We investigate the ideal generated by Jacobian determinants. We generalize Freudenburg’s Lemma to the case of r polynomials in n variables over an arbitrary unique factorization domain. This lemma is useful in reformulations of the Jacobian Conjecture.
Let 1 < p < infinity be a fixed exponent. Suppose that (f(n))(n >= 0), (g(n))(n >= 0) are real-valued martingales satisfying f(0) equivalent to x, g(0 )equivalent to( )y, parallel to(f(n))(n >= 0)parallel to(p )= F and g(n)-g(n-1 )= v(n)(f(n)-f(n-1)), n = 1, 2, ... , for some predictable sequence (v(n))(n >= 1) taking values in [-1, 1]. The purpose of this paper is to determine the optimal (i.e. the smallest) constant B= B-1,B-p(x, y, F) for which parallel to(g(n))(n >= 0)parallel to(1) <= B-1,B-p(x, y, F).
Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be a nilpotent subgroup of G. Then there exists v is an element of V such that |CH(v)| <= (|H|/p)1/p, where CH(v) is the centralizer of v in H and p is the smallest prime divisor of |H|.
Let f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish some asymptotic estimates for the quantities X psi f(x) := n <= x fx]) X Lambda(n)f , pi f(x) := n p <= x (fx]) f p for x -> infinity, where Lambda(n) is the von Mangoldt function and [t] is the integer part of t is an element of R. These generalise or sharpen some recent results of Saito-Suzuki-Takeda-Yoshida. As an application, we show that X 1 similar to x ->infinity p <= x, [x/p] is prime (X p 1 p(p + 1) x log x
Let Phi be an Orlicz function and L Phi (X, Sigma, & micro;) be the corresponding Orlicz space on a non-atomic, sigma-finite, complete measure space (X, Sigma, & micro;). We describe the local uniform convexity of Orlicz spaces endowed with the s-norm and discuss the weak and compact variants of this property. Also, we derive some results in approximation theory, concerning best approximations and farthest points. Thus, our study provides a comprehensive generalization of several results that have been obtained for Orlicz spaces with the Orlicz norm and the Luxemburg norm.
In this paper, motivated by a recent work of Z.-W. Sun and by using algebraic properties of Jacobi sums and the theory of almost circulant matrices posed by H.-L. Wu and L.-Y. Wang, we determine the explicit value of the determinant of the matrix Yq,d(psi) = [psi(si + dsj) + psi(si-dsj)]2 <= i,j <=(q-1)/2, where q equivalent to 3 (mod 4) is a prime power, d is a non-zero element over the finite field Fq,psi is a non-trivial multiplicative character of Fq and s1 = 1, s2, ..., s(q-1)/2 are all non-zero squares over Fq.
Answering questions raised by Leonetti and by Rinc & oacute;n-Villamizar and Uzc & aacute;tegui-Aylwin we characterize ideals I subset of P(omega) such that c0,I is complemented in t infinity as exactly those ideals for which the space KI = Stone(P(omega)/I) is approximable, i.e., the unit ball of the space M(KI) of signed Radon measures on KI is separable in the weak & lowast; topology.
We prove a combinatorial identity, which implies a connection between an identity of Milne and an identity of Glass and Ng concerning the hook length formula. As applications, we obtain some new formulas on partitions, factorial numbers, rising factorial numbers, falling factorial numbers, and Catalan numbers.
We study pedal curves for planar curves, possibly with singularities, called frontal curves. In particular, we analyze the types of singularities on pedal curves for a given frontal in terms of the geometrical properties.
We show that under the Continuum Hypothesis, the topological group of all homeomorphisms of the Cech-Stone remainder of omega with the G(delta)-topology is a universal object for all P-groups of weight at most c.
