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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1answer
22 views

integrate over a contour

Please help me with this: Find the value of $\displaystyle\int_{\gamma}\frac{e^z}{z-Logz}dz$, $\gamma$ is the positively oriented contour consisting of four vertices at $2, 4, 4+3i, 2+3i$
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0answers
28 views

Comparison of the remainder of a series and its general term

Let $\sum\limits_{n = 0}^{+ \infty} u_{n}$ be a convergent series, such that $\forall n , u_{n} > 0$ My question is : Under which conditions can we find a constant $C > 0$ such that $$\forall ...
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3answers
87 views

Multivariable limit $\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^2}{|x|^3+|y|^3}=0$

I need to prove $$\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^2}{|x|^3+|y|^3}=0$$ I sort of know how to do it using polar coordinates, but I was trying to find an upper bound. Any ideas? I also wouldn't ...
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4answers
4k views

Finding one sided limits algebraically

I was wondering what the best method was for proving this limit algebraically: $$\lim_{x \to 1}\frac{3x^4-8x^3+5}{x^3-x^2-x+1}$$ I know the answer to this question is ; $$\lim_{x \to 1^+}\frac{3x^4-8x^...
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1answer
115 views

Boolean Closure and Borel sets

Denote the boolean closure of a family of sets $\mathcal S$ by $\mathcal B(\mathcal F)$, then in a metric space it is well known that $\mathcal B(\mathcal F) = \mathcal B(\mathcal G) = \mathcal B(\...
2
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2answers
286 views

What is the difference between diffeomorphism and isomorphism?

What is the difference between diffeomorphisms and isomorphisms? I know isomorphisms already from my abstract algebra/group theory course, and now I'm studying analysis on (sub)manifolds, where this ...
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1answer
44 views

Is this series G(1/n) convergent or divergent given G(x)?

Suppose $G(x)=\int_0^x\sin{\left(e^s-1\right)}ds$ Does the series $\sum_{n=1}^{\infty}G(\frac{1}{n})$ converge or diverge? I'm not sure how to go about solving this; however in our notes it says ...
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2answers
23 views

Show that for any $s \in S$ where $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}$, we have $d|s$ where $d$ is the smallest element

Explain why the set $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}, s = ma+ nb $} has a smallest element, call it d, so we know there exists $x,y \in \mathbb{Z}$ such that $d = ax + by$, and ...
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5answers
113 views

How to find $\lim_{x\to\infty}{\frac{e^x}{x^a}}$?

$$\lim_{x\to\infty}{\frac{e^x}{x^a}}$$ For $a\in \Bbb R,$ find this limit. I would say for $a\ge 0$ lim is equal to $\lim_{x\to\infty}{\frac{e^x}{a!x^0}=\infty}$ (from L'Hopital). For $a<0$, lim ...
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0answers
42 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
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1answer
77 views

Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
4
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2answers
65 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as $T(h)...
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1answer
218 views

Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
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0answers
33 views

Logistic regression eye treacting data (need model)

I have two sets of time course data, they are for an eye-tracking study. The data is 20 100ms chunks, one category being percent fixations for canonical sentences, and the other being percent looks ...
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1answer
44 views

How do we take the integral of $\int3x^{-2}log_a^{2x}dx$?

$\int3x^{-2}\log_{a}^{2x}dx$ $\quad$ a>0 I solve this integration by parts. (not sure though) is there another way to solve this? $ u=\log_{a}^{2x}, dv=3x^{-2}dx$ $ -\log_{a}^{2x}3x^{-1}+\int3x^{-...
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0answers
74 views

Integration question measure theory

For the function $$ f(x) = \begin{cases} \infty & \text{if $x=0$} \\ 1/x & \text{if $x \in \mathbb{Q} \smallsetminus 0$} \\ 0 & \text{Otherwise} ...
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0answers
127 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in L^p(\...
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2answers
39 views

Proof that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$

How do I prove that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$. I am not sure which type of proof to use, for example induction with two variables. The graphs suggest C can always be ...
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1answer
55 views

Show that if $f\colon\mathbb{R}^2\to\mathbb{R}$ is $C^2$, then any nonempty $\omega$-limit for the equation $x'=\nabla f(x)$ is a critical point.

