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Evaluate the integral

$$\int_0^\infty \frac{x - \sin x}{x^3}\, dx$$

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Suppose $f : (a,b) \to \Bbb R$ is a convex function. Show that $f$ is differentiable at all but countably many points and the derivative is nondecreasing.

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Compute the center of the ring $M_n(\Bbb C)$ of all $n\times n$ complex matrices.

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Suppose $(X, \mathscr{O}_X)$ is a locally ringed space. Show that for all $\mathscr{O}_X$-modules $\mathscr{F}$ and $\mathscr{G}$ of finite type, $\operatorname{Supp}(\mathscr{F}\otimes_{\mathscr{O}_X} \mathscr{G}) = \operatorname{Supp}(\mathscr{F})\cap \operatorname{Supp}(\mathscr{G})$.

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Problem of the Week #291 - Jul 03, 2019]]>

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Show that $$\int_0^\infty \left[\frac{J_1(t)}{t}\right]^2\, dt = \frac{4}{3\pi}$$

where $J_1$ is the Bessel function of the first kind of order one.

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Let $(X,\mathcal{M}, \mu)$ be a positive measure space, and let $\{E_n\}$ be a sequence of sets in $\mathcal{M}$ such that $\displaystyle\lim_n \mu(E_n) = 0$. Prove that if $1 \le p \le \infty$, then for all $f\in \mathscr{L}^p(X,\mathcal{M},\mu)$, $\displaystyle\lim_n \int_{E_n} f\, d\mu = 0$.

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Problem of the Week #289 - May 21, 2019]]>

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Let $X$ be a finite set with more than one element, and $G$ be a finite group acting transitively on $X$. Show that some element of $G$ is free of fixed points.

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1. Show that if $G$ is an abelian group and $p$ is a prime such that $px = 0$ for all $x\in G$, then $G$ has the structure of a vector space over $\Bbb Z/p\Bbb Z$.

2. If $S$ is a bounded linear operator on a Banach space $X$, show that the spectral radius of $S$ is the infimum of $\|S^n\|^{1/n}$, as $n$ ranges over the positive integers.

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Remember to read the...

Problem of the Week #286 - Mar 05, 2019]]>

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Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a Radon measure on $X$. Recall that the complement of the support of $\mu$ is the union of all open subsets of $X$ of $\mu$-measure zero. Show that the support of $\mu$ is the set of all $x\in X$ such that for all compactly supported continuous functions $f : X\to [0,1]$ with $f(x) > 0$, the integral $\int_X f\, d\mu > 0$.

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Problem of the Week #287 - Apr 09, 2019]]>

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Evaluate the integral $$\int_0^\infty \frac{\cos(ax)}{\cosh b + \cosh x}\, dx$$ where $a$ is real and $b > 0$.

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If $M$ and $N$ are topological manifolds with boundary of dimensions $m$ and $n$, respectively, show that $M \times N$ is an $(m+n)$-manifold with boundary $\partial M \times N \cup M \times \partial N$.

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Problem of the Week #284 - Jan 07, 2019]]>

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Evaluate the integral

$$\int_{-1}^1 \left(\frac{1-x}{1+x}\right)^{\!\!a} \frac{dx}{(x - b)^2}$$

where $0 < a < 1$ and $b > 1$.

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Suppose $X$ is a closed connected orientable manifold of dimension $2n$. Prove that if the homology group $H_{n-1}(X)$ is torsion free, then $H_n(X)$ is also torsion free.

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Given a complex Borel measure $\mu$ on the torus $\Bbb T^1$, define the Fourier coefficients of $\mu$ by $\hat{\mu}(n) := \int_{\Bbb T} e^{-2\pi i nx}\, d\mu(x)$, $n\in \Bbb Z$. Show that if the sequence $(\hat{\mu}(n))\in \ell^1(\Bbb Z)$, then $\mu$ has a Radon-Nikyodym derivative with respect to the Lebesgue measure on $\Bbb T$.

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Problem of the Week #281 - Oct 30, 2018]]>

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Let $f : X \to Y$ be a closed map of topological spaces such that the fibers $f^{-1}(y)$ are compact for every $y\in Y$. Prove that $f$ is a proper map, i.e., $f^{-1}(K)$ is a compact subset of $X$ for every compact subset $K$ of $Y$.

