel phase-analysis method is proposed. To get the fringe order of a fringe image, the amplitude-modulation fringe pattern is carried out, which is combined with the phase-shift method. The primary phase value is obtained by a phase-shift algorithm, and the fringe-order information is encoded in the amplitude-modulation fringe pattern. Different from other methods, the amplitude-modulation fringe identifies the fringe order by the amplitude of the fringe pattern. In an amplitude-modulation fringe pattern, each fringe has its own amplitude; thus, the order information is integrated in one fringe pattern, and the absolute fringe phase can be calculated correctly and quickly with the amplitude-modulation fringe image. The detailed algorithm is given, and the error analysis of this method is also discussed. Experimental results are presented by a full-field shape measurement system where the data has been processed using the proposed algorithm. (c) 2010 Optical Society of America. 2. Broadband metasurface holograms: toward complete phase and amplitude engineering. Science.gov (United States) Wang, Qiu; Zhang, Xueqian; Xu, Yuehong; Gu, Jianqiang; Li, Yanfeng; Tian, Zhen; Singh, Ranjan; Zhang, Shuang; Han, Jiaguang; Zhang, Weili 2016-09-12 As a revolutionary three-dimensional imaging technique, holography has attracted wide attention for its ability to photographically record a light field. However, traditional phase-only or amplitude-only modulation holograms have limited image quality and resolution to reappear both amplitude and phase information required of the objects. Recent advances in metasurfaces have shown tremendous opportunities for using a planar design of artificial meta-atoms to shape the wave front of light by optimal control of both its phase and amplitude. Inspired by the concept of designer metasurfaces, we demonstrate a novel amplitude-phase modulation hologram with simultaneous five-level amplitude modulation and eight-level phase modulation. Such a design approach seeks to turn the perceived disadvantages of the traditional phase or amplitude holograms, and thus enable enhanced performance in resolution, homogeneity of amplitude distribution, precision, and signal-to-noise ratio. In particular, the unique holographic approach exhibits broadband characteristics. The method introduced here delivers more degrees of freedom, and allows for encoding highly complex information into designer metasurfaces, thus having the potential to drive next-generation technological breakthroughs in holography. 3. Digital services using quadrature amplitude modulation (QAM) over CATV analog DWDM system Science.gov (United States) Yeh, JengRong; Selker, Mark D.; Trail, J.; Piehler, David; Levi, Israel 2000-04-01 Dense Wavelength Division Multiplexing (DWDM) has recently gained great popularity as it provides a cost effective way to increase the transmission capacity of the existing fiber cable plant. For a long time, Dense WDM was exclusively used for baseband digital applications, predominantly in terrestrial long haul networks and in some cases in metropolitan and enterprise networks. Recently, the performance of DWDM components and frequency-stabilized lasers has substantially improved while the costs have down significantly. This makes a variety of new optical network architectures economically viable. The first commercial 8- wavelength DWDM system designed for Hybrid Fiber Coax networks was reported in 1998. This type of DWDM system utilizes Sub-Carrier Multiplexing (SCM) of Quadrature Amplitude Modulated (QAM) signals to transport IP data digital video broadcast and Video on Demand on ITU grid lightwave carriers. The ability of DWDM to provide scalable transmission capacity in the optical layer with SCM granularity is now considered by many to be the most promising technology for future transport and distribution of broadband multimedia services. 4. A Millimetre-Wave Cuboid Solid Immersion Lens with Intensity-Enhanced Amplitude Mask Apodization Science.gov (United States) Yue, Liyang; Yan, Bing; Monks, James N.; Dhama, Rakesh; Wang, Zengbo; Minin, Oleg V.; Minin, Igor V. 2018-03-01 Photonic jet is a narrow, highly intensive, weak-diverging beam propagating into a background medium and can be produced by a cuboid solid immersion lens (SIL) in both transmission and reflection modes. Amplitude mask apodization is an optical method to further improve the spatial resolution of a SIL imaging system via reduction of waist size of photonic jet, but always leading to intensity loss due to central masking of the incoming plane wave. In this letter, we report a particularly sized millimetre-wave cuboid SIL with the intensity-enhanced amplitude mask apodization for the first time. It is able to simultaneously deliver extra intensity enhancement and waist narrowing to the produced photonic jet. Both numerical simulation and experimental verification of the intensity-enhanced apodization effect are demonstrated using a copper-masked Teflon cuboid SIL with 22-mm side length under radiation of a plane wave with 8-mm wavelength. Peak intensity enhancement and the lateral resolution of the optical system increase by about 36.0% and 36.4% in this approach, respectively. 5. Elementary amplitudes from full QCD and the stochastic vacuum model International Nuclear Information System (INIS) Martini, A.F.; Menon, M.J. 2002-01-01 In a previous work, making use of the gluon gauge-invariant two-point correlation function determined from lattice QCD in the quenched approximation and the stochastic vacuum model, we determined the elementary (parton-parton) scattering amplitude in the momentum transfer space. In this communication we compute the elementary amplitude from new lattice QCD calculations that include the effects of dynamical fermions (full QCD). The main conclusion is that the inclusion of dynamical fermions leads to a normalized elementary amplitude that decreases more quickly with the momentum transfer than that in the quenched approximation. (author) 6. Renormalization Scale-Fixing for Complex Scattering Amplitudes Energy Technology Data Exchange (ETDEWEB) Brodsky, Stanley J.; /SLAC; Llanes-Estrada, Felipe J.; /Madrid U. 2005-12-21 We show how to fix the renormalization scale for hard-scattering exclusive processes such as deeply virtual meson electroproduction by applying the BLM prescription to the imaginary part of the scattering amplitude and employing a fixed-t dispersion relation to obtain the scale-fixed real part. In this way we resolve the ambiguity in BLM renormalization scale-setting for complex scattering amplitudes. We illustrate this by computing the H generalized parton distribution at leading twist in an analytic quark-diquark model for the parton-proton scattering amplitude which can incorporate Regge exchange contributions characteristic of the deep inelastic structure functions. 7. Calculation and modular properties of multi-loop superstring amplitudes International Nuclear Information System (INIS) Danilov, G S 2012-01-01 Multi-loop superstring amplitude is calculated in the conventional gauge where Grassmann moduli are carried by the 2D gravitino field. Generally, instead of the modular symmetry, the amplitudes hold the symmetry under modular transformations added by relevant transformations of the 2D local supersymmetry. If a number of loops are larger than 3, the integration measures are not modular forms. In this case the expression for the amplitude contains an integral over the bound of the fundamental region of the modular group. (paper) 8. Amplitude Modulation in the δ Sct star KIC 7106205 Directory of Open Access Journals (Sweden) Bowman Dominic. M. 2015-01-01 Full Text Available The δ Sct star KIC 7106205 showed amplitude modulation in a single p mode, whilst all other p and g modes remained stable in amplitude and phase over 1470 d of the Kepler dataset. The data were divided into 30 time bins of equal length and a series of consecutive Fourier transforms was calculated. A fixed frequency, calculated from a least-squares fit of all data, allowed amplitude and phase for every mode in each time bin to be tracked. The missing p mode energy was not transferred to any other visible modes. 9. Corrections to the large-angle scattering amplitude International Nuclear Information System (INIS) Goloskokov, S.V.; Kudinov, A.V.; Kuleshov, S.P. 1979-01-01 High-energy behaviour of scattering amplitudes is considered within the frames of Logunov-Tavchelidze quasipotential approach. The representation of scattering amplitude of two scalar particles, convenient for the study of its asymptotic properties is given. Obtained are corrections of the main value of scattering amplitude of the first and the second orders in 1/p, where p is the pulse of colliding particles in the system of the inertia centre. An example of the obtained formulas use for a concrete quasipotential is given 10. Direct Calculation of the Scattering Amplitude Without Partial Wave Analysis Science.gov (United States) Shertzer, J.; Temkin, A.; Fisher, Richard R. (Technical Monitor) 2001-01-01 Two new developments in scattering theory are reported. We show, in a practical way, how one can calculate the full scattering amplitude without invoking a partial wave expansion. First, the integral expression for the scattering amplitude f(theta) is simplified by an analytic integration over the azimuthal angle. Second, the full scattering wavefunction which appears in the integral expression for f(theta) is obtained by solving the Schrodinger equation with the finite element method (FEM). As an example, we calculate electron scattering from the Hartree potential. With minimal computational effort, we obtain accurate and stable results for the scattering amplitude. 11. Amplitude-to-frequency converter of radioisotope instruments International Nuclear Information System (INIS) Demchenkov, V.P.; Korobkov, I.N. 1988-01-01 An amplitude-to-frequency converter designed for signal processing of radioisotope relay devices is descibed. The basic elements of the converter are a scaling amplifier, an analog-to-digital converter, a code-to-frequency converter, a null-organ, a delay unit and a clock-pulse generator. The designed amplitude-to-frequency converter takes into account a prior information about the signal shape of the energy spectrum. The converter processes input pulses of 0.10 V amplitude and duration more than 2μs. The energy channel number is 64 12. Gluon cascades and amplitudes in light-front perturbation theory International Nuclear Information System (INIS) Cruz-Santiago, C.A.; Staśto, A.M. 2013-01-01 We construct the gluon wave functions, fragmentation functions and scattering amplitudes within the light-front perturbation theory. Recursion relations on the light-front are constructed for the wave functions and fragmentation functions, which in the latter case are the light-front analogs of the Berends–Giele recursion relations. Using general relations between wave functions and scattering amplitudes it is demonstrated how to obtain the maximally-helicity violating amplitudes, and explicit verification of the results is based on simple examples. 13. Wavelength-stepped, actively mode-locked fiber laser based on wavelength-division-multiplexed optical delay lines Science.gov (United States) Lee, Eunjoo; Kim, Byoung Yoon 2017-12-01 We propose a new scheme for an actively mode-locked wavelength-swept fiber laser that produces a train of discretely wavelength-stepped pulses from a short fiber cavity. Pulses with different wavelengths are split and combined by standard wavelength division multiplexers with fiber delay lines. As a proof of concept, we demonstrate a laser using an erbium doped fiber amplifier and commercially available wavelength-division multiplexers with wavelength spacing of 0.8 nm. The results show simultaneous mode-locking at three different wavelengths. Laser output parameters in time domain, optical and radio frequency spectral domain, and the noise characteristics are presented. Suggestions for the improved design are discussed. 14. On the interaction of small-scale linear waves with nonlinear solitary waves Science.gov (United States) Xu, Chengzhu; Stastna, Marek 2017-04-01 In the study of environmental and geophysical fluid flows, linear wave theory is well developed and its application has been considered for phenomena of various length and time scales. However, due to the nonlinear nature of fluid flows, in many cases results predicted by linear theory do not agree with observations. One of such cases is internal wave dynamics. While small-amplitude wave motion may be approximated by linear theory, large amplitude waves tend to be solitary-like. In some cases, when the wave is highly nonlinear, even weakly nonlinear theories fail to predict the wave properties correctly. We study the interaction of small-scale linear waves with nonlinear solitary waves using highly accurate pseudo spectral simulations that begin with a fully nonlinear solitary wave and a train of small-amplitude waves initialized from linear waves. The solitary wave then interacts with the linear waves through either an overtaking collision or a head-on collision. During the collision, there is a net energy transfer from the linear wave train to the solitary wave, resulting in an increase in the kinetic energy carried by the solitary wave and a phase shift of the solitary wave with respect to a freely propagating solitary wave. At the same time the linear waves are greatly reduced in amplitude. The percentage of energy transferred depends primarily on the wavelength of the linear waves. We found that after one full collision cycle, the longest waves may retain as much as 90% of the kinetic energy they had initially, while the shortest waves lose almost all of their initial energy. We also found that a head-on collision is more efficient in destroying the linear waves than an overtaking collision. On the other hand, the initial amplitude of the linear waves has very little impact on the percentage of energy that can be transferred to the solitary wave. Because of the nonlinearity of the solitary wave, these results provide us some insight into wave-mean flow 15. Wavelength stabilized high pulse power laser diodes for automotive LiDAR Science.gov (United States) Knigge, A.; Klehr, A.; Wenzel, H.; Zeghuzi, A.; Fricke, J.; Maaßdorf, A.; Liero, A.; Tränkle, G. 2018-03-01 Diode lasers generating optical pulses with high peak power and lengths in the nanosecond range are key components of systems for free-space communication, metrology, material processing, spectroscopy, and light detection and ranging (LiDAR) as needed for object detection and autonomous driving. Automotive LiDAR systems demand additionally a good beam quality and low wavelength shift with temperature due to the wide operating temperature span. We present here internally wavelength stabilized lasers emitting ns optical pulses from an emission aperture between 30 μm and 100 μm with peak powers of tens of Watts at wavelengths around 905 nm. The vertical structure based on AlGaAs (confinement and cladding layers) and InGaAs (active quantum well) is especially optimized for pulsed operation with respect to the implementation of a surface Bragg grating with a high reflectivity. The fabricated 6 mm long distributed Bragg reflector (DBR) broad area (BA) lasers are electrically driven by an in-house developed high-speed unit generating 3 to 10 ns long nearly rectangular shaped current pulses with amplitudes of up to 250 A. Such lasers emit optical pulses with a peak power of more than 30 W at 95 A pulse current up to a temperature of 85°C with a wavelength shift as low as 65 pm/K and a lateral beam propagation factor less than 10. The influence of the lateral aperture width and the pulse length on the beam quality will be shown. A monolithic integration of 3 DBR BA lasers on a single chip whose emission can be combined into a single beam raises the output power to more than 100 W. 16. A pulse amplitude discriminator with very low-power consuming International Nuclear Information System (INIS) Deng Changming; Liu Zhengshan; Zhang Zhiyong; Cheng Chang 2000-01-01 A low-power pulse amplitude discriminator is described. The discriminator circuit is mainly composed of an integrated voltage comparator, MAX921, and owns the characters of very low-power and low operating voltage 17. Some tree-level string amplitudes in the NSR formalism International Nuclear Information System (INIS) Becker, Katrin; Becker, Melanie; Melnikov, Ilarion V.; Robbins, Daniel; Royston, Andrew B. 2015-01-01 We calculate tree level scattering amplitudes for open strings using the NSR formalism. We present a streamlined symmetry-based and pedagogical approach to the computations, which we first develop by checking two-, three-, and four-point functions involving bosons and fermions. We calculate the five-point amplitude for massless gluons and find agreement with an earlier result by Brandt, Machado and Medina. We then compute the five-point amplitudes involving two and four fermions respectively, the general form of which has not been previously obtained in the NSR formalism. The results nicely confirm expectations from the supersymmetric F 4 effective action. Finally we use the prescription of Kawai, Lewellen and Tye (KLT) to compute the amplitudes for the closed string sector. 18. Amplitude chimeras and chimera death in dynamical networks International Nuclear Information System (INIS) Zakharova, Anna; Kapeller, Marie; Schöll, Eckehard 2016-01-01 We find chimera states with respect to amplitude dynamics in a network of Stuart- Landau oscillators. These partially coherent and partially incoherent spatio-temporal patterns appear due to the interplay of nonlocal network topology and symmetry-breaking coupling. As the coupling range is increased, the oscillations are quenched, amplitude chimeras disappear and the network enters a symmetry-breaking stationary state. This particular regime is a novel pattern which we call chimera death. It is characterized by the coexistence of spatially coherent and incoherent inhomogeneous steady states and therefore combines the features of chimera state and oscillation death. Additionally, we show two different transition scenarios from amplitude chimera to chimera death. Moreover, for amplitude chimeras we uncover the mechanism of transition towards in-phase synchronized regime and discuss the role of initial conditions. (paper) 19. Stora's fine notion of divergent amplitudes International Nuclear Information System (INIS) Várilly, Joseph C.; Gracia-Bondía, José M. 2016-01-01 Stora and coworkers refined the notion of divergent quantum amplitude, somewhat upsetting the standard power-counting recipe. This unexpectedly clears the way to new prototypes for free and interacting field theories of bosons of any mass and spin. 20. Euclidean to Minkowski Bethe-Salpeter amplitude and observables International Nuclear Information System (INIS) Carbonell, J.; Frederico, T.; Karmanov, V.A. 2017-01-01 We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude - or alternatively the light-front wave function - as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function. (orig.) 1. Laser beam complex amplitude measurement by phase diversity. Science.gov (United States) Védrenne, Nicolas; Mugnier, Laurent M; Michau, Vincent; Velluet, Marie-Thérèse; Bierent, Rudolph 2014-02-24 The control of the optical quality of a laser beam requires a complex amplitude measurement able to deal with strong modulus variations and potentially highly perturbed wavefronts. The method proposed here consists in an extension of phase diversity to complex amplitude measurements that is effective for highly perturbed beams. Named camelot for Complex Amplitude MEasurement by a Likelihood Optimization Tool, it relies on the acquisition and processing of few images of the beam section taken along the optical path. The complex amplitude of the beam is retrieved from the images by the minimization of a Maximum a Posteriori error metric between the images and a model of the beam propagation. The analytical formalism of the method and its experimental validation are presented. The modulus of the beam is compared to a measurement of the beam profile, the phase of the beam is compared to a conventional phase diversity estimate. The precision of the experimental measurements is investigated by numerical simulations. 2. Mimicking an amplitude damping channel for Laguerre Gaussian Modes CSIR Research Space (South Africa) Dudley, Angela L 2010-10-01 Full Text Available An amplitude damping channel for Laguerre-Gaussian (LG) modes is presented. Experimentally the action of the channel on LG modes is in good agreement with that predicted theoretically.... 3. Singularities in four-body final-state amplitudes International Nuclear Information System (INIS) 1978-01-01 Like three-body amplitudes, four-body amplitudes have subenergy threshold singularities over and above total-energy singularities. In the four-body problem we encounter a new type of subenergy singularity besides the usual two- and three-body subenergy threshold singularities. This singularity will be referred to as ''independent-pair threshold singularity'' and involves pair-subenergy threshold singularities in each of the two independent pair subenergies in four-body final states. We also study the particularly interesting case of resonant two- and three-body interactions in the four-body isobar model and the rapid (singular) dependence of the isobar amplitudes they generate in the four-body phase space. All these singularities are analyzed in the multiple-scattering formalism and it is shown that they arise from the ''next-to-last'' rescattering and hence may be represented correctly by an approximate amplitude which has that rescattering 4. Bessel–Gauss resonator with internal amplitude filter CSIR Research Space (South Africa) Litvin, IA 2008-05-01 Full Text Available The authors investigate a conventional resonator configuration, using only spherical curvature optical elements, for the generation of Bessel–Gauss beams. This is achieved through the deployment of a suitable amplitude filter at a Fourier plane... 5. Analytic computations of massive one-loop amplitudes International Nuclear Information System (INIS) Badger, Simon; Yundin, Valery; Sattler, Ralf 2010-06-01 We show some new applications of on-shell methods to calculate compact helicity amplitudes for t anti t production through gluon fusion. The rational and mass renormalisation contributions are extracted from two independent Feynman diagram based approaches. (orig.) 6. Euclidean to Minkowski Bethe-Salpeter amplitude and observables Energy Technology Data Exchange (ETDEWEB) Carbonell, J. [Universite Paris-Sud, IN2P3-CNRS, Institut de Physique Nucleaire, Orsay Cedex (France); Frederico, T. [Instituto Tecnologico de Aeronautica, DCTA, Sao Jose dos Campos (Brazil); Karmanov, V.A. [Lebedev Physical Institute, Moscow (Russian Federation) 2017-01-15 We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude - or alternatively the light-front wave function - as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function. (orig.) 7. The zerology of kaon-nucleon forward scattering amplitudes International Nuclear Information System (INIS) Dumbrajs, O. 1981-01-01 It has been realized for a long time that zeros of the forward kaon-nucleon scattering amplitudes are useful in correlating different low and high-energy scattering parameters and in providing a consistency test of available data. The simplest possibility of exploring zeros is to evaluate the ordinary dispersion relations in the complex energy plane. The more natural way of bringing zeros of amplitudes into play is to consider either one of the more sophisticated forms of dispersion relations: i) phase dispersion relations, ii) inverse-amplitude dispersion relations, iii) logarithmic dispersion relations, or to apply the maximum modulus theorem and a factorization theorem. The author concentrates on the use of logarithmic dispersion relations because this approach seems to be the most convenient one for future extensions to nonforward scattering data analyses based on the zeros of the amplitude. (Auth.) 8. MULTI-WAVELENGTH RADIO CONTINUUM EMISSION STUDIES OF DUST-FREE RED GIANTS Energy Technology Data Exchange (ETDEWEB) O' Gorman, Eamon; Harper, Graham M. [School of Physics, Trinity College Dublin, Dublin 2 (Ireland); Brown, Alexander [Center for Astrophysics and Space Astronomy, University of Colorado, 389 UCB, Boulder, CO 80309 (United States); Drake, Stephen [NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States); Richards, Anita M. S. [Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL (United Kingdom) 2013-10-01 Multi-wavelength centimeter continuum observations of non-dusty, non-pulsating K spectral-type red giants directly sample their chromospheres and wind acceleration zones. Such stars are feeble emitters at these wavelengths, however, and previous observations have provided only a small number of modest signal-to-noise measurements slowly accumulated over three decades. We present multi-wavelength Karl G. Jansky Very Large Array thermal continuum observations of the wind acceleration zones of two dust-free red giants, Arcturus (α Boo: K2 III) and Aldebaran (α Tau: K5 III). Importantly, most of our observations of each star were carried out over just a few days, so that we obtained a snapshot of the different stellar atmospheric layers sampled at different wavelengths, independent of any long-term variability. We report the first detections at several wavelengths for each star including a detection at 10 cm (3.0 GHz: S band) for both stars and a 20 cm (1.5 GHz: L band) detection for α Boo. This is the first time single (non-binary) luminosity class III red giants have been detected at these continuum wavelengths. Our long-wavelength data sample the outer layers of α Boo's atmosphere where its wind velocity is approaching (or possibly has reached) its terminal value and the ionization balance is becoming frozen-in. For α Tau, however, our long-wavelength data are still sampling its inner atmosphere, where the wind is still accelerating probably due to its lower mass-loss rate. We compare our data with published semi-empirical models based on ultraviolet data, and the marked deviations highlight the need for new atmospheric models to be developed. Spectral indices are used to discuss the possible properties of the stellar atmospheres, and we find evidence for a rapidly cooling wind in the case of α Boo. Finally, we develop a simple analytical wind model for α Boo based on our new long-wavelength flux measurements. 9. MULTI-WAVELENGTH RADIO CONTINUUM EMISSION STUDIES OF DUST-FREE RED GIANTS International Nuclear Information System (INIS) O'Gorman, Eamon; Harper, Graham M.; Brown, Alexander; Drake, Stephen; Richards, Anita M. S. 2013-01-01 Multi-wavelength centimeter continuum observations of non-dusty, non-pulsating K spectral-type red giants directly sample their chromospheres and wind acceleration zones. Such stars are feeble emitters at these wavelengths, however, and previous observations have provided only a small number of modest signal-to-noise measurements slowly accumulated over three decades. We present multi-wavelength Karl G. Jansky Very Large Array thermal continuum observations of the wind acceleration zones of two dust-free red giants, Arcturus (α Boo: K2 III) and Aldebaran (α Tau: K5 III). Importantly, most of our observations of each star were carried out over just a few days, so that we obtained a snapshot of the different stellar atmospheric layers sampled at different wavelengths, independent of any long-term variability. We report the first detections at several wavelengths for each star including a detection at 10 cm (3.0 GHz: S band) for both stars and a 20 cm (1.5 GHz: L band) detection for α Boo. This is the first time single (non-binary) luminosity class III red giants have been detected at these continuum wavelengths. Our long-wavelength data sample the outer layers of α Boo's atmosphere where its wind velocity is approaching (or possibly has reached) its terminal value and the ionization balance is becoming frozen-in. For α Tau, however, our long-wavelength data are still sampling its inner atmosphere, where the wind is still accelerating probably due to its lower mass-loss rate. We compare our data with published semi-empirical models based on ultraviolet data, and the marked deviations highlight the need for new atmospheric models to be developed. Spectral indices are used to discuss the possible properties of the stellar atmospheres, and we find evidence for a rapidly cooling wind in the case of α Boo. Finally, we develop a simple analytical wind model for α Boo based on our new long-wavelength flux measurements 10. Multi-wavelength Radio Continuum Emission Studies of Dust-free Red Giants Science.gov (United States) O'Gorman, Eamon; Harper, Graham M.; Brown, Alexander; Dranke, Stephen; Richards, Anita M. S. 2013-01-01 Multi-wavelength centimeter continuum observations of non-dusty, non-pulsating K spectral-type red giants directly sample their chromospheres and wind acceleration zones. Such stars are feeble emitters at these wavelengths, however, and previous observations have provided only a small number of modest signal-to-noise measurements slowly accumulated over three decades. We present multi-wavelength Karl G. Jansky Very Large Array thermal continuum observations of the wind acceleration zones of two dust-free red giants, Arcturus (alpha Boo: K2 III) and Aldebaran (alpha Tau: K5 III). Importantly, most of our observations of each star were carried out over just a few days, so that we obtained a snapshot of the different stellar atmospheric layers sampled at different wavelengths, independent of any long-term variability. We report the first detections at several wavelengths for each star including a detection at 10 cm (3.0 GHz: S band) for both stars and a 20 cm (1.5 GHz: L band) detection for alpha Boo. This is the first time single (non-binary) luminosity class III red giants have been detected at these continuum wavelengths. Our long-wavelength data sample the outer layers of alpha Boo's atmosphere where its wind velocity is approaching (or possibly has reached) its terminal value and the ionization balance is becoming frozen-in. For alpha Tau, however, our long-wavelength data are still sampling its inner atmosphere, where the wind is still accelerating probably due to its lower mass-loss rate. We compare our data with published semi-empirical models based on ultraviolet data, and the marked deviations highlight the need for new atmospheric models to be developed. Spectral indices are used to discuss the possible properties of the stellar atmospheres, and we find evidence for a rapidly cooling wind in the case of alpha Boo. Finally, we develop a simple analytical wind model for alpha Boo based on our new long-wavelength flux measurements. 11. Optofluidic intracavity spectroscopy for spatially, temperature, and wavelength dependent refractometry Science.gov (United States) Kindt, Joel D. A microfluidic refractometer was designed based on previous optofluidic intracavity spectroscopy (OFIS) chips utilized to distinguish healthy and cancerous cells. The optofluidic cavity is realized by adding high reflectivity dielectric mirrors to the top and bottom of a microfluidic channel. This creates a plane-plane Fabry-Perot optical cavity in which the resonant wavelengths are highly dependent on the optical path length inside the cavity. Refractometry is a useful method to determine the nature of fluids, including the concentration of a solute in a solvent as well as the temperature of the fluid. Advantages of microfluidic systems are the easy integration with lab-on-chip devices and the need for only small volumes of fluid. The unique abilities of the microfluidic refractometer in this thesis include its spatial, temperature, and wavelength dependence. Spatial dependence of the transmission spectrum is inherent through a spatial filtering process implemented with an optical fiber and microscope objective. A sequence of experimental observations guided the change from using the OFIS chip as a cell discrimination device to a complimentary refractometer. First, it was noted the electrode structure within the microfluidic channel, designed to trap and manipulate biological cells with dielectrophoretic (DEP) forces, caused the resonant wavelengths to blue-shift when the electrodes were energized. This phenomenon is consistent with the negative dn/dT property of water and water-based solutions. Next, it was necessary to develop a method to separate the optical path length into physical path length and refractive index. Air holes were placed near the microfluidic channel to exclusively measure the cavity length with the known refractive index of air. The cavity length was then interpolated across the microfluidic channel, allowing any mechanical changes to be taken into account. After the separation of physical path length and refractive index, it was of interest 12. Aging increases compensatory saccade amplitude in the video head impulse test Directory of Open Access Journals (Sweden) Eric R Anson 2016-07-01 13. The method of contour rotations and the three particle amplitudes International Nuclear Information System (INIS) Brinati, J.R. 1980-01-01 The application of the method of contour rotations to the solution of the Faddeev-Lovelace equations and the calculation of the break-up and stripping amplitudes in a system of three distinct particles is reviewed. A relationship between the masses of the particles is obtained, which permits the break-up amplitude to be calculated from a single iteration of the final integral equation. (Author) [pt 14. Cosmological constraints on the amplitude of relic gravitational waves International Nuclear Information System (INIS) Novosyadlij, B.; Apunevich, S. 2005-01-01 The evolution of the amplitude of relic gravitational waves (RGW) generated in early Universe has been analyzed. The analytical approximation is presented for angular power spectrum of cosmic microwave background anisotropies caused by gravitational waves through Sachs-Wolfe effect. The estimate of the most probable value for this amplitude was obtained on the basis of observation data on cosmic microwave background anisotropies from COBE, WMAP and BOOMERanG experiments along with large-scale structure observations 15. Amplitude Modulated Sinusoidal Signal Decomposition for Audio Coding DEFF Research Database (Denmark) Christensen, M. G.; Jacobson, A.; Andersen, S. V. 2006-01-01 In this paper, we present a decomposition for sinusoidal coding of audio, based on an amplitude modulation of sinusoids via a linear combination of arbitrary basis vectors. The proposed method, which incorporates a perceptual distortion measure, is based on a relaxation of a nonlinear least......-squares minimization. Rate-distortion curves and listening tests show that, compared to a constant-amplitude sinusoidal coder, the proposed decomposition offers perceptually significant improvements in critical transient signals.... 16. Multiple pole in the electron--hydrogen-atom scattering amplitude International Nuclear Information System (INIS) Amusia, M.Y.; Kuchiev, M.Y. 1982-01-01 It is demonstrated that the amplitude for electron--hydrogen-atom forward scattering has the third-order pole at the point E = -13.6 eV, E being the energy of the incident electron. The coefficients which characterize the pole are calculated exactly. The invalidity of the Born approximation is proved. The contribution of the pole singularity to the dispersion relation for the scattering amplitude is discussed 17. Factorization of the six-particle multi-Regge amplitude International Nuclear Information System (INIS) Moen, I.O. 1975-01-01 It is shown that factorization of the multi-Regge contribution to the six-particle amplitude follows from the complex-helicity-plane structure, the Steinmann relations, and extended unitarity. The six-particle multi-Regge amplitude also satisfies some new discontinuity relations which are interpreted as resulting from the interplay of singularities required by the Gram-determinant constraint in four-dimensional space-time 18. Controlling quantum interference in phase space with amplitude OpenAIRE Xue, Yinghong; Li, Tingyu; Kasai, Katsuyuki; Okada-Shudo, Yoshiko; Watanabe, Masayoshi; Zhang, Yun 2017-01-01 We experimentally show a quantum interference in phase space by interrogating photon number probabilities (n?=?2, 3, and 4) of a displaced squeezed state, which is generated by an optical parametric amplifier and whose displacement is controlled by amplitude of injected coherent light. It is found that the probabilities exhibit oscillations of interference effect depending upon the amplitude of the controlling light field. This phenomenon is attributed to quantum interference in phase space a... 19. Fatigue life assessment under multiaxial variable amplitude loading International Nuclear Information System (INIS) Morilhat, P.; Kenmeugne, B.; Vidal-Salle, E.; Robert, J.L. 1996-06-01 A variable amplitude multiaxial fatigue life prediction method is presented in this paper. It is based on a stress as input data are the stress tensor histories which may be calculated by FEM analysis or measured directly on the structure during the service loading. The different steps of he method are first presented then its experimental validation is realized for log and finite fatigue lives through biaxial variable amplitude loading tests using cruciform steel samples. (authors). 9 refs., 7 figs 20. Global Well-Posedness of the Boltzmann Equation with Large Amplitude Initial Data Science.gov (United States) Duan, Renjun; Huang, Feimin; Wang, Yong; Yang, Tong 2017-07-01 The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new {L^∞_xL^1v\\cap L^∞_{x,v}} approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted {L^∞} norm under some smallness condition on the {L^1_xL^∞_v} norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the {L^∞_{x,v}} norm with explicit rates of convergence are also studied.	{"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": 0.8865125179290771, "perplexity": 2596.6899421792077}, "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-2020-16/segments/1585371700247.99/warc/CC-MAIN-20200407085717-20200407120217-00494.warc.gz"}
https://www.physicsforums.com/threads/sho-on-a-conveyor-belt.168604/	# SHO on a conveyor belt • Start date #### cbjewelz 3 0 This one is killing me. 1. Homework Statement A SHO is resting on a horizontal surface with which it has static and kinetic friction coefficients us and uk . We have k for the spring constant and m for the mass. The surface is actually part of a conveyor belt that is maintained at a speed u whatever the behavior of the SHO. (speed is to the right) see figure Assume that the mass of the SHO is initially at rest at its equilibrium position. The motion can be divided into distinct regimes, depending upon whether there is slipping or not. You are to begin with the first regime. (a) Derive expressions for the displacement and velocity as a function of time. (b) Derive the condition under which the SHO will oscillate sinu- soidally forever. Consider the situation under which the mass can experience static friction. (c) At what time t1 does the SHO lose its sinusoidal behavior and sticks? Express t1 in terms of the given parameters. (d) What is the displacement at the point when the mass sticks? Ex- press your answer in terms of u, ?k , g, and the angular frequency ?0 of the SHO. (e) What is the motion of mass when it sticks? (f ) What is the displacement when the mass begins to slip? 2. The attempt at a solution I'm really only having trouble with the first part. I'm thinking that the conveyor belt picks up the mass due to static friction and carries it with constant speed until kx overcomes static friction at which point there is slipping, etc. Is this correct? and if so how do we explain how it is initially put into motion because we cannot have an instantaneous force brining the mass to u. Thanks #### Attachments • 4.7 KB Views: 265 Last edited: Related Introductory Physics Homework Help News on Phys.org #### cbjewelz 3 0 I don't know why this was moved to the introductory physics forum. This problem involves multiple differential equations that must be solved with considerations of slipping and frictional forces included. #### denverdoc 957 0 From the reading of the problem, I don't think it says what you suggest for parts 1) & 2) instead, that the mass is slipping constantly. #### tim_lou 682 1 I don't think problems like these can be solved using the usual differential equation method. rather, you should try to explain the motion and connect the pieces together. first imagine the mass is stationary, it will move at constant speed with the conveyor belt as long as the force exerted by the spring doesn't exceed the static friction. once the spring is pulled hard enough, you'll get some motion. sooner or later, the mass will come around, at some instant, it will move at the same velocity as the conveyor belt, and static friction kicks in, depending on the force at that instance, you will have a couple different situations. you just have to solve for the motions individually and paste them together. Last edited: • Posted Replies 11 Views 5K • Posted Replies 1 Views 3K • Posted Replies 6 Views 2K • Posted Replies 8 Views 2K • Posted Replies 1 Views 2K • Posted Replies 8 Views 8K • Posted Replies 1 Views 2K • Posted Replies 5 Views 4K ### Physics Forums Values We Value Quality • Topics based on mainstream science • Proper English grammar and spelling We Value Civility • Positive and compassionate attitudes • Patience while debating We Value Productivity • Disciplined to remain on-topic • Recognition of own weaknesses • Solo and co-op problem solving	{"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": 0.8384888768196106, "perplexity": 1186.212957327392}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986666467.20/warc/CC-MAIN-20191016063833-20191016091333-00192.warc.gz"}
https://www.kullabs.org/class-7/optional-maths-7/transformation-2/transformation?type=quiz	Transformation Subject: Optional Maths Overview This note gives the information of congruent transformation and reflection using coordinates. Transformation Congruent Transformation Congruent transformation changes in position but no change in the size of an object is called congruent transformation .It is also called isometric transformation .Here, Reflection and Rotation are the examples of congruent transformation. ReflectIon A transformation in which a geometric figure is reflected across a line , creating a mirror image . That line is called the axis of a reflection . Let's see the object given below and try jot down conclusion from the figure . FIGURE Conclusion 1. The image lies on the opposite side of the object with respect to the mirror (axis of reflection ) 2. The object distance is the distance between the mirror to the object is equal to the image distance is the distance between the mirror to the image . 3. Object size and image size are equal . A rule which shifts an object is the reflection to the objects of same form each being at an equal distance from a fixed line .And this fixed line is known as mirror or axis of reflection . Reflection using coordinates As we know, the coordinate system has been well used .In coordinate plane, we can reflect the objects. Here, the coordinates of geometrical figures formed after the reflection about x-axis and y-axis . X-axis and Y-axis are known as the axis of reflection . x-axis as the axis of reflection Look at the following figure, and learn to find the coordinates of the image of a point in different quadrants when x-axis is the axis of reflection . From the above figure, when a geometrical figure is reflected about x-axis as the axis of reflection, the x-coordinate of the image remains the same and the sign of y-coordinate of the image is changed . FIGURE y-axis as the axis of reflection: Look the following figure and learn the coordinate of the image of a point in different quadrants when y-axis is the axis of reflection .FIGURE Look the following figure and learn to find the coordinates of the image of a point in different quadrants when y-axis is the axis of reflection . FIGURE Rotation A transformation in which a plane figure turns around a fixed center point . In other words , one point on the plane the center of rotation , is fixed and everything else on the plane rotates about that point by a given angle . Study the following figure and find out the idea of rotation of a point. FIGURE In the same way, each vertex of a geometrical figure are rotated through a certain angle in a certain direction about a given center of rotation , this figure is rotated through the same angle in the same direction about the same centre of rotation .To rotate this geometrical figure, these three conditions are required ; 1. Centre of rotation 2.Angle of rotation 3. Direction of rotation A figure can be rotated in two directions. 1. Anti-clockwise direction ( positive direction ) Here, a figure rotates in opposite hand of a clock in an anti-clockwise direction. 1. Clockwise direction ( Negative direction ) Here, In the clockwise direction, we rotate a figure in the same direction of the rotation of the hand of a clock. Now learn how to draw the image of a figure when it is rotated through the given angle in the given direction about a given centre of rotation. Rotation through 900 in anti-clockwise direction FIGURE 1.The dotted line of each vertex of figure should join to the centre of rotation 2.Draw 900 at o in each dotted lines with the help of a protractor in an anti-clockwise direction . 3.Cut off 0A’ = 0A, 0B’ = 0B and 0C’ = 0C. 4.Now, join A’, B’, and C’ . Rotation through 900 in clockwise direction FIGURE Rotation of geometrical figure using coordinates Look the following figure and learn to find out the coordinates of the image of a point when it is rotated through 900 and 1800 in anti-clockwise and clockwise directions about the centre of rotation at the origin . Rotation through 900 in anti-clockwise about origin FIGURE Rotation through 900 in clockwise about origin FIGURE In clockwise, a point is rotated through 90 in a direction about the origin as the centre of rotation, x and y-coordinates are exchanged by making the sign of y-coordinate just opposite. Rotation through 1800 in anti- clockwise and clockwise about origin Here a point rotated through 1800 about the origin , then the coordinates of the image are the same in both directions. Learn the following figure . FIGURE In anti-clockwise, a point is rotated through 900 or in a clockwise direction about the origin as the centre of rotation ,x and y-coordinates the image remains the same just by changing their signs. Things to remember • Congruent transformation change in position but no change in size of an objects . • Fixed line is called the mirror or axis of reflection . • Congruent transformation also known as isometric transformation. • It includes every relationship which established among the people. • There can be more than one community in a society. Community smaller than society. • It is a network of social relationships which cannot see or touched. • common interests and common objectives are not necessary for society. Transformations Solution: According to the question, A (3, -3) → A ' (-3, -3) B( 4, 2) → B ' (-4, 3) C(1, 3) → C ' (-1, 3) So, the coordinates of the vertices of image ΔA'B'C' are A' (-3, -3), B(-4, 3) and C' (-1, 3) Solution: According to the given information, A(-3, 4) → A' (-3, -4) B(3,2) → B' (3, -2) C(5,5) → C' (5, -5) So, the coordinates of the vertices of image ΔA'B'C' are A' (-3, -4), B' (3, -2) and c' (5, -5). Solution Here, A(2,4) translated by ($\frac{-5}{2}$) and then ( 2-5, 4 +2) = (-3,6) , i.e C is obtained. The distance from B to D, N to M and C are parallel as well as equal. So, M maps N and D maps B. Solution: According to the question, A (-2, 3) → A ' (-2, -3) B( 1, 2) → B ' (1, -2) C(3, 3) → C ' (3, -3) So, the coordinates of the vertices of image ΔA'B'C' are A' (-2, -3), B(1, -2) and C' (3, -3) Solution: According to the question, A (2, -3) → A ' (-2,-3) B( 2, 1) → B ' (-2, 1) C(3, 2) → C ' (-3, 2) So, the coordinates of the vertices of image ΔA'B'C' are A' (-2, -3), B(-2, 3) and C' (-3, 2) Solution: According to the question, A(-1, -2) → A ' (-1, 2) B( 3, 2) → B ' (3, -2) C(1, 2) → C ' (1, -2) So, the coordinates of the vertices of image ΔA'B'C' are A' (-1, -2), B(3, -2) and C' (1, -2) Solution: Join each of the vertex to the centre of rotation. Draw angle of 90O at O with OC, OA and OB in an anticlockwise direction. Cut OA = OA' , OC' = OC' and OB = OB' Join A' B' and C' ΔA'B'C' is image of ΔABC Solution P(x,y) → P' (-y, x) A(-3, 4) → A(4, 3) B(4,3) →B(-4, 3) Solution: According to the question, When ΔABC is rotated through 90º in an anti-clockwise direction about the origin. A(3,2)→ A' (-3, 2) B' (2,1)→B' (-2, 1) C(2,3) →C' (-2, 3) So, the coordinates of the vertices of image ΔA'B'C are A (-3, 2), B(-2, 1) and C(-2, 3). Transformation is the process of changing the object either in position or in size or in both. Reflection is a rule which shifts the same object to the object form of each being at an equal distance from the fixed line. The fixed point is known as the centre of rotation. Reflection and Rotation are the examples of congruent transformation. In the clockwise direction (Negative direction), we rotate a figure in the same direction of the rotation.	{"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": 0.7558577060699463, "perplexity": 1262.7062524211528}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710503.24/warc/CC-MAIN-20221128102824-20221128132824-00545.warc.gz"}
https://www.lernapehlivan.com/publication/positive-integers-represented-by-regular-primitive-positive-definite-integral-ternary-quadratic-forms/	# Positive Integers Represented by Regular Primitive Positive-Definite Integral Ternary Quadratic Forms ### Abstract In 1997 Jagy, Kaplansky and Schiemann determined that there are at most 913 (classes of) primitive, positive-definite, integral ternary quadratic forms $ax^2 + by^2 + cz^2 + dyz + ezx + fxy$ which are regular. In this paper the positive integers represented by these 913 ternary forms are given. Publication In INTEGERS, Volume 19 (2019), A45	{"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": 0.911078155040741, "perplexity": 2446.6302060189437}, "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/1593655890181.37/warc/CC-MAIN-20200706191400-20200706221400-00083.warc.gz"}
http://mathhelpforum.com/geometry/26245-pythagorean-theorem-print.html	# Pythagorean Theorem. • January 16th 2008, 05:50 PM overduex Pythagorean Theorem. Help! My sister needs help, I don't know how to explain this to her, can anyone please explain how?? Directions: Determine whether each triangle with the given side lengths is a right triangle. Write yes or no. EX: 10 ft, 24 ft, 25 ft Thanks! • January 16th 2008, 06:07 PM mr fantastic Quote: Originally Posted by overduex Help! My sister needs help, I don't know how to explain this to her, can anyone please explain how?? Directions: Determine whether each triangle with the given side lengths is a right triangle. Write yes or no. EX: 10 ft, 24 ft, 25 ft Thanks! Does $10^2 + 24^2 = 25^2$? Note: The hypotenuse is the longest side of a right triangle. • January 16th 2008, 06:30 PM OnMyWayToBeAMathProffesor triangle so, we can see it is not a right triangle, because if it were, the Pythagorean theorem would work, but it does not, so it is not a right triangle. it would be if the side were 10, 24 and 26.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9390985369682312, "perplexity": 663.3165114106839}, "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-2014-41/segments/1412037662910.19/warc/CC-MAIN-20140930004102-00364-ip-10-234-18-248.ec2.internal.warc.gz"}
http://adaptive-networks.wikidot.com/paper:ren2006	Ren2006 # Interplay between evolutionary game and network structure: the coevolution of social net, cooperation and wealth J. Ren, X. Wu, W.-X. Wang, G. Chen, B.-H. Wang arXiv:physics/0605250 We study the interplay between evolutionary game and network structure and show how the dynamics of the game affect the growth pattern of the network and how the evolution of the network influence the cooperative behavior in the game. Simulation results show that the payoff-based preferential attachment mechanism leads to the emergence of a scale-free structural property, $P(k)\sim k^{-\gamma}$. Moreover, we investigate the average path length and the assortative mixing features. The obtained results indicate that the network has small-world and positive assortative behaviors, which are consistent with the observations of some real social networks. In parallel, we found that the evolution of the underlying network structure effectively promotes the cooperation level of the game. We also investigate the wealth distribution obtained by our model, which is consistent with the Pareto law in the real observation. In addition, the analysis of the generated scale-free network structure is provided for better understanding the evolutionary dynamics of our model. This paper on arXiv page revision: 4, last edited: 03 Dec 2009 16:38 Materials on this page may be the property of the authors and/or journals named.	{"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": 0.9703553318977356, "perplexity": 694.3366464222125}, "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/1508187820700.4/warc/CC-MAIN-20171017033641-20171017053641-00187.warc.gz"}
https://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00006.html	Re: Replacement Rule with Sqrt in denominator • To: mathgroup at smc.vnet.net • Subject: [mg114332] Re: Replacement Rule with Sqrt in denominator • From: Richard Fateman <fateman at cs.berkeley.edu> • Date: Wed, 1 Dec 2010 02:12:17 -0500 (EST) On 11/30/2010 8:22 AM, Murray Eisenberg wrote: > It's not a bug (and not really a "feature", either): > > FullForm[1/Sqrt[x]] > Power[x,Rational[-1,2]] > > Thus there is no "Sqrt[x] really in 1/Sqrt[x] (the latter being just a > quick input form), just as there is no "I" in 1+2I. The bug is not in the representation. It is in the rule processing. After all, a rule replacing Sqrt[x] by y could also replace (x)^(-1/2) by y^(-1). That would "fix" the "feature". > > On 11/30/2010 4:05 AM, Richard Fateman wrote: >> On 11/20/2010 3:11 AM, kj wrote: >> >>> >>> I don't know a way to write the replacement rule so that it works >>> for both Sqrt[x] and 1/Sqrt[x], but maybe someone else does. >>> >>> ~kj >>> >> See >> www.cs.berkeley.edu/~fateman/papers/better-rules.pdf >> >> which describes how one can automatically produce rules that overcome >> this bug in Mathematica. oops. I mean rules that overcome this >> "feature" in Mathematica. >> >> RJF >> >> > To quote from the above paper... The argument that Mathematica already does the right thing by \literally" matching expressions loses much of its force by observing how frequently mathematicians report its behavior as erroneous. This report by kj is just another incident. You could argue "the customer is always wrong", but you don't really have to do that. See the referenced paper. RJF • Prev by Date: Re: Re-virginating Manipulates? • Next by Date: Re: C compiler error Mathematica 8 and Cuda • Previous by thread: Re: Replacement Rule with Sqrt in denominator • Next by thread: Re: Replacement Rule with Sqrt in denominator	{"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": 0.9248678684234619, "perplexity": 15805.506071996339}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337731.82/warc/CC-MAIN-20221006061224-20221006091224-00102.warc.gz"}
https://tivadardanka.com/blog/probability-is-logic/	# Probability is logic "Probability is the logic of science." There is a deep truth behind this conventional wisdom: probability is the mathematical extension of logic, augmenting our reasoning toolkit with the concept of uncertainty. In-depth exploration of probabilistic thinking incoming. Our journey ahead has three stops: 1. an introduction to mathematical logic, 2. a touch of elementary set theory, 3. and finally, understanding probabilistic thinking. First things first: mathematical logic. ## Mathematical logic 101 In logic, we work with propositions. A proposition is a statement that is either true or false, like "it's raining outside" or "the sidewalk is wet". These are often abbreviated as variables, such as $A = \text{"it's raining outside"}.$ We can formulate complex propositions from smaller building blocks with logical connectives. For example, consider the proposition "if it is raining outside, then the sidewalk is wet". This is the combination of two propositions, connected by the implication connective. There are four essential connectives: • NOT ($\neg$), also known as negation, • AND ($\land$), • OR ($\lor$), • THEN ($\to$), also known as implication. Connectives are defined by the truth values of the resulting propositions. For instance, if A is true, then NOT A is false; if A is false, then NOT A is true. Denoting true by 1 and false by 0, we can describe connectives with truth tables. Here is the one for negation ($\neg$). AND ($\land$) and OR ($\lor$) connect two propositions. $A \land B$ is true if both $\textstyle A$ and $\textstyle B$ are true, and $A \lor B$ is true if either one is. The implication connective THEN ($\to$) formalizes the deduction of a conclusion $\textstyle B$ from a premise $\textstyle A$. By definition, $A \to B$ is true if $\textstyle B$ is true, or both $\textstyle A$ and $\textstyle B$ are false. An example: if "it's raining outside", THEN "the sidewalk is wet". Science is just the collection of complex propositions like "if X is a closed system, THEN the entropy of X cannot decrease". (As the 2nd law of thermodynamics states.) The entire body of scientific knowledge is made of $A \to B$ propositions. In practice, our thinking process is the following: "I know that $A \to B$ is true and $\textstyle A$ is true. Therefore, $\textstyle B$ must be true as well." This is called modus ponens, the cornerstone of scientific reasoning. (If you don't understand modus ponens, take a look at the truth table of the $\to$ connective, a few paragraphs above. The case when $A \to B$ is true and $\textstyle A$ is true is described by the very first row, which can only happen if $\textstyle B$ is true as well.) ## Set theory = logic (more or less) Logical connectives can be translated to the language of sets. Union ($\cup$) and intersection ($\cap$), two fundamental operations, are particularly relevant to us. Notice how similar the symbols for AND ($\land$) and intersection ($\cap$) are? This is not an accident. By definition, any element $\textstyle x$ is the element of $A \cap B$ if and only if ($\textstyle x$ is an element of $\textstyle A$) AND ($\textstyle x$ is an element of $\textstyle B$). Similarly, union corresponds to the OR connective, as the figure below shows. What's most important for us is that the implication connective THEN ($\to$) corresponds to the "subset of" relation, denoted by the $\subseteq$ symbol. Now that we understand how to formulate scientific truths as "premise $\to$ conclusion" statements and see how this translates to sets, we are finally ready to talk about probability. ## Probability as logic What is the biggest flaw of mathematical logic? That we rarely have all the information to decide if a proposition is true or false. Consider the following: "it'll rain tomorrow". During the rainy season, all we can say is that rain is more likely, but tomorrow can be sunny as well. Probability theory generalizes classical logic by measuring truth on a scale between 0 and 1, where 0 is false and 1 is true. If the probability of rain tomorrow is 0.9, it means that rain is significantly more likely, but not absolutely certain. Instead of propositions, probability operates on events. In turn, events are represented by sets. For example, if I roll a dice, the event "the result is less than five" is represented by the set $A = {1, 2, 3, 4}$. In fact, $P(A) = 4/6$. ($\textstyle P$ denotes the probability of an event.) As discussed earlier, the logical connectives AND and OR correspond to basic set operations: AND is intersection, OR is union. This translates to probabilities as well. How can probability be used to generalize the logical implication? A "probabilistic $A \to B$" should represent the likelihood of $\textstyle B$, given that $\textstyle A$ is observed. This is formalized by conditional probability. At the deepest level, the conditional probability $P(B \| A)$ is the mathematical formulation of our belief in the hypothesis $\textstyle B$, given empirical evidence $\textstyle A$. A high $P(B \| A)$ makes $\textstyle B$ more likely to happen, given that $\textstyle A$ is observed. On the other hand, a low $P(B \| A)$ makes $\textstyle B$ less likely to happen when $\textstyle A$ occurs as well. This is why probability is called the logic of science. To give you a concrete example, let's go back to the one mentioned earlier: the rain and the wet sidewalk. For simplicity, denote the events by \begin{align} A &= \text{"the sidewalk is wet"}, \\ B &= \text{"it's raining outside"}. \end{align} The sidewalk can be wet for many reasons, say the neighbor just watered the lawn. Yet, the primary cause of a wet sidewalk is rain, so $P(B \| A)$ is close to 1. If somebody comes in and tells you that the sidewalk is wet, it is safe to infer rain. Probabilistic inference like the above is the foundation of machine learning. For instance, the output of (most) classification models is the distribution of class probabilities, given an observation. To wrap up, here is how Maxwell — the famous physicist — thinks about probability. "The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man's mind. — James Clerk Maxwell" By now, you can fully understand what Maxwell meant.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 57, "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": 0.9402099847793579, "perplexity": 638.87613178071}, "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/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00361.warc.gz"}
https://davidlindell.com/publications/spad-transient	### Reconstructing transient images from single-photon sensors Matthew O'Toole, Felix Heide, David B. Lindell, Kai Zang, Steven Diamond, Gordon Wetzstein Capturing and reconstructing transient images with single-photon avalanche diodes (SPAD) at interactive rates. ### Abstract Computer vision algorithms build on 2D images or 3D videos that capture dynamic events at the millisecond time scale. However, capturing and analyzing “transient images” at the picosecond scale—i.e., at one trillion frames per second—reveals unprecedented information about a scene and light transport within. This is not only crucial for time-of-flight range imaging, but it also helps further our understanding of light transport phenomena at a more fundamental level and potentially allows to revisit many assumptions made in different computer vision algorithms. In this work, we design and evaluate an imaging system that builds on single photon avalanche diode (SPAD) sensors to capture multi-path responses with picosecond-scale active illumination. We develop inverse methods that use modern approaches to deconvolve and denoise measurements in the presence of Poisson noise, and compute transient images at a higher quality than previously reported. The small form factor, fast acquisition rates, and relatively low cost of our system potentially makes transient imaging more practical for a range of applications. Color Image Transient Image Statue of David Fiber Pepsi Bottle Fruit Foam Box Resolution Chart Fiber: CVPR 2017 ### Acknowledgments The authors are grateful to Edoardo Charbon, Pierre-Yves Cattin, and Samuel Burri for providing the LinoSPAD sensor used in this work and continued support of it. O’Toole is supported by the Government of Canada through the Banting Postdoctoral Fellowships program, and Wetzstein is supported by a National Science Foundation CAREER award (IIS 1553333), a Terman Faculty Fellowship, a Sloan Fellowship, the Center for Automotive Research at Stanford (CARS), and by the KAUST Office of Sponsored Research through the Visual Computing Center CCF grant. ### Citation	{"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": 0.24353304505348206, "perplexity": 5076.180334217997}, "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/1664030335504.37/warc/CC-MAIN-20220930212504-20221001002504-00439.warc.gz"}
http://proceedings.mlr.press/v70/baram17a.html	# End-to-End Differentiable Adversarial Imitation Learning Nir Baram, Oron Anschel, Itai Caspi, Shie Mannor ; Proceedings of the 34th International Conference on Machine Learning, PMLR 70:390-399, 2017. #### Abstract Generative Adversarial Networks (GANs) have been successfully applied to the problem of policy imitation in a model-free setup. However, the computation graph of GANs, that include a stochastic policy as the generative model, is no longer differentiable end-to-end, which requires the use of high-variance gradient estimation. In this paper, we introduce the Model-based Generative Adversarial Imitation Learning (MGAIL) algorithm. We show how to use a forward model to make the computation fully differentiable, which enables training policies using the exact gradient of the discriminator. The resulting algorithm trains competent policies using relatively fewer expert samples and interactions with the environment. We test it on both discrete and continuous action domains and report results that surpass the state-of-the-art.	{"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": 0.8098111748695374, "perplexity": 1611.4774826426037}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514355.90/warc/CC-MAIN-20181021203102-20181021224602-00390.warc.gz"}
