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.     

On the stability of characterizations by an identical distribution property

Let X ₁, X ₂, … , X _n be i.i.d. random variables with common distribution F, and let b ₁, b ₂, … , b _n be real coefficients such that ∑ b _j ² = 1. We prove that F is close to the normal distribution in the Lévy metric whenever the distribution of the linear statistic ∑ b _j X _j is close to F.     

A bounded N-tuplewise independent and identically distributed counterexample to the CLT

Summary. A sequence of random variables X ₁,X ₂,X ₃,… is said to be N-tuplewise independent if X i ₁,X i ₂,…,X i _N are independent whenever (i ₁,i ₂,…,i _N ) is an N-tuple of distinct positive integers. For any fixed N∈ℤ⁺, we construct a sequence of bounded identically distributed N-tuplewise independent random variables which fail to satisfy the central limit theorem. Received: 17 May 1996 / In revised form: 28 January 1998     

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.     

Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

Uniform bound for asymptotic normality of discounted <Emphasis Type="Italic">m</Emphasis>-dependent random variables using Heinrich's method

We estimate the difference | F_Zv(x) - F(x) | \left| {{F_{{Z_v}}}(x) - \Phi (x)} \right| , where F_Zv(x) {F_{{Z_v}}}(x) is the distribution function of normalized series Z _v = B ⁻¹ _v Σ^∞ _j=0 v ^j X _j with B ² _v = \mathbb E \mathbb {E} (Σ^∞ _j=0 v ^j X _j ) > 0 and the discount factor v, 0 < v < 1; X ₀,X ₁,X ₂,… is a sequence of m-dependent random variables, and Φ(x) is the standard normal distribution function. In a particular case, the obtained upper bound is of order O((1−v)^1/2).     

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝ ^d of cardinality (d+1)t, partitioned into t-point subsets C ₁,C ₂,…,C _d+1 (which we think of as color classes; e.g., the points of C ₁ are red, the points of C ₂ blue, etc.), there exist r disjoint sets R ₁,R ₂,…,R _r ⊆C that are rainbow, meaning that |R _i ∩C _j |≤1 for every i,j, and whose convex hulls all have a common point.     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.     

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.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

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.     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

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 region of values of the system {ƒ(z1), …, ƒ(zn)} in the class of typically real functions. II

The paper studies the region of values D_m,1(T) of the system {ƒ(z₁), ƒ(z₂), …, ƒ(zm), ƒ(r)}, m e 1, where z_j (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a₂z²+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set D_m,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication, the region of values of ƒ(z_m) in the subclass of functions from the class T with prescribed values ƒ(z_k) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005.     

The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

Let S ⊂ ℝⁿ be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x₁ (M₁) ∪ … ∪ x_r (M_r) where each M_j is connected, x_j: M_j → V_j is a classical minimal immersion into an affine subspace V_j of ℝⁿ, and the subspaces V₁,…, V_r are pairwise orthogonal. Here we prove that if M_j is orientable, then x_j (M_j) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.     

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.     

On the stability of characterizations of the exponential distribution

Summary It is shown that if the distribution of min {X ₁/a₁, X₂/a₂,…, X_N/a_N} is close to that ofX ₁, then the distribution is close to the exponential distribution. The paper was presented at the Conference on Mathematical Statistics and Probability Theory held at the New Delhi Centre of the Indian Statistical Institute, December, 1980. The Institute of Statistical Mathematics     

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.     

Potential Theoretic Approach to Rendezvous Numbers

We analyze relations between various forms of energies (reciprocal capacities), the transfinite diameter, various Chebyshev constants and the so-called rendezvous or average number. The latter is originally defined for compact connected metric spaces (X,d) as the (in this case unique) nonnegative real number r with the property that for arbitrary finite point systems {x ₁, …, x _n } ⊂ X, there exists some point x ∈ X with the average of the distances d(x,x _j ) being exactly r. Existence of such a miraculous number has fascinated many people; its normalized version was even named “the magic number” of the metric space. Exploring related notions of general potential theory, as set up, e.g., in the fundamental works of Fuglede and Ohtsuka, we present an alternative, potential theoretic approach to rendezvous numbers.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.