Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 27

The resultant of two homogeneous polynomials is homogeneous

I haven't been able to find a proof for this theorem in the literature:Let $f,gin k[x_0,dots,x_k]$ be homogeneous polynomials, of degree $m$ and $n$ respectively. Then $R_{x_0}(f,g)$ is homogeneous of...

Asked on 12/21/2021

1 answer

Real Analysis Methodologies to Prove $-int_x^infty frac{cos(t)}{t},dt=gamma+log(x)+int_0^x frac{cos(t)-1}{t},dt$ for $x>0$

MOTIVATION:In this question, the OP asked to prove the Cosine Integral identity $$-int_x^infty frac{cos(t)}{t},dt=gamma+log(x)+int_0^x frac{cos(t)-1}{t},dt$$ where $gamma $ is the Euler-Mascheroni constant. However,...

Asked on 12/21/2021 by Mark Viola

1 answer

Determine the $lambda in mathbb{R}$ for which this integral converges

Consider the cuspoidal cubic given by $x^2 - y^3 =0$ in $mathbb{C}^2$. The log-canonical threshold of the cuspoidal cubic is determined by finding the largest value of ...

Asked on 12/21/2021

2 answer

How to prove that there is an infinite number of primitive pythagorean triples such as $b=a+1$ and $2 | a$?

I need to prove that there is an infinite primitive pythagorean triples such as $b=a+1$ and $2 | a$ but I don't know how. I know that ...

Asked on 12/21/2021

2 answer

$int_0^1frac{lnleft(1-tright)ln^3 t}{2-t}dt$

From results and techniques of Integrating $int_0^1 frac{ln(1+x)ln^3 x}{1+x},dx$ with restricted techniques i think i can prove that: $displaystyle int_0^1frac{lnleft(1-tright)ln^3 t}{2-t}dt=-frac{3}{16}zeta(5)-frac{21}{4}zeta(2)zeta(3)+frac{9}{4}zeta(4)ln 2-frac{3}{20}ln^5 2+18 text{Li}_5left(frac{1}{2}right)$ I'm wondering if this...

Asked on 12/21/2021 by FDP

1 answer

Convergence of $sum_{n=1}^infty frac{(-1)^n}{x^2-n^2}$ in the reals and in the complex numbers

The goal is to find the values of $xinmathbb{R}$ (or $zinmathbb{C}$) such that the series $$sum_{n=1}^infty frac{(-1)^n}{x^2-n^2}$$ converges. The symbols $mathbb{R},mathbb{C}$ exclude $lbrace...

Asked on 12/21/2021

2 answer

Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $(R,mathfrak m, k)$ be a local complete Gorenstein ring of dimension $d$. Let $M,N$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $d$) ...

Asked on 12/21/2021 by user521337

1 answer

Is the set ${x in mathbb R^n : d(x, M) = c}$ a smooth manifold for a small constant $c$ when $M$ is a smooth manifold embedded in $mathbb R^n$?

I am a beginner of differential geometry. Let $M$ be a smooth manifold embedded in $mathbb R^n$ and consider thesubset$$ ...

Asked on 12/21/2021 by Yoshimi Saito

1 answer

Topological proof that a continuous injective real function on an interval has a continuous inverse

The text Topology and Groupoids by Brown states the following on page 8 in Chapter 1:1.2.4 (Inverse rule.) Let $f colon A to mathbb{R}$ be a real function which...

Asked on 12/21/2021

3 answer

Uniform motion question

A biker and a skater set out at 4:30 PM from the same point, headed in the same direction. The biker is travelling at a rate of 15 km/hr faster...

Asked on 12/21/2021 by Simran

3 answer

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