All Questions
5,857 questions
1
vote
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759
views
meromorphic extension of a function
Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in \...
- 1,111
3
votes
0
answers
375
views
An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only
The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...
- 128k
2
votes
1
answer
116
views
Bounding a function with second moments
Let $f(x,y)$ be a non-negative function with $x,y \in \mathbb R^3$ that satisfies
$$
I_1(f) := \iint_{\mathbb R^3 \times \mathbb R^3 } f(x,y) \, dx \ dy < \infty
$$
and
$$
I_2(f) := \iint_{\...
- 183
4
votes
1
answer
383
views
Horizontal Sobolev space on Carnot group
This question is connected with my previous: Heisenberg group: function without vertical derivative.
Here I am trying to look from another side: what is a difference between Sobolev space and ...
- 399
5
votes
1
answer
1k
views
Possible to find a set of log-concave functions with log-concave sums?
While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
- 51
2
votes
2
answers
224
views
"Then obviously..." (a bound on strongly differential functions on an interval)
On the fourth page of their 1967 paper Local Behavior of Solutions of Quasilinear Parabolic Equations, Aaronson and Serrin comment: "Consider a strongly differentiable function $w$ of the real ...
- 363
3
votes
0
answers
652
views
Derivatives of $O$-regular varying functions are $O$-regular varying functions?
The Monotone Density Theorem for regularly varying functions says, in essence:
Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ well-...
- 825
3
votes
0
answers
160
views
integral with simple approximation. But why?
I have the following integral
$$g(x_0) = \int_{-\infty}^{\infty} \frac{1}{(1+x^2)^{3/4}}\frac{1}{(1+(x+x_0)^2)^{3/4}}\exp\left(-\frac{2\pi i}{\lambda}\left[\sqrt{1+x^2}-\sqrt{1+(x+x_0)^2} \right] \...
2
votes
1
answer
383
views
Continuous real function on germs
Let $C_0^{m,n}$ be the space of germs of continuous maps from $\mathbb{R}^m$ to $\mathbb{R}^n$, located at $0\in\mathbb{R}^m$, with the usual inductive limit topology. One can also consider $C_0^{m,n}$...
- 21.5k
1
vote
2
answers
692
views
Can we extend an a.e. Lipschitz map defined on a closed subset of R^N to the whole space so that it is still a.e. Lipschitz?
I have the following question. Let $A$ be a metrically oriented $n$-dimensional subset of $\mathbb{R}^N$ and $f$ a continuous map from $A$ to $\mathbb{R}^M$. We know that $\operatorname{Lip} f < +\...
- 1,881
-1
votes
1
answer
4k
views
Lipschitz condition on the first derivative of a function? [closed]
If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?
- 101
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
0
votes
2
answers
144
views
Optimization function of two variables
Let $A, B, C, D \in \mathbb{R^*_+}$.
Is it possible to solve
$$
\max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)}
$$
The KKT conditions give for an extrema $(x^*,y^*)$
...
user83947
1
vote
1
answer
603
views
The smallest altitude amongst the triangles formed by points in the unit circle
Let $S$ be a finite set of points inside the unit circle. Consider all possible triangles formed by three distinct points in $S$, and among all such triangles find the smallest altitude. Denote this ...
- 19
3
votes
1
answer
1k
views
Integral inequality for convex function
Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac{1}{b-a} \int_a^b u(...
- 33
6
votes
0
answers
210
views
Generalized singular numbers and the Haagerup $L^p$ spaces
Let $M$ be a semi-finite von Neumann algebra with a trace $\tau$.Let $S(M)$ be the algebra of all affiliated operators measurable with respect to $M$.
The $L^p$ norm on $M$ is given by
\begin{...
- 361
-2
votes
1
answer
92
views
An inequality between two real-valued concave functions
Can anyone help me prove the following inequality? Thanks!
- 9
5
votes
1
answer
327
views
Convergence in energy of bounded (semi)subharmonic functions
Consider a sequence $(f_n)$ of functions in the flat torus $T^d$ converging Lebesgue-a.e. to a limit function $f$.
Assume that:
1) $|f_n|(x)\leq 1$ for every $n,x$
2) $\Delta f_n\geq -1$ in the ...
- 660
4
votes
1
answer
1k
views
Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
- 3,630
1
vote
0
answers
105
views
Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 1,385
1
vote
0
answers
128
views
determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 31
3
votes
2
answers
2k
views
The extension of smooth function
If $U$ is a bounded domain in $\mathbb R^n$ whose boundary is smooth, and $f$ is a smooth function on $U$ whose partial derivatives of all orders have a continuous extensions to $\bar U$. For an ...
- 1,441
2
votes
1
answer
383
views
Hardy space, Lebesgue space for $p<1$,
We denote $\mathcal D'(\mathbb R^n)$ the space of distributions, and $\mathcal D(\mathbb R^n)$ the space of smooth, compactly supported functions.
Let $\rho\in \mathcal D'(\mathbb R^n)$ such that ...
- 630
6
votes
2
answers
930
views
reverse mathematics strength of "Lipschitz functions are somewhere differentiable"
What is the reverse mathematics strength of
"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
(...
user5810
0
votes
1
answer
163
views
$\int_0^t f(s)\,dB_s$ normally distributed, mean and variance
Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...
