1
vote
0answers
62 views

We have $ f(x) = \sum_{n \geq 1} \frac{(x-1)^n}{n}$ prove that $f(x) = -\ln(2-x)$.

I am having problems with the following exercise, I have solved the first two parts of the exercise but I am unsure about the last part. I have the following power series $$f(x) = \sum_{n \geq 1} ...
1
vote
0answers
38 views

What is this quantity?

I wonder whether there is a closed form for $$-\sum_{k=1}^{\infty}\frac{\Delta^{k}\pi(x)}{k!}(-x)_k$$ where $\pi(x)$ is the prime-counting function and $(x)_k$ is the falling factorial. In other ...
1
vote
0answers
49 views

Integrate the Dirichlet distribution over finite intervals of each random

Is there a close form solution for the integration of the dirichlet distribution over finite intervals of each random variable set up over the simplex. For example $$\int_{a}^{b}\int_{c}^{d} ...
1
vote
0answers
41 views

Ellipticity of an operator in Gunther's proof of the isometric embedding

In Deane Yang's notes about Gunther's proof of the celebrated isometric embedding theorem, at the end it is stated that $v$ inherits the regularity of $h$ because the operator $I-Q_0(v,\cdot)$ is ...
1
vote
0answers
41 views

Solving linear constrained optimization problem

So I have the following constrained optimization problem to optimize a circuit (electrical engineering) that I am working on: Minimize the following expression (power dissipation): $$I_{B1}(V - C_1) ...
1
vote
0answers
41 views

Consider a first order quasi linear equation under various Cauchy Data

Consider the first order Quasi linear PDE given by $$uu_x+u_y=1$$ As usual $$J=\begin{vmatrix} f'(s_o) & a(f(s_0),g(s_0),h(s_0))\\ g'(s_o) & b(f(s_0),g(s_0),h(s_0)) \end{vmatrix}$$ The ...
1
vote
0answers
49 views

How to prove these properties of step function?

My book(Differential Calculus by Amit M. Agarwal) lists these properties of step-function: $$[x] = \left[\frac{x+1}{2}\right] + \left[\frac{x}{2}\right] \\\\\\\ \left[\frac{n+1}{2}\right] ...
1
vote
0answers
82 views

differentiating a smooth function defined by an integral

Suppose we define a function by the integral $$ f(x) = \int_{-\infty}^{\infty}g(x,y) dy $$Suppose we know that $f(x)$ is smooth. Does this mean that necessarily $$ ...
1
vote
0answers
39 views

Is it right definition of divergence at a point?

In my text Let D be the domain of function $f$ and $a$ be a cluster point of $D$. '$f$ goes to infinity at a point $a$' is $\forall Y, \exists \delta >0, \forall x \in D : ...
1
vote
0answers
51 views

Euler discretization for a trending Ornstein-Uhlenbeck process Matlab Code

I'm working on a matlab code for a trending OU process. I could easily find lot of material and reference for a stationary OU process and how to discretize it using a euler scheme like: ...
1
vote
0answers
16 views

How to estimate the occurrence time of an event for scheduling purposes precisely?

I working on a system in which a number of requests arrive in random time. Consider the following timeline: In the above example, we have a certain number of components: ...
1
vote
0answers
57 views

Complex analysis cycle

I am reading "Complex Variables" written by R. B. Ash & W. P. Novinger, and in the 3rd chapter I've got stuck. I have questions concerning the following definition. Let $\gamma_1, \gamma_2, ...
1
vote
0answers
116 views

Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map ...
1
vote
0answers
43 views

If the Galois Group of a Field Extension is Abelian

Let $f(x)$ be irreducible over a field $k$ characteristic zero, with splitting field $E$, and $\alpha$ and $\beta$ be the roots of $f$ in $E$. If the galois group of $E/k$ is abelian prove that ...
1
vote
0answers
54 views

Monte Carlo Integration via Ray Casting

Let's suppose we have a 2D line segment $S$ at $y=0$ and extending from $x=-h$ to $x=+h$. We define a function $f(x)=1$ for every point $x$ of $S$. We wish to integrate $f$ over $S$ with Monte Carlo ...
1
vote
0answers
43 views

