Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2016-05-06T13:34:29Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/2381570Question on Littlewood-Paley trichotomysamhttp://mathoverflow.net/users/584992016-05-06T12:48:13Z2016-05-06T12:48:13Z
<p>In proving the product estimate, we need the Littlewood-Paley trichotomy. See <a href="http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps" rel="nofollow">http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps</a>.</p>
<p>In the decomposition
$$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f P_{k''} g)$$</p>
<p>The relation is obtained in analysis of the support of $D'=2^{k'-1} \le |\xi| \le 2^{k'+1}$,$D''=2^{k''-1} \le |\xi| \le 2^{k''+1}$. The note says $P_{k'} f P_{k''}g$ has Fourier support in $D'+D''$, what does it mean? I think it seems one need convolution to get the support for $|\xi|$. But I am unable to carry it out and derive the decomposition. How could we choose $k' \le k-5$ and find $k'' \in [k-3,k+3]$ to be the intersection region with $2^{k-1} \le |\xi| \le 2^{k+1}$?</p>
http://mathoverflow.net/q/2381560Proof of partial derivative of a distributionuser3183950http://mathoverflow.net/users/913142016-05-06T12:41:26Z2016-05-06T12:41:26Z
<p>I wanted to proof the formula for the partial derivative of a distribution, but I have an error on my result and I don't understand where I am wrong.</p>
<p>Here is my proof :</p>
<p>I assume that my function f is discontinuous on a surface. I call "Disc" ensemble of (y,z) where on a given x there is a discontinuity.</p>
<p>$$ < \frac{ \partial [f]}{\partial x} , \phi > = - \int dx dy dz f(x,y,z) \frac{ \partial \phi}{\partial x} = - \int_{Disc \times \mathbb{R} } dx dy dz f(x,y,z) \frac{ \partial \phi}{\partial x} - \int_{( \mathbb{R^2} \backslash Disc ) \times \mathbb{R}} dx dy dz f(x,y,z) \frac{ \partial \phi}{\partial x} $$</p>
<p>And
$$\int_{Disc \times \mathbb{R} } dx dy dz f(x,y,z) \frac{ \partial \phi}{\partial x} = -\int dy \int_{Disc} dz \int_{\mathbb{R}} dx \frac{ \partial f}{\partial x}(x,y,z) \phi(x,y,z) + \int dy \int_{Disc} dz (f(x^+(y,z),y,z)-f(x^-(y,z),y,z))\phi(x(y,z),y,z) $$</p>
<p>The second term come from the integration by part (because here, f is discontinuous).</p>
<p>$$ \int_{( \mathbb{R^2} \backslash Disc ) \times \mathbb{R}} dx dy dz f(x,y,z) \frac{ \partial \phi}{\partial x}=-\int_{( \mathbb{R^2} \backslash Disc ) \times \mathbb{R}} dx dy dz \frac{ \partial f}{\partial x}(x,y,z) \phi(x,y,z)$$</p>
<p>Outside "Disc", everything is smooth, so I don't have another term.</p>
<p>Finally, I have :</p>
<p>$$< \frac{ \partial [f]}{\partial x} , \phi >=-\int \int \int dx dy dz \frac{ \partial f}{\partial x}(x,y,z) \phi(x,y,z) + \int dy \int_{Disc} dz (f(x^+(y,z),y,z)-f(x^-(y,z),y,z))\phi(x(y,z),y,z) $$</p>
<p>So :
$$ < \frac{ \partial [f]}{\partial x} , \phi >=< [\frac{ \partial f}{\partial x}] , \phi > + \int dy \int_{Disc} dz (f(x^+(y,z),y,z)-f(x^-(y,z),y,z))\phi(x(y,z),y,z) $$</p>
<p>My second term is then not what I should have and I don't understand where I am wrong.</p>
<p>Could you help me ?</p>
<p>Thanks !</p>
http://mathoverflow.net/q/2381550Passing motivic decompositions from rational to algebraic equivalencenxirhttp://mathoverflow.net/users/512512016-05-06T12:39:42Z2016-05-06T12:39:42Z
<p>It is well known that there are several adequate equivalence relations for algebraic cycle (see <a href="https://en.wikipedia.org/wiki/Adequate_equivalence_relation" rel="nofollow">https://en.wikipedia.org/wiki/Adequate_equivalence_relation</a> for a list including definitions).</p>
<p>The classic category of motives $\mathcal{M}_k$ over a field $k$, defined by Grothendieck is based on choosing rational equivalence for $\sim$. But one can also choose other equivalence relations for $\sim$, and thus get a motivic category $\mathcal{M}^\sim_k$ , which might have different properties (being Tannakian for example).</p>
<p>Assume we have a motivic decomposition of a smooth, projective variety $X$ over $k$,</p>
<p>$M(X) = \bigoplus_{i\in I} M_i$,</p>
<p>in $\mathcal{M}_k$. </p>
<p>1.Is it known how the decomposition of $X$ in $\mathcal{M}^\sim_k$ will differ from the above?</p>
<p>1.1.Is it true, that one will always have less or at most the same number of summands?</p>
<p>2.Can you give a specific example (aside from projective spaces)?</p>
<p>I am interested in the case for $\sim$ to be $alg$, but every example is welcome, considering how little is probably known.</p>
<p>To give more context. I am trying to find out more about criteria for the motive of a variety to be not decomposable. So lifting and descent properties are, what i am really interested in.</p>
http://mathoverflow.net/q/2381533Physical meaning of the Lebesgue measureuser21820http://mathoverflow.net/users/500732016-05-06T11:57:32Z2016-05-06T13:21:44Z
<h2>Question (informal)</h2>
<blockquote>
<p>Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the Jordan measure? Specifically, is there a Jordan non-measurable but Lebesgue-measurable subset of Euclidean space that has physical meaning? If not, then is there a Jordan measurable set that has no physical meaning?</p>
</blockquote>
