Cokernel bundles and Fibonacci bundles

We are interested in those bundles C on ?^N which admit a resolution of the form 0 → ?^s ? E ?^t ? F → C → 0. In this paper we prove that, under suitable conditions on (E, F), a generic bundle with this form is either simple or canonically decomposable. As applications we provide an easy criterion for the stability of such bundles on ?² and we prove the stability when E = ??, F = ??(1) and C is an exceptional bundle on ?^N for N ≥ 2. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surfaces of order three with a peak. II

In [1], we presented a theory of surfaces of order three in real projective three-spaceP ³. In the present paper, we prove that a surfaceF of order three with a peak contains one, two or three lines and there are four types of suchF based upon the configuration of these lines. We describe eachF by determining the existence and the distribution of elliptic, parabolic and hyperbolic points; that is, points ofF not lying on any line contained inF.     

Gauss‐Green theorem for weakly differentiable vector fields,sets of finite perimeter,and balance laws

We analyze a class of weakly differentiable vector fields F : ?ⁿ → ?ⁿ with the property that F ∈ L^∞ and div F is a (signed) Radon measure. These fields are called bounded divergence‐measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence‐measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss‐Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure μ that is absolutely continuous with respect to ??^{N ? 1} on ?^N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure‐theoretic interior of the set with respect to the measure ||μ||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss‐Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N ? 1)‐dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure‐valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc.     

An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Let B be the unit ball in C ⁿ and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ _n,α that provide a way to extend a locally univalent function ? ? H(U) to a locally univalent mapping F_α ? H(B), where α ? (0,1]. If ? is normalized univalent, then F_α can be imbedded in a Loewner chain. Also if ? ? S?, then F_α is starlike. We show that if ? belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping F_α satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ _n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C ⁿ (for n ≤ 2.)     

Uniform Boundedness of Norms of Convex and Nonconvex Processes

The lower limit F:? ⁿ ? ^m of a sequence of closed convex processes F _ν:? ⁿ ? ^m is again a closed convex process. In this note, we prove the following uniform boundedness principle: if F is nonempty-valued everywhere, then there is a positive integer ν₀ such that the tail {F _ν}_ν≥ν0 is “uniformly bounded” in the sense that the norms ‖ F _ν‖ are bounded by a common constant. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.     

Dirichlet duality and the nonlinear Dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?ⁿ. In our approach the equation is replaced by a subset F ? Sym²(?ⁿ) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999     

Size ramsey numbers of stars versus 4‐chromatic graphs

We investigate the asymptotics of the size Ramsey number î(K_1,nF), where K_1,n is the n‐star and F is a fixed graph. The author 11 has recently proved that r?(K_1,n,F)=(1+o(1))n² for any F with chromatic number χ(F)=3. Here we show that r?(K_1,n,F)≤ n²+o(n²), if χ (F) ≥ 4 and conjecture that this is sharp. We prove the case χ(F)=4 of the conjecture, that is, that r?(K_1,n,F)=(4+o(1))n² for any 4‐chromatic graph F. Also, some general lower bounds are obtained. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 220–233, 2003     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

The (F*F) 1/4-method for the transmission problem in two-dimensional linear elasticity

In this article we present an inversion algorithm for the determination of the shape of a two-dimensional penetrable obstacle from knowledge of the elastic field generated by an incident plane compressional and shear wave. In particular, Kirsch's improved variant of the linear sampling method, the so called (F ^* F?)^1/4-method is extended to the elastic case. A mathematical analysis that reveals the compactness and normality of the far-field operator is presented. Finally, numerical results are presented showing the robustness of the (F ^* F?)^1/4-method with respect to noise.     

Some Results on Systems of Finite Sets That Satisfy a Certain Intersection Condition

Let S be a finite set, and fix K>2. Let F be a family of subsets of S with the property that whenever A₁,...,A_k are sets in F, not necessarily distinct, and A₁ ? ? ? A_k = ?, then A₁ ? ? ? A_k = S. We prove here that the maximum size of such a family is 2^|S|?1 + 1. If we require that the sets A₁,...,A_k be distinct, then the maximum size of F is again 2^|S|?1 + 1, provided that |S| ≥ log₂(K?2)+3.     

