Infinite Set Research Articles

Martini Indices, Fisher Indices, and Consistency-in-Aggregation

In this article we consider the price and quantity indices introduced in the nineties by the Italian statistician Marco Martini (1944–2002). It turns out that Fisher indices can be considered as members of the infinite set of Martini indices. In particular attention is paid to the question to what extent these indices exhibit Consistency-in-Aggregation. This throws new light on the behavior of Fisher indices.

Orbit-finite linear programming

An infinite set is orbit-finite if, up to permutations of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.

On elements of prescribed norm in maximal orders of a quaternion algebra

Abstract Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p, $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$ -module by elements with norm in S. The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.

Infinite unrestricted sumsets of the form B+B$B+B$ in sets with large density

AbstractFor a set , we characterize the existence of an infinite set and such that , where , in terms of the density of the set . Specifically, when the lower density or the upper density , the existence of such a set and is assured. Furthermore, whenever or , we show that the shift is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three‐coloring of the natural numbers that does not contain a monochromatic for any infinite set and number .

1/c deformations of AdS3 boundary conditions and the Dym hierarchy

Abstract This work introduces a novel family of boundary conditions for AdS3 General Relativity, constructed through a polynomial expansion in negative integer powers of the Brown-Henneaux central charge. The associated dynamics is governed by the Dym hierarchy of integrable equations. It is shown that the infinite set of Dym conserved charges generates an abelian asymptotic symmetry group. Additionally, these boundary conditions encompass black hole solutions, whose thermodynamic properties are examined.

Strictly Conservative Neural Implicits

AbstractWe describe a method to convert 3D shapes into neural implicit form such that the shape is approximated in a guaranteed conservative manner. This means the input shape is strictly contained inside the neural implicit or, alternatively, vice versa. Such conservative approximations are of interest in a variety of applications, including collision detection, occlusion culling, or intersection testing. Our approach is the first to guarantee conservativeness in this context of neural implicits. We support input given as mesh, voxel set, or implicit function. Adaptive affine arithmetic is employed in the neural network fitting process, enabling the reasoning over infinite sets of points despite using a finite set of training data. Combined with an interior point style optimization approach this yields the desired guarantee.

The mean number of $2$-torsion elements in the class groups of cubic orders

We determine the mean number of 2-torsion elements in class groups of cubic orders, when such orders are enumerated by discriminant. Specifically, we prove that when isomorphism classes of totally real (resp., complex) cubic orders are enumerated by discriminant, the average 2 -torsion in the class group is 1 + \frac{1}{4}\times \frac{\zeta(2)}{\zeta(4)} (resp., 1 + \frac{1}{2}\times \frac{\zeta(2)}{\zeta(4)} ). In particular, we find that the average 2 -torsion in the class group increases when one ranges over all orders in cubic fields instead of restricting to the subfamily of rings of integers of cubic fields, where the average 2 -torsion in the class group was first determined in work of Bhargava to be \frac{5}{4} (resp., \frac{3}{2} ).By work of Bhargava–Varma, proving this result amounts to obtaining an asymptotic count of the number of “reducible” \operatorname{SL}_{3}(\mathbb{Z}) -orbits on the space \mathbb{Z}^{2} \otimes_{\mathbb{Z}}\operatorname{Sym}^{2} \mathbb{Z}^{3} of 3 \times 3 symmetric integer matrices having bounded invariants and satisfying local conditions. In this paper, we resolve the generalization of this orbit-counting problem where the dimension 3 is replaced by any fixed odd integer N \geq 3 . More precisely, we determine asymptotic formulas for the number of reducible \operatorname{SL}_{N}(\mathbb{Z}) -orbits on \mathbb{Z}^{2} \otimes_{\mathbb{Z}}\operatorname{Sym}^{2} \mathbb{Z}^{N} satisfying general infinite sets of congruence conditions.

Infinite distributive laws for the lateral operations on a Riesz space

We prove analogues of the well known infinite distributive laws for the lateral infima and suprema instead of order ones. The proofs are more involved than that for the original laws. We show that one of the laws holds true whenever both sides of the equality are well defined. The other one is false in general, even if both sides are well defined, but true for finite sets. The proofs of two laws are completely different. The question of under what assumptions on the Riesz space and objects involved in, the second distributive law is valid for infinite sets, remains unsolved.

Model-driven development for functional correctness of avionics systems: a verification framework for SysML specifications

AbstractCurrently, the most widespread software quality assurance methods in the avionics domain are semi-automated reviews and testing. However, their effort grows disproportionately to the size of the system under development. Also, these methods cannot achieve exhaustive coverage due to the complexity of today’s avionics systems and their potentially infinite set of combinations of possible inputs and system states. Furthermore, the later software issues are detected in the development process, the more expensive it is to fix them. To overcome these issues, a model-driven verification approach for modeling and analyzing avionics systems in early phases of the development is presented. To this end, semantics is given to SysML v2 models by a mapping to a theorem prover encoding. The development of a dedicated SysML v2 profile supporting event-driven data flow specifications, the encoding of corresponding structures in the theorem prover Isabelle, and a generator creating theorems from SysML v2 models are presented. The approach is evaluated by formally proving a representative liveness property of a hierarchical system model from the avionics domain. Since liveness properties can be negated only by infinite data sequences and thus cannot be covered exhaustively by testing, this case study demonstrates the added value for meeting typical safety requirements in the avionics domain. The results can be transferred from avionics to other domains, as well.

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

A sufficient and necessary condition for infinite orthogonal sets on some Moran measures

A sufficient and necessary condition for infinite orthogonal sets on some Moran measures

Automating Unrealizability Logic: Hoare-Style Proof Synthesis for Infinite Sets of Programs

Automated verification of all members of a (potentially infinite) set of programs has the potential to be useful in program synthesis, as well as in verification of dynamically loaded code, concurrent code, and language properties. Existing techniques for verification of sets of programs are limited in scope and unable to create or use interpretable or reusable information about sets of programs. The consequence is that one cannot learn anything from one verification problem that can be used in another. Unrealizability logic (UL), proposed by Kim et al. as the first Hoare-style proof system to prove properties over sets of programs (defined by a regular tree grammar), presents a theoretical framework that can express and use reusable insight. In particular, UL features nonterminal summaries ---inductive facts that characterize recursive nonterminals (analogous to procedure summaries in Hoare logic). In this work, we design the first UL proof synthesis algorithm, implemented as Wuldo. Specifically, we decouple the problem of deciding how to apply UL rules from the problem of synthesizing/checking nonterminal summaries by computing proof structure in a fully syntax-directed fashion. We show that Wuldo, when provided nonterminal summaries, can express and prove verification problems beyond the reach of existing tools, including establishing how infinitely many programs behave on infinitely many inputs. In some cases, Wuldo can even synthesize the necessary nonterminal summaries. Moreover, Wuldo can reuse previously proven nonterminal summaries across verification queries, making verification 1.96 times as fast as when summaries are instead proven from scratch.

Vanishing conditions for higher order couplings in heterotic theories

For compactifications of heterotic string theory, we elucidate simple cohomological conditions that lead to the vanishing of superpotential n-point couplings for all n. These results generalize some vanishing theorems for Yukawa couplings that have previously appeared in the literature to all higher orders. In some cases, these results are enough to show that certain fields do not appear in the perturbative superpotential at all. We illustrate our discussion with a number of concrete examples. In some cases, our results can be confirmed by showing that symmetries indeed forbid the couplings that vanish. In many, however, no such symmetries are known to exist and, as such, the infinite sets of vanishing couplings that are found are surprising from a four-dimensional perspective. Published by the American Physical Society 2024

The Baker-Coon-Romans N-point amplitude and an exact field theory limit of the Coon amplitude

We study the N-point Coon amplitude discovered first by Baker and Coon in the 1970s and then again independently by Romans in the 1980s. This Baker-Coon-Romans (BCR) amplitude retains several properties of tree-level string amplitudes, namely duality and factorization, with a q-deformed version of the string spectrum. Although the formula for the N-point BCR amplitude is only valid for q > 1, the four-point case admits a straightforward extension to all q ≥ 0 which reproduces the usual expression for the four-point Coon amplitude. At five points, there are inconsistencies with factorization when pushing q < 1. Despite these issues, we find a new relation between the five-point BCR amplitude and Cheung and Remmen’s four-point basic hypergeometric amplitude, placing the latter within the broader family of Coon amplitudes. Finally, we compute the q → ∞ limit of the N-point BCR amplitudes and discover an exact correspondence between these amplitudes and the field theory amplitudes of a scalar transforming in the adjoint representation of a global symmetry group with an infinite set of non-derivative single-trace interaction terms. This correspondence at q = ∞ is the first definitive realization of the Coon amplitude (in any limit) from a field theory described by an explicit Lagrangian.

Stability analysis of SnO2 surfaces using First-Principles computational methods

Abstract This study focuses on investigating the structural and electronic properties of the most stable faces of SnO2 using the Quantum Espresso software. Structural properties were found applying the generalized gradient approximation (GGA-PBE) with ultrasoft pseudopotentials and plane wave basis sets. Methodology involved determining the cutoff of the set of plane waves, selecting appropriate k-points to represent the first Brillouin zone, and optimized the lattice parameters (a and c). Models were generated for each studied surface, with the most optimal being the one with the lowest energy. These results are based on the periodicity of infinite sets of surfaces, increasing the gaps of the vacuum layers. As a result of the calculations, a discrepancy of less than 1% was observed in the lattice parameters compared to previous publications. Furthermore, it was found that the (110) is the most stable surface, in agreement with results published using VASP.

Information disturbance tradeoff in bidirectional QKD

Making use of the Quantum Network formalism of Phys. Rev. A, 82 (2010) 062 305, we present the case for quantum networks with finite outcomes, more specifically one which could distinguish only between specific unitary operators in a given basis for operators. Despite its simplicity, we proceed to build a network derived from the optimal strategy in Phys. Rev. A, 82 (2010) 062 305 and show that the information-disturbance tradeoff in distinguishing between two operators acting on qubits, selected from mutually unbiased unitary bases is equal to the case of estimating an operator selected randomly from the set of SU(2) based on the Haar measure. This suggests that such strategies in distinguishing between mutually unbiased operators is not any easier than estimating an operator derived from an infinite set. We then show how this network can be used as a natural attack strategy against a bidirectional quantum cryptographic protocol.

Robust flat bands in twisted trilayer graphene moiré quasicrystals

Moiré structures formed by twisting three layers of graphene with two independent twist angles present an ideal platform for studying correlated quantum phenomena, as an infinite set of angle pairs is predicted to exhibit flat bands. Moreover, the two mutually incommensurate moiré patterns among the twisted trilayer graphene (TTG) can form highly tunable moiré quasicrystals. This enables us to extend correlated physics in periodic moiré crystals to quasiperiodic systems. However, direct local characterization of the structure of the moiré quasicrystals and of the resulting flat bands are still lacking, which is crucial to fundamental understanding and control of the correlated moiré physics. Here, we demonstrate the existence of flat bands in a series of TTGs with various twist angle pairs and show that the TTGs with different magic angle pairs are strikingly dissimilar in their atomic and electronic structures. The lattice relaxation and the interference between moiré patterns are highly dependent on the twist angles. Our direct spatial mappings, supported by theoretical calculations, reveal that the localization of the flat bands exhibits distinct symmetries in different regions of the moiré quasicrystals.

Двоїстий алгоритм пошуку простих чисел на відрізках великих розмірностей

A matrix model of subsequence of natural numbers with a multiplicative basis from the first prime numbers is proposed. The matrix model is a square matrix, where vector columns are arithmetic progressions with difference and number of progressions equal to product of base elements. By crossing out arithmetic progressions with first terms which are multiples of base elements, we obtain a symmetric sparse matrix that contains all the prime numbers of subsequence of natural numbers, except for the base ones, which increases the density of prime numbers in sparse matrices. Sparse matrices are not formed clearly, only the vector of first terms of arithmetic progressions is formed. Properties of sparse matrices have been proven. Formulas that speed up the calculation of compound numbers in arithmetic progressions have been derived, the structures of vector elements of first terms of arithmetic progressions have been determined, the connectivity of symmetric parts of sparse matrices has been investigated. With the expansion of the base the number of pairs of elements with the difference equal to the power of two («twins», «fours», etc.) increases in the vector of first terms. This is a necessary condition for the existence of constants for which linear equations of two variables can have an infinite set of solutions in prime numbers. The irregularity of distribution of prime numbers in subsequences of natural numbers is related to the structure of elements of the vector of first terms. An algorithm for finding prime numbers on segments of large dimensions with a parallel calculation process has been built. The proposed algorithm is binary to algorithms for sifting subsequences of natural numbers by prime divisors. In these algorithms, it is not possible to parallelize the calculation process, since screening procedure requires storing the numerical information from the preceding steps (vector model of array processing). The binary algorithm calculates compound numbers in each pair of arithmetic progressions with symmetric first terms simultaneously, using only vector of the first terms of arithmetic progressions, which makes possible processing of large-dimensional arrays

Infinite Family of Integrable Sigma Models Using Auxiliary Fields.

We introduce a class of 2D sigma models which are parametrized by a function of one variable. In addition to the physical field g, these models include an auxiliary field v_{α} which mediates interactions in a prescribed way. We prove that every theory in this family is classically integrable, in that it possesses an infinite set of conserved charges in involution, which can be constructed from a Lax representation for the equations of motion. This class includes the principal chiral model (PCM) and all deformations of the PCM by functions of the energy-momentum tensor.

A note on the cardinality of definable families of sets in o‐minimal structures

AbstractWe prove that any definable family of subsets of a definable infinite set in an o‐minimal structure has cardinality at most . We derive some consequences in terms of counting definable types and existence of definable topological spaces.