Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Generalized rough approximations in

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call -inner and -outer approximations respectively ( being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic by means of step functions from the hypercube [0,1]^k and taking value in an arbitrary (finite) subset of . Then, we will show that if a fuzzy set is definable as truth table of a formula of , then both its -inner and -outer approximation are definable as truth table of formulas of . Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.     

On connected line sets of antiflag class in

We investigate the partial linear spaces, fully embedded in an affine space with the property that for every antiflag {p,L}, the number of lines through p intersecting L is either 0,α, or q. Besides some general results we prove a complete classification of those geometries fully embedded in an affine plane of order q and of the connected geometries with 1<α<q, fully embedded in .     

Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

Weyl sums in with digital restrictions

Let be a finite field and consider the polynomial ring . Let . A function , where G is a group, is called strongly Q-additive, if f(AQ+B)=f(A)+f(B) holds for all polynomials with degB<degQ. We estimate Weyl sums in restricted by Q-additive functions. In particular, for a certain character E we study sums of the form

where is a polynomial with coefficients contained in the field of formal Laurent series over and the range of P is restricted by conditions on f_i(P), where f_i (1ir) are Q_i-additive functions. Adopting an idea of Gel'fond such sums can be rewritten as sums of the form

with . Sums of this shape are treated by applying the kth iterate of the Weyl–van der Corput inequality and studying higher correlations of the functions f_i. With these Weyl sum estimates we show uniform distribution results.     

Varieties of modules for

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety , that is the set of pairs of (n×n)-matrices (A,B) such that A²=B²=[A,B]=0, is equidimensional. can be identified with the ‘variety of n-dimensional modules’ for , or equivalently, for k[X,Y]/(X²,Y²). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic >2. We also prove that if e²=0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of as a direct sum of indecomposable components of varieties of -modules.     

Chromatic Turán problems and a new upper bound for the Turán density of

We consider a new type of extremal hypergraph problem: given an r-graph and an integer k≥2 determine the maximum number of edges in an -free, k-colourable r-graph on n vertices.Our motivation for studying such problems is that it allows us to give a new upper bound for an old Turán problem. We show that a 3-graph in which any four points span at most two edges has density less than , improving previous bounds of due to de Caen [D. de Caen, Extension of a theorem of Moon and Moser on complete subgraphs, Ars Combin. 16 (1983) 5–10], and due to Mubayi [D. Mubayi, On hypergraphs with every four points spanning at most two triples, Electron. J. Combin. 10 (10) (2003)].     

-structure on the deformation complex of a morphism

-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is the extension of a -algebra by an associative algebra. Actions of -algebras on associative and -algebras are analyzed; extensions of -algebras by associative and -algebras that they act upon are constructed. The resulting -algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the -algebra on the deformation complex of the corresponding diagram algebra.     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

On Feferman’s operational set theory

We study and some of its most important extensions primarily from a proof-theoretic perspective, determine their consistency strengths by exhibiting equivalent systems in the realm of traditional set theory and introduce a new and interesting extension of which is conservative over .     

Cohomology and deformations of the infinite-dimensional filiform Lie algebra

Denote the infinite-dimensional -graded Lie algebra defined by basis e_i, i1, and relations [e₁,e_i]=e_i+1 for all i2. We compute in this article the bracket structure on , and in relation to this, we establish that there are only finitely many true deformations of in each non-positive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.     

On the -fuzzy topological spaces

As a natural generalization of fuzzy metric spaces due to George and Veeramani [George A, Veeramani P. On some result in fuzzy metric space. Fuzzy Sets Syst 1994;64:395–9], the present author defined the notion of -fuzzy metric spaces. In this paper we prove some known results of metric spaces including Uniform continuity theorem and Ascoli–Arzela theorem for -fuzzy metric spaces. We also prove that every -fuzzy metric space has a countably locally finite basis and use this result to conclude that every -fuzzy metric space is metrizable.     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

The implicitization problem for

We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in , based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions.Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of , and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.     

Classification of marginally trapped Lorentzian flat surfaces in and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in with light-like mean curvature vector.     

Minimum Lee weights of Type II codes over

Among binary images of a Type II code over with respect to the trace-orthogonal bases, it has been unknown whether the minimum Hamming weights depend on the choice of bases or not; the minimum Hamming weights of the binary images of known Type II codes have been independent of the choice of bases.In this paper we give a complete list of Type II codes over of length 8 up to a certain equivalence. This list shows us an example of a Type II code of which minimum Lee weight depends on the choice of a basis.     

Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.