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Expected value of stochastic process

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad$$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
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Dirac functional embedding

I got the following set up: Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...
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Bernstein sets of large cardinality

A metrizable space $X$ will be called a generalized Bernstein set is every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$. It is well-known that the real line contains ...
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$TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?

Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$. Define $M := K_{n,n} /SU(n)$. Using ...
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History of the notation for substitution

One of the very common notations for syntactic substitution is $[\ /\ ]$. There seems to be a large inconsistency in the literature about its use. Many write $[t/x]$ for substitute $t$ for $x$ (e.g. ...
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For the action of $S_n$ on $\mathbb C^n$ the elementary symmetric polynomials generate the ring of polynomial invariants. What are the generators for the action of $S_n$ on $\mathbb C^n \otimes \... 0answers 52 views Find equidistant points on surface of sphere [on hold] Given a sphere, find the maximum number of points can be placed on the surface of the sphere such that all are equidistant from each other. I am not at all good in maths. Please let me know what can ... 1answer 128 views Left adjoint to Double Nerve? The well known nerve functor from small categories to simplicial sets has a left adjoint, namely the fundamental category functor. Does the double nerve functor$N^2:2Cat\rightarrow sSSet$from 2-... 1answer 110 views Evaluation of sum of factorials Is there an evaluation of this sum (possibly involving gamma functions)?$k$and$n$are natural numbers and$x$is real with$0<x<1\$. $$\sum_{\substack{k=0\\n-k\text{ even}}}^n \frac{(-1)^{(n-... 0answers 35 views Approximate unit in C*-algebra with additional properties In the book about K-theory (a friendly approach) by Wegge and Olsen I came across the following notion: in Lemma 16.4 authors assume that the C^*-algebra A possess "(commuting) approximate unit (... 1answer 206 views G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3? G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3? [on hold] As summarized in the title, suppose there is an isomorphism between G_1 \times G_2 and G_3, is it always true that BG_1 \times BG_2 is homotopy equivalent to BG_3? If it is not always true ... 1answer 52 views Concentration of matrix norms under random projection. Let X be a given matrix of dimension p \times q. Let G be a s \times p dimensional matrix of standard normal/Gaussian random variables. Are there cases where one can been able to quantify ... 0answers 46 views Do the normal forms for braid groups really conceal information about better than randomly applying the braid relations? In braid based cryptography, one typically wants to conceal the way a certain braid b has been obtained. One therefore puts b into some normal form. Since every braid has a unique normal form, the ... 0answers 98 views What is the smallest density of a metrizable space without countable separation? A Tychonoff space X is defined to have countable separation if some (equivalently, any) compactification bX of X contains a countable family \mathcal U of open sets such that for any points x\... 1answer 72 views If a Weyl element preserves a root, then it has a representative which preserves the root space? Let G be a reductive group defined over a field F. Let \Sigma be the set of roots of G with respect to a Borel subgroup B=TU with torus T. Let W=N_G(T)/T be the Weyl group of G. For \... 0answers 25 views Basic Definition and Notations in RWRE [on hold] From the definition of Zeitouni's lecture notes on RWRE: (V, E) is a special graph, and N_v:= \{k \in V: (v,k) \in E\} is the neighborhood of v \in V. \Omega = \prod_{v \in V} M_1(N_v) ... 0answers 108 views Ring structure on cohomology of groups Assume that G is a finite group and that A is an arbitrary G-module. Then we know that can define the cohomology groups of G with coefficients in A in the usual way and we denote the latter ... 0answers 19 views Closed form formula for fill rate given a discrete distribution? [on hold] I'm wondering whether there is a closed form way to obtain good estimates for fill rate given a discrete distribution of demand. I created a simple monte carlo simulation to see if I could see any ... 