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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6answers
2k views

smooth functions or continuous

When we say a function is smooth? Is there any difference between smooth function and continuous function? If they are the same, why sometimes we say f is smooth and sometimes f is continuous? Please ...
0
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1answer
246 views

Show that there exists unique real solution

Show that there exists unique real numbers $a$ and $b$ satisfying $$3\sin a-2\cos b=6a-12,$$ $$\cos a + 3\sin b=6b+6.$$ Thank you!
5
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2answers
88 views

Expressing $\frac{d}{dt}\left(\int_{D(t)}u(x,t)dx\right)-\int_{D(t)}u_t(x,t)dx$ as a surface integral?

the following question was the last problem on the Fall 2010 qualifying exam at UCLA. Define $D(t)=\{x^2+y^2\leq r^2(t)\}\subseteq\mathbb{R}^2$ where $r(t)\colon\mathbb{R}\to\mathbb{R}$ is ...
7
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2answers
195 views

Definition of weak solutions from geometrical point of view

Why are weak solutions defined like: A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu:=\operatorname{div}(A\nabla u)+b\cdot\nabla u+cu=f+\operatorname{div}F, \;\text{in } \Omega $$ if ...
3
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2answers
74 views

Differentiability at a point when you actually want in a neighborhood

Let $f$ be a real-valued function on an open interval $I$ containing $c$. If $f$ is differentiable at $c$, and $(x_{n})$ and $(y_{n})$ are sequences in $I$ such that $x_{n}<c<y_{n}$ and ...
2
votes
2answers
549 views

strong maximum principle - harmonic function

Consider the following the theorem in the classical PDE book of Evans( chapter 2 ) : (part of the strong maximum principle) Let $U$ a open set in $R^n$ and $u \in C^2 (U) \cap C(\overline{U})$, with ...
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1answer
734 views

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that ...
2
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0answers
257 views

Bounded Lipschitz Metric on Space of Positive Measures

The bounded Lipschitz metric ($d_{BL}$) metrizes the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ $$d(\mu, \nu) = ...
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0answers
54 views

Showing $\frac{d}{dx}\left(\frac{f(x)}{1 + cf(x)}\right) \rightarrow 0$ as $c \rightarrow \infty$

The problem I am working on is as follows: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuously differentiable, periodic of period 1, and nonnegative. Show that ...
4
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2answers
203 views

Question about finding the norm of a bounded linear operator

Let H be a Hilbert space. Suppose $(i_k)_1^\infty$ is a complete orthonormal sequence in H. Let $a_k \in \mathbb{C}$ for $k \in \mathbb{N}$. Assume there is a bounded linear operator $T:H \rightarrow ...
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1answer
83 views

proving boundedness of a linear functional

Let $f$ be a linear functional on a normed vector space V and $f^{-1}(\{0\})$ is closed. Prove that $f$ is bounded
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1answer
471 views

closed subspace of normed vector space

Is every finite dimensional subspace of a normed vector space closed? If yes, please prove it or else give a counter example.
1
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1answer
131 views

Hahn Banach theorem with no dominating sublinear functional

Let $V$ be a vector space and $M$ be subspace of it. If $f$ is a linear functional on $M$, is it possible to extend it to the whole space $V$? If we have a sublinear functional $p$ on $V$ dominating ...
4
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2answers
118 views

Am I right in my conclusions about these series?

I'm trying to decide if these series converge or diverge: $$\sum_{n=1}^{\infty} (-1)^n \left(\frac{2n + 100 }{3n + 1 }\right)^n $$ Here $\lim_{n\to\infty} \left(\frac{2n + 100 }{3n + 1 }\right)^n ...
4
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2answers
89 views

How to decide convergence or otherwise of these series?

How to decide whether the following three series converge or diverge? $$\sum_{n=2}^{\infty} \frac{(-1)^n}{\sqrt{n} + (-1)^n} $$ By the limit comparison with the divergent series ...
0
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1answer
77 views

how to prove an identity related to $\int_0^\infty\sin(x^{1+a})dx$?

i have made some experiments in maple evaluating the integral $$\int_0^\infty\sin(x^{1+a})dx$$ and the computer give me the following result ...
7
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1answer
177 views

$f''(x)+e^xf(x)=0$ , prove $f(x)$ is bounded

Differentiable function in $\mathbb{R}$ for which $f''(x) + e^x f(x)=0$ for every $x$. Prove that $f(x)$ is bounded as $x \rightarrow +\infty$ I have tried a few stuff but they didnt work out, for ...
1
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1answer
76 views

Why is the case $1/2<\rho\leq 1$ trivial in proving the following inequality?

