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).     

Dirichlet series with functional equations and arithmetical identities

Exploiting the functional equation of Hecke-type associated with a function satisfying a modular relation with a residual function as developed in Bochner (J Indian Math Soc 16:99–102, 1952), Chandrasekharan and Narasimhan (Ann Math 74:1–23, 1961) derived the equivalence of the functional equation to two arithmetical identities. Hawkins and Knopp (Contemp Math 143:451–475, 1993) showed the equivalence of the functional equation to modular integrals with rational period functions of weight 2k, \(k \in \mathbb {Z}^+\) on the theta group \(\Gamma _\vartheta \). The aim of the current work is to show that results analogous to those of Chandrasekharan and Narasimhan can be developed in the Hawkins and Knopp context, but with respect to the full modular group \(\Gamma (1)\), rather than the theta group \(\Gamma _\vartheta \).     

On geometric properties of the generating function for the Ramanujan sequence

The Ramanujan sequence \(\{\theta _{n}\}_{n \ge 0}\), defined as \(\theta _{0}= {1}/{2}\), \({n^{n}} \theta _{n}/{n !} = {e^{n}}/{2} - \sum _{k=0}^{n-1} {n^{k}}/{k !}\, \), \(n \ge 1\), has been studied on many occasions and in many different contexts. Adell and Jodrá (Ramanujan J 16:1–5, 2008) and Koumandos (Ramanujan J 30:447–459, 2013) showed, respectively, that the sequences \(\{\theta _{n}\}_{n \ge 0}\) and \(\{4/135 - n \cdot (\theta _{n}- 1/3 )\}_{n \ge 0}\) are completely monotone. In the present paper, we establish that the sequence \(\{(n+1) (\theta _{n}- 1/3 )\}_{n \ge 0}\) is also completely monotone. Furthermore, we prove that the analytic function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} / n^{\alpha }\) is universally starlike for every \(\alpha \ge 1\) in the slit domain \(\mathbb {C}\setminus [1,\infty )\). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by Ruscheweyh et al. (Israel J Math 171:285–304, 2009), namely that the function \((\theta _{1}- 1/3 )^{-1}\sum _{n=1}^{\infty } (\theta _{n}- 1/3 ) z^{n} \) is universally convex.     

Rulesets for Beatty games

We describe a ruleset for a 2-pile subtraction game with P-positions \(\{(\lfloor \alpha n \rfloor ,\lfloor \beta n \rfloor ) : n \in \mathbb Z_{\ge 0} \}\) for any irrational \(1< \alpha < 2\), and \(\beta \) such that \(1/\alpha +1/\beta = 1\). We determine the \(\alpha \)’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.     

On the Rationality of the Spectrum

Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.     

Improved Moser–Trudinger inequality of Tintarev type in dimension <Emphasis Type="Italic">n</Emphasis> and the existence of its extremal functions

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$

for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)

     

Global Solutions to a Nonlocal Fisher-KPP Type Problem

We consider a nonlocal Fisher-KPP reaction-diffusion model arising from population dynamics, consisting of a certain type reaction term \(u^{\alpha} ( 1-\int_{\varOmega}u^{\beta}dx ) \), where \(\varOmega\) is a bounded domain in \(\mathbb{R}^{n}(n \ge1)\). The energy method is applied to prove the global existence of the solutions and the results show that the long time behavior of solutions heavily depends on the choice of \(\alpha\), \(\beta\). More precisely, for \(1 \le\alpha <1+ ( 1-2/p ) \beta\), where \(p\) is the exponent from the Sobolev inequality, the problem has a unique global solution. Particularly, in the case of \(n \ge3\) and \(\beta=1\), \(\alpha<1+2/n\) is the known Fujita exponent (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966). Comparing to Fujita equation (Fujita in J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 13:109–124, 1966), this paper will give an opposite result to our nonlocal problem.     