http://en.wikipedia.org/wiki/Galaxy_rotation_curve	Galaxy rotation curve Rotation curve of a typical spiral galaxy: predicted (A) and observed (B). The discrepancy between the curves can be accounted for by adding a dark matter component to the galaxy.[1] The rotation curve of a disc galaxy (also called a velocity curve) is a plot of the magnitude of the orbital velocities (i.e., the speeds) of visible stars or gas in that galaxy versus their radial distance from that galaxy's center, typically rendered graphically as a scatter plot. A general feature of the galaxy rotation curves that have been obtained through measurement to date is that the orbital speed of stars and gas is almost constant as far from the galactic centre as can be measured (line B in the illustration): that is, stars are observed to revolve around the centre of the galaxy at almost the same speed over a large range of distances from the centre of the galaxy. If disc galaxies had mass distributions similar to the observed distributions of stars and gas then the orbital speed would decline at increasing distances (dotted line A in illustration) in the same way as do other systems with most of their mass in the centre, such as the Solar System or the moons of Jupiter. The rotation curves of Spiral galaxies are also known to be asymmetric. The observational data from each side of a galaxy are generally averaged. Therefore, only highly asymmetric cases are well known. But smaller asymmetries are also reported.[2] RC asymmetry appears to be normal rather than exceptional.[3] The galaxy rotation problem is the discrepancy between observed galaxy rotation curves and the theoretical prediction, assuming a centrally-dominated mass associated with the observed luminous material. When masses of galaxies are calculated solely from the luminosities and mass-to-light ratios in the disk, and if core portions of spiral galaxies are assumed to approximate to those of stars, the masses derived from the kinematics of the observed rotation and the law of gravity do not match. This discrepancy can be accounted for by postulating a large amount of dark matter that permeates the galaxy and extends into the galaxy's halo. Though dark matter is by far the most accepted explanation for the resolution to the galaxy rotation problem, other proposals have been offered with varying degrees of success. Of the possible alternatives, the most notable is Modified Newtonian Dynamics (MOND), which involves modifying the laws of gravity.[4] History and description of the galaxy rotation problem In 1932 Jan Hendrik Oort became the first to report measurements that the stars in the Solar neighborhood moved faster than expected when a mass distribution based upon visible matter was assumed, but this measurement was later determined to be essentially erroneous.[5] In 1933, Fritz Zwicky postulated "missing mass" to account for the orbital velocities of galaxies in clusters. In 1939, Horace Babcock reported in his PhD thesis measurements of the rotation curve for Andromeda which suggested that the mass-to-luminosity ratio increases radially.[6] He, however, attributed it to either absorption of light within the galaxy or modified dynamics in the outer portions of the spiral and not to any form of missing matter. In 1959, Louise Volders demonstrated that spiral galaxy M33 does not spin as expected according to Keplerian dynamics.[7] Following this, in the late 1960s and early 1970s, Vera Rubin, a young astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington worked with a new sensitive spectrograph that could measure the velocity curve of edge-on spiral galaxies to a greater degree of accuracy than had ever before been achieved.[8] Together with fellow staff-member Kent Ford, Rubin announced at a 1975 meeting of the American Astronomical Society the discovery that most stars in spiral galaxies orbit at roughly the same speed,[9] which implied that their mass densities were uniform well beyond the location with most of the stars (the galactic bulge), a result independently found in 1978.[10] Rubin presented her results in an influential paper in 1980.[11] These results suggest that either Newtonian gravity does not apply universally or that, conservatively, upwards of 50% of the mass of galaxies was contained in the relatively dark galactic halo. Met with skepticism, Rubin insisted that the observations were correct. Based on Newtonian mechanics and assuming, as was originally thought, that most of the mass of the galaxy had to be in the galactic bulge near the center, matter (such as stars and gas) in the disk portion of a spiral should orbit the center of the galaxy similar to the way in which planets in the solar system orbit the sun, i.e. where the average orbital speed of an object at a specified distance away from the majority of the mass distribution would decrease inversely with the square root of the radius of the orbit (the dashed line in Fig. 1). Observations of the rotation curve of spirals, however, do not bear this out. Rather, the curves do not decrease in the expected inverse square root relationship but are "flat", i.e. outside of the central bulge the speed is nearly a constant (the solid line in Fig. 1). It is also observed that galaxies with a uniform distribution of luminous matter have a rotation curve that slopes up from the center to the edge, and most low-surface-brightness galaxies (LSB galaxies) rotate with a rotation curve that slopes up from the center, indicating little core bulge. The rotation curves can be explained if there is a substantial amount of matter permeating the galaxy that is not emitting light in the mass-to-light ratio of the central bulge. The material responsible for the extra mass was dubbed, "dark matter", the existence of which was first posited in the 1930s by Jan Oort in his measurements of the Oort constants and Fritz Zwicky in his studies of the masses of galaxy clusters, though these proposals were left unexplored until after Rubin's work was accepted as correct. The existence of non-baryonic cold dark matter (CDM) is today a major feature of the Lambda-CDM model that describes the cosmology of the universe. Halo density profiles In order to accommodate a flat rotation curve, a density profile for galactic environs must be different than one that is centrally concentrated. Newton's version of Kepler's Third Law states that the radial density profile ρ(r) equals $\rho(r) = \frac{3 [v(r)]^2}{4 \pi G r^2}$ where v(r) is the radial orbital velocity profile and G is the gravitational constant. This profile closely matches the expectations of a singular isothermal sphere profile where if v(r) is approximately constant then the density $\rho \sim r^{-2}$ to some inner "core radius" where the density leveled off to a constant. Observations did not comport with such a simple profile as reported by Navarro, Frenk, and White in a seminal 1996 paper: If more massive halos were indeed associated with faster rotating disks and so with brighter galaxies, a correlation would be expected between the luminosity of binary galaxies and the relative velocity of their components. Similarly, there should be a correlation between the velocity of a satellite galaxy relative to its primary and the rotation velocity of the primary's disk. No such correlations are apparent in existing data.[12] The authors then remarked, as did a few others before them, that a "gently changing logarithmic slope" for a density profile could also accommodate approximately flat rotation curves over large scales. They wrote down the famous Navarro–Frenk–White profile which is consistent both with N-body simulations and observations given by $\rho (r)=\frac{\rho_0}{\frac{r}{R_s}\left(1~+~\frac{r}{R_s}\right)^2}$ where the central density, ρ0, and the scale radius, Rs, are parameters that vary from halo to halo. In part because the slope of the density profile diverges at the center, other alternative profiles have been proposed, for example, the Einasto profile which has exhibited as good or better agreement with certain dark matter halo simulations.[13][14] Further investigations The rotational dynamics of galaxies are, in fact, extremely well characterized by their position on the Tully-Fisher relation which shows that for spiral galaxies that rotational velocity is uniquely related to its total luminosity with essentially no scatter. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity and then extrapolate its rotation curve from its location on the Tully-Fisher diagram. Likewise, knowing the rotational velocity of a spiral galaxy is an excellent indication of its luminosity. Thus the amplitude of the galaxy rotation curve is related to the galaxy's visible mass. While precise fitting bulge, disk, and halo density profiles is a rather complicated process, it is straightforward to model the observables of rotating galaxies through this relationship.[15] So, while state-of-the-art cosmological and galaxy formation simulations of dark matter with normal baryonic matter included can be matched to galaxy observations, there is not yet any straightforward explanation as to why the scaling relationship exists as observed.[16][17] Additionally, detailed investigations of the rotation curves of low-surface-brightness galaxies (LSB galaxies) in the 1990s[18] and of their position on the Tully-Fisher relation[19] showed that LSB galaxies had to have dark matter haloes that are more extended and less dense than those of HSB galaxies and thus surface brightness is related to the halo properties. Such dark matter-dominated dwarf galaxies may hold the key to solving the dwarf galaxy problem of structure formation. Additionally, analysis of the centres of low surface brightness galaxies showed that the shape of the rotation curves in the centre of dark-matter dominated systems, indicated a profile that differed from the NFW spatial mass distribution profile.[20] This so-called cuspy halo problem of cold dark matter is requires detail modeling and understanding of the feedback mechanisms in the innermost regions of galaxies.[21] That dark matter theory continues to be supported as an explanation for galaxy rotation curves is because the evidence for dark matter is not solely derived from these curves. It has been uniquely successful in simulating the formation of the large scale structure seen in the distribution of galaxies and in explaining the dynamics of groups and clusters of galaxies.[22] Dark matter also correctly predicts the results of gravitational lensing observations, see especially the Bullet Cluster. Left: A galaxy with a rotation curve as predicted before the effects of dark matter were known. Right: A galaxy with a flat rotation curve. Alternatives to dark matter There are a number of attempts to solve the problem of galaxy rotation curves without invoking dark matter. One of the most discussed alternatives is MOND (Modified Newtonian Dynamics), originally proposed by Mordehai Milgrom as a phenomenological explanation back in 1983 but which has been seen to have predictive power in accounting for galaxy rotation curves. This posits that the physics of gravity changes at large scale but, until recently, was not a relativistic theory. However, this changed with the development by Jacob Bekenstein of the tensor–vector–scalar gravity (TeVeS) theory,[4][23] enabling gravitational lensing to be covered by the theory. Another, similar, alternative is the relativistic modified gravity (MOG) theory, also called scalar–tensor–vector gravity (STVG), of John Moffat.[24] Brownstein and Moffat [25] applied MOG and MOND to the question of galaxy rotation curves, and demonstrated excellent fits to a large sample of over 100 low-surface-brightness (LSB), high surface brightness (HSB) and dwarf galaxies.[26] Each galaxy rotation curve fit was made without dark matter, using only the available photometric data (stellar matter and visible gas) and a two-parameter mass distribution model which made no assumption regarding the mass to light ratio. The MOG results were compared to MOND and were nearly indistinguishable right out to the edge of the rotation curve data, where MOND predicts a forever flat rotation curve, but MOG predicts an eventual return to the familiar inverse-square gravitational force law. Although these alternatives are not considered by the astronomical community to be as convincing as the dark matter model,[27][28] gravitational lensing studies have been proposed as the means to separate the predictions of the different theories. Indeed, gravitational lensing by the Bullet Cluster was reported to provide the best current evidence for the nature of dark matter[29][30] and to provide "evidence against some of the more popular versions of Modified Newtonian Dynamics (MOND)" as applied to large galactic clusters.[31] In retort, Milgrom, the original proposer of MOND, posted a rejoinder online[32] that claims MOND correctly accounts for the dynamics of galaxies outside of galaxy clusters, and removes the need for most dark matter in clusters, leaving twice as much matter as is visible, which Milgrom expects to be simply unseen ordinary matter rather than exotic cold dark matter. Some Quantum Gravity theories also give alternative explanations, see alternative theories to dark matter. Footnotes 1. ^ "The generally accepted explanation of the mass discrepancy is the proposal that spiral galaxies consist of a visible component surrounded by a more massive and extensive dark component .." is stated in the introduction of the article: K.G. Begeman, A.H. Broeils, R.H.Sanders (1991). "Extended rotation curves of spiral galaxies: dark haloes and modified dynamics". Monthly Notices of the Royal Astronomical Society 249: 523–537. Bibcode:1991MNRAS.249..523B. available online at the Smithsonian/NASA Astrophysics Data System. Also Figure 1 of the article has numerous galactic rotation curves qualitatively similar to this one. 2. ^ Shane, W. W. and Bieger-Smith, G. P., 1966, “The galactic rotation curve derived from observations of neutral hydrogen”, Bull. Astron. Inst. Netherlands’’, 18, 263. 3. ^ Jog, C. J., 2002, “Large-scale asymmetry of rotation curves in lopsided spiral galaxies”, A&A’’,391,471 and references therein. 4. ^ a b For an extensive discussion of the data and its fit to MOND see Mordehai Milgrom (2007). "The MOND Paradigm". arXiv:0801.3133 [astro-ph]. This paper is a talk presented at the XIX Rencontres de Blois "Matter and energy in the Universe: from nucleosynthesis to cosmology". 5. ^ Kuijken K., Gilmore G., 1989a, MNRAS, 239, 651 6. ^ Babcock, H, 1939, “The rotation of the Andromeda Nebula”, Lick Observatory bulletin ; no. 498 7. ^ L. Volders. "Neutral hydrogen in M 33 and M 101". Bulletin of the Astronomical Institutes of the Netherlands 14 (492): 323–334. 8. ^ V. Rubin, W. K. Ford, Jr (1970). "Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions". Astrophysical Journal 159: 379. Bibcode:1970ApJ...159..379R. doi:10.1086/150317. 9. ^ Rubin, V. C., Thonnard, N., and Ford, W. K., Jr., 1978, ApJ 225 L107-L111, http://adsabs.harvard.edu/abs/1978ApJ...225L.107R 10. ^ A. Bosma, "The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types", PhD Thesis, Rijksuniversiteit Groningen, 1978, available online at the Nasa Extragalactic Database 11. ^ V. Rubin, N. Thonnard, W. K. Ford, Jr, (1980). "Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)". Astrophysical Journal 238: 471. Bibcode:1980ApJ...238..471R. doi:10.1086/158003. 12. ^ Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M. (May 10, 1996). "The Structure of Cold Dark Matter Halos". The Astrophysical Journal 463: 563. arXiv:astro-ph/9508025. Bibcode:1996ApJ...462..563N. doi:10.1086/177173. 13. ^ Merritt, David; Graham, Alister; Moore, Benjamin; Diemand, Jurg; Terzić, Balsa (20 December 2006). "Empirical Models for Dark Matter Halos". The Astronomical Journal 132 (6): 2685–2700. arXiv:astro-ph/0509417. Bibcode:2006AJ....132.2685M. doi:10.1086/508988. 14. ^ Merritt, David; et al. (May 2005). "A Universal Density Profile for Dark and Luminous Matter?". The Astrophysical Journal 624 (2): L85–L88. arXiv:astro-ph/0502515. Bibcode:2005ApJ...624L..85M. doi:10.1086/430636. 15. ^ Reliance on Indirect Evidence Fuels Dark Matter Doubts: Scientific American 16. ^ Weinberg, David H.; et, al. (2008). "Baryon Dynamics, Dark Matter Substructure, and Galaxies". The Astrophysical Journal 678 (1): 6–21. Bibcode:2008ApJ...678....6W. doi:10.1086/524646. Retrieved 13 September 2012. 17. ^ Duffy, Alan R.; al., et (2010). "Impact of baryon physics on dark matter structures: a detailed simulation study of halo density profiles". Monthly Notices of the Royal Astronomical Society 405 (4): 2161–2178. arXiv:1001.3447. Bibcode:2010MNRAS.405.2161D. doi:10.1111/j.1365-2966.2010.16613.x. Retrieved 13 September 2012. 18. ^ W. J. G. de Blok, S. McGaugh (1997). "The dark and visible matter content of low surface brightness disc galaxies". Monthly Notices of the Royal Astronomical Society 290: 533–552. arXiv:astro-ph/9704274. Bibcode:1997MNRAS.290..533D. doi:10.1093/mnras/290.3.533. available online at the Smithsonian/NASA Astrophysics Data System 19. ^ M. A. Zwaan, J. M. van der Hulst, W. J. G. de Blok, S. McGaugh (1995). "The Tully-Fisher relation for low surface brightness galaxies: implications for galaxy evolution". Monthly Notices of the Royal Astronomical Society 273: L35–L38. arXiv:astro-ph/9501102. Bibcode:1995MNRAS.273L..35Z. available online at the Smithsonian/NASA Astrophysics Data System 20. ^ W. J. G. de Blok, A. Bosma (2002). "High-resolution rotation curves of low surface brightness galaxies". Astronomy & Astrophysics 385 (3): 816–846. arXiv:astro-ph/0201276. Bibcode:2002A&A...385..816D. doi:10.1051/0004-6361:20020080. available online at the Smithsonian/NASA Astrophysics Data System 21. ^ de Blok, W. G. The Core Cusp Problem. "Dwarf Galaxy Cosmology" special issue of Advances in Astrophysics. 2009. [1]. 22. ^ Peter, Annika H. G. Dark Matter: A Brief Review. Proceedings of Science. 2012. 23. ^ J. D. Bekenstein (2004). "Relativistic gravitation theory for the modified Newtonian dynamics paradigm". Physical Review D 70 (8): 083509. arXiv:astro-ph/0403694. Bibcode:2004PhRvD..70h3509B. doi:10.1103/PhysRevD.70.083509. 24. ^ J. W. Moffat (2006). "Scalar tensor vector gravity theory". Journal of Cosmology and Astroparticle Physics 3 (03): 4. arXiv:gr-qc/0506021. Bibcode:2006JCAP...03..004M. doi:10.1088/1475-7516/2006/03/004. 25. ^ http://www.arxiv.org/abs/astro-ph/0506370 astro-ph/0506370 26. ^ J. R. Brownstein and J. W. Moffat (2006). "Galaxy Rotation Curves Without Non-Baryonic Dark Matter". Astrophysical Journal 636 (2): 721. arXiv:astro-ph/0506370. Bibcode:2006ApJ...636..721B. doi:10.1086/498208. 27. ^ BBC - Science & Nature - Horizon 28. ^ Chandra Press Room :: Chandra Casts Cloud On Alternative Theory :: October 22, 2002 29. ^ M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman, C. Jones, S. Murray, and W. Tucker. Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56. arXiv:astro-ph/0309303. Bibcode:2004ApJ...606..819M. doi:10.1086/383178. 30. ^ M. Markevitch, S. Randall, D. Clowe, A. Gonzalez and M. Bradac (16–23 July 2006). "Dark Matter and the Bullet Cluster". 36th COSPAR Scientific Assembly. Beijing, China. abstract only 31. ^ http://hea-www.harvard.edu/LUNCH_TALKS/abstracts.html lunch-time talk at Harvard University by Scott Randall on 31 May 2006, abstract only 32. ^ The Bullet Cluster (Milgrom)	{"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": 3, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9312254190444946, "perplexity": 2204.39099993727}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223202548.14/warc/CC-MAIN-20140423032002-00028-ip-10-147-4-33.ec2.internal.warc.gz"}
http://math-mprf.org/journal/articles/id887/	Dynamic Critical Behavior of Cluster Algorithms for 2D Ashkin-Teller and Potts Models #### J. Salas 2001, v.7, Issue 1, 55-74 ABSTRACT We study the dynamic critical behavior of two algorithms: the Swendsen - Wang algorithm for the two-dimensional Potts model with $q=2,3,4$ and a Swendsen - Wang-type algorithm for the two-dimensional symmetric Ashkin - Teller model on the self-dual curve. We find that the Li - Sokal bound on the autocorrelation time $\tau_{{int},{\cal E}} \geq \hbox{const} \times C_H$ is almost, but not quite sharp. The ratio $\tau_{{\rm int},{\cal E}}/C_H$ appears to tend to infinity either as a logarithm or as a small power (0.05 < p < 0.12). We also show that the exponential autocorrelation time $\tau_{{\rm exp},{\cal E}}$ is proportional to the integrated autocorrelation time $\tau_{{\rm int},{\cal E}}$. Keywords: Ising model,Ashkin - Teller model,Monte Carlo,Swendsen - Wangalgorithm,cluster algorithm,dynamic critical exponent,Li - Sokal bound	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9591851234436035, "perplexity": 2430.2162063896735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737645.2/warc/CC-MAIN-20200808110257-20200808140257-00551.warc.gz"}
http://mathhelpforum.com/discrete-math/217129-combinatorial-identity-pascal-s-triangle-hexagon-identity.html	# Math Help - Combinatorial Identity, Pascal's Triangle, Hexagon Identity 1. ## Combinatorial Identity, Pascal's Triangle, Hexagon Identity Latex is pretty sweet! just learned how to use this, so for this question Prove the following combinatorial identity if ${1}\leq k. This identity is known as the hexagon identity and relates terms in Pascal's Triangle. $\left(\begin{array}{cc}n-1\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n\\k+1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k\end{array}\right)$ = $\left(\begin{array}{cc}n-1\\k\end{array}\right)$ $\left(\begin{array}{cc}n\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k+1\end{array}\right)$ where do i even begin to solve this? The Pascal's Triangle i kind of understand but the rest is news to me. 2. ## Re: Combinatorial Identity, Pascal's Triangle, Hexagon Identity Originally Posted by zhengcl86 Latex is pretty sweet! just learned how to use this, so for this question Prove the following combinatorial identity if ${1}\leq k. This identity is known as the hexagon identity and relates terms in Pascal's Triangle. $\left(\begin{array}{cc}n-1\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n\\k+1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k\end{array}\right)$ = $\left(\begin{array}{cc}n-1\\k\end{array}\right)$ $\left(\begin{array}{cc}n\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k+1\end{array}\right)$ where do i even begin to solve this? The Pascal's Triangle i kind of understand but the rest is news to me. Use the identity $\left(\begin{array}{cc}n\\k\end{array}\right)$ $=\frac{n!}{(n-k)!k!}$ and this becomes trivially true by permutation of the denominators of the factors. 3. ## Re: Combinatorial Identity, Pascal's Triangle, Hexagon Identity Originally Posted by Gusbob Use the identity $\left(\begin{array}{cc}n\\k\end{array}\right)$ $=\frac{n!}{(n-k)!k!}$ and this becomes trivially true by permutation of the denominators of the factors. is that 1<k<n? 4. ## Re: Combinatorial Identity, Pascal's Triangle, Hexagon Identity Originally Posted by zhengcl86 Prove the following combinatorial identity if ${1}\leq k. This identity is known as the hexagon identity and relates terms in Pascal's Triangle. $\left(\begin{array}{cc}n-1\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n\\k+1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k\end{array}\right)$ = $\left(\begin{array}{cc}n-1\\k\end{array}\right)$ $\left(\begin{array}{cc}n\\k-1\end{array}\right)$ $\left(\begin{array}{cc}n+1\\k+1\end{array}\right)$ The LHS is $\frac{(n-1)!}{(k-1)!(n-k)!}\cdot\frac{n!}{(k+1)!(n-k-1)!}\cdot\frac{(n+1)!}{(k!)(n-k+1)!}$ REARRANGE: $\frac{(n-1)!}{(k)!(n-k-1)!}\cdot\frac{n!}{(k-1)!(n-k+1)!}\cdot\frac{(n+1)!}{(k+1)!(n-k)!}$ 5. ## Re: Combinatorial Identity, Pascal's Triangle, Hexagon Identity Originally Posted by Plato The LHS is $\frac{(n-1)!}{(k-1)!(n-k)!}\cdot\frac{n!}{(k+1)!(n-k-1)!}\cdot\frac{(n+1)!}{(k!)(n-k+1)!}$ REARRANGE: $\frac{(n-1)!}{(k)!(n-k-1)!}\cdot\frac{n!}{(k-1)!(n-k+1)!}\cdot\frac{(n+1)!}{(k+1)!(n-k)!}$ is the rearranged part the answer? i kinda have a similar answer but a couple faults on the 3rd spot where n+1 / (k+1) (n+1-k-1) somehow i simplified it down to (k-n)! (n-k-1)! = (n-k+1)! (n+1-k-1)! 6. ## Re: Combinatorial Identity, Pascal's Triangle, Hexagon Identity Originally Posted by zhengcl86 is the rearranged part the answer? i kinda have a similar answer but a couple faults on the 3rd spot where n+1 / (k+1) (n+1-k-1) somehow i simplified it down to (k-n)! (n-k-1)! = (n-k+1)! (n+1-k-1)! ${\color{red}\frac{(n-1)!}{(k)!(n-k-1)!}}\cdot{\color{blue}\frac{n!}{(k-1)!(n-k+1)!}}\cdot\color{green}\frac{(n+1)!}{(k+1)!(n-k)!}$ ${\color{red}\binom{n-1}{k}}\cdot{\color{blue}\binom{n}{k-1}}\cdot{\color{green}\binom{n+1}{k+1}}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 31, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9102818965911865, "perplexity": 1912.4705674789425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826759.85/warc/CC-MAIN-20160723071026-00006-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.lmfdb.org/knowledge/show/cmf.dimension	show · cmf.dimension all knowls · up · search: The dimension of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms. The dimension of a newform refers to the dimension of the $\Q$-vector space spanned by its Galois conjugates, equivalently, its orbit under the Hecke operators. This is equal to the degree of its coefficient field (as an extension of $\Q$). The relative dimension of $S_k^{\rm new}(N,\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newforms $f\in S_k^{\rm new}(N,\chi)$. Authors: Knowl status: • Review status: reviewed • Last edited by Andrew Sutherland on 2019-01-30 16:01:38 Referred to by: History: Differences	{"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": 0.9515941143035889, "perplexity": 315.3174798750879}, "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-2020-05/segments/1579250607314.32/warc/CC-MAIN-20200122161553-20200122190553-00336.warc.gz"}
http://mathhelpforum.com/advanced-algebra/79365-no-simple-groups-order-48-a.html	# Thread: No simple groups of order 48 1. ## No simple groups of order 48 Prove there are no simple groups of order 48 Ok, here's what i have so far. $48=2^43$ $n_{2} = 1$ mod $2$ & $n_{2} \| 3$ $n_{2} = 1$ or $3$ $n_{3} = 1$ mod $3$ & $n_{3} \| 16$ $n_{3} = 1, 4,$ or $16$ So we must eliminate these cases to get $n_{2} = 1$ or $n_{3} = 1$: $(n_{2}, n_{3})=(3,4)$ $(n_{2}, n_{3})=(3,16)$ How do I eliminate these cases? 2. Originally Posted by frankdent1 Prove there are no simple groups of order 48 Ok, here's what i have so far. $48=2^43$ $n_{2} = 1$ mod $2$ & $n_{2} \| 3$ $n_{2} = 1$ or $3$ $n_{3} = 1$ mod $3$ & $n_{3} \| 16$ $n_{3} = 1, 4,$ or $16$ So we must eliminate these cases to get $n_{2} = 1$ or $n_{3} = 1$: $(n_{2}, n_{3})=(3,4)$ $(n_{2}, n_{3})=(3,16)$ How do I eliminate these cases? This problem is a little involved. Consider the Sylow 2-subgroups we know that there are 1 or 3 such Sylow subgroups. If there was only one Sylow 2-subgroup then there is nothing to prove because it is invariant under conjugation and so it is normal. Thus, we will assume there are 3 Sylow 2-subgroups. Let $H,K$ be two of these three Sylow subgroups. Notice that $\|H\|,\|K\| = 16$ and $\|HK\|\|H\cap K\| = \|H\|\|K\|$. Since $HK\subseteq G \implies \|HK\| \leq 48$, therefore $\|H\cap K\|$ cannot be less than or equal to $4$. This forces $\|H\cap K\|=8$ by applying Lagrange's theorem. Since $H\cap K$ is of index $2$ in $H$ and $K$ it means $H\cap K \triangleleft H, H\cap K\triangleleft K$. Therefore, the normalizer, $N(H\cap K)$ must contain both $H$ and $K$. By Lagrange's theorem it means $\|N(H\cap K)\|>16$ is a multiple of $16$ and a divisor of $\|G\|=48$. The only possibility is $N(H\cap K) = G \implies H\cap K\triangleleft G$ and so $G$ is not simple.	{"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": 50, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9671304225921631, "perplexity": 107.78345066556136}, "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-2017-04/segments/1484560284429.99/warc/CC-MAIN-20170116095124-00160-ip-10-171-10-70.ec2.internal.warc.gz"}
http://www.sceneadvisor.com/Oregon/maximum-error-of-estimation.html	Computer Repair and Solutions / Web Design and Development Address Klamath Falls, OR 97601 (530) 255-4874 # maximum error of estimation Chiloquin, Oregon Relative error in the volume is calculated by dividing the error by the total volume. Another area of inferential statistics is sample size determination. Equivalent Systems Solving of System of Two Equation with Two Variables. Well... Source(s): Experience ไนหกเก ไฐสศฐศ ไ๐จฐร ไเหฮฐไ๐โ＇ร ＆ ไเววไฐ＇ร ไฐพพกร · 5 years ago 0 Thumbs up 0 Thumbs down Comment Add a comment Submit · just now Report Abuse The maximum maximum-likelihood share\|improve this question asked Mar 2 '14 at 19:55 Stefan 16816 What $60$ stands for? –Alecos Papadopoulos Mar 2 '14 at 20:05 Do you have a If you're in need of an estimate of the variance of the front leg length of red-eyed tree frogs, you'll probably be able to find it in a research paper reported Table of Contents Υπενθύμιση αργότερα Έλεγχος Υπενθύμιση απορρήτου από το YouTube, εταιρεία της Google Παράβλεψη περιήγησης GRΜεταφόρτωσηΣύνδεσηΑναζήτηση Φόρτωση... Επιλέξτε τη γλώσσα σας. Κλείσιμο Μάθετε περισσότερα View this message in English Το the t-value on the right side of the equation depends on n: That's not particularly helpful given that we are trying to findn! What is the maximum error in using this value of the radius to compute the volume of the sphere? After all, scientific research is typically not done in a vacuum. Mitch Keller 6.099 προβολές 6:22 95% confidence margin of error - Διάρκεια: 1:51. For example, if we change the sample variance tos2= 82, then the necessary sample sizes for variouserrorsεand confidence levels(1−α) become: Factors Affecting the Sample Size If we take a look back The radius of a sphere was measured and found to be 20 cm with a possible error in measurement of at most 0.01 cm. Margin of error occurs whenever a population is incompletely sampled." If you have the maximum error of estimation, then you have the large amount of random sampling error Good luck! A level of confidence is the probability that the interval estimate will contain the parameter. So, the maximum error in the calculated volume is about 50.27\ cm^3. Show all work. (D) For this same sample of n = 45, what is the width of the confidence interval around the population mean? Intuitively, the Fisher information indicates the steepness of the curvature of the log-likelihood surface around the MLE, and so the amount of 'information' that $y$ provides about $\theta$. The basic confidence interval for a symmetric distribution is set up to be the point estimate minus the maximum error of the estimate is less than the true population parameter which Area in Tails Since the level of confidence is 1-alpha, the amount in the tails is alpha. You can only upload photos smaller than 5 MB. Let me say that again: Statistics are calculated, parameters are estimated. Please log in or register to use bookmarks. For this reason, statisticians like to give an interval estimate which is a range of values used to estimate the parameter. Since $\hat{\alpha}$ is just a fixed real number, I don't see in what way it could have a standard error. Linked 0 Obtaining Uncertainity from MLE 4 Confidence interval and sample size multinomial probabilities Related 4Maximum Likelihood Estimation2Maximum Likelihood estimation of a function1Maximum Likelihood Estimation and Standard Errors3Stuck on a maximum N(e(s(t))) a string Is it possible for NPC trainers to have a shiny Pokémon? Point Estimates There are two types of estimates we will find: Point Estimates and Interval Estimates. Show all work. (E) Given this same confidence level and standard deviation, find n if E = 3.5. (Always round to the nearest whole person.) Show all work. up vote 13 down vote favorite 9 I'm a mathematician self-studying statistics and struggling especially with the language. Mathispower4u 5.720 προβολές 6:44 Maximum Error in Trapezoidal Rule & Simpson's Rule READ DESCRIPTION - Διάρκεια: 20:13. Is there a difference between u and c in mknod Open git tracked files inside editor Can an umlaut be written as a line in handwriting? De Moivre's Formula Converting Proper Fraction into Infinite Periodic Decimal Converting Infinite Periodic Decimal into Proper Fraction Number Plane.Cartesian Coordinate System in the Plane and Space Coordinate Line Polar Coordinate System In these cases, the statistics can't be used since the sample hasn't been taken yet. Function y=ln(x) Raising Binomial to the Natural Power (Newton's Binom Formula) Rational Fraction and its Basic Property Reducing of Rational Fractions Reducing Rational Fractions to the Common Denominator Definition of Trigonometric A confidence interval is an interval estimate with a specific level of confidence. Public huts to stay overnight around UK Make an ASCII bat fly around an ASCII moon Uploading a preprint with wrong proofs UV lamp to disinfect raw sushi fish slices Wardogs You can see an example of this generalization from some of the numbers generated in that last example: (2) As the confidence level(1−α)100% increases, thenecessary sample size increases. Function y=e^x. So, if there is a confidence level which isn't given above, all you need to do to find it is divide the confidence level by two, and then look up the A confidence interval is an interval estimate with a specific level of confidence. Show all your calculations. Also notice - if you look at the student's t distribution, the top row is a level of confidence, and the bottom row is the z-score. In general, when making sample size calculations such at this one, it is a good idea to change all of the factors to see what the "cost" in sample size is Same with radius. Khan Academy 52.701 προβολές 9:18 95% Confidence Interval - Διάρκεια: 9:03. share\|improve this answer edited Mar 4 '14 at 1:32 answered Mar 2 '14 at 21:09 Alecos Papadopoulos 30k151122 +1 for distinguishing between $\hat{\alpha}$ and $\sqrt{n}(\hat{\alpha} - \alpha)$ -- certainly Volume of sphere is V=4/3pir^3. For a $\mathrm{Pareto}(\alpha,y_0)$ distribution with a single realization $Y = y$, the log-likelihood where $y_0$ is known: \begin{aligned} \mathcal{L}(\alpha\|y,y_0) &= \log \alpha + \alpha \log y_0 - (\alpha + 1) The larger the margin of error, the less faith one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population. This formula will work for means and proportions because they will use the Z or T distributions which are symmetric. We talked about problems of obtaining the value of the parameter earlier in the course when we talked about sampling techniques. The point estimate is the single best value. Definition.The sample size necessary for estimating a population mean μ with(1−α)100%confidence anderror no larger thanεis: $$n = \dfrac{(z^2_{\alpha/2})s^2}{\epsilon^2}$$ Typically, the hardest part of determining the necessary sample size is finding s2, Since error is very small we can write that Delta y ~~dy, so error in measurement is differential of the function. Answer by stanbon(72917) (Show Source): You can put this solution on YOUR website! Hope that helps. Link to this page: maximum error of the estimate Facebook Twitter Feedback My bookmarks ? So $\hat \alpha = h(\mathbf X)$ and $\hat \alpha(\mathbf X = \mathbf x) = 4.6931$ for $\mathbf x = \{14,\,21,\,6,\,32,\,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": 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": 0.887445867061615, "perplexity": 1108.291160529542}, "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-2019-09/segments/1550247499009.48/warc/CC-MAIN-20190221031117-20190221053117-00245.warc.gz"}
http://www.anyamartin.com/site/b9a8fa-unsolvable-math-problem-solved	# unsolvable math problem solved Haselgrove, C. B. Haselgrove 1958, smallest counterexample found by Tanaka 1980). however in my current state of mind I’m far more inclined to the pragmatic “well you can show if it’s true for any particular n you care about, close enough” – think I’m losing my youthful inquisitiveness! When he ran into the classroom several minutes late, he found three equations written on the blackboard. The Bieberbach conjecture (L. de Branges 1985). Snopes and the Snopes.com logo are registered service marks of Snopes.com. It's a bit like trying to predict earthquakes, in that we have only rough probabilities to go by. $\sigma(n)\leq H_n+\ln(H_n)e^{H_n}$ In 2002, Jeffrey Lagarias proved that this problem is equivalent to the Riemann Hypothesis, a famous question about the complex roots of the Riemann zeta function. Schuler’s published version was a bit garbled and exaggerated but essentially correct. Autom., Lang. The trick is finding integers that work for all equations, or the numbers for x, y, and z that will all equal k. Over the years, scientists have solved for nearly every integer between 0 and 100. If A+ B= C. And A, B, C, x, y, and z … Bull. He later worked for the Air Force, took a position with the RAND Corporation as a research mathematician in 1952, became professor of operations research at Berkeley in 1960, and joined the faculty of Stanford University in 1966, where he taught and published as a professor of operations research until the 1990s. The #1 tool for creating Demonstrations and anything technical. If it is even, calculate $$n/2$$. Around 1950 I received a letter from Abraham Wald enclosing the final galley proofs of a paper of his about to go to press in the Annals of Mathematical Statistics. The four-color theorem (Appel and Haken 1977ab and Appel et al. 25, 403-412, 1991. The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. It’s even more interesting that there are nonconstructive proofs for strategies. Here's a Numberphile video explaining why this problem has proved to be so tricky: Earlier this year, Andrew Booker of the University of Bristol spent weeks with a supercomputer to finally arrive at a solution for 33. famous problems which have recently been solved include: 1. He stopped and said, “Hey, George — I was visiting in Indiana recently and heard a sermon about you in church. I typically ask students to write a three-page reflective essay about their experience with the homework in the course, and almost all of the students talk about working on the open problems. The health benefits of sunlight: Can vitamin D help beat covid-19? Solved Problems. I am not a mathematician but I have a question about the The Collatz Conjecture. Hales, T. C. "The Honeycomb Conjecture." Again, most mathematicians believe that the answer to this question is yes, but a proof remains elusive. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole. In these frustrated magnets, spins often flip around randomly in a way that, it turns out, is a useful model of other disordered systems including financial markets. 278-283). Appel, K. and Haken, W. "Every Planar Map is Four-Colorable, II: Reducibility." Albers, Donald J. and Constance Reid. I wish that 30 years ago I had studied under you. [sic]. Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of July, 2020:[7], The seventh problem, the Poincaré conjecture, has been solved;[13] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved. 237, 108-121, 1977b. This is one problem that is worth more than just prestige. Many mathematicians have tried – and failed – to resolve the matter, including Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan. The search for the origin of life: From panspermia to primordial soup. Mathematika 5, When I use these problems for in-class work, I will typically pose the problem to the students without telling them it is unsolved, and then reveal the full truth after they have been working for fifteen minutes or so. And this particular version is all the more interesting for being based on a real-life incident! The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings? The unsolved question about this process is: If you start from any positive integer, does this process always end by cycling through $$1,4,2,1,4,2,1,\ldots$$? Several dozens of unresolved problems for Combinatorial Geometry are depicted in the book, Many unresolved problems for Graph theory are depicted in the article, The list of several unresolved problems converning, This page was last edited on 24 November 2020, at 00:55. I have been teaching Discrete Math (including some number theory) at the University of Alabama for years and one of the problems in our book has to do with making change. It also impacts popular movie culture—remember "Good Will Hunting" and more recently "Gifted. Illinois J. The origin of that minister’s sermon can be traced to another Lutheran minister, the Reverend Schuler [sic] of the Crystal Cathedral in Los Angeles. The four-color theorem (Appel and Haken "Ring-Theoretic Properties of Certain Hecke Algebras." This question was first asked by Paul Erdős and Ernst Strauss in 1948, hence its name, and mathematicians have been working hard on it ever since. University of California Student Solves Unsolvable Math Problems. “On the Non-Existence of Tests of ‘Student’s’ Hypothesis Having Power Functions Independent of Sigma.” 5 Simple Math Problems No One Can Solve. It also forces students to think outside of the box and try things they wouldn’t normally do. When the pendulum is prodded at an almost constant rate though, the mathematics falls apart. "I feel relieved," Booker says of breaking the 65-year old puzzle first set down at Cambridge in a press statement. A year later, when I began to worry about a thesis topic, Neyman just shrugged and told me to wrap the two problems in a binder and he would accept them as my thesis. 137-152, 1985. One of my favorite assignments for students in undergraduate mathematics courses is to have them work on unsolved math problems. In Math as in Dance, Don’t Miss a Step, or Else You May Fall, Clay Mathematics Foundation will reward you with \$1,000,000, mathematical practice standards in the Common Core, particularly harmful for women studying mathematics, Bridging Cultures: An Iranian Woman from an Historically Black College Teaching in a Prison in the US, The Choice to go Asynchronous: Discussion Board Based IBL, MATH ON THE BORDER: Working with unaccompanied migrant children in Federal custody, Reflecting on mathematics as the art of giving the same name to different things (Part 2): Averages finite and continuous, THE ZOOM ROOM: Vignette and Reflections About Online Teaching, Active Learning in Mathematics Series 2015. Things like air passing over an aircraft wing or water flowing out of a tap. Math. This is fantastic and inspiring! There are plenty of mathematical expressions that have no exact solution. The answers took over a million hours to compute. Hales, T. C. "A Computer Verification of the Kepler Conjecture." If the class of atomic models of a complete first order theory is, Is every infinite, minimal field of characteristic zero, Does a finitely presented homogeneous structure for a finite relational language have finitely many. New York: W. W. Norton, 1989. 3. It promotes persistence and the pursuit of understanding which is critical for any student who is studying mathematics to understand and acquire those qualities. The student turned in his test paper and left. S. Kitaev. By doing this, the students get to experience the shift in perspective that comes when what appears to be a simple problem in arithmetic suddenly becomes a near-impossibility. This entry was posted in Uncategorized. Bookmark the permalink.	{"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": 0.5374565720558167, "perplexity": 979.4653024306577}, "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/1623487608856.6/warc/CC-MAIN-20210613131257-20210613161257-00452.warc.gz"}
http://mathhelpforum.com/calculus/179000-taylor-development-print.html	# Taylor development • Apr 29th 2011, 03:43 PM Ulysses Taylor development Hi there. I have this exercise which I'm trying to solve now. It says: Using that $\displaystyle\sum_{n=0}^{\infty}x^n=(1-x)^{-1}$ find one Taylor development for the function $f(x)=\ln(1-x)$ $f^1(x)=\displaystyle\frac{-1}{(1-x)},f^2(x)=\displaystyle\frac{-1}{(1-x)^2},f^3(x)=\displaystyle\frac{-2}{(1-x)^3},f^4(x)=\displaystyle\frac{-6}{(1-x)^4},f^5(x)=\displaystyle\frac{-24}{(1-x)^5}$ And then: $\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac {f^b(x_0)(x-x_0)^n}{n!}=-\displaystyle\frac{(x-x_0)}{(1-x_0)}-\displaystyle\frac{(x-x_0)^2}{2(1-x_0)^2}-\displaystyle\frac{2(x-x_0)^3}{6(1-x_0)^3}-\displaystyle\frac{6(x-x_0)^4}{24(1-x_0)^4}-\displaystyle\frac{24(x-x_0)^5}{120(1-x_0)^5}+\ldots+-\displaystyle\frac{(x-x_0)^n}{n(1-x_0)^n}$ I have two problems with this. In the first place, the general expression that I've found (which is probably wrong) doesn't work for n=0, it does for the others values of n. I thought of starting the summation at 1, but I'm not sure if this is valid. In the second place I don't know how to use the relation the problem gives at the beginning. I can see that I have (1-x_0) for every term, but I couldn't make it fit inside the summation. So this is what I got: $\displaystyle\sum_{n=1}^{\infty}-\displaystyle\frac{(x-x_0)^n}{n(1-x_0)^n}$ Bye there, thanks for your help and suggestions. • Apr 29th 2011, 03:49 PM TheEmptySet You didn't use the hint:) Notice that $f(x)=\ln(1-x) \implies f'(x)=\frac{-1}{1-x}$ But now by the hint we know this is a geometric series and gives $f'(x)=\frac{-1}{1-x}=-\sum_{n=0}^{\infty}x^n$ But we want the power series for f not its derivative so lets integrate to get $\int f'(x)dx =-\int \sum_{n=0}^{\infty}x^ndx=-\sum_{n=0}^{\infty}\int x^ndx$ This gives $f(x)=C-\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}$ and since $f(0)=0 \implies C=0$ $f(x)=-\sum_{n=0}^{\infty}\frac{x^{n+1}}{n+1}$ • Apr 29th 2011, 03:54 PM Ulysses Very nice. Thank you. I would never realized of it :P BTW, the problem is implicitly telling that the function is taken on an interval where its uniformly convergent, right? • Apr 29th 2011, 04:12 PM TheCoffeeMachine Oops, I thought wrong; sorry. :) • Apr 29th 2011, 04:14 PM Ulysses I think not but now I don't need 'em anyway :P • Apr 29th 2011, 04:18 PM TheCoffeeMachine Quote: Originally Posted by Ulysses I think not but now I don't need 'em anyway :P Yeah, I missed that you accounted for the sign alternations in the denominators! :P • Apr 29th 2011, 04:52 PM TheEmptySet Quote: Originally Posted by Ulysses Very nice. Thank you. I would never realized of it :P BTW, the problem is implicitly telling that the function is taken on an interval where its uniformly convergent, right? Yes the geometric series is uniformly convergent on \|x\|<1 • Apr 30th 2011, 06:58 AM Ulysses Thank to all of you. Now I'm trying to solve the same exercise, but with the function $f(x)=(1+4x^2)^{-1}$ I've tried to make some algebraic work to get the expression I'm looking for, $(1-x)^{-1}$ but I didn't get it. I've made the first derivative, and a few more, but I think its just necessary the first, as before. I get: $f'(x)=\diplaystyle\frac{-8x}{(1+4x^2)^2}$ I thought of making $f(x)=\diplaystyle\frac{1}{(1+4x^2)}= \diplaystyle\frac{1-4x^2}{(1+4x^2)(1-4x^2)}=\diplaystyle\frac{1-4x^2}{(1-16x^2)}$ So, when I take the derivative in this last form I get: $f'(x)=\diplaystyle\frac{-256x^3}{(1-16x^4)^2}$ I'm not sure how to proceed. • Apr 30th 2011, 07:11 AM TheEmptySet Quote: Originally Posted by Ulysses Thank to all of you. Now I'm trying to solve the same exercise, but with the function $f(x)=(1+4x^2)^{-1}$ I've tried to make some algebraic work to get the expression I'm looking for, $(1-x)^{-1}$ but I didn't get it. I've made the first derivative, and a few more, but I think its just necessary the first, as before. I get: $f'(x)=\diplaystyle\frac{-8x}{(1+4x^2)^2}$ I thought of making $f(x)=\diplaystyle\frac{1}{(1+4x^2)}= \diplaystyle\frac{1-4x^2}{(1+4x^2)(1-4x^2)}=\diplaystyle\frac{1-4x^2}{(1-16x^2)}$ So, when I take the derivative in this last form I get: $f'(x)=\diplaystyle\frac{-256x^3}{(1-16x^4)^2}$ I'm not sure how to proceed. It is just function composition. Let $y=-4x^2$ Then we know that $f(x)=\frac{1}{1+4x^2}=\frac{1}{1-y}=f(y)$ But this is just a geometric series so we get $f(y)=\frac{1}{1-y}=\sum_{n=0}^{\infty}y^n$ Now just plug back in $y=-4x^2$ $f(x)=\sum_{n=0}^{\infty}(-4x^2)^n=\sum_{n=0}^{\infty}(-1)^n4^nx^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": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.955000638961792, "perplexity": 551.3160101958684}, "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/1498128320593.91/warc/CC-MAIN-20170625221343-20170626001343-00040.warc.gz"}
http://papers.nips.cc/paper/6066-f-gan-training-generative-neural-samplers-using-variational-divergence-minimization	# NIPS Proceedingsβ ## f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization [PDF] [BibTeX] [Supplemental] [Reviews] ### Abstract Generative neural networks are probabilistic models that implement sampling using feedforward neural networks: they take a random input vector and produce a sample from a probability distribution defined by the network weights. These models are expressive and allow efficient computation of samples and derivatives, but cannot be used for computing likelihoods or for marginalization. The generative-adversarial training method allows to train such models through the use of an auxiliary discriminative neural network. We show that the generative-adversarial approach is a special case of an existing more general variational divergence estimation approach. We show that any $f$-divergence can be used for training generative neural networks. We discuss the benefits of various choices of divergence functions on training complexity and the quality of the obtained generative models.	{"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": 0.9029598236083984, "perplexity": 705.6101216108733}, "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/1492917122933.39/warc/CC-MAIN-20170423031202-00448-ip-10-145-167-34.ec2.internal.warc.gz"}
https://jnnp.bmj.com/content/70/3/421.8	Article Text Texture of the Nervous System of Man and the Vertebrates by Santiago Ramon y Cajal. Volume 1. 1. ALASTAIR COMPSTON ## Statistics from Altmetric.com Texture of the Nervous System of Man and the Vertebrates by Santiago Ramon y Cajal. Volume 1. Translated and edited by pedro pasikand tauba pasik (Pp v1, 631. DM289, $US189.). Published by: Springer-Verlag, New York, 1999. 84-07-00200-3 (complete series 3-211-83056-1). Everything we know about structure, function, and physiology in the nervous system at the cellular level, in health and disease, evolves from the concept that organisation is through the connectivity of functionally independent neurons and their processes. Santiago Ramon y Cajal distinguished neurons from glia; showed the variability of dendritic arborisations and axon terminations; established that axon cylinders end freely but form contacts; conceived that the nerve impulse is conducted between axons, dendrites, and the cell body of neighbouring neurons; had the concepts of trophism and tropism; and following Rudolph Virchow, regarded the cell as the unit of all biological systems. His most detailed studies were of the cerebellum but, in time, no part of the brain and spinal cord went unexplored. His great synthesis was to settle debate on the neuron theory. His descriptions were supplemented by beautiful drawings based on Golgi stains. He and Golgi were jointly awarded the Nobel prize for medicine in 1906. They disagreed publicly during the lectures in Stockholm. Cajal is the most significant neuroscientist of the 20th century—Sir Charles Sherrington being his only serious competitor. They met only once when Sherrington hosted Cajal's stay in London to deliver the 1894 Royal Society Croonian lecture. During the visit, Cajal was arrested as a vagrant at Cambridge railway station when visiting the provinces to receive an honorary doctorate. Cajal and Sherrington fell over themselves to outpraise each other. Sherrington on Cajal: “He is the greatest anatomist the nervous system has ever known . . .he solved at a stroke the direction of nerve currents in their travel through the brain and spinal cord . . .it was a step of genius to study the embryonic nervous system.“ Between 1880 and 1933, Cajal wrote 288 scientific publications including 22 monographs. Much of his work remains untranslated from the original Spanish and hence unread. But the sustained admiration for Cajal's writings and their contemporary relevance for neuroscience is now matched by a welcome revival in publishing his works.Textura del sistema nervioso del hombre y de los vertebrados was published from Madrid in three volumes (1897, 1898, and 1904). It was updated by Cajal with new text and illustrations for the translation into French asHistologie du systeme nerveux de l'homme et des vertebres by Dr Leon Azoulay (2 volumes: 1909-11). The complete French edition was first translated into English by Neely Swanson and Larry Swanson as Histology of the Nervous System of Man and Vertebrates (Oxford University Press, 1995). Now the original Spanish text is undergoing translation by Pedro Pasik and Tauba Pasik as Texture of the Nervous System of Man and the Vertebrates. The first of three volumes appeared in 1999; the other two are promised for 2000. The advertising flysheet champions Cajal's discovery of growth cones, chemoattractant substances, dendritic spines, and cortical interneurons and claims absolute authority over both the French and first English editions. It boasts illustrations based on original reproductions of drawings archived in the Cajal Institute in Madrid (the evidence is in the Museo-Cajal-Madrid stamp on many figures) with very little copied from previously published editions. Facts and citations are corrected from Cajal's original text and authenticated against contemporary sources. In which edition should the discerning Cajal reader invest? When complete, the Springer set will cost DM850/£330/$550 compared with £150 for the two Oxford volumes. The difference is worth paying. The English-Spanish text is authentic: compare “the nervous system represents the ultimate boundary in the evolution of living matter, and the most complicated machinery of noblest activities that Nature has to offer“ (English-Spanish) with “countless modifications during evolution have provided living matter with an instrument of unparalleled complexity and remarkable functions: the nervous system, the most highly organised structure in the animal kingdom“ (English-French); or “it appears that with this [chemotactic] hypothesis we have shed light into a dark cave, when in reality we have explored only the entrance, from which its imposing abyss appears even more distant and black“ (English-Spanish) versus “the theory of chemotaxis we advanced . . .initially appeared to be pure conjecture with no hope of verification, although recently it has gained experimental support“(English-French). The text is authoritative and the production lavish. Pedro and Tauba Pasik include, and readily identify, translation of material added by Cajal for the French edition between 1904 and 1909. Original footnotes are retained but the citations are modernised and gathered in a single section completing the English-Spanish text. The lack of an index will be put right when volume three is published. The illustrations are incomparably better in the Springer than the Oxford volume[s]. The line drawings are much more crisp; the original figures of methylene blue staining reproduce poorly as black and white (Oxford) but some of their polychromatic figures are more subtle. Volume one deals with the general principles of organisation in the nervous system and Cajal's methods, the details of neuronal structure and the spinal cord. Volumes two and three will complete the medulla and pons, cerebellum, midbrain, diencephalon including the retina, cortex, and autonomic nervous system. The original Spanish and French editions are very expensive and virtually unobtainable. For the historian, physician, or scientist who studies neuroscience, whether or not to invest in the Springer set is simply not an issue—even if you already have the Oxford edition. Both are magnificent publishing achievements . . .but the Pasiks are on course to produce the definitive English language edition of the definitive Cajal. View Abstract ## Request Permissions If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.	{"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": 0.3027845323085785, "perplexity": 4054.1846221147116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737238.53/warc/CC-MAIN-20200808021257-20200808051257-00430.warc.gz"}
http://chrisfmathsphysicsmusic.blogspot.com/2014/	## Saturday, 29 November 2014 ### True Patriotism or Nationalism This rant is a lot less objective than I usually am but I am getting really sick of politicians and the press appealing to a bogus sense of patriotism and I despair of parties that should know better such as the :Labour party for abandoning core values of the basis of socialism such as that what ever country you come from shouldn't matter as to how you are treated. Dr Johnson a person, who given my instintively left wing sympathies I would not have had any sympathies with, summed the current situation up precisely, when he claimed that patriotism was the last refuge of a scoundrel. It seems to me that both Alex Salmond and his successor Nicola Sturgeon and Nigel Farage are all scoundrels in that respect. Their so called arguments are essentially appeals to the lowest denominator of tribal instincts. None of them appeal to what has actually made Britain great (and I use that word delibrately as opposed to Scotish or English) the Britain that makes me proud is the contribution to the intellectual and aesthetic contribution to the world not the flag waving nationalism of those who thought it right to display a saltire during the referendum or a St George cross in rochester. No, the Britain that makes me proud is the scientific acheivements of Newton, James Clerk Maxwell, Charles Darwin, Crick and Watson, and Fleming and Peter Higgs, the mathematical achievements of Napier, Hamilton, and Wiles, The poetry of Shakespeare,. Dryden, Milton, Pope, Byron Shelly Keats Coleridge, Wordswith and Burns. The novels of Austen,. the Bronte sisters, Charles Dickens, Robert Louis Stevenson, Thomas Hardy and George Orwell. The philosophy of Locke, Berkely Hume, J S Mill, and Bertrand Russell. The music of Purcell, Handel (who spennt most of his compostional life here), Elgar, Vaughan Williams, Benjamin Britten. (And I am sure there are plenty of others who I have missed) How many of those idiots who feel it necessary to drap a Saltire or a St George cross outside their windows even know who most of the people I have mentioned above are or even appreciate what the afoerementioned people have achieved. No it's all a bogus appeal to the lowest common denominator of tribal instinct to enable the scoundrels who dress themselves up in the nationalist flag of whatever country be it Scotland or England without any appreciation of the people in their countries who have achieved lasting achievements which will transcend the vagaries of the current political situation, as opposed to an appeal to the lowest common denominator of flag waving nationalism of the worst kind. If that makes me snobbish I don't give a monkeys. ## Sunday, 23 November 2014 ### Piano Grade 1 result Pass This is just a short post to let those lknow that I passed my Grade 1 piano exam with 110 marks. A respectable pass but nothing great. I think I'll have to postpone my booking for the Usher hall :) My main problem is that I find it difficult to maintain a steady pulse and two hand co-ordination this is only going to get more difficult as time progresses. I am handicapped by not having a natural sense of rhythm which some people seem to have. I was never good at dancing, I don't buy the idea as some people seem to think that you either have it or you don't. I believe firmly that by training you can gradually improve. Although I do find trying to follow a metronome really tedious and boring. Obviously it's more difficult but I believe you can actually get there. I must have improved my sense of basic pulse from last time as I was able to do the hand clapping example without any problem whereas at my last resit. I was unable to do so. I am now faced with a dilemma, as the ABRSM syllabus for grade 2 has changed but you are allowed to do the past syllabus for the next time round which would be in March. I have been studying the three pieces on and off and am reasonably confident about two of them, so I could put myself forward for the grade 2 exam but unless I get my rhythm sorted out then it would be marginal whether or not I pass. I will make the decision in the next two months or so On another topic I want to make a few remarks about Duncan's comment to my last post that we don't know anything about the nature of dark matter or dark energy. That's not entirely true, we know their equations of state and also their ratios. Also we can use the Friedmann equations to predict the future expansion rate of the universe. In a way the situation is similar to the classical ideas of gravity, electromagnetic fields and entropy. It was never possible to get an idea of what these entities actually were however just like dark matter and dark energy their effects on other objects were able to be predicted and measured. To some extent it doesn't really matter that we don't know what dark energy or dark matter actually are because as inputs to cosmological models their effects on the universe can be calculated. Just as say knowing that the gravitational force between two bodies obeyed an inverse square law enabled planetary motions to be predicted. I would say this is more important, given the lack of experimental evidence, than a fruitless attempt to undestand the true nature given the lack of experimental evidence as to the nature of dark matter or dark energy. It seems to me that physics has made great strides in developing our understanding of nature works, without getting bogged down in trying to understand the actual nature of some of the important entities that it has used. I'll leave the speculation to those cleverer than me in the mean time I want to focus on the uses of physics which is more than enough to keep us busy for a long time. ## Thursday, 6 November 2014 Hi first of all apologies for not posting for a while. I guess without the pressure of OU TMA's there is not that much to post about Anyway today I resat my grade 1 piano exam. I felt a lot more comfortable than before and think I've done enough to pass but it won't be a great one. So how do I feel slightly ambivalent to say the least. I know on a good day I can knock off the pieces set for grade 1 no problem, but in a different environment it's not so easy. I still had problems adjusting to the feel of the piano in the church as opposed to my home one. Maybe I should ask the church if I can spend a few hours practisinng on their piano. So I stumbled on most of the tasks I was asked to do but unlike last time I was able to recover. Is that enough to pass who knows hopefully yes but for me piano playing isn't a natural thing. I'll persist in practice and see how far I can go, until I really do hit a brick wall. As for other stuff I have been trying to understand the cosmological revolution of the late 1990's where it was established that the current universe is a) Accelerating b) Appear's to consist of 70% dark energy and 30% matter of which 26% is so called dark matter and only 4% is baryonic matter c) Flat I hope to finish my write up of this by the start of the new year until then watch this space Best wishes Chris ## Sunday, 24 August 2014 ### Orbitals In another round of the 'quantum wars' on the OU fora we have got round to discusssing orbitals and what they mean. It wasn't till I started thinking about it that I realised the usual picture of an electron orbiting the nucleus in an orbit of fixed radius like a mini-solar system, and that if it loses energy it 'jumps' from one to another is totally misleading to say the least. The latter view is definitely true of the Bohr model and I guess most people carry this over to their thinking when (if) they study quantum physics further. However it is just not correct, When you solve Schrodinger's equation for the hydrogen atom what, one ends up with for each energy level is a 3 dimensional probability density function which gives the probability of finding an electron in a given region. This means that the electron in a given orbital can be anywhere in principle allowed by the particular orbital (better called probability density function) so there is a small but finite probability that it could be 1 m, 10m and so on away from the nucleus. Of course it will be closer to the average of the probabiity density function. The crucial point is however that when the electron either gains or loses energy that loss or gain in energy is fixed (quantised). However the electron does not 'jump' from one orbital to another what happens is that the probability density function changes in accordance with the appropriate function for that energy level. Thus the misleading picture given by the Bohr model is totally inadequate to do justice to the picture presented by the solution to Schrodinger's equation. Matter's aren't helped by the depiction of the energy level diagrams with their pictures of arrows going from one energy level to another giving the impression that the particle is actually jumping from one energy level to another. But those 'jumps' are changes in energy not position. Finally it is important to remember that those pictures of orbitals shown in chemistry or physics textbooks are pictures of probability density functions. The usual convention when drawing the boundary surfaces is to draw the boundary marking off the region where the electron is likely to be found 95% of the time they do not represent the fixed distance of the electron from the nucleus of the atom. 95% of the time the electron will be inside the so called 'shell' but there is also a 5% probability that it will be outside the shell. Here are some pretty pictures of hydrogen atom probability density functions for you you to drool over. https://search.yahoo.com/search?ei=utf-8&fr=ytff1-yff22&p=Pictures%20of%20hydrogen%20atom%20orbitals%20&type= And here is the first experimental observation of the probability density function of the hydrogen atom http://io9.com/the-first-image-ever-of-a-hydrogen-atoms-orbital-struc-509684901 ## Monday, 4 August 2014 ### Yet More Quantum madness the so called Cheshire cat For those like me, who despair at the increasing tendency to the irrationality of the world, even in science. The recently published claim that it has been possible to separate out the spin component of a quantum object from it's spatial component can only cause one to wonder whether the world has gone mad. For those who aren't aware this BBC post is typical of the hype http://www.bbc.co.uk/news/science-environment-28543990 and the original paper is here http://www.nature.com/ncomms/2014/140729/ncomms5492/full/ncomms5492.html So the claim at first appears to be by a special arrangement of beam splitters it is possible to disembody a particles spin from it's location. So that in one path only the location of the particle can be determined on another part it's magnetic moment which seems to have magically been disentangled from the particle. A literal interpretation of this is just not credible. First of all spin has no spatial properties so it can't move anywhere. Spin is a vector which is allowed quantum mechanically to take a number of fixed directions usually up or down. it is a property of a quantum object. So if the authors of the paper are claiming what the hype says they are then that would require a rewriting of the whole of quantum mechanics of spin and as far as I can tell the authors are not seeking to challenge the standard quantum mechanical concept of spin. So what exactly have the authors shown? The key to interpreting the experiment is the concept of quantum superposition there are essentially two interpretations The first beloved of the popular literature, and it would seem journalists who want to peddle the alleged mysterious aspects of quantum mechanics, is called the ontic (short for ontological) interpretation. The ontic view is that quantum superpositions are real namely that a cat really is in limbo between being alive or dead or in the case of interferometer experiments each particle really does travel down both paths simultaneously or in the two slit experiment passes through both slits simultaneously. The solution to Schrodinger's equation is a real wave and something physical (even though it doesn't actually affect anything (for example exert a force on a particle as an electric fiield does), For an N body problem, it is some sort of field in 3N + 1 configuration space as opposed to our 3+1 dimensional space time. Also when a measurement is made the solution to Schrodinger's eqaution collapses into one of the states. This is usually called wave-packet collapse but it really should be called the solution to Schrodinger's equation collapse. On the ontic view, it is only by measurement that a system jumps from it's state of limbo of possible states into a definite state. Extrapolating (slightly unfairly I admit) this means the moon really isn't there when one looks at it. It would seem from the language used by the authors talking of the wavefunction travelling down both paths simultaneously they subscribe to the ontic view. On this view in this experiment, it really does seem that the spin part of a qauntum objects solution to Schrodinger's equation can be detached from it's spatial part. But just what disembodied spin means hasn't really been explained and if it really was disembodied then all the text books on quantum mechanics would have to be rewritten. Conversely from an epistemic (statistical) point of view, the solution to Schrodinger's eqaution is a probability amplitude who's modulus squared gives when suitably normalised gives rise to a probability density function. The Born interpretation, on this view the superposition of states represents the lack of knowledge of an observer about a given situation and nothing physical. As the solution to Schrodinger's equation is primarily a statistical quantity, applying it to work out the behaviour of one particle is just as meaningful, as trying to predict the roll of a single dice or a spin of a roulette wheel. When the solution to Schrodinger's equation 'collapses' all that happens is that one of the possibilities is realised. Albeit in order to get the correct probabilities for a given situation one has to use complex numbers which accounts (mathematically at least) for all the interference effects associated with many quantum phenomenon. I have discussed how this works for two state systems in previous posts http://chrisfmathsphysicsmusic.blogspot.co.uk/search?q=Two+state+ Where I also point out that the Dirac notation can also be applied to classical systems. But in order to explain many quantum phenomenon, one has to extend our notion of probability to include complex numbers. On this epistemic\statistical view then when a beam of particles passes through a beam splitter, a single particle, does not travel through both paths simultaneously, but the solution to Schrodinger's equation represents the probability amplitude that it will traverse one path or the other. On this view in the Cheshire Cat experiment, if the neutron passes one path it's location, will be measured on the other it's magnetic moment will be measured. There is no disembodiment of the spin part of the neutron's solution to Schrodinger's equation from it's spatial part. That realtively prosaic account of course won't attract attention and so would be ignored by most people. The epistemic view and the ontic view, as they are interpreting the same mathematical object, namely the Solution to Schrodinger's equation cannot be empirically distinguished from each other. However given the more modest claims of the epistemic view vis a vis our notions of quantum wierdness I know which one I prefer. All the epistemic (statistical view) is asking is that we extend our notions of probability to include complex numbers. In contrast the ontic view would claim that wavepacket collapse is real, that spin's can be dismbodied from their constituent particles and that act's of measurement do create reality. Of course some people would find the epistemic\statistical view unsatisfactory, accusations of instrumentalism etc but if there are no hidden variables then there can be nothing behind the statistics. I really do have to ask why is there such a hunger for irrational explanations of quantum phenomenon. I really do despair some times. ## Tuesday, 17 June 2014 ### Grade 1 piano exam result Well I just failed got 94 out of 150 When it came to the exam I just freaked out Whilst I don't want to make excuses the fact that the piano was an actual piano as opposed to a digital one meant it was a totally different experience to my normal clavinova. And even in the warm up room there was only a digital piano. So despite having gone through the routine 2 or 3 times in the warm up room I was totally unprepared and it was a downward spiral I fluffed a note on the first scale that made me tense and nervous then I fluffed the first piece and the second and so on. It was about the most ghastly ten minutes I've spent in a long time Ok so I became dis-orientated but at least I know what it is to expect and I would hope next time around I'll do much better In the mean time I'm plowing my own furrow with the help of stuff on the internet https://www.webpianoteacher.com/ His approach is amazing he writes out the notes on a white board and copying them onto a bit of paper definitely helps in the initial stages of learning a pieve. Going to try and learn the first movement of the moonlight sonataover the next few weeks, following his approach here is the first lesson (which is free) For budding pianists his site is definitely the best ## Monday, 2 June 2014 Well the culmination of almost two years since I started getting an interest in learning the piano has arrived I have my grade 1 piano exam tomorrow. I feel reasonably confident I'll pass but I'm not sure I will get to either merit 130/150 or distinction 140/150 It's kind of weird really, to get an idea of what I'll be doing here is a link to the ABRSM site which shows the sort of thing I'll be asked to do I hope the examiner is as kind as the examiner in the video Anyway I'll let you know how I get on For interest the pieces I've chosen are Fugue No 4 Alec Rowley Thursday R R Bennett and if you do a serch for grade 1 piano pieces on you tube you should be able to hear what they sound like Good luck to anyone taking ABRSM exams tomorrow whatever your instrument. ## Thursday, 22 May 2014 ### dangers of OU fora Hi this is more of a warning to myself and others about the dangers of the OU fora. I have effectively wasted a large amount of mine and other peoples time getting involved in a long running dispute which has been continuing for about 2 years with my bete noir. Those who read the fora especially the philosophy ones will have been the victim of this dispute. It's not important what the actual details are but safe to say my sparring partner is one of those people who evades the issue, keeps twisting and turning and so makes it difficult to answer his points. Anyway as I say I have wasted so much time getting sucked in to the so called debate. that I really don't want to discuss it further. I will do my best to ignore this person and I want to apologise to any one who got sucked into the debate ## Monday, 28 April 2014 ### Do Virtual particles exist Well been having a ding dong with my bete noir on the physics forums about whether or not virtual particles exist. In my opinion they don't but unfortunately the popular physics literature abounds with such ideas and so the myth is generated that they actually do exist. A really good overview of why they can't really be said to exist is given here http://www.mat.univie.ac.at/~neum/physfaq/topics/virtual I'll just summarise the salient points What do we mean by a virtual particle ? Let's start by describing what they are not they are not tempory resonances detected at places like CERN those are real particles. A virtual particle is one which is said to be exchanged during a particle interaction, the story goes because of the Uncertainty principle applied to energy and time. $$dE dT >= \frac{h}{2\pi}$$ In a particle interaction there is the possibility of a temporal violation of the conservation of energy and so a particle can be formed which then gets immediately absorbed by the other particle. The classic example is QED where electrons are said to exchange a virtual photon between them when they scatter off each other. And because of the uncertainty principle a positron can be seen equivalent to an electron moving backwards in time. Those phrases should be enough to raise the suspicions of any one who seeks a rational basis for physics. It's all a bit of fudge really. The whole idea of virtual particle exchange particles is an artefact of the use of Feynman diagrams. A Feynman diagram is a term in a perturbation series expansion because we have no other tool to calculate with. Essentially in quantum field theory one has a set of propagaters (Green Functions) representing the free particles and a set of vertices to represent the interaction between them.. A Feynaman diagram is essentially a graphical representation of a complicated mathematical expression representing the interaction. The simplest are the Tree diagrams essentially an H combined by putting two vertices together. This gives a second order contribution to the scattering process the external legs of the H represent the electron or other particle and the internal line represents the 'exchanged photon' But the internal line can also split to give more complicated diagrams involving loops such as ----O--- or even ----O-----O---- that would be the virtual photon dissociating into an electron and a positron and recombining to form a photon and so forth. The total contribution to a Feynman diagram expansion is a sum over all possible diagrams the contribution to which decreases as the number of loops increases. Whilst the Feynman diagram expansion is intuitive it should be stressed we are not talking about actual processes. If we were then the infinite series would have to be summed up. As a literal interpretation that would mean that in each particle interaction there would be an infinite number of processes that occur and so the interaction could just not happen. So a virtual particle is simply an internal line in a Feynman diagram nothing more, it is an artefact of the need to use perturbation theory and so cannot be real. Yes electrons and photons interact with each other and yes the use of Feynman diagrams gives rise to some of the most accurate results in physics for at least QED anyway. But I do find it worrying that what is essentially a metaphor is taken literally, a bit like fundamentalist religion I guess. Hopefully sooner rather than later someone will find a way to solve Quantum Field theory non-perturbatively and the Feynman picture will vanish until that day however we are stuck with it. ## Wednesday, 2 April 2014 ### ABRSM Grade 5 Theory result Well just got my result for the ABRSM grade 5 theory exam equates to 82% which is a merit, feeling reasonably chuffed, put myself forward for grade 1 piano in June doubt whether I'll get such a good result still 2 months to focus. More later ## Sunday, 30 March 2014 ### Saylor ECON 101 Microeconomics As my followers will know I have embarked on the excellent (and free !!) courses from Saylor. http://www.saylor.org/ I'm currently studying Microeconomics. 