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In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries are i.i.d. copies of a random variable $\xi \in \mathbb{Z}_p$. We consider the minimal number of random entries of $X_n$ required for the cokernel of $X_n$ to converge to the Cohen--Lenstra distribution. When $\xi$ is given by the Haar measure, we prove a lower bound of the number of random entries and prove its converse-type result using random regular bipartite multigraphs. When $\xi$ is a general random variable, we determine the minimal number of random entries. Let $M_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ with $k$-step stairs of zeros and the other entries given by independent random $\epsilon$-balanced variables valued in $\mathbb{Z}_p$. We prove that the cokernel of $M_n$ converges to the Cohen--Lenstra distribution under a mild assumption. This extends Wood's universality theorem on random $p$-adic matrices., Comment: 48 pages

In [GTZ08, GTZ12], the following result was established: given polynomials $f,g\in\mathbb{C}[x]$ of degrees larger than $1$, if there exist $\alpha,\beta\in\mathbb{C}$ such that their corresponding orbits $\mathcal{O}_f(\alpha)$ and $\mathcal{O}_g(\beta)$ (under the action of $f$, respectively of $g$) intersect in infinitely many points, then $f$ and $g$ must share a common iterate, i.e., $f^m=g^n$ for some $m,n\in\mathbb{N}$. If one replaces $\mathbb{C}$ with a field $K$ of characteristic $p$, then the conclusion fails; we provide numerous examples showing the complexity of the problem over a field of positive characteristic. We advance a modified conjecture regarding polynomials $f$ and $g$ which admit two orbits with infinite intersection over a field of characteristic $p$. Then we present various partial results, along with connections with another deep conjecture in the area, the dynamical Mordell-Lang conjecture., Comment: 13 pages, comments welcome

In this paper, we prove the converse of the dynamical Mordell--Lang conjecture in positive characteristic: For every subset $S \subseteq \mathbb{N}_0$ which is a union of finitely many arithmetic progressions along with finitely many $p$-sets of the form $\left \{ \sum_{j=1}^{m} c_j p^{k_jn_j} : n_j \in \mathbb{N}_0 \right \}$ ($c_j \in \mathbb{Q}$, $k_j \in \mathbb{N}_0$), there exist a split torus $X = \mathbb{G}_m^k$ defined over $K=\overline{\mathbb{F}_p}(t)$, an endomorphism $\Phi$ of $X$, $\alpha \in X(K)$ and a closed subvariety $V \subseteq X$ such that $\left \{ n \in \mathbb{N}_0 : \Phi^n(\alpha) \in V(K) \right \} = S$., Comment: 6 pages

Automatic Speech Recognition (ASR) still face challenges when recognizing time-variant rare-phrases. Contextual biasing (CB) modules bias ASR model towards such contextually-relevant phrases. During training, a list of biasing phrases are selected from a large pool of phrases following a sampling strategy. In this work we firstly analyse different sampling strategies to provide insights into the training of CB for ASR with correlation plots between the bias embeddings among various training stages. Secondly, we introduce a neighbourhood attention (NA) that localizes self attention (SA) to the nearest neighbouring frames to further refine the CB output. The results show that this proposed approach provides on average a 25.84% relative WER improvement on LibriSpeech sets and rare-word evaluation compared to the baseline., Comment: Accepted for IEEE ASRU 2023

On-device Automatic Speech Recognition (ASR) models trained on speech data of a large population might underperform for individuals unseen during training. This is due to a domain shift between user data and the original training data, differed by user's speaking characteristics and environmental acoustic conditions. ASR personalisation is a solution that aims to exploit user data to improve model robustness. The majority of ASR personalisation methods assume labelled user data for supervision. Personalisation without any labelled data is challenging due to limited data size and poor quality of recorded audio samples. This work addresses unsupervised personalisation by developing a novel consistency based training method via pseudo-labelling. Our method achieves a relative Word Error Rate Reduction (WERR) of 17.3% on unlabelled training data and 8.1% on held-out data compared to a pre-trained model, and outperforms the current state-of-the art methods., Comment: Accepted for IEEE ASRU 2023

