Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

Waring’s problem in prime numbers

The existence of a constant V(n) such that any sufficiently large natural number can be represented as a sum of nth degrees of primes in total quantity not exceeding this constant is proved.     

Parametric stability in Robe’s problem

In this work we have given a Hamiltonian formulation to Robe’s problem, obtaining again the classic results. We have computed the resonances existing in the circular case and obtained some information with regard to the linear stability of the central equilibrium of Robe’s problem in the elliptic case. In some critical cases we have constructed, in the parameter plane, the boundary curves that separate the regions of stability and instability.     

Students’ reasoning about relationships between variables in a real-world problem

This paper reports on part of an investigation of fifteen second-semester calculus students’ understanding of the concept of parametric function. Employing APOS theory as our guiding theoretical perspective, we offer a genetic decomposition for the concept of parametric function, and we explore students’ reasoning about an invariant relationship between two quantities varying simultaneously with respect to a third quantity when described in a real-world problem, as such reasoning is important for the study of parametric functions. In particular, we investigate students’ reasoning about an adaptation of the popular bottle problem in which they were asked to graph relationships between (a) time and volume of the water, (b) time and height of the water, and (c) volume and height of the water. Our results illustrate that several issues make reasoning about relationships between variables a complex task. Furthermore, our findings indicate that conceiving an invariant relationship, as it relates to the concept of parametric function, is nontrivial, and various complimentary ways of reasoning are favorable for developing such a conception. We conclude by making connections between our results and our genetic decomposition.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s(n) denote the number of solutions of the equation \({n= x^2 + y_1^k + y_2^k + \cdots + y_s^k}\) in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s(n) is obtained when k ≥ 3 and s ≥ 2^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s (n) denote the number of solutions of the equation n = x² + y₁^k + y₂^k + ?+ y_s^k{n= x^2 + y_1^k + y_2^k + \cdots + y_s^k} in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s (n) is obtained when k ≥ 3 and s ≥ 2 ^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2 ^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

To Favard’s theory for functional equations

We obtain some conditions for existence of almost periodic solutions for almost periodic functional equations in a Banach space which do not use the H-classes of these equations.     

Plateau’s problem in Finsler 3-space

We explore a connection between the Finslerian area functional based on the Busemann–Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau’s problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.     

The decidability of craig’s interpolation property in well-composed J-logics

Under study are the extensions of Johansson??s minimal logic J. We find sufficient conditions for the finite approximability of J-logics in dependence on the form of their axioms. Using these conditions, we prove the decidability of Craig??s interpolation property (CIP) in well-composed J-logics. Previously all J-logics with weak interpolation property (WIP) were described and the decidability of WIP over J was proved. Also we establish the decidability of the problem of amalgamability of well-composed varieties of J-algebras.     

New dynamical variables in Einstein’s theory of gravity

We describe an alternative formalism for Einstein’s theory of gravity. The role of dynamical variables is played by a collection of ten vector fields f _μ ^A , A = 1,..., 10. The metric is a composite variable, g _μν = f _μ ^A f _ν ^A . The proposed scheme may lead to further progress in a theory of gravity where Einstein’s theory is to play the role of an effective theory, with Newton’s constant appearing by introducing an anomalous Green’s function.     

Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Chowla’s cosine problem

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.     

Logarithmic Hesse’s problem

We show that if a Laurent polynomial on the coordinate ring of the complex algebraic torus on n variables has vanishing logarithmic Hessian, then up to an automorphism of the torus, the Laurent polynomial depends on at most n−1 variables.     

On Kulikov’s problem

Kulikov has given an étale morphism of degree d > 1 which is surjective modulo codimension two with X simply connected, settling his generalized jacobian problem. His method reduces the problem to finding a hypersurface and a subgroup of index d generated by geometric generators. By contrast we show that if D has simple normal crossings away from a set of codimension three and meets the hyperplane at infinity transversely, then necessarily d = 1. Received: 21 November 2006     

Reich’s problem concerning Halpern’s convergence

We discuss Halpern’s convergence for nonexpansive mappings in Hilbert spaces. We prove that one of the conditions in [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486–491] is the weakest sufficient condition among the conditions known to us. We also improve a necessary condition, which is close to Wittmann’s. This is one step to solve the problem raised by Reich in 1974 and 1983. Received: 15 July 2008     

Richardson’s iterative method for surface interpolation

Richardson’s iterative method has been used for approximating interpolating surfaces. We propose efficient modifications for tensor product surfaces that require less computational cost and storage, as well as faster convergence. Also, we consider some aspects of the triangular case together with some counter examples.     

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

On Andrunakievich’s chain and Koethe’s problem

In 1969 Andrunakievich asked whether one gets a ring without nonzero nil left ideals from an arbitrary ring R by factoring out the ideal A(R) which is the sum of all nil left ideals of R. Recently, it was shown that this problem is equivalent to Koethe’s problem. In this context one may consider the chain of ideals, which starts with A ₁(R) = A(R) ⊆ A ₂(R), where A ₂(R)/A ₁(R) = A(R/A ₁(R)), and extends by repeating this process. We study the properties of this chain and show that, assuming a negative solution of Koethe’s problem, this chain can terminate at any given ordinal number.     

On mockenhoupt’s conjecture in the Hardy-Littlewood majorant problem

The Hardy-Littlewood majorant problem has a positive answer only for even integer exponents p, while there are counterexamples for all p {ie91-1} 2?. Montgomery conjectured that even among the idempotent polynomials there must exist counterexamples, i.e. there exist a finite set of characters and some ± signs with which the signed character sum has larger p ^th norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, Mockenhaupt conjectured that even the classical {ie91-2} three-term character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0, 1, 2, i.e. in the range 0 < p < 6, p {ie91-3} 2?. Continuing this work here we demonstrate that even k = 3, 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G′²/G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation