Search

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω−⊂ℝ3$$ {\Omega}^{-}\subset {\mathrm{\mathbb{R}}}^3 $$, has a compact boundary surface S=∂Ω−$$ S=\partial {\Omega}^{-} $$, which is divided into two disjoint parts, the Dirichlet part SD$$ {S}_D $$ and the Neumann part SN$$ {S}_N $$, where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L2$$ {L}_2 $$‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space [W2,loc1(Ω−)]3$$ {\left[{W}_{2, loc}^1\left({\Omega}^{-}\right)\right]}^3 $$ and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the Lp$$ {L}_p $$‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses Cα$$ {C}^{\alpha } $$‐Hölder continuity property in the closed domain Ω−‾$$ \overline{\Omega^{-}} $$ with α=12−ε$$ \alpha =\frac{1}{2}-\varepsilon $$, where ε>0$$ \varepsilon >0 $$ is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In this paper, the coupled linear theory of thermoelasticity for nanomaterials with triple porosity is considered in which the combination of Darcy's law and the volume fraction concept for three levels of pores (macro-, meso- and micropores) is provided. The 3D basic boundary value problems (BVPs) of steady vibrations of this theory are formulated and these BVPs are investigated using the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, the formula of integral representation of regular vectors is obtained. The surface (single-layer and double-layer) and volume potentials are introduced and their basic properties are given. Some useful singular integral operators are defined for which Noether's theorems are valid. The symbolic determinants and indexes of these operators are calculated. The BVPs of steady vibrations are reduced to the equivalent singular integral equations. Finally, with the help of the potential method, we prove the existence theorems for classical solutions of the aforementioned BVPs. [ABSTRACT FROM AUTHOR]

In the present paper we investigate three-dimensional dynamical mixed boundary-transmission problems for composed body consisting of two adjacent anisotropic elastic components having a common interface surface. The two contacting elastic components are subject to different mathematical models: Green–Lindsay’s model of generalized thermo-electro-magneto-elasticity in one component and Green–Lindsay’s model of generalized thermo-elasticity in the other one. We assume that the composed elastic structure under consideration contains an interfacial crack. We prove the uniqueness and existence theorems in appropriate function spaces. [ABSTRACT FROM AUTHOR]

The dichromatic number χ→(D) $\overrightarrow{\chi }(D)$ of a digraph D $D$ is the minimum number of colours needed to colour the vertices of a digraph such that each colour class induces an acyclic subdigraph. A digraph D $D$ is k $k$‐dicritical if χ→(D)=k $\overrightarrow{\chi }(D)=k$ and each proper subdigraph H $H$ of D $D$ satisfies χ→(H)

The Dirichlet, Neumann and mixed type interaction dynamical problems between thermo-elastic and thermo-piezo-elastic bodies are studied. The model under consideration is based on the Green–Naghdi theory of thermo-piezo-electricity without energy dissipation. This theory allows the thermal waves to propagate only with a finite speed. Using the Laplace transform, potential theory and the method of boundary pseudodifferential equations, the existence and uniqueness of solutions is proved and their smoothness is analyzed. [ABSTRACT FROM AUTHOR]

The paper studies three-dimensional interaction crack type boundary-transmission problems of pseudo-oscillations between thermo-elastic and thermo-piezo-elastic bodies taking microrotations into account. The model under consideration is based on the Green–Naghdi theory of thermo-piezo-electricity without energy dissipation. This theory permits the thermal waves to propagate only with a finite speed. The system of partial differential equations of pseudo-oscillations is obtained from the corresponding dynamical model by the Laplace transform. Using the potential theory and the method of boundary pseudodifferential equations, we prove the existence, uniqueness and regularity of solutions. [ABSTRACT FROM AUTHOR]

