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In the context of constant--dimension subspace codes, an important problem is to determine the largest possible size $A_q(n, d; k)$ of codes whose codewords are $k$-subspaces of $\mathbb{F}_q^n$ with minimum subspace distance $d$. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant--dimension subspace codes for many parameters, including $A_q(10, 4; 5)$, $A_q(12, 4; 4)$, $A_q(12, 6, 6)$ and $A_q(16, 4; 4)$., 17 pages; construction for A_(10,4;5) was flawed

Minimal rank-metric codes or, equivalently, linear cutting blocking sets are characterized in terms of the second generalized rank weight, via their connection with evasiveness properties of the associated $q$-system. Using this result, we provide the first construction of a family of $\mathbb{F}_{q^m}$-linear MRD codes of length $2m$ that are not obtained as a direct sum of two smaller MRD codes. In addition, such a family has better parameters, since its codes possess generalized rank weights strictly larger than those of the previously known MRD codes. This shows that not all the MRD codes have the same generalized rank weights, in contrast to what happens in the Hamming metric setting., 31 pages; published in Annali di Matematica Pura ed Applicata

In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an \begin{document}$ n $\end{document}-dimensional vector space over the finite field \begin{document}$ {\mathbb F_{q}} $\end{document}. All these codes admit the general linear group \begin{document}$ {{{{\rm{GL}}}}}(n,q) $\end{document} as a transitive automorphism group.

Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

Several classes of near-MDS codes of ${\rm PG}(3,q)$ are described. They are obtained either by considering the intersection of an elliptic quadric ovoid and a Suzuki-Tits ovoid of a symplectic polar space ${\cal W}(3, q)$ or starting from the $q+1$ points of a twisted cubic of ${\rm PG}(3, q)$. As a by-product two classes of complete caps of ${\rm PG}(4,q)$ of size $2q^2-q \pm \sqrt{2q} + 2$ are exhibited.

This paper is devoted to finding out some realization of the concept of b-metric like space. First, we attain a fixed point for two fuzzy mappings satisfying a suitable requirement of contractiveness. Subsequently, we apply such a result to graphic contractions. Also, we attain a unique solution for a system of integral equations, and lastly we give an application to ensure that there exists a common bounded solution of a suitable functional equation in dynamic programming.

In [2] and [18] are presented the first two families of maximum scattered F q -linear sets of the projective line PG ( 1 , q n ) . More recently in [22] and in [5] , new examples of maximum scattered F q -subspaces of V ( 2 , q n ) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the F q -linear sets presented in [22] and in [5] , for n = 6 , 8 , are new. Also, for q odd, q ≡ ± 1 , 0 ( mod 5 ) , we present new examples of maximum scattered F q -linear sets in PG ( 1 , q 6 ) , arising from trinomial polynomials, which define new F q -linear MRD-codes of F q 6 × 6 with dimension 12, minimum distance 5 and left idealiser isomorphic to F q 6 .

A set (Formula presented.), (Formula presented.) is a super-Vandermonde set if (Formula presented.) for (Formula presented.). We determine the structure of super-Vandermonde sets of size (Formula presented.) (almost small) and size (Formula presented.) (almost large).

The purpose of this paper is to study the solution of two systems of nonlinear integral equations via fixed point results in a complete dislocated b-metric space. Also the notion of graphic contractions on a closed set for two families of graph dominated multivalued mappings is introduced. Our results generalize some previous results in the existing literature.

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant dimension subspace codes. We also construct two new non-equivalent $(6, (q^3-1)(q^2+q+1), 4; 3)_q$ constant dimension subspace orbit-codes.

We provide sufficient and necessary conditions for the coefficients of a $q$-polynomial $f$ over $\mathbb{F}_{q^n}$ which ensure that the number of distinct roots of $f$ in $\mathbb{F}_{q^n}$ equals the degree of $f$. We say that these polynomials have maximum kernel. As an application we study in detail $q$-polynomials of degree $q^{n-2}$ over $\mathbb{F}_{q^n}$ which have maximum kernel and for $n\leq 6$ we list all $q$-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary $q$-polynomial. Analogous results are proved for $q^s$-polynomials as well, where $\gcd(s,n)=1$., Comment: Revised version, final version to appear in Finite Feilds and Their Applications. We added an Appendix with some more details regarding calculations and a proof for Theorem 2.2 to make the paper self-contained

In this article, we study the approximation of common zeros of non-self inverse strongly monotone operators defined on a closed convex subset C of a Hilbert space H. For a non-self family of operators, we introduce an iterative algorithm without relying on projections. Approximation of common fixed points for finite families of non-self strict pseudo-contractions in the sense of Browder-Petryshyn is also obtained. The novelty of our algorithm is that the coefficients are not given a priori and no assumptions are made on them, but they are constructed step by step in a natural way.

