Recent Questions - Mathematics Stack Exchange |
- Applying window to a signal
- Embarrassingly basic calculus question
- help solving this partial differential equation
- Calculating the limit of a Wasserstein distance of two SDE's
- Will the L'Hopital's Rule work here, when it agrees with the derivation of the rule as I know it?
- Linear Programming Maximization Problem
- Solutions to $4^\alpha+1=q^\beta$
- Stolz Cesaro type limit
- Writing a grammar that creates a specific language from a given grammar in Chomsky normal form
- How do I quantify the amount of information in the following expression?
- What is the condition for the saddle point of a function of three variables?
- Can Holder Constant and Coefficient be related?
- Find the distance between two towers/building
- Rayleigh-Quotient eigenvalue problem
- pushforward of cohomology via a projection
- What can we say about the boundedness of second derivative of a function from that of its first derivative? [closed]
- Heat equation in plane polar coordinates with zero initial condition
- Limit of a function that has different formats when $x$ has different values
- Prove $ \operatorname{erfi}(x) \in O(e^{x^2})$
- Use quantifiers and predicates with more than one vari- able to express these statements.
- $f(z) = w$ has n solutions $\forall w \in \mathbb{C} \iff f$ is a polynomial of degree $n$
- Is the norm of a non-invertible ideal the gcd of the norm of its elements?
- Laplace Transform of Frequency Decay: $\sin(2\pi t \cdot f_0 \cdot e^{-t})u(t)$
- How many pairs of points at least on the unit sphere can be used to determine a rotation matrix in $\mathbb{R}^n$ uniquely?
- Uniqueness of Fokker-Planck solution
- Is the highest weight space of $V^{\otimes k}$ an irreducible representation of $S_k$?
- Is the convergence of these two series equivalent? (They come from Khinchin's theorem and the Duffin-Schaeffer conjecture.)
- Space of lipschitz functions form a Banach space
- Density function for large numbers with the property $\omega(n)=k\ $?
| Posted: 15 Apr 2022 09:34 AM PDT I want to apply a window to a signal so I can apply the Stockwell Transform. Am using the following formula $$ v(t) = \sum_{t'} u(t') W(t'-t) $$ Is this the way to do it as I am getting confused on whether to use $\;u(t') W(t'-t)\;$ or $\;u(t') W(t-t')$. |
| Embarrassingly basic calculus question Posted: 15 Apr 2022 09:33 AM PDT Let $f: \mathbb{R} \to \mathbb{R}$ be infinitely differentiable with $f \ge 0, f' \le 0, f'' \ge 0, f''' \le 0$, etc.. Prove that $f\cdot f'' \ge (f')^2$. I know this is true, and I do have a proof via Bernstein's theorem on totally monotone functions and Cauchy-Schwarz. However, I expect there to be a much simpler, and probably short and slick, proof. I would appreciate if someone found that proof. Maybe only the conditions on $f,f',f'',f'''$ are needed, idk. |
| help solving this partial differential equation Posted: 15 Apr 2022 09:33 AM PDT I can't solve this partial differential equation, $$y F_{x}+x F_y[1- (x^2 -y^2)(2h(x) y \sqrt{x^2-y^2 } +f(x) y^2 + 2 g(x))]=0$$ where $F(x,y)$ is a differentiable function of two-variables and $f(x)$, $g(x)$ and $h(x)$ are differentiable functions of one variable x. My attempt: $$\frac{dx}{y}=\frac{dy}{x[1- (x^2 -y^2)(2h(x) y \sqrt{x^2-y^2 } +f(x) y^2 + 2 g(x)]}, $$ then $$\frac{dy}{dx}= \frac{x}{y}[1- (x^2 -y^2)(2h(x) y \sqrt{x^2-y^2 } +f(x) y^2 + 2 g(x)]. $$ $$\frac{d}{dx}(\frac{y}{\sqrt{x^2-y^2}})= -2h(x) x^3 -f(x) x^3 \frac{y}{\sqrt{x^2- y^2}} - \frac{2g(x)x^3}{y \sqrt{x^2- y^2}}. $$ I am trying to find a function that help me to solve this problem. please, can anyone help me? |
