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curious little things - aggregated feedsenMath Overflow Recent Questions: Magnitude and distribution of largest prime factor?
https://mathoverflow.net/questions/334159/magnitude-and-distribution-of-largest-prime-factor
<p><a href="https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem" rel="nofollow noreferrer">Erdos-Kac law</a> state a typical number of magnitude <span class="math-container">$n$</span> has <span class="math-container">$\log\log n$</span> primes.</p>
<blockquote>
<p>What is magnitude and distribution of largest prime factor of typical magnitude <span class="math-container">$n$</span> natural number?</p>
</blockquote>Sun, 16 Jun 2019 19:06:48 -0600Math Overflow Recent Questions: Explicit bivariate quadratic polynomials where Coppersmith is better than standard solver?
https://mathoverflow.net/questions/334158/explicit-bivariate-quadratic-polynomials-where-coppersmith-is-better-than-standa
<p><a href="http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf" rel="nofollow noreferrer">http://www.numbertheory.org/pdfs/general_quadratic_solution.pdf</a> gives a general method to solve quadratic bivariate diophantine equation while <a href="https://en.wikipedia.org/wiki/Coppersmith_method" rel="nofollow noreferrer">Coppersmith introduced a method to solve bivariate polynomials</a> which work provably and have been shown to break <span class="math-container">$RSA$</span> system if half of low significant bits of either <span class="math-container">$P$</span> or <span class="math-container">$Q$</span> are known.</p>
<p>The equation that comes out is <span class="math-container">$$(2^ku+v)(2^ku'+v')=PQ$$</span> where if we assume <span class="math-container">$v$</span> is known. Then <span class="math-container">$vv'\equiv PQ\bmod 2^k$</span> gives <span class="math-container">$v'$</span>.</p>
<p>So we have a quadratic diophantine equation <span class="math-container">$$2^kuu'+(uv'+u'v)=\frac{PQ-vv'}{2^k}.$$</span></p>
<p>Why do I need Coppersmith's method to solve this? Can't a regular diophantine solver work here and so are there explicit polynomials where Coppersmith is better than standard solver in bivariate quadratic case?</p>Sun, 16 Jun 2019 18:29:51 -0600Math Overflow Recent Questions: Deligne's Mixed Hodge Theory
https://mathoverflow.net/questions/334156/delignes-mixed-hodge-theory
<p>Deligne constructs Mixed Hodge Structures (MHS) on the cohomology, <span class="math-container">$H^{*}(X)$</span>, of an algebraic variety <span class="math-container">$X$</span>, in his papers Hodge II and Hodge III. Really the question below is rather vague, and is motivated by my desire to separate the construction into two parts, the linear algebraic and the geometric. </p>
<p>Much of the input is subtle linear/ homological algebra, ie deals with the target category of the construction (namely MHS). The geometric content, ie that dealing with the source category, seems to be really about approximating varieties by smooth projective such, and it's this I'd like to focus on.</p>
<p>If <span class="math-container">$U$</span> is a smooth variety the construction proceeds via an embedding <span class="math-container">$U\rightarrow X$</span> into a smooth projective <span class="math-container">$X$</span>, so that the complement <span class="math-container">$D$</span> is normal crossings. One might imagine that this then determines what the MHS should be, at least if we demand Gysin sequences, however <span class="math-container">$D$</span> need not be smooth itself, and so we don't have a construction in this case. The construction for something of the form <span class="math-container">$D$</span> is gotten by taking a simplicial resolution of <span class="math-container">$D$</span>, basically by taking the Cech nerve of the resolution of <span class="math-container">$D$</span> given by the disjoint union of its components.</p>
