1
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0answers
45 views

Supermodularity and n-increasingness

Let $\geq$ be the usual partial order over $\mathbb{R}^n$ (i.e. if $x,y\in\mathbb{R}^n$, $x\geq y$ iff $ x_i\geq y_i \forall i=1,\dots,n$). Definition 1: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$. ...
1
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0answers
62 views

Conditions on $f$,$g$ such that $\int f(x) - g(x)\,\mathrm{d}x$ converges given $f \sim g$

Let $f$, and $g$ be two functions. I am trying to study under what conditions the integral $$ \int_C f(x) - g(x)\,\mathrm{d}x $$ converges. Where $C$ is some open, half open or closed interval. ...
1
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0answers
42 views

Divisors of Fermat Curves.

I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of ...
1
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0answers
191 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
1
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0answers
54 views

MLE of discrete uniform distribution

Assume that $X$ is a discrete random variable with uniform distribution on the set $\{1,2,3,\ldots, N\}$, where $N$ is an unknown positive integer. Find the MLE $\hat{N}_k$ of $N$, assuming that ...
1
vote
0answers
40 views

Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$ x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)} $$ for ...
1
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0answers
42 views

Assistance solving $x'(t)=t-x(t)^2$

I'm taking a second level ODE class and for part of some problem I need to solve a nonlinear first-order differential equation, but I've never worked with nonlinear problems before (there was no ...
1
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0answers
56 views

Conditional convergence of $\sum _{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} \log(1+a_n)$

In the Ahlfors Complex Analysis book section on infinite products, there is a result that the the series $\sum _{n=1}^{\infty} |a_n|$ converges exactly when $\sum _{n=1}^{\infty} | \log(1+a_n)|$ does. ...
1
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0answers
48 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
1
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0answers
16 views

Question about integrals of an unkown function and then differentiating to it

In my thesis I encountered the following problem: I have an unknown function y(x) and I need to calculate the following combination of integrating and differentiating: ...
1
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0answers
19 views

General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
1
vote
0answers
23 views

Why is tree traversal the fastest ray-box method?

I'm learning ray tracing (the problem of intersecting a ray, aka a vector, against a 3D box defined by a max and a min point) and I'm wondering: why is a tree traversal (e.g. bounding volume ...
1
vote
0answers
46 views

Pointwise convergence to $\ln(x)$

I came up with a stepfunction on $(0,1]$: $s_n = \sum_{i=1}^{n} \ln (\frac{i}{n}) \chi_{(\frac{i-1}{n},\frac{i}{n}]} $, where $\chi$ denotes the characteristic function. I need to show that this ...
1
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0answers
19 views

Multiple measurements per person per treatment

Suppose I wish to assess reaction time of individuals before and after treatment. Now to analyse the results I could use a paired t-test or if I had additional treatments, I could use a repeated ...
1
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0answers
41 views

Implications of convergence in probability?

Let $||.||$ indicate the Euclidean norm. Let $\theta_0$ be a specific value of the parameter $\theta \in \Theta\subseteq \mathbb{R}^d$ and $G_n$ be a random vector-valued function $$ G_n : \Theta ...
1
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0answers
43 views

matrix power after multiplying row by factor

Consider a Matrix $M$ with non-negative entries and it's power $M^n$. Assume we know every entry of $M^n$ denoted by $m_{ij;n}$. Now we multiply the $i$-th row of $M$ by a positive factor $\alpha$ ...
1
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0answers
49 views

Solve Poisson's equation

I want to solve Poissons equation $$ C=\nabla^2 v $$ where $C$ is a constant and v my variable. I want to solve over some 2D domain D with the boundary condition that v is zero on the edge. How does ...
1
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0answers
59 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
1
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0answers
68 views

Calculating the limit with integrals of Bessel function

I have $\alpha,\alpha_{0}$ complex numbers. Now I would like to calculate the following: $$\lim\limits_{\epsilon\rightarrow ...
1
vote
0answers
60 views

How to find the axis of rotation needed to rotate a $ 3d$ vector to another $3d$ vector?

