Commutator equations in free groups

Letf ₁, …,f _n be free generators of a free groupF. We consider the equation [z ₁, …,z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, …,z _n]_ω and [f ₁, …,f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solutionz ₁=r₁, …,z _z=r _n we have <r ₁, …,r _n>=〈f ₁, …f _n〉.     

Bounds on the number of cycles of length three in a planar graph

LetG be ap-vertex planar graph having a representation in the plane with nontriangular facesF ₁,F ₂, …,F _r. Letf ₁,f ₂, …,f _r denote the lengths of the cycles bounding the facesF ₁,F ₂, …,F _r respectively. LetC ₃(G) be the number of cycles of length three inG. We give bounds onC ₃(G) in terms ofp,f ₁,f ₂, …,f _r. WhenG is 3-connected these bounds are bounds for the number of triangles in a polyhedron. We also show that all possible values ofC ₃(G) between the maximum and minimum value are actually achieved. This research was supported in part by the U.S.A.F. Office of Scientific Research, Systems Command, under Grant AFOSR-76-3017 and the National Science Foundation under Grant ENG79-09724.     

A criterion for regular sequences

LetR be a commutative noetherian ring and ƒ₁, …, ƒ_r ∃ R. In this article we give (cf. the Theorem in §2) a criterion for ƒ₁, …, ƒ_r to be regular sequence for a finitely generated module overR which strengthens and generalises a result in [2]. As an immediate consequence we deduce that if V(g ₁, …,g _r ) ⊆ V(ƒ₁, …, ƒ_r) in SpecR and if ƒ₁, …, ƒ_r is a regular sequence inR, theng ₁, …,g _r is also a regular sequence inR.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

The inverse Galois problem over formal power series fields

Consider a valuation ringR of a discrete Henselian field and a positive integerr. LetF be the quotient field of the ringR[[X ₁, …,X _r ]]. We prove that every finite group occurs as a Galois group overF. In particular, ifK ₀ is an arbitrary field andr≥2, then every finite group occurs as a Galois group overK ₀((X ₁, …,X _r )). The work on this paper started when the author was an organizer of a research group on the Arithmetic of Fields in the Institute for Advanced Studies at the Hebrew Univesity of Jerusalem in 1991–92. It was partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Reducing system of parameters and the Cohen-Macaulay property

Let R be a local ring and let (x ₁, …, x _r) be part of a system of parameters of a finitely generated R-module M, where r < dim_R M. We will show that if (y ₁, …, y _r) is part of a reducing system of parameters of M with (y ₁, …, y _r) M = (x ₁, …, x _r) M then (x ₁, …, x _r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V _R(x ₁, …, x _r) with dim_R R/P = dim_R M − r the localization M _P of M at P is an r-dimensional Cohen-Macaulay module over R _P. Furthermore, we will show that M is a Cohen-Macaulay module iff y _d is a non zero divisor on M/(y ₁, …, y _d−1) M, where (y ₁, …, y _d) is a reducing system of parameters of M (d:= dim_R M).     

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF ₄, andH _r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH _R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF ₄ over the ringA = ℤ[u ^1/2,u ^-1/2] of Laurent polynomials in an indeterminateu ^1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.     

Estimates for the best asymmetric approximations of asymmetric classes of functions

We obtain asymptotically sharp estimates for the best (α, β) -approximations of the classes W_{1;\upgamma, \updelta} ^r W_{1;\upgamma, \updelta }^r with natural r by algebraic polynomials in the mean.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

