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For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

This paper investigates the transitivity of the relation defined by degeneration of finitely generated modules over an associative algebra. It is proved in this paper that if L degenerates to M and M degenerates to N, then $$L^{\oplus e}$$ degenerates to $$N^{\oplus e}$$ for some (but explicitly given) integer $$e>0$$ .

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions $$\Gamma $$ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps, there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements. In this paper, we present new frameworks for combining descent and negative curvature directions: alternating two-step approaches and dynamic step approaches. The aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on (estimated) upper-bounding models of the objective function. A consequence of this aspect is that our frameworks can, in theory, employ fixed stepsizes, which makes the methods readily translatable from deterministic to stochastic settings. For deterministic problems, we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps. We also show that gains can be made in a stochastic setting in cases when a standard stochastic-gradient-type method might make slow progress.

In this paper, we derive closed-form formulas for moments of the asset price in the Barndorff-Nielsen and Shephard (BN–S) stochastic volatility model. We also present similar results where the market is driven by a BN–S-type stochastic process. It is shown that in both cases the results depend on the cumulant transform of the background driving Levy process for the models. In this paper, we have also obtain various approximate expressions for the expected value of the square-root process for the shifted asset price with respect to the BN–S model.

The aim of this paper is to consider the relation between $$\mathsf {Lie}$$-isoclinism and isomorphism of two pairs of Leibniz algebras. We show that, unlike the absolute case for finite dimensional Lie algebras, these concepts are not identical, even if the pairs of Leibniz algebras are $$\mathsf {Lie}$$-stem. Moreover, throughout the paper, we provide some conditions under which $$\mathsf {Lie}$$-isoclinism and isomorphism of $$\mathsf {Lie}$$-stem Leibniz algebras are equal. In order to get this equality, the concept of factor set is studied as well.

In this paper we improve the bounds for the Carath\'eodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and $[0,1]^n$. We also treat moment problems with small gaps. We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarrow\mathbb{R}$ which needs at least $(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$ atoms $l_{x_i}$. Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\rightarrow\mathbb{R}$ which need to be extended to the worst case degree $4d$, $\tilde{L}:\mathbb{R}[x_1,\dots,x_n]_{\leq 4d}\rightarrow\mathbb{R}$, in order to have a flat extension., Comment: The first version contained the Carath\'eodory numbers from Hilbert functions and the shape reconstruction from derivatives of moments. The second part part extended and the paper is split into two papers: "Carath\'eodory numbers from Hilbert functions" and "Shape and Gaussian Mixture Reconstruction from Derivatives of Moments"

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages

This is the first of a series of two papers in which we present a solution to Manin’s Real Multiplication program (Manin in: Laudal and Piene (eds) The Legacy of Niels Henrik Abel, Springer, Berlin, 2004) —an approach to Hilbert’s 12th problem for real quadratic extensions of $$\mathbb Q$$ —in positive characteristic, using quantum analogs of the modular invariant and the exponential function. In this first paper, we treat the problem of Hilbert class field generation. If $$k=\mathbb F_{q}(T)$$ and $$k_{\infty }$$ is the analytic completion of k, we introduce the quantum modular invariant $$\begin{aligned} j^\mathrm{qt}: k_{\infty }\multimap k_{\infty } \end{aligned}$$ as a multivalued, discontinuous modular invariant function. Then if $$K=k(f)\subset k_{\infty }$$ is a real quadratic extension of k and f is a fundamental unit, we show that the Hilbert class field $$H_{\mathcal {O}_{K}}$$ (associated to $$\mathcal {O}_{K}=$$ integral closure of $$\mathbb F_{q}[T]$$ in K) is generated over K by the product of the multivalues of $$j^\mathrm{qt}(f)$$ .

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.

In this paper we use the blow-up surgery introduced in [G] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [G]. Consequently we gain more flexibility which allows to construct new volume-preserving partially hyperbolic flows as well as new examples which are not fiberwise Anosov. The latter are produced by starting with the geodesic flow on complex hyperbolic manifold which admits a totally geodesic complex curve. Then by performing the slow-down first and the blow-up second we obtain a new (volume-preserving) partially hyperbolic flows., Comment: 16 pages. This is part II. Part I being arXiv:1609.05925

In this paper, we introduce a new class of curves that we call helical extension curve in Euclidean space. Then we investigate the differential geometric properties of the helical extension curves. The main result of the paper is that the helical extension curve of a helix is a plane curve. Moreover, we give several simple characterizations of helical extension curve of special curves such as spherical curve.