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Problem of the Week #280 - Oct 16, 2018]]>

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Show that if $f$ is an entire function with $\lim\limits_{z\to \infty} \dfrac{\operatorname{Re}f(z)}{z} = 0$, then $f$ is constant.

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Let $(X_n)_{n \in \Bbb N}$ be a sequence of positive i.i.d. random variables such that $E[\ln X_n]$ is a constant finite positive number $\mu$. Show that if $$T_n := \prod_{i = 1}^n X_i^{1/n}\quad (n = 1,2,3,...)$$ then $(T_n)_{n\in \Bbb N}$ converges in probability to $e^{\mu}$.

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Problem of the Week #278 - Sep 11, 2018]]>

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Why is every cofibration injective?

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For $p$ a prime integer, compute the order of the special linear group $SL_n(\Bbb F_p)$, where $\Bbb F_p$ is the field with $p$ elements.

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Show that in a compact Hausdorff space, any countable collection of dense open sets has dense intersection.

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Let $X$ and $Y$ be normal topological spaces. Suppose $A$ is a closed subset of $X$ and $f : A \to Y$ is a continuous map. Prove that the adjunction space $X \cup_f Y$ is normal.

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Let $F$ be the field of fractions of a unique factorization domain $A$, and let $L$ be an algebraic extension field of $F$. Fix $c\in L$. Prove that the kernel of the evaluation map $\operatorname{ev}_c : A[x] \to L$ is principal.

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Problem of the Week #273 - Jul 03, 2018]]>

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Find a general solution of the nonlinear differential equation

$$\left(\frac{dy}{dt}\right)^{\!\!2} - y\frac{d^2y}{dt^2} = -1$$

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Suppose $a$ is a fixed complex number in the open unit disk $\Bbb D$. Consider the holomorphic mapping $\phi : \Bbb D \to \Bbb D$ given by $\phi(z) := (z - a)/(1 - \bar{a}z)$. Find, with proof, the average value of $\left\lvert\frac{d\phi}{dz}\right\rvert^2$ over $\Bbb D$, i.e., the integral $$\frac{1}{\pi}\iint_{\Bbb D} |\phi'(x + yi)|^2\, dx\, dy$$

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Problem of the Week #271 - May 29, 2018]]>

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Prove that for all vector fields $X$, $Y$, and $Z$ on a smooth manifold, their Lie derivatives $\mathscr{L}_X$, $\mathscr{L}_Y$, and $\mathscr{L}_Z$ satisfies Jacobi’s identity $$[\mathscr{L}_X,[\mathscr{L}_Y,\mathscr{L}_Z]] + [\mathscr{L}_Y, [\mathscr{L}_Z,\mathscr{L}_X]] + [\mathscr{L}_Z, [\mathscr{L}_X, \mathscr{L}_Y]] = 0$$

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Problem of the Week #270 - May 15, 2018]]>

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Suppose $f$ is analytic on a simple closed contour $c$ in the complex plane. Prove $\displaystyle\int_c \overline{f(z)}f’(z)\, dz$ is purely imaginary.

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Let $D_n(t) = \sum\limits_{\lvert k\rvert \le n} e^{2\pi i kt}$ for $t\in [-.5, .5]$. Show that if $n \ge 2$, there are positive constants $A$ and $B$ independent of $n$, such that

$$A \le \frac{1}{\log n}\int_{-.5}^{.5} |D_n(t)|\, dt \le B$$

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Problem of the Week #268 - Apr 03, 2018]]>

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If $n$ is a positive integer, evaluate

$$\int_{0}^\infty \frac{dx}{1 + x^n}$$

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Let $\Bbb Z

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Problem of the Week #266 - Feb 27, 2018]]>

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Let $F : (0, \infty)\times (0,\infty) \to \Bbb R$ be given by

$$F(\alpha, \beta) = \int_0^\infty \frac{\cos(\alpha x)}{x^4 + \beta^4}\, dx$$ Show that $$\frac{F(\alpha,\beta)}{F(\beta,\alpha)} = \frac{\alpha^3}{\beta^3}$$ as long as there is no positive integer $n$ such that $\alpha = \dfrac{(4n-1)\pi\sqrt{2}}{4\beta}$.

