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In this paper, we construct new self-dual codes from a construction that involves a unique combination; \begin{document}$ 2 \times 2 $\end{document} block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings \begin{document}$ \mathbb{F}_2 $\end{document} , \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document} . Using extensions and neighbours of codes, we construct \begin{document}$ 32 $\end{document} new self-dual codes of length \begin{document}$ 68 $\end{document} . We construct 48 new best known singly-even self-dual codes of length 96.

A Boolean function in \begin{document}$ n $\end{document} variables is 2-rotation symmetric if it is invariant under even powers of \begin{document}$ \rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1) $\end{document} , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of \begin{document}$ \rho^2 $\end{document} to a single monomial. If the quartic MRS 2-function in \begin{document}$ 2n $\end{document} variables has a monomial \begin{document}$ x_1 x_q x_r x_s $\end{document} , then we use the notation \begin{document}$ {2-}(1,q,r,s)_{2n} $\end{document} for the function. A detailed theory of equivalence of quartic MRS 2-functions in \begin{document}$ 2n $\end{document} variables was given in a \begin{document}$ 2020 $\end{document} paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called \begin{document}$ mf1 $\end{document} and \begin{document}$ mf2 $\end{document} in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions \begin{document}$ {2-}(1,q,r,s)_{2n} $\end{document} (with \begin{document}$ q say), \begin{document}$ n = s, s+1, \ldots $\end{document} can be shown to satisfy. This problem was solved for the \begin{document}$ mf1 $\end{document} case only in the \begin{document}$ 2020 $\end{document} paper. Using new ideas about "short" functions, Cusick and Cheon found formulas for the \begin{document}$ mf2 $\end{document} weights in a \begin{document}$ 2021 $\end{document} sequel to the \begin{document}$ 2020 $\end{document} paper. In this paper the actual recursions for the weights in the \begin{document}$ mf2 $\end{document} case are determined.

In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The Matching Cut problem is the problem of deciding whether a given graph has a matching cut. Let H be a fixed undirected graph. A vertex coloring of an undirected input graph G is said to be an H - Free Coloring if none of the color classes contain H as an induced subgraph. The H - Free Chromatic Number of G is the minimum number of colors required for an H - Free Coloring of G . Both The Matching Cut problem and the H - Free Coloring problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of f ( | | φ | | , t ) n O ( 1 ) , where | | φ | | is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. The dependency of f ( | | φ | | , t ) on | | φ | | can be as bad as a tower of exponentials. In this paper, we provide explicit combinatorial FPT algorithms for Matching Cut problem and H - Free Coloring problem, parameterized by the tree-width of G . The single exponential FPT algorithm for the Matching Cut problem answers an open question posed by Kratsch and Le (2016). The techniques used in the paper are also used to provide an FPT algorithm for a variant of H -free coloring, where H is forbidden as a subgraph (not necessarily induced) in the color classes of G .

An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star graph $S_n$ consists of $n!$ vertices, each labelled with a distinct permutation of $[n]$. Two vertices are adjacent if their labels differ exactly in the first and one other position. $S_n$ is an $(n-1)$-regular, vertex-transitive graph with diameter $\lfloor 3(n-1)/2 \rfloor$. Orientations of $S_n$, called unidirectional star graphs and distributed routing protocols over them were studied by Day and Tripathi [Information Processing Letters (1993)] and Fujita [The First International Symposium on Computing and Networking (CANDAR 2013)]. Fujita showed that the (directed) diameter of this unidirectional star graph $\overrightarrow{S_n}$ is at most $\lceil{5n/2}\rceil + 2$. In this paper, we propose a new distributed routing algorithm for the same $\overrightarrow{S_n}$ analysed by Fujita, which routes a packet from any node $s$ to any node $t$ at an undirected distance $d$ from $s$ using at most $\min\{4d+4, 2n+4\}$ hops. This shows that the (directed) diameter of $\overrightarrow{S_n}$ is at most $2n+4$. We also show that the diameter of $\overrightarrow{S_n}$ is at least $2n$ when $n \geq 7$, thereby showing that our upper bound is tight up to an additive factor., Full version of the paper to be presented in CALDAM 2020

The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all $$R^{K}_{{\varvec{0}}}$$ -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an $$ER^{K}$$ -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature.

