Maximal Set Research Articles

Enhancing some separability criteria via equiangular tight frames

Abstract Improving the detection ability of quantum entanglement is one of the es-
sential tasks in quantum computing and quantum information. Finite tight frames
play a fundamental role in a wide variety of areas, and generally, each application
requires a specific class of frames and is closely related to quantum measurement. It
is worth noting that a maximal set of complex equiangular vectors is closely related
to a symmetric informationally complete measurement. Hence, our goal in this work
is to propose a series of separability criteria assigned to a finite tight frame and some
well-known inequalities in different quantum systems, respectively. In addition, some
tighter criteria to detect entanglement for many-body quantum states are presented in
arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detec-
tion criteria is illustrated through some detailed examples.

A boolean-algebraic approach to completeness for normal modal predicate logics

This paper introduces an innovative methodology for demonstrating completeness for normal modal predicate logics. Traditional proofs typically involve constructing canonical models, wherein possible worlds are defined as maximal consistent sets possessing specific properties, with a heavy reliance on the Barcan Formula to affirm the existence of these worlds. Our approach deviates from the classical method by utilizing Boolean algebras and ultrafilters to construct models. Unlike conventional methods, our construction of possible worlds does not depend on the Barcan Formula; rather, these properties are ensured by Tarski’s Lemma. Furthermore, our proof distinguishes itself from other Boolean-algebraic completeness proofs in two key respects: it employs Kripke semantics instead of algebraic semantics and exclusively relies on ultrafilters, thereby offering a more concise approach. This methodology facilitates a natural extension from modal propositional logic to modal predicate logics and circumvents the added complexity of Q-filters. In our model, the equivalence class of each theorem of a normal modal predicate logic is a member of all worlds, while the equivalence class of each non-theorem is a member of some worlds. Consequently, the structure of these worlds ensures that non-theorems are false in the model.

On the Maximal Output Admissible Set for Linear Distributed Systems

On the Maximal Output Admissible Set for Linear Distributed Systems

The solution of super-scale DEA models based on preference character

The basic relationships and intrinsic laws in the data envelopment analysis (DEA) are the basis for designing fast solutions for big data DEA models. By delving into these relationships, it becomes feasible to address super-scale DEA problems more swiftly, aligning with the need for timeliness in real-world efficiency evaluations and reducing time costs. This paper leverages the intrinsic characteristics of DEA to formulate a DEA model scale reduction approach, offering reliable theoretical support for solving super- scale DEA problems. Initially, the partially ordered set theory to prove the relationship between the efficient decision making units (DMUs) of DEA and the maximal element of the partially ordered set. Building on this, a DEA model based on maximal element set (DEA-PS) is constructed based on the theory of partially ordered set theory, and the equivalence between DEA-PS model and the original DEA model is proved. Furthermore, it is proved that the dominant theorem exists in the solution of multiple classical DEA models, and a method of super-scale DEA models based on dominant theorem is given. The conclusions of this paper theoretically reveal the relationship between efficient DMUs and maximal elements, from this relationship, two key insights emerge: (1) Regardless of how huge the number of DMU is, the proposed method in this paper can realize the scale reduction of the DEA model quickly, (2) The algorithm has good compatibility with all current big data DEA model algorithms.

Discovering API usage specifications for security detection using two-stage code mining

An application programming interface (API) usage specification, which includes the conditions, calling sequences, and semantic relationships of the API, is important for verifying its correct usage, which is in turn critical for ensuring the security and availability of the target program. However, existing techniques either mine the co-occurring relationships of multiple APIs without considering their semantic relationships, or they use data flow and control flow information to extract semantic beliefs on API pairs but difficult to incorporate when mining specifications for multiple APIs. Hence, we propose an API specification mining approach that efficiently extracts a relatively complete list of the API combinations and semantic relationships between APIs. This approach analyzes a target program in two stages. The first stage uses frequent API set mining based on frequent common API identification and filtration to extract the maximal set of frequent context-sensitive API sequences. In the second stage, the API relationship graph is constructed using three semantic relationships extracted from the symbolic path information, and the specifications containing semantic relationships for multiple APIs are mined. The experimental results on six popular open-source code bases of different scales show that the proposed two-stage approach not only yields better results than existing typical approaches, but also can effectively discover the specifications along with the semantic relationships for multiple APIs. Instance analysis shows that the analysis of security-related API call violations can assist in the cause analysis and patch of software vulnerabilities.

On Complexity Bounds for the Maximal Admissible Set of Linear Time-Invariant Systems

On Complexity Bounds for the Maximal Admissible Set of Linear Time-Invariant Systems

On Some Operators in Anti-Topological Spaces

This article introduces some anti-topological operators and provides various examples and results regarding the new operators. Specifically, We introduce the anti-frontier operator and examine essential properties of such concept. Furthermore, we introduce anti-accumulation points and anti-exterior of sets. We also introduce minimal and maximal anti-open sets and their dual. We show that every anti-open and anti-closed set are both minimal and maximal.

