Order 2m Research Articles

Blocks with Defect Group $\boldsymbol{Q_{2^n}\times C_{2^m}}$ and $\boldsymbol{SD_{2^n}\times C_{2^m}}$

We determine the numerical invariants of blocks with defect group \(Q_{2^n}\times C_{2^m}\) and \(SD_{2^n}\times C_{2^m}\), where \(Q_{2^n}\) denotes a quaternion group of order 2n, \(C_{2^m}\) denotes a cyclic group of order 2m, and \(SD_{2^n}\) denotes a semidihedral group of order 2n. This generalizes Olsson’s results for m = 0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).

High Order Stable Finite Difference Methods for the Schrödinger Equation

In this paper we extend the Summation-by-parts-simultaneous approximation term (SBP-SAT) technique to the Schrodinger equation. Stability estimates are derived and the accuracy of numerical approximations of interior order 2m, m=1,2,3, are analyzed in the case of Dirichlet boundary conditions. We show that a boundary closure of the numerical approximations of order m lead to global accuracy of order m+2. The results are supported by numerical simulations.

Existence of normal bimagic squares

In this paper we provide a construction of normal bimagic squares by means of a magic pair of orthogonal general bimagic squares. It is shown that a normal bimagic square of order mn exists for all positive integers m,n such that m,n∉{2,3,6} and m≡n(mod2), and a normal bimagic square of order 4m exists if and only if m≥2.

THE SYMMETRIC GENUS OF 2-GROUPS

AbstractLet G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

Defining Sets and Critical Sets in (0,1)‐Matrices

AbstractIf D is a partially filled‐in (0, 1)‐matrix with a unique completion to a (0, 1)‐matrix M (with prescribed row and column sums), we say that D is a defining set for M. If the removal of any entry of D destroys this property (i.e. at least two completions become possible), we say that D is a critical set for M. In this note, we show that the complement of a critical set for a (0, 1)‐matrix M is a defining set for M. We also study the possible sizes (number of filled‐in cells) of defining sets for square matrices M with uniform row and column sums, which are also frequency squares. In particular, we show that when the matrix is of even order 2m and the row and column sums are all equal to m, the smallest possible size of a critical set is precisely m2. We give the exact structure of critical sets with this property. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 253–266, 2013

Butterfly Graphs with Shell Orders m and 2m+1 are Graceful

A graceful labelling of an un directed graph G with n edges is a one-one function from the set of vertices V(G) to the set {0, 1, ,2, . . ., n} such that the induced edge labels are all distinct. An induced edge label is the absolute difference between the two end vertex labels. A shell graph is defined as a cycle Cn with (n - 3) chords sharing a common end point called the apex . A double shell is one vertex union of two shells. A bow graph is defined to be a double shell in which each shell has any order. In this paper we define a butterfly graph as a bow graph with exactly two pendant edges at the apex and we prove that all butterfly graphs with one shell of order m and the other shell of order (2m + 1) are graceful.

Characterizations of Hardy spaces associated to higher order elliptic operators

In this paper, the authors first show that the classical Hardy space H1(Rn) can be characterized by the non-tangential maximal functions and the area integrals associated with the semigroups e−tP and e−tP, respectively, where P is an elliptic operator with real constant coefficients of homogeneous order 2m (m⩾1). Moreover, the authors also prove that H1(Rn) can be characterized by the Riesz transforms ∇mP−1/2 if and only if m is an odd integer. In the main part of this paper, the authors develop a theory of Hardy space associated with L, where L is a higher order divergence form elliptic operator with complex bounded measurable coefficients. The authors set up a molecular Hardy space HL1(Rn) and give its characterizations by area integrals related to the semigroups e−tL and e−tL, respectively. Finally, authors give the (HL1,L1) boundedness of Riesz transforms, square functions and maximal functions associated with L.

On the asymptotic [formula omitted]-structure of invariant differential operators on symplectic manifolds

We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of Sn, which extends to an action of Sn+1. We study this structure viewing n as a parameter, in the sense of Deligneʼs category. For manifolds of dimension 2d, we show that the isotypic part of this space of ⩽2d+1-th tensor powers of the reflection representation h=Cn of Sn+1 is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of ⩽4-th tensor powers of the reflection representation.We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational fractions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from Z/(n+1) to Sn+1 where n is viewed as a parameter.We show that the space of invariant operators of order 2m has polynomial dimension in n of degree equal to 2m, while the part not coming from Poisson polynomials has polynomial dimension of degree ⩽2m−3. We use this to compute asymptotics of the dimension of invariant operators. We also give new bounds on the order of invariant operators for a fixed n.We apply this to the Poisson and Hochschild homology associated to the singularity C2dn/Sn+1. Namely, the canonical surjection from HP0(OC2dn/Sn+1,OC2dn) to grHH0(Weyl(C2dn)Sn+1,Weyl(C2dn)) (the Brylinski spectral sequence in degree zero) restricts to an isomorphism in the aforementioned isotypic part h⊗⩽2d+1, and also in h⊗⩽4. We prove a partial converse. Finally, the kernel of the entire surjection has dimension on the order of 1n3 times the dimension of the homology group.

