Harmonic functions on the real hyperbolic ball II Hardy‐Sobolev and Lipschitz spaces

In this paper, we pursue the study of harmonic functions on the real hyperbolic ball started in [13]. Our focus here is on the theory of Hardy‐Sobolev and Lipschitz spaces of these functions. We prove here that these spaces admit Fefferman‐Stein like characterizations in terms of maximal and square functionals. We further prove that the hyperbolic harmonic extension of Lipschitz functions on the boundary extend into Lipschitz functions on the whole ball. In doing so, we exhibit differences of behaviour of derivatives of harmonic functions depending on the parity of the dimension of the ball and on the parity of the order of derivation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees

We consider conditioned Galton–Watson trees and show asymptotic normality of additive functionals that are defined by toll functions that are not too large. This includes, as a special case, asymptotic normality of the number of fringe subtrees isomorphic to any given tree, and joint asymptotic normality for several such subtree counts. Another example is the number of protected nodes. The offspring distribution defining the random tree is assumed to have expectation 1 and finite variance; no further moment condition is assumed. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 57–101, 2016     

The Height of Increasing Trees

Increasing trees have been introduced by Bergeron, Flajolet, and Salvy [1]. This kind of notion covers several well-know classes of random trees like binary search trees, recursive trees, and plane oriented (or heap ordered) trees. We consider the height of increasing trees and prove for several classes of trees (including the above mentioned ones) that the height satisfies EH _n ~ γlogn (for some constant γ > 0) and Var H _n =  O(1) as n → ∞. The methods used are based on generating functions. This research was supported by the Austrian Science Foundation FWF, project S9604, that is part of the Austrian National Research Network "Analytic Combinatorics and Probabilistic Number Theory".     

The Wiener Index of simply generated random trees

Asymptotics are obtained for the mean, variance, and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton–Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 337–358, 2003     

The cut‐tree of large trees with small heights

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017     

Analysis of three graph parameters for random trees

We consider three basic graph parameters, the node‐independence number, the path node‐covering number, and the size of the kernel, and study their distributional behavior for an important class of random tree models, namely the class of simply generated trees, which contains, e.g., binary trees, rooted labeled trees, and planted plane trees, as special instances. We can show for simply generated tree families that the mean and the variance of each of the three parameters under consideration behave for a randomly chosen tree of size n asymptotically ～μn and ～νn, where the constants μ and ν depend on the tree family and the parameter studied. Furthermore we show for all parameters, suitably normalized, convergence in distribution to a Gaussian distributed random variable. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

In this paper, we consider a class of asymptotically linear second‐order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.     

Half plane Toeplitz operators with generating functions having homogeneous discontinuities

We consider Toeplitz operators over the discrete half plane with generating functions admitting discontinuities of special kind which we call homogenous ones. The main results are the criteria of invertibility and one‐sided invertibility of such operators. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A representation on the labeled rooted forests

We consider the conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. By interpreting these objects in terms of labeled rooted forests, we give a characterization of the labeled rooted trees whose S_n orbit afford the sign representation. Applications to rook theory are offered.     

Ancestors and descendants in evolving k‐tree models

We consider several random graph models based on k‐trees, which can be generated by applying the probabilistic growth rules “uniform attachment”, “preferential attachment”, or a “saturation”‐rule, respectively, but which also can be described in a combinatorial way. For all of these models we study the number of ancestors and the number of descendants of nodes in the graph by carrying out a precise analysis which leads to exact and limiting distributional results. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 465–489, 2014     

Stability of general virus dynamics models with both cellular and viral infections and delays

We consider general virus dynamics model with virus‐to‐target and infected‐to‐target infections. The model is incorporated by intracellular discrete or distributed time delays. We assume that the virus‐target and infected‐target incidences, the production, and clearance rates of all compartments are modeled by general nonlinear functions that satisfy a set of reasonable conditions. The non‐negativity and boundedness of the solutions are studied. The existence and stability of the equilibria are determined by a threshold parameter. We use suitable Lyapunov functionals and apply LaSalle's invariance principle to prove the global asymptotic stability of the all equilibria of the model. We confirm the theoretical results by numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Limit laws for the Randić index of random binary tree models

We investigate the Randić index of random binary trees under two standard probability models: the one induced by random permutations and the Catalan (uniform). In both cases the mean and variance are computed by recurrence methods and shown to be asymptotically linear in the size of the tree. The recursive nature of binary search trees lends itself in a natural way to application of the contraction method, by which a limit distribution (for a suitably normalized version of the index) is shown to be Gaussian. The Randić index (suitably normalized) is also shown to be normally distributed in binary Catalan trees, but the methodology we use for this derivation is singularity analysis of formal generating functions.     