Carsten Thomassen and I proved that in any graph G , the number of cycles containing a specified edge as well as all the vertices of odd degree is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Shunichi Toida and where all vertices have odd degree it is Andrew Thomason’s extension of Smith’s Theorem. Our proof was not algorithmic. In this paper, I extend Thomason’s algorithm to one which, in a non-eulerian graph, finds a second cycle containing a specified edge and all the odd-degree vertices. Kenneth Berman extended Thomason’s Theorem to spanning trees: he proved that if T is a spanning tree such that the degree of every vertex in G − E ( T ) is odd, then there is an even number of spanning trees of G with the same degree as T at each vertex. In this paper, I give a common generalization of all of these theorems to a parity theorem about “good” trees in general graphs. My proof provides an algorithm for finding a second “good” tree.

We consider the model Y = X + ξ where Y is observable, ξ is a noise random variable with density f ξ , X has an unknown mixed density such that P ( X = X c ) = 1 − p , P ( X = a ) = p with X c being continuous and p ∈ ( 0 , 1 ) , a ∈ R . Typically, in the last decade, the model has been widely considered in a number of papers for the case of fully known quantities a , f ξ . In this paper, we relax the assumptions and consider the parametric error ξ ∼ σ N ( 0 , 1 ) with an unknown σ > 0 . From i.i.d. copies Y 1 , … , Y m of Y we will estimate ( σ , p , a , f X c ) where f X c is the density of X c . We also find the lower bound of convergence rate and verify the minimax property of established estimators.

For k ≥ 2 , spanning trees T 1 , T 2 , … , T k in a graph G are called to be completely independent if for any two distinct vertices x and y , the paths connecting them in T 1 , T 2 , … , T k are pairwise openly disjoint. We call such spanning trees T 1 , T 2 , … , T k the completely independent spanning trees (CIST for short). Recently, Hong and Zhang (2020) found that a sufficient condition for Hamiltonian graphs also suffices the existence of two CISTs. That is, if G is a graph with n vertices, | N ( x ) ∪ N ( y ) | ≥ n 2 and | N ( x ) ∩ N ( y ) | ≥ 3 for every two non-adjacent vertices x , y of G and n ≥ 5 , then G has two CISTs. More recently, Qin et al. (2020) proposed that the restriction on the number of vertices in the statement should be n ≥ 8 , and then pointed out that the Claim 1 in Hong’s paper is not always true for general case, which was corrected by presenting an amendment. However, there still exists a flaw in the corresponding revised proof (see details from the second part of our paper). Accordingly, we give a new amendment to correct the proof.

Given a graph H and an integer k ≥ 1 , the Gallai–Ramsey number G R k ( H ) is defined to be the minimum integer n such that every k -edge coloring of the complete graph K n contains either a rainbow (all different colored) triangle or a monochromatic copy of H . In this paper, we study Gallai–Ramsey numbers for graphs with chromatic number three such as K m for m ≥ 2 , where K m is a kipas with m + 1 vertices obtained from the join of K 1 and P m , and a class of graphs with five vertices, denoted by ℋ . We first study the general lower bound of such graphs and propose a conjecture for the exact value of G R k ( K m ) . Then we give a unified proof to determine the Gallai–Ramsey numbers for many graphs in ℋ and obtain the exact value of G R k ( K 4 ) for k ≥ 1 . Our outcomes not only indicate that the conjecture on G R k ( K m ) is true for m ≤ 4 , but also imply several results on G R k ( H ) for some H ∈ ℋ which are proved individually in different papers.

The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex spanning tree T and an additional set L of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T cup F. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O(log^2 n)-competitive algorithm and showed an Omega(log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O(log n)-competitive algorithm for online WTAP, which is tight up to constant factors.

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α =1/2. Several realization problems are also generated in this paper.

