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).

learn more… | top users | synonyms (1)

0
votes
1answer
7 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
1
vote
2answers
22 views

Prove that $f(x,y)$ is continuous in $(0,0)$

Prove that $f(x,y)$ is continuous in $(0,0)$, where \begin{equation} f(x,y) = \begin{cases} \frac{x^2y}{x^4+y^2}, & (x,y)\neq 0\\ 0, & (x,y) = (0,0) \end{cases} \end{equation} The solution I ...
-1
votes
0answers
22 views

A problem of Taylor series

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
0
votes
0answers
4 views

Bipartite graph matching partitioning using clustering algorithm

I am identifying information from a document using bipartite graph model now I have to extract that information which are closely matched. hence I want to use clustering technique to group the data ...
0
votes
2answers
23 views

The Fourier transform of functions with compact support is differentiable.

1) How can I prove that if $f(x)$ is a continous function with compact support (let's say $f(x)=0$ $\forall x\in B(0,R)^c$), then its Fourier transform $\hat{f}(\xi)$ is differentiable? 2) Is there ...
5
votes
0answers
38 views

How to prove this integral [duplicate]

Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1) $$
2
votes
2answers
91 views

How to prove this problem [on hold]

Let $f$ be a real valued continuous function on $[0,\infty]$ such that $$ \lim \limits_{x\to\infty}\left(f(x)+\int_{0}^{x}f(t)dt\right) $$ exists. Prove that $$ \lim \limits_{x\to\infty}f(x)=0 $$
2
votes
1answer
28 views

Relation between runge domain and polynomial convexity

Are these concepts the same? Just to state the definitions Definition 1 A domain $\Omega \in \mathbb{C}^n$ is a Runge domain if every function $f \in H(\Omega)$ can be approximated, uniformly on ...
1
vote
1answer
21 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
3
votes
2answers
42 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
4
votes
2answers
39 views

dominated convergence for functions $\mathbb R^n\to\mathbb R^m$?

I do know the dominated convergence theorem for functions $f:\mathbb R^n\to\mathbb R$. Now let $U\subset\mathbb R^n$ and $f: U\to\mathbb R^m$. Is there any dominated convergence theorem for ...
1
vote
1answer
40 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
-3
votes
1answer
32 views

A problem of the limit of a serie [on hold]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Calculate the following limit: Thanks!
0
votes
1answer
12 views

Prove $f\in \mathscr{R}(\alpha)$ and $\int_a^b f\ d\alpha = f(s)$ with the following conditions.

If $a<s<b$, $f$ is bounded on $[a,b]$, $f$ is continuous at $s$, and $\alpha(x)=I(x-s)$, then prove that: $$f\in \mathscr{R}(\alpha)$$ and $$\int_a^b f\ d\alpha = f(s)$$ $I$ is a unit step ...
-5
votes
0answers
63 views

A not very easy problem… [on hold]

I leave a challenge, a derivative problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - \alpha x \sin x = \mathcal{O}(x^4), \text{ as } x\to 0 $$
1
vote
1answer
18 views

Compactness of a convex collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $(-\epsilon,\epsilon)$ on $S_0$ while ...
0
votes
0answers
29 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
0
votes
0answers
28 views

How is this convex set compact as well?

Given $\epsilon\in(0,1)$, supposing we have a collection $\mathscr{C}(\epsilon)$ of polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ takes on value $0$ on $S_0$ while being in range ...
3
votes
1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
0
votes
0answers
18 views

Proving Euler's theorem for homogeneous functions.

This problem is from Apostol's Mathematical Analysis. Let $f$ be defined on an open set $S$ in $R^n$. We say that $f$ is homogeneous of degree $p$ over $S$ if $f(\lambda x)=\lambda ^p f(x)$ for every ...
0
votes
1answer
19 views

Relationship between centralization and floating-point arithmetic

Suppose I have a problem of least squares where I have $n$ independents regressor variables $x_i$, and suppose I applied the process of centralization and scaling to this variable through ...
0
votes
0answers
21 views

How do I compute this metric projection?

