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http://mathhelpforum.com/differential-geometry/172878-subsequence-converging-p.html	# Math Help - subsequence converging to p... 1. ## subsequence converging to p... Let (an) be a sequence of real numbers, and let E = {an : n ∈ N} be the range of (an). Prove that (an) has a subsequence converging to p iff either p appears infinitely many times in (an) or p is an accumulation point of E. For this if and only if is it showing that either it appears infinitely many times or p is an accumulation point of E? If $p$ appears infinitely often you can just take the subsequence to be that point repeated. If not, then for every $\varepsilon>0$ there exists some point of the sequence different from $p$ in $B_\varepsilon(p)$ and so in particular $B_{\varepsilon}\cap P\supsetneq\{p\}$. Since $\varepsilon$ was arbitrary the conclusion follows.	{"extraction_info": {"found_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": 0.9723858833312988, "perplexity": 231.06707138484032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011017001/warc/CC-MAIN-20140305091657-00098-ip-10-183-142-35.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/38193/interpretation-of-the-second-incompleteness-theorem?sort=oldest	# Interpretation of the Second Incompleteness Theorem For simplicity, let me pick a particular instance of G\"odel's Second Incompleteness Theorem: ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that ZFC is consistent. (Here ZFC can be replaced by any other sufficiently good, sufficiently strong set of axioms, but this is not the issue here.) This theorem has been interpreted by many as saying "we can never know whether mathematics is consistent" and has encouraged many people to try and prove that ZFC (or even PA) is in fact inconsistent. I think a mainstream opinion in mathematics (at least among mathematician who think about foundations) is that we believe that there is no problem with ZFC, we just can't prove the consistency of it. A comment that comes up every now and then (also on mathoverflow), which I tend to agree with, is this: () "What do we gain if we could prove the consistency of (say ZFC) inside ZFC? If ZFC was inconsistent, it would prove its consistency just as well." In other words, there is no point in proving the consistency of mathematics by a mathematical proof, since if mathematics was flawed, it would prove anything, for instance its own non-flawedness. Hence such a proof would not actually improve our trust in mathematics (or ZFC, following the particular instance). Now here is my question: Does the observation () imply that the only advantage of the Second Incompleteness Theorem over the first one is that we now have a specific sentence (in this case Con(ZFC)) that is undecidable, which can be used to prove theorems like "the existence of an inacessible cardinal is not provable in ZFC"? In other words, does this reduce the Second Incompeteness Theorem to a mere technicality without any philosophical implication that goes beyond the First Incompleteness Theorem (which states that there is some sentence $\phi$ such that neither $\phi$ nor $\neg\phi$ follow from ZFC)? - In the case of inaccessible cardinals, you can bypass the second incompleteness theorem in the following sense. If an inaccessible exists, then there is a least inaccessible $\kappa$, and its existence is not provable because $V_\kappa$ is a model of ZF+"there is no inaccessible". (I like to call this Zermelo's incompleteness theorem, because he proposed the argument in 1928, before Goedel.) – John Stillwell Sep 9 '10 at 16:45 @John: But in $V_\kappa$ there are still plenty of (countable, transitive) models of ${\sf ZFC}+$there is an inaccessible'', so this sense of incompleteness is certainly weaker. – Andres Caicedo Sep 9 '10 at 16:49 @Stefan: Do you think the first theorem already invalidates Hilbert's program? – Andres Caicedo Sep 9 '10 at 16:49 About (): If we could prove the consistency of ZFC inside ZFC we would have shown the inconsistency of ZFC - and thus gained interesting information, isn't it? – Peter Arndt Sep 9 '10 at 17:17 @Andres: I was thinking that observation () already invalidates Hilberts program to some extent (the consistency part). The First Incompleteness Theorem takes care of another issue: No reasonable system of axioms for mathematics is complete. But Andreas below has a real point. – Stefan Geschke Sep 9 '10 at 19:59 For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the consistency of ZFC, then a proof of Con(ZFC) carried out in ZFC would indeed be of little use. But a proof of Con(ZFC) carried out in a more reliable system, like Peano arithmetic or primitive recursive arithmetic, would (before G"odel) have been useful, and I think this is what Hilbert was hoping for. G"odel's second incompleteness theorem tells us that this sort of thing can't happen (unless even the more reliable system is inconsistent). - The answer is the following observation due to Hilbert: If we can prove the consistency of $ZFC$ using elementary methods, then any elementary theorem of $ZFC$ has an elementary proof, i.e. we don't need ideal/abstract objects like sets or real number for dealing with concrete/finite objects like numbers. The importance of Godel's theorems is not that $ZFC$ can't prove its own consistency but rather the weaker result that elementary methods (assuming that listing these methods is easy, i.e. recursively enumerable) cannot prove all elementary results, in other words, we need abstract objects even for doing elementary number theory. Hilbert wanted to show that although abstract objects are helpful for elementary mathematics in practice, they are not essential and can be avoided (at least in theory) if needed. But Godel's first incompleteness theorem already shows that this is not true. (Here elementary can arguably be identified with unbounded-quantifier-free formulas or $\Pi_1$ sentences.) - The fact that the second incompleteness theorem refers to consistency is important for several applications, both philosophical and mathematical. Philosophically, the second incompleteness theorem is what lets us know that we cannot, in general, prove the existence of a (set) model of ZFC within ZFC itself. This is a fundamental obstruction to naive methods of proving relative consistency results. We cannot show, for example, that the continuum hypothesis is unprovable in ZFC by constructing a set model of ZFC where CH fails using methods that themselves can be formalized in ZFC. Philosophically, this says we should not be surprised that the relative consistency results that we do have require methods that cannot be formalized within ZFC. Second, there are some theorems (perhaps less well known) that leverage the second incompleteness theorem to prove the existence of special kinds of models. These are mathematical results, not philosophical ones. Theorem (Harvey Friedman). Let $S$ be an effective theory of second-order arithmetic that contains the theory ACA0. If there is a countable ω-model of $S$, then there is a countable $\omega$-model of $S$ + "there is no countable $\omega$-model of $S$." The proof proceeds by showing that, if the conclusion fails, a certain effective theory obtained from $S$ is consistent and proves its own consistency. The type of model constructed by the theorem is useful for proving that certain systems of second-order arithmetic are not the same. - I thought that Cohen's proof is formalizable in $ZFC$ in the form "if $ZFC$ is consistent, then $ZFC+\lnot CH$ is consistent". – Kaveh Sep 9 '10 at 17:36 Nice example! That's also what I was pointing to in my comment to the question. If you manage to carry out a consistency proof in a setting where you shouldn't be able to (by the 2nd incompleteness theorem), then you have a contradiction - which is valuable information about your hypotheses. – Peter Arndt Sep 9 '10 at 17:37 Carl, Is the proof of Harvey's result basically a use of $\Sigma^1_1$ absoluteness? I've proved versions of this in class for stronger theories, so perhaps I'm overlooking some technicality at the level of second order arithmetic. – Andres Caicedo Sep 9 '10 at 18:13 @Kaveh: In my terminology, that approach proves the consistency of "ZFC + X" in the stronger theory ZFC + Con(ZFC). – Carl Mummert Sep 9 '10 at 19:49 @Andres: the proof does use the downward absoluteness of $\Pi^1_1$ formulas. There is a proof in Simpson's book on second-order arithmetic if you're interested in looking it up. The reason ACAo is needed is to reason about the satisfaction predicate of a countable $\omega$-model coded as a real. Strangely, every $\omega$-model of WKLo does contain a real that codes a countable $\omega$-model of WKLo, so the assumption of ACAo isn't trivial. – Carl Mummert Sep 9 '10 at 19:57 While it's not directly a philosophical benefit, the Second Incompleteness Theorem is quite useful for giving concrete unprovability results: if we want to prove that theory T does not prove theorem X, it suffices to show that X implies the consistency of T. For instance, Harvey Friedman has a number of results showing that some theorem implies the well-foundedness of some ordinal notation, where the ordinal notation, in turn, is known to imply the consistency (indeed, 1-consistency) of the theory. - John H Conway proves and discusses the incompleteness theorem is his badass wolf prize lectures: http://www.math.princeton.edu/facultypapers/Conway/ Anyone who hasn't seen these talks is missing out. - On a Mac, I'm able to play the video of these lectures using the free software VLC: videolan.org/vlc – Dan Ramras Sep 10 '10 at 0:43 There is another nice consequence of the Goedel first incompleteness theorem. Indeed by proving that there exists an undecidable sentence, the theorem is offering a formal proof of the consistency of ZFC (if it were not consistent then it would prove whatever). The only problem is that it is doing so inside ZFC, so the proof is not really worth because it would carry on also if ZFC were inconsistent. I think this is also related to your sentence "a mainstream opinion in mathematics ... is that we believe that there is no problem with ZFC". - The consistency of the theory is a hypothesis in the incompleteness theorems; they don't establish the consistency themselves. – Carl Mummert Sep 10 '10 at 11:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9288308024406433, "perplexity": 474.4544638649377}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440644065318.20/warc/CC-MAIN-20150827025425-00257-ip-10-171-96-226.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/54969/relationship-between-integrable-1-plane-fields-and-a-one-parameter-group-of-diff	# Relationship between integrable 1-plane fields and a one parameter group of diffeomorphisms A 1-plane field on a manifold is a smooth choice of a 1 dimensional subspace of the tangent space at every point. If this plane field is integrable then there is an associated 1-dimensional foliation on the manifold that has the plane field as its tangent space. Parameterizing the leaves of the foliation would seem to give a one parameter group of diffeomorphisms. Likewise, to every one parameter group of diffeomorphisms there is an associated vector field (which could be thought of as a smooth 1-plane field), and at any point flowing as long as one could in both directions would seem to give a leaf. The union of these leaves would then seem to give a foliation. Can someone confirm if this line of reasoning is essentially correct. These ideas seem to be the same except for foliations don't come with a parameterization. I'm guessing there is a natural way to parameterize using the charts though. - The leaves may not be coherently orientable. So your plane field (really a line field) may not lift to a vector field, let alone a 1-parameter family of diffeomorphisms. You can construct non-orientable line fields in $\mathbb R^2 \setminus \{0\}$.	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8539835214614868, "perplexity": 191.07008849154676}, "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-2014-42/segments/1413507446525.24/warc/CC-MAIN-20141017005726-00369-ip-10-16-133-185.ec2.internal.warc.gz"}
http://wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=2011&number=1672	WIAS Preprint No. 1672, (2011) # On a singularly perturbed initial value problem in case of a double root of the degenerate equation Authors • Butuzov, Valentin F. • Nefedov, Nikolai N. • Recke, Lutz • Schneider, Klaus 2010 Mathematics Subject Classification • 34A12 34E05 34E15 Keywords • singularly perturbed first order ordinary differential equation, initial value problem, boundary layer, double root of degenerate equation, asymptotic expansion Abstract We study the initial value problem of a singularly perturbed first order ordinary differential equation in case that the degenerate equation has a double root. We construct the formal asymptotic expansion of the solution such that the boundary layer functions decay exponentially. This requires a modification of the standard procedure. The asymptotic solution will be used to construct lower and upper solutions guaranteeing the existence of a unique solution and justifying its asymptotic expansion.	{"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": 0.9124411344528198, "perplexity": 722.8785930340803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00082.warc.gz"}
http://math.stackexchange.com/questions/286406/graph-homomorphism-and-cliques	Graph Homomorphism and cliques Let $G$ and $H$ be two graphs. It is known that if there is a homomorphism from $G$ to $H$, then $\omega(G) \leq \omega(H)$ where $\omega(G)$ is the clique number of $G$. When does the converse hold and when does it fail? - A trivial example of the converse's failure would be to take $G$ to be a complete graph and $H$ to be an edgeless graph. – Austin Mohr Jan 25 '13 at 4:02 It should be $\omega(G) \leq \omega(H)$, not $\geq$. – polkjh Jan 25 '13 at 7:38 @polkjh Look at observation $2.6$ here (mast.queensu.ca/~ctardif/articles/ghss.pdf) – Turbo Jan 25 '13 at 7:57 I think that is a mistake. Suppose there is a homomorphism from $G$ to $H$. Adding edges and vertices to $H$ retains the homomorphism from $G$ to $H$. So we can increase $\omega(H)$ indefinitely and still have a homomorphism from $G$ to $H$, which contradicts that $\omega(H)$ cannot be more than $\omega(G)$. – polkjh Jan 25 '13 at 8:07 2 Answers If $H$ is a subgraph of $G$, then there is a homomorphism $H\to G$. But clearly $\omega(H)$ can even be zero, while $\omega(H)$ could be as large as $\|V(G)\|$. For the converse, there is no hope. As an example consider triangle-free graphs, with $\omega=2$. For each positive integer $m$, there are triangle-free graphs with chromatic number greater than $m$, and for such graphs there is no homomorphism to $K_m$. - Let $H=K_n$. Then existence of homomorphism from $G$ to $H$ implies $G$ is $n$-colorable. But $\omega(G) \leq n$ does not imply that $G$ is $n$-colorable (consider an odd cycle and $n=2$). So atleast for complete graphs $H$, the converse holds only if $\omega(H) \geq \chi(G)$ (chromatic number). This is not necessary for general graphs $H$. But I doubt there is any simple characterization of when the converse holds for generel $H$. - I think if a homomorphism from $G$ to $H = K_{n}$ exists, then $\omega(G) \geq n$ (mast.queensu.ca/~ctardif/articles/ghss.pdf observation $2.6$) – Turbo Jan 25 '13 at 8:07 I think that might be a typo in the paper. Look at my comment above. – polkjh Jan 25 '13 at 8:09 It might help if you try to prove the statement yourself (homomorphism from $G$ to $H$ implies $\omega(G) \leq \omega(H)$). It is not too difficult. – polkjh Jan 25 '13 at 8:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9641934633255005, "perplexity": 152.89554679610728}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049275835.98/warc/CC-MAIN-20160524002115-00125-ip-10-185-217-139.ec2.internal.warc.gz"}
https://jeremykun.com/2013/05/15/properties-of-morphisms/?shared=email&msg=fail	Properties of Morphisms This post is mainly mathematical. We left it out of our introduction to categories for brevity, but we should lay these definitions down and some examples before continuing on to universal properties and doing more computation. The reader should feel free to skip this post and return to it later when the words “isomorphism,” “monomorphism,” and “epimorphism” come up again. Perhaps the most important part of this post is the description of an isomorphism. Isomorphisms, Monomorphisms, and Epimorphisms Perhaps the most important paradigm shift in category theory is the focus on morphisms as the main object of study. In particular, category theory stipulates that the only knowledge one can gain about an object is in how it relates to other objects. Indeed, this is true in nearly all fields of mathematics: in groups we consider all isomorphic groups to be the same. In topology, homeomorphic spaces are not distinguished. The list goes on. The only way to do determine if two objects are “the same” is by finding a morphism with special properties. Barry Mazur gives a more colorful explanation by considering the meaning of the number 5 in his essay, “When is one thing equal to some other thing?” The point is that categories, more than existing to be a “foundation” for all mathematics as a formal system (though people are working to make such a formal system), exist primarily to “capture the essence” of mathematical discourse, as Mazur puts it. A category defines objects and morphisms, but literally all of the structure of a category lies in its morphisms. And so we study them. The strongest kind of morphism we can consider is an isomorphism. An isomorphism is the way we say two objects in a category are “the same.” We don’t usually care whether two objects are equal, but rather whether some construction is unique up to isomorphism (more on that when we talk of universal properties). The choices made in defining morphisms in a particular category allow us to strengthen or weaken this idea of “sameness.” Definition: A morphism $f : A \to B$ in a category $\mathbf{C}$ is an isomorphism if there exists a morphism $g: B \to A$ so that both ways to compose $f$ and $g$ give the identity morphisms on the respective objects. That is, $gf = 1_A$ and $fg = 1_B$. The most basic (usually obvious, but sometimes very deep) question in approaching a new category is to ask what the isomorphisms are. Let us do this now. In $\mathbf{Set}$ the morphisms are set-functions, and it is not hard to see that any two sets of equal cardinality have a bijection between them. As all bijections have two-sided inverses, two objects in $\mathbf{Set}$ are isomorphic if and only if they have the same cardinality. For example, all sets of size 10 are isomorphic. This is quite a weak notion of “sameness.” In contrast, there is a wealth of examples of groups of equal cardinality which are not isomorphic (the smallest example has cardinality 4). On the other end of the spectrum, a poset category $\mathbf{Pos}_X$ has no isomorphisms except for the identity morphisms. The poset categories still have useful structure, but (as with objects within a category) the interesting structure is in how a poset category relates to other categories. This will become clearer later when we look at functors, but we just want to dissuade the reader from ruling out poset categories as uninteresting due to a lack of interesting morphisms. Consider the category $\mathbf{C}$ of ML types with ML functions as morphisms. An isomorphism in this category would be a function which has a two-sided inverse. Can the reader think of such a function? Let us now move on to other, weaker properties of morphisms. Definition: A morphism $f: A \to B$ is a monomorphism if for every object $C$ and every pair of morphisms $g,h: C \to A$ the condition $fg = fh$ implies $g = h$. The reader should parse this notation carefully, but truly think of it in terms of the following commutative diagram: Whenever this diagram commutes and $f$ is a monomorphism, then we conclude (by definition) that $g=h$. Remember that a diagram commuting just means that all ways to compose morphisms (and arrive at morphisms with matching sources and targets) result in an identical morphism. In this diagram, commuting is the equivalent of claiming that $fg = fh$, since there are only two nontrivial ways to compose. The idea is that monomorphisms allow one to “cancel” $f$ from the left side of a composition (although, confusingly, this means the cancelled part is on the right hand side of the diagram). The corresponding property for cancelling on the right is defined identically. Definition: A morphism $f: A \to B$ is an epimorphism if for every object $C$ and every pair of morphism $g,h: B \to C$ the condition $gf = hf$ implies $g = h$. Again, the relevant diagram. Whenever $f$ is an epimorphism and this diagram commutes, we can conclude $g=h$. Now one of the simplest things one can do when considering a category is to identify the monomorphisms and epimorphisms. Let’s do this for a few important examples. Monos and Epis in Various Categories In the category $\mathbf{Set}$, monomorphisms and epimorphisms correspond to injective and surjective functions, respectively. Lets see why this is the case for monomorphisms. Recall that an injective function $f$ has the property that $f(x) = f(y)$ implies $x=y$. With this property we can show $f$ is a monomorphism because if $f(g(x)) = f(h(x))$ then the injective property gives us immediately that $g(x) = h(x)$. Conversely, if $f$ is a monomorphism and $f(x) = f(y)$, we will construct a set $C$ and two convenient functions $g, h: C \to A$ to help us show that $x=y$. In particular, pick $C$ to be the one point set $\left \{ c \right \}$, and define $g(c) = x, h(c) = y$. Then as functions $fg = fh$. Indeed, there is only one value in the domain, so saying this amounts to saying $f(x) = fg(c) = fh(c) = f(y)$, which we know is true by assumption. By the monomorphism property $g = h$, so $x = g(c) = h(c) = y$. Now consider epimorphisms. It is clear that a surjective map is an epimorphism, but the converse is a bit trickier. We prove by contraposition. Instead of now picking the “one-point set,” for our $C$, we must choose something which is one element bigger than $B$. In particular, define $g, h : B \to B'$, where $B'$ is $B$ with one additional point $x$ added (which we declare to not already be in $B$). Then if $f$ is not surjective, and there is some $b_0 \in B$ which is missed by $f$, we define $g(b_0) = x$ and $g(b) = b$ otherwise. We can also define $h$ to be the identity on $B$, so that $gf = hf$, but $g \neq h$. So epimorphisms are exactly the surjective set-maps. There is one additional fact that makes the category of sets well-behaved: a morphism in $\mathbf{Set}$ is an isomorphism if and only if it is both a monomorphism and an epimorphism. Indeed, isomorphisms are set-functions with two-sided inverses (bijections) and we know from classical set theory that bijections are exactly the simultaneous injections and surjections. A warning to the reader: not all categories are like this! We will see in a moment an example of a nontrivial category in which isomorphisms are not the same thing as simultaneous monomorphisms and epimorphisms. The category $\mathbf{Grp}$ is very similar to $\mathbf{Set}$ in regards to monomorphisms and epimorphisms. The former are simply injective group homomorphisms, while the latter are surjective group homomorphisms. And again, a morphisms is an isomorphism if and only if it is both a monomorphism and an epimorphism. We invite the reader to peruse the details of the argument above and adapt it to the case of groups. In both cases, the hard decision is in choosing $C$ when necessary. For monomorphisms, the “one-point group” does not work because we are constrained to send the identity to the identity in any group homomorphism. The fortuitous reader will avert their eyes and think about which group would work, and otherwise we suggest trying $C = \mathbb{Z}$. After completing the proof, the reader will see that the trick is to find a $C$ for which only one “choice” can be made. For epimorphisms, the required $C$ is a bit more complex, but we invite the reader to attempt a proof to see the difficulties involved. Why do these categories have the same properties but they are acquired in such different ways? It turns out that although these proofs seem different in execution, they are the same in nature, and they follow from properties of the category as a whole. In particular, the “one-point object” (a singleton set for $\mathbf{Set}$ and $\mathbb{Z}$ for $\mathbf{Grp}$) is more categorically defined as the “free object on one generator.” We will discuss this more when we get to universal properties, but a “free object on $n$ generators” is roughly speaking an object $A$ for which any morphism with source $A$ must make exactly $n$ “choices” in its definition. With sets that means $n$ choices for the images of elements, for groups that means $n$ consistent choices for images of group elements. On the epimorphism side, the construction is a sort of “disjoint union object” which is correctly termed a “coproduct.” But momentarily putting aside all of this new and confusing terminology, let us see some more examples of morphisms in various categories. Our recent primer on rings was well-timed, because the category $\mathbf{Ring}$ of rings (with identity) is an example of a not-so-well-behaved category. As with sets and groups, we do have that monomorphisms are equivalent to injective ring homomorphisms, but the argument is trickier than it was for groups. It is not obvious which “convenient” object $C$ to choose here, since maps $\mathbb{Z} \to R$ yield no choices: 1 maps to 1, 0 maps to 0, and the properties of a ring homomorphism determine everything else (in fact, the abelian group structure and the fact that units are preserved is enough). This makes $\mathbb{Z}$ into what’s called an “initial object” in $\mathbf{Ring}$; more on that when we study universal properties. In fact, we invite the reader to return to this post after we talk about the universal property of polynomial rings. It turns out that $\mathbb{Z}[x]$ is a suitable choice for $C$, and the “choice” made is where to send the indeterminate $x$. On the other hand, things go awry when trying to apply analogous arguments to epimorphisms. While it is true that every surjective ring homomorphism is an epimorphism (it is already an epimorphism in $\mathbf{Set}$, and the argument there applies), there are ring epimorphisms which are not surjections! Consider the inclusion map of rings $i : \mathbb{Z} \to \mathbb{Q}$. The map $i$ is not surjective, but it is an epimorphism. Suppose $g, h : \mathbb{Q} \to R$ are two parallel ring morphisms, and they agree on $\mathbb{Z}$ (they will always do so, since there is only one ring homomorphism $\mathbb{Z} \to R$). Then $g,h$ must also agree on $\mathbb{Q}$, because if $p,q \in \mathbb{Z}$ with $q \neq 0$, then $\displaystyle g(p/q) = g(p)g(q^{-1}) = g(p)g(q)^{-1} = h(p)h(q)^{-1} = h(p/q)$ Because the map above is also an injection, the category of rings is a very naturally occurring example of a category which has morphisms that are both epimorphisms and monomorphisms, but not isomorphisms. There are instances in which monomorphisms and epimorphisms are trivial. Take, for instance any poset category. There is at most one morphism between any two objects, and so the conditions for an epimorphism and monomorphism vacuously hold. This is an extreme example of a time when simultaneous monomorphisms and epimorphisms are not the same thing as isomorphisms! The only isomorphisms in a poset category are the identity morphisms. The inspection of epimorphisms and monomorphisms is an important step in the analysis of a category. It gives one insight into how “well-behaved” the category is, and picks out the objects which are special either for their useful properties or confounding trickery. This reminds us of a quote of Alexander Grothendieck, one of the immortal gods of mathematics who popularized the use of categories in mainstream mathematics. A good category containing some bad objects is preferable to a bad category containing only good objects. I suppose the thesis here is that the having only “good” objects yields less interesting and useful structure, and that one should be willing to put up with the bad objects in order to have access to that structure. Next time we’ll jump into a discussion of universal properties, and start constructing some programs to prove that various objects satisfy them. Until then! 4 thoughts on “Properties of Morphisms” 1. DR6 There is something I don’t understand. If homo-, epi-, and isomorphisms are the closest thing to equality that is used in categories, what does the “=” mean in the definitions then? It seems to be used always on morphisms, how is equality between them defined then? Like • The concept of an isomorphism is the only thing that replaces equality, and we have to be clear here: isomorphism is the best concept of equality we have for objects. Morphisms, on the other hand, are just elements of sets, so equality is what you think it is (on the other hand, computing equality of morphisms is difficult or impossible for most categories). One can also call two objects “equal,” and it still means what you think it does (as in they are identically the same thing). The point is that equality is far too strict a condition to be useful. For instance, in the category of types (in your favorite programming language) there could conceivably be a type for Big-Endian integers and Little-Endian integers. They would be isomorphic as types (there is an obvious isomorphism between them), but not equal as types. This makes things awkward if we want to think of “integers” as a unique type. One of the major paradigm shifts of category theory is to accomodate this: all of the constructions we will define in category theory are only unique up to isomorphism. As such, by saying isomorphism is the best replacement for equality, I don’t mean “closest to equality” but rather “most appropriate.” Like 2. Michael Maloney Hey Jeremy, I think you have a few small typos above. In your paragraph beginning with “Why do these categories have the same properties but they are acquired in such different ways?”, you have some LaTeX that went awry. Later, when talking about the category of rings, you had “There are ring epimorphisms which are not isomorphisms!” I believe you meant they are not surjections 🙂 I think it also bears mentioning that not all epic monos are isos. That fact really surprised me (and bothered me a bit) when I first learned it! Regarding poset categories, I think you can make a very general statement about “boring things” in mathematics. Boring things are often very useful as sanity checks. If a statement fails for the boring things, then it surely can’t hold for more interesting ones 🙂 (And in this particular case, you can use any poset category to prove that the existence of a non-iso epimono). Keep up the good work. This is a nice-looking primer series so far 🙂 Like • Thanks for the tips. I mentioned that there are epis and monos that are not isos (I said this for posets and the example Z -> Q). I think that poset categories are extremely useful not so much as being nice counterexamples (though you pointed out that they are and I wholeheartedly agree with the sentiment) but in how they are used to build more complicated constructions. For instance, in computational topology an A-persistence object is a functor from a poset category A to a category B, and if A and B satisfy some mild properties then the category of A-persistence objects (for a fixed B) is an abelian category. Also, though I believe most people in denotational semantics don’t phrase things in categorical terms, domains and Scott domains are poset categories with some extra properties (e.g. having limits, though I can’t remember the others off the top of my head).	{"extraction_info": {"found_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": 97, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9284564256668091, "perplexity": 276.83201515064087}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141182794.28/warc/CC-MAIN-20201125125427-20201125155427-00510.warc.gz"}
https://www.physicsforums.com/threads/stokes-theorem.556005/	# Stokes Theorem 1. Dec 1, 2011 ### berm 1. The problem statement, all variables and given/known data Use stoke's theorem to calculate integral of F dot dr over C where F(x,y,z) = (-Z^2,y^2,x^2) and C is the curve of intersection of the place -y+z=0 and the parabloid z=x^2+y^2 2. Relevant equations 3. The attempt at a solution found the curl of -2x-2xj, but coudlnt figure out how to calculate the integral 2. Dec 1, 2011 ### Pythagorean can you set up the integral for us and tell us exactly what your problem is? 3. Dec 2, 2011 ### HallsofIvy Staff Emeritus '-2x-2xj' makes no sense. And if you intended '(-2x-2x)j', that is 0. Since the curve is given as the intersection of the two surfaces, you can use Stoke's theorem on either surface. I imagine the plane would be easier. Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Similar Discussions: Stokes Theorem	{"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": 0.940789520740509, "perplexity": 1889.8455047776424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806317.75/warc/CC-MAIN-20171121055145-20171121075145-00364.warc.gz"}
https://stats.stackexchange.com/questions/270731/churn-risk-and-lifetime-period-relationship	Churn risk and lifetime period relationship I am exploring churn and lifetime modeling. From what I see on various online material, the lifetime period is defined as 1 / churn-risk. For example, if the churn risk for a 12-month period is estimated to be .3, the lifetime is calculated to 3.3 years by 1 / .3. Is there a mathematical proof of this way of modeling the lifetime period? Intuitively, 1 - churn risk sounds more correct (probability of not churning). Where does the assumption that churn risk increase over time come from? Is there a distribution formula that is derived to 1 / churn risk? This comes from the geometric distribution. If the probability of churning in any given year is $p$ (where $p$ is constant across years, and years are independent) then the expected number of years until you churn is $1/p$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9544864296913147, "perplexity": 1036.8116462199077}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027319915.98/warc/CC-MAIN-20190824063359-20190824085359-00144.warc.gz"}
http://sparkandshine.net/tags/%E6%89%80%E6%9C%89%E6%A0%BC/	## How to pronounce words ending in S The pronunciation of words ending in S depends on the final consonant, i.e., the final sound.	{"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": 0.9478379487991333, "perplexity": 2161.3235334950646}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257647530.92/warc/CC-MAIN-20180320185657-20180320205657-00733.warc.gz"}
http://clay6.com/qa/7536/prove-that-the-points-representing-the-complex-numbers-left-7-5-mathit-righ	# Prove that the points representing the complex numbers $\left ( 7+5\mathit{i} \right )$, $\left ( 5+2\mathit{i} \right )$, $\left ( 4+7\mathit{i} \right )$ and $\left ( 2+4\mathit{i} \right )$ form a parallelogram. (Plot the points and use midpoint formula). Toolbox: • If $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ are represented by the points $A(x_2,y_2),B(x_2,y_2)$ on the complex Argand plane,then their sum $(x_1+x_2)+i(y_1+y_2)$ is represented by the point $C(x_1+x_2,y_1+y_2)$ which completes the parallelogram $OACB$. Step 1: Let $z_1=(7+5i),z_2=(5+2i),z_3=(4+7i),z_4=(2+4i)$ be represented by the points $A,B,C,D$ in the Argand plane. Plotting the points $A(7,5),B(5,2),C(4,7),D(2,4)$ we have: Step 2: The diagonals of the quadrilateral $DBAC$ are $DA$ and $BC$. Th midpoint of the diagonal $DA$ is $\big(\large\frac{7+2}{2},\frac{4+5}{2}\big)$ i.e $(\large\frac{9}{2},\frac{9}{2}\big)$ and the midpoint of the diagonal $BC$ is $\big(\large\frac{5+4}{2},\frac{2+7}{2}\big)$ i.e $\big(\large\frac{9}{2},\frac{9}{2}\big)$. The diagonals bisect each other. Therefore $DBAC$ is a parallelogram. edited Jun 10, 2013	{"extraction_info": {"found_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": 0.9420106410980225, "perplexity": 216.0366170323751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1524125945037.60/warc/CC-MAIN-20180421051736-20180421071736-00589.warc.gz"}
http://link.springer.com/chapter/10.1007%2F978-3-642-03040-6_61	Chapter Advances in Neuro-Information Processing Volume 5507 of the series Lecture Notes in Computer Science pp 501-508 # Particle Swarm Optimization and Differential Evolution in Fuzzy Clustering • Fengqin YangAffiliated withCollege of Computer Science and Technology, Jilin University • , Changhai ZhangAffiliated withCollege of Computer Science and Technology, Jilin University • , Tieli SunAffiliated withCollege of Computer Science, Northeast Normal University * Final gross prices may vary according to local VAT. ## Abstract Fuzzy clustering helps to find natural vague boundaries in data. The fuzzy c-means (FCM) is one of the most popular clustering methods based on minimization of a criterion function because it works fast in most situations. However, it is sensitive to initialization and is easily trapped in local optima. Particle swarm optimization (PSO) and differential evolution (DE) are two promising algorithms for numerical optimization. Two hybrid data clustering algorithms based the two evolution algorithms and the FCM algorithm, called HPSOFCM and HDEFCM respectively, are proposed in this research. The hybrid clustering algorithms make full use of the merits of the evolutionary algorithms and the FCM algorithm. The performances of the HPSOFCM algorithm and the HDEFCM algorithm are compared with those of the FCM algorithm on six data sets. Experimental results indicate the HPSOFCM algorithm and the HDEFCM algorithm can help the FCM algorithm escape from local optima.	{"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": 0.8662996292114258, "perplexity": 1743.2637182950436}, "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/1461861754936.62/warc/CC-MAIN-20160428164234-00197-ip-10-239-7-51.ec2.internal.warc.gz"}
http://eprints.maths.manchester.ac.uk/928/	# Rational Cherednik algebras and Hilbert schemes Gordon, I. and Stafford, J.T. (2005) Rational Cherednik algebras and Hilbert schemes. Advances in Mathematics, 198 (1). pp. 222-274. ISSN 0001-8708 Let $H_c$ be the rational Cherednik algebra of type $A_{n-1}$ with spherical subalgebra $U_c = e H_c e$. Then $U_c$ is filtered by order of differential operators, with associated graded ring $\mbox{gr} U_c = \mathbb{C} [ \mathfrak{h} \oplus \mathfrak{h}^]$ where $W$ is the $n$th symmetric group. We construct a filtered $\mathbb{Z}$-algebra $b$ such that, under mild conditions on $c$: • the category $B$-qgr of graded noetherian $B$-modules modulo torsion is equivalent to $U_c$-mod; • the associated graded $\mathbb{Z}$-algebra $\mbox{gr}B$ has $\mbox{gr}B-lqgr \simeq \coh \Hilb(n)$ This can be regarded as saying that $U_c$ simultaneously gives a non-commutative deformation of $\mathfrak{h} \oplus \mathfrak{h}^ / W$ and of its resolution of singularities $\Hilb(n) \rightarrow \mathfrak{h} \oplus \mathfrak{h}^*$. As we show elsewhere, this result is a powerful tool for studying the representation theory of $H_c$ and its relationship to $\Hilb(n)$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9582827091217041, "perplexity": 279.12183128496224}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202506.45/warc/CC-MAIN-20190321072128-20190321094128-00368.warc.gz"}
http://en.wikipedia.org/wiki/Steady_state	In systems theory, a system in a steady state has numerous properties that are unchanging in time. This means that for those properties p of the system, the partial derivative with respect to time is zero: $\frac{\partial p}{\partial t} = 0$ The concept of steady state has relevance in many fields, in particular thermodynamics, economics, and engineering. Steady state is a more general situation than dynamic equilibrium. If a system is in steady state, then the recently observed behavior of the system will continue into the future. In stochastic systems, the probabilities that various states will be repeated will remain constant. In many systems, steady state is not achieved until some time after the system is started or initiated. This initial situation is often identified as a transient state, start-up or warm-up period. While a dynamic equilibrium occurs when two or more reversible processes occur at the same rate, and such a system can be said to be in steady state, a system that is in steady state may not necessarily be in a state of dynamic equilibrium, because some of the processes involved are not reversible. For example: The flow of fluid through a tube or electricity through a network could be in a steady state because there is a constant flow of fluid, or electricity. Conversely, a tank being drained or filled with fluid is a system in transient state, because its volume of fluid changes with time. ## Applications ### Economics A steady state economy is an economy of relatively stable size. It features stable population and stable consumption that remain at or below carrying capacity. The term typically refers to a national economy, but it can also be applied to the economic system of a city, a region, or the entire planet. Note that Robert Solow and Trevor Swan applied the term steady state a bit differently in their economic growth model. Their steady state occurs when investment equals depreciation, and the economy reaches equilibrium, which may occur during a period of growth. ### Electronics In electronics, steady state is an equilibrium condition of a circuit or network that occurs as the effects of transients are no longer important. Steady state determination is an important topic, because many design specifications of electronic systems are given in terms of the steady-state characteristics. Periodic steady-state solution is also a prerequisite for small signal dynamic modeling. Steady-state analysis is therefore an indispensable component of the design process. In some cases, it is useful to consider constant envelope vibration – vibration that never settles down to motionlessness, but continues to move at constant amplitude – a kind of steady-state condition. ### Chemistry In chemistry, thermodynamics, and other chemical engineering, a steady state is a situation in which all state variables are constant in spite of ongoing processes that strive to change them. For an entire system to be at steady state, i.e. for all state variables of a system to be constant, there must be a flow through the system (compare mass balance). One of the simplest examples of such a system is the case of a bathtub with the tap open but without the bottom plug: after a certain time the water flows in and out at the same rate, so the water level (the state variable being Volume) stabilizes and the system is at steady state. Of course the Volume stabilizing inside the tub depends on the size of the tub, the diameter of the exit hole and the flowrate of water in. Since the tub can overflow, eventually a steady state can be reached where the water flowing in equals the overflow plus the water out through the drain. A steady state flow process requires conditions at all points in an apparatus remain constant as time changes. There must be no accumulation of mass or energy over the time period of interest. The same mass flow rate will remain constant in the flow path through each element of the system.[1] Thermodynamic properties may vary from point to point, but will remain unchanged at any given point.[2] ### Electrical engineering It is the ability of electrical machine or power system to regain its original/previous state is called Steady state stability. [3] The stability of a system refers to the ability of a system to return to its steady state when subjected to a disturbance. As mentioned before, power is generated by synchronous generators that operate in synchronism with the rest of the system. A generator is synchronized with a bus when both of them have same frequency, voltage and phase sequence. We can thus define the power system stability as the ability of the power system to return to steady state without losing synchronism. Usually power system stability is categorized into Steady State, Transient and Dynamic Stability Steady State Stability studies are restricted to small and gradual changes in the system operating conditions. In this we basically concentrate on restricting the bus voltages close to their nominal values. We also ensure that phase angles between two buses are not too large and check for the overloading of the power equipment and transmission lines. These checks are usually done using power flow studies. Transient Stability involves the study of the power system following a major disturbance. Following a large disturbance the synchronous alternator the machine power (load) angle changes due to sudden acceleration of the rotor shaft. The objective of the transient stability study is to ascertain whether the load angle returns to a steady value following the clearance of the disturbance. The ability of a power system to maintain stability under continuous small disturbances is investigated under the name of Dynamic Stability (also known as small-signal stability). These small disturbances occur due random fluctuations in loads and generation levels. In an interconnected power system, these random variations can lead catastrophic failure as this may force the rotor angle to increase steadily. ### Mechanical engineering When a periodic force is applied to a mechanical system, it will typically reach steady state after going through some transient behavior. This is often observed in vibrating systems, such as a clock pendulum, but can happen with any type of stable or semi-stable dynamic system. The length of the transient state will depend on the initial conditions of the system. Given certain initial conditions a system may be in steady state from the beginning. ### Physiology Homeostasis (from Greek: ὅμοιος, hómoios, "similar"; and στάσις, stásis, "standing still") is the property of a system, either open or closed, that regulates its internal environment and tends to maintain a stable, constant condition. Typically used to refer to a living organism, the concept came from that of milieu interieur that was created by Claude Bernard and published in 1865. Multiple dynamic equilibrium adjustment and regulation mechanisms make homeostasis possible. ### Fiber optics In fiber optics, "steady state" is a synonym for equilibrium mode distribution.[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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8631858825683594, "perplexity": 515.0219762068239}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1386163051992/warc/CC-MAIN-20131204131731-00081-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.ideals.illinois.edu/handle/2142/24081	## Files in this item FilesDescriptionFormat application/pdf Traiteur_Justin.pdf (2Mb) (no description provided)PDF ## Description Title: A short-term ensemble wind-speed forecasting system for wind power applications Author(s): Traiteur, Justin J. Advisor(s): Roy, Somnath B. Department / Program: Atmospheric Sciences Discipline: Atmospheric Sciences Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: M.S. Genre: Thesis Subject(s): Short-term wind speed forecasting Bayesian Model Averaging Weather Research and Forecasting Single-Column Model (WRF-SCM) Abstract: Accurate short-term wind speed forecasts for utility-scale wind farms will be crucial for the U.S. Department of Energy’s (DOE) goal of providing 20% of total power from wind by 2030. For typical pitch-controlled wind turbines, power output varies as the cube of wind speed over a significant portion of the power output curve. Therefore, small improvements in wind-speed forecasts would constitute much larger improvements in wind power forecasts. In addition, communicating the level of uncertainty in these wind speed forecasts will allow the industry to better quantify the level of financial risk inherent with these forecasts. In this study, a computationally efficient and accurate forecasting system is developed. This system uses a 21-member ensemble of the Weather Research and Forecasting Single-Column Model (WRF-SCM V3.1.1) to generate a probability distribution function (PDF) of 1-hour forecasts at a 90m height location in West/Central Illinois. The WRF-SCM ensemble was initialized by the 20 km Rapid update Cycle (RUC) 00h forecast and perturbed by both perturbations in the initial conditions and physics options. The PDF was calibrated using Bayesian Model Averaging (BMA) where the individual forecasts were weighted according to their performance. This combination of a mesoscale numerical weather prediction ensemble system and Bayesian statistics allowed for both accurate prediction of 1-hour wind speed forecasts and their level of uncertainty. Issue Date: 2011-05-25 URI: http://hdl.handle.net/2142/24081 Rights Information: Copyright 2011 Justin J. Traiteur Date Available in IDEALS: 2011-05-25 Date Deposited: 2011-05	{"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": 0.845949649810791, "perplexity": 4951.146883507142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1443737942301.73/warc/CC-MAIN-20151001221902-00179-ip-10-137-6-227.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/687622/prove-that-the-limit-of-two-consecutive-fibonacci-numbers-exists	# Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate] Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove that the limit does in fact exist? Is there more than one method? I do not think this is a duplicate. Please can someone explicitly show how to split the odd and even terms of this ratio sequence into two sequences- one monotonic increasing and one monotonic decreasing- and given that all ratios are between 1 and 2, show that the limit exists and we do not oscillate forever. Most of the question has been answered. I have shown that there are two subsequences- one increasing and bounded above by 2 and one decreasing, bounded below by 1. Using the fact that the limit exists, I can show it has value $\phi$. But how can I show the limit is the same for both subsequences?! • I would start with the fact that the limit is obviously bounded above by $2$ (can you see this?), and show it is monotonic. – preferred_anon Feb 23 '14 at 21:12 • A simple way is to use the "Binet" formula for $F_n$. – André Nicolas Feb 23 '14 at 21:15 • Daniel Littlewood- thank you. The only problem is that the sequence of ratios oscillates to the limit, right? If I could say it was monotonic increasing and bounded above then I would know what to do, or monotonic decreasing and bounded below, but that isn't the case. Andre Nicolas- Thanks for that answer. The Binet formula is quite incredible! I've seen that proof and would like to use that as a second solution. I would also really like to be able to show the limit exists using only real analysis if possible. – Guest Feb 23 '14 at 21:17 • @preferred_anon only monotonic considering even or odd terms. – qwr Feb 15 '19 at 17:15 By Cassini's Identity: $$\left\|\frac{F_{n+1}}{F_{n}}-\frac{F_{n}}{F_{n-1}}\right\|=\left\|\frac{F_{n+1}F_{n-1}-F_{n}^{2}}{F_{n}F_{n-1}}\right\|=\left\|\frac{(-1)^{n}}{F_{n}F_{n-1}}\right\| \to 0$$ Proof of Cassini's Identity: $$F_{n+1}F_{n-1}-F_{n}^{2}\\ =(F_{n}+F_{n-1})F_{n-1}-F_{n}^{2}\\ =F_{n-1}F_{n}-F_{n}^{2}+F_{n-1}^{2}\\ =-(F_{n}(F_{n}-F_{n-1})-F_{n-1}^{2})\\ =-(F_{n}F_{n-2}-F_{n-1}^{2})$$ You can fill in the rest by induction. • I like this a lot. I haven't heard of Cassini's Identity before. Could you please link to a proof or derivation of it? – Guest Feb 23 '14 at 21:45 • I've sketched a proof in the answer - is this clear enough? – preferred_anon Feb 23 '14 at 22:13 • Yes this is good, thank you! What I really want to do is split the sequence of ratios into two sequences- a monotonic increasing one and a monotonic decreasing one and go from there. But I don't know how to do it... – Guest Feb 23 '14 at 22:14 • Simply replace $n$ by $2n$. Clearly when $n$ is even the difference is positive and when $n$ is odd the difference is negative! – preferred_anon Feb 23 '14 at 23:13 • Yes, I did that. 2n and 2n+1 for even and odd respectively. But then what? – Guest Feb 23 '14 at 23:15 Hint: Consider $\frac{F_n}{F_{n-1}}=1+\frac{F_{n-2}}{F_{n-1}}$. Let $f_n=\frac{F_n}{F_{n-1}}$. We have $$f_n=1+\frac{1}{f_{n-1}}.$$ Let $f(x)=1+\frac{1}{x}$ for $1<x<2$. Show if $x<\phi$, then $x<f(x)<\phi$ (We also have $\phi<f(x)<x$ for $x>\phi$, but it is irrelevant to our concern.). • Thank you for this answer. This is how I was able to show that the limit was $\phi$ if it exists, but how do you know $f_n$ exists and is a real number as $n \to \infty$? – Guest Feb 23 '14 at 21:19 • Thanks for the edit. Would you be able to do the next step? Sorry- I am still unclear on how this would prove the existence of the limit given its oscillation? – Guest Feb 23 '14 at 21:44 • @Guest If you showed my claim, then $f_1<f_2<f_3\cdots \le \phi$. – Ma Ming Feb 23 '14 at 22:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9414756894111633, "perplexity": 190.0217986773205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1593657142589.93/warc/CC-MAIN-20200713033803-20200713063803-00330.warc.gz"}
https://www.physicsforums.com/threads/temperature-of-a-electron.238344/	# Temperature of a electron? 1. Jun 2, 2008 ### eminent_youtom Hi there, I was wondering from few weeks that what is a temperature? According to the classical idea, temperature is caused by the vibration (oscillation) of molecules and atoms. So, is it appropriate to ask the temperature of a single electron or few electrons? till what level temperature exists? what actually is temperature? what happens to the atoms, especially electrons, when we decrease the temperature of the system to the absolute zero, like if we cool it to near absolute zero temeprature? 2. Jun 2, 2008 ### kanato Temperature is a statistical quantity. The formal definition is dS/dE, the change in entropy with respect to energy holding volume and particle number constant. Typically, a practical definition comes from the fact that the atoms, molecules, or whatever particles in your system have average kinetic energy <KE> = 3/2 kT, where k is Boltzmann's constant. (This does not apply in the case of a highly degenerate fermi gas, such as electrons in a solid.) Note too that the average specifically means to average over the KE of all the particles in your system, so it is not correct to take the KE of a random particle in the sample, multiply by 2/(3k) and call that its temperature. In summary, it does not make sense to ask what the temperature of a single electron, atom, molecule, or even a handful of such particles. So you ask, at what scale does temperature 'kick in' where it makes sense to talk about temperature? Probably around a few million particles, although I don't think there is any set rule. I would say that if you had some large system, and you cut in in half, and whatever means you had for measuring the temperature gives you the same result for both halves without large fluctuations, then you're ok talking about temperature. Classically speaking, if you lower the temperature to absolute zero, the atoms stop moving. This is not actually the case in quantum mechanics, because zero point motion would be left (you can think of it as an expression of the uncertainty principle; if you knew that p=0 from zero temperature, and the position was fixed, that would violate the UP, so things have to keep moving). Electrons don't change their behavior much at low temperatures, thanks to the exclusion principle. Basically, they are required to fill available states from the lowest up, with only one to each state. The highest filled state defines the energy scale for electrons, in temperature units it's several(/ten-) thousand K. At any temperature that's a fair amount below this temperature, they act basically the same. So cooling to absolute zero from room temperature (300K) does virtually nothing to them. 3. Jun 2, 2008 ### Mapes Kanato, I believe you mean T = dE/dS (great answer otherwise). 4. Jun 3, 2008 ### kanato Heh whoops, yeah, that's what I meant :) 5. Jun 3, 2008 ### eminent_youtom Does energy has a temperature? i am talking about the pure energy without any mass. otherwise if there are masses, certainly energy is gonna to increase the temperature of the masses. The main reason behind my question is the temperature of blackholes. Stephen Hawking argued that pair of virtual particles are created near the event horizon [quantum fluctuation due to uncertainty principle], due to which blackholes are radiating energy. Which means blackholes have some temperature. It is widely accepted fact. HOWEVER, blackholes are such a system where spacetime fabric is wrapped in such a way that, tremendous masses are squeezed in a small volume. i.e. all the atoms and particle are squeezed together, where even atom doesn't exists in a pure atomic form. According to NASA, http://www.gsfc.nasa.gov/scienceques2002/20030516.htm" [Broken] now my question is, how is it possible to have vibration of atoms in blackholes? what is allowing them to vibrate (if they exists)? Although the mathematics behind the temperature of blackholes are well estlablished, couldnt it be just a theoritical idea? because, particles are almost freezed inside the blackhole. so far, have scientists detected any electromagnetic wave radiated by blackholes? Last edited by a moderator: May 3, 2017 6. Jun 3, 2008 ### kanato I don't know what pure energy is, without mass, if it's not light. A system of photons does have temperature. Temperature is related to the energy and entropy of a system, so if the system has internal degrees of freedom (different ways to distribute the energy to the particles inside the system) then it has a temperature. You question on black holes is better directed at another board, probably the High Energy, Nuclear, Particle Physics board, or the Cosmology board. I think the interior structure of a black hole is probably something that remains highly speculative at this point, but I don't know. 7. Jun 3, 2008 ### Gokul43201 Staff Emeritus Yes. To my knowledge, the x-ray signature (caused by ionization of collapsing atoms) of black holes is one of the easiest ways to detect them.	{"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": 0.8136731386184692, "perplexity": 704.427398047955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1524125945940.14/warc/CC-MAIN-20180423085920-20180423105920-00224.warc.gz"}
http://cs.stackexchange.com/questions/22735/sandwiching-languages	# Sandwiching Languages I am studying for my algorithms final and came across the following problem: Find three languages $L_1 \subset L_2 \subset L_3$ over the same alphabet such that $L_2 \in P$ and $L_1,L_3$ are undecidable. I am having trouble coming up with an example of three such languages. My first thought was to use a form of the halting problem for both $L_1$ and $L_3$ since that is pretty much the only undecidable language I know and am familiar with. I was thinking of perhaps coming up with something of the form \begin{align} L_1 &= \{M \mid \text{$M$ is a Turing machine that starts with 00 and halts}\}\\ L_2 &= \{M \mid \text{$M$ is a Turing machine that starts with a 00}\}\\ L_3 &= ? \end{align} but this doesn't seem to be working. Any ideas are appreciated! - Hint: Let $L_3 = L_2 \cup M$, where $M$ is a language similar to $L_1$ and disjoint from $L_2$. Hint: Your approach can be completed and you need to change only one word in the description of $L_1$ to get a description for $L_3$ that works. I am not sure if I see what word I should change, but I might have solved it anyway. Let $L_1,L_2$ be defined as above. Then simply define $L_3 = \{M \mid M \text{ is a Turing machine that starts with 00, or starts with 01 and halts}\}$. – nacho Mar 18 at 0:48	{"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": 0.995720386505127, "perplexity": 134.66492992091958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500822560.65/warc/CC-MAIN-20140820021342-00433-ip-10-180-136-8.ec2.internal.warc.gz"}
https://www.fernuni-hagen.de/ti/forschung/publikationen/BDFKKPSW16.shtml	# Veröffentlichung Titel: Improved Approximation Algorithms for Box Contact Representations AutorInnen: Michael A. Bekos Thomas C. van Dijk Martin Fink Philipp Kindermann Stephen G. Kobourov Sergey Pupyrev Joachim Spoerhase Alexander Wolff Kategorie: Artikel in Zeitschriften erschienen in: Algorithmica, Vol. 77, no. 3, 2017, pp. 902-920 Abstract: We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (CROWN) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. CROWN is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, MAX-CROWN, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.	{"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": 0.8718858957290649, "perplexity": 1029.4413544760575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304915.53/warc/CC-MAIN-20220126041016-20220126071016-00601.warc.gz"}
https://de.zxc.wiki/wiki/Gleichung	# equation Oldest printed equation (1557), in today's notation "14x + 15 = 71" In mathematics, an equation is understood to mean a statement about the equality of two terms , which is symbolized with the help of the equal sign ("="). Formally, an equation has the form ${\ displaystyle T_ {1} = T_ {2}}$, where the term is called the left side and the term is called the right side of the equation. Equations are either true or satisfied (for example ) or false (for example ). If at least one of the terms of variables depends, is just a form of expression before; whether the equation is true or false then depends on the specific values used. The values of the variables for which the equation is satisfied are called solutions of the equation. If two or more equations are given, one also speaks of a system of equations${\ displaystyle T_ {1}}$${\ displaystyle T_ {2}}$${\ displaystyle 1 = 1}$${\ displaystyle 1 = 2}$${\ displaystyle T_ {1}, T_ {2}}$, a solution to it must satisfy all equations simultaneously. ## Types of equations Equations are used in many contexts; accordingly there are different ways of dividing the equations according to different aspects. The respective classifications are largely independent of one another; an equation can fall into several of these groups. For example, it makes sense to speak of a system of linear partial differential equations. ### Classification according to validity #### Identity equations Equations can be generally valid, i.e. they can be true by inserting all variable values from a given basic set or at least from a previously defined subset thereof. The general validity can either be proved with other axioms or it can itself be assumed as an axiom. Examples are: • the Pythagorean theorem : is true for right triangles , if the side opposite the right angle ( hypotenuse ) and the cathetus denote${\ displaystyle a ^ {2} + b ^ {2} = c ^ {2}}$${\ displaystyle c}$${\ displaystyle a, b}$ • the associative law : is true for all natural numbers and generally for any elements of a group (as an axiom)${\ displaystyle (a + b) + c = a + (b + c)}$ ${\ displaystyle a, b, c}$${\ displaystyle a, b, c}$ • the first binomial formula : is true for all real numbers${\ displaystyle (a + b) ^ {2} = a ^ {2} + 2ab + b ^ {2}}$ ${\ displaystyle a, b}$ • the Euler's identity : is true for all real${\ displaystyle e ^ {i \ varphi} = \ cos \ left (\ varphi \ right) + i \ sin \ left (\ varphi \ right)}$${\ displaystyle \ varphi}$ In this context one speaks of a mathematical proposition or law. To distinguish between equations that are not generally valid, the congruence sign ("≡") is used for identities instead of the equal sign. #### Determining equations Often one task is to determine all variable assignments for which the equation becomes true. This process is known as solving the equation . To distinguish between identity equations, such equations are referred to as determining equations . The set of variable assignments for which the equation is true is called the solution set of the equation. If the solution set is the empty set , the equation is called unsolvable or unsatisfiable. Whether an equation is solvable or not can depend on the basic set considered, for example: • the equation is unsolvable as an equation over the natural or the rational numbers and has the solution set as an equation over the real numbers${\ displaystyle x ^ {2} = 2}$${\ displaystyle \ lbrace {\ sqrt {2}}, - {\ sqrt {2}} \ rbrace}$ • the equation is unsolvable as an equation over the real numbers and has the solution set as an equation over the complex numbers${\ displaystyle x ^ {2} = - 2}$${\ displaystyle \ lbrace {\ sqrt {2}} i, - {\ sqrt {2}} i \ rbrace}$ In the case of determining equations, variables sometimes appear that are not sought but are assumed to be known. Such variables are called parameters . For example, the formula for solving the quadratic equation is ${\ displaystyle x ^ {2} + px + q \; = \; 0}$ for unknown unknowns and given parameters and${\ displaystyle x}$${\ displaystyle p}$${\ displaystyle q}$ ${\ displaystyle x_ {1,2} \; = \; - {\ frac {p} {2}} \ pm {\ sqrt {{{\ frac {p ^ {2}} {4}} - q}}}$. If you insert one of the two solutions into the equation, the equation is transformed into an identity, i.e. it becomes a true statement for any choice of and . For here the solutions are real, otherwise complex. ${\ displaystyle x_ {1}, x_ {2}}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle 4q \ leq p ^ {2}}$ #### Equations of definition Equations can also be used to define a new symbol. In this case, the symbol to be defined is written on the left, and the equal sign is often replaced by the definition sign (“: =”) or written over the equal sign “def”. For example, which is derivative of a function at a position by ${\ displaystyle f}$${\ displaystyle x_ {0}}$ ${\ displaystyle f '(x_ {0}): = \ lim _ {x \ to x_ {0}} {\ frac {f (x) -f (x_ {0})} {x-x_ {0}} }}$ Are defined. In contrast to identities, definitions are not statements; so they are neither true nor false, just more or less useful. ### Division on the right #### Homogeneous equations A defining equation of form ${\ displaystyle T (x) = 0}$ is called a homogeneous equation . If a function is , the solution is also called the zero of the function. Homogeneous equations play an important role in the solution structure of linear systems of equations and linear differential equations . If the right side of an equation is not equal to zero, the equation is said to be inhomogeneous. ${\ displaystyle T}$${\ displaystyle x}$ #### Fixed point equations A defining equation of form ${\ displaystyle T (x) = x}$ is called the fixed point equation and its solution is called the fixed point of the equation. Fixed point theorems give more precise information about the solutions of such equations . ${\ displaystyle x}$ #### Eigenvalue problems A defining equation of form ${\ displaystyle T (x) = \ lambda x}$ is called the eigenvalue problem, where the constant (the eigenvalue) and the unknown (the eigenvector) are sought together. Eigenvalue problems have diverse areas of application in linear algebra, for example in the analysis and decomposition of matrices , and in areas of application, for example structural mechanics and quantum mechanics . ${\ displaystyle \ lambda}$${\ displaystyle x \ neq 0}$ ### Classification according to linearity #### Linear equations An equation is called linear if it is in the form ${\ displaystyle T \ left (x \ right) = a}$ can be brought, where the term is independent of and the term is linear in , so ${\ displaystyle a}$${\ displaystyle x}$${\ displaystyle T (x)}$${\ displaystyle x}$ ${\ displaystyle T \ left (\ lambda x + \ mu y \ right) = \ lambda T \ left (x \ right) + \ mu T \ left (y \ right)}$ applies to coefficients . It makes sense to define the appropriate operations, so it is necessary that and are from a vector space and the solution is sought from the same or a different vector space . ${\ displaystyle \ lambda, \ mu}$${\ displaystyle T (x)}$${\ displaystyle a}$ ${\ displaystyle V}$${\ displaystyle x}$${\ displaystyle W}$ Linear equations are usually much easier to solve than nonlinear ones. The superposition principle applies to linear equations : The general solution of an inhomogeneous equation is the sum of a particulate solution of the inhomogeneous equation and the general solution of the associated homogeneous equation. Because of the linearity there is at least one solution to a homogeneous equation. If a homogeneous equation has a unique solution, then a corresponding inhomogeneous equation also has at most one solution. A related but much more in-depth statement in functional analysis is Fredholm's alternative . ${\ displaystyle x = 0}$ #### Nonlinear equations Nonlinear equations are often differentiated according to the type of nonlinearity. In school mathematics in particular , the following basic types of non-linear equations are dealt with. ##### Algebraic equations If the equation term is a polynomial , one speaks of an algebraic equation. If the polynomial is at least degree two, the equation is called nonlinear. Examples are general quadratic equations of form ${\ displaystyle ax ^ {2} + bx + c = 0}$ or cubic equations of form ${\ displaystyle ax ^ {3} + bx ^ {2} + cx + d = 0}$. There are general solution formulas for polynomial equations up to degree four . ##### Fractional equations If an equation contains a fraction term in which the unknown occurs at least in the denominator , one speaks of a fraction equation, for example ${\ displaystyle {\ frac {x + 2} {x ^ {2} +3}} = {\ frac {2} {x + 1}}}$. By multiplying by the main denominator, in the example , fractional equations can be reduced to algebraic equations. Such a multiplication is usually not an equivalence conversion and a case distinction must be made, in the example the fraction equation is not included in the definition range . ${\ displaystyle (x ^ {2} +3) (x + 1)}$${\ displaystyle x = -1}$ ##### Root equations In the case of root equations, the unknown is at least once under a root , for example ${\ displaystyle {\ sqrt {x}} = 1-x}$ Root equations are special power equations with an exponent . Root equations can be solved by a root is isolated and then the equation with the root exponent (in the example ) potentiates is. This process is repeated until all roots are eliminated. Increasing to the power of an even-numbered exponent does not represent an equivalence conversion and therefore in these cases a corresponding case distinction must be made when determining the solution. In the example, squaring leads to the quadratic equation , the negative solution of which is not in the definition range of the output equation . ${\ displaystyle {\ tfrac {1} {n}}}$${\ displaystyle n}$${\ displaystyle n = 2}$${\ displaystyle x = (1-x) ^ {2}}$ ##### Exponential equations In exponential equations , the unknown appears at least once in the exponent , for example: ${\ displaystyle 2 ^ {3x + 2} = 4 ^ {x + 1}}$ Exponential equations can be solved by taking logarithms . Conversely, logarithmic equations - i.e. equations in which the unknown occurs as a number (argument of a logarithm function) - can be solved by exponentiation . ##### Trigonometric equations If the unknowns appear as an argument of at least one angle function , one speaks of a trigonometric equation, for example ${\ displaystyle \ sin (x) = \ cos (x)}$ The solutions to trigonometric equations are generally repeated periodically , unless the solution set is limited to a certain interval , for example . Alternatively, the solutions can be parameterized by an integer variable . For example, the solutions to the above equation are given as ${\ displaystyle [0.2 \ pi)}$${\ displaystyle k}$ ${\ displaystyle x = {\ frac {\ pi} {4}} + \ pi k}$ with .${\ displaystyle k \ in \ mathbb {Z}}$ ### Classification according to unknown unknowns #### Algebraic equations In order to distinguish equations in which a real number or a real vector is searched for from equations in which, for example, a function is searched, the term algebraic equation is sometimes used, but this term is not restricted to polynomials . However, this way of speaking is controversial. #### Diophantine equations If one looks for integer solutions of a scalar equation with integer coefficients, one speaks of a Diophantine equation. An example of a cubic Diophantine equation is ${\ displaystyle 2x ^ {3} -x ^ {2} -8x = -4}$, of the integers that satisfy the equation, here the numbers . ${\ displaystyle x \ in \ mathbb {Z}}$${\ displaystyle x = \ pm 2}$ #### Difference equations If the unknown is a consequence , one speaks of a difference equation. A well-known example of a second order linear difference equation is ${\ displaystyle x_ {n} -x_ {n-1} -x_ {n-2} = 0}$, whose solution for starting values and the Fibonacci sequence is. ${\ displaystyle x_ {0} = 0}$${\ displaystyle x_ {1} = 1}$ ${\ displaystyle 1,2,3,5,8,13, \ ldots}$ #### Functional equations If the unknown of the equation is a function that occurs without derivatives, one speaks of a functional equation. An example of a functional equation is ${\ displaystyle f (x + y) = f (x) f (y)}$, whose solutions are precisely the exponential functions . ${\ displaystyle f (x) = a ^ {x}}$ #### Differential equations If a function is sought in the equation that occurs with derivatives, one speaks of a differential equation. Differential equations are very common when modeling scientific problems. The highest occurring derivative is called the order of the differential equation. One differentiates: ${\ displaystyle f '(x) + xf (x) = 0}$ • partial differential equations in which partial derivatives occur according to several variables, for example the linear transport equation of the first order ${\ displaystyle {\ frac {\ partial f (x, t)} {\ partial t}} + {\ frac {\ partial f (x, t)} {\ partial x}} = 0}$ {\ displaystyle {\ begin {aligned} {\ ddot {x}} _ {1} & = 2x_ {1} \ lambda \\ {\ ddot {x}} _ {2} & = 2x_ {2} \ lambda - 1 \\ 0 & = x_ {1} ^ {2} + x_ {2} ^ {2} -1 \ end {aligned}}} ${\ displaystyle {\ rm {d}} S_ {t} = rS_ {t} {\ rm {d}} t + \ sigma S_ {t} {\ rm {d}} W_ {t}}$ #### Integral equations If the function you are looking for occurs in an integral, one speaks of an integral equation. An example of a linear integral equation of the 1st kind is ${\ displaystyle \ int _ {0} ^ {x} (xt) f (t) ~ \ mathrm {d} t = x ^ {3}}$. ## Chains of equations If there are several equal signs in one line, one speaks of an equation chain . In an equation chain, all expressions separated by equal signs should have the same value. Each of these expressions must be considered separately. For example, is the equation chain ${\ displaystyle 17 + 3 = 20/2 = 10 + 7 = 17}$ wrong, because broken down into individual equations it leads to wrong statements. For example, it is true ${\ displaystyle 17 + 3 = 40/2 = 10 + 10 = 20}$. Chains of equations can be meaningfully interpreted , especially because of the transitivity of the equality relation. Chains of equations often appear together with inequalities in estimates , for example for${\ displaystyle n \ geq 3}$ ${\ displaystyle 2n ^ {2} = n ^ {2} + n ^ {2} \ geq n ^ {2} + 3n> n ^ {2} + 2n + 1 = (n + 1) ^ {2}}$. ## Systems of equations Often several equations that must be fulfilled at the same time are considered and several unknowns are searched for at the same time. ### Systems of linear equations A system of equations - that is, a set of equations - is called a system of linear equations if all equations are linear. For example is {\ displaystyle {\ begin {aligned} x + y + z & = 5 \\ 2x-z & = 13 \ end {aligned}}} a system of linear equations consisting of two equations and three unknowns and . If both the equations and the unknowns are combined into tuples , an equation system can also be understood as a single equation for an unknown vector . In linear algebra, for example, a system of equations is written as a vector equation ${\ displaystyle x, y}$${\ displaystyle z}$ ${\ displaystyle A \ cdot {\ vec {x}} = {\ vec {b}}}$ with a matrix , the unknown vector and the right hand side , where is the matrix-vector product . In the example above are ${\ displaystyle A}$${\ displaystyle {\ vec {x}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle (\ cdot)}$ ${\ displaystyle A = {\ begin {pmatrix} 1 & 1 & 1 \\ 2 & 0 & -1 \ end {pmatrix}}}$, and .${\ displaystyle {\ vec {x}} = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}}$${\ displaystyle {\ vec {b}} = {\ begin {pmatrix} 5 \\ 13 \ end {pmatrix}}}$ ### Nonlinear systems of equations Systems of equations whose equations are not all linear are called nonlinear systems of equations. For example is ${\ displaystyle \ left \ {{\ begin {array} {rcl} 3x ^ {2} + 2xy & = & 1 \\\ sin (x) \ cdot \ ln (y) & = & e ^ {x} \ end {array }} \ right.}$ a nonlinear system of equations with the unknowns and . There are no generally applicable solution strategies for such systems of equations. Often one only has the possibility to determine approximate solutions with the help of numerical methods. A powerful approximation method is, for example, the Newton method . ${\ displaystyle x}$${\ displaystyle y}$ A rule of thumb states that the same number of equations as there are unknowns are required for a system of equations to be uniquely solvable. However, this is actually only a rule of thumb, it is valid to a certain extent for real equations with real unknowns because of the main theorem on implicit functions . ## Solving equations ### Analytical solution As far as possible, one tries to find the exact solution of a determining equation. The most important tools here are equivalence transformations , through which one equation is gradually transformed into other equivalent equations (which have the same set of solutions) until an equation is obtained whose solution can be easily determined. ### Numerical solution Many equations, especially those from scientific applications, cannot be solved analytically. In this case one tries to calculate an approximate numerical solution on the computer. Such procedures are dealt with in numerical mathematics . Many nonlinear equations can be solved approximately by approximating the nonlinearities occurring in the equation linearly and then solving the resulting linear problems (for example using Newton's method ). For other problem classes, for example when solving equations in infinite-dimensional spaces, the solution is sought in suitably chosen finite-dimensional subspaces (for example in the Galerkin method ). ### Qualitative analysis Even if an equation cannot be solved analytically, it is still often possible to make mathematical statements about the solution. In particular, we are interested in the question of whether a solution exists at all, whether it is unique, and whether it depends continuously on the parameters of the equation. If this is the case, one speaks of a correctly posed problem . 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https://robert-williams.org/2020/11/	# Monthly Archives: November 2020 ## Follow up: GKL similarity and social choice The previous post discusses a “distance minimizing” way of picking a compromise between agents with a diverse set of utilities. If you measure distance (better: divergence) between utility functions by square Euclidean distance, then a utilitarian compromise pops out. I wanted now to discuss briefly a related set of results (I’m grateful for pointers and discussion with Richard Pettigrew here, though he’s not to blame for any goofs I make along the way). The basic idea here is to use a different distance/divergence measure between utilities, and look at what happens. One way to regard what follows is as a serious contender (or serious contenders) for measuring similarity of utilities. But another way of looking at this is as an illustration that the choice of similarity I made really has significant effects. I borrowed the square Euclidean distance analysis of similarity from philosophical discussions of similarity of belief states. And the rival I now discuss is also prominent in that literature (and is all over the place in information theory). It is (generalized) Kullback-Leibler relative entropy (GKL), and it gets defined, on a pair of real valued vectors U,V in this way: $D_{KL}(U,V):=\sum_{p\in P} U(p)\log \frac{U(p)}{V(p)} - U(p)+V(p)$ Note that when the vectors are each normalized to the same quantity, the sum of U(p) over all p is equal to the sum of V(p) over all p, and so two latter summands cancel. In the more general case, they won’t. Kullback-Leibler relative entropy is usually applied with U and V being probability functions, which are normalized, so you normally find it in the form where it is a weighted sum of logs. Notoriously, GKL is not symmetric: the distance from U to V can be different from the distance to U from V. This matters; more anon. (One reason I’m a little hesitant with using this as a measure of similarity between utilities in this context is the following. When we’re using it to measure similarity between beliefs or probability functions, there’s a natural interpretation of it as the expectation from U’s perspective of difference between the log of U and the log of V. But when comparing utilities rather than probabilities means we can’t read the formula this way. It feels to me a bit more of a formalistic enterprise for that reason. Another thing to note is that taking logs is well defined only when the relevant utilities are positive, which again deserves some scrutiny. Nevertheless….) What happens when we take GKL as a distance (divergence) measure, and then have a compromise between a set of utilities by minimizing total sum distance from the compromise point to the input utilities? This article by Pettigrew gives us the formal results that speak to the question. The key result is that the compromise utility $U_C$ that emerges from a set of m utility functions $U_i$ is the geometrical mean: $U_C(p)= (\prod_{i\in A} U_i(p))^{\frac{1}{m}}$. Where the utilitarian compromise utilities arising from squared euclidean distance similarity look to the sum of individual utilities, this compromise looks at the product of individual utilities. It’s what’s called in the social choice literature a symmetrical “Nash social welfare function” (that’s because it can be viewed as a special case of a solution to a bargaining game that Nash characterized: the case where the “threat” or “status quo” point is zero utility for all). It has some interesting and prima facie attractive features—it prioritizes the worse off, in that a fixed increment of utility will maximize the product of everyone’s utilities if awarded to someone who has ex ante lowest utility. It’s also got an egalitarian flavour, in that you maximize the product of a population’s utilities by dividing up total utility evenly among the population (contrast utilitarianism, where you can distribute utility in any old way among a population and get the same overall sum, and so any egalitarian features of the distribution of goods have to rely on claims about diminishing marginal utility of those goods; which by the same token leaves us open to “utility monsters” in cases where goods have increasing utility for one member of the population). Indeed, as far as I can tell, it’s a form of prioritarianism, in that it ranks outcomes by way of a sum of utilities which are discounted by the application of a concave function (you preserve the ranking of outcomes if you transform the compromise utility function by a monotone increasing function, and in this case we can first raise it to the mth power, and then take logs, and the result will be the sum of log utilities. And since log is itself a concave function this meets the criteria for prioritarianism). Anyway, the point here is not to evaluate Nash social welfare, but to derive it. The formal result is proved in the Pettigrew paper, as a corollary to a very general theorem. Under the current interpretation that theorem also has the link between squared Euclidean distance and utilitarianism of the previous post as another special case. However, it might be helpful to see how the result falls out of elementary minimization (it was helpful for me to work through it, anyway, so I’m going to inflict it on you). So we start with the following characterization, where A is the set of agents whose utilities we are given: $U_C=\textsc{argmin}_X \sum_{i\in A} D_{KL}(X,U_i)$ To find this we need to find X which makes this sum minimal (P being the set of n propositions over which utilities are defined, and A being the set of m agents): $\sum_{i\in A} \sum_{p\in P} X(p)\log \frac{X(p)}{U_i(p)} - X(p)+U_i(p)$ Rearrange as a sum over p: $\sum_{p\in P} \sum_{i\in A} X(p)\log \frac{X(p)}{U_i(p)} - X(p)+U_i(p)$ Since we can assign each X(p) independently of the others, we minimize this sum by minimizing each summand. Fixing p, and writing $x:=X(p)$ and $u_i:=U_i(p)$, our task now is to find the value of u which minimizes the following: $\sum_{i\in A} x\log \frac{x}{u_i} - x+u_i$ We do this by differentiating and setting the result to zero. The result of differentiating (once you remember the product rule and that differentiating logs gives you a reciprocal) is: $\sum_{i\in A} \log \frac{x}{u_i}$ But a sum of logs is the log of the product, and so the condition for minimization is: $0=\log \frac{x^m}{\prod_{i\in A}u_i}$ Taking exponentials we get: $1=\frac{x^m}{\prod_{i\in A}u_i}$ That is: $x^m=\prod_{i\in A}u_i$ Unwinding the definitions of the constants and variables gets us the geometrical mean/Nash social welfare function as promised. So that’s really neat! But there’s another question to ask here (also answered in the Pettigrew paper). What happens if we minimize sum total distance, not from the compromise utility to each of the components, but from the components to the compromise? Since GKL distance/divergence is not symmetric, this could give us something different. So let’s try it. We swap the positions of the constant and variables in the sums above, and the task becomes to minimize the following: $\sum_{i\in A} u_i\log \frac{x}{u_i} - u_i+x$ When we come to minimize this by differentiating, we no longer have a product of functions in x to differentiate with respect to x. That makes the job easier, and ends up with us with the constraint: $\sum_{i\in A} 1-\frac{u_i}{x}$ Rearranging we get: $x= \frac{1}{n} \sum_{i\in A} u_i$ and we’re back to the utilitarian compromise proposal again! (That is, this distance-minimizing compromise delivers the arithmetical mean rather than the geometrical mean of the components). Stepping back: what we’ve seen is that if you want to do distance-minimization (similarity-maximization, minimal-mutilation) compromise on cardinal utilities then the precise way distance you choose really matters. Go for squared euclidean distance and you get utilitarianism dropping out. Go for the log distance of the GKL, and you get either utilitarianism or the Nash social welfare rule dropping out, depending on the “direction” in which you calculate the distances. These results are the direct analogues of results that Pettigrew gives for belief-pooling. If we assume that the way of measuring similarity/distance for beliefs and utilities should be the same (as I did at the start of this series of posts) then we may get traction on social welfare functions through studying what is reasonable in the belief pooling setting (or indeed, vice versa). ## From desire-similarity to social choice In an earlier post, I set out proposal for measuring distance or (dis)similarity between desire-states (if you like, between utility functions defined over a vector of propositions). That account started with the assumption that we measured strength of desire by real numbers. And the proposal was to measure the (dis)similarity between desires by the squared euclidean distance between the vectors of desirability at issue. If $\Omega$ is the finite set of n propositions at issue, we characterize similarity like this: $d(U,V)= \sum_{p\in\Omega} (U(p)-V(p))^2$ In that earlier post, I linked this idea to “value” dominance arguments for the characteristic equations of causal decision theory. Today, I’m thinking about compromises between the desires of a diverse set of agents. The key idea here is to take a set A of m utility functions $U_i$, and think about what compromise utility vector $U_C$ makes sense. Here’s the idea: we let the compromise $U_C$ be that utility vector which is closest overall to the inputs, where we measure overall closeness simply by adding up the distance between it and the input utilities $U_i$. That is: $U_C = \textsc{argmin}_X \sum_i d(X,U_i)$ So what is the X which minimizes the following? $\sum_{p\in\Omega} \sum_{i\in A} (X(p)-U_i(p))^2$ Rearranging: $\sum_{i\in A} \sum_{p\in\Omega}(X(p)-U_i(p))^2$ This is a sum of m summands, each of which is positive. So you find the minimum value by minimizing each summand. And to minimize the ith summand we differentiate and set the result to zero: $\sum_{p\in\Omega}(X(p)-U_i(p))=0$ This gives us the following value of X(p): $X(p)=\frac{\sum_{i\in A}U_i(p)}{m}$ This tells us exactly what value $U_C$ must assign to p. It must be the average utility assigned to p of the m input functions. Suppose our group of agents is faced with a collective choice between a number of options. Then one option O is strictly preferred to the other options according to the compromise utility $U_C$ just in case the average utility the agents assign to it is greater than the average utility the agents assign to any other option. (In fact, since the population is fixed when evaluating each option, we can ignore the fact we’re taking averages—O is preferred exactly when the sum total of utilities assigned to it across the population is greater than for any other). So the procedure for social choice “choose according to the distance-mimimizing compromise function” is the utilitarian choice procedure. That’s really all I want to observe for today. A couple of finishing up notes. First, I haven’t found a place where this mechanism for compromise choice is set out and defended (I’m up for citations though, since it seems a natural idea). Second, there is at least an analogous strategy already in the literature. In Gaertner’s A Primer in Social Choice Theory he discusses (p.112) the Kemeny procedure for social choice, which works on ordinal preference rankings over options, and proceeds by finding that ordinal ranking which is “closest” to a profile of ordinal rankings of the options by a population. Closeness is here measured by the Kemeny metric, which counts the number of pairwise preference reversals required to turn one ranking into the other. Some neat results are quoted: a Condorcet winner (the option that would win against all others in a purality vote) if it exists is always top of the Kemeny compromise ranking. As the Kemeny compromise ranking stands to the Kemeny distance metric over sets of preference orderings, so the utilitarian utility function stands to the square-distance divergence over sets of cardinal utility functions. I’ve been talking about all this as if every aspect of utility functions were meaningful. But (as discussed in recent posts) some disagree. Indeed, one very interesting argument for utilitarianism has as a premise that utility functions are invariant under level-changes—i.e the utility function U and the utility function V represent the same underlying desire-state if there is a constant $a$ such that for each proposition p, $U(p)=V(p)+a$ (see Gaertner ch7). Now, it seems like the squared euclidean similarity measure doesn’t jive with this picture at all. After all, if we measure the squared Euclidean distance between U and V that differ by a constant, as above, we get: $\sum_{p\in\Omega}(V(p)-U(p))^2=\sum_{p\in\Omega}(U(p)+a-U(p))^2=n.a^2$ On the one hand, on the picture just mentioned, these are supposed to be two representations of the same underlying state (if level-boosts are just a “choice of unit”) and on the other hand, they have positive dissimilarity by the distance measure I’m working with. Now, as I’ve said in previous posts, I’m not terribly sympathetic to the idea that utility functions represent the same underlying desire-state when they’re related by a level boost. I’m happy to take the verdict of the squared euclidean similarity measure literally. After all, it was only one argument for utilitarianism as a principle of social choice that required the invariance claim–the reverse implication may not hold. In this post we have, in effect, a second independent argument for utilitarianism as a social choice mechanism that starts from a rival, richer preference structure. But what if you were committed to the level-boosting invariance picture of preferences? Well, really what you should be thinking about in that case is equivalence classes of utility functions, differing from each other solely by a level-boost. What we’d really want, in that case, is a measure of distance or similarity between these classes, that somehow relates to the squared euclidean distance. One way forward is to find a canonical representative of each equivalence class. For example, one could choose the member of a given equivalence class that is closest to the null utility vector–from a given utility function U, you find its null-closest equivalent by subtracting a constant equal to the average utility it assigns to propositions: $U_0=U-\frac{\sum_{p\in\Omega} U(p)}{n}$. Another way to approach this is to look at the family of squared euclidean distances between level-boosted equivalents of two given utility functions. In general, these distances will take the form $\sum_{p\in Omega} ((U(p)-\alpha) -(V(p) -\beta))^2=\sum_{p\in \Omega} (U(p)-V(p) -\gamma)^2$ (Where $\gamma=\alpha-\beta$.) You find the minimum element in this set of distances (the closest the two equivalence classes come to each other) by differentiating with respect to gamma and setting the result to zero. That is: $0=\sum_{p\in Omega} (U(p)-V(p) -\gamma)$, which rearranging gives: $\gamma=\frac{\sum_{p\in \Omega} (U(p)-V(p))}{n}=\frac{\sum_{p\in \Omega} U(p)}{n}-\frac{\sum_{p\in \Omega} V(p))}{n}$ Working backwards, set $\alpha:=\frac{\sum_{p\in \Omega} U(p)}{n}$ and $\beta:=\frac{\sum_{p\in \Omega} V(p))}{n}$, and we have defined two level boosted variants of the original U and V which minimize the distance between the classes of which they are representatives (in the square-euclidean sense). But note these level boosted variants are just $U_0$ and $V_0$. That is: minimal distance (in the square-euclidean sense) between two equivalence classes of utility functions is achieved by looking at the squared euclidean distance between the representatives of those classes that are closest to the null utility. This is a neat result to have in hand. I think the “minimum distance between two equivalence classes” is better motivated than simply picking arbitrary representatives of the two families, if we want a way of extending the squared-Euclidean measure of similarity to utilities which are assumed to be invariant under level boosts. But this last result shows that we can choose (natural) representatives of the equivalence classes generated and measure the distance between them to the same effect. It also shows us that the social choice compromise which minimizes distance between families of utility can be found by (a) using the original procedure above for finding the utility function $U_C$ selected as a minimum-distance compromise between the reprentative of each family of utility functions; and (b) selecting the family of utility functions that are level boosts of $U_C$. Since the level boosts wash out of the calculation of the relative utilities of a set of options, all the members of the $U_C$ family will agree on which option to choose from a given set. I want to emphasize again: my own current view is that the complexity intoduced in the last few paragraphs is unnecessary (since my view is that utilities that differ by constant factors from one another represent distanct desire-states). But I think you don’t have to agree with me on this matter to use the minimum distance compromise argument for utilitarian social choice. ## How emotions might constrain interpretation Joy is appropriate when you learn that something happens that you really really want. Despair is appropriate when you learn that something happens that you really really don’t want to happen. Emotional indifference is appropriate when you learn that something happens which you neither want nor don’t want–which is null for you. And there are grades of appropriate emotional responses—from joy to happiness, to neutrality, to sadness, to despair. I take it that we all know the differences in the intensity of the feeling in each case, and have no trouble distinguishing the valence as positive or negative. More than just level and intensity of desire matters to the appropriateness of an emotional response. You might not feel joy in something you already took for granted, for example. Belief-like as well as desire-like states matter when we assess an overall pattern of belief/desire/emotional states as to whether they “hang together” in an appropriate way–whether they are rationally coherent. But levels and intensities of desire obviously matter (I think). Suppose you were charged with interpreting a person about whose psychology you knew nothing beforehand. I tell you what they choose out the options facing them in a wide variety of circumstances, in response to varying kinds of evidence. This is a hard task for you, even given the rich data, but if you assumed the personal is rational you could make progress. But if all you did was attribute beliefs and desires which (structrurally) rationalize the choices and portray the target as responding rationally to the evidence, then there’d be a distintive kind of in-principle limit built into the task. If you attributed utility and credences which make the target’s choices maximize expected utility, and evolve by conditionalization on evidence, then you’d get a fix on what the target prefers to what, but not, in any objective sense, how much more they prefer one thing to another, or whether they are choosing x over y because x is the “lesser or two evils” or the “greater of two goods”. If you like, think of two characters facing the same situation–an enthusiast who just really likes the way the world is going, but mildly prefers some future developments to others, and the distraught one, who thinks the world has gone to the dogs, but regards some future developments as even worse than others. You can see how the the choice-dispositions of the two given the same evidence could match despite their very different attitudes. So given only information about the choice-dispositions of a target, you wouldn’t know whether to interpret the target as an enthusiast or their distraight friend. While the above gloss is impressionistic, it reflects a deep challenge to the attempt to operationalize or otherwise reduce belief-desire psychology to patterns of choice-behaviour. It receives its fullest formal articulation in the claim that positive affine transformations of a utility function will preserve the “expected utility property”. (Any positive monotone transformation of a utility function will preserve the same ordering over options. The mathetically interesting bit here is that the positive affine transformations of utility function guarantee that the pattern between preferences over outcomes and preferences over acts that bring about those outcomes, mediated by credences in the act-outcome links, are all preserved). One reaction to this in-principle limitation is to draw the conclusion that really, there are no objective facts about the level of desire we each have in an outcome, or how much more desirable we find one thing than another. A famous consequence of drawing that conclusion is that no objective sense could be made out of questions like: do I desire this pizza slice more or less than you do? Or questions like: does the amount by which I desire the pizza more than the crisps exceed the amount you desire the pizza more than the crisps? And clearly if desires aren’t “interpersonally comparable” in this sort of ways, certain ways of appealing to them within accounts of how its appropriate to trade off one person’s desires against another’s won’t make sense. A Rawlsian might say: if there’s pizza going spare, give it to the person for whom things are going worst (for whom the current situation, pre-pizza, is most undesirable). A utilitarian might say: if everyone is going to get pizza or crisps, and everyone prefers pizza to crisps, give the pizza to the person who’ll appreciate it the most (i.e. prefers pizza over crisps more than anyone else). If the whole idea of interpersonal comparisons of level and differences of desirability are nonsense, however, then those proposals write cheques that the metaphysics of attitudes can’t pay. (As an aside, it’s worth noting at this point that you could have Rawlsian or utilitarian distribution principles that work with quantities other than desire—some kind of objective “value of the outcome for each person”. It seems to me that if the metaphysics of value underwrites interpersonally comparable quantities like the levels of goodness-for-Sally for pizza, and goodness-difference-between-pizza-and-crisps-for-Harry, then the metaphysics of desires should be such that Sally and Harry’s desire-state will, if tuned in correctly, reflect these levels and differences.) It’s not only the utilitarian and Rawlsian distribution principles (framed in terms of desires) that have false metaphysical presuppositions if facts about levels and differences in desire are not a thing. Intraindividual ties between intensities of emotional reaction and strength of desire, and between type of emotional reaction and valence of desire, will have false metaphysical presuppositions if facts about an individual’s desire are invariant under affine tranformation. Affine transformations can change the “zero point” on the scale on which we measure desirability, and shrink or grow the differences between desirabilities. But we can’t regard zero-points or strengths of gaps as merely projections of the theorist (“arbitrary choices of unit and scale”) if we’re going to tie to them to real rational constraints on type and intensity of emotional reaction. However. Suppose in the interpretive scenario I gave you, you knew not only the choice-behaviour of your target in a range of varying evidential situations, but also their emotional responses to the outcome of their acts. Under the same injunction to find a (structurally) rationalizing interpretation of the target, you’d now have much more to go on. When they have emotional reactions rationally linked to indifference, you would attribute a zero-point in the level of desirability. When an outcome is met with joy, and another with mere happiness, you would attribute a difference in desire (of that person, for that outcome) that makes sense of both. Information about emotions, together with an account of the rationality of emotions, allow us to set the scale and unit in interpreting an individual, in a way choice-behaviour alone struggles to. As a byproduct, we would then have a epistemic path to interpersonal comparability of desires. And in fact, this looks like an epistemic path that’s pretty commonly available in typical interpersonal situations–the emotional reactions of others are not more difficult to observe than the intentions with which they act or the evidence that is available to them. Emotions, choices and a person’s evidence are all interestingly epistemically problematic, but they are “directly manifestable” in a way that contrasts with the beliefs and desires that mesh with them. The epistemic path suggests a metaphysical path to grounding levels and relative intensities of desires. Just as you can end up with a metaphysical argument against interpersonal comparability of desires by commiting oneself to grounding facts about desires in patterns of choice-behaviour, and then noting the mathematical limits of that project, you can get, I think, a metaphysical vindication of interpersonal comparabiilty of desire by including in the “base level facts” upon which facts about belief and desire are grounded facts about, type, intensity and valence of intentional emotional states. As a result, the metaphysical presuppositions of the desire-based Rawlsian and utilitarian distribution principles are met, and our desires have the structure necessary to capture and reflect level and valence of any good-for-x facts that might feature in a non-desire based articulation of those kind of principles. In my book The Metaphysics of Representation I divided the task of grounding intentionality into three parts. First, grounding base-level facts about choice and perceptual evidence (I did this by borrowing from the teleosemantics literature). Then grounding belief-desire intentional facts in the base-level facts, via a broadly Lewisian metaphysical form of radical interpretation. (The third level concerned representational artefacts like words, but needn’t concern us here). In these terms, what I’m contemplating is to add intensional emotional states to the base level, using that to vindicate a richer structure of belief and desire. Now, this is not the only way to vindicate levels and strength of desires (and their interpersonal comparability) in this kind of framework. I also argue in the book that the content-fixing notion of “correct interpretation” should use a substantive conception of “rationality”. The interpreter should not just select any old structurally-rationalizing interpretation of their target, but will go for the one that makes them closest to an ideal, where the ideal agent responds to their reasons appropriately. If an ideal agent’s strength and levels of desire are aligned, for example, to the strength and level of value-for-the-agent present in a situation, then this gives us a principled way to select between choice-theoretically equivalent interpretations of a target, grounding choices of unit and scale and interpersonal comparisons. I think that’s all good! But I think that including emotional reactions as constraining factors in interpretation can help motivate the hypothesis that there will be facts about the strength and level of desire of the ideal agent, and gives a bottom-up data-based constraint on such attributions that complements the top-down substantive-rationality constraint on attributions already present in my picture. I started thinking about this topic with an introspectively-based conviction that of course there are facts about how much I want something, and whether I want it or want it not to happen. I still think all this. But I hope that I’ve now managed to identify how those convinctions to their roles in a wider theoretical edifice–their rational interactions with obvious truths about features of our emotional lives, the role of these in distribution principles, which give a fuller sense of what is at stake if we start denying that the metaphysics of attitudes has this rich structure. I can’t see much reason to go against this, unless you are in the grip of a certain picture of how attitudes get metaphysically grounded in choice-behaviour. And I like a version of that picture! But I’ve also sketched how the very links to emotional states give you a version of that kind of metaphysical theory that doesn’t have the unwelcome, counterintuitive consequences its often associated with.	{"extraction_info": {"found_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": 44, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.830536425113678, "perplexity": 957.4017597104006}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374217.78/warc/CC-MAIN-20210306004859-20210306034859-00298.warc.gz"}
https://cs.stackexchange.com/questions/54355/brzozowski-algebraic-method-for-nfa	# Brzozowski algebraic method for NFA Currently I have a graph (basically, a state graph) in Scala which is similar to an NFA • some nodes have multiple in/outgoing edges • single start state • there might be multiple final states • a state can contain self loops where a symbol is consumed • no epsilon transitions What i'm trying to do is generate a regular expression from this graph. I've found the following topic using Brzozowski algebraic method: https://cs.stackexchange.com/q/2392 However, it seems to be for DFA's, so my question: can I use Brzozowski algebraic method OR Transitive closure method with my NFA, or do i need conversion first? I need an algorithm that can be done by a computer to automatically generate a regex from a given graph. • Have you looked at the method? Does is rely on determinism anywhere? Some of the other methods at that question you link are explicitly said to work for nondeterministic automata. – Raphael Mar 12 '16 at 16:49 • Your method is for NFA's yes, but i need an algorithm to solve it computationally, without "prolog style" languages. The other methods do not state anything about whether they are suitable for NFA or not, or am i overlooking things? – Captain Obvious Mar 13 '16 at 11:19 • Well, maybe the authors did not feel the need to state anything explicitly, which would indicate that NFAs work as input. Have you looked at the methods? Try to find out for yourself? Do the methods need determinism at any time? – Raphael Mar 13 '16 at 17:50 • Hey Raphael, thanks for your time. Yes, I did look at them and I don't see anything special that might fail the algorithm, but in the end I'm not sure and searching google didn't really help me to validate that it can be applied to an NFA. In some cases my graph is a DFA, but sometimes I have 2 edges with the same symbol, i.e. 2 choices instead of 1 which makes it an NFA. Any clarification is appreciated if i'm wrong – Captain Obvious Mar 13 '16 at 18:37 • You can not "assume" anything. You can check and make sure, i.e. prove what you need. I did not look at the methods closely myself, so I can not say one or the other with certainty. – Raphael Mar 15 '16 at 13:51 the system can be written as \begin{align} X_1 &= (a^b + a)X_2 + b^X_3 \\ X_2 &= (a+b)X_1 + bX_3 + 1 \\ X_3 &= aX_1 + aX_2 + 1 \end{align} Replacing $X_3$ by $aX_1 + aX_2 + 1$, and observing that $a + b^a = b^a$, we obtain the equivalent system \begin{align} X_1 &= (a^b + a)X_2 + b^(aX_1 + aX_2 + 1) = b^aX_1 + (a^b + b^a)X_2 + b^* \\ X_2 &= (a+b)X_1 + b(aX_1 + aX_2 + 1) + 1 = (a + b + ba)X_1 + baX_2 + b + 1\\ X_3 &= aX_1 + aX_2 + 1 \end{align} We deduce from the second equation $$X_2 = (ba)^((a + b + ba)X_1 + b + 1)$$ and replacing $X_2$ by its value in the first equation, we obtain \begin{align} X_1 &= b^aX_1 + (a^b + b^a)(ba)^((a + b + ba)X_1 + b + 1) + b^\\ &= (b^a + (a^b + b^a)(ba)^(a + b + ba))X_1 + (a^b + b^a)(ba)^(b + 1) + b^ \end{align} Finally, the language recognised by the automaton is $$X_1 = \bigl(b^a + (a^b + b^a)(ba)^(a + b + ba)\bigr)^[(a^b + b^a)(ba)^(b + 1) + b^]$$ since $1$ is the unique initial state. Now, if you want to implement this algorithm, you may end up with more complicated regular expressions. For instance, the sentence "observing that $a + b^a = b^a$" might be difficult to implement. Other useful simplifications like $a^* + 1 = a^*$ or even $K + K = K$ or $K + L = L + K$, where $K$ and $L$ are regular expressions, are also not easy to implement and require more sophisticated rewriting systems.	{"extraction_info": {"found_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": 0.983152449131012, "perplexity": 665.6608170485997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1600400198213.25/warc/CC-MAIN-20200920125718-20200920155718-00481.warc.gz"}