Standard properties of trigonemetric functions

You have $\sin(\frac{\pi}{6})= \frac{1}{2}$ and $\sin(\frac{5\pi}{6})= \frac{1}{2}$ and the interval [$\frac{\pi}{6},\frac{5\pi}{6}$] has length $\geq 1$ This is used as I'm sure most will be familiar ...
1
vote
0answers
59 views

Asymptotic form of Whittaker function

I am working with Whittaker functions for a project and have no experience with asymptotic analysis - how is the following expression, for $\kappa \rightarrow \infty$ through the real numbers ...
1
vote
0answers
37 views

Leray sheaf being constant - what does it mean in terms of singular cohomology?

Let me first admit that I know next to nothing about sheaf cohomology, but I might have encountered a good reason to learn it. Suppose that I have a fibration $F \to E \to B$ and I know that its ...
1
vote
0answers
39 views

How to show that a seperatrix exists for the Fisher-KPP equation

We have the Fisher-KPP equation: $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + ru(1-u)$ We can reduce this to a second order ODE: $cu_{\xi} = u_{\xi\xi}+u(1-u)$ where $\xi = ...
1
vote
0answers
33 views

subsheaf of free sheaves

Let $X$ be an irreducible nodal curve, $E:=\oplus_{i=1}^r \mathcal{O}_X$ be a free sheaf on $X$ and $F \subset E$ a (coherent) subsheaf. Is it possible to write $F$ as a direct sum of subsheaves ...
1
vote
0answers
61 views

Pre-College Algebra Book

I am looking for a high school/ pre-college level Algebra book that is self contained for self-study. Nothing special, I don't want a book about number theory, but a book in preparation of high school ...
1
vote
0answers
23 views

Can we have extension of Mercer theorem to interpolation?

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
1
vote
0answers
46 views

Is it known whether or not the 'Hauptvermutung' is true for finite simplicial complexes in $\mathbb{R}^4$?

If I have two finite simplicial 4-complexes embedded linearly in $\mathbb{R}^4$ (as in all the lines and faces are straight and flat and there are only a finite number of 4-simplices) do they have a ...
1
vote
0answers
18 views

Change of Variable for Time Invariance Check

I've been studying for a signals and systems class coming this fall and can't figure out how the following change of variable is being applied according to standard definition: $$T[x(t-\sigma)] = ...
1
vote
0answers
53 views

If there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$, can we prove that $f\in L$?

I know, if $f\in L$ (the set of all Lebesgue integrable functions), then there exists a sequence $(\phi_n)$ of step functions such that $\phi_n \longrightarrow f$ almost everywhere on $[a,b]$ and ...
1
vote
0answers
100 views

Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
1
vote
0answers
20 views

Question about the sign of a certain sum

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to ...
1
vote
0answers
52 views

Riemann-Roch for nodal curves

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ ...
1
vote
0answers
33 views

Looking for a Formula for ROI, couldn't get an answer in Finance

This is honestly a pretty simple problem, but for whatever reason I am not able to pull it all together. I was talking theoretically with a friend and neither of us can nail down the maths so I coming ...
1
vote
0answers
56 views

Solving quadratic congruences

System of equation is : $$ x^2 \equiv 2 \mod 3 $$ $$ x^2 \equiv 4 \mod 5 $$ So, if first equation doesn't have solution what should I do with it?
1
vote
0answers
99 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
1
vote
0answers
30 views

Lebesgue characterization of Baire class 1 functions

Lebesgue's characterization of Baire class one functions on $\mathbb R$ is the following: $f:\mathbb R \rightarrow \mathbb R$ is Baire class one iff for all $r\in \mathbb R$ $\{ x: f > r \}$ and ...
1
vote
0answers
27 views

Show $Y(t)=X^{(1)}(t)-X^{(2)}(t)$ and $\lim_{t\to\infty} \mathbb{E}Y^2(t)=0$ , for $dX^{(i)}=\mu X^{(i)}dt+\sigma X^{(i)}dW$

I am trying to solve this exercise which my professor has "solved" (he says what the result but not how he gets it). This is in a problem sheet which is about the Euler-Maruyama scheme. What I get ...
1
vote
0answers
59 views

A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
1
vote
0answers
5k views

trivial and non-trivial solution of homogeneous equation

Suppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 ...
1
vote
0answers
27 views

How to affirm equivalent event?