<p>I'd like then to say very roughly that the construction suggests that we should replace the category of varieties, <span class="math-container">$Var_{\mathbb{C}}$</span>, with something like the category of simplicial objects in "closed inclusions of smooth projective varieties" ie level wise closed inclusions of simplicial smooth projective varieties. I'll call this category <span class="math-container">$C$</span> for now. I would love if there was a notion of weak equivalence in <span class="math-container">$C$</span>, such that there was an inclusion of categories <span class="math-container">$Var_{\mathbb{C}}\rightarrow C_{loc} $</span>, where the RHS denotes the localized category. I believe objects of <span class="math-container">$C$</span> should give rise to chain complexes of MHS and I want the weak equivalences to be taken to quasi-isomorphisms. Nb one must still check that the MHS corresponding to a variety is in the abelian categroy of MHS. </p>Sun, 16 Jun 2019 17:04:14 -0600Math Overflow Recent Questions: Eigenvalues of cyclic stochastic matrices
https://mathoverflow.net/questions/334154/eigenvalues-of-cyclic-stochastic-matrices
<p>Let's consider the following <span class="math-container">$n \times n$</span> cyclic stochastic matrix</p>
<p><span class="math-container">$$ M= \begin{pmatrix}
0 & a_2 & & & &b_n \\\
b_1 & 0& a_3& &&& \\\
& b_2 & 0& \ddots & & \\\
& &\ddots&\ddots &a_{n-1} & \\\
& && &0 &a_n \\\
a_1 & & & &b_{n-1} &0
\end{pmatrix}
$$</span></p>
<p>such that <span class="math-container">$\forall i$</span>, <span class="math-container">$a_i,\,b_i$</span> are positive real number, <span class="math-container">$a_i+b_i = 1$</span> and all other component of the matrix are zeros. This is a cyclic matrix in the sense that the associated graph is cyclic.</p>
<p>From the Perron-Frobenius theorem, the eigenvalues <span class="math-container">$\lambda$</span> of such matrix all belong to the unit circle.
<span class="math-container">$$(\Re \lambda )^2 + (\Im \lambda )^2 \leq 1 $$</span></p>
<p>From numerical explorations, I believe that all eigenvalues of <span class="math-container">$M$</span> belong to the ellipse
<span class="math-container">$$(\Re \lambda )^2 + \frac{(\Im \lambda )^2}{(\tanh p)^2} \leq 1 $$</span></p>
<p>where <span class="math-container">$p$</span> denote <span class="math-container">$p = \frac{1}{2}\ln \frac{\sqrt[n]{\prod_i a_i}}{\sqrt[n]{\prod_i b_i}}$</span>, assumed to be positive, otherwise inverse <span class="math-container">$a_i$</span> and <span class="math-container">$b_i$</span>.</p>
<p>One of the extremal case is the symmetric case <span class="math-container">$a_i=b_i$</span> where <span class="math-container">$p=0$</span> and all eigenvalues are real. The equality is reached in the uniform case of all <span class="math-container">$a_i$</span> to being equal to some value and all <span class="math-container">$b_i$</span> being equal to another value, the matrix being then a circulant matrix.</p>
<p>I can already prove that the imaginary part of the eigenvalue is bounded by <span class="math-container">$\tanh p$</span> (see below), but I am unable to extend the prove to include the real part.
