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curious little things - aggregated feedsenMath Overflow Recent Questions: Find conditions over $U$ such that optimization over a convex cone generated by $U$ is equal to optimization over $U$
https://mathoverflow.net/questions/279367/find-conditions-over-u-such-that-optimization-over-a-convex-cone-generated-by
<p><strong><em>Reading several pappers to prepare my thesis I found the following problem:</em></strong></p>
<p>We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[2pt]
\text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{1}
$$</p>
<p>where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ and $\mathcal{A}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ are linear functions non-zero, $b\in\mathbb{R}^{m}$, $\mathcal{C}$ is a convex cone in $\mathbb{R}^{n}$ and $K$ is a closed convex cone in $\mathbb{R}^{m}$.</p>
<p><strong>Question:</strong> I need to find conditions over a set $U\subset\mathbb{R}^{n}$ such that if $\mathcal{C}$ is the convex cone generated by $U$, then problem $(1)$ is equivalent to the following problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in U} & f(x) \\[2pt]
\text{s.t.} & \mathcal{A}x-b \in K \end{array} \right. \tag{2}
?$$</p>Tue, 22 Aug 2017 20:15:19 -0600Math Overflow Recent Questions: Martingale Approach for upper bound on time until n deaths
https://mathoverflow.net/questions/279366/martingale-approach-for-upper-bound-on-time-until-n-deaths
<p>suppose we have $n$ Geometric$(p)$ random variables $X_1,\dots,X_n$, and let $Y_t$ be the number of these random variable still alive at time $t$, i.e.</p>
<p>$Y_t = \sum_{i=1}^n \mathbb{1}\{X_i \geq t\}$</p>
<p>I'm interested in upper bounding the expected first time that $Y_t = 0$ using a martingale approach. Here's an example argument that doesn't quite work: we can easily show that</p>
<p>$M_t = \log(Y_t) - \log(n(1-p)^t)$</p>
<p>is a supermartingale. Now we can apply the optional stopping theorem for any stopping time $\tau$ that occurs almost surely to get a bound</p>
<p>$\mathbb{E}[\tau] \leq \frac{1}{\log(\frac{1}{1-p})}(\log(n) - \mathbb{E}[\log(Y_{\tau})])$</p>
<p>However, if we let $\tau$ be the first time that all our geo random variable die, i.e. the first time that $Y_t = 0$, then we can't get anything useful from this bound, since $\mathbb{E}[\log(Y_{\tau})] = -\infty$. I'm wondering if there's a modification of this argument that works? Or if anyone knows alternate approaches? Note that this time is the same as the expected maximum of $n$ geometric random variables. But I'm specifically wondering about a martingale approach (just out of curiosity).</p>Tue, 22 Aug 2017 19:47:59 -0600Math Overflow Recent Questions: Hessian of a multiple variables function
https://mathoverflow.net/questions/279365/hessian-of-a-multiple-variables-function
<p>I have a function $f(X,Y)$ w.r.t. to two sets of variables $X$ and $Y$. $X = \{x_1,x_2,\dots,x_n\}$ and $Y = \{y_1,y_2,\dots,y_m\}$ where $n$ and $m$ are finite. What is the Hessian matrix of this function?</p>
<p>Here is my thought:
\begin{bmatrix}
\frac{\partial^2 f}{\partial x_1 \partial x_1}& \frac{\partial^2 f}{\partial x_1 \partial x_2}&\dots&\frac{\partial^2 f}{\partial x_1 \partial x_n} &\dots & \frac{\partial^2 f}{\partial x_1 \partial y_1} &\dots& \frac{\partial^2 f}{\partial x_1 \partial y_m} \\
\frac{\partial^2 f}{\partial x_2 \partial x_1}& \frac{\partial^2 f}{\partial x_2 \partial x_2}&\dots&\frac{\partial^2 f}{\partial x_2 \partial x_n} &\dots & \frac{\partial^2 f}{\partial x_2 \partial y_1} &\dots& \frac{\partial^2 f}{\partial x_2 \partial y_m}\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\
\frac{\partial^2 f}{\partial y_1 \partial x_1}& \frac{\partial^2 f}{\partial y_1 \partial x_2}&\dots&\frac{\partial^2 f}{\partial y_1 \partial x_n} &\dots & \frac{\partial^2 f}{\partial y_1 \partial y_1} &\dots& \frac{\partial^2 f}{\partial y_1 \partial y_m}\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots&\dots\\
\frac{\partial^2 f}{\partial y_m \partial x_1}& \frac{\partial^2 f}{\partial y_m \partial x_2}&\dots&\frac{\partial^2 f}{\partial y_m \partial x_n} &\dots & \frac{\partial^2 f}{\partial y_m \partial y_1} &\dots& \frac{\partial^2 f}{\partial y_m \partial y_m}
\end{bmatrix}</p>
<p>Please correct me if I am wrong.</p>Tue, 22 Aug 2017 19:44:21 -0600Math Overflow Recent Questions: Does a quotient group $G/N$ have a natural action on the regular representation of $G$?
