New Solvable Variants of the Goldfish Many‐Body Problem

A technique to manufacture solvable variants of the “goldfish” many‐body problem is introduced, and several many‐body problems yielded by it are identified and discussed, including cases featuring multiperiodic or isochronous dynamics.     

More,or less,trivial matrix functional equations

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1) case, such as \({\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }\); others—such as \({\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }\)—possess nontrivial solutions only in the matrix case (N > 1), namely their scalar (N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two (N × N)-matrices \({\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}\) satisfy the triplet of functional equations written above—and nontrivial examples of such matrices are exhibited— then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read \({\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }\).     

Globally generated vector bundles on the Segre threefold with Picard number two

We classify globally generated vector bundles on with small first Chern class, i.e. , . Our main method is to investigate the associated smooth curves to globally generated vector bundles via the Hartshorne–Serre correspondence.     

On topological filtrations of groups

The (weak) geometric simple connectivity and the quasi-simple filtration are topological notions of manifolds, which may be defined for discrete groups too. It turns out that they are equivalent for finitely presented groups, but the main problem is the absence of examples of groups which do not satisfy them. In this note we study some algebraic classes of groups with respect to these properties.     

Novel solvable many-body problems

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type (“acceleration equals forces”) which determine the motion of points in the complex plane. These models are solvable, namely their con?guration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena.     

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N “particles”) at any later time t can be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N² arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.     

Amplification phenomena of Casimir force fluctuations on close scatterers coupled via a coherent fermionic fluid

We study the mechanical actions affecting close scatterers immersed in a coherent fermionic fluid. Using a scattering field theory, we theoretically analyse the single-scatterer and the two-scatterer case. Concerning the single-scatterer case, we find that a net force affects the scatterer dynamics only in non-equilibrium condition, i.e. imposing the presence of a non-vanishing particle current flowing through the system. The force fluctuation (variance) is instead not negligible both in equilibrium and in non-equilibrium conditions. Concerning the two-scatterer case, an attractive fluid-mediated Casimir force is experienced by the scatterers at small spatial separation, while a decaying attractive/repulsive behavior as a function of the scatterer separation is found. Furthermore, the Casimir force fluctuations acting on a given scatterer in close vicinity of the other present an oscillating behavior reaching a long distance limit comparable to the noise level of the single-scatterer case. The relevance of these findings is discussed in connection with fluctuation phenomena in low-dimensional nanostructures and cold atoms systems.     

Screening of matrix metalloproteinases available from the protein data bank: insights into biological functions, domain organization, and zinc binding groups

A total of 142 matrix metalloproteinase (MMP) X-ray crystallographic structures were retrieved from the Protein Data Bank (PDB) and analyzed by an automated and efficient routine, developed in-house, with a series of bioinformatic tools. Highly informative heat maps and hierarchical clusterograms provided a reliable and comprehensive representation of the relationships existing among MMPs, enlarging and complementing the current knowledge in the field. Multiple sequence and structural alignments permitted better location and display of key MMP motifs and quantification of the residue consensus at each amino acid position in the most critical binding subsites of MMPs. The MMP active site consensus sequences, the C-alpha root-mean-square deviation (RMSd) analysis of diverse enzymatic subsites, and the examination of the chemical nature, binding topologies, and zinc binding groups (ZBGs) of ligands extracted from crystallographic complexes provided useful insights on the structural arrangements of the most potent MMP inhibitors.     

Enthalpy/entropy driven activation of the first interquinone electron transfer in bacterial photosynthetic reaction centers embedded in vesicles of physiologically important phospholipids

The thermodynamics and kinetics of light-induced electron transfer in bacterial photosynthetic RCs are sensitive to physiologically important lipids (phosphatidylcholine, cardiolipin and phosphatidylglycerol) in the environment. The analysis of the temperature-dependence of the rate of the P(+)Q(A)(-)Q(B)-->P(+)Q(A)Q(B)(-) interquinone electron transfer revealed high enthalpy change of activation in zwitterionic or neutral micelles and vesicles and low enthalpy change of activation in vesicles constituted of negatively charged phospholipids. The entropy change of activation was compensated by the changes of enthalpy, thus the free energy change of activation ( approximately 500 meV) did not show large variation in vesicles of different lipids.     

Fullerenes: multitask components in molecular machinery

Molecular machines are molecular-scale devices that carry out predetermined tasks derived from molecular motion. This Minireview illustrates how fullerenes can be used as multitask building blocks in molecular machinery, providing new perspectives for fullerenes. Indeed, C(60) can be applied as a photo- and electroactive stopper owing to its size, as a probe for molecular motion as a result of its well-defined physicochemical properties, and to induce motion through pi-pi interactions. Such molecular motion can be employed to modulate light-driven electron-transfer events, extending the potential applications of molecular machines to the typical fields of application of fullerenes.