Hyperconnected Research Articles

Shell-model representations of the microscopic version of the Bohr–Mottelson collective model

The structure of the irreducible collective spaces of the group Sp(12, R), which many-particle nuclear states are classified according to the chain Sp(12, R) ⊃ U(6) ⊃ SO(6) ⊃ SU pn (3) ⨂ SO(2) ⊃ SO(3) of the proton–neutron symplectic model (PNSM), is considered in detail. This chain of the PNSM was recently shown to correspond to a microscopic shell-model version of the Bohr–Mottelson collective model. The construction of the relevant shell-model representations of the Sp(12, R) group along this chain is considered for three nuclei with varying collective properties and from different mass regions. It is shown that the SU pn (3) basis states of the Sp(12, R) representations belonging to the SO(6) irreps with seniority υ ≥ υ 0, with υ 0 denoting the maximal seniority SO(6) irrep contained in the Sp(12, R) bandhead, are always Pauli allowed, but organized in a different way into different SO(6) shells. This is in contrast to the case of filling the levels of the standard three-dimensional harmonic oscillator and using the plethysm operation. Although the SU pn (3) multiplets within υ < υ 0 are not all Pauli forbidden, it is safe to discard them. The results obtained in the present work are important for the practical application of the microscopic version of the Bohr–Mottelson collective model.

𝐾𝐾-duality for self-similar groupoid actions on graphs

We extend Nekrashevych’s K K KK -duality for C ∗ C^* -algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid ( G , E ) (G,E) acting faithfully on a finite directed graph E E , we associate two C ∗ C^* -algebras, O ( G , E ) \mathcal {O}(G,E) and O ^ ( G , E ) \widehat {\mathcal {O}}(G,E) , to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in K K KK -theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author.

Improved effective dipole moment model for axially symmetric [formula omitted] molecules: Parallel [formula omitted] bands

The new improved effective dipole moment model is derived for molecules of the axial C3v symmetry. The obtained result is, on the one hand, of the same order of efficiency but much simpler and clearer in applications in comparison with previous models derived on the basis of the irreducible tensorial sets theory and, on the other hand, mathematically more correct in comparison with concepts like the Herman–Walles function. Symmetric ro–vibrational bands of the mentioned molecule types are considered in the present study with taking into account the fourth order centrifugal corrections to the main effective dipole moment operator both for”allowed“ and ”forbidden“ transitions.

Compact convergence, deformation of the $L^{2}$-$\overline{\partial}$-complex and canonical $K$-homology classes

Let (X,\gamma) be a compact, irreducible Hermitian complex space of complex dimension m and with \dim(\mathrm{sing}(X))=0 . Let (F,\tau)\rightarrow X be a Hermitian holomorphic vector bundle over X , and let us denote by \overline{\eth}_{F,m,\mathrm{abs}} the rolled-up operator of the maximal L^{2} - \overline{\partial} -complex of F -valued (m,\bullet) -forms. Let \pi:M\rightarrow X be a resolution of singularities, g a metric on M , E:=\pi^{*}F and \rho:=\pi^{*}\tau . In this paper, under quite general assumptions on \tau , we prove the following equality of analytic K -homology classes [\overline{\eth}_{F,m,\mathrm{abs}}]=\pi_{*}[\overline{\eth}_{E,m}] , with \overline{\eth}_{E,m} the rolled-up operator of the L^{2} - \overline{\partial} -complex of E -valued (m,\bullet) -forms on M . Our proof is based on functional analytic techniques developed in Kuwae and Shioya (2003) and provides an explicit homotopy between the even unbounded Fredholm modules induced by \overline{\eth}_{F,m,\mathrm{abs}} and \overline{\eth}_{E,m} .

On Fuzzy Baire-Separated Spaces and Related Concepts

In this paper, a new class of fuzzy topological spaces, namely fuzzy Baire-separated spaces is introduced in terms of fuzzy Baire sets. Several characterizations of fuzzy Baire-separated spaces are established. It is shown that fuzzy Baire sets lie between disjoint fuzzy P-sets and fuzzy F&amp;lt;sub&amp;gt;σ&amp;lt;/sub&amp;gt;- sets in a fuzzy Baire-separated space. Conditions under which fuzzy topological spaces become fuzzy Baire-separated spaces are established. Fuzzy nowhere dense sets are fuzzy closed sets in fuzzy nodec spaces and subsequently a question will arise. Which fuzzy topological spaces [other than fuzzy hyperconnected spaces, fuzzy globally disconnected spaces] have fuzzy closed sets with fuzzy nowhere denseness? For this, fuzzy topological spaces having fuzzy closed sets with fuzzy nowhere denseness are identified in this paper. It is verified that fuzzy ultraconnected spaces are non fuzzy Baire -separated spaces. The means, by which fuzzy weakly Baire space become fuzzy Baire -separated spaces and in turn fuzzy Baire - separated spaces become fuzzy seminormal spaces, are obtained. There are scope in this paper for exploring the inter-relations between fuzzy Baire spaces and Baire -separated spaces.