The aim of this article is to investigate the parameter sensitivity of the (Non-Linear) Unsteady Vortex Lattice Method-Vortex Particle Method [(NL-)UVLM-VPM] with Particle Strength Exchange-Large Eddy Simulations (PSE-LES) method on a lower Reynolds number rotor. The previous work detailed the method, but introduced parameters whose influence were not investigated. Most importantly, the Vreman model coefficient was chosen arbitrarily and was not suitable to ensure stability for this lower Reynolds number rotor simulation. In addition, the previous work presented a consistency study where geometry and time discretization were refined simultaneously. The present article starts with a comparative literature review of potential methods used to solve the aerodynamics of an isolated hovering rotor. This review highlights the differences in modeling, discretizations, sensitivity analysis, validation cases, and the results chosen by the different studies. Then, a transparent and thorough parametric study of the method is presented alongside discussions of the observed results and their physical interpretation regarding the flow. The sensitivity analysis is performed for the three free parameters of UVLM, namely Vatistas core size, the geometry and the temporal discretizations, and then for the three additional parameters introduced by UVLM-VPM, which are the Vreman model coefficient, the particle spacing, and the conversion time. The effect of different databases in the non-linear coupling is also shown. The method is shown to be consistent with both geometry and temporal refinements. It is also consistent with the expected behavior of the different parameters change, including the numerical stability that depends on the strength of the LES diffusion controlled by the Vreman model coefficient. The effect of discretization refinement presented here not only shows the integrated coefficients where different errors can cancel each other, but also looks at their convergence and where relevant, the distributed loads and tip singularity position. Finally, the aerodynamics results of the method are compared for different databases and with higher fidelity Unsteady Reynolds Averaged Navier–Stokes (URANS) 3D results on a lower Reynolds number rotor. [ABSTRACT FROM AUTHOR]

In this paper, the coupled linear quasi-static theory of elasticity for materials with double porosity is considered in which the concepts of Darcy's law and volume fractions are proposed. The system of general governing equations is expressed in terms of the displacement vector field, the changes of the volume fractions of pores and fissures, and the changes of the fluid pressures in pores and fissures networks. By virtue of Green's identity, the uniqueness theorems of the basic internal and external boundary value problems (BVPs) are proved. The fundamental solution of the system of steady vibration equations in the theory under consideration is constructed. Then, the surface and volume potentials are constructed and their basic properties are given. Some useful singular integral operators are studied. Finally, on the basis of these results the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

This paper concerns the coupled linear quasi-static theory of thermoelasticity for materials with double porosity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. By virtue of Green's identity the uniqueness theorems for classical solutions of the internal and external quasi-static boundary value problems (BVPs) are proved. The fundamental solution of the system of steady vibration equations in the considered theory is constructed and its basic properties are established. Then, the surface and volume potentials are presented and their basic properties are given. Finally, on the basis of these results the existence theorems for classical solutions of the above mentioned BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

In this paper, the coupled linear theory of viscoelasticity for materials with a triple porosity is developed and the basic boundary value problems (BVPs) of steady vibrations are investigated. The governing systems of equations of motion and steady vibrations are presented, the fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Green’s identities for bounded and unbounded domains are established and the existence and uniqueness theorems for classical solutions of the foregoing BVPs are proved by using the potential method and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

We prove that every triangle‐free 4‐critical graph G satisfies e(G) ≥ 5v (G)+2/3 . This result gives a unified proof that triangle‐free planar graphs are 3‐colourable, as well as that graphs of girth at least five which embed in either the projective plane, torus or Klein Bottle are 3‐colourable, which are results of Grötzsch, Thomassen, and Thomas and Walls. Our result is nearly the best possible, as Davies has constructed triangle‐free 4‐critical graphs G such that e(G) ≥ 5v (G)+4/3 . To prove this result, we prove a more general result characterizing sparse 4‐critical graphs with few vertex‐disjoint triangles. [ABSTRACT FROM AUTHOR]

We consider a special approach to investigate a mixed boundary value problem (BVP) for the Lamé system of elasticity in the case of three-dimensional bounded domain Ω ⊂ R 3 , when the boundary surface S = ∂ Ω is divided into two disjoint parts, S D and S N , where the Dirichlet and Neumann type boundary conditions are prescribed respectively for the displacement vector and stress vector. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations which do not contain neither extensions of the Dirichlet or Neumann data, nor the Steklov–Poincaré type operator. Moreover, the right hand sides of the resulting pseudodifferential system are vectors coinciding with the Dirichlet and Neumann data of the problem under consideration. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space [ W 2 1 (Ω) ] 3 and representability of solutions in the form of linear combination of the single layer and double layer potentials with densities supported respectively on the Dirichlet and Neumann parts of the boundary. Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that the operator is invertible in the L p -based Besov spaces with 4 3 < p < 4 , which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ‾ with α = 1 2 − ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

