Eine Bemerkung zum grossen Satz von Desargues in affinen Hjelmslev-Ebenen

An affine Hjelmslev-plane is called desarguesian if and only if the translations form a transitive group and for each pair t₁ and t₂ of translations, where each trace of t₁ is a trace or t₂, there exists a trace preserving endomorphism, which maps t₁ in t₂. The purpose of this paper is to charakterize desarguesian affine Hjelmslev-planes by means of a condition, which corresponds to the theorem of Desargues in ordinary affine planes, including the case, that each line contains only three classes of neighbour points. This case was omitted in [5].     

Gauss periods as constructions of low complexity normal bases

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343–366); in particular, optimal normal bases are Gauss periods of type (n, 1) for any characteristic and of type (n, 2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type (n, t) for all n and t = 3, 4, 5 over any finite field and give a slightly weaker result for Gauss periods of type (n, 6). In addition, we give some general results on the so-called cyclotomic numbers, which are intimately related to the structure of Gauss periods. We also present the general form of a normal basis obtained by the trace of any normal basis in a finite extension field. Then, as an application of the trace construction, we give upper bounds on the complexity of the trace of a Gauss period of type (n, 3).     

On the noncommutative residue and the heat trace expansion on conic manifolds

 Given a cone pseudodifferential operator P we give a full asymptotic expansion as t → 0 ⁺ of the trace Tr Pe ^-tA , where A is an elliptic cone differential operator for which the resolvent exists on a suitable region of the complex plane. Our expansion contains log t and new (log t)² terms whose coefficients are given explicitly by means of residue traces. Cone operators are contained in some natural algebras of pseudodifferential operators on which unique trace functionals can be defined. As a consequence of our explicit heat trace expansion, we recover all these trace functionals. Received: 12 November 2001 / Revised version: 26 June 2002 Mathematics Subject Classification (2000): Primary 58J35; Secondary 35C20, 58J42     

Normal Spreads

In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t – 1, q) into the Grassmannian of the subspaces of rank t of PG(2t – 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt – 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt – 1, q) by a cap V _{r, t} of PG(r ^t – 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r – 1), and that the collineation group of PG(r ^t – 1, q) stabilizing V _{r, t} acts 2-transitively on V _{r, t} . In particular, we prove that V _{3, 2} is the union of q ² – q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q ²) is represented by a hyperplane section U of V _{3, 2}. For q 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q ⁺ (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q ²). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino.     

Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

The operator e^–tA and its trace Tr e^–tA, for t > 0, are investigated in the case when A is an elliptic differential operator on a manifold with conical singularities. Under a certain spectral condition (parameter–ellipticity) we obtain a full asymptotic expansion in t of the heat trace as t → 0⁺. As in the smooth compact case, the problem is reduced to the investigation of the resolvent (A–λ)^–1. The main step consists in approximating this family by a parametrix of A – λ constructed within a suitable parameter–dependent calculus.     

Long time behaviour of the solution to non-linear Kraichnan equations

We consider the solution of a nonlinear Kraichnan equation with a covariance kernel k and boundary condition H(t, t)=1. We study the long time behaviour of H as the time parameters t, s go to infinity, according to the asymptotic behaviour of k. This question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non-commutative processes satisfying a linear differential equation, but also naturally shows up in the study of the so-called response function and aging properties of the dynamics of some disordered spin systems.     

Q-Splines

The classical weighted spline introduced by Ph. Cinquin (1981), (see also K. Salkauskas (1984) and T.A. Foley (1986)) consists in minimizing _a ^b w(t)(x(t))² dt under the conditionsx(t _i )=y _i ,i=1,...,n, where the functionw is piecewise constant on the subdivisiona<t ₁<t ₂<...<t _n <b. The solution is a cubic spline, but it is notC ². We consider here the minimization of

Embeddings of non-commutative Lp-spaces into non-commutative L1-spaces, 1 < p < 2

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomorphic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative L_p(N, t) L_p(N, \tau) -spaces defined by a finite trace on a finite von Neumann algebra. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.     

Forbidding Complete Hypergraphs as Traces

Let 2 ≤q ≤min{p, t − 1} be fixed and n → ∞. Suppose that is a p-uniform hypergraph on n vertices that contains no complete q-uniform hypergraph on t vertices as a trace. We determine the asymptotic maximum size of in many cases. For example, when q = 2 and p∈{t, t + 1}, the maximum is , and when p = t = 3, it is for all n≥ 3. Our proofs use the Kruskal-Katona theorem, an extension of the sunflower lemma due to Füredi, and recent results on hypergraph Turán numbers. Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship. Research supported in part by NSA grant H98230-06-1-0140. Part of the research conducted while his working at University of Illinois at Chicago as a NSF VIGRE postdot.     

