Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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130 views

Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz.

Let $U$ and $V$ be normed linear spaces over $\mathbb{R}$, and $L : U \mapsto V$ a linear function. Prove that if $L(S)$ is bounded, where $S$ is the unit sphere of $U$, then $L$ is Lipschitz. There ...
13
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5answers
1k views

Must a monotone function have a monotone derivative?

If a function is differentiable and monotone on the interval $(a, b)$, then its derivative is also monotone on $(a, b)$. How do you prove this statement is wrong? Can you please provide an ...
3
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1answer
136 views

Does there exist a differentiable function $\Bbb R^2$ to $\Bbb R$ with certain partial derivative properties?

Does there exist a counter-example to the following claim: For a function $f: \mathbb{R}^2 \to \mathbb{R}$, if: $D_1f$ exists in some ball around the origin and is continuous at the origin (but not ...
13
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1answer
200 views

If $\lambda_n \sim \mu_n$, is it true that $\sum \exp(-\lambda_n x) \sim \sum \exp(-\mu_n x)$ as $x \to 0$?

If $\lambda_n,\mu_n \in \mathbb{R}$, $\lambda_n \sim \mu_n$ as $n \to +\infty$, and $\mu_n \to +\infty$ as $n \to +\infty$, is it true that $$ \sum_{n=1}^{\infty} \exp(-\lambda_n x) \sim ...
2
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2answers
201 views

Proof of Integral and Limes

I need an idea for the following statement: $f: [a,b]\to\mathbb{R}$ continuous and $x\in (a,b)$ $\implies$ $\lim\limits_{h \rightarrow 0}{\frac{1}{2h} \int_{x-h}^{x+h} f(y) \, dy = f(x) }$ I just ...
2
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2answers
341 views

Open Balls and Continuity

When we define continuity using open balls, we define $$\forall \epsilon >0, \exists\delta>0:f(B_\delta(a))\subset B_\epsilon(f(a))$$ Let us consider everything unspecified to be in ...
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1answer
106 views

Simplifying an integral by introducing an additional parameter.

This is one of my homework tasks this week. Calculate the integral $$I = \int_0^\infty dx\ x^3 e^{-x}$$ by introducing an additional parameter $\lambda$ and rewriting the exponential function as ...
5
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1answer
581 views

Difference between Heine-Borel Theorem and Bolzano-Weierstrass Theorem

It's a very basic (may be a trivial) question but what is the exact difference, if any, between Heine Borel Theorem and Bolzano Weierstrass Theorem. It is true that one (Heine Borel) can be proved ...
2
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1answer
80 views

Mathematical Analysis integral inequality

Let $f$ be a nonnegative continuous function on $[0,1]$ , and $f$ nondecrease. Then for any $0<\alpha<\beta<1$ , we have ...
2
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2answers
102 views

A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other

The functionals $$ \phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t $$ define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$. a) Show that $(\phi_n)$ converges ...
2
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1answer
69 views

Integral. $\int_{0}^{s}{\lambda \cdot |t|^{q-2}\cdot t\text{dt}}$

How can I calculate the following integral : $$\int_{0}^{s}{\lambda \cdot |t|^{q-2}\cdot t\text{dt}}$$ $q \geq6$, $\lambda >0$ and $ s \in \mathbb{R}$. It's hard for me to calcualte this ...
11
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1answer
92 views

Existence of a specific reordering bijection

Please consider a bijection $g:\mathbb{N}\rightarrow\mathbb{N}$ with following properties: For all real series $(a_n)_{n\geq1}$, convergence of $\sum_{n=1}^{\infty}a_n$ implies convergence of ...
3
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1answer
138 views

Ultrafilters and measurability

Consider a compact metric space $X$, the sigma-algebra of the boreleans of $X$, a sequence of measurable maps $f_n: X \to\Bbb R$ and an ultrafilter $U$. Take, for each $x \in X$, the $U$-limit, say ...
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0answers
74 views

Spectrum of laplacian in a parallelogram

Is the spectrum of the laplacian on an arbitrary parallelogram with dirichlet boundary conditions known?
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5answers
1k views

Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
4
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3answers
230 views

Given a real valued $C^1$ function $f$, show there exists a continuous vector-valued function $F$ with $f(X) = X \cdot F(X)$

Assume $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ is a function with continuous first order partial derivatives such that $f(0)=0$. Show there exists a continuous function $F:\mathbb{R}^{n}\rightarrow ...
4
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3answers
197 views

$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$ c0nvergence

$$\sum_{n=2}^\infty \frac{1}{(\ln\, n)^2}$$ The series converge? Please verify my solution below
2
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1answer
86 views

