Let n be a positive integer. A collection 𝒮 of subsets of [n]={1,… ,n} is called symmetric if X∈𝒮 implies X^*∈𝒮 , where X^*:={i∈ [n]:n-i+1∉ X} . As the main results of this paper, one shows that in each of the three types of separation relations: strong, weak and chord ones, the following “purity phenomenon” takes place: all inclusion-wise maximal symmetric separated collections in 2^[n] have the same cardinality. These give “symmetric versions” of well-known results on the purity of usual strongly, weakly and chord separated collections of subsets of [n], and in the case of weak separation, this extends a result due to Karpman on the purity of symmetric weakly separated collections in ( [ [n]; n/2 ]) for n even.

For positive integers n,n', we give a combinatorial characterization for the set of quadratic inequalities on minors that are valid for all n× n' totally nonnegative matrices. This is obtained as a consequence from our earlier results on stable quadratic identities on minors of matrices generated by flows in planar graphs via Lindström's construction.

We propose versions of higher Bruhat orders for types B and C. This is based on a theory of higher Bruhat orders of type A and their geometric interpretations (due to Manin–Shekhtman, Voevodskii–Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes.

The concept of sequential choice functions is introduced and studied. This concept applies to the reduction of the problem of stable matchings with sequential workers to a situation where the workers are linear.

We carried out an experimental ultrasonic study of polyhydric alcohols with the general chemical formula CnHn+2(OH)n with an increasing number of OH groups: glycerol (n = 3), erythritol (n = 4), xylitol (n = 5), sorbitol (n = 6). The baric and temperature dependences of the elastic characteristics of these substances in the crystalline and glassy states were studied both under isothermal compression up to 1 GPa and during the isobaric heating of 77-295 K. For glycerol, glasses were obtained at different cooling rates, glass-liquid transitions were studied at different pressures. All the studied glasses have lower elastic moduli than the same substances in the crystalline state at the same pressure-temperature conditions. We obtained a cascade of glass-supercooled liquid-crystal transitions during heating of glassy erythritol. In the series of substances with n = 3, 4, 5 the bulk moduli show a tendency to decrease with increasing n. However, sorbitol (n = 6) unexpectedly has the highest elastic moduli among the studied substances in both the glassy and crystalline states. The studied glassformers show a general tendency to increase the glass transition temperature Tg and the fragility coefficient m with increasing n.

The paper presents the results of investigation of the dislocation structure, the magnitude of the curvature–torsion of the crystal lattice, the amplitudes of the internal stress fields, as well as the redistribution of carbon atoms in zones of stable localization of deformations depending on the state of samples made of structural steel 20. Samples were studied without operation, after operation without destruction, and with destruction. The samples were stretched at the same speed until a stable deformation localization zone appeared, after which the loading was stopped. To study the dislocation structure, we used the method of transmission electron microscopy on thin foils. It has been shown that deformation of samples made of structural steel 20 in the direction “initial” → “not destroyed” → “destroyed” sample in all morphological components of the structure, as well as in the material as a whole, gradually leads to a fragmentation of the structure down to a nanocrystalline structure in individual areas of the material. Dislocations move to fragment boundaries, and the scalar dislocation density ρ decreases. At the same time, internal shear stresses σs decrease. The change in local stresses σl is of a different nature, namely, the plastic component of local stresses σ _l^pl decreases, and the elastic component σ _l^el increases sharply. It has been established that deformation of steel 20 leads to the destruction of cementite particles located at the boundaries of dislocation fragments (the volume fraction decreases) and the formation of carbides inside the fragments (their volume fraction increases). Carbon from destroyed cementite particles at the boundaries of fragments is spent on the formation of cementite particles inside the fragments, of crystal lattice defects, and the formation of carbides in new morphological structures (micro- and nanograins).

