Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

In this paper we consider vector fields in R3 that are invariant under a suitable symmetry and that posses a “generalized heteroclinic loop” L formed by two singular points (e+ and e −) and their invariant manifolds: one of dimension 2 (a sphere minus the points e+ and e −) and one of dimension 1 (the open diameter of the sphere having endpoints e+ and e −). In particular, we analyze the dynamics of the vector field near the heteroclinic loop L by means of a convenient Poincar´e map, and we prove the existence of infinitely many symmetric periodic orbits near L. We also study two families of vector fields satisfying this dynamics. The first one is a class of quadratic polynomial vector fields in R3, and the second one is the charged rhomboidal four body problem.

Orthogonal protection of peptides and peptoids for cyclization by the thiol-ene reaction and conjugation

Cyclic peptides and peptoids were prepared using the thiolene Michael-type reaction. The linear precursors were provided with additional functional groups allowing for subsequent conjugation: an orthogonally protected thiol, a protected maleimide, or an alkyne. The functional group for conjugation was placed either within the cycle or in an external position. The click reactions employed for conjugation with suitably derivatized nucleoside or oligonucleotides were either cycloadditions (Diels-Alder, Cu(I)-catalyzed azide-alkyne) or the same Michael-type reaction as for cyclization.

Synchronization of symmetric chaotic systems

This paper contains a study of the synchronization by homogeneous nonlinear driving of systems that are symmetric in phase space. The main consequence of this symmetry is the ability of the response to synchronize in more than just one way to the driving systems. These different forms of synchronization are to be understood as generalized synchronization states in which the motions of drive and response are in complete correlation, but the phase space distance between them does not converge to zero. In this case the synchronization phenomenon becomes enriched because there is multistability. As a consequence, there appear multiple basins of attraction and special responses to external noise. It is shown, by means of a computer simulation of various nonlinear systems, that: (i) the decay to the generalized synchronization states is exponential, (ii) the basins of attraction are symmetric, usually complicated, frequently fractal, and robust under the changes in the parameters, and (iii) the effect of external noise is to weaken the synchronization, and in some cases to produce jumps between the various synchronization states available

Orthogonal protecting groups in the synthesis of tryptophanyl-hexahydropyrroloindoles

The synthesis of various polycyclic systems containing a C3aNi bond between a hexahydropyrrolo[2,3-b]indole and an indole tryptophan is described here. A series of experiments were performed to determine the best combination of five orthogonal protecting groups and the best reaction conditions for formation of said bond, which is a common feature among many recently discovered marine natural products.

Coifman-aallokkeet ja skaalaussuotimen muodostus differentiaalievoluut ion avulla

Ortogonaalisen M-kaistaisen moniresoluutioanalyysin matemaattiset perusteet esitetään yksityiskohtaisesti. Coifman-aallokkeiden määritelmä yleistetään dilaatiokertoimelle M ja nollasta poikkeavalle häviävien momenttien keskukselle.Funktion approksimointia näytepisteistä aallokkeiden avulla pohditaan ja erityisesti esitetään approksimaation asymptoottinen virhearvio Coifman-aallokkeille. Skaalaussuotimelle osoitetaan välttämättömät ja riittävät ehdot, jotka johtavat yleistettyihin Coifman-aallokkeisiin. Moniresoluutioanalyysin tiheys todistetaansuoraan Lebesguen integraalin määritelmään perustuen yksikön partitio-ominaisuutta käyttäen. Todistus on riittävä sellaisenaan avaruudessa L2(Wd) käyttämättä Fourier-tason ominaisuuksia tai ehtoja. Mallatin algoritmi johdetaan M-kaistaisille aallokkeille ja moniuloitteisille signaaleille. Algoritmille esitetään myös rekursiivinen muoto. Differentiaalievoluutioalgoritmin avulla ratkaistaan Coifman-aallokkeisiin liittyvien skaalaussuotimien kertoimien arvoja useille skaalausfunktiolle. Approksimaatio- ja kuvanpakkausesimerkkejä esitetään menetelmien havainnollistamiseksi. Differentiaalievoluutioalgoritmin avulla etsitään myös referenssikuville optimoitu skaalaussuodin. Löydetty suodin on regulaarinen ja erittäinsymmetrinen.