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Remember to read the...

Problem of the Week #265 - Feb 13, 2018]]>

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Let $G$ be a compact Lie group, and let $V$ be a finite-dimensional representation of $G$. Prove that if $\chi$ is the character associated with $V$, then $\int_G \chi(g)\, dg = \operatorname{dim}(V^G)$ where $V^G\subset V$ is the subspace of $G$-invariants of $V$.

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Problem of the Week #264 - Jan 23, 2018]]>

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Prove that a compact differentiable surface homeomorphic to the real projective plane has a point at which the Gaussian curvature is positive.

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Consider the open unit disk $\Bbb D\subset \Bbb C$ with Riemannian metric $ds^2 = \dfrac{\lvert dz\rvert^2}{(1 - \lvert z\rvert^2)^2}$. Find a formula for the (Riemannian) distance between two points in $\Bbb D$, and use it to find the distance between $-\frac{1}{2}e^{i\pi/4}$ and $\frac{1}{2}e^{i\pi/4}$.

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Problem of the Week #262 - Dec 19, 2017]]>

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If $\phi : A \to B$ is a local homeomorphism from a compact space $A$ to a connected Hausdorff space $B$, show that $\phi$ is surjective and the fibers of $\phi$ over the points of $B$ are finite sets.

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Problem of the Week #261 - Dec 05, 2017]]>

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Suppose $\mu$ is a finite Borel measure on $\Bbb R^n$. Define the maximal function of $\mu$ by $$\mathcal{M}\mu(x) = \sup_{0 < r < \infty} \frac{\mu(B(x;r))}{m(B(x;r))}\quad (x\in \Bbb R^n)$$ Here, $m$ denotes the Lebesgue measure on $\Bbb R^n$. Show that if $\mu$ is mutually singular with respect to $m$ (i.e., $\mu \perp m$), then $\mathcal{M}\mu = \infty$ a.e. $[\mu]$.

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Problem of the Week #260 - Nov 21, 2017]]>

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Given commutative rings with unity $R$ and $S$, let $\phi : R \to S$ be a morphism of rings. It induces a morphism $\phi^* : \operatorname{Spec}(S) \to \operatorname{Spec}(R)$ of prime spectra such that $\phi^*(\mathfrak{q}) = \phi^{-1}(\mathfrak{q})$ for all $\mathfrak{q}\in \operatorname{Spec}(S)$. Show that if $X$ is a finitely generated $R$-module, the support of $S\otimes_R X$ is the inverse image of the support of $X$ under the induced map $\phi^*$...

Problem of the Week #259 - Oct 31, 2017]]>

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Find the norm of the linear operator $T: \mathscr{L}^p(0, \infty) \to \mathscr{L}^p(0, \infty)$ defined by the equation

$$(Tf)(x) = \frac{1}{x}\int_0^x f(t)\, dt$$

Here it is assumed that $1 < p < \infty$.

Note: The space $\mathscr{L}^p(0,\infty)$ consists of all Lebesgue integrable functions $f : (0,\infty) \to \Bbb R$ such that $\|f\|_p < \infty$. For $p < \infty$, $\|f\|_p := \left(\int_0^\infty \lvert f(x)\rvert^p\, dx\right)^{1/p}$.

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Remember...

Problem of the Week #258 - Oct 17, 2017]]>

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Using contour integration, prove

$$\int_0^\infty \sin(x^\alpha)\, dx = \sin\left(\frac{\pi}{2\alpha}\right)\,\Gamma\!\left(1 + \frac{1}{\alpha}\right),\quad \alpha > 1.$$

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Prove that if $n > 0$, an even map between $n$-spheres has even homological degree.

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Let $\Bbb D$ be the open unit disc in the complex plane, and let $f$ be a continuous complex function on $\partial\Bbb D$. Consider the function

$$F(re^{i\phi}) \,\dot{=}\, \frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})\frac{1-r^2}{1-2r\cos(\theta-\phi) + r^2}\, d\theta\quad (re^{i\phi}\in \Bbb D)$$

Prove $F$ is harmonic on $\Bbb D$, and that for all $z_0\in \partial \Bbb D$, $\lim\limits_{z\to z_0} F(z) = f(z_0)$.

