Minimum of sums of floor function over unit square
Define $$\varphi_0(x,y):=\sum_{j=1}^{3}(\lfloor y \rfloor+\lfloor\eta_0x-y\rfloor-\lfloor y-\eta_j x\rfloor-\lfloor(\eta_0-\eta_j)x-y\rfloor-2\lfloor\eta_j...
View ArticleHelp with filling in the details to show that $\lim\limits_{n\to\infty}...
So we have,$$\begin{align}\lim_{n\to\infty} \sum_{k=1}^{n}\left(\frac{k}{n}\right)^n &= \lim_{n\to\infty} \sum_{j=0}^{n-1}\bigg(\frac{n-j}{n}\bigg)^n \\&= \lim_{n\to\infty}...
View ArticleRegarding uniform convergence of series
For $x \in [-1, 1],$ define two sequences as follows$$f_n(x)=(-1)^n \frac{x^2+n}{n^2} \quad \text{and} \quad g_n(x)=(-1)^n \frac{x^2+n^2}{n^3}.$$Then show that $\Sigma_{n \geq 1}f_n$ and $\Sigma_{n...
View ArticleConvergence of sequence related to Euler-Mascheroni Constant [closed]
Define $(x_n)$ as a sequence satisfying: $x_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n$ for all $n\geq 1$.Let $\lim x_n = c$ . Prove that $\frac{1}{2n+1}<x_n-c<\frac{1}{2n}$ for all...
View ArticleOn the definition of lacunary set
When I read the paper titled "Maximal operators associated to sets of directions of Hausdorff and Minkowski dimension zero" by Paul Hagelstein, I am confused by the definition of lacunary set. Let...
View ArticlePartition of unity and euclidean factor control
Problem: If $U$ is open subset of $X\times \mathbb{R}^n$, $X$ paracompact, then there exists map $\lambda\colon X\rightarrow (0,1]$ so that if $(x,v)\in X\times \mathbb{R}^n$ such that...
View ArticleEstimating piecewise constant in time functions independently of a time step.
Let $h=\frac{T}{N}>0$ be a time step and initial data $f_{0}\in H^{1}(\Omega)^{3}$, $g_{0}\in L^{2}(\Omega)^{3\times 3}$, assume now we have two sequences $f_k\in H^{1}(\Omega)^{3}$ and $g_{k}\in...
View Article$\sum \sum \frac{1}{m n^2 + n m^2}$ has no closed form? [duplicate]
Consider$$A = \sum_{n>0} \sum_{m>0} \dfrac{1}{m n^2 + n m^2}$$$A$ has no closed form ?Notice similar ones do for instance :$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = -...
View ArticleIf $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x)...
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$"$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss of...
View ArticleProve that the set of Algebraic Numbers is countable.
Please evaluate the following, and help complete the proof. Let $R$ be the set of polynomials with rational coefficients, then $$R=\{P(x)|a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0, a_i\in \mathbb{Q}\}$$Given...
View ArticleContinuous strictly increasing function with derivative infinity at a measure...
Let $E\subset [0,1]$ with $\mu(E)=0$. Does there exist a continuous, strictly increasing function $f$ on $[0,1]$ so that $f'(x)=\infty$ for all $x\in E$ (in Lebesgue sense)?I think there exist such a...
View ArticleExistence of solution of variational problema
Given the function $f(\xi)=\sqrt{1+\|\xi\|^2}$ with $\xi \in \mathbb{R}^n$, define de functional$$E(u)=\int_\Omega f(\nabla u(x))\,dx$$with $\Omega \subset \mathbb{R}^n$ open, bounded and having...
View ArticleHow to prove that one normed vector space is the completion of another one?...
Suppose we are given two normed vector spaces $Y$ and $X$. $Y$ is said to be the completion of $X$ if $Y$ consists of all points in $X$ and all Cauchy sequences are given a limit belonging to $Y$.How...
View ArticleEvaluating $\int_0^1 \frac{x\arctan(x)}{1-x^2} \log^2 \left( \frac{1 + x^2...
Amongst the integrals involving products of arctan and logarithms in the numerator, the integral below,$$\int_0^1 \frac{x\arctan(x)}{1-x^2} \log\left( \frac{1 + x^2 }{2} \right)...
View ArticleStrong-convexity parameter dimension dependency for $1/2$-Tsallis-entropy...
Let $K \in \mathbb{N}$ and let $f:(0,1)^K \to \mathbb{R}$ be the function $x \mapsto - 2 \sum_{k=1}^K \sqrt{x_k}$ (that, apart from constants, it is the $1/2$-Tsallis entropy). I'm trying to figure out...
