Recent Questions - MathOverflowmost recent 30 from mathoverflow.net2014-08-28T03:26:20Zhttp://mathoverflow.net/feedshttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://mathoverflow.net/q/1795561Interpolation between L^1 and Sobolev SpaceTBShttp://mathoverflow.net/users/575582014-08-28T00:34:36Z2014-08-28T00:34:36Z
<p>Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that </p>
<p>$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ||f||_{L^1({\mathbb R})}^{1-\frac{\alpha}{\beta}} ||D^\beta f||_{L^2({\mathbb R})}^{\frac{\alpha}{\beta}}$ </p>
<p>for $0 < \alpha \leq \beta$ arbitrary (i.e., fractional)?
This is a special case of a Gagliardo-Nirenberg-Sobolev inequality, but Nirenberg's 1959 proof in <em>Ann. Scuola Norm. Sup. Pisa</em> only holds for integer $\alpha, \beta$. There are many authors who prove slightly different GNS inequalities, but I need this exact one and it won't budge. One can play with Nirenberg's idea to remove the constraint that $\beta$ is an integer. Knowing that, you can work even harder and get $1 \leq \alpha \leq \beta$. But for $0 < \alpha < 1$, the proof breaks seemingly irreparably. It's also unfortunately true that the $L^1$ breaks Littlewood-Paley proofs (at least without some replacement for Nirenberg's magic).</p>
<p>Has anyone seen this particular family of inequalities anywhere? If it's a folk theorem, what's the trick?</p>
http://mathoverflow.net/q/1795551Generalization of a theorem of Øystein Ore in group theory?Sébastien Palcouxhttp://mathoverflow.net/users/345382014-08-28T00:25:32Z2014-08-28T01:02:59Z
<p><strong>Theorem</strong> (<a href="http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077490633" rel="nofollow">Øystein Ore, 1938</a>): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is <a href="http://en.wikipedia.org/wiki/Distributive_lattice" rel="nofollow">distributive</a>.<br>
<em>Proof</em>: see below. </p>
<p>Let $(H \subset G)$ be an inclusion of finite groups and $\mathcal{L}(H \subset G)$ its lattice of intermediate subgroups.<br>
I would like to generalize the above theorem of Øystein Ore to the inclusions of finite groups, i.e. find an equivalent formulation of the following property $(D)$, in term of "cyclic" notions. $$(D) \ \ \ \ \mathcal{L}(H \subset G) \text{ is distributive } $$ <em>Definition</em>: An inclusion of finite groups $(H \subset G)$ is <strong>cyclic</strong> if it checks $(C_0)$.<br>
$$(C_0) \ \ \ \exists g \in G \text{ such that } \langle H,g \rangle = G$$ <em>Examples</em>: The maximal inclusions are cyclic, and $(\{e\} \subset G)$ is cyclic iff $G$ is cyclic.</p>
<p><strong>Question</strong>: $(D)$ $\Rightarrow$ $(C_0)$ ? </p>
<p><em>Remark</em>: I've checked that it's true for index $[G:H]<32$ and $\vert G \vert \le 10^4$ with <a href="http://www.gap-system.org/" rel="nofollow">GAP</a>.<br>
The converse is false: $(S_2 \subset S_4)$ is a cyclic inclusion with a non-distributive lattice. </p>
<hr>
<p><em>Bonus question</em>: How complete $(C_0)$ for having an equivalence with $(D)$? </p>
<p>I've tried all the following completions of strictly increasing strongness: </p>
<ul>
<li>$(C_1) \ $ $\forall K$, $H \le K \le G$, $\exists g \in G$ such that $\langle H,g \rangle = K$ </li>
<li>$(C_2) \ $ $\forall K$, $H \le K \le G$, $\exists g \in G$, $\exists n \ge 0$ such that $\langle H,g \rangle = G$ and $\langle H,g^n \rangle = K$ </li>
<li>$(C_3) \ $ $\exists g \in G$, $\forall K$, $H \le K \le G$, $\exists n \ge 0$ such that $\langle H,g^n \rangle = K$ </li>
</ul>
<p><em>Conjecture</em>: $(D)$ is strictly between $(C_1)$ and $(C_3)$.<br>
I've checked it for index $[G:H] < 32$ and $\vert G \vert \le 10^3$. Moreover $(D)$ is <em>not orderable</em> with $(C_2)$. </p>
