Problems in Mathematics

The problems are listed in increasing order of difficulty. When a problem is simply a mathematical statement, the reader is supposed to supply a proof. Answers are given (or will be given) to all of the problems. This is mostly for quality control; the answers allow contributors other than the initial writer of the problem to check the validity of the problems. In other words, the reader is strongly discouraged from seeing the answers before they successfully solve the problems themselves.

Commutative algebra

Problem: A finite integral domain is a field. Template:HideNseek

Problem: A polynomial has integer values for sufficiently large integer arguments if and only if it is a linear combination (over $𝐙$) of binomial coefficients $(\genfrac{}{}{0ex}{}{t}{n})$. Template:HideNseek

Problem: An integral domain is a PID if its prime ideals are principal. (Hint: apply Zorn's lemma to the set S of all non-principal prime ideals.) Template:HideNseek

Problem: A ring is noetherian if and only if its prime ideals are finitely generated. (Hint: Zorn's lemma.) Template:HideNseek

Problem: Every nonempty set of prime ideals has a minimal element with respect to inclusion.

Problem: If an integral domain A is algebraic over a field F, then A is a field. Template:HideNseek

Problem: Every two elements in a UFD have a gcd.

Problem: If $f\in A[X]$ is a unit, then $f-a_0$ is nilpotent, where $a_0=f(0)$ is the constant term of f. Template:HideNseek

Problem: The nilradical and the Jacobson radical of $A[X]$ coincide. Template:HideNseek

Problem: Let A be a ring such that every ideal not contained in its nilradical contains an element e such that $e^2=e\ne 0$. Then the nilradical and the Jacobson radical of $A$ coincide. Template:HideNseek

Problem: $f\in A[[X]]$ is a unit if and only if the constant term of f is a unit.

Real analysis

Problem: $\sqrt{3}+2^{1/3}$ is irrational. Template:HideNseek

Problem: Is ${\sqrt{2}}^{\sqrt{2}}$ irrational?

Problem: Compute $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sin x}{x}$

Problem: If $\underset{x\to c}{\lim }f(x)+f^{\prime }(x)$ exists, then $\underset{x\to c}{\lim }f(x)$ exists and $\underset{x\to c}{\lim }{f}^{\prime }(x)=0$ Template:HideNseek

Problem: Let $f:ℝ\to [0,+\mathrm{\infty })$ nonvanishing and such that $f(x)f^{″}(x)\ge 0$, then $\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x{)}^{2}dx=+\mathrm{\infty }$

Problem Let $X$ be a complete metric space, and $f:X\to X$ be a function such that $f\circ f$ is a contraction. Then $f$ admits a fixed point. Template:HideNseek

Problem Let $X$ be a compact metric space, and $f:X\to X$ be such that

$d(f(x),f(y))<d(x,y)$

for all $x\ne y\in X$. Then $f$ admits a unique fixed point. (Do not use Banach's fixed point theorem.) Template:HideNseek

Problem Let $f:{ℝ}^2\to {ℝ}^2$ be such that

$d(f(x),f(y))\ge Ad(x,y),A>1$

then $f$ admits a unique fixed point.

Problem Let $X$ be a compact metric space, and $f:X\to X$ be a contraction. Then

$\bigcap _n^{\mathrm{\infty }}{f}^n(X)$

consists of exactly one point. Template:HideNseek

Problem: Every closed subset of ${𝐑}^n$ is separable. Template:HideNseek

Problem: Any connected nonempty subset of $𝐑$ either consists of a single point or contains an irrational number. Template:HideNseek

Problem: Let $f:𝐑\to 𝐑$ be a bounded function. $f$ is continuous if and only if $f$ has closed graph.

Problem: Let $f:𝐑\to 𝐑$ be a homeomorphism, then $f$ is monotone.

