On Gibbs measures of P-adic Potts model on the cayley tree

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ? _p) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ? p and p ≥ 3 (resp. q ? 2², p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ? p if p ≥ 3 and q ? 2² if p = 2.     

On a factor associated with the unordered phase of λ-model on a cayley tree

In this paper we consider nearest-neighbours models, where the spin takes values in the set Φ = {η₁, η₁, …, η_q} and is assigned to the vertices of the Cayley tree Γ^k. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine a type of von Neumann algebra generated by the GNS construction associated with the unordered phase of the λ-model. We give also some physical applications of the obtained result.     

Weak deflection angle and shadow cast by the charged-Kiselev black hole with cloud of strings in plasma

In this study, the gravitational deflection angle of photons in the weak field limit (or the weak deflection angle) and shadow cast by the electrically charged and spherically symmetric static Kiselev black hole (BH) in the string cloud background are investigated. The influences of the BH charge Q, quintessence parameter γ, and string cloud parameter a on the weak deflection angle are studied using the Gauss-Bonnet theorem, in addition to studying the influences on the radius of photon spheres and size of the BH shadow in the spacetime geometry of the charged-Kiselev BH in string clouds. Moreover, we study the effects of plasma (uniform and non-uniform) on the weak deflection angle and shadow cast by the charged-Kiselev BH surrounded by the clouds of strings. In the presence of a uniform/nonuniform plasma medium, an increase in the string cloud parameter a increases the deflection angle α. In contrast, a decrease in the BH charge Q decreases the deflection angle. Further, we observe that an increase in the BH charge Q causes a decrease in the size of the shadow of the BH. We notice that, with an increase in the values of the parameters γ and a, the size of the BH shadow increases, and therefore, the intensity of the gravitational field around the charged-Kiselev BH in string clouds increases. Thus, the gravitational field of the charged-Kiselev BH in the string cloud background is stronger than the field produced by the pure Reissner-Nordstrom BH. Moreover, we use the data released by the Event Horizon Telescope (EHT) collaboration, for the supermassive BHs M87* and Sgr A*, to obtain constraints on the values of the parameters γ and a.     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

Quantum effects on the black hole shadow and deflection angle in the presence of plasma

In this study, the optical properties of a renormalization group improved (RGI) Schwarzschild black hole (BH) are investigated in a plasma medium. Beginning with the equations of motion in a plasma medium, we aim to present the modifications in the shadow radius of the RGI BH. To this end, we compute the deflection angle of light in the weak gravity regime for uniform and non-uniform plasma media. Importantly, owing to the plasma media, we discover that the equations of motion for light obtained from the radiating and infalling/rest gas have to be modified. This, in turn, changes and modifies the expression for the intensity observed far away from the BH. Finally, we obtain the shadow images for the RGI BH for different plasma models. Although quantum effects change the background geometry, such effects are minimal, and practically detecting these effects using the current technology based on supermassive BH shadows is impossible. The parameter Ω encodes the quantum effects, and in principle, one expects such quantum effects to play significant roles only for very small BHs. However, the effects of plasma media can play an important role in the optical appearance of BHs, as they affect and modify the equations of motion.     

Thermodynamics via Inducing

We consider a Turing-type reaction-diffusion system involving quadratic and cubic nonlinearities and numerically investigate the role of nonlinear terms in producing spots, stripes, labyrinths, hexagonal arrangement of spots, blotches and transitions among them. From our numerical experiments performed on a square domain with zero-flux boundary conditions, we observe that the system displays a form of multistability for which different stable spatial distribution of concentrations appear for a same set of control parameters depending upon the initial conditions. For varying values of model parameters both in the first- and the second-stage of simulations, we obtain a number of transition states that are found to be sensitive on the relative strength of the quadratic and cubic coupling terms. We obtain a graphical relationship among such model parameters at which the transitions take place.

     

On the Uniqueness of Gibbs Measures for <Emphasis Type="Italic">p</Emphasis>-Adic Nonhomogeneous λ-Model on the Cayley Tree

We consider a nearest-neighbor p-adic -model with spin values ±1 on a Cayley tree of order k 1. We prove for the model there is no phase transition and as well as being unique, the p-adic Gibbs measure is bounded if and only if p 3. If p=2, then we find a condition which guarantees the nonexistence of a phase transition. Besides, the results are applied to the p-adic Ising model and we show that for the model there is a unique p-adic Gibbs measure.     

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

In the present paper the Ising model with competing binary (J) and binary (J₁) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.     

On mixing and completely mixing properties of positive -contractions of finite von Neumann algebras

Akcoglu and Suchaston proved the following result: Let be a positive contraction. Assume that for the sequence converges weakly in . Then either or there exists a positive function , such that . In the paper we prove an extension of this result in a finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative -space has no nonzero positive invariant element, then its mixing property implies the completely mixing property.