<p><em>Remark</em>: All these notions and questions can be extended to the <a href="http://en.wikipedia.org/wiki/Subfactor" rel="nofollow">theory of subfactors</a> by using the coproduct on the $2$-boxes space (see <a href="http://mathoverflow.net/q/179179/34538">here</a> and <a href="http://mathoverflow.net/q/179188/34538">there</a>).</p>
<hr>
<p><strong>Proof</strong> of the theorem of Øystein Ore: </p>
<p>Suppose first that $\mathcal{L}(G)$ is distributive and let $a,b \in G$. We have to show that $\langle a,b \rangle$ is cyclic. Because $\langle a \rangle \cap \langle b \rangle$ is centralized by $a$ and $b$, $\langle a \rangle \cap \langle b \rangle \leq Z(\langle a,b \rangle)$. Also,$\langle ab \rangle \cup \langle a \rangle = \langle a,b \rangle = \langle ab \rangle \cup \langle b \rangle$, and then by distributivity $$ \langle ab \rangle \cup (\langle a \rangle \cap \langle b \rangle) = (\langle ab \rangle \cup \langle a \rangle) \cap (\langle ab \rangle) \cup \langle b \rangle) = \langle a,b \rangle $$
So, $\langle a,b \rangle / \langle a \rangle \cap \langle b \rangle \simeq \langle a,b \rangle /(\langle a,b \rangle \cap (\langle a \rangle \cap \langle b \rangle) )$ is cyclic, and then $\langle a,b \rangle$ is abelian, as cyclic extension of a central subgroup. By the structure of finitely generated abelian groups, there are $c,d \in G$ such that $\langle a,b \rangle = \langle c \rangle \times \langle d \rangle$. As we've already shown, $\langle c,d \rangle / \langle c \rangle \cap \langle d \rangle$ is cyclic. Because $\langle c \rangle \cap \langle d \rangle = 1$, $\langle a,b \rangle = \langle c,d \rangle $ is cyclic. </p>
<p>Now suppose that $G$ is cyclic and let $A,B,C \in \mathcal{L}(G)$. Because $G$ is abelian,<br>
we just need to verify that $A(B \cap C) = AB \cap AC$. Clearly, $A(B \cap C) \leq AB \cap AC$.<br>
Let $x \in AB \cap AC$, then $x=ab=a'c$ with $a,a' \in A$, $b \in B$ and $c \in C$.<br>
Because $G$ is cyclic, $\exists g \in G$ such that $\langle a,a',b,c \rangle = \langle g \rangle$. Next, $ab=a'c$ implies that $\langle g \rangle = (A \cap \langle g \rangle)(B \cap \langle g \rangle)=(A \cap \langle g \rangle)(C \cap \langle g \rangle)$. If one of the three subgroups $A \cap \langle g \rangle$, $B \cap \langle g \rangle$, $C \cap \langle g \rangle$ is trivial, then $x=b=c \in B \cap C$ or $x \in A$. In each case, $x \in A(B \cap C)$.<br>
So suppose that all these subgroups are non-trivial and let $n,r,s$ the respective indices of $A \cap \langle g \rangle$, $B \cap \langle g \rangle$, $C \cap \langle g \rangle$ in $\langle g \rangle$. So $(n,r) = 1 = (n,s)$, and then $(n,rs) = 1$ and so $\langle g \rangle = \langle g^n \rangle \langle g^{rs} \rangle = (A \cap \langle g \rangle)= (B \cap C \cap \langle g \rangle) \leq A(B \cap C)$.<br>
Again, it follows that $x \in A(B \cap C)$. So $A(B \cap C) = AB \cap AC$ as expected. $\square$</p>
http://mathoverflow.net/q/179553-3Paritial order help? [on hold]Baton Coléhttp://mathoverflow.net/users/575552014-08-27T23:50:51Z2014-08-27T23:50:51Z
<p>Let (A,≼A) and (B,≼B) be partially ordered sets. Define C = A×B and define the relation ≼' on C by (a,b)≼'(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′).</p>
<p>(a) Prove that ≼' is a partial order on C.</p>
<p>$[(a,b) \preccurlyeq$ $(a',b')] \wedge [(a',b') \preccurlyeq$ $(a'',b'')] \iff$
$[(a\preceq_A a') \wedge (b\preceq_B b')] \wedge[(a'\preceq_A a'') \wedge (b'\preceq_B b'')] \iff$ $ [(a\preceq_A a') \wedge (a'\preceq_A a'')] \wedge[(b\preceq_B b') \wedge (b'\preceq_B b'')] \Rightarrow $<br>
$[(a\preceq_A a'') \wedge (b\preceq_B b'')] \iff $
$[(a,b) \preccurlyeq$ $(a'',b'')]$</p>
<p>(b) Prove that if a is a maximal element of (A,≼A) and if b is a maximal element of (B,≼B) then (a, b) is a maximal element of (C, ≼').</p>
<p>I do not know how?</p>
http://mathoverflow.net/q/1795511Are linear algebraic groups rigid?Qfwfqhttp://mathoverflow.net/users/47212014-08-27T23:35:51Z2014-08-28T01:56:27Z
