0
votes
0answers
27 views

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 ...
0
votes
0answers
30 views

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 ...
0
votes
0answers
19 views

Bad coditioned games

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 ...
0
votes
0answers
77 views

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) $Q$ ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
42 views

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 ...
0
votes
0answers
66 views

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 ...
0
votes
0answers
48 views

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 ...
0
votes
0answers
34 views

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 ...
0
votes
0answers
54 views

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 ...
0
votes
0answers
9 views

Normal distribution light-lifetime question

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 ...
0
votes
0answers
41 views

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 ...
0
votes
0answers
35 views

Basic graph sketching

Sketch $y(x) = \dfrac{e^{(1/2)x}}{1+e^x}$, so far I've found a stationary point at x = 0, and $y(x) \approx e^{(-1/2)x} $, so as $ x \to +\infty $ $y(x) \to 0 $ but as $x \to -\infty$ surely $y(x) \to ...
0
votes
0answers
54 views

A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
0
votes
0answers
35 views

relation of dimension of a finite module to the dimension of the underlying ring

In my previous question intuition in definition of the dimension of a finite module, the author of the given answer writes "the Hilbert function of the module roughly grows like a scalar times the ...
0
votes
0answers
57 views

Yoneda's lemma to prove $f^*(\tilde M) \cong \widetilde{B\otimes _A M}$

Let $M$ be an $A$-module. and let $\phi : A \rightarrow B$ be a ring homomorphism. Let $f$ be the corresponding morphism of scheme from $SpecB$ to $Spec A$. Then prove that $f^*(\tilde M) \cong ...
0
votes
0answers
25 views

show that there exists $\theta_k$

Good Afternoon! I am working on a problem whose solution requires, as an intermediate step, to show that there exists $\theta_k$ such that $\sum_{k=2}^n[log(k) - log(k-\frac{1}{2})]=\sum_{k=2}^n ...
0
votes
0answers
18 views

Non-zero solutions to $B(p+p^2+p^3+\ldots)=Ap$

Let $\mathbf{p}$ be a vector of length $N$ with $0\le p\le 1$. $A$ is a non-negative matrix (not necessarily symmetric) with zero diagonal entries. $B$ is a diagonal matrix with positive entries. ...
0
votes
0answers
69 views

Probability of a car accident given that a man is driving (Bayes formula)

I came across the following exercise illustrating the Bayes formula: Let's consider the events: F = {female driver}, G = {male driver}, and A = {car accident}. We also know that ...
0
votes
0answers
44 views

How to make a proof of this

I want to prove: If $i \in \mathbb R \setminus \mathbb Q$ and ${p \over q}$ is nearer to $i$ than ${p'\over q'}$ then $q>q'$. Here $p,q$ and $p',q'$ do not share divisors. My idea was to use $$ ...
0
votes
0answers
49 views

Complex integral with non holomorphic point on the curve of integration

How do I solve the following integral where I have a point where the function isn't holomorphic on a point of the curve of integration, like the following? $$\int_{|z-1|=3} ...
0
votes
0answers
35 views

Evaluate the surface integral

Let S = $\{(x,y,z) \in R^3$ |$ x^2+y^2+z^2=1\}$.Let V = $(v_1,v_2,v_3)$ be solenoidal vector field on $R^3$. Evalute: $$\int_S [x(x+v_1(x,y,z)) + y(y+v_2(x,y,z)) + z(z+v_3(x,y,z))]dS$$
0
votes
0answers
32 views

Some estimation concerning bivariate normal distribution

I found the following identity: let $\Phi(t)$ be the distribution function of a standard normal random variable and $F(x,y)$ the distribution function of a bivariate normal random vector $(X_1,X_2)$ ...
0
votes
0answers
65 views

Reckless manipulation of differentials

In single variable calculus, aswell as in applied calculus courses such as mechanics, we constantly prove theorems and solve problems by cancelling, multiplying and factoring differential expressions ...
0
votes
0answers
105 views

Question on derivative

If $F_2(u)=\frac12 (Au,u)$ where $A$ is a continuous and self adjoint operator and $\eta$ the flow défini l'o.d.e $$ \begin{cases} \displaystyle\eta '(s)=- \frac{A\eta(s)}{||A\eta (s)||}\\ \eta(0)=u ...
0
votes
0answers
28 views

Confidence interval for Laplace scale parameter with location parameter unknown

Let $x_1,\dots,x_n$ be i. i. d. Laplace random values with location parameter $a$ and scale parameter $b$. The moment estimator for scale parameter $b$ is $b=\sqrt{0.5\tilde S^2}$, where $\tilde S$ ...
0
votes
0answers
58 views

