Symmetric periodic orbits near a heteroclinic loop in R3 formed by two singular points, a semistable periodic orbit and their invariant manifolds

In this paper we consider C1 vector fields X in R3 having a “generalized heteroclinic loop” L which is topologically homeomorphic to the union of a 2–dimensional sphere S2 and a diameter connecting the north with the south pole. The north pole is an attractor on S2 and a repeller on . The equator of the sphere is a periodic orbit unstable in the north hemisphere and stable in the south one. The full space is topologically homeomorphic to the closed ball having as boundary the sphere S2. We also assume that the flow of X is invariant under a topological straight line symmetry on the equator plane of the ball. For each n ∈ N, by means of a convenient Poincar´e map, we prove the existence of infinitely many symmetric periodic orbits of X near L that gives n turns around L in a period. We also exhibit a class of polynomial vector fields of degree 4 in R3 satisfying this dynamics.

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.

A Generalized Model Study of Scour Around Bridge Piers and Abutments, HR-24, 1953

Four classes of variables are apparent in the problem of scour around bridge piers and abutments--geometry of piers and abutments, stream-flow characteristics, sediment characteristics, and geometry of site. The laboratory investigation, from its inception, has been divided into four phases based on these classes. In each phase the variables in three of the classes are held constant and those in the pertinent class are varied. To date, the first three phases have been studied. Typical scour bole patterns related to the geometry of the pier or abutment have been found. For equilibrium conditions of scour with uniform sand, the velocity of flow and the sand size do not appear to have any measurable effects on the depth of scour. This result is especially encouraging in the search for correlation between model and prototype since it would indicate that, primarily, only the depth of flow might be involved in the scale effect. The technique of model testing has been simplified, therefore, because rate of sediment transportation does not need to be scaled. Prior to the establishment of equilibrium conditions, however, depths of scour in excess of those for equilibrium conditions have been found. A concept of active scour as an imbalance between sediment transport capacity and rate of sediment supply has been used to explain the laboratory observations.

Fourier transform convolution integrals applied to generalized Born molecular volume.

Generalized Born methods are currently among the solvation models most commonly used for biological applications. We reformulate the generalized Born molecular volume method initially described by (Lee et al, 2003, J Phys Chem, 116, 10606; Lee et al, 2003, J Comp Chem, 24, 1348) using fast Fourier transform convolution integrals. Changes in the initial method are discussed and analyzed. Finally, the method is extensively checked with snapshots from common molecular modeling applications: binding free energy computations and docking. Biologically relevant test systems are chosen, including 855-36091 atoms. It is clearly demonstrated that, precision-wise, the proposed method performs as good as the original, and could better benefit from hardware accelerated boards.

Performance Analysis of Two Quantum Reaction Dynamics Codes: Time-Dependent and Time-Independent Strategies

The computer simulation of reaction dynamics has nowadays reached a remarkable degree of accuracy. Triatomic elementary reactions are rigorously studied with great detail on a straightforward basis using a considerable variety of Quantum Dynamics computational tools available to the scientific community. In our contribution we compare the performance of two quantum scattering codes in the computation of reaction cross sections of a triatomic benchmark reaction such as the gas phase reaction Ne + H2+ %12. NeH++ H. The computational codes are selected as representative of time-dependent (Real Wave Packet [ ]) and time-independent (ABC [ ]) methodologies. The main conclusion to be drawn from our study is that both strategies are, to a great extent, not competing but rather complementary. While time-dependent calculations advantages with respect to the energy range that can be covered in a single simulation, time-independent approaches offer much more detailed information from each single energy calculation. Further details such as the calculation of reactivity at very low collision energies or the computational effort related to account for the Coriolis couplings are analyzed in this paper.

Synthesis of F-18-labeled cyclic-[RGDfK] peptides for PET imaging of angiogenesis

Objectives: αvβ3 integrin is of great interest for tumor targeting because of its high concentration in tumor tissue. It recognizes ligands containing an arginine-glycine-aspartate motif (RGD), and a number of RGD-containing peptides have been developed as PET imaging probes of angiogenesis. We synthesized a series of 18F-labeled cyclic-[RGDfK] peptides for in vivo imaging of αvβ3 expression. Our F-18 labeled prosthetic groups were attached to the αvβ3 ligand via click chemistry, and the reaction conditions (time, temperature, solvent and pH) were optimized by using single modified amino acids.Methods: Seven amino acids were selected considering their different biochemical properties (polarity, total charge, presence of aromatic ring and heteroatom). All the amino acids were modified by the introduction of azido moiety to allow the interaction with alkyne prosthetic groups. Once the conditions of the click chemistry were optimized, the prosthetic groups were also coupled with the cyclic-[RGDfK] exhibiting an azido function. 4- Trimethylammonium-nitrobenzene triflate was used as precursor for the radiosynthesis of the prosthetic groups. The fluorination was carried out with K2CO3/K2.2.2 in CH3CN at 95 oC, and the nitro group was reduced with NaBH4 and Pd/C in MeOH. The resulting 18F-aniline was subsequently coupled to alkynoic acids to yield the final F-18 labeled prosthetic groups. Finally, the prosthetic groups were attached to the peptides via Huisgen's cycloaddition. Figure 1. F-18 labeled αvβ3 ligand.Results: Our new prosthetic groups were successfully clicked to the modified amino acids and to the cyclic- [RGDfK], and the reactions were almost quantitative within 1 to 3.5 h. The pH of the reaction did not influence the reaction kinetic and yield. The four steps of the F-18 labeling were completely automated providing the final products in quantities and yields practical for PET imaging. IC50 values of our ligands for αvβ3 and α5β1 demonstrated a high selectivity of our compounds towards αvβ3, as well as the negligible effect of the prosthetic groups on the affinity of the ligand to its receptor, as confirmed by the prediction of the molecular modeling.Conclusions: We have successfully synthesized novel F-18 labeled prosthetic groups, as well as novel PET imaging probes of αvβ3 expression. The reaction conditions of the Huisgen's cycloaddition were optimized with selected modified amino acids, and subsequently transposed to the cyclic-[RGDfK] peptide. IC50 data demonstrate that our 18F-labeled ligands were selective for αvβ3. In vivo microPET/CT studies in tumor bearing mice are underway.