In this paper, we study the combinatorial relations between the cokernels $\text{cok}(A_n+px_iI_n)$ ($1 \le i \le m$) where $A_n$ is an $n \times n$ matrix over the ring of $p$-adic integers $\mathbb{Z}_p$, $I_n$ is the $n \times n$ identity matrix and $x_1, \cdots, x_m$ are elements of $ \mathbb{Z}_p$ whose reductions modulo $p$ are distinct. For a positive integer $m \le 4$ and given $x_1, \cdots, x_m \in \mathbb{Z}_p$, we determine the set of $m$-tuples of finitely generated $\mathbb{Z}_p$-modules $(H_1, \cdots, H_m)$ for which $(\text{cok}(A_n+px_1I_n), \cdots, \text{cok}(A_n+px_mI_n)) = (H_1, \cdots, H_m)$ for some matrix $A_n$. We also prove that if $A_n$ is an $n \times n$ Haar random matrix over $\mathbb{Z}_p$ for each positive integer $n$, then the joint distribution of $\text{cok}(A_n+px_iI_n)$ ($1 \le i \le m$) converges as $n \rightarrow \infty$., Comment: 30 pages

We study the joint distribution of random abelian and non-abelian groups. In the abelian case, we prove several universality results for the joint distribution of the multiple cokernels for random $p$-adic matrices. In the non-abelian case, we compute the joint distribution of random groups given by the quotients of the free profinite group by random relations. In both cases, we generalize the known results on the distribution of the cokernels of random $p$-adic matrices and random groups. Our proofs are based on the observation that mixed moments determine the joint distribution of random groups, which extends the works of Wood for abelian groups and Sawin for non-abelian groups., Comment: 29 pages, to appear in J. Algebra

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers $\mathcal{O}$ of a quadratic extension $K$ of $\mathbb{Q}_p$. For each positive integer $n$, let $X_n$ be a random $n \times n$ Hermitian matrix over $\mathcal{O}$ whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of $X_n$ always converges to the same distribution which does not depend on the choices of $X_n$ as $n \rightarrow \infty$ and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood in the case of the ring of integers of a quadratic extension of $\mathbb{Q}_p$., Comment: 31 pages, to appear in TAMS

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in \mathbb{Z}_p[t]$ be monic polynomials whose reductions modulo $p$ in $\mathbb{F}_p[t]$ are distinct and irreducible. We determine the limit of the joint distribution of the cokernels $\text{cok} (P_1(A)), \cdots, \text{cok}(P_l(A))$ for a random $n \times n$ matrix $A$ over $\mathbb{Z}_p$ with respect to Haar measure as $n \rightarrow \infty$. By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels $\text{cok}(A)$ and $\text{cok}(A+B_n)$ become independent as $n \rightarrow \infty$, where $B_n$ is a fixed $n \times n$ matrix over $\mathbb{Z}_p$ for each $n$ and $A$ is a random $n \times n$ matrix over $\mathbb{Z}_p$., Comment: 18 pages, to appear in Forum Math

In this paper we count the number $N_3^{\text{tor}}(X)$ of $3$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that $N_3^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \frac{\log 2 + \varepsilon}{\log \log X}}$, and this upper bound can be improved to $N_3^{\text{tor}}(X) \ll X (\log X)^4 \log \log X$ under the Cohen-Lenstra heuristics for $p=3$. We also prove that for $67$ out of $72$ conjugacy classes of finite nontrivial subgroups of $\operatorname{GL}_3(\mathbb{Z})$, Malle's conjecture for tori over $\mathbb{Q}$ holds up to a bounded power of $\log X$ under the Cohen-Lenstra heuristics for $p=3$ and Malle's conjecture for quartic $A_4$-fields., Comment: 33 pages, to appear in J. Number Theory