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Problem of the Week #255 - Aug 15, 2017]]>

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Show that the tensor algebra of $\Bbb Z/n\Bbb Z$ is isomorphic to $\Bbb Z[x]/(nx)$.

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Determine the value of the definite integral

$$\int_0^\infty \frac{dt}{(1+t^2)t^{1/2}}$$

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Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that

1. $\Bbb E[f(|X_n|)]$ is uniformly bounded

2. $\dfrac{f(x)}{x}\to \infty$ as $x\to \infty$

Show that $(X_n)$ is uniformly integrable.

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Problem of the Week #252 - Jun 27, 2017]]>

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Let $f : M \to M$ be a self-map of a smooth manifold $M$. Prove that the graph of $f$ is transversal to the diagonal of $M$ if and only if the fixed points of $M$ are nondegenerate, i.e., for all fixed points $p$, $+1$ is not an eigenvalue of $df_p$.

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Problem of the Week #251 - Jun 13, 2017]]>

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Consider a strictly increasing sequence of natural numbers $(n_k)_{k = 1}^\infty$, and suppose $X$ is the subset of $[0,2\pi]$ consisting of all $x$ such that the sequence $(\sin(n_k x))_{k = 1}^\infty$ is convergent. Prove $X$ has Lebesgue measure zero.

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Problem of the Week #250 - May 23, 2017]]>

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Suppose $Z$ is a standard Gaussian random variable. Prove $\Bbb P(\lvert Z\rvert \ge z) = O[\exp(-z^2/2)]$ as $z\to \infty$.

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Prove that $$\int_{-\pi}^{\pi}\ln\lvert 1 - e^{it}\rvert\, dt = 0$$

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Let $f : \Bbb S^1\subset \Bbb C \to \Bbb C$ be a continuous map. Show that if $f$ is continuously differentiable on $\Bbb S^1$, then its Fourier coefficient sequence $\{\hat{f}_n\}_{n\in \Bbb Z}$ belongs to $\ell^1(\Bbb Z)$.

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Problem of the Week #247 - Apr 11, 2017]]>

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Prove that if $\Bbb R$ is homeomorphic to a cartesian product $A\times B$, then either $A$ or $B$ is a singleton.

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Consider the Lebesgue space $L^1(\Bbb R)$ as an algebra with product given by convolution. Prove that $L^1(\Bbb R)$ is isomorphic as an algebra to an ideal in the algebra $M(\Bbb R)$ of complex Borel measures on $\Bbb R$, and identify the ideal. Note the product in $M(\Bbb R)$ is given by convolution of measures.

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Problem of the Week #245 - Mar 07, 2017]]>

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Let $(\mathcal{C}, \partial)$ be a chain complex of abelian groups. Suppose $f, g : \mathcal{C} \to \mathcal{C}$ are homotopic chain maps. Construct an explicit chain homotopy between the $n$-fold compositions $f^n$ and $g^n$.

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Problem of the Week #244 - Feb 21, 2017]]>

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Consider the normed space $\mathcal{M}(X)$ of all complex regular Borel measures on a locally compact Hausdorff space $X$, with total variation norm $\|\mu\| := \lvert \mu\rvert (X)$, for all $\mu\in \mathcal{M}(X)$. Prove that $\mathcal{M}(X)$ is a Banach space.

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Problem of the Week #243 - Feb 07, 2017]]>

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Suppose $\Gamma$ is a finite group of homeomorphisms of a Hausdorff space $M$ such that every non-identity element of $\Gamma$ is fixed point free. Show that $\Gamma$ acts on $M$ properly discontinuously.

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Problem of the Week #242 - Jan 24, 2017]]>

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Consider an analytic map $f : \Bbb D \to \Bbb C$ such that $f(z) = \sum\limits_{n = 0}^\infty a_n z^n$ for all $z\in \Bbb D$. Prove that $f$ is injective, provided

$$\sum_{n = 2}^\infty n\lvert a_n\rvert < \lvert a_1\rvert.$$

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Problem of the Week #241 - Jan 10, 2017]]>

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Prove that a homeomorphism of the closed unit disk onto itself must map $S^1$ onto $S^1$.