101 http://www.saylor.org/courses/econ101/ as I want to expand my limited knowledge about this subject. I've just finished the first three units The Saylor course\provides a free pdf course book specially commisioned from the course which you can download and also links to various on line videos such as those provided by the Kahn academy and a few others. The aim of Saylor courses is to utilise on line resources to provide the equivalent of a college or university education in the subject area at the appropriate level and also to provide some form of assessment usually a two hour final exam. These are slowly being recognised by outside institutions and this trend can only grow. Here is a review of the first three units 1) On production possibility curves and their implication for trade. A production possibility curve represents the total amount of goods or time one can spend on a given activity. Suppose for example I can spend 20 hours a week studying and I'm studying either music or maths. I can spend all the time studying maths or all the time studying music. However if I do so I wont progress on maths if I focus solely on music and vice-versa. Suppose I have an exam in both subjects coming up and I decide that an hour studying music will get me 2% extra marks whilst an hour studying maths (as I'm quite good at that already) will only get me 1% extra marks and suppose I can at present average 50% for music and 60% for a maths exam then I have to do a trade off. Clearly I have to spend more time on music than maths. to bump up the marks. There is a greater opportunity cost in maths as for each hour I spend studying maths I give up 2 marks in the music exam whilst for each hour I spend studying maths I give up only 1% maths. The study of this leads to some quite interesting results in terms of trade (due to Ricardo) namely that although one country may have an absolute advantage over another in producing two or more goods both countries can expand the amount of goods by specialisation in one form of good. . 2) This covered the subject of Supply and Demand and what causes it's shift. Also more contentiously the arguments for abandoning rent controls, or a minimum wage or taxation are given. On the neo-classical model, (Adam Smith's free hand) the market is generated by each person selfishly pursuing his or own self interests eventually an equilibrium price will emerge where the amount demanded by consumers is equal to the amount suppliers are willing to produce. In the usual and fairly simplistic way as prices rise demand falls and as prices rise supply increases and the equilibrium price is at the intersection of the two curves normally represented by two straight lines. Setting a floor allegedly interferes with this equilibrium and causes a shortage in the supply or at least so the argument goes. However this is based on a number of dubious assertions. Namely that there is a stable equilibrium price something impossible when more than one good is considered. It ignores the time delay inherent in a change in price and it would seem to justify a fairly obnoxious right wing agenda. It's a bit worrying that right wing institutions such as the institute for fiscal studies, have based what is a totally unfair set of policies on a fairly naive view of markets. Fortunately there is an antidote to all this given in this wonderful book by Stephen Keen http://www.amazon.com/Debunking-Economics-Revised-Expanded-Dethroned/dp/1848139926 and for a detailed critique of the current state of economic theory it can't be bettered. Although I would prefer a more mathematical account. Stephen Keen was one of the few people to predict the current recession by modelling the economy using ideas from chaos theory . I hope to understand his models a lot better in the coming years watch this space. Unit 3 continues the effects of supply and demand by studyiong what is called consumer and producer surplus. This is essentially the total area made between the supply and demand curves and the equilibrium point and the price axis (the y axis). If government taxes goods then the supply curve shifts upwards and the area decreases. Also so it is claimed a band is taken out of the area which is government revenue. But this leads to another area called 'dead weight loss'. The terminology is really quite perjorative first of all we are not just consumers and producers. Secondly all this argument ignores the benefits of taxation in terms of provision of public services, health, public transport etc. It is claimed that as these are 'free' and do not respect private rights these will necessarily be inferior to anything the private sector can produce. Nothing is mentioned of the fact that the private sector rips off people as their concern is profit. But in more general terms the picture of humanity depicted by neo-classical economists is really quite depressing. Yes we all like a good bargain, but do we have the time to seek out the bargain. Even if you do get a bargain you might not get the quality of goods offered is not likely to be optimum or it might cause you inconvenience. For example I regularly buy books at bargain price from Amazon but about 1 times in 10 I never receive the goods as the company does not use regular mail service but some fly by night courier company who seem incapable of putting their depots in a reasonable place accessible to people after work. I also note that for a government which is allegedly based on free-market principles they have not failed to increase taxes on goods, Perhaps George Osborne should do a course similar to economics 101. I may have given the impression that I'm not enjoying this course that is wrong. I'm actually enjoying finding out where the arguments that are regularly given in the press and right wing media come from and I always have Stephen Keen to help me take the wilder claims of neo-classical economics with a pinch of salt. It is worrying that what is essentially an ideology is being presented as scientific fact. The coursre as it stands isn't that mathematical but more meatier courses await. As far as assessment goes this isn't quite ideal you are referred to tests on a micro-economics book published by Pearson and sometimes this covers topics not covered yet. There will be a final exam which you can take when you feel you are ready. You have to get 70% or over and then you get a certificate,but you can resit as many times as you like with a two week gap. As the course is free (in contradiction to the principles that would seem to underly the ideology of neo-classical economics ) then one can hardly complain and I thank Saylor for saving me £750 and rising in OU fees. I really would recommend people to take up Saylor courses Next post I'll review the other Saylor course I'm doing namely calculus of a single variable. Bye for now. ## Wednesday, 5 March 2014 ### ABRSM grade 5 theory exam post match analysis Grade 5 Theory Post match analysis Well today I sat my first ABRSM theory exam. This was grade 5 theory the questions tend to form a fairly set pattern of questions with a few minor variations Question 1 first part is usually a question on rhythm either as in the case today putting bar lines in various tricky combinations of quavers, semiquavers and hemisemiquavers or alternatively (slightly harder) deducing the time signature as it varies across the melody. Think I got this correct Question 1 second part gives a passsage and you have to ask various questions about identifying chords quite often they throw a wobbly in that they will sneak a treble clef in the bass line or vice versa. This question seeemed relatively straightforward so again think I got most of this correct. Question 2 can either be a question identifying 5 intervals or rewriting a piece in open score in short form or vice versa. The latter although straightforward provided you ensure that the tenor part in open score is usually wrtten an octave higher that it would sound is really tedious fortunately the question was of the former type and provided you are careful making sure that you take into account the key signature can be reduced to modulo 12 arithmetic something I'll expand on in another post. Question 3 is usually a series of questions on a piece of music identifying various terms such as Adagio or more obscure one's such as Zart (German for play it tenderly yeah right sounds more like fart :) ) still . Fortunately the terms weren't that difficult but I interpreted sforzando as play the note suddenly loud as opposed to forced. So probably lost two easy marks there. Question 4 is usually a transpositon which can be quite tricky still following my fail safe method outlined a few posts ago I feel I got this one correct . We had to transpose a piece in F major up a major second. Once you realise that the new key is G major and that Eb transposes to F natural whilst E transposes up to F# but the key signature of G major takes this into account so you don't need to put the acccidental in then this is straightforward but tedious as they say in the books. I tend to leave this question till second last Question 5 is a question on various scales so you are given a starting note and you had to write the ascending harmonic or melodic minor on that note. This time the note was the lower F on a bass clef it's relative major is Ab and it was the ascending harmonic minor that you had to write. This is the same as the notes of Ab major but the leading note in this case Gb is raised a semitone to G natural. The second part asked us to identify a key with 5 sharps (B major) and write a descending scale starting from B easy once you have identified the correct key. So again I think I got most of this out Question 6 you had to complete a melody given a few starting notes. Normally you have to make sure you reach the dominant note at the end of the first phrase and find a way back to the tonic note in this case D. In between I tend to use a logical chord progession either IV or VI chords unfortunately my melodies tend on the whole to be a bit boring essentially ascending or descending scales with an occasional leap. Still if the melody is logical harmonically then I can't see the problem. Also you have to add dynamics and tempo directions. Who knows how much I got say 10 out of 15 Question 7 the one I find the easiest and the one I normally do first namely identify appropriate chords to fit a melody. You only have 4 to choose from namely 1, II, IV and V and usually this is quite straightforward once you have identified the key of the piece. So overall take off a few marks for untidyness, and the melody I feel I'm reasonably confident of getting a merit ie > 80 % maybe distinction but one can never realise if you have made a silly mistake fingers crossed I haven't but this is very much a subjective view point so who knows. Do I feel that doing grade 5 theory is worth it after having studied music at allegedly degree level via the open university. You betcha the grade 5 exams are good at checking you know the basics and I feel a lot more confident about how scales are constructed and how to identify intervals. Yes the OU is conceptually more advanced than grade 5 but it didn't really reinforce the basics. So they are complementary and I feel the better for it. Bring on grade 6 (by the end of the year hopefully) ## Monday, 17 February 2014 ### Hello Saylor Bye Bye OU A week ago searching for alternatives to the Open University courses on logic a search led to this wonderful institution http://www.saylor.org/ This offers a range of Free Online courses which are equivalent to courses studied at American Universities. Unlike coursera or Future learn which are examples of MOCC's the courses available can be accessed at anytime. Also they can be put together to earn the equivalent of a full degree provided you stay the course. What is really impressive are the maths courses http://www.saylor.org/majors/mathematics/ Having looked at the course content I think it's fair to say that they are at least equivalent to the OU and some such Abstract Algebra II and Real Analysis II go further than the OU do. I have always wanted to understand politics and economics better and so am aiming to do at least the core sections in these disciplines. http://www.saylor.org/majors/economics/ I've registered for microeconomics and also calculus of a single variable I which has a really good explanation of the epsilon delta definition of continuity. The great thing is that you can take your time there is no scrambling to complete a TMA Anyway hopefully I can zip through a lot of the maths courses in the next two years I'maiming for 1 module in maths every three months or so. The great thing is that all this is free and the courses are being recognised as equivalent to having studied equivalent to that in an American universities. It can only be a matter of time before this becomes more and more widespread. When the OU is becoming more and more inflexible and restricted in the options they offer and the prices they charge unless it get's it act together institutions like Saylor will rightly supersede them. As a bit of light relief I'm also doing the Future learn course on the Higg's boson https://www.futurelearn.com/courses/higg The lectures seem a bit more advanced than the assessment. the second lecture on the derivation of conservation laws such as momentum from imposing translational invariance on Newton's law is really neat. However the assessment is confined to a few multiple choice questions on general principles. Anyway it's hello Saylor the way to go. ## Sunday, 19 January 2014 ### ABRSM Grade 5 Theory Transposition by Numbers One of the most difficult parts of the grade 5 theory exam is to get the intervals correct OK we can probably tell that it's a fifth but is it a perfect fifth or a diminished fifth or what? Similarly with transposition again yes transposing a piece up or down by a minor 3rd say means that the notes are displaced up or down a line or space but what about the accidentals, even worse what if it asks for a key signature change. A while ago when I was first studying music with the OU I hit on the idea that a lot of the confusion could be clarified by using numbers instead of letters. This works really well for intervals and transposition First number the notes of the Chromatic scale starting with middle C as zero So we have the table Db Eb Gb Ab Bb C C# D D# E F F# G G# A A# B 0 1 2 3 4 5 6 7 8 9 10 11 Each number is the number of semitones above middle C that the corresponding note is. Then for transposition for grade 5 the main ones are Up or down a major 2nd (2 semitones) simply add or subract 2 modulo 12 to the numbers then use the first line to work out the notes Up or down a minor 3rd (3 semitones) simply add or subtract 3 modulo 12 to the numbers then use the first line to work out the notes Finally up or down a perfect 5th (7 semitones) So for example suppose we want to transpose a given melody down a perfect 5th we subtract 7 then add 12 if the new number is less than zero or subtract 12 if the new number is greater than 12. Db Eb Gb Ab Bb C C# D D# E F F# G G# A A# B 0 1 2 3 4 5 6 7 8 9 10 11 Performing the arithmetic gives for C 0 -> -7 then add 12 to get 5 (you only have to do this once) then just start from 5 to give Db Eb Gb Ab Bb C C# D D# E F F# G G# A A# B 0 1 2 3 4 5 6 7 8 9 10 11 5 6 7 8 9 10 11 0 1 2 3 4 So that we see eg that C goes down to F (5) or Ab goes to D (2) and so forth. So writing this table out and doing the appropriate arithmetic will give you a fail safe method of transposing accurately especially for those tricky accidentals. Of course you have to remember if the original melody has a given key signature to take into account the key notes. Thus for Bb major the accidentals are Bb and Eb so every time you see a B or an E remember these are Bb and Eb This also works for key signature changes so suppose I start in Eb and I want to go up a major second Eb is 3 according to the table add 2 to give me 5 the key signature is now that for F major ie simply with Bb In the next post I will show how a similar technique can be used for intervals ## Tuesday, 14 January 2014 ### On line piano tuition for Grade 1 and above Well tomorrow I take my first official piano lesson with the aim of doing grade 1 by June. I have been practicing on and off for about 18 months now and I can't say it has been smooth sailing to say the least. Inded I felt in a bit of a rut as I was playing However over Christmas I found these amazing on line videos by Alison Sparrow who has some really useful tips Plus being very attractive to look at. Could be the Nigella Lawson of piano and violin teaching. Anyway her tip on practicing really helped me get to grips with three of the Grade 1 pieces. When you try and set up a practice session the temptation is to try to and play the piece as a whole with the inevitable stumbling. Alison's method for which I cannot thank her enough is to break it down. At grade 1 the pieces are usually 16 bars what Alison suggests is play the first 4 bar phrase 4 times. Then play the second 4 bar phrase 4 times then play the first and second phrases together 4 times. Then move onto the third phrase play that 4 times, then the 4th phrase 4 times. Then put the 3rd and fourth phrases together 4 times. Then finally play the piece 4 times repeat this for about 2 weeks and you should have the piece under your belt. Anyway it certainly seems to have helped me. This is so obvious when pointed out to you but not at all obvious when you are practicing on your own. Another good website I have found is that of Shawn Cheeks especially his 'boot camp' sight reading course. Shawn points out that most musicians rely on their ear and memory. What this misses is actually engaging with the music as it is written as his career progressed he found it more and more difficult to tackle the more difficult pieces. So he decided to go back to basics and really learn the music. Have a look at his introduction on You tube. Again I have been following his boot camp and can see the improvement. Certainly spelling out the notes whilst learning a new piece is really helpful His philosophy is outlined here I've also put myself forward for grade 5 music theory this march What i haven't done is any number theory or logic. I have given myself 4 days this weekend to complete the assignment if i don't then I will probably quit I can't really say I'm enjoying the course OK I haven't really put much effort into learning it. Right now though my musical interests are dominanting if I can achieve grade 5 and 6 theory and grade 1 and 2 piano by the end of the year and get back into general I guess the non stop deadline of assignments exams, followed by going straight back to other assignments over the past two yeara has taken it's toll there is a small chance I'll continue but I can't see it. ## Friday, 3 January 2014 ### Instrumentalism or Why we should Shut up and calculate. Prior to the festive season, I got myself in one of my perenial debates on the physics forum about the meaning or not of quantum mechanics. My main protaganist will be well known to those who follow the debates. He claims to endorse the Copenhagen Interpretation but also denies that he is an instrumentalist. A somewhat inconsistent opinion in my view. Anyone it’s not my job to help him see the contradictions in his position. I do want to make a defence of instrumentalism however, that is some what missed by self styled philosophers of physics, but is in fact the current practice of most physicists. OK what is instrumentalism ? essentially it is the view that the main aim of science is to provide empirically adequate models of nature without bothering to much about how the underlying concepts used to make the predictions correspond to reality. What do I mean by empirically adequate, it is essentiallly the condition that the predictions of the theory when instantiated in a concrete model give reasonable agreement with experiment. For the quantative sciences such as physics this makes the theory testable or at least a given model of a given phenomenon predicitable. If the model does not give reasonable agreement with experiment then one can try and make the model more accurate by including more terms in the model or trying another approach. It is important to make a distinction between theories and models this distinction is often blurred. A theory is a set of general principles in physics, there are about 8 sets of general principles which have been discovered Classical physics Newton’s Laws of motion. The macroscopic laws of thermodynamics. Maxwell’s equations. Modem Physics Special relativity. General relativiry. Non relativistic quantum mechanics. Statistical physics. Quantum field theory. Note I do not include speculative theories such as superstrings because so far there is not one concrete prediction that has come out of it. At this stage it is ‘Not even wrong’ In order to describe natural phenomenon, one takes one of the above set of principles appropriate to the phenomenon in question. Then with the aid of mathematics and empirical information, such as the masses of particles involved sets up the appropriate equations and solves them either analytically or for complicated problems one has to resort to computers. So for example to model the properties of stars as they collapse, one needs a combination of Statistical physics and General relativity along with an appropriate equation of state. To model planetary motion one would use either Newton’s laws of motion or for say mercury one has to resort to General relativity. A simple model would neglect the interactions between the planets a more complicated model would include these. Deciding on what approximations are appropriate is a necessary skill of a good physicist. As we apply the above general principles to more and more phenomenon we gradually begin to understand how nature works and whats more with the aid of mathematics can predict how systems will behave. We can predict the energy levels of a molecule or solid. We can predict the orbits of planets, we can predict the decay rates or scattering cross sections of particles. We can even predict the rate of expansion of the universe from it’s early stages by a combination of Einstein’s general theory of relativity. relativistic statistical physics and a knowledge of the basic particles involved. All of this is so obvious to a practicing physicist, but so called self styled philosophers of science aren’t happy with this. For them the aim of science is not to make concrete predictions of phenomenon but to describe reality as it is initself. They want to concentrate on the meaning of the general principles but therein lies a problem because some of the concepts used can be quite obscure. For example in classical physics the nature of gravitation remained obscure all one could say about it was that it obeyed an inverse square law. In thermodynamics the concept of entropy also remained obscure although it had a perfectly precise meaning in terms of a measure of heat transfer . Also whilst Maxwell’s equations involved electrical and magetic fields their real nature remained obscure and all sorts of weird and wonderful ideas about the ‘real nature’ of an electric field involving vortices and eddy currents in the Aether were prevalent at the time. However more concrete these models appeared rather than Maxwell’s equations they didn’t really add much to the understanding of electromagnetism and these were ultimately shown to be wrong as the Aether was proven not to exist. The situation in intepreting the nature of an electric field so exasperated Helmholtz that when asked what Maxwell’s theory was he replied that Maxwell’s theory was Maxwell’s equations. Quite similar to the attitude which I favour of ‘Shut up and calculate’ when it comes to interpreting quantum mechanics. My protaganist in the OU debates on the fora doesn’t like the above view for him and many others like him such as Karl Popper this reduces physics to engineering (as if that were a bad thing). Well I’ve got news for him and Karl Popper and other anti-instrumentalists most of what is published in physics journals today is an application of one of the 8 above sets of general principles to model a given phenomenon. Indeed it is only by continuing the process of detailed modelling that we understand how nature works. If that’s ‘just’ engineering so what ? The rest arguing about the nature of gravitation, the wave function, whether or not particles are real or waves are, really doesn’t move us forward and to some extent it doesn’t matter, as I can still use the general principles coupled with empirical information to make concrete predictions about natural phenomenon. What other means of understanding nature do we have ? It is only by shutting up and calculating that we will get any where. The attitude prevalent amongst certain people that one should always be looking for more new general principles is misguided, the need for new general principles will be forced upon us when we are able to probe nature at more and more higher energies or shorter length scales.. Until then we should all shut up and calculate.	{"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": 0.5832383036613464, "perplexity": 887.5264804699107}, "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-2018-22/segments/1526794867254.84/warc/CC-MAIN-20180525235049-20180526015049-00219.warc.gz"}
https://www.physicsforums.com/threads/do-neutron-stars-have-a-minimum-volume.909648/	# B Do neutron stars have a minimum volume? 1. Mar 30, 2017 ### nitsuj Do neutron stars have a minimum volume? Anything "in the way" of perhaps baseball sized neutron star? Or would something like that be an impossibility? How neat to have one in a laboratory...or not lol 2. Mar 30, 2017 ### Staff: Mentor You need enough mass to maintain the pressure required for a neutron star. It might be possible to make artificial neutron stars a bit smaller than natural ones, but that still means something like 1 solar mass. In terms of size: More massive neutron stars are actually a bit smaller. But if you make them too massive, they collapse to a black hole. => all neutron stars have a radius of a few kilometers 3. Mar 30, 2017 ### nitsuj So much for my neutron baseballs baseball company. :D such an interesting phenomenon , I imagine all the cool physicists study such massive bodies. lol But no neutron stars in the "lab". Apparently some rotate at as much as 70% of c!! Angular momentum anyone? a few km radius "ball of immense energy" of pretty much every form in extremes lol I wonder how fast they dissipate all but their (rest) mass. Last edited: Mar 30, 2017 4. Mar 31, 2017 ### Staff: Mentor Not that much - smaller than the star that collapsed. Large v, but small r doesn't make a large product vr. They slow down in their rotation over time and they get colder. If they don't catch enough mass to convert to a black hole, they will still spin quite rapidly when they are cold. 5. Mar 31, 2017 ### nitsuj lol with all do respect you are making light of an object 12 km wide of some of the most dense "pile" of matter spinning at the better part of c. I appreciate, perhaps in a different way than you that the angular momentum is less in this stage of a star, but in the context of the size of the object.....it could fit "on" earth... I can actually visualize the size of it, I've walked that distance....to think an "extreme" astronomical phenomena could be "familiar" in any way is unusual for me, particularly size. curious if we could play with such dense matter that gravity wouldn't be so elusive from a measurement perspective. 6. Mar 31, 2017 ### Staff: Mentor Well, compared to everyday objects or planets the angular momentum is huge, of course. Compared to other objects with a similar mass as the neutron star it is not. Even if a smaller amount with that density would be stable, you couldn't play with it, everything large enough to be visible would just crush the ground and fall towards the core of Earth. 7. Apr 3, 2017 ### nikkkom Fast rotation of NS does not strike me as particularly amazing, considering other properties on NS. Such as magnetic fields with astounding energy density, more than energy density in a form of "usual matter" around us. Just think about that. Antimatter is often thought of as "ultimate energy storage" in terms of density, ~1000 times denser than U235. But here, we have just the "boring old" electromagnetic field, in vacuum, tangled up so tightly, it has much, much more energy per unit volume than antimatter (at STP) would! Magnetic reconnections on the Sun cause solar flares. This is basically magnetic field jumping into a less energetic configuration. Magnetic reconnections on a NS are far, far, FAR scarier. They happen in microseconds, they release energy equivalent to many tons of mass, and (since it's EM field) they easily convert this energy into EM waves, meaning gamma-rays. The "empty space" filled with magnetic field suddenly turns into a gamma-ray inferno. Everyone knows about immense gravity on the surface, but just how immense it is, it's hard to visualize. On the surface, a test object would fall one meter in about one millisecond - and attain velocity of several thousand km/s! If a material baryonic object (a rock) would fall on NS from infinity, it would heat on impact to the temperature on the order of trillion kelvins! 8. Apr 3, 2017 ### nitsuj It was the magnetism and rotation that got me thinking more about them....the very magnetic ones will have the spin slowed remarkably by interactions, from there the idea all of this in a space 12km wide was just so wow! Second was about the tidal effects of such a tiny tiny object. I wonder what is the most dense energy we've reached ("massive / macroscopic") Another neat consideration is the repulsive forces of the particles, the gravity containing it and the balance struck. Particularly for a "cold" or "dead" one. Apparently at the LHC they've found possible "(quark &) gluon plasma", to which inserted in the article was a comment from a scientist that said "Besides black holes, there's nothing denser that we're creating." I wonder if that is the same as him saying, quark-gluon plasma is the last step before a black hole. Last edited: Apr 3, 2017 9. Apr 3, 2017 ### nitsuj Coolest experiment ever! 10. Apr 4, 2017 ### Staff: Mentor The quark gluon plasma created at the LHC is hotter than the core of neutron stars. It should also have a higher density. The cores of neutron stars have a density slightly higher than the nuclei of heavy elements in regular matter.	{"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": 0.8197216987609863, "perplexity": 1897.1182594935442}, "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/1521257647556.43/warc/CC-MAIN-20180321004405-20180321024405-00771.warc.gz"}
http://mathhelpforum.com/calculus/194827-area-integral-circle.html	# Math Help - area integral on a circle 1. ## area integral on a circle any help for this ... Given the function f : $f(x,y)=(4-x^2)^{-y^2/2}$ which is on a circle area $D={x^2+y^2\leq 4} $ with center (0,0) and radius of 2. compute $\int \int_{D}^{.}f(x,y)$ 2. ## Re: area integral on a circle What kind of help do you want? Do you know how to integrate at all? Integrating over a disc is a fairly standard problem in Calculus. Do you have any idea how to find the limits of integration in xy coordinates? What about polar coordinates? 3. ## Re: area integral on a circle yes i would like the solution please. 4. ## Re: area integral on a circle any help ????????? 5. ## Re: area integral on a circle Originally Posted by kotsos yes i would like the solution please. You won't get the solution here. You can get hints, but you are expected to do your OWN work! As was suggested, try converting everything to polars.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9898112416267395, "perplexity": 1120.0806844930598}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538423.10/warc/CC-MAIN-20140416005218-00156-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.semanticscholar.org/paper/New-horizons-in-cosmology-with-spectral-distortions-Chluba-Abitbol/2cb8c03a31994f98d42f3fc1baa69db479d6eb90	# New horizons in cosmology with spectral distortions of the cosmic microwave background @article{Chluba2019NewHI, title={New horizons in cosmology with spectral distortions of the cosmic microwave background}, author={Jens Chluba and Maximilian Abitbol and Nabila Aghanim and Yacine Ali-Haimoud and Marcelo A. Alvarez and Kaustuv Basu and Boris Bolliet and Carlo Burigana and Paolo Bernardis and Jacques Delabrouille and Emanuela Dimastrogiovanni and Fabio Finelli and Dale J. Fixsen and Luke Hart and Carlos Hernandez-Monteagudo and J. Colin Hill and Alan Kogut and Kazunori Kohri and Julien Lesgourgues and Bruno Maffei and John C. Mather and Suvodeep Mukherjee and Subodh P. Patil and Andrea Ravenni and Mathieu Remazeilles and Aditya Rotti and Jos{\'e} Alberto Rubi{\~n}o-Mart{\'i}n and Joseph I. Silk and Rashid Sunyaev and E. R. Switzer}, journal={arXiv: Cosmology and Nongalactic Astrophysics}, year={2019} } • J. Chluba, E. Switzer • Published 4 September 2019 • Physics • arXiv: Cosmology and Nongalactic Astrophysics Voyage 2050 White Paper highlighting the unique science opportunities using spectral distortions of the cosmic microwave background (CMB). CMB spectral distortions probe many processes throughout the history of the Universe. Precision spectroscopy, possible with existing technology, would provide key tests for processes expected within the cosmological standard model and open an enormous discovery space to new physics. This offers unique scientific opportunities for furthering our understanding… Thermalization of large energy release in the early Universe • Physics • 2020 Spectral distortions of the cosmic microwave background (CMB) provide a unique tool for learning about the early phases of cosmic history, reaching deep into the primordial Universe. At redshifts $z Constraining the inflationary potential with spectral distortions • Physics • 2020 Measuring spectral distortions (SDs) of the cosmic microwave background (CMB) will provide new constraints on previously unexplored scales of the primordial power spectrum, allowing us to extend the The synergy between CMB spectral distortions and anisotropies • Physics • 2020 Spectral distortions and anisotropies of the CMB provide independent and complementary probes to study energy injection processes in the early universe. Here we discuss the synergy between these Measuring the Hubble Constant from the Cooling of the CMB Monopole • Physics The Astrophysical Journal • 2020 The cosmic microwave background (CMB) monopole temperature evolves with the inverse of the cosmological scale factor, independent of many cosmological assumptions. With sufficient sensitivity, Unlocking the synergy between CMB spectral distortions and anisotropies Measurements of the cosmic microwave background (CMB) spectral distortions (SDs) will open a new window on the very early universe, providing new information complementary to that gathered from CMB New physics from the polarized light of the cosmic microwave background Cosmology requires new physics beyond the Standard Model of elementary particles and ﬁelds. What is the fundamental physics behind dark matter and dark energy? What generated the initial ﬂuctuations The COSmic Monopole Observer (COSMO) The COSmic Monopole Observer (COSMO) is an experiment to measure low-level spectral distortions in the isotropic component of the Cosmic Microwave Background (CMB). Deviations from a pure blackbody Bridging the gap: spectral distortions meet gravitational waves • Physics Monthly Notices of the Royal Astronomical Society • 2021 Gravitational waves (GWs) have the potential to probe the entirety of cosmological history due to their nearly perfect decoupling from the thermal bath and any intervening matter after emission. In Sensitivity forecasts for the cosmological recombination radiation in the presence of foregrounds • Physics • 2020 The cosmological recombination radiation (CRR) is one of the inevitable$\Lambda$CDM spectral distortions of the cosmic microwave background (CMB). While it shows a rich spectral structure across Primordial nucleosynthesis constraints on high-z energy releases • Physics • 2020 The cosmic microwave background (CMB) spectrum provides tight constraints on the thermal history of the universe up to$z \sim 2\times 10^6\$. At higher redshifts thermalization processes become very ## References SHOWING 1-10 OF 557 REFERENCES Spectral Distortions of the CMB as a Probe of Inflation, Recombination, Structure Formation and Particle Physics Following the pioneering observations with COBE in the early 1990s, studies of the cosmic microwave background (CMB) have focused on temperature and polarization anisotropies. CMB spectral Distinguishing different scenarios of early energy release with spectral distortions of the cosmic microwave background Deviations of the cosmic microwave background (CMB) frequency spectrum from a pure blackbody tell an exciting story about the thermal history of our Universe. In this paper, we illustrate how well Prospects for Measuring Cosmic Microwave Background Spectral Distortions in the Presence of Foregrounds • Physics • 2017 Measurements of cosmic microwave background spectral distortions have profound implications for our understanding of physical processes taking place over a vast window in cosmological history. Detecting the cosmological recombination signal from space • Physics • 2015 Spectral distortions of the CMB have recently experienced an increased interest. One of the inevitable distortion signals of our cosmological concordance model is created by the cosmological Which spectral distortions does ΛCDM actually predict Ever refined cosmological measurements have established the ΛCDM concordance model, with the key cosmological parameters being determined to per cent-level precision today. This allows us to make The Cosmic Microwave Background and the Ionization History of the Universe • Physics • 2006 Details of how the primordial plasma recombined and how the Universe later reionized are currently somewhat uncertain. This uncertainty can restrict the accuracy of cosmological parameter Using CMB spectral distortions to distinguish between dark matter solutions to the small-scale crisis • Physics • 2017 The dissipation of small-scale perturbations in the early universe produces a distortion in the blackbody spectrum of cosmic microwave background photons. In this work, we propose to use these Probing the Universe after Cosmological Recombination through the Effect of Neutral Lithium on the Microwave Background Anisotropies I show that neutral lithium couples strongly to the cosmic microwave background (CMB) through its 6708 Å resonant transition after it recombines at z ~ 500. At observed wavelengths of ≲335 μm, the Another look at distortions of the Cosmic Microwave Background spectrum • Physics • 2016 We review aspects of Cosmic Microwave Background (CMB) spectral distortions which do not appear to have been fully explored in the literature. In particular, implications of recent evidences of A Hydrodynamical Approach to CMB mu-distortions • Physics • 2012 Spectral distortion of the cosmic microwave background provides a unique op- portunity to probe primordial perturbations on very small scales by performing large-scale measurements. We discuss in a	{"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": 0.4605866074562073, "perplexity": 4361.193477166524}, "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/1659882572127.33/warc/CC-MAIN-20220815024523-20220815054523-00640.warc.gz"}
https://ryansblog.xyz/post/c40301cd-e240-4417-9283-c88bd3e746a9	# Machine Learning - Reading Notes on Bayesian Optimization ### Brief Description Bayesian optimization is a technique to perform optimization tasks when the objective function is unknown or hard to evaluate. A function is hard to evaluate if it takes time to get the results or if it's expensive to evaluate. Here we use the general sense of the term function: a process that generates outputs from inputs. For example, simulations and drug experiments can both be viewed as functions. The primary goal of an optimization task is to find the optimal value of the function. If the objective function is unknown, there isn't anything we can do about it. We can evaluate the function at different points and select pick the best value among the results or we assume the function takes a certain form so that we can get a theoretical analytic result. ### Surrogate and Acquisition Function Therefore, there are two aspects of the optimization problem when it involves an unknown objective function: 1. Which form should we use to approximate the unknown objective function? 2. How to select the points at which we want to evaluate the function? Remember the function is hard to evaluate and we want to minimize the number of function evaluations. In Bayesian optimization, we assume the vector $$[f(x_1), f(x_2), ..., f(x_n)]$$ follows a multivariate normal distribution and we select the next data point to evaluate based on acquisition functions. Now let's take a look at the while loop. At the beginning of the loop, we have • n evaluation results: $$f(x_1), f(x_2), ..., f(x_n)$$ • A (prior) distribution of the value $$f(x)$$. (denoted by $$\mathbb{P}_n(f(x))$$). • A distribution of the optimum $$x^$$. (denoted by $$\mathbb{P}_n(x^)$$). At the end of the loop, we have • A new evaluation result $$f(x_{n+1})$$. • A (posterior) distribution of the value $$f(x)$$. (denoted by $$\mathbb{P}_n(f(x))$$). • A new distribution of the optimum $$x^$$. (denoted by $$\mathbb{P}_n(x^)$$). The important point is that at any time, we have the distribution of the value $$f(x)$$ for any $$x$$ in the domain space. The next question is how to select the next data point $$x_{n+1}$$ to evaluate. In Bayesian optimization, the next data point is the maximizer of the acquisition function. There are three common acquisition functions used in Bayesian optimization: • Expected Improvement • Entropy Search #### Expected Improvement As we are evaluating the function at more data points, the known optimal value is increasing. Suppose the current optimum is $$x_n^{}$$, according to the Expected Improvement acquisition function, we should select the next data point to evaluate so that it can maximize the expected increase of the optimal value. At step $$n$$, we have the distribution of $$f(x)$$ for all $$x$$. Therefore, we can make an educational guess: the maximizer of the unknown function is the data point $$x^{}$$ where $$\mathbb{P}_n(x^)$$ has the best mean denoted by $$\mu_{n}^$$. According to the Knowledge Gradient method, we should select the next data point so that it can maximize the expected increase of this value: $$KG_n(x) = E_n[\mu_{n+1}^* - \mu_{n}^* \| x_{n+1} = x]$$ #### Entropy Search As we mentioned earlier, we also have the distribution of the optimum $$\mathbb{P}_n(x^*)$$ at each step. Because entropy represents uncertainty, the idea of Entropy search is that we should select the next data point so that it can maximize the reduction of the entropy of the optimum. ----- END ----- Welcome to join reddit self-learning community. Want some fun stuff?	{"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": 0.8721107840538025, "perplexity": 292.60191110041995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710662.60/warc/CC-MAIN-20221128203656-20221128233656-00805.warc.gz"}
http://mathhelpforum.com/pre-calculus/28777-product-dot-vectors.html	1. ## Product Dot Vectors Given "a" and "b" unit vectors, a) if the angle between them is 60 degrees, calculate (6a + b) product dot (a - 2b) Now use $\cos (60^ \circ ) = \frac{{a \cdot b}}{{\left\\| a \right\\|\left\\| b \right\\|}} = a \cdot b$ to finish.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9902402758598328, "perplexity": 1974.9645228338811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501173761.96/warc/CC-MAIN-20170219104613-00549-ip-10-171-10-108.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/79788/is-it-possible-to-find-the-digit-sum-of-n-n-in-mathbbn-text-and-n	# Is it possible to find the digit sum of $n!$ ($n \in \mathbb{N} \text{ and } n \le100$) without actually computing the factorial? Is it possible to find the digit sum of $n!$ ($n \in \mathbb{N} \text{ and } n \le100$) without actually computing the factorial? I faced this problem in quantitative aptitude test which asks to find the sum of the digits of $25!$, I was wondering if there is any paper pencil approach to deduce the sum from counting the number of each digits in the factorial without actually computing the whole thing. - – Rudy the Reindeer Nov 7 '11 at 8:34 In the present form of the question, the answer is "yes, obviously"; there are lots of ways of doing that -- for instance you can take out all factors of $2$ and $5$ and use only the excess of one of them over the other; then you've chopped off the terminal $0$s and never compute $n!$. So to make this an interesting question you'll have to make some quantitative specification how much easier the method should be than actually computing the factorial. – joriki Nov 7 '11 at 8:56 You can find the digit sum mod nine very easily: any natural number $m$ is congruent to its digit sum mod 9. If $n\geq 6$ then $9 \| n!$ so the digit sum has to be zero mod 9... – m_t_ Nov 7 '11 at 9:12 @ mt_:I think it's called digital root, but I don't want that here. – VelvetThunder Nov 7 '11 at 10:09 By "without actually computing the whole thing", I'm going to assume you mean "without actually computing factorial in the traditional, brute force, way", i.e. "show any possible shortcut".... let's take a look at 25! because some of the paper-and-pencil approaches will apply to higher numbers as well... First let's write out the factors of the numbers from 2 to 25: 2,3,22,5,23,7,222,33,25,11,223,13,27,35,2222,17,233,19,225,37,211,23,2223,55 Counting up the factors, we have 22 2s, ten 3s, etc, that is, 25! can be written as: 2^22 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23 As joriki says, we can toss tens, so let's toss out 6 5s and 6 2s: 2^16 * 3^10 * 7^3 * 11^2 * 13 * 17 * 19 * 23 We can make it still easier on ourselves by factoring out some numbers that are easier for us to work with on paper. For example, it's easy to multiply by 1001, and 1001 is 71113, so let's separate those 3 factors out: 2^16 * 3^10 * 7^2 * 11 * 17 * 19 * 23 * 1001 Likewise 99 is easy to work with, so let's get that out: 2^16 * 3^8 * 7^2 * 17 * 19 * 23 * 1001 * 99 And 98 is only a little harder than 99 is: 2^15 * 3^8 * 17 * 19 * 23 * 1001 * 99 * 98 And finally, 102 is also just two away from 100: 2^14 * 3^7 * 19 * 23 * 1001 * 99 * 98 * 102 Come to think of it, 98 * 102 is 9,996, just four away from 10,000: 2^14 * 3^7 * 19 * 23 * 1001 * 99 * 9,996 And multiplying by 9 is easier than 3 so: 2^14 * 3 * 9^3 * 19 * 23 * 1001 * 99 * 9,996 We're out of tricks at this point, so let's go for it. Let's do the big primes now: 19 * 23 = 437. Then multiply by 9 3 times: 4370 - 437 = 3933 39330 - 3933 = 35397 353970 - 35397 = 318573 And multiply by 3 once: 318573 * 3 = 955719 Then double this 14 times. (Hope I'm not the only one who finds it easy to double numbers?) 1911438 3822876 7645752 15291504 30583008 61166016 122332032 244664064 489328128 978656256 1957312512 3914625024 7829250048 15658500096 The only factors we have left are 1001, 99, and 9,996. Let's do that last one first... it's 10,000-4, so we'll double two more times: 31317000192 62634000384 And then subtract: 156585000960000 - 62634000384 =156522366959616 Multiply by 99: 15652236695961600 - 156522366959616 =15495714329001984 And finally multiply by 1001: 15495714329001984000 + 15495714329001984 =15511210043330985984 Finally, you can add up those digits. 1+5+5+1+1+2+1+0+0+4+3+3+3+0+9+8+5+9+8+4=72 as mt_ mentions, we expect the result to be evenly divisible by 9, so 72 sounds quite plausible. p.s. Note to the stackexchange sysops: the trend on this question seems to be to post comments (not answers) to the original question. I would gladly do that, but since I still have a low rep, I am barred from posting comments. - Good, but you can use LaTeX here (a substantial subset thereof). Encase with dollars or double dollars. use \cdot for star (times). Not too hard, looks great too. :) – J. W. Perry Jan 16 '14 at 8:48	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.791878342628479, "perplexity": 238.64192494757987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146302.25/warc/CC-MAIN-20160205193906-00314-ip-10-236-182-209.ec2.internal.warc.gz"}
https://www.ias.ac.in/listing/bibliography/boms/Sashikala_Ojha	• Sashikala Ojha Articles written in Bulletin of Materials Science • Photoluminescence, thermally stimulated luminescence and electron paramagnetic resonance studies of U6+ doped BaSO4 U6+ doped BaSO4 samples were synthesized by precipitation route. PL, TL and EPR investigations of 𝛾 and self 𝛼 irradiated samples were carried out. PL spectra of these samples give structured broad band peaking around 518 nm with five vibronic bands centred around 498.4, 516.0, 533.7, 554.0 and 575.1 nm, respectively and the average frequency of symmetric stretching of O=U=O in the ground electronic state was found to be 674 cm−1. Trap level spectroscopic studies of U doped BaSO4 give glow peaks at 411, 488 and 512 K, respectively and their spectral characteristics are typical of UO$_{2}^{2+}$ emission. EPR studies of 𝛾-irradiated U6+:BaSO4 sample have shown the presence of sulphoxy centred radicals like SO$_{4}^{−}$ and SO$_{3}^{−}$ in addition to OH$^{\bullet}$, O$_{3}^{−}$ and SH2−. TSL peaks at 411 and 488 K were correlated with thermal destruction of SO$_{4}^{−}$ and SO$_{3}^{−}$ radicals. • # Bulletin of Materials Science Volume 45, 2022 All articles Continuous Article Publishing mode • # Dr Shanti Swarup Bhatnagar for Science and Technology Posted on October 12, 2020 Prof. Subi Jacob George — Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bengaluru Chemical Sciences 2020	{"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": 0.6445669531822205, "perplexity": 12364.038233124647}, "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/1652662522741.25/warc/CC-MAIN-20220519010618-20220519040618-00724.warc.gz"}
http://blogs.scienceforums.net/ajb/?p=1478&cpage=1	## I could do with $100,000…but Scott Aaronson has offered$100,000 to anyone that can show that viable quantum computers are fundamentally impossible. The question is not a practical one, there seems many technical difficulties that have prevented the primitive quantum processors being scaled up to working quantum computers. The question is deeper than this and asks is our understanding of quantum mechanics right? What is true is that the scaling up of quantum computers cannot be forbidden by some obvious or trivial reason within quantum mechanics. So the bet is to come up with some convincing reason why practical scaled up quantum computers cannot be realised. Scott Aaronson’s blog IEEE Spectrum Scott Aaronson’s homepage ### 2 Responses to “I could do with \$100,000…but” 1. Xittenn says: I would first have to ask for a definition of ‘quantum computer’. I mean technically solid state is already quantum in its nature being that the switching mechanism employs quantum tunneling. Regardless I hope there isn’t a winner as the ideas presented are meant to increase our present state of technological advancement. If someone does find a valid reason to refute the concepts, I hope at the very least it increases our knowledge in such a way as to create an avenue that does achieve what the original research had intended to. 2. Uncle Al says: A scaled quantum computer will decay by dissipation while it is being configured, running, and being read. Whether that is a fundamental or technological limit on problem size is debatable. Somebody must discover or synthesize a real world usable, deeply state-stable vs. temperature q-bit. As nobody knows what what that is or how to do it, managed research will magnificently fail in creating it. Discovery is, by definition, off PERT-chart and off allocated budget. Discovery disciplines management rewarded for enforcing process.	{"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": 0.49176040291786194, "perplexity": 1382.1766153412646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645330816.69/warc/CC-MAIN-20150827031530-00166-ip-10-171-96-226.ec2.internal.warc.gz"}
http://bima.astro.umd.edu/nemo/manual/node37.html	Next: 5.1.2 Integration using hackcode1 Up: 5.1 N-body experiments Previous: 5.1 N-body experiments Contents Index ## 5.1.1 Setting it up In Chapter we already used mkplummer to create a Plummer model; here we shall use the program mkommod ("MaKe an Osipkov-Merritt MODel") to make two random N-body realizations of a King model with dimensionless central potential and 100 particles each. The small number of particles is solely for the purpose of getting results within reasonable time. Adjust it to whatever you can afford on your CPU and test your patience (see Appendix benchmarks). 1% mkommod in=$NEMODAT/k7isot.dat out=tmp1 nbody=100 seed=280158 These models are produced in so-called RMS-units in which the gravitational constant G=1, the total mass M=1, and binding energy E=-1/2. In case you would like virial units (E=-1/4)5.2the models have to be rescaled using snapscale: 2% snapscale in=tmp1 out=tmp1s rscale=2 "vscale=1/sqrt(2.0)" In the case that your user interface was not compiled with the NEMOINP5.3directive, the vscale expression has to be calculated by you, i.e. vscale=0.707107. Also note the use of the quotes in the expression, to prevent the shell to give special meaning to the parenthesis, which are shell meta characters. The second galaxy is made in a similar way, with a different seed of course: 3% mkommod in=$NEMODAT/k7isot.dat out=tmp2 nbody=100 seed=130159 This second galaxy needs to be rescaled too, if you want virial units: 4% snapscale in=tmp2 out=tmp2s rscale=2 "vscale=1/sqrt(2.0)" We then set up the collision by stacking the two snapshots, albeit with a relative displacement. The program snapstack was exactly written for this purpose: 5% snapstack in1=tmp1s in2=tmp2s out=i001.dat \ deltar=4,0,0 deltav=-1,0,0 The galaxies are initially separated by 4 unit length and approaching each other with a velocity consistent with infall from infinity (parabolic encounter). The particles assembled in the data file i001.dat are now ready to be integrated. To look at the initials conditions we could use: 6% snapplot i001.dat xrange=-5:5 yrange=-5:5 which is displayed in Figure . Next: 5.1.2 Integration using hackcode1 Up: 5.1 N-body experiments Previous: 5.1 N-body experiments Contents Index (c) Peter Teuben	{"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": 0.8323237299919128, "perplexity": 12428.527892133254}, "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/1558232257156.50/warc/CC-MAIN-20190523063645-20190523085645-00148.warc.gz"}
http://math.stackexchange.com/users/28372/julian-rosen	# Julian Rosen less info reputation 928 bio website math.uwaterloo.ca/~j2rosen location Waterloo, Ontario age 30 member for 2 years, 8 months seen yesterday profile views 442 I am a postdoc in math at the University of Waterloo 70 The $100$th derivative of $(x^2 + 1)/(x^3 - x)$ 33 Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function? 24 Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$ 24 How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion? 16 Random point uniform on a sphere # 5,495 Reputation +50 If the number $x$ is algebraic, then $x^2$ is also algebraic +10 Find the probability that a matrix is diagonalizable +25 In what topological abelian groups does convergence to zero imply summability? +10 The $100$th derivative of $(x^2 + 1)/(x^3 - x)$ # 7 Questions 11 A field without a canonical square root of $-1$ 9 Homogeneous polynomials in two variables taking integer values 3 Algorithm for finding generators of an ideal 3 Demonstrating a coin is not fair with as few flips as possible 3 Power series identity [duplicate] # 88 Tags 178 calculus × 11 38 sequences-and-series × 6 75 derivatives × 2 38 functional-equations × 3 66 real-analysis × 9 37 functions × 2 59 abstract-algebra × 13 35 integration × 5 40 group-theory × 6 34 ring-theory × 6 # 11 Accounts Mathematics 5,495 rep 928 MathOverflow 2,516 rep 1821 Puzzling 1,233 rep 219 English Language Learners 186 rep 2 Physics 126 rep 4	{"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": 0.881088137626648, "perplexity": 1614.1440351536714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768741.35/warc/CC-MAIN-20141217075248-00091-ip-10-231-17-201.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/trigonometry/CLONE-68cac39a-c5ec-4c26-8565-a44738e90952/chapter-2-acute-angles-and-right-triangles-section-2-5-further-applications-of-right-triangles-2-5-exercises-page-87/23	# Chapter 2 - Acute Angles and Right Triangles - Section 2.5 Further Applications of Right Triangles - 2.5 Exercises - Page 87: 23 2202.6 ft #### Work Step by Step $\angle A = 90 ^{\circ} -58^{\circ} 22' = 31^{\circ} 38'$ $AB = 2587 ft$ [in a right-angles trinagle] Apply cosine definition $\cos (31^{\circ} 38') =\frac{x}{2587}$ $x\approx 2202.6 ft$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	{"extraction_info": {"found_math": 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": 0.40483254194259644, "perplexity": 2025.3498828394204}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684226.55/warc/CC-MAIN-20191018154409-20191018181909-00532.warc.gz"}
http://mathematica.stackexchange.com/users/91/whuber?tab=reputation&sort=post	whuber Reputation 17,526 Top tag Next privilege 20,000 Rep. Access 'trusted user' tools 1 37 101 Impact ~243k people reached • 23 helpful flags • 1,794 votes cast # 17,526 Reputation 10 yesterday +10 02:08 upvote Insert $+$, $-$, $\times$, $/$, $($, $)$ into $123456789$ to make it equal to $100$ 10 May 3 +10 01:28 upvote LinearModelFit with millions of rows 10 May 2 +10 02:10 upvote How to punch a hole in some 3D distribution of points 10 Apr 30 10 Apr 16 10 Apr 13 10 Apr 4 10 Mar 25 10 Mar 19 10 Mar 18 10 Mar 17 20 Mar 16 10 Mar 14 10 Mar 13 10 Feb 27 20 Feb 23 10 Feb 17 10 Feb 11 10 Feb 8 10 Feb 6 15 Feb 3 20 Jan 29 10 Jan 27 10 Jan 23 10 Jan 19 10 Jan 11 10 Jan 6 10 Jan 2 10 Dec 30 '15 10 Dec 18 '15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6242924332618713, "perplexity": 12974.43062791847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461861812410.28/warc/CC-MAIN-20160428164332-00206-ip-10-239-7-51.ec2.internal.warc.gz"}
https://iwaponline.com/jh/article/14/3/772/3180/Lattice-Boltzmann-method-for-2D-flows-in	In order to improve efficiency and accuracy, while maintaining an ease of modeling flows with the lattice Boltzmann approach in domains having complex geometry, a method for modeling equations of 2D flow in curvilinear coordinates has been developed. Both the transformed shallow water equations and the transformed 2D Navier-Stokes equations in the horizontal plane were synchronized with the equilibrium distribution function and the force term in the rectangular lattice. Since the solution of these equations takes place in the classical rectangular lattice environment, boundary conditions are modeled in the standard form of already existing simple methods (bounce-back), not requiring any additional functions. Owing to this and to the fact that the proposed method ensures a more accurate fitting of equations, even to domains of interest having complex geometry, the accuracy of solution is significantly increased, while the simplicity of the standard lattice Boltzmann approach is maintained. For the shallow water equations transformed in curvilinear coordinates, the proposed procedure is verified in three different hydraulic problems, all characterized by complex geometry. This content is only available as a PDF.	{"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": 0.8552079200744629, "perplexity": 327.633869072668}, "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/1579251801423.98/warc/CC-MAIN-20200129164403-20200129193403-00213.warc.gz"}
https://www.meritnation.com/ask-answer/question/34-the-system-of-linear-equations-x-yz-0-xyz-0-x-yz-0-has-a/straight-lines/10838230	# 34. The system of linear equations x+λy−z=0 λx−y−z=0 x+y−λz=0 has a non-trivial solution for : (1) exactly one value of λ. (2) exactly two values of λ. (3) exactly three values of λ. (4) infinitely many values of λ. dear student regards • 1 What are you looking for?	{"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": 0.8379561305046082, "perplexity": 3689.2576012925138}, "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-2023-14/segments/1679296943555.25/warc/CC-MAIN-20230320175948-20230320205948-00595.warc.gz"}
http://mathematica.stackexchange.com/questions/15387/threading-distribute/20586	I wish to use Distribute[p.(q+r)+m] to get p.q+p.r+m. However, Distribute apparently does not thread through all parts of the expression. What should I do instead? - Is Distribute[#] & /@ (p.(q + r) + m) acceptable ? – b.gatessucks Nov 29 '12 at 9:26 @b.gatessucks This is not good because if there is a function like Log[a+b] inside the expression, the distribute would expand that too, yielding Log[a]+Log[b]. – QuantumDot Jan 8 at 20:06 @QuantumDot That would be a different question, too. – b.gatessucks Jan 8 at 20:10 Oh, that's true. On second thought, never mind. – QuantumDot Jan 8 at 20:24 You don't give a lot of context about how the target expression is generated (interactively or via code). The documentation for Distribute indicates that it won't automatically map down through an expression. But, one way to do this comes to mind. Via code, using ReplaceAll and a replacement rule: expr=p.(q+r)+m expr /. d : HoldPattern[Dot[_, Plus[_, _]]] :> Distribute[d] (* m + p.q + p.r *) - You could also define special symbols that distribute themselves when they get combined with Plus under a Dot operation. Here I denote those symbols with a hat: Clear[OverHat]; OverHat /: Dot[x___OverHat, Plus[y__OverHat], z___OverHat] := Then you can enter your expression like this: OverHat[m] + OverHat[p].(OverHat[q] + OverHat[r]) where the OverHat is more easily input using the shortcut Ctrl7+^ to get this: - p.(q + r) + m /. Dot -> Composition[Distribute, Dot]	{"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": 0.4711446464061737, "perplexity": 5557.526363803052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345773230/warc/CC-MAIN-20131218054933-00083-ip-10-33-133-15.ec2.internal.warc.gz"}
https://arxiv.org/abs/physics/0403088	physics (what is this?) # Title:Minkowski space-time: a glorious non-entity Authors:Harvey R. Brown (Oxford), Oliver Pooley (Oxford) Abstract: Einstein distinguished between principle'' and constructive'' theories in physics, and although he thought the latter were more explanatory than the former, he regarded his 1905 formulation of special relativity theory as a principle theory. Some have claimed that Minkowski space-time can serve as the deep structure within a constructive reformulation of special relativity. We criticise this claim, and argue that a satisfactory constructive version of special relativity must involve a dynamical treatment of the forces holding together the constitutive parts of moving rods and clocks. Comments: 18 pages, written for Petkov (ed.), The Ontology of Spacetime (in preparation) Subjects: History and Philosophy of Physics (physics.hist-ph) Cite as: arXiv:physics/0403088 [physics.hist-ph] (or arXiv:physics/0403088v1 [physics.hist-ph] for this version) ## Submission history From: Oliver Pooley [view email] [v1] Wed, 17 Mar 2004 16:19:36 UTC (23 KB)	{"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": 0.44356122612953186, "perplexity": 6685.6964596522175}, "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/1542039746398.20/warc/CC-MAIN-20181120130743-20181120152743-00143.warc.gz"}
http://math.stackexchange.com/questions/40320/proof-that-the-largest-eigenvalue-of-a-stochastic-matrix-is-1	# Proof that the largest eigenvalue of a stochastic matrix is 1 The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to 1) is 1. Wikipedia marks this as a special case of the Perron-Frobenius theorem... but I wonder if there is a simpler (more direct) way to demonstrate this result? - Here's a really elementary proof (which is a slight modification of Fanfan's answer to a question of mine). As Calle shows, it is easy to see that the eigenvalue $1$ is obtained. Now, suppose $Ax = \lambda x$ for some $\lambda > 1$. Since the rows of $A$ are nonnegative and sum to $1$, each element of vector $Ax$ is a convex combination of the components of $x$, which can be no greater than $x_{max}$, the largest component of $x$. On the other hand, at least one element of $\lambda x$ is greater than $x_{max}$, which proves that $\lambda > 1$ is impossible. - Thank you, simple and self-contained, that is what I had in mind! – koletenbert May 24 '11 at 6:59 Say $A$ is a $n \times n$ row stochastic matrix. Now: $$A \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix} = \begin{pmatrix} \sum_{i=1}^n a_{1i} \\ \sum_{i=1}^n a_{2i} \\ \vdots \\ \sum_{i=1}^n a_{ni} \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}$$ Thus the eigenvalue 1 is attained. To show that the this is the largest eigenvalue you can use the Gershgorin circle theorem. Take row $k$ in $A$. The diagonal element will be $a_{kk}$ and the radius will be $\sum_{i\neq k} \|a_{ki}\| = \sum_{i \neq k} a_{ki}$ since all $a_{ki} \geq 0$. This will be a circle with its center in $a_{kk} \in [0,1]$, and a radius of $\sum_{i \neq k} a_{ki} = 1-a_{kk}$. So this circle will have 1 on its perimeter. This is true for all Gershgorin circles for this matrix (sine $k$ was taken arbitrarily). Thus, since all eigenvalues lie in the union of the Gershgorin circles, all eigenvalues $\lambda_i$ satisfy $\|\lambda_i\| \leq 1$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9333098530769348, "perplexity": 113.20109694685236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701147492.21/warc/CC-MAIN-20160205193907-00229-ip-10-236-182-209.ec2.internal.warc.gz"}
http://mathematica.stackexchange.com/questions/18380/automatic-recovery-after-crash?answertab=active	# Automatic recovery after crash I want Mathematica to periodically autosave my notebooks to the temp folder. And in case of front-end crash i want to see the dialog after restart with the following choices: • recover from autosaved drafts • discard drafts and open last manually saved versions • start a new session Ideally i would like to make a fully automated solution that doesn't requre to execute any commands each time i open a new notebook. Do you have any suggestions how to implement this functionality? There is a hidden NotebookAutoSave option which saves notebook at every evaluation, but it has several disadvantages. It overwrites manually saved notebook and if i accidentally delete an important code (and evalute something) i'm unable to restore from the manually saved file. In OS X a built-in backup system can help, but when notebook is saved after each evaluation i get too many versions. There is also no possibility to distinguish manually saved version from all that autosaves in the Time Machine. Another problem with NotebookAutoSave is that it is only triggered by the evaluation. It doesn't help when i write code, open built-in help and front-end crashes during help browsing process. - Related (possible duplicates): mathematica.stackexchange.com/q/8761/5, mathematica.stackexchange.com/q/6435/5 – The Toad Jan 24 '13 at 21:56 Motivated by the same concerns, I wrote the following code and added it to my init.m file. There are two main entries you'll want to change to use this. The global variable $SwapDirectory is where the swap files are saved (by swap file, I mean it in the VIm sense; an "extra" copy of your notebook, separate from your manually saved copy that periodically saves any new work). The swap files are organized within the swap directory in a directory structure which "mirrors" their original file locations, and have ".swp" appended to their file names. The other variable you might want to change is the number of seconds between autosaves, indicated by the "300" (corresponding to 5 minutes) near the bottom of the code below. At the appropriate times, this code will (automatically in the background) save swap files for ALL open notebooks, unless they are unmodified from their manually-saved versions (this exception makes the code more efficient, and more importantly, prevents the storage of swap files for documentation notebooks, for example). Some things to note: 1) the Mathematica Put command seems to have trouble writing to network drives, even when offline access is enabled. Therefore, it is probably best to choose a SwapDirectory that is on your local machine. 2) Within SwapDirectory, you should create a sub-directory called "Recovery". This is where the AutoSaveSwap routine will make an initial save of any notebooks for which there is NO existing manual save location. 3) I have not designed an automatic dialog like you were hoping for for the front-end, but recovering any given notebook is easy. Simply evaluate RecoverSwap["filePath"] where "filePath" is a string representing the filePath of the MANUALLY-SAVED copy of the file (i.e., not the file that was created by AutoSave). This will then pop up a window containing the most recent auto-saved version of the file. The manually saved version is NEVER overwritten, unless you explicitly choose to do so. Once the recovered version pops up, you can save it whereever you like, or discard it at your discretion. 4) You should probably add this code to the KERNEL version of init.m ($UserBaseDirectory/Kernel/init.m) rather than the frontend version... this way, if you quit and restart the kernel, the autosave feature will also restart. On the other hand, this means that you must evaluate at least one expression after each start or restart to begin auto-saving. Once this initial evaluation is done, you do NOT need to have evaluated a cell for it to be backed up (unlike the built-in autosave utility). Hope this helps someone! Feel free to respond with any questions, suggestions, or requests for improvement you may have. And, if you find this post useful, upvotes would be most appeciated! Take care. $SwapDirectory= "C:\\Users\\pacoj\\Swap Files\\"; SaveSwap[nb_NotebookObject]:=Module[ {fileName, swapFileName, nbout, nbdir, nbdirout, recoveryDir}, If[ ! SameQ[Quiet[NotebookFileName[nb]],$Failed], (* if the notebook is already saved to the file system ) fileName = Last[ FileNameSplit[ NotebookFileName[nb]] ]; swapFileName = fileName <> ".swp"; nbdir = Rest[FileNameSplit @ NotebookDirectory[nb]]; nbdirout= FileNameJoin[ FileNameSplit[$SwapDirectory]~Join~nbdir]<>"\\"; If[!DirectoryQ[nbdirout], CreateDirectory[nbdirout]]; nbout = NotebookGet[nb]; Put[nbout, nbdirout <> swapFileName], ( else, if the file has never been saved, save as untitled *) recoveryDir=$SwapDirectory <> "Recovery\\"; fileName= ("WindowTitle" /. NotebookInformation[nb])<>".nb"; NotebookSave[nb, recoveryDir <> fileName] ] ]; RecoverSwap::noswp= "swap file 1 not found in expected location"; RecoverSwap[nbfilename_String]:=Module[ {fileName, swapFileName, nbin, nbdir, nbdirout}, fileName= Last[ FileNameSplit[ nbfilename] ]; swapFileName= fileName <> ".swp"; nbdir= Most[ Rest[FileNameSplit @ nbfilename] ]; nbdirout= FileNameJoin[ FileNameSplit[\$SwapDirectory]~Join~nbdir]<>"\\"; If[ FileNames[swapFileName, {nbdirout}] == {}, Message[RecoverSwap::noswp,nbdirout <> swapFileName]; Return[], nbin= Get[nbdirout <> swapFileName]; NotebookPut[nbin] ] ];	{"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": 0.39806613326072693, "perplexity": 6899.820531913013}, "config": {"markdown_headings": true, "markdown_code": false, "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-2015-27/segments/1435375090887.26/warc/CC-MAIN-20150627031810-00294-ip-10-179-60-89.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/low-load-factor-and-voltage-drops.630764/	# Low load factor and Voltage drops 1. Aug 23, 2012 ### rozan977 My lecturer, in the notes he provided to us mentioned that low load factor of a distribution transformer results in higher system losses and voltage drops. I thought the by higher system losses he means the iron losses in transformer which is comparably higher in no-load and lower load conditions. But I just couldn't figure out how low load factor causes voltage drops in system??? 2. Aug 24, 2012 ### uart He's probably referring to the effect of reactive current causing comparatively much more voltage drop in upstream reactive impedance. Because the series impedance of the power system is primarily inductive, then the voltage drops due to reactive current tends to subtract directly, whereas those from real current components add vectorally at right angles. For example, say you had a load taking 1.0 pu reactive current and the series impedance was j0.1 pu. The required generator voltage is 1.1 pu (the voltage drop is in phase so adds algebraically). Now consider the same case but with 1.0 pu real current being supplied. The required generator voltage is now only 1.005 pu (volt drop adds at 90 degrees). 3. Aug 25, 2012 ### I_am_learning uart, I think you are confusing 'Load factor' with 'power factor'. @rozan977, I think the lecturer is trying to explain what would happen in a distribution system already in existence if the Load factor goes low. In such case, assuming same average monthly consumption, a low load factor means, high currents for shorter period of time, (as compared to the currents being distributed in time, in case of higher load factor). Higher currents obviously results in higher voltage drop (during the high current period). If we do the maths, we can also see that, when currents are higher for short period and less for longer period (instead of evenly spreading out), the power loss in the line will be higher. for example, For a building, the current is 10A for whole 24 hours (load factor of 1) Then its daily energy Loss = 10^2R24 In another building, the current is 20A for 12hrs and 0A for 12hrs (same average daily energy). This gives load factor of 0.5. Then its daily energy loss = 20^2R12, which is double of the previous case. 4. Aug 25, 2012 ### uart No, actually I think you're confusing load factor with something like duty factor or duty cycle (the ratio of average to peak). The load factor of a transformer (at any given point in time) is simply the operating power (at that time) as a ratio or percentage of it's rated power. Since the magnetizing reactance draws approximately constant reactive current (independent of load) then a low load factor implies a worse power factor. So no, I'm not confusing them, they are in fact related. Last edited: Aug 25, 2012 5. Aug 25, 2012 ### I_am_learning In the book "ELECTRIC POWER DISTRIBUTION EQUIPMENT AND SYSTEMS", Author T.A. Short says 6. Aug 25, 2012 ### uart The definition I've got here is: Edit. Just googled a few papers on the subject and some authors are using the definition of "current power/rated power" while others are using "average/peak". So yeah, take your pick. I guess the op will have to let use know what particular definition of "load factor" he's using. Last edited: Aug 25, 2012 Similar Discussions: Low load factor and Voltage drops	{"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": 0.9058482050895691, "perplexity": 2597.9883586868414}, "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/1508187825473.61/warc/CC-MAIN-20171022222745-20171023002745-00362.warc.gz"}
