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.

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

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.)

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

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.

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.

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

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.

Problem: $$f \in AX$$ 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.

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

Problem: Compute $$\int_{-\infty}^\infty {\sin x \over x}$$

Problem: If $$\lim_{x \to c} f(x) + f'(x)$$ exists, then $$\lim_{x \to c} f(x)$$ exists and $$\lim_{x \to c} f'(x) = 0$$

Problem: Let $$f:\R \to [0,+\infty)$$ nonvanishing and such that $$f(x)f''(x)\geq0$$, then $$\int_{-\infty}^{+\infty}f(x)^2dx=+\infty$$

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.

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.)

Problem Let $$f:\R^2 \to \R^2$$ be such that
 * $$d(f(x), f(y)) \geq A d(x, y), \qquad A>1$$

then $$f$$ admits a unique fixed point.

Problem Let $$X$$ be a compact metric space, and $$f:X \to X$$ be a contraction. Then
 * $$\bigcap_n^\infty f^n(X)$$

consists of exactly one point.

Problem: Every closed subset of $$\mathbf{R}^n$$ is separable.

Problem: Any connected nonempty subset of $$\mathbf{R}$$ either consists of a single point or contains an irrational number.

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

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

Problem Let $$f:[0,1]^2 \to \mathbf{R}$$ be a continuous function. Then
 * $$g(x) = \sup \{ f(x, y) | y \in [0, 1] \} \quad (x \in [0, 1]) $$

is continuous.

Problem Let $$f, g: \mathbf{R} \to \mathbf{R}$$ 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.

Problem Suppose $$f: \mathbf{R} \to \mathbf{R}$$ is uniformly continuous. Then there are constants $$a, b$$ such that:
 * $$|f(x)| \le a|x| + b$$

for all $$x \in \mathbf{R}$$.

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.

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].

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 $$A^n = I$$

Problem Let A be a real symmetric positive-definite matrix and b some fixed vector. Let $$\phi(x) = \langle Ax, x \rangle - 2 \langle x, b \rangle$$. Then $$Az = b$$ if and only if $$\phi(z) \le \phi(x)$$

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

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

Problem''Let $$S, T$$ be square matrices of the same size. Then $$ST$$ and $$TS$$ have the same eigenvalues.''

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

Problem Let $$A$$ be a square matrix over complex numbers. A is a real symmetric matrix if and only if
 * $$\langle Ax, x \rangle$$

is real for every x.

Problem Suppose the square matrix $$a_{ij}$$ satisfies:
 * $$|a_{ii}| > \sum_{j \ne i} |a_{ij}|$$

for all $$i$$. Then $$A$$ is invertible.

Problem Let $$T, S \in \operatorname{End}(V)$$. 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?

Problem: Let $$T, S$$ be linear operators on $$V$$. Then
 * $$\operatorname{dim}\operatorname{ker}(TS) \le \operatorname{dim}\operatorname{ker}(S) + \operatorname{dim}\operatorname{ker}(T)$$

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

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

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

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

Problem: Let A be a normal matrix. Then:
 * $$\|A\| = \max_{ |x|=1 } |(Ax \mid x)| = \sup_{\lambda \in \operatorname{Sp}(A)} |\lambda|$$

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

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

Problem: $$T \mapsto \sup_{\|x\|=1} (Tx \mid x)$$ is a norm for bounded operators T on a "complex" Hilbert space.