View ArticleSummable derivative implies absolutely continuous
Theorem 7.21 in 'Real and Complex Analysis' of Rudin says: '' If $f:[a,b]\longrightarrow \mathbb{R}$ is a derivable function and if $f'\in L^1([a,b])$, then $f$ is absolutely continuous ''7.21 Theorem...
View ArticleDirichlet problem for upper half plane
The Dirichlet problem I read is as follows:If $f$ is an integrable function, find a function $u$ such that for $x \in \mathbb{R}, y>0$ \begin{align}u_{xx} + u_{yy} & =0 \\\lim_{y \to 0^+} u(x,y)...
View ArticleSince $\rm{DL_i(n)}$ has an only algorithmic structure, is it possible that...
I have a problem that I am completely stuck on.Let$$f_1(n)^{\rm DL_1(n)}\equiv g_1(n) \!\!\!\!\!\pmod {h_1(n)}$$and$$f_2(n)^{\rm DL_2(n)}\equiv g_2(n) \!\!\!\!\!\pmod {h_2(n)}$$The functions $f_i(n),...
View ArticlePerfect set and binary representation
Consider a perfect set $P$ in $[0,1]$ and look at the elements in binary representation. Consider an element $y= 0.y_1y_2y_3.... \in P$, is it possible to find an $K$ such that for any $k>K$ there...
View ArticleEquivalence of The Cut Property and the Axiom of Completeness
To preface the incoming text wall (that I do apologize for), please note that I have read all of the similar questions and their answers on this topic, and none of them have answered the questions I...
View ArticleEquivalence of Field Norms
This is from Koblitz p-adic Numbers, p-adic Analysis, and Zeta-Functions chapter 1 exercise 5.Suppose $||\cdot||_1$ and $||\cdot||_2$ are equivalent field norms (a sequence is cauchy in $||\cdot||_1$...
View ArticleMeaning of stronger hypotheses
This is from An Introduction to Banach Space Theory, by Robert E. Megginson.Theorem 4.3.6Suppose that $(x_n)$ is a sequence in a Banach space. Then $(x_n)$ is a basic sequence equivalent to the...
View ArticleNonnegative function such that $sup_{n\in\mathbb{N}}...
Can we define a function $f:X\mapsto [0,\infty)$ for $X=\{\frac{1}{n}\mid n\in\mathbb{N}\}\cup\{0\}$ such that $sup_{n\in\mathbb{N}}...
View ArticleCan a continuous nowhere differentiable function be uniformly continuous and...
I came up with the following problems myself(a) Give an example of a continuous nowhere differentiable function $f:\mathbb{R}\to\mathbb{R}$ that is bounded and not uniformly continuous.(b) Give an...
View ArticleClik here to view.
Does $f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin(x/k)$ have infinitely many...
I am investigating the properties of the function $f(x)$ defined for $x \in \mathbb{C}$ by the series:$$f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin\left(\frac{x}{k}\right)$$This function was the subject...
View ArticleConvergence of Fourier series under scaling at the origin
I’m working on the following problem:Let $f$ and $g$ be $2\pi$-periodic integrable functions such that in some neighborhood of $0$ one has$$g(x) = f(a x)$$for a fixed constant $a\neq 0$. Prove that the...
View ArticleIf this operator is an isometry, does that imply it is not compact?
Suppose we have a right-shift operator $T: l^{\infty} \rightarrow l^{\infty}$ given by$$\displaystyle Tx = (0, x_1, x_2, ...)$$for any sequence $x = (x_1, x_2, ...) \in l^{\infty}.$ Then by computing...
View ArticleHow to show that in a normed vector space over $\mathbb{R}$ or $\mathbb{C}$...
What I have:Call $W$ the nonempty closed and open subset of $X$ and let $y\in X\backslash W$.If $\inf\|y-x\|_{x\in W}=0$ then we have a sequence $x_1,x_2,...$ of elements of $W$ such that...
View ArticleConfusion regarding partial sums and the Cauchy series definition.
I'm trying to learn more about series and have recently been introduced to the notion of partial sums and Cauchy series. The problem is that the author keeps writing partial sums without stating which...
View ArticleClik here to view.
total set in the dual space
A set $A$ in a normed space $X$ (respectively, $A \subset X^*$) is called total (resp. $w^*$- total) if $span A$ (:= the linear span of $A$) is dense in $X$ (resp. $w^*$-dense in $X^*$) i.e.Suppose $M$...
View Article