Generalized Hénon Mappings and Foliation by Injective Brody Curves

We consider a finite composition of generalized Hénon mappings \({\mathfrak {f}}:{\mathbb {C}}^2\rightarrow {\mathbb {C}}^2\) and its Green function \({\mathfrak {g}}^+:{\mathbb {C}}^2\rightarrow {\mathbb {R}}_{\ge 0}\) (see Sect. 2). It is well known that each level set \(\{{\mathfrak {g}}^+=c\}\) for \(c>0\) is foliated by biholomorphic images of \({\mathbb {C}}\) and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in \(\mathbb {P}^2\) (see Sect. 4). We also study the behavior of the level sets of \({\mathfrak {g}}^+\) near infinity.     

Invariant generalized ideal classes – structure theorems for <Emphasis Type="BoldItalic">p</Emphasis>-class groups in <Emphasis Type="BoldItalic">p</Emphasis>-extensions

We give, in sections 2 and 3, an english translation of: Classes généralisées invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: From Theory to Practice, SMM, Springer-Verlag, 2nd corrected printing 2005. We recall, in section 4, some structure theorems for finite \(\mathbb {Z}_{p}[G]\)-modules (\(G \simeq \mathbb {Z}/p\,\mathbb {Z}\)) obtained in: Sur les ?-classes d’idéaux dans les extensions cycliques relatives de degré premier ?, Annales de l’Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a p-class group in a cyclic extension of degree p. In section 1973, we apply this to the study of the structure of relative p-class groups of Abelian extensions of prime to p degree, using the Thaine–Ribet–Mazur–Wiles–Kolyvagin ‘principal theorem’, and the notion of ‘admissible sets of prime numbers’ in a cyclic extension of degree p, from: Sur la structure des groupes de classes relatives, Annales de l’Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the p-ramification theory (as dual form of non-ramification theory) and which have become standard in a p-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.     

Log-Sobolev inequalities for semi-direct product operators and applications

In this paper, we prove some type of logarithmic Sobolev inequalities (with parameters) for operators in semi-direct product forms (see Sect. 1 for a precise definition). This generalizes the tensorization procedure for this type of inequalities and allows to deal with some operators with varying coefficients. We provide many examples of applications and obtain ultracontractive bounds for some of these operators by using appropriate Hardy’s type inequalities necessary for our method. This theory is developed in the setting of carré du champ with diffusion property.     

Analytical discussion for the mixed integral equations

This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation in a Banach space. Banachs fixed point theorem is used to prove the existence and uniqueness of the solution. To find the numerical solution, the integral equation is reduced to a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.     

Condition Length and Complexity for the Solution of Polynomial Systems

Smale’s 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale in Mathematical problems for the next century, American Mathematical Society, Providence, 2000). The main progress on Smale’s problem is Beltrán and Pardo (Found Comput Math 11(1):95–129, 2011) and Bürgisser and Cucker (Ann Math 174(3):1785–1836, 2011). In this paper, we will improve on both approaches and prove an interesting intermediate result on the average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán and Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser and Cucker (2011), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of Bürgisser and Cucker (2011) but relies only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser and Cucker (2011). We build on methods developed in Armentano et al. (2014).     

On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation \(iu_t+u_{xx}-|u|^2u=0\). As the first step local well-posedness in the modulation space \(M_{2,p}\) (\(2\le p<\infty \)) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global well-posedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of \(M_{2,p}\), and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).     

A Characterisation of Some $$\mathbf {}$$-Like Logics

In Béziau (Log Log Philos 15:99–111, 2006) a logic \(\mathbf {Z}\) was defined with the help of the modal logic \(\mathbf {S5}\). In it, the negation operator is understood as meaning ‘it is not necessary that’. The strong soundness–completeness result for \(\mathbf {Z}\) with respect to a version of Kripke semantics was also given there. Following the formulation of \(\mathbf {Z}\) we can talk about \(\mathbf {Z}\)-like logics or Beziau-style logics if we consider other modal logics instead of \(\mathbf {S5}\)—such a possibility has been mentioned in [1]. The correspondence result between modal logics and respective Beziau-style logics has been generalised for the case of normal logics naturally leading to soundness–completeness results [see Marcos (Log Anal 48(189–192):279–300, 2005) and Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 34(4):229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski (Bull Sect Log 37(3–4):185–196, 2008), (Bull Sect Log 38(3–4):189–203, 2009) some partial results for non-normal cases are given. In the present paper we try to give similar but more general correspondence results for the non-normal-worlds case. To achieve this aim we have to enrich original Beziau’s language with an additional negation operator understood as ‘it is necessary that not’.     

A relationship between the space of orthomorphisms and the centre of a vector lattice revisited

In (Chil et al. Positivity, 2014), the authors claim to give a counterexample to the main result, about Wickstead’s question, in a recent paper of Toumi (see Theorem 3, When orthomorphisms are in the ideal center, Positivity 18(3):579–583, 2014). In this note we show that their example is consistent with the main result of Toumi and not a counterexample.     