The aim of this paper is to study the existence of solutions for critical Schrodinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: $$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$ where $$M:[0, \infty )\rightarrow [0, \infty )$$ is a continuous function, $$(-\Delta )_p^{s}$$ is the fractional p-Laplacian, $$00$$ is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into $$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$ Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

This paper is concerned with a Delsarte-type extremal problem. Denote by$${\mathcal {P}}(G)$$P(G)the set of positive definite continuous functions on a locally compact abelian groupG. We consider the function class, which was originally introduced by Gorbachev,$$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$G(W,Q)G=f∈P(G)∩L1(G):f(0)=1,suppf+⊆W,suppf^⊆Qwhere$$W\subseteq G$$W⊆Gis closed and of finite Haar measure and$$Q\subseteq {\widehat{G}}$$Q⊆G^is compact. We also consider the related Delsarte-type problem of finding the extremal quantity$$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$D(W,Q)G=sup∫Gf(g)dλG(g):f∈G(W,Q)G.The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem$${\mathcal {D}}(W,Q)_G$$D(W,Q)G. The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where$$G={\mathbb {R}}^d$$G=Rd. So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$-Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ωin two-dimensional domains $$\Omega $$Ω. In particular, this paper presents a dense family $$\mathcal {A}$$A of simply-connected two-dimensional domains with analytic boundaries such that, for each $$\Omega \in \mathcal {A}$$Ω∈A, the nodal set of the eigenfunction $$\phi _{\sigma _j}$$ϕσj “is not dense at scale $$\sigma _j^{-1}$$σj-1”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains $$\Omega \in \mathcal {A}$$Ω∈A, the nodal sets of the eigenfunctions $$\phi _{\sigma _j}$$ϕσj associated with the eigenvalue $$\sigma _j$$σj have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $$\Omega \in \mathcal {A}$$Ω∈A there is a value $$r_1>0$$r1>0 such that for each j there is $$x_j\in \Omega $$xj∈Ω such that $$\phi _{\sigma _j}$$ϕσj does not vanish on the ball of radius $$r_1$$r1 around $$x_j$$xj.

In the present paper, we define generalized skew semi-invariant submersion and give an example for such submersions. Necessary and sufficient conditions are given to obtain the integrability and totally geodesicness of the distributions, which are mentioned in the definition of generalized skew semi-invariant submersion. The geometry of the fibers is investigated. At the end of the paper, the generalized skew semi-invariant submersion is considered with parallel canonical structures and some results are obtained.

The link between Vandiver’s conjecture and Gauss sums is well known since the papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459, 1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Angles and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the p-th cyclotomic field (Sur la p-ramification abelienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order p of $${\mathbb {F}}_\ell ^\times $$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in §5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as $$\ell $$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.

This paper concerns both norm estimation and pointwise approximation for the Bochner–Riesz means of an arbitrary Morrey function on $${{{{\mathbb {R}}}}^n}$$—Theorems 1.1 and 1.2 for $$L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$—thereby generalizing the corresponding results for $$L^p({{{{\mathbb {R}}}}^n})$$ in Stein (Acta Math 100:93–147, 1958) and Carbery et al. (J Lond Math Soc 38:513–524, 1988). As a side note, this paper also establishes Lemma 4.1 of Tomas–Stein type—if $$f\in L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$ under $$ 2^{-1}(n+1)

This paper continues ongoing investigations into questions initiated by seventeenth-century experiments of Mariotte on mutual attractions and repulsions of floating objects, later reformulated mathematically by Laplace in the context of idealized surface tension theory. The characteristic nonlinearity in the governing equations imposes restrictions on behavior of solutions as graphs in the Laplace model, to the effect that a priori relations connecting height and inclination of a presumed surface interface occur, severely restricting the heights that can be achieved by solution curves. We develop below some relevant consequences of that phenomenon. The initial section provides general (perhaps inadequate) background, characterizes global qualitative behavior of all solutions of the underlying equations, and elucidates the significant distinctions in behavior that occur for solutions over an interval that extends unboundedly in one base direction. The section next following presents a non-existence result ensuing from the limitations imposed on such solutions (this result is elaborated in the Appendix). Next follow individual properties of general solutions. The remainder of the paper addresses the forces arising between partially submerged bodies. Theorem 6 offers an alternative derivation of a discovery due to Aspley, He, and McCuan, which displays a remarkable identification of net horizontal force with a first integral of the basic equations. These relations simplify significantly some earlier literature; additionally, as indicated in Section 5, they open the way to consideration of a much larger class of partially immersed objects than appears in the Laplace formulation. We complete Section 5 with an illustrative example in a particular case, indicative of a general study currently underway.

In this paper, we show the existence of a weak solution to the Maxwell–Stokes type equation with a potential satisfying the Dirichlet condition, under the hypothesis that the domain has no holes, using a version of the de Rham lemma that was proved in our previous paper. We also give the regularity of weak solutions.