| Calculating the limit of a Wasserstein distance of two SDE's Posted: 15 Apr 2022 09:33 AM PDT I am trying to prove that: $\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with $$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$ $$dZ_t = -h(Z_t)dt+\sqrt(\frac{2}{\beta})dB_t$$ I know that the Wasserstein distance in this case is given by: $$W_2(\mu, \nu) := \iint|\theta-\theta'|^2\zeta(d\theta d\theta'))^\frac{1}{2}$$ However, I am not too sure where to go from here. |
| Will the L'Hopital's Rule work here, when it agrees with the derivation of the rule as I know it? Posted: 15 Apr 2022 09:36 AM PDT Say I have to find the limit for: $$\lim_{x\rightarrow a}\frac{f(x)}{g(x)}$$ Such that $$g(a)=0≠f(a)$$ Multiplying the numerator and denominator by $g(x)$, I get$$\lim_{x\rightarrow a}\frac{f(x)g(x)}{g(x)^2}$$ and I get the indeterminate form $\frac 00$. So can I use the L'Hopital's Rule to find the limit here? From 3B1B's video about L'Hopital's Rule at time stamp 14:30, the rule works as the value of the numerator graph and denominator graph at given value which $x$ approaches is $0$, so I can just divide their derivatives. So will the L'Hopital's Rule work here, considering I can just derive the rule from such arrangement of the graphs? |
| Linear Programming Maximization Problem Posted: 15 Apr 2022 09:19 AM PDT Consider This Linear Program: Maximize {x}, Subject To The Constraints: -x+y <= 1; x+y <= 4; x-y <= 2; x,y >= 0 When maximizing this problem, does Z=x+0y ? |
| Solutions to $4^\alpha+1=q^\beta$ Posted: 15 Apr 2022 09:18 AM PDT Problem. Solve the equation $$ 4^\alpha+1=q^\beta $$ for prime number $q$ and positive integers $\alpha,\beta$. Background. This question arose in the context of the following question:
Or more specifically,
Since this question might be related to Mersenne primes, which is currently an unsolved problem in number theory, here I only search for the most accurate description of the solutions, like "$q$ can be a Mersenne prime" or "$\beta$ must be odd". My attempt. $4^\alpha+1\equiv2\pmod q$, but Fermat's little theorem tells us $q^\beta\equiv1\pmod3$ whenever $q>3$ and $\beta$ is even. Therefore $\beta$ must be odd. If $\alpha$ is odd then $5\vert4^\alpha+1$, therefore $q=5$. Assume $\alpha$ is even and $\alpha=2\alpha_1$. If $\alpha_1$ is odd then $q=17$. Similarly we finally get to $q=5,17,11,13,\cdots$. Moreover we have $r:=\delta_{q^\beta}(4)\vert q^{\beta-1}(q-1)=\varphi(q^\beta)$, which means $q\vert r$ or $r\vert(q-1)$. Here $\delta_p(a)$ denotes the order of $a$ modulo $p$. Are there any further observations? |
| Posted: 15 Apr 2022 09:17 AM PDT Given ${k(a_{k+1}-a_k)}_{k≥1}$ is bounded and ${\lim_{n\to \infty} {\frac{a_1+a_2+....+a_n}{n}}}=a$ Prove that $${\lim_{n\to \infty} a_n}=a$$ From ${k(a_{k+1}-a_k)}_{k≥1}$ is bounded, I have ${a_n}$ is bounded since Adding all, ${\exists m,M\epsilon \Bbb R}$ such that $$m≤{(a_{2}-a_1)}≤M$$ $$m≤{2(a_{3}-a_2)}≤M$$ $$m≤{3(a_{3}-a_2)}≤M$$ $$.........$$ $$m≤{n(a_{n+1}-a_n)}≤M$$ we have $nm≤na_{n+1}-(a_1+a_2+....+a_n)≤nM\implies$ $m≤a_{n+1}-{\frac{a_1+a_2+....+a_n}{n}}≤M$, since ${\lim_{n\to \infty} {\frac{a_1+a_2+....+a_n}{n}}}=a$ so for a $${\epsilon}=1, {\exists N\epsilon\Bbb N}$$ such that $$m+a-1≤a_{n+1}≤M+a+1$$ $\forall n≥N$ so $a_n$ is bounded. Then I stuck what next? Please can anyone help me without limsup or liminf use. Thanks in advance |
| Writing a grammar that creates a specific language from a given grammar in Chomsky normal form Posted: 15 Apr 2022 09:15 AM PDT Given a grammar in Chomsky normal form that creates a language L over alphabet Sigma, that the letter z doesn't belong to Sigma, without the empty string. I need to write a context-free grammar that creates the following L' language: L' = {a_1za_2za_3z...a_nz | n >= 1 and for every i, a_i is in Sigma, and exists b_1,b_2,...,b_n in Sigma such that a_1b_1...a_nb_n belongs to L} (you can also see this picture for L' definition) So I think every rule A -> a in the given Chomsky normal form grammar should be changed to A -> az but I am not sure if this is enough. |
