Questions tagged [analysis]

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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Pullback of a constant coefficient form

Let $\omega=\sum_{I}C_Idx_I$, where $I=(i_1,\dots,i_n)$ is a multi-index and $C_I$ constants, be an $n$-form in $\mathbb{R}^m$, with constant coefficients. Here $dx_I$ means $dx_{i_1}\wedge\dots\wedge ...
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$L^{\infty}([a,b])$ and the Lebesgue-Stieltjes integral

Let $[a,b]$ be a compact interval in $\mathbb{R}$, and $L^{\infty}([a,b])$ the space of all lebesgue measurable functions on $[a,b]$ essentially bounded; my question is whether these functions are ...
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Definition of Riemann-Stieltjes integral

In Measure and Integral by Jaroslav Lukeš, Jan Malý, they define the Riemann-Stieltjes integral for continuous functions as: where $\varphi $ is nondecreasing on $\mathbb{R}$. My first question is ...
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How to show that $||\int_{a}^{b}f(t) dt||_2 \leq \int _{a}^{b}||f(t)||_2 dt $?

Let $f: [a,b] \to \mathbb{R}^2$ be continuous . Why is $||\int_{a}^{b}f(t) dt||_2 \leq \int _{a}^{b}||f(t)||_2 dt $ $\quad$ (the first integral is defined componentwise and $||.||_2$ is the ...
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What fails in the maximum value principle if the domain is not connected?

We have discussed the maximum value principle: Let $\Omega \subset \Bbb{C}$ be an open connected subset and $f:\Omega \rightarrow \Bbb{C}$ be an analytic function. Assume that there exists $z_0\in \...
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Determinate $d^{k}f_{c}$ for function $f(x,y)$ with k and c given

In my Calculus 3 class I have been given following problem: Determinate: $d^{k}f_{c}$ for function $f(x,y)$ where $c = (c1,c2)$ and $k = 2$. My problem is that I have no clue what does $d^{k}f_{c}$ ...
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Evaluating the value of |$f'(x)$| [duplicate]

Let $f \in C^2[0,2]$ and $|f(x)| \le 1, |f''(x)|\le1$ for all $x \in[0,1]$ Prove that $|f'(x)|\le2 $ for all $x \in [0,2]$ I tried Taylor expansion to evaluate $|f'(x)|$, but it seems to work only ...
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Is the convolution of $L^2$ functions continuous? [duplicate]

Is the answer to the following question positive? The convolution of two $L^2(\mathbb R)$ functions is continuous I briefly recall it here: Take $f$ and $g$ $\in L^2(\mathbb R)$, then I want to show ...
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1 answer
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Explanation about this probabilistic proof that leads to $log_RN$

I am reading a proof based on (probabilistic) analysis on how a specific data structure performs for the case of search miss. The base assumption is that the probability that each of the $N$ keys in a ...
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Extrapolate impression values from total impressions.

Im looking to extrapolate impression values for specific page level URLs however I only have the total impressions for a domain, and the ranking of the URLs (where the closer you are to 1 the higher ...
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0 answers
30 views

Multiple integrals identity

I am trying to solve a problem from my homework and I am one step away of getting it done. Let $ a< b$ two real numbers and consider the multiple integral: $$I = \int_{a}^{b}dx_{1}\int_{a}^{x_{1}}...
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Prove that $e^{\frac{1}{n}} \le 1 + \frac{1}{n - 1}$ [duplicate]

How to prove that: $$e^{\frac{1}{n}} \le 1 + \frac{1}{n - 1}$$ where $n\in\mathbb{N}, n \ge 2$. I can't seem to find any connection from description after limit and this equation. This is part of the ...
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Two Power Series Identify on an Open interval not Containing Zero

I searched a lot for a convincing answer for this question but failed to find one (That is formally complete). I wonder if the following claim is true, and if so, for a formal proof. Claim: Let there ...
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Show that $ \frac{x^2}{(x^2+y^2)^{3/4}} $ is continuous on $\mathbb{R}^2$.

