The equational complexity of Lyndon’s algebra

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.     

On Linnik’s almost Goldbach theorem

It is proved that every sufficiently large even integer can be represented as a sum of two primes and at most 2250 powers of 2. Project supported partially by the National Natural Science Foundation of China (Grant No. 19771029).     

A modification of Polin’s variety

In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locally finite equational classes which satisfy some nontrivial congruence identity. Received October 9, 1998; accepted in final form January 19, 1999.     

Sunyer-i-Balaguer’s almost elliptic functions and Yosida’s normal functions

We study the properties of two classes of meromorphic functions in the complex plane. The first one is the class of almost elliptic functions in the sense of Sunyer-i-Balaguer. This is the class of meromorphic functions f such that the family {f(z + h)} _h∈ℂ is normal with respect to the uniform convergence in the whole complex plane. Given two sequences of complex numbers, we provide sufficient conditions for themto be zeros and poles of some almost elliptic function. These conditions enable one to give (for the first time) explicit non-trivial examples of almost elliptic functions. The second class was introduced by K. Yosida, who called it the class of normal functions of the first category. This is the class of meromorphic functions f such that the family {f(z + h)} _h∈ℂ is normal with respect to the uniform convergence on compacta in the complex plane and no limit point of the family is a constant function. We give necessary and sufficient conditions for two sequences of complex numbers to be zeros and poles of some normal function of the first category and obtain a parametric representation for this class in terms of zeros and poles.     

Birkhoff’s variety theorem in many sorts

For many-sorted algebras, Garrett Birkhoff’s characterization of equational classes is proved to generalize in case of finitely many sorts. For infinitely sorted algebras, closure under directed unions needs to be added.     

Bochner’s theorem and Stepanov almost periodic functions

Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.      

Dye’s theorem in the almost continuous category

Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G _δ subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures.     

On the Shimura variety having Mumford’s fake projective plane as a connected component

We give an explicit construction of a unitary Shimura surface that has Mumford’s fake projective plane as one of its connected components. Moreover, as a byproduct of the construction, we show that Mumford’s fake projective place has a model defined over the 7th cyclotomic field.     

A variety of Euler’s sum of powers conjecture

Czechoslovak Mathematical Journal - We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $$left{{matrix{{n = {a_1} + {a_2} + ldots + {a_{s -...     

Free algebras of a unary variety with Mal’tsev’s operation that satisfies the Pixley conditions

In this paper we consider the variety V P of algebras with one unary and one ternary operation p that satisfies the Pixley identities, provided that operations are permutable. We describe the structure of a free algebra of the variety V P and study the structure of unary reducts of free algebras. We prove the solvability of the word problem in free algebras and the uniqueness of a free basis; we also describe groups of automorphisms of free algebras. Similar results are obtained for free algebras of a subvariety of the variety V P defined by the identities p(p(x, y, z), y, z) = p(x, y, z) and p(x, y, p(x, y, z)) = p(x, y, z).     

Hua’s theorem with nine almost equal prime variables

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as

Hua’s theorem for five almost equal prime squares

Let p_i, 1≦ i ≦ 5, be prime numbers. It is proved that every sufficiently large integer N that satisfies N ≡ 5(mod 24) can be written as N = p²₁ + p²₂ + p²₃ + p²₄ + p²₅, where Received: 10 February 2005     

Approximation of almost Sahoo–Riedel’s points by Sahoo–Riedel’s points

In this paper, we present an extention of Hyers–Ulam stability of Sahoo–Riedel’s points for real-valued differentiable functions on [a, b] and then we obtain stability results of Flett’s points for functions in the class of continuously differentiable functions on [a, b] with f′(a) = f′(b).     

Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s

The authors derive best possible maximum principles for some combinations of solutions of a class of fully nonlinear elliptic PDEs and their gradients. These maximum principles are then applied to establish various inequalities of interest in the theory of Weingarten surfaces.     

Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s

The authors derive best possible maximum principles for some combinations of solutions of a class of fully nonlinear elliptic PDEs and their gradients. These maximum principles are then applied to establish various inequalities of interest in the theory of Weingarten surfaces.     

Hausdorff’s forgotten proof that almost all numbers are normal

Mathematische Semesterberichte - In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in base 2. We generalize this proof to show that almost all...     

Adomian decomposition method is a special case of Lyapunov’s artificial small parameter method

The Adomian decomposition method and Lyapunov’s artificial small parameter method are two popular analytic methods for solving nonlinear differential equations. In this short paper, we show that the Adomian decomposition method can be derived from the more general Lyapunov’s artificial small parameter method.     

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.     

Fefferman’s mapping theorem on almost complex manifolds in complex dimension two

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds (see Theorem 1.1). The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.Mathematics Subject Classification (2000): 32H02,53C15     

Bohr’s equivalence relation in the space of Besicovitch almost periodic functions

The Ramanujan Journal - Based on Bohr’s equivalence relation which was established for general Dirichlet series, in this paper we introduce a new equivalence relation on the space of almost...