Combinatorics of Labelled Parallelogram polyominoes

We obtain explicit formulas for the enumeration of labelled parallelogram polyominoes. These are the polyominoes that are bounded, above and below, by north-east lattice paths going from the origin to a point (k,n). The numbers from 1 and n (the labels) are bijectively attached to the $n$ north steps of the above-bounding path, with the condition that they appear in increasing values along consecutive north steps. We calculate the Frobenius characteristic of the action of the symmetric group S_n on these labels. All these enumeration results are refined to take into account the area of these polyominoes. We make a connection between our enumeration results and the theory of operators for which the intergral Macdonald polynomials are joint eigenfunctions. We also explain how these same polyominoes can be used to explicitly construct a linear basis of a ring of SL_2-invariants.Comment: 25 pages, 9 figure

Homomorphisms between Solomon's descent algebras

In a previous paper (see A. Garsia and C. Reutenauer (Adv. in Math. 77, 1989, 189–262)), we have studied algebraic properties of the descent algebras Σn, and shown how these are related to the canonical decomposition of the free Lie algebra corresponding to a version of the Poincaré-Birkhoff-Witt theorem. In the present paper, we study homomorphisms between these algebras Σn. The existence of these homomorphisms was suggested by properties of some directed graphs that we constructed in the previous paper (reference above) describing the structure of the descent algebras. More precisely, examination of the graphs suggested the existence of homomorphisms Σn→Σn−s and Σn→Σn+s. We were then able to construct, for any s (0<s<n), a surjective homomorphism Δs: Σn→Σn−s and an embedding Γs:Σn−s→Σn, which reflects these observations. The homomorphisms Δs may also be defined as derivations of the free associative algebra Q〈t1,t2,…> which sends ti on ti−s, if one identifies the basis element D⊆S of Σn with some word (coding S) on the alphabet T={t1, t2,…}. We show that this mapping is indeed a homomorphism, using the combinatorial description of the multiplication table of Σn given in the previous paper (reference above)

Random Walk with Shrinking Steps: First Passage Characteristics

We study the mean first passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like $\lambda^{n}$ with $\lambda\leq1$ . We also present, for pedagogical purposes, a continuum system with a diffusion constant decaying exponentially in continuous time. Qualitatively both systems are alike in their global properties. However, the discrete case shows very rich mathematical structure, depending on the value of the shrinking parameter, such as self-repetitive and fractal-like structure for the first passage characteristics. The results we present show that the most important quantitative behavior of the discrete case is that the support of the distribution function evolves in time in a rather complicated way in contrast to the time independent lattice structure of the ordinary random walker. We also show that there are critical values of $\lambda$ defined by the equation $\lambda^{K}+2\lambda^{P}-2=0$ with $\{K,N\}\in{\mathcal N}$ where the mean first passage time undergo transitions.Comment: Major Re-Editing of the article. Conclusions unaltere

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. Essential deformations are classified by the Harrison component of Hochschild cohomology, that vanishes on smooth manifolds and reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian $*$-products. This paper begins a study of abelian quantization on plane curves over \Crm, being algebraic varieties of the form R2/I where I is a polynomial in two variables; that is, abelian deformations of the coordinate algebra C[x,y]/(I). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co-)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves C[x,y]/(I), but the cohomology depends on the local algebra of the singularity of I at the origin.Comment: 21 pages, LaTex format. To appear in Letters Mathematical Physic

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

R-parity Conserving Supersymmetry, Neutrino Mass and Neutrinoless Double Beta Decay

We consider contributions of R-parity conserving softly broken supersymmetry (SUSY) to neutrinoless double beta (\znbb) decay via the (B-L)-violating sneutrino mass term. The latter is a generic ingredient of any weak-scale SUSY model with a Majorana neutrino mass. The new R-parity conserving SUSY contributions to \znbb are realized at the level of box diagrams. We derive the effective Lagrangian describing the SUSY-box mechanism of \znbb-decay and the corresponding nuclear matrix elements. The 1-loop sneutrino contribution to the Majorana neutrino mass is also derived. Given the data on the \znbb-decay half-life of $^{76}$Ge and the neutrino mass we obtain constraints on the (B-L)-violating sneutrino mass. These constraints leave room for accelerator searches for certain manifestations of the 2nd and 3rd generation (B-L)-violating sneutrino mass term, but are most probably too tight for first generation (B-L)-violating sneutrino masses to be searched for directly.Comment: LATEX, 29 pages + 4 (uuencoded) figures appende

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Effect of the ethinylestradiol/norelgestromin contraceptive patch on body composition. Results of bioelectrical impedance analysis in a population of Italian women

<p>Abstract</p> <p>Background</p> <p>As weight gain is one of the most frequently cited reasons for not using and for discontinuing hormonal contraceptives, in an open-label, single-arm, multicentre clinical study we evaluated the effect of the ethinylestradiol/norelgestromin contraceptive patch (EVRA, Janssen-Cilag International, Belgium) on body composition using bioelectrical impedance analysis (BIA).</p> <p>Methods</p> <p>Body weight and impedance vector components (resistance (R) and reactance (Xc), at 50 kHz frequency, Akern-RJL Systems analyzer) were recorded before entry, after 1, 3 and 6 months in 182 Italian healthy women aged 29 yr (18 to 45), and with BMI 21.8 kg/m²(16 to 31). Total body water (TBW) was estimated with a BIA regression equation. Vector BIA was performed with the RXc mean graph method and the Hotelling's T²test for paired and unpaired data.</p> <p>Results</p> <p>After 6 months body weight increased by 0.64 kg (1.1%) and TBW increased by 0.51 L (1.7%). The pattern of impedance vector displacement indicated a small increase in soft tissue hydration (interstitial gel fluid). Body composition changes did not significantly differ among groups of previous contraceptive methods. Arterial blood pressure did not significantly change over time.</p> <p>Conclusion</p> <p>After 6 months of treatment with the ethinylestradiol/norelgestromin contraceptive patch we found a minimal, clinically not relevant, increase in body weight less than 1 kg that could be attributed to an adaptive interstitial gel hydration. This fluctuation is physiological as confirmed by the lack of any effect on blood pressure. This could be useful in increasing women's choice, acceptability and compliance of the ethinylestradiol/norelgestromin contraceptive patch.</p