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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2022, 5 Real analysis, Stolz–Cesàro theorem, decreasing sequences

Suppose ${𝑎_{𝑛}}$ and ${𝑏_{𝑛}}$ are monotone strictly decreasing sequences of positive real numbers converging to zero. Prove that if there exists a limit

lim_{𝑛 \to \infty} \frac{𝑎_{𝑛 + 1} - 𝑎_{𝑛}}{𝑏_{𝑛 + 1} - 𝑏_{𝑛}} = 𝐴,

then

lim_{𝑛 \to \infty} \frac{𝑎_{𝑛}}{𝑏_{𝑛}} = 𝐴 .

UC Irvine Fall 2022, 6 Real analysis, local diffeomorphisms, Jacobians

Denote

𝑈 \subset ℝ^{2}, 𝑈 = {(𝑥, 𝑦) : 𝑦 > | 𝑥 |} .

Is it true that the map

𝑓 (𝑥, 𝑦) = (\int_{𝑦}^{𝑥} 𝑒^{𝑡^{2}} d 𝑡, \int_{𝑦}^{𝑥} 𝑒^{- 𝑡^{2}} d 𝑡)

is a local diffeomorphism near every point $(𝑥, 𝑦) \in 𝑈$ ? Explain your answer.

UC Irvine Fall 2022, 7 Real analysis, surface integrals, ellipsoid, divergence theorem

Let

Σ = {(𝑥, 𝑦, 𝑧) \in ℝ^{3} : \frac{𝑥^{2}}{𝑎^{2}} + \frac{𝑦^{2}}{𝑏^{2}} + \frac{𝑧^{2}}{𝑐^{2}} = 1} .

Compute the integral

\int_{Σ} 𝑥^{3} d 𝑦 d 𝑧 + 𝑦^{3} d 𝑧 d 𝑥 + 𝑧^{3} d 𝑥 d 𝑦 .

UC Irvine Fall 2022, 8 Real analysis, uniform continuity, multivariable functions

Prove or disprove that

𝑓 (𝑥, 𝑦) = ln (1 + 𝑥^{2} + 𝑦^{2}) + sin (𝑥^{2})

is uniformly continuous on $ℝ^{2}$ .

UC Irvine Fall 2022, 9 Real analysis, compactness, Cauchy-type conditions, counterexamples

Let $(𝑀, 𝑑)$ be a bounded metric space that has the following property: for any sequence ${𝑥_{𝑛}} \subset 𝑀$ with

𝑑 (𝑥_{𝑛}, 𝑥_{𝑛 + 1}) \leq \frac{1}{𝑛},

there exists a convergent subsequence. Does it imply that $(𝑀, 𝑑)$ is compact? Prove or give a counterexample.

Stanford Fall 2023, 1 Real analysis, Fourier series, density, Borel measures

Let $𝜇$ be a finite Borel measure on the unit circle $⊤ = {𝑧 \in ℂ : | 𝑧 | = 1}$ such that

lim_{𝑛 \to \infty} \int_{⊤} 𝑧^{𝑛} d 𝜇 (𝑧) = 0 .

Prove that for any $𝑓 : ⊤ \to ℂ$ , $𝑓 \in 𝐿^{1} (⊤, 𝜇)$ , we have

lim_{𝑛 \to \infty} \int_{⊤} 𝑧^{𝑛} 𝑓 (𝑧) d 𝜇 (𝑧) = 0 .

Stanford Fall 2023, 1 Real analysis, tempered distributions, Fourier transform, distributions, weak-star topology

Let $𝒮︀ (ℝ^{𝑛})$ and ${𝒮︀}^{'} (ℝ^{𝑛})$ denote the spaces of Schwartz functions and tempered distributions, respectively.

(a) For every $𝑡 > 0$ , define $𝑃_{𝑡} : 𝒮︀ (ℝ^{𝑛}) \to 𝒮︀ (ℝ^{𝑛})$ by $(𝑃_{𝑡} 𝑓) (𝑥) = 𝑓 (𝑡 𝑥)$ . Show that $𝑃_{𝑡}$ extends (weak-*) continuously to a map .

(b) We say that $𝑢 \in {𝒮︀}^{'} (ℝ^{𝑛})$ is homogeneous of degree $𝑑$ if for all $𝑡 > 0$ . Show that if $𝑢 \in {𝒮︀}^{'} (ℝ^{𝑛})$ is homogeneous of degree $𝑑$ , then its Fourier transform is homogeneous of degree $- 𝑛 - 𝑑$ .

(c) Show that if $𝑢 \in {𝒮︀}^{'} (ℝ^{𝑛})$ is a compactly supported distribution which is homogeneous of some degree $𝑑$ , then $𝑢$ is a differentiated delta distribution, and $𝑑$ is an integer $\leq - 𝑛$ .

UC Irvine Fall 2023, 1 Real analysis, L^p norms, limit as p to infinity, essential supremum

Let $𝑓$ be a positive continuous function on $[0, 1]$ . Prove that

lim_{𝑝 \to \infty} ∥ 𝑓 \underset{𝐿^{𝑝 ([0, 1])}}{∥} = ∥ 𝑓 \underset{𝐿^{\infty} ([0, 1])}{∥} .

Stanford Fall 2023, 2 Real analysis, Fourier series, convolution, density

Fix $𝑓 \in 𝐿^{1} (⊤)$ , where $⊤ = ℝ / 2 𝜋 ℤ$ . For each $𝜏 \in ⊤$ , define $𝑓_{𝜏}$ , a translate of $𝑓$ by $𝜏$ , using $𝑓_{𝜏} (𝑥) = 𝑓 (𝑥 - 𝜏)$ . Consider the sets ${ℬ︀}_{𝑓}$ and ${ℳ︀}_{𝑓}$ defined by

{ℬ︀}_{𝑓} = {\sum_{𝑖 = 1}^{𝑁} 𝑎_{𝑖} 𝑓_{𝜏_{𝑖}} (𝑥) : 𝑁 \in ℕ, 𝑎_{𝑖} \in ℝ, 𝜏_{𝑖} \in ⊤}, {ℳ︀}_{𝑓} = {𝑓 * 𝑔 : 𝑔 \in 𝐿^{1} (⊤)} .

Show that , where the overlines denote the $𝐿^{1} (⊤)$ -closures.

Stanford Fall 2023, 2 Real analysis, interpolation inequalities

For $𝑓 \in 𝐶^{2} (ℝ)$ , let $𝑀_{𝑘} = {sup}_{𝑥} | 𝑓^{(𝑘)} (𝑥) |$ .

(a) Prove that $𝑀_{1} \leq 2 \sqrt{𝑀_{0} 𝑀_{2}}$ .

(b) Show that if equality holds in (a), then $𝑓$ is a constant function.

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