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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4 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
1
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1answer
20 views

Mean value formula integrals

Let $f: B(0,R) \rightarrow \mathbb{R}$ be a continuous function. Then I was wondering whether $$\frac{1}{\text{area}(\partial B(0,r))} \int_{\partial B(0,r)} (f(x)-f(0)) dS(x) \rightarrow_{r ...
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0answers
8 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
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0answers
8 views

upper semi continuity and closeness

Let $F:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be set-valued mapping, under which assumptions closeness of $F$ implies upper semi continuity?
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0answers
18 views

Sobolev space exercise1 [on hold]

Let $B_{1}(0) \subseteq \mathbb{R}^{n}$ and $f(x)=|x|^{\gamma}$ with $\gamma >0$, what $\gamma $ verified that $f \in W^{1,p} (B_{1}(0))$?
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1answer
23 views

Modifying a bijective function.

If $f$ is a bijection from $X$ to $\{i \in \mathbb{N}: i < n\}$ and I define a function $$g:X -\{x\} \rightarrow \{i \in \mathbb{N}: i < n-1\}$$ such that $g(y) = f(y)$ if $f(y) < f(x)$ and ...
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3answers
468 views

Is there anything wrong with this proposed proof of the irrationality of Euler's constant?

Let $\{\lambda_n\}$ be the sequence given by $H_n - \ln n$. We claim that $\lambda_n$ is irrational for every integer $n>1$ and justify this by the following argument: Assume that $\lambda_k$ is ...
4
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0answers
38 views

Spivak's smooth partition of unity [duplicate]

You are right for your link But In your address, There is not any solution for this question and somebody had said that $f$ is redandant without that present even a reason or one proof or a rational ...
0
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1answer
29 views

Problem about a multivariable calculus

Decide for which of the functions $F:\mathbb R^3\to\mathbb R^3$ given below , there exists a function $f:\mathbb R^3 \to \mathbb R$ such that $(\nabla f)(x)=F(x)$. (A) ...
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3answers
48 views

How to find this type limit which has polynomial in sqrt?

I have no idea to find the below limit $$\lim_{n \rightarrow +\infty}\frac{2\sqrt{9n^2+20n+10}-6n-5}{\sqrt{9n^2+20n+10}-3n-5}=?$$
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0answers
32 views

Heine Borel theorem application

In pg 161 of Stroock and Varadhan Book Multidimensional Diffusion processes, one reads I don't understand the term $$\max_{1 \leq m \leq M}\{|t- s| + |y-x|: t>s \text{ and } (t,y) \notin V_m ...
2
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2answers
35 views

“Scalar product” of two Lp spaces

I was reading the book A. Lasotta and M. C. Mackey, "Chaos, Fractals, and Noise: Stochastic Aspects of Dynamic", Springer, 1991 On page 27, they defined a ``scalar product'' as follows. Let ...
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1answer
54 views

Are all derivatives of sinc function bounded on real axis?

It seems that all derivatives of $sinc$ function ($sinc(x)=sin(x)/x$) are bounded on real axis. Is it true or no?
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1answer
17 views

Piecewise $C_1$ and piecewise continuous

I would appreciate if the following questions could be clarified with your help. If a function is piecewise $C_1$, does this imply that it's also piecewise continuous? If a function is piecewise ...
0
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0answers
37 views

How to prove $x^nx^m = x ^{n + m}$ where $n$ or $m$ are negative.

In my text book I am given some properties of exponentiation, one of them being $$x^nx^m = x ^{n + m} \text{ where } x,y \text{ are rational and } n,m \text{ are natural.} $$ which I have completed ...
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1answer
27 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
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0answers
28 views

Uniformly boundedness of convolutions

Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$. Let $X_1,\dots,X_n$ ...
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0answers
48 views

$ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ [on hold]

I need help, I dont understad how it do $ \frac{\partial}{\partial x}(\frac{\partial f(x,y,z)}{\partial x}+\frac{\partial f(x,y,z)}{\partial y}\frac{\partial y(x,z)}{\partial x})=0$ please please ...
1
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1answer
29 views

If partial derivatives w.r.t. x and y are equal at each point (x,y) then which options are correct?

Let, $f$ be a function on $\mathbb R^2$ such that $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)$ for all $(x,y)\in \mathbb R^2$. Then which is(/are) correct? ...
0
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1answer
29 views

A function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous.

I'm having a confusion over the veracity of the statement that a function that satisfies the Intermediate Value Theorem and takes each value only finitely many times is continuous. I've seen from a ...
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1answer
39 views

$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne
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1answer
34 views

$f:[a,b]\to \mathbb R$ is continuous , has a finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?

If $f:[a,b]\to \mathbb R$ is a continuous function having finite number of local maxima and minima ; then how to prove that $f$ is bounded variation on $[a,b]$ ?
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1answer
8 views

Change of Basis in Canonical Correlation Analysis

I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation. How does the following change of basis works? The equation doesn't even ...
1
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1answer
21 views

Characterizing connected sets of $\mathbb R^n$ is terms of differentiable maps for which zero derivative everywhere implies constant

Let $U $ be an open subset of $ \mathbb R^n$ ; then how to prove that $U$ is connected iff for every differentiable function $f:U \to \mathbb R$ , $\nabla f(x)=0 \implies f $ is constant on $U$ ?
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0answers
10 views

How to rescale parameters?

