All Questions
5,909 questions
21
votes
3
answers
2k
views
Felix Klein on mean value theorem and infinitesimals
This is a reference request prompted by some intriguing comments made by Felix Klein.
In 1908, Felix Klein formulated a criterion of what it would take for a theory of infinitesimals to be ...
- 16.6k
1
vote
1
answer
115
views
Well-definedness for a singular integral
Let $T_\alpha$ be a singular integral operator defined by
$$
T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds
$$
for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$.
...
- 16.2k
2
votes
1
answer
125
views
Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$
I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
CommunityBot
- 1
51
votes
2
answers
5k
views
A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
- 105k
2
votes
1
answer
182
views
Proof of existence and uniqueness of solution to f(c)=0
I have a function $f:R^n_+\rightarrow R^n$ for which I want to show the following:
$$\exists c\in R^n_+ \quad \forall i,j\,\,f_i(c)=f_j(c)$$
Where $f_i (c)$ are the different coordinates of $f$.
$f$ ...
- 56.9k
9
votes
1
answer
635
views
De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions
Context: In formulating problems for secondary school mathematics teachers (and students) about absolute value functions, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \...
CommunityBot
- 1
2
votes
1
answer
84
views
How to choose function $\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$?
Can we expect to choose a function $f:\mathbb R \to \mathbb R$ (nonzero compactly supported) so that
$\sum_{m\in \mathbb Z} (-1)^m f(x+m) f(x-m+n)=0$ for all $x\in \mathbb R$ and $n\in \mathbb Z$?...
- 39.1k
10
votes
1
answer
329
views
Is there a triangle which makes dense set of angles by drawing medians?
This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
- 125
1
vote
0
answers
126
views
identity involving spectral functions
Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true :
$$ A^* f(AA^*) = f(A^* A) A^*$$
- 2,118
6
votes
1
answer
213
views
A one-dimensional integral minimization problem
Let $\mathscr F$ be the collection of smooth functions $f \colon
\mathbb R \to \mathbb R$ such that
$f \in C^\infty_c(\mathbb R)$, with $\text{supp } f \subset [-1,1]$;
$\int_0^1 x f(x) dx ...
- 128k
7
votes
2
answers
998
views
Uniform continuity of heat semigroup
I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It ...
- 16.2k
7
votes
1
answer
1k
views
A variational problem - some guidance
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.
Let
$\...
CommunityBot
- 1
5
votes
2
answers
1k
views
An example of an open discontinuous function
Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the ...
- 4,786
1
vote
1
answer
633
views
Does sequence almost sure convergence imply almost sure convergence?
This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\...
- 23.2k
7
votes
4
answers
3k
views
Upper bound of the expectation of sum of the absolute value pairs
We have two arrays $A,B$ of length $n$. All values are i.i.d drawn from same distribution on $[0,1]$. If we sort $A,B$ in non-decreasing order and let $A_{(i)},B_{(i)}$ denote the i-th value in the ...
- 128k
4
votes
0
answers
289
views
A uniform Riemann sum approximation of the integral of the Fejer kernels
Let $F_N(t)$ denote the Fejer kernel
$$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$
Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
- 136
4
votes
1
answer
731
views
when is an eigenvalue differentiable with respect to a parameter? [duplicate]
Let say we have a symmetric matrix $A(\omega)$ depending smoothly on some variables $\omega \in \Omega$ with $\Omega \subset \mathbb{R}^d$ a $d$-dimensional parameterspace (this means the eigenvalues ...
- 12.7k
2
votes
0
answers
257
views
The nonlinear operator defined as the commutator of a matrix and a nonlinear operator
In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
- 620
2
votes
0
answers
65
views
Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 623
3
votes
1
answer
876
views
Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
- 24.2k
1
vote
0
answers
114
views
density of fractal measures
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
- 11
2
votes
1
answer
93
views
Lipschitz bound on semigroups
Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$
Now if you think about ...
- 13k
5
votes
1
answer
279
views
Mixed norm inequality
Suppose we have a product space $(X_1\times X_2,\mu_1\otimes\mu_2)$, with finite measures $\mu_1,\mu_2$ and $p>1$.
Is there a possibility that an inequality of this form holds on the product space?
...
- 128k
1
vote
0
answers
45
views
Shifting Sobolev norms in a hyperbolic estimate
Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...
- 63.9k
3
votes
1
answer
172
views
Infimum of an integral functional involving a symmetric matrix
I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
...
- 128k
2
votes
1
answer
151
views
Rademacher‘s Theorem and Baire category
Is there a variant of Rademacher‘s Theorem where the smallness of the points of non-differentiability is measured in terms of Baire category instead of measure?
More precisely, let X be a separable ...
- 42.8k
9
votes
3
answers
563
views
Non-uniqueness of flow for divergence free vector fields
I am looking for a (possibly time-dependent) vector field $v: [0,1] \times \mathbb R_x^d \to
\mathbb R^d$ such that
$\text{div}_x v = 0$ ;
$v$ has more than one (measure-preserving) flow,...
- 151
2
votes
1
answer
347
views
Convergence rate for $L^2$ convergence
Let $f \in L^2(\mathbb R)$ then it is well-known that
$$ \widetilde{f}(x):=\sum_{n \in \mathbb Z} \frac{1}{\varepsilon}\int_{[n\varepsilon,(n+1)\varepsilon]} f(s) \ ds 1_{[n\varepsilon,(n+1)\...