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If \(T:C\rightarrow H\) is a non-self and k-strict pseudocontractive mapping, we can define a map \(v:C\rightarrow \mathbb {R}\) by \(\, v(x):=\inf \{\lambda \ge 0:\lambda x+(1-\lambda )Tx\in C\}.\) Then, for a fixed \(x_{0}\in C\) and for \(\alpha _{0}:=\max \{k,v(x_{0})\},\) we define the Krasnoselskii–Mann algorithm \(x_{n+1}=\alpha _{n}x_{n}+(1-\alpha _{n})Tx_{n},\) where \(\alpha _{n+1}=\max \{\alpha _{n},v(x_{n+1})\}.\) So, here the coefficients \(\alpha _{n}\) are not chosen a priori, but built step by step. We prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ⁡ ( 5 , q 2 ) {\operatorname{PG}(5,q^{2})} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, whose associated semifield has center containing 𝔽 q {\mathbb{F}_{q}} . Equivalently, we classify, up to isotopy, commutative semifields of order q 6 {q^{6}} , for q 2 > 2 ⋅ 3 8 {q^{2}>2\cdot 3^{8}} odd, with middle nucleus containing 𝔽 q 2 {\mathbb{F}_{q^{2}}} and center containing 𝔽 q {\mathbb{F}_{q}} .

A matrix approach to approximating solutions of variational inequalities in Hilbert spaces is introduced. This approach uses two matrices: one for iteration process and the other for regularization. Ergodicity and convergence (both weak and strong) are studied. Our methods combine new or well-known iterative methods (such as the original Mann’s method) with regularized processes involved regular matrices in the sense of Toeplitz.

Let H be a Hilbert space, (Wn)nN a suitable family of map- pings, S a nonexpansive mapping and D a strongly monotone operator. We are interested in the strong convergence of the general scheme

By exploring some geometry of Segre varieties and Veronese varieties, new families of non linear maximum rank distance codes and optimal constant rank codes are provided.

The maximal cliques of the graph NU ( 4 , q 2 ) related to the Hermitian surface of PG ( 3 , q 2 ) and of the graph NO ± ( 2 n + 2 , q ) , q even, n ≥ 1 , are classified.

We introduce the concept of fundamental sequence for a finite graded poset X which is also a discrete dynamical model. The concept of fundamental sequence is a refinement of the concept of parallel convergence time for these models. We compute the parallel convergence time and the fundamental sequence when X is the finite lattice $P(n,r)$ of all the signed integer partitions ${a}_{r},\dots ,{a}_{1},{b}_{1},\dots ,{b}_{n-r}$ such that $r\ge {a}_{r}\ge \cdots \ge {a}_{1}\ge 0\ge {b}_{1}\ge \cdots \ge {b}_{n-r}\ge -(n-r)$ , where $n\ge r\ge 0$ , and when X is the sublattice $P(n,d,r)$ of all the signed integer partitions of $P(n,r)$ having exactly d nonzero parts.

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, 2015), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, 2015) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof.

We designed cylindrical AlGaAs-on-aluminium-oxide all-dielectric nanoantennas with magnetic dipole resonance at near-infrared wavelengths. Our choice of material system offers a few crucial advantages with respect to the silicon-oninsulator platform for operation around 1.55μm wavelength: absence of two-photon absorption, high χ(2) nonlinearity, and the perspective of a monolithic integration with a laser. We analyzed volume second-harmonic generation associated to a magnetic dipole resonance in these nanoantennas, and we predict a conversion efficiency exceeding 10-3 with 1GW/cm2 of pump intensity.

An implicit algorithm for finding common fixed points of an uncountable family of nonexpansive mappings is proposed. A new inexact iteration method is also proposed for countable family of nonexpansive mappings. Several strong convergence theorems based on our main results are established in the setting of Banach spaces. Both algorithms are applied for finding zeros of accretive operators and for solving convex minimization, split feasibility and equilibrium problems.

