Fitzpatrick functions,cyclic monotonicity and Rockafellar’s antiderivative

Several deeper results on maximal monotone operators have recently found simpler proofs using Fitzpatrick functions. In this paper, we study a sequence of Fitzpatrick functions associated with a monotone operator. The first term of this sequence coincides with the original Fitzpatrick function, and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties. It is shown that for any maximal cyclically monotone operator, the pointwise supremum of the sequence of Fitzpatrick functions is closely related to Rockafellar’s antiderivative. Several examples are explicitly computed for the purpose of illustration. In contrast to Rockafellar’s result, a maximal 3-cyclically monotone operator need not be maximal monotone. A simplified proof of Asplund’s observation that the rotation in the Euclidean plane by π/n$\pi /n$ is n$n$-cyclically monotone but not (n+1)$(n+1)$-cyclically monotone is provided. The Fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail. We conclude with a new proof of Moreau’s result concerning the convexity of the set of proximal mappings.     

Contractive mappings and existence of cycle times for a monotone and homogeneous function

Given a monotone and homogeneous self-mapping f$f$ of the n$n$-dimensional positive cone, a family of contractive mappings is used to define an equivalence relation in the index set, as well as a total order among the equivalence classes. Then, it is shown (i) that the cycle times are well-defined at each index belonging to the maximal and minimal classes, and (ii) that the cycle times of f$f$ exist at every index whenever a weak convexity condition is satisfied.     

Global attractivity of positive periodic solutions for nonlinear impulsive systems

In this paper, the existence, uniqueness and global attractivity of positive periodic solutions for nonlinear impulsive systems are studied. Firstly, existence conditions are established by the method of lower and upper solutions. Then uniqueness and global attractivity are obtained by developing the theories of monotone and concave operators. And lastly, the method and the results are applied to the impulsive n$n$-species cooperative Lotka–Volterra system and a model of a single-species dispersal among n$n$-patches.     

An alternative approach to well-posedness of a class of differential inclusions in Hilbert spaces

We show the well-posedness of initial value problems for differential inclusions of a certain type using abstract perturbation results for maximal monotone operators in Hilbert spaces. For this purpose the time derivative is established in an exponentially weighted _L2$L_2$ space. The problem of well-posedness then reduces to show that the sum of two maximal monotone operators in time and space is again maximal monotone. The theory is exemplified by three inclusions describing phenomena in mathematical physics involving hysteresis.     

A new condition for maximal monotonicity via representative functions

In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S+A^∗TA$S+A^{\ast }TA$, where S:X?X^∗$S:X?X^{\ast }$, T:Y?Y^∗$T:Y?Y^{\ast }$ are two maximal monotone operators, A:X→Y$A:X\to Y$ is a linear continuous mapping and X,Y$X,Y$ are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.     

Fixed point theorems for mixed monotone operators with PPF dependence

In this paper, fixed point theorems for mixed monotone operators are investigated by weakening the requirements in the contractive assumption and strengthening the metric space utilized with a partial order, taking the domain space C[[a,b],E]$C[[a,b],E]$ different from the range E$E$ and using the method of reflection operators.     

Lifted infinitesimal holomorphic representation for the n-dimensional complex hyperbolic ball and for Cartan domains of type I

The Laplacian and Ornstein–Uhlenbeck operators on the finite dimensional complex ball are obtained from the infinitesimal holomorphic representation of the group U(n,1)$U(n,1)$. We compare the invariant measures for these operators with the unitarizing measures of the discrete series representation. Then with Hua differential calculus, we show how to extend the results to domains with matrix elements.     

Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e.  , up to a specified combinatorial factor, the monotone Hurwitz number in genus g$g$ with ramification specified by a given partition is a polynomial indexed by g$g$ in the parts of the partition.     

The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n$n$-polytopes with the smallest number of flags are found, for every rank n>1$n>1$. With a few small exceptions, the smallest regular n$n$-polytopes come from a family of ‘tight’ polytopes with 2⋅4ⁿ⁻¹$2\cdot 4^{n-1}$ flags, one for each n$n$, with Schläfli symbol {4∣4∣?∣4}$\{4\mid 4\mid ?\mid 4\}$. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.     

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p$p$-Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ>0$\lambda >0$ varies in a half-line.     

The Quicksort process

Quicksort on the fly returns the input of n$n$ reals in increasing natural order during the sorting process. Correctly normalized the running time up to returning the l$l$-th smallest out of n$n$ seen as a process in l$l$ converges weakly to a limiting process with path in the space of cadlag functions.     

Approximate martingale estimating functions for stochastic differential equations with small noises

An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter ε$\epsilon$ from discrete time observations at n$n$ regularly spaced time points k/n$k/n$, k=0,1,…,n$k=0,1,\dots ,n$. We show asymptotic efficiency of an M$M$-estimator derived from the approximate martingale estimating function as ε→0$\epsilon \to 0$ and n→∞$n\to \infty$ simultaneously.     

Non-homogeneous random walks on a semi-infinite strip

We study the asymptotic behaviour of Markov chains (_Xn,_ηn)$(X_n,{\eta }_n)$ on _Z+×S${\mathbb{Z}}_{+}\times S$, where _Z+${\mathbb{Z}}_{+}$ is the non-negative integers and S$S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of _Xn$X_n$, and that, roughly speaking, _ηn${\eta }_n$ is close to being Markov when _Xn$X_n$ is large. This departure from much of the literature, which assumes that _ηn${\eta }_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for _Xn$X_n$ given _ηn${\eta }_n$. We give a recurrence classification in terms of increment moment parameters for _Xn$X_n$ and the stationary distribution for the large- X$X$ limit of _ηn${\eta }_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between _Xn$X_n$ (rescaled) and _ηn${\eta }_n$. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on _Z+${\mathbb{Z}}_{+}$ (the case where S$S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where _ηn${\eta }_n$ tracks an internal state of the system.     

Intertwining operators and the restriction of representations of rank-one orthogonal groups

We give a complete classification of intertwining operators (breaking symmetry operators  ) between spherical principal series representations of O(n+1,1)$O(n+1,1)$ and O(n,1)$O(n,1)$ together with explicit formulae of the distribution kernels. Further we use this to determine the breaking symmetry operators between their irreducible composition factors.     

Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent

The class of coalescent processes with simultaneous multiple collisions (Ξ$\Xi$-coalescents) without proper frequencies is considered. We study the asymptotic behavior of the external branch length, the total branch length and the number of mutations on the genealogical tree as the sample size n$n$ tends to infinity. The limiting random variables arising are characterized via exponential integrals of the subordinator associated with the frequency of singletons of the coalescent. The proofs are based on decompositions into external and internal branches. The asymptotics of the external branches is treated via the method of moments. The internal branches do not contribute to the limiting variables since the number _Cn$C_n$ of collisions for coalescents without proper frequencies is asymptotically negligible compared to n$n$. The results are applied to the two-parameter Poisson–Dirichlet coalescent indicating that this particular class of coalescent processes in many respects behaves approximately as the star-shaped coalescent.