- 128k
28
votes
7
answers
5k
views
Rolle's theorem in n dimensions
This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
- 1,169
1
vote
1
answer
79
views
Feasible set for linear constraints on monotonic functions
Consider functions $f_i(x)$ that map $\mathbb{R}^n$ to $\mathbb{R}$ for $i\in{1,\dots,k}$. Assume these functions are monotonically increasing in their arguments and continuous everywhere Also, one ...
- 128k
1
vote
0
answers
136
views
Reference for Existence and uniqueness of an Integro-Differential Equation
I have an Integro-Differential Equation (IDE) of the following form:
$$
x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds,
$$
I have found this classical reference, but the IDEs considered therein ...
- 121
0
votes
0
answers
126
views
$p$-volume of $n$-dimensional hyper-ellipsoids
I read that the unit hypersphere has maximum volume for dimension five and would like to generalize this result. (If you think that integrating over an $n$-dimensional $p$-hyper-ellipsoid area ($x_1^...
5
votes
1
answer
3k
views
Operator norm vs spectral radius for positive matrices
I believe the following statement should be true but somehow I don't see an argument:
For every integer $d>1$ there exists a constant $C=C(d)>1$ such that whenever $A$ is a $d \times d$ matrix ...
6
votes
2
answers
633
views
Interpolation space between $L^1\cap L^2$ and $L^1$
In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
- 17.2k
4
votes
1
answer
203
views
If $T_1T_2 = T_2T_1$, why $r(T_1 + T_2) \leq r(T_1) + r(T_2)$?
Let $T_1$ and $T_2$ be two bounded linear operators in a complex banach space $X$.
If $T_1T_2 = T_2T_1$, I want to know how to show that
$$
r(T_1+T_2) \leq r(T_1) + r(T_2),
$$
where $r(A)$ ...
- 17.2k
1
vote
0
answers
143
views
The average order of Mobius function in different intervals
Let $\mu$ be the Mobius function. Davenport proved that for any $\alpha\in \mathbb{R}$, for any $N\in \mathbb{N}$ and for any $A>0$, there exists a constant $C_A$ such that
$$
\left|\sum_{n=1}^{N} \...
- 675
0
votes
1
answer
3k
views
Summation of $\log n/ \log(\log n)$
Given $h>0$, I would like to estimate the following summation by some function $f(N)$:
$$
S_N=\sum_{n=2}^{N} \frac{\log n}{\log^h(\log n)}=O(f(N)).
$$
Obviously, we see that
$$
S_N>\sum_{n=2}^...
- 26.9k
3
votes
2
answers
137
views
What are some applications of Dilation Structures(idempotent right quasi-groups) from Emergent Algebra?
According to the following Journal Articles, there are these structures called Dilation Structures that are formalised in Emergent Algebras, examined in the case of metric spaces with dilations, and ...
- 785
9
votes
1
answer
460
views
Summing moments and Riemann zeta values
Let $d\mu_n(x)=\cos^{2n}x\,dx$ and consider the averages of moments
$$\alpha_n=\frac{\int_0^{\pi/2}x^4d\mu_n(x)}{\int_0^{\pi/2}d\mu_n(x)}.$$
Then, I have encountered a curious evaluation
$$\sum_{n=1}^{...
- 188k
9
votes
0
answers
569
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 4,713
2
votes
0
answers
246
views
Decay rate of least eigenvalue of Gram matrices
Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$:
$$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$
In ...
- 141
2
votes
0
answers
60
views
Gronwall type inequality involving iterated integrals
Let $p(t), a(t)$ be non-negative, continuous functions on $[0,T]$. Suppose that we have:
$$p(t) \leq a(t) + C \int_0^t du e^{-\kappa(t-u)}p(u) \int_0^u ds e^{-\kappa(u-s)} p(s),$$
where $\kappa, C >...
- 31
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.
$$
p_n = n \...
- 655
3
votes
2
answers
179
views
Is $\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}}{t\log(1-t)}dt$ bounded by a constant for large $y$?
For $p\in (0,1)$ and $\epsilon>0$ a small enough constant, consider the function $f:\mathbb{N}\to\mathbb{R}$ given by
$$f(y)=\int_0^\epsilon \frac{\left(\frac{p-t}{p(1-t)}\right)^y-(1-t)^{y(1-p)/p}...
- 128k
2
votes
1
answer
178
views
Differentiability of the logarithmic potential
Assume $\mu$ is a measure supported on a real finite interval $[a,b]$, and let
$$p_\mu(z)=\int\log|z-t|d\mu(t),$$
denote the logarithmic potential associated to $\mu$. Are there (possibly simple) ...
- 3,085
3
votes
1
answer
431
views
Can I approximate a function of bounded variation with orthogonal polynomial?
Let function $u\in BV(\Omega)$ be a function of bounded variation and $\Omega\subset \mathbb R^2$ be a smooth domain. I know it is possible to approximate function $u$ with polynomials, i.e.,
$$
u = \...
- 28k
5
votes
1
answer
211
views
Pointwise convergence in functional calculus
Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \...
- 4,729
5
votes
0
answers
166
views
global estimate for biharmonic function
My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions
Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,...
- 914
1
vote
1
answer
131
views
Convergence of $L^p$ of approximation
Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$
Then, we may define the coefficients
$$\alpha_{ij}= \frac{1}{\...
- 128k
7
votes
3
answers
843
views
Distance function to $\Omega\subset\mathbb{R}^n$ differentiable at $y\notin\Omega$ implies $\exists$ unique closest point
I am trying to show the following two statements are true:
(1) For any nonempty set $\Omega\subset\mathbb{R}^n$, the set $B$ consisting of points $y\notin\Omega$ where there is not a unique closest ...
- 1,676