Modules with a“nice”endomorphism ring and a new characterization of semisimple modules and rings

In this work we introduce some classes of modules whose endomorphism rings have some of the properties of the endomorphism rings of vector spaces. Then we apply these notions to obtain new characterizations of semisimple rings and modules.     

Modeling of the dynamic phenomena “stick-slip” and “whirl” of a drill string demonstrator

Vibrations of drill strings in oil and gas well bores can reduce drilling performance or lead to damage of drill string components. These vibrations are neither optically observable nor measurable because of the geological formation. Hence usually simulations are used to understand the dynamic behavior. The test stand “OSTrator” adds more insight to this problem. The demonstrator represents a scaled drill rig granting direct access to the complex dynamic behavior of the real drill string. Two phenomena are of special interest: the stick-slip effect, a torsional oscillation, and the whirl effect, a periodic torsional-lateral movement with permanent wall contact. Since the OSTrator was built for showcase purposes, it has to be piloted in and out Stick-Slip and Whirl domains by a control algorithm. Therefore efficient multi-body-systems able to represent these two effects in faster than real time are demanded. In this presentation, the modeling of the stick-slip and the whirl effect are discussed. The models should be able to predict these effects and supply data to avoid or confront them. Key questions are the modeling of the wall contact and the resistance at the drill bit. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).     

On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Critical sets and two-dimensional topologically mixing attractors of a pair of coupled “Tent maps”

Endomorphisms of the plane generated by coupling two identical one-dimensional unimodal piece-wise linear maps are proved to have two-dimensional topologically mixing attractors. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 654–658, May, 1999.     

On Sums of Two Squares of Primes and a Cube of a Prime

In this paper we are concerned with the exceptional set of the sum of two squares of primesand a cube of a prime P;+p;+p;.Noting that竹三1 or 3(mod 6)is a necessary conditionfor the solvability of the equation n=P}+P;+P;(see【1]),we define E(N)=Card{n:n≤N,礼∈三and n≠P;+Pi+P;for any Pi,1≤i s 3), (1)where三={n:n三1 or 3(mod 6)). This and similar problems have been studied by a number of authors.In 1937,Davenportand Heilbronn[2J proved that if后2 3 is an odd integer then almost all posi…     

Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H₁,...,H_k}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H₁,...,H_k)-free graph,denoted by ex(n,F) or ex(n,{H₁,H₂,...,H_k}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons...     

Erratum to “On a Classical Theorem on the Diameter and Minimum Degree of a Graph”

The original version of the article was published in[1]. Unfortunately, the original version of this article contains a mistake:in Theorem 6.2 appears that β(n, Δ)=(n-Δ+5)/4 but the correct statement is β(n, Δ)=(n-Δ+4)/4. In this erratum we correct the theorem and give the correct proof.     

Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system

A conic linear system is a system of the form?P(d): find x that solves b - Ax∈C _Y , x∈C _X ,? where C _X and C _Y are closed convex cones, and the data for the system is d=(A,b). This system is“well-posed” to the extent that (small) changes in the data (A,b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be the smallest change in the data Δd=(ΔA,Δb) for which the system P(d+Δd) is “ill-posed”, i.e., d+Δd is in the intersection of the closure of feasible and infeasible instances d’=(A’,b’) of P(·). Renegar also defined the “condition measure” of the data instance d as C(d):=∥d∥/ρ(d), and showed that this measure is a natural extension of the familiar condition measure associated with systems of linear equations. This study presents two categories of results related to ρ(d), the distance to ill-posedness, and C(d), the condition measure of d. The first category of results involves the approximation of ρ(d) as the optimal value of certain mathematical programs. We present ten different mathematical programs each of whose optimal values provides an approximation of ρ(d) to within certain constants, depending on whether P(d) is feasible or not, and where the constants depend on properties of the cones and the norms used. The second category of results involves the existence of certain inscribed and intersecting balls involving the feasible region of P(d) or the feasible region of its alternative system, in the spirit of the ellipsoid algorithm. These results roughly state that the feasible region of P(d) (or its alternative system when P(d) is not feasible) will contain a ball of radius r that is itself no more than a distance R from the origin, where the ratio R/r satisfies R/r≤c ₁ C(d), and such that r≥ and R≤c ₃ C(d), where c ₁,c ₂,c ₃ are constants that depend only on properties of the cones and the norms used. Therefore the condition measure C(d) is a relevant tool in proving the existence of an inscribed ball in the feasible region of P(d) that is not too far from the origin and whose radius is not too small. Received November 2, 1995 / Revised version received June 26, 1998?Published online May 12, 1999     