The aim of this article is to investigate the parameter sensitivity of the (Non-Linear) Unsteady Vortex Lattice Method-Vortex Particle Method [(NL-)UVLM-VPM] with Particle Strength Exchange-Large Eddy Simulations (PSE-LES) method on a lower Reynolds number rotor. The previous work detailed the method, but introduced parameters whose influence were not investigated. Most importantly, the Vreman model coefficient was chosen arbitrarily and was not suitable to ensure stability for this lower Reynolds number rotor simulation. In addition, the previous work presented a consistency study where geometry and time discretization were refined simultaneously. The present article starts with a comparative literature review of potential methods used to solve the aerodynamics of an isolated hovering rotor. This review highlights the differences in modeling, discretizations, sensitivity analysis, validation cases, and the results chosen by the different studies. Then, a transparent and thorough parametric study of the method is presented alongside discussions of the observed results and their physical interpretation regarding the flow. The sensitivity analysis is performed for the three free parameters of UVLM, namely Vatistas core size, the geometry and the temporal discretizations, and then for the three additional parameters introduced by UVLM-VPM, which are the Vreman model coefficient, the particle spacing, and the conversion time. The effect of different databases in the non-linear coupling is also shown. The method is shown to be consistent with both geometry and temporal refinements. It is also consistent with the expected behavior of the different parameters change, including the numerical stability that depends on the strength of the LES diffusion controlled by the Vreman model coefficient. The effect of discretization refinement presented here not only shows the integrated coefficients where different errors can cancel each other, but also looks at their convergence and where relevant, the distributed loads and tip singularity position. Finally, the aerodynamics results of the method are compared for different databases and with higher fidelity Unsteady Reynolds Averaged Navier–Stokes (URANS) 3D results on a lower Reynolds number rotor.

In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 , when the boundary surface S = ∂ ⁡ Ω is divided into two disjoint parts S D and S N where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ (Ω) . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p -based Besov spaces, which under appropriate boundary data implies C α -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ with α = 1 2 - ε , where ε > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

Recently, Kostochka and Yancey [7] proved that a conjecture of Ore is asymptotically true by showing that every k -critical graph satisfies | E (G) | ≥ ⌈ (k 2 − 1 k − 1) | V (G) | − k (k − 3) 2 (k − 1) ⌉. They also characterized [8] the class of graphs that attain this bound and showed that it is equivalent to the set of k -Ore graphs. We show that for any k ≥ 33 there exists an ε > 0 so that if G is a k -critical graph, then | E (G) | ≥ (k 2 − 1 k − 1 + ε) | V (G) | − k (k − 3) 2 (k − 1) − (k − 1) ε T (G) , where T (G) is a measure of the number of disjoint K k − 1 and K k − 2 subgraphs in G. This also proves for k ≥ 33 the following conjecture of Postle [12] regarding the asymptotic density: For every k ≥ 4 there exists an ε k > 0 such that if G is a k -critical K k − 2 -free graph, then | E (G) | ≥ (k 2 − 1 k − 1 + ε k) | V (G) | − k (k − 3) 2 (k − 1). As a corollary, our result shows that the number of disjoint K k − 2 subgraphs in a k -Ore graph scales linearly with the number of vertices and, further, that the same is true for graphs whose number of edges is close to Kostochka and Yancey's bound. [ABSTRACT FROM AUTHOR]

In this paper, the linear model of thermoelastic double-porosity materials is proposed in which the coupled phenomenon of the concepts of Darcy's law and the volume fractions is considered. The basic systems of governing equations of motion and steady vibrations are obtained. Then, the boundary value problems of steady vibrations are investigated by means of the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, on the basis of Green's identities and the surface and volume potentials the existence and uniqueness theorems for the classical solutions of the internal and external basic boundary value problems of the considered theory are proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special approach to investigate a three-dimensional mixed boundary value problem (BVP) for an anisotropic Helmholtz type equation which is a second order elliptic partial differential equation containing a complex parameter ν. The boundary surface S = @ of a domain under consideration, = R3, is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of linear combination of the corresponding single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP to a system of pseudodifferential equations. It is shown that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W1 2 (). Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the Lp-based Besov spaces, which under appropriate boundary data implies C-Höolder continuity of the solution to the mixed BVP in the closed domain with α = 1/2 -, where" > 0 is an arbitrarily small number. [ABSTRACT FROM AUTHOR]