https://www.physicsforums.com/threads/limiting-value-of-the-velocity.223583/	# Limiting value of the velocity 1. Mar 22, 2008 ### alice137 1. The problem statement, all variables and given/known data This problem concerns the mathematical treatment of a simple model of an electric toy car of mass m, which is initially stationary. The bat- teries in the car can be considered as an electrical power source with constant power P. Find the limiting value of the velocity at very large times and comment on whether your result seems reasonable. 2. Relevant equations On calculating I got v=sqrt(2Pt/m). 3. The attempt at a solution As t -> Infinity, so does v-> Infinity. But that does not makes sense. Where am I wrong. Please help. Last edited: Mar 22, 2008 2. Mar 22, 2008 ### alice137 Another part of the question was: Find the limiting value of the acceleration at very large and very small times and comment on whether your results seem reason- able. On calaulating I got a=sqrt(p/2mt). As t -> 0, a -> Infinity. Again That doesn't makes sense. 3. Mar 26, 2008 ### tiny-tim Hi alice! What makes you think you're wrong? The question obviously expects the answer not to be make sense. You're asked to comment on why correct maths doesn't seem to agree with reality! What do you think … ? 4. Mar 26, 2008 ### alice137 Well I personally belive that the answer for acceleration makes sense. Initially the car might have acquired some velocity in a small time. I think the high acceleration is plausible. For velocity, it cannot increase without bounds as velocity of a material object is limited by Relativity. But if we see the time dependent relation of acceleration and see that as t-> infinity, a->0, which means that v-> some constant. But I cannot find that constant based on these arguments. 5. Mar 26, 2008 ### tiny-tim Hi alice! At relativistic speeds, the mass m gets bigger, and the original formula is probably wrong also. No, you're wrong - there's no problem there. a->0 does not have the same consequences as a = 0: there's no reason why v should -> constant. √n -> ∞, but √(n+1) - √n -> 0 (for example, √1,000,001 - √1,000,000 ~ .0005). 6. Mar 26, 2008 ### alice137 What do you insist then; v should increase without bounds? 7. Mar 26, 2008 ### tiny-tim No, but the momentum can increase without bounds, even though v is always less than c. (momentum = mv/√(1 - v^2/c^2).) 8. Mar 26, 2008 ### alice137 Then I suppose that the answer to the first question should be v-> c at very large times even although we cannot see it directly from the equation. I guess we need to derive another relativistic expression for very large times. Right! Similar Discussions: Limiting value of the velocity	{"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": 0.9201623201370239, "perplexity": 1792.7009072094065}, "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/1492917121665.69/warc/CC-MAIN-20170423031201-00388-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.openimpulse.com/blog/products-page/product-category/laser-diode-module-red/	# Laser Diode Module Mini laser diode module, ideal for many applications like intrusion detection, laser projectors, optic listeners and optical wireless communications. SKU: SLI7861771221,CQY16582451925 Price: 0.17 $Old Price: 0.99$ Product in stock SSL Certificate Quantity # Overview Mini laser diode module, ideal for many applications like intrusion detection, laser projectors, optic listeners and optical wireless communications. It contains both the laser diode and the focal lens, being ready for immediate use in your projects. ## Specifications • Power: 5 mW • Wavelength: 650 nm • Forward Voltage: 3 V • Current: < 20 mA • Working Life: 1000 hours Note: The laser diode can be easily damaged by over voltage or over current. Warning: Please take special care when using this product and do not aim it to the eyes !	{"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": 0.25218603014945984, "perplexity": 11337.409852912839}, "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/1610703521987.71/warc/CC-MAIN-20210120182259-20210120212259-00692.warc.gz"}
https://webwork.maa.org/moodle/mod/forum/discuss.php?d=2900	WeBWorK Problems Inequalities with two variables in answers? by Paul Seeburger - Number of replies: 11 I am trying to write a series of problems for multivariable calculus that ask the student to enter the domain and range of a function of two variables. The range is no problem as it uses interval notation, but the domain is harder since I want my students to use set-builder notation. For example, f(x, y) = (x+y)/(x-y) would have domain: { (x, y) : x <> y } and range (-inf, inf). f(x, y) = arcsin(y - x) would have domain: { (x, y) : x - 1 <= y <= x + 1 } and range [-pi/2, pi/2]. Ideally I would love to be able to require students to enter the whole set-builder notation, but for now I would be happy if I could check only the condition involving an inequality in two variables. Is there any way to do this? Perhaps if I were to write my own checker? Or would I need to create a whole new Context? Has anyone worked on this? It would be useful also in algebra when we solve linear inequalities in two variables or absolute value inequalities, etc. This is something I AM able to do with MyMathLab, Pearson's commercial online homework system. Thanks! Paul Edit: I found a way to do this that seems to work, but it is using the older answer checkers. See below: $domain = "x-1 <= y <= x+1"; ANS( str_cmp($domain,'remove_whitespace')); I suppose this should even work with the whole set-builder notation. But is there a better way to do this kind of thing using a MathObject? Paul Re: Inequalities with two variables in answers? by Davide Cervone - Ideally I would love to be able to require students to enter the whole set-builder notation, but for now I would be happy if I could check only the condition involving an inequality in two variables. Is there any way to do this? WeBWorK doesn't currently have a way to do this. It's inequality notation is basically an alternative for describing intervals, so they must have a variable on one side and a number on the other. They are not general inequalities of multiple variables. It would be possible to extend the current inequalities context to make them work with a variable on one side and a formula (in other variables) on the other, so you could get things like x-1 <= y <= x+1, or y < x^2+ 1 but not y - 1 < x^2 or x^2 + y^2 < 10. I do not know a reasonable (numerical) way to check a student answer for something like x^2 + y^2 < 10 in general. The crucial values are where x^2 + y^2 = 10, and this is a level set, which is notoriously hard to work with numerically, especially when the level set has singularities. One possibility would be to use Newton's method to find a bunch of solutions to the equality and make sure the student answer also has those as solutions, and then check a bunch of random points to see that the signs match for student and professor's answers. But the numerics for this are pretty delicate. The parserImplicitEquation.pl does something like this for equalities like x^2 + y^2 = 10, so that could be a starting point, but it is not a particularly reliable macro package, so I would be hesitant about that. But people are using it, so perhaps it would work for you. Do you have any suggestions about how such an answer could be checked numerically in a reliable way? Perhaps if I were to write my own checker? Or would I need to create a whole new Context? An answer checker is not going to be enough, since you also need to extend the context to allow the inequalities, so that means writing your own context. And that is not as easy as I would like it to be. To handle set-builder notation would also require a new context. But without a reliable way to test multi-variable inequalities, I don't think it is work the effort. About using string comparisons for this, you are aware, of course, that this means the student must enter the answer in exactly the format you did, and that means correct answers will be marked incorrect. E.g., if you have { (x,y) : y < 2x + 1 } and the student writes { (x,y) : y < 2x + 1 } or { (x,y) : y < 1 + 2x } then they will be marked incorrect. Personally, I think that is worse that rewriting the question to use the mathematical tools that you have available. For example, give them "{ (x,y) : y < [_________] }" where they have to fill in the blank with a formula. I know it doesn't force them to write the set themselves, but I would rather they be marked correct when they write the correct formula in an unexpected way than be forced to write set notation with the formula in a particular order. I don't like string checking for such answers for this reason, and would not recommend that approach. Re: Inequalities with two variables in answers? by Paul Seeburger - Thank you for your detailed response, Davide! I can see that there are a lot of issues to consider here that I had not anticipated. I think an improved inequalities macro would be quite useful in a variety of topics. Of course the domain of a function of two variables is one. Can the current inequalities context handle compound inequalities like the following? 2x + 1 <= -5 or 2x + 1 >= 5 Hopefully it can. I would like to port some problems I have written for MyMathLab over to WebWork someday to help students step through the process of solving absolute value inequalities. In the meantime though, I will try my hand at using the contextArbitraryString.pl macro that you told me how to use in your response to my other post. You said: It would be possible to extend the current inequalities context to make them work with a variable on one side and a formula (in other variables) on the other, so you could get things like x-1 <= y <= x+1, or y < x^2+ 1 Would it be difficult to do this? Is this automatically acceptable to the inequalities context? I am guessing not. What would it involve to try to extend it to handle these possibilities? That would help with some of my examples. Now, concerning the issue of multiple ways to represent the correct answer condition. I stress the importance of making the form of this condition as clear as possible, and although I certainly would accept { (x,y) : y < 2x + 1 } for { (x,y) : y < 2x + 1 } I would not want to accept { (x,y) : y + 2x < 1 } which will not work anyway right now. =) Ideally I would want to be able to give a hint to make the condition more clear by isolating y. I hesitate still to use: "{ (x,y) : y < [_________] }" since it seems to lead them way more than I want to in this kind of problem. There is so much variety in the types of conditions that arise in this context. I want them to be able to identify whether the clearest condition can be obtained by isolating (solving for) y, by putting the inequality in circle form, ellipse form, hyperbola form, or perhaps just leaving it in its original form (e.g., xy > 0). Thanks again, Davide! Paul Re: Inequalities with two variables in answers? by Davide Cervone - Can the current inequalities context handle compound inequalities like the following? 2x + 1 <= -5 or 2x + 1 >= 5 The inequalities context currently requires a variable on one side and a number on the other. So it would not accept 2x+1 <= -5; you would have to enter x <= -3. But you can enter things like x <= -3 or x >= 2 (which is implemented as a union of intervals internally). You said: It would be possible to extend the current inequalities context to make them work with a variable on one side and a formula (in other variables) on the other, so you could get things like x-1 <= y <= x+1, or y < x^2+ 1 Would it be difficult to do this? With the limitation that you must have a variable on one side and a formula (that doesn't include that variable) on the other, it can be done with some changes to the current context. But it means you would have to enter y < x^2 + 1 not y - 1 < x^2, a restriction that I don't like. Is this automatically acceptable to the inequalities context? No, as you suspected. What would it involve to try to extend it to handle these possibilities? It is not something that someone not familiar with the internal details of contexts would want to attempt. The inequalities context is a particularly complicated one, and not the place to begin with modifications to contexts. In any case, I have most of the work done already (as an experiment to see if it could be done). Re: Inequalities with two variables in answers? by Paul Seeburger - Thanks so much for your replies, Davide! Can the current inequalities context handle compound inequalities like the following? 2x + 1 <= -5 or 2x + 1 >= 5 The inequalities context currently requires a variable on one side and a number on the other. So it would not accept 2x+1 <= -5; you would have to enter x <= -3. But you can enter things like x <= -3 or x >= 2 (which is implemented as a union of intervals internally). This seems to be a case where the equivalent answers are not being accepted yet, even when there is only one variable. Is this correct? It would be great to have the ability to grade the statement above as the correct way to rewrite \|2x + 1\| >= 5 without absolute value symbols, but I would really want to require the form shown, i.e., accepting either 2x + 1 <= -5 or 2x + 1 >= 5 or 2x + 1 >= 5 or 2x + 1 <= -5 My hope is to guide students through the steps I require them to show on paper to solve these absolute value inequalities in my intermediate algebra class (as I currently do in MyMathLab with a series of problems I designed there). Hopefully this makes sense, and hopefully there will be an acceptable way to do this. It appears to me that the contextArbitraryString.pl you told me about may well allow me to do this, although as you point out, it is not yet the ideal solution. Thanks again, Davide! Paul Re: Inequalities with two variables in answers? by Davide Cervone - This seems to be a case where the equivalent answers are not being accepted yet, even when there is only one variable. Is this correct? You are correct that this does not allow the equivalent answers. But there are two important features that affect this: first, it is easy to say what the required format is (a variable on one side and a number on the other), so the question can tell you what is required, and the context reports the problem if you enter something that isn't in that form. (The original requirement for the inequality context was to describe intervals, not do to general inequalities.) What bothers me about your approach is that I don't see a reasonable way to describe the required form ("clearest" or "simplest" form isn't precise enough) so that a student is clear about what is acceptable and what isn't, and that your techniques aren't going to tell him. The difference between the forms that you accept and those that you don't are going to see arbitrary. Which of 44 different forms for y = (1+3x)/2 in our other discussion should be acceptable? There are more that I didn't include, as well. So how do you describe the form you are willing to accept, and how do you guide the student to that form? You say you accept 2x + 1 <= -5 or 2x + 1 >= 5 or 2x + 1 >= 5 or 2x + 1 <= -5 -5 >= 2x + 1 or 2x + 1 >= 5 or -5 >= 2x + 1 or 5 <= 2x + 1 or 1 + 2x <= -5 or 1 + 2x >= 5 or any of a multitude of other permutations. And if not, why not? It appears to me that the contextArbitraryString.pl you told me about may well allow me to do this, although as you point out, it is not yet the ideal solution. Very far from ideal, in my opinion. But I guess I've said enough about that, so am not going to continue the thread any further. Re: Inequalities with two variables in answers? by Paul Seeburger - Thanks, Davide! For this question, I would like to accept any of the options you listed above, but I would want the student to not yet simplify the answer, so I can grade them as they complete the steps (and provide some feedback along the way as to how they are doing). Let me give an example of what I am trying to do in a number of different problem types. This problem will be scaffolded to help them through the process of solving it using the particular method I require my students to show on paper. I already do this in MyMathLab and it works quite well. The requirements in MyMathLab are very similar to what I have to work with using contextArbitraryString.pl, but actually in WebWork I can provide a lot of helpful guidance and feedback I could not do in MyMathLab. Ex. \| 2x + 1 \| >= 5 1. They would choose the best statement of four including the correct answer: "The value of 2x + 1 is 5 or more units from 0 on the number line." 2. They would then need to select the correct number line graph representation of this statement from a multiple choice list of generated numberlines. 3. Then they would be asked to use the numberline reference graph to rewrite the absolute inequality without absolute value symbols. This is where I would need the above mentioned expression to be graded. That is: 2x + 1 <= -5 or 2x + 1 >= 5 In MyMathLab, I have been pretty strict about this form, but I can see that in WebWork I can actually allow a lot more possibilities as long as I list them carefully. 4. Then I would ask the student to solve this inequality and enter the correct simplified version in set-builder notation provided for them: x <= -3 or x >= 2 WebWork Interval context would accept this last answer, correct? I may still use my own checker here though, since I ask my students to enter this sort of condition so that it reads from left to right on the numberline. I hope this scaffolding/tutorial approach I am trying to use to develop new problems in WebWork makes some sense. Thanks again, Davide! I do understand where you are coming from on this and I appreciate your help! Paul Re: Inequalities with two variables in answers? by Davide Cervone - 4. ... WebWork Interval context would accept this last answer, correct? No, but the Inequalities context would. Re: Inequalities with two variables in answers? by D. Brian Walton - Paul, I have been thinking about problems of a similar nature, in that I want students to "work through" problems rather than just give an answer. One such a problem was to require students to compute a limit "step-by-step" where they needed to specify a particular limit rule. Each step must be either (1) an elementary limit (e.g., limit of a constant or limit of a power of x) or (2) an arithmetic combination of a limit (e.g., sum, difference, product, quotient) of formulas whose limits had previously been computed. The answer checker was a complex multi-answer checker that could only verify if each line satisfied the requirements specified and applied the rules appropriately and that ultimately answered the question of interest as the last step. It could not check if the student avoided unnecessary extra steps, only if every step was logically valid. The instructions for the students was also quite lengthy and involved a javascript popup window to explain how to indicate how the student was to specify which rules were being used. Most students still seemed to require an explanation in class as well to use such a problem. I have been thinking that a better approach, if it really had to be done, was to create a javascript shell that actually implemented a nicer interface and enforced logical rules and then simply reported some type of summary to WeBWorK for validation that the problem was completed correctly. I have not yet tried such a thing. Anyway, there seem to be ways to impose a structure on WeBWorK to implement these types of problems, but they are fairly challenging and not the foundational philosophy that WeBWorK is built on. - Brian In reply to D. Brian Walton Re: Inequalities with two variables in answers? by Paul Seeburger - Thanks, Brian! What is the foundational philosophy that WebWork is built on? (at least as it pertains to this idea of helping students learn the steps as well as the answers to solving problems) I assumed the basic philosophy is to help students learn mathematics through practice of meaningful and varied example problems. I don't see this approach to be too difficult to implement for many types of problems. What you described with limits would certainly be much more difficult, but the example I described above concerning absolute value inequalities I already successfully implemented in MyMathLab (with improved student success on tests), and now that I am creating problems in WebWork, I am finding that it will actually be much easier to do this kind of thing here, once I figure out some of the basic techniques and coding issues. Since we require our students to show clear work on tests and graded worksheets, and we tell them that we are grading their work, not their answers only, I think we really need to find a better way to evaluate and train our students in online homework systems than to just ask for final answers. For example, I don't think there is much value in asking students to solve word problems in WebWork, if all we ask for is a numerical answer. We should ask for some sort of indication of a correct set-up or variable definition, a correct equation if possible, and certainly units on their answer whenever appropriate. In Calculus, I don't see much use in asking students to evaluate definite integrals in WebWork if we are just asking for the numerical answer AND we don't even care if it is the exact answer. Most students will just use their calculator to approximate the integral rather than using any real calculus to work it out. In definite integrals, I believe we need to at least ask them for the antiderivative expression to make the problem more meaningful in this context. I also believe we should always ask for the exact answer to definite integrals. Is there something that would cause this to be a problem? We are training our students to solve problems in a particular way by the way we ask our questions and by the answers we accept as correct. Students will almost always tend toward the method that gives them a correct answer with the least amount of time and thought. I expect you agree with me on much of this, but I am curious if this idea has been discussed already in the WebWork community. Thanks again, Brian! Paul Re: Inequalities with two variables in answers? by D. Brian Walton - Paul, I agree with what you said. My point was more along the lines that WeBWorK is not very designed to check student's thinking process, which is what I was presuming from your original example regarding the absolute value inequality. My interpretation of the WeBWorK philosophy is that it is designed to have students submit answers that are well-defined mathematical objects (numbers, formulas, intervals, etc.) and compare that to a pre-calculated (either predetermined or algorithmically generated) and give a student Yes/No answers on the correctness of that answer. Furthermore, the underlying machinery is designed in a way to allow the student to enter the answer in a much more flexible way than providing a list of approved answers. Any answer that is mathematically correct is supposed to be acceptable. Special contexts were later created to facilitate certain conditions such as requiring simplified polynomials or to exclude the use of various functions. Your example with the absolute value inequality is problematic because the student's answer is a class of mathematical object that has not yet been implemented (i.e., a logical statement with free variables). However, it is composed of elements that are implemented. For example, if you were to construct a problem that involved multiple parts, you could ask for the student to fill in the blanks: [------](expression) >= [---](number) and/or(popup-list) [------](expression) < [---](number) This allows the student to construct the statement you were desiring, but the structure of the problem imposes the form. I implemented some similar problems based on rewriting expressions involving absolute value as piecewise formulas, similar to the following form (hopefully the forum reformatting doesn't make this unreadable): \|x^2-5x\| = { [-----](formula) if x >= [-----](number) { [-----](formula) if x <= [-----](number) Along the lines of what you were discussing, this could be extended to include an earlier step: \|x^2-5x\| = { [-----](formula) if [-----](formula) >= 0 { [-----](formula) if [-----](formula) < 0 Such problems make it so that you are only checking the formulas, not the inequality itself. My experience with these problems suggest that students do not pay attention to the structure of the problem as it is presented on WeBWorK. They are only focused on finding the formula/answer that goes in the answer box. I'm not sure that WeBWorK can teach students to practice the metacognition required to think at this higher level. - Brian Re: Inequalities with two variables in answers? by Davide Cervone - I stress the importance of making the form of this condition as clear as possible, and although I certainly would accept { (x,y) : y < 2*x + 1 } for { (x,y) : y < 2x + 1 } I would not want to accept { (x,y) : y + 2x < 1 } I wouldn't want to accept that either, but I would want to accept { (x,y) : y - 2x < 1 } which describes the same set. :-) I understand your desire, but "clear" is subjective. While you might think that breaking down the relationship between x and y (a two-dimensional view) is the clearest, it might be clearer to me to think of when a certain multivariable function is below a certain value (a three-dimensional view). Unless you are very explicit (in the problem) about how your answer is to be expressed, I think you are asking for trouble to try to restrict the form of the answer that you accept. For example, if you simply ask for the domain of the function and accept { (x,y) : y < 2x + 1 } but not { (x,y) : y - 2x < 1 }, then as a student I would be pretty upset. After all, these are the same sets. If there is some other information you are after (e.g., the particular format of an inequality), then perhaps you should not be asking about the set, but about that inequality. I hesitate still to use: "{ (x,y) : y < [_________] }" since it seems to lead them way more than I want to in this kind of problem. Yes, I understand that. But since you have to be clear about the form you want anyway, and are going to give them a hint when they don't do this, I don't understand the pedagogical value in having them possibly enter sets that are correct and forcing them to rewrite them in another form. The reason I suggested this form was not because I didn't think it would be nice to have them write the set-buildner notation, but because this is a way to get the problem you want right now with the tools available. It is a way that works, that gives appropriate feedback, and that marks correct answers correct and incorrect answers incorrect. That is not going to be the case for your string-based checking (see our other conversation about string checking).	{"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": 0.570919930934906, "perplexity": 478.4447064702303}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572127.33/warc/CC-MAIN-20220815024523-20220815054523-00554.warc.gz"}
https://www.physicsforums.com/threads/relation-between-thermal-expansion-and-density-in-polymers.824881/	# Relation between thermal expansion and density in Polymers 1. Jul 26, 2015 ### sjumesh Hello all, Is there any direct relation ship exists between the thermal expansion and density of polymer material (eg. PMMA, Teflon) ? Like, Lorentz- lorentz law relates directly the refractive index and density of the polymer. similarly any laws relating (thermal expansion and density of polymer)or ( thermal expansion or refractive index) exists ? 2. Jul 26, 2015 ### Staff: Mentor Yes. The coefficient of volumetric thermal expansion of any material is related to its density by: $$\alpha=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_P$$ Chet 3. Jul 26, 2015 ### sjumesh Dear Chestermiller, At first, thank you for your message. Yes, i was asking for these equations. And, also is there any relationship between the thermal expansion and refractive index of material. Any equation relating thermal expansion and refractive index of material. Can you suggest me any books or materials to study. regards, Umesh.S 4. Jul 26, 2015 ### Staff: Mentor I 'M not aware of such a rrlationship, but I'm not an expert. Any text on polymer science should answer your question. Chet 5. Jul 26, 2015 ### sjumesh Dear Chestermiller, i will look in the polymer science for further details. regards, Umesh.S	{"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": 0.9542091488838196, "perplexity": 3433.065946322937}, "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/1508187825575.93/warc/CC-MAIN-20171023035656-20171023055656-00438.warc.gz"}
https://winscp.net/forum/viewtopic.php?t=14031	# Topic "How use Put command for uploading only new files" Author Message Pepper Guest Hi. I wrote script which must upload some files to my FTP. In which I use PUT command. How I can say to PUT command don't upload the already existing files in remote directory? Before WinSCP I excellent used ncftpput tool, but it don't understand russian utf-8 file names. And if I try upload same files of local and remote directory ncftpput say: "remote file appears to be the same as the local file, upload is not necessary." - is what I need. But WinSCP in this case still upload files - do unnecessary work. (Sorry for my English.) Advertisements Pepper Guest In other words, I need a command (or combination command/switches) like: "synchronize remote" but for files (not for folders) Thanks! martin [View user's profile] Site Admin Joined: 2002-12-10 Posts: 25748 Location: Prague, Czechia You can do synchronize remote -filemask=filename to restrict synchronization to a single file. https://winscp.net/eng/docs/scriptcommand_synchronize There's also this outstanding request: https://winscp.net/tracker/show_bug.cgi?id=666 Pepper Guest martin wrote: You can do synchronize remote -filemask=filename to restrict synchronization to a single file. https://winscp.net/eng/docs/scriptcommand_synchronize There's also this outstanding request: https://winscp.net/tracker/show_bug.cgi?id=666 Thank you, for reply and outstanding request. synchronize remote -filemask=filename - I can't use. Awaiting we can solve this problem in the next release WinSCP? martin [View user's profile] Site Admin Joined: 2002-12-10 Posts: 25748 Location: Prague, Czechia Pepper wrote: synchronize remote -filemask=filename - I can't use. Why? _________________ Martin Prikryl Pepper Guest martin wrote: Pepper wrote: synchronize remote -filemask=filename - I can't use. Why? I use WinSCP in script mode. I run it from batch: WinSCP.exe /console /script="C:\wscpcmd.txt" /parameter %prm% wscpcmd.txt: open ftp://... -passive=on -timeout=10 -rawsettings Utf=2 option batch on option confirm off option reconnecttime 20 synchronize remote "C:\to_upload\%1%" /Public close exit where is %prm% can be single file or folder, and I don't know exists this file/folder at FTP or not. If I upload folder to FTP - I don't know which files in this folder already exists, because I can't use -filemask. (In this example sсript will be problem if %prm% contain single file name (not folder).) Pepper Guest Or, if I use put for uploading single files with switch -filemask for example: put -resume -transfer=automatic -filemask="\|my_file.zip" "my_file.zip" can be situation where local file will be newer than remote, and it not be uploaded, because file mask switch present. martin [View user's profile] Site Admin Joined: 2002-12-10 Posts: 25748 Location: Prague, Czechia If the parameter can be folder or file, you just need to make the include mask accept both files and folders: synchronize remote "C:\to_upload\" /Public -filemask="%1%;%1%\" _________________ Martin Prikryl Advertisements You can post new topics in this forum ## What is WinSCP? It is award-winning SFTP client, SCP client, FTPS client and FTP client integrated into one software program for file transfer to FTP server or secure SFTP server. [More] And it's free!	{"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": 0.5359305739402771, "perplexity": 25295.005416664273}, "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-2017-13/segments/1490218189090.69/warc/CC-MAIN-20170322212949-00602-ip-10-233-31-227.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/195501-gamma-function-limit-involving-stirlings-formula.html	## gamma function limit involving stirlings formula Hello, I am trying to show the following: $\lim_{x\rightarrow\infty}\frac{\Gamma (x+c)}{x^c \Gamma (x)}=1$, where $c\in\mathbb{R}$. Recall Stirling's formula: $\lim_{x\rightarrow\infty}\frac{\Gamma (x+1)}{(x/e)^x \sqrt{2\pi x}}=1$. Any suggestions on how to prove the former limit using this formula? I have tried the change of variables $t=x(c+u)$, but got stuck. Any help would be greatly appreciated.	{"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": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9890424013137817, "perplexity": 102.63405877885744}, "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/1518891812788.42/warc/CC-MAIN-20180219191343-20180219211343-00541.warc.gz"}
https://labs.tib.eu/arxiv/?author=Abdusalam%20Abdukerim	• ### Particle Physics Constraints on Self-Interacting Dark Matter from PandaX-II Experiment(1802.06912) Feb. 19, 2018 hep-ph, hep-ex Self-Interacting Dark Matter (SIDM) is a leading candidate to solve the puzzles of the cold dark matter paradigm on galactic scales. We present a particle-physics study on SIDM models in PandaX-II, a direct detection experiment in China JinPing underground Laboratory. We use data collected in 2016 and 2017 runs, corresponding to a total exposure of 54 ton day, the largest published data set of its kind to date. Strong combined limits are set on the mass of the dark-force mediator, its mixing with the standard model particles, and the mass of dark matter. Together with considerations from the Big-Bang Nucleosynthesis, our results put tight constraints on SIDM models. • ### Exploring the dark matter inelastic frontier with 79.6 days of PandaX-II data(1708.05825) Oct. 3, 2017 hep-ph, hep-ex, astro-ph.CO We report here the results of searching for inelastic scattering of dark matter (initial and final state dark matter particles differ by a small mass splitting) with nucleon with the first 79.6-day of PandaX-II data (Run 9). We set the upper limits for the spin independent WIMP-nucleon scattering cross section up to a mass splitting of 300 keV/c$^2$ at two benchmark dark matter masses of 1 and 10 TeV/c$^2$. • ### Dark Matter Results From 54-Ton-Day Exposure of PandaX-II Experiment(1708.06917) We report a new search of weakly interacting massive particles (WIMPs) using the combined low background data sets in 2016 and 2017 from the PandaX-II experiment in China. The latest data set contains a new exposure of 77.1 live day, with the background reduced to a level of 0.8$\times10^{-3}$ evt/kg/day, improved by a factor of 2.5 in comparison to the previous run in 2016. No excess events were found above the expected background. With a total exposure of 5.4$\times10^4$ kg day, the most stringent upper limit on spin-independent WIMP-nucleon cross section was set for a WIMP with mass larger than 100 GeV/c$^2$, with the lowest exclusion at 8.6$\times10^{-47}$ cm$^2$ at 40 GeV/c$^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": 0.9075117111206055, "perplexity": 1985.7527588537073}, "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/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00107.warc.gz"}
https://wenbinyugroup.github.io/ivabs/prevabs/contents/guide/pre_component.html	# Components# Listing 37 Input syntax for components.# 1<component name="" type="" depend="">...</component> Specification • <component> - Root element for the definition of each component. • name - Name of the component. • type - Type of the component. Choose one from ‘laminate’ and ‘fill’. Default is ‘laminate’. • depend - List of name of dependent components, delimited by commas. ## Laminate-type component# For a cross section in PreVABS, laminates are created as segments. A segment is a unique combination of a base line and a layup. Segments are connected through different ways as shown in Fig. 35. According to this, segments can be grouped into components. The rule of thumb is: if two segments are connected in the first two ways (‘V1’ and ‘V2’), then they belong to one component; if they are connected as the third way (‘T’), then they should be put into different components, and component 2 will be created after the finish of component 1. A schematic plot of a segment is shown in Fig. 36. The base line provides a reference for the position and direction of the layup. Layers can be laid to the left or right of the base line. The direction is defined as one’s left or right, assuming one is walking along the direction of the base line. All segments definitions are stored in the main input file. The complete format will be discussed later since the overall configurations are also included in this file, which will be explained in Section: Other input settings. The input syntax for laminate-type components are given in Listing 38. Each component can have multiple segments. There are two ways to define segments. ### DEFINITION 1: Define segments individually# Use a base line and a single layup to create the segment. In this way, the layup covers the entire of the base line. Each <segment> element has one attribute, name, and two child elements, <baseline> and <layup>. The <layup> element has another attribute direction (see Fig. 36). Joint of two connecting segments can be changed from ‘V2’ (default) to ‘V1’ by using a <joint> element. It requires the names of two segments, delimited by a comma (‘,’) and an attribute style specifying the joint type. ### DEFINITION 2: Define segments collectively# Use a base line and multiple layups to create multiple segments. Each layup can be assigned to a portion of the base line, using a beginning and an ending locations. These locations are normalized parametric positions on the base line. The beginning location must be smaller than the ending one. If the line is open, the location can only be a number between 0 and 1. If the line is closed, the location can be any number, even negative, as long as the length is not greater than 1. Then PreVABS will split the base line, combine layups and create segments automatically. An example is provided below (Fig 37, Fig 38, Listing 39). Listing 38 Input syntax for the laminate-type components.# 1<component name="cmp_surface"> 2 <segment name="sgm_1"> 3 <baseline> baseline1 </baseline> 4 <layup direction="right"> layup1 </layup> 5 </segment> 6 <segment name="sgm_2"> 7 <baseline>...</baseline> 8 <layup>...</layup> 9 </segment> 10 ... 11 <joint style="2"> sgm_1,sgm_2 </joint> 12 ... 13</component> 14 15 16<component name="cmp_web"> 17 <segment> 18 <baseline> baseline2 </baseline> 19 <layup direction="left"> layup2 </layup> 20 </segment> 21 ... 22</component> 23 24 25<component name="..."> 26 <segments> 27 <baseline> base_line_3_name </baseline> 28 <layup_side> left </layup_side> 29 <layup> layup_1_name </layup> 30 <layup begin="0.1"> layup_2_name </layup> 31 <layup end="0.7"> layup_3_name </layup> 32 <layup begin="0.2" end="0.9"> layup_4_name </layup> 33 ... 34 </segments> 35</component> #### Example# Listing 39 Example input.# 1<component name="..."> 2 <segments> 3 <baseline> l </baseline> 4 <layup> layup1 </layup> 5 <layup begin="0.2" end="0.75"> layup2 </layup> 6 <layup begin="-0.1" end="0.5"> layup3 </layup> 7 <layup begin="-0.6" end="-0.2"> layup4 </layup> 8 </segments> 9</component> Specification DEFINITION 1 • <segment> - Root element of the definition of the segment. • name - Name of the segment. • <baseline> - Name of the base line defining this segment. • <layup> - Name of the layup defining this segment. • direction - Direction of layup. Choose one from ‘left’ and ‘right’. Default is ‘left’. • <joint> - Names of two segments delimited by a comma (‘,’) that will be joined. • style - Style of the joint. Choose one from ‘1’ and ‘2’. Default is ‘1’. DEFINITION 2 • <segments> - Root element of the definition. • <baseline> - Name of the base line definint these segments. • <layup_side> - Direction of the following layups. Choose one from ‘left’ and ‘right’. Default is ‘left’. • <layup> - Name of the layup. • begin - Normalized parametric beginning location of the layup on the base line. Defualt is ‘0.0’. • end - Normalized parametric ending location of the layup on the base line. Defualt is ‘1.0’. ## Fill-type component# Besides creating laminates, user can use one material to fill a region. A typical usage is a nose mass in an airfoil type cross section. A schematic plot is shown in Fig. 39. The key to this type of component is the indication of the fill region. There are two things to pay attention to. First, make sure that the boundary of the region is well-defined. The region can be surrounded by a number of components. User can also create some extra base lines as new boundaries. Second, locate the region where the material will be filled into. Then can be done by using a point inside the region, or if there are extra base lines, user can indicate the fill side with respect to one of the lines, similar to defining layup side in a segment. Fill-type components are also defined in the main input file. A template is shown in Listing 40. A fill-type component is indicated by the attribute type="fill". A <material> child element is required. A <baseline> element is optional and is used to create extra boundaries. This sub-element has one attribute fillside, which can be either left or right. A <location> element is used to store the name of a point that is inside the desired fill region, and is also optional. Listing 40 Input syntax for the fill-type components.# 1<component name="cmp_fill_1" type="fill"> 2 <location> point_fill </location> 3 <material> material1 </material> 4</component> 5 6 7<component name="cmp_fill_2" type="fill"> 8 <baseline> bsl </baseline> 9 <location> point_fill </location> 10 <material> material1 </material> 11 <mesh_size at="p1,p2"> 0.1 </mesh_size> 12</component> 13 14 15<component name="cmp_fill_3" type="fill"> 16 <baseline fillside="right"> bsl_3 </baseline> 17 <material> material1 </material> 18</component> ### Local mesh size# Besides the mesh size set globally in the main input file, filling type components can be assigned local mesh sizes. This is usually for the purpose of reducing the total number of elements and computational cost. This feature is based on the embedded objects (points and lines) provided by Gmsh. Hence, one or more points need to be specified as the ‘seed’ of local mesh sizes. <mesh_size at="p1,p2">0.1</mesh_size> The mesh size will vary gradually from the local to the global setting. Hence, if only one point is used, then the local mesh size will only affect a circular region. If multiple points are used, then lines will be created by connecting points seqentially and assigned the local mesh size as well. An example is provided below (Fig 40, Fig 41, Listing 41). Listing 41 Example input.# 1<component name="filling 1" type="fill" depend="..."> 2 <location> A </location> 3 <material> material1 </material> 4 <mesh_size at="A"> 0.2 </mesh_size> 5</component> 6 7<component name="filling 2" type="fill" depend="..."> 8 <location> B </location> 9 <material> material2 </material> 10 <mesh_size at="B,C"> 0.2 </mesh_size> 11</component> Specification • <material> - Name of the material to be filled. Required. • <location> - Name of the point located in the fill region. Optional. • <baseline> - Name of the base line defining part or complete boundary. Optional. • fillside - Side of the fill with respect to the base line. Optional. • <theta1> - Rotating angle in degree about the $$x_1$$ axis. Optional. Default is 0 degree. • <theta3> - Rotating angle in degree about the $$y_3$$ axis. Optional. Default is 0 degree. • <mesh_size> - Local mesh size. Optional. • at - A list of names of points where the local mesh size will be assgined. Required. ## Components dependency# If two components are connected in the ‘T’ type (Fig. 35), then the order of creation of the two components must be specified. This is done by specifying the dependency. For the case shown in the figure, the creation of the thin vertical component (purple) is dependent on the creation of the thick horizontal component (blue). Hence, in the definition of the purple component, a depend attribute should be specified as shown below. Listing 42 Dependency.# 1<component name="cmp_blue" type="laminate">...</component> 2<component name="cmp_purple" type="laminate" depend="cmp_blue">...</component> If a component is dependent on multiple components, their names shoule all be listed, delimited by comma.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3849959075450897, "perplexity": 4404.061411666365}, "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/1669446710808.72/warc/CC-MAIN-20221201085558-20221201115558-00390.warc.gz"}
https://codereview.stackexchange.com/questions/212337/leetcode-three-sum-in-python	# Leetcode Three Sum in Python Here's my solution for the Leet Code's Three Sum problem -- would love feedback on (1) code efficiency and (2) style/formatting. This is Python 3. Problem: Given an array nums of n integers, are there elements a, b, c in nums such that a + b + c = 0? Find all unique triplets in the array which gives the sum of zero. Note: The solution set must not contain duplicate triplets. Example: Given array nums = [-1, 0, 1, 2, -1, -4], A solution set is: [ [-1, 0, 1], [-1, -1, 2] ] My solution: def threeSum(nums): """ :type nums: List[int] :rtype: List[List[int]] """ def _twoSum(nums_small, target): ''' INPUT: List of numeric values, int target OUTPUT: list: pair(s) from nums_small that add up to target and -1target ''' nums_dict = {} return_lst = [] for indx, val in enumerate(nums_small, target): if target - val in nums_dict: return_lst.append([val, target - val, -1target]) nums_dict[val] = indx return return_lst return_dict = {} for indx, val in enumerate(nums): for solution in _twoSum(nums[:indx] + nums[indx+1:], -1*val): if sorted(solution) not in return_dict.values(): return_dict[str(indx)+str(solution)] = sorted(solution) # ... hacky way of storing multiple solutions to one index ... return list(return_dict.values()) • What Python version did you write this for? – Mast Jan 28 '19 at 7:01 ## Code "When all you have is a hammer, everything looks like a nail." Your tool chest seems to have dict, which you are using to implement a container which doesn't contain any duplicates. There is a name for that tool. It is called as set. Study it; you will find it very useful. Consider: nums_dict = {} for indx, val in enumerate(nums_small, target): if target - val in nums_dict: pass nums_dict[val] = indx You are never using the value indx; you are just storing it and never retrieving it. You could just as easily write: nums_dict = {} for val in nums_small: if target - val in nums_dict: pass nums_dict[val] = True which avoids the enumerate(), with the odd initial value, and the indices which did nothing. But this is still storing the values True, and never retrieving them. You don't need to do this, if you use a set: nums_dict = set() for val in nums_small: if target - val in nums_dict: pass You again use a dict for return_dict to ensure you don't have any duplicate solutions. Actually, this code is a worse that the former, because you are using in return_dict.values(), so instead of an $$\O(1)\$$ key lookup, you're doing an $$\O(n)\$$ search through the entire list of values! If that wasn't bad enough, you are sorting solutions twice: once to see if they already exist in return_dict, and a second time to add it into the dictionary if it wasn't found. Saving the sorted solution would avoid the second sort. You could replace return_dict with a set as well, but there is a catch, you can't add a list into a set, because the items in a set must be hashable to a constant value but lists are unhashable (because they can be mutated). You can get around this by using tuples, which can't be mutated, so they can be hashed. return_dict = set() for index, val in ...: for solution in ...: sorted_solution = tuple(sorted(solution)) return list(return_dict) The above returns a list of tuples. If you need a list of lists, that can be easily obtained as well, using list comprehension ... return [ list(solution) for solution in return_dict ] ... which is another useful tool for your tool chest. ## Algorithm Using _twoSum() to solve the threeSum() problem is good reuse of code. Kudos. However, looping over all indices and for each index, constructing a new list omitting just that one value from the list nums[:indx] + nums[indx+1:] is extremely inefficient. Since "Leetcode" is a puzzle solving site, I won't give you the better algorithm, but I will say that splitting nums into len(nums) different lists is not the way to go. I would split it into 2 lists, or maybe 3 lists tops. Good luck! • It took me a while to realize that you are talking about a dictionary (dict) when saying map and not the built-in function map... Jan 28 '19 at 11:54 • @Graipher The dangers of too many languages, and RUI. Fixed. Jan 28 '19 at 14:23 • Ah, thanks for reminders on tuples @AJNeufeld Jan 31 '19 at 16:21	{"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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2726876437664032, "perplexity": 2418.6836703738873}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585246.50/warc/CC-MAIN-20211019074128-20211019104128-00554.warc.gz"}
https://www.ias.ac.in/listing/bibliography/jess/A._P._Singh	• A P Singh Articles written in Journal of Earth System Science • Magmatic underplating beneath the Rajmahal Traps: Gravity signature and derived 3-D configuration The early Cretaceous thermal perturbation beneath the eastern continental margin of the Indian shield resulted in the eruption of the Rajmahal Traps. To understand the impact of the magmatic process that originated in the deep mantle on the lower crustal level of the eastern Indian shield and adjoining Bengal basin the conspicuous gravity anomalies observed over the region have been modelled integrating with available geophysical information. The 3-D gravity modelling has delineated 10–15 km thick high-density (ρ = 3.02 g/cm3) accreted igneous layer at the base of the crust beneath the Rajmahal Traps. Thickness of this layer varies from 16 km to the west of the Rajmahal towards north to about 12 km near Kharagpur towards south and about 18 km to the east of the Raniganj in the central part of the region. The greater thickness of the magmatic body beneath the central part of the region presents itself as the locus of the potential feeder channel for the Rajmahal Traps. It is suggested that the crustal accretion is the imprint of the mantle thermal perturbation, over which the eastern margin of the eastern Indian shield opened around 117 Ma ago. The nosing of the crustal accretion in the down south suggests the possible imprint of the subsequent magmatic intrusion along the plume path. • Spatial variation of the aftershock activity across the Kachchh Rift Basin and its seismotectonic implications We analyzed 3365 relocated aftershocks with magnitude of completeness (Mc) ≥ 1.7 that occurred in the Kachchh Rift Basin (KRB) between August 2006 and December 2010. The analysis of the new aftershock catalogue has led to improved understanding of the subsurface structure and of the aftershock behaviour. We characterized aftershock behaviour in terms of 𝑎-value, 𝑏-value, spatial fractal dimension ($D_s$), and slip ratio (ratio of the slip that occurred on the primary fault and that of the total slip). The estimated 𝑏-value is 1.05, which indicates that the earthquake occurred due to active tectonics in the region. The three dimensional 𝑏-value mapping shows that a high 𝑏-value region is sandwiched around the 2001 Bhuj mainshock hypocenter at depths of 20–25 km between two low 𝑏-value zones above and below this depth range. The $D_s$-value was estimated from the double-logarithmic plot of the correlation integral and distance between hypocenters, and is found to be 2.64 ± 0.01, which indicates random spatial distribution beneath the source zone in a two-dimensional plane associated with fluid-filled fractures. A slip ratio of about 0.23 reveals that more slip occurred on secondary fault systems in and around the 2001 Bhuj earhquake (Mw 7.6) source zone in KRB. • # Journal of Earth System Science Volume 129, 2020 All articles Continuous Article Publishing mode • # Editorial Note on Continuous Article Publication Posted on July 25, 2019	{"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": 0.7033306360244751, "perplexity": 4457.929256160003}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400249545.55/warc/CC-MAIN-20200926231818-20200927021818-00288.warc.gz"}
http://math.stackexchange.com/questions/512906/why-does-there-exist-a-unique-quotient-topology-that-makes-a-given-surjective-ma	# Why does there exist a unique quotient topology that makes a given surjective map a quotient map? I'm reading Munkres on the quotient topology, pg. 138 It says that: If X is a topological space and A is a set and if $p: X \to A$ is a surjective map , then there exists exactly one topology on A relative to which p is a quotient map. It then goes on to say that this topology is defined by letting the open sets be those subsets U of A such that $p^{-1}(U)$ is open in X. I understand why this is a topology, and I understand why this makes the p map a quotient map. I just don't understand why this is the only topology to make p a quotient map. I understand that it may be a minimal topology to make p a quotient map. It was my understanding that being a quotient map is a rather weak condition compared to others such as being an open map (when surjectivity is assumed). So couldn't we have defined a topology on A that makes p into an open map, whereby it would then be a quotient map? This topology would be finer than the one given above and could be defined by having the open sets U be those where there exists an open set V in X such that p(V) = U. Is there any substantive difference here or would they essentially be the same topologies? - Isn't quotient map defined as a map where $U$ is open iff $p^{-1}(U)$ is open? – Stefan Hamcke Oct 3 '13 at 2:26 @StefanH. yes, thank you. I just remembered that open map implies quotient map only when the map is both surjective and continuous. I had forgotten the continuous part, which explains my confusion. – user96787 Oct 3 '13 at 2:32 The final topology on $A$ with respect to $p$ is the finest topology for which $p$ is continuous. Every subset of $A$ that may be open (whose preimage under $p$ is open) is open in the final topology. Since a quotient map is among other things continuous, you cannot take a strictly finer topology on $A$ and have $p$ still be a quotient map.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9570218920707703, "perplexity": 146.16826458613096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701161718.0/warc/CC-MAIN-20160205193921-00265-ip-10-236-182-209.ec2.internal.warc.gz"}
https://sunlimingbit.wordpress.com/2017/04/03/newton-tensor/	## Newton tensor Suppose ${A:V\rightarrow V}$ is a symmetric endomorphism of vector space ${V}$, ${\sigma_k}$ is the ${k-}$th elementary symmetric function of the eigenvalue of ${A}$. Then $\displaystyle \det(A+tI)=\sum_{k=0}^n \sigma_k t^{n-k}$ One can define the ${k-}$th Newton transformation as the following $\displaystyle \det(A+tI)(A+tI)^{-1}=\sum_{k=0}^{n-1}T_k(A)t^{n-k-1}$ This means $\displaystyle \det(A+tI)=\sum_{k=0}^{n-1}T_k(A+tI)t^{n-k-1}$ $\displaystyle =T_0 t^n+\sum_{k=0}^{n-2}(A\cdot T_k(A)+T_{k+1}(A))t^{n-k-1}+T_{n-1}(A)$ By comparing coefficients of ${t}$, we get the relations of ${T_k}$ $\displaystyle T_0=1,\quad A\cdot T_k(A)+T_{k+1}(A)=\sigma_{k+1}I,\, 0\leq k\leq n-2\quad T_{n-1}(A)=\sigma_n$ Induction shows $\displaystyle T_{k}(A)=\sigma_kI-\sigma_{k-1}A+\cdots+(-1)^kA^k$ For example $\displaystyle T_1(A)=\sigma_1I-A$ $\displaystyle T_2(A)=\sigma_2-\sigma_1A+A^2$ One of the important property of Newton transformation is that: Suppose ${F(A)=\sigma_k(A)}$, then $\displaystyle F^{ij}=\frac{\partial F}{\partial A_{ij}}=T_{k-1}^{ij}(A)$ The is because $\displaystyle \frac{\partial }{\partial A_{ij}}\det(A+tI)=\det(A+tI)((A+tI)^{-1})_{ij}.$ If ${A\in \Gamma_k}$, then ${T_{k-1}(A)}$ is positive definite and therefore ${F}$ is elliptic. Remark: Hu, Z., Li, H. and Simon, U. . Schouten curvature functions on locally conformally flat Riemannian manifolds. Journal of Geometry, 88(1${-}$2), (2008), 75${-}$100.	{"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": 24, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9990612864494324, "perplexity": 951.8249061033496}, "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/1508187823478.54/warc/CC-MAIN-20171019212946-20171019232946-00362.warc.gz"}
http://lt-jds.jinr.ru/record/72008?ln=en	/ hep-ex CERN-PH-2017-162 Search for a scalar partner of the top quark in the jets plus missing transverse momentum final state at $\sqrt{s}$=13 TeV with the ATLAS detector Pages: 57 Abstract: A search for pair production of a scalar partner of the top quark in events with four or more jets plus missing transverse momentum is presented. An analysis of 36.1 fb$^{-1}$ of $\sqrt{s}$=13 TeV proton--proton collisions collected using the ATLAS detector at the LHC yields no significant excess over the expected Standard Model background. To interpret the results a simplified supersymmetric model is used where the top squark is assumed to decay via $\tilde{t}_1 \rightarrow t^{()} \tilde\chi^0_1$ and $\tilde{t}_1\rightarrow b\tilde\chi^\pm_1 \rightarrow b W^{()} \tilde\chi^0_1$, where $\tilde\chi^0_1$ ($\chi^\pm_1$) denotes the lightest neutralino (chargino). Exclusion limits are placed in terms of the top-squark and neutralino masses. Assuming a branching ratio of 100\% to $t \tilde\chi^0_1$, top-squark masses in the range 450--950 GeV are excluded for $\tilde\chi^0_1$ masses below 160 GeV. In the case where $m_{\tilde{t}_1}\sim m_t+m_{\tilde\chi^0_1}$, top-squark masses in the range 235--590 GeV are excluded. Note: Temporary entry; 57 pages in total, author list starts on page 40, 12 figures, 14 tables, submitted to JHEP, all figures including auxiliary figures are available at http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/SUSY-2016-15 Total numbers of views: 1335 Numbers of unique views: 918	{"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": 0.9920218586921692, "perplexity": 2390.5503931385333}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806569.66/warc/CC-MAIN-20171122103526-20171122123526-00302.warc.gz"}
https://jwst-pipeline.readthedocs.io/en/stable/jwst/pipeline/calwebb_spec3.html	# calwebb_spec3: Stage 3 Spectroscopic Processing¶ Config calwebb_spec3.cfg Class Spec3Pipeline Stage 3 processing for spectroscopic observations is intended for combining the calibrated data from multiple exposures (e.g. a dither/nod pattern) into a single combined 2D or 3D spectral product and a combined 1D spectrum. Before being combined, the exposures may receive additional corrections for the purpose of background matching and subtraction, as well as outlier rejection. The steps applied by the calwebb_spec3 pipeline are shown below. This pipeline is intended for non-TSO spectra only. TSO spectral data should be processed using the calwebb_tso3 pipeline. Instrument/Mode NIRSpec MIRI NIRISS NIRCam Step FS MOS IFU FS MRS SOSS WFSS WFSS master_background exp_to_source mrs_imatch outlier_detection resample_spec cube_build extract_1d combine_1d Notice that NIRCam and NIRISS WFSS, as well as NIRISS SOSS data, receive only minimal processing by calwebb_spec3. WFSS 2D input data are reorganized into source-based products by the exp_to_source step (see below), have 1D extracted spectra produced for each source, and then the 1D spectra for each source are combined into a final 1D spectrum. NIRISS SOSS inputs do not go through the exp_to_source step, because they contain data for a single source. Hence the only processing that they receive is to extract a 1D spectrum from each input and then combine those spectra into a final 1D spectrum. This type of processing is intended only for NIRISS SOSS exposures that are not obtained in TSO mode. TSO mode NIRISS SOSS exposures should be processed with the calwebb_tso3 pipeline. ## Arguments¶ The calwebb_spec3 pipeline does not have any optional arguments. ## Inputs¶ ### 2D calibrated data¶ Data model File suffix _cal The inputs to calwebb_spec3 should be in the form of an ASN file that lists the multiple exposures to be processed into combined output products. The individual exposures should be calibrated the (“_cal”) products from calwebb_spec2 processing. The member list for each product in the ASN file can also contain exposures of dedicated background targets, which are intended for use in the master_background step. These input exposures must be the “x1d” products (extracted 1-D spectra) of the background target(s) and are usually the “x1d” files produced by the calwebb_spec2 pipeline. They must be listed in the ASN file with “exptype” values of “background” in order to be correctly identified as background exposures. See the master_background for more details. ## Outputs¶ ### Source-based calibrated data¶ Data model MultiExposureModel File suffix _cal For NIRSpec fixed-slit, NIRSpec MOS, and NIRCam and NIRISS WFSS, which have a defined set of slits or sources, the data from the input calibrated exposures is reorganized by the exp_to_source step so that all of the instances of data for a particular source/slit are contained in a single product. These are referred to as “source-based” products, as opposed to the input exposure-based products. The source-based collections of data are saved in intermediate files, one per source/slit. The root names of the source-based files contain the source ID as an identifier and use the same “_cal” suffix as the input calibrated exposure files. An example source-based file name is “jw00042-o001_s0002_niriss_gr150r_f150w_cal.fits”, where “s0002” is the source id. The reorganized sets of data are sent to subsequent steps to process and combine all the data for one source at a time. ### CR-flagged exposures¶ Data model ImageModel File suffix _crf If the outlier_detection step is applied, a new version of each input calibrated exposure is created, in which the DQ array has been updated to flag pixels detected as outliers. These files use the “_crf” (CR-Flagged) product type suffix and also includes the association candidate ID as a new field in the original product root name, e.g. “jw96090001001_03101_00001_nrs2_o001_crf.fits.” ### 2D resampled and combined spectral data¶ Data model DrizProductModel File suffix _s2d When processing non-IFU modes, a resampled/rectified 2D product of type “_s2d” is created containing the rectified and combined data for a given slit/source, which is the output of the resample_spec step. ### 3D resampled and combined spectral data¶ Data model IFUCubeModel File suffix _s3d When processing IFU exposures, a resampled and combined 3D IFU cube product created by the cube_build step is saved as an “_s3d” file. ### 1D extracted spectral data¶ Data model MultiSpecModel File suffix _x1d All types of inputs result in a 1D extracted spectral data product, which is saved as a “_x1d” file, and is normally the result of performing the extract_1d step on the combined “_s2d” or “_s3d” product. For NIRCam and NIRISS WFSS, as well as NIRISS SOSS data, the extract_1d is performed on the individual unresampled 2D cutout images, resulting in multiple 1-D spectra per source in a “_x1d” product. Those spectra are combined using the subsequent combine_1d step (see below). ### 1D combined spectral data¶ Data model CombinedSpecModel File suffix _c1d For NIRCam and NIRISS WFSS, as well as NIRISS SOSS data, the combine_1d combines the multiple 1-D spectra for a given source into a final spectrum, which is saved as a “_c1d” product.	{"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": 0.1621048003435135, "perplexity": 6126.654570972796}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738351.71/warc/CC-MAIN-20200808194923-20200808224923-00410.warc.gz"}
http://math.stackexchange.com/users/23327/malice?tab=summary	# Malice less info reputation 2 bio website location age member for 2 years, 3 months seen Nov 5 '13 at 14:29 profile views 27 # 7 Questions 4 A question on modular arithmetic 1 Minimizing Sum of Product 0 Parametric Cubic Spline interpolation 0 Spline interpolation on $n$ dimension 0 Questions about the definition of group actions # 30 Reputation +20 A question on modular arithmetic +5 Minimizing Sum of Product This user has not answered any questions # 8 Tags 0 optimization × 3 0 matrices 0 linear-programming × 2 0 algorithms 0 interpolation × 2 0 group-theory 0 numerical-methods 0 modular-arithmetic # 5 Accounts Stack Overflow 101 rep 110 Mathematics 30 rep 2 Geographic Information Systems 30 rep 4 Cryptography 13 rep 3 Cross Validated 1 rep 1	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.617762565612793, "perplexity": 4629.585426622417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609535745.0/warc/CC-MAIN-20140416005215-00591-ip-10-147-4-33.ec2.internal.warc.gz"}
https://socratic.org/questions/how-do-you-solve-and-graph-abs-2c-1-7	Algebra Topics # How do you solve and graph abs(2c-1)<=7? Jan 11, 2018 $- 3 \setminus \le c \setminus \le 4$ #### Explanation: $\| 2 c - 1 \| \setminus \le 7$ Consider a general inequality with the absolute value: $\| f \left(x\right) \| \setminus \le a$ By definition this is equivalent to solving the following inequalities: $f \left(x\right) \setminus \le a$ and $f \left(x\right) \setminus \ge - a$ In our specific case this means: $2 c - 1 \setminus \le 7$ $2 c - 1 \setminus \ge - 7$ So let's solve the first one: $2 c - 1 \setminus \le 7$ $\setminus \textcolor{b l u e}{\left(1\right)} + 2 c - 1 \setminus \le 7 + \setminus \textcolor{b l u e}{\left(1\right)}$ $2 c \setminus \le 8$ $\setminus \frac{2}{2} c \setminus \le \setminus \frac{8}{2}$ $c \setminus \le 4$ Now the second: $2 c - 1 \setminus \ge - 7$ $2 c \setminus \ge - 6$ $c \setminus \ge - 3$ This is the graph of the first inequality: graph{x <= 4 [-6.244, 6.243, -3.12, 3.123]} And this of the second: graph{x => - 3 [-4.163, 1.997, -1.455, 1.624]} So the solution and the final graph are the common part of the two inequalities: $- 3 \setminus \le c \setminus \le 4$ ##### Impact of this question 257 views around the world You can reuse this answer	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 16, "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": 0.9563008546829224, "perplexity": 3326.6814951999213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370494349.3/warc/CC-MAIN-20200329140021-20200329170021-00221.warc.gz"}