<p>The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's possible to deform the group structure on a fixed variety (that admits at least a structure of an algebraic group). </p>
http://mathoverflow.net/q/1795500irreducible etale cover of a blowupmatthewhttp://mathoverflow.net/users/389522014-08-27T23:15:41Z2014-08-28T03:00:33Z
<p>Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.</p>
<p>Do there exist smooth morphisms of schemes $X_i\rightarrow Y_i$ such that $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and form an etale cover of $W$?
[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]</p>
http://mathoverflow.net/q/1795491second smallest eigenvalue Laplacian - submodular set functionImanhttp://mathoverflow.net/users/575542014-08-27T22:11:50Z2014-08-27T22:11:50Z
<p>Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ is a submodular set function?</p>
http://mathoverflow.net/q/1795482Symmetry on a sphereuser45673http://mathoverflow.net/users/456732014-08-27T21:20:22Z2014-08-27T21:42:13Z
<p>Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least $C$ ($u(Rx)$ is a rotation of $u$ on $S^2$). Such function can't be "too" non-symmetric. I wonder what kind of symmetry results the above condition implies. </p>
<p>If $u$ is a monotone axially symmetric function, then nonempty components of $\{x\in S^2: u(x)> u(Rx)\}$ are open hemispheres. So axially symmetric functions do satisfy the condition. Does the above condition necessarily imply that $u$ is axially symmetric? If not, can we deduce a weaker notion of symmetry? </p>
http://mathoverflow.net/q/1795470Is a inverse limit of indecomposable again indecomposable?Chicohttp://mathoverflow.net/users/410352014-08-27T21:08:54Z2014-08-28T00:27:42Z
<p>In truth, I do not need in the general case.</p>
<p>Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the <em>category of complexes</em> over $\mathrm{Mod}\ R$.</p>
<p>If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X_i^\bullet, \varphi_{i,j}\colon X_j^\bullet \to X_i^\bullet\}$ is an <em>inverse system</em> over $\mathbb{N}$ such that $X_i^\bullet \in \mathrm{Comp}(R)$ is indecomposable, for all $i \in \mathbb{N}$, so is indecomposable $\varprojlim X_i^\bullet$?</p>
<p><strong>Edit.</strong> Of course, I'm assuming $\varphi_{i,j} \neq 0$ for all $j\geq i \in \mathbb{N}$.</p>
<hr>
<p><strong>Remark.</strong> A category $\mathcal{C}$ is <em>complete</em> if $\varprojlim A_i$ exists in $\mathcal{C}$ for every inverse system $\{A_i , \varphi_{i,j}\}$ in $\mathcal{C}$. The categories $\mathcal{C} = \mathrm{Mod} \ R$ and $\mathcal{C} = \mathrm{Comp}(R)$ are examples of complete categories.</p>
http://mathoverflow.net/q/179546-3Euclidean geometry problems for thirs year students [on hold]peterhttp://mathoverflow.net/users/575502014-08-27T21:02:37Z2014-08-27T21:02:37Z
<p>Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that involve not complicated proofs. </p>
http://mathoverflow.net/q/1795430Simple monotone differential operatorspeterhttp://mathoverflow.net/users/575502014-08-27T20:35:55Z2014-08-27T20:35:55Z
<p>Where one may find any reference to lemmas the following kind:</p>
<p>If x(t) is C1 in [0,T], x(0)>0, dx/dt + c(t)x(t) >0 in [0,T] then x(t) > 0 in [0,T].</p>
<p>There is a version with weak inequalities. </p>
<p>This lemma is also provable for bounded variation functions (BV), if one imposes a restriction that the jumps must be positive. </p>
<p>I need to quote lemmas of this type. I will be grateful if you could provide me with appropriate references, thanks.</p>
http://mathoverflow.net/q/1795422Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$And85http://mathoverflow.net/users/573482014-08-27T20:30:29Z2014-08-27T23:41:42Z
<p>Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?</p>