| How do I quantify the amount of information in the following expression? Posted: 15 Apr 2022 09:15 AM PDT Ok, so lets have N = factorial(9999999999) The number N is mindbogglingly huge and yet, can be represented very neatly and compactly as factorial(9999999999). I have two questions:
I thought I could perhaps write and compile a program to get the factorial of a number N and compare the (size of the executable + size of 9999999999) with the number of bits used to store N. But, this would vary with the language used and there's also stuff in the executable other than the factorial definition. So, how do I quantify the amount of information that something like factorial(x) comprises? PS plz excuse the dumb question. I just find it fascinating that such large numbers can have such compact representations. |
| What is the condition for the saddle point of a function of three variables? Posted: 15 Apr 2022 09:29 AM PDT For a function $f(x,y)$ of two real variables $x$ and $y$, a point $(x_0,y_0)$ is a saddle point if the determinant of the Hessian matrix $$[f_{xx}f{yy}-(f_{xy})^2]_{x=x_0,y=y_0}<0.$$ If we are given a function $f(x,y,z)$ of three real variables $x,y$ and $z$, is there a similar criterion for a point $(x_0,y_0,z_0)$ to be a saddle point in terms of the determinant of Hessian matrix? |
| Can Holder Constant and Coefficient be related? Posted: 15 Apr 2022 09:08 AM PDT A function is Holder continuous if there exists $L,\alpha>0$ such that for all $x,y$ we have $$ |f(x) - f(y)|\leq L |x-y|^\alpha $$ My question is let's say there exists $\alpha\in (0,1)$ such that for all $x,y$ we have, $$ |f(x)-f(y)| \leq 2^{1-\alpha}|x-y|^\alpha $$ or more generally, $$ |f(x)-f(y)| \leq C(\alpha)|x-y|^\alpha, $$ Is this still considered to be Holder continuous despite the coefficient depending on the Holder constant? It appears that it should be Holder continuous but I feel like there is something silly I am missing that causes a problem. |
| Find the distance between two towers/building Posted: 15 Apr 2022 09:08 AM PDT Mr Gray stood on top of the CN tower and spotted a Casa lama, his angle of depression is 6 degrees. He then turned around 110 degree and found CNE. His angle of depression to CNE is about 9 degree. The height of CN tower is 553 m above the ground. Calculate the distance between casa lama and CNE. I Know how to measure angle of depression and that this question would consists of 2 triangles but I am confused how to use that 110 degree in calculation. What is that angle?? Please help me with this question and I really prefer if somebody draw a proper triangle and locate that 110 degree |
| Rayleigh-Quotient eigenvalue problem Posted: 15 Apr 2022 09:10 AM PDT It says in my script: For $A\in\mathbb{R}^{n\times n}$ $$\|Ax-R_A(x)x\|_2=min_{\lambda \in \mathbb{R}}\|Ax-\lambda x\|_2$$ in which $R_A(x)=\frac{x^TAx}{x^Tx}, x\in\mathbb{R}^n, x\neq 0$ is Rayleigh-quotient and $$\|\lambda-R_A(x)\|_2 \leq 2\|A\|_2\|v-x\|_2 \forall x\in\mathbb{R}^n, \|x\|_2=1$$ in which $\lambda\in\mathbb{R}$ is a eigenvalue and $v, \|v\|_2=1$ its eigenvector. It says the proofs are pretty easy, but I don´t know how to start.. Any help is greatly appreciated! |
| pushforward of cohomology via a projection Posted: 15 Apr 2022 09:14 AM PDT Let $X,Y$ be two smooth closed manifolds and consider the projection $p:X\times Y\rightarrow Y$. I want to understand the pushforward $$p_*:H^k(X\times Y,\mathbb{Z})\rightarrow H^k(Y,\mathbb{Z})$$ By our setting, we would like to use the Poincare duality (the idea c.f. this anwer) $$H^k(X\times Y,\mathbb{Z})\cong H_{n+m-k}(X\times Y,\mathbb{Z})$$ where we assume $n=\dim{X}$ and $m=\dim{Y}$. But, with this idea, we will get a map $$H^k(X\times Y,\mathbb{Z})\cong H_{n+m-k}(X\times Y,\mathbb{Z})\rightarrow H_{n+m-k}( Y,\mathbb{Z})\rightarrow H^{k-n}(Y,\mathbb{Z})$$ which is not what we expect (e.g. when $n=m$, this map is not so meaningful). So I want to know what is the correct way to define and understand such a map (must we use Thom-isomorphisms?). |
