$$\sum_{K=0}^\infty \frac{(-1)^K}{(K+1)(2K+1)3^K}$$ I have no idea on how to tackle this problem.the series doesn't look like a telescopic series I tried splitting into partial reactions(without disturbing the 3^k But it's of no use I tried to multiply the whole series with 3 and then subtract th...

Let $B$ be a Banach space, let $(X,\mathcal{A})$ be a measurable space, and let $\mu:\mathcal{A}\to B$ be a vector-valued measure. In general, if $B$ doesn't have the Radon-Nikodym property, it is not true that we can express $\mu$ as $$ \mu(A) = \int_A f(x)\,m(dx) $$ for some (real-valued) measu...

My question is about separation of $\displaystyle \int _0 ^2 f(x) dx$ defined by $f(x) = \left\{\begin{aligned} &x^2 ,\ x \in [0,1]\\ &x^4+4 ,\ x \in (1,2] \end{aligned} \right.$ Of course, we should write in the following way $\displaystyle \int _0 ^1 f(x) dx + \displaystyle \int _1 ^2 f(x) dx=\...

I read in Gallauer's notes on infinity categories, in "Warning 1.9." without proof, that the geometric realization of a simplicial set does not determine it and wanted to make sure I understand why. A proof of this statement should somehow construct a pair of non-isomorphic simplicial sets having...

Suppose $A$ is an abelian group and $$0\to A\to H\to G\to 0$$ is exact. Does it follow that that this SES is isomorphic to one of the form $$ 0\to A\to H'\to G\to 0$$ such that $A$ is contained in the center $Z(H')$ of $H'$? I ask for the following reason: According to this article, there's a bi...

This was from a past exam we were given. I am stuck, but here is what I have so far. To find the area of EFGH, we can find the individual coordinates of E,F,G,H so we can find the distance between each vertex and finally compute the area. According to hint, I will centre $B(0,0)$. Since $|AB| =2...

Let $D:X\to Y$ be a Fredholm operator, so that its index $\dim\ker (D)-\dim \text{coker}(D)$ is defined. We can view $D$ as a complex $\mathfrak{D}:0\to X\xrightarrow{D} Y\to 0$, and for a chain map $T:\mathfrak{D}\to \mathfrak{D}$, i.e. a commutative diagram $\require{AMScd}$ \begin{CD} X @>{D}>>...

I am wondering if there is a central limit theorem for array of dependent random variables. Suppose we have array of random variables $\{X_{nm},n=1,2,\dots,m=1,2,\dots,r_n\}$ where $X_{nm}$ are identically distributed, within each array $X_{nm}$ are dependent but independent across arrays (i.e., ...

Consider the following generalized linear combination of trigonometric polynomials $$ P_{a,v}(t) = \sum_{k=1}^n (a_k + i\langle v_k,t\rangle) e^{i \langle x_k, t\rangle} $$ with $a=(a_1,\dots, a_n)\in \mathbb{R}^n$ and $v_k,t,x_k \in \mathbb{R}^d$ with pairwise distinct $x_k$. I need the followin...

Let $N^a$, $N^b$ be two jump process with stochastic intensity process $(\lambda^a_t)_{t\in\mathbb{R}}$, $(\lambda^b_t)_{t\in\mathbb{R}}$ (the lambdas are $\mathcal{F}_t$ -adapted). Let $N$ defined by : $N_t := N^a_t + N^b_t$. Now define $\Delta N_t = N_t - N_{t-} \in \{0, 1\} a.s.$ if $N$ jumps ...

I'm trying to solve Exercise 4.6 from Albiac and Kalton's book "Topics in Banach space theory". The exercise is as follows: (a) If $K$ is extremally disconnected, show that for every bounded lower semicontinuous function $f$, the upper semicontinuous regularization $$\tilde{f}(s) = \inf \{g(s) :...

Let $E\to M$ be an oriented rank 3 real vector bundle over a smooth 4-manifold $M$ and suppose $p_1(E)=0$. Why does this imply that the structure group of $E\to M$ can be reduced to a finite group $G$? I merely know that two $SO(3)$-bundles over a 4-manifold are isomorphic if and only if they hav...