<p>I am interested in partucular in the case of elliptic curve (as in the end of section 3 of Skinner-Urban's work).</p>
<p>Thank you!</p>
http://mathoverflow.net/q/179541-3How to compute difference between two mathematical forms? [on hold]hahahttp://mathoverflow.net/users/575492014-08-27T20:18:54Z2014-08-27T20:18:54Z
<p>Please notice that this question focuses on the form, eg. 3SinX and 4SinX are the identical forms.</p>
http://mathoverflow.net/q/1795400Prime order elements in $GL(n,\mathbb{Z})$Maxim Stykowhttp://mathoverflow.net/users/241562014-08-27T20:17:09Z2014-08-27T20:52:49Z
<p>What is known about the elements of prime order $p$ in $GL(n,\mathbb{Z})$? All I know at this point is that $GL(n,\mathbb{Z})$ has an element of prime order $p$ iff $p-1\leq n$ and that there are only finitely many conjugacy classes of such elements (Minkowsky).</p>
http://mathoverflow.net/q/179535-1Is real analytic function good enough (see problem)? [on hold]Braslavhttp://mathoverflow.net/users/573562014-08-27T19:20:20Z2014-08-27T19:38:32Z
<p>Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then necessary $f(t)=0$ for all $t \in \mathbb{R}$? </p>
http://mathoverflow.net/q/179534-4Theorem related to Goldbach's Conjecture [on hold]Ricardo Barcahttp://mathoverflow.net/users/575462014-08-27T18:53:44Z2014-08-27T18:53:44Z
<p>Theorem. Let $x > 4$ be an even number. Let $P_x$ be the set of primes less than $\sqrt{x}$. Let $p$ be a prime less than $x$. Therefore, if $p \not\equiv x \pmod{q}$ for every $q \in P_x$, then $x-p$ is a prime or $x-p=1$.</p>
<p>This theorem can be proved by contradiction, assuming that $c = x - p$ is a composite number.</p>
<p>Example: Let $x=40$. Then, we have $P_{x=40} = \{ p \, | \, p < \sqrt{40} \} = \{ 2, 3, 5 \}$. We proceed to test $x=29$, as follows</p>
<p>\begin{align*}
29 \not\equiv 40 \pmod{2},
\end{align*}</p>
<p>\begin{align*}
29 \not\equiv 40 \pmod{3},
\end{align*}</p>
<p>\begin{align*}
29 \not\equiv 40 \pmod{5},
\end{align*}</p>
<p>and then, by the preceding theorem, we have that $40 = 29 + 11$ is the sum of two primes.</p>
<p>Someone can tell me if this theorem is a well known result from Number Theory?. Thank you very much.</p>
http://mathoverflow.net/q/179531-1Translation of the paper titled " Quelques remarques sur les groupes de Lie algebriques reels" [on hold]ejjuhttp://mathoverflow.net/users/266312014-08-27T18:02:49Z2014-08-27T18:02:49Z
<p>Is there any English translation of the following paper which is written in French.</p>
<p><a href="http://projecteuclid.org/euclid.jmsj/1260975679" rel="nofollow">http://projecteuclid.org/euclid.jmsj/1260975679</a></p>
http://mathoverflow.net/q/1795301Quadratic Gauss sums: Explicit determinations?PatWernhttp://mathoverflow.net/users/575442014-08-27T17:55:53Z2014-08-27T18:19:31Z
<p>Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet character modulo $q$, not necessarily primitive? Here $e(x)$ denotes $e^{2\pi ix}$.</p>
<p>Thanks in advance.</p>
http://mathoverflow.net/q/1795271Cubic Cayley (undirected) graphsRobin Saundershttp://mathoverflow.net/users/43362014-08-27T15:59:52Z2014-08-27T20:10:56Z
<p>The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) involutions, or (B) they are each other's inverses and of order >2. Note that</p>
<ul>
<li>non-isomorphic Cayley graphs can be isomorphic as graphs, so that the same cubic graph may arise as a Cayley graph of both types A and B;</li>
<li>a single group can have non-isomorphic cubic Cayley graphs, so that the same group may have Cayley graphs of both types A and B;</li>
<li>in particular, it might even be possible for a single group to give rise to non-isomorphic cubic Cayley graphs which are isomorphic as graphs.</li>
</ul>
<p>What is known about the characterizations of, or relationship between, the two types A and B of cubic Cayley graphs, and in particular the situations in which they can coincide - either in the sense of being isomorphic as graphs, or of coming from the same group (or possibly both)?</p>