| Posted: 15 Apr 2022 09:10 AM PDT Suppose I have a function $f\in L^{2}[0,1)$, which has a bounded first order derivative and also $f'\in L^{2}[0,1)$. Then what can we say about the boundedness of $f'' $ ? |
| Heat equation in plane polar coordinates with zero initial condition Posted: 15 Apr 2022 09:20 AM PDT The problem is to solve $\displaystyle\frac{\partial u}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial u}{\partial r} \right) - \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}} = F$ on a unit radius disk subject to the conditions $\begin{align} u(t,1,\theta) = 0 && u(0,r,\theta) = 0, \end{align} $ and $u$ bounded as $r \rightarrow 0$. The source function $F$ is continuous, bounded, and obeys the same conditions as $u$. My Attempt I began by solving the homogeneous problem, $\displaystyle\frac{\partial u}{\partial t}-\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial u}{\partial r} \right) - \frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\theta^{2}} = 0$ using separation of variables, taking $u(t,r,\theta) = T(t)R(r)\Theta(\theta)$. For the angular part, this produced a Sturm-Liouville problem: $\begin{align} \Theta'' - \alpha\Theta = 0, && \Theta(\theta) = \Theta(\theta + 2\pi). \end{align} $ The eigenfunctions for this are $\begin{align} \Theta_{n}(\theta) = A_{n}\cos(n\theta) + B_{n}\sin(n\theta), && \alpha_{n}=-n^2, && n = 0,1,2,... \end{align} $ Using these eigenvalues and substituting into the radial ODE gives, after some rearranging and substitution, the ODE $\displaystyle x^{2}\frac{d^{2}B}{dx^{2}} + x\frac{dB}{dx} + \left( x^{2} - n^{2} \right)B = 0,$ where $B(x) = R(r)$, and $x = \mu r$. The bounded solution is the Bessel function of the first kind. The eigenvalues are $-\mu^{2} = -x_{nm}^{2}$, where $x_{nm}$ is the $m$th zero of the $n$th Bessel function of the first kind. Finally, solving the temporal ODE yields $T_{n}(t) = D_{n}e^{-x_{nm}^{2}t},$ for some constants $D_{n}$. By the superposition principle, the general solution is $\displaystyle u(t,r,\vartheta) = \sum_{n = 0,\, m=1}^{\infty}{J_{n}\left( x_{nm}r \right)e^{- x_{nm}^{2}t}\left\lbrack A_{nm}\cos\left( n\theta \right) + B_{nm}\sin\left( n\theta \right) \right\rbrack}.$ Attempting to impose the initial condition $u(0,r,\theta) = 0$ just gives the trivial solution. I suspect this is because the source function $F$ has not yet been accounted for. I considered introducing a particular solution, but this shouldn't be necessary as the boundary conditions are homogeneous. I have no idea how to proceed. Solving The Inhomogeneous Equation Solving the inhomogeneous equation for the source function $F$ could be done with an eigenfunction expansion. How is this done in the case of two spatial dimensions $r$ and $\theta$? Would the eigenfunction expansions be in the form of a Fourier-Bessel series? |
| Limit of a function that has different formats when $x$ has different values Posted: 15 Apr 2022 09:14 AM PDT $$f(x)=\begin{cases} 1/(x-2)^2, &\text{if } x<2 \\ 3, &\text{if } x=2 \\ 4/(x-2)^2 &\text{if } x>2 \end{cases}$$ So: $\lim_{x\to 2} f(x) = ?$ |
| Prove $ \operatorname{erfi}(x) \in O(e^{x^2})$ Posted: 15 Apr 2022 09:29 AM PDT How can we prove: $$ \operatorname{erfi}(x) \in O (e^{x^2}) $$ Samely we can prove: $$ \lim_{ x \rightarrow \infty } \frac {\int_0^x e^{t^2} \, dt} {e^{x^2}} = 0 $$ Also it is so awkward. The integral of a function should be greater than or equal to that function but here it seems like it is the opposite. |
| Use quantifiers and predicates with more than one vari- able to express these statements. Posted: 15 Apr 2022 09:14 AM PDT I am stuck on a question in Discrete Mathematics (7th ed.) by Kenneth H Rosen.