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Problem of the Week #240 - Jan 03, 2017]]>

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Let $1 < p < \infty$, and let $(f_n)$ be a sequence of real-valued functions in $\mathscr{L}^p(-\infty, \infty)$ which converges pointwise a.e. to zero. Show that if $\|f_n\|_p$ is uniformly bounded, then $(f_n)$ converges weakly to zero in $\mathscr{L}^p(-\infty,\infty)$.

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Problem of the Week #239 - Dec 27, 2016]]>

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Give two different proofs of the following result:

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Let $Z$ be the center of a finite group $G$. Prove that there are at most $(G : Z)$ elements in each conjugacy class of $G$.

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Show that if $f$ is an entire function such that $\int_{-\infty}^\infty \int_{-\infty}^\infty \lvert f(x + yi)\rvert^2\, dx\, dy < \infty$, then $f$ is identically zero.

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Let $R$ be a commutative ring. If $N$ and $P$ are submodules of an $R$-module $M$ such that $M/N$ and $M/P$ are Artinian, show that $M/(N\cap P)$ is Artinian.

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Let $n$ be a positive integer, and let $\Bbb S^n \to \Bbb S^n$ be a fixed point free continuous map. Show that the map's homological degree is $(-1)^{n+1}$.

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If $\omega$ is a two-form on the four-sphere, is $\omega^2$ (i.e., $\omega \wedge \omega$) nowhere vanishing?

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Let $R$ be a commutative ring with unity. Show that if $S$ is multiplicatively closed in $R$ and if every $R$-module is flat, then every $S^{-1}R$-module is flat.

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Consider a sequence of real numbers $(x_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty \lvert x_n y_n\rvert$ converges for every real sequence $(y_n)_{n = 1}^\infty$ such that $\sum\limits_{n = 1}^\infty y_n^2$ converges. Prove that $\sum\limits_{n = 1}^\infty x_n^2$ converges.

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Problem of the Week #231 - Nov 01, 2016]]>

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Show that if $(X,\mathcal{M},\mu)$, $(Y,\mathcal{N},\nu)$ are finite measure spaces, $1 < p < \infty$, and $K$ is a measurable function on $X\times Y$, there is a bounded integral operator $I(K) : \mathscr{L}^p(\nu) \to \mathscr{L}^p(\mu)$ given by

$$I(K)(f) :x \mapsto \int_Y K(x,y)\,f(y)\, d\nu(y)\quad (f\in \mathscr{L}^p(\mu)),$$

provided that the kernel $K$ satisfies the conditions $\sup_x \int_Y \lvert K(x,y)\rvert\, d\nu(y) < \infty$ and $\sup_y...

Problem of the Week #230 - Oct 25, 2016]]>

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Let $(X,\mathscr{O})$ be a ringed space. Suppose $\mathscr{F}$ is an invertible sheaf over $\mathscr{O}$. That is, $\mathscr{F}$ is a rank one locally free module over $\mathscr{O}$. Prove that there is an isomorphism between the tensor sheaf $\mathscr{F}\otimes_\mathcal{O}\check{\mathscr{F}}$ and structure sheaf $\mathscr{O}$, where $\check{\mathscr{F}} = \operatorname{Hom}_X(\mathscr{F},\mathscr{O})$.

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Remember to read the...

Problem of the Week #229 - Oct 18, 2016]]>

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Call an

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Remember to read the...

Problem of the Week #228 - Oct 11, 2016]]>

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Let $\mathscr{F} \overset{\eta}{\to} \mathscr{G}$ be a morphism of sheaves over a topological space $X$. Prove that quotient sheaf $\mathscr{F}/\operatorname{ker}(\eta)$ is isomorphic to the image sheaf $\operatorname{im}(\eta)$.

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Problem of the Week #227 - Oct 04, 2016]]>

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Let $(X,\mu)$ be a measure space, $f\in \mathcal{L}^1(\mu)$, and $\phi_n\in \mathcal{L}^1(\mu)$ such that $\sup_{n,t}\lvert \phi_n(t)\rvert \le 1$ and $\|\phi_n\|_1 \to 0$ as $n\to \infty$. Show that $\|f\phi_n\|_1 \to 0$ as $n\to \infty$.