Filtering of Finite-State Time-Nonhomogeneous Markov Processes, a Direct Approach

A filtering equation is derived for P(x _t =x|y _s ,s∈[0,t]) for a continuous-time finite-state two-component time-nonhomogeneous cadlag Markov process z _t =(x _t ,y _t ) . The derivation is based on some new ideas in the filtering theory and does not require any knowledge of stochastic integration. Accepted 10 August 1999. Online publication 13 November 2000.     

Codimensions and trace codimensions of matrices are asymptotically equal

The codimensionsc _n and the trace codimensionst _n of thek×k matrices are asymptotically equal: lim _n→∞(t _n /c _n )=1. Sincet _n ≅q(n)·k ²ⁿ whereq(x) is a known rational function, this asymptotically givesc _n . This has applications to the codimensions of Capelli identities.     

Zinbiel algebras under <Emphasis Type="Italic">q</Emphasis>-commutators

An algebra with the identity t ₁(t ₂ t ₃) = (t ₁ t ₂+t ₂ t ₁)t ₃ is called Zinbiel. For example, ℂ[x] under the multiplication is Zinbiel. Let a ○ _q b = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A ⁽⁻¹⁾ = (A, ○₋₁) satisfies the identities t ₁ t ₂ = −t ₂ t ₁ and (t ₁ t ₂)(t ₃ t ₄) + (t ₁ t ₄)(t ₃ t ₂) = jac(t ₁, t ₂, t ₃)t ₄ + jac(t ₁, t ₄, t ₃)t ₂, where jac(t ₁, t ₂, t ₃) = (t ₁ t ₂)t ₃ + (t ₂ t ₃)t ₁ + (t ₃ t ₁)t ₂. We find basic identities for q-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if q ² ≠ 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 57–78, 2005.     

FINITE DIMENSIONAL REPRESENTATIONS OF u-HOPF ALGEBRAS

Splitting trace modules of u-Hopf algebras are characterized. As an example, all finite dimensional splitting trace modules of the restricted form U_t (sl ₂) are classified. The semisimplicity of a representation V from that of tensor product V ? M in _H M for u-Hopf algebra H is given.     

Three new dicyclic solutions of (21, 42, 20, 10, 9)-designs

We give three dicyclic solutions of a (21, 42, 20, 10, 9)-design which are new. Though the parameters of this design belong to the series (4t + 1, 8t + 2, 4t, 2t, 2t − 1), 4t + 1 = 21 is not a prime power.     

Ring-Theoretic Properties of PvMDs

We extend to Prüfer v-multiplication domains some distinguished ring-theoretic properties of Prüfer domains. In particular, we consider the t##-property, the t-radical trace property, w-divisoriality, and w-stability.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

Existential closure of block intersection graphs of infinite designs having finite block size and index

In this article we study the n‐existential closure property of the block intersection graphs of infinite t‐(v, k, λ) designs for which the block size k and the index λ are both finite. We show that such block intersection graphs are 2‐e.c. when 2?t?k ? 1. When λ = 1 and 2?t?k, then a necessary and sufficient condition on n for the block intersection graph to be n‐e.c. is that n?min{t, ?(k ? 1)/(t ? 1)? + 1}. If λ?2 then we show that the block intersection graph is not n‐e.c. for any n?min{t + 1, ?k/t? + 1}, and that for 3?n?min{t, ?k/t?} the block intersection graph is potentially but not necessarily n‐e.c. The cases t = 1 and t = k are also discussed. © 2011 Wiley Periodicals, Inc. J Combin Designs 19: 85–94, 2011     

Inactivity times of coherent systems with dependent components under periodical inspections

If we know that a coherent system has failed before a time t, the inactivity time is the period (from t) in which the system has been broken. If T is the system lifetime, the inactivity time at t is (t?T|T<t). Under periodical inspections, we may typically know that the system was working at a time t₁, but that is broken at another time t₂>t₁. Under this assumption, we obtain representations for the reliability function of the system inactivity time (t₂?T|t₁<T<t₂). We consider both the cases of independent and dependent components. Similar representations are obtained under other assumptions with partial information about component failures at times t₁ and t₂. These representations are used to compare stochastically the inactivity times under different assumptions. Some illustrative examples are provided.