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We construct a continuous nowhere monotone and nondifferentiable function depending on a single parameter q0 ϵ (0; 1). For functions from this continual class, we describe their structural, variational, fractal, and integrodifferential properties.

We consider a class of two-dimensional integral equations in ℝ2 with monotone nonlinear terms and prove the existence of a nonnegative bounded solution. We study the asymptotic behavior of this solution with respect to each variable. The result is illustrated by an example.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

International audience; Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

We establish conditions for the existence of a class of monotone solutions of the second-order differential equations with regularly and rapidly varying nonlinearities and the asymptotic representations of these solutions as t ↑ ω (ω ≤ + ∞).

We study various classes of nonlinear equations containing operators of potential type (Riesz potential). By the method of monotone operators in the Lebesgue spaces of real-valued functions Lp(a, b) we prove global theorems on the existence, uniqueness, estimates, and methods of construction of their solutions. We present applications that illustrate the results obtained.

Some new characterizations of the exponential distribution in the class of distributions with monotone hazard rate are obtained. The characterizations are formulated in terms of expectations of order statistics. This is a post-peer-review, pre-copyedit version of an article published in Journal of Mathematical Sciences. Locked until 18 September 2019 due to copyright restrictions. The final authenticated version is available online at: https://doi.org/10.1007/s10958-018-4041-4.

We characterize linear transformations on matrices over commutative antinegative semirings that are monotone with respect to minus, star, and sharp partial orders.

The Polya–Szego inequality is considered for a monotone rearrangement with integrand depending on the rearrangement variable. The inequality is proved for integrands having polynomial growth.

This survey is concerned with structural characteristics of “suns” in normed linear spaces. Special attention is paid to connectedness and monotone path-connectedness of suns. We address both direct theorems of the geometric approximation theory, in which approximative properties of sets are derived from their structural characteristics, and inverse theorems, in which from approximative characteristics of sets one derives their structural properties.

We study congruences on the semigroup (ℤ lex ) of monotone injective partial self-maps of the set of L n × lexℤ with cofinite domains and images, where L n × lexℤ is the lexicographic product of an n -element chain and a set of integers with ordinary linear order. The structure of the sublattice of congruences on (ℤ lex ) contained in the least group congruence is described.

Necessary and sufficient conditions for L p -integrability with power weight of a function f rep- resented by the double series with respect to a multiplicative system with generalized monotone coefficients are obtained. These conditions are given in terms of the coefficients or their second mixed differences. In addition, the integrability of the difference quotient f (x,y) � f (x, 0) � f (0,y )+ f (0, 0) /(xy) is studied.

For every real number γ ≥ 1 we denote by K↓(γ) the least possible value of the constant term of an even nonnegative trigonometric polynomial with monotone coefficients such that all its coefficients, save for the constant term, are not lesser than 1 and the sum of these coefficients equals γ. In this paper, the asymptotic estimate of K↓(γ) is found and some extremal problems on the minimum of the constant term of an even nonnegative trigonometric polynomial are studied.

By the method of potential monotone operators we prove global theorems on existence, uniqueness, and ways to find a solution for different classes of nonlinear discrete equations of convolution type with kernels of special form both in weighted and in weightless real spaces l p . Using the property of potentiality of the operators under consideration, in the case of space l 2 and in the case of a weighted space l p (ϱ) with a generic weight ϱ, we prove that a discrete equation of convolution type with an odd power nonlinearity has a unique solution and it (the main result) can be found by gradient method.

The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.

The paper is concerned with solarity of intersections of suns with bars (in particular, with closed balls and extreme hyperstrips) in normed linear spaces. A sun in a finite-dimensional (BM)-space (in particular, in l 1(n)) is shown to be monotone path connected. A nonempty intersection of an m-connected set (in particular, a sun in a two-dimensional space or in a finite-dimensional (BM)-space) with a bar is shown to be a monotone path-connected sun. Similar results are obtained for boundedly compact subsets of infinite-dimensional spaces. A nonempty intersection of a monotone path-connected subset of a normed space with a bar is shown to be a monotone path-connected α-sun.

We study the discrete spectrum of the Dirichlet problem for the Laplace operator on the cross of two strips with widths 1 and H which are perpendicular to each other. We verify that for any parameter H > 0 the discrete spectrum consists of the only point μ1 H while the function H ↦ μ1 H is strictly monotone decreasing. We consider other cross-shaped junctions of quantum waveguides and, in particular, construct asymptotics of eigenvalues as H → +0.