Given a positive integer n and an r-uniform hypergraph F, the Turán number ex(n, F ) is the maximum number of edges in an F -free r-uniform hypergraph on n vertices. The Turán density of F is defined as π(F)=limn→∞ex(n,F)(rn)\pi \left( F \right) = {\lim _{n \to \infty }}{{ex\left( {n,F} \right)} \over {\left( {_r^n} \right)}}. The Lagrangian density of F is π (F ) = sup{r!λ(G) : G is F -free}, where λ(G) is the Lagrangian of G. Sidorenko observed that π(F ) ≤ π (F ), and Pikhurko observed that π(F ) = π (F ) if every pair of vertices in F is contained in an edge of F . Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. For example, in the paper [A hypergraph Turán theorem via Lagrangians of intersecting families, J. Combin. Theory Ser. A 120 (2013) 2020–2038], Hefetz and Keevash studied the Lagrangian densitiy of the 3-uniform graph spanned by {123, 456} and the Turán number of its extension. In this paper, we show that the Lagrangian density of the 3-uniform graph spanned by {123, 234, 456} achieves only on K53K_5^3 . Applying it, we get the Turán number of its extension, and show that the unique extremal hyper-graph is the balanced complete 5-partite 3-uniform hypergraph on n vertices.

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction: { ( − Δ ) p s u + ( − Δ ) q s u + V ( e x ) ( | u | p − 2 u + | u | q − 2 u ) = h ( u ) + | u | r − 2 u in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 , in R N , } where e is a positive parameter, 0 s 1 , 2 ⩽ p q N / s , ( − Δ ) t s ( t ∈ { p , q } ) is the fractional t-Laplace operator, while V : R N ↦ R and h : R ↦ R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r = q s ⁎ : = N q / ( N − s q ) or r > q s ⁎ . The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that e is small enough and in close relationship with the set where the potential V attains its minimum.

In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper normal subgroup $N$ with the property that the automorphism group of the digraph fixes each $N$-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup., Comment: 27 pages

Let G be a connected plane graph with vertex set V and edge set E. For X ∈ {V, E, V ∪ E}, two elements of X are facially adjacent in G if they are incident elements, adjacent vertices, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A coloring of G is facial with respect to X if there is a coloring of elements of X such that facially adjacent elements of X receive different colors. A facial coloring of G is odd if for every face f and every color c, either no element or an odd number of elements incident with f is colored by c. In this paper we investigate odd facial colorings of trees. The main results of this paper are the following: (i) Every tree admits an odd facial vertex-coloring with at most 4 colors; (ii) Only one tree needs 6 colors, the other trees admit an odd facial edge-coloring with at most 5 colors; and (iii) Every tree admits an odd facial total-coloring with at most 5 colors. Moreover, all these bounds are tight.

In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Belt\'an, U. Etayo, J. Marzo and J. Ortega-Cerd\`a, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer., Comment: 21 pages

An approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by$\mathcal{J}_{\nu , \lambda }^{\mu }$Jν,λμ, where$\mu >0$μ>0and$\lambda ,\nu \in \mathbb{C\ }$λ,ν∈Cget increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμby applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, themth derivative of$\mathcal{J}_{\nu ,\lambda }^{\mu }$Jν,λμis considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions$\mathcal{I}_{\nu ,\lambda }^{\mu }$Iν,λμdefined by$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$Iν,λμ(z)=i−2λ−νJν,λμ(iz)of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for$\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$Iν,λμ(z)are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

Fibonacci cubes are induced subgraphs of hypercube graphs obtained by restricting the vertex set to those binary strings which do not contain consecutive 1s. This class of graphs has been studied extensively and generalized in many different directions. Induced subgraphs of the hypercube on binary strings with restricted runlengths as vertices define Fibonacci-run graphs. These graphs have the same number of vertices as Fibonacci cubes, but fewer edges and different graph theoretical properties. Basic properties of Fibonacci-run graphs are presented in a companion paper, while in this paper we consider the nature of the degree sequences of Fibonacci-run graphs. The generating function we obtain is a refinement of the generating function of the degree sequences, and has a number of corollaries, obtained as specializations. We also obtain several properties of Fibonacci-run graphs viewed as a partially ordered set, and discuss its embedding properties.

International audience; In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent $a$ of type (species) $i$ interacts with an agent $b$ of type (species) $j$ with $a$ as the initiator, then $b$'s type becomes $i$ with probability $P_{ij}$. In such an interaction, we think of $a$ as the predator, $b$ as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models andgeneralize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random.We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an $n$-agent population converges to some absorbing state in time polynomial in $n$, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but ``control'' convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a ``coin-flip consensus'' in a distributed system. Some of our techniques may be of independent value.

Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, $$\mathcal {A}(G)=\lbrace \alpha \in Aut(G) :x\alpha (x)=\alpha (x)x~~for~ all~ x\in G\rbrace $$ is equal to the group of all central automorphisms, $$Aut_{c}(G)$$ , except only for five ones. Also, we determined the structure of $$Aut_{c}(G)$$ and $$\mathcal {A}(G)$$ for these five groups. Using these results, in this paper, we find the structure of $$\mathcal {A}(G)=Aut_{c}(G)$$ for the remaining 2-groups of almost maximal class. Also, we prove the following results

In this paper, we prove some new characterizations of two weighted functions u and v in norm inequalities of Hardy’s type, in the context of dynamic inequalities on time scales $${\mathbb {T}}$$ . These norm inequalities studied the boundedness of the operator of Hardy’s type between the weighted spaces $$ L_{v}^{p}({\mathbb {T}})$$ and $$L_{u}^{q}(\mathbb {T)}$$ . The paper covers the different cases when $$1

In the paper, Wisła (J Math Anal Appl 483(2):123659, 2020, 10.1016/j.jmaa.2019.123659), it was proved that the classical Orlicz norm, Luxemburg norm and (introduced in 2009) p-Amemiya norm are, in fact, special cases of the s-norms defined by the formula $$\left\| x\right\| _{\Phi ,s}=\inf _{k>0}\frac{1}{k}s\left( \int _T \Phi (kx)d\mu \right) $$ x Φ , s = inf k > 0 1 k s ∫ T Φ ( k x ) d μ , where s and $$\Phi $$ Φ are an outer and Orlicz function respectively and x is a measurable real-valued function over a $$\sigma $$ σ -finite measure space $$(T,\Sigma ,\mu )$$ ( T , Σ , μ ) . In this paper the strict monotonicity, lower and upper uniform monotonicity and uniform monotonicity of Orlicz spaces equipped with the s-norm are studied. Criteria for these properties are given. In particular, it is proved that all of these monotonicity properties (except strict monotonicity) are equivalent, provided the outer function s is strictly increasing or the measure $$\mu $$ μ is atomless. Finally, some applications of the obtained results to the best dominated approximation problems are presented.

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

The quadratic matrix equation C ∗ X ∗ A X C + C ∗ X ∗ B + B ∗ X C + D = 0 , where A ∈ ℂ n × n , B ∈ ℂ n × p , C ∈ ℂ m × p and D ∈ ℂ p × p are given complex matrices with A ≥ 0 and D ∗ = D , and X ∈ ℂ n × m is a variable matrix to be determined, was first discussed by Fujioka and Hara (1994). The purpose of this paper is to provide an alternative approach to solve this matrix equation. The necessary and sufficient conditions for the equation to have solutions are presented and the general solution of the equation is provided when the stated conditions are satisfied. In contrast to the method proposed by Fujioka and Hara, the strategy adopted in this paper is from the special case to the general one, and the results seem to be more compact.

In this paper we study L p Marcinkiewicz-Zygmund type inequalities c 1 ∑ 1 ≤ j ≤ N w j | g ( x j ) | p ≤ ‖ g ‖ L p ( K ) p ≤ c 2 ∑ 1 ≤ j ≤ N w j | g ( x j ) | p for general exponential sums of the form g ( x ) = ∑ 1 ≤ j ≤ n a j e 〈 λ j , x 〉 , x , λ j ∈ R d , a j ∈ R . One of the main results of the paper asserts that when 1 ≤ p ∞ , K = [ a , b ] and the exponents satisfy relations λ j + 1 − λ j ≥ ϵ , 1 ≤ j ≤ n − 1 , max 1 ≤ j ≤ n ⁡ | λ j | ≤ Λ then Marcinkiewicz-Zygmund type inequalities hold for certain point sets of cardinality N ≤ c n ln 1 p + 1 ⁡ Λ ϵ . Since the dependence of cardinality on the parameters ϵ , Λ appears only in the logarithmic term, this bound is “almost” degree and separation independent. Moreover, the discrete meshes x j and weights w j = x j + 1 − x j are given explicitly and they are universal in the sense that they work for any exponential sum as above. This result will rely on some new degree independent Bernstein-Markov type inequality for exponential sums. Moreover we will extend our considerations to multivariate exponential sums. In addition, it will be shown that much stronger results hold for exponential sums with nonnegative coefficients.