The minimum identifying code graphs

Let G be a graph and B(u) be the set of u with all of its neighbors in G. A set S of vertices is called an identifying code of G if, for every pair of distinct vertices u and v, both B(u)∩S and B(v)∩S are nonempty and distinct. A minimum identifying code of a graph G is an identifying code of G with minimum cardinality and M(G) is the cardinality of a minimum identifying code for G. A minimum identifying code graph G of order n is a graph with M(G)=⌈log2(n+1)⌉ having the minimum number of edges. Moncel (2006) [5] constructed minimum identifying code graphs of order 2m−1 for m≥2 and left the same problem but for arbitrary order open. In this paper, we aimed at the construction of connected minimum identifying code graphs in order to solve this problem for integer order greater than or equal to 4. Furthermore, we discussed some related properties.

Augmentation quotients for complex representation rings of dihedral groups

Denote by Dm the dihedral group of order 2m. Let ℛ(Dm) be its complex representation ring, and let Δ(Dm) be its augmentation ideal. In this paper, we determine the isomorphism class of the n-th augmentation quotient Δn(Dm)/Δn+1(Dm) for each positive integer n.

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m − 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.

PAPERS PUBLISHED IN ARCHAEOLOGY, ETHNOLOGY & ANTHROPOLOGY OF EURASIA IN 2011

This paper presents innovative contributions to the fields of cardinal spline interpolation and subdivision. In particular, it unifies cardinal Br-spline fundamental functions for interpolation that are made of r=ML+1 (L∈N∪{0}) distinct pieces between each pair of interpolation nodes and are featured by the properties of C2M−2 smoothness, approximation order 2M and support width 2M(r+1)r, with the basic limit functions of a special class of non-stationary subdivision schemes of arity M.After introducing a general result, we focus our attention on the subclass of fourth-order accurate, C2 smooth Br-splines with maximum width of the compact support 6. The binary subdivision scheme yielding these fundamental functions outperforms the existing interpolatory schemes and seems to be the most adequate starting point to obtain compactly supported fundamental (spline) functions for local interpolation over quadrilateral and triangular meshes.

Arc-regular cubic graphs of order four times an odd integer

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein---Walter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2,q)) containing PSL(2,q) for some odd prime-power q. Also a construction is given for infinitely many such graphs--namely a family of Cayley graphs for the groups PSL(2,p 3) where p is an odd prime; the smallest of these has order 9828.

On pointed Hopf algebras over dihedral groups

Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provides for each dihedral group infinitely many non-trivial new examples.

Partial Latin Squares Are Avoidable

A square array is avoidable if for each set of n symbols there is an n × n Latin square on these symbols which differs from the array in every cell. The main result of this paper is that for m ≥ 2 any partial Latin square of order 4m − 1 is avoidable, thus concluding the proof that any partial Latin square of order at least 4 is avoidable.

Levi quasivarieties of exponent p s

For an arbitrary class M of groups, L(M) denotes a class of all groups G the normal closure of any element in which belongs to M; qM is a quasivariety generated by M. Fix a prime p, p ≠ 2, and a natural number s, s ≥ 2. Let qF be a quasivariety generated by a relatively free group in a class of nilpotent groups of class at most 2 and exponent p s , with commutator subgroups of exponent p. We give a description of a Levi class generated by qF. Fix a natural number n, n ≥ 2. Let K be an arbitrary class of nilpotent groups of class at most 2 and exponent 2 n , with commutator subgroups of exponent 2. Assume also that for all groups in K, elements of order 2 m , 0 < m < n, are contained in the center of a given group. It is proved that a Levi class generated by a quasivariety qK coincides with a variety of nilpotent groups of class at most 2 and exponent 2 n , with commutator subgroups of exponent 2.

INFLUENCE OF STRONGLY CLOSED 2-SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

AbstractLet H ≤ K be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever ag ∈ K, where a ∈ H, g ∈ G, then ag ∈ H. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order 2m (and 4 if m = 1) of a 2-Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory13(2) (2010), 267–276).

THE REAL GENUS OF 2-GROUPS II

Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that if G is a group of order 2m that has real genus congruent to 5 (mod 8), then either G has exponent 2m~3 and a dihedral subgroup of index 4 or else the exponent of G is 2m~2. It follows that there are at most 53 isomorphism types of 2-groups with real genus congruent to 5 (mod 8). A consequence of this bound is that almost all positive integers that are the real genus of a 2-group are congruent to 1 (mod 8). We also prove that almost all positive integers that are the real genus of a 2-group with Frattini-class 2 are in a set that has density zero in the positive integers.

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of $\Lambda_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles

We consider the homogeneous Dirichlet problem in the unit disk K ⊂ R 2 for a general typeless differential equation of any even order 2m, m ≥ 2, with constant complex coefficients whose characteristic equation has multiple roots ± i. For each value of multiplicity of the roots i and – i, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order.