В работе представлены результаты исследования дислокационной структуры, величины кривизны-кручения кристаллической решетки, амплитуд полей внутренних напряжений, а также перераспределения атомов углерода в зонах устойчивой локализации деформаций в зависимости от состояния образцов из конструкционной стали 20. Исследовались образцы без эксплуатации, после эксплуатации без разрушения и с разрушением. Образцы растягивали с одинаковой скоростью до появления устойчивой зоны локализации деформации, после чего нагружение останавливали. Для исследования дислокационной структуры в работе использовали метод просвечивающей электронной микроскопии на тонких фольгах. Показано, что деформация образцов из конструкционной стали 20 в направлении «исходный» ® «не разрушенный» ® «разрушенный» во всех морфологических составляющих структуры, а также в целом по материалу постепенно приводит к измельчению структуры вплоть в отдельных участках материала до нанокристаллической. Дислокации перемещаются на границы фрагментов, скалярная плотность дислокаций r уменьшается. Одновременно уменьшаются и внутренние напряжения сдвига sЛ. Изменение локальных напряжений sд носит иной характер, а именно, пластическая составляющая локальных напряжений уменьшается, а упругая резко возрастает. Установлено, что деформация стали 20 приводит к разрушению частиц цементита, расположенных на границах дислокационных фрагментов (объемная доля уменьшается), и образованию карбидов внутри фрагментов (объемная доля их увеличивается). Углерод из разрушенных частиц цементита на границах фрагментов идет на образование частиц цементита внутри фрагментов, на дефекты кристаллической решетки и образование карбидов в новых морфологических структурах (микро- и нанозернах).

The use of acoustic and magnetic methods of non-destructive testing to detect zones of stable localization of deformation in order to assess and predict the performance of long-term equipment is of scientific and practical interest at present. A structural–mechanical criterion was developed that reflects the relationships between the structural and substructural states, internal stress fields and stable localization of deformations with the characteristics of non-destructive tests in the metal of long-term equipment made of structural 0.2 C steel and heat-resistant 0.12C-1Cr-1Mo-1V steel. The values of the structural–mechanical criteria Ks.-m for structural 0.2 C steel and for heat-resistant 0.12C-1Cr-1Mo-1V steel, corresponding to the moment of stable localization of deformation, are established. At the same time, it is recommended to replace the checked equipment nodes due to the exhaustion of the resource. The proposed and justified approach to assessing and predicting the performance and residual life of long-term power equipment, based on the identified relationships between the structural and substructural states, internal stress fields and stable localization of deformations with the characteristics of non-destructive tests and the calculation of the structural–mechanical criterion, was applied at a number of power plants in the Kemerovo region—Kuzbass. A methodology was developed for evaluating the residual life, based on the identification and use of relationships between structural and substructural states, internal stress fields and stable localization of deformations with the characteristics of non-destructive tests and the calculation of a structural–mechanical criterion.

Luders deformation in an aluminum-magnesium alloy was studied in a wide range of loading rates. It was found that the deformation is always accompanied by the propagation of deformation fronts, which move either continuously or intermittently depending on the strain rate. The kinetics of fronts is determined by the reaction of active deformable media to an external mechanical action and is determined by the autowave nature of plastic deformation.

The authors studied the nature of mobile fronts of localized deformation that generate and propagate during deformation of metastable austenitic-martensitic TRIP steel VNS9-Sh along the entire length of the loading curve from the yield point to fracture. A joint research of the nature of the deformation fronts movement and kinetics of the magnetic phase accumulation made it possible to establish that the fronts under consideration are the fronts of the thermoelastic phase transformation of metastable austenite into martensite. This transformation is realized firstly by formation of the Chernov–Lüders bands and then the Portevin–Le Chatelier bands. Both processes are consistent with staging of the deformation curve, which contains a pseudo-plateau, a section with an increasing hardening coefficient, and a section with a decreasing hardening coefficient. It is shown that the deformation-induced phase transformation corresponds to the fronts propagating on the pseudo-plateau and on the section of loading curve with an increasing hardening coefficient. The Portevin–Le Chatelier bands, which are formed in the section of the loading diagram with a decreasing hardening coefficient, are not associated with “austenite-martensite” transformation and have a twin nature. The kinetics of thermoelastic transformation fronts, as well as deformation fronts in materials with a shear mechanism of shaping, can be described in terms of the autowave concept. On the yield plateaus, the phase transformation occurs through generation and propagation of localized plasticity switching autowaves. In the section with an increasing hardening coefficient, it continues through generation and movement of excitation autowaves. The propagation regions of excitation autowaves are limited in the sample space. They are set by the zones of origin and annihilation of primary switching autowaves which were formed on the yield plateau.