I saw a result that says: Given a nonzero vector $a$ and the convex set $K:=\{y\in H: \langle a,y\rangle =\alpha, \alpha \in \mathbb{R}\}$ a hyperplane, then $$P_Kx=x-\frac{\langle ...
0
votes
2answers
28 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
-1
votes
0answers
36 views

Show that subspace metric induces subspace topology [on hold]

Let $(X,d)$ be a metric space, let $\tau$ be the topology on $X$ induced by $d$ and $A \subset X$. Define $d_A: A \times A \to \mathbb R$ as $d_A(a,b)=d(a,b) \forall a,b \in A$ . Show that $d_A$ ...
1
vote
1answer
31 views

Show that $\tau_A$ is a topology on $A$

Let $(X,\tau)$ be a topological space and $A \subset X$. Let $\tau_A$={$A \cap U: U \in \tau$}. Show that $\tau_A$ is a topology on $A$. I know that I need to prove three properties to prove ...
2
votes
1answer
56 views

Please check my demonstration of de l'hopital's rule

I have demostrate the de l'hopital theorem but in some steps I'm not 100% sure; The theorem I demostrate is for: $\lim_{x\rightarrow a+} \frac{f'(x)}{g'(x)}=L \implies\lim_{x\rightarrow a+} ...
-1
votes
0answers
31 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
-1
votes
2answers
31 views

Show that $d_f$ is a metric on $X$ [on hold]

Let $(X,d)$ be a metric space, and let $f: X \to X$ be a bijection. Define $$d_f: X \times X \to \mathbb R $$ as $d_f(x,y)=d(f(x),f(y))$ $\forall x,y \in X$ Show that $d_f$ is ...
-6
votes
1answer
39 views

how to find all roots of a polynomial function? [on hold]

Let $f(x)$ be a polynomial of degree 5 and $f(|x|)$ has nine real roots, then how many real roots does $f(x)$ has? Five real roots Four positive roots One negative root Nothing can be said in ...
7
votes
2answers
47 views

limit of function $\sin(x \ln x)/x$ as $x\rightarrow 0$

I am trying to find $\lim \limits_{x \to 0} \frac{\sin(x \space \ln(x))}{x}$. I believe I have solved it using the squeeze theorem to determine: $\frac{-1}{x} \leq \frac{\sin(x \space ln(x))}{x} \leq ...
1
vote
1answer
21 views

using logical quantifiers to write that f approaching infinity DOES NOT tend to infinity

Is this the same as writing that the limit of f as f approaches $\infty$ is L? i.e.: $\forall \space \epsilon > 0 \space \exists \space c \space \forall \space x>c : |f(x) - L|< \epsilon$
0
votes
0answers
29 views

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak's Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. ...
0
votes
0answers
17 views

Condition for all derivatives to be L-Lipschitz

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with infinitely many derivatives and let us use the notation $$ f^{(n)}(x)=\frac{\mathrm{d}^nf(x)}{\mathrm{d}x^n}. $$ Assume that $f^{(n)}$ is ...
1
vote
1answer
12 views

Why is a continuous function of a Riemann-Stieltjes Integrable function, also Riemann-Stieltjes Integrable?

Suppose $f\in \mathscr{R}(\alpha)$ on $[a,b]$, $m\leq f\leq M$, $\phi$ is continuous on $[m,M]$, and $h(x)=\phi(f(x))$ on $[a,b]$. Then prove that $h(x)\in\mathscr{R}(\alpha)$ on $[a,b]$.
1
vote
0answers
28 views

Proving that $\phi$ is orthogonal to the harmonic forms given $\int\phi \;d\mathrm{vol}$.

I want to prove that given a connected closed (compact without boundary) oriented Riemmanian manifold $(M,g)$, the condition $$\int_M \varphi \;\mathrm{d}\mathrm{vol}_g=0$$ implies that $\int \varphi ...
2
votes
1answer
13 views

Given a function $f$ defined in $R^2$. Let $F(r,\theta)=f(r\cos\theta,r\sin\theta).$ Verify a formula of the modulus of the gradient.