I'm studying Elliptic Partial Differential Equations by Q. Han and F. Lin. In Lemma 1.41 is given the elliptic equation $D_j(a_{ij}D_i u)=0$ where the coefficient matrix $(a_{ij})$ is constant ...
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1answer
33 views

Example of $\{b_{n}\}$ such that $\sum_{n = 1}^{\infty}b_{n}b_{n + 1} < \infty$ but $\sum_{n = 1}^{\infty}(b_{n + 1} - b_{n})^{2} = \infty$

What is an example of a sequence of positive numbers $\{b_{n}\}$ such that $\sum_{n = 1}^{\infty}b_{n}b_{n + 1} < \infty$ but $\sum_{n = 1}^{\infty}(b_{n + 1} - b_{n})^{2} = \infty$?
7
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1answer
152 views

Integrals of matrix functions

I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ...
6
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3answers
492 views

The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is ...
2
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0answers
182 views

Gagliardo Nirenberg Sobolev inequality

Assume that $f$ satisfies the equality in the Gagliardo Nirenberg Sobolev inequality for the best constant. What can be said about $f$?
0
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1answer
93 views

Upper semicontinuity , a analysis problem.

I have a upper semicontinuous function, $L:X\to\mathbb{R}$, and exists $x\in X$ with the following property: There exists a neighborhood $U$ of $x$ such that for every $\delta>0$, there exists ...
4
votes
5answers
197 views

Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded.

Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded. I have been thinking about this. And I have come to the ...
1
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1answer
390 views

Runge-Kutta method for multiple springs

If we have a spring attached to a wall with an object on the other side, the differential equations describing the system are: $$x'=v$$ $$v'=-\frac{k}{m}x-\frac{b}{m}v$$ Where: x is position of the ...
2
votes
1answer
122 views

Problem of proofs

I've been away from math for a long time ,and while I was trying to relearn it using Courant and Fritz 's booknon calculus,I loved the explanations but I couldn't solve any exercices(they're almost ...
2
votes
1answer
37 views

Question about putting an upper bound on a particular operator

So according to Wikipedia, given V(f)(t) = $\int_0^t f(s)ds$ where $f(s) \in L^2 (0,1)$ and $t \in (0,1)$. They say that $||V|| = \frac{2}{\pi}$ and I have seen the proof of this on a MSE post. The ...
3
votes
3answers
164 views

The set $A=\{(x, y)\in \mathbb{R}^2:|x|=|y|\}$ is connected

Two disjoint sets $A$ and $B$, neither empty, are said to be mutually separated if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is ...
6
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3answers
733 views

Uncountable set with exactly one limit point

Is there any uncountable subset of $\mathbb{C}$ with exactly one limit point?
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1answer
66 views

Optimisation Problem about convex curve

A smooth closed curve C is said to be convex if it lies wholly to one side of each tangent . Show that for the triangle of minimum area circumscribed about C that each side is tangent to C at its ...
2
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1answer
172 views

theorem 1 chapter 2 - Evans PDE

My doubt is about the proof of the theorem 1 section 2.2.1 of the evans pde classic book. My doubt: Consider the function $$\Phi(x) = \begin{cases} - \frac{1}{2 \pi} \log |x| & \text{if $n= ...
2
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1answer
68 views

Convergence in $L^P(E)$ implying convergence in measure requires $m(E) < \infty$?

It's a common analysis problem to show that if $m(E) < \infty$ to show that convergence in $L^P(E)$ implies convergence in measure. However, I don't see where the necessity of $m(E) < \infty$ ...
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2answers
86 views

Compactness of the set $(0, 1)$ as a subset of $\mathbb{R}^2$?

An open ball in $\mathbb{R}^2$, centered at the point $(1/2, 0)$ and of radius $1/2$ covers the segment $(0,1)$. The open ball thus forms a finite cover of $(0,1)$, implying that $(0,1)$ is a compact ...
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2answers
101 views

Continuity on a neighbourhood of a point

If a function $f$ is continuous at $x$, then there exists a neighbourhood of $x$, on which $f$ is continuous. Is it true of false?
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1answer
130 views

Convex functions and Hahn-Banach application

Let $Z$ be a convex subset of a real vector space, and $f:Z \to \mathbb{R}^m$ be such that every component $f_i:Z \to \mathbb{R}$ is a convex function. Let $S:\mathbb{R}^m \to \mathbb{R}$ be defined ...
2
votes
4answers
77 views

How can I prove that such sequence exists?

