Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subcentrality of restrictions of boundary measures on state spaces of C1-algebras

Let F be a closed face of the weak¹ compact convex state space of a unital C¹-algebra A. The author has already shown that F is a Choquet simplex if and only if p_φ^Fπ_φ(A)″p_φ^F is abelian for any φ in F with associated cyclic representation ($\text{H}$_φ,π_φ,ξ_φ), where p_φ^F is the orthogonal projection of $\text{H}$_φ onto the subspace spanned by vectors η defining vector states a → 〈π_φ(a)η, η)〉 lying in F. It is shown here that if B is a C¹-subalgebra of A containing the unit and such that ξ_φ is cyclic in $\text{H}$_φ for π_φ(B) for any φ in F, then the boundary measures on F are subcentral as measures on the state space of B if and only if p_φ^F(π_φ(A), π_φ(B)′)″p_φ^F is abelian for all φ in F. If A is separable, this is equivalent to the condition that any state in F with (π_φ(A)′ ∩ π_φ(B)″) one-dimensional is pure. Taking A to be the crossed product of a discrete C¹-dynamical system (B, G, α), these results generalise known criteria for the system to be G-central.     

Generalization of a theorem of Erdős and Rényi on Sidon sequences

Erd?s and Rényi claimed and Vu proved that for all h ≥ 2 and for all ? > 0, there exists g = g_h(?) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ? x^1/h‐?. We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies g_h(?) ? ?^?1, improving the bound g_h(?) ? ?^?h+1 obtained by Vu. Finally we use the “alteration method” to get a better bound for g₃(?), obtaining a more precise estimate for the growth of B₃[g] sequences. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of Lie Derivations on Prime Rings

Let ? be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ? satisfies L([A, B]) = [L(A), B] + [A, L(B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L(A) = ?(A) + h(A) for all A ∈ ?, where ? is an additive derivation on ? and h is an additive map into its center.     

On Galois Algebras Satisfying the Fundamental Theorem

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ?∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.     

Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T₁,T₂:A→B are surjective mappings that satisfy σπ(T₁(f)T₂(g))∩σπ(fg)≠∅ for all f,g∈A, then T₁(f)T₂(1)=T₁(1)T₂(f) for all f∈A, and the map f?T₁(f)T₂(1) is an isometric algebra isomorphism.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

Projective MV-algebras and rational polyhedra

Let M be an n-generator projective MV-algebra. Then there is a rational polyhedron P in the n-cube [0, 1] ⁿ such that M is isomorphic to the MV-algebra M(P){{\rm{\mathcal {M}}}(P)} of restrictions to P of the McNaughton functions of the free n-generator MV-algebra. P necessarily contains a vertex vP of the n-cube. We characterize those polyhedra contained in the n-cube such that M(P){{\mathcal {M}}(P)} is projective. In particular, if the rational polyhedron P is a union of segments originating at some fixed vertex vP of the n-cube, then M(P){{\mathcal {M}}(P)} is projective. Using this result, we prove that if A = M(P){A = {\mathcal {M}}(P)} and B = M(Q){B = {\mathcal {M}}(Q)} are projective, then so is the subalgebra of A × B given by {(f, g) | f(v _P ) = g(v _Q ), and so is the free product A \coprod B{A \coprod B} .     

Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

Let Π be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation ~ such that (A,~) is the regular tree with d +1 edges meeting at each vertex. Let $\text{g}$(π)be the simplex of corresponding Markov random fields. We show that if Π satisfies a ‘boundedness’ condition then $\text{g}$(π).We further study the structure of $\text{g}$(π) when Π is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, ~) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for $\text{g}$(π) and for $\text{g}$(π)to have precisely one member.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

On the Gröbner complexity of matrices

In this paper we show that if for an integer matrix A the universal Gröbner basis of the associated toric ideal I_A coincides with the Graver basis of A, then the Gröbner complexity u(A) and the Graver complexity g(A) of its higher Lawrence liftings agree, too. In fact, if the universal Gröbner basis of I_A coincides with the Graver basis of A, then also the more general complexities u(A,B) and g(A,B) agree for arbitrary B. We conclude that for the matrices A_3×3 and A_3×4, defining the 3×3 and 3×4 transportation problems, we have u(A_3×3)=g(A_3×3)=9 and u(A_3×4)=g(A_3×4)≥27. Moreover, we prove that u(A_a,b)=g(A_a,b)=2(a+b)/gcd(a,b) for positive integers a,b and .     

