Intersection patterns and incidence theorems

Let $A$ and $B$ be sets in a finite vector space. In this paper, we study the magnitude of the set $A\cap f(B)$, where $f$ runs through a set of transformations. More precisely, we will focus on the cases that the set of transformations is given by orthogonal matrices or orthogonal projections. One of the most important contributions of this paper is to show that if $A, B\subset \mathbb{F}_q^d$ satisfy some natural conditions then, for almost every $g\in O(d)$, there are at least $\gg q^d$ elements $z\in \mathbb{F}_q^d$ such that $$|A\cap (g(B)+z)| \sim \frac{|A||B|}{q^d}.$$ This infers that $|A-gB|\gg q^d$ for almost every $g\in O(d)$. In the flavor of expanding functions, with $|A|\le |B|$, we also show that the image $A-gB$ grows exponentially. In two dimensions, the result simply says that if $|A|=q^x$ and $|B|=q^y$, as long as $0<x\le y<2$, then for almost every $g\in O(2)$, we can always find $\epsilon=\epsilon(x, y)>0$ such that $|A-gB|\gg |B|^{1+\epsilon}$. To prove these results, we need to develop a new and robust incidence bound between points and rigid motions by using a number of techniques including algebraic methods and discrete Fourier analysis. Our results are essentially sharp in odd dimensions.Comment: Submitted version. 40 page

Orientable smooth manifolds are essentially quasigroups

We introduce an $n$-dimensional analogue of the construction of tessellated surfaces from finite groups first described by Herman and Pakianathan. Our construction is functorial and associates to each $n$-ary alternating quasigroup both a smooth, flat Riemannian $n$-manifold which we dub the open serenation of the quasigroup in question, as well as a topological $n$-manifold (the serenation of the quasigroup) which is a subspace of the metric completion of the open serenation. We prove that every connected orientable smooth manifold is serene, in the sense that each such manifold is a component of the serenation of some quasigroup. We prove some basic results about the variety of alternating $n$-quasigroups and note connections between our construction and topics including Latin hypercubes, Johnson graphs, and Galois theory

Combinatorics of Euclidean spaces over finite fields

Let $\mathbb{F}_{q}$ be the finite field with an odd prime power $q$. In this paper, we study various combinatorial properties related to non-degenerate quadratic spaces over finite fields. First, we investigate the Euclidean poset $E_{n}(q)$, which consists of all subspaces of $(\mathbb{F}_{q}^{n},\text{Euc}_{n})$ that have an orthonormal basis, where $\text{Euc}_{n}(\mathbf{x}):=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$. Using this poset structure, we show that the number of $k$-dimensional subspaces of $(\mathbb{F}_{q}^{n},\text{Euc}_{n})$ that have an orthonormal basis behaves like the binomial coefficient, which we call the Euclidean-binomial coefficient $\binom{n}{k}_{q}^{\perp}$ for $k=1,\cdots,n$. The main purpose of this paper is to study its various combinatorial properties.Comment: 21 pages, comments are welcom

Diophantine tuples and multiplicative structure of shifted multiplicative subgroups

We provide a substantial improvement on a recent result by Dixit, Kim, and Murty on the upper bound of $M_k(n)$, the largest size of a generalized Diophantine tuple with property $D_k(n)$, that is, each pairwise product is $n$ less than a $k$-th power. In particular, we show $M_k(n)=o(\log n)$ for a specially chosen sequence of $k$ and $n$ tending to infinity, breaking the $\log n$ barrier unconditionally. One innovation of our proof is a novel combination of Stepanov's method and Gallagher's larger sieve. One main ingredient in our proof is a non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress towards a conjecture of S\'{a}rk\"{o}zy on multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in \mathbb{F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in $\mathbb{F}_p$ non-trivially.Comment: 48 pages, 1 figur

Towards End-to-End Generative Modeling of Long Videos with Memory-Efficient Bidirectional Transformers

Autoregressive transformers have shown remarkable success in video generation. However, the transformers are prohibited from directly learning the long-term dependency in videos due to the quadratic complexity of self-attention, and inherently suffering from slow inference time and error propagation due to the autoregressive process. In this paper, we propose Memory-efficient Bidirectional Transformer (MeBT) for end-to-end learning of long-term dependency in videos and fast inference. Based on recent advances in bidirectional transformers, our method learns to decode the entire spatio-temporal volume of a video in parallel from partially observed patches. The proposed transformer achieves a linear time complexity in both encoding and decoding, by projecting observable context tokens into a fixed number of latent tokens and conditioning them to decode the masked tokens through the cross-attention. Empowered by linear complexity and bidirectional modeling, our method demonstrates significant improvement over the autoregressive Transformers for generating moderately long videos in both quality and speed. Videos and code are available at https://sites.google.com/view/mebt-cvpr2023

Cellular plasticity and immune microenvironment of malignant pleural effusion are associated with EGFR-TKI resistance in non-small-cell lung carcinoma

Malignant pleural effusion (MPE) is a complication of lung cancer that can be used as an alternative method for tissue sampling because it is generally simple and minimally invasive. Our study evaluated the diagnostic potential of non-small-cell lung carcinoma (NSCLC)-associated MPE in terms of understanding tumor heterogeneity and identifying response factors for EGFR tyrosine kinase inhibitor (TKI) therapy. We performed a single-cell RNA sequencing analysis of 31,743 cells isolated from the MPEs of 9 patients with NSCLC (5 resistant and 4 sensitive to EGFR TKI) with EGFR mutations. Interestingly, lung epithelial precursor-like cells with upregulated GNB2L1 and CAV1 expression were enriched in the EGFR TKI-resistant group. Moreover, GZMK upregulated transitional effector T cells, and plasmacytoid dendritic cells were significantly enriched in the EGFR TKI-resistant patients. Our results suggest that cellular plasticity and immunosuppressive microenvironment in MPEs are potentially associated with the TKI response of patients with EGFR-mutated NSCLC

Assessment of spatial tumor heterogeneity using CT growth patterns estimated by tumor tracking on 3D CT volumetry of multiple pulmonary metastatic nodules.

PurposeOur purpose was to assess the differences in growth rates of multiple pulmonary metastatic nodules using three-dimensional (3D) computed tomography (CT) volumetry and propose a concept of CT spatial tumor heterogeneity.Materials and methodsWe manually measured the largest diameter of metastatic pulmonary nodules on chest CT scans, and calculated the 3D maximum diameter and the volume using a semi-automated 3D CT volumetry of each nodule. The tumor response was assessed according to the revised RECIST 1.1. We defined a nodule as an outlier based on 1.5 times growth during follow-up. The CT spatial tumor heterogeneity was statistically analyzed by the "minimum combination t-test method" devised in our study.ResultsOn manual measurement, the tumor response category was stable disease (SD) in all 10 patients. Of them, total 155 metastatic nodules (4-52 nodules per patient) were segmented using the 3D CT volumetry. In the 3D maximum diameter, 9 patients had SD except for one patient with partial response in the two selected nodules; for the volume, all 10 patients were SD. For the 3D maximum diameter, six patients had at least one outlier; whereas five patients had the outlier on the volume measurement. Six patients were proven to have overall CT spatial tumor heterogeneity.ConclusionsThe spatial tumor heterogeneity determined in a CT parametric approach could be statistically assessed. In patients with CT spatial heterogeneity, tumors with different growth rates may be neglected when the nodules are assessed according to the current guideline