Principal Divisors of rational function

Let $X$ be a nonsingular curve. Let $k$ be algebraically closed field. If $f$ is a rational function on $X,$ then the principal divisors of $f$ is defined by $(f) = \sum_{P\in X}\nu_{P}(f)P,$ where ...
0
votes
0answers
38 views

Ideals, closure

Suppose $x\in R$, where $R$ is a commutative ring with unity, and $I$ is an ideal of $R$. Suppose further that $rx\in I$ for every $r\in R$. Does this imply that $x \in I$? (if so why?) Thanks for any ...
0
votes
0answers
33 views

Approach to store result of intersecting $n$ planes

The equation of the plane is ax + by + cz + d = 0, where (a,b,c) is the plane's normal, and d is the distance to the origin. This means that every point (x,y,z) ...
0
votes
0answers
41 views

Proving that a cyclic permutation has subgroups

This is my own work, I'm "self-learning" and trying to give Algebra some much needed attention. I've defined a permutation as follows: Let X be the set {1,2,3,...,n} Let f be a permutation on X. ...
0
votes
0answers
33 views

Diffie-Hellman key exchange problem

I would like to ask about a little help about a Diffie Hellman key exchange problem with polynomials. Until now i've solved such problems with numbers, so the algorithm is clear for me. The confusing ...
0
votes
0answers
27 views

Conics with rational points

Let C be a conic over a field k, so $C = V_+(F) \subset \mathbb{P}^2_k$, where $F(x,y,z)$ is homogenous of degree 2. I am trying to show that if C contains a rational point P and is regular at P, then ...
0
votes
0answers
25 views

Rearranging a formula for $P$

$E=2.5\cdot P\cdot\left(1-\frac{x}{P}\right)^{0.286}$ I am trying to solve the equation for $P$. I cannot get passed the step to factor $P$ and get it by itself.
0
votes
0answers
51 views

Compact notation for expression

Consider the following expression: $$(a^2+b^2)^2 = a^4 + b^4 + 2a^2b^2$$ $$(a^2+b^2+c^2)^2 = a^4 + b^4 + c^4 + 2a^2b^2+2a^2c^2 +2b^2c^2$$ Or this one, in which you cannot group terms: $$(a+b)(a^3+b^3) ...
0
votes
0answers
61 views

Why does this matrix has only zeros and ones in the diagonal?

If $T$ and $M$ are $3\times 3$ invertible matrices. Consider the matrix $L=T^tMT$. I know this matrix is diagonalizable, because it's symmetric. I would like to know why if the field $k$ in the ...
0
votes
0answers
34 views

Travelling salesman problem on a general metric

We do know that the travelling salesman problem is NP-equivalent for $(\mathbb{R},d_{|\cdot|})$, but e. g. for the discrete metric on a set where the distance is zero if two elements are identical and ...
0
votes
0answers
26 views

Doubts on Conjugate and Biconjugate Gradient Method

I am not able to prove that $r^t_iAd_j=0$ for $j\neq i-1$, given $r^t_i$ the $i$-th residual $b-Ax_i$ and $d_j$ the $j$-th $A$-conjugate direction ...
0
votes
0answers
23 views

Wilson score for a range of ratings?

(Note: I did look at similarly worded questions and none seem to answer this one.) Given this scenario: People can rate 0 or more items on a scale of 0-X. I am trying to find the Wilson score ...
0
votes
0answers
31 views

Checking if estimator is biased

Estimator $S$ for a small sample $n<10$ is very uneffective. The samples of few elements are, however, being used in the production controls. In this case, standard deviation is approximated ...
0
votes
0answers
62 views

A estimate for the L1 norm of the Dirichlet kernel

Consider the classical Dirichlet kernel given by $D_{N}(x) = \displaystyle\frac{sin((2N + 1)\pi x)}{sin(\pi x)}$ for $x\neq [-\frac{1}{2} , \frac{1}{2}]-\{0\}$ and $D_N(0) = 2N +1$ . Define $L_N = ...
0
votes
0answers
55 views

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 ...
0
votes
0answers
61 views

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 ...
0
votes
0answers
20 views

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 $r_n(t)$ ...
0
votes
0answers
52 views

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. ...
0
votes
0answers
69 views

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 ...
0
votes
0answers
33 views

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 ...
0
votes
0answers
35 views

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 ...
0
votes
0answers
12 views

small deviations along hyperbolic G-orbit quasi-geodesics

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 ...
0
votes
0answers
28 views

Modeling a discrete probabilistic time evolution (in matlab?)

Good day, I am trying to model the following situation. There is a system, which can be in three states. State M1, M2 and M3. From state M1, it can go into state 2 with probability p1. From state ...

15 30 50 per page