I am familiar with alternating series in the form $\sum_{n=1}^{\infty} (-1)^na_n$ where the first term is negative and $\sum_{n=1}^{\infty} (-1)^{n+1}a_n$ where the second term is negative. I know these series alternate with every other term being negative. My question is: does a way to make the ...

Can a connected planar graph have 10 vertices and edges? is this possible? Using Euler’s formula, $V − E + F = 2$. $10 − 10 + F = 2$, Therefore $F = 2$. Do I also need to use this formula: $2E$ $\geq$ $3F$? or Do I use $E \leq 3v-6$? I'm a little lost if this type of graph is possible or not and ...

Prove or Disprove: Is there a connected planar graph with an odd number of faces where every vertex has a degree of 6? I know, Theorem: In a connected planar graph where each vertex has the same degree of 6, the number of faces cannot be odd. Proof: Let G be a connected planar graph with every ve...

I'm searching the specific value of constant C in Proposition 2.6.1 regarding the sums of independent sub-Gaussian random variables, as presented in Vershynin's "High-Dimensional Probability" : https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-book.pdf. Proposition 2.6.1 : Let $X1, . . . , X...

Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. Define $$\hat G_n(t)=P_{F_n}\Bigg{(}\sqrt{n}\Big{[}T(F_n^*)-T(F_n)\Big{]}\le t\Bigg{)}\\G_n(t)=P_{F_n}\Bigg{(...

In Artur Avila and Jairo Bochi's lecture notes (see here: http://mat.puc-rio.br/~jairo/docs/trieste.pdf) in section 3.1 they deal with Lyapunov exponents of products of random i.i.d. matrices. Let $\{Y_{i}\}$ be a squence of random independent matrices in $SL(2,\mathbb{R})$ (i.e. real matrices 2x...

The actual question states the following; "Find the mass of a Quater-Disc (in terms of R), in the first quadrant, of radius 'R' if density varies as D = xy" My first thought was somehow turning this problem into another one in hyperbolic coordinates and integrating for the transformation of a cir...

When using the formula θ = arc length/ radius we should be getting a ratio between the same physical quantities(length) so then why does it have a unit radian(rad)? Or is the ratio the unit?

I need help understanding the following: In particular, I need help understanding how $P$ is defined. What is $\mathbb{1}\otimes i^* $ and then the map pointing upwards into $H^*(X,R)$? I understand that $P$ is just the composition of these two maps, but I don't understand what these two maps ar...

Is there a subset $S$ of the rationals between 0 and 1 $$S \subset \mathbb{Q} \cap[0,1] \stackrel{\text{def}}= [0,1]_\mathbb{Q} $$ that is dense in $[0,1]_\mathbb{Q}$, in the sense that $$\forall q_1, q_2 \in [0,1]_\mathbb{Q}, q_1<q_2,\quad\exists s\in S, q_1<s<q_2$$ has the property that given...

I'm studying Field Extension Theory and Galois Theory, and unlike (for example) studying Linear Algebra, the concepts feel very much like manipulation of symbols, and I don't feel to have a very strong intuitive judge inside me to say when a theorem is true or not. Besides, I have to present this...

Say we have a collection $\Omega$ of sets, and a statement $P(s)$ which can be either true or false depending on the input set $s$. Say we take two arbitrary sets $a$ and $b$ in $\Omega$ and that for all $a,b\in\Omega$, $P(a) \land P(b) \implies P(a \cap b)$. Does this then imply that $\displayst...

My professor gives me the theorem on non- diagonalisable matrices: Let a matrix the $A \in M_{n\times n}(\mathbb{R}).$ $A$ has $k$ independent eigen vectors $\Leftrightarrow$ A is similar to $$ \begin{pmatrix} \Lambda & B \\ 0 & C \end{pmatrix} $$ where $$\Lambda= diag(\lambda_1,\lambda_2....