Surjective Nash maps between semialgebraic sets

In this work we study the existence of surjective Nash maps between two given semialgebraic sets S and T. Some key ingredients are: the irreducible components Si⁎ of S (and their intersections), the analytic path-connected components Tj of T (and their intersections) and the relations between dimensions of the semialgebraic sets Si⁎ and Tj. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps.The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: a full characterization of the semialgebraic subsetsS⊂Rnthat are the image of the closed unit ballB‾mofRmcentered at the origin under a Nash mapf:Rm→Rn. The necessary and sufficient conditions that must satisfy such a semialgebraic set S are: it is compact, connected by analytic paths and has dimensiond≤m.Two remarkable consequences of the latter result are the following: (1) pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension d are Nash images ofB‾d, and (2) compact semialgebraic sets of dimension d are projections of non-singular algebraic sets of dimension d whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions).

On soft parametric somewhat-open sets and applications via soft topologies

In this work, we adopt a new approach to study a new class of soft sets depending on the generalizations of open subsets in the parametric topological spaces. We first define the class of soft parametric somewhat-open sets and explore its basic features. We illustrate this class represents a proper extension of soft open and soft somewhat-open sets under a full soft topology. We derive the next formula1+∏η∈H(|Θη|−1)≤|Ϝ|≤1+(2|U|−1)|H|, which determines the lower and upper bounds of the cardinality number Ϝ of the family of soft parametric somewhat-open subsets of a soft topological space (U,Θ,H), where Θη is a parametric topology inspired by Θ. Then, we introduce two novel kinds of soft compact and Lindelöf spaces inspired by the class of soft parametric somewhat-open sets and explain the relations between them with the aid of some counterexamples. We also examine the navigation of these spaces between soft and parametric (classical) structures and supply the necessary conditions that guarantee some directions. In the end, we introduce the concept of soft ps-connected spaces and give some of its equivalent descriptions. Furthermore, we prove the identity between this concept and soft hyperconnected spaces and show that the existence of a somewhat connected (parametric) space is used to confirm the possession of a soft ps-connected property.

Three-Dimensional Rotational Averaging Using Irreducible Sets of Linearly Independent Fundamental Isotropic Cartesian Tensors: A Computational Approach.

The theoretical formulation of linear and nonlinear molecular spectroscopies applied to isotropic samples (e.g., liquid or gas solutions) goes through a fundamental step known as the rotational averaging of Cartesian tensors. Rotational averaging of Cartesian tensors is a mathematical procedure from which the expressions for the rotationally invariant observables (e.g., rates or intensities), associated with a given spectroscopic process, can be found. In this work, the mathematical/computational procedure for finding the rotational averages of Cartesian tensors of any rank n, which is based on the use of the fundamental isotropic Cartesian tensors (FICTs), is discussed. Moreover, for the first time, a heuristic computational method for finding a set of linearly independent FICTs is proposed. The procedure has been tested for 2 ≤ n ≤ 12, where most of the linear and nonlinear molecular spectroscopies apply (e.g., one-photon and multiphoton absorption, emission, electronic circular dichroism, Raman optical activity, coherent and incoherent mth-harmonic generation, etc.). Finally, it is shown how this computational procedure can be extended for n > 12.

A general categorical reflection for various completions of [formula omitted]-closure spaces and [formula omitted]-ordered sets

In this paper, we consider a full subcategory S of the category CLQ of stratified Q-closure spaces, satisfying some conditions. We construct a reflection from the category CLQ onto S, called the S-completion, which encompasses a variety of completions for Q-closure spaces, Q-cotopological spaces, and Q-ordered sets. The notion of a fuzzy subset system Z on a category C of structured sets is introduced. By considering fuzzy subset systems Z on the category CLQ, we introduce the concept of a Z-convergence Q-closure space, as a generalization of sober/bounded sober/monotone convergence Q-cotopological spaces. We show that the category S is exactly a category of Z-convergence spaces for some fuzzy subset system Z. And, in the case that Z consists of irreducible fuzzy sets, the S-completion can be restricted to the setting of Q-cotopological spaces. The sobrification, bounded sobrification, D-completion of Q-cotopological spaces are special cases of the S-completion. After establishing the connections between fuzzy subset systems on CLQ and those on the category ORDQ of Q-ordered sets, we provide a general approach to generating various completions of Q-ordered sets from the S-completion of corresponding Q-closure spaces. Completions of Q-ordered sets with respect to different types of fuzzy directedness can be obtained directly in this useful framework.