Stability of Dirac concentrations in an integro-PDE model for evolution of dispersal

We consider an integro-PDE model from evolutionary biology. The solution \(u_\epsilon (x,\alpha )\) is structured by two variables \(x\in D \subset {\mathbb {R}}^k\) and \(\alpha \in (\underline{\alpha },{\overline{\alpha }})\subset \subset {\mathbb {R}}_+\). The diffusion coefficient in the x direction depends on \(\alpha \) and the diffusion coefficient in the \(\alpha \) direction is a constant \(\epsilon ^2\). A special feature of this model is the appearance of the integral \({\hat{u}}_\epsilon (x)\) of the solution in the \(\alpha \) variable, which can be viewed as an infinite dimensional parameter of the problem. In a previous work, the existence of a steady state that exhibits Dirac-concentration in one of the variables yet remains regular in the other variables was proved independently by Lam and Lou (J Funct Anal 272:1755–1790, 2017) and by Perthame and Souganidis (Math Model Nat Phenom 11:154–166, 2016). In this paper, we tackle the long-time dynamics of solutions. When the environment function is non-constant, we show that the steady state is linearly stable by considering the corresponding nonlocal eigenvalue problem. Uniqueness of steady state is obtained from the stability result via a degree argument. When the environment function is a constant, the global asymptotic stability result is obtained. This problem can be regarded as a competition of infinitely many species parameterized by \(\alpha \). As with the competition model for three or more species, the integro-PDE model does not generate a monotone dynamical system so that it is necessary to consider all (real or complex) eigenvalues in determining its linear stability.     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Let V be a locally bounded measurable function on \({\mathbb {R}}^d\) such that \(\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x\) is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form

$$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$

Taking \(\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}\) with \(0<\alpha <2\) and \(\delta \geqslant 0\), we get new conclusions for (exponentially) tempered fractional Dirichlet forms, which not only complete our recent work (Chen and Wang in Stoch Process Their Appl 124:123–153, 2014; Wang and Wang in J Theor Probab 28:423–448, 2015), but also improve the main result in Mouhot et al. (J Math Pures Appl 95:72–84, 2011).

     

Weakly horospherically convex hypersurfaces in hyperbolic space

In Bonini et al. (Adv Math 280:506–548, 2015), the authors develop a global correspondence between immersed weakly horospherically convex hypersurfaces \(\phi :M^n \rightarrow \mathbb {H}^{n+1}\) and a class of conformal metrics on domains of the round sphere \(\mathbb {S}^n\). Some of the key aspects of the correspondence and its consequences have dimensional restrictions \(n\ge 3\) due to the reliance on an analytic proposition from Chang et al. (Int Math Res Not 2004(4):185–209, 2004) concerning the asymptotic behavior of conformal factors of conformal metrics on domains of \(\mathbb {S}^n\). In this paper, we prove a new lemma about the asymptotic behavior of a functional combining the gradient of the conformal factor and itself, which allows us to extend the global correspondence and embeddedness theorems of Bonini et al. (2015) to all dimensions \(n\ge 2\) in a unified way. In the case of a single point boundary \(\partial _{\infty }\phi (M)=\{x\} \subset \mathbb {S}^n\), we improve these results in one direction. As an immediate consequence of this improvement and the work on elliptic problems in Bonini et al. (2015), we have a new, stronger Bernstein type theorem. Moreover, we are able to extend the Liouville and Delaunay type theorems from Bonini et al. (2015) to the case of surfaces in \(\mathbb {H}^{3}\).     

Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation \({{\mathrm{div}}}(\sigma \nabla u) = 0\) in dimension two, in the case when the essential range of \(\sigma \) consists of only two elliptic matrices, i.e., \(\sigma \in \{\sigma _1, \sigma _2\}\) a.e. in \(\Omega \). In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices \(\sigma _1\) and \(\sigma _2\), exponents \(p_{\sigma _1,\sigma _2}\in (2,+\infty )\) and \(q_{\sigma _1,\sigma _2}\in (1,2)\) have been found so that if \(u\in W^{1,q_{\sigma _1,\sigma _2}}(\Omega )\) is solution to the elliptic equation then \(\nabla u\in L^{p_{\sigma _1,\sigma _2}}_{\mathrm{weak}}(\Omega )\) and the optimality of the upper exponent \(p_{\sigma _1,\sigma _2}\) has been proved. In this paper we complement the above result by proving the optimality of the lower exponent \(q_{\sigma _1,\sigma _2}\). Precisely, we show that for every arbitrarily small \(\delta \), one can find a particular microgeometry, i.e., an arrangement of the sets \(\sigma ^{-1}(\sigma _1)\) and \(\sigma ^{-1}(\sigma _2)\), for which there exists a solution u to the corresponding elliptic equation such that \(\nabla u \in L^{q_{\sigma _1,\sigma _2}-\delta }\), but \(\nabla u \notin L^{q_{\sigma _1,\sigma _2}}\). The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.     