0answers 56 views A rash guess about distribution of primes based on meager empirical evidence? Elementary number theory is a field in which imbeciles can ask questions that experts cannot answer (and I wonder if discrete geometry is a similar subject in that respect?) and herewith I submit ... 1answer 128 views Resolution of the ideal of the Abel-Jacobi image of a curve? Let C be a complex curve of genus g\ge 2 and let a\colon C\to J(C) be the Abel-Jacobi map. Is there a finite resolution of the ideal \mathcal I_{a(C)} whose terms are sums line bundles of the ... 0answers 23 views function mapping odd numbers to counting numbers [on hold] Mapping even numbers to counting numbers is straight forward. Without introducing any other variable: i = 0, 2, 4, 6, . . . if i > 0: count = i/2 what about ... 0answers 26 views similarity metric for geometries [on hold] I'm searching for a method to calculate the degree of similarity of two given geometries. These geometries can be of any type and can have an arbitrary shape. For the sake of simplicity, I primarily ... 0answers 150 views Can some exotic sphere be diffeomorphically embedded into some R^n? Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some R^n? How does such an embedding (if it exists) look like? I.e., what are the equations for a particular embedding? ... 0answers 56 views A property of minimal prime ideals [on hold] Let R be a commutative ring with 1, and let p be a minimal prime ideal of R. If p\subseteq I_1+ I_2, where I_1 and I_2 are two ideals of R, can we deduce that p\subseteq I_1  or ... 1answer 62 views Exact formula for computing n-step transition probability of random walks with self-transitions Consider a semi-infinite random walks X_n, n=0,1,2,\ldots, whose state space is a set of consecutive integers and whose one-step transition probabilities are P_{ij}=\mathrm{Pr}\{X_{n+1}=j|X_n=i\}... 1answer 126 views Bijection modeling isomorphism of infinite-dimensional vector spaces Let T : V \to W be an isomorphism of vector spaces with bases B_V and B_W, which may be of any cardinality. Does there exist a bijection f : B_V \to B_W such that, for each b_V \in B_V,... 1answer 153 views Cambridge Mathematical Tripos papers from late 19th century Are the scanned images of Cambridge Mathematical Tripos papers from late 19th century available anywhere on Internet? 1answer 63 views Variation of Radon transform for probability measures on \mathbb C Let \mu be a probability measure on \mathbb C. For z \in \mathbb C, let$$f^z \colon \mathbb C \to \mathbb R_{\geq 0}$$be the function f^z(\lambda) = |\lambda - z|. Consider now the family (\... 1answer 138 views Could I affirm that f is not identically 0? Consider the following situation: Let \Omega =l^{\infty}(\mathbb{R}) be the space of all bounded sequences of real numbers. We will consider in \Omega the metric:  d(x,y)=\sum_{i\geq 1}\frac{|... 0answers 40 views When is a functorial coverage a sheaf, and what universal property does it have? In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category \mathcal{C}. Quoting verbatim, a coverage on \mathcal{C} is a function assigning to each object A of \... 0answers 70 views Measures on a unit sphere of a Hilbert space Consider a real separable infinite-dimensional Hilbert space H. Let S=\{h\in H\mid \|h\|=1\} be a unit sphere in H. What are the most natural measures on S? Is there a (Borel) measure \mu on ... 0answers 54 views Reference quest: variety of lines and variety of planes Let X\subset \mathbb P_{\mathbb C}^n be a smooth projective variety, F(X)\subset G(2,n+1) its Fano variety of lines and$$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$... 0answers 65 views Ellipticity of Bott-Chern Laplacian I want to prove that Bott-Chern Laplacian$$\tilde{\Delta}_{BC}^{p,q}=(\partial\bar\partial)(\partial\bar\partial)^*+(\partial\bar\partial)^*(\partial\bar\partial)+(\bar\partial^*\partial)(\bar\...
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I'm looking for any graduate programs related to graph theory or combinatorics in canada like in waterloo or simon fraser universities. any other suggestions?
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getting papers published when you're not affiliated to a university [on hold]

I graduated in Maths 20 years ago, spent a long time away from the subject and recently returned to it. I work entirely alone right now but after a refresher phase, I'm starting to look at some very ...