Scalar, mathematical definition

Physicists often refer to complex scalar product spaces as Hilbert spaces. The formal mathematical definition of a Hilbert space requires more than just the existence of a complex scalar product the space must be closed a.k.a. complete in a certain technical sense. Because every scalar product space is a subset of some Hilbert space, the discrepancy in terminology between mathematicians and physicists does not have dire consequences. However, in this text, to avoid discrepancies with other mathematics textbooks, we will use complex scalar product. ... 

The criteria (9.23), (9.24), and (9.26) are all rather obvious properties of Euclidean geometry. All of these properties can be traced back to mathematical properties of the scalar product (R Ry), the key structure-maker of a metric space. We therefore wish to determine whether a proposed definition of scalar product satisfies these criteria, and thus guarantees that M is a Euclidean space. 

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

There are two competing and equivalent nomenclature systems encountered in the chemical literature. The description of data in terms of ways is derived from the statistical literature. Here a way is constituted by each independent, nontrivial factor that is manipulated with the data collection system. To continue with the example of excitation-emission matrix fluorescence spectra, the three-way data is constructed by manipulating the excitation-way, emission-way, and the sample-way for multiple samples. Implicit in this definition is a fully blocked experimental design where the collected data forms a cube with no missing values. Equivalently, hyphenated data is often referred to in terms of orders as derived from the mathematical literature. In tensor notation, a scalar is a zeroth-order tensor, a vector is first order, a matrix is second order, a cube is third order, etc. Hence, the collection of excitation-emission data discussed previously would form a third-order tensor. However, it should be mentioned that the way-based and order-based nomenclature are not directly interchangeable. By convention, order notation is based on the structure of the data collected from each sample. Analysis of collected excitation-emission fluorescence, forming a second-order tensor of data per sample, is referred to as second-order analysis, as compared with the three-way analysis just described. In this chapter, the way-based notation will be arbitrarily adopted to be consistent with previous work. 

Mathematical optimization deals with determining values for a set of unknown variables x, X2, , x , which best satisfy (optimize) some mathematical objective quantified by a scalar function of the unknown variables, F(xi, X2, , xn). The function F is termed the objective function bounds on the variables, along with mathematical dependencies between them, are termed constraints. Constraint-based analysis of metabolic systems requires definition of the constraints acting on biochemical variables (fluxes, concentrations, enzyme activities) and determining appropriate objective functions useful in determining the behavior of metabolic systems. 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

The examples in this section show various forms of math atical expressions representing system properties, from simple scalars to exponential functions. For taking into account these discrepancies, the use of the operator symbolism is a convenient means of representing various mathematical forms without having to specify them. By nsing the state variable generalized symbols, the definitions of the three constitutive properties are as follows ... 

First the concept of Lyapunov functions will be introduced. A Lyapunov function, Y x t)), is a positive scalar that depends on the system s state. By definition, the time derivative of a Lyapunov function is non-positive. Mathematically these conditions can be described by ... 

The mathematical conception of an independent definition of geometric subjects (as reaction paths) in the configuration space starts with the idea of an analogous transformation of the coordinates as well as the angle relations in the new system. The distortion of equipotential lines in the new system should be compensated by an inverse distortion of the scalar product defining the angles ... 

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