In this article, we propose a modified Korpelevich's method for solving variational inequalities. Under some mild assumptions, we show that the suggested method converges strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.

An equilibrium theory is developed in Hadamard manifolds. The existence of equilibrium points for a bifunction is proved under suitable conditions, and applications to variational inequality, fixed point and Nash equilibrium problems are provided. The convergence of Picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points.

Kantor and Williams (Trans Am Soc 356:895---938, 2004) introduced a family of non-desarguesian symplectic semifields of even order and studied a number of structures connected with such semifields; namely, symplectic spreads, orthogonal spreads and Z 4-linear codes. Also, they provided equivalence results concerning such objects, although under certain field restrictions. In this article we will succeed in removing such hypotheses.

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.

Starting by a finite family of mappings, we define the concept of procedure with Lipschitzian dependence of the coefficients. We give seven concrete examples of such procedures and prove the strong convergence of two viscosity methods.

Let and be two integers such that ; we denote by the minimum [maximum] number of the nonnegative partial sums of a sum , where are real numbers arbitrarily chosen in such a way that of them are nonnegative and the remaining are negative. We study the following two problems:which are the values ofandfor eachand,?ifis an integer such that, can we findreal numbers,such thatof them are nonnegative and the remainingare negative with,such that the number of the nonnegative sums formed from these numbers is exactly?

We characterize the relative compactness of subsets of the space \({\mathcal{BC}^m([0,+\infty [;E)}\) of bounded and m-differentiable functions defined on [0, +∞[ with values in a Banach space E. Moreover, we apply this characterization to prove the existence of solutions of a boundary value problem in Banach spaces.

In this paper, we present iteration schemes to weakly and strongly approximate common fixed points of a finite family of a class of strict pseudocontractions.

In this paper, we introduce a new iterative scheme for finding a common element of the set of common solutions of a finite family of equilibrium problems with relaxed monotone mappings, of the set of common solutions of a finite family of variational inequalities and of the set of common fixed points of an infinite family of nonexpansive mappings in a Hilbert space. Strong convergence for the proposed iterative scheme is proved. As an application, we solve a multi-objective optimization problem using the result of this paper. Our results improve and extend the corresponding ones announced by others.

In this paper, we study rank two semifields of order q^6 that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) [12] for q=1(mod3). Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with q=1(mod3); for any q=2^2^h, such that h=1(mod3) and for any q=3^h with h@?0(mod3). Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields.

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Mainge [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence...

An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances of parameter selections. Applications in hierarchical minimization problems are also included.

We use strongly pseudocontraction to regularize the following ill-posed monotone variational inequality: finding a point with the property such that , where , are two pseudocontractive self-mappings of a closed convex subset of a Hilbert space with the set of fixed points . Assume the solution set of (VI) is nonempty. In this paper, we introduce one implicit scheme which can be used to find an element . Our results improve and extend a recent result of (Lu et al. 2009).

In this paper we show that starting from a symplectic semifield spread $${\mathcal{S}}$$ of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of $${\mathcal{S}}$$ , and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s 2, s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings { T n } with common fixed points, an equilibrium function G , a contraction f with coefficient 0 α 1 and a strongly positive linear bounded operator A with coefficient γ ¯ > 0 . Let 0 γ γ ¯ α . We define a suitable Mann type algorithm which strongly converges to the unique solution of the minimization problem min x ∈ C 1 2 〈 A x , x 〉 − h ( x ) , where h is a potential function for γ f and C is the intersection of the equilibrium points and the common fixed points of the sequence { T n } .

In this paper, we introduce an Ishikawa implicit iterative process with errors for a finite family of N asymptotically nonexpansive mappings as follows: x n = ( 1 - α n - γ n ) x n - 1 + α n T i ( n ) k ( n ) y n + γ n u n , y n = ( 1 - β n - δ n ) x n + β n T i ( n ) k ( n ) x n + δ n v n , n ⩾ 1 , where, for any n ∈ N fixed, k(n) − 1 denotes the quotient of the division of n by N and i(n) the rest, i.e. n = ( k ( n ) - 1 ) N + i ( n ) , i ( n ) ∈ { 1 , … , N } . The sequences {αn}, {βn}, {γn}, {δn} are four real sequences in [0, 1] satisfying αn + γn ⩽ 1 and βn + δn ⩽ 1 for all n ⩾ 1, {un}, {vn} are two bounded sequences and x0 is a given point. In the setting of uniformly convex Banach spaces we give some results of weak and strong convergence of the above iterative process. The results presented here are situated on the line of research of the corresponding results of Sun [J. Math. Anal. Appl. 286 (1) (2003) 351–358], Osilike [J. Math. Anal. Appl. 294 (1) (2004) 73–81], Chang et al. [J. Math. Anal. Appl. 313 (1) (2006) 273–283], Gu [J. Math. Anal. Appl. 329 (2) (2007) 766–776], Huang and Noor [Appl. Math. Comput. 190 (1) (2007) 356–361], Su and Qin [Appl. Math. Comput. 186 (1) (2007) 271–278), Zhou et al. [Appl. Math. Comput. 173 (1) (2006) 196–212] and some others.