<p>If you understand my question as it is, great! If not, in the subsequent sections I will set up as clear definitions as I can so that this question is <strong>not opinion-based</strong> and has a correct answer that is one of the following:</p>
<ol>
<li><p>Yes, some Jordan non-measurable subset of Euclidean space has physical meaning.</p></li>
<li><p>No, there is no physically meaningful interpretation of Jordan non-measurable sets (in Euclidean space), but at least Jordan measurable sets do have physical meaning.</p></li>
<li><p>No, even the collection of Jordan measurable sets is not wholly physically meaningful.</p></li>
</ol>
<p>In all cases, the answer must be justified. What counts as justification for (1) would be clear from the below definitions. As for (2), it is enough if the theorems in present scientific knowledge can be proven in some formal system in which every constructible set is Jordan measurable, or at least I would like citations of respected scientists who make this claim and have not been disproved. Similarly for (3), there must be some weaker formal system which does not even permit an embedding of Jordan sets but which suffices for the theorems in present scientific knowledge!</p>
<h2>Definitions</h2>
<p>Now what do I mean by <strong>physical meaning</strong>? A statement about the world has physical meaning if and only if it is empirically verified, so it must be of the form:</p>
<blockquote>
<p>For every object X in the collection C, X has property P.</p>
</blockquote>
<p>For example:</p>
<blockquote>
<p>For every particle X, its speed measured in any reference frame does not exceed the speed of light.</p>
</blockquote>
<p>By <strong>empirical verification</strong> I mean that you can test the statement on a large number of instances (that cover the range of applicability well). This is slightly subjective but all scientific experiments follow it. Of course empirical verification does not imply truth, but it is not possible to empirically prove anything, which is why I'm happy with just empirical evidence, and I also require empirical verification only up to the precision of our instruments.</p>
<p>I then define that a <strong>mathematical structure $M$ has physical meaning</strong> if and only if $M$ has a physically meaningful interpretation, where an <strong>interpretation</strong> is defined to be an embedding (structure-preserving map) from $M$ into the world. Thus a physically meaningful interpretation would be an interpretation where all the statements that correspond to structure preservation have physical meaning (in the above sense).</p>
<p>Finally, I allow approximation in the embedding, so $M$ is still said to have <strong>(approximate) physical meaning</strong> if the embedding is approximately correct under some asymptotic condition.</p>
<p>For example:</p>
<blockquote>
<p>The structure of $V = \mathbb{R}^3$ has an (approximate) physically meaningful interpretation as the points in space as measured simultaneously in some fixed reference frame centred on Earth.</p>
</blockquote>
<p>One property of this vector-space is:</p>
<blockquote>
<p>$\forall u,v \in V\ ( |u|+|v| \ge |u+v| )$.</p>
</blockquote>
<p>Which is indeed empirically verified for $|u|,|v| \approx 1$, which essentially says that it is correct for all position vectors of everyday length (not too small and not too big). The approximation of this property can be written precisely as the following pair of sentences:</p>
<blockquote>
<p>$\forall ε>0\ ( \exists δ>0\ ( \forall u,v \in V\ ( |u|-1 < δ \land |v|-1 < δ \rightarrow |u|+|v| \ge |u+v|-ε ) ) )$.</p>
</blockquote>
<p>This notion allows us to classify scientific theories such as Newtonian mechanics or special relativity as approximately physically meaningful, even when they fail in the case of large velocities or large distances respectively.</p>
<h2>Question (formal)</h2>
<blockquote>
<p>Does the structure of Jordan measurable subsets of $\mathbb{R}^3$ have (approximate) physical meaning? This is a 3-sorted first-order structure, with one sort for the points and one sort for the Jordan sets and one sort for $\mathbb{R}$, which function as both scalars and measure values.</p>
<p>If so, is there a proper extension of the Jordan measure on $\mathbb{R}^3$ that has physical meaning? More specifically, the domain for the sort of Jordan sets as defined above must be extended, and the other two sorts must be the same, and the original structure must embed into the new one, and the new one must satisfy non-negativity and finite additivity. Bonus points if the new structure is a substructure of the Lebesgue measure. Maximum points if the new structure is simply the Lebesgue measure!</p>
<p>If not, is there a proper substructure of the Jordan measure on $\mathbb{R}^3$ such that its theory contains all the theorems in present scientific knowledge (under suitable translation; see (*) below)? And what is an example of a Jordan set that is not an element in this structure?</p>
</blockquote>
<h2>Remarks</h2>
