A measurable-group-theoretic solution to von Neumann’s problem

We give a positive answer, in the measurable-group-theory context, to von Neumann’s problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors. D. Gaboriau’s research was supported by CNRS. R. Lyons’ research was supported partially by NSF grant DMS-0705518 and Microsoft Research.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

A note on Furstenberg’s filtering problem

This note gives a positive answer to an old question in elementary probability theory that arose in Furstenberg’s seminal article “Disjointness in Ergodic Theory.” As a consequence, Furstenberg’s filtering theorem holds without any integrability assumption.     

On Hilbert’s solution of Waring’s problem

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x ₁ ^k + x ₂ ^k + … + x _g ^k , where the x _i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.     

Numerical method for an approximate solution to Bulgakov’s problem

A great deal of the attention devoted to controlled system dynamics optimization in the literature is given to Bulgakov’s familiar problem of the accumulation of perturbation in a linear system. A more general case of this problem is considered in this work. An approximate method for calculating the value characterizing maximum dispersion is discussed.     

Vinogradow’s solution of Waring’s problem: Hypothesis K of Hardy and Littlewood

Proceedings - Mathematical Sciences -     

Curry’s Formalism as Structuralism

In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written then, although one of them was not published until 1951. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s views by referring to some of his later writings on the subject. I claim that Curry’s philosophy was not what is now usually called formalism, but is really a form of structuralism.     

A remark on the Abel’s and Dirichlet’s criterions concerning generalizations to monotonicity

The Abel’s and Dirichlet’s criterions for convergence of series in analysis are very basic classical results and both require the monotonicity condition. In this note we show that the monotonicity condition in these criterions can be generalized to RBV condition, while cannot be generalized to quasimonotonicity.     

Recurrent approach to Blaschke’s problem

We have obtained a recurrence formula $I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}We have obtained a recurrence formula I_n+3 = \frac4(n+3)p(n+4)VI_n+1I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1} for integro-geometric moments in the case of a circle with the area V, where I_n = ò\nolimits_{K ?G}sⁿd GI_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G, while in the case of a convex domain K with the perimeter S and area V the recurrence formula

In+3 = \frac8(n+3)V2(n+1)(n+4)p[\fracd In+1d V - \fracIn+1S \fracd Sd V ] I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big]      13. Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem      D. V. Kostin 《Mathematical Notes》2008,83(1-2):48-56 We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.      14. Mikhlin’s problem on Carnot groups      N. N. Romanovskii 《Siberian Mathematical Journal》2008,49(1):155-165 We consider one class of singular integral operators over the functions on domains of Carnot groups. We prove the L p boundedness, 1 < p > ∞, for the operators of this class. Similar operators over the functions on domains of Euclidean space were considered by Mikhlin.      15. On Karatsuba’s problem related to Gram’s law      M. A. Korolev 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):156-166 A number of new results related to Gram’s law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram’s law.      16. A Borel solution to the Horn–Tarski problem      Stevo Todorcevic 《Acta Mathematica Hungarica》2014,142(2):526-533 We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition.      17. On Andrunakievich’s chain and Koethe’s problem      M. A. Chebotar  P. -H. Lee  E. R. Puczyłowski 《Israel Journal of Mathematics》2010,180(1):119-128 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.      18. A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’      《Operations Research Letters》1988,7(3):117-125 Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.      19. The solution of graham’s greatest common divisor problem      M. Szegedy 《Combinatorica》1986,6(1):67-71 The following conjecture of R. L. Graham is verified: Ifn≧n 0, wheren 0 is an explicitly computable constant, then for anyn distinct positive integersa 1,a 2, ...,a n we have a i /(a i ,a j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a i ,a j ) stands for the greatest cnmmon divisor ofa i anda j .      20. Linear algebra-based solution of the gambler’s ruin problem      Greg Orosi 《International Journal of Mathematical Education in Science & Technology》2017,48(1):107-111 In this paper, we derive the result of the classical gambler’s ruin problem using elementary linear algebra. Moreover, the pedagogical advantage of the derivation is briefly discussed.

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

Mikhlin’s problem on Carnot groups

We consider one class of singular integral operators over the functions on domains of Carnot groups. We prove the L _p boundedness, 1 < p > ∞, for the operators of this class. Similar operators over the functions on domains of Euclidean space were considered by Mikhlin.     

On Karatsuba’s problem related to Gram’s law

A number of new results related to Gram’s law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram’s law.     

A Borel solution to the Horn–Tarski problem

We describe a Borel poset satisfying the σ-finite chain condition but failing to satisfy the σ-bounded chain condition.     

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.     

A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’

Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.     

The solution of graham’s greatest common divisor problem

The following conjecture of R. L. Graham is verified: Ifn≧n ₀, wheren ₀ is an explicitly computable constant, then for anyn distinct positive integersa ₁,a ₂, ...,a _n we have a _i /(a _i ,a _j ) ≧ ≧n, and equality holds only in two trivial cases. Here (a _i ,a _j ) stands for the greatest cnmmon divisor ofa _i anda _j .     

Linear algebra-based solution of the gambler’s ruin problem

In this paper, we derive the result of the classical gambler’s ruin problem using elementary linear algebra. Moreover, the pedagogical advantage of the derivation is briefly discussed.