Problem Let $f:[0,1{]}^2\to 𝐑$ be a continuous function. Then

$g(x)=\sup \{f(x,y)|y\in [0,1]\}(x\in [0,1])$

is continuous. Template:HideNseek

Problem Let $f,g:𝐑\to 𝐑$ be continuous functions such that: $f(g(x))=g(f(x))$ for every $x$. The equation $f(f(x))=g(g(x))$ has a solution if and only if $f(x)=g(x)$ has one. Template:HideNseek

Problem Suppose $f:𝐑\to 𝐑$ is uniformly continuous. Then there are constants $a,b$ such that:

$|f(x)|\le a|x|+b$

for all $x\in 𝐑$. Template:HideNseek

Problem Let X be a compact metric space, and $f:X\to X$ be an isometry: i.e., $d(f(x),f(y))=d(x,y)$. Then f is a bijection. Template:HideNseek

Problem Let $p_n$ be a sequence of polynomials with degree ≤ some fixed D. If $p_n$ converges pointwise to 0 on [0, 1], then $p_n$ converges uniformly on [0, 1]. Template:HideNseek

Problem On a closed interval a monotone function has at most countably many discontinuous points.

Problem Prove that in Rⁿ the relation $B_r(x)\supset B_s(y)$ implies r > s and find a metric space when the implication doesn't hold.

Linear algebra

Throughout the section $V$ denotes a finite-dimensional vector space over the field of complex numbers.

Problem Given an $n$, find a matrix with integer entries such that $A\ne I$ but $A^n=I$ Template:HideNseek

Problem Let A be a real symmetric positive-definite matrix and b some fixed vector. Let $\varphi (x)=⟨Ax,x⟩-2⟨x,b⟩$. Then $Az=b$ if and only if $\varphi (z)\le \varphi (x)$ Template:HideNseek

Problem If $\mathrm{tr}(AB)=0$ for all square matrices $B$, then $A=0$ Template:HideNseek

Problem Let x be a square matrix over a field of characteristic zero. If $\mathrm{tr}(x^k)=0$ for all $k>0$, then $x$ is nilpotent. Template:HideNseek

ProblemLet $S,T$ be square matrices of the same size. Then $ST$ and $TS$ have the same eigenvalues. Template:HideNseek

Problem Let $S,T$ be square matrices of the same size. Then $ST$ and $TS$ have the same eigenvalues with same multiplicity. Template:HideNseek

Problem Let $A$ be a square matrix over complex numbers. A is a real symmetric matrix if and only if

$⟨Ax,x⟩$

is real for every x. Template:HideNseek

Problem Suppose the square matrix $a_{ij}$ satisfies:

$|a_{ii}|>\sum _{j\ne i}|a_{ij}|$

for all $i$. Then $A$ is invertible. Template:HideNseek

Problem Let $T,S\in \mathrm{End}(V)$. If $V$ is finite-dimensional, then prove $TS$ is invertible if and only if $ST$ is invertible. Is this also true when $V$ is infinite-dimensional? Template:HideNseek

Problem: Let $T,S$ be linear operators on $V$. Then

$\dim \ker (TS)\le \dim \ker (S)+\dim \ker (T)$

Problem Every matrix (over an arbitrary field) is similar to its transpose. Template:HideNseek

Problem Every nonzero eigenvalue of a skew-symmetric matrix is pure imaginary.

Problem If the transpose of a matrix $A$ is zero, then $A$ is similar to a matrix with the main diagonal consisting of only zeros.

Problem $\mathrm{rank}(A^n)-\mathrm{rank}(A^{n-1})\le \mathrm{rank}(A^{n+1})-\mathrm{rank}(A^n)$ for any square matrix $A$.

Problem: Every square matrix is similar to an upper-triangular matrix. Template:HideNseek

Problem: Let A be a normal matrix. Then $A^{*}$ is a polynomial in A.

Problem: Let A be a normal matrix. Then:

$\Vert A\Vert =\underset{|x|=1}{\max }|(Ax\mid x)|=\underset{\lambda \in \mathrm{Sp}(A)}{\sup }|\lambda |$

Problem: Let A be a square matrix. Then $A\to 0$ (in operator norm) if and only if the spectral radius of $A<1$ Template:HideNseek

Problem: Let A be a square matrix. Then $\Vert A\Vert =\Vert A^{*}A{\Vert }^{1/2}$

Problem: $T\mapsto \underset{\Vert x\Vert =1}{\sup }(Tx\mid x)$ is a norm for bounded operators T on a "complex" Hilbert space. Template:HideNseek