Theories defined by non-Hermitian PT-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. In this paper the conventional Bell basis is discussed using the CPT-inner product for a general two-state system in PT-symmetric quantum theory. The conventional Bell basis has many applications in quantum information theory because of the mutual orthogonality, normlization, and maximally entangled degree of each vector in the basis. This paper analyzes four properties of the Bell basis in PT-symmetric quantum theory: CPT-orthogonality, CPT-norm, CPT-entanglement degree and CPT-mutual unbiased property with other maximally entangled bases. The conventional Hermitian Bell states are not CPT-normalized and are not maximally entangled states in PT-symmetric quantum theory. The orthogonality still holds for four of the six pairs of Bell states under the CPT-inner product. Furthermore, the mutually unbiased property with another maximally entangled basis is preserved at the level of twenty five percent.

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.

Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most $$n+m$$. If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly $$n+m$$. As in the previous paper, the results are proved using Leibniz difference operators.

This paper resolves the difficulty in using Almon’s technique in the estimation of distributed lag coefficients when the researchers run into the problem of multicollinearity. Multicollinearity arising in the analysis leads us to the biased estimation methods, one of which is Almon ridge estimator. In consideration of two-parameter ridge solution of Lipovetsky and Conklin (Appl Stoch Mod Bus Ind 21:525–540, 2005), we offer Almon two-parameter ridge estimator, which has advantage over Almon estimator as well as Almon ridge estimator with regard to mean square error criterion. Besides, an improvement in coefficient of multiple determination is observed for the new estimator relative to the Almon ridge estimator. An illustrative application of the mentioned estimators is examined by widely used Almon data to support the theoretical results. Additionally, a Monte Carlo experiment is performed to investigate the performance of the estimators in detail. © 2018, Shiraz University. Firat University Scientific Research Projects Management Unit: FBA-2018-10169 This paper was supported by Çukurova University Scientific Research Projects Unit Project Number: FBA-2018-10169.

Let $$\mathcal{\widetilde{O}}$$ (B) be the category of (open) subcategories of a topological groupoid B: In this paper we study Cat-valued sheaves over category $$\mathcal{\widetilde{O}}$$ (B): The paper introduces a notion of categorical union, such that the categorical union of subcategories is a subcategory. We use this definition of categorical unions to define a categorical cover of a topological category. Instead of assuming a Grothendieck topology, we define Cat-valued sheaves in terms of the categorical cover defined in this paper. The main result is the following. For a fixed category C, the categories of local functorial sections from B to C define a Catvalued sheaf on $$\mathcal{\widetilde{O}}$$ (B): Replacing C with a categorical group G; we find a CatGrp-valued sheaf on $$\mathcal{\widetilde{O}}$$ (B): We also relate and distinguish our construction with the notion of stacks.

This paper is concerned with the qualitative properties of viscosity solutions to a class of Hamilton-Jacobi equations (HJEs) in Banach spaces. Specifically, based on the concept of $\beta $ -derivative (Deville et al. 1993), we establish the existence, uniqueness and stability of $\beta $ -viscosity solutions for a class of HJEs in the form $u+H(x,u,Du)= 0$ . The obtained results in this paper extend earlier works in the literature, for example, Crandall and Lions (J. Funct. Anal. 62, 379–398, 1985, J. Funct. Anal., 65, 368–405, 1986) and Deville et al. (1993).

Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

This paper is devoted to the analysis of a simple Lotka–Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J Math Biol 76:697–754, 2018b) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper, we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and $$n-1$$ predators. The jth predator eats the $$j-1$$ st species and is eaten by the $$j+1$$ st predator; this way each species only interacts with at most two other species—the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann Appl Probab 28:1893–1942, 2018a). Unlike most of the results available in the literature, we provide an in-depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analyzing trophic cascades in a plant–herbivore–predator system and providing persistence/extinction criteria for food chains of length $$n\le 4$$ .

In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$ where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form $$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$ where αj and $$ {\alpha}_j^{\prime } $$ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: • all the main ideas of the construction are presented; • all the technicalities of the proof in the general setting [8] are avoided.

Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s examples of continuum many pairwise non-isomorphic separable II1 factors are in fact pairwise non-elementarily equivalent. Their proof proceeded by showing that any ultrapowers of any two distinct McDuff examples are not isomorphic. In a paper by the first two authors of this paper, Ehrenfeucht–Fra¨isse games were used to find an upper bound on the quantifier complexity of sentences distinguishing the McDuff examples, leaving it as an open question to find concrete sentences distinguishing the McDuff factors. In this paper, we answer this question by providing such concrete sentences.

The purpose of this paper is to give some new Karush–Kuhn–Tucker-type necessary optimality conditions for nonsmooth semi-infinite problems. Moreover, we present some suitable examples for our results. The paper is organized by Frechet and Mordukhovich subdifferentials.