The kinetics of Chernov–Lüders and Portevin–Le Chatelier deformation processes at the yield plateau in an aluminum-magnesium alloy is analyzed. It is established that the deformation is localized in moving deformation fronts. In general, the movement of the fronts proceeds discretely and only in the phase of abrupt unloading of the sample. Continuous movement of deformation fronts is possible if the rate of recovery of the operating stresses applied by test device is greater than or equal to the rate of stress decline controlled by internal processes at a lower structural-scale level. The kinetics of the motion of deformation fronts, both in the Chernov–Lüders and Portevin–Le Chatelier processes, can be described within the autowave concept of plastic flow, and the fronts themselves represent excitation autowaves.

The paper analyzes the elastoplastic transition in Fe–0.025 wt. % C at a temperature of 296–503 K and strain rate of 6.67·10−6–3.33·10−3 s−1. The analysis shows that the lower yield stress increases by a power law with increasing the strain rate, and that its rate sensitivity decreases linearly with increasing the test temperature. At temperatures lower than 393 K, the rate sensitivity of the lower yield stress is normal, and at 393–503 K, it is zero. In the range 393–503 K, the kinetics of the Lüders bands is changed from steady to discrete, and the higher the strain rate, the higher the temperature of this transition. Using the available data on the dynamics of dislocations and diffusion of interstitial impurities in the test alloy, it is demonstrated that the kinetics of Lüders bands are controlled by the effect of dynamic strain aging. If the arrest time of mobile dislocations tw at barriers which are overcome via thermal activation is comparable with the precipitation time of interstitial atoms ta at these dislocations, the motion of a Lüders band is discrete, and the band represents an excitation wave of localized plasticity; its refractory period is determined by the time of dynamic strain aging. If ta >> tw, the band moves monotonically and represents a switching autowave. The results of the analysis suggest that the effect of serrated yielding at the lower temperature boundary of blue brittleness can be suppressed by increasing the strain rate. When the arrest time of dislocations tw decreases, the comparability of tw and ta is broken, and no excitation autowave is formed. The data reported in the paper can be used to develop warm rolling technologies for materials with a sharp elastoplastic transition.

The elastic properties of glass α- and β-modifications of benzophenone (C6H5)2CO are determined for the first time by the ultrasonic method at high pressures up to 1 GPa and in the temperature range of 77 K< T < 293 K. Four states of benzophenone are experimentally observed in the investigated temperature range of 77-293 K: glass, supercooled liquid, and α- and β-crystalline phases. The boundaries of phase transitions during isobaric heating are determined. The bulk and shear moduli B and G, Poisson's ratio σ, and density ρ are calculated. Glassy benzophenone has much lower elastic moduli than the α-modification at 77 K (the shear modulus G of glass is about half the value for the crystal). The crystalline α-modification demonstrates a strong softening of both moduli with an increase in the temperature in the range of 77-293 K. Heating of glass leads to the formation of a very viscous supercooled liquid, which crystallizes into the metastable β-modification. A further increase in the temperature leads to a monotropic phase transition β → α.

We consider a hypergraph (I, C), with possible multiple (hyper)edges and loops, in which the vertices i ∈ I are interpreted as agents, and the edges c ∈ C as contracts that can be concluded between agents. The preferences of each agent i concerning the contracts where i takes part are given by use of a choice function fi possessing the so-called path independent property. In this general setup we introduce the notion of stable network of contracts. The paper contains two main results. The first one is that a general problem on stable systems of contracts for (I, C, f) is reduced to a set of special ones in which preferences of agents are described by use of so-called weak orders, or utility functions. However, for a special case of this sort, the stability may not exist. Trying to overcome this trouble when dealing with such special cases, we introduce a weaker notion of metastability for systems of contracts. Our second result is that a metastable system always exists.