I tried with polar coordinates, I found the limit is infinite (which is not true the graph show it's 0). I tried to majorate with something who has for limit 0 but still impossible. with polar ...
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An asymptotic in the hint of Stein's Complex Analysis Chapter 6 Exercise 12(a).

The entire statement of Stein's Complex Analysis Chapter 6 Exercise 12(a) is Show that $1/\lvert\Gamma(s)\rvert$ is not $O(e^{c\lvert s\rvert})$ for any $c>0$. And the hint of this problem is If $...
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How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$

I have stumbled across this triple integral $$\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$$ where $$\Omega =\left\{(x,y,z)\in{\cal{R}}^3\ \bigg| \ \frac{x^2}{...
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1 answer
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Understanding Real analyticity

I'm going to state my assumption of the definition of Real analyticity, and how I understand it based on my current understanding. Please tell me if they are correct or not and please help me ...
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Is this generalized metric $d_t(a,b):X\times X\longrightarrow [0,\infty)$ a continous function?

Consider the following generalized metric space where triangle inequality has been extended. Let $X$ be a non-empty set and $P, Q, R: X \times X \rightarrow[1, \infty)$. A function $d_{t}: X \times X \...
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How to apply the Stieljes transform of $\mu$ show that $L_g(z)=\frac{E[YX]S(z)}{1-E[X^2]S(z)} $?

Let probability measure $\mu$ be semi-circle law, that is for $x\in [-2,2]$, $$ \frac{d\mu}{dx}=\frac{1}{2\pi}\sqrt{4-x^2}. $$ Let $S: R\setminus[-2,2]\to R$ be the Stieljes transform of $\mu$, i.e., $...
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How to prove that there exists $C_1>0$ such that $ \max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\le 2^nC^nC_1^n\frac{T^n}{n!} $?

I have two iterated update upper bound for two sequences of continuous functions $\{f_n\}$ and $\{g_n\}$, that is there exists $C>0$ such that $$ \|f_{n+1}-f_n\|_{\infty}\le C\int_0^T|f_n(s)-f_{n-1}...
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Is this integral convolution continuous?

The following question is related to the one in Is this convolution continuous? Let $0 \leq \xi \leq T$, $Y_0 \in L^2([-T, T])$, $a_1 \in L^2([0,T])$ and consider the following function: $$ Y_{1}(\xi)...
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Convex compact set in $\mathbb{R}^n$ where, given any point in it, the result of replacing two of its coordinates with their mean lies in the set.

Let $X$ be a nonempty compact convex subset of $\mathbf{R}^n$. Suppose this subset has the following property: for every $x = (x_1, \dots, x_n) \in X$, for every $1 \le i< j \le n$, $$({x_1}, \...
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Solve the following differential equation $t + \frac{f(t) f'(t) \sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f(t)}{\sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f'(t)} = const.$

I have to find a real function $f(t)$ which makes the following $s$ constant : $$s = t + \frac{f(t) f'(t) \sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f(t)}{\sqrt{n^{2}+n^{2}f'(t)^{2}-1}-f'(t)}$$ where $n$ is a ...
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A bound on a function involving exponentials [closed]

Does there exist constants $c,x_0>0$ such that $$\sqrt{e^{1/x}-1}\left(1-\sqrt{e^{-1/x}}\right)\leqslant \frac{c}{x^2}$$ for all $x>x_0$? So far I have only been able to show that this is $O(x^{...
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Is there a sequence of real numbers with no subsequential limits? [closed]

I'm taking these definitions from Walter Rudin's Principles of Mathematical Analysis: Let $\{a_n\}$ be a sequence of real numbers. A subsequence of $\{ a_n\}$ is a set $\{a_{n_k}\}_{k=1}^{\infty}$ ...
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Counting measure has no Lebesgue decomposition proof verification

