Methods and means of asymutal-invariant muller matrix polyarimetry of optical and anisotropic biological layers

Objectives: The work is aimed at the theoretical substantiation and experimental development of the azimuthally invariant polarimetry method of partially depolarizing optical anisotropic biological layers on the basis of coordinate Muller-matrix mapping of histological sections for differential diagnostics of changes in optical anisotropy, which are associated with the emergence of pathological states. Results: The method of azimuthally invariant Muller-matrix mapping of optically anisotropic samples of the myocardium is proposed and grounded. The values of azimuthally invariant matrix element, superposition of matrix elements and the magnitude of the matrix vector distributions are obtained. Conclusion: The magnitude of the 1st-4th order statistical moments’ dependences, which characterize the distribution of the Muller-matrix invariant (MMI) of the histological sections of the myocardium are determined. The study of the possibility of differentiating causes of death due to ischemic heart disease (IHD) and acute coronary insufficiency (ACI) conducted from the standpoint of evidence-based medicine.

Asymptotic symmetries and charges at null infinity: from low to high spins

Weinberg’s celebrated factorisation theorem holds for soft quanta of arbitrary integer spin. The same result, for spin one and two, has been rederived assuming that the infinite-dimensional asymptotic symmetry group of Maxwell’s equations and of asymptotically flat spaces leave the S-matrix invariant. For higher spins, on the other hand, no such infinite-dimensional asymptotic symmetries were known and, correspondingly, no a priori derivation of Weinberg’s theorem could be conjectured. In this contribution we review the identification of higher-spin supertranslations and superrotations in D = 4 as well as their connection to Weinberg’s result. While the procedure we follow can be shown to be consistent in any D, no infinite-dimensional enhancement of the asymptotic symmetry group emerges from it in D > 4, thus leaving a number of questions unanswered.

A generalization of Turaev’s virtual string cobracket and self-intersections of virtual strings

Previously we defined an operation [Formula: see text] that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that [Formula: see text] gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that [Formula: see text] gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings [Formula: see text] such that [Formula: see text] gives a formula for the minimal self-intersection number [Formula: see text]. Finally, we compare the bound given by [Formula: see text] to a bound given by Turaev’s based matrix invariant [Formula: see text], and construct an example that shows the bound on the minimal self-intersection number given by [Formula: see text] is sometimes stronger than the bound [Formula: see text].

FIXED-POINT INTERACTIONS AND RENORMALIZATION GROUP INVARIANCE IN THE TWO-NUCLEON SYSTEM

We study the fixed-point interactions and the renormalization group invariance for an effective nucleon–nucleon (NN) interaction in the leading-order (LO) chiral effective field theory (ChEFT) renormalized within the framework of the subtracted kernel method (SKM) approach. By solving a nonrelativistic Callan–Symanzik (NRCS) equation we show how the driving term evolves with the subtraction scale to keep the T-matrix invariant. We calculate the fixed-point interaction from the driving term and compare the results obtained with and without its evolution through the NRCS equation.

A SEQUENCE OF DEGREE ONE VASSILIEV INVARIANTS FOR VIRTUAL KNOTS

For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite-dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one double-point.

A MATRIX INVARIANT OF CURVES IN S2

Let k: S1 → S2 be a generic immersion with n double points. We present an algorithm that assigns to k a partitioned n × n matrix over Z/2Z, and show that k gives rise to an orthogonal decomposition of (Z/2Z)n. We discuss a connection between this decomposition and the trip matrix of an alternating knot diagram produced from k.