Subharmonic Solutions with Prescribed Minimal Period of a Forced Pendulum Equation with Impulses

We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014).     

On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes

Stochastic convexity and its applications are very important in mathematics and probability (Aequationes Mathematicae 20:184–197, 1980). There are two well-known inequalities for convex stochastic processes: Jensen’s inequality and Hermite–Hadamard’s inequality. Recently, Hafiz (Stoch Anal Appl 22:507–523, 2004) has provided fractional calculus for some stochastic processes. The problem is how to formulate these inequalities for stochastic processes in the class of fractional calculus and that is what is done in this paper. Our results generalize the corresponding ones in the literature.     

Two Kneser–Poulsen-type inequalities in planes of constant curvature

We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase. This generalizes the theorem on the monotonicity of the perimeter of the convex hull of a finite set under contractions, proved in the Euclidean plane by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3]. We also prove that the area of the intersection of finitely many disks in the hyperbolic plane does not decrease after such a contractive rearrangement. The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].     

A viscosity iterative technique for split variational inclusion and fixed point problems between a Hilbert space and a Banach space

The main purpose of this paper is to introduce a viscosity-type iterative algorithm for approximating a common solution of a split variational inclusion problem and a fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common solution of a split variational inclusion problem and a fixed point problem for a multivalued quasi-nonexpansive mapping between a Hilbert space and a Banach space. Furthermore, we applied our results to study a split convex minimization problem. Also, a numerical example of our result is given. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13, 759–775, 2012), Moudafi (J. Optim. Theory Appl. 150, 275–283, 2011), Takahashi and Yao (Fixed Point Theory Appl. 2015, 87, 2015), and a host of other important results in this direction.     

On the congruence subgroup problem for branch groups

We prove here an energy estimate for the Cauchy problem for hyperbolic equations with double characteristic, which contains both effectively and non-effectively points (see L. Hörmander [3] and R. Melrose [8]) in a unique framework.     

Mixed-integer quadratic programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73–77 [17]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299–304 [1], von zur Gathen and Sieveking in Proc Am Math Soc 72:155–158 [18], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161–172. Springer [9], Papadimitriou in J Assoc Comput Mach 28:765–768 [15]).     

A Geometric Proof of Bourgain’s $$L^2$$ Estimate of Maximal Operators Along Analytic Vector Fields

Bourgain (A remark on the maximal function associated to an analytic vector field. Analysis at Urbana, Cambridge University Press, Cambridge, 1989) proved that the maximal operator associated with an analytic vector field is bounded on \(L^2\). In the present paper, we give a geometric proof of Bourgain’s result by using the tools developed by Lacey and Li in (Trans Am Math Soc 358(9):4099–4117, 2006) and (Mem Am Math Soc 205 (965):viii+72, 2010).     

Refinements of mean-square stochastic integral inequalities on convex stochastic processes

Recently, in the class of convex stochastic processes, Kotrys (Aequat Math 83:143–151, 2012; Aequat Math 86:91–98, 2013) proposed upper and lower bounds of mean-square stochastic integrals by using Hermite–Hadamard inequality. This paper shows that these bounds can be refined. Our results extend and refine the corresponding ones in the literature. Finally, an open problem for further investigations is given.     

The Spectral Theorem for Unitary Operators Based on the S-Spectrum

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [32], [33], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem.     

Instanton Floer homology and contact structures

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or without boundary—defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifold to properties of its Stein fillings. Our construction is inspired by a reformulation of a similar invariant in the monopole Floer setting defined by Baldwin and Sivek (arXiv:1403.1930, 2014).     