https://physics.stackexchange.com/questions/635069/why-is-there-no-kinetic-term-in-the-hamiltonian-of-the-ising-model	# Why is there no kinetic term in the Hamiltonian of the Ising model? I am used to the Hamiltonian formalism in the context of (quantum) field theory, where as far as I can remember it always has the form of a kinetic term + a potential term. For me the absence of kinetic terms means a theory without dynamics. In Wikipedia the Hamiltonian of the Ising Model reads: $$H(\sigma) = \sum_{i,j} J_{ij} \sigma_i \sigma_j + \sum_j h_j \sigma_j\,, \tag{1}$$ where the first term corresponds to interactions and where the sum runs over nearest neighbors, so I suppose $$i \neq j$$. The second term corresponds to an external potential. Why is there no kinetic term? Can the system evolve in time, e.g. by seeing the 2d Ising Model as a (Euclidean) (1+1)d model? How should I picture this Hamiltonian as a system in my head? • Somehow I think that the (1+1)d suggestion is not correct, and that instead I should see the temperature as the inverse time. Is that true? That still does not explain the absence of kinetic term I suppose. – Pxx May 9, 2021 at 10:19 • Related: math.stackexchange.com/questions/124706/… (moreover: "inverse temperature=time" is the Wick rotation, see my answer below). May 9, 2021 at 11:55 I always thought the Ising model as the simplest model for magnetism in which spins (atoms, electrons or particles with spin to be precise) are in a static lattice, and the model only considers the spin-spin interaction. It's static in the sense that they are in a fixed lattice, but there is dynamics since spins do evolve in time. If you which to study something else that just a fixed lattice, then search for itinerant magnetism. The Hubbard model is one of the simplest models in which spins aren't just glued and can jump between the lattice spaces. • No I am happy with the Ising model, but how can I see this time dependence? More explicitly, how can I compute the spin $\sigma_i$ after a time $t$ given an initial configuration $S$ that specifies all the initial spins? On the wikipedia page they talk about the correlators $\langle \sigma_i \sigma_j \rangle$, but that's not time-dependent is it? – Pxx May 9, 2021 at 11:13 • You have to solve the Schrödinger equation, using the Ising hamiltonian. Instead of a continuous wavefunction like when solving a particle in a box, here the functions are spinors. Each $\sigma$ is a linear combination of up and down eigenfunctions, so the full solution will be of the type: $\sigma_i(t) = A_i(t) \left \| \uparrow \right \rangle + B_i(t) \left \|\downarrow \right \rangle$, for each $i$. In these representation, the dynamics is in determining the functions $A_i$ and $B_i$. May 10, 2021 at 0:42 • @FrancisLecuak you are talking about a quantum model, while the OP is about the classical Ising model, which does not have an intrinsic dynamics (or a Schroedinger equation). I try to address these points in my answer below. May 10, 2021 at 8:54 Just a complement to the other answers, as they do not seem to touch on this issue. People have of course been studying the dynamics of Ising models for a long time, but they usually (always?) use stochastic dynamics rather than a Hamiltonian evolution. That is, they consider a Markov chain on the set of configurations having the Gibbs measure as stationary distribution. There are many choices. For instance, in order to describe a non-conservative dynamics (evolving magnetization), Glauber dynamics is often used, while a conservative dynamics (fixed magnetization) can be studied using Kawasaki dynamics. See, for instance, these notes (Sections 6.2 and 6.4, respectively) or this book (Chapter 8) for more information. With such models, one can then study all sorts of dynamical aspects: the approach to equilibrium, metastability, etc. You probably have in mind a dynamical "Hamiltonian system", which is typically defined over a phase space (typically position-momentum). In this case it makes sense to talk about potential and kinetic energy. Here you are doing equilibrium statistical mechanics. The $$H$$ of the Ising model is not the full Hamiltonian of the underlying dynamical system. In fact, at this basic level, the Ising model is not defined over a "phase space", but rather over a "configuration space" that is the space of all spin configurations. There is no explicit time (this is equilibrium thermodynamics/statistical mechanics), and there are no canonically conjugate variables. From an abstract point of view (useful in optimization problems), the function $$H$$ of the Ising model may be called "cost of the configuration". If you really want to introduce time, then you have to perform a Wick rotation (but by doing this you end up with a quantum system, see also this). In this case you will be able to "see" a discrete time derivative in the Hamiltonian, which can be used to define a sort of kinetic energy. However, this would not solve your conceptual problem, as the Wick rotation is just a formal way to map quantum systems into thermodynamic systems. In other words: there is no way to compute the $$i$$-spin after a time $$t$$ given an initial configuration that specifies all the initial spins within the framework of equilibrium statistical mechanics. At equilibrium the temporal dynamics of each spin is ruled by thermal fluctuations, which description is beyond the formulation you are referring to in your question. Edit: here I am not saying that every statistical mechanical model does not have a kinetic term. However, are some statistical mechanical models that are not defined on a phase space, but rather on a configuration space. I also completely agree with the nice answer of @Qmechanic. You can add to $$H$$ a "kinetic" term $$\sum_i^N \sigma_i ^2/2 = N/2$$, which is just an irrelevant constant! However, this does not answer to the more "philosophical" part of OP's question. I believe that, in the end, the answer is: not all statistical mechanics models stem from an underlying microscopic Hamiltonian dynamical model (despite they have a so-called "Hamiltonian"). After all, the degrees of freedom appearing in the irrelevant kinetic term $$N/2$$ are still the $$\sigma_i$$, and not some canonically conjugate variables. • I see, thanks for your answer. It would be nice if the person who put you a -1 could explain its downvote. I have one follow-up question: in principal you could imagine the same system (nearest-neighbors interaction) but in non-thermal equilibrium, right? So you start with a configuration that does not minimize $H$ and let it evolve until it reaches equilibrium. – Pxx May 9, 2021 at 12:09 • I have no idea, probably it has been down voted because of a misunderstanding (you can upvote it if you find it useful): I am NOT saying that EVERY statistical mechanics model has no kinetic energy. I am referring to the Ising one. The Ising model has no kinetic energy, it accounts for the spin-spin or spin-external field interactions. May 9, 2021 at 12:33 • For you second question: yes, you can do that. It's the idea beyond the Monte Carlo simulation of the Ising model. Note, however, that it is a "fake" dynamical evolution which is used to sample configurations from the Gibbs ensemble (typically by using the Metropolis algorithm)! It is not an evolution coming from a dynamical Hamiltonian. May 9, 2021 at 12:33 An underlying dynamics is implicit in the thermodynamic treatment. If there were no dynamics, there couldn't be an approach to equilibrium or spontaneous changes in the system. In brief, the system would not get thermodynamic equilibrium, which is the prerequisite for thermodynamics. However, in the classical case, kinetic energy plays a separate role from potential energy, and provided the system is Hamiltonian$$^$$, one can deal with the potential energy term separately. To stress an important point, since the kinetic energy term in the Hamiltonian of a classical system contributes with an additive analytical term in the thermodynamic limit, it does not influence the possibility of phase transitions. Just to add an explicit indication to the OP, I recall that there is an explicit version of the Ising model equipped with deterministic dynamics: Creutz, M. (1986). Deterministic Ising dynamics. Annals of physics, 167(1), 62-72. $$^$$ Of course, it is possible to introduce stochastic dynamics, as indicated by @YvanVelenik, and there is quite a large literature about that. However, the introduction of stochastic dynamics is not necessary to answer the original question. In classical statistical mechanics (as opposed to quantum stat-phys), the dynamical (kinetic energy) part of the Hamiltonian decouples from the interaction (potential) part, since both momentum and position are independent. The same is true for classical spin systems: their thermodynamics is independent from the dynamics, which has to be added by hand. Depending on the choice of dynamics (an example of which is Glauder dynamics), the dynamics exponents at the thermal phase transition changes, see the review by Hohenberg and Halperin. Of course, classical spin models are microscopically coming from a corresponding quantum model. However, it is well known that thermal phase transitions of quantum systems are described by static classical models (here meaning described by the classical Ising model without dynamics). How to find the classical dynamics of a quantum spin model at a thermal transition is still quite an open question, since computing the dynamics of a quantum system at a phase transition is very hard. For what its worth, recall that a kinetic term in the Hamiltonian for a rigid body in classical mechanics is $$H=\sum_{i=1}^3\frac{L_i^2}{2I_i}$$. Or more abstractly: A kinetic term of the form $$H=\sum_{i=1}^n\frac{p_i^2}{2m_i}$$. In that sense a kinetic term for the Ising model would be $$\propto \sum_i\sigma^2_i$$, which is just a constant that can be dropped, since the spin is $$\sigma_i\in\{\pm 1\}$$. • Thanks for your answer! I am not sure I follow, the first term in $(1)$ is of the form $\sigma_i \sigma_j$ with $i \neq j$ and not of the form $\sigma_i \sigma_i$... Could you explain in a bit more detail how the analogy with rigid bodies work? – Pxx May 9, 2021 at 11:15 • I updated the answer. May 9, 2021 at 11:37 • I disagree with the answer. The thermodynamics of classical spin models is always defined without dynamics, independent of the constraint (take for instance a Blum-Cappel model, where $\sigma=-1,0,1$. The true reason is that the dynamics and static part of a classical model decouple in the partition function (as is easier to see for a collection of particles interacting with some potential).	{"extraction_info": {"found_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": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8747110962867737, "perplexity": 278.9656766364071}, "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-2023-14/segments/1679296943637.3/warc/CC-MAIN-20230321064400-20230321094400-00731.warc.gz"}
https://math.stackexchange.com/questions/1987075/why-is-the-preimage-orientation-given-by-a-transversal-map-smooth	# Why is the preimage orientation given by a transversal map smooth? On page 100 of Differential Topology, Guillemin & Pollack define, given a smooth map $f: X \rightarrow Y$ between an orientedmanifold with boundary and an oriented boundaryless manifold and a submanifold $Z$ of $Y$ (also boundaryless and oriented) such that $f \pitchfork Z$ and $\partial f \pitchfork Z$, the preimage orientation for the manifold $S=f^{-1} (Z)$. What he does is first orient $df_{s}(N_s(S,X))$ (with $N_s(S,X)$ being the orthogonal complement in $T_s(X)$ of $T_s(S)$) so that $df_s(N_s(S,X)) \oplus T_{f(s)}(Z)=T_{f(s)}(Y)$, and then orient $N_s(S,X)$ and $S$ so that $df_s\|_{N_s(S,X)}$ is an orientation preserving isomorphism and $N_s(S,X) \oplus T_s(S) = T_s(X)$ My question is how can I show that this orientation is smooth (in the sense that around each $s \in S$ there is a smooth chart of S that preserves the orientation). I managed to do this in the case that $s \in \text{Int}(S)$ but my argument doesn't seem to translate at all to the case $s \in \partial S$.	{"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": 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": 0.9148473143577576, "perplexity": 127.16961268706963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1560627998238.28/warc/CC-MAIN-20190616122738-20190616144738-00402.warc.gz"}
https://meta.stackexchange.com/questions/334703/difference-between-revisions-of-a-list-is-shown-as-complete-rewrite-of-the-list	# Difference between revisions of a list is shown as complete rewrite of the list I've come across a strange display of a difference between two revisions of a list in a post, where in a newer revision only one bullet point was changed: the "inline" and "side-by-side" differences look as if the whole list has been completely rewritten. See the following screenshots of this page at ru.SO (the place where I first found it; not sure how to search for edited lists on other SE sites). ## Side-by-side diff Inline diff shows basically the same problem as side-by-side one, only the large red wall of text is above the green one. • And it only seems to have started around revision 39 (I tried some of the ones before 39, and the diffs were correctly shown) – muru Oct 8 at 9:12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.824081301689148, "perplexity": 1404.0059415328553}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1570986697439.41/warc/CC-MAIN-20191019164943-20191019192443-00177.warc.gz"}
http://www.shelovesmath.com/algebra/beginning-algebra/direct-inverse-and-joint-variation/	Direct, Inverse, Joint and Combined Variation This section covers: When you start studying algebra, you will also study how two (or more) variables can relate to each other specifically. The cases you’ll study are: • Direct Variation, where both variables either increase or decrease together • Inverse or Indirect Variation, where when one of the variables increases, the other one decreases • Joint Variation, where more than two variables are related directly • Combined Variation, which involves a combination of direct or joint variation, and indirect variation These sound like a lot of fancy math words, but it’s really not too bad. Here are some examples of direct and inverse variation: • Direct: The number of dollars I make varies directly (or you can say varies proportionally) with how much I work. • Direct: The length of the side a square varies directly with the perimeter of the square. • Inverse: The number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of). • Inverse: The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running. • Inverse: My GPA may vary directly inversely with the number of hours I watch TV. Direct or Proportional Variation When two variables are related directly, the ratio of their values is always the same. So as one goes up, so does the other, and if one goes down, so does the other. Think of linear direct variation as a “y = mx” line, where the ratio of y to x is the slope (m). With direct variation, the y-intercept is always 0 (zero); this is how it’s defined. Direct variation problems are typically written: → y = kx where k is the ratio of y to x (which is the same as the slope or rate). Some problems will ask for that k value (which is called the constant of variation or constant of proportionality – it’s like a slope!); others will just give you 3 out of the 4 values for x and y and you can simply set up a ratio to find the other value. I’m thinking the k comes from the word “constant” in another language. (I’m assuming in these examples that direct variation is linear; sometime I see it where it’s not, like in a Direct Square Variation where $y=k{{x}^{2}}$. There is a word problem example of this here.) Remember the example of making $10 an hour at the mall (y = 10x)? This is an example of direct variation, since the ratio of how much you make to how many hours you work is always constant. We can also set up direct variation problems in a ratio, as long as we have the same variable in either the top or bottom of the ratio, or on the same side. This will look like the following. Don’t let this scare you; the subscripts just refer to the either the first set of variables , or the second . Direct Variation Word Problem: So we might have a problem like this: The value of y varies directly with x, and y = 20 when x = 2. Find y when x = 8. (Note that this may be also be written “y is proportional to x, and y = 20 when x = 2. Find y when x = 8.”) Solution: We can solve this problem in one of two ways, as shown. We do these methods when we are given any three of the four values for x and y. Formula Method: Proportion Method: It’s really that easy. Can you see why the proportion method can be the preferred method, unless you are asked to find the k constant in the formula? Again, if the problem asks for the equation that models this situation, it would be “y = 10x“. Direct Variation Word Problem: The amount of money raised at a school fundraiser is directly proportional to the number of people who attend. Last year, the amount of money raised for 100 attendees was$2500. How much money will be raised if 1000 people attend this year? Solution: Let’s do this problem using both the Formula Method and the Proportion Method: Direct Variation Word Problem: Brady bought an energy efficient washing machine for her new apartment. If she saves about 10 gallons of water per load, how many gallons of water will she save if she washes 20 loads of laundry? Solution: Let’s do this with the proportion model: See how similar these types of problems are to the Proportions problems we did earlier? Direct Square Variation Word Problem Again, a Direct Square Variation is when y is proportional to the square of x, or $y=k{{x}^{2}}$. Let’s work a word problem with this type of variation: If y varies directly with the square of x, and if y = 4 when x = 3, what is y when x = 2? Solution: Let’s do this with the formula method and the proportion method: Inverse or Indirect Variation Inverse or Indirect Variation is refers to relationships of two variables that go in the opposite direction. Let’s supposed you are comparing how fast you are driving (average speed) to how fast you get to your school. You might have measured the following speeds and times: (Note that means “approximately equal to”). Do you see how when the x variable goes up, the y goes down, and when you multiply the x with the y, we always get the same number? (Note that this is different than a negative slope, since with a negative slope, we can’t multiply the x’s and y’x to get the same number). So the formula for inverse or indirect variation is: → where k is always the same number. (Note that you could also have an Indirect Square Variation or Inverse Square Variation, like we saw above for a Direct Variation. This would be of the form $y=\frac{k}{{{{x}^{2}}}}\text{ or }{{x}^{2}}y=k$.) Here is a sample graph for inverse or indirect variation. This is actually a type of Rational Function (function with a variable in the denominator) that we will talk about in the Rational Expressions and Functions section here. Inverse Variation Word Problem: So we might have a problem like this: The value of y varies inversely with x, and y = 4 when x = 3. Find x when y = 6. The problem may also be worded like this: Let = 3, = 4, and = 6. Let y vary inversely as x. Find . Solution: We can solve this problem in one of two ways, as shown. We do these methods when we are given any three of the four values for x and y. Formula Method: Product Rule Method: Inverse Variation Word Problem: For the Choir fundraiser, the number of tickets Allie can buy is inversely proportional to the price of the tickets. She can afford 15 tickets that cost $5 each. How many tickets can Allie buy if each cost$3? Solution: Let’s use the product method: “Work” Inverse Proportion Word Problem: Here’s a more advanced problem that uses inverse proportions in a “work” word problem; we’ll see more “work problems” here in the Systems of Linear Equations Section and here in the Rational Functions and Equations Section. If 16 women working 7 hours day can paint a mural in 48 days, how many days will it take 14 women working 12 hours a day to paint the same mural? Solution: The three different values are inversely proportional; for example, the more women you have, the less days it takes to paint the mural, and the more hours in a day the women paint, the less days they need to complete the mural: You might be asked to look at functions (equations or points that compare x’s to unique y’s – we’ll discuss later in the Algebraic Functions section) and determine if they are direct, inverse, or neither: Joint Variation and Combined Variation Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let’s do a joint variation problem: Supposed x varies jointly with y and the square root of z. When x = –18 and y = 2, then z = 9. Find y when x = 10 and z = 4. Let’s set this up like we did with direct variation, find the k, and then solve for y: Combined variation involves a combination of direct or joint variation, and indirect variation. Since these equations are a little more complicated, you probably want to plug in all the variables, solve for k, and then solve back to get what’s missing. Here is the type of problem you may get: (a) y varies jointly as x and w and inversely as the square of zFind the equation of variation when y = 100, x = 2, w = 4, and z = 20. (b) Then solve for y when x = 1, w = 5, and z = 4. Let’s solve: Combined Variation Word Problem: The volume of wood in a tree (V) varies directly as the height (h) and inversely as the square of the girth (g). If the volume of a tree is 144 cubic meters when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters? Solution: Combined Variation Word Problem: The average number of phone calls per day between two cities has found to be jointly proportional to the populations of the cities, and inversely proportional to the square of the distance between the two cities. The population of Charlotte is about 1,500,000 and the population of Nashville is about 1,200,000, and the distance between the two cities is about 400 miles. The average number of calls between the cities is about 200,000. (a) Find the k and write the equation of variation. (b) The average number of daily phone calls between Charlotte and Indianapolis (which has a population of about 1,700,000) is about 134,000. Find the distance between the two cities. Solution: This one looks really tough, but it’s really not that bad if you take it one step at a time: Joint Variation Word Problem: The area of a triangle is jointly related to the height and the base. If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area? Solution: We probably know the equation for the area of a triangle to be $A=\frac{1}{2}bh$, (b = base and h = height) so we can think of the area having a joint variation with b and h, with $k=\frac{1}{2}$. So let’s do the math for this problem; we can just keep the variable k in the problem: Combined Variation Word Problem: y varies jointly with ${{x}^{3}}$ and z, and varies inversely with ${{r}^{2}}$. What is the effect on y when x is doubled and r is halved? Solution: Since we want x to double and r to be halved, we can just put in the new “values” and see what happens to y. Make sure to put them in parentheses, and “push the exponents through”: One word of caution: I found a variation problem in an SAT book that stated something like this: “If x varies inversely with y and varies directly with z, and if y and z are both 12 when x = 3, what is the value of y + z when x = 5”. I found that I had to solve it setting up two variation equations with two different k‘s (otherwise you can’t really get an answer). So watch the wording of the problems. 🙁 Here is how I did this problem: We’re doing really difficult problems now – but see how, if you know the rules, they really aren’t bad at all? Learn these rules, and practice, practice, practice! For more practice, try the problem below. You can type in more problems (by hitting the “x” to clear this problem), or click on the “?” to drill down for example problems. If you click on the “View Steps” box on the widget’s answer screen, you will go to the Mathway site, where you can register for a free seven-day trial of the full version (steps included) of the software. You can even get math worksheets. You can also go to the Mathway site here, where you can register for the trial, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy! < On to Introduction to the Graphing Display Calculator (GDC). I’m proud of you for getting this far! You are ready! 64 thoughts on “Direct, Inverse, Joint and Combined Variation” 1. Hi. I just want to add that formulas such as for direct variation are often valid in the real world for a limited domain only. For example as you stretch a rubber band, the distance stretched is directly proportional to the force of stretching it – until it breaks. Recipes (not that I cook) may say something like “for two portions, allow twice as much cooking time,” although that direct proportion may not continue to hold for any number of portions. And you wouldn’t double the temperature for two portions. 2. I do appreciate your explanation to studentsparents(like me),etc..and I can say that I am able to manage and understand,in order to help my kids in colleges. Thanks a lot 3. I love the way the information is formatted and easy to understand. Keep up the good work Girls need Math! 6. it is very interested it has help me with my exame 7. So… what if y varies inversely with both x and z? Would the equation to figure that out be y=k/xz ? • Great question! Yes, that is how I’d set it up. Do have the actual problem though? It might be set up with two different “k”s – I’ve seen this too. Thanks, Lisa • I believe that would actually be joint variation, since no indirect (inverse) variation is involved. Lisa 8. The information is very helpful. May I request from any one from here,whether the COMBINED/JOIN VARIATION is included in IGCSE /O-Level Examinations? Thank you. Thankaraj. • I’m sorry, I don’t know anything about the IGCSE exam. Sorry I can’t help you. Lisa 9. Thanks A Lot For This 🙂 It Is A Big Help In My Project ^^ 10. thanks….. this is big help to my project…. it really helps me =D Direct variation, y=kx And indirect relations… Do indirect relations apply to all odd negative functions? Everything that starts high and ends low? I think sometimes people confuse relations with correlations. I work with both science and math to ensure correct vocabulary… Any help with these terms is appreciated! • Thanks for writing! Yes, you are correct, relations are confused with correlations. Here is what I found for a definition of “indirect relations”: The relationship between two variables which move in opposite directions; when one of the variables increases the other variable decreases. So this would just imply a negative slope, if the relation is linear. So I think you are correct in saying that indirect relations apply to all odd negative functions. For correlations, here is a good definition from the same source: Degree and type of relationship between any two or more quantities (variables) in which they vary together over a period; for example, variation in the level of expenditure or savings with variation in the level of income. A positive correlation exists where the high values of one variable are associated with the high values of the other variable(s). A ‘negative correlation’ means association of high values of one with the low values of the other(s). Correlation can vary from +1 to -1. Any more thoughts? Lisa 12. Great! Could you please explain the last problem you found on an SAT exam where you had to solve for two values of K? Manuel • Sure – and I’ll add that to the webpage. x = ksub1/y, x = ksub2z. 3 = ksub1/12, so ksub1 = 36. So x = 36/y for the first equation. 3 = ksub212, so ksub2 = 1/4. So x = (1/4)z for the second equation. Putting in 5 for x, we get y = 36/5, and z = 20. So y + z = 27.2. Does that make sense? Lisa 13. Hello Lisa, I have been using your examples to understand direct, inverse and joint variation, however I came across this problem below and I don’t understand the question. I hope you can help me get some sense of the right answer. What is the relation between the variables in the equation x^4/y=7? I thought that the answer would be: x^4 varies directly as y after rearranging the equation to x^4=7y. However, the correct answer is Y varies directly as x^4. I though that since x^4 was equal to k times y (x^4=ky), that X^4 would varied directly as y. My question is, when do we know if x or y varies as the other one. Thank you! • Good question! The way I’ve always thought about it is the variable that starts out (for example, y varies directly to …) should be just 1 variable with a coefficient of 1. So it should be like y, x, etc, and not x^4. So your answer is correct – just not good “math grammar”. Does that make sense? The other thing that’s puzzling about this problem though is that a direct variation typically is linear; in this case it isn’t. I need to do more research on this; I notice on the web that some say it has to be linear, others not. Hope this helps. Lisa • Okay, it does make sense. I will use your advise and see if I get some correct answers. Thank you very much!! 14. Hi, I just want to clarify this problem 🙁 In my math class, we are currently on this topic, but we came across this problem that confused us: 1.) y= xz/z – Joint Variation 2.) y= 2x/z – Combined Variation May I ask what is the difference that made no.1 a Joint Variation since it included an indirect variation within the problem. I also asked my teacher and he said that the only difference was that in no.1, since the (k) was in variable, it had to be considered a Joint Variation, while in no.2, since the k was (2) we considered that it should be Combined Variation, and as I looked at your examples and other sites for cross reference and I also had to look at my book for confirmation but I could only see that Joint Variation is similar to Direct Variation but dealing with multiple variables, while Combined Variation would have a Direct/Joint Variation paired with an Inverse Variation, I am currently confused by this and would like to clarify this to my teacher and my classmates as well or is only an error from my teacher?, an explanation would be grateful 🙂 Alviiiin • Hello – thanks for writing! I was under the impression that JOINT variation is in the form y = kyz, where there are NO variables in the denominator. Combined variation has variables in the numerator AND denominator, such as y = kx/z. The k’s are always on top (numerator) in both cases. Does that make sense? I’m not sure about your teacher’s explanation. Lisa 15. i didnt understand the last problem when y & z =12 but the equation y=k₂z you plugged the value of x which is equals to 3 • Thank you so much – you are absolutely correct – I had the problem wrong. I fixed it – can you check it? Can you find more problems? haha Thanks again, Lisa 16. what is the meaning of direct square variation? 17. Okay, used all the information that was provided by you but somehow I still don’t think my answer is correct for this particular problem that I have. Above in your example on combined variation I used the formula to find k=0.03 or 0.0262 , now that I have that my answer is 80028/25. I must have a whole number to determine the amount of phone calls made daily to from Dallas (702,000)to Little Rock(38,000) these cities are 500 miles apart. I am not sure where I am making my mistake here… 80028/25= (0.03)(702,000)(38,000)/(500)^2 : c=(k)(P1)(P2)/d^2 18. I still dont get it 19. If y varies inversely with the square of x and x varies directly with z. What the relationship between y and z? • This is a good one! I’d set this up like this: y = k/x^2, x = zk (different k). So then we have y = k/(zk)^2, or y = k/(z^2k^2). You can take the k on top and the other k squared on bottom to make a new constant and put it all on top, so we get y = k/z^2. So I’d say y varies inversely as the square of z. Does that make sense? I’m not 100% sure though 🙂 Lisa 20. How can v prove this x varies directly to b then prove that b varies directly to a • Thanks for writing! I assume that you mean if x varies directly to b, then b varies directly to x. If x varies directly to b, then there exists a constant k where x = kb. Then we can solve for b: b = x/k, or b = x * (1/k). 1/k is still a constant, so b varies directly to x. Does that make sense? Lisa 21. The Area of a triangle is jointly related with the height and the base. If the base is increased by 40% and the base is decreased by 20%, what will be the percentage change in the Area? • Thanks for writing – this is a good problem and I think I’ll add one like it! Here’s how I’d do it: A = khb (actually k = 1/2). If the base is increased by 40% and the base (do you mean height?) is decreased by 20%, we’ll have A = k(1.4b)(.8h) = 1.12khb. So the new Area will be increased by 12%. Does that make sense? You can put real numbers in to see how it works! Lisa 22. Hello There! Can you add More Examples of Direct , Joint , Combined , and Inverse Variation? Now? Thankyou! 🙂 23. The table below shows a direct variation. What is the value of m? x y 12 24 18 m 24 48 A.2 B.24 C.30 D.36 • Thanks for writing! m would be 36, since the y values are twice the x values. So the direct variation would y = 2x. Does that make sense? Lisa 24. The quantity of y is partly constant and partly varies as the square of x . write down the relationship between x and y when x=1and y=11 and when x=2;y=5 . Find the value of y when x=4 • Hello! Here’s how I’d do this: y = kx^2. so 11 = k(1)^2, so k = 11 (value of y when x = 4 would be y = 16k or y = 11(4)^2 = 176. For other relationship, 5 = k(2)^2, so k = 5/4 (value of y when x = 4 is y = 16k or y = (5/4)(4)^2 = 20. Does that make sense? Lisa 25. Does ancient civilization use these Variations (Joint, inverse)? In what kind (like constructions or something)? By the Way this helped me! 26. Sir please I want you to be my online mathematics teacher.	{"extraction_info": {"found_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": 7, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8030651807785034, "perplexity": 752.48650845355}, "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/1471982292944.18/warc/CC-MAIN-20160823195812-00099-ip-10-153-172-175.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/249893/fields-where-at-a-and-at-a	# Fields where $A^t=A$ and $A^t=-A$ Are there fields other $Z_2$ where there are matrices other than the zero matrix which are both symmetric and anti-symmetric at the same time? ( $Z_2$ is {0,1} with modulo 2 addition and multiplication ) - You are asking for fields with characteristic 2. – akkkk Dec 3 '12 at 11:15 @akkkk: Why did you delete your answer? – wj32 Dec 3 '12 at 11:17 @wj32: I don't know, I wanted to add that the only field satisfying that was $F_2$, but then I realized that's not true so I figured I'm not an expert after all ;) – akkkk Dec 3 '12 at 11:18 Any field with characteristic 2 has this property. As you may recall, the characteristic of a ring $R$ is the smallest positive number $n$ such that $\sum_1^n 1_R=0$. Concretely, let $A$ be a matrix over a field of characteristic 2, then $A^T$ also is, but $A^T+A^T=2A^T=0$ so $A^T$ is an additive inverse for $A^T$, which we also denote as $A^T=-A^T$. Examples of other fields of characteristic 2 are: rational functions over $F_2$, the algebraic closure of $F_2$, and Laurent series over $F_2$. - What does that mean? – Robert S. Barnes Dec 3 '12 at 11:19 @RobertS.Barnes: It means that $1 + 1 = 0$, where $1$ is the multiplicative identity in the field. – wj32 Dec 3 '12 at 11:19 Or to put it another way, a field has characteristic 2 if and only if it has ${\bf Z}_2$ as a subfield. – Gerry Myerson Dec 3 '12 at 11:44 The notation $F_2$ indicates a field of characteristic 2? – Robert S. Barnes Dec 3 '12 at 12:42 I'm sorry, no, $F_2$ is the field of order 2, so just what you call $\mathbf{Z}_2$. – akkkk Dec 3 '12 at 13:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9289600253105164, "perplexity": 299.27416171406617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1469257823802.12/warc/CC-MAIN-20160723071023-00091-ip-10-185-27-174.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/how-to-find-max-velocity-in-a-spring-mass-system.919939/	# how to find max velocity in a spring-mass system? Tags: 1. Jul 11, 2017 ### Helly123 1. The problem statement, all variables and given/known data Two masses connected with a spring with contant k. The string streched by l . Find the max velocity of mass m! M2 ___spring___ M1 M2--stretched by x2--____spring____--x1--M1 l = x1+ x2 2. Relevant equations F = k.l Mass1.x1 = mass2.x2 (x= displacement?) a=w^2 x v = wx Ep + Ek1 = Ep + Ek2 Ep = 1/2kx^2 Vmax at equilibrium = $\sqrt{\frac{A^2k}{m}}$ 3. The attempt at a solution kl = m.a a = k.l/m While l = x1 + x2 After i get a, i can find w, and can find v My question is, why can't i use Vmax at equilibrium = $\sqrt{\frac{A^2k}{m}}$ ? The A = x1 m1x1 = m2x2 X1 = m2.(l-x1)/m1 X1=m2.l /(m1+m2) Substitute x1 to A. But i get different answer. 2. Jul 11, 2017 ### TSny Is k the effective spring constant for m1's motion? Hint: Write Hooke's law in terms of x1 rather than L. Last edited: Jul 11, 2017 3. Jul 12, 2017 ### Helly123 Ah, i get it.. The Hooke's law = k.x1 F=m.a m1.a = kx1 K= m1.a/x1 Then substitute k to equation. So, when we use total constant the displacement is the total L. While we use only partly displacement the k constant different one? 4. Jul 12, 2017 ### TSny Yes. The force felt by each mass is F = kL where L is the total amount of stretch of the spring from equilibrium. Using some of your equations in your first post, you should be able to express L in terms of just x1. Then you can identify the effective spring constant k1 for mass m1. 5. Jul 12, 2017 ### J Hann If the system can move freely then isn't the total momentum always zero? 6. Jul 12, 2017 ### TSny Yes. Draft saved Draft deleted Similar Discussions: how to find max velocity in a spring-mass system?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9207543730735779, "perplexity": 3424.2952680278554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102663.36/warc/CC-MAIN-20170816212248-20170816232248-00519.warc.gz"}
http://math.stackexchange.com/questions/57233/stalk-of-coherent-sheaf-vanishing	# Stalk of coherent sheaf vanishing is the following true: if I have a coherent sheaf $F$ on a noetherian scheme $X$ with a point $x$ and the stalk $F_x$ is zero, then there is a neighborhood $U$ of $x$, such that the restriction of $F$ to $U$ is zero? Thank you - Yes. For the locality of the problem, you can assume that $X$ is affine: $X = \mathrm{Spec} A$ and $F = \tilde{M}$, where $A$ is a noetherian ring and $M$ is a finite $A$-module. Let $P$ a prime such that $M_P = 0$. Let $\{ x_1, \dots, x_n \}$ a set of generators of $M$ as an $A$-module. Then exist $s_i \in A \setminus P$ such that $s_i x_i = 0$. Pick $s = s_1 \cdots s_n$, then $F \vert_{D(s)} = 0$.	{"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": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981743693351746, "perplexity": 39.27967531760229}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1466783392527.68/warc/CC-MAIN-20160624154952-00071-ip-10-164-35-72.ec2.internal.warc.gz"}
https://brilliant.org/practice/why-vector-spaces/	### Linear Algebra with Applications Waves have a lot of nice properties that make them easy to analyze. So when a scientist or engineer needs to analyze a signal that's not exactly a wave, they often approximate it as a wave. It turns out that vector spaces show us the "best" possible way to do this! In this quiz, we'll sketch the basic ideas and fill the details in later. Note on Notation: $(f+g)(t)$ means "evaluate $f +g$ at $t$," not "multiply $f + g$ by $t.$" # Why Vector Spaces? Any signal of interest to a scientist is a real-valued function $f$ of time $t.$ Let's call the set of all such functions $F.$ Waves live in $F,$ too, but they're special since they repeat. Put all the waves $\big($i.e. functions obeying $w(t+1) = w(t) \big)$ in a subset called $W.$ We can add functions and scalar-multiply with the following rules: $(f+g)(t) = f(t) + g(t) \hspace{0.5cm} \text{and} \hspace{0.5cm} (c f)(t) = c f(t).$ Which of the following statements is true? Note: For reference, you can find the definition of a vector space here. # Why Vector Spaces? Both $F$ and $W$ are vector spaces, so we can rephrase our original problem of approximating a signal by a wave this way: What vector in the subspace $W$ is as close to $\|f\rangle \in F$ as possible? We call $W$ a subspace because it is a space that's also a subset. There's a way of solving this problem based solely on vector ideas. That means we can develop a strategy for solving it in, say, the plane (which is easier to visualize) and carry what we learn there over to the space $F!$ # Why Vector Spaces? In the picture below, the point plays the role of the signal, the line plays the role of $W,$ and the Cartesian plane corresponds to $F.$ Before we can take the analogy too seriously, we have to prove that the Cartesian plane is, in fact, a vector space. The plane is made up of all real number pairs $(x_{1},x_{2}),$ so we can define addition and scalar multiplication naturally as \begin{aligned} (x_{1} , x_{2}) + (y_{1} , y_{2}) & = (x_{1} +y_{1}, x_{2} + y_{2}) \\ c (x_{1},x_{2}) & = (c x_{1} , c x_{2} ). \end{aligned} With these definitions, is the Cartesian plane a vector space? # Why Vector Spaces? With $(x_{1} , x_{2} ) + (y_{1},y_{2})= (x_{1}+y_{1}, x_{2} + y_{2})$ and $c(x_{1},x_{2}) = (c x_{1}, c x_{2} ),$ the Cartesian plane is a vector space. As a set, the line in the picture below consists of all pairs $(x_{1}, m x_{1} )$ where $m$ is the slope. It inherits addition and scalar multiplication from the plane, but to be a vector subspace, it has to be closed under these operations. This means • the sum of any two points on the line is another point on the line; • scaling a point on the line produces another point on the line. Select the correct statement from the list of options. # Why Vector Spaces? So far, our analogy works quite well: both the Cartesian plane $($playing the role of $F)$ and the line $($playing the role of $W)$ are vector spaces. Now, we need to decide how to tell if the point $(a,b)$ (representing the signal) is "close" to a point $(x_{1} , m x_{1} )$ on the line. The natural way to do this is to use the distance between $(a,b)$ and $(x_{1} , m x_{1} ),$ which is the length of the line segment connecting them. The interactive below lets you pick a point on the line with slope $m =0.5.$ It then computes the distance between it and $(a,b) = (2,3).$ (We choose these numbers just to be concrete!) Is there a smallest possible distance? # Why Vector Spaces? With a bit more work, we can actually show that the line segment minimizing the distance is perpendicular to the original line. So here's the strategy we should import to the signal problem: • Define distances between vectors in $F$ and $W. \\\\ \text{} \\\\$ • Minimize the distance from $W$ to the signal $\|f \rangle \in F.$ It'll take some work, but in later chapters we'll show there's a purely vector-based way to do these things! # Why Vector Spaces? We can't fill in the details here, but we can certainly peek ahead and see the result of applying this strategy to a slightly simpler problem. Suppose instead that $F$'s functions are defined on the smaller interval $[0,1],$ and $W$ is built up from the waves $\big \| \sin\left( 2 \pi t \right) \big\rangle , \big \| \sin\left( 4 \pi t \right) \big\rangle , \big \| \sin\left(6 \pi t \right) \big\rangle , \ldots, \big \| \sin\left( 2 \pi n t \right) \big\rangle,$where $n$ is a positive integer. The interactive below shows the result of approximating a step function by a combination of these sine waves: The more waves we use, the better, but even with just a few waves the signal approximation is pretty good. The scheme we outlined in this quiz applies to many real-world problems from physics and statistics, as we'll see. This is just one example of how useful the vector space concept can be. ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 54, "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": 0.9148823618888855, "perplexity": 349.3687508800896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1590347422803.50/warc/CC-MAIN-20200602033630-20200602063630-00567.warc.gz"}
http://www.educator.com/studyguide/math/the-limit-of-a-function/	# The Limit of a Function ## Formal Definition Let f(x) be a function defined on an interval that contains x=a. This function may, but does not have to be defined for input x=a. Then we say that: $\lim \limits_{x \to a} = L$ if for every arbitrarily small number there is some number (exists) such that $\|f(x) - L < g\|$ whenever $0 < \|x-a\| < \delta$ So what does this long and complicated definition tell us? Let’s find out. Take a look at the following graph: This graph represents illustration of the definition above. 1. We first pick an band around the number L on the yaxis. 2. Then we determine a δ band around the number a on the xaxis so that for all x-values (excluding x=a ) inside the δ band, the corresponding yvalues lie inside the δ band. In other words, we first pick a prescribed closeness (δ) to L . Then we get close enough (δ) to a so that all the corresponding yvalues fall inside the band. If a δ > 0 can be found for each value of δ > 0, then we have proven that L is the correct limit. If there is a single ε > 0 for which this process fails, then the limit L has been incorrectly computed, or the limit does not exist. Still not clear enough? Don’t worry, it gets better. ## Intuitive definition Let’s take a look at this function: $f(x) = \frac{(x^2-1)}{x-1}$ What is the value of this function if x=1 ? $f(1) = \frac{(1^2-1)}{1-1}$ $f(1) = \frac{0}{0}$ So here we got the indeterminate. In other words, we can’t evaluate f(1) because 0 divided by 0 is not a defined value. What we can do is take a look at what happens if we approach input x=1 closely. For example, for x = 0.5 we have: $f(0.5) = \frac{(0.5^2-1)}{0.5-1}$ $f(0.5) = 1.5$ If we approach x=1 closer and closer and evaluate we get these results: x 0.9 0.99 0.999 f(x) 1.9 1.99 1.999 Let’s study the results from the table. As x gets closer to value 1 (x approaches 1), the evaluation of a function gets closer to value 2 (f(x) approaches 2). What this tells us is that, although we couldn’t evaluate f(x) for x=1, we can assume it is going to be 2 but this is not mathematically correct answer. So, what do we do? Here’s where the limit of a function comes in handy. Using limit, we can say that the limit of f(x), as x approaches 1, is 2. This gives us a more intuitive definition of a limit. Definition:If value of a function f(x) approaches L when input x approaches a then we say that L is the limit of a function f(x) at point x=a. If a function has limit at point x=a then we say that function converges at that point. Otherwise (if this limit does not exist) we say that function diverges at point x=a. ## Testing sides, left-hand and right-hand limits Functions can be defined in many ways. Some functions have weird graphs where value y jumps from one point to another. Take a look at these two graphs for example: In the left graph, when input x approaches x=1 from the left side then the value of function approaches 3. We write this as $\lim \limits_{x \to 1-} f(x) = 3$ Notice that in the index we have that small ‘-‘ (minus) sign. This is how we tell that it’s the left-hand limit (approaching from the left side). For the right-hand limit it will be the plus (+) in the index. So let’s see what happens in this same graph when x approaches x=1 from the right side. We see that value of a function approaches 1, which means that $\lim \limits_{x \to 1+} f(x) = 1$ So the right-hand limit is 1. We see that left-hand limit and right-hand limit are not the same. When this is the case, the limit of function is not defined, or: $\lim \limits_{x \to 1} f(x) = \text{undefined}$ Now let’s take a look at the other graph (the one on the right). In this graph, as input x approaches x=1 from the left side the value of a function f(x) approaches 1, so the left-hand limit is 1. When x approaches x=1 from the right side the function f(x) again approaches 1, so the right-hand limit is also 1. When left-hand limit and right-hand limit are the same then we say that ‘normal’ limit (or just limit) of the function is that value. So in our case limit of a function is 1, or: Since: $\lim \limits_{x \to 1-} f(x) = 1$ And: $\lim \limits_{x \to 1+} f(x) = 1$ Then: $\lim \limits_{x \to 1} f(x) = 1$ What you need to remember is: • If left-hand limit and right-hand limit are different at some point then the limit of a function does not exist at that point • If both left-hand limit and right-hand limit are the same then the limit of a function is that value ## To infinity and beyond… When dealing with limits we often need to deal with infinite valules. There are two cases. First, what happens when input x approaches infinity. The second is what happens when f(x) approaches infinity. Take a look at this function: $f(x)=\frac{1}{x-1}$ Let’s approach x=1 from both sides: $\lim \limits_{x \to 1-} \frac{1}{x-1} = \frac{1}{0-} = - \infty$ $\lim \limits_{x \to 1+} \frac{1}{x-1} = \frac{1}{0+} = + \infty$ We see that when we approach x=1 from left side the value of f(x) gets infinitely small (negative infinity), and if we approach from the right side the value of f(x) gets infinitely large (positive infinity). This helps us graph the function: 1. First draw a vertical line through x=1 (this is called vertical asymptote). 2. Then, sketch a function such that it approaches that line but never touches it. When approaching from the left go down (-∞), and when approaching from the right go up (+∞) Now we need to examine what happens when x approaches infinity (x → ±∞). To do this we need to know this simple rule: $\lim \limits_{x \to \infty} \frac{k}{x} = 0$ This rule is true for any real value k. Let’s get back to the example and using this rule investigate what happens when x approaches infinity values. $\lim \limits_{x \to \infty+} \frac{1}{x-1} =\frac{1}{\infty+} = 0$ $\lim \limits_{x \to \infty-} \frac{1}{x-1} =\frac{1}{\infty-} = 0$ This also helps us sketch the graph: 1. First draw a horizontal line at y=0 (which is the same as x-axis and this is called horizontal asymptote). 2. Then, sketch our function such that when x → +∞ (goes as far right as possible) the function approaches x-axis. Do the same thing on the left side. Here is the graph: Got it? Great! So next time you see a graph of a function try and think about what happens when x approaches infinity values. How do we use limits to determine that behaviour? What if f(x) approaches infinity values? This type of thinking helps you get a better feel for limits and better understanding of them. ## Methods to evaluate limits There are four basic methods to evaluate limits. #1. Substitute x with the value it approaches. This is the simplest method but it is rarely applicable. Here’s an example: $\lim \limits_{x \to 6} \frac{1}{x-3} = \frac{1}{6-3} = \frac{1}{3}$ Simple right? But the problem is that using this method you often get indeterminate such as 0/0. #2. Factoring. Consider this example $\lim \limits_{x \to 1} \frac{x^2 -2x + 1}{x-1}$ If we just put 1 instead of x we get 0/0. So we try factoring the numerator to get: $\lim \limits_{x \to 1} \frac{(x-1)(x-1)}{(x-1)} = \lim \limits_{x \to 1} x-1$ Now we just put the value in to get: $\lim \limits_{x \to 1} x -1 = 1 - 1 =0$ So the limit is 0. #3. Multiplying by conjugate. The conjugate of expression is the same expression except the sign in the middle is changed, for example conjugate of a+b is a-b. This method often helps when we have fractions with radicals: $\lim \limits_{x \to 4} \frac{2-\sqrt{x}}{4-x}$ Multiply both sides of the fraction by conjugate of numerator: $\lim \limits_{x \to 4} \frac{2-\sqrt{x}}{4-x} \times \frac{2+\sqrt{x}}{2+\sqrt{x}}$ Now use difference of squares formula which is a² – b² = (a – b) (a + b) to simplify the numerator: $\lim \limits_{x \to 4} \frac{4-x}{(4-x)(2+\sqrt{x})} = \lim \limits_{x \to 4}\frac{1}{2+\sqrt{x}}$ Put in the value x=4: $\lim \limits_{x \to 4} \frac{4-x}{(4-x)(2+\sqrt{x})} = \frac{1}{2+\sqrt{x}} = \frac{1}{4}$ #4 Degree of a rational function. Rational function is of form $f(x) = \frac{P(x)}{Q(x)}$ By finding out the degree of a function we can easily determine if limit is 0,+∞,-∞ . Example: $\lim \limits_{x \to \infty} \frac{x^3}{x-1}$ Divide both sides of the fraction by largest degree of x: $\lim \limits_{x \to \infty} \frac{\frac{x^3}{x^3}}{\frac{x}{x^3}-\frac{1}{x^3}}$ Now in the numerator we have 1 (when canceled out) and in the denominator, if we let x approach infinity, we get 1/∞ – 1/∞ = 0 – 0 = 0. So what we have in the end is 1/0 and that is +∞, which is our limit. For more examples check out some of the lessons in our AP Calculus AB course. We cover everything from Limits of a Function to Areas Between Curves.	{"extraction_info": {"found_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": 31, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.955579936504364, "perplexity": 429.1774779398578}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1550247515149.92/warc/CC-MAIN-20190222094419-20190222120419-00191.warc.gz"}
https://www.arxiv-vanity.com/papers/1506.02915/	# Mittag-Leffler Analysis II: Application to the fractional heat equation Martin Grothaus Florian Jahnert January 28, 2021 ###### Abstract Mittag-Leffler analysis is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which generalizes the powerful theory of Gaussian analysis and in particular white noise analysis. In this paper we further develop the Mittag-Leffler analysis by characterizing the convergent sequences in the distribution space. Moreover we provide an approximation of Donsker’s delta by square integrable functions. Then we apply the structures and techniques from Mittag-Leffler analysis in order to show that a Green’s function to the time-fractional heat equation can be constructed using generalized grey Brownian motion (ggBm) by extending the fractional Feynman-Kac formula [Sch92]. Moreover we analyse ggBm, show its differentiability in a distributional sense and the existence of corresponding local times. Keywords: Non-Gaussian analysis, generalized functions, generalized grey Brownian motion, time-fractional heat equation Mathematics Subject Classification (2010): Primary: 46F25, 60G22. Secondary: 26A33, 33E12. ## 1 Introduction The Mittag-Leffler analysis is part of the field of research which tries to transfer the concepts and results known from Gaussian analysis to a non-Gaussian setting. First steps in this direction were made in [Ito88] using the Poisson measure. This approach was generalized with the help of biorthogonal systems, called Appell systems, see e.g. [Dal91, ADKS96, KSWY98], which are suitable for a wide class of measures including the Gaussian measure and the Poisson measure [KSS97]. Two properties of the measure are essential: An analyticity condition of its Laplace transform and a non-degeneracy or positivity condition (see also [KK99]). In [KSWY98] a test function space and a distribution space is constructed and the distributions are characterized using an integral transform and spaces of holomorphic functions on locally convex spaces. The corresponding results in Gaussian analysis, see e.g. [KLP96, KLS96, GKS99], paved the way for applying successfully the abstract theory to numerous concrete problems such as intersection local times for Brownian motion [dFHSW97] as well as for fractional Brownian motion [DODS08, OSS11] and stochastic partial differential equations, see e.g. [GKU99, GKS00] or the monograph [HØUZ96]. In Mittag-Leffler analysis, the measures , , are defined via their characteristic function given by Mittag-Leffler functions. It is shown in [GJRdS15] that the Mittag-Leffler measures belong to the class of measures for which Appell systems exist. Furthermore the (weak) integrable functions with values in the distribution space are characterized and a distribution is constructed which is a generalization of Donsker’s delta in Gaussian analysis. We further develop the theory of Mittag-Leffler analysis in this work by characterizing the convergent sequences in the distribution space . This is done in Section 2, where we first repeat the main steps for the construction of a Mittag-Leffler analysis from [GJRdS15] and we introduce the test function space and the distribution space . In this way we give an approximation of Donsker’s delta by a sequence of Bochner integrals in the Hilbert space . In Section 3, we demonstrate that generalized grey Brownian motion (short ggBm) , , can be defined in the setting of Mittag-Leffler analysis using the probability space , where are the tempered distributions, denotes the cylinder algebra and , , are the Mittag-Leffler measures. This stochastic process was analysed for example in [Sch90, MM09] in a slightly different setting. In Section 4 we establish the connection of ggBm to the fractional heat equation. This equation has already been studied for example in [SW89, Mai95, Koc90, EK04]. In [OB09] the solution to the time-fractional heat equation is interpreted as the density of the composition of certain stochastic processes. We obtain a time-fractional heat kernel by extending the fractional Feynman-Kac formula [Sch92] to localized (Dirac delta) initial values and justify in this way that ggBm plays the same role for the time-fractional heat equation as Brownian motion does for the usual heat equation. In Section 5 we apply the characterizations from Mittag-Leffler analysis to show differentiability of in and we prove the existence of ggBm local times using Donsker’s delta, see also [dSE15]. • Giving a characterization of convergent sequences in the distribution space , see Theorem 2.3. • Providing an approximation of Donsker’s delta by a sequence of Bochner integrals, see Theorem 2.3. • Constructing the Green’s function to time-fractional heat equations by establishing a time-fractional Feynman-Kac formula, see Theorem 4 and Remark 4. • Showing differentiability of generalized grey Brownian motion in a distributional sense, see Theorem 5. • Constructing local times of grey Brownian motions, see Theorem 5.1. ## 2 Mittag-Leffler Analysis ### 2.1 Prerequisites In this section, we repeat the construction of Mittag-Leffler measures as probability measures on a conuclear space from [Sch92]. First we need to collect some facts about nuclear triples used in this paper. For details see e.g. [Sch71, RS72]. Let be a real separable Hilbert space with inner product and corresponding norm . Let be a nuclear space which is continuously and densely embedded in and let be its dual space. The canonical dual pairing between and is denoted by , and by identifying with its dual space via the Riesz isomorphism we get the inclusions . In particular for , . We assume that can be represented by a countable family of Hilbert spaces as follows: For each let be a real separable Hilbert space with norm such that continuously and the inclusion is a Hilbert Schmidt operator. It is no loss of generality to assume on and , . The space is assumed to be the projective limit of the spaces , that is and the topology on is the coarsest locally convex topology such that all inclusions are continuous. This also gives a representation of in terms of an inductive limit: Let be the dual space of with respect to and let the dual pairing between and be denoted by as well. Then is a Hilbert space and we denote its norm by . It follows by general duality theory that , and we may equip with the inductive topology, that is the finest locally convex topology such that all inclusions are continuous. We end up with the following chain of dense and continuous inclusions: \cN⊂\cHp+1⊂\cHp⊂\cH⊂\cH−p⊂\cH−(p+1)⊂\cN′. We also use tensor products and complexifications of these spaces. In the following we always identify for with . The notation is kept for the norm and denotes the bilinear dual pairing. {remark} is a perfect space, i.e. every bounded and closed set in is compact. As a consequence strong and weak convergence coincide in both and , see page 73 in [GV64] and Section I.6.3 and I.6.4 in [GS68]. {example} Consider the white noise setting, where is the space of Schwartz test functions, and are the tempered distributions. is dense in and can be represented as the projective limit of certain Hilbert spaces , , with norms denoted by , see e.g. [Kuo96]. Thus the white noise setting is an example for the nuclear triple described above. ### 2.2 The Mittag-Leffler measure As Mittag-Leffler measures , , we denote the probability measures on whose characteristic functions are given via Mittag-Leffler functions. The Mittag-Leffler function was introduced by Gösta Mittag-Leffler in [ML05] and we also consider a generalization first appeared in [Wim05]. {definition} For the Mittag-Leffler function is an entire function defined by its power series \rmEβ(z):=∞∑n=0znΓ(βn+1),z∈\C. Here denotes the well-known Gamma function which is an extension of the factorial to complex numbers such that for . Furthermore we define for the following entire function of Mittag-Leffler type, see also [EMOT55], \rmEβ,γ(z):=∞∑n=0znΓ(βn+γ),z∈\C. The Mittag-Leffler function is an entire function and thus absolutely convergent on compact sets. Hence we may interchange sum and derivative and calculate the derivative of : {lemma} For the derivative of the Mittag-Leffler function , , it holds \rmd\rmdz\rmEβ(z)=\rmEβ,β(z)β,z∈\C. The mapping is completely monotonic for , see [Pol48, Fel71]. This is sufficient to show that \cN∋ξ↦\rmEβ\lb−\halb\labξ,ξ\rab\rb∈\R is a characteristic function on , see [Sch92]. Using the theorem of Bochner and Minlos, see e.g. [BK95], the following probability measures on , equipped with its cylindrical -algebra, can be defined: {definition} For the Mittag-Leffler measure is defined to be the unique probability measure on such that for all ∫\cN′exp(i\labω,ξ\rab)\rmdμβ(ω)=\rmEβ\lb−12\labξ,ξ\rab\rb. The corresponding spaces of complex-valued functions are denoted by for with corresponding norms . By we define the expectation of . In [Sch92] all moments of are calculated: {lemma} Let and . Then ∫\cN′\labω,ξ\rab2n+1\rmdμβ(ω)=0and∫\cN′\labω,ξ\rab2n\rmdμβ(ω)=(2n)!Γ(βn+1)2n\labξ,ξ\rabn. In particular . {remark} Using Lemma 2.2 it is possible to define for as the -limit of , where is a sequence in converging to in . {lemma} For it holds that ∫\cN′\labω,ξ\rab\labω,η\rab\rmdμβ=1Γ(β+1)\labξ,η\rab. ###### Proof. By definition we have for each ∫\cN′exp\lbi\labω,a1ξ+a2η\rab\rb\rmdμβ=∞∑n=0(−1)n2nΓ(βn+1)\laba1ξ+a2η,a1ξ+a2η\rabn. Applying the operator to both sides and evaluate at we get the result. Interchanging of sum and derivative is possible since the Mittag-Leffler function is entire. Further for all it holds \abs∂∂a1i\labω,η\rabexp\lbi\labω,a1ξ+a2η\rab\rb ≤\abs\labω,η\rab\e\absa2\labω,η\rab\abs\labω,ξ\rab\e\absa1\abs\labω,ξ\rab ≤\e2\abs\labω,ξ\rab\e2\abs\labω,η\rab:=g(ω). by Lemma 4.1 in [GJRdS15]. Hence interchanges with the integral. A similar argument yields that also the integral and interchange. ∎ {remark} It is shown in [GJRdS15] that does not admit a chaos expansion if . This means that there is no system of polynomials , , on fulfilling the following properties simultaneously: 1. [label=()] 2. for some monic polynomial on or of degree . 3. in for . 4. in if in . This is in contrast to the case where the Mittag-Leffler measure is a Gaussian measure. It is a well-known fact in Gaussian analysis that such a system of orthogonal polynomials, the Wick-ordered polynomials, always exists, see e.g. [Kuo96, HKPS93]. ### 2.3 Distributions and Donsker’s delta Instead of using a system of orthogonal polynomials, it was proposed in [GJRdS15] to use Appell systems, compare [KSWY98]. These are biorthogonal systems allowing to construct a test function and a distribution space. As shown in [GJRdS15] the measures , , satisfy the following: 1. The measure has an analytic Laplace transform in a neighborhood of zero, i.e. the mapping \cN\C∋θ↦lμβ(φ):=∫\cN′exp(\labω,θ\rab)\rmdμβ(ω)=\rmEβ\lb\halb\labθ,θ\rab\rb∈\C is holomorphic in a neighborhood of zero. 2. For any nonempty open subset it holds that . We introduce the space of smooth polynomials on , denoted by and consisting of finite linear combinations of functions of the form , where and . Every smooth polynomial has a representation φ(ω)=N∑n=0\labω\otn,φ(n)\rab, where and is a finite sum of elements of the form , . We equip with the natural topology such that the mapping φ=∞∑n=0\lab⋅\otn,φ(n)\rab↔→φ=\lcbφ(n):n∈\N\rcb becomes a topological isomorphism from to the topological direct sum of tensor powers , i.e. \cP(\cN′)≃∞⨁n=0\cN\wotn\C (note that only for finitely many ). Then we introduce the space as the dual space of with respect to , i.e. \cP(\cN′)⊂L2(μβ)⊂\cP′μβ(\cN′) and the dual pairing between and is a bilinear extension of the scalar product on by \ddpfφμβ=(f,¯¯¯¯φ)L2(μβ),φ∈\cP(\cN′),f∈L2(μβ). Note that (A1) ensures that is dense, see [Sko]. Then it is possible to construct an Appell System generated by the measure . We will only give the main definitions, for further details and proofs we refer to [GJRdS15, KSWY98]. For each and we construct such that the Appell polynomials \lx@paragraphsignμβ=\lcb\labPμβn(⋅),φ(n)\rab∣∣φ(n)∈\cN^⊗n\C,n∈\N\rcb, give a representation for by φ=N∑n=0\labPμβn(⋅),φ(n)\rab (1) for suitable and . Every distribution is of the form Φ=∞∑n=0Qμβn(Φ(n)), (2) where is from the -system \Qμβ=\lcbQμβn(Φ(n))∣∣Φ(n)∈\lb\cN^⊗n\C\rb′,n∈\N\rcb. Furthermore the dual pairing of a distribution and a test function is given by \ddpQμβn\lbΦ(n)\rb\labPμβm,φ(m)\rabμβ=δm,nn!\labΦ(n),φ(n)\rab,n,m∈\N0, (3) for and . With the help of the Appell system a test function and a distribution space can now be constructed, see [KSWY98], which results in the following chain of spaces (\cN)1μβ⊂(\cHp)1q,μβ⊂L2(μβ)⊂(\cH−p)−1−q,μβ⊂(\cN)−1μβp,q∈\N. Here, denotes the completion of with respect to , given by \normφ2p,q,μβ:=N∑n=0(n!)22nq\absφ(n)2p,p,q∈\N0,φ∈\cP(\cN′). There are such that is topologically embedded in for all , see [KK99]. Moreover, the set of -exponentials is total in . Here the neighborhood of zero is defined by . Since the -exponentials have the series expansion it holds that for each the norm is given by \normeμβ(θ,⋅)2p,q,μβ=∞∑n=02−nq=(1−2−q)−1. (4) By we denote the set of all for which is finite, where \normΦ2−p,−q,μ:=∞∑n=02−qn\absΦ(n)2−p,p,q∈\N0. It holds that is the dual of and the test function space is defined as the projective limit of . This is a nuclear space which is continuously embedded in . Moreover it turns out that the test function space is the same for all measures satisfying (A1) and (A2), thus we will just use the notation . The space of distributions is the inductive limit of and is the dual of with respect to and the dual pairing between a distribution as in (2) with a test function as in (1) is given by \ddpΦφμβ=∞∑n=0n!\labΦ(n),φ(n)\rab. We shall use the same notation for the dual pairing between and . As in [GJRdS15] we introduce the -transform of on a suitable neighborhood of zero by SμβΦ(θ)=\ddpΦeμβ(θ,⋅)μβ=1\rmEβ\lb\halb\labθ,θ\rab\rb\ddpΦ\e\lab⋅,θ\rabμβ,θ∈\cU. Furthermore we define the -transform of at by TμβΦ(θ)=\ddpΦexp(i\lab⋅,θ\rabμβ. For as in (2) we have SμβΦ(θ)=∞∑n=0\labΦ(n),θ⊗n\rab,θ∈\cU. (5) The relation between - and -transform is given by, see [GJRdS15] TμβΦ(θ)=lμβ(iθ)SμβΦ(iθ),θ∈\cU. (6) The space can be characterized via the -transform using spaces of holomorphic functions on . By we denote the space of all holomorphic functions at zero. Let and be holomorphic on a neighborhood of zero, respectively. We identify and if there is a neighborhood and such that for all . is the union of the spaces \lcbF∈\Hol∣∣np,l,∞(F)=sup\absθp≤2−l\absF(θ)<∞\rcb,p,l∈\N and carries the inductive limit topology. The following theorem is proven in [KSWY98]: {theorem} The -transform is a topological isomorphism from to . Moreover, if for all with and if with and such that then and \normΦ−p′,q,μβ≤np,l,∞(F)(1−ρ)−1/2. As a corollary from the characterization theorem [GJRdS15] proves a result which describes the integrable mappings (in a weak sense) with values in : {theorem} Let be a measure space and for all . Let be an appropriate neighborhood of zero and such that: 1. [label=()] 2. is measurable for all . 3. for all . Then there exists such that for all SμβΨ(θ)=∫TSμβΦt(θ)\rmdν(t). We denote by and call it the weak integral of . Another consequence of Theorem 2.3 characterizes the convergent sequences in : {theorem} Let be a sequence in . Then converges strongly in if and only if there exist with the following two properties: 1. [label=()] 2. is a Cauchy sequence for all . 3. is holomorphic on and there is a constant such that \absSμβΦn(θ)≤C for all and for all . ###### Proof. First assume that and hold. From (5) and by Theorem 2.3 it follows that there exist such that \normΦn−p′,−q′,μβ ≤np,q,∞(SμβΦn)(1−ρ)−1/2≤C(1−ρ)−1/2. (7) Because of (i) together with (7) and since the -exponentials are a total set in we conclude that the sequence is a Cauchy sequence for all by approximating with -exponentials. Indeed, for choose from the linear span of the -exponentials such that and choose large enough such that . Then \abs\ddpΦn−Φmφμβ≤\abs\ddpΦn−Φme+\normΦn+Φm−p′,−q′,μβ\normφ−ep′,q′,μβ<ε. Thus the mapping , , is well-defined, linear and continuous since \absΦ(φ)≤liminfn→∞\normΦn−p′,−q′,μβ\normφp′,q′,μβ≤C(1−ρ)−1/2\normφp′,q′,μβ. Hence . This shows that converges weakly to in since is reflexive. Finally we use that in the dual space of a nuclear space strong and weak convergence coincide, see Remark 2.1. Hence converges strongly to in . Conversely let converge strongly to some . Strong convergence implies that there exist such that in as . Thus is obviously fulfilled for all . The strong convergence also implies that there exists such that . Thus we have for all and for all that \abs(SμβΦn)(θ)≤\normΦn−p,−q,μβ\normeμβ(θ;⋅)p,q,μβ≤K(1−2−q)−1/2<∞, see (4). This shows . ∎ {remark} Because of (6) is holomorphic if and only is holomorphic. Thus the characterization theorems 2.3, 2.3 and 2.3 also hold if the -transform is replaced by the -transform. The characterization theorems have already successfully been applied in Mittag-Leffler analysis. A generalization of Donsker’s delta from Gaussian analysis could be constructed in via Theorem 2.3 and its -transform is calculated, see [GJRdS15]. In fact, for and Donsker’s delta is defined as a weak integral in the sense of Theorem 2.3 by δa(\lab⋅,η\rab)=12π∫\Rexp\lbix(\lab⋅,η\rab−a)\rb\rmdx∈\lb\cN\rb−1μβ. (8) We prove in the following theorem that the weak integral in (8) can be approximate by a sequence of Bochner integrals from . Thus Donsker’s delta can be represented as a limit of square integrable functions. {theorem} For and it holds that δa(\lab⋅,η\rab)=limn→∞12π∫n−n\eix(\lab⋅,η\rab−a)\rmdx in (\cN)−1μβ. ###### Proof. Set , . First note that as a Bochner integral since ∫n−n\normexp\lbix(\lab⋅,η\rab−a)\rbL2(μβ)\rmdx=∫n−n1\rmdx=2n<∞. We have for the -transform of that \lbTμβΦn\rb(θ)=12π∫n−n\e−ixa\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab)\rmdx, see Proposition 5.6 in [GJRdS15]. Now it holds that the integrand \1[−n,n](x)\e−ixa\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab) converges pointwisely for each to \e−ixa\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab) as and it is bounded by . It was shown in Proposition 5.2 in [GJRdS15] that there is a neighborhood of zero and a constant such that ∫\R\abs\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab)\rmdx Applying dominated convergence we see that converges as to 12π∫\R\e−ixa\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab)\rmdx. Moreover for all it holds \abs\lbTμβΦn\rb(θ)≤12π∫\R\abs\rmEβ(−\halbx2\labη,η\rab−\halb\labθ,θ\rab−x\labθ,η\rab)\rmdx Now the assertion follows by applying Theorem 2.3. ∎ ## 3 Grey Noise Analysis ### 3.1 Basic Definitions The main ideas of grey noise analysis go back to Schneider in [Sch90]. He constructed grey Brownian motion on a concrete probability space. Further details were given amongst others by Mura and Mainardi in [MM09] and also in [Kuo96, KS93]. We now want to point out that grey noise analysis is a special case of Mittag-Leffler analysis, where the spaces and are chosen in a suitable way as follows. Consider the space of Schwartz test functions equipped with the following scalar product (ξ,η)α=C(α)∫\R\absx1−α¯¯¯¯¯¯¯¯¯¯¯~ξ(x)~η(x)\rmdx,ξ,η∈\cS(\R). Here, and . The notation stands for the Fourier transform of , which is defined by ~η(x)=(\cFη)(x)=1√2π∫\Rη(t)\eitx\rmdt,x∈\R. By we denote the norm coming from . Define the Hilbert space to be the abstract completion of with respect to . [Sch92] gives an orthonormal system	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9949928522109985, "perplexity": 483.2324357659246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585768.3/warc/CC-MAIN-20211023193319-20211023223319-00465.warc.gz"}
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Please wait. # Chapter 2 Fractions. ## Presentation on theme: "Chapter 2 Fractions."— Presentation transcript: Chapter 2 Fractions Learning Unit Objectives #2 Fractions Learning Unit Objectives Types of Fractions and Conversion Procedures LU2.1 Recognize the three types of fractions Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions Convert fractions to lowest and highest terms Learning Unit Objectives #2 Fractions Learning Unit Objectives LU2.2 Adding and Subtraction of Fractions Add like and unlike fractions Find the least common denominator (LCD) by inspection and prime numbers Subtract like and unlike fractions Add and subtract mixed numbers with the same or different denominators Learning Unit Objectives #2 Fractions Learning Unit Objectives LU2.3 Multiplying and Dividing Fractions Multiply and divide proper fractions and mixed numbers Use the cancellation method in the multiplication and division of fractions Types of Fractions Proper Numerator Proper fractions have a value less than 1; its numerator is smaller than its denominator. 3, 4, 12, 11 Denominator Types of Fractions Improper Numerator Improper Fractions have a value equal to or greater than 1; its numerator is equal or greater than its denominator. 19, 9, 13, 42 Denominator Types of Fractions Mixed numbers are the sum of a whole number greater than zero and a proper fraction Mixed Numbers 5, , , 1 9 4 2 15 Converting Improper Fractions to Whole or Mixed Numbers 15 2 Steps 1. Divide the numerator by the denominator 2. a. If you have no remainder, the quotient is a whole number 2 b. If you have a remainder, the quotient is a mixed number = 1 3 R 1 15 1 = 3 Converting Mixed Numbers to Improper Fractions Mixed Numbers 3 Steps 1. Multiply the denominator of the fraction by the whole number. 2. Add the product from Step 1 to the numerator of the old fraction. 3 Place the total from Step 2 over the denominator of the old fraction to get the improper fraction. 1 8 6 (8 x 6) = 48 (8 x 6) = 48 = 49 49 8 Reducing Fractions to Lowest Terms by Inspection / / Find the lowest whole number that will divide evenly into the numerator and denominator = = Finding the Greatest Common Divisor 24 30 1 24 30 24 6 Step 1. Divide the numerator into the denominator Step 2. Divide the remainder in Step 1 into the divisor of Step 1 4 24 Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0 24 / 30 / = Divisibility Tests = = = = = = = Sum of the digits is divisible by 3 Last two digits can be divided by 4 The number is even and 3 will divide into the sum of the digits Last digit is 0,2,4,6,8 Last digit is 0 or 5 The last digit is 0 = (40) (60) = = = = = 3 + 6 = 9 / 3 = 3 6 + 9 = 15 / 3 = 5 = Raising Fractions to Higher Terms When Denominator is Known 2 Steps 1. Divide the new denominator by the old denominator to get the common number that raises the fraction to higher terms. 2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator. 4 = ? 4 7 28 28 4 x 4 = 16 16 28 Adding Like Fractions Add the numerators and place the total over the denominator If the total of your numerators is the same as your original denominator, convert your answer to a whole number; if the total is larger than your original denominator, convert your answer to a mixed number + = -b + = = 1 Least Common Denominator (LCD) The smallest nonzero whole number into which ALL denominators will divide evenly. + 7 42 21 What is the least common denominator? Adding Unlike Fractions 4 Steps 1. Find the LCD 2. Change each fraction to a like fraction with the LCD. 3. Add the numerators and place the total over the LCD. 4. If necessary, reduce the answer to lowest terms. + + + = 47 Prime Numbers A whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number. Examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 Adding Mixed Numbers 4 4 3 Steps 1. Add the fractions. 2. Add the whole numbers. 3. Combine steps 1 & 2. Be sure you do not have an improper fraction in your final answer. If necessary, reduce the answer to lowest terms. 6 6 + 7 + 7 = 1 Step 1 = Step 2 18 = 18 Step 3 Subtracting Like Fractions Step Subtract the numerators and place the total over the denominator Step 2 - If necessary, reduce the answer to lowest terms / / - = = Subtracting Unlike Fractions Step 1. Find the LCD Step 2. Raise the fraction to its equivalent value. Step 3. Subtract the numerators and place the answer over the LCD. Step 4. If necessary, reduce the answer to lowest terms. - 38 = 19 - Subtracting Mixed Numbers When Borrowing is Not Necessary Step 1. Subtract fractions, making sure to find the LCD. Step 2. Subtract whole numbers. Step 3. Reduce the fractions to lowest terms. 1 8 6 6 6 Subtracting Mixed Numbers When Borrowing is Necessary Step 1. Make sure the fractions have the LCD. Step 2. Borrow from the whole number. Step 3. Subtract whole numbers and fractions. Step 4. Reduce the fractions to lowest terms. 3 4 3 3 2 -1 -1 -1 1 Multiplying Proper Fractions Step 1. Multiply the numerator and the denominators Step 2. Reduce the answer to lowest terms x x = = Multiplying Mixed Numbers Multiply the numerator and denominators Convert the mixed numbers to improper fractions Reduce the answer to lowest terms 1 2 3 X 1 = X = = 1 Dividing Proper Fractions Invert (turn upside down) the divisor (the second fraction) Multiply the fractions Reduce the answer to lowest terms . . = X = Dividing Mixed Numbers Convert all mixed numbers to improper fractions Invert the divisor and multiply Reduce the answer to lowest terms 8 X 2 = X = = 3 Problem 2-31: Solution: 2 3 16 8 x = 3 16 15 1 5⅓ ounces 3 4 2 6 12 1 Cream cheese: 3 4 2 6 12 1 X = = cup Sugar: 1 2 3 6 x = = cup Vanilla: 2 3 4 2 x = 3 4 1 1⅓ teaspoon Butter: Problem 2-38: Solution: 115 + 66 + 106 + 110 = 397 = 398 feet 4 8 2 1 Problem 2-46: Solution: 1 X \$8 = x \$8 = \$12 1 2 3 \$12 x 6 = \$72 Problem 2-56: Solution: 9 8 3 + 5 + 6 + 4 = 18 = 19 1 4 2 - 19 4 2 8 1 = = 19 1 4 2 23 4 2 8 1 Days left Similar presentations Ads by Google	{"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": 0.8601713180541992, "perplexity": 613.6754905946614}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743105.25/warc/CC-MAIN-20181116152954-20181116174954-00215.warc.gz"}
https://www.codeproject.com/Articles/4047091/The-Math-behind-Neural-Networks-Part-1-The-Rosenbl	14,928,120 members Articles / Artificial Intelligence / Neural Networks Article Posted 12 May 2019 33.1K views 60 bookmarked # The Math behind Neural Networks: Part 1 - The Rosenblatt Perceptron Rate me: A try it yourself guide to the basic math behind perceptrons ## Introduction A lot of articles introduce the perceptron showing the basic mathematical formulas that define it, but without offering much of an explanation on what exactly makes it work. And surely it is possible to use the perceptron without really understanding the basic math involved with it, but is it not also fun to see how all this math you learned in school can help you understand the perceptron, and in extension, neural networks? I also got inspired for this article by a series of articles on Support Vector Machines, explaining the basic mathematical concepts involved, and slowly building up to the more complex mathematics involved. So that is my intention with this article and the accompaning code: show you the math envolved in the preceptron. And, if time permits, I will write articles all the way up to convolutional neural networks. Of course, when explaining the math, the question is: where do you start and when do you stop explaining? There is some math involved that is rather basic, like for example what is a vector?, what is a cosine?, etc… I will assume some basic knowledge of mathematics like you have some idea of what a vector is, you know the basics of geometry, etc… My assumptions will be arbitraty, so if you think i’m going too fast in some explanations just leave a comment and I will try to expand on the subject. So, let us get started. ### The Series 1. The Math behind Neural Networks: Part 1 - The Rosenblatt Perceptron 2. The Math behind Neural Networks: Part 2 - The ADALINE Perceptron 3. The Math behind Neural Networks: Part 3 - Neural Networks 4. The Math behind Neural Networks: Part 4 - Convolutional Neural Networks ### Setting some bounds A perceptron basically takes some input values, called “features” and represented by the values $x_1, x_2, ... x_n$ in the following formula, multiplies them by some factors called “weights”, represented by $w_1, w_2, ... w_n$, takes the sum of these multiplications and depending on the value of this sum outputs another value $o$: $o = f(w_1x_1 + w_2x_2 + ... + w_ix_i + ... + w_nx_n)$ There are a few types of perceptrons, differing in the way the sum results in the output, thus the function $f()$ in the above formula. In this article we will build on the Rosenblatt Perceptron. It was one of the first perceptrons, if not the first. During this article I will simply be using the name “Perceptron” when referring to the Rosenblatt Perceptron We will investigate the math envolved and discuss its limitations, thereby setting the ground for the future articles. ## The basic math formula for the Rosenblatt Perceptron $f(x) = \begin{cases} 1 & \text{if } w_1x_1 + w_2x_2 + ... + w_ix_i + ... + w_nx_n > b\\ 0 & \text{otherwise} \end{cases}$ So, what the perceptron basically does is take some linear combination of input values or features, compare it to a threshold value $b$, and return 1 if the threshold is exceeded and zero if not. The feature vector is a group of characteristics describing the objects we want to classify. In other words, we classify our objects into two classes: a set of objects with characteristics (and thus a feature vector) resulting in in output of 1, and a set of objects with characteristics resulting in an output of 0. If you search the internet on information about the perceptron you will find alternative definitions which define the formula as follows: $f(x) = \begin{cases} +1 & \text{if } w_1x_1 + w_2x_2 + ... + w_ix_i + ... + w_nx_n > b\\ -1 & \text{otherwise} \end{cases}$ We will see further this does not affect the workings of the perceptron Lets digg a little deeper: ## Take a linear combination of input values Remember the introduction. In it we said the perceptron takes some input value $[x_1, x_2, ..., x_i, ..., x_n]$, also called features, some weights $[w_1, w_2, ..., w_i, ..., w_n]$, multiplies them with each other and takes the sum of these multiplications: $w_1x_1 + w_2x_2 + ... + w_ix_i + ... + w_nx_n$ This is the definition of a Linear Combination: it is the sum of some terms multiplied by constant values. In our case the terms are the features and the constants are the weights. If we substitute the subscript by a variable $i$, then we can write the sum as $\sum_{i=1}^{n} w_ix_i$ This is called the Capital-sigma notation, the $\sum$ represents the summation, the subscript $_{i=1}$ and the superscript $^{n}$ represent the range over which we take the sum and finally $w_ix_i$ represents the “things” we take the sum of. Also, we can see all $x_i$ and all $w_i$ as so-called vectors: \begin{aligned} \mathbf{x}&=[x_1, x_2, ..., x_i, ..., x_n]\\ \mathbf{w}&=[w_1, w_2, ..., w_i, ..., w_n] \end{aligned} In this, $n$ represents the dimension of the vector: it is the number of scalar elements in the vector. For our discussion, it is the number of characteristics used to describe the objects we want to classify. In this case, the summation is the so-called dot-product of the vectors: $\mathbf{w} \cdot \mathbf{x}$ About the notation: we write simple scalars (thus simple numbers) as small letters, and vectors as bold letters. So in the above $x$ and $w$ are vectors and $x_i$ and $w_i$ are scalars: they are simple numbers representing the components of the vector. ## Oooh, hold your horses! You say what? A ‘Vector’ ? Ok, I may have gone a little too fast there by introducing vectors and not explaining them. ### Definition of a Vector To make things more visual (which can help but isn’t always a good thing), I will start with a graphical representation of a 2-dimensional vector: The above point in the coordinate space $\mathbb{R}^2$ can be represented by a vector going from the origin to that point: $\mathbf{a} = (a_1, a_2), \text{ in }\mathbb{R}^2$ We can further extend this to 3-dimensional coordinate space and generalize it to n-dimensional space: $\mathbf{a} = (a_1, a_2, ..., a_n), \text{ in }\mathbb{R}^n$ In text (from Wikipedia): A (Euclidean) Vector is a geometric object that has a magnitude and a direction #### The Magnitude of a Vector The magnitude of a vector, also called its norm, is defined by the root of the sum of the squares of it’s components and is written as $\lvert\lvert{\mathbf{a}}\lvert\lvert$ In 2-dimensions, the definition comes from Pythagoras’ Theorem: $\lvert\lvert{\mathbf{a}}\lvert\lvert = \sqrt{(a_1)^2 + (x_2)^2}$ Extended to n-dimensional space, we talk about the Euclidean norm: $\lvert\lvert{\mathbf{a}}\lvert\lvert = \sqrt{a_1^2 + a_2^2 + ... + a_i^2 + ... + a_n^2} = \sqrt{\sum_{i=1}^{n} a_i^2}$ Try it yourself: Vector Magnitude interactive #### The Direction of a Vector The direction of a 2-dimensional vector is defined by its angle to the positive horizontal axis: $\theta =\tan^{-1}(\frac{a_2}{a_1})$ This works well in 2 dimensions but it doesn't scale to multiple dimensions: for example in 3 dimensions, in what plane do we measure the angle? Which is why the direction cosines where invented: this is a new vector taking the cosine of the original vector to each axis of the space. $(\cos(\alpha_1), \cos(\alpha_2), ..., \cos(\alpha_i), ..., \cos(\alpha_n))$ We know from geometry that the cosine of an angle is defined by: $\cos(\alpha) = \frac{\text{adjacent}}{\text{hypothenuse}}$ So, the definition of the direction cosine becomes $(\frac{a_1}{\lvert\lvert{\mathbf{a}}\lvert\lvert}, \frac{a_2}{\lvert\lvert{\mathbf{a}}\lvert\lvert}, ..., \frac{a_i}{\lvert\lvert{\mathbf{a}}\lvert\lvert}, ..., \frac{a_n}{\lvert\lvert{\mathbf{a}}\lvert\lvert})$ This direction cosine is a vector $\mathbf{v}$ with length 1 in the same direction as the original vector. This can be simply determined from the definition of the magnitude of a vector: \begin{aligned} \lvert\lvert{\mathbf{v}}\lvert\lvert&=\sqrt{(\frac{a_1}{\lvert\lvert{\mathbf{a}}\lvert\lvert})^2 + (\frac{a_2}{\lvert\lvert{\mathbf{a}}\lvert\lvert})^2 + ... + (\frac{a_i}{\lvert\lvert{\mathbf{a}}\lvert\lvert})^2 + ... + (\frac{a_n}{\lvert\lvert{\mathbf{\mathbf{a}}}\lvert\lvert})^2}\\ &=\sqrt{\frac{(a_1)^2+(a_2)^2+...+(a_i)^2+...+(a_n)^2}{\lvert\lvert{\mathbf{a}}\lvert\lvert^2}}\\ &=\frac{\sqrt{(a_1)^2+(a_2)^2+...+(a_i)^2+...+(a_n)^2}}{\lvert\lvert{\mathbf{a}}\lvert\lvert}\\ &=\frac{\lvert\lvert{\mathbf{a}}\lvert\lvert}{\lvert\lvert{\mathbf{a}}\lvert\lvert}\\ &=1\\ \end{aligned} This vector with length 1 is also called the unit vector. Try it yourself: Vector Direction interactive ### Operations with Vectors #### Sum and difference of two Vectors Say we have two vectors: \begin{aligned} \mathbf{a} &= (a_1, a_2, ..., a_n), \text{ in }\mathbb{R}^n\\ \mathbf{b} &= (b_1, b_2, ..., b_n), \text{ in }\mathbb{R}^n \end{aligned} The sum of two vectors is the vector resulting from the addition of the components of the orignal vectors. \begin{aligned} \mathbf{c} &= \mathbf{a} + \mathbf{b}\\ &= (a_1 + b_1, a_2 + b_2, ..., a_n + b_n) \end{aligned} Try it yourself: Sum of vectors interactive The difference of two vectors is the vector resulting from the differences of the components of the original vectors: \begin{aligned} \mathbf{c} &= \mathbf{a} - \mathbf{b}\\ &= (a_1 - b_1, a_2 - b_2, ..., a_n - b_n) \end{aligned} Try it yourself: Difference of vectors interactive #### Scalar multiplication Say we have a vector $\mathbf{a}$ and a scalar $\lambda$ (a number): \begin{aligned} \mathbf{a} &= (a_1, a_2, ..., a_n), \text{ in }\mathbb{R}^n\\ \lambda \end{aligned} A vector multiplied by a scalar is the vector resulting of the multiplication of each component of the original vector by the scalar: \begin{aligned} \mathbf{c} &= \lambda \mathbf{a}\\ &= (\lambda a_1, \lambda a_2, ..., \lambda a_n) \end{aligned} Try it yourself: Scalar Multiplication for vectors interactive #### Dot product The dot-product is the scalar (a real number) resulting of taking the sum of the products of the corresponding components of two vectors: ### Behaviour of the Rosenblat Perceptron Because the formula of the perceptron is basically a hyperplane, we can only classify things into two classes which are lineary seperable. A first class with things above the hyper-plane and a second class with things below the hyper-plane. ### Formalising some things: a few definitions We’ve covered a lot of ground here, but without using a lot of the lingo surrounding perceptrons, neural networks and machine learning in general. There was already enough lingo with the mathematics that I didn’t want to bother you with even more definitions. However, once we start diving deeper we’ll start uncovering some pattern / structure in the way we work. At that point, it will be interesting to have some definitions which allow us to define steps in this pattern. So, here are some definitions: Feed forward single layer neural network What we have now is a feed forward single layer neural network: Neural Network A neural network is a group of nodes which are connected to each other. Thus, the output of certain nodes serves as input for other nodes: we have a network of nodes. The nodes in this network are modelled on the working of neurons in our brain, thus we speak of a neural network. In this article our neural network had one node: the perceptron. Single Layer In a neural network, we can define multiple layers simply by using the output of preceptrons as the input for other perceptrons. If we make a diagram of this we can view the perceptrons as being organised in layers in which the output of a layer serves as the input for the next layer. Feed Forward This stacking of layers on top of each other and the output of previous layers serving as the input for next layers results in feed forward networks. There is no feedback of upper layers to lower layers. There are no loops. For our single perceptron we also have no loops and thus we have a feed forward network. Integration function The calculation we make with the weight vector w and the feature vector x is called the integration function. In the Rosenblatt perceptron the integration function is the dot-product. Bias The offset b with which we compare the result of the integration function is called the bias. Activation function (transfer function) The output we receive from the perceptron based on the calculation of the integration function is determined by the activation function. The activation function for the Rosenblatt perceptron is the Heaviside step function. Supervised learning Supervised learning is a type of learning in which we feed samples into our algorithm and tell it the result we expect. By doing this the neural network learns how to classify the examples. After giving it enough samples we expect to be able to give it new data which it will automatically classify correctly. The opposite of this is Unsupervised learning in which we give some samples but without the expected result. The algorithm is then able to classify these examples correctly based on some common properties of the samples. There are other types of learning like reïnforcement learning which we will not cover here. Online learning The learning algorithm of the Rosenblatt preceptron is an example of an online learning algorithm: with each new sample given the weight vector is updated; The opposite of this is batch learning in which we only update the weight vector after having fed all samples to the learning algorithm. This may be a bit abstract here but we’ll clarify this in later articles. ## What is wrong with the Rosenblatt perceptron? The main problem of the Rosenblatt preceptron is its learning algorithm. Allthough it works, it only works for linear seperable data. If the data we want to classify is not linearily seperable, then we do not really have any idea on when to stop the learning and neither do we know if the found hyperplane somehow minimizes the wrongly classified data. Also, let’s say we have some data which is linearily seperable. There are several lines which can seperate this data: We would like to find the hyperplane which fits the samples best. That is, we would like to find a line similar to the following: There are of course mathematical tools which allow us to find this hyperplane. They basically all define some kind of error function and then try to minimize this error. The error function is typically defined as a function of the desired output and the effective output just like we did above. The minimization is done by calculating the derivative of this error function. And herein is the problem for the Rosenblatt preceptron. Because the output is defined by the Heaviside Step function and this function does not have a derivative, because it is not continuous, we cannot have a matematically rigourous learning method. If the above is gong a little to fast, don’t panic. In the next article about the ADALINE perceptron we’ll dig deeper into error functions and derivation. ## References ### Javascript libraries used in the Try it yourself pages For the SVG illustrations I use the well known D3.js library For databinding Knockout.js is used Mathematical formulas are displayed using MathJax ### Vector Math The inspiration for writing this article and a good introduction to vector math: SVM - Understanding the math - Part 2 Some wikipedia articles on the basics of vectors and vector math: Euclidean vector Magnitude Direction cosine An understandable proof of why the dot-product is also equal to he product of the length of the vectors with the cosine of the angle between the vectors: Proof of dot-product ### Hyperplanes and Linear Seperability Two math stackexchange Q&A’s on the equation of a hyperplane: Hyperplane equation intuition / geometric interpretation Why is the product of a normal vector and a vector on the plane equal to the equation of the plane? ### Convexity Definition of convexity: Convex set Discussing convexity, we also discussed Line segments: Line segment Proving a half-plane is convex: How do I prove that half a plane is convex? A more in depth discussion of convexity: Lecture 1 Convex Sets ### Perceptron Wikipedia on the perceptron: Perceptron Another explanation of the perceptron: The Simple Perceptron A Peceptron is a special kind of linear classifier Following article as an interesting view on what they call the duality of input and weight-space: 3. Weighted Networks – The Perceptron ### Perceptron Learning Following article gives another intuitive explanation on why the learning algorithm works: Perceptron Learning Algorithm: A Graphical Explanation Of Why It Works An animated gif of the perceptron learning rule: Perceptron training without bias ### Convergence of the learning algorithm This YouTube video presents a very understandable proof: Lec-16 Perceptron Convergence Theorem A written version of the same proof can be found in this pdf: CHAPTER 1 Rosenblatt’s Perceptron By the way, there is much more inside that pdf then just the proof. ## History • Version 1.0: initial release (12 May 2019) • Version 1.1: added sourcecode for the try-it-yourself links (22 May 2019) ## Share Software Developer (Senior) Belgium No Biography provided First Prev Next Message Closed 1-Jun-21 0:44 Member 15225740 1-Jun-21 0:44 Message Closed 11-May-21 22:26 Abubaker Sadique 11-May-21 22:26 The Math behind Neural Networks Member 85621609-Jun-20 8:15 Member 8562160 9-Jun-20 8:15 What a resource! astodola22-Feb-20 3:57 astodola 22-Feb-20 3:57 5 stars Twiggy Ramirezz24-Nov-19 19:02 Twiggy Ramirezz 24-Nov-19 19:02 My vote of 5 Member 1459438014-Nov-19 2:15 Member 14594380 14-Nov-19 2:15 Excellent! Chris Maunder30-Aug-19 8:03 Chris Maunder 30-Aug-19 8:03 Excellent Article rob.evans5-Jul-19 19:30 rob.evans 5-Jul-19 19:30 My vote of 5 maj00011-Jun-19 23:57 maj000 11-Jun-19 23:57 My vote of 5 KarstenK11-Jun-19 1:19 KarstenK 11-Jun-19 1:19 Re: My vote of 5 den2k8811-Jun-19 3:34 den2k88 11-Jun-19 3:34 Your Vector explanation is good den2k8811-Jun-19 0:15 den2k88 11-Jun-19 0:15 Re: Your Vector explanation is good Serge Desmedt17-Jun-19 18:56 Serge Desmedt 17-Jun-19 18:56 Last Visit: 31-Dec-99 18:00 Last Update: 17-Jun-21 22:20 Refresh 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": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8871241211891174, "perplexity": 991.1839020436215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1623487635920.39/warc/CC-MAIN-20210618073932-20210618103932-00284.warc.gz"}
http://jultika.oulu.fi/Record/isbn978-951-42-8916-3	### Ionospheric D-region studies by means of active heating experiments and modelling Saved in: Author: Kero, Antti1,2 Organizations: 1University of Oulu, Faculty of Science, Department of Physical Sciences 2University of Oulu, Sodankylä Geophysical Observatory Format: ebook Version: published version Access: open Online Access: PDF Full Text (PDF, 2.1 MB) Language: English Published: 2008 Publish Date: 2008-11-17 Thesis type: Doctoral Dissertation Defence Note: Academic Dissertation to be presented, with the permission of the Faculty of Science of the University of Oulu, for public discussion in Polaria, Sodankylä, on 8th December, 2008, at 12 o’clock noon. Description: # Abstract Powerful radio waves can heat an electron gas via collisions between free electrons and neutral particles. Since the discovery of the Luxembourg effect in 1934, this effect is known to take place in the D-region ionosphere. According to theoretical models, the EISCAT Heating facility is capable of increasing the electron temperature by a factor of 5–10 in the D region, depending mostly on the electron density profile. Various indirect evidence for the existence of the D-region heating effect has been available, including successful modification of ionospheric conductivities and mesospheric chemistry. However, an experimental quantification of the electron temperature at its maximum in the heated D-region ionosphere has been missing. In particular, incoherent scatter (IS) radars should be able to observe directly plasma parameters, such as the electron temperature, although the heated D-region ionosphere is not a trivial target because of low electron density, and hence, small signal-to-noise ratio (SNR). In this thesis, Papers I and III present unique estimates for heated D-region electron temperatures based on IS measurements. It turned out that the theoretical predictions of the electron temperature generally agree with the few existing observations, at least at the altitudes of the maximum heating effect. Quite in contrast, when the D-region heating effect on the cosmic radio noise absorption was verified for the first time by the statistical data analysis presented in Paper II, the absorption enhancements due to heating were found to be an order of magnitude smaller than model results. The reason for this discrepancy remains still as open question, although one possible explanation is provided by the electron-temperature dependent ion chemistry, which was not taken into account in the modelling. The significance of the heating-induced ion chemistry effect in the D-region was investigated in Paper IV. There the heating-induced negative ion formation is proposed as a potential explanation for the observed modulation of Polar Mesosphere Winter Echo (PMWE) power. see all Series: Sodankylä geophysical observatory publications ISSN: 1456-3673 ISSN-L: 1456-3673 ISBN: 978-951-42-8916-3 ISBN Print: 978-951-42-8915-6 Issue: 102 Subjects:	{"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": 0.9117633104324341, "perplexity": 2733.555511354518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1560627999291.1/warc/CC-MAIN-20190620230326-20190621012326-00252.warc.gz"}
https://par.nsf.gov/biblio/10276387-centrality-dependence-production-nuclear-modification-pb-collisions-sqrt-s_-mathrm-nn-tev	Centrality dependence of J/ψ and ψ(2S) production and nuclear modification in p-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ = 8.16 TeV A bstract The inclusive production of the J/ ψ and ψ (2S) charmonium states is studied as a function of centrality in p-Pb collisions at a centre-of-mass energy per nucleon pair $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 8 . 16 TeV at the LHC. The measurement is performed in the dimuon decay channel with the ALICE apparatus in the centre-of-mass rapidity intervals − 4 . 46 < y cms < − 2 . 96 (Pb-going direction) and 2 . 03 < y cms < 3 . 53 (p-going direction), down to zero transverse momentum ( p T ). The J/ ψ and ψ (2S) production cross sections are evaluated as a function of the collision centrality, estimated through the energy deposited in the zero degree calorimeter located in the Pb-going direction. The p T -differential J/ ψ production cross section is measured at backward and forward rapidity for several centrality classes, together with the corresponding average 〈 p T 〉 and $$\left\langle {p}_{\mathrm{T}}^2\right\rangle$$ p T 2 values. The nuclear effects affecting the production of both charmonium states are studied using the nuclear modification factor. In the p-going direction, a suppression of the production of both charmonium states is more » Authors: ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more » Award ID(s): Publication Date: NSF-PAR ID: 10276387 Journal Name: Journal of High Energy Physics Volume: 2021 Issue: 2 ISSN: 1029-8479 1. A bstract The production cross-sections of J/ψ mesons in proton-proton collisions at a centre-of-mass energy of $$\sqrt{s}$$ s = 5 TeV are measured using a data sample corresponding to an integrated luminosity of 9 . 13 ± 0 . 18 pb − 1 , collected by the LHCb experiment. The cross-sections are measured differentially as a function of transverse momentum, p T , and rapidity, y , and separately for J/ψ mesons produced promptly and from beauty hadron decays (nonprompt). With the assumption of unpolarised J/ψ mesons, the production cross-sections integrated over the kinematic range 0 < pmore » 2. A bstract The production of prompt D 0 , D + , and D *+ mesons was measured at midrapidity (\| y \| < 0.5) in Pb–Pb collisions at the centre-of-mass energy per nucleon–nucleon pair $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV with the ALICE detector at the LHC. The D mesons were reconstructed via their hadronic decay channels and their production yields were measured in central (0–10%) and semicentral (30–50%) collisions. The measurement was performed up to a transverse momentum ( p T ) of 36 or 50 GeV/c depending on the D meson species andmore » 3. A bstract A measurement of inclusive, prompt, and non-prompt J/ ψ production in p-Pb collisions at a nucleon-nucleon centre-of-mass energy $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV is presented. The inclusive J/ ψ mesons are reconstructed in the dielectron decay channel at midrapidity down to a transverse momentum p T = 0. The inclusive J/ ψ nuclear modification factor R pPb is calculated by comparing the new results in p-Pb collisions to a recently measured proton-proton reference at the same centre-of-mass energy. Non-prompt J/ ψ mesons, which originate from the decay of beauty hadrons, are separated frommore » 4. A bstract Measurement of Z-boson production in p-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 8 . 16 TeV and Pb-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV is reported. It is performed in the dimuon decay channel, through the detection of muons with pseudorapidity − 4 < η μ < − 2 . 5 and transverse momentum $${p}_{\mathrm{T}}^{\mu }$$ p T μ > 20 GeV/ c in the laboratory frame. The invariant yield and nuclear modification factor are measured for opposite-sign dimuons with invariant mass 60 < m μμ < 120more » 5. A bstract The inclusive J/ ψ elliptic ( v 2 ) and triangular ( v 3 ) flow coefficients measured at forward rapidity (2 . 5 < y < 4) and the v 2 measured at midrapidity (\| y \| < 0 . 9) in Pb-Pb collisions at $$\sqrt{s_{\mathrm{NN}}}$$ s NN = 5 . 02 TeV using the ALICE detector at the LHC are reported. The entire Pb-Pb data sample collected during Run 2 is employed, amounting to an integrated luminosity of 750 μ b − 1 at forward rapidity and 93 μ b − 1 at midrapidity.more »	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9974820017814636, "perplexity": 2827.5959420719996}, "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-2022-33/segments/1659882572220.19/warc/CC-MAIN-20220816030218-20220816060218-00485.warc.gz"}
