Linear independence measures for infinite products

Let f be an entire transcendental function with rational coefficients in its power series about the origin. Further, let f satisfy a functional equation f(qz)= (z−c)f(z)+Q(z) with and some particular c∈ℚ. Then the linear independence of 1,f(α), f(−α) over ℚ for non-zero α∈ℚ is proved, and a linear independence measure for these numbers is given. Clearly, for Q= 0 the function f can be written as an infinite product. Received: 19 September 2000 / Revised version: 14 March 2001     

Realisierbarkeit von darstellungen endlicher gruppen in einheitswurzelkörpern

Zusammenfassung LetD:G→GL(n,C) be an irreducible linear representation of a finite groupG with the characterX. IfD is realizible in Q(ξ _m ) and Q(ξ _m′ ) we give a condition for then realizability ofD in Q(ξ_(m′)). If the degreen is a prime ≠ 2, we show thatD realizible in Q(ξ _f ), wheref is the conductor of the abelian extensionQ(X)/Q.     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998     

Intrinsic metric preserving maps on partially ordered groups

In [11] Pap proved that a surjective mapf from an abelian lattice ordered groupG ₁ onto an abelian Archimedean lattice ordered group G₂ which preserves non-zero intrinsic metricsd ₁, andd ₂ onG ₁ andG ₂, respectively (i.e.d ₁(x,y)=d₁(z, t) implies d₂(f(x)f(y))= d₂(f(z),f(t))) and satisfiesf(0)=0 is a homomorphism and put the question whether that assertion is true in the case that G₂ is a non-Archimedean lattice ordered group. In this paper it is proved that a surjective map from an abelian directedG ₁ onto a directed group G₂ such thatf(0)=0 is a homomorphism if ¦x –y ¦=¦z – t¦ implies ¦f(x) –f(y)¦=¦f(z) –f(t)¦ and it is shown that the answer to the question of Pap is positive.Presented by M. Henriksen.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Convergence of Rational Interpolants with Preassigned Poles

Let Ω be a domain in the extended complex plane such that ∞∈Ω . Further, let K= C / Ω and, for each n , let Q _n be a monic polynomial of degree n with all its zeros in K . This paper is concerned with whether (Q _n ) can be chosen so that, if f is any holomorphic function on Ω and P _n is the polynomial part of the Laurent expansion of Q _n f at ∞ , then (P _n /Q _n ) converges to f locally uniformly on Ω . It is shown that such a sequence (Q _n ) can be chosen if and only if either K has zero logarithmic capacity or Ω is regular. January 21, 1999. Date accepted: August 17, 1999.     

On the bounded version of Hilbert's tenth problem

 The paper establishes lower bounds on the provability of 𝒟=NP and the MRDP theorem in weak fragments of arithmetic. The theory I ⁵ E ₁ is shown to be unable to prove 𝒟=NP. This non-provability result is used to show that I ⁵ E ₁ cannot prove the MRDP theorem. On the other hand it is shown that I ¹ E ₁ proves 𝒟 contains all predicates of the form (∀i≤|b|)P(i,x)^Q(i,x) where ^ is =, <, or ≤, and I ⁰ E ₁ proves 𝒟 contains all predicates of the form (∀i≤b)P(i,x)=Q(i,x). Here P and Q are polynomials. A conjecture is made that 𝒟 contains NLOGTIME. However, it is shown that this conjecture would not be sufficient to imply 𝒟=N P. Weak reductions to equality are then considered as a way of showing 𝒟=NP. It is shown that the bit-wise less than predicate, ≤₂, and equality are both co-NLOGTIME complete under FDLOGTIME reductions. This is used to show that if the FDLOGTIME functions are definable in 𝒟 then 𝒟=N P. Received: 13 July 2001 / Revised version: 9 April 2002 / Published online: 19 December 2002 Key words or phrases: Bounded Arithmetic – Bounded Diophantine Complexity     

On inner dilatations of the mappings with unbounded characteristic

For the mappings f:D ? D￠,  D,  D￠ ì \mathbbRⁿ f:D \to D',\,\,D,\,\,D' \subset {\mathbb{R}^n} , n ≥ 2, satisfying certain geometric conditions in the fixed domain D, we have proved estimates of the form K _I (x, f) ≤ Q(x) almost everywhere, where K _I (x, f) is the inner dilatation of f at a point x, and Q(x) is a fixed real-valued function responsible for the “control” over a distortion of the families of curves in D at a mapping f.     