Semiclassical Spectral Asymptotics on Foliated Manifolds

We consider a (hypo)elliptic pseudodifferential operator A_h on a closed foliated manifold (M,ℱ), depending on a parameterh > 0, of the form A_h = A+h^mB, where A is a formally self–adjoint tangentially elliptic operator of orderμ > 0 with the nonnegative principal symbol and B is a formally self–adjoint classical pseudodi.erential operator of orderm > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if μ < m, and its transversal principal symbol is positive, if μ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function N_h(λ) of the operator A_h when h tends to 0 and λ is constant.     

Leibniz 2-Algebras and Twisted Courant Algebroids

Let Y be an integral projective curve whose singularities are of type A_k, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= p_a(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h⁰(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ? P ^g.     

The conformal plate buckling equation

The linear equation Δ²u = 1 for the infinitesimal buckling under uniform unit load of a thin elastic plate over ?² has the particularly interesting nonlinear generalization Δ_g²u = 1, where Δ_g = e^?2u Δ is the Laplace‐Beltrami operator for the metric g = e^2ug₀, with g₀ the standard Euclidean metric on ?². This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order Δu(x)+K_g(x) exp(2u(x)) = 0 and Δ K_g(x) + exp(2u(x)) = 0, with x ∈ ?², describing a conformally flat surface with a Gauss curvature function K_g that is generated self‐consistently through the metric's conformal factor. We study this conformal plate buckling equation under the hypotheses of finite integral curvature ∫ K_g exp(2u)dx = κ, finite area ∫ exp(2u)dx = α, and the mild compactness condition K₊ ∈ L¹(B₁(y)), uniformly w.r.t. y ∈ ?². We show that asymptotically for |x|→∞ all solutions behave like u(x) = ?(κ/2π)ln |x| + C + o(1) and K(x) = ?(α/2π) ln|x| + C + o(1), with κ ∈ (2π, 4π) and . We also show that for each κ ∈ (2π, 4π) there exists a K^* and a radially symmetric solution pair u, K, satisfying K(u) = κ and maxK = K^*, which is unique modulo translation of the origin, and scaling of x coupled with a translation of u. © 2001 John Wiley & Sons, Inc.     

Notes on flatness and the quot functor on rings

For flat modules M over a ring A we study the similarities between the three statements,dim _k (P) ( _k (P)? _A M =dfor all prime ideals P of A, the Ap-module M _p is free of rank d for all prime ideals P of A, and M is a locally free J4-module of rank d. We have particularly emphasized the case when there is an>l-algebra B, essentially of finite type, and M is a finitely generated B-module.     

Norm conditions for real-algebra isomorphisms between uniform algebras

Let A and B be uniform algebras. Suppose that α ≠ 0 and A ₁ ⊂ A. Let ρ, τ: A ₁ → A and S, T: A ₁ → B be mappings. Suppose that ρ(A ₁), τ(A ₁) and S(A ₁), T(A ₁) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖_∞ = ‖ρ(f)τ(g) − α‖_∞ for all f, g ∈ A ₁, S(e ₁)⁻¹ ∈ S(A ₁) and S(e ₁) ∈ T(A ₁) for some e ₁ ∈ A ₁ with ρ(e ₁) = 1, then there exists a real-algebra isomorphism $ \tilde S $ \tilde S : A → B such that $ \tilde S $ \tilde S (ρ(f)) = S(e ₁)⁻¹ S(f) for every f ∈ A ₁. We also give some applications of this result.