<p>I hope this question is not too vague.</p>
http://mathoverflow.net/q/1794877A metric space of geometric shapesErel Segal Halevihttp://mathoverflow.net/users/344612014-08-27T07:11:52Z2014-08-27T17:48:46Z
<p>My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:</p>
<ul>
<li>Families of similar shapes, such as squares, are closed in this metric. Also more general families, such as the family of 2-fat objects, are closed in this metric.</li>
<li>Converging sequences of non-overlapping shapes, converge to non-overlapping limits.</li>
<li>Every continuous measure on a converging sequence, converges to the measure of the limit.</li>
</ul>
<p>I tried the Hausdorff distance, which is a metric on the space of closed sets, but found out that it doesn't say much about measures.</p>
<p>I tried the Symmetric distance (defined as the area of the symmetric difference), but found out that it is only a pseudo-metric. I tried to make it a metric by restricting the underlying space of shapes, but found out that <a href="http://math.stackexchange.com/questions/909056/how-to-make-the-symmetric-distance-a-metric">it is tricky even when only polygons are considered</a>. I thought of converting the pseudo-metric to a metric on equivalence classes and then selecting a representative shape from each equivalence class, but <a href="http://math.stackexchange.com/questions/845610/representative-elements-in-the-symmetric-difference-metric">found no simple way to do this selection</a>.</p>
<p>I thought of defining a new metric which is the maximum of the Hausdorff distance and the Symmetric distance and enjoy the best of the two worlds, but at that point, it began to feel like reinventing the wheel. Surely I am not the first who needs a metric between plain geometric shapes.</p>
<p>So my question is:</p>
<p>Is there a paper or a book that explicitly studies the topic of metrics between shapes in the plane, not in the context of general topology but with attention to the specific geometric properties?</p>
http://mathoverflow.net/q/1794801Exponential Sum BoundMayank Pandeyhttp://mathoverflow.net/users/409832014-08-27T03:15:04Z2014-08-28T01:03:02Z
<p>In
<code>http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf</code>, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
$$\Phi(\alpha) = \sum_{h\le 6H}\sum_{P<x\le 2P}e(\alpha h(3x^2 + 3xh + h^2))$$
They then assert that $$\int_{0}^1\left|\Phi(\alpha)\right|^4d\alpha\ll H^3P^{2 + \epsilon}$$
for all $\epsilon>0$</p>
<p>However, I don't have access to the paper mentioned in which this fact is proven. Can someone show me a proof of this? It is supposed to be along the lines of the proof of Hua's Lemma, but I can't see how this is the case.</p>
http://mathoverflow.net/q/17937110What arrangement of unit cubes minimizes surface area?Benjamin Dickmanhttp://mathoverflow.net/users/229712014-08-25T17:55:42Z2014-08-28T01:24:25Z
<p><strong>Question A.</strong> How does one arrange $n$ unit cubes to minimize surface area?</p>
<p><strong>Question B.</strong> How does one arrange $n$ unit cubes to form a rectangular prism of minimal surface area?</p>
<p>Various curricular materials discuss this problem for a specific number of cubes, but I have not seen the general case broached. For an example of an exploration at the pre-secondary level, see <strong><a href="http://illuminations.nctm.org/Lesson.aspx?id=2009">this instructional plan</a></strong> from the <em>National Council of Teachers of Mathematics</em>. A recent example of the problem <em>posed</em> in generality can be found on <strong><a href="http://blog.mrmeyer.com/2014/purposeful-practice-dandy-candies/">this math education blog-post</a></strong> as well as <a href="http://www.math.wm.edu/~ckli/mathed/problems.pdf#3"><strong>here</strong></a>.</p>