Semistable abelian varieties with good reduction outside 15

We show that there are no non-zero semi-stable abelian varieties over ${{\bf Q}(\sqrt{5})}$ with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X ₀(15).     

Q-universal varieties of bounded lattices

A quasivariety K of algebraic systems of finite type is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.? It is known that, for every variety K of (0, 1)-lattices, if K contains a finite nondistributive simple (0, 1)-lattice, then K is Q-universal, see [3]. The opposite implication is obviously true within varieties of modular (0, 1)-lattices. This paper shows that in general the opposite implication is not true. A family (A _i : i < 2ω) of locally finite varieties of (0, 1)-lattices is exhibited each of which contains no simple non-distributive (0, 1)-lattice and each of which is Q-universal. Received July 19, 2001; accepted in final form July 11, 2002.     

On solutions of the Beltrami equation

In this paper we study the existence and uniqueness of solutions of the Beltrami equationf _-z (z) =Μ(z)f _z (z), whereΜ(z) is a measurable function defined almost everywhere in a plane domain ‡ with ‖ΜΜ∞ = 1-Here the partialsf _z andf _z of a complex valued functionf _z exist almost everywhere. In case ‖Μ‖∞ ≤9 < 1, it is well-known that homeomorphic solutions of the Beltrami equation are quasiconformal mappings. In case ‖Μ‖∞= 1, much less is known. We give sufficient conditions onΜ(z) which imply the existence of a homeomorphic solution of the Beltrami equation, which isACL and whose partial derivativesf _z andf _z are locally inL ^q for anyq < 2. We also give uniqueness results. The conditions we consider improve already known results.     

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f_i:X → X_i, i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J_X, the Prym variety P(f₁, f₂) of the pair (f₁, f₂). In some cases we can compute the type of the restriction of the canonical principal polarization of J_X. We obtain 2 families of Prym-Tyurin varieties of exponent 6. Received: 2 September 2004     

Extremal probabilities for Gaussian quadratic forms

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf _Q P(Q≤x)=P(χ² _n /n≤x), where χ _n ² is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z ₁ ,Z ₂ are independent standard Gaussian rv's, then sup_0≤λ≤1/2 P{λZ ₁ ² +(1−λ)Z ₂ ² ≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes. Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003 Research supported by NSA Grant MDA904-02-1-0091 Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10     

On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

We consider complex-valued functions f∈L ¹(ℝ₊), where ℝ₊:=[0,∞), and prove sufficient conditions under which the sine Fourier transform [^(f)]_s\hat{f}_{s} and the cosine Fourier transform [^(f)]_c\hat{f}_{c} belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.     

Jensen’s functional equation on the symmetric group S<Subscript>n</Subscript>

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy ⁻¹) =  2f(x) and f(xy) + f(y ⁻¹ x) =  2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group GL_n(R), R=\mathbbZ,\mathbbR{{GL_n(R), R=\mathbb{Z},\mathbb{R}}} , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group S _n , the group of all solutions of these functional equations coincides with the group of all homomorphisms from (S _n , ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

On Lefschetz series

Let R = ⊕ _i=0^∞ R ⁱ be a connected graded commutative algebra over the field ℚ of rational numbers, and let f be a graded endomorphism of R. In this paper, we show that the Lefschetz series of f can be computed directly from the induced linear map Q(f) on the ℚ vector space of indecomposables of R. We give an explicit algorithm to compute the Lefschetz series of f from Q(f). The main tool we used is the graded algebra version of Gr?bner basis theory. At the end of this paper, some examples and applications are given.     

A note on the existence of singular integrals

Abstract. In this note the existence of a singular integral operator T acting on Lipo(R“) spacesis studied. Suppose