<p>A related question is what integrals have physical meaning. I believe many applied mathematicians consider Riemann integrals to be necessary, but I'm not sure what proportion consider extensions of that to be necessary for describing physical systems. I understand that the Lebesgue measure is an elegant extension and has nice properties such as the dominated convergence theorem, but my question focuses on whether 'pathological' sets that are not Jordan measurable actually 'occur' in the physical world. Therefore I'm not looking for the most elegant theory that proves everything we want, but for a (multi-sorted) structure whose domains actually have physical existence.</p>
<p>The fact that we do not know the <strong>true</strong> underlying structure of the world does not prevent us from postulating embeddings from a mathematical model into it. For a concrete example, the standard model of PA has physical meaning via the ubiquitous embedding as binary strings in some physical medium like computer storage, with arithmetic operations interpreted as the physical execution of the corresponding programs. I think most logicians would accept that this claim holds (at least for natural numbers below $2^{1024}$). Fermat's little theorem, which is a theorem of PA, and its consequences for RSA, has certainly been empirically verified by the entire internet's use of HTTPS, and of course there are many other theorems of PA underlying almost every algorithm used in software!</p>
<p>Clearly also, this notion of embedding is not purely mathematical but has to involve natural language, because that is what we currently use to describe the real world. But as can be seen from the above example, such translation does not obscure the obvious intended meaning, which is facilitated by the use of (multi-sorted) first-order logic, which I believe is sufficiently expressive to handle most aspects of the real world (see the below note).</p>
<p>(*) Since the 3-sorted structure of the Jordan measure essentially contains the second-order structure of the reals and much more, I think that all the theorems of real/complex analysis that have physical meaningfulness can be suitably translated and proven in the associated theory, but if anyone thinks that there are some empirical facts about the world that cannot be suitably translated, please state them explicitly, which would then make the answer to the last subquestion a "no".</p>
http://mathoverflow.net/q/2381520A question about the definition of complete dg Lie algebras in a paper of Lazarev and MarklDaniel Robert-Nicoudhttp://mathoverflow.net/users/441342016-05-06T11:24:19Z2016-05-06T11:24:19Z
<p>In their paper <a href="http://arxiv.org/abs/1305.1037" rel="nofollow">Disconnected Rational Homotopy Theory</a>, Lazarev and Markl give the following definition (page 23):</p>
<blockquote>
<p><strong>Definition:</strong> A <em>complete differential graded Lie algebra</em> is an inverse limit of finite-dimensional, nilpotent differential graded Lie algebras (dgla).</p>
</blockquote>
<p>The limit here is taken in the category of dglas, which is complete (and cocomplete). However, a couple of lines later, they show that complete dglas are pronilpotent assuming, apparently without loss of generality, that any complete dgla is a <em>sequential</em> limit of finite-dimensional nilpotent dglas, which I strongly believe is a more stringent condition that being an arbitrary limit. So what am I missing? Is the definition incorrect as stated (in the sense that we allow only some special kinds of limits), is the proof that complete dglas are pronilpotent wrong, or is there some implication or equivalence between various kinds of limits in the category of dglas that I don't know about?</p>
<p>In particular, I am interested to the model structure that they put on the category of complete dglas a bit later in the paper (page 36). We need the category to be complete and cocomplete, and if we can take arbitrary limits then completeness is obvious, while if we only allow e.g. sequential limits I'm not sure it holds.</p>
<p>I will be grateful for any contribution.</p>
<p><strong>Remark:</strong> Cross-posted from <a href="http://math.stackexchange.com/q/1772980/60713">math.SE</a>.</p>
http://mathoverflow.net/q/2381510One problem about tower stabilityAlexei Fedotovhttp://mathoverflow.net/users/44752016-05-06T11:06:49Z2016-05-06T11:06:49Z
<p>Some years ago i asked myself a question that i still can not answer. Here it is. Let be given a tower consisting of finite homogeneous equal to each other cubic blocks staying one on another. What is the condition for stability of such tower? First one can consider 2-dimensional analogue of this problem where 2-dimensional tower is staying on a real line. </p>
http://mathoverflow.net/q/238150-1Connected Lie subgroups of SU(2) and SU(3) [on hold]Quickhttp://mathoverflow.net/users/913082016-05-06T10:43:51Z2016-05-06T11:02:27Z
<p>Two questions:</p>
<p>Why is every connected subgroup of SU(2) closed?</p>
<p>Can we find a non closed connected subgroup of SU(3)?</p>
http://mathoverflow.net/q/2381480Functional equation: Can we find a function that satisfies this equation?Peterhttp://mathoverflow.net/users/913072016-05-06T10:21:15Z2016-05-06T12:49:20Z
<p>$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$</p>
