Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 64

Computing $2 binom{n}{0} + 2^2 frac{binom{n}{1}}{2} + 2^3 frac{binom{n}{2}}{3} + cdots + 2^{n+1} frac{binom{n}{n}}{n+1}$

How can I compute the sum $2 binom{n}{0} + 2^2 frac{binom{n}{1}}{2} +2^3 frac{binom{n}{2}}{3} + cdots + 2^{n+1} frac{binom{n}{n}}{n+1}$? I think I should expand $(1+ sqrt{2})^n$ or something...

Asked on 11/29/2021 by RaduV

4 answer

Let $Lin End(V)$ with $L(V)=W$. Then $Tr(L)=Tr(L|_W)$

Let $V$ be a finite dimensional vector space and $W$ a subspace. Let $L$ be an endomorphism with image is $W$. Then $Tr(L)=Tr(L|_W)$ where ...

Asked on 11/29/2021

1 answer

What's nonsensical about this definition of the order of a meromorphic function?

According to this math.stackexchange.com answer, the following definition of Huybrechts in his book Complex Geometry is nonsensical:Let $X$ be a complex manifold. Let $Y subset X$ be...

Asked on 11/29/2021

1 answer

Let $lambda$ be a real eigenvalue of matrix $AB$. Prove that $|lambda| > 1$.

Let $A$ and $B$ be real symmetric matrices with all eigenvalues strictly greater than 1. Let$lambda$ be a real eigenvalue of matrix $AB$. Prove that...

Asked on 11/29/2021

1 answer

Integral of a product of Bessel functions of the first kind

I want to do this integral $H(rho)=int_{0}^{infty} J_1(2 pi Lr)J_0(2pi rho r)dr$, where $J_1$ and $J_0$ are Bessel functions of the first kind and $Lin mathbb{R}$...

Asked on 11/29/2021 by user740332

1 answer

Connectives in George Tourlakis' Mathematical Logic

In page 10 of Mathematical Logic, Tourlakis says that "Readers who have done some elementary course in logic, or in the context of a programming course, may have learned that...

Asked on 11/29/2021 by Darvid

1 answer

Let $T=int_{0}^{x}f(y)dt$. Find eigenvalues and range of $T+T^*$

Consider $T=int_{0}^{x}f(y)dt$ as a map from $L^2[0,M]$ to $L^2[0,M]$ find eigenvalues and range of $T+T^*$I am pretty sure the solution is simple but I would like...

Asked on 11/29/2021

0 answer

Find the smallest eigenvalue of $G=[ exp(-(x_i-x_j )^2]_{i,j}$ for ${bf x}=[x_1,dots,x_n]$

Consider a sequence ${x_1,...,x_n }$ such that $b=max_i |x_i|$ and $d_{min}=min_{ij: i neq j} |x_i-x_j|$. We assume that $b<infty$ and $d_{min}>0$. Can we...

Asked on 11/29/2021

2 answer

Proof of convergence of $sum_{n=1}^{infty}frac{(-1)^{lfloor nsqrt{2}rfloor}}{n}$

I had this task long long time ago in a calculus class. I remember it had a very elegant solution using nice property of $text{Spec}(sqrt{2})$,where$$text{Spec}(alpha)={lflooralpha...

Asked on 11/29/2021 by kubus

1 answer

Inverting product of non-square matrices?

I am working on an Optimization problem and I need to show that $$AB(B^TAB)^{-1} B^T = I_n$$ where $A$ is $n times n$ and invertible, $B$...

Asked on 11/29/2021

1 answer

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