The Riesz space structure of an Abelian W*-algebra

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M₀ be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M₀, and an elementary proof is given of the fact that a positive self-adjoint transformation in M₀ has a unique positive square root in M₀. It is then shown that when the algebraic operations are suitably defined, then M₀ becomes a commutative algebra. If ReM₀ denotes the set of all self-adjoint elements of M₀, then it is proved that ReM₀ is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM₀ which characterizes the normal integrals on the order dense ideals of ReM₀. It is then shown that ReM₀ may be identified with the extended order dual of ReM, and that ReM₀ is perfect in the extended sense.

Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

Relativistic quark models and the current algebra

In this thesis we are concerned with finding representations of the algebra of SU(3) vector and axial-vector charge densities at infinite momentum (the "current algebra") to describe the mesons, idealizing the real continua of multiparticle states as a series of discrete resonances of zero width. Such representations would describe the masses and quantum numbers of the mesons, the shapes of their Regge trajectories, their electromagnetic and weak form factors, and (approximately, through the PCAC hypothesis) pion emission or absorption amplitudes.

We assume that the mesons have internal degrees of freedom equivalent to being made of two quarks (one an antiquark) and look for models in which the mass is SU(3)-independent and the current is a sum of contributions from the individual quarks. Requiring that the current algebra, as well as conditions of relativistic invariance, be satisfied turns out to be very restrictive, and, in fact, no model has been found which satisfies all requirements and gives a reasonable mass spectrum. We show that using more general mass and current operators but keeping the same internal degrees of freedom will not make the problem any more solvable. In particular, in order for any two-quark solution to exist it must be possible to solve the "factorized SU(2) problem," in which the currents are isospin currents and are carried by only one of the component quarks (as in the K meson and its excited states).

In the free-quark model the currents at infinite momentum are found using a manifestly covariant formalism and are shown to satisfy the current algebra, but the mass spectrum is unrealistic. We then consider a pair of quarks bound by a potential, finding the current as a power series in 1/m where m is the quark mass. Here it is found impossible to satisfy the algebra and relativistic invariance with the type of potential tried, because the current contributions from the two quarks do not commute with each other to order 1/m³. However, it may be possible to solve the factorized SU(2) problem with this model.

The factorized problem can be solved exactly in the case where all mesons have the same mass, using a covariant formulation in terms of an internal Lorentz group. For a more realistic, nondegenerate mass there is difficulty in covariantly solving even the factorized problem; one model is described which almost works but appears to require particles of spacelike 4-momentum, which seem unphysical.

Although the search for a completely satisfactory model has been unsuccessful, the techniques used here might eventually reveal a working model. There is also a possibility of satisfying a weaker form of the current algebra with existing models.

Forma canónica de Kronecker

Este libro trata de explicar con claridad y sencillez la forma canónica de Kronecker de haces de matrices para la relación de equivalencia estricta. El tema es importante para los ingenieros, físicos, químicos, economistas y otros científicos que estudian sistemas lineales con control, por lo que una introducción asequible y rigurosa se echa de menos. También esperamos que el libro sea de utilidad para los matemáticos en un segundo curso de álgebra lineal como complemento natural del estudio de la forma canónica de Jordan. La forma canónica de Kronecker es llamada igualmente de Weierstrass-Kronecker, ya que Weierstrass desarrolla la teoría de los divisores elementales y Kronecker la de los índices minimales. Desde un punto de vista epistemológico e histórico deben relacionarse estas teorías con el estudio geométrico de los haces de cónicas y cuádricas para la formación del estudiante de matemáticas. Este libro no intenta establecer estas conexiones. Al lector que desee proseguir en los precedentes históricos le recomendamos el libro sobre historia de las matemáticas de Bourbaki y también artículos de Robert Thompson, Frank Uhlig y otros en la revista Linear Algebra and Its Applications en los años 1980.

All their eggs in one basket: a rocky reef nursery for the longnose skate (Raja rhina Jordan & Gilbert, 1880) in the southern California Bight

Skates (family Rajidae) are oviparous and lay tough, thick-walled eggs. At least some skate species lay their eggs in spatially restricted nursery grounds where embryos develop and hatch (Hitz, 1964; Hoff, 2007). After hatching, neonates may quickly leave the nursery grounds (Hoff, 2007). Egg densities in these small areas may be quite high. As an example, in the eastern Bering Sea, a site <2 km2 harbored eggs of Alaska skate (Bathyraja parmifera) exceeding 500,000/km2. All skate nursery grounds have been identified over soft sea floors (Lucifora and García, 2004; Hoff, 2007).

Species of the rougheye rockfish complex: resurrection of Sebastes melanostictus (Matsubara, 1934) and a redescription of Sebastes aleutianus (Jordan and Evermann, 1898)(Teleostei: Scorpaeniformes)

The widespread and commercially important rougheye rockfish, Sebastes aleutianus (Jordan and Evermann, 1898), has been considered a single variable species, with light- and dark-colored forms, found on the outer continental shelf and upper slope of the North Pacific Ocean. Genetic analysis of 124 specimens verified the presence of two species in new specimens collected from Alaska to Oregon, and the two species were analyzed for distinguishing color patterns and morphological characters. Characters distinguishing the two were extended to an analysis of 215 additional formalin-fixed specimens representing their geographic ranges. Sebastes aleutianus is pale, often has dark mottling on the dorsum in diffuse bands, and does not have distinct dark spots on the spinous dorsal fin; it ranges from the eastern Aleutian Islands and southeastern Bering Sea to California. Sebastes melanostictus (Matsubara, 1934), the blackspotted rockfish, ranges from central Japan, through the Aleutian Islands and Bering Sea, to southern California. It is darker overall and spotting is nearly always present on the spinous dorsal fin. Sebastes swifti (Evermann and Goldsborough, 1907) is a synonym of S. aleutianus; S. kawaradae (Matsubara, 1934) is a synonym of S. melanostictus. The subgenus Zalopyr is restricted to S. aleutianus and S. melanostictus. Nomenclatural synonymies, diagnoses, descriptions, and distributions are provided for each species.

Length-weight relationship of fishes from coral reefs along the coastline of Jordan (Gulf of Aqaba)

The parameters a and b of the length-weight relationship of the form W = a.Lb were estimated for 15 fish species caught along the coastline of Jordan in the Gulf of Aqaba. The sampling was carried out between July 1999 and January 2001. Data from 1 000 fish individuals (identified to eight families and 15 species) were used for this purpose.

Stanford University’s John Otterbein Snyder: Student, Collaborator, and Colleague of David Starr Jordan and Charles Henry Gilbert

John Otterbein Snyder (1867–1943) was an early student of David Starr Jordan at Stanford University and subsequently rose to become an assistant professor there. During his 34 years with the university he taught a wide variety of courses in various branches of zoology and advised numerous students. He eventually mentored 8 M.A. and 4 Ph.D. students to completion at Stanford. He also assisted in the collection of tens of thousands of fish specimens from the western Pacific, central Pacific, and the West Coast of North America, part of the time while stationed as “Naturalist” aboard the U.S. Fish Commission’s Steamer Albatross (1902–06). Although his early publications dealt mainly with fish groups and descriptions (often as a junior author with Jordan), after 1910 he became more autonomous and eventually rose to become one of the Pacific salmon, Oncorhynchus spp., experts on the West Coast. Throughout his career, he was especially esteemed by colleagues as “a stimulating teacher,” “an excellent biologist,” and “a fine man.