In this paper we count the number $N_n^{\text{tor}}(X)$ of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor of the associated character is bounded by $X$. This can be understood as a generalization of counting number fields of given degree by discriminant. We suggest a conjecture on the asymptotics of $N_n^{\text{tor}}(X)$ and prove that this conjecture follows from Malle's conjecture for tori over $\mathbb{Q}$. We also prove that $N_2^{\text{tor}}(X) \ll_{\varepsilon} X^{1 + \varepsilon}$, and this upper bound can be improved to $N_2^{\text{tor}}(X) \ll_{\varepsilon} X (\log X)^{1 + \varepsilon}$ under the assumption of the Cohen-Lenstra heuristics for $p=3$., Comment: 20 pages, exposition improved

In this paper we improve the upper bound of the number $N_{K, n}(X)$ of degree $n$ extensions of a number field $K$ with absolute discriminant bounded by $X$. This is achieved by giving a short $\mathcal{O}_K$-basis of an order of an extension $L$ of $K$. Our result generalizes the best known upper bound on $N_{\mathbb{Q}, n}(X)$ by Lemke Oliver and Thorne to all number fields $K$. Precisely, we prove that $N_{K, n}(X) \ll_{K, n} X^{c (\log n)^2}$ for an explicit constant $c$ independent on $K$ and $n$. We also improve the upper bound of the number of maximal arithmetic subgroups in certain connected semisimple Lie groups., Comment: This paper has been withdrawn by the author due to a critical error

We propose an adapter based multi-domain Transformer based language model (LM) for Transformer ASR. The model consists of a big size common LM and small size adapters. The model can perform multi-domain adaptation with only the small size adapters and its related layers. The proposed model can reuse the full fine-tuned LM which is fine-tuned using all layers of an original model. The proposed LM can be expanded to new domains by adding about 2% of parameters for a first domain and 13% parameters for after second domain. The proposed model is also effective in reducing the model maintenance cost because it is possible to omit the costly and time-consuming common LM pre-training process. Using proposed adapter based approach, we observed that a general LM with adapter can outperform a dedicated music domain LM in terms of word error rate (WER)., Comment: This paper is accepted for presentation at IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE ICASSP), 2021

Given a positive integer $r$ and a prime power $q$, we estimate the probability that the characteristic polynomial $f_{A}(t)$ of a random matrix $A$ in $\mathrm{GL}_{n}(\mathbb{F}_{q})$ is square-free with $r$ (monic) irreducible factors when $n$ is large. We also estimate the analogous probability that $f_{A}(t)$ has $r$ irreducible factors counting with multiplicity. In either case, the main term $(\log n)^{r-1}((r-1)!n)^{-1}$ and the error term $O((\log n)^{r-2}n^{-1})$, whose implied constant only depends on $r$ but not on $q$ nor $n$, coincide with the probability that a random permutation on $n$ letters is a product of $r$ disjoint cycles. The main ingredient of our proof is a recursion argument due to S. D. Cohen, which was previously used to estimate the probability that a random degree $n$ monic polynomial in $\mathbb{F}_{q}[t]$ is square-free with $r$ irreducible factors and the analogous probability that the polynomial has $r$ irreducible factors counting with multiplicity. We obtain our result by carefully modifying Cohen's recursion argument in the matrix setting, using Reiner's theorem that counts the number of $n \times n$ matrices with a fixed characteristic polynomial over $\mathbb{F}_{q}$., Comment: 19 pages. A conjecture in the previous draft has been resolved and its proof is included, and another author has been added

In this paper, we prove that the number $B(p,g)$ of isomorphism classes of abelian varieties over a prime field $\mathbb{F}_p$ of dimension $g$ has a lower bound $p^{\frac{1}{2} g^2 (1+o(1))}$ as $g \rightarrow \infty$. This is the first nontrivial result on the lower bound of $B(p,g)$. We also improve the upper bound $2^{34g^2} p^{\frac{69}{4} g^2 (1+o(1))}$ of $B(p,g)$ given by Lipnowski and Tsimerman (Duke Math. J. 167:3403-3453, 2018) to $p^{\frac{45}{4} g^2(1+o(1))}$., Comment: 20 pages, to appear in IMRN

In this paper, we present a new on-device automatic speech recognition (ASR) system based on monotonic chunk-wise attention (MoChA) models trained with large (> 10K hours) corpus. We attained around 90% of a word recognition rate for general domain mainly by using joint training of connectionist temporal classifier (CTC) and cross entropy (CE) losses, minimum word error rate (MWER) training, layer-wise pre-training and data augmentation methods. In addition, we compressed our models by more than 3.4 times smaller using an iterative hyper low-rank approximation (LRA) method while minimizing the degradation in recognition accuracy. The memory footprint was further reduced with 8-bit quantization to bring down the final model size to lower than 39 MB. For on-demand adaptation, we fused the MoChA models with statistical n-gram models, and we could achieve a relatively 36% improvement on average in word error rate (WER) for target domains including the general domain., Comment: Accepted and presented at the ASRU 2019 conference

In this paper, we investigate the asymptotic behavior of the number $s_q(g)$ of isogeny classes of simple abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$. We prove that the logarithmic asymptotic of $s_q(g)$ is the same as the logarithmic asymptotic of the number $m_q(g)$ of isogeny classes of all abelian varieties of dimension $g$ over $\mathbb{F}_q$. We also prove that $$ \limsup_{g \rightarrow \infty} \frac{s_q(g)}{m_q(g)}=1. $$ This suggests that there are much more simple isogeny classes of abelian varieties over $\mathbb{F}_q$ of dimension $g$ than non-simple ones for sufficiently large $g$, which can be understood as the opposite situation to a main result of Lipnowski and Tsimerman (Duke Math 167:3403-3453, 2018)., Comment: 9 pages, to appear in Math. Z

In this paper, we generalize the arithmetic Chern-Simons theory to regular flat separated schemes of finite type over rings of integers of number fields by applying the duality theorems for arithmetic schemes., Comment: 10 pages. v2: minor corrections

The goal of this paper is two-fold: we generalize the arithmetic Chern-Simons theory over totally imaginary number fields studied in [Kim15, CKK+16] to arbitrary number fields (with real places) and provide new examples of non-trivial arithmetic Chern-Simons invariant with coefficient $\mathbb{Z}/n\mathbb{Z}$ $(n \geq 2)$ associated to a non-abelian gauge group. The main idea for the generalization is to use cohomology with compact support (see [Mil06]) to deal with real places. Before the results of this paper, non-trivial examples were limited to some non-abelian gauge group with coefficient $\mathbb{Z}/2\mathbb{Z}$ in [CKK+16] and the abelian cyclic gauge group with coefficient $\mathbb{Z}/n\mathbb{Z}$ in [BCG+18]. Our non-trivial examples (with non-abelian gauge group and general coefficient $\mathbb{Z}/n\mathbb{Z}$) will be given by a simple twisting argument based on examples of [BCG+18]., Comment: 21 pages, exposition improved

In this note, we characterize all functions $f : \mathbb{N} \rightarrow \mathbb{C}$ such that $f(x_1^2+ \cdots + x_k^2)=f(x_1)^2+ \cdots + f(x_k)^2$, where $k \geq 3$ and $x_1, \cdots, x_k$ are positive integers., Comment: 3 pages

In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and sufficient condition. Also we give some sufficient conditions for each $n \geq 2$ where $a_i$ ($1 \leq i \leq m$) are pairwisely coprime., Comment: 5 pages

In this note, we prove that for every two positive integers $m \geq n \geq 9$, there exist $n$ positive rational numbers whose product is 1 and sum is $m$., Comment: 4 pages

In this paper, we consider the following problem: For every positive integer $r \geq 2$, find all positive integers $n$ such that for every semigroup of order $\geq n$ in which $x^r=x$ for every element $x$ has a subsemigroup of order $n$., Comment: 15 pages