Linear Statements Research Articles

Features of application of the Lanchester-type mathematical models in stochastic formulation when assessing the realities of air-land battle

This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations.

Students’ overreliance on linearity in economic thinking: An exploratory study at the tertiary level

In mathematics education, a vast amount of research has shown that students tend to rely on linearity in situations that are not linear at all. Several researchers in economics and economics education acknowledge people's overreliance on linearity as a potential source of misperception of economic phenomena, but within these fields this tendency has never been studied empirically in a systematic way. This paper is a first attempt to fill this gap in the research literature. The paper consists of two main parts. First, we provide an overview of instances of, and comments on, people's overreliance on linearity in the economics (education) literature. Second, we present an empirical study with business economics students who were confronted with correct and incorrect linear statements about micro- and macroeconomic situations. Our results show that even tertiary level students over-rely on linearity when analyzing such statements. We also find that this phenomenon is affected by the economic domain and by the way statements are formulated.

Linear time-dependent constraints programming with MSVL

This paper investigates specifying and solving linear time-dependent constraints with an interval temporal logic programming language MSVL. To this end, linear constraint statements involving linear equality and non-strict inequality are first defined. Further, the time-dependent relations in the constraints are specified by temporal operators, such as [InlineEquation not available: see fulltext.]. Thus, linear time-dependent constraints can be smoothly incorporated into MSVL. Moreover, to solve the linear constraints within MSVL by means of reduction, the operational semantics for linear constraints is given. In particular, semantic equivalence rules and transition rules within a state are presented, which enable us to reduce linear equations, inequalities and optimization problems in a convenient way. Besides, the operational semantics is proved to be sound. Finally, a production scheduling application is provided to illustrate how our approach works in practice.

Dynamic interaction of bridges and high-speed trains

Dynamic interaction of railway bridges and rolling stocks has been the subject of numerous studies for a rather long time [1–20]. The framework of railway bridges experiences significant loads from high-speed trains. Therefore, new methods for studying the interaction between high-speed trains and railway bridges are constantly developed in many countries [10–19]. The method proposed in [9, 16, 18, 19] for beams, rod and combined systems, bridge girders, and railway structures takes into account any required number of shapes in the deflection decomposition and has the resolving system of equations with unconditionally stable integration scheme and minimum number of unknowns (dynamic reaction of the wheels). The bridge dynamics problems of large dimension [16–20] are usually solved by the method of direct integration of equations (with the use of beam boundary elements), but the method proposed here is based on the use of the results of the modal analysis of the structure, which is very convenient, with the capabilities of the modern computational complexes taken into account, for determining the set of natural modes and frequencies of vibrations of any spatial linearly deformable bridge structures. Just as in [16–20] for problems with moving load, the method permits obtaining the resolving system of equations with minimum number of unknowns for a high degree of spatial digitation of the bridge framework and any necessary number of natural modes. Moreover, the step procedures proposed in [19, 20] to solve the differential equations of mechanical system vibrations permit using the car model as a moving load in the same way in the linear and nonlinear statements. In the proposed method, there are in fact two objects; namely, the set of bridge spans and the rolling stock (stocks) are considered as two super-elements (two substructures) interacting with each other if one of them is moving. The proposed method permits determining the stress-strain state of the bridge and the rolling stock traditionally after determining the dynamic pressures exerted by the wheeled carts on the bridge. The modern approaches to calculating the bridge-rolling stock interaction [10–17] are based on the wide use of computers with application of the finite element method and method of stepwise integration of equations describing the systemmotion. For example, in [10, 13] the iteration approach is used in problems of large dimension when the rolling stock-bridge system is decomposed. In [12], the explicit scheme of integration of differential equations is used to compensate for the large dimension of the problem. In [14], the beam finite elements are used to analyze a single-span bridge of box section in the span structure modeling. In [14], at the stage of time-digitation (by the Newton method), after the “rolling stock-bridge” system is decomposed into subsystems, the constraint reactions are omitted but the large dimension of the problem is preserved. Thus, the problem of large dimension in problems of bridge dynamics is often solved approximately with a certain accuracy of the solution.

BOUNDARY-VALUE PROBLEMS OF THE THEORY OF THIN AND NONTHIN ORTHOTROPIC SHELLS WITH ACCOUNT OF NONLINEARLY ELASTIC PROPERTIES AND LOW SHEAR RIGIDITY OF COMPOSITE MATERIALS

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.

Elastoplastic state of nonshallow flexible ellipsoidal shells under concentric loading

The elastoplastic stress-strain state of flexible ellipsoidal shells under concentric loading is investigated by the solution of nonlinear boundary-value problems. Numerical results are presented for open shells in the case of a free (unsupported) edge and a shell with a central opening (the periphery of the opening is supported by a thin ring). Solutions of the problems in linear and nonlinear statements are analyzed. Numerical results for ellipsoidal and spherical shells are compared.

