Problem on Function of Prime Number.
Let $f:(1,\infty)\to(0,\infty)$ be a continuous function such that for every $n\in\mathbb{N}$, $f(n)$ is the smallest prime factor of $n$. Then which of the following is/are correct?a)...
View ArticleProve that if $f$ is continuous, and has two asymptotes, then it is uniformly...
The exercise is the following:Let $f:\mathbb{R} \rightarrow \mathbb{R} $ continuous such that$\lim_{x\rightarrow+\infty} f(x) = \ell_1$ and $\lim_{x\rightarrow-\infty} f(x) = \ell_2$ for some certain...
View ArticleFunctional equation $f(xf(y)) + f(yf(x)) = 1 + f(x + y) \space$ [B6 Putnam...
How do we find all smooth functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that$$f(xf(y)) + f(yf(x)) = 1 + f(x + y)$$for all $x, y > 0$?By comparing degrees, the only polynomial solution is $f(x)...
View ArticleSmooth case of Jordan Brouwer Separation Theorem; proving there is more than...
I'm having trouble proving something.I am using smooth degree theory to prove the Jordan Brouwer Separation Theorem for a smooth compact connected oriented $n$ dimensional surface $M \subset...
View ArticleLinking Fourier Coefficients of periodic functions
Let $\tau\in (0,1)$ and assume that we have a $\tau$-periodic function $$f_1(t) = \sum\limits_{k\in\mathbb{Z}} a^1_k e^{\frac{2\pi i k}{\tau}t},$$a $(1-\tau)$-periodic function$$f_2(t) =...
View ArticleWhat does it mean for a curve to be another parametrization of a path?
I think I am somewhat struggling understanding paths. I have a curve $\phi$ defined as closed and simple which gives the path $C_\phi$. There is another curve $\psi$ that is defined as another...
View Articlecontinuous extension and smooth extension of a function
Let $X$ be a metric space. Let $E$ be a subset of $X$.(1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such that...
View ArticleOn surface of $C = \{ x^2+y^2=1,0
On the surface of the cylinder $C = \{ x^2 + y^2 = 1, \ 0<z<1 \}$ there is subset A that is measurable in terms of $\lambda_2$.Additionally, we know that: $A_t = A \cap \{z <t \} ,...
View ArticleThe set of all ellipsoids $\mathcal{E}(A)$ contained in a bounded open set...
We call $A\subset\Bbb R^d$ a convex body if $A$ is a convex, non-empty, open, and bounded. The open unit ball in $\Bbb R^d$ is denoted by $B_d$. In Tao-Vu's book, they say:Define an ellipsoid to be any...
View ArticleIf a Radon measure is a tempered distribution, does it integrate all Schwartz...
The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a nonnegative Radon measure $\mu$ on $\mathbb{R}^n$. Let...
View ArticleLet $f:\mathbb{R}^n\to \mathbb{R}$ be a $C^1$ function, if $|\nabla f|=1,$ is...
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is a $C^1$ function, I wonder whether or not $f$ is surjective provided that $|\nabla f(x)|=1$ for any $x\in\mathbb{R}^n$ ?When $n=1,$ the condition $|f^\prime|=1$...
View ArticleTorsion subgroup of elliptic curve via birational transformation
Compute the torsion subgroup of $Y^2=X(X-1)(X-2)$.The solution given by my instructor is to observe that the birational transformation $(X,Y)\mapsto (X-1,Y)$ takes the curve to $Y^2=X(X+1)(X-1)=X^3-X$...
View ArticleHelp me calculate the triple summation
ProblemWe consider$$\sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\xi_i)(x_{j\nu}-\xi_j) = \sum_{\nu=1}^n \sum_{i=1}^k \sum_{j=1}^k a_{ij}(x_{i\nu}-\bar{x}_i)(x_{j\nu}-\bar{x}_j) - 2n...
View ArticleEquation in space of complex numbers
Giving an equation: $$z^2-(m+2)z+4(m-1)=0$$I need to find the number of all the integer $m$ such that this equation has two complex solutions $z_1$, $z_2$ that satisfy: $$\vert...
View ArticleUniformly continuous derivative implies existence of limit
Let $f \in C^1([0, +\infty))$. Suppose that $\lim_{x \rightarrow +\infty} f(x)=L$ and $f'$ is uniformly continuous.Show that $$\lim_{x \rightarrow +\infty} f'(x) + f(x)=L$$I tried to apply L'Hospital's...
View ArticleLet f : R → R be differentiable at x = 0. Define g(x) = f(x^2 ). Show that g...
Let $f : \mathbb{R} → \mathbb{R}$ be differentiable at $x = 0$. Define $g(x) = f(x^2)$. Showthat $g$ is differentiable at $0 $(by the chain rule).
View ArticleQuotient of decreasing bounded sequences is definely increasing
Suppose that $\{a_k\}_{k\in\mathbb{N}}$ is a bounded sequence of real numbers with $a_k\in[0,1]$, $\forall k\in\mathbb{N}$. Suppose that $a_k$ is decreasing ($a_k > a_{k+1})$ and that...
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Strange substitution made in a paper to find asymptotics
In the quoted section from this paper, why is the author able to "substitute this result into Eq. (2.1)"? This should hold for $z$ large. But not everything on the contour is large. Why can the author...
View ArticleMeaning of the method of characteristics at crossing points of characteristic...
Given a system of linear PDE's of first order such as $w^1_t + cw^1_x = 0,w^2_t - cw^2_x =0,$ one usually uses the method of characteristics to find the solution $(w^1(x,t),w^2(x,t))$ on each point of...
View ArticleLet f monotone function in $\mathbb{R}$. Then $x_n$ Cauchy implies $f(x_n)$...
I have to prove is true the following, given $f:\mathbb{R}\to\mathbb{R}$, f monotone in $\mathbb{R}$:$\forall x_n$ sequence in $\mathbb{R}$, if $(x_n)$ is a Cauchy sequence, then $f(x_n)$ is a Cauchy...
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