http://tex.stackexchange.com/tags/color/new	# Tag Info 2 I redefine the shaded environment from framed to have a frame, then define a note environment, based on ntheorem with option framed, as a shaded theorem: \documentclass[english]{article} \usepackage[utf8]{inputenc} \usepackage{babel} \usepackage{geometry}% \usepackage[x11names]{xcolor} \usepackage{fourier, erewhon} \usepackage{lipsum} \usepackage{... 3 Very close to your code fragment, just replaces \fbox by \fcolorbox from the xcolor package: \documentclass{article} \usepackage{framed} \usepackage{xcolor} \usepackage{graphicx} \definecolor{note}{rgb}{1 1 0.5} \makeatletter \renewenvironment{framed}{% \def\FrameCommand##1{\hskip\@totalleftmargin \fboxsep=\FrameSep\fcolorbox{black}{note}{##... 5 Here's a tcolorbox version with sidebyside for the logo. Please note that such a box is not breakable! \documentclass{article} \usepackage[svgnames,dvipsnames,x11names]{xcolor} \usepackage[most]{tcolorbox} \usepackage[tikz]{bclogo} \newtcolorbox{framed}[1][]{ colframe=lightgray, colback=yellow!40!white, enhanced jigsaw, sharp corners, lower ... 1 Adding \usepackage[hidelinks]{hyperref} before \usepackage[numbered]{bookmark} Gets rid of the color boxes. However, my Figure numbers and captions are vertically offset from each other by about half a line... 0 I'm trying to use \newcommand{\myref}[1]{{\color{red}\ref{#1}}} in a very similar case, but it works only compiling once. After that, calling the bibliography and then compiling other two times the color effect introduced by \newcommand disappears! Any suggestions for this behavior? 4 For colours you have to use \arrayrulecolor{…} at the beginning of a row. For the thickness of \midrules, you can change the value of\lightrulewidth or change the thickness of a specific rule with the optional argument. Demo: \documentclass{article} \usepackage{booktabs} \usepackage[x11names, table]{xcolor} \setlength\lightrulewidth{0.3pt} \begin{document}... 4 You can use \arrayrulecolor{<colour>} from colortbl to change the colour, and \specialrule{<width>}{<above>}{<below>} to set a rule of specific dimensions: \documentclass{article} \usepackage[table]{xcolor} \usepackage{booktabs} \begin{document} \begin{tabular}{ c c c } \toprule A & B \\ \midrule Q & M ... 0 This is not the full solution yet -- for some reason the ToC color is wrong if the next \section does not have a \subsection etc. The solution uses xcntperchap to track the number of images per section, subsection and subsubsection. If this number is zero, the ToC color is changed. \documentclass[a4paper, hidelinks]{article} \usepackage{xcolor} \... -1 You can get this effect by alternating row and column color like following: \newcolumntype{g}{>{\columncolor{Gray}}c} \begin{table}[ht] \centering \begin{tabular}{c\|g\|g\|g\|g\|g\|g\|g} \hline \rowcolor{white} &col1 &col2 &col3 &col4 & col5 &col6 &col7\\ \hline \rowcolor{white} row1& \ra & \ra & \ra & \ra & \ra &... 7 The warning "pop empty color stack" is usually caused by color stack commands out of order. A typical scenario is the use of low level box set command \setbox instead of LaTeX box set macros (\sbox, environment lrbox, ...). When LaTeX sets a color, then the color reset command is issued after the current group. Therefore LaTeX box macros add a group layer ... 10 Well what it means is that's you've corrupted the colour stack. Unlike font changes colour is not an intrinsic attribute of a tex box, so colour changes are managed by nodes marking the change of state whatsit, Originally it was very hard to maintain colour correctly over page breaks, as the macro package needed to normalise colour in the page headings and ... 1 You're using wrongly \flushleft \newcommand{\insertImg}[3]{% \begin{figure}[h] \centering \includegraphics[height=50mm]{#1} \caption{#2}\label{#1} \raggedright Źródło: Na podstawie \cite{#3} \end{figure} } \newcommand{\insertImgSetSize}[4]{% \begin{figure}[h] \centering \includegraphics[height=#2mm]{#1} \caption{#3}\label{#1} ... 2 The bug is fixed in KOMA-Script Version 3.21 which is the current version on CTAN, in TeX Live 2016 and in MiKTeX. \documentclass{scrartcl}[2016/06/14] \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{xcolor} \usepackage{scrlayer-scrpage} \addtokomafont{section}{\color{red}} \addtokomafont{pageheadfoot}{\color{blue}} \begin{document} ... \\... 0 Here is my approach, not even as elegant as other answers, but using TiKZ Matrix. \documentclass{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{gillius2} % Stolen from Bernard answer \renewcommand\familydefault{\sfdefault} \usepackage[dvipsnames]{xcolor} %always declare the xcolor package before tikz \usepackage{tikz} \... 3 Here is a solution, playing with the parameters of booktabs: \documentclass{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{gillius2} \usepackage{geometry} \usepackage[table, x11names]{xcolor} \usepackage{booktabs, hhline} \usepackage{etoolbox} \newcommand\thickvrule[1][DarkOrange2]{\ifblank{#1}{}{\color{#1}}\vrule width 2pt} \... 2 You can create something like the following using the following code. The code is annotated to explain it a bit, but you'll need to look at the manuals for tikzmark and/or TikZ for further details. The latter is very long, but you obviously don't need most of it and the bits used here are simple. \documentclass{article} \usepackage[table]{xcolor} \... 1 use \leavevmode\color so the paragraph starts before the colour, or perhaps better if you do not want line breaking in that first column use c not C for the first column. 1 \documentclass[10pt,a4paper]{article} \usepackage[margin=3cm]{geometry} \usepackage{float} \frenchspacing \usepackage{hyperref} \usepackage{graphicx} \usepackage{paralist} \usepackage{color} \hbadness=10000 %.........................................................% \begin{document} \begin{flushleft} \textcolor[rgb]{0.00,0.00,1.00}{\underline{Anum Naeem ... 2 Let's assume the theorem in question has been defined with a package such as amsthm or ntheorem and that it's been given the "label" thm:abc. Two suggestions: Since you're loading the hyperref package, you could use the command \autoref{thm:abc} to cross-reference the theorem. In addition to loading the hyperref package, you could load the cleveref ... 9 You could shade it if you make the paths closed. This is based on Mark Wibrow's code, as quoted in the question. % answer to http://tex.stackexchange.com/questions/314918/generate-coloured-fancy-delaunay-patterns-in-tikz, modifying code by Mark Wibrow at http://tex.stackexchange.com/a/260652/ \documentclass[tikz,border=10pt,multi,rgb]{standalone} % xcolor ... 1 4 The \renewcommand{\ch@p} does not survive the typesetting phase of the chapter title in the ToC and as such, section ToC entries use the wrong color setting. \renewcommand is not global (unless being specified directly in the preamble or in the document body outside of nested groups. If the colour settings should be persistent, use \gdef\ch@p instead. Now ... 3 You should use \leavevmode before the color command. This will avoid that a page break can occur between the color and the title of the section. \documentclass{scrartcl} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{xcolor} \usepackage{scrlayer-scrpage} \addtokomafont{section}{\leavevmode\color{red}} \addtokomafont{pageheadfoot}{\color{... 1 This problem has something to do with different colour models. There was a question somewhere on this side which explains it in detail, but unfortunately I can not find it right now. In short: a workaround is to use a colour defined in RGB for the mixing. \documentclass[xcolor=dvipsnames]{beamer} \usetheme{Frankfurt} \beamertemplatenavigationsymbolsempty ... 1 the problem caused in \foreach, also look at \foreach not behaving in axis environment to solve the problem I use (during using colormaps) \edef\temp{\noexpand\addplot[color of colormap={\l}, solid, mark=none,domain=-1:1,samples=100]{\kx};} \temp instead of \addplot[color of colormap={\l}, solid, mark=none,domain=-1:1,samples=100]{\kx}; ... 9 Use two columns instead of one: \documentclass[xcolor=table]{beamer} \usepackage{siunitx,booktabs,multirow} \begin{document} \begin{frame} \centering \begin{tabular}{@{}lrccc@{}} \toprule \multicolumn{2}{@{}c}{\multirow{2}{}{\huge Parameters}} & \multicolumn{3}{c}{Levels}\\ \cmidrule{3-5} & & 1 & 2 &... 2 To get a clear explanation of why it does not work you probably need to dig deep into beamer. My guess is that \item is broken from the text after it, which should be displayed on the same slide. One way to solve the issue is to keep it together in a command. Below is a version where \citem{color}{text} print a bullet in color followed by text. The starred ... 0 Thanks that works perfectly! I put it in a language define: \lstdefinelanguage{BL233} { basicstyle = \ttfamily, morestring=[b][\color{stringcol}]", showstringspaces=false, literate = *{0}{{\textcolor{numbercol}{\ttfamily 0}}}1 % numbers } Also lets me define a second language expansion that explains the single letter macro commands: eg ... 1 You could use \alert<>{} to specify the slides on which the text should be shown 'alerted'. By default, this happens to be red. For example (and simplifying the code and MWE a bit): \documentclass{beamer} \mode<presentation> { \usetheme{Madrid} \usecolortheme[RGB={0,0,0}]{structure} } \usepackage{amsmath,amsfonts} \usepackage{tikz} \begin{... 2 A basic installation of MiKTeX means that only a part of the complete MiKTeX Distribution is installed. If you need more there is an automatic way, that MiKTeX installs needed missing packages by its own (You have to activate this while installation). Of course you need then a running internet connection to get the missing packages loaded. That is the ... 0 We can only guess what have you got on your computer. There are at least two methods that should help. Find All programs -> MikTeX 2.9 -> Maintenance (Admin) -> Settings -> General -> Reresh FNDB. Copy xcolor.sty to the working path. The first one is more general, but if you quickly need only a few packages to use, the second method may be a temporary ... 1 A preliminary solution, working for the transparency so far only. \documentclass[a4paper,landscape]{article} \usepackage[outline]{contour} \usepackage{xcolor} \usepackage{graphicx} %\usepackage{transparent} \usepackage{background} \usepackage{eso-pic} \newcommand\BackgroundPic{% \includegraphics[% width=\paperwidth, keepaspectratio%, ]{london3.png}... 4 This can be achieved with tcolorbox, one of my favourite packages (;-)) \documentclass{article} \usepackage[most]{tcolorbox} % Define a new box for this \newtcolorbox[auto counter]{MyBox}[2][]{% title={MyBox \thetcbcounter\ #2}, boxsep=1mm, colback=white!40!yellow, breakable, #1 } \usepackage{blindtext} \begin{document} \begin{MyBox}{My nice ... Top 50 recent answers are included	{"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": 0.9592618346214294, "perplexity": 6782.698938105312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783408840.13/warc/CC-MAIN-20160624155008-00143-ip-10-164-35-72.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/243350/subspaces-of-l-p2p-infty/243356	# Subspaces of $L_{p}(2<p<\infty)$ Let $p>2$ and $X$ a subspace of $L_{p}$. Then Kadec and Pelczynski proved that either $X$ is isomorphic to $l_{2}$ or $X$ contains a subspace isomorphic to $l_{p}$. Question: if $X$ is isomorphic to $l_{2}$, does $X$ contain a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$? Yes. That follows, e.g., from the Krivine-Maurey theory of stable spaces even if it was known before their work. For $p<2$ you get from their theory, and more or less classical considerations, Aldous' theorem that every subspace of $L_p$ contains for every $\epsilon > 0$ a subspace that is $1+\epsilon$-isomorphic to $\ell_r$ for some $p\le r \le 2$. • Let $1<p<2$. Let $X$ be a subspace of $L_{p}$ that contains no isomorphic copy of $l_{p}$. Does $X$ contain for every $\epsilon>0$ a subspace that is $(1+\epsilon)$-isomorphic to $l_{2}$? – Dongyang Chen Jun 30 '16 at 0:26 • No. $L_p$ contains a subspace isometric to $\ell_r$, for every $p<r<2$. This was first observed by M. Kadec and follows easily from the existence and properties of $r$ stable random variables, due to P. Levy. – Gideon Schechtman Jun 30 '16 at 8:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9864632487297058, "perplexity": 96.79613512274078}, "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/1620243989812.47/warc/CC-MAIN-20210515035645-20210515065645-00344.warc.gz"}
http://learn.district196.org/course/view.php?id=2508	## Topic outline • ### General What is American Studies? The U.S. History curriculum in this school district is divided into two parts. Part 1 is taught in the earlier grades, and part 2 is taught in the 7th grade. The 7th grade curriculum focuses on the 20th Century America. Our course is taught sequentially. We begin with Reconstruction after the Civil War and conclude with America in a Changing World. I look forward to working with you and your son or daughter in making this year a very positive and rewarding learning experience. If you ever have any questions or concerns, please contact me. Mr. Slattery Room 217 E-mail: jere.slattery@district196.org Online Textbook: https://www.pearsonsuccessnet.com User Name: Am Studies Student (Case Sensitive)	{"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": 0.16937240958213806, "perplexity": 3740.321239549827}, "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/1505818687606.9/warc/CC-MAIN-20170921025857-20170921045857-00717.warc.gz"}
https://www.physicsforums.com/threads/zero-divided-by-zero.29213/	# Zero divided by Zero ? 1. Jun 4, 2004 ### Physics is Phun Zero divided by Zero ??? I am wondering why 0/0 = indeterminate my reasoning is that anything divided by itself is 1 right? 2/2=1 1/1=1 so wouln't lim x->0 x/x = 1 if so the why wouldn't 0/0=1 I am not trying to argue that 0/0 is not indeterminate. I accept that it is, but I am still have to wonder, why? 2. Jun 4, 2004 ### mathman Try x2/x or x/x2, as x goes to 0. In both cases, you have 0/0, but the first limit is 0 and the second is infinite. That's why 0/0 is indeterminate. 3. Jun 4, 2004 ### Icarus More generally, a = b/c is defined to mean that b = ac. Given b and c with c not zero, there is a unique a such that b = ac. For any b other than zero, there is no a for which b = a0. So b/0 does not exist. Finally that leaves b = c = 0: But 0 = a0 for every a. This is why 0/0 is indeterminant: you can get anything at all from a 0/0 situation. In addition to the two situations "Mathman" has supplied, limx->0 ax/x = a for any number a. Indeed, if 0/0 = 1, then all derivatives would be constant, and all functions linear! Last edited: Jun 7, 2004 4. Jun 7, 2004 ### HallsofIvy Staff Emeritus The difficulty with you logic is that you can't use "anything divided by itself is 1" to PROVE that something (in particular 0) divided by itself is 1! You COULD use "any non-zero number divided by itself is 1" to conclude that the limit of x/x as x-> 0 is 1 but unfortunately, as mathman pointed out, other sequences that wind up as "0/0" give other limits. (Actually, it is not "unfortunate". If 0/0 were equal to 1 calculus would be trivial and useless! 5. Jun 7, 2004 ### phoenixthoth here's how i explain it sometimes. consider balls in boxes. 8/4=2 because it would take 2 boxes, 4 balls each, to total 8 balls. in general a/b=c if it takes c boxes, b balls each, to total a balls. now consider 0/0. the question is this: how many boxes, 0 balls each, would it take to total 0 balls? well, it could be 1, yes, but it might as well be a million: a million boxes, 0 balls each, would total 0 balls. incidentally, consider 1/0. how many boxes, 0 balls each, would it take to total 1 ball? 6. Jun 18, 2004 Let's say that 0/0=X. Then, 0=0X. Well, there's more than one solution to that equation. In fact, there's infinitely many, and thus, indeterminant. Same thing with Infinity/Infinity; Infinity=InfinityX has infinitely many solutions, thus indeterminant. 7. Jun 18, 2004 ### Zurtex Sorry but your proof is flawed, you have assumed that 0/0 = 1 to show that 0/0 = any number. This is would suggest all numbers = 1. 8. Jun 18, 2004 ### arildno Now, I would like to comment your reasoning here, that any number divided by itself is 1. (It is a very common reasoning, but flawed nonetheless) Let's review a few axioms concerning the real numbers: 1. There is a real number 0, so that for any real number a, we have a+0=a (This can be regarded as the definition of the real number commonly known as 0, we may prove, with a few other axioms, that only a single 0 exists.) 2. Given any real numbers a,b,c we have a distributive law connecting addition and multiplication: a(b+c)=ab+ac 3. For any real number a, there exists a real number (-a), which has the property: a+(-a)=0 ((-a) is called the additive inverse of a; we may show that for any a, only one additive inverse exists; and we also have: (-(-a))=a, that is the additive inverse of the additive inverse of a is a itself) given real numbers a,b,c, we have: a+b+c=a+(b+c) That is, by axiom, the order in which we sum together numbers are irrelevant. (The notation a+b+c means: Add b to a, then add c to to the result, whereas a+(b+c) means add the sum of b and c to a) Now, 1,2,3,4 , along with "trivial" axioms about how equalities can be manipulated, are enough to prove that for any a, we have a0=0. Let z=a0 Since, 0=0+0, we may manipulate our equation into: z=a0=a(0+0)=a0+a0=z+z Or we have: z=z+z We then have: z+(-z)=z+z+(-z) Or: 0=z+0, which means a0=0 for any choice of real number a! But that means, if you want to define a real number 1/a such that a(1/a) equals 1, you cannot have a real number 1/0 (so that 0(1/0)=1), because we have proven that no such real number can exist! Hence, the 0/0 (=0*(1/0)) expression should be seen a danger signal which tells you to proceed with extreme care in order to avoid meaningless results..	{"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": 0.9373030066490173, "perplexity": 1185.322666625131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049278091.17/warc/CC-MAIN-20160524002118-00014-ip-10-185-217-139.ec2.internal.warc.gz"}
https://link.springer.com/chapter/10.1007%2F978-3-319-16133-4_13	# Effects of Water Flow on Energy Consumption and Travel Times of Micro-Ferries for Energy-Efficient Transport over Water • M. Burger • B. De Schutter Chapter Part of the Operations Research/Computer Science Interfaces Series book series (ORCS, volume 58) ## Abstract Controlling the transport of water by adjusting water flows in rivers and canals, inevitably will have an effect on the transport over water by vessels as well. We will discuss the effect of flowing water on scheduling micro-ferries (small autonomous water-taxis) using the least amount of energy, while aiming at satisfying customer demands with respect to pick-up times. This trade-off will be made by optimizing the assignment of micro-ferries to customers in a specific order, and by searching for the best travel speeds. The interplay between controlling transport of water and scheduling transport over water will become clear by the explicit relation between the speed of the water (influenced by water management) on travel times and energy consumption, derived in this chapter. It is shown that on average the travel times (and thereby the energy consumption) will increase with increasing magnitudes of the current. Hence, decisions made on water management have a direct effect on the performance of the transport system, and the interests of both parties should be taken into account to obtain a well-functioning water transport system. ## Keywords Energy Consumption Travel Time Total Energy Consumption Travel Speed Inertial Reference Frame These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. ## Notes ### Acknowledgements This work is supported by the European Union 7th Framework Program [FP7/2007-2013] under grant agreement no. 257462 HYCON2 Network of Excellence, and TUD COST Action TU1102 Towards Autonomic Road Transport Support Systems (ARTS). ## References 1. 1. Bektaş T, Laporte G. The pollution-routing problem. Transp Res B Methodol. 2011;45(8):1232–50. 2. 2. Bektas T. The multiple traveling salesman problem: an overview of formulations and solution procedures. Omega 2006;34(3):209–19. 3. 3. Boyd SP, Vandenberghe L. Convex optimization. 8th ed. United Kingdom: Cambridge University Press; 2010.Google Scholar 4. 4. Burger M. Exact and compact formulation of the fixed-destination travelling salesman problem by cycle imposement through node currents. In: Proceedings of the 2013 international conference on operations research, Rotterdam, the Netherlands, Sept 2013. New York: Springer.Google Scholar 5. 5. Burger M, De Schutter B. Energy-efficient transportation over flowing water. In: Proceedings of the 10th IEEE international conference on networking, sensing and control, Evry, France, Apr 2013. IEEE. pp. 226–31.Google Scholar 6. 6. Burger M, De Schutter B, Hellendoorn H. An improved method for solving micro-ferry scheduling problems. In: Proceedings of the 1st European symposium on quantitative methods in transportation systems, Lausanne, Switzerland, Sept 2012. pp. 164–71.Google Scholar 7. 7. Burger M, De Schutter B, Hellendoorn H. Micro-ferry scheduling problem with charging and embarking times. In:Proceedings of the 13th IFAC symposium on control in transportation systems, Sofia, Bulgaria, Sept 2012.Google Scholar 8. 8. Burger M, De Schutter B, Hellendoorn H. Micro-ferry scheduling problem with time windows. In: Proceedings of the 2012 American control conference, Montréal, Canada, June 2012. IEEE. pp. 3998–4003.Google Scholar 9. 9. Conejo AJ, Castillo E, Mínguez R, García-Bertrand R. Decomposition techniques in mathematical programming. 1st ed. Berlin, Germany: Springer; 2006.Google Scholar 10. 10. Daganzo CF. Fundamentals of transportation and traffic operations. 3rd ed. United Kingdom: Pergamon Press; 1997.Google Scholar 11. 11. Fossen TI. Marine control systems - guidance, navigation and control of ships, rigs and underwater vehicles. 1st ed. Trondheim, Norway: Marine Cybernetics; 2002.Google Scholar 12. 12. Kara I, Kara BY, Yetis MK. Energy minimizing vehicle routing problem. In: Dress A, Xu Y, Zhu B, editors. Combinatorial optimization and applications. Lecture notes in computer science, vol. 4616. Berlin/Heidelberg: Springer; 2007. pp. 62–71. 13. 13. Kulkarni RV, Bhave PR. Integer programming formulations of vehicle routing problems. Eur J Oper Res. 1985; 20(1):58–67. 14. 14. Laporte G. The traveling salesman problem: an overview of exact and approximate algorithms. Eur J Oper Res. 1992;59(2):231–47. 15. 15. Miller CE, Tucker AW, Zemlin RA. Integer programming formulation of traveling salesman problems. J ACM. 1960;7(4):326–29. 16. 16. Savelsbergh MWP, Sol M. The general pickup and delivery problem. Transp Sci. 1995;29(1):17–29. 17. 17. Taha HA. Operations research: an introduction. 4th ed. New York: Macmillan Publishing Company; 1987.Google Scholar 18. 18. Toth P, Vigo D, editors. The vehicle routing problem. Monographs on discrete mathematics and applications. Philadelphia: SIAM; 2002.Google Scholar 19. 19. Xiao Y, Zhao Q, Kaku I, Xu Y. Development of a fuel consumption optimization model for the capacitated vehicle routing problem. Comput Oper Res. 2012;39(7):1419–31.	{"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": 0.8266109824180603, "perplexity": 7546.024546368558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738858.45/warc/CC-MAIN-20200811235207-20200812025207-00465.warc.gz"}
https://www.assignmentexpert.com/homework-answers/economics/macroeconomics/question-46628	64 573 Assignments Done 99,2% Successfully Done In September 2018 # Answer to Question #46628 in Macroeconomics for Sem Question #46628 questions: 1. what is the term recession refers to? 2. what is the most volatile component of aggregate demand? 3. Assume that the Marginal propensity to consume(MPC) is 0.75, and the investment spending rises by $25 billion. By how much will the real GDP change? 4. What is the equation for the velocity f money? 5. if Namibians suddenly decide to hold more cash for carrying on transactions and for precautionary reasons, what is more likely to happen? 6. Supply-side fiscal policy is generally enacted through what? 7. The Phillips curve describes the relationship between what and what? 8. Phillips curve is an illustration of what? 9. In the crude equation of exchange where MV=PY, what is likely to happen? 10. A bill of exchange is a money market financial security that is traded where? 11. What is the most important conclusions from the keynesian liquidity preference theory? 12. Where changes in the growth of money supply in an economy can originate from? Expert's answer 1. what is the term recession refers to? In economics, a recession is a business cycle contraction. It is a general slowdown in economic activity. Macroeconomic indicators such as GDP (gross domestic product), investment spending, capacity utilization, household income, business profits, and inflation fall, while bankruptcies and the unemployment rate rise. 2. what is the most volatile component of aggregate demand? The most volatile component of aggregate demand is investment. 3. Assume that the Marginal propensity to consume(MPC) is 0.75, and the investment spending rises by$25 billion. By how much will the real GDP change? S = s*Y, s = 1 - c, so change in Y = 25/(1 - 0.75) = \$100 billion. 4. What is the equation for the velocity f money? MV = PQ, so velocity V = PQ/M, where P - price, Q - quantity, M - money base. 5. if Namibians suddenly decide to hold more cash for carrying on transactions and for precautionary reasons, increase in the inflation rate is more likely to happen. 6. Supply-side fiscal policy is generally enacted through an increase in government spending. 7. The Phillips curve describes the relationship between inflation and unemployment. 8. Phillips curve is an illustration of an inverse correlation between an economy's unemployment rate and inflation rate 9. In the crude equation of exchange where MV=PY, mistakes in real money base calculation are likely to happen. 10. A bill of exchange is a money market financial security that is usually traded internationally. 11. What is the most important conclusions from the keynesian liquidity preference theory? The LIQUIDITY PREFERENCE THEORY is the idea that investors demand a premium for securities with longer maturities, which entail greater risk, because they would prefer to hold cash, which entails less risk. The more liquid an investment, the easier it is to sell quickly for its full value. Because interest rates are more volatile in the short term, the premium on short- versus medium-term securities will be greater than the premium on medium- versus long-term securities. 12. Changes in the growth of money supply in an economy can originate from change in the reserve requirement rate. Need a fast expert's response? Submit order and get a quick answer at the best price for any assignment or question with DETAILED EXPLANATIONS!	{"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": 0.35945242643356323, "perplexity": 4131.992077662118}, "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/1537267156460.64/warc/CC-MAIN-20180920101233-20180920121633-00470.warc.gz"}
https://meangreenmath.com/2017/02/21/my-favorite-one-liners-part-21/	# My Favorite One-Liners: Part 21 In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Sometimes, just every once in a blue moon, something in mathematics doesn’t appear right to students at first glance. For example, take the common notation $(a,b)$ What does this symbol mean? Sadly, it depends on the context. Sometimes, it means a point in the Cartesian plane whose first coordinate is $a$ and whose second coordinate is $b$. Other times, it could mean the set $\{x : a < x < b\}$, or the interval between $a$ and $b$ that does not contain the endpoints. You’d think that, by now, mathematicians would’ve figure out a way to not denote these two completely different things with the same symbol. Indeed, I’ve seen textbooks that use $]a,b[$ to denote the open interval between $a$ and $b$ to avoid this duplication; however, this notation hasn’t been widely adopted by the mathematical community. So here’s my quip when something like this comes up. Sometimes, a young child will come crying to her parents to complain about the injustices in the world, and the child may be right. But all the parent can say is, “Sorry, sweetheart, but sometimes life isn’t fair.” And I’ll act this out, talking to an imaginary child as I look down to the floor. To complete the quip, I’ll then turn to my class and conclude, “Sorry, sometimes life isn’t fair.” It doesn’t make much sense, but we’re stuck with it.	{"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": 9, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8711459040641785, "perplexity": 693.3886533056695}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947033.92/warc/CC-MAIN-20180424174351-20180424194351-00527.warc.gz"}
https://www.physicsforums.com/threads/a-second-order-nonlinear-ordinary-differential-equation.387607/	# A second order nonlinear ordinary differential equation • Start date • #1 Lightfuzz 15 0 How, if possible, could I solve the equation: x''x=((x')^2)/2? Thanks. • #2 Gold Member 1,609 531 If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant. • #3 hamster143 908 2 Try to find such a pair of real numbers a & b that d/dt (x^a x'^b) is proportional to x"x-((x')^2)/2. • #4 Lightfuzz 15 0 If you are allowed to guess, then try guess a very often used function that when differentiated gives a scaled version of itself, i.e. a function that satisfy x' = a x, with a being a constant. I've tried that but is there a way to actually solve it without guessing? • #5 5,054 2,054 Equations like this, which do not contain the independent variable, are called autonomous systems. They are easily solved with the substitution x'' = x' dx'/dx . Then collect the terms in x' and x and integrate them both. This will give you x' in terms of x, which you can then integrate a second time. • #6 Phrak 4,265 2 Do you only require a numerical solution given some particular boundry conditions? • #7 gato_ 116 0 rearrange: $$\frac{1}{2x}=\frac{\ddot{x}}{x}=-\frac{d}{dx}(\frac{1}{\dot{x}})$$ $$dt=(C-ln(\sqrt{x}))dx =d[x(C'-ln(\sqrt{x}))]$$ • Last Post Replies 2 Views 228 • Last Post Replies 4 Views 1K • Last Post Replies 3 Views 1K • Last Post Replies 7 Views 802 • Last Post Replies 1 Views 406 • Last Post Replies 10 Views 1K • Last Post Replies 3 Views 774 • Last Post Replies 12 Views 1K • Last Post Replies 7 Views 1K • Last Post Replies 1 Views 712	{"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": 0.9164595603942871, "perplexity": 1483.6331593976722}, "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/1669446711013.11/warc/CC-MAIN-20221205064509-20221205094509-00784.warc.gz"}
https://www.semanticscholar.org/paper/Rational-vertex-operator-algebras-are-finitely-Dong-Zhang/06a11a694d7960c9923f2dd080526ce3f1b7fbc1	# Rational vertex operator algebras are finitely generated @article{Dong2008RationalVO, title={Rational vertex operator algebras are finitely generated}, author={Chongying Dong and Wei Zhang}, journal={arXiv: Quantum Algebra}, year={2008} } • Published 13 June 2008 • Mathematics • arXiv: Quantum Algebra 8 Citations ### SOME FINITE PROPERTIES FOR VERTEX OPERATOR SUPERALGEBRAS • Mathematics • 2011 Vertex operator superalgebras are studied and various results on rational Vertex operator superalgebras are obtained. In particular, the vertex operator super subalgebras generated by the weight 1/2 ### Lattice Subalgebras of Strongly Regular Vertex Operator Algebras We prove a sharpened version of a conjecture of Dong–Mason about lattice subalgebras of a strongly regular vertex operator algebra V, and give some applications. These include the existence of a ### Parafermion vertex operator algebras • Mathematics • 2011 This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie ### The varieties of Heisenberg vertex operator algebras • Mathematics • 2015 For a vertex operator algebra V with conformal vector ω, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and ### Vertex operator algebras and weak Jacobi forms • Mathematics • 2011 Let $V$ be a strongly regular vertex operator algebra. For a state $h \in V_1$ satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions ### Moduli spaces of conformal structures on Heisenberg vertex algebras • Mathematics • 2018 This paper is a continuation to understand Heisenberg vertex algebras in terms of moduli spaces of their conformal structures. We study the moduli space of the conformal structures on a Heisenberg ## References SHOWING 1-10 OF 19 REFERENCES ### Some finiteness properties of regular vertex operator algebras Abstract We give a natural extension of the notion of the contragredient module for a vertex operator algebra. By using this extension we prove that for regular vertex operator algebras, Zhu's C 2 ### Rational vertex operator algebras and the effective central charge • Mathematics • 2002 We establish that the Lie algebra of weight 1 states in a (strongly) rational vertex operator algebra is reductive, and that its Lie rank 1 is bounded above by the effective central charge c~. We ### Automorphism groups and derivation algebras of finitely generated vertex operator algebras • Mathematics • 2001 We investigate the general structure of the automorphism group and the Lie algebra of derivations of a finitely generated vertex operator algebra. The automorphism group is isomorphic to an algebraic ### Twisted representations of vertex operator algebras • Mathematics • 1998 This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. ### A Z_2-orbifold model of the symplectic fermionic vertex operator superalgebra We construct an irrational C_2-cofinite vertex operator algebra associatted to a finite dimensional vector space with a nondegenerate skew-symmetric bilinear form. We also classify its equivalence ### Certain Generating Subspaces for Vertex Operator Algebras • Mathematics • 1998 Abstract Minimal generating subspaces of “weak PBW type” for vertex operator algebras are studied and a procedure is developed for finding such subspaces. As applications, some results on generalized ### Holomorphic vertex operator algebras of small central charge • Mathematics • 2002 We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we ### Integrability of C_2-cofinite vertex operator algebras • Mathematics • 2006 The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie ### Integrability of C2-cofinite vertex operator algebras • Mathematics • 2006 The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions (C2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie ### Regularity of Rational Vertex Operator Algebras • Mathematics • 1995 Rational vertex operator algebras, which play a fundamental role in rational conformal field theory (see [BPZ] and [MS]), single out an important class of vertex operator algebras. Most vertex	{"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": 0.8457456231117249, "perplexity": 1205.1971889372328}, "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/1664030337322.29/warc/CC-MAIN-20221002115028-20221002145028-00521.warc.gz"}
https://share.cocalc.com/share/61e842a0d231fd22977e0bb9656da9bdb3b1200d/Crypytography/00_Basics.html?viewer=download	# Overview¶ The best way to learn is by doing. In cryptography, we need to deal with big number which may not have native language support. The earliest comprehensive library is http://read.pudn.com/downloads85/sourcecode/crypt/325570/examples/onb/DH.example/onb.c__.htm and http://www.amazon.com/Implementing-Elliptic-Cryptography-Michael-Rosing/dp/1884777694. There are plenty of c library, like Miracl, PBC, OpenssL, Nacl, libsodium, etc. On the other hand, the low level routine like gmp, ntl, flint, gnugp are also popular. We will however focus on full open source LibTom project http://www.libtom.net/ and ocassionally use Miracl or others. The decision of choosing LibTom is its code is written for novice or student and it has accompanied book http://shop.oreilly.com/product/9781597491129.do and https://goo.gl/p2QWJt. Of course there is open tex file for each subprojects and a complete book called Tommath.pdf in the archieve file. Though it has built-in support for long integer. The cryptography modules inside python are not as plentiful as other fields like machine learning or network secruity (see https://lwn.net/Articles/595790/). To the best of our knowledge, the sympy provides somewhat complete cryptography tools http://docs.sympy.org/dev/_modules/sympy/crypto/crypto.html. If you would like to get a flavor of crypto, sage may be a good choice, and there are relative plenty source for it http://www.sagemath.org/library-publications.html#books. However, in this tutorial I will use pycrtpo and cryptography instead since it is directly builds upon python, and also contains a collections of great toolkit. See https://www.dlitz.net/software/pycrypto/doc/ and https://cryptography.io/en/latest/ Other python biding like pynacl may also be helpful, keep track with it !! ## Pycrypto¶ How can we do encryption (only pairing x) In [10]: from Crypto.Cipher import AES plaintext = 'Martinet is god!' encobj = AES.new(key, AES.MODE_ECB) ciphertext = encobj.encrypt(plaintext) print ciphertext.encode('hex') # For read friendly 292138362368e0cdd76508781b3f3b82 You might have noticed that the length of hex-encoded ciphertext is 32 characters. By encoding it in hex, we've doubled the size because each binary character is represented by two hex characters, thus the actual ciphertext is only 16 characters The common encoding way in cryptography is base64 (which is the same as many other applications), see http://blog.rlr-uk.com/2011/03/base64-encoding-is-not-cryptography.html for the reason You may notice that we could not change the size of key and plaintext arbitrary in this example, we will relax this constraint by padding and utilizing so called KDF (key derivation function) in later sections In [11]: import binascii ciphertext = binascii.unhexlify('292138362368e0cdd76508781b3f3b82') decobj = AES.new(key, AES.MODE_ECB) plaintext = decobj.decrypt(ciphertext) # Resulting plaintext print plaintext Martinet is god! ## Cryptography¶ Let's see another well-documented toolbox cryptography (exactly the name!!) In [14]: from cryptography.hazmat.primitives.ciphers import ( Cipher, algorithms, modes ) from cryptography.hazmat.backends import default_backend plaintext = 'Martinet is god!' encryptor = Cipher(algorithms.AES(key),modes.ECB(),backend=default_backend()).encryptor() ciphertext = encryptor.update(plaintext) + encryptor.finalize() print ciphertext.encode('hex') 292138362368e0cdd76508781b3f3b82 In [17]: decryptor = Cipher(algorithms.AES(key), modes.ECB(), backend=default_backend()).decryptor() plaintext = decryptor.update(ciphertext) + decryptor.finalize() print plaintext Martinet is god! In [ ]:	{"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": 0.44383618235588074, "perplexity": 6962.466823692456}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347399830.24/warc/CC-MAIN-20200528170840-20200528200840-00119.warc.gz"}
https://quant.stackexchange.com/questions/15740/sharpe-ratio-for-loans	# Sharpe Ratio for loans I am trying to calculate the sharpe ratio for a set of loans. These loans have already matured and I know if they were good or not: grades are the different grades of the loans. interest rate is the floating point value: 12.12 13.11 7.90 8.90 15.31 14.33 11.14 16.29 17.27 10.99 this is the python code that tries to compute the sharpe ratio for each grade. for grade,name in enumerate(grades): print name, grade if pd.isnull(name): continue grade = float(grade) positive = (df_loans_completed.ix[s_grade_stats.groups[(grade, True)]]['interest_rate'] - R) negative = (df_loans_completed.ix[s_grade_stats.groups[(grade, False)]]['interest_rate'] - R) sigma = positive.append(negative).std() avg_return = positive.mean() print "sharpe:", (avg_return -R) / sigma • should I consider the fact that I know which loans returned an interest positive and which did not ?	{"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": 0.9398927092552185, "perplexity": 4249.520683416192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738816.7/warc/CC-MAIN-20200811150134-20200811180134-00273.warc.gz"}
https://www.quantumstudy.com/fx-tan2-x-where-denotes-the-greatest-integer-function-then/	# $f(x) = [tan^2 x]$ where [.] denotes the greatest integer function. Then Q: Let $f(x) = [tan^2 x]$ where [.] denotes the greatest integer function. Then (A) $\lim_{h \rightarrow 0} f(x)$ does not exist (B) f(x) is continuous at x = 0 (C) f(x) is not differentiable at x = 0 (D) f'(0) = 1 Sol: $\lim_{h \rightarrow 0} [tan^2 (0+h)] = \lim_{h \rightarrow 0} [tan^2 (0-h)]= [tan^2 (0)] = 0$ So, f(x) is continuous at x = 0. Since f(x) = 0 in the neighbourhood of 0, f'(0) = 0. Hence (B) is the correct 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": 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": 0.9991743564605713, "perplexity": 220.78844394585917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057598.98/warc/CC-MAIN-20210925052020-20210925082020-00180.warc.gz"}
http://mathhelpforum.com/calculus/130225-horizontal-asymptote-problem-print.html	# Horizontal Asymptote Problem • February 22nd 2010, 06:26 PM bobsanchez Horizontal Asymptote Problem I understand fairly well how to find horizontal asymptotes, but this function is really messing with me. I don't think it has any, but I want to be sure. If it does have some, I'm sure I'm missing some basic step. Anyways, here it is: f(x) = 3x^3 - x +1 / x+3 What would be the horizontal asymptote(s) if they even exist for this function? • February 22nd 2010, 06:39 PM drumist Is this the correct function? $f(x) = \frac{3x^3-x+1}{x+3}$ To determine the end-behavior (i.e., what happens at the extreme "left" and extreme "right" of the graph), you can take the largest power term from each the numerator and denominator, and calculate the limit like this: $\lim_{x\to+\infty} \frac{3x^3-x+1}{x+3} = \lim_{x\to+\infty} \frac{3x^3}{x} = \lim_{x\to+\infty} 3x^2 = +\infty$ You should also check the left side if it's not a simple rational function. In this case it's not necessary, but just to be thorough: $\lim_{x\to-\infty} \frac{3x^3-x+1}{x+3} = \lim_{x\to-\infty} \frac{3x^3}{x} = \lim_{x\to-\infty} 3x^2 = +\infty$ Since the end-behavior goes to infinity on both sides, there aren't any horizontal asymptotes. • February 22nd 2010, 06:42 PM bobsanchez Quote: Originally Posted by drumist Is this the correct function? $f(x) = \frac{3x^3-x+1}{x+3}$ To determine the end-behavior (i.e., what happens at the extreme "left" and extreme "right" of the graph), you can take the largest power term from each the numerator and denominator, and calculate the limit like this: $\lim_{x\to+\infty} \frac{3x^3-x+1}{x+3} = \lim_{x\to+\infty} \frac{3x^3}{x} = \lim_{x\to+\infty} 3x^2 = +\infty$ You should also check the left side if it's not a simple rational function. In this case it's not necessary, but just to be thorough: $\lim_{x\to-\infty} \frac{3x^3-x+1}{x+3} = \lim_{x\to-\infty} \frac{3x^3}{x} = \lim_{x\to-\infty} 3x^2 = +\infty$ Since the end-behavior goes to infinity on both sides, there aren't any horizontal asymptotes. Okay, I thought so, just wanted to make sure I wasn't missing something obvious. Thanks for the help, I appreciate it. :)	{"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": 0.9377875924110413, "perplexity": 558.63882040602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737893676.56/warc/CC-MAIN-20151001221813-00158-ip-10-137-6-227.ec2.internal.warc.gz"}
http://93138588.com/science/mathematics/fractional_calculus/	Fractional Calculus Computer Algebra System math software Home \| index \| math \| software \| calculator \| handbook \| help \| forum \| contact \| 中文 # Fractional Calculus 分数阶微积分 ## 1. Input Fractional Calculus into Calculator Input sin(0.5,x) for the 0.5th order derivative of sin(x), which is denoted by sin^((0.5))(x). It is differrent from power of sin(x) as sin(x)^2. sin(0.5,x) can be computed and plot. d(sin(x),x,0.5) computes the 0.5th order derivative of sin(x). If you want to hold it (not evaluate), use the derivative holder ds(y,x,0.5). ## 3. Difference between Caputo definition and the Riemann-Liouville (R-L) definition The Caputo (Davison-Essex) and the Riemann-Liouville definitions are different in the following aspect: in the Caputo formula, differentiation is performed first, then integration; but in the R-L formula it is the other way around. The Caputo definition has advantages over the R-L definition: 1. The Caputo definition implemented maps constants to zero, imitating integer order differentiation, while the R-L definition does not. 2. The Caputo definition requires initial value in fractional differential equation is the same as in differential equations, while the R-L definition does not. These properties of the Caputo definition make it suitable to work with initial value problems for fractional differential equations. So the Caputo definition is used here.	{"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": 0.9401422142982483, "perplexity": 2661.6017510902025}, "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/1508187824931.84/warc/CC-MAIN-20171022003552-20171022023552-00453.warc.gz"}
http://davidruescas.com/2016/01/	Probability of an election tie Sparked by recent events in politics, a lot of debate and controversy has occurred on the Spanish blogosphere around a simple question of probability: What is the probability that a Yes/No election with 3030 voters results in a tie? Before suggesting answers, let me make it clear that the main controversy has occurred when trying to answer this question in its barest form, without any additional information besides its simplest formulation above. To make it doubly clear, this is all the information that defines the problem: 1) There are 3030 voters that can vote Yes or No. 2) Yes and No votes are treated as Bernoulli trials. We model a series of Bernoulli trials with a binomial distribution. It has two parameters, the number of events, and the probability of success for each event: X ~ Bin(n, p) Our question is answered by this piece-wise function P(tie) = P(X = n/2) if n is even 0 otherwise All we need to do is plug in the parameters and we’re done. We’ve been given n = 3030 in our problem definition. But wait a minute, what about p? The problem definition states that votes are Bernoulli trials, but we know nothing about p! In order to create intuition for the situation, let me pose two related questions. What is the probability of getting 5 heads in 10 trials when tossing a fair coin? What is the probability of getting 5 heads in 10 trials when tossing a coin where the only thing we know about the coin is that it can land heads or tails? In the first question we have been given information about the coin we are tossing, which we input into the binomial model. In the second question we know nothing about the coin, and therefore nothing about the second parameter p. This is precisely the case with our election problem. So how do we proceed? In both cases the answer is the same, we must construct a version of the binomial that allows us to represent this state of information. The beta-binomial probability distribution comes to the rescue. From wikipedia: In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. I hope something rang a bell when you saw the word “unknown” above, this is exactly our situation. What we do, therefore, is to construct a non-informative prior over p that represents our lack of information about said parameter. In the beta-binomial distribution this prior takes the form of a Beta distribution, and the usual choice as non-informative prior is Beta(1, 1), with alpha = beta = 1. You can see how this choice of prior favors no values of p: Having represented our state of knowledge about p as the choice of prior Beta(1, 1), and given that the parameter n is 3030, we now have all the ingredients to calculate things in a way that is consistent with our problem definition. We do this by using the probability mass function of the beta binomial: We therefore want (since 3030 is even): P(X = 1515) = 1/3031 Does that fraction seem funny? That value is precisely one divided by the total number of possible election results. You can see this by considering that results can go from [Yes 0 – 3030 No] all the way up to [Yes 3030 – 0 No]. And in fact, using our beta-binomial model with Beta(1, 1) all these results are given the same probability: 1/3031. This should come as no surprise, given that as we’ve said repeatedly, the problem definition is such that we know nothing about the election, we have no way to favor one result over the other. You can see this below, the probability for all results is 1/3031 = 0.00033. The p = 0.5 mistake In spite of all of the above, most of the people that analyzed our problem got another result, not 1/3031, but instead 0.0145. This corresponds to calculating a binomial distribution with p = 0.5: X ~ Binomial(3030, 0.5) P(X=1515) = 0.01449 How did they get to assuming p = 0.5? Well, it seems that those who went this route did not know about the beta-binomial distribution, and the beta prior that allows us to represent knowledge about p. Without these tools they made an unwarranted assumption: that the lack of information about p is equivalent to 100% certainty that p is 0.5. The source of that confusion is an insidious “coincidence”: The probability of heads and tails for an unknown coin “happens” to be exactly the same as that for a coin which we know with 100% probability that it is fair. Let me restate that P(head for a single coin toss we know nothing about) = 0.5 P(head for a single coin toss we know 100% to be fair) = 0.5 Because the value is the same, it’s easy to jump to the conclusion that a series of coin tosses for a coin that we know nothing about is treated the same way as a series of coin tosses for which we know for sure that the coin is fair! Unfortunately the above coincidence does not reappear: P(n successes for coin tosses we know nothing about) P(n successes for coin tosses we know 100% to be fair) To illustrate that setting p=0.5 (or any other point value) represents zero uncertainty about the value of p, let’s plot a few graphs for priors and probabilities of ties. This will show how our non-informative prior Beta(1, 1) progressively approximates the p=0.5 assumption as we reduce its uncertainty. • alpha = beta = 1 (non-informative) probability of tie = 1 / 3031 = 0.00033 • alpha = beta = 10 With Beta(10, 10) the probability of tie increases from 1/3031 to 0.0012: probability of tie = 0.0012 • alpha = beta = 100 probability of tie = 0.0036 • alpha = beta = 10000 probability of tie = 0.0135 • alpha = beta = 5×10^5 probability of tie = 0.01447 A formal version of the above trend is: As the variance of our prior tends to zero, the probability of a tie tends to 0.1449 (the value obtained with p = 0.5) How can we interpret p? Given the assumption that voter choices are Bernoulli trials, what can be said about the significance of p? We can offer an interpretation, although this won’t change any of our results above. Having said this, consider that p describes how likely it is that a voter selects Yes or No in an election. If we interpret that a voter chooses Yes or No depending on his/her preferences and the content of the question posed in the election, then p represents the relationship between the election content and the set of voter’s preferences. Saying that we don’t know anything about the election is like saying that we know nothing about the voter’s preferences and question posed to them. If, for example, we knew that the question was about animal rights, and we also knew that the set of voters were animal activists, then we’d probably have a high value of p. Conversely if we asked gun supporters about gun control, p would be low. And if we asked a generally controversial question to the general population, we’d have p around 0.5. Unless we have prior information that rules out or penalizes certain combinations of elections questions and preferences, we must use a non-informative prior for p, such as Beta(1,1).	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8734742403030396, "perplexity": 675.9018103890807}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662519037.11/warc/CC-MAIN-20220517162558-20220517192558-00541.warc.gz"}
https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/f07/f07gsf.html	NAG FL Interfacef07gsf (zpptrs) ▸▿ Contents Settings help FL Name Style: FL Specification Language: 1Purpose f07gsf solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides, $AX=B ,$ where $A$ has been factorized by f07grf, using packed storage. 2Specification Fortran Interface Subroutine f07gsf ( uplo, n, nrhs, ap, b, ldb, info) Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: ap() Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,) Character (1), Intent (In) :: uplo #include <nag.h> void f07gsf_ (const char uplo, const Integer n, const Integer nrhs, const Complex ap[], Complex b[], const Integer ldb, Integer info, const Charlen length_uplo) The routine may be called by the names f07gsf, nagf_lapacklin_zpptrs or its LAPACK name zpptrs. 3Description f07gsf is used to solve a complex Hermitian positive definite system of linear equations $AX=B$, the routine must be preceded by a call to f07grf which computes the Cholesky factorization of $A$, using packed storage. The solution $X$ is computed by forward and backward substitution. If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{H}}Y=B$ and then $UX=Y$. If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{H}}X=Y$. 4References Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore 5Arguments 1: $\mathbf{uplo}$Character(1) Input On entry: specifies how $A$ has been factorized. ${\mathbf{uplo}}=\text{'U'}$ $A={U}^{\mathrm{H}}U$, where $U$ is upper triangular. ${\mathbf{uplo}}=\text{'L'}$ $A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular. Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$. 2: $\mathbf{n}$Integer Input On entry: $n$, the order of the matrix $A$. Constraint: ${\mathbf{n}}\ge 0$. 3: $\mathbf{nrhs}$Integer Input On entry: $r$, the number of right-hand sides. Constraint: ${\mathbf{nrhs}}\ge 0$. 4: $\mathbf{ap}\left(\right)$Complex (Kind=nag_wp) array Input Note: the dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$. On entry: the Cholesky factor of $A$ stored in packed form, as returned by f07grf. 5: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Complex (Kind=nag_wp) array Input/Output Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$. On entry: the $n×r$ right-hand side matrix $B$. On exit: the $n×r$ solution matrix $X$. 6: $\mathbf{ldb}$Integer Input On entry: the first dimension of the array b as declared in the (sub)program from which f07gsf is called. Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$. 7: $\mathbf{info}$Integer Output On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6). 6Error Indicators and Warnings ${\mathbf{info}}<0$ If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated. 7Accuracy For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where • if ${\mathbf{uplo}}=\text{'U'}$, $\|E\|\le c\left(n\right)\epsilon \|{U}^{\mathrm{H}}\|\|U\|$; • if ${\mathbf{uplo}}=\text{'L'}$, $\|E\|\le c\left(n\right)\epsilon \|L\|\|{L}^{\mathrm{H}}\|$, $c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$ where $\mathrm{cond}\left(A,x\right)={‖\|{A}^{-1}\|\|A\|\|x\|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\|{A}^{-1}\|\|A\|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$. Forward and backward error bounds can be computed by calling f07gvf, and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling f07guf. 8Parallelism and Performance f07gsf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. f07gsf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information. The total number of real floating-point operations is approximately $8{n}^{2}r$. This routine may be followed by a call to f07gvf to refine the solution and return an error estimate. The real analogue of this routine is f07gef. 10Example This example solves the system of equations $AX=B$, where $A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i )$ and $B= ( 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i ) .$ Here $A$ is Hermitian positive definite, stored in packed form, and must first be factorized by f07grf. 10.1Program Text Program Text (f07gsfe.f90) 10.2Program Data Program Data (f07gsfe.d) 10.3Program Results Program Results (f07gsfe.r)	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 73, "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": 0.9807451367378235, "perplexity": 2694.250990118207}, "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-2021-25/segments/1623487613453.9/warc/CC-MAIN-20210614201339-20210614231339-00504.warc.gz"}
http://tug.org/pipermail/xy-pic/2001-October/000019.html	# [Xy-pic] Controlloing the dimensions of xy boxes Luigi Santocanale luigis@cpsc.ucalgary.ca Sat, 27 Oct 2001 17:14:21 -0600 This is a multi-part message in MIME format. --------------88E24D4B5EEB358D99E014B4 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Hi everybody, I have several difficulties controlling the dimensions produced by the environments \xy\endxy. The following is an example of what I mean (Latex file follows, dvi-file in attachment). There are two different graph escapes which are supposed to draw the objects of a categorical commutative square. However, even if very similar, they behave in a very different manner. Have you got any suggestion ? Thanks, Luigi %%% Begin Latex File \documentclass{article} \usepackage[all]{xy} %% Draw a square, 1st attempt \newgraphescape{s}[8]{ []+{#1}="1" ( :@{{}{}{}}[d(#5)r(#6)]+{#2}="2" :@{{}{}{}}[d(#8)r(#7)]+{#4}="4", :@{{}{}{}}[d(#8)r(#7)]+{#3}="3" ) } \newgraphescape{S}[6]{ []!s{#1}{#2}{#3}{#4}{#5}{0}{#6}{0} } \begin{document} %% Draw a square, 2nd corrected attempt \newgraphescape{t}[8]{ [] :@{{}{}{}}[]+{#1}="1" %% This line the only difference ( :@{{}{}{}}[d(#5)r(#6)]+{#2}="2" :@{{}{}{}}[d(#8)r(#7)]+{#4}="4", :@{{}{}{}}[d(#8)r(#7)]+{#3}="3" ) } \newgraphescape{T}[6]{ []!t{#1}{#2}{#3}{#4}{#5}{0}{#6}{0} } $$\fbox{\xy\xygraph{ [] !S{a}{b}{c}{d}{1}{-1} }\endxy} \hspace{2em} \fbox{\xy\xygraph{ [] !T{a}{b}{c}{d}{1}{-1} }\endxy}$$ \end{document} %%% End of Latex File -- Luigi Santocanale Department of Computer Science, University of Calgary. Tel: (403) 220-5106, fax:(403) 284-4707. http://www.cpsc.ucalgary.ca/~luigis/ --------------88E24D4B5EEB358D99E014B4 Content-Type: application/x-dvi; name="fbox.dvi" Content-Transfer-Encoding: base64 Content-Disposition: inline; filename="fbox.dvi" 9wIBg5LAHDsAAAAAA+gbIFRlWCBvdXRwdXQgMjAwMS4xMC4yNzoxNzA0iwAAAAEAAAAAAAAA	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9739267826080322, "perplexity": 15433.603200112577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813088.82/warc/CC-MAIN-20180220185145-20180220205145-00781.warc.gz"}
http://math.stackexchange.com/questions/154872/uniform-integrability-after-composition	# Uniform Integrability after composition Let $\mu$ be a finite measure on $X \subseteq \mathbb{R}^n$. Consider the Uniformly Integrable family $\{ f_n(\cdot) \}_{n \in \mathbb{N}}$ of functions $f_n : X \rightarrow \mathbb{R}_{\geq 0}$. Consider a continuous, strictly-increasing function $\phi:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $\phi(0)=0$ such that $\phi(f_n(\cdot))$ is integrable for all $n$. 1. Is the family $\{ \phi( f_n(\cdot) ) \}_{n \in \mathbb{N}}$ Uniformly Integrable as well? 2. More generally, with functions $g_n : X \rightarrow \mathbb{R}^n$ such that the family $\{ \|g_n(\cdot)\| \}_{n \in \mathbb{N}}$ is Uniformly Integrable, and a continuous $\psi: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$, is $\{ \psi(g_n(\cdot)) \}_{n \in \mathbb{N}}$ Uniformly Integrable? - no, they are not even necessarily integrable,e,g,$\phi(x) = x^2$ – mike Jun 6 '12 at 23:44 Why? Counterexample? If it is not the case, under which conditions we have that UI is preserved? – Adam Jun 6 '12 at 23:52 I see your comment. Of course, this is a trivial condition that has to be satisfied. I'll update the question accordingly. – Adam Jun 7 '12 at 0:00 Consider $n=1$ and $f_j=j\chi_{(0,\frac 1{j^2})}$. Fix $R>0$, when $j\geq R$ we have $$\int_{\|f_j\|\geq R}\|f_j\|dP\leq \sqrt{\mu(\|f_j\|\geq R)}\leq \frac 1R,$$ hence $\{f_j\}$ are uniformly integrable, and in $L^3$. We have $\lVert f_j\rVert_{L^3}=j\left(\int_{(0,\frac 1{j^2})}1\right)^{1/3}=j^{1/3}$, hence the family $\{f_n^3\}$ is not uniformly integrable. What additional assumptions should be made on $\phi$ to get that $\{ \phi(f_n(\cdot)) \}_n$ are UI? – Adam Apr 2 at 9:17 Moreover, in your example, are the functions $\phi(f_n(\cdot))$ integrable? – Adam Apr 2 at 9:23 Yes, $\phi(f_n)$ is bounded and the measure space is finite. For conditions about preservation of uniform integrability, I presume you already know de la Vallée Poussin's theorem, so it could be a start. – Davide Giraudo Apr 2 at 9:24 Yes, I do know that and I am not clear about your comment. In the de la Vallée Poussion Theorem, we have a scalar function $G$ such that $\lim_{x \rightarrow \infty} G(x)/x = \infty$. Here we want to use like $\|f\|$ is place of $G$, but it depends on the parameter $n$. – Adam Apr 2 at 9:37	{"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": 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": 0.8953741788864136, "perplexity": 421.9412527814858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164024169/warc/CC-MAIN-20131204133344-00044-ip-10-33-133-15.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/44592/does-f-sim-g-imply-f-asymp-g-in-certain-conditions	# Does $f \sim g$ imply $f \asymp g$ in certain conditions? I got a good answer to this question over on MathOverflow a while ago. Harald Hanche-Olsen claimed that, if $f, g: D\to \mathbb{R}^+$, then $$f(x) \sim g(x) \implies f(x) \asymp g(x) \qquad \qquad ()$$ holds whenever $D\subseteq\mathbb{C}$ is closed in $\mathbb{C}$, and is false whenever it is not closed. However, something bugs me about this. Most instances I've seen this, it does hold when $D$ is the strictly positive reals, which I believe is not closed in $\mathbb{C}$. So this is my question. Does $()$ hold when $f, g: \mathbb{R}^+ \to \mathbb{R}^+$? That is, when $f$ and $g$ are positive real-valued and defined on the positive reals? (Edit: For a detailed explanation of which definitions I am using, see the link. I believe they are standard.) - Claimed in the sense of proved. – Did Jun 10 '11 at 17:48 No, for basically the same reason it fails for other non-closed sets. Just because $\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=1$, it doesn't mean that $\lim_{x\rightarrow 0} \frac{f(x)}{g(x)}$ even exists. So you could, for example, take $f(x) = \frac{1}{x}$ and $g(x) = \frac{1}{x+1}$. Then certainly $\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}=1$, but there is no $A$ such that $f(x) < Ag(x)$ for all $x$, since $f$ has a vertical asymptote at 0.	{"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": 0.9728570580482483, "perplexity": 121.38284376228059}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645323734.78/warc/CC-MAIN-20150827031523-00066-ip-10-171-96-226.ec2.internal.warc.gz"}
https://linearalgebras.com/solution-abstract-algebra-exercise-3-2-12.html	If you find any mistakes, please make a comment! Thank you. ## Given a subgroup of a group, the numbers of left and right cosets are equal Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.12 Let $G$ be a group and $H \leq G$. Prove that the map $x \mapsto x^{-1}$ sends each left coset to a right coset (of $H$), and hence $\|H \backslash G\| = \|G/H\|$. Solution: (This is a slightly different approach.) Define $\psi : G \rightarrow H \backslash G$ by $\psi(x) = Hx^{-1}$. Suppose $y^{-1}x \in H$; then $y^{-1} \in Hx^{-1}$. and we have $$Hy^{-1} = Hx^{-1},$$ so $\psi(x) = \psi(y)$. By Lemma 1 to Exercise 3.2.10, this induces a mapping $\varphi : G/H \rightarrow H \backslash G$ given by $$\varphi(xH) = Hx^{-1}.$$ $\psi$ is clearly surjective, so $\varphi$ is surjective. Now suppose $\varphi(x) = \varphi(y)$; then $Hx^{-1} = Hy^{-1}$, so that in particular $y^{-1} \in Hx^{-1}$ and hence $$y^{-1}x \in H\Longrightarrow xH = yH.$$ Hence $\varphi$ is a bijection. #### Linearity This website is supposed to help you study Linear Algebras. Please only read these solutions after thinking about the problems carefully. Do not just copy these 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": 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": 0.980154275894165, "perplexity": 159.15699645699408}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104277498.71/warc/CC-MAIN-20220703225409-20220704015409-00774.warc.gz"}
https://learn.careers360.com/medical/questions/?page=2	## Filters Sort by : Q The number of sigma $(\sigma)$ and pi$(\pi)$ bonds in pent-2-en-4-yne is: • Option 1) 10 $\sigma$ bonds and 3 $\pi$ bonds • Option 2) $\sigma$ bonds and 5 $\pi$ bonds • Option 3) 11 $\sigma$ bonds and 2 $\pi$ bonds • Option 4) 13 $\sigma$ bonds and no $\pi$ bonds Option 1 Structure of pent-2-en-4-yne Here there are 10 bonds and bonds And 3 bonds and Option 1)10 bonds and 3 bondsOption 2)8 bonds and 5 bondsOption 3)11 bonds and 2 bondsOption 4)13 bonds and no bonds An alkane "A" on reaction with O3 and Zn-H2O gives propane and ethanal in equimolar ratio. Addition of HCl to Alkane "A" gives "B" as the major product. The structure of the product "B" is • Option 1) • Option 2) • Option 3) • Option 4) This oxidation of alkene occurs at the double bond. Thus, propane & ethanol must combine to give the alkene (Propanone) Now this reacts with HCl to form B as follows: ... Which of the following acts as a circuit protection device ? • Option 1) conductor • Option 2) inductor • Option 3) switch • Option 4) fuse fuse Option 1)conductor Option 2)inductor Option 3)switch Option 4)fuse In which of the following devices , the eddy current effect is not used ? • Option 1) induction furnace • Option 2) magnetic braking in train • Option 3) electromagnet • Option 4) electric heater electric heater Option 1)induction furnace Option 2)magnetic braking in train Option 3)electromagnet Option 4)electric heater Match the Xenon compound in Column-I with its structure in Column-II and assign the correct code: Column-I Column-II (a) XeF4 (i) pyramidal (b) XeF6 (ii) Square planar (c) XeOF4 (iii) distorted octahedral (d) XeO3 (iv) square pyramidal Code: (a) (b) (c) (d) • Option 1) (i) (ii) (iii) (iv) • Option 2) (ii) (iii) (iv) (i) • Option 3) (ii) (iii) (i) (iv) • Option 4) (iii) (iv) (i) (ii) XeF4 it is square planar. XeF6 it is distorted octahedral because of the lone pair present. XeOF4 Its is square pyramidal & its hybridization is sp3d2 XeO3 Its hybridization is sp3 & its shape is pyramidal. (a) - (ii), (b) - (iii), (c) - (iv), (d) - (i) Option 2 is correct.Option 1)(i) (ii) (iii) (iv)Option 2)(ii) (iii) (iv) (i)Option 3)(ii) (iii) (i) ... Pick the wrong answer in the context with rainbow • Option 1) When the light rays undergo two internal reflection in a water drop , a secondary rainbow is formed • Option 2) The order of colors is reversed in the secondary rainbow • Option 3) An observer can see a rainbow when his front is towards the sun • Option 4) Rainbow is a combined effect of dispersion , refreaction and reflection of sunlight Rainbow can seen when back is towards the Sun.Option 1)When the light rays undergo two internal reflection in a water drop , a secondary rainbow is formed Option 2)The order of colors is reversed in the secondary rainbow Option 3)An observer can see a rainbow when his front is towards the sun Option 4)Rainbow is a combined effect of dispersion , refreaction and reflection of sunlight A particle moving with velocity $\vec V$ is acted by three forces shown by the vector triangle PQR . The velocity of the particle will : • Option 1) increase • Option 2) derease • Option 3) remain constant • Option 4) change acording to the smallest force QR Option 1) increase Option 2) derease Option 3) remain constant Option 4) change acording to the smallest force QR Which mixture of the solution will lead to the formation of negatively charged colloidal $[AgI]I^-$ sol. ? • Option 1) 50 mL of 1 M $AgNO_3$ + 50 mL of 1.5 M $KI$ • Option 2) 50 mL of 1 M $AgNO_3$ + 50 mL of 2 M $KI$ • Option 3) 50 mL of 2 M $AgNO_3$ + 50 mL of 1.5 M $KI$ • Option 4) 50 mL of 0.1 M $AgNO_3$ + 50 mL of 0.1 M $KI$ Option 2 Negatively charged colloidal need 2 mol of Option 1)50 mL of 1 M + 50 mL of 1.5 M Option 2)50 mL of 1 M + 50 mL of 2 M Option 3)50 mL of 2 M + 50 mL of 1.5 M Option 4)50 mL of 0.1 M + 50 mL of 0.1 M Which of the following is incorrect statement? • Option 1) PbF4 is covalent in nature • Option 2) SiCl4 is easily hydrolysed • Option 3) GeX4 (X = F, Cl, Br, I) is more stable than GeX2 • Option 4) SnF4 is ionic in nature (i) PbF4 is not covalent in nature. Because Pb atom is very & F atom is very small, so according to Fajan's rule, PbF4 is not covalent. (ii) SiCl4 is not a very stable compound because of steric hindrance thus can be hydrolysed very easily (iii) GeX4 is more stable than GeX2 as in GeX4 all the orbitals are fully filled. (iv) SnF4 is ionic in nature. Because F atom is very small & Sn atom is... Two parallel infinite line charges with linear charges densities $+ \lambda C/ m \: \: and \: \: - \lambda C/ m$ are placed at a distance of 2R in free space . what is the electric field mid way between the two line charges ? • Option 1) zero • Option 2) $\frac{2 \lambda }{\pi \epsilon _0 R} N/C$ • Option 3) $\frac{ \lambda }{\pi \epsilon _0 R} N/C$ • Option 4) $\frac{\lambda }{2\pi \epsilon _0 R} N/C$ Option 1) zero Option 2) Option 3) Option 4) What is the correct electronic configuration of the central atom in $K_6[Fe(CN)_6]$ based on crystal field theory? • Option 1) $t_{2g}^4e_{g}^2$ • Option 2) $t_{2g}^6e_{g}^0$ • Option 3) $e^3t_2^3$ • Option 4) $e^4t_2^2$ Splitting occurs as follows for octahedral For Fe2+ (d6), electron distribution occurs as follows: is the correct distribution. Option 2 is correct. Option 1)Option 2)Option 3)Option 4) Two similar thin equiconvex lenses , of focal length f each , are kept coaxially in contact with each other such that the focal length of the combination is $F_1$ . When the space between the two lenses is filled with glycerin then the equivalent focal length is $F_2$ . The ratio $F_1 : F_2$ • Option 1) 2 : 1 • Option 2) 1 : 2 • Option 3) 2 : 3 • Option 4) 3 : 4 Option 1)2 : 1 Option 2)1 : 2 Option 3)2 : 3 Option 4)3 : 4 The non-essential amino acid among the following is: • Option 1) valine • Option 2) leucine • Option 3) alanine • Option 4) lysine Non essential amino acids are those which produce in the human body itself. Essential amino acids are those which do not produce in the human body & we have to take them from outside sources. Alanine is the non-essential amino acid. Option 3 is correct.Option 1)valineOption 2)leucineOption 3)alanineOption 4)lysine For an ideal solution, the correct option is: • Option 1) $\Delta_{mix}S = 0$ at constant T and P • Option 2) $\Delta_{mix}V \neq 0$ at constant T and P • Option 3) $\Delta_{mix}H = 0$ at constant T and P • Option 4) $\Delta_{mix}G = 0$ at constant T and P For an ideal solution, the Gibbs free energy is always negative, thus their mixing is always spontaneous. The volume change of the mixing of ideal solutions is zero. An ideal solution is one in which the enthalpy of the solution is zero at constant T and P Option 3 is correct.Option 1) at constant T and POption 2) at constant T and POption 3) at constant T and POption 4) at constant T and P For the cell reaction $2Fe^{3+}(aq)+ 2I^{-}(aq) \rightarrow 2Fe^{2+}(aq) + I_2(aq)$ $E^0_{cell} = 0.24V$ at 298K. The standard Gibbs energy $(\Delta_r G^0)$ of the cell reaction is: [Given that Faraday constant F = 96500 C mol-1] • Option 1) -46.32 kJ mol-1 • Option 2) -23.16 kJ mol-1 • Option 3) 46.32 kJ mol-1 • Option 4) 23.16 kJ mol-1 We know that Option 1 is correctOption 1)-46.32 kJ mol-1Option 2)-23.16 kJ mol-1Option 3)46.32 kJ mol-1Option 4)23.16 kJ mol-1 Enzymes that utilize ATP in phosphate transfer require an alkaline earth metal (M) as the cofactor M is: • Option 1) Be • Option 2) Mg • Option 3) Ca • Option 4) Sr Option 2 Mg is the alkaline earth metal as the co-factor. (FACT)Option 1)BeOption 2)MgOption 3)CaOption 4)Sr A compound is formed by cation C and anion A. The Anions form hexagonal close packed lattice and the cations occupy 75% of the octahedral void. The formula of the compound is • Option 1) $C_2A_3$ • Option 2) $C_3A_2$ • Option 3) $C_3A_4$ • Option 4) $C_4A_3$ Option 3 Anions: No of ions per unit in HCP = 6 Cations: No of octahedral voids in HCP = 6 So, 75% of 6 = 4.5 Option 1)Option 2)Option 3)Option 4) A gas at 350K and 15 bar has molar volume 20 percent smaller than that for an ideal gas under the same conditions. The correct option about the gas and its compressibility factor (Z) is: • Option 1) Z>1 and attractive forces are dominant • Option 2) Z>1 and repulsive forces are dominant • Option 3) Z<1 and attractive forces are dominant • Option 4) Z<1 and repulsive forces are dominant Option 3 Molar volume of gas is 20% smaller than that of the ideal gas. Hence Z < 1 and attractive forces are dominant. When Z < 1: Gas shows negative deviation and is more compressible than expected from ideal behaviour. Z < 1 corresponds to attractive forces between molecules.Option 1)Z>1 and attractive forces are dominantOption 2)Z>1 and repulsive forces are dominantOption 3)Z<1... • Option 1) nylon-6,6 • Option 2) nylon 2 - nylon 6 • Option 3) nylon-6 • Option 4) Buna-S Option 2 Biodegradable polymer is Nylon-2-Nylon-6 monomers: Glycine, aminocaproic acid Option 1)nylon-6,6Option 2)nylon 2 - nylon 6Option 3)nylon-6Option 4)Buna-S For the chemical reaction $N_2(g) + 3H_2(g)\leftrightharpoons 2NH_3(g)$ the correct option is: • Option 1) $-\frac{1}{3}\frac{d[H_2]}{dt} = -\frac{1}{2}\frac{d[NH_3]}{dt}$ • Option 2) $-\frac{d[N_2]}{dt} = {2}\frac{d[NH_3]}{dt}$ • Option 3) $-\frac{d[N_2]}{dt} = \frac{1}{2}\frac{d[NH_3]}{dt}$ • Option 4) ${3}\frac{d[H_2]}{dt} ={2}\frac{d[NH_3]}{dt}$ For a chemical reaction: The rate is always given as: for the reaction: , Thus, Option 3 is correct.Option 1)Option 2)Option 3)Option 4) Exams Articles Questions	{"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": 48, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8918765783309937, "perplexity": 14312.387851622494}, "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/1573496669868.3/warc/CC-MAIN-20191118232526-20191119020526-00457.warc.gz"}
https://www.goanywhere.com/forum/single-file-move-rename-506	## Single file move/rename Posts: 1 Joined: Wed Oct 26, 2011 1:11 pm Post by Zhivago » Tue Mar 25, 2014 7:59 am I have a instance where a file is posted on our FTP server with a name like: FSE-MAINES_mmddyyyy_hhmmss.txt I need to place this file in one folder (on another server) as FSE.TXT and a copy of the file in an archive folder (on the same server as the FSE.TXT file). Not sure how to grab FSE-MAINES_mmddyyyy_hhmmss.txt with a wildcard and do the move or copy with the rename. Support Specialist Posts: 592 Joined: Tue Jul 17, 2012 2:12 pm Post by Support_Rick » Fri Mar 28, 2014 5:15 pm Here's some example XML that you could use to get your desired results: Code: Select all <createFileList label="Get list of Files" fileListVariable="MyFileList" version="1.0"> <fileset dir="/Path/To/My/File"> <wildcardFilter> <include pattern="FSE-*" /> </wildcardFilter> </fileset> </createFileList> <copy label="Copy to Server" sourceFile="${MyFileList}" destFile="\\Server\Path\to\Destination\FSE.txt" whenFileExists="overwrite" version="1.0" /> <copy label="Copy to Archive" sourceFile="${MyFileList}" destFile="\\Server\Path\to\Archive\FSE.txt" whenFileExists="overwrite" version="1.0" />	{"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": 0.5622044205665588, "perplexity": 14849.118734006519}, "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-2020-16/segments/1585371858664.82/warc/CC-MAIN-20200409122719-20200409153219-00444.warc.gz"}
https://brilliant.org/discussions/thread/prove-it-9/	× # Prove it! prove Hadamard's inequality for any matrix $$\left[ { a }_{ n }^{ i } \right]$$ $${ det }\left[ { a }_{ n }^{ i } \right] \le\displaystyle\prod _{ n=1 }^{ k }{ (\displaystyle\sum _{ i=1 }^{ k }{ \left[ { a }_{ n }^{ i } \right] ) } }$$ Hint:Differential calculus Note:all the entries in the matrix are non-negative Note by Hummus A 1 year, 9 months ago MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$...$$ or $...$ to ensure proper formatting. 2 \times 3 $$2 \times 3$$ 2^{34} $$2^{34}$$ a_{i-1} $$a_{i-1}$$ \frac{2}{3} $$\frac{2}{3}$$ \sqrt{2} $$\sqrt{2}$$ \sum_{i=1}^3 $$\sum_{i=1}^3$$ \sin \theta $$\sin \theta$$ \boxed{123} $$\boxed{123}$$ Sort by: For the above inequality to be true, it is required that the all entries of the matrix are non-negative. Assuming that to be the case, we can upper-bound the determinant of the matrix $$A$$ by the permanent of the matrix $$A$$ to obtain $\text{det}(A) \stackrel{(a)}{\leq} \text{per}(A) = \sum_{\sigma \in \pi} \prod_{n=1}^{k}a_{n \sigma({n})}\stackrel{(b)}{\leq} \prod_{n=1}^{k}(\sum_{i=1}^{k}a_{i,n}),$ where $$\pi$$ is the set of all permutations on $$n$$ elements and the inequalities (a) and (b) follows from the non-negativity of the elements. - 1 year, 9 months ago nice short proof! and yes i think that the entries should all be non-negative i'll mention that in the note - 1 year, 9 months ago Also, the LHS should be $$\text{det(A)}$$ as I proved, instead of $$\text{det(A)}^2$$. As a counter-example, take $A=\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}$ - 1 year, 9 months ago you're right i don't know why i thought it was the square of it i'll edit the note - 1 year, 9 months ago	{"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": 0.9971283674240112, "perplexity": 2884.587394878412}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806715.73/warc/CC-MAIN-20171123012207-20171123032207-00376.warc.gz"}
