querent

Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2024, 5 Real analysis, convolution, identity elements, nonexistence proofs

Show that there does not exist a function $𝑔 \in 𝐿^{1} (ℝ)$ such that

𝑓 * 𝑔 = 𝑓

for all $𝑓 \in 𝐿^{1} (ℝ)$ . Here

𝑓 * 𝑔 (𝑥) = \int_{ℝ} 𝑓 (𝑦) 𝑔 (𝑥 - 𝑦) d 𝑦

is the convolution of $𝑓$ and $𝑔$ .

UC Irvine Fall 2024, 6 Real analysis, heat kernel inequalities, partial derivatives, convexity

Let $𝑁 \in 𝐶^{\infty} (ℝ \times (0, \infty))$ satisfy

𝑁 (𝑝 + 𝑎, 𝑡 + 𝑎^{2}) + 𝑁 (𝑝 - 𝑎, 𝑡 + 𝑎^{2}) \geq 2 𝑁 (𝑝, 𝑡)

for all $𝑎, 𝑝 \in ℝ$ and $𝑡 > 0$ . Show that

UC Irvine Fall 2024, 6 Real analysis, Hardy–Littlewood maximal function, weak-type inequalities

Let $𝑓 \in 𝐿^{1} (ℝ)$ , and for all $𝑧 \in ℝ$ define

𝑔 (𝑧) ≔ sup_{𝑟 > 0} \frac{1}{2 𝑟} \int_{𝑧 - 𝑟}^{𝑧 + 𝑟} | 𝑓 (𝑥) | d 𝑥 .

Prove that, for any $𝑡 > 0$ ,

| {𝑧 : 𝑔 (𝑧) > 𝑡} | \leq \frac{5}{𝑡} ∥ 𝑓 \underset{𝐿^{1} (ℝ)}{∥} .

UC Irvine Fall 2024, 7 Real analysis, Green's theorem, line integrals

Let $𝑄$ be the square with vertices $(1, 1), (1, - 1), (- 1, 1), (- 1, - 1)$ , and let $𝐶$ be its boundary oriented counterclockwise. Evaluate

\int_{𝐶} \frac{- 𝑦}{𝑥^{2} + 𝑦^{2}} d 𝑥 + \frac{𝑥}{𝑥^{2} + 𝑦^{2}} d 𝑦 .

UC Irvine Fall 2024, 8 Real analysis, Gaussian integrals, multivariable calculus

Calculate

\int_{ℝ^{2}} 𝑒^{- 3 𝑥^{2} + 2 𝑥 𝑦 - 3 𝑦^{2}} d 𝑥 d 𝑦 .

UC Irvine Fall 2024, 9 Real analysis, metric spaces, separability, completeness

Give an example of a path connected bounded complete metric space that is not separable.

Stanford Fall 2025, 1 Real analysis, Banach spaces, spectral theory, bounded operators

Let $𝑋$ be a Banach space and $𝑇 : 𝑋 \to 𝑋$ be a bounded linear operator.

(a) Suppose $𝑇^{𝑛} = 𝐼$ for some $𝑛 \in ℤ_{\geq 2}$ . Prove that $𝜆 \in 𝜎 (𝑇) \Rightarrow 𝜆^{𝑛} = 1$ . Moreover, if $𝜆^{𝑛} \neq 1$ , write down an expression for ${(𝑇 - 𝜆 𝐼)}^{- 1}$ .

(b) Suppose $‖ 𝑇^{𝑛} ‖ < 1$ for some $𝑛 \in ℤ_{\geq 2}$ . Prove that $𝐼 - 𝑇$ is invertible.

Stanford Fall 2025, 1 Real analysis, Hilbert space geometry, weak convergence

Let $𝐻$ be a Hilbert space. Suppose ${𝑥_{𝑛}}_{𝑛 = 1}^{\infty} \subset 𝐻$ is a sequence such that $𝑥_{𝑛} ⇀ 𝑥$ for some $𝑥 \in 𝐻$ in the weak topology.

(a) Let $𝑦_{𝑛} = \frac{1}{𝑛} (𝑥_{1} + \dots + 𝑥_{𝑛})$ . Prove that $𝑦_{𝑛} ⇀ 𝑥$ in the weak topology.

(b) Is it always true that $𝑦_{𝑛} \to 𝑥$ in norm? Prove this or give a counterexample.

UC Irvine Fall 2025, 1 Real analysis, uniform convergence, Riemann integrals, limits

Let $𝑓_{𝑛} : [0, 1] \to ℝ$ be a sequence of continuous functions that converges uniformly to $𝑓$ . Show that

lim_{𝑛 \to \infty} \int_{0}^{1} \sqrt{1 + 𝑓_{𝑛} {(𝑥)}^{2}} d 𝑥 = \int_{0}^{1} \sqrt{1 + {𝑓 (𝑥)}^{2}} d 𝑥 .

UC Irvine Fall 2025, 1 Real analysis, measurable functions, positivity, integral bounds

Let $𝑓$ be a measurable function on $[0, 1]$ such that $𝑓 (𝑥) > 0$ for all $𝑥 \in [0, 1]$ , and assume that

\int_{[0, 1]} 𝑓 (𝑥) d 𝑥 < \infty .

Let $0 < 𝛾 < 1$ . Prove that

inf_{| 𝐸 | = 𝛾} \int_{𝐸} 𝑓 (𝑥) d 𝑥 > 0,

where the infimum is taken over all measurable subsets $𝐸 \subset [0, 1]$ of Lebesgue measure $𝛾$ .

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