Functional Limit Theorems for Dynamical Systems with Correlated Maximal Sets

In order to obtain functional limit theorems for heavy-tailed stationary processes arising from dynamical systems, one needs to understand the clustering patterns of the tail observations of the process. These patterns are well described by means of a structure called the pilling process introduced recently in the context of dynamical systems. So far, the pilling process has been computed only for observable functions maximised at a single repelling fixed point. Here, we study richer clustering behaviours by considering correlated maximal sets, in the sense that the observable is maximised in multiple points belonging to the same orbit, and we work out explicit expressions for the pilling process when the dynamics is piecewise linear and expanding (1-dimensional and 2-dimensional).

Optimal Knowledge Distillation through Non-Heuristic Control of Dark Knowledge

In this paper, a method is introduced to control the dark knowledge values also known as soft targets, with the purpose of improving the training by knowledge distillation for multi-class classification tasks. Knowledge distillation effectively transfers knowledge from a larger model to a smaller model to achieve efficient, fast, and generalizable performance while retaining much of the original accuracy. The majority of deep neural models used for classification tasks append a SoftMax layer to generate output probabilities and it is usual to take the highest score and consider it the inference of the model, while the rest of the probability values are generally ignored. The focus is on those probabilities as carriers of dark knowledge and our aim is to quantify the relevance of dark knowledge, not heuristically as provided in the literature so far, but with an inductive proof on the SoftMax operational limits. These limits are further pushed by using an incremental decision tree with information gain split. The user can set a desired precision and an accuracy level to obtain a maximal temperature setting for a continual classification process. Moreover, by fitting both the hard targets and the soft targets, one obtains an optimal knowledge distillation effect that mitigates better catastrophic forgetting. The strengths of our method come from the possibility of controlling the amount of distillation transferred non-heuristically and the agnostic application of this model-independent study.

Informationally overcomplete measurements from generalized equiangular tight frames

Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

Hierarchy Elements in an Almost Distributive Lattice

In this paper, we introduce hierarchy elements in an almost distributive lattice with respect to a non-empty set and obtain some of their algebraic properties. We characterize initial segments, ideals, and maximal sets in almost distributive lattices in terms of hierarchy sets and prove that the class of hierarchy sets forms a distributive lattice, which is not an induced sublattice. Also, we characterize hierarchy sets using compatible sets in an almost distributive lattice.

GRIEVOUS: your command-line general for resolving cross-dataset genotype inconsistencies.

Harmonizing variant indexing and allele assignments across datasets is crucial for data integrity in cross-dataset studies such as multi-cohort genome-wide association studies, meta-analyses, and the development, validation, and application of polygenic risk scores. Ensuring this indexing and allele consistency is a laborious, time-consuming, and error-prone process requiring a certain degree of computational proficiency. Here, we introduce GRIEVOUS, a command-line tool for cross-dataset variant homogenization. By means of an internal database and a custom indexing methodology, GRIEVOUS identifies, formats, and aligns all biallelic single nucleotide polymorphisms (SNPs) across all summary statistic and genotype files of interest. Upon completion of dataset harmonization, GRIEVOUS can also be used to extract the maximal set of biallelic SNPs common to all datasets. GRIEVOUS and all supporting documentation and tutorials can be found at https://github.com/jvtalwar/GRIEVOUS. It is freely and publicly available under the MIT license and can be installed via pip.

Generating m-Ary Gray Codes and Related Algorithms

In this work, we systematize several implementations of the Gray code over an alphabet with m≥2 elements, which we present in C code so that they can be used directly after copying from the text. We consider two variants—reflected and modular (or shifted) m-ary Gray codes. For both variants, we present the ranking and unranking functions, as well as algorithms for generating only a part of the code, more precisely the codewords between two given vectors. Finally, we give algorithms that generate a maximal set of non-proportional vectors of length n over the given alphabet in a Gray code.

Comparative evaluation of the effects of different torque settings on dentinal crack formation using single- and multi-file system: An in vitro study.

To improve efficiency, biomechanical preparation in root canal treatment is shifting from manual SS to nickel-titanium (NiTi) rotary devices. While multi-file NiTi systems entail crack and fracture issues, modern single-file systems address these concerns. The aim of this study was to evaluate and compare the effects of different torque settings on dentinal crack formation using single-file systems (SFS) (One Curve [OC]) and multi-file systems (ProTaper Next [PTN]) at different levels of the tooth. The study was conducted on 45 freshly extracted human mandibular premolars divided into groups: OC at minimal and maximal torque, PTN at minimal and maximal torque, and a control group. After canal preparation, teeth were horizontally sectioned at 3, 6, and 9 mm from the apex, and then examined for cracks using a stereomicroscope. This was analyzed using Chi-square test. PTN group: Highest crack rates at the middle (55.6%) and apical (77.8%) thirds with maximum torque; OC group: Highest rates at the middle (22.2%) with minimal torque and apical (11.1%) with maximum torque. Maximal torque settings had more incidence of cracks compared to minimal torque settings. It can be stated that SFS (OC) produced less cracks compared to multi-file system (PTN) at both minimal and maximal torque settings.