https://worldwidescience.org/topicpages/s/symmetric+simple+exclusion.html	#### Sample records for symmetric simple exclusion 1. Parallel coupling of symmetric and asymmetric exclusion processes International Nuclear Information System (INIS) Tsekouras, K; Kolomeisky, A B 2008-01-01 A system consisting of two parallel coupled channels where particles in one of them follow the rules of totally asymmetric exclusion processes (TASEP) and in another one move as in symmetric simple exclusion processes (SSEP) is investigated theoretically. Particles interact with each other via hard-core exclusion potential, and in the asymmetric channel they can only hop in one direction, while on the symmetric lattice particles jump in both directions with equal probabilities. Inter-channel transitions are also allowed at every site of both lattices. Stationary state properties of the system are solved exactly in the limit of strong couplings between the channels. It is shown that strong symmetric couplings between totally asymmetric and symmetric channels lead to an effective partially asymmetric simple exclusion process (PASEP) and properties of both channels become almost identical. However, strong asymmetric couplings between symmetric and asymmetric channels yield an effective TASEP with nonzero particle flux in the asymmetric channel and zero flux on the symmetric lattice. For intermediate strength of couplings between the lattices a vertical-cluster mean-field method is developed. This approximate approach treats exactly particle dynamics during the vertical transitions between the channels and it neglects the correlations along the channels. Our calculations show that in all cases there are three stationary phases defined by particle dynamics at entrances, at exits or in the bulk of the system, while phase boundaries depend on the strength and symmetry of couplings between the channels. Extensive Monte Carlo computer simulations strongly support our theoretical predictions. Theoretical calculations and computer simulations predict that inter-channel couplings have a strong effect on stationary properties. It is also argued that our results might be relevant for understanding multi-particle dynamics of motor proteins 2. National simple: Constitutionality Analysis of Exclusions Sectorial Directory of Open Access Journals (Sweden) 2016-06-01 Full Text Available The Constitution defines the favored legal treatment for small businesses without making any explicit exception, including and especially for tax obligations. Nevertheless, all the laws, which have introduced tax benefits guided by this higher provision, have discriminated small companies due to the economic sector of activity. Known as “National Simple” and introduced by the Complementary Law No. 123/06, the current legislation did not extend its benefits to small production units of a number of industries, such as the automotive industry, the passenger transport industry, the energy industry and the industry of manufacture of weapons, beverages and tobacco products. By demonstrating the mistakes of the arguments in favor of such exclusions, the article holds up that none of these provisions meets constitutional standards. Furthermore, based on a critical analysis of the Positive Law, it is shown that the hidden desire behind the exclusions was to keep the economic sectors of high profitability under control of big corporations to the detriment of smaller initiatives. 3. Two-channel totally asymmetric simple exclusion processes International Nuclear Information System (INIS) Pronina, Ekaterina; Kolomeisky, Anatoly B 2004-01-01 Totally asymmetric simple exclusion processes, consisting of two coupled parallel lattice chains with particles interacting with hard-core exclusion and moving along the channels and between them, are considered. In the limit of strong coupling between the channels, the particle currents, density profiles and a phase diagram are calculated exactly by mapping the system into an effective one-channel totally asymmetric exclusion model. For intermediate couplings, a simple approximate theory, that describes the particle dynamics in vertical clusters of two corresponding parallel sites exactly and neglects the correlations between different vertical clusters, is developed. It is found that, similarly to the case of one-channel totally asymmetric simple exclusion processes, there are three stationary state phases, although the phase boundaries and stationary properties strongly depend on inter-channel coupling. Extensive computer Monte Carlo simulations fully support the theoretical predictions 4. Shocks induced by junctions in totally asymmetric simple exclusion processes under periodic boundary condition Energy Technology Data Exchange (ETDEWEB) Sun, Xiaoyan, E-mail: [email protected] [College of Physics and Electronic Engineering, Guangxi Teacher Education University, Nanning 530001 (China); Xie, Yanbo [Department of Modern Physics, University of Science and Technology of China, Hefei 230026 (China); He, Zhiwei [College of Science, China Agricultural University, Beijing 100083 (China); Wang, Binghong [Department of Modern Physics, University of Science and Technology of China, Hefei 230026 (China) 2011-07-11 This Letter investigates a totally asymmetric simple exclusion process (TASEP) with junctions in a one-dimensional transport system. Parallel update rules and periodic boundary condition are adopted. Two cases corresponding to different update rules are studied. The results show that the stationary states of system mainly depend on the selection behavior of particle at the bifurcation point. -- Highlights: → For no preference case, the system exists three stationary phases. → For preference case, the system exists five stationary phases. → The road lengths have not qualitative influence on the fundamental diagram. 5. SIMPLE MODELS OF THREE COUPLED PT -SYMMETRIC WAVE GUIDES ALLOWING FOR THIRD-ORDER EXCEPTIONAL POINTS Directory of Open Access Journals (Sweden) Jan Schnabel 2017-12-01 Full Text Available We study theoretical models of three coupled wave guides with a PT-symmetric distribution of gain and loss. A realistic matrix model is developed in terms of a three-mode expansion. By comparing with a previously postulated matrix model it is shown how parameter ranges with good prospects of finding a third-order exceptional point (EP3 in an experimentally feasible arrangement of semiconductors can be determined. In addition it is demonstrated that continuous distributions of exceptional points, which render the discovery of the EP3 difficult, are not only a feature of extended wave guides but appear also in an idealised model of infinitely thin guides shaped by delta functions. 6. Characteristics of the asymmetric simple exclusion process in the presence of quenched spatial disorder Science.gov (United States) Ebrahim Foulaadvand, M.; Chaaboki, Sanaz; Saalehi, Modjtaba 2007-01-01 We investigate the effect of quenched spatial disordered hopping rates on the characteristics of the asymmetric simple exclusion process with open boundaries both numerically and by extensive simulations. Disorder averages of the bulk density and current are obtained in terms of various input and output rates. We study the binary and uniform distributions of disorder. It is verified that the effect of spatial inhomogeneity is generically to enlarge the size of the maximal-current phase. This is in accordance with the mean-field results obtained by Harris and Stinchcombe [Phys. Rev. E 70, 016108 (2004)]. Furthermore, we obtain the dependence of the current and the bulk density on the characteristics of the disorder distribution function. It is shown that the impact of disorder crucially depends on the particle input and out rates. In some situations, disorder can constructively enhance the current. 7. Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model Directory of Open Access Journals (Sweden) Nikolay M. Bogoliubov 2009-04-01 Full Text Available The basic model of the non-equilibrium low dimensional physics the so-called totally asymmetric exclusion process is related to the 'crystalline limit' (q → ∞ of the SU_q(2 quantum algebra. Using the quantum inverse scattering method we obtain the exact expression for the time-dependent stationary correlation function of the totally asymmetric simple exclusion process on a one dimensional lattice with the periodic boundary conditions. 8. Signal optimization in urban transport: A totally asymmetric simple exclusion process with traffic lights. Science.gov (United States) Arita, Chikashi; Foulaadvand, M Ebrahim; Santen, Ludger 2017-03-01 We consider the exclusion process on a ring with time-dependent defective bonds at which the hopping rate periodically switches between zero and one. This system models main roads in city traffics, intersecting with perpendicular streets. We explore basic properties of the system, in particular dependence of the vehicular flow on the parameters of signalization as well as the system size and the car density. We investigate various types of the spatial distribution of the vehicular density, and show existence of a shock profile. We also measure waiting time behind traffic lights, and examine its relationship with the traffic flow. 9. Asymmetric simple exclusion process with position-dependent hopping rates: Phase diagram from boundary-layer analysis. Science.gov (United States) Mukherji, Sutapa 2018-03-01 In this paper, we study a one-dimensional totally asymmetric simple exclusion process with position-dependent hopping rates. Under open boundary conditions, this system exhibits boundary-induced phase transitions in the steady state. Similarly to totally asymmetric simple exclusion processes with uniform hopping, the phase diagram consists of low-density, high-density, and maximal-current phases. In various phases, the shape of the average particle density profile across the lattice including its boundary-layer parts changes significantly. Using the tools of boundary-layer analysis, we obtain explicit solutions for the density profile in different phases. A detailed analysis of these solutions under different boundary conditions helps us obtain the equations for various phase boundaries. Next, we show how the shape of the entire density profile including the location of the boundary layers can be predicted from the fixed points of the differential equation describing the boundary layers. We discuss this in detail through several examples of density profiles in various phases. The maximal-current phase appears to be an especially interesting phase where the boundary layer flows to a bifurcation point on the fixed-point diagram. 10. Causal symmetric spaces CERN Document Server Olafsson, Gestur; Helgason, Sigurdur 1996-01-01 This book is intended to introduce researchers and graduate students to the concepts of causal symmetric spaces. To date, results of recent studies considered standard by specialists have not been widely published. This book seeks to bring this information to students and researchers in geometry and analysis on causal symmetric spaces.Includes the newest results in harmonic analysis including Spherical functions on ordered symmetric space and the holmorphic discrete series and Hardy spaces on compactly casual symmetric spacesDeals with the infinitesimal situation, coverings of symmetric spaces, classification of causal symmetric pairs and invariant cone fieldsPresents basic geometric properties of semi-simple symmetric spacesIncludes appendices on Lie algebras and Lie groups, Bounded symmetric domains (Cayley transforms), Antiholomorphic Involutions on Bounded Domains and Para-Hermitian Symmetric Spaces 11. Effect of self-deflection on a totally asymmetric simple exclusion process with functions of site assignments Science.gov (United States) Tsuzuki, Satori; Yanagisawa, Daichi; Nishinari, Katsuhiro 2018-04-01 This study proposes a model of a totally asymmetric simple exclusion process on a single-channel lane with functions of site assignments along the pit lane. The system model attempts to insert a new particle to the leftmost site at a certain probability by randomly selecting one of the empty sites in the pit lane, and reserving it for the particle. Thereafter, the particle is directed to stop at the site only once during its travel. Recently, the system was determined to show a self-deflection effect, in which the site usage distribution biases spontaneously toward the leftmost site, and the throughput becomes maximum when the site usage distribution is slightly biased to the rightmost site. Our exact analysis describes this deflection effect and show a good agreement with simulations. 12. Inter-particle gap distribution and spectral rigidity of the totally asymmetric simple exclusion process with open boundaries International Nuclear Information System (INIS) Krbalek, Milan; Hrabak, Pavel 2011-01-01 We consider the one-dimensional totally asymmetric simple exclusion process (TASEP model) with open boundary conditions and present the analytical computations leading to the exact formula for distance clearance distribution, i.e. probability density for a clear distance between subsequent particles of the model. The general relation is rapidly simplified for the middle part of the one-dimensional lattice. Both the analytical formulas and their approximations are compared with the numerical representation of the TASEP model. Such a comparison is presented for particles occurring in the internal part as well as in the boundary part of the lattice. Furthermore, we introduce the pertinent estimation for the so-called spectral rigidity of the model. The results obtained are sequentially discussed within the scope of vehicular traffic theory. 13. Crankshafts: using simple, flat C2h-symmetric molecules to direct the assembly of chiral 2D nanopatterns. Science.gov (United States) Zhou, Hui; Wuest, James D 2013-06-18 Linear D2h-symmetric bisisophthalic acids 1 and 2 and related substances have well-defined flattened structures, high affinities for graphite, and strong abilities to engage in specific intermolecular interactions. Their adsorption produces characteristic nanopatterns that reveal how 2D molecular organization can be controlled by reliable interadsorbate interactions such as hydrogen bonds when properly oriented by molecular geometry. In addition, the behavior of these compounds shows how large-scale organization can be obstructed by programming molecules to associate strongly according to competing motifs that have similar stability and can coexist smoothly without creating significant defects. Analogous new bisisophthalic acids 3a and 4a have similar associative properties, and their unique C2h-symmetric crankshaft geometry gives them the added ability to probe the poorly understood effect of chirality on molecular organization. Their adsorption shows how nanopatterns composed predictably of a single enantiomer can be obtained by depositing molecules that can respect established rules of association only by accepting neighbors of the same configuration. In addition, an analysis of the adsorption of crankshaft compounds 3a and 4a and their derivatives by STM reveals directly on the molecular level how kinetics and thermodynamics compete to control the crystallization of chiral compounds. In such ways, detailed studies of the adsorption of properly designed compounds on surfaces are proving to be a powerful way to discover and test rules that broadly govern molecular organization in both 2D and 3D. 14. A simple method for purification of lipopolysaccharides from E. coli 55:B5 using size exclusion chromatography International Nuclear Information System (INIS) Perdomo, Rolando; Montero, Vivian 2006-01-01 Several methods for the extraction of endotoxin or lipopolysaccharide from Gram negative bacteria have been described. However, the product is often contaminated with nucleic acids or proteins in a proportion depending on the extraction method used. Molecular and immunological studies require further purification of the raw LPS. We present here, a simple method for the purification of raw LPS obtained by the standard hot phenol-water procedure using size exclusion chromatography in Sepharose CL-6B. We demonstrated that the using of DNAse and RNAse treatment of the sample before the chromatographic step is necessary to abrogate the nucleic acid contamination in the LPS fraction. The spectrophotometric properties of the pure LPS were verified, supporting the immediate online detection of the LPS and oligonucleotides fractions spectrophotometrically at 206 nm. The mobile phase used (NaCl 0.2 M) do not absorb at 206 nm while maintains the LPS aggregates and therefore, allows the separation of the LPS fraction from the oligoribonucleotide and desoxioligoribonucleotide fractions. The yield of pure LPS was around 98%. Chemical and biological characterizations were conducted in order to assess the feasibility of the procedure developed. (Author) 15. Axially symmetric U-O-Ln- and U-O-U-containing molecules from the control of uranyl reduction with simple f-block halides International Nuclear Information System (INIS) Arnold, Polly L.; Cowie, Bradley E.; Suvova, Marketa; Zegke, Markus; Love, Jason B.; Magnani, Nicola; Colineau, Eric; Griveau, Jean-Christophe; Caciuffo, Roberto 2017-01-01 The reduction of U"V"I uranyl halides or amides with simple Ln"I"I or U"I"I"I salts forms highly symmetric, linear, oxo-bridged trinuclear U"V/Ln"I"I"I/U"V, Ln"I"I"I/U"I"V/Ln"I"I"I, and U"I"V/U"I"V/U"I"V complexes or linear Ln"I"I"I/U"V polymers depending on the stoichiometry and solvent. The reactions can be tuned to give the products of one- or two-electron uranyl reduction. The reactivity and magnetism of these compounds are discussed in the context of using a series of strongly oxo-coupled homo- and heterometallic poly(f-block) chains to better understand fundamental electronic structure in the f-block. (copyright 2017 Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim) 16. Axially symmetric U-O-Ln- and U-O-U-containing molecules from the control of uranyl reduction with simple f-block halides Energy Technology Data Exchange (ETDEWEB) Arnold, Polly L.; Cowie, Bradley E.; Suvova, Marketa; Zegke, Markus; Love, Jason B. [EaStCHEM School of Chemistry, University of Edinburgh (United Kingdom); Magnani, Nicola; Colineau, Eric; Griveau, Jean-Christophe; Caciuffo, Roberto [European Commission, Directorate for Nuclear Safety and Security, Joint Research Centre, Karlsruhe (Germany) 2017-08-28 The reduction of U{sup VI} uranyl halides or amides with simple Ln{sup II} or U{sup III} salts forms highly symmetric, linear, oxo-bridged trinuclear U{sup V}/Ln{sup III}/U{sup V}, Ln{sup III}/U{sup IV}/Ln{sup III}, and U{sup IV}/U{sup IV}/U{sup IV} complexes or linear Ln{sup III}/U{sup V} polymers depending on the stoichiometry and solvent. The reactions can be tuned to give the products of one- or two-electron uranyl reduction. The reactivity and magnetism of these compounds are discussed in the context of using a series of strongly oxo-coupled homo- and heterometallic poly(f-block) chains to better understand fundamental electronic structure in the f-block. (copyright 2017 Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim) 17. Large deviation principle for one-dimensional random walk in dynamic random environment : attractive spin-flips and simple symmetric exclusion NARCIS (Netherlands) Avena, L.; Hollander, den W.Th.F.; Redig, F.H.J. 2009-01-01 Consider a one-dimensional shift-invariant attractive spin-ip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In [2] we proved a law 18. Large deviation principle for one-dimensional random walk in dynamic random environment: attractive spin-flips and simple symmetric exclusion NARCIS (Netherlands) Avena, L.; Hollander, den W.Th.F.; Redig, F.H.J. 2010-01-01 Consider a one-dimensional shift-invariant attractive spin-flip system in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied sites has a local drift to the right but on vacant sites has a local drift to the left. In previous work we 19. Symmetric textures International Nuclear Information System (INIS) Ramond, P. 1993-01-01 The Wolfenstein parametrization is extended to the quark masses in the deep ultraviolet, and an algorithm to derive symmetric textures which are compatible with existing data is developed. It is found that there are only five such textures 20. Fluorescence Exclusion: A Simple Method to Assess Projected Surface, Volume and Morphology of Red Blood Cells Stored in Blood Bank Directory of Open Access Journals (Sweden) Camille Roussel 2018-05-01 Full Text Available Red blood cells (RBC ability to circulate is closely related to their surface area-to-volume ratio. A decrease in this ratio induces a decrease in RBC deformability that can lead to their retention and elimination in the spleen. We recently showed that a subpopulation of “small RBC” with reduced projected surface area accumulated upon storage in blood bank concentrates, but data on the volume of these altered RBC are lacking. So far, single cell measurement of RBC volume has remained a challenging task achieved by a few sophisticated methods some being subject to potential artifacts. We aimed to develop a reproducible and ergonomic method to assess simultaneously RBC volume and morphology at the single cell level. We adapted the fluorescence exclusion measurement of volume in nucleated cells to the measurement of RBC volume. This method requires no pre-treatment of the cell and can be performed in physiological or experimental buffer. In addition to RBC volume assessment, brightfield images enabling a precise definition of the morphology and the measurement of projected surface area can be generated simultaneously. We first verified that fluorescence exclusion is precise, reproducible and can quantify volume modifications following morphological changes induced by heating or incubation in non-physiological medium. We then used the method to characterize RBC stored for 42 days in SAG-M in blood bank conditions. Simultaneous determination of the volume, projected surface area and morphology allowed to evaluate the surface area-to-volume ratio of individual RBC upon storage. We observed a similar surface area-to-volume ratio in discocytes (D and echinocytes I (EI, which decreased in EII (7% and EIII (24%, sphero-echinocytes (SE; 41% and spherocytes (S; 47%. If RBC dimensions determine indeed the ability of RBC to cross the spleen, these modifications are expected to induce the rapid splenic entrapment of the most morphologically altered RBC 1. A Conceptually Simple Modeling Approach for Jason-1 Sea State Bias Correction Based on 3 Parameters Exclusively Derived from Altimetric Information Directory of Open Access Journals (Sweden) Nelson Pires 2016-07-01 Full Text Available A conceptually simple formulation is proposed for a new empirical sea state bias (SSB model using information retrieved entirely from altimetric data. Nonparametric regression techniques are used, based on penalized smoothing splines adjusted to each predictor and then combined by a Generalized Additive Model. In addition to the significant wave height (SWH and wind speed (U10, a mediator parameter designed by the mean wave period derived from radar altimetry, has proven to improve the model performance in explaining some of the SSB variability, especially in swell ocean regions with medium-high SWH and low U10. A collinear analysis of scaled sea level anomalies (SLA variance differences shows conformity between the proposed model and the established SSB models. The new formulation aims to be a fast, reliable and flexible SSB model, in line with the well-settled SSB corrections, depending exclusively on altimetric information. The suggested method is computationally efficient and capable of generating a stable model with a small training dataset, a useful feature for forthcoming missions. 2. Holographic Spherically Symmetric Metrics Science.gov (United States) Petri, Michael The holographic principle (HP) conjectures, that the maximum number of degrees of freedom of any realistic physical system is proportional to the system's boundary area. The HP has its roots in the study of black holes. It has recently been applied to cosmological solutions. In this article we apply the HP to spherically symmetric static space-times. We find that any regular spherically symmetric object saturating the HP is subject to tight constraints on the (interior) metric, energy-density, temperature and entropy-density. Whenever gravity can be described by a metric theory, gravity is macroscopically scale invariant and the laws of thermodynamics hold locally and globally, the (interior) metric of a regular holographic object is uniquely determined up to a constant factor and the interior matter-state must follow well defined scaling relations. When the metric theory of gravity is general relativity, the interior matter has an overall string equation of state (EOS) and a unique total energy-density. Thus the holographic metric derived in this article can serve as simple interior 4D realization of Mathur's string fuzzball proposal. Some properties of the holographic metric and its possible experimental verification are discussed. The geodesics of the holographic metric describe an isotropically expanding (or contracting) universe with a nearly homogeneous matter-distribution within the local Hubble volume. Due to the overall string EOS the active gravitational mass-density is zero, resulting in a coasting expansion with Ht = 1, which is compatible with the recent GRB-data. KAUST Repository Jiang, Haiyong 2016-04-11 We present an automatic algorithm for symmetrizing facade layouts. Our method symmetrizes a given facade layout while minimally modifying the original layout. Based on the principles of symmetry in urban design, we formulate the problem of facade layout symmetrization as an optimization problem. Our system further enhances the regularity of the final layout by redistributing and aligning boxes in the layout. We demonstrate that the proposed solution can generate symmetric facade layouts efficiently. © 2015 IEEE. KAUST Repository Jiang, Haiyong; Yan, Dong-Ming; Dong, Weiming; Wu, Fuzhang; Nan, Liangliang; Zhang, Xiaopeng 2016-01-01 We present an automatic approach for symmetrizing urban facade layouts. Our method can generate a symmetric layout through minimally modifying the original input layout. Based on the principles of symmetry in urban design, we formulate facade layout symmetrization as an optimization problem. Our method further enhances the regularity of the final layout by redistributing and aligning elements in the layout. We demonstrate that the proposed solution can effectively generate symmetric facade layouts. KAUST Repository Jiang, Haiyong; Dong, Weiming; Yan, Dongming; Zhang, Xiaopeng 2016-01-01 We present an automatic algorithm for symmetrizing facade layouts. Our method symmetrizes a given facade layout while minimally modifying the original layout. Based on the principles of symmetry in urban design, we formulate the problem of facade layout symmetrization as an optimization problem. Our system further enhances the regularity of the final layout by redistributing and aligning boxes in the layout. We demonstrate that the proposed solution can generate symmetric facade layouts efficiently. © 2015 IEEE. KAUST Repository Jiang, Haiyong 2016-02-26 We present an automatic approach for symmetrizing urban facade layouts. Our method can generate a symmetric layout through minimally modifying the original input layout. Based on the principles of symmetry in urban design, we formulate facade layout symmetrization as an optimization problem. Our method further enhances the regularity of the final layout by redistributing and aligning elements in the layout. We demonstrate that the proposed solution can effectively generate symmetric facade layouts. 7. The Symmetric Rudin-Shapiro Transform DEFF Research Database (Denmark) Harbo, Anders La-Cour 2003-01-01 A method for constructing spread spectrum sequences is presented. The method is based on a linear, orthogonal, and symmetric transform given as the Rudin-Shapiro transform (RST), which is in many respects quite similar to the Haar wavelet packet transform. The RST provides the means for generatin...... large sets of spread spectrum signals. This presentation provides a simple definition of the symmetric RST that leads to a fast N log(N) and numerically stable implementation of the transform.... 8. The Symmetric Rudin-Shapiro Transform DEFF Research Database (Denmark) Harbo, Anders La-Cour 2003-01-01 A method for constructing spread spectrum sequences is presented. The method is based on a linear, orthogonal, symmetric transform, the Rudin-Shapiro transform (RST), which is in many respects quite similar to the Haar wavelet packet transform. The RST provides the means for generating large sets...... of spread spectrum signals. This presentation provides a simple definition of the symmetric RST that leads to a fast N log(N) and numerically stable implementation of the transform.... 9. On Symmetric Polynomials OpenAIRE Golden, Ryan; Cho, Ilwoo 2015-01-01 In this paper, we study structure theorems of algebras of symmetric functions. Based on a certain relation on elementary symmetric polynomials generating such algebras, we consider perturbation in the algebras. In particular, we understand generators of the algebras as perturbations. From such perturbations, define injective maps on generators, which induce algebra-monomorphisms (or embeddings) on the algebras. They provide inductive structure theorems on algebras of symmetric polynomials. As... 10. Symmetric cryptographic protocols CERN Document Server Ramkumar, Mahalingam 2014-01-01 This book focuses on protocols and constructions that make good use of symmetric pseudo random functions (PRF) like block ciphers and hash functions - the building blocks for symmetric cryptography. Readers will benefit from detailed discussion of several strategies for utilizing symmetric PRFs. Coverage includes various key distribution strategies for unicast, broadcast and multicast security, and strategies for constructing efficient digests of dynamic databases using binary hash trees. • Provides detailed coverage of symmetric key protocols • Describes various applications of symmetric building blocks • Includes strategies for constructing compact and efficient digests of dynamic databases 11. Centrioles in Symmetric Spaces OpenAIRE Quast, Peter 2011-01-01 We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles. 12. A symmetrical rail accelerator International Nuclear Information System (INIS) Igenbergs, E. 1991-01-01 This paper reports on the symmetrical rail accelerator that has four rails, which are arranged symmetrically around the bore. The opposite rails have the same polarity and the adjacent rails the opposite polarity. In this configuration the radial force acting upon the individual rails is significantly smaller than in a conventional 2-rail configuration and a plasma armature is focussed towards the axis of the barrel. Experimental results indicate a higher efficiency compared to a conventional rail accelerator 13. Symmetric eikonal expansion International Nuclear Information System (INIS) Matsuki, Takayuki 1976-01-01 Symmetric eikonal expansion for the scattering amplitude is formulated for nonrelativistic and relativistic potential scatterings and also for the quantum field theory. The first approximations coincide with those of Levy and Sucher. The obtained scattering amplitudes are time reversal invariant for all cases and are crossing symmetric for the quantum field theory in each order of approximation. The improved eikonal phase introduced by Levy and Sucher is also derived from the different approximation scheme from the above. (auth.) 14. Implementation of mutual exclusion in VHDL NARCIS (Netherlands) Boersma, M.V.; Benders, L.P.M.; Stevens, M.P.J.; Wilsey, P.A.; Rhodes, D. 1994-01-01 In VHDL it is difficult to implement mutual exclusion at an abstract level since atomic actions are required. A local status model and an arbiter model are presented to achieve mutual exclusion in VHDL. Shared data, protected by a mutual exclusion mechanism, cannot be modelled as a simple, resolved 15. Symmetric configurations highlighted by collective quantum coherence Energy Technology Data Exchange (ETDEWEB) Obster, Dennis [Radboud University, Institute for Mathematics, Astrophysics and Particle Physics, Nijmegen (Netherlands); Kyoto University, Yukawa Institute for Theoretical Physics, Kyoto (Japan); Sasakura, Naoki [Kyoto University, Yukawa Institute for Theoretical Physics, Kyoto (Japan) 2017-11-15 Recent developments in quantum gravity have shown the Lorentzian treatment to be a fruitful approach towards the emergence of macroscopic space-times. In this paper, we discuss another related aspect of the Lorentzian treatment: we argue that collective quantum coherence may provide a simple mechanism for highlighting symmetric configurations over generic non-symmetric ones. After presenting the general framework of the mechanism, we show the phenomenon in some concrete simple examples in the randomly connected tensor network, which is tightly related to a certain model of quantum gravity, i.e., the canonical tensor model. We find large peaks at configurations invariant under Lie-group symmetries as well as a preference for charge quantization, even in the Abelian case. In future study, this simple mechanism may provide a way to analyze the emergence of macroscopic space-times with global symmetries as well as various other symmetries existing in nature, which are usually postulated. (orig.) 16. Multiparty symmetric sum types DEFF Research Database (Denmark) Nielsen, Lasse; Yoshida, Nobuko; Honda, Kohei 2010-01-01 This paper introduces a new theory of multiparty session types based on symmetric sum types, by which we can type non-deterministic orchestration choice behaviours. While the original branching type in session types can represent a choice made by a single participant and accepted by others...... determining how the session proceeds, the symmetric sum type represents a choice made by agreement among all the participants of a session. Such behaviour can be found in many practical systems, including collaborative workflow in healthcare systems for clinical practice guidelines (CPGs). Processes...... with the symmetric sums can be embedded into the original branching types using conductor processes. We show that this type-driven embedding preserves typability, satisfies semantic soundness and completeness, and meets the encodability criteria adapted to the typed setting. The theory leads to an efficient... 17. Counting with symmetric functions CERN Document Server Mendes, Anthony 2015-01-01 This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas. The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions. Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions. Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4. The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya’s enu... 18. Symmetric Tensor Decomposition DEFF Research Database (Denmark) Brachat, Jerome; Comon, Pierre; Mourrain, Bernard 2010-01-01 We present an algorithm for decomposing a symmetric tensor, of dimension n and order d, as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables...... of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation...... of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions and for detecting the rank.... 19. Revisiting the Optical PT-Symmetric Dimer Directory of Open Access Journals (Sweden) José Delfino Huerta Morales 2016-08-01 Full Text Available Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of PT -symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical PT -symmetric dimer, a two-waveguide coupler where the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry-based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar N-waveguide couplers that are the optical realization of the Lorentz group in 2 + 1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of the Ehrenfest theorem. 20. Distributed Searchable Symmetric Encryption NARCIS (Netherlands) Bösch, C.T.; Peter, Andreas; Leenders, Bram; Lim, Hoon Wei; Tang, Qiang; Wang, Huaxiong; Hartel, Pieter H.; Jonker, Willem Searchable Symmetric Encryption (SSE) allows a client to store encrypted data on a storage provider in such a way, that the client is able to search and retrieve the data selectively without the storage provider learning the contents of the data or the words being searched for. Practical SSE schemes 1. Symmetric waterbomb origami. Science.gov (United States) Chen, Yan; Feng, Huijuan; Ma, Jiayao; Peng, Rui; You, Zhong 2016-06-01 The traditional waterbomb origami, produced from a pattern consisting of a series of vertices where six creases meet, is one of the most widely used origami patterns. From a rigid origami viewpoint, it generally has multiple degrees of freedom, but when the pattern is folded symmetrically, the mobility reduces to one. This paper presents a thorough kinematic investigation on symmetric folding of the waterbomb pattern. It has been found that the pattern can have two folding paths under certain circumstance. Moreover, the pattern can be used to fold thick panels. Not only do the additional constraints imposed to fold the thick panels lead to single degree of freedom folding, but the folding process is also kinematically equivalent to the origami of zero-thickness sheets. The findings pave the way for the pattern being readily used to fold deployable structures ranging from flat roofs to large solar panels. 2. Symmetric modular torsatron Science.gov (United States) Rome, J.A.; Harris, J.H. 1984-01-01 A fusion reactor device is provided in which the magnetic fields for plasma confinement in a toroidal configuration is produced by a plurality of symmetrical modular coils arranged to form a symmetric modular torsatron referred to as a symmotron. Each of the identical modular coils is helically deformed and comprise one field period of the torsatron. Helical segments of each coil are connected by means of toroidally directed windbacks which may also provide part of the vertical field required for positioning the plasma. The stray fields of the windback segments may be compensated by toroidal coils. A variety of magnetic confinement flux surface configurations may be produced by proper modulation of the winding pitch of the helical segments of the coils, as in a conventional torsatron, winding the helix on a noncircular cross section and varying the poloidal and radial location of the windbacks and the compensating toroidal ring coils. 3. Spherically symmetric charged compact stars Energy Technology Data Exchange (ETDEWEB) Maurya, S.K. [University of Nizwa, Department of Mathematical and Physical Sciences, College of Arts and Science, Nizwa (Oman); Gupta, Y.K. [Jaypee Institute of Information Technology University, Department of Mathematics, Noida, Uttar Pradesh (India); Ray, Saibal [Government College of Engineering and Ceramic Technology, Department of Physics, Kolkata, West Bengal (India); Chowdhury, Sourav Roy [Seth Anandaram Jaipuria College, Department of Physics, Kolkata, West Bengal (India) 2015-08-15 In this article we consider the static spherically symmetric metric of embedding class 1. When solving the Einstein-Maxwell field equations we take into account the presence of ordinary baryonic matter together with the electric charge. Specific new charged stellar models are obtained where the solutions are entirely dependent on the electromagnetic field, such that the physical parameters, like density, pressure etc. do vanish for the vanishing charge. We systematically analyze altogether the three sets of Solutions I, II, and III of the stellar models for a suitable functional relation of ν(r). However, it is observed that only the Solution I provides a physically valid and well-behaved situation, whereas the Solutions II and III are not well behaved and hence not included in the study. Thereafter it is exclusively shown that the Solution I can pass through several standard physical tests performed by us. To validate the solution set presented here a comparison has also been made with that of the compact stars, like RX J 1856 - 37, Her X - 1, PSR 1937+21, PSRJ 1614-2230, and PSRJ 0348+0432, and we have shown the feasibility of the models. (orig.) 4. Symmetric vectors and algebraic classification International Nuclear Information System (INIS) Leibowitz, E. 1980-01-01 The concept of symmetric vector field in Riemannian manifolds, which arises in the study of relativistic cosmological models, is analyzed. Symmetric vectors are tied up with the algebraic properties of the manifold curvature. A procedure for generating a congruence of symmetric fields out of a given pair is outlined. The case of a three-dimensional manifold of constant curvature (''isotropic universe'') is studied in detail, with all its symmetric vector fields being explicitly constructed 5. Representations of locally symmetric spaces International Nuclear Information System (INIS) Rahman, M.S. 1995-09-01 Locally symmetric spaces in reference to globally and Hermitian symmetric Riemannian spaces are studied. Some relations between locally and globally symmetric spaces are exhibited. A lucid account of results on relevant spaces, motivated by fundamental problems, are formulated as theorems and propositions. (author). 10 refs 6. Filtering microfluidic bubble trains at a symmetric junction. Science.gov (United States) Parthiban, Pravien; Khan, Saif A 2012-02-07 We report how a nominally symmetric microfluidic junction can be used to sort all bubbles of an incoming train exclusively into one of its arms. The existence of this "filter" regime is unexpected, given that the junction is symmetric. We analyze this behavior by quantifying how bubbles modulate the hydrodynamic resistance in microchannels and show how speeding up a bubble train whilst preserving its spatial periodicity can lead to filtering at a nominally symmetric junction. We further show how such an asymmetric traffic of bubble trains can be triggered in symmetric geometries by identifying conditions wherein the resistance to flow decreases with an increase in the number of bubbles in the microchannel and derive an exact criterion to predict the same. 7. Symmetric extendibility of quantum states OpenAIRE Nowakowski, Marcin L. 2015-01-01 Studies on symmetric extendibility of quantum states become especially important in a context of analysis of one-way quantum measures of entanglement, distilabillity and security of quantum protocols. In this paper we analyse composite systems containing a symmetric extendible part with a particular attention devoted to one-way security of such systems. Further, we introduce a new one-way monotone based on the best symmetric approximation of quantum state. We underpin those results with geome... 8. A symmetric safety valve International Nuclear Information System (INIS) Burtraw, Dallas; Palmer, Karen; Kahn, Danny 2010-01-01 How to set policy in the presence of uncertainty has been central in debates over climate policy. Concern about costs has motivated the proposal for a cap-and-trade program for carbon dioxide, with a 'safety valve' that would mitigate against spikes in the cost of emission reductions by introducing additional emission allowances into the market when marginal costs rise above the specified allowance price level. We find two significant problems, both stemming from the asymmetry of an instrument that mitigates only against a price increase. One is that most important examples of price volatility in cap-and-trade programs have occurred not when prices spiked, but instead when allowance prices collapsed. Second, a single-sided safety valve may have unintended consequences for investment. We illustrate that a symmetric safety valve provides environmental and welfare improvements relative to the conventional one-sided approach. 9. Exact axially symmetric galactic dynamos Science.gov (United States) Henriksen, R. N.; Woodfinden, A.; Irwin, J. A. 2018-05-01 We give a selection of exact dynamos in axial symmetry on a galactic scale. These include some steady examples, at least one of which is wholly analytic in terms of simple functions and has been discussed elsewhere. Most solutions are found in terms of special functions, such as associated Lagrange or hypergeometric functions. They may be considered exact in the sense that they are known to any desired accuracy in principle. The new aspect developed here is to present scale-invariant solutions with zero resistivity that are self-similar in time. The time dependence is either a power law or an exponential factor, but since the geometry of the solution is self-similar in time we do not need to fix a time to study it. Several examples are discussed. Our results demonstrate (without the need to invoke any other mechanisms) X-shaped magnetic fields and (axially symmetric) magnetic spiral arms (both of which are well observed and documented) and predict reversing rotation measures in galaxy haloes (now observed in the CHANG-ES sample) as well as the fact that planar magnetic spirals are lifted into the galactic halo. 10. Symmetric q-Bessel functions Directory of Open Access Journals (Sweden) Giuseppe Dattoli 1996-05-01 Full Text Available q analog of bessel functions, symmetric under the interchange of q and q^ −1 are introduced. The definition is based on the generating function realized as product of symmetric q-exponential functions with appropriate arguments. Symmetric q-Bessel function are shown to satisfy various identities as well as second-order q-differential equations, which in the limit q → 1 reproduce those obeyed by the usual cylindrical Bessel functions. A brief discussion on the possible algebraic setting for symmetric q-Bessel functions is also provided. 11. Integrable dissipative exclusion process: Correlation functions and physical properties Science.gov (United States) Crampe, N.; Ragoucy, E.; Rittenberg, V.; Vanicat, M. 2016-09-01 We study a one-parameter generalization of the symmetric simple exclusion process on a one-dimensional lattice. In addition to the usual dynamics (where particles can hop with equal rates to the left or to the right with an exclusion constraint), annihilation and creation of pairs can occur. The system is driven out of equilibrium by two reservoirs at the boundaries. In this setting the model is still integrable: it is related to the open XXZ spin chain through a gauge transformation. This allows us to compute the full spectrum of the Markov matrix using Bethe equations. We also show that the stationary state can be expressed in a matrix product form permitting to compute the multipoints correlation functions as well as the mean value of the lattice and the creation-annihilation currents. Finally, the variance of the lattice current is computed for a finite-size system. In the thermodynamic limit, it matches the value obtained from the associated macroscopic fluctuation theory. 12. Conformally symmetric traversable wormholes International Nuclear Information System (INIS) Boehmer, Christian G.; Harko, Tiberiu; Lobo, Francisco S. N. 2007-01-01 Exact solutions of traversable wormholes are found under the assumption of spherical symmetry and the existence of a nonstatic conformal symmetry, which presents a more systematic approach in searching for exact wormhole solutions. In this work, a wide variety of solutions are deduced by considering choices for the form function, a specific linear equation of state relating the energy density and the pressure anisotropy, and various phantom wormhole geometries are explored. A large class of solutions impose that the spatial distribution of the exotic matter is restricted to the throat neighborhood, with a cutoff of the stress-energy tensor at a finite junction interface, although asymptotically flat exact solutions are also found. Using the 'volume integral quantifier', it is found that the conformally symmetric phantom wormhole geometries may, in principle, be constructed by infinitesimally small amounts of averaged null energy condition violating matter. Considering the tidal acceleration traversability conditions for the phantom wormhole geometry, specific wormhole dimensions and the traversal velocity are also deduced 13. Spherically symmetric self-similar universe Energy Technology Data Exchange (ETDEWEB) Dyer, C C [Toronto Univ., Ontario (Canada) 1979-10-01 A spherically symmetric self-similar dust-filled universe is considered as a simple model of a hierarchical universe. Observable differences between the model in parabolic expansion and the corresponding homogeneous Einstein-de Sitter model are considered in detail. It is found that an observer at the centre of the distribution has a maximum observable redshift and can in principle see arbitrarily large blueshifts. It is found to yield an observed density-distance law different from that suggested by the observations of de Vaucouleurs. The use of these solutions as central objects for Swiss-cheese vacuoles is discussed. 14. Geometrodynamics of spherically symmetric Lovelock gravity International Nuclear Information System (INIS) Kunstatter, Gabor; Taves, Tim; Maeda, Hideki 2012-01-01 We derive the Hamiltonian for spherically symmetric Lovelock gravity using the geometrodynamics approach pioneered by Kuchar (1994 Phys. Rev. D 50 3961) in the context of four-dimensional general relativity. When written in terms of the areal radius, the generalized Misner-Sharp mass and their conjugate momenta, the generic Lovelock action and Hamiltonian take on precisely the same simple forms as in general relativity. This result supports the interpretation of Lovelock gravity as the natural higher dimensional extension of general relativity. It also provides an important first step towards the study of the quantum mechanics, Hamiltonian thermodynamics and formation of generic Lovelock black holes. (fast track communication) 15. Mesotherapy for benign symmetric lipomatosis. Science.gov (United States) Hasegawa, Toshio; Matsukura, Tomoyuki; Ikeda, Shigaku 2010-04-01 Benign symmetric lipomatosis, also known as Madelung disease, is a rare disorder characterized by fat distribution around the shoulders, arms, and neck in the context of chronic alcoholism. Complete excision of nonencapsulated lipomas is difficult. However, reports describing conservative therapeutic measures for lipomatosis are rare. The authors present the case of a 42-year-old man with a diagnosis of benign symmetric lipomatosis who had multiple, large, symmetrical masses in his neck. Multiple phosphatidylcholine injections in the neck were administered 4 weeks apart, a total of seven times to achieve lipolysis. The patient's lipomatosis improved in response to the injections, and he achieved good cosmetic results. Intralesional injection, termed mesotherapy, using phosphatidylcholine is a potentially effective therapy for benign symmetric lipomatosis that should be reconsidered as a therapeutic option for this disease. 16. Looking for symmetric Bell inequalities OpenAIRE Bancal, Jean-Daniel; Gisin, Nicolas; Pironio, Stefano 2010-01-01 Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell e... 17. On Split Lie Algebras with Symmetric Root Systems ... and any I j a well described ideal of , satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of is characterized and it is shown that is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected. 18. On split Lie algebras with symmetric root systems ideal of L, satisfying [Ij ,Ik] = 0 if j = k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected. Keywords. Infinite dimensional Lie ... 19. Hard exclusive QCD processes Energy Technology Data Exchange (ETDEWEB) Kugler, W. 2007-01-15 Hard exclusive processes in high energy electron proton scattering offer the opportunity to get access to a new generation of parton distributions, the so-called generalized parton distributions (GPDs). This functions provide more detailed informations about the structure of the nucleon than the usual PDFs obtained from DIS. In this work we present a detailed analysis of exclusive processes, especially of hard exclusive meson production. We investigated the influence of exclusive produced mesons on the semi-inclusive production of mesons at fixed target experiments like HERMES. Further we give a detailed analysis of higher order corrections (NLO) for the exclusive production of mesons in a very broad range of kinematics. (orig.) 20. Droplet Traffic Control at a simple T junction Science.gov (United States) Panizza, Pascal; Engl, Wilfried; Colin, Annie; Ajdari, Armand 2006-03-01 A basic yet essential element of every traffic flow control is the effect of a junction where the flow is separated into several streams. How do pedestrians, vehicles or blood cells divide when they reach a junction? How does the outcome depend on their density? Similar fundamental questions hold for much simpler systems: in this paper, we have studied the behaviour of periodic trains of water droplets flowing in oil through a channel as they reach a simple, locally symmetric, T junction. Depending on their dilution, we observe that the droplets are either alternately partitioned between both outlets or sorted exclusively into the shortest one. We show that this surprising behaviour results from the hydrodynamic feed-back of drops in the two outlets on the selection process occurring at the junction. Our results offer a first guide for the design and modelling of droplet traffic in complex branched networks, a necessary step towards parallelized droplet-based lab-on-chip'' devices. 1. Harmonic analysis on symmetric spaces CERN Document Server Terras, Audrey This text explores the geometry and analysis of higher rank analogues of the symmetric spaces introduced in volume one. To illuminate both the parallels and differences of the higher rank theory, the space of positive matrices is treated in a manner mirroring that of the upper-half space in volume one. This concrete example furnishes motivation for the general theory of noncompact symmetric spaces, which is outlined in the final chapter. The book emphasizes motivation and comprehensibility, concrete examples and explicit computations (by pen and paper, and by computer), history, and, above all, applications in mathematics, statistics, physics, and engineering. The second edition includes new sections on Donald St. P. Richards’s central limit theorem for O(n)-invariant random variables on the symmetric space of GL(n, R), on random matrix theory, and on advances in the theory of automorphic forms on arithmetic groups. 2. Looking for symmetric Bell inequalities International Nuclear Information System (INIS) Bancal, Jean-Daniel; Gisin, Nicolas; Pironio, Stefano 2010-01-01 Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-Massar-Popescu type. 3. Symmetric normalisation for intuitionistic logic DEFF Research Database (Denmark) Guenot, Nicolas; Straßburger, Lutz 2014-01-01 We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus......, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric... 4. Diagrams for symmetric product orbifolds International Nuclear Information System (INIS) Pakman, Ari; Rastelli, Leonardo; Razamat, Shlomo S. 2009-01-01 We develop a diagrammatic language for symmetric product orbifolds of two-dimensional conformal field theories. Correlation functions of twist operators are written as sums of diagrams: each diagram corresponds to a branched covering map from a surface where the fields are single-valued to the base sphere where twist operators are inserted. This diagrammatic language facilitates the study of the large N limit and makes more transparent the analogy between symmetric product orbifolds and free non-abelian gauge theories. We give a general algorithm to calculate the leading large N contribution to four-point correlators of twist fields. 5. Looking for symmetric Bell inequalities Energy Technology Data Exchange (ETDEWEB) Bancal, Jean-Daniel; Gisin, Nicolas [Group of Applied Physics, University of Geneva, 20 rue de l' Ecole-de Medecine, CH-1211 Geneva 4 (Switzerland); Pironio, Stefano, E-mail: [email protected] [Laboratoire d' Information Quantique, Universite Libre de Bruxelles (Belgium) 2010-09-24 Finding all Bell inequalities for a given number of parties, measurement settings and measurement outcomes is in general a computationally hard task. We show that all Bell inequalities which are symmetric under the exchange of parties can be found by examining a symmetrized polytope which is simpler than the full Bell polytope. As an illustration of our method, we generate 238 885 new Bell inequalities and 1085 new Svetlichny inequalities. We find, in particular, facet inequalities for Bell experiments involving two parties and two measurement settings that are not of the Collins-Gisin-Linden-Massar-Popescu type. 6. Symmetric autocompensating quantum key distribution Science.gov (United States) Walton, Zachary D.; Sergienko, Alexander V.; Levitin, Lev B.; Saleh, Bahaa E. A.; Teich, Malvin C. 2004-08-01 We present quantum key distribution schemes which are autocompensating (require no alignment) and symmetric (Alice and Bob receive photons from a central source) for both polarization and time-bin qubits. The primary benefit of the symmetric configuration is that both Alice and Bob may have passive setups (neither Alice nor Bob is required to make active changes for each run of the protocol). We show that both the polarization and the time-bin schemes may be implemented with existing technology. The new schemes are related to previously described schemes by the concept of advanced waves. 7. Exclusive Dealing and Entry OpenAIRE João Leão 2008-01-01 This paper examines the use of exclusive dealing agreements to prevent the entry of rival firms. An exclusive dealing agreement is a contract between a buyer and a seller where the buyer commits to buy a good exclusively from the seller. One main concern of the literature is to explain how an incumbent seller is able to persuade the buyers to sign an exclusive dealing agreement that deters the entry of a more efficient rival seller. We propose a new explanation when the buyers are downstream ... 8. Krein signature for instability of PT-symmetric states Science.gov (United States) Chernyavsky, Alexander; Pelinovsky, Dmitry E. 2018-05-01 Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT-symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature. 9. Symmetric relations of finite negativity NARCIS (Netherlands) Kaltenbaeck, M.; Winkler, H.; Woracek, H.; Forster, KH; Jonas, P; Langer, H 2006-01-01 We construct and investigate a space which is related to a symmetric linear relation S of finite negativity on an almost Pontryagin space. This space is the indefinite generalization of the completion of dom S with respect to (S.,.) for a strictly positive S on a Hilbert space. 10. Tilting-connected symmetric algebras OpenAIRE Aihara, Takuma 2010-01-01 The notion of silting mutation was introduced by Iyama and the author. In this paper we mainly study silting mutation for self-injective algebras and prove that any representation-finite symmetric algebra is tilting-connected. Moreover we give some sufficient conditions for a Bongartz-type Lemma to hold for silting objects. 11. Symmetric group representations and Z OpenAIRE 2017-01-01 We discuss implications of the following statement about the representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation, and every nonnegative integer appears infinitely often as a Littlewood-Richardson coefficient and as a Kronecker coefficient. 12. Symmetric Key Authentication Services Revisited NARCIS (Netherlands) Crispo, B.; Popescu, B.C.; Tanenbaum, A.S. 2004-01-01 Most of the symmetric key authentication schemes deployed today are based on principles introduced by Needham and Schroeder [15] more than twenty years ago. However, since then, the computing environment has evolved from a LAN-based client-server world to include new paradigms, including wide area 13. Quantum systems and symmetric spaces International Nuclear Information System (INIS) Olshanetsky, M.A.; Perelomov, A.M. 1978-01-01 Certain class of quantum systems with Hamiltonians related to invariant operators on symmetric spaces has been investigated. A number of physical facts have been derived as a consequence. In the classical limit completely integrable systems related to root systems are obtained 14. The symmetric longest queue system NARCIS (Netherlands) van Houtum, Geert-Jan; Adan, Ivo; van der Wal, Jan 1997-01-01 We derive the performance of the exponential symmetric longest queue system from two variants: a longest queue system with Threshold Rejection of jobs and one with Threshold Addition of jobs. It is shown that these two systems provide lower and upper bounds for the performance of the longest queue 15. Strong orientational coordinates and orientational order parameters for symmetric objects International Nuclear Information System (INIS) Haji-Akbari, Amir; Glotzer, Sharon C 2015-01-01 Recent advancements in the synthesis of anisotropic macromolecules and nanoparticles have spurred an immense interest in theoretical and computational studies of self-assembly. The cornerstone of such studies is the role of shape in self-assembly and in inducing complex order. The problem of identifying different types of order that can emerge in such systems can, however, be challenging. Here, we revisit the problem of quantifying orientational order in systems of building blocks with non-trivial rotational symmetries. We first propose a systematic way of constructing orientational coordinates for such symmetric building blocks. We call the arising tensorial coordinates strong orientational coordinates (SOCs) as they fully and exclusively specify the orientation of a symmetric object. We then use SOCs to describe and quantify local and global orientational order, and spatiotemporal orientational correlations in systems of symmetric building blocks. The SOCs and the orientational order parameters developed in this work are not only useful in performing and analyzing computer simulations of symmetric molecules or particles, but can also be utilized for the efficient storage of rotational information in long trajectories of evolving many-body systems. (paper) 16. Symmetric imaging findings in neuroradiology International Nuclear Information System (INIS) Zlatareva, D. 2015-01-01 Full text: Learning objectives: to make a list of diseases and syndromes which manifest as bilateral symmetric findings on computed tomography and magnetic resonance imaging; to discuss the clinical and radiological differential diagnosis for these diseases; to explain which of these conditions necessitates urgent therapy and when additional studies and laboratory can precise diagnosis. There is symmetry in human body and quite often we compare the affected side to the normal one but in neuroradiology we might have bilateral findings which affected pair structures or corresponding anatomic areas. It is very rare when clinical data prompt diagnosis. Usually clinicians suspect such an involvement but Ct and MRI can reveal symmetric changes and are one of the leading diagnostic tool. The most common location of bilateral findings is basal ganglia and thalamus. There are a number of diseases affecting these structures symmetrically: metabolic and systemic diseases, intoxication, neurodegeneration and vascular conditions, toxoplasmosis, tumors and some infections. Malformations of cortical development and especially bilateral perisylvian polymicrogyria requires not only exact report on the most affected parts but in some cases genetic tests or combination with other clinical symptoms. In the case of herpes simplex encephalitis bilateral temporal involvement is common and this finding very often prompt therapy even before laboratory results. Posterior reversible encephalopathy syndrome (PReS) and some forms of hypoxic ischemic encephalopathy can lead to symmetric changes. In these acute conditions MR plays a crucial role not only in diagnosis but also in monitoring of the therapeutic effect. Patients with neurofibromatosis type 1 or type 2 can demonstrate bilateral optic glioma combined with spinal neurofibroma and bilateral acoustic schwanoma respectively. Mirror-image aneurysm affecting both internal carotid or middle cerebral arteries is an example of symmetry in 17. Definition of Exclusion Zones Using Seismic Data Science.gov (United States) Bartal, Y.; Villagran, M.; Ben Horin, Y.; Leonard, G.; Joswig, M. - In verifying compliance with the Comprehensive Nuclear-Test-Ban Treaty (CTBT), there is a motivation to be effective, efficient and economical and to prevent abuse of the right to conduct an On-site Inspection (OSI) in the territory of a challenged State Party. In particular, it is in the interest of a State Party to avoid irrelevant search in specific areas. In this study we propose several techniques to determine exclusion zones', which are defined as areas where an event could not have possibly occurred. All techniques are based on simple ideas of arrival time differences between seismic stations and thus are less prone to modeling errors compared to standard event location methods. The techniques proposed are: angular sector exclusion based on a tripartite micro array, half-space exclusion based on a station pair, and closed area exclusion based on circumferential networks. 18. Parity-Time Symmetric Photonics KAUST Repository Zhao, Han 2018-01-17 The establishment of non-Hermitian quantum mechanics (such as parity-time (PT) symmetry) stimulates a paradigmatic shift for studying symmetries of complex potentials. Owing to the convenient manipulation of optical gain and loss in analogy to the complex quantum potentials, photonics provides an ideal platform for visualization of many conceptually striking predictions from the non-Hermitian quantum theory. A rapidly developing field has emerged, namely, PT symmetric photonics, demonstrating intriguing optical phenomena including eigenstate coalescence and spontaneous PT symmetry breaking. The advance of quantum physics, as the feedback, provides photonics with brand-new paradigms to explore the entire complex permittivity plane for novel optical functionalities. Here, we review recent exciting breakthroughs in PT symmetric photonics while systematically presenting their underlying principles guided by non-Hermitian symmetries. The potential device applications for optical communication and computing, bio-chemical sensing, and healthcare are also discussed. 19. Homotheties of cylindrically symmetric static spacetimes International Nuclear Information System (INIS) 1998-08-01 In this note we consider the homotheties of cylindrically symmetric static spacetimes. We find that we can provide a complete list of all metrics that admit non-trivial homothetic motions and are cylindrically symmetric static. (author) 20. Maximally Symmetric Composite Higgs Models. Science.gov (United States) Csáki, Csaba; Ma, Teng; Shu, Jing 2017-09-29 Maximal symmetry is a novel tool for composite pseudo Goldstone boson Higgs models: it is a remnant of an enhanced global symmetry of the composite fermion sector involving a twisting with the Higgs field. Maximal symmetry has far-reaching consequences: it ensures that the Higgs potential is finite and fully calculable, and also minimizes the tuning. We present a detailed analysis of the maximally symmetric SO(5)/SO(4) model and comment on its observational consequences. 1. Social Exclusion Anxiety DEFF Research Database (Denmark) Søndergaard, Dorte Marie 2017-01-01 Social exclusion anxiety is a term which builds on a social-psychological concept of human beings as existentially dependent on social embeddedness. This entry explores the concept in relation to bullying among children, which is a widespread and serious problem in schools and institutions. Social...... exclusion anxiety and longing for belonging are both central aspects of the affects and processes that enact and challenge social groups. Social exclusion anxiety should not be confused with ‘social phobia’, which is a concept within clinical psychology that focuses on the individual and refers to a phobic...... psychological condition. Social exclusion anxiety instead points to a distributed affect which circulates and smolders in all social groups. This is the result of an ever-present risk of someone being judged unworthy to belong to, or deemed not a legitimate participant in, a social group. Such anxiety may... 2. On symmetric structures of order two Directory of Open Access Journals (Sweden) Michel Bousquet 2008-04-01 Full Text Available Let (ω n 0 < n be the sequence known as Integer Sequence A047749 http://www.research.att.com/ njas/sequences/A047749 In this paper, we show that the integer ω n enumerates various kinds of symmetric structures of order two. We first consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects. 3. Communication: Symmetrical quasi-classical analysis of linear optical spectroscopy Science.gov (United States) Provazza, Justin; Coker, David F. 2018-05-01 The symmetrical quasi-classical approach for propagation of a many degree of freedom density matrix is explored in the context of computing linear spectra. Calculations on a simple two state model for which exact results are available suggest that the approach gives a qualitative description of peak positions, relative amplitudes, and line broadening. Short time details in the computed dipole autocorrelation function result in exaggerated tails in the spectrum. 4. Simple machines CERN Document Server Graybill, George 2007-01-01 Just how simple are simple machines? With our ready-to-use resource, they are simple to teach and easy to learn! Chocked full of information and activities, we begin with a look at force, motion and work, and examples of simple machines in daily life are given. With this background, we move on to different kinds of simple machines including: Levers, Inclined Planes, Wedges, Screws, Pulleys, and Wheels and Axles. An exploration of some compound machines follows, such as the can opener. Our resource is a real time-saver as all the reading passages, student activities are provided. Presented in s 5. Baryon symmetric big bang cosmology International Nuclear Information System (INIS) Stecker, F.W. 1978-01-01 It is stated that the framework of baryon symmetric big bang (BSBB) cosmology offers our greatest potential for deducting the evolution of the Universe because its physical laws and processes have the minimum number of arbitrary assumptions about initial conditions in the big-bang. In addition, it offers the possibility of explaining the photon-baryon ratio in the Universe and how galaxies and galaxy clusters are formed. BSBB cosmology also provides the only acceptable explanation at present for the origin of the cosmic γ-ray background radiation. (author) 6. Symmetric functions and orthogonal polynomials CERN Document Server Macdonald, I G 1997-01-01 One of the most classical areas of algebra, the theory of symmetric functions and orthogonal polynomials has long been known to be connected to combinatorics, representation theory, and other branches of mathematics. Written by perhaps the most famous author on the topic, this volume explains some of the current developments regarding these connections. It is based on lectures presented by the author at Rutgers University. Specifically, he gives recent results on orthogonal polynomials associated with affine Hecke algebras, surveying the proofs of certain famous combinatorial conjectures. 7. Immanant Conversion on Symmetric Matrices Directory of Open Access Journals (Sweden) Purificação Coelho M. 2014-01-01 Full Text Available Letr Σn(C denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C -> Σn (C satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB = dχ·(Φ(Α + αΦ(Β for all matrices A,В ε Σ„(С and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С. 8. The Pauli Exclusion Principle his exclusion principle, the quantum theory was a mess. Moreover, it could ... This is a function of all the coordinates and 'internal variables' such as spin, of all the ... must remain basically the same (ie change by a phase factor at most) if we ... 9. Exclusive Production at CMS CERN Document Server Walczak, Marek 2016-01-01 I briefly introduce so-called central exclusive production. I mainly focus on the example analyses that have been performed in the CMS experiment at CERN. I conclude with ideas and perspectives for future work that will be done during Run 2 of the LHC. I pay special attention to the ultraperipheral collisions. 10. Ombuds' Corner: Social exclusion CERN Document Server Vincent Vuillemin 2012-01-01 In this special video edition of the Ombuds' Corner, Vincent Vuillemin takes a look at a social exclusion at CERN. Please note that the characters and situations appearing in this work are fictitious, and any resemblance to real persons or events is purely coincidental. Contact the Ombuds Early! 11. Social exclusion of children NARCIS (Netherlands) Annette Roest; Anne Marike Lokhorst; Cok Vrooman 2010-01-01 Original title: Sociale uitsluiting bij kinderen. Combating social exclusion of children is a subject that has received growing attention in Dutch government policy in recent years. To date, however, no analysis has been performed to ascertain the extent and origins of this phenomenon. This 12. An iteration for indefinite and non-symmetric systems and its application to the Navier-Stokes equations Energy Technology Data Exchange (ETDEWEB) Wathen, A. [Oxford Univ. (United Kingdom); Golub, G. [Stanford Univ., CA (United States) 1996-12-31 A simple fixed point linearisation of the Navier-Stokes equations leads to the Oseen problem which after appropriate discretisation yields large sparse linear systems with coefficient matrices of the form (A B{sup T} B -C). Here A is non-symmetric but its symmetric part is positive definite, and C is symmetric and positive semi-definite. Such systems arise in other situations. In this talk we will describe and present some analysis for an iteration based on an indefinite and symmetric preconditioner of the form (D B{sup T} B -C). 13. Probabilistic cloning of three symmetric states International Nuclear Information System (INIS) Jimenez, O.; Bergou, J.; Delgado, A. 2010-01-01 We study the probabilistic cloning of three symmetric states. These states are defined by a single complex quantity, the inner product among them. We show that three different probabilistic cloning machines are necessary to optimally clone all possible families of three symmetric states. We also show that the optimal cloning probability of generating M copies out of one original can be cast as the quotient between the success probability of unambiguously discriminating one and M copies of symmetric states. 14. Simple prostatectomy Science.gov (United States) ... Han M, Partin AW. Simple prostatectomy: open and robot-assisted laparoscopic approaches. In: Wein AJ, Kavoussi LR, ... M. is also a founding member of Hi-Ethics and subscribes to the principles of the Health ... 15. Spherically symmetric scalar field collapse 2013-03-01 Mar 1, 2013 ... The very recent interest in scalar field collapse stems from a cosmological ... The objective of the present investigation is to explore the collapsing modes of a simple ..... The authors thank the BRNS (DAE) for financial support. 16. Classification of symmetric toroidal orbifolds Energy Technology Data Exchange (ETDEWEB) Fischer, Maximilian; Ratz, Michael; Torrado, Jesus [Technische Univ. Muenchen, Garching (Germany). Physik-Department; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany) 2012-09-15 We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield N{>=}1 supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as Z{sub 3}, Z{sub 4}, Z{sub 6}-I etc. and 358 with non-Abelian point groups such as S{sub 3}, D{sub 4}, A{sub 4} etc. We also briefly explore the properties of some orbifolds with Abelian point groups and N=1, i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking. 17. Nonlinear PT-symmetric plaquettes International Nuclear Information System (INIS) Li Kai; Kevrekidis, P G; Malomed, Boris A; Günther, Uwe 2012-01-01 We introduce four basic two-dimensional (2D) plaquette configurations with onsite cubic nonlinearities, which may be used as building blocks for 2D PT-symmetric lattices. For each configuration, we develop a dynamical model and examine its PTsymmetry. The corresponding nonlinear modes are analyzed starting from the Hamiltonian limit, with zero value of the gain–loss coefficient, γ. Once the relevant waveforms have been identified (chiefly, in an analytical form), their stability is examined by means of linearization in the vicinity of stationary points. This reveals diverse and, occasionally, fairly complex bifurcations. The evolution of unstable modes is explored by means of direct simulations. In particular, stable localized modes are found in these systems, although the majority of identified solutions are unstable. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper) 18. PT-symmetric planar devices for field transformation and imaging International Nuclear Information System (INIS) Valagiannopoulos, C A; Monticone, F; Alù, A 2016-01-01 The powerful tools of transformation optics (TO) allow an effective distortion of a region of space by carefully engineering the material inhomogeneity and anisotropy, and have been successfully applied in recent years to control electromagnetic fields in many different scenarios, e.g., to realize invisibility cloaks and planar lenses. For various field transformations, it is not necessary to use volumetric inhomogeneous materials, and suitably designed ultrathin metasurfaces with tailored spatial or spectral responses may be able to realize similar functionalities within smaller footprints and more robust mechanisms. Here, inspired by the concept of metamaterial TO lenses, we discuss field transformations enabled by parity-time (PT) symmetric metasurfaces, which can emulate negative refraction. We first analyze a simple realization based on homogeneous and local metasurfaces to achieve negative refraction and imaging, and we then extend our results to arbitrary PT-symmetric two-port networks to realize aberration-free planar imaging. (paper) 19. Simple unification International Nuclear Information System (INIS) Ponce, W.A.; Zepeda, A. 1987-08-01 We present the results obtained from our systematic search of a simple Lie group that unifies weak and electromagnetic interactions in a single truly unified theory. We work with fractionally charged quarks, and allow for particles and antiparticles to belong to the same irreducible representation. We found that models based on SU(6), SU(7), SU(8) and SU(10) are viable candidates for simple unification. (author). 23 refs 20. Relativistic fluids in spherically symmetric space International Nuclear Information System (INIS) Dipankar, R. 1977-12-01 Some of McVittie and Wiltshire's (1977) solutions of Walker's (1935) isotropy conditions for relativistic perfect fluid spheres are generalized. Solutions are spherically symmetric and conformally flat 1. "Exclusive Dealing Contract and Inefficient Entry Threat" OpenAIRE Noriyuki Yanagawa; Ryoko Oki 2008-01-01 This paper examines the effects of exclusive dealing contracts in a simple model with manufacturers-distributors relations. We consider entrants in both manufacturing and distribution sectors. It is well-known that a potential entry threat is welfare increasing under homogenous price competition, even though the potential entrant is less productive. This paper reexamines this intuition by employing the above model. We show that the entry threat of a less-productive manufacturer is welfare dec... 2. Exclusive processes in QCD International Nuclear Information System (INIS) Mueller, A.H. 1981-01-01 In this talk I concentrate on purely exclusive processes. In Sec. II form factors and exclusive decays of heavy quarkonium states will be discussed. In Sec. III elastic wide angle elastic scattering will be considered with emphasis placed on the energy dependence for a fixed angle. The x → 1 limit of structure functions is discussed in Sec. IV. This is a limit which matches on, in a rather complicated way, with transition form factors. In Sec. V the idea of intrinsic charm is considered, mostly from a conceptual viewpoint as to its definition and possible existence. In Sec. VI there is a brief discussion of calculations of matrix elements which occur in deeply inelastic scattering by use of a bag model. In Sec. VII wee parton cancellations and Sudakov corrections for μ-pair production are considered. Sec. VIII concerns soft particle production and the mutliplicity of hadrons in a jet. (orig./HSI) 3. The psychology of exclusivity Directory of Open Access Journals (Sweden) Troy Jollimore 2008-02-01 Full Text Available Friendship and romantic love are, by their very nature, exclusive relationships. This paper suggests that we can better understand the nature of the exclusivity in question by understanding what is wrong with the view of practical reasoning I call the Comprehensive Surveyor View. The CSV claims that practical reasoning, in order to be rational, must be a process of choosing the best available alternative from a perspective that is as detached and objective as possible. But this view, while it means to be neutral between various value-bearers, in fact incorporates a bias against those value-bearers that can only be appreciated from a perspective that is not detached—that can only be appreciated, for instance, by agents who bear long-term commitments to the values in question. In the realm of personal relationships, such commitments tend to give rise to the sort of exclusivity that characterizes friendship and romantic love; they prevent the agent from being impartial between her beloved’s needs, interests, etc., and those of other persons. In such contexts, I suggest, needs and claims of other persons may be silenced in much the way that, as John McDowell has suggested, the temptations of immorality are silenced for the virtuous agent. 4. Comprehensive asynchronous symmetric rendezvous algorithm in ... Meenu Chawla 2017-11-10 Nov 10, 2017 ... Simulation results affirm that CASR algorithm performs better in terms of average time-to-rendezvous as compared ... process; neighbour discovery; symmetric rendezvous algorithm. 1. .... dezvous in finite time under the symmetric model. The CH ..... CASR algorithm in Matlab 7.11 and performed several. 5. Signed Young Modules and Simple Specht Modules OpenAIRE Danz, Susanne; Lim, Kay Jin 2015-01-01 By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent inter... 6. Symmetric splitting of very light systems International Nuclear Information System (INIS) Grotowski, K.; Majka, Z.; Planeta, R. 1985-01-01 Fission reactions that produce fragments close to one half the mass of the composite system are traditionally observed in heavy nuclei. In light systems, symmetric splitting is rarely observed and poorly understood. It would be interesting to verify the existence of the symmetric splitting of compound nuclei with A 12 C + 40 Ca, 141 MeV 9 Be + 40 Ca and 153 MeV 6 Li + 40 Ca. The out-of-plane correlation of symmetric products was also measured for the reaction 186 MeV 12 C + 40 Ca. The coincidence measurements of the 12 C + 40 Ca system demonstrated that essentially all of the inclusive yield of symmetric products around 40 0 results from a binary decay. To characterize the dependence of the symmetric splitting process on the excitation energy of the 12 C + 40 C system, inclusive measurements were made at bombarding energies of 74, 132, 162, and 185 MeV 7. Baryon symmetric big bang cosmology Science.gov (United States) Stecker, F. W. 1978-01-01 Both the quantum theory and Einsteins theory of special relativity lead to the supposition that matter and antimatter were produced in equal quantities during the big bang. It is noted that local matter/antimatter asymmetries may be reconciled with universal symmetry by assuming (1) a slight imbalance of matter over antimatter in the early universe, annihilation, and a subsequent remainder of matter; (2) localized regions of excess for one or the other type of matter as an initial condition; and (3) an extremely dense, high temperature state with zero net baryon number; i.e., matter/antimatter symmetry. Attention is given to the third assumption, which is the simplest and the most in keeping with current knowledge of the cosmos, especially as pertains the universality of 3 K background radiation. Mechanisms of galaxy formation are discussed, whereby matter and antimatter might have collided and annihilated each other, or have coexisted (and continue to coexist) at vast distances. It is pointed out that baryon symmetric big bang cosmology could probably be proved if an antinucleus could be detected in cosmic radiation. 8. Substring-Searchable Symmetric Encryption Directory of Open Access Journals (Sweden) Chase Melissa 2015-06-01 Full Text Available In this paper, we consider a setting where a client wants to outsource storage of a large amount of private data and then perform substring search queries on the data – given a data string s and a search string p, find all occurrences of p as a substring of s. First, we formalize an encryption paradigm that we call queryable encryption, which generalizes searchable symmetric encryption (SSE and structured encryption. Then, we construct a queryable encryption scheme for substring queries. Our construction uses suffix trees and achieves asymptotic efficiency comparable to that of unencrypted suffix trees. Encryption of a string of length n takes O(λn time and produces a ciphertext of size O(λn, and querying for a substring of length m that occurs k times takes O(λm+k time and three rounds of communication. Our security definition guarantees correctness of query results and privacy of data and queries against a malicious adversary. Following the line of work started by Curtmola et al. (ACM CCS 2006, in order to construct more efficient schemes we allow the query protocol to leak some limited information that is captured precisely in the definition. We prove security of our substring-searchable encryption scheme against malicious adversaries, where the query protocol leaks limited information about memory access patterns through the suffix tree of the encrypted string. 9. Exclusion and authorization International Nuclear Information System (INIS) Cooper, J.R. 2003-01-01 'Everyone in the world is exposed to radiation from natural and artificial sources. Any realistic system of radiological protection must have a clearly defined scope if it is not to apply to the whole of mankind's activities'. This quote, from ICRP Publication 60 (ICRP, 1991), remains apposite. The main tool for defining scope is the concept of exclusion: situations, sources or exposures that are excluded from the system of radiological protection are, to all intents and purposes, ignored. Sources and exposures that are not excluded are within the scope of the system of protection and by inference within regulatory systems implementing ICRP recommendations. These sources and exposures should be subject to appropriate authorization by the relevant regulatory authority. In order to avoid excessive regulatory procedures, however, provisions should be made for granting an exemption in cases where it is clear that regulatory provisions are unnecessary. Exemption is a regulatory tool intended to facilitate efficient use of regulatory resources. Nevertheless, the regulatory act of granting exemptions is, in itself, a form of authorization and the material or situation so exempted remains within the regulatory system. This distinction between exclusion and exemption is an important one. Historically, the concept of exclusion has been applied to sources or exposures that are essentially unamenable to control because of their widespread nature. The usually quoted examples are cosmic radiation at ground level and 40 K in the body. Clearly, many exposures from natural sources could fall into this category. The challenges are firstly to establish a sound basis for deciding which should be excluded and which should be controlled, and secondly to see if the concept could or should be applied to artificial sources and exposures. These two questions are the subject of this paper. (author) 10. Uniqueness of flat spherically symmetric spacelike hypersurfaces admitted by spherically symmetric static spacetimes Science.gov (United States) 2007-11-01 It is known that spherically symmetric static spacetimes admit a foliation by flat hypersurfaces. Such foliations have explicitly been constructed for some spacetimes, using different approaches, but none of them have proved or even discussed the uniqueness of these foliations. The issue of uniqueness becomes more important due to suitability of flat foliations for studying black hole physics. Here, flat spherically symmetric spacelike hypersurfaces are obtained by a direct method. It is found that spherically symmetric static spacetimes admit flat spherically symmetric hypersurfaces, and that these hypersurfaces are unique up to translation under the timelike Killing vector. This result guarantees the uniqueness of flat spherically symmetric foliations for such spacetimes. 11. Social exclusion anxiety DEFF Research Database (Denmark) Søndergaard, Dorte Marie 2014-01-01 . The concepts I work with are the need for belonging, social exclusion anxiety and the production of contempt and dignity by both children and adults. I develop a new definition of bullying, drawing upon Judith Butler’s (1999) concept of ‘abjection’ as well as Karen Barad’s concept of ‘intra-acting forces......’ (Barad 2007). My definition in this chapter contributed to the shorter definition of bullying in the Introduction (see page XX), but it is more fully developed here in relation to the types of mechanisms and processes involved. Barad’s term ‘intra-action’ helps draw attention to the mutually... 12. The Topology of Three-Dimensional Symmetric Tensor Fields Science.gov (United States) Lavin, Yingmei; Levy, Yuval; Hesselink, Lambertus 1994-01-01 We study the topology of 3-D symmetric tensor fields. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector field topology. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. First, we introduce a new method for locating 3-D degenerate points. We then extract the topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Finally, we demonstrate the use of tensor field topology for the interpretation of the two-force Boussinesq problem. 13. Social exclusion and education Directory of Open Access Journals (Sweden) Jokić Vesna 2009-01-01 Full Text Available Social exclusion is a process whereby certain individuals are pushed to the edge of society and prevented from participating fully by virtue of their poverty, or lack of basic competencies and lifelong learning opportunities or as a result of discrimination. This distances them from job, income and education opportunities as well as social and community networks and activities. Quality education (conditions and access/accessibility/availability is one of the factors that significantly influence the reduced social exclusion. In other words, education has is key role key role in ensuring social inclusion (equal opportunities and active social participation. At the same time, education and lifelong learning is established as the basis for achieving the goals of sustainable economic development (economy based on knowledge and to achieve social cohesion. Quality education is a prerequisite for progress, development and well-being of the community. Conditions and accessibility to education have become priorities of national reforms in most European countries. The subject of this paper is the educational structure of population of Serbia and the accessibility of education. The analysis covers the educational structure with regard to age, gender and type of settlement (city and other/villages settlements. 14. Stationary states of a PT symmetric two-mode Bose–Einstein condensate International Nuclear Information System (INIS) Graefe, Eva-Maria 2012-01-01 The understanding of nonlinear PT symmetric quantum systems, arising for example in the theory of Bose–Einstein condensates in PT symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy models at hand. In the present paper the stationary states of a simple toy model of a PT symmetric system previously introduced in [1, 2] are investigated. The model can be interpreted as a simple description of a Bose–Einstein condensate in a PT symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straightforward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper) 15. Particle in a box in PT-symmetric quantum mechanics and an electromagnetic analog Science.gov (United States) Dasarathy, Anirudh; Isaacson, Joshua P.; Jones-Smith, Katherine; Tabachnik, Jason; Mathur, Harsh 2013-06-01 In PT-symmetric quantum mechanics a fundamental principle of quantum mechanics, that the Hamiltonian must be Hermitian, is replaced by another set of requirements, including notably symmetry under PT, where P denotes parity and T denotes time reversal. Here we study the role of boundary conditions in PT-symmetric quantum mechanics by constructing a simple model that is the PT-symmetric analog of a particle in a box. The model has the usual particle-in-a-box Hamiltonian but boundary conditions that respect PT symmetry rather than Hermiticity. We find that for a broad class of PT-symmetric boundary conditions the model respects the condition of unbroken PT symmetry, namely, that the Hamiltonian and the symmetry operator PT have simultaneous eigenfunctions, implying that the energy eigenvalues are real. We also find that the Hamiltonian is self-adjoint under the PT-symmetric inner product. Thus we obtain a simple soluble model that fulfills all the requirements of PT-symmetric quantum mechanics. In the second part of this paper we formulate a variational principle for PT-symmetric quantum mechanics that is the analog of the textbook Rayleigh-Ritz principle. Finally we consider electromagnetic analogs of the PT-symmetric particle in a box. We show that the isolated particle in a box may be realized as a Fabry-Perot cavity between an absorbing medium and its conjugate gain medium. Coupling the cavity to an external continuum of incoming and outgoing states turns the energy levels of the box into sharp resonances. Remarkably we find that the resonances have a Breit-Wigner line shape in transmission and a Fano line shape in reflection; by contrast, in the corresponding Hermitian case the line shapes always have a Breit-Wigner form in both transmission and reflection. 16. The symmetric extendibility of quantum states International Nuclear Information System (INIS) Nowakowski, Marcin L 2016-01-01 Studies on the symmetric extendibility of quantum states have become particularly important in the context of the analysis of one-way quantum measures of entanglement, and the distillability and security of quantum protocols. In this paper we analyze composite systems containing a symmetric extendible part, with particular attention devoted to the one-way security of such systems. Further, we introduce a new one-way entanglement monotone based on the best symmetric approximation of a quantum state and the extendible number of a quantum state. We underpin these results with geometric observations about the structures of multi-party settings which posses substantial symmetric extendible components in their subspaces. The impossibility of reducing the maximal symmetric extendibility by means of the one-way local operations and classical communication method is pointed out on multiple copies. Finally, we state a conjecture linking symmetric extendibility with the one-way distillability and security of all quantum states, analyzing the behavior of a private key in the neighborhood of symmetric extendible states. (paper) 17. Averaging in spherically symmetric cosmology International Nuclear Information System (INIS) Coley, A. A.; Pelavas, N. 2007-01-01 The averaging problem in cosmology is of fundamental importance. When applied to study cosmological evolution, the theory of macroscopic gravity (MG) can be regarded as a long-distance modification of general relativity. In the MG approach to the averaging problem in cosmology, the Einstein field equations on cosmological scales are modified by appropriate gravitational correlation terms. We study the averaging problem within the class of spherically symmetric cosmological models. That is, we shall take the microscopic equations and effect the averaging procedure to determine the precise form of the correlation tensor in this case. In particular, by working in volume-preserving coordinates, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. We find that the correlation tensor in a Friedmann-Lemaitre-Robertson-Walker (FLRW) background must be of the form of a spatial curvature. Inhomogeneities and spatial averaging, through this spatial curvature correction term, can have a very significant dynamical effect on the dynamics of the Universe and cosmological observations; in particular, we discuss whether spatial averaging might lead to a more conservative explanation of the observed acceleration of the Universe (without the introduction of exotic dark matter fields). We also find that the correlation tensor for a non-FLRW background can be interpreted as the sum of a spatial curvature and an anisotropic fluid. This may lead to interesting effects of averaging on astrophysical scales. We also discuss the results of averaging an inhomogeneous Lemaitre-Tolman-Bondi solution as well as calculations of linear perturbations (that is, the backreaction) in an FLRW background, which support the main conclusions of the analysis 18. Linac design algorithm with symmetric segments International Nuclear Information System (INIS) Takeda, Harunori; Young, L.M.; Nath, S.; Billen, J.H.; Stovall, J.E. 1996-01-01 The cell lengths in linacs of traditional design are typically graded as a function of particle velocity. By making groups of cells and individual cells symmetric in both the CCDTL AND CCL, the cavity design as well as mechanical design and fabrication is simplified without compromising the performance. We have implemented a design algorithm in the PARMILA code in which cells and multi-cavity segments are made symmetric, significantly reducing the number of unique components. Using the symmetric algorithm, a sample linac design was generated and its performance compared with a similar one of conventional design 19. Symmetric nuclear matter with Skyrme interaction International Nuclear Information System (INIS) Manisa, K.; Bicer, A.; Atav, U. 2010-01-01 The equation of state (EOS) and some properties of symmetric nuclear matter, such as the saturation density, saturation energy and incompressibility, are obtained by using Skyrme's density-dependent effective nucleon-nucleon interaction. 20. Performance limitations of translationally symmetric nonimaging devices Science.gov (United States) Bortz, John C.; Shatz, Narkis E.; Winston, Roland 2001-11-01 The component of the optical direction vector along the symmetry axis is conserved for all rays propagated through a translationally symmetric optical device. This quality, referred to herein as the translational skew invariant, is analogous to the conventional skew invariant, which is conserved in rotationally symmetric optical systems. The invariance of both of these quantities is a consequence of Noether's theorem. We show how performance limits for translationally symmetric nonimaging optical devices can be derived from the distributions of the translational skew invariant for the optical source and for the target to which flux is to be transferred. Examples of computed performance limits are provided. In addition, we show that a numerically optimized non-tracking solar concentrator utilizing symmetry-breaking surface microstructure can overcome the performance limits associated with translational symmetry. The optimized design provides a 47.4% increase in efficiency and concentration relative to an ideal translationally symmetric concentrator. 1. Symmetrical parahiliar infiltrated, cough and dyspnoea International Nuclear Information System (INIS) 2004-01-01 It is the case a patient to who is diagnosed symmetrical parahiliar infiltrated; initially she is diagnosed lymphoma Hodgkin, treaty with radiotherapy and chemotherapy, but the X rays of the thorax demonstrated parahiliars and paramediastinals infiltrated 2. Introduction to left-right symmetric models International Nuclear Information System (INIS) Grimus, W. 1993-01-01 We motivate left-right symmetric models by the possibility of spontaneous parity breaking. Then we describe the multiplets and the Lagrangian of such models. Finally we discuss lower bounds on the right-handed scale. (author) 3. A cosmological problem for maximally symmetric supergravity International Nuclear Information System (INIS) German, G.; Ross, G.G. 1986-01-01 Under very general considerations it is shown that inflationary models of the universe based on maximally symmetric supergravity with flat potentials are unable to resolve the cosmological energy density (Polonyi) problem. (orig.) 4. Theorem on axially symmetric gravitational vacuum configurations Energy Technology Data Exchange (ETDEWEB) Papadopoulos, A; Le Denmat, G [Paris-6 Univ., 75 (France). Inst. Henri Poincare 1977-01-24 A theorem is proved which asserts the non-existence of axially symmetric gravitational vacuum configurations with non-stationary rotation only. The eventual consequences in black-hole physics are suggested. 5. When push comes to shove: Exclusion processes with nonlocal consequences Science.gov (United States) Almet, Axel A.; Pan, Michael; Hughes, Barry D.; Landman, Kerry A. 2015-11-01 Stochastic agent-based models are useful for modelling collective movement of biological cells. Lattice-based random walk models of interacting agents where each site can be occupied by at most one agent are called simple exclusion processes. An alternative motility mechanism to simple exclusion is formulated, in which agents are granted more freedom to move under the compromise that interactions are no longer necessarily local. This mechanism is termed shoving. A nonlinear diffusion equation is derived for a single population of shoving agents using mean-field continuum approximations. A continuum model is also derived for a multispecies problem with interacting subpopulations, which either obey the shoving rules or the simple exclusion rules. Numerical solutions of the derived partial differential equations compare well with averaged simulation results for both the single species and multispecies processes in two dimensions, while some issues arise in one dimension for the multispecies case. 6. Symmetric Imidazolium-Based Paramagnetic Ionic Liquids Science.gov (United States) 2017-11-29 Charts N/A Unclassified Unclassified Unclassified SAR 14 Kamran Ghiassi N/A 1 Symmetric Imidazolium-Based Paramagnetic Ionic Liquids Kevin T. Greeson...NUMBER (Include area code) 29 November 2017 Briefing Charts 01 November 2017 - 30 November 2017 Symmetric Imidazolium-Based Paramagnetic Ionic ... Liquids K. Greeson, K. Ghiassi, J. Alston, N. Redeker, J. Marcischak, L. Gilmore, A. Guenthner Air Force Research Laboratory (AFMC) AFRL/RQRP 9 Antares 7. Pion condensation in symmetric nuclear matter International Nuclear Information System (INIS) Kabir, K.; Saha, S.; Nath, L.M. 1987-09-01 Using a model which is based essentially on the chiral SU(2)xSU(2) symmetry of the pion-nucleon interaction, we examine the possibility of pion condensation in symmetric nucleon matter. We find that the pion condensation is not likely to occur in symmetric nuclear matter for any finite value of the nuclear density. Consequently, no critical opalescence phenomenon is expected to be seen in the pion-nucleus interaction. (author). 20 refs 8. Pion condensation in symmetric nuclear matter Science.gov (United States) Kabir, K.; Saha, S.; Nath, L. M. 1988-01-01 Using a model which is based essentially on the chiral SU(2)×SU(2) symmetry of the pion-nucleon interaction, we examine the possibility of pion condensation in symmetric nucleon matter. We find that the pion condensation is not likely to occur in symmetric nuclear matter for any finite value of the nuclear density. Consequently, no critical opalescence phenomenom is expected to be seen in the pion-nucleus interaction. 9. Marginal Stability Diagrams for Infinite-n Ballooning Modes in Quasi-symmetric Stellarators International Nuclear Information System (INIS) Hudson, S.R.; Hegna, C.C.; Torasso, R.; Ware, A. 2003-01-01 By perturbing the pressure and rotational-transform profiles at a selected surface in a given equilibrium, and by inducing a coordinate variation such that the perturbed state is in equilibrium, a family of magnetohydrodynamic equilibria local to the surface and parameterized by the pressure gradient and shear is constructed for arbitrary stellarator geometry. The geometry of the surface is not changed. The perturbed equilibria are analyzed for infinite-n ballooning stability and marginal stability diagrams are constructed that are analogous to the (s; alpha) diagrams constructed for axi-symmetric configurations. The method describes how pressure and rotational-transform gradients influence the local shear, which in turn influences the ballooning stability. Stability diagrams for the quasi-axially-symmetric NCSX (National Compact Stellarator Experiment), a quasi-poloidally-symmetric configuration and the quasi-helically-symmetric HSX (Helically Symmetric Experiment) are presented. Regions of second-stability are observed in both NCSX and the quasi-poloidal configuration, whereas no second stable region is observed for the quasi-helically symmetric device. To explain the different regions of stability, the curvature and local shear of the quasi-poloidal configuration are analyzed. The results are seemingly consistent with the simple explanation: ballooning instability results when the local shear is small in regions of bad curvature. Examples will be given that show that the structure, and stability, of the ballooning mode is determined by the structure of the potential function arising in the Schroedinger form of the ballooning equation 10. Generalized exclusion and Hopf algebras International Nuclear Information System (INIS) Yildiz, A 2002-01-01 We propose a generalized oscillator algebra at the roots of unity with generalized exclusion and we investigate the braided Hopf structure. We find that there are two solutions: these are the generalized exclusions of the bosonic and fermionic types. We also discuss the covariance properties of these oscillators 11. Evaluating Alternatives to Exclusive "He." Science.gov (United States) Todd-Mancillas, William R. A study was conducted to determine the effects on reading comprehension of the use of the exclusive pronoun "he" and more or less contrived alternatives. Subjects, 358 students enrolled in an introduction to human communication at a large northeastern university, read three different forms of the same essay. One essay form exclusively used "he,"… 12. The Pauli exclusion principle origin, verifications and applications CERN Document Server Kaplan, Ilya G 2017-01-01 This is the first scientific book devoted to the Pauli Exclusion Principle, which is a fundamental principle of quantum mechanics and is permanently applied in chemistry, physics, molecular biology and in physical astronomy. However, while the principle has been studied for more than 90 years, rigorous theoretical foundations still have not been established and many unsolved problems remain. Following an introduction and historical survey, this book discusses the still unresolved questions around this fundamental principle. For instance, why, according to the Pauli Exclusion Principle, are only symmetric and antisymmetric permutation symmetries for identical particles realized, while the Schrödinger equation is satisfied by functions with any permutation symmetry? Chapter 3 covers possible answers to this, while chapter 4 presents effective and elegant methods for finding the Pauli-allowed states in atomic, molecular and nuclear spectroscopy. Chapter 5 discusses parastatistics and fractional statistics, dem... 13. The Point Zoro Symmetric Single-Step Procedure for Simultaneous Estimation of Polynomial Zeros Directory of Open Access Journals (Sweden) Mansor Monsi 2012-01-01 Full Text Available The point symmetric single step procedure PSS1 has R-order of convergence at least 3. This procedure is modified by adding another single-step, which is the third step in PSS1. This modified procedure is called the point zoro symmetric single-step PZSS1. It is proven that the R-order of convergence of PZSS1 is at least 4 which is higher than the R-order of convergence of PT1, PS1, and PSS1. Hence, computational time is reduced since this procedure is more efficient for bounding simple zeros simultaneously. 14. Crossing-symmetric solutions to low equations International Nuclear Information System (INIS) McLeod, R.J.; Ernst, D.J. 1985-01-01 Crossing symmetric models of the pion-nucleon interaction in which crossing symmetry is kept to lowest order in msub(π)/msub(N) are investigated. Two iterative techniques are developed to solve the crossing-symmetric Low equation. The techniques are used to solve the original Chew-Low equations and their generalizations to include the coupling to the pion-production channels. Small changes are found in comparison with earlier results which used an iterative technique proposed by Chew and Low and which did not produce crossing-symmetric results. The iterative technique of Chew and Low is shown to fail because of its inability to produce zeroes in the amplitude at complex energies while physical solutions to the model require such zeroes. We also prove that, within the class of solutions such that phase shifts approach zero for infinite energy, the solution to the Low equation is unique. (orig.) 15. PT symmetric Aubry–Andre model International Nuclear Information System (INIS) Yuce, C. 2014-01-01 PT symmetric Aubry–Andre model describes an array of N coupled optical waveguides with position-dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of quasi-periodicity for small number of lattice sites. We obtain the Hofstadter butterfly spectrum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserve the total intensity. - Highlights: • We show that PT symmetric Aubry–Andre model may have real spectrum. • We show that the reality of the spectrum depends sensitively on the degree of disorder. • We obtain the Hofstadter butterfly spectrum for PT symmetric Aubry–Andre model. • We discuss that phase transition from extended to localized states exists 16. PT symmetric Aubry–Andre model Energy Technology Data Exchange (ETDEWEB) 2014-06-13 PT symmetric Aubry–Andre model describes an array of N coupled optical waveguides with position-dependent gain and loss. We show that the reality of the spectrum depends sensitively on the degree of quasi-periodicity for small number of lattice sites. We obtain the Hofstadter butterfly spectrum and discuss the existence of the phase transition from extended to localized states. We show that rapidly changing periodical gain/loss materials almost conserve the total intensity. - Highlights: • We show that PT symmetric Aubry–Andre model may have real spectrum. • We show that the reality of the spectrum depends sensitively on the degree of disorder. • We obtain the Hofstadter butterfly spectrum for PT symmetric Aubry–Andre model. • We discuss that phase transition from extended to localized states exists. 17. All-optical symmetric ternary logic gate Science.gov (United States) 2010-09-01 Symmetric ternary number (radix=3) has three logical states (1¯, 0, 1). It is very much useful in carry free arithmetical operation. Beside this, the logical operation using this type of number system is also effective in high speed computation and communication in multi-valued logic. In this literature all-optical circuits for three basic symmetrical ternary logical operations (inversion, MIN and MAX) are proposed and described. Numerical simulation verifies the theoretical model. In this present scheme the different ternary logical states are represented by different polarized state of light. Terahertz optical asymmetric demultiplexer (TOAD) based interferometric switch has been used categorically in this manuscript. 18. Symmetry theorems via the continuous steiner symmetrization Directory of Open Access Journals (Sweden) L. Ragoub 2000-06-01 Full Text Available Using a new approach due to F. Brock called the Steiner symmetrization, we show first that if $u$ is a solution of an overdetermined problem in the divergence form satisfying the Neumann and non-constant Dirichlet boundary conditions, then $Omega$ is an N-ball. In addition, we show that we can relax the condition on the value of the Dirichlet boundary condition in the case of superharmonicity. Finally, we give an application to positive solutions of some semilinear elliptic problems in symmetric domains for the divergence case. 19. The Axially Symmetric One-Monopole International Nuclear Information System (INIS) Wong, K.-M.; Teh, Rosy 2009-01-01 We present new classical generalized one-monopole solution of the SU(2) Yang-Mills-Higgs theory with the Higgs field in the adjoint representation. We show that this solution with θ-winding number m = 1 and φ-winding number n = 1 is an axially symmetric generalization of the 't Hooft-Polyakov one-monopole. We construct this axially symmetric one-monopole solution by generalizing the large distance asymptotic solutions of the 't Hooft-Polyakov one-monopole to the Jacobi elliptic functions and solving the second order equations of motion numerically when the Higgs potential is vanishing. This solution is a non-BPS solution. 20. Symmetric splitting of very light systems International Nuclear Information System (INIS) Grotowski, K.; Majka, Z.; Planeta, R. 1984-01-01 Inclusive and coincidence measurements have been performed to study symmetric products from the reactions 74--186 MeV 12 C+ 40 Ca, 141 MeV 9 Be+ 40 Ca, and 153 MeV 6 Li+ 40 Ca. The binary decay of the composite system has been verified. Energy spectra, angular distributions, and fragment correlations are presented. The total kinetic energies for the symmetric products from these very light composite systems are compared to liquid drop model calculations and fission systematics 1. The theory of spherically symmetric thin shells in conformal gravity Science.gov (United States) Berezin, Victor; Dokuchaev, Vyacheslav; Eroshenko, Yury The spherically symmetric thin shells are the nearest generalizations of the point-like particles. Moreover, they serve as the simple sources of the gravitational fields both in General Relativity and much more complex quadratic gravity theories. We are interested in the special and physically important case when all the quadratic in curvature tensor (Riemann tensor) and its contractions (Ricci tensor and scalar curvature) terms are present in the form of the square of Weyl tensor. By definition, the energy-momentum tensor of the thin shell is proportional to Diracs delta-function. We constructed the theory of the spherically symmetric thin shells for three types of gravitational theories with the shell: (1) General Relativity; (2) Pure conformal (Weyl) gravity where the gravitational part of the total Lagrangian is just the square of the Weyl tensor; (3) Weyl-Einstein gravity. The results are compared with these in General Relativity (Israel equations). We considered in detail the shells immersed in the vacuum. Some peculiar properties of such shells are found. In particular, for the traceless ( = massless) shell, it is shown that their dynamics cannot be derived from the matching conditions and, thus, is completely arbitrary. On the contrary, in the case of the Weyl-Einstein gravity, the trajectory of the same type of shell is completely restored even without knowledge of the outside solution. 2. Thermoacoustic focusing lens by symmetric Airy beams with phase manipulations Science.gov (United States) Liu, Chen; Xia, Jian-Ping; Sun, Hong-Xiang; Yuan, Shou-Qi 2017-12-01 We report the realization of broadband acoustic focusing lenses based on two symmetric thermoacoustic phased arrays of Airy beams, in which the units of thermoacoustic phase control are designed by employing air with different temperatures surrounded by rigid insulated boundaries and thermal insulation films. The phase delays of the transmitted and reflected units could cover a whole 2π interval, which arises from the change of the sound velocity of air induced by the variation of the temperature. Based on the units of phase control, we design the transmitted and reflected acoustic focusing lenses with two symmetric Airy beams, and verify the high self-healing focusing characteristic and the feasibility of the thermal insulation films. Besides, the influences of the bending angle of the Airy beam on the focusing performance are discussed in detail. The proposed acoustic lens has advantages of broad bandwidth (about 4.8 kHz), high focusing performance, self-healing feature, and simple structure, which enable it to provide more schemes for acoustic focusing. It has excellent potential applications in acoustic devices. 3. Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion International Nuclear Information System (INIS) 2011-01-01 Consider a non-symmetric generalized diffusion X(⋅) in ℝ d determined by the differential operator A(x) = -Σ ij ∂ i a ij (x)∂ j + Σ i b i (x)∂ i . In this paper the diffusion process is approximated by Markov jump processes X n (⋅), in homogeneous and isotropic grids G n ⊂ℝ d , which converge in distribution in the Skorokhod space D([0,∞),ℝ d ) to the diffusion X(⋅). The generators of X n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d≥3 can be applied to processes for which the diffusion tensor {a ij (x)} 11 dd fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X n (⋅). For piece-wise constant functions a ij on ℝ d and piece-wise continuous functions a ij on ℝ 2 the construction and principal algorithm are described enabling an easy implementation into a computer code. 4. Spherical aberration correction with threefold symmetric line currents. Science.gov (United States) Hoque, Shahedul; Ito, Hiroyuki; Nishi, Ryuji; Takaoka, Akio; Munro, Eric 2016-02-01 It has been shown that N-fold symmetric line current (henceforth denoted as N-SYLC) produces 2N-pole magnetic fields. In this paper, a threefold symmetric line current (N3-SYLC in short) is proposed for correcting 3rd order spherical aberration of round lenses. N3-SYLC can be realized without using magnetic materials, which makes it free of the problems of hysteresis, inhomogeneity and saturation. We investigate theoretically the basic properties of an N3-SYLC configuration which can in principle be realized by simple wires. By optimizing the parameters of a system with beam energy of 5.5keV, the required excitation current for correcting 3rd order spherical aberration coefficient of 400 mm is less than 1AT, and the residual higher order aberrations can be kept sufficiently small to obtain beam size of less than 1 nm for initial slopes up to 5 mrad. Copyright © 2015 Elsevier B.V. All rights reserved. 5. Problems of Chernobyl Exclusion Zone International Nuclear Information System (INIS) Kholosha, V.Yi. 2014-01-01 The collection comprises the results of researches and design activity in the ChNPP exclusion zone, aimed at the development of technologies, equipment and devices for radioactive waste management and ChNPP accident clean-up, at studying the composition and structure of the Exclusion zone soil activity solid bearers, form transformation of the fission products of fuel fallout radionuclide composition in the ChNPP near zone, the spatial distribution of radionuclides and other radioecological issues.. Much attention is paid to medical and biological aspects of the accident influence on the flora, fauna and people's health, labour conditions and incidence of the workers of the Exclusion zone 6. Small diameter symmetric networks from linear groups Science.gov (United States) Campbell, Lowell; Carlsson, Gunnar E.; Dinneen, Michael J.; Faber, Vance; Fellows, Michael R.; Langston, Michael A.; Moore, James W.; Multihaupt, Andrew P.; Sexton, Harlan B. 1992-01-01 In this note is reported a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter. Some of the constructions are in the range of current potential engineering significance. The constructions are Cayley graphs of linear groups obtained by experimental computation. 7. Sobolev spaces on bounded symmetric domains Czech Academy of Sciences Publication Activity Database Engliš, Miroslav Roč. 60, č. 12 ( 2015 ), s. 1712-1726 ISSN 1747-6933 Institutional support: RVO:67985840 Keywords : bounded symmetric domain * Sobolev space * Bergman space Subject RIV: BA - General Mathematics Impact factor: 0.466, year: 2015 http://www.tandfonline.com/doi/abs/10.1080/17476933. 2015 .1043910 8. Cuspidal discrete series for semisimple symmetric spaces DEFF Research Database (Denmark) Andersen, Nils Byrial; Flensted-Jensen, Mogens; Schlichtkrull, Henrik 2012-01-01 We propose a notion of cusp forms on semisimple symmetric spaces. We then study the real hyperbolic spaces in detail, and show that there exists both cuspidal and non-cuspidal discrete series. In particular, we show that all the spherical discrete series are non-cuspidal. (C) 2012 Elsevier Inc. All... 9. Exact solutions of the spherically symmetric multidimensional ... African Journals Online (AJOL) The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic ... 10. Super-symmetric informationally complete measurements Energy Technology Data Exchange (ETDEWEB) Zhu, Huangjun, E-mail: [email protected] 2015-11-15 Symmetric informationally complete measurements (SICs in short) are highly symmetric structures in the Hilbert space. They possess many nice properties which render them an ideal candidate for fiducial measurements. The symmetry of SICs is intimately connected with the geometry of the quantum state space and also has profound implications for foundational studies. Here we explore those SICs that are most symmetric according to a natural criterion and show that all of them are covariant with respect to the Heisenberg–Weyl groups, which are characterized by the discrete analog of the canonical commutation relation. Moreover, their symmetry groups are subgroups of the Clifford groups. In particular, we prove that the SIC in dimension 2, the Hesse SIC in dimension 3, and the set of Hoggar lines in dimension 8 are the only three SICs up to unitary equivalence whose symmetry groups act transitively on pairs of SIC projectors. Our work not only provides valuable insight about SICs, Heisenberg–Weyl groups, and Clifford groups, but also offers a new approach and perspective for studying many other discrete symmetric structures behind finite state quantum mechanics, such as mutually unbiased bases and discrete Wigner functions. 11. Harmonic maps of the bounded symmetric domains International Nuclear Information System (INIS) Xin, Y.L. 1994-06-01 A shrinking property of harmonic maps into R IV (2) is proved which is used to classify complete spacelike surfaces of the parallel mean curvature in R 4 2 with a reasonable condition on the Gauss image. Liouville-type theorems of harmonic maps from the higher dimensional bounded symmetric domains are also established. (author). 25 refs 12. On isotropic cylindrically symmetric stellar models International Nuclear Information System (INIS) Nolan, Brien C; Nolan, Louise V 2004-01-01 We attempt to match the most general cylindrically symmetric vacuum spacetime with a Robertson-Walker interior. The matching conditions show that the interior must be dust filled and that the boundary must be comoving. Further, we show that the vacuum region must be polarized. Imposing the condition that there are no trapped cylinders on an initial time slice, we can apply a result of Thorne's and show that trapped cylinders never evolve. This results in a simplified line element which we prove to be incompatible with the dust interior. This result demonstrates the impossibility of the existence of an isotropic cylindrically symmetric star (or even a star which has a cylindrically symmetric portion). We investigate the problem from a different perspective by looking at the expansion scalars of invariant null geodesic congruences and, applying to the cylindrical case, the result that the product of the signs of the expansion scalars must be continuous across the boundary. The result may also be understood in relation to recent results about the impossibility of the static axially symmetric analogue of the Einstein-Straus model 13. The Mathematics of Symmetrical Factorial Designs The Mathematics of Symmetrical Factorial Designs. Mausumi Bose (nee Sen) obtained her MSc degree in. Statistics from the Calcutta. University and PhD degree from the Indian Statistical. Institute. She is on the faculty of the Indian. Statistical Institute. Her main field of research interest is design and analysis of experiments. 14. Symmetric intersections of Rauzy fractals \| Sellami \| Quaestiones ... African Journals Online (AJOL) In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is re ection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is ... 15. Fourier inversion on a reductive symmetric space NARCIS (Netherlands) Ban, E.P. van den 1999-01-01 Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we 16. A viewpoint on nearly conformally symmetric manifold International Nuclear Information System (INIS) Rahman, M.S. 1990-06-01 Some observations, with definition, on Nearly Conformally Symmetric (NCS) manifold are made. A number of theorems concerning conformal change of metric and parallel tensors on NCS manifolds are presented. It is illustrated that a manifold M = R n-1 x R + 1 , endowed with a special metric, is NCS but not of harmonic curvature. (author). 8 refs 17. Harmonic analysis on reductive symmetric spaces NARCIS (Netherlands) Ban, E.P. van den; Schlichtkrull, H. 2000-01-01 We give a relatively non-technical survey of some recent advances in the Fourier theory for semisimple symmetric spaces. There are three major results: An inversion formula for the Fourier transform, a Palley-Wiener theorem, which describes the Fourier image of the space of completely supported 18. Fourier transforms on a semisimple symmetric space NARCIS (Netherlands) Ban, E.P. van den; Schlichtkrull, H. 1994-01-01 Let G=H be a semisimple symmetric space, that is, G is a connected semisimple real Lie group with an involution ?, and H is an open subgroup of the group of xed points for ? in G. The main purpose of this paper is to study an explicit Fourier transform on G=H. In terms of general representation 19. Fourier transforms on a semisimple symmetric space NARCIS (Netherlands) Ban, E.P. van den; Carmona, J.; Delorme, P. 1997-01-01 Let G=H be a semisimple symmetric space, that is, G is a connected semisimple real Lie group with an involution ?, and H is an open subgroup of the group of xed points for ? in G. The main purpose of this paper is to study an explicit Fourier transform on G=H. In terms of general representation 20. Capacity Bounds and Mapping Design for Binary Symmetric Relay Channels Directory of Open Access Journals (Sweden) Majid Nasiri Khormuji 2012-12-01 Full Text Available Capacity bounds for a three-node binary symmetric relay channel with orthogonal components at the destination are studied. The cut-set upper bound and the rates achievable using decode-and-forward (DF, partial DF and compress-and-forward (CF relaying are first evaluated. Then relaying strategies with finite memory-length are considered. An efficient algorithm for optimizing the relay functions is presented. The Boolean Fourier transform is then employed to unveil the structure of the optimized mappings. Interestingly, the optimized relay functions exhibit a simple structure. Numerical results illustrate that the rates achieved using the optimized low-dimensional functions are either comparable to those achieved by CF or superior to those achieved by DF relaying. In particular, the optimized low-dimensional relaying scheme can improve on DF relaying when the quality of the source-relay link is worse than or comparable to that of other links. 1. Waterbomb base: a symmetric single-vertex bistable origami mechanism International Nuclear Information System (INIS) Hanna, Brandon H; Lund, Jason M; Magleby, Spencer P; Howell, Larry L; Lang, Robert J 2014-01-01 The origami waterbomb base is a single-vertex bistable origami mechanism that has unique properties which may prove useful in a variety of applications. It also shows promise as a test bed for smart materials and actuation because of its straightforward geometry and multiple phases of motion, ranging from simple to more complex. This study develops a quantitative understanding of the symmetric waterbomb base's kinetic behavior. This is done by completing kinematic and potential energy analyses to understand and predict bistable behavior. A physical prototype is constructed and tested to validate the results of the analyses. Finite element and virtual work analyses based on the prototype are used to explore the locations of the stable equilibrium positions and the force–deflection response. The model results are verified through comparisons to measurements on a physical prototype. The resulting models describe waterbomb base behavior and provide an engineering tool for application development. (paper) 2. Exclusive Rights and State Aid DEFF Research Database (Denmark) Ølykke, Grith Skovgaard 2017-01-01 Exclusive rights are granted in order to regulate markets as one of several possible tools of public intervention. The article considers the role of State aid law in the regulation of exclusive rights. Whereas the right of Member States to organise markets as monopolies and the choice of provider...... are regulated by free movement rules and Article 106 TFEU, State aid law regulates the terms of the right to ensure that the beneficiary is not granted an economic advantage. Exclusive rights may be granted on various terms: for a payment, in combination with compensation or as compensation. The two former...... kinds of terms are regulated under State aid law which requires market terms. The granting of exclusive rights as compensation is analysed on the basis of the Eventech judgment, and it is found that when no financial transaction is included in the grant, it resembles a decision to organise a market... 3. Exclusive processes at Jefferson Lab There is no clear guidance from theory as to the limits of the transition region; .... behavior in exclusive photoreactions with hadrons in the final state at large t may provide .... The planned medium acceptance detector (MAD) system in Hall A. 4. Central Exclusive Production at LHCb CERN Document Server INSPIRE-00106463 2015-01-01 Central Exclusive Production is a unique QCD process in which particles are produced via colourless propagators. Several results have been obtained at LHCb for the production of single charmonia, pairs of charmonia, and single bottomonia. 5. Central Exclusive Production at LHCb CERN Document Server AUTHOR\|(INSPIRE)INSPIRE-00392425 2017-01-01 The LHCb detector, with its excellent momentum resolution and flexible trigger strategy, is ideally suited for measuring particles produced exclusively. In addition, a new system of forward shower counters has been installed upstream and downstream of the detector, and has been used to facilitate studies of Central Exclusive Production. Such measurements of integrated and differential cross-section in both Run 1 and Run 2 of the LHC, are summarised here. 6. Exclusive Territories and Manufacturers’ Collusion OpenAIRE Salvatore Piccolo; Markus Reisinger 2010-01-01 This paper highlights the rationale for exclusive territories in a model of repeated interaction between competing supply chains. We show that with observable contracts exclusive territories have two countervailing effects on manufacturers' incentives to sustain tacit collusion. First, granting local monopolies to retailers distributing a given brand softens inter- and intrabrand competition in a one-shot game. Hence, punishment profits are larger, thereby rendering deviation more profitable.... 