Narrow Results

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,

In my talk at the Seventh QCD Workshop held in Villefranche in January 2003 I showed that a non-Hermitian Hamiltonian H possessing an unbroken symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator , which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate is cumbersome in quantum mechanics and impossible in quantum field theory. I describe here an alternative method for calculating directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method gives the operator in quantum field theory. The operator is a new time-independent observable in -symmetric quantum field theory.

The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

The role of dissipation with respect to a microscopic superposition of quantum states is investigated by means of master equations. This has implications for the study of the emergence of classicality from the quantum level. In particular, it illustrates why it is difficult to observe a macroscopic quantum state. The role of the environment is assumed by the measuring apparatus. A pure state is reduced to a mixture in the pointer basis of the system by means of the interaction with the apparatus. It is the intention that this type of analysis will have applications to experiments which are designed to better understand the environmental-assisted invariance formulation of quantum mechanics.

In this paper, a novel method to express Chinese characters mathematically is presented based on the knowledge of the structure of Chinese characters. Each Chinese character can be denoted by a mathematical expression in which the operands are components of Chinese characters and the operators are the location relations between the components. Five hundred five components are selected and 6 operators are defined to express all the Chinese characters successfully. These mathematical expressions of Chinese characters are simple, natural, and can be operated like the common mathematical expression of numbers. It makes Chinese information processing much simpler than before. This theory has been applied successfully in fonts automation, Chinese information transmission among different platforms and different operating systems on Internet, and knowledge discovery of the structure of Chinese characters. It can also be applied extensively to many areas such as typesetting, advertising, packing design, virtual library, network transmission, pattern recognition and Chinese mobile communication.

The role of the geometry of prefractal interfaces in Laplacian transport is analyzed through its "harmonic geometrical spectrum." This spectrum summarizes the properties of the Dirichlet-to-Neumann operator associated with these geometries. Numerical analysis shows that very few eigenmodes contribute significantly to the macroscopic response of the system. The hierarchical spatial frequencies of these particular modes correspond to the characteristic length scales of the interface. From this result, a simplified analytical model of the response of self-similar interfaces is developed. This model reproduces the classical low and high frequency asymptotic limits and gives an approximate constant phase angle behavior for the intermediate frequency region. It also provides an analytical description for the crossovers between these regimes and for their dependency on the order of the prefractal interface. In this frame, it is shown that the properties of any generation prefractal can be deduced from the properties of the fractal generator, which are easy to reach numerically.

Comonotone maxitivity and minitivity of Sugeno integral can be seen as commuting of the Sugeno integral and max, resp. min operator for comonotone functions. In the paper, we look for the other operators commuting with the Sugeno integral when comonotone functions are considered. We give full description of all such commuting operators, which can be classified into 18 classes.

In this paper, we develop some new aggregation operators with pure linguistic information including the uncertain pure linguistic weighted geometric mean (UPLWGM) operator, the induced uncertain pure linguistic ordered weighted geometric mean (IUPLOWGM) operator, and the induced uncertain pure linguistic hybrid geometric mean (IUPLHGM) operator. These developed aggregation operators are very suitable to deal with the situation where the input arguments are represented in uncertain pure linguistic variables. Also, as a more general type of aggregation operator, the IUPLHGM operator is based on the UPLWGM and IUPLOWGM operators, and it can reflect the importance degrees of both the given uncertain linguistic variables and their ordered positions. Moreover, in the situations where the information about all the attribute weights, the attribute values and the expert weights are expressed in the form of linguistic labels variables, we develop an approach based on the IUPLHGM operator for multiple attribute group decision making with pure linguistic information. Finally, an application of the developed approach to group decision making problem regarding the selection of investments is given. Also, we present a comparative analysis with other related decision making methods to demonstrate the effectiveness of the developed approach.

We define the concept of "semicoprime" for preradicals and for submodules and we prove some properties that relate both of them. For any ring we define the ultrasocle preradical as a certain join of maximal semicoprime preradicals. It defines a kind of primary decomposition on modules. We compare the greatest semicoprime preradical, the meet of all unipotent preradicals, the socle preradical, and the ultrasocle preradical. We characterize rings which are finite product of simple rings in terms of some of these preradicals. We study the least semicoprime preradical above any preradical and we prove some of its properties. Using "Amitsur constructions" we define some related operators and prove some of their properties.

Any semigroup S can be embedded into a semigroup, denoted by ΨS, having some remarkable properties. For general semigroups there is a close relationship between local submonoids of S and of ΨS. For a number of usual semigroup properties , we prove that S and ΨS simultaneously satisfy or not. For a regular semigroup S, the relationship of S and ΨS is even closer, especially regarding the natural partial order and Green's relations; in addition, every element of ΨS is a product of at most four idempotents. For completely regular semigroups S, the relationship of S and ΨS is still closer. On the lattice of varieties of completely regular semigroups regarded as algebras with multiplication and inversion, by means of ΨS, we define an operator, denoted by Ψ. We compare Ψ with some of the standard operators on and evaluate it on a small sublattice of .

We prove that the class of positive operators from $L_{\infty }(\mu )$ to $Y$ has the Bishop-Phelps-Bollobás property for any positive measure $\mu$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. The same result also holds for the pair $(c_0,Y)$ for any uniformly monotone Banach lattice $Y.$ Further we show that these results are optimal in case that $Y$ is strictly monotone.

The goal of this work is to quantitatively assess the relationship between the reported restenosis rates and stent induced arterial stress or strain parameters through finite element method. The impact of three stent designs (Palmaz–Schatz stent, Express stent, and Multilink Vision stent) on the arterial stress distributions were characterized. The influences of initial stent deployment location, stent-tissue friction, and plaque properties on the arterial stresses were also investigated. Higher arterial stresses were observed at the proximal end of the plaque. The Multilink–Vision stent induced lesser stress concentrations due to the high stiffness of the Cobalt Chromium material and thinner strut thickness. The stent-induced arterial stress concentrations were positively correlated with the reported in-stent restenosis rates, with a correlation coefficient of 0.992. Stent deployment initiated at the center of the lumen led to less arterial stress variation, while deployment closer to the thinner edge of the plaque causes higher arterial stresses. The friction between the stent and tissue was found to contribute to larger stress alternations for the plaque only. Increased plaque stiffness resulted in a reduced arterial stress concentration and clinical restenosis rate. Results presented herein suggested that arterial stresses serve as a comprehensive index factor to predict the occurrence of in-stent restenosis, which will facilitate the new stent design and surgical planning.

This study aims at investigating $L^B$-valued general fuzzy automata, for simplicity, $L^B$-valued GFA, with respect to their algebraic properties and based on t-norm/t-conorm and general implicators, where $L$ stands for residuated lattice and $B$ is a set of propositions about the GFA, in which its underlying structure is a complete infinitely distributive lattice. Specifically, we introduce the concepts of $L^B$-valued operators with t-norm and $L^B$-valued operators with t-conorm. We also examine the relationships between the $L^B$-valued operators with t-norm and $L^B$-valued operators with t-conorm. Finally, we introduce the concepts of $L^B$-valued operators with implicator and study the relationships between the $L^B$-valued operators with the implicator and the $L^B$-valued operators with t-norm/t-conorm. To clarify the notions and the results obtained in this study, some examples are submitted as well.

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct modulo powers of 3. In this paper, we focus on congruences for pod(n) modulo 5. For example, for α ≥ 2 and n ≥ 0, we have