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x)=x\sin(x)$. Prove that, for all c $\in \mathbb{R}$, there is a sequence $x_n$ $\in \mathbb{R}$ with $\displaystyle \lim_{n \to \infty}x_n = ...
6
votes
1answer
108 views

Describe all holomorphic functions.

Problem: Describe the class of all holomorphic functions on $\mathbb{C}-\{0\}$ such that $$\sup_{(x,y)\neq (0,0)}\frac{|f(x+iy)|}{|\log(x^2+y^2)|}<\infty.$$ Attempt at a solution: Let $z=x+iy$, ...
0
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0answers
110 views

What does this notation regarding integral equation kernels and norms mean?

I am attempting to understand what types of kernels the standard theory of Fredholm Type-2 integral equations applies to, but I've never taken a course in analysis. Basically, given a kernel, ...
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0answers
67 views

Daniell approach to $\int \sigma \mbox{ } d\mu \geq 0$ clearification

Can someone help me clarify the following proof. Theorem: If $\sigma \in C_0$ and $\sigma \geq 0$ a.e.$(\mu)$, then $\int \sigma \mbox{ } d\mu \geq 0$ Proof: The inequality is equivalent to proving ...
1
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1answer
64 views

det function in concave

Let $f(A)=(\det(A))^{\frac{1}{n}}$. And assume domain of $f$ is space of positive semi definite symmetric $n\times n$ matrices with real entries. Show that $f$ is concave: $$f((1-t)A+tB)) \ge ...
2
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0answers
42 views

Why is the partial derivative of this fuction locally bounded?

We have a function for $x_i, t_i >0$ $$|f(x_1,t_1)-f(x_0,t_0)| \leq C (|t_1-t_0|^{1/2} + |x_1-x_0|)$$ Why does this mean $f_t$ is locally bounded? $f$ is non-increasing and convex in $x$. $f$ is ...
0
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1answer
337 views

How to integrate over an arbitrarily positioned spherical cap in spherical coordinates

If you want to integrate over the SURFACE of a spherical cap that is positioned in the way it is on wikipedia, this is rather simple. since it has azimuthal symmetry you get a factor $2\pi$ and for ...
5
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2answers
131 views

Question about a particular linear operator

Let A be a linear operator. $A: L^2(0,1) \rightarrow L^2(0,1)$ given by $Ag(a) = \int_0^a(a-x)g(x)dx$ where $a \in (0,1)$. This is the integral operator, and we know ||A|| < 1 which is easy to ...
5
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2answers
222 views

Is there any value of zeta that is an integer?

Is there any value which we can substitute for $s$ in $\zeta (s)$ such that $$\sum_{n=1}^{\infty }n^{-s}\in \mathbb{Z}$$
1
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1answer
346 views

Show that the Fourier transform of a radial function $ L^1 (\mathbb{R}) $ is also radial

How do I prove that the Fourier transform of a radial function $ f \in L^1 (\mathbb{R}) $ is also radial function? I tried by polar coordinates but I dont got.
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1answer
140 views

Calculus prequisites book

Everytime I try read a calculus textbook I find that my books (serge lang and gelfand's )didn't cover a subject well (like say minimum of a quadratic polynomial) ...I need a recommendation for a ...
2
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2answers
215 views

Do integral with Legendre Polynomials

Is it possible to integrate this analytically: $$ \int_{0}^{2\pi} P_l(\cos(\theta-\theta')) P_l(\cos(\theta)) \sin(\theta) d\theta$$ I mean the integral would be pretty easy by using the ...
0
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2answers
114 views

Is the function of two strictly concave functions also concave?

This may be a trivial question to most, but here we go: I have two strictly concave functions, say $f(x)$ and $g(x)$. From this can I say that a function of those two functions, $h[f(x), g(x)]$, is ...
2
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2answers
103 views

Is $H_0^1([a,b]) \subset C([a,b],\mathbb{R})$?

i have a small question : how to see that $H_0^1([a,b])\subset C([a,b],\mathbb{R})$? Please Thank you
4
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1answer
146 views

Why they use the word “abstract” for naming mathematical fields of study?

I've found some books with titles such as abstract analysis - but I don't understand why they choose such word. For me it seems quite vague and perhaps misleading - consider the examples: Real ...