This paper introduces the expanded real numbers as an ordered subring of hyperreal number field that does not contain any infinitesimals, and defines set all integrable functions from to numbers. allows identify Dirac delta with a special hyperreal-valued function variable: thus defined is general tool, applicable for mathematical modeling physical systems in which infinitely high densities occ...

The article deals with problems of testing implementations of mathematical functions working with floating-point numbers. It considers current standards’ requirements to such implementations and demonstrates that those requirements are not sufficient for correct operation of modern systems using sophisticated mathematical modeling. Correct rounding requirement is suggested to guarantee preserva...

1) The facility with numbers and calculations, 2) The art of avoiding unnecessary calculations, 3) Memorizing formulas and theorems, 4) Understanding and finding proofs, 5) Solving problems with well-described procedures, 6) Solving problems for which there are no well-described procedures, 7) Understanding abstract mathematical concepts, 8) Creating new mathematical concepts, and clarifying an...

The article concerns problems of formulating standard requirements to implementations of mathematical functions working with floating-point numbers and conformance test development for them. Inconsistency and incompleteness of available standards in the domain is demonstrated. Correct rounding requirement is suggested to guarantee preservation of all important properties of functions and to sup...

1. BASIC PROBABILITY THEORY 1.1 Measure-theoretical foundation of probability theory 1.2 Independence 1.3 Borel measurable functions 1.4 Mathematical expectatlon 1.5 Characteristic functions 1.6 Random functions 2. CONVERGENCE 2.1 Heak and strong convergence of random variables 2.2 Convergence of mathematical expectations 2.3 Convergence of distributions 2.4 Central limit theoreras 2.5 Further ...

Most of the researches in the domain of fuzzy number comparisons serve the fuzzy number ordering purpose. For making a comparison between two fuzzy numbers, beyond the determination of their order, it is needed to derive the magnitude of their order. In line with this idea, the concept of inequality is no longer crisp however it becomes fuzzy in the sense of representing partial belonging or de...

Every physical measuring needs a finite, different from zero measurement time and provides information in form of the choice of a measurement result from all possible measurement results. If infinitely many (different) measurement results would be possible, the choice of a measurement result could deliver an infinite quantity of information. But the results of physical measurings (of finite dur...

Primary immune responses generate short-term effectors and long-term protective memory cells. The delineation of the genealogy linking naive, effector, and memory cells has been complicated by the lack of phenotypes discriminating effector from memory differentiation stages. Using transcriptomics and phenotypic analyses, we identify Bcl2 and Mki67 as a marker combination that enables the tracki...

This paper proposes a new plant/controller design integration framework that seeks the optimal pair of the plant and the controller achieving the best possible closed-loop performance in H2 control. The framework is further equipped with a symbolic-numeric hybrid optimization approach to effectively search the optimum. The first step of the suggested approach relies on an algebraic approach to ...

The study of hydraulic structures makes systematic use of mathematical models in order to verify their behaviour. On-line use of these models to synthesise predictive control permits basing control on almost perfect knowledge of every aspect of the process. Achieving this aim requires good management of the embedded numeric model and the incorporation of an efficient resetting procedure. A simp...