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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 provide an abstract aproach to the study of classes of multiple summing multilinear operators between Banach spaces. The main purpose is unify the study of several known classes and results, for example multiple $(p, q_1,\dots, q_n)$-summing operators, multiple mixing $(s, q, p)$-summing operators and multiple strong $(s, q, p)$-mixing summing operators. We also define new class of multiple summing multilinear operator that are particular cases of our construction and, therefore, satisfy the results proved in the paper., Comment: 19 pages

[EN] In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient A and initial conditions X-0 and X-1. In a previous study (Calbo et al. in Comput Math Appl 61(9):2782-2792, 2011), a mean square convergent power series solution on (-1/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X-0 and X-1, independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an L-p solution (1, This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.

In this paper, we show that a generalized Sasakian space form of dimension >3 is either of constant sectional curvature, or a canal hypersurface in Euclidean or Minkowski spaces, or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold, or locally a warped product of a real line and a generalized complex space form, or an $${\alpha}$$ -Sasakian space form, or it is of five dimension and admits an $${\alpha}$$ -Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension >5 is obtained. A local classification of Riemannian manifolds of quasi constant sectional curvature of dimension >3 is also given in this paper.

We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by Garcia-Rio and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Aahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O'Neill's book and Falcitelli, Ianus and Pastore's book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu's book to obtain some characterizations and to give a semi-Riemannian version of Fischer's (resp. Aahin's) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method., Comment: This is a preprint of a paper whose final and definite form will be published in Mediterr. J. Math

In this paper, we investigate boundedness, polar decomposition and spectral decomposition of weighted conditional expectation type operators on \(L^2(\Sigma )\). Some examples are provided to illustrate concrete application of the main results of the paper.

In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

In symmetric cryptography, the round functions used as building blocks for iterated block ciphers are obtained as the composition of different layers acting as a sequence of bijective transformations providing global increasing complexity. The study of the conditions on such layers which make the group generated by the round functions of a block cipher a primitive group has been addressed in the past years, both in the case of Substitution Permutation Networks and Feistel Networks, giving to block cipher designers the recipe to avoid the imprimitivity attack, which exploits the invariance of some subspaces during the encryption. In the case of Lai–Massey schemes, where both Substitution Permutation Network and Feistel Network features are combined, the resistance against imprimitivity attacks has been a long-standing open problem. In this paper we consider a generalization of such a scheme and we prove its resistance against the imprimitivity attack. Our solution is obtained as a consequence of a more general result in which the problem of proving the primitivity of a generalized Lai–Massey scheme is reduced to the simpler one of proving the primitivity of the group generated by the round functions of a strictly related Substitution Permutation Network. We prove how this implies a reduction in the computational cost of invariant-subspace search.

Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann–Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.

The notions of r-Bell polynomials and their generalization, the r-Dowling polynomials are due to Mező and Cheon, Jung. Recently, Nyul and Rácz defined the r-Lah polynomials, which are close relatives of r-Bell polynomials. In the present paper, we introduce the Dowling type generalization of r-Lah polynomials, the r-Dowling–Lah polynomials. We give a comprehensive study of them using the results of the author and Nyul on r-Whitney–Lah numbers, which are the coefficients of these polynomials.

In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen [Monotonicity properties, inequalities, and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283 (2016), 385--396.]., 10 pages, accepted for publication in Mediterranean Journal of Mathematics

In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpi\'nski gasket in $\mathbb{R}^2$. Parameters $\lambda, \gamma, \alpha, \beta$ are real and $11$ we show the existence of at least two nontrivial weak solutions to the system of equations for some $(\lambda,\gamma) \in \mathbb{R}^2.$, Comment: 25 pages, 2 figures

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated with the function be dense. The study is done also with the help of the generalized frame operator associated with a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed, if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.

In this paper, we consider the existence of positive solutions for a singular elliptic problem involving an asymptotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison techniques, we show the existence, uniqueness, non-existence, and regularity of the solutions. We also obtain a bifurcation-type result.

We consider the nonlinear elliptic equation with quadratic convection $$ -\Delta u + g(u) |\nabla u|^2=\lambda f(u) $$ - Δ u + g ( u ) | ∇ u | 2 = λ f ( u ) in a smooth bounded domain $$ \Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N ($$ N \ge 3$$ N ≥ 3 ) with zero Dirichlet boundary condition. Here, $$ \lambda $$ λ is a positive parameter, $$ f:[0, \infty ):(0\infty ) $$ f : [ 0 , ∞ ) : ( 0 ∞ ) is a strictly increasing function of class $$C^1$$ C 1 , and g is a continuous positive decreasing function in $$ (0, \infty ) $$ ( 0 , ∞ ) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution $$u^*$$ u ∗ . A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function $$ h(t)=f(t)e^{-\int _0^t g(s)ds}$$ h ( t ) = f ( t ) e - ∫ 0 t g ( s ) d s , nor that the functions $$ gh/h'$$ g h / h ′ or $$ h'' h/h'^2$$ h ′ ′ h / h ′ 2 admit a limit at infinity.

[EN] In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function., Luis Bernal-Gonzalez, Maria del Carmen Calderon-Moreno, and Jose A. Prado-Bassas have been supported by the Plan Andaluz de Investigacion de la Junta de Andalucia FQM-127 Grant P08-FQM-03543 and by MCINN Grant PGC2018-098474-B-C21. Marina Murillo-Arcila is supported by MEC, Grant MTM2016-75963-P, and PID2019-105011GB-I00.

We show that if a homogeneous polynomial $f$ in $n$ variables has Waring rank $n+1$, then the corresponding projective hypersurface $f=0$ has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of $f$. We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5., v4: We have added Lemma 2.2, which is necessary for a clear proof of the unicity of the integer $k$ in Theorem 2.4. We also attribute Proposition 4.1 to Neriman Tokcan, since we were informed that it appears as Theorem 3.1 in her paper 'On the Waring rank of binary forms', arXiv:1610.09065

In this paper, we consider the curves on the surface in the unit 3-sphere. For a regular curve on a surface in the unit 3-sphere, we have a moving frame along the curve which is called a spherical Darboux frame. We induce two special vector fields along the curve with respect to the spherical Darboux frame and investigate the singularities of extrinsic flat great circular surfaces associated with these vector fields.

This paper is concerned with the existence and uniqueness, and Ulam--Hyers stabilities of solutions of nonlinear impulsive $\varphi$--Hilfer fractional differential equations. Further, we investigate the dependence of the solution on the initial conditions, order of derivative and the functions involved in the equations. The outcomes are acquired in the space of weighted piecewise continuous functions by means of fixed point theorems and the generalized version of Gronwall inequality., Comment: 24

In this paper, we extend the classical Korovkin theorems to the framework of comonotone additive, sublinear, and monotone operators. Based on the theory of Choquet capacities, several concrete examples illustrating our results are also discussed.

In this paper, we define the Dunford–Henstock–Kurzweil and the Dunford–McShane integrals of Banach space-valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space $${\mathbb {R}}^{m}$$ . We will show that the new integrals are “natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock–Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.

We introduce and study a new functional which was motivated by our paper on the Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, R\u{a}dulescu and Repov\v{s}, Nonlinearity 31 (2018), 1518-1534). We also study the eigenvalue problem for equations involving this new functional.

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian., Comment: All comments are welcome

In this paper, we classify filiform associative algebras of degree p over a field of characteristic zero. Moreover, over an algebraically closed field of characteristic zero, we also classify filiform nilpotent associative algebras and naturally graded quasi-filiform nilpotent associative algebras, described through the characteristic sequence C(A) = (n- 2 , 1 , 1) or C(A) = (n- 2 , 2).

Inspired by the results in a recent paper by Galloway and Vega (Lett Math Phys 108(10):2285–2292, 2018), we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field in a globally hyperbolic space-time with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik’s splitting conjecture. In particular, we give a complementary result to the main theorem in [11].

[EN] We provide structural criteria for some finite factorised groups G = AB when the conjugacy class sizes in G of certain pi-elements in A boolean OR B are either pi-numbers or pi'-numbers, for a set of primes pi. In particular, we extend for products of groups some earlier results, M. J. Felipe is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain). A. Martínez-Pastor is supported by Proyecto MTM 2014-54707-C3-1-P, Ministerio de Economía, Industria y Competitividad (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain). The results in this paper are part of the V. M. Ortiz Sotomayor Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). The authors are also supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovación y Universidades

In this paper, we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. The assumption on the time-dependent coefficients for the damping terms means that these coefficients are summable and nonnegative. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato’s lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case, an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p - q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of semilinear not-damped wave equations with the same kind of nonlinearities.

We give a correction to Theorem 1.2 in a previous paper [Mediterr. J. Math. (2018) 15:227]. Two examples are given to explain the corrected conclusion.

