TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

TO THE QUESTION OF ANALYSIS OF EQUATIONS OF MOTION OF A RIGID BODY DURING THE MECHANICAL OSCILATIONS

Depending on the current position of the mass in different areas of the spring deformation during the oscillation process the values that determines the natural frequency of free continuous oscillations have opposite signs. It is defined by the change in the direction of acceleration of the mass in these areas, which makes it possible to determine a single inhomogeneous differential equation of the oscillation process in different areas of the movement of the mass. When the oscillation amplitude is much less than the static position of the mass, this inhomogeneous differential equation represents a homogeneous differential equation of free undamped oscillations.

ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

About unbounded complex operators

The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.

Differential heterogenesis and the emergence of semiotic function

In this study, we analyse the notion of “differential heterogenesis” proposed by Deleuze and Guattari on a morphogenetic perspective. We propose a mathematical framework to envisage the emergence of singular forms from the assemblages of heterogeneous operators. In opposition to the kind of differential calculus that is usually adopted in mathematical-physical modelling, which tends to assume a homogeneous differential equation applied to an entire homogeneous region, heterogenesis allows differential constraints of qualitatively different kinds in different points of space and time. These constraints can then change in time, opening the possibility for new kinds of differential dynamics and the emergence of distinct entities and forms. Formally, we show that operators with different phase spaces can be assembled on the basis of a result of Rothschild & Stein (1976. Hypoelliptic differential operators and nilpotent groups. Acta Mathematica 137. 247–320). Furthermore, operators with different dynamics can be assembled by means of a partition of the unit. After stating the concept of differential heterogenesis in terms of contemporary mathematics, we show that this construction sheds light on the constitution of the semiotic function. In fact, both the Merleau-Pontian and the Deleuzian approaches share a common conceptualisation of the semiotic function and its emergence in terms of a morphodynamics of heterogeneous assemblages with a divergent actualisation. This divergent actualisation allows the co-constitution of various expression and content planes. Finally, we show that the divergent actualisation can be interpreted as the directions of principal eigenvectors of the actualized flow.

About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

On coefficient bounds of some new subfamilies of close-to-convex functions of complex order related to generalized differential operator

We aim to estimate coefficient inequalities for some new subfamilies of close-to-convex functions, which are here, defined by generalized differential operator and Cauchy–Euler type non-homogeneous differential equation. The results presented here would extend, unify and improve some recent results in literature.