- 43
5
votes
4
answers
526
views
Existence of an arbitrary Small positive continuous real Valued Function
Let $(X,\tau)$ be a Tychonoff Topological space.
For each $x\in X$ consider an arbitrary positive real number $\epsilon_x>0$. Is There a continuous real valued function $f:X\rightarrow \mathbb{...
- 1,788
5
votes
2
answers
273
views
Smooth convex extensibility of combination of two line segments
This is a refined version of my earlier question Convex extensibility of combination of two lines.
Is there a smooth function $f:[0,1]\times [0,1]\rightarrow\mathbb R$ such
that for all $x\in [0,...
- 24.8k
2
votes
0
answers
519
views
Regularity in PDE theory
I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
Let $U$ be connected,open and bounded in $\mathbb{R}^n$ ...
- 231
0
votes
1
answer
321
views
Is the span of those vectors dense in $\ell_2$?
For all $x \in \mathbb{R}^n$ and $\alpha \in \mathbb{Z}_{\geq 0}^n$ let $x^\alpha=x_1^{\alpha_1} \cdots x_n^{\alpha_n}$. Let $$\ell^2=\{z=(z_\alpha)_{\alpha \in \mathbb{Z}_{\geq 0}^n}:\, z_{\alpha} \...
- 3,031
1
vote
1
answer
260
views
Nepero game (by Yacov Perelman)
I have already posted this question time before on stackexchange, but didn't receive a definitive solution.
So this is the game: consider a positive integer number $n$ and divide it in a finite ...
- 11
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
3
votes
1
answer
336
views
Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points
I am currently working on a problem where I have to minimize a $m$-strongly convex function
$$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$
over a bounded integer lattice,
$$L = \mathbb{Z}^n \cap [-...
- 379
1
vote
1
answer
156
views
Step 2 of The Strichartz's Estimates in Cazenave's Book
My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates.
The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...
- 13
0
votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617
2
votes
0
answers
73
views
Inequality concerning integral w.r.t. Hölder functions [closed]
I didn't get any reaction on that question on Math.stackexchange.
So I hope to get a hint from the experts.
I have to prove the following inequality.
Let $\nu$ be a function on $[0,1]$ with one ...
- 115
2
votes
0
answers
259
views
How to analytically evaluate this n-dimensional iterated integral?
I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral
$$\int_{-\infty}^{+\infty}dx_1\int_{-\...
- 161
2
votes
0
answers
563
views
The functional equation of Hofstadter's Q sequence
Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and
$Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything
on this sequence has been proved -- not even that $Q(n)$ is well-...
- 19.6k
1
vote
2
answers
183
views
Convergence of Sobolev functions near the boundary
Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$.
Question 1: Can we ...
- 169
1
vote
1
answer
91
views
Design measure, which cannot be factorized as a product of measures
Let $\mathcal{S}_x$ and $\mathcal{S}_y$ be a finite discrete sets, such that
$$
0 < |\mathcal{S}_x| < \infty, \qquad 0 < |\mathcal{S}_y| < \infty, \qquad \mathcal{S}_x \cap \mathcal{S}_y =...
- 181
1
vote
2
answers
318
views
Finite interpolation by nondecreasing indefinitely differentiable functions in a finite-dimensional space
Some time ago, I asked about inite interpolation by
a nondecreasing polynomial here at Finite interpolation by a nondecreasing polynomial. This turned out to be an already solved problem; it also ...
- 3,595
1
vote
0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
1
vote
1
answer
131
views
Maximum of gradient of convex functions [closed]
The question comes from the page 472, Elliptic partial differential equations of second order/ David Gilbarg, Neil S. Trudinger. In one dimension it's obviously true, but it seems more involved in ...
- 31
3
votes
1
answer
124
views
Injectivity of vector functions: Numerical Verification
Problem Setup
Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact.
The function $f(\cdot)$ can only be evaluated numerically.
...
- 33
2
votes
1
answer
1k
views
On an eigenvalue inequality
Let $\lambda_1 (\cdot)$ be the larger absolute value
eigenvalue of a $2\times2$ matrix and $\lambda_2 (\cdot)$
the smaller absolute value eigenvalue of a $2\times2$ matrix, i.e.
$|\lambda_1 (\cdot)| \...
- 29
2
votes
1
answer
160
views
Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?
The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...
- 10.1k
3
votes
2
answers
295
views
Finding a simpler "local" lower bound for a rational function
I have obtained as the expression for some quantity the following gargantuan formula:
$$ \frac{k^8 + 3k^7 + 8k^6 + 3k^5 - 16k^4 - 32k^3 + 63k^2 - 34k + 6}{k^6 + 3k^5 + 6k^4 - 24k^2 + 21k - 5}$$.
...
- 6,962
5
votes
1
answer
218
views
Existence of doubling non-Polish metric measure spaces
Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is doubling if there exists a constant $C \geq 1$ such that $0 < \...
user14166
2
votes
1
answer
800
views
A question about Skorokhod metric
I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
- 1,835
2
votes
1
answer
8k
views
Example of function of bounded variation but not absolutely continuous. [closed]
I know that every absolutely continuous functions are of bounded(finite) variations but converse need not be true. and the cantor function is well-known example of function of bounded variation which ...
- 123
3
votes
1
answer
133
views
Restrictions on spectral measure
Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C_c(\mathbb{R})$ to $$T_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\...
- 587