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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12 views

Prove that $F_1$ and $F_2$ are continuous and that $\int_{\gamma_1}F_1(z) dz = \int_{\gamma_2}F_2(z) dw$

Let $\Omega_1, \Omega_2 \subseteq \mathbb{C}$ and let $\gamma_1: [a,b] \to \Omega_1$, $\gamma_2: [c,d] \to \Omega_2$ be paths. Let $f$ be a continuous function defined on $\gamma_1 \times \gamma_2$ ...
3
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0answers
39 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
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0answers
8 views

Basic examples of functions in Hörmander class

The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with $$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq ...
4
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1answer
91 views

Proving convergence for a series.

Let there be given that $\displaystyle\sum_{n=1}^{\infty}a_{n}$ is a convergent series and $a_{i}$ is not negative for every $i\in\mathbb{N}$. And i want to prove that this series is also convergent ...
4
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1answer
127 views

Proving existence of at least one root

The function $f:\mathbb{R}\to\mathbb{R}$, is continuous and $a>0$. How can I prove that there is at least one root of this equation: $f(x)=f(\sqrt{|x^2-a|})$
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2answers
43 views

Finding all complex roots of this equation

So i have this equation: $z^5-4z^4+11z^3+12z^2-42z+52=0 \text{ for }z\in\Bbb{C}$ One root is: $z=1+i$ That gives us also the 2nd root. $z=1-i$ But i am stuck with how to get other 3. I thought i ...
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1answer
31 views

prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$

$\displaystyle\lim_{x \to a}f(x) = \lim_{x \to a}g(x)$ exist and $\displaystyle\lim_{x \to a}(h_1(x)g(x)+h_2(x)f(x))$ exist prove that $\displaystyle\lim_{x \to a}(h_1(x)+h_2(x))f(x)$ exist I would ...
3
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1answer
23 views

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space?

Which subsets of a given real linear space $V$ are open unit balls with respect to some norm on the space? My textbook (Metric Spaces - Michael Searcoid) gives the following hints to answer the above ...
1
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1answer
43 views

How to rigorously establish this limit of sums

Assuming that $$\lim_{n}\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)=\int_{\mathbb{R}} f(u)g(u)\mathsf du,$$ (where $f$ is $C^2$ and $g$ and $g_n$ are probability distribution functions) I ...
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15 views

Representing numbers by quasilexicographic ordered strings, formula for size of conversion between different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
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1answer
20 views

Egorov's theorem and Baire class $1$ function

Suppose $f$ is Baire class 1. Then there exists $f_n$ each one is continuous and that $f_n \to f$. By Egorov's theorem, a measurable $\mu(B) < \varepsilon$, and $(f_n)$ converges to $f$ uniformly ...
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0answers
24 views

Constructing uniform convergence.

Suppose $f_n\rightarrow f$ point wise and each $f_n$ is continuous. Can we construct a sequence of continuous functions $(g_n)$ that converges to $f$ uniformly and each $g_n$ is continuous?
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4answers
74 views

Prove that there is no strictly increasing and surjective function from $\mathbb Q$ into $\mathbb N$

Let $f: \mathbb{Q} \rightarrow \mathbb{N}$ be such that for $x<y$ in $\mathbb{Q}$ one has $f(x) < f(y)$ in $\mathbb{N}$. Prove that $f$ is not surjective. I tried a proof by ...
4
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0answers
25 views

When does interchangibility of limit and Riemann integral imply uniform convergence?

Let $\{f_n\}$ be a sequence of real-valued functions defined on an interval $[a,b]$ such that each $f_n$ is Riemann integrable, $\{f_n\}$ converges point-wise to $f$, $f$ is Riemann integrable and ...
4
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2answers
64 views

$n$-th derivative of $(x^2-1)^n$ has distinct real roots in $[-1,1]$.