7. Exclusion statistics and integrable models International Nuclear Information System (INIS) Mashkevich, S. 1998-01-01 The definition of exclusion statistics, as given by Haldane, allows for a statistical interaction between distinguishable particles (multi-species statistics). The thermodynamic quantities for such statistics ca be evaluated exactly. The explicit expressions for the cluster coefficients are presented. Furthermore, single-species exclusion statistics is realized in one-dimensional integrable models. The interesting questions of generalizing this correspondence onto the higher-dimensional and the multi-species cases remain essentially open 8. Weakly Interacting Symmetric and Anti-Symmetric States in the Bilayer Systems Science.gov (United States) Marchewka, M.; Sheregii, E. M.; Tralle, I.; Tomaka, G.; Ploch, D. We have studied the parallel magneto-transport in DQW-structures of two different potential shapes: quasi-rectangular and quasi-triangular. The quantum beats effect was observed in Shubnikov-de Haas (SdH) oscillations for both types of the DQW structures in perpendicular magnetic filed arrangement. We developed a special scheme for the Landau levels energies calculation by means of which we carried out the necessary simulations of beating effect. In order to obtain the agreement between our experimental data and the results of simulations, we introduced two different quasi-Fermi levels which characterize symmetric and anti-symmetric states in DQWs. The existence of two different quasi Fermi-Levels simply means, that one can treat two sub-systems (charge carriers characterized by symmetric and anti-symmetric wave functions) as weakly interacting and having their own rate of establishing the equilibrium state. 9. A simple application of the Newman-Penrose spin coefficient formalism International Nuclear Information System (INIS) Davis, T.M. 1976-01-01 As a simple application of the Newman-Penrose spin coefficient formalism, useful for beginners, the vacuum symmetry (Schwarzschild) solution is found. The calculations also show that all spherically symmetric metrics are Petrov type D. (author) 10. Dynamics of non-Markovian exclusion processes International Nuclear Information System (INIS) Khoromskaia, Diana; Grosskinsky, Stefan; Harris, Rosemary J 2014-01-01 Driven diffusive systems are often used as simple discrete models of collective transport phenomena in physics, biology or social sciences. Restricting attention to one-dimensional geometries, the asymmetric simple exclusion process (ASEP) plays a paradigmatic role to describe noise-activated driven motion of entities subject to an excluded volume interaction and many variants have been studied in recent years. While in the standard ASEP the noise is Poissonian and the process is therefore Markovian, in many applications the statistics of the activating noise has a non-standard distribution with possible memory effects resulting from internal degrees of freedom or external sources. This leads to temporal correlations and can significantly affect the shape of the current-density relation as has been studied recently for a number of scenarios. In this paper we report a general framework to derive the fundamental diagram of ASEPs driven by non-Poissonian noise by using effectively only two simple quantities, viz., the mean residual lifetime of the jump distribution and a suitably defined temporal correlation length. We corroborate our results by detailed numerical studies for various noise statistics under periodic boundary conditions and discuss how our approach can be applied to more general driven diffusive systems. (paper) 11. Dynamics of non-Markovian exclusion processes Science.gov (United States) Khoromskaia, Diana; Harris, Rosemary J.; Grosskinsky, Stefan 2014-12-01 Driven diffusive systems are often used as simple discrete models of collective transport phenomena in physics, biology or social sciences. Restricting attention to one-dimensional geometries, the asymmetric simple exclusion process (ASEP) plays a paradigmatic role to describe noise-activated driven motion of entities subject to an excluded volume interaction and many variants have been studied in recent years. While in the standard ASEP the noise is Poissonian and the process is therefore Markovian, in many applications the statistics of the activating noise has a non-standard distribution with possible memory effects resulting from internal degrees of freedom or external sources. This leads to temporal correlations and can significantly affect the shape of the current-density relation as has been studied recently for a number of scenarios. In this paper we report a general framework to derive the fundamental diagram of ASEPs driven by non-Poissonian noise by using effectively only two simple quantities, viz., the mean residual lifetime of the jump distribution and a suitably defined temporal correlation length. We corroborate our results by detailed numerical studies for various noise statistics under periodic boundary conditions and discuss how our approach can be applied to more general driven diffusive systems. 12. Women in Chernobyl Exclusion Zone International Nuclear Information System (INIS) Balashevska, Y.; Kireev, S.; Navalikhin, V. 2015-01-01 Today, 29 years after the Chernobyl accident, the Exclusion Zone still remains an areal unsealed radiation source of around 2600 km"2. It is not just a gigantic radioactive waste storage facility (the amount of radioactive waste accumulated within the Zone, except for the Shelter, is estimated at about 2.8 million m"3), but also a unique research and engineering platform for biologists, radiologists, chemists and physicists. Taking into account the amount of the radionuclides released during the accident, it becomes quite understood that the radiological environment in the Exclusion Zone is far from favorable. However, among the Exclusion Zone personnel who numbers 5000, there are female workers. The poster represents the results of the research performed among the female employees of the largest enterprise of the Exclusion Zone, “Chornobyl Spetskombinat”. The survey was performed with the view to knowing what makes women work in the most radioactively contaminated area in Europe, and what their role is, to revealing their fears and hopes, and to estimating the chances of the brave women of Chernobyl Exclusion Zone to succeed in their careers. (author) 13. RELIGIOUS EXCLUSIVITY AND PSYCHOSOCIAL FUNCTIONING. Science.gov (United States) Gegelashvili, M; Meca, A; Schwartz, S J 2015-01-01 In the present study we sought to clarify links between religious exclusivity, as form of intergroup favoritism, and indices of psychosocial functioning. The study of in group favoritism has generally been invoked within Social Identity Theory and related perspectives. However, there is a lack of literature regarding religious exclusivity from the standpoint of social identity. In particular, the ways in which religious exclusivity is linked with other dimensions of religious belief and practice, and with psychosocial functioning, among individuals from different religious backgrounds are not well understood. A sample of 8545 emerging-adult students from 30 U.S. universities completed special measures. Measure of religious exclusivity was developed and validated for this group. The results suggest that exclusivity appears as predictor for impaired psychosocial functioning, low self-esteem and low psychosocial well-being for individuals from organized faiths, as well as for those identifying as agnostic, atheist, or spiritual/nonreligious. These findings are discussed in terms of Social Identity Theory and Terror Management Theory (TMT). 14. Representations of the infinite symmetric group CERN Document Server Borodin, Alexei 2016-01-01 Representation theory of big groups is an important and quickly developing part of modern mathematics, giving rise to a variety of important applications in probability and mathematical physics. This book provides the first concise and self-contained introduction to the theory on the simplest yet very nontrivial example of the infinite symmetric group, focusing on its deep connections to probability, mathematical physics, and algebraic combinatorics. Following a discussion of the classical Thoma's theorem which describes the characters of the infinite symmetric group, the authors describe explicit constructions of an important class of representations, including both the irreducible and generalized ones. Complete with detailed proofs, as well as numerous examples and exercises which help to summarize recent developments in the field, this book will enable graduates to enhance their understanding of the topic, while also aiding lecturers and researchers in related areas. 15. Symmetric, discrete fractional splines and Gabor systems DEFF Research Database (Denmark) Søndergaard, Peter Lempel 2006-01-01 In this paper we consider fractional splines as windows for Gabor frames. We introduce two new types of symmetric, fractional splines in addition to one found by Unser and Blu. For the finite, discrete case we present two families of splines: One is created by sampling and periodizing the continu......In this paper we consider fractional splines as windows for Gabor frames. We introduce two new types of symmetric, fractional splines in addition to one found by Unser and Blu. For the finite, discrete case we present two families of splines: One is created by sampling and periodizing...... the continuous splines, and one is a truly finite, discrete construction. We discuss the properties of these splines and their usefulness as windows for Gabor frames and Wilson bases.... 16. Overlap-free symmetric D 0 Lwords Directory of Open Access Journals (Sweden) Anna Frid 2001-12-01 Full Text Available A D0L word on an alphabet Σ={0,1,…,q-1} is called symmetric if it is a fixed point w=φ(w of a morphism φ:Σ * → Σ * defined by φ(i= t 1 + i t 2 + i … t m + i for some word t 1 t 2 … t m (equal to φ(0 and every i ∈ Σ; here a means a mod q. We prove a result conjectured by J. Shallit: if all the symbols in φ(0 are distinct (i.e., if t i ≠ t j for i ≠ j, then the symmetric D0L word w is overlap-free, i.e., contains no factor of the form axaxa for any x ∈ Σ * and a ∈ Σ. 17. Young—Capelli symmetrizers in superalgebras† Science.gov (United States) Brini, Andrea; Teolis, Antonio G. B. 1989-01-01 Let Supern[U [unk] V] be the nth homogeneous subspace of the supersymmetric algebra of U [unk] V, where U and V are Z2-graded vector spaces over a field K of characteristic zero. The actions of the general linear Lie superalgebras pl(U) and pl(V) span two finite-dimensional K-subalgebras B and [unk] of EndK(Supern[U [unk] V]) that are the centralizers of each other. Young—Capelli symmetrizers and Young—Capelli -symmetrizers give rise to K-linear bases of B and [unk] containing orthogonal systems of idempotents; thus they yield complete decompositions of B and [unk] into minimal left and right ideals, respectively. PMID:16594014 18. Factored Facade Acquisition using Symmetric Line Arrangements KAUST Repository Ceylan, Duygu 2012-05-01 We introduce a novel framework for image-based 3D reconstruction of urban buildings based on symmetry priors. Starting from image-level edges, we generate a sparse and approximate set of consistent 3D lines. These lines are then used to simultaneously detect symmetric line arrangements while refining the estimated 3D model. Operating both on 2D image data and intermediate 3D feature representations, we perform iterative feature consolidation and effective outlier pruning, thus eliminating reconstruction artifacts arising from ambiguous or wrong stereo matches. We exploit non-local coherence of symmetric elements to generate precise model reconstructions, even in the presence of a significant amount of outlier image-edges arising from reflections, shadows, outlier objects, etc. We evaluate our algorithm on several challenging test scenarios, both synthetic and real. Beyond reconstruction, the extracted symmetry patterns are useful towards interactive and intuitive model manipulations. 19. Commutative curvature operators over four-dimensional generalized symmetric Directory of Open Access Journals (Sweden) 2014-12-01 Full Text Available Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds. 20. Strangeness Suppression of qq ¯ Creation Observed in Exclusive Reactions Science.gov (United States) Mestayer, M. D.; Park, K.; Adhikari, K. P.; Aghasyan, M.; Pereira, S. Anefalos; Ball, J.; Battaglieri, M.; Batourine, V.; Bedlinskiy, I.; Biselli, A. S.; Boiarinov, S.; Briscoe, W. J.; Brooks, W. K.; Burkert, V. D.; Carman, D. S.; Celentano, A.; Chandavar, S.; Charles, G.; Colaneri, L.; Cole, P. L.; Contalbrigo, M.; Cortes, O.; Crede, V.; D'Angelo, A.; Dashyan, N.; De Vita, R.; Deur, A.; Djalali, C.; Doughty, D.; Dupre, R.; Alaoui, A. El; Fassi, L. El; Elouadrhiri, L.; Eugenio, P.; Fedotov, G.; Fleming, J. A.; Forest, T. A.; Garillon, B.; Garçon, M.; Ghandilyan, Y.; Gilfoyle, G. P.; Giovanetti, K. L.; Girod, F. X.; Goetz, J. T.; Golovatch, E.; Gothe, R. W.; Griffioen, K. A.; Guegan, B.; Guidal, M.; Hakobyan, H.; Hanretty, C.; Hattawy, M.; Holtrop, M.; Hughes, S. M.; Hyde, C. E.; Ilieva, Y.; Ireland, D. G.; Jiang, H.; Jo, H. S.; Joo, K.; Keller, D.; Khandaker, M.; Kim, A.; Kim, W.; Koirala, S.; Kubarovsky, V.; Kuleshov, S. V.; Lenisa, P.; Levine, W. I.; Livingston, K.; Lu, H. Y.; MacGregor, I. J. D.; Mayer, M.; McKinnon, B.; Meyer, C. A.; Mirazita, M.; Mokeev, V.; Montgomery, R. A.; Moody, C. I.; Moutarde, H.; Movsisyan, A.; Camacho, C. Munoz; Nadel-Turonski, P.; Niccolai, S.; Niculescu, G.; Niculescu, I.; Osipenko, M.; Ostrovidov, A. I.; Pappalardo, L. L.; Paremuzyan, R.; Peng, P.; Phelps, W.; Pisano, S.; Pogorelko, O.; Pozdniakov, S.; Price, J. W.; Protopopescu, D.; Puckett, A. J. R.; Raue, B. A.; Rimal, D.; Ripani, M.; Rizzo, A.; Rosner, G.; Roy, P.; Sabatié, F.; Saini, M. S.; Schott, D.; Schumacher, R. A.; Simonyan, A.; Sokhan, D.; Strauch, S.; Sytnik, V.; Tang, W.; Tian, Ye; Ungaro, M.; Vernarsky, B.; Vlassov, A. V.; Voskanyan, H.; Voutier, E.; Walford, N. K.; Watts, D. P.; Wei, X.; Weinstein, L. B.; Wood, M. H.; Zachariou, N.; Zhang, J.; Zhao, Z. W.; Zonta, I.; CLAS Collaboration 2014-10-01 We measured the ratios of electroproduction cross sections from a proton target for three exclusive meson-baryon final states: ΛK+, pπ0, and nπ+, with the CLAS detector at Jefferson Lab. Using a simple model of quark hadronization, we extract qq ¯ creation probabilities for the first time in exclusive two-body production, in which only a single qq ¯ pair is created. We observe a sizable suppression of strange quark-antiquark pairs compared to nonstrange pairs, similar to that seen in high-energy production. 1. Strangeness suppression of qq creation observed in exclusive reactions. Science.gov (United States) Mestayer, M D; Park, K; Adhikari, K P; Aghasyan, M; Pereira, S Anefalos; Ball, J; Battaglieri, M; Batourine, V; Bedlinskiy, I; Biselli, A S; Boiarinov, S; Briscoe, W J; Brooks, W K; Burkert, V D; Carman, D S; Celentano, A; Chandavar, S; Charles, G; Colaneri, L; Cole, P L; Contalbrigo, M; Cortes, O; Crede, V; D'Angelo, A; Dashyan, N; De Vita, R; Deur, A; Djalali, C; Doughty, D; Dupre, R; El Alaoui, A; El Fassi, L; Elouadrhiri, L; Eugenio, P; Fedotov, G; Fleming, J A; Forest, T A; Garillon, B; Garçon, M; Ghandilyan, Y; Gilfoyle, G P; Giovanetti, K L; Girod, F X; Goetz, J T; Golovatch, E; Gothe, R W; Griffioen, K A; Guegan, B; Guidal, M; Hakobyan, H; Hanretty, C; Hattawy, M; Holtrop, M; Hughes, S M; Hyde, C E; Ilieva, Y; Ireland, D G; Jiang, H; Jo, H S; Joo, K; Keller, D; Khandaker, M; Kim, A; Kim, W; Koirala, S; Kubarovsky, V; Kuleshov, S V; Lenisa, P; Levine, W I; Livingston, K; Lu, H Y; MacGregor, I J D; Mayer, M; McKinnon, B; Meyer, C A; Mirazita, M; Mokeev, V; Montgomery, R A; Moody, C I; Moutarde, H; Movsisyan, A; Camacho, C Munoz; Nadel-Turonski, P; Niccolai, S; Niculescu, G; Niculescu, I; Osipenko, M; Ostrovidov, A I; Pappalardo, L L; Paremuzyan, R; Peng, P; Phelps, W; Pisano, S; Pogorelko, O; Pozdniakov, S; Price, J W; Protopopescu, D; Puckett, A J R; Raue, B A; Rimal, D; Ripani, M; Rizzo, A; Rosner, G; Roy, P; Sabatié, F; Saini, M S; Schott, D; Schumacher, R A; Simonyan, A; Sokhan, D; Strauch, S; Sytnik, V; Tang, W; Tian, Ye; Ungaro, M; Vernarsky, B; Vlassov, A V; Voskanyan, H; Voutier, E; Walford, N K; Watts, D P; Wei, X; Weinstein, L B; Wood, M H; Zachariou, N; Zhang, J; Zhao, Z W; Zonta, I 2014-10-10 We measured the ratios of electroproduction cross sections from a proton target for three exclusive meson-baryon final states: ΛK(+), pπ(0), and nπ(+), with the CLAS detector at Jefferson Lab. Using a simple model of quark hadronization, we extract qq creation probabilities for the first time in exclusive two-body production, in which only a single qq pair is created. We observe a sizable suppression of strange quark-antiquark pairs compared to nonstrange pairs, similar to that seen in high-energy production. 2. Irreducible complexity of iterated symmetric bimodal maps Directory of Open Access Journals (Sweden) J. P. Lampreia 2005-01-01 Full Text Available We introduce a tree structure for the iterates of symmetric bimodal maps and identify a subset which we prove to be isomorphic to the family of unimodal maps. This subset is used as a second factor for a ∗-product that we define in the space of bimodal kneading sequences. Finally, we give some properties for this product and study the ∗-product induced on the associated Markov shifts. 3. A symmetric Roos bound for linear codes NARCIS (Netherlands) Duursma, I.M.; Pellikaan, G.R. 2006-01-01 The van Lint–Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound 4. Symmetric voltage-controlled variable resistance Science.gov (United States) Vanelli, J. C. 1978-01-01 Feedback network makes resistance of field-effect transistor (FET) same for current flowing in either direction. It combines control voltage with source and load voltages to give symmetric current/voltage characteristics. Since circuit produces same magnitude output voltage for current flowing in either direction, it introduces no offset in presense of altering polarity signals. It is therefore ideal for sensor and effector circuits in servocontrol systems. 5. Resistor Networks based on Symmetrical Polytopes Directory of Open Access Journals (Sweden) Jeremy Moody 2015-03-01 Full Text Available This paper shows how a method developed by Van Steenwijk can be generalized to calculate the resistance between any two vertices of a symmetrical polytope all of whose edges are identical resistors. The method is applied to a number of cases that have not been studied earlier such as the Archimedean polyhedra and their duals in three dimensions, the regular polytopes in four dimensions and the hypercube in any number of dimensions. 6. Symmetric vs. asymmetric punishment regimes for bribery OpenAIRE Engel, Christoph; Goerg, Sebastian J.; Yu, Gaoneng 2012-01-01 In major legal orders such as UK, the U.S., Germany, and France, bribers and recipients face equally severe criminal sanctions. In contrast, countries like China, Russia, and Japan treat the briber more mildly. Given these differences between symmetric and asymmetric punishment regimes for bribery, one may wonder which punishment strategy is more effective in curbing corruption. For this purpose, we designed and ran a lab experiment in Bonn (Germany) and Shanghai (China) with exactly the same... 7. Symmetric scrolled packings of multilayered carbon nanoribbons Science.gov (United States) Savin, A. V.; Korznikova, E. A.; Lobzenko, I. P.; Baimova, Yu. A.; Dmitriev, S. V. 2016-06-01 Scrolled packings of single-layer and multilayer graphene can be used for the creation of supercapacitors, nanopumps, nanofilters, and other nanodevices. The full atomistic simulation of graphene scrolls is restricted to consideration of relatively small systems in small time intervals. To overcome this difficulty, a two-dimensional chain model making possible an efficient calculation of static and dynamic characteristics of nanoribbon scrolls with allowance for the longitudinal and bending stiffness of nanoribbons is proposed. The model is extended to the case of scrolls of multilayer graphene. Possible equilibrium states of symmetric scrolls of multilayer carbon nanotribbons rolled up so that all nanoribbons in the scroll are equivalent are found. Dependences of the number of coils, the inner and outer radii, lowest vibrational eigenfrequencies of rolled packages on the length L of nanoribbons are obtained. It is shown that the lowest vibrational eigenfrequency of a symmetric scroll decreases with a nanoribbon length proportionally to L -1. It is energetically unfavorable for too short nanoribbons to roll up, and their ground state is a stack of plane nanoribbons. With an increasing number k of layers, the nanoribbon length L necessary for creation of symmetric scrolls increases. For a sufficiently small number of layers k and a sufficiently large nanoribbon length L, the scrolled packing has the lowest energy as compared to that of stack of plane nanoribbons and folded structures. The results can be used for development of nanomaterials and nanodevices on the basis of graphene scrolled packings. 8. Is the Universe matter-antimatter symmetric International Nuclear Information System (INIS) Alfven, H. 1976-09-01 According to the symmetric cosmology there should be antimatter regions in space which are equally as large as the matter regions. The regions of different kind are separated by Leidenfrost layers, which may be very thin and not observable from a distance. This view has met resistance which in part is based on the old view that the dilute interstellar and intergalactic medium is more or less homogeneous. However, through space research in the magnetosphere and interplanetary space we know that thin layers, dividing space into regions of different magnetisation, exist and based on this it is concluded that space in general has a cellular structure. This result may break down the psychological resistance to the symmetric theory. The possibility that every second star in our galaxy consists of antimatter is discussed, and it is shown that this view is not in conflict with any observations. As most stars are likely to be surrounded by solar systems of a structure like our own, it is concluded that collisions between comets and antistars (or anticomets and stars) would be rather frequent. Such collisions would result in phenomena of the same type as the observed cosmic γ-ray bursts. Another support for the symmetric cosmology is the continuous X-ray background radiation. Also many of the observed large energy releases in cosmos are likely to be due to annihilation 9. Problems of Chernobyl exclusion zone International Nuclear Information System (INIS) 1994-01-01 The collection reflects the results of researches and test-design activities in the exclusion area of the Chernobyl NPP directed to elaborate the equipment and devices for scientific researches and elimination of the accident after effects at the Chernobyl NPP and to study composition and structure of solid-phase bearers of the activity in the soil of the exclusion area, form transformation of decay products, radionuclide composition of the fuel precipitation in the nearest zone of the Chernobyl NPP. Special attention is paid to medical-biological problems of the accident after effects influence on flora, fauna and human health, labour conditions and sick rate of people working in the exclusion area 10. Unified Treatment of a Class of Spherically Symmetric Potentials: Quasi-Exact Solution International Nuclear Information System (INIS) 2016-01-01 We investigate the Schrödinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra of sl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory of sl(2) Lie algebra. 11. On the harmonic starlike functions with respect to symmetric ... African Journals Online (AJOL) In the present paper, we introduce the notions of functions harmonic starlike with respect to symmetric, conjugate and symmetric conjugate points. Such results as coefficient inequalities and structural formulae for these function classes are proved. Keywords: Harmonic functions, harmonic starlike functions, symmetric points, ... 12. Problems of Chernobyl exclusion zone International Nuclear Information System (INIS) 1996-01-01 The collection comprises the results of researches and design activity in the ChNPP exclusion zone with the aim to develop technology, equipment and instruments for RAW management and accident clean-up, studying of the composition and structure of the activity solid bearers in the soil of the exclusion zone and transformation of the radionuclides in the nearest zone of ChNPP. Much attention is paid to medical and biological problems of the accident influence on the flora, fauna and people's health labour conditions and incidence of the people involved 13. Problems of Chornobyl Exclusion Zone International Nuclear Information System (INIS) Kashparov, V.A. 2009-01-01 The collection comprises the results of researches and design activity in the ChNPP exclusion zone with the aim to develop technology, equipment and instruments for RAW management and accident clean-up, studying of the composition and structure of the activity solid bearers in the soil of the exclusion zone and transformation of the radionuclides in the nearest zone of ChNPP. Much attention is paid to medical and biological problems of the accident influence on the flora, fauna and people's health, labour conditions and incidence of the people involved. 14. Exclusion statistics and integrable models International Nuclear Information System (INIS) Mashkevich, S. 1998-01-01 The definition of exclusion statistics that was given by Haldane admits a 'statistical interaction' between distinguishable particles (multispecies statistics). For such statistics, thermodynamic quantities can be evaluated exactly; explicit expressions are presented here for cluster coefficients. Furthermore, single-species exclusion statistics is realized in one-dimensional integrable models of the Calogero-Sutherland type. The interesting questions of generalizing this correspondence to the higher-dimensional and the multispecies cases remain essentially open; however, our results provide some hints as to searches for the models in question 15. Dual formulation of covariant nonlinear duality-symmetric action of kappa-symmetric D3-brane Science.gov (United States) Vanichchapongjaroen, Pichet 2018-02-01 We study the construction of covariant nonlinear duality-symmetric actions in dual formulation. Essentially, the construction is the PST-covariantisation and nonlinearisation of Zwanziger action. The covariantisation made use of three auxiliary scalar fields. Apart from these, the construction proceed in a similar way to that of the standard formulation. For example, the theories can be extended to include interactions with external fields, and that the theories possess two local PST symmetries. We then explicitly demonstrate the construction of covariant nonlinear duality-symmetric actions in dual formulation of DBI theory, and D3-brane. For each of these theories, the twisted selfduality condition obtained from duality-symmetric actions are explicitly shown to match with the duality relation between field strength and its dual from the one-potential actions. Their on-shell actions between the duality-symmetric and the one-potential versions are also shown to match. We also explicitly prove kappa-symmetry of the covariant nonlinear duality-symmetric D3-brane action in dual formulation. 16. Computer Code for Interpreting 13C NMR Relaxation Measurements with Specific Models of Molecular Motion: The Rigid Isotropic and Symmetric Top Rotor Models and the Flexible Symmetric Top Rotor Model Science.gov (United States) 2017-01-01 top rotor superimposes an effective correlation time, τe, onto a symmetric top rotor to account for internal motion. 2. THEORY The purpose...specifically describe how simple 13C relaxation theory is used to describe quantitatively simple molecular 3 motions. More-detailed accounts ...of nuclear magnetic relaxation can be found in a number of basic textbooks (i.e., Farrar and Becker, 1971; Fukushima and Roeder, 1981; Harris, 1986 17. A simple and realistic triton wave function International Nuclear Information System (INIS) 1980-01-01 We propose a simple triton wave function that consists of a product of three correlation operators operating on a three-body spin-isospin state. This wave function is formally similar to that used in the recent variational theories of nuclear matter, the main difference being in the long-range behavior of the correlation operators. Variational calculations are carried out with the Reid potential, using this wave function in the so-called 'symmetrized product' and 'independent pair' forms. The triton energy and density distributions obtained with the symmetrized product wave function agree with those obtained in Faddeev and other variational calculations using harmonic oscillator states. The proposed wave function and calculational methods can be easily generalized to treat the four-nucleon α-particle. (orig.) 18. Exclusive processes in quantum chromodynamics International Nuclear Information System (INIS) Brodsky, S.J.; Lepage, G.P. 1981-06-01 Large momentum transfer exclusive processes and the short distance structure of hadronic wave functions can be systematically analyzed within the context of perturbative QCD. Predictions for meson form factors, two-photon processes γγ → M anti M, hadronic decays of heavy quark systems, and a number of other related QCD phenomena are reviewed 19. Exclusive meson production at COMPASS CERN Document Server Pochodzalla, Josef; Moinester, Murray; Piller, Gunther; Sandacz, Andrzej; Vanderhaeghen, Marc; Pochodzalla, Josef; Mankiewicz, Lech; Moinester, Murray; Piller, Gunther; Sandacz, Andrzej; Vanderhaeghen, Marc 1999-01-01 We explore the feasibility to study exclusive meson production (EMP) in hard muon-proton scattering at the COMPASS experiment. These measurements constrain the off-forward parton distributions (OFPD's) of the proton, which are related to the quark orbital contribution to the proton spin. 20. A simple finite element method for linear hyperbolic problems International Nuclear Information System (INIS) Mu, Lin; Ye, Xiu 2017-01-01 Here, we introduce a simple finite element method for solving first order hyperbolic equations with easy implementation and analysis. Our new method, with a symmetric, positive definite system, is designed to use discontinuous approximations on finite element partitions consisting of arbitrary shape of polygons/polyhedra. Error estimate is established. Extensive numerical examples are tested that demonstrate the robustness and flexibility of the method. 1. Dijet rates with symmetric Et cuts International Nuclear Information System (INIS) Banfi, Andrea; Dasgupta, Mrinal 2004-01-01 We consider dijet production in the region where symmetric cuts on the transverse energy, E t , are applied to the jets. In this region next-to-leading order calculations are unreliable and an all-order resummation of soft gluon effects is needed, which we carry out. Although, for illustrative purposes, we choose dijets produced in deep inelastic scattering, our general ideas apply additionally to dijets produced in photoproduction or gamma-gamma processes and should be relevant also to the study of prompt di-photon E t spectra in association with a recoiling jet, in hadron-hadron processes. (author) 2. Covariant, chirally symmetric, confining model of mesons International Nuclear Information System (INIS) Gross, F.; Milana, J. 1991-01-01 We introduce a new model of mesons as quark-antiquark bound states. The model is covariant, confining, and chirally symmetric. Our equations give an analytic solution for a zero-mass pseudoscalar bound state in the case of exact chiral symmetry, and also reduce to the familiar, highly successful nonrelativistic linear potential models in the limit of heavy-quark mass and lightly bound systems. In this fashion we are constructing a unified description of all the mesons from the π through the Υ. Numerical solutions for other cases are also presented 3. Symmetric Logic Synthesis with Phase Assignment OpenAIRE Benschop, N. F. 2001-01-01 Decomposition of any Boolean Function BF_n of n binary inputs into an optimal inverter coupled network of Symmetric Boolean functions SF_k (k \\leq n) is described. Each SF component is implemented by Threshold Logic Cells, forming a complete and compact T-Cell Library. Optimal phase assignment of input polarities maximizes local symmetries. The "rank spectrum" is a new BF_n description independent of input ordering, obtained by mapping its minterms onto an othogonal n \\times n grid of (transi... 4. Elastic energy for reflection-symmetric topologies International Nuclear Information System (INIS) Majumdar, A; Robbins, J M; Zyskin, M 2006-01-01 Nematic liquid crystals in a polyhedral domain, a prototype for bistable displays, may be described by a unit-vector field subject to tangent boundary conditions. Here we consider the case of a rectangular prism. For configurations with reflection-symmetric topologies, we derive a new lower bound for the one-constant elastic energy. For certain topologies, called conformal and anticonformal, the lower bound agrees with a previous result. For the remaining topologies, called nonconformal, the new bound is an improvement. For nonconformal topologies we derive an upper bound, which differs from the lower bound by a factor depending only on the aspect ratios of the prism 5. Nanotribology of Symmetric and Asymmetric Liquid Lubricants Directory of Open Access Journals (Sweden) 2010-03-01 Full Text Available When liquid molecules are confined in a narrow gap between smooth surfaces, their dynamic properties are completely different from those of the bulk. The molecular motions are highly restricted and the system exhibits solid-like responses when sheared slowly. This solidification behavior is very dependent on the molecular geometry (shape of liquids because the solidification is induced by the packing of molecules into ordered structures in confinement. This paper reviews the measurements of confined structures and friction of symmetric and asymmetric liquid lubricants using the surface forces apparatus. The results show subtle and complex friction mechanisms at the molecular scale. 6. Unary self-verifying symmetric difference automata CSIR Research Space (South Africa) Marais, Laurette 2016-07-01 Full Text Available stream_source_info Marais_2016_ABSTRACT.pdf.txt stream_content_type text/plain stream_size 796 Content-Encoding ISO-8859-1 stream_name Marais_2016_ABSTRACT.pdf.txt Content-Type text/plain; charset=ISO-8859-1 18th... International Workshop on Descriptional Complexity of Formal Systems, 5 - 8 July 2016, Bucharest, Romania Unary self-verifying symmetric difference automata Laurette Marais1,2 and Lynette van Zijl1(B) 1 Department of Computer Science, Stellenbosch... 7. Characterisation of an AGATA symmetric prototype detector International Nuclear Information System (INIS) Nelson, L.; Dimmock, M.R.; Boston, A.J.; Boston, H.C.; Cresswell, J.R.; Nolan, P.J.; Lazarus, I.; Simpson, J.; Medina, P.; Santos, C.; Parisel, C. 2007-01-01 The Advanced GAmma Tracking Array (AGATA) symmetric prototype detector has been tested at University of Liverpool. A 137 Ce source, collimated to a 2 mm diameter, was scanned across the front face of the detector and data were acquired utilising digital electronics. Pulse shapes from a selection of well-defined photon interaction positions have been analysed to investigate the position sensitivity of the detector. Furthermore, the application of the electric field simulation software, Multi Geometry Simulation (MGS) to generate theoretical pulse shapes for AGATA detectors has been presented 8. How Symmetrical Assumptions Advance Strategic Management Research DEFF Research Database (Denmark) Foss, Nicolai Juul; Hallberg, Hallberg 2014-01-01 We develop the case for symmetrical assumptions in strategic management theory. Assumptional symmetry obtains when assumptions made about certain actors and their interactions in one of the application domains of a theory are also made about this set of actors and their interactions in other...... application domains of the theory. We argue that assumptional symmetry leads to theoretical advancement by promoting the development of theory with greater falsifiability and stronger ontological grounding. Thus, strategic management theory may be advanced by systematically searching for asymmetrical... 9. Characterisation of an AGATA symmetric prototype detector Energy Technology Data Exchange (ETDEWEB) Nelson, L. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom)]. E-mail: [email protected]; Dimmock, M.R. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom)]. E-mail: [email protected]; Boston, A.J. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom)]. E-mail: [email protected]; Boston, H.C. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom); Cresswell, J.R. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom); Nolan, P.J. [Oliver Lodge Laboratory, University of Liverpool, Oxford Street, Liverpool L69 7ZE (United Kingdom); Lazarus, I. [CCLRC Daresbury Laboratory, Daresbury, Warrington WA4 4AD (United Kingdom); Simpson, J. [CCLRC Daresbury Laboratory, Daresbury, Warrington WA4 4AD (United Kingdom); Medina, P. [Institut de Recherches Subatomiques, Strasbourg BP28 67037 (France); Santos, C. [Institut de Recherches Subatomiques, Strasbourg BP28 67037 (France); Parisel, C. [Institut de Recherches Subatomiques, Strasbourg BP28 67037 (France) 2007-04-01 The Advanced GAmma Tracking Array (AGATA) symmetric prototype detector has been tested at University of Liverpool. A {sup 137}Ce source, collimated to a 2 mm diameter, was scanned across the front face of the detector and data were acquired utilising digital electronics. Pulse shapes from a selection of well-defined photon interaction positions have been analysed to investigate the position sensitivity of the detector. Furthermore, the application of the electric field simulation software, Multi Geometry Simulation (MGS) to generate theoretical pulse shapes for AGATA detectors has been presented. 10. Soft theorems for shift-symmetric cosmologies Science.gov (United States) Finelli, Bernardo; Goon, Garrett; Pajer, Enrico; Santoni, Luca 2018-03-01 We derive soft theorems for single-clock cosmologies that enjoy a shift symmetry. These so-called consistency conditions arise from a combination of a large diffeomorphism and the internal shift symmetry and fix the squeezed limit of all correlators with a soft scalar mode. As an application, we show that our results reproduce the squeezed bispectrum for ultra-slow-roll inflation, a particular shift-symmetric, nonattractor model which is known to violate Maldacena's consistency relation. Similar results have been previously obtained by Mooij and Palma using background-wave methods. Our results shed new light on the infrared structure of single-clock cosmological spacetimes. 11. Pion condensation in symmetric nuclear matter International Nuclear Information System (INIS) Shamsunnahar, T.; Saha, S.; Kabir, K.; Nath, L.M. 1991-01-01 We have investigated the possibility of pion condensation in symmetric nuclear matter using a model of pion-nucleon interaction based essentially on chiral SU(2) x SU(2) symmetry. We have found that pion condensation is not possible for any finite value of the density. Consequently, no critical opalescence phenomenon is likely to be seen in pion-nucleus scattering nor is it likely to be possible to explain the EMC effect in terms of an increased number of pions in the nucleus. (author) 12. Baryon symmetric big-bang cosmology Energy Technology Data Exchange (ETDEWEB) Stecker, F.W. 1978-04-01 The framework of baryon-symmetric big-bang cosmology offers the greatest potential for deducing the evolution of the universe as a consequence of physical laws and processes with the minimum number of arbitrary assumptions as to initial conditions in the big-bang. In addition, it offers the possibility of explaining the photon-baryon ratio in the universe and how galaxies and galaxy clusters are formed, and also provides the only acceptable explanation at present for the origin of the cosmic gamma ray background radiation. 13. Baryon symmetric big-bang cosmology International Nuclear Information System (INIS) Stecker, F.W. 1978-04-01 The framework of baryon-symmetric big-bang cosmology offers the greatest potential for deducing the evolution of the universe as a consequence of physical laws and processes with the minimum number of arbitrary assumptions as to initial conditions in the big-bang. In addition, it offers the possibility of explaining the photon-baryon ratio in the universe and how galaxies and galaxy clusters are formed, and also provides the only acceptable explanation at present for the origin of the cosmic gamma ray background radiation 14. Simple Kidney Cysts Science.gov (United States) ... Solitary Kidney Your Kidneys & How They Work Simple Kidney Cysts What are simple kidney cysts? Simple kidney cysts are abnormal, fluid-filled ... that form in the kidneys. What are the kidneys and what do they do? The kidneys are ... 15. On the Relations between the Attacks on Symmetric Homomorphic Encryption over the Residue Ring Directory of Open Access Journals (Sweden) Alina V. Trepacheva 2017-06-01 Full Text Available The paper considers the security of symmetric homomorphic cryptosystems (HC over the residue ring. The main task is to establish an equivalence between ciphertexts only attack (COA and known plaintexts attack (KPA for HC. The notion of reducibility between attacks and sufficient condition of reducibility from COA to KPA are given for this purpose. The main idea is: to prove reducibility from COA to KPA we need to find a function over residue ring being efficiently computable and having a small image size comparing with the size of residue ring. The study of reducibility existence is important since it allows to understand better the security level of symmetric HC proposed in literature. A vulnerability against KPA has been already found for the majority of these HC. Thus the reducibility presence can demonstrate that cryptosystems under the study are not secure even against COA, and therefore they are totally insecure and shouldn’t be used in practice. We give an example of reducibility from COA to KPA for residue ring being a simple field. Based on this example we show an efficient COA on one symmetric HC for small field. Also we separately consider the case of residue ring composed using number n being hard-to-factor. For such n an efficient algorithm to construct an efficiently computable function with small image is unknown so far. So further work related to cryptanalysis of existing symmetric HC will be directed into study of functions properties over residue rings modulo numbers hard for factorization. 16. Parametric amplification and bidirectional invisibility in PT -symmetric time-Floquet systems Science.gov (United States) Koutserimpas, Theodoros T.; Alù, Andrea; Fleury, Romain 2018-01-01 Parity-time (PT )-symmetric wave devices, which exploit balanced interactions between material gain and loss, exhibit extraordinary properties, including lasing and flux-conserving scattering processes. In a seemingly different research field, periodically driven systems, also known as time-Floquet systems, have been widely studied as a relevant platform for reconfigurable active wave control and manipulation. In this article, we explore the connection between PT -symmetry and parametric time-Floquet systems. Instead of relying on material gain, we use parametric amplification by considering a time-periodic modulation of the refractive index at a frequency equal to twice the incident signal frequency. We show that the scattering from a simple parametric slab, whose dynamics follows the Mathieu equation, can be described by a PT -symmetric scattering matrix, whose PT -breaking threshold corresponds to the Mathieu instability threshold. By combining different parametric slabs modulated out of phase, we create PT -symmetric time-Floquet systems that feature exceptional scattering properties, such as coherent perfect absorption (CPA)-laser operation and bidirectional invisibility. These bidirectional properties, rare for regular PT -symmetric systems, are related to a compensation of parametric amplification due to multiple scattering between two parametric systems modulated with a phase difference. 17. Investigating the exclusive protoproduction of dileptons at high energies International Nuclear Information System (INIS) 2008-01-01 Using the high energy color dipole approach, we study the exclusive photoproduction of lepton pairs γN→γ(→l + l - )N (with N=p, A). We use simple models for the elementary dipole-hadron scattering amplitude that captures main features of the dependence on atomic number A, on energy and on momentum transfer t. This investigation is complementary to conventional partonic description of timelike Compton scattering, which considers quark handbag diagrams at leading order in α s and simple models of the relevant generalized parton distributions. These calculations are input in electromagnetic interactions in pp and AA collisions to measured at the LHC. 18. The exclusion problem in seasonally forced epidemiological systems. Science.gov (United States) 2015-02-21 The pathogen exclusion problem is the problem of finding control measures that will exclude a pathogen from an ecological system or, if the system is already disease-free, maintain it in that state. To solve this problem we work within a holistic control theory framework which is consistent with conventional theory for simple systems (where there is no external forcing and constant controls) and seamlessly generalises to complex systems that are subject to multiple component seasonal forcing and targeted variable controls. We develop, customise and integrate a range of numerical and algebraic procedures that provide a coherent methodology powerful enough to solve the exclusion problem in the general case. An important aspect of our solution procedure is its two-stage structure which reveals the epidemiological consequences of the controls used for exclusion. This information augments technical and economic considerations in the design of an acceptable exclusion strategy. Our methodology is used in two examples to show how time-varying controls can exploit the interference and reinforcement created by the external and internal lag structure and encourage the system to 'take over' some of the exclusion effort. On-off control switching, resonant amplification, optimality and controllability are important issues that emerge in the discussion. Copyright © 2014 Elsevier Ltd. All rights reserved. 19. Problems of Chernobyl exclusion zone International Nuclear Information System (INIS) 1996-01-01 The collection comprises the results of researches and design activity in the ChNPP exclusion zone with the aim to develop technology, equipment and instruments for RAW management and accident clean-up, studying of the composition and structure of the activity solid bearers in the soil of the exclusion zone and transformation of the radionuclides in the nearest zone of ChNPP. Much attention is paid to medical and biological problems of the accident influence on the flora, fauna and people's health, labour conditions and incidence of the people involved. The collection comprises the information for scientists, experts, postgraduates and students in gaged in ecology, radioecology, nuclear engineering, radiology, radiochemistry and radiobiology 20. Electroweak Baryogenesis in R-symmetric Supersymmetry Energy Technology Data Exchange (ETDEWEB) Fok, R.; Kribs, Graham D.; Martin, Adam; Tsai, Yuhsin 2013-03-01 We demonstrate that electroweak baryogenesis can occur in a supersymmetric model with an exact R-symmetry. The minimal R-symmetric supersymmetric model contains chiral superfields in the adjoint representation, giving Dirac gaugino masses, and an additional set of "R-partner" Higgs superfields, giving R-symmetric \\mu-terms. New superpotential couplings between the adjoints and the Higgs fields can simultaneously increase the strength of the electroweak phase transition and provide additional tree-level contributions to the lightest Higgs mass. Notably, no light stop is present in this framework, and in fact, we require both stops to be above a few TeV to provide sufficient radiative corrections to the lightest Higgs mass to bring it up to 125 GeV. Large CP-violating phases in the gaugino/higgsino sector allow us to match the baryon asymmetry of the Universe with no constraints from electric dipole moments due to R-symmetry. We briefly discuss some of the more interesting phenomenology, particularly of the of the lightest CP-odd scalar. 1. Exclusive photoreactions on light nuclei International Nuclear Information System (INIS) Maruyama, K. 1989-08-01 The mechanism of photon absorption on light nuclei in the Δ-resonance region is discussed. The present status of experimental results is briefly summarized. A recent data from 1.3-GeV Tokyo ES using a π sr spectrometer is introduced. Exclusive measurements of the photodisintegration of 3 He and 4 He may be a clear way to identify 2N, 3N and 4N absorptions. (author) 2. Gender, Marginalisation and Social Exclusion DEFF Research Database (Denmark) D. Munk, Martin The paper is focused on the fact that marginalisation and social exclusion are gender-related in the EU. Even when boys and girls experience the same kinds of strain and social inheritance, they react socially different. Likewise women and men are marginalised in different ways. The differing...... access to the five ressources: cultural, financial, mental, social and powerrelated resources is highlighted. It is demonstrated how gender involves living in different realities, and requires different solutions to create equal possibilities.... 3. What Is a Simple Liquid? Directory of Open Access Journals (Sweden) Trond S. Ingebrigtsen 2012-03-01 Full Text Available This paper is an attempt to identify the real essence of simplicity of liquids in John Locke’s understanding of the term. Simple liquids are traditionally defined as many-body systems of classical particles interacting via radially symmetric pair potentials. We suggest that a simple liquid should be defined instead by the property of having strong correlations between virial and potential-energy equilibrium fluctuations in the NVT ensemble. There is considerable overlap between the two definitions, but also some notable differences. For instance, in the new definition simplicity is not a direct property of the intermolecular potential because a liquid is usually only strongly correlating in part of its phase diagram. Moreover, not all simple liquids are atomic (i.e., with radially symmetric pair potentials and not all atomic liquids are simple. The main part of the paper motivates the new definition of liquid simplicity by presenting evidence that a liquid is strongly correlating if and only if its intermolecular interactions may be ignored beyond the first coordination shell (FCS. This is demonstrated by NVT simulations of the structure and dynamics of several atomic and three molecular model liquids with a shifted-forces cutoff placed at the first minimum of the radial distribution function. The liquids studied are inverse power-law systems (r^{-n} pair potentials with n=18,6,4, Lennard-Jones (LJ models (the standard LJ model, two generalized Kob-Andersen binary LJ mixtures, and the Wahnstrom binary LJ mixture, the Buckingham model, the Dzugutov model, the LJ Gaussian model, the Gaussian core model, the Hansen-McDonald molten salt model, the Lewis-Wahnstrom ortho-terphenyl model, the asymmetric dumbbell model, and the single-point charge water model. The final part of the paper summarizes properties of strongly correlating liquids, emphasizing that these are simpler than liquids in general. Simple liquids, as defined here, may be 4. Rotor current transient analysis of DFIG-based wind turbines during symmetrical voltage faults International Nuclear Information System (INIS) Ling, Yu; Cai, Xu; Wang, Ningbo 2013-01-01 Highlights: • We theoretically analyze the rotor fault current of DFIG based on space vector. • The presented analysis is simple, easy to understand. • The analysis highlights the accuracy of the expression of the rotor fault currents. • The expression can be widely used to analyze the different levels of voltage symmetrical fault. • Simulation results show the accuracy of the expression of the rotor currents. - Abstract: The impact of grid voltage fault on doubly fed induction generators (DFIGs), especially rotor currents, has received much attention. So, in this paper, the rotor currents of based-DFIG wind turbines are considered in a generalized way, which can be widely used to analyze the cases under different levels of voltage symmetrical faults. A direct method based on space vector is proposed to obtain an accurate expression of rotor currents as a function of time for symmetrical voltage faults in the power system. The presented theoretical analysis is simple and easy to understand and especially highlights the accuracy of the expression. Finally, the comparable simulations evaluate this analysis and show that the expression of the rotor currents is sufficient to calculate the maximum fault current, DC and AC components, and especially helps to understand the causes of the problem and as a result, contributes to adapt reasonable approaches to enhance the fault ride through (FRT) capability of DFIG wind turbines during a voltage fault 5. A simple fluxgate magnetometer using amorphous alloys International Nuclear Information System (INIS) Ghatak, S.K.; Mitra, A. 1992-01-01 A simple fluxgate magnetometer is developed using low magnetostrictive ferromagnetic amorphous alloy acting as a sensing element. It uses the fact that the magnetization of sensing element symmetrically magnetized by a sinusoidal field contains even harmonic components in presence of dc signal field H and the amplitude of the second harmonic component of magnetization is proportional to H. The sensitivity and linearity of the magnetometer with signal field are studied for parallel configuration and the field ranging from 10 nT to 10 μT can be measured. The functioning of the magnetometer is demonstrated by studying the shielding and flux-trapping phenomena in high-Tc superconductor. (orig.) 6. Geometric inequalities for axially symmetric black holes International Nuclear Information System (INIS) Dain, Sergio 2012-01-01 A geometric inequality in general relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr-Newman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse; they are closely related with the cosmic censorship conjecture. Axially symmetric black holes are the natural candidates to study these inequalities because the quasi-local angular momentum is well defined for them. We review recent results in this subject and we also describe the main ideas behind the proofs. Finally, a list of relevant open problems is presented. (topical review) 7. A symmetric bipolar nebula around MWC 922. Science.gov (United States) Tuthill, P G; Lloyd, J P 2007-04-13 We report regular and symmetric structure around dust-enshrouded Be star MWC 922 obtained with infrared imaging. Biconical lobes that appear nearly square in aspect, forming this "Red Square" nebula, are crossed by a series of rungs that terminate in bright knots or "vortices," and an equatorial dark band crossing the core delimits twin hyperbolic arcs. The intricate yet cleanly constructed forms that comprise the skeleton of the object argue for minimal perturbation from global turbulent or chaotic effects. We also report the presence of a linear comb structure, which may arise from optically projected shadows of a periodic feature in the inner regions, such as corrugations in the rim of a circumstellar disk. The sequence of nested polar rings draws comparison with the triple-ring system seen around the only naked-eye supernova in recent history: SN1987A. 8. Minimal Left-Right Symmetric Dark Matter. Science.gov (United States) Heeck, Julian; Patra, Sudhanwa 2015-09-18 We show that left-right symmetric models can easily accommodate stable TeV-scale dark matter particles without the need for an ad hoc stabilizing symmetry. The stability of a newly introduced multiplet either arises accidentally as in the minimal dark matter framework or comes courtesy of the remaining unbroken Z_{2} subgroup of B-L. Only one new parameter is introduced: the mass of the new multiplet. As minimal examples, we study left-right fermion triplets and quintuplets and show that they can form viable two-component dark matter. This approach is, in particular, valid for SU(2)×SU(2)×U(1) models that explain the recent diboson excess at ATLAS in terms of a new charged gauge boson of mass 2 TeV. 9. Design and Analysis of Symmetric Primitives DEFF Research Database (Denmark) Lauridsen, Martin Mehl . In the second part, we delve into the matter of the various aspects of designing a symmetric cryptographic primitive. We start by considering generalizations of the widely acclaimed Advanced Encryption Standard (AES) block cipher. In particular, our focus is on a component operation in the cipher which permutes...... analyze and implement modes recommended by the National Institute of Standards and Technology (NIST), as well as authenticated encryption modes from the CAESAR competition, when instantiated with the AES. The data processed in our benchmarking has sizes representative to that of typical Internet traffic...... linear cryptanalysis. We apply this model to the standardized block cipher PRESENT. Finally, we present very generic attacks on two authenticated encryption schemes, AVALANCHE and RBS, by pointing out severe design flaws that can be leveraged to fully recover the secret key with very low complexity... 10. Quasiaxially symmetric stellarators with three field periods International Nuclear Information System (INIS) Garabedian, P.; Ku, L. 1999-01-01 Compact hybrid configurations with two field periods have been studied recently as candidates for a proof of principle experiment at the Princeton Plasma Physics Laboratory. This project has led us to the discovery of a family of quasiaxially symmetric stellarators with three field periods that have significant advantages, although their aspect ratios are a little larger. They have reversed shear and perform better in a local analysis of ballooning modes. Nonlinear equilibrium and stability calculations predict that the average beta limit will be at least as high as 4% if the bootstrap current turns out to be as big as that expected in comparable tokamaks. The concept relies on a combination of helical fields and bootstrap current to achieve adequate rotational transform at low aspect ratio. copyright 1999 American Institute of Physics 11. Primordial two-component maximally symmetric inflation Science.gov (United States) Enqvist, K.; Nanopoulos, D. V.; Quirós, M.; Kounnas, C. 1985-12-01 We propose a two-component inflation model, based on maximally symmetric supergravity, where the scales of reheating and the inflation potential at the origin are decoupled. This is possible because of the second-order phase transition from SU(5) to SU(3)×SU(2)×U(1) that takes place when φ≅φcinflation at the global minimum, and leads to a reheating temperature TR≅(1015-1016) GeV. This makes it possible to generate baryon asymmetry in the conventional way without any conflict with experimental data on proton lifetime. The mass of the gravitinos is m3/2≅1012 GeV, thus avoiding the gravitino problem. Monopoles are diluted by residual inflation in the broken phase below the cosmological bounds if φcUSA. 12. Lovelock black holes with maximally symmetric horizons Energy Technology Data Exchange (ETDEWEB) Maeda, Hideki; Willison, Steven; Ray, Sourya, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Centro de Estudios CientIficos (CECs), Casilla 1469, Valdivia (Chile) 2011-08-21 We investigate some properties of n( {>=} 4)-dimensional spacetimes having symmetries corresponding to the isometries of an (n - 2)-dimensional maximally symmetric space in Lovelock gravity under the null or dominant energy condition. The well-posedness of the generalized Misner-Sharp quasi-local mass proposed in the past study is shown. Using this quasi-local mass, we clarify the basic properties of the dynamical black holes defined by a future outer trapping horizon under certain assumptions on the Lovelock coupling constants. The C{sup 2} vacuum solutions are classified into four types: (i) Schwarzschild-Tangherlini-type solution; (ii) Nariai-type solution; (iii) special degenerate vacuum solution; and (iv) exceptional vacuum solution. The conditions for the realization of the last two solutions are clarified. The Schwarzschild-Tangherlini-type solution is studied in detail. We prove the first law of black-hole thermodynamics and present the expressions for the heat capacity and the free energy. 13. Polyhomogeneous expansions from time symmetric initial data Science.gov (United States) Gasperín, E.; Valiente Kroon, J. A. 2017-10-01 We make use of Friedrich’s construction of the cylinder at spatial infinity to relate the logarithmic terms appearing in asymptotic expansions of components of the Weyl tensor to the freely specifiable parts of time symmetric initial data sets for the Einstein field equations. Our analysis is based on the assumption that a particular type of formal expansions near the cylinder at spatial infinity corresponds to the leading terms of actual solutions to the Einstein field equations. In particular, we show that if the Bach tensor of the initial conformal metric does not vanish at the point at infinity then the most singular component of the Weyl tensor decays near null infinity as O(\\tilde{r}-3\\ln \\tilde{r}) so that spacetime will not peel. We also provide necessary conditions on the initial data which should lead to a peeling spacetime. Finally, we show how to construct global spacetimes which are candidates for non-peeling (polyhomogeneous) asymptotics. 14. From Symmetric Glycerol Derivatives to Dissymmetric Chlorohydrins Directory of Open Access Journals (Sweden) Gemma Villorbina 2011-03-01 Full Text Available The anticipated worldwide increase in biodiesel production will result in an accumulation of glycerol for which there are insufficient conventional uses. The surplus of this by-product has increased rapidly during the last decade, prompting a search for new glycerol applications. We describe here the synthesis of dissymmetric chlorohydrin esters from symmetric 1,3-dichloro-2-propyl esters obtained from glycerol. We studied the influence of two solvents: 1,4-dioxane and 1-butanol and two bases: sodium carbonate and 1-butylimidazole, on the synthesis of dissymmetric chlorohydrin esters. In addition, we studied the influence of other bases (potassium and lithium carbonates in the reaction using 1,4-dioxane as the solvent. The highest yield was obtained using 1,4-dioxane and sodium carbonate. 15. Bidding behavior in a symmetric Chinese auction Directory of Open Access Journals (Sweden) Mauricio Benegas 2015-01-01 Full Text Available This paper purposes a symmetric all-pay auction where the bidders compete neither for an object nor the object itself but for a lottery on receive. That lottery is determined endogenously through the bids. This auction is known as chance auction or more popularly as Chinese auction. The model considers the possibility that for some bidders the optimal strategy is to bid zero and to rely on luck. It showed that bidders become less aggressive when the lottery satisfies a variational condition. It was also shown that luck factor is decisive to determine if the expected payoff in Chinese auction is bigger or smaller than expected payoff in standard all-pay auction. 16. Canonical quantization of static spherically symmetric geometries International Nuclear Information System (INIS) Christodoulakis, T; Dimakis, N; Terzis, P A; Doulis, G; Grammenos, Th; Melas, E; Spanou, A 2013-01-01 The conditional symmetries of the reduced Einstein–Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two ''scale factors'' 17. Cryptanalysis of Some Lightweight Symmetric Ciphers DEFF Research Database (Denmark) Abdelraheem, Mohamed Ahmed Awadelkareem Mohamed Ahmed In recent years, the need for lightweight encryption systems has been increasing as many applications use RFID and sensor networks which have a very low computational power and thus incapable of performing standard cryptographic operations. In response to this problem, the cryptographic community...... on a variant of PRESENT with identical round keys. We propose a new attack named the Invariant Subspace Attack that was specifically mounted against the lightweight block cipher PRINTcipher. Furthermore, we mount several attacks on a recently proposed stream cipher called A2U2....... of the international standards in lightweight cryptography. This thesis aims at analyzing and evaluating the security of some the recently proposed lightweight symmetric ciphers with a focus on PRESENT-like ciphers, namely, the block cipher PRESENT and the block cipher PRINTcipher. We provide an approach to estimate... 18. Cosmic ray antimatter and baryon symmetric cosmology Science.gov (United States) Stecker, F. W.; Protheroe, R. J.; Kazanas, D. 1982-01-01 The relative merits and difficulties of the primary and secondary origin hypotheses for the observed cosmic-ray antiprotons, including the new low-energy measurement of Buffington, et al. We conclude that the cosmic-ray antiproton data may be evidence for antimatter galaxies and baryon symmetric cosmology. The present bar P data are consistent with a primary extragalactic component having /p=/equiv 1+/- 3.2/0.7x10 = to the -4 independent of energy. We propose that the primary extragalactic cosmic ray antiprotons are most likely from active galaxies and that expected disintegration of bar alpha/alpha ban alpha/alpha. We further predict a value for ban alpha/alpha =/equiv 10 to the -5, within range of future cosmic ray detectors. 19. Symmetric alignment of the nematic matrix between close penetrable colloidal particles International Nuclear Information System (INIS) Teixeira, P I C; Barmes, F; Cleaver, D J 2004-01-01 A simple model is proposed for the liquid crystal matrix surrounding 'soft' colloidal particles whose separation is much smaller than their radii. We use our implementation of the Onsager approximation of density-functional theory (Chrzanowska et al 2001 J. Phys.: Condens. Matter 13 4715) to calculate the structure of a nanometrically thin film of hard Gaussian overlap particles of elongations κ = 3 and 5, confined between two solid walls. The penetrability of either substrate can be tuned independently to yield symmetric or hybrid alignment. Comparison with Monte Carlo simulations of the same system (Cleaver and Teixeira 2001 Chem. Phys. Lett. 338 1, Barmes and Cleaver 2004 in preparation) reveals good agreement in the symmetric case 20. Reaction mechanism for the symmetric breakup of 24Mg following an interaction with 12C International Nuclear Information System (INIS) Gyapong, G.J.; Watson, D.L.; Catford, W.N.; Clarke, N.M.; Bennett, S.J.; Freer, M.; Fulton, B.R.; Jones, C.D.; Leddy, M.; Murgatroyd, J.T.; Rae, W.D.M.; Simmons, P. 1994-01-01 Data on the yield of the symmetric breakup of 24 Mg as a function of beam energy are presented and compared with detailed calculations of the energy dependence. The 24 Mg states seen in symmetric breakup agree with previously observed breakup states having spin and parities J π =4 + ,(6 + ),8 + . The data allow the variations of yield for indivual states to be judged, as the beam energy is varied. The variation in the yield of the 4 + states is compared in detail with calculations assuming several possible compound nuclear or direct reaction mechanisms. It is concluded that a massive ( 12 C) transfer or a simple statistical compound process are unlikely mechanisms, but that each of several other mechanisms is consistent with the data. ((orig.)) 1. Symmetrization of the beam-beam interaction in an asymmetric collider International Nuclear Information System (INIS) Chin, Y.H. 1990-07-01 This paper studies the idea of symmetrizing both the lattice and the beams of an asymmetric collider, and discusses why this regime should be within the parametric reach of the design in order to credibly ensure its performance. Also examined is the effectiveness of a simple compensation method using the emittance as a free parameter and that it does not work in all cases. At present, when there are no existing asymmetric colliders, it seems prudent to design an asymmetric collider so as to be similar to a symmetric one (without relying on a particular theory of the asymmetric beam-beam interaction that has not passed tests of fidelity). Nevertheless, one must allow for the maximum possible flexibility and freedom in adjusting those parameters that affect luminosity. Such a parameter flexibility will be essential in tuning the collider to the highest luminosity 2. Factors influencing knowledge and practice of exclusive ... African Journals Online (AJOL) Factors influencing knowledge and practice of exclusive breastfeeding in Nyando ... The overall objective of this study was to determine factors influencing the ... EBF and its benefits), pre lacteal feeds and exclusive breastfeeding consistency. 3. Symmetric Topological Phases and Tensor Network States Science.gov (United States) Jiang, Shenghan Classification and simulation of quantum phases are one of main themes in condensed matter physics. Quantum phases can be distinguished by their symmetrical and topological properties. The interplay between symmetry and topology in condensed matter physics often leads to exotic quantum phases and rich phase diagrams. Famous examples include quantum Hall phases, spin liquids and topological insulators. In this thesis, I present our works toward a more systematically understanding of symmetric topological quantum phases in bosonic systems. In the absence of global symmetries, gapped quantum phases are characterized by topological orders. Topological orders in 2+1D are well studied, while a systematically understanding of topological orders in 3+1D is still lacking. By studying a family of exact solvable models, we find at least some topological orders in 3+1D can be distinguished by braiding phases of loop excitations. In the presence of both global symmetries and topological orders, the interplay between them leads to new phases termed as symmetry enriched topological (SET) phases. We develop a framework to classify a large class of SET phases using tensor networks. For each tensor class, we can write down generic variational wavefunctions. We apply our method to study gapped spin liquids on the kagome lattice, which can be viewed as SET phases of on-site symmetries as well as lattice symmetries. In the absence of topological order, symmetry could protect different topological phases, which are often referred to as symmetry protected topological (SPT) phases. We present systematic constructions of tensor network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries. 4. The radiation chemistry of symmetric aliphatic polyesters International Nuclear Information System (INIS) Babanalbandi, A.; Hill, D.J.T.; Pomery, P.J.; Whittaker, A.K. 1996-01-01 5. FFLP problem with symmetric trapezoidal fuzzy numbers Directory of Open Access Journals (Sweden) 2015-04-01 Full Text Available The most popular approach for solving fully fuzzy linear programming (FFLP problems is to convert them into the corresponding deterministic linear programs. Khan et al. (2013 [Khan, I. U., Ahmad, T., & Maan, N. (2013. A simplified novel technique for solving fully fuzzy linear programming problems. Journal of Optimization Theory and Applications, 159(2, 536-546.] claimed that there had been no method in the literature to find the fuzzy optimal solution of a FFLP problem without converting it into crisp linear programming problem, and proposed a technique for the same. Others showed that the fuzzy arithmetic operation used by Khan et al. (2013 had some problems in subtraction and division operations, which could lead to misleading results. Recently, Ezzati et al. (2014 [Ezzati, R., Khorram, E., & Enayati, R. (2014. A particular simplex algorithm to solve fuzzy lexicographic multi-objective linear programming problems and their sensitivity analysis on the priority of the fuzzy objective functions. Journal of Intelligent and Fuzzy Systems, 26(5, 2333-2358.] defined a new operation on symmetric trapezoidal fuzzy numbers and proposed a new algorithm to find directly a lexicographic/preemptive fuzzy optimal solution of a fuzzy lexicographic multi-objective linear programming problem by using new fuzzy arithmetic operations, but their model was not fully fuzzy optimization. In this paper, a new method, by using Ezzati et al. (2014’s fuzzy arithmetic operation and a fuzzy version of simplex algorithm, is proposed for solving FFLP problem whose parameters are represented by symmetric trapezoidal fuzzy number without converting the given problem into crisp equivalent problem. By using the proposed method, the fuzzy optimal solution of FFLP problem can be easily obtained. A numerical example is provided to illustrate the proposed method. 6. Axially symmetric Lorentzian wormholes in general relativity International Nuclear Information System (INIS) Schein, F. 1997-11-01 The field equations of Einstein's theory of general relativity, being local, do not fix the global structure of space-time. They admit topologically non-trivial solutions, including spatially closed universes and the amazing possibility of shortcuts for travel between distant regions in space and time - so-called Lorentzian wormholes. The aim of this thesis is to (mathematically) construct space-times which contain traversal wormholes connecting arbitrary distant regions of an asymptotically flat or asymptotically de Sitter universe. Since the wormhole mouths appear as two separate masses in the exterior space, space-time can at best be axially symmetric. We eliminate the non-staticity caused by the gravitational attraction of the mouths by anchoring them by strings held at infinity or, alternatively, by electric repulsion. The space-times are obtained by surgically grafting together well-known solutions of Einstein's equations along timelike hypersurfaces. This surgery naturally concentrates a non-zero stress-energy tensor on the boundary between the two space-times which can be investigated by using the standard thin shell formalism. It turns out that, when using charged black holes, the provided constructions are possible without violation of any of the energy conditions. In general, observers living in the axially symmetric, asymptotically flat (respectively asymptotically de Sitter) region axe able to send causal signals through the topologically non-trivial region. However, the wormhole space-times contain closed timelike curves. Because of this explicit violation of global hyperbolicity these models do not serve as counterexamples to known topological censorship theorems. (author) 7. Gravitational collapse and topology change in spherically symmetric dynamical systems Energy Technology Data Exchange (ETDEWEB) Csizmadia, Peter; Racz, Istvan, E-mail: [email protected], E-mail: [email protected] [RMKI H-1121 Budapest, Konkoly Thege Miklos ut 29-33 (Hungary) 2010-01-07 A new numerical framework, based on the use of a simple first-order strongly hyperbolic evolution equations, is introduced and tested in the case of four-dimensional spherically symmetric gravitating systems. The analytic setup is chosen such that our numerical method is capable of following the time evolution even after the appearance of trapped surfaces, more importantly, until the true physical singularities are reached. Using this framework, the gravitational collapse of various gravity-matter systems is investigated, with particular attention to the evolution in trapped regions. It is verified that, in advance of the formation of these curvature singularities, trapped regions develop in all cases, thereby supporting the validity of the weak cosmic censor hypothesis of Penrose. Various upper bounds on the rate of blow-up of the Ricci and Kretschmann scalars and the Misner-Sharp mass are provided. In spite of the unboundedness of the Ricci scalar, the Einstein-Hilbert action was found to remain finite in all the investigated cases. In addition, important conceptual issues related to the phenomenon of topology changes are discussed. 8. Mechanisms and Management of Diabetic Painful Distal Symmetrical Polyneuropathy Science.gov (United States) Tesfaye, Solomon; Boulton, Andrew J.M.; Dickenson, Anthony H. 2013-01-01 Although a number of the diabetic neuropathies may result in painful symptomatology, this review focuses on the most common: chronic sensorimotor distal symmetrical polyneuropathy (DSPN). It is estimated that 15–20% of diabetic patients may have painful DSPN, but not all of these will require therapy. In practice, the diagnosis of DSPN is a clinical one, whereas for longitudinal studies and clinical trials, quantitative sensory testing and electrophysiological assessment are usually necessary. A number of simple numeric rating scales are available to assess the frequency and severity of neuropathic pain. Although the exact pathophysiological processes that result in diabetic neuropathic pain remain enigmatic, both peripheral and central mechanisms have been implicated, and extend from altered channel function in peripheral nerve through enhanced spinal processing and changes in many higher centers. A number of pharmacological agents have proven efficacy in painful DSPN, but all are prone to side effects, and none impact the underlying pathophysiological abnormalities because they are only symptomatic therapy. The two first-line therapies approved by regulatory authorities for painful neuropathy are duloxetine and pregabalin. α-Lipoic acid, an antioxidant and pathogenic therapy, has evidence of efficacy but is not licensed in the U.S. and several European countries. All patients with DSPN are at increased risk of foot ulceration and require foot care, education, and if possible, regular podiatry assessment. PMID:23970715 9. Symmetric co-movement between Malaysia and Japan stock markets Science.gov (United States) Razak, Ruzanna Ab; Ismail, Noriszura 2017-04-01 The copula approach is a flexible tool known to capture linear, nonlinear, symmetric and asymmetric dependence between two or more random variables. It is often used as a co-movement measure between stock market returns. The information obtained from copulas such as the level of association of financial market during normal and bullish and bearish markets phases are useful for investment strategies and risk management. However, the study of co-movement between Malaysia and Japan markets are limited, especially using copulas. Hence, we aim to investigate the dependence structure between Malaysia and Japan capital markets for the period spanning from 2000 to 2012. In this study, we showed that the bivariate normal distribution is not suitable as the bivariate distribution or to present the dependence between Malaysia and Japan markets. Instead, Gaussian or normal copula was found a good fit to represent the dependence. From our findings, it can be concluded that simple distribution fitting such as bivariate normal distribution does not suit financial time series data, whose characteristics are often leptokurtic. The nature of the data is treated by ARMA-GARCH with heavy tail distributions and these can be associated with copula functions. Regarding the dependence structure between Malaysia and Japan markets, the findings suggest that both markets co-move concurrently during normal periods. 10. Two updating methods for dissipative models with non symmetric matrices International Nuclear Information System (INIS) Billet, L.; Moine, P.; Aubry, D. 1997-01-01 In this paper the feasibility of the extension of two updating methods to rotating machinery models is considered, the particularity of rotating machinery models is to use non-symmetric stiffness and damping matrices. It is shown that the two methods described here, the inverse Eigen-sensitivity method and the error in constitutive relation method can be adapted to such models given some modification.As far as inverse sensitivity method is concerned, an error function based on the difference between right hand calculated and measured Eigen mode shapes and calculated and measured Eigen values is used. Concerning the error in constitutive relation method, the equation which defines the error has to be modified due to the non definite positiveness of the stiffness matrix. The advantage of this modification is that, in some cases, it is possible to focus the updating process on some specific model parameters. Both methods were validated on a simple test model consisting in a two-bearing and disc rotor system. (author) 11. Tartrazine exclusion for allergic asthma. Science.gov (United States) Ardern, K D; Ram, F S 2001-01-01 Tartrazine is the best known and one of the most commonly used food additives. Food colorants are also used in many medications as well as foods. There has been conflicting evidence as to whether tartrazine causes exacerbations of asthma with some studies finding a positive association especially in individuals with cross-sensitivity to aspirin. To assess the overall effect of tartrazine (exclusion or challenge) in the management of asthma. A search was carried out using the Cochrane Airways Group specialised register. Bibliographies of each RCT was searched for additional papers. Authors of identified RCTs were contacted for further information for their trials and details of other studies. RCTs of oral administration of tartrazine (as a challenge) versus placebo or dietary avoidance of tartrazine versus normal diet were considered. Studies which focused upon allergic asthma, were also included. Studies of tartrazine exclusion for other allergic conditions such as hay fever, allergic rhinitis and eczema were only considered if the results for subjects with asthma were separately identified. Trials could be in either adults or children with asthma or allergic asthma (e.g. sensitivity to aspirin or food items known to contain tartrazine). Study quality was assessed and data abstracted by two reviewers independently. Outcomes were analysed using RevMan 4.1.1. Ninety abstracts were found, of which 18 were potentially relevant. Six met the inclusion criteria, but only three presented results in a format that permitted analysis and none could be combined in a meta-analysis. In none of the studies did tartrazine challenge or avoidance in diet significantly alter asthma outcomes. Due to the paucity of available evidence, it is not possible to provide firm conclusions as to the effects of tartrazine on asthma control. However, the six RCTs that could be included in this review all arrived at the same conclusion. Routine tartrazine exclusion may not benefit most patients 12. Exclusive electroproduction of pion pairs International Nuclear Information System (INIS) Warkentin, N.; Schaefer, A.; Diehl, M.; Ivanov, D. Yu. 2007-01-01 We investigate electroproduction of pion pairs on the nucleon in the framework of QCD factorization for hard exclusive processes. We extend previous analyses by taking the hard-scattering coefficients at next-to-leading order in α s . The dynamics of the produced pion pair is described by two-pion distribution amplitudes, for which we perform a detailed theoretical and phenomenological analysis. In particular, we obtain constraints on these quantities by comparing our results with measurements of angular observables that are sensitive to the interference between two-pion production in the isoscalar and isovector channels. (orig.) 13. Exclusion Bounds for Extended Anyons Science.gov (United States) Larson, Simon; Lundholm, Douglas 2018-01-01 We introduce a rigorous approach to the many-body spectral theory of extended anyons, that is quantum particles confined to two dimensions that interact via attached magnetic fluxes of finite extent. Our main results are many-body magnetic Hardy inequalities and local exclusion principles for these particles, leading to estimates for the ground-state energy of the anyon gas over the full range of the parameters. This brings out further non-trivial aspects in the dependence on the anyonic statistics parameter, and also gives improvements in the ideal (non-extended) case. 14. Surfactant-aided size exclusion chromatography NARCIS (Netherlands) Horneman, D.A.; Wolbers, M.; Zomerdijk, M.; Ottens, M.; Keurentjes, J.T.F.; Wielen, van der L.A.M. 2004-01-01 The flexibility and selectivity of size exclusion chromatog. (SEC) for protein purifn. can be modified by adding non-ionic micelle-forming surfactants to the mobile phase. The micelles exclude proteins from a liq. phase similar to the exclusion effect of the polymer fibers of the size exclusion 15. Magnetospectroscopy of symmetric and anti-symmetric states in double quantum wells Science.gov (United States) Marchewka, M.; Sheregii, E. M.; Tralle, I.; Ploch, D.; Tomaka, G.; Furdak, M.; Kolek, A.; Stadler, A.; Mleczko, K.; Zak, D.; Strupinski, W.; Jasik, A.; Jakiela, R. 2008-02-01 The experimental results obtained for magnetotransport in the InGaAs/InAlAs double quantum well (DQW) structures of two different shapes of wells are reported. A beating effect occurring in the Shubnikov-de Haas (SdH) oscillations was observed for both types of structures at low temperatures in the parallel transport when the magnetic field was perpendicular to the layers. An approach for the calculation of the Landau level energies for DQW structures was developed and then applied to the analysis and interpretation of the experimental data related to the beating effect. We also argue that in order to account for the observed magnetotransport phenomena (SdH and integer quantum Hall effect), one should introduce two different quasi-Fermi levels characterizing two electron subsystems regarding the symmetry properties of their states, symmetric and anti-symmetric ones, which are not mixed by electron-electron interaction. 16. Stationary axially symmetric exterior solutions in the five-dimensional representation of the Brans-Dicke-Jordan theory of gravitation International Nuclear Information System (INIS) Bruckman, W. 1986-01-01 The inverse scattering method of Belinsky and Zakharov is used to investigate axially symmetric stationary vacuum soliton solutions in the five-dimensional representation of the Brans-Dicke-Jordan theory of gravitation, where the scalar field of the theory is an element of a five-dimensional metric. The resulting equations for the spacetime metric are similar to those of solitons in general relativity, while the scalar field generated is the product of a simple function of the coordinates and an already known scalar field solution. A family of solutions is considered that reduce, in the absence of rotation, to the five-dimensional form of a well-known Weyl-Levi Civita axially symmetric static vacuum solution. With a suitable choice of parameters, this static limit becomes equivalent to the spherically symmetric solution of the Brans-Dicke theory. An exact metric, in which the Kerr-scalar McIntosh solution is a special case, is given explicitly 17. Implementation of exclusive truck facilities Energy Technology Data Exchange (ETDEWEB) Fekpe, E. [Battelle Memorial Inst., Columbus, OH (United States). Transportation Market Sector 2007-07-01 This paper discussed the issue of highway congestion, safety, and efficiency in freight movement on highways, with particular reference to the challenge of supporting increasing capacity demand from truck traffic. Innovative and practical solutions are needed to address the growing need for more efficient freight movement while maintaining acceptable levels of safety on highways. The concept of exclusive truck facilities (ETFs) is becoming an attractive option as a feasible strategy to help stabilize traffic flow, reduce congestion, improve safety, enhance transportation system management, improve access to freight facilities, and improve efficiency in freight movement along corridors of national importance. ETFs can either be truck only lanes or truckways. Passenger cars may not use ETFs. However, the use of ETFs could involve high costs of construction, maintenance, and acquisition of additional right of way. A cost-benefit analysis was performed for alternative ETF configurations under different traffic and site characteristics. A set of criteria was then proposed for identifying suitable locations for exclusive truck lanes. It was proposed that ETFs are economically feasible at locations with traffic volume of 100,000 vehicles per day or more and with a truck volume of at least 25 per cent of the traffic. In addition, the rate of truck-involved fatal crashes and level of service should be used to prioritize preliminary candidate locations that satisfy the traffic criteria. Consideration should also be given to the existence of freight terminals, ports, processing centers or regional distribution centres that are close to highways. 11 refs., 1 tab. 18. Exclusive scattering off the deuteron Energy Technology Data Exchange (ETDEWEB) Amrath, D. 2007-12-15 Exclusive processes are a special class of processes giving insight into the inner structure of hadrons. In this thesis we consider two exclusive processes and compute their total cross sections as well as the beam charge and beam polarization asymmetries for different kinematical constraints. These calculations o er the opportunity to get access to the nonperturbative GPDs. Theoretically they can be described with the help of models. The rst process we investigate contains a GPD of the pion, which is basically unknown so far. We include different models and make predictions for observables that could in principle be measured at HERMES at DESY and CLAS at JLab. The second process we consider is electron-deuteron scattering in the kinematical range where the deuteron breaks up into a proton and a neutron. This can be used to investigate the neutron, which cannot be taken as a target due to its lifetime of approximately 15 minutes. For the calculation of the electron-deuteron cross section we implement models for the proton and neutron GPDs. Once there are experimental data available our calculations are ready for comparison. (orig.) 19. Is the world simple or complicated CERN Document Server Barrow, John D 1998-01-01 Stop some particle physicists in the street and they will soon be trying to persuade you that the world is altogether simple and symmetrical. But stop a biologist, an economist, or a social scientist and they will tell you quite the opposite: the world is a higgledy-piggledy collection of complexities that owes little to symmetry and displays precious little simplicity. So who is right : is the world really complicated or is it simple ? We shall look at the reasoning that leads to these different conclusions, show why we got different answers to our question, and look at some of the recent developments that have taken place in the study of systems from sand-piles to music on the border between order and chaos. We shall also look at some of the connections between our aesthetic sensibilities and the structure of scientific theories. 20. Crossing simple resonances International Nuclear Information System (INIS) Collins, T. 1985-08-01 A simple criterion governs the beam distortion and/or loss of protons on a fast resonance crossing. Results from numerical integrations are illustrated for simple sextupole, octupole, and 10-pole resonances 1. Crossing simple resonances Energy Technology Data Exchange (ETDEWEB) Collins, T. 1985-08-01 A simple criterion governs the beam distortion and/or loss of protons on a fast resonance crossing. Results from numerical integrations are illustrated for simple sextupole, octupole, and 10-pole resonances. 2. Exclusive Breastfeeding Determinants in Breastfeeding Mother Directory of Open Access Journals (Sweden) Ika Mustika 2017-04-01 Full Text Available Exclusive breastfeeding until 6 month is very important for baby. The proportion of mothers who exclusively breastfeed their babies up to 6 months remains low. Factors influencing the exclusive breastfeeding namely sociodemograph factors , factors pre / post delivery , and psychosocial factors. This aims of this study to identify determinant factors of exclusive breastfeeding on mother. This research method is a systematic review , by analyzing the various studies on exclusive breastfeeding. There are 17 studies. The results obtained occupational factors most studied with significant results ( median OR = 1.265 . Psychosocial factors that have significant relationship is support of her husband (average OR = 4.716 and family support ( average OR = 1.770 . Conclusions : factors influencing the exclusive breastfeeding is occupational factor. Socialization and support from people nearby, health workers, and all parties is needed for exclusive breastfeeding for six months can be achieved. 3. Entangling capabilities of symmetric two-qubit gates Com- putational investigation of entanglement of such ensembles is therefore impractical for ... the computational complexity. Pairs of spin-1 ... tensor operators which can also provide different symmetric logic gates for quantum pro- ... that five of the eight, two-qubit symmetric quantum gates expressed in terms of our newly. 4. SUSY formalism for the symmetric double well potential symmetric double well potential barrier we have obtained a class of exactly solvable potentials subject to moving boundary condition. The eigenstates are also obtained by the same technique. Keywords. SUSY; moving boundary condition; exactly solvable; symmetric double well; NH3 molecule. PACS Nos 02.30.Ik; 03.50. 5. A New Formulation for Symmetric Implicit Runge-Kutta-Nystrom ... African Journals Online (AJOL) In this paper we derive symmetric stable Implicit Runge-Kutta –Nystrom Method for the Integration of General Second Order ODEs by using the collocation approach.The block hybrid method obtained by the evaluation of the continuous interpolant at different nodes of the polynomial is symmetric and suitable for stiff intial ... 6. Crossing symmetric solution of the Chew-Low equation International Nuclear Information System (INIS) McLeod, R.J.; Ernst, D.J. 1982-01-01 An N/D dispersion theory is developed which solves crossing symmetric Low equations. The method is used to generate crossing symmetric solutions to the Chew-Low model. We show why the technique originally proposed by Chew and Low was incapable of producing solutions. (orig.) 7. Sparse symmetric preconditioners for dense linear systems in electromagnetism NARCIS (Netherlands) Carpentieri, Bruno; Duff, Iain S.; Giraud, Luc; Monga Made, M. Magolu 2004-01-01 We consider symmetric preconditioning strategies for the iterative solution of dense complex symmetric non-Hermitian systems arising in computational electromagnetics. In particular, we report on the numerical behaviour of the classical incomplete Cholesky factorization as well as some of its recent 8. Stability of transparent spherically symmetric thin shells and wormholes International Nuclear Information System (INIS) Ishak, Mustapha; Lake, Kayll 2002-01-01 The stability of transparent spherically symmetric thin shells (and wormholes) to linearized spherically symmetric perturbations about static equilibrium is examined. This work generalizes and systematizes previous studies and explores the consequences of including the cosmological constant. The approach shows how the existence (or not) of a domain wall dominates the landscape of possible equilibrium configurations 9. Coupled dilaton and electromagnetic field in cylindrically symmetric ... The dilaton black hole solutions have attracted considerable attention for the ... theory and study the corresponding cylindrically symmetric spacetime, where .... where Йm and Йe are integration constants to be interpreted later as the ..... feature is apparent for the cylindrically symmetric spacetime in the presence of the dila-. 10. Radon transformation on reductive symmetric spaces: support theorems NARCIS (Netherlands) Kuit, J.J.\|info:eu-repo/dai/nl/313872589 2011-01-01 In this thesis we introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and study some of their properties. In particular we obtain a generalization of Helgason's support theorem for the horospherical transform on a Riemannian symmetric space. 11. New approach to solve symmetric fully fuzzy linear systems In this paper, we present a method to solve fully fuzzy linear systems with symmetric coefﬁcient matrix. The symmetric coefﬁcient matrix is decomposed into two systems of equations by using Cholesky method and then a solution can be obtained. Numerical examples are given to illustrate our method. 12. Synthesis & Characterization of New bis-Symmetrical Adipoyl ... African Journals Online (AJOL) Full Title: Synthesis and Characterization of New bis-Symmetrical Adipoyl, Terepthaloyl, Chiral Diimido-di-L-alanine Diesters and Chiral Phthaloyl-L-alanine Ester of Tripropoxy p-tert-Butyl Calix[4]arene and Study of Their Hosting Ability for Alanine and Na+. Bis-symmetrical tripropoxy p-tert-butyl calix[4]arene esters were ... 13. FACES WITH LARGE DIAMETER ON THE SYMMETRICAL TRAVELING SALESMAN POLYTOPE NARCIS (Netherlands) SIERKSMA, G; TIJSSEN, GA This paper deals with the symmetric traveling salesman polytope and contains three main theorems. The first one gives a new characterization of (non)adjacency. Based on this characterization a new upper bound for the diameter of the symmetric traveling salesman polytope (conjectured to be 2 by M. 14. Symmetric metamaterials based on flower-shaped structure International Nuclear Information System (INIS) Tuong, P.V.; Park, J.W.; Rhee, J.Y.; Kim, K.W.; Cheong, H.; Jang, W.H.; Lee, Y.P. 2013-01-01 We proposed new models of metamaterials (MMs) based on a flower-shaped structure (FSS), whose “meta-atoms” consist of two flower-shaped metallic parts separated by a dielectric layer. Like the non-symmetric MMs based on cut-wire-pairs or electric ring resonators, the symmetrical FSS demonstrates the negative permeability at GHz frequencies. Employing the results, we designed a symmetric negative-refractive-index MM [a symmetric combined structure (SCS)], which is composed of FSSs and cross continuous wires. The MM properties of the FSS and the SCS are presented numerically and experimentally. - Highlights: • A new designed of sub-wavelength metamaterial, flower-shaped structure was proposed. • Flower-shaped meta-atom illustrated effective negative permeability. • Based on the meta-atom, negative refractive index was conventionally gained. • Negative refractive index was demonstrated with symmetric properties for electromagnetic wave. • Dimensional parameters were studied under normal electromagnetic wave 15. How to model mutually exclusive events based on independent causal pathways in Bayesian network models OpenAIRE Fenton, N.; Neil, M.; Lagnado, D.; Marsh, W.; Yet, B.; Constantinou, A. 2016-01-01 We show that existing Bayesian network (BN) modelling techniques cannot capture the correct intuitive reasoning in the important case when a set of mutually exclusive events need to be modelled as separate nodes instead of states of a single node. A previously proposed ‘solution’, which introduces a simple constraint node that enforces mutual exclusivity, fails to preserve the prior probabilities of the events, while other proposed solutions involve major changes to the original model. We pro... 16. efficient and convenient synthesis of symmetrical carboxylic African Journals Online (AJOL) Preferred Customer The product can be isolated by a simple extraction with organic solvent, and the catalyst ... reactant, either dissolved in water (liquid-liquid) or present in solid state ... solution was stirred for 16 h at 35 °C followed by filtration and washing with ... 17. Symmetric weak ternary quantum homomorphic encryption schemes Science.gov (United States) Wang, Yuqi; She, Kun; Luo, Qingbin; Yang, Fan; Zhao, Chao 2016-03-01 Based on a ternary quantum logic circuit, four symmetric weak ternary quantum homomorphic encryption (QHE) schemes were proposed. First, for a one-qutrit rotation gate, a QHE scheme was constructed. Second, in view of the synthesis of a general 3 × 3 unitary transformation, another one-qutrit QHE scheme was proposed. Third, according to the one-qutrit scheme, the two-qutrit QHE scheme about generalized controlled X (GCX(m,n)) gate was constructed and further generalized to the n-qutrit unitary matrix case. Finally, the security of these schemes was analyzed in two respects. It can be concluded that the attacker can correctly guess the encryption key with a maximum probability pk = 1/33n, thus it can better protect the privacy of users’ data. Moreover, these schemes can be well integrated into the future quantum remote server architecture, and thus the computational security of the users’ private quantum information can be well protected in a distributed computing environment. 18. Skyrmions and vector mesons: a symmetric approach International Nuclear Information System (INIS) Caldi, D.G. 1984-01-01 We propose an extension of the effective, low-energy chiral Lagrangian known as the Skyrme model, to one formulated by a non-linear sigma model generalized to include vector mesons in a symmetric way. The model is based on chiral SU(6) x SU(6) symmetry spontaneously broken to static SU(6). The rho and other vector mesons are dormant Goldstone bosons since they are in the same SU(6) multiplet as the pion and other pseudoscalars. Hence the manifold of our generalized non-linear sigma model is the coset space (SU(6) x SU(6))/Su(6). Relativistic effects, via a spin-dependent mass term, break the static SU(6) and give the vectors a mass. The model can then be fully relativistic and covariant. The lowest-lying Skyrmion in this model is the whole baryonic 56-plet, which splits into the octet and decuplet in the presence of relativistic SU(6)-breaking. Due to the built-in SU(6) and the presence of vector mesons, the model is expected to have better phenomenological results, as well as providing a conceptually more unified picture of mesons and baryons. 29 references 19. Randomized Symmetric Crypto Spatial Fusion Steganographic System Directory of Open Access Journals (Sweden) Viswanathan Perumal 2016-06-01 Full Text Available The image fusion steganographic system embeds encrypted messages in decomposed multimedia carriers using a pseudorandom generator but it fails to evaluate the contents of the cover image. This results in the secret data being embedded in smooth regions, which leads to visible distortion that affects the imperceptibility and confidentiality. To solve this issue, as well as to improve the quality and robustness of the system, the Randomized Symmetric Crypto Spatial Fusion Steganography System is proposed in this study. It comprises three-subsystem bitwise encryption, spatial fusion, and bitwise embedding. First, bitwise encryption encrypts the message using bitwise operation to improve the confidentiality. Then, spatial fusion decomposes and evaluates the region of embedding on the basis of sharp intensity and capacity. This restricts the visibility of distortion and provides a high embedding capacity. Finally, the bitwise embedding system embeds the encrypted message through differencing the pixels in the region by 1, checking even or odd options and not equal to zero constraints. This reduces the modification rate to avoid distortion. The proposed heuristic algorithm is implemented in the blue channel, to which the human visual system is less sensitive. It was tested using standard IST natural images with steganalysis algorithms and resulted in better quality, imperceptibility, embedding capacity and invulnerability to various attacks compared to other steganographic systems. 20. Triple symmetric key cryptosystem for data security Science.gov (United States) Fuzail, C. Md; Norman, Jasmine; Mangayarkarasi, R. 2017-11-01 As the technology is getting spreads in the macro seconds of speed and in which the trend changing era from human to robotics the security issue is also getting increased. By means of using machine attacks it is very easy to break the cryptosystems in very less amount of time. Cryptosystem is a process which provides the security in all sorts of processes, communications and transactions to be done securely with the help of electronical mechanisms. Data is one such thing with the expanded implication and possible scraps over the collection of data to secure predominance and achievement, Information Security is the process where the information is protected from invalid and unverified accessibilities and data from mishandling. So the idea of Information Security has risen. Symmetric key which is also known as private key.Whereas the private key is mostly used to attain the confidentiality of data. It is a dynamic topic which can be implemented over different applications like android, wireless censor networks, etc. In this paper, a new mathematical manipulation algorithm along with Tea cryptosystem has been implemented and it can be used for the purpose of cryptography. The algorithm which we proposed is straightforward and more powerful and it will authenticate in harder way and also it will be very difficult to break by someone without knowing in depth about its internal mechanisms. 1. Experimental pseudo-symmetric trap EPSILON International Nuclear Information System (INIS) Skovoroda, A.A.; Arsenin, V.V.; Dlougach, E.D.; Kulygin, V.M.; Kuyanov, A.Yu.; Timofeev, A.V.; Zhil'tsov, V.A.; Zvonkov, A.V. 2001-01-01 Within the framework of the conceptual project 'Adaptive Plasma EXperiment' a trap with the closed magnetic field lines 'Experimental Pseudo-Symmetric trap' is examined. The project APEX is directed at the theoretical and experimental development of physical foundations for stationary thermonuclear reactor on the basis of an alternative magnetic trap with tokamak-level confinement of high β plasma. The fundamental principle of magnetic field pseudosymmetry that should be satisfied for plasma to have tokamak-like confinement is discussed. The calculated in paraxial approximation examples of pseudosymmetric curvilinear elements with poloidal direction of B isolines are adduced. The EPSILON trap consisting of two straight axisymmetric mirrors linked by two curvilinear pseudosymmetric elements is considered. The plasma currents are short-circuited within the curvilinear element what increases the equilibrium β. The untraditional scheme of MHD stabilization of a trap with the closed field lines by the use of divertor inserted into axisymmetric mirror is analyzed. The experimental installation EPSILON-OME that is under construction for experimental check of divertor stabilization is discussed. The possibility of ECR plasma production in EPSILON-OME under conditions of high density and small magnetic field is examined. (author) 2. Left-right symmetric superstring supergravitation International Nuclear Information System (INIS) Burova, M.V.; Ter-Martirosyan, K.E. 1988-01-01 A left-right (L-R) symmetric model of four-dimensional supergravitation with a SO(10) gauge group obtained as the low-energy limit is superstring theory is considered. The spectrum of the gauge fields and their interactions are in agreement with the Weinberg-Salam theory. In addition, the model includes heavy W R ± and Z μ ' bosons. Beside the N g =3 generations of the 16-plets the SO(10) model includes the fragments of such generations which play the role of Higgs particles and also scalar chiral filds, the number of which exceeds by one the number of generations. As a result the neutrinos of each generation obtain a stable small Majorana mass. It is shown that the scalar field potential leads to spontaneous violation of the SU(2) R group and L-R symmetry and at low energies the standard Weinberg-Salam theory appears. However, reasonable values of X bosons masses M x and sun 2 Θ W (Θ W is the Weinberg angle) can be obtained in the model only in the case of high mass scale M R ∼10 10 -10 12 GeV of the right group SU(2) R violation 3. Symmetric charge transfer cross section of uranium International Nuclear Information System (INIS) Shibata, Takemasa; Ogura, Koichi 1995-03-01 Symmetric charge transfer cross section of uranium was calculated under consideration of reaction paths. In the charge transfer reaction a d 3/2 electron in the U atom transfers into the d-electron site of U + ( 4 I 9/2 ) ion. The J value of the U atom produced after the reaction is 6, 5, 4 or 3, at impact energy below several tens eV, only resonant charge transfer in which the product atom is ground state (J=6) takes place. Therefore, the cross section is very small (4-5 x 10 -15 cm 2 ) compared with that considered so far. In the energy range of 100-1000eV the cross section increases with the impact energy because near resonant charge transfer in which an s-electron in the U atom transfers into the d-electron site of U + ion. Charge transfer cross section between U + in the first excited state (289 cm -1 ) and U in the ground state was also obtained. (author) 4. Dynamics of an exclusion process with creation and annihilation International Nuclear Information System (INIS) Juhasz, Robert; Santen, Ludger 2004-01-01 We examine the dynamical properties of an exclusion process with creation and annihilation of particles in the framework of a phenomenological domain-wall theory, by scaling arguments and by numerical simulation. We find that the length and the time scales are finite in the maximum current phase for finite creation and annihilation rates as opposed to the algebraically decaying correlations of the totally asymmetric simple exclusion process (TASEP). Critical exponents of the transition to the TASEP are determined. The case where bulk creation and annihilation rates vanish faster than the inverse of the system size N is also analysed. We point out that shock localization is possible even for rates proportional to N -a , 1 < a < 2 5. Two-body relativistic scattering with an O(1,1)-symmetric square-well potential International Nuclear Information System (INIS) Arshansky, R.; Horwitz, L.P. 1984-01-01 Scattering theory in the framework of a relativistic manifestly covariant quantum mechanics is applied to the relativistic analog of the nonrelativistic one-dimensional square-well potential, a two-body O(1,1)-symmetric hyperbolic square well in one space and one time dimension. The unitary S matrix is explicitly obtained. For well sizes large compared to the de Broglie wavelength of the reduced motion system, simple formulas are obtained for the associated sequence of resonances. This sequence has equally spaced levels and constant widths for higher resonances, and linearly increasing widths for lower-lying levels 6. Z4-symmetric factorized S-matrix in two space-time dimensions International Nuclear Information System (INIS) Zamolodchikov, A.B. 1979-01-01 The factorized S-matrix with internal symmetry Z 4 is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. The S-matrix cotains the resonance poles naturally. The simple formal relation between the general factorized S-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation the Z 4 -symmetric S-matrix corresponds to the Baxter transfer matrix itself. (orig.) 7. Mixed boson-fermion description of correlated electrons: Fluctuation corrections in the symmetric treatment International Nuclear Information System (INIS) Vicente Alvarez, J.J.; Balseiro, C.A.; Ceccatto, H.A. 1995-07-01 We consider the introduction of fluctuation corrections to saddle- point results in the symmetric treatment of a mixed pseudofermion-boson representation of correlated electrons. In our calculations we avoid the complications of working in the discrete imaginary-time formulation of the functional integral, a procedure recently advocated in the literature as mandatory for this problem. For a simple two-site model our approach leads to approximate results in remarkable agreement with the exact ones, and without the spurious nonanalyticities of other similar treatments. (author). 14 refs, 2 figs 8. Physical aspects of pseudo-Hermitian and PT-symmetric quantum mechanics International Nuclear Information System (INIS) 2004-01-01 For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT-symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h, the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT-symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT-symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics 9. Iterative solution of the Grad-Shafranov equation in symmetric magnetic coordinates International Nuclear Information System (INIS) Brambilla, Marco 2003-01-01 The inverse Grad-Shafranov equation for axisymmetric magnetohydrodynamic equilibria is reformulated in symmetric magnetic coordinates (in which magnetic field lines look 'straight', and the geometric toroidal angle is one of the coordinates). The poloidally averaged part of the equilibrium condition and Ampere law takes the form of two first-order ordinary differential equations, with the two arbitrary flux functions, pressure and force-free part of the current density, as sources. The condition for the coordinates to be flux coordinates, and the poloidally varying part of the equilibrium equation are similarly transformed into a set of first-order ordinary differential equations, with coefficients depending on the metric, and explicitly solved for the radial derivatives of the coefficients of the Fourier representation of the Cartesian coordinates in the poloidal angle. The derivation exploits the existence of Boozer-White coordinates, but does not require to find these coordinates explicitly; on the other hand, it offers a simple recipe to perform the transformation to Boozer-White coordinates, if required. Use of symmetric flux coordinates is advantageous for the formulation of many problems of equilibrium, stability, and wave propagation in tokamak plasmas, since these coordinates have the simplest metric of their class. It is also shown that in symmetric flux coordinates the Lagrangian equations of the drift motion of charged particles are automatically solved for the time derivatives, with right-hand sides closely related to the coefficients of the inverse Grad-Shafranov equation 10. Comparison of eigensolvers for symmetric band matrices. Science.gov (United States) Moldaschl, Michael; Gansterer, Wilfried N 2014-09-15 We compare different algorithms for computing eigenvalues and eigenvectors of a symmetric band matrix across a wide range of synthetic test problems. Of particular interest is a comparison of state-of-the-art tridiagonalization-based methods as implemented in Lapack or Plasma on the one hand, and the block divide-and-conquer (BD&C) algorithm as well as the block twisted factorization (BTF) method on the other hand. The BD&C algorithm does not require tridiagonalization of the original band matrix at all, and the current version of the BTF method tridiagonalizes the original band matrix only for computing the eigenvalues. Avoiding the tridiagonalization process sidesteps the cost of backtransformation of the eigenvectors. Beyond that, we discovered another disadvantage of the backtransformation process for band matrices: In several scenarios, a lot of gradual underflow is observed in the (optional) accumulation of the transformation matrix and in the (obligatory) backtransformation step. According to the IEEE 754 standard for floating-point arithmetic, this implies many operations with subnormal (denormalized) numbers, which causes severe slowdowns compared to the other algorithms without backtransformation of the eigenvectors. We illustrate that in these cases the performance of existing methods from Lapack and Plasma reaches a competitive level only if subnormal numbers are disabled (and thus the IEEE standard is violated). Overall, our performance studies illustrate that if the problem size is large enough relative to the bandwidth, BD&C tends to achieve the highest performance of all methods if the spectrum to be computed is clustered. For test problems with well separated eigenvalues, the BTF method tends to become the fastest algorithm with growing problem size. 11. Survival and transmission of symmetrical chromosomal aberrations International Nuclear Information System (INIS) Savage, J.R.K. 1979-01-01 The interaction between the lesions to produce chromosomal structural changes may be either asymmetrical (A) or symmetrical (S). In A, one or more acentric fragments are always produced, and there may also be the mechanical separation problems resulting from bridges at anaphase, while S-changes never produce fragment, and pose no mechanical problem in cell division. If A and S events occur with equal frequency, it might be an indication that they are truly the alternative modes of lesion interaction. Unstimulated lymphocytes were irradiated with 2.68 Gy 250 kV X-ray, and metaphases were sampled at 50 h after the stimulation. Preparations were complete diploid cells, and any obvious second division cells were rejected. So far as dermal repair and fibroblast functions are concerned, aberration burden seems to have little consequence from the view-point of the long-term survival in vivo. Large numbers of aberrations (mainly S translocation and terminal deletion) were found in the samples taken up to 60 years after therapy. Skin biopsies were removed 1 day and 6 months after irradiation and cultured. In irradiated cells, reciprocal translocations dominated, followed by terminal deletions, then inversions, while no chromosome-type aberration was seen in the control cells. a) The relative occurrence of A : S changes, b) long-term survival in vivo, c) the possibility of in vivo repair, and d) some unusual features of translocation found in Syrian hamsters are reviewed. The relevance or importance of major S events is clearly dependent upon the cells, the tissues or the organisms in which they occur. (Yamashita, S.) 12. Exclusive Higgs production at the LHC Energy Technology Data Exchange (ETDEWEB) Dechambre, Alice [Universite de Liege, Institut d' Astrophysique et de Geophysique, Allee du 6 aout, 17 - Bat. B5c, B-4000 Liege 1 - Sart-Tilman (Belgium); Staszewski, Rafal [IRFU/SPP, CEA-Saclay, bat. 141, 91191 Gif-sur-Yvette Cedex (France); Henryk Niewodniczanski, Institute of Nuclear Physics - PAN, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Krakow (Poland); Royon, Christophe [IRFU/SPP, CEA-Saclay, bat. 141, 91191 Gif-sur-Yvette Cedex (France) 2010-07-01 After a brief description of the models of exclusive diffractive Higgs production, we first evaluate the theoretical uncertainties that affect the calculation of exclusive cross section (jets, Higgs...). In addition, in view of the recent measurement of exclusive di-jet at CDF and the new implementation of the corresponding cross section in FPMC (Forward Physics Monte-Carlo), we developed an analysis strategy that can be used to narrow down these uncertainties with the help of early LHC measurement. (authors) 13. Negotiations and Exclusivity Contracts for Advertising OpenAIRE Anthony Dukes; Esther Gal–Or 2003-01-01 Exclusive advertising on a given media outlet is usually profitable for an advertiser because consumers are less aware of competing products. However, for such arrangements to exist, media must benefit as well. We examine conditions under which such exclusive advertising contracts benefit both advertisers and media outlets (referred to as ) by illustrating that exclusive equilibria arise in a theoretical model of the media, advertisers, and consumers who participate in both the product and me... 14. Inclusive education and social exclusion Directory of Open Access Journals (Sweden) Maria Luisa Bissoto 2013-01-01 Full Text Available The aim of this paper is critically examining assumptions underlying the Inclusive Education concept, arguing that this can only be effectively considered when understood in a broader context of social inclusion and exclusion. Methodologically, this article relies on international documents and bibliographic references about Inclusive Education, that have been chosen by systematize and characterize different social and educational inclusive practices, encouraging the elaboration of a general overview on this topic. The results of this analysis conclude that it is essential for Inclusive Education that educational institutions review their goals and reasons of social existence. In the concluding remarks it is argued that education is better understood as the act of encouraging and welcoming the efforts of individuals in their attempts to engage in social networking, which sustains life. This includes the acceptance of other reality interpretations and understanding that educational action cannot be restricted by the walls of institutions. It requires the participation of the whole community. Action perspectives likely to promote social inclusion and inclusive education are suggested. 