Kisin and Pappas constructed integral models of Hodge-type Shimura varieties with parahoric level structure at $p>2$, such that the formal neighbourhood of a mod~$p$ point can be interpreted as a deformation space of $p$-divisible group with some Tate cycles (generalising Faltings' construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod~$p$ points of Kisin-Pappas integral models with parahoric level structure; namely, we obtain the dimension of central leaves and the almost product structure of Newton strata. In the case of hyperspecial level strucure (i.e., in the good reduction case), our main results were already obtained by Hamacher, and the result of this paper holds for ramified groups as well., 33 pages; section 2.5 added to fill in the gap in the earlier version

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring., Comment: 46 pages

The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velazquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.

The concept of Takagi–Landsberg’s fractal surface is applied in this paper for constructing a parametric model of a hyperbolic paraboloid (hypar) shell structure using the Midpoint Displacement Method (MDM) based on the Iterated Function System (IFS) and controlled by the relative size value (w), a factor of fractal dimension. This method of generating a parametric model of a hypar is applied to create a domain of non-integer dimensions through which the hypar surface passes through textural changes, thus transforming the smooth hypar surface from its two-dimensional shape to a higher but non-integer-dimensional irregular surface that results in the changes of structural behavior. This paper briefly compares the structural behavior between the regular hypar and the fractal-based irregular hypar, and also searches the optimal shape of the hypar in terms of minimum deformation from the collection of its regular version and its different levels of irregular versions.

This paper presents an alternative steady-state solution to the discrete-time Geo/G/1/N + 1 queueing system using roots. The analysis has been carried out for a late-arrival system using the imbedded Markov chain method, and the solutions for the early arrival system have been obtained from those of the late-arrival system. Using roots of the associated characteristic equation, the distributions of the numbers in the system at various epochs are determined. We find a unified approach for solving both finite- and infinite- buffer systems. We investigate the measures of effectiveness and provide numerical illustrations. We establish that, in the limiting case, the results thus obtained converge to the results of the continuous-time counterparts. The applications of discrete-time queues in modeling slotted digital computer and communication systems make the contributions of this paper relevant.

Many researchers have been interested in the concept of statistical convergence because of the fact that it is stronger than the classical convergence. Also, the concepts of statistical equal convergence and equi-statistical convergence are more general than the statistical uniform convergence. In this paper we define a new type of statistical convergence by using the notions of equi-statistical convergence and statistical equal convergence to prove a Korovkin type theorem. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were demonstrated by earlier authors. After, we present an example in support of our definition and result presented in this paper. Finally, we also compute the rates of statistical equi-equal convergence of sequences of positive linear operators.

Scientific problems often feature observational data received in the form $$w_1=l_1(f),\ldots $$ , $$w_m=l_m(f)$$ of known linear functionals applied to an unknown function f from some Banach space $$\mathcal {X}$$ , and it is required to either approximate f (the full approximation problem) or to estimate a quantity of interest Q(f). In typical examples, the quantities of interest can be the maximum/minimum of f or some averaged quantity such as the integral of f, while the observational data consists of point evaluations. To obtain meaningful results about such problems, it is necessary to possess additional information about f, usually as an assumption that f belongs to a certain model class $$\mathcal {K}$$ contained in $$\mathcal {X}$$ . This is precisely the framework of optimal recovery, which produced substantial investigations when the model class is a ball in a smoothness space, e.g., when it is a unit ball in Lipschitz, Sobolev, or Besov spaces. This paper is concerned with other model classes described by approximation processes, as studied in DeVore et al. [Data assimilation in Banach spaces, (To Appear)]. Its main contributions are: (1) designing implementable optimal or near-optimal algorithms for the estimation of quantities of interest, (2) constructing linear optimal or near-optimal algorithms for the full approximation of an unknown function using its point evaluations. While the existence of linear optimal algorithms for the approximation of linear functionals Q(f) is a classical result established by Smolyak, a numerically friendly procedure that performs this approximation is not generally available. In this paper, we show that in classical recovery settings, such linear optimal algorithms can be produced by constrained minimization methods. We illustrate these techniques on several examples involving the computation of integrals using point evaluation data. In addition, we show that linearization of optimal algorithms can be achieved for the full approximation problem in the important situation where the $$l_j$$ are point evaluations and $$\mathcal {X}$$ is a space of continuous functions equipped with the uniform norm. It is also revealed how quasi-interpolation theory enables the construction of linear near-optimal algorithms for the recovery of the underlying function.

We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation–coagulation equations.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

In this paper we study non-compact self-shrinkers first in general codimension and then in codimension 1. We respectively prove some vanishing theorems giving rise to rigidity of the self-shrinker and then estimates involving the higher order mean curvatures for the oriented case. The paper ends with some results on their index when considered as appropriate $$\bar f$$ -minimal hypersurfaces.

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals.

In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.