Small Littlewood–Richardson coefficients

We develop structural insights into the Littlewood–Richardson graph, whose number of vertices equals the Littlewood–Richardson coefficient \(c_{\lambda ,\mu }^{\nu }\) for given partitions \(\lambda \), \(\mu \), and \(\nu \). This graph was first introduced in Bürgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639–1681, 2013), where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood–Richardson coefficient: We design an algorithm for the exact computation of \(c_{\lambda ,\mu }^{\nu }\) with running time \(\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )\), where \(\lambda \), \(\mu \), and \(\nu \) are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge t\) whose running time is \(\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )\). Even the existence of a polynomial-time algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge 2\) is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that \(c_{\lambda ,\mu }^{\nu }=2\) implies \(c_{M\lambda ,M\mu }^{M\nu } = M+1\) for all \(M \in \mathbb {N}\). Here, the stretching of partitions is defined componentwise.     

On Some Properties of a Class of Fractional Stochastic Heat Equations

We consider nonlinear parabolic stochastic equations of the form \(\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }\) on the ball \(B(0,\,R)\), where \(\dot{\xi }\) denotes some Gaussian noise and \(\sigma \) is Lipschitz continuous. Here \(\mathcal {L}\) corresponds to a symmetric \(\alpha \)-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on \(\sigma \), we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).     

New constructions of permutation polynomials of the form $$x^rh\left( x^{q-1}\right) $$ over $${\mathbb F}_{q^2}$$Fq2

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\), where \(q=2^k\), \(h(x)=1+x^s+x^t\) and \(r, k>0, s, t\) are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing the corresponding fractional polynomials permute a smaller set \(\mu _{q+1}\), where \(\mu _{q+1}:=\{x\in \mathbb {F}_{q^2} : x^{q+1}=1\}\). Motivated by these results, we characterize the permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\) such that \(h(x)\in {\mathbb F}_q[x]\) is arbitrary and q is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over \({\mathbb F}_{q^2}\) into that of showing permutations over a small subset S of a proper subfield \({\mathbb F}_{q}\), which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form \(x^rh\left( x^{q-1}\right) \) over \({\mathbb F}_{q^2}\). Moreover, we can explain most of the known permutation trinomials, which are in Ding et al. (SIAM J Discret Math 29:79–92, 2015), Gupta and Sharma (Finite Fields Appl 41:89–96, 2016), Li and Helleseth (Cryptogr Commun 9:693–705, 2017), Li et al. (New permutation trinomials constructed from fractional polynomials, arXiv: 1605.06216v1, 2016), Li et al. (Finite Fields Appl 43:69–85, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017) over finite field with even characteristic.     

Taut contact circles and bi-contact metric structures on three-manifolds

Geiges and Gonzalo (Invent. Math. 121:147–209 1995, J. Differ. Geom. 46:236–286 1997, Acta. Math. Vietnam 38:145–164 2013) introduced and studied the notion of taut contact circle on a three-manifold. In this paper, we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. We characterize the existence of a taut contact metric circle and of a bi-contact metric structure. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, a simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: SU(2), \({\widetilde{SL}}(2,{\mathbb {R}})\), \({\widetilde{E}}(2)\), E(1, 1). Moreover, we obtain a classification of three-manifolds which admit a Cartan structure \((\eta _1,\eta _2)\) with the so-called Webster function \({\mathcal {W}}\) constant along the flow of \(\xi _1\) (equivalently \(\xi _2\)). Finally, we study the metric cone, i.e., the symplectization, of a bi-contact metric three-manifold. In particular, the notion of bi-contact metric structure is related to the notions of conformal symplectic couple (in the sense of Geiges (Duke Math. J. 85:701–711 1996)) and symplectic pair (in the sense of Bande and Kotschick (Trans. Am. Math. Soc. 358(4):1643–1655 2005)).     

Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Let \(\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\) be the n-dimensional Heisenberg group, \(Q=2n+2\) be the homogeneous dimension of \(\mathbb {H}^{n}\). We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group \(\mathbb {H}^{n}\). Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space \({ HW}^{1,Q}(\mathbb {H}^{n}) \) on the entire Heisenberg group \(\mathbb {H}^{n}\). Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in \(\mathbb {H}^{n}\) by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space \(\mathbb {H}^n\) by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polyá–Szegö inequality. Due to the absence of the Polyá–Szegö inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on \(\mathbb {H}^{n}\):

$$\begin{aligned} -\mathrm {div}\left( \left| \nabla _{\mathbb {H}}u\right| ^{Q-2} \nabla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$

with nonlinear terms f of maximal exponential growth \(\exp (\alpha t^{\frac{Q}{Q-1}})\) as \(t\rightarrow +\infty \). All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

