Order 2n Research Articles

Rogue Waves in the Nonlinear Schrödinger, Kadomtsev–Petviashvili, Lakshmanan–Porsezian–Daniel and Hirota Equations

We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order 2N. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers PN as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree N(N+1) in x and t by an exponential depending on time t and depending on 2N−2 real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on 2N−2 real parameters. We present different examples of rogue waves for the LPD and Hirota equations.

Transition of the semiclassical resonance widths across a tangential crossing energy-level

We consider a 1D 2×2 matrix-valued operator (1.1) with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order n, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order 2n. Below this level, they have no intersection, which suggests exponentially small widths of resonances (see e.g., [1,2]), while above this level, on the contrary, they intersect at two points, which implies a polynomial order of the widths as proved in [3]. We prove that the transition of the resonance widths near the crossing level is described in terms of a generalized Airy function. This result generalizes [4] to the tangential crossing and [3] to the crossing at a turning point. Our approach is based on the computation of the microlocal transfer matrix at the crossing point between the incoming and outgoing microlocal solutions.

Off-diagonally symmetric domino tilings of the Aztec diamond of odd order

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order 2n−1 are equal when the boundary defect is at the kth position and the (2n−k)th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Characteristic polynomial of maximum and minimum matrix of square power graph of dihedral group of order 2n with odd natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Laplacian polynomial of square power graph of dihedral group of order 2n with even natural number n

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

The Structure of the Generalized Cayley Graph Cay_m (〖 D〗_2n,S) of Valency Three

Suppose that G is a finite group and S is a non-empty subset of G such that e∉S and S^(-1)⊆S. Suppose that Cay(G,S) is the Cayley graph whose vertices are all elements of G and two vertices x and y are adjacent if and only if xy^(-1)∈S. In this paper,we introduce the generalized Cayley graph denoted by Cay_m (G,S) that is a graph with vertex set consists of all column matrices X_m which all components are in G and two vertices X_m and Y_m are adjacent if and only if X_m [(Y_m )^(-1) ]^t∈M(S), where 〖Y_m〗^(-1) is a column matrix that each entry is the inverse of similar entry of Y_m and M(S) is m×m matrix with all entries in S , [Y^(-1) ]^t is the transpose of Y^(-1) and m≥1. We aim to determine the structure of Cay_m (G,S) when G is the dihedral group of order 2n and | S |= 3 for every m≥2, n≥3.

Power Cayley Graphs of Dihedral Groups with Certain Order

Combination of the concepts of power graph and Cayley graph associated to groups has led to the introduction to two new variations of Cayley graph known as the union power Cayley graph and the intersection power Cayley graph. The set of vertices for both graphs consist of the elements of a finite group G. Consider any inverse-closed subset S of G, two vertices x and y are adjacent in the union power Cayley graph if 〖xy〗^(-1)∈S or if either one is an integral power of the other. Furthermore, x and y are adjacent in the intersection power Cayley graph if 〖xy〗^(-1)∈S and if either one is an integral power of the other. In this paper, the generalization of the union power Cayley graphs and the intersection power Cayley graphs of the dihedral groups with order 2n, for n≥3 and n=P^m;P is prime and m is a natural number, relative to a specific subset containing rotation elements in the groups is found. In addition, properties of these graphs including the clique numbers, vertex chromatic numbers, girths and diameters are computed. Finally, the characteristics of the graphs, whether they are connected, regular, complete, and planar are also determined.

Closeness Energy of Non-Commuting Graph for Dihedral Groups

This paper focuses on the non-commuting graph for dihedral groups of order 2n, D2n, where n ≥ 3. We show the spectrum and energy of the graph corresponding to the closeness matrix. The result is that the obtained energy is always twice its spectral radius and is never an odd integer. Moreover, it is classified as hypoenergetic.

Simulations of Cayley graphs of dihedral group

Let Γ be a finite group with identity element e and let S ⊆ Γ − {e} which is inverse-closed, i.e., S = S−1 := {s−1 : s ∈ S}. An undirected Cayley graph on a group Γ with connection set S, denoted by Cay(Γ, S), is a graph with vertex set Γ and edges xy for all pairs x,y ∈ Γ such that xy−1 ∈ S. The dihedral group of order 2n, denoted by D2n, is a group generated by two elements a and b subject to the three relations: an = e, b2 = e, and bab−1 = a−1. In this paper, we investigate the Cayley graphs of the dihedral group for some connection sets S satisfying |S| = 1, 2, 3 by doing simulations using Wolfram Mathematica.

Acentralizers of some Finite Groups

Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.

A note on weighted sums of Vietoris’ sequence

The aim of this paper is to formulate some different sums adapted to the Brousseau sum for Vietoris’ numbers. For this purpose, we firstly give our auxiliary results concerning the sequence of Vietoris’ numbers of order 2𝑛 and 2𝑛 + 1. Moreover, we consider alternating summations and some other weighted sums for these rational number sequence and derive them.

Numerical treatment of Burgers' equation based on weakly L-stable generalized time integration formula with the NSFD scheme

In this study, we present a weakly L-stable convergent time integration formula of order N1>3 (N1∈Z is odd) to solve Burgers' equation. The time integration formula for the initial value problem w′(t)=f(t,w),w(t0)=η0 is formulated using backward explicit Taylor series approximation of order (N1−1) and Hermite approximation polynomial of order (N1−2). We convert the Burgers' equation into the initial value problem using the nonstandard finite difference scheme for spatial derivatives and implement the derived integration formula. The nonstandard finite difference scheme makes it possible to choose several denominator functions. The method's convergence, stability and truncation error are also discussed. To show the correctness and effectiveness of the proposed technique, we present numerical solutions, ‖e‖2 and ‖e‖∞ error norms in several tables and figures. Additionally, the numerical outcomes are compared with the results of some existing techniques and exact solutions.

