Operator Algebras Research Articles

Modular implementation of the linear- and cubic-scaling orbital minimization methods in electronic structure codes using atomic orbitals.

We present a code modularization approach to design efficient and massively parallel cubic- and linear-scaling solvers for electronic structure calculations using atomic orbitals. The modular implementation of the orbital minimization method, in which linear algebra and parallelization issues are handled via external libraries, is demonstrated in the SIESTA code. The distributed block compressed sparse row (DBCSR) and scalable linear algebra package (ScaLAPACK) libraries are used for algebraic operations with sparse and dense matrices, respectively. The MatrixSwitch and libOMM libraries, recently developed within the Electronic Structure Library, facilitate switching between different matrix formats and implement the energy minimization. We show results comparing the performance of several cubic-scaling algorithms, and also demonstrate the parallel performance of the linear-scaling solvers, and their supremacy over the cubic-scaling solvers for insulating systems with sizes of several hundreds of atoms.

Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras

Abstract The residual finite-dimensionality of a ${\textrm{C}}^{\ast }$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of when residual finite-dimensionality of an operator algebra is inherited by its maximal ${\textrm{C}}^{\ast }$-cover, which we establish in many cases of interest.

Identifikasi Kesalahan Siswa Dalam Menyelesaikan Soal Tipe Hots (Higher Order Thinking Skill) Materi Operasi Aljabar Di SMP Muhammadiyah 4 Palembang

Many students still need to improve, causing low student learning outcomes. This study aimed to find out what mistakes students made in solving the Hots (Higher Order Thinking Skill) type questions on algebraic operations material at SMP Muhammadiyah 4 Palembang. This research is a qualitative descriptive study. The data collection technique is in the form of giving as many as five test questions. Based on the research results, the mistakes made by students in solving HOTS (Higher Order Thinking Skill) type questions in algebraic material are conceptual and procedural errors. Concept errors occur in each item, with the percentage of errors that occur, namely item number 1 at 33.3%, number 2 at 23.3%, number 3 at 16.7%, number 4 at 16.7%, number 5 at 26.7%. The average percentage of students' conceptual errors is 23.34%. Procedural errors occur in each item, with the percentage of errors that occur, namely item number 1 of 10%, 2 of 20%, 3 of 33.3%, 4 of 26.7%, and 5 of 16.7 %. The average percentage of procedural errors made by students is 21.34%.

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

Geometric Aggregation Operators for Solving Multicriteria Group Decision-Making Problems Based on Complex Pythagorean Fuzzy Sets

The Complex Pythagorean fuzzy set (CPyFS) is an efficient tool to handle two-dimensional periodic uncertain information, which has various applications in fuzzy modeling and decision making. It is known that the aggregation operators influence decision-making processes. Algebraic aggregation operators are the important and widely used operators in decision making techniques that deal with uncertain problems. This paper investigates some complex Pythagorean fuzzy geometric aggregation operators, such as complex Pythagorean fuzzy weighted geometric (CPyFWG), complex Pythagorean fuzzy ordered weighted geometric (CPyFOWG), complex Pythagorean fuzzy hybrid geometric (CPyFHG), induced complex Pythagorean fuzzy ordered weighted geometric (I-CPyFOWG), and induced complex Pythagorean fuzzy hybrid geometric (I-CPyFHG), and their structure properties, such as idempotency, boundedness, and monotonicity. In addition, we compare the proposed model with their existing models, such as complex fuzzy set and complex intuitionistic fuzzy set. We analyze an example involving the selection of an acceptable location for hospitals in order to demonstrate the effectiveness, appropriateness, and efficiency of the novel aggregation operators.

P-adic vertex operator algebras

We postulate axioms for a chiral half of a nonarchimedean 2-dimensional bosonic conformal field theory, that is, a vertex operator algebra in which a p-adic Banach space replaces the traditional Hilbert space. We study some consequences of our axioms leading to the construction of various examples, including p-adic commutative Banach rings and p-adic versions of the Virasoro, Heisenberg, and the Moonshine module vertex operator algebras. Serre p-adic modular forms occur naturally in some of these examples as limits of classical 1-point functions.

Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25

We classify all two-dimensional, unitary, rational conformal field theories with two primaries, central charge $c<25$, and arbitrary Wronskian index. In mathematical parlance, we classify all strongly regular vertex operator algebras (VOAs) with central charge $c<25$ and exactly two simple modules. We find that any such theory is either one of the Mathur-Mukhi-Sen (MMS) theories $\mathsf{A}_{1,1}$, $\mathsf{G}_{2,1}$, $\mathsf{F}_{4,1}$, or $\mathsf{E}_{7,1}$, or it is a coset of a chiral algebra with one primary operator (also known as a holomorphic VOA) by such an MMS theory. By leveraging existing results on the classification of holomorphic VOAs, we are able to explicitly enumerate all of the aforementioned cosets and compute their characters. This leads to 123 theories, most of which are new. We emphasize that our work is a bona fide classification of RCFTs, not just of characters. Our techniques are general, and we argue that they offer a promising strategy for classifying chiral algebras with low central charge beyond two primaries.

Quasitriangular operator algebras

We give characterizations of quasitriangular operator algebras along the line of Voiculescu's characterization of quasidiagonal C⁎-algebras.

S3-permutation orbifolds of Virasoro vertex algebras

In this paper, a continuation of [24], we investigate the S3-orbifold subalgebra of (Vc)⊗3, that is, we consider the S3-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras Vc. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of c. More precisely, we show that this vertex algebra is of type (2,4,62,82,9,102,11,123).We also investigate two prominent examples of simple S3-orbifold algebras corresponding to central charges c=12 (Ising model) and c=−225 (i.e. (2,5)-minimal model). We prove that the former is a new unitary W-algebra of type (2,4,6,8) and the latter is isomorphic to the affine simple W-algebra of type G2 at non-admissible level −196. We also provide another version of this isomorphism using the affine W-algebra of type G2 coming from a subregular nilpotent element.

Strong 3-skew commutativity preserving maps on prime rings with involution

ABSTRACT Let be a unital prime -ring containing a nontrivial symmetric idempotent. For , the 3-skew commutator is defined by . Let be a surjective map. It is shown that satisfies for all if and only if there exists with such that for all . Where I is the unit of and is the symmetric center of . This result then is applied to some operator algebras. Communicated by Igor Klep

Convergence of sewing conformal blocks

In a recent work [On factorization and vector bundles of conformal blocks from vertex algebras, preprint (2019), arXiv:1909.04683], Damiolini et al. showed that for a [Formula: see text]-cofinite rational vertex operator algebra [Formula: see text], sheaves of conformal blocks are locally free and satisfy the factorization property. In this paper, we prove that if [Formula: see text] is [Formula: see text]-cofinite, the sewing of conformal blocks is convergent. This proves a conjecture proposed by Zhu [Global vertex operators on Riemann surfaces, Commun. Math. Phys. 165(3) (1994) 485–531] and Huang [Some open problems in mathematical two-dimensional conformal field theory, preprint (2016), arXiv:1606.04493], generalizing previous results on the convergence of products, iterates, and traces (but not pseudo-traces) of vertex operators and intertwining operators in [Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9(1) (1996) 237–302; Y. Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Algebra 100(1–3) (1995) 173–216; Y. Z. Huang, Differential equations, duality and modular invariance, Commun. Contemp. Math. 7(05) (2005) 649–706; Y. Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011), arXiv:1110.1929] and on the convergence of projective factors related to the central charge of the Virasoro algebra in [Y. Z. Huang, Two-Dimensional Conformal Geometry and Vertex Operator Algebras, Vol. 148 (Springer Science &amp; Business Media, 1997)].

Geometric tool kit for higher-spin gravity (Part I): An introduction to the geometry of differential operators

This paper provides an introduction to the algebra and geometry of differential operators and jet bundles. Its point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalizations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.

Robust hardware implementation of a block-cipher scheme based on chaos and biological algebraic operations

Robust hardware implementation of a block-cipher scheme based on chaos and biological algebraic operations

A Structure Preserving Shift-Invert Infinite Arnoldi Algorithm for a Class of Delay Eigenvalue Problems with Hamiltonian Symmetry

This work considers a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axes. We propose a method to iteratively approximate the eigenvalues closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end, our method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a “real skew-Hamiltonian”-like structure. Furthermore, it is proven that the Krylov subspace constructed by applying this operator satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that, even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, SIAM J. Sci. Comput., 22 (2001), pp. 1905–1925] to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite-dimensional linear algebra operations. The performance of the resulting numerical algorithm is verified for two example problems: the first example illustrates the advantage of the proposed approach in preserving purely imaginary eigenvalues when working in finite precision, while the second one demonstrates its applicability to a large scale problem.