Find data to perform regression analysis

I'm trying to find some data (two continuous variables that I believe are correlated) online for which I can perform a regression anaylsis, my assignment sheet says: The data may be found anywhere ...
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1answer
25 views

Sketching Semi-Cubical Graphs

I'm having trouble sketching these kind of graphs, for example, I have the equation: $$(1-y_2^2)^3=\frac{9}{4}y_1^2$$ Note: the map is over the real numbers. Could somebody please give me some ...
2
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0answers
45 views

Determinant of Jacobi of $C^3$ function

If I have a function $f:\mathbb{R}^n \to \mathbb{R}^n$ that is $C^3$, then the elements of the Jacobian matrix is $C^2.$ Is the determinant of the Jacobian matrix also $C^2$?
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1answer
199 views

Permitted value of epsilon in Discrete Metric Space

If we define a Metric Space with Discrete Metric, say $(\mathbb{R},d)$. Then whenever we talk about epsilon or delta, such as talking about neighbouringhood, limitpoint, can we take epsilon to be not ...
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0answers
88 views

Compute significance of Kendall tau-b?

I have so-far tried all ways of computing kendall tau significance (where there are ties) described here. However, none of them works good, even for relatively large vectors. I think the problem is ...
2
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1answer
56 views

Question about a theorem of integration

In Darboux integration, If $f$ is a bounded function on $[a,b]$, then $L(f)\le U(f)$ where $L(f),U(f)$ are the lower and upper Darboux integral. My question is why it requires $f$ to be bounded?
4
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1answer
165 views

Is the zero set of a non zero real valued harmonic function discrete?

It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete. By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real ...
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2answers
1k views

Triangle Inequality for supremum metric

Edited Heavily Here all functions are from $[0,1]$ to $\mathbb{R}$ and are bounded. Prove the following Triangle inequality in following case: ...
2
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4answers
141 views

Understanding Set Notation

I'm having some trouble understanding a definition and explanation in my textbook Introduction to Analysis by Edward Gaughan 5th edition. The book begins with some preliminary information about sets ...
1
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2answers
107 views

How to express this convolution by the sum of integrals

If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,1]\\ f_{2}\left(x\right), & x\in[1,\sqrt{5}]\\ 0, & \mbox{elsewhere} \end{cases}$$ what does the piecewise-defined function ...
2
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1answer
50 views

Multivariable Jacobian Problem: tranpose of the Jacobian times $f(x)$ is zero

Here's a little problem I ran across and could use a hint or two on: Let $U \subset \mathfrak{R}^{n}$ be open, $f:U \rightarrow \mathfrak{R}^{m}$ differential on $U$ and satisfying $\|f(X)\|=1$ on ...
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2answers
229 views

Why trigonometric polynomials form an inner product space.

Trigonometric polynomial is defined as a function $$ f(x)=\sum_{n=1}^ka_ne^{i\lambda_nx}$$ for some positive integer $k$, complex coefficients $\{a_n\}$ and real coefficients $\{\lambda_n\}$. Let TP ...
4
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3answers
213 views

The value of $(2^n+3^n+4^n)^{1/n}$ as $n \rightarrow \infty?$

I was thinking about the following problem: How can i find the value of $(2^n+3^n+4^n)^{1/n}$ as $n \rightarrow \infty?$ Can someone point me in the right direction? Thanks in advance for your time. ...
2
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1answer
83 views

Is there a homeomorphism from the space of sequences to [0,1]?

If I consider the norm for the space of sequences of digits {0-9} to mimic the norm for real numbers. $|\left\{x_n\right\}| = \sum_{n=1}^{\infty} \frac{x_n}{10^n}$ shouldn't I now have a space ...
2
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1answer
120 views

uniform continuity of $f$ on $(a,b) \implies \exists \lim_{x \to a^+} f(x)$?

Does $\lim_{x \to a^+} f(x)$ necessarily exist for all $f$ uniformly continuous on $(a,b)$? This has no relation whatsoever to homework. It occurred to me after looking at another question (which ...
2
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1answer
44 views

Invertible idempotent in a C-star algebra question

Let $J$ be an idempotent element in a unital $C^*$ algebra. Why is $I+(J-J^*)(J^*-J)$ invertible? I have been trying to show that $\|(J-J^*)(J^*-J)\|<1$, but I could not do this.
2
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1answer
294 views

Similar proof of Peano's Existence Theorem

As many of you will know, Peano's theorem states that if $f(x,y)$ is continuous and bounded in the strip $T: |x-x_0| \le a, |y|\le\infty $. Then the intitial value problem $y'=f(x,y), y(x_0)=y_0$, ...
2
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1answer
68 views

How to prove that an algebraic structure can be embedded into another?