<p>My own attempt at combing the literature did not produce anything of import. From the two dimensional perspective, I see a <strong><a href="http://math.stackexchange.com/q/214234/37122">question</a></strong> in a somewhat similar vein on MSE about unit squares, and another 2D <strong><a href="http://mathoverflow.net/q/118411/22971">question</a></strong> on MO, but neither seems applicable to the questions above.</p>
<p>Moreover, there are some related results (recently relayed to me by a geometer-mathematician) in:</p>
<blockquote>
<p>Williams, W., & Thompson, C. (2008). The Perimeter of a Polyomino and the Surface Area of a Polycube. <em>The College Mathematics Journal</em>, 233-237.</p>
</blockquote>
<p>For example, see the excerpt below:</p>
<p><img src="http://i.stack.imgur.com/OLFGn.jpg" alt="enter image description here"></p>
<p>One might also check the <strong><a href="http://oeis.org/A193416">OEIS sequence</a></strong> recommended by Robert Israel (or found by <strong><a href="https://www.google.com/search?q=polycube%20minimal%20surface%20area">googling</a></strong>), though I cannot vouch for its validity (cf. e.g. the remarks of its contributor <strong><a href="https://oeis.org/wiki/User:Juan_Barajas_Martin">here</a></strong>).</p>
<p>Non-trivial bounds, algorithmic approaches, or other relevant remarks are all welcome!</p>
http://mathoverflow.net/q/1772676Clifford algebras for quadratic modules over ringed spacesMatthias Wendthttp://mathoverflow.net/users/508462014-07-28T12:32:34Z2014-08-27T20:03:08Z
<p>What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ringed, but I am willing to assume that $2$ is invertible in the sheaf of rings if that helps. Searching mathematical reviews revealed a somewhat related paper: </p>
<ul>
<li>B. Auslander. The Brauer group of a ringed space. J. Alg. 4 (1966), 220-273. </li>
</ul>
<p>This paper develops some of the relevant math, but does not talk about the Clifford algebra. I would expect that there is some paper from around the same period defining the Clifford algebra for quadratic modules over a ringed space, but I could not find anything. </p>
<p>I would also be interested if there are papers writing down the Clifford invariant mapping from the Witt group to the Brauer group in the general setting of ringed spaces. </p>
http://mathoverflow.net/q/1771983Norm of triangular truncation operator on rank deficient matricesgalaashttp://mathoverflow.net/users/564942014-07-27T16:56:03Z2014-08-27T21:24:29Z
<p>Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ A\|\le\frac{\ln n}{\pi}\|A\|$$
where $\circ$ is the hadamard product. </p>
<p>For example, a proof of above was given in <a href="http://library.isical.ac.in/jspui/bitstream/10263/4956/1/Pinching,%20Trimming,%20Truncating,%20and%20Averaging%20of%20Matrices.pdf" rel="nofollow">this paper</a> </p>
<p>If $A$ is of rank $r<n$, a simple inequality yields:
$$\|T\circ A\|\le\|A\|_F\le \sqrt r\|A\|$$
where $\|\cdot\|_F$ is the Frobenius norm</p>
<p>Therefore for rank deficient $A$, the bound can be smaller.<br>
I was wondering if this $\sqrt r$ is sharp enough, is there an example to show
$$\|T\circ A\|/\|A\|\to c\sqrt r$$
For A of varying size but fixed rank.<br>
Or is it possible to improve $\sqrt r$ to something even smaller, like the logarithm in general case? </p>
<p><strong>Edit:</strong><br>
There're some misunderstanding about my question, I re-organized it as follow:<br>
Let $A_{n\times n}$ of rank $r$, where $$\frac{\ln n}{\pi}>\sqrt r$$
Let $n$ grow in someway but keep $r$ fixed, is it possible to construct an example to show
$$\frac{\|T\circ A\|}{\|A\|}\sim O(\sqrt r)$$
or to show the ratio is actually $O(\ln r)$?</p>
