Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Ulam Stability of n-th Order Delay Integro-Differential Equations

In this paper, the Ulam stability of an n-th order delay integro-differential equation is given. Firstly, the existence and uniqueness theorem of a solution for the delay integro-differential equation is obtained using a Lipschitz condition and the Banach contraction principle. Then, the expression of the solution for delay integro-differential equation is derived by mathematical induction. On this basis, we obtain the Ulam stability of the delay integro-differential equation via Gronwall–Bellman inequality. Finally, two examples of delay integro-differential equations are given to explain our main results.

Gravitational Radius in view of Existence and Uniqueness Theorem

Talking about a black hole, one has in mind the process of unlimited self-compression of gravitating matter with a mass greater than critical. With a mass greater than the critical one, the elasticity of neutron matter cannot withstand gravitational compression. However, compression cannot be unlimited, because with increasing pressure, neutrons turn into some other “more elementary” particles. These can be bosons of the Standard Model of elementary particles. The wave function of the condensate of neutral bosons at zero temperature is a scalar field. If instead of the constraint det gik < 0 we use a weaker condition of regularity (all invariants of the metric tensor gik are finite), then there is a regular static spherically symmetric solution to Klein-Gordon and Einstein equations, claiming to describe the state to which the gravitational collapse leads. With no restriction on total mass. In this solution, the metric component grr changes its sign twice: g rr (r) = 0 at r=rg and r=rh > rg . Between these two gravitational radii the signature of the metric tensor gik is (+, +, -, -). Gravitational radius rg inside the gravitating body ensures regularity in the center. Within the framework of the phenomenological model “λψ4 ”, relying on the existence and uniqueness theorem, the main properties of a collapsed black hole are determined. At r = rg a regular solution to Klein-Gordon and Einstein equations exists, but it is not a unique one. Gravitational radius rg is the branch point at which, among all possible continuous solutions, we have to choose a proper one, corresponding to the problem under consideration. We are interested in solutions that correspond to a finite mass of a black hole. It turns out that the density value of bosons is constant at r < rg. It depends only on the elasticity of a condensate, and does not depend on the total mass. The energy-momentum tensor at r ⩽ rg corresponds to the ultra relativistic equation of state p = ɛ/3. In addition to the discrete spectrum of static solutions with a mass less than the critical one (where grr < 0 does not change sign), there is a continuous spectrum of equilibrium states with grr(r) changing sign twice, and with no restriction on mass. Among the states of continuous spectrum, the maximum possible density of bosons depends on the mass of the condensate and on the rest mass of bosons. The rest energy of massive Standard Model bosons is about 100 GeV. In this case, for the black hole in the center of our Milky Way galaxy, the maximum possible density of particles should not exceed 3 × 1081 cm-3.

ESTIMATION AND UNIQUENESS OF THE GOMPERTZ FORCE OF MORTALITY IN TERMS OF THE MODAL AGE AT DEATH

The initial level of mortality and the rate at which mortality rises with age are generally expressed in terms of the Gompertz force of mortality (hazard function). In their paper, James W. Vaupel and others define the Gompertz force of mortality as the rate at which mortality rises with age and the modal age at death. In this paper we estimate the Gompertz force of mortality and prove uniqueness theorem.

The Uniqueness of Solution to the Inverse Eigenvalue Problem for an Arrow-shaped Generalised Jacobi Matrix

This paper studies the inverse eigenvalue problem for an arrow-shaped generalised Jacobi matrix, inverting matrices through two eigen-pairs. In the paper, the existence and uniqueness of the solution to the problem are discussed, and mathematical expressions as well as a numerical example are given. Finally, the uniqueness theorem of its matrix is established by mathematical derivation.

Uniqueness theorem in coupled strain gradient elasticity with mixed boundary conditions

The equilibrium equations and the traction boundary conditions are evaluated on the basis of the condition of the stationarity of the Lagrangian for coupled strain gradient elasticity. The quadratic form of strain energy can be written as a function of the strain and the second gradient of displacement and contains a fourth-, a fifth- and a sixth-order stiffness tensor $${\mathbb {C}}_4$$ C 4 , $${\mathbb {C}}_5$$ C 5 and $${\mathbb {C}}_6$$ C 6 , respectively. Assuming invariance under rigid body motions the balance of linear and angular momentum is obtained. The uniqueness theorem (Kirchhoff) for the mixed boundary value problem is proved for the case of the coupled linear strain gradient elasticity (novel). To this end, the total potential energy is altered to be presented as an uncoupled quadratic form of the strain and the modified second gradient of displacement vector. Such a transformation leads to a decoupling of the equation of the potential energy density. The uniqueness of the solution is proved in the standard manner by considering the difference between two solutions.

On properties of h-differentiable functions

Research in the theory of functions of an h-complex variable is of interest in connection with existing applications in non-Euclidean geometry, theoretical mechanics, etc. This article is devoted to the study of the properties of h-differentiable functions. Criteria for h-differentiability and h-holomorphy are found, formulated and proved a theorem on finite increments for an h-holomorphic function. Sufficient conditions for h-analyticity are given, formulated and proved a uniqueness theorem for h-analytic functions.

On some properties of p-holomorphic and p-analytic function

In this article the relationship between the conditions of p-differentiability, p-holomorphycity, and the existence of the derivative of a function of a p-complex variable is considered. The general form of a p-holomorphic function is found. The sufficient conditions for p-analyticity and local invertibility are obtained. The open mapping theorem and the principle of maximum of the norm for a p-holomorphic function and the uniqueness theorem are proved.