The phenomenon of the strain-rate sensitivity of metallic materials has been a topic of interest since the first mechanical tests at different strain rates were performed. The problem of its theoretical description appeared simultaneously. Despite the significant number of studies covering this issue, it is necessary to rule out a few drawbacks of previously reported models, which is the goal of this work. Herein, an extension of the elastic–viscoplastic model to a generalized state of stress is proposed while aiming to describe the strain rate sensitivity of Armco-iron samples that were pulled in tension within the framework of the finite-difference method. A mathematical model was formulated using equivalent stress and strain, which alleviated the complexity of the relaxation-type constitutive equations. The critical shear stress (CSS) function describes S-type instability with a single equation. The plastic strain rate was calculated based on the well-known Orowan equation, which is related to dislocation dynamics. In addition, the model took the material’s microstructure into account based on the design of a representative volume element (RVE) using the step-by-step packing (SSP) method. The results of the modeling were compared with the available experimental data and were found to satisfactorily correlate with them. The results suggest that the misfit error between the model and experimental data did not exceed 10% in the range of strain rates under study, which is a reliable outcome.

The paper studies complementary choice functions, i.e. monotonic and consistent choice functions. Such choice functions were introduced and used in the work [ 13 ] for investigation of matchings with complementary contracts. We provide three (universal) ways of constructing such choice functions: through pre-topologies, as direct images of completely complementary (or pre-ordered) choice functions, and with the help of supermodular set-functions.

Reasons for the existence of “fast” sound at terahertz frequencies in various liquids have been analyzed. It has been shown that the fast sound speed is described well by the conventional formula from the theory of elasticity V_l = ((B(ω ) + 4/3G(ω ))/ρ)^1/2 , where ρ is the density of a liquid and B(ω ) and G(ω ) are the bulk and shear moduli at the frequency ω, respectively. The excess of the speed of fast sound over the speed of normal sound in “normal” liquids is 10–20 G(ω ) at high frequencies, and vanishes on the Frenkel line. At the same time, the huge excess (50–120 B(ω ) . Anomalously low relaxing bulk moduli were studied in our previous works for many oxide and chalcogenide glasses near smeared pressure-induced phase transitions. In anomalous liquids, smeared phase transitions also occur in a wide temperature and pressure region, which sharply reduces the bulk moduli and speeds of sound. Thus, the record large difference between speeds of fast and normal sound in anomalous liquids is due not to anomalously fast sound but to the fact that normal sound in such liquids is anomalously “slow” and bulk moduli are anomalously low. Ultrasonic studies of low- and high-density amorphous water ices show that their bulk moduli are indeed a factor of 4–5 higher than the bulk modulus of water. In addition, because of smeared phase transitions, the heat capacities of water and tellurium melt are a factor of 1.5–2 higher than those for normal liquids; i.e., anomalous liquids are characterized not only by an anomalous (nonmonotonic) behavior but also by anomalous magnitudes of physical quantities for most of the available measurement methods. A similar anomalous increase in the compressibility and heat capacity is observed for all fluids in the close vicinity of the liquid–gas critical point. In this case, anomalously fast sound is observed at terahertz frequencies, which is also due to a sharp increase in the bulk modulus B(ω ) at high frequencies. At the same time, high compressibility and heat capacity, as well as a large excess of the speed of fast sound over the speed of normal sound, for anomalous liquids and glasses near smeared phase transitions are not necessarily due to the proximity of critical points and occur in any scenario of the smeared phase transition.

A review of works devoted to the theory of stable matchings or, more generally, of stable networks of contracts is given. A set (network) of contracts is called stable if no coalition has an available contract that gives all coalition members strictly more than the proposed set. In a special case, this concept was introduced in 1962 by Gale and Shapley and has since gone a long way in its development both theoretically (theorems, structures, and algorithms) and in the field of applications in economics, physics, biology, and mathematics.

The paper proposes a model of general economic equilibrium, when some firms can influence prices by their actions. Their primitives (that is, the description of production, distribution, and consumption) of the proposed model are the same as those of the Arrow–Debreu model. The difference lies in the behavior of oligopolistic firms: it is assumed that they (as in the original Cournot model) assign their production plans. The main innovation is the description of the decision-making process on production plans by oligopolistic firms. In previous models, it was assumed that oligopolistic firms seek to maximize profits at current prices. Here, a more natural assumption is made that oligopolistic firms make decisions about production plans through a vote of shareholders. It is shown how the Arrow–Debreu model can be modified to account for this circumstance. The concept of Condorcet–Cournot–Walras equilibrium is introduced, combining the ideas of these three classics. The question of the existence of equilibria is discussed. A simplified version of the model is also considered, in which each firm is owned by one agent. The simplification is that there is no need to turn to the voting.