Asymptotic Version Research Articles

Scaling laws and exact results in turbulence *

In this note, we address the validity of certain exact results from turbulence theory in the deterministic setting. The main tools, inspired by the work of Duchon and Robert (2000 Nonlinearity 13 249–55) and Eyink (2003 Nonlinearity 16 137), are a number of energy balance identities for weak solutions of the incompressible Euler and Navier–Stokes equations. As a consequence, we show that certain weak solutions of the Euler and Navier–Stokes equations satisfy deterministic versions of Kolmogorov’s 45 , 43 , 415 laws. We apply these computations to improve a recent result of Hofmanova et al (2023 arXiv:2304.14470), which shows that a construction of solutions of forced Navier–Stokes due to Bruè et al (2023 Commun. Pure Appl. Anal.) and exhibiting a form of anomalous dissipation satisfies asymptotic versions of Kolmogorov’s laws. In addition, we show that the globally dissipative 3D Euler flows recently constructed by Giri et al (2023 arXiv:2305.18509) satisfy the local versions of Kolmogorov’s laws.

Semi-c-periodicity, c-uniform recurrence and almost automorphy in the complex plane

This paper is devoted to develop the concepts of semi-c-periodicity, c-uniform recurrence and almost automorphy for functions defined on vertical strips in the complex plane, where c is a non-zero complex number. As an extension of the study performed for functions defined on the real axis, this work aims to investigate the main properties of these classes of functions and establish their connections with the more known class of c-almost periodic functions defined on vertical strips. In fact, we resolve an open problem which was raised in 2020 for the real case. Additionally, we also use this approach to introduce an asymptotic version of these new classes of functions.

Стационарная вращательно-симметричная конвекция при изотахофорезе в круговой цилиндрической области с недеформируемой свободной непроницаемой боковой поверхностью

The results of a study of the isotachophoresis process convective stability are presented. Isotachophoresis is the method of separation of a multicomponent mixture to individual components by an external electric field. It is assumed that the process of isotachophoresis occur in an infinite vertical circular cylinder with free side boundaries. The problem of the stationary (monotonic, neutral) concentration gravitational convection occurrence is transform to the study of the solution stability for the problem linearized on the solution of the original problem corresponding to the mechanical equilibrium of the final stage of the isotachophoresis process, in which the zones of the individual components of the mixture move with the identical constant velocity in a stationary flow. It is indicated that the main contribution to the stability/instability of the equilibrium is made by concentration gradients in the vicinity of the boundaries between the zones of individual components. Due to the complexity of constructing an exact solution corresponding to mechanical equilibrium, the exact solution replaced by an asymptotic solution (piecewise constant concentration distribution), which, in turn, is an exact solution of the asymptotic version of the transport under action of an electric field component equations (diffusion-free model). The choice of piecewise constant mechanical equilibrium leads to concentration gradients in the form of delta functions and a linear boundary value problem with delta-like coefficients to determine the critical parameters of instability. An exact solution of the problem (a system of ordinary differential equations with boundary conditions) and, using the method of one-dimensional perturbations, an exact dispersion relation for determining critical parameters (in the case of vanishingly small diffusion) is constructed.

Colorful Matchings

Suppose a committee consisting of three members is tasked with matching candidates to different positions. However, all of the committee members disagree on the job placement for every candidate, i.e., every candidate is matched to three different positions according to three committee members. All three committee members are competitive and want to push through as many of their placements as possible. Can they find a compromise which allows each committee member to be responsible for a third of all the candidate placements? In this paper we will consider an asymptotic version of this question and several other variants of a similar problem. As an application we will consider an embedding question—which hypertrees does a large Steiner system always contain?

A pair degree condition for Hamiltonian cycles in 3-uniform hypergraphs

AbstractWe prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$ -uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to Rödl, Ruciński, and Szemerédi. For graphs, Chvátal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chvátal’s theorem was taken by Pósa, who improved on Dirac’s tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle.In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$ -uniform hypergraphs by proving a $3$ -uniform analogue of Pósa’s result. In particular, our result strengthens the asymptotic version of the result by Rödl, Ruciński, and Szemerédi.

C5 ${C}_{5}$ is almost a fractalizer

AbstractWe determine the maximum number of induced copies of a 5‐cycle in a graph on vertices for every . Every extremal construction is a balanced iterated blow‐up of the 5‐cycle with the possible exception of the smallest level where for , the Möbius ladder achieves the same number of induced 5‐cycles as the blow‐up of a 5‐cycle on eight vertices. This result completes the work of Balogh, Hu, Lidický, and Pfender, who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method, but we use a new and more sophisticated approach which allows us to extend its use to small graphs.

TVMM: an R package for testing hypothesis on mean vectors

TThe Multivariate Tests for the Vector of Means (TVMM) package has six functions: T2O function, which computes Hotelling original’s T² test; T2Boot function, which computes the parametric bootstrap version of the original T2 test; T2RobustBoot function, which computes the robust parametric bootstrapversion of the T2Boot test; LRTTrace function, which computes the asymptotic version of the likelihood ratio test (LRT) using the trace operator; LRTTBoot function, which computes the parametric bootstrap version of the LRTTrace and the LRTTRBoot, which computes the robust version of the LRTTBoot. The alternative test versions of the LRT have the advantage of being valid for high-dimensional data. We describe the methods and illustrate the use of the TVMM package with real data on soil properties.

Asymptotic ω-Primality of Finitely Generated Cancelative Commutative Monoids

The computation of ω-primality has been object of study, mainly, for numerical semigroups due to its multiple applications to the Factorization Theory. However, its asymptotic version is less well known. In this work, we study the asymptotic ω-primality for finitely generated cancelative commutative monoids. By using discrete geometry tools and the Python programming language we present an algorithm to compute this parameter. Moreover, we improve the proof of a known result for numerical semigroups.

An asymptotic version of Cobham’s theorem

An asymptotic version of Cobham’s theorem

A Class of Bayesian Lower Bounds for Parameter Estimation Via Arbitrary Test-Point Transformation

In this paper, a new class of global mean-squared-error (MSE) lower bound for Bayesian parameter estimation is derived. First, it is shown that under the non-Bayesian framework, the Hammersley-Chapman-Robbins (HCR) for the problem of single-source parameter estimation, is related to the corresponding ambiguity function. This result is achieved by judicious choice of signal test-point. This result implies that optimal shift test-points may be parameter-dependent, but this approach cannot be imitated in test-point based Bayesian bounds, where the test-points should be independent of the random parameters. Based on this observation, a new class Bayesian MSE lower bound is derived. In the proposed class, the shift test-points in the Cauchy-Schwarz based bounds are substituted with arbitrary transformations. The proposed class generalizes the Weiss-Weinstein bound (WWB) to arbitrary transformations. For the problem of single source parameter estimation, a scaling transformation for the signal test-point is proposed based on the structure of the optimal test-point in the HCR bound in the non-Bayesian framework. The test-point scale parameters are analytically optimized and a closed-form expression for the bound, which depends on the ambiguity function, is derived. For the problem of direction-of-arrival (DOA) estimation using a linear array, a nonlinear transformation for the parameter of interest is proposed. In the problems of frequency estimation and DOA estimation, the proposed bound accurately predicts the threshold phenomenon of the maximum <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a-posteriori</i> probability estimator, and is much tighter than the WWB in both the threshold prediction and the asymptotic performance. In addition, an asymptotic version of the bound was derived and it is analytically shown to be tighter than the Bayesian Cramér-Rao bound.

Maximum spread of graphs and bipartite graphs

Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.

A generalization of Hardy’s operator and an asymptotic Müntz–Szász Theorem

We prove an asymptotic version of the Muntz–Szasz theorem, and use it to prove that all monomial operators are vanishing preserving.

Generalized binomials in fractional calculus

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities, including an adapted version of the Pascal's rule. We then investigate the associated generating functions, for which we establish a recursive, combinatorial and integral formulation. From this, we derive an asymptotic version of the Binomial Theorem. A combinatorial and asymptotic analysis of some finite sums completes the paper.

A NOTE ON AN ASYMPTOTIC VERSION OF A PROBLEM OF MAHLER

AbstractWe prove that for any infinite sets of nonnegative integers $\mathcal {A}$ and $\mathcal {B}$ , there exist transcendental analytic functions $f\in \mathbb {Z}\{z\}$ whose coefficients vanish for any indexes $n\not \in \mathcal {A}+\mathcal {B}$ and for which $f(z)$ is algebraic whenever z is algebraic and $|z|&lt;1$ . As a consequence, we provide an affirmative answer for an asymptotic version of Mahler’s problem A.

Superconnection and family Bergman kernels

We establish an asymptotic version of Bismut’s local family index theorem for the Bergman kernel. The key idea is to use the superconnection as in the local family index theorem.

Short Proofs of Rainbow Matchings Results

Abstract A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back to the work of Euler on Latin squares and has been the focus of extensive research ever since. Many conjectures in this area roughly say that “every edge coloured graph of a certain type contains a rainbow matching using every colour.” In this paper we introduce a versatile “sampling trick,” which allows us to asymptotically solve some well-known conjectures and to obtain short proofs of old results. In particular: $\bullet $ We give the first asymptotic proof of the “non-bipartite” Aharoni–Berger conjecture, solving two conjectures of Aharoni, Berger, Chudnovsky, and Zerbib. $\bullet $ We give a very short asymptotic proof of Grinblat’s conjecture (first obtained by Clemens, Ehrenmüller, and Pokrovskiy). Furthermore, we obtain a new asymptotically tight bound for Grinblat’s problem as a function of edge multiplicity of the corresponding multigraph. $\bullet $ We give the first asymptotic proof of a 30-year-old conjecture of Alspach. $\bullet $ We give a simple proof of Pokrovskiy’s asymptotic version of the Aharoni–Berger conjecture with greatly improved error term.

The Barban–Davenport–Halberstam theorem for a restricted set of arithmetic progressions

We consider the asymptotic version of the Barban–Davenport–Halberstam theorem in the form proved first by Hooley. Instead of averaging the expected error term over all reduced residue classes to modules in a given range, here we only consider those reduced arithmetic progressions that satisfy an additional congruence condition and provide an asymptotic approximation.

Common Complements of Linear Subspaces and the Sparseness of MRD Codes

We consider the problem of estimating the number of common complements of a family of subspaces over a finite field, in terms of the cardinality of the family and its intersection structure. We derive upper and lower bounds for this number, along with their asymptotic versions as the field size tends to infinity. We use these bounds to describe the general behavior of common complements with respect to sparsity and density, showing that the decisive property is whether or not the number of spaces to be complemented is negligible with respect to the field size. The proof techniques are based on the study of isolated vertices in certain bipartite graphs. By specializing our results to matrix spaces, we answer an open question in coding theory, proving that MRD codes in the rank metric are sparse for all parameter sets as the field grows, with only very few exceptions. We also investigate the density of MRD codes as their column length tends to infinity, obtaining a new asymptotic bound. Using properties of the Euler function from number theory, we then show that our bound improves on known results for most parameter sets. We conclude the paper by establishing two structural properties of the density function of rank-metric codes.

Instance-based learning using the half-space proximal graph

• We propose a parameter-free instance-based classifier , the HSP classifier. • Our accuracy outperforms kNN while keeping its simplicity and complexity. • We also present a probabilistic and asymptotic versions of the HSP improving both the time and the original proposal’s accuracy. • The same weigthing methods that improves kNN, also improves the HSP and its asymptotic variants The primary example of instance-based learning is the k -nearest neighbor rule (kNN), praised for its simplicity and the capacity to adapt to new unseen data and toss away old data. The main disadvantages often mentioned are the classification complexity , which is O ( n ) , and the estimation of the parameter k , the number of nearest neighbors to be used. The use of indexes at classification time lifts the former disadvantage, while there is no conclusive method for the latter. This paper presents a parameter-free instance-based learning algorithm using the Half-Space Proximal (HSP) graph. The HSP neighbors simultaneously possess proximity and variety concerning the center node. To classify a given query, we compute its HSP neighbors and apply a simple majority rule over them. In our experiments, the resulting classifier bettered K N N for any k in a battery of datasets. This improvement sticks even when applying weighted majority rules to both kNN and HSP classifiers. Surprisingly, when using a probabilistic index to approximate the HSP graph and consequently speeding-up the classification task , our method could improve its accuracy in stark contrast with the kNN classifier, which worsens with a probabilistic index.

Array BP-XOR Codes for Hierarchically Distributed Matrix Multiplication

A novel fault-tolerant computation technique based on array Belief Propagation (BP)-decodable XOR (BP-XOR) codes is proposed for distributed matrix-matrix multiplication. The proposed scheme is shown to be configurable and suited for modern hierarchical compute architectures such as Graphical Processing Units (GPUs) equipped with multiple nodes, whereby each has many small independent processing units with increased core-to-core communications. The proposed scheme is shown to outperform a few of the well--known earlier strategies in terms of total end-to-end execution time while in presence of slow nodes, called $stragglers$. This performance advantage is due to the careful design of array codes which distributes the encoding operation over the cluster (slave) nodes at the expense of increased master-slave communication. An interesting trade-off between end-to-end latency and total communication cost is precisely described. In addition, to be able to address an identified problem of scaling stragglers, an asymptotic version of array BP-XOR codes based on projection geometry is proposed at the expense of some computation overhead. A thorough latency analysis is conducted for all schemes to demonstrate that the proposed scheme achieves order-optimal computation in both the sublinear as well as the linear regimes in the size of the computed product from an end-to-end delay perspective.