First of all, I am a maths newby and never got any education on rescaling parameters on whatsoever. The knowledge that I have is based on what I know from mathematical research papers and as ...
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0answers
21 views

Help with a definition involving multiple suprema/infima

I have trouble understanding a definition that comes up in a proof of the Prokhorov theorem. Let $E$ be a Polish space and $M$ a set of probability measures on the Borel $\sigma$-algebra on $E$. From ...
0
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0answers
7 views

Weight change of other criteria in sensitivity analysis

I want to conduct a simple senstivity analysis as described here (page 45). Let's assume I have three criteria: C1, C2 and C3. I weighted them 50%, 30%, 20%. Now I want to drive the sensitivity ...
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1answer
24 views

$P$ is a monic polynomial of degree $n$ , then which are correct?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ is a real number. Then which of the following statements are necessarily correct ? If $n$ ...
0
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1answer
27 views

If $f\in C^2(\mathbb R)$ then $M_1^2 \le 2M_0 M_2$, where $M_k = \text {sup}_x |(d/dx)^k f(x)|$ for $k=0,1,2.$

I wanna prove this problem. I tried it with Mean Value Theorem but cannot proceed to any plausible result. So could I have some hints?
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1answer
58 views

Find the minimum value of $P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}$

Let $x,y,z$ be positive real numbers such that $x^3+y^3+z^3=3$. Find the minimum value of $$P=\frac{1}{2-x}+\frac{1}{2-y}+\frac{1}{2-z}.$$ I think that we need to show that $\dfrac{1}{2-x} \ge ...
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1answer
34 views

What does bounded partial derivatives exactly mean?

This might be a naive question, but if I give myself a continuously differentiable function $f$ from $\mathbb{R}^n$ to $\mathbb{R}$ which is said to have bounded partial derivatives, does this mean ...
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0answers
23 views

Why is n-th Fréchet derivative symmetric?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$. Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function. Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is ...
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60 views
+100

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
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1answer
38 views

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$ I was hoping that someone would maybe be familiar to this $w$ function that is stated, because this is the only ...
2
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3answers
148 views

Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
0
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1answer
26 views

Continuous functions with domain in the Natural Numbers

Can functions with domain in the Natural Numbers be continuous? In the high school, it is teached an intuitive notion of continuous functions: functions which will always appear as an "unbroken ...
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0answers
17 views

Construction of a function with linear start and horizontal asymptote equal to 1

I need a function which starts linearly at x=0 (with parameter settable slope !) and approaches horizontal asymptote of y=1 if x goes to infinity. Also the functions convergence speed to the asymptote ...
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0answers
16 views

Banach$^*-$ algebra for two different multiplication

Let $B$ be a Banch $^*-$ algebra and we say $f$ is positive linear functional on $B$ if $f(xx^*)\geq 0$ for all $x\in B.$ Let $B$ be a Banach algebra with two different multiplication operations, ...
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1answer
33 views

Properties of decreasing sequence of Lebesgue measurable sets.

I'm trying to prove a property of Lebesgue measure sets that says: If the $A_{k}$'s are measurable and $A_{1} \supset A_{2} \supset A_{3} \supset \ldots,$ and if $\lambda (A_{1}) < \infty, $ then ...
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2answers
30 views

Prove that If $\lim_{n \to \infty} |x_n-x_{n+p}| = 0$ for all $p \geq 1$, then $\{x_n\}$ is Cauchy sequence ??? [on hold]

Let $\{x_n\}$ be a sequence in $\mathbb{R}$. If $\lim_{n \to \infty} |x_n-x_{n+p}| = 0$ for all $p \geq 1$, then $\{x_n\}$ is a Cauchy sequence.
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1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
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0answers
61 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
2
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0answers
39 views

Example of Measure of non-compactness?

I can't understand the following example of measure of non-compactness, which was given in a research article. Definition: A nonnegative function $\phi$ defined on the bounded subsets of $X$ will ...
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0answers
29 views

Can we make $S_n \to \delta_x$ for $S_n$ an exponential polynomial?

Consider $f_\lambda: \Bbb{R}_+ \to \Bbb{R}_+$,$$f_\lambda(t) = e^{-\lambda t}$$ Now consider the finite linear combinations of these functions (exponential polynomials) $$ S(t) = \sum_{i = 1}^N ...
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2answers
50 views

Proof that Right hand and Left hand derivatives always exist for convex functions.

Definition A function $f$ is convex on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have $$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$ While reading the ...
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1answer
19 views

finding sup,inf, max,min of $A\cap B$ and $A\cup B$, if they exist and proving that $A\cap B$ is bounded.

As the title says i am trying to find and prove inf,sup, min, max if they exist for $A\cup B$ and $A\cap B$. And then prove that $A\cap B$ is bounded. Which will actually be easy, after i find all ...
1
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1answer
72 views

How to prove by Mathematical Induction. [duplicate]

I want to know how to prove this inequality by mathematical induction: $a_k's$ are nonnegative numbers. Prove that$$a_1a_2\cdots a_n\leq \left(\frac{a_1+a_2+\cdots+a_n}{n}\right)^n.$$ In the inductive ...
0
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2answers
28 views

finding for which $c\in \mathbb{R}$ sequence converges

so i am trying to find for which $c\in\mathbb{R}$ this sequence converges: $a_{1}=c$ and $a_{n+1}=1+\frac{a_{n}^2}{4}$ So i got the basic idea how to do this. First i found the candidate for limit: ...
0
votes
1answer
25 views

Finding a function such that $(n-2)/2 + f(n-1) \leq f(n)$

By bounding a certain quantity defined on real numbers by $f(n)$ I derived the following inequality arising from an inductive argument. $ (n-2)/2 + f(n-1) \leq f(n).$ A solution to the above ...
0
votes
1answer
38 views

$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...