In the present paper, the coupled linear model of elastic triple-porosity materials is considered in which the coupled phenomenon of the concepts of Darcy's law and the volume fractions of three levels of pores (macro-, meso- and micropores) is proposed. The basic boundary value problems of steady vibrations are investigated by means of the potential method (boundary integral equation method) and the theory of singular integral equations. In particular, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions and Green's identities are obtained. The basic properties of the surface and volume potentials are established. On the basis of Green's identities and the properties of these potentials, the existence and uniqueness theorems for the classical solutions of the basic boundary value problems of the theory under consideration are proved. [ABSTRACT FROM AUTHOR]

In the paper, we study Dirichlet, Neumann and mixed type interaction problems of pseudo-oscillations between thermo-elastic and thermo-piezo-elastic bodies. The model under consideration is based on the Green-Haghdi theory of thermo-piezo-electricity without energy dissipation. This theory allows the thermal waves to propagate only with a finite speed. Using the potential theory and the method of boundary pseudodifferential equations, we prove the existence and uniqueness of solutions and analyze their smoothness. [ABSTRACT FROM AUTHOR]

The change of meshing stiffness of gears is caused by the change of meshing position and meshing number between gears when the gears are running.The change of meshing stiffness reflects the change of meshing process of gears.When pitting occurs,the meshing stiffness of gears will change,which will lead to the change of vibration law of gearbox.This change can be used to judge whether the internal fault of gearbox is serious or not.According to the actual situation of pitting,pitting pits are set as elliptical sections.The meshing stiffness under pitting failure is calculated by potential energy method,and the influence of pitting location,size and depth on meshing stiffness is analyzed.The results show that pitting has no effect on the meshing stiffness of the non-pitting area under the gear teeth during the meshing process.The closer the pitting position is to the top of the gear teeth,the greater the impact on the meshing stiffness above the pitting gear teeth.The deeper the pitting pit is,the larger the area is,the more the meshing stiffness decreases,and the more obvious the influence of the area change on the meshing stiffness changes.

A theoretical formulation is presented for the displacements, temperature and stresses in a 3 D transversely isotropic thermoelastic solid under arbitrary mechanical loads and heat flux. By using two scalar potential functions, 3 D patch-load, patch-heat source, point-load and point-heat source Green's functions are given for the displacements, temperature and stresses. Basic governing equations of coupled thermoelasticity in transversely isotropic solids are presented in the cylindrical coordinates system. By means of two scalar potential functions, the four coupled governing equations for this class of problems are uncoupled into a 6th order and a 2nd order partial differential equations. Applying the Fourier expansions and Hankel transforms, the angular and radial variables are suppressed and 6th and 2nd order ordinary differential equations are obtained. By solving the ordinary differential equations, the potential functions, displacements, temperature and stresses are obtained in the Fourier-Hankel transformed space. By applying the Hankel inversion theorem, all responses are presented in the physical time-spatial space. To prove the validity of the analytical and numerical solutions, our results are compared with existing solutions reported in the literature. Numerical results are also included to show the effects of material anisotropy, elastic and thermal properties; and frequency of excitation on the responses. [ABSTRACT FROM AUTHOR]

We investigate mixed type boundary–transmission problems of the generalized thermo‐electro‐magneto‐elasticity theory for complex elastic anisotropic layered structures containing interfacial cracks. This type of problems is described mathematically by systems of partial differential equations with appropriate transmission and boundary conditions for six dimensional unknown physical field (three components of the displacement vector, electric potential function, magnetic potential function, and temperature distribution function). We apply the potential method and the theory of pseudodifferential equations and prove uniqueness and existence theorems of solutions to different type mixed boundary–transmission problems in appropriate Sobolev spaces. We analyze smoothness properties of solutions near the edges of interfacial cracks and near the curves where different type boundary conditions collide. [ABSTRACT FROM AUTHOR]