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N 0 ( m , n ) denote the number of odd Durfee symbols of n with odd rank m, and N 0 ( r , m ; n ) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N 0 ( r , 12 ; n ) by utilizing some identities involving Appell-Lerch sums m ( x , q , z ) and a universal mock theta function g ( x , q ) . Based on these formulas, we determine the signs of N 0 ( r , 12 ; 4 n + t ) − N 0 ( s , 12 ; 4 n + t ) for all 0 ≤ r , s ≤ 6 and 0 ≤ t ≤ 3 . In particular, we prove that N 0 ( 2 , 12 ; 4 n + 1 ) = N 0 ( 4 , 12 ; 4 n + 1 ) . Moreover, let D k 0 ( n ) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D 2 0 ( 8 n + s ) and D 3 0 ( 16 n + t ) are even with s ∈ { 4 , 6 } and t ∈ { 1 , 9 , 11 , 13 } which were confirmed by Wang. In this paper, we found new congruences for D k 0 ( n ) . In particular, for k = 2 or 3, we give characterizations of n such that D k 0 ( n ) is odd and prove that D k 0 ( n ) take even values with probability 1 for n ≥ 0 .

Congruences for total number parts functions associated to ranks of partitions are recently studied in many papers. In particular, Chan, Osburn and the author proved congruences for N T ‾ ( b , k , n ) (resp. N T 2 ‾ ( b , k , n ) ) which denotes the total number of parts in the overpartitions of n with D-rank (resp. M 2 -rank) congruent to b modulo k. In this paper, we replace these congruences by equalities.

In this paper, based on the theory of \begin{document}$ \mathbb{Z}_{4} $\end{document}-valued quadratic forms we propose several classes of generalized Boolean bent functions over \begin{document}$ \mathbb{Z}_{4} $\end{document}, and new families of codebooks are constructed from these functions. The codebooks constructed in this paper are nearly optimal with respect to the Welch bound, and their parameters are new. Furthermore, some Boolean bent functions are also derived.

In this paper, we generalize the notion of self-orthogonal codes to \begin{document}$ \sigma $\end{document}-self-orthogonal codes over an arbitrary finite ring. Then, we study the \begin{document}$ \sigma $\end{document}-self-orthogonality of constacyclic codes of length \begin{document}$ p^s $\end{document} over the finite commutative chain ring \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}, where \begin{document}$ p $\end{document} is a prime, \begin{document}$ u^2 = 0 $\end{document} and \begin{document}$ \sigma $\end{document} is an arbitrary ring automorphism of \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. We characterize the structure of \begin{document}$ \sigma $\end{document}-dual code of a \begin{document}$ \lambda $\end{document}-constacyclic code of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. Further, the necessary and sufficient conditions for a \begin{document}$ \lambda $\end{document}-constacyclic code to be \begin{document}$ \sigma $\end{document}-self-orthogonal are provided. In particular, we determine all \begin{document}$ \sigma $\end{document}-self-dual constacyclic codes of length \begin{document}$ p^s $\end{document} over \begin{document}$ \mathbb F_{p^m} + u \mathbb F_{p^m} $\end{document}. In the end of this paper, when \begin{document}$ p $\end{document} is an odd prime, we extend the results to constacyclic codes of length \begin{document}$ 2 p^s $\end{document}.

Quaternary sequences with optimal autocorrelation property are preferred in applications. Cyclotomic classes of order 4 are widely used in the constructions of quaternary sequences due to the convenience of defining a quaternary sequence with the cyclotomic classes of order 4 as its support set. Recently, several classes of optimal quaternary sequences of period \begin{document}$ 2p $\end{document}, which are all closely related to the cyclotomic classes of order 4 with respect to \begin{document}$ \mathbb{Z}_p $\end{document} were introduced in the literature. However, less attention has been paid to the equivalence between these known results. In this paper, we introduce the unified form of this kind of quaternary sequences to classify these known results and then conclude the unified forms of these optimal quaternary sequences. By doing this, we disclose the relationship between the optimal quaternary sequences derived from different methods in the literature on one hand. And on the other hand, when the new obtained optimal quaternary sequence period is \begin{document}$ 2p $\end{document} and the cyclotomic classes of order 4 are involved, the methods and the results given in this paper can be used to identify if the sequence is new or not.

