Fixpoints and Bounded Fixpoints for Complex Objects

We investigate a query language for complex-object databases, which is designed to (1) express only tractable queries, and (2) be as expressive over flat relations as first order logic with fixpoints. The language is obtained by extending the nested relational algebra NRA with a bounded fixpoint operator. As in the flat case, all PTime computable queries over ordered databases are expressible in this language. The main result consists in proving that this language is a conservative extension of the first order logic with fixpoints, or of the while-queries (depending on the interpretation of the bounded fixpoint: inflationary or partial). The proof technique uses indexes, to encode complex objects into flat relations, and is strong enough to allow for the encoding of NRA with unbounded fixpoints into flat relations. We also define a logic based language with fixpoints, the nested relational calculus , and prove that its range restricted version is equivalent to NRA with bounded fixpoints

Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra

Three loop ladder and $V$-topology diagrams contributing to the massive operator matrix element $A_{Qg}$ are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable $N$ and the dimensional parameter $\varepsilon$. Given these representations, the desired Laurent series expansions in $\varepsilon$ can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of $N$ are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of $V$-topologies.Comment: 110 pages Latex, 4 Figure

Complexity Begets Crosscutting, Dooms Hierarchy (Another Paper on Natural Kinds)

There is a perennial philosophical dream of a certain natural order for the natural kinds. The name of this dream is ‘the hierarchy requirement’ (or ‘assumption’ or ‘thesis’). According to this postulate, proper natural kinds form a taxonomy which is both unique (i.e., there is only one taxonomy of such natural kinds) and traditional (i.e., said taxonomy consists of nested relations between specific and then more general kinds, each kind occupying one and only one particular place within that framework of relations). Here I demonstrate that complex scientific objects exist: objects which generate different systems of scientific classification, produce myriad legitimate alternatives amongst the nonetheless still natural kinds, and make the hierarchical dream impossible to realize, except at absurdly great cost. Philosophical hopes for a certain order in nature cannot be fulfilled. Natural kinds crosscut one another, ubiquitously so, and this crosscutting spells the end of the hierarchical dream

Complexity Begets Crosscutting, Dooms Hierarchy (Another Paper on Natural Kinds)

There is a perennial philosophical dream of a certain natural order for the natural kinds. The name of this dream is ‘the hierarchy requirement’ (or ‘assumption’ or ‘thesis’). According to this postulate, proper natural kinds form a taxonomy which is both unique (i.e., there is only one taxonomy of such natural kinds) and traditional (i.e., said taxonomy consists of nested relations between specific and then more general kinds, each kind occupying one and only one particular place within that framework of relations). Here I demonstrate that complex scientific objects exist: objects which generate different systems of scientific classification, produce myriad legitimate alternatives amongst the nonetheless still natural kinds, and make the hierarchical dream impossible to realize, except at absurdly great cost. Philosophical hopes for a certain order in nature cannot be fulfilled. Natural kinds crosscut one another, ubiquitously so, and this crosscutting spells the end of the hierarchical dream

On Link Homology Theories from Extended Cobordisms

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel-Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any EQFT based on a Z_2-extension of the embedded cobordism category which coincides with Khovanov after reducing the coefficients modulo 2, gives rise to a link invariant homology theory isomorphic to those of Khovanov.Comment: Lots of figure

User Defined Types and Nested Tables in Object Relational Databases

Bernadette Byrne, Mary Garvey, ‘User Defined Types and Nested Tables in Object Relational Databases’, paper presented at the United Kingdom Academy for Information Systems 2006: Putting Theory into Practice, Cheltenham, UK, 5-7 June, 2006.There has been much research and work into incorporating objects into databases with a number of object databases being developed in the 1980s and 1990s. During the 1990s the concept of object relational databases became popular, with object extensions to the relational model. As a result, several relational databases have added such extensions. There has been little in the way of formal evaluation of object relational extensions to commercial database systems. In this work an airline flight logging system, a real-world database application, was taken and a database developed using a regular relational database and again using object relational extensions, allowing the evaluation of the relational extensions.Peer reviewe

An Evaluation of Physical Disk I/Os for Complex Object Processing

In order to obtain the performance required for nonstandard database environments, a hierarchical complex object model with object references is used as a storage structure for complex objects. Several storage models for these complex objects, as well as a benchmark to evaluate their performance, are described. A cost model for analytical performance evaluation is developed, and the analytical results are validated by means of measurements on the DASDBS, complex object storage system. The results show which storage structures for complex objects are the most efficient under which circumstance

Thermal out-of-time-order correlators, KMS relations, and spectral functions

We describe general features of thermal correlation functions in quantum systems, with specific focus on the fluctuation-dissipation type relations implied by the KMS condition. These end up relating correlation functions with different time ordering and thus should naturally be viewed in the larger context of out-of-time-ordered (OTO) observables. In particular, eschewing the standard formulation of KMS relations where thermal periodicity is combined with time-reversal to stay within the purview of Schwinger-Keldysh functional integrals, we show that there is a natural way to phrase them directly in terms of OTO correlators. We use these observations to construct a natural causal basis for thermal n-point functions in terms of fully nested commutators. We provide several general results which can be inferred from cyclic orbits of permutations, and exemplify the abstract results using a quantum oscillator as an explicit example.Comment: 36 pages + appendices. v2: minor changes + refs added. v3: minor changes, published versio