What is the general method to prove that an algebraic structure can be embedded into another? If this is too general a question, how can I prove that $\mathbb{Z}$, or $\mathbb{Q}$ can be embedded in ...
2
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0answers
126 views

Lipschitz Vectors

I am trying to understand why a vector valued function where it is Lipschitz in each dimension, is also lipschitz. In particular, I have a probability density function $p(x)= \prod p_{j}(x_{j})$ ...
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2answers
2k views

Illustration Proof that every sequence of real numbers has monotone subsequence

While proving every sequence of real numbers has a monotone subsequence, we take two cases, either there are infinitely many "peaks" or else "finitely many" peaks. However, I am unable to grasp from ...
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1answer
80 views

Analytic extension in several variables

Let $D \subset \mathbb{C}^{n}$ be an open domain and $K \subset D$ be a compact subset such that $D - K$ is simply connected. Let furthermore $f$ be a analytic function defined on $D-K$. Is it ...
4
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1answer
487 views

Metrizability of weak convergence by the bounded Lipschitz metric

Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = ...
2
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2answers
376 views

Let $X$ and $Y$ be Banach spaces, show that if they are isomorphic, then $X$ is reflexive iff $Y$ is reflexive.

I want to show that if $X$ and $Y$ are two Banach spaces, and $T : X \to Y$ is an isomorphism, then $$ X \textrm{ reflexive} \iff Y \textrm{ reflexive}. $$ I saw several proofs, but I cannot ...
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1answer
163 views

How to prove $ \sum_{n=1}^{\infty}\left|\frac{a_{1}+\cdots+a_{n}}{n}\right|^{p}\leq\left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}|a_{n}|^{p} $

Let define $(a_n)_{n\geq1}$ as real series. Prove, that $$ \sum_{n=1}^{\infty}\left|\frac{a_{1}+\cdots+a_{n}}{n}\right|^{p}\leq\left(\frac{p}{p-1}\right)^{p}\sum_{n=1}^{\infty}|a_{n}|^{p} $$ (*) ...
5
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1answer
290 views

Complex infinite sum convergence problem

Suppose that the complex infinite sum $ \sum_{n=1}^{\infty}(-1)^{n}Z_n$ converges. Define $A \subset \mathbb{C}$ by $A=\{{z\in\mathbb{C}\mid\exists ...
2
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2answers
90 views

Complex function inequality

Let $\lambda \in (0, 1)$ . Suppose, that function $f : \mathbb{C} \rightarrow \mathbb{C}$ satisfy the inequality $|f(u) - f(v)| \leq \lambda|u-v|$ Prove, that for all $a \in \mathbb{C}$ $z = f(z) + ...
2
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1answer
158 views

Convergence test with polynomials

Let $P, Q \colon \mathbb{R} \rightarrow \mathbb{R}$ are polynomials, and $Q(n) \neq 0$ for $n\in\mathbb{N}$ Suppose, that $\deg (P) < \deg (Q)$. Prove, that infinite sum ...
3
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1answer
101 views

Index of a Fredholm Operator on Paths

I'm a novice to analysis but I need to understand the following example. Any help would be greatly appreciated. This might be of interest to some because it gives a way of quantifying changes in ...
2
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1answer
181 views

A question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis

I have a question on Corollary of Lusin's Theorem in Rudin's Real and Complex analysis (3rd edition, page 56). Here Rudin explicitly requires that $|f| \leq 1$. But I can not see why this requirement ...
5
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0answers
144 views

Linear algebra estimates

Here is something that has been troubleing me lately. I don't know if it is true of not. I suspect it is. Suppose that $A,B$ are two $n \times n$ matrices with complex entries. $A^t = A$, $\bar B^t = ...
4
votes
1answer
103 views

Trading localisation for regularity

When reading about Schrödinger's fundamental solution in 1D, $$u(t,x)=\frac{1}{\sqrt{4\pi it}} \int_\mathbb{R} u_0(y) e^{\frac{i(x-y)^2}{4t}}dy$$ the author says thus Schrödinger evolution is ...
3
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1answer
154 views

calculating a multivariate integral via level sets

I'm considering the possibility of calculating an integral of the form $\int_{S_n} f(x_1,\dots,x_n) dx_1\dots dx_n$ via level sets, where $S_n$ is the domain of integration. In my problem everything ...
2
votes
1answer
173 views

Proving a theorem about continuity property with contradiction.

Proving that if $f$ is continuous on an interval $I$, any $a,b\in I,a<b$ and for any $y$,where $f(a) < y < f(b)$, there exist a $x\in (a,b)$ s.t $f(x)=y$. I have seen a prove using the ...