15. Perturbative QCD and exclusive processes International Nuclear Information System (INIS) Bennett, J.; Hawes, F.; Zhao, M.; Zyla, P. 1991-01-01 The authors discuss perturbation theory as applied to particle physics calculations. In particle physics one is generally interested in the scattering amplitude for a system going from some initial state to a final state. The intermediate state or states are unknown. To get the scattering amplitude it is necessary to sum the contributions from processes which pass through all possible intermediate states. Intermediate states involve the exchange of intermediate vector bosons between the particles, and with this interaction is associated a coupling constant α. Each additional boson exchange involves an additional contribution of α to the coupling. If α is less than 1, one can see that the relative contribution of higher order processes is less and less important as α falls. In QCD the gluons serve as the intermediate vector bosons exchanged by quarks and gluons, and the interaction constant is not really a constant, but depends upon the distance between the particles. At short distances the coupling is small, and one can assume perturbative expansions may converge rapidly. Exclusive scattering processes, as opposed to inclusive, are those in which all of the final state products are detected. The authors then discuss the application of perturbative QCD to the deuteron. The issues of chiral conservation and color transparancy are also discussed, in the scheme of large Q 2 interations, where perturbative QCD should be applicable 16. Exclusive processes in pp collisions in CMS OpenAIRE da Silveira, Gustavo G.; Collaboration, for the CMS 2013-01-01 We report the results on the searches of exclusive production of low- and high-mass pairs with the Compact Muon Solenoid (CMS) detector in proton-proton collisions at $\\sqrt{s}$ = 7 TeV. The analyses comprise the central exclusive $\\gamma\\gamma$ production, the exclusive two-photon production of dileptons, $e^{+}e^{-}$ and $\\mu^{+}\\mu^{-}$, and the exclusive two-photon production of $W$ pairs in the asymmetric $e^{\\pm}\\mu^{\\mp}$ decay channel. No diphotons candidates are observed in data and ... 17. A unified approach for the synthesis of symmetrical and unsymmetrical dibenzyl ethers from aryl aldehydes through reductive etherification Directory of Open Access Journals (Sweden) J. Sembian Ruso 2016-05-01 Full Text Available In this paper, we describe a simple and convenient conversion of aryl aldehydes to symmetrical dibenzyl ethers through reductive etherification. Similarly, unsymmetrical dibenzyl ether was obtained from aryl aldehyde and TES-protected benzyl alcohol. Triethyl silane with catalytic amount of InCl3 was found to be an efficient condition for the reductive etherification. Moreover, it exhibits remarkable functional group compatibility with yield ranging from good to excellent. 18. Radon transformation on reductive symmetric spaces:Support theorems DEFF Research Database (Denmark) Kuit, Job Jacob 2013-01-01 We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms. A reductive symmetric space is a homogeneous space G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open sub...... is based on the relation between the Radon transform and the Fourier transform on G/H, and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.... 19. Nilpotent orbits in real symmetric pairs and stationary black holes International Nuclear Information System (INIS) Dietrich, Heiko; De Graaf, Willem A.; Ruggeri, Daniele; Trigiante, Mario 2017-01-01 In the study of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determine the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of (SL 2 (R)) 4 acting on the fourth tensor power of the natural 2-dimensional SL 2 (R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) 20. Color symmetrical superconductivity in a schematic nuclear quark model DEFF Research Database (Denmark) Bohr, Henrik; Providencia, C.; da Providencia, J. 2010-01-01 In this letter, a novel BCS-type formalism is constructed in the framework of a schematic QCD inspired quark model, having in mind the description of color symmetrical superconducting states. In the usual approach to color superconductivity, the pairing correlations affect only the quasi-particle...... states of two colors, the single-particle states of the third color remaining unaffected by the pairing correlations. In the theory of color symmetrical superconductivity here proposed, the pairing correlations affect symmetrically the quasi-particle states of the three colors and vanishing net color... 1. Highly-dispersive electromagnetic induced transparency in planar symmetric metamaterials. Science.gov (United States) Lu, Xiqun; Shi, Jinhui; Liu, Ran; Guan, Chunying 2012-07-30 We propose, design and experimentally demonstrate highly-dispersive electromagnetically induced transparency (EIT) in planar symmetric metamaterials actively switched and controlled by angles of incidence. Full-wave simulation and measurement results show EIT phenomena, trapped-mode excitations and the associated local field enhancement of two symmetric metamaterials consisting of symmetrically split rings (SSR) and a fishscale (FS) metamaterial pattern, respectively, strongly depend on angles of incidence. The FS metamaterial shows much broader spectral splitting than the SSR metamaterial due to the surface current distribution variation. 2. Geometric characteristics of aberrations of plane-symmetric optical systems International Nuclear Information System (INIS) Lu Lijun; Deng Zhiyong 2009-01-01 The geometric characteristics of aberrations of plane-symmetric optical systems are studied in detail with a wave-aberration theory. It is dealt with as an extension of the Seidel aberrations to realize a consistent aberration theory from axially symmetric to plane-symmetric systems. The aberration distribution is analyzed with the spot diagram of a ray and an aberration curve. Moreover, the root-mean-square value and the centroid of aberration distribution are discussed. The numerical results are obtained with the focusing optics of a toroidal mirror at grazing incidence. 3. Path integral representation of the symmetric Rosen-Morse potential International Nuclear Information System (INIS) Duru, I.H. 1983-09-01 An integral formula for the Green's function of symmetric Rosen-Morse potential is obtained by solving path integrals. The correctly normalized wave functions and bound state energy spectrum are derived. (author) 4. The geometrical theory of diffraction for axially symmetric reflectors DEFF Research Database (Denmark) Rusch, W.; Sørensen, O. 1975-01-01 The geometrical theory of diffraction (GTD) (cf. [1], for example) may be applied advantageously to many axially symmetric reflector antenna geometries. The material in this communication presents analytical, computational, and experimental results for commonly encountered reflector geometries... 5. Symmetric Pin Diversion Detection using a Partial Defect Detector (PDET) International Nuclear Information System (INIS) Sitaraman, S.; Ham, Y.S. 2009-01-01 Since the signature from the Partial Defect Detector (PDET) is principally dependent on the geometric layout of the guide tube locations, the capability of the technique in detecting symmetric diversion of pins needs to be determined. The Monte Carlo simulation study consisted of cases where pins were removed in a symmetric manner and the resulting signatures were examined. In addition to the normalized gamma-to-neutron ratios, the neutron and gamma signatures normalized to their maximum values, were also examined. Examination of the shape of the three curves as well as of the peak-to-valley differences in excess of the maximum expected in intact assemblies, indicated pin diversion. A set of simulations with various symmetric patterns of diversion were examined. The results from these studies indicated that symmetric diversions as low as twelve percent could be detected by this methodology 6. Systems of Differential Equations with Skew-Symmetric, Orthogonal Matrices Science.gov (United States) Glaister, P. 2008-01-01 The solution of a system of linear, inhomogeneous differential equations is discussed. The particular class considered is where the coefficient matrix is skew-symmetric and orthogonal, and where the forcing terms are sinusoidal. More general matrices are also considered. 7. A Paley-Wiener theorem for reductive symmetric spaces NARCIS (Netherlands) Ban, E.P. van den; Schlichtkrull, H. 2006-01-01 Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized. 8. Report on the Dynamical Evolution of an Axially Symmetric Quasar ... retical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. ... galaxies and non-axially symmetric quasar models (see Papadopoulos & Caranicolas. 9. first principles derivation of a stress function for axially symmetric African Journals Online (AJOL) HOD governing partial differential equations of linear isotropic elasticity were reduced to the solution of the biharmonic ... The stress function was then applied to solve the axially symmetric ..... [1] Borg S.K.: Fundamentals of Engineering Elasticity,. 10. Symmetrization of mathematical model of charge transport in semiconductors Directory of Open Access Journals (Sweden) Alexander M. Blokhin 2002-11-01 Full Text Available A mathematical model of charge transport in semiconductors is considered. The model is a quasilinear system of differential equations. A problem of finding an additional entropy conservation law and system symmetrization are solved. 11. An algebraic approach to the non-symmetric Macdonald polynomial International Nuclear Information System (INIS) Nishino, Akinori; Ujino, Hideaki; Wadati, Miki 1999-01-01 In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them 12. Hardware Realization of Chaos Based Symmetric Image Encryption KAUST Repository Barakat, Mohamed L. 2012-01-01 This thesis presents a novel work on hardware realization of symmetric image encryption utilizing chaos based continuous systems as pseudo random number generators. Digital implementation of chaotic systems results in serious degradations 13. Experimental technique of calibration of symmetrical air pollution ... Based on the inherent property of symmetry of air pollution models, a Symmetrical Air Pollution. Model ... process is in compliance with air pollution regula- ..... Ground simulation is achieved through MATLAB package which is based on least-. 14. Hardware Realization of Chaos-based Symmetric Video Encryption KAUST Repository 2013-01-01 This thesis reports original work on hardware realization of symmetric video encryption using chaos-based continuous systems as pseudo-random number generators. The thesis also presents some of the serious degradations caused by digitally 15. Invariant subspaces in some function spaces on symmetric spaces. II International Nuclear Information System (INIS) Platonov, S S 1998-01-01 Let G be a semisimple connected Lie group with finite centre, K a maximal compact subgroup of G, and M=G/K a Riemannian symmetric space of non-compact type. We study the problem of describing the structure of closed linear subspaces in various function spaces on M that are invariant under the quasiregular representation of the group G. We consider the case when M is a symplectic symmetric space of rank 1 16. Symmetric coupling of four spin-1/2 systems Science.gov (United States) Suzuki, Jun; Englert, Berthold-Georg 2012-06-01 We address the non-binary coupling of identical angular momenta based upon the representation theory for the symmetric group. A correspondence is pointed out between the complete set of commuting operators and the reference-frame-free subsystems. We provide a detailed analysis of the coupling of three and four spin-1/2 systems and discuss a symmetric coupling of four spin-1/2 systems. 17. Multiple symmetrical lipomatosis (Madelung's disease) - a case report International Nuclear Information System (INIS) Vieira, Marcelo Vasconcelos; Abreu, Marcelo de; Furtado, Claudia Dietz; Silveira, Marcio Fleck da; Furtado, Alvaro Porto Alegre; Genro, Carlos Horacio; Grazziotin, Rossano Ughini 2001-01-01 Multiple symmetrical lipomatosis (Madelung's disease) is a rare disorder characterized by deep accumulation of fat tissue, involving mainly the neck, shoulders and chest. This disease is associated with heavy alcohol intake and it is more common in men of Mediterranean origin. This disease can cause severe aesthetic deformities and progressive respiratory dysfunction. We report a case of a patient with multiple symmetrical lipomatosis and describe the clinical and radiological features of this disorder. (author) 18. Symmetrized neutron transport equation and the fast Fourier transform method International Nuclear Information System (INIS) Sinh, N.Q.; Kisynski, J.; Mika, J. 1978-01-01 The differential equation obtained from the neutron transport equation by the application of the source iteration method in two-dimensional rectangular geometry is transformed into a symmetrized form with respect to one of the angular variables. The discretization of the symmetrized equation leads to finite difference equations based on the five-point scheme and solved by use of the fast Fourier transform method. Possible advantages of the approach are shown on test calculations 19. 10 CFR 1009.4 - Exclusions. Science.gov (United States) 2010-01-01 ... 10 Energy 4 2010-01-01 2010-01-01 false Exclusions. 1009.4 Section 1009.4 Energy DEPARTMENT OF ENERGY (GENERAL PROVISIONS) GENERAL POLICY FOR PRICING AND CHARGING FOR MATERIALS AND SERVICES SOLD BY DOE § 1009.4 Exclusions. This part shall not apply when the amount to be priced or charged is... 20. Fighting poverty and exclusion through social investment DEFF Research Database (Denmark) Kvist, Jon The fight against poverty and social exclusion is at the heart of the Europe 2020 strategy for smart, sustainable and inclusive growth. With more than 120 million people in the EU at risk of poverty or social exclusion, EU leaders have pledged to bring at least 20 million people out of poverty an... 1. 18 CFR 1308.3 - Exclusions. Science.gov (United States) 2010-04-01 ... 18 Conservation of Power and Water Resources 2 2010-04-01 2010-04-01 false Exclusions. 1308.3... General Matters § 1308.3 Exclusions. (a) This part does not apply to any TVA contract which does not contain a disputes clause. (b) Except as otherwise specifically provided, this part does not apply to any... 2. Subspace exclusion zones for damage localization DEFF Research Database (Denmark) Bernal, Dionisio; Ulriksen, Martin Dalgaard 2018-01-01 , this is exploited in the context of structural damage localization to cast the Subspace Exclusion Zone (SEZ) scheme, which locates damage by reconstructing the captured field quantity shifts from analytical subspaces indexed by postulated boundaries, the so-called exclusion zones (EZs), in a model of the structure... 3. Critical properties of some aliphatic symmetrical ethers International Nuclear Information System (INIS) Nikitin, Eugene D.; Popov, Alexander P.; Bogatishcheva, Nataliya S. 2014-01-01 Highlights: • Critical properties of simple aliphatic ethers were measured. • The ethers decompose at near-critical temperatures. • Pulse-heating method with short residence times was used. -- Abstract: The critical temperatures T c and the critical pressures p c of dihexyl, dioctyl, and didecyl ethers have been measured. According to the measurements, the coordinates of the critical points are T c = (665 ± 7) K, p c = (1.44 ± 0.04) MPa for dihexyl ether, T c = (723 ± 7) K, p c = (1.19 ± 0.04) MPa for dioctyl ether, and T c = (768 ± 8) K, p c = (1.03 ± 0.03) MPa for didecyl ether. All the ethers studied degrade chemically at near-critical temperatures. A pulse-heating method applicable to measuring the critical properties of thermally unstable compounds has been used. The times from the beginning of a heating pulse to the moment of reaching the critical temperature were from 0.06 to 0.46 ms. The short residence times provide little decomposition of the substances in the course of the experiments. The critical properties of the ethers investigated in this work have been discussed together with those of methyl to butyl ethers. The experimental critical constants of the ethers have been compared with those estimated by the group-contribution methods of Wilson and Jasperson and Marrero and Gani. The Wilson/Jasperson method provides a better estimation of the critical temperatures and pressures of simple aliphatic ethers in comparison with the Marrero/Gani method if reliable normal boiling temperatures are used in the method of Wilson and Jasperson 4. Simple models of equilibrium and nonequilibrium phenomena International Nuclear Information System (INIS) Lebowitz, J.L. 1987-01-01 This volume consists of two chapters of particular interest to researchers in the field of statistical mechanics. The first chapter is based on the premise that the best way to understand the qualitative properties that characterize many-body (i.e. macroscopic) systems is to study 'a number of the more significant model systems which, at least in principle are susceptible of complete analysis'. The second chapter deals exclusively with nonequilibrium phenomena. It reviews the theory of fluctuations in open systems to which they have made important contributions. Simple but interesting model examples are emphasised 5. Exact stationary state for an asymmetric exclusion process with fully parallel dynamics NARCIS (Netherlands) Gier, J.C.\|info:eu-repo/dai/nl/170218430; Nienhuis, B. The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation 6. Simple WZW currents International Nuclear Information System (INIS) Fuchs, J. 1990-08-01 A complete classification of simple currents of WZW theory is obtained. The proof is based on an analysis of the quantum dimensions of the primary fields. Simple currents are precisely the primaries with unit quantum dimension; for WZW theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible. (author). 19 refs.; 2 tabs 7. Cotangent bundles over all the Hermitian symmetric spaces International Nuclear Information System (INIS) Arai, Masato; Baba, Kurando 2016-01-01 We construct the N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. In order to construct them we use the projective superspace formalism which is an N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N =1 superfields, once the Kähler potentials of the base manifolds are obtained. Starting with N = 1 supersymmetric Kähler nonlinear sigma models on the Hermitian symmetric spaces, we extend them into the N = 2 supersymmetric models by using the projective superspace formalism and derive the general formula for the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. We apply to the formula for the non-compact Hermitian symmetric space E 7 /E 6 × U(1) 1 . (paper) 8. Optomechanically induced absorption in parity-time-symmetric optomechanical systems Science.gov (United States) Zhang, X. Y.; Guo, Y. Q.; Pei, P.; Yi, X. X. 2017-06-01 We explore the optomechanically induced absorption (OMIA) in a parity-time- (PT -) symmetric optomechanical system (OMS). By numerically calculating the Lyapunov exponents, we find out the stability border of the PT -symmetric OMS. The results show that in the PT -symmetric phase the system can be either stable or unstable depending on the coupling constant and the decay rate. In the PT -symmetric broken phase the system can have a stable state only for small gain rates. By calculating the transmission rate of the probe field, we find that there is an inverted optomechanically induced transparency (OMIT) at δ =-ωM and an OMIA at δ =ωM for the PT -symmetric optomechanical system. At each side of δ =-ωM there is an absorption window due to the resonance absorption of the two generated supermodes. Comparing with the case of optomechanics coupled to a passive cavity, we find that the active cavity can enhance the resonance absorption. The absorption rate at δ =ωM increases as the coupling strength between the two cavities increases. Our work provides us with a promising platform for controlling light propagation and light manipulation in terms of PT symmetry, which might have potential applications in quantum information processing and quantum optical devices. 9. A cascaded three-phase symmetrical multistage voltage multiplier International Nuclear Information System (INIS) Iqbal, Shahid; Singh, G K; Besar, R; Muhammad, G 2006-01-01 A cascaded three-phase symmetrical multistage Cockcroft-Walton voltage multiplier (CW-VM) is proposed in this report. It consists of three single-phase symmetrical voltage multipliers, which are connected in series at their smoothing columns like string of batteries and are driven by three-phase ac power source. The smoothing column of each voltage multiplier is charged twice every cycle independently by respective oscillating columns and discharged in series through load. The charging discharging process completes six times a cycle and therefore the output voltage ripple's frequency is of sixth order of the drive signal frequency. Thus the proposed approach eliminates the first five harmonic components of load generated voltage ripples and sixth harmonic is the major ripple component. The proposed cascaded three-phase symmetrical voltage multiplier has less than half the voltage ripple, and three times larger output voltage and output power than the conventional single-phase symmetrical CW-VM. Experimental and simulation results of the laboratory prototype are given to show the feasibility of proposed cascaded three-phase symmetrical CW-VM 10. Strategy as simple rules. Science.gov (United States) Eisenhardt, K M; Sull, D N 2001-01-01 11. Robust Visual Tracking via Exclusive Context Modeling KAUST Repository Zhang, Tianzhu; Ghanem, Bernard; Liu, Si; Xu, Changsheng; Ahuja, Narendra 2015-01-01 appearances as linear combinations of dictionary templates that are updated dynamically. Learning the representation of each particle is formulated as an exclusive sparse representation problem, where the overall dictionary is composed of multiple {group 12. Exclusive hadronic processes and color transparency It is known that at asymptotically large momentum transfer certain exclusive hadronic ... indicates that the Brodsky–Lepage factorization scheme fails, independent of ..... A basic feature of -initiated reactions is that most events are knocked out. 13. Exclusion, exemption, clearance European Union approach International Nuclear Information System (INIS) Janssens, A. 1997-01-01 The presentation overviews the following issues: Euratom Basic Safety Standards; administrative requirements; radiation protection of the population. Scope of the Standards: natural radiation sources; exclusion. Exemption; Clearance; Import of radioactive scrap metal 14. Exclusion of pneumothorax by radionuclide lung scan International Nuclear Information System (INIS) Weiss, P.E. 1986-01-01 A case is reported in which ventilation lung imaging was useful in excluding a large pneumothorax. This technique may be helpful in patients with emphysema in whom exclusion of pneumothorax by radiographic criteria might be difficult 15. Imaging partons in exclusive scattering processes Energy Technology Data Exchange (ETDEWEB) Diehl, Markus 2012-06-15 The spatial distribution of partons in the proton can be probed in suitable exclusive scattering processes. I report on recent performance estimates for parton imaging at a proposed Electron-Ion Collider. 16. Nonlinear Cross-Diffusion with Size Exclusion KAUST Repository Burger, Martin; Di Francesco, Marco; Pietschmann, Jan-Frederik; Schlake, Bä rbel 2010-01-01 The aim of this paper is to investigate the mathematical properties of a continuum model for diffusion of multiple species incorporating size exclusion effects. The system for two species leads to nonlinear cross-diffusion terms with double 17. Exclusion, Violence, and Community Responses in Central ... International Development Research Centre (IDRC) Digital Library (Canada) Personal 2015-05-13 May 13, 2015 ... similar conditions of social exclusion, different levels of violence can be explained because communities capacities to face violence. • Methodology: ... in El Salvador. • Mix of quantitative and qualitative techniques of research. 18. Exclusive processes at high momentum transfer CERN Document Server 2002-01-01 This book focuses on the physics of exclusive processes at high momentum transfer and their description in terms of generalized parton distributions, perturbative QCD, and relativistic quark models. It covers recent developments in the field, both theoretical and experimental. 19. The "symmetrical turn" in the study of colective action Directory of Open Access Journals (Sweden) Israel Rodríguez Giralt 2008-11-01 Full Text Available Conceptualising and understanding forms of collective action is one of the historic preoccupations of social thought. Good evidence of this can be found in the long line of disputes and polemics that runs through the history of thought about these social phenomena. It shows the difficulty social sciences have faced, and continue to face, when it comes to defining, explaining and delineating a phenomena as ephemeral and liminal as this one.In this context, I propose that a discussion of the implications an STS focus could have for the analysis of contemporary collective action. The main hypothesis I develop states that the conceptual and methodological baggage that goes with the Actor-Network theory (ANT, and its shaping into what has been called the 'symmetrical turn' in the social sciences, become a fundamental resource for renewing and enriching the analysis of collective action. For this, I will bring together two main contributions: its alternative understanding of social action (to explain the social it is necessary to leave the exclusive concern with social relations aside and take into account the non-human actors, such as the technical procedures in which they are involved; and its original definition of the “collective” (the collective is basically an aggregate of humans and non-humans, without predefined borders, it is just the relational product created by the constant and precarious commitment between heterogeneous elements. Both contributions, I affirm, allow the opening of an interesting discussion about agency and the possibility of articulating a new theory of collective action that differs from the dominant traditions in that it considers and assumes the heterogeneous and relational character of all social actors, and, as such, it also assumes that all social action is the emergent effect, the interactive product of those hybrid collectives in action. To give an example of the fertility of this approach, I focus on an 20. Model of reversible vesicular transport with exclusion International Nuclear Information System (INIS) Bressloff, Paul C; Karamched, Bhargav R 2016-01-01 A major question in neurobiology concerns the mechanics behind the motor-driven transport and delivery of vesicles to synaptic targets along the axon of a neuron. Experimental evidence suggests that the distribution of vesicles along the axon is relatively uniform and that vesicular delivery to synapses is reversible. A recent modeling study has made explicit the crucial role that reversibility in vesicular delivery to synapses plays in achieving uniformity in vesicle distribution, so called synaptic democracy (Bressloff et al 2015 Phys. Rev. Lett. 114 168101). In this paper we generalize the previous model by accounting for exclusion effects (hard-core repulsion) that may occur between molecular motor-cargo complexes (particles) moving along the same microtubule track. The resulting model takes the form of an exclusion process with four internal states, which distinguish between motile and stationary particles, and whether or not a particle is carrying vesicles. By applying a mean field approximation and an adiabatic approximation we reduce the system of ODEs describing the evolution of occupation numbers of the sites on a 1D lattice to a system of hydrodynamic equations in the continuum limit. We find that reversibility in vesicular delivery allows for synaptic democracy even in the presence of exclusion effects, although exclusion does exacerbate nonuniform distributions of vesicles in an axon when compared with a model without exclusion. We also uncover the relationship between our model and other models of exclusion processes with internal states. (paper) 1. Association between Exclusive Breastfeeding and Child Development Directory of Open Access Journals (Sweden) Ghaniyyatul Khudri 2016-03-01 Full Text Available Background: Child development highly correlates with child’s quality. The fastest child development period is during the first three years, also called golden period. This research was aimed to discover correlation between exclussive breastfeeding and child development in Cipacing Village Jatinangor, district of Sumedang. Methods: This research was conducted using cross-sectional method in thirteen Pos Pelayanan Terpadu (Posyandu Cipacing Village in Jatinangor. One hundred and two children aged 12−24 months with their caregiver were recruited as respondents by using cluster sampling method. Hist ory of exclusive breastfeeding was assessed with questionnaire while child development status was assesed with Kuesioner Pra Skrining Perkembangan (KPSP in September 2013 after informed consent was obtained. Chi-square test analysis was performed to determine correlation between exclusive breastfeeding and child development status. Results: Overall, children in Cipacing Village had non-exclusive breastfeeding history (83.3%, and only 16.7% respondents had exclusive breastfeeding history. Meanwhile, 89.2% of children had normal development status, and 10.8% had delayed development status. Statistic analysis using chi-square test in the level of 95% confidence between exclusive breastfeeding and child development showed p=0.686 and odds ratio 2.133. Conclusions: There is no significant relationship between history of exclusive breastfeeding and child development status. 2. Decomposition of a symmetric second-order tensor Science.gov (United States) Heras, José A. 2018-05-01 In the three-dimensional space there are different definitions for the dot and cross products of a vector with a second-order tensor. In this paper we show how these products can uniquely be defined for the case of symmetric tensors. We then decompose a symmetric second-order tensor into its ‘dot’ part, which involves the dot product, and the ‘cross’ part, which involves the cross product. For some physical applications, this decomposition can be interpreted as one in which the dot part identifies with the ‘parallel’ part of the tensor and the cross part identifies with the ‘perpendicular’ part. This decomposition of a symmetric second-order tensor may be suitable for undergraduate courses of vector calculus, mechanics and electrodynamics. 3. Solitons in PT-symmetric potential with competing nonlinearity International Nuclear Information System (INIS) Khare, Avinash; Al-Marzoug, S.M.; Bahlouli, Hocine 2012-01-01 We investigate the effect of competing nonlinearities on beam dynamics in PT-symmetric potentials. In particular, we consider the stationary nonlinear Schrödinger equation (NLSE) in one dimension with competing cubic and generalized nonlinearity in the presence of a PT-symmetric potential. Closed form solutions for localized states are obtained. These solitons are shown to be stable over a wide range of potential parameters. The transverse power flow associated with these complex solitons is also examined. -- Highlights: ► Effect of competing nonlinearities on beam dynamics in PT-symmetric potentials. ► Closed form solutions for localized states are. ► The transverse power flow associated with these complex solitons is also examined. 4. Nilpotent orbits in real symmetric pairs and stationary black holes Energy Technology Data Exchange (ETDEWEB) Dietrich, Heiko [School of Mathematical Sciences, Monash University, VIC (Australia); De Graaf, Willem A. [Department of Mathematics, University of Trento, Povo (Italy); Ruggeri, Daniele [Universita di Torino, Dipartimento di Fisica (Italy); INFN, Sezione di Torino (Italy); Trigiante, Mario [DISAT, Politecnico di Torino (Italy) 2017-02-15 In the study of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determine the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of (SL{sub 2}(R)){sup 4} acting on the fourth tensor power of the natural 2-dimensional SL{sub 2}(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) 5. Tourist Demand Reactions: Symmetric or Asymmetric across the Business Cycle? Science.gov (United States) Bronner, Fred; de Hoog, Robert 2017-09-01 Economizing and spending priorities on different types of vacations are investigated during two periods: an economic downturn and returning prosperity. Two nation-wide samples of vacationers are used: one during a downturn, the other one at the start of the recovery period. Through comparing the results, conclusions can be drawn about symmetric or asymmetric tourist demand across the business cycle. The main summer holiday has an asymmetric profile: being fairly crisis-resistant during a recession and showing considerable growth during an expansion. This does not apply to short vacations and day trips, each having a symmetric profile: during a recession they experience substantial reductions and during expansion comparable growth. So when talking about tourist demand in general , one cannot say that it is symmetric or asymmetric across the business cycle: it depends on the type of vacation. Differences in tourist demand are best explained by the role of Quality-of-Life for vacationers. 6. Symmetric spaces and the Kashiwara-Vergne method CERN Document Server Rouvière, François 2014-01-01 Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit of Kashiwara and Vergne's or... 7. Rings with involution whose symmetric elements are central Directory of Open Access Journals (Sweden) Taw Pin Lim 1980-01-01 Full Text Available In a ring R with involution whose symmetric elements S are central, the skew-symmetric elements K form a Lie algebra over the commutative ring S. The classification of such rings which are 2-torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies [[x,y],z]=f(y,zx−f(z,xy. If S is a field of char ≠2, f≠0 and dimK>1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K′ is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim≤2. 8. Kinetic-energy distribution for symmetric fission of 236U International Nuclear Information System (INIS) Brissot, R.; Bocquet, J.P.; Ristori, C.; Crancon, J.; Guet, C.R.; Nifenecker, H.A.; Montoya, M. 1980-01-01 Fission fragment kinetic-energy distributions have been measured at the Grenoble high-flux reactor with the Lohengrin facility. Spurious events were eliminated in the symmetric region by a coherence test based on a time-of-flight measurement of fragment velocities. A Monte-Carlo calculation is then performed to correct the experimental data for neutron evaporation. The difference between the most probable kinetic energy in symmetric fission and the fission in which the heavy fragment is 'magic' (Zsub(H)=50) is found to be approximately =30 MeV. The results suggest that for the symmetric case the total excitation energy available at scission is shared equally among the fragments. (author) 9. The discrete dynamics of symmetric competition in the plane. Science.gov (United States) Jiang, H; Rogers, T D 1987-01-01 We consider the generalized Lotka-Volterra two-species system xn + 1 = xn exp(r1(1 - xn) - s1yn) yn + 1 = yn exp(r2(1 - yn) - s2xn) originally proposed by R. M. May as a model for competitive interaction. In the symmetric case that r1 = r2 and s1 = s2, a region of ultimate confinement is found and the dynamics therein are described in some detail. The bifurcations of periodic points of low period are studied, and a cascade of period-doubling bifurcations is indicated. Within the confinement region, a parameter region is determined for the stable Hopf bifurcation of a pair of symmetrically placed period-two points, which imposes a second component of oscillation near the stable cycles. It is suggested that the symmetric competitive model contains much of the dynamical complexity to be expected in any discrete two-dimensional competitive model. 10. Bound states for non-symmetric evolution Schroedinger potentials Energy Technology Data Exchange (ETDEWEB) Corona, Gulmaro Corona [Area de Analisis Matematico y sus Aplicaciones, Universidad Autonoma Metropolitana-Azcapotalco, Atzcapotzalco, DF (Mexico)). E-mail: [email protected] 2001-09-14 We consider the spectral problem associated with the evolution Schroedinger equation, (D{sup 2}+ k{sup 2}){phi}=u{phi}, where u is a matrix-square-valued function, with entries in the Schwartz class defined on the real line. The solution {phi}, called the wavefunction, consists of a function of one real variable, matrix-square-valued with entries in the Schwartz class. This problem has been dealt for symmetric potentials u. We found for the present case that the bound states are localized similarly to the scalar and symmetric cases, but by the zeroes of an analytic matrix-valued function. If we add an extra condition to the potential u, we can determine these states by an analytic scalar function. We do this by generalizing the scalar and symmetric cases but without using the fact that the Wronskian of a pair of wavefunction is constant. (author) 11. Solution of generalized shifted linear systems with complex symmetric matrices International Nuclear Information System (INIS) Sogabe, Tomohiro; Hoshi, Takeo; Zhang, Shao-Liang; Fujiwara, Takeo 2012-01-01 We develop the shifted COCG method [R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, T. Fujiwara, Linear algebraic calculation of Green’s function for large-scale electronic structure theory, Phys. Rev. B 73 (165108) (2006) 1–9] and the shifted WQMR method [T. Sogabe, T. Hoshi, S.-L. Zhang, T. Fujiwara, On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems, Electron. Trans. Numer. Anal. 31 (2008) 126–140] for solving generalized shifted linear systems with complex symmetric matrices that arise from the electronic structure theory. The complex symmetric Lanczos process with a suitable bilinear form plays an important role in the development of the methods. The numerical examples indicate that the methods are highly attractive when the inner linear systems can efficiently be solved. 12. Simple Finite Sums KAUST Repository Alabdulmohsin, Ibrahim M. 2018-03-07 We will begin our treatment of summability calculus by analyzing what will be referred to, throughout this book, as simple finite sums. Even though the results of this chapter are particular cases of the more general results presented in later chapters, they are important to start with for a few reasons. First, this chapter serves as an excellent introduction to what summability calculus can markedly accomplish. Second, simple finite sums are encountered more often and, hence, they deserve special treatment. Third, the results presented in this chapter for simple finite sums will, themselves, be used as building blocks for deriving the most general results in subsequent chapters. Among others, we establish that fractional finite sums are well-defined mathematical objects and show how various identities related to the Euler constant as well as the Riemann zeta function can actually be derived in an elementary manner using fractional finite sums. 13. Simple Finite Sums KAUST Repository Alabdulmohsin, Ibrahim M. 2018-01-01 We will begin our treatment of summability calculus by analyzing what will be referred to, throughout this book, as simple finite sums. Even though the results of this chapter are particular cases of the more general results presented in later chapters, they are important to start with for a few reasons. First, this chapter serves as an excellent introduction to what summability calculus can markedly accomplish. Second, simple finite sums are encountered more often and, hence, they deserve special treatment. Third, the results presented in this chapter for simple finite sums will, themselves, be used as building blocks for deriving the most general results in subsequent chapters. Among others, we establish that fractional finite sums are well-defined mathematical objects and show how various identities related to the Euler constant as well as the Riemann zeta function can actually be derived in an elementary manner using fractional finite sums. 14. Symmetric vs. asymmetric stem cell divisions: an adaptation against cancer? Directory of Open Access Journals (Sweden) Leili Shahriyari Full Text Available Traditionally, it has been held that a central characteristic of stem cells is their ability to divide asymmetrically. Recent advances in inducible genetic labeling provided ample evidence that symmetric stem cell divisions play an important role in adult mammalian homeostasis. It is well understood that the two types of cell divisions differ in terms of the stem cells' flexibility to expand when needed. On the contrary, the implications of symmetric and asymmetric divisions for mutation accumulation are still poorly understood. In this paper we study a stochastic model of a renewing tissue, and address the optimization problem of tissue architecture in the context of mutant production. Specifically, we study the process of tumor suppressor gene inactivation which usually takes place as a consequence of two "hits", and which is one of the most common patterns in carcinogenesis. We compare and contrast symmetric and asymmetric (and mixed stem cell divisions, and focus on the rate at which double-hit mutants are generated. It turns out that symmetrically-dividing cells generate such mutants at a rate which is significantly lower than that of asymmetrically-dividing cells. This result holds whether single-hit (intermediate mutants are disadvantageous, neutral, or advantageous. It is also independent on whether the carcinogenic double-hit mutants are produced only among the stem cells or also among more specialized cells. We argue that symmetric stem cell divisions in mammals could be an adaptation which helps delay the onset of cancers. We further investigate the question of the optimal fraction of stem cells in the tissue, and quantify the contribution of non-stem cells in mutant production. Our work provides a hypothesis to explain the observation that in mammalian cells, symmetric patterns of stem cell division seem to be very common. 15. Niche-independent symmetrical self-renewal of a mammalian tissue stem cell. Directory of Open Access Journals (Sweden) Luciano Conti 2005-09-01 Full Text Available Pluripotent mouse embryonic stem (ES cells multiply in simple monoculture by symmetrical divisions. In vivo, however, stem cells are generally thought to depend on specialised cellular microenvironments and to undergo predominantly asymmetric divisions. Ex vivo expansion of pure populations of tissue stem cells has proven elusive. Neural progenitor cells are propagated in combination with differentiating progeny in floating clusters called neurospheres. The proportion of stem cells in neurospheres is low, however, and they cannot be directly observed or interrogated. Here we demonstrate that the complex neurosphere environment is dispensable for stem cell maintenance, and that the combination of fibroblast growth factor 2 (FGF-2 and epidermal growth factor (EGF is sufficient for derivation and continuous expansion by symmetrical division of pure cultures of neural stem (NS cells. NS cells were derived first from mouse ES cells. Neural lineage induction was followed by growth factor addition in basal culture media. In the presence of only EGF and FGF-2, resulting NS cells proliferate continuously, are diploid, and clonogenic. After prolonged expansion, they remain able to differentiate efficiently into neurons and astrocytes in vitro and upon transplantation into the adult brain. Colonies generated from single NS cells all produce neurons upon growth factor withdrawal. NS cells uniformly express morphological, cell biological, and molecular features of radial glia, developmental precursors of neurons and glia. Consistent with this profile, adherent NS cell lines can readily be established from foetal mouse brain. Similar NS cells can be generated from human ES cells and human foetal brain. The extrinsic factors EGF plus FGF-2 are sufficient to sustain pure symmetrical self-renewing divisions of NS cells. The resultant cultures constitute the first known example of tissue-specific stem cells that can be propagated without accompanying 16. Enhanced Sensitivity of Anti-Symmetrically Structured Surface Plasmon Resonance Sensors with Zinc Oxide Intermediate Layers Directory of Open Access Journals (Sweden) Nan-Fu Chiu 2013-12-01 Full Text Available We report a novel design wherein high-refractive-index zinc oxide (ZnO intermediary layers are used in anti-symmetrically structured surface plasmon resonance (SPR devices to enhance signal quality and improve the full width at half maximum (FWHM of the SPR reflectivity curve. The surface plasmon (SP modes of the ZnO intermediary layer were excited by irradiating both sides of the Au film, thus inducing a high electric field at the Au/ZnO interface. We demonstrated that an improvement in the ZnO (002 crystal orientation led to a decrease in the FWHM of the SPR reflectivity curves. We optimized the design of ZnO thin films using different parameters and performed analytical comparisons of the ZnO with conventional chromium (Cr and indium tin oxide (ITO intermediary layers. The present study is based on application of the Fresnel equation, which provides an explanation and verification for the observed narrow SPR reflectivity curve and optical transmittance spectra exhibited by (ZnO/Au, (Cr/Au, and (ITO/Au devices. On exposure to ethanol, the anti-symmetrically structured showed a huge electric field at the Au/ZnO interface and a 2-fold decrease in the FWHM value and a 1.3-fold larger shift in angle interrogation and a 4.5-fold high-sensitivity shift in intensity interrogation. The anti-symmetrically structured of ZnO intermediate layers exhibited a wider linearity range and much higher sensitivity. It also exhibited a good linear relationship between the incident angle and ethanol concentration in the tested range. Thus, we demonstrated a novel and simple method for fabricating high-sensitivity, high-resolution SPR biosensors that provide high accuracy and precision over relevant ranges of analyte measurement. 17. Angularly symmetric splitting of a light beam upon reflection and refraction at an air-dielectric plane boundary. Science.gov (United States) Azzam, R M A 2015-12-01 Conditions for achieving equal and opposite angular deflections of a light beam by reflection and refraction at an air-dielectric boundary are determined. Such angularly symmetric beam splitting (ASBS) is possible only if the angle of incidence is >60° by exactly one third of the angle of refraction. This simple law, plus Snell's law, leads to several analytical results that clarify all aspects of this phenomenon. In particular, it is shown that the intensities of the two symmetrically deflected beams can be equalized by proper choice of the prism refractive index and the azimuth of incident linearly polarized light. ASBS enables a geometrically attractive layout of optical systems that employ multiple prism beam splitters. CERN Document Server Katz, Abbott 2011-01-01 Get the most out of Excel 2010 with Excel 2010 Made Simple - learn the key features, understand what's new, and utilize dozens of time-saving tips and tricks to get your job done. Over 500 screen visuals and clear-cut instructions guide you through the features of Excel 2010, from formulas and charts to navigating around a worksheet and understanding Visual Basic for Applications (VBA) and macros. Excel 2010 Made Simple takes a practical and highly effective approach to using Excel 2010, showing you the best way to complete your most common spreadsheet tasks. You'll learn how to input, format, CERN Document Server Mazo, Gary 2011-01-01 If you have a Droid series smartphone - Droid, Droid X, Droid 2, or Droid 2 Global - and are eager to get the most out of your device, Droids Made Simple is perfect for you. Authors Martin Trautschold, Gary Mazo and Marziah Karch guide you through all of the features, tips, and tricks using their proven combination of clear instructions and detailed visuals. With hundreds of annotated screenshots and step-by-step directions, Droids Made Simple will transform you into a Droid expert, improving your productivity, and most importantly, helping you take advantage of all of the cool features that c 20. Clusters in simple fluids International Nuclear Information System (INIS) Sator, N. 2003-01-01 This article concerns the correspondence between thermodynamics and the morphology of simple fluids in terms of clusters. Definitions of clusters providing a geometric interpretation of the liquid-gas phase transition are reviewed with an eye to establishing their physical relevance. The author emphasizes their main features and basic hypotheses, and shows how these definitions lead to a recent approach based on self-bound clusters. Although theoretical, this tutorial review is also addressed to readers interested in experimental aspects of clustering in simple fluids 1. Association "Les Simples" OpenAIRE Thouzery, Michel 2014-01-01 Fondée par les producteurs du Syndicat Inter-Massifs pour la Production et l’Économie des Simples (S.I.M.P.L.E.S), l’association base son action sur la recherche et le maintien d’une production de qualité (herboristerie et préparations à base de plantes) qui prend en compte le respect de l’environnement et la pérennité des petits producteurs en zone de montagne. Actions de formation Stages de découverte de la flore médicinale sauvage, Stages de culture et transformation des plantes médicinale... 2. A simple electron multiplexer International Nuclear Information System (INIS) Dobrzynski, L; Akjouj, A; Djafari-Rouhani, B; Al-Wahsh, H; Zielinski, P 2003-01-01 We present a simple multiplexing device made of two atomic chains coupled by two other transition metal atoms. We show that this simple atomic device can transfer electrons at a given energy from one wire to the other, leaving all other electron states unaffected. Closed-form relations between the transmission coefficients and the inter-atomic distances are given to optimize the desired directional electron ejection. Such devices can be adsorbed on insulating substrates and characterized by current surface technologies. (letter to the editor) 3. Some curvature properties of quarter symmetric metric connections International Nuclear Information System (INIS) Rastogi, S.C. 1986-08-01 A linear connection Γ ji h with torsion tensor T j h P i -T i h P j , where T j h is an arbitrary (1,1) tensor field and P i is a 1-form, has been called a quarter-symmetric connection by Golab. Some properties of such connections have been studied by Rastogi, Mishra and Pandey, and Yano and Imai. In this paper based on the curvature tensor of quarter-symmetric metric connection we define a tensor analogous to conformal curvature tensor and study some properties of such a tensor. (author) 4. Symmetric bends how to join two lengths of cord CERN Document Server Miles, Roger E 1995-01-01 A bend is a knot securely joining together two lengths of cord (or string or rope), thereby yielding a single longer length. There are many possible different bends, and a natural question that has probably occurred to many is: "Is there a 'best' bend and, if so, what is it?"Most of the well-known bends happen to be symmetric - that is, the two constituent cords within the bend have the same geometric shape and size, and interrelationship with the other. Such 'symmetric bends' have great beauty, especially when the two cords bear different colours. Moreover, they have the practical advantage o 5. Norm estimates of complex symmetric operators applied to quantum systems International Nuclear Information System (INIS) Prodan, Emil; Garcia, Stephan R; Putinar, Mihai 2006-01-01 This paper communicates recent results in the theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schroedinger operators. In particular, we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schroedinger operators with spectral gaps. Second, we provide new ways of evaluating the resolvent norm for Schroedinger operators appearing in the complex scaling theory of resonances 6. Exploring plane-symmetric solutions in f(R) gravity Energy Technology Data Exchange (ETDEWEB) Shamir, M. F., E-mail: [email protected] [National University of Computer and Emerging Sciences, Department of Sciences and Humanities (Pakistan) 2016-02-15 The modified theories of gravity, especially the f(R) gravity, have attracted much attention in the last decade. This paper is devoted to exploring plane-symmetric solutions in the context of metric f(R) gravity. We extend the work on static plane-symmetric vacuum solutions in f(R) gravity already available in the literature [1, 2]. The modified field equations are solved using the assumptions of both constant and nonconstant scalar curvature. Some well-known solutions are recovered with power-law and logarithmic forms of f(R) models. 7. Characterization of Generalized Young Measures Generated by Symmetric Gradients Science.gov (United States) De Philippis, Guido; Rindler, Filip 2017-06-01 This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer-Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The "local" proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti's rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences. 8. Integrability and symmetric spaces. II- The coset spaces International Nuclear Information System (INIS) Ferreira, L.A. 1987-01-01 It shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a fundamental Poisson bracket relation, and consequently charges involution, is that it must be a symmetric space. The conditions a hamiltonian, or any function of the canonical variables, has to satisfy in order to commute with these charges are studied. It is shown that, for the case of non compact symmetric space, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities. (author) [pt 9. Color-symmetric superconductivity in a phenomenological QCD model DEFF Research Database (Denmark) Bohr, Henrik; Providencia, C.; Providencia, J. da 2009-01-01 In this paper, we construct a theory of the NJL type where superconductivity is present, and yet the superconducting state remains, in the average, color symmetric. This shows that the present approach to color superconductivity is consistent with color singletness. Indeed, quarks are free...... in the deconfined phase, but the deconfined phase itself is believed to be a color singlet. The usual description of the color superconducting state violates color singletness. On the other hand, the color superconducting state here proposed is color symmetric in the sense that an arbitrary color rotation leads... 10. (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms International Nuclear Information System (INIS) Klimyk, A U; Patera, J 2007-01-01 We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are eigenfunctions of the Laplace operator on the corresponding fundamental domains satisfying certain boundary conditions. To symmetric and antisymmetric multivariate exponential functions there correspond Fourier transforms. There are three types of such Fourier transforms: expansions into the corresponding Fourier series, integral Fourier transforms and multivariate finite Fourier transforms. Eigenfunctions of the integral Fourier transforms are found 11. Positive projections of symmetric matrices and Jordan algebras DEFF Research Database (Denmark) Fuglede, Bent; Jensen, Søren Tolver 2013-01-01 An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model.......An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model.... 12. Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems International Nuclear Information System (INIS) Scott, D.S.; Ward, R.C. 1981-01-01 Methods are presented for computing eigenpairs of the quadratic lambda-matrix, M lambda 2 + C lambda + K, where M, C, and K are large and sparse, and have special symmetry-type properties. These properties are sufficient to insure that all the eigenvalues are real and that theory analogous to the standard symmetric eigenproblem exists. The methods employ some standard techniques such as partial tri-diagonalization via the Lanczos Method and subsequent eigenpair calculation, shift-and- invert strategy and subspace iteration. The methods also employ some new techniques such as Rayleigh-Ritz quadratic roots and the inertia of symmetric, definite, quadratic lambda-matrices 13. Determination of symmetrical index for 3H in river waters International Nuclear Information System (INIS) Jankovic, M.; Todorovic, D.; Jankovic, B.; Nikolic, J.; Sarap, N. 2011-01-01 The paper presents the results of determining the symmetric index, which describes the magnitude of the tritium content changes with time, for samples of Sava and Danube river waters and Mlaka creek water. The results cover the period from 2003 to 2008. It was shown that the value of the symmetric index is the highest for Mlaka samples, which is in accordance with the fact that in these samples the highest concentration of tritium was found in comparison with samples of the Sava and Danube. [sr 14. Flat synchronizations in spherically symmetric space-times International Nuclear Information System (INIS) 2010-01-01 It is well known that the Schwarzschild space-time admits a spacelike slicing by flat instants and that the metric is regular at the horizon in the associated adapted coordinates (Painleve-Gullstrand metric form). We consider this type of flat slicings in an arbitrary spherically symmetric space-time. The condition ensuring its existence is analyzed, and then, we prove that, for any spherically symmetric flat slicing, the densities of the Weinberg momenta vanish. Finally, we deduce the Schwarzschild solution in the extended Painleve-Gullstrand-LemaItre metric form by considering the coordinate decomposition of the vacuum Einstein equations with respect to a flat spacelike slicing. 15. A simple application of the Newman-Penrose spin coefficient formalism International Nuclear Information System (INIS) Davis, T.M. 1976-01-01 A simple application of the Newman-Penrose formalism is given which can be used by those learning this theory. The spherically symmetric Einstein-Maxwell problem for the Reissner-Nordstrom solution is solved using the Newman-Penrose formalism. The calculation is carried out using a Minkowski null tetrad. (author) 16. Improved self-exclusion program: preliminary results. Science.gov (United States) Tremblay, Nicole; Boutin, Claude; Ladouceur, Robert 2008-12-01 The gambling industry has offered self-exclusion programs for quite a long time. Such measures are designed to limit access to gaming opportunities and provide problem gamblers with the help they need to cease or limit their gambling behaviour. However, few studies have empirically evaluated these programs. This study has three objectives: (1) to observe the participation in an improved self-exclusion program that includes an initial voluntary evaluation, phone support, and a mandatory meeting, (2) to evaluate satisfaction and usefulness of this service as perceived by self-excluders, (3) to measure the preliminary impact of this improved program. One hundred sixteen self-excluders completed a questionnaire about their satisfaction and their perception of the usefulness during the mandatory meeting. Among those participants, 39 attended an initial meeting. Comparisons between data collected at the initial meeting and data taken at the final meeting were made for those 39 participants. Data showed that gamblers chose the improved self-exclusion program 75% of the time; 25% preferred to sign a regular self-exclusion contract. Among those who chose the improved service, 40% wanted an initial voluntary evaluation and 37% of these individuals actually attended that meeting. Seventy percent of gamblers came to the mandatory meeting, which was a required condition to end their self-exclusion. The majority of participants were satisfied with the improved self-exclusion service and perceived it as useful. Major improvements were observed between the final and the initial evaluation on time and money spent, consequences of gambling, DSM-IV score, and psychological distress. The applicability of an improved self-exclusion program is discussed and, as shown in our study, the inclusion of a final mandatory meeting might not be so repulsive for self-excluders. Future research directives are also proposed. 17. Simple Driving Techniques DEFF Research Database (Denmark) 2002-01-01 -like language. Our aim is to extract a simple notion of driving and show that even in this tamed form it has much of the power of more general notions of driving. Our driving technique may be used to simplify functional programs which use function composition and will often be able to remove intermediate data... 18. A Simple Tiltmeter Science.gov (United States) Dix, M. G.; Harrison, D. R.; Edwards, T. M. 1982-01-01 Bubble vial with external aluminum-foil electrodes is sensing element for simple indicating tiltmeter. To measure bubble displacement, bridge circuit detects difference in capacitance between two sensing electrodes and reference electrode. Tiltmeter was developed for experiment on forecasting seismic events by changes in Earth's magnetic field. 19. A Simple Hydrogen Electrode Science.gov (United States) Eggen, Per-Odd 2009-01-01 This article describes the construction of an inexpensive, robust, and simple hydrogen electrode, as well as the use of this electrode to measure "standard" potentials. In the experiment described here the students can measure the reduction potentials of metal-metal ion pairs directly, without using a secondary reference electrode. Measurements… 20. Structure of simple liquids International Nuclear Information System (INIS) Blain, J.F. 1969-01-01 The results obtained by application to argon and sodium of the two important methods of studying the structure of liquids: scattering of X-rays and neutrons, are presented on one hand. On the other hand the principal models employed for reconstituting the structure of simple liquids are exposed: mathematical models, lattice models and their derived models, experimental models. (author) [fr 1. Simple mathematical fireworks International Nuclear Information System (INIS) De Luca, R; Faella, O 2014-01-01 Mathematical fireworks are reproduced in two dimensions by means of simple notions in kinematics and Newtonian mechanics. Extension of the analysis in three dimensions is proposed and the geometric figures the falling tiny particles make on the ground after explosion are determined. (paper) 2. simple sequence repeat (SSR) African Journals Online (AJOL) In the present study, 78 mapped simple sequence repeat (SSR) markers representing 11 linkage groups of adzuki bean were evaluated for transferability to mungbean and related Vigna spp. 41 markers amplified characteristic bands in at least one Vigna species. The transferability percentage across the genotypes ranged ... 3. A Simple Wave Driver Science.gov (United States) Temiz, Burak Kagan; Yavuz, Ahmet 2015-01-01 This study was done to develop a simple and inexpensive wave driver that can be used in experiments on string waves. The wave driver was made using a battery-operated toy car, and the apparatus can be used to produce string waves at a fixed frequency. The working principle of the apparatus is as follows: shortly after the car is turned on, the… 4. A novel sandwich differential capacitive accelerometer with symmetrical double-sided serpentine beam-mass structure International Nuclear Information System (INIS) Xiao, D B; Li, Q S; Hou, Z Q; Wang, X H; Chen, Z H; Xia, D W; Wu, X Z 2016-01-01 This paper presents a novel differential capacitive silicon micro-accelerometer with symmetrical double-sided serpentine beam-mass sensing structure and glass–silicon–glass sandwich structure. The symmetrical double-sided serpentine beam-mass sensing structure is fabricated with a novel pre-buried mask fabrication technology, which is convenient for manufacturing multi-layer sensors. The glass–silicon–glass sandwich structure is realized by a double anodic bonding process. To solve the problem of the difficulty of leading out signals from the top and bottom layer simultaneously in the sandwich sensors, a silicon pillar structure is designed that is inherently simple and low-cost. The prototype is fabricated and tested. It has low noise performance (the peak to peak value is 40 μg) and μg-level Allan deviation of bias (2.2 μg in 1 h), experimentally demonstrating the effectiveness of the design and the novel fabrication technology. (paper) 5. Stationary bound-state massive scalar field configurations supported by spherically symmetric compact reflecting stars Energy Technology Data Exchange (ETDEWEB) 2017-12-15 It has recently been demonstrated that asymptotically flat neutral reflecting stars are characterized by an intriguing no-hair property. In particular, it has been proved that these horizonless compact objects cannot support spatially regular static matter configurations made of scalar (spin-0) fields, vector (spin-1) fields and tensor (spin-2) fields. In the present paper we shall explicitly prove that spherically symmetric compact reflecting stars can support stationary (rather than static) bound-state massive scalar fields in their exterior spacetime regions. To this end, we solve analytically the Klein-Gordon wave equation for a linearized scalar field of mass μ and proper frequency ω in the curved background of a spherically symmetric compact reflecting star of mass M and radius R{sub s}. It is proved that the regime of existence of these stationary composed star-field configurations is characterized by the simple inequalities 1 - 2M/R{sub s} < (ω/μ){sup 2} < 1. Interestingly, in the regime M/R{sub s} << 1 of weakly self-gravitating stars we derive a remarkably compact analytical equation for the discrete spectrum {ω(M,R_s, μ)}{sup n=∞}{sub n=1} of resonant oscillation frequencies which characterize the stationary composed compact-reflecting-star-linearized-massive-scalar-field configurations. Finally, we verify the accuracy of the analytically derived resonance formula of the composed star-field configurations with direct numerical computations. (orig.) 6. A prospective multi-centric open clinical trial of homeopathy in diabetic distal symmetric polyneuropathy. Science.gov (United States) Nayak, Chaturbhuja; Oberai, Praveen; Varanasi, Roja; Baig, Hafeezullah; Ch, Raveender; Reddy, G R C; Devi, Pratima; S, Bhubaneshwari; Singh, Vikram; Singh, V P; Singh, Hari; Shitanshu, Shashi Shekhar 2013-04-01 To evaluate homeopathic treatment in the management of diabetic distal symmetric polyneuropathy. A prospective multi-centric clinical observational study was carried out from October 2005 to September 2009 by Central Council for Research in Homeopathy (CCRH) (India) at its five institutes/units. Patients suffering from diabetes mellitus (DM) and presenting with symptoms of diabetic polyneuropathy (DPN) were screened, investigated and were enrolled in the study after fulfilling the inclusion and exclusion criteria. Patients were evaluated by the diabetic distal symmetric polyneuropathy symptom score (DDSPSS) developed by the Council. A total of 15 homeopathic medicines were identified after repertorizing the nosological symptoms and signs of the disease. The appropriate constitutional medicine was selected and prescribed in 30, 200 and 1 M potency on an individualized basis. Patients were followed up regularly for 12 months. Out of 336 patients (167 males and 169 females) enrolled in the study, 247 patients (123 males and 124 females) were analyzed. All patients who attended at least three follow-up appointments and baseline curve conduction studies were included in the analysis.). A statistically significant improvement in DDSPSS total score (p = 0.0001) was found at 12 months from baseline. Most objective measures did not show significant improvement. Lycopodium clavatum (n = 132), Phosphorus (n = 27) and Sulphur (n = 26) were the medicines most frequently prescribed. Adverse event of hypoglycaemia was observed in one patient only. This study suggests homeopathic medicines may be effective in managing the symptoms of DPN patients. Further studies should be controlled and include the quality of life (QOL) assessment. Copyright © 2013 The Faculty of Homeopathy. Published by Elsevier Ltd. All rights reserved. 7. ''Follow that quark!'' (and other exclusive stories) International Nuclear Information System (INIS) Carroll, A.S. 1987-01-01 Quarks are considered to be the basic constituents of matter. In a series of recent experiments, Carroll studied exclusive reactions as a means of determining the interactions between quarks. Quantum Chromo-dynamics (QCD) is the modern theory of the interaction of quarks. This theory explains how quarks are held together via the strong interaction in particles known as hadrons. Hadrons consisting of three quarks are called baryons. Hadrons made up of a quark and an antiquark are called mesons. In his lecture, Carroll describes what happens when two hadrons collide and scatter to large angles. The violence of the collision causes the gluons that bind the quarks in a particular hadron to temporarily lose their grip on particular quarks. Quarks scramble toward renewed unity with other quarks, and they undergo rearrangement, which generally results in additional new particles. A two-body exclusive reaction has occurred when the same number of particles exist before and after the collisions. At large angles these exclusive reactions are very rare. The labels on the quarks known as flavor enable the experimenter to follow the history of individual quarks in detail during these exclusive reactions. Carroll describes the equipment used in the experiment to measure short distance, hard collisions at large angles. The collisions he discusses occur when a known beam of mesons or protons collide with a stationary proton target. Finally, Carroll summarizes what the experiments have shown from the study of exclusive reactions and what light some of their results shed on the theory of QCD 8. Bilaterally symmetric Fourier approximations of the skull outlines of ... Present work illustrates a scheme of quantitative description of the shape of the skull outlines of temnospondyl amphibians using bilaterally symmetric closed Fourier curves. Some special points have been identified on the Fourier fits of the skull outlines, which are the local maxima, or minima of the distances from the ... 9. PEO nanocomposite polymer electrolyte for solid state symmetric Physical and electrochemical properties of polyethylene oxide (PEO)-based nanocomposite solid polymer electrolytes (NPEs) were investigated for symmetric capacitor applications. Nanosize fillers, i.e., Al2O3 and SiO2 incorporated polymer electrolyte exhibited higher ionic conductivity than those with filler-free composites ... 10. Symmetric approximations of the Navier-Stokes equations International Nuclear Information System (INIS) Kobel'kov, G M 2002-01-01 A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as ε→0 of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established 11. Duality, phase structures, and dilemmas in symmetric quantum games International Nuclear Information System (INIS) Ichikawa, Tsubasa; Tsutsui, Izumi 2007-01-01 Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD, and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of the players are provided 12. SUSY formalism for the symmetric double well potential Using first- and second-order supersymmetric Darboüx formalism and starting with symmetric double well potential barrier we have obtained a class of exactly solvable potentials subject to moving boundary condition. The eigenstates are also obtained by the same technique. 13. Initial value formulation for the spherically symmetric dust solution International Nuclear Information System (INIS) Liu, H. 1990-01-01 An initial value formulation for the dust solution with spherical symmetry is given explicitly in which the initial distributions of dust and its velocity on an initial surface are chosen to be the initial data. As special cases, the Friedmann universe, the Schwarzschild solution in comoving coordinates, and a spherically symmetric and radially inhomogeneous cosmological model are derived 14. Coupled dilaton and electromagnetic field in cylindrically symmetric ... An exact solution is obtained for coupled dilaton and electromagnetic ﬁeld in a cylindrically symmetric spacetime where an axial magnetic ﬁeld as well as a radial electric ﬁeld both are present. Depending on the choice of the arbitrary constants our solution reduces either to dilatonic gravity with pure electric ﬁeld or to that ... 15. PT-Symmetric Waveguides and the Lack of Variational Techniques Czech Academy of Sciences Publication Activity Database Krejčiřík, David 2012-01-01 Roč. 73, č. 1 (2012), s. 1-2 ISSN 0378-620X Institutional support: RVO:61389005 Keywords : Robin Laplacian non-self-adjoint boundary conditions * complex symmetric operator * PT-symmetry * waveguides * discrete and essential spectra Subject RIV: BA - General Mathematics Impact factor: 0.713, year: 2012 16. Confining but chirally symmetric dense and cold matter International Nuclear Information System (INIS) Glozman, L. Ya. 2012-01-01 The possibility for existence of cold, dense chirally symmetric matter with confinement is reviewed. The answer to this question crucially depends on the mechanism of mass generation in QCD and interconnection of confinement and chiral symmetry breaking. This question can be clarified from spectroscopy of hadrons and their axial properties. Almost systematical parity doubling of highly excited hadrons suggests that their mass is not related to chiral symmetry breaking in the vacuum and is approximately chirally symmetric. Then there is a possibility for existence of confining but chirally symmetric matter. We clarify a possible mechanism underlying such a phase at low temperatures and large density. Namely, at large density the Pauli blocking prevents the gap equation to generate a solution with broken chiral symmetry. However, the chirally symmetric part of the quark Green function as well as all color non-singlet quantities are still infrared divergent, meaning that the system is with confinement. A possible phase transition to such a matter is most probably of the first order. This is because there are no chiral partners to the lowest lying hadrons. 17. Technical report: Electric field in not completely symmetric systems International Nuclear Information System (INIS) Vila, F. 1994-08-01 In this paper it is studied theoretically the electric field in the not completely symmetric system earthed metallic sphere-uniformly charged dielectric plan, for sphere surface points situated in the plan that contains sphere's center and vertical symmetry axe of dielectric plan. (author). 11 refs, 1 fig 18. Symmetrical waveguide devices fabricated by direct UV writing DEFF Research Database (Denmark) Færch, Kjartan Ullitz; Svalgaard, Mikael 2002-01-01 Power splitters and directional couplers fabricated by direct UV writing in index matched silica-on-silicon samples can suffer from an asymmetrical device performance, even though the UV writing is carried out in a symmetrical fashion. This effect originates from a reduced photosensitivity... 19. On the axially symmetric equilibrium of a magnetically confined plasma International Nuclear Information System (INIS) Lehnert, B. 1975-01-01 The axially symmetric equilibrium of a magnetically confined plasma is reconsidered, with the special purpose of studying high-beta schemes with a purely poloidal magnetic field. A number of special solutions of the pressure and magnetic flux functions are shown to exist, the obtained results may form starting-points in a further analysis of physically relevant configurations. (Auth.) 20. Symmetric structures of coherent states in superfluid helium-4 International Nuclear Information System (INIS) 1981-02-01 Coherent States in superfluid helium-4 are discussed and symmetric structures are assigned to these states. Discrete and continuous series functions are exhibited for such states. Coherent State structure has been assigned to oscillating condensed bosons and their inter-relations and their effects on the superfluid system are analysed. (author) 1. Spectra of PT -symmetric Hamiltonians on tobogganic contours The term PT -symmetric quantum mechanics, although defined to be of a much broader use, was coined in tight connection with C. Bender's analysis of one- ... on the other hand, the other members of the family were strange Hamiltonians with imaginary potentials which do not appear physical at all. The aim of the. 2. Symmetrical and asymmetrical growth restriction in preterm-born children NARCIS (Netherlands) Bocca-Tjeertes, Inger; Bos, Arend; Kerstjens, Jorien; de Winter, Andrea; Reijneveld, Sijmen OBJECTIVE: To determine how symmetric (proportionate; SGR) and asymmetric (disproportionate; AGR) growth restriction influence growth and development in preterms from birth to 4 years. METHODS: This community-based cohort study of 810 children comprised 86 SGR, 61 AGR, and 663 non-growth restricted 3. Perception of the Symmetrical Patterning of Human Gait by Infants. Science.gov (United States) Booth, Amy E.; Pinto, Jeannine; Bertenthal, Bennett I. 2002-01-01 Two experiments tested infants' sensitivity to properties of point-light displays of a walker and a runner that were equivalent regarding the phasing of limb movements. Found that 3-, but not 5-month-olds, discriminated these displays. When the symmetrical phase-patterning of the runner display was perturbed by advancing two of its limbs by 25… 4. Rotationally symmetric numerical solutions to the sine-Gordon equation DEFF Research Database (Denmark) Olsen, O. H.; Samuelsen, Mogens Rugholm 1981-01-01 We examine numerically the properties of solutions to the spherically symmetric sine-Gordon equation given an initial profile which coincides with the one-dimensional breather solution and refer to such solutions as ring waves. Expanding ring waves either exhibit a return effect or expand towards... 5. Symmetrical Womanhood: The Educational Ideology of Activism at Wellesley. Science.gov (United States) Palmieri, Patricia Ann 1995-01-01 The ideology of higher education for women at Wellesley College in the late 19th and early 20th centuries is discussed in the context of feminism and the women's suffrage movement. "Symmetrical womanhood," a concept emphasizing balance of traditional roles and intellectual and community involvement, was a goal of Wellesley faculty of… 6. Normalizations of Eisenstein integrals for reductive symmetric spaces NARCIS (Netherlands) van den Ban, E.P.; Kuit, Job 2017-01-01 We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the \\sigma-minimal principal series. In addition, we obtain related Eisenstein integrals, but with 7. Analytic families of eigenfunctions on a reductive symmetric space NARCIS (Netherlands) Ban, E.P. van den; Schlichtkrull, H. 2000-01-01 In harmonic analysis on a reductive symmetric space X an important role is played by families of generalized eigenfunctions for the algebra D (X) of invariant dierential operators. Such families arise for instance as matrix coeÆcients of representations that come in series, such as the (generalized) 8. Whittaker Vector of Deformed Virasoro Algebra and Macdonald Symmetric Functions Science.gov (United States) Yanagida, Shintarou 2016-03-01 We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof, we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra. 9. Plane Symmetric Cosmological Model with Quark and Strange ... Keywords. f(R,T) theory of gravity—plane symmetric space-time—quark and strange quark matter—constant deceleration parameter. 1. Introduction. Modern astrophysical observations point out that present expansion of the Universe is an accelerated epoch. The most fascinating evidence for this is found in measurements ... 10. Separator-Integrated, Reversely Connectable Symmetric Lithium-Ion Battery. Science.gov (United States) Wang, Yuhang; Zeng, Jiren; Cui, Xiaoqi; Zhang, Lijuan; Zheng, Gengfeng 2016-02-24 A separator-integrated, reversely connectable, symmetric lithium-ion battery is developed based on carbon-coated Li3V2(PO4)3 nanoparticles and polyvinylidene fluoride-treated separators. The Li3V2(PO4)3 nanoparticles are synthesized via a facile solution route followed by calcination in Ar/H2 atmosphere. Sucrose solution is used as the carbon source for uniform carbon coating on the Li3V2(PO4)3 nanoparticles. Both the carbon and the polyvinylidene fluoride treatments substantially improve the cycling life of the symmetric battery by preventing the dissolution and shuttle of the electroactive Li3V2(PO4)3. The obtained symmetric full cell exhibits a reversible capacity of ≈ 87 mA h g(-1), good cycling stability, and capacity retention of ≈ 70% after 70 cycles. In addition, this type of symmetric full cell can be operated in both forward and reverse connection modes, without any influence on the cycling of the battery. Furthermore, a new separator integration approach is demonstrated, which enables the direct deposition of electroactive materials for the battery assembly and does not affect the electrochemical performance. A 10-tandem-cell battery assembled without differentiating the electrode polarity exhibits a low thickness of ≈ 4.8 mm and a high output voltage of 20.8 V. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. 11. Is PT -symmetric quantum theory false as a fundamental theory? Czech Academy of Sciences Publication Activity Database Znojil, Miloslav 2016-01-01 Roč. 56, č. 3 (2016), s. 254-257 ISSN 1210-2709 R&D Projects: GA ČR GA16-22945S Institutional support: RVO:61389005 Keywords : quantum mechanics * PT-symmetric representations of observables * masurement outcomes Subject RIV: BE - Theoretical Physics 12. A New Symmetrical Unit for Breakwater Armour : First Tests NARCIS (Netherlands) Salauddin, M.; Broere, A.; Van der Meer, J.W.; Verhagen, H.J.; Bijl, E. 2015-01-01 A new and symmetrical single layer armour unit, the crablock, has been designed in the UAE. One breakwater was reconstructed with crablock, but very limited testing had been performed. Just to become more acquainted with this new unit, pre-competitive research at a university has been performed, 13. Helically symmetric experiment, (HSX) goals, design and status International Nuclear Information System (INIS) Anderson, F.S.B.; Almagri, A.F.; Anderson, D.T.; Matthews, P.G.; Talmadge, J.N.; Shohet, J.L. 1995-01-01 HSX is a quasi-helically symmetric (QHS) stellarator currently under construction at the Torsatron-Stellarator Laboratory of the University of Wisconsin-Madison. This device is unique in its magnetic design in that the magnetic field spectrum possesses only a single dominant (helical) component. This design avoids the large direct orbit losses and the low-collisionality neoclassical losses associated with conventional stellarators. The restoration of symmetry to the confining magnetic field makes the neoclassical confinement in this device analogous to an axisymmetric q=1/3 tokamak. The HSX device has been designed with a clear set of primary physics goals: demonstrate the feasibility of construction of a QHS device, examine single particle confinement of injected ions with regard to magnetic field symmetry breaking, compare density and temperature profiles in this helically symmetric system to those for axisymmetric tokamaks and conventional stellarators, examine electric fields and plasma rotation with edge biasing in relation to L-H transitions in symmetric versus non-symmetric stellarator systems, investigate QHS effects on 1/v regime electron confinement, and examine how greatly-reduced neoclassical electron thermal conductivity compares to the experimental χ e profile. 3 refs., 4 figs., 1 tab 14. On the random geometry of a symmetric matter antimatter universe International Nuclear Information System (INIS) Aldrovandi, R.; Goto, M. 1977-05-01 A statistical analysis is made of the randon geometry of an early symmetric matter-antimatter universe model. Such a model is shown to determine the total number of the largest agglomerations in the universe, as well as of some special configurations. Constraints on the time development of the protoagglomerations are also obtained 15. A summary view of the symmetric cosmological model International Nuclear Information System (INIS) Aldrovandi, R. 1975-01-01 A brief analysis of cosmological models is done, beginning with the standard model and following with the symmetric model of Omnes. Some attempts have been made for the phase transition in thermal radiation at high temperatures, to the annihilation period and to coalescence. One model with equal amounts of matter and antimatter seems to be reasonable [pt 16. Compactons in PT-symmetric generalized Korteweg–de Vries ... ... Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Pramana – Journal of Physics; Volume 73; Issue 2. Compactons in P T -symmetric generalized Korteweg–de Vries equations. Carl M Bender Fred Cooper Avinash Khare Bogdan Mihaila Avadh Saxena. Volume 73 Issue 2 August 2009 ... 17. New algorithms for the symmetric tridiagonal eigenvalue computation Energy Technology Data Exchange (ETDEWEB) Pan, V. [City Univ. of New York, Bronx, NY (United States)]\|[International Computer Sciences Institute, Berkeley, CA (United States) 1994-12-31 The author presents new algorithms that accelerate the bisection method for the symmetric eigenvalue problem. The algorithms rely on some new techniques, which include acceleration of Newtons iteration and can also be further applied to acceleration of some other iterative processes, in particular, of iterative algorithms for approximating polynomial zeros. 18. Strain-induced formation of fourfold symmetric SiGe quantum dot molecules. Science.gov (United States) Zinovyev, V A; Dvurechenskii, A V; Kuchinskaya, P A; Armbrister, V A 2013-12-27 The strain field distribution at the surface of a multilayer structure with disklike SiGe nanomounds formed by heteroepitaxy is exploited to arrange the symmetric quantum dot molecules typically consisting of four elongated quantum dots ordered along the [010] and [100] directions. The morphological transition from fourfold quantum dot molecules to continuous fortresslike quantum rings with an increasing amount of deposited Ge is revealed. We examine key mechanisms underlying the formation of lateral quantum dot molecules by using scanning tunneling microscopy and numerical calculations of the strain energy distribution on the top of disklike SiGe nanomounds. Experimental data are well described by a simple thermodynamic model based on the accurate evaluation of the strain dependent part of the surface chemical potential. The spatial arrangement of quantum dots inside molecules is attributed to the effect of elastic property anisotropy. 19. Spherically symmetric models with pressure: separating expansion from contraction and generalizing TOV condition CERN Document Server Mimoso, José Pedro; Mena, Filipe C 2010-01-01 We investigate spherically symmetric perfect fluid spacetimes and discuss the existence and stability of a dividing shell separating expanding and collapsing regions. We perform a 3+1 splitting and obtain gauge invariant conditions relating the intrinsic spatial curvature of the shells to the ADM mass and to a function of the pressure which we introduce and that generalises the Tolman-Oppenheimer-Volkoff equilibrium condition. We analyse the particular cases of the Lema\\^itre-Tolman-Bondi dust models with a cosmological constant as an example of a $\\Lambda$-CDM model and its generalization to contain a central perfect fluid core. These models provide simple, but physically interesting illustrations of our results. 20. Applying exclusion likelihoods from LHC searches to extended Higgs sectors International Nuclear Information System (INIS) Bechtle, Philip; Heinemeyer, Sven; Staal, Oscar; Stefaniak, Tim; Weiglein, Georg 2015-01-01 LHC searches for non-standard Higgs bosons decaying into tau lepton pairs constitute a sensitive experimental probe for physics beyond the Standard Model (BSM), such as supersymmetry (SUSY). Recently, the limits obtained from these searches have been presented by the CMS collaboration in a nearly model-independent fashion - as a narrow resonance model - based on the full 8 TeV dataset. In addition to publishing a 95 % C.L. exclusion limit, the full likelihood information for the narrowresonance model has been released. This provides valuable information that can be incorporated into global BSM fits. We present a simple algorithm that maps an arbitrary model with multiple neutral Higgs bosons onto the narrow resonance model and derives the corresponding value for the exclusion likelihood from the CMS search. This procedure has been implemented into the public computer code HiggsBounds (version 4.2.0 and higher). We validate our implementation by cross-checking against the official CMS exclusion contours in three Higgs benchmark scenarios in the Minimal Supersymmetric Standard Model (MSSM), and find very good agreement. Going beyond validation, we discuss the combined constraints of the ττ search and the rate measurements of the SM-like Higgs at 125 GeV in a recently proposed MSSM benchmark scenario, where the lightest Higgs boson obtains SM-like couplings independently of the decoupling of the heavier Higgs states. Technical details for how to access the likelihood information within HiggsBounds are given in the appendix. The program is available at http:// higgsbounds.hepforge.org. (orig.) 1. Propagation of symmetric and anti-symmetric surface waves in aself-gravitating magnetized dusty plasma layer with generalized (r, q) distribution Science.gov (United States) Lee, Myoung-Jae; Jung, Young-Dae 2018-05-01 The dispersion properties of surface dust ion-acoustic waves in a self-gravitating magnetized dusty plasma layer with the (r, q) distribution are investigated. The result shows that the wave frequency of the symmetric mode in the plasma layer decreases with an increase in the wave number. It is also shown that the wave frequency of the symmetric mode decreases with an increase in the spectral index r. However, the wave frequency of the anti-symmetric mode increases with an increase in the wave number. It is also found that the anti-symmetric mode wave frequency increases with an increase in the spectral index r. In addition, it is found that the influence of the self-gravitation on the symmetric mode wave frequency decreases with increasing scaled Jeans frequency. Moreover, it is found that the wave frequency of the symmetric mode increases with an increase in the dust charge; however, the anti-symmetric mode shows opposite behavior. 2. Testing the Pauli Exclusion Principle for Electrons International Nuclear Information System (INIS) Marton, J; Berucci, C; Cargnelli, M; Ishiwatari, T; Bartalucci, S; Bragadireanu, M; Curceanu, C; Guaraldo, C; Iliescu, M; Pietreanu, D; Piscicchia, K; Ponta, T; Vidal, A Romero; Scordo, A; Sirghi, D L; Bertolucci, S; Matteo, S Di; Egger, J-P; Laubenstein, M; Milotti, E 2013-01-01 One of the fundamental rules of nature and a pillar in the foundation of quantum theory and thus of modern physics is represented by the Pauli Exclusion Principle. We know that this principle is extremely well fulfilled due to many observations. Numerous experiments were performed to search for tiny violation of this rule in various systems. The experiment VIP at the Gran Sasso underground laboratory is searching for possible small violations of the Pauli Exclusion Principle for electrons leading to forbidden X-ray transitions in copper atoms. VIP is aiming at a test of the Pauli Exclusion Principle for electrons with high accuracy, down to the level of 10 −29 – 10 −30 , thus improving the previous limit by 3–4 orders of magnitude. The experimental method, results obtained so far and new developments within VIP2 (follow-up experiment at Gran Sasso, in preparation) to further increase the precision by 2 orders of magnitude will be presented 3. Digital exclusion in higher education contexts DEFF Research Database (Denmark) Khalid, Md. Saifuddin; Pedersen, Mette Jun Lykkegaard 2016-01-01 The integration and adoption of digital technologies have enabled improvements in the quality of and inclusion in higher education. However, a significant proportion of the population has either remained or become digitally excluded. This systematic literature review elucidates the factors...... underlying the concepts of “digital exclusion” and the “digital divide” in higher education. The identified factors are grouped into three categories: social exclusion (i.e., low income, ICT-avoidance as the norm, lack of motivation and commitment, and physical or mental disability), digital exclusion (i.......e., lack of hardware devices and Internet services) and accessibility (which include the division between rural and urban areas, as well as disparities in ICT literacy and information literacy). These factors are multi-tiered and overlapping. Studies on the digital divide, digital exclusion, and barriers... 4. Exclusive vector meson production at HERA Energy Technology Data Exchange (ETDEWEB) Szuba, Dorota [Hamburg University, Hamburg (Germany); Collaboration: H1 Collaboration; ZEUS Collaboration 2013-04-15 The exclusive photoproduction of {Upsilon} has been studied with the ZEUS detector in ep collisions at HERA. The exponential slope, b, of the \|t\|-dependence of the cross section, where t is the squared four-momentum transfer at the proton vertex, has been measured. This constitutes the first measurement of the \|t\|-dependence of the {gamma}p{yields}{Upsilon}p cross section. The differential crosssections as a function of t at lower energies of {gamma}p centre-of-mass has been studied in exclusive diffractive photoproduction of J/{psi} mesons with the H1 detector. The exclusive electroproduction of two pions has been measured by the ZEUS experiment. The two-pion invariant-mass distribution is interpreted in terms of the pion electromagnetic form factor, assuming that the studied mass range includes the contributions of the {rho}, {rho} Prime and . {rho}'' vector-meson states. 5. Exclusive vector meson production at HERA International Nuclear Information System (INIS) Szuba, Dorota 2013-01-01 The exclusive photoproduction of Υ has been studied with the ZEUS detector in ep collisions at HERA. The exponential slope, b, of the \|t\|-dependence of the cross section, where t is the squared four-momentum transfer at the proton vertex, has been measured. This constitutes the first measurement of the \|t\|-dependence of the γp→Υp cross section. The differential crosssections as a function of t at lower energies of γp centre-of-mass has been studied in exclusive diffractive photoproduction of J/ψ mesons with the H1 detector. The exclusive electroproduction of two pions has been measured by the ZEUS experiment. The two-pion invariant-mass distribution is interpreted in terms of the pion electromagnetic form factor, assuming that the studied mass range includes the contributions of the ρ, ρ′ and . ρ'' vector-meson states. 6. Complexity is simple! Science.gov (United States) Cottrell, William; Montero, Miguel 2018-02-01 In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in [1]. 7. Unicameral (simple) bone cysts. Science.gov (United States) 2006-09-01 Since their original description by Virchow, simple bone cysts have been studied repeatedly. Although these defects are not true neoplasms, simple bone cysts may create major structural defects of the humerus, femur, and os calcis. They are commonly discovered incidentally when x-rays are taken for other reasons or on presentation due to a pathologic fracture. Various treatment strategies have been employed, but the only reliable predictor of success of any treatment strategy is the age of the patient; those being older than 10 years of age heal their cysts at a higher rate than those under age 10. The goal of management is the formation of a bone that can withstand the stresses of use by the patient without evidence of continued bone destruction as determined by serial radiographic follow-up. The goal is not a normal-appearing x-ray, but a functionally stable bone. 8. Teenage pregnancy and exclusive breastfeeding rates. Science.gov (United States) Puapompong, Pawin; Raungrongmorakot, Kasem; Manolerdtewan, Wichian; Ketsuwan, Sukwadee; Wongin, Sinutchanan 2014-09-01 Teenage pregnancy is an important health issue globally and in Thailand Younger age mothers decide on the breastfeeding practices ofthe first 6-month. To find the rates of 6-month exclusive breastfeeding practices of teenage mothers and compare them with the rates of 6-month exclusive breastfeeding practices in mothers who are 20 years of age or more. Three thousand five hundred sixty three normal, postpartum women, who delivered without complications at the HRH Princess Maha Chakri Sirindhorn Medical Center in the Nakhon Nayok Province between 2010 and2013 were included in this study. At the second daypostpartum, the data of latch scores and the data of the practice of exclusive breastfeeding were collected Telephone follow-ups on the seventh, fourteenth, and forty-fifth postpartum days and at the second, fourth, and sixth month postpartum month were collected and used for exclusive breastfeeding data following discharge. Demographic data included the maternal age, parity, gestational age, marital status, occupation, religion, route ofdelivery, estimated blood loss, body mass index, nipple length, and the childs birth weight. The collected data was analyzed by the t-test, Chi-square, and odds ratio with 95% confidence interval. The percentage of teenage pregnancies was at 14.8% (527 cases). On postpartum day 2, the percentage of latch scores of 8 or less was 66.4%. At the seventh, fourteenth, and forty-fifth day and at the second, fourth, and sixth months postpartum, the exclusive breastfeeding rates were 88.5, 78.5, 57.6, 43.1, 32.9, and27.0%, respectively. Comparison of the 6-month exclusive breastfeeding rates between teenage mothers and mothers 20 years ofage or older were not statistically significant (pteenage mothers was at 27.0% and had no significant differences from the rates of mothers 20 years of age or more. 9. Barriers to Exclusive Breastfeeding among Urban Mothers Directory of Open Access Journals (Sweden) Lazina Sharmin 2016-05-01 Full Text Available Background: Breastfeeding is the unique source of nutrition and it plays an important role in the growth, development and survival of the infants. The initiation of breastfeeding within one hour and continuation of only breast milk up to six months ensure maximum benefits. The prevalence of exclusive breastfeeding in Bangladesh is 56% which is low. We designed this study to find out the factors influencing the duration of breastfeeding in Bangladeshi population. Objective: To study the factors influencing noncompliance to exclusive breastfeeding. Materials and Methods: This cross sectional study was conducted in Dhaka Shishu Hospital during the period January to June 2011. It includes 125 infant (1–12 months-mother pairs randomly selected from the inpatient and outpatient departments of Dhaka Shishu Hospital. Mother-infant pairs were divided into two groups based on continuation of only breastfeeding up to six months. Outcomes were compared between two groups. Results: In this study exclusive breastfeeding was found in 27.2% and nonexclusive breastfeeding was in 72.8% cases. It was found that in most cases (40% termination of breastfeeding was at 3--4 months. The study revealed that insufficient milk production due to poor position and attachment, social factors such as influence of husband and other family members, joining to service etc act as barrier to exclusive breastfeeding. Mass media and advice from health professionals had a higher influence on lower rate of exclusive breastfeeding. Women who were multiparous, housewives were more likely to maintain optimal breastfeeding. Conclusion: The present study reveals some important factors contributing to low rate of exclusive breastfeeding in Bangladesh. 10. Lambda Polarization in Exclusive Electro- and Photoproduction at CLAS International Nuclear Information System (INIS) Mestayer, M.D. 2003-01-01 The CLAS collaboration at JLab has recent results on Λ polarization for both electroproduction and photoproduction of K + Λ exclusive states. I note the striking phenomenological trends in the data and discuss the underlying physics which might give rise to these phenomena; both in the context of an e.ective Lagrangian formalism, where the degrees of freedom are intermediate mesons and baryons, and also in the context of a simple quark picture. The quark model argument leads to the conclusion that the s and (bar s) quarks are produced with spins anti-aligned, in apparent contradiction to the popular 3 P 0 model of quark pair creation in which the pair is created with vacuum quantum numbers (J=0 and positive parity), i.e. in an S=1, L=1, J=0 angular momentum state 11. Totally asymmetric exclusion processes with particles of arbitrary size CERN Document Server Lakatos, G 2003-01-01 The steady-state currents and densities of a one-dimensional totally asymmetric exclusion process (TASEP) with particles that occlude an integer number (d) of lattice sites are computed using various mean-field approximations and Monte Carlo simulations. TASEPs featuring particles of arbitrary size are relevant for modelling systems such as mRNA translation, vesicle locomotion along microtubules and protein sliding along DNA. We conjecture that the nonequilibrium steady-state properties separate into low-density, high-density, and maximal current phases similar to those of the standard (d = 1) TASEP. A simple mean-field approximation for steady-state particle currents and densities is found to be inaccurate. However, we find local equilibrium particle distributions derived from a discrete Tonks gas partition function yield apparently exact currents within the maximal current phase. For the boundary-limited phases, the equilibrium Tonks gas distribution cannot be used to predict currents, phase boundaries, or ... CERN Document Server Carter, Roger 1991-01-01 Information Technology: Made Simple covers the full range of information technology topics, including more traditional subjects such as programming languages, data processing, and systems analysis. The book discusses information revolution, including topics about microchips, information processing operations, analog and digital systems, information processing system, and systems analysis. The text also describes computers, computer hardware, microprocessors, and microcomputers. The peripheral devices connected to the central processing unit; the main types of system software; application soft CERN Document Server Murphy, Patrick 1982-01-01 Modern Mathematics: Made Simple presents topics in modern mathematics, from elementary mathematical logic and switching circuits to multibase arithmetic and finite systems. Sets and relations, vectors and matrices, tesselations, and linear programming are also discussed.Comprised of 12 chapters, this book begins with an introduction to sets and basic operations on sets, as well as solving problems with Venn diagrams. The discussion then turns to elementary mathematical logic, with emphasis on inductive and deductive reasoning; conjunctions and disjunctions; compound statements and conditional 14. 78 FR 55687 - Notice of Intent To Grant an Exclusive, Partially Exclusive or Non-Exclusive License of the... Science.gov (United States) 2013-09-11 ... DEPARTMENT OF DEFENSE Department of the Army Notice of Intent To Grant an Exclusive, Partially..., 2012 Entitled ''Tie-Down and Jack Fitting Assembly for Helicopter'' AGENCY: Department of the Army, [email protected]us.army.mil . SUPPLEMENTARY INFORMATION: The patent application relates to the aviation platforms... 15. Exclusive B Decays to Charmonium Final States Energy Technology Data Exchange (ETDEWEB) Barrera, Barbara 2000-10-13 We report on exclusive decays of B mesons into final states containing charmonium using data collected with the BABAR detector at the PEP-II storage rings. The charmonium states considered here are J/{psi}, {psi}(2S), and {chi}{sub c1}. Branching fractions for several exclusive final states, a measurement of the decay amplitudes for the B{sup 0} {yields} J/{psi} K* decay, and measurements of the B{sup 0} and B{sup +} masses are presented. All of the results we present here are preliminary. 16. Exclusive hadronic and nuclear processes in QCD International Nuclear Information System (INIS) Brodsky, S.J. 1985-12-01 Hadronic and nuclear processes are covered, in which all final particles are measured at large invariant masses compared with each other, i.e., large momentum transfer exclusive reactions. Hadronic wave functions in QCD and QCD sum rule constraints on hadron wave functions are discussed. The question of the range of applicability of the factorization formula and perturbation theory for exclusive processes is considered. Some consequences of quark and gluon degrees of freedom in nuclei are discussed which are outside the usual domain of traditional nuclear physics. 44 refs., 7 figs 17. Exclusion Statistics in Conformal Field Theory Spectra International Nuclear Information System (INIS) Schoutens, K. 1997-01-01 We propose a new method for investigating the exclusion statistics of quasiparticles in conformal field theory (CFT) spectra. The method leads to one-particle distribution functions, which generalize the Fermi-Dirac distribution. For the simplest SU(n) invariant CFTs we find a generalization of Gentile parafermions, and we obtain new distributions for the simplest Z N -invariant CFTs. In special examples, our approach reproduces distributions based on 'fractional exclusion statistics' in the sense of Haldane. We comment on applications to fractional quantum Hall effect edge theories. copyright 1997 The American Physical Society 18. SIMPLE ESTIMATOR AND CONSISTENT STRONGLY OF STABLE DISTRIBUTIONS Directory of Open Access Journals (Sweden) Cira E. Guevara Otiniano 2016-06-01 Full Text Available Stable distributions are extensively used to analyze earnings of financial assets, such as exchange rates and stock prices assets. In this paper we propose a simple and strongly consistent estimator for the scale parameter of a symmetric stable L´evy distribution. The advantage of this estimator is that your computational time is minimum thus it can be used to initialize intensive computational procedure such as maximum likelihood. With random samples of sized n we tested the efficacy of these estimators by Monte Carlo method. We also included applications for three data sets. 19. Applications of exact traveling wave solutions of Modified Liouville and the Symmetric Regularized Long Wave equations via two new techniques Science.gov (United States) Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar 2018-06-01 In this current work, we employ novel methods to find the exact travelling wave solutions of Modified Liouville equation and the Symmetric Regularized Long Wave equation, which are called extended simple equation and exp(-Ψ(ξ))-expansion methods. By assigning the different values to the parameters, different types of the solitary wave solutions are derived from the exact traveling wave solutions, which shows the efficiency and precision of our methods. Some solutions have been represented by graphical. The obtained results have several applications in physical science. 20. Exclusive breastfeedingand postnatal changes in maternal ... African Journals Online (AJOL) To evaluate the impact of exclusive breastfeeding (EBFing) practice on maternal anthropometry during the first 6months of birth. Measurement of weight, height, triceps skin-fold thickness (TST), and mid-arm circumference (MAC) was carried out in a matched cohort of women practicing EBFing and those using other ... 1. Determinants of exclusive breastfeeding practices in Ethiopia ... African Journals Online (AJOL) Background: Despite the demonstrated benefits of breast milk, the prevalence of breastfeeding, in-particular exclusive breastfeeding (EBF), in many developing countries including Ethiopia is lower than the international recommendation of EBF for the first six months of life. Objective: To assess the practice of EBF and ... 2. 40 CFR 503.6 - Exclusions. Science.gov (United States) 2010-07-01 ... treatment of domestic sewage in a treatment works. (i) Drinking water treatment sludge. This part does not... water or ground water used for drinking water. (j) Commercial and industrial septage. This part does not... DISPOSAL OF SEWAGE SLUDGE General Provisions § 503.6 Exclusions. (a) Treatment processes. This part does... 3. Sexism and Permanent Exclusion from School Science.gov (United States) Carlile, Anna 2009-01-01 Focussing on narratives collected during a two year participant observation research project in the children's services department of an urban local authority, this article addresses the intersection between incidents of permanent exclusion from school and assumptions made on the basis of a young person's gender. The article considers gendered… 4. Knowledge, Attitude and Practice Towards Exclusive Breast ... African Journals Online (AJOL) Knowledge, Attitude and Practice Towards Exclusive Breast-Feeding At Jimma, Ethiopia. Teklebrhan Tema. Abstract. No abstract - Available on PDF. Full Text: EMAIL FREE FULL TEXT EMAIL FREE FULL TEXT · DOWNLOAD FULL TEXT DOWNLOAD FULL TEXT · AJOL African Journals Online. HOW TO USE AJOL. 5. Exclusive production of W pairs in CMS CERN Document Server INSPIRE-00002838 2014-01-01 We report the results on the search for exclusive production of $W$ pairs in the LHC with data collected by the Compact Muon Solenoid detector in proton-proton collisions at $\\sqrt{s}$~=~7~TeV. The analysis comprises the two-photon production of a $W$ pairs, ${pp\\to p\\,W^{+}W^{-}\\,p\\to p\\,\ 6. Exclusive processes in pp collisions in CMS CERN Document Server Da Silveira, Gustavo Gil 2013-01-01 We report the results on the searches of exclusive production of low- and high-mass pairs with the Compact Muon Solenoid (CMS) detector in proton-proton collisions at$\\sqrt{s}$= 7 TeV. The analyses comprise the central exclusive$\\gamma\\gamma$production, the exclusive two-photon production of dileptons,$e^{+}e^{-}$and$\\mu^{+}\\mu^{-}$, and the exclusive two-photon production of$W$pairs in the asymmetric$e^{\\pm}\\mu^{\\mp}$decay channel. No diphotons candidates are observed in data and an upper limit on the cross section is set to 1.18 pb with 95% confidence level for$E_{T}(\\gamma)>$5.5 GeV and$\|\\eta(\\gamma)\|$5.5 GeV and$\|\\eta(e)\|$11.5 GeV,$p_{\\textrm{T}}(\\mu)>$4 GeV and$\|\\eta(\\mu)\|$4 GeV,$\|\\eta(\\mu)\|$20 GeV. Moreover, the study of the tail of the dilepton transverse momentum distribution resulted in model-independent upper limits for the anomalous quartic gauge couplings, which are of the order of 10$^{-4}$. 7. Exclusive production of$W$pairs in CMS OpenAIRE Da Silveira, Gustavo Gil; CMS 2014-01-01 We report the results on the search for exclusive production of$W$pairs in the LHC with data collected by the Compact Muon Solenoid detector in proton-proton collisions at$\\sqrt{s}$~=~7~TeV. The analysis comprises the two-photon production of a$W$pairs,${pp\\to p\\,W^{+}W^{-}\\,p\\to p\\,\ 8. Starvation-free mutual exclusion with semaphores NARCIS (Netherlands) The standard implementation of mutual exclusion by means of a semaphore allows starvation of processes. Between 1979 and 1986, three algorithms were proposed that preclude starvation. These algorithms use a special kind of semaphore. We model this so-called buffered semaphore rigorously and provide 9. Deadlocks and dihomotopy in mutual exclusion models DEFF Research Database (Denmark) Raussen, Martin 2005-01-01 spaces, the directed ($d$-spaces) of M.Grandis and the flows of P. Gaucher. All models invite to use or modify ideas from algebraic topology, notably homotopy. In specific semaphore models for mutual exclusion, we have developed methods and algorithms that can detect deadlocks and unsafe regions and give... 10. Counterfactual overdetermination vs. the causal exclusion problem. Science.gov (United States) Sparber, Georg 2005-01-01 This paper aims to show that a counterfactual approach to causation is not sufficient to provide a solution to the causal exclusion problem in the form of systematic overdetermination. Taking into account the truthmakers of causal counterfactuals provides a strong argument in favour of the identity of causes in situations of translevel, causation. 11. Factors associated with exclusive breastfeeding among mothers ... African Journals Online (AJOL) Conclusion: This study could help mothers, Ministry of Health and other nongovernmental organisations working with child health programmes, in likely interventions and supporting the ongoing child survival programmes, by taking appropriate steps in enhancing exclusive breastfeeding. As mothers attend antenatal and ... 