Uniform Convexity and Convergence of a Sequence of Sets in a Complete Geodesic Space

In this paper, we first introduce two new notions of uniform convexity on a geodesic space, and we prