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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A supremum/Inf property?

Can something justify this equality to me? $$\sup_y \{ \langle y,x\rangle + \inf_z\{f(z) - \langle y,z\rangle \} \} = \sup_y \{ \inf_z \{ f(z) - \langle y,x - z\rangle\} \}$$ I don't understand how you can just put the inner product put inside…
Lemon
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question about compactness proof in complete metric space!

I cannot solve this: Let $X$ be a complete metric space. Suppose that for any $r> 0$ there are finite points $x_1, x_2, \dots ,x_n$ such that $N_r(x_1),\dots, N_r(x_n)$ cover $X$. Show that $X$ is compact.
user143299
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Using the chain rule in $\mathbb{R}^n$

Suppose that the function $\psi:\mathbb{R}^2 \to \mathbb{R}$ is continuously differentiable. Define the function $g:\mathbb{R}^2 \to \mathbb{R}$ by $g(s,t) = \psi(s^2t,s)$ for $(s,t) \in \mathbb{R}^2$. Find $\frac{\partial g}{\partial s}(s,t)$. Here…
Zachary Luety
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Integraion of the function $1/r$ over a sphere in $\mathbb{R}^3$

Assume in $\mathbb{R}^3$ there is a sphere $S=S(A,R)$ centered at a point $A$ with radius $R$, and $|A|=a$, where $|A|$ is the Euclid norm of $A$. Now let a $X$ be a uniform distributed random point on $S$, we want to calculate $$E\frac{1}{|X|}.$$ …
zemora
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How to setup a sequence of functions?

I have a function $F(x,t)=\int_0^t f(s,x)ds$ and I want to see if I can write $$\frac{\partial F(x,t)}{\partial x}=\int_0^t \frac{\partial f(s,x)}{\partial x}ds$$ So, I basically want to know if I can pass the limit of the derivative towards the…
The Mighty Algernon
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Showing two sets have touching boundaries (function sets)

Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous on $[0,1]$ and differentiable on $(0,1)$. Suppose that $f(0)< 0 < f(1)$ and $f'(x) \neq 0$ for every $x \in (0,1)$. Let $S_{1} = \{ x \in [0,1]: f(x) > 0\}$ and $S_{2} = \{x \in [0,1]: f(x) <…
DRich
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Implicit Function Theorem

Lets $F(x, y, z):\mathbb{R}^{3}\to\mathbb{R}^{1}$, and lets $F_y\neq 0$ and $F_z\neq 0$ in some neighborhood $V$ of $(x_0, y_0, z_0)$. Am I right that: $$\frac{\partial F}{\partial x}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial…
Aspirin
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Existence of a differentiable path passing through 3 points in $R^{n}$.

Show that, given any three points of $R^{n}$, there is a differentiable path through these three points. I'm having difficulty solving this problem. I'm trying to solve this problem as follows: Let $a,b,c \in R^{n}$. Taking two straight paths…
user137153
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upper and lower limits of sequences

Suppose that $t_n\leq s_n$ for all $n\geq N_0$, and $\{s_n\}$ converges to s. Prove that lim sup $t_n\leq s$. I want to somehow use the fact that lim inf $t_n\leq$ lim inf $s_n$ and lim sup $t_n \leq$ lim inf $s_n$ but I don't know what to do from…
blubberbrot
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Regularity of a function between two paraboloids tangents

I know that the regularity of a continuous function $u$ between two paraboloids tangents in a neighbourhood of a point $x_0$ is $C^{1,1}$. I'd like to see for example, how to prove that $u$ is differentiable at $x_0$.
user29999
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inequality-why is it like that?

I saw the solution of an exercise and there it is used the following inequality: $$e^{-(n-1)x} \leq e^{-{(n-1)}} ,\forall x \in [0,+\infty)$$ Why is it like that? I haven't understood it.. $$$$ The exercise is: Let $f_n:[0,+\infty) \to \mathbb{R},…
evinda
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Calculating similarity of gaps between integers in a set

First off I should state that I'm not a mathematician, I'm a programmer (Python, Javascript). But I thought this was more of a mathematical question than a programming one, so I'm asking it here. I want to write a program that accepts a simple…
BigglesZX
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integration of product of even and odd function

I have a problem like this: Let $f:[-a,a]\to\mathbb R$ be a continuous function where $a>0$. If $f$ satisfies that $$\int_{-a}^a f(x)g(x)dx=0$$ for every integrable even function $g:[-a,a]\to\mathbb R,$ show that $f$ is an odd function. My…
homegrown
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Analysis over finite fields

At the beginning of my study in analysis I learned something about convergence of sequences for example, metric spaces and so forth... Most of the time we considered metric spaces $(\mathbb{K}, d), \mathbb{K} \in \{\mathbb{R},\mathbb{C} \}$ with a…
RedRose
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Necessary Condition-Counterexample

Let $f_n:I \to \mathbb{R}$ a sequence of functions,that does not get zero at any point.We suppose that $f_n \to f$ uniformly and that $\exists M>0$ such that $|f(x)| \geq M, \forall x \in I$.Then $\frac{1}{f_n} \to \frac{1}{f}$ uniformly. To show…
evinda
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