My solution is $$\exists x \forall y \exists z B(x,y,z),$$ where $B(x,y,z)$ means "student x is in room y in building z", and the universe of discourse x:{all students in the class} , y:{all rooms}, z:{all buildings}. |
| $f(z) = w$ has n solutions $\forall w \in \mathbb{C} \iff f$ is a polynomial of degree $n$ Posted: 15 Apr 2022 09:14 AM PDT Here is a conjecture that I came up with yesterday. Conjecture: Suppose that $f:\mathbb{C} \to \mathbb{C}$ is entire. Then $f(z) = w$ has $n$ solutions $\forall w \in \mathbb{C} \iff f $ is a polynomial of degree $n$. The converse follows immediately from the Fundamental Theorem of Algebra, but the forward implication is a lot more difficult. And after thinking about it more, it seems to me that the forward implication is probably not true. Unfortunately, I can't immediately think of a counterexample. I thought a little about the approach I might take to prove this. Suppose $f$ is as above. Then $f(z)=0$ has $n$ solutions, so we can write $f(z)=p(z)g(z)$ where $p$ is a polynomial of degree $n$ and $g$ is entire with $g\neq 0$. If $g$ is a constant, then we are done. Otherwise, by Liouville and Casaroti-Weierstrass, $g$ is unbounded and gets arbitrarily close to any value in $\mathbb{C}$. We can also write $g=e^h$ for some $h$ entire. (There is also Picard's theorem, but I am reluctant to use this as I don't know the proof.) The above gets somewhere, but I am not sure it leads to a proof without another insight. If the conjecture is false, then it definitely doesn't lead to a proof. So, is the conjecture true or false? And if it is true, am I on the right track to prove it? (I would appreciate small hints much more than a full answer.) |
| Is the norm of a non-invertible ideal the gcd of the norm of its elements? Posted: 15 Apr 2022 09:23 AM PDT Let $K$ be an imaginary quadratic number field and $\mathcal{O}$ a (possibly non-maximal) order inside $K$. Consider $I$ to be an $\mathcal{O}$-ideal. The norm $N(I)$ is given by the index $[\mathcal{O}:I]$. I'm reading a paper that states without proof that $$N(I)=\gcd(\{N(\alpha) \mid \alpha \in I\})$$ I've been trying to prove this statement. It's clear that $N(I) \mid \gcd(\{N(\alpha) \mid \alpha \in I\})$. Also, $N(I)\in I$, so $\gcd(\{N(\alpha) \mid \alpha \in I\}) \mid N(I)^2$. And this is where I get stuck. So, I started to wonder if this is actually true even for non-invertible ideals. For example, consider $\mathcal{O}=\mathbb{Z}[\sqrt{-3}]$ and the singular prime ideal $\mathfrak{p}=(2,1+\sqrt{-3})$ with norm 2. With the aid of a computer algebra system, I computed an expression for the norm of a generic element in $\mathfrak{p}$ and it's always divisible by 4. So, $\gcd(\{N(\alpha) \mid \alpha \in \mathfrak{p}\})=4$, while $N(\mathfrak{p})=2$. Hence, the equality cannot be true for every $\mathcal{O}$-ideal. My question is, do you know if this equality is true in the case of invertible ideals? If so, could you point me to a proof or a reference? Thanks! |
