All Questions
1,487 questions with no upvoted or accepted answers
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113
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Maximizing a parametric integral over the unit sphere
I am trying to compute the nonnegative quantity
$$
\underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty
$$
where $\...
- 1,538
1
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0
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160
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Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
1
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0
answers
597
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What is $T T^*$ argument?
During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/...
- 159
1
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0
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76
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Geometric series involving the Laguerre polynomials
Let put $\alpha=5$ and $x=3$. Consider the following set given by
$$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$
Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
1
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0
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134
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Number of solutions to a diophantine equation
Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$.
Define the proportion
$$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...
- 43.2k
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0
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70
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Examples of reaction-diffusion systems with analytical solutions
I want to study how some numerical schemes work on $2$-dimensional reaction-diffusion systems on rectangles with Neumann Boundary conditions and I search for a while for a problem of the form:
$$\...
- 1,759
1
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0
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110
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Zeroth homology of the complement of a closed set
Suppose $F$ is a closed set in $\mathbb{R}^n$ with $n>1$.
Are there some known conditions that must be imposed on $F$ so that its complement in $\mathbb{R}^n$ has a finite number of components? ...
- 411
1
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0
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72
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Compute surface Sobolev norm using local coordinate
For a bounded $\Omega\subset \mathbb{R}^n$ with Lipschitz boundary, there are various definitions of fractional Sobolev spaces (a.k.a. Sobolev-Slobodeckij spaces) on $\partial \Omega$, either by using ...
- 503
1
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0
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72
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Elliptic systems with two dimensions
Let $ \Omega\subset\mathbb{R}^2 $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq 2 $ and $ 1\leq\...
- 1,219
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0
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75
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Existence of solutions to $\alpha(s)=\mathbb P[Y_s>0] + \int_0^s \dot{\alpha}(t)\mathbb P[Y^{t,0}_s>0] dt$
Let $\alpha:\mathbb R_+\to\mathbb R_+$ be a "nice" function with $\alpha(0)=1$. Define the process
$$Y_t=Y_0+t+\int_0^t\frac{dW_u}{1+\alpha(u)},\quad \forall t\ge 0,$$
where $Y_0>0$ has a ...
- 1,334
1
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0
answers
85
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Boundary estimates for elliptic systems
Let $ \Omega\subset\mathbb{R}^d $ is a $ C^{1,\eta} $ domian with $ 0<\eta<1 $. Assume that $ A(y)=(a_{ij}^{\alpha\beta}(y)) $ is a matrix valued function, where $ 1\leq i,j\leq d $ and $ 1\leq\...
- 1,219
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0
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99
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Estimate on integral with logarithmic weight
Let $f:\mathbb R^n \to \mathbb R$. What (minimal) assumptions are needed, in addition to $f \in L^1 \cap L^\infty$, to have
$$
\int_{\mathbb R^{2n}} \frac{|f(x)-f(y)|}{(|x-y|+\epsilon)^{n+1}} \, dx \, ...
- 839
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0
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76
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Representing a function in terms of higher order differences
I want to write a function in terms of its mollification and higher order
forward differences. Given a function $u:\mathbb{R}\rightarrow\mathbb{R}$ and
$h>0$, we set $u_{h}(x):=\frac{1}{h}u\left( \...
- 411
1
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0
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76
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Error estimates for orthogonal polynomial approximation
tl;dr: Are there explicit bounds for the approximation error by orthogonal polynomials?
There are various ways to formulate this question more precisely, so want I emphasize up front that this is a ...
- 19
1
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0
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259
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Divergence of conformal Killing vector fields on $S^2$ and the spherical harmonics
Can anyone think of a conformal Killing vector field $W$ on $S^2$ with the round metric that is not Killing such that its divergence is $L^2$-orthogonal to the spherical harmonics with $\ell = 1$?
One ...
- 969
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0
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100
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Exponential decay of a random matrix falling into a ball
Let $A=U\Sigma V^T\in\mathbb{R}^{n\times n}$ be a random matrix defined in the following way: $U,V$ are uniformly distributed on the orthogonal group $O(n)$, $\Sigma$ is a diagonal matrix such that ...
- 1,495
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67
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Solution to recurrence relation from integro-differential dynamical system?
Consider the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1}
\end{equation}
such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
- 79
1
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0
answers
98
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Two definitions of Sobolev spaces and the trace theorem
Let $M=[0,\infty) \times S^2$. We have the regular regular Sobolev space $H^1(M)$.
