Depolarizing differential Mueller matrices

The evolution of a polarized beam can be described by the differential formulation of Mueller calculus. The nondepolarizing differential Mueller matrices are well known. However, they only account for 7 out of the 16 independent parameters that are necessary to model a general anisotropic depolarizing medium. In this work we present the nine differential Mueller matrices for general depolarizing media, highlighting the physical implications of each of them. Group theory is applied to establish the relationship between the differential matrix and the set of transformation generators in the Minkowski space, of which Lorentz generators constitute a particular subgroup.     

Accelerated Electrophotocatalytic C(sp3)−H Heteroarylation Enabled by an Efficient Continuous-Flow Reactor**

Electrophotocatalytic transformations are garnering attention in organic synthesis, particularly for accessing reactive intermediates under mild conditions. Moving these methodologies to continuous-flow systems, or flow ElectroPhotoCatalysis (f-EPC), showcases potential for scalable processes due to enhanced irradiation, increased electrode surface, and improved mixing of the reaction mixture. Traditional methods sequentially link photochemical and electrochemical reactions, using flow reactors connected in series, yet struggle to accommodate reactive transient species. In this study, we introduce a new flow reactor concept for electrophotocatalysis (EPC) that simultaneously utilizes photons and electrons. The reactor is designed with a transparent electrode and employs cost-effective materials. We used this technology to develop an efficient process for electrophotocatalytic heteroarylation of C(sp³)−H bonds. Importantly, the same setup can also facilitate purely electrochemical and photochemical transformations. This reactor represents a significant advancement in electrophotocatalysis, providing a framework for its application in flow for complex synthetic transformations.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

Static bending of granular beam: exact discrete and nonlocal solutions

Meccanica - This study is an attempt towards a better understanding of the length scale effects on the bending response of the granular beams. To this aim, a unidimensional discrete granular chain...     

Static and dynamic behaviour of nonlocal elastic bar using integral strain-based and peridynamic models

The static and dynamic behaviour of a nonlocal bar of finite length is studied in this paper. The nonlocal integral models considered in this paper are strain-based and relative displacement-based nonlocal models; the latter one is also labelled as a peridynamic model. For infinite media, and for sufficiently smooth displacement fields, both integral nonlocal models can be equivalent, assuming some kernel correspondence rules. For infinite media (or finite media with extended reflection rules), it is also shown that Eringen's differential model can be reformulated into a consistent strain-based integral nonlocal model with exponential kernel, or into a relative displacement-based integral nonlocal model with a modified exponential kernel. A finite bar in uniform tension is considered as a paradigmatic static case. The strain-based nonlocal behaviour of this bar in tension is analyzed for different kernels available in the literature. It is shown that the kernel has to fulfil some normalization and end compatibility conditions in order to preserve the uniform strain field associated with this homogeneous stress state. Such a kernel can be built by combining a local and a nonlocal strain measure with compatible boundary conditions, or by extending the domain outside its finite size while preserving some kinematic compatibility conditions. The same results are shown for the nonlocal peridynamic bar where a homogeneous strain field is also analytically obtained in the elastic bar for consistent compatible kinematic boundary conditions at the vicinity of the end conditions. The results are extended to the vibration of a fixed–fixed finite bar where the natural frequencies are calculated for both the strain-based and the peridynamic models.     

Pourquoi, pour qui enseigner les mathématiques?

The evolution of education in Belgium is described from 1830 to our days. The mastery of education has always been a subject of disputes between the political forces in presence. With the flow of years, more attention is paid to social considerations and the necessity of a more democratic educational system arises. Simultaneously the pedagogical ideas change and active methods of teaching are promoted, although not always used. Different schoolsystems still coexist. However, except for philosophical and religious education, the differences between them tend to diminish. This is in particular the case for mathematics teaching whose evolution is also described with some details.     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.