# All Questions

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### Relating the R-Transform in Free Probability to noncommutative group representations

In traditional (commutative) probability theory, sums of random variables correspond to convolution of distribution functions, which plays well with the Fourier Transform. In free (noncommutative) ...
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### tessellation of an arbitrary shape

Is there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate by some shapes?
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### Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity? $$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$ Here, $|x|$ denotes the pointwise absolute ...
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### Cayley graph of dihedral group is isomorphic to which kind of graphs? [on hold]

Let D_{2n}= be dihedral group of order 2n. Also let D_{2n}= in which 1\ notin S=S^{-1}. In this case Cay(D_{2n}, S) is isomorphic to which kind of graphs? This is my conjecture that this graph is ...
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### Find the joint density function?

Assume that $X_t$ is the OU process , i.e, $dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$. Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$. I want ...
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### Branching process and process stochastic

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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### Necessary and sufficient condition for order relations which are realizable as subsets of real numbers [duplicate]

Is there any simple necessary and sufficient condition for a totally ordered set $S$ to be realizable as a subset of $\mathbb{R}$?
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### What is the formula to calculate the total sum of all radiuses of circles that can be contained in a square [on hold]

What is the formula to calculate the total (maximum possible) sum of all radiuses of circles that can be contained in a square considering that the square dimensions are fixed and that the circles are ...
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### Loss of Kinetic Energy [on hold]

can somebody help me with how we walked from ((m1u12)/2)+(m2u22)/2) to ((m12u12 + m22u22 + m1m2(u12+u22))/2(m1+m2) . probably a walk through of how the loss of kinetic energy was gotten to be ...
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### Independent Generic Curves in the Projective Plane

I'm trying to read M. Nagata's paper On the Fourteenth Problem of Hilbert and I've run into some trouble understanding the definitions he's using. The setup is as follows: let $\pi$ be a prime field ...
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### Weight shrinking in linear regression by L2 regularization [on hold]

Quoting Prof. Bengio from his Deep Learning text (http://www.iro.umontreal.ca/~bengioy/dlbook/regularization.html), $w = (X^{T}X + \alpha I)^{-1}X^{T}y$ We can see L2 regularization causes ...
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### Definition of Ito Integral

In Karatzas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable and adapted processes ($f(t,\omega)$), the authors say that there ...
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### Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
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### When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with ...
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### Topic in functional analysis [on hold]

i'm a graduate student and i like an analysis. What are current research topics in the functional analysis especially in geometry of Banach spaces? I would like to read about them.
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### Topology of ring of global sections of finite union of affinoid opens in a rigid analytic space

Let $X$ be a rigid analytic space over a non-Archimedean field $k$. If $U_1,\ldots,U_n\subseteq X$ are affinoid opens, then it's usually not clear whether or not the admissible open ...
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### How to write an abstract for a math paper? [on hold]

How would you go about writing an abstract for a Math paper? I know that an abstract is supposed to "advertise" the paper. However, I do not really know how to get started. Could someone tell me how ...
For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$. I know that for $m=2$, there are some applications of finding shortest paths (or distance ...
Let us work on a bounded domain $\Omega$ with Neumann BCs, with $(\varphi_k, \lambda_k)$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Given \$u \in H^{\frac ...