Convergent Approaches for the Synthesis of the Anti-tumoral Peptide, Kahalalide F. Study of Orthogonal Protecting Groups

Kahalalide compounds are peptides that are isolated from a Hawaiian herbivorous marine species of mollusc, Elysia rufescens, and its diet, the green alga Bryopsis sp. Kahalalide F and its synthetic analogues are the most promising compounds of the Kahalalide family because they show anti-tumoral activity. Linear solid-phase syntheses of Kahalalide F have been reported. Here we describe several new improved synthetic routes based on convergent approaches with distinct orthogonal protection schemes for the preparation of Kahaladide analogues. These strategies allow a better control and characterization of the intermediates because more reactions are performed in solution. Five derivatives of Kahalalide F were synthesized using several convergent approaches.

From symmetric glycerol derivatives to dissymmetric chlorohydrins

The anticipated worldwide increase in biodiesel production will result in an accumulation of glycerol for which there are insufficient conventional uses. The surplus of this by-product has increased rapidly during the last decade, prompting a search for new glycerol applications. We describe here the synthesis of dissymmetric chlorohydrin esters from symmetric 1,3-dichloro-2-propyl esters obtained from glycerol. We studied the influence of two solvents: 1,4-dioxane and 1-butanol and two bases: sodium carbonate and 1-butylimidazole, on the synthesis of dissymmetric chlorohydrin esters. In addition, we studied the influence of other bases (potassium and lithium carbonates) in the reaction using 1,4-dioxane as the solvent. The highest yield was obtained using 1,4-dioxane and sodium carbonate.

Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.

Orthogonal protecting groups in the synthesis of tryptophanyl-hexahydropyrroloindoles

The synthesis of various polycyclic systems containing a C3a-Ni bond between a hexahydropyrrolo[2,3-b]indole and an indole tryptophan is described here. A series of experiments were performed to determine the best combination of five orthogonal protecting groups and the best reaction conditions for formation of said bond, which is a common feature among many recently discovered marine natural products.

The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$.

Symmetric or asymmetric gasoline prices? A meta-analysis approach

The analysis of price asymmetries in the gasoline market is one of the most studied in the energy economics literature. Nevertheless, the great variability of results makes it very difficult to extract conclusive results on the existence or not of asymmetries. This paper shows through a meta-analysis approach how the industry segment analysed, the quality and quantity of data, the estimator and the model used may explain this heterogeneity of results.

Convergent Approaches for the Synthesis of the Anti-tumoral Peptide, Kahalalide F. Study of Orthogonal Protecting Groups

Kahalalide compounds are peptides that are isolated from a Hawaiian herbivorous marine species of mollusc, Elysia rufescens, and its diet, the green alga Bryopsis sp. Kahalalide F and its synthetic analogues are the most promising compounds of the Kahalalide family because they show anti-tumoral activity. Linear solid-phase syntheses of Kahalalide F have been reported. Here we describe several new improved synthetic routes based on convergent approaches with distinct orthogonal protection schemes for the preparation of Kahaladide analogues. These strategies allow a better control and characterization of the intermediates because more reactions are performed in solution. Five derivatives of Kahalalide F were synthesized using several convergent approaches.

Orthogonal function method in modelling paper web quality variation

This work is devoted to the analysis of signal variation of the Cross-Direction and Machine-Direction measurements from paper web. The data that we possess comes from the real paper machine. Goal of the work is to reconstruct the basis weight structure of the paper and to predict its behaviour to the future. The resulting synthetic data is needed for simulation of paper web. The main idea that we used for describing the basis weight variation in the Cross-Direction is Empirical Orthogonal Functions (EOF) algorithm, which is closely related to Principal Component Analysis (PCA) method. Signal forecasting in time is based on Time-Series analysis. Two principal mathematical procedures that we used in the work are Autoregressive-Moving Average (ARMA) modelling and Ornstein–Uhlenbeck (OU) process.

Instability of a square sheet under symmetric biaxial loading

An early experiment found that a square rubber sheet under symmetric biaxial loading may not remain square. This curious result has been one of the most instructive examples in finite elasticity. Here thermodynamic considerations are used to analyze this instability.