12. 46 CFR 504.4 - Categorical exclusions. Science.gov (United States) 2010-10-01 ... FEDERAL MARITIME COMMISSION GENERAL AND ADMINISTRATIVE PROVISIONS PROCEDURES FOR ENVIRONMENTAL POLICY ANALYSIS § 504.4 Categorical exclusions. (a) No environmental analyses need be undertaken or environmental... foreign country. (19) Action taken on special docket applications pursuant to § 502.271 of this chapter... 13. 10 CFR 830.2 - Exclusions. Science.gov (United States) 2010-01-01 ... by the Department of Transportation; (d) Activities conducted under the Nuclear Waste Policy Act of... ENERGY NUCLEAR SAFETY MANAGEMENT § 830.2 Exclusions. This part does not apply to: (a) Activities that are regulated through a license by the Nuclear Regulatory Commission (NRC) or a State under an Agreement with... 14. Testing the exclusivity effect in location memory. Science.gov (United States) Clark, Daniel P A; Dunn, Andrew K; Baguley, Thom 2013-01-01 There is growing literature exploring the possibility of parallel retrieval of location memories, although this literature focuses primarily on the speed of retrieval with little attention to the accuracy of location memory recall. Baguley, Lansdale, Lines, and Parkin (2006) found that when a person has two or more memories for an object's location, their recall accuracy suggests that only one representation can be retrieved at a time (exclusivity). This finding is counterintuitive given evidence of non-exclusive recall in the wider memory literature. The current experiment explored the exclusivity effect further and aimed to promote an alternative outcome (i.e., independence or superadditivity) by encouraging the participants to combine multiple representations of space at encoding or retrieval. This was encouraged by using anchor (points of reference) labels that could be combined to form a single strongly associated combination. It was hypothesised that the ability to combine the anchor labels would allow the two representations to be retrieved concurrently, generating higher levels of recall accuracy. The results demonstrate further support for the exclusivity hypothesis, showing no significant improvement in recall accuracy when there are multiple representations of a target object's location as compared to a single representation. 15. Bitcoin and Beyond: Exclusively Informational Money NARCIS (Netherlands) Bergstra, J.A.; de Leeuw, K. 2013-01-01 The famous new money Bitcoin is classified as a technical informational money (TIM). Besides introducing the idea of a TIM, a more extreme notion of informational money will be developed: exclusively informational money (EXIM). The informational coins (INCOs) of an EXIM can be in control of an agent 16. Urban violence and exclusion in the DRC International Development Research Centre (IDRC) Digital Library (Canada) support, children from impoverished households, many of whom are uneducated, are adding to the ... The goal of this study was to identify the dynamic interplay among poverty/exclusion ... The lack of public lighting and access points to water exposes girls to .... work together to develop more inclusive economic and social. 17. 40 CFR 68.126 - Exclusion. Science.gov (United States) 2010-07-01 ... ACCIDENT PREVENTION PROVISIONS Regulated Substances for Accidental Release Prevention § 68.126 Exclusion. Flammable Substances Used as Fuel or Held for Sale as Fuel at Retail Facilities. A flammable substance... substance is used as a fuel or held for sale as a fuel at a retail facility. [65 FR 13250, Mar. 13, 2000] ... 18. The Exclusive Pursuit of Social Inclusion Science.gov (United States) Goodson, Ivor 2005-01-01 Despite its best intentions, social exclusion has grown rather than diminished under New Labour's education policies. In order to understand this, Ivor Goodson argues that we need to engage with the history of the formal curriculum and the long and continuing fight over what counts as proper knowledge. Taking science and environmental science as… 19. 27 CFR 8.51 - Exclusion, in general. Science.gov (United States) 2010-04-01 ..., DEPARTMENT OF THE TREASURY LIQUORS EXCLUSIVE OUTLETS Exclusion § 8.51 Exclusion, in general. (a) Exclusion, in whole or in part occurs: (1) When a practice by an industry member, whether direct, indirect, or... tie or link between the industry member and retailer or by any other means of industry member control... International Nuclear Information System (INIS) Lira, Ignacio 2013-01-01 An inductive strategy is proposed for teaching dimensional analysis to second- or third-year students of physics, chemistry, or engineering. In this strategy, Buckingham's theorem is seen as a consequence and not as the starting point. In order to concentrate on the basics, the mathematics is kept as elementary as possible. Simple examples are suggested for classroom demonstrations of the power of the technique and others are put forward for homework or experimentation, but instructors are encouraged to produce examples of their own. (paper) CERN Document Server Murphy, Patrick 1982-01-01 Applied Mathematics: Made Simple provides an elementary study of the three main branches of classical applied mathematics: statics, hydrostatics, and dynamics. The book begins with discussion of the concepts of mechanics, parallel forces and rigid bodies, kinematics, motion with uniform acceleration in a straight line, and Newton's law of motion. Separate chapters cover vector algebra and coplanar motion, relative motion, projectiles, friction, and rigid bodies in equilibrium under the action of coplanar forces. The final chapters deal with machines and hydrostatics. The standard and conte CERN Document Server Wooldridge, Susan 2013-01-01 Data Processing: Made Simple, Second Edition presents discussions of a number of trends and developments in the world of commercial data processing. The book covers the rapid growth of micro- and mini-computers for both home and office use; word processing and the 'automated office'; the advent of distributed data processing; and the continued growth of database-oriented systems. The text also discusses modern digital computers; fundamental computer concepts; information and data processing requirements of commercial organizations; and the historical perspective of the computer industry. The CERN Document Server Deane, Sharon 2003-01-01 ASP Made Simple provides a brief introduction to ASP for the person who favours self teaching and/or does not have expensive computing facilities to learn on. The book will demonstrate how the principles of ASP can be learned with an ordinary PC running Personal Web Server, MS Access and a general text editor like Notepad.After working through the material readers should be able to:* Write ASP scripts that can display changing information on a web browser* Request records from a remote database or add records to it* Check user names & passwords and take this knowledge forward, either for their 4. Theory of simple liquids CERN Document Server Hansen, Jean-Pierre 1986-01-01 This book gives a comprehensive and up-to-date treatment of the theory of ""simple"" liquids. The new second edition has been rearranged and considerably expanded to give a balanced account both of basic theory and of the advances of the past decade. It presents the main ideas of modern liquid state theory in a way that is both pedagogical and self-contained. The book should be accessible to graduate students and research workers, both experimentalists and theorists, who have a good background in elementary mechanics.Key Features* Compares theoretical deductions with experimental r 5. View-tolerant face recognition and Hebbian learning imply mirror-symmetric neural tuning to head orientation Science.gov (United States) Leibo, Joel Z.; Liao, Qianli; Freiwald, Winrich A.; Anselmi, Fabio; Poggio, Tomaso 2017-01-01 SUMMARY The primate brain contains a hierarchy of visual areas, dubbed the ventral stream, which rapidly computes object representations that are both specific for object identity and robust against identity-preserving transformations like depth-rotations [1, 2]. Current computational models of object recognition, including recent deep learning networks, generate these properties through a hierarchy of alternating selectivity-increasing filtering and tolerance-increasing pooling operations, similar to simple-complex cells operations [3, 4, 5, 6]. Here we prove that a class of hierarchical architectures and a broad set of biologically plausible learning rules generate approximate invariance to identity-preserving transformations at the top level of the processing hierarchy. However, all past models tested failed to reproduce the most salient property of an intermediate representation of a three-level face-processing hierarchy in the brain: mirror-symmetric tuning to head orientation [7]. Here we demonstrate that one specific biologically-plausible Hebb-type learning rule generates mirror-symmetric tuning to bilaterally symmetric stimuli like faces at intermediate levels of the architecture and show why it does so. Thus the tuning properties of individual cells inside the visual stream appear to result from group properties of the stimuli they encode and to reflect the learning rules that sculpted the information-processing system within which they reside. PMID:27916522 6. High performance symmetric supercapacitor based on zinc hydroxychloride nanosheets and 3D graphene-nickel foam composite International Nuclear Information System (INIS) Khamlich, S.; Abdullaeva, Z.; Kennedy, J.V.; Maaza, M. 2017-01-01 Highlights: • A symmetric supercapacitor device based on NiF-G/ZHCNs composite materials electrodes was successfully fabricated. • The fabricated device exhibited high specific areal capacitance of 222 mF cm"−"2 at 1.0 mA cm"−"2 current density. • Excellent cycling performance with 96% specific capacitance retention after 5000 cycles was achieved. - Abstract: In this work, zinc hydroxychloride nanosheets (ZHCNs) were deposited on 3d graphene-nickel foam (NiF-G) by employing a simple hydrothermal synthesis method to form NiF-G/ZHCNs composite electrode materials. The fabricated NiF-G/ZHCNs electrode revealed a well-developed pore structures with high specific surface area of 119 m"2 g"−"1, and used as electrode materials for symmetric supercapacitor with aqueous alkaline electrolyte. The specific areal capacitance and electron charge transfer resistance (R_c_t) were 222 mF cm"−"2 (at current density of 1.0 mA cm"−"2) and 1.63 Ω, respectively, in a symmetric two-electrode system. After 5000 cycles with galvanostatic charge/discharge, the device can maintain 96% of its initial capacitance under 1.0 mA cm"−"2 and showed low R_c_t of about 9.84 Ω. These results indicate that NiF-G/ZHCNs composite is an excellent electrode material for electrochemical energy storage devices. 7. Probabilistic simple sticker systems Science.gov (United States) Selvarajoo, Mathuri; Heng, Fong Wan; Sarmin, Nor Haniza; Turaev, Sherzod 2017-04-01 A model for DNA computing using the recombination behavior of DNA molecules, known as a sticker system, was introduced by by L. Kari, G. Paun, G. Rozenberg, A. Salomaa, and S. Yu in the paper entitled DNA computing, sticker systems and universality from the journal of Acta Informatica vol. 35, pp. 401-420 in the year 1998. A sticker system uses the Watson-Crick complementary feature of DNA molecules: starting from the incomplete double stranded sequences, and iteratively using sticking operations until a complete double stranded sequence is obtained. It is known that sticker systems with finite sets of axioms and sticker rules generate only regular languages. Hence, different types of restrictions have been considered to increase the computational power of sticker systems. Recently, a variant of restricted sticker systems, called probabilistic sticker systems, has been introduced [4]. In this variant, the probabilities are initially associated with the axioms, and the probability of a generated string is computed by multiplying the probabilities of all occurrences of the initial strings in the computation of the string. Strings for the language are selected according to some probabilistic requirements. In this paper, we study fundamental properties of probabilistic simple sticker systems. We prove that the probabilistic enhancement increases the computational power of simple sticker systems. 8. Simple stochastic simulation. Science.gov (United States) Schilstra, Maria J; Martin, Stephen R 2009-01-01 Stochastic simulations may be used to describe changes with time of a reaction system in a way that explicitly accounts for the fact that molecules show a significant degree of randomness in their dynamic behavior. The stochastic approach is almost invariably used when small numbers of molecules or molecular assemblies are involved because this randomness leads to significant deviations from the predictions of the conventional deterministic (or continuous) approach to the simulation of biochemical kinetics. Advances in computational methods over the three decades that have elapsed since the publication of Daniel Gillespie's seminal paper in 1977 (J. Phys. Chem. 81, 2340-2361) have allowed researchers to produce highly sophisticated models of complex biological systems. However, these models are frequently highly specific for the particular application and their description often involves mathematical treatments inaccessible to the nonspecialist. For anyone completely new to the field to apply such techniques in their own work might seem at first sight to be a rather intimidating prospect. However, the fundamental principles underlying the approach are in essence rather simple, and the aim of this article is to provide an entry point to the field for a newcomer. It focuses mainly on these general principles, both kinetic and computational, which tend to be not particularly well covered in specialist literature, and shows that interesting information may even be obtained using very simple operations in a conventional spreadsheet. 9. Nonstandard jump functions for radically symmetric shock waves International Nuclear Information System (INIS) Baty, Roy S.; Tucker, Don H.; Stanescu, Dan 2008-01-01 Nonstandard analysis is applied to derive generalized jump functions for radially symmetric, one-dimensional, magnetogasdynamic shock waves. It is assumed that the shock wave jumps occur on infinitesimal intervals and the jump functions for the physical parameters occur smoothly across these intervals. Locally integrable predistributions of the Heaviside function are used to model the flow variables across a shock wave. The equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the physical parameters for two families of self-similar flows. It is shown that the microstructures for these families of radially symmetric, magnetogasdynamic shock waves coincide in a nonstandard sense for a specified density jump function. 10. Random matrix ensembles for PT-symmetric systems International Nuclear Information System (INIS) Graefe, Eva-Maria; Mudute-Ndumbe, Steve; Taylor, Matthew 2015-01-01 Recently much effort has been made towards the introduction of non-Hermitian random matrix models respecting PT-symmetry. Here we show that there is a one-to-one correspondence between complex PT-symmetric matrices and split-complex and split-quaternionic versions of Hermitian matrices. We introduce two new random matrix ensembles of (a) Gaussian split-complex Hermitian; and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary sizes. We conjecture that these ensembles represent universality classes for PT-symmetric matrices. For the case of 2 × 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues, the one-level densities and the level spacings in the case of real eigenvalues. (fast track communication) 11. Synthesis of novel symmetrical macrocycle via oxidative homocoupling of bisalkyne Energy Technology Data Exchange (ETDEWEB) Kamalulazmy, Nurulain; Hassan, Nurul Izzaty [School of Chemical Sciences and Food Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan (Malaysia) 2014-09-03 A novel symmetrical macrocycle has been synthesised via oxidative homocoupling of bisalkyne, diprop-2-ynyl pyridine-2,6-dicarboxylate mediated by copper (I) iodide (CuI) and 4-dimethylaminopyridine (DMAP). The precursor compound was synthesised from 2,6-pyridine dicarbonyl dichloride and propargyl alcohol in the presence of triethylamine. The reaction mixture was stirred overnight and further purified via column chromatograpy with 76% yield. Single crystal for X-ray study was obtained by recrystallization from acetone. Subsequently, a symmetrical macrocycle was synthesised from oxidative homocoupling of precursor compound in open atmosphere. The crude product was purified by column chromatography to furnish macrocycle compound with 5% yield. Both compounds were characterised by IR, {sup 1}H and {sup 13}C NMR and mass spectral techniques. The unusual conformation of the bisalkyne and twisted conformation of designed macrocycle has influence the percentage yield. This has been studied thoroughly by X-ray crystallography and electronic structure calculations. 12. Information Retrieval and Criticality in Parity-Time-Symmetric Systems. Science.gov (United States) Kawabata, Kohei; Ashida, Yuto; Ueda, Masahito 2017-11-10 By investigating information flow between a general parity-time (PT-)symmetric non-Hermitian system and an environment, we find that the complete information retrieval from the environment can be achieved in the PT-unbroken phase, whereas no information can be retrieved in the PT-broken phase. The PT-transition point thus marks the reversible-irreversible criticality of information flow, around which many physical quantities such as the recurrence time and the distinguishability between quantum states exhibit power-law behavior. Moreover, by embedding a PT-symmetric system into a larger Hilbert space so that the entire system obeys unitary dynamics, we reveal that behind the information retrieval lies a hidden entangled partner protected by PT symmetry. Possible experimental situations are also discussed. 13. Symmetrical and overloaded effect of diffusion in information filtering Science.gov (United States) Zhu, Xuzhen; Tian, Hui; Chen, Guilin; Cai, Shimin 2017-10-01 In physical dynamics, mass diffusion theory has been applied to design effective information filtering models on bipartite network. In previous works, researchers unilaterally believe objects' similarities are determined by single directional mass diffusion from the collected object to the uncollected, meanwhile, inadvertently ignore adverse influence of diffusion overload. It in some extent veils the essence of diffusion in physical dynamics and hurts the recommendation accuracy and diversity. After delicate investigation, we argue that symmetrical diffusion effectively discloses essence of mass diffusion, and high diffusion overload should be published. Accordingly, in this paper, we propose an symmetrical and overload penalized diffusion based model (SOPD), which shows excellent performances in extensive experiments on benchmark datasets Movielens and Netflix. 14. EXCEPTIONAL POINTS IN OPEN AND PT-SYMMETRIC SYSTEMS Directory of Open Access Journals (Sweden) Hichem Eleuch 2014-04-01 Full Text Available Exceptional points (EPs determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2 × 2 matrix that is characteristic either of open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment on the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features. 15. Biophysical information in asymmetric and symmetric diurnal bidirectional canopy reflectance Science.gov (United States) Vanderbilt, Vern C.; Caldwell, William F.; Pettigrew, Rita E.; Ustin, Susan L.; Martens, Scott N.; Rousseau, Robert A.; Berger, Kevin M.; Ganapol, B. D.; Kasischke, Eric S.; Clark, Jenny A. 1991-01-01 The authors present a theory for partitioning the information content in diurnal bidirectional reflectance measurements in order to detect differences potentially related to biophysical variables. The theory, which divides the canopy reflectance into asymmetric and symmetric functions of solar azimuth angle, attributes asymmetric variation to diurnal changes in the canopy biphysical properties. The symmetric function is attributed to the effects of sunlight interacting with a hypothetical average canopy which would display the average diurnal properties of the actual canopy. The authors analyzed radiometer data collected diurnally in the Thematic Mapper wavelength bands from two walnut canopies that received differing irrigation treatments. The reflectance of the canopies varied with sun and view angles and across seven bands in the visible, near-infrared, and middle infrared wavelength regions. Although one of the canopies was permanently water stressed and the other was stressed in mid-afternoon each day, no water stress signature was unambiguously evident in the reflectance data. 16. Implications of the Cosmological Constant for Spherically Symmetric Mass Distributions Science.gov (United States) Zubairi, Omair; Weber, Fridolin 2013-04-01 In recent years, scientists have made the discovery that the expansion rate of the Universe is increasing rather than decreasing. This acceleration leads to an additional term in Albert Einstein's field equations which describe general relativity and is known as the cosmological constant. This work explores the aftermath of a non-vanishing cosmological constant for relativistic spherically symmetric mass distributions, which are susceptible to change against Einstein's field equations. We introduce a stellar structure equation known as the Tolman-Oppenhiemer-Volkoff (TOV) equation modified for a cosmological constant, which is derived from Einstein's modified field equations. We solve this modified TOV equation for these spherically symmetric mass distributions and obtain stellar properties such as mass and radius and investigate changes that may occur depending on the value of the cosmological constant. 17. Maximal slicing of D-dimensional spherically symmetric vacuum spacetime International Nuclear Information System (INIS) Nakao, Ken-ichi; Abe, Hiroyuki; Yoshino, Hirotaka; Shibata, Masaru 2009-01-01 We study the foliation of a D-dimensional spherically symmetric black-hole spacetime with D≥5 by two kinds of one-parameter families of maximal hypersurfaces: a reflection-symmetric foliation with respect to the wormhole slot and a stationary foliation that has an infinitely long trumpetlike shape. As in the four-dimensional case, the foliations by the maximal hypersurfaces avoid the singularity irrespective of the dimensionality. This indicates that the maximal slicing condition will be useful for simulating higher-dimensional black-hole spacetimes in numerical relativity. For the case of D=5, we present analytic solutions of the intrinsic metric, the extrinsic curvature, the lapse function, and the shift vector for the foliation by the stationary maximal hypersurfaces. These data will be useful for checking five-dimensional numerical-relativity codes based on the moving puncture approach. 18. Thermal properties of self-gravitating plane-symmetric configuration Energy Technology Data Exchange (ETDEWEB) Hara, T; Ikeuchi, S [Kyoto Univ. (Japan). Dept. of Physics; Sugimoto, D 1976-09-01 As a limiting case of rotating stars, thermal properties of infinite plane-symmetric self-gravitating gas are investigated. Such a configuration is characterized by surface density of the plane instead of stellar mass. In the Kelvin contraction, temperature of the interior decreases, if the surface density is kept constant. If the accretion of matter takes place, or if the angular momenta are transferred outward, the surface density will increase. In this case, the temperature of the interior may increase. When a nuclear burning is ignited, it is thermally unstable in most cases, even when electrons are non-degenerate. This thermal instability is one of the essential differences of the plane-symmetric configuration from the spherical star. Such instabilities are computed for different cases of nuclear fuels. This type of nuclear instability is the same phenomenon as thermal instability of a thin shell burning in a spherical star. 19. Continuous symmetric reductions of the Adler-Bobenko-Suris equations International Nuclear Information System (INIS) Tsoubelis, D; Xenitidis, P 2009-01-01 Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three-point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi 'generating partial differential equations' is established and an auto-Baecklund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1 δ=0 members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents 20. Exotic fermions in the left-right symmetric model International Nuclear Information System (INIS) Choi, J.; Volkas, R.R. 1992-01-01 A systematic study is made of non-standard fermion multiplets in left-right symmetric models with gauge group SU(3) x SU(2) L x SU(2) R x U(1) BL . Constraints from gauge anomaly cancellation and invariance of Yukawa coupling terms are used to define interesting classes of exotic fermions. The standard quark lepton spectrum of left-right symmetric models was identified as the simplest member of an infinite class. Phenomenological implications of the next simplest member of this class are then studied. Classes of exotic fermions which may couple to the standard fermions through doublet Higgs bosons were also considered, then shown that some of these exotics may be used to induce a generalised universal see-saw mechanism. 12 refs., 1 tab 1. Long-term repetition priming with symmetrical polygons and words. Science.gov (United States) Kersteen-Tucker, Z 1991-01-01 In two different tasks, subjects were asked to make lexical decisions (word or nonword) and symmetry judgments (symmetrical or nonsymmetrical) about two-dimensional polygons. In both tasks, every stimulus was repeated at one of four lags (0, 1, 4, or 8 items interposed between the first and second stimulus presentations). This paradigm, known as repetition priming, revealed comparable short-term priming (Lag 0) and long-term priming (Lags 1, 4, and 8) both for symmetrical polygons and for words. A shorter term component (Lags 0 and 1) of priming was observed for nonwords, and only very short-term priming (Lag 0) was observed for nonsymmetrical polygons. These results indicate that response facilitation accruing from repeated exposure can be observed for stimuli that have no preexisting memory representations and suggest that perceptual factors contribute to repetition-priming effects. 2. Admissible perturbations and false instabilities in PT -symmetric quantum systems Science.gov (United States) Znojil, Miloslav 2018-03-01 One of the most characteristic mathematical features of the PT -symmetric quantum mechanics is the explicit Hamiltonian dependence of its physical Hilbert space of states H =H (H ) . Some of the most important physical consequences are discussed, with emphasis on the dynamical regime in which the system is close to phase transition. Consistent perturbation treatment of such a regime is proposed. An illustrative application of the innovated perturbation theory to a non-Hermitian but PT -symmetric user-friendly family of J -parametric "discrete anharmonic" quantum Hamiltonians H =H (λ ⃗) is provided. The models are shown to admit the standard probabilistic interpretation if and only if the parameters remain compatible with the reality of the spectrum, λ ⃗∈D(physical ) . In contradiction to conventional wisdom, the systems are then shown to be stable with respect to admissible perturbations, inside the domain D(physical ), even in the immediate vicinity of the phase-transition boundaries ∂ D(physical ) . 3. PT-symmetric ladders with a scattering core Energy Technology Data Exchange (ETDEWEB) D' Ambroise, J. [Department of Mathematics, Amherst College, Amherst, MA 01002-5000 (United States); Lepri, S. [CNR – Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del piano 10, I-50019 Sesto Fiorentino (Italy); Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (Italy); Malomed, B.A. [Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978 (Israel); Kevrekidis, P.G. [Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305 (United States) 2014-08-01 We consider a PT-symmetric chain (ladder-shaped) system governed by the discrete nonlinear Schrödinger equation where the cubic nonlinearity is carried solely by two central “rungs” of the ladder. Two branches of scattering solutions for incident plane waves are found. We systematically construct these solutions, analyze their stability, and discuss non-reciprocity of the transmission associated with them. To relate the results to finite-size wavepacket dynamics, we also perform direct simulations of the evolution of the wavepackets, which confirm that the transmission is indeed asymmetric in this nonlinear system with the mutually balanced gain and loss. - Highlights: • We model a PT-symmetric ladder system with cubic nonlinearity on two central rungs. • We examine non-reciprocity and stability of incident plane waves. • Simulations of wavepackets confirm our results. 4. The inverse spatial Laplacian of spherically symmetric spacetimes International Nuclear Information System (INIS) Fernandes, Karan; Lahiri, Amitabha 2017-01-01 We derive the inverse spatial Laplacian for static, spherically symmetric backgrounds by solving Poisson’s equation for a point source. This is different from the electrostatic Green function, which is defined on the four dimensional static spacetime, while the equation we consider is defined on the spatial hypersurface of such spacetimes. This Green function is relevant in the Hamiltonian dynamics of theories defined on spherically symmetric backgrounds, and closed form expressions for the solutions we find are absent in the literature. We derive an expression in terms of elementary functions for the Schwarzschild spacetime, and comment on the relation of this solution with the known Green function of the spacetime Laplacian operator. We also find an expression for the Green function on the static pure de-Sitter space in terms of hypergeometric functions. We conclude with a discussion of the constraints of the electromagnetic field. (paper) 5. Entanglement of polar symmetric top molecules as candidate qubits. Science.gov (United States) Wei, Qi; Kais, Sabre; Friedrich, Bretislav; Herschbach, Dudley 2011-10-21 Proposals for quantum computing using rotational states of polar molecules as qubits have previously considered only diatomic molecules. For these the Stark effect is second-order, so a sizable external electric field is required to produce the requisite dipole moments in the laboratory frame. Here we consider use of polar symmetric top molecules. These offer advantages resulting from a first-order Stark effect, which renders the effective dipole moments nearly independent of the field strength. That permits use of much lower external field strengths for addressing sites. Moreover, for a particular choice of qubits, the electric dipole interactions become isomorphous with NMR systems for which many techniques enhancing logic gate operations have been developed. Also inviting is the wider chemical scope, since many symmetric top organic molecules provide options for auxiliary storage qubits in spin and hyperfine structure or in internal rotation states. © 2011 American Institute of Physics 6. Solving the generalized symmetric eigenvalue problem using tile algorithms on multicore architectures KAUST Repository Ltaief, Hatem; Luszczek, Piotr R.; Haidar, Azzam; Dongarra, Jack 2012-01-01 This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric 7. Complex group algebras of the double covers of the symmetric and alternating group DEFF Research Database (Denmark) Bessenrodt, Christine; Nguyen, Hung Ngoc; Olsson, Jørn Børling 2015-01-01 We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras......We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras... 8. Non-Abelian behavior of α bosons in cold symmetric nuclear matter International Nuclear Information System (INIS) Zheng Hua; Bonasera, Aldo 2011-01-01 The ground-state energy of infinite symmetric nuclear matter is usually described by strongly interacting nucleons obeying the Pauli exclusion principle. We can imagine a unitary transformation which groups four nonidentical nucleons (i.e., with different spin and isospin) close in coordinate space. Those nucleons, being nonidentical, do not obey the Pauli principle, thus their relative momenta are negligibly small (just to fulfill the Heisenberg principle). Such a cluster can be identified with an α boson. But in dense nuclear matter, those α particles still obey the Pauli principle since are constituted of fermions. The ground state energy of nuclear matter α clusters is the same as for nucleons, thus it is degenerate. We could think of α particles as vortices which can now braid, for instance making 8 Be which leave the ground state energy unchanged. Further braiding to heavier clusters ( 12 C, 16 O,...) could give a different representation of the ground state at no energy cost. In contrast d-like clusters (i.e., N=Z odd-odd nuclei, where N and Z are the neutron and proton number, respectively) cannot describe the ground state of nuclear matter and can be formed at high excitation energies (or temperatures) only. We show that even-even, N=Z, clusters could be classified as non-Abelian states of matter. As a consequence an α condensate in nuclear matter might be hindered by the Fermi motion, while it could be possible a condensate of 8 Be or heavier clusters. 9. Identifying compatibility of lithium salts with LiFePO4 cathode using a symmetric cell Science.gov (United States) Tong, Bo; Wang, Jiawei; Liu, Zhenjie; Ma, Lipo; Zhou, Zhibin; Peng, Zhangquan 2018-04-01 The electrochemical performance of lithium-ion batteries is dominated by the interphase electrochemistry between the electrolyte and electrode materials. A multitude of efforts have been dedicated to the solid electrolyte interphase (SEI) formed on the anode. However, the interphase on the cathode, namely the cathode electrolyte interphase (CEI), is left aside, partially due to the fact that it is hard to single out the CEI considering the complicated anode-cathode inter-talk. Herein, a partially delithiated lithium iron phosphate (Li0.25FePO4) electrode is used as the anode. Owing to a high voltage plateau (≈3.45 V vs. Li/Li+), negligible reduction reactions of electrolyte occur on the L0.25FePO4 anode. Therefore, the CEI can be investigated exclusively. Using a LiFePO4\|Li0.25FePO4 symmetric cell configuration, we scrutinize the compatibility of the electrolytes containing a wide spectrum of lithium salts, Li[(FSO2)(Cm F2m+1SO2)N] (m = 0, 1, 2, 4), with the LiFePO4, in both cycling and calendar tests. It is found that the Li[(FSO2)(n-C4F9SO2)N] (LiFNFSI)-based electrolyte exhibits the highest compatibility with LiFePO4. 10. Procrustes Problems for General, Triangular, and Symmetric Toeplitz Matrices Directory of Open Access Journals (Sweden) Juan Yang 2013-01-01 Full Text Available The Toeplitz Procrustes problems are the least squares problems for the matrix equation AX=B over some Toeplitz matrix sets. In this paper the necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions of the Toeplitz Procrustes problems when the unknown matrices are constrained to the general, the triangular, and the symmetric Toeplitz matrices, respectively. The algorithms are designed and the numerical examples show that these algorithms are feasible. 11. Superfield Lax formalism of supersymmetric sigma model on symmetric spaces International Nuclear Information System (INIS) Saleem, U.; Hassan, M. 2006-01-01 We present a superfield Lax formalism of the superspace sigma model based on the target space G/H and show that a one-parameter family of flat superfield connections exists if the target space G/H is a symmetric space. The formalism has been related to the existence of an infinite family of local and non-local superfield conserved quantities. A few examples have been given to illustrate the results. (orig.) 12. Two-parametric PT-symmetric quartic family International Nuclear Information System (INIS) Eremenko, Alexandre; Gabrielov, Andrei 2012-01-01 We describe a parametrization of the real spectral locus of the two-parametric family of PT-symmetric quartic oscillators. For this family, we find a parameter region where all eigenvalues are real, extending the results of Dorey et al (2007 J. Phys. A: Math Theor. 40 R205–83) and Shin (2005 J. Phys. A: Math. Gen. 38 6147–66; 2002 Commun. Math. Phys. 229 543–64). (paper) 13. Quantum cloning of mixed states in symmetric subspaces International Nuclear Information System (INIS) Fan Heng 2003-01-01 Quantum-cloning machine for arbitrary mixed states in symmetric subspaces is proposed. This quantum-cloning machine can be used to copy part of the output state of another quantum-cloning machine and is useful in quantum computation and quantum information. The shrinking factor of this quantum cloning achieves the well-known upper bound. When the input is identical pure states, two different fidelities of this cloning machine are optimal 14. Research on Characteristics of New Energy Dissipation With Symmetrical Structure Science.gov (United States) Ming, Wen; Huang, Chun-mei; Huang, Hao-wen; Wang, Xin-fang 2018-03-01 Utilizing good energy consumption capacity of arc steel bar, a new energy dissipation with symmetrical structure was proposed in this article. On the base of collection experimental data of damper specimen Under low cyclic reversed loading, finite element models were built by using ANSYS software, and influences of parameter change (Conduction rod diameter, Actuation plate thickness, Diameter of arc steel rod, Curved bars initial bending) on energy dissipation performance were analyzed. Some useful conclusions which can lay foundations for practical application were drawn. 15. Some problems in operator theory on bounded symmetric domains Czech Academy of Sciences Publication Activity Database Engliš, Miroslav 2004-01-01 Roč. 81, č. 1 (2004), s. 51-71 ISSN 0167-8019. [Representations of Lie groups, harmonic analysis on homogeneous spaces and quantization. Leiden, 07.12.2002-13.12.2002] R&D Projects: GA ČR GA201/03/0041 Institutional research plan: CEZ:AV0Z1019905 Keywords : homogeneous multiplication operators * bounded symmetric domains Subject RIV: BA - General Mathematics Impact factor: 0.354, year: 2004 16. ${ \\mathcal P }{ \\mathcal T }$-symmetric interpretation of unstable effective potentials CERN Document Server Bender, Carl M.; Mavromatos, Nick E.; Sarkar, Sarben 2016-01-01 The conventional interpretation of the one-loop effective potentials of the Higgs field in the Standard Model and the gravitino condensate in dynamically broken supergravity is that these theories are unstable at large field values. A ${ \\mathcal P }{ \\mathcal T }$-symmetric reinterpretation of these models at a quantum-mechanical level eliminates these instabilities and suggests that these instabilities may also be tamed at the quantum-field-theory level. 17. Dp spaces on bounded symmetric domains of Cn International Nuclear Information System (INIS) Shi Jihuai. 1989-06-01 In this paper, the space D p (Ω) of functions holomorphic on bounded symmetric domain of C m is defined. We prove that H p (Ω) is contained in D p (Ω) if 0 p (Ω) is contained in H p (Ω) if p ≥2, and both inclusions are proper. Further we find that some theorems on H p (Ω) can be extended to the wider class D p (Ω) for 0 < p ≤ 2. (author). 12 refs 18. A time-symmetric Universe model and its observational implication International Nuclear Information System (INIS) Futamase, T.; Matsuda, T. 1987-01-01 A time-symmetric closed-universe model is discussed in terms of the radiation arrow of time. The time symmetry requires the occurrence of advanced waves in the recontracting phase of the Universe. The observational consequences of such advanced waves are considered, and it is shown that a test observer in the expanding phase can observe a time-reversed image of a source of radiation in the future recontracting phase 19. Time-symmetric universe model and its observational implication Energy Technology Data Exchange (ETDEWEB) Futamase, T.; Matsuda, T. 1987-08-01 A time-symmetric closed-universe model is discussed in terms of the radiation arrow of time. The time symmetry requires the occurrence of advanced waves in the recontracting phase of the Universe. We consider the observational consequences of such advanced waves, and it is shown that a test observer in the expanding phase can observe a time-reversed image of a source of radiation in the future recontracting phase. 20. Weaving and neural complexity in symmetric quantum states Science.gov (United States) Susa, Cristian E.; Girolami, Davide 2018-04-01 We study the behaviour of two different measures of the complexity of multipartite correlation patterns, weaving and neural complexity, for symmetric quantum states. Weaving is the weighted sum of genuine multipartite correlations of any order, where the weights are proportional to the correlation order. The neural complexity, originally introduced to characterize correlation patterns in classical neural networks, is here extended to the quantum scenario. We derive closed formulas of the two quantities for GHZ states mixed with white noise.	{"extraction_info": {"found_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": 0.8192079067230225, "perplexity": 2439.3765570614937}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593586.54/warc/CC-MAIN-20180722194125-20180722214125-00100.warc.gz"}
http://mathhelpforum.com/discrete-math/214561-proof-combination-equation-print.html	# Proof of combination equation Printable View • Mar 10th 2013, 08:21 PM Paulo1913 Proof of combination equation Hi, I am having trouble trying to show the following proof: C(2n,n)-C(2n,n-1)=(1/(n+1))*C(2n,n). C(2n,n) is equal to 2n!/(n!^2), and I have worked out C(2n,n-1) to be (2n)!/((n-1)!(n+1)! but I am not totally sure about that one, and if that is correct, I cannot seem to simplify it into the form required. Thanks for any help • Mar 11th 2013, 04:42 AM veileen Re: Proof of combination equation $\textup{C}_{2n}^n-\textup{C}_{2n}^{n-1}=\textup{C}_{2n}^n-\frac{(2n)!}{(n-1)!\cdot (n+1)!}=$ $=\textup{C}_{2n}^n-\frac{n\cdot (2n)!}{n\cdot (n-1)!\cdot n! \cdot (n+1)}=\textup{C}_{2n}^n-\frac{n\cdot (2n)!}{n!\cdot n! \cdot (n+1)}=$ $=\textup{C}_{2n}^n-\frac{n}{n+1}\cdot \frac{ (2n)!}{n!\cdot n!}=\textup{C}_{2n}^n-\frac{n}{n+1}\textup{C}_{2n}^n=$ $=\left (1- \frac{n}{n+1} \right )\textup{C}_{2n}^n=\frac{n+1-n}{n+1}\textup{C}_{2n}^n=\frac{1}{n+1}\textup{C}_{ 2n}^n$	{"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": 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": 0.9794853925704956, "perplexity": 1589.4291560556267}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423787.24/warc/CC-MAIN-20170721162430-20170721182430-00668.warc.gz"}
https://cs.stackexchange.com/questions/16585/shift-and-or-multiplication-operation	# Shift-and-or multiplication operation Continuing in the same vein as Carry-free multiplication operation, a followup question is as follows (differences in bold): Let $r = p \oplus q$ be an operation similar to multiplication, but slightly simpler: when expressed via long-multiplication the columns aren't summed up, but rather or'd (not xor) together. Nothing is carried. $$\left[\begin{matrix} &&p_n & ... & p_i & ... & p_2 & p_1 \\ &&q_n & ... & q_i & ... & q_2 & q_1 & \otimes\\ \hline\\ &&q_1 \cdot p_n & ... & q_1 \cdot p_i & ... & q_1 \cdot p_2 & q_1 \cdot p_1\\ &q_2 \cdot p_n & ... & q_2 \cdot p_i & ... & q_2 \cdot p_2 & q_2 \cdot p_1\\ &&&&&&&...\\ q_i \cdot p_n & ... & q_i \cdot p_i & ... & q_i \cdot p_2 & q_i \cdot p_1 & \stackrel{i}{\leftarrow} &&{\bigvee} \\ \hline \\ \\r_{2n}& ... & r_i & ... &r_4& r_3 & r_2 &r_1 & = \end{matrix} \right]$$ Using the long-multiplication-style formulation, this takes $\mathcal O\left(\max\left(\left\|p\right\|,\left\|q\right\|\right)^2\right)=\mathcal O\left(\left\|r\right\|^2\right)$ time. Can we do better? Perhaps we can reuse some existing multiplication algorithms, or even better. • this is very similar to a problem known as "boolean convolution" in savages models of computation book.... – vzn Oct 30 '13 at 17:41 • @vzn can you link it here again? – Realz Slaw Oct 30 '13 at 17:42 • Have you done any thinking about whether Karatsuba, FFT, etc. methods apply to this operation as well? That'd be the first thing I would try. – D.W. Oct 30 '13 at 17:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.8638671040534973, "perplexity": 1742.8919671134788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573289.83/warc/CC-MAIN-20190918131429-20190918153429-00310.warc.gz"}
https://www.physicsforums.com/threads/rearranging-equations.82758/	# Homework Help: Rearranging equations. 1. Jul 20, 2005 ### Serj In 19 days I will be taking a physics class. I have already taken two algebra classes yet I do not know how to rearrange equations. What are the rules for rearranging an equation. F=ma, how do you rearrange it so you know what m= ? 2. Jul 20, 2005 ### HallsofIvy You "undo" whatever is done to a. In F= ma, a is multiplied by m. To "undo" that, you do the opposite of "multiply by m"- you divide by m. Dividing both sides of the equation by m, F/m= (ma)/m= a so a= F/m. 3. Jul 20, 2005 ### quasar987 mmmh.. by 'algebra' I guess you mean 'arithmetics' ? The key idea is that F, m and a represent numbers, like 1, 2, 3¼, pi³,...etc. And for any number 'n', there exists another number 'm' such that nm = 1. This number m is of course, the inverse of n: m = 1/n. So if for exemple, we want to isolate m in F=ma, we want to multiply both sides by the inverse of a, like so $$F=ma \Leftrightarrow \frac{1}{a}F=m\frac{a}{a} \Leftrightarrow \frac{F}{a} = m$$ 4. Jul 20, 2005 ### Serj t=(Vf-Vi)/a if I wanted to find "a" I would multiply both sides by a, yes? but that would leave me with at=Vf-Vi right? and "a" would not be isolated on one side of the =. what am I doing wrong? 5. Jul 20, 2005 ### quasar987 Continue. Divide both sides by t. 6. Jul 20, 2005 ### TD In these type of problems you only need to know two fundamental properties of equations. Equation remain equivalent under two certain operations. a) An eqaution remains equivalent when you add the same number to both sides (this may be negative if you wish to 'substract'). Example, we want a out of: $$a + b = c \Leftrightarrow a + b - b = c - b \Leftrightarrow a = c - b$$ b) An eqaution remains equivalent when you multiply both sides with the same factor ($$\ne 0$$) Example, we want a out of: $a\frac{b}{c} = d \Leftrightarrow a\frac{b}{c}\frac{c}{b} = d\frac{c}{b} \Leftrightarrow a = \frac{{dc}}{b}$ That's all you need here 7. Jul 20, 2005 ### Serj thanks everyone 8. Jul 20, 2005 ### EnumaElish An eqaution remains equivalent when you multiply both sides with the same factor, including 0. Generally, an eqaution remains equivalent when you apply an invertible (strictly increasing or strictly decreasing) function to both sides: a = b implies Log(a) = Log(b); a = sqrt(b) implies a2 = (sqrt(b))2 = b. In general, if a = f(b) and g is the inverse of f, then g(a) = g(f(b)) = b. Last edited: Jul 20, 2005 9. Jul 20, 2005 ### TD I thought that in equivalent equations, the solutions have to remain the same. In that case, you have to exclude 0, no? 10. Jul 20, 2005 ### Serj ok I have the equation a=(V2-V1)/t , And I want to make it V2=? . it would be aV2= (-V1)/t ,is it good so far? but i dont know what to do with "a" 11. Jul 20, 2005 ### EnumaElish You have to pay attention to the parantheses. a = (V2 - V1)/t at = V2 - V1 at + V1 = V2 12. Jul 20, 2005 ### EnumaElish I see what you're saying. I am splitting hairs when I point out that strictly speaking, a = b is preserved under multiplication with zero, which is not covered under your (or, the)definition of equivalent equations. Last edited: Jul 20, 2005 13. Jul 20, 2005 ### Serj why did you out everything except V2 on the other side of the = instead of putting V2 were "a" was and put "a" on the other side? why did you multiply "t" and "a"? if "t" was the numerator, would you still multiply it with "a"? 14. Jul 21, 2005 ### ek I remember in grade 11 my chem teacher taught us some little triangle method for the equation n=cv to solve for each quantity. The the next year in Phys12 I sit down at my desk the first day and written on my desk is the little triangle cheat method and someone commented below it "morons use triangles". I always got a kick out of that. How old are you and what grade are you in? I didn't take a physics class until grade 10 and we were doing stuff like this in grade 7/8. I find it odd that you would be taking a physics class without knowing these algebra fundamentals. 15. Jul 21, 2005 ### ek why did you out everything except V2 on the other side of the = instead of putting V2 were "a" was and put "a" on the other side? It's the same thing. V = at and at = V are the same thing. why did you multiply "t" and "a"? You multiply both sides by t. The t's cancel on the right side and you're left with at = V2 - V1 if "t" was the numerator, would you still multiply it with "a"? No, then you would divide it out. Divide both sides by t and you're left with a/t = V2-V1 (Which obviously isn't a valid equation) 16. Jul 21, 2005 ### EnumaElish ek's explanation is right: a = (V2 - V1)/t (V2 - V1)/t = a t(V2 - V1)/t = ta V2 - V1 = ta V2 - V1 + V1 = ta + V1 V2 = V1 + ta 17. Aug 13, 2005 ### Serj I've got a problem I don't know how to rearrange. d=Vt+1/2 at^2 ,i'm supposed to find out what t equals. d/V=Vt/V+1/2 at^2 d/V=t +1/2 at^2 2d/V=t +21/2 at^2 2 d/V=t + at^2 (2 d/V)/a=t + at^2/a (suare root of)(2 d/V)/a=t + (square root of)t^2 (suare root of)(2 d/V)/a=t +t ((suare root of)(2 d/V)/a)/2=2t/2 ((suare root of)(2 d/V)/a)/2=t What did I do wrong? how do I fix it so Im not dividing fractions 18. Aug 13, 2005 ### Learning Curve What do you mean "at power of two"? Do you mean the whole expression is to the power of 2? And is Vt one variable or two? 19. Aug 13, 2005 ### VietDao29 You are wrong at the second line. You have: $$a = b + c$$ $$\Leftrightarrow \frac{a}{d} = \frac{b + c}{d} = \frac{b}{d} + \frac{c}{d}$$ We divide both sides by d <> 0. Anyway, to find t from: $$d = vt + \frac{1}{2}a t ^ 2$$ $$\Leftrightarrow \frac{a}{2}t ^ 2 + vt - d = 0$$ t is the unknown. Can you solve: $$\alpha x ^ 2 + \beta x + \gamma = 0$$ for x? $$\alpha , \beta , \gamma$$ are already known. Viet Dao, 20. Aug 13, 2005 ### Learning Curve (ax^2)/a+bx-bx+y-y=(-bx-y)/a x^2=(-bx-y)/a square root both sides x= square root of (-bx-y)/a That's really messy and I don't know if thats how you would do it. Share this great discussion with others via Reddit, Google+, Twitter, or Facebook	{"extraction_info": {"found_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": 0.8386745452880859, "perplexity": 2290.9162662462213}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039743732.41/warc/CC-MAIN-20181117185331-20181117211331-00253.warc.gz"}
https://www.physicsforums.com/threads/2-d-kinematics-distance-of-rocket-from-launch-pad.633564/	# 2-D Kinematics: Distance of Rocket from Launch Pad 1. ### JohnSwine 2 1. The problem statement, all variables and given/known data A model rocket is launched from rest with an upward acceleration of 6.00m/s^2 and, due to a strong wind, a horizontal acceleration of 1.50m/s^2. How far is the rocket from the launch pad 6.00s later when the rocket engine runs out of fuel? 2. Relevant equations Not sure.. 3. The attempt at a solution I've been trying to figure this out, and I'm still not sure how to approach it. 2. ### azizlwl 973 Sketch a velocity vs. time graph. You will see that from zero the velocity increases with the rate 6(m/s)/s for 6 secs. Then it slows down to with rate of g untl zero velocity where it is at the top of the flight Next it will continue with negative velocity(acceleration g) until it reaches the ground. The net total area is equal to zero. The horizontal vector remains constant. Taking total flight time, you can multiply this to horizontal velocity to find the answer. 3. ### ehild 11,798 You certainly know that you can describe the motion of a rocket as if it moved independently both in horizontal (x) and vertical (y) direction. Both motion happens with uniform acceleration. You certainly learnt how the displacement changes with time during a motion with uniform acceleration? write up the equations both for x and y directions. ehild Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook Similar discussions for: 2-D Kinematics: Distance of Rocket from Launch Pad	{"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": 0.9218999147415161, "perplexity": 1155.9113494617816}, "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-2015-11/segments/1424936463093.76/warc/CC-MAIN-20150226074103-00183-ip-10-28-5-156.ec2.internal.warc.gz"}
http://www.sciforums.com/threads/fundamental-confusions-of-calculus.112396/page-8#post-2903805	# Fundamental confusions of calculus Discussion in 'Physics & Math' started by arfa brane, Feb 11, 2012. Not open for further replies. 1. ### PeteIt's not rocket surgeryRegistered Senior Member Messages: 10,167 Hi rpenner, and thanks. I don't understand the subscript notation. Does it just mean that a,b,c,z are constants? Last edited: Feb 14, 2012 3. ### rpennerFully WiredValued Senior Member Messages: 4,833 Yes, that was my intent 5. ### PeteIt's not rocket surgeryRegistered Senior Member Messages: 10,167 Thanks, Notation isn't my strong point... which is one of the reasons confusion abounds 7. ### TachBannedBanned Messages: 5,265 I simply wanted to generate for you a case that : -helps you understand the difference between total and partial derivative -helps you understand what happens when not all variables in the function description are functions of the variable (t in this case) you are taking the derivatives This is the partial derivative wrt t. This is the total derivative wrt t. Last edited: Feb 14, 2012 8. ### PeteIt's not rocket surgeryRegistered Senior Member Messages: 10,167 I appreciate your generous effort, but forgive me if I am not confident in your teaching ability in this area. I'm generally inclined to trust the combined credentials of temur, rpenner, przyk, and AlphaNumeric, supported by textbooks and online material. No, that's another partial derivative. In that derivative z is an independent variable, treated as constant. Taking a total derivative would mean treating z as a function of t, not as a constant. See rpenner's post (also said by przyk, temur, David Metzler in the video, and possibly others): Last edited: Feb 14, 2012 9. ### AlphaNumericFully ionizedRegistered Senior Member Messages: 6,702 You failed to show how my use of the definition was false, despite having been asked more than once. You failed to respond to my question about what the canonical momentum for $T = \frac{1}{2}m\dot{q}^{2}$ is, because it would mean admitting that $p \equiv \frac{\partial}{\partial \dot{q}}\frac{1}{2}m\dot{q}^{2} = m\dot{q}$, which contradicts your claims about partial derivatives and also your claim you understand Hamiltonian mechanics while holding this view. You are deliberately avoided addressing simple, relevant, direct questions which are based on definitions and found throughout the literature, as well as basic lecture notes. I would lock this thread but it seems Rpenner and Pete are in the middle of something. You have one last chance. If you respond without addressing my direct questions then this thread will definitely be locked and you given a warning for trolling. You've already had more last chances than you deserve. 10. ### TachBannedBanned Messages: 5,265 You got it totally backwards, let's try it one more time: partial derivative means that if $f=f(t,u)$ where $u=u(t)$ then $\frac{\partial f}{\partial t}$ is the partial derivative while $\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u}\frac{du}{dt}$ is the total derivative.Partial derivative means differentiation only wrt the explicit variable (t in the example), total derivative means partial derivative PLUS differentiation (via chain rule) through all variables that are a function of the variable considered. Therefore, contrary to your claims, I know exactly what I am doing and the method described (it is textbook, really) produces the correct answer: So, $p=\frac{\partial T}{\partial \dot{q}}=m\dot{q}$ The video that Pete linked in is very good, in this respect. I can track your error to the fact that you failed repeatedly to note that in my example, $f=3 \theta +u+v$ and you kept considering that $f=3 \theta +sin^2(\theta)+v$. So, you are getting the wrong partial derivative. Not a big error, but please don't try to pin it on me, own it. Last edited: Feb 14, 2012 11. ### przyksquishyValued Senior Member Messages: 3,203 And? $x$ and $y$ are also not the bug's coordinates. They're general coordinates on the plane. For the purpose of calculating the bug's temperature from $T(t,\,x,\,y,\,z)$, you have to set $x(t) = f(t)$, $y(t) = g(t)$, and $z(t) = 0$. Then, and only then, are you taking a total derivative. There is a difference between saying $\frac{\mathrm{d}T}{\mathrm{d}t} \,=\, \frac{\partial T}{\partial t} \,+\, \frac{\partial T}{\partial x} \, \frac{\mathrm{d}x}{\mathrm{d}t} \,+\, \frac{\partial T}{\partial y} \, \frac{\mathrm{d}y}{\mathrm{d}t}$ because $\frac{\mathrm{d}z}{\mathrm{d}t} = 0$, which is a total derivative, and $\Bigl( \frac{\partial T}{\partial t} \Bigr)_{z} \,=\, \frac{\partial T}{\partial t} \,+\, \frac{\partial T}{\partial x} \, \frac{\mathrm{d}x}{\mathrm{d}t} \,+\, \frac{\partial T}{\partial y} \, \frac{\mathrm{d}y}{\mathrm{d}t}$ (for lack of better notation), because you're leaving $z$ independent, in which case you're still taking a partial derivative, specifically of the function $(t,\,z) \,\mapsto\, T(t,\, x(t),\, y(t),\, z) \,.$ To calculate the bug's temperature from $T$, you have to set $z(t) = 0$, and you're therefore in the former case. And you're calculating a total derivative only because you set $z(t) = 0$. Do you understand this? Because it's the point we've been trying to get through to you. 12. ### TachBannedBanned Messages: 5,265 Once again, I am not taking the bug temperature. I am taking the temperature of an arbitrary point in space. 13. ### przyksquishyValued Senior Member Messages: 3,203 If you're just doing this then you're not calculating any sort of derivative at all. You're just evaluating $T(t,\,x,\,y,\,z)$ at some arbitrary location and time. So what the hell are you even talking about? 14. ### Guest254Valued Senior Member Messages: 1,056 This really should have ended by now. Tach doesn't know what he's talking about and these two posts cement that fact. No ifs or buts -- he doesn't understand elementary calculus and these two posts aptly demonstrate this unfortunate truth. However, I don't think I want the thread to end. It's strangely amusing to watch the train wreck unfold! 15. ### TachBannedBanned Messages: 5,265 I simply constructed a function for Pete to calculate the total derivative. 16. ### TachBannedBanned Messages: 5,265 Of course, in your trolling haste, you failed to notice that $f(\theta,u,v)=3 \theta+u+v$ where $u=sin^2(\theta)$ and $v=ln(x)$. So, what is $\frac{\partial f}{\partial \theta}$, Guest? Did your degree skirt over partial derivatives? Last edited: Feb 14, 2012 17. ### rpennerFully WiredValued Senior Member Messages: 4,833 Your reference is non-normative. Normative references do not support your assertions and slurs. It's not clear that this reference about a "bug crawling on a plate" supports your assertions, let alone your slurs. http://www.wolframalpha.com/input/?i=D[ 3 theta + (sin theta)^2 + ln(x), theta ] http://www.wolframalpha.com/input/?...+ (sin theta)^2 + ln(x) with respect to theta (The show steps button is nice) Moving the goal posts from $f(\theta,x) = 3 \theta + \sin^2 \theta + \ln x$ to $f(\theta, u, v) = 3 \theta + u + v$ serves no pedagogical purpose. 18. ### TachBannedBanned Messages: 5,265 I haven't moved any goal-posts, $f=f(\theta,u(\theta),v(x))$ since the beginning, you arrived to this thread late. Here is a normative reference I gave earlier. As for Guest, he has a long history of trolling my posts. He would have had an excuse , except that I have shown the $f=f(\theta,u(\theta),v(x))$ repeatedly. So, he has no excuse. Come to think of it, neither have you since I take it that you understand perfectly the significance of $f=f(\theta,u(\theta),v(x))$. 19. ### TrippyALEA IACTA ESTStaff Member Messages: 10,890 See, rightly or wrongly, that's not how I would perform the partial derivative, I, and from what I gather, everybody else who is disagreeing with you, would not re-define the equation that way. If I was given: $f(\theta,x)=3 \theta+sin^2(\theta)+ln(x)$ And asked to find the partial derivatives, I would do it using: $f(\theta)=3 \theta+sin^2(\theta)+k$ {where k=ln(x)} And: $f(x)=k+ln(x)$ {where k=$3 \theta+sin^2(\theta)$} And if I was asked to find the partial derivative $\frac{\partial f}{\partial \theta}$ using the chain rule, I would start with: $f(\theta)=3 \theta+sin^2(\theta)+k$ {where k=ln(x)} Then I would set: $u=sin(\theta)$ Then I would find: $\frac{df}{du}$ for the equation $f(\theta)=3 \theta+u^2+k$ And find: $\frac{du}{d \theta}$ for $u=sin(\theta)$ And then proceed to evaluate: $\frac{\partial f}{\partial \theta} = \frac{df}{du}\frac{du}{d \theta}$ But that, to me, seems unneccessary, when the initial problem is relatively uncomplicated. It seemd to me that what you've done is redefined the problem as being: $f(\theta,x)=g(\theta)+h(\theta)+i(x)$ And then come to the conclusion that $f'(\theta)=g'(\theta)$ while neglecting to evaluate $h'(\theta)$ Last edited: Feb 14, 2012 20. ### TachBannedBanned Messages: 5,265 Sure, IF you were given the above. But what you are being given is $f(\theta, u(\theta),v)=3\theta+u+v$ where $u(\theta)=sin^2(\theta)$, $v(x)=ln(x)$ The total derivative will be the same but the partial derivative differs , depending on what you consider that has been given as $f$. 21. ### TrippyALEA IACTA ESTStaff Member Messages: 10,890 No, that's exactly what we were given: Post #18 If you meant something else, perhaps you should be more careful in the future. 22. ### RJBeeryNatural PhilosopherValued Senior Member Messages: 4,222 Refusing to acknowledge the error only makes the clanger that much louder, Tach; especially in light or your hyper-critical, condescending nature. Live by the sword... 23. ### TrippyALEA IACTA ESTStaff Member Messages: 10,890 There's a greater irony in there.	{"extraction_info": {"found_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": 2, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8030859231948853, "perplexity": 1884.641805046221}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1656103347800.25/warc/CC-MAIN-20220628020322-20220628050322-00485.warc.gz"}
http://docmadhattan.fieldofscience.com/2012/08/note-on-theory-of-hypergeometric.html	### Note on the theory of hypergeometric functions In the few last days I have search for more links about Mary Frances Winston, and I find the paper written by Mary when she was at Göttingen. So I try to translate(1) it using Google. I hope that this version is enough clear. Riemann defined in its basic treatise the related $P$-functions first of all such as, whose exponents differ by integers, while the exponents are subject only to the condition that their sum is equal to 1: $\alpha' + \alpha'' + \beta' + \beta'' + \gamma' + \gamma'' = 1 \qquad (1)$ and that none of the differences $\pm (\alpha' - \alpha''), \; \pm (\beta' - \beta''), \; \pm (\gamma' - \gamma'')$ should be an integer. Meanwhile, the analytical definition provided herein is not the essence of the relationship. Rather all of the following developments about Riemann's related funcrions rest thereon, if only related functions should have the same monodromy. Now, Prof. Klein in his lectures about the hypergeometric function in the winter 1893-94, essentially noted that the latter is generally a result of the aforementioned analytical definition and that a special case exists, which is an exception. It is those exponents' system in which there exists the relation: $\pm (\alpha' - \alpha'') \pm (\beta' - \beta'') \pm (\gamma' - \gamma'') = 2k + 1 \qquad (2)$ where $k$ is an arbitrary integer. Here are the P-functions: $P \begin{pmatrix} 0 & \infty & 1 & \\ \alpha' + a' & \beta' + b' & \gamma' + c' & x \\ \alpha'' + a'' & \beta''+ b'' & \gamma'' + c'' & \end{pmatrix} \qquad (3)$ (where $a'$, $a''$, $b'$, $b''$, $c'$, $c''$ are integers of zero-sum) divide into two separate bands such that only the P-functions of the individual band are related to each other, i.e. have the same monodromy. I was busy trying to find the analytical features of these two bands, and so the Riemann's initial analytical definition of the complete set, that fits all cases included in the Riemann treatise. My result is this: It finds a relation instead of (2), so the left side of (1) breaks down in two integer triplets $\alpha' + \beta' + \gamma' \qquad \text{and} \qquad \alpha'' + \beta'' + \gamma''$ On of such triplet (due to (1)) is necessarily positive, the other null or negative. We assume the definite expression for the triplet $(\alpha' + \beta' + \gamma')$ to be positive. The function (3) is given by the $P$-function if and only if the corresponding triplet $\alpha' + a' + \beta' + b' + \gamma' + c'$ is also positive. The proof is very simple. It is sufficient to establish the singular points $0$, $\infty$, $1$ corresponding fundamental branches of the P-function in the form of hypergeometric series and then to make the comparison. For example, for $x = 0$we have: $P^{(\alpha')} = x^{\alpha'} \cdot (1-x)^{\gamma'} \cdot F (\alpha' + \beta' + \gamma', \alpha' + \beta'' + \gamma', 1 + \alpha' - \alpha'', x)$ $P^{(\alpha'')} = x^{\alpha''} \cdot (1-x)^{\gamma''} \cdot F (\alpha'' + \beta' + \gamma'', \alpha'' + \beta'' + \gamma'', 1 + \alpha'' - \alpha', x)$ and here we see immediately that the one or other of the $F$-series breaks that occur (and therefore represents a rational integral function of $x$), according as $(\alpha' + \beta' + \gamma') \qquad \text{or} \qquad (\alpha'' + \beta'' + \gamma'')$ It is a null or negative integer. This is the essence. Here I don't discuss further details. Winston, F.M. (1895). Eine Bemerkung zur Theorie der hypergeometrischen Function, Mathematische Annalen, 46 (1) 160. DOI: 10.1007/BF02096208 (Göttinger Digitalisierungszentrum \| Academic Search) (1) It seems that an english version of the paper exists, but I cannot find it. #### 1 comment: 1. You have an incredible amount and quality of information on your site! I am going to link to your site from my own, at Math Concepts Explained. Would you be open to including my site in your list of MathBlogs? Cheers, Shaun Markup Key: - <b>bold</b> = bold - <i>italic</i> = italic - <a href="http://www.fieldofscience.com/">FoS</a> = FoS	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8916705846786499, "perplexity": 573.6500719806717}, "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-2018-22/segments/1526794867041.69/warc/CC-MAIN-20180525043910-20180525063910-00418.warc.gz"}
https://www.physicsforums.com/threads/convolution-homework-involving-impulse-functions.596310/	Homework Help: Convolution Homework Involving Impulse Functions 1. Apr 13, 2012 martnll2 1. The problem statement, all variables and given/known data How do you do a convolution of two functions containing only impulses? 2. Relevant equations Say you have 2 functions to convolve, f1 and f2. I can't do the impulse symbol, so lets call it q. Say f1 = 2q(t+1) + 2q(t-4) and f2 = q(t-3) What is f1 convolved with f2? Or how do you do it? f(t) convolved with y(t) = h(t) F(w)Y(w) = H(w) 3. The attempt at a solution So I thought the way to do this is to just add the two functions together, but I am unsure. 2. Apr 13, 2012 RoshanBBQ Do you know the impulse sampling rule? Let a(t) be another impulse. It works the same way. Set up the integrals for convolution and use this rule as needed. $$\int\limits_{-\infty}^{\infty} \delta(t-t_0)a(t)dt=a(t_0)$$ More generally, the limits of integration can be from b to c as long as t_0 is in [b,c]. Otherwise, the integral is zero. 3. Apr 13, 2012 martnll2 Thanks, I'll try it out after studying for this exam and let you know how it goes :D	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9615437984466553, "perplexity": 1154.3953319970396}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583511216.45/warc/CC-MAIN-20181017195553-20181017221053-00409.warc.gz"}
https://labs.tib.eu/arxiv/?author=T.%20DeGrand	• ### The Nc dependencies of baryon masses: Analysis with Lattice QCD and Effective Theory(1404.2301) April 8, 2014 hep-ph, nucl-th, hep-lat Baryon masses at varying values of Nc and light quark masses are studied with Lattice QCD and the results are analyzed in a low energy effective theory based on a combined framework of the 1/Nc and Heavy Baryon Chiral Perturbation Theory expansions. Lattice QCD results for Nc = 3, 5 and 7 obtained in quenched calculations, as well as results for unquenched calculations for Nc = 3, are used for the analysis. The results are consistent with a previous analysis of Nc = 3 Lattice QCD results, and in addition permit the determination of sub-leading in 1/Nc effects in the spin-flavor singlet component of the baryon masses as well as in the hyperfine splittings. • ### Improving meson two-point functions by low-mode averaging(hep-lat/0409056) Sept. 13, 2004 hep-lat Some meson correlation functions have a large contribution from the low lying eigenmodes of the Dirac operator. The contribution of these eigenmodes can be averaged over all positions of the source. This can improve the signal in these channels significantly. We test the method for meson two-point functions. • ### Lattice Calculation of Heavy-Light Decay Constants with Two Flavors of Dynamical Quarks(hep-lat/0206016) Sept. 5, 2002 hep-ph, hep-lat We present results for $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios in the presence of two flavors of light sea quarks ($N_f=2$). We use Wilson light valence quarks and Wilson and static heavy valence quarks; the sea quarks are simulated with staggered fermions. Additional quenched simulations with nonperturbatively improved clover fermions allow us to improve our control of the continuum extrapolation. For our central values the masses of the sea quarks are not extrapolated to the physical $u$, $d$ masses; that is, the central values are "partially quenched." A calculation using "fat-link clover" valence fermions is also discussed but is not included in our final results. We find, for example, $f_B = 190 (7) (^{+24}_{-17}) (^{+11}_{-2}) (^{+8}_{-0})$ MeV, $f_{B_s}/f_B = 1.16 (1) (2) (2) (^{+4}_{-0})$, $f_{D_s} = 241 (5) (^{+27}_{-26}) (^{+9}_{-4}) (^{+5}_{-0})$ MeV, and $f_{B}/f_{D_s} = 0.79 (2) (^{+5}_{-4}) (3) (^{+5}_{-0})$, where in each case the first error is statistical and the remaining three are systematic: the error within the partially quenched $N_f=2$ approximation, the error due to the missing strange sea quark and to partial quenching, and an estimate of the effects of chiral logarithms at small quark mass. The last error, though quite significant in decay constant ratios, appears to be smaller than has been recently suggested by Kronfeld and Ryan, and Yamada. We emphasize, however, that as in other lattice computations to date, the lattice $u,d$ quark masses are not very light and chiral log effects may not be fully under control. • ### Light hadron properties with improved staggered quarks(hep-lat/0208041) Aug. 22, 2002 hep-lat Preliminary results from simulations with 2+1 dynamical quark flavors at a lattice spacing of 0.09 fm are combined with earlier results at a=0.13 fm. We examine the approach to the continuum limit and investigate the dependence of the pseudoscalar masses and decay constants as the sea and valence quark masses are separately varied. • ### Heavy-light decay constants with three dynamical flavors(hep-lat/0110072) Oct. 12, 2001 hep-lat We present preliminary results for the heavy-light leptonic decay constants in the presence of three light dynamical flavors. We generate dynamical configurations with improved staggered and gauge actions and analyze them for heavy-light physics with tadpole improved clover valence quarks. When the scale is set by $m_\rho$, we find an increase of approximately 23% in $f_B$ with three dynamical flavors over the quenched case. Discretization errors appear to be small (of order 3% or less) in the quenched case but have not yet been measured in the dynamical case. • ### Perturbation Theory for Fat-link Fermion Actions(hep-lat/9909083) Sept. 10, 1999 hep-lat We discuss weak coupling perturbation theory for lattice actions in which the fermions couple to smeared gauge links. The normally large integrals that appear in lattice perturbation theory are drastically reduced. Even without detailed calculation, it is easy to determine to good accuracy the scale of the logarithms that appear in cases where an anomalous dimension is present. We describe several 1-loop examples for fat-link Wilson and clover fermions. including the additive mass shift, the relation between the lattice and MSbar quark masses, and the axial current renormalization factor ($Z_A$) for light-light and static-light currents. • ### Investigating and Optimizing the Chiral Properties of Lattice Fermion Actions(hep-lat/9810061) Aug. 4, 1999 hep-lat We study exceptional modes of both the Wilson and the clover action in order to understand why quenched clover spectroscopy suffers so severely from exceptional configurations. We show that, in contrast to the case of the Wilson action, a large clover coefficient can make the exceptional modes extremely localized and thus very sensitive to short distance fluctuations. We describe a way to optimize the chiral behavior of Wilson-type lattice fermion actions by studying their low energy real eigenmodes. We find a candidate action, the clover action with fat links with a tuned clover term. We present a calculation of spectroscopy and matrix elements at Wilson gauge coupling beta=5.7. When compared to simulations with the standard (nonperturbatively improved) clover action at small lattice spacing, the action shows good scaling behavior, with an apparent great reduction in the number of exceptional configurations. • ### Simple Observables from Fat Link Fermion Actions(hep-lat/9903006) March 2, 1999 hep-lat A comparison is made of the (quenched) light hadron spectrum and of simple matrix elements for a hypercubic fermion action (based on a fixed point action) and the clover action, both using fat links, at a lattice spacing a= 0.18 fm. Renormalization constants for the naive and improved vector current and the naive axial current are computed using Ward identities. The renormalization factors are very close to unity, and the spectroscopy of light hadrons and the pseudoscalar and vector decay constants agree well with simulations at smaller lattice spacings (and with experiment). • ### Contents of Lattice 98 Proceedings(hep-lat/9811023) Nov. 27, 1998 hep-lat We give here a compilation of papers presented at Lattice 98 (XVI Intl. Symposium on Lattice Field Theory, Boulder, Colorado, USA, 13-18 July 1998). The contents are in html form with clickable links to the papers that exist on the hep-lat archives. We hope that this will make it easier to access the presentations at the conference. Comments on, and corrections to, this compilation should be sent to [email protected]. Version 3: some more additions to the list. • ### Heavy-Light Decay Constants: Conclusions from the Wilson Action(hep-lat/9809109) Oct. 19, 1998 hep-lat We report on the results of a MILC collaboration calculation of $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios. We discuss the most important errors in more detail than we have elsewhere. • ### Lattice Determination of Heavy-Light Decay Constants(hep-ph/9806412) Oct. 19, 1998 hep-ph, hep-lat We report on the MILC collaboration's calculation of $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$, and their ratios. Our central values come from the quenched approximation, but the quenching error is estimated from $N_F=2$ dynamical staggered lattices. We use Wilson light valence quarks and Wilson and static heavy quarks. We find, for example, $f_B=157 \pm 11 {}^{+25}_{-9} {}^{+23}_{-0} \MeV$, $f_{B_s}/f_B = 1.11 \pm 0.02 {}^{+0.04}_{-0.03} \pm 0.03$, $f_{D_s} = 210 \pm 9 {}^{+25}_{-9} {}^{+17}_{-1} \MeV$ and $f_{B}/f_{D_s} = 0.75 \pm 0.03 {}^{+0.04}_{-0.02} {}^{+0.08}_{-0.00}$, where the errors are statistical, systematic (within the quenched approximation), and systematic (of quenching), respectively. • ### Exotic meson spectroscopy from the clover action at beta = 5.85 and 6.15(hep-lat/9809087) Sept. 14, 1998 hep-lat We repeat our original simulations of the hybrid meson spectrum using the clover action, as a check on lattice artifacts. Our results for the 1-+ masses do not substantially change. We present preliminary results for the wave function of the 1-+ state in Coulomb gauge. • ### Structure of the QCD Vacuum As Seen By Lattice Simulations(hep-lat/9801037) Jan. 27, 1998 hep-lat This talk is a review of our studies of instantons and their properties as seen in our lattice simulations of SU(2) gauge theory. We have measured the topological susceptibility and the size distribution of instantons in the QCD vacuum. We have also investigated the properties of quarks moving in instanton background field configurations, where the sizes and locations of the instantons are taken from simulations of the full gauge theory. By themselves, these multi-instanton configurations do not confine quarks, but they induce chiral symmetry breaking. • ### Heavy-Light Decay Constants from Wilson and Static Quarks(hep-lat/9709142) Sept. 29, 1997 hep-lat MILC collaboration results for \fB, \fBs, \fD, \fDs and their ratios are presented. These results are still preliminary, but the analysis is close to being completed. Sources of systematic error, both within the quenched approximation and from quenching itself, are estimated. We find, for example, $f_B=153\pm 10 {}^{+36}_{-13} {}^{+13}_{-0} MeV$, and $f_{B_s}/f_B = 1.10 \pm 0.02 {}^{+0.05}_{-0.03} {}^{+0.03}_{-0.02}$, where the errors are statistical, systematic (within the quenched approximation), and systematic (of quenching), respectively. The extrapolation to the continuum and the chiral extrapolation are the largest sources of error. Present central values are based on linear chiral extrapolations; a shift to quadratic extrapolations would raise $f_B$ by $\approx20$ MeV and make the error within the quenched approximation more symmetric. • ### Scaling Tests of Some Lattice Fermion Actions(hep-lat/9709052) Sept. 16, 1997 hep-lat I describe studies of quenched spectroscopy, using crude approximations to fixed point actions for fermions interacting with SU(3) gauge fields. These actions have a hypercubic kinetic term and a complicated lattice anomalous magnetic moment term. They show improved scaling compared to the conventional Wilson action. • ### B Mixing on the Lattice: $f_B$, $f_{B_s}$ and Related Quantities(hep-ph/9709328) Sept. 12, 1997 hep-ph, hep-lat The MILC collaboration computation of heavy-light decay constants is described. Results for $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios are presented. These results are still preliminary, but the analysis is close to being completed. Sources of systematic error, both within the quenched approximation and from quenching itself, are estimated, although the latter estimate is rather crude. A sample of our results is: $f_B=153 \pm 10 {}^{+36}_{-13} {}^{+13}_{-0} MeV$, $f_{B_s}/f_B = 1.10 \pm 0.02 {}^{+0.05}_{-0.03} {}^{+0.03}_{-0.02}$, and $f_{B}/f_{D_s} = 0.76 \pm 0.03 {}^{+0.07}_{-0.04} {}^{+0.02}_{-0.01}$, where the errors are statistical, systematic (within the quenched approximation), and systematic (of quenching), respectively. The largest source of error comes from the extrapolation to the continuum. The second largest source is the chiral extrapolation. At present, the central values are based on linear chiral extrapolations; a shift to quadratic extrapolations would for example raise $f_B$ by $\approx 20$ MeV and thereby make the error within the quenched approximation more symmetric. • ### Lattice Gauge Theory for QCD(hep-ph/9610391) Oct. 17, 1996 hep-ph, hep-lat These lectures provide an introduction to lattice methods for nonperturbative studies of Quantum Chromodynamics. Lecture 1 (Ch. 2): Basic techniques for QCD and results for hadron spectroscopy using the simplest discretizations; lecture 2 (Ch. 3): improved actions''--what they are and how well they work lecture 3 (Ch. 4): SLAC physics from the lattice: structure functions, the mass of the glueball, heavy quarks and $\alpha_s(M_Z)$, and $B-\bar B$ mixing. • ### Nonperturbative Quantum Field Theory on the Lattice(hep-th/9610132) Oct. 17, 1996 hep-th, hep-lat These lectures provide an introduction to lattice methods for nonperturbative studies of quantum field theories, with an emphasis on Quantum Chromodynamics. Lecture 1 (Ch. 2): gauge field basics Lecture 2 (Ch. 3): Abelian duality with a lattice regulator (Ch. 4): simple lattice intuition Lecture 3 (Ch. 5): standard methods (and results) for hadron spectroscopy Lecture 4 (Ch. 6): bare actions and physics Lecture 5 (Ch. 7): two case studies, mass of the glueball and $\alpha_s(M_Z)$ • ### Update on $f_B$(hep-lat/9608092) Aug. 16, 1996 hep-ph, hep-lat We describe the current status of the MILC collaboration computation of $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios. Progress over the past year includes: better statistics and plateaus at $\beta=6.52$ (quenched), $\beta=5.6$ ($N_F=2$) and $\beta=5.445$ ($N_F=2$), new runs with a wide range of dynamical quark masses at $\beta=5.5$ ($N_F=2$), an estimate of the systematic errors due to the chiral extrapolation, and an improved analysis which consistently takes into account both the correlations in the data at every stage and the systematic effects due to changing fitting ranges. • ### Fixed-point action for fermions in QCD(hep-lat/9608056) Aug. 12, 1996 hep-lat We report our progress constructing a fixed-point action for fermions interacting with SU(3) gauge fields. • ### $f_B$ quenched and unquenched(hep-lat/9509045) Sept. 14, 1995 hep-ph, hep-lat Results for $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$, and their ratios are presented. High statistics quenched runs at $\beta=5.7$, $5.85$, $6.0$, and $6.3$, plus a run still in progress at $\beta=6.52$ make possible a preliminary extrapolation to the continuum. The data allows good control of all systematic errors except for quenching, although not all of the error estimates have been finalized. Results from configurations which include effects of dynamical quarks show a significant deviation from the quenched results and make possible a crude estimate of the quenching error. • ### Fixed point actions for SU(3) gauge theory(hep-lat/9508024) Aug. 23, 1995 hep-lat We summarize our recent work on the construction and properties of fixed point (FP) actions for lattice $SU(3)$ pure gauge theory. These actions have scale invariant instanton solutions and their spectrum is exact through 1--loop, i.e. in their physical predictions there are no $a^n$ nor $g^2 a^n$ cut--off effects for any $n$. We present a few-parameter approximation to a classical FP action which is valid for short correlation lengths. We perform a scaling test of the action by computing the quantity $G = L \sqrt{\sigma(L)}$, where the string tension $\sigma(L)$ is measured from the torelon mass $\mu = L \sigma(L)$, on lattices of fixed physical volume and varying lattice spacing $a$. While the Wilson action shows scaling violations of about ten per cent, the approximate fixed point action scales within the statistical errors for $1/2 \ge aT_c$. • ### The classically perfect fixed point action for SU(3) gauge theory(hep-lat/9506030) June 27, 1995 hep-lat In this paper (the first of a series) we describe the construction of fixed point actions for lattice $SU(3)$ pure gauge theory. Fixed point actions have scale invariant instanton solutions and the spectrum of their quadratic part is exact (they are classical perfect actions). We argue that the fixed point action is even 1--loop quantum perfect, i.e. in its physical predictions there are no $g^2 a^n$ cut--off effects for any $n$. We discuss the construction of fixed point operators and present examples. The lowest order $q {\bar q}$ potential $V(\vec{r})$ obtained from the fixed point Polyakov loop correlator is free of any cut--off effects which go to zero as an inverse power of the distance $r$. • ### Non--perturbative tests of the fixed point action for SU(3) gauge theory(hep-lat/9506031) June 27, 1995 hep-lat In this paper (the second of a series) we extend our calculation of a classical fixed point action for lattice $SU(3)$ pure gauge theory to include gauge configurations with large fluctuations. The action is parameterized in terms of closed loops of link variables. We construct a few-parameter approximation to the classical FP action which is valid for short correlation lengths. We perform a scaling test of the action by computing the quantity $G = L \sqrt{\sigma(L)}$ where the string tension $\sigma(L)$ is measured from the torelon mass $\mu = L \sigma(L)$. We measure $G$ on lattices of fixed physical volume and varying lattice spacing $a$ (which we define through the deconfinement temperature). While the Wilson action shows scaling violations of about ten per cent, the approximate fixed point action scales within the statistical errors for $1/2 \ge aT_c \ge 1/6$. Similar behaviour is found for the potential measured in a fixed physical volume. • ### Status and prospect for determining $f_B$, $f_{B_s}$, $f_{B_s} / f_B$ on the lattice(hep-ph/9503336) March 15, 1995 hep-ph, hep-lat Preliminary results from the MILC collaboration for $f_B$, $f_{B_s}$, $f_D$, $f_{D_s}$ and their ratios are presented. We compute in the quenched approximation at $\beta=6.3$, 6.0 and 5.7 with Wilson light quarks and static and Wilson heavy quarks. We attempt to quantify all systematic errors other than quenching, and have a first indication of the size of quenching errors.	{"extraction_info": {"found_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": 0.9354485869407654, "perplexity": 1445.4062185746952}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038464065.57/warc/CC-MAIN-20210417222733-20210418012733-00406.warc.gz"}
https://math.libretexts.org/Courses/Monroe_Community_College/MTH_104_Intermediate_Algebra/8%3A_Roots_and_Radicals/8.2%3A_Simplify_Expressions_with_Roots	$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ # 8.1: Simplify Expressions with Roots $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ Learning Objectives By the end of this section, you will be able to: • Simplify expressions with roots • Estimate and approximate roots • Simplify variable expressions with roots Before you get started, take this readiness quiz. 1. Simplify: a. $$(−9)^{2}$$ b. $$-9^{2}$$ c. $$(−9)^{3}$$ If you missed this problem, review Example 2.21. 2. Round $$3.846$$ to the nearest hundredth. If you missed this problem, review Example 1.34. 3. Simplify: a. $$x^{3} \cdot x^{3}$$ b. $$y^{2} \cdot y^{2} \cdot y^{2}$$ c. $$z^{3} \cdot z^{3} \cdot z^{3} \cdot z^{3}$$ If you missed this problem, review Example 5.12. ## Simplify Expressions with Roots In Foundations, we briefly looked at square roots. Remember that when a real number $$n$$ is multiplied by itself, we write $$n^{2}$$ and read it '$$n^{2}$$ squared’. This number is called the square of $$n$$, and $$n$$ is called the square root. For example, $$13^{2}$$ is read "$$13$$ squared" $$169$$ is called the square of $$13$$, since $$13^{2}=169$$ $$13$$ is a square root of $$169$$ Definition $$\PageIndex{1}$$: Square and Square Root of a Number Square If $$n^{2}=m$$, then $$m$$ is the square of $$n$$. Square Root If $$n^{2}=m$$, then $$n$$ is a square root of $$m$$. Notice $$(−13)^{2} = 169$$ also, so $$−13$$ is also a square root of $$169$$. Therefore, both $$13$$ and $$−13$$ are square roots of $$169$$. So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, $$\sqrt{m}$$, which denotes the positive square root of $$m$$. The positive square root is also called the principal square root. We also use the radical sign for the square root of zero. Because $$0^{2}=0, \sqrt{0}=0$$. Notice that zero has only one square root. Definition $$\PageIndex{2}$$: Square Root Notation $$\sqrt{m}$$ is read "the square root of $$m$$." If $$n^{2}=m$$, then $$n=\sqrt{m}$$, for $$n\geq 0$$. We know that every positive number has two square roots and the radical sign indicates the positive one. We write $$\sqrt{169}=13$$. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $$-\sqrt{169}=-13$$. Example $$\PageIndex{1}$$ Simplify: 1. $$\sqrt{144}$$ 2. $$-\sqrt{289}$$ Solution: a. $$\sqrt{144}$$ Since $$12^{2}=144$$. $$12$$ b. $$-\sqrt{289}$$ Since $$17^{2}=289$$ and the negative is in front of the radical sign. $$-17$$ Exercise $$\PageIndex{1}$$ Simplify: 1. $$-\sqrt{64}$$ 2. $$\sqrt{225}$$ 1. $$-8$$ 2. $$15$$ Exercise $$\PageIndex{2}$$ Simplify: 1. $$\sqrt{100}$$ 2. $$-\sqrt{121}$$ 1. $$10$$ 2. $$-11$$ Can we simplify $$-\sqrt{49}$$? Is there a number whose square is $$-49$$? $$($$___$$)^{2}=-49$$ Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to $$\sqrt{-49}$$. The square root of a negative number is not a real number. Example $$\PageIndex{2}$$ Simplify: 1. $$\sqrt{-196}$$ 2. $$-\sqrt{64}$$ Solution: a. $$\sqrt{-196}$$ There is no real number whose square is $$-196$$. $$\sqrt{-196}$$ is not a real number. b. $$-\sqrt{64}$$ The negative is in front of the radical. $$-8$$ Exercise $$\PageIndex{3}$$ Simplify: 1. $$\sqrt{-169}$$ 2. $$-\sqrt{81}$$ 1. not a real number 2. $$-9$$ Exercise $$\PageIndex{4}$$ Simplify: 1. $$-\sqrt{49}$$ 2. $$\sqrt{-121}$$ 1. $$-7$$ 2. not a real number So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots. Let’s review some vocabulary first. $$\begin{array}{ll}{\text { We write: }} & {\text { We say: }} \\ {n^{2}} & {n \text { squared }} \\ {n^{3}} & {n \text { cubed }} \\ {n^{4}} & {n \text { to the fourth power }} \\ {n^{5}} & {n \text { to the fifth power }}\end{array}$$ The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube. It will be helpful to have a table of the powers of the integers from $$−5$$ to $$5$$. See Figure 8.1.2 Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of $$−2$$ to help you see this. We will now extend the square root definition to higher roots. Definition $$\PageIndex{3}$$: Nth Root of a Number If $$b^{n}=a$$, then $$b$$ is an $$n^{th}$$ root of $$a$$. The principal $$n^{th}$$ root of $$a$$ is written $$\sqrt[n]{a}$$. The $$n$$ is called the index of the radical. Just like we use the word ‘cubed’ for $$b^{3}$$, we use the term ‘cube root’ for $$\sqrt[3]{a}$$. We can refer to Figure 8.1.2 to help find higher roots. \begin{aligned} 4^{3} &=64 & \sqrt[3]{64}&=4 \\ 3^{4} &=81 & \sqrt[4]{81}&=3 \\(-2)^{5} &=-32 & \sqrt[5]{-32}&=-2 \end{aligned} Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers. ## Properties of $$\sqrt[n]{a}$$ When $$n$$ is an even number and • $$a \geq 0$$, then $$\sqrt[n]{a}$$ is a real number. • $$a<0$$, then $$\sqrt[n]{a}$$ is not a real number. When $$n$$ is an odd number, $$\sqrt[n]{a}$$ is a real number for all the values of $$a$$. We will apply these properties in the next two examples. Example $$\PageIndex{3}$$ Simplify: 1. $$\sqrt[3]{64}$$ 2. $$\sqrt[4]{81}$$ 3. $$\sqrt[5]{32}$$ Solution: a. $$\sqrt[3]{64}$$ Since $$4^{3}=64$$. $$4$$ b. $$\sqrt[4]{81}$$ Since $$(3)^{4}=81$$. $$3$$ c. $$\sqrt[5]{32}$$ Since $$(2)^{5}=32$$. $$2$$ Exercise $$\PageIndex{5}$$ Simplify: 1. $$\sqrt[3]{27}$$ 2. $$\sqrt[4]{256}$$ 3. $$\sqrt[5]{243}$$ 1. $$3$$ 2. $$4$$ 3. $$3$$ Exercise $$\PageIndex{6}$$ Simplify: 1. $$\sqrt[3]{1000}$$ 2. $$\sqrt[4]{16}$$ 3. $$\sqrt[5]{243}$$ 1. $$10$$ 2. $$2$$ 3. $$3$$ In this example be alert for the negative signs as well as even and odd powers. Example $$\PageIndex{4}$$ Simplify: 1. $$\sqrt[3]{-125}$$ 2. $$\sqrt[4]{16}$$ 3. $$\sqrt[5]{-243}$$ Solution: a. $$\sqrt[3]{-125}$$ Since $$(-5)^{3}=-125$$. $$-5$$ b. $$\sqrt[4]{16}$$ Think, $$(?)^{4}=-16$$. No real number raised to the fourth power is negative. Not a real number. c. $$\sqrt[5]{-243}$$ Since $$(-3)^{5}=-243$$. $$-3$$ Exercise $$\PageIndex{7}$$ Simplify: 1. $$\sqrt[3]{-27}$$ 2. $$\sqrt[4]{-256}$$ 3. $$\sqrt[5]{-32}$$ 1. $$-3$$ 2. not real 3. $$-2$$ Exercise $$\PageIndex{8}$$ Simplify: 1. $$\sqrt[3]{-216}$$ 2. $$\sqrt[4]{-81}$$ 3. $$\sqrt[5]{-1024}$$ 1. $$-6$$ 2. not real 3. $$-4$$ ## Estimate and Approximate Roots When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the $$\sqrt{4}=2$$, what is the value of $$\sqrt{21}$$ or $$\sqrt[3]{50}$$? In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation. To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of $$\sqrt{11}$$, we see $$11$$ is between perfect square numbers $$9$$ and $$16$$, closer to $$9$$. Its square root then will be between $$3$$ and $$4$$, but closer to $$3$$. Similarly, to estimate $$\sqrt[3]{91}$$, we see $$91$$ is between perfect cube numbers $$64$$ and $$125$$. The cube root then will be between $$4$$ and $$5$$. Example $$\PageIndex{5}$$ Estimate each root between two consecutive whole numbers: 1. $$\sqrt{105}$$ 2. $$\sqrt[3]{43}$$ Solution: a. Think of the perfect square numbers closest to $$105$$. Make a small table of these perfect squares and their squares roots. $$\sqrt{105}$$ Locate $$105$$ between two consecutive perfect squares. $$100<\color{red}105 \color{black} <121$$ $$\sqrt{105}$$ is between their square roots. $$10< \color{red}\sqrt{105}< \color{black}11$$ b. Similarly we locate $$43$$ between two perfect cube numbers. $$\sqrt[3]{43}$$ Locate $$43$$ between two consecutive perfect cubes. $$\sqrt[3]{43}$$ is between their cube roots. Exercise $$\PageIndex{9}$$ Estimate each root between two consecutive whole numbers: 1. $$\sqrt{38}$$ 2. $$\sqrt[3]{93}$$ 1. $$6<\sqrt{38}<7$$ 2. $$4<\sqrt[3]{93}<5$$ Exercise $$\PageIndex{10}$$ Estimate each root between two consecutive whole numbers: 1. $$\sqrt{84}$$ 2. $$\sqrt[3]{152}$$ 1. $$9<\sqrt{84}<10$$ 2. $$5<\sqrt[3]{152}<6$$ There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the $$\sqrt{x}$$ key on your calculator. To find a cube root, or any root with higher index, you will use the $$\sqrt[y]{x}$$ key. When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is $$≈$$ and it is read ‘approximately’. Suppose your calculator has a $$10$$ digit display. You would see that $$\sqrt{5} \approx 2.236067978$$ rounded to two decimal places is $$\sqrt{5} \approx 2.24$$ $$\sqrt[4]{93} \approx 3.105422799$$ rounded to two decimal places is $$\sqrt[4]{93} \approx 3.11$$ How do we know these values are approximations and not the exact values? Look at what happens when we square them: \begin{aligned}(2.236067978)^{2} &=5.000000002 &(3.105422799)^{4}&=92.999999991 \\(2.24)^{2} &=5.0176 & (3.11)^{4}&=93.54951841 \end{aligned} Their squares are close to $$5$$, but are not exactly equal to $$5$$. The fourth powers are close to $$93$$, but not equal to $$93$$. Example $$\PageIndex{6}$$ Round to two decimal places: 1. $$\sqrt{17}$$ 2. $$\sqrt[3]{49}$$ 3. $$\sqrt[4]{51}$$ Solution: a. $$\sqrt{17}$$ Use the calculator square root key. $$4.123105626 \dots$$ Round to two decimal places. $$4.12$$ $$\sqrt{17} \approx 4.12$$ b. $$\sqrt[3]{49}$$ Use the calculator $$\sqrt[y]{x}$$ key. $$3.659305710 \ldots$$ Round to two decimal places. $$3.66$$ $$\sqrt[3]{49} \approx 3.66$$ c. $$\sqrt[4]{51}$$ Use the calculator $$\sqrt[y]{x}$$ key. $$2.6723451177 \ldots$$ Round to two decimal places. $$2.67$$ $$\sqrt[4]{51} \approx 2.67$$ Exercise $$\PageIndex{11}$$ Round to two decimal places: 1. $$\sqrt{11}$$ 2. $$\sqrt[3]{71}$$ 3. $$\sqrt[4]{127}$$ 1. $$\approx 3.32$$ 2. $$\approx 4.14$$ 3. $$\approx 3.36$$ Exercise $$\PageIndex{12}$$ Round to two decimal places: 1. $$\sqrt{13}$$ 2. $$\sqrt[3]{84}$$ 3. $$\sqrt[4]{98}$$ 1. $$\approx 3.61$$ 2. $$\approx 4.38$$ 3. $$\approx 3.15$$ ## Simplify Variable Expressions with Roots The odd root of a number can be either positive or negative. For example, But what about an even root? We want the principal root, so $$\sqrt[4]{625}=5$$. But notice, How can we make sure the fourth root of $$−5$$ raised to the fourth power is $$5$$? We can use the absolute value. $$\|−5\|=5$$. So we say that when $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. This guarantees the principal root is positive. Definition $$\PageIndex{4}$$: Simplifying Odd and Even Roots For any integer $$n\geq 2$$, when the index $$n$$ is odd $$\sqrt[n]{a^{n}}=a$$ when the index $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$ We must use the absolute value signs when we take an even root of an expression with a variable in the radical. Example $$\PageIndex{7}$$ Simplify: 1. $$\sqrt{x^{2}}$$ 2. $$\sqrt[3]{n^{3}}$$ 3. $$\sqrt[4]{p^{4}}$$ 4. $$\sqrt[5]{y^{5}}$$ Solution: a. We use the absolute value to be sure to get the positive root. $$\sqrt{x^{2}}$$ Since the index $$n$$ is even, $$\sqrt[n]{a^{n}}=\|a\|$$. b. This is an odd indexed root so there is no need for an absolute value sign. $$\sqrt[3]{m^{3}}$$ Since the index is $$n$$ is odd, $$\sqrt[n]{a^{n}}=a$$. $$m$$ c. $$\sqrt[4]{p^{4}}$$ Since the index $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. $$\|p\|$$ d. $$\sqrt[5]{y^{5}}$$ Since the index $$n$$ is odd, $$\sqrt[n]{a^{n}}=a$$. $$y$$ Exercise $$\PageIndex{13}$$ Simplify: 1. $$\sqrt{b^{2}}$$ 2. $$\sqrt[3]{w^{3}}$$ 3. $$\sqrt[4]{m^{4}}$$ 4. $$\sqrt[5]{q^{5}}$$ 1. $$\|b\|$$ 2. $$w$$ 3. $$\|m\|$$ 4. $$q$$ Exercise $$\PageIndex{14}$$ Simplify: 1. $$\sqrt{y^{2}}$$ 2. $$\sqrt[3]{p^{3}}$$ 3. $$\sqrt[4]{z^{4}}$$ 4. $$\sqrt[5]{q^{5}}$$ 1. $$\|y\|$$ 2. $$p$$ 3. $$\|z\|$$ 4. $$q$$ What about square roots of higher powers of variables? The Power Property of Exponents says $$\left(a^{m}\right)^{n}=a^{m \cdot n}$$. So if we square $$a^{m}$$, the exponent will become $$2m$$. $$\left(a^{m}\right)^{2}=a^{2 m}$$ Looking now at the square root. $$\sqrt{a^{2 m}}$$ Since $$\left(a^{m}\right)^{2}=a^{2 m}$$. $$\sqrt{\left(a^{m}\right)^{2}}$$ Since $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. $$\left\|a^{m}\right\|$$ So $$\sqrt{a^{2 m}}=\left\|a^{m}\right\|$$. We apply this concept in the next example. Example $$\PageIndex{8}$$ Simplify: 1. $$\sqrt{x^{6}}$$ 2. $$\sqrt{y^{16}}$$ Solution: a. $$\sqrt{x^{6}}$$ Since $$\left(x^{3}\right)^{2}=x^{6}$$. $$\sqrt{\left(x^{3}\right)^{2}}$$ Since the index $$n$$ is even $$\sqrt{a^{n}}=\|a\|$$. $$\left\|x^{3}\right\|$$ b. $$\sqrt{y^{16}}$$ Since $$\left(y^{8}\right)^{2}=y^{16}$$. $$\sqrt{\left(y^{8}\right)^{2}}$$ Since the index $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. $$y^{8}$$ In this case the absolute value sign is not needed as $$y^{8}$$ is positive. Exercise $$\PageIndex{15}$$ Simplify: 1. $$\sqrt{y^{18}}$$ 2. $$\sqrt{z^{12}}$$ 1. $$\|y^{9}\|$$ 2. $$z^{6}$$ Exercise $$\PageIndex{16}$$ Simplify: 1. $$\sqrt{m^{4}}$$ 2. $$\sqrt{b^{10}}$$ 1. $$m^{2}$$ 2. $$\|b^{5}\|$$ The next example uses the same idea for higher roots. Example $$\PageIndex{9}$$ Simplify: 1. $$\sqrt[3]{y^{18}}$$ 2. $$\sqrt[4]{z^{8}}$$ Solution: a. $$\sqrt[3]{y^{18}}$$ Since $$\left(y^{6}\right)^{3}=y^{18}$$. $$\sqrt[3]{\left(y^{6}\right)^{3}}$$ Since $$n$$ is odd, $$\sqrt[n]{a^{n}}=a$$. $$y^{6}$$ b. $$\sqrt[4]{z^{8}}$$ Since $$\left(z^{2}\right)^{4}=z^{8}$$. $$\sqrt[4]{\left(z^{2}\right)^{4}}$$ Since $$z^{2}$$ is positive, we do not need an absolute value sign. $$z^{2}$$ Exercise $$\PageIndex{17}$$ Simplify: 1. $$\sqrt[4]{u^{12}}$$ 2. $$\sqrt[3]{v^{15}}$$ 1. $$\|u^{3}\|$$ 2. $$v^{5}$$ Exercise $$\PageIndex{18}$$ Simplify: 1. $$\sqrt[5]{c^{20}}$$ 2. $$\sqrt[6]{d^{24}}$$ 1. $$c^{4}$$ 2. $$d^{4}$$ In the next example, we now have a coefficient in front of the variable. The concept $$\sqrt{a^{2 m}}=\left\|a^{m}\right\|$$ works in much the same way. $$\sqrt{16 r^{22}}=4\left\|r^{11}\right\|$$ because $$\left(4 r^{11}\right)^{2}=16 r^{22}$$. But notice $$\sqrt{25 u^{8}}=5 u^{4}$$ and no absolute value sign is needed as $$u^{4}$$ is always positive. Example $$\PageIndex{10}$$ Simplify: 1. $$\sqrt{16 n^{2}}$$ 2. $$-\sqrt{81 c^{2}}$$ Solution: a. $$\sqrt{16 n^{2}}$$ Since $$(4 n)^{2}=16 n^{2}$$. $$\sqrt{(4 n)^{2}}$$ Since the index $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. $$4\|n\|$$ b. $$-\sqrt{81 c^{2}}$$ Since $$(9 c)^{2}=81 c^{2}$$. $$-\sqrt{(9 c)^{2}}$$ Since the index $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$. $$-9\|c\|$$ Exercise $$\PageIndex{19}$$ Simplify: 1. $$\sqrt{64 x^{2}}$$ 2. $$-\sqrt{100 p^{2}}$$ 1. $$8\|x\|$$ 2. $$-10\|p\|$$ Exercise $$\PageIndex{20}$$ Simplify: 1. $$\sqrt{169 y^{2}}$$ 2. $$-\sqrt{121 y^{2}}$$ 1. $$13\|y\|$$ 2. $$-11\|y\|$$ This example just takes the idea farther as it has roots of higher index. Example $$\PageIndex{11}$$ Simplify: 1. $$\sqrt[3]{64 p^{6}}$$ 2. $$\sqrt[4]{16 q^{12}}$$ Solution: a. $$\sqrt[3]{64 p^{6}}$$ Rewrite $$64p^{6}$$ as $$\left(4 p^{2}\right)^{3}$$. $$\sqrt[3]{\left(4 p^{2}\right)^{3}}$$ Take the cube root. $$4p^{2}$$ b. $$\sqrt[4]{16 q^{12}}$$ Rewrite the radicand as a fourth power. $$\sqrt[4]{\left(2 q^{3}\right)^{4}}$$ Take the fourth root. $$2\|q^{3}\|$$ Exercise $$\PageIndex{21}$$ Simplify: 1. $$\sqrt[3]{27 x^{27}}$$ 2. $$\sqrt[4]{81 q^{28}}$$ 1. $$3x^{9}$$ 2. $$3\|q^{7}\|$$ Exercise $$\PageIndex{22}$$ Simplify: 1. $$\sqrt[3]{125 q^{9}}$$ 2. $$\sqrt[5]{243 q^{25}}$$ 1. $$5p^{3}$$ 2. $$3q^{5}$$ The next examples have two variables. Example $$\PageIndex{12}$$ Simplify: 1. $$\sqrt{36 x^{2} y^{2}}$$ 2. $$\sqrt{121 a^{6} b^{8}}$$ 3. $$\sqrt[3]{64 p^{63} q^{9}}$$ Solution: a. $$\sqrt{36 x^{2} y^{2}}$$ Since $$(6 x y)^{2}=36 x^{2} y^{2}$$ $$\sqrt{(6 x y)^{2}}$$ Take the square root. $$6\|xy\|$$ b. $$\sqrt{121 a^{6} b^{8}}$$ Since $$\left(11 a^{3} b^{4}\right)^{2}=121 a^{6} b^{8}$$ $$\sqrt{\left(11 a^{3} b^{4}\right)^{2}}$$ Take the square root. $$11\left\|a^{3}\right\| b^{4}$$ c. $$\sqrt[3]{64 p^{63} q^{9}}$$ Since $$\left(4 p^{21} q^{3}\right)^{3}=64 p^{63} q^{9}$$ $$\sqrt[3]{\left(4 p^{21} q^{3}\right)^{3}}$$ Take the cube root. $$4p^{21}q^{3}$$ Exercise $$\PageIndex{23}$$ Simplify: 1. $$\sqrt{100 a^{2} b^{2}}$$ 2. $$\sqrt{144 p^{12} q^{20}}$$ 3. $$\sqrt[3]{8 x^{30} y^{12}}$$ 1. $$10\|ab\|$$ 2. $$12p^{6}q^{10}$$ 3. $$2x^{10}y^{4}$$ Exercise $$\PageIndex{24}$$ Simplify: 1. $$\sqrt{225 m^{2} n^{2}}$$ 2. $$\sqrt{169 x^{10} y^{14}}$$ 3. $$\sqrt[3]{27 w^{36} z^{15}}$$ 1. $$15\|mn\|$$ 2. $$13\left\|x^{5} y^{7}\right\|$$ 3. $$3w^{12}z^{5}$$ Access this online resource for additional instruction and practice with simplifying expressions with roots. • Simplifying Variables Exponents with Roots using Absolute Values ## Key Concepts • Square Root Notation • $$\sqrt{m}$$ is read ‘the square root of $$m$$’ • If $$n^{2}=m$$, then $$n=\sqrt{m}$$, for $$n≥0$$. Figure 8.1.1 • The square root of $$m$$, $$\sqrt{m}$$, is a positive number whose square is $$m$$. • nth Root of a Number • If $$b^{n}=a$$, then $$b$$ is an $$n^{th}$$ root of $$a$$. • The principal $$n^{th}$$ root of $$a$$ is written $$\sqrt[n]{a}$$. • $$n$$ is called the index of the radical. • Properties of $$\sqrt[n]{a}$$ • When $$n$$ is an even number and • $$a≥0$$, then $$\sqrt[n]{a}$$ is a real number • $$a<0$$, then $$\sqrt[n]{a}$$ is not a real number • When $$n$$ is an odd number, $$\sqrt[n]{a}$$ is a real number for all values of $$a$$. • Simplifying Odd and Even Roots • For any integer $$n≥2$$, • when $$n$$ is odd $$\sqrt[n]{a^{n}}=a$$ • when $$n$$ is even $$\sqrt[n]{a^{n}}=\|a\|$$ • We must use the absolute value signs when we take an even root of an expression with a variable in the radical. ### Glossary square of a number If $$n^{2}=m$$, then $$m$$ is the square of $$n$$. square root of a number If $$n^{2}=m$$, then $$n$$ is a square root of $$m$$.	{"extraction_info": {"found_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": 0.9315385222434998, "perplexity": 497.8044094533987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301217.83/warc/CC-MAIN-20220119003144-20220119033144-00598.warc.gz"}
https://www.physicsforums.com/threads/basic-integral.891380/	# B Basic integral 1. Oct 30, 2016 ### Stephanus Dear PF Forum, I'd like to study integral. But I just realize that I'm lack of basic integral. Let y=x2 Here is the graph. So, 1. dx is the distance between the red vertical lines? But it's very, very, very small distance. 2. f(x) * dx is the yellow area? 3. $\int_0^2 x^2\,dx$ is the green (plus yellow) area? 4. And one more thing. The blue line function can be derived from the derivative of f(x)? For example the derivative of x2 is 2x. So a line that crosses (x,f(x)) where x=1 is 2. The blue line function is y = 2x. If it crosses (x,f(x)) where x = 2, then the blue line function is y = 4x? Thank you very much 2. Oct 30, 2016 ### Staff: Mentor We usually call this $\Delta x$. If $x_i$ is a number in the subinterval at the base of the yellow region, then $f(x_i)\Delta x$ is the area of a rectangle that is approximately equal to the area of the yellow region. The smaller the value of $\Delta x$, the better the approximation is. Here we are approximating a roughly trapezoidal shape by one with a rectangular shape. Yes The blue line doesn't "cross" the curve -- it touches it at the point of tangency. No. The slope of the tangent line (what you're calling the blue line) is 4, but if you draw this line you'll see that the tangent line intersects the y-axis below the origin. The line whose slope is 4, that is tangent to the graph of y = x2 at (2, 4), has this equation: y = 4x - 4. 3. Oct 30, 2016 ### FactChecker EDIT: Sorry. I took a break before submitting and this is mostly a repeat of @Mark44's answer above. Yes. Yes. But there is a small range of x values in that short dx, so you pick one, say x0, in that range and use f(x0)) * dx as the approximation of the yellow area. As dx gets smaller, the approximation gets better. Yes. The derivative f(x0) only gives you the slope of the line tangent at x0. So figuring out the constant term of the line is still required. Except for the constant. In general, the equation of the tangent line is f'(x0) * (x-x0) + f(x0) 4. Oct 30, 2016 ### Stephanus Yes, I think it's just semantic here. I usually see f(u) du or F(v) dv. But delta is more appropriate I think. The difference, right? Thanks. Trapezoidal shape looks very similar to rectangle. Thanks. Thanks Yes, I should have drawn it more clearly. What I was going to draw is that the blue line did touch the curve. It touches not crosses. Thanks Yes, I should have drawn it below the origin. Thanks. So it is slope? In high school we call it gradient. THanks. 5. Oct 30, 2016 ### Stephanus Thanks @FactChecker . I suspect that all the answer should be yes. But I need confirmation. I'm going to read some calculus source. Thanks. 6. Oct 30, 2016 ### Staff: Mentor Here the meanings are the same, but the term "gradient" is also used in functions of two or more variables, and in that context, the gradient of f (denoted $\nabla f$) is a vector that consists of the partial derivatives of f. For example, for a function f(x, y, z), $\nabla f$ is the vector $<\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}>$ . 7. Oct 30, 2016 ### FactChecker I would only add that the gradient vector points in the direction of fastest increase of the function and its magnitude is the slope in that direction. Draft saved Draft deleted Similar Discussions: Basic integral	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9202291369438171, "perplexity": 647.25779099849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105326.6/warc/CC-MAIN-20170819070335-20170819090335-00097.warc.gz"}
https://physics.stackexchange.com/questions/541876/can-the-equation-of-total-maximum-amplitude-a-n-sqrta-12a-222a-1a-2-cos	Can the equation of total maximum amplitude $A_n=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)}$ be used though the waves are not in the same line Let $$S_1$$ and $$S_2$$ placed in the same point be the source of two waves which are propagating in the same line, also the phase differernce between the two waves $$\Delta\phi=0$$. Equation of the two waves is given by $$y_1=A_1\sin(\omega t-kx)$$ and $$y_2=A_2\sin(ωt-kx)$$ respectively. Now at the distance $$x_1$$ from the sources, the equations of SHM of a particle become \begin{align} y_1=A_1\sin(\omega t-kx_1) \quad \text{(for wave 1)} \\ y_2=A_2\sin(\omega t-kx_1) \quad \text{(for wave 2)} \end{align} the resultant equation of SHM is given by just adding the two equation $$y_n=A_1\sin(\omega t-kx_1)+A_2\sin(\omega t-kx_2)$$ As written in my book the equation can repressented like $$y_n=A_n\sin(\omega t-kx_1-\theta)$$ where $$A_n$$ is the net maximum displacement due to the two waves and $$\theta$$ is phase difference. To find $$A_n$$ and $$\theta$$, we treat $$A_1$$ and $$A_2$$ as vector and consider the angle between them is same with the phase difference of two SHM $$\Delta\phi$$ . According to the above $$A_n =\sqrt{A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)}.$$ The equation is good one when the waves are in the same line. Because if $$\Delta\phi=0$$ then the displacement for the each waves just add up and give the total displacement which also can be find by the above equation $$A_n= \sqrt{(A_1^2+A_2^2+2A_1A_2\cos(0)}= \sqrt{A_1^2+A_2^2+2A_1A_2} =A_1+A_2$$ Also the formula is effective when the waves are in the same line and phase of two SHM differ by $$\pi$$ as here the total displacement is the subtraction of two displacement due to the individual waves. I believe that the equation is valid for any other cases where the waves are in the same line though I do not find any reason why the angle between $$A_1$$ and $$A_2$$ displacement would be equal to the phase difference of two SHM due to the waves. Though I have seen in the above two cases, it is undoutlessly applicable as when the phase difference is 0 the directions of the two displacement are same and when the phase difference is $$\pi$$, we can subtract the minor displacement from the mazor as the direction of the displacements are opposite . Those two cases show that we can take $$A_1$$ and $$A_2$$ as vector , also phase difference $$\Delta\phi$$ can be taken as angle between $$A_1$$ and $$A_2$$ . But when we think about any other cases where the phase difference is not $$0$$ or $$\pi$$ but the ange between the displacement is either $$0$$ or $$\pi$$ (when the particles go in the same direction the angle is $$0$$ and when they go opposite the angle is $$\pi$$) (note- I am assuming the waves are in the same line) Then in those cases , why do we use the phase difference $$\Delta\phi$$ as the angle between $$A_1$$ and $$A_2$$ insteat of $$0$$ and $$\pi$$. Another problem with the equation i find when i think such a case where the waves are not in the same line. Let the equation of two waves be $$y_1=A_1\sin(\omega t-kx)$$ and $$y_2=A-2\sin(\omega t-kx)$$ respectively. Now the two waves superpose at point $$P$$ with the angle $$\pi/2$$ means the waves are perpendicular with each other. let the distance travelled by the first wave to reach point $$P$$ be equal to distance travelled by the second wave to reach point $$P$$. If the distance is $$x_1$$ then the equation of of SHM of a particle on point $$P$$ (the point of superposition) become \begin{align} y_1=A_1\sin(\omega t-kx_1) \quad \text{(for wave 1)} \\ y_2=A_2\sin(\omega t-kx_1) \quad \text{(for wave 2)} \end{align} We can see clearly that the phase difference between two SHM $$\Delta\phi$$ is $$0$$ as the path difference $$∆x$$ is $$0$$. So according to my book the equation of resultant SHM is given by \begin{align} y_n&=y_1+y_2 \\ &=A_1\sin(\omega t-kx_1)+A_2\sin(\omega t-kx_2) \\ &=A_n\sin(\omega t-kx_1-θ) \end{align} And \begin{align} A_n&=\sqrt{(A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)} \\ &=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(0)} \quad \text{(as phase difference is 0)} \\ &=\sqrt{A_1^2+A_2^2+2A_1A_2)} \\ &=A_1+A_2 \end{align} But if we imagine the the situation we will find the angle between $$A_1$$ and $$A_2$$ is $$\pi/2$$ as the waves superpose by the angle of $$\pi/2$$. So the value of $$A_n$$ should be equal to \begin{align} A_n&= \sqrt{A_1^2+A_2^2+2A_1A_2\cos(π/2)} \\ &=\sqrt{A_1^2+A_2^2} \quad \text{(as two SHM are same phase so when y_1=A_1, y_2=A_2)} \end{align} That does not match with the above which i got using my book formula. • Can the equation of total maximum amplitude $$Aₙ=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)}$$ be used though the waves are not in the same line. • So let say you have two waves $A_{1}e^{j\mathbf{k1}.\mathbf{r}-\omega t}$, $A_{2}e^{j\mathbf{k2}.\mathbf{r}-\omega t}$, with $\vert \mathbf{k_{1}} \vert=\vert \mathbf{ k_{2} }\vert$=k, the angle $(\mathbf{k_{1}},\mathbf{k_{2}})= \theta$, what exactly do you want to know? Apr 6, 2020 at 15:17 For your first question I assume what you want is a derivation of your formula. What you want is to express the real part of $$A_{1}e^{j(-kx+\omega t)}+A_{2}e^{j(-kx+\omega t+\phi)}$$ as new expression which is the real part of $$A_{n}e^{-j(kx+\omega t+ \psi)}$$ so you may simply write: $$A_{1}e^{j(kx-\omega t)}+A_{2}e^{j(kx-\omega t+\phi)} = (A_{1} +A_{2}e^{j\phi})e^{j(kx-\omega t)}$$ Now since the complex $$(A_{1} +A_{2}e^{j\phi})=(A_{1}+A_{2}\cos(\phi)+ j A_{2}sin(\phi))$$ its modulus is indeed $$A_{n}=\sqrt{A_{1}^{2} +A_{2}^{2}+ 2A_{1}A_{2}\cos(\phi)}$$ and its argument is $$\psi=\left( \frac {A_{2}\sin(\phi)}{A_{1}+A_{2}\cos(\phi)} \right)$$. For your second question, beware that the angle difference in the phasors representing your waves cannot be identified with the angle between the vectors $$\mathbf{k_{1}}$$ and $$\mathbf{k_{2}}$$ (I assume $$\vert \mathbf{k_{1}} \vert = \vert \mathbf{k_{2}} \vert =k )$$ that is, the angle between the vectors defining the direction of propagation of 2 waves $$A_{1}e^{j(\mathbf{k_{1}}\mathbf{r} -\omega t)}$$ and $$A_{1}e^{j(\mathbf{k_{2}}\mathbf{r} -\omega t)}$$. Assuming you look at the field in a particular direction such that $$\mathbf{r}=r\mathbf{i}$$, and denoting $$\theta_{1}$$ the angle between $$\mathbf{i}$$ and $$\mathbf{k_{1}}$$, $$\theta_{2}$$ the angle between $$\mathbf{i}$$ and $$\mathbf{k_{2}}$$, we have: $$\mathbf{k_{1}}.\mathbf{r}= k\cos(\theta_{1})r=k_{1}r$$ $$\mathbf{k_{2}}.\mathbf{r}= k\cos(\theta_{2})r=k_{2}r$$ The resulting field along $$\mathbf{i}$$ is: $$A_{1}e^{j(k_{1}r -\omega t)} + A_{2}e^{j(k_{2}r -\omega t)}$$ So the difference of phase between the two phasors is not constant, it is equal to $$(k_{2}-k_{1})r$$ and depends on $$r$$, your formula may thus not be so useful in this case. Of course in the particular case when $$\theta_{1} = \theta_{2}=\theta$$ then $$k_1= k_2= k\cos(\theta)$$ and the resulting fied simply writes: $$(A_{1}+A_{2})e^{j(kcos(\theta)r -\omega t)}$$ Hope it helps.	{"extraction_info": {"found_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": 88, "wp-katex-eq": 0, "align": 5, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9986749887466431, "perplexity": 204.48604361112237}, "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/1652662604794.68/warc/CC-MAIN-20220526100301-20220526130301-00667.warc.gz"}
https://www.global-sci.org/intro/article_detail/cicp/7916.html	Volume 2, Issue 3 Potts Model with q States on Directed Barabási-Albert Networks Commun. Comput. Phys., 2 (2007), pp. 522-529. Published online: 2007-02 Preview Full PDF 435 3540 Export citation Cited by • Abstract On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S =1 was seen to show a spontaneous magnetisation. In this case, a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We also obtained for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 for the directed Barabási-Albert network. Theses results are different from the results obtained for the same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is of first-order. • Keywords Monte Carlo simulation, Ising, networks, disorder. • AMS Subject Headings • BibTex • RIS • TXT @Article{CiCP-2-522, author = {}, title = {Potts Model with q States on Directed Barabási-Albert Networks}, journal = {Communications in Computational Physics}, year = {2007}, volume = {2}, number = {3}, pages = {522--529}, abstract = { On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S =1 was seen to show a spontaneous magnetisation. In this case, a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We also obtained for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 for the directed Barabási-Albert network. Theses results are different from the results obtained for the same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is of first-order. }, issn = {1991-7120}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cicp/7916.html} } TY - JOUR T1 - Potts Model with q States on Directed Barabási-Albert Networks JO - Communications in Computational Physics VL - 3 SP - 522 EP - 529 PY - 2007 DA - 2007/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cicp/7916.html KW - Monte Carlo simulation, Ising, networks, disorder. AB - On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S = 1/2 was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. However, on these networks the Ising model spin S =1 was seen to show a spontaneous magnetisation. In this case, a first-order phase transition for values of connectivity z = 2 and z = 7 is well defined. On these same networks the Potts model with q=3 and 8 states is now studied through Monte Carlo simulations. We also obtained for q = 3 and 8 states a first-order phase transition for values of connectivity z = 2 and z = 7 for the directed Barabási-Albert network. Theses results are different from the results obtained for the same model on two-dimensional lattices, where for q=3 the phase transition is of second order, while for q=8 the phase transition is of first-order. F. W. S. Lima. (2020). Potts Model with q States on Directed Barabási-Albert Networks. Communications in Computational Physics. 2 (3). 522-529. doi: Copy to clipboard The citation has been copied to your clipboard	{"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": 0.9042944312095642, "perplexity": 1201.004849316244}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780055601.25/warc/CC-MAIN-20210917055515-20210917085515-00341.warc.gz"}
