Bounding sequence extremal functions with formations

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has no subsequence isomorphic to $u$. For every sequence $u$ define $\mathit{fw}(u)$, the formation width of $u$, to be the minimum $s$ for which there exists $r$ such that there is a subsequence isomorphic to $u$ in every $(r, s)$-formation. We use $\mathit{fw}(u)$ to prove upper bounds on $\mathit{Ex}(u, n)$ for sequences $u$ such that $u$ contains an alternation with the same formation width as $u$. We generalize Nivasch's bounds on $\mathit{Ex}((ab)^{t}, n)$ by showing that $\mathit{fw}((12 \ldots l)^{t})=2t-1$ and $\mathit{Ex}((12\ldots l)^{t}, n) =n2^{\frac{1}{(t-2)!}\alpha(n)^{t-2}\pm O(\alpha(n)^{t-3})}$ for every $l \geq 2$ and $t\geq 3$, such that $\alpha(n)$ denotes the inverse Ackermann function. Upper bounds on $\mathit{Ex}((12 \ldots l)^{t} , n)$ have been used in other papers to bound the maximum number of edges in $k$-quasiplanar graphs on $n$ vertices with no pair of edges intersecting in more than $O(1)$ points. If $u$ is any sequence of the form $a v a v' a$ such that $a$ is a letter, $v$ is a nonempty sequence excluding $a$ with no repeated letters and $v'$ is obtained from $v$ by only moving the first letter of $v$ to another place in $v$, then we show that $\mathit{fw}(u)=4$ and $\mathit{Ex}(u, n) =\Theta(n\alpha(n))$. Furthermore we prove that $\mathit{fw}(abc(acb)^{t})=2t+1$ and $\mathit{Ex}(abc(acb)^{t}, n) = n2^{\frac{1}{(t-1)!}\alpha(n)^{t-1}\pm O(\alpha(n)^{t-2})}$ for every $t\geq 2$.Comment: 25 page

Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms

We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic points. This settles Smale's tenth problem in the special case of area-preserving diffeomorphisms of closed surfaces.Comment: v4: A typo in the abstract is corrected and a few more expository changes are made. 68 page

Periodic points of rational area-preserving homeomorphisms

An area-preserving homeomorphism isotopic to the identity is rational if its rotation vector is a real multiple of a rational class. Rationality can always be achieved by $C^\infty$-small perturbations. We show that any area-preserving homeomorphism of a compact surface of genus at least two, which is isotopic to the identity and rational, is either the identity or has periodic points of unbounded minimal period. We show a similar result, using a theorem of Oxtoby-Ulam, when the map is only assumed to preserve a Borel probability measure with full support. We also discuss maps which are not isotopic to the identity and lower genus surfaces. The proofs of the main results rely on periodic Floer homology.Comment: 13 pages + reference

Symplectic dynamics: Invariant measures, closing lemmas, and equidistribution

In this thesis, we develop and apply techniques in symplectic geometry and gauge theory to study symplectic dynamical systems. The first part devises a method to construct invariant measures of Hamiltonian flows from pseudoholomorphic curves, and applies it to show broad classes of these systems are not uniquely ergodic. The second part studies quantitative invariants from Periodic Floer homology, in particular using Seiberg–Witten theory to give a precise accounting of their high-degree asymptotics. The main dynamical application is a proof that a generic area-preserving diffeomorphism of a compact surface has an equidistributed sequence of periodic orbits

Contact homology and the strong closing lemma for ellipsoids

We prove a conjecture of Irie, stating that a strong version of the smooth closing lemma holds for Reeb flows on ellipsoids in any dimension. The proof involves analyzing a constrained cobordism map in contact homology, using holomorphic intersection theory.Comment: 56 pages, 10 figures, comments welcome

Coincidences among skew Grothendieck polynomials

The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons

Kinetic analysis of a human chorionic gonadotropin- epitope-paratope interaction

Kinetics of protein-protein or ligand-ligate interaction has predominantly been studied by optical spectroscopy (particularly fluorescence) and surface plasmon resonance biosensors. Almost all such studies are based on association kinetics between ligand-ligate and suffer from certain methodological and interpretational limitations. Therefore, kinetic analyses of dissociation data of such interactions become indispensable. In the present investigation, the radiolabeled human chorionic gonadotropin- (125IhCG) was employed as a probe and nitrocellulose (NC) as a solid support to immobilize monoclonal antibody (MAb) G1G10.1. The NC-G1G10.1-125IhCG complex (NCcom) was prepared and the dissociation of radiolabeled hCG was carried out in the presence of excess unlabeled ligate. From the experimental dissociation data under varying ionic strength, dissociation constants (k-1), association constants (k+1) and affinity constants (ka) were calculated. The values obtained were utilized in exploring the amino acid residues constituting an epitopic region of hCG involved in interaction with the complementary paratope on MAb G1G10.1. Kinetic data of the present study supported our recently published findings [using single step-solid phase radioimmunoassay (SS-SPRIA)] that the core region of hCG epitope consists of Arg (94,95) and Asp (99) while a Lys (104) and a His (106) are in proximity to the core epitopic region. Based on the results of present investigation, we conclude that dissociation kinetics coupled with SS-SPRIA unequivocally provides considerable insight into the study of ligand-ligate interactions and epitope analysis

