On ∞-quasivarieties

We introduce the notion of an ∞-quasivariety and characterize ∞-quasivarieties as classes closed with respect to certain operators.     

Derivations On a C∞-ring

The main purpose of this paper is to provide several results on objects lying between differential geometry and algebraic geometry such as C^∞-rings and derivations on a C^∞-ring. A C^∞-ring is defined as a set with operations by C^∞-functions on Euclidean spaces. A derivation on a C^∞-ring is defined in two method, an ?-derivation and a C^∞-derivation. The main result of this paper is to show that any ?-derivation is a C^∞-derivation for some classes of C^∞-rings (Theorems 3.1, 3.2).     

On the Grothendieck construction for ∞-categories

We provide, among other things: (i) a Bousfield–Kan formula for colimits in ∞-categories (generalizing the 1-categorical formula for a colimit as a coequalizer of maps between coproducts); (ii) ∞-categorical generalizations of Barwick–Kan's Theorem B_n and Dwyer–Kan–Smith's Theorem C_n (regarding homotopy pullbacks in the Thomason model structure, which themselves vastly generalize Quillen's Theorem B); and (iii) an articulation of the simultaneous and interwoven functoriality of colimits (or dually, of limits) for natural transformations and for pullback along maps of diagram ∞-categories.     

On the Structure of ∞-Harmonic Maps

Let H ∈ C ²(? ^N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E _∞(u, Ω) = ‖H(Du)‖ _{L ^∞(Ω)} defined on maps u: Ω ? ? ⁿ  → ? ^N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? ^N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

On the cohomology of Banach A∞-module over admissible Banach A∞-algebra

In this paper we are concerned with Banach A_∞-module M over admissible Banach A_∞-algebra A. We give some properties of admissible modules and algebras. We study the cohomology of the complex C_∞(A, M). We show that the vanishing of cohomology of this complex in certain dimensions implies to the existence of the A_∞-module structure.     

On the Multiplicity of a C^∞-differentiable Function-germ

Risler ＆ Trotman in 1997 proved that the multiplicity of an analytic function germ is left-right lipschitz invariant, which provided a partial answer to Zariski conjecture. In this note, based on the recent work of Comte, Milman ＆ Trotman, we generalize the work of them to prove that the multiplicity of a C^∞ differentiable function germ is also left-right lipschitz invariant.     

On Minimal Projections in l∞ (n)

 In this paper we present an estimate of the relative projection constant for a particular class of subspaces of of codimension two. In some cases the exact value of will be calculated. Also Theorem 2.5 from [11] will be generalized. (Received 21 December 1998)     

On complemented copies of c 0 and ℓ ∞

We furnish examples of pairs of Banach spaces X, Y so that none of c ₀ and l _∞ live inside X ^? and Y, but they embed complementably into the space DP(X,Y) of the Dunford–Pettis operators from X into Y.     

On a class of closed prime ideals in H ∞

Gorkin and Mortini introduced the concept of k-hulls, k(x), of points x in M(H ^∞)????D, and studied the ideal structures of H ^∞ and H ^∞ +C. They posed a problem for which x∈ M(H ^∞)????D the set I(k(x)) is a closed prime ideal. In this article, we give a partial answer for sparse points x.     

On some properties of the metric subalgebras ofℓ ∞

The objects studied are the subalgebras of which contain c_o. These are isometrically isomorphic to the algebras C( ) where is a compactification of a discrete denumerable set N . It is shown: 1) If is metric then there is a projection of norm 1, P: C( ) C( ) with kernel c_o defined by PF = f o where is a retraction of onto = – N . 2) If is metric, then the group of homeomorphisms of is isomorphic to a complete group of permutations of the natural numbers . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ("complete" means "no outer automorphisms, trivial center").     

On the SIG-Dimension of Trees Under the L ∞-Metric

Let ${{\mathcal P},}$ where ${|{\mathcal P}| \geq 2,}$ be a set of points in d-dimensional space with a given metric ρ. For a point ${p \in {\mathcal P},}$ let r _p be the distance of p with respect to ρ from its nearest neighbor in ${{\mathcal P}.}$ Let B(p,r _p ) be the open ball with respect to ρ centered at p and having the radius r _p . We define the sphere-of-influence graph (SIG) of ${{\mathcal P}}$ as the intersection graph of the family of sets ${\{B(p,r_p)\ | \ p\in {\mathcal P}\}.}$ Given a graph G, a set of points ${{\mathcal P}_G}$ in d-dimensional space with the metric ρ is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of ${{\mathcal P}_G.}$ It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L _∞-metric in some space of finite dimension. The SIG-dimension under the L _∞-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L _∞-metric. It is denoted by SIG _∞(G). We study the SIG-dimension of trees under the L _∞-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447–460, 2003). Let T be a tree with at least two vertices. For each ${v\in V(T),}$ let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as ${\alpha(T) = \max_{x \in V(T)}}$ leaf-degree(x). Let ${ S = \{v\in V(T)\|\}}$ leaf-degree{(v) = α}. If |S| = 1, we define β(T) = α(T) ? 1. Otherwise define β(T) = α(T). We show that for a tree ${T, SIG_\infty(T) = \lceil \log_2(\beta + 2)\rceil}$ where β = β (T), provided β is not of the form 2 ^k ? 1, for some positive integer k ≥ 1. If β = 2 ^k ? 1, then ${SIG_\infty (T) \in \{k, k+1\}.}$ We show that both values are possible.     

The Non-Singularity and Regularity of GP-Vˊ-Rings

In this paper,we introduce a non-trivial generalization of ZI-rings-quasi ZI-rings.A ring R is called a quasi ZI-ring,if for any non-zero elements a,b ∈ R,ab = 0 implies that there exists a positive integer n such that an = 0 and anRbn = 0.The non-singularity and regularity of quasi ZI,GP-Vˊ-rings are studied.Some new characterizations of strong regular rings are obtained.These effectively extend some known results.     

On finite element approximation in theL ∞-norm of variational inequalities

Summary We are interested in the approximation in theL -norm of variational inequalities with non-linear operators and somewhat irregular obstacles. We show that the order of convergence will be the same as that of the equation associated with the non-linear operator if the discrete maximum principle is verified.     

On the asymptotic behavior of the bestL ∞-approximation by polynomials

LetE _n (f) denote the sup-norm-distance (with respect to the interval [?1, 1]) betweenf and the set of real polynomials of degree not exceedingn. For functions likee ^x , cosx, etc., the order ofE _n (f) asn→∞ is well known. A typical result is $$2^{n - 1} n!E_{n - 1} (e^x ) = 1 + 1/4n + O(n^{ - 2} ).$$ It is shown in this paper that 2 ^n?1 n!E _n?1(e ^x ) possesses a complete asymptotic expansion. This result is contained in the more general result that for a wide class of entire functions (containing, for example, exp(cx), coscx, and the Bessel functionsJ _k (x)) the quantity $$2^{n - 1} n!E_{n - 1} \left( f \right)/f^{(n)} \left( 0 \right)$$ possesses a complete asymptotic expansion (providedn is always even (resp. always odd) iff is even (resp. odd)).