# All Questions

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### Expected Number of Support Vectors in Vapnik's book

In Vapnik's book "Statistical Learning Theory", Theorem 10.5 states that - for a Support Vector Machine - the expected probability of error (of the optimal hyperplane) is upper bounded by $1/(l+1)$ ...
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### A connection between nonplanar complete graphs and the alternating group?

I didn't get any response on MSE so I though I'd give this a try here (my question on MSE). I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...
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### Maximal ideals in function spaces [migrated]

Hello friends of mathematics, i have got a question. In the lecture we proved that the maximal ideals of $C(X)$ are the sets of functions which vanishes on a closed subset of $X$. But now i will look ...
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### basic probability question [closed]

How many ways are there for three married couples to arrange themselves in a queue for the cinema, assuming spouses always stand together. would that be 6x5x4? my notes have no solutions!
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### Logic without disjunction

I'm working on a small proof system, which in principle only has equality as predicate. This equality has some axioms which make them sort of a structural equality: two terms are equal iff their ...
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### The uniqueness of inhomogeneous wave equation with Neumann BC [closed]

I got stuck with problem (c), when sigma < 0, it seems that the solution is unique, but how to prove it?
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### Quartic surfaces with separate tritangents

Let $S$ be a smooth quartic surface in $\mathbb{P}^3$, given by an equation $F=0$. Through a general point $p$ of $S$ pass two tritangents, that is, lines with a contact of order 3 at $p$. This ...
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### Derivative of Riemann Zeta at nontrivial zeros

I would like to know whether the real part of the first derivative of the Zeta function at the non trivial zeros of Zeta is stricly positive and if so, is there a proof for it. Also, are there tables ...
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### Completion of abelian topological groups

During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by ...
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### Global sections for torus fiber bundle

Let us consider the following situation: $\pi:X\rightarrow B$ is a locally trivial fibration between smooth manifolds, its fiber being a torus $T$. My question is two-fold: 1) what is the obstruction ...
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### Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight ...
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### Uniform Law Of Iterated Logarithm for VC classes

Kenneth Alexander proved a uniform Law Of Iterated logarithm for Vapnik-Chervonenkis classes in the article Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm (Ann. ...
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### centralisers of maximal split tori

Suppose that $G$ is a reductive group defined over a field $k$ which is not quasisplit. Suppose that $S$ is a maximal $k$-split torus. Let $\mathcal{L}(S)$ be the centraliser of $S$ and ...
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### Does the derivative of the largest eigenvalue and its associated eigenvector exist? [migrated]

Let $\boldsymbol{A}(t):\mathbb{R} \rightarrow \mathbb{C}^{n\times n}$ be a rank-deficient Hermitian function-valued matrix (its entries are analytic functions). Furthermore let $\lambda_1(t)$ (which ...
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### Covering a set of points by bounded geometric object/objects

1) Let $S$ be a set of $n$ points in $R^d$. Now, given a bounded geometric object $G$, the problem is to check whether $S$ can be contained in $G$. 2) Also, in general setting, the problem is to ...
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### Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
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### Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$

I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO ...
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### fraction power of operators in $C_0$ semigroup

Let $X$ be Banach space, and $\{Z(t)\}_{t\geq 0}\subseteq B(X)$ be the $C_0$-Semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}_{t\geq 0}$. A ...
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### How many ways do we have to prove that a mapping is open?

Given a continuous mapping $f$ between Euclidean domains (or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? ...
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### Hurwitz integers represented as sums of two squares of Hurwitz integers

I wonder if there exists a characterisation of Hurwitz integers which are represented as sums of two squares of Hurwitz integers, up to multiplication by a unit. And if so, could you please point to a ...
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### Probability of a random walk to hit x before y? [closed]

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map $$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first ...
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### finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$. Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that ...
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Already done this problem in class but I was absent. I cannot figure out the problem. Could someone please do it out in detail so I can see how this problem is done? We would like to prove that for ...
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### Stratifications and Filtrations of the Affine Grassmannian

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that ...
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### Quantum Cartan matrices and Coxeter elements

Let $\Gamma$ be a bipartite graph, with the vertices partitioned into disjoint sets $\Pi_1$ and $\Pi_2$. Let $W$ be the associated Weyl group, with Coxeter generators $\{s_i\}_{i\in \Gamma}$. Let ...
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### Large powers with modulo [closed]

How do I calculate the value of $712^{197}$ mod $31$? I need it to calculate $712^{197}$ mod $1333$. I know $1333=31 \times 43$. So I am trying to calculate $712^{197}$ mod $31$ and $712^{197}$ mod ...
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### How many 2L-bit numbers are the product of two L-bit numbers?

If I multiply two integers $x, y$ in $[0,2^L)$, I get an integer in $[0,2^{2L})$. Clearly, this map from $[0,2^L) \times [0,2^L) \to [0,2^{2L})$ is not bijective. I am interested in the size of ...
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### Computation of large modulo value [closed]

For a problem on Miller-Rabin's primality test, I need to calculate $36^{333}$ mod $1333$. How do I calculate this value? Any help will be greatly appreciated.
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### Is there interesting structure in the space of Voronoi diagrams?

Given a finite set $X = \{x_1,\dots,x_N\}$ of points in $\mathbb{R}^n$, let $V_j(X)$ be the interior of the Voronoi polytope corresponding to $x_j$, i.e., the interior of the set of points closer to ...