     

Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For \(k,m,n\in {\mathbb {N}}\), we consider \(n^k\times n^k\) random matrices of the form

$$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$

where \(\tau _{\alpha }\), \(\alpha \in [m]\), are real numbers and \({\mathbf {y}}_\alpha ^{(j)}\), \(\alpha \in [m]\), \(j\in [k]\), are i.i.d. copies of a normalized isotropic random vector \({\mathbf {y}}\in {\mathbb {R}}^n\). For every fixed \(k\ge 1\), if the Normalized Counting Measures of \(\{\tau _{\alpha }\}_{\alpha }\) converge weakly as \(m,n\rightarrow \infty \), \(m/n^k\rightarrow c\in [0,\infty )\) and \({\mathbf {y}}\) is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of \({\mathcal {M}}_{n,m,k}({\mathbf {y}})\) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For \(k=2\), we define a subclass of good vectors \({\mathbf {y}}\) for which the centered linear eigenvalue statistics \(n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ \) converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.

     

Performance of convex underestimators in a branch-and-bound framework

The efficient determination of tight lower bounds in a branch-and-bound algorithm is crucial for the global optimization of models spanning numerous applications and fields. The global optimization method \(\alpha \)-branch-and-bound (\(\alpha \)BB, Adjiman et al. in Comput Chem Eng 22(9):1159–1179, 1998b, Comput Chem Eng 22(9):1137–1158, 1998a; Adjiman and Floudas in J Global Optim 9(1):23–40, 1996; Androulakis et al. J Global Optim 7(4):337–363, 1995; Floudas in Deterministic Global Optimization: Theory, Methods and Applications, vol. 37. Springer, Berlin, 2000; Maranas and Floudas in J Chem Phys 97(10):7667–7678, 1992, J Chem Phys 100(2):1247–1261, 1994a, J Global Optim 4(2):135–170, 1994), guarantees a global optimum with \(\epsilon \)-convergence for any \(\mathcal {C}^2\)-continuous function within a finite number of iterations via fathoming nodes of a branch-and-bound tree. We explored the performance of the \(\alpha \)BB method and a number of competing methods designed to provide tight, convex underestimators, including the piecewise (Meyer and Floudas in J Global Optim 32(2):221–258, 2005), generalized (Akrotirianakis and Floudas in J Global Optim 30(4):367–390, 2004a, J Global Optim 29(3):249–264, 2004b), and nondiagonal (Skjäl et al. in J Optim Theory Appl 154(2):462–490, 2012) \(\alpha \)BB methods, the Brauer and Rohn+E (Skjäl et al. in J Global Optim 58(3):411–427, 2014) \(\alpha \)BB methods, and the moment method (Lasserre and Thanh in J Global Optim 56(1):1–25, 2013). Using a test suite of 40 multivariate, box-constrained, nonconvex functions, the methods were compared based on the tightness of generated underestimators and the efficiency of convergence of a branch-and-bound global optimization algorithm.     

Second Order Riesz Transforms on Multiply–Connected Lie Groups and Processes with Jumps

We study a class of combinations of second order Riesz transforms on Lie groups \(\mathbb {G}=\mathbb {G}_{x} \times \mathbb {G}_{y}\) that are multiply connected, composed of a discrete abelian component \(\mathbb {G}_{x}\) and a compact connected component \(\mathbb {G}_{y}\). We prove sharp L^p estimates for these operators, therefore generalizing previous results (Volberg, A., Nazarov, F.: St Petersburg Math J. 15 (14), 563–573 2004; Domelevo, K., Petermichl, S.: Adv. Math. 262, 932–952 2014; Bañuelos, R., Baudoin, F.: Potential Anal. 38(4), 1071–1089 2013). We construct stochastic integrals with jump components adapted to functions defined on the semi-discrete set \(\mathbb {G}_{x} \times \mathbb {G}_{y}\). We show that these second order Riesz transforms applied to a function may be written as conditional expectation of a simple transformation of a stochastic integral associated with the function. The analysis shows that Itô integrals for the discrete component must be written in an augmented discrete tangent plane of dimension twice larger than expected, and in a suitably chosen discrete coordinate system. Those artifacts are related to the difficulties that arise due to the discrete component, where derivatives of functions are no longer local. Previous representations of Riesz transforms through stochastic integrals in this direction do not consider discrete components and jump processes.