In this paper, we prove some results concerning the loxodromes on an invariant surface in a three-dimensional Riemannian manifold, a part of which generalizes classical results about loxodromes on rotational surfaces in $${{\mathbb {R}}}^3$$. In particular, we show how to parametrize a loxodrome on an invariant surface of $${\mathbb {H}}^2\times {{\mathbb {R}}}$$ and $${\mathbb {H}}_3$$, and we exhibit the loxodromes of some remarkable minimal invariant surfaces of these spaces. In addition, we give an explicit description of the loxodromes on an invariant surface with constant Gauss curvature.

In this paper, we study the hypercyclic composition operators on weighted Banach spaces of functions defined on discrete metric spaces. We show that the only such composition operators act on the "little" spaces. We characterize the bounded composition operators on the little spaces, as well as provide various necessary conditions for hypercyclicity.

In this paper, we will consider matrices with entries in the space of operators $\mathcal{B}(H)$, where $H$ is a separable Hilbert space, and consider the class of (left or right) Schur multipliers that can be approached in the multiplier norm by matrices with a finite number of diagonals. We will concentrate on the case of Toeplitz matrices and of upper triangular matrices to get some connections with spaces of vector-valued functions., arXiv admin note: text overlap with arXiv:1810.07819

The paper deals with semilinear evolution equations in Banach spaces. By means of linear control terms, the controllability problem is investigated and the solutions satisfy suitable nonlocal conditions. The Cauchy multi-point condition and the mean value condition are included in the present discussion. The final configuration is always achieved with a control with minimum norm. The results make use of fixed point techniques; two different approaches are proposed, depending on the use of norm or weak topology in the state space. The discussion is completed with some applications to dynamics of diffusion processes.

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $$(f,b): M^n \rightarrow X^n$$ with control map $$q: X^n \rightarrow B$$ to complete controlled surgery is an element $$\sigma ^c (f, b) \in H_n (B, \mathbb {L})$$ , where $$M^n, X^n$$ are topological manifolds of dimension $$n \ge 5$$ . Our proof uses essentially the geometrically defined $$\mathbb {L}$$ -spectrum as described by Nicas (going back to Quinn) and some well-known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map $$H_n (B, \mathbb {L}) \rightarrow L_n (\pi _1 (B))$$ in terms of forms in the case $$n \equiv 0 (4)$$ . Finally, we explicitly determine the canonical map $$H_n (B, \mathbb {L}) \rightarrow H_n (B, L_0)$$ .

The Waldschmidt constant \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\) of a radical ideal I in the coordinate ring of \({\mathbb {P}}^N\) measures (asymptotically) the degree of a hypersurface passing through the set defined by I in \({\mathbb {P}}^N\). Nagata’s approach to the 14th Hilbert Problem was based on computing such constant for the set of points in \({\mathbb {P}}^2\). Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore, the research focuses on looking for accurate bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\). In the paper, we deal with \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\), the Waldschmidt constant for s very general lines in \({\mathbb {P}}^3\). We prove that \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2s-1}\rfloor \) holds for all s, whereas the much stronger bound \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2.5 s}\rfloor \) holds for all s but \(s=4\), 7 and 10. We also provide an algorithm which gives even better bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\), very close to the known upper bounds, which are conjecturally equal to \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\) for s large enough.

In this paper, we first give new generalizations for third-order Jacobsthal $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and third-order Jacobsthal-Lucas $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$ sequences for Jacobsthal and Jacobsthal-Lucas numbers. Considering these sequences, we define the matrix sequences which have elements of $\{J_{n}^{(3)}\}_{n\in \mathbb{N}}$ and $\{j_{n}^{(3)}\}_{n\in \mathbb{N}}$. Then we investigate their properties., Comment: 9 pages

In this work we prove the existence of solutions for a class of generalized Choquard equations involving the $$\Delta _\Phi $$ -Laplacian operator. Our arguments are essentially based on variational methods. One of the main difficulties in this approach is to use the Hardy–Littlewood–Sobolev inequality for nonlinearities involving N-functions. The methods developed in this paper can be extended to wide classes of nonlinear problems driven by nonhomogeneous operators.