Equivalent Black volatilities

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σ _B in terms of today's forward price F ₀ ≡ F(0), the strike K of the option, and the time to expiry t_ex . The price of any call or put can then be calculated simply by substituting this implied volatility into Black's formula. For example, for a power law (constant elasticity of variance) model dF(t)=aF^β dW(t) we obtain σ _B = a/f _aυ ^1? β {1 + (1?β)(2+β)/24 (F ₀ ? K/f _aυ)² + (1 ? β)²/24 a ² t_ex /f _aυ ^2?2β +…} where f _aυ = ½(F ₀ + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.     

Baer subplanes generated by collineations between pencils of lines

In this paper we define a degenerateC _F-set in PG (2,q ²) as the set of points of intersection of corresponding lines under a suitable collineation between two pencils of lines with vertices two distinct pointsA andB mapping the lineA ∨B onto itself. We prove that every such a set is the union of the lineA ∨B and a Baer subplane and vice versa every Baer subplane can be seen as a subset of a degenerateC _F-set.     

Spectral self-concordant functions in the space of two-by-two symmetric matrices

We show that given a two-variable, symmetric, ?-self-concordant function f, the spectral function F = f ○ λ is 2(1 + 3?)-self-concordant. Correspondingly, if f is ?-self-concordant barrier, then 4(1 + 3?)² F is a 4(1 + 3?)²?-self-concordant barrier.     

ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

The phase transition in random horn satisfiability and its algorithmic implications

Let c > 0 be a constant, and Φ be a random Horn formula with n variables and m = c · 2ⁿ clauses, chosen uniformly at random (with repetition) from the set of all nonempty Horn clauses in the given variables. By analyzing PUR, a natural implementation of positive unit resolution, we show that lim_n→∞ Pr(Φ is satisfiable) = 1 ? F(e^?c), where F(x) = (1 ? x)(1 ? x²)(1 ? x⁴)(1 ? x⁸) …. Our method also yields as a byproduct an average‐case analysis of this algorithm. Published 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 483–506, 2002     

Maximal triangle-free graphs with restrictions on the degrees

We investigate the problem that at least how many edges must a maximal triangle-free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n ? 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence c_k → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cn^?, 1/2 < ? < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n - 1)^1/2 is impossible in a maximal triangle-free graph.)     

Bounded factorization property for Fréchet spaces

An operator T ∈ L(E, F) factors over G if T = RS for some S ∈ L(E, G) and R ∈ L(G, F); the set of such operators is denoted by L^G(E, F). A triple (E, G, F) satisfies bounded factorization property (shortly, (E, G, F) ∈ ???) if L^G(E, F) ? LB(E, F), where LB(E, F) is the set of all bounded linear operators from E to F. The relationship (E, G, F) ∈ ??? is characterized in the spirit of Vogt's characterisation of the relationship L(E, F) = LB(E, F) [23]. For triples of K?othe spaces the property ??? is characterized in terms of their K?othe matrices. As an application we prove that in certain cases the relations L(E, G₁) = LB(E, G₁) and L(G₂, F) = LB(G₂, F) imply (E, G, F) ∈ ??? where G is a tensor product of G₁ and G₂.     

Finiteness of integral values for the ratio of two linear recurrences

Let {F(n)} _{n ∈ N} ,{G(n)} _{n ∈ N} , be linear recurrent sequences. In this paper we are concerned with the well-known diophantine problem of the finiteness of the set ? of natural numbers n such that F(n)/G(n) is an integer. In this direction we have for instance a deep theorem of van der Poorten; solving a conjecture of Pisot, he established that if ? coincides with N, then {F(n)/G(n)} _{n ∈ N} is itself a linear recurrence sequence. Here we shall prove that if ? is an infinite set, then there exists a nonzero polynomial P such that P(n)F(n)/G(n) coincides with a linear recurrence for all n in a suitable arithmetic progression. Examples like F(n)=2 ⁿ -2, G(n)=n+2 ⁿ +(-2) ⁿ , show that our conclusion is in a sense best-possible. In the proofs we introduce a new method to cope with a notorious crucial difficulty related to the existence of a so-called dominant root. In an appendix we shall also prove a zero-density result for ? in the cases when the polynomial P cannot be taken a constant. Oblatum 12-XI-2001 & 31-I-2002?Published online: 29 April 2002