0
votes
0answers
5 views

Help with method of characteristics question.

I'm trying to solve the partial differential equation, $$ \frac{\partial F}{\partial t} = (z-1)\left(kF - d \frac{\partial F}{\partial z}\right) $$ where $k$ and $d$ are constants greater then $0$, ...
0
votes
0answers
4 views

Division algorithm variations

Derive this version of the division algorithm . For integers a, b with b ≠ 0 there exist unique integers q and r that satisfy a = qb + r, where -1/2|b| < r ≤ 1/2 b. I started off with letting a ...
1
vote
0answers
8 views

Arguing a stationary distribution exists

I am trying to show that there exists a stationary distribution when $q>p$ for the Markov process with one-step transition matrix $$ \begin{bmatrix} q & p & 0 & 0 & ...
0
votes
0answers
8 views

Transposing matrix when differentiating it

Hi so I am trying to understand the solution of linear regression with matrices (found at the following link) and an confused about how on page 10 he says the derivative of 2Y'XB with respect to B is ...
-1
votes
0answers
7 views

Identity Substitution in Polyadic Singular Sentences

Let the sentence "Yash loves Priya" be symbolized as Lyp. Let the sentence "Priya is Dr.Lingnurkar" be symbolized as p=l. The identity substitution rule is a=b,ϕa,/∴ϕb. In my textbook, ...
0
votes
0answers
11 views

transform integral to differential equations

I found a similar system of integral equations in a paper. It says that it can be solved by differentiating and then using standard techniques. My question is, how can I differentiate such a system in ...
-3
votes
0answers
11 views

Chord and tangent properties

In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle BCD. Prove that diagonal BD is parallel to the tangent to the point circle at point A.
2
votes
1answer
27 views

What can we conclude about the function $f$?

Let $f$ be a scalar field, $f:\mathbb R^n\to\mathbb R$. Suppose there is an $n$-ball $B(a;r)$ centered at $a$ with radius $r$ and a fixed vector $y\in\mathbb R^n$ such that $f'(x;y)=0$ for every ...
-4
votes
1answer
42 views

Why is this statement true?

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be the function $f(x,y)=(y-x^2)(y-2x^2).$ Why is this statement true: $t\mapsto f(t\xi)$ has in $t=0$ a local minimum for every $\xi\in\mathbb{R}^n$
1
vote
1answer
16 views

Intersection of two lines in complex numbers given four points

How to find the point of intersection of two lines, given four points, two of which are on each line, in complex numbers? Thank you!
0
votes
2answers
57 views

Set Theory (Definition of a set)

A set is defined as the collection of well-defined and distinct objects. This implies that a member of a set can't be repetitive in the set. Now when we discuss groups in Group Theory, if we check the ...
0
votes
1answer
23 views

Finding an equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$

How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface. While reading the solution of ...
0
votes
0answers
10 views

Heat equation with fourier transformation

I want to understand a solution from an exercise where we should find a solution of the heat equation: $$\frac{\partial u(x,t)}{\partial t}=\sum_{j=1}^{n}\frac{\partial^2 u(x,t)}{\partial x_j^2} $$ ...
0
votes
0answers
19 views

Compactness of operators

Let $X$ be a separable Hilbert space. Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, which is compact. Let $B$ be an operator in $X$, boundedly invertible, that is ...
-4
votes
1answer
19 views

No of ways of selecting r objects from n distinct objects, allowing repeated selections

I'm self studying discrete math from a books which states the formula for No of ways of selecting r objects from n distinct objects, allowing repeated selections as $C(n+r-1, r)$. I couldn't ...
0
votes
0answers
6 views

Generating bernoulli variables for different lambda's

In order to generate $M$ paths of length $N$ I have to generate Bernoulli variables. In Matlab I used: Q=binornd(1,lambda,L,N); Now I want to generate this for a sequence of values lf lambda, but I ...
0
votes
2answers
18 views

Question on how a matrix is calculated from an example

I have the following laplacian matrix given to me in a textbook. In the textbook, the matrix calculation is always done from the 3 x 3 matrix (the methods I learnt makes me cut the matrix further ...
0
votes
1answer
24 views

Proving a relation between inradius ,circumradius and exradii in a triangle

Prove that in a triangle $$r^2+r_1^2+r_2^2+r_3^2=16R^2-(a^2+b^2+c^2)$$ where the symbols have their usual meanings. I am looking for a smaller or elegant proof using trigonometry. A geometric proof ...
1
vote
0answers
19 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
4
votes
1answer
39 views

How to find the shift that minimizes the difference between two vectors?

I am looking for a efficient way to find the value of k that minimizes $\sum(s_t - b_{t+k})^2$ where $s$ and $b$ are N-dimensional vectors and the values are wrapped around like this: $b_{t+k} := ...
4
votes
1answer
10 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
0
votes
1answer
47 views

confusion about square root

I understand by convention ,if $\sqrt{x^2}=a$ , a is defined to be the positive root of x or the principal square root. but what does this mean for exponential equations- does $x^{0.5}=-5$ have no ...
-1
votes
1answer
16 views

Create formula to get a value

I have clients whichs pay a monthly fee. I would like to know the accumulative total for a given quantity of month for a given fee. Plus I get 2 new clients every month. Let's say a client pays ...
0
votes
0answers
33 views

A question of odds

Consider an experiment with four possible outcomes, and suppose that the quoted odds for the first three of these outcomes are as follows. What must be the odds against outcome 4 if ...
0
votes
0answers
6 views

multi-objective reduction of a given set

I have a set of arguments $v_k$. Each argument has a set of two different numeric values $x_{ak} \in [0,\infty]$ and $x_{bk} \in [0,\infty]$ associated to it. The set $V$ contains all $v_k$s. I’m now ...
1
vote
1answer
32 views

$T:X \to Y$ bounded linear map and $X$ separable implies $Y$ is separable?