Agler-Commutant Lifting on an Annulus

This note presents a commutant lifting theorem (CLT) of Agler type for the annulus \({\mathbb A}\) . Here the relevant set of test functions are the minimal inner functions on \({\mathbb A}\) —those analytic functions on \({\mathbb A}\) which are unimodular on the boundary and have exactly two zeros in \({\mathbb A}\) —and the model space is determined by a distinguished member of the Sarason family of kernels over \({\mathbb A}\) . The ideas and constructions borrow freely from the CLT of Ball et al. (Indiana Univ Math J 48(2):653–675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) for the polydisc, and Ambrozie and Eschmeier (A commutant lifting theorem on analytic polyhedra. Topological algebras, their applications, and related topics, 83108, Banach Center Publications, vol 67. Polish Academy of Sciences, Warsaw, 2005) for the ball in \({\mathbb C^n}\) , as well as generalizations of the de Branges–Rovnyak construction like found in Agler (On the representation of certain holomorphic functions defined on a polydisc. Topics in operator theory: Ernst D. Hellinger memorial volume, operator theory: advances and applications, vol 48. Birkhäuser, Basel, pp 47–66, 1990) and Ambrozie et al. (J Oper Theory 47(2):287–302, 2002). It offers a template for extending the result in McCullough and Sultanic (Complex Anal Oper Theory 1(4):581–620, 2007) to infinitely many test functions. Among the needed new ingredients is the formulation of the factorization implicit in the statement of the results in Ball et al. (Indiana Univ Math J 48(2):653–675, 1999) and Archer (Unitary dilations of commuting contractions. PhD thesis, University of Newcastle, 2004) and McCullough and Sultanic (Complex Anal Oper Theory 1(4):581–620, 2007) in terms of certain functional Hilbert spaces of Hilbert space valued functions.     

On the bidomain problem with FitzHugh–Nagumo transport

The bidomain problem with FitzHugh–Nagumo transport is studied in the \(L_p\!-\!L_q\)-framework. Reformulating the problem as a semilinear evolution equation, local well-posedness is proved in strong as well as in weak settings. We obtain solvability for initial data in the critical spaces of the problem. For dimension \(d\le 4\), by means of energy estimates and a recent result of Serrin type, global existence is shown. Finally, stability of spatially constant equilibria is investigated, to the result that the stability properties of such equilibria are the same as for the classical FitzHugh–Nagumo system in ODE’s. These properties of the bidomain equations are obtained combining recent results on the bidomain operator (Hieber and Prüss in Theory for the bidomain operator, submitted, 2018), on critical spaces for parabolic evolution equations (Prüss et al in J Differ Equ 264:2028–2074, 2018), and qualitative theory of evolution equations.     

On some problems of tensor calculus. I

Basic definitions of linear algebra and functional analysis are given. In particular, the definitions of a semigroup, group, ring, field, module, and linear space are given [1, 2, 3, 6]. A local theorem on the existence of homeomorphisms is stated. Definitions of the inner r-product, local inner product of tensors whose rank is not less than r, and of local norm of a tensor [22] are also given. Definitions are given and basic theorems and propositions are stated and proved concerning the linear dependence and independence of a system of tensors of any rank. Moreover, definitions and proofs of some theorems connected with orthogonal and biorthonormal tensor systems are given. The definition of a multiplicative basis (multibasis) is given and ways of construction bases of modules using bases of modules of smaller dimensions. In this connection, several theorems are stated and proved. Tensor modules of even orders and problems on finding eigenvalues and eigentensors of any even rank are studied in more detail than in [22]. Canonical representations of a tensor of any even rank are given. It is worth while to note that it was studied by the Soviet scientist I. N. Vekua, and an analogous problem for the elasticity modulus tensor was considered by the Polish scientist Ya. Rikhlevskii in 1983–1984.     

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of \(\mathbb R^m\) were studied in Aliprantis et al. (Macroecon Dyn 10(1):77–99, 2006), Aliprantis et al. (J Econometrics 136(2):431–456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr\(\ddot{\mathrm{a}}\)ge Algebra Geom 43:297–302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in \(C(\mathbb R^m)\), while piecewise affine functions are sequentially order dense in \(C(\mathbb R^m)\). This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)     

Set-valued loss-based risk measures

In this paper, we introduce a new class of set-valued risk measures, named set-valued convex loss-based risk measures. Representation results are provided. This new class can be considered as a set-valued extension of those introduced by Cont et al. (Stat Risk Model Appl Finance Insur 30(2):133–167, 2013) and Chen et al. (Positivity, 2017). Finally, examples are also given to illustrate the set-valued convex loss-based risk measures.     

