Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action ofSL _n (K) on the affine spaceX = V^⊕m ⊕ (V*)^⊕q whereV = K ⁿ ,K an algebraically closed field of arbitrary characteristic andm,q > n. We construct a ‘standard monomial’’ basis for the ring of invariantsK[X] ^SL _n ^(K). As a consequence, we deduce thatK[X] ^SL _n ^(K) is Cohen-Macaulay. We also present the first and second fundamental theorems forSL _n (K)- actions.     

A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

To any finite symmetric subsetR ⊂ O₃ corresponds a Hecke operatorT _R on L²(S ²) which leaves the eigenspaces ℋ_n (n ≥ 0) of the Laplacian invariant. We compute the trace ofT _R | ℋ_n and prove that the sum of the positive eigenvalues ofT _R on ⊕_k=0 ^n-1ℋ_k prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator , and renormalizing the traces ofT _{R m} | ℋ _n slightly, we obtain sequences of Fourier coefficients of modular forms on Γ₀(4). Dedicated to R. Remmert on the occasion of his seventieth birthday     

Jacobson radical of skew polynomial rings and skew group rings

LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R _σ[x])={Σ_ir_i x ⁱ:r₀∈I∩J(R]), r _i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R _Σ[x])|s= (ii)J(R _σ<x>)=(J(R _σ<x>)∩R)_σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.     

Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

Nonstandard arithmetic of iterated polynomials

LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈Z[X], we denote byE(φ) the set of all integersa with φ ^m (a) =φ ⁿ (a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈Z[X] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ ^m (a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ ^k (T))=0 .     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

Multipliers for logarithmic Cauchy integrals in the ball

Let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥ 1. Let K \mathcal{K} ₀(n) denote the class of functions defined for z ∈ B _n as a constant plus the integral of the kernel log(1/(1 −〈z, ζ〉)) against a complex Borel measure on the sphere {ζ ∈ \mathbbC \mathbb{C} ⁿ ,: |ζ| = 1}. Properties of holomorphic functions g such that fg ∈ K \mathcal{K} ₀(n) for all f ∈ K \mathcal{K} ₀(n) are studied. The extended Cesàro operators are investigated on the spaces K \mathcal{K} ₀(n), n ≥ 1. Bibliography: 15 titles.     

On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.     

On a method for interpolating functions on chaotic nets

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x| ^m form odd,K(x)=|x| ^m In |x| form even (x∈ℝ ⁿ ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝ ⁿ . We construct a function of the form

Sobolev functions whose inner trace at the boundary is zero

Let Ω⊂R ⁿ be an arbitrary open set. In this paper it is shown that if a Sobolev functionf∈W ^1,p (Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, thenf is weakly zero on ϖΩ in the sense thatf∈W ₀ ^1,p (Ω).     

On invariant additive subgroups

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)⁻¹ ⊆A for allt ∈R such thatt ²=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C ₂, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C ₂ andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα ² Z ⊆ {β ²:β ∈Z}.     

There are no graded domains with GK dimension strictly between 2 and 3

Let K be a field, and let R=⊕_n∈NR_n be a finitely generated, graded K-algebra which is a domain. It is shown that R cannot have Gelfand-Kirillov dimension strictly between 2 and 3. Mathematics Subject Classification (2000) 16D90, 16P40, 16S80, 16W50     

Derivations with power central values on Lie ideals in prime rings

Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n ₁ ⩾ 0, n ₂ ⩾ 0, n ₃ ⩾ 0, (u ⁿ¹ [d(u), u]u ⁿ²) ⁿ³ ∈ Z(R) for all u ∈ U, then R satisfies S ₄, the standard identity in four variables.     

Approximation of Bochner-Riesz means of conjugate Fourier integrals below the critical index

In this paper we study the approximation on set of full measure for functions in Sobolev spaces L _m ¹ (R ⁿ) (m∈ℕ) by Bochner-Riesz means of conjugate Fourier integrals below the critical index. A theorem concerning the precise approximation orders with relation to the number m of space L _m ¹ (R ⁿ) and the index of Bochner-Riesz means is obtained. Supported by NNSFC.     

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

LetX(-ϱB ^m ×C ⁿ be a compact set over the unit sphere ϱB ^m such that for eachz∈ϱB ^m the fiberX _z ={ω∈C ⁿ ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC ⁿ . The polynomial hull ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z ₀,w ₀)∈Int there exists a smooth up to the boundary analytic discF:Δ→B ^m ×C ⁿ with the boundary inX such thatF(0)=(z ₀,w ₀). This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.