We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of mixed boundary-transmission problems for composed elastic structures consisting of two contacting anisotropic bodies occupying two threedimensional adjacent regions with a common contacting interface, being a proper part of their boundaries. It is assumed that the contacting elastic bodies are subject to different mathematical models. In particular, we consider Green-Lindsay's model of generalized thermo-electro-magnetoelasticity in one elastic component, while in the other one, we considered Gree Lindsay's model of generalized thermo-elasticity. The interaction of the thermo-mechanical and electro-magneticffelds in the composed piecewise elastic structure is described by the fully coupled systems of partial differential equations of pseudo-oscillations, obtained from the corresponding dynamical models by the Laplace transform. These systems are equipped with the appropriate mixed boundary-transmission conditions which cover the conditions arising in the case of interfacial cracks. Using the potential method and the theory of pseudodifferential equations on manifolds with a boundary, the uniqueness and existence theorems in suitable function spaces are proved, the regularity of solutions is analyzed and singularities of the corresponding thermo-mechanical and electro-magneticffelds near the interfacial crack edges are characterized. The explicit expressions for the stress singularity exponents are derived and it is shown that they depend essentially on the material parameters. A special class of composed elastic structures is considered, where the so-called oscillating stress singularities do not occur. [ABSTRACT FROM AUTHOR]

For each integer k \geq 2, we determine a sharp bound on mad(G) such that V (G) can be partitioned into sets I and Fk, where I is an independent set and G[Fk] is a forest in which each component has at most k vertices. For each k we construct an infinite family of examples showing our result is the best possible. Our results imply that every planar graph G of girth at least 9 (resp., 8, 7) has a partition of V (G) into an independent set I and a set F such that G[F] is a forest with each component of order at most 3 (resp., 4, 6). Hendrey, Norin, and Wood asked for the largest function g(a, b) such that if mad(G) < g(a, b), then V (G) has a partition into sets A and B such that mad(G[A]) < a and mad(G[B]) < b. They specifically asked for the value of g(1, b), i.e., the case when A is an independent set. Previously, the only values known were g(1, 4/3) and g(1, 2). We find g(1, b) whenever 4/3 < b < 2. [ABSTRACT FROM AUTHOR]

In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green's identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 {\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍ p s ⁢ (𝒞), 1/p < s < 1 + 1/p, 1 < p < ∞. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain Ωα ⊂ ℝ2 of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2 × 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ+ in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731]. [ABSTRACT FROM AUTHOR]

Statistical arbitrage methods identify mispricings in securities with the goal of building portfolios which are weakly correlated with the market. In pairs trading, an arbitrage opportunity is identified by observing relative price movements between a pair of two securities. By simultaneously observing multiple pairs, one can exploit different arbitrage opportunities and increase the performance of such methods. However, the use of a large number of pairs is difficult due to the increased probability of contradictory trade signals among different pairs. In this paper, we propose a novel portfolio construction method based on preference relation graphs, which can reconcile contradictory pairs trading signals across multiple security pairs. The proposed approach enables joint exploitation of arbitrage opportunities among a large number of securities. Experimental results using three decades of historical returns of roughly 500 stocks from the S&P 500 index show that the portfolios based on preference relations exhibit robust returns even with high transaction costs, and that their performance improves with the number of securities considered. • Contradictory signals may occur in pairs trading with a large number of securities. • Preference relation graphs are proposed to reconcile such signals. • A total ordering among securities is used to build robust portfolios. • The proposed approach benefits from larger sets of securities. [ABSTRACT FROM AUTHOR]

The solution of transportation problem for unmanned multiobject systems is described. The factor for calculating of the cost matrix taking into account the specifics of multi-object systems of unmanned vehicles is chosen.