For $n=1,2,3,\ldots$, let $$f(x) = (x^2-1)^n .$$ Show that the $n$-th derivative $f^{(n)}$ has distinct real roots in $[-1,1]$. I have no idea about the problem. Could I have a hint?
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2answers
820 views

How can I obtain this division's limit without using derivatives?

$$\lim_{y\to 0} \frac{y}{\cos(\frac{\pi}{2}(1+y))}$$ Can anybody help me? I can use basic properties of limits, and some of those basic known limits. I know it would be easier with derivatives, but i ...
6
votes
4answers
91 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
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1answer
22 views

product of two sequences [on hold]

Let $X$ be a Banach space. Let $C$ be nonempty,closed and convex subset of $X$. Let $x_n$ be a convergent sequence in C and $t_n$ a convergent sequence in $\mathbb{R}^+$. Is it true that $t_nx_n$ ...
0
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1answer
32 views

Explain about proof

Let $0 \leq R_1 \leq R_2 \leq \infty$ and let $f$ be holomorphic in the annulus $R_1 < |z - z_0| < R_2 $. Then, for any $r_1, r_2, z $ such that $R_1 < r_1 <|z-z_0| < r_2 < R_2$, we ...
4
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1answer
25 views

Estimates for parabolic vs elliptic PDE

Elliptic and parabolic PDE share many properties. They each, for example, have an associated maximum principle and their value at any point depends on the entirety of the boundary data. I have been ...
2
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43 views

Variable coefficient wave equation

Consider the equation $$u_{tt} - f(x)^{2}u_{xx} + u_{t} = 0$$ for $(x,t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0) = 0$ and $u_{t}(x,0) = 0$ for all $x \in \mathbb{R}$. Furthermore, suppose ...
0
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2answers
32 views

Proving the last part of Nested interval property implying Axiom of completeness

I took a non-empty set A that is bounded above. And I went on with the regular algorithm, which either gave us a LUB or gave us an infinite chain of nested intervals $I_1$ $\supseteq$ $I_2$ ...
4
votes
2answers
177 views

Compact support vs. vanishing at infinity?

Consider the two sets $$ C_0 = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \lim_{|x|\to \infty} f(x) = 0\}$$ $$ C_c = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous ...
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0answers
20 views

Check / Improve a Proof - Convex functional on convex domain is continuous throughout its interior

I am working through Luenberger's Vector Space Methods for Optimization book. I am working on a corollary left to the reader, and I'm not quite sure that my proof is correct/the sharpest proof out ...
2
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3answers
81 views

$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$

Prove that for all $\xi \in \mathbb{C}$, $$\int_{-\infty}^{\infty}e^{-\pi x^2}\cdot e^{-2\pi ix\xi}dx = e^{\pi\xi^2}$$ I don't really know how to compute this integral. Can you please help me?
0
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1answer
28 views

Characterization of contraction mapping

Let $T$ be a mapping from $\mathbb{R}^n \to \mathbb{R}^n$. Fix $x^\star \in \mathbb{R}^n$, and suppose that the Jacobian matrix of $T(x) $ at $x = x^\star$is symmetric. Then, I know that if all the ...
1
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1answer
37 views

Is this correct for Rudin exercise 3.7? Prove the series is convergent

This is Baby Rudin exercise 7 of Chapter 3. Prove that the convergence of $\sum{a_n}$ implies the convergence of $\sum{\sqrt {a_k} \over k}$ if $a_n \le 0$. Proof: I will attempt to show that the ...
0
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2answers
12 views

On the existence of a particular type of real sequence of functions

Does there exist a sequence of real valued functions $\{f_n\}$ with domain $\mathbb R$ which is uniformly convergent ( on some subset of $\mathbb R$ ) to a continuous function and such that each $f_n$ ...
1
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2answers
24 views

denseness of polynomials in bounded borel measurable functions

Let $K\subseteq \mathbb{R}$ be compact, consider $B(K)$ the set of all bounded borel measurable functions $f:K\to \mathbb{C}$ and endow $B(K)$ with the uniform norm, so you obtain a Banach space. My ...
1
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1answer
23 views

On the projection onto the image set of an $m\times n$ matrix

I came accross as statement that: "If $K$ is the image set of an $m\times n$ matrix $A$ with full column rank, then $$P_Kx=A(A^TA)^{-1}A^Tx."$$ How do I verify this? I know that the inequality ...
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2answers
78 views

Improper rational/trig integral comes out to $\pi/e$ [on hold]

During my studying to integration I find this integration. So I tried to prove but I got stuk. So I need help in this integration. $$\displaystyle\int_{-\infty}^{\infty} \frac{x \sin (x)}{1+x^2} ...
3
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1answer
35 views

the sequence of derivative cannot satisfy $|f^{(n)}(z_0)| > n!n^n$

Let $f: \Omega \to \mathbb{C}$. Prove that for any $z_0 \in \Omega$, the sequence of derivatives cannot satisfy $|f^{(n)}(z_0)| > n!n^n$ In this problem, I intend to prove by contradiction, and I ...
5
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2answers
57 views

