Categorical Structures in Non-Commutative Measure Theory.

In this paper, we investigate effect algebras and base normed spaces from the categorical point of view. We prove that the category of effect algebras is complete and cocomplete as well as the category of base normed spaces is complete, and discuss the contravariant functor from the category of effect algebras to the category of base normed spaces.

Prime factors

We use Voiculescu’s free probability theory to prove the existence of prime factors, hence answering a longstanding problem in the theory of von Neumann algebras.

On representations of finite type

Representations of the (infinite) canonical anticommutation relations and the associated operator algebra, the fermion algebra, are studied. A “coupling constant” (in (0,1]) is defined for primary states of “finite type” of that algebra. Primary, faithful states of finite type with arbitrary coupling are constructed and classified. Their physical significance for quantum thermodynamical systems at high temperatures is discussed. The scope of this study is broadened to include a large class of operator algebras sharing some of the structural properties of the fermion algebra.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

Inositol 1,4,5-trisphosphate receptor subtype 3 in pancreatic islet cell secretory granules revisited.

It has been reported that the inositol 1,4,5-trisphosphate receptor subtype 3 is expressed in islet cells and is localized to both insulin and somatostatin granules [Blondel, O., Moody, M. M., Depaoli, A. M., Sharp, A. H., Ross, C. A., Swift, H. & Bell, G. I. (1994) Proc. Natl. Acad. Sci. USA 91, 7777-7781]. This subcellular localization was based on electron microscope immunocytochemistry using antibodies (affinity-purified polyclonal antiserum AB3) directed to a 15-residue peptide of rat inositol trisphosphate receptor subtype 3. We now show that these antibodies cross-react with rat, but not human, insulin. Accordingly, the anti-inositol trisphosphate receptor subtype 3 (AB3) antibodies label electron dense cores of mature (insulin-rich) granules of rat pancreatic beta cells, and rat granule labeling was blocked by preabsorption of the AB3 antibodies with rat insulin. The immunostaining of immature, Golgi-associated proinsulin-rich granules with AB3 antibodies was very weak, indicating that cross-reactivity is limited to the hormone and not its precursor. Also, the AB3 antibodies labeled pure rat insulin crystals grown in vitro but failed to stain crystals grown from pure human insulin. By immunoprecipitation, the antibodies similarly displayed a higher affinity for rat than for human insulin. We could not confirm the labeling of somatostatin granules using AB3 antibodies.

Equações de onda generalizadas e quantização funtorial para teorias de campo escalar livre

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre.

1800, Catalogus eorum qui in Universitate Harvardiana, Cantabrigiae, in republica Massachusettensi, ab anno MDCXLII, ad annum MDCCC, alicujus gradû laureâ donati sunt.

Marbled paper cover. Includes a biographical note on Joshua Moody (Harvard AB 1707), and a few additional annotations. Asterisks added next to the names of alumni who died after the Catalogue's publication.

Letter of recommendation for Amos Main from Joseph Moodey to Nathan Prince, 1725 July 2

Handwritten letter sent by Joseph Moody, schoolmaster in York, to Harvard Tutor Nathan Prince recommending student Amos Main for acceptance to the College. In the letter, Moody requests Prince give Main an examination for admission, with the caveat that though Main has been studying Latin and Greek he has a difficult home life and is "somewhat Raw; yet I hope you'l wink at it." The letter, dated July 2, 1725, is written on a folded folio-sized leaf; there are handwritten notes about Massachusetts towns on the verso.

Lowell, Massachusetts, 1850 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: Plan of the city of Lowell, Massachusetts, from actual surveys by Sidney & Neff. It was published by S. Moody in 1850. Scale [ca. 1:3,450]. The image inside the map neatline is georeferenced to the surface of the earth and fit to the Massachusetts State Plane Coordinate System, Mainland Zone (in Feet) (Fipszone 2001) coordinate system. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. This map shows features such as roads, railroads, canals, drainage, public buildings, schools, churches, parks, industry locations (e.g. mills, factories, etc.), private buildings with names of property owners, and more. Relief shown by hachures. Includes also illustrations of local buildings in margins.This layer is part of a selection of digitally scanned and georeferenced historic maps from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of originators, ground condition dates, scales, and map purposes.

Marshfield, Massachusetts, 1838 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: Map of Marshfield, Mass., surveyed by John Ford Jr. ; on stone by J.E. Moody. It was published by Thomas Moore's Lithography in 1838. Scale [1:19,800]. The image inside the map neatline is georeferenced to the surface of the earth and fit to the Massachusetts State Plane Coordinate System, Mainland Zone (in Feet) (Fipszone 2001). All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, or other information associated with the principal map. This map shows features such as roads, railroads, drainage, public buildings, schools, churches, cemeteries, industry locations (e.g. mills, factories, mines, etc.), private buildings with names of property owners, town boundaries and more. Relief is shown by hachures. This layer is part of a selection of digitally scanned and georeferenced historic maps of Massachusetts from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of regions, originators, ground condition dates (1755-1922), scales, and purposes. The digitized selection includes maps of: the state, Massachusetts counties, town surveys, coastal features, real property, parks, cemeteries, railroads, roads, public works projects, etc.

A tiny spark /

Poems.

Rings of operators.

"Prepared with the assistance of a grant from the Research Corporation."

Linear operators and their spectra with applications to quantum mechanics.

From lectures given at the New York university Institute for mathematica and mechanics, by R. Cournat and others.