I am working through exercises in Rudin RCA but I have some questions about whether my justification is valid as it differs from other posts on the site. Let $\mu$ be the Lebesgue measure on $(0,1)$ ...
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converges and determine the limit $a_{n+1} / a_n < q$ [closed]

Let $(a_n)$ be a sequence in $\mathbb{Q}$ with $a_n > 0$ for all $n \in\mathbb{N}$, and let $q \in\mathbb{Q}$ be a number with $0 < q < 1$. Furthermore let $$ \frac{a_{n+1}}{a_n} \le q\tag1 $$...
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2 votes
2 answers
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For a convex compact set $K$ in $R^n$, if $x \in \partial K,\ \lambda x $ is not in $K$ if $\lambda>1.$

Is it true that for a convex compact set $K$ in $\mathbb{R}^n$ that contains $0$ in its interior, if $x \in \partial K$ , then $\lambda x\not\in K$ for $\lambda > 1$? This is something that feels ...
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Fractional powers of the Laplacian

I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for $$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}....
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4 votes
2 answers
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Is a continuous function on compact convex set where the boundary is mapped to the set a self mapping?

Let $K ⊂ R^n$ be convex and compact with $0$ in the interior of $K$. Let $f ∈ C(K, R^n)$ with $f(∂K) ⊂ K$. If this is the case, do we in fact have $f(K) \subset K$. It is probably not the case that ...
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What does "contained continuously" mean? As in, "metric space $X$ is contained continuously in metric space $Y$".

What we mean by the "Contained continuously" terminology?! Definition: Let $H$ be a Hilbert space and $E$ is an open bounded subset of $\mathbb[C]$. The Hilbert space of measurable ...
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Prove that even doubly periodic function satisfies a differential equation

Let $f(z)$ be analytic in $\mathbb{C}\setminus\{m+ni:m,n\in\mathbb{Z}\}$. Assume that $f(z)=f(-z)$ , $f(z)=f(z+m+ni)$ and $f$ has a pole of order $2$ at $0$. Prove that there exist numbers $a_0,a_1,...
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Proof of convergence of random variables in $L^p$ via convergence in probability and uniform integrability.

Consider the following proposition. Part (i) i have no problem with. Its the proof of part (ii) that (because of my lack of knowledge of advanced measure theory) am having trouble in understanding. ...
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2 votes
2 answers
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Proof verification: If $f:[0,1]^2\mapsto\mathbb{R}$ is continuous, then $f$ is not injective

If $f:[0,1]^2\mapsto\mathbb{R}$ is continuous, then $f$ is not injective I start by ascerting that, since $[0,1]^2$ is compact, then there are $x_1,x_2\in[0,1]^2$ such that for every $x\in [0,1]^2$ $...
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When we calculate the irr, how to decide the initial range of irr?

For the cash flow $C_i$, initial investment costs $P$ which is negative, the number of time period $i$, when we calculate the irr $r$, how to decide the initial range of $r$? $$P + \sum_{i=1}^{n}\frac{...
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4 votes
3 answers
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For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$?

For what values of $p>0$ is $\lim_{n\rightarrow\infty}\int_0^n\frac{(1-\frac{x}{n})^ne^x}{n^p}dx=0$? My thoughts: We know that $(1-\frac{x}{n})^n\leq e^x$, so the numerator is $\leq e^{2x}$. So, ...
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For a compactly supported, continuously differentiable function $f$, do we have $\frac{f(x+h)-f(x)}{h}\to f'(x)$ uniformly?

For a compactly supported, continuously differentiable function $f$, do we have $\frac{f(x+h)-f(x)}{h}\to f'(x)$ uniformly? (Euclidean space) I think this is true, since compact supported-ness is ...
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Continuity of Taylor remainder for a multivariate $C^1$ function

Suppose I have a function jointly $C^1$ in two variables. By Taylor's theorem I can expand in one of the variables, say the second, and get a remainder term $R(x,y,h) = \frac{f(x,y+h) - f(x,y)}{h} - \...
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39 views

Area integral of a complex function over $\mathbb{C}$

I am trying to compute the Lebesgue integral of $K(z,w)=(1+z\overline{w})^{n-1}$ over the region $\Omega = \{z: 0<\text{Re}\{z\}<1, 0<\text{Im}\{z\}<1 \}$ with respect to the measure $\...
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3 votes
2 answers
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Does the set of convex combination of points in Cantor set contains a non empty open interval?

