Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

Let $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

On the Domains of Bessel Operators

We consider the Schrödinger operator on the halfline with the potential $$(m^2-\frac{1}{4})\frac{1}{x^2}$$ ( m 2 - 1 4 ) 1 x 2 , often called the Bessel operator. We assume that m is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $$|\mathrm{Re}(m)|<1$$ | Re ( m ) | < 1 and of its unique closed realization for $$\mathrm{Re}(m)>1$$ Re ( m ) > 1 coincide with the minimal second-order Sobolev space. On the other hand, if $$\mathrm{Re}(m)=1$$ Re ( m ) = 1 the minimal second-order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

R. KALMAN `S PROBLEM ABOUT FIBONACCI `S NUMBERS

In this paper authors are considered the R. Kalman`s problem about of Fibonacci numbers. An overview of research methods for control theory systems in two concepts “state space” and the “input-output” mapping is presented. In this paper, we consider the problem of R. Kalman on Fibonacci numbers, which consists in the following. R. Kalman's problem on Fibonacci numbers is considered, which is as follows. Fibonacci numbers form a minimal Realization. The authors of the article formulated a theorem, which was given the name of the outstanding American Scientist R. Kalman. The proof of the theorem is very cumbersome, therefore, authors proved it using an example when the Fibonacci numbers are obtained on the basis of the application of the B. Ho`s algorithm. B. Ho is a purple of R. Kalman. In this paper, the algorithm of B. Ho is given, which allows one to find the parameters of the initial linear deterministic system. Based on these parameters, we find the initial Fibonacci numbers. Thus, Fibonacci numbers are closely related to the problem of linear deterministic implementation and to B. Ho's algorithm.

Split SIMPs with decays

We discuss a minimal realization of the strongly interacting massive particle (SIMP) framework. The model includes a dark copy of QCD with three colors and three light flavors. A massive dark photon, kinetically mixed with the Standard Model hypercharge, maintains kinetic equilibrium between the dark and visible sectors. One of the dark mesons is necessarily unstable but long-lived, with potential impact on CMB observables. We show that an approximate “isospin” symmetry acting on the down-type quarks is an essential ingredient of the model. This symmetry stabilizes the dark matter and allows to split sufficiently the masses of the other states to suppress strongly their relic abundances. We discuss for the first time the SIMP cosmology with sizable mass splittings between all meson multiplets. We demonstrate that the SIMP mechanism remains efficient in setting the dark matter relic density, while CMB constraints on unstable relics can be robustly avoided. We also consider the phenomenological consequences of isospin breaking, including dark matter decay. Cosmological, astrophysical, and terrestrial probes are combined into a global picture of the parameter space. In addition, we outline an ultraviolet completion in the context of neutral naturalness, where confinement at the GeV scale is generic. We emphasize the general applicability of several novel features of the SIMP mechanism that we discuss here.

Invariant Manifolds in Human Joint Angle Analysis During Walking Gait

The complex dynamics of human gait is yet to be completely understood. Researchers have quantified stability of walking gait using Floquet multipliers as well as Lyapunov exponents. In this article, we utilize the techniques and tools from dynamical system theory and invariant manifolds to map the gait data onto a time invariant representation of a dynamical system. As an example, the complex behavior of the joint angle during walking was studied using a conformal mapping approach that transformed the time periodic system into a time invariant linear system. Time-delay embedding was used to reconstruct the dynamics of the original gait system with time series kinematic data. This minimal realization of the system was used to construct a Single Degree of Freedom (SDOF) oscillator. The time evolution of the linear oscillatory system was mapped back using the conformal mapping derived using Lyapunov-Floquet Theory. This algorithm was verified for walking gait kinematics data for two healthy human subjects. A comparison was drawn between the phase space behavior of the original time periodic system and the remapped time invariant system. The two systems showed good correlation. The algorithm resulted in a well correlated phase space representation.