continuous points of increasing function which has non-negative increments
I'm struggling to show that an increasing and right-continuous function $F:\mathbb{R}^d\to [0,1]$ whic has non-negative increments doesn't have "too many discontinuous points".Details are below:Let...
View ArticleA global implicit function theorem on a minimisation problem
Let $g(x,y)\in C^1(\mathbb{R}^n\times\mathbb{R}^n,\mathbb{R})$ such that the least eigenvalue of $\nabla_{yy} g(x,y)$ is always greater than $1$, $\nabla_{yy} g(x,y)$ is invertible and $\nabla_{yy}...
View ArticleSemianalytic vs Analytic Partition
I am trying to figure out the difference between a semianalytic and an analytic partion.My definition is as follow:Let $U \subset \mathbb{R}^D$Let $h=\{h_1, \dotsi , h_n\}$ with $h_i$ real-valued...
View ArticleDefinition of the periodic $L^p$ space on torus
In his Real Analysis, Folland uses the notation $L^p({\mathbb T}^n)$ (where $\mathbb{T}^n$ denotes the n-dimensional torus) is used before Hausdorff measure is introduced. (See for instance Chapter 8:...
View Articlespectrum of weighted sum of projection operators
Let $P_i$ be a collection of projection operators on some Hilbert space $H$, so that $P_iP_j=0$, and so that $\sum_i P_i=Id$, where $Id$ is the identity operator. Assume that there exists a bounded...
View ArticleCovering interval by prescribed open sets.
Consider $I=[0,1]$. Assume a priori we are given a collection of open sets $(U_{t})_{t\in [0,1]}$, $U_t$ open in $I$ such that $I=\bigcup_{t\in I}U_t$. Here, $U_t\owns t$.Problem is: Show that there...
View ArticleProof that $\frac{1}{2}[e^{ix}+e^{-ix}]$ has a zero? [closed]
In Rory's answer to this question, it is stated that"He then derives all the usual properties (both trigonometric and analytic) of sine and cosine in just two pages. All his proofs are simple and...
View ArticleTheory real analysis 2 [closed]
We say a uniformly continuous function is uniformly continuous: state this more precisely and prove it. If we let $E$ be dense subset of a metric space $X$, and if we let $f : E \to Y$ be uniformly...
View ArticleQuestion on coordinate dependencies in transition map
This may be a stupid question. Suppose some open set $U$ on a manifold is described by local coordinates $(x,y)$. Suppose furthermore there is another coordinate chart $(\tilde{x}, \tilde{y})$ such...
View ArticleCan I simplify multiple inequality cases into a single absolute inequality?
I'm not an expert and I haven't delt with these kinds of inequalities for a while, so I'm hoping someone can explain if my thought process is reliable and will suffice to prove the results I'm looking...
View Articlehow to show that $1/(1+x^2)$ is a contraction?
I am trying to show that $1/(1+x^2)$ is a contraction but I cannot find the contraction factor. So far I got: $\lvert...
View ArticleMaking Formal Substitution to a Complex Power Series
This problem asks me to:"Making the formal substitution $z-a=(z-z_0)+(z_0-a)$' in the power series $\sum_{i=0}^{\infty}A_n(z-a)^n$ and gathering like terms, obtain a series...
View ArticleEvaluate $\lim_{n\to \infty} \sum_{k=0}^n \frac{\sqrt {kn}}{n}$
I'm not sure which would be the best way to compute this limit. As you might have observed, if you expand the infinite sum and rearrange some terms you get:$$\lim_{n\to \infty}\frac{\sqrt...
View ArticleContinuity of a map between $\ell_1\times \ell_1$ and $\mathbf R$
Let $f\colon \ell_1\to \mathbf R$ be defined by $f((a_n)) = \sum_{n\geq 1} \frac{g(na_n)}{n}$, where $g\colon\mathbf R\to \mathbf R$ is defined by $g(x) = \mathrm{ln}(1+x^2)$.I am trying to show that...
View ArticleGlobal optimum in 2D
Suppose a function $f(x,y)$ is defined and differentiable anywhere in $\mathbb{R}^2$. Suppose also $f(0,0)$ is a local maximum with second-derivative test (namely, locally concave down, or,...
View ArticleThe variation of an absolutely continuous function that is of finite...
I want to prove the following:Proposition$\quad$If $F:\mathbb{R}\to\mathbb{R}$ is absolutely continuous and of finite variation, then $V_F$ is absolutely continuous.Here is my attempt so far:Let...
View ArticleCredible reference for concave functions?
I'm working on programming models stemming from convex analysis ideas for various data-sets.The functions I'm working with tend to be downwawrd facing cones, which is to say if you imagine for instance...
View ArticleDo the set of all integers have the least upper bound property?
I was thinking about this and it seems to me the answer is no, but a book I was reading says that the majorum of any set of integers is an element of that set.And if that's the case does it mean that...
View ArticleConditions on real function $f$ for $\int_{\mathbb{R}} [f(x+t)-f(x)]dx=t$ for...
Suppose $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)\to 0$ as $x\to-\infty$, $f(x)\to 1$ as $x\to \infty$, and $$g(t):=\int_{\mathbb{R}} [f(x+t)-f(x)]~dx$$ exists for all real $t$. Does it follow that...
View ArticleHow to make a common operation function that will convert 011 to 1010 and 000...
For a problem. Very tricky!There are 12 operations: a. Addition of 1 from the left (Example: 000 -> 1000) b. Addition of 0 from the left (Example: 111 -> 0111) c. Addition of 1 from the...
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