| Laplace Transform of Frequency Decay: $\sin(2\pi t \cdot f_0 \cdot e^{-t})u(t)$ Posted: 15 Apr 2022 09:10 AM PDT For a sine wave of exponentially decaying frequency, is the Laplace Transform numerically solvable? If not, why? One example could be: $$\sin(2\pi t \cdot f_0 \cdot e^{-t})u(t)$$ where $f_0$ is the frequency starting at $t=0$, and $u(t)$ is the unit-step function. I cannot find this equation on any Laplace tables, although there are real-world dynamic systems for which this would be helpful. Other variations might be:
|
| Posted: 15 Apr 2022 09:20 AM PDT A rotation matrix map the unit sphere $\mathbb{S}^n$ onto itself, we want to identify the rotation matrix $R$ by pairs of points $x,x'\in \mathbb{S}^n$, such that $$ Rx = x' $$ for $n = 2$, one pair of points is enough, for the general case, is it true that $n-1$ pair of points are enough? for $n = 3$, using the axis-angle representation of rotation matrix, the axis of rotation passing through the center of the unit sphere and intersecting the unit sphere with $e$, then $d(e,x) = d(e,x')$, that is $e$ must be on the circle that is the intersection of the unit sphere with the perpendicular plane of the segment $xx'$ passing through its middle point. using another pair of points can found another circle, and the point $e$ is the intersection of the two circles. so two pair of points should be enough. |
| Uniqueness of Fokker-Planck solution Posted: 15 Apr 2022 09:30 AM PDT I am interested in the Fokker-Planck equation of the evolution of a probability distribution, which is described by $$ \partial_t p(x,t) = \mathbf{\nabla}\cdot \big( \nabla E(x)\, p(x,t) + \nabla p(x,t) \big) $$ This stochastic process has the steady-state $ p_{ss}(x) \propto e^{-E(x)}$. My question is, whether or not the steady-state is unique (up to a constant factor) when we impose periodic boundary (i.e., say $x\in [-1,1]^d$ and $E$ is periodic). |
| Is the highest weight space of $V^{\otimes k}$ an irreducible representation of $S_k$? Posted: 15 Apr 2022 09:21 AM PDT Let $V$ be $n$-dimensional vector space over $\mathbb{C}$ with given action of $GL_n$. Let $W_\lambda$ be the highest weight space of weight $\lambda$ in $V^{\otimes k}$. Is $W_\lambda$ an irreducible $S_k$ representation? This should follow from Schur-Weyl duality, but I don't see why. |
| Posted: 15 Apr 2022 09:34 AM PDT I am trying to wrap my head around two theorems of Diophantine approximation: Khinchin's theorem and the Duffin and Schaeffer conjecture. To the best of my understanding, here is what they say: Khinchin's theorem. Let $\psi$ be a function $\mathbb{N} \to \mathbb{R}^+$ such that $q\psi(q)$ is non-increasing. Call $\alpha$ "$\psi$-approximable" if there exist infinitely many $\frac{p}{q}$ such that $|\alpha - \frac{p}{q}| < \frac{\psi(q)}{q}$. (Let's call this inequality (⋆).)
Duffin and Schaeffer conjecture. Let $\psi$ be any function $\mathbb{N} \to \mathbb{R}_{\geq 0}$. Call $\alpha$ "$\psi_r$-approximable" ($r$ for reduced) if there exist infinitely many reduced $\frac{p}{q}$ such that (⋆) holds.