We also have the space $H^1\bigg([0,\infty); H^1(S^2)\bigg)$. Are those two spaces the same? Does one contain the ...
- 969
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260
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A sufficient condition for multiple differentiability of a function of several variables
While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1:
Let $f: \mathbb{R}^n\to\mathbb{R}$, $\...
1
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0
answers
34
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$L^p$-continuity for discrete linear causal systems
Let $p \in [1, +\infty)$, $(b_0(n)), \dots (b_m(n)), (a_1(n)), \dots, (a_m(n))$ suitable sequences of real numbers and consider the map $\phi: \ell^p \to \ell^p$, $x \mapsto y$ defined by:
\begin{...
- 41
1
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0
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142
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Fourier transform of the Bochner-Riesz multipliers
How to obtain the decay of Fourier transform of the Bochner-Riesz multipliers? For $\lambda>0$ define:
$$
\hat{m_{\lambda}}(x)=\int\limits_{\mathbb{R}^d} (1-|\xi|^2)_{+}^{\lambda}e^{2\pi i x\cdot \...
- 351
1
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0
answers
38
views
Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
- 47
1
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0
answers
87
views
Computation of the trace of a convolution operator
I try to understand a particular step on page two from the research paper "A multidimensional analog of a limit theorem of G. Szegö".
https://iopscience.iop.org/article/10.1070/...
- 197
1
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0
answers
77
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Divergence between random variables after transformation
Let $X$ and $Y$ be random variables with laws $\mu_X$, $\mu_Y$ and $d$ be some $f$-divergence (e.g. KL, total variation, Hellinger). Writing $d(X,Y)$ for the divergence between $\mu_X$ and $\mu_Y$, ...
- 19
1
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0
answers
87
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Estimating the impact of replacing a negative exponential by a truncation of its Taylor series in an integral
Let $f(x)$ be a smooth function that takes both positive and negative values and suppose there exists an increasing sequence of positive numbers $R_i$ diverging to $\infty$ such that
$$\lim_{i \...
- 81
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0
answers
161
views
What is the necessary and sufficient condition for a chain rule hold?
Assume that $f: [0,+\infty) \to [0,+\infty)$ is a $C^1$, increasing, and concave function with $f(0)=0$. Let $g:[0,+\infty) \to [0, +\infty)$ be a non-increasing function.
My question is that, does ...
- 11
1
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0
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186
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Lipschitz continuity of an implicit function generated by a monotonic and Lipschitz multivariate function
Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ satisfying the following conditions:
$z=F(x,y)$ is Lipschitz continuous w.r.t. $(x,y)$;
Given $x$, $F(x,y)$ is non-...
- 611
1
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0
answers
184
views
A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$
Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
1
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0
answers
46
views
Boundary estimates for Neumann derivative of solution to Laplacian equation with Dirchlet boundary data
Let $\Omega \subset \mathbb{R}^n$ be a smooth domain. Consider the following Laplacian equation with Dirichlet boundary condition.
\begin{equation}
\begin{cases}
\Delta u=0\quad &\mbox{in $\Omega$}...
- 1,350
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0
answers
117
views
Is a "global period" similar to a "local period"?
Let $v\in\mathcal{C}^1(\mathbb{R}^n,\mathbb{R}^n)$ $(n\geq 2)$ a vector field, such that the set $E=\{v=0\}$ is a manifold of dimension $n-2$. Assume that for every $x\in\mathbb{R}^n-E$, the ...
- 449
1
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0
answers
88
views
Density of $C^k$-functions with Lipschitz partial derivatives
Let $N$ and $M$ be complete Riemannian manifolds, of respective dimension $n$ and $m$ with $n,m\geq 1$. Let $C^{k,1}_b(N,M)$ be set of all bounded continuous functions $f:N\rightarrow M$ for which ...
- 5,405
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0
answers
145
views
Integrability conditions imply existence of potential
I'm looking for a proof of the following well-known theorem:
If $f$ is a continuously differentiable vector field in a simply connected region $G\subset \mathbb{R}^n$ which satisfies the ...
- 2,160
1
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0
answers
81
views
Finding the K-means of the normal distribution
Let $K\in\mathbb N^+$ be a parameter.
Given a distribution $q$ over the real numbers, K-means clustering aims to find $K$ centroids $c_1,\ldots,c_k\in\mathbb R$ that minimize
$$
\int_{-\infty}^\infty ...