We characterize additive maps on the matrix algebra over an arbitrary field with three or more elements that are monotone with respect to the \( \mathop{\leq}\limits^{\#} \)- and \( \mathop{\leq}\limits^{\mathrm{cn}} \)-orders and build some examples of nonadditive monotone maps.

In this article, the first steps towards mathematical modelling of time-related musical structures are taken, and the algebraic structure of musical time relations is elaborated starting from a perceptive point of view. A basic characterization of fundamental properties of perceived time relations and their interpretations regarding musical context are given, and some mathematical properties of the proposed definitions are examined. Stemming from musical motivation, a category is found whose objects are finite strict (partially) ordered sets and whose morphisms are weakly monotone and reflect the strict order of the codomain. The category is found to have initial and terminal objects, equalizers, and coequalizers but fails to have binary products or coproducts.

We consider the problem of local and global solvability of a nonlinear problem with a free (unknown) line of discontinuity of initial data for a hyperbolic system of first-order quasilinear equations with two independent variables. For the problem to be globally solvable, one should additionally require that the coefficients and free terms be monotone and of constant sign.

The paper investigates bijective maps on the set of matrices of index one that are monotone with respect to the order induced by the group inverse matrix. Bibliograhy: 33 titles.

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.

An analog of the classical Couette problem for the motion of an incompressible viscoelastic Kelvin–Voigt medium in a gap between two coaxial cylinders is considered. The external cylinder is fixed, whereas the internal cylinder is rotating either forcedly with a constant velocity or inertially. The unique solvability is proved in both cases. An analogy between the propagation of acoustic cylindrical waves in a viscous gas and transversal waves in a Kelvin–Voigt medium moving along circular trajectories is discussed. The asymptotic behavior of solutions is also studied. It is shown that the limit regime can be monotone or oscillatory in dependence on the parameters of the problem. Bibliography: 11 titles. Illustrations: 2 figures.

A subset M of a normed linear space X is said to be R-weakly convex (R > 0 is fixed) if the intersection (D R (x, y) \ {x, y}) ∩ M is nonempty for all x, y ∈ M, 0 0 if and only if M is a disjoint union of monotonically path-connected suns in C(Q), the Hausdorff distance between each pair of the components of M being at least 2R.

We deal with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model. Bibliography: 24 titles.

We have proposed a modification of the methods for solving the system of integral equations [M. Ya. Leonov and N. Yu. Shvaiko, “Complex plane deformation,” Dokl. Akad. Nauk SSSR, 159, No. 2, 1007–1010 (1964); N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005)]. These equations describe the development of plane plastic deformation for simple and complex loading processes. A characteristic feature of these equations lies in the presence of unknown functions both under the integral sign and in the integration limits. We have written analytical solutions for monotone deformation and in a small neighborhood of an angular point of the loading trajectory. For arbitrary piecewise smooth trajectories, we have reduced this problem to the Cauchy problem for a first-order differential equation with known initial conditions. The results obtained simplify significantly the construction of constitutive equations $ {\dot{\sigma }_{mn}} \sim {\dot{\varepsilon }_{mn}} $ and their use in applied problems of the theory of plasticity as compared with [N. Yu. Shvaiko, “On the theory of slip with smooth and singular loading surfaces,” Mat. Metody Fiz.-Mekh. Polya, 48, No. 3, 129–137 (2005); N. Yu. Shvaiko, Complex Loading and Problems of Stability [in Russian], Izd. DGU, Dnepropetrovsk (1989)].

A curve formed by inversion of a logarithmic spiral is called a double logarithmic spiral. The curves in this family possess the following property: there always exists such a spiral with continuous and monotone curvature satisfying any possible boundary conditions (endpoints, tangents, and curvatures). The problem of constructing a spiral with continuous curvature and prescribed curvature elements at the endpoints is thus solved. Bibliography: 6 titles.

In this paper, the structure of the set of threshold functions and complexity problems are considered. The notion of the signature of a threshold function is defined. It is shown that if a threshold function essentially depends on all of its variables, then the signature of this function is unique. The set of threshold functions is partitioned into classes with equal signatures. A theorem characterizing this partition is proved. The importance of the class of monotone threshold functions is emphasized. The complexity of transferring one threshold function specified by a linear form into another is examined. It is shown that in the worst case this transfer would take exponential time. The structure of the set of linear forms specifying the same threshold function is also examined. It is proved that for any threshold function this set of linear forms has a unique basis in terms of the operation of addition of linear forms. It is also shown that this basis is countable.

