Does the integral converge $\int\limits_{0}^{1}\frac{\cos\left ( t^{-2}...
Investigate the convergence and absolute convergence of the integral at $a\in \mathbb{R}$$$I=\int\limits_{0}^{1}\frac{\cos\left ( t^{-2} \right )}{\left ( 2-t^2\cos\left ( t^{-2} \right ) \right...
View ArticleReal analysis supremum of a set [closed]
Define supremum of a set $S\subset R$ and show that the $\sup S =\frac{1}{2}$where $S=\{\frac{n}{2n+1} \mid n\in N\}$.
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Convergence of the sequence of the orthocenters, incenters and centroids of a...
Now asked on MO here.Given a triangle $A_1B_1C_1$ from the triangle $A_nB_nC_n$ construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is the incenter...
View ArticleWhy do bounded domains with smooth boundary have finitely many connected...
Let $\Omega \subset \mathbb{R}^n$ be a bounded (not necessarily connected) set with $C^2$-boundary $\partial \Omega$. Then $\Omega$ can only have finitely many connected components.
View ArticleQuestion regarding compact and convex sets
Let $A,B\subseteq \mathbb{R}^{n}$ be compact sets, with $A\cap B\neq \emptyset$, such that $A\cup B$ is convex. Also let $L=\{\lambda(t)=(1-t)x+ty \, : \, 0\leq t\leq 1\}$ for $x\in A$ and $y\in B$. Is...
View ArticleProve that $f(x+y)=f(x)+f(y)$, and $f(x\cdot y) = f(x)\cdot f(y)$ implies...
Q: Prove that $f(x+y)=f(x)+f(y)$, and $f(x\cdot y) = f(x)\cdot f(y)$ implies $f:\mathbb{R} \to \mathbb{R}'$ is a bijective mapping that preserves order.Original question ($\S$2.2/23 in Mathematical...
View ArticleA bounded open set that contains infinitely many open sets but not their...
Let $V$ be a bounded open set of $\mathbb{R}^{n}$ ($n>1$). Can there exists an INFINITE sequence of disjoint open sets $(C_{n})$, all included in the closure of $V$ and such that the boundary of...
View Articleintegrals over dyadic intervals implies function is constant
For each natural number $n\geq 1$, consider the dyadic intervals partitioning the unit interval $$I_n^1=[0,1/2^{n}], ~~J_n^1=[1/2^{n},2\cdot 1/2^{n}]$$$$I_n^2=[2\cdot 1/2^{n},3\cdot 1/2^{n}],...
View ArticleQuestion about Caratheodory's Theorem applied on unbounded sets
Caratheodory's Theorem: Suppose $D$ is a bounded simply connected region in the plane and $\partial D$ is a simple closed curve. Let $\phi : D \to \mathbb{D}$ be a conformal equivalence. Then $\phi$...
View ArticleChange the order of integrals..
$$\int_0^1dx\int_0^{1-x}dy\int_0^{x+y}f(x,y,z)dz$$to$$\int dz\int dx \int f(x,y,z) dy$$Not sure if to split the shape into two shapes, or do it directly. Either way i would like to know how its done..
View Articlethe series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly...
To show that the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$.My Try: Let us consider $u_n(x) =\cos(nx), x...
View ArticleDoes square-integrability imply quartic-integrability? [closed]
Consider a smooth function $f$ that is square integrable, i.e.$$\int_{\mathbb{R}} dx \, f^2(x) < \infty$$Does square integrability of $f$ imply integrability of some higher powers of $f$, e.g. does...
View ArticleA differentiable and unbounded function with infinitely many critical points,...
I'm searching for a function $f: \Bbb R^2 \rightarrow \Bbb R$ which has following properties: Both $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist and are continuous in $\Bbb...
View ArticleExplanation for proof of the continuity of the function in $t$ defined as the...
Statement: Let $f:[a,b]\times\mathbb{R} \to \mathbb{R}, (x,t) \mapsto f(x,t)\in\mathbb{R}$ be a continuous function. Then$$F(t) := \int_a^b f(x,t)dx$$is continuous.I understand the proof for this up...
View ArticleEvery divergence-free vector field generated from skew-symmetric matrix
Let $[a_{i,j}(x_1,\ldots,x_n)]$ be a skew-symmetric $n\times n$ matrix of functions $a_{i,j}\in C^\infty(\mathbb{R}^n)$. The vector field $$v=\sum\left(\dfrac{\partial}{\partial...
View ArticleShow that $f(x+y)=f(x)+f(y)$,$f(xy)=f(x)f(y)$ says that $f:\mathbb...
We are asked in Zorich Mathematical Analysis 2.2 to prove the following things:Show that if $(\mathbb{R},+,\cdot,\le)$ and $(\mathbb{R'},+,\cdot,\le)$ are two models of real numbers (i.e. are complete...
View ArticleThe legendre transform reaches the supremum it's defined with.
As is known, the Legendre transform g of a convex function f is defined as :$g(p)=\sup_{x \in \mathbb{R}^n} p(x)-f(x)$With $p\in\mathcal{L}(\mathbb{R}^n,\mathbb{R})$.However, it can be shown that the...
View ArticleQuestion about extension of harmonic function on punctured unit disk
TheoremIf $u: \mathbb{D'} = \mathbb{D} \setminus \{0\} \to \mathbb{R}$ is harmonic and bounded, then $u$ extends to a function harmonic in $\mathbb{D}$.In the next proof $\Pi^+$ is the upper...
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$7.24$, Real and Complex analysis, W. Rudin, Case 2.
These definitions are necessary:There is the theorem:If$(a)$$V$ is open in $R^{k}$.$(b)$$T : V \to R^{k}$ is continuous, and$(c)$$T$ is differentiable at some point $x \in V$, then $$ \lim_{r \to 0}...
View ArticlePrecise characterization of simple functions belong to $L^p$.
Let $(X,\mathcal{M},\mu)$ be a measure space e and let $p\in [1,\infty)$ be a real number. I would like to characterize the simple functions that are in $L^p(X,\mathcal{M},\mu)$.It's easy to prove that...
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