Four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups

We construct four infinite families of chiral 3-polytopes of type {4,8}, with 1024m4, 2048m4, 4096m4 and 8192m4 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n≥10. (On the other hand, no chiral polytopes of type {4,8} exist for n≤9.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss (2006) [15] and a problem proposed by Pellicer (2012) [13].

Syzygies for the vector invariants of the dihedral group

The dihedral group D2n of order 2n acts on the space of m-tuples of vectors from its defining 2-dimensional representation. The corresponding algebra of polynomial invariants has a natural structure as a module over the general linear group GLm(C). Therefore the ideal of relations between the generators of the algebra of invariants can be treated as a GL-ideal (i.e. an ideal stable with respect to the appropriate action of GLm(C)). It is shown that this GL-ideal is generated by relations depending on no more than 3 vector variables. A minimal GL-ideal generating system is found for the cases of m=2 and an arbitrary n, and for the cases of an arbitrary m and n=4, n=5, and n=6.

On the existence and uniqueness of a positive solution to a boundary value problem for a nonlinear ordinary differential equation of 4n order

The paper considers a two-point boundary value problem with homogeneous boundary conditions for a single nonlinear ordinary differential equation of order 4n. Using the well-known Krasnoselsky theorem on the expansion (compression) of a cone, sufficient conditions for the existence of a positive solution to the problem under consideration are obtained. To prove the uniqueness of a positive solution, the principle of compressed operators was invoked. In conclusion, an example is given that illustrates the fulfillment of the obtained sufficient conditions for the unique solvability of the problem under study.

Sumsets in dicyclic groups Q 4 n and Um,n

Let G be an arbitrary group. For any subsets A and B of G, let A * B = { a * b ; a ∈ A , b ∈ B } where ` * ’ is the binary operation on G. By μ G ( r , s ) , we denote the minimum cardinality of the set A * B , where | A | = r and | B | = s . In 2003, Eliahou et al. proposed a conjecture that for every finite group G of order g and every pair of integers r , s with 1 ≤ r , s ≤ g , k G ( r , s ) ≤ μ G ( r , s ) , where k G ( r , s ) = min d ∈ H ( G ) { ( ⌈ r d ⌉ + ⌈ s d ⌉ − 1 ) d } and H ( G ) is the set of orders of finite subgroups of G. In 2003, Eliahou et al. verified the conjecture for dihedral group D q of index q = p v . In this paper, we determine the upper bound for the size of μ Q 4 n ( r , s ) and μ U m , n ( r , s ) , where Q 4 n = 〈 a , b : a 2 n = e , b 2 = a n , a b = b a − 1 〉 is a dicyclic group of order 4n and U m , n = 〈 a , b : a 2 n = e , b m = e , a b a − 1 = b − 1 〉 and verify the conjecture proposed by Eliahou et al. [7] for Q 4 n for n to be a prime power.

Hamilton cycles passing through a matching in a bipartite graph with high degree sum

Let G be a balanced bipartite graph of order 2n with partite sets X and Y, and M be a matching of k edges in G. Let σ1,1(G)=min⁡{dG(x)+dG(y):x∈X,y∈Y,xy∉E(G)}. Conditions on σ1,1(G) for the existence of a Hamilton cycle passing through M have been studied by many researchers, but the threshold is not known to be sharp for 2n3≤k≤n−2.In this paper we prove that G has a Hamilton cycle passing through all the edges of M if σ1,1(G)≥2n−k and k≤n−2. Moreover, we show that the lower bound on σ1,1(G) is sharp for every k with 2n3≤k≤n−2.

Univariate Gauss quadrature for structural modelling of tissues and materials with distributed fibres

This work presents a method for computing the averaged free energy and constitutive relations in hyperelastic material models with distributed fibres, as they apply to soft fibre-reinforced materials and biological tissues. While these models are currently implemented through either spherical cubature of the fibre free energy or its Taylor series, we here propose a new method based on a univariate Gauss quadrature rule with integration points and weights informed by the statistical moments of the distribution of fibre stretch. As an intrinsic property, the new approach separates the integration of the fibre constitutive law from the integration of the orientation distribution, the latter leading to structural tensors of even order. Provided the latter 2n−1 tensors are computed accurately up to tensorial order 2(2n−1), the method integrates exactly any polynomial of order 2n−1 that agrees with the fibre law at the n integration points. After formally introducing the quadrature method for generally non-affine fibre deformations and arbitrary order, we focus on the important special case of affine fibre kinematics and discuss the rules with n≤3 integration points, for which the corresponding positions and weights are determined analytically. At a computational cost comparable to the existing approaches, the new method does not require the fibre law to be analytic and can thus robustly deal with piece-wise definitions of the fibre energy, in contrast to Taylor-series approaches, and it does not induce additional anisotropy as it can occur with spherical cubature rules. The 3-point rule is further investigated and illustrated in numerical examples relating to soft collagenous tissues based on a Fortran implementation of the method suitable for use in finite element analyses.

Williamson type Hadamard matrices with circulant components

It has been shown that Williamson Hadamard matrices of order 4n do not exist for n=35,47,53,59. We consider the class of Williamson type matrices with circulant (but not necessarily symmetric) components when n=p is an odd prime or n=pq is the product of distinct odd primes each of which is a generator modulo the other, showing that any such matrix may be derived from a Williamson matrix by a common cyclic shift of the components. In particular, this means that there are no Williamson type matrices with circulant components for n=35,47,53,59.