The problem of measuring the quantity of output in the input–output model by W. Leontief in modeling the trends in economic reproduction of nations and regions

Subject. This article deals with the problem of constructing a Leontief's input–output matrix. Objectives. The article aims to determine the rules for constructing a Leontief's input–output matrix on the basis of data on production time and quantity of product output. Methods. For the study, I used the methods of logical and mathematical analyses. Results. The article formulates the rules for constructing a Leontief's input–output matrix, taking into account differences in the time of production, quantity of output, as well as the conditions for the reproduction of the resources expended. It summarizes these rules for the J. von Neumann model. Conclusions. The proposed approach to the analysis of the material mechanism of economic reproduction defines the relationship between the quantitative and cost characteristics of the production and consumption of products and resources. This relationship opens up new opportunities for the application of input–output models to create simple and accurate algorithms for identifying and predicting the macroeconomic trends.

Yuri Manin on Biomolecules as Quantum Computers

In 1980, Russian mathematician Yuri Manin published Computable and Uncomputable. On pages 14 and 15 of his introduction, Manin suggests that “Molecular biology furnishes examples of the behavior of natural (not engineered by humans) systems which we have to describe in terms initially devised for discrete automata.” Manin then describes the remarkable energy efficiency of naturally occurring biomolecular processes such as DNA replication. He proposes modeling such behaviors in terms of unitary rotations in a finite-dimensional Hilbert space. The decomposition of such systems then corresponds to the tensor product decomposition of the state space, that is, to quantum entanglement. Manin’s initial focus on biological molecules as examples of highly energy-efficient quantum automata is unique among quantum computing’s founding figures since both he and other early leaders quickly moved to the then-new and exciting concept of von Neumann automata. The von Neumann formalism reinterpreted molecular quantum computing in terms of qubits, which made it possible to imagine the power of quantum computing as not much more than a superposition of virtual binary computers. This paper provides the original excerpt of Manin’s molecular computing argument. A useful analytical feature of Manin’s pre-von-Neumann model of quantum computation is its openness to new formalisms that avoid accidentally making classical physics dominant over the quantum world by expressing quantum states only in terms of concepts such as automata that assume extreme classical precision and complexity.

A Modified Dynamical Model of Cosmology I Theory

Wheeler (1964) had formulated Mach’s principle as the boundary condition for general relativistic field equations. Here, we use this idea and develop a modified dynamical model of cosmology based on imposing Neumann boundary condition on cosmological perturbation equations. Then, it is shown that a new term appears in the equation of motion, which leads to a modified Poisson equation. In addition, a modified Hubble parameter is derived due to the presence of the new term. Moreover, it is proved that, without a cosmological constant, such a model has a late time-accelerated expansion with an equation of state converging to w < − 1 . Also, the luminosity distance in the present model is shown to differ from that of the Λ C D M model at high redshifts. Furthermore, it is found that the adiabatic sound speed squared is positive in radiation-dominated era and then converges to zero at later times. Theoretical implications of the Neumann boundary condition have been discussed, and it is shown that, by fixing the value of the conjugate momentum (under certain conditions), one could derive a similar version of modified dynamics. In a future work, we will confine the free parameters of the Neumann model based on hype Ia Supernovae, Hubble parameter data, and the age of the oldest stars.

Stable Computer Control Algorithm of Von Neumann Model

The problem of computer control algorithm for the singular Von Neumann input-output model is researched. A kind of new mathematic method is applied to study the singular systems without converting them into general systems. A kind of stability condition under which the singular input-output model is admissible is proved with the form of linear matrix inequality. Based on this, a new state feedback stability criterion is established. Then the formula of a desired state feedback controller is derived.