I'm kind of struggling with an exercise I found in a book about Poincaré-Bendixon theory and I would like some help. The exercise is precisely what I wrote on the title: I have to show that if $f\...
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1answer
66 views

Is it possible that $f$ is differentiable?

Let $$f=\left\{\begin{matrix} 0,0 \leq x <1\\ 1,x=1 \end{matrix}\right.$$ This function is not continuous at $x=1$.Is it possible that $f$ is differentiable?
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157 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
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2answers
299 views

Something that isn't continuous can be proven to be continuous (so it is continuous - definitions - but doesn't look it!)

I'm sorry to post this, either I am right and it is continuous, or because I am on $\mathbb{Q}$ not $\mathbb{R}$ that saying "if that delta works, any smaller delta will!" (which can be proven by by ...
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1answer
36 views

Want to show that a solution to some ODE is unique.

So here is my problem, I just found the solution $x(t)=\frac{1}{t^2+1}$ to the following differential equation,$$ \dot{x}=-2tx^2,\;x(0)=1$$ Now I would like to show that my solution is unique and I ...
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1answer
44 views

What norm on $\mathbb C (z)$

There are several different ways to define a norm on the space of polynomials $\mathbb C [z]$. For example, $\|p\| = \sup_{|z|\le 1}|p(z)|$ defines a norm. If $\mathbb C (z)$ denotes the field of ...
4
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1answer
105 views

Are pseudo(micro)-local operators pseudodifferential?

$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the ...
0
votes
1answer
349 views

Isometry is not surjective

According to the definition I am using, an isometry is a mapping $f:X \rightarrow Y$ between two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$: $$ d_{Y}(f(a),f(b)) = d_{X}(a,b) $$ for all $a,b \in X $ I ...
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2answers
41 views

Show that abs(p(x))=e^x has a solution

How could I show that $$|P(x)| = e^x$$ Has a real solution, whereas $P(x)$ is a polynomial not identically zero?
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2answers
81 views

Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$

Prove that for all $x,y>0$ the following inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$ is true. It smells like Jensen inequality, but all I can get is that $\frac{x+y}{2}ln(x) + \frac{x+y}{2} ln(...
0
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2answers
122 views

Proving a if a property holds for a dense set then it holds on the field that the set is a subset of.

I am currently studying for my analysis exam and have come across this question, I can't seem to grasp the idea of a "dense set" especially with the definition given in the question. When I read it, ...
0
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1answer
26 views

question about line integral in a strength field

Let the field strength $\bar{F}(x,y) = e^xy ·\vec{i} , ((x,y) \in R^2)$. How can I prove, without doing any calculations, that the line integer of $\bar{F}$ along the segment joining $(2,0)$ with $(2,...
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2answers
44 views

How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$

How to show $(s+t)^p\le 2^{p-1}(s^p + t^p)$ for $p\gt1$ I know how to prove for $s+t=1$ by $\min\lbrace t^p+(1-t)^p\rbrace=2^{1-p}$. But I do not know to how to generalize for $s+t\gt0$. Could you ...
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1answer
85 views

Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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0answers
39 views

$C^\infty$ function which is constant on two intervals and implicitly defined inbetween

My textbook states without further reference that there exists a $C^\infty$ function $\alpha:\mathbb{R}\to\mathbb{R}$, such that $|\alpha'(t)| < K$ where $K>2$, and $$ \alpha(t) = \begin{cases} ...
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1answer
38 views

Question about relative singular homology groups

I know that the sphere $S^{\infty}$ is contractible, but why if $H$ is a Hilbert space then we have $$H_q(H,S^{\infty})=0, q\in \mathbb{N}?$$ Please help me Thank you
0
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1answer
55 views

How to get a parametrization for a line?