<p>Is there a function $f$ from $R^{\infty}$ to $R^{\infty}$ that satisfies this equation for all natural ${\alpha}$ and ${\beta}$ ?</p>
<p>I already know that any function that has $f\left( \overrightarrow {0}\right)= \overrightarrow {0}$ satisfies the equation, so are there any other functions that satisfy the equation?</p>
<p>Thank you in advance!</p>
<p>*My phrasing for this being a "funtional equation" was flawed. All I really wanted to know was the existance of a function that satisfies the equation above.</p>
http://mathoverflow.net/q/2381472Effective divisor in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-classesIgnacio Barroshttp://mathoverflow.net/users/497422016-05-06T09:39:05Z2016-05-06T09:39:05Z
<p>Does anybody knows an effective class in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-coefficients? The standard references; Logan, Farkas or Brill-Noether divisors have all non-negative coefficients for the $\psi$-classes.</p>
<p>It would be much appreciated. Thanks .</p>
http://mathoverflow.net/q/2381451A question on uniformly corepresented functorRonhttp://mathoverflow.net/users/582032016-05-06T09:08:17Z2016-05-06T09:08:17Z
<p>Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly corepresented. Is the fiber product $\mathcal{F} \times_{h_M} h_U$ corepresented by $U$?</p>
<p>Any good reference on uniform corepresented functor will be most helpful.</p>
<p>PS. Here $k$ is an algebraically closed field</p>
http://mathoverflow.net/q/2381440How to solve these simultaneous equations? [on hold]Hanjun koohttp://mathoverflow.net/users/913032016-05-06T08:41:39Z2016-05-06T08:53:10Z
<p>Find the all possible pairs of $x,y,z$ so that</p>
<blockquote>
<p>1) $y^3 = x^3 + 9x^2 - 9x +8$<br>
2) $y^2 = z^3 + 17$<br>
Note that, $x,y,z$ are all positive integers.</p>
</blockquote>
<p>I've tried many ways to solve this problem but all failed.</p>
<p>For example, I tried to divide into the cases where $y$ is an even number or not.</p>
<p>My second attempt was via factorization. But I failed again.</p>
<p>How can I solve this?</p>
<p>Any hint or answer will be appreciated. :)</p>
http://mathoverflow.net/q/2381430Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean PlaneManfred Weishttp://mathoverflow.net/users/313102016-05-06T08:34:58Z2016-05-06T08:34:58Z
<p>I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the $[0,1)\times [0,1)$ square and then simply count the number of occurences of the figures I am interested in (a simple example would be counting convex vs non-convex quadrilaterals, i.e. Sylveser's Four Point Problem).</p>
<p>Now I suspect, that using a finite region instead of the infinite plane leads to some bias near the boundary of the region. </p>
<p>My idea to reduce that bias somewhat would be to </p>
<ul>
<li><p>introduce a wrap around, i.e. by making the point set periodic in $x$- and in $y$-direction. </p></li>
<li><p>iterate over all sample points $p_i:=(x_i,y_i)\in [0,1)\times[0,1)$ </p></li>
<li><p>augment $p_i$ with points $q\ \in\ [x_i,x_i+1)\times[y_i,y_i+1)\setminus p\quad $ to generate e.g. a sample of four distint points from the random points. </p></li>
<li><p>check the properties of the determined subset of the points and update the statistics accordingly</p></li>
</ul>
<blockquote>
<p><strong>Questions:</strong> </p>
<ul>
<li><p>does the "wrap-around" of point sets give better approximations of the statistics of infinite points, when compared to ordinary iteration over e.g. the 4-tuples of the unit square. </p></li>
<li><p>are there better ways of mimicking an infinite, random discrete point set that is uniformly distributed over the entire Euclidean plane with finite set of points?</p></li>
</ul>
</blockquote>
http://mathoverflow.net/q/2381426Union-closed family generated by n 2-setsPanurgehttp://mathoverflow.net/users/828402016-05-06T08:33:33Z2016-05-06T11:12:43Z
<p>I asked this question on <a href="http://math.stackexchange.com/questions/1772398/union-closed-family-generated-by-n-2-sets">Stackexchange</a>, but I got no answer, so I ask it here.</p>
<p>Let us define a $2$-set as a set with exactly $2$ elements. For a natural number $n$, let $l(n)$ denote the least possible number of members of a union-closed family of sets generated by $n$ distinct $2$-sets. I'm interested in a useful formula or minoration for $l(n)$.</p>
<p>There are easy majorations, for example $l({r\choose 2}) \leq 2^{r} - 1 - r$ and (for $r \geq 1$) $l({r\choose 2}+1) \leq 2^{r} + 2^{r-1} - 1 - r$, and these upper bounds seem to be exact values for small values of $r$, but I would avoid the task of handling this question if there is literature about it. Do you know ? Thanks in advance.</p>
<p>Note : there is a similar question here :
<a href="http://mathoverflow.net/questions/75585/kruskal-katona-type-question-for-union-closed-families-of-sets">Kruskal-Katona type question for union-closed families of sets</a></p>
<p>but not identical.</p>
http://mathoverflow.net/q/238141-4significance level from 0.10 to 0.05, increase or decrease? [on hold]H Wuhttp://mathoverflow.net/users/913022016-05-06T08:22:39Z2016-05-06T08:22:39Z
<p>A trend test show the trend is downward (alpha<0.10). Then the trend is still downward, but alpha<0.05. Then I describe as follows:
1. The statistical significance is increasing.
2. The statistical significance is decreasing.
3. The significance level is increasing.
4. The significance level is decreasing.
Which is right?</p>
http://mathoverflow.net/q/2381400Invariance of Gauss-Bonnet theorem with respect to connection?Ayanhttp://mathoverflow.net/users/255162016-05-06T07:58:40Z2016-05-06T07:58:40Z
<p>I am stuck with a basic understanding of the generalized (and even the ordinary version of) Gauss-Bonnet theorem. For a compact 2-dimensional Riemannian manifold $M$ with boundary $\partial M$, let $K$ be the Gaussian curvature of $M$ and $k_g$, the geodesic curvature of $\partial M$. Then</p>
<p>$$\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M),$$</p>
<p>where $\chi(M)$ is the Euler characteristic of $M$. My questions are:</p>
<ol>
<li><p>The Gaussian curvature and the geodesic curvature are functions of the connection that one puts on $M$, and in the standard version of the theorem, we usually put the induced Euclidean connection on the 2-manifold $M$ from its embedding space $\mathbb{R}^3$; whereas, the right hand side of the above equation is a topological invariant of $M$, and, thus, is independent of any connection that we adorn $M$ with. Then the left hand side, as well, should be independent of the connection. How is this invariance with respect to the connection on $M$ is concealed in the left hand side integrals?</p></li>
<li><p>I have only seen the statement of the generalized Gauss-Bonnet theorem (from the book "From Calculus to Cohomology") and I guess it partially answers my question, but I don't have a clear understanding of its underpinning. The generalized version says that, for any 2$n$-dimensional compact oriented smooth manifold $M$,</p></li>
</ol>
<p>$$\int_M Pf\bigg(\frac{-F^{\nabla}}{2\pi}\bigg)=\chi(M)$$</p>
<p>holds, where $F^\nabla$ is the curvature associated with <em>any metric connection</em> on the tangent bundle of $M$. Here, $Pf:\mathfrak{so}_{2n}\to\mathbb{R}$ is something called the Pfaffian and is defined on the space of skew-symmetric matrices.</p>
<p>So the definition of the Pfaffian must be the answer to my question. So how does the Pfaffian make the left hand side invariant with respect to the connection? Why do we require the evenness of the dimension and orientation of $M$? And finally, why do we require a metric-compatible (perhaps, torsion free) connection in the first place?</p>
<p>A detailed explanation would be much appreciated. I have just started reading on Gauss-Bonnet theorem and I guess my query lies at the heart of the underlying philosophy of this gem theorem of differential topology.</p>
<p>Looking forward to a detailed explanation or references on this particular explanation.</p>
<p>(I think partial answer to my question is in Prof. Bryant's answer to this <a href="http://mathoverflow.net/questions/84521/a-question-on-generalized-gauss-bonnet-theorem">A question on Generalized Gauss-Bonnet Theorem</a>.)</p>
http://mathoverflow.net/q/2381283What are the sense and reference of the propositions $R$$\notin$$R$,$R$$\in$$R$, where $R$={x|x$\notin$x} in Frege's Grundgesetze?Thomas Benjaminhttp://mathoverflow.net/users/205972016-05-06T05:21:36Z2016-05-06T11:13:34Z
<p>In their paper, "Frege's New Science" (Notre Dame Journal of Formal Logic, Vol. 41,No. 3, 2000), Antonelli and May give the following quote of Frege, from his paper "Uber die Grundlagen der Geometrie," (<em>Jahresbericht der Deutschen Mathematiker Vereinigung</em>, vol. 15 (1906), pp. 293-309, 377-403, 423-30. English translation by E-H. W. Kluge in _Gottlob Frege, <em>On the Foundations of Geometry and Formal Theories of Arithmetic</em>, Yale University Press, 1971, pg. 69):</p>
<blockquote>
<p>What I call a proposition <em>tout court</em> or a real proposition is a group of of signs that expresses a thought [has a sense--my comment]; however, whatever only has the grammatical form [i.e. being a well-formed formula--my comment] of a proposition I call a pseudo-proposition.</p>
</blockquote>
<p>Since it is well-known that</p>
<blockquote>
<p>$R$$\notin$$R$ $\Leftrightarrow$ $R$$\in$$R$ (this is actually true if both $R$$\notin$$R$ and $R$$\in$$R$ are false, but then this seems to contradict the Law of Excluded Middle)</p>
</blockquote>
<p>can one rightly attribute either a reference (since they are 'propositions', their reference must be, according to Frege, either The True, or The False) or a sense (what would that be, exactly) to '$R$$\notin$$R$' or '$R$$\in$$R$'? And if not, are '$R$$\notin$$R$', '$R$$\in$$R$' "pseudopropositions"?</p>
<p>Furthermore, since</p>
<blockquote>
<p>$R$$\notin$$R$ $\Leftrightarrow$ $R$$\in$$R$</p>
</blockquote>
<p>can be derived from Basic Law $V$, shouldn't Basic law $V$ be restricted to apply only to "real propositions"? And how might that be done in the context of the <em>Grundegesetze</em>? </p>
http://mathoverflow.net/q/2381273Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebraStevenhttp://mathoverflow.net/users/887882016-05-06T05:01:50Z2016-05-06T13:05:32Z
<p>Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.</p>
<p>Tilting module theory play an important role in the study $\mathcal{O}$. They are sometimes projective covers and hence injective since the duality functor preserve tilting modules. Is it true that every tilting module has simple socle?</p>
<p>Thanks!</p>
http://mathoverflow.net/q/2381201What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?Jeff http://mathoverflow.net/users/912732016-05-06T01:16:14Z2016-05-06T12:58:09Z
<p>Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?</p>
<p>Existing results:</p>
<p>It has been known that $E|E(X|\mathcal{A})-E(X)|\le 2\alpha(\sigma(X),\mathcal{A})$, where $\alpha(\sigma(X),\mathcal{A})$ is the $\alpha$ mixing coefficient between $\sigma(X),\mathcal{A}$, and the $\sigma(X)$ is the $\sigma$-algebra generated by $X$. This result shows that $|E(X|\mathcal{A})-E(X)|=O_P(\alpha(\sigma(X),\mathcal{A}))$. </p>
<p>In practice, the above upper bound is too loose, and we need an exponential type upper bound to strengthen it. Suppose we can prove the following
$$
E\phi(|E(X|\mathcal{A})-E(X)|/C)\le\alpha(\sigma(X),\mathcal{A}),
$$
where $\phi(x)=\exp(x^2)-1$. Then one can argue that<br>
$$
|E(X|\mathcal{A})-E(X)|=O_P(\sqrt{\log(1+\alpha(\sigma(X),\mathcal{A})}).