Spectral structure of Moran Sierpinski-type measure on R2

Let Mn=diag[3pn,3qn] with pn,qn⩾1 for all n⩾1 and let D={(0,0)t,(1,0)t,(0,1)t} . One can generate a Borel probability measure μ{Mn},D=δM1−1D∗δ(M2M1)−1D∗δ(M3M2M1)−1D∗⋯. Such measure μ{Mn},D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L2(μ{Mn},D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L2(μ{Mn},D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L2(μ{Mn},D) or not.

Graphs obtained by disjoint unions and joins of cliques and stable sets

We consider the set of graphs that can be constructed from a one-vertex graph by repeatedly adding a clique or a stable set linked to all or none of the vertices added in previous steps. This class of graphs contains various well-studied graph families such as threshold, domishold, co-domishold and complete multipartite graphs, as well as graphs with linear clique-width at most 2. We show that it can be characterized by three forbidden induced subgraphs as well as by properties involving maximal stable sets and minimal dominating sets. We also give a simple recognition algorithm and formulas for the computation of the stability and domination numbers of these graphs.

On parametric semidefinite programming with unknown boundaries

In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the a priori unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this a priori unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.

Lowness properties for strong reducibilities and the computational power of maximal sets

We introduce the notion of eventually uniformly weak truth table array computable (e.u.wtt-a.c.) sets. As our main result, we show that a computably enumerable (c.e.) set has this property iff it is weak truth table ( wtt-) reducible to a maximal set. Moreover, in this equivalence we may replace maximal sets by quasi-maximal sets, hyperhypersimple sets or dense simple sets and we may replace wtt-reducibility by identity-bounded Turing reducibility (or any intermediate reducibility). Here, a set A is e.u.wtt-a.c. if there is an effective procedure which, for any given partial wtt-functional Φ ˆ, yields a computable approximation g ( x , s ) of the domain of Φ ˆ A together with a computable indicator function k ( x , s ) and a computable order h ( x ) such that, once the indicator becomes positive, i.e., k ( x , s ) = 1, the number of the mind changes of the approximation g on x after stage s is bounded by h ( x ) where, for total Φ ˆ A , the indicator eventually becomes positive on almost all arguments x of Φ ˆ A . In addition to our main result, we show several properties of the computably enumerable e.u.wtt-a.c. sets. For instance, the class of these sets is closed downwards under wtt-reductions and closed under join. Moreover, we relate this class to – and separate it from – well known classes in the literature. On the one hand, the class of the wtt-degrees of the c.e. e.u.wtt-a.c. sets is strictly contained in the class of the array computable c.e. wtt-degrees. On the other hand, every bounded low set is e.u.wtt-a.c. but there are e.u.wtt-a.c. c.e. sets which are not bounded low. Here a set A is bounded low if A † ⩽ wtt ∅ † , i.e., if A † is ω-c.a., where A † is the wtt-jump of A (Anderson, Csima and Lange (Archive for Mathematical Logic 56(5–6) (2017) 507–521)). Finally, we prove that there is a strict hierarchy within the class of the bounded low c.e. sets A depending on the order h that bounds the number of mind changes of a computable approximation of A † , and we show that there exists a Turing complete set A such that A † is h-c.a. for any computable order h with h ( 0 ) > 0.

Frame set for Gabor systems with Haar window

We describe the full structure of the frame set for the Gabor system G(g;α,β):={e−2πimβ⋅g(⋅−nα):m,n∈Z} with the window being the Haar function g=−χ[−1/2,0)+χ[0,1/2). This is the first compactly supported window function for which the frame set is represented explicitly.The strategy of this paper is to introduce the piecewise linear transformation M on the unit circle, and to provide a complete characterization of structures for its (symmetric) maximal invariant sets. This transformation is related to the famous three gap theorem of Steinhaus which may be of independent interest. Furthermore, a classical criterion on Gabor frames is improved, which allows us to establish a necessary and sufficient condition for the Gabor system G(g;α,β) to be a frame, i.e., the symmetric invariant set of the transformation M is empty.

Dimension in team semantics

AbstractWe introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique maximal and a unique minimal set. In addition to considering dimensions of particular families of sets, we study the behavior of dimensions under operators that map families of sets to new families of sets. We identify natural sufficient criteria for such operators to preserve the growth class of the dimensions. We apply the theory of our dimensions for proving new hierarchy results for logics with team semantics. To this end we associate each atom with a natural notion or arity. First, we show that the standard logical operators preserve the growth classes of the families arising from the semantics of formulas in such logics. Second, we show that the upper dimension of $k+1$ -ary dependence, inclusion, independence, anonymity, and exclusion atoms is in a strictly higher growth class than that of any k-ary atoms, whence the $k+1$ -ary atoms are not definable in terms of any atoms of smaller arity.