Mathematics Stack Exchange
Mathematics Stack Exchange

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Singular Points in Different Areas of Mathematics

What is the relationship between singular points of algebraic curves (as described here or here), singular points of ode's (as described here or here) and singular points in complex analysis (as described here or here)? They seem like three…
bolbteppa
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$\lim_{n\rightarrow \infty} \frac{b_n}{a_n} = 0 $ prove that $\lim_{n\rightarrow \infty} a_n = \infty$ if $\lim_{n\rightarrow \infty} b_n = \infty$

I have this question: For $\large \ a_n > 0 $ and $\large \ b_n > 0 $, and $\large \lim_{n\rightarrow \infty} \frac{b_n}{a_n} = 0 $ prove that $\lim_{n\rightarrow \infty} a_n = \infty$ if $\lim_{n\rightarrow} b_n = \infty$ One way I thought I can…
Logarithm
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$f(X \cap Y) \subset f(X) \cap f(Y) $

In class we had the following function which I intend to prove for my own peace of mind. Let $M$ and $N$ be sets and $f: M \longrightarrow N$ a function: \begin{align}f: P(M) &\longrightarrow P(N) \tag{P denotes Powerset} \\ X & \longmapsto \lbrace…
Spaced
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The mapping of a Set onto the characteristic function is a bijection

Let $X$ be a Set. For all Subsets $ A \subset X$ the characteristic function of A is defined as: \begin{align} \chi_A(x)= \begin{cases} 1 \iff x \in A \\ 0 \iff x \notin A \end{cases} \end{align} Let $\lbrace0,1\rbrace^X$ be the Set of functions $ X…
Spaced
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2 answers

Properties of a surjective local diffeomorphism

Assume that $f:\mathbb{R}^N\to\mathbb{R}^N$ is a surjective function and in addition suppose that $f$ is a local diffeomorphism. Take two points in the image of $f$, let's say, $f(x),f(y)$ with $f(x)\neq f(y)$. Is it possible to find a continuous…
Tomás
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De la Vallée Poussin’s Theorem

I try to understand De la Vallée Poussin’s Theorem. I don't understand the orange bit: I really don't see why they need to have same sign. Any help would be appreciated :)
Kasper
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Prove $\sup(AB) = \sup(A)\sup(B)$

I know $\sup(AB) = \sup(A)\sup(B)$ if $A$ and $B$ are nonnegative, but what if the assumption is dropped that $A$ and $B$ are nonnegative. Does this change the answer?
MDW
  • 366
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How prove it such that the sequence $\{L_{f}(\phi_{j})\}$ does not tend to zero.

Let $f(x)=e^{x^2}$, For $\phi\in D(R)$,define $L_{f}(\phi)=\int f\phi dx$; construct a sequence of function $\{\phi_{j}\}$ in $D$ that tends to zero in $S$ but such that the sequence $\{L_{f}(\phi_{j})\}$ does not tend to zero. where $S$ is meaning…
math110
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Orthogonal projection on subspace spanned by $\{ f_m := e^{imx}\mid |m|\leq n \}$

Consider $C([a,b], \mathbb{C})$, the space of continuous complex-valued functions on $[a,b]$. Define the complex inner product on $C([a,b], \mathbb{C})$ as $$\langle f,g\rangle=\int_a^bf(t)\overline{g(t)} \, dt .$$ Take $[a,b] = [0, 2 \pi]$.…
Jeroen
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Finding the nearest number to zero among $\sin1, \sin2, \sin3, \ldots, \sin n$ (in radians)

How can we find the nearest number to zero among $$\sin1, \sin2, \sin3, \ldots, \sin n$$ (where $n$ is a natural number, and the functions are based on radians)?
Setareh
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How can I prove $f_M(x) = \min(f(x),M)$ is lower semi-continuous?

My question is: Suppose that $f:X\rightarrow \mathbb{R}$ is lower semicontinuous and M is a real number. Define $f_M:X\rightarrow\mathbb{R}$ by $$ f_M(x) = \min(f(x),M). $$ Prove that $f_M$ is lower semicontinuous. The definition that I am using…
RDizzl3
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A function with zero integral and zero integral with exponential kernel

Is there any nontrivial function, say $f\in C_0^\infty([0,\infty))$ such that $$\int_0^\infty f(x)\,dx=\int_0^\infty e^{-x}f(x)\,dx=0?$$ My guess is no (i.e. $f\equiv 0$), but am not really sure. I could show there is no such $f$ within a subset in…
Jingeon An-Lacroix
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union of infinitely many bounded sets is not bounded

Why is a union of infinitely many bounded sets not necessarily bounded, please? In addition, what condition can we add to make this union bounded, please?
user92699
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$h(x) = \begin{cases} e^{-(1/x)} \sec{x} & \text{for} \quad x \in (0,\pi/2) \\ 0 & \text{for} \quad x \leq 0, \end{cases} $ is of class $C^{\infty}$.

How do I prove that $h:(-\pi/2,\pi/2) \rightarrow [0,\infty)$, $$h(x) = \begin{cases} e^{-(1/x)} \sec{x} & \text{for} \quad x \in (0,\pi/2) \\ 0 & \text{for} \quad x \leq 0, \end{cases} $$ is $C^{\infty}$ on $(-\pi/2,\pi/2)$? By definition, a…
user38268
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Question on convex hull

Let $(x_n)_n$ be a countable subset of $C$ that is dense in $C$; For every $n$ let $C_n=conv\lbrace x_1, x_2, . . . , x_n \rbrace $ ($C\subset E$nonempty convex set, $E$ a finite-dimensional normed space) How to prove that $C_n$ is compact and that…
Vrouvrou
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