Regular Partition Research Articles

Regular partitions of gentle graphs

Szemeredi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as ``gentle'' classes.

Lacunary eta-quotients modulo powers of primes

An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Okounkov, and Han.

A general method to obtain the spectrum and local spectra of a graph from its regular partitions

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, a method that gives all the spectrum, and also the local spectra, of a graph from the quotient matrices of some of its regular partitions, is proposed. Moreover, from such partitions, the $C$-local multiplicities of any class of vertices $C$ is also determined, and some applications of these parameters in the characterization of completely regular codes and their inner distributions are described. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.

A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results.In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment.

Analysis of HDG method for the reaction-diffusion equations

In this paper, we study and analyse the hybridized discontinuous Galerkin method for the reaction-diffusion problem. We present a mixed formulation based on some operators with discontinuous piecewise polynomials in a shape regular partition. We prove the well posedness of the mixed scheme and establish some error estimates with optimal order of convergence. Finally, we give some numerical simulations to confirm the convergence results and to validate the mixed formulation.

Congruences for $\ell$-regular partitions and bipartitions

Define F ( q ) : = ∑ n = − ∞ ∞ ( − 1 ) δ n ( a n + b ) q ( c n 2 + d n ) ∕ 2 , which includes Ramanujan’s theta function as a special case. We establish a dissection identity for this function, and use it to derive congruence properties for the coefficients of F ( q ) . As an application we deduce several infinite families of congruences for ℓ -regular partitions and ℓ -regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo 5 .

Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs P(n) and to a recent known family of mixed graphs Γ(d,n,r) (having edges with and without direction). As a byproduct, the existence of perfect codes in P(n) allows us to give a lower bound for the multiplicity of its eigenvalue −1.

Occupancy-Map-Based Rate Distortion Optimization and Partition for Video-Based Point Cloud Compression

The video-based point cloud compression (V-PCC) is the state-of-the-art dynamic point cloud compression method developed by the Moving Pictures Experts Group (MPEG). It projects the point cloud patch by patch to its bounding box and organizes all the patches into a video to utilize the efficient video coding framework. However, the unoccupied pixels among different patches will lead to inefficiency of the video compression. First, the unoccupied pixels are treated equal to the occupied pixels in the rate distortion optimization process. However, the unoccupied pixels are useless for the reconstructed quality of the point cloud. Second, the edges of the occupied and unoccupied pixels can divide a coding unit into arbitrary shapes. Consequently, they are not well-characterized by the regular partitions supported in the video coding framework. Therefore, we propose using occupancy-map-based rate distortion optimization and partition to deal with these two problems. First, the occupancy map is used as a mask to ignore the distortions of the unoccupied pixels when calculating the rate distortion cost of a specified block. This strategy is applied to both intra prediction, inter prediction, and the sample adaptive offset to boost the performance. Second, we propose an occupancy-map-based partition to divide the occupied pixels from different patches and the unoccupied pixels into different prediction units (PUs). The motions of occupied PUs are then predicted using the auxiliary information. The unoccupied PU is finally padded using the approach that generates the geometry frames. The proposed algorithms are implemented in the V-PCC and High Efficiency Video Coding reference software. The experimental results show that the algorithms can individually contribute significant performance improvements compared with the V-PCC. Additionally, their combination can achieve even more bitrate savings than the sum of the individual algorithms.

Boolean lattices in finite alternating and symmetric groups

Abstract Given a group G and a subgroup H, we let $\mathcal {O}_G(H)$ denote the lattice of subgroups of G containing H. This article provides a classification of the subgroups H of G such that $\mathcal {O}_{G}(H)$ is Boolean of rank at least $3$ when G is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilisers of chains of regular partitions, and the other arises by taking stabilisers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices related to the dual Ore’s theorem and to a problem of Kenneth Brown.

On the perfectness of minimal regular partitions of the edge set of the $n$-dimensional cube

On the perfectness of minimal regular partitions of the edge set of the $n$-dimensional cube

A characterization of finite arithmetic progressions via regular partitions of quadrilaterals

ABSTRACTIn this article, we will give a geometric interpretation of certain finite arithmetic progressions. For this purpose, we will introduce the concept of the n-regular partition of a quadrilateral.

On the Perfectness of Minimal Regular Partitions of the Edge Set of the n-Dimensional Cube

On the Perfectness of Minimal Regular Partitions of the Edge Set of the n-Dimensional Cube

Congruences for the coefficients of the powers of the Euler Product

Let $$p_k(n)$$ be given by the series expansion of the k-th power of the Euler Product $$\prod _{n=1}^{\infty }(1-q^n)^k=\sum _{n=0}^{\infty }p_k(n)q^{n}$$. By investigating the properties of the modular equations of the second and the third order under the Atkin U-operator, we determine the generating functions of $$p_{8k}(2^{2\alpha } n +\frac{k(2^{2\alpha }-1)}{3})$$$$(1\le k\le 3)$$ and $$p_{3k}(3^{2\beta }n+\frac{k(3^{2\beta }-1)}{8})$$$$(1\le k\le 8)$$ in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m, we obtain infinite families of congruences for $$p_k(n)$$ modulo any $$m\ge 2$$, where $$1\le k\le 24$$ and 3|k or 8|k. Based on these congruences for $$p_k(n)$$, infinite families of congruences for many partition functions such as the overpartition function, t-core partition functions and $$\ell $$-regular partition functions are easily obtained.

A Tight Bound for Hyperaph Regularity

The hypergraph regularity lemma—the extension of Szemeredi’s graph regularity lemma to the setting of k-uniform hypergraphs—is one of the most celebrated combinatorial results obtained in the past decade. By now there are several (very different) proofs of this lemma, obtained by Gowers, by Nagle–Rodl–Schacht–Skokan and by Tao. Unfortunately, what all these proofs have in common is that they yield regular partitions whose order is given by the k-th Ackermann function. We show that such Ackermann-type bounds are unavoidable for every $$k \ge 2$$ , thus confirming a prediction of Tao. Prior to our work, the only result of this type was Gowers’ famous lower bound for graph regularity.

Infinitely Many Congruences for k-Regular Partitions with Designated Summands

Andrews et al. (Acta Arith. 105:51–66, 2002) introduced and studied the partition function PD(n), the number of partitions of n with designated summands. Recently, congruences involving the number of $$\ell $$ -regular partitions with designated summands, denoted by $$PD_{\ell }(n)$$ , have been explored for specific fixed values of $$\ell $$ . In this paper, we provide several families containing infinitely many congruences for $$PD_k(n)$$ for various values of k.

You Can't See Me: Anonymizing Graphs Using the Szemerédi Regularity Lemma

Complex networks gathered from our online interactions provide a rich source of information that can be used to try to model and predict our behavior. While this has very tangible benefits that we have all grown accustomed to, there is a concrete privacy risk in sharing potentially sensitive data about ourselves and the people we interact with, especially when this data is publicly available online and unprotected from malicious attacks. k-anonymity is a technique aimed at reducing this risk by obfuscating the topological information of a graph that can be used to infer the nodes' identity. In this paper we propose a novel algorithm to enforce k-anonymity based on a well-known result in extremal graph theory, the Szemerédi regularity lemma. Given a graph, we start by computing a regular partition of its nodes. The Szemerédi regularity lemma ensures that such a partition exists and that the edges between the sets of nodes behave almost randomly. With this partition, we anonymize the graph by randomizing the edges within each set, obtaining a graph that is structurally similar to the original one yet the nodes within each set are structurally indistinguishable. We test the proposed approach on real-world networks extracted from Facebook. Our experimental results show that the proposed approach is able to anonymize a graph while retaining most of its structural information.

A concise proof to the spectral and nuclear norm bounds through tensor partitions

Abstract On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

Introducing the BITIX: The Bitcoin Fear Gauge

We acquire a unique dataset of high-frequency traded prices for bitcoin call and put options from the Deribit cryptocurrency derivatives exchange, by 15-minute sampling via the application programming interface. We use these prices to construct a term structure of bitcoin implied volatility indices using a variance swap fair-value formula that is employed by the CBOE for the VIX, an index commonly referred to as the ‘investor fear gauge’ for the US stock market. Employing over 7 million option prices, we construct the bitcoin implied volatility indices with maturities from one week to three months, sampled every 15 minutes from March 2019 to March 2020. We discuss the features of the index and the associated bitcoin variance risk premia, with three different regular time partitions for realised variance, viz. 15-minutes, hourly and daily. We also examine the relationship between bitcoin’s 30-day realised variance, volatility index and variance risk premium with their equivalent for US equities, oil, gold, the USD/EUR exchange rate and the 10-year US treasury note.

A sparse regular approximation lemma

We introduce a new variant of Szemeredi's regularity which we call the sparse regular approximation lemma (SRAL). The input to this is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity already suffices for proving the graph removal and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

Efficient Removal Without Efficient Regularity

Obtaining an efficient bound for the triangle removal lemma is one of the most outstanding open problems of extremal combinatorics. Perhaps the main bottleneck for achieving this goal is that triangle-free graphs can be highly unstructured. For example, triangle-free graphs might have only regular partitions (in the sense of Szemeredi) of tower-type size. And indeed, essentially all the graph properties $$\mathcal{P}$$ for which removal lemmas with reasonable bounds were obtained, are such that every graph satisfying $$\mathcal{P}$$ has a small regular partition. So in some sense, a barrier for obtaining an efficient removal lemma for property $$\mathcal{P}$$ was not having an efficient regularity lemma for graphs satisfying $$\mathcal{P}$$ . In this paper we consider the property of being induced C4-free, which also suffers from the fact that a graph might satisfy this property but still have only regular partitions of tower-type size. By developing a new approach for this problem we manage to overcome this barrier and thus obtain a merely exponential bound for the induced C4 removal lemma. We thus obtain the first efficient removal lemma that does not rely on an efficient version of the regularity lemma. This is the first substantial progress on a problem raised by Alon in 2001, and more recently by Alon, Conlon and Fox.