Let $T:X \to Y$ be a bounded linear map between Banach spaces. Suppose that $X$ is separable. Is it true that $Y$ has to be separable? I think yes, since the map is continuous it takes the ...
0
votes
1answer
23 views

Each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$ when $f$ is additive

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an additive function such that $f(1)=0$. Then for each non-empty interval $I \subset \mathbb{R}$ we have $f(I)=f(\mathbb{R})$. Since $f$ is additive ...
3
votes
1answer
15 views

Properties of trace-class operators

Let $X$ be a separable Hilbert space (real or complex). Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, and suppose $B\in\mathcal{L}\left(X\right)$, which is of trace-class. ...
1
vote
0answers
4 views

The uniqueness of solution of an equation that involves CDFs

I have two monotone CDFs $F(x)$ and $G(x)$. The functions are symmetric in a sense that $F(x)=1-G(1-x)$, $f(x)=g(1-x)$. I am trying to show that equation $xF(2x)+(1-x)G(2x)=1/n$, $n\geq2$ has a unique ...
1
vote
0answers
7 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
0
votes
1answer
25 views

NOT bounded functions that satisfy a condition.

I am looking for not bounded functions that satisfy a condition. Let $dx$ be a Lebsegue measure on $\mathbb{R}$. Define \begin{align} \mu(A):=\int_{A}\frac{1}{\sqrt ...
1
vote
0answers
13 views

Fourier Cosine Expansion of Piecewise Continuous Function

Hi I am trying to represent this following function: $$f(x)=\begin{cases} 35.6236 + 0.161087e^{59.9842x},0\leq x < 0.1 \\ 35.6236 + 0.161087e^{59.9842 (-x + 0.2)},0.1\leq x \leq 0.2 \\ ...
1
vote
0answers
14 views

Finite flat pushforward of a constant sheaf

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also ...
2
votes
1answer
14 views

Multicategories with out-arities

Basically, my question is: Why the emphasis on domains in the notion of multicategory? I will now give the formal framework to state it correctly. Passing from categories to multicategories ...
0
votes
2answers
30 views

Calculating a characteristic function two different ways gives contradictory results. Why?

I am trying to calculate a characteristic function directly and via the conditional distributions. I get contradictory results: Let $X$ and $Y$ be random variables defined on the same probability ...
0
votes
2answers
48 views

Solving this inequality with integral

We have function $f:\mathbb{R}-\{2 \}\to\mathbb{R}$ $$f(x)=\frac{x^2}{x-2}$$ Show that $8\le\int\limits _3^4f\left(x\right)dx\le9$ I solved the definite integral and got $\int\limits ...
1
vote
0answers
22 views

Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$.

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...
0
votes
0answers
10 views

A function for non-linear animation steps (large in the middle, small at the ends)

In a word game for Android I animate movement of letter tiles (for example when user selects "shuffle tiles" or "return tiles from game board" in menu) in a linear way (they have constant velocities) ...
0
votes
1answer
20 views

Determine relation of $x$ and $y$ from results

I can't seem to determine the relation between $x$ and $y$ for this problem. All of the previous ones I have done have been doable simply by eye-balling the relation between $x$ and $y$, but here I am ...
2
votes
0answers
27 views

What's the differences between multi variable and vector calculus

This is a conceptual question. If we use vector calculus and multi variable calculus as synonym, will it be completely wrong? If so what topics does multi variable calculus have but vector ...
-7
votes
1answer
58 views

I have written proof of Goldbach's conjecture How do I register it? [on hold]

I have written proof of Goldbach's conjecture How do I register it? Please email the appropriate steps and magazines. Thanks
1
vote
1answer
37 views

Why is this true: $1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$

In my lecture notes, the following is written: $$1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$$ as $\varepsilon \rightarrow 0$ and $n$ some fixed constant (non-negative ...
2
votes
3answers
57 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
0
votes
0answers
10 views

An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
0
votes
0answers
17 views

Series expansion of reciprocal function of Generalized Exponential Integral

Generalized Exponential Integral of order p has a series expansion http://dlmf.nist.gov/8.19.10 Is there a series expansion of the reciprocal function ?
1
vote
0answers
20 views

Example comparing Riemann's and Lebesgue's methods of integration

It is well known that a function which is Riemann integrable is also Lebesgue integrable, and both integrations result in the same value. Question: Can one give an example of a Riemann integrable ...
0
votes
0answers
12 views

Type of critical value

Let $f(x,y,z) = (x-y)^2 + e^{z^2}$. Is it correct that the origin is a critical value of $f$ that is a saddle point? I get for the Hessian matrix $\begin {pmatrix} 2 & & \\ & 2 & \\ ...
1
vote
0answers
25 views

Estimating the “size” of the mathematical research literature

The other day I was telling one of my friends that mathematics, as a living science, possesses quite an extensive research literature. How extensive then, she asked. Unfortunately, I didn't have ...
0
votes
0answers
11 views

FFT procedure for evaluationg a polynomial at $N$ Fourier points

The following is the recursive FFT procedure of Algorithm for evaluationg a polynomial of length $N$ at $N$ Fourier points. Algorithm (FFT - fast Fourier transform). Input arguments. $ \ ...
0
votes
2answers
18 views

Radius of convergence of series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$

Is it correct that the convergence radius of the series $\sum^\infty_{n=0} 3^{-n} (2 \pi)^{-n} (\arctan n)^n x^n$ equals $12$?

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