Evaluation of the free radical scavenging activities of ellagic acid and ellagic acid peracetate by epr spectrometry

The purpose of this study was to examine the free radical scavenging and antioxidant activities of ellagic acid (EA) and ellagic acid peracetate (EAPA) by measuring their reactions with the radicals, 2,2-diphenyl-1-picrylhydrazyl and galvinoxyl using EPR spectroscopy. We have also evaluated the influence of EA and EAPA on the ROS production in L-6 myoblasts and rat liver microsomal lipid peroxidation catalyzed by NADPH. The results obtained clearly indicated that EA has tremendous ability to scavenge free radicals, even at concentration of 1 µM. Interestingly even in the absence of esterase, EAPA, the acetylated product of EA, was also found to be a good scavenger but at a relatively slower rate. Kinetic studies revealed that both EA and EAPA have ability to scavenge free radicals at the concentrations of 1 µM over extended periods of time. In cellular systems, EA and EAPA were found to have similar potentials for the inhibition of ROS production in L-6 myoblasts and NADPH-dependent catalyzed microsomal lipid peroxidation

Molecular dissection of an hCG-β epitope using single-step solid phase radioimmunoassa

Background: Peptides and proteins have both sequence-specific (contiguous) and conformation-specific (discontiguous) epitopes. Sequence-specific epitopes are delineated by peptide approach and other robust methods like competition assays, gene expression assays, synthetic peptide library based assays etc. Available methods for delineation of conformation-specific epitopes are cumbersome (X-ray crystallography etc.), time consuming and require costly sophisticated equipments. Hence, there is a need to develop a simple method for identification and mapping of conformation-specific epitopes. Method: In the single-step solid phase radioimmunoassay (SS-SPRIA), an immunochemical bridge of ‘mouse IgG-anti-mouse IgG’ was prepared in the polypropylene wells followed by adsorption with hCG specific monoclonal antibody (MAb) $G_1G_{10.}1$. The extent of competitive inhibition in binding ability of $^{125}IhCG-β$ with chemically or enzymatically modified hCG-β to immobilized MAb $G_1G_{10.}1$ in comparison to hCG-β standards was utilized to identify the epitopic amino acid involved in epitope–paratope interaction. Results: Data clearly suggest that the epitope under investigation consisted of Arg (94, 95) and Asp (99) at the core region with a Lys (104) and a His (106) in the proximity and absence of chymotrypsin susceptible Phe or Tyr in this region. Conclusion: The data of SS-SPRIA revealed the 93–100 loop of amino acid sequence, as the core region of conformation-specific epitope of hCG-β at or near the receptor-binding region. Hence, SS-SPRIA seems to be a simple method for identification and mapping of conformation-specific epitopes

The Competence of 7,8-Diacetoxy-4-Methylcoumarinand other Polyphenolic Acetates in Mitigating the Oxidative Stress and their Role in Angiogenesis.

The potential role of polyphenolic acetate (PA) in causing diverse biological and pharmacological actions has been well studied in our laboratory. Our investigations, for the first time, established the role of calreticulin transacetylase (CRTAase) in catalyzing the acetylation of nitric oxide synthase (NOS) by Pas leading to robust activation of NOS. 7, 8-Diacetoxy-4-methylcoumarin (DAMC) and other acetoxycoumarins augmented the expression of thioredoxin (TRX) and vascular endothelial growth factor (VEGF) in human peripheral blood mononuclear cells (PBMCs). These findings substantiated our earlier observations that DAMC was a superb inducer of angiogenesis. The enhanced expression of thioredoxin reductase (TRXR) and diminished expression of thioredoxin interacting protein (TRXIP) leading to increased expression and activity of TRX in PBMCs due to the action of DAMC was revealed by real time RT-PCR analysis. The possible activation of TRX due to acetylation was confirmed by the fact that TRX activity of PBMCs was enhanced by variousacetoxycoumarins in tune with their affinities to CRTAase as substrates. DAMC caused enhanced production of NO by way of acetylation of NOS as mentioned above and thereby acted as an inducer of VEGF. Real time RT-PCR and VEGF ELISA results also revealed the overexpression of TRX. DAMC and other PAs were found to reduce the oxidative stress in cells as proved by significant reduction of intracellular ROS levels. Thus, the crucial role of TRX in DAMC-induced angiogenesis with the involvement of VEGF was established