0
votes
0answers
39 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
1
vote
1answer
53 views

System of ODE - Solution

I have a system of ODE to solve $$ A_{5 \times 5}\ddot{q}(t)_{5 \times 1}+ B_{5 \times 5}\dot{q}(t)_{5 \times 1}+ C_{5 \times 1} =0\tag 1$$ Given Data $A,B,C$ are constants.We know what is ...
0
votes
0answers
12 views

Definite integral - Finding an equivalent form

I have the following definite integral $ \int_{0}^{L} {\psi(t) }_{1 \times 5}{A(s)}_{5 \times 5}(\psi(t) _{1 \times 5})^{T} {B(s)}_{5 \times 5} ds \tag 1 $ Given data All dimensions are ...
1
vote
1answer
36 views

If a real polynomial of degree $n\gt 1$ has a root of modulus exceeding all others, is that one a real root?

Suppose $a_nx^n+\ldots+a_1x+a_0=0\; (a_n\in \mathbb{R})$ has $n$ distinct roots $r_1,r_2,\ldots, r_n$ (no multiple roots), and if $\exists r_k$ s.t. $\forall r_i\in\{r_1,r_2\cdots r_n\}-\{r_k\}$, ...
2
votes
0answers
60 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
2
votes
2answers
84 views

Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
0
votes
0answers
42 views

ODE with multiple simple conditions $f'(x)=f(x)(Ax+D ) $

I have an ODE to solve . The main issue is,in addition to solving it I have to keep some conditions too in the solution of f(x).. I am bit confused regarding how to deal with it. Equation is given ...
1
vote
1answer
37 views

Inner product of functions as integration

I am trying to teach my self some linear algebra in preparation for a module in machine learning. I am using Gilbert Strang's text Introduction to Linear Algebra and am having some difficulties. My ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
2
votes
4answers
91 views

What is the geometric meaning of this integral?

In my math book, there is an exercise where the task is to compute the following integral and to interpret the result geometrically: $$\int_0^\pi\cos mx \cos nx \ dx$$ where $m,n \in \mathbb{N}$. ...
3
votes
1answer
31 views

A difficult integral over matrix exponents - is there an analytical solution?

In short, I'm trying to find some method, apart from numerically integrating, to find the value of \begin{equation} \bar{X}_t = \int_0^t e^{A^T (t-s)} Q e^{As} ds, \end{equation} where $Q$ is ...
2
votes
0answers
73 views

An integral problem related to matrix determinant

I am stuck in an integral problem: ...
2
votes
2answers
186 views

Prove that the Laplace trasform is a Linear trasformation

Could you help me prove that the Laplace Trasform is a Linear trasformation? Thank you.
2
votes
2answers
115 views

Any way to simplify the matrix integral $\int_{0}^{\infty}e^{A z}e^{B z}dz$ if A and B do not commute but are diagonalizable>

Define $A$ and $B$ square matrices where all eigenvalues are $< 0$ for both, and there is no eigenvalue multiplicity. Completely diagonalizable, etc. But assume that $A$ and $B$ do not commute. ...
4
votes
1answer
235 views

Gaussian Matrix Integral

I need your help to solve this exercise : Let $S$ be a symmetric Hermitian matrix $N\times N$ : $S=(s_{ij})$ with $s_{ij}=s_{ji}$. When $\langle s_{ij}s_{kl}\rangle\neq 0$ What $$\int ...
0
votes
3answers
59 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
0
votes
4answers
65 views

Integral Problem $\sin^6 x$. [closed]

What is the integral of $\sin^6 x$? Can some one publish it with the method it would be really helpful. Specially I want the trig identities.
0
votes
1answer
54 views

Polynomial Basis with zero integral

I have a question and please I need help. First all, this is the context: Let $p : T\rightarrow\mathbb{R}$ a polynomial of degree $\leq k$ with null measure condition onto T, that is $$\int_T p\ =\ ...
1
vote
0answers
35 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
2
votes
1answer
84 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
0
votes
1answer
68 views

Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
1
vote
1answer
85 views
+100

Surjectivity of an integration map

N.B.: Thanks to studiosus answer I realised I should ask for more conditions or otherwise the answer is straightforwardly wrong. I rechecked my problem and added new assumptions that I boldface. ...
0
votes
1answer
35 views

First-order linear differential equation

I have this question, and the working out below is as far as I can get: $$ x \frac{dy}{dx} - y = y^2 \\ p(x) = -\frac{1}{x} \\ q(x) = \frac{y^2}{x} \\ u(x)= e^{\int -\frac{1}{x}dx} \rightarrow ...
1
vote
1answer
36 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
2
votes
1answer
62 views

Prove that $DT = I_v$, $TD \neq I_v$, where $D$ = differentiation operator and $T$ is integration

Let $V$ be the linear space of all real polys $p(x)$. Let $D$ denote the differentiation operator, and let $T$ the integration operator that maps each polynomial $p$ onto the polynomial $q$ given by ...
0
votes
1answer
35 views

how can get simple function

how can get $y$ simple function of $x$ from : $$ \frac{2 \sqrt{y} \sqrt{y-b} \log \left(\sqrt{y-b}+\sqrt{y}\right)}{\sqrt{\frac{A y (y-b)}{b}}}+\text{constant} =x$$ where : $$ y=y_0 at x=0$$ and ...
0
votes
1answer
29 views

Appreciate help with solving a probability density function for its constant term

I am using StackOverflow a lot for asking and answering programming related questions, and I hope it is appropriate if I'd ask my question below on here on this sister-site. If not, please let me know ...
0
votes
1answer
139 views