The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of \(\mathrm {Cay}(S,C)\) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of \(\mathrm {Cay}(S,C)\) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when \(\mathrm {Cay}(S,C)\) is color-endomorphism vertex-transitive [Theorem 3.4].     

Short-Time Existence of the Möbius-Invariant Willmore Flow

In this article the author proves existence and uniqueness of a smooth short-time solution of the “Möbius-invariant Willmore flow” Eq. (9) starting in a \(C^{\infty }\)-smooth immersion \(F_0\) of a fixed smooth compact torus \({\varSigma }\) into \(\mathbb {R}^n\) without umbilic points. Hence, for some sufficiently small \(T^*>0\) there exists a unique smooth family \(\{f_t\}\) of smooth immersions of the torus \({\varSigma }\) into \(\mathbb {R}^n\), with \(f_0=F_0\), which solve the evolution Eq. (9) for \(t \in [0,T^*]\) and whose tracefree parts \(A^{0}_{f_t}(x)\) of their second fundamental forms do not vanish in any \((x,t) \in {\varSigma }\times [0,T^*]\). The right-hand side of Eq. (9) has the specific property that any family \(\{f_t\}\) of umbilic-free \(C^4\)-immersions \(f_t:{\varSigma }\longrightarrow \mathbb {R}^n\) solves Eq. (9) if and only if its composition \({\varPhi }(f_t)\) with any applicable Möbius-transformation \({\varPhi }\) of \(\mathbb {R}^n\) solves Eq. (9) as well.     

Virtual boundaries of Hadamard spaces with admissible actions of higher rank

In this paper, we prove that every conformal minimal immersion of an open Riemann surface into \({\mathbb {R}}^n\) for \(n\ge 5\) can be approximated uniformly on compacts by conformal minimal embeddings (see Theorem 1.1). Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into \({\mathbb {R}}^5\) (see Theorem 1.2). One of our main tools is a Mergelyan approximation theorem for conformal minimal immersions to \({\mathbb {R}}^n\) for any \(n\ge 3\) which is also proved in the paper (see Theorem 5.3).     

Relative <Emphasis Type="Italic">t</Emphasis>-designs in binary Hamming association scheme <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, 2)

A relative t-design in the binary Hamming association schemes H(n, 2) is equivalent to a weighted regular t-wise balanced design, i.e., certain combinatorial t-design which allows different sizes of blocks and a weight function on blocks. In this paper, we study relative t-designs in H(n, 2), putting emphasis on Fisher type inequalities and the existence of tight relative t-designs. We mostly consider relative t-designs on two shells. We prove that if the weight function is constant on each shell of a relative t-design on two shells then the subset in each shell must be a combinatorial \((t-1)\)-design. This is a generalization of the result of Kageyama who proved this under the stronger assumption that the weight function is constant on the whole block set. Using this, we define tight relative t-designs for odd t, and a strong restriction on the possible parameters of tight relative t-designs in H(n, 2). We obtain a new family of such tight relative t-designs, which were unnoticed before. We will give a list of feasible parameters of such relative 3-designs with \(n \le 100\), and then we discuss the existence and/or the non-existence of such tight relative 3-designs. We also discuss feasible parameters of tight relative 4-designs on two shells in H(n, 2) with \(n \le 50\). In this study we come up with the connection on the topics of classical design theory, such as symmetric 2-designs (in particular 2-\((4u-1,2u-1,u-1)\) Hadamard designs) and Driessen’s result on the non-existence of certain 3-designs. We believe Problems 1 and 2 presented in Sect. 5.2 open a new way to study relative t-designs in H(n, 2). We conclude our paper listing several open problems.     

Hedonic coalition formation games with variable populations: core characterizations and (im)possibilities

We consider hedonic coalition formation games with variable sets of agents and extend the properties competition sensitivity and resource sensitivity (introduced by Klaus, Games Econ Behav 72:172–186, 2011, for roommate markets) to hedonic coalition formation games. Then, we show that on the domain of solvable hedonic coalition formation games, the Core is characterized by coalitional unanimity and Maskin monotonicity (see also Takamiya, Maskin monotonic coalition formation rules respecting group rights. Niigata University, Mimeo, 2010, Theorem 1). Next, we characterize the Core for solvable hedonic coalition formation games by unanimity, Maskin monotonicity, and either competition sensitivity or resource sensitivity (Corollary 2). Finally, and in contrast to roommate markets, we show that on the domain of solvable hedonic coalition formation games, there exists a solution not equal to the Core that satisfies coalitional unanimity, consistency, competition sensitivity, and resource sensitivity (Example 2).     

Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

The main object of study in this paper is the double holomorphic Eisenstein series \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) having two complex variables \(\mathbf{s}=(s_1,s_2)\) and two parameters \(\mathbf{z}= (z_1,z_2)\) which satisfies either \(\mathbf{z}\in (\mathfrak {H}^+)^2\) or \(\mathbf{z}\in (\mathfrak {H}^-)^2\), where \(\mathfrak {H}^{\pm }\) denotes the complex upper and lower half-planes, respectively. For \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\), its transformation properties and asymptotic aspects are studied when the distance \(|z_2-z_1|\) becomes both small and large under certain natural settings on the movement of \(\mathbf{z}\in (\mathfrak {H}^{\pm })^2\). Prior to the proofs our main results, a new parameter \(\eta \), which plays a pivotal role in describing our results, is introduced in connection with the difference \(z_2-z_1\). We then establish complete asymptotic expansions for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) when \(\mathbf{z}\) moves within the poly-sector either \((\mathfrak {H}^+)^2\) or \((\mathfrak {H}^-)^2\), so as to \(\eta \rightarrow 0\) through \(|\arg \eta |<\pi /2\) in the ascending order of \(\eta \) (Theorem 1). This further leads us to show that counterpart expansions exist for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in the descending order of \(\eta \) as \(\eta \rightarrow \infty \) through \(|\arg \eta |<\pi /2\) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) in closed forms for integer lattice point \(\mathbf{s}\in \mathbb {Z}^2\) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of \(\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})\) at \(\mathbf{s}\in \mathbb {Z}^2\) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases \(2\pi (1, z_j)\), \(j=1,2\). The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.     

On a rigidity result of singularities

The aim of this note is to prove, in the spirit of a rigidity result for isolated singularities of Schlessinger see Schlessinger (Invent Math 14:17–26, 1971) or also Kleiman and Landolfi (Compositio Math 23:407–434, 1971), a variant of a rigidity criterion for arbitrary singularities (Theorem 2.1 below). The proof of this result does not use Schlessinger’s Deformation Theory [Schlessinger (Trans Am Math Soc 130:208–222, 1968) and Schlessinger (Invent Math 14:17–26, 1971)]. Instead it makes use of Local Grothendieck-Lefschetz Theory, see (Grothendieck 1968, Éxposé 9, Proposition 1.4, page 106) and a Lemma of Zariski, see (Zariski, Am J Math 87:507–536, 1965, Lemma 4, page 526). I hope that this proof, although works only in characteristic zero, might also have some interest in its own.     

Root and critical point behaviors of certain sums of polynomials

It is known that no two roots of the polynomial equation

$$\begin{aligned} \begin{aligned} \prod _{j=1}^n (x-r_j) + \prod _{j=1}^n (x+r_j) =0, \end{aligned} \end{aligned}$$

(1)

where \(0 < r_1 \le r_2 \le \cdots \le r_n\), can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for \(0< h< r_k\), the roots of

$$\begin{aligned} (x-r_k-h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^n(x-r_j) + (x+r_k+h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array} }^n (x+r_j) = 0 \end{aligned}$$

(2)

and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.

     

A new look at a smart polling model

We analyze a Markovian smart polling model, which is a special case of the smart polling models studied in the work of Boon et al. (Queueing Syst 66:239–274, 2010), as well as a generalization of the gated M / M / 1 queue considered in Resing and Rietman (Stat Neerlandica 58:97–110, 2004). We first derive tractable expressions for the stationary distribution (when it exists) as well as the Laplace transforms of the transition functions of this polling model—while further assuming the system is empty at time zero—and we also present simple necessary and sufficient conditions for ergodicity of the smart polling model. Finally, we conclude the paper by briefly explaining how these techniques can be used to study other interesting variants of this smart polling model.     

A Generalization of the Univalence on the Boundary Theorem with Applications to Sobolev Mappings

We give a generalization of a known theorem from classical complex analysis, namely the univalence on the boundary theorem. We apply this result to obtain some univalence conditions for Sobolev mappings \(f\in C({\overline{D}},{\mathbb {R}}^n)\bigcap W_{loc}^{1,q}(D,{\mathbb {R}}^n)\) which are injective on \(\partial D\), in connection with a known result of Ball from (Proc R Soc Edinb Sect A 88(3–4):315–328, 1981) modeling nonlinear elasticity.     

A remark on the result of Radu Miculescu and Alexandru Mihail

In this note, we show that the assumption of continuity considered in the recent result of Miculescu and Mihail (J Fixed Point Theory Appl, doi: 10.1007/s11784-017-0411-7, 2017) can be relaxed further. We also observe that the power convex contraction introduced by Miculescu and Mihail (see condition (1.1)) provides one more solution to the open question of Rhoades (Contemp Math 72:233–245, 1988] regarding existence of a contractive definition which is strong enough to generate a fixed point but does not force the mapping to be continuous at the fixed point.     

Comments on: Queueing models for the analysis of communication systems

The paper under discussion is a well-written exposition on the performance modeling of communication systems by discrete-time queueing systems, and their analysis. It basically consists of two parts: a review of the literature, focusing on the modelling of information streams and on scheduling disciplines (Sects. 2, 3), and a demonstration of some key methods for the analysis of discrete-time queueing systems, focusing on a particular two-class discrete-time queue with correlated arrivals and two priority classes (Sects. 4–6). In Sect. 1 of the present note, we make some introductory comments. In Sect. 2, realizing that the literature review in Bruneel et al. (TOP, 2014) is authoritative and extensive, we focus on a few adjacent topics which fall outside the scope of Bruneel et al. (TOP, 2014) but which in our view may also be of some interest. Finally, in Sect. 3, we discuss the analysis in Sects. 4–6 of Bruneel et al. (TOP, 2014).