Generalization of the Optical Theorem to an Arbitrary Multipole Excitation of a Particle near a Transparent Substrate

In the present paper, the generalization of the optical theorem to the case of a penetrable particle deposited near a transparent substrate that is excited by a multipole of an arbitrary order and polarization has been derived. In the derivation we employ classic Maxwell’s theory, Gauss’s theorem, and use a special representation for the multipole excitation. It has been shown that the extinction cross-section can be evaluated by the calculation of some specific derivatives from the scattered field at the position of the multipole location, in addition to some finite integrals which account for the multipole polarization and the presence of the substrate. Finally, the present paper considers some specific examples for the excitation of a particle by an electric quadrupole.

Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.

Modelling purposeful processes based on the geometric representation of their trajectories

The possibilities of studying processes based on the geometric representation of their trajectories in a multidimensional space of factors and using projections of trajectories on planes suitable for analysis are discussed. A connection is established between such constructions and the spiral and snail of development. Based on the proposed projection, a development snail is defined as a special representation format of the purposeful process, expanding the possibilities of analysis of modelling results.

A priori estimates of the mass of the burnt materials in rooms of buildings in the integral mathematical model of the initial stage of the fire

The integral mathematical model of the initial stage of a fire in the premises of buildings focuses on the problem of estimating the mass of burnt materials assuming that the data on the rate of combustion of combustible materials are not complete. A priori estimates of the mass of burnt materials are found, which do not use data on the combustible material combustion rate. To derive these estimates, a special representation of the modulus of the increment (fall) of oxygen density in a room with a fire in the initial stage was preliminarily found. Two approaches to the derivation of a priori estimates of the mass of burnt materials have been implemented - using the upper and lower estimates of the modulus of the increment (fall) of the oxygen density. A comparison of a priori estimates obtained with different approaches is performed. Research can be useful at assessment of ecological situation and consequences of the fire.

Spatial Data Mining, Spatial Data Warehousing, and Spatial OLAP

Data mining is one of the current vital fields for all types of data including spatial data. An example of useful extracted patterns from spatial data is to describe changes in metropolitan poverty rates based on city distances from major highways. Geospatial is a term that has recently been gaining in popularity; moreover, many applications on geospatial have different uses in different fields such as geo-tagging, geospatial technology, and geo-fencing. Analyzing spatial data is considered a complex task due to its details as it is related to locations with a special representation such as longitude and latitude. Other attributes are involved in the description of objects which can be analyzed using different data mining techniques. In this chapter, a demonstration of the basic information is performed considering spatial data and spatial data mining including all aspects such as the different type of data, different methods of analysis, different mining techniques, and other related topics.

Markoff Numbers

The Markoff equation is the diophantine equation x2 +y2 +z2 = 3xyz. A solution is called a Markoff triple. The main result in this chapter is a bijection between lower Christoffel words and Markoff triples. The bijection uses several ingredients: a special representation of the free monoid into SL2(N), the so-called Fricke relations, which relate the traces of two matrices in SL2, their product and their commutator (an equation reminiscent of the Markoff equation, as noted first by Harvey Cohn). Another lemma describes the socalled Markoff moves: they relate Markoff triples each to another. The chapter ends with a statement of the famous Frobenius conjecture: it asks whether the parametrization of Markoff numbers (that is, components of a Markoff triple), which is surjective by the theorem, is also injective.

Spatial Data Mining, Spatial Data Warehousing, and Spatial OLAP

Data mining is one of the current vital fields for all types of data including spatial data. An example of useful extracted patterns from spatial data is to describe changes in metropolitan poverty rates based on city distances from major highways. Geospatial is a term that has recently been gaining in popularity; moreover, many applications on geospatial have different uses in different fields such as geo-tagging, geospatial technology, and geo-fencing. Analyzing spatial data is considered a complex task due to its details as it is related to locations with a special representation such as longitude and latitude. Other attributes are involved in the description of objects which can be analyzed using different data mining techniques. In this chapter, a demonstration of the basic information is performed considering spatial data and spatial data mining including all aspects such as the different type of data, different methods of analysis, different mining techniques, and other related topics.

Minimality and onsetless syllables in Zinza

A dispreference for onsetless syllables is often manifested by processes of consonant insertion or vowel deletion which eliminate onsetless syllables, and it is sometimes manifested by disallowing prosodic distinctions which onsetful syllables make, e.g. onsetless syllables may resist bearing tone or stress. This paper investigates the interaction between minimality and the structure of onsetless syllables in the Bantu language Zinza, documenting a previously unobserved onsetless effect which is partially conditioned by minimality. In Zinza, word-initial vowels of disyllables are lengthened, so /o-ljá/ becomes [óó-lja] ‘you eat’ (compare [o-líma] ‘you cultivate’ and [bá-lja] ‘they eat’, with no lengthening). These facts test previous theories of onsetless effects (Downing 1993, 1998, Odden 1995, Orie 2000). I argue that a special representation of onsetless syllables is not warranted, and does not explain this pattern of lengthening. A constraint against word-initial short onsetless syllables interacting with word minimality suffices to explain this lengthening.