Operators in Krein Space

We study self-adjoint operators in Krein space. Our goal is to show that there is a relationship between the following classes of operators: operators with a compact “corner,” definitizable operators, operators of classes (H) and K(H), and operators of class D _κ ⁺.     

Uniqueness of the group operation on the lattice of order-automorphisms of the real line

A(R) is the lattice-ordered group (l-group) of all order-automorphisms of the real lineR, with the usual pointwise order and “of course” with composition as the group operation. In fact, what other choices are there for a group operation having the same identity that would give anl-group? Composition in the reverse order would work. But there are no other choices — the group operation can be recognized in the lattice. Several classes of abelianl-groups having a unique group operation have been found by Conrad and Darnel, but this is the first non-abelian example having the minimum of two group operations. “Conversely”, Holland has shown that for the groupA(R) under composition, the only lattice orderings yielding anl-group are the pointwise order and its dual. These results also hold for the rational lineQ.     

Exponential box splines

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZ ^s and a vector of constants λ ?R ⁿ, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofR ⁿ—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).     

Projection matrices revisited: a potential-growth indicator and the merit of indication

The mathematics of matrix models for age- and/or stage-structured population dynamics substantiates the use of the dominant eigenvalue λ ₁ of the projection matrix L as a measure of the growth potential, or of adaptation, for a given species population in modern plant or animal demography. The calibration of L = T +F on the “identified-individuals-of-unknown-parents” kind of empirical data determines precisely the transition matrix T, but admits arbitrariness in the estimation of the fertility matrix F. We propose an adaptation principle that reduces calibration to the maximization of λ ₁(L) under the fixed T and constraints on F ensuing from the data and expert knowledge. A theorem has been proved on the existence and uniqueness of the maximizing solution for projection matrices of a general pattern. A conjugated maximization problem for a “potential-growth indicator” under the same constraints has appeared to be a linear-programming problem with a ready solution, the solution testing whether the data and knowledge are compatible with the population growth observed.     

On Dirichlet Type Problems of Polynomial Dirac Equations with Boundary Conditions

Let ${{\bf D}_{\bf x} := \sum_{i=1}^n \frac{\partial}{\partial x_i} e_i}$ be the Euclidean Dirac operator in ${\mathbb{R}^n}$ and let P(X) = a _m X ^m + . . . + a ₁ X +  a ₀ be a polynomial with real coefficients. Differential equations of the form P(D _x )u(x) = 0 are called homogeneous polynomial Dirac equations with real coefficients. In this paper we treat Dirichlet type problems of the a slightly less general form P(D _x )u(x) = f(x) (where the roots are exclusively real) with prescribed boundary conditions that avoid blow-ups inside the domain. We set up analytic representation formulas for the solutions in terms of hypercomplex integral operators and give exact formulas for the integral kernels in the particular cases dealing with spherical and concentric annular domains. The Maxwell and the Klein–Gordon equation are included as special subcases in this context.     

Transformations of discrete closure systems

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Because, for many arbitrary operators α, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems. Our primary interest is in functions f that map power sets 2 ^U into power sets 2 ^U′, which are called transformations. Functions over continuous domains are usually characterized in terms of open sets. When the domains are discrete, closed sets seem more appropriate. In particular, we consider monotone transformations which are “continuous”, or “closed”. These can be used to establish criteria for asserting that “the closure of a transformed image under f is equal to the transformed image of the closure”. Finally, we show that the categories MCont and MClo of closure systems with morphisms given by the monotone continuous transformations and monotone closed transformations respectively have concrete direct products. And the supercategory Clo of MClo whose morphisms are just the closed transformations is shown to be cartesian closed.     

Analysis of Basis Pursuit via Capacity Sets

Finding the sparsest solution α for an under-determined linear system of equations D α=s is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of α that allow its recovery using ? ₁-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of D called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given D, called the capacity sets. We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of D α=s.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Adjoining an Identity to a Reduced Archimedean f-ring, II: Algebras

In “Part I” (presented at Ord05 (Oxford, MS)), we have discussed, for reduced archimedean f-rings, the canonical extension of such a ring, A, to one with identity, uA, and the class U of u-extendable maps (i.e., homomorphisms which lift over the u’s to identity preserving homomorphisms). We showed that U is a category and u becomes a functor from U which is a monoreflection; the maps in U were characterized. This paper addresses the interaction between our functor u, and v , the vector lattice monoreflection in archimedean ?-groups (due to Conrad and Bleier). In short, v restricts to a monoreflection of reduced archimedean f-rings into reduced archimedean f-algebras, ψ ∈ U if and only if v ψ ∈ U, and vu is a monoreflection into reduced archimedean f-algebras with identity. This work was motivated by the question put to us by G. Buskes at Ord05: what maps are o-extendable; i.e., extend over the orthomorphism rings? (The orthomorphism ring oA is a unital extension of uA, and any o-extendable map lies in U.) While a complete answer seems quite complicated (if not hopelessly out of reach), here we shall identify a class of objects D for which oD = vuD and all maps from D lie in U, hence any map from D to a reduced archimedean f-algebra is o-extendable.     

