Algorithmic randomness of continuous functions

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}We investigate notions of randomness in the space of continuous functions on . A probability measure is given and a version of the Martin-L?f test for randomness is defined. Random continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. Research partially supported by the National Science Foundation grants DMS 0532644 and 0554841 and 00652732. Thanks also to the American Institute of Mathematics for support during 2006 Effective Randomness Workshop; Remmel partially supported by NSF grant 0400307; Weber partially supported by NSF grant 0652326. Preliminary version published in the Third International Conference on Computability and Complexity in Analysis, Springer Electronic Notes in Computer Science, 2006.     

Space complexity of Abelian groups

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism.      

A sensitive capillary GC-MS method for analysis of topiramate from plasma obtained from single-dose studies

Topiramate (Topamax®) is an antiepileptic medication used as adjunctive and monotherapy in patients with epilepsy and for migraine prophylaxis. A GC‐MS assay was developed that was capable of detecting topiramate plasma concentrations following a single rectal or oral dose administration. Topiramate plasma samples were prepared by solid‐phase extraction and were quantified by GC‐MS analysis. The topiramate standard curves were split from 0.1 to 4 µg/mL and from 4 to 40 µg/mL in order to give a more accurate determination of the topiramate concentration. The accuracy of the standards ranged from 94.6 to 107.3% and the precision (%CV) ranged from 1.0 to 5.3% for both curves at all concentrations. The %CV for quality controls was <7.6%. The assay is both accurate and precise and will be used to complete future pharmacokinetic studies. Copyright © 2012 John Wiley & Sons, Ltd.     

A bijective proof of the generating function for the number of reverse plane partitions via lattice paths

Recently, Hillman and Grassl gave a bijective proof for the generating function for the number of reverse plane partitions of a fixed shape λ. We give another bijective proof for this generating function via completelv different methods. Our bijection depends on a lattice path coding of reverse plane partitions and a new method for constructing bisections out of certain pairs of involutions due to Garsia and Milne.     

Multiplying Schur functions

A new combinatorial rule for expanding the product of Schur functions as a sum of Schur functions is formulated. The rule has several advantages over the Littlewood-Richardson rule (D. E. Littlewood and A. R. Richardson, Philos. Trans. Roy. Soc. London Ser. A233 (1934), 49–141). First this rule allows for direct computation of the expansion of the product of any number of Schur functions, not just the product of two Schur functions. Also, the rule is easily stated and is well suited to computer implementation. It is shown that the rule implies the Littlewood-Richardson rule and gives a combinatorial proof that the coefficient of S_λ in the product S_μS_ν equals the coefficient of S_ν in the expansion of the skew Schur function S_λ/μ. The rule is derived from some results proved independently by A. P. Hillman and R. M. Grassl (J. Combin. Comput. Sci. Systems5 (1980), 305–316) and by D. White (J. Combin. Theory Ser. A30 (1981), 237–247) on the Robinson-Schensted-Knuth correspondence.     

Bijective proofs of some classical partition identities

A bijective proof of a general partition theorem is given which has as direct corollaries many classical partition theorems due to Euler, Glaisher, Schur, Andrews, Subbarao, and others. It is shown that the bijective proof specializes to give bijective proofs of these classical results and moreover the bijections which result often coincide with bijections which have occurred in the literature. Also given are some sufficient conditions for when two classes of words omitting certain sequences of words are in bijection.     

A computational and combinatorial exposé of plethystic calculus

In recent years, plethystic calculus has emerged as a powerful technical tool for studying symmetric polynomials. In particular, some striking recent advances in the theory of Macdonald polynomials have relied heavily on plethystic computations. The main purpose of this article is to give a detailed explanation of a method for finding combinatorial interpretations of many commonly occurring plethystic expressions, which utilizes expansions in terms of quasisymmetric functions. To aid newcomers to plethysm, we also provide a self-contained exposition of the fundamental computational rules underlying plethystic calculus. Although these rules are well-known, their proofs can be difficult to extract from the literature. Our treatment emphasizes concrete calculations and the central role played by evaluation homomorphisms arising from the universal mapping property for polynomial rings.     

Quasi-symmetric functions and up–down compositions

Carlitz (1973) [5] and Rawlings (2000) [13] studied two different analogues of up–down permutations for compositions with parts in {1,…,n}. Cristea and Prodinger (2008/2009) [7] studied additional analogues for compositions with unbounded parts. We show that the results of Carlitz, Rawlings, and Cristea and Prodinger on up–down compositions are special cases of four different analogues of generalized Euler numbers for compositions. That is, for any s≥2, we consider classes of compositions that can be divided into an initial set of blocks of size s followed by a block of size j where 0≤j≤s−1. We then consider the classes of such compositions where all the blocks are strictly increasing (weakly increasing) and there are strict (weak) decreases between blocks. We show that the weight generating functions of such compositions w=w₁?w_m, where the weight of w is , are always the quotients of sums of quasi-symmetric functions. Moreover, we give a direct combinatorial proof of our results via simple involutions.