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=DBSHBB_2004_v41n3_461	HOMOMORPHISMS BETWEEN C-ALGEBRAS ASSOCIATED WITH THE TRIF FUNCTIONAL EQUATION AND LINEAR DERIVATIONS ON C-ALGEBRAS Title & Authors HOMOMORPHISMS BETWEEN C-ALGEBRAS ASSOCIATED WITH THE TRIF FUNCTIONAL EQUATION AND LINEAR DERIVATIONS ON C-ALGEBRAS Park, Chun-Gil; Hou, Jin-Chuan; Abstract It is shown that every almost linear mapping h : A\longrightarrowB of a unital $\small{C^{}}$ -algebra A to a unital $\small{C^{}}$ -algebra B is a homomorphism under some condition on multiplication, and that every almost linear continuous mapping h : A\longrightarrowB of a unital $\small{C^{}}$ -algebra A of real rank zero to a unital $\small{C^{}}$ -algebra B is a homomorphism under some condition on multiplication. Furthermore, we are going to prove the generalized Hyers-Ulam-Rassias stability of -homomorphisms between unital $\small{C^{}}$ -algebras, and of C-linear -derivations on unital $\small{C^{}}$ -algebras./ -algebras. Keywords homomorphism;C-algebra of real rank zero;linear derivation;stability.; Language English Cited by 1. ON A GENERALIZED TRIF'S MAPPING IN BANACH MODULES OVER A C-ALGEBRA,;; 대한수학회지, 2006. vol.43. 2, pp.323-356 2. d-ISOMETRIC LINEAR MAPPINGS IN LINEAR d-NORMED BANACH MODULES,;; 대한수학회지, 2008. vol.45. 1, pp.249-271 3. STABILITY OF THE CAUCHY FUNCTIONAL EQUATION IN BANACH ALGEBRAS,;; Korean Journal of Mathematics, 2009. vol.17. 1, pp.91-102 4. APPROXIMATE BI-HOMOMORPHISMS AND BI-DERIVATIONS IN C-TERNARY ALGEBRAS,;; 대한수학회보, 2010. vol.47. 1, pp.195-209 5. ON THE STABILITY OF BI-DERIVATIONS IN BANACH ALGEBRAS,;; 대한수학회보, 2011. vol.48. 5, pp.959-967 6. STABILITY OF THE JENSEN TYPE FUNCTIONAL EQUATION IN BANACH ALGEBRAS: A FIXED POINT APPROACH,;;;; Korean Journal of Mathematics, 2011. vol.19. 2, pp.149-161 7. THE STABILITY OF LINEAR MAPPINGS IN BANACH MODULES ASSOCIATED WITH A GENERALIZED JENSEN MAPPING,; 충청수학회지, 2011. vol.24. 2, pp.287-301 8. GENERALIZED (θ, ø)-DERIVATIONS ON BANACH ALGEBRAS,;; Korean Journal of Mathematics, 2014. vol.22. 1, pp.139-150 1. GENERALIZED (θ, ø)-DERIVATIONS ON BANACH ALGEBRAS, Korean Journal of Mathematics, 2014, 22, 1, 139 2. Approximately Linear Mappings in Banach Modules over a C-algebra, Acta Mathematica Sinica, English Series, 2007, 23, 11, 1919 3. Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms betweenC -algebras, Mathematische Nachrichten, 2008, 281, 3, 402 4. Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces, Acta Mathematica Sinica, English Series, 2006, 22, 6, 1789 5. Stability of the Jensen-Type Functional Equation inC∗-Algebras: A Fixed Point Approach, Abstract and Applied Analysis, 2009, 2009, 1 6. Isomorphisms Between Quasi-Banach Algebras, Chinese Annals of Mathematics, Series B, 2007, 28, 3, 353 7. Approximate Cubic ∗-Derivations on Banach ∗-Algebras, Abstract and Applied Analysis, 2012, 2012, 1 8. Cubic derivations on Banach algebras, Acta Mathematica Vietnamica, 2013, 38, 4, 517 9. Homomorphisms between JC–algebras and Lie C–algebras, Acta Mathematica Sinica, English Series, 2005, 21, 6, 1391 10. On the Cauchy–Rassias stability of a generalized additive functional equation, Journal of Mathematical Analysis and Applications, 2008, 339, 1, 372 11. The Pexiderized Apollonius–Jensen type additive mapping and isomorphisms betweenC-algebras, Journal of Difference Equations and Applications, 2008, 14, 5, 459 12. STABILITY OF THE JENSEN TYPE FUNCTIONAL EQUATION IN BANACH ALGEBRAS: A FIXED POINT APPROACH, Korean Journal of Mathematics, 2011, 19, 2, 149 13. WITHDRAWN: Stability of the Cauchy–Jensen functional equation in -algebras: A fixed point approach, Journal of Mathematical Analysis and Applications, 2007 14. Jordan∗-Derivations onC∗-Algebras andJC∗-Algebras, Abstract and Applied Analysis, 2008, 2008, 1 15. ON APPROXIMATE n-ARY DERIVATIONS, International Journal of Geometric Methods in Modern Physics, 2011, 08, 03, 485 16. Generalized additive mapping in Banach modules and isomorphisms between C-algebras, Journal of Mathematical Analysis and Applications, 2006, 314, 1, 150 17. The Stability of a Quadratic Functional Equation with the Fixed Point Alternative, Abstract and Applied Analysis, 2009, 2009, 1 18. The N-Isometric Isomorphisms in Linear N-Normed C-Algebras, Acta Mathematica Sinica, English Series, 2006, 22, 6, 1863 References 1. J. Funct. Anal., 1991. vol.99. pp.131-149 2. J. Math. Anal. Appl., 1994. vol.184. pp.431-436 3. J. London Math. Soc., 1988. vol.37. 2, pp.294-316 4. Math. Scand., 1985. vol.57. pp.249-266 5. Elementary Theory, 1983. 6. J. Math. Anal. Appl., 2003. vol.278. pp.93-108 7. Proc. Amer. Math. Soc., 1978. vol.72. pp.297-300 8. J. Math. Anal. Appl., 2002. vol.272. pp.604-616	{"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": 6, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.761202335357666, "perplexity": 4316.991400236207}, "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-2017-30/segments/1500549425193.20/warc/CC-MAIN-20170725122451-20170725142451-00007.warc.gz"}
https://svn.haxx.se/tsvnusers/archive-2006-03/0102.shtml	# Problem with special characters in password with ntlm auth From: Yves Glodt <y.glodt_at_sitasoftware.lu> Date: 2006-03-07 18:10:53 CET Hello, I have set up an Subversion server using the following software: - Apache 2.0.55 - SVN 1.3.0 - mod_auth_sspi 1.0.4-rc - Openssl 0.9.8a - TortoiseSVN 1.3.2, Build 5840 - 32 Bit I connect to it from Tortoisesvn via https, and it works well, except one thing.... It seems if a user has a special character in his password, he is always rejected with the following error (from apache errorlog): (OS 1326)Logon failure: unknown user name or bad password. : user MYDOMAIN\\yves: authentication failure for "/". Special character is e.g. any of the following: é \$ ' ! What makes me post here is the fact that I can login with my user/password through firefox, it only seems to fail with Tortoisesvn... Someone has an idea where this could come from? Is there another way to debug this? Best regards, Yves --------------------------------------------------------------------- To unsubscribe, e-mail: users-unsubscribe@tortoisesvn.tigris.org For additional commands, e-mail: users-help@tortoisesvn.tigris.org Received on Tue Mar 7 18:09:20 2006 This is an archived mail posted to the TortoiseSVN Users mailing list.	{"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": 0.829145073890686, "perplexity": 16311.037106879918}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613780.89/warc/CC-MAIN-20170530031818-20170530051818-00154.warc.gz"}
https://publiclab.org/notes/shubham8705/07-01-2020/what-does-covid-19-have-to-do-with-the-climate-and-environmental-crisis	# What does COVID-19 have to do with the climate and environmental crisis by shubham8705 \| 01 Jul 06:26 shubham8705 was awarded the Watchdog Barnstar by ektopyrotic for their work in this research note. The appearance of zoonotic diseases (viruses that are transmitted from animals to humans) is not new to this time, although it does seem to be on the rise. Research suggests these have quadrupled in the past 50 years. And a look at this young 21st century seems sufficient evidence, given that four have already occurred: Severe Acute Respiratory Syndrome (SARS), avian influenza (H5N1), swine (H1N1) and the current COVID-19. In all cases, they were viruses unique to animal populations that mutated, invaded a human organism, and then spread as new pathogens among the world population. The common bond between all of them? The human being. This was explained by David Quammen, author of " Spillover: Animal Infections and the Next Human Pandemic" A wild species can contain 50 viruses unknown to humans. They do not affect them because they evolved with them. But that is not our case. When their habitat is intervened, when the forests where these species live are cut down, when they are taken out of there to sell them as commodities and consume them, the natural barriers that exist between them and us are also broken, exposing ourselves to new viruses. Although one does not live near the disappeared or fragmented forest or the markets where wild fauna and flora are commercialized, the consequences come, as currently demonstrated by COVID-19 and pandemics such as HIV, SARS or yellow fever. This is explained by the United Nations Environment Program (UN Environment):"By changing the use of land for settlements, agriculture, logging or industries and their associated infrastructures, the habitat of the animals has been fragmented or invaded. Natural buffer zones, which normally separate humans from wildlife, have been destroyed, and bridges have been created for pathogens to pass from animals to people. " According to its 2016 Borders report, 75%of all emerging infectious diseases in humans are of animal origin and closely related to the health of ecosystems. Zoonoses thrive when there are changes in the environment, in animal or human hosts, or in the pathogens themselves. "In the past century, the combination of population growth and declining ecosystems and biodiversity has led to unprecedented opportunities that have facilitated the transfer of pathogens from animals to people. On average, a new infectious disease emerges in humans every four months, "he said. # Deforestation and change in land use Forests are essential ecosystems for life on Earth. We know. These regulate water, conserve the soil and the atmosphere and provide countless services that satisfy needs, not to mention the multitude of living beings (some of us known, others not yet) that make these environments their home. However, human intervention in them has been devastating. Since the Industrial Revolution until now, a third of the Earth's forests have been removed, according to the latest report from the Intergovernmental Platform on Biodiversity and Ecosystem Services(IPBES), released last year. In other words, at present, the world's forest area is 68%from the pre-industrial level estimated. And there's more: 75% has been damaged from the planet's surface. Today, only 25% is free from substantial impacts caused by human activities and the percentage would drop to 10% in 2050. How is this related to the emergence of new viruses? Devoid of their natural habitat, wildlife is forced to change its distribution, to migrate in search of food, to come into contact with other species to compete for scarce resources and to approach human populations in search of spaces to survive. Ricardo Baldi, scientist of the National Council of Scientific and Technical Research of Argentina ( Conicet), as explained: "Deforestation is linked to more than 30% of registered outbreaks of diseases, such as Ebola and Zika, in the last 30 years. The loss of habitat makes the animals stay closer to human populations, also generating contact opportunities and, therefore, the appearance of new zoonoses ". According to a studyIn the Amazon, an increase in deforestation of around 4% over a three-year period increased the incidence of malaria by almost 50% as mosquitoes that transmit the disease flourished in recently deforested areas. And is that, by invading and fragmenting tropical forests - habitats of innumerable animal species and, within them, unknown and potentially new viruses - there is a risk of releasing viruses from their natural hosts, which can jump to humans. An example of this is HIV, which possibly crossed from chimpanzees to people in the 1920s, when hunters killed and ate them in Africa. # Biodiversity protects us "Ecosystems are inherently resilient and adaptable, and by supporting the existence of diverse species, they help regulate disease. The more biodiverse an ecosystem is, the more difficult it is for a pathogen to spread rapidly, " explains UN Environment. In other words, biodiversity protects us. The lower the biodiversity, the greater the possibility that the agents that go around in nature behave like pathogens and we will relive the current pandemic situation with new zoonotic diseases. "Genetic diversity provides a natural source of disease resistance among animal populations. For example, intensive livestock farming often produces genetic similarities within herds and herds, increasing the susceptibility of these animals to the spread of pathogens from wildlife, "states the United Nations agency. "Biodiverse areas allow disease-transmitting vectors to feed on a wide variety of hosts, some of which are less effective reservoirs of pathogens. Conversely, when pathogens are found in less biodiverse areas, transmission can be amplified, as has been demonstrated in the case of West Nile virus and Lyme disease. " A decade ago, scientists already estimated that between 150 and 200 species of plants, insects, birds, and mammals go extinct every 24 hours.. This is almost 1000 times the "natural" or "bottom" rate. Currently, according to IPBES, nearly 1 million animal and plant species are in danger of extinction, many in the coming decades and more than ever in the history of humanity. Added to this is the illegal or poorly regulated wildlife trade, a millionaire business (that moves between 8000 and 20,000 million euros a year) comparable to arms or drug trafficking, and which today ---with the COVID-19 pandemic , which is estimated to have originated in a Chinese market that sold these species, it needs to begin to be recognized as a public health issue since it facilitates zoonotic jumps. It is worth mentioning that the first cases of SARSwere associated with contact with caged civets in a market and some Ebola cases are believed to be in Central Africa they were transferred from animal hosts to humans when infected gorilla meat was consumed. Annually, as estimated by the World Wide Fund for Nature(WWF, for its acronym in English), this trade is responsible for the death of an average of 100 tigers, 30,000 elephants, more than 1,000 rhinos and more than 100,000 pangolins. To this are added, for example, the 1.5 million live birds that are marketed per year and the up to 440,000 tons of medicinal plants that are transported illegally in that same period. # Air pollution and COVID-19: a bad combination Currently 9 out of 10 peoplethey breathe air that the World Health Organization (WHO) considers too polluted, causing some 4.2 million premature deaths by year. A Harvard studyprovided the first clear evidence that exposure to air pollution by particles over many years can strongly influence the chances of living or dying from COVID-19. "For every small increase in air pollution during that time, there is a substantial increase in death from COVID-19," said Aaron Bernstein, pediatrician and director of the Harvard Center for Climate, Health and the Global Environment. , at a press conference on April 16. He spoke virtually alongside WHO's Dr. Maria Neira and former head of the United States Environmental Protection Agency (EPA), Gina McCarthy, and the full video can be viewed here.. Before the current health crisis, scientists had already shown that polluted air was associated with increased mortality during the 2002 SARS pandemic, another strain of coronavirus.that killed 774 people and infected 8890. According to research published in 2003, a person living in an area with high air pollution was more than twice as likely to die from SARS. In Chinese cities with high or moderate air pollution, the mortality rate was 8.9% and 7.49%, respectively, compared to areas with low air pollution, where it was 4%. # How to wash your hands without water? Another environmental factor that exacerbates the risk of contracting COVID-19 is the lack of access to water and sanitation. It is clear: those who do not have water to wash their hands and maintain personal hygiene will have difficulty practicing even basic safety measures. Today, around 3 billion peoplethey lack access to basic hand washing facilities. In the least developed countries, 22% of the sanitary facilities do not have water service, 21% do not have a sanitation service and 22% do not have waste management. Ensuring access to water means facing a series of interconnected environmental challenges. No less important is the climate crisis, a multiplier of threatswhich will increase the pressure on water resources, as higher temperatures cause an increase in both droughtslike desertificationand threaten mountain glaciers, which are important freshwater reservoirs. # Worse with the climate crisis Beyond the current crisis, the climate emergency is also causing new risksof infectious diseases, changing its geographical distribution and seasonal behavior. As a result, our ability to predict and prepare for new outbreaks of infectious disease can be weakened. And the impact will not be the same for everyone. "The sudden consequences of climate change disproportionately affect people with fewer resources, increasing their vulnerability and amplifying the possibilities of spreading zoonotic diseases," explains UN Environment. Infectious diseases spread in many ways. Two of the mainthey are human-to-human, such as COVID-19, and vector vectors transmitted by other animals, such as mosquitoes and fleas. By changing weather patterns and extreme events, the climate crisis will have an impact on vector diseases, altering the population, scope and survival of the animals that carry them. Among the most important examples are dengue and malaria, both spread by mosquitoes. Both diseases are enormous public health challenges and their transmission is affected by climatic factors.such as temperature, humidity and rainfall. Malaria killed more than 400,000 peoplein 2018, 67% were children under 5 years old. About half the world's population is at risk of contracting dengue fever and an estimated 100 to 400 million infections occur each year. Although the symptoms are mostly mild, it can develop into a serious illness, requiring medical attention, and is one of the leading causes of hospitalization and death. in Asia and Latin America. The climate crisis is making conditions more favorable for both. Data dating back to the 1950s shows that 9 of the 10 best years for dengue transmission occurred since 2000. For its part, the climate suitability for malaria transmission in the highlands of Africa for 2012-2107 was 30% higherthan the baseline of the 1950s. Future climate change may make conditions even more suitable for malaria, although such predictions are difficult to make given the complexity. For human-transmitted diseases, such as COVID-19, the picture is more complicated. The climate crisis does not change the geographic reach of humans as clearly as it does for animals and insects. However, it could play a role in some cases. In the United States, for example, research suggests that warmer winters caused by the climate emergency could bring more severe flu seasons, with the risk that warmer weather could promote sustained transmission, leading to the flu "season" last all year. By adopting the Paris Agreementin 2015, almost 200 countries pledged to prevent the global mean temperature from rising above 2 ° C from pre-industrial levels by the end of the century and to do everything possible to limit this warming to 1.5 ° C. To do this, greenhouse gas (GHG) emissions from human activity should be reduced 45% from what they were in 2010 before 2030. Today, we are not on track to achieve this goal, but there is still a window of time (increasingly shorter) to do it. Today, mitigating and adapting to the climate emergency is more urgent than ever; not only because of the risks of future pandemics that greater global warming could bring, but also because measures to prevent it also contribute to avoiding new health crises. Reducing air pollution by cutting fossil fuels is a tool to improve public health. Phasing out fossil fuels could prevent 3.6 million premature deaths each year only for outdoor air pollution and 5.6 million if it includes pollution from agriculture and households. This, in turn, can also bring secondary benefits to the water supply. For example, in 2013, the total water consumption of the coal industry was estimated at 22.7 billion cubic meters meters per year, enough to satisfy the most basic water needs of 1.2 billion people. The transition to energy based on sources that need little water, such as wind and solar, as well as the application of energy efficiency, would help limit the climate crisis and water stress . # The day after tomorrow: economic stimuli The effects of COVID-19 are felt beyond our health and economy. In recent months, with the brake on the unprecedented activity that social distancing measures brought as a consequence, we have witnessed (virtual or face-to-face) cleaner airs and waters, more celestial skies and the return of species, among others. Wonderfully resilient nature shows us that there is not much we need to do (or not do) for it to flourish again. However, when the crisis subsides, the imperative will be economic recovery. The question is what it will be like, where governments and industries will focus, and if we will learn anything from this pandemic and focus on building a healthier and more equitable world for all. Because, as David Quammen pointed out, if we want to avoid future coronaviruses, we must radically change our production and consumption patterns to reduce our interference from the natural world. As he recently told the BBC"There is no 'natural world', it is a false and artificial phrase. There is a world, and human beings are part of it along with viruses and chimpanzees and bats." One of the keys here is the multi-million dollar economic stimulus packages that governments are deploying in the face of the pandemic and those that they will deploy to restart the economies of each country and region as the worst of this happens. In a joint statement, the leaders of the European Union pointed out that, after the immediate response to the health crisis and jobs in danger, the economic recovery will take place in accordance with the objectives of its Green Pact, whose objective is to conserve, protect and improve community "natural capital" and protect the health and well-being of its citizens from environmental risks and impacts, all in a "fair and inclusive" spirit. Among the voices supporting cleaner, non-fossil stimulus packages, focused on climate and biodiversity, are the World Bank. and Michael Liebreich of Bloomberg New Energy Finance. The Secretary General of the United Nations, António Guterres, is also making his voice heardin this aspect. On Earth Day, ask to unite to demand a healthy and resilient future for people and the planet. Since last year I had made a call to end the $5.2 trillion in fossil fuel subsidies, that new coal plants are not built and that financial markets go from dirty to clean. And you are not alone. The Secretary-General of the Organization for Economic Cooperation and Development(OECD) Ángel Gurría told the BBCthat "the most important intergenerational responsibility is to preserve the planet" and urged governments to focus on "other struggles that we have to wage". In turn, in a column for The Economist, former Bank of England Governor Mark Carney said the climate remains "the big test" for leaders, asking them, after COVID-19, to focus on delivering "a planet fit for our grandchildren to live in. " Even the G20 Finance Ministers' meeting, which took place last week, focused on the same issue in a joint statement..pdf). In line with stimuli to mitigate not only the economic crisis but also the post-COVID-19 climate crisis, the International Renewable Energy Agency (Irena) published a new report showing that the decarbonisation of the energy system it benefits short-term recovery, while creating resilient and inclusive economies and societies. ( Here the full report). Although the path to deeper decarbonization requires a total energy investment of up to$ 130 billion, the socioeconomic benefits of such investment would be considerable on issues such as employment and economic growth, the report reveals. The world certainly has a chance to get out of COVID-19 in a better way how it treated the planet in pre-pandemic times. If you have any queries feel free to contact us at: Sarkari Exam Result	{"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": 0.15355226397514343, "perplexity": 2938.5494638231526}, "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/1610703509973.34/warc/CC-MAIN-20210117051021-20210117081021-00719.warc.gz"}
http://mathhelpforum.com/algebra/166026-do-we-just-multiply-everything.html	Thread: Do we just multiply everything? 1. Do we just multiply everything? Picture, because i don't remember doing something like this in my life. The set of 2 by 2 matrices forms a vector space with scalar multiplication defined by $u\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix} ua & ub \\ uc & ud\end{bmatrix}$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9959704279899597, "perplexity": 195.20674438053862}, "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-2016-50/segments/1480698544672.33/warc/CC-MAIN-20161202170904-00243-ip-10-31-129-80.ec2.internal.warc.gz"}
http://en.wikipedia.org/wiki/Capacitor	# Capacitor "Capacitive" redirects here. For the term used when referring to touchscreens, see capacitive sensing. Capacitor Miniature low-voltage capacitors (next to a cm ruler) Electronic symbol 4 electrolytic capacitors of different voltages and capacitance A capacitor (originally known as a condenser) is a passive two-terminal electrical component used to store energy electrostatically in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors (plates) separated by a dielectric (i.e., insulator). The conductors can be thin films of metal, aluminum foil or disks, etc. The "nonconducting" dielectric acts to increase the capacitor's charge capacity. A dielectric can be glass, ceramic, plastic film, air, paper, mica, etc. Capacitors are widely used as parts of electrical circuits in many common electrical devices. Unlike a resistor, an ideal capacitor does not dissipate energy. Instead, a capacitor stores energy in the form of an electrostatic field between its plates. When there is a potential difference across the conductors (e.g., when a capacitor is attached across a battery), an electric field develops across the dielectric, causing positive charge (+Q) to collect on one plate and negative charge (−Q) to collect on the other plate. If a battery has been attached to a capacitor for a sufficient amount of time, no current can flow through the capacitor. However, if an accelerating or alternating voltage is applied across the leads of the capacitor, a displacement current can flow. An ideal capacitor is characterized by a single constant value for its capacitance. Capacitance is expressed as the ratio of the electric charge (Q) on each conductor to the potential difference (V) between them. The SI unit of capacitance is the farad (F), which is equal to one coulomb per volt (1 C/V). Typical capacitance values range from about 1 pF (10−12 F) to about 1 mF (10−3 F). The capacitance is greater when there is a narrower separation between conductors and when the conductors have a larger surface area. In practice, the dielectric between the plates passes a small amount of leakage current and also has an electric field strength limit, known as the breakdown voltage. The conductors and leads introduce an undesired inductance and resistance. Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems, they stabilize voltage and power flow.[1] ## History In October 1745, Ewald Georg von Kleist of Pomerania, Germany, found that charge could be stored by connecting a high-voltage electrostatic generator by a wire to a volume of water in a hand-held glass jar.[2] Von Kleist's hand and the water acted as conductors, and the jar as a dielectric (although details of the mechanism were incorrectly identified at the time). Von Kleist found that touching the wire resulted in a powerful spark, much more painful than that obtained from an electrostatic machine. The following year, the Dutch physicist Pieter van Musschenbroek invented a similar capacitor, which was named the Leyden jar, after the University of Leiden where he worked.[3] He also was impressed by the power of the shock he received, writing, "I would not take a second shock for the kingdom of France."[4] Daniel Gralath was the first to combine several jars in parallel into a "battery" to increase the charge storage capacity. Benjamin Franklin investigated the Leyden jar and came to the conclusion that the charge was stored on the glass, not in the water as others had assumed. He also adopted the term "battery",[5][6] (denoting the increasing of power with a row of similar units as in a battery of cannon), subsequently applied to clusters of electrochemical cells.[7] Leyden jars were later made by coating the inside and outside of jars with metal foil, leaving a space at the mouth to prevent arcing between the foils.[citation needed] The earliest unit of capacitance was the jar, equivalent to about 1.11 nanofarads.[8] Leyden jars or more powerful devices employing flat glass plates alternating with foil conductors were used exclusively up until about 1900, when the invention of wireless (radio) created a demand for standard capacitors, and the steady move to higher frequencies required capacitors with lower inductance. More compact construction methods began to be used, such as a flexible dielectric sheet (like oiled paper) sandwiched between sheets of metal foil, rolled or folded into a small package. Early capacitors were also known as condensers, a term that is still occasionally used today, particularly in high power applications, like automotive systems. The term was first used for this purpose by Alessandro Volta in 1782, with reference to the device's ability to store a higher density of electric charge than a normal isolated conductor.[9] ## Theory of operation Main article: Capacitance ### Overview Charge separation in a parallel-plate capacitor causes an internal electric field. A dielectric (orange) reduces the field and increases the capacitance. A simple demonstration of a parallel-plate capacitor A capacitor consists of two conductors separated by a non-conductive region.[10] The non-conductive region is called the dielectric. In simpler terms, the dielectric is just an electrical insulator. Examples of dielectric media are glass, air, paper, vacuum, and even a semiconductor depletion region chemically identical to the conductors. A capacitor is assumed to be self-contained and isolated, with no net electric charge and no influence from any external electric field. The conductors thus hold equal and opposite charges on their facing surfaces,[11] and the dielectric develops an electric field. In SI units, a capacitance of one farad means that one coulomb of charge on each conductor causes a voltage of one volt across the device.[12] An ideal capacitor is wholly characterized by a constant capacitance C, defined as the ratio of charge ±Q on each conductor to the voltage V between them:[10] $C= \frac{Q}{V}$ Because the conductors (or plates) are close together, the opposite charges on the conductors attract one another due to their electric fields, allowing the capacitor to store more charge for a given voltage than if the conductors were separated, giving the capacitor a large capacitance. Sometimes charge build-up affects the capacitor mechanically, causing its capacitance to vary. In this case, capacitance is defined in terms of incremental changes: $C= \frac{\mathrm{d}Q}{\mathrm{d}V}$ ### Hydraulic analogy In the hydraulic analogy, a capacitor is analogous to a rubber membrane sealed inside a pipe. This animation illustrates a membrane being repeatedly stretched and un-stretched by the flow of water, which is analogous to a capacitor being repeatedly charged and discharged by the flow of charge. In the hydraulic analogy, charge carriers flowing through a wire are analogous to water flowing through a pipe. A capacitor is like a rubber membrane sealed inside a pipe. Water molecules cannot pass through the membrane, but some water can move by stretching the membrane. The analogy clarifies a few aspects of capacitors: • The current alters the charge on a capacitor, just as the flow of water changes the position of the membrane. More specifically, the effect of an electric current is to increase the charge of one plate of the capacitor, and decrease the charge of the other plate by an equal amount. This is just as when water flow moves the rubber membrane, it increases the amount of water on one side of the membrane, and decreases the amount of water on the other side. • The more a capacitor is charged, the larger its voltage drop; i.e., the more it "pushes back" against the charging current. This is analogous to the fact that the more a membrane is stretched, the more it pushes back on the water. • Charge can flow "through" a capacitor even though no individual electron can get from one side to the other. This is analogous to the fact that water can flow through the pipe even though no water molecule can pass through the rubber membrane. Of course, the flow cannot continue in the same direction forever; the capacitor will experience dielectric breakdown, and analogously the membrane will eventually break. • The capacitance describes how much charge can be stored on one plate of a capacitor for a given "push" (voltage drop). A very stretchy, flexible membrane corresponds to a higher capacitance than a stiff membrane. • A charged-up capacitor is storing potential energy, analogously to a stretched membrane. ### Energy of electric field Work must be done by an external influence to "move" charge between the conductors in a capacitor. When the external influence is removed, the charge separation persists in the electric field and energy is stored to be released when the charge is allowed to return to its equilibrium position. The work done in establishing the electric field, and hence the amount of energy stored, is[13] $W = \int_0^Q V \mathrm{d}q = \int_0^Q \frac{q}{C} \mathrm{d}q = {1 \over 2} {Q^2 \over C} = {1 \over 2} C V^2 = {1 \over 2} VQ$ Here Q is the charge stored in the capacitor, V is the voltage across the capacitor, and C is the capacitance. In the case of a fluctuating voltage V(t), the stored energy also fluctuates and hence power must flow into or out of the capacitor. This power can be found by taking the time derivative of the stored energy: $P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{2} CV^2\right) = C V(t) \frac{\mathrm{d}V}{\mathrm{d}t}$ ### Current–voltage relation The current I(t) through any component in an electric circuit is defined as the rate of flow of a charge Q(t) passing through it, but actual charges—electrons—cannot pass through the dielectric layer of a capacitor. Rather, one electron accumulates on the negative plate for each one that leaves the positive plate, resulting in an electron depletion and consequent positive charge on one electrode that is equal and opposite to the accumulated negative charge on the other. Thus the charge on the electrodes is equal to the integral of the current as well as proportional to the voltage, as discussed above. As with any antiderivative, a constant of integration is added to represent the initial voltage V(t0). This is the integral form of the capacitor equation:[14] $V(t) = \frac{Q(t)}{C} = \frac{1}{C}\int_{t_0}^t I(\tau) \mathrm{d}\tau + V(t_0)$ Taking the derivative of this and multiplying by C yields the derivative form:[15] $I(t) = \frac{\mathrm{d}Q(t)}{\mathrm{d}t} = C\frac{\mathrm{d}V(t)}{\mathrm{d}t}$ The dual of the capacitor is the inductor, which stores energy in a magnetic field rather than an electric field. Its current-voltage relation is obtained by exchanging current and voltage in the capacitor equations and replacing C with the inductance L. ### DC circuits A simple resistor-capacitor circuit demonstrates charging of a capacitor. A series circuit containing only a resistor, a capacitor, a switch and a constant DC source of voltage V0 is known as a charging circuit.[16] If the capacitor is initially uncharged while the switch is open, and the switch is closed at t0, it follows from Kirchhoff's voltage law that $V_0 = v_\text{resistor}(t) + v_\text{capacitor}(t) = i(t)R + \frac{1}{C}\int_{t_0}^t i(\tau) \mathrm{d}\tau$ Taking the derivative and multiplying by C, gives a first-order differential equation: $RC\frac{\mathrm{d}i(t)}{\mathrm{d}t} + i(t) = 0$ At t = 0, the voltage across the capacitor is zero and the voltage across the resistor is V0. The initial current is then I(0) =V0/R. With this assumption, solving the differential equation yields \begin{align} I(t) &= \frac{V_0}{R} e^{-\frac{t}{\tau_0}} \\ V(t) &= V_0 \left( 1 - e^{-\frac{t}{\tau_0}}\right) \end{align} where τ0 = RC is the time constant of the system. As the capacitor reaches equilibrium with the source voltage, the voltages across the resistor and the current through the entire circuit decay exponentially. The case of discharging a charged capacitor likewise demonstrates exponential decay, but with the initial capacitor voltage replacing V0 and the final voltage being zero. ### AC circuits Impedance, the vector sum of reactance and resistance, describes the phase difference and the ratio of amplitudes between sinusoidally varying voltage and sinusoidally varying current at a given frequency. Fourier analysis allows any signal to be constructed from a spectrum of frequencies, whence the circuit's reaction to the various frequencies may be found. The reactance and impedance of a capacitor are respectively \begin{align} X &= -\frac{1}{\omega C} = -\frac{1}{2\pi f C} \\ Z &= \frac{1}{j\omega C} = -\frac{j}{\omega C} = -\frac{j}{2\pi f C} \end{align} where j is the imaginary unit and ω is the angular frequency of the sinusoidal signal. The −j phase indicates that the AC voltage V = ZI lags the AC current by 90°: the positive current phase corresponds to increasing voltage as the capacitor charges; zero current corresponds to instantaneous constant voltage, etc. Impedance decreases with increasing capacitance and increasing frequency. This implies that a higher-frequency signal or a larger capacitor results in a lower voltage amplitude per current amplitude—an AC "short circuit" or AC coupling. Conversely, for very low frequencies, the reactance will be high, so that a capacitor is nearly an open circuit in AC analysis—those frequencies have been "filtered out". Capacitors are different from resistors and inductors in that the impedance is inversely proportional to the defining characteristic; i.e., capacitance. A capacitor connected to a sinusoidal voltage source will cause a displacement current to flow through it. In the case that the voltage source is V0cos(ωt), the displacement current can be expressed as: $I = C \frac{dV}{dt} = -\omega {C}{V_\text{0}}\sin(\omega t)$ At sin(ωt) = -1, the capacitor has a maximum (or peak) current whereby I0 = ωCV0. The ratio of peak voltage to peak current is due to capacitive reactance (denoted XC). $X_C = \frac{V_\text{0}}{I_\text{0}} = \frac{V_\text{0}}{\omega C V_\text{0}} = \frac{1}{\omega C}$ XC approaches zero as ω approaches infinity. If XC approaches 0, the capacitor resembles a short wire that strongly passes current at high frequencies. XC approaches infinity as ω approaches zero. If XC approaches infinity, the capacitor resembles an open circuit that poorly passes low frequencies. The current of the capacitor may be expressed in the form of cosines to better compare with the voltage of the source: $I = - {I_\text{0}}{\sin({\omega t}}) = {I_\text{0}}{\cos({\omega t} + {90^\circ})}$ In this situation, the current is out of phase with the voltage by +π/2 radians or +90 degrees (i.e., the current will lead the voltage by 90°). ### Laplace circuit analysis (s-domain) When using the Laplace transform in circuit analysis, the impedance of an ideal capacitor with no initial charge is represented in the s domain by: $Z(s) = \frac{1}{sC}$ where • C is the capacitance, and • s is the complex frequency. ### Parallel-plate model Dielectric is placed between two conducting plates, each of area A and with a separation of d The simplest capacitor consists of two parallel conductive plates separated by a dielectric (such as air) with permittivity ε . The model may also be used to make qualitative predictions for other device geometries. The plates are considered to extend uniformly over an area A and a charge density ±ρ = ±Q/A exists on their surface. Assuming that the width of the plates is much greater than their separation d, the electric field near the centre of the device will be uniform with the magnitude E = ρ/ε. The voltage is defined as the line integral of the electric field between the plates $V= \int_0^d E\,\mathrm{d}z = \int_0^d \frac{\rho}{\varepsilon}\,\mathrm{d}z = \frac{\rho d}{\varepsilon} = \frac{Qd}{\varepsilon A}$ Solving this for C = Q/V reveals that capacitance increases with area of the plates, and decreases as separation between plates increases. $C = \frac{\varepsilon A}{d}$ The capacitance is therefore greatest in devices made from materials with a high permittivity, large plate area, and small distance between plates. A parallel plate capacitor can only store a finite amount of energy before dielectric breakdown occurs. The capacitor's dielectric material has a dielectric strength Ud which sets the capacitor's breakdown voltage at V = Vbd = Udd. The maximum energy that the capacitor can store is therefore $E = \frac{1}{2}CV^2=\frac{1}{2} \frac{\varepsilon A}{d} (U_d d)^2 = \frac{1}{2} \varepsilon A d U_d^2$ We see that the maximum energy is a function of dielectric volume, permittivity, and dielectric strength per distance. So increasing the plate area while decreasing the separation between the plates while maintaining the same volume has no change on the amount of energy the capacitor can store. Care must be taken when increasing the plate separation so that the above assumption of the distance between plates being much smaller than the area of the plates is still valid for these equations to be accurate. In addition, these equations assume that the electric field is entirely concentrated in the dielectric between the plates. In reality there are fringing fields outside the dielectric, for example between the sides of the capacitor plates, which will increase the effective capacitance of the capacitor. This could be seen as a form of parasitic capacitance. For some simple capacitor geometries this additional capacitance term can be calculated analytically.[17] It becomes negligibly small when the ratio of plate area to separation is large. Several capacitors in parallel ### Networks For capacitors in parallel Capacitors in a parallel configuration each have the same applied voltage. Their capacitances add up. Charge is apportioned among them by size. Using the schematic diagram to visualize parallel plates, it is apparent that each capacitor contributes to the total surface area. $C_\mathrm{eq}= C_1 + C_2 + \cdots + C_n$ For capacitors in series Several capacitors in series Connected in series, the schematic diagram reveals that the separation distance, not the plate area, adds up. The capacitors each store instantaneous charge build-up equal to that of every other capacitor in the series. The total voltage difference from end to end is apportioned to each capacitor according to the inverse of its capacitance. The entire series acts as a capacitor smaller than any of its components. $\frac{1}{C_\mathrm{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n}$ Capacitors are combined in series to achieve a higher working voltage, for example for smoothing a high voltage power supply. The voltage ratings, which are based on plate separation, add up, if capacitance and leakage currents for each capacitor are identical. In such an application, on occasion, series strings are connected in parallel, forming a matrix. The goal is to maximize the energy storage of the network without overloading any capacitor. For high-energy storage with capacitors in series, some safety considerations must be applied to ensure one capacitor failing and leaking current will not apply too much voltage to the other series capacitors. Series connection is also sometimes used to adapt polarized electrolytic capacitors for bipolar AC use. See electrolytic capacitor#Designing for reverse bias. Voltage distribution in parallel-to-series networks. To model the distribution of voltages from a single charged capacitor $\left( A \right)$ connected in parallel to a chain of capacitors in series $\left( B_\text{n} \right)$ : \begin{align} (volts) A_\mathrm{eq} &= A\left(1 - \frac{1}{n + 1}\right) \\ (volts) B_\text{1..n} &= \frac{A}{n} \left(1 - \frac{1}{n + 1}\right) \\ A - B &= 0 \end{align} Note: This is only correct if all capacitance values are equal. The power transferred in this arrangement is: $P = \frac{1}{R} \cdot \frac{1}{n + 1} A_\text{volts} \left( A_\text{farads} + B_\text{farads} \right)$ ## Non-ideal behavior Capacitors deviate from the ideal capacitor equation in a number of ways. Some of these, such as leakage current and parasitic effects are linear, or can be assumed to be linear, and can be dealt with by adding virtual components to the equivalent circuit of the capacitor. The usual methods of network analysis can then be applied. In other cases, such as with breakdown voltage, the effect is non-linear and normal (i.e., linear) network analysis cannot be used, the effect must be dealt with separately. There is yet another group, which may be linear but invalidate the assumption in the analysis that capacitance is a constant. Such an example is temperature dependence. Finally, combined parasitic effects such as inherent inductance, resistance, or dielectric losses can exhibit non-uniform behavior at variable frequencies of operation. ### Breakdown voltage Main article: Breakdown voltage Above a particular electric field, known as the dielectric strength Eds, the dielectric in a capacitor becomes conductive. The voltage at which this occurs is called the breakdown voltage of the device, and is given by the product of the dielectric strength and the separation between the conductors,[18] $V_{\text{bd}}= E_{\text{ds}} d$ The maximum energy that can be stored safely in a capacitor is limited by the breakdown voltage. Due to the scaling of capacitance and breakdown voltage with dielectric thickness, all capacitors made with a particular dielectric have approximately equal maximum energy density, to the extent that the dielectric dominates their volume.[19] For air dielectric capacitors the breakdown field strength is of the order 2 to 5 MV/m; for mica the breakdown is 100 to 300 MV/m; for oil, 15 to 25 MV/m; it can be much less when other materials are used for the dielectric.[20] The dielectric is used in very thin layers and so absolute breakdown voltage of capacitors is limited. Typical ratings for capacitors used for general electronics applications range from a few volts to 1 kV. As the voltage increases, the dielectric must be thicker, making high-voltage capacitors larger per capacitance than those rated for lower voltages. The breakdown voltage is critically affected by factors such as the geometry of the capacitor conductive parts; sharp edges or points increase the electric field strength at that point and can lead to a local breakdown. Once this starts to happen, the breakdown quickly tracks through the dielectric until it reaches the opposite plate, leaving carbon behind and causing a short (or relatively low resistance) circuit. The results can be explosive as the short in the capacitor draws current from the surrounding circuitry and dissipates the energy.[21] The usual breakdown route is that the field strength becomes large enough to pull electrons in the dielectric from their atoms thus causing conduction. Other scenarios are possible, such as impurities in the dielectric, and, if the dielectric is of a crystalline nature, imperfections in the crystal structure can result in an avalanche breakdown as seen in semi-conductor devices. Breakdown voltage is also affected by pressure, humidity and temperature.[22] ### Equivalent circuit Two different circuit models of a real capacitor An ideal capacitor only stores and releases electrical energy, without dissipating any. In reality, all capacitors have imperfections within the capacitor's material that create resistance. This is specified as the equivalent series resistance or ESR of a component. This adds a real component to the impedance: $R_\text{C}= Z + R_\text{ESR} = \frac{1}{j\omega C} + R_\text{ESR}$ As frequency approaches infinity, the capacitive impedance (or reactance) approaches zero and the ESR becomes significant. As the reactance becomes negligible, power dissipation approaches PRMS = VRMS² /RESR. Similarly to ESR, the capacitor's leads add equivalent series inductance or ESL to the component. This is usually significant only at relatively high frequencies. As inductive reactance is positive and increases with frequency, above a certain frequency capacitance will be canceled by inductance. High-frequency engineering involves accounting for the inductance of all connections and components. If the conductors are separated by a material with a small conductivity rather than a perfect dielectric, then a small leakage current flows directly between them. The capacitor therefore has a finite parallel resistance,[12] and slowly discharges over time (time may vary greatly depending on the capacitor material and quality). ### Q factor The quality factor (or Q) of a capacitor is the ratio of its reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the capacitor, the closer it approaches the behavior of an ideal, lossless, capacitor. The Q factor of a capacitor can be found through the following formula: $Q = \frac{X_C}{R_C}=\frac{1}{\omega C R_C}$ Where: • $\omega$ is frequency in radians per second, • $C$ is the capacitance, • $X_C$ is the capacitive reactance, and • $R_C$ is the series resistance of the capacitor. ### Ripple current Ripple current is the AC component of an applied source (often a switched-mode power supply) whose frequency may be constant or varying. Ripple current causes heat to be generated within the capacitor due to the dielectric losses caused by the changing field strength together with the current flow across the slightly resistive supply lines or the electrolyte in the capacitor. The equivalent series resistance (ESR) is the amount of internal series resistance one would add to a perfect capacitor to model this. Some types of capacitors, primarily tantalum and aluminum electrolytic capacitors, as well as some film capacitors have a specified rating value for maximum ripple current. • Tantalum electrolytic capacitors with solid manganese dioxide electrolyte are limited by ripple current and generally have the highest ESR ratings in the capacitor family. Exceeding their ripple limits can lead to shorts and burning parts. • Aluminum electrolytic capacitors, the most common type of electrolytic, suffer a shortening of life expectancy at higher ripple currents. If ripple current exceeds the rated value of the capacitor, it tends to result in explosive failure. • Ceramic capacitors generally have no ripple current limitation and have some of the lowest ESR ratings. • Film capacitors have very low ESR ratings but exceeding rated ripple current may cause degradation failures. ### Capacitance instability The capacitance of certain capacitors decreases as the component ages. In ceramic capacitors, this is caused by degradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the most significant aging factors, while the operating voltage has a smaller effect. The aging process may be reversed by heating the component above the Curie point. Aging is fastest near the beginning of life of the component, and the device stabilizes over time.[23] Electrolytic capacitors age as the electrolyte evaporates. In contrast with ceramic capacitors, this occurs towards the end of life of the component. Temperature dependence of capacitance is usually expressed in parts per million (ppm) per °C. It can usually be taken as a broadly linear function but can be noticeably non-linear at the temperature extremes. The temperature coefficient can be either positive or negative, sometimes even amongst different samples of the same type. In other words, the spread in the range of temperature coefficients can encompass zero. See the data sheet in the leakage current section above for an example. Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound waves resulting in a microphonic effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing AC current. Some dielectrics also generate piezoelectricity. The resulting interference is especially problematic in audio applications, potentially causing feedback or unintended recording. In the reverse microphonic effect, the varying electric field between the capacitor plates exerts a physical force, moving them as a speaker. This can generate audible sound, but drains energy and stresses the dielectric and the electrolyte, if any. ### Current and voltage reversal Current reversal occurs when the current changes direction. Voltage reversal is the change of polarity in a circuit. Reversal is generally described as the percentage of the maximum rated voltage that reverses polarity. In DC circuits, this will usually be less than 100% (often in the range of 0 to 90%), whereas AC circuits experience 100% reversal. In DC circuits and pulsed circuits, current and voltage reversal are affected by the damping of the system. Voltage reversal is encountered in RLC circuits that are under-damped. The current and voltage reverse direction, forming a harmonic oscillator between the inductance and capacitance. The current and voltage will tend to oscillate and may reverse direction several times, with each peak being lower than the previous, until the system reaches an equilibrium. This is often referred to as ringing. In comparison, critically damped or over-damped systems usually do not experience a voltage reversal. Reversal is also encountered in AC circuits, where the peak current will be equal in each direction. For maximum life, capacitors usually need to be able to handle the maximum amount of reversal that a system will experience. An AC circuit will experience 100% voltage reversal, while under-damped DC circuits will experience less than 100%. Reversal creates excess electric fields in the dielectric, causes excess heating of both the dielectric and the conductors, and can dramatically shorten the life expectancy of the capacitor. Reversal ratings will often affect the design considerations for the capacitor, from the choice of dielectric materials and voltage ratings to the types of internal connections used.[24] ### Dielectric absorption Capacitors made with some types of dielectric material show "dielectric absorption" or "soakage". On discharging a capacitor and disconnecting it, after a short time it may develop a voltage due to hysteresis in the dielectric. This effect can be objectionable in applications such as precision sample and hold circuits. ### Leakage Leakage is equivalent to a resistor in parallel with the capacitor. Constant exposure to heat can cause dielectric breakdown and excessive leakage, a problem often seen in older vacuum tube circuits, particularly where oiled paper and foil capacitors were used. In many vacuum tube circuits, interstage coupling capacitors are used to conduct a varying signal from the plate of one tube to the grid circuit of the next stage. A leaky capacitor can cause the grid circuit voltage to be raised from its normal bias setting, causing excessive current or signal distortion in the downstream tube. In power amplifiers this can cause the plates to glow red, or current limiting resistors to overheat, even fail. Similar considerations apply to component fabricated solid-state (transistor) amplifiers, but owing to lower heat production and the use of modern polyester dielectric barriers this once-common problem has become relatively rare. ### Electrolytic failure from disuse Electrolytic capacitors are conditioned when manufactured by applying a voltage sufficient to initiate the proper internal chemical state. This state is maintained by regular use of the equipment. If a system using electrolytic capacitors is unused for a long period of time it can lose its conditioning, and will generally fail with a short circuit when next operated, permanently damaging the capacitor. To prevent this in tube equipment, the voltage can be slowly brought up using a variable transformer (variac) on the mains, over a twenty or thirty minute interval. Transistor equipment is more problematic as such equipment may be sensitive to low voltage ("brownout") conditions, with excessive currents due to improper bias in some circuits.[citation needed] ## Capacitor types Main article: Types of capacitor Practical capacitors are available commercially in many different forms. The type of internal dielectric, the structure of the plates and the device packaging all strongly affect the characteristics of the capacitor, and its applications. Values available range from very low (picofarad range; while arbitrarily low values are in principle possible, stray (parasitic) capacitance in any circuit is the limiting factor) to about 5 kF supercapacitors. Above approximately 1 microfarad electrolytic capacitors are usually used because of their small size and low cost compared with other technologies, unless their relatively poor stability, life and polarised nature make them unsuitable. Very high capacity supercapacitors use a porous carbon-based electrode material. ### Dielectric materials Capacitor materials. From left: multilayer ceramic, ceramic disc, multilayer polyester film, tubular ceramic, polystyrene, metalized polyester film, aluminum electrolytic. Major scale divisions are in centimetres. Most types of capacitor include a dielectric spacer, which increases their capacitance. These dielectrics are most often insulators. However, low capacitance devices are available with a vacuum between their plates, which allows extremely high voltage operation and low losses. Variable capacitors with their plates open to the atmosphere were commonly used in radio tuning circuits. Later designs use polymer foil dielectric between the moving and stationary plates, with no significant air space between them. In order to maximise the charge that a capacitor can hold, the dielectric material needs to have as high a permittivity as possible, while also having as high a breakdown voltage as possible. Several solid dielectrics are available, including paper, plastic, glass, mica and ceramic materials. Paper was used extensively in older devices and offers relatively high voltage performance. However, it is susceptible to water absorption, and has been largely replaced by plastic film capacitors. Plastics offer better stability and aging performance, which makes them useful in timer circuits, although they may be limited to low operating temperatures and frequencies. Ceramic capacitors are generally small, cheap and useful for high frequency applications, although their capacitance varies strongly with voltage and they age poorly. They are broadly categorized as class 1 dielectrics, which have predictable variation of capacitance with temperature or class 2 dielectrics, which can operate at higher voltage. Glass and mica capacitors are extremely reliable, stable and tolerant to high temperatures and voltages, but are too expensive for most mainstream applications. Electrolytic capacitors and supercapacitors are used to store small and larger amounts of energy, respectively, ceramic capacitors are often used in resonators, and parasitic capacitance occurs in circuits wherever the simple conductor-insulator-conductor structure is formed unintentionally by the configuration of the circuit layout. Electrolytic capacitors use an aluminum or tantalum plate with an oxide dielectric layer. The second electrode is a liquid electrolyte, connected to the circuit by another foil plate. Electrolytic capacitors offer very high capacitance but suffer from poor tolerances, high instability, gradual loss of capacitance especially when subjected to heat, and high leakage current. Poor quality capacitors may leak electrolyte, which is harmful to printed circuit boards. The conductivity of the electrolyte drops at low temperatures, which increases equivalent series resistance. While widely used for power-supply conditioning, poor high-frequency characteristics make them unsuitable for many applications. Electrolytic capacitors will self-degrade if unused for a period (around a year), and when full power is applied may short circuit, permanently damaging the capacitor and usually blowing a fuse or causing failure of rectifier diodes (for instance, in older equipment, arcing in rectifier tubes). They can be restored before use (and damage) by gradually applying the operating voltage, often done on antique vacuum tube equipment over a period of 30 minutes by using a variable transformer to supply AC power. Unfortunately, the use of this technique may be less satisfactory for some solid state equipment, which may be damaged by operation below its normal power range, requiring that the power supply first be isolated from the consuming circuits. Such remedies may not be applicable to modern high-frequency power supplies as these produce full output voltage even with reduced input. Tantalum capacitors offer better frequency and temperature characteristics than aluminum, but higher dielectric absorption and leakage.[25] Polymer capacitors (OS-CON, OC-CON, KO, AO) use solid conductive polymer (or polymerized organic semiconductor) as electrolyte and offer longer life and lower ESR at higher cost than standard electrolytic capacitors. A Feedthrough is a component that, while not serving as its main use, has capacitance and is used to conduct signals through a circuit board. Several other types of capacitor are available for specialist applications. Supercapacitors store large amounts of energy. Supercapacitors made from carbon aerogel, carbon nanotubes, or highly porous electrode materials, offer extremely high capacitance (up to 5 kF as of 2010) and can be used in some applications instead of rechargeable batteries. Alternating current capacitors are specifically designed to work on line (mains) voltage AC power circuits. They are commonly used in electric motor circuits and are often designed to handle large currents, so they tend to be physically large. They are usually ruggedly packaged, often in metal cases that can be easily grounded/earthed. They also are designed with direct current breakdown voltages of at least five times the maximum AC voltage. ### Structure Capacitor packages: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. Major scale divisions are cm. The arrangement of plates and dielectric has many variations depending on the desired ratings of the capacitor. For small values of capacitance (microfarads and less), ceramic disks use metallic coatings, with wire leads bonded to the coating. Larger values can be made by multiple stacks of plates and disks. Larger value capacitors usually use a metal foil or metal film layer deposited on the surface of a dielectric film to make the plates, and a dielectric film of impregnated paper or plastic – these are rolled up to save space. To reduce the series resistance and inductance for long plates, the plates and dielectric are staggered so that connection is made at the common edge of the rolled-up plates, not at the ends of the foil or metalized film strips that comprise the plates. The assembly is encased to prevent moisture entering the dielectric – early radio equipment used a cardboard tube sealed with wax. Modern paper or film dielectric capacitors are dipped in a hard thermoplastic. Large capacitors for high-voltage use may have the roll form compressed to fit into a rectangular metal case, with bolted terminals and bushings for connections. The dielectric in larger capacitors is often impregnated with a liquid to improve its properties. Capacitors may have their connecting leads arranged in many configurations, for example axially or radially. "Axial" means that the leads are on a common axis, typically the axis of the capacitor's cylindrical body – the leads extend from opposite ends. Radial leads might more accurately be referred to as tandem; they are rarely actually aligned along radii of the body's circle, so the term is inexact, although universal. The leads (until bent) are usually in planes parallel to that of the flat body of the capacitor, and extend in the same direction; they are often parallel as manufactured. Small, cheap discoidal ceramic capacitors have existed since the 1930s, and remain in widespread use. Since the 1980s, surface mount packages for capacitors have been widely used. These packages are extremely small and lack connecting leads, allowing them to be soldered directly onto the surface of printed circuit boards. Surface mount components avoid undesirable high-frequency effects due to the leads and simplify automated assembly, although manual handling is made difficult due to their small size. Mechanically controlled variable capacitors allow the plate spacing to be adjusted, for example by rotating or sliding a set of movable plates into alignment with a set of stationary plates. Low cost variable capacitors squeeze together alternating layers of aluminum and plastic with a screw. Electrical control of capacitance is achievable with varactors (or varicaps), which are reverse-biased semiconductor diodes whose depletion region width varies with applied voltage. They are used in phase-locked loops, amongst other applications. ## Capacitor markings Most capacitors have numbers printed on their bodies to indicate their electrical characteristics. Larger capacitors like electrolytics usually display the actual capacitance together with the unit (for example, 220 μF). Smaller capacitors like ceramics, however, use a shorthand consisting of three numbers and a letter, where the numbers show the capacitance in pF (calculated as XY × 10Z for the numbers XYZ) and the letter indicates the tolerance (J, K or M for ±5%, ±10% and ±20% respectively). Additionally, the capacitor may show its working voltage, temperature and other relevant characteristics. ### Example A capacitor with the text 473K 330V on its body has a capacitance of 47 × 103 pF = 47 nF (±10%) with a working voltage of 330 V. The working voltage of a capacitor is the highest voltage that can be applied across it without undue risk of breaking down the dielectric layer. ## Applications This mylar-film, oil-filled capacitor has very low inductance and low resistance, to provide the high-power (70 megawatt) and high speed (1.2 microsecond) discharge needed to operate a dye laser. ### Energy storage A capacitor can store electric energy when disconnected from its charging circuit, so it can be used like a temporary battery, or like other types of rechargeable energy storage system.[26] Capacitors are commonly used in electronic devices to maintain power supply while batteries are being changed. (This prevents loss of information in volatile memory.) Conventional capacitors provide less than 360 joules per kilogram of energy density, whereas a conventional alkaline battery has a density of 590 kJ/kg. In car audio systems, large capacitors store energy for the amplifier to use on demand. Also for a flash tube a capacitor is used to hold the high voltage. ### Pulsed power and weapons Groups of large, specially constructed, low-inductance high-voltage capacitors (capacitor banks) are used to supply huge pulses of current for many pulsed power applications. These include electromagnetic forming, Marx generators, pulsed lasers (especially TEA lasers), pulse forming networks, radar, fusion research, and particle accelerators. Large capacitor banks (reservoir) are used as energy sources for the exploding-bridgewire detonators or slapper detonators in nuclear weapons and other specialty weapons. Experimental work is under way using banks of capacitors as power sources for electromagnetic armour and electromagnetic railguns and coilguns. ### Power conditioning A 10 millifarad capacitor in an amplifier power supply Reservoir capacitors are used in power supplies where they smooth the output of a full or half wave rectifier. They can also be used in charge pump circuits as the energy storage element in the generation of higher voltages than the input voltage. Capacitors are connected in parallel with the power circuits of most electronic devices and larger systems (such as factories) to shunt away and conceal current fluctuations from the primary power source to provide a "clean" power supply for signal or control circuits. Audio equipment, for example, uses several capacitors in this way, to shunt away power line hum before it gets into the signal circuitry. The capacitors act as a local reserve for the DC power source, and bypass AC currents from the power supply. This is used in car audio applications, when a stiffening capacitor compensates for the inductance and resistance of the leads to the lead-acid car battery. #### Power factor correction A high-voltage capacitor bank used for power factor correction on a power transmission system In electric power distribution, capacitors are used for power factor correction. Such capacitors often come as three capacitors connected as a three phase load. Usually, the values of these capacitors are given not in farads but rather as a reactive power in volt-amperes reactive (var). The purpose is to counteract inductive loading from devices like electric motors and transmission lines to make the load appear to be mostly resistive. Individual motor or lamp loads may have capacitors for power factor correction, or larger sets of capacitors (usually with automatic switching devices) may be installed at a load center within a building or in a large utility substation. ### Suppression and coupling #### Signal coupling Main article: capacitive coupling Polyester film capacitors are frequently used as coupling capacitors. Because capacitors pass AC but block DC signals (when charged up to the applied dc voltage), they are often used to separate the AC and DC components of a signal. This method is known as AC coupling or "capacitive coupling". Here, a large value of capacitance, whose value need not be accurately controlled, but whose reactance is small at the signal frequency, is employed. #### Decoupling Main article: decoupling capacitor A decoupling capacitor is a capacitor used to protect one part of a circuit from the effect of another, for instance to suppress noise or transients. Noise caused by other circuit elements is shunted through the capacitor, reducing the effect they have on the rest of the circuit. It is most commonly used between the power supply and ground. An alternative name is bypass capacitor as it is used to bypass the power supply or other high impedance component of a circuit. Decoupling capacitors need not always be discrete components. Capacitors used in these applications may be built in to a printed circuit board, between the various layers. These are often referred to as embedded capacitors.[27] The layers in the board contributing to the capacitive properties also function as power and ground planes, and have a dielectric in between them, enabling them to operate as a parallel plate capacitor. #### High-pass and low-pass filters Further information: High-pass filter and Low-pass filter #### Noise suppression, spikes, and snubbers Further information: High-pass filter and Low-pass filter When an inductive circuit is opened, the current through the inductance collapses quickly, creating a large voltage across the open circuit of the switch or relay. If the inductance is large enough, the energy will generate a spark, causing the contact points to oxidize, deteriorate, or sometimes weld together, or destroying a solid-state switch. A snubber capacitor across the newly opened circuit creates a path for this impulse to bypass the contact points, thereby preserving their life; these were commonly found in contact breaker ignition systems, for instance. Similarly, in smaller scale circuits, the spark may not be enough to damage the switch but will still radiate undesirable radio frequency interference (RFI), which a filter capacitor absorbs. Snubber capacitors are usually employed with a low-value resistor in series, to dissipate energy and minimize RFI. Such resistor-capacitor combinations are available in a single package. Capacitors are also used in parallel to interrupt units of a high-voltage circuit breaker in order to equally distribute the voltage between these units. In this case they are called grading capacitors. In schematic diagrams, a capacitor used primarily for DC charge storage is often drawn vertically in circuit diagrams with the lower, more negative, plate drawn as an arc. The straight plate indicates the positive terminal of the device, if it is polarized (see electrolytic capacitor). ### Motor starters Main article: motor capacitor In single phase squirrel cage motors, the primary winding within the motor housing is not capable of starting a rotational motion on the rotor, but is capable of sustaining one. To start the motor, a secondary "start" winding has a series non-polarized starting capacitor to introduce a lead in the sinusoidal current. When the secondary (start) winding is placed at an angle with respect to the primary (run) winding, a rotating electric field is created. The force of the rotational field is not constant, but is sufficient to start the rotor spinning. When the rotor comes close to operating speed, a centrifugal switch (or current-sensitive relay in series with the main winding) disconnects the capacitor. The start capacitor is typically mounted to the side of the motor housing. These are called capacitor-start motors, that have relatively high starting torque. Typically they can have up-to four times as much starting torque than a split-phase motor and are used on applications such as compressors, pressure washers and any small device requiring high starting torques. Capacitor-run induction motors have a permanently connected phase-shifting capacitor in series with a second winding. The motor is much like a two-phase induction motor. Motor-starting capacitors are typically non-polarized electrolytic types, while running capacitors are conventional paper or plastic film dielectric types. ### Signal processing The energy stored in a capacitor can be used to represent information, either in binary form, as in DRAMs, or in analogue form, as in analog sampled filters and CCDs. Capacitors can be used in analog circuits as components of integrators or more complex filters and in negative feedback loop stabilization. Signal processing circuits also use capacitors to integrate a current signal. #### Tuned circuits Capacitors and inductors are applied together in tuned circuits to select information in particular frequency bands. For example, radio receivers rely on variable capacitors to tune the station frequency. Speakers use passive analog crossovers, and analog equalizers use capacitors to select different audio bands. The resonant frequency f of a tuned circuit is a function of the inductance (L) and capacitance (C) in series, and is given by: $f = \frac{1}{2 \pi \sqrt{LC}}$ where L is in henries and C is in farads. ### Sensing Main article: capacitive sensing Most capacitors are designed to maintain a fixed physical structure. However, various factors can change the structure of the capacitor, and the resulting change in capacitance can be used to sense those factors. Changing the dielectric: The effects of varying the characteristics of the dielectric can be used for sensing purposes. Capacitors with an exposed and porous dielectric can be used to measure humidity in air. Capacitors are used to accurately measure the fuel level in airplanes; as the fuel covers more of a pair of plates, the circuit capacitance increases. Changing the distance between the plates: Capacitors with a flexible plate can be used to measure strain or pressure. Industrial pressure transmitters used for process control use pressure-sensing diaphragms, which form a capacitor plate of an oscillator circuit. Capacitors are used as the sensor in condenser microphones, where one plate is moved by air pressure, relative to the fixed position of the other plate. Some accelerometers use MEMS capacitors etched on a chip to measure the magnitude and direction of the acceleration vector. They are used to detect changes in acceleration, in tilt sensors, or to detect free fall, as sensors triggering airbag deployment, and in many other applications. Some fingerprint sensors use capacitors. Additionally, a user can adjust the pitch of a theremin musical instrument by moving their hand since this changes the effective capacitance between the user's hand and the antenna. Changing the effective area of the plates: Capacitive touch switches are now used on many consumer electronic products. ### Oscillators Further information: Hartley oscillator Example of a simple oscillator that requires a capacitor to function A capacitor can possess spring-like qualities in an oscillator circuit. In the image example, a capacitor acts to influence the biasing voltage at the npn transistor's base. The resistance values of the voltage-divider resistors and the capacitance value of the capacitor together control the oscillatory frequency. ## Hazards and safety Capacitors may retain a charge long after power is removed from a circuit; this charge can cause dangerous or even potentially fatal shocks or damage connected equipment. For example, even a seemingly innocuous device such as a disposable camera flash unit powered by a 1.5 volt AA battery contains a capacitor which may be charged to over 300 volts. This is easily capable of delivering a shock. Service procedures for electronic devices usually include instructions to discharge large or high-voltage capacitors, for instance using a Brinkley stick. Capacitors may also have built-in discharge resistors to dissipate stored energy to a safe level within a few seconds after power is removed. High-voltage capacitors are stored with the terminals shorted, as protection from potentially dangerous voltages due to dielectric absorption. Some old, large oil-filled paper or plastic film capacitors contain polychlorinated biphenyls (PCBs). It is known that waste PCBs can leak into groundwater under landfills. Capacitors containing PCB were labelled as containing "Askarel" and several other trade names. PCB-filled paper capacitors are found in very old (pre-1975) fluorescent lamp ballasts, and other applications. Capacitors may catastrophically fail when subjected to voltages or currents beyond their rating, or as they reach their normal end of life. Dielectric or metal interconnection failures may create arcing that vaporizes the dielectric fluid, resulting in case bulging, rupture, or even an explosion. Capacitors used in RF or sustained high-current applications can overheat, especially in the center of the capacitor rolls. Capacitors used within high-energy capacitor banks can violently explode when a short in one capacitor causes sudden dumping of energy stored in the rest of the bank into the failing unit. High voltage vacuum capacitors can generate soft X-rays even during normal operation. Proper containment, fusing, and preventive maintenance can help to minimize these hazards. High-voltage capacitors can benefit from a pre-charge to limit in-rush currents at power-up of high voltage direct current (HVDC) circuits. This will extend the life of the component and may mitigate high-voltage hazards. ## References 1. ^ Bird, John (2010). Electrical and Electronic Principles and Technology. Routledge. pp. 63–76. ISBN 9780080890562. Retrieved 2013-03-17. 2. ^ Williams, Henry Smith. "A History of Science Volume II, Part VI: The Leyden Jar Discovered". Retrieved 2013-03-17. 3. ^ Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s. John Wiley & Sons. p. 23. ISBN 9780780311930. Retrieved 2013-03-17. 4. ^ Houston, Edwin J. (1905). Electricity in Every-day Life. P. F. Collier & Son. p. 71. Retrieved 2013-03-17. 5. ^ Isaacson, Walter (2003). Benjamin Franklin: An American Life. Simon and Schuster. p. 136. ISBN 9780743260848. Retrieved 2013-03-17. 6. ^ Franklin, Benjamin (1749-04-29). "Experiments & Observations on Electricity: Letter IV to Peter Collinson" (PDF). p. 28. Retrieved 2009-08-09. 7. ^ Morse, Robert A. (September 2004). "Franklin and Electrostatics—Ben Franklin as my Lab Partner" (PDF). Wright Center for Science Education. Tufts University. p. 23. Retrieved 2009-08-10. "After Volta’s discovery of the electrochemical cell in 1800, the term was then applied to a group of electrochemical cells" 8. ^ "eFunda: Glossary: Units: Electric Capacitance: Jar". eFunda. Retrieved 2013-03-17. 9. ^ "Sketch of Alessandro Volta". The Popular Science Monthly (New York: Bonnier Corporation): 118–119. May 1892. ISSN 0161-7370. 10. ^ a b Ulaby, p.168 11. ^ Ulaby, p.157 12. ^ a b Ulaby, p.169 13. ^ Hammond, Percy (1964). Electromagnetism for Engineers: An Introductory Course. The Commonwealth and International Library of Science, Technology, Engineering and Liberal Studies. Applied Electricity and Electronics Division 3. Pergamon Press. pp. 44–45. 14. ^ Dorf, p.263 15. ^ Dorf, p.260 16. ^ "Capacitor charging and discharging". All About Circuits. Retrieved 2009-02-19. 17. ^ Pillai, K. P. P. (1970). "Fringing field of finite parallel-plate capacitors". Proceedings of the Institution of Electrical Engineers 117 (6): 1201–1204. doi:10.1049/piee.1970.0232. edit 18. ^ Ulaby, p.170 19. ^ Pai, S. T.; Qi Zhang (1995). Introduction to High Power Pulse Technology. Advanced Series in Electrical and Computer Engineering 10. World Scientific. ISBN 9789810217143. Retrieved 2013-03-17. 20. ^ Dyer, Stephen A. (2004). Wiley Survey of Instrumentation and Measurement. John Wiley & Sons. p. 397. ISBN 9780471221654. Retrieved 2013-03-17. 21. ^ Scherz, Paul (2006). Practical Electronics for Inventors (2nd ed.). McGraw Hill Professional. p. 100. ISBN 9780071776448. Retrieved 2013-03-17. 22. ^ Bird, John (2007). Electrical Circuit Theory and Technology. Routledge. p. 501. ISBN 9780750681391. Retrieved 2013-03-17. 23. ^ "Ceramic Capacitor Aging Made Simple". Johanson Dielectrics. 2012-05-21. Retrieved 2013-03-17. 24. ^ "The Effect of Reversal on Capacitor Life" (PDF). Engineering Bulletin 96-004. Sorrento Electronics. November 2003. Retrieved 2013-03-17. 25. ^ Guinta, Steve. "Ask The Applications Engineer – 21". Analog Devices. Retrieved 2013-03-17. 26. ^ Miller, Charles. Illustrated Guide to the National Electrical Code, p. 445 (Cengage Learning 2011). 27. ^ Alam, Mohammed; Michael H. Azarian; Michael Osterman; Michael Pecht (2010). "Effectiveness of embedded capacitors in reducing the number of surface mount capacitors for decoupling applications". Circuit World 36 (1): 22.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 31, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7144142985343933, "perplexity": 1577.6688578073383}, "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/1405997857710.17/warc/CC-MAIN-20140722025737-00141-ip-10-33-131-23.ec2.internal.warc.gz"}
