# All Questions

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### A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another metric $d_{f}$ on $Y$ With $$d_{f}(a,b)=Hd(f^{-1}(x), f^{-1}(y))$$ ...
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### Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series $$E(z,s)=\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},$$ where $z=x+iy$. We think of $E(z,s)$ as a function on ...
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I'm reading through T. Taos book on random matrices (to be found here), and it is frequent to make without-loss-of-generality certain reductions when proving theorems as for example Hoeffdings ...
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### Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed? What is the distribution of euclidean norm of complex Nagakami? Cheers!
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### Help in finding distribution of the following function of random variable

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$ $$X_2 \sim \mathcal{CN}(0,\sigma)$$ If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...
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### What is induction to n^(n-1) >= n! for n=9,10 [on hold]

Can somebody help me? How do I prove n^(n-1) >= n! for n=9,10..... by induction?
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### A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint): Taking $A$ to be a point and iterating this collapsing lemma, this ...
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### Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...
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### Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Suppose we have a (not necessarily square) matrix $A$ whose entries are $\in \{0, 1\}$, and for all pairs of columns $x, y$ the entries of $x - y$ are either all non-negative or all non-positive. Then ...
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### Dimension of Hilbert scheme?

I have a subvariety $V \subset X$, and I want to compute the dimension of the connected component of $\textrm{Hilb}(X)$ containing $[V]$. I can give explicit deformations of $V$ showing that the ...
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### Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?

Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is ...
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### Spectral sequence and HOM functor

I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules ...
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### Number of graphs with n vertices and k edges up to isomorphy [duplicate]

Is there a way to figure out the number of individual graphs with n vertices and k edges up to isomorphy? I'm particularly looking at graphs with: n = 25, k = 50 n = 50, k = 170 n = 100, k = 700
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### Rate of convergence in narrow convergence

Does anyone help me in the following question? I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
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### Bounding a function pointwise from bound on the expectation [migrated]

Suppose we have a Lipschitz continuous positive function with bounded expectation Eg < a. What can be said about the point-wise bound ?
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### What are the minimal degrees of the real and imaginary part of an algebraic complex number?

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...
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### An alternative Generalization of The Riemann Zeta Function [on hold]

If we consider the Riemann Zeta Function as follows $$\zeta(s)=\underset{n=1}{\overset{\infty}{\sum}}\cfrac{1}{n^{s}}=\cfrac{1}{1^{s}}+\cfrac{1}{2^{s}}+\cfrac{1}{3^{s}}+\cdots.$$ could we generalize ...
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It is well known that any operad on a nice monoidal category induces a monad. I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...
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### What is God's number for the WrapSlide puzzle?

WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
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### First-order nonlinear ordinary differential eqauation [on hold]

can someone help me to solve this equation? I have been trying a few methods. Thanks. y'=(y/x)*((xy + 1)/(xy - 1))
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### Wave equation with linear coefficients

The following pde came up in a physics problem: $$(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),$$ A,B,C,D are fixed ...
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### Looking for the manuscript “Uniform polytopes” by N. Johnson

The manuscript Uniform Polytopes (1991) by Norman Johnson is cited in the wikipedia page on uniform polytopes (http://en.wikipedia.org/wiki/Uniform_polytope). Is there an electronic copy of this ...
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### Intuitive explanation of multi-objective optimization of $z = (1-x) \times (1-y)$

I have managed to reduce an optimization problem in the domain of distributed systems into the following formula: $$\Gamma = (1-\varepsilon_\alpha) \times (1-\varepsilon_\beta)$$ in which both ...
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### Functional representation of adapted jointly measurable stochastic processes

It seems like the question stated here in MSE has no answer yet and seems therefore for me to be not of a basic question type. For this reason I move it to MO. Let $X_t : \Omega \to E, \ t \geq 0$ be ...
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### Meaning of $g_d^r$ in algebraic geometry

As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? ...
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### Projective dimension of a sub-ideal

Let $\mathbf{k}$ be a field, and let $S=\mathbf{k}[x_1,x_2,\ldots,x_n]$. Let $I\subset J$ be finitely generated monomial ideals in $S$. Is it true that the projective dimension of $I$ is either ...
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### K-Permutations with forbidden numbers [on hold]

This question has some references to programming and not as many mathematical terms as you might like, but I think it's more appropriate in a mathematics forum. Introduction (Skip if you are ...
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### Which finite p-groups occur as commutators of finite p-groups?

Let $p$ be a prime number. For which finite $p$-groups $H$ there is a finite $p$-group $G$ such that $[G,G] \cong H$?