Optimal boundary control of glass cooling processes

In this paper, an optimal control problem for glass cooling processes is studied. We model glass cooling using the SP1 approximations to the radiative heat transfer equations. The control variable is the temperature at the boundary of the domain. This results in a boundary control problem for a parabolic/elliptic system which is treated by a constrained optimization approach. We consider several cost functionals of tracking‐type and formally derive the first‐order optimality system. Several numerical methods based on the adjoint variables are investigated. We present results of numerical simulations illustrating the feasibility and performance of the different approaches. Copyright © 2004 John Wiley & Sons, Ltd.     

Counting trees using symmetries

We prove a new formula for the generating function of multitype Cayley trees counted according to their degree distribution. Using this formula we recover and extend several enumerative results about trees. In particular, we extend some results by Knuth and by Bousquet-Mélou and Chapuy about embedded trees. We also give a new proof of the multivariate Lagrange inversion formula. Our strategy for counting trees is to exploit symmetries of refined enumeration formulas: proving these symmetries is easy, and once the symmetries are proved the formulas follow effortlessly. We also adapt this strategy to recover an enumeration formula of Goulden and Jackson for cacti counted according to their degree distribution.     

Statistical field theory of a nonadditive system

Based on quantum field methods, we develop a statistical theory of complex systems with nonadditive potentials. Using the Martin-Siggia-Rose method, we find the effective system Lagrangian, from which we obtain evolution equations for the most probable values of the order parameter and its fluctuation amplitudes. We show that these equations are unchanged under deformations of the statistical distribution while the probabilities of realizing different phase trajectories depend essentially on the nonadditivity parameter. We find the generating functional of a nonadditive system and establish its relation to correlation functions; we introduce a pair of additive generating functionals whose expansion terms determine the set of multipoint Green’s functions and their self-energy parts. We find equations for the generating functional of a system having an internal symmetry and constraints. In the harmonic approximation framework, we determine the partition function and moments of the order parameter depending on the nonadditivity parameter. We develop a perturbation theory that allows calculating corrections of an arbitrary order to the indicated quantities.     

Limit laws for embedded trees: Applications to the integrated superBrownian excursion

We study three families of labeled plane trees. In all these trees, the root is labeled 0 and the labels of two adjacent nodes differ by 0,1, or ?1. One part of the paper is devoted to enumerative results. For each family, and for all j ∈ ?, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than j; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled j; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled at least, j. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity. The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size n with the uniform distribution and study the following random variables: M_n, the largest label occurring in a (random) tree; X_n(j), the number of nodes labeled j; and X(j), the number of nodes labeled j or more. We obtain limit laws for scaled versions of these random variables. Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion. In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas) and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On the Boolean algebras of definable sets in weakly o‐minimal theories

We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ω‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ω‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inheritance of Smoothness by Extremal Functions in Bergman Spaces $$A_p$$ for $$0

Burchaev  Kh. Kh.  Ryabykh  G. Yu. 《Mathematical Notes》2021,110(1-2):167-185

Asymptotic formulae for eigenvalues and eigenfunctions of q‐Sturm‐Liouville problems

We investigate the asymptotic behavior of the eigenvalues and the eigenfunctions of q‐Sturm‐Liouville eigenvalue problems. For this aim we study the asymptotic behavior of q‐trigonometric functions as well as fundamental sets of solutions of the associated second order q‐difference equation. As in classical Sturm‐Liouville theory, the eigenvalues behave like zeros of q‐trigonometric functions and the eigenfunctions behave like q‐trigonometric functions. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the internal path length of d‐dimensional quad trees

It is proved that the internal path length of a d‐dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the m‐ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999