This paper contains a review of the authors’ results in the theory of algorithm complexity. The results described concern methods for obtaining lower bounds (containing almost all exponential lower bounds on monotone complexity of monotone functions), synthesis of asymptotically optimal functional networks, minimization of Boolean functions, and the problem of solving Boolean equations.

We propose a test for the proportional hazards (Cox) model which is oriented against wide classes of alternatives including monotone hazard ratios and crossings of survival functions and can be used when data are left truncated and right censored. The limit distribution of the test statistics is derived. Bibliography: 20 titles.

The paper treats plane curves with monotone curvature {spirals). Some familiar facts of the geometry of spirals are generalized by way of removing requirements that the curvature be continuous and have constant sign. Necessary and sufficient conditions for the existence of a spiral with given boundary conditions are considered. The existence of bipolar equation of a spiral is also discussed. Bibliography: 8 titles.

A one-dimensional monotone follower control problem with a nonconvex Lagrangian is considered. The control problem consists in tracking a standard Wiener process by an adapted nondecreasing process starting at 0. The verification theorem for the problem is presented. The optimal control and the value function are explicitly defined. For some values of parameters of the problem, it is shown that the value function belongs to C2. An interesting feature of the optimally controlled state process is that for some initial states it has jumps at times other than the inital time.

We characterize linear transformations on the matrix algebra over an arbitrary field with characteristic not equal to 2 that are monotone with respect to the {ie832-03}-or {ie832-04}-order.

We study a multi-valued function {ie078-01} associating with the temperature t the volume part of one of the phases of the equilibrium distribution in the phase transition problem of continuum mechanics. We establish the closedness of the graph of this function, the monotone decrease, and the localization of variability zones of its upper and lower envelopes. Bibliography: 6 titles.

The main object of study is the space of all monotone continuous functions CM(X) on a connected Tychonoff space X endowed with the topology of pointwise (CM p (X)) or uniform (CM(X)) convergence. Technical questions concerning restriction and extension of monotone functions are considered in Sec. 2. Conditions for CM(X) to separate the points of X and for CM(X) to contain only constant functions are found in Sec. 3. In Sec. 4, the linear structure of CM(X) is studied and all linear subspaces of CM(X) for a certain class of spaces X are described. In Sec. 5, conditions under which CM(X) is closed and nowhere dense in C p (X) and C(X) are determined. The metrizability of CM p (X) is considered in Sec. 6; necessary and sufficient metrizability conditions for various classes of spaces X are obtained. In Sec. 7, criteria for σ-compactness and the Hurewicz property in the class of spaces CM p (X) are given.

An optimization model of a one-dimensional transportation network is considered under the condition that the distribution of sources and sinks (for example, the population and workplaces in urban planning problems) is expressed in terms of finite Borel measures ϕ+ and ϕ− of the same total masses, whereas the unknowns are one-dimensional Federer-Fleming currents T and S modeling transportation of people with or without the use of a public transportation network respectively. The choice of the network must provide the minimum for the cost of mass transfer from sources to sinks (for example, the transportation of people to workplaces) together with the cost of construction and exploitation of the network. Thus, the problem is to minimize the functional $$(T,S) \mapsto A\mathbb{M}^\alpha (T) + B\mathbb{M}^\alpha (S) + H(\mathbb{M}^0 (S))$$ among pairs of flat chains of finite mass such that $$\partial (T + S) = \varphi ^ + - \varphi ^ - ,$$ where A is the cost of transportation with the use of the network, B is the cost of transportation without the use of the network, H is a given monotone nondecreasing function characterizing the cost of construction and exploitation of the network, α ∈ (0, 1) is a given parameter, and \(\mathbb{M}^\delta \) is the δ-mass of the current. To prove the existence theorem, we develop a mathematical tool based on representations of one-dimensional currents in terms of measures on a space of Lipschitz paths. Bibliography: 18 titles. Illustration: 1 figure.

