Mathematical operator linear operators

In die HMO approximation, the n-electron wave function is expressed as a linear combination of the atomic orbitals (for the case in which the plane of the molecule coincides with the x-y plane). Minimizing the total rt-electron energy with respect to the coefficients leads to a series of equations from which the atomic coefficients can be extracted. Although the mathematical operations involved in solving the equation are not... 

H. W. Kuhn and A. W. Tucker, Non-linear Programming, in J. Neyman, ed., Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951 Thomas L. Saaty, Mathematical Methods of Operations Research, McGraw-Hill Book Co., New York, 1959. 

The wave mechanics discussed in Chapter 2 is a linear theory. In order to develop the theory in a more formal manner, we need to discuss the properties of linear operators. An operator 4 is a mathematical entity that transforms a function ip into another function 0... 

We first state the postulates succinctly and then elaborate on each of them with particular regard to the mathematical properties of linear operators. The postulates are as follows. 

We may only raise these questions. The development of the main mathematical tools allowing their study (theories of Markoff chains and of linear operators in Hilbert space) may, however, allow us to hope for progress in the coming years. ... 

Kristoffersen, T.K. (2005) Deviation measures in linear two-stage stochastic programming. Mathematical Methods of Operations Research, 62, 255. 

Molecular orbitals will be very irregular three-dimensional functions with maxima near the nuclei since the electrons are most likely to be found there and falling off toward zero as the distance from the nuclei increases. There will also be many zeros defining nodal surfaces that separate phase changes. These requirements are satisfied by a linear combination of atom-centered basis functions. The basis functions we choose should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei since, when the electron is close to one atom and far from the others, its distribution will resemble an AO of that atom. And yet they should be simple enough that mathematical operations required in the solution of the Fock equations can actually be carried out efficiently. The first requirement is easily satisfied by choosing hydrogenic AOs as a basis... 

In its full generality, our third fundamental quantum-mechanical assumption says that the same kind of decomposition is possible with base states of the position observable, the momentum observable or indeed any observable. In other words, every observable has a complete set of base states. Typically the information about the base states and the value of the observable on each base state is collected into a mathematical object called a self-adjoint linear operator. The base states are the eigenvectors and the corresponding values of the observable are the eigenvalues. For more information about this point of view, see [RS, Section VIII.2]. 

Not only is this notation more succinct, but it also suggests the sophisticated (and correct ) point of view that is itself a mathematical object worthy of study. In fact, dy is a linear operator for more details on this topic, see [Si,... 

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

Proposition 6.11 implies that irreducible representations are the identifiable basic building blocks of all finite-dimensional representations of compact groups. These results can be generalized to infinite-dimensional representations of compact groups. The main difficulty is not with the representation theory, but rather with linear operators on infinite-dimensional vector spaces. Readers interested in the mathematical details ( dense subspaces and so on) should consult a book on functional analysis, such as Reed and Simon [RS],... 

Perturbation theory of linear operators (in Japanese), Sugaku (Mathematics) 2 (1950), 201-208. 

Following Ref. [73] we describe first the steps before we comment on the mathematical results. Let us start with an isometric semigroup, G(f) t > 0, appropriately defined in the Hilbert space fi. If there exists a contractive semigroup SG t > 0 (defined on fi) and an invertible linear operator A, with its domain and range both dense in k, such that... 

Some comments about nonlinearities in the Hamiltonian may be added here. The case we are considering here is called scalar nonlinearity (in the mathematical literature it is also called nonlocal nonlinearity ) [7] this means that the operators are of the form P(u) = (An, u)Bu where A, B are linear operators and<.,.>is the inner product in a Hilbert space. The literature on scalar nonlinearities applied to chemical problems is quite scarce (we cite here a few papers [2,8]) but the results justified by this approach are of universal use in solvation methods. 

See e.g. T. Kato, Perturbation of Linear Operators, (Springer Verlag, Berlin 1966) M. Read and B. Simon, Methods of Modern Mathematical Physcs, (Academic Press, New York 1972). 

The function (1) may be linearized if, through any set of mathematical operations, equation 1 may be transformed into... 

MOGP is based on the more traditional optimisation method genetic programming (GP), which is a type of GA [53,54]. The main difference between GP and a GA is in the chromosome representation in a GA an individual is usually represented by a fixed-length linear string, whereas in GP individuals are represented by treelike structures hence, they can vary in shape and size as the population undergoes evolution. The internal nodes of the tree, typically represent mathematical operators, and the terminal nodes, typically represent variables and constant values thus, the chromosome can represent a mathematical expression as shown in Fig. 4. 

Quantum mechanics involves two distinct sets of hypotheses—the general mathematical scheme of linear operators and state vectors with its associated probability interpretation and the commutation relations and equations of motion for specific dynamical systems. It is the latter aspect that we wish to develop, by substituting a single quantum dynamical principle for the conventional array of assumptions based on classical Hamiltonian dynamics and the correspondence principle. 

The mathematical formalism jofitjuantum mechanics is expressed in terms of linear operators, which rep resent the observables of a system, acting on a state vector which is a linear superposition of elements of an infinitedimensional linear vector space called Hilbert space. We require a knowledge of just the basic properties and consequences of the underlying linear algebra, using mostly those postulates and results that have direct physical consequences. Each state of a quantum dynamical system is exhaustively characterized by a state vector denoted by the symbol T >. This vector and its complex conjugate vector Hilbert space. The product clT ), where c is a number which may be complex, describes the same state. 

The backbone of Monte Carlo simulation is the ability to generate random numbers because random numbers form the basis of the random draws from a probability distribution. Computers, because they are based on rules, algorithms, and mathematical operations, cannot generate truly random numbers. Instead, random numbers start from some point in the algorithm, called the seed, and proceed in a linear, predictable manner but when examined in the short term appear to be random. It... 

Fourier Transform is a mathematical linear operation that decomposes a function into a continuous spectrum of its frequency components as a sum of sinusoids and cosinusoids. 

Most of the readers will probably be trained within differential calculus, with linear algebra, or with statistics. All the mathematical operations needed in these disciplines are by far more complex than that single one needed in partial order. The point is that operating without numbers may appear somewhat strange. The book aims to reduce this uncomfortable strange feeling. 

The measurement of differential pressure will be an inherently linear operation, subject to an exponential lag of typically about 0.5 seconds. The mathematical description is thus ... 

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