Nonprobability Gibbs measures for the HC model with a countable set of spin values for a “wand”-type graph on a Cayley tree

Theoretical and Mathematical Physics - We study Gibbs measures for the HC model with a countable set $$\mathbb Z$$ of spin values and a countable set of parameters (i.e., with the activity function...     

Gibbs measures for the HC Blume–Capel model with countably many states on a Cayley tree

Theoretical and Mathematical Physics - We study the Blume–Capel model with a countable set $$mathbb Z$$ of spin values and a force $$Jin mathbb R$$ of interaction between the nearest...     

Extremality of Gibbs Measures for the $$HC$$ -Blume–Capel Model on the Cayley Tree

Mathematical Notes - In this paper, we consider translation-invariant Gibbs measures (TIGMs) for the $$HC$$ -Blume–Capel model in case of “wands” with chemical potential with...     

Existence of a Phase Transition for the Potts p-adic Model on the Set ℤ

We consider the Potts model on the set in the field Q _p of p-adic numbers. The range of the spin variables (n), , in this model is . We show that there are some values q=q(p) for which phase transitions occur.     

Translation-invariant <Emphasis Type="Italic">p</Emphasis>-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

We consider the p-adic Ising–Vannimenus model on the Cayley tree of order k = 2. This model contains nearest-neighbor and next-nearest-neighbor interactions. We investigate the model using a new approach based on measure theory (in the p-adic sense) and describe all translation-invariant p-adic quasi-Gibbs measures associated with the model. As a consequence, we can prove that a phase transition exists in the model. Here, “phase transition” means that there exist at least two nontrivial p-adic quasi-Gibbs measures such that one is bounded and the other is unbounded. The methods used are inapplicable in the real case.     

Bounds on the vertex–edge domination number of a tree

A vertex–edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex–edge domination number of a graph G  , denoted by γ_ve(T)${\gamma }_{\mathrm{ve}}(T)$, is the minimum cardinality of a vertex–edge dominating set of G. We prove that for every tree T   of order n?3$n?3$ with l leaves and s   support vertices, we have (n−l−s+3)/4?γ_ve(T)?n/3$(n-l-s+3)/4?{\gamma }_{\mathrm{ve}}(T)?n/3$, and we characterize the trees attaining each of the bounds.     

Ground States and Phase Transition of the λ Model on the Cayley Tree

We consider the λ model, a generalization of the Potts model, with spin values {1, 2, 3} on the order-two Cayley tree. We describe the model ground states and prove that translation-invariant Gibb measures exist, which means that a phase transition exists. We establish that two-periodic Gibbs measures exist.     

Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

Gibbs phenomenon and its removal for a class of orthogonal expansions

We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.     

A note on “hierarchical models for multi-project planning and scheduling”

We consider the multi-mode resource-constrained project scheduling problem. The focus of our analysis is on an algorithm recently proposed by Speranza and Vercellis for finding makespan minimal solutions. The correctness of the algorithm is examined. By counterexamples we illustrate that the algorithm does not generally find (existing) optimal solutions.     

Stability on a cone in terms of two measures for impulsive differential equations with “supremum”

The stability of nonlinear impulsive differential equations with “supremum” is studied. A special type of stability, combining two different measures and a dot product, is defined. The definition is a generalization of several types of stability known in the literature. Razumikhin’s method as well as a comparison method for scalar impulsive ordinary differential equations have been employed.     

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Given k identical salesmen, where k ? 2 is a constant independent of the input size, the min–max k-traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree-shaped network, so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized. The problem is known to be NP-hard even when k = 2. In this paper, we have developed a pseudo-polynomial time exact algorithm for this problem with any constant k ? 2, closing a question that has remained open for a decade. Along with this, we have further developed a (1 + ?)-approximation algorithm for any ? > 0.     

Fisher information and α-connections for a class of transformational models

In this paper we use invariant theory for representations of groups in order to get an indirect method of computing the Fisher metric and the α-connections of transformational models parameterized by symmetric spaces. Among others these models include the Von Mises–Fisher model, the Hyperboloid model, the multivariable zero mean normal model with determinant one covariant matrix and the Wishard model.     

Polycondenser’s capacity and the module of a family of vector measures

The paper considers the capacity of a polycondenser and the module of a family of measures associated with the class of curves connecting the polycondenser’s plates. It is proved that these quantities are equal. Also a relation between the polycondenser’s capacity and the module of a family of vector measures associated with the class of functions admissible for the polycondenser’s capacity is established. The results obtained generalize the earlier results by M. Ohtsuka and H. Aikawa.