Finite p-Groups All of Whose Proper Quotient Groups are Abelian or Inner-Abelian

In this article, finite p-groups all of whose proper quotient groups are abelian or inner-abelian are classified. As a corollary, finite p-group all of whose proper quotient groups are abelian, and finite p-groups all of whose proper sections are abelian or inner-abelian are also classified.     

Some new thin sets of integers in harmonic analysis

We randomly construct various subsets A of the integers which have both smallness and largeness properties. They are small since they are very close, in various senses, to Sidon sets: the continuous functions with spectrum in Λ have uniformly convergent series, and their Fourier coefficients are in ℓ_p for all p > 1; moreover, all the Lebesgue spaces L _Λ ^q are equal forq < +∞. On the other hand, they are large in the sense that they are dense in the Bohr group and that the space of the bounded functions with spectrum in Λ is nonseparable. So these sets are very different from the thin sets of integers previously known.     

Planes and processes

This article presented to Combinatorics 2006 is a survey of finite projective planes and the processes used to construct them. All non-translation planes are described, fundamental processes in translation planes are defined and some of these are used to connect semi-field flocks with symplectic spreads. Hermitian ovoids are connected to extensions of derivable nets, and three types of ‘lifting’ methods are discussed. Furthermore, hyperbolic fibrations and ‘regulus-inducing’ central collineation groups are connected to flocks of quadratic cones. Finally, hyper-reguli and multiple hyper-regulus replacement are considered.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

Functions of a matrix and Krylov matrices

For a given nonderogatory matrix A, formulas are given for functions of A in terms of Krylov matrices of A. Relations between the coefficients of a polynomial of A and the generating vector of a Krylov matrix of A are provided. With the formulas, linear transformations between Krylov matrices and functions of A are introduced, and associated algebraic properties are derived. Hessenberg reduction forms are revisited equipped with appropriate inner products and related properties and matrix factorizations are given.     

On the integral equation formulations of some 2D contact problems

Two integral equations, representing the mechanical response of a 2D infinite plate supported along a line and subject to a transverse concentrated force, are examined. The kernels of the integral operators are of the type (x−y)ln|x−y| and (x−y)²ln|x−y|. In spite of the fact that these are only weakly singular, the two equations are studied in a more general framework, which allows us to consider also solutions having non-integrable endpoint singularities. The existence and uniqueness of solutions of the equations are discussed and their endpoint singularities detected.Since the two equations are of interest in their own right, some properties of the associated integral operators are examined in a scale of weighted Sobolev type spaces. Then, new results on the existence and uniqueness of integrable solutions of the equations that in some sense are complementary to those previously obtained are derived.     

Construction of Cartesian authentication codes from unitary geometry

In the present paper several constructions of Cartesian authentication codes from unitary geometry over finite fields are presented and their size parameters are computed. Assuming that the encoding rules are chosen according to a uniform probability distribution, the probabilities P _I and P _S of a successful impersonation attack and a successful substitution attack, respectively, of these codes are also computed. Moreover, those codes so constructed, for which P _I and P _S are nearly optimal, are also determined.     

Generalized canonical analysis for time series

Canonical correlation analysis is shown to be equivalent to the problem of estimating a linear regression matrix, B₀, of less than full rank. After reparameterizing B₀ some estimates of the new parameters, obtained by solving an eigenvalue problem and closely related to canonical correlations and vectors, are found to be consistent and efficient when the residuals are mutually independent. When the residuals are generated by an autocorrelated, stationary time series these estimates are still consistent and obey a central limit theorem, but they are no longer efficient. Alternative, more general estimates are suggested which are efficient in the presence of serial correlation. Asymptotic theory and iterative computational procedures for these estimates are given. A likelihoodratio test for the rank of B₀ is seen to be an extension of a familiar test for canonical correlations.     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

On torsionless and dense acts

Torsionless acts over a monoid S are investigated, in particular torsionless factor acts of 2 -free and 1 -free acts. Monoids over which free or projective acts are torsionless and vice versa are characterized. Some necessary conditions for torsionless acts to be principally weakly flat, weakly flat or strongly flat are given. First results on dense acts are mentioned and several examples, mostly on the basis of cofree acts, are presented to illustrate these concepts. August 15, 2000     

Generalizations of M-matrices which may not have a nonnegative inverse

Generalizations of M-matrices are studied, including the new class of GM-matrices. The matrices studied are of the form sI-B with B having the Perron-Frobenius property, but not necessarily being nonnegative. Results for these classes of matrices are shown, which are analogous to those known for M-matrices. Also, various splittings of a GM-matrix are studied along with conditions for their convergence.     

On left (θ, ϕ)-derivations in BCI-algebras

The notion of (regular) left (θ, ϕ)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. Conditions for a left (θ, ϕ)-derivation to be regular are provided. The concepts of a d_(θ,ϕ)-invariant left (θ, ϕ)-derivation and θ-ideal are introduced, and their relations are discussed. Furthermore, some more interesting results are established.     

On a signless Laplacian spectral characterization of -shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified.     

Topological nearrings whose additive groups are Euclidean

Let (G,+) be a group with a locally compact Hausdorff topology for which the binary operation + is continuous. Those, binary operation * onG for which (G, +, *) is a topological nearring are described. In the case whereG is abelian, those binary operations * for which (G, +, *) is a topological ring are also described. Versions of these results are then obtained in the special case where the group is the topological Euclideann-group,R ⁿ. A family of binary operations * for which (R ⁿ, +, *)_is a topological nearring is then investigated in some detail. Most of these nearrings turn out to be planar. Their ideals are completely determined and we characterize those nearrings which are simple. The multiplicative semi-groups (R ⁿ, *) of these nearrings are then investigated. Green's relations are completely determined and it is shown that a number of familiar properties of semigroups are equivalent for these particular semigroups. Finally, all those binary operations * for which (R, +, *) is a topological nearring are completely described. It is determined when any two of these nearrings are isomorphic and for each of these nearrings, its automorphism group, is completely determined.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

Orthogonality connected with integral means and characterizations of inner product spaces

Kikianty and Dragomir (Math Inequal Appl 13:1–32, 2010) introduced the p?HH norms on the Cartesian square of a normed space, which are equivalent, but are geometrically different, to the well-known p-norms. In this paper, notions of orthogonality in terms of the 2?HH norm are introduced; and their properties are studied. Some characterizations of inner product spaces are established, as well as a characterization of strictly convex spaces.     

Quasi-symmetric 2, 3, 4-designs

Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.