An Extended Faber Operator

Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

Critical sets in 3-space

Given a non-empty compact set C ?R ³, is C the set of critical points for some smooth proper functionf :R ³ →R ₊? In this paper we prove that the answer is “yes” for Antoine’s Necklace and most but not all tame links.     

Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene

We prove that many (non-associative) topological division algebrasD of dimensionn ∈ N over the centreK do not yield topological affine or projective planes (of Lenz-Barlotti type V) in contrast to the results of SKORNJAKOV [20], SALZMANN [18] and [19], GRUNDHÖFER [7], HARTMANN [11] and RINK [17] concerning projective planes coordinatized by compact or special topological ternary fields. In particular, this holds for every non-trivial and non-archimedian valuation topology ofK distinct from the order topology ifK is a real-closed field, and if the division algebraD =K ⁿ carries the product topology.     

On the complete cd-index of a Bruhat interval

We study the non-negativity conjecture of the complete cd-index of a Bruhat interval as defined by Billera and Brenti. For each cd-monomial M we construct a set of paths, such that if a “flip condition” is satisfied, then the number of these paths is the coefficient of the monomial M in the complete cd-index. When the monomial contains at most one d, then the condition follows from Dyer’s proof of Cellini’s conjecture. Hence the coefficients of these monomials are non-negative. We also relate the flip condition to shelling of Bruhat intervals.     

Spin geometry and grand unification

Gravitation becomes unified with quantum mechanics when we recognize that the spacetime tetrads and the matter fields of Fermions are the integral and half-integral spin representations of theEinstein group, E, the global extension of the Poincaré group to a curved spacetimeM. There are8 fundamental spinor representations of theE group, interchanged byP, T, andC: the degree-one maps of spin space overM. Tensor products of2 spinor fields buildClifford vectors or 1 forms, e.g. the spacetime tetrads. It takes tensor products of all8 spinor fields to build a natural 4 form; in particular, ourE-invariant Lagrangian density . We propose a simple form for : the8-spinor factorization of theMaurer-Cartan 4-form, Ω⁴. Thespin connections Ω_α step off the conjoined left and right internalgl (2, ?) phase increments over aspacetime incremente _α. Our actionS _g measures the covering number of the spinor phases over spacetimeM∪D _J; theD _J aresingular domains or caustics, whereJ=1, 2, and 3 chiral pairs of spin waves cross. Here, the massive Dirac equations emerge to govern the mass scattering that keep the “null zig-zags” of a bispinor particle confined to a timelike worldtube. We identify the coupled envelopes of 1, 2, and 3 chiral bispinor pairs as the leptons, mesons, and hadrons, respectively. These source topologically —nontrivialgl (2,C) phase distributions in the far-field region, which appear aseffective vector potentials. Their vorticities are thespin curvatures, whose Hermitian parts —thegravitational curvatures —specify how our spacetime manifoldM must expand and curve to accommodate such anholonomic differentials. The anti-Hermitian parts reproduce the standard electroweak and strong fields, together with their actions. also contains some new cross terms between electroweak potentials and gravitational curvatures. Do these signal a failure of unification, or predict new phenomena?     

Quelques proprietes des systemes generateurs minimaux des monoides

In this paper we give some results about minimal generating systems of a monoïd M. The main tool is a relation denoted “S” which is finer than the relation “J”.     

Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles

The pressure function P(A, s) plays a fundamental role in the calculation of the dimension of “typical” self-affine sets, where A = (A ₁, …,A _k ) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical” self-affine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general sub-additive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles.     

On a class of sofic affine invariant subsets of the 2-torus related to an Erd?s problem

Let 1 < β <?2 be a real number and G be the closed projection on the 2-torus of the (modified) Rademacher graph in base β. The smallest compact containing G and left invariant by the diagonal endomorphism ${(x,y)\mapsto(2x,\beta y)}$ (mod 1) is denoted by K. For β a simple Parry number of PV-type, K is proved to be a sofic affine invariant set with a fractal geometry closed to the one of G. When β is the golden number, we prove the uniqueness of the measure with full Hausdorff dimension on K.     

Generation and Enumeration of Some Classes of Interval Orders

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2?+?2. We provide a recursive method for the unique generation of interval orders of size n?+?1 from those of size n, extending the technique presented by El-Zahar (1989) and then re-obtain the enumeration of this class, as done in Bousquet-Melou et al. (2010). As a consequence we provide a method for the enumeration of several subclasses of interval orders, namely AV(2?+?2, N), AV(2?+?2, 3?+?1), AV(2?+?2, N, 3?+?1). In particular, we prove that the first two classes are enumerated by the sequence of Catalan numbers, and we establish a bijection between the two classes, based on the cardinalities of the principal ideals of the posets.