A real interval vector/matrix is an array whose entries are real intervals. In this paper, we consider the real linear interval equations \begin{document}$ \bf{Ax} = \bf{b} $\end{document} with \begin{document}$ {{\bf{A}} }$\end{document}, \begin{document}$ \bf{b} $\end{document} respectively, denote an interval matrix and an interval vector. The aim of the paper is to study the numerical solution of the linear interval equations for various classes of coefficient interval matrices. In particular, we study the convergence of interval AOR method when the coefficient interval matrix is either interval strictly diagonally dominant matrices, interval \begin{document}$ L $\end{document}-matrices, interval \begin{document}$ M $\end{document}-matrices, or interval \begin{document}$ H $\end{document}-matrices.

In this paper, we present two constructions of low-hit-zone frequen-cy-hopping sequence (LHZ FHS) sets. The constructions in this paper generalize the previous constructions based on \begin{document}$ m $\end{document}-sequences and \begin{document}$ d $\end{document}-form functions with difference-balanced property, and generate several classes of optimal LHZ FHS sets and LHZ FHS sets with optimal periodic partial Hamming correlation (PPHC).

In this paper, permutation polynomials of the form x d + L ( x s ) over finite fields F q 3 are investigated, where q = 2 m and m is a positive integer. By means of the iterative method, the multivariate method and the resultant elimination, six classes of permutation trinomials are obtained from certain integers d, s and linearized polynomials L ( x ) . It is also showed that the permutations proposed in this paper are quasi-multiplicative inequivalent to all known permutation trinomials over F q 3 .

For convex body M and star body Q in R n , we introduce a new concept of the ( p , q ) -projection body Π p , q ( M , Q ) , as an extension of L p -projection body Π p M . For p ≥ 1 and convex bodies K , L in R n , the L p -Shephard problem asks whether Π p K ⊆ Π p L implies that | K | n − p ≤ | L | n − p ? Here | M | is the volume of M. It has shown in [21] , [22] , [24] that the answer is affirmative when n ≤ 2 and negative when n ≥ 3 . In this paper, we consider a variant of the L p -Shephard problem via ( p , q ) -projection bodies, i.e., if Π p K ⊆ Π p , q ( M , Q ) ⊆ Π p L , for any fixed Q and q ∈ R , can we compare the volume of K , M and L? The problem includes two parts with comparing | M | and | L | when Π p , q ( M , Q ) ⊆ Π p L , and comparing | K | and | M | when Π p K ⊆ Π p , q ( M , Q ) . This paper answers two parts of the question by connecting with L p -projection body, and Levy's embedding L p . Note that in the case of q = n (or Q = M ) the question is exactly the classic L p -Shephard problem.

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

We investigate the density of compactly supported smooth functions in the Sobolev space W k , p on complete Riemannian manifolds. In the first part of the paper, we extend to the full range p ∈ [ 1 , 2 ] the most general results known in the Hilbertian case. In particular, we obtain the density under a quadratic Ricci lower bound (when k = 2 ) or a suitably controlled growth of the derivatives of the Riemann curvature tensor only up to order k − 3 (when k > 2 ). To this end, we prove a gradient regularity lemma that might be of independent interest. In the second part of the paper, for every n ≥ 2 and p > 2 we construct a complete n -dimensional manifold with sectional curvature bounded from below by a negative constant, for which the density property in W k , p does not hold for any k ≥ 2 . We also deduce the existence of a counterexample to the validity of the Calderon–Zygmund inequality for p > 2 when Sec ≥ 0 , and in the compact setting we show the impossibility to build a Calderon–Zygmund theory for p > 2 with constants only depending on a bound on the diameter and a lower bound on the sectional curvature.

Denote by C α ( D ) the space of the functions f on the unit disk D which are Holder continuous with the exponent α, and denote by C 1 , α ( D ) the space which consists of differentiable functions f such that their derivatives are in the space C α ( D ) . Let C be the Cauchy transform of Dirichlet problem. In this paper, we obtain the norm estimates of ‖ C ‖ L p → L q , where 3 / 2 p 2 and q = p / ( p − 1 ) . Suppose g ∈ L p ( D ) and f = G [ g ] is the Green potential of g. By using Sobolev embedding theorem, we show that if 1 p ≤ 2 , then f ∈ C μ ( D ) , where μ = 2 − 2 / p . We also show that if 2 p ∞ , then f ∈ C 1 , ν ( D ) , where ν = 1 − 2 / p . Finally, for the case p = ∞ , we show that f is not necessarily in C 1 , 1 ( D ) , but its gradient, i.e., | ∇ f | is Lipschitz continuous with respect to the pseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [9] .