The Norm of the Laplace Operator Restriction to a Homogeneous Polynomial Space

We investigate the restriction Δ _r,μ of the Laplace operator Δ onto the space of r-variate homogeneous polynomials F of degree μ. In the uniform norm on the unit ball of ℝ ^r , and with the corresponding operator norm, ‖Δ _r,μ F‖≤‖Δ _r,μ ‖⋅‖F‖ holds, where, for arbitrary F, the ‘constant’ ‖Δ _r,μ ‖ is the best possible. We describe ‖Δ _r,μ ‖ with the help of the family T _μ (σ x), , of scaled Chebyshev polynomials of degree μ. On the interval [−1,+1], they alternate at least (μ−1)-times, as the Zolotarev polynomials do, but they differ from them by their symmetry. We call them Zolotarev polynomials of the second kind, and calculate ‖Δ _r,μ ‖ with their help. We derive upper and lower bounds, as well as the asymptotics for μ→∞. For r≥5 and sufficiently large μ, we just get ‖Δ _r,μ ‖=(r−2)μ(μ−1). However, for 2≤r≤4 or lower values of μ, the result is more complicated. This gives the problem a particular flavor. Some Bessel functions and the φcot φ-expansion are involved.      

On a region of values in the class of typically real functions

The paper studies the regions of values of the systems {f(z₁), f(r₁), f(r₂),…, f(r_n)} and {f(r₁), f(r₂),…, f (r_n)}, where n ⁥ 2; z₁ is an arbitrary fixed point of the disk U = {z: |z| < 1} with Im z₁ ≠ 0; r_j are fixed numbers, 0 < r_j < 1, j = 1, 2,…, n; f ∈ T, and the class T consists of the functions f(z), f(0) = 0, f′(0) = 1, regular in the disk U and satisfying the condition Im f(z) · Imz > 0 for Im z ≠ 0. As an implication, the region of values of f(z₁) in the subclass of functions f ∈ T with prescribed values f(r_j) (j = 1, 2,…, n) is determined. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 5–16.     

Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

Let M be a generic CR manifold in \BbbC^m+d\Bbb{C}^{m+d} of codimension d, locally given as the common zero set of real-valued functions r ¹,…,r ^d . Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r ¹,…,r ^d of third order.     

Regression of polynomial statistics on the sample mean and natural exponential families

A polynomial Q = Q(X ₁, …, X _n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α ₁(F), …, α _m (F)), where q(u ₁, …, u _m ) is a polynomial in u ₁, …, u _m and α _j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X ₁ + … + X n) = 0 holds if and only if q(α ₁(θ), …, α _m (θ)) ≡ 0, where α _j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X ₁ + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _θ(x) = ∫_−∞ ^x e ^θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.     

Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Г_r,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c₁; c₂,…, c_n, there exist integersx ₁,x₂,…,x_n satisfying 0 < Q(x₁+c₁,x₂ + c₂,…,x_n + c_n) ≤(Г|Z > |)^1/n. All the values of Г_r,n—r are known except for г_1,4. Earlier it was shown that 8 ≤Г_1,4 ≤16. Here we improve the upper bound to get Г_1,4 < 12.     

First cohomology group of rank two Witt algebra to its Larsson modules

Let U = ℂ², Γ = ℤ², and let ℂ[x ₁^±1, x ₂^±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x ₁^±1, x ₂^±1], which is spanned by elements of the form D(u, r) = x ^r (u ₁ d ₁ + u ₂ d ₂), u = (u ₁, u ₂) ∈ U, r ∈ Γ, where d ₁ and d ₂ are the degree derivations of ℂ[x ₁^±1, x ₂^±1]. The image of gl ₂-module V under Larsson functor F ^α , denoted by W = F ^α (V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H ¹(L,W).     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

On the monodromy group for the contact problem for plane curves

Summary Fixr≥2,N=r(r+3)/2 andN smooth plane curvesA ₁…,A _N with degA _i>-2 fori=l,…,N. Then the monodromy group for the plane curves of degreer tangent tog _iA_i, g_i∈PGL(3), is the full symmetric group.

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Riassunto Sianor≥2,N=r(r+3)/2 eA ₁…,A _N curve piane lisce di grado almeno 2. Si dimostra che la monodromia per le curve piane di grador tangenti ag ₁ A _i,g _i∈PGL(3), è il gruppo simmetrico.

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Supported in part by NATO junior fellowship at M.I.T.     

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.