Questions tagged [integral-kernel]
The integral-kernel tag has no usage guidance.
91
questions
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Is the heat kernel of a manifold $p$-integrable?
If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
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0
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68
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the design of kernel function and integral transform
I read a solution of an integral inequality.
The solution uses condition $$f(1)=f(0)=f'(0)=0$$ to derive that
$$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\...
1
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1
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119
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Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
1
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0
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59
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
0
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0
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46
views
Relation between Kernel density estimation and Reproducing kernel Hilbert space?
The procedure of kernel density estimation using a kernel $K$ is very similar to the construction of an RKHS from the kernel $K(x,y) = K(x-y).$ However, this viewpoint is not mentioned every place I ...
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49
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Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?
[Question originally posted here but maybe it is more suitable for this site.]
The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar ...
10
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572
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Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
3
votes
1
answer
268
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How to find the inverse of a product of two integral equations
Problem
I am trying to invert an equation of the form:
$R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$
where $0\leq l_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
2
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1
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242
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Rate of convergence of Fejer kernel to the Dirac delta function
This seems like something one might find in a book so I would be grateful for any references you think may be helpful.
I am interested in the rate at which of a function integrated against the $N$th ...
2
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1
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144
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Numerical methods for integral eigenvalue equation
I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is ...
1
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1
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158
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On an integral equation
Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...
1
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0
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123
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Heat kernel coefficients for Laplacian in instanton background
The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...
1
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0
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54
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$L^2$ norm of a kernel with a variable width
Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
2
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0
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43
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When can convolutional integral operators be sampled
Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...
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0
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77
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On an integral equation of Volterra type
Consider the following integral equation
$$f(t) = A(t) + \int_0^t K(s,t)f'(s)ds,\quad \forall t\ge 0,\label{1}\tag{$\ast$}$$
where $A: \mathbb R_+\to [0,1]$ and $K: \{(s,t): 0\le s\le t\}\to[0,1]$ are ...
2
votes
1
answer
424
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Structure of the inverse of a Fredholm integral operator of the second kind
NOTE: Cross-posted on Mathematics Stack Exchange
I am trying to solve an equation of the form
$$ (\mathbb{I} + K)\phi = f $$
where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\...
1
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0
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79
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Is there any way this property of semigroups can be satisfied?
Suppose you have the heat semigroup $(S(t))_{t>0}$, such that
$$S(t)u(x) = (4\pi t)^{-n/2}\int_{\mathbb{R}^n}e^{-|x-y|^2/4t}u(y)dy.$$
The semigroup has the property that
$$S(t)S(s)u(x) = S(t+s)u(x)....
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53
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Properties of a kernel convolution $K'(x,y) = \int_X\int_X K_0(x,a)K(a,b)K_0(b,y)d\mu(a)d\mu(b)$ where $K$ and $K_0$ are kernels on $(X,\mu)$
Let $(X,\mu)$ be a probability measure space and $K:X \times X \to \mathbb R$ be a (psd) kernel on $X$. Let $K_0$ be another kernel on $X$ and defined a new kernel $\widetilde K$ on $X$ by
$$
\...
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0
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44
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Knowledge on weighted integral operators?
There are tons of books and a huge literature on the properties of the following integral operator:
\begin{equation}
T(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)dx,
\end{equation}
where $K(x,z)$ is, say, ...
3
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0
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160
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Radial-energy decomposition of a velocity field in 2D: is anyone able to show that the lemma below is true (…or false)?
In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma.
Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity
$$\omega=\...
1
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0
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93
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Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls
Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
3
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0
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240
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Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
5
votes
0
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352
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Nonlinear variation of constants formula
Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
3
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163
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Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions
Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
3
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308
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Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?
When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
4
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463
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Calculation of an inverse Mellin transform
Let $z \in C$ and consider the following integral equation:
$$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$...
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166
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Unique solution for 2$\times$2 Fredholm integral equations system
Consider the following system of Fredholm integral equations with constant kernel matrix
$$
f(x)=K(x)\int_{0}^{1}f(s)ds
$$
where $K(x)\in C([0,1];M_{2\times 2}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\...
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275
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Solution set of integral equation/ Kernel of linear operator
I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int_0^1 \int_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in ...
2
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263
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Existence of integral kernel
I know the following statement ture.
Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$.
Then, $T$ has the integral kernel $...