I also try to play with the Brauer theorem about oval of Cassini exposed into [Horn & Johnson, Matrix Analysis], but it did not get me anywhere</p>
<p>Do you have any hints or suggestions to prove the inclusion of the eigenvalue into the ellipse?</p>
<hr>
<p>Proof for the imaginary part:</p>
<p>Denote <span class="math-container">$z$</span> the left eigenvector associated with eigenvalue <span class="math-container">$\lambda$</span>, we have from the eigenvalue equation <span class="math-container">$\lambda z = z M $</span>,
<span class="math-container">$$\forall i,\quad \lambda = a_i \frac{z_{i-1}}{z_i} + b_i\frac{z_{i+1}}{z_i} = \frac{a_i}{a_i+b_i} \frac{z_{i-1}}{z_i} + \frac{b_i}{a_i+b_i} \frac{z_{i+1}}{z_i} $$</span>,
where <span class="math-container">$i+1$</span> and <span class="math-container">$i-1$</span> ar evaluated modulo <span class="math-container">$n$</span>, ad the second equality follow from <span class="math-container">$a_i+b_i=1$</span>.</p>
<p>By taking the product of the imaginary part of all previous equation and denoting <span class="math-container">$p_i= \ln \sqrt{\frac{a_i}{bi}}$</span> , we get
<span class="math-container">$$ \Im \lambda = \sqrt{\prod_i \,a_i b_i \Im \frac{z_{i-1}}{z_i} \Im \frac{z_{i+1}}{z_i} }\prod_i \frac{\sinh (p_i+\frac{1}{2}\ln \Im\frac{ z_{i+1} }{z_{i}} \Im\frac{z_{i} }{z_{i-1}} )}{\cosh p_i} \leq \prod_i \frac{\sinh (p_i+\frac{1}{2}\ln \Im\frac{ z_{i+1} }{z_{i}} \Im\frac{z_{i} }{z_{i-1}} )}{\cosh p_i}$$</span>
The inequality use that <span class="math-container">$ \prod_i \Im \frac{z_{i-1}}{z_i}\leq 1$</span>. The concavity of <span class="math-container">$\ln \sinh$</span> and the convexity of <span class="math-container">$\ln \cosh$</span>, give the result
<span class="math-container">$$ \Im \lambda \leq \tanh p.$$</span></p>Sun, 16 Jun 2019 16:32:45 -0600Math Overflow Recent Questions: About $K$-rectification of increasing Tableaux
https://mathoverflow.net/questions/334152/about-k-rectification-of-increasing-tableaux
<p>Let <span class="math-container">$T$</span> be a standard Young Tableaux on <span class="math-container">$[n]$</span>. Denote the RSK algorithm <span class="math-container">$\text{RSK}(w)=(P(T),Q(T))$</span> for <span class="math-container">$w\in\mathfrak{S}_n$</span>, where <span class="math-container">$P(T)$</span> is the Schencted insertion Tableaux.</p>
<p>For <span class="math-container">$1\leq i\leq j\leq n$</span>. Let <span class="math-container">$T_{[i,j]}$</span> be the skew SYT by restricting <span class="math-container">$T$</span> to the segament <span class="math-container">$[i,j]$</span>. For a skew shape <span class="math-container">$Y$</span>, define the rectification of Y, <span class="math-container">$\text{Rect}(Y)$</span> to be applying <em>jeu de taquin</em> on <span class="math-container">$Y$</span> to obtain a standard shape. See section 2.1 of <a href="https://reader.elsevier.com/reader/sd/pii/S0097316505002013?token=B3981B70322E704ABFB6C25E0CE2F63F92CD0340D50644A85E67EB3EE647C4B415B5B1E042CE5E5FE2A08210F3341AA8" rel="noreferrer">this paper</a>, in which <span class="math-container">$\text{Rect}$</span> is denoted as <span class="math-container">$\text{std}$</span>.</p>
<p>It is well known that</p>
<blockquote>
<p>For <span class="math-container">$w\in \mathfrak{S}_n$</span>, <span class="math-container">$T\in \text{SYT}_n$</span>. If <span class="math-container">$P(w)=T$</span>, then
<span class="math-container">$$\text{Rect}(T_{[i,j]})=P(w_{[i,j]})$$</span>
for all <span class="math-container">$[i,j]\subseteq [n]$</span>, where <span class="math-container">$w_{[i,j]}$</span> means restricting the permutation to the subalphabet <span class="math-container">$[i,j]$</span>, e.g. <span class="math-container">$126534_{[2,5]}=2534$</span>.</p>
</blockquote>
<p>My question is: is there a <span class="math-container">$K$</span>-theoretic analog of this property, in terms of <a href="https://arxiv.org/abs/math/0601514" rel="noreferrer">Hecke insertion</a>, <a href="https://arxiv.org/abs/0705.2915" rel="noreferrer"><span class="math-container">$K$</span>-jeu-de-taquin</a> of increasing tableaux?</p>
<p>Specifically. We define <span class="math-container">$K$</span>-rectification by replacing jdt with <span class="math-container">$K$</span>-jdt, and denote <span class="math-container">$K$</span>-<span class="math-container">$P(w)$</span> the Hecke-insertion tableau of the word <span class="math-container">$w$</span>.</p>
<blockquote>
<p>Let <span class="math-container">$T$</span> be an increasing Tableau (of alphabet <span class="math-container">$[n]$</span>) and <span class="math-container">$Y=T_{[1,i]}$</span> such that <span class="math-container">$Y$</span> is an SYT and <span class="math-container">$1\cdots i\notin T\backslash Y$</span> (so that there is no ambiguity).