https://cs.technion.ac.il/events/view-event.php?evid=682	Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work ~\cite{Yekhanin08} Yekhanin constructs a $3$-query LDC with sub-exponential length of size $\exp(\exp(O(\frac{\log n}{\log\log n})))$. However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give the first unconditional constant query LDC construction with subexponantial codeword length. In addition our construction reduces codeword length. We give construction of $$3$$-query LDC with codeword length $\exp(\exp(O(\sqrt{\log n \log \log n })))$. Our construction also could be extended to higher number of queries.	{"extraction_info": {"found_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": 0.8223901391029358, "perplexity": 411.20038657043943}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00732.warc.gz"}
http://math.stackexchange.com/questions/783436/find-the-value-of-tan100-tan125-tan100-tan125?answertab=oldest	Find the value of: tan(100) + tan(125) + tan(100)•tan(125) The question given was - solve all the equations that are possible to solve without a calculator. (This is a part of it) Please try to solve this question (obviously without a calculator) if it's possible. :) thanks in advance. All values are in degrees - We are mathematicians, how do you even know the value exists ;-) – dcs24 May 6 at 11:24 Using calculator the answer is 1. I am not able to reach it. :\| – Harshal Gajjar May 6 at 11:27 Hence, the answer is 1. – Awesome May 6 at 11:27 (-1) for that, Awesome. – Harshal Gajjar May 6 at 11:28 Hint: What is $\tan 225^{\circ}$, and use the addition formula for tan. – Macavity May 6 at 11:29 Hint: $\tan(\alpha + \beta)=\dfrac{\tan(\alpha)+\tan(\beta)}{1-\tan(\alpha)\tan(\beta)}$ and $\tan(225^\circ)=1$.	{"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": 0.8666567802429199, "perplexity": 1330.3655352081253}, "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-2014-23/segments/1405997894275.63/warc/CC-MAIN-20140722025814-00085-ip-10-33-131-23.ec2.internal.warc.gz"}
https://www.netzerowatch.com/james-hansen-admits-global-warming-slowing-falling-ipcc-projections/	# James Hansen Admits Global Warming Slowing, Falling Below IPCC Projections \| The Hockey Schtick Is James Hansen becoming a skeptic of the IPCC? A paper published today by James Hansen has some startling admissions, including * the effect [forcing] of man-made greenhouse gas emissions has fallen below IPCC projections, despite an increase in man-made CO2 emissions exceeding IPCC projections * the growth rate of the greenhouse gas forcing has “remained below the peak values reached in the 1970s and early 1980s, has been relatively stable for about 20 years, and is falling below IPCC (2001) scenarios (figure 5).” * the airborne fraction of CO2 [the ratio of observed atmospheric CO2 increase to fossil fuel CO2 emissions] has decreased over the past 50 years [figure 3], especially after the year 2000 * Hansen believes the explanation for this conundrum is CO2 fertilization of the biosphere from “the surge of fossil fuel use, mainly coal.” * “the surge of fossil fuel emissions, especially from coal burning, along with the increasing atmospheric CO2 level is ‘fertilizing’ the biosphere, and thus limiting the growth of atmospheric CO2.” * “the rate of global warming seems to be less this decade than it has been during the prior quarter century” According to “coal death train” Hansen, However, it is the dependence of the airborne fraction on fossil fuel emission rate that makes the post-2000 downturn of the airborne fraction particularly striking. The change of emission rate in 2000 from 1.5% yr-1 to 3.1% yr-1 (figure 1), other things being equal, would have caused a sharp increase of the airborne fraction (the simple reason being that a rapid source increase provides less time for carbon to be moved downward out of the ocean’s upper layers). We suggest that the huge post-2000 increase of uptake by the carbon sinks implied by figure 3 is related to the simultaneous sharp increase in coal use (figure 1). We suggest that the surge of fossil fuel use, mainly coal, since 2000 is a basic cause of the large increase of carbon uptake by the combined terrestrial and ocean carbon sinks. One mechanism by which fossil fuel emissions increase carbon uptake is by fertilizing the biosphere via provision of nutrients essential for tissue building, especially nitrogen, which plays a critical role in controlling net primary productivity and is limited in many ecosystems.” So is the new data we present here good news or bad news, and how does it alter the ‘Faustian bargain’? At first glance there seems to be some good news. First, if our interpretation of the data is correct, the surge of fossil fuel emissions, especially from coal burning, along with the increasing atmospheric CO2 level is ‘fertilizing’ the biosphere, and thus limiting the growth of atmospheric CO2Also, despite the absence of accurate global aerosol measurements, it seems that the aerosol cooling effect is probably increasing based on evidence of aerosol increases in the Far East and increasing ‘background’ stratospheric aerosols. Both effects work to limit global warming and thus help explain why the rate of global warming seems to be less this decade than it has been during the prior quarter century.” ## CLIMATE FORCING GROWTH RATES: DOUBLING DOWN ON OUR FAUSTIAN BARGAIN #### OPEN ACCESS James Hansen, Pushker Kharecha and Makiko Sato Perspective This is a Perspective for the article 2012 Environ. Res. Lett. 7 044035 Rahmstorf et al ‘s (2012) conclusion that observed climate change is comparable to projections, and in some cases exceeds projections, allows further inferences if we can quantify changing climate forcings and compare those with projections. The largest climate forcing is caused by well-mixed long-lived greenhouse gases. Here we illustrate trends of these gases and their climate forcings, and we discuss implications. We focus on quantities that are accurately measured, and we include comparison with fixed scenarios, which helps reduce common misimpressions about how climate forcings are changing. Annual fossil fuel CO2 emissions have shot up in the past decade at about 3% yr-1, double the rate of the prior three decades (figure 1). The growth rate falls above the range of the IPCC (2001) ‘Marker’ scenarios, although emissions are still within the entire range considered by the IPCC SRES (2000). The surge in emissions is due to increased coal use (blue curve in figure 1), which now accounts for more than 40% of fossil fuel CO2 emissions. Figure 1. CO2 annual emissions from fossil fuel use and cement manufacture, an update of figure 16 of Hansen (2003) using data of British Petroleum (BP 2012) concatenated with data of Boden et al (2012). The resulting annual increase of atmospheric CO2 (12-month running mean) has grown from less than 1 ppm yr-1 in the early 1960s to an average ~2 ppm yr-1 in the past decade (figure 2). Although CO2 measurements were not made at sufficient locations prior to the early 1980s to calculate the global mean change, the close match of global and Mauna Loa data for later years suggests that Mauna Loa data provide a good approximation of global change (figure 2), thus allowing a useful estimate of annual global change beginning with the initiation of Mauna Loa measurements in 1958 by Keeling et al(1973). Figure 2. Annual increase of CO2 based on data from the NOAA Earth System Research Laboratory (ESRL 2012). CO2change and global temperature change are 12-month running means of differences for the same month of consecutive years. Nino index (Nino3.4 area) is 12-month running mean. Both temperature indices use data from Hansen et al (2010). Annual mean CO2 amount in 1958 was 315 ppm (Mauna Loa) and in 2012 was 394 ppm (Mauna Loa) and 393 ppm (Global). Full story	{"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": 0.8182697296142578, "perplexity": 2858.2202264058446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1659882571538.36/warc/CC-MAIN-20220812014923-20220812044923-00162.warc.gz"}
http://math.stackexchange.com/questions/411534/prove-t-has-at-most-two-distinct-eigenvalues	# Prove $T$ has at most two distinct eigenvalues The question is from Axler's Linear Algebra text. The $\mathcal{L}(V)$ stands for the space of linear operators on the vector space $V$. Suppose that V is a complex vector space with dim $V=n$ and $T \in \mathcal{L}(V)$ is such that $$\text{null} \ T^{n-2} \neq \text{null} \ T^{n-1}$$ Prove that $T$ has at most two distinct eigenvalues. I fist thought of solving this by contradiction. That is, I thought, suppose there were three distinct eigenvalues. Then, there would be an equation like $$(x-\lambda_1I)^{d_1}(x-\lambda_2I)^{d_2}(x-\lambda_3I)^{d_3}$$ where the $d_i$'s are positive integers that sum to dim $V$. Call this polynomial $q(x)$ the characteristic poly. Thus, by Cayley's theorem, $$q(T)=(T-\lambda_1I)^{d_1}(T-\lambda_2I)^{d_2}(T-\lambda_3I)^{d_3}=0$$ Then multiplying out and setting dim $V = d_1+d_2+d_3 = n$, I could then get a poly with the various powers of $T$ (this is a little tricky to write down). In particular, I wanted to see the powers $n-2$ and $n-1$. I thought, ok, so now, rewrite the poly in terms of each of these and using the fact that $\text{null} \ T^{n-2} \neq \text{null} \ T^{n-1}$ you get some vector $v \in V$ such that, $$T^{n-1}v = (\text{poly}_1)v = 0$$ but $$T^{n-2}v = (\text{poly}_2)v = k \neq 0$$ I can think of some interesting things about $(\text{poly})_1$ and $(\text{poly})_2$, in particular, each has the monic term $T^n$. At this point, I'm not sure any of this helped. Well, anyways, I'm sure someone has a much better approach! Thanks in advance to anyone who read this. - Hints: consider the chain of subspaces $\{0\}=\ker T^0\subseteq\ker T\subseteq \ker T^2\subseteq \ldots...$ and think about what happens if $\ker T^{k-1}=\ker T^k$ at some point. Then prove that the assumption $\mbox{null} T^{n-2}\neq \mbox{null} T^{n-1}$ yields $\mbox{null} T^{n-1}=n-1$ or $\mbox{null} T^{n-1}=n$. In the latter case $T^{n-1}=0$, so it is easy to conclude. In the former case, take a basis of $\ker T^{n-1}$, complete it into a basis of $V$, and consider the matrix of $T$ in the latter. Note: there is nothing special about $\mathbb{C}$ with this approach. It could be over any field. - thanks a lot. quick question, the dim$\mbox{null}T^{n-1} = n-1 \text{or} n$? I was trying to figure that out at first but kept tripping up on this point. Is it because if, as you said, $\ker T^{k-1} = \ker T^k$ at some point, then the dim of the kernel of $T^n$ will equal also... said differently then since $\ker T^{k-1} = \ker T^k = \cdots = \ker T^n$, then dim$V = \text{dim}ker T + \text{dim} im T$, you can deduce that the above equals $n-1$ or $n$? – IQ472 Jun 5 '13 at 0:48 @IQ472 As you observed, if it is equal at some point, it is stationary. So the sequence can only become stationary after $n-1$. And before that, the inclusions were strict, the dimension was increasing by $1$ at least at each step. – 1015 Jun 5 '13 at 0:50 Suppose $T(v) = \lambda v$ where $\lambda \ne 0$. Then $v$ is not part of the nullspace of $T^{n-1}$. Take a basis $\mathscr B_1$ for the nullspace of $T^{n-1}$ and add $v$ to that basis. Call this linearly independent set $\mathscr B$. We want to prove that $\mathscr B$ is a basis for $V$. $T$ is nilpotent on $\operatorname{span}( \mathscr B_1 )$, so $T^{\|\mathscr B_1\|} = 0$ But $T^{n-2} \ne 0$, so $\|\mathscr B_1\| > n-2$. So $\mathscr B$ is a linearly independent set of order $n$ and is thus a basis. The only eigenvalue for a nilpotent matrix is $0$ (use a change of basis to make the matrix upper triangular), so the only eigenvalues for $T$ are $0$ and $\lambda$. - Thanks so much, your approach clarified exactly how to go about this problem. Again thanks! – IQ472 Jun 5 '13 at 0:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9839022159576416, "perplexity": 98.57809846203193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637899041.10/warc/CC-MAIN-20141030025819-00180-ip-10-16-133-185.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/54076/notation-for-a-representable-functor?sort=oldest	# Notation for a representable functor For an object $X$ of a category, $h_X$ is the contravariant functor represented by $X$, i.e. $h_X = Hom(-,X)$. Question a) Who invented this notation? (My guess: Grothendieck) b) Is there a special reason why the letter $h$ was chosen? Is it in an abbreviation for "homomorphism"? - I dunno, but I think the more interesting case is $h^X:=Hom(X,-)$, where it's a superscript because it is contravariant in $X$. – Harry Gindi Feb 2 '11 at 9:49 @Harry: Offtopic. – Martin Brandenburg Feb 2 '11 at 10:54 Hey, don't be rude about it! If my comment is off topic, then your entire question is most certainly so. – Harry Gindi Feb 2 '11 at 17:22 b) Quite possibly is a shortcut for $Hom$. Sometimes the letter $y$ is used (for Yoneda). The trouble is when you are considering the representable functor defined over several categories, e.g. a category and a subcategory. Bonus: If you, instead of considering contravariant functors $\mathrm{Sch}^{o} \to \mathrm{Set}$, use covariant functors $\mathrm{Aff} \to \mathrm{Set}$ the notation used in EGA is $h_X^{o}$. Perhaps the reason is that Yoneda's map is contravariant in this case. Of course the notation $h_X$ is used extensively in EGA, but where can you find evidence that that it is Grothendieck's invention? – Martin Brandenburg Feb 2 '11 at 10:53 Further evidence: The notation is already on SGA 3 and 4. There are several exposés by Grothendieck in Henri Cartan's seminar from 1960/61 in which he explains his point of view of Teichmüller's space through representable functors in the analytical category and he uses the notation $h_X$. I don't think anyone else was using these ideas at that time. Cartan's seminar is available at numdam.org. – Leo Alonso Feb 2 '11 at 11:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9737007021903992, "perplexity": 802.8075675206328}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1410657135558.82/warc/CC-MAIN-20140914011215-00236-ip-10-234-18-248.ec2.internal.warc.gz"}
http://cbse-notes.blogspot.in/2011/12/class-10-physics-light-numerical-and.html	## Saturday, December 10, 2011 ### Class 10 - Physics - Light - Numerical and Questions Q1: When an object is placed at a distance of 60 cm from a convex mirror, the magnification produced is 1/2. Where should the object be placed to get a magnification of 1/3? Q2: When an object is kept at a distance of 60cm from a concave mirror, the magnification is 1/2. Where should the object be placed to get magnification of 1/3? Answer: (Hint: steps are same as in the above question, except it is concave mirror, m = -1/2. Compute f in first case i.e. f = -20 cm and the compute u in case II i.e. u = 80 cm) Q3(CBSE Board): A concave lens made of material of refractive index (n1) is kept in a medium of refractive index (n2). A parallel beam of light is incident on the lens. Complete the path of the rays of light emerging from a concave lens if: (a) n1 > n2 (b) n1 = n2 (c) n1 < n2 Q4: A convex lens made of material of refractive index (n1) is kept in a medium of refractive index (n2). A parallel beam of light is incident on the lens. Complete the path of the rays of light emerging from a concave lens if: (a) n1 > n2 (b) n1 = n2 (c) n1 < n2 Q5: Which of the two has a great power? A lens of short focal length or a lens of large focal length? Answer: lens of short focal length. Q6: When a convex lens and a concave lens of equal focal lengths are placed side by side, what will be the equivalent power and the combined focal length. Answer: Equivalent Power = 0 and Combined focal length = infinity. Q7: When a plane mirror is rotated by an angle $\theta$, what will be the angle of rotation of the reflected ray for the fixed incident ray? Answer: The reflected ray will be rotated twice w.r.t to the angle of rotation of mirror. i.e. $2\theta$ Consider the following figure: As shown, mirror M is rotated by an angle $\phi$. Let I be incident ray, N be normal and R is reflected ray. Before rotation, let the incident angle be $\theta$ After rotation, normal moves by angle $\phi$, $\therefore \text{incident angle } \angle {ION'} = \theta + \phi \newline \angle{ROR'} = \angle{IOR'} - \angle{IOR} = 2(\theta + \phi) - 2\theta = 2\phi$ Q8: With respect to air the refractive index of kerosene is 1.44 and that of diamond is 2.42. Calculate the refractive index of diamond with respect to kerosene. 2. In Q3 the last n1<n2 should'nt be a convex lens??? 1. they have said "if" so we can accept that 3. i cud hav lookd it earlier 1. This comment has been removed by the author. 4. In Q2, u cannot be (+)80.... u has to be -ive 1. this is very true but it is said that in hint apply that u is if positive so what will be the answer u should read the question nicely 5. Questions are too easy........post sumthing intresting 6. Thank you very much. Found this post very useful 7. thanks it is very useful for me!	{"extraction_info": {"found_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": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.837249219417572, "perplexity": 702.3337709171824}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1484560280668.34/warc/CC-MAIN-20170116095120-00574-ip-10-171-10-70.ec2.internal.warc.gz"}
https://mathoverflow.net/questions/242551/short-time-asymptotics-for-brownian-motion-on-a-compact-manifold	Short time asymptotics for Brownian motion on a compact manifold Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in this context, quasi-isometric means that $c_1 g_{\mathbb{R}^n} \leq g \leq c_2 g_{\mathbb{R}^n}$ on $B(p, r)$ and $B(q, s)$. Now, consider a compact set $K \subset B(p, r)$ and its image $K'$ inside $B(q, s)$ under the coordinate chart. It seems intuitively that the probability of a Brownian particle (corresponding to the Brownian motion on $M$) starting at $p$ and hitting $K$ within small time $t$ is comparable to the probability of a Brownian particle (corresponding to the Brownian motion in $\mathbb{R}^n$) starting at $q$ hitting $K'$ within (the same) small time $t$. How can one prove this? To rephrase, is there a precise estimate to fit the following heuristic: if the metric is locally slightly perturbed, then the Brownian motion for small times is also slightly perturbed? Edit: An issue seems to be that there needs to be a global argument somewhere. For example, one somehow needs to say that the contribution from paths which go far out and then come back again in short time is negligible in both cases. This seems very believable intuitively, as a Brownian motion effectively travels a distance $\sim t^2$ in time $\sim t$, but again, I am uncertain how to formalize this.	{"extraction_info": {"found_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": 0.9678924679756165, "perplexity": 167.11989276638383}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313589.19/warc/CC-MAIN-20190818022816-20190818044816-00266.warc.gz"}
https://brilliant.org/discussions/thread/math-and-science-book-review-2018/	# Math and Science Book Review 2018 A Course in Modern Mathematical Physics by Peter Szekeres Rating 10/10 Publisher Cambridge University Press This book is more than just textbook. It is a comprehensive journey of advance mathematics and how it applies to modern physics. Want to learn the pillars of mathematics? This book is for you! Want to learn modern physics the proper way? This book is for you! My advice is to start with the math: sets, logic, groups, Hilbert space, tensors, and topology. Once the advanced math is mastered, move onto the modern physics theories. Mathematics in 10 Lessons by Jerry P. King Rating 9/10 Publisher Prometheus Books I really liked the choice of the math topics and the order it is presented in this book. A true tour of the basics of math. Starting with logic, it naturally progresses to sets, algebra, geometry, all the way to calculus. What's great is that several important formulas and proofs are explicitly presented! My only problem with the book is the lack of depth, but I think the author plans remove many details due to his intended audience. Great book nonetheless! A Student's Guide to Waves by Daniel Fleisch and Laura Kinnaman Rating 10/10 Publisher Cambridge University Press Daniel Fleisch is one of my favourite science writers. Why? His books are not condescendingly advanced/tricky or written for the layman. Perfect for math and physics students. I would even say that if every physics major reads Daniel Fleisch's books, they truly understand math and physics. In this book, you learn about waves and its applications. I particularly liked the sections on electromagnetic waves and quantum wavefunctions. I would recommend Daniel Fleisch's books over Richard Feynman! Calculus for Biology and Medicine by Claudia Neuhauser Rating 9/10 The title of this book does not serve the book justice! The author or perhaps the publisher should rename it Mathematics for Biology and Medicine. I find this textbook surprisingly complete in its treatment of calculus, linear algebra, probability and statistics, and even applied differential equations. I really liked using this book as a reference for tutoring! Note by Steven Zheng 1 year, 5 months ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. 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: What level of math education do you need for "A Course in Modern Mathematical Physics"? - 1 year, 5 months ago - 1 year, 5 months ago I agree with you that starting with mathematics is right. Because mathematics is a basic subject from which students start studying in schools. Also student start with subject of writing, they learn to write without using the service PapersOwl which can write essay for students instead of them. If a student learns to write self-consciously, then he or she will not be helped by writing even a complicated paper afterward. - 10 months, 1 week ago	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "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": 0.8594452738761902, "perplexity": 2438.967059524299}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579250608295.52/warc/CC-MAIN-20200123041345-20200123070345-00244.warc.gz"}
https://www.physicsforums.com/threads/discrepancy-between-my-work-and-books-static-fluid-pressure.694886/	# Discrepancy between my work and book's [Static Fluid/Pressure] 1. Jun 1, 2013 ### jeff.berhow The book (Example 13-3): My attempt: My confusion is in why they are starting their height at 1.0 meters above the surface of the water and then getting a y_2 value of 2.0 meters. That's the discrepancy that's giving me a different value than the book's. It never says in the question to do that, and I'm baffled as to how I am supposed to know. Thanks in advance for the help. 2. Jun 1, 2013 ### TSny Hello. In P = ρgh, h denotes the depth below the surface of the water. h is also your integration variable. What is the value of h at the top of the window? At the bottom of the window? [EDIT: Note that the window is submerged below the surface of the water.] 3. Jun 1, 2013 ### jeff.berhow h = 0.0 meters at the top and h = 1.0 meters at the bottom of the window. 4. Jun 1, 2013 ### TSny The widow is embedded in the side of the aquarium with the top of the window 1.0 m below the surface of the water. 5. Jun 1, 2013 ### jeff.berhow Ah, I see. I wish they would explicitly state these things in the question as it seems I am unable to derive them from the pictures, haha. Well, they don't write the books for me! Thanks TSny! I will approach this problem again. Do you think I can keep my origin at the top of the window and change my bounds to -1m to +1m? It seems this will still give me an incorrect value and I may have to change my origin to the top of the water tank. 6. Jun 1, 2013 ### TSny Suppose you introduce a y-axis going vertically downward. You can certainly choose the origin of y to be at the top of the window. Then you would need to think about how to express the depth h in terms of y. [EDIT: You could then integrate from y = 0 to y = 1.] 7. Jun 1, 2013 ### jeff.berhow Alright, it seems you have nailed the source of confusion on the head. I will have to redraw my diagram and rewrite my integral thinking about the relation of h and y. edit: I was way overthinking this. I reintegrated with new bounds and it comes out perfectly. Thanks a bunch, TSny! Last edited: Jun 1, 2013 Draft saved Draft deleted Similar Discussions: Discrepancy between my work and book's [Static Fluid/Pressure]	{"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": 0.8843621611595154, "perplexity": 737.8200272388962}, "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-2017-34/segments/1502886105712.28/warc/CC-MAIN-20170819182059-20170819202059-00248.warc.gz"}
https://www.physicsforums.com/threads/absolute-space-and-time-in-einsteins-general-theory-of-relativity.4905/	# Absolute Space and Time in Einstein's General Theory of Relativity 1. Aug 18, 2003 ### Eugene Shubert The Special Theory of Relativity, we teach our students, did away with Absolute Space and Absolute Time, leaving us with no absolute motion or rest, and also no absolute time order. General Relativity is viewed as extending the "relativity of motion" applicable to curved spacetimes, and General Relativity's most probable models of our actual spacetimes (the big-bang models) appear to re-introduce a privileged "cosmic" time order, and a definite sense of absolute rest. In particular, some of the same kinds of effects whose absence led to rejection of Newtonian absolute space are present in these models of GTR. End of quote. Colloquium for 13-NOV-97 Abstract, UCR. See http://physics.ucr.edu/Active/Abs/abstract-13-NOV-97.html [Broken] I'm delighted that common sense is finally being recognized in the physics community. When do you think it will be realized that an absolute time order precludes the possibility of anything falling into a black hole? http://www.everythingimportant.org/relativity/simultaneity.htm Last edited by a moderator: May 1, 2017 2. Aug 18, 2003 ### FZ+ It does? That's news to me. Wanna point out how before you run wild with this? 3. Aug 19, 2003 ### Eugene Shubert The colloquium started without me. I don't even know the academician who delivered the talk. The posted sketch seems eminently believable, based on the elementary global theorem of the second link. If you believe the analysis is false, how then do you resolve the paradox of the great illumination? 4. Aug 19, 2003 ### Eugene Shubert Let's start with the basics. A frame is an absolute frame of reference if it is physically distinguished from all other frames and if it is totally independent of the distribution of matter/energy in the universe. The simplest and easiest to conceive universe that has an absolute frame of reference is the cylinder SxR. The physics of the SxR universe is dealt with at length in these links: http://cornell.mirror.aps.org/abstract/PRD/v8/i6/p1662_1 [Broken] http://qcd.th.u-psud.fr/page_perso/Uzan/fileps/art_2002_ullp_ejp23.pdf http://www.everythingimportant.org/relativity/simultaneity.htm The reader is required to comprehend trivial model universes or advanced mathematics. The space part of SxR is a circle S. Let an observer in the circle universe be at rest in her own "inertial frame of reference." Let her pick a positive and negative direction. Suppose she has a nice watch on her wrist to note the time. Other than just looking nice and being able to see how old she's getting, let her do experiments. Let t1 be the time she measures for a photon to circumnavigate the universe in the positive direction. Let t2 be the time she measures for a photon to circumnavigate the universe in the negative direction. Being only a one-dimensional creature she is still smart enough to realize the impossibility of all frames agreeing on a frame independent law of light propagation. Consequently, if t1 doesn't equal t2, then she is moving at some velocity v with respect to an absolute frame of reference. Isn't it obvious, based on the global theorem, that the velocity v is given by the equation: t1/t2 = (c+v)/(c-v) ? Last edited by a moderator: May 1, 2017 5. Aug 19, 2003 ### Broken There will never be any time difference to the observer. The only time difference that will be measured will be from the person sitting in your "absolute frame of reference". Everyone in other frames will get different time differences and they will all think they are in what you claim to be the "absolute frame of reference". No one will agree on anything. This is an argument I've seen raised many times and refuted each time. Also, from what I've just seen of "www.everythingimportant.com", it doesn't look like a reliable scientific reference. Broken 6. Aug 19, 2003 ### Hurkyl Staff Emeritus Actually, that's incorrect. For example, assume the obvious structure on SxR, and suppose in the "absolute" frame of reference it takes light one year to circumnavigate the universe. Now, if an observer travelling 0.5c in the clockwise direction (WRT the "absolute" frame) emits a clockwise travelling photon, it will be 1.73 years before he sees it again, yet only 0.58 years for an emitted counterclockwise travelling photon. (as computed by actually drawing SxR and the worldlines, and directly computing the proper time observed by the observer) However, the everythingimportant website has its problems; it all starts to go downhill with the statement "It’s obvious from the notion of measure that the distance around the universe is the same in both directions"; it's not even obvious how to measure the distance around the universe, let alone obvious that it must be the same. One way to do this is to integrate spatial displacement dx along a circumnavigating worldline (or any circumnavigating path, for that matter) in a particular globally minowski frame. The "absolute" reference frame will indeed find, when measuring in this way, that the distance around the universe will be a constant no matter what path is taken. However, in every other reference frame, the measured distance around the universe will vary depending on the chosen path. In particular, for the previously mentioned observer, if he used the worldlines of the two photons to measure the distance around the univesre, he'd find it the clockwise-travelling photon underwent a spatial displacement of 1.73 light-years, and the counterclockwise-travelling photon underwent a spatial displacement of 0.58 light-years (results that are unsurprisingly consistent with the speed of light). I don't see any justification in calling the "absolute" reference frame absolute; it does have one particular property that the others do not, but that's it. Anyways, the section "A Global Theorem" depends crucially on there being a well-defined "distance around the universe" in any reference frame, but such a thing has not been demonstrated to exist. 7. Aug 19, 2003 ### Eugene Shubert Hurkyl, You got the ratio correct but the numbers wrong. t1 = 1.5 and t2 = .5 Use the transformation equations: x'=Y(v)(x-vt) t'=t/Y(v) Y(v)=1/sqrt(1-v^2/c^2) With these equations and the following rules you'll see that my system is free of contradictions. LENGTH CONTRACTION Let d be the distance around the universe in the absolute frame of reference. Let d' be the distance around the universe according to a "stationary" observer in a moving frame of reference. Then d'=Y(v)d THE LAW OF LIGHT PROPAGATION To track the motion of light rays in a moving frame of reference, use the equation, distance =rate*time and take special note of the positive and negative directions: C(v) is the velocity of light in the positive direction. -> C(v) is the velocity of light in the negative direction. <- C(v) = (Y(v)^2)(c-v) -> C(v) = (Y(v)^2)(c+v) <- Be bold. Assume that distance in a moving frame makes sense. (I made that assumption). It then follows that "distance around the universe is the same in both directions." More importantly, can you prove that there are any contradictions in my interpretation of relativity on SxR? Eugene Shubert http://www.everythingimportant.org/relativity/simultaneity.htm Last edited: Aug 20, 2003 8. Aug 20, 2003 ### Broken The observer was described as being in the motion frame. Just terminology against convention in the original stated problem. Broken 9. Aug 20, 2003 ### Hurkyl Staff Emeritus I didn't use any transformation. I computed the proper time experienced by the moving observer using the formula for proper time: (c dτ)2 = (c dt)2 - dx2 If you draw the worldlines (using the "absolute" coordinates), while waiting for the clockwise travelling photon to return, the moving observer undergoes a spatial displacement Δx = 1 light-year, and undergoes a (coordinate) temporal displacement of Δt = 2 years, so plugging into the formula for proper time you get an elapsed proper time Δτ = [squ]3. For the counter-clockwise travelling photon, Δx = 1/3, Δt = 2/3, and Δτ = 1/[squ]3 Which brings us to your transformation equations; they do not preserve the form of the metric. (t', x') is not an inertial reference frame. Basically, you have thrown away the whole of relativity just so you could have a constant "distance around the universe" in every reference frame.	{"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": 0.8146716952323914, "perplexity": 1093.0357919651833}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1521257645069.15/warc/CC-MAIN-20180317120247-20180317140247-00611.warc.gz"}
http://arxiv.org/abs/1112.2327	hep-th (what is this?) # Title: New Approach to Nonperturbative Quantum Mechanics with Minimal Length Uncertainty Authors: Pouria Pedram Abstract: The existence of a minimal measurable length is a common feature of various approaches to quantum gravity such as string theory, loop quantum gravity and black-hole physics. In this scenario, all commutation relations are modified and the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). Here, we present a one-dimensional nonperturbative approach to quantum mechanics with minimal length uncertainty relation which implies X=x to all orders and P=p+(1/3)\beta p^3 to first order of GUP parameter \beta, where X and P are the generalized position and momentum operators and [x,p]=i\hbar. We show that this formalism is an equivalent representation of the seminal proposal by Kempf, Mangano, and Mann and predicts the same physics. However, this proposal reveals many significant aspects of the generalized uncertainty principle in a simple and comprehensive form and the existence of a maximal canonical momentum is manifest through this representation. The problems of the free particle and the harmonic oscillator are exactly solved in this GUP framework and the effects of GUP on the thermodynamics of these systems are also presented. Although X, P, and the Hamiltonian of the harmonic oscillator all are formally self-adjoint, the careful study of the domains of these operators shows that only the momentum operator remains self-adjoint in the presence of the minimal length uncertainty. We finally discuss the difficulties with the definition of potentials with infinitely sharp boundaries. Comments: 11 pages, 5 figures Subjects: High Energy Physics - Theory (hep-th) Journal reference: Phys. Rev. D 85, 024016 (2012) DOI: 10.1103/PhysRevD.85.024016 Cite as: arXiv:1112.2327 [hep-th] (or arXiv:1112.2327v2 [hep-th] for this version) ## Submission history From: Pouria Pedram [view email] [v1] Sun, 11 Dec 2011 07:15:20 GMT (459kb) [v2] Fri, 13 Jan 2012 09:59:37 GMT (460kb)	{"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": 0.8949588537216187, "perplexity": 919.4515500394821}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982296721.55/warc/CC-MAIN-20160823195816-00228-ip-10-153-172-175.ec2.internal.warc.gz"}
http://math.stackexchange.com/help/badges/9?page=3	# Help Center > Badges > Autobiographer Completed all user profile fields. Awarded 13043 times Awarded jul 21 at 16:11 to Awarded jul 21 at 8:35 to Awarded jul 21 at 8:30 to Awarded jul 21 at 7:59 to Awarded jul 21 at 5:04 to Awarded jul 20 at 22:40 to Awarded jul 20 at 21:53 to Awarded jul 20 at 16:05 to Awarded jul 20 at 15:47 to Awarded jul 20 at 13:16 to Awarded jul 20 at 5:52 to Awarded jul 20 at 4:26 to Awarded jul 19 at 22:44 to Awarded jul 19 at 21:29 to Awarded jul 19 at 20:34 to Awarded jul 19 at 14:49 to Awarded jul 19 at 13:53 to Awarded jul 19 at 7:37 to Awarded jul 18 at 22:50 to Awarded jul 18 at 20:26 to Awarded jul 18 at 18:26 to Awarded jul 18 at 17:03 to Awarded jul 18 at 10:35 to Awarded jul 18 at 9:54 to Awarded jul 18 at 8:24 to Awarded jul 18 at 1:31 to Awarded jul 17 at 18:17 to Awarded jul 17 at 15:38 to Awarded jul 17 at 14:50 to Awarded jul 17 at 14:40 to Awarded jul 17 at 13:44 to Awarded jul 17 at 13:05 to Awarded jul 17 at 12:43 to Awarded jul 17 at 12:28 to Awarded jul 17 at 11:54 to Awarded jul 17 at 11:38 to Awarded jul 17 at 10:37 to Awarded jul 17 at 7:51 to Awarded jul 17 at 6:27 to Awarded jul 17 at 4:50 to Awarded jul 17 at 2:00 to Awarded jul 16 at 21:02 to Awarded jul 16 at 19:17 to Awarded jul 16 at 17:55 to Awarded jul 16 at 16:36 to Awarded jul 16 at 15:26 to Awarded jul 16 at 12:48 to Awarded jul 16 at 12:28 to Awarded jul 16 at 10:33 to Awarded jul 16 at 5:22 to Awarded jul 16 at 4:31 to Awarded jul 16 at 4:17 to Awarded jul 16 at 2:13 to Awarded jul 16 at 1:52 to Awarded jul 15 at 20:54 to Awarded jul 15 at 18:39 to Awarded jul 15 at 15:29 to Awarded jul 15 at 13:57 to Awarded jul 15 at 11:45 to Awarded jul 15 at 9:29 to	{"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": 0.9475999474525452, "perplexity": 2487.7087473643883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1406510274967.3/warc/CC-MAIN-20140728011754-00252-ip-10-146-231-18.ec2.internal.warc.gz"}
https://searxiv.org/search?author=Bhanu%20Prakash%20Reddy	Searching Arxiv, refresh for possibly better results. ### Results for "Bhanu Prakash Reddy" total 1806took 0.10s StRE: Self Attentive Edit Quality Prediction in WikipediaJun 11 2019Wikipedia can easily be justified as a behemoth, considering the sheer volume of content that is added or removed every minute to its several projects. This creates an immense scope, in the field of natural language processing towards developing automated ... More ECHO: An Adaptive Orchestration Platform for Hybrid Dataflows across Cloud and EdgeJul 04 2017The Internet of Things (IoT) is offering unprecedented observational data that are used for managing Smart City utilities. Edge and Fog gateway devices are an integral part of IoT deployments to acquire real-time data and enact controls. Recently, Edge-computing ... More Understanding Stokes-Einstein Relation in Supercooled Liquids using Random PinningFeb 01 2016Breakdown of Stokes-Einstein relation in supercooled liquids is believed to be one of the hallmarks of glass transition. The phenomena is studied in depth over many years to understand the microscopic mechanism without much success. Recently it was found ... More Neutrino Scattering in Strangeness-Rich Stellar MatterDec 07 1995Dec 08 1995We calculate neutrino cross sections from neutral current reactions in dense matter containing hyperons. We show that Sigma- hyperons give significant contributions. To lowest order, the contributions from the neutral Lambda and Sigma0, which have zero ... More Superdense star in a space-time with minimal lengthJan 07 2019In this paper we generalise core-envelope model of superdense star to a non-commutative space-time and study the modifications due to the existence of a minimal length, predicted by various approaches to quantum gravity. We first derive Einstein's field ... More Neutrino Scattering in a Newly Born Neutron StarOct 15 1996Oct 16 1996We calculate neutrino cross sections from neutral current reactions in the dense matter encountered in the evolution of a newly born neutron star. Effects of composition and of strong interactions in the deleptonization and cooling phases of the evolution ... More Can Evolutionary Sampling Improve Bagged Ensembles?Oct 03 2016Perturb and Combine (P&C) group of methods generate multiple versions of the predictor by perturbing the training set or construction and then combining them into a single predictor (Breiman, 1996b). The motive is to improve the accuracy in unstable classification ... More Effect of pinning on the yielding transition of amorphous solidsAug 29 2018Aug 30 2018Using numerical simulations, we have studied the yielding response, in the athermal quasi static limit, of a model amorphous material having inclusions in the form of randomly pinned particles. We show that, with increasing pinning concentration, the ... More Neutrinos from Protoneutron Stars: A Probe of Hot and Dense matterAug 03 1995Neutrino processes in dense matter play a key role in the dynamics, deleptonization and early cooling of hot protoneutron stars formed in the gravitational collapse of massive stars. Here we calculate neutrino mean free paths from neutrino-hyperon interactions ... More Hierarchical Role-Based Access Control with Homomorphic Encryption for Database as a ServiceMar 02 2016Database as a service provides services for accessing and managing customers data which provides ease of access, and the cost is less for these services. There is a possibility that the DBaaS service provider may not be trusted, and data may be stored ... More Mining Interesting Trivia for Entities from WikipediaOct 11 2015Trivia is any fact about an entity, which is interesting due to any of the following characteristics - unusualness, uniqueness, unexpectedness or weirdness. Such interesting facts are provided in 'Did You Know?' section at many places. Although trivia ... More Reconstructing Self Organizing Maps as Spider Graphs for better visual interpretation of large unstructured datasetsDec 24 2012Self-Organizing Maps (SOM) are popular unsupervised artificial neural network used to reduce dimensions and visualize data. Visual interpretation from Self-Organizing Maps (SOM) has been limited due to grid approach of data representation, which makes ... More Neutron stars as probes of extreme energy density matterApr 07 2014Neutron stars have long been regarded as extra-terrestrial laboratories from which we can learn about extreme energy density matter at low temperatures. In this article, I highlight some of the recent advances made in astrophysical observations and related ... More Quark matter and the astrophysics of neutron starsApr 02 2007Some of the means through which the possible presence of nearly deconfined quarks in neutron stars can be detected by astrophysical observations of neutron stars from their birth to old age are highlighted. Neutron Stars and the EOSJul 01 2013Implications of recently well-measured neutron star masses, particularly near and above 2 solar masses, for the equation of state (EOS) of neutron star matter are highlighted. Model-independent upper limits to thermodynamic properties in neutron stars, ... More Measures of Fault Tolerance in Distributed Simulated AnnealingDec 13 2012Dec 30 2012In this paper, we examine the different measures of Fault Tolerance in a Distributed Simulated Annealing process. Optimization by Simulated Annealing on a distributed system is prone to various sources of failure. We analyse simulated annealing algorithm, ... More Transformation formulas in Quantum CohomologyOct 03 2002The article discusses an action of the center of G on the quantum cohomology of G/P's constructed geometrically. It is shown how to recover Bertram's Quantum Schubert Calculus from this action, and also a refinement of a formula of Fulton and Woodward ... More Extremal rays in the Hermitian eigenvalue problemMay 30 2017Nov 16 2017The Hermitian eigenvalue problem asks for the possible eigenvalues of a sum of $n\times n$ Hermitian matrices, given the eigenvalues of the summands. The regular faces of the cones $\Gamma_n(s)$ controlling this problem have been characterized in terms ... More Comparative analysis of ADTCP and M-ADTCP: Congestion Control Techniques for improving TCP performance over Ad-hoc NetworksSep 12 2012Identifying the occurrence of congestion in a Mobile Ad-hoc Network (MANET) is a major task. 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More Universal dynamics of dilute and semidilute solutions of flexible linear polymersJan 21 2019Mar 01 2019Experimental observations and computer simulations have recently revealed universal aspects of polymer solution behaviour that were previously unknown. This progress has been made possible due to developments in experimental methodologies and simulation ... More Derivations of the Lie algebra of strictly block upper triangular matricesMar 25 2016Let ${\mathcal N}$ be the Lie algebra of all $n\times n$ strictly block upper triangular matrices over a field ${\mathbb F}$ relative to a given partition. In this paper, we give an explicit description of all derivations of ${\mathcal N}$. Flying in Two DimensionsOct 16 2011Diversity and specialization of behavior in insects is unmatched. Insects hop, walk, run, jump, row, swim, glide and fly to propel themselves in a variety of environments. We have uncovered an unusual mode of propulsion of aerodynamic flight in two dimensions ... 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We show, by constructing ... More On finite generation of the section ring of the determinant of cohomology line bundleJun 28 2016Jul 26 2016For $C$ a stable curve of arithmetic genus $g\ge 2$, and $\mathcal{D}$ the determinant of cohomology line bundle on $\operatorname{Bun}_{\operatorname{SL}(r)}(C)$, we show the section ring for the pair $(\operatorname{Bun}_{\operatorname{SL}(r)}(C), \mathcal{D})$ ... More A novel quantum grid search algorithm and its applicationMar 18 2019Jun 03 2019In this paper we present a novel quantum algorithm, namely the quantum grid search algorithm, to solve a special search problem. Suppose $k$ non-empty buckets are given, such that each bucket contains some marked and some unmarked items. In one trial ... More Unparticle physics at hadron collider via dilepton productionMay 31 2007Oct 24 2007The scale invariant unparticle physics recently proposed by Georgi could manifest at low energies as non integral number d_U of invisible particles. 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More Baryonic contributions to e+e- yields in a hydrodynamic model of Pb+Au collisions at the SPSJul 16 1999We analyze e+e- yields from matter containing baryons in addition to mesons using a hydrodynamic approach to describe Pb+Au collisions at 158 A GeV/c. We use two distinctly different e+e- production rates to provide contrast. Although the presence of ... More Turning Bitcoins into the Best-coinsDec 23 2014In this paper we discuss Bitcoin, the leader among the existing cryptocurrencies, to analyse its trends, success factors, current challenges and probable solutions to make it even better. In the introduction section, we discuss the history and working ... More Neutral Triple Vector Boson Production in Randall-Sundrum Model at the LHCJul 31 2015Nov 04 2015In this paper, triple neutral electroweak gauge boson production processes, viz. \gamma\gamma\gamma, \gamma\gamma Z, \gamma ZZ and ZZZ productions merged to 1-jet have been studied at the leading order in QCD in the context of Randall-Sundrum model at ... More Quantum Alternation: Prospects and ProblemsNov 05 2015We propose a notion of quantum control in a quantum programming language which permits the superposition of finitely many quantum operations without performing a measurement. This notion takes the form of a conditional construct similar to the IF statement ... 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More The Pervasive Role of the Nuclear Symmetry Energy in the Structure and Evolution of Neutron StarsDec 10 2008The multifaceted role of the density dependent nuclear symmetry energy in the nuclear astrophysics involving neutron stars is highlighted. Efforts toward a model independent determination of the dense matter equation state through a deconstruction of ... More GCD Computation of n IntegersJul 25 2014Greatest Common Divisor (GCD) computation is one of the most important operation of algorithmic number theory. In this paper we present the algorithms for GCD computation of $n$ integers. We extend the Euclid's algorithm and binary GCD algorithm to compute ... More On the existence and characterization of the maxent distribution under general moment inequality constraintsJun 05 2005A broad set of sufficient conditions that guarantees the existence of the maximum entropy (maxent) distribution consistent with specified bounds on certain generalized moments is derived. Most results in the literature are either focused on the minimum ... More Simultaneous polarization squeezing in polarized N photon state and diminution on a squeezing operationMar 28 2016We study polarization squeezing of a pure photon number state which is obviously polarized but the mere change in the basis of polarization leads to simultaneous polarization squeezing in all the components of Stokes operator vector except those falling ... More Alteration in Non-Classicality of Light on Passing Through a Linear Polarization Beam SplitterMar 28 2016Apr 22 2016We observe the polarization squeezing in the mixture of a two mode squeezed vacuum and a simple coherent light through a linear polarization beam splitter. Squeezed vacuum not being squeezed in polarization, generates polarization squeezed light when ... More Interaction Strictly Improves the Wyner-Ziv Rate-distortion functionJan 15 2010Jun 01 2010In 1985 Kaspi provided a single-letter characterization of the sum-rate-distortion function for a two-way lossy source coding problem in which two terminals send multiple messages back and forth with the goal of reproducing each other's sources. Yet, ... More An Auxiliary 'Differential Measure' for $SU(3)$May 27 1996May 29 1996A 'differential measure' is used to cast our calculus for the group $SU(3)$ into a form similar to Schwinger's boson operator calculus for the group $SU(2)$. It is then applied to compute (i) the inner product between the basis states and (ii) an algebraic ... More Secret Key and Private Key Constructions for Simple Multiterminal Source ModelsNov 13 2005This work is motivated by recent results of Csiszar and Narayan (IEEE Trans. on Inform. Theory, Dec. 2004), which highlight innate connections between secrecy generation by multiple terminals and multiterminal Slepian-Wolf near-lossless data compression ... More Secret Key and Private Key Constructions for Simple Multiterminal Source ModelsAug 12 2010We propose an approach for constructing secret and private keys based on the long-known Slepian-Wolf code, due to Wyner, for correlated sources connected by a virtual additive noise channel. Our work is motivated by results of Csisz\'ar and Narayan which ... More Polarised and Unpolarised Charmonium Production at Higher Orders in vAug 28 1997We study the unpolarised and polarised hadro-production of charmonium in non-relativistic QCD (NRQCD) at low transverse momentum, including sufficiently higher orders in the relative velocity, $v$, so as to study the ratio of $\chi_{c1}$ and $\chi_{c2}$ ... More Adjacency Matrix and Energy of the Line Graph of $Γ(\mathbb{Z}_n)$Jun 23 2018Let $\Gamma(\mathbb{Z}_n)$ be the zero divisor graph of the commutative ring $\mathbb{Z}_n$ and $L(\Gamma(\mathbb{Z}_n))$ be the line graph of $\Gamma(\mathbb{Z}_n)$. In this paper, we discuss about the neighborhood of a vertex, the neighborhood number ... More Variation of Weyl modules in $p$-adic familiesDec 04 2015Jul 26 2016Given a Weil-Deligne representation with coefficients in a domain, we prove the rigidity of the structures of the Frobenius-semisimplifications of the Weyl modules associated to its pure specializations. Moreover, we show that the structures of the Frobenius-semisimplifications ... More Bulk Viscosity of Interacting HadronsJun 30 2009Sep 16 2009We show that first approximations to the bulk viscosity $\eta_v$ are expressible in terms of factors that depend on the sound speed $v_s$, the enthalpy, and the interaction (elastic and inelastic) cross section. The explicit dependence of $\eta_v$ on ... More Adaptive Scheduling in Real-Time Systems Through Period AdjustmentDec 14 2012Real time system technology traditionally developed for safety critical systems, has now been extended to support multimedia systems and virtual reality. A large number of real-time application, related to multimedia and adaptive control system, require ... More Proceedings 9th Workshop on Quantum Physics and LogicJul 29 2014This volume contains the proceedings of the ninth workshop on Quantum Physics and Logic (QPL2012) which took place in Brussels from the 10th to the 12th of October 2012. QPL2012 brought together researchers working on mathematical foundations of quantum ... More Valuation Networks and Conditional IndependenceMar 06 2013Valuation networks have been proposed as graphical representations of valuation-based systems (VBSs). 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More The multiplicative eigenvalue problem and deformed quantum cohomologyOct 11 2013Nov 01 2013We construct deformations of the small quantum cohomology rings of homogeneous spaces G/P, and obtain an irredundant set of inequalities determining the multiplicative eigenvalue problem for the compact form K of G. Trajectory optimization using quantum computingJan 14 2019We present a framework wherein the trajectory optimization problem (or a problem involving calculus of variations) is formulated as a search problem in a discrete space. A distinctive feature of our work is the treatment of discretization of the optimization ... More e+e- yields in Pb+Au collisions at 158 AGeV/c: Assessment of baryonic contributionsDec 10 1998Jan 27 1999Using a hydrodynamic approach to describe Pb+Au collisions at 158 AGeV/c, we analyze e+e- yields from matter containing baryons in addition to mesons. We employ e+e- production rates from two independent calculations, which differ both in their input ... 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https://www.gradesaver.com/textbooks/math/algebra/algebra-2-1st-edition/chapter-2-linear-equations-and-functions-2-8-graph-linear-inequalities-in-two-variables-2-8-exercises-skill-practice-page-136/32	## Algebra 2 (1st Edition) Plugging in the given values into the inequality: $0.25(\frac{4}{3})-0=\frac{1}{3}\leq1$, thus $(\frac{4}{3},0)$ is not a solution. $0.25(\frac{2}{3})-(-4)=\frac{25}{6}\gt$, thus $(\frac{2}{3},-4)$ is a solution.	{"extraction_info": {"found_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": 0.9997557997703552, "perplexity": 325.1977424117181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1664030337415.12/warc/CC-MAIN-20221003101805-20221003131805-00322.warc.gz"}
http://slideplayer.com/slide/714937/	Cooperative Transmit Power Estimation under Wireless Fading Murtaza Zafer (IBM US), Bongjun Ko (IBM US), Ivan W. Ho (Imperial College, UK) and Chatschik. Presentation on theme: "Cooperative Transmit Power Estimation under Wireless Fading Murtaza Zafer (IBM US), Bongjun Ko (IBM US), Ivan W. Ho (Imperial College, UK) and Chatschik."— Presentation transcript: Cooperative Transmit Power Estimation under Wireless Fading Murtaza Zafer (IBM US), Bongjun Ko (IBM US), Ivan W. Ho (Imperial College, UK) and Chatschik Bisdikian (IBM US) Problem Synopsis Node T is a wireless transmitter with unknown Tx power P, and unknown location (x,y) Nodes {m 1,…, m N } are monitors that measure received power {p i } Goal – given {p i } and {(x i,y i )} (monitor locations), estimate unknown P (and also unknown location (x,y)) m2m2 P (x,y) m3m3 mNmN p N (x N,y N ) p 3 (x 3,y 3 ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) m1m1 T Problem Synopsis Sensor Networks – Event detection – {m 1,…, m N } are sensors, and T is the source point of an event – Goal – detect important events, eg: bomb explosion, based on measured power Wireless Ad-hoc Networks – physical layer monitoring – {m 1,…, m N } monitor a wireless network – Goal – detect maximum transmit power violation; i.e. detect misbehaving/mis- configured nodes, signal jamming m2m2 P (x,y) m3m3 mNmN p N (x N,y N ) p 3 (x 3,y 3 ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) m1m1 T Applications Blind estimation – no prior knowledge (statistical or otherwise) of the location or transmission power of T Talk Overview Power propagation model – Lognormal fading Deterministic Case – geometrical insights Single/two monitor scenario Multiple monitor scenario Stochastic Case Maximum Likelihood (ML) estimate Asymptotic optimality of ML estimate Numerical Results Conclusion Power Propagation model Lognormal fading P i = received power at monitor i d i = distance between the transmitter and monitor i α = attenuation factor, (α > 1) k = normalizing constant H i = lognormal random variable W i – unknown to the monitor – represents the aggregated effect of randomness in the environment; eg: multi-path fading didi PiPi T mimi P Deterministic Case Power propagation model: T 1 Monitor 1 P P1P1 d1d1 best estimate of transmit power: P* P 1 Single monitor measurement (no fading/random noise in power measurements) Deterministic Case Monitor 2 Note: d 1, d 2 are unknown Monitor 1 P P1P1 P2P2 d 12 d1d1 d2d2 2 T 1 Simple Cooperation: P* max(P 1, P 2 ) Q: Can we do better? Locus of T, Two monitor scenario Eqn (1) Eqn (2) Equation of a circle Deterministic Case Two monitor scenario P achieves lower bound, 2 1 T (x 1, y 1 )(x 2, y 2 ) P1P1 P2P2 T T T T T (x, y) x θ where, center of circle Deterministic Case Multiple monitor scenario With multiple monitors – diversity in measurements System of equations with unknowns (x,y,P) We should be able to solve these equations to obtain exact P ? Answer: Yes and No !! Deterministic Case 1 2 (x r, y r ) d r,1 d r,2 T (x, y) 3 4 d1d1 d2d2 Theorem: There is a unique solution (P, x, y) except when the monitors are placed on an arc of a circle or a straight line that does not pass through the actual transmitter location. Proof: A location (x, y) is a solution if and only if it satisfies d 1 /d 2 =c 1, …, d N-1 /d N = c N-1 The actual location (x r, y r ) is one solution; thus d r,1 /d r,2 =c 1, …, d r,N-1 /d r,N = c N-1 There exists another solution at (x, y) if and only if, d r,1 /d r,2 = d 1 /d 2, …; equivalently, T Deterministic Case 1 2 (x r, y r ) d r,1 d r,2 T (x, y) 3 4 d1d1 d2d2 Observation: Without transmit power information, and if monitors lie on an arc of a circle, even with infinite monitors and no fading, the transmission power (and transmitter location) cannot be uniquely determined. T Theorem: There is a unique solution (P, x, y) except when the monitors are placed on an arc of a circle or a straight line that does not pass through the actual transmitter location. Deterministic Case Multiple monitor scenario 12 Corollary 1: Two monitors always has multiple solutions Deterministic Case Multiple monitor scenario 13 Corollary 1: Two monitors always has multiple solutions Counter-intuitive Insight: For any regular polygon placement of monitors the transmission power cannot be uniquely determined !! Corollary 2: Three monitors as a triangle always has multiple solutions 2 Conversely: For all non-circular placement of monitors, transmission power can be uniquely determined. Talk Overview Power propagation model – Lognormal fading Deterministic Case – geometrical insights Single/two monitor scenario Multiple monitor scenario Stochastic Case ML estimate Asymptotic optimality of ML estimate Numerical Results Conclusion Stochastic Case m1m1 P (x,y) m2m2 mNmN p N (x N,y N ) p 2 (x 2,y 2 ) p 1 (x 1,y 1 ) Let z i = ln(p i ), Let Z = ln(P), and ML estimate (Z,x,y) is the value that maximizes the joint probability density function The joint probability density function Maximum Likelihood Estimate T Power attenuation model Stochastic Case Theorem: The ML estimate for N monitor case is given as, (x,y) is the solution to the minimization above, where the objective function is sample variance of {ln(p i d i α )} distance between some location (x,y) and monitor i distance between estimated Tx. location (x,y) and monitor i P is proportional to the geometric mean of {p i (d* i ) α } Stochastic Case What happens when N increases ? more number of measurements of received power increase in the spatial diversity of measurements Does the transmission power estimate improve ? Answer: Yes !! ; Estimator is asymptotically optimal Stochastic Case Asymptotic optimality as N increases Random Monitor Placement N monitors placed i.i.d. randomly in a bounded region Г Each monitor makes an independent measurement of the received power Random placement is such that it is not a distribution over an arc of a circle Let P N * be the estimated transmit power using the results presented earlier Theorem: As N increases the estimated transmit power converges to the actual power P almost surely, Numerical Results Synthetic data set –N = 2 to 20 monitors placed uniformly at random in a disk of radius R = 40. –Received power is generated by i.i.d. lognormal fading model for each monitor. –Performance measured: averaged over estimation for 1000 transmitter locations. Empirical data set –Sensor network measurement data by N. Patwari. –Total 44 wireless devices; each device transmits at -37.47 dBm; received powers are measured between all pairs of devices –The data is statistically shown to fit well to the lognormal fading model = 2.3, and dB = 3.92. –Randomly chosen N=3,4,…,10 monitors out of 44 devices. Numerical Results Performance metric –The above metric measures the average mean-square dB error Estimators –MLE-Coop-fmin ML estimate with fminsearch in MATLAB for location estimation –MLE-Coop-grid ML estimation with location estimation by dividing region into grid points –MLE-ideal ML estimate by assuming that the transmitter location is magically known –MLE-Pair ML estimate is obtained by considering only monitor pairs Average taken over all the pair-wise estimates Numerical Results Synthetic data set Empirical data set (MLE-Coop-grid) Conclusion Blind estimation of transmission power – Studied estimators for deterministic and stochastic signal propagation – Utilized spatial diversity in measurements – Obtained asymptotically optimal ML estimate – Presented numerical results quantifying the performance Geometrical insights – Two-monitor estimation was equivalent to locating the transmitter on a certain unique circle – If monitors are placed on a arc of a circle, the transmission power cannot be determined with full accuracy (even with infinite monitors) Download ppt "Cooperative Transmit Power Estimation under Wireless Fading Murtaza Zafer (IBM US), Bongjun Ko (IBM US), Ivan W. Ho (Imperial College, UK) and Chatschik." 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http://www.waynetippetts.com/?tag=green-jacket	## Posts Tagged ‘Green Jacket’ ### Paris – Street Life Wednesday, November 19th, 2014 Tags: , , , , , , , , , , , , , , , Posted in Street Style \| No Comments » ### London – Tom Tuesday, June 17th, 2014 Tags: , , , , , , , , , , , , Posted in Street Style \| No Comments » ### London – Street Life Monday, May 27th, 2013 Tags: , , , , , , , , , , , , , , , , , Posted in Street Style \| No Comments » ### Paris – Green Lite Saturday, December 1st, 2012 Tags: , , , , , , , , , , , , , , , Posted in Street Style \| 1 Comment » ### London – LFW – Susie lau-layered Sunday, October 30th, 2011 Susie’s approach too autumn layering-with the two jackets, especially the tartan on top of the green, the Burgundy red shoes, and the paisley patterned shirt. What is there not to like! The dark blue skirt holds everything together nicely, with a pop of colour provided by the orange belt. What a wonderfully creative combination of […] Tags: , , , , , , , , , , , , , , , , , Posted in Street Style \| 1 Comment » ### London – Pop-Up Friday, August 26th, 2011 Amiee was looking for a part time job before starting University in September, when I spotted her in Covent Garden. Loving the colour pop of Amiee’s Topshop mini dress. Tags: , , , , , , , , , , , , , , , , , , , , , Posted in Street Style \| 2 Comments »	{"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": 0.9412476420402527, "perplexity": 3919.0717318630896}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397744.64/warc/CC-MAIN-20160624154957-00121-ip-10-164-35-72.ec2.internal.warc.gz"}
https://crypto.stackexchange.com/questions/78372/understanding-a-security-proof	# Understanding a security proof I am reading the paper "RingCT 3.0 for Blockchain Confidential Transaction: Shorter Size and Stronger Security" (https://eprint.iacr.org/2019/508.pdf), an improvement to Ring Confidential Transactions. However, I am having trouble understanding the first security proof (page 17, the definition is in page 8). Can someone help me out, please? In particular I'm looking for some intution on the proof (with a reduced amount of mathematical notation).	{"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": 0.8426293134689331, "perplexity": 1296.2095759790075}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107874637.23/warc/CC-MAIN-20201021010156-20201021040156-00350.warc.gz"}
https://www.math.princeton.edu/events/motivic-galois-groups-and-periods-2013-02-12t213002	# Motivic Galois groups and periods - Joseph Ayoub , Zurich University Fine Hall 322 We explain how to associate a universal pro-algebraic group to the Betti realization functor from the triangulated category of motives over a subfield of $\mathbb{C}$. We then give a concrete description of the torsor of isomorphisms between the Betti realization and the de Rham realization. If time permits, some applications to periods will be sketched.	{"extraction_info": {"found_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": 0.8880350589752197, "perplexity": 595.7520789625894}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267868135.87/warc/CC-MAIN-20180625150537-20180625170537-00193.warc.gz"}
http://www.quantumphysicslady.org/glossary/quantum-mechanics/	Home » quantum mechanics # quantum mechanics « Back to Glossary Index In quantum mechanics, physicists study how tiny particles behave. These include atoms, the components of atoms like electrons and protons, and other tiny particles like photons of light. These tiny particles follow the laws of quantum mechanics and are called “quantum particles.” In the early 1900’s, physicists started experimenting with atoms and molecules. They discovered that atoms and their components follow different laws from those of ordinary objects like tables and chairs, so-called “macroscopic” objects. The mathematical laws governing the movements and forces among tables and chairs, “classical mechanics,” were first articulated by Isaac Newton in the 1600’s. For example, Newton developed the mathematical law of the amount of force exerted by a chair when thrown at a table: Force = Mass times Acceleration. However, when physicists of the early 1900’s used the equations of classical mechanics to predict the results of experiments with electrons and tiny particles of light, their predictions turned out to be erroneous. After considerable dismay, confusion, and fumbling around, they came up with quantum theory. By the 1920’s and 30’s, they were able to articulate the laws of quantum theory with mathematical equations. The mathematical version of this field then became known as “quantum mechanics” on analogy with “classical mechanics.” Quantum mechanics is also often called “quantum physics.” a term which doesn’t emphasize the mathematical aspect of the field. The term “quantum physics” can also be applied to the early years before the mathematical equations of quantum mechanics were developed. Quantum mechanics describes a different reality from the one that we experience daily. It’s a sublevel of reality that co-exists with ours and underlies it. I’ll call it “Quantumland.” In Quantumland, an electron is a “wave” that instantaneously turns into a particle the moment that we detect it. “Wave” is in quotes because it’s not a wave like a water wave or other waves that we encounter at the macroscopic level. The mathematical description of a quantum wave is quite different from that of a water wave. The mathematical equations that describe quantum particles don’t fit how we think mathematics should describe things in the physical universe. ## Quantumland In Quantumland, quantum particles behave in a number of odd ways: • As mentioned, electrons, photons, and other quantum particles can act like a kind of wave at one moment and a particle the next. This transition is demonstrated in the above video. • Quantum particles, in certain regards, appear to behave with true randomness. The properties of individual particles, such as their position when detected, do not appear to be determined by causal events. However, the probability that a collection of particles will manifest particular properties can be predicted. For example, the percentage of particles that will be detected in particular positions can be predicted. • Some of the properties of quantum particles, such as momentum, are not defined until the particles interact with some other part of the physical universe. For example, the direction of travel of a photon of light appears to be many directions at the same time, as if smelling all the possibilities. When detected, that is, absorbed by an electron, the photon will be found in a definite position, one in accord with the straight-line travel. • Quantum particles appear to cause effects prior in time (retro-causality). • Quantum particles such as electrons can disappear from one side of a barrier and instantaneously appear on the other. This, despite having insufficient energy for this maneuver and without actually passing through the barrier. This is “quantum tunneling.” • Particles which have interacted together are “entangled.” The behaviors of entangled particles are correlated despite any amount of distance separating them. For example, two entangled electrons sitting halfway across the universe from each other will always coordinate their spins. If one begins to spin clockwise, the other will begin to spin counterclockwise. According to the Theory of Special Relativity, which prohibits faster than light communication, this cannot be due to signaling each other. It appears to be due to a simultaneity of behavior which physicists can describe but not explain. • Particles, known as “virtual particles,” can appear with no material source nor energy source and disappear almost instantaneously. These particles might be thought to violate the Law of Conservation of Energy. But the violation occurs so briefly that the mathematics of quantum mechanics, specifically, the Heisenberg Uncertainty Principle, is adequate to describe this behavior. ## Interpretations of Quantum Mechanics Most physicists agree on the key mathematical equations of quantum mechanics. But physicists have not reached agreement on the implications of quantum mechanics for the nature of quantum particles and the nature of reality. Physicists have developed about 15 different interpretations which assign meaning to quantum physics, describing both how quantum particles behave and the nature of reality. Quantumland is a term coined by Dr. Ruth E. Kastner, one of the developers of the Transactional Interpretation of quantum mechanics. Ruth E. Kastner, Understanding Our Unseen Reality, Solving Quantum Riddles; Imperial College Press, 2015, London. *The mathematical description of a quantum wave is called a “wave function.” A wave function can include imaginary numbers. Imaginary numbers are based on the square root of negative 1. You are right if you’re wondering what number squared (multiplied times itself) equals negative 1. There is no such number. The square root of negative 1 does not exist and is called an “imaginary number.” Nevertheless, imaginary numbers are needed when describing quantum waves. This is only one of the issues that besets our ability to conceptualize the mathematics of quantum waves and to consider these waves real. Another issue is that the quantum wave describes probabilities as to how the particle will behave rather than describing definite properties. This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"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": 0.837562620639801, "perplexity": 536.9918236853898}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1634323585305.53/warc/CC-MAIN-20211020090145-20211020120145-00632.warc.gz"}
https://www.physicsforums.com/threads/elevator-problem-help.47667/	# Homework Help: Elevator problem - help 1. Oct 13, 2004 ### vtech elevator problem - urgent help!!! The cable of the 1800 kg elevator snaps when the elevator is at rest at the first floor, where the cab bottom is a distance d = 3.7 m above a cushioning spring whose spring constant is k = 0.15 MN/m. A safety device clamps the elevator against guide rails so that a constant frictional force of 4.4 kN opposes the motion of the elevator. (image attached - i don't know how to make it embedded, sorry) (a) Find the speed of the elevator just before it hits the spring._____ m/s (b) Find the maximum distance x that the spring is compressed.______ m (c) Find the distance that the elevator will bounce back up the shaft._____m (d) Using conservation of energy, find the approximate total distance that the elevator will move before coming to rest._____m -------------------- alrite let's see.... a) i've gotten the first answer by dKinetic + dPotGrav (or Ug) + E thermal (from friction) = 0 from conservation of energy.... solve for V final from the kinetic part... 7.37ms-1 b) i believe it should be -1/2mv^2 - mgS + 1/2kS^2 + F(of friction)S = 0 but i'm not sure if I'm getting the signs wrong... should my answer be negative because its compressing or not?.... not getting the right answer... help please!!!! c) now i need my result from b) as x and is it 1/2kx^2 = mg(d up) + Friction(s +d up) solve for (d up).... Note: not sure if should just use answer from b) times the friction force or is it friction (d up)... instead of what my equation says....????? d) to be honest i'm sorry but i have no clue to this one... and without b) and c)... i'm having trouble understanding it... i hope some1 can help me out???!!! thanx in advance!!! #### Attached Files: • ###### 08_62.gif File size: 8.7 KB Views: 195 2. Oct 13, 2004 ### Randomphyre Hmm, Im stuck on c) and d) too. but i can help you on b) you have it right but use the velocity you got in a) and if you have a TI-86 or higher just use solver to find distance from the quadratic equation you get. I used: .5kS^2=.5mv^2-(force of friction)S+mgS 3. Oct 13, 2004 ### vtech question? i solved the quadratic, but is it the negative s because its in compression or not?... i got either 0.90 or -0.72... i'm thinking its the positive... but i only have 1 try left... which one did u use? just wanna make sure!!! 4. Oct 13, 2004 ### Randomphyre It is .9 but your equations for C and D dont work (I did the same thing as you did in C and the answer was wrong) so try it yourself and if it is still wrong were gonna have to brainstorm. 5. Oct 13, 2004 ### vtech actually, ur right.... friction only acts during d up... my original equation had friction acting at (s + d up) it should be: 1/2kx^2 = mg(d up) + Friction(d up) 6. Oct 13, 2004 ### vtech other ? hey randomphyre, i was wondering... have u gotten around to ch8 #66 part (c)... any clues???? its the crates falling on the conveyor belt... energy supplied by motor??? 7. Oct 13, 2004 ### Randomphyre haha can get all the other parts except that. Its got me stumped too. 8. Oct 13, 2004 ### vtech did u get d) for the elevator??? Ugrav (as a function of S + dup) = Uspring (as a function of S) + F(friction) L solve for L.... i thought it made sense... but i didn't get the right answer!... wut do u think??	{"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": 0.8574414849281311, "perplexity": 2919.0475430543956}, "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/1539583511173.7/warc/CC-MAIN-20181017111301-20181017132801-00539.warc.gz"}
https://www.studypug.com/uk/uk-a-level-maths/trigonometry/30-60-90-special-right-triangles	# Solving expressions using 30-60-90 special right triangles ## What is a 30-60-90 triangle? One of the two special right triangles you’ll be facing in trigonometry is the 30-60-90 triangle. The other one is the 45 45 90 triangle. These triangles are special triangles because the ratio of their sides are known to us so we can make use of this information to help us in right triangle trigonometry problems. In the case of the 30-60-90 triangle, their side’s ratios are 1 : 2 : sqrt3. We’ll prove that this is true first so that you can more easily remember the triangle’s properties. A 30-60-90 triangle is actually half of an equilateral triangle. Triangle ABC shown here is an equilateral triangle. Since it's equilateral, each of its 3 angles are 60 degrees respectively. Its sides are also equal. If we draw a line AD down the middle to bisect angle A into two 30 degree angles, you can now see that the two new triangles inside our original triangle are 30-60-90 triangles. We know that BD is equalled to DC, which is also equalled to half of AB since AB was originally equalled to BC before it was split in half. So therefore, the ratio of BD : AB is 1 : 2. Since we have a hypotenuse due to the 90 degree angle that exists inside our triangle, we can use the pythagorean theorem to figure out the last side: AD. 1^2 + AD ^2 = 2^2 AD ^2 = 4 - 1 And there you have it! You now know how all the sides of the 30-60-90 triangles came about. Let’s move on to solving right triangles with our knowledge on the sides’ ratios. ## Solve for 30 60 90 triangle Here we have a 30-60-90 special right triangle, with the three interior angles of 30, 60, 90 degrees. We know that the length of each side in this triangle is in a fixed ratio. We can now use the ratio to solve the following problem. Question 1: If ?=30? , find the exact value of the following expression: sin 2? Solution: Since ? = 30?, sin 2? = sin 60?. We know that sin ? = oppositehypotenuse Since we are looking for the value of sin 60?, we are going to look at the 60? angle in the special triangle and find out the opposite side and hypotenuse of the angle. And we found that the opposite side = 3 and the hypotenuse = 2. Then, we put the numbers into sin 60? = oppositehypotenuse. This gives us the final answer of 32 Question 2: If ?=60? , find the exact value of the following expression: 4sin3?2 Solution: First, we put ?=60? into 4sin3?2. We get 4sin3(602)=4sin3(30) From the special triangle, we know that sin 30? = 12 Then, we can now solve the cubic root: 4(12)3=4(18) A good online resource to help you check your work when it comes to 30-60-90 triangles (or even 45 45 90 isosceles right triangle ones!) can be found here. You can also play along with this online scalable 30-60-90 triangle to see how the sides change while staying in the 1 : 2 : sqrt3 ratio. ### Solving expressions using 30-60-90 special right triangles #### Lessons $\sin 30^\circ = { 1 \over 2}$ $\cos 30^\circ = {{\sqrt 3} \over 2}$ $\tan 30^\circ = {1 \over \sqrt 3}$ $\sin 60^\circ= {{\sqrt3} \over 2}$ $\cos 60^\circ= {1 \over 2}$ $\tan 60^\circ = {{\sqrt 3} \over 1}$ • 1. What are special triangles? • 2. If $\theta = 30^\circ$, find the exact value of the following expressions. a) $\sin 2 \theta$ b) $\cos^2 2 \theta$ c) ${{\tan^2 \theta} \over 2}$ • 3. If $\theta = 60^\circ$, find the exact value of the following expressions. a) $4 \sin^3 {\theta \over 2}$ b) ${2 \over {{3 \cos^2}{ \theta \over 2}}}$ c) $3 \tan^3 {\theta \over 2}$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 32, "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": 0.9064271450042725, "perplexity": 496.2048593436429}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1518891814827.46/warc/CC-MAIN-20180223174348-20180223194348-00224.warc.gz"}
https://captaincalculator.com/math/percent/percentage-decrease-calculator/	# Percentage Decrease Calculator LAST UPDATE: April 8th, 2018 ## How to Calculate Percentage Decrease Percentage decrease is found by dividing the decrease by the starting number, then multiplying that result by 100%. Note: the percent decrease measures FROM the first value. A decrease from 50 of 25 is a change of 50% (25 is the difference between the two numbers, and 25 is 50% of 50)). If that decease were looked at the other way around, it would be a different percentage change (25 increase by 25 to 50 is an increase of 100%). ## Example Decrease from 50 by 25: Decrease from 75 by 25: Decrease from 0.0443 by 0.0001 ## What is a percentage? A percentage is a number expressed as a fraction of 100. If a number is 100% (100 percent), then it is a “whole” – the same as one. If a number is 50%, then it is a half – the same as 0.5 or 1/2. If a number is 400%, then it is 4 times, the same as 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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 4, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8755035400390625, "perplexity": 862.6723379492117}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1539583510749.55/warc/CC-MAIN-20181016113811-20181016135311-00070.warc.gz"}
https://www.physicsforums.com/threads/circumscribing-an-ellipse.79699/	# Circumscribing an ellipse 1. Jun 20, 2005 ### wisredz Let P(x,a) and Q(-x,a) be two points on the upper half of the ellipse $$\frac{x^2}{100}+\frac{(y-5)^2}{25}=1$$ centered at (0,5). A triangle RST is formed by using the tangent lines to the ellipse at Q and P. Show that the area of the triangle is $$A(x)=-f'(x)[x-\frac{f(x)}{f'(x)}]^2$$ where y=f(x) is the function representing the upper half of the ellipse. Last edited: Jun 20, 2005 2. Jun 20, 2005 ### StatusX Have you been able to figure out where the third line of the triangle intersects the ellipse? 3. Jun 20, 2005 ### wisredz I never thought of that but that point isn't even related to f(x). But the point is (0,0). I don't understand how to use that point. The triangle's R corner is on the y axis. The other two are on the x axis. 4. Jun 20, 2005 ### StatusX Ok, so just find its base and height. Remember that f'(x) is the slope of the line tangent to the ellipse, so you can use it to find where that line intersects the x and y axes. 5. Jun 20, 2005 ### wisredz Well, it all get's messed up because I do not know what to do when trying to find an equation for one of the edges. That is because I have f'(x) in terms of x and I have the point Q(x,a) and when I try it everyting gets messed up. What should I do now? I had already tried until this point but I always get lost right here... 6. Jun 20, 2005 ### StatusX a=f(x), and point Q is at (x,f(x)). Did you misunderstand this part? Just draw everything. The line passes through Q and has a slope of f'(x). You need the x and y intercepts to get the base and height of the triangle. 7. Jun 20, 2005 ### wisredz The point-slope equation for the tangent passing through the point $Q(x_0,f(x_0))$ would be $y-f(x_0)=f'(x)(x-x_0)$ right? But when I give x the value of 0 I get $x_0$ as the y intersection. Am I doing something wrong here? 8. Jun 20, 2005 ### StatusX The y intercept is the y-value when x=0, so plug in 0 for x and solve for y. 9. Jun 21, 2005 ### wisredz Yeah, I know that. But when I plug in x=0, f'(x)=0. So the right hand side of the point slope equation becomes 0. from here $y=f(x_0)$, which is quite impossible by the figure drawn in the book. Btw, I'll give f(x) and f'(x) in case that you may spot an error in the calculations. $$f(x)= \frac {(\sqrt(100-x^2)}{2}+5$$ $$f'(x)= \frac {-x}{2*\sqrt(100-x^2)}$$ I actually graphed these functions and everything seems to be alright... 10. Jun 21, 2005 ### StatusX that's f'(x0), right? 11. Jun 22, 2005 ### wisredz Thanks a lot, I did it now. My mistake was not using f'(x_0) but instead f'(x). Thanks a lot again...	{"extraction_info": {"found_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": 0.8676594495773315, "perplexity": 684.3489427064225}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218186841.66/warc/CC-MAIN-20170322212946-00045-ip-10-233-31-227.ec2.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/1102.0197/	# Device-independent witnesses of genuine multipartite entanglement Jean-Daniel Bancal, Nicolas Gisin, Yeong-Cherng Liang, Stefano Pironio Group of Applied Physics, University of Geneva, Switzerland Laboratoire d’Information Quantique, Université Libre de Bruxelles, Belgium June 23, 2022 ###### Abstract We consider the problem of determining whether genuine multipartite entanglement was produced in an experiment, without relying on a characterization of the systems observed or of the measurements performed. We present an -partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states, but which can be violated by -partite entangled states, such as GHZ states. In contrast to traditional entanglement witnesses, the violation of this inequality implies that the state is not biseparable independently of the Hilbert space dimension and of the measured operators. Violation of this inequality does not imply, however, genuine multipartite non-locality. We show more generically how the problem of identifying genuine tripartite entanglement in a device-independent way can be addressed through semidefinite programming. The generation of multipartite entanglement is a central objective in experimental quantum physics. For instance, entangled states of fourteen ions and six photons have recently been produced ion ; photon . In any such experiment, a typical question arises: How can we be sure that genuine -partite entanglement was present? A state is said to be genuinely -partite entangled if it is not biseparable, that is, if it cannot be prepared by mixing states that are separable with respect to some partition. Consider for instance the tripartite case: a state is said to be biseparable if it admits a decomposition ρbs=∑kρkAB⊗ρkC+∑kρkAC⊗ρkB+∑kρkBC⊗ρkA, (1) where the weight of each individual state in the mixture has been included in its normalization; a state that cannot be written as above is genuinely tripartite entangled. Determining whether genuine -partite entanglement was produced in an experiment represents a difficult problem that has attracted much attention recently (see e.g. review ; multent ). The usual approach consists of measuring a witness of genuine multipartite entanglement, or of doing the full tomography of the state followed by a direct analysis of the reconstructed density matrix. Such approaches, however, not only rely on the observed statistics to conclude about the presence of entanglement, but also require a detailed characterization of the systems observed and of the measurements performed. Consider for instance the following witness of genuine tripartite entanglement: M=X1X2X3−X1Y2Y3−Y1X2Y3−Y1Y2X3, (2) where and are the Pauli spin observables in the and direction for particle . For any biseparable three-qubit state addendum . Thus if we measure three spin- particles in the and direction and find an average value , we can conclude that the state exhibits genuine tripartite entanglement. Suppose, however, that the measurement carries a slight (possibly unnoticed) bias towards the direction. That is, instead of measuring , we actually measure . Then it is not difficult to see, all other measurements being ideal, that the biseparable state , where , yields which is strictly larger than 2 for any . Thus, unless we measure all particles exactly along the and directions, we can no longer conclude that observing implies genuine tripartite entanglement. Importantly, this is not a unique feature of the above witness, but rather all conventional witnesses are, to some extent, susceptible to such systematic errors that are seldom taken into account. Furthermore, tomography and usual entanglement witnesses typically assume that the dimension of the Hilbert space is known. For instance, in a typical experiment demonstrating, say, entanglement between four ions, we usually view each ion as a two-level system. But an ion is a complex object with many degrees of freedom (position, vibrational modes, internal energy levels, etc). How do we know, given the inevitable imperfections of the experiment, that it is justified to treat the relevant Hilbert space of each ion as two-dimensional and how does this simplification affects our conclusions about the entanglement present in the system lutk ? Even if it is justified to view each ion as a qubit, is entanglement between four systems really necessary to reproduce the measurement data, or could they be reproduced with fewer entangled systems if qutrits were manipulated instead? These remarks motivate the introduction of entanglement witnesses that are able to guarantee that a quantum system exhibits (-partite) entanglement, without relying on the types of measurements performed, the precision involved in their implementation, or on assumptions about the relevant Hilbert space dimension. We call such witnesses, device-independent entanglement witnesses (DIEW). This type of approach was already considered in Refs. seevinck ; diew . Note that other solutions to the above problems are possible, such as entanglement witnesses tolerating a certain misalignement in the measurement apparatuses uff or the characterization of realistic measurement apparatuses through squashing maps lutk . These types of more specific approaches, however, still require some partial characterization of the system and measurement apparatuses, which is not necessary when using DIEWs. Any DIEW is a Bell inequality (i.e. a witness of nonlocality). Indeed, the violation of a Bell inequality implies the presence of entanglement, and any measurement data that does not violate any Bell inequality can be reproduced using quantum states that are fully separable R.F.Werner:PRA:1989 . The violation of a Bell inequality is thus a necessary and sufficient condition for the detection of entanglement in a device-independent (DI) setting. This observation is the main insight behind DI quantum cryptography collective ; Pironio10 , where the presence of entanglement is the basis of security. The relation between DIEW for genuine -partite entanglement and witnesses of multipartite nonlocality is more subtle. While there exist Bell inequalities that detect genuine -partite nonlocality svet ; collins ; svetd , not every DIEW for -partite entanglement corresponds to such a Bell inequality. Consider for instance, the expression (2). If no assumptions are made on the type of systems observed and measurements performed, the inequality corresponds to Mermin’s Bell-type inequality mermin , i.e., a value necessarily reveal non-locality, hence entanglement. Moreover, a value guarantees genuine tripartite entanglement collins ; seevinck . The Mermin expression (2) can thus be used as a tripartite DIEW. Yet, it cannot be used as a Bell inequality for genuine tripartite non-locality, since a simple model involving communication between two parties only already achieves the algebraic maximum collins . The objectives of this paper are to formalize the concept of DIEW for genuine multipartite entanglement and initiate a systematic study that goes beyond the early examples given in seevinck ; diew . Following this line, we start by introducing the notion of quantum biseparable correlations. We then present a simple DIEW for -partite entanglement which is stronger for GHZ (Greenberger-Horne-Zeilinger) states than all the inequalities introduced in seevinck ; diew . In the case , we also provide a general method for determining whether given correlations reveal genuine tripartite entanglement and apply it to GHZ and W states. Apart from yielding practical criteria for the characterization of entanglement in a multipartite setting, our results also clarify the relation between device-independent multipartite entanglement and mulitpartite nonlocality. 1. Biseparable quantum correlations. For simplicity of exposition, let us consider an arbitrary tripartite system (the following discussion easily generalizes to the -party case). To characterize in a DI way its entanglement properties, we consider a Bell-type experiment: on each subsystem, one of possible measurements is performed, yielding one of possible outcomes. We adopt a black-box description of the experiment and represent the measurements on each of the three subsystems by classical labels (corresponding, e.g., to the values of macroscopic knobs on the measurement apparatuses) and denote the corresponding classical outcomes . The correlations obtained in the experiment are characterized by the joint probabilities of finding the triple of outcomes given the measurement settings . We say that the correlations are biseparable quantum correlations if they can be reproduced through local measurements on a biseparable state . That is, if there exist a biseparable quantum state (1) in some Hilbert space , measurement operators , , and (which without loss of generality we can take to be projections satisfying and ), such that (3) If given quantum correlations are not biseparable, they necessarily arise from measurements on a genuinely tripartite entangled state, and this conclusion is independent of any assumptions on the type of measurements performed or on the Hilbert space dimension. Equivalently, biseparable quantum correlations can be defined as those that can be written in the form P(abc\|xyz)=∑kPkQ(ab\|xy)PkQ(c\|z)+∑kPkQ(ac\|xz)PkQ(b\|y)+∑kPkQ(bc\|yz)PkQ(a\|x), (4) where and correspond, respectively, to arbitrary two-party and one-party quantum correlations, i.e., they are of the form and for some unormalized quantum states , and measurement operators , , (and similarly for the other terms in (4)). Note that here the measurement operators for different bi-partitions do not need to be the same (though this can always be achieved as shown in Appendix D). That any correlations of the form (3) is of the form (4) is immediate using the definition (1) of biseparable states. Conversely, it is easy to see that any correlations of the form (4) is of the form (3), see Appendix A. Let denote the set of tripartite quantum correlations and the set of biseparable quantum correlations. From (4), it is clear that is convex and that its extremal points are of the form where is an extremal point of the set of bipartite quantum correlations and an extremal point of the set of single-party correlations (the extreme points of are actually classical, deterministic points). Since the set is convex, it can be entirely characterized by linear inequalities. Those linear inequalities separating from correspond to DIEWs for genuine tripartite entanglement. Since has an infinite number of extremal points, there exist an infinite number of such inequalities. Note that the set of local correlations is contained in . This implies that any DIEW for genuine tripartite entanglement is a Bell inequality (though not necessarily a tight Bell inequality). Note also that the decomposition (4) corresponds to a Svetlichny-type decomposition svet where all bipartite factors are restricted to be quantum, whereas less restrictive constraints (or even none in Svetlichny original definition ) are imposed on these bipartite terms in the definitions of multipartite non-locality svet . It follows that the set of genuinely bipartite non-local correlations is larger than the set of biseparable quantum correlations as illustrated in Fig. 1. Thus while any Bell inequality detecting genuine tripartite non-locality is a DIEW for genuine tripartite entanglement, the converse is not necessarily true. All these observations extend to the -party case. 2. A DIEW for -partite entanglement. We now present a DIEW for parties, where each party performs a measurement and obtains an outcome . We denote the correlator associated to the measurement settings , i.e. the expectation value , where denotes an -tuple of outcomes. Let be the sum of correlators for which the measurement settings of the parties sum up to . Let be a function such that and taking successively the values on the integers . Then the inequality In=3n∑k=nfk−nEkn≤2×3n−3/2 (5) is satisfied by all biseparable quantum correlations, and is thus a DIEW for genuine -partite entanglement. The proof of this statement is based on the decomposition (4) and is given in Appendix B. The Svetlichny bound associated to the expression , on the other hand, is easily found to be (see Appendix Bremark . We now illustrate how this DIEW can be used to detect genuine multipartite entanglement. For this, let us consider the noisy GHZ state characterized by the visibility . Carrying out the measurements In the case , the DIEW (5) takes the form . It therefore involves only 18 expectation values, compared to for a full tomography of a three-qubit system. Let us stress, however, that contrary to usual entanglement witnesses is not restricted to two-dimensional Hilbert spaces, even though it uses observables with binary outcomes. For instance, if all parties perform the measurements with on the three-qutrit state , then , showing that the state is genuinely tripartite-entangled. 3. General characterization of biseparable quantum correlations in the case . Though the DIEW (5) seems particularly well adapted to GHZ states, we cannot expect a single nor a finite set of DIEW to completely characterize the biseparable region, as illustrated in Fig. 1. It is thus desirable to derive a general method to decide whether arbitrary correlations are biseparable. Here we show how the semi-definite programming (SDP) techniques introduced in Navascues07 ; hierarchy can be used to certify that the correlations observed in an experiment are genuinely tripartite entangled. Our approach is based on the observation that the tensor product separation at the level of states in the definition (1) of biseparable states can be replaced by a commutation relation at the level of operators. Specifically, let denotes the three possible partitions of the parties into two groups. Then, are biseparable quantum correlations if and only if there exist three arbitrary (not necessarily biseparable) states and three sets of measurement operators such that P(abc\|xyz)=∑str[Msa\|x⊗Msb\|y⊗Msc\|zρs], (6) where measurement operators corresponding to an isolated party commute, i.e., , and similarly for the other partitions. The equivalence between (3) and (6) is established in Appendix D. The problem of determining whether given correlations are biseparable thus amounts to finding a set of operators satisfying a finite number of algebraic relations (the projection defining relations of the type , and the commutation relations mentioned above) such that (6) holds. Such a problem is a typical instance of the SDP approach introduced in Navascues07 ; hierarchy (see details in Appendix D). Specifically, it follows from the results of hierarchy that one can define an infinite hierarchy of criteria that are necessarily satisfied by any correlations of the form (6) and which can be tested using SDP. If given correlations do not satisfy one of these criteria, we can conclude that they reveal genuinely tripartite entanglement. Further, it is possible in this case to derive an associated DIEW from the solution of the dual SDP. Modulo a technical assumption, it can be shown that the hierarchy of SDP criteria is complete, that is, if given correlations are not biseparable this will necessarily show-up at some finite step in the hierarchy. 4. Application to GHZ and W states. Using finite levels of this hierarchy and optimizing over the possible measurements, we investigated the minimal visibilities above which the GHZ state and the W state exhibit correlations that are not biseparable (and thus reveal genuine tripartite entanglement) in the case of two and three measurement settings per party. Our results are summarized in Table 1. For GHZ states, the reported visibility for three measurements per party correspond to the threshold required to violate the DIEW (5), suggesting that this DIEW is optimal in this case. In the case of two measurements per party, we could not lower the visibility below the threshold , which corresponds to the visibility required to violate the DIEW based on Mermin expression (2) and the DIEWs introduced in seevinck ; diew . Note, however, that for the GHZ state violates Svetlichny’s inequality svet and thus exhibits genuine tripartite nonlocality. Thus for GHZ state the DIEWs introduced in seevinck ; diew do not improve over what can already be concluded using the standard notion of tripartite non-locality. On the other hand, our numerical explorations suggest that the visibilities for GHZ states with three measurements and for W states with two measurements cannot be attained using the notion of genuine tripartite non-locality, illustrating the interest of the weaker notion of DIEW. Dicussion. To conclude, we comment on some possible directions for future research. First of all, note that by identifying the measurement settings “” with and “” with , the two-setting DIEW based on Mermin expression (2) can be written as , which is of the same general form as the three-setting DIEW (5). This suggests that the DIEWs based on (2) and (5) actually form part of a larger family of -settings DIEWs. This question deserves further investigation. A second problem is to derive simple DIEWs that are adapted to W states and that can in particular reproduce the threshold visibilities obtained in Table 1. Finally, we have shown a practical method to characterize three-partite biseparable correlations using SDP. It would be interesting to understand how to generalize these results to the -partite case. A possibility would be to combine the approach of Navascues07 ; hierarchy with the symmetric extensions introduced in terhal . This question will be investigated elsewhere. #### Acknowledgments. This work was supported by the Swiss NCCRs QP and QSIT, the European ERC-AG QORE, and the Brussels-Capital region through a BB2B grant. ## Appendix A Biseparable quantum correlations Here, we show that the definitions (3) and (4) of biseparable correlations are equivalent. For the sake of generality, we prove this equivalence in the -partite scenario. Let denote a subset of containing from one to elements, and let denote the complementary subset. The pair thus represent a partition of the parties into two non-empty groups. Let be an (unormalized) -partite state that is product across the partition . An -partite state is then biseparable if it can be written as a mixture ρbs=∑t∑ktρktt⊗ρktt′. (7) A state that cannot be written in the above form is genuinely -partite entangled. Let be an -partite probability distribution characterizing a Bell experiment with measurement settings and measurement outcomes . We say that the correlations described by are biseparable quantum correlations iff P(¯a\|¯x)=tr[Ma1\|x1⊗…⊗Man\|xnρbs] (8) for some biseparable state and measurement operators . Equivalently, as we will see, biseparable quantum correlations can be defined as those that can be written in the form P(¯a\|¯x)=∑t∑ktPktQ(¯at\|¯xt)PktQ(¯at′\|¯xt′), (9) where we write and for the outcome and measurement settings of the parties belonging to the subset and where are arbitrary (unnormalized) quantum correlations for the parties in . That any correlations of the form (8) is of the form (9) is immediate using the definition (7) of biseparable states. Conversely, (9) can be rewritten in the form P(¯a\|¯x)=∑t∑kttr[⨂i∈nMktai\|xiρktt⊗ρktt′]. (10) This gives a representation that is almost of the form (8), except that a different set of measurement operators corresponds to each term in the decomposition (7) of the biseparable state . This can be fixed by introducing local ancillas on each system acting as labels that indicate which term in the decomposition is considered: defining , we finally get , and thus any correlations of the form (10) can be written as in Eq. (8). Let denote the set of quantum correlations defined for the parties belonging to and let denote the set of biseparable quantum correlations, which is a subset of the -partite quantum correlations. From (9), it is clear that is convex and that its extreme points are of the form where is an extremal point of the set . ## Appendix B DIEWs for genuine n-partite entanglement Here we derive the maximum value that the expression , see Eq. (5), can take for measurements made on a biseparable quantum state (the biseparable bound), and for Svetlichny models (the Svetlichny bound). From (9), it follows that the biseparable bound is obtained by maximizing over all correlations of the form P(¯a\|¯x)=∑t∑ktPktQ(¯at\|¯xt)PktQ(¯at′\|¯xt′). (11) In a Svetlichny model, on the other hand, one considers correlations of the form P(¯a\|¯x)=∑t∑ktPkt(¯at\|¯xt)Pkt(¯at′\|¯xt′), (12) i.e. no constraints (apart from positivity and normalization) are imposed on the joint terms . But more refined models à la Svetlichny can be introduced, see e.g. pirbancsca , the more constraining one being the one where the joint terms are assumed to be no-signalling. We will compute the bound on assuming these no-signalling constraints and see that even in this case there exists a gap between the quantum biseparable and Svetlichny bounds (note also that the no-signalling bound that we will derive actually coincides with the more general unconstrained Svetlichny bound because is an inequality involving only full -partite correlators). We will now prove the biseparable bound given in Eq. (5) for using induction on the number of parties . Let us start with the first step of the induction, which consists of showing that the inequality holds for . By linearity of in the probabilities, convexity of the decomposition (4) for tripartite biseparable quantum correlations and the fact that is invariant under any permutation of the parties, it is sufficient to prove the bound for correlations of the form . Moreover, it is sufficient to consider the case where is extremal, i.e., one where every is determined unambiguously as a function of . To this end, let us label the eight distinct deterministic assignments of outcome for given input by , where the deterministic (quantum) probability distributions are such that if and otherwise. Substituting these eight possible strategies into [see (5)] gives eight different bipartite Bell expressions for the system : I3,γ=6∑k=2gγ(k)Ek2 (13) where . Determining the biseparable bound of therefore amounts to finding the maximum of the above Bell expressions over arbitrary bipartite quantum correlations , i.e., finding the corresponding Tsirelson bounds. It can be easily verified that the eight expressions are either vanishing or equivalent, under relabelling of inputs and outputs, to two times the 3-input chained Bell inequality Braunstein90 . The biseparable bound on is thus equal to the two times the Tsirelson bound of the 3-input chained Bell inequality Wehner06 , i.e., , which is in agreement with Eq. (5). Next, we need to show that whenever inequality (5) holds for parties, it must also hold for parties. By linearity of in the probabilities and convexity of the decomposition (11), it is sufficient to compute the bound over all product correlations of the form . Moreover, since is symmetric under any permutation of parties, it is sufficient to consider the biseparations , where . With obvious notation, we write the corresponding correlations as . Bayes’ rule and the fact that quantum correlations satisfy the no-signaling principle allow us to write the quantum correlations for parties to as . With straightforward algebra, it can be seen that the quantity for this type of distribution can be rewritten as Ekn+1=3∑xn+1=11∑an+1=−1an+1PQ(an+1\|xn+1)× Ek−xn+1n(an+1,xn+1) (14) where the -partite quantity depends on the values of and since it is evaluated on the distribution . Inserting this expression in the definition , we obtain In+1=3∑xn+1=11∑an+1=−1an+1PQ(an+1\|xn+1)Ixn+1n(an+1,xn+1) (15) where we have defined (note that is different from zero only if ) and write to remind that is evaluated on the distribution which depends on and . Using the fact that and that the probabilities are bounded between and , it follows from (15) that In+1≤3∑xn+1=1\|Ixn+1n(an+1,xn+1)\|. (16) Let denote the maximal value of the quantity taken by any biseparable correlations. Since the distribution is biseparable, we clearly have that , independently of the values of and . We thus find In+1≤3∑j=1max±Ijn. (17) An important point to note now is that for all are equivalent to , i.e., one can obtain the expression for by starting from any of and applying a different labeling for the inputs and/or outputs. To see that this is the case, we first note from the definition of that . In other words, we can obtain by starting from and applying the mapping to each of the . Clearly, the same argument also demonstrates the equivalence between and . Next, note that if the following mappings are applied to the definition of , namely and except for in which case , then we can also obtain the expression for from , thus showing their equivalence. We thus have that and thus In+1≤3maxIn. (18) Now, by the induction hypothesis we know that for all biseparable correlations . It then follows from Eq. (18) that , which completes the proof. Note that, following the above reasoning, any witness for parties and inputs per party which can be written in the form , where is a function satisfying , can be generalized to more parties. Note also that the above biseparable bound for the witness (5) is tight, as it can always be achieved by performing the following local measurements {cosϕ(xi)σx+sinϕ(xi)σy: i=1,…,n−1σz: i=n, (19) with on the biseparable state \|ψn⟩=\|GHZn−1⟩⊗\|0⟩. (20) Finally, let us note that the same procedure as the one detailed above can be followed to obtain the (no-signalling) Svetlichny bound of . The only difference is that in the first step of the induction proof, we must compute the no-signalling bound of the inequality instead of the quantum biseparable bound. As in the quantum biseparable case, this reduces to computing twice the (bipartite) no-signalling bound of the 3-input chained Bell inequality, which gives and thus . To show that this bound is tight consider a strategy where the -th party is separated from the rest and always outputs 1. Using an analysis similar to the one leading to Eq. (16), we can then see that the remaining parties are playing an effective game defined by which can be shown to be equivalent to twice the expression . It is a simple exercise to show that the algebraic maximum of is and that this is always achievable by players that are constrained only by the no-signalling principle. Therefore, with this particular strategy, one achieves , which saturates the bound derived above. ## Appendix C Biseparable model for projective measurements on the tripartite GHZ state Here we present a biseparable model that simulates von Neumann measurements on noisy tripartite GHZ states ρ=V\|000+111⟩⟨000+111\|2+(1−V)\openone8 (21) of visibility . Clearly this allows to simulate states with as well, by mixing this model with another one in which all parties produce uniformly random outcomes. The model is presented as a protocol in which a source distributes quantum states and random variables to the parties. All parties can share the random variables, but since only biseparable states may be used in the model, not more than two parties at a time can share a quantum state. After distribution, the parties receive their respective measurement directions , or belonging to the Bloch sphere, and measure their quantum system and/or process the information they received accordingly in order to produce binary outcomes . We check that the outcomes produced by the model are identical to the ones found when measuring state (21) in the given bases. ### c.1 The model Before the parties receive their inputs, a common source chooses a party at random, say Charlie (), and sends him the vector →λ=(sinαcosβ,sinαsinβ,cosα), (22) uniformly chosen on the sphere . The source then also provides the two other parties, Alice and Bob in this case, with the quantum state \|ΦAB⟩=cosα2\|00⟩+sinα2e−iβ\|11⟩. (23) Moreover, the source sends to all parties the signs , which are independently and identically distributed with . At the time of measurement, Alice and Bob measure their system according to and and get outcomes , while Charlie calculates . The parties then output respectively , and . ### c.2 Correlations obtained by the model Here we compare the correlations that are created when Alice, Bob and Charlie apply the biseparable model above to the ones that they would get by measuring state (21). Since the model treats equally each party, its correlations are symmetric under exchange of the parties and we only need to check the following three relations: ⟨A⟩ =0 (24) ⟨AB⟩ =12cosθxcosθy (25) ⟨ABC⟩ =12sinθxsinθysinθzcos(φx+φy+φz). (26) Here we parametrized the parties’ measurements in terms of spherical coordinates: →x=(sinθxcosφx,sinθxsinφx,cosθx) (27) and similarly for and . For definiteness, in the following we use the brackets to express the expectation value of a quantity with respect to a quantum state, and the bar ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯f(α,β)=14π∫2π0dβ∫π0dαsinαf(α,β) (28) for the average of a function over the random variable and . We always start by considering that Charlie is the special party chosen by the source, to whom the hidden vector is sent, and perform the symetrization afterwards. Symmetrized quantities are indexed by the symbol . For simplicity we average over the depolarization signs only at the end of the calculation. #### Single-party expectation value One can easily verify that the outcomes of each party is locally random according to the model, in agreement with equation (24). #### Two-party correlators Let us first calculate correlations , and for the case in which Charlie receives the vector , and average over the choice of the party being alone afterwards. The first term is easily found from equation (23), which gives ⟨ab⟩=cosθxcosθy+sinαsinθxsinθycos(β+φx+φy) (29) and thus . To compute the correlation , it is useful to write Alice’s state as ρA=trB(\|ΦAB⟩⟨ΦAB\|)=cos2α2\|0⟩⟨0\|+sin2α2\|1⟩⟨1\|. (30) The expectation value for Alice’s outcome is then . Concerning Charlie, his outcome is totally determined by and . For simplicity we assume that , but the other situation can be treated similarly. In this case, for , Charles always has , and for , he always has . Now for , we can write the interval of for which the product , and its complement, with . We thus have the three following cases: 1. . In this case , which gives . 2. . In this case , which gives . 3. . In this case one has where χI(β)={1if β∈I0if β∉I (31) is the indicator function. In total, after integration over and , this gives . Similarly one can check that , and so once averaged over the choice of the party being alone, the bipartite correlations are given by ¯¯¯¯¯¯¯¯¯¯⟨ab⟩↺=23cosθxcosθy. (32) Finally, we need to apply the signs , , . Since the expectation value of the product of two signs is , it follows that overall the correlation between two outcomes produced by the model is ⟨AB⟩=12cosθxcosθy, (33) in agreement with equation (25). #### Three-party correlators We now proceed to calculate the tripartite correlations that are created by the model. Let us consider the three preceding cases separately again: 1. . In this case , so as given by equation (29). 2. . In this case , so . 3. . In this case one has . In total, the tripartite correlation are found after integration over and to be ¯¯¯¯¯¯¯¯¯¯¯¯⟨abc⟩=12sinθxsinθysinθzcos(φx+φy+φz), (34) which does not depend on which party is alone. One can check that the application of the signs has no influence on the tripartite correlations since they cancel out, and so . The model thus reproduces the expected correlations (26). ## Appendix D Characterizing biseparable correlations through SDP in the tripartite case (n=3) It has been shown in Navascues07 how to define a hierarchy of SDP that characterizes the set of bipartite quantum correlations. Note that biseparable quantum correlations can be written as a finite sum of such bipartite quantum correlations using the fact that each in (4) can be taken to be extremal and using the fact that there are a finite number of such extremal points corresponding to classical, deterministic strategies. It therefore follows that biseparable quantum correlations can be characterized using a finite number of the SDP hierarchies introduced in Navascues07 . Note however that the number of single-party deterministic strategies, and thus the number of terms in Eq. (4), grows exponentially with the number of measurement settings, making such an approach impractical even for small problems. Here we introduce an alternative SDP approach that has better scaling properties. Our approach is based on the observation that the tensor product separation at the level of states can be replaced by a commutation relation at the level of operators. Specifically, let denote the three possible partitions of the parties into two groups. 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https://www.toktol.com/notes/section/888/maths/functions-and-graphs/graphs	Use adaptive quiz-based learning to study this topic faster and more effectively. # Graphs ## Local extrema Given a function $f$, a point $\xo$ is • a local maximum if, for all $x$ around $\xo$, $$f(x)\le f(\xo);$$ • a local minimum if, for all $x$ around $\xo$, $$f(x)\ge f(\xo).$$ An extremum is either a minimum or a maximum. A local extremum is sometimes called a turning point or a relative extremum. $0$ is a minimum of $x^2$ and a maximum of $-x^2$. At a local extremum, the tangent must be horizontal $$f'(\xo)=0.$$ At a local maximum $\xo$, $f$ "increases" to its left, so $f'(\xo)\ge 0$, and "decreases" to its right, so $f'(\xo)\le 0$. Thus, $f'(\xo)=0$. Local minimum (A), inflection point (B), local maximum (C) of a function ## Stationary points A stationary point $\xo$ of a function $f$ has horizontal tangent $$f'(\xo) = 0$$ A point of inflection is a stationary point that is not an extremum. In practice, a stationary point $\xo$ is • a local maximum if $f''(\xo)\lt 0$, • a local minimum if $f''(\xo) \gt 0$, • a point of inflection if $f''$ changes sign around $\xo$. $0$ is a stationary point of $$f_1(x) = x^2,\quad f_2(x)= x^3,\quad f_3(x) = -x^2.$$ It is minimum of $f_1$ ($f''_1(0)= 2$), a maximum of $f_3$ ($f''_3(0)= -2$) and an inflection point of $f_2$ ($f''_2(x) = 6x$). We show that, when $f''(\xo)\lt 0$, $\xo$ is a local maximum. $f'$ is decreasing and $0$ at $\xo$. On the left of $\xo$, $f'$ is positive, so $f$ is increasing, so $f(x)\le f(\xo)$. Similarly, on its right, $f(x)\le f(\xo)$. Local minimum (A), inflection point (B), local maximum (C) of a function ## Asymptotes An asymptote is a straight line tangent to a function at infinity. • A vertical asymptote is a line of the form $x=a$ when $$\lim_{x\to \pm a}f(x) = \pm\infty.$$ The function has an infinite limit at $a$ when $x$ approaches $a$ either from below ($x\to a-$) or from above ($x\to a+$). • A horizontal asymptote is a line of the form $y=b$ when $$\lim_{x\to \pm\infty}f(x) = b.$$ The function has a limit when $x$ goes to infinity. • An oblique asymptote is a line $y = ax + b$ ($a\ne 0$) when $$\lim_{x\to\pm\infty}\big(f(x)-ax\big) = b.$$ $f(x) = 2(x-1)/x$ has two asymptotes: $x = 0$ and $y=2$. Asymptotes: Vertical $(x=0)$ horizontal $(y=1)$ and oblique $(y=x-1)$ ## Graph Transformations 1 The graph of modified functions can often be deduced from simple geometrical transformations of the graph of the original function. To deduce the transformation, we often consider how a few points on the graph are modified from the original graph and then extrapolate. We explain which modified function corresponds to the appropriate transformation on the graph of $f$. $a\ne 0$ is a fixed number. • Horizontal translation of $a$: $f(x-a)$ • Vertical translation of $a$: $f(x)+a$ • Horizontal scaling of ratio $1/a$: $f(ax)$. It is a stretch or dilation if $\vert a\vert \lt 1$ and a compression if $\vert a\vert \gt 1$. • Vertical scaling of ratio $1/a$: $af(x)$ • Horizontal reflection (along vertical axis): $f(-x)$ • Vertical reflection (along horizontal axis): $-f(x)$ • Rotation of angle $\pi$ about the origin:$-f(-x)$ Transformation of graph of $f(x)$ Translation $f(x-1)$ horizontal scaling $f(2x)$ vertical scaling $2f(x)$ horizontal reflection $f(-x)$ vertical reflection $-f(x)$ symmetry $-f(-x)$ ## Graph Transformations 2 It is often useful to be able to apply more elaborate graph transformations. The methodology is unchanged: draw the function, see how it changes a few points and extrapolate to get the full graph. • $f(\vert x\vert)$: Horizontal reflection of the graph of $f$ for $x\ge0$. • $\vert f(x)\vert$: Vertical reflection of the negative part of the graph of $f$. • $1/f(x)$: roots of $f$ (solutions of $f(x)=0$) are transformed into vertical asymptotes; monotonicity is transformed into inverse monotonicity (if $f$ is increasing, $1/f$ is decreasing); when $\vert f(x)\vert = 1$, graphs $f$ and $1/f$ intersect. • $y^2=f(x)$: when $f(x)\lt 0$, $y$ is undefined; if $f(x)\ge 0$, $y$ is the graph superposition of the graphs of the functions $\sqrt{f(x)}$ and $-\sqrt{f(x)}$; monotonicity of $\sqrt{f(x)}$ and $f$ is same; when $f(x) = 1$, graphs $f$ and $\sqrt{f}$ intersect. Transformation of graph of $\sin x$. 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http://blitiri.blogspot.com/2015_10_01_archive.html	## 25 October 2015 ### The average of five people It has been said that “You are the average of the five people you spend the most time with.” By a relative of the mean value theorem (used e.g. in electrostatics) it then follows that there are no local extrema in society. ### Varying the counter ion changes the kinetics, but not the final structure of colloidal gels Our paper just got published in the Journal of Colloid and Interface Science ! Isotropic fluids only have two viscosities, intervening in shear and extensional deformations. For anisotropic media, such as nematic liquid crystals, more coefficients are needed, as shown by Mięsowicz in the late '30s: three shear viscosities, labeled $$\eta_1$$ to $$\eta_3$$, a fourth one $$\eta_{12}$$ introduced later by Helfrich and a rotational viscosity, $$\gamma_1$$. We do not worry here about extensional deformations. The whole topic was put on a solid theoretical basis in the '60s by Leslie and Ericksen [brief and clear presentation here] who introduced six coefficients ($$\alpha_1$$ to $$\alpha_6$$), only five of which are independent. As one can expect from dimensional analysis, the two sets of viscosities, $$\left \lbrace \eta_i, \gamma _1 \right \rbrace$$ and $$\left \lbrace \alpha_j \right \rbrace$$ are linearly related.	{"extraction_info": {"found_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": 0.8864154815673828, "perplexity": 1143.616731967759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1518891813187.88/warc/CC-MAIN-20180221004620-20180221024620-00366.warc.gz"}
