0
votes
0answers
5 views

derivative of domain of integration

Suppose $f(x,y)$ is a function $\mathbb{R}^2\times \mathbb{R}^2 \to\mathbb{R}$ and $\Omega(x)$ is a family of compact regions of the plane whose boundary curve $\gamma(s,x)$ varies smoothly in $x$. I ...
1
vote
0answers
13 views

Erasing numbers from circle and writing down sum

There are $50$ copies of the number $1$, and $50$ copies of the number $-1$, written alternately in a circle. In each step, we pick an arbitrary number, write down the sum of the number and its two ...
2
votes
2answers
13 views

Does being a local minimum imply a positive definite hessian?

If $p\in R^{m}$ is a local minimum of $F:R^{m}\rightarrow R$, then can we conclude that $\dfrac{\partial ^2F}{\partial x \partial x'}[p]$ is positive definite?
0
votes
0answers
8 views

Exponential bound and differentiability

Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ be s.t. $$ \left|f\left(x\right)\right|\leq ce^{ax^{2}} $$ for positive constants $a,c>0$. Suppose $f$ is differentiable. Under what conditions can we ...
2
votes
1answer
18 views

Could “$\infty$” be understood by taking the reciprocals of the Hypereal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
2
votes
1answer
29 views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
-2
votes
2answers
36 views

If $\sqrt{n}+ 8= n+1$, what is $n$?

If $\sqrt{n}+ 8= n+1$, what is $n$? Please show as many steps as possible so I can understand the process.
0
votes
0answers
11 views

Who can find mistakes in this calculation process?

I compute the conditional Separman's rho using the following method, but I do not know whether it is right? Who can tell me whether it is right? Thanks.
0
votes
2answers
36 views

How to prove $\displaystyle \lim_{n\longrightarrow\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$?

How to prove that $\displaystyle \lim_{n\longrightarrow\infty}\sum_{k=1}^{n}\frac{n+k}{n^2+k}=\frac{3}{2}$? I suppose some bounds are nedded, but the ones I have found are not sharp enough (changing ...
1
vote
1answer
14 views

When does a continuous function defined on a non-compact closed and bounded convex set has a fixed point?

Is there any result in fixed point theory which will give the existence of a fixed point for a continuous function defined on a non-compact, closed and bounded convex set?
3
votes
1answer
11 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
0
votes
0answers
17 views

Interpretation of parametrization

Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function. Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$ ...
1
vote
1answer
39 views

Explanation of an integral formula for the expectation of $(X_1-X_2)(Y_1-Y_2)$

I do not understand the proof of this expression. Who can explain it to me using simpler words? I do not understand the following black part:
3
votes
2answers
46 views

Problems whose first solutions had been using Calculus but later was shown to be done by n0n-Calculus methods

I was wondering about mathematical problems whose first published solutions was obtained by using methods of Calculus but later was shown (or known) to be solvable by using non-Calculus methods. ...
1
vote
2answers
33 views

How find the smallest $m$ such this $|A|=n,|B|=m,A\subseteq B$

let $n\ge 5$ Find the smallest $m$, there exist two set $A,B$(with element is postive integer )such following two condition: $|A|=n,|B|=m,A\subseteq B$ $\forall x,y(x\neq y)\in B$, have $x+y\in B$ ...
1
vote
2answers
16 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is an ...
-4
votes
0answers
26 views

How to find area of a curve?

Here is the equation of a curve: $ a^2 \cdot y^2 = x^3(2a - v) $. Now I want to find the whole area of the curve . How can I find the whole area?
5
votes
1answer
19 views

inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
3
votes
0answers
19 views

Multiple integral involving product of gamma function

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
0
votes
1answer
19 views

Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
3
votes
0answers
19 views

Area of weird circumcenter triangle equals area of medial triangle

Let $X$, $Y$, $Z$ be the midpoints of sides $BC$, $AC$, $AB$ respectively in triangle $ABC$. Let $O_{A}$, $O_{B}$, and $O_{C}$ be the circumcenters of triangles $AZX$, $BXY$, and $CYZ$ respectively. ...
1
vote
0answers
14 views

Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
1
vote
0answers
14 views

Shifting integration variables

I'm not sure how to pose this question precisely, but I'll try. I'm trying to see what happens when you have an integral of the form $\int \mathrm{d}x \,f(x-g(z))$ and you try and write it as $\int ...
0
votes
2answers
13 views

Show that the number of subsets of $S_1 \cup \dots \cup S_t$ that contain at most one element from each $S_i$ is $(a_1 + 1)(a_2 + 1) \dots (a_t + 1)$.

I found this problems on Aigner's: A course in enumeration: 1.1 We are given $t$ disjoint sets $S_i$ with $|Si| = a_i$. Show that the number of subsets of $S_1 \cup \dots \cup S_t$ that contain ...
1
vote
2answers
47 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
1
vote
0answers
56 views

Why are quaternions useful? [duplicate]

What I mean is why are they used basically where they are used? Listing some advantages of using them would be better. I am taking a mechanics course where a teacher mentioned them in a discussion ...
0
votes
0answers
10 views

Matroid Isomorphism Definition

I'm working though Welsh's Matroid Theory work, and he very casually mentions matroid isomorphisms in the first chapter but I don't think I like his statement. He says that two matroids ...
1
vote
0answers
38 views

How prove this complex inequality with same as (2014 china CMO) Cauchy-Schwarz inequality

let $r$ is give numbers,let $z_{1},z_{2},\cdots,z_{n}$ such $|z_{i}-1|\le r,i=1,2,\cdots,n,r\in(0,1)$ show that ...
2
votes
1answer
33 views

What is an affine space?

