Fredholm Properties of the Difference of Orthogonal Projections in a Hilbert Space

Buckholtz (Proc. Amer. Math. Soc. 128 (2000), 1415–1418) gave necessary and sufficient conditions for the invertibility of the difference of two orthogonal projections in a Hilbert space. We generalize this result by investigating when the difference of such projections is a Fredholm operator, and give an explicit formula for its Fredholm inverse.     

Viscosity solutions of Isaacs' equations and differential games with Lipschitz controls

It is demonstrated that the upper and lower values of a two-person, zero-sum differential game solve the respective upper and lower Isaacs' equations in the viscosity sense (introduced by Crandall and Lions (Trans. Amer. Math. Soc. 277 (1983), 1–42). Since such solutions are unique, this yields a fairly simple proof that the game has value should the minimax condition hold. As a further application of viscosity techniques, a new and simpler proof that the upper and lower values can be approximated by the values of certain games with Lipschitz controls is given.     

Supplement to multiplier theorems

A multiplier theorem in (J. Combinatorial Theory Ser. A, in press) is extended to cyclic group divisible difference sets (GDDSs) of small size. A multiplier theorem for abelian difference sets in (Proc. Amer. Math. Soc.68 (1978), 375–379) is extended to abelian GDDSs. A remark on the existence of cyclic affine planes is made based on a previously proved multiplier theorem.     

Differentials of fuzzy functions

In this paper the Rådström embedding theorem (Proc. Amer. Math. Soc.3 (1952), 165) is generalized and is used to define the concept of the differential of a fuzzy function.     

A “simple” rectangular puzzle

In this note we describe a dynamic, d-dimensional toroidal puzzle, the duplication puzzle. It was first introduced in [Rosenfeld, M., A dynamic puzzle, Amer. Math. Monthly 98 (1991), 22–24]. We calculate some optimal solutions for the 2-dimensional toroidal puzzle, for higher dimensional tori and demonstrate how they might be used to improve known lower bounds of the Shannon Capacity of odd cycles.     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

Some extensions of Carlitz on cyclic sums and generating functions

Recently, the author (Proc. Amer. Math. Soc., 57 1976, 271–275) derived two theorems involving double series, which gave as a consequence new and known generating functions for the Jacobi polynomial. The method of proof differed from that of previous workers. Using an extension of this procedure, we present in this paper two theorems for double and m-dimensional series which generalize our previous work. These formulas also yield new generating functions for the Jacobi polynomial and extend some formulas of Carlitz (Boll. U.M.I. (3), 16 1961, 150–155) and others. A feature of this work is the inclusion of the Jacobi polynomial within the framework of m-dimensional cyclic sums, thus generalizing a main result of Carlitz (SIAM Rev., 6 1964, 20–30).     

Non-Liouville numbers and a Theorem of Hörmander

It is shown in this paper that Theorem 1 of [G. H. Meisters, “Translation-invariant linear forms and a formula for the Dirac measure,” J. Functional Analysis 8 (1971), 173–188] can be deduced from a very general result of Lars Hörmander, namely, Theorem 1 of “Generators for some rings of analytic functions” [Bull. Amer. Math. Soc.73 (1967), 943–949]. However, Hörmander's theorem is evidently not applicable in several other cases where Meisters'-type results have been obtained (e.g., Theorem 1 of G.H. Meisters and Wolfgang M. Schmidt, “Translation-invariant linear forms on L²(G) for compact abelian groups G,” J. Functional Analysis11 (1972), 407–424).     

Groups of Permutation Polynomials over Finite Fields

LetFbe a finite field. We apply a result of Thierry Berger (1996,Designs Codes Cryptography,7, 215–221) to determine the structure of all groups of permutations onFgenerated by the permutations induced by the linear polynomials and any power map which induces a permutation onF. This generalizes a result of Leonard Carlitz (1953,Proc. Amer. Math. Soc.,4, 538).     

Rational connectivity and sections of families over curves

A “pseudosection” of the total space X of a family of varieties over a base variety B is a subvariety of X whose general fiber over B is rationally connected. We prove a theorem which is a converse, in some sense, of the main result of [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90]: a family of varieties over B has a “pseudosection” if its restriction to each one-parameter subfamily has a “pseudosection” (which, due to [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90], holds if and only if each one-parameter subfamily has a section). This is used to give a negative answer to a question posed by Serre to Grothendieck: There exists a family of O-acyclic varieties (a family of Enriques surfaces) parametrized by P¹ with no section.     

Stability of a pexiderial functional equation in random normed spaces

The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72:297–300, 1978. Recently, the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation

$$f(x+y)+f(x-y)=2f(x)+2f(y)$$     

A lower bound for the first eigenvalue of an elliptic operator

Oscillatory behavior of the solutions of the nth-order delay differential equation L_nx(t) + q(t)f(x[g(t)]) = 0 is discussed. The results obtained are extensions of some of the results by Kim (Proc. Amer. Math. Soc.62 (1977), 77–82) for y⁽ⁿ⁾ + py = 0.     

Canonical partition theorems for parameter sets

A canonical (i.e., unrestricted) version of the partition theorem for k-parameter sets of Graham and Rothschild (Trans. Amer. Math. Soc.159 (1971), 257–291) is proven. Some applications, e.g., canonical versions, of the Rado-Folkman-Sanders theorem and of the partition theorem for finite Boolean algebras are given. Also the Erdös-Rado canonization theorem (J. London Math. Soc.25 (1950), 249–255) turns out to be an immediate corollary.     

A free boundary arising from a model for nerve conduction

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions (Trans. Amer. Math. Soc.277 (1983), 1–42) introduced the class of “viscosity” solutions of these equations and proved uniqueness within this class. This paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness.     

On the stability of <Emphasis Type="Italic">θ</Emphasis>-derivations on <Emphasis Type="Italic">JB</Emphasis>*-triples

We introduce the concept of θ-derivations on JB*-triples and prove the Hyers–Ulam-Rassias stability of θ-derivations on JB*-triples. We deal with the Hyers-Ulam-Rassias stability that was first introduced by Th. M. Rassias in the paper “On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300”. The first author was supported by Korea Research Foundation Grant KRF-2005-070-C00009.     

On a result of Pazy concerning the asymptotic behaviour of nonexpansive mappings

In 1971, Pazy [Israel J. Math. 9 (1971), 235–240] presented a beautiful trichotomy result concerning the asymptotic behaviour of the iterates of a nonexpansive mapping. In this note, we analyze the fixedpoint- free case in more detail. Our results and examples give credence to the conjecture that the iterates always converge cosmically. The relationship to recent work by Lins [Proc. Amer. Math. Soc. 137 (2009), 2387–2392] is also discussed.     

The entropy condition and the admissibility of shocks

The author proposed (Trans. Amer. Math. Soc.199 (1974), 89–112) the extended entropy condition (E) and solved the Riemann problem for general 2 × 2 conservation laws. The Riemann problem for 3 × 3 gas dynamics equations was treated by the author (J. Differential Equations18 (1975), 218–231). In this paper we justify condition (E) by the viscosity method in the spirit of Gelfand [Uspehi Mat. Nauk14 (1959), 87–158]. We show that a shock satisfies condition (E) if and only if the shock is admissible, that is, it is the limit of progressive wave solutions of the associated viscosity equations. For the “genuinely nonlinear” 2 × 2 conservation laws, Conley and Smoller [Comm. Pure Appl. Math.23 (1970), 867–884] proved that a shock satisfies Lax's shock inequalities [cf. Comm. Pure Appl. Math.14 (1957), 537–566] if and only if it is admissible. In this paper, we consider systems that are not necessarily genuinely nonlinear.     

On n-skein isomorphisms of graphs

H. Whitney [Amer. J. Math.54 (1932), 150–168] proved that edge isomorphisms between connected graphs with at least five vertices are induced by isomorphisms and that circuit isomorphisms between 3-connected graphs are induced by isomorphisms. R. Halin and H. A. Jung [J. London Math. Soc.42 (1967), 254–256] generalized these results by showing that for n ≥ 2, n-skein isomorphisms between (n+1)-connected graphs are induced by isomorphisms. In this paper we show that for n ≥ 2, n-skein isomorphisms between 3-connected graphs having (n+1)-skeins are induced by isomorphisms.     

A unifying Radon-Nikodym theorem for vector measures

It is a known fact that certain derivation bases from martingales with a directed index set. On the other hand it is also true that the strong convergence of certain abstract martingales is a consequence of the Radon-Nikodym theory for vector measures (cf. Uhl, J. J., Jr., Trans. Amer. Math. Soc.145 1969, 271–285). Many other connections and applications of the latter theory with multidimensional problems in stochastic processes and representation theory are known (cf. Dinculeanu, N., Studia Math.25 1965, 181–205; Dinculeanu, N., and Foias, C., Canad. J. Math.13 1961, 529–556; Rao, M. M., Ann. Mat. pura et applicata76 1967, 107–132; Rybakov, V. I., Izv. Vys?. U?ebn. Zaved. Matematika19 1968, 92–101; Rybakov, V. I., Dokl. Akad. Nauk SSSR180 1968, 620–623). Starting from various vantage points, many authors have proposed several hypotheses for establishing abstract Radon-Nikodym theorems. In view of the great interest and importance of this problem in the areas mentioned above, it is natural to obtain a unifying result with a general enough hypothesis to deduce the various forms of the Radon-Nikodym theorem for vector measures. This should illuminate the Radon-Nikodym theory for vector measures and stimulate further work in abstract martingale problems. In this paper the first problem is attacked, leaving the martingale part and other applications for another study.The main result (Theorem 7 of Section 2) provides the desired unification and from if the Dunford-Pettis theorem, the Phillips theorem and several others are obtained. As martingale-type arguments are constantly present, a careful reader may note the easy translation of the hypothesis to the martingale convergence problem but we treat only the Radon-Nikodym problem using the language of measure theory and linear analysis.     

On multiple sum and product sets of finite sets of integers

Let $\text{A?}\text{Z}$ be a finite set of integers of cardinality |A|=N?2. Given a positive integer k, denote kA (resp. A^(k)) the set of all sums (resp. products) of k elements of A. We prove that for all b>1, there exists k=k(b) such that max(|kA|,|A^(k)|)>N^b. This answers affirmably questions raised in Erd?s and Szemerédi (Stud. Pure Math., 1983, pp. 213–218), Elekes et al. (J. Number Theory 83 (2) (2002) 194–201) and recently, by S. Konjagin (private communication). The method is based on harmonic analysis techniques in the spirit of Chang (Ann. Math. 157 (2003) 939–957) and combinatorics on graphs. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 337 (2003).