I have two vectors $(a,b,c)$ and $(d,e,f)$. How can I find the axis of rotation needed to rotate the first vector to be parallel to the other vector? Thanks
1
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0answers
23 views

joint density random variables with a set of equations

There are $n$ equations: $f_i(x_1,x_2,...,x_n,e_i)=0$, $i$ from $1$ to $n$, where $e_i$ are independent random variables whose expectations are all $0$. $x_i$ are random variables. Suppose the map $e ...
1
vote
0answers
59 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
1
vote
0answers
40 views

Non-Intersecting up-right lattice paths and standard Young Tableaux

Consider the Lattice $\mathbb{Z}^2$ and an initial set of points with coordinates $(0,u_1)$, $(0,u_2)$, $\cdots$ $(0,u_n)$, final set of points $(m,v_1),(m,v_2),\cdots,(m,v_n)$, where $v_i,u_i$ are ...
1
vote
0answers
84 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
1
vote
0answers
34 views

Generators of Special Linear Matrix ???

This is a simple question, anyone can help: Can one generate this matrix $A_1$ or $A_2$ or $A_3$ from two matrices $B$, $C$ and their inverse ($B^{-1}$, $C^{-1}$): $$ A_1=\begin{pmatrix} 0& ...
1
vote
0answers
90 views

Cramer's rule and understanding Area/Volume

I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a ...
1
vote
0answers
27 views

About analytic functions with nonnegative real part in the right-hand plane

I am reading some lecture notes where the following claim is made. Suppose $h(s)$ is analytic in the region $\{{\rm Re}(z) > 0 \}$, $h(\omega) \in \mathbb{R}$ for all $\omega \in \mathbb{R}$, and ...
1
vote
0answers
51 views

Finding a particular integral basis of the cyclotomic field

Let $\zeta_{39}$ be a primitive $39$th root of unity. How can I prove that all the conjugates of $\zeta_{39}$ form an integral basis of $\mathbb{Q}(\zeta_{39})$? This is from the paper "Cyclotomic ...
1
vote
0answers
70 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
1
vote
0answers
71 views

Chu-Vandermonde-like combinatorial identity

I am looking for a simple combinatorial proof of the binomial identity: $$\sum_{j=0}^n \binom{2j}{j}\binom{2n-2j}{n-j} = 4^n.\tag{1}$$ The standard way I know is to exploit the generating function: ...
1
vote
0answers
76 views

Transition probabilities with drift-diffusion SDE with compound poisson process

I am looking to find a transition kernel in the following control scenario: Let $X_t = \mu \int\limits_0^t X_t \,dt + \sigma \int\limits_0^t X_t \,dB_t + Y_t$, where $X_t$ is some state process ...
1
vote
0answers
25 views

Name of decision method in which probability of taking an action is exactly past successes / past attempts, while alternative actions normalize

The probability of choosing among options $X_1$, $X_2$, $X_3$, $...X_n$ is initially uniform, i.e. $P(X_j)=1/n$. On choosing $X_j$, either success or failure will occur (with unknown probabilities, ...
1
vote
0answers
34 views

Hatcheck Experiment $n=2k$

I'm sure the famous hatcheck experiment is well known. However I stumbled upon an interesting question. Consider the experiment with $n=2k$ hats, in other words an even number of hats. Find the ...
1
vote
0answers
35 views

basis theorem in holomorphic tangent space

I know that if $(x^1, \cdots, x^n)$ is a local coordinate system in a manifold $M$ then $\{\frac{\partial}{\partial x^1},\cdots, \frac{\partial}{\partial x^n}\}$ forms a basis for the tangent space ...
1
vote
0answers
282 views

Is it possible to convert calculate the percentile and t-scores from mean and standard deviation?