I am having trouble understanding what an affine space is. I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say: "The upshot is that, even in the affine plane, one can ...
0
votes
0answers
6 views

Left and Right Discrete Maximal Functions

Define the uncentered maximal function $$\widetilde{M}f(n)=\sup_{s,r\in\mathbb{Z}^{+}}\dfrac{1}{s+r+1}\sum_{k=-r}^{k=s}\left|f(n+k)\right|,$$ where $\mathbb{Z}^{+}=\left\{0,1,2,\ldots\right\}$. Define ...
2
votes
1answer
25 views

probability circle determined by chord determined by two random points is enclosed in bigger circle

Two points $A$ and $B$ are chosen uniformly at random from the interior of a circle $X_1$. Let $X_2$ be the circle whose diameter is the segment $AB$. What is the probability that $X_2$ is contained ...
1
vote
1answer
15 views

Triangles formed by line segments in a square

There is a square, denoted by points A, B, C, and D. There are 30 distinct points located inside the square (call these $A_2, A_3, A_4, ... A_{31}$. Non-intersecting segments $A_iA_j$ vertices are ...
2
votes
0answers
12 views

Distinct integers with $a=\text{lcm}(|a-b|,|a-c|)$ and permutations

Do there exist three pairwise different integers $a,b,c$ such that $$a=\text{lcm}(|a-b|,|a-c|), b=\text{lcm}(|b-a|,|b-c|), c=\text{lcm}(|c-a|,|c-b|)?$$ None of the integers can be $0$, because the ...
1
vote
3answers
48 views

Seemingly Simple Integration: $x/(x-1)$

I am currently working on some advanced engineering math but this seemingly simple integral has me stuck. Someone please show me how to derive it. It is part of a far bigger more complex problem in ...
3
votes
5answers
73 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
1
vote
1answer
12 views

How to calculate 2-d plane from 3 4-d points?

I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron). The 3-d cross-sections will be calculated as: x+y+z+w=c C is a constant that I will vary to get different ...
1
vote
0answers
15 views

Ability to View Answers in LaTex [migrated]

Is there an option or can one be implemented so that new users like myself can view the "source code" of others answers. Obviously there are many tutorials in which we can find the correct commands, ...
0
votes
1answer
16 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
1
vote
2answers
33 views

Health Risk Probability

Question: An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0.1 that a woman ...
2
votes
1answer
20 views

Find the continuous function such that the Riemann integrable is the same

Find all functions $f$ such that $f$ is continuous on $[0,1]$ and $\int_0^x f(t) dt = \int_x^1 f(t) dt$ for every x $\in (0,1)$ I can't think of any function that would satisfy this property! ...
1
vote
4answers
37 views

How to prove this equation by induction?

I am trying to prove this equation by mathematical induction $$f_{n+1}f_{n-1} = f_{n}^{2}+(-1)^n$$ is true where $f_{n} = $ the nth number in the Fibonacci sequence. I don't quite get how to do this ...
-1
votes
0answers
13 views

Differential equation to space state excercise

This is a "back of chapter" excercise which im trying to solve, my answer doesnt match the solution printed on the book, I want to write the equation in state space matrix form without using the ...
2
votes
1answer
33 views

Chance of winning a game of hearts with four players

I play hearts with a computer game program. The game is set up so that four people are playing the game. The question is: What are the mistakes, if any, with assuming that the probability of winning a ...
-4
votes
0answers
36 views

complex problem in linear algebra [on hold]

Let $A$ be an $n$ by $n$ matrix. Let $D$ be an $n$ by $n$ diagonal matrix with distinct diagonal entries, and let $u$ be an $n$ by $1$ column vector with all non-zero entries. Let $Aq=\lambda q$ with ...
2
votes
1answer
19 views

Can a low-rank matrix set have nonempty interior?

The answer to this question may be super simple, but it is very not obvious to me. Consider the space $S^n$ of symmetric $n\times n$ matrices. Consider $T\subset S$ the set of rank $n-1$ matrices. ...
0
votes
1answer
27 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
-7
votes
1answer
41 views

How to describe the Cartesian product $\mathbb{R} × \mathbb{R}$? [on hold]

Let $\mathbb{R}$ denote the set of all real numbers. Describe $\mathbb{R} × \mathbb{R}$. (This is a self-answered question).
1
vote
0answers
8 views

Solution to truncated renewal function

Let's begin with some theory on the renewal process. In a renewal process $N(t)$, let $t$ denote the interarrival time, and $f(t)$ and $F(t)$ denote the PDF and CDF respectively. Let $M(t)=E[N(t)]$, ...
4
votes
0answers
37 views

Proving that $T$:$(x_1,…,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},…,\frac {x_n+x_1}{2})$ leads to nonintegral components

Start with $n$ paiwise different integers $x_1,x_2,...,x_n,(n>2)$ and repeat the following step: $T$:$(x_1,...,x_n) \rightarrow (\frac {x_1+x_2}{2},\frac {x_2+x_3}{2},...,\frac {x_n+x_1}{2})$ ...
-4
votes
0answers
27 views

Let A,B be nxn matrices such that detA Not equal to 0, but detB = 0: Show [on hold]

Let $A$, $B$ be $n\times n$ matrices such that $\det A \neq 0$, but $\det B = 0$. Show $\|A-B\|_2 \geq(\|A^{-1}\|_2)^{-1}$.

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