From the textbook "Probability, Statistics and Random Processes for Electrical Engineering "3rd edition p. 154 A binary transmission system sends a '$0$' bit by transmitting a $-v$ voltage ...
1
vote
0answers
47 views

sum of 3 sine waves, with uniformly distributed frequency and phase

To my surprise, the sum of three sine waves of similar frequencies and random phase doesn't necessarily has its maximum at 3 which I simply assumed to be correct.* This question arise from a ...
1
vote
0answers
18 views

Optimal set of generators of conformal group in 2D

Can we write Lorentz transformations and dilations in terms of translations and special conformal transformations? In V. Kac's book "Vertex algebra for beginners" 2nd edition, on p.7, Kac writes that ...
1
vote
0answers
38 views

Combinations within a Combination

Someone did it for me $20$ Years ago. It was using $12$ Numbers in this example $1$ to $12$ and there were $42$ sets of $6$ numbers. In that sample each of the $12$ numbers ONLY appears $21$ times, ...
1
vote
0answers
26 views

Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star ...
1
vote
0answers
21 views

Minimum vertex cover of vertex disjoint odd holes and antiholes

I am interested in knowing whether the minimum vertex cover of a graph that can be written as the union of vertex-disjoint odd holes and odd antiholes can be found exactly, in polynomial time. I could ...
1
vote
0answers
52 views

Algebraic Multiplicity and Geometric Multiplicity

Problem Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and suppose the linear transformation $T:V\rightarrow V$ has eigenvalue $\lambda_0$ with algebraic multiplicity $n_0=3$. (a) ...
1
vote
0answers
54 views

What is the probability to pass through $1\le m\le n$ vertices of an $n$-sided polygon after $t$ seconds?

Suppose a flea is on a vertex of an $n$-sided polygon. It stays still for exactly one second, and then jumps instantly to an adiacent vertex. Let us assume it has no memory of its previous jumps and ...
1
vote
0answers
42 views

How can I blow-up a smooth projective surface with certain conditions?

Let $X$ be a smooth projective surface and $K$ a canonical divisor on $X$. Suppose $V$ and $W$ are subspaces of $H^0(X, \mathcal{O}_X(nK))$ (for $n$ large). Q: How can we blow-up $X$ to obtain $\pi: ...
1
vote
0answers
57 views

Why does the uniform probability function have the y value it does? (1/b-a)

It seems very clear in the discrete case. Rolling a die for example, you have a discrete valued function $f(x)= \frac{1}{6}$, $x \in \{1,2,3,4,5,6\}$. The sum of all the values is 1. In the ...
1
vote
0answers
25 views

Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
1
vote
0answers
83 views

Origins of Kalman filter Algorithms in his paper in 1960

Concerning Kalman's original paper published in 1960, "A New Approach to Linear Filtering and Prediction Problems", it seems the majority is to show the orthogonal projection is the optimal estimation ...
1
vote
0answers
13 views

Indexing interactions between and withing entities

I'm trying to create/find an index to compare/order systems with multiple entities based on the diversity of the interaction between the entities. Assume you have few systems of entities that can ...
1
vote
0answers
36 views

Integrate a Dirichlet Posterior over a finite interval

How do you integrate (analytically or numerically) a Dirichlet Posterior over a finite interval or union of finite intervals of events/rvs $x_i$
1
vote
0answers
40 views

Can I “squeeze” the x-axis when I solve a diff. eq?

I am trying to solve a (rather ugly) differential equation numerically. (If you're curious, the equation is $\frac{3}{2}\left(\frac{\partial_x f(x)}{f(x)}\right)^2+\frac{\partial_x ...

15 30 50 per page