I need to produce a table that will allow you to look up various types of scores for a test using the number of correct answers like this: ...
1
vote
0answers
33 views

Quantitative version of Lévy's continuity theorem

Lévy's continuity theorem implies that if the sequence of characteristic functions $(\varphi_n)_n$ of a sequence of random variables $(X_n)_n$ converges pointwise to the characteristic function ...
1
vote
0answers
114 views

Double integral - failure of Fubini's theorem

I want to calculate the integral $$I = \int_{-\infty}^{+\infty} dy \int_1^{\infty}dx x^2 \frac{x^4+10x^2 y^2 -15 y^4}{(x^2 + y^2)^4}.$$ If I perform the $x$ integration first, I obtain $$I = ...
1
vote
0answers
37 views

Commutative monoidal category free over singleton? Useful in proving coherence?

The proof of coherence in monoidal categories in CWM is based on the existence of a monoidal category free over a singleton. Denoting this category by ...
1
vote
0answers
39 views

uniqueness of antiderivative up to constants

Statement: Let $[a.b]$ be a compact interval of positive length, and let $f,g:[a,b]\rightarrow \mathbb{R}$ be differentiable functions. Show that $f'(x) = g'(x)$ for every $x\in [a,b]$ if and only if ...
1
vote
0answers
244 views

Shadow angle calculation for solar tracking application

Shadow Length Dear all, *I am looking for relationship Between Lmin and solar radiation angle.I know Here in above link they provided relation. But i don't know how to calculate it. x- modules ...
1
vote
0answers
40 views

is following model stationary?

I am interested if following model is stationary,model is represented by following formula $$ x(n) = \sum_{p=1}^{P} a_p \cos(2\pi f_pn + \phi_p) + \epsilon(n) $$ $n$ is changing from $1$ to $N$, I ...
1
vote
0answers
61 views

what is the example of $x_n$

$x_n\gt0,$ $x_n^\frac1n $ converges, but $\frac{x_{n+1}}{x_n}$ diverges. what is the example of this? and how to prove '$x_n\gt0,$ $\frac{x_{n+1}}{x_n}$converges$\to x_n^\frac1n $ ...
1
vote
0answers
459 views

Using SVD in PCA for image compression

I found some help material and guided by it tried to implement PCA using SVD in MAtlab for image compression. I did it in this way: ...
1
vote
0answers
35 views

Why is the restricted nullcone a variety?

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $(\mathfrak{g},[\cdot,\cdot],(\cdot)^{[p]})$ be a finite-dimensional restricted Lie algebra. Define the restricted ...
1
vote
0answers
44 views

How does one change the top number in a summation?

Sorry I do not know the correct term (I am guessing "upper limit"). Here is what I mean. $$\sum\limits_{i=1}^{\color{red}{17}}\frac{2i}{i+3}$$ The $17$ is what I am talking about as "the top number". ...
1
vote
0answers
37 views

Simple question about sup of a set

LEt $X \subseteq \mathbb{R}^n$ be bounded, and let $f: X \to \mathbb{R}$ be a function. Are we allow to perform the following operation? $$ \sup_{x \in X} f(x) + \sup_{y \in X} f(x) = \sup_{x \in X} ...
1
vote
0answers
64 views

Game theory Zero Sum Game Proof

Zerosum games. A coailitional game with transferable payoff is zerosum if $v(S) + v(N - S) = v(N)$ for every coalition $S$; it is additive if $v(S) + v(T) = v(S \cup T)$ for all disjoint $S$ and $T$. ...
1
vote
0answers
51 views

What is the probability that the first of the two chosen numbers from 1 to 10 is a prime and the second is greater than 3?

A person may choose any two numbers from 1 to 10 (repetitions are not allowed). What is the probability that the first number is a prime number and the second number is greater than 3? (assume that 1 ...
1
vote
0answers
48 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
1
vote
0answers
48 views

Finding an orthogonal basis outside the intersection of two subspaces.

Let $A_1 \in \mathbb{C}^{10 \times 200}$, $A_2 \in \mathbb{C}^{10\times 200}$ and let $G_1 \in \mathbb{C}^{200\times 190}$ represent an orthogonal basis for $N(A_1)$, $G_2 \in \mathbb{C}^{200\times ...

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