Abstracts of papers in the Journal of Symbolic Computation

We consider linear problems in fields, ordered fields, discretely valued fields (with finite residue field or residue field of characteristic zero) and fields with finitely many independent orderings and discrete valuations. Most of the fields considered will be of characteristic zero. Formally, linear statements about these structures (with parameters) are given by formulas of the respective first-order language, in which all bound variables occur only linearly. We study symbolic algorithms ( linear elimination procedures ) that reduce linear formulas to linear formulas of a very simple form, i.e. quantifier-free linear formulas, and algorithms ( linear decision procedures ) that decide whether a given linear sentence holds in all structures of the given class. For all classes of fields considered, we find linear elimination procedures that run in double exponential space and time. As a consequence, we can show that for fields (with one or several discrete valuations), linear statements can be transferred from characteristic zero to prime characteristic p , provided p is double exponential in the length of the statement. (For similar bounds in the non-linear case, see Brown, 1978.) We find corresponding linear decision procedures in the Berman complexity classes [EQUATION] for d = 1,2. In particular, all these procedures run in exponential space. The technique employed is quantifier elimination via Skolem terms based on Ferrante & Rackoff (1975). Using ideas of Fischer & Rabin (1974). Berman (1977), Füer (1982), we establish lower bounds for these problems showing that our upper bounds are essentially tight. For linear formulas with a bounded number of quantifiers all our algorithms run in polynomial time. For linear formulas of bounded quantifier alternation most of the algorithms run in time [EQUATION] for fixed λ.

The complexity of linear problems in fields

We consider linear problems in fields, ordered fields, discretely valued fields (with finite residue field or residue field of characteristic zero) and fields with finitely many independent orderings and discrete valuations. Most of the fields considered will be of characteristic zero. Formally, linear statements about these structures (with parameters) are given by formulas of the respective first-order language, in which all bound variables occur only linearly. We study symbolic algorithms ( linear elimination procedures ) that reduce linear formulas to linear formulas of a very simple form, i.e. quantifier-free linear formulas, and algorithms ( linear decision procedures ) that decide whether a given linear sentence holds in all structures of the given class. For all classes of fields considered, we find linear elimination procedures that run in double exponential space and time. As a consequence, we can show that for fields (with one or several discrete valuations), linear statements can be transferred from characteristic zero to prime characteristic p , provided p is double exponential in the length of the statement. (For similar bounds in the non-linear case, see Brown, 1978 .) We find corresponding linear decision procedures in the Berman complexity classes ∪ c ∈ N S T A ( * , 2 c n , d n ) for d = 1, 2. In particular, all hese procedures run in exponential space. The technique employed is quantifier elimination via Skolem terms based on Ferrante & Rackoff (1975) . Using ideas of Fischer & Rabin (1974) , Berman (1977) , Fürer (1982), we establish lower bounds for these problems showing that our upper bounds are essentially tight. For linear formulas with a bounded number of quantifiers all our algorithms run in polynomial time. For linear formulas of bounded quantifier alternation most of the algorithms run in time 2 O ( n k ) for fixed k .

Abstracts of papers in the Journal of Symbolic Computation

We consider linear problems in fields, ordered fields, discretely valued fields (with finite residue field or residue field of characteristic zero) and fields with finitely many independent orderings and discrete valuations. Most of the fields considered will be of characteristic zero. Formally, linear statements about these structures (with parameters) are given by formulas of the respective first-order language, in which all bound variables occur only linearly. We study symbolic algorithms (linear elimination procedures) that reduce linear formulas to linear formulas of a very simple form, i.e. quantifier-free linear formulas, and algorithms (linear decision procedures) that decide whether a given linear sentence holds in all structures of the given class. For all classes of fields considered, we find linear elimination procedures that run in double exponential space and time. As a consequence, we can show that for fields (with one or several discrete valuations), linear statements can be transferred from characteristic zero to prime characteristic p, provided p is double exponential in the length of the statement. (For similar bounds in the non-linear case, see Brown, 1978.) We find corresponding linear decision procedures in the Berman complexity classes [EQUATION] for d = 1,2. In particular, all these procedures run in exponential space. The technique employed is quantifier elimination via Skolem terms based on Ferrante & Rackoff (1975). Using ideas of Fischer & Rabin (1974). Berman (1977), Fuer (1982), we establish lower bounds for these problems showing that our upper bounds are essentially tight. For linear formulas with a bounded number of quantifiers all our algorithms run in polynomial time. For linear formulas of bounded quantifier alternation most of the algorithms run in time [EQUATION] for fixed λ.

Statistische betrachtungen zur frage der abhängigkeit der nitrifikation von bodentemperatur und bodenfeuchtigkeit

By means of linear, linear-transformed and non-linear statements, the relationship is discussed between soil temperature and soil moisture on the one side and nitrification in the upper layers of a bare soil on the other. The results of the methods are compared. It is established that a non-linear model gives more information than any other. By means of the linear statements only the influence of soil temperature is discovered, whereas the non-linear model gives the combined action of soil temperature and soil moisture on the formation of NO 3-N. The non-linear model is based on a known model of the weather-yield statistics. By means of a square function, a hyperplane is fitted on the observed data and is resolved into components, resulting in a response hyperboloid, of which the surfaces of equal response are conic sections. By means of known formulae from analytic geometry, the Eigen-values and Eigen-vectors of the square form are calculated from the coefficients of the components to determine the type of the conic sections and they are used for criteria of the effectiveness of the parameters. The Eigen-vectors show that soil temperature and soil moisture are independent in their influence on the nitrification, while the Eigen-values indicate that the efficiency of the parameters is almost identical. The amount of NO 3-N increased with rising soil temperature. In the relation to soil moisture there is an optimum near the field capacity. Falling soil temperature in autumn and winter results in a decrease of NO 3-N, but only towards a fixed minimum. Calculations by use of the non-linear model result in a minimum of nearly 2 p.p.m., which is in conformity with common observations. A considerable increase of NO 3-N in spring, indicative of the beginning of micro-organic activity, results only after the soil temperature exceeds 5°C, even if the soil moisture is near the optimum. All results are valid only according the material used and are therefore to be regarded more qualitatively than quantitatively.