Regularity of extended conjugate graphs of finite groups

<abstract><p>The extended conjugate graph associated to a finite group $ G $ is defined as an undirected graph with vertex set $ G $ such that two distinct vertices joined by an edge if they are conjugate. In this article, we show that several properties of finite groups can be expressed in terms of properties of their extended conjugate graphs. In particular, we show that there is a strong connection between a graph-theoretic property, namely regularity, and an algebraic property, namely nilpotency. We then give some sufficient conditions and necessary conditions for the non-central part of an extended conjugate graph to be regular. Finally, we study extended conjugate graphs associated to groups of order $ pq $, $ p^3 $, and $ p^4 $, where $ p $ and $ q $ are distinct primes.</p></abstract>

Applications of a New q -Difference Operator in Janowski-Type Meromorphic Convex Functions

The main aim of the present article is the introduction of a new differential operator in q -analogue for meromorphic multivalent functions which are analytic in punctured open unit disc. A subclass of meromorphic multivalent convex functions is defined using this new differential operator in q -analogue. Furthermore, we discuss a number of useful geometric properties for the functions belonging to this class such as sufficiency criteria, coefficient estimates, distortion theorem, growth theorem, radius of starlikeness, and radius of convexity. Also, algebraic property of closure is discussed of functions belonging to this class. Integral representation problem is also proved for these functions.

Transmuting off-shell CHY integrals in the double-cover framework

In this paper, by defining off-shell amplitudes as off-shell CHY integrals, and redefining the longitudinal operator, we demonstrate that the differential operators which link on-shell amplitudes for a variety of theories together link off-shell amplitudes in a similar manner. Based on the algebraic property of the differential operator, we also generalize three relations among color-ordered on-shell amplitudes, including the color-ordered reversed relation, the photon decoupling relation, the Kleiss–Kuijf relation, to off-shell ones. The off-shell CHY integrals are chosen to be in the double-cover framework, thus, as a by product, our result also provides a verification for the double-cover construction.

Quantum Markov chains: A unification approach

In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on general local algebras, possessing a generic algebraic property. We stress that this kind of algebras includes both Boson and Fermi algebras. Our main results concern two reconstruction theorems for quantum Markov chains and for quantum Markov states. Namely, we illustrate the results through examples.

An elementary proof of the Eichler–Selberg trace formula

We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group{\mathrm{SL}_{2}(\mathbb{Z})}, using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

Automatic Inference of Term Equivalence in Term Rewriting Systems

In this paper we propose a parametric technique to automatically infer algebraic property-oriented specifications from Term Rewriting Systems. Namely, given the source code of a TRS we infer a specification which consists of a set of most general equations relating terms that rewrite, for all possible instantiations, to the same set of normal forms.The semantic-based inference method that we propose can cope with non-constructor-based TRSs, and considers non-ground terms. Particular emphasis is posed to avoid the generation of “redundant” equations that can be a logical consequence of other ones.To experiment on the validity of our proposal we have considered an instance of the method employing a novel (condensed) semantics for left-linear TRSs and we have implemented a “proof of concept” prototype in Haskell which is available online.

Quantum key distribution without the wavefunction

A well-known feature of quantum mechanics is the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over any classical communication channel. It is demonstrated that this quantum key distribution allows a much more general and abstract access than commonly thought. The results include some generalizations of the Hilbert space version of quantum key distribution, but are based upon a general nonclassical extension of conditional probability. A special state-independent conditional probability is identified as origin of the superior security of quantum key distribution; this is a purely algebraic property of the quantum logic and represents the transition probability between the outcomes of two consecutive quantum measurements.

Uncovering multiloci-ordering by algebraic property of Laplacian matrix and its Fiedler vector

Motivation: The loci-ordering, based on two-point recombination fractions for a pair of loci, is the most important step in constructing a reliable and fine genetic map.Results: Using the concept from complex graph theory, here we propose a Laplacian ordering approach which uncovers the loci-ordering of multiloci simultaneously. The algebraic property for a Fiedler vector of a Laplacian matrix, constructed from the recombination fraction of the loci-ordering for 26 loci of barley chromosome IV, 846 loci of Arabidopsisthaliana and 1903 loci of Malus domestica, together with the variable threshold uncovers their loci-orders. It offers an alternative yet robust approach for ordering multiloci.Availability and implementation : Source code program with data set is available as supplementary data and also in a software category of the website (http://biophysics.dgist.ac.kr)Contact: [email protected] or [email protected] information: Supplementary data are available at Bioinformatics online.