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.     

Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. \(\sqrt{\mu}\) and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.     

Reliability problems in multiple path-shaped facility location on networks

In this paper we study a location problem on networks that combines three important issues: (1) it considers that facilities are extensive, (2) it handles simultaneously the location of more than one facility, and (3) it incorporates reliability aspects related to the fact that facilities may fail. The problem consists of locating two path-shaped facilities minimizing the expected service cost in the long run, assuming that paths may become unavailable and their failure probabilities are known in advance. We discuss several aspects of the computational complexity of problems of locating two or more reliable paths on graphs, showing that multifacility path location–with and without reliability issues–is a difficult problem even for 2 facilities and on very special classes of graphs. In view of this, we focus on trees and provide a polynomial time algorithm that solves the 2 unreliable path location problem on tree networks in $O\left(n^2\right)$ time, where $n$ is the number of vertices.     

On the generalization of the Salem test

The assertion that the Salem test [5] for the uniform convergence of a trigonometric Fourier series is improvable, is proved. In particular, an example of a continuous function, which does not fulfill the condition of the Salem test but satisfies the condition of the generalized Salem test [10], is constructed.Besides, the theorem which improves Golubov’s [3,4] result for continuous functions of two variables, is given.     

On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9].     

A parametric simplex algorithm for linear vector optimization problems

In this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (the Evans–Steuer) algorithm (Math Program 5(1):54–72, 1973). Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne (Vector optimization with infimum and supremum. Springer, Berlin, 2011), that is, it finds a subset of efficient solutions that allows to generate the whole efficient frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczyński and Vanderbei (Econometrica 71(4):1287–1297, 2003). The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm [Benson’s algorithm in Hamel et al. (J Global Optim 59(4):811–836, 2014)], that also provides a solution in the sense of Löhne (2011), and with the Evans–Steuer algorithm (1973). The results show that for non-degenerate problems the proposed algorithm outperforms Benson’s algorithm and is on par with the Evans–Steuer algorithm. For highly degenerate problems Benson’s algorithm (Hamel et al. 2014) outperforms the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than the Evans–Steuer algorithm.     

Set-valued Brownian motion

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach spaces X. The present paper is an application of the paper (Labuschagne et al. in Quaest Math 30(3):285–308, 2007) in which an embedding result is obtained which considers also the ordered structure of the family of compact convex subsets of a Banach space X and of Grobler and Labuschagne (J Math Anal Appl 423(1):797–819, 2015; J Math Anal Appl 423(1):820–833, 2015) in which these processes are considered in f-algebras. Moreover, in the space of continuous functions defined on a Stonian space, a direct Levy’s result follows.     

The Calabi metric for the space of Kähler metrics

Given any closed Kähler manifold we define, following an idea by Calabi (Bull. Am. Math. Soc. 60:167–168, 1954), a Riemannian metric on the space of Kähler metrics regarded as an infinite dimensional manifold. We prove several geometrical features of the resulting space, some of which we think were already known to Calabi. In particular, the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the Cauchy and the Dirichlet problems for the geodesic equation are given.     

Generalized Control Systems in the Space of Probability Measures

In this paper we formulate a time-optimal control problem in the space of probability measures. The main motivation is to face situations in finite-dimensional control systems evolving deterministically where the initial position of the controlled particle is not exactly known, but can be expressed by a probability measure on \(\mathbb {R}^{d}\). We propose for this problem a generalized version of some concepts from classical control theory in finite dimensional systems (namely, target set, dynamic, minimum time function...) and formulate an Hamilton-Jacobi-Bellman equation in the space of probability measures solved by the generalized minimum time function, by extending a concept of approximate viscosity sub/superdifferential in the space of probability measures, originally introduced by Cardaliaguet-Quincampoix in Cardaliaguet and Quincampoix (Int. Game Theor. Rev. 10, 1–16, 2008). We prove also some representation results linking the classical concept to the corresponding generalized ones. The main tool used is a superposition principle, proved by Ambrosio, Gigli and Savaré in Ambrosio et al. [3], which provides a probabilistic representation of the solution of the continuity equation as a weighted superposition of absolutely continuous solutions of the characteristic system.     

Efficient importance sampling in low dimensions using affine arithmetic

Despite the development of sophisticated techniques such as sequential Monte Carlo (Del Moral et al. in J R Stat Soc Ser B 68(3):411–436, 2006), importance sampling (IS) remains an important Monte Carlo method for low dimensional target distributions (Chopin and Ridgway in Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation, 32:64–87, 2017). This paper describes a new technique for constructing proposal distributions for IS, using affine arithmetic (de Figueiredo and Stolfi in Numer Algorithms 37(1–4):147–158, 2004). This work builds on the Moore rejection sampler (Sainudiin in Machine interval experiments, Cornell University, Ithaca, 2005; Sainudiin and York in Algorithms Mol Biol 4(1):1, 2009) to which we provide a comparison.     

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).     

Improved local convergence analysis of the Gauss–Newton method under a majorant condition

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97–118, 2008), Chen and Li (Appl Math Comput 170:686–705, 2005), Chen and Li (Appl Math Comput 324:1381–1394, 2006), Ferreira (J Comput Appl Math 235:1515–1522, 2011), Ferreira and Gonçalves (Comput Optim Appl 48:1–21, 2011), Ferreira and Gonçalves (J Complex 27(1):111–125, 2011), Li et al. (J Complex 26:268–295, 2010), Li et al. (Comput Optim Appl 47:1057–1067, 2004), Proinov (J Complex 25:38–62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines: new directions in pure, applied and computational mathematics 185–196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123–134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.     

