All Questions
5,871 questions
1
vote
1
answer
3k
views
In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives
I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem.
"Bochner's theorem states that a
positive ...
- 11
19
votes
3
answers
1k
views
functions from Q to itself with derivative zero
Let $f: {\bf Q} \rightarrow {\bf Q}$ be a "${\bf Q}$-differentiable" function whose "${\bf Q}$-derivative" is constantly zero; that is, for all $x \in {\bf Q}$ and all $\epsilon > 0$ in ${\bf Q}$, ...
- 19.7k
4
votes
1
answer
261
views
Minimizing action squared versus action
I have a very basic question in the calculus of variations:
Suppose I want to minimize the functional
$$A[r, r'] = \int_\Omega L(r, r') dx $$
When is it possible to say that extremals of $A$ agree ...
user7807
0
votes
1
answer
721
views
Pointwise limit at Lebesgue's point
Dear MOs,
I am sorry if this problem is too elementary for someone. I just want to get confirmation.
Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by the Lebesgue ...
- 1,649
3
votes
2
answers
1k
views
convergence of infimum
I have a question during my intership. Given a convergent sequence of continuous et convex functions $\{f_n(x)\}$ defined in $\mathbb{R}^M$. These functions are uniformly Lipschitz continuous which ...
- 69
6
votes
1
answer
3k
views
Proving the interior of a dual cone is the set of vectors whose inner product is strictly positive on the cone
Apologies for posting such a simple question to mathoverflow. I've have been stuck trying to solve this problem for some time and have posted this same query to math.stackexchange (but have received ...
- 283
1
vote
1
answer
496
views
Convergence of Difference Quotients
Let $\gamma_{\varepsilon} \rightharpoonup \gamma$ in $W^{1,\infty}(0,1)$. Then for any fixed $s \in \mathbb (0,1)$ does the limit $\lim_{\varepsilon \rightarrow 0} \frac{\gamma_{\varepsilon}(s\...
- 213
4
votes
1
answer
471
views
Ask for theory about the weighted L^2(R^d) space.
Dear MOs,
I am now considering the following norm:
$$
||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:.
$$
where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
- 1,649
0
votes
1
answer
612
views
Calculating a distributional derivative
Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
- 213
0
votes
1
answer
238
views
A property of a quasiperiodic function
Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.
Does the limit of
$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$
Exist ...
- 213
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
464
views
smooth families of analytic functions
My question is essentially whether taking partial derivatives of a smooth family of analytic functions yields again a smooth family of analytic functions.
The precise question is the following:
Let $...
- 31
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
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 161
3
votes
0
answers
227
views
Mesh for 3d dungeons game. [closed]
Hallo, I look for some F: R^2->R height function which would generate the Speleothem ceiling http://en.wikipedia.org/wiki/Speleothem for 3d game taking place in dungeons/caves.
The function might be ...
- 139
0
votes
1
answer
3k
views
Is the sum sin(n) bounded? [closed]
I wonder whether the sequence $s_{n} = \sum_{k = 1}^{n} \sin k$ is bounded.
The answer seems no, but I have no idea how to prove this from the irrationality of $\pi$.
- 1
2
votes
2
answers
408
views
Higher order partial derivatives and global regularity.
Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous.
Is it true that $f_{xy}$ exists and continuous?
Is it true that $f_{yx}$ ...
- 105
6
votes
1
answer
2k
views
Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?
I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...
- 476
13
votes
2
answers
2k
views
Asymptotics of the n-th prime using the gamma function
In the paper http://rgmia.org/papers/v8n2/eepnt.pdf, the author proves that proves an explicit inequality on prime numbers using the gamma function and as a corollary, he showed that.
$$
p_n = n \...
- 5,470
7
votes
3
answers
709
views
On the inequality $\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\right)^2$
I'm have some difficulties in bounding the following inequality:
I want to find a c as small as possible s.t.
$$\sum_{i=1}^nx_i^4\sum_{i=1}^nx_i^2 -\sum_{i=1}^nx_i^6 \leq c\left(\sum_{i=1}^nx_i^3\...
- 191
1
vote
0
answers
827
views
Question about Riemann integral and total variation [closed]
Let $g$ be Riemann integrable on $[a,b]$, $f(x)=\int_a^xg(t)dt$ for $x∈[a,b]$.
How to show that the total variation of $f$ is equal to $∫_a^b|g(x)|dx$?
- 133
3
votes
1
answer
352
views
Integral Equation with "convolution"
I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...
- 31
0
votes
1
answer
905
views
Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
- 81
8
votes
3
answers
1k
views
On the set of divergence to infinity for sequences of positive continuous functions
Hi,
I have asked this question on math.stackexchange but it has not received much attention, so I ask it here.
This question is partly motivated by this one, which contains an example of a sequence $...
- 2,154
0
votes
1
answer
939
views
Asymptotic equivalence for functions with zeros
I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$.
...
- 2,480
0
votes
0
answers
92
views
Lower bound for double sums with power law decay terms.
This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion.
The motivation to ask here if the inequality below ...
- 2,044
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
- 24.9k
4
votes
1
answer
882
views
What is the domain of the "average operator"?
