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Steven Duplij

Publications and source records attributed to Steven Duplij.

At least 37 records · Page 2Linked to original sources
arXiv2021

Polyadic braid operators and higher braiding gates

Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.

quant-ph↗
arXiv2021

Higher braid groups and regular semigroups from polyadic-binary correspondence

In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and $R$-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, $n$-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

math.GR↗
arXiv2020

Double-graded quantum superplane

A $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalisation of the quantum superplane is proposed and studied. We construct a bicovariant calculus on what we shall refer to as the \emph{double-graded quantum superplane}. The commutation rules between the coordinates, their differentials and partial derivatives are explicitly given. Furthermore, we show that an extended version of the double-graded quantum superplane admits a natural Hopf $\mathbb{Z}_2^2$-algebra structure.

math.QA↗
arXiv2020

Structure of unital 3-fields

We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring $\mathcal{R}$ having $\mathbb{Z}\diagup 2\mathbb{Z}$ as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of $\mathcal{R}$. We include a number of illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.

math.RA↗
arXiv2020

Membership deformation of commutativity and obscure $n$-ary algebras

A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and $\varepsilon $-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" $\varepsilon$-Lie algebras and Weyl algebras. Further, the above constructions are extended to $n$-ary algebras for which the projective representations and $\varepsilon $-commutativity are studied.

math.RA↗
arXiv2020

Graded medial $n$-ary algebras and polyadic tensor categories

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or $\varepsilon$-commutativity) we introduce almost mediality ("commutativity-to-mediality" ansatz). Higher graded twisted products and "deforming" brackets (being the medial analog of Lie brackets) are defined. Toyoda's theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an $n$-ary tensor product as an additional multiplication with $n-1$ associators of the arity $2n-1$ satisfying a $\left( n^{2}+1\right) $-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called "groupal") is defined: they are close to monoidal categories, but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in $n$-ary groups). The arity-nonreducible $n$-ary braiding is introduced and the equation for it is derived, which for $n=2$ coincides with the Yang-Baxter equation. Then, analogously to the first part of the paper, we introduce "medialing" instead of braiding and construct "medialed" polyadic tensor categories.

math.RA↗
arXiv2019

On Lagrangian and non-Lagrangian conformal-invariant nonlinear electrodynamics

A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.

hep-th↗
arXiv2019

Double-Graded Supersymmetric Quantum Mechanics

A quantum mechanical model that realizes the $ \mathbb{Z}_2 \times \mathbb{Z}_2$-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.

math-ph↗
arXiv2019

Polyadic Hopf algebras and quantum groups

This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the $R$-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and $M$-matrix, for which the compatibility equations are found.

math.RA↗
arXiv2018

Polyadic systems, representations and quantum groups

Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.

math.RT↗
arXiv2017

Polyadic integer numbers and finite (m,n)-fields

The polyadic integer numbers, which form a polyadic ring, are representatives of a fixed congruence class. The basics of polyadic arithmetic are presented: prime polyadic numbers, the polyadic Euler function, polyadic division with a remainder, etc. are introduced. Secondary congruence classes of polyadic integer numbers, which become ordinary residue classes in the "binary limit", and the corresponding finite polyadic rings are defined. Polyadic versions of (prime) finite fields are introduced. These can be zeroless, zeroless and nonunital, or have several units; it is even possible for all of their elements to be units. There exist non-isomorphic finite polyadic fields of the same arity shape and order. None of the above situations is possible in the binary case. It is conjectured that a finite polyadic field should contain a certain canonical prime polyadic field, defined here, as a minimal finite subfield, which can be considered as a polyadic analogue of GF(p).

math.RA↗
arXiv2017

Arity shape of polyadic algebraic structures

Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. Relations between operations arising from the structure definitions, however, lead to restrictions which determine their possible arity shapes and lead us to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections are introduced, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators. Another application is connected with number theory, and it is shown that congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced (see Definition 6.16), and Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat's last theorem are formulated. For nonderived polyadic ring operations (on polyadic numbers) neither of these statements holds, and counterexamples are given. Finally, a procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov's theorem to the Tarry-Escott problem is presented.

math.RA↗
arXiv2017

Conformal symmetry transformations and nonlinear Maxwell equations

We make use of the conformal compactification of Minkowski spacetime $M^{\#}$ to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime $[M^{\#}]^{-1}$ obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying $M^{\#}$ with the projective light cone in $(4+2)$-dimensional spacetime, we write two independent conformal-invariant functionals of the $6$-dimensional Maxwellian field strength tensors -- one bilinear, the other trilinear in the field strengths -- which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.

hep-th↗
arXiv2015

Triander: A new program for visual analysis of nucleotide sequences

The Triander program is an interactive software package for nucleotide sequence visualization. The program was developed using the freeware Pascal RAD IDE Lazarus, and its source code and binaries compiled for Windows are freely accessible at http://www.icbge.org.ua/eng/Triander . Triander can produce four types of plots. It is possible to build three DNA walks independently for each nucleotide position in triplets. The use of nucleotide vectors with unequal modulus leads to a significant reduction in the visual information lost in DNA walks. The program can be used in the investigation of the fine structure of sequences, to find standard patterns in them and to locate nontrivial regions for further detailed analysis.

q-bio.GN↗
arXiv2015

Cho-Duan-Ge decomposition of QCD in the constraintless Clairaut-type formalism

We apply the recently derived constraintless Clairaut-type formalism to the Cho-Duan-Ge decomposition in SU(2) QCD. We find nontrivial corrections to the physical equations of motion and that the contribution of the topological degrees of freedom is qualitatively different from that found by treating the monopole potential as though it were dynamic. We also find alterations to the field commutation relations that undermine the particle interpretation in the presence of the chromomonopole condensate.

hep-th↗
arXiv2014

A "q-deformed" generalization of the Hosszu-Gluskin theorem

In this paper a new form of the Hosszú-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszú-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszú-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.

math.RA↗
arXiv2014

Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space. In this case the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.

math-ph↗