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Yufei Zhao

Publications and source records attributed to Yufei Zhao.

At least 73 records · Page 4Linked to original sources
arXiv2024

Spectral non-concentration near the top for unimodular random graphs

In recent work on equiangular lines, Jiang, Tidor, Yuan, Zhang, and Zhao showed that a connected bounded degree graph has sublinear second eigenvalue multiplicity. More generally they show that there cannot be too many eigenvalues near the top of the spectrum. We extend this result to infinite unimodular random graphs. As a corollary, the spectral distribution of the adjacency operator cannot have an atom at the top. For an infinite regular expander, we deduce that the singularity of the spectral measure at the top satisfies $μ_G[(1-θ)ρ,ρ] \lesssim θ^c$ for some constant $c>0$, where $ρ$ is the spectral radius of the adjacency operator of the graph. This implies new general estimates on the return probabilities of random walks.

math.PR↗
arXiv2023

Uniacute Spherical Codes

A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a fundamental question in discrete geometry and has been extensively investigated for various choices of $L$. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to $L = \{-α, α\}$, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that $N_L(d) = O_L(d)$ for $L = [-1, -β] \cup \{α\}$ with $α,β> 0$ (we call such $L$-codes "uniacute"), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a "uniform bound" showing $\limsup_{d\to\infty} N_L(d)/d \le 2p$ for $L = [-1, -β] \cup \{α\}$ and $p = \lfloor α/β\rfloor + 1$. For which $(α,β)$ is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate $p$-block structure. We also formulate a notion of "modular codes," which we conjecture to be optimal in high dimensions.

math.CO↗
arXiv2023

Efficient Multi-Pair IoT Communication with Holographically Enhanced Meta-Surfaces Leveraging OAM Beams: Bridging Theory and Prototype

Meta-surfaces, also known as Reconfigurable Intelligent Surfaces (RIS), have emerged as a cost-effective, low power consumption, and flexible solution for enabling multiple applications in Internet of Things (IoT). However, in the context of meta-surface-assisted multi-pair IoT communications, significant interference issues often arise amount multiple channels. This issue is particularly pronounced in scenarios characterized by Line-of-Sight (LoS) conditions, where the channels exhibit low rank due to the significant correlation in propagation paths. These challenges pose a considerable threat to the quality of communication when multiplexing data streams. In this paper, we introduce a meta-surface-aided communication scheme for multi-pair interactions in IoT environments. Inspired by holographic technology, a novel compensation method on the whole meta-surface has been proposed, which allows for independent multi-pair direct data streams transmission with low interference. To further reduce correlation under LoS channel conditions, we propose a vortex beam-based solution that leverages the low correlation property between distinct topological modes. We use different vortex beams to carry distinct data streams, thereby enabling distinct receivers to capture their intended signal with low interference, aided by holographic meta-surfaces. Moreover, a prototype has been performed successfully to demonstrate two-pair multi-node communication scenario operating at 10 GHz with QPSK/16-QAM modulation.

eess.SP↗
arXiv2023

Multi-mode OAM Convergent Transmission with Co-divergent Angle Tailored by Airy Wavefront

Wireless backhaul offers a more cost-effective, time-efficient, and reconfigurable solution than wired backhaul to connect the edge-computing cells to the core network. As the amount of transmitted data increases, the low-rank characteristic of Line-of-Sight (LoS) channel severely limits the growth of channel capacity in the point-to-point backhaul transmission scenario. Orbital Angular Momentum (OAM), also known as vortex beam, is considered a potentially effective solution for high-capacity LoS wireless transmission. However, due to the shortcomings of its energy divergence and the specificity of multi-mode divergence angles, OAM beams have been difficult to apply in practical communication systems for a long time. In this work, a novel multi-mode convergent transmission with co-scale reception scheme is proposed. OAM beams of different modes can be transmitted with the same beam divergent angle, while the wavefronts are tailored by the ring-shaped Airy compensation lens during propagation, so that the energy will converge to the same spatial area for receiving. Based on this scheme, not only is the Signal-to-Noise Ratio (SNR) greatly improved, but it is also possible to simultaneously receive and demodulate OAM channels multiplexed with different modes in a limited space area. Through prototype experiments, we demonstrated that 3 kinds of OAM modes are tunable, and different channels can be separated simultaneously with receiving power increasing. The measurement isolations between channels are over 11 dB, which ensures a reliable 16-QAM multiplexing wireless transmission demo system. This work may explore the potential applications of OAM-based multi-mode convergent transmission in LoS wireless communications.

cs.IT↗
arXiv2023

Chiral charge density wave and backscattering-immune orbital texture in monolayer 1T-TiTe2

Non-trivial electronic states are attracting intense attention in low-dimensional physics. Though chirality has been identified in charge states with a scalar order parameter, its intertwining with charge density waves (CDW), film thickness and the impact on the electronic behaviors remain less well understood. Here, using scanning tunneling microscopy, we report a 2 x 2 chiral CDW as well as a strong suppression of the Te-5p hole-band backscattering in monolayer 1T-TiTe2. These exotic characters vanish in bilayer TiTe2 with a non-CDW state. Theoretical calculations approve that chirality comes from a helical stacking of the triple-q CDW components and therefore can persist at the two-dimensional limit. Furthermore, the chirality renders the Te-5p bands an unconventional orbital texture that prohibits electron backscattering. Our study establishes TiTe2 as a promising playground for manipulating the chiral ground states at the monolayer limit and provides a novel path to engineer electronic properties from an orbital degree.

cond-mat.mtrl-sci↗
arXiv2023

Nearly all $k$-SAT functions are unate

We prove that $1-o(1)$ fraction of all $k$-SAT functions on $n$ Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer $k$ and as $n \to \infty$. This resolves a conjecture by Bollobás, Brightwell, and Leader from 2003.

math.CO↗
arXiv2023

Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset of $n$ vertices where all of the edges between them receive a common color. If $n$ is fixed and $\frac{s}{r}$ is less than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is known to grow exponentially in $r$, while if $\frac{s}{r}$ is greater than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is bounded. Here we prove bounds for $R(n;r,s)$ in the intermediate range where $\frac{s}{r}$ is close to $1 - \frac{1}{n-1}$ by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.

math.CO↗
arXiv2023

Quantum Unique Ergodicity for Cayley graphs of quasirandom groups

A finite group $G$ is called $C$-quasirandom (by Gowers) if all non-trivial irreducible complex representations of $G$ have dimension at least $C$. For any unit $\ell^{2}$ function on a finite group we associate the quantum probability measure on the group given by the absolute value squared of the function. We show that if a group is highly quasirandom, in the above sense, then any Cayley graph of this group has an orthonormal eigenbasis of the adjacency operator such that the quantum probability measures of the eigenfunctions put close to the correct proportion of their mass on subsets of the group that are not too small.

math.SP↗
arXiv2023

Human Behavior in the Time of COVID-19: Learning from Big Data

Since the World Health Organization (WHO) characterized COVID-19 as a pandemic in March 2020, there have been over 600 million confirmed cases of COVID-19 and more than six million deaths as of October 2022. The relationship between the COVID-19 pandemic and human behavior is complicated. On one hand, human behavior is found to shape the spread of the disease. On the other hand, the pandemic has impacted and even changed human behavior in almost every aspect. To provide a holistic understanding of the complex interplay between human behavior and the COVID-19 pandemic, researchers have been employing big data techniques such as natural language processing, computer vision, audio signal processing, frequent pattern mining, and machine learning. In this study, we present an overview of the existing studies on using big data techniques to study human behavior in the time of the COVID-19 pandemic. In particular, we categorize these studies into three groups - using big data to measure, model, and leverage human behavior, respectively. The related tasks, data, and methods are summarized accordingly. To provide more insights into how to fight the COVID-19 pandemic and future global catastrophes, we further discuss challenges and potential opportunities.

cs.CY↗
arXiv2023

Graphs with high second eigenvalue multiplicity

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree $n$-vertex graph? The best known upper bound is $O(n/\log\log n)$. The previously known best known lower bound is on the order of $n^{1/3}$ (for infinitely many $n$), coming from Cayley graphs on $\text{PSL}(2,q)$. Here we give constructions showing a lower bound on the order of $\sqrt{n/\log n}$. We also construct Cayley graphs with second eigenvalue multiplicity at least $n^{2/5}-1$. Earlier techniques show that there are at most $O(n/\log\log n)$ eigenvalues (counting multiplicities) within $O(1/\log n)$ of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

math.CO↗
arXiv2022

Joints tightened

In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $\binom{k}{d-1}$ lines and their $d$-wise intersections to form $\binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.

math.CO↗
arXiv2022

Cayley graphs that have a quantum ergodic eigenbasis

We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $n$ for which the sum of the dimensions of its irreducible representations is $o(n)$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis.

math.SP↗
arXiv2022

Spherical two-distance sets and eigenvalues of signed graphs

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting.

math.CO↗
arXiv2022

On the number of error correcting codes

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - C_q\sqrt{n \log n}$ (for sufficiently large constant $C_q$), where $H_q(n, t) = q^n / V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$.

math.CO↗
arXiv2022

Enumerating k-SAT functions

How many $k$-SAT functions on $n$ boolean variables are there? What does a typical such function look like? Bollobás, Brightwell, and Leader conjectured that, for each fixed $k \ge 2$, the number of $k$-SAT functions on $n$ variables is $(1+o(1))2^{\binom{n}{k} + n}$, or equivalently: a $1-o(1)$ fraction of all $k$-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for $k=2$. The conjecture was confirmed for $k=2$ by Allen and $k=3$ by Ilinca and Kahn. We show that the problem of enumerating $k$-SAT functions is equivalent to a Turán density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollobás--Brightwell--Leader conjecture for $k=4$ by solving the corresponding Turán density problem. Our solution applies a recent result of Füredi and Maleki on the minimum triangular edge density in a graph of given edge density. In an appendix (by Nitya Mani and Edward Yu), we further confirm the $k=5$ case of the conjecture via a brute force computer search.

math.CO↗
arXiv2022

Joints of varieties

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. More generally, we prove the same bound when the set of $N$ planes is replaced by a set of 2-dimensional algebraic varieties of total degree $N$, and a joint is a point that is regular for three varieties whose tangent planes at that point are not all contained in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects, relating the degree of a polynomial and its orders of vanishing on a given set of points on a variety.

math.CO↗
arXiv2022

Extension complexity of low-dimensional polytopes

Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension $d$, the extension complexity of a random $d$-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic $n$-vertex polygon (whose vertices lie on a circle) has extension complexity at most $24\sqrt n$. This bound is tight up to the constant factor $24$. Finally, we show that there exists an $n^{o(1)}$-dimensional polytope with at most $n$ vertices and extension complexity $n^{1-o(1)}$. Our theorems are proved with a range of different techniques, which we hope will be of further interest.

math.CO↗