# All Questions

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### Bounded Operator, Proof

Let $V$ be continuous, $V\geq0$ and $V\rightarrow \infty$ as $||x||\rightarrow \infty$. Define $H:=-\Delta+V$. I want to show that $$G:=\big((-\Delta)^{\frac{1}{2}}+1\big)(H+1)^{-\frac{1}{2}}$$ is ...
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### Are stably similar matrices similar?

Let $R$ be a ring and let $Gl_n(R)$ denote the set of invertible $n$ by $n$ matrices. Two matrices $A,B\in Gl_n(R)$ are called similar, if there exists another $P\in Gl_n(R)$ such that ...
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Given a bimatricial game $(A,B)$, what can I say about distances between its nash equilibria and the nash equilibria of another perturbed game $(A + \delta A, B + \delta B)$. In other words, do ...
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### Braid Group Abelianization

I'm a physics students, facing for the first time homotopy and homology stuff and I need a little help. I'm dealing with a path connected topological space (actually a differentiable manifold) $Q$ ...
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### What is the sample space for multiple random experiments

Say we flip a coin $n$ times and define, for the i-th coin flip, the event $H_i$ for "we get Heads" and $T_i$ for "we get Tails". For example, we might be interested in studying the event "The first ...
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### Different fields of work in different fields of mathematics?

I'd like to get some insight into what sorts of jobs one can expect from the different branches of mathematics degrees. The B.sc. degrees I am familiar with are: Pure mathematics. Applied ...
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### Result is not answer key's answer, Need to find where the problem is

So in my unit review I have come across the question: Determine the distance from point $A(-2,1,1)$ to the line with the equation $\vec{r} = (3,0,-1) + t (1,1,2),tER$ So I let $B = (3,0,-1)$ and ...
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### How to solve affine Linear Matrix Inequlaity in MATLAB?

I wanted to solve following linear matrix inequality $F(h(t))<0$ where $$F(h(t)) = A(h(t))'P−C'R +PA(h(t))−R'C$$ Matrix $F$ is affine in $h(t)$ and $P, R$ are matrix variables and $C$ is matrix ...
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### Auto-intersection of a line on a smooth cubic surface

Can someone help me with the following idea? I think that i made a mistake: Let $X$ be a smooth surface of degree $d$ in $\mathbb{P}^3$ and $L$ denote the divisor class of a line on $X$. We have ...
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### References for Hidden Markov Chains

I'm looking for some nice introductions to Hidden Markov Chains. Preferably some that begin from the basic definitions. I would like some of these references to be papers published in journals. Any ...
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### Vector cross product question

Let a and b be the position vetors of two points A and B, and let p be the position vector of another point P. Consider the line through the points A and B and show that the shortest distance ...
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I have this in the practice questions for my upcoming exam, and we have just learned about Normal distribution today. The lifetime of a street lightbulb is $X ~ N(1,000,40,000)$ hours. Find the ...
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### If $b > 1$ and $B(r)$ is the set of all numbers $b^t$, where $t$ is rational and $t \leq r$, prove that $b^r = \sup B(r)$ where $r$ is rational.

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. As a follow-up to this question, Rudin asks ...
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### Finding the components of the Riemannian tensor given the components of a metric.

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
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### Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
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### Find error in a stochastic algorithm

I have $n$ cities, and I want to simulate the transition of people between these cities according to some rules (not all the cities are connected). Each city have $m_n(t)$ citizens and a rate $r_n(t)$ ...
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### Theorem of Poincaré-Bendixson

We can't deduce existence of a closed orbit from the theorem, when there is a fixed point in the region R. Is the fixed point not an attractor, we can cut out a small circle around the fixed point. ...
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### Trains with cuisenaire rods

I'd like to discuss some questions that seem to be not known and/or not studied so far (I will warmly thank you if I'm wrong and you can cite any reference on the topic). As a starting point, the ...
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### Proving an elementary abelian group has the same number of subgroups of order $k$ as of index $k$.

I was thinking of a nice way to show that the elementary abelian group of order $p^n$ has the same number of subgroups of order $p^k$ and of index $p^k$, for every $k$ between $0$ and $n$. Since such ...
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### Generalized distributive law

Let $p,q,r,C_{ij}$ be formulae in propositional logic, or even simply symbols, i'm only interested in notations. Distributive law says: $$p\vee (q\wedge r)=(p\vee q)\wedge (p\vee r)$$ I want to ...
Let $A$ and $B$ be two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$. If $p$ is a point in the hyperbolic plane, we can consider the broken geodesic ray $\gamma_A$ described by the vertices ...