https://www.andlearning.org/factorial-formula/	Connect with us # What is Factorial? Factorial Formula, Function, Number, Equation Published on By ### What is Factorial? The concept of Factorials was given in Algebra and further, they can be used for statistics, permutation, combinations, and to calculate the probabilities. Wherever you see the exclamation symbol(!) behind an integer, this is a factorial. ## Factorial Formula The Factorial formula is generally defined as the product of given number with all lowest vale numbers. This is denoted by Factorial (!) exclamation symbol. This is also taken as the series of number in descending order. The factorial formula is generally required in permutation and combinations to calculate the probabilities. For the integer greater than or equal to one, the factorial formula is mathematical is generalized as given below. n! = n (n – 1) (n – 2) (n – 3) … (3)(2)(1) If p = 0, then p! = 1 by convention. $\LARGE n! \; = 1 \; \times 2 \; \times 3 \; \times ….. \; \times \; (n-1) \; \times n$ Factorials can be defined for all integers whose values are taken as greater than zero. For the integer greater than or equal to one, the factorial would be the multiplication of all lowest numbers. In case of non-negative integers, the Factorial formula is not applicable. ### Factorial of a Number Calculation of factorial of a number is easy. For example, 4! In mathematics is “six factorials.” Based on the formula it could be elaborated as – 4! = 4 x 3 x 2 x 1 = 24 Thus, Factorial of a number 4 is 24 here. ### Factorial Function From the arithmetic point of view, Factorial functions are needed to showcase fixed-sized integers and avoids the overflows of data. They are frequently used for complex computer problems and difficult to calculate manually without a proper understanding of the concept. If this is not possible to calculate the factorial of a number directly then it is generally broken down into parts to make the calculation easy. There are a variety of mathematical applications where factorial can be calculated directly but still, you should know the basics to solve real-world problems quickly on your fingertips. Take an example, if you are preparing for competitive exams then do automatic computer programs are fruitful there? Obviously, No! Here, you should know the Factorial concept in detail, and its basic formulas, factorial function, factorial equations etc. Also, do practice a number of questions to solve difficult problems with ease. They are the part of higher studies too where students are planning to join mathematics course during their graduation or post-graduation studies. ### Factorial equation Factorial equations are the expressions in mathematics where basic formulas are used to compute the final outcome. For n factorial, there is n number of possible ways of arranging a number. They are into existence since 12th centuries and used for a plenty of daily applications too. The notation of factorial was suggested by a popular French mathematician Christian Kramp in 1808. In certain case, the definition of a factorial function can also be extended to non-integer arguments while keeping important properties the same. This is generally studied in advanced mathematics for analysis of concepts.	{"extraction_info": {"found_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": 0.8963356614112854, "perplexity": 571.9779980598587}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1585370524043.56/warc/CC-MAIN-20200404134723-20200404164723-00092.warc.gz"}
http://mathoverflow.net/questions/53901/ping-pong-and-free-group-factors	Ping Pong and Free Group Factors This question concerns alternative characterizations of free group factors. The ping pong lemma is a well-known criteria for the freeness of a group. I've often wondered if there is a ping pong like criterion that can be used to determine if a given type $II_{1}$ factor is a free group factor, e.g. a ping pong-like criterion for the action of the factor on some Hilbert space. Question: Is there a ping pong lemma analogue for group von Neumann algebras? - I'm not even aware of any criterion at the purely vN level which characterizes the class of free group factors within the class of all group vN algebras -- but there are people reading MO who would no much more about this than I do. (Of course that class might consist of only one vN alg up to isomorphism, which is one reason I'm slightly pessimistic.) – Yemon Choi Jan 31 '11 at 18:25 Indeed, Yemon, I think this is the case. One other such criterion has recently been proposed by Popa: whether the "free flip" can be connected to the identity. (This is even stronger in that it goes beyond characterizing the free group factors within group von Neumann algebras...) – Jon Bannon Jan 31 '11 at 18:45 .."this" above being that no such abstract criterion is yet available. I would also be very surprised if my question were accessible. The idea is that if there is a decomposition of the Hilbert space on which the von Neumann algebra acts (I'm thinking something like Voiculescu's "free product" of Hilbert spaces) then one could in some setting deduce that the acting factor must be an interpolated free group factor. (This is kind of crazy, but who knows?) – Jon Bannon Jan 31 '11 at 18:51 Also, we should emphasize that this question has little to do with whether free group factors are isomorphic. It asks only whether a factor is a free group factor...so somehow may avoid the issue Yemon is concerned with. – Jon Bannon Jan 31 '11 at 21:25 Well, isn't this (vaguely) the kind of problem which motivated Murray and von Neumann and others to look at Property P, Property Gamma, and others? Maybe someone who knows about l^2-Betti numbers can tell us if something along those lines might be useful... – Yemon Choi Feb 1 '11 at 3:27 I am guessing that the answer is "yes" if you interpret the question in the following way. Let $A_i$ be some subalgebras of a von Neumann algebra $(M,\tau)$ and assume that there are mutually orthogonal Hilbert subspaces $H_i$ of $H=L^2(M)$ so that for all $i$, $x (H\ominus H_i) \subset H_i$ whenever $x\in A_i$ with $\tau(x)=0$. Let us also assume that $1 \perp \oplus_i H_i$ (probably this is not necessary). Then if $y = x_1 \dots x_n$ with $x_j \in A_{i(j)}$, $i(1)\neq i(2)$, $i(2)\neq i(3)$, etc. and $\tau (x_j) = 0$, we have: $x_n 1 \in H_{i(n)}$ since $1\in H\ominus H_i$; $x_{n-1} x_n 1 \in H_{i(n-1)}$ since $x_n 1 \in H_{i(n)} \subset H\ominus H_{i(n-1)}$ (because $i(n)\neq i(n-1)$ and so $H(i(n))\perp H(i(n-1))$; $x_{n-2} x_{n-1} x_n 1 \in H_{i(n-2)}$ since $x_{n-1} x_n 1 \in H_{i(n-1)}\subset H\ominus H_{i(n-2)}$, etc. Thus We get that $x_1\dots x_n 1 \in H_{i(1)} \perp 1$, so that $\tau(y)=0$. It follows that $A_1,\dots,A_n$ are freely independent. (Conversely, if $M$ is generated by $A_1,\dots,A_n$ and they are free inside of $M$, then $L^2(M) = \mathbb{C}1 \oplus \oplus_k \oplus_{j_1\neq j_2, j_2\neq j_3,\dots} L^2_0(A_{j_1})\otimes \cdots \otimes L^2_0(A_{j_k})$, where $L^2_0(A_j) = \{1\}^\perp \cap L^2(A_j)$. Then you can take $H_j = \oplus_k \oplus_{j_1\neq j_2, j_2\neq j_3,\dots; j_1= j} L^2_0(A_{j_1})\otimes \cdots \otimes L^2_0(A_{j_k})$ and then $H_j$ are orthogonal and $H\ominus H_j$ is taken to $H_j$ by any $x\in A_j$ with $\tau(x)=0$). If you now make some assumption (e.g. that $A_j$ are finite-dimensional, abelian or hyperfinite) then it follows from Ken Dykema's results (see e.g. his paper on Interpolated free group factors in Duke Math J.) that the von Neumann algebra they generate inside of $M$ is an interpolated free group factor. This is similar to the assumption you have put on the group (since the subgroup generated by a single element in the ping-pong lemma is necessarily abelian). On the other hand, you raise the much bigger question of whether there exists some criterion that singles out free group factors -- just as the various functional-analytical criteria were shown by Connes to be equivalent to hyperfiniteness. Unfortunately, not much in known in this direction (note that a similar question exists on the ergodic equivalence side of things: is there a functional-analytic way of recognizing treeable actions? Or Bernoulli actions of free groups?) - Thank you for the answer. This is precisely the sort of interpretation I thought may be possible. – Jon Bannon Feb 1 '11 at 12:58 Especially, I like the comments regarding ergodic equivalence in the last paragraph. – Jon Bannon Feb 3 '11 at 22:11	{"extraction_info": {"found_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": 0.847390353679657, "perplexity": 292.5712327669408}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701153323.32/warc/CC-MAIN-20160205193913-00237-ip-10-236-182-209.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/151086-extreme-points-choquet-theorem.html	# Thread: extreme points and choquet theorem 1. ## extreme points and choquet theorem define the set A to be the functions $\displaystyle f:[0,1]\rightarrow[-1,1]$ such that f is measurable and its integral is $\displaystyle \intop _0 ^1 fdt = 0$. let $\displaystyle \varphi(f)=\intop_0 ^1 e^{\alpha f(t)} dt$. I need to show that $\displaystyle \varphi(f) \leq e^{\alpha^2/2}$ for all $\displaystyle f \in A$, using the inequality $\displaystyle \frac{e^t+e^{-t}}{2} \leq e^{t^2/2}$, and using the extreme points of A. I know how to prove that the extreme points are $\displaystyle \chi_B -\chi _{B^c}$ where B is a set of measure 1/2, and it is easy to see that for these type of functions the inequality holds. so now I want to show that the maximum of $\displaystyle \varphi$ is attained on these extreme points. One way I thought of proving this, is by showing that $\displaystyle \varphi$ is continuous, and so has a maximum, because A is compact in the weak* topology. Then I can use the fact that the function is convex to show that the maximum must be on the extreme points. The other way is somehow use choquet theorem that connects between the functions in A to its extreme points. don't really know how to continue from here, so any help would be appreciated 2. Use Jensen's inequality on $\displaystyle \frac{e^t+e^{-t}}{2} \leq e^{t^2/2}$. I think that would get you where you need to go.	{"extraction_info": {"found_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": 0.9751690030097961, "perplexity": 91.94057269711271}, "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/1526794863410.22/warc/CC-MAIN-20180520112233-20180520132233-00074.warc.gz"}
http://math.stackexchange.com/questions/422517/how-to-calculate-equation-of-normal-to-the-curve-which-is-parallel-to-another-li	# How to calculate equation of normal to the curve which is parallel to another line Find the equation of the normal to the curve y=3x^2-2x-1 which is parallel to the line y=x-3 . Hi I'm having trouble figuring out when to use which gradients as initially the gradient you get is one which you hen convert into -1 and then finally use the gradient of one again. The answer is y=x-17/12 or 12y=12x-17 (same thing) - A normal at some point is perpendicular to the tangent line at that point. The tangent line's slope at any point is $$y'=6x-2\;\;\text{and we want this to be perpendicular to a line with slope}\;\;1\implies$$ $$6x-2=-1\implies x=\frac16$$ and since $$y\left(\frac16\right)=3\frac1{36}-2\frac16-1=\frac1{12}-\frac13-1=-\frac54\implies \left(\frac16\,,\,-\frac54\right)$$ is the tangency point , so the normal there is $$y+\frac54=x-\frac16\implies y=x-\frac{17}{12}$$ -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 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": 0.8771622180938721, "perplexity": 423.02663791844225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413558067467.26/warc/CC-MAIN-20141017150107-00156-ip-10-16-133-185.ec2.internal.warc.gz"}
http://mathhelpforum.com/algebra/19438-factoring-imperfect-squares.html	1. Factoring Imperfect squares Well, my question is how. Here's one from the sheet, I have the answers already, but everyone knows the answers don't matter in math >.> 1) $x^4 + x^2 + 25$ 2) $2x^4 + 8$ If you think these are both done with a pretty similar method, then you don't have to show me both. They just look different. 2. Originally Posted by Freaky-Person 1) $x^4 + x^2 + 25$ $x^4+x^2+25=(x^2+5)^2-9x^2,$ now factorise the difference of two perfect squares. 3. Originally Posted by Freaky-Person Well, my question is how. Here's one from the sheet, I have the answers already, but everyone knows the answers don't matter in math >.> 1) $x^4 + x^2 + 25$ 2) $2x^4 + 8$ If you think these are both done with a pretty similar method, then you don't have to show me both. They just look different. I'm not sure what you mean by "factoring imperfect squares." Neither of these expressions factor over the integers, nor do they factor over the rationals. In fact, neither of them factor over the reals, either. By solving for the zeros of the polynomials I can tell you that $x^4 + x^2 + 25 = \left ( x - \left ( \frac{3}{2} + i \cdot \frac{\sqrt{11}}{2} \right ) \right ) \left ( x - \left ( \frac{3}{2} - i \cdot \frac{\sqrt{11}}{2} \right ) \right )$ $\left ( x - \left ( -\frac{3}{2} + i \cdot \frac{\sqrt{11}}{2} \right ) \right ) \left ( x - \left ( -\frac{3}{2} - i \cdot \frac{\sqrt{11}}{2} \right ) \right )$ You can get this by setting $x^4 + x^2 + 25 = 0$ and putting the factors in terms of $(x - r_1)(x - r_2)....$ where $r_1, r_2, ...$ are the zeros of the polynomial. -Dan 4. Originally Posted by Krizalid $x^4+x^2+25=(x^2+5)^2-9x^2,$ now factorise the difference of two perfect squares. Cool! Way to show me up! (Again. ) -Dan 5. Originally Posted by Freaky-Person 2) $2x^4 + 8$ $2(x^4+4)$ It remains to factorise $x^4+4=(x^2+2)^2-4x^2$, it's the same as above. Originally Posted by topsquark I'm not sure what you mean by "factoring imperfect squares." I get it like "complete the square". Cheers, K. 6. miss me? okay, you've all been a TON of help, but then I stumbled onto a grevious problem! (oh noez) In the following equation $4b^4 - 13b^2 + 1$ it factors to $(2b^2 - 3b - 1)(2b^2 + 3b - 1)$ and I'm wondering why it's MINUS 1 and not PLUS 1 7. Originally Posted by Freaky-Person okay, you've all been a TON of help, but then I stumbled onto a grevious problem! (oh noez) In the following equation $4b^4 - 13b^2 + 1$ it factors to $(2b^2 - 3b - 1)(2b^2 + 3b - 1)$ and I'm wondering why it's MINUS 1 and not PLUS 1 Because -1 X -1 = 1. I know 1 X 1 also = 1, but if you multiply out $(2b^2 - 3b + 1)(2b^2 + 3b + 1)$ it does not equal $4b^4 - 13b^2 + 1$ 8. Originally Posted by WWTL@WHL Because -1 X -1 = 1. I know 1 X 1 also = 1, but if you multiply out $(2b^2 - 3b + 1)(2b^2 + 3b + 1)$ it does not equal $4b^4 - 13b^2 + 1$ I shall now try and STUMP YOU!! RAWR!! $m^4 - 19m^2 + 9$ factors to $(m^2 - 5m + 3)(m^2 + 5m + 3)$ LOOK!! There be a PLUS now!! It's magic!! The wizard did it!! 9. I'm scared. 10. Originally Posted by Freaky-Person $m^4 - 19m^2 + 9$ Mmm... what's the problem with this? $m^4-19m^2+9=(m^2+3)^2-25m^2$, the rest is trivial. 11. Originally Posted by Krizalid Mmm... what's the problem with this? $m^4-19m^2+9=(m^2+3)^2-25m^2$, the rest is trivial. look up up uuuuup! Cause I wrote a question like this one, but it ended up with the 3rd term being NEGATIVE!! So I was like ZOMG! WHY!?! =3 12. Originally Posted by Freaky-Person Cause I wrote a question like this one, but it ended up with the 3rd term being NEGATIVE!! So I was like ZOMG! WHY!?! Why not? What's so surprising about the 3rd term being negative? Is it the fact that you haven't seen it negative before? , , , factorization of imperfect sqaures Click on a term to search for related topics.	{"extraction_info": {"found_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": 28, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9231363534927368, "perplexity": 1096.6502419987169}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806543.24/warc/CC-MAIN-20171122084446-20171122104446-00217.warc.gz"}
https://quant.stackexchange.com/questions/37942/stochastic-calculus-how-to-test-for-dependency-of-random-variables/37944	# Stochastic Calculus: How to test for dependency of random variables If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as: $$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process. For $s<t$, Will $X_s$ and $X_s-X_t$ then be independent? My intuition says it will be independent because the stochastic $W_t-W_s$ and $Ws$ is independent by definition of the Wiener proces. However, I can't prove this. • Do you intend to calculate the Itô integral of $g$? – Raskolnikov Jan 27 '18 at 4:47 It is equivalent to ask whether $X_s$ and $X_t-X_s$ are equivalent. Now $X_t-X_s=\int_s^t f(u)dW_u$ and $X_s=\int_0^s f(u)dW_u$. But we have a definition of the Itō integral like: $$\int_s^t f(u)dW_u = \lim_{n\to\infty} \sum_{i=1}^{n} f(x^{(n)}_i)(W_{x^{(n)}_{i+1}}-W_{x^{(n)}_{i}})$$ for suitable partitions $s=x_1^{(n)}<\dots<x_{n+1}^{(n)}=t$ and all these increments are $W_{x_{i+1}}-W_{x_{i}}$ are independent of $\{W_u\}_{u\le s}$.	{"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": 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": 0.9859956502914429, "perplexity": 131.54047593196378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526888.75/warc/CC-MAIN-20190721040545-20190721062545-00138.warc.gz"}
http://mathhelpforum.com/differential-equations/119294-implicit-form-question.html	1. ## Implicit form question Hi, I am able to find the general solution in implicit form for single line equations but I'm really having trouble with this & would appreciate some assistance. $\frac{dy}{dx}=\frac{6y^\frac{2}{3} sin x}{(2 - cos x)^3}$ Thanks 2. Did you try separating variables?. $\frac{1}{6}\int y^{\frac{-2}{3}}dy=\int\frac{sin(x)}{(2-cos(x))^{3}}dx$ Integrate both sides and don't forget the constant on the right. 3. Originally Posted by galactus Did you try separating variables?. $\frac{1}{6}\int y^{\frac{-2}{3}}dy=\int\frac{sin(x)}{(2-cos(x))^{3}}dx$ Integrate both sides and don't forget the constant on the right. Thank you & I'll give it a go now. 4. Okay so integrating both sides gives me this: $\frac{y^5/3}{10}=-\frac{1}{2(2-cos x)^2} + c$ Is this the implicit form or do I need to go further? I have an initial condition $y(0)=1$ which I presume I can solve for c & then substitute back into this equation? 5. Yes, use your initial condition $\frac{1}{2}y^{\frac{1}{3}}=\frac{-1}{2(cos(x)-2)^{2}}+C$ $y^{\frac{1}{3}}=\frac{-1}{(cos(x)-2)^{2}}+2C$ $y=\left(\frac{-1}{(cos(x)-2)^{2}}+C_{1}\right)^{3}$ 6. Originally Posted by galactus $\frac{1}{2}y^{\frac{1}{3}}=\frac{-1}{2(cos(x)-2)^{2}}+C$ $y^{\frac{1}{3}}=\frac{-1}{(cos(x)-2)^{2}}+2C$ $y=\left(\frac{-1}{(cos(x)-2)^{2}}+C_{1}\right)^{3}$ Right I can see how you've done that (including the initial fraction for y which I left undone!) but where does the $+C_{1}$ come from? 7. Originally Posted by bigroo Right I can see how you've done that (including the initial fraction for y which I left undone!) but where does the $+C_{1}$ come from? I'm still not sure about that $+C_{1}$ so I've left it as $+2C$. Maybe someone will tell me otherwise. Also why did the denominator reverse? Substituting back into general equation for $y(0)=1$: $1=\left(\frac{-1}{(2-cos(0))^{2}}+2C\right)^{3}$ $C=1$ Hence $y=\left(\frac{-1}{(cos(x)-2)^{2}}+2\right)^{3}$ Would this satisfy my original question now or have I made it explicit? Thanks	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9831620454788208, "perplexity": 444.11854535787774}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1471982296721.55/warc/CC-MAIN-20160823195816-00084-ip-10-153-172-175.ec2.internal.warc.gz"}
http://thevirtuosi.blogspot.com/2011/12/report-from-trenches-cms-grad-student.html	## Tuesday, December 13, 2011 ### Report from the Trenches: A CMS Grad Student's Take on the Higgs Hi folks. It's been an embarrassingly long time since I last posted, but today's news on the Higgs boson has brought me out of hiding. I want to share my thoughts on today's announcement from the CMS and ATLAS collaborations on their searches for the Higgs boson. I'm a member of the CMS collaboration, but these are my views and don't represent those of the collaboration. The upshot is that ATLAS sees a 2.3 sigma signal for a Higgs boson at 126 GeV. CMS sees a 1.9 sigma excess around 124 GeV. CERN is being wishy-washy about whether or not this is actually a discovery. After all the media hype leading up to the announcement, this is somewhat disappointing, but maybe not too surprising. First of all, what does a 2 sigma signal mean? The significance corresponds to the probability of seeing a signal as large or larger than the observed one given only background events. That is, what's the chance of seeing what we saw if there is no Higgs boson? You can think of the significance in terms of a Normal distribution. The probability of the observation corresponds to the integral of the tails of the Normal distribution from the significance to infinity. For those of you in the know, this is just 1 minus the CDF evaluated at the significance. For a 2 sigma observation, this corresponds to about 5%. For both experiments, there was a 5% chance of observing the signal they observed or bigger if the Higgs boson doesn't exist. In medicine, this would be considered an unqualified success. So why is CERN being so cagey? In particle physics we require at least 3 sigma before we even consider something interesting, and 5 sigma to consider it an unambiguous discovery. The reasons why the burden of proof is so much higher in particle physics than in other fields aren't entirely clear to me. I suspect is has to do with the relative ease of running the collider a little longer compared to recruiting more human test subjects, to use medicine as an example. Given what I've just told you that we need a 3 sigma significance in particle physics, why is everyone so excited about a couple of 2 sigma results? Well, the first reason is that both results show bumps at approximately the same Higgs mass. Although it's not rigorous, you can get a rough idea of what the significance of the combined results are by adding the significances in quadrature. This gives us about 2.8 sigma. Higher, but still not up to the magic number of 3. The explanation for the excitement that is most compelling brings us to Bayesian statistics. The paradigm of Bayesian statistics says that our belief in something given new information is the product of our prior beliefs and a term which updates them based on the new information. Physicists have long expected to find a Higgs boson with a mass around 120 GeV. So our prior degree of belief is pretty high. Thus, it doesn't take as much to convince us (or me anyway) that we have observed the Higgs boson. In contrast, consider the OPERA collaboration's measurement of neutrinos going faster than the speed of light. This claims to be a 6 sigma result, but no one expected to find superluminal neutrinos, so our (or at least my) prior for this is much lower. (Aside: If the OPERA result is wrong, it is likely due to a systematic effect rather than a statistical one. Nevertheless, I stand by my point.) The final thing that excites me about this observation is that what we've seen is completely consistent with what we would expect to see from the Standard Model. Forgetting about significances for the moment, when the CMS experiment fits for the Higgs boson mass, they find a cross section that agrees very well with that predicted by the Standard Model. In the plot below, you're interested in the masses where the black line is near 1. The ATLAS experiment actually sees more signal than one would expect. This is likely just a statistical fluctuation, and explains why the ATLAS result has a higher significance. In conclusion, while CERN is being non-committal, in my opinion, we have seen the first hints of the Higgs boson. This is mostly due to my high personal prior that there the Higgs boson exists around the observed mass. Unfortunately, Bayesian priors are for the most part a qualitative thing. Thus, ATLAS and CMS are sticking to the hard numbers, which say that what we have looks promising, but is not yet anything to get excited about. I'll close by reminding you all to take this all with a grain of salt. There is every possibility that this is just a fluctuation. I'll remind you that at the end of last summer, CMS and ATLAS both showed a 3 sigma excess around 140 GeV, which went away just a month later at the next conference. So let's cross our fingers that next year's data will give us a definitive answer on this question. By the way, if anyone wants to know more, fire away in the comments. I'll do my best. #### 1 comment: 1. Do running the collider a little longer could result in same Higgs mass at different sigma signal.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8669894933700562, "perplexity": 431.5282368485101}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1487501170404.1/warc/CC-MAIN-20170219104610-00476-ip-10-171-10-108.ec2.internal.warc.gz"}
https://manasataramgini.wordpress.com/2019/01/22/an-apparition-of-mordell/	## An apparition of Mordell Consider the equation: $y^2=x^3+k$ where $k$ is a positive integer 1, 2, 3… For a given $k$, will the above equation have integer solutions and, if yes, what are they and how many? We have heard of accounts of people receiving solutions to scientific or mathematical problems in their dreams. We have never had any such dream; in fact we get most of our scientific or mathematical insights when we are either in a semi-awake but conscious reverie or at the peak of our alertness. However, we on rare occasions we have had dreams which present mathematical matters. On the night between the 18th and 19th of Jan 2019 we had such dream. It was a long dream which featured human faces we do not recall seeing in real life: we forgot their role in the dream on waking. However, what we remembered of the dream was the striking and repeated appearance of the above equation along with its solutions for several $k$. On waking, we distinctly recall seeing the cases of $k=1, 8, 9$ though there were many more in the dream. There was a degree of discomfort from the dream for in the groggy state of waking from it we knew that some of these solutions had slipped away. So, at the first chance we got, we played a bit with this equation this on our laptop. We should mention that we have not previously played with this equation and have given it little if any thought before: we glanced at it when we had previously written about the cakravala but really did not give it any further consideration then or thereafter. Hence, we were charmed by its unexpected and strong appearance in our dream. This equation is known as Mordell’s equation after the mathematician who started studying it intensely about a century ago. The equation itself was know before him to a French mathematician Bachet and is sometimes given his name. It has apparently been widely studied by modern mathematicians and they know a lot about it. As a mathematical layman we are not presenting any of that discussion here but simply record below our elementary exploration of it. Figure 1. These equations define a class of elliptic curves and by definition given the square term they are symmetric about the $x$-axis (Figure 1). Since, we are here only looking at the real plane, the curve is only defined starting from $y=0, x=-\sqrt[3]{k}$ where it cuts the $x$-axis. From there on it opens symmetrically towards $\infty$ in the direction of the positive $x$. So, essentially we are looking for the lattice points which lie on this curve. Given the square term, we will get symmetric pairs of solutions about the $x$-axis (Figure 1). In geometric terms, it might be viewed as the problem of which squares and cubes with sides of integer units are inter-convertible by the addition of $k$ units. Mordell had pointed out about a century ago that these equations might have either no integer solutions or only a finite number of them. Let is consider a concrete example with $k=1$: $y^2=x^3+1$. One can get three trivial solutions right away: by setting $x=-1$ we get $y=0$. Similarly, setting $x=0$, we get $y=\pm 1$. Further, we see that if $x=2$ we get $y=\pm 3$. Thus, we can write the solutions as $(x,y)$ pairs: (-1,0); (0,1); (0, -1); (2,3); (2,-3): a total of 5 unique integer solutions. Can there be any more solutions than these? To get an intuitive geometric feel for this we first observe how these solutions sit on the curve (Figure 1). We notice that the 3 distinct ones are on the same straight line (also applies to their mirror images via a mirrored line). This is a important property of elliptic curves (being cubic curves) that allows us to understand the situation better. Figure 2. In Figure 2, we consider the curve $y^2=x^3+1$ in greater detail. We observe that if we have two integer solutions points $P$ and $Q$ and we connect them we get the third one $R_1$. Thus, if we have two solutions, in this case the trivial ones, we can easily find the third one by drawing a line through them and seeing where it cuts the elliptic curve. The point where it cuts it gives a further solution. We also see from the figure that joining symmetric solutions like points $Q$, $Q^\prime$ will not yield any further solutions because the resultant line will be parallel to the $y$-axis. Thus, we might get an additional solution only if the slope of the line joining the 2 prior solutions is neither $0$ (coincident with the $x$-axis) nor $\infty$ (parallel to $y$-axis). Hence, we may ask: now that we have $R_1$ can we get a further solution? We can see that joining $Q^\prime$ to $R_1$ yields a line that will never again cut the curve $y^2=x^3+1$. Thus, we can geometrically see that there can be no more than the 5 above solutions we obtained. In algebraic language the slope of the line joining the first two solutions can be written as: $m=\dfrac{y_P-y_Q}{x_P-x_Q}$ From this one can calculate the coordinates of the third point $R$ as: $x_R=m^2-x_P-x_Q, \; y_R=y_P+m(x_R-x_P)$ Thus, in Figure 2, if we were to join $Q^\prime$ to $R_1$ we have: $m=2$; thus for the “new point” we get: $x=4-0-2=2, \; y=-1+2(2-0)=3$. We simply get back $R_1$ indicating that there are no further integer solutions than the 5 we have. One may also see a parallel between this procedure of obtaining a third integer solution by joining two points and the process of obtaining a composite number by multiplying two prime numbers. If we know the two starting points it is easy to get the third but if we were to only know the third point for a large number getting its precursors would be a difficult task. This relates to the use of elliptic curves as an alternative for primes in cryptography (see our earlier note on the use of prime numbers in the same). Figure 3. In the case of $y^2=x^3+1$ the joining procedure terminated after the first new point we got but can there be cases where this yields more points? To see an example of this let us consider $y^2=x^3+9$. One can get a trivial solution by simply placing $x=0$ to get $y=\pm 3$. Further, it is also easy to see that by taking $x=-2$ we get $y=\pm 1$. Thus, we get four points that we may call $P$, $P^\prime$, $Q$, $Q^\prime$(Figure 3 panel 1). By joining $P$ to $Q$ we get a further point $R_1$. Similarly, joining $P^\prime$ to $Q$ or $Q^\prime$ to $R_1$ we get yet another point $R_2$. We can likewise obtain their mirror images $R^\prime_1$ and $R^\prime_2$ (Figure 3). Finally, by joining $R^\prime_1$ to $R_2$ we get yet another point $R_3$ and likewise we can get its mirror image point $R^\prime_3$. Beyond this the joining procedure yields no further points. Thus, we are left with a total of 10 integer solutions for $y^2=x^3+9$: (-2,1); (0,3); (3,6); (6,15); (40,253); (-2,-1); (0,-3); (3,-6); (6,-15); (40,-253) Figure 4. We can then systematically explore the solutions for all $k=1..1000$. If we plot all solutions and zoom in close to origin we find a dense clustering of the solutions forming a swallow-tail like structure whose outline is an integer approximation of an elliptic curve (Figure 4). Figure 5. We further find that some $k$ have no integer solutions at all. There is a formal way to use modulos and factorization to prove this for particular $k$. The sequence of $k$ for which no integer solutions exist can be computationally obtained and goes as: 6, 7, 11, 13, 14, 20, 21, 23, 29, 32… Figure 5 shows how the $n^{th}$ term of this sequence grows. We find empirically that it appears to be bounded by or at least approximated by the shape of a scaled form of the logarithmic integral: $y=\pi^2 \textrm{Li}(x)$. Whether this is true or what the significance of it may be remains unknown to us. Figure 6. We can also look at the sequence defined by the number of integer solutions by $k$. This is plotted in Figure 6 and remarkably shows a structure with no obvious regularity. A closer look allows us to discern the following patterns: 1) The most common number of solutions is 0. For $k=1..1000$ this happens with a probability of 0.549, i.e. more than half the times there are no integer solutions for Mordell’s equation. The next most frequent number of solutions is 2. This happens with a probability of 0.306 in this range. These are the cases when you just have two symmetric solutions differing in the sign of their $y$ value. 2) An odd number of solutions is obtained only when $k$ is a perfect cube. This is because only in this case we get the unpaired solution of the form $(-\sqrt[3]{k},0)$. The cubic powers of 2 are particular rich in solutions. E.g. $k=2^9=512$ yields 9 solutions: (-8,0); (-7,13); (4,24); (8,32); (184,2496); (-7,-13); (4,-24); (8,-32); (184,-2496). 3) If $k$ is a perfect square then for $k$, $k+1$ and $k-1$ we will have at least 2 solutions: for $k$ we have $(0,\pm\sqrt{k})$; for $k-1$ we have $(1,\pm\sqrt{k})$; for $k+1$ we have $(-1,\pm\sqrt{k})$. This would predict that, taken together, perfect square $k$ and their two immediate neighbors on either side would have a higher average number of solutions than an equivalent number of $k$ drawn at random in the same range. This is found to be the case empirically (Figure 7). Figure 7. The mean number of solutions of the square $k$ and their immediate neighbors is 4.108696 (red line in Figure 7) as opposed to the mean number of solutions of 1.522 (black line) for all $k=1..1000$. The former is 10.42 standard deviations away from the mean for equivalently sized samples drawn randomly from $k=1..1000$. 4) Further some squares and square-neighbors show what we call a lucky cubic conjunction (LCC), i.e. they generate a significantly larger number of perfect squares when summed with cubes than other numbers. One such square showing a LCC is $15^2=225$. It shows the record number of solutions (26) for $k=1..1000$: (-6,3); (-5,10); (0,15); (4,17); (6,21); (10,35); (15,60); (30,165); (60,465); (180,2415); (336,6159); (351,6576); (720114, 611085363); (-6,-3); (-5,-10); (0,-15); (4,-17); (6,-21); (10,-35); (15,-60); (30,-165); (60,-465); (180,-2415); (336,-6159); (351,-6576); (720114, -611085363). One can right away see that: $-6^3+225=3^2\\ -5^3+225=10^2\\ 4^3+225=17^2\\ 6^3+225=21^2\\ 10^3+225=35^2$ and so on. A square neighbor with a LCC is $17=16+1$ which has 8 cubic conjunctions leading to its 16 solutions: (-2,3); (-1,4); (2,5); (4,9); (8,23); (43,282); (52,375); (5234,378661); (-2,-3); (-1,-4); (2,-5); (4,-9); (8,-23); (43,-282); (52,-375); (5234,-378661). $1025=32^2+1$ is an even more monstrous square neighbor with a LCC outside the range that we systematically explored. This number has a whopping 16 cubic conjunctions giving rise to 32 solutions: (-10,5); (-5,30); (-4,31); (-1,32); (4,33); (10,45); (20,95); (40,255); (50,355); (64,513); (155,1930); (166,2139); (446,9419); (920,27905); (3631,218796); (3730,227805); (-10,-5); (-5,-30); (-4,-31); (-1,-32); (4,-33); (10,-45); (20,-95); (40,-255); (50,-355); (64,-513); (155,-1930); (166,-2139); (446,-9419); (920,-27905); (3631,-218796); (3730,-227805). In my computational exploration of these elliptic curves I am yet to find any other that out does 1025. 5) While $k$ which are squares and square neighbors have at least 2 solutions guaranteed, in principle a non-square or non-square neighbor number can show a LCC and give rise to a large number of solutions. One such as $k=297$ which shows 9 cubic conjunctions to give 18 solutions: (-6,9); (-2,17); (3,18); (4,19); (12,45); (34,199); (48,333); (1362,50265); (93844,28748141); (-6,-9); (-2,-17); (3,-18); (4,-19); (12,-45); (34,-199); (48,-333); (1362,-50265); (93844,-28748141). $k=873$ also shows a similar LCC, again with 9 cubic conjunctions. Outside the range we systematically explored, we found $k=2089$ to show a remarkable LCC with 14 conjunctions yielding 28 solutions: (-12,19); (-10,33); (-4,45); (3,46); (8,51); (18,89); (60,467); (71,600); (80,717); (170,2217); (183,2476); (698,18441); (9278,893679); (129968,46854861); (-12,-19); (-10,-33); (-4,-45); (3,-46); (8,-51); (18,-89); (60,-467); (71,-600); (80,-717); (170,-2217); (183,-2476); (698,-18441); (9278,-893679); (129968,-46854861). This is the second highest number of solutions we have seen for the Mordell’s equations we have studied. A reader might explore these and see if he can find a $k$ with bigger number of solution This entry was posted in Scientific ramblings and tagged , , , , , , , , , , , . 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": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 106, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9001230597496033, "perplexity": 568.9051270104701}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655896374.33/warc/CC-MAIN-20200708031342-20200708061342-00152.warc.gz"}
https://mathoverflow.net/questions/267459/induced-factorization-system-as-a-pullback-in-bf-cat	# Induced factorization system as a pullback in $\bf Cat$ $\require{AMScd}$If $\mathcal{X}$ is a category and $I$ a small category, the category of functors $\mathcal{X}^I$ inherits a (orthogonal) factorization system for each (orthogonal) factorization system on $\mathcal{X}$, defining the two classes objectwise. It seems to me that I can define this factorization system "formally" in this way. Call $(\cal A,B)$ the factorization system on $\mathcal X$. Then there is a pullback diagram $$\begin{CD} \mathcal{A}^I @>>> \mathcal{A}^{\|I\|} \\ @VVV @VVp_V\\\ \mathcal{X}^I @>>j^> \mathcal{X}^{\|I\|} \end{CD}$$ where $j : \|I\|\hookrightarrow I$ is the inclusion of the discrete subcategory of $I$ into $I$ itself, $j^$ is the induced functor, and $\mathcal{A}$ is regarded a the nonfull subcategory of $\mathcal{X}$ on the arrows of $\cal A$ and $p_$ comes from the obvious functor $\mathcal A \to \mathcal X$. Is this rewriting correct? If by $\mathcal{A}^I$ you mean the non-full subcategory of $\mathcal{X}^I$ corresponding to the left class of the induced factorization system (which is not the functor category of $I$ into $\mathcal{A}$), then yes, it does fit into such a pullback square. This doesn't construct the whole factorization system however. • ${\cal A}^I$ is precisely what you said (the notation was ambiguous, I must admit). Why it doesn't construct the FS? Once you have a class you get the other by orthogonality. – Fosco Apr 18 '17 at 16:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9839195609092712, "perplexity": 214.21378543749353}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496669431.13/warc/CC-MAIN-20191118030116-20191118054116-00040.warc.gz"}
https://www.physicsforums.com/threads/solid-conducting-sphere.177283/	# Solid conducting sphere 1. Jul 16, 2007 ### kiwikahuna 1. The problem statement, all variables and given/known data Consider a solid conducting sphere with a radius 1.5 cm and charge -4.4pC on it. There is a conducting spherical shell concentric to the sphere. The shell has an inner radius 3.7 cm and outer radius 5.1 cm and a net charge 27.4 pC on the shell. A) denote the charge on the inner surface of the shell by Q'2 and that on the outer surface of the shell by Q ''2 . Find the charge Q''2. Answer in units of pC. B) Find the magnitude of the electric field at point P, midway between the outer surface of the solid conducting sphere and the inner surface of the conducting spherical shell. Answer in units N/C. C) Find the potential V at point P. Assume the potential at r = infinity. Answer in units of volt. 2. Relevant equations E =kQ/r^2 V = kQ/r 3. The attempt at a solution I've figured out parts A and B but I'm struggling with Part C. I used the equation V = kQ/r where Q = -4.4e-12C ; k = 8.98755e9 and r = 0.026 m My answer (-1.52 V) is wrong but I have no idea why. Please help if you can. 2. Jul 17, 2007 ### chanvincent We know the total potential is merely the sum of its part, i.e. $$V_{total}=V_1+V_2+V_3+...$$ In your solution, I can see you have only considered the influence of the center solid conducting sphere, but you have not included the influence made by the spherical shell. In other words, you are missing a term in your solution... 3. Jul 17, 2007 ### kiwikahuna So we are only looking at the charge from the center of the solid conducting sphere and the inner charge of the shell? The inner charge of the shell would be the same charge as the charge from the center except its sign would be opposite. The charge from the center is -4.4e-12 C and the inner charge of the shell is +4.4e-12 C So should it be... V1 = (9e9) (-4.4e-12) / 0.026 m V2 = (9e9) (4.4e-12) / 0.026 m But then if you add V1 and V2 together, the total potential would be zero? 4. Jul 17, 2007 ### ice109 what is the answer? btw kiwikahuna you have a private message Last edited: Jul 17, 2007 5. Jul 17, 2007 ### kiwikahuna Unfortunately I don't know what the right answer is. Thanks for letting me in on the PM, ice. I hardly ever check it. ^_^ Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Similar Discussions: Solid conducting sphere	{"extraction_info": {"found_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": 0.9127503037452698, "perplexity": 832.7809399084953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891817523.0/warc/CC-MAIN-20180225225657-20180226005657-00527.warc.gz"}
https://stats.stackexchange.com/questions/230730/negative-binomial-for-underdispersed-data	# negative binomial for underdispersed data? I've read in several places that a negative-binomial model is a reasonable alternative to a Poisson regression when the latter shows overdispersion. However, none of the several sources I read said whether it is also an improvement over a Poisson that shows underdispersion. So is a negative-binomial worth considering for count data where the variance doesn't equal the mean? Or more specifically, only when the variance is larger than the mean? The variance of a variable $y$ with negative binomial distribution with expectation $\mathrm{E}(y) = \mu > 0$ and dispersion parameter $\theta > 0$ is $\mathrm{Var}(y) = \mu + \frac{1}{\theta} \cdot \mu^2$. Thus, the negative binomial distribution is always overdispersed with $\mathrm{Var}(y) > \mu$ and only reaches equidispersion for $\theta \rightarrow \infty$ (i.e., the Poisson distribution). If you do not need a full likelihood model, a quasi-Poisson model with estimated dispersion parameter (which can be $< 1$) may be useful as well.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 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": 0.9634106755256653, "perplexity": 418.01318507167974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573258.74/warc/CC-MAIN-20190918065330-20190918091330-00490.warc.gz"}
http://math.eretrandre.org/tetrationforum/showthread.php?mode=threaded&tid=210&pid=2527	• 0 Vote(s) - 0 Average • 1 • 2 • 3 • 4 • 5 Universal uniqueness criterion II bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 11/16/2008, 05:27 PM (This post was last modified: 11/18/2008, 11:32 AM by bo198214.) Lets summarize what we have so far: Proposition. Let $S$ be a vertical strip somewhat wider than $1$, i.e. $S=\{z\in\mathbb{C}: x_1-\epsilon<\Re(z) for some $x_1\in\mathbb{R}$ and $\epsilon>0$. Let $D=\mathbb{C}\setminus (-\infty,x_0]$ for some $x_0 and let $G\subseteq G'$ be two domains (open and connected) for values, and let $F$ be holomorphic on $G'$. Then there is at most one function $f$ that satisifies (1) $f$ is holomorphic on $D$ and $f(S)\subseteq G\subseteq f(D)=G'$ (2) $f$ is real and strictly increasing on $\mathbb{R}\cap S$ (3) $f(z+1)=F(f(z))$ for all $z\in D$ and $f(x_1)=y_1$ (4) There exists an inverse holomorphic function $f^{-1}$ on $G$, i.e. a holomorphic function such that $f(f^{-1}(z))=z=f^{-1}(f(z))$ for all $z\in G$. Proof. Let $g,h$ be two function that satisfy the above conditions. Then the function $\delta(z)=g^{-1}(h(z))$ is holomorphic on $S$ (because $h(S)\subseteq G$ and (4)) and satisfies $h(z)=g(\delta(z))$. By (3) and (4) $\delta(z+1)=g^{-1}(F(h(z)))=g^{-1}(F(g(\delta(z))))=g^{-1}(g(\delta(z)+1))=\delta(z)+1$ and $\delta(0)=0$. So $\delta$ can be continued from $S$ to an entire function and is real and strictly increasing on the real axis. $\delta(\mathbb{C})=\mathbb{C}$ by our previous considerations. By Big Picard every real value of $\delta$ is taken on infinitely often if $\delta$ is not a polynomial, but every real value is only taken on once on the real axis, thatswhy still $\delta(\mathbb{C}\setminus\mathbb{R})=\mathbb{C}$. But this is in contradiction to $g^{-1}:G\to D=\mathbb{C}\setminus [x_0,\infty)$. So $\delta$ must be a polynomial that takes on every real value at most once. This is only possible for $\delta(x)=x+c$ with $c=0$ because $\delta(0)=0$.$\boxdot$. In the case of tetration one surely would chose $x_0=-2$ and $x_1=0$ or $x_1=-1$. However I am not sure about the domain $G$ which must contain $f(S)$ and hence give some bijection $f:G\leftrightarrow S'$, with some $S'\supseteq S$. Of course in the simplest case one just chooses $G=f(S)$ if one has some function $f$ in mind already. However then we can have a different function $f_2$ with $f(S)\neq f_2(S)$ but our intention was to have a criterion that singles out other solutions. So we need an area $G$ on which every slog should be defined at least and satisfy $\text{sexp}(\text{slog})=\text{id}=\text{slog}(\text{sexp})$ as well as $\text{sexp}(S)\subseteq G$. « Next Oldest \| Next Newest » Messages In This Thread Universal uniqueness criterion II - by bo198214 - 11/16/2008, 05:27 PM Simplified Universal Uniqueness Criterion - by bo198214 - 11/16/2008, 06:13 PM RE: Simplified Universal Uniqueness Criterion - by Kouznetsov - 11/19/2008, 03:14 AM RE: Simplified Universal Uniqueness Criterion - by bo198214 - 11/19/2008, 01:16 PM RE: Universal uniqueness criterion II - by Kouznetsov - 11/17/2008, 01:27 AM RE: Universal uniqueness criterion II - by bo198214 - 11/17/2008, 02:45 PM RE: Universal uniqueness criterion II - by Kouznetsov - 11/18/2008, 01:25 AM RE: Universal uniqueness criterion II - by bo198214 - 11/18/2008, 11:31 AM RE: Universal uniqueness criterion II - by Kouznetsov - 11/19/2008, 02:59 AM Possibly Related Threads... 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http://www.research.lancs.ac.uk/portal/en/publications/freezein-dark-matter-from-a-subhiggs-mass-clockwork-sector-via-the-higgs-portal(8a84c9d5-fbf7-4505-befe-a89435144372).html	Home > Research > Publications & Outputs > Freeze-in dark matter from a sub-Higgs mass clo... ### Electronic data • PhysRevD.98 Final published version, 314 KB, PDF document ## Freeze-in dark matter from a sub-Higgs mass clockwork sector via the Higgs portal Research output: Contribution to journalJournal article Published Article number 123503 15/12/2018 Physical Review D 12 98 9 Published 7/12/18 English ### Abstract The clockwork mechanism allows extremely weak interactions and small mass scales to be understood in terms of the structure of a theory. A natural application of the clockwork mechanism is to the freeze-in mechanism for dark matter production. Here we consider a Higgs portal freeze-in dark matter model based on a scalar clockwork sector with a mass scale which is less than the Higgs boson mass. The dark matter scalar is the lightest scalar of the clockwork sector. Freeze-in dark matter is produced by the decay of thermal Higgs bosons to the clockwork dark matter scalars. We show that the mass of the dark matter scalar is typically in the 1-10 keV range and may be warm enough to have an observable effect on perturbation growth and Lyman-$\alpha$ observations. Clockwork Higgs portal freeze-in models have a potentially observable collider phenomenology, with the Higgs boson decaying to missing energy in the form of pairs of long-lived clockwork sector scalars, plus a distribution of different numbers of quark and lepton particle-antiparticle pairs. The branching ratio to different numbers of quark and lepton pairs is determined by the clockwork sector parameters (the number of clockwork scalars $N$ and the clockwork charge $q$), which could therefore be determined experimentally if such Higgs decay modes are observed. In the case of a minimal Standard Model observable sector, the combination of nucleosynthesis and Lyman-$\alpha$ constraints is likely to exclude on-shell Higgs decays to clockwork scalars, although off-shell Higgs decays would still be possible. On-shell Higgs decays to clockwork scalars can be consistent with cosmological constraints in simple extensions of the Standard Model with light singlet scalars.	{"extraction_info": {"found_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": 0.9791005849838257, "perplexity": 1693.7245145230336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145654.0/warc/CC-MAIN-20200222054424-20200222084424-00256.warc.gz"}
https://math.stackexchange.com/questions/600424/prove-that-the-gaussian-integers-ring-is-a-euclidean-domain	# Prove that the Gaussian Integer's ring is a Euclidean domain I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far. To be a Euclidean domain means that there is a defined application (often called norm) that verifies this two conditions: • $\forall a, b \in \mathbb{Z}[i] \backslash {0} \hspace{2 mm} a \mid b \hspace{2 mm} \rightarrow N(a) \leq N (b)$ • $\forall a, b \in \mathbb{Z}[i] \hspace{2 mm} b \neq 0 \rightarrow \exists c,r \in \mathbb{Z}[i] \hspace {2 mm}$ so that $\hspace{2 mm} a = bc + r \hspace{2 mm} \text{and} \hspace{2 mm} (r = 0 \hspace{2 mm} \text{or} \hspace{2 mm} r \neq 0 \hspace{2 mm} N(r) \lt N (b) )$ I have that the application meant to be the "norm" goes: $N(a +b i) = a^2 + b^2$, and I've managed to prove the first condition, given that N is a multiplicative function, but I can not find a way to prove the second condition. I've search for a similar question but I have not found any so far, please redirect me if there's already a question about this and forgive me for my poor use of latex. Let $a=\alpha_1+\alpha_2 i, b=\beta_1+\beta_2i$ where $\alpha_1,\alpha_2,\beta_1,\beta_2\in\Bbb Z$. Then $$\frac ab=\frac{\alpha_1+\alpha_2i}{\beta_1+\beta_2i}=\frac{(\alpha_1+\alpha_2i)(\beta_1-\beta_2i)}{N(b)}=\frac{(\alpha_1\beta_1+\alpha_2\beta_2)-(\alpha_1\beta_2-\alpha_2\beta_1)i}{N(b)}$$ By a modified form of the division algorithm on the integers, $\exists q_1,q_2,r_1,r_2\in\Bbb Z$ such that \begin{align}\alpha_1\beta_1+\alpha_2\beta_2&=N(b)q_1+r_1\\\alpha_1\beta_2-\alpha_2\beta_1&=N(b)q_2+r_2\end{align} Where $-\frac12N(b)\le r_\ell\le\frac12N(b)$. Then our quotient is $q=q_1-q_2i$ and our remainder is $r=r_1-r_2i$. Then $\frac ab=\frac{N(b)q+r}{N(b)}$ or $$a=bq-\frac{r}{\overline b}$$ By closure, $\frac{r}{\overline b}\in\Bbb Z[i]$, so $\frac{r}{\overline b}$ is the remainder. $$N\left(\frac{r}{\overline b}\right)=N\left(\overline {b^{-1}}\right)N(r)=N(b)^{-1}N(r)$$ While $N(r)=r_1^2+r_2^2\le2\left(\frac12N(b)\right)^2=\frac12N(b)^2$. Thus the remainder satisfies $$N\left(\frac{r}{\overline b}\right)\le \frac12N(b)^{-1}N(b)^2=\frac12N(b)$$ • Ok, as I commented to Dietrich, your answer follows the same logic as the proposition he talked about, the idea of using the division of $\mathbb{Q}[i]$ to get the structure of the remainder and verify the conditions, I'll look into it carefully and try to understand the whole proof, thank you :D – Rmongeca Dec 10 '13 at 11:24 • Why is $\frac{r}{\bar b}\in \Bbb Z[i]$? – Hrit Roy May 10 '19 at 15:23 • @HritRoy r/b = bq-a, and both a, b, and q are elements of Z[i]. – Alex Li Nov 19 '19 at 15:06 • Shouldn't $-\frac{r}{\overline{b}}$ be considered the remainder, since I believe the convention is that $a = bq + r$? – del42z Dec 5 '19 at 22:25 • $\frac{b}{N(b)} = \frac{1}{\bar{b}}$ so it's $a = bq + \frac{r}{\bar{b}}$ instead of minus. That said, this is one of the worst written proofs I have ever read. – AMRO Jan 3 at 21:24 For example : $\alpha=3+2i$ and $\beta=-10+6i$ Consider a circle of radius $\sqrt{3^2+2^2}$ by centered at $\beta$, and then move from $\alpha$ to $\beta$ ! $$\beta=\overrightarrow{AB}+\overrightarrow{BD}+\overrightarrow{DE}+\overrightarrow{Ef}+\overrightarrow{FG}+\overrightarrow{GH}+(-1-i) =$$ $$\alpha+i\alpha-\alpha+i\alpha-\alpha+i\alpha+(-1-i)=$$ $$(-1+3i)(3+2i)+(-1-i)$$ It's easy to generalize this idea to get a complete proof. And by this way it's easy to see why $q$ and $r$ in $\beta = q\alpha+r$ are not necessarily unique. Because you can move form $\alpha$ into the circle around the $\beta$, by different ways! Here is a different geometric proof. We have two Gaussian integers $a$ and $b$, and we have to prove that there exists a Gaussian integer $z$ such that $$\|az-b\|<\|a\|$$ Well, let's consider the set $A=\{az\mid z\in\mathbb Z[i]\}$. What does it look like? Writing $z=x+yi$, we see that $az=xa+y(ai)$. But $ai$ is just $a$, rotated clockwise by $90$ degrees, so $a$ and $ai$ form a pair of orthogonal vectors of length $\|a\|$. We now see that our set $A$ is a square lattice, it is the set of gridpoints of a square grid of mesh size $\|a\|$. We have to prove that at least one of these gridpoints is inside the open disc of radius $\|a\|$ centered at $b$. But in fact, something slightly stronger is true: given a square grid of mesh size $s$, any disc of radius $s$ must contain a grid point, no matter where it's placed. This is visually obvious: such a disc has diameter $2s$ whereas a square has a diameter of only $\sqrt 2 s$. There clearly isn't enough room to place a disc without overlapping a gridpoint. • Very nice proof. +1. – Anacardium Oct 23 at 19:14 The norm is $N(a+bi)=a^2+b^2$, and a proof is in many books on number theory. I recommend Ireland and Rosen, "A classical Introduction to Modern Number Theory, Proposition $1.4.1$, or http://homepage.univie.ac.at/dietrich.burde/papers/burde_37_comm_alg.pdf, Proposition $1.1.12$ for $\mathbb{Z}[\sqrt{-2}]$, $\mathbb{Z}[i]$, $\mathbb{Z}[\sqrt{2}]$, and $\mathbb{Z}[\sqrt{3}]$. • Ok, I've read the proposition you indicate. Basically it uses the same idea as the answer @TimRatigan wrote, using the division defined in $\mathbb{Q}[i]$ to get the structure of the remainder, and then verify the condition for the remainder, I'll look into it and try to understand it thoroughly, thank you :D – Rmongeca Dec 10 '13 at 11:17 ## Here is an Elegant Proof It is well known that $$(\mathbb{Z}[i]=\{a+bi \mid a,b \in \mathbb{Z}\},+,\cdot )$$ in integral domain. Consider, $$N:\Bbb Z[i] \to \Bbb N$$ defined by $$\color{blue}{N(z) = z\bar{z}=\|z\|^2 =a^2+b^2~~~ \text{for}~~z= a+ib.}$$ We want to show that $$N$$ define an integral function for our Ring $$\Bbb Z[i].$$ • $$N(0) = 0$$ and $$N(z) \gt 0$$ for $$z\neq 0.$$ • $$z,w,q\in \Bbb Z[i]\setminus\{0\}$$ such that $$z=wq$$ i.e $$w\|z$$ we have $$N(q) \gt 0 \implies N(q) \ge 1$$ since $$N(q) \in \Bbb N$$ Then, $$N(w) \le N(w)N(q) = \|w\|^2\|q\|^2 = \|wq\|^2 =N(wq) =N(z)$$ So, if $$w\|z$$ then $$N(w) \le N(z)$$. • Now we want to show the Euclidean division property. we use the following Lemma: for every $$x\in \Bbb R$$ there exists a unique $$u\in \Bbb Z$$ such that $$\color{blue}{ \|x-u\|\le \frac{1}{2}}$$ Proof: Let denote by $$\lfloor \ell \rfloor$$ is the floor of $$\ell$$. Then We know that $$\lfloor x+\frac{1}{2} \rfloor\le x+\frac{1}{2} \lt \lfloor x+\frac{1}{2}\rfloor +1\implies-\frac{1}{2}\le x -\lfloor x+\frac{1}{2} \rfloor\lt \frac{1}{2}$$ Taking $$u= \lfloor x+\frac{1}{2}$$ The unicity follows from the unicity follows from the unicity of the floor. Now let $$z,w\in \Bbb Z[i]\setminus\{0\}$$ then $$\frac{z}{w}$$ can be written as $$\color{red}{ \frac{z}{w}= x+iy :=\frac{z\bar{w}}{\|w\|^2}~~~~~ \text{with }~~~x,y\in \Bbb Q.}$$ From the Lemma there exist $$u,v\in \Bbb Z$$ such that $$\color{blue}{ \|x-u\|\le \frac{1}{2}~~~~\text{and}~~~\|y-v\|\le \frac{1}{2}}.$$ Then,we can write $$\color{red}{ \frac{z}{w}= x+iy = q +t~~~~~ \text{with }~~~q\in \Bbb Z, t\in \Bbb Q.}$$ Where, $$\color{blue}{q= u+iv ~~\text{and}~~~t =x-u+i(y-v)}$$. Then we have, $$\color{blue}{ z= qw +tw \implies r:= tw = z-qw \in \Bbb Z.}$$ Hence $$\color{red}{z= qw +r}$$ with $$q,r \in \Bbb Z$$ with $$r=tw$$ where we have, $$\color{blue}{ t =x-u+i(y-v),~~~\|x-u\|\le \frac{1}{2}~~~~\text{and}~~~\|y-v\|\le \frac{1}{2}}$$ Which means that, $$N(t) =\|x-u\|^2+\|y-v\|^2\le \frac{1}{2}$$ Therefore, $$N(r) =N(tw) =N(t)N(w) \le \frac12N(w) \lt N(w)$$ That is $$\color{red}{N(r) \lt N(w).}$$ conclusion N is divivion for the Ring $$\Bbb Z[i].$$ • Uniqueness in your lemma doesn't hold unless $\leq$ is switched to $<$. Counterexample: $x = 2.5,u_1 = 2, u_2 = 3$, then $\|x-u_1\| = \|x-u_2\| = 1/2 \leq 1/2$ but $u_1 \neq u_2$. When the inequality is changed to a strict inequality it can be shown to hold via the triangle inequality and the fact that the distance between two integers cannot be less than one. You don't appear to use uniqueness in this proof though so I believe its a fairly inconsequential error. – David Reed Nov 6 '19 at 6:02 • Of course, thank you for the remark. the strict inequality appears in the proof. – Guy Fsone Nov 6 '19 at 17:39 For the ring of Gaussian integers $$\Bbb{Z}[i] = \left\{ a + bi \mid a, b \in \Bbb{Z} \right\},$$ use the norm $$N(a + bi) = a^2 + b^2.$$ • Yes, I wanted to put this norm, but I got confused and wrote it wrong. Already corrected, thanks. – Rmongeca Dec 10 '13 at 11:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 39, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9743121862411499, "perplexity": 203.80531308626982}, "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-2020-50/segments/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00502.warc.gz"}
http://mathhelpforum.com/number-theory/152746-positive-solutions-linear-diophantine-equation-print.html	Positive Solutions to Linear Diophantine Equation • Aug 3rd 2010, 10:24 PM roninpro Positive Solutions to Linear Diophantine Equation I've been looking at the following problem, involving Diophantine equations: Let $a,b,c\geq 0$ such that $\gcd(a,b)=1$ and $c\geq (a-1)(b-1)$. Show that there exist nonnegative integers $s,t$ so that $as+bt=c$. I know that the solutions to a linear Diophantine equation are given by $x=x_0+bt, y=y_0-at$, where $(x_0,y_0)$ is a particular solution and $t$ is an integer parameter. If I want these to be positive simultaneously, I need to have $\dfrac{-x_0}{b}\leq t\leq \dfrac{y_0}{a}$. In other words, there needs to be an integer in the interval $\left[\dfrac{-x_0}{b}, \dfrac{y_0}{a}\right]$. From here, I cannot seem to put bounds on the endpoints of the interval. I would appreciate any insight you may have on the issue. • Aug 3rd 2010, 11:11 PM chiph588@ • Aug 3rd 2010, 11:22 PM roninpro Thanks. I looked into the link provided in the Yahoo answer and was pleasantly surprised to see that the proof given was geometric. I myself spent some time trying to handle the problem geometrically (with a similar approach), but I couldn't get it to work. I should be more careful next time!	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9409744739532471, "perplexity": 162.75463619672038}, "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-2016-44/segments/1476988719463.40/warc/CC-MAIN-20161020183839-00027-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathhelpforum.com/differential-geometry/226730-isometry.html	1. ## isometry Exercice: "Show that a map F:R^3->R^3 is an isometry if and only if there is an orthogonal matrix M (3 by 3 real matrix) and a vector v in R^3 such that F(x)=Mx +v. Prove also that arc length, curvature and torsion are preserved under isometries." I realy dont know how to start...could you help me starting? Thanks! 2. ## Re: isometry Originally Posted by Pipita Exercice: "Show that a map F:R^3->R^3 is an isometry if and only if there is an orthogonal matrix M (3 by 3 real matrix) and a vector v in R^3 such that F(x)=Mx +v. Prove also that arc length, curvature and torsion are preserved under isometries." I realy dont know how to start...could you help me starting? Thanks! An Isometry F is such that given y1=Fx1, y2=Fx2, then d(y1,y2) = d(x1,x2) where d() is the metric on your space. The usual Euclidean metric if not otherwise specified. so suppose Fx = Mx + v, i.e. a potential scaling/rotation and a translation. It should be pretty clear that the translation has no effect on the length. So you just have to show what properties M must have to preserve lengths. In the other direction suppose you have an isometry, then d(y1,y2)=d(x1,x2) and you can work out from there what linear transformations satisfy this and thus derive M and v. The rest of it is just chasing the transformation through the process of deriving arc length, curvature, torsion. 3. ## Re: isometry Originally Posted by romsek An Isometry F is such that given y1=Fx1, y2=Fx2, then d(y1,y2) = d(x1,x2) where d() is the metric on your space. The usual Euclidean metric if not otherwise specified. so suppose Fx = Mx + v, i.e. a potential scaling/rotation and a translation. It should be pretty clear that the translation has no effect on the length. So you just have to show what properties M must have to preserve lengths. In the other direction suppose you have an isometry, then d(y1,y2)=d(x1,x2) and you can work out from there what linear transformations satisfy this and thus derive M and v. The rest of it is just chasing the transformation through the process of deriving arc length, curvature, torsion. yeah i know the definition of isometry but i really dont know how to start this proof...it seems it misses some info 4. ## Re: isometry You are not giving a lot of information here. What "seems it misses some info"? Are you looking for a description for how to set up a proof? That is simple. You are asked to prove an if and only if statement. So, you assume that F(x) = Mx+v for some orthogonal matrix M and some vector v. Prove that F is an isometry. Next, suppose F is an isometry, and prove the existence of M and v. What exactly is the "missed info"?	{"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": 0.9551417827606201, "perplexity": 623.0675614554348}, "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-2018-17/segments/1524125946256.50/warc/CC-MAIN-20180423223408-20180424003408-00396.warc.gz"}
https://www.physicsforums.com/threads/interesting-question.105861/	# Interesting Question 1. Jan 5, 2006 Hey guys, u know how Euclid proved that primes r infinite. Now knowing that primes r infinite, if we take some primes p1, p2, p3,.....,pn then will p1p2.......pn(+/-)1 always be prime? 2. Jan 5, 2006 ### Muzza No. 5 - 1 and 5 + 1 are not prime, for example. 3. Jan 5, 2006 ### HallsofIvy Staff Emeritus Since 5 is not a product of primes, I don't see how that is relevant. The point was that Euclid showed that, if there exist only a finite number of primes, p1, p2, ..., pn, then their product plus 1 is not divisible by any of p1,..., pn and so is either prime itself or is divisible by a prime not in the original list- in either case a contradiction. Aditya89's question was whether that must, in fact, always be a prime. I don't believe so but I don't see a proof offhand. 4. Jan 5, 2006 ### Muzza It's perfectly relevant. Let n = 1 and p_1 = 5. But if you for some odd reason believe that n = 1 is forbidden, consider 2357 - 1 and 23571113 + 1. Last edited: Jan 5, 2006 5. Jan 5, 2006 ### mathwonk This brings up a point that comes up whenever you have to teach the prime factorization theorem. I have started saying something like " every integer greater than 1, is either prime or is a product of primes," because so many students fail to grok that a product of n factors can be prime if n = 1. this somewhat clunky statement can also help them when they have to use it, as sometimes the first case is to assume their number is prime, and the second case is to assume it is not, but is a product of primes. 6. Jan 5, 2006 ### shmoe If you want infinitely many counter examples, take any two odd primes (or actually any number of odd primes), p1, p2, then p1p2+/-1 will be even and greater than 2, hence composite. 7. Jan 5, 2006 ### vaishakh No, the thing meant is you can refer to this set {2,3,5,7,11,13,17,....} and by now I think you can predict the rest of the set. The thing to be proved is that if we find the product of some first n consecutive terms in this set, and add or subtract, the number we get NEED NOT BE A PRIME. 8. Jan 5, 2006 ### vaishakh Sorry, add or subtract "1", the number we get need not be a prime. But anyway Aditya wants us to prove that they are always prime. 9. Jan 5, 2006 ### Muzza In my world at least, "some primes" is not synonymous with "consecutive primes". Even that can't be done. See the first reply in this thread. Last edited: Jan 5, 2006 10. Jan 5, 2006 ### shmoe It's maybe worth pointing out that the Euclid proof does not need to say anything about the n primes you start with. They don't need to be consecutive. They don't need to be ordered. They don't need to be the first n. They don't even need to be unique. Take any n primes, multiply them, add 1, this new number must be divisible by a prime that you didn't start with, though it need not be prime itself. This guarantees that for any finite list of primes, you can always add one more. pi (really $$p_i$$) is often used to denote the nth prime in number theory, but not always, so you can usually expect an explicit statement to this effect. If someone writes "some primes p1, p2, ..., pn" there's no reason to assume any other meaning than some collection of n numbers that are all prime. 11. Jan 6, 2006 ### vaishakh You don't read Aditya's post. Read my post and prove or disprove it if you can. Interpret whatever you want to it, Just as Shmoe showed its very easy if you neglect two, by getting even numbers you are again reaching a composite number. But I think proving what I mean is quite difficult. I want to see if you can and moreover I am also interested the way to see how such proofs are framed in number theory because I have just now passed tenth grade and has no much knowldege on how to frame differant types proofs in number theory. 12. Jan 6, 2006 ### Muzza If anything, you haven't read MY post (if you had, you surely would have noticed that I even quoted from Aditya's post). I did disprove it, in the third reply to this thread. I gave the counterexamples 2357 - 1 and 23571113 + 1. Last edited: Jan 6, 2006 13. Jan 6, 2006 ### shmoe He did. I'm in complete agreement with Muzza and how he interpreted the OP, I was just adding some more info. You should look again at what I said about the $$p_i$$ notation. Even if you wanted to make the assumption that $$p_i$$ is the ith prime, a single value of n where p1p2...pn+1 is not prime would disprove this interpretation of the OP's question (likewise for the - case), which asked if it was always* prime (Muzza did this). 14. Jan 8, 2006 ### HallsofIvy Staff Emeritus Sorry Muzza, I see your point now. Aditya89's first post referred to Euclid's proof of the fact that there are an infinite number of primes and I didn't notice his reference to "some primes". Your counterexample clearly shows that the answer is "no". I'm still not clear if the product p1p2...pn+ 1 where the product is of all primes less than or equal to pn is necessarily prime. I suspect it isn't but don't see a proof. 15. Jan 8, 2006 ### shmoe He gave a counterexample to this as well, 235711*13+1. 16. Jan 15, 2006 ### benorin Here's how it goes: There are infinitely many primes Here's how it goes: Theorem There are infinitely many primes. proof: Suppose there are only infinitely many primes. Let $p_1,p_2,\ldots , p_n$ be the list of all prime numbers. Put $m=p_1p_2\ldots p_n+1$. Since m is clearly larger than 1, so m is either prime, or a product of primes. If m is prime, then this gives a contradiction as m is clearly larger than any prime on the above list. If m is not prime, then it is a product of primes. Let q denote such a prime (this would be on the list.) Then m is divisible by q, but m is not divisible by any $p_i$ for m divided by $p_i$ gives a quotient of $\frac{p_1p_2\ldots p_n+1}{p_i}$ and a remainder of 1. I coppied this proof out of How to Prove It: A Structured Approach by Daniel J. Velleman. 17. Jan 15, 2006 ### mathwonk this is un productive, but maybe that does not matter since someone wonders abnout it? 18. Feb 15, 2006 ### Canute For any series of consecutive primes p1xp2...pn the product N of this series +/-1 is not necessarily prime. This is because between pn and N there are primes which may be factors of N +/-1. Or is this not what was being asked? Similar Discussions: Interesting Question	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9259722828865051, "perplexity": 591.0547632964457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1505818688932.49/warc/CC-MAIN-20170922093346-20170922113346-00671.warc.gz"}
https://www.physicsforums.com/threads/chemical-reaction.111171/	# Chemical Reaction 1. Feb 18, 2006 ### PPonte CuSO4(aq) + Fe(s) --> FeSO4(aq) + Cu(s) FeSO4(aq) + Cu(s) --> CuSO4(aq) + Fe(s) Why does the iron plate becomes brown? I have no precise idea, but I think it might be related to the loss of electrons of the iron plate. Could someone help, please? 2. Feb 18, 2006 ### PPonte My idea was so stupid. I am almost sure that it is because the iron is going into solution and the copper is coming out of solution forming the brown thing in the iron plate. Am i right now? 3. Feb 18, 2006 ### GCT yep -----------------------------------------------------------------------	{"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": 0.9039324522018433, "perplexity": 3246.832110823441}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542455.45/warc/CC-MAIN-20161202170902-00411-ip-10-31-129-80.ec2.internal.warc.gz"}
http://harvard.voxcharta.org/2012/03/13/x-ray-dips-in-the-seyfert-galaxy-fairall-9-compton-thick-comets-or-a-failed-radio-galaxy/	We investigate the spectral variability of the Seyfert galaxy Fairall 9 using almost 6 years of monitoring with the Rossi X-ray Timing Explorer (RXTE) with an approximate time resolution of 4 days. We discover the existence of pronounced and sharp dips in the X-ray flux, with a rapid decline of the 2--20 keV flux of a factor 2 or more followed by a recovery to pre-dip fluxes after ~10 days . These dips skew the flux distribution away from the commonly observed log-normal distribution. Dips may result from the eclipse of the central X-ray source by broad line region (BLR) clouds, as has recently been found in NGC 1365 and Mrk 766. Unlike these other examples, however, the clouds in Fairall 9 would need to be Compton-thick, and the non-dip state is remarkably free of any absorption features. A particularly intriguing alternative is that the accretion disk is undergoing the same cycle of disruption/ejection as seen in the accretion disks of broad line radio galaxies (BLRGs) such as 3C120 but, for some reason, fails to create a relativistic jet. This suggests that a detailed comparison of Fairall 9 and 3C120 with future high-quality data may hold the key to understanding the formation of relativistic jets in AGN.	{"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": 0.9151127934455872, "perplexity": 1924.2089132224871}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738662321.82/warc/CC-MAIN-20160924173742-00182-ip-10-143-35-109.ec2.internal.warc.gz"}
http://physics.stackexchange.com/questions/32485/effects-of-a-non-lorentz-invariant-vacuum-state	# Effects of a non-Lorentz-invariant vacuum state I'm here asking about real or though experiments (i.e., physical effects) where, at least in principle, one can see some consequence of a non-Lorentz-invariant vacuum state in an otherwise Poincare invariant theory. Let me develop the question. Assume a theory in which the Hamiltonian (that closes Poincare algebra with the rest of generators) $H$ acts non-trivially on the vacuum state $\|0>$, where 'non-trivially' simply means: $$H\|0> = E \, \|0>$$ with $E$ a positive constant. Since the Hamiltonian transforms as the temporal component of a 4-vector, the vacuum state is not Lorentz invariant. Therefore, the theory is not Lorentz invariant (it is usually claimed that this is an additional condition besides the Poincare algebra). However, I'm not able to see any consequence of this fact. I think that this does not affect any cross-section or decay rate. I think that one can redefine the Hamiltonian so that $H'=H-E$ and $H'\|0>=0 \,$ (this does not affect the Poincare algebra if one also redefines the boost generators properly). I know this seems obvious (this redefinition is usually done in canonical quantization when one doesn't adopt normal ordering), but I've just read this in this forum: http://physics.stackexchange.com/a/8360/10522 However, in special relativity, energy is the time component of a 4-vector and it matters a great deal whether it is zero or nonzero. In particular, the energy of the empty Minkowski space has to be exactly zero because if it were nonzero, the state wouldn't be Lorentz-invariant: Lorentz transformations would transform the nonzero energy (time component of a vector) to a nonzero momentum (spatial components). One has a family of vacuum states related by Poincare transformations which are unitary, I don't think this is a problem... what do you think? Added: 1) I'm not thinking about a Poincare invariant Lagrangian with a potential of the form $(A^2(x)-v)^2$, where $A_{\mu}(x)$ is a vectorial field that acquires a vacuum expectation value $v$. Assume that every field has zero vev. 2) I'm looking for Lorentz violating effects instead of vacuum energy effects unless you argue that these vacuum energy effects (Lamb shift, spontaneous emission, etc.) break Lorentz symmetry. - – Qmechanic Jul 20 '12 at 22:06 I think it is not exactly the same question. The Hamiltonian of my theory is the special-relativistic one with one constant added. I'm going to edit the question to be more clear. – drake Jul 20 '12 at 22:16 The formulation of this question assumes that it is impossible to have a vacuum state where $\langle0\|H\|0\rangle > 0$ without violating Lorentz invariance. This is not true. Generally, when there is energy density in the vacuum, you have the appropriate pressure to keep Lorentz invariance, because the stress tensor ends up proportional to $g_{\mu\nu}$ is Lorentz invariant. In infinite space, the energy would be infinite, since it's a finite energy density. It you cut off the theory in a big box to regulate the energy, you do get a Lorentz breaking total energy in the vacum, but the breaking of Lorentz invariance is only coming from the fact that there are walls or identifications which pick out a Lorentz frame. If you boost the box, you have a momentum in the box, but that's just because the pressures on the edges of the boosted box are not balanced anymore, so that there is a net momentum coming in from the box walls, or if it is periodic, you have a net momentum from the moving periodic boundaries. So it is not true that vacuum energy breaks Lorentz invariance. This is a little counterintuitive, because we are used to energies being localized in a particle, so that pressure can't be constantly moving momentum around. In the vacuum, the energy is everywhere, and you can have pressure stresses which keep the whole formulation Lorentz invariant. This is why the vacuum energy infinity in field theories is said lead to cosmological constant renormalization, not Lorentz breaking. - Nice answer and good point the one of the proportionality between energy-momentum tensor and the Minkowski metric (upvote for you), however I'm not sure your answer is correct (maybe I'm missing something). I know that a box would break Lorentz, so let's assume that the volume is infinite and the energy momentum tensor is proportional to Minkowski metric and their components are finite. – drake Jul 23 '12 at 17:42 Continuation: To have a Lorentz invariant theory one needs an invariant vacuum state so $B_i \|0>=0$, where $B_i$ is the boost generator. And thus one should have $<0\|H\|0>=0$ since Poincare algebra contains $[B_i, P_j] \sim H \delta _{ij}$. In others words, if vacuum state carries energy-momentum vector $(E,0)$ (with $E$ different from zero), the vacuum cannot be Lorentz invariant. This argument suggests that (renormalized?) vacuum energy must be zero in a Lorentz invariant theory. – drake Jul 23 '12 at 17:43 One more point: the fact that the energy-momentum tensor (=stress tensor) be proportional to the Minkowski metric is a consequence of having a Poincare invariant action functional, but it doesn't say anything about the symmetries of the vacuum state, right? – drake Jul 23 '12 at 17:58 @drake: One could also have $\langle 0 \| H 0 \rangle = \pm\infty$ consistently--- that's the constant energy density and a cosmological constant. Of course you can't have a finite energy--- the space is infinite, you have a finite density. You are right that if it is any finite value, then it must be zero, but it's not deep at all. – Ron Maimon Jul 23 '12 at 18:46 Dear Ron, $B_i^{\dagger}=B_i$, $P_i\|0>$ finite, $B_i\|0>=0$, $[B_i, P_j]=i\,H \delta_{ij}$ $\implies$ $<0\|H\|0>=0$. Just take the vacuum expectation value of the commutator. – drake Jul 23 '12 at 19:01 In axiomatic quantum field theory, it is assumed that there is a unique Poincare invariant (projective) state. An arbitrary normalized representative $\|0\rangle$ of this state is called the vacuum state. Poincare invariance implies that for all $x$, the state $e^{x\cdot P/\hbar}\|0\rangle$ is a multiple of $\|0\rangle$. This implies that $P\|0\rangle=p\|0\rangle$ for some $p$. Redefining $P$ as $P-p$ gives another representation of the Poincare group in which the vacuum state has zero 4-momentum, as usually assumed. On the other hand, Rn Maimon considers a situation not covered by axiomatic QFT, where a state $\|0\rangle$ he calles (in my view mistakenly) the vacuum state is not in the domain of the translation group. Thus $e^{x\cdot P/\hbar}\|0\rangle$ is (fro nonzero $x$) not a state vector in the physical Hilbert space (but a proper distribution), the vacuum expectation value of the momentum vector does not exist, and the usual covariance arguments break down. As I understand him, he takes this to be the situation in an infinite universe with finite local momentum density but infinite total energy. However, in such a situation, Poincare invariance is no longer relevant. Indeed, in quantum gravity, there doesn't seem to be an observer-independent notion of a vacuum state, due to the Unruh effect. Instead opne has a family of Hadamard states that, taken together, replace the vacuum state of flat QFT. This family as a whole is indeed Poincare invariant, even invariant under the group of volume-preserving diffeomorphisms. -	{"extraction_info": {"found_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": 0.976489245891571, "perplexity": 320.90784676265963}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1418802770371.28/warc/CC-MAIN-20141217075250-00084-ip-10-231-17-201.ec2.internal.warc.gz"}
http://mathhelpforum.com/geometry/35584-height-rectangular-prism-print.html	# height of a rectangular prism • April 22nd 2008, 03:17 PM jwein4492 height of a rectangular prism I need to find the height of a rectangle prism the measurments are the area of base =20 sq. cm and the volume is 40 cm3 • April 22nd 2008, 05:00 PM Mathstud28 Quote: Originally Posted by jwein4492 I need to find the height of a rectangle prism the measurments are the area of base =20 sq. cm and the volume is 40 cm3 $V_{prism}=l\cdot{w}\cdot{h}$ $A_{base}l\cdot{w}=20$ make as substitution $V_{prism}=40=20\cdot{h}\Rightarrow{h=2}$ • April 22nd 2008, 05:04 PM Jen Quote: Originally Posted by jwein4492 I need to find the height of a rectangle prism the measurments are the area of base =20 sq. cm and the volume is 40 cm3 The volume of a rectangular prism is v=lwh We are given the area of the base which takes care of the length and width so now we have, $40cm^3=20cm^2*h$ So to isolate the height we divide both sides by $20cm^2$ which gives us, $\frac{40cm^3}{20cm^2}=h=2cm$	{"extraction_info": {"found_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": 0.8521549105644226, "perplexity": 1320.8507081645869}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1440644068184.11/warc/CC-MAIN-20150827025428-00323-ip-10-171-96-226.ec2.internal.warc.gz"}
http://kitada.com/List/time/199907/0061.html	# [time 478] Parallel translation, etc.. part IV Matti Pitkanen ([email protected]) Sat, 24 Jul 1999 10:40:08 +0300 (EET DST) This like Terminator, The return of Terminator I, II,....ad infinitum! In part IV we end up with the proposal that the replament of NP-computability with quantum computatibility by a infinite computer (the universe) with infinite computing time might be the TGD:eish version of computability(;-) > [MP] > > > > The point is that I do not identify observations as points of > > > > spacetime or configuration space. They are not active 'events'. > > > > Quantum jumps between quantum states= quantum histories are events and one > > > > cannot localize them to anywhere (one can of course, identify > > > > these events as pairs of possible initial and final quantum histories > > > > so that one can speak about the space of all possible experiences). > [SPK] > > > I say that observations are co-inductively related posets of points (as > > > an abstraction since true infinitesimal points can not be distiguised > > > from each other as they can not encode any information!!!), not > > > individual points per say. > [MP] > > I undestand that you great idea is identify observations with geometrical > > structures, 'posets of points of space'. I also parametrize > > the set of all possible observations: not as posets but as > > allowed quantum history pairs: but this parametrization tells > > anything about content of observation: it is just labelling: the only > > thing that matters that this naming scheme is one-to-one. I believe > > that the content of observation/cs experience cannot be expressed by any > > mathematical formula. > > Sure, but when we construct intricate geometrical model we are doing > just that! We are attempting to express the content of observations/cs > experience with a mathemathical formula! SO long as we understand that > the "model" is not the "thing" we are ok. > But what differentiates the model from the thing? Does this difference mean that mathematical formula does not characterize the observation completely? > > > To call them "space-time framings" speaks to > > > the fact that we always make observations in terms of a M^4 frame... The > > > quantum jumps are more epiphenomena that objective, but the restriction > > > that observers can only communicate effectively about M^4 framings that > > > do not logically contradict each other follows from CE! The > > > schizophrenic is an example of an observer that is attempting to > > > communicate about a M^4 framing that is logically inconsistent with > > > another's! > > > > > But there is the notion of observer. You take it as granted. I take > > observation fundamental. > > No! I, like you, take observation as fundamental! I just am being > explicit about the fact that what each observer has a framing of their > observations is not an a priori given, it is a construction! Thus, with > Pratt I say "cognito, ergo eram", I think therefore I was.... > I express it more technically: cogito, sequence of quantum jumps without any gap between existed (;-). Or even more precisely: a cascade like generation of selves within me occurred. > > > > I see no problems with Heisenberg's uncertainty relations: informational > > > > time development operator U reduces at QM limit to Schrodinger equation. > > > > Metric, etc.. classical gauge fields are not quantized in TGD. > > > > Neither spacetime coordinates are quantized. There is > > > > absolutely no quantization, only classical geometry of > > > > infinite-dimensional configuration space and classical spinor fields of > > > > configuration space. Oscillator operator algebras etc are geometrized in > > > > this approach. > > > > > > I see the evolution of information in terms of the evolution of the LSs > > > as they interact. Thus the act of the Universe 'experiencing itself' is > > > an ongoing process. It is what "concurrent computation" is all about is > > > a fundamental sense! Again, this is why I find Peter's work so useful! > > > :-) > > > > > > > Here here agree completely. But 'universe=experience about universe' > > is where I cannot follow you. This is simply too strong assumption and > > leads to the hopeless attempt of writing formula for the contents > > of cs experience. > > The Universe is not experience about the Universe! I apologize if I did > not express this well. The Universe is mere Existence of All that > exists, everything simultaneously. Yes. I understood this from above. This would be for me the space of all universes, quantum histories. >Experiences are \epsilon-consistent > information structures (Complete Atomic Boolean Algebras are examples) > that require finite material structure to be actualized. An observation > is an actualization, it is the "enbodiment" of the information. Again I follow: quantum jumps, actualizations, mean hopping around space of quantum histories. > Descartes was incorrect in his dualism, because he assumed that mind and > matter were invariant substances, I, with Pratt, propose that they are > "acts", and acts can only be actual in time. Existence has no time > associated, thus it does not generate local experience; it is the local > actualities of LSs that do that. :-) > > snip > [MP] > > > > I just saw a paper in which it was shown that divergence problem is > > > > not solved by noncommutatitivy of the spacetime coordinates. There is also > > > > problem with the loss of general coordinate invariance. One must assume > > > > special coordinates and very high symmetries if one wants > > > > special coordinates. > > > > > > Could you give me the reference of this paper? I can order a copy from > > > the library! :-) > > > > It was paper by my 'boss' Masud-Chaichian and Peter Presnajder and third > > theoretician. They constructed noncommutative QFT in two-dimensional case. > > For cylinder it worked but for more general case they found infinities. > > They also suggested generalization of results to higher dimensional case. > > I do not have the paper here but I could ask for bibliodata. > > Thanks! ;-) I try to find the paper next week. > > snip > > > [MP] > > > > God of Singularity concept is based on traditional concept of > > > > psychological time. Also the question why everything has not > > > > happened is created by the same concept of psychological time. > > > > > > Yes, but here the error is in the tacit (subconscious) assumption that > > > the psychological time of one person is one and the same of that of > > > another! The clocking by the QM propagator of the LS defines the > > > individual time of the LS, thus representing psychological time very > > > well! > > > > Also that. But the real blunder is the identification > > is the assumption that contents of cs experience correspond to > > time=constant snapshot. > > Neurophysiologists tell us that this is not the case. Consdier music > > as example. We are able to experience frequencies, which > > is nonlocal concept with respect to time. > > Yes, that is why I use an M^4 to frame an observation, there are both > spatial and temporal non-localities involved. This is also why we can > use a RW metric to model how the space-time configurations of a single > observation are distributed! This speaks to concurrence, we are not able > to experience points, we experience hyper-surfaces! So, we agree here! > :-) Not quite! Experiencing of mere hypersurfaces would not make possible experiening of frequencies: complete localization in time means by uncertainty principle of Fourier Analysis means that there is not frequency information. Cognition must be time nonlocal if it is to give some information about what will happen and happened. This is why cognitive spacetime sheets made possible by the classical nondeterminism of Kahler action are so crucial for TGD. > > > The "everything has not already happened" notion is indeed related to > > > psychological time, but in the sense stated above. "Everything" much > > > included all possible "actual" experiences, and obviously, these involve > > > NP-complete computational issues! This later notion is at the heart of > > > my argument. > > > > > > > I regard this concept as badly wrong. In TGD framework subjective time > > > > corresponds to quantum jumps and there can be no first quantum jump. > > > > > > I am not communicating my notion since your are not aware of the > > > NP-completeness problem! Karl Svozil's papers point the way! We agree > > > that "there can be no first quantum jump"! The ideas of co-induction > > > and related issues involved are in Peter's papers... > > > > I have studied Peter's papers (rather technical!). My view is that cannot > > start from so technical concept like NP-completeness in building model > > for universe: the reason is that I do not believe that universe computes > > itself into existence: it just exists! Even more, it is able to replace > > itself with a new one again and again and do also some computation > > besides that! > > Matti, NP-Completeness is not merely a technical concept! It is a > fundamental problem! How does the Universe calculate the minimum energy > configuration of a protein molecule? How does the Universe figure out > the most stable orbits in a stellar system? How is it that soap bubbles > always cover the most volume with the least surface? How is it that a > quasi-crystal can grow at all? How is it that Lagrangians are calculated > by the Universe? All of these questions are aspects of the > NP-Completteness problem! But why universe should calculate it? Even for modelling purposes in some remote psychological future and even at subjective distance of infinitely many quantum jumps? And how should universe calculate itself to existence: does the hardware used belong to universe. This like Munchausen trick: logical impossibility. Quantum jump replaces the computation (in classical sense as I understand). Quantum jump is what allows quasi-chrystal to grow! In the initial universe quasi-chrystal cannot grow but by quantum jumps one ends up to the universe where quasi-christal has grown. > What is interesting is that it has been proven that if there exists a > finite computational scheme that can compute a given example of an > NP-complete problem, this scheme or algorithm can be transformed in > polynomial time into a scheme to compute any other NP-Complete problem. > But this, I think, only works for situations that can be modeled (or > "simulated") by Turing Machines. Peter's work shows us that most of the > computations that occur are not TM simulatable and thus we need to look > at this more closely. Perhaps it is not an accident that quantum jump can be regarded at general level as infinitely long quantum computation. Psi_i corresponds to initial state of quantum computer. UPsi_i corresponds to the final state of qcomputer after infinitely long calculation and UPsi_i-->Psi_f means halting of quantum computation and emerges of the result of computation as conscious experience. What about NP completeness problem when one introduces infinitely large quantum computers calculating infinitely long time? Can nondeterministic computations help. What about sequences of quantum computations each lasting infinitely long time?: these are suggested by the notion of self. Thought as a cascade of quantum jumps creating hierarchy of subselves of self? [To avoid confusions: the calculation time has nothing to do with the experience psychological time]. > [MP] > > > > This requirement plus p-adic evolution > > > > as gradual statistical increase of p-adic prime of the universe > > > > immediately leads to the requirement that > > > > also infinite p-adic primes are possible and that recent universe > > > > must correspond to infinite prime. Every moment of consciousness > > > > decomposes to infinite number of subexperiences with values of > > > > psychological time ranging from zero to infinity. What we really 'know' is > > > > that local arrow of psychological > > > > time exists: if one is satisfied with this then paradoxes disappear. > > > > Universe becomes 4-dimensional living being getting conscious > > > > information about its entire 4-dimensional body in every quantum jump. > > > > Cognitive spacetime sheets are the sensory organs of this infinitely > > > > large 4-dimensional living system. > [SPK] > > > The notion of "gradual statistical increase of p-adic prime of the > > > universe" is given in my thinking in the sense that the overall > > > concurrent interactions of LSs are modelable in this way. It would be a > > > representation of the accumulated "experience of the Universe", but with > > > the caveat that it is information that can not be gotten in finite time > > > or finite energy with arbitrary accuracy. The operator formalisms that > > > Schommers talks about, I believe, is useful to us in thinking about > > > this! > > > > > > > Yes: I have grasped the universe computes itself into existence philosophy > > but.... > > Umm, I do not say that "the universe computes itself into existence"! I > say that the individual experiences of Local Systems (using Hitoshi's > definition of LSs) are given in terms of space-times framings. This > follow from the distinction that I make between "existence" (qua CE) and > "actuality" which is a "local notion" that represents the subjective > experiences (observations, measurements, etc.) of an LS given any > particular moment of their local time. Thus I say that the Universe > experiences itself by the acts of observation of the finite LSs, which > are considered computations of NP-complete problems. LS:s as computationas of quantum-computable problems? This would be TGD inspired computationalism!(;-). > The key argument is that nothing can "happen" unless a price is paid. > Existence in-itself does not require the generation of equilibria. It > is at equilibria with respect to itself, that is why it merely exists. > It does not change, it has no duration or extension or any other > properties other that mere existence. It is the grundlagen! I do not > associate any space or time properties to it, those are the properties > of the observations of Local Systems, not the Universe itself. > > > > > > > [SPK] > > > > > > This notion is very different from Hitoshi's idea, but perhaps the > > > > > > difference is due to the different ways that time is treated. > > > > > > I still see these as complementary! You see space-times as a priori > > > > > > surfaces, subsets of the totality U that are connected by quantum jumps > > > > > > "in time", Hitoshi, as I understand, sees space-times as the "clocked" > > > > > > poset of observations of LS, which are a priori quantum mechanical > > > > > > systems existing tenselessly as subsets of the totality U. > > > > > > Thus you are proposing space-times as a priori and Hitoshi > > > > > > proposes quantum local systems as a priori, this is a chicken-egg > > > > > > complementarity! We need to see that this is just a matter of > > > > > > perspective! > > > [MP] > > > > I have the feeling that this is not a matter of perspective. Our > > > > basic philosophies are different. > > > > > > In a way, yes. You are a Platonist, and I something different. I think > > > that Plato's Idea Reality is the Universe in-itself, but as Kant > > > argued well, is not knowable in-itself. I see it as Existence itself! > > > All experiences, measurements, observations, qualia or what ever, are > > > not given directly by their mere existence (as all exist in the > > > ontological sense), but have finite properties given by the interactions > > > between the finite subsets or "facets" or LSs of the Universe. We can > > > only observe shadow, consciousness is not capable of knowling the > > > "in-itself"! > > > > Funny thing, reading this I find that I agree completely. But somewhere > > the differences emerge: computationalism is one of the division lines. > > And this is exactly why we must discuss the notions we have about > computation! > Well. Replacing NP-computability with quantum-computability is what TGD would suggest. Best, MP This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:36:57 JST	{"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": 0.9308829307556152, "perplexity": 4113.2957151891105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887746.35/warc/CC-MAIN-20180119045937-20180119065937-00242.warc.gz"}
http://rainnic.altervista.org/index.php/en/content/beamer-presentation-template?language_content_entity=en	A beamer presentation template Submitted by Nicola Rainiero on 2012-12-05 (last updated on 2016-08-18) I publish the template which I used for the presentation of my Thesis, it could still be useful for write any types of presentation. Latex and Beamer gave me the faculty to focus my main attention on contents, avoiding the definition of color scheme and style. Beside the PDF output works in each PC and projector. How many times have you seen some presentations with errors in pagination of contents and with wrong colors for text and background? Sometimes it happens because changes the final version of the program for the presentation or the author didn't test his work on the projector. You can avoid these problems using Beamer, a LaTeX class for creating slides for presentations. In a few words these are some features1: • easy management of transition effects; • sectioning of your contents like a LaTeX document standard; • possibility to use other LaTeX packages (for example you can embed 3D object as I explained in this article); • wide choice of themes and color scheme (as documented in the following gallery); • in every slides you can navigate the own presentation and the buttons are not invasive; • on the background you can put a TOC of your exposition, where corresponding item changes the color when it is activated; The template It follows my simple TeX template with some comment of explaination. You can print the slides (4 slide for page) commenting the first row and removing the comment in the second one. Here is the zip file: Presentation_Template_TeX.zip. While at this link you can see my original and full presentation. \documentclass[11pt,xcolor={dvipsnames}]{beamer} % presentation output % \documentclass[11pt,xcolor={dvipsnames},handout]{beamer} % Beamer printout % xcolor allows to use many new colors with \usecolortheme \mode{ \usetheme{Warsaw} % Here is a gallery with other themes: % http://deic.uab.es/~iblanes/beamer_gallery/ \usecolortheme[named=OliveGreen]{structure} % Others: OliveGreen, Brown, Sepia, RawSienna, \setbeamercovered{transparent} \setbeamercolor{block title example}{fg=white,bg=Blue} \setbeamercolor{block body example}{fg=black,bg=Blue!10} \setbeamercolor{postit}{fg=black,bg=OliveGreen!20} \setbeamercolor{postit2}{fg=yellow,bg=OliveGreen} % \setbeamercolor{NEW_STYLE_NAME}{fg=COLOR_FOREGROUNG,bg=COLOR_BACKGROUNG} } %% Setting for Beamer printout % reference: http://mathoverflow.net/questions/5893/beamer-printout \usepackage{pgfpages} \mode{ \usetheme{default} \setbeamercolor{background canvas}{bg=Black!5} \pgfpagesuselayout{4 on 1}[a4paper,portrait,border shrink=2.5mm] % 4 slide in one page } %% Setting for Beamer printout \usepackage[italian]{babel} \usepackage[latin1]{inputenc} \usepackage{times} \usepackage[T1]{fontenc} \usepackage{graphics} \graphicspath{{images/}} % all the graphics files will go in the subdirectory images \usepackage{numprint} % with this one \np{1000} becomes 1 000 \usepackage{mathcomp} \usepackage{gensymb} % with this one \numprint[\textcelsius]{20} becomes 20°C \newcommand{\ud}{\mathop{}\ \mathrm{d}} % with this one \ud{x} becomes dx \usepackage{mathtools} \DeclarePairedDelimiter{\abs}{\lvert}{\rvert} % to define absolute value (mathtools is required) \hypersetup{ pdfsubject={UNIVERSITY, DEPARTMENT}, pdfauthor={NAME SURNAME}, pdfkeywords={KEY1, KEY2, KEY3, etc.}, pdfpagemode=FullScreen, % once opened it goes in fullscreen modality %citecolor=black, %filecolor=black, %urlcolor=black } \usepackage[absolute,overlay]{textpos} \setlength{\TPHorizModule}{1mm} \setlength{\TPVertModule}{1mm} %%%% A NEW COMMAND TO FIX LOGO POSITION (x,y) in mm \newcommand{\MyLogo}{% \begin{textblock}{14}(2.0,0.1) % \pgfuseimage{logo} \includegraphics[height=1.15cm, angle=0]{logo} \end{textblock} } %%%% A NEW COMMAND TO FIX LOGO POSITION (x,y) in mm %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \author[NAME SURNAME] {NAME SURNAME} \institute[INSTITUTE NAME] { COMPLETE NAME OF THE INSTITUTE\\ OTHER INFORMATION \\ DEPARTMENT NAME \\[0.5cm] SUPERVISOR\\ Prof. Eng. \textbf{NAME SURNAME}\\[0.25cm] CORRELATOR\\ Eng. \textbf{NAME SURNAME}\\ } \date{December 2, 2012} %\logo{\includegraphics[height=1.5cm, angle=0]{logo}} % To have a logo on each page... BAD RESULT!! %\titlegraphic{\includegraphics[height=1.4cm, angle=0]{logo}} % To have an imagie on title page %%%% TO HAVE A TOC ON EVERY SLIDE %\AtBeginSubsection[] %{ % \begin{frame}{Sommario} % \tableofcontents[currentsection,currentsubsection] % \tableofcontents[currentsection] % \tableofcontents % \end{frame} %} %%%% TO HAVE A TOC ON EVERY SLIDE \begin{document} \transduration{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% TITLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \transdissolve \MyLogo \begin{center} % \includegraphics[height=1.5cm, angle=0]{unipd} \titlepage \end{center} \end{frame} %%%% TOC \begin{frame}{Contents} \transboxin \MyLogo %\tableofcontents[pausesections,part=1] \tableofcontents \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%% FIRST SECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{TITLE OF SECTION 1} \subsection{TITLE OF SUBSECTION 1.1} \begin{frame}{TITLE OF FRAME 1} \transboxin %\transblindshorizontal % type of transition effect \MyLogo \begin{center} \includegraphics[width=.3\textwidth]{image} \begin{block}{Block title} Description of this block. Description of this block. Description of this block. Description of this block. \end{block} \end{center} \end{frame} \begin{frame}{TITLE OF FRAME 2} %\transblindshorizontal \MyLogo \begin{center} \includegraphics[width=.3\textwidth]{image} Description of this block. Description of this block. Description of this block. Description of this block. \\ \vspace{0.8cm} \end{center} \end{frame} \begin{frame}{TITLE OF FRAME 3} %\transglitter \MyLogo \begin{columns} \column{.6\textwidth} \includegraphics[width=.6\textwidth]{image} \column{.4\textwidth} \begin{block}{Block title} Description of this block. Description of this block. Description of this block. Description of this block. \end{block} \bigskip \bigskip \begin{block}{Block title} Description of this block. Description of this block. Description of this block. Description of this block. \end{block} \end{columns} \end{frame} \begin{frame}{TITLE OF FRAME 4} %\transglitter \MyLogo \begin{columns} \column{.6\textwidth} \includegraphics<1>[width=.5\textwidth]{image1} \includegraphics<2>[width=.5\textwidth]{image} \includegraphics<3>[width=.5\textwidth]{image3} \column{.4\textwidth} \begin{itemize} \end{itemize} \end{columns} \end{frame} \begin{frame}{TITLE OF FRAME 5} %\transdissolve \MyLogo \begin{columns} \column{.4\textwidth} \begin{itemize} \item Text text text \item Text text text \item Text text text \item Text text text \end{itemize} \column{.6\textwidth} \includegraphics [width=.5\textwidth]{image} \end{columns} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame}{TITLE OF FRAME 6} \framesubtitle{SUBTITLE OF FRAME 6} %\transdissolve \MyLogo \begin{columns} \column{.25\textwidth} \begin{align} &= \dfrac{\abs{Q_2}}{\abs{Q_2}-Q_1} \\ \end{align} \column{.75\textwidth} \includegraphics[width=.5\textwidth]{image1} \end{columns} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{TITLE OF SUBSECTION 1.2} \begin{frame}{SUBTITLE OF FRAME 7} %\transblindshorizontal \MyLogo \begin{enumerate} \item<1-> Text text text text text text text text text text text text text text text text text text text text text text text text text text; \item<2-> text text text text text text text text text text text text text text text text text text text text text text text text text; \item<3-> text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text. \end{enumerate} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{TITLE OF SUBSECTION 1.3} \begin{frame}{SUBTITLE OF FRAME 8} %\transblindshorizontal \MyLogo \begin{center} \includegraphics[width=.2\textwidth]{image1} \end{center} \pause \begin{itemize} \item text text text text text text text text text text \item text text text text \item text text text text text text text text text text text text \item text text text text \end{itemize} \end{beamercolorbox} \end{frame} \subsection{TITLE OF SUBSECTION 1.4} \begin{frame}{SUBTITLE OF FRAME 9} \transdissolve \MyLogo \begin{itemize} \item<1-> Text text text text text text text text text text text text text text text text text text; \item<2-> text text text text text text text text text text text text text text text text text text text text text text text text text; \item<3-> text text text text text text text text text text. \end{itemize} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%% SECOND SECTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{TITLE OF SECTION 2} \subsection{TITLE OF SUBSECTION 2.1} \begin{frame}{SUBTITLE OF FRAME 10} \transboxin \MyLogo \begin{itemize} \item<1-> text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text text \numprint[m]{100} text text text text $\numprint[]{4} \div \numprint[kW_t]{7}$ \end{beamercolorbox} \item<2-> text text text text text text text text text text text text text text text text text text text text \item<3-> text text text text text text text text text text text text text text text text text text text text \end{itemize} \end{frame} % % ... % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LAST FRAME %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{frame} \transboxin \MyLogo \vspace{1.0cm} \end{beamercolorbox} \pause \end{frame} \end{document} Related Content: Nicola Rainiero A civil geotechnical engineer with the ambition to facilitate own work with free software for a knowledge and collective sharing. Also, I deal with green energy and in particular shallow geothermal energy. I have always been involved in web design and 3D modelling.	{"extraction_info": {"found_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": 0.9803392887115479, "perplexity": 1521.1649200848024}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527518.38/warc/CC-MAIN-20190419081303-20190419103303-00327.warc.gz"}
https://socratic.org/questions/how-do-you-solve-the-polynomial-inequality-and-state-the-answer-in-interval-nota-3	Precalculus Topics # How do you solve the polynomial inequality and state the answer in interval notation given 4x^4>=3x+1? $\left[1 , + \infty\right)$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 1, "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": 0.997009813785553, "perplexity": 994.8068844557927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 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/1579251778168.77/warc/CC-MAIN-20200128091916-20200128121916-00246.warc.gz"}
https://iitutor.com/factorising-difference-of-squares-non-monic-coefficients/	# Factorising Difference of Squares: Non-Monic Coefficients ## Transcript All right! We’ve got x squared. But, what about 4? Try to change this whole thing to a square number guys. What squared is 4? 4 is two squared, isn’t it? And 9, 9 is 3 squared, so I can make this one, 2x, the whole thing squared because it’s 2 squared, x squared. So I can change this to 2x, the whole thing squared and this one is 3 squared, so this time we have 2x and 3, so we go 2x plus 3, 2x minus 3, okay? Get the idea? So, that’s what we do. The same kind of thing, what squared is 16? 4, 4 squared is 16. And 25 is 5 squared. So, this one guys, this one will be 4 squared and x squared, isn’t it? Which is 4x the whole thing squared? So you make it into 1 and 25 is 5 squared, so it’s going to be 4x and 5 that we’re going to be using, so it’s 4x plus 5, 4x minus 5, okay? So, it’s very very easy as long as you can identify the squares. Just be careful when you find the squares. A fraction, but it’s okay. We do the same kind of thing. All right! This one, think about it like this guys. 4x squared over 9. It’s 4 over 9x squared, isn’t it? So 4 over 9, what squared is that? 4 over 9, it’s 4 is 2 squared, 9, it’s 3 squared and we’ve got the x squared. See how they’re all squares, so you can make that into 1 by changing it to 2x over 3, the whole thing squared, okay? So you just need one bracket with one square and you know that 25 is 5 squared and 16 is 4 squared, so we just have 5 on 4 squared, just make it square minus a square like that. So this is what we concentrate and this is what we concentrate on, so it’s 2x oh sorry 2x over 3 plus 5 on 4 times 2x over 3 minus 5 over 4, okay? So it’s a bit more confusing when you have fractions but just keep it the same method. Trying to look at the squares of both numerator and 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": 0.8098242878913879, "perplexity": 704.9724701207547}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943746.73/warc/CC-MAIN-20230321193811-20230321223811-00339.warc.gz"}

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