https://mathoverflow.net/questions/279362/does-a-quotient-group-g-n-have-a-natural-action-on-the-regular-representation
<p>Let $G$ be a group. I am happy to assume niceties such as finite and abelian, but perhaps it is not necessary to answer my question.</p>
<p>Consider the $|G|$-dimensional vector space $V$ (over some nice field $K$, possibly even $\mathbb C$ if need be) whose basis $\{v_g\}_{g \in G}$ is labeled by the group elements of $G$. We know any subgroup $H \subset G$ has a natural action on this vector space: any $h \in H$ defines a linear map $h \cdot v_g = v_{hg}$.</p>
<p>I was now wondering: take a normal subgroup $N \trianglelefteq G$. Is there a natural action of $G/N$ on $V$? I would imagine something in the line of $[g'] \cdot v_g = k \; v_g$ where $k\in K$, but not sure what $k$ should be. Possibly something like ``$k = -1$ if $[g'] = [g]$ and $k=1$ otherwise''?</p>
<p>(For those interested in the more general context I am wondering about: given a (central) extension $1 \to A \to E \to B \to 1$, do $A$ and $B$ have a natural action on the regular representation of $E$?)</p>Tue, 22 Aug 2017 18:29:57 -0600Math Overflow Recent Questions: Quadratic character base
https://mathoverflow.net/questions/279359/quadratic-character-base
<p>What is the use of Quadratic Character Base in Sieving specifically Lattice Sieving?</p>
<p>I know it is used to make sure that the final combination is constructed in such a way that its is a square in Z[alpha] (on the algebraic side) but can someone illustrate in detail?</p>Tue, 22 Aug 2017 16:57:42 -0600Math Overflow Recent Questions: separable variable method came from ? [on hold]
https://mathoverflow.net/questions/279358/separable-variable-method-came-from
<p>can anybody know why in separable variable we need assumption that our function is made from product of our variable , not added results or qoutient.</p>
<p>Y(a,b,c)=A(a).B(b).C(c).</p>
<p>why we can assume that ? where is came from ? (philosophy) </p>
<p>who was found this technique ?</p>Tue, 22 Aug 2017 16:43:43 -0600Math Overflow Recent Questions: Partial Differential Equations, Schwartz and Hilbert space [on hold]
https://mathoverflow.net/questions/279357/partial-differential-equations-schwartz-and-hilbert-space
<p>Define a Hilbert space $H_j$ by </p>
<p>$$H_j = (u \in L^2(R^d): \int (1+ x_j^2)|u|^2 dx + \int (1+ \xi_j^2)| \widehat {u} |^2 < \infty).$$</p>
<p>The square of the norm in $H_j$ is the sum of the integrals in brackets.</p>
<p><strong>I need help how to prove that:</strong></p>
<p>The Schwartz space $S$ is dense in $H_j$. </p>
<p>$S(R^d)= (u \in C^{\infty}: \alpha, \beta \in N^d, sup_{x \in R^d} |x^{\alpha} \partial^{\beta} u(x)|< \infty ) $</p>Tue, 22 Aug 2017 16:20:44 -0600Math Overflow Recent Questions: What does homomorphism between languages mean to the correspoding Turing Machines?