In this part of the paper, we investigate the structure of an arbitrary measure μ supported by a polyhedral cone C in R d in the case where the decumulative distribution function gμ of the measure μ satisfies certain continuity conditions. If a face Γ of the cone C satisfies appropriate conditions, the restriction μ|Γint of the measure μ to the interior part of Γ is proved to be absolutely continuous with respect to the Lebesgue measure λΓ on the face Γ. Besides, the density of the measure μ|Γint is expressed as the derivative of the function gμ multipied by a constant. This result was used in the first part of the paper to find the finite-dimensional distributions of a monotone random field on a poset. Bibliography: 6 titles.

For an arbitrary poset H and measure ρ on H × R (where R is the real axis), we construct a monotone decreasing stochastic field ηρ and compute its finite-dimensional distributions. In the case where H is a Λ-semilattice and the measure ρ satisfies additional conditions, we compute various characteristics of the field ηρ such as the expectation of the field value at a point, variance of the field value at a point, and correlation function of the field. The described construction of random fields gives a new method for constructing positive definite functions on posets. Bibliography: 6 titles.

Homogenization theorems are obtained for nonlinear second-order elliptic operators occurring in some models of media with double porosity. Basic homogenization properties are proved by the method of two-scale convergence combined with the technique of p-connectedness.

In this paper, limit cycles of polynomial dynamical systems are studied. For the global analysis of bifurcations of limit cycles, we use the Wintner–Perko termination principle. Monotone families of limit cycles and rotated vector fields and limit-cycle problems for quadratic systems are also discussed.

We investigate the almost sure asymptotic behavior of increments of sums of i.i.d. random variables over increasing runs in an associate sequence. The Shepp law, the Erdős–Renyi law, and the Csorgő–Reveesz law are obtained for increments of sums over increasing runs formed by random variables taking their values in a fixed interval. Bibliography: 17 titles.

We prove the existence and the uniqueness of a weak solution to the mixed boundary problem for the elliptic-parabolic equation $$\begin{gathered} \partial _t b(u) - {\text{div\{ }}\left| {\delta {\text{(}}u{\text{)}}} \right|^{m - 2} \delta {\text{(}}u{\text{)\} = }}f{\text{(}}x,t{\text{),}} \hfill \\ \delta {\text{(}}u{\text{): = }}\nabla u + k(b(u))\vec e,{\text{ }}\left| {\vec e} \right| = 1,m >1, \hfill \\ \end{gathered} $$ with a monotone nondecreasing continuous function b. Such equations arise in the theory of non-Newtonian filtration as well as in the mathematical glaciology. Bibliography: 16 titles.

In the article, we study the problem of existence of an optimal control for monotone nonlinear distributed systems with a generalized effect. The proposed methods allow one to investigate monotone nonlinear systems with point, impulsive, point-impulsive, and moving controls as well as their generalizations.

We establish conditions under which the relation M(x, F) ∼ Μ(x, F) ∼ m(x, F) holds except for a small set, as ¦x¦→ +∞ for an entire function F(z) of several complex variables z ∃ ℂ (p≥2) represented by a Dirichlet series, where M(x, F) = sup{¦F(x+iy¦: y ∃ ℝp}, m(x, F) = inf{¦F(x+iy)¦: y ∃ ℝp} Μ(x, F) being the maximal term of the Dirichlet series, and x ∃ ℝp.

Suppose that the Lindelof conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where e(t) is a monotone decreasing function, $$\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$$ . Then it is proved that for |t|≥T0 the disk $$\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $$ contains at most 20v log |t| zeros of ζ(s) if $$\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $$ . There exists an absolute constant A such that for |t|≥T1 the disk $$\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $$ contains at least one zero of ζ(s). Bibliography: 2 titles.

In this paper we give a characterization of the classMSS of all one-dimensional d.f.'s G satisfying the functional equation Gα(x)=G(L(x)), where α∈(0, 1] andL(x) is strictly monotone.

A wide class of reliability theory models or lifetime data can be described as follows. Assume that the lifetime distribution function is F(t, θ)=F0(λ(θ)t), where θ is the parameter characterizing some inner properties of a product and λ(θ) is an unknown increasing function. The paper deals with methods of estimation of λ(θ) from the sample (ti,θi),i = 1, ...,n, for the case of exponentialF0.

We study the problem of the existence of the optimal control of monotone nonlinear distributed systems with generalized interaction. The method proosed allows one to study monotone nonlinear systems with point, impulse, point impulse, and nonstationary controls as well as with their generalizations. Bibliography: 2 titles.