- 618
1
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0
answers
112
views
A problem related to analytic function
Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...
user245973
1
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0
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70
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Extension of the Kelvin transform
Suppose $B=B(y,r)$ is ball in $\mathbb{R}^m$ ($m\geq2$), and $u$ a superharmonic function on a neighborhood of the closure $\overline{B}$ of $B$. We know that the Kelvin transform of $u$ with respect ...
- 411
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0
answers
70
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An inequality for a recursively defined sequence of numbers
Consider an arbitrary sequence $(x_n)_{n \in \mathbb{N}} \subseteq \mathbb{R}$ and $r \in \mathbb{R}$ with $r > 2$.
Set $y_0 = 1$ and $z_0 = 0$ and for $n \in \mathbb{N}$ recursively define
$$y_n = ...
- 111
1
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0
answers
35
views
Prove monotonicity of a system of difference equations
I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as ...
- 389
1
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0
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121
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Lipschitz hypersurface
I asked this already on Math SE. Maybe this definition is not quite common, but I'm asking myself what a Lipschitz hypersurface is. Intuitively this is a hypersurface which can locally be parametrized ...
- 173
1
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0
answers
76
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Regarding an integrability property of Schwartz class function
Let $f\in\mathcal{S}(\mathbb{R^n})$, Schwartz class, satisfying
$$\int_{\mathbb{R}^n}|f(x)|e^{g(||x||)}dx<\infty, $$
where $g:[0,\infty)\to[0,\infty)$ be an increasing function satisfying $\int_0^\...
- 99
1
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0
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56
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Monotonically increasing and bounded function is in $BV_{loc}$?
For any $n\in \mathbb{N}$ let $f_n:\mathbb{R}\to [0,1]$ be monotonically increasing and $\lim_{x\to -\infty} f_n(x)=0$ and $\lim_{x\to \infty} f_n(x)=1$. It follows $f_n$ is differentiable a.e..
I'm ...
- 173
1
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0
answers
85
views
Superharmonicity of the distance function
Suppose $V$ is a convex open proper subset of $\mathbb{R}^m$ ($m\geq2$). It is known that the function $u(x)=$dist$(x,\partial V)$ is superharmonic on $V$. Is there a similar result without $V$ being ...
- 411
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0
answers
119
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May sequential continuity of a map on compact sets fail to admit extrema?
Let $X$ be a compact topological space. Is there an example of a sequentially upper-semicontinuous function $f: X \rightarrow \mathbb{R}$ that does not admit a maximum point in $X$?
My very rough ...
- 215
1
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0
answers
38
views
Uniform boundedness of Green Function
This post has already been on StackExchange but hasn't received any answers. Since its origin is a research paper, I thought that maybe this forum might be a better place for it. If you disagree ...
- 121
1
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0
answers
61
views
An open set whose complement is non-thin at infinity
Let $x^*$ designate the inverse of a point $x\in\mathbb{R}^m$ under the Kelvin transformation with respect to the circle of center 0 and radius 1. Recall that
$$x^*=|x|^{-2}x.$$
For a set $E$, we set $...
- 411
1
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0
answers
79
views
Convergence mode with inputs and functions varying in tandem
Given a sequence $(f_n)$ of functions between metric spaces, let's say that $f_n$ "converges flexibly" to $f$ if, whenever $x_n \to x$ is a convergent sequence of inputs, it follows that $...
- 143
1
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0
answers
258
views
Cut-off function and fractional Laplacian
Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and
$$
|\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s}...
- 99
1
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0
answers
144
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Liouville theorem for elliptic equation with advection term
How can one prove that any $L^2$ solution of
$$ - \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N $$
is zero if $a(x)$ is a divergence-free vector field such that
$\int |\...
- 99
1
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0
answers
65
views
Normalizing constants preserve metric entropy
Suppose $\mathcal{F}=\left\{f\in L^2([a,b]): 0<\underline{c}\leq f\leq\overline{c} \right\}$. Consider the following transformation
$$\tilde{\mathcal{F}} := \left\{\frac{f}{\int f d\mu}: f\in \...
- 11
1
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0
answers
122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
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0
answers
33
views
Existence theory for first order scalar discontinuous ODE
Consider the scalar i.v.p. in ${\mathbb R}$
$$
x'=f(t,x), \; t\in[0,T], \; x(0)=x_0,
$$
where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R} $, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$...
- 31