https://mathoverflow.net/questions/279354/what-does-homomorphism-between-languages-mean-to-the-correspoding-turing-machine
<p>According to <a href="http://www.sciencedirect.com/science/article/pii/0304397585900180" rel="nofollow noreferrer">the article</a>: every c.e.language over $\Sigma^*$can be formed by homomorphism from a Dyck language over $\Sigma^{'}$ intersection with a minimal linear language over $\Sigma^{'}$ to the Kleene closure $\Sigma^*$ over a alphabet.</p>
<p>Now we know, intersection between languages is parallel to series connection of the corresponding Turing Machines. Then what does homomorphism between languages mean to the correspoding Turing Machines? </p>
<p>Partially it seemingly means merge of computational paths, </p>Tue, 22 Aug 2017 15:25:53 -0600Math Overflow Recent Questions: Divisibility property of integers congruent to quadratic residues
https://mathoverflow.net/questions/279353/divisibility-property-of-integers-congruent-to-quadratic-residues
<p>If $a^2 \equiv b \bmod p$ then can we expect divisibility properties of $a$ and $b$ to be different than what a 'normal' behavior would indicate?</p>
<p>Pick a perfect square $a^2$ and a non-square $b$ in interval $[2^{m-1},2^m]$. Pick $t=O((\log m)^c)$ different primes $p_1,\dots,p_t\in[m,2m]$ where $c\in\Bbb R_+$ is fixed.</p>
<p>The number of pairs of $(a^2,b)\in[2^{m-1},2^m]\times[2^{m-1},2^m]$ such that $a^2\equiv b\bmod p_i$ and $a^2\equiv b\bmod (p_i+\Delta_j)$ where $j\in\{1,\dots,t'\}$ and $\Delta_1,\Delta_2$ are distinct fixed integers from $\{1,\dots,t\}$ is heuristically at most $\frac{2^{2m}}{(\min(a,b)^{(t'+1)(\log m)^c}}$.</p>
<p>Is it possible there could be at most only $\frac{2^{2m}}{(\min(a,b)^{2^{(t'+1)}(\log m)^c}}$ such pairs?</p>
<blockquote>
<p>What is the fastest growing function $f(t', m)$ of $t',m$ that provides correct estimate in form of $\frac{2^{2m}}{(\min(a,b)^{f(t', m)}}$ such pairs?</p>
</blockquote>Tue, 22 Aug 2017 15:21:22 -0600Math Overflow Recent Questions: An explicit representation for polynomials generated by a power of $x/\sin(x)$
https://mathoverflow.net/questions/279348/an-explicit-representation-for-polynomials-generated-by-a-power-of-x-sinx
<p>The coefficients $d_{k}(n)$ given by the power series
$$\left(\frac{x}{\sin x}\right)^{n}=\sum_{k=0}^{\infty}d_{k}(n)\frac{x^{2k}}{(2k)!}$$
are polynomials in $n$ of degree $k$. First few examples:
$$d_{0}(n)=1,\quad d_{1}(n)=\frac{n}{3}, \quad d_{2}(n)=\frac{2 n}{15}+\frac{n^2}{3}, \quad d_{3}(n)=\frac{16n}{63}+\frac{2 n^{2}}{3}+\frac{5n^3}{9}.$$
<b>Question:</b> Is there an explicit formula for the coefficients of polynomials $d_{k}(n)$?</p>
<p><b>Remark:</b> I am aware of their connection with the Bernoulli polynomials of higher order $B_{n}^{(a)}(x)$. Namely, one has $d_{k}(n)=(-4)^{k}B_{2k}^{(n)}(n/2)$. This formula and several other alternative expressions can be found in the book of N. E. Norlund (Springer, 1924) but none of them seems to be very helpful.</p>Tue, 22 Aug 2017 14:22:30 -0600