$$
Clearly, this largely improve the first upper bound to a sharper level.</p>
<p>My question is how to prove this?</p>
http://mathoverflow.net/q/2381181On the numerical range of non-self adjoint Gaussian matrixgondolfhttp://mathoverflow.net/users/49872016-05-06T01:06:36Z2016-05-06T11:03:56Z
<p>For a complex $n \times n$ matrix $A$, its numerical range is the set</p>
<p>$$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$</p>
<p>We can further define the smallest absolute value of the numbers in the numerical range as
$$r(A) = \inf\ \{ |\lambda| : \lambda \in W(A) \} = \inf_{\|x\|=1} |\langle Ax, x \rangle|.$$</p>
<p>$M=(m_{l,k})_{n\times n}$ is called non-self adjoint Gaussian random matrix if $m_{l,k}$ are i.i.d. standard complex Gaussians $~N(0,1/n)+iN(0,1/n)$.</p>
<p>We are interested in the probability density function of $r(M)$ as $M$ being distributed as non-self adjoint Gaussian random matrix.</p>
<p>What is the probability that $r(M)$ is larger than $1-\epsilon$ for some given constant $\epsilon$?</p>
http://mathoverflow.net/q/2381090Simulated Annealing algorithm for MINLP [on hold]MR Meyqanihttp://mathoverflow.net/users/912812016-05-05T20:53:44Z2016-05-06T10:11:43Z
<p>In the objective function of a mathematical programming model,we have an expression like this:
$$
\biggl(\biggl|X\biggl| \biggl) . Q
$$
in which both X and Q are continuous variables, and $||$ indicates the absolute value function.
It's clear that above expression cannot be transformed to a linear one,using Linearization techniques.Because of this,I decided to solve my MINLP model by applying the Simulated Annealing algorithm.My question is,has any research been done so far to solve MINLPs with SA without linearzing them first?</p>
http://mathoverflow.net/q/2381021random consecutive decreasing subset/chain in point processuser3760541http://mathoverflow.net/users/662512016-05-05T19:05:36Z2016-05-06T07:55:45Z
<p>In my study of percolation system, I encounter a very interesting problem. I tried to map it into well-studied permutation problem but not very successful... I debrief it as follows:<br>
imagine you have a unit square and a sequence of uniform points $\{X_i\}_{i>=0}$ is constructed in the square. You can understand it as a stochastic process.<br>
Next, let us define a consecutive decreasing chain: assume we have a consecutive subsequence $\{X_j\}_{j=i}^{i+m}$, there exists a permutation of the subscripts $\pi :j\rightarrow\pi(j)\in\{i,...,i+m\}$ and if $j\neq j'$, then $\pi(j)\neq \pi(j')$ such that $X_{\pi(i)}\succ X_{\pi(i+1)}\succ ...\succ X_{\pi(i+m)}$, where $X=(x,y)\succ X'=(x',y')$ stands for $x>x'$ and $y<y'$. Then we call $\{X_j\}_{j=i}^{i+m}$ forms a consecutive decreasing chain.
By above definition, we can further define a stopping time sequence $\{T_k\}_{k\ge 0}$ in this way: $T_0=0$ and $T_k=\max\{j:\{X_j\}_{j=T_{k-1}}^{j-1}\ is\ a\ consecutive\ decreasing\ chain\}$
I am interested in the behavior of $T_{k+1}-T_k$ for any fixed $k$, i.e., the length of the $kth$ consecutive decreasing chain. Various questions could be asked about it such as expectation, typical value, variance and all associated asymptotic values.</p>
http://mathoverflow.net/q/2380973Solving algebraic recurrence relations on a cyclic graphDavid R. MacIverhttp://mathoverflow.net/users/49592016-05-05T17:59:21Z2016-05-06T09:53:17Z
<p>I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms:</p>
<ol>
<li>$p_i = 0$.</li>
<li>$p_i = 1$</li>
<li>$p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all distinct.</li>
<li>$p_i = q p_j + (1 - q) p_k$ for some other $j, k$ and $0 < q < 1$</li>
</ol>
<p>(i.e. these are activation probabilities for distributions of boolean valued random variables, each of which is either a constant, a mixture of two others or a conjunction of two others)</p>
<p>In general there may be and probably are cycles in the equations where the definition of $i$ depends on the definition of $j$ depends on the definition of $i$, etc. </p>
<p>For a given set of equations, I would like to determine:</p>
<ol>
<li>If there is a unique solution to these equations</li>
<li>A closed form for it, or at least an algorithm for producing an exact answer, if there is one.</li>
</ol>
<p>(I am not optimistic about the second, but it sure would be nice)</p>
<p>I'd also appreciate literature pointers if this refers to a common class of things that I just don't know the term for.</p>
<p>In practice I will probably just solve this approximately as an iterative solution - the multiplicative terms suggest that it will tend to converge to a fixed point pretty fast - but I would like to know if there's a better way.</p>
http://mathoverflow.net/q/2380865Semi-continuity of intersection numbersGiuliohttp://mathoverflow.net/users/488662016-05-05T15:26:54Z2016-05-06T12:57:15Z
<p>I always trusted the following quite vague statement:</p>
