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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10 views

Intuition for visualising dense monotonic discontinuous function

My question is about the function defined in Rudin 4.31, mentioned by this question: Remark 4.31 in Baby Rudin: How to verify these points? I'm having trouble trying to visualise what such a ...
2
votes
2answers
32 views

Limit $\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\sin x^2+y^2}$

Find the limit of: $$\lim_{\left(x,y\right) \rightarrow \left(0,0\right)} \frac{x^3+y^3}{\left(\sin x^2 \right)+y^2}$$ How to find this limit? What is the most straightforward method?
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0answers
16 views

Find minimum distance between the plane and the beginning of Cartesian plane.

Find minimum distance between the plane: $S=\{\left(x,y,z\right) \in \mathbb{R}^3: x+yz=2012 \}$ and the beginning of Cartesian plane $(0,0,0)$. I want to minimize this with use of lagrange's ...
1
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0answers
27 views

Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to x_0} f(t)=f(x_0)$

The problem: Suppose $f:[a,b]\to R$ and define $g: [a,b]\to R$ as follows: $g(x)=\sup \{f(t):a\le t\le x \}$ Prove that $g$ has a limit at $x_0$ if $f$ has a limit at $x_0$ and $\lim_{t\to ...
5
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1answer
67 views

A possible dumb question about derivative

I was solving some differentiation problems when I found the function $$g(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}.$$ So I thought: If I define the function $f:\mathbb{R_{x>0}}\to \mathbb{R}$ as ...
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2answers
25 views

On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
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0answers
14 views

Is the following derivative also differentiable with respect to $n$?

Let $f(x)$ be the $n-th$ derivative with respect to $x$ of $x \exp (n-1) log (x-1)$ evaluated at $x=1$. Is $f(x)$ differentiable with respect to $n$ ?
4
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3answers
84 views

Is the $n$th derivative of a continuous function also continuous?

Consider a differentiable (and hence continuous) function of order $n-1$. Is the $n$th derivative of such a function always continuous? As an example, is the $n-th$ derivative of the function $f(s) = ...
0
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1answer
19 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
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0answers
17 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
5
votes
2answers
72 views

Can $\mathbb R$ be partitioned into a countable number of dense subsets with same cardinality?

Is it possible to partition $\mathbb R$ into an countable number of disjoint dense subsets with the same cardinality? Furthermore, is it possible to partition the reals into an uncountable ...
2
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1answer
26 views

Every ordered field that has the least upper bound property is isomorphic to the real number system.

Okay, so here's a theorem from Rudin: "Every ordered field that has the least upper bound property is isomorphic to the real number system." Here's a definition: "Ordered fields are isomorphic if ...
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0answers
19 views

Big $O$ question for While and For loops [on hold]

I have to find the exact $O(N)$ for these instructions, not just the order of magnitude. I'm not getting any of the answers provided for me. I know the first loop is $O(3N+2)$. The declaration of ...
2
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1answer
38 views

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $

Show that $ \int_0^2 e^{x^2-x} dx \in [2e^{-1/4},2e^2] $ If $f(x)\leq g(x)$ for $x\in[a,b]$ then $\int^b_af(x)dx\leq \int^b_ag(x)dx$ if $x\in [0,2]$ then $x^2-x\leq x$, so $$0 \leq \int_0^2 ...
2
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6answers
57 views

Limit of $\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$

Limit of $$\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$$ I know (thanks to wolfram) it is equal to $y^2$, but i do not know how to show that.
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2answers
29 views

Find all points such that function has all partial derivatives in that point.

Find all points $(x,y) \in \mathbb{R}^2$ such that function has all partial derivatives in that point.$$ f(x,y) = \begin{cases} \frac{\sin(xy^2)}{y} &\mbox{if } y>0 \\ xy^2 & \mbox{if } y ...
1
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2answers
23 views

Finding a minimum of a function, measuring the sum of the squares of distance from some points of the $\mathbb{R}^n$

Given are a finite number of points $a_1, ..., a_m \in \mathbb{R}^n$. Consider the sum of the squares of distance: $$f(x) = \sum_{k=1}^m ||x-a_k||^2, x \in \mathbb{R}^n$$ with $||.||$ being the ...
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2answers
39 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
0
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0answers
12 views

Solutions for the dependency problem

Currently I read about the dependency problem of interval arithmetic. Mainly it's the problem that in the equation $X-X$ for $X$ being an interval the following is calculated: $$X-X=\{x-y:x\in X, y\in ...
1
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2answers
48 views

Calculate double integral $\iint_A \sin (x+y) dxdy$

Calculate double integral $$\iint_A \sin (x+y) dxdy$$ where: $$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$$ How to calculate that? $x+y$ in sin is confusing as i do not ...
3
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2answers
41 views

Double integral $\int\int_A y dx dy$

Calculate Double integral $$\iint_A y dxdy$$ where: $$A=\{(x,y)\in\mathbb{R}^2 : x^2+y^2\le4, y \ge 0 \}$$ I do not know what would be the limit of integration if i change this to polar coordinates. ...
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votes
1answer
26 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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votes
2answers
30 views

A Crucial Observation On Li's Criterion for the Riemann Hypothesis?

