Let $f:[0,\infty)\to[0,\infty)$ be a smooth, strictly monotonically increasing function. Consider the inequality $$ \qquad \quad f(n) \cdot \frac{n-1}{n^2} > a \qquad (n\in\{2,3,4,...\};\ a>0). \qquad (*) $$ Assume that inequality $(*)$ holds for $n=2$. Notice that $\frac{n-1}{n^2}$ is strictly m...

I'm trying to define isomorphism in a logical way. Is the following statement true for Isomorphism's definition? Let $\langle S,⋆\rangle$ and $\langle S',⋆'\rangle$ be Algebraic Structures. These two structures are isomorphic if and only if: $$\left( \exists \phi : S \rightarrow S' \right) \left...

I refer to M.Cohen for my definitions. I am trying to understand the notion of "torsion" defined by $$ \tau(C) = \prod_{i = 0}^m[b_i b_{i-1}/ c_i]^{{-1}^{(i+1)}} $$ In particular, if I were to consider $C, C'$ to be within the same acyclic chain complex, then I should get $$ \tau(C') = \tau(C) \...

One of the isomorphism theorems states $(HN) \ / \ N \cong H \ / \ (H\cap N)$. I am confused about the first part $(HN) \ / \ N$, and whether it is equivalent to $H \ / \ N$. Every element in $HN$ is $hn$ for some $h \in H, n \in N$. So then every coset in $(HN) \ / \ N$ can be expressed as $\{hn...

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) obtaining $(v_{1},\cdots,v_{n})$. Now I'd like to swap position $i$ and $i+1$ in $ B = (b_1,\cdots,b_...

I know the typical parameterization of an ellipse is (a cos(t), b sin(t)). However, this is not an arc-length parameterization and I can't make the usual techniques for getting an arc-length parameterization to work since there is square root in the integrand that doesn't simplify if a is not equ...

My teacher given me the definition of the canonical spectrum theorem which is given below: Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and set of eigen values,$\sigma(A)$={$\lambda_1$,$\lambda_2$........,$\lambda_k$}. $A$ is diagonalisable $\Leftrightarrow A = \sum{\lambda_ip_i}$ such tha...

Is there any reference on the Feynman-Kac theorem of the weak solution of parabolic PDEs? So far I can only find the one for classical solution.

This is a problem from Linear Algebra Done Right 4th edition Chapter 7B (This is an open access book available from Professor Sheldor Axler's website). I am self-studying and need some help. The problem goes like this: T is a operator on a finite-dimensional vector space over F (field). (1) Suppo...

I don't understand what I found when calculating the expected value of a card game. A deck contains 40 cards. 8 of them are red cards and 32 of them are blue cards. At the start of the game, 5 cards are drawn to be the starting hand. The question is to find the expected value of drawing red cards...

Suppose I have a triangular $d\times d$ matrix $A$ with constant rows normalized to have Euclidean norm 1. Empirically it appears that operator norm (largest singular value) of such matrix is $\sqrt{d \log 2}$, for large $d$, why? For instance, for $d=5$ $$A=\left( \begin{array}{ccccc} 1 & 0 & 0 &...

I have an amazing ancient problem collection book by Chistyakov. I have managed to solve 165 problems of 248 (as of now, i. e. of time of current question posting); the remaining ones are really hard. If you permit, I would like to ask help for problem #100. One must find the legs (= catheti) of ...

I am interested in the following integral: $$f(x)=\int dt \phi(t) [x+\sigma t]_+^n $$ where $\phi(x)=e^{-x^2/2}/\sqrt{2\pi}$ is a standard Gaussian distribution, $[x]_+=max(x,0)$ is a rectification, $n\ge0$ an exponent, and $\sigma>0$ the magnitude of the noise. Denoting $\Phi(x)=\int_{-\infty}^x...

Let $((0,1], \mathcal{B},\mu)$ be a measure space. Let $\{A_i\}_{i\ge 1}$ be a sequence of Borel measurable sets so that $\mu(A_i)\ge c$ for all $i\ge 1$ and some universal constants $c\in (0,1)$. Show that for all $k=1,2,3,\dots$ and $\epsilon>0$, there exists $i_1<i_2<\dots<i_k$ so that $$ \mu(...

Let $(X_t)_{t\geq0}$ be a stochastic process on some probability sprace $(\Omega, \mathcal{F}, P)$. Then for $s < t$, we define the $\textit{increment}$ of the process, $X_t - X_s$ over the interval $[s, t]$. The process $(X_t)_{t\geq0}$ has $\textit{independent increments}$ if, for every set of...

Since every group is the homomorphic image of a free group, and every module is the homomorphic image of a free module, do we have an analogous result for the rings? To take into account of nonunital rings, I wonder if we can always start from an ideal, or at least a subring, of a free algebra, a...

I have been told to calculate $$ \int_0^{2\pi}\frac{1}{2+2\text{sin}(\theta)} d\theta $$ I set $z = e^{i \theta}$ so parameterising by the unit circle and ended up with $$ \int_C \frac{1}{(z+i)^2} dz $$ My issue is that $-i$ is not in the interior of the unit circle so I can't apply the cauchy in...

Find the minimum of $y=f(x)=\dfrac{x^3}{x-6}$ for $x>6$ I can solve the question using derivatives but I have no any idea how to do it without them. Using derivatives, we find $x=9$ and $y_{min}=243$.

$$u_{n} = \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)} + \frac{1}{3\cdot(n-2)} + \dots + \frac{1}{n\cdot1}.$$ Show that, $\lim_{n\rightarrow\infty} u_n = 0$. The only approach I can see is either finding $nu_{n}$ or $(n+1)u_{n}$ and seeing that: $$(n+1) \ u_{n+1} = (1+\frac{1}{n}) + (\frac{1}{2} + ...

Let's take a Vitali set in a model of ZFC, then map its elements to the corresponding reals in the Solovay model and consider them as a set. We get a Vitali set in the Solovay model while it shouldn't exist there. Where is the flaw in that argument? Would that collection of reals simply not const...