https://testbook.com/question-answer/the-components-in-the-circuit-shown-below-are-idea--5ec937ccf60d5d024b99389f	# The components in the circuit shown below are ideal. If the op-amp is in positive feedback and the input voltage Vi is a sine wave of amplitude 1 V, the output voltage V0 is Free Practice With Testbook Mock Tests ## Options: 1. a non-inverted sine wave of 2 V amplitude 2. an inverted sine wave of 1 V amplitude 3. a square wave of 5 V amplitude 4. a constant of either +5 V or -5 V. ### Correct Answer: Option 4 (Solution Below) This question was previously asked in GATE EC 2020 Official Paper ## Solution: Analysis: The given circuit is a Schmitt trigger circuit. It is redrawn as: Applying KCL at the non-investing terminal, we get: $$\frac{{{V^ + } - {V_i}}}{{1k}} + \frac{{{V^ + } - {V_0}}}{{1k{\rm{\Omega }}}} = 0$$ V+ - Vi + V+ - V0 = 0 $${V^ + } = \frac{{{V_i} + {V_0}}}{2}$$ ---(1) Case 1: When V0 = 5V (+Vsat) From equation (1), we can write: $${V^ + } = \frac{{5 + {V_0}}}{2} = 2.5 + \frac{{{V_0}}}{2}$$ For a positive peak of +1V for Vi, V+ will be: V+ = 2.5 + 0.5 V+ = 3V Since V- = 0 V The difference in the non-inverting and inverting terminal of Op-Amp is: Vd = V+ - V- Vd = 3 V Since the Op-Amp amplifies the difference in the input voltage, the output will remain constant at +Vsat (5V) as Vd = + 3V (Positive) Now for a negative peak of -1V for Vin, V+ will be: V+ = 2.5 – 0.5 V+ = 2V Vd = 2 – 0 Vd = 2V The difference is positive for both the positive and negative peak of Vi, The output will remain at + 5V (+Vsat), irrespective of the input. Case 2: When V0 = -5V (-Vsat) Again apply KCL at the non-investing terminal, we get: $$\frac{{{V^ + } - \left( { - 5} \right)}}{{1k}} + \frac{{{V^ + } - {V_i}}}{{1k}} = 0$$ $${V^ + } = \frac{{{V_i} - 5}}{2}$$ $${V^ + } = \; - 2.5 + \frac{{{V_i}}}{2}$$ Now, for a positive peak of Vi at 1V, V+ will be: V+ = -2.5 + 0.5 V+ = -2V Vd = V+ - V- Vd = -2V Since the difference is negative, the output will remain at -Vsat Similarly, for a negative peak of Vi at -1V, V+, will be, V+ = -2.5 – 0.5 V+ = -3V Vd = -3V Source the difference is negative the output will remain at -Vsat We can conclude that the output will either be +Vsat or -Vsat	{"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": 0.7214021682739258, "perplexity": 7019.580734246534}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046152144.81/warc/CC-MAIN-20210726152107-20210726182107-00367.warc.gz"}
http://cms.math.ca/cmb/kw/Hamiltonian	location: Publications → journals Search results Search: All articles in the CMB digital archive with keyword Hamiltonian Expand all Collapse all Results 1 - 4 of 4 1. CMB Online first Dimassi, Mouez Semi-classical asymptotics for SchrÃ¶dinger operator with oscillating decaying potential We study the distribution of the discrete spectrum of the SchrÃ¶dinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$. Keywords:periodic SchrÃ¶dinger operator, semi-classical asymptotics, effective Hamiltonian, asymptotic expansion, spectral shift functionCategories:81Q10, 35P20, 47A55, 47N50, 81Q15 2. CMB 2015 (vol 58 pp. 651) Tang, Xianhua Ground State Solutions of Nehari-Pankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$ This paper is concerned with the following elliptic system of Hamiltonian type $\left\{ \begin{array}{ll} -\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ -\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{array} \right.$ where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity. We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system. Especially, our method is applicable for the case when $W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{\|\alpha_iu+\beta_iv\|}g_i(x, t)t\mathrm{d}t +\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x, t)t\mathrm{d}t,$ where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne 0$ and $a_j\gt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every $x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$. Keywords:Hamiltonian elliptic system, superlinear, ground state solutions of Nehari-Pankov type, strongly indefinite functionalsCategories:35J50, 35J55 3. CMB 2009 (vol 52 pp. 416) Malik, Shabnam; Qureshi, Ahmad Mahmood; Zamfirescu, Tudor Hamiltonian Properties of Generalized Halin Graphs A Halin graph is a graph $H=T\cup C$, where $T$ is a tree with no vertex of degree two, and $C$ is a cycle connecting the end-vertices of $T$ in the cyclic order determined by a plane embedding of $T$. In this paper, we define classes of generalized Halin graphs, called $k$-Halin graphs, and investigate their Hamiltonian properties. Keywords:$k$-Halin graph, Hamiltonian, Hamiltonian connected, traceableCategories:05C45, 05C38 4. CMB 2001 (vol 44 pp. 323) Schuman, Bertrand Une classe d'hamiltoniens polynomiaux isochrones Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S]. Keywords:Hamiltonian system, normal forms, resonance, linearizationCategories:34C20, 58F05, 58F22, 58F30 top of page \| contact us \| privacy \| site map \|	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8039572238922119, "perplexity": 2414.5366657189093}, "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/1471982292151.8/warc/CC-MAIN-20160823195812-00165-ip-10-153-172-175.ec2.internal.warc.gz"}
https://physics.stackexchange.com/questions/219573/time-dilation-on-satellites-due-to-gr	# Time dilation on Satellites due to GR I am trying to determine the time dilation onboard a satellite (say GPS @ 20,000km) w.r.t an observer on the earth. I have already determined the special relativity component using: $$t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}}$$ And I got the correct answer for the time dilation simply due to relative motion (7 microseconds after 24 hrs). Im not sure if determining the component due to GR is this simple, but my first attempt was to evaluate it using the equation to determiine gravitational time dilation outside a non rotating sphere in a circular orbit using the Schwarzschild metric. $$t' = t\sqrt{1-\frac{3GM}{rc^2}}$$ I dont seem to be getting the correct answer (45 microseconds after 24 hrs). Any ideas? The equation you quote: $$t' = t\sqrt{1-\frac{3GM}{rc^2}} \tag{1}$$ gives the time relative to an observer at infinity. You want the time relative to an observer on the Earth's surface. You need to calculate: $$t_\text{satellite} = t\sqrt{1-\frac{3GM}{r_\text{satellite}c^2}}$$ and: $$t_\text{Earth} = t\sqrt{1-\frac{2GM}{r_\text{Earth}c^2}}$$ where $r_\text{Earth}$ is the radius of the Earth and $r_\text{satellite}$ is the radius of the satellite's orbit (measured from the centre of the Earth). The relative time dilation is then the ratio of these two times. Note that equation (1) combines the special and general relativity contributions to the time dilation i.e. it includes both the gravitational time dilation and the effect of the orbital velocity. The observer on the Earth's surface isn't in a circular orbit so the equation is slightly different (a factor of 2 in the square root rather than 3). Incidentally, when the gravitational field is weak (like the Earth's) we can use the weak field approximation for time dilation: $$\frac{dt_B}{dt_A} = \sqrt{1 - \frac{2(\Phi_A - \Phi_B)}{c^2}} \tag{1}$$ The quantity $\Phi_A - \Phi_B$ is the difference in the Newtonian gravitational potential energy between $A$ and $B$, and $dt_B/dt_A$ is the time dilation of $B$'s clock relative to $A$'s clock. • The observer on earth is in a couple of different orbits - once around the Sun, and once every 24 hours around the center of Earth. The former orbit is "shared" by the satellite, while the latter is "very slow". Is it worth computing the correction needed for the latter (which is dependent on latitude)? – Floris Nov 20 '15 at 8:48 • @Floris: no :-) – John Rennie Nov 20 '15 at 8:49 • See also Sagnac distortion. It results in an error "on the order of hundreds of nanoseconds, or tens of meters in position". – Floris Nov 20 '15 at 8:56 • @Floris No (well, of course it depends on your application): look at John's equation (1): if the Newtonian approximation is mildly affected by an effect, then the error in a Newtonian potential difference becomes that error divided by $c^2$ in the weak field time dilation factor. – Selene Routley Nov 20 '15 at 9:02 • @Floris: If someone wants to post this as a question I'd be happy to give a detailed answer. It's pretty straightforward as you just integrate the metric for constant $r$ and $\theta$ and $\phi = \omega tr\sin\theta$. But I think the calculation is unnecessary here. – John Rennie Nov 20 '15 at 9:31	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9611490368843079, "perplexity": 316.7613390402474}, "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-2020-40/segments/1600400188049.8/warc/CC-MAIN-20200918155203-20200918185203-00016.warc.gz"}
https://www.deepdyve.com/lp/ou_press/on-the-asymptotic-efficiency-of-approximate-bayesian-computation-FRQadh1gRG	# On the asymptotic efficiency of approximate Bayesian computation estimators On the asymptotic efficiency of approximate Bayesian computation estimators SUMMARY Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large data limit. Our key assumption is that the data are summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is of the same dimension as the parameter vector, $$p$$, and that any summary statistic of higher dimension can be reduced, through a linear transformation, to dimension $$p$$ in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an importance sampling algorithm that samples from the approximate Bayesian computation posterior affects the accuracy of estimators. We give conditions on the importance sampling proposal distribution such that the variance of the estimator will be of the same order as that of the maximum likelihood estimator based on the summary statistics used. This suggests an iterative importance sampling algorithm, which we evaluate empirically on a stochastic volatility model. 1. Introduction Many statistical applications involve inference about models that are easy to simulate from, but for which it is difficult, or impossible, to calculate likelihoods. In such situations it is possible to use the fact that we can simulate from the model to enable us to perform inference. There is a wide class of such likelihood-free methods of inference, including indirect inference (Gouriéroux & Ronchetti, 1993; Heggland & Frigessi, 2004), the bootstrap filter (Gordon et al., 1993), simulated methods of moments (Duffie & Singleton, 1993), and synthetic likelihood (Wood, 2010). We consider a Bayesian version of these methods, termed approximate Bayesian computation. This involves defining an approximation to the posterior distribution in such a way that it is possible to sample from this approximate posterior using only the ability to sample from the model. Arguably the first approximate Bayesian computation method was that of Pritchard et al. (1999), and these methods have been popular within population genetics (Beaumont et al., 2002), ecology (Beaumont, 2010) and systems biology (Toni et al., 2009). More recently, there have been applications to areas including stereology (Bortot et al., 2007), finance (Peters et al., 2011) and cosmology (Ishida et al., 2015). Let $$K({x})$$ be a density kernel, scaled, without loss of generality, so that $$\max_{{x}}K({x})=1$$. Further, let $$\varepsilon>0$$ be a bandwidth. Denote the data by $${Y}_{\rm obs}=(\,y_{\rm obs,1},\ldots,y_{{\rm obs},n})$$. Assume we have chosen a finite-dimensional summary statistic $${s}_{n}({Y})$$, and write $${s}_{\rm obs}={s}_{n}({Y}_{\rm obs})$$. If we model the data as a draw from a parametric density, $$f_{n}({y} \mid{\theta})$$, and assume a prior $$\pi({\theta})$$, then we define the approximate Bayesian computation posterior as $$\pi_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon)\propto\pi({\theta})\int f_{n}({s}_{\rm obs}+\varepsilon{v} \mid {\theta})K({v})\,{\rm d}{v},\label{eq:piABC}$$ (1) where $$f_{n}({s} \mid {\theta})$$ is the density for the summary statistic implied by $$f_{n}({y} \mid {\theta})$$. Let $$f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)=\int f_{n}({s}_{\rm obs}+\varepsilon{v} \mid {\theta})K({v})\,{\rm d}{v}$$. This framework encompasses most implementations of approximate Bayesian computation. In particular, the use of the uniform kernel corresponds to the popular rejection-based rule (Beaumont et al., 2002). The idea is that $$f_{\rm ABC}({s}_{\rm obs}\,{\mid}\, {\theta},\varepsilon)$$ is an approximation of the likelihood. The approximate Bayesian computation posterior, which is proportional to the prior multiplied by this likelihood approximation, is an approximation of the true posterior. The likelihood approximation can be interpreted as a measure of how close, on average, the summary $${s}_{n}$$ simulated from the model is to the summary for the observed data, $${s}_{\rm obs}$$. The choices of kernel and bandwidth determine the definition of closeness. By defining the approximate posterior in this way, we can simulate samples from it using standard Monte Carlo methods. One approach, which we will focus on later, uses importance sampling. Let $$K_{\varepsilon}({x})=K({x}/\varepsilon)$$. Given a proposal density $$q_{n}({\theta})$$, a bandwidth $$\varepsilon$$, and a Monte Carlo sample size $$N$$, an importance sampler would proceed as in Algorithm 1. The set of accepted parameters and their associated weights provides a Monte Carlo approximation to $$\pi_{\rm ABC}$$. If we set $$q_{n}({\theta})=\pi({\theta})$$ then this is just a rejection sampler. In practice sequential importance sampling methods are often used to learn a good proposal distribution (Beaumont et al., 2009). Algorithm 1. Importance and rejection sampling approximate Bayesian computation. Step 1. Simulate $${\theta}_{1},\ldots,{\theta}_{N}\sim q_{n}({\theta})$$. Step 2. For each $$i=1,\ldots,N$$, simulate $${Y}^{(i)}=\big\{\,y_{1}^{(i)},\ldots,y_{n}^{(i)}\big\}\sim f_{n}(\,y\mid {\theta}_{i})$$. Step 3. For each $$i=1,\ldots,N$$, accept $${\theta}_{i}$$ with probability $$K_{\varepsilon}\big\{{s}_{n}^{(i)}-{s}_{\rm obs}\big\}$$, where $${s}_{n}^{(i)}={s}_{n}\{{Y}^{(i)}\}$$, and define the associated weight as $$w_{i}=\pi({\theta}_{i})/q_{n}({\theta}_{i})$$. There are three choices in implementing approximate Bayesian computation: the choice of summary statistic, the choice of bandwidth, and the Monte Carlo algorithm. For importance sampling, the last of these involves specifying the Monte Carlo sample size, $$N$$, and the proposal density, $$q_{n}({\theta})$$. These, roughly, relate to three sources of approximation. To see this, note that as $$\varepsilon\rightarrow0$$ we would expect (1) to converge to the posterior given $${s}_{\rm obs}$$ (Fearnhead & Prangle, 2012). Thus the choice of summary statistic governs the approximation, or loss of information, between using the full posterior distribution and using the posterior given the summary. The value $$\varepsilon$$ then affects how close the approximate Bayesian computation posterior is to the posterior given the summary. Finally there is Monte Carlo error from approximating the approximate Bayesian computation posterior with a Monte Carlo sample. The Monte Carlo error is affected not only by the Monte Carlo algorithm, but also by the choices of summary statistic and bandwidth, which together affect the probability of acceptance in Step 3 of Algorithm 1. Having a higher-dimensional summary statistic, or a smaller value of $$\varepsilon$$, will tend to reduce this acceptance probability and hence increase the Monte Carlo error. This work studies the interaction between the three sources of error, when the summary statistics obey a central limit theorem for large $$n$$. We are interested in the efficiency of approximate Bayesian computation, where by efficiency we mean that an estimator obtained from running Algorithm 1 has the same rate of convergence as the maximum likelihood estimator for the parameter given the summary statistic. In particular, this work is motivated by the question of whether approximate Bayesian computation can be efficient as $$n\rightarrow\infty$$ if we have a fixed Monte Carlo sample size. Intuitively this appears unlikely. For efficiency we will need $$\varepsilon\rightarrow0$$ as $$n\rightarrow\infty$$, and this corresponds to an increasingly strict condition for acceptance. Thus we may imagine that the acceptance probability will necessarily tend to zero as $$n$$ increases, and we will need an increasing Monte Carlo sample size to compensate for this. However, our results show that Algorithm 1 can be efficient if we choose a proposal distribution with a suitable scale and location and appropriately heavy tails. If we use such a proposal distribution and have a summary statistic of the same dimension as the parameter vector, then the posterior mean of approximate Bayesian computation is asymptotically unbiased with a variance that is $$1+O(1/N)$$ times that of the estimator maximizing the likelihood of the summary statistic. This is similar to asymptotic results for indirect inference (Gouriéroux & Ronchetti, 1993; Heggland & Frigessi, 2004). Our results also lend theoretical support to methods that choose the bandwidth indirectly by specifying the proportion of samples that are accepted, as this leads to a bandwidth which is of the optimal order in $$n$$. We first prove a Bernstein–von Mises-type theorem for the posterior mean of approximate Bayesian computation. This is a nonstandard convergence result, as it is based on the partial information contained in the summary statistics. For related convergence results see Clarke & Ghosh (1995) and Yuan & Clarke (2004), though these do not consider the case where the dimension of the summary statistic is larger than that of the parameter. Dealing with this case introduces extra challenges. Our convergence result for the posterior mean of approximate Bayesian computation has practically important consequences. It shows that any $$d$$-dimensional summary with $$d>p$$ can be projected to a $$p$$-dimensional summary statistic without any loss of information. Furthermore it shows that using a summary statistic of dimension $$d>p$$ can lead to an increased bias, so the asymptotic variance can be reduced if the optimal $$p$$-dimensional projected summary is used instead. If a $$d$$-dimensional summary is used, with $$d>p$$, it suggests choosing the variance of the kernel to match the variance of the summary statistics. This paper adds to a growing literature on the theoretical properties of approximate Bayesian computation. Initial results focused on comparing the bias of approximate Bayesian computation to the Monte Carlo error, and studying how these depend on the choice of $$\varepsilon$$. The convergence rate of the bias is shown to be $$O(\varepsilon^{2})$$ in various settings (e.g., Barber et al., 2015). This can then be used to consider how the choice of $$\varepsilon$$ should depend on the Monte Carlo sample size so as to balance bias and Monte Carlo variability (Blum, 2010; Barber et al., 2015; Biau et al., 2015). There has also been work on consistency of approximate Bayesian computation estimators. Marin et al. (2014) consider consistency when performing model choice and Frazier et al. (2016) consider consistency for parameter estimation. The latter work, which appeared after the first version of this paper, includes a result on the asymptotic normality of the posterior mean that is similar to our Theorem 1, albeit under different conditions, and also gives results on the asymptotic form of the posterior obtained using approximate Bayesian computation. This shows that for many implementations of approximate Bayesian computation, the posterior will overestimate the uncertainty in the parameter estimate that it gives. Finally, a number of papers have looked at the choice of summary statistics (e.g., Wegmann et al., 2009; Blum, 2010; Prangle et al., 2014). Our Theorem 1 gives insight into this. As mentioned above, this theorem shows that, in terms of minimizing the asymptotic variance, we should use a summary statistic of the same dimension as the number of parameters. In particular it supports the suggestion in Fearnhead & Prangle (2012) of having one summary per parameter, with that summary approximating the maximum likelihood estimator for that parameter. 2. Notation and set-up Denote the data by $${Y}_{\rm obs}=(\,y_{{\rm obs},1},\ldots,$$$$y_{{\rm obs},n})$$, where $$n$$ is the sample size and each observation $$y_{{\rm obs},i}$$ can be of arbitrary dimension. We make no assumption directly on the data, but make assumptions on the distribution of the summary statistics. We consider the asymptotics as $$n\rightarrow\infty$$, and denote the density of $${Y}_{\rm obs}$$ by $$\,f_{n}({y} \mid {\theta})$$, where $${\theta}\in\mathcal{P}\subset\mathbb{R}^p$$. We let $${\theta}_{0}$$ denote the true parameter value and $$\pi({\theta})$$ its prior distribution. For a set $$A$$, let $$A^{\rm c}$$ be its complement with respect to the whole space. We assume that $${\theta}_{0}$$ is in the interior of the parameter space, and that the prior is differentiable in a neighbourhood of the true parameter: Condition 1. There exists some $$\delta_{0}>0$$ such that $$\mathcal{P}_{0}=\{{\theta}:\|{\theta}-{\theta}_{0}\|<\delta_{0}\}\subset\mathcal{P}$$, $$\pi({\theta})\in C^{1}(\mathcal{P}_{0})$$ and $$\pi({\theta}_{0})>0$$. To implement approximate Bayesian computation we will use a $$d$$-dimensional summary statistic, $${s}_{n}({Y})\in\mathbb{R}^{d}$$, such as a vector of sample means of appropriately chosen functions. We assume that $${s}_{n}({Y})$$ has a density function which depends on $$n$$, and we denote this by $$f_{n}({s} \mid {\theta})$$. We will use the shorthand $${S}_{n}$$ to denote the random variable with density $$f_{n}({s} \mid {\theta})$$. In approximate Bayesian computation we use a kernel, $$K({x})$$, with $$\max_{{x}}K({x})=1$$, and a bandwidth $$\varepsilon>0$$. As we vary $$n$$ we will often wish to vary $$\varepsilon$$, and in these situations we denote the bandwidth by $$\varepsilon_{n}$$. For Algorithm 1 we require a proposal distribution, $$q_{n}({\theta})$$, and allow this to depend on $$n$$. We assume the following conditions on the kernel, which are satisfied by all commonly-used kernels, Condition 2. The kernel satisfies (i) $$\int{v} K({v})\,{\rm d}{v}=0$$; (ii) $$\int\prod_{k=1}^{l}v_{i_{k}}K({v})\,{\rm d}{v}<\infty$$ for any coordinates $$(v_{i_{1}},\ldots,v_{i_{l}})$$ of $${v}$$ and $$l\leq p+6$$; (iii) $$K({v})\propto\bar{K}(\\|{v}\\|_{\Lambda}^{2})$$ where $$\\|{v}\\|_{\Lambda}^{2}={v}^{{\rm T}}\Lambda{v}$$ with $$\Lambda$$ a positive-definite matrix, and $$K({v})$$ is a decreasing function of $$\\|{v}\\|_{\Lambda}$$; (iv) $$K({v})=O\{\exp(-c_1 \\|v\\|_{\Gamma}^{-1})\}$$ for some $$\alpha_{1}>0$$ and $$c_{1}>0$$ as $$\\|{v}\\|_{\Lambda}\rightarrow\infty$$. For a real function $$g({x})$$ denote its $$k$$th partial derivative at $${x}={x}_{0}$$ by $$D_{x_{k}}g({x}_{0})$$, the gradient function by $$D_{{x}}g({x}_{0})$$ and the Hessian matrix by $$H_{{x}}g({x}_{0})$$. To simplify notation, $$D_{\theta_{k}}$$, $$D_{{\theta}}$$ and $$H_{{\theta}}$$ are written as $$D_{k}$$, $$D$$ and $$H$$, respectively. For a series $$x_{n}$$ we use the notation that for large enough $$n$$, $$x_{n}=\Theta(a_{n})$$ if there exist constants $$m$$ and $$M$$ such that $$0<m<\|x_{n}/a_{n}\|<M<\infty$$, and $$x_{n}=\Omega(a_{n})$$ if $$\|x_{n}/a_{n}\|\rightarrow\infty$$. For two square matrices $$A$$ and $$B$$, we say that $$A\leq B$$ if $$B-A$$ is semipositive definite, and $$A<B$$ if $$B-A$$ is positive definite. Our theory will focus on estimates of some function $${h}(\theta)$$ of $$\theta$$, which satisfies differentiability and moment conditions that will control the remainder terms in Taylor expansions. Condition 3. The $$k$$th coordinate of $${h}({\theta})$$, $$h_{k}({\theta})$$, satisfies (i) $$h_{k}({\theta})\in C^{1}(\mathcal{P}_{0})$$; (ii) $$D_{k}h({\theta}_{0})\neq0$$; and (iii) $$\int h_{k}({\theta})^{2}\pi({\theta})\,{\rm d}{\theta}<\infty$$. The asymptotic results presuppose a central limit theorem for the summary statistic. Condition 4. There exists a sequence $$a_{n}$$, with $$a_{n}\rightarrow\infty$$ as $$n\rightarrow\infty$$, a $$d$$-dimensional vector $${s}({\theta})$$ and a $$d\times d$$ matrix $$A({\theta})$$ such that for all $${\theta}\in\mathcal{P}_0$$, $a_{n}\{{S}_{n}-{s}({\theta})\}\rightarrow \mathcal{N}\{0,A({\theta})\}, \quad {n\rightarrow\infty},$ with convergence in distribution. We also assume that $$s_{\rm obs}\rightarrow s(\theta_0)$$ in probability. Furthermore, assume that (i) $${s}({\theta})\in C^{1}(\mathcal{P}_{0})$$ and $$A({\theta})\in C^{1}(\mathcal{P}_{0})$$, and $$A({\theta})$$ is positive definite for $${\theta}\in\mathcal{P}_{0}$$; (ii) for any $$\delta>0$$ there exists a $$\delta'>0$$ such that $$\\|s(\theta)-s(\theta_0)\\|>\delta'$$ for all $$\theta$$ satisfying $$\\|\theta-\theta_0\\|>\delta$$; (iii) $$I({\theta})=D{s}({\theta})^{{\rm T}}A^{-1}({\theta})D{s}({\theta})$$ has full rank at $${\theta}={\theta}_{0}$$. Under Condition 4, $$a_{n}$$ is the rate of convergence in the central limit theorem. If the data are independent and identically distributed, and the summaries are sample means of functions of the data or of quantiles, then $$a_{n}=n^{1/2}$$. In most applications the data will be dependent, but if summaries are sample means (Wood, 2010), quantiles (Peters et al., 2011; Allingham et al., 2009; Blum & François, 2010) or linear combinations thereof (Fearnhead & Prangle, 2012), then a central limit theorem will often still hold, though $$a_n$$ may increase more slowly than $$n^{1/2}$$. Part (ii) of Condition 4 is required for the true parameter to be identifiable given only the summary of data. The asymptotic variance of the summary-based maximum likelihood estimator for $${\theta}$$ is $$I^{-1}({\theta}_{0})/a_{n}^{2}$$. Condition (iii) ensures that this variance is valid at the true parameter. We next require a condition that controls the difference between $$f_{n}({s}\mid {\theta})$$ and its limiting distribution for $${\theta}\in\mathcal{P}_{0}$$. Let $$\mathcal{N}({x};{\mu},\Sigma)$$ be the normal density at $${x}$$ with mean $${\mu}$$ and variance $$\Sigma$$. Define $$\tilde{f}_{n}({s} \mid {\theta})=\mathcal{N}\{{s};{s}({\theta}),A({\theta})/a_{n}^{2}\}$$ and the standardized random variable $$W_{n}({s})=a_{n}A({\theta})^{-1/2}\{{s}-{s}({\theta})\}$$. Let $$\tilde{f}_{W_{n}}({w} \mid {\theta})$$ and $$f_{W_{n}}({w} \mid {\theta})$$ be the densities of $$W_{n}({s})$$ when $${s}\sim\tilde{f}_{n}({s} \mid {\theta})$$ and $$f_{n}({s} \mid {\theta})$$ respectively. The condition below requires that the difference between $$f_{W_{n}}({w} \mid {\theta})$$ and its Edgeworth expansion $$\tilde{f}_{W_{n}}({w} \mid {\theta})$$ be of $$o(a_n^{-2/5})$$ and can be bounded by a density with exponentially decreasing tails. This is weaker than the standard requirement, $$o(a_n^{-1})$$, for the remainder in the Edgeworth expansion. Condition 5. There exists $$\alpha_{n}$$ satisfying $$\alpha_{n}/a_n^{2/5}\rightarrow\infty$$ and a density $$r_{\rm max}({w})$$ satisfying Condition 2(ii) and (iii) where $$K({v})$$ is replaced with $$r_{\rm max}({w})$$, such that $$\sup_{{\theta}\in\mathcal{P}_{0}}\alpha_{n}\| f_{W_{n}}({w} \mid {\theta})-\tilde{f}_{W_{n}}({w} \mid {\theta})\|\leq c_{3}r_{\rm max}({w})$$ for some positive constant $$c_{3}$$. The following condition further assumes that $$f_{n}(s \mid \theta)$$ has exponentially decreasing tails with rate uniform in the support of $$\pi(\theta)$$. Condition 6. The following statements hold: (i) $$r_{\rm max}(w)$$ satisfies Condition 2(iv); and (ii) $$\sup_{\theta\in\mathcal{P}_{0}^{\rm c}}f_{W_{n}}(w \mid \theta)=O\{\exp(-c_2 \\|w\\|^{\alpha_2})\}$$ as $$\\|w\\|\rightarrow\infty$$ for some positive constants $$c_{2}$$ and $$\alpha_{2}$$, and $$A(\theta)$$ is bounded in $$\mathcal{P}$$. 3. Posterior mean asymptotics We first ignore any Monte Carlo error, and focus on the ideal estimator of the true posterior mean from approximate Bayesian computation. This is the posterior mean, $${h}_{\rm ABC}$$, where ${h}_{\rm ABC}={E}_{\pi_{\rm ABC}}\{{h}({\theta})\mid {s}_{\rm obs}\}= \int {h}({\theta}) \pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_n)\text{.}$ This estimator depends on $$\varepsilon_n$$, but we suppress this from the notation. As an approximation to the true posterior mean $${E}\{{h}({\theta})\mid {Y}_{\rm obs}\}$$, $${h}_{\rm ABC}$$ contains errors from the choice of bandwidth $$\varepsilon_{n}$$ and summary statistic $${s}_{\rm obs}$$. To understand the effects of these two sources of error, we derive results for the asymptotic distributions of $${h}_{\rm ABC}$$ and the likelihood-based estimators, including the summary-based maximum likelihood estimator and the summary-based posterior mean, where we consider randomness solely due to the randomness of the data. Let $$T_{{\rm obs}}=a_{n}A(\theta_{0})^{-1/2}\{s_{\rm obs}-s(\theta_{0})\}$$. Theorem 1. Assume Conditions 1–6. (i) Let $$\hat{{{\theta}}}_{_{{ \rm MLES}}}=\mathop{\arg\max}\limits_{\theta\in\mathcal{P}}\log f_{n}({s}_{\rm obs}\mid {\theta})$$. For $${h}_{{s}}={h}(\hat{{{\theta}}}_{_{{ \rm MLES}}})$$ or $${E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, $a_{n}\{{h}_{{s}}-{h}({\theta}_{0})\}\rightarrow \mathcal{N}\{0,D{h}({\theta}_{0})^{\mathrm{T}}I^{-1}({\theta}_{0})D{h}({\theta}_{0})\},\quad n\rightarrow\infty,$ with convergence in distribution. (ii) Define $$c_{\infty}= \lim_{n\rightarrow\infty}a_{n}\varepsilon_{n}$$. Let $$Z$$ be the weak limit of $$T_{\rm obs}$$, which has a standard normal distribution, and let $$R(c_{\infty},Z)$$ be a random vector with mean zero that is defined in the Supplementary Material. If $$\varepsilon_{n}=o(a_{n}^{-3/5})$$, then $a_{n}\{h_{{\rm ABC}}-h(\theta_{0})\}\rightarrow Dh(\theta_{0})^{\mathrm{T}}\{I(\theta_{0})^{-1/2}Z+R(c_\infty,Z)\}, \quad n\rightarrow\infty,$ with convergence in distribution. If either $$\varepsilon_{n}=o(a_{n}^{-1})$$, $$d=p$$, or the covariance matrix of $$K(v)$$ is proportional to $$A(\theta_{0})$$, then $$R(c_{\infty},Z)=0$$. For other cases, the variance of $$I(\theta_{0})^{-1/2}Z+R(c_{\infty},Z)$$ is no less than $$I^{-1}(\theta_{0})$$. Theorem 1 (i) shows the validity of posterior inference based on the summary statistics. Regardless of the sufficiency and dimension of $${s}_{\rm obs}$$, the posterior mean based on the summary statistics is consistent and asymptotically normal with the same variance as the summary-based maximum likelihood estimator. Denote the bias of approximate Bayesian computation, $${h}_{\rm ABC}-{E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, by $$\mbox{bias}_{\rm ABC}$$. The choice of bandwidth affects the size of the bias. Theorem 1(ii) indicates two regimes for the bandwidth for which the posterior mean of approximate Bayesian computation has good properties. The first case is when $$\varepsilon_{n}$$ is $$o(1/a_{n})$$. For this regime the posterior mean of approximate Bayesian computation always has the same asymptotic distribution as that of the true posterior given the summaries. The other case is when $$\varepsilon_{n}$$ is $$o(a_n^{-3/5})$$ but not $$o(n^{-1})$$. We obtain the same asymptotic distribution if either $$d=p$$ or we choose the kernel variance to be proportional to the variance of the summary statistics. In general for this regime of $$\varepsilon_n$$, $${h}_{\rm ABC}$$ will be less efficient than the summary-based maximum likelihood estimator. When $$d>p$$, Theorem 1(ii) shows that $$\mbox{bias}_{\rm ABC}$$ is nonnegligible and can increase the asymptotic variance. This is because the leading term of $$\mbox{bias}_{\rm ABC}$$ is proportional to the average of $${v}={s}-{s}_{\rm obs}$$, the difference between the simulated and observed summary statistics. If $$d>p$$, the marginal density of $${v}$$ is generally asymmetric, and thus is no longer guaranteed to have a mean of zero. One way to ensure that there is no increase in the asymptotic variance is to choose the variance of the kernel to be proportional to the variance of the summary statistics. The loss of efficiency we observe in Theorem 1(ii) for $$d>p$$ gives an advantage for choosing a summary statistic with $$d=p$$. The following proposition shows that for any summary statistic of dimension $$d>p$$ we can find a new $$p$$-dimensional summary statistic without any loss of information. The proof of the proposition is trivial and hence omitted. Proposition 1. Assume the conditions of Theorem 1. If $$d>p$$, define $$C=D{s}({\theta}_{0})^{\mathrm{T}}A({\theta}_{0})^{-1}$$. The $$p$$-dimensional summary statistic $$C{S}_{n}$$ has the same information matrix, $$I({\theta})$$, as $${S}_{n}$$. Therefore the asymptotic variance of $${h}_{\rm ABC}$$ based on $$C{s}_{\rm obs}$$ is smaller than or equal to that based on $${s}_{\rm obs}$$. Theorem 1 leads to the following natural definition. Definition 1. Assume that the conditions of Theorem 1 hold. Then the asymptotic variance of $${h}_{\rm ABC}$$ is ${\small{\text{AV}}}_{{h}_{\rm ABC}}=\frac{1}{a_{n}^{2}}D{h}({\theta}_{0})^{\mathrm{T}}I_{\rm ABC}^{-1}({\theta}_{0})D{h}({\theta}_{0})\text{.}$ 4. Asymptotic properties of rejection and the importance sampling algorithm 4.1. Asymptotic Monte Carlo error We now consider the Monte Carlo error involved in estimating $${h}_{\rm ABC}$$. Here we fix the data and consider solely the stochasticity of the Monte Carlo algorithm. We focus on Algorithm 1. Remember that $$N$$ is the Monte Carlo sample size. For $$i=1,\ldots,N$$, $${\theta}_{i}$$ is the proposed parameter value and $$w_{i}$$ is its importance sampling weight. Let $$\phi_{i}$$ be the indicator that is 1 if and only if $$\theta_{i}$$ is accepted in Step 3 of Algorithm 1 and let $$N_{\rm acc}=\sum_{i=1}^{N}\phi_{i}$$ be the number of accepted parameters. Provided $$N_{\rm acc}\geq1$$, we can estimate $${h}_{\rm ABC}$$ from the output of Algorithm 1 with ${\hat{h}}=\sum_{i=1}^{N}{h}({\theta}_{i})w_{i}\phi_{i}\Big/\sum_{i=1}^{N}w_{i}\phi_{i}\text{.}$ Define the acceptance probability $p_{{\rm acc},q}=\int q({\theta})\int f_{n}({s}\mid {\theta})K_{\varepsilon}({s}-{s}_{\rm obs})\,\mbox{d}{s}\,\mbox{d}{\theta}$ and the density of the accepted parameter $q_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon)=\frac{q_{n}({\theta})f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)}{\int q_{n}({\theta})f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)\,{\rm d}\theta}\text{.}$ Finally, define \begin{align} &\Sigma_{{\rm IS},n}=E_{\pi_{\rm ABC}}\left\{({h}({\theta})-{h}_{\rm ABC})^{2}\frac{\pi_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon_{n})}{q_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon_{n})}\right\}\!,\quad \Sigma_{{\rm ABC},n}=p_{{\rm acc},q_{n}}^{-1}\Sigma_{{\rm IS},n},\label{ISABC_var} \end{align} where $$\Sigma_{{\rm IS},n}$$ is the importance sampling variance with $$\pi_{\rm ABC}$$ as the target density and $$q_{\rm ABC}$$ as the proposal density. Note that $$p_{{\rm acc},q_{n}}$$ and $$\Sigma_{{\rm IS},n}$$, and hence $$\Sigma_{{\rm ABC},n}$$, depend on $${s}_{\rm obs}$$. Standard results give the following asymptotic distribution of $${\hat{h}}$$. Proposition 2. For a given $$n$$ and $${s}_{\rm obs}$$, if $${h}_{\rm ABC}$$ and $$\Sigma_{{\rm ABC},n}$$ are finite, then ${N}^{1/2}({\hat{h}}-{h}_{\rm ABC})\rightarrow \mathcal{N}(0,\Sigma_{{\rm ABC},n}),$ in distribution as $$N\rightarrow\infty$$. This proposition motivates the following definition. Definition 2. For a given $$n$$ and $${s}_{\rm obs}$$, assume that the conditions of Proposition 2 hold. Then the asymptotic Monte Carlo variance of $${\hat{h}}$$ is $${\small{\text{MCV}}}_{{\hat{h}}}={N}^{-1}\Sigma_{{\rm ABC},n}$$. 4.2. Asymptotic efficiency We have defined the asymptotic variance as $$n\rightarrow\infty$$ of $${h}_{\rm ABC}$$, and the asymptotic Monte Carlo variance as $$N\rightarrow\infty$$ of $${\hat{h}}$$. The error of $${h}_{\rm ABC}$$ when estimating $${h}({\theta}_{0})$$ and the Monte Carlo error of $${\hat{h}}$$ when estimating $${h}_{\rm ABC}$$ are independent, which suggests the following definition. Definition 3. Assume the conditions of Theorem 1, and that $${h}_{\rm ABC}$$ and $$\Sigma_{{\rm ABC},n}$$ are bounded in probability for any $$n$$. Then the asymptotic variance of $${\hat{h}}$$ is ${\small{\text{AV}}}_{{\hat{h}}}=\frac{1}{a_{n}^{2}}{h}({\theta}_{0})^{\mathrm{T}}I_{\rm ABC}^{-1}({\theta}_{0})D{h}({\theta}_{0})+\frac{1}{N}\Sigma_{{\rm ABC},n}\text{.}$ We can interpret the asymptotic variance of $${\hat{h}}$$ as a first-order approximation to the variance of our Monte Carlo estimator for both large $$n$$ and $$N$$. We wish to investigate the properties of this asymptotic variance, for large but fixed $$N$$, as $$n\rightarrow\infty$$. The asymptotic variance itself depends on $$n$$, and we would hope it would tend to zero as $$n$$ increases. Thus we will study the ratio of $${\small{\text{AV}}}_{{\hat{h}}}$$ to $${\small{\text{AV}}}_{{ \rm MLES}}$$, where, by Theorem 1, the latter is $$a_{n}^{-2}{h}({\theta}_{0})^{\mathrm{T}}I^{-1}({\theta}_{0})D{h}({\theta}_{0})$$. This ratio measures the efficiency of our Monte Carlo estimator relative to the maximum likelihood estimator based on the summaries; it quantifies the loss of efficiency from using a nonzero bandwidth and a finite Monte Carlo sample size. We will consider how this ratio depends on the choice of $$\varepsilon_{n}$$ and $$q_{n}({\theta})$$. Thus we introduce the following definition. Definition 4. For a choice of $$\varepsilon_{n}$$ and $$q_{n}({\theta})$$, we define the asymptotic efficiency of $${\hat{h}}$$ as ${\small{\text{AE}}}_{{\hat{h}}}=\lim_{n\rightarrow\infty}\frac{{\small{\text{AV}}}_{{ \rm MLES}}}{{\small{\text{AV}}}_{{\hat{h}}}}\text{.}$ If this limiting value is zero, we say that $${\hat{h}}$$ is asymptotically inefficient. We will investigate the asymptotic efficiency of $${\hat{h}}$$ under the assumption of Theorem 1 that $$\varepsilon_{n}=o(a_{n}^{-3/5})$$. We shall see that the convergence rate of the importance sampling variance $$\Sigma_{{\rm IS},n}$$ depends on how large $$\varepsilon_{n}$$ is relative to $$a_n$$, and so we further define $$a_{n,\varepsilon}=a_n$$ if $$\lim_{n\rightarrow\infty} a_n\varepsilon_n<\infty$$ and $$a_{n,\varepsilon}=\varepsilon_n^{-1}$$ otherwise. If our proposal distribution in Algorithm 1 is either the prior or the posterior, then the estimator is asymptotically inefficient. Theorem 2. Assume the conditions of Theorem 1. (i) If $$q_{n}({\theta})=\pi({\theta})$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d-p})$$ and $$\Sigma_{{\rm IS},n}=\Theta_{\rm p}(a_{n,\varepsilon}^{-2})$$. (ii) If $$q_{n}({\theta})=\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d})$$ and $$\Sigma_{{\rm IS},n}=\Theta_{\rm p}(a_{n,\varepsilon}^{p})$$. In both cases $${\hat{h}}$$ is asymptotically inefficient. The result in part (ii) shows a difference from standard importance sampling settings, where using the target distribution as the proposal leads to an estimator with no Monte Carlo error. The estimator $${\hat{h}}$$ is asymptotically inefficient because the Monte Carlo variance decays more slowly than $$1/a_{n}^{2}$$ as $$n\rightarrow\infty$$. However, this is caused by different factors in each case. To see this, consider the acceptance probability of a value of $$\theta$$ and the corresponding summary $${s}_{n}$$ simulated in one iteration of Algorithm 1. This acceptance probability depends on $$\frac{{s}_{n}-{s}_{\rm obs}}{\varepsilon_{n}}=\frac{1}{\varepsilon_{n}}\left[\{{s}_{n}-{s}({\theta})\}+\{{s}({\theta})-{s}({\theta}_{0})\}+\{{s}({\theta}_{0})-{s}_{\rm obs}\}\right],\label{eq:accp}$$ (2) where $${s}({\theta})$$, defined in Condition 4, is the limiting value of $${s}_{n}$$ as $$n\rightarrow\infty$$ if data are sampled from the model for parameter value $$\theta$$. By Condition 4, the first and third bracketed terms within the square brackets on the right-hand side are $$O_{\rm p}(a_{n}^{-1})$$. If we sample $$\theta$$ from the prior, the middle term is $$O_{\rm p}(1)$$, and thus (2) will blow up as $$\varepsilon_{n}$$ goes to zero. Hence $$p_{{\rm acc},\pi}$$ goes to zero as $$\varepsilon_{n}$$ goes to zero, which causes the estimate to be inefficient. If we sample from the posterior, then by Theorem 1 we expect the middle term to also be $$O_{\rm p}(a_{n}^{-1})$$. Hence (2) is well behaved as $$n\rightarrow\infty$$, and $$p_{{\rm acc},\pi}$$ is bounded away from zero, provided either $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$ or $$\varepsilon_{n}=\Omega(a_{n}^{-1})$$. However, if we use $$\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$ as a proposal distribution, the estimates are still inefficient due to an increasing variance of the importance weights: as $$n$$ increases the proposal distribution is more and more concentrated around $${\theta}_{0}$$, while $$\pi$$ does not change. 4.3. Efficient proposal distributions Consider proposing the parameter value from a location-scale family, i.e., our proposal is of the form $$\sigma_{n}\Sigma^{1/2}{X}+\mu_{n}$$, where $${X}\sim q(\cdot)$$, $$E({X})=0$$ and $$\mbox{var}({X})=I_{p}$$. This defines a general form of proposal density, where the centre $$\mu_{n}$$, the scale rate $$\sigma_{n}$$, the scale matrix $$\Sigma$$ and the base density $$q(\cdot)$$ all need to be specified. We will give conditions under which such a proposal density results in estimators that are efficient. Our results are based on an expansion of $$\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$. Consider the rescaled random variables $${t}=a_{n,\varepsilon}({\theta}-{\theta}_{0})$$ and $${v}=\varepsilon_{n}^{-1}({s}-{s}_{\rm obs})$$. Recall that $${T}_{\rm obs}=a_{n}A({\theta}_{0})^{-1/2}\{{s}_{\rm obs}-{s}({\theta}_{0})\}$$. Define an unnormalized joint density of $${t}$$ and $${v}$$ as $g_{n}({t},{v};\tau)=\begin{cases} \begin{array}{@{}ll} \mathcal{N}\big[\{D{s}({\theta}_{0})+\tau\}{t};a_{n}\varepsilon_{n}{v}+A({\theta}_{0})^{1/2}{T}_{\rm obs},A({\theta}_{0})\big]K({v}), & a_{n}\varepsilon_{n}\rightarrow c<\infty,\\[5pt] \mathcal{N}\left[\{D{s}({\theta}_{0})+\tau\}{t};{v}+\frac{1}{a_{n}\varepsilon_{n}}A({\theta}_{0})^{1/2}{T}_{\rm obs},\frac{1}{a_{n}^{2}\varepsilon_{n}^{2}}A({\theta}_{0})\right]K({v}), & a_{n}\varepsilon_{n}\rightarrow\infty, \end{array}\end{cases}$ and further define $$g_{n}({t};\tau)=\int g_{n}({t},{v};\tau)\,{\rm d}{v}$$. For large $$n$$, and for the rescaled variable $${t}$$, the leading term of $$\pi_{\rm ABC}$$ is then proportional to $$g_{n}({t};0)$$. For both limits of $$a_{n}\varepsilon_{n}$$, $$g_{n}({t};\tau)$$ is a continuous mixture of normal densities with the kernel density determining the mixture weights. Our main theorem requires conditions on the proposal density. First, it requires that $$\sigma_{n}=a_{n,\varepsilon}^{-1}$$ and that $$c_{\mu}=\sigma_{n}^{-1}({\mu}_{n}-{\theta}_{0})$$ is $$O_{\rm p}(1)$$. This ensures that under the scaling of $$t$$, as $$n\rightarrow\infty$$, the proposal is not increasingly overdispersed compared to the target density, and the acceptance probability can be bounded away from zero. Second, the proposal distribution is required to be sufficiently heavy-tailed. Condition 7. There exist positive constants $$m_{1}$$ and $$m_{2}$$ satisfying $$m_{1}^{2}I_{p}<Ds(\theta_{0})^{\mathrm{T}}Ds(\theta_{0})$$ and $$m_{2}I_{d}<A(\theta_{0})$$, $$\alpha\in(0,1)$$, $$\gamma\in(0,1)$$ and $$c\in(0,\infty)$$ such that for any $$\lambda>0$$, \begin{align} &\sup_{t\in\mathbb{R}^{p}}\frac{\mathcal{N}(t;0,m_{1}^{-2}m_{2}^{-2}\gamma^{-1})}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty, \quad \sup_{t\in\mathbb{R}^{p}}\frac{\bar{K}^{\alpha}(\\|\lambda t\\|^2)}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty,\quad \sup_{t\in\mathbb{R}^{p}}\frac{\bar{r}_{\rm max}(\\|m_{1}m_{2}{\gamma}^{1/2}t\\|^2)}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty,& \end{align} where $$\bar{r}_{\rm max}(\cdot)$$ satisfies $$r_{\rm max}(v)=\bar{r}_{\rm max}(\\|v\\|_{\Lambda}^2)$$, and for any random series $$c_{n}$$ in $$\mathbb{R}^{p}$$ satisfying $$c_{n}=O_{\rm p}(1)$$, $\sup_{t\in\mathbb{R}^{p}}\frac{q(t)}{q(t+c_{n})}=O_{\rm p}(1)\text{.}$ If we choose $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$, the Monte Carlo importance sampling variance for the accepted parameter values is $$\Theta(a_{n}^{-2})$$ and has the same order as the variance of the summary-based maximum likelihood estimator. Theorem 3. Assume the conditions of Theorem 1. If the proposal density $$q_{n}({\theta})$$ is $\beta\pi({\theta})+(1-\beta)\frac{1}{\sigma_{n}^{p}\|\Sigma\|^{1/2}}q\left\{\sigma_{n}^{-1}\Sigma^{-1/2}({\theta}-{\mu}_{n})\right\}\!,$ where $$\beta\in(0,1)$$, $$q(\cdot)$$ and $$\Sigma$$ satisfy Condition 7, $$\sigma_{n}=a_{n,\varepsilon}^{-1}$$ and $$c_{\mu}$$ is $$O_{\rm p}(1)$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d})$$ and $$\Sigma_{{\rm IS},n}=O_{\rm p}(a_{n,\varepsilon}^{-2})$$. Then if $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$, $${\small{\text{AE}}}_{{\hat{h}}}=\Theta_{\rm p}(1)$$. Furthermore, if $$d=p$$, $${\small{\text{AE}}}_{{\hat{h}}}=1-K/(N+K)$$ for some constant $$K$$. The mixture with $$\pi({\theta})$$ here is to control the importance weight in the tail area (Hesterberg, 1995). It is not clear whether this is needed in practice, or is just a consequence of the approach taken in the proof. Theorem 3 shows that with a good proposal distribution, if the acceptance probability is bounded away from zero as $$n$$ increases, the threshold $$\varepsilon_{n}$$ will have the preferred rate $$\Theta(a_{n}^{-1})$$. This supports using the acceptance rate to choose the threshold based on aiming for an appropriate proportion of acceptances (Del Moral et al., 2012; Biau et al., 2015). In practice, $$\sigma_{n}$$ and $$\mu_{n}$$ need to be adaptive to the observations since they depend on $$n$$. For $$q(\cdot)$$ and $$\Sigma$$, the following proposition gives a practical suggestion that satisfies Condition 7. Let $$T(\cdot;\gamma)$$ be the multivariate $$t$$ density with degree of freedom $$\gamma$$. The following result says that it is theoretically valid to choose any $$\Sigma$$ if a $$t$$ distribution is chosen as the base density. Proposition 3. Condition 7 is satisfied for $$q(\theta)=T({\theta};\gamma)$$ with any $$\gamma>0$$ and any $$\Sigma$$. Proof. The first part of Condition 7 follows as the $$t$$-density is heavy tailed relative to the normal density, $$\bar{K}(\cdot)$$ and $$\bar{r}_{\rm max}(\cdot)$$. The second part can be verified easily. □ 4.4. Iterative importance sampling Taken together, Theorem 3 and Proposition 3 suggest proposing from the mixture of $$\pi({\theta})$$ and a $$t$$ distribution with the scale matrix and centre approximating those of $$\pi_{\rm ABC}({\theta})$$. We suggest the following iterative procedure, similar in spirit to that of Beaumont et al. (2009). Algorithm 2. Iterative importance sampling approximate Bayesian computation. Input a mixture weight $$\beta$$, a sequence of acceptance rates $$\{p_{k}\}$$, and a location-scale family. Set $$q_1({\theta})=\pi({\theta})$$. For $$k=1,\ldots,K$$: Step 1. Run Algorithm 1 with simulation size $$N_{0}$$, proposal density $$\beta\pi({\theta})+(1-\beta)q_{k}({\theta})$$ and acceptance rate $$p_{k}$$, and record the bandwidth $$\varepsilon_{k}$$. Step 2. If $$\varepsilon_{k-1}-\varepsilon_{k}$$ is smaller than some positive threshold, stop. Otherwise, let $$\mu_{k+1}$$ and $$\Sigma_{k+1}$$ be the empirical mean and variance matrix of the weighted sample from Step 1, and let $$q_{k+1}({\theta})$$ be the density with centre $$\mu_{k+1}$$ and variance matrix $$2\Sigma_{k+1}$$. Step 3. If $$q_{k}({\theta})$$ is close to $$q_{k+1}({\theta})$$ or $$K=K_{\rm max}$$, stop. Otherwise, return to Step $$1$$. After the iteration stops at the $$K$$th step, run Algorithm 1 with the proposal density $$\beta\pi({\theta})+(1-\beta)q_{K+1}({\theta})$$, $$N-KN_{0}$$ simulations and $$p_{K+1}$$. In this algorithm, $$N$$ is the number of simulations allowed by the computing budget, $$N_{0}<N$$ and $$\{p_{k}\}$$ is a sequence of acceptance rates, which we use to choose the bandwidth. The maximum value $$K_{\rm max}$$ of $$K$$ is set such that $$K_{\rm max}N_{0}=N/2$$. The rule for choosing the new proposal distribution is based on approximating the mean and variance of the density proportional to $$\pi(\theta)f_{\rm ABC}({s}_{\rm obs}\mid \theta,\varepsilon)^{1/2}$$, which is optimal (Fearnhead & Prangle, 2012). It can be shown that these two moments are approximately equal to the mean and twice the variance of $$\pi_{\rm ABC}({\theta})$$ respectively. For the mixture weight, $$\beta$$, we suggest a small value, and use $$0.05$$ in the simulation study below. 5. Numerical examples 5.1. Gaussian likelihood with sample quantiles This example illustrates the results in § 3 with an analytically tractable problem. Assume that the observations $${Y}_{\rm obs}=(\,y_{1},\ldots,y_{n})$$ follow the univariate normal distribution $$\mathcal{N}(\mu,\sigma)$$ with true parameter values $$(1,2^{1/2})$$. Consider estimating the unknown parameter $$(\mu,\sigma)$$ with the uniform prior in the region $$[-10,10]\times[0,10]$$ using Algorithm 1. The summary statistic is $$\{\exp(\hat{q}_{\alpha_1}/2),\ldots,\exp(\hat{q}_{\alpha_d}/2)\}$$ where $${\hat{q}}_{\alpha}$$ is the sample quantile of $${Y}_{\rm obs}$$ for probability $$\alpha$$. We present results for $$n=10^{5}$$. Smaller sizes from $$10^{2}$$ to $$10^{4}$$ show similar patterns. The probabilities $$\alpha_{1},\ldots,\alpha_{d}$$ for calculating quantiles are selected with equal intervals in $$(0,1)$$, and $$d=2,9$$ and $$19$$ were tested. In order to investigate the Monte Carlo error-free performance, $$N$$ is chosen to be large enough that the Monte Carlo errors were negligible. We compare the performances of the approximate Bayesian computation estimator $${\hat{\theta}}$$, the maximum likelihood estimator based on the summary statistics and the maximum likelihood estimator based on the full dataset. Since the dimension reduction matrix $$C$$ in Proposition 1 can be obtained analytically, the performance of $${\hat{\theta}}$$ using the original $$d$$-dimensional summary is compared with that using the two-dimensional summary. The results of mean square error are presented in Fig. 1. Fig. 1. View largeDownload slide Mean square errors, MSE, of point estimates for $$200$$ datasets. Point estimates compared include $$\theta_{\rm ABC}$$ using the original summary statistic (solid) and the transformed summary statistic (dashed), the dimension of which is reduced to $$2$$ according to Proposition 1, and the maximum likelihood estimates based on the original summary statistic (dotted) and the full dataset (dash-dotted). Fig. 1. View largeDownload slide Mean square errors, MSE, of point estimates for $$200$$ datasets. Point estimates compared include $$\theta_{\rm ABC}$$ using the original summary statistic (solid) and the transformed summary statistic (dashed), the dimension of which is reduced to $$2$$ according to Proposition 1, and the maximum likelihood estimates based on the original summary statistic (dotted) and the full dataset (dash-dotted). The phenomena implied by Theorem 1 and Proposition 1 can be seen in this example, together with the limitations of these results. First, $${E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, equivalent to $${\hat{\theta}}$$ with small enough $$\varepsilon$$, and the maximum likelihood estimator based on the same summaries have similar accuracy. Second, when $$\varepsilon$$ is small, the mean square error of $${\hat{\theta}}$$ equals that of the maximum likelihood estimator based on the summary. When $$\varepsilon$$ becomes larger, for $$d>2$$ the mean square error increases more quickly than for $$d=2$$. This corresponds to the impact of the additional bias when $$d>p$$. For all cases, the two-dimensional summary obtained by projecting the original $$d$$ summaries is, for small $$\varepsilon$$, as accurate as the maximum likelihood estimator given the original $$d$$ summaries. This indicates that the lower-dimensional summary contains the same information as the original one. For larger $$\varepsilon$$, the performance of the reduced-dimension summaries is not stable, and is in fact worse than the original summaries for estimating $$\mu$$. This deterioration is caused by the bias of $${\hat{\theta}}$$, which for larger $$\varepsilon$$ is dominated by higher-order terms in $$\varepsilon$$ which could be ignored in our asymptotic results. 5.2. Stochastic volatility with ar$$(1)$$ dynamics We consider a stochastic volatility model from Sandmann & Koopman (1998) for the de-meaned returns of a portfolio. Denote this return for the $$t$$th time period by $$y_t$$. Then $x_{t} =\phi x_{t-1}+\eta_{t},\ \eta_{t}\sim \mathcal{N}(0,\sigma_{\eta}^{2}); \quad y_{t} =\bar{\sigma}\exp(x_t/2)\xi_{n},\ \xi_{t}\sim \mathcal{N}(0,1),$ where $$\eta_{t}$$ and $$\xi_{t}$$ are independent, and $$x_t$$ is a latent state that quantifies the level of volatility for time period $$t$$. By the transformation $$y_{t}^{}=\log y_{t}^{2}$$ and $$\xi_{t}^{}=\log\xi_{t}^{2}$$, the observation equation in the state-space model can be transformed to \begin{equation} y_{n}^{} =2\log\bar{\sigma}+x_{n}+\xi_{n}^{},\quad \exp(\xi_{n}^{})\sim\chi_{1}^{2}, \end{equation} which is linear and non-Gaussian. Approximate Bayesian computation can be used to obtain an off-line estimator for the unknown parameters of this model. Here we illustrate the effectiveness of iteratively choosing the importance proposal for large $$n$$ by comparing with rejection sampling. In the iterative algorithm, a $$t$$ distribution with five degrees of freedom is used to construct $$q_{k}$$. Consider estimating the parameter $$(\phi,\sigma_{\eta},\log\bar{\sigma})$$ under a uniform prior in the region $$[0,1)\times[0.1,3]\times[-10,-1]$$. The setting with the true parameter $$(\phi,\sigma_{\eta},\log\bar{\sigma})=(0.9,0.675,-4.1)$$ is studied. We use a three-dimensional summary statistic that stores the mean, variance and lag-one autocovariance of the transformed data. If there were no noise in the state equation for $$\xi_{n}^{}$$, then this would be a sufficient statistic of $${Y}^{}$$, and hence is a natural choice for the summary statistic. The uniform kernel is used in the accept-reject step. We evaluate rejection sampling and iterative importance sampling methods on data of length $$n=100,500,2000$$ and $$10\,000$$, and use $$N=40\,000$$ Monte Carlo simulations. For iterative importance sampling, the sequence $$\{p_{k}\}$$ has the first five values decreasing linearly from $$5\%$$ to $$1\%$$, and later values being $$1\%$$. We further set $$N_0=2000$$ and $$K_{\rm max}=10$$. For the rejection sampler, acceptance probabilities of both $$5\%$$ and $$1\%$$ were tried and $$5\%$$ was chosen as it gave better performance. The simulation results are shown in Fig. 2. Fig. 2. View largeDownload slide Comparisons of rejection (solid) and iterative importance sampling (dashed) versions of approximate Bayesian computation. For each $$n$$, the logarithm of the average mean square error, MSE, across $$100$$ datasets is reported. For each dataset, the Monte Carlo sample size is $$40\,000$$. Ratios of mean square errors of the two methods are given in the table, and smaller values indicate better performance of iterative importance sampling. For each method in the plots, a line is fitted to the connected lines and the slope is reported in the table. Smaller values indicate faster decrease of the mean square error. Fig. 2. View largeDownload slide Comparisons of rejection (solid) and iterative importance sampling (dashed) versions of approximate Bayesian computation. For each $$n$$, the logarithm of the average mean square error, MSE, across $$100$$ datasets is reported. For each dataset, the Monte Carlo sample size is $$40\,000$$. Ratios of mean square errors of the two methods are given in the table, and smaller values indicate better performance of iterative importance sampling. For each method in the plots, a line is fitted to the connected lines and the slope is reported in the table. Smaller values indicate faster decrease of the mean square error. For all parameters, iterative importance sampling shows increasing advantage over rejection sampling as $$n$$ increases. For larger $$n$$, the iterative procedure obtains a centre for proposals closer to the true parameter and a bandwidth that is smaller than those used for rejection sampling. These contribute to the more accurate estimators. It is easy to estimate $$\log\bar{\sigma}$$, since the expected summary statistic $$\tilde{E}({Y}^{})$$ is roughly linear in $$\log\bar{\sigma}$$. Thus iterative importance sampling has less of an advantage over rejection sampling when estimating this parameter. 6. Discussion Our results suggest that one can obtain efficient estimates using approximate Bayesian computation with a fixed Monte Carlo sample size as $$n$$ increases. Thus the computational complexity of approximate Bayesian computation will just be the complexity of simulating a sample of size $$n$$ from the underlying model. Our results on the Monte Carlo accuracy of approximate Bayesian computation considered the importance sampling implementation given in Algorithm 1. If we do not use the uniform kernel, then there is a simple improvement on this algorithm, which absorbs the accept-reject probability within the importance sampling weight. A simple Rao–Blackwellization argument then shows that this leads to a reduction in Monte Carlo variance, so our positive results about the scaling of approximate Bayesian computation with $$n$$ will also immediately apply to this implementation. Similar positive Monte Carlo results are likely to apply to Markov chain Monte Carlo implementations of approximate Bayesian computation. A Markov chain Monte Carlo version will be efficient provided the acceptance probability does not degenerate to zero as $$n$$ increases. However, at stationarity it will propose parameter values from a distribution close to the approximate Bayesian computation posterior density, and Theorems 2 and 3 suggest that for such a proposal distribution the acceptance probability will be bounded away from zero. Whilst our theoretical results suggest that point estimates based on approximate Bayesian computation have good properties, they do not suggest that the approximate Bayesian computation posterior is a good approximation to the true posterior. In fact, Frazier et al. (2016) show that it will overestimate uncertainty if $$\varepsilon_n=O(a_n^{-1})$$. However, Li & Fearnhead (2018) show that using regression methods (Beaumont et al., 2002) to post-process approximate Bayesian computation output can lead to both efficient point estimation and accurate quantification of uncertainty. Acknowledgement This work was supported by the U.K. Engineering and Physical Sciences Research Council. Supplementary material Supplementary material available at Biometrika online contains proofs of the main results. References Allingham D. , King R. A. R. & Mengersen K. L. ( 2009 ). Bayesian estimation of quantile distributions. Statist. Comp. 19 , 189 – 201 . 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Google Scholar CrossRef Search ADS © 2018 Biometrika Trust This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrika Oxford University Press # On the asymptotic efficiency of approximate Bayesian computation estimators , Volume Advance Article (2) – Jan 20, 2018 15 pages Publisher Oxford University Press ISSN 0006-3444 eISSN 1464-3510 D.O.I. 10.1093/biomet/asx078 Publisher site See Article on Publisher Site ### Abstract SUMMARY Many statistical applications involve models for which it is difficult to evaluate the likelihood, but from which it is relatively easy to sample. Approximate Bayesian computation is a likelihood-free method for implementing Bayesian inference in such cases. We present results on the asymptotic variance of estimators obtained using approximate Bayesian computation in a large data limit. Our key assumption is that the data are summarized by a fixed-dimensional summary statistic that obeys a central limit theorem. We prove asymptotic normality of the mean of the approximate Bayesian computation posterior. This result also shows that, in terms of asymptotic variance, we should use a summary statistic that is of the same dimension as the parameter vector, $$p$$, and that any summary statistic of higher dimension can be reduced, through a linear transformation, to dimension $$p$$ in a way that can only reduce the asymptotic variance of the posterior mean. We look at how the Monte Carlo error of an importance sampling algorithm that samples from the approximate Bayesian computation posterior affects the accuracy of estimators. We give conditions on the importance sampling proposal distribution such that the variance of the estimator will be of the same order as that of the maximum likelihood estimator based on the summary statistics used. This suggests an iterative importance sampling algorithm, which we evaluate empirically on a stochastic volatility model. 