This paper studies finite sample minimax optimal randomization schemes and estimation schemes in estimating parameters including the average treatment effect, when treatment effects are heterogeneous. A randomization scheme is a distribution over a group of permutations of a given treatment assignment vector. An estimation scheme is a joint distribution over assignment vectors, linear estimators, and permutations of assignment vectors. The key element in the minimax problem is that the worst case is over a class of distributions of the data which is invariant to a group of permutations. First, I show that given any assignment vector and any estimator, the uniform distribution over the same group of permutations, namely the complete randomization scheme, is minimax optimal. Second, under further assumptions on the class of distributions and the objective function, I show the minimax optimal estimation scheme involves completely randomizing an assignment vector, while the optimal estimator is the difference-in-means under complete invariance and a weighted average of within-block differences under a block structure, and the number of treated units is determined by the Neyman allocation.

Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width \begin{document}$ Z $\end{document} and zero odd-period correlation zone (ZOCZ) width \begin{document}$ Z_{odd} = Z $\end{document} by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.

The Langberg-Medard multiple unicast conjecture claims that for any strongly reachable \begin{document}$ k $\end{document} -pair network, there exists a multi-flow with rate \begin{document}$ (1,1,\dots,1) $\end{document} . In this paper, we examine an optimization problem \begin{document}$ \mathcal P_{\mathcal S_k} $\end{document} such that its optimal value \begin{document}$ \mathcal O_{\mathcal S_k} $\end{document} naturally gives a lower bound on the multi-flow rate for the strongly reachable \begin{document}$ k $\end{document} -pair network. We first prove \begin{document}$ \lim_{k\rightarrow \infty}\mathcal O_{\mathcal S_k} = \frac{9}{8} $\end{document} and then show that the multi-flow \begin{document}$ \mathcal C^*_k $\end{document} constructed in our previous work is asymptotically optimal for \begin{document}$ \mathcal P_{\mathcal S_k} $\end{document} and optimal if and only if \begin{document}$ k = 1, 2, 6, 10 $\end{document} .

Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting subject of study in recent years. The construction of optimal cyclic codes over finite fields is important as they have maximal minimum distance once the length and dimension are given. In this paper, we present two classes of new optimal ternary cyclic codes \begin{document}$ \mathcal{C}_{(2,v)} $\end{document} by using monomials \begin{document}$ x^2 $\end{document} and \begin{document}$ x^v $\end{document} for some suitable \begin{document}$ v $\end{document} and explain the novelty of the codes. Furthermore, the weight distribution of \begin{document}$ \mathcal{C}_{(2,v)}^{\perp} $\end{document} for \begin{document}$ v = \frac{3^{m}-1}{2}+2(3^{k}+1) $\end{document} is determined.

In 2002 Mauduit and Sarkozy started to study finite sequences of \begin{document}$ k $\end{document} symbols \begin{document}$ E_{N} = \left(e_{1},e_{2},\cdots,e_{N}\right)\in \mathcal{A}^{N}, $\end{document} where \begin{document}$ \mathcal{A} = \left\{a_{1},a_{2},\cdots,a_{k}\right\}(k\in \mathbb{N},k\geq 2) $\end{document} is a finite set of \begin{document}$ k $\end{document} symbols. Later many pseudorandom sequences of \begin{document}$ k $\end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the \begin{document}$ k $\end{document} -ary Sidel'nikov sequences with length \begin{document}$ q-1 $\end{document} by using the estimates for certain character sums with exponential function, where \begin{document}$ q $\end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size \begin{document}$ \phi(q-1) $\end{document} of \begin{document}$ k $\end{document} -ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that \begin{document}$ k = o(q^{\frac{1}{4}}) $\end{document} , where \begin{document}$ \phi $\end{document} denotes Euler's totient function.

In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.

This paper mainly study \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document} -additive codes. A Gray map from \begin{document}$ \mathbb{Z}_{2}^{\alpha}\times\mathbb{Z}_{4}^{\beta}[u] $\end{document} to \begin{document}$ \mathbb{Z}_{4}^{\alpha+2\beta} $\end{document} is defined, and we prove that is a weight preserving and distance preserving map. A MacWilliams-type identity between the Lee weight enumerator of a \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document} -additive code and its dual is proved. Some properties of one-weight \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document} -additive codes and two-weight projective \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document} -additive codes are discussed. As main results, some construction methods for one-weight and two-weight \begin{document}$ \mathbb{Z}_{2}\mathbb{Z}_{4}[u] $\end{document} -additive codes are studied, meanwhile several examples are presented to illustrate the methods.

