Mathematics Questions, Problems & Solutions : TransWikia.com ~ Page 58

Radioactivity formula using differential equations?

After $12$ days, a $40$ gram substance decays to $9.3$ grams. If the decay rate is proportional to the amount of substance left, how much of the...

Asked on 12/03/2021 by Mitali Mittal

3 answer

What is the algebraic interpretation of a contracted product?

Suppose you have a linear reductive group $G$ acting on an algebraic variety $X$. Let $Pleq G$ be an algebraic subgroup and let $Y subseteq X$...

Asked on 12/03/2021 by James Steele

0 answer

$f:[0,1]rightarrow[0,1]$, measurable, and $int_{[0,1]}f(x)dx=yimplies m{x:f(x)>frac{y}{2}}geqfrac{y}{2}$.

Question: Suppose $f:[0,1]rightarrow[0,1]$ is a measurable function such that $int_0^1f(x)dx=y$. Prove that $m{x:f(x)>frac{y}{2}}geqfrac{y}{2}$. My thoughts: Since we have $int_0^1f(x)dx=yimplies frac{1}{2}int_0^1f(x)dx=frac{y}{2}$. Now, I was hoping...

Asked on 12/03/2021

1 answer

Evaluating an integral with a division by $0$ issue

Problem: Evaluate the following integral.$$ int_{0}^{4} frac{dx}{sqrt{4-x}} $$ Answer: Let $u = 4 - x$. We have:begin{align*}du &= -dx \int_{0}^{4} frac{dx}{sqrt{4-x}} &= int_{4}^{0}...

Asked on 12/01/2021

3 answer

Evaluating $sumlimits_{i=lceil frac{n}{2}rceil}^inftybinom{2i}{n}frac{1}{2^i}$

Let $a_n = sumlimits_{i=lceil frac{n}{2}rceil}^inftybinom{2i}{n}frac{1}{2^i}$. Prove that $a_{n+1} = 2a_n + a_{n-1}$. I have tried considering the derivatives of $frac{1}{1-x^2}=1+x^2+x^4+...$, and although this may work, it certainly...

Asked on 12/01/2021 by Maxim Enis

1 answer

Lie group theory's connection to fractional calculus?

My friend and I were discussing the following identity:$$M = expbig{(}log(M)big{)} = expBig{(}frac{log(M)}{2}Big{)} expBig{(}frac{log(M)}{2}Big{)}$$ for any $M$ in a Lie group and...

Asked on 12/01/2021

0 answer

Solution to autonomous differential equation with locally lipscitz function

As I was learning about the following theorem and its proof from the book Nonlinear Systems by H. K. Khalil, I encountered a difficulty in grasping some parts of the...

Asked on 12/01/2021

1 answer

Let $A,Bin M_n (mathbb R), lambdain sigma(B), alpha in sigma (B)$ and $$ be inner product show $=lambda||x||^2+alpha$

Let $A,Bin M_n (mathbb R), lambdain sigma (A), alpha in sigma (B)$ and $<x,y>$ be inner product show $<Ax+By,x>=lambda||x||^2+alpha<x,y>$ where $x,yin mathbb R^n$ are eigenvectors associated...

Asked on 12/01/2021

0 answer

Is an automorphism a function or a group?

this is my first time on the Stack Exchange so I apologize if I went about asking this improperly! I was working on a problem set that asked to compute...

Asked on 12/01/2021 by Josh Charleston

2 answer

Fundamental Theorem of Projective Geometry for Finite Fields

I was trying to understand the proof that the automorphism group of a projective plane over a finite field $F_q$ (denoted $PG(2, q)$) is $PGamma L(3,...

Asked on 12/01/2021 by am2000

0 answer

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