“Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom

We construct solutions of analogues of a time-dependent Schrödinger equation corresponding to an isomonodromic polynomial Hamiltonian of a Garnier system with two degrees of freedom. The solutions are determined by solutions of linear differential equations whose compatibility condition is the given Garnier system. With explicit substitutions, these solutions reduce to solutions of the Belavin–Polyakov–Zamolodchikov equations with four times and two spatial variables.     

New degrees of freedom for high-order Whitney approximations of Darcy’s flows

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. We propose a high-order discretisation based on Whitney finite elements, namely, Raviart-Thomas finite elements of degree r +?1 for the discharge and discontinuous piecewise polynomial finite elements of degree r for the pressure, with r ≥?0. We comment on the use of new degrees of freedom that have a clear physical meaning, the so-called weights on the small simplices, for the involved discharge and pressure fields. We describe a new numerical strategy to solve the discrete problem based on a tree-cotree block-decomposition of the unknowns that is natural when considering these new degrees of freedom. Preliminary numerical tests in two dimensions confirm the stability of the adopted method and the effectiveness of the new degrees of freedom.

     

Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λ_j,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ_j is of the order of λ^d/2, where d, the number of degrees of freedom, is less than two.     

Modules with Nakayama’s property

Modules M _A with Nakayama’s property are studied. In particular, for a right invariant ring A, it is proved that all right A-modules satisfy Nakayama’s property if and only if the ring A is right perfect.     

“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H ₁(z, t, q ₁, q ₂, p ₁, p ₂) corresponding to the second equation P ₁ ² in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P ₁ ² with respect to z. This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian H ₂(z, t, q ₁, q ₂, p ₁, p ₂) of a Hamiltonian system with respect to t compatible with P ₁ ² . A similar situation occurs for the P ₂ ² equation in the Painlevé II hierarchy.     

One modification of Gomory’s algorithm

The problem of constructing optimum cuts in the Gomory method for integer linear programming (ILP) problems is considered. Algorithms for finding cuts are described, and their complexity is assessed. The modified Gomory algorithm is compared to a standard Gomory algorithm.     

Properties of two Shannon’s ciphers

In 1949 Shannon published the famous paper “Communication theory of secrecy systems” where he briefly described two ciphers, but did not investigate their properties. In this note we carry out information-theoretical analysis of these ciphers. In particular, we propose estimations of the cipher equivocation and the probability of correct deciphering without key.     

Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around the ‘gap’ aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the first case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Research of T. Leiger supported by Estonian Science Foundation Grant 5376.     

A superadditive property of Hadamard’s gamma function

Hadamard’s gamma function is defined by

Properties of solutions of two second-order differential equations with the Painlevé property

Theoretical and Mathematical Physics - We consider a Hamiltonian system equivalent to the Painlevé II equation with respect to one component and to the Painlevé XXXIV equation with...     

Expressing One’s Mind

Remarks such as ‘I am in pain’ and ‘I think that it’s raining’ are puzzling, since they seem to literally describe oneself as being in pain or having a particular thought, but their conditions of use tend to coincide with unequivocal expressions of pain or of that thought. This led Wittgenstein, among others, to treat such remarks as expressing, rather than as reporting, one’s mental states. Though such expressivism is widely recognized as untenable, Bar-On has recently advanced a neo-expressivist view, on which such remarks exhibit characteristics of both expressions of mental states and reports of those states. I argue against any attempt to see such remarks as both reporting and expressing the same mental states, and that a correct account rests on distinguishing the truth conditions of such remarks from their conditions of use.     

Uniform distribution of generalized Kakutani’s sequences of partitions

We consider a generalized version of Kakutani’s splitting procedure where an arbitrary starting partition π is given and in each step all intervals of maximal length are split into m parts, according to a splitting rule ρ. We give conditions on π and ρ under which the resulting sequence of partitions is uniformly distributed.     

On Classes of Generalized Subreducts of Tarski’s Relation Algebras with One Diophantine Binary Operation

Mathematical Notes - In the paper, systems of elementary axioms are found for classes of groupoids and ordered groupoids of binary relations with Diophantine operations that are generalized...     

Schoenberg’s polynomial B-splines of odd degrees: A brief review of application

A brief overview of applications of Schoenberg’s polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the found Schoenberg B-spline of 15th degree are presented. High-quality approximations of smooth functions with an infinite Fourier transform by functions with a finite Fourier transform are found.     

Developing the concept of linear function: one student’s experiences with dynamic physical models

Developing a view of functions as systematic processes involving co-variation among variables has been identified as a goal for mathematics learners at the pre-university level. In this investigation I examined the processes used by an eighth-grade student to interpret linear functions originating in dynamic physical models and the processed he used to link his interpretations to tables, equations, and graphs. The student deepened his understanding of functions by generalizing his view of multiplication beyond that of products resulting from the multiplication of individual factors or from repeated addition. He was enabled to do this by building links between graphic and tabular representations of the functions generated from his exploration with dynamic physical models and by comparing tables of different linear functions. This paper suggests that the development of a student’s reasoning about functions originating in dynamic physical models can be interpreted in terms of generalized multiplicative processes that may occur thorough mapping variations and that a student who interprets such functions as generalized multiplicative processes may use notational variations to generate representing equations.     

A unification of two refinements of Euler’s partition theorem

We obtain a unification of two refinements of Euler’s partition theorem respectively due to Bessenrodt and Glaisher. A specialization of Bessenrodt’s insertion algorithm for a generalization of the Andrews-Olsson partition identity is used in our combinatorial construction.     

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.     

On the Bose–Maslov statistics in the case of infinitely many degrees of freedom

We find the asymptotics of the counting function of elements of an additive arithmetical semigroup for the case of an exponential counting function of prime generators, which has a natural interpretation in terms of Bose statistics as well as in the problem of counting the number of Gaussian packets on decorated graphs.