In 1997, Xian Jin Li formulated an interesting criterion whose validity is completely equivalent to the Riemann Hypothesis, namely: Define the real number $(n-1)!\lambda_n$ to be the $n-th$ ...
0
votes
1answer
21 views

Problem: conservative and not conservative $F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right)$

I don't know how I can solve this problem: Consider $$F=\left( y+\frac{y}{x^2+y^2}, x-\frac{x}{x^2+y^2}\right).$$ Proving that $F$ is not conservative in $\mathbb{R}^2-(0,0)$ but is conservative ...
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0answers
5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
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1answer
14 views

Classify the growth of functions and find a more general growth function

The following function $f(t,x):[0,T]\times R\mapsto R$ such that $\int^T_0|f(t,0)|^2 d t<\infty$, where $0<T<\infty$. If $f(t,x)$ satisfies $|f(t,x)|\leq Ax+B$ for each $x\in R$ and $A, B$ ...
1
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1answer
59 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
0
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0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
3
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0answers
40 views

Ramanujan Infinity sum functional equations

i was reading about the mellin transform ans i found the following $$\sum _{k=1}^{\infty } \left(\frac{e^{-k x}}{e^{-2 k x}+1}-\frac{\pi \text{sech}\left(\frac{\pi ^2 k}{x}\right)}{2 ...
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0answers
28 views

What powers of $|x|$ belong to $L^1$?

Prove that $|x|^ {−qp} \in L^{1}(U)$, where $U=B_{1}(0)\subset \mathbb{R}^{n}$. I think I could use polar coordinates to facilitate the work but not sure if it is useful.
1
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1answer
17 views

uniform continuity

Let $F(s,y)$ be uniformly continuous in $[a,b] \times B$, where $B \subset R^n$ is a closed subset. Assume $x_k \rightarrow x$ in $C[a,b]$ with $x_k(t) \in B$ and prove $$\int_a^b F(s,x_k(s)) ds ...
7
votes
2answers
102 views

Real analytic functions

I'm writing because I don't know the usefulness of real analytic functions. I mean, I know that analyticity is something more respect differentiable ($C^\infty$ function), but I don't have in mind a ...
0
votes
1answer
22 views

Exercise: Uniform Boundedness Principle and Double dual

Let $X$ be a normed vector space and $(x_{n})$ be a sequence in $X$. Show that if the sequence $f(x_{n})$ is bounded for every $f \in X^{\ast}$, then there exists $C > 0$ such that $\|x_{n}\| < ...
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0answers
24 views

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$

Check pointwise convergence and uniform convergence of $f_n(x) = n^3x\exp(-nx^2)$ on $[0,1]$ Pointwise convergence: $$ \lim_{n\rightarrow\infty} f_n(x) = \lim_{n\rightarrow\infty} ...
4
votes
3answers
55 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
2
votes
2answers
34 views

Injectivity of the function $x||x||$ on $\mathbb R^n$

Let , $f:\mathbb R^n\to \mathbb R^n$ be a function defined by $f(x)=x||x||^2$ for $x\in \mathbb R^n$. Then , which are correct ? (A) $f$ is one-one. (B) $f$ has an inverse. Here $f$ is not a ...
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0answers
21 views

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continous on $(0,\pi)$

Check whether the sum of the series $\sum^{\infty}_{n=1}\frac{\sin(nx)}{nx}\cos\frac{x}{n}$ is continuous on $(0,\pi)$ I think about showing the uniform convergence of $$ f_k: \mathbb (0, \pi) ...
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0answers
9 views

Reference of integral on differential manifolds and conformal aplications

I need goods and fast reference about integral of differential manifolds, more precisely about results of change variable but not with differential forms. I need goods and fast reference about ...
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1answer
58 views

If $f$ is differentiable and $f'$ is bounded then relation between upper sum , lower sum and the integral

Let , $f:\mathbb R\to \mathbb R$ be a differentiable function such that $f'$ is bounded. Given a closed and bounded interval $[a,b]$ and partition $P=\{a=a_0<a_1<\cdots <a_n=b\}$ of $[a,b]$ . ...
0
votes
2answers
27 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...
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0answers
25 views

Find $C^1$ class function such that

Given: $$g: \mathbb{R}^3 \rightarrow \mathbb{R}, g(x,y,z)=z^3-3xyz-x-8$$ Decide whether in the neighbourhood of the point $(x,y)=(0,0)$ there exist $C^1$ class function $z=z(x,y)$, such that ...
2
votes
0answers
43 views

symplectic structure on $S^2$

i was looking for a symplectic structure on the $S^2 $. Originally i considered the Poisson-Structure of a rigid body, which was given by $\{F,G\}=\langle \Pi, \nabla F \times \nabla G \rangle$, for ...
0
votes
1answer
19 views

Wheeden-Zygmund exercise

Define $\limsup_{k \rightarrow \infty}a_k$ and $\liminf_{k \rightarrow \infty}a_k$ by $$\limsup_{k \rightarrow \infty}a_k = \lim_{j\rightarrow \infty}b_j = \inf_{j}\{\sup_{k\geq j}a_k \} $$ ...
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0answers
26 views

Calculating Combinations / Permutations [on hold]

How do I calculate the number of outcomes as a whole of a series of individual tests with there own outcomes? For example, the best description I could think of would be: There are 10 tests and each ...
0
votes
1answer
15 views

Find maxima and minima of the function

Given: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f\left(x,y \right)=-x^4+x^3-3x^2y+3xy^2-y^3$$ Find all points where gradient is equal to zero. Decide whether in those points function has either maxima ...
0
votes
1answer
33 views

Boolean Algebra, stuck

I'm having trouble simplifying this Boolean Algebra equation. Can anyone help? XY'Z + X'Y'Z + XYZ + XY'Z
6
votes
0answers
42 views
+50

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
vote
1answer
34 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 ...
0
votes
0answers
11 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}) : ...
-1
votes
0answers
9 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?