Recursive Construction Research Articles

Almost affinely disjoint subspaces and covering Grassmannian codes

A family of k-dimensional subspaces of Fqn with pairwise trivial intersection is called L-almost affinely disjoint (AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Liu et al. introduced the notion of AAD family, and investigated the lower and upper bounds of maximal such sets and conjectured that for any k, n and a large enough L=L(n,k), there exists an [n,k,L]q-AAD family with size qn−2k in [Finite Fields Appl. 75 (2021), 101879]. Etzion and Zhang introduced covering Grassmannian codes (CGCs) for generalized combination networks [IEEE Trans. Inf. Theory, 65 (2019), 4131–4142.]. An α-(n,k,δ)qc covering Grassmannian code C is a set of k-dimensional subspaces of Fqn, such that every set of α codewords of C spans a subspace of dimension at least k+δ. In this paper, we give a construction of AAD families and CGCs based on maximum rank metric codes and caps in projective geometries, and a recursive construction based on maximum rank metric codes. As a consequence, we prove that Liu et al.'s conjecture is still true for k≥3 and n=4k, and improve lower bounds on maximum sizes of AAD families and lower bounds on maximum sizes of 3-(n,k,2k)qc covering Grassmannian codes.

Mean Lipschitz–Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz–Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions, we show that rescaled limits of these mean values exist as the parallel radius tends to 0 . Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

Functional directed acyclical graphs for scattering amplitudes in perturbation theory

I describe a mathematical framework for the efficient processing of the very large sets of Feynman diagrams contributing to the scattering of many particles. I reexpress the established numerical methods for the recursive construction of scattering elements as operations on compact abstract data types. This allows efficient perturbative computations in arbitrary models, as long as they can be described by an effective, not necessarily local, Lagrangian.

Recursive definition and two edge-disjoint Hamiltonian cycles of bubble-sort star graphs

The bubble-sort star graph is an interconnection network for multiprocessor systems. Recursive structure is important for interconnection networks, since it can reduce the complex cases into simple cases, and it can keep good properties independent of dimensions. Many algorithms use the idea of recursive construction. Edge-disjoint Hamiltonian cycles not only provide the basis of all-to-all broadcasting algorithm for networks but also provide a replaceable Hamiltonian cycle for transmission when the other Hamiltonian cycle contains faulty edges in an interconnection network. In this paper, we propose the recursive definition of bubble-sort star graphs. Then as an application, we obtain two edge-disjoint Hamiltonian cycles of bubble-sort star graphs.

THE INTERSECTION PROBLEM FOR RESOLVABLE BALANCED INCOMPLETE BLOCK DESIGNS WITH SMALL ORDERS

The intersection problem is one of the basic problems in combinatorial design theorys. In this paper, two cases of resolvable balanced incomplete block designs of size four are investigated. As the basic step of the study on the intersection problem for resolvable balanced incomplete block designs, we have determined some values of the intersection numbers. It will play an important role on the further recursive constructions.

A binary embedding of the stable line-breaking construction

We embed Duquesne and Le Gall’s stable tree into a binary compact continuum random tree (CRT) in a way that solves an open problem posed by Goldschmidt and Haas. This CRT can be obtained by applying a recursive construction method of compact CRTs as presented in earlier work to a specific distribution of a random string of beads, i.e. a random interval equipped with a random discrete measure. We also express this CRT as a tree built by replacing all branch points of a stable tree by i.i.d. copies of a Ford CRT, each rescaled by a factor intrinsic to the stable CRT.

Faster Montgomery multiplication and Multi-Scalar-Multiplication for SNARKs

The bottleneck in the proving algorithm of most of elliptic-curve-based SNARK proof systems is the Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview of a variant of the Pippenger MSM algorithm together with a set of optimizations tailored for curves that admit a twisted Edwards form. We prove that this is the case for SNARK-friendly chains and cycles of elliptic curves, which are useful for recursive constructions. Our contribution is twofold: first, we optimize the arithmetic of finite fields by improving on the well-known Coarsely Integrated Operand Scanning (CIOS) modular multiplication. This is a contribution of independent interest that applies to many different contexts. Second, we propose a new coordinate system for twisted Edwards curves tailored for the Pippenger MSM algorithm.Accelerating the MSM over these curves is critical for deployment of recursive proof< systems applications such as proof-carrying-data, blockchain rollups and blockchain light clients. We implement our work in Go and benchmark it on two different CPU architectures (x86 and arm64). We show that our implementation achieves a 40-47% speedup over the state-of-the-art implementation (which was implemented in Rust). This MSM implementation won the first place in the ZPrize competition in the open division “Accelerating MSM on Mobile” and will be deployed in two real-world applications: Linea zkEVM by ConsenSys and probably Celo network.

Linear and circular single‐change covering designs revisited

AbstractA single‐change covering design (SCCD) is a ‐set and an ordered list of blocks of size where every pair from must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single‐change covering design, or more simply, a single‐change covering design. A single‐change covering design is circular when the first and last blocks also differ by one element. A single‐change covering design is minimum if no other smaller design can be constructed for a given . In this paper, we use a new recursive construction to solve the existence of circular SCCD() for all and three residue classes of circular SCCD() modulo 16. We solve the existence of three residue classes of SCCD modulo 16. We prove the existence of circular SCCD, for all , using difference methods.

Simulating first-order phase transition with hierarchical autoregressive networks.

We apply the hierarchical autoregressive neural network sampling algorithm to the two-dimensional Q-state Potts model and perform simulations around the phase transition at Q=12. We quantify the performance of the approach in the vicinity of the first-order phase transition and compare it with that of the Wolff cluster algorithm. We find a significant improvement as far as the statistical uncertainty is concerned at a similar numerical effort. In order to efficiently train large neural networks we introduce the technique of pretraining. It allows us to train some neural networks using smaller system sizes and then employ them as starting configurations for larger system sizes. This is possible due to the recursive construction of our hierarchical approach. Our results serve as a demonstration of the performance of the hierarchical approach for systems exhibiting bimodal distributions. Additionally, we provide estimates of the free energy and entropy in the vicinity of the phase transition with statistical uncertainties of the order of 10^{-7} for the former and 10^{-3} for the latter based on a statistics of 10^{6} configurations.

Recursion in the classical limit and the neutron-star Compton amplitude

We study the compatibility of recursive techniques with the classical limit of scattering amplitudes through the construction of the classical Compton amplitude for general spinning compact objects. This is done using BCFW recursion on three-point amplitudes expressed in terms of the classical spin vector and tensor, and expanded to next-to-leading-order in ћ by using the heavy on-shell spinors. Matching to the result of classical computations, we find that lower-point quantum contributions are, in general, required for the recursive construction of classical, spinning, higher-point amplitudes with massive propagators. We are thus led to conclude that BCFW recursion and the classical limit do not commute. In possession of the classical Compton amplitude, we remove non-localities to all orders in spin for opposite graviton helicities, and to fifth order in the same-helicity case. Finally, all possible on-shell contact terms potentially relevant to black-hole scattering at the second post-Minkowskian order are enumerated and written explicitly.

On the Cauchy problem for the Hartree approximation in quantum dynamics

We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schrödinger equations for the evolution of product state approximations. They are a prominent example for dimension reduction in the context of the time-dependent Dirac–Frenkel variational principle. Our main result addresses a general setting with smooth potentials where the nonlinear coupling cannot be considered as a perturbation. The proof uses a recursive construction that is inspired by the standard approach for the Cauchy problem associated to symmetric quasilinear hyperbolic equations. We also discuss the case of Coulomb potentials, though treated differently (using Strichartz estimates and a classical fixed point argument).

Block-Graceful Designs

In this article, we adapt the edge-graceful graph labeling definition into block designs and define a block design V , B with V = v and B = b as block-graceful if there exists a bijection f : B ⟶ 1,2 , … , b such that the induced mapping f + : V ⟶ Z v given by f + x = ∑ x ∈ A A ∈ B f A mod v is a bijection. A quick observation shows that every v , b , r , k , λ − BIBD that is generated from a cyclic difference family is block-graceful when v , r = 1 . As immediate consequences of this observation, we can obtain block-graceful Steiner triple system of order v for all v ≡ 1 mod 6 and block-graceful projective geometries, i.e., q d + 1 − 1 / q − 1 , q d − 1 / q − 1 , q d − 1 − 1 / q − 1 − BIBDs. In the article, we give a necessary condition and prove some basic results on the existence of block-graceful v , k , λ − BIBDs. We consider the case v ≡ 3 mod 6 for Steiner triple systems and give a recursive construction for obtaining block-graceful triple systems from those of smaller order which allows us to get infinite families of block-graceful Steiner triple systems of order v for v ≡ 3 mod 6 . We also consider affine geometries and prove that for every integer d ≥ 2 and q ≥ 3 , where q is an odd prime power or q = 4 , there exists a block-graceful q d , q , 1 − BIBD. We make a list of small parameters such that the existence problem of block-graceful labelings is completely solved for all pairwise nonisomorphic BIBDs with these parameters. We complete the article with some open problems and conjectures.

More results on large sets of Kirkman triple systems

The existence of large sets of Kirkman triple systems (LKTSs) is one of the best-known open problems in combinatorial design theory. Steiner quadruple systems with resolvable derived designs (RDSQSs) play an important role in the recursive constructions of LKTSs. In this paper, we introduce a special combinatorial structure $$\hbox {RDSQS}^{*}(v)$$ and use it to present a construction for RDSQS(4v). As a consequence, some new infinite families of LKTSs are given.

MUREN: MUltistage Recursive Enhanced Network for Coal-Fired Power Plant Detection

The accurate detection of coal-fired power plants (CFPPs) is meaningful for environmental protection, while challenging. The CFPP is a complex combination of multiple components with varying layouts, unlike clearly defined single objects, such as vehicles. CFPPs are typically located in industrial districts with similar backgrounds, further complicating the detection task. To address this issue, we propose a MUltistage Recursive Enhanced Detection Network (MUREN) for accurate and efficient CFPP detection. The effectiveness of MUREN lies in the following: First, we design a symmetrically enhanced module, including a spatial-enhanced subnetwork (SEN) and a channel-enhanced subnetwork (CEN). SEN learns the spatial relationships to obtain spatial context information. CEN provides adaptive channel recalibration, restraining noise disturbance and highlighting CFPP features. Second, we use a recursive construction set on top of feature pyramid networks to receive features more than once, strengthening feature learning for relatively small CFPPs. We conduct comparative and ablation experiments in two datasets and apply MUREN to the Pearl River Delta region in Guangdong province for CFPP detection. The comparative experiment results show that MUREN improves the mAP by 5.98% compared with the baseline method and outperforms by 4.57–21.38% the existing cutting-edge detection methods, which indicates the promising potential of MUREN in large-scale CFPP detection scenarios.

New Constructions of Optimal Optical Orthogonal Codes Based on Partitionable Sets and Almost Partitionable Sets

Variable-weight optical orthogonal codes (OOCs) were introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, partitionable sets and almost partitionable sets are used to construct variable-weight OOCs. Four new recursive constructions for optimal cyclic packing are presented. Consequently, new infinite classes of optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(v,W,1,Q)$ </tex-math></inline-formula> -OOCs are obtained.

Point-missing s-resolvable t-designs: infinite series of 4-designs with constant index

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the left( {begin{array}{c}v +1 kend{array}}right) k-sets chosen from a (v+1)-set X into (v+1) mutually disjoint s-(v,k,delta ) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing (t-1)-resolvable t-(v,k,lambda ) design leads to the existence of a t-(v,k+1,lambda ') design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4-(3^n,5,5), 4-(3^n+2,5,5) and 4-(2^n+1,5,5), for n ge 2, and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.

Ordered covering arrays and upper bounds on covering codes

AbstractThis work shows several direct and recursive constructions of ordered covering arrays (OCAs) using projection, fusion, column augmentation, derivation, concatenation, and Cartesian product. Upper bounds on covering codes in Niederreiter–Rosenbloom–Tsfasman (shorten by NRT) spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.

On a pair of orthogonal golf designs

A collection of n−2 mutually disjoint idempotent symmetric quasigroups of order n is called a golf design. Two golf designs of order n are said to be orthogonal, denoted by OG(n), if for any idempotent symmetric quasigroup in one golf design, an orthogonal mate can be found in the other golf design, and no two idempotent symmetric quasigroups in one golf design share a common orthogonal mate. In this paper, we give the first three examples of OG(n) for n∈{13,15,17} using starter adder method. We also present a recursive construction and apply it to prove that an OG(13n+2) exists for any positive integer n.

Symmetric generating functions and Euler–Stirling statistics on permutations

We present (bi-)symmetric generating functions for the joint distributions of Euler–Stirling statistics on permutations, including the number of descents (des), inverse descents (ides), the number of left-to-right maxima (lmax), the number of right-to-left maxima (rmax) and the number of left-to-right minima (lmin). We also show how they recover the classical symmetric generating function for permutations due to Carlitz, Roselle and Scoville (1966).Our proofs exploit three different recursive constructions of inversion sequences, bijections on the multiple equidistributions of Euler–Stirling statistics over permutations and transformation formulas of basic hypergeometric series. We further establish a new quadruple equidistribution of Euler–Stirling statistics over inversion sequences, as progress towards a conjecture proposed by Schlosser and the author (2020).

Mobility decisions, economic dynamics and epidemic.

We propose a model, which nests a susceptible-infected-recovered-deceased (SIRD) epidemic model into a dynamic macroeconomic equilibrium framework with agents' mobility. The latter affect both their income and their probability of infecting and being infected. Strategic complementarities among individual mobility choices drive the evolution of aggregate economic activity, while infection externalities caused by individual mobility affect disease diffusion. The continuum of rational forward-looking agents coordinates on the Nash equilibrium of a discrete time, finite-state, infinite-horizon Mean Field Game. We prove the existence of an equilibrium and provide a recursive construction method for the search of an equilibrium(a), which also guides our numerical investigations. We calibrate the model by using Italian experience on COVID-19 epidemic and we discuss policy implications.