On the Semicircular Law of Large-Dimensional Random Quaternion Matrices

It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erd?s in Russ Math Surv 66(3):507–626, 2011; Erd?s in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erd?s et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.     

Depth first search in claw-free graphs

Optimization problems concerning the vertex degrees of spanning trees of connected graphs play an extremely important role in network design. Minimizing the number of leaves of the spanning trees is NP-hard, since it is a generalization of the problem of finding a hamiltonian path of the graph. Moreover, Lu and Ravi (The power of local optimization: approximation algorithms for maximum-leaf spanning tree (DRAFT), CS-96-05, Department of Computer Science, Brown University, Providence, 1996) showed that this problem does not even have a constant factor approximation, unless \(\hbox {P}=\hbox {NP}\), thus properties that guarantee the existence of a spanning tree with a small number of leaves are of special importance. In this paper we are dealing with finding spanning trees with few leaves in claw-free graphs. We prove that all claw-free graphs have a DFS-tree, such that the leaves different from the root have no common neighbour, generalizing a theorem of Kano et al. (Ars Combin 103:137–154, 2012). The result also implies a strengthening of a result of Ainouche et al. (Ars Combin 29C:110–121, 1990).     

Absolute Cesàro Series Spaces and Matrix Operators

In this paper we derive a series space \(\vert C_{\lambda,\mu} \vert _{k}\) using the well known absolute Cesàro summability \(\vert C_{\lambda,\mu} \vert _{k}\) of Das (Proc. Camb. Philol. Soc. 67:321–326, 1970), compute its \(\beta\)-dual, give some algebraic and topological properties, and characterize some matrix operators defined on that space. So we generalize some results of Bosanquet (J. Lond. Math. Soc. 20:39–48, 1945), Flett (Proc. Lond. Math. Soc. 7:113–141, 1957), Mehdi (Proc. Lond. Math. Soc. (3)10:180–199, 1960), Mazhar (Tohoku Math. J. 23:433–451, 1971), Orhan and Sar?göl (Rocky Mt. J. Math. 23(3):1091–1097, 1993) and Sar?göl (Commun. Math. Appl. 7(1):11–22, 2016; Math. Comput. Model. 55:1763–1769, 2012).     

Symmetry axioms and perceived ambiguity

Since at least de Finetti (Annales de l’Institut Henri Poincare 7:1–68, 1937), preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper, we explore (1) the relationship between the symmetry assumption of Klibanoff et al. (KMS) (Econometrica 82:1945–1978, 2014) and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures, shown by KMS (2014) to reflect only perceived ambiguity, and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. (J Econ Theory 118:133–173, 2004), Nehring (Ambiguity in the context of probabilistic beliefs, working paper, 2001, Bernoulli without Bayes: a theory of utility-sophisticated preference, working paper, 2007) and Ghirardato and Siniscalchi (A more robust definition of multiple priors, working paper, 2007, Econometrica 80:2827–2847, 2012). This Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature [including that in KMS (2014)] are equivalent. In KMS (2014), we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. Here, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be the same under symmetry. This condition is rather stringent. Only when it is satisfied may the Bewley set be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.     

Prediction of catastrophes in space over time

Predicting rare events, such as high level up-crossings, for spatio-temporal processes plays an important role in the analysis of the occurrence and impact of potential catastrophes in, for example, environmental settings. Designing a system which predicts these events with high probability, but with few false alarms, is clearly desirable. In this paper an optimal alarm system in space over time is introduced and studied in detail. These results generalize those obtained by de Maré (Ann. Probab. 8, 841–850, 1980) and Lindgren (Ann. Probab. 8, 775–792, 1980, Ann. Probab. 13, 804–824, 1985) for stationary stochastic processes evolving in continuous time and are applied here to stationary Gaussian random fields.     

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.     

On manifolds of tensors of fixed TT-rank

Recently, the format of TT tensors (Hackbusch and Kühn in J Fourier Anal Appl 15:706–722, 2009; Oseledets in SIAM J Sci Comput 2009, submitted; Oseledets and Tyrtyshnikov in SIAM J Sci Comput 31:5, 2009; Oseledets and Tyrtyshnikov in Linear Algebra Appl 2009, submitted) has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor \({U \in \mathbb{R}^{n_1\times \cdots\times n_d}}\) and for the manifold of tensors of TT-rank \({\underline{r}}\) . As a first result, we prove that the TT (or compression) ranks r _i of a tensor U are unique and equal to the respective separation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that the set \({\mathbb{T}}\) of TT tensors of fixed rank \({\underline{r}}\) locally forms an embedded manifold in \({\mathbb{R}^{n_1\times\cdots\times n_d}}\) , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices (Conte and Lubich in M2AN 44:759, 2010), we introduce certain gauge conditions to obtain a unique representation of the tangent space \({\mathcal{T}_U\mathbb{T}}\) of \({\mathbb{T}}\) and deduce a local parametrization of the TT manifold. The parametrisation of \({\mathcal{T}_{U}\mathbb{T}}\) is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems (Lubich in From quantum to classical molecular dynamics: reduced methods and numerical analysis, 2008). We conclude with remarks on those applications and present some numerical examples.     

A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

In this short note, we generalized an energy estimate due to Malchiodi–Martinazzi (J Eur Math Soc 16:893–908, 2014) and Mancini–Martinazzi (Calc Var 56:94, 2017). As an application, we used it to reprove existence of extremals for Trudinger–Moser inequalities of Adimurthi–Druet type on the unit disc. Such existence problems in general cases had been considered by Yang  (Trans Am Math Soc 359:5761–5776, 2007; J Differ Equ 258:3161–3193, 2015) and Lu–Yang (Discrete Contin Dyn Syst 25:963–979, 2009) by using another method.