$\mathcal{C}$ denote the cantor middle third set. $$\mathcal{C}_t=\{(1-t)x+ty : x, y\in \mathcal{C} \}$$ $\mathcal{C}_0=\mathcal{C}_1=\mathcal{C}$ and we can prove that that $\mathcal{C}$ contains no ...
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1 answer
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Is My Solution Valid?

Question from the 1999 Bulgarian Math Olympiad: Find all pairs $(x,y)\in\mathbb{Z}$ satisfying $$x^3=y^3+2y^2+1$$ My first approach was to take the cube root of both sides: $$x=\sqrt[3]{y^3+2y^2+1}$$ ...
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1 vote
1 answer
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Union of spheres is a null set

It is proven that the sphere $S(0^{[k]};a)$ is a null set in $R^k$ for every $a \in R$. Question: Let $C \subseteq R$ be a null set. Prove that $\bigcup_{a \in C} S(0^{[k]};a)$ is a null set in $R^k$. ...
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Time-Series Analysis

Consider a linear dynamic system: $$ y(t) = \theta^{T}x(t) + e(t) $$ In this formula, $y(t) \in \mathcal{R} , \theta \in \mathcal{R}^d ,x(t) \in \mathcal{R}^d ,e(t) \in N(0,\lambda(t))$ And they are ...
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1 answer
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integration by parts on a fourier transformation

Given is a function $$ \dfrac{1}{2 \pi} \int\limits_{- \infty}^{+ \infty} \mathrm{e}^{-itx}\dfrac{f(t)}{g(-t/h)} \, dt.$$ In the paper I'm working with it's written that by integration by parts, we ...
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Selection of a regular function in range of matrix

Let $A$ be a $m \times n$ matrix and $f \in C^k([0,1], \mathbb{R}^m)$ ($k \in \mathbb{N}$) such that $f(x) \in \mathrm{Range} \, A$ for almost every $x \in [0,1]$. Can we find a function $g \in C^k([0,...
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1 vote
2 answers
32 views

Show that if $0 < \frac{1}{j},\frac{1}{k} \leq \frac{1}{N} \leq 1$, then $|\frac{1}{j}-\frac{1}{k}| \leq \frac{1}{N}$. ($j,k,N \in \mathbb{N}^+$))

I am working through Tao's analysis I book and I am trying to prove that the sequence $a_n = \frac{1}{n}$ is a Cauchy sequence (Proposition 5.1.11). I understand the proof, but I am having trouble ...
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What real analysis wants to convey and how it developed? [closed]

Most of us have studied real analysis as undergrad. But i cannot understand the main convincing point to study real analysis and how to summerise it. I know this question is little tricky and everyone ...
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2 votes
0 answers
27 views

What does such notations $\mathbb{E}_{X\sim \mu}$ mean?

I read a notation on a paper about statistics and machine learning ("High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models" by Tengyuan Liang, Subhabrata Sen, and Pragya ...
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Why can we say that the limits of $f_n$ exist and $f_n\to f$ a.s. by Arzela-Ascoli's theorem?

I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging. On page 123. For any time $T>0$, define a sequence of function $f_n(t,t)\in C([0.T]\times [0,T], \mathbb{R})$...
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  • 1,688
0 votes
2 answers
28 views

Continuity a function of several real variables

The goal is to study the continuity of the following function: $f(x, y)= \begin{cases}x^{2}y, & \text { si }|x|<y \\ y, & \text { otherwise. }\end{cases}$ $D_{f}=\left\{(x, y) \in \mathbb{R}...
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