Factorization invariants of the additive structure of exponential Puiseux semirings

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μn)n≥0 on a Polish space X, the normalized sequence (μn/μn(X))n≥0 agrees with the marginal predictive distributions of some random process (Xn)n≥1. Moreover, μn=μn−1+RXn, n≥1, where x↦Rx is a random transition kernel on X; thus, if μn−1 represents the contents of an urn, then Xn denotes the color of the ball drawn with distribution μn−1/μn−1(X) and RXn—the subsequent reinforcement. In the case RXn=WnδXn, for some non-negative random weights W1,W2,…, the process (Xn)n≥1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (Xn)n≥1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Dirichlet Polynomials and Entropy

A Dirichlet polynomial d in one variable y is a function of the form d(y)=anny+⋯+a22y+a11y+a00y for some n,a0,…,an∈N. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy H(d) of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by L(d)=2H(d). On the other hand, we will define a rig homomorphism h:Dir→Rect from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is R⩾0×R⩾0 and whose additive structure involves the weighted geometric mean; we write h(d)=(A(d),W(d)), and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula A(d)=L(d)W(d) holds for any Dirichlet polynomial d. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism h applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.

Tasks of inter-project unification of spacecraft on-board systems for remote sensing of the Earth

The article discusses and presents the formulation of problems of inter-project unification of on-board systems in the development of modifications of spacecraft that are part of space systems for remote sensing of the Earth. It is shown that when developing a complex of advanced space systems, it is possible to partially combine unified on-board systems and finished products, which, under given constraints, provides a minimum of total costs. The formulation of the main task of inter-project unification of spacecraft for remote sensing of the Earth using finished products and partially unified on-board systems and a special case of the problem — conducting an economically justified inter-project unification from completely unified on-board systems (aggregates) of promising modifications of spacecraft is given. The initial data and limitations for solving the main and particular problems are determined. The tasks are presented in a deterministic setting. The concept of optimality of the choice of areas of unification of each on-board system is formulated, which is characterized by the minimum of a criterion having an additive structure, this is the total economic effect for areas of unification. It is believed that the analysis of the results of solving the problems of inter-project unification in the development of promising modifications of spacecraft will reveal the directions of inter-project unification of those on-board systems for which it is most appropriate; to formulate the fundamental principles of modernization of space systems and creation of modifications of spacecraft for remote sensing of the Earth in the planned period.

The Effect of Mean-Reverting Processes in the Pricing of Options in the Energy Market: An Arithmetic Approach

In this paper we study the effect that mean-reverting components in the arithmetic dynamics of electricity spot price have on the price of a call option on a swap. Our model allows for seasonal effects, spikes, and negative values of the price of electricity. We show that for sufficiently large delivery periods of the swap contract, the error that one makes by neglecting some of the mean-reverting processes affecting the spot price evolution converges to zero. The decay rate is explicitly calculated. This is achieved by exploiting the additive structure of the electricity price process in order to determine an explicit closed-form formula for the price of the call on a swap. The theoretical analysis is then illustrated via a numerical example.

Some special structures of S*and A* semirings

Objectives: The main objective of this research article is to study the semiring structures, we have majorly focused on the constrains under which the structures of S*and A* semirings are additively and/or multiplicatively idempotent. We have also concentrated on the study of structures of totally ordered S* and A* semirings. Methods: We have imposed singularity, cancellation property, Integral Multiple Property (IMP) and some other constrains on both semirings. Findings: when we imposed totally ordered condition on these two semirings we observed that the additive structure takes place as a maximum addition. Applications: The proposed idempotents have wide applications to computer science, dynamical and logical systems, cryptography, graph theory and artificial intelligence.

The Twelve Tones

Serialism retains its cachet as one of the most severe, learned styles to have been developed in the last one hundred years. Adès seldom uses serialism on its own, but rather in concert with other compositional techniques. For example, he opens The Four Quarters (2010) with a striking juxtaposition of serialism and isorhythm. In a somewhat different vein, Adès relies on what he calls a “magnetic” approach to serialism in Polaris (2010), which artfully masks intricate row relations behind a gradually additive structure. Taken as a whole, Adès’s serial works demonstrate his comfort with not only the history and techniques of serialism, but also the potential for certain strict compositional styles to achieve particular affective ends in diverse contexts.

Mod-two cohomology rings of alternating groups

We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.