Finitely additive representation of spaces

Let be any atomless and countably additive probability measure on the product space with the usual σ-algebra. Then there is a purely finitely additive probability measure λ on the power set of a countable subset such that can be isometrically isomorphically embedded as a closed subspace of L^p(λ). The embedding is strict. It is also ‘canonical,’ in the sense that it maps simple and continuous functions on to their restrictions to T.     

(Semi-)Riemannian geometry of (para-)octonionic projective planes

We use reduced homogeneous coordinates to construct and study the (semi-)Riemannian geometry of the octonionic (or Cayley) projective plane , the octonionic projective plane of indefinite signature , the para-octonionic (or split octonionic) projective plane and the hyperbolic dual of the octonionic projective plane . We also show that our manifolds are isometric to the (para-)octonionic projective planes defined classically by quotients of Lie groups.     

Propagating sharp group homology decompositions

A collection of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either the subgroups in or their centralizers or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset which imply comparison results.     

Configurations in abelian categories. II. Ringel–Hall algebras

This is the second in a series on configurations in an abelian category . Given a finite poset (I,), an (I,)-configuration (σ,ι,π) is a finite collection of objects σ(J) and morphisms ι(J,K) or in satisfying some axioms, where J,KI. Configurations describe how an object X in decomposes into subobjects.The first paper defined configurations and studied moduli spaces of (I,)-configurations in , using the theory of Artin stacks. It showed well-behaved moduli stacks of objects and configurations in exist when is the abelian category coh(P) of coherent sheaves on a projective scheme P, or mod- of representations of a quiver Q.Write for the vector space of -valued constructible functions on the stack . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication * on using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that is a -algebra. We also study representations of , the Lie subalgebra of functions supported on indecomposables, and other algebraic structures on .Then we generalize all these ideas to stack functions , a universal generalization of constructible functions, containing more information. When Extⁱ(X,Y)=0 for all and i>1, or when for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in .     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

On the Brown–Shields conjecture for cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function to be cyclic, i.e. for to be dense in . This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in iff it is outer and its zero set (defined appropriately) is of capacity zero.     

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 .     

Topological complexity is a fibrewise L–S category

Topological complexity of a space B is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version of topological complexity with an additional condition: in a robot motion planning, a motion must be stasis if the initial and the terminal states are the same. Our main goal is to show the equalities and , where d(B)=B×B is a fibrewise pointed space over B whose projection and section are given by the canonical projection to the second factor and the diagonal. In addition, our method in studying fibrewise L–S category is able to treat a fibrewise space with singular fibres.     

Limits of small functors

For a small category enriched over a suitable monoidal category , the free completion of under colimits is the presheaf category . If is large, its free completion under colimits is the -category of small presheaves on , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on .     

Levels of multi-continued fraction expansion of multi-formal Laurent series

The multi-continued fraction expansion of a multi-formal Laurent series is a sequence pair consisting of an index sequence and a multi-polynomial sequence . We denote the set of the different indices appearing infinitely many times in by H_∞, the set of the different indices appearing in by H₊, and call |H_∞| and |H₊| the first and second levels of , respectively. In this paper, it is shown how the dimension and basis of the linear space over F(z) (F) spanned by the components of are determined by H_∞ (H₊), and how the components are linearly dependent on the mentioned basis.     

On the complexity of matrix reduction over finite fields

In this paper we study the complexity of matrix elimination over finite fields in terms of row operations, or equivalently in terms of the distance in the Cayley graph of generated by the elementary matrices. We present an algorithm called striped matrix elimination which is asymptotically faster than traditional Gauss–Jordan elimination. The new algorithm achieves a complexity of row operations, and operations in total, thanks to being able to eliminate many matrix positions with a single row operation. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices. Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from , with n and q small. Finally we consider an extension from finite fields to finite semifields of the matrix reduction problem. We give a conjecture on the behaviour of a natural analogue of GL_n for semifields and prove this for a certain class of semifields.     

Inheritance of hyper-duality in imprimitive Bose–Mesner algebras

We prove the following result concerning the inheritance of hyper-duality by block and quotient Bose–Mesner algebras associated with a hyper-dual pair of imprimitive Bose–Mesner algebras. Let and denote Bose–Mesner algebras. Suppose there is a hyper-duality ψ from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to . Also suppose that is imprimitive with respect to a subset of Hadamard idempotents, so is dual imprimitive with respect to the subset of primitive idempotents, where is the formal duality associated with ψ. Let denote the block Bose–Mesner algebra of on the block containing p, and let denote the quotient Bose–Mesner algebra of with respect to . Then there is a hyper-duality from the subconstituent algebra of with respect to p to the subconstituent algebra of with respect to .     

Stationary twists and energy minimizers on a space of measure preserving maps

Let be a bounded Lipschitz domain, a suitably quasiconvex integrand and consider the energy functional

over the space of measure preserving maps

In this paper we discuss the question of existence of multiple strong local minimizers for over . Moreover, motivated by their significance in topology and the study of mapping class groups, we consider a class of maps, referred to as twists, and examine them in connection with the corresponding Euler–Lagrange equations and investigate various qualitative properties of the resulting solutions, the stationary twists. Particular attention is paid to the special case of the so-called p-Dirichlet energy, i.e., when .     

Formal extension and quotients of the invariant holonomic system

Let be a semisimple Lie algebra and a Cartan subalgebra of . Fix . Let be the invariant holonomic system (see [R. Hotta, M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984) 327–358]). First we investigate its formal extension . In the sequel we calculate the characteristic variety of some simple quotients of and its Fourier transform .     

Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

Irreducible complex skew-Berger algebras

Irreducible skew-Berger algebras , i.e. algebras spanned by the images of the linear maps satisfying the Bianchi identity, are classified. These Lie algebras can be interpreted as irreducible complex Berger superalgebras contained in .     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

The Burnside ring of fusion systems

Given a saturated fusion system on a finite p-group S we define a ring modeled on the Burnside ring of finite groups. We show that these rings have several properties in common. When is the fusion system of G we describe the relationship between these rings.