Effect of a measuring instrument in the “Bose condensate” of a classical gas in a phase transition and in experiments with negative pressure

We systematically present a new approach to classical thermodynamics using asymptotic distributions from number theory that generalize the Bose-Einstein distribution. We justify the transition to the liquid state, the thermodynamics of fluids, and also the behavior of liquids in the region of negative pressures We present a comparison with experimental data.     

Random walk of a “drunk company”

We study the collective behavior of a system of Brownian agents each of which moves orienting itself to the group as a whole. This system is the simplest model of the motion of a “united drunk company.” For such a system, we use the functional integration technique to calculate the probability of transition from one point to another and to determine the time dependence of the probability density to find a member of the “drunk company” near a given point. It turns out that the system exhibits an interesting collective behavior at large times and this behavior cannot be described by the simplest mean-field-type approximation. We also obtain an exact solution in the case where one of the agents is “sober” and moves along a given trajectory. The obtained results are used to discuss whether such systems can be described by different theoretical approaches.     

On a Result of Pandey and Bhargava

Let S denote the class of functions f(z) which are analytic in the unit disc D={z:|z|<1} and normalized by the conditions f(0)=0=f′(0)-1. Let K(μ)and S (m,M ) be the subclasses of S and satisfying the conditions     

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

Singularity of the “swallow-tail” type and the glass-glass transition in a system of collapsing hard spheres

In the mode-coupling approximation, we consider the transition to the glass state in a system of collapsing hard spheres (a system with the hard-core potential to which a repulsive step is added). We propose an approximation for the structure factor of the system, which we use to construct the phase diagram of the transition to the glass state. We show that there exists a maximum on the liquid-glass curve corresponding to the reentrant transition to the glass state in the system. In the framework of the proposed model, we consider bifurcations of solutions of the equations describing the transition to the glass state and show that there exist bifurcations of the “swallow-tail” type corresponding to the glass-glass transition.     

Classical analogue of a quantum Schwarzschild black hole: “Standard model” and beyond

We build a model in which the main global properties of classical and semiclassical black holes become local: these are the event horizon, “no-hair,” temperature, and entropy. Our construction is based on the features of a quantum collapse, discovered when studying some particular quantum black hole models. But our model is purely classical, and this allows using the Einstein equations and classical (local) thermodynamics self-consistently and, in particular, solving the “puzzle of log 3.”     

Normalizer of a group of “diagonal” automorphisms in algebras over a commutative ring

Let S be a faithful algebra over commutative ring R. It is assumed that S is additively generated by its invertible elements. It is shown that the nomalizer of subgroup Aut(S_s) of group Aut(S_R) coincides with the semidirect product Aut(S_S) Aut(S/R),where the second factor is the group of all ring automorphisms of ring S identical on R.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 5–8, 1991.     

Numbers of the Equitype and Quasi-Equitype Decompositions of a Permutation

In the paper“The equitype and quasi-equitype decompositions of arbitrarypermutation by permutations of order 2 or 3”,we proved that any even permu-tation can be expressed as a product of two equitype involutions and any odd per-mutation can be expressed as a product of two quasi-equitype involutions. Meanwh-ile,we gave the condition on t such that an even permutation has(2~t)equitypedecompositions and the condition on t such that an odd permutation has(2;2~t)quasi-equitype decompositions.In this paper,we give the numbers of the equitypeand quasi-equitype decompositions of a permutation.The main results are