arXiv subjects

Yufei Zhao

Publications and source records attributed to Yufei Zhao.

At least 91 records · Page 5Linked to original sources

arXiv2022

Equiangular lines with a fixed angle

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < α< 1$. Let $N_α(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\arccos α$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-α)/(2α)$. If $k < \infty$, then $N_α(d) = \lfloor k(d-1)/(k-1) \rfloor$ for all sufficiently large $d$, and otherwise $N_α(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$ for every integer $k\ge 2$ and all sufficiently large $d$. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

math.CO↗

arXiv2022

Removal lemmas and approximate homomorphisms

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $ε>0$ there exists $M$ such that every triangle-free graph $G$ has an $ε$-approximate homomorphism to a triangle-free graph $F$ on at most $M$ vertices (here an $ε$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $ε|V(G)|^2$ edges of $G$ are mapped to edges of $F$). One consequence of our results is that the least possible $M$ in the triangle-free lemma grows faster than exponential in any polynomial in $ε^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal.

math.CO↗

arXiv2021

A counterexample to the Bollobás-Riordan conjectures on sparse graph limits

Bollobás and Riordan, in their paper "Metrics for sparse graphs," proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom $C_4$-counts), but with no limit expressible as a kernel.

math.CO↗

arXiv2021

Exploring a planet, revisited

How should we place $n$ great circles on a sphere to minimize the furthest distance between a point on the sphere and its nearest great circle? Fejes Tóth conjectured that the optimum is attained by placing $n$ circles evenly spaced all passing through the north and south poles. This conjecture was recently proved by Jiang and Polyanskii. We present a short simplification of Ortega-Moreno's alternate proof of this conjecture.

math.MG↗

arXiv2021

The regularity method for graphs with few 4-cycles

We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: * Every $n$-vertex graph with no 5-cycle can be made triangle-free by deleting $o(n^{3/2})$ edges. * For $r \geq 3$, every $n$-vertex $r$-graph with girth greater than $5$ has $o(n^{3/2})$ edges. * Every subset of $[n]$ without a nontrivial solution to the equation $x_1 + x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$.

math.CO↗

arXiv2021

Patterns without a popular difference

Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(α^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p : p \in P\}$, where $α= |A|/N^r$? Green showed that all 3-point $P \subseteq \mathbb{Z}$ have the above property. Green and Tao showed that 4-point sets of the form $P = \{a, a+b, a+c, a+b+c\} \subseteq \mathbb{Z}$ also have the property. We show that no other sets have the above property. Furthermore, for various $P$, we provide new upper bounds on the number of translates of $d \cdot P$ that one can guarantee to find.

math.CO↗

arXiv2021

Which graphs can be counted in $C_4$-free graphs?

For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $ε$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property.

math.CO↗

arXiv2021

Routes to realize the axion-insulator phase in MnBi$_{2}$Te$_{4}$(Bi$_{2}$Te$_{3}$)$_{n}$ family: a perspective

Axion, first postulated as a hypothetical particle in high-energy physics, is now extended to describe a novel topological magnetoelectric effect derived from the Chern-Simons theory in condensed matter systems. The recent discovered intrinsic magnetic topological insulators MnBi2Te4 and its derivatives have attracted great attention because of their potential as a material platform to realize such a quantized axion field. Since the magnetic exchange gap can bring the half-quantized anomalous Hall effect at the surface, an axion insulator manifests as quantum anomalous Hall and zero Hall plateau effects in the thin films. However, many puzzles about this material family remain elusive yet, such as the gapless surface state and the direct experimental evidence of the axion insulator. In this Perspective, we discuss the preconditions, manifestations and signatures of the axion-insulator phase, in the context of the development of the natural magnetic topological heterostructure MnBi2Te4(Bi2Te3)n family with various intriguing quantum phenomena. Recent theoretical and experimental efforts regarding the intrinsic magnetic topological insulators are summarized here to pave the way for this phenomenally developing field.

cond-mat.mtrl-sci↗

arXiv2021

On anisotropic non-Lipschitz restoration model: lower bound theory and convergent algorithm

For nonconvex and nonsmooth restoration models, the lower bound theory reveals their good edge recovery ability, and related analysis can help to design convergent algorithms. Existing such discussions are focused on isotropic regularization models, or only the lower bound theory of anisotropic model with a quadratic fidelity. In this paper, we consider a general image recovery model with a non-Lipschitz anisotropic composite regularization term and an $\ell_q$ norm ($1\leq q<+\infty$) data fidelity term. We establish the lower bound theory for the anisotropic model with an $\ell_1$ fidelity, which applies to impulsive noise removal problems. For the general case with $1\leq q<+\infty$, a support inclusion analysis is provided. To solve this non-Lipschitz composite minimization model, we are then motivated to introduce a support shrinking strategy in the iterative algorithm and relax the support constraint to a thresholding support constraint, which is more computationally practical. The objective function at each iteration is also linearized to construct a strongly convex subproblem. To make the algorithm more implementable, we compute an approximation solution to this subproblem at each iteration, but not exactly solve it. The global convergence result of the proposed inexact iterative thresholding and support shrinking algorithm with proximal linearization is established. The experiments on image restoration and two stage image segmentation demonstrate the effectiveness of the proposed algorithm.

math.NA↗

arXiv2021

$K_4$-intersecting families of graphs

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that the maximum size of a triangle-intersecting family of graphs on $n$ vertices has size at most $2^{\binom{n}{2} - 3}$, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to $K_t$-intersecting families. We prove these conjectures for $t \in \{3,4\}$, showing that if $\mathcal F_1$ and $\mathcal F_2$ are families of graphs on $n$ labeled vertices such that for any $G_1 \in \mathcal F_1$ and $G_2 \in \mathcal F_2$, $G_1 \cap G_2$ contains a $K_t$, then $\lvert \mathcal F_1 \rvert \lvert \mathcal F_2 \rvert \le 4^{\binom{n}{2} - \binom{t}{2}}$, with equality if and only if $\mathcal F_1 = \mathcal F_2$ consists of all graphs that contain some fixed $K_t$. We also establish a stability result. More generally, "$G_1 \cap G_2$ contains a $K_t$" can be replaced by "$G_1$ and $G_2$ agree on a non-$(t-1)$-colorable graph."

math.CO↗

arXiv2021

Koopmans' theorem as the mechanism of nearly gapless surface states in self-doped magnetic topological insulators

The magnetization-induced gap at the surface state is widely believed as the kernel of magnetic topological insulators (MTIs) because of its relevance to various topological phenomena, such as the quantum anomalous Hall effect and the axion insulator phase. However, if the magnetic gap exists in an intrinsic MTI, such as MnBi$_{2}$Te$_{4}$, still remains elusive, with significant discrepancies between theoretical predictions and various experimental observations. Here, including the previously overlooked self-doping in real MTIs, we find that in general a doped MTI prefers a ground state with a gapless surface state. We use a simple model based on Koopmans' theorem to elucidate the mechanism and further demonstrated it in self-doped MnBi$_{2}$Te$_{4}$/(Bi$_{2}$Te$_{3}$)$_{n}$ family through first-principles calculations. Our work shed light on the design principles of MTIs with magnetic gaps by revealing the critical role of doping effects in understanding the delicate interplay between magnetism and topology.

cond-mat.mtrl-sci↗

arXiv2021

Spectral signatures of the surface anomalous Hall effect in magnetic axion insulators

The topological surface states of magnetic topological systems, such as Weyl semimetals and axion insulators, are associated with unconventional transport properties such as nonzero or half-quantized surface anomalous Hall effect. Here we study the surface anomalous Hall effect and its spectral signatures in different magnetic topological phases using both model Hamiltonian and first-principles calculations. We demonstrate that by tailoring the magnetization and interlayer electron hopping, a rich three-dimensional topological phase diagram can be established, including three types of topologically distinct insulating phases bridged by Weyl semimetals, and can be directly mapped to realistic materials such as MnBi2Te4/(Bi2Te3)n systems. Among them, we find that the surface anomalous Hall conductivity in the axion-insulator phase is a well-localized quantity either saturated at or oscillating around e2/2h, depending on the magnetic homogeneity. We also discuss the resultant chiral hinge modes embedded inside the side surface bands as the potential experimental signatures for transport measurements. Our study is a significant step forward towards the direct realization of long-sought axion insulators in realistic material systems.

cond-mat.mes-hall↗

arXiv2021

The cylindrical width of transitive sets

We show that for every $1 \le k \le d/(\log d)^C$, every finite transitive set of unit vectors in $\mathbb{R}^d$ lies within distance $O(1/\sqrt{\log (d/k)})$ of some codimension $k$ subspace, and this distance bound is best possible. This extends a result of Ben Green, who proved it for $k=1$.

math.MG↗

arXiv2020

Common and Sidorenko Linear Equations

A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\mathbb{F}_q^n$ is asymptotically (as $n \to \infty$) at least the number expected in a random two-coloring. The linear equation is Sidorenko if the number of solutions in any dense subset of $\mathbb{F}_q^n$ is asymptotically at least the number expected in a random set of the same density. In this paper, we characterize those linear equations which are common, and those which are Sidorenko. The main novelty is a construction based on choosing random Fourier coefficients that shows that certain linear equations do not have these properties. This solves problems posed in a paper of Saad and Wolf.

math.CO↗

arXiv2020

Cayley graphs without a bounded eigenbasis

Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon--Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor's result to nonabelian groups.

math.CO↗

arXiv2020

On the upper tail problem for random hypergraphs

The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős--Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, i.e., the first order asymptotics of the log-probability that the number of copies of fixed subgraph $H$ in a sparse Erdős--Rényi random $k$-uniform hypergraph exceeds its expectation by a constant factor. This conjecture turns out to be significantly more intricate compared to the case for graphs. We verify our conjecture when the fixed subgraph $H$ being counted is a clique, as well as when $H$ is the 3-uniform 6-vertex 4-edge hypergraph consisting of alternating faces of an octahedron, where new techniques are required.

math.CO↗

arXiv2020

Hypergraph expanders of all uniformities from Cayley graphs

Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.

math.CO↗

arXiv2020

Testing linear-invariant properties

Fix a prime $p$ and a positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there exists an oblivious tester for this property with one-sided error and constant query complexity. Furthermore, a property is proximity oblivious-testable (PO-testable) if the test is also independent of the proximity parameter $ε$. It is known that a number of natural properties such as linearity and being a low degree polynomial are PO-testable. These properties are examples of linear-invariant properties, meaning that they are preserved under linear automorphisms of the domain. Following work of Kaufman and Sudan, the study of linear-invariant properties has been an important problem in arithmetic property testing. A central conjecture in this field, proposed by Bhattacharyya, Grigorescu, and Shapira, is that a linear-invariant property is testable if and only if it is semi subspace-hereditary. We prove two results, the first resolves this conjecture and the second classifies PO-testable properties. (1) A linear-invariant property is testable if and only if it is semi subspace-hereditary. (2) A linear-invariant property is PO-testable if and only if it is locally characterized. Our innovations are two-fold. We give a more powerful version of the compactness argument first introduced by Alon and Shapira. This relies on a new strong arithmetic regularity lemma in which one mixes different levels of Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer, Hatami, Hatami, and Lovett by removing the bounded complexity restriction in their work. Our second innovation is a novel recoloring technique called patching. This Ramsey-theoretic technique is critical for working in the linear-invariant setting and allows us to remove the translation-invariant restriction present in previous work.

math.CO↗