In this paper we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson-Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces., Comment: 15 pages

There is a mistake in the last subsection (5.3 Totally geodesic null hypersurfaces in Robertson–Walker spaces) of the paper [1]

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given., final version, accepted by MJOM

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization., 20 pages. The paper "Bousfield localisations along Quillen bifunctors and applications" (arXiv:1411.0500v1) has been divided into two parts: "Towers and fibered products of model structures", which is this arXiv submission, and "Bousfield localisations along Quillen bifunctors" (arXiv:1411.0500v2)

The mixed braid groups $B_{2,n}, \ n \in \mathbb{N}$, with two fixed strands and $n$ moving ones, are known to be related to the knot theory of certain families of $3$-manifolds. In this paper we define the mixed Hecke algebra $\mathrm{H}_{2,n}(q)$ as the quotient of the group algebra ${\mathbb Z}\, [q^{\pm 1}] \, B_{2,n}$ over the quadratic relations of the classical Iwahori-Hecke algebra for the braiding generators. We furhter provide a potential basis $\Lambda_n$ for $\mathrm{H}_{2,n}(q)$, which we prove is a spanning set for the $\mathbb{Z}[q^{\pm 1}]$-additive structure of this algebra. The sets $\Lambda_n,\ n \in \mathbb{Z}$ appear to be good candidates for an inductive basis suitable for the construction of Homflypt-type invariants for knots and links in the above $3$-manifolds., Comment: 10 pages, 3 figures

In this paper, we shall discuss the family of biharmonic mappings for which maximum principle holds. As a consequence of our study, we present Schwarz Lemma for the family of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings., Comment: 16 pages; 10 figures; The article is with a journal

Links in lens spaces may be defined to be equivalent by ambient isotopy or by diffeomorphism of pairs. In the first case, for all the combinatorial representations of links, there is a set of Reidemeister-type moves on diagrams connecting isotopy equivalent links. In this paper we provide a set of moves on disk, band and grid diagrams that connects diffeo equivalent links: there are up to four isotopy equivalent links in each diffeo equivalence class. Moreover, we investigate how the diffeo equivalence relates to the lift of the link in the $3$-sphere: in the particular case of oriented primitive-homologous knots, the lift completely determines the knot class in $L(p,q)$ up to diffeo equivalence, and thus only four possible knots up to isotopy equivalence can have the same lift., 25 pages, 26 figures

We prove in this paper the existence of dense linear subspaces in the classical holomorphic Lipschitz spaces in the disc all of whose non-null functions are nowhere differentiable at the boundary. Infinitely generated free algebras as well as infinite dimensional Banach spaces consisting of Lipschitz functions enjoying the mentioned property almost everywhere on the boundary are also exhibited. It is also investigated the algebraic size of the family of functions in the disc algebra that either do not preserve Borel sets on the unit circle or possess the Cantor boundary behavior on the disc.

In a previous paper (Antic et al., three-dimensional CR submanifolds of the nearly Kahler $$\mathbb {S}^{3}\times \mathbb {S}^{3}$$ , 2017), the authors together with L. Vrancken initiated the study of 3-dimensional CR submanifolds of the nearly Kahler homogeneous $$\mathbb {S}^3\times \mathbb {S}^3$$ . As is shown by Butruille, this is one of only four homogeneous 6-dimensional nearly Kahler manifolds. Besides its almost complex structure J, it also admits a canonical almost product structure P, see (Moruz and Vrancken, Publ Inst Math 2018) and (Bolton et al., Tohoku Math J 67:1–17, 2015). Along a proper 3-dimensional CR submanifold, the tangent space of $$\mathbb {S}^3\times \mathbb {S}^3$$ can be naturally split as the orthogonal sum of three 2-dimensional vector bundles $$\mathcal {D}_1$$ , $$\mathcal {D}_2$$ and $$\mathcal {D}_3$$ . We study the CR submanifolds in relation with the behavior of the almost product structure on these vector bundles.

In this paper, we study the following fractional Navier boundary value problem $$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$ where $$\alpha ,\beta \in (0,1]$$ such that $$\alpha +\beta >1$$ , $$D^{\beta }$$ and $$D^{\alpha }$$ stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that $$a+b>0$$ . The function g is a nonnegative continuous function in $$[0,\infty )$$ that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.

In this paper, we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $$K_3(\alpha ,\beta ,\gamma )$$ and $$D_t(\alpha ,\beta ,\gamma )$$ . We also compute the derivations under some restrictions of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras. All superalgebras are considered over a field of the characteristic 0.

In this paper, we give sufficient conditions for the existence and uniqueness of asymptotically ω-periodic solutions for a nonlinear differential equation with piecewise constant argument in a Banach space via asymptotically ω-periodic functions in the Stepanov sense. This is done using the Banach fixed point Theorem.

This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of $$\mathbb {R}^{n}.$$ We show that, under suitable conditions on the initial data, the solution exists globally in time. Then, we prove the general energy decay of global solutions by applying a lemma of Martinez, which allows us to get our decay result for a class of relaxation functions wider than that usually considered.