Then is it always true that:
<span class="math-container">$$K\text{-Rect}(T\backslash Y)=K\text{-}P(w_{[i+1,n]})$$</span>?
where <span class="math-container">$w$</span> is the row-reading word of <span class="math-container">$T$</span>, or even all words such that <span class="math-container">$K\text{-}P(w)=T$</span>.</p>
</blockquote>
<p>For example, Let <span class="math-container">$T=\begin{matrix}1&2&4\\3&5&6\\4&6&9 \end{matrix},Y=\begin{matrix}1&2\\3&\end{matrix}$</span>. We have
<span class="math-container">$$K\text{-Rect}\left(\begin{matrix}*&*&4\\*&5&6\\4&6&9 \end{matrix} \right)=\begin{matrix}4&5&6\\5&9&\\6&& \end{matrix}$$</span>
The row reading word of <span class="math-container">$T$</span> is <span class="math-container">$w=\mathfrak{row}(T)=469356124$</span>, and
<span class="math-container">$$K\text{-}P(w_{[4,9]})=K\text{-}P(469564)= \begin{matrix}4&5&6\\5&9&\\6&& \end{matrix}$$</span></p>
<blockquote>
<p>Thanks!</p>
</blockquote>Sun, 16 Jun 2019 15:37:32 -0600Math Overflow Recent Questions: An explicit formula for characteristic polynomial of matrix tensor product
https://mathoverflow.net/questions/334148/an-explicit-formula-for-characteristic-polynomial-of-matrix-tensor-product
<p>Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of roots of P,Q. </p>
<p>I guess its coefficients could be expressed through coefficients of P and Q.
But I don't know the explicit formula and I cannot find it. I also failed to find it out myself -- I tried different approaches. Maybe it should be that characteristic polynomial, maybe resultant of some form, but..</p>
<p>I hope this is done by someone already.</p>Sun, 16 Jun 2019 14:31:19 -0600Math Overflow Recent Questions: Does the isometry group of a closed simple smooth curve in the plane constrain its perimeter^2/area ratio?