I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$
whenever the limit ...
- 225
0
votes
3
answers
404
views
Some Questions about zero-dimensional subsets of the unit interval related to cantor set
Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
- 1,788
2
votes
1
answer
182
views
represented as a series of periodic function
Is there any necessary and sufficient condition for function $f$ such that:
$f(x)=\sum_{k=1}^{\infty} f_k(x)$ for all $x \in \mathbb{R}$,where $(f_n )_{n=1}^{\infty}$ is a sequence of periodic ...
- 407
0
votes
1
answer
857
views
Is Jordan outer measure finitely additive on positively separated sets in $\mathbb{R^n}$?
I am trying to argue that exterior measure has nice properties that Jordan outer measure doesn't have. One of them is finite additivity, but I can't find a simple way to show Jordan outer measure is ...
1
vote
1
answer
350
views
Strong convergence in reflecxive Banach space
Let $(X, \|\cdot\|)$ be an Banach space. Assume that a sequence $f_n \rightarrow f$ weakly in $X$, and $\|f_n\| \rightarrow \|f\|$ as $n \rightarrow \infty$. It's known that if $X$ is a uniformly ...
- 425
0
votes
3
answers
302
views
Vector "product" diagonalization
Consider a vector space $V$ (with dimension $n+1$ and elements $v$) on which a (commutative and associative) "product" $\odot$ taking $V\odot V\rightarrow V$ is defined, and an $1$ element $v_0$ ...
- 4,829
32
votes
1
answer
17k
views
The gimbal lock shows up in my quaternions
I suspect this is a bit basic for mathoverflow, seeing I'm still just an undergraduate
I've been playing around with quaternions as means to eliminate the gimbal lock. From what I understand, one ...
46
votes
4
answers
8k
views
Why could Mertens not prove the prime number theorem?
We know that
$$
\sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x)
$$
where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy
$$
\sum_{p \le x}\frac{1}{p} = \ln\ln ...
- 5,470
12
votes
1
answer
1k
views
Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are "sort of increasing" or "sort of decreasing" (as defined below)?
Is the following true?
If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
- 167
1
vote
1
answer
223
views
f_n(x,p) converge uniformly to nice f(x,p); do zeros of f_n(.,p) converge uniformly to zeros of f(.,p)?
Fix compact intervals $X, P \subseteq \mathbb{R}$.
Let $f_n : X \times P \to \mathbb{R}$ be a sequence of $C^2$ functions converging uniformly to a $C^2$ function $f$. The first and second ...
- 1,068
3
votes
0
answers
289
views
How well do continuously differentiable functions behave from R^2 to R^2 ?
The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question.
In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...
- 529
8
votes
2
answers
2k
views
Divergent series expansion in Apéry's proof of the irrationality of $\zeta(2)$ and $\zeta(3)$
UPDATE. I am now making this a CW in the hope someone can improve the content of this question and/or correct the text.
This is a concise version of this math.SE question of mine. I've got an answer ...
2
votes
0
answers
131
views
Bounding an integral with a small parameter by log
I have been working through Erdos & Yau's `Linear Boltzmann equation as the weak coupling limit of a random Schrodinger Equation,'
(arXiv link: http://arxiv.org/abs/math-ph/9901020), and for an ...
- 21
4
votes
2
answers
1k
views
Reducing system of equations involving Erf, Error Function
I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...
- 123
1
vote
2
answers
474
views
Chebyshev's Theorem
Hi,
I´m looking for Chebyshev´s theorem which says that the inequality $|x(k)-y|<3/k$ has infinitely many solutions, where $x(k)=x_0+k\alpha \pmod 1$, $\alpha$ is an irrational number, and $x_0,y\...
- 11
10
votes
2
answers
6k
views
Who was the first to formulate the inverse function theorem?
Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$.
...
- 103
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
689
views
Partitions of an interval
This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there.
Specifically, consider "partitions" ...
- 2,358
30
votes
4
answers
2k
views
is f a polynomial provided that it is "partially" smooth?
Let $f$ be a $C^\infty$ function on $(c,d)$ ,and
let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$.
Suppose for each $n\in ...
- 407
1
vote
1
answer
771
views
A question about the tail of an absolutely integrable function
Assume $X$ is a measure space and $f : X \to [0,\infty]$ is an absolutely integrable function (that is $\int_X f \; d \mu < \infty$). This question is about the asymptotic behaviour of the function ...
- 3,830
4
votes
2
answers
3k
views
Chain rule for fractional laplacian
Does anyone know a formula of chain rule for fractional laplacian?
say we take the fractional laplacian of order a on function $g(U(x))$ $x\in \mathbb{R}^2$, $U \in \mathbb{R}$, $g \colon \mathbb{R} \...
- 41
2
votes
1
answer
271
views
Solvability of the quaternion equations of the Quaternion rationals
So, Let K be the non-commutative field or division ring of the Quaternions. We will consider a sub-field of it L=Q[i,j] in other words, L=Q+Qi+Qj+Qk. Now let us consider the ring of skew polynomials ...
4
votes
2
answers
977
views
Articles with examples of Darboux functions without fixed points
A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \...
- 2,222