1. Introduction Many statistical applications involve inference about models that are easy to simulate from, but for which it is difficult, or impossible, to calculate likelihoods. In such situations it is possible to use the fact that we can simulate from the model to enable us to perform inference. There is a wide class of such likelihood-free methods of inference, including indirect inference (Gouriéroux & Ronchetti, 1993; Heggland & Frigessi, 2004), the bootstrap filter (Gordon et al., 1993), simulated methods of moments (Duffie & Singleton, 1993), and synthetic likelihood (Wood, 2010). We consider a Bayesian version of these methods, termed approximate Bayesian computation. This involves defining an approximation to the posterior distribution in such a way that it is possible to sample from this approximate posterior using only the ability to sample from the model. Arguably the first approximate Bayesian computation method was that of Pritchard et al. (1999), and these methods have been popular within population genetics (Beaumont et al., 2002), ecology (Beaumont, 2010) and systems biology (Toni et al., 2009). More recently, there have been applications to areas including stereology (Bortot et al., 2007), finance (Peters et al., 2011) and cosmology (Ishida et al., 2015). Let $$K({x})$$ be a density kernel, scaled, without loss of generality, so that $$\max_{{x}}K({x})=1$$. Further, let $$\varepsilon>0$$ be a bandwidth. Denote the data by $${Y}_{\rm obs}=(\,y_{\rm obs,1},\ldots,y_{{\rm obs},n})$$. Assume we have chosen a finite-dimensional summary statistic $${s}_{n}({Y})$$, and write $${s}_{\rm obs}={s}_{n}({Y}_{\rm obs})$$. If we model the data as a draw from a parametric density, $$f_{n}({y} \mid{\theta})$$, and assume a prior $$\pi({\theta})$$, then we define the approximate Bayesian computation posterior as $$\pi_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon)\propto\pi({\theta})\int f_{n}({s}_{\rm obs}+\varepsilon{v} \mid {\theta})K({v})\,{\rm d}{v},\label{eq:piABC}$$ (1) where $$f_{n}({s} \mid {\theta})$$ is the density for the summary statistic implied by $$f_{n}({y} \mid {\theta})$$. Let $$f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)=\int f_{n}({s}_{\rm obs}+\varepsilon{v} \mid {\theta})K({v})\,{\rm d}{v}$$. This framework encompasses most implementations of approximate Bayesian computation. In particular, the use of the uniform kernel corresponds to the popular rejection-based rule (Beaumont et al., 2002). The idea is that $$f_{\rm ABC}({s}_{\rm obs}\,{\mid}\, {\theta},\varepsilon)$$ is an approximation of the likelihood. The approximate Bayesian computation posterior, which is proportional to the prior multiplied by this likelihood approximation, is an approximation of the true posterior. The likelihood approximation can be interpreted as a measure of how close, on average, the summary $${s}_{n}$$ simulated from the model is to the summary for the observed data, $${s}_{\rm obs}$$. The choices of kernel and bandwidth determine the definition of closeness. By defining the approximate posterior in this way, we can simulate samples from it using standard Monte Carlo methods. One approach, which we will focus on later, uses importance sampling. Let $$K_{\varepsilon}({x})=K({x}/\varepsilon)$$. Given a proposal density $$q_{n}({\theta})$$, a bandwidth $$\varepsilon$$, and a Monte Carlo sample size $$N$$, an importance sampler would proceed as in Algorithm 1. The set of accepted parameters and their associated weights provides a Monte Carlo approximation to $$\pi_{\rm ABC}$$. If we set $$q_{n}({\theta})=\pi({\theta})$$ then this is just a rejection sampler. In practice sequential importance sampling methods are often used to learn a good proposal distribution (Beaumont et al., 2009). Algorithm 1. Importance and rejection sampling approximate Bayesian computation. Step 1. Simulate $${\theta}_{1},\ldots,{\theta}_{N}\sim q_{n}({\theta})$$. Step 2. For each $$i=1,\ldots,N$$, simulate $${Y}^{(i)}=\big\{\,y_{1}^{(i)},\ldots,y_{n}^{(i)}\big\}\sim f_{n}(\,y\mid {\theta}_{i})$$. Step 3. For each $$i=1,\ldots,N$$, accept $${\theta}_{i}$$ with probability $$K_{\varepsilon}\big\{{s}_{n}^{(i)}-{s}_{\rm obs}\big\}$$, where $${s}_{n}^{(i)}={s}_{n}\{{Y}^{(i)}\}$$, and define the associated weight as $$w_{i}=\pi({\theta}_{i})/q_{n}({\theta}_{i})$$. There are three choices in implementing approximate Bayesian computation: the choice of summary statistic, the choice of bandwidth, and the Monte Carlo algorithm. For importance sampling, the last of these involves specifying the Monte Carlo sample size, $$N$$, and the proposal density, $$q_{n}({\theta})$$. These, roughly, relate to three sources of approximation. To see this, note that as $$\varepsilon\rightarrow0$$ we would expect (1) to converge to the posterior given $${s}_{\rm obs}$$ (Fearnhead & Prangle, 2012). Thus the choice of summary statistic governs the approximation, or loss of information, between using the full posterior distribution and using the posterior given the summary. The value $$\varepsilon$$ then affects how close the approximate Bayesian computation posterior is to the posterior given the summary. Finally there is Monte Carlo error from approximating the approximate Bayesian computation posterior with a Monte Carlo sample. The Monte Carlo error is affected not only by the Monte Carlo algorithm, but also by the choices of summary statistic and bandwidth, which together affect the probability of acceptance in Step 3 of Algorithm 1. Having a higher-dimensional summary statistic, or a smaller value of $$\varepsilon$$, will tend to reduce this acceptance probability and hence increase the Monte Carlo error. This work studies the interaction between the three sources of error, when the summary statistics obey a central limit theorem for large $$n$$. We are interested in the efficiency of approximate Bayesian computation, where by efficiency we mean that an estimator obtained from running Algorithm 1 has the same rate of convergence as the maximum likelihood estimator for the parameter given the summary statistic. In particular, this work is motivated by the question of whether approximate Bayesian computation can be efficient as $$n\rightarrow\infty$$ if we have a fixed Monte Carlo sample size. Intuitively this appears unlikely. For efficiency we will need $$\varepsilon\rightarrow0$$ as $$n\rightarrow\infty$$, and this corresponds to an increasingly strict condition for acceptance. Thus we may imagine that the acceptance probability will necessarily tend to zero as $$n$$ increases, and we will need an increasing Monte Carlo sample size to compensate for this. However, our results show that Algorithm 1 can be efficient if we choose a proposal distribution with a suitable scale and location and appropriately heavy tails. If we use such a proposal distribution and have a summary statistic of the same dimension as the parameter vector, then the posterior mean of approximate Bayesian computation is asymptotically unbiased with a variance that is $$1+O(1/N)$$ times that of the estimator maximizing the likelihood of the summary statistic. This is similar to asymptotic results for indirect inference (Gouriéroux & Ronchetti, 1993; Heggland & Frigessi, 2004). Our results also lend theoretical support to methods that choose the bandwidth indirectly by specifying the proportion of samples that are accepted, as this leads to a bandwidth which is of the optimal order in $$n$$. We first prove a Bernstein–von Mises-type theorem for the posterior mean of approximate Bayesian computation. This is a nonstandard convergence result, as it is based on the partial information contained in the summary statistics. For related convergence results see Clarke & Ghosh (1995) and Yuan & Clarke (2004), though these do not consider the case where the dimension of the summary statistic is larger than that of the parameter. Dealing with this case introduces extra challenges. Our convergence result for the posterior mean of approximate Bayesian computation has practically important consequences. It shows that any $$d$$-dimensional summary with $$d>p$$ can be projected to a $$p$$-dimensional summary statistic without any loss of information. Furthermore it shows that using a summary statistic of dimension $$d>p$$ can lead to an increased bias, so the asymptotic variance can be reduced if the optimal $$p$$-dimensional projected summary is used instead. If a $$d$$-dimensional summary is used, with $$d>p$$, it suggests choosing the variance of the kernel to match the variance of the summary statistics. This paper adds to a growing literature on the theoretical properties of approximate Bayesian computation. Initial results focused on comparing the bias of approximate Bayesian computation to the Monte Carlo error, and studying how these depend on the choice of $$\varepsilon$$. The convergence rate of the bias is shown to be $$O(\varepsilon^{2})$$ in various settings (e.g., Barber et al., 2015). This can then be used to consider how the choice of $$\varepsilon$$ should depend on the Monte Carlo sample size so as to balance bias and Monte Carlo variability (Blum, 2010; Barber et al., 2015; Biau et al., 2015). There has also been work on consistency of approximate Bayesian computation estimators. Marin et al. (2014) consider consistency when performing model choice and Frazier et al. (2016) consider consistency for parameter estimation. The latter work, which appeared after the first version of this paper, includes a result on the asymptotic normality of the posterior mean that is similar to our Theorem 1, albeit under different conditions, and also gives results on the asymptotic form of the posterior obtained using approximate Bayesian computation. This shows that for many implementations of approximate Bayesian computation, the posterior will overestimate the uncertainty in the parameter estimate that it gives. Finally, a number of papers have looked at the choice of summary statistics (e.g., Wegmann et al., 2009; Blum, 2010; Prangle et al., 2014). Our Theorem 1 gives insight into this. As mentioned above, this theorem shows that, in terms of minimizing the asymptotic variance, we should use a summary statistic of the same dimension as the number of parameters. In particular it supports the suggestion in Fearnhead & Prangle (2012) of having one summary per parameter, with that summary approximating the maximum likelihood estimator for that parameter. 2. Notation and set-up Denote the data by $${Y}_{\rm obs}=(\,y_{{\rm obs},1},\ldots,$$$$y_{{\rm obs},n})$$, where $$n$$ is the sample size and each observation $$y_{{\rm obs},i}$$ can be of arbitrary dimension. We make no assumption directly on the data, but make assumptions on the distribution of the summary statistics. We consider the asymptotics as $$n\rightarrow\infty$$, and denote the density of $${Y}_{\rm obs}$$ by $$\,f_{n}({y} \mid {\theta})$$, where $${\theta}\in\mathcal{P}\subset\mathbb{R}^p$$. We let $${\theta}_{0}$$ denote the true parameter value and $$\pi({\theta})$$ its prior distribution. For a set $$A$$, let $$A^{\rm c}$$ be its complement with respect to the whole space. We assume that $${\theta}_{0}$$ is in the interior of the parameter space, and that the prior is differentiable in a neighbourhood of the true parameter: Condition 1. There exists some $$\delta_{0}>0$$ such that $$\mathcal{P}_{0}=\{{\theta}:\|{\theta}-{\theta}_{0}\|<\delta_{0}\}\subset\mathcal{P}$$, $$\pi({\theta})\in C^{1}(\mathcal{P}_{0})$$ and $$\pi({\theta}_{0})>0$$. To implement approximate Bayesian computation we will use a $$d$$-dimensional summary statistic, $${s}_{n}({Y})\in\mathbb{R}^{d}$$, such as a vector of sample means of appropriately chosen functions. We assume that $${s}_{n}({Y})$$ has a density function which depends on $$n$$, and we denote this by $$f_{n}({s} \mid {\theta})$$. We will use the shorthand $${S}_{n}$$ to denote the random variable with density $$f_{n}({s} \mid {\theta})$$. In approximate Bayesian computation we use a kernel, $$K({x})$$, with $$\max_{{x}}K({x})=1$$, and a bandwidth $$\varepsilon>0$$. As we vary $$n$$ we will often wish to vary $$\varepsilon$$, and in these situations we denote the bandwidth by $$\varepsilon_{n}$$. For Algorithm 1 we require a proposal distribution, $$q_{n}({\theta})$$, and allow this to depend on $$n$$. We assume the following conditions on the kernel, which are satisfied by all commonly-used kernels, Condition 2. The kernel satisfies (i) $$\int{v} K({v})\,{\rm d}{v}=0$$; (ii) $$\int\prod_{k=1}^{l}v_{i_{k}}K({v})\,{\rm d}{v}<\infty$$ for any coordinates $$(v_{i_{1}},\ldots,v_{i_{l}})$$ of $${v}$$ and $$l\leq p+6$$; (iii) $$K({v})\propto\bar{K}(\\|{v}\\|_{\Lambda}^{2})$$ where $$\\|{v}\\|_{\Lambda}^{2}={v}^{{\rm T}}\Lambda{v}$$ with $$\Lambda$$ a positive-definite matrix, and $$K({v})$$ is a decreasing function of $$\\|{v}\\|_{\Lambda}$$; (iv) $$K({v})=O\{\exp(-c_1 \\|v\\|_{\Gamma}^{-1})\}$$ for some $$\alpha_{1}>0$$ and $$c_{1}>0$$ as $$\\|{v}\\|_{\Lambda}\rightarrow\infty$$. For a real function $$g({x})$$ denote its $$k$$th partial derivative at $${x}={x}_{0}$$ by $$D_{x_{k}}g({x}_{0})$$, the gradient function by $$D_{{x}}g({x}_{0})$$ and the Hessian matrix by $$H_{{x}}g({x}_{0})$$. To simplify notation, $$D_{\theta_{k}}$$, $$D_{{\theta}}$$ and $$H_{{\theta}}$$ are written as $$D_{k}$$, $$D$$ and $$H$$, respectively. For a series $$x_{n}$$ we use the notation that for large enough $$n$$, $$x_{n}=\Theta(a_{n})$$ if there exist constants $$m$$ and $$M$$ such that $$0<m<\|x_{n}/a_{n}\|<M<\infty$$, and $$x_{n}=\Omega(a_{n})$$ if $$\|x_{n}/a_{n}\|\rightarrow\infty$$. For two square matrices $$A$$ and $$B$$, we say that $$A\leq B$$ if $$B-A$$ is semipositive definite, and $$A<B$$ if $$B-A$$ is positive definite. Our theory will focus on estimates of some function $${h}(\theta)$$ of $$\theta$$, which satisfies differentiability and moment conditions that will control the remainder terms in Taylor expansions. Condition 3. The $$k$$th coordinate of $${h}({\theta})$$, $$h_{k}({\theta})$$, satisfies (i) $$h_{k}({\theta})\in C^{1}(\mathcal{P}_{0})$$; (ii) $$D_{k}h({\theta}_{0})\neq0$$; and (iii) $$\int h_{k}({\theta})^{2}\pi({\theta})\,{\rm d}{\theta}<\infty$$. The asymptotic results presuppose a central limit theorem for the summary statistic. Condition 4. There exists a sequence $$a_{n}$$, with $$a_{n}\rightarrow\infty$$ as $$n\rightarrow\infty$$, a $$d$$-dimensional vector $${s}({\theta})$$ and a $$d\times d$$ matrix $$A({\theta})$$ such that for all $${\theta}\in\mathcal{P}_0$$, $a_{n}\{{S}_{n}-{s}({\theta})\}\rightarrow \mathcal{N}\{0,A({\theta})\}, \quad {n\rightarrow\infty},$ with convergence in distribution. We also assume that $$s_{\rm obs}\rightarrow s(\theta_0)$$ in probability. Furthermore, assume that (i) $${s}({\theta})\in C^{1}(\mathcal{P}_{0})$$ and $$A({\theta})\in C^{1}(\mathcal{P}_{0})$$, and $$A({\theta})$$ is positive definite for $${\theta}\in\mathcal{P}_{0}$$; (ii) for any $$\delta>0$$ there exists a $$\delta'>0$$ such that $$\\|s(\theta)-s(\theta_0)\\|>\delta'$$ for all $$\theta$$ satisfying $$\\|\theta-\theta_0\\|>\delta$$; (iii) $$I({\theta})=D{s}({\theta})^{{\rm T}}A^{-1}({\theta})D{s}({\theta})$$ has full rank at $${\theta}={\theta}_{0}$$. Under Condition 4, $$a_{n}$$ is the rate of convergence in the central limit theorem. If the data are independent and identically distributed, and the summaries are sample means of functions of the data or of quantiles, then $$a_{n}=n^{1/2}$$. In most applications the data will be dependent, but if summaries are sample means (Wood, 2010), quantiles (Peters et al., 2011; Allingham et al., 2009; Blum & François, 2010) or linear combinations thereof (Fearnhead & Prangle, 2012), then a central limit theorem will often still hold, though $$a_n$$ may increase more slowly than $$n^{1/2}$$. Part (ii) of Condition 4 is required for the true parameter to be identifiable given only the summary of data. The asymptotic variance of the summary-based maximum likelihood estimator for $${\theta}$$ is $$I^{-1}({\theta}_{0})/a_{n}^{2}$$. Condition (iii) ensures that this variance is valid at the true parameter. We next require a condition that controls the difference between $$f_{n}({s}\mid {\theta})$$ and its limiting distribution for $${\theta}\in\mathcal{P}_{0}$$. Let $$\mathcal{N}({x};{\mu},\Sigma)$$ be the normal density at $${x}$$ with mean $${\mu}$$ and variance $$\Sigma$$. Define $$\tilde{f}_{n}({s} \mid {\theta})=\mathcal{N}\{{s};{s}({\theta}),A({\theta})/a_{n}^{2}\}$$ and the standardized random variable $$W_{n}({s})=a_{n}A({\theta})^{-1/2}\{{s}-{s}({\theta})\}$$. Let $$\tilde{f}_{W_{n}}({w} \mid {\theta})$$ and $$f_{W_{n}}({w} \mid {\theta})$$ be the densities of $$W_{n}({s})$$ when $${s}\sim\tilde{f}_{n}({s} \mid {\theta})$$ and $$f_{n}({s} \mid {\theta})$$ respectively. The condition below requires that the difference between $$f_{W_{n}}({w} \mid {\theta})$$ and its Edgeworth expansion $$\tilde{f}_{W_{n}}({w} \mid {\theta})$$ be of $$o(a_n^{-2/5})$$ and can be bounded by a density with exponentially decreasing tails. This is weaker than the standard requirement, $$o(a_n^{-1})$$, for the remainder in the Edgeworth expansion. Condition 5. There exists $$\alpha_{n}$$ satisfying $$\alpha_{n}/a_n^{2/5}\rightarrow\infty$$ and a density $$r_{\rm max}({w})$$ satisfying Condition 2(ii) and (iii) where $$K({v})$$ is replaced with $$r_{\rm max}({w})$$, such that $$\sup_{{\theta}\in\mathcal{P}_{0}}\alpha_{n}\| f_{W_{n}}({w} \mid {\theta})-\tilde{f}_{W_{n}}({w} \mid {\theta})\|\leq c_{3}r_{\rm max}({w})$$ for some positive constant $$c_{3}$$. The following condition further assumes that $$f_{n}(s \mid \theta)$$ has exponentially decreasing tails with rate uniform in the support of $$\pi(\theta)$$. Condition 6. The following statements hold: (i) $$r_{\rm max}(w)$$ satisfies Condition 2(iv); and (ii) $$\sup_{\theta\in\mathcal{P}_{0}^{\rm c}}f_{W_{n}}(w \mid \theta)=O\{\exp(-c_2 \\|w\\|^{\alpha_2})\}$$ as $$\\|w\\|\rightarrow\infty$$ for some positive constants $$c_{2}$$ and $$\alpha_{2}$$, and $$A(\theta)$$ is bounded in $$\mathcal{P}$$. 3. Posterior mean asymptotics We first ignore any Monte Carlo error, and focus on the ideal estimator of the true posterior mean from approximate Bayesian computation. This is the posterior mean, $${h}_{\rm ABC}$$, where ${h}_{\rm ABC}={E}_{\pi_{\rm ABC}}\{{h}({\theta})\mid {s}_{\rm obs}\}= \int {h}({\theta}) \pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_n)\text{.}$ This estimator depends on $$\varepsilon_n$$, but we suppress this from the notation. As an approximation to the true posterior mean $${E}\{{h}({\theta})\mid {Y}_{\rm obs}\}$$, $${h}_{\rm ABC}$$ contains errors from the choice of bandwidth $$\varepsilon_{n}$$ and summary statistic $${s}_{\rm obs}$$. To understand the effects of these two sources of error, we derive results for the asymptotic distributions of $${h}_{\rm ABC}$$ and the likelihood-based estimators, including the summary-based maximum likelihood estimator and the summary-based posterior mean, where we consider randomness solely due to the randomness of the data. Let $$T_{{\rm obs}}=a_{n}A(\theta_{0})^{-1/2}\{s_{\rm obs}-s(\theta_{0})\}$$. Theorem 1. Assume Conditions 1–6. (i) Let $$\hat{{{\theta}}}_{_{{ \rm MLES}}}=\mathop{\arg\max}\limits_{\theta\in\mathcal{P}}\log f_{n}({s}_{\rm obs}\mid {\theta})$$. For $${h}_{{s}}={h}(\hat{{{\theta}}}_{_{{ \rm MLES}}})$$ or $${E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, $a_{n}\{{h}_{{s}}-{h}({\theta}_{0})\}\rightarrow \mathcal{N}\{0,D{h}({\theta}_{0})^{\mathrm{T}}I^{-1}({\theta}_{0})D{h}({\theta}_{0})\},\quad n\rightarrow\infty,$ with convergence in distribution. (ii) Define $$c_{\infty}= \lim_{n\rightarrow\infty}a_{n}\varepsilon_{n}$$. Let $$Z$$ be the weak limit of $$T_{\rm obs}$$, which has a standard normal distribution, and let $$R(c_{\infty},Z)$$ be a random vector with mean zero that is defined in the Supplementary Material. If $$\varepsilon_{n}=o(a_{n}^{-3/5})$$, then $a_{n}\{h_{{\rm ABC}}-h(\theta_{0})\}\rightarrow Dh(\theta_{0})^{\mathrm{T}}\{I(\theta_{0})^{-1/2}Z+R(c_\infty,Z)\}, \quad n\rightarrow\infty,$ with convergence in distribution. If either $$\varepsilon_{n}=o(a_{n}^{-1})$$, $$d=p$$, or the covariance matrix of $$K(v)$$ is proportional to $$A(\theta_{0})$$, then $$R(c_{\infty},Z)=0$$. For other cases, the variance of $$I(\theta_{0})^{-1/2}Z+R(c_{\infty},Z)$$ is no less than $$I^{-1}(\theta_{0})$$. Theorem 1 (i) shows the validity of posterior inference based on the summary statistics. Regardless of the sufficiency and dimension of $${s}_{\rm obs}$$, the posterior mean based on the summary statistics is consistent and asymptotically normal with the same variance as the summary-based maximum likelihood estimator. Denote the bias of approximate Bayesian computation, $${h}_{\rm ABC}-{E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, by $$\mbox{bias}_{\rm ABC}$$. The choice of bandwidth affects the size of the bias. Theorem 1(ii) indicates two regimes for the bandwidth for which the posterior mean of approximate Bayesian computation has good properties. The first case is when $$\varepsilon_{n}$$ is $$o(1/a_{n})$$. For this regime the posterior mean of approximate Bayesian computation always has the same asymptotic distribution as that of the true posterior given the summaries. The other case is when $$\varepsilon_{n}$$ is $$o(a_n^{-3/5})$$ but not $$o(n^{-1})$$. We obtain the same asymptotic distribution if either $$d=p$$ or we choose the kernel variance to be proportional to the variance of the summary statistics. In general for this regime of $$\varepsilon_n$$, $${h}_{\rm ABC}$$ will be less efficient than the summary-based maximum likelihood estimator. When $$d>p$$, Theorem 1(ii) shows that $$\mbox{bias}_{\rm ABC}$$ is nonnegligible and can increase the asymptotic variance. This is because the leading term of $$\mbox{bias}_{\rm ABC}$$ is proportional to the average of $${v}={s}-{s}_{\rm obs}$$, the difference between the simulated and observed summary statistics. If $$d>p$$, the marginal density of $${v}$$ is generally asymmetric, and thus is no longer guaranteed to have a mean of zero. One way to ensure that there is no increase in the asymptotic variance is to choose the variance of the kernel to be proportional to the variance of the summary statistics. The loss of efficiency we observe in Theorem 1(ii) for $$d>p$$ gives an advantage for choosing a summary statistic with $$d=p$$. The following proposition shows that for any summary statistic of dimension $$d>p$$ we can find a new $$p$$-dimensional summary statistic without any loss of information. The proof of the proposition is trivial and hence omitted. Proposition 1. Assume the conditions of Theorem 1. If $$d>p$$, define $$C=D{s}({\theta}_{0})^{\mathrm{T}}A({\theta}_{0})^{-1}$$. The $$p$$-dimensional summary statistic $$C{S}_{n}$$ has the same information matrix, $$I({\theta})$$, as $${S}_{n}$$. Therefore the asymptotic variance of $${h}_{\rm ABC}$$ based on $$C{s}_{\rm obs}$$ is smaller than or equal to that based on $${s}_{\rm obs}$$. Theorem 1 leads to the following natural definition. Definition 1. Assume that the conditions of Theorem 1 hold. Then the asymptotic variance of $${h}_{\rm ABC}$$ is ${\small{\text{AV}}}_{{h}_{\rm ABC}}=\frac{1}{a_{n}^{2}}D{h}({\theta}_{0})^{\mathrm{T}}I_{\rm ABC}^{-1}({\theta}_{0})D{h}({\theta}_{0})\text{.}$ 4. Asymptotic properties of rejection and the importance sampling algorithm 4.1. Asymptotic Monte Carlo error We now consider the Monte Carlo error involved in estimating $${h}_{\rm ABC}$$. Here we fix the data and consider solely the stochasticity of the Monte Carlo algorithm. We focus on Algorithm 1. Remember that $$N$$ is the Monte Carlo sample size. For $$i=1,\ldots,N$$, $${\theta}_{i}$$ is the proposed parameter value and $$w_{i}$$ is its importance sampling weight. Let $$\phi_{i}$$ be the indicator that is 1 if and only if $$\theta_{i}$$ is accepted in Step 3 of Algorithm 1 and let $$N_{\rm acc}=\sum_{i=1}^{N}\phi_{i}$$ be the number of accepted parameters. Provided $$N_{\rm acc}\geq1$$, we can estimate $${h}_{\rm ABC}$$ from the output of Algorithm 1 with ${\hat{h}}=\sum_{i=1}^{N}{h}({\theta}_{i})w_{i}\phi_{i}\Big/\sum_{i=1}^{N}w_{i}\phi_{i}\text{.}$ Define the acceptance probability $p_{{\rm acc},q}=\int q({\theta})\int f_{n}({s}\mid {\theta})K_{\varepsilon}({s}-{s}_{\rm obs})\,\mbox{d}{s}\,\mbox{d}{\theta}$ and the density of the accepted parameter $q_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon)=\frac{q_{n}({\theta})f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)}{\int q_{n}({\theta})f_{\rm ABC}({s}_{\rm obs}\mid {\theta},\varepsilon)\,{\rm d}\theta}\text{.}$ Finally, define \begin{align} &\Sigma_{{\rm IS},n}=E_{\pi_{\rm ABC}}\left\{({h}({\theta})-{h}_{\rm ABC})^{2}\frac{\pi_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon_{n})}{q_{\rm ABC}({\theta} \mid {s}_{\rm obs},\varepsilon_{n})}\right\}\!,\quad \Sigma_{{\rm ABC},n}=p_{{\rm acc},q_{n}}^{-1}\Sigma_{{\rm IS},n},\label{ISABC_var} \end{align} where $$\Sigma_{{\rm IS},n}$$ is the importance sampling variance with $$\pi_{\rm ABC}$$ as the target density and $$q_{\rm ABC}$$ as the proposal density. Note that $$p_{{\rm acc},q_{n}}$$ and $$\Sigma_{{\rm IS},n}$$, and hence $$\Sigma_{{\rm ABC},n}$$, depend on $${s}_{\rm obs}$$. Standard results give the following asymptotic distribution of $${\hat{h}}$$. Proposition 2. For a given $$n$$ and $${s}_{\rm obs}$$, if $${h}_{\rm ABC}$$ and $$\Sigma_{{\rm ABC},n}$$ are finite, then ${N}^{1/2}({\hat{h}}-{h}_{\rm ABC})\rightarrow \mathcal{N}(0,\Sigma_{{\rm ABC},n}),$ in distribution as $$N\rightarrow\infty$$. This proposition motivates the following definition. Definition 2. For a given $$n$$ and $${s}_{\rm obs}$$, assume that the conditions of Proposition 2 hold. Then the asymptotic Monte Carlo variance of $${\hat{h}}$$ is $${\small{\text{MCV}}}_{{\hat{h}}}={N}^{-1}\Sigma_{{\rm ABC},n}$$. 4.2. Asymptotic efficiency We have defined the asymptotic variance as $$n\rightarrow\infty$$ of $${h}_{\rm ABC}$$, and the asymptotic Monte Carlo variance as $$N\rightarrow\infty$$ of $${\hat{h}}$$. The error of $${h}_{\rm ABC}$$ when estimating $${h}({\theta}_{0})$$ and the Monte Carlo error of $${\hat{h}}$$ when estimating $${h}_{\rm ABC}$$ are independent, which suggests the following definition. Definition 3. Assume the conditions of Theorem 1, and that $${h}_{\rm ABC}$$ and $$\Sigma_{{\rm ABC},n}$$ are bounded in probability for any $$n$$. Then the asymptotic variance of $${\hat{h}}$$ is ${\small{\text{AV}}}_{{\hat{h}}}=\frac{1}{a_{n}^{2}}{h}({\theta}_{0})^{\mathrm{T}}I_{\rm ABC}^{-1}({\theta}_{0})D{h}({\theta}_{0})+\frac{1}{N}\Sigma_{{\rm ABC},n}\text{.}$ We can interpret the asymptotic variance of $${\hat{h}}$$ as a first-order approximation to the variance of our Monte Carlo estimator for both large $$n$$ and $$N$$. We wish to investigate the properties of this asymptotic variance, for large but fixed $$N$$, as $$n\rightarrow\infty$$. The asymptotic variance itself depends on $$n$$, and we would hope it would tend to zero as $$n$$ increases. Thus we will study the ratio of $${\small{\text{AV}}}_{{\hat{h}}}$$ to $${\small{\text{AV}}}_{{ \rm MLES}}$$, where, by Theorem 1, the latter is $$a_{n}^{-2}{h}({\theta}_{0})^{\mathrm{T}}I^{-1}({\theta}_{0})D{h}({\theta}_{0})$$. This ratio measures the efficiency of our Monte Carlo estimator relative to the maximum likelihood estimator based on the summaries; it quantifies the loss of efficiency from using a nonzero bandwidth and a finite Monte Carlo sample size. We will consider how this ratio depends on the choice of $$\varepsilon_{n}$$ and $$q_{n}({\theta})$$. Thus we introduce the following definition. Definition 4. For a choice of $$\varepsilon_{n}$$ and $$q_{n}({\theta})$$, we define the asymptotic efficiency of $${\hat{h}}$$ as ${\small{\text{AE}}}_{{\hat{h}}}=\lim_{n\rightarrow\infty}\frac{{\small{\text{AV}}}_{{ \rm MLES}}}{{\small{\text{AV}}}_{{\hat{h}}}}\text{.}$ If this limiting value is zero, we say that $${\hat{h}}$$ is asymptotically inefficient. We will investigate the asymptotic efficiency of $${\hat{h}}$$ under the assumption of Theorem 1 that $$\varepsilon_{n}=o(a_{n}^{-3/5})$$. We shall see that the convergence rate of the importance sampling variance $$\Sigma_{{\rm IS},n}$$ depends on how large $$\varepsilon_{n}$$ is relative to $$a_n$$, and so we further define $$a_{n,\varepsilon}=a_n$$ if $$\lim_{n\rightarrow\infty} a_n\varepsilon_n<\infty$$ and $$a_{n,\varepsilon}=\varepsilon_n^{-1}$$ otherwise. If our proposal distribution in Algorithm 1 is either the prior or the posterior, then the estimator is asymptotically inefficient. Theorem 2. Assume the conditions of Theorem 1. (i) If $$q_{n}({\theta})=\pi({\theta})$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d-p})$$ and $$\Sigma_{{\rm IS},n}=\Theta_{\rm p}(a_{n,\varepsilon}^{-2})$$. (ii) If $$q_{n}({\theta})=\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d})$$ and $$\Sigma_{{\rm IS},n}=\Theta_{\rm p}(a_{n,\varepsilon}^{p})$$. In both cases $${\hat{h}}$$ is asymptotically inefficient. The result in part (ii) shows a difference from standard importance sampling settings, where using the target distribution as the proposal leads to an estimator with no Monte Carlo error. The estimator $${\hat{h}}$$ is asymptotically inefficient because the Monte Carlo variance decays more slowly than $$1/a_{n}^{2}$$ as $$n\rightarrow\infty$$. However, this is caused by different factors in each case. To see this, consider the acceptance probability of a value of $$\theta$$ and the corresponding summary $${s}_{n}$$ simulated in one iteration of Algorithm 1. This acceptance probability depends on $$\frac{{s}_{n}-{s}_{\rm obs}}{\varepsilon_{n}}=\frac{1}{\varepsilon_{n}}\left[\{{s}_{n}-{s}({\theta})\}+\{{s}({\theta})-{s}({\theta}_{0})\}+\{{s}({\theta}_{0})-{s}_{\rm obs}\}\right],\label{eq:accp}$$ (2) where $${s}({\theta})$$, defined in Condition 4, is the limiting value of $${s}_{n}$$ as $$n\rightarrow\infty$$ if data are sampled from the model for parameter value $$\theta$$. By Condition 4, the first and third bracketed terms within the square brackets on the right-hand side are $$O_{\rm p}(a_{n}^{-1})$$. If we sample $$\theta$$ from the prior, the middle term is $$O_{\rm p}(1)$$, and thus (2) will blow up as $$\varepsilon_{n}$$ goes to zero. Hence $$p_{{\rm acc},\pi}$$ goes to zero as $$\varepsilon_{n}$$ goes to zero, which causes the estimate to be inefficient. If we sample from the posterior, then by Theorem 1 we expect the middle term to also be $$O_{\rm p}(a_{n}^{-1})$$. Hence (2) is well behaved as $$n\rightarrow\infty$$, and $$p_{{\rm acc},\pi}$$ is bounded away from zero, provided either $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$ or $$\varepsilon_{n}=\Omega(a_{n}^{-1})$$. However, if we use $$\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$ as a proposal distribution, the estimates are still inefficient due to an increasing variance of the importance weights: as $$n$$ increases the proposal distribution is more and more concentrated around $${\theta}_{0}$$, while $$\pi$$ does not change. 4.3. Efficient proposal distributions Consider proposing the parameter value from a location-scale family, i.e., our proposal is of the form $$\sigma_{n}\Sigma^{1/2}{X}+\mu_{n}$$, where $${X}\sim q(\cdot)$$, $$E({X})=0$$ and $$\mbox{var}({X})=I_{p}$$. This defines a general form of proposal density, where the centre $$\mu_{n}$$, the scale rate $$\sigma_{n}$$, the scale matrix $$\Sigma$$ and the base density $$q(\cdot)$$ all need to be specified. We will give conditions under which such a proposal density results in estimators that are efficient. Our results are based on an expansion of $$\pi_{\rm ABC}({\theta}\mid {s}_{\rm obs},\varepsilon_{n})$$. Consider the rescaled random variables $${t}=a_{n,\varepsilon}({\theta}-{\theta}_{0})$$ and $${v}=\varepsilon_{n}^{-1}({s}-{s}_{\rm obs})$$. Recall that $${T}_{\rm obs}=a_{n}A({\theta}_{0})^{-1/2}\{{s}_{\rm obs}-{s}({\theta}_{0})\}$$. Define an unnormalized joint density of $${t}$$ and $${v}$$ as $g_{n}({t},{v};\tau)=\begin{cases} \begin{array}{@{}ll} \mathcal{N}\big[\{D{s}({\theta}_{0})+\tau\}{t};a_{n}\varepsilon_{n}{v}+A({\theta}_{0})^{1/2}{T}_{\rm obs},A({\theta}_{0})\big]K({v}), & a_{n}\varepsilon_{n}\rightarrow c<\infty,\\[5pt] \mathcal{N}\left[\{D{s}({\theta}_{0})+\tau\}{t};{v}+\frac{1}{a_{n}\varepsilon_{n}}A({\theta}_{0})^{1/2}{T}_{\rm obs},\frac{1}{a_{n}^{2}\varepsilon_{n}^{2}}A({\theta}_{0})\right]K({v}), & a_{n}\varepsilon_{n}\rightarrow\infty, \end{array}\end{cases}$ and further define $$g_{n}({t};\tau)=\int g_{n}({t},{v};\tau)\,{\rm d}{v}$$. For large $$n$$, and for the rescaled variable $${t}$$, the leading term of $$\pi_{\rm ABC}$$ is then proportional to $$g_{n}({t};0)$$. For both limits of $$a_{n}\varepsilon_{n}$$, $$g_{n}({t};\tau)$$ is a continuous mixture of normal densities with the kernel density determining the mixture weights. Our main theorem requires conditions on the proposal density. First, it requires that $$\sigma_{n}=a_{n,\varepsilon}^{-1}$$ and that $$c_{\mu}=\sigma_{n}^{-1}({\mu}_{n}-{\theta}_{0})$$ is $$O_{\rm p}(1)$$. This ensures that under the scaling of $$t$$, as $$n\rightarrow\infty$$, the proposal is not increasingly overdispersed compared to the target density, and the acceptance probability can be bounded away from zero. Second, the proposal distribution is required to be sufficiently heavy-tailed. Condition 7. There exist positive constants $$m_{1}$$ and $$m_{2}$$ satisfying $$m_{1}^{2}I_{p}<Ds(\theta_{0})^{\mathrm{T}}Ds(\theta_{0})$$ and $$m_{2}I_{d}<A(\theta_{0})$$, $$\alpha\in(0,1)$$, $$\gamma\in(0,1)$$ and $$c\in(0,\infty)$$ such that for any $$\lambda>0$$, \begin{align} &\sup_{t\in\mathbb{R}^{p}}\frac{\mathcal{N}(t;0,m_{1}^{-2}m_{2}^{-2}\gamma^{-1})}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty, \quad \sup_{t\in\mathbb{R}^{p}}\frac{\bar{K}^{\alpha}(\\|\lambda t\\|^2)}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty,\quad \sup_{t\in\mathbb{R}^{p}}\frac{\bar{r}_{\rm max}(\\|m_{1}m_{2}{\gamma}^{1/2}t\\|^2)}{q\{\Sigma^{-1/2}(t{-}c)\}}<\infty,& \end{align} where $$\bar{r}_{\rm max}(\cdot)$$ satisfies $$r_{\rm max}(v)=\bar{r}_{\rm max}(\\|v\\|_{\Lambda}^2)$$, and for any random series $$c_{n}$$ in $$\mathbb{R}^{p}$$ satisfying $$c_{n}=O_{\rm p}(1)$$, $\sup_{t\in\mathbb{R}^{p}}\frac{q(t)}{q(t+c_{n})}=O_{\rm p}(1)\text{.}$ If we choose $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$, the Monte Carlo importance sampling variance for the accepted parameter values is $$\Theta(a_{n}^{-2})$$ and has the same order as the variance of the summary-based maximum likelihood estimator. Theorem 3. Assume the conditions of Theorem 1. If the proposal density $$q_{n}({\theta})$$ is $\beta\pi({\theta})+(1-\beta)\frac{1}{\sigma_{n}^{p}\|\Sigma\|^{1/2}}q\left\{\sigma_{n}^{-1}\Sigma^{-1/2}({\theta}-{\mu}_{n})\right\}\!,$ where $$\beta\in(0,1)$$, $$q(\cdot)$$ and $$\Sigma$$ satisfy Condition 7, $$\sigma_{n}=a_{n,\varepsilon}^{-1}$$ and $$c_{\mu}$$ is $$O_{\rm p}(1)$$, then $$p_{{\rm acc},q_{n}}=\Theta_{\rm p}(\varepsilon_{n}^{d}a_{n,\varepsilon}^{d})$$ and $$\Sigma_{{\rm IS},n}=O_{\rm p}(a_{n,\varepsilon}^{-2})$$. Then if $$\varepsilon_{n}=\Theta(a_{n}^{-1})$$, $${\small{\text{AE}}}_{{\hat{h}}}=\Theta_{\rm p}(1)$$. Furthermore, if $$d=p$$, $${\small{\text{AE}}}_{{\hat{h}}}=1-K/(N+K)$$ for some constant $$K$$. The mixture with $$\pi({\theta})$$ here is to control the importance weight in the tail area (Hesterberg, 1995). It is not clear whether this is needed in practice, or is just a consequence of the approach taken in the proof. Theorem 3 shows that with a good proposal distribution, if the acceptance probability is bounded away from zero as $$n$$ increases, the threshold $$\varepsilon_{n}$$ will have the preferred rate $$\Theta(a_{n}^{-1})$$. This supports using the acceptance rate to choose the threshold based on aiming for an appropriate proportion of acceptances (Del Moral et al., 2012; Biau et al., 2015). In practice, $$\sigma_{n}$$ and $$\mu_{n}$$ need to be adaptive to the observations since they depend on $$n$$. For $$q(\cdot)$$ and $$\Sigma$$, the following proposition gives a practical suggestion that satisfies Condition 7. Let $$T(\cdot;\gamma)$$ be the multivariate $$t$$ density with degree of freedom $$\gamma$$. The following result says that it is theoretically valid to choose any $$\Sigma$$ if a $$t$$ distribution is chosen as the base density. Proposition 3. Condition 7 is satisfied for $$q(\theta)=T({\theta};\gamma)$$ with any $$\gamma>0$$ and any $$\Sigma$$. Proof. The first part of Condition 7 follows as the $$t$$-density is heavy tailed relative to the normal density, $$\bar{K}(\cdot)$$ and $$\bar{r}_{\rm max}(\cdot)$$. The second part can be verified easily. □ 4.4. Iterative importance sampling Taken together, Theorem 3 and Proposition 3 suggest proposing from the mixture of $$\pi({\theta})$$ and a $$t$$ distribution with the scale matrix and centre approximating those of $$\pi_{\rm ABC}({\theta})$$. We suggest the following iterative procedure, similar in spirit to that of Beaumont et al. (2009). Algorithm 2. Iterative importance sampling approximate Bayesian computation. Input a mixture weight $$\beta$$, a sequence of acceptance rates $$\{p_{k}\}$$, and a location-scale family. Set $$q_1({\theta})=\pi({\theta})$$. For $$k=1,\ldots,K$$: Step 1. Run Algorithm 1 with simulation size $$N_{0}$$, proposal density $$\beta\pi({\theta})+(1-\beta)q_{k}({\theta})$$ and acceptance rate $$p_{k}$$, and record the bandwidth $$\varepsilon_{k}$$. Step 2. If $$\varepsilon_{k-1}-\varepsilon_{k}$$ is smaller than some positive threshold, stop. Otherwise, let $$\mu_{k+1}$$ and $$\Sigma_{k+1}$$ be the empirical mean and variance matrix of the weighted sample from Step 1, and let $$q_{k+1}({\theta})$$ be the density with centre $$\mu_{k+1}$$ and variance matrix $$2\Sigma_{k+1}$$. Step 3. If $$q_{k}({\theta})$$ is close to $$q_{k+1}({\theta})$$ or $$K=K_{\rm max}$$, stop. Otherwise, return to Step $$1$$. After the iteration stops at the $$K$$th step, run Algorithm 1 with the proposal density $$\beta\pi({\theta})+(1-\beta)q_{K+1}({\theta})$$, $$N-KN_{0}$$ simulations and $$p_{K+1}$$. In this algorithm, $$N$$ is the number of simulations allowed by the computing budget, $$N_{0}<N$$ and $$\{p_{k}\}$$ is a sequence of acceptance rates, which we use to choose the bandwidth. The maximum value $$K_{\rm max}$$ of $$K$$ is set such that $$K_{\rm max}N_{0}=N/2$$. The rule for choosing the new proposal distribution is based on approximating the mean and variance of the density proportional to $$\pi(\theta)f_{\rm ABC}({s}_{\rm obs}\mid \theta,\varepsilon)^{1/2}$$, which is optimal (Fearnhead & Prangle, 2012). It can be shown that these two moments are approximately equal to the mean and twice the variance of $$\pi_{\rm ABC}({\theta})$$ respectively. For the mixture weight, $$\beta$$, we suggest a small value, and use $$0.05$$ in the simulation study below. 5. Numerical examples 5.1. Gaussian likelihood with sample quantiles This example illustrates the results in § 3 with an analytically tractable problem. Assume that the observations $${Y}_{\rm obs}=(\,y_{1},\ldots,y_{n})$$ follow the univariate normal distribution $$\mathcal{N}(\mu,\sigma)$$ with true parameter values $$(1,2^{1/2})$$. Consider estimating the unknown parameter $$(\mu,\sigma)$$ with the uniform prior in the region $$[-10,10]\times[0,10]$$ using Algorithm 1. The summary statistic is $$\{\exp(\hat{q}_{\alpha_1}/2),\ldots,\exp(\hat{q}_{\alpha_d}/2)\}$$ where $${\hat{q}}_{\alpha}$$ is the sample quantile of $${Y}_{\rm obs}$$ for probability $$\alpha$$. We present results for $$n=10^{5}$$. Smaller sizes from $$10^{2}$$ to $$10^{4}$$ show similar patterns. The probabilities $$\alpha_{1},\ldots,\alpha_{d}$$ for calculating quantiles are selected with equal intervals in $$(0,1)$$, and $$d=2,9$$ and $$19$$ were tested. In order to investigate the Monte Carlo error-free performance, $$N$$ is chosen to be large enough that the Monte Carlo errors were negligible. We compare the performances of the approximate Bayesian computation estimator $${\hat{\theta}}$$, the maximum likelihood estimator based on the summary statistics and the maximum likelihood estimator based on the full dataset. Since the dimension reduction matrix $$C$$ in Proposition 1 can be obtained analytically, the performance of $${\hat{\theta}}$$ using the original $$d$$-dimensional summary is compared with that using the two-dimensional summary. The results of mean square error are presented in Fig. 1. Fig. 1. View largeDownload slide Mean square errors, MSE, of point estimates for $$200$$ datasets. Point estimates compared include $$\theta_{\rm ABC}$$ using the original summary statistic (solid) and the transformed summary statistic (dashed), the dimension of which is reduced to $$2$$ according to Proposition 1, and the maximum likelihood estimates based on the original summary statistic (dotted) and the full dataset (dash-dotted). Fig. 1. View largeDownload slide Mean square errors, MSE, of point estimates for $$200$$ datasets. Point estimates compared include $$\theta_{\rm ABC}$$ using the original summary statistic (solid) and the transformed summary statistic (dashed), the dimension of which is reduced to $$2$$ according to Proposition 1, and the maximum likelihood estimates based on the original summary statistic (dotted) and the full dataset (dash-dotted). The phenomena implied by Theorem 1 and Proposition 1 can be seen in this example, together with the limitations of these results. First, $${E}\{{h}({\theta})\mid {s}_{\rm obs}\}$$, equivalent to $${\hat{\theta}}$$ with small enough $$\varepsilon$$, and the maximum likelihood estimator based on the same summaries have similar accuracy. Second, when $$\varepsilon$$ is small, the mean square error of $${\hat{\theta}}$$ equals that of the maximum likelihood estimator based on the summary. When $$\varepsilon$$ becomes larger, for $$d>2$$ the mean square error increases more quickly than for $$d=2$$. This corresponds to the impact of the additional bias when $$d>p$$. For all cases, the two-dimensional summary obtained by projecting the original $$d$$ summaries is, for small $$\varepsilon$$, as accurate as the maximum likelihood estimator given the original $$d$$ summaries. This indicates that the lower-dimensional summary contains the same information as the original one. For larger $$\varepsilon$$, the performance of the reduced-dimension summaries is not stable, and is in fact worse than the original summaries for estimating $$\mu$$. This deterioration is caused by the bias of $${\hat{\theta}}$$, which for larger $$\varepsilon$$ is dominated by higher-order terms in $$\varepsilon$$ which could be ignored in our asymptotic results. 5.2. Stochastic volatility with ar$$(1)$$ dynamics We consider a stochastic volatility model from Sandmann & Koopman (1998) for the de-meaned returns of a portfolio. Denote this return for the $$t$$th time period by $$y_t$$. Then $x_{t} =\phi x_{t-1}+\eta_{t},\ \eta_{t}\sim \mathcal{N}(0,\sigma_{\eta}^{2}); \quad y_{t} =\bar{\sigma}\exp(x_t/2)\xi_{n},\ \xi_{t}\sim \mathcal{N}(0,1),$ where $$\eta_{t}$$ and $$\xi_{t}$$ are independent, and $$x_t$$ is a latent state that quantifies the level of volatility for time period $$t$$. By the transformation $$y_{t}^{}=\log y_{t}^{2}$$ and $$\xi_{t}^{}=\log\xi_{t}^{2}$$, the observation equation in the state-space model can be transformed to \begin{equation} y_{n}^{} =2\log\bar{\sigma}+x_{n}+\xi_{n}^{},\quad \exp(\xi_{n}^{})\sim\chi_{1}^{2}, \end{equation} which is linear and non-Gaussian. Approximate Bayesian computation can be used to obtain an off-line estimator for the unknown parameters of this model. Here we illustrate the effectiveness of iteratively choosing the importance proposal for large $$n$$ by comparing with rejection sampling. In the iterative algorithm, a $$t$$ distribution with five degrees of freedom is used to construct $$q_{k}$$. Consider estimating the parameter $$(\phi,\sigma_{\eta},\log\bar{\sigma})$$ under a uniform prior in the region $$[0,1)\times[0.1,3]\times[-10,-1]$$. The setting with the true parameter $$(\phi,\sigma_{\eta},\log\bar{\sigma})=(0.9,0.675,-4.1)$$ is studied. We use a three-dimensional summary statistic that stores the mean, variance and lag-one autocovariance of the transformed data. If there were no noise in the state equation for $$\xi_{n}^{}$$, then this would be a sufficient statistic of $${Y}^{}$$, and hence is a natural choice for the summary statistic. The uniform kernel is used in the accept-reject step. We evaluate rejection sampling and iterative importance sampling methods on data of length $$n=100,500,2000$$ and $$10\,000$$, and use $$N=40\,000$$ Monte Carlo simulations. For iterative importance sampling, the sequence $$\{p_{k}\}$$ has the first five values decreasing linearly from $$5\%$$ to $$1\%$$, and later values being $$1\%$$. We further set $$N_0=2000$$ and $$K_{\rm max}=10$$. For the rejection sampler, acceptance probabilities of both $$5\%$$ and $$1\%$$ were tried and $$5\%$$ was chosen as it gave better performance. The simulation results are shown in Fig. 2. Fig. 2. View largeDownload slide Comparisons of rejection (solid) and iterative importance sampling (dashed) versions of approximate Bayesian computation. For each $$n$$, the logarithm of the average mean square error, MSE, across $$100$$ datasets is reported. For each dataset, the Monte Carlo sample size is $$40\,000$$. Ratios of mean square errors of the two methods are given in the table, and smaller values indicate better performance of iterative importance sampling. For each method in the plots, a line is fitted to the connected lines and the slope is reported in the table. Smaller values indicate faster decrease of the mean square error. Fig. 2. View largeDownload slide Comparisons of rejection (solid) and iterative importance sampling (dashed) versions of approximate Bayesian computation. For each $$n$$, the logarithm of the average mean square error, MSE, across $$100$$ datasets is reported. For each dataset, the Monte Carlo sample size is $$40\,000$$. Ratios of mean square errors of the two methods are given in the table, and smaller values indicate better performance of iterative importance sampling. For each method in the plots, a line is fitted to the connected lines and the slope is reported in the table. Smaller values indicate faster decrease of the mean square error. For all parameters, iterative importance sampling shows increasing advantage over rejection sampling as $$n$$ increases. For larger $$n$$, the iterative procedure obtains a centre for proposals closer to the true parameter and a bandwidth that is smaller than those used for rejection sampling. These contribute to the more accurate estimators. It is easy to estimate $$\log\bar{\sigma}$$, since the expected summary statistic $$\tilde{E}({Y}^{})$$ is roughly linear in $$\log\bar{\sigma}$$. Thus iterative importance sampling has less of an advantage over rejection sampling when estimating this parameter. 6. Discussion Our results suggest that one can obtain efficient estimates using approximate Bayesian computation with a fixed Monte Carlo sample size as $$n$$ increases. Thus the computational complexity of approximate Bayesian computation will just be the complexity of simulating a sample of size $$n$$ from the underlying model. Our results on the Monte Carlo accuracy of approximate Bayesian computation considered the importance sampling implementation given in Algorithm 1. If we do not use the uniform kernel, then there is a simple improvement on this algorithm, which absorbs the accept-reject probability within the importance sampling weight. A simple Rao–Blackwellization argument then shows that this leads to a reduction in Monte Carlo variance, so our positive results about the scaling of approximate Bayesian computation with $$n$$ will also immediately apply to this implementation. Similar positive Monte Carlo results are likely to apply to Markov chain Monte Carlo implementations of approximate Bayesian computation. A Markov chain Monte Carlo version will be efficient provided the acceptance probability does not degenerate to zero as $$n$$ increases. However, at stationarity it will propose parameter values from a distribution close to the approximate Bayesian computation posterior density, and Theorems 2 and 3 suggest that for such a proposal distribution the acceptance probability will be bounded away from zero. Whilst our theoretical results suggest that point estimates based on approximate Bayesian computation have good properties, they do not suggest that the approximate Bayesian computation posterior is a good approximation to the true posterior. In fact, Frazier et al. (2016) show that it will overestimate uncertainty if $$\varepsilon_n=O(a_n^{-1})$$. However, Li & Fearnhead (2018) show that using regression methods (Beaumont et al., 2002) to post-process approximate Bayesian computation output can lead to both efficient point estimation and accurate quantification of uncertainty. Acknowledgement This work was supported by the U.K. Engineering and Physical Sciences Research Council. Supplementary material Supplementary material available at Biometrika online contains proofs of the main results. References Allingham D. , King R. A. R. & Mengersen K. L. ( 2009 ). Bayesian estimation of quantile distributions. Statist. Comp. 19 , 189 – 201 . 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DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month \$360/year Save searches from PubMed Create lists to Export lists, citations	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 6, "x-ck12": 0, "texerror": 0, "math_score": 0.9872127175331116, "perplexity": 299.7125720419718}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583515041.68/warc/CC-MAIN-20181022113301-20181022134801-00348.warc.gz"}
https://mathoverflow.net/questions/178579/canonical-metric-on-grassmann-manifold/178585	# Canonical Metric on Grassmann Manifold I was curious and quite clueless as to how we can equip the Grassmann Manifold with a canonical metric - I have yet to find anything upon this subject. • The Grassmanian is a homogeneous space for the orthogonal group (unitary group in the complex case) and hence inherits a natural metric. Aug 14, 2014 at 23:28 • If you want an explicit formula, see mathoverflow.net/questions/141483/… Aug 15, 2014 at 1:46 • See montefiore.ulg.ac.be/systems/Publi/Grass_geom.pdf, for the Grassmann manifold of $p$-planes in $\mathbf{R}^n$. – user62675 Aug 15, 2014 at 2:33 • The space $\mathrm{Gr}(p,V)$ of $p$-dimensional subspaces in a vector space $V$ does not have a nontrivial canonical metric. There is no Riemannian metric on this space that is invariant under the natural action of $\mathrm{Aut}(V)=\mathrm{GL}(V)$. Upon fixing an additional structure on $V$, namely, a positive definite inner product $q$, there is a Riemannian metric on this space that is invariant under the natural action of $\mathrm{Aut}(V,q)=\mathrm{O}(q)$, unique up to a constant multiple. This multiple can be uniquely determined by specifying the total volume or diameter, for example. Aug 15, 2014 at 11:18 • @PaulSiegel , Aaron asked canonical metric not natural metric, ;) . See my answer – user21574 Jan 18, 2016 at 16:16 Since Grassmannian $Gr(n,m)=SO(n+m)/SO(n)\times SO(m)$ is a homogeneous manifold, you can take any Riemannian metric, and average with $SO(n+m)$-action. Then you show that an $SO(n+m)$-invariant metric is unique up to a constant. This is easy, because the tangent space $T_VGr(n,m)$ (tangent space to a plane $V\subset W$) is $Hom(V,V^\bot)$, and your metric must be $SO(V)\times SO(V^\bot)$-invariant. Such a metric is unique (up to a constant multiplier), which follows, e.g., from Schur's lemma. • he asking canonical metric, – user21574 Jan 18, 2016 at 16:26 • @HassanJolany This metric is actually symmetric and unique up to a constant multiple, as stated above. Any normalisation you wish to choose makes it canonical. Jan 26, 2016 at 12:40 • what is the your definition about canonical metric,? – user21574 Jan 26, 2016 at 15:25 • To repeat what Sebastian already said, the question refers to Grassmannians, which most of us would interpret as real Grassmannians, i.e., $G(n,k) =$ the space of $k$-linear subspaces in an $n$-dimensional real vector space. In general there is no complex structure on such a space, for example if the Grassmannian is odd-dimensional. However, it is a homogeneous space, and the standard meaning of a "canonical metric" on a homogeneous space is one that is invariant under the group action. Jan 26, 2016 at 16:33 • I didn't know the book of Besse, I just checked it. It seems it had long history and I didn't know anything about it. – user21574 Jan 26, 2016 at 17:47 In fact, $Gr(n,m)$ with its canonical metric induced from an Euclidean structure is one of the few spaces where you can write down solutions of the geodesic equations explicitly by a formula and write down a formula for the geodesic distance. See the following paper for this • MR1856419 Neretin, Yurii A. On Jordan angles and the triangle inequality in Grassmann manifolds. Geom. Dedicata 86 (2001), no. 1-3, 81–92. • So is there any exception to what Yurii has shown? Should any invariant metric on the grassmannian look like a symmetric norm function of Jordan angles? May 14, 2015 at 21:53 • Since it is an irreducible symmetric space of compact type, there is not a lot of choice: any invariant metric is a constant multiple of this one. May 15, 2015 at 4:39 • Great!...do you mind providing a reference? i.e. uniqueness of invariant Finsler metric...I'm kinda slow...I am an engineer... May 15, 2015 at 8:38 • Misha Verbitsky's answer gives you a proof in 3 lines. Helgason: "Differential Geometry, Symmetric Spaces, and Lie groups" is a standard reference, but by far too much for this simple question. As a Finsler metric it is not unique: any power of the Riemannian norm is an invariant Finsler metric. May 15, 2015 at 8:54 • Oh, by the way, is it possible to characterize all invariant Finsler metrics in some way? May 15, 2015 at 10:08 A nice geometric way of endowing a Grassmann manifold with a metric (understood here as a distance, and not directly as a Riemannian metric) is to use the Hausdorff distance for subsets of the round sphere. Consider $V$ a real vector space of dimension $n$ endowed with an inner product, and let $Gr_k(V)$ be the Grassmannian of $k$-planes on $V$. Let $x,y\in Gr_k(V)$, and denote by $S_x$ and $S_y$ the subspheres of the unit sphere $S_V=\{v\in V:\\|v\\|=1\}$ defined by intersecting it with the subspaces $x$ and $y$ respectively. Then $S_x,S_y\subset S_V$ are closed subsets and the distance between $x,y\in Gr_k(V)$ can be defined as the Hausdorff distance between these sets: $$dist(x,y)=dist_H(S_x,S_y).$$ Note that this distance does not coincide with the symmetric space distance on $Gr_k(V)=SO(n)/SO(k)SO(n-k)$. As explained in the paper of Neretin in Michor's answer, the symmetric space distance is computed in terms of the principal angles between two subspaces (namely, it is the square root of the sum of the squares of these angles); while the above distance defined in terms of the Hausdorff distance of closed sets in the unit sphere measures exactly the largest principal angle between subspaces, ignoring all the other smaller principal angles.	{"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": 0.9282058477401733, "perplexity": 249.81803825797567}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662541747.38/warc/CC-MAIN-20220521205757-20220521235757-00368.warc.gz"}
https://www.bartleby.com/questions-and-answers/the-formation-of-krypton-from-rubidum-decay-is-a-result-of-______.-a-alpha-emission-b-beta-emission-/f03bf681-84df-43b4-8415-567e17132f80	The formation of krypton from rubidum decay is a result of ______. A) alpha emission B) beta emission C) positron emission D) electron capture E) neutron capture Question The formation of krypton from rubidum decay is a result of ______. A) alpha emission B) beta emission C) positron emission D) electron capture E) neutron capture	{"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": 0.9858661890029907, "perplexity": 26138.663393065755}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141706569.64/warc/CC-MAIN-20201202083021-20201202113021-00285.warc.gz"}
http://math.stackexchange.com/questions/357272/how-can-i-prove-that-xy-leq-x2y2/357275	# How can I prove that $xy\leq x^2+y^2$? How can I prove that $xy\leq x^2+y^2$ for all $x,y\in\mathbb{R}$ ? - you can even prove $\|2xy\|<x^2+y^2$ by using some basic identities. – Maesumi Apr 10 '13 at 16:27 Note that you will need $(x, y) \neq (0, 0)$, to show the strict inequality. – Calvin Lin Apr 10 '13 at 16:28 You can easily prove that $\|xy\|\leq \left(\mbox{max}\{\|x\| , \|y\| \}\right)^2$. – N. S. Apr 11 '13 at 17:50 You also need $x$ and $y$ to be real, since it is false for $x = y = i$. – marty cohen Apr 15 '13 at 14:32 Is this a competition for the most complicated answer? 0_o What the hell is wrong with you, people? – Alexei Averchenko Apr 15 '13 at 15:24 \begin{align} 0\leq (x-y)^2 \implies & 0\leq x^2-2xy +y^2 \\ \implies & 2xy\leq x^2+y^2 \\ \implies & xy\leq {x^2+y^2} \end{align} Update. This proves I'm not assuming that $xy \leq 2xy$! Clearly this is only valid for non-negative $xy$. Note that if $xy\leq 0$ if, only if, $2xy\leq 0$. - If you remove the divisions by $2$ and write $xy\leq 2xy$ instead, I will happily upvote. The only price to pay is to assume $xy\geq 0$. But if $xy<0$, there is nothing to prove. But don't feel any pressure! – 1015 Apr 10 '13 at 17:04 @julien Nice sugestion. – MathOverview Apr 10 '13 at 23:32 I kept my word, +1. – 1015 Apr 10 '13 at 23:49 @Calvin Lin , because you gave me a downvote? – MathOverview Apr 11 '13 at 1:04 Maybe that's because you didn't say at the beginning that you assumed $xy\geq 0$. For otherwise $xy\leq 2xy$ is no true. Also, $xy\leq 2xy$ does not explicitly appear. Sorry if you got one downvote because of me... I still think your answer is the best here. – 1015 Apr 11 '13 at 1:08 $$x^2+y^2-xy=\frac{x^2}{2}+\frac{y^2}{2}+\frac{(x-y)^2}{2}$$ - This is imo one very, very nice hint. +1 – DonAntonio Apr 10 '13 at 16:41 Indeed. +1. – Did Apr 10 '13 at 16:48 Quite an elegant hint! – Barranka Apr 10 '13 at 22:28 Use polar coordinates: $$x=r \cos \theta,y=r \sin \theta$$ Your inequality becomes $$r^2 \cos \theta \sin \theta \leq r^2$$ which is pretty much trivial. - +1 cos i like this unnecessary overkill :D I don't know if this, or the question is more elementary... – Lost1 Apr 10 '13 at 16:47 But you make both $x,y$ bound . ie. less than $r$ and even related to each other . $x^2+y^2=r^2$ .They become the co-ordinates of a circle! – ABC Apr 10 '13 at 16:49 @exploringnet I don't see a problem. Any point in the plane has its polar coordinates. The radius $r$ is definitely not constant! – user1337 Apr 10 '13 at 17:06 @exploringnet and they are. This relationship doesn't make them dependent. – user1337 Apr 10 '13 at 17:13 @exploringnet: They're not dependent; it's not determined that $x = r/2$ until $y$ is known. – LarsH Apr 11 '13 at 7:58 Though there are quite a few proofs already given, I'd like to add a visual one. - Why so much whitespace? – Ruslan Apr 10 '13 at 17:19 @Ruslan: Sorry, I should have cropped it. I used Latex with pstricks and generated a page of output. – Shaun Ault Apr 10 '13 at 17:54 Thank you to whomever took the time to crop that image for me! – Shaun Ault Apr 11 '13 at 17:13 Nice. ${}{}{}{}$ – The Chaz 2.0 Apr 11 '13 at 17:50 How do you prove $y\le x$ says that half the rectangle fits withing half the square? – maxuel Jan 16 at 10:02 another interesting answers - maybe the most interesting: $$x^2+y^2-xy=\left(x-\frac{y}{2}\right)^2+\frac{3y^2}{4} \geq 0.$$ - just the sum of two squares --simple and elegant .. – Halil Duru Apr 11 '13 at 5:35 Very nice !+1 for you. – MathOverview Apr 11 '13 at 17:51 For $xy<0$, this is trivial You can do better by proving $2xy \leq x^2 + y^2$. Move $2xy$ to the right handside and factorise - You'd want a $\leq$ sign instead. – Calvin Lin Apr 10 '13 at 16:27 @CalvinLin yes I do :D thanks – Lost1 Apr 10 '13 at 16:32 A proof with only words: assume $x$ and $y$ are positive, if one is negative this is trivial. Then one of $x$ or $y$ is smallest, the other largest. Assume $y$ is the largest (else switch). Then $xy$ is clearly less than $y^2$, and adding $x^2$ doesn't change that! - I like proofs with only words. :-) This answer has been the most accessible one for me. You might want to mention the case where $x$ and $y$ are both negative, though. – LarsH Apr 11 '13 at 7:42 $x^2+y^2=AC^2$ $\sin \angle BAC= \dfrac{BC}{AC}$ $\sin \angle BCA= \dfrac{AB}{AC}$ $\sin \angle BCA \cdot \sin \angle BAC <1 \implies \dfrac{xy}{x^2+y^2} <1$. - $$(x^2 - xy + y^2)(x + y) = x^3 + y^3.$$ $x^3 + y^3$ and $x + y$ have the same sign: $x \leq -y$ iff $x^3 \leq -y^3$. Therefore, $x^2 - xy + y^2 \geq 0$. About the special case $x = -y$: it’s not really that special, but for a technical reason that goes beyond precalculus. Since non-trivial Zariski open sets are dense in the Euclidean topology, and polynomial maps are continuous between Euclidean topologies, all inequalities of the form $f(x_1, \ldots, x_n) \geq 0$ (where $f$ is a polynomial) that hold on a Zariski open set must hold everywhere, because $[0, +\infty)$ is closed and its preimage must thus contain the closure of said Zariski open set, i.e. the whole space. Therefore, such apparent special cases can be safely ignored as long as they’re Zariski closed and the inequality is not strict. - Technically, this needs the $x=-y$ case to be treated separately. – ronno May 9 '13 at 3:35 @ronno Not really, because $X := \mathbb{R}^2 \setminus \{x = -y\}$ is dense in $\mathbb{R}^2$, and $[0, +\infty)$ is closed in $\mathbb{R}$, therefore its pullback along $(x, y) \mapsto x^2 - xy + y^2$ must be $\operatorname{cl}X = \mathbb{R}^2$. Of course it is too technical for a precalculus-level question. – Alexei Averchenko Aug 14 '14 at 20:18 Another one - if $xy \le 0$ it is trivial, so let $xy > 0$. Then we have $$1 \le \dfrac{x}{y}+\dfrac{y}{x}$$ Now for any positive number, either it or its reciprocal must exceed $1$, unless both are $1$. - If $xy$ is negative, then the statement is obvious since $xy < 0 \leq x^2 + y^2$. Otherwise, $xy$ is non-negative, and we can show that $xy \leq 2xy \leq x^2+y^2$, where the latter follows from the trivial inequality $(x-y)^2 \geq 0$. - Just look at this picture below: - Trying to make sure I understand how $a, b$ relate to $x, y$. Are we to assume that $b = max(x, y)$ and $a = min(x, y)$? (And that $x, y \geq 0$.) – LarsH Apr 11 '13 at 8:02 Intuitive Approach if $\|x\|\le \|y\|$ then we have $$x\cdot y \le y\cdot y \le y.y + x\cdot x$$ $$\Rightarrow x\cdot y \le x^2+y^2\tag1$$ Using Symmetry $(1)$ holds if $\|x\|\ge \|y\|$ - This is clearly a consequence of the concavity of the logarithm and the monotonicity of the exponential. Since $xy\leq \|x\|\|y\|$, we can assume that $x >0$ and $y > 0$. Then we have $$xy = \exp(\ln(xy)) = \exp(\ln x + \ln y) = \exp(\frac12\ln x^2 + \frac12 \ln y^2) \leq \exp(\ln(\frac{x^2}2 + \frac{y^2}2)) =\frac{x^2}2 + \frac{y^2}2 \leq x^2 + y^2.$$ QED - A different approach, $$A.M[x_i^n]\ge(A.M[x_i])^n\ge GM[x_1]^n$$ ie. AM of terms with $n$th power is greater than $n$th power of AM ie. greater than $n$th power of GM . So,$$\dfrac{x^2+y^2}2\ge(\dfrac{x+y}2)^2\ge(xy)$$ and so,$$(x^2+y^2)\ge xy$$ And for $xy<0$ the inequality is obvious. - Technique 1:$$\sqrt{x^2y^2} = xy \le \dfrac{x^2 + y^2}{2}\le x^2 + y^2$$ Technique 2:$$(x - y)^2 \ge 0 \implies x^2 +y^2 - 2xy \ge 0 \implies x^2 + y^2 \ge 2xy\ge xy$$ Technique 3 (my favorite): There's a statement $\dfrac{y}{x} + \dfrac{x}{y} \ge 2$ with many classical proofs (which I'd not state here). We can write the inequality as follows:$$\dfrac{y}{x}+\dfrac{x}{y} \ge 1$$Divide both sides by $xy$.$$\dfrac{1}{x^2} + \dfrac{1}{y^2} \ge \dfrac{1}{xy}$$Rewrite.$$\dfrac{x^2 + y^2}{x^2y^2} \ge \dfrac{xy}{x^2y^2}$$And finally...$$x^2 + y^2 \ge xy$$ - $$x^2+y^2-xy=\frac{(2x-y)^2+3y^2}4=\frac{(2x-y)^2+(\sqrt3y)^2}4$$ Now, the square of any real numbers is $\ge0$ So, $(2x-y)^2+(\sqrt3y)^2\ge0,$ the equality occurs if each $=0$ - Thanks for the down-vote. I could find a better way to prove it. – lab bhattacharjee May 10 '13 at 14:18 Dear lab bhattacharjee, to be honest, I should admit that it is me who downvoted your answer earlier. I did the downvote not because your answer is not good, but personally I don't encourage users with high reputation like you to pay much attention to trivial questions like this one. In my opinion, supporting such questions will diminish the quality of this website. By the way, I also downvoted lots of other answers(to this/other post) provided by users with high reputation for the same reason. In view of your serious attitude to downvotes, I have to cancel my downvote. Sorry for bothering. – 23rd May 10 '13 at 15:57 Assume $x$ and $y$ are positive. Draw the right triangle with vertices $(0,0), (x,0), (x,y)$. Draw the circle around $(0,0)$ that circumscribes the triangle. Reflect the triangle in the $x$-axis. Reflect the two triangles in the $y$-axis too, for a better result. - Let $y=kx$ then $x xk \leq x^2 + x^2k^2$ since $k\leq k^2+1$ [in view of $0\leq (k-1/2)^2+3/4]$ And the ineq. clearly holds when $x=0$ . $\hspace {33mm} \blacksquare$ - $$(x-y)^2=x^2+y^2-2xy\\ (x-y)^2+2xy=x^2+y^2\\ 2xy\leq x^2+y^2$$ Therefore $xy\leq x^2+y^2$. Hence the proof. - First, $$a^2+3b^2\geq 0$$ Now, $$a^2+2ab+b^2+a^2-2ab+b^2\geq a^2-b^2$$ Thus, $$(a+b)^2+(a-b)^2\geq (a+b)(a-b)$$ Let $a+b=x, a-b=y$, i.e., $a=\frac{x+y}{2}, b=\frac{x-y}{2}$ which is one-to-one correspondence between $(x,y)$ and $(a,b)$, then, $$x^2+y^2\geq xy$$ Q.E.D - $(x - y)^2 \geq 0$ $x^2 + y^2 -2xy \geq 0$ $x^2 + y^2 \geq 2xy > xy$ upate Happy Green Kid is right below, the case for xy < 0 needs to be considered: so, continuing from: $x^2 + y^2 \geq 2xy$ if $xy > 0:$ $x^2 + y^2 \geq 2xy > xy$ else $x^2 + y^2 > 0 > xy$ in both cases $x^2 + y^2 > xy$ - If $xy < 0$ then $2xy < xy$ – Happy Green Kid Naps Apr 11 '13 at 13:49 If xy is negative, then it is trivial. If both x and y are negative, then let x := -x and y := -y. This means that we can safely assume that both x and y are positive. Square each sides yields $$x^2y^2 \le x^4 + 2x^2y^2 + y^4$$ , which trivially holds. - Let $y>=x. xy<=x^2+y^2$. Divide both sides by $y$, and you get $x<=(x^2/y)+y$. If $x$ and $y$ are both positive numbers, this is clearly true because we said from the beginning that $y>=x$. - Say $y\ne 0$ then divide both sides $xy\le x^2+y^2$ by $y^2$; let $z=\frac{x}{y}$; then we need $z\le 1+z^2$ or equivalently $z^2-z+1\ge 0$; completing the square gives $(z-\frac{1}{2})^2+\frac{3}{4}\ge 0$ which is clear. - My viral 5 cents: $$xy\le x^2+y^2$$ rearrange it $$0\le x^2+y^2 - xy \implies$$ $$0\le \frac{x^2 + y^2 + (x-y)^2}{2} \implies$$ $$0\le x^2 + y^2 + (x-y)^2, \forall x,y \in \mathbb{R}$$ finally, let's prove $$a^2 \ge 0, \forall a \in \mathbb{R}$$ $\mathbb{R}$ is unitary ring $\implies$ has an inverse element: $$a + (-a) = 0 \implies a = -(-a)$$ $$0 = 0 \cdot b = (a + (-a)) \cdot b = a \cdot b + (-a) \cdot b = 0\implies$$ $a \cdot b$ and $(-a) \cdot b$ are inverse to each other, but in turn, inverse element for $a \cdot b$ is $-(a \cdot b) \implies$ $$(-a) \cdot b = -(a \cdot b)$$ or $$(-a) \cdot (-a) = -(a \cdot (-a)) = -(-a) \cdot a = a \cdot a$$ thus $$x \cdot x + y \cdot y + (x - y) \cdot (x - y)$$ let $z = x -y$ and we will have $$x \cdot x + y \cdot y + z \cdot z = x^2 + y^2 + z^2 \ge 0, \forall x,y,z \in \mathbb{R} \implies \\ \\x^2+y^2 + (x-y)^2 \ge 0, \forall x,y \in \mathbb{R} \implies$$ $$xy\le x^2+y^2, \forall x,y \in \mathbb{R}$$ $$Q.E.D.$$ - for downvote: if it isn't a plain racism :) please show me my faults to learn – rook Apr 12 '13 at 10:59 (-1) since this is just wrong: $a^2 \geq b^2$ does not imply $a \geq b$. – TMM Apr 15 '13 at 13:50 @TMM, you're right, possibly there is another kind of problem - is to prove that the squaring in certain circumstances didn't broke an inequality sign – rook Apr 15 '13 at 14:48 This is really not different (now) than the earlier highly upvoted answers. – mixedmath Apr 16 '13 at 17:35 $$xy\le x^2+y^2$$ Move xy to the right $$\implies 0 \le x^2 - xy + y^2$$ This becomes the prove that $$x^2 - xy + y^2$$ is always positive AKA greater or equal to zero we can rewrite it as $$0 \le x^2 - 2(\dfrac{1}{2} xy) + y^2$$ Which simly means that we can write it as: $$0 \le x^2 - 2(\dfrac{1}{2} xy) + y^2$$ Our equation has the form: $$a^2 - 2(ab) + b^2 = (a - b)^2$$ By factorization we get: $$0 \le 2\left(\sqrt{\dfrac{1}{2}}(x) - \sqrt{\dfrac{1}{2}}(y)\right)^2$$ Since $$2$$ is positive $$2\left(\sqrt{\dfrac{1}{2}}(x) - \sqrt{\dfrac{1}{2}}(y)\right)^2$$ is positive as the square of a number so: $$2\left(\sqrt{\dfrac{1}{2}}(x) - \sqrt{\dfrac{1}{2}}(y)\right)^2$$ Can only be positive meaning: $$0 \le 2\left(\sqrt{\dfrac{1}{2}}(x) - \sqrt{\dfrac{1}{2}}(y)\right)^2$$ Is always true. -	{"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": 16, "x-ck12": 0, "texerror": 0, "math_score": 0.938210666179657, "perplexity": 533.2594201599491}, "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/1448398446218.95/warc/CC-MAIN-20151124205406-00185-ip-10-71-132-137.ec2.internal.warc.gz"}
http://physicshelpforum.com/electricity-magnetism/3714-magnitude-magnetic-field.html	Physics Help Forum magnitude of magnetic field Electricity and Magnetism Electricity and Magnetism Physics Help Forum Dec 7th 2009, 12:23 PM #1 Junior Member Join Date: Dec 2009 Posts: 8 magnitude of magnetic field A current I is uniformly distributed over the cross section of a long, straight wire of radius a = 2 mm. At the surface of the wire, the magnitude of the magnetic field is B = 8 mT. The magnitude of the magnetic field 5 mm from the axis is......?? Dec 14th 2009, 01:47 PM #2 Senior Member Join Date: Apr 2008 Posts: 815 Originally Posted by alex83 A current I is uniformly distributed over the cross section of a long, straight wire of radius a = 2 mm. At the surface of the wire, the magnitude of the magnetic field is B = 8 mT. The magnitude of the magnetic field 5 mm from the axis is......?? Use Ampere's law to reach the expression of the magnetic field at any point outside the wire and you're done. __________________ Isaac If the problem is too hard just let the Universe solve it. Tags field, magnetic, magnitude Thread Tools Display Modes Linear Mode Similar Physics Forum Discussions Thread Thread Starter Forum Replies Last Post Legato Electricity and Magnetism 1 Jan 19th 2010 08:10 PM theramblingmark Electricity and Magnetism 1 Jan 15th 2010 08:03 PM van-hilst Advanced Electricity and Magnetism 2 Dec 26th 2009 11:37 AM tag16 Electricity and Magnetism 1 Sep 27th 2009 05:23 PM dragon8438 Advanced Electricity and Magnetism 1 Mar 3rd 2009 07:02 PM	{"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": 0.9306700229644775, "perplexity": 1123.3591539646184}, "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-2018-43/segments/1539583513844.48/warc/CC-MAIN-20181021094247-20181021115747-00473.warc.gz"}
https://mathsmadeeasy.co.uk/ks3-maths-revision-cards-answers/g2/	# G2 Question: Plot the graph of $y-2x=-3$ (hint: rearrange the equation so it’s in 𝑦=𝑚𝑥+𝑐 form!) Answer: Let’s rearrange this equation. Subtract 1 from both sides: $y-2x=-3$ $y=2x-3$ Using this equation we find the value of $y$. To do this we create a table, show below. Now we have the following coordinates $(-1,-5) , (0,-3), (1, -1), (2,1), (3,3)$	{"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": 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": 5, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8794478178024292, "perplexity": 682.4794253141519}, "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/1593657151761.87/warc/CC-MAIN-20200714212401-20200715002401-00285.warc.gz"}
http://eprint.iacr.org/2012/028/20120122:042721	## Cryptology ePrint Archive: Report 2012/028 Houssem MAGHREBI and Emmanuel PROUFF and Sylvain GUILLEY and Jean-Luc DANGER Abstract: One protection of cryptographic implementations against side-channel attacks is the masking of the sensitive variables. In this article, we present a first-order masking that does not leak information when the registers change values according to some specific (and realistic) rules. This countermeasure applies to all devices that leak a function of the distance between consecutive values of internal variables. In particular, we illustrate its practicality on both hardware and software implementations. Moreover, we introduce a framework to evaluate the soundness of the new first-order masking when the leakage slightly deviates from the rules involved to design the countermeasure. It reveals that the countermeasure remains more efficient than the state-of-the-art first-order masking if the deviation from the ideal model is equal to a few tens of percents, and that it is as good as a first-order Boolean masking even if the deviation is $50$\%. Category / Keywords: implementation / First-order masking, leakage in distance, leakage-free countermeasure Note: Paper to be published at CT-RSA 2012, with some corrections in the construction of the $F$ functions (in Sec. 4.1).	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6922139525413513, "perplexity": 1594.1286524456807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823805.20/warc/CC-MAIN-20160723071023-00291-ip-10-185-27-174.ec2.internal.warc.gz"}
http://aas.org/archives/BAAS/v31n5/aas195/736.htm	AAS 195th Meeting, January 2000 Session 118. Supermassive Black Holes Oral, Saturday, January 15, 2000, 10:00-11:30am, Centennial I and II ## [118.05] STIS Observations of the Center of M32 M. Valluri (Chicago), C.L. Joseph, D. Merritt, R. Olling (Rutgers), STIS Team We present an analysis of the HST/STIS data for the nucleus of M32. We find a very steep rise in the stellar velocity dispersions inside of ~0.2'', from \sigma~50 km/s at 0.2'' to \sigma~160 km/s at 0.02''. Our data are the first to clearly resolve this feature, which is strong evidence for a supermassive dark object. The distribution of line-of-sight stellar velocities (LOSVD) is strongly non-Gaussian near the galaxy center, with extended wings like those expected from stars near a BH. Because the stellar motions at these small radii are dominated by the central dark object rather than by the combined gravitational force of the other stars, we are able to place much stronger constraints on the mass of the black hole than in previous studies with lower spatial resolution. We present estimates based on a three-integral axisymmetric modelling algorithm for the mass of the black hole and the orbital kinematics of the stars. Support for this work was provided to the STIS IDT by NASA. [Previous] \| [Session 118] \| [Next]	{"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": 0.8572738766670227, "perplexity": 3180.7967711067186}, "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/1429246660743.58/warc/CC-MAIN-20150417045740-00163-ip-10-235-10-82.ec2.internal.warc.gz"}