Given a function $f$ defined in $R^2$. Let $$F(r,\theta)=f(r\operatorname{cos}\theta,r\operatorname{sin}\theta).$$ Verify the formula $$|\nabla f(r\operatorname{cos}\theta, ...
1
vote
1answer
40 views

Show that S is closed but not compact

Show that $S$={$(x,y,z)\in \mathbb R^3: x^3+y^4-z^2=1$} is closed but not compact where $\mathbb R^3$ is the usual topology. Can anyone explain how to go about answering this? I have to show that ...
0
votes
1answer
15 views

Can we deduce if a set is measurable, given a measurable function and a measurable space?

Let $f(x):X\rightarrow Y $, where $X$ is a measurable space. Suppose that $f$ is measurable. Let $E$ be a subset of $X$. Now, suppose that $f(E)$ is closed or clopen. Can we deduce that $E$ is a ...
0
votes
1answer
12 views

Why does a Hermitian operator with singleton spectrum have to be scalar?

One proof of Schur's lemma proceeds by showing that a Hermitian intertwining operator of an irreducible representation (of a topological group on a Hilbert space) has a spectrum that contains only one ...
2
votes
2answers
59 views

A property for an ODE

$2\leq n\in\mathbb{N}$. I have no idea how to show that there is a unique solution $y\in C^1([0,T))$ of the ODE \begin{eqnarray} \begin{cases} y'(t)=(1+y(t)^2)\left(1-\dfrac{n-1}{t}y(t)\right)\ \ \ ...
0
votes
1answer
22 views

Are maximal intervals of open nonempty sets always equal?

Let $O\subset\mathbb{R}$ be an open nonempty interval. Define for every $x\in O$: $$a_x = \inf\{a\in\mathbb{R}\mid(a,x]\subset O\}$$ $$b_x = \sup\{b\in\mathbb{R}\mid[x,b)\subset O\}$$ $$I_x = (a_x, ...
0
votes
0answers
9 views

Consistent estimators/convergene in probability and slutsky

Let $m_n$ be a consistent estimator of $g(\vec\alpha)$ where $\vec\alpha = (\alpha_1,\cdots,\alpha_k)\in \mathbb{R}^k$ and $v_n$ be a consistent estimator of $f(\alpha_1,g(\vec\alpha))$. Suppose that ...
1
vote
1answer
44 views

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$

Let $A,B \subset \mathbb{R}^n$ open sets, $A\subset B$. How can I set a function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$, $C^{\infty}$, such that: \begin{equation*} f(x)=0, \textrm{ if } x\in ...
4
votes
3answers
46 views

Find an example of a sequence not in $l^1$ satisfying certain boundedness conditions.

This question is about getting a concrete example for this question on bounded holomorphic functions posed by @user122916 (something that he really expected as explained in the comments). Give an ...
3
votes
1answer
21 views

Equivalency of the set of real numbers to the set of all continuous real functions?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(0,1)$ and also equivalent to the set of all points in $\mathbb{R}^2$. I have seen a claim in a book ...
0
votes
1answer
21 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
3
votes
2answers
29 views

Equivalency of real numbers and points in the plane?

I understand that the set of real numbers is equivalent to the set of real numbers in the interval $(-1,1)$ by simply using $arctan$ function. However, I do not know how to find a one-to-one mapping ...
1
vote
1answer
15 views

define $f(\mathbf{x})=f_1(x_1)+\cdots +f_n(x_n)$. Show that $f$ has a differential at each point of an n-dimensional interval.

Given $n$ real-valued functions $f_1, \dots, f_n$, defined and having finite derivatives in the interval $(a,b)$. For each $\mathbf{x}$ in the $n$-dimensional interval $$S=\{(x_1,\dots ,x_n)\mid a\lt ...
4
votes
0answers
35 views

$\phi_{\epsilon} \ast \mu \rightarrow \mu$?

Let $\phi$ be a non-negative function on $\mathbb{R}$ with $\int_{\mathbb{R}} \phi = 1$. Define $\phi_{\epsilon}(x)=\epsilon^{-1}\phi(\epsilon^{-1}x)$ for $x \in \mathbb{R}, \epsilon > 0$. For $f ...
2
votes
2answers
50 views

Homeomorphism between the set of invertible matrices and itself

Consider the set of invertible $n \times n$-matrices $GL_n(\mathbb{R}) = \{A \in M_{n \times n}(R) \mid A\text{ is invertible}\}$. I now want to prove that the transformation $$f: A \mapsto A^{-1}$$ ...