https://mathoverflow.net/questions/334146/does-the-isometry-group-of-a-closed-simple-smooth-curve-in-the-plane-constrain-i
<p>Let <span class="math-container">$C$</span> be a simple closed smooth curve delimitating a bounded domain <span class="math-container">$D$</span> in the euclidean plane of isometry group <span class="math-container">$G$</span> and of given area <span class="math-container">$A$</span>. Does the minimal possible ratio <span class="math-container">$\dfrac{P^{2}}{A}$</span> where <span class="math-container">$P$</span> is the perimeter hence the total length of <span class="math-container">$C$</span> decrease when <span class="math-container">$G$</span> runs over a sequence <span class="math-container">$(G_{i})_{i>0}$</span> of groups such that <span class="math-container">$i<j$</span> implies <span class="math-container">$G_{i}$</span> is a strict subgroup of <span class="math-container">$G_{j}$</span>?</p>
<p>Edit: as exposed in comments, even a tiny perturbation of the circle provides a counter example. So suppose further that the ratios of the lengths of any two mutually orthogonal symmetry axes are integers less than a given positive constant <span class="math-container">$K$</span>. Does the modified question have an affirmative answer?</p>Sun, 16 Jun 2019 14:04:38 -0600Math Overflow Recent Questions: Identifying a determinantal condition
https://mathoverflow.net/questions/334145/identifying-a-determinantal-condition
<p>Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?</p>
<blockquote>
<p><strong>Condition.</strong> For a fixed <span class="math-container">$n > 0$</span>, all the <span class="math-container">$2 \times 2$</span> minors of the matrix
<span class="math-container">$$
\begin{bmatrix}
1 & x & \dotsm & x^n \\\
1 & f & \dotsm & f^n
\end{bmatrix}
$$</span>
are linearly independent over <span class="math-container">$\Bbb{Z}$</span>, where <span class="math-container">$f: \Bbb{R} \to \Bbb{R}$</span> and <span class="math-container">$f \neq 0,x$</span>.</p>
</blockquote>
<p>In other words, I would like to characterise the functions <span class="math-container">$f$</span> for which <span class="math-container">$x^if^j - x^jf^i$</span>, with <span class="math-container">$0 \leq i < j \leq n$</span>, are linearly independent over <span class="math-container">$\Bbb{Z}$</span>.</p>Sun, 16 Jun 2019 13:02:45 -0600Math Overflow Recent Questions: A conjecture concerning symmetric convex sets
https://mathoverflow.net/questions/334143/a-conjecture-concerning-symmetric-convex-sets
<p>Let's suppose that <span class="math-container">$S \subset \mathbb{R}^n$</span> is convex and symmetric so:</p>
<p><span class="math-container">\begin{equation}
x \in S \iff -x \in S \tag{1}
\end{equation}</span></p>
<p>Now, if we define the radius of <span class="math-container">$S$</span> as <span class="math-container">$R$</span> such that:</p>
<p><span class="math-container">\begin{equation}
R = \sup_{x \in S} \lVert x \rVert \tag{2}
\end{equation}</span></p>
<p>and use (2) to define:</p>
<p><span class="math-container">\begin{equation}
V = \{x \in S: \lVert x \rVert = R\} \tag{3}
\end{equation}</span></p>
<p>then I conjecture that: </p>
<p><span class="math-container">\begin{equation}
S = \text{conv}(V) \tag{*}
\end{equation}</span></p>
<p>I have worked out special cases of this problem within the context of high-dimensional probability but I suspect that it's generally true. </p>
<p>Might there be a theorem which guarantees this result? </p>Sun, 16 Jun 2019 12:33:22 -0600Math Overflow Recent Questions: Limitations on method of Lagrange multipliers
https://mathoverflow.net/questions/334138/limitations-on-method-of-lagrange-multipliers
<p>My general question is this:</p>
<p>What are the conditions (if any) such that the method of Lagrange multipliers will <strong>NOT</strong> find all the critical points of a differentiable function?</p>
<p>To give some context to this very general question, for</p>
<p><span class="math-container">$$f(x, y, z) = 600xy + 900xz + 900yz \text{ subject to } xyz = 486$$</span></p>
<p>I confirmed a minimum at (9, 9, 6) using a Lagrange multiplier. That method also indicated that was the only critical point. However, Wolfram found an approximation to an <strong>additional minimum</strong>, which looks valid.</p>
<p>So I am confused. My best guess at an explanation is that although the function is everywhere differentiable, the constraint is not continuous everywhere. But that is a pure guess.</p>
<p>To get the full context behind my question, please look at the following thread at a math homework site where I volunteer:</p>
<p><a href="https://www.freemathhelp.com/forum/threads/maximum-minimum-in-multivariable-functions.116663/" rel="nofollow noreferrer">https://www.freemathhelp.com/forum/threads/maximum-minimum-in-multivariable-functions.116663/</a> </p>Sun, 16 Jun 2019 11:14:01 -0600