We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup (T(s))s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common element z of the set of the equilibrium points and the set of common fixed points of (T(s))s≥0. Such common element z is the unique solution of a variational inequality, which is the optimality condition for a minimization problem.

In this paper, we suggest and analyze a hybrid algorithm for pseudo-contractive mappings in Hilbert spaces. Further, we prove the strong convergence of the proposed iterative algorithm for Lipschitz pseudo-contractive mappings in Hilbert spaces.

By using a modification of the hybrid method in mathematical programming, we give some new algorithm to find out a fixed point of mappings which are in some sense non-expansive. Our results hold in reflexive, strictly convex, smooth Banach spaces with the Kadec˘–Klee property.

Let H be a Hilbert space. Consider on H a sequence of nonexpansive mappings { T n } with common fixed points, a finite family of equilibrium functions { G i } i = 1 , … , K , a contraction f with coefficient 0 α 1 and a strongly positive linear bounded operator A with coefficient γ ¯ > 0 . Let 0 γ γ ¯ / α . Assuming there are common equilibrium points of the family { G i } i = 1 , … , K which are also fixed points for { T n } , we define a suitable sequence which strongly converges to the unique such point which also satisfies the variational inequality 〈 ( A − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 for all the x in the intersection of the equilibrium points and the common fixed points of the sequence { T n } .

In this paper, we introduce a modified Mann iterative process for approximating a common fixed point of a finite family of strict pseudo-contractions in Hilbert spaces. We establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the recent ones announced by many others.

Let C be a nonempty closed convex subset of a real Hilbert space H. Let Q:C→C be a fixed contraction and S,T:C→C be two nonexpansive mappings such that Fix(T)≠∅. Consider the following two-step iterative algorithm: $$\begin{array}{@{}rll}x_{n+1}&=&\alpha_{n}Qx_{n}+(1-\alpha_{n})Ty_{n},\\[1.5mm]y_{n}&=&\beta_{n}Sx_{n}+(1-\beta_{n})x_{n},\quad n\geq0.\end{array}$$ It is proven that under appropriate conditions, the above iterative sequence {x n } converges strongly to $\tilde{x}\in \mathrm{Fix}(T)$ which solves some variational inequality depending on a given criterion S, namely: find $\tilde{x}\in H$ ; $0\in (I-S)\tilde{x}+N_{\mathrm{Fix}(T)}\tilde{x}$ , where N Fix(T) denotes the normal cone to the set of fixed points of T.

In [G. Marino, O. Polverino, R. Trombetti, On F"q-linear sets of PG(3,q^3) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769-788] it has been proven that there exist six non-isotopic families F"i (i=0,...,5) of semifields of order q^6 with left nucleus F"q"^"3 and center F"q, according to the different geometric configurations of the associated F"q-linear sets. In this paper we first prove that any semifield of order q^6 with left nucleus F"q"^"3, right and middle nuclei F"q"^"2 and center F"q is isotopic to a cyclic semifield. Then, we focus on the family F"4 by proving that it can be partitioned into three further non-isotopic families: F"4^(^a^), F"4^(^b^), F"4^(^c^) and we show that any semifield of order q^6 with left nucleus F"q"^"3, right and middle nuclei F"q"^"2 and center F"q belongs to the family F"4^(^c^).

We show that the generalized quadrangle W"3(q) for odd q has exponentially many 12(q+1)-ovoids, thus implying that the generalized quadrangle Q(4,q) has exponentially many hemisystems for odd q. For q even, we show that W"3(q) has m-ovoids for all integers m, 1=

We give new sharper valuations for the optimal retraction constant in some Banach spaces.