arXiv subjects

Tian Yao

Publications and source records attributed to Tian Yao.

17 recordsLinked to original sources

arXiv2026

On Erd\H{o}s--Ko--Rado and Hilton--Milner Theorems for Direct Products

We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erd\H{o}s--Ko--Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{155} (2018), 493--502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [\emph{J. Combin. Theory Ser. A} \textbf{156} (2018), 44--60] asking for a classification of all extremal families, including the possible non-shifted maximizers.

math.CO↗

arXiv2026

Non-trivial cross-$t$-intersecting families for vector spaces with the maximum sum of sizes

Let $V$ be an $n$-dimensional vector space over a finite field. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are non-empty families of $k$-subspaces and $\ell$-subspaces of $V$, respectively. They are said to be cross-$t$-intersecting if $\dim(F\cap G)\geq t$ for any $F\in\mathcal{F}$ and $G\in \mathcal{G}$, and are further called non-trivial if $\dim(\cap_{F\in\mathcal{F}}F)<t$ and $\dim(\cap_{G\in\mathcal{G}}G)<t$. In this paper, we characterize the non-trivial cross-$t$-intersecting families with the maximum sum of sizes. When $t=1$, our result serves as the $q$-analog of the theorems in [9,11].

math.CO↗

arXiv2026

Extremal $t$-intersecting families for finite sets with $t$-covering number at least $t+2$

Let $\mathcal{F}\subseteq{[n]\choose k}$ be a $t$-intersecting family. Define the $t$-covering number $\tau_t(\mathcal{F})$ of $\mathcal{F}$ as the minimum size of a subset $S$ of $[n]$ with $|S\cap F|\geqslant t$ for each $F\in\mathcal{F}$. In this paper, we characterize $\mathcal{F}$ for which $|\mathcal{F}|$ takes the maximum value under the condition that $\tau_t(\mathcal{F})\geqslant t+2$ and $n$ is sufficiently large, thereby generalizing two results by Frankl.

math.CO↗

arXiv2026

$s$-almost cross-$t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F} _{q} $, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are said to be cross-$t$-intersecting if $\dim(F\cap G)\ge t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result.

math.CO↗

arXiv2026

$s$-almost $t$-intersecting families for finite sets

A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\left|\bigcap_{F\in\mathcal{F}} F\right|<t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.

math.CO↗

arXiv2026

Multi-channel multi-speaker transformer for speech recognition

With the development of teleconferencing and in-vehicle voice assistants, far-field multi-speaker speech recognition has become a hot research topic. Recently, a multi-channel transformer (MCT) has been proposed, which demonstrates the ability of the transformer to model far-field acoustic environments. However, MCT cannot encode high-dimensional acoustic features for each speaker from mixed input audio because of the interference between speakers. Based on these, we propose the multi-channel multi-speaker transformer (M2Former) for far-field multi-speaker ASR in this paper. Experiments on the SMS-WSJ benchmark show that the M2Former outperforms the neural beamformer, MCT, dual-path RNN with transform-average-concatenate and multi-channel deep clustering based end-to-end systems by 9.2%, 14.3%, 24.9%, and 52.2% respectively, in terms of relative word error rate reduction.

cs.SD↗

arXiv2025

Cross-intersection theorems for uniform partitions of finite sets

A set partition is $c$-uniform if every block has size $c$. Two families of $c$-uniform partitions of a finite set are said to be cross $t$-intersecting if two partitions from different families share at least $t$ blocks. In this paper, we establish some product-type extremal results for such cross $t$-intersecting families. Our results yield an Erdős-Ko-Rado theorem and a Hilton-Milner theorem for uniform set partitions. Additionally, cross $t$-intersecting families with the maximum sum of their sizes are also characterized.

math.CO↗

arXiv2025

$s$-almost cross-$t$-intersecting families for finite sets

Two families $\mathcal{F}$ and $\mathcal{G}$ of $k$-subsets of an $n$-set are called $s$-almost cross-$t$-intersecting if each member in $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members in $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we characterize the $s$-almost cross-$t$-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the $s$-almost cross-$t$-intersecting families which are not cross-$t$-intersecting.

math.CO↗

arXiv2024

$s$-almost $t$-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in \mathcal{F}$. We say $\mathcal{F}$ is $s$-almost $t$-intersecting if for each $F\in \mathcal{F}$ there are at most $s$ members $F^{\prime}$ of $\mathcal{F}$ such that $\dim(F\cap F^{\prime})<t$. In this paper, we prove that $s$-almost $t$-intersecting families with maximum size are $t$-intersecting. We also consider $s$-almost $t$-intersecting families which are not $t$-intersecting, and characterize such families with maximum size for $(s,t)\neq(1,1)$. The result for $1$-almost $1$-intersecting families provided by Shan and Zhou is generalized.