Bounds on the smallest sets of quantum states with special quantum nonlocality

An orthogonal set of states in multipartite systems is called to be strong quantum nonlocality if it is locally irreducible under every bipartition of the subsystems \cite{Halder19}. In this work, we study a subclass of locally irreducible sets: the only possible orthogonality preserving measurement on each subsystems are trivial measurements. We call the set with this property is locally stable. We find that in the case of two qubits systems locally stable sets are coincide with locally indistinguishable sets. Then we present a characterization of locally stable sets via the dimensions of some states depended spaces. Moreover, we construct two orthogonal sets in general multipartite quantum systems which are locally stable under every bipartition of the subsystems. As a consequence, we obtain a lower bound and an upper bound on the size of the smallest set which is locally stable for each bipartition of the subsystems. Our results provide a complete answer to an open question (that is, can we show strong quantum nonlocality in Cd1&amp;#x2297;Cd1&amp;#x2297;&amp;#x22EF;&amp;#x2297;CdN for any di&amp;#x2265;2 and 1&amp;#x2264;i&amp;#x2264;N?) raised in a recent paper \cite{Shi22N}. Compared with all previous relevant proofs, our proof here is quite concise.

Effective Dipole Moment Model for Axially Symmetric C3v Molecules: Application to the Precise Study of Absolute Line Strengths of the ν6 Fundamental of CH335Cl.

The effective dipole moment model for molecules of axial C3v symmetry is derived on the basis of the symmetry properties of a molecule which, on the one hand, is of the same order of efficiency (but much simpler and clearer in applications) as the analogous models derived on the basis of the irreducible tensorial sets theory, and, on the other hand, mathematically more correct in comparison with concepts like the Herman-Walles function used in the models. As an application of the general results obtained, we discuss high-resolution infrared spectra of CH335Cl, recorded with the Zürich prototype ZP2001 (Bruker IFS125 HR) Fourier transform infrared spectrometer at a resolution of 0.001 cm-1 and analyzed in the region of 880-1190 cm-1 (ν6 bending fundamental centered at ν0 = 1018.070790 cm-1). Absolute strengths of more than 2800 transitions (2081 lines) were obtained from the fit of their shapes both with Voigt and Hartmann-Tran profiles, and parameters of the effective dipole moment of the ν6 band were determined by the computer code SYMTOMLIST (SYMmetric TOp Molecules: LIne STrengths), created on the basis of a derived theoretical model. As the first step of the analysis of the experimental data, assignments of the recorded lines were made. A total of 5124 transitions with Jmax = 68, Kmax = 21 were assigned to the ν6 band. The weighted fit of 2077 upper energy values obtained from the experimentally recorded transitions was made with a Hamiltonian which takes into account different types of ro-vibrational effects in doubly degenerate vibrational states of the C3v-symmetric molecule. As the result, a set of 25 fitted parameters was obtained which reproduces the initial 2077 upper "experimental" ro-vibrational energy values with a root mean square deviation drms=4.7×10-5 cm-1. At the second step of the analysis, the computer code SYMTOMLIST was used for determination of the parameters of the derived effective dipole moment model. Six effective dipole moment parameters were obtained from the weighted fit procedure which reproduces absolute experimental strengths of the 2804 initial experimental transitions with a relative drms=3.4%.

On the microscopic shell-model version of the Bohr-Mottelson collective model

The recently proposed microscopic shell-model version of the Bohr-Mottelson (BM) collective model is considered in more detail in the coordinate representation. The latter possesses a clear and transparent physical meaning, which reveals several features of the new version of the collective model missed in the previous formulation. The relationship to the original BM model is considered, along with the relationships between the different limiting submodels of the microscopic version of the BM model, which closely resemble the relationships of the original Wilets-Jean and rotor models. The kinematically correct many-particle wave functions of the microscopic version of the BM model, conserving the experimentally observed integrals of motion, are shown to consist of collective irrotational-flow and intrinsic components in accordance with the original BM unified model. The general BM Hamiltonian is obtained as a contraction limit of the microscopic many-particle nuclear Hamiltonian, or, alternatively, by restricting the latter to the scalar irreducible collective space.