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Problem of the Week #226 - Sep 27, 2016]]>

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Suppose $X$ is a compact Hausdorff space. Show that there is homeomorphism between $X$ and the collection $Y$ of maximal ideals in $C(X,\Bbb R)$, the space of continuous real-valued functions on $X$ (here, $Y$ is topologized with the Zariski topology).

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Problem of the Week #225 - Sep 20, 2016]]>

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Let $(X,\mu)$ be a positive measure space. For $0 < p < \infty$, why is the mapping $\mathcal{L}^p(X,\mu) \to \mathcal{L}^1(X,\mu)$ sending $f$ to $\lvert f\rvert^p$, continuous?

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Let $G$ be a group with finite index subgroups $H$ and $K$. Suppose $H$ and $K$ have relatively prime indices in $G$. Why must $G$ be the internal direct product of $H$ and $K$?

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Evaluate the integral

$$\int_{-\infty}^\infty \frac{\ln^2\lvert x\rvert}{x^2+1}\, dx$$

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Suppose $M$ is a smooth path-connected manifold. Consider the differential form

$$\nu = \Re\left\{\frac{1}{2\pi i} \frac{dz}{z}\right\}$$

which generates $H^1_{dR}(\Bbb C^\times)$, the first de Rham cohomology of $\Bbb C^\times$. Show that every smooth map $f : M \to \Bbb C^\times$ can be lifted to smooth map $M\to \Bbb C$ via the exponential map, provided that the image of $\nu$ under $f^* : H^1_{dR}(\Bbb C^\times) \to H^1_{dR}(M)$ is zero.

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Problem of the Week #221 - Aug 23, 2016]]>

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Let $f : \Bbb R^n \to \Bbb R$ be a function such that $f$ and its maximal function $\mathcal{M}f$ belong to $\mathcal{L}^1(\Bbb R^n)$. Show that $f(x) = 0$ for almost every $x\in \Bbb R^n$.

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Problem of the Week #220 - Aug 16, 2016]]>

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Let $A$ and $B$ be nonsingular $n\times n$-matrices over a field $\Bbb k$. Show that for all but finitely many $x\in \Bbb k$, $xA + B$ is nonsingular.

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Let $F$ be an entire function for which there exists $t > 0$ such that $\lvert F(z)\rvert = O(\exp(\lvert z\rvert^t))$ as $\lvert z\rvert \to \infty$. Show that there is a constant $M > 0$ such that for all $n$ sufficiently large, $$\lvert F^{(n)}(0)\rvert \le Mn!\left(\frac{et}{n}\right)^{n/t}$$

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Problem of the Week #218 - Aug 02, 2016]]>

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Prove

$$\lim_{n\to \infty} \int_0^{\pi/(2n)} \frac{\sin 2nx}{\sin x}\, dx = \int_0^\pi \frac{\sin x}{x}\, dx.$$

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Let $p$ be a prime greater than $3$. Compute the sum of the quadratic residues in $\Bbb Z/p\Bbb Z$.

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Does there exist a real-valued function on $\Bbb R$ that is discontinuous only on the irrationals?

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Give an example of a unit of the integral group ring $\Bbb Z[S_3]$ that is not of the form $1x$ for some $x\in S_3$.

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Let $A$ be a commutative ring with unity. Prove that a polynomial $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ over $A$ is a unit in $A[x]$ if and only if $a_0$ is an $A$-unit and $a_1,\ldots, a_n$ are nilpotent in $A$.

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Problem of the Week #213 - Jun 28, 2016]]>

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Evaluate the infinite series

$$\sum_{n = 1}^\infty \frac{(-1)^{n+1} n^2}{n^3 + 1}$$

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Show that the complex function

$$F(z) = \frac{1}{\pi}\int_0^1 \int_{-\pi}^\pi \frac{r}{re^{i\theta} + z}\, d\theta\, dr$$

is anti-holomorhpic (i.e., the conjugate $\bar{F}$ is holomorphic) in the open unit disc, $\Bbb D$, and holomoprhic in complement $\Bbb C \setminus \bar{\Bbb D}$ of the closed unit disc.