<p>If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in disc, then the intersection number $D_1(t)\cdots D_k(t)$ can only increase under specialisation of $t$.</p>
<p>Without any flatness assumption on $D_i(t)$ and $X_t$ this is false, as shown in the post
<a href="http://mathoverflow.net/questions/176064/semicontinuity-of-degree-of-fibers-for-a-proper-map">Semicontinuity of degree of fibers for a proper map</a></p>
<p>I guess, under an appropiate fltaness assumption, my belief follows from the standard semi-continuity theorem (Hartshorne III.11). However, I was not able to find any precise statement/reference.</p>
<p>I would like to have a precise statment about the semi-continuity of intersection numbers, with possibly a reference in the literature. </p>
<p>This should also fit with the question
<a href="http://mathoverflow.net/questions/42796/more-upper-lower-semi-continuous-functions-in-algebraic-geometry">More upper/lower semi-continuous functions in (algebraic) geometry?</a></p>
http://mathoverflow.net/q/2380812Field of definition of an algebraic setGe Tanglihttp://mathoverflow.net/users/703602016-05-05T13:29:20Z2016-05-06T08:26:11Z
<p>I find this definition in Silverman's book, The Arithmetic of Elliptic Curves:
an algebraic set(in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in $K[X]=K[x_1,...x_n]$.</p>
<p>But what if at the beginning we are given polynomials in $K[X]$ and the algebraic set is defined by these polynomials? I mean, can the ideal of this algebraic set be generated by polynomials in $K[X]$?</p>
<p>I already know that this is equal to the following question:</p>
<p>If $I\subset \bar K[X]$ can be generated by elements in $K[X]$, can $r(I)$ be generated by elements in $K[X]$?</p>
<p>I find it not so easy as I thought. Can anyone lead me out? Thanks!</p>
http://mathoverflow.net/q/2380651globally well-defined holomorphic vector field on a curve $y^N = x^2 - z^2$sealiyhttp://mathoverflow.net/users/912522016-05-05T09:33:00Z2016-05-06T08:48:59Z
<p>Let us start with a multiple cover $C$ of the $x$-plane branched at $z$ and $-z$, and so described by an equation $y^N = x^2 - z^2$. </p>
<p>For $N=2$, it is known that there are globally-defined holomorphic vector fields on $C$ that are of the form $V_n = u^{n+1} \frac{d}{du} $ where $u= x \pm y$ for any $n\in \mathbb{Z}$. </p>
<p>Can we also construct such well-defined vector fields on $C$ for any $N$? </p>
<p>Naively, they can be generalized to $V_n = u^{n+1} \frac{d}{du} $ where $u= \left(x^{\frac 2N} - w^i y\right)^{\frac N2}$ for $i=0,1,..,N-1$ and $w$ is the N-th root of unity. Am I missing something here? </p>
<p>I am a theoretical physicist, and not good at math.
Need your help.</p>
<p>Thank you very much in advance. </p>
http://mathoverflow.net/q/2380440Branches of the tetration functionjames.nixonhttp://mathoverflow.net/users/782492016-05-04T20:16:42Z2016-05-06T12:07:32Z
<p>Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the construction, but it is there in the cosmos of function space.) </p>
<p>Let $z= x+iy$</p>
<p>$$F:\mathbb{C}_{x>-2} \to \mathbb{C}$$</p>
<p>$$F(0) = 1$$</p>
<p>$$\eta^{F(z)} = F(z+1)$$</p>
<p>$$F:(-2,\infty) \to (-\infty,e)$$</p>
<p>$$\frac{d}{dz}F(z) \neq 0$$</p>
<p>$$\frac{d}{dx}F(x) > 0$$</p>
<p>This is tetration base $\eta$, and is a valuable function for many reasons. We are sticking to the regular iteration method.</p>
<p>Now my question is difficult to state but revolves around extending this function to a larger domain than the half plane $\Re(z) > -2$. It is rather trivial to show $$\log_\eta(F(z)) = e\log(F(z)) = F(z-1) + e2\pi i k$$ for an arbitrary branch of $\log$, for some $k \in \mathbb{Z}$--locally about a point $z$.</p>
<p>What I'm trying to understand is if the following is an extension of $F$ for $|\arg(z-2)| < \pi$. Namely if $F$ extends everywhere to the complex plane excepting $y=0,x\le-2$. It is difficult to state but the proof of an extension goes as follows.</p>
<p>$F(-1) =0$ and therefore $\log(F(z))$ is not holomorphic in a neighborhood of $z=-1$ so that $F(z)$ is not holomorphic for $z$ in a neighborhood of $-2$ necessarily. So that the singularity at $-2$ is a logarithmic singularity--a branch point if you will.</p>
<p>Using the functional equation of $F$: $\eta^{F(z)} = F(z+1)$ and by taking the implicit equation</p>
<p>$$\eta^{\mu} - F(z-2)=0$$</p>
<p>for $0 < x < 1+\epsilon,\,y>0$ and arbitrary $\epsilon > 0$, we have a bunch of functions $\mu$ defined locally about each point $z = x+ i y$. Now $\mu + e2\pi i k $ is equally so a plausible function by the periodicity of $\eta^z$.</p>
<p>What grounds do we need so that we can say there is $\mu$ holomorphic for all $0 < x < 1 + \epsilon,\,\,y>0$? As in, what do we need to show in order to say this covering of the domain with analytic functions $\mu$ about each point $z$ forms an analytic function on the entire domain?</p>
