A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

Two Structurally Similar Maize Cytosolic Superoxide Dismutase Genes, Sod4 and Sod4A, Respond Differentially to Abscisic Acid and High Osmoticum1

The maize (Zea mays) superoxide dismutase genes Sod4 and Sod4A are highly similar in structure but each responds differentially to environmental signals. We examined the effects of the hormone abscisic acid (ABA) on the developmental response of Sod4 and Sod4A. Although both Sod4 and Sod4A transcripts accumulate during late embryogenesis, only Sod4 is up-regulated by ABA and osmotic stress. Accumulation of Sod4 transcript in response to osmotic stress is a consequence of increased endogenous ABA levels in developing embryos. Sod4 mRNA is up-regulated by ABA in viviparous-1 mutant embryos. Sod4 transcript increases within 4 h with ABA not only in developing embryos but also in mature embryos and in young leaves. Sod4A transcript is up-regulated by ABA only in young leaves, but neither Sod4 nor Sod4A transcripts changed in response to osmotic stress. Our data suggest that in leaves Sod4 and Sod4A may respond to ABA and osmotic stress via alternate pathways. Since the Sod genes have a known function, we hypothesize that the increase in Sod mRNA in response to ABA is due in part to ABA-mediated metabolic changes leading to changes in oxygen free radical levels, which in turn lead to the induction of the antioxidant defense system.

A germ-line Tsc1 mutation causes tumor development and embryonic lethality that are similar, but not identical to, those caused by Tsc2 mutation in mice

Tuberous sclerosis (TS) is characterized by the development of hamartomas in various organs and is caused by a germ-line mutation in either TSC1 or TSC2 tumor suppressor genes. From the symptomatic resemblance among TS patients, involvement of TSC1 and TSC2 products in a common pathway has been suggested. Here, to analyze the function of the Tsc1 product, we established a line of Tsc1 (TSC1 homologue) knockout mouse by gene targeting. Heterozygous Tsc1 mutant (Tsc1+/−) mice developed renal and extra-renal tumors such as hepatic hemangiomas. In these tumors, loss of wild-type Tsc1 allele was observed. Homozygous Tsc1 mutants died around embryonic days 10.5–11.5, frequently associated with neural tube unclosure. As a whole, phenotypes of Tsc1 knockout mice resembled those of Tsc2 knockout mice previously reported, suggesting that the presumptive common pathway for Tsc1 and Tsc2 products may also exist in mice. Notably, however, development of renal tumors in Tsc1+/− mice was apparently slower than that in Tsc2+/− mice. The Tsc1 knockout mouse described here will be a useful model to elucidate the function of Tsc1 and Tsc2 products as well as pathogenesis of TS.

Similar antigenic surfaces, rather than sequence homology, dictate T-cell epitope molecular mimicry.

Molecular mimicry, normally defined by the level of primary-sequence similarities between self and foreign antigens, has been considered a key element in the pathogenesis of autoimmunity. Here we describe an example of molecular mimicry between two overlapping peptides within a single self-antigen, both of which are recognized by the same human self-reactive T-cell clone. Two intervening peptides did not stimulate the T-cell clone, even though they share nine amino acids with the stimulatory peptides. Molecular modeling of major histocompatibility complex class II-peptide complexes suggests that both of the recognized peptides generate similar antigenic surfaces, although these are composed of different sets of amino acids. The molecular modeling of a peptide shifted one residue from the stimulatory peptide, which was recognized in the context of the same HLA molecule by another T-cell clone, generated a completely different antigenic surface. Functional studies using truncated peptides confirmed that the anchor residues of the two "mimicking" epitopes in the HLA groove differ. Our results show, for two natural epitopes, how molecular mimicry can occur and suggest that studies of potential antigenic surfaces, rather than sequence similarity, are necessary for analyzing suspected peptide mimicry.

The silver gene of Drosophila melanogaster encodes multiple carboxypeptidases similar to mammalian prohormone-processing enzymes.

The silver (svr) gene of Drosophila melanogaster is required for viability, and severe mutant alleles result in death prior to eclosion. Adult flies homozygous or hemizygous for weaker alleles display several visible phenotypes, including cuticular structures that are pale and silvery in color due to reduced melanization. We have identified and cloned the DNA encoding the svr gene and determined the sequence of several partially overlapping cDNAs derived from svr mRNAs. The predicted amino acid sequence of the polypeptides encoded by these cDNAs indicates that the silver proteins are members of the family of preprotein-processing carboxypeptidases that includes the human carboxypeptidases E, M, and N. One class of svr mRNAs is alternatively spliced to encode at least two polyproteins, each of which is composed of two carboxypeptidase domains.

Identification of a plant serine-arginine-rich protein similar to the mammalian splicing factor SF2/ASF.

We show that the higher plant Arabidopsis thaliana has a serine-arginine-rich (SR) protein family whose members contain a phosphoepitope shared by the animal SR family of splicing factors. In addition, we report the cloning and characterization of a cDNA encoding a higher-plant SR protein from Arabidopsis, SR1, which has striking sequence and structural homology to the human splicing factor SF2/ASF. Similar to SF2/ASF, the plant SR1 protein promotes splice site switching in mammalian nuclear extracts. A novel feature of the Arabidopsis SR protein is a C-terminal domain containing a high concentration of proline, serine, and lysine residues (PSK domain), a composition reminiscent of histones. This domain includes a putative phosphorylation site for the mitotic kinase cyclin/p34cdc2.

Robust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape

A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.

E-polynomials of the SL(2, C)-character varieties of surface groups

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2, C), for any possible holonomy around the puncture. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behavior of the E-polynomial under fibrations.

On the existence of fractal strings whose set of dimensions of fractality is not perfect

In this paper we give an example of a nonlattice self-similar fractal string such that the set of real parts of their complex dimensions has an isolated point. This proves that, in general, the set of dimensions of fractality of a fractal string is not a perfect set.

Privileged Regions in Critical Strips of Non-lattice Dirichlet Polynomials

This paper shows, by means of Kronecker’s theorem, the existence of infinitely many privileged regions called r -rectangles (rectangles with two semicircles of small radius r ) in the critical strip of each function Ln(z):= 1−∑nk=2kz , n≥2 , containing exactly [Tlogn2π]+1 zeros of Ln(z) , where T is the height of the r -rectangle and [⋅] represents the integer part.

Numerical resolution of Emden's equation using Adomian polynomials

Purpose: In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium. Design/methodology/approach: In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known. Findings: Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal. Originality/value: The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.