For an odd prime \begin{document}$ p $\end{document} and positive integers \begin{document}$ m $\end{document} and \begin{document}$ \ell $\end{document} , let \begin{document}$ \mathbb{F}_{p^m} $\end{document} be the finite field with \begin{document}$ p^{m} $\end{document} elements and \begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document} . Thus \begin{document}$ R_{\ell,m} $\end{document} is a finite commutative non-chain ring of order \begin{document}$ p^{2^{\ell} m} $\end{document} with characteristic \begin{document}$ p $\end{document} . In this paper, we aim to construct quantum codes from skew constacyclic codes over \begin{document}$ R_{\ell,m} $\end{document} . First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

In this paper, we study the structure of duadic codes of an odd length \begin{document}$ n $\end{document} over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} , \begin{document}$ u^2 = 0 $\end{document} , (more generally over \begin{document}$ \mathbb{Z}_{q}+u\mathbb{Z}_{q} $\end{document} , \begin{document}$ u^2 = 0 $\end{document} , where \begin{document}$ q = p^r $\end{document} , \begin{document}$ p $\end{document} a prime and \begin{document}$ (n, p) = 1 $\end{document} ) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} are determined. We present a sufficient condition for abelian codes of the same length over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} to have the same minimum Hamming distance. A new Gray map over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} is defined, and it is shown that the Gray image of an abelian code over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} is an abelian code over \begin{document}$ \mathbb{Z}_4 $\end{document} . We have obtained five new linear codes of length \begin{document}$ 18 $\end{document} over \begin{document}$ \mathbb{Z}_4 $\end{document} from duadic codes of length \begin{document}$ 9 $\end{document} over \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} through the Gray map and a new map from \begin{document}$ \mathbb{Z}_4+u\mathbb{Z}_4 $\end{document} to \begin{document}$ \mathbb{Z}_4^2 $\end{document} .

In this paper, we are concerned about bounds and constructions of optimal \begin{document}$ 2 $\end{document} -D \begin{document}$ (n\times m,3,2,1) $\end{document} -optical orthogonal codes. The exact number of codewords of an optimal \begin{document}$ 2 $\end{document} -D \begin{document}$ (n\times m,3,2,1) $\end{document} -optical orthogonal code is determined for \begin{document}$ n = 2 $\end{document} , \begin{document}$ m\equiv 1 \pmod{2} $\end{document} , and \begin{document}$ n\equiv 1 \pmod{2} $\end{document} , \begin{document}$ m\equiv 1,3,5 \pmod{12} $\end{document} , and \begin{document}$ n\equiv 4 \pmod{6} $\end{document} , \begin{document}$ m\equiv 8 \pmod{16} $\end{document} .

Let \begin{document}$ \mathbb{F}_{q} $\end{document} be a finite field with character \begin{document}$ p $\end{document} . In this paper, the multiplicative group \begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document} is decomposed into a mutually disjoint union of \begin{document}$ \gcd(6l^mp^n,q-1) $\end{document} cosets over subgroup \begin{document}$ $\end{document} , where \begin{document}$ \xi $\end{document} is a primitive element of \begin{document}$ \mathbb{F}_{q} $\end{document} . Based on the decomposition, the structure of constacyclic codes of length \begin{document}$ 6l^mp^n $\end{document} over finite field \begin{document}$ \mathbb{F}_{q} $\end{document} and their duals is established in terms of their generator polynomials, where \begin{document}$ p\neq{3} $\end{document} and \begin{document}$ l\neq{3} $\end{document} are distinct odd primes, \begin{document}$ m $\end{document} and \begin{document}$ n $\end{document} are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length \begin{document}$ 6l^mp^n $\end{document} over \begin{document}$ \mathbb{F}_{q} $\end{document} .

In this paper, for each of six families of three-valued \begin{document}$ m $\end{document} -sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring \begin{document}$ R = \mathbb{F}_2+u\mathbb{F}_2 $\end{document} , where \begin{document}$ u^2 = 0. $\end{document} The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are \begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $\end{document} for some explicit \begin{document}$ r. $\end{document}