Let A and H be two cocommutative Hopf algebras such that A is an H-bimodule Hopf algebra. Suppose that R:A→ A is a linear map and B is a Rota-Baxter operator of H. In this paper we will characterize the Rota-Baxter operators on the L-R smash product A♮ H and give the necessary and sufficient conditions to make B a Rota-Baxter operator of A♮ H. Then we will consider the dual case, and construct a Rota-Baxter co-operator on the L-R smash coproduct C⋉ H, where C and H are commutative Hopf algebras and C is an H-bicomodule Hopf algebra.
Let f (x) is an element of Z[x] be an Nth degree polynomial that is monic and irreducible over Q. We say that f (x) is monogenic if {1, 0, 02, ..., 0N-1} is a basis for the ring of integers of Q(0), where f (0) = 0. We say that f (x) is cyclic if the Galois group of f (x) over Q is the cyclic group of order N. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences.
A right R-module M over a ring R is said to be FI-extending if any fully invariant submodule of M is essential in a direct summand of M. We prove that if R has ACC on the right annihilators, then RR is FI-extending if, and only if, every f.g. projective module over R is FI-extending. This is an affirmative answer to the question raised by Birkenmeier-Park-Rizvi [Comm. Algebra 30 (2002), 1833-1852].
We investigate translation length functions for two-generated groups acting by isometries on Λ-trees, where Λ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of any element of the group, under certain assumptions on the translation lengths of its generators and their products. Our approach is purely combinatorial and uses only the defining axioms of pseudo-lengths. As shown by Parry, pseudo-lengths coincide with the translation length functions for actions on Λ-trees. Furthermore, we prove that, under certain conditions on four elements α,β,γ,δ∈Λ, there exists a unique pseudo-length on the free group F(a,b) assigning these values to a, b, ab, ab−1, respectively. Applications include results on properly discontinuous actions and discrete free groups of isometries. We also develop an algorithmic approach to studying translation length functions arising from free actions on R-trees. Based on this, we state a conjecture that would lead to a description of Aut(F2)-orbits in the Culler–Vogtmann outer space.
This paper introduces and studies the Ehrhart spectrum of a set E subset of Zr, defined as the set of all Ehrhart polynomials of simplices with vertices in E, generalizing the notion of volume spectrum. We show that for any E subset of Zr with positive upper Banach density, there is some n is an element of Z such that the Ehrhart spectrum of nZr is contained in the Ehrhart spectrum of E, generalizing an earlier result by the first and third authors for the volume spectrum of E.
We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem dealing with the absence of the Lavrentiev phenomenon, a cornerstone issue in the calculus of variations. We point out some inconsistencies in the original proof by providing a counterexample and supply the result with a new, concise, and complete reasoning. In the appendix, we also provide additional details to supplement the original proof.
We define the numbers of level crossings by a c & agrave;dl & agrave;g (RCLL) real function x: [0, +infinity) -* R and, in analogy to the work of Bertoin and Yor (2014), we prove that for x with locally finite total variation, these numbers are densities of relevant occupation measures associated with x. Next, depending on the regularity of x and f : R -* R, we derive change of variable formulas, which may be seen as analogs of the It & ocirc; or Tanaka- Meyer formulas. Some of these formulas were already given by Bertoin and Yor (2014), but we also present some generalizations.
Several weakenings of the T_2 property for topological spaces, including k-Hausdorff, KC, weakly Hausdorff, semi-Hausdorff, RC, and US, have been studied by mathematicians. Here we provide a complete survey of how these properties do or do not relate to one another, including several new results to fill in the gaps in the existing literature, motivated by the use of a community-maintained database of topological spaces and their properties.
We study linear recurrence and weak mixing of a two-parameter family of interval translation maps T alpha,beta for the subset of parameter space where T alpha,beta has a Cantor attractor. For this class, there is a procedure similar to the Rauzy induction which acts as a dynamical system G on parameter space, which was used previously to decide whether T alpha,beta has an attracting Cantor set, and if so, whether T alpha,beta is uniquely ergodic. In this paper we use properties of G to decide whether T alpha,beta is linearly recurrent or weakly mixing.
