Are affine functions strongly quasiconvex?
Let $C \subset \mathbb R^2$ be such that $C$ only contains pairs of the form $(p,1-p)$, $p\in [0,1]$. Clearly $C$ is convex and closed. Let $v \in V$, and define$$f(u) = \langle v,u\rangle.$$Let...
View Article$\int_{[0,t]} |f(s)| \, ds \leq \int_{[0,\infty]}e^{M(t-s)}...
For any Lebesgue measurable function $f$, and $M>0$, if $\operatorname{sup}_{s\geq 0}|f(s)|e^{-Ms}<\infty$. I wonder if the following inequality hold $$\int_{[0,t]} |f(s)| \, ds \leq...
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Proving legendre transform exists locally if $f$ twice continuously...
BackgroundsLegendre transform is defined as follows:And the problem I am trying to solve follows:My thoughtsI searched up to acquired that the matrix concerned is a hessian matrix.Also, the previous...
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What is the probability that the absolute value of the roots of a polynomial...
Note: Posted in MO since it is unanswered in MSE.Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we can divide each...
View ArticleUniform Convergence and ODE
The following problem appeared on a past exam at my institution:Suppose that for each $n\in\mathbb{N}$, $u_n:\mathbb{R}\rightarrow\mathbb{R}$ is a differentiable function satisfying...
View ArticleIs $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ true for non-analytic smooth...
Does $f(a+b\epsilon)=f(a)+b\epsilon f'(a)$ remain true for non-analytic smooth functions of dual number argument? Where $ \epsilon^2=0$, $\epsilon \neq 0$ and $a,b \in \mathbb R$. I found proofs based...
View Article2 Commuting, Continuous mappings from a closed interval onto itself doesn't...
The question is as such:If two continuous mappings $f$ and $g$ of a closed interval into itself commute, that is, $f\circ g=g\circ f$, then they do not always have a common fixed point.-- Zorich...
View ArticleContinuous mapping on a compact metric space is uniformly continuous
I am struggling with this question:Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$. Thanks for your...
View ArticleUniformly equicontinuous
Let $X$ and $Y$ be metric space and $a \in X$. A family $A$ of functions from $X$ to $Y$ is said to be equicontinuous at $a$ if for any $\epsilon >0$ there exists a $\delta >0$ such that...
View ArticleProving the Unit Sphere without the North Pole is Homeomorphic to the Plane
I'm having trouble proving that the real plane and unit sphere with the north pole removed are homeomorphic.Even considering the function that maps from the sphere to the plane, I can't seem to...
View ArticleWhat Lipschitz functions can be represented as an accumalating integral?
In class I saw that for a Riemann integrable function $f:[a, b] \to \Bbb R$, the function $F(x) = \int_a^x f(t)dt $ is continuous and is even Lipschitz (There is a constant $M>0$ such that for all...
View ArticleIs it possible to show the convexity of $f(x)$ by removing one of the hypotheses
I just solved an exercises which stated to show $f$ is strictly convex in $(-1, 1)$, knowing that $f: (-1, 1) \to \mathbb{R}$ is: continuous, positive and at least twice differentiable everywhere. Also...
View ArticleSkipping CDF for sampling
i am writing a thesis and am sadly stuck...I am trying to sample from a distribution of the form $$a*\exp(-2 \pi^2 x^2)(x^{d-1})$$Now my instinct was to "simply" calculate the CDF and sample that way,...
View ArticleReferences for multivariable calculus
Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with...
View ArticleHow is the integral of a simple function well-defined in Folland?
I am reading Folland's real analysis text, section 2.2 on integration of nonnegative functions. I am stuck at the definition of the integral of a simple function and how to show it is well-defined....
View ArticleOn the Convexity and Concavity of a Function
Let $p\in [1,\infty)$. Define $f:[0,\infty)\to\mathbb{R}$ as$$f(t):=\left(1+t^{\frac{1}{p}}\right)^p+\left|1-t^{\frac{1}{p}}\right|^p,\text{ for all }t\in [0,\infty).$$Prove that, $f$ is convex when...
View ArticleReal Analysis Limit Problem with Continuity and Existence of Limit [duplicate]
I was working on this problem and I cannot manage to solve it.Let $f : \mathbb{R} \to \mathbb{R}$ be continuous. For all $\alpha > 0$, the sequence $f(\alpha), f(2\alpha), f(3\alpha), \ldots \to 0$....
View ArticleProve that $f'(0) = 0$ for Function Inequality Within Absolute Values [closed]
Assume $f$ is a function that is differentiable around $x=0$ and that:$$\ |f(\frac{x}{5})| \leq \frac{1}{8}|f(x)|$$Show that $f'(0) = 0$Unsure how prove this?
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producing a formula that encompasses the idea of range partition for...
I'm an IB student who currently just finished Math 31 HL (calculus grade 12). For an Internal assessment math paper(due in 2 days) I wanted to try and derive a formula that works like the Riemann summ...
View ArticleIf $A$ and $B$ are continuous linear operators from $F$ into $E$, $|A-B|
Let $E$ be a Banach space and $F$ a normed space. Assume that $A$ and $B$ are continuous linear operators from $F$ into $E$, $|A-B|<1 $, and $A$ is invertible. Then I want to show that $B$ is...
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