<p>I assume the result requires some knowledge in Riemann surfaces (something I am a beginner in), and I assume it is not trivial to show this result. Multivalued functions have never been my strong suit, and so I hope the answer can be found in simpler terms than some advanced argument from complex manifolds.</p>
<p>If $\mu$ is holomorphic in this domain, there is only one $\mu$ that agrees with $F(z-3)$ for $1 < x < 1 + \epsilon$ by the identity theorem. And therefore we have found an extension of $F(z)$. Repeating this procedure eventually shows $F(z)$ is holomorphic for all $y>0$, and an equal procedure works for $y<0$. This in the end shows $F(z)$ is holomorphic everywhere excluding $y=0,x\le-2$--i.e: when $|\arg(z-2)|<\pi$.</p>
http://mathoverflow.net/q/2377673First to note/document the relation between permutohedra and multiplicative inversionTom Copelandhttp://mathoverflow.net/users/121782016-04-30T21:51:25Z2016-05-06T08:25:31Z
<p>The relation between the refined face numbers of the <a href="http://en.wikipedia.org/wiki/Permutohedron" rel="nofollow">permutohedra</a> and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in the OEIS entries <a href="http://oeis.org/A049019" rel="nofollow">A049019</a> and <a href="http://oeis.org/A133314" rel="nofollow">A133314</a>. What are some early references for this relation?</p>
http://mathoverflow.net/q/2302850Normed space between $H^{0+}$ and $L^2$Capublancahttp://mathoverflow.net/users/545522016-02-05T14:33:06Z2016-05-06T10:57:34Z
<p>In the space $\in L^2(\mathbb{R}^3)$, consider the following condition.
$$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$
Of course if $f\in H^s(\mathbb{R}^3)$ for some $s>0$, condition (*) is satisfied. I`m asking if there is some Banach space $X\subset L^2$ such that</p>
<p>1) Every $f\in X$ satisfies condition (*)</p>
<p>2) $H^s\subset X$ for every $s>0$</p>
<p>3) I can interpolate between $X$ and $L^1(\mathbb{R}^3)$</p>
<p>Thank you for any suggestion</p>
http://mathoverflow.net/q/755858Kruskal-Katona type question for union-closed families of setsPeter Hegartyhttp://mathoverflow.net/users/176372011-09-16T09:10:10Z2016-05-06T08:50:51Z
<p>Question : Let $n,k$ be two positive integers with $n \geq k$. Let $\mathcal{F}$ be a family of $C(n,k)$ sets, each of size $k$, and let $\langle\mathcal{F}\rangle$ denote the union-closed family generated by $\mathcal{F}$, i.e.: $\langle\mathcal{F}\rangle$ consists of all those sets which can be expressed as a union of members of $\mathcal{F}$. Must it be the case that
\begin{equation}
|\langle\mathcal{F}\rangle| \geq \sum_{j=k}^{n} C(n,j),
\end{equation}
with equality if and only if $\mathcal{F}$ consists of all $k$-element subsets of an $n$-set ?</p>
<p>It is easy to see that if the inequality holds (whatever about uniqueness), then it implies that, for any union-closed family $\mathcal{G}$ and non-negative integer $m$, if $|\mathcal{G}| \geq 2^{m}$ then the average size, let's denote it $w(\mathcal{G})$, of a member of $\mathcal{G}$ is at least $m/2$. This is, in turn, a special case of a result of Reimer [1] that, for any union-closed family $\mathcal{G}$ one has $w(\mathcal{G}) \geq \frac{1}{2} \log_{2} |\mathcal{G}|$. Indeed I had conjectured the same result and in thinking about it was led to the above question, before I recently became aware of Reimer's proof, which is a beautiful piece of work !</p>
<p>One can obviously try to generalise my question to an arbitrary number of generating $k$-sets, perhaps along the lines of the Kruskal-Katona theorem for shadows ?</p>
<p>[1] D. Reimer, An average set size theorem, Combin. Probab. Computing 12 (2003), 89-93. </p>
http://mathoverflow.net/q/661831de Rham cohomology and flat vector bundlesSpinorbundlehttp://mathoverflow.net/users/6752009-11-23T21:28:47Z2016-05-06T11:38:26Z
<p>I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed forms in $\Omega^k(M)$ modulo the set of exact forms. The coboundary operator is given by the exterior derivative.</p>
<p>Let now $E \rightarrow M$ be a vector bundle with connection $\nabla^E$ over $M$, and consider the $E$-valued $k$-forms on $M$: $\Omega^k(M,E)=\Gamma(\Lambda^k TM^\ast \otimes E)$.<br>
If $E$ is a <strong>flat</strong> vector bundle, we get a coboundary operator $d^{\nabla^E}$ (since $d^{\nabla^E} \circ d^{\nabla^E} = R^{\nabla^E}=0$, with $R^{\nabla^E}$ being the curvature) and we can define </p>
<p>
$$H_{dR}^{k}(M,E) := \frac{ker \quad d^{\nabla^E}|_{\Omega^k(M,E)}}{im \quad d^{\nabla^E}|_{\Omega^{k-1}(M,E)}}$$
</p>
<p>So my question: Is this somehow useful? I mean can one use this definition to make some statements about $M$ or $E$ or whatever? Or is the restriction of $E$ to be a flat vector bundle somehow disturbing? Or is this completely useless?</p>