The algebra of thin measurable operators is directly finite

Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\mathcal{H}$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$-algebra of all $\tau$-compact operators and $T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$-algebra of all operators $X=A+\lambda I$ with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. It is proved that every operator of $T(\mathcal{M},\tau)$ that is left-invertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$. It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\mathcal{B} (\mathcal{H})$. For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$, the inclusion $\mu(t; Q)\in \{0\}\bigcup [1, +\infty)$ holds for all $t&amp;gt;0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010.

Analysis of TOPSIS techniques based on bipolar complex fuzzy N‐soft setting and their applications in decision‐making problems

AbstractA novel model termed a bipolar complex fuzzy N‐soft set (BCFN‐SS) is initiated for tackling information that involves positive and negative aspects, the second dimension, and parameterised grading simultaneously. The theory of BCFN‐SS is the generalisation of two various theories, that is, bipolar complex fuzzy (BCF) and N‐SS. The invented model of BCFN‐SS helps decision‐makers to cope with the genuine‐life dilemmas containing BCF information along with parameterised grading at the same time. Further, various algebraic operations, including the usual type of union, intersection, complements, and a few others types, are invented. Certain primary operational laws for BCFN‐SS are also invented. Moreover, a technique for order preference by similarity to the ideal solution (TOPSIS) approach is devised in the setting of BCFN‐SS for managing strategic decision‐making (DM) dilemmas containing BCFN‐SS information. Keeping in mind the usefulness and benefits of the TOPSIS approach, two various types of TOPSIS approaches in the environment of BCFN‐SS are devised and then a numerical example for exposing the usefulness of the devised TOPSIS approach is interpreted. To disclose the prominence and benefits of the devised work, the devised approaches with numerous prevailing work are compared.

Local nets of unbounded operator algebras

Two rigorous mathematical foundations for relativistic quantum field theory are local nets of von Neumann algebras and Wightman field theory. We recall that local nets of von Neumann algebras are local nets of bounded operator algebras, whereas Wightman field theory makes use of unbounded linear operators. In this paper, we introduce local nets of unbounded operator algebras and show how these can be connected with Wightman field theory. This is motivated by the fact that observables in quantum mechanics are unbounded operators in general.

Split Casimir operator for simple Lie algebras in the cube of ad-representation and Vogel parameters

In this paper, we constructed characteristic identities for the 3-split (polarized) Casimir operators of simple Lie algebras in the adjoint representations [Formula: see text] and deduced a certain class of subrepresentations in [Formula: see text]. The projectors onto invariant subspaces for these subrepresentations were directly constructed from the characteristic identities for the 3-split Casimir operators. For all simple Lie algebras, universal expressions for the traces of higher powers of the 3-split Casimir operators were found and dimensions of the subrepresentations in [Formula: see text] were calculated. All our formulas are in agreement with the universal description of (irreducible) subrepresentations in [Formula: see text] for simple Lie algebras in terms of the Vogel parameters.

Mini-Workshop: Recent Developments in Representation Theory and Mathematical Physics

This mini-workshop was devoted to foster the interactions between mathematicians and mathematical physicists who are working on questions related to representation theory. This includes for example the representation theory of supergroups, vertex operator algebras and quantum groups. Another focus was on link and manifold invariants and TQFTs.

From Moyal deformations to chiral higher-spin theories and to celestial algebras

We study the connection of Moyal deformations of self-dual gravity and self-dual Yang-Mills theory to chiral higher-spin theories, and also to deformations of operator algebras in celestial holography. The relation to Moyal deformations illuminates various aspects of the structure of chiral higher-spin theories. For instance, the appearance of the self-dual kinematic algebra in all the theories considered here leads via the double copy to vanishing tree-level scattering amplitudes. Regarding celestial holography, the Moyal deformation of self-dual gravity was recently shown to lead to the loop algebra of W∧, and we obtain here the analogous deformation of a Kac-Moody algebra corresponding to Moyal-deformed self-dual Yang-Mills theory. We also introduce the celestial algebras for various chiral higher-spin theories.