1
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0
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67
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Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix
Let $\phi:[-1,1] \to \mathbb R$ be a function such that
$\phi$ is $\mathcal C^\infty$ on $(-1,1)$.
$\phi$ is continuous at $\pm 1$.
For concreteness, and if it helps, In my specific problem I have $\...
2
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1
answer
621
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General strategy for studying the decay of eigenvalues of kernel integral operators
Disclaimer. Please, be patient, I'm here to learn functional analysis...
Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite ...
4
votes
1
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266
views
Inverting convolutions over finite intervals
There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral ...
1
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0
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63
views
Approximate identities on the unit disk and going beyond a power series' radius of convergence
Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...
3
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1
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295
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Gradient condition implies Hörmander condition
We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition
$$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
1
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0
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102
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Is this a positive definite kernel?
Under which conditions on the function :
\begin{array}{l|rcl}
K : & \mathbb R^+ & \longrightarrow & (0, 1)\\
&t & \longmapsto & K(t) \end{array}
is the symmetric ...
1
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0
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88
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How does the constant bound on Hormander's $L^2$ estimate for non-degenerate phase depend on the cut-off function?
A classical estimate, due to Hormander, assets that the integral operator
$$Tu = \int_{\mathbb{R}^{d}} e^{i\varphi(x,y)/h} a(x,y) u(x) dx, \ \ a \in \mathcal{C}_{c}(\mathbb{R}^{2d}), \ \ \varphi \in \...
1
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1
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616
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Injectivity of an integral operator
Consider the operator
$$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$
with $k\in L^2((0,1)\times(0,1)).$
I want to know under what assumption the kernel is reduced to zero. i....
5
votes
1
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609
views
Boundedness of integral operators on spaces of continuous functions
Consider a standard integral operator $T$ formally defined by
$$
Tf(x):=\int_{K} k(x,y)f(y)dy,\qquad x\in K,
$$
where $K$ is a locally compact metric measure space. It is immediate to see that the ...
3
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0
answers
108
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Generalizing the heat kernel approach
I notice a way of solving equations that goes roughly like this:
Want to solve $Tu=v$, i.e. hoping to find "$T^{-1}(v)$".
$T^{-1}$ might not exist "on the nose", so instead we consider $\int_0^\infty ...
1
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0
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148
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Existence of continuous integral kernel
Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$).
A ...
3
votes
1
answer
549
views
Eigenfunctions and eigenvalues of an operator defined by a certain integral
Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by
$ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
1
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0
answers
67
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Angle between Fleming-Viot type 3-particle system
Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
2
votes
0
answers
263
views
Convolution and approximate heat kernel
I have the following question: Let $A, B\in C^\infty((0,\infty)\times M\times M)$, where $M$ is a compact manifold. Define
\begin{align}
(A*B)(t,x,y) = \int_0^t\int_M A(t-s,x,z)B(s,z,y)dzds
\end{align}...
2
votes
0
answers
112
views
Solving Fredholm integral equation in Lp
I have a very simple integral equation
$$
f(x) - \lambda \int_a^be^{x-y}f(y)dy=1
$$
which is Fredholm of the 2nd kind, with separable kernel. It is needed to find the values of $\lambda\in\mathbb R$, ...
4
votes
1
answer
501
views
Trace-class properties of integral operator
Let $k$ be a smooth, compactly supported function defined on $\mathbb{R}^{2}$ and let $Op(k)$ denote the integral operator on $L^{2}(\mathbb{R})$ defined as $$Op(k)f(\cdot)=\int_{\mathbb{R}}k(y,\cdot)...
0
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0
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69
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Looking for example of integral transformations that preserve number of zeros
Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros.
I am looking for non-trivial examples of integral transformation
\begin{align}
g(x)= \int f(t) h(t,x) dt
\end{align}
such that $f$ and $g$...
2
votes
0
answers
57
views
Integral equation with kernel defined in a rectangle
Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...
2
votes
0
answers
44
views
A special integral equation of Volterra type
Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation:
$$
f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0
$$
My question is : under what condition ...
4
votes
1
answer
281
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integral kernel function for the SU(N) group
It is well know that the Haar probability measure for the $U(N)$ group, given by
$$
\begin{align}
dX_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 ...
3
votes
1
answer
301
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Which utility functions are linearly transformed by normal perturbations?
I'll ask this question as pure economics, as pure math, and showing the translation.
Economics (micro):
Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...