http://mathoverflow.net/q/1684950Third order central moment of a positive linear combination of log-normal random variablesShravan Mohanhttp://mathoverflow.net/users/513912014-05-29T05:07:08Z2014-08-27T22:25:17Z
<p>What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables? </p>
<p>It seems to be a common notion that the skewness of random variables with longer tails to the right is positive. Is it correct? If so, how do you prove it? </p>
http://mathoverflow.net/q/1639340Fixed point problem with a monotone vector as a fixed point?TomHhttp://mathoverflow.net/users/498312014-04-21T11:26:32Z2014-08-28T02:42:29Z
<p>Suppose $F : [0,1]^n \to [0,1]^n$ is continuously differentiable and $0 < \frac{\partial F_1}{\partial x_i} \leq \dots \leq \frac{\partial F_n}{\partial x_i} < \beta < 1$ for all $i =1,\dots,n$. Conjecture: there exists unique $x^* = F(x^*)$, and moreover, $x_1^* \leq \dots \leq x_n^*$.</p>
<p>Proof of the first part is quite straightforward: one can easily verify that $F$ is a contraction mapping and then apply the contraction mapping theorem. I would need some help with the second claim.</p>
http://mathoverflow.net/q/1633820Probability generating function zero implies random variable is infinitemathjungehttp://mathoverflow.net/users/496032014-04-14T22:00:59Z2014-08-27T20:24:27Z
<p>Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to the root of a graph by an interacting particle system. We prove $V = \infty$ a.s. by showing that $f$ satisfies a recurrence relation</p>
<p>$$f(x) =\frac{x+2}{3}f\Bigl(\frac{x+1}{2}\Bigr)^2
+\frac{x+1}{3}f\Bigl(\frac{x}{2}\Bigr)\biggl(1
-f\Bigl(\frac{x+1}{2}\Bigr)\biggr)$$</p>
<p>which through analytic methods we prove can only be satisfied when $f \equiv 0$ on $[0,1)$. Though our technique works we are somewhat baffled and are hoping to, in our upcoming paper, give some context by providing examples of this type of argument occurring in probability literature.</p>
<p>So, the question is are there other examples of proving a r.v. is a.s. infinite by proving the generating function is identically zero?</p>
http://mathoverflow.net/q/1598970Homotopy with non piece-wise linear boundaryJuergenhttp://mathoverflow.net/users/480182014-03-09T23:52:21Z2014-08-28T00:25:17Z
<p>in the middle of a long proof I encounter the following problem.</p>
<p>Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand $E$ as a set of probability functions.)</p>
<p>Let $x^\dagger$ be the probability function in $E$ which has maximum Shannon entropy:
i.e. $\{x^\dagger\}=\arg\sup_{\vec x\in E}-\sum_{i=1}^nx_i\cdot \log(x_i)$. This function is well-known to be unique. In the case I am interested in, I can assume that $x^\dagger_i>0$ for all $i$.</p>
<p>For all $k\in\mathbb N$ let $c_k(i)$ be an $n$-tuple of numbers such that for all $1\leq i\leq n$ it holds that $\lim_k c_k(i)=\log(x_i)$.</p>
<p>I need to show the following:
there exists a sequence $(q_k)_{k\in\mathbb N}$ with $q_k\in E$ such that</p>
<ol>
<li>$q_k \in \arg \sup_{\vec x\in E}-\sum_{i=1}^n x_i c_k(i)$ and</li>
<li>$\lim_k q_k = x^\dagger$.</li>
</ol>
<p>The main problem is that $\arg \sup_{\vec x\in E}-\sum_{i=1}^n x_i c_k(i)$ may contain more than one element.</p>
<p>So, if I replace $E$ by a closed, convex set with an open (i.e. non-empty) interior $U_\epsilon(E)\subset\mathbb R^n$ with a boundary which is nowhere piece-wise linear,
then $\arg \sup_{\vec x\in U_\epsilon(E)}-\sum_{i=1}^n x_i c_k(i)$
has a unique solution in $U_\epsilon(E)$. (This is a general fact about linear optimisation problems; or so I hope :))</p>
<p>These maxima will then obtain, in general, not for probability functions. But I think I can handle this.</p>
<p>What I need for my proof is the following: Given a fixed closed and convex set $E$ as above I need to construct sets $U_\epsilon(E)$
such that</p>
<ol>
<li>$U_\epsilon(E)$ is closed,</li>
<li>$U_\epsilon(E)$ varies continuously with $\epsilon>0$,</li>