Distributivity of a segmentation lattice

Closure spaces, namely the finite ones, with closed singletons are studied on the level of segmentations — partitions of the space into closed subsets. Segmentations form a lattice and we study spaces for which this lattice is distributive. Studying these spaces may help understanding mathematical background for segmentation of a digital image. A crucial notion is that of connectively irreducible sets which can be defined in any finite closure space. The paper provides several equivalent conditions for segmentational distributivity in terms of triples of closed sets, connected systems of closed sets, property of induced closure operator on down-sets of connectively irreducible sets, and finally by restriction (or disability) of existence of certain sublattices.

Blow‐ups and modifications of lcK spaces

AbstractIn this article, we prove that the blow‐up of an lcK space along a subspace that verifies certain conditions is lcK if and only if is induced gcK, generalizing a theorem of Ornea–Verbitsky–Vuletescu to singular locally irreducible spaces. We also show that even if modifications of lcK spaces are not always of lcK type, they always admit quasi‐lcK metrics.

On α-Fuzzy Soft Irreducible Spaces

We define fuzzy soft irreducible sets, a-fuzzy soft irreducible sets in fuzzy soft topological spaces and study the properties including (fuzzy soft continuity; fuzzy soft homeomorphism and fuzzy soft topological properties) on a-fuzzy soft irreducible sets.

Attributes Reduction on SE-ISI Concept Lattice for an Incomplete Context Using Object Ranking

The formal concept of lattice plays a vital role in knowledge discovery. Reduction of the attribute has many applications in machine learning technology and data mining fields. In this paper, we introduce an object ranking concept to define a consistency set and the reduction of the attributes by structural features. An incomplete information system works on the three-way concepts using the SE-ISI Context. The granular was emphasized with join (meet) irreducible sets using the object ranking concepts. A dual operator is defined based on the object ranking concepts and its properties and conditions are verified. Hence, this elaborates on the four kinds of reduction of the attributes. The ordered pairs give the knowledge of the attributes that deal with the interval set of both the approximation of rough set theory concerning the objects. Therefore, the relationship between four kinds of reduction of the attribute was appropriate to access the consistency set using the object ranking concepts by some of the theorems and examples.

Character degrees of normally monomial 𝑝-groups of maximal class

AbstractA finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

Submaximality on Bigeneralized Topological Spaces

In this article, in a bigeneralized topological space, we introduce a new space namely, (s, v)-bigeneralized submaximal space, and analyze its nature. Also, the characterization theorem for a (s, v)-bigeneralized submaximal space, image and preimage of (s, v) bigeneralized submaximal is a (s, v)-bigeneralized submaximal space under (µ, η)-open, (µ, η)-continuous map, respectively are proved. Further, the relationship between hyperconnected space and submaximal space in a pairwise bigeneralized submaximal space is given.

On the preferred extensions of argumentation frameworks: Bijections with naive sets

This paper deals with the problem of finding the preferred extensions of an argumentation framework by means of a bijection with the naive sets of another framework. First, we consider the case where an argumentation framework is naive-bijective: its naive sets and preferred extensions are equal. Recognizing naive-bijective argumentation frameworks is hard, but we show that it is tractable for frameworks with bounded in-degree. Next, we give a bijection between the preferred extensions of an argumentation framework being admissible-closed (the intersection of two admissible sets is admissible) and the naive sets of another framework on the same set of arguments. On the other hand, we prove that identifying admissible-closed argumentation frameworks is coNP-complete. At last, we introduce the notion of irreducible self-defending sets as those that are not the union of others. It turns out there exists a bijection between the preferred extensions of an argumentation framework and the naive sets of a framework on its irreducible self-defending sets. Consequently, the preferred extensions of argumentation frameworks with some lattice properties can be listed with polynomial delay and polynomial space.

Closure operators, irreducibility, Urysohn's lemma, and Tietze extension theorem for proximity spaces

In this paper, we introduce two notions of closure in the category of proximity spaces which satisfy (weak) hereditariness, productivity, and idempotency, and we characterize each of $T_{i}, i=0,1,2$, proximity spaces by using these closure operators and show how these subcategories are related. Furthermore, we characterize the irreducible proximity spaces and investigate the relationship among each of irreducible, connected and $T_{i}, i=1,2$, proximity spaces. Finally, we present Tietze extension theorem and Urysohn's lemma for proximity spaces.