I am currently studying for some exams about line integrals. And I am struggling to get the parametrization for given problems. For example the line of the following Ellipse $ {x^2 \over a^2} + {y^2 \...
2
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1answer
52 views

Is there a way to interpret summation by parts as integration by parts with counting measure?

I find it difficult to remember the different forms of summation by parts: where the indices begin, end, whether to take forward/backward differences, etc. For example, Wikipedia has one form $$\sum_{...
0
votes
1answer
52 views

Is norm of a differentiable function continuous?

The following argument is from my class notes. "Suppose $\gamma(s): I\rightarrow \mathbb R^3$ is a regular curve. Fix $t_0$ and define $s(t) = \int_{t_0}^t \|\gamma'(t)\|dt$. That is, $s(t)$ is the ...
0
votes
1answer
59 views

Convex continuous functions [duplicate]

\begin{equation} w \left (\dfrac{x+y}{2} \right ) \le \dfrac{1}{2}(w(x) +w(y)) \quad \mbox{for all} \quad x,y \in \Omega, \end{equation} is a sufficient condition to a continuous function $w \in C^0(...
2
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1answer
365 views

Alternatives to Rudin

I'm taking an advanced calculus class this semester and we've been using Rudin's Principles of Mathematical Analysis. I was wondering if anyone could suggest some good analysis textbooks aimed toward ...
2
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1answer
52 views

Interesting question about functions

I saw the following question and I would like to share. I don't know the answer. Suppose that the function $f:\Bbb{N}\to\Bbb{N}$ has the property $f(f(n))<f(n+1)$ for any $n\in\Bbb{N}$. Prove that ...
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1answer
84 views

Polygons circumscribed to a circle with the smallest circumference

Let $n \geq 3$. Prove that among all $n$-polygons circumscribed to a (particular) circle, the regular $n$-polygon has the smallest circumference. It shouldn't be difficult, but I don't even know how ...
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2answers
260 views

Strange Property of Ultrametric Spaces and Metric Completion

The following property of ultrametric spaces seems quite strange: (No new values of the metric after completion) Let $x_1, x_2, \ldots$ be a sequence in $X$ converging to $x \in X$. Suppose $a \in ...
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0answers
88 views

Integration by parts on a sum of a product of partial derivatives

Let $\Omega$ be a region $\subset \subset \mathbb{R}^{n}$, and suppose that both $u_{\infty}$ and $\varphi$ $\in C_{0}^{\infty}(\Omega)$. Then, I want to perform integration by parts on the following:...
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1answer
64 views

Is every separately continuous function on $R^2$ continuous?

My friend asked me this question a few days ago. I felt it's not right but couldn't find a single counterexample. Any comment is appreciated.
3
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1answer
40 views

Count PI with analytical methods

Is there a differential equation which can be used to count the value of pi? I was able to describe pi only with sequences based on polygons with infinite corners. I think I'll need a continuous ...
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2answers
61 views

Can an integral be proved to have a finite value if an upper bound of the integrand has a finite value for improper integrals?

Can we say $ \int_{0}^{\infty} f(x) \text{dx} < \infty$ if $\exists \quad g(x) : \quad g(x)\geq f(x)\; \forall x \in \mathbb{R}$ and $ \int_{0}^{\infty} g(x) \text{dx}$ is finite. If yes, ...
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0answers
24 views

Verifying an algebra

Consider the metric space of bounded, continuous functions from $[0,1]$ to $\mathbb{R}$, and a subset $E$: \begin{equation} E=\left\{\sum_{i=0}^N a_ie^{b_ix}\mid a_i,b_i\in\mathbb{R}\right\} \end{...
2
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0answers
42 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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0answers
43 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to $\lambda$....
2
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1answer
63 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to $\lambda$....