A problem related to integration in $L^1$

If $f\in L^1[0, 1]$ and $\int_{0}^1 x^nf(x)=0$ for all $n = 0,1,2,...$then prove that $f$ is identically zero almost everywhere. This would be very easier to prove if $f$ were continuous on $[0, 1]$ ...
0
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0answers
18 views

Sensitivity Analysis: Calculating Allowable increase/Decrease for a Binding Constraint

Let say we have the following equations: Objective Function = $7T+5C$ Contraints $3T + 4C \le 2400$ $2T + C \le 1000$ $C \le 450$ $T\le 100$ How would we calculate the allowable increase and ...
0
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2answers
29 views

Open cover with no finite subcovers for the set [0, ∞)

I am trying to find an open cover with no finite subcovers for the set $[0, \infty)$ I am thinking union from $n=1$ to $\infty$ of the sets $(0,n)$ Does this work or does this give me $(0,\infty)$? ...
3
votes
1answer
89 views

When does $\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$

Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$ I originally wanted to try to prove it by showing $$\lim \prod^N ...
2
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1answer
18 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...
0
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1answer
23 views

Fourier expansion of absolute value of a periodic function

For an arbitrary periodic function p(x), whose period and Fourier expansion might have been known in advance, how can we get the Fourier expansion/coefficients of |p(x)| from them? Or, if possible, ...
4
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2answers
44 views

Why can the elements of $L^\infty$ be approximated in $L^p$ by $C^1$-functions?

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $f\in L^\infty(\Omega)$. From which theorem does the existence of $(f_k)_{k\in\mathbb N}\subseteq C^1(\overline\Omega)$ with ...
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0answers
16 views

f:Lebesgue measurable function ⇆ ∀ε>0, ∃g:continuous function s.t. λ({x|f(x)≠g(x)})<ε

my friend told me this non-obvious prop. I think false,but I can't understand. Does anyone solve this problem?
3
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1answer
49 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
0
votes
1answer
22 views

conditions on Fourier Transform of derivative

At page 445 of Myint-U's Linear Partial Differential Equations (4th Ed), Fourier Tranform of derivative is defined as: Let $f$ be a continuous and piecewise smooth in $(-\infty, \infty)$. Let $f(x)$ ...
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1answer
44 views

Proving an equivalent definition of the $\lim_{x\to a}f(x)$ exists [duplicate]

Prove that the following statements are equivalent. (a) $\lim_{x\to a}f(x)$ exists (b) Given $\epsilon \gt 0$, there is a $\delta \gt 0$ such that if $0\lt |x-a| \lt \delta, 0\lt |y-a| \lt \delta$, ...
5
votes
1answer
59 views

Derivation of Gradshteyn and Ryzhik integral 3.876.1 (in question)

In the Gradshteyn and Ryzhik Table of Integrals, the following integral appears (3.876.1, page 486 in the 8th edition): \begin{equation} \int_0^{\infty} \frac{\sin (p \sqrt{x^2 + a^2})}{\sqrt{x^2 + ...
0
votes
4answers
30 views

what is the subset of $\Bbb{R}^2$ that corresponds to set of complex numbers $z$ such that $|z|\leq 1$ [on hold]

I am trying to understand what is the subset of $\Bbb{R}^2$ that corresponds to the set of complex numbers $z$ such that $|z|\leq 1$ Can you help?
1
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2answers
33 views

Measure space and measurable function

Let $f :\mathbb R\rightarrow \mathbb R$ is a continuous function then the set $\{x \in \mathbb R : \mu ((f^{-1}(x)) >0 \}$ has a zero measure. I think in the case, if f is a step function this ...
0
votes
3answers
42 views

Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
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votes
2answers
26 views

$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [on hold]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
0
votes
3answers
12 views

$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
1
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2answers
43 views

Sum over values of auxiliary function gets arbitrary big, justification

Let $f : \mathbb N_{>0} \to \mathbb R_{\ge 0}$ be a function satisfying $\sum_{n=1}^{\infty} 2^{-f(n)} = \infty$ (like $f(n) = \log n$). Define $$ F(n) = \left\lfloor \log_2\left( \sum_{i=1}^n ...