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Problem of the Week #211 - Jun 14, 2016]]>

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Evaluate the abelianization of the fundamental group of the $n$-fold connected sum $\underbrace{\Bbb RP^2\, \# \cdots \#\, \Bbb RP^2}_{n}$.

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Define the logarithm of an $n\times n$ matrix $A$ by the power series

$$\sum_{k = 1}^\infty \frac{(-1)^{k-1}(A - I)^k}{k}$$

which converges for $\|A - I\| < 1$ (the standard matrix norm is being used here). Prove that for all $n\times n$ matrices $A$ and $B$ with $\|A - I\| < 1$, $\|B - I\| < 1$, $\|AB - I\| < 1$, and $AB = BA$,

$$\log(AB) = \log A + \log B$$

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Problem of the Week #209 - May 31, 2016]]>

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Let $R$ be a ring such that for all $r\in R$, $r^3 = r$. Prove $R$ is commutative.

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Prove the following theorem of Tate: If $\phi : X\to Y$ is a mapping of real Banach spaces such that for some positive number $M$, $\lvert \phi(x + y) - \phi(x) - \phi(y)\rvert \le M$ ($x,y\in X$), then there is a unique additive mapping $\psi : X \to Y$ such that $\psi - \phi$ is bounded on $X$.

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Problem of the Week #207 - May 17, 2016]]>

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By method of contour integration, find the values of the integrals

$$\int_{-\infty + i\alpha}^{\infty + i\alpha} e^{-x^2}\, dx$$

for all $\alpha \ge 0$.

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Let $q$ be a complex number with $\lvert q \rvert < 1$. Show that

$$\prod_{n = 1}^\infty (1 - q^n) \sum_{n = -\infty}^\infty q^{n+2n^2} = \prod_{n = 1}^\infty (1 - q^{2n})^2$$

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Problem of the Week #205 - May 3, 2016]]>

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Why must a group with cyclic automorphism group be abelian?

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Show that the holomorphic mappings on a compact connected Riemann surface are constant.

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Let $u$ be an $H^1(\Bbb R^d)$-solution of the semi-linear PDE

$$-\Delta u + au = b|u|^{\alpha}u\quad (a > 0,\, \alpha > 0,\, b\in \Bbb R)$$

Derive the Pohozaev identity

$$(d - 2)\int_{\Bbb R^d} \lvert \nabla_xu\rvert^2\, dx + da\int_{\Bbb R^d} \lvert u\rvert^2\, dx = \frac{2bd}{\alpha + 2}\int_{\Bbb R^d} \lvert u\rvert^{\alpha + 2}$$

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Problem of the Week #202 - April 12, 2016]]>

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Let $\Bbb D$ denote the open unit disc in the complex plane. Given a holomorphic function $f$ on $\Bbb D$, define

$$N_p(f) := \sup_{0 < r < 1} \left[\frac{1}{2\pi}\int_{-\pi}^\pi \lvert f(re^{i\theta})\rvert^p\, d\theta\right]^{1/p},\quad 0 < p < \infty$$

The $p$th

If $f\in H^p(\Bbb D)$ and $1\le p <...

Problem of the Week #200 - Tuesday, March 29, 2016]]>

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Problem of the Week #199 - March 22, 2016]]>

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Let $\Lambda :\Bbb R \to \Bbb R$ be a mapping such that for all bounded measurable mappings $f : [0,1]\to \Bbb R$,

$$\Lambda\left(\int_0^1 f(x)\, dx\right) \le \int_0^1 \Lambda(f(x))\, dx.$$

Show that $\Lambda$ is a convex mapping.

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Problem of the Week #198 - March 15, 2016]]>

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Let $D$ be a division ring. Show that if $D$ is not simultaneously of characteristic two and commutative, then $D$ is generated by products of perfect squares.

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Evaluate the integral

$$\int_0^1 \frac{t^{-1/2}(1-t)^{-1/4}}{(16 - 7t)^{5/4}}\, dt.$$

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An abelian group $G$ is called

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Problem of the Week #195 - February 23, 2016]]>