Find the area of $y = x^2$ without calculus

Is it possible to find the area bounded by $x=0, y=1, y=x^2$ without using calculus? I know the answer is $\frac{2}{3}$, easily found with integration.
0
votes
0answers
23 views

Gaussian for Grassmann variables

Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix. I want to generalize the following $$I(A) =\int d\theta_1d\theta_2~ ...
36
votes
1answer
384 views

Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
2
votes
1answer
61 views

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$ where $a\in[0,1]$. The mean-value theorem is $\frac{f(y)-f(x)}{y-x}=\nabla ...
2
votes
3answers
527 views

Volume of $n$ dimensional ellipsoid

Let $c_1,c_2,...,c_n$ be positive constants. Consider the $n$ dimensional ellipsoid given by $\{(x_1,...,x_n)|\sum_{k=1}^n\frac{x_k^2}{c_k^2}<1\}$. Prove that it's $n$ dimensional volume is ...
0
votes
1answer
71 views

Fredholm integral equation of first kind

I want to solve the Fredholm integral equation of first kind: $$ \int_L K(x,y)U(y)dy = f(x) $$ in these equation the function $U(y)$ is the unknown and the so-called kernel $K$ and the right hand side ...
1
vote
1answer
29 views

Showing a set is orthonormal using an integral

Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle ...
1
vote
2answers
73 views

linear algebra foundation of Riemann integrals

Let $V$ be the vector space of real functions $f\colon [a,b]\to \mathbb R$ and let $X$ be the set of characteristic (indicatrix) functions of subintervals: $X=\{\mathbb 1_I\colon I\subset [a,b] $ ...
0
votes
3answers
49 views

Using the result of a derivative, find the integral?

I'm just a little bit unsure of this question: "Show that: $d/dx (\sin x)^3=3\cos x(\sin x)^2$ and hence use this result to find the integral (limits of $pi/2$ and $0$) of $(\sin x)^2\cos x.$" Is it ...
2
votes
2answers
98 views

Evaluation Integral: Method of Partial Fractions - Can the system of equations always be solved?

I've been studying the method of partial fractions for evaluation integrals. So far every example and exercise have been fairly straight forward, but I still have some unanswered questions: 1) The ...
1
vote
0answers
119 views

(When) is this matrix positive definite?

I have a symmetric $n \times n$ matrix (say, $M$) with $[i,j]$ element \begin{equation} M_{[i,j]} = \int_{\mathbb{R}} [p_i(z)-g(z)][p_j(z)-g(z)]~dz, \end{equation} where $p_i(\cdot), p_j(\cdot),$ ...
0
votes
0answers
103 views

Prove that the determinant of the Jacobian transformation from the interval [-1,1] to any line segment is given by:

This is just for fun...it’s part of a larger numerical application where I’m applying a quadrature rule to a line integral. I’m fairly certain my answer is correct but have no idea how to prove it. ...
2
votes
1answer
39 views

Prove lower bound of integral

I have a continuous function $h:[a,b]\rightarrow\Bbb C$. Let $$M=\sup_{x\in [a,b]}|h(x)|$$ I need to find function $f\in L^2[a,b]$ with: $${||f||}^2=\int_{a}^b|f(x)|^2dx=1$$ such that: $$ ...
0
votes
1answer
64 views

Generalization of integral using special function

Is it possible to find a closed formula for the following integrals for any $k\in \mathbb{N}^{*}$: $$ I=\int_{0}^{1} \left[\ln\left(\frac{p}{1-p}\right)\right]^{k}p^{2}dp ,\quad J=\int_{0}^{1} ...
-3
votes
1answer
44 views

Computation of integral [closed]

I want to compute this integral: \begin{equation*} J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp \end{equation*} It will be great if you can detail the proof. I tryed to do change of variable it does not ...
3
votes
0answers
114 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
1
vote
2answers
67 views

How is the norm of a partition related to the norm of a vector?

Just finished a course in linear algebra, where the norm of a vector essentially was described as the length of the vector. In calculus, we just started talking about the definite integral of a ...
0
votes
1answer
139 views

Integral operator

The space $C$ of continuous functions $f(u)$ on the interval $[0, 1]$ is one of many infinite-dimensional analogues of $\mathbb{R}^n$ , and continuous functions $A(u, v)$ on the square $0 \leq u, v ...
2
votes
2answers
94 views

Find the fallacy in using the Cauchy–Schwarz inequality

Let $\int_{a}^{b}\frac{f(x)}{x}dx=k$, wherein $f(x),a,b,k$ are positive. According to the Cauchy–Schwarz inequality: $\int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ...
1
vote
0answers
39 views

near- rank - deficiency property

I am trying to understand low rank approximations and when I was reading up on it, it is stated that one of the methods of getting the low rank approximation is the near-rank-deficiency property holds ...
0
votes
1answer
120 views

How to simulate IMU data using position and orientation?

I want to make a simulator to verify that my imu algorithm is working. I am given: $p_0$ - starting position $p_1$ - final position $q_0$ - starting orientation $q_1$ - final orientation I want ...
2
votes
1answer
85 views

Self-adjoint operator in the space of twice continuously differentiable functions

There is a problem in the textbook with which I am having difficulties. Prove that operator $A$: $Ay=xy''+y'$ defined on space of twice continuously differentiable functions (scalar product is ...
0
votes
1answer
101 views

Need to expand $\nabla$ and $\Delta$ included term

In this equation, they used $\nabla$ and $\Delta$, I need to expand them to understand this equation. More to see in this article http://arxiv.org/abs/0802.3525 in equation (15) \begin{eqnarray} ...