<li>the interior of $U_\epsilon(E)$ is open for $\epsilon>0$,</li>
<li>$\{(1+\epsilon) x^\dagger\} = \arg \sup_{\vec x\in U_\epsilon(E)} -\sum_{i=1}^n ,x_i\log(x^\dagger_i)$,</li>
<li>the boundary of $U_\epsilon(E)$ is nowhere piece-wise linear and</li>
<li>$\lim_{\epsilon\rightarrow 0}U_\epsilon(E)=E$.</li>
</ol>
<p>The last limit is taken over all strictly positive $\epsilon$.</p>
<p>@4: It is well-known that $\{ x^\dagger\} = \arg \sup_{\vec x\in E} -\sum_{i=1}^n ,x_i\log(x^\dagger_i)$.</p>
<p>So, I can simply assume that there exists a homotopy which gives me what I need or do I have to/can I prove the existence of such a homotopy.</p>
<p>All help much appreciated.</p>
http://mathoverflow.net/q/1457442How to find the coefficients of a poor-converging series?Igor Kotelnikovhttp://mathoverflow.net/users/414852013-10-24T14:28:04Z2014-08-27T19:24:28Z
<p>I have the series</p>
<blockquote>
<p>$\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$</p>
</blockquote>
<p>and the boundary conditions </p>
<blockquote>
<p>$\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n)=1.$</p>
</blockquote>
<p>Here $J_\nu$ stands for a Bessel function and the coefficients $a_n$ are unknown. It can be readily found from the above BC at $r=0$ that</p>
<blockquote>
<p>$a_0=1.$</p>
</blockquote>
<p>A simple solution can be found for $\Phi=\pi/2$, namely</p>
<blockquote>
<p>$a_0=1$ and $a_{n}=2\times(-1)^n$ for $n>1$.</p>
</blockquote>
<p>However, I want to solve this problem for $\Phi=3\pi/4$. It is not even evident that a solution exists for $\Phi=3\pi/4$. Indeed, if one expands the Bessel functions into the Taylor series and equate to $0$ the coefficients before different powers of $x$ then one obtains that $a_n=0$ for all $n$. In any case, these problems seems to be ill-posed. I computed a numerical solution for this problem using different mehods (see e.g. <a href="http://mathoverflow.net/posts/145154/edit">Inverse problems for an asymptotic series which depends on a parameter</a>) but still not sure that I did that well. Could someone direct me to a correct way?</p>
<p>Finally I need to compute the Laplace inverse for </p>
<blockquote>
<p>$\Gamma(7/3)p^{-7/3}\psi(r,\theta;p)$</p>
</blockquote>
<p>which is an electrostatic potential outside a rectangular corner of a homogeneous beam of charged particles provided that the so called Pierce electrodes compensate an electrostatic repulsion of the particles in the directions transversal to direction of the beam propagation.</p>
http://mathoverflow.net/q/1330307Defining a topology in the Power SetJoaquín Moragahttp://mathoverflow.net/users/315242013-06-07T04:32:27Z2014-08-27T20:40:46Z
<p>I have the follwing question:</p>
<ol>
<li><p>Given a topological space $T$ is possible in general to give a topology to $2^T$ (the power set of $T$) such that this topology in $2^T$ is related to $T$.</p></li>
<li><p>If the answer in general is no, are there conditions over $T$ to do this?.</p></li>
<li><p>Im interseted to know if given a topological space $T$ and a topology on $2^T$ that is induced by the topology on $T$ one can know some topological properties of $2^T$ knowing that of $T$.</p></li>
</ol>
http://mathoverflow.net/q/498202What is the geometric meaning of the third derivative of a function at a point? [closed]AJAYhttp://mathoverflow.net/users/116692010-12-18T19:03:29Z2014-08-27T17:45:31Z
<p>What is the geometric meaning of the third derivative of a function at a point?</p>
<p>This question is now asked on the sister site: <a href="http://math.stackexchange.com/questions/14841/what-is-the-meaning-of-the-third-derivative-of-a-function-at-a-point">http://math.stackexchange.com/questions/14841/what-is-the-meaning-of-the-third-derivative-of-a-function-at-a-point</a>.
If you have an interesting answer to contribute, please do it there!</p>