(Note that $\phi(q)$ is Euler's totient function.) Now, suppose that $\psi$ is a function satisfying the conditions of Khinchin's theorem. In this case, an irrational number $\alpha$ is $\psi$-approximable if and only if $\alpha$ is $\psi_r$-approximable. Here's the reason. In this case, $\psi$ is decreasing, so if $\frac{km}{kn}$ satisfies (⋆) then so does $\frac{m}{n}$. And if $\frac{m}{n}$ satisfies (⋆), then for $k$ large enough $\frac{km}{kn}$ will not satisfy (⋆). So the only way infinitely many fractions can satisfy (⋆) is if infinitely many reduced fractions do. Here is what this seems to imply: for such functions $\psi$, the series $\sum_{q = 1}^\infty \psi(q)$ converges if and only if the series $\sum_{q = 1}^\infty \frac{\phi(q)\psi(q)}{q}$ converges. Indeed, $\sum_{q = 1}^\infty \psi(q)$ converges $\iff$ a.e. $\alpha$ is not $\psi$-approximable $\iff$ a.e. $\alpha$ is not $\psi_r$-approximable $\iff$ $\sum_{q = 1}^\infty \frac{\phi(q)\psi(q)}{q}$ converges. Here is my question. Is it really true that, if $\psi: \mathbb{N} \to \mathbb{R}^+$ is a function with the property that $q\psi(q)$ is non-increasing, then $\sum_{q = 1}^\infty \psi(q)$ converges $\iff $ $\sum_{q = 1}^\infty \frac{\phi(q)\psi(q)}{q}$ converges? Is there some "easy" way to see why? One direction is obvious, since $\psi(q)$ is always larger than $\frac{\phi(q)\psi(q)}{q}$. But the other direction seems interesting and surprising. For example, if we plug in $\psi(q) = \frac{1}{q\log q\log\log q}$, then we have that $\sum_{q = 2}^\infty \frac{1}{q\log q\log\log q}$ diverges, so the above would say that $\sum_{q = 2}^\infty \frac{\phi(q)}{q^2\log q\log\log q}$ diverges, too. (I've started at $q = 2$ to avoid dividing by 0.) Thanks for your help! Note. I have no advanced knowledge of either number theory or analysis. So if you could make your answer(s) user-friendly, I'd appreciate it. :-) Edit: $\sum_{q = 1}^\infty \psi(q)$ converges $\iff$ a.e. $\alpha$ is not $\psi$-approximable $\iff$ a.e. $\alpha$ is not $\psi_r$-approximable $\iff$ $\sum_{q = 1}^\infty \frac{\phi(q)\psi(q)}{q}$ converges. I didn't have those "not"s before. Edit 2: Changed to say "In this case, an irrational number $\alpha$ is $\psi$-approximable if and only if $\alpha$ is $\psi_r$-approximable." If $\alpha$ is rational, $\psi$-approximability and $\psi_r$-approximability are not equivalent. |
| Space of lipschitz functions form a Banach space Posted: 15 Apr 2022 09:24 AM PDT Let $X$ be a Banach space. Show that $L=\{f:X\to\mathbb{R}: f \mbox{ is Lipschitz}, f(0) = 0\}$ with the norm $$||f||_{Lip_0} = \sup\left\{\frac{|f(x)-f(y)|}{||x-y||}, x\neq y\in X\right\}$$ is a Banach space. I've found Banach space of p-Lipschitz functions but I did not understand the proof given. I have a few questions first. Which norm is $||x-y||$? So I need to prove that every Cauchy sequence in $L$ converges to an element of $L$, right? In other words, $\forall \epsilon>0$ there exists $n_0$ such that $m,n>n_0\implies ||f_m-f_n||_{Lip_0}<\epsilon$ $$ ||f_m-f_n||_{Lip_0} = \sup\left\{\frac{|(f_m-f_n)(x)-(f_m-f_n)(y)|}{||x-y||}, x\neq y\in X\right\} = \sup\left\{\frac{|f_m(x)-f_m(y)|}{||x-y||}+\frac{f_n(y)-f_n(x)}{||x-y||}, x\neq y\in X\right\}$$ both $f_m$ and $f_n$ are Lipschitz so they're continous, which means something I don't know what. |
| Density function for large numbers with the property $\omega(n)=k\ $? Posted: 15 Apr 2022 09:11 AM PDT For large numbers, there is a density function which is the probability that a number "near" $n$ is prime. This is $\large \frac{1}{\ln(n)}$ This allows to estimate the number of primes in the interval, lets say, $[10^{49},10^{49}+10^{10}]$ without calculating them.
|
| You are subscribed to email updates from Recent Questions - Mathematics Stack Exchange. To stop receiving these emails, you may unsubscribe now. | Email delivery powered by Google |
| Google, 1600 Amphitheatre Parkway, Mountain View, CA 94043, United States | |
No comments:
Post a Comment