This paper concerns with the coupled linear theory of viscoelasticity for porous materials. In this theory the coupled phenomenon of the concepts of Darcy's law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions and its basic properties are presented. Green's identities are obtained and the uniqueness theorems for the regular (classical) solutions of the BVPs of steady vibrations are proved. The surface and volume potentials are constructed and the basic properties of these potentials are given. Finally, the existence theorems for classical solutions of the BVPs of steady vibrations are proved by means of the potential method and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

This article is concerned with the coupled linear quasi-static theory of thermoporoelasticity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourie's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of general governing equations is expressed in terms of the displacement vector field, the changes of the volume fraction of pores, the fluid pressure in pore network and temperature. The fundamental solution of the system of quasi-static equations in the considered theory is constructed and its basic properties are established. On the basis of Green's formulas the uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

A multigraph G is near-bipartite if V (G) can be partitioned as I, F such that I is an independent set and F induces a forest. We prove that a multigraph G is near-bipartite when 3|W| - 2|E(G[W])| ≥ 1 for every W ⊆ V (G), and G contains no K4 and no Moser spindle. We prove that a simple graph G is near-bipartite when 8|W| -5| E(G[W])| ≥ - 4 for every W ⊆ V (G), and G contains no subgraph from some finite family H. We also construct infinite families to show that both results are the best possible in a very sharp sense. [ABSTRACT FROM AUTHOR]

The purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain Ωα ⊂ ℝ2 of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces ℍsp(Ωα), s > 1/p , 1 < p < ∞. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev-Slobodečkii space on a semi-infinite axis핎s−1/p p (ℝ+), which is ofMellin convolution type. Applying the recent results onMellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 (2016), no. 2, 707-731], explicit conditions of the unique solvability of this BIE in the Sobolev-Slobodečkii 핎rp(ℝ+) and Bessel potential ℍrp(ℝ+) spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 (2013), no. 3, 439-467], but there were made fatal errors. In the present paper, we correct these results. [ABSTRACT FROM AUTHOR]

This paper concerns the coupled linear theory of elasticity for isotropic porous materials. In this theory the coupled phenomena of the concepts of Darcy's law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established and Green's identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single layer and double layer) and volume potentials are constructed and the basic properties of these potentials are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]

In this paper, the basic boundary value problems of stationary oscillations for thermoelastic isotropic microstretch materials with microtemperatures and microrotation, are studied. To this end, there are constructed the matrix of fundamental solutions of the corresponding system of partial differential equations, and the mapping properties of the corresponding volume and layer potential operators are investigated. With the help of the potential method, the basic boundary value problems are reduced to the corresponding system of singular integral equations, and the existence theorems of solutions are proved. [ABSTRACT FROM AUTHOR]

The paper deals with the basic boundary value problems of statics for thermoelastic isotropic microstretch materials with microtemperatures and microrotation. For the homogeneous system of partial differential equations of statics, the fundamental matrix is constructed explicitly in terms of elementary functions. By means of the fundamental matrix is constructed the corresponding volume and layer potentials, and their mapping properties are investigated. The basic Dirichlet and Neumann type boundary value problems are reduced to the corresponding system of singular integral equations, and the existence theorems of solutions are proved. [ABSTRACT FROM AUTHOR]

The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.

Sparse superposition codes, or sparse regression codes, constitute a new class of codes, which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity, as one of the code parameters goes to infinity. Furthermore, we provide a closed-form formula for the algorithmic threshold of the underlying code ensemble in terms of Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing. [ABSTRACT FROM AUTHOR]

We investigate the mixed Dirichlet-Neumann boundary value problems for the Laplace-Beltrami equation on a smooth bounded surface with a smooth boundary in non-classical setting in the Bessel potential space for , . To the initial BVP we apply a quasi-localization and obtain a model BVP for the Laplacian. The model mixed BVP on the half plane is investigated by the potential method and is reduced to an equivalent system of Mellin convolution equations in Sobolev-Slobodečkii space. Boundary integral equations are ivestigated in both Bessel potential and Sobolev-Slobodečkii spaces. The symbol of the obtained system is written explicitly and is responsible for the Fredholm properties and the index of the system. An explicit criterion for the unique solvability of the initial BVP in the non-classical setting is derived as well. [ABSTRACT FROM AUTHOR]

In this paper the linear theory of triple porosity thermoelasticity is considered and the basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, Green's identities are obtained and the Sommerfeld–Kupradze type radiation conditions are established. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are given. The BVPs are reduced to the always solvable singular integral equations for which Fredholm's theorems are valid. Finally, the existence and uniqueness theorems for classical solutions of the basic BVPs of steady vibrations are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations. [ABSTRACT FROM AUTHOR]