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Let \({\alpha}\) be a bounded linear operator in a Banach space \({\mathbb{X}}\), and let A be a closed operator in this space. Suppose that for \({\Phi_1, \Phi_2}\) mapping D(A) to another Banach space \({\mathbb{Y}}\), \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) are generators of strongly continuous semigroups in \({\mathbb{X}}\). Assume finally that \({A_{|{\rm ker}\, \Phi_\text{a}}}\), where \({\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}\) and \({\beta = I_\mathbb{X} - \alpha}\), is a generator also. In the case where \({\mathbb{X}}\) is an L ¹-type space, and \({\alpha}\) is an operator of multiplication by a function \({0 \le \alpha \le 1}\), it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of \({A_{|{\rm ker}\, \Phi_1}}\) with probability \({\alpha (x)}\) and is subject to the rules of \({A_{|{\rm ker}\, \Phi_2}}\) with probability \({\beta (x)= 1 - \alpha (x)}\). We provide an approximation (a singular perturbation) of the semigroup generated by \({A_{|{\rm ker}\, \Phi_\text{a}}}\) by semigroups built from those generated by \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.     

A mass transportation approach for Sobolev inequalities in variable exponent spaces

In this paper we provide a proof of the Sobolev–Poincaré inequality for variable exponent spaces by means of mass transportation methods, in the spirit of Cordero-Erausquin et al. (Adv Math 182(2):307–332, 2004). The importance of this approach is that the method is flexible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result of Fan (J Math Anal Appl 339(2):1395–1412, 2008) by obtaining an explicit dependence of the exponent in the constant.     

Asymptotic Expansion for the Distribution Density Function of the Compound Poisson Process in Large Deviations

The paper is devoted to obtaining the asymptotic expansion and determination of the structure of the remainder term taking into consideration large deviations in the Cramér zone for the distribution density function of the standardized compound Poisson process. Following Deltuvien? and Saulis (Acta Appl Math 78:87–97, 2003. doi: 10.1023/A:1025783905023; Lith Math J 41:620–625, 2001) and Saulis and Statulevi?ius [Limit theorems for large deviations. Mathematics and its applications (Soviet Series), vol 73, pp 154–187, Kluwer, Dordrecht, 1991], the solution to the problem is achieved by first using a general lemma presented by Saulis (see Lemma 6.1 in Saulis and Statulevi?ius 1991, p. 154) on the asymptotic expansion for the density function of an arbitrary random variable with zero mean and unit variance and combining methods for cumulants and characteristic functions. By taking into consideration the large deviations in the Cramér zone for the density function of the standardized compound Poisson process, the result for the asymptotic expansion extends the asymptotic expansions for the density function of the sums of non-random number of summands (Deltuvien? and Saulis 2003, 2001).     

<Emphasis Type="Italic">l</Emphasis><Subscript>1</Subscript>-<Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript> regularization of split feasibility problems

Numerous problems in signal processing and imaging, statistical learning and data mining, or computer vision can be formulated as optimization problems which consist in minimizing a sum of convex functions, not necessarily differentiable, possibly composed with linear operators and that in turn can be transformed to split feasibility problems (SFP); see for example Censor and Elfving (Numer. Algorithms 8, 221–239 1994). Each function is typically either a data fidelity term or a regularization term enforcing some properties on the solution; see for example Chaux et al. (SIAM J. Imag. Sci. 2, 730–762 2009) and references therein. In this paper, we are interested in split feasibility problems which can be seen as a general form of Q-Lasso introduced in Alghamdi et al. (2013) that extended the well-known Lasso of Tibshirani (J. R. Stat. Soc. Ser. B 58, 267–288 1996). Q is a closed convex subset of a Euclidean m-space, for some integer m ≥ 1, that can be interpreted as the set of errors within given tolerance level when linear measurements are taken to recover a signal/image via the Lasso. Inspired by recent works by Lou and Yan (2016), Xu (IEEE Trans. Neural Netw. Learn. Syst. 23, 1013–1027 2012), we are interested in a nonconvex regularization of SFP and propose three split algorithms for solving this general case. The first one is based on the DC (difference of convex) algorithm (DCA) introduced by Pham Dinh Tao, the second one is nothing else than the celebrate forward-backward algorithm, and the third one uses a method introduced by Mine and Fukushima. It is worth mentioning that the SFP model a number of applied problems arising from signal/image processing and specially optimization problems for intensity-modulated radiation therapy (IMRT) treatment planning; see for example Censor et al. (Phys. Med. Biol. 51, 2353–2365, 2006).