Let $E$ be a topological vector space and let us consider a property $\mathcal P$. We say that the subset $M$ of $E$ formed by the vectors in $E$ which satisfy $\mathcal P$ is $\mu$-lineable (for certain cardinal $\mu$, finite or infinite) if $M \cup \{0\}$ contains an infinite dimensional linear space of dimension $\mu$. In 1966 V. Gurariy provided a non-constructive proof of the $\aleph_0$-lineability of the set of {\em Weierstrass' Monsters} (continuous nowhere differentiable functions on $\mathbb{R}$). Here we provide the first constructive proof of the ${\mathfrak c}$-lineability of this set (where $\mathfrak{c}$ denotes the continuum). Of course, this result is the best possible in terms of dimension.

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.

In this article we study the Darboux integrability of the polynomial differential systems \[ \dot x = y-x^2, \quad \dot y=z-x, \quad \dot z= -d-ax -by-cz. \] This system comes from the study of a Hopf bifurcation in slow-fast systems with two slow variables and one fast variable. The tools used here for studying the Darboux integrability can be applied to arbitrary polynomial differential systems in $\mathbb R^3$.

We constructively prove a decomposition theorem for order bounded homogeneous orthogonally-additive polynomials on archimedean vector lattices. In this way, we constructively generalize results of Sundaresan (1991), D. Pérez-Garcia, and I. Villanueva, (2005), Y. Benyamini, S. Lassalle and J.~L.~G.~Llavona (2006), D. Carando, S. Lassalle and I. Zalduendo (2006 and 2012).

The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation \begin{equation*} A_1XB_1+A_2\overline{X}B_2+C_1YD_1+C_2\overline{Y}D_2=E, \end{equation*} over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches.

In this paper, Hermite-Hadamard type inequalities involving Hadamard fractional integrals via convex functions are studied. An important integral identity and new Hermite-Hadamard type integral inequalities involving Hadamard fractional integrals are also presented. Some applications to special means of real numbers are given.

Recently, Leindler [2] introduced the sequences of Head Bounded Variation ($HBVS$) and the sequences of Rest Bounded Variation ($RBVS$), which are nontrivial generalizations of nondecreasing sequences and nonincreasing sequences respectively. In the present paper, we generalize a classical result of Mazhar([1]) on the approximation by means of Fourier Stieltjes series by using the $HBVS$ and $RBVS.$

There are insertion-type characterizations in pointfree topology that extend well known insertion theorems in point-set topology for all relevant higher separation axioms with one notable exception: complete regularity. In this paper we fill this gap. The situation reveals to be an interesting and peculiar one: contrarily to what happens with all the other higher separation axioms, the extension to the pointfree setting of the classical insertion result for completely regular spaces characterizes a formally weaker class of frames introduced in this paper (called \emph{completely c-regular frames}). The fact that any compact sublocale (quotient) of a completely regular frame is a $C$-sublocale ($C$-quotient) is obtained as a corollary.

We study recurrence properties for the nonautonomous discrete system given by a sequence $(f_n)_{n=1}^\infty$ of continuous selfmaps on a compact metric space. In particular, our attention is paid to the case when the sequence $(f_n)_{n=1}^\infty$ converges uniformly to a map $f$ or forms an equicontinuous family. In the first case we investigate the structure and behavior of an $\omega$-limit set of $(f_n)$ by a dynamical property of the limit map $f$. We also present some examples of $(f_n)$ and $f$ on the closed interval: (a) $\omega(x, (f_n)) \smallsetminus \Omega(f)\not=\emptyset$ for some point $x$; or (b) the set of periodic points of $f$ is closed and for some point $x$, $\omega(x, (f_n))$ is infinite. In the second case we create a perturbation of $(f_n)$ whose nonwandering set has small measure.

We are concerned with the following semilinear elliptic equation $\Delta u=\lambda f\left( x,u\right) $ in $D,$ subject to some Dirichlet conditions, where $\lambda \geq 0$ is a parameter and $D$ is a smooth domain in $\mathbb{R}^{n}\left( n\geq 3\right) $. Under some appropriate assumptions on the nonnegative nonlinearity term $f\left( x,u\right) ,$ we show the existence of a positive bounded solution for the above semilinear elliptic equation. Our approach is based on Schauder's fixed point Theorem.

Lorentz surfaces with parallel mean curvature vector in $\mathbb{E}_2^4$ have been classified in [14]. In this paper, we continue to classify Lorentz surfaces with parallel mean curvature vector in pseudo-Riemannian space forms $S_2^4(1)$ and $H_2^4(-1)$. Consequently, we achieve the complete classification of Lorentz surfaces with parallel mean curvature vector in 4-dimensional neutral indefinite space form with index 2.

Our starting point is a very simple one, namely that of a set $\mathcal{L} _{4}$ of four mutually skew lines in $\operatorname*{PG}(7,2).$ Under the natural action of the stabilizer group $\mathcal{G}(\mathcal{L}_{4} )<\operatorname*{GL}(8,2)$ the $255$ points of $\operatorname*{PG}(7,2)$ fall into four orbits $\omega_{1},\omega_{2},\omega_{3},\omega_{4},$ of respective lengths $12,54,108,81.$ We show that the $135$ points $\in\omega_{2}\cup \omega_{4}$ are the internal points of a hyperbolic quadric $\mathcal{H}_{7}$ determined by $\mathcal{L}_{4},$ and that the $81$-set $\omega_{4}$ (which is shown to have a sextic equation) is an orbit of a normal subgroup $\mathcal{G} _{81}\cong(Z_{3})^{4}$ of $\mathcal{G}(\mathcal{L}_{4}).$ There are $40$ subgroups $\cong(Z_{3})^{3}$ of $\mathcal{G}_{81},$ and each such subgroup $H<\mathcal{G}_{81}$ gives rise to a decomposition of $\omega_{4}$ into a triplet $\{\mathcal{R}_{H},\mathcal{R}_{H}^{\prime},\mathcal{R}_{H} ^{\prime\prime}\}$ of $27$-sets. We show in particular that the constituents of precisely $8$ of these $40$ triplets are Segre varieties $\mathcal{S} _{3}(2)$ in $\operatorname*{PG}(7,2).$ This ties in with the recent finding 225-239 --- that each $\mathcal{S}=\mathcal{S}_{3}(2)$ in $\operatorname*{PG} (7,2)$ determines a distinguished $Z_{3}$ subgroup of $\operatorname*{GL} (8,2)$ which generates two sibling copies $\mathcal{S}^{\prime},\mathcal{S} ^{\prime\prime}$ of $\mathcal{S}.$

A family $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of sets covering a set $E$ is called a resolution for $E$ if $A_{\alpha }\subseteq A_{\beta }$ whenever $\alpha \leq \beta $. A locally convex space (lcs) $E$ is said to belong to class $\mathfrak{G}$ if there is a resolution $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ for $(E^{\prime},\sigma (E^{\prime },E))$ such that each sequence in any $A_{\alpha }$ is equicontinuous. The class $\mathfrak{G}$ contains `almost all' useful locally convex spaces (including $(LF)$-spaces and $(DF)$-spaces). We show that$\;\left( i\right) $ every semi-reflexive lcs $E$ in class $\mathfrak{G}$ is a Lindelöf $\Sigma $-space in the weak topology (this extends a corresponding result of Preiss-Talagrand for WCG Banach spaces) and the weak* dual of $E$ is both $K$-analytic and has countable tightness, $\left( ii\right) $ a barrelled space $E$ has a weakly compact resolution if and only if $E$ is weakly $K$-analytic, and $\left( iii\right) $ if $E$ is barrelled or bornological then $E^{\prime }$ has a weak* compact resolution if and only if it is weak* $K$-analytic. As an additional consequence we provide another approach to show that the weak* dual of a quasi-barrelled space in class $\mathfrak{G}$ is $K$-analytic. These results supplement earlier work of Talagrand, Preiss, Cascales, Ferrando, Kąkol, López Pellicer and Saxon.