math.CO↗

arXiv2024

The Database and Benchmark for the Source Speaker Tracing Challenge 2024

Voice conversion (VC) systems can transform audio to mimic another speaker's voice, thereby attacking speaker verification (SV) systems. However, ongoing studies on source speaker verification (SSV) are hindered by limited data availability and methodological constraints. This paper presents the Source Speaker Tracking Challenge (SSTC) on STL 2024, which aims to fill the gap in the database and benchmark for the SSV task. In this study, we generate a large-scale converted speech database with 16 common VC methods and train a batch of baseline systems based on the MFA-Conformer architecture. In addition, we introduced a related task called conversion method recognition, with the aim of assisting the SSV task. We expect SSTC to be a platform for advancing the development of the SSV task and provide further insights into the performance and limitations of current SV systems against VC attacks. Further details about SSTC can be found in https://sstc-challenge.github.io/.

eess.AS↗

arXiv2024

More on $r$-cross $t$-intersecting families for vector spaces

Let $V$ be a finite dimensional vector space over a finite field. Suppose that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross $t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine the extremal structure when $\prod_{i=1}^r|\mathscr{F}_i|$ is maximum under the condition that $\dim(\bigcap_{F\in\mathscr{F}_i}F)<t$ for each $i$.

math.CO↗

arXiv2024

The maximum sum of sizes of non-empty cross $t$-intersecting families

Let $[n]:=\lbrace 1,2,\ldots,n \rbrace$, and $M$ be a set of positive integers. Denote the family of all subsets of $[n]$ with sizes in $M$ by $\binom{\left[n\right]}{M}$. The non-empty families $\mathcal{A}\subseteq\binom{\left[n\right]}{R}$ and $\mathcal{B}\subseteq \binom{\left[n\right]}{S}$ are said to be cross $t$-intersecting if $|A\cap B|\geq t$ for all $A\in \mathcal{A}$ and $B\in \mathcal{B}$. In this paper, we determine the maximum sum of sizes of non-empty cross $t$-intersecting families, and characterize the extremal families. Similar result for finite vector spaces is also proved.

math.CO↗

arXiv2022

Cross $t$-intersecting families for symplectic polar spaces

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the collection of all $k$-dimensional totally isotropic subspace in $\mathscr{P}$. Let $\mathscr{F}_1\subset\mathscr{P}_{m_1}$ and $\mathscr{F}_2\subset\mathscr{P}_{m_2}$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-dimensional totally isotropic subspace. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.

math.CO↗

arXiv2022

Cross $t$-intersecting families for finite affine spaces

Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by $\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and $\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-flats in $AG(n,\mathbb{F}_q)$. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.

math.CO↗

arXiv2021

Large non-trivial $t$-intersecting families for signed sets

For positive integers $n,r,k$ with $n\ge r$ and $k\ge2$, a set $\{(x_1,y_1),(x_2,y_2),\dots,(x_r,y_r)\}$ is called a $k$-signed $r$-set on $[n]$ if $x_1,\dots,x_r$ are distinct elements of $[n]$ and $y_1\dots,y_r\in[k]$. We say a $t$-intersecting family consisting of $k$-signed $r$-sets on $[n]$ is trivial if each member of this family contains a fixed $k$-signed $t$-set. In this paper, we determine the structure of large maximal non-trivial $t$-intersecting families. In particular, we characterize the non-trivial $t$-intersecting families with maximum size for $t\ge2$, extending a Hilton-Milner-type result for signed sets given by Borg.

math.CO↗

arXiv2020

Non-trivial $t$-intersecting families for symplectic polar spaces

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the set of all $m$-dimensional subspaces in $\mathscr{P}$. We say a $t$-intersecting subfamily of $\mathscr{P}_m$ is trivial if there exists a $t$-dimensional subspace contained in each member of this family. In this paper, we determine the structure of maximum sized non-trivial $t$-intersecting subfamilies of $\mathscr{P}_m$.

math.CO↗

arXiv2020

Extremal $t$-intersecting families for direct products

In this paper, by shifting technique we study $t$-intersecting families for direct products where the ground set is divided into several parts. Assuming the size of each part is sufficiently large, we determine all extremal $t$-intersecting families for direct products. We also prove that every largest $t$-intersecting subfamily of a more general family introduced by Katona is trivial under certain conditions.

math.CO↗