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We consider a random field $\\varphi \\colon \\{1,\\ldots ,N\\}\\rightarrow {\\Bbb R}$ as a model for a linear chain attracted to the defect line φ = 0, that is, the x-axis. The free law of the field is specified by the density $\\text{exp}(-\\Sigma _{i}V(\\Delta \\varphi _{i}))$ with respect to the Lebesgue measure on ${\\Bbb R}^{N}$, where Δ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value $\\varepsilon _{c}^{a}$ such that when $\\varepsilon >\\varepsilon _{c}^{a}$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\\varepsilon <\\varepsilon _{c}^{a}$ (delocalization). The two critical values are nontrivial and distinct: $0<\\varepsilon _{c}^{{\\rm p}}<\\varepsilon _{c}^{{\\rm w}}<\\infty $, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\\varepsilon =\\varepsilon _{c}^{{\\rm p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\\varepsilon =\\varepsilon _{c}^{{\\rm w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.

We give further results on the Perron-Frobenius theorem for tensors, generalize other theorems from matrices to tensors, and give an equivalent condition for nonnegative irreducible tensors. [PUBLICATION ABSTRACT]

Let ${\\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\\mathbb {R}}^n$. Then ${\\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\\mathcal {D}}$.

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.[PUBLICATION ABSTRACT]

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.

Given a self-similar set K defined from an iterated function system$\\Gamma =(\\gamma _{1},\\ldots ,\\gamma _{d})$and a set of functions$H=\\{h_{i}:K\\to \\mathbb {R}\\}_{i=1}^{d}$satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras$\\mathcal {O}_{\\Gamma }$and their Toeplitz extensions$\\mathcal {T}_{\\Gamma }$. We then characterize the KMS states for this action. For each$\\beta \\in (0,\\infty )$, there is a Ruelle operator$\\mathcal {L}_{H,\\beta }$, and the existence of KMS states at inverse temperature$\\beta $is related to this operator. The critical inverse temperature$\\beta _{c}$is such that$\\mathcal {L}_{H,\\beta _{c}}$has spectral radius 1. If$\\beta <\\beta _{c}$, there are no KMS states on$\\mathcal {O}_{\\Gamma }$and$\\mathcal {T}_{\\Gamma }$; if$\\beta =\\beta _{c}$, there is a unique KMS state on$\\mathcal {O}_{\\Gamma }$and$\\mathcal {T}_{\\Gamma }$which is given by the eigenmeasure of$\\mathcal {L}_{H,\\beta _{c}}$; and if$\\beta>\\beta _{c}$, including$\\beta =\\infty $, the extreme points of the set of KMS states on$\\mathcal {T}_{\\Gamma }$are parametrized by the elements of K and on$\\mathcal {O}_{\\Gamma }$by the set of branched points.

We establish a Perron–Frobenius-type theorem for the subdominant eigenvalue of Möbius monotone transition matrices defined on partially ordered state spaces. This result extends the classical work of Keilson and Kester, where they considered stochastically monotone transition matrices in a totally ordered setting. Furthermore, we show that this subdominant eigenvalue is the geometric ergodicity rate.

This article constructs all the anisotropic spaces of four-dimensional numbers in which the commutative and associative operations of addition and multiplication are defined. In this case, so-called “zero divisors” appear in these spaces. The structures of zero divisors in each space are described and their properties are investigated. It is shown that there are two types of zero divisors and they form a two-dimensional subspace of the four-dimensional space. A space of 4 × 4 matrices is constructed that is isomorphic to the space of four-dimensional numbers. The concept of the spectrum of a four-dimensional number is introduced and a bijective mapping between four-dimensional numbers and their spectra is constructed. Thanks to this, methods for solving linear and quadratic equations in four-dimensional spaces are developed. It is proven that a quadratic equation in a four-dimensional space generally has four roots. The concept of the spectral norm is introduced in the space of four-dimensional numbers and the equivalence of the spectral norm to the Euclidean norm is proved.