Vector Lattice Powers: f-Algebras and Functional Calculus

Let s ∈ {2.3,…} and E be an Archimedean vector lattice. We prove that there exists a unique pair (E ^? ,?), where E ^? is an Archimedean vector lattice and ?:E× ··· ×E (s times) → E ^? is a symmetric lattice s-morphism, such that for every Archimedean vector lattice F and every symmetric lattice s-morphism T:E × ··· × E (s times) → F, there exists a unique lattice homomorphism T ^? :E ^?  → F such that T = T ^? ○?. We give two approaches to construct (E ^? ,?) based on f-algebras and functional calculus, respectively, provided that E is also uniformly complete.     

Root Sublattices and Torus-Restriction of Prehomogeneous Vector Spaces Associated with Dynkin Quivers

For a Dynkin quiver Γ with r vertices, a subset S of the vertices of Γ, and an r-tuple d = (d⁽¹⁾, d⁽²⁾,…, d^(r)) of positive integers, we define a “torus-restricted” representation (G_S, R _d (Γ)) in natural way. Here we put G_S = G₁ × G₂ × … ×G_r, where each G_i is either SL(d⁽ⁱ⁾) or GL(d⁽ⁱ⁾) according to S containing i or not. In this paper, for a prescribed torus-restriction S, we give a necessary and sufficient condition on d that R _d (Γ) has only finitely many G_S-orbits. This can be paraphrased as a condition whether or not d is contained in a certain lattice spanned by positive roots of Γ. We also discuss the prehomogeneity of (G_S, R _d (Γ)).     

Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs†

Let Q be a m × m real matrix and f _j  : ? → ?, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x _j and f _j (x _j ), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph 𝒢 is just any linear operator D:C ⁰(𝒢) → C ⁰(𝒢), where C ⁰(𝒢) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V × ? → ?, one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C ⁰(𝒢). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (𝒢, d), where 𝒢 is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ?₂. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ?₂ u = F(u), for the metric d = 1.     

MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL ON RBMO

Given a positive Radon measure μ on R^d satisfying the linear growth condition μ（B（x,r））≤C0r^n,x∈R^d,r〉0,（1） where n is a fixed number and O〈n≤d. When d-1〈n,it is proved that if Tt,N1=0,then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a. e. on R^d.     

Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ? = ?F, R, ρ? be a system language. Given a class of ?-systems K and an ?-algebraic system A = ?SEN,?N,F??, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K _A of A-systems in K as the collection of all members of K of the form 𝔄 = ? SEN,?N,F?,R ^𝔄 ?, for some set of relation systems R ^𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K _A and global properties connecting K _A and K _A′, whenever there exists an ?-morphism ? F,α? : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K _A is an algebraic closure system, for every ?-algebraic system A, provided that K is closed under subsystems and reduced products.     

Mechanistic explanation of integral calculus

The anatomic features of filaments, drawn through graphs of an integral F(x) and its derivative f(x), clarify why integrals automatically calculate area swept out by derivatives. Each miniscule elevation change dF on an integral, as a linear measure, equals the magnitude of square area of a corresponding vertical filament through its derivative. The sum of all dF increments combine to produce a range ΔF on the integral that equals the exact summed area swept out by the derivative over that domain. The sum of filament areas is symbolized ∫f(x)dx, where dx is the width of any filament and f(x) is the ordinal value of the derivative and thus, the intrinsic slope of the integral point dF/dx. dx displacement widths, and corresponding dF displacement heights, along the integral are not uniform and are determined by the intrinsic slope of the function at each point. Among many methods that demonstrate why integrals calculate area traced out by derivatives, this presents the physical meaning of differentials dx and dF, and how the variation in each along an integral curve explicitly computes area at any point traced by the derivative. This area is the filament width dx times its height, the ordinal value of the derivative function f(x), which is the tangent slope dF/dx on the integral. This explains thoroughly but succinctly the precise mechanism of integral calculus.     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Random Measures and Applications

Abstract

A mapping Z(·) from a δ-ring ?₀(?) into the vector space of random variables L ^p (P) is a vector-valued measure if it is σ-additive in the metric of its range. It is a vector measure if the range is a Banach space and a random measure if also its values are independent on disjoint sets. An important reason for this study is to construct integrals relative to such Zs, which typically do not have finite variation. For this, it is essential to find a controlling (σ-finite) measure for Z that is not available if 0 <p < 1, and here the random measure is taken to be p-stable and utilize properties of infinitely divisible distributions. In the case of p = 2, Z(·) induces a bimeasure, and if p > 2 is an integer it induces a polymeasure, either of which need not be (signed) measures on product spaces. Important applications lead to all these possibilities. In all those cases, a detailed analysis of vector-valued set functions is presented, with special focus for the cases of 0 <p < 1 and p = 2 where probability and Bochner's L ^{2, 2} boundedness plays a key role. Specialization if Z is stationary, harmonizable, and/or isotropic are discussed using the group structure of ? ⁿ , n ≥ 1, extending it for an lca group G. If Z is Banach valued or a quasi-martingale measure, methods of obtaining integrals are outlined in the last section, and open problems motivated by applications are pointed out at various places.     

Properties of the Fiber Cone of Ideals in Local Rings

Abstract

For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ _J  : F _R (I) → F _R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ _J  ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G ₊(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface.     

Powers of Elements and Monomial Ideals #

All monomial ideals I ? k[X ₀,…, X _d ] are classified which satisfy the following condition: If f ∈ I with f ⁿ  ∈ I ⁿ⁺¹ for some n, then f ∈ (X ₀,…, X _d ) I.     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.     

Fiber Coefficients and Depth of Fiber Cones

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When r(I | K)<∞, we give a lower bound and an upper bound for f ₁(I). Under the above assumption on r(I | K) and depth G(I) ≥ d ? 1, we also provide a characterization, in terms of f ₁(I), of the condition depth F _K (I) ≥ d ? 1.     

Strong Cesàro summability of double Fourier integrals

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on . Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Properties of Modules and Rings Relative to Some Matrices

Let R be a ring and  _β×α(R) (?   _β×α(R)) the set of all β × α full (row finite) matrices over R where α and β ≥ 1 are two cardinal numbers. A left R-module M is said to be “injective relative” to a matrix A ? ?   _β×α(R) if every R-homomorphism from R ^(β) A to M extends to one from R ^(α) to M. It is proved that M is injective relative to A if and only if it is A-pure in every module which contains M as a submodule. A right R-module N is called flat relative to a matrix A ?  _β×α(R) if the canonical map μ: N? R ^(β) A → N ^α is a monomorphism. This extends the notion of (m, n)-flat modules so that n-projectivity, finitely projectivity, and τ-flatness can be redefined in terms of flatness relative to certain matrices. R is called left coherent relative to a matrix A ?  _β×α(R) if R ^(β) A is a left R-ML module. Some results on τ-coherent rings and (m, n)-coherent rings are extended.