# All Questions

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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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.
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### 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 ...
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### 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?
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### 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?
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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! ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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. ...