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Mathematics : Recent Questions and Answers (Page 47)

Find answers to your questions about Mathematics or help others by answering their Mathematics questions.

Financial Mathematics : Annuity loan with a different first period?

Edit:Typo in $LaTeX$ transcription. Edit 2: Please note that my assumption here is wrong. The correct equation is available in my separate answer. When trying to teach myself some...

Asked on 12/08/2021

2 answer

Complete DVR containing a field isomorphic to residue field

Let $(R, mathfrak m, k)$ be a ($mathfrak m$-adically) complete DVR containing $kcong R/mathfrak m$. Also assume $k$ is algebraically closed. Then, is...

Asked on 12/08/2021 by uno

2 answer

Positive semi-definite real matrix with unit diagonal

Give an example of a $ntimes n$ positive semi-definite real matrix $Min mathbb{R}^{n times n}$, such that the following two conditions hold:the eigenvalues $lambda_1, dots, lambda_n$ of...

Asked on 12/08/2021 by fq00

1 answer

Prove $prod_{nge0}^{ }frac{left(n+aright)left(n+bright)}{left(n+cright)left(n+dright)}=frac{Gamma(c)Gamma(d) }{ Gamma(a)Gamma(b)}$

How to prove the following identity? $$prod_{nge0}^{ }frac{left(n+aright)left(n+bright)}{left(n+cright)left(n+dright)}tag{$a,b,c,d in mathbb R$}=frac{Gamma(c)Gamma(d) }{ Gamma(a)Gamma(b)}$$ For $a+b=c+d$.The product can be rewritten as :$$prod_{nge0}^{ }frac{left(n+aright)left(n+bright)}{left(n+cright)left(n+dright)}=prod_{nge0}^{ }frac{n^{2}+left(a+bright)n+ab}{n^{2}+left(c+dright)n+cd}$$$$=prod_{nge0}^{ }frac{n^{2}+left(a+bright)n+ab}{n^{2}+left(a+bright)n+cd}=prod_{nge0}^{...

Asked on 12/08/2021

1 answer

Closed-form expression for $prod_{n=0}^{infty}frac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}$?

I have recently come across this infinite product, and I was wondering what methods I could use to express the product in closed-form (if it is even possible): $$prod_{n=0}^{infty}dfrac{(4n+3)^{1/(4n+3)}}{(4n+5)^{1/(4n+5)}}=dfrac{3^{1/3}}{5^{1/5}}cdot...

Asked on 12/08/2021 by Harukr

1 answer

Examples of topological Immersion which are not embedding

It is said that topological immersion is locally 1-1 while embedding is globally 1-1, so we can say topological immersion is locally 'embedding'? (this also means except for a few...

Asked on 12/08/2021

0 answer

Is $sin(frac{1}{|z|})$ holomorphic on $Bbb C-{0}$?

$f(z)=sin(frac{1}{|z|})$, $zin Bbb C-{0}$. $$frac{partial f}{partialbar{z}}=frac{partial sin(frac{1}{|z|})}{partialbar{z}}=frac{partial}{partial bar{z}}sin(frac{1}{(zbar z)^{1/2}})=cos(frac{1}{|z|})(2^{-1}z)(zbar{z})^{frac{-3}{2}} neq 0$$ So, $f(z)$ is not holomorphic on $Bbb C-{0}$. Am I correct?...

Asked on 12/08/2021

2 answer

Show that $V=Z(x;T)oplus Z(y;T)$ and the $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors implies that $V$ is cyclic

Provided that $V=Z(x;T)oplus Z(y;T)$ where $Z(v;T)$ denotes the cyclic subspace and the corresponding $T$-annihilators $mu_{T,x},,mu_{T,y}$ do not share any common divisors, show that $V$ is...

Asked on 12/08/2021

2 answer

What is exponential map in differential geometry

Is $exp_{q}(v)$ a projection of point $q$ to some point $q'$ along the geodesic whose tangent (right?) at $q$ is the vector $v$? And so...

Asked on 12/06/2021

1 answer

Question on cardinality of sets

Let $a=bigcup_{iin I_a}a_i$ and $b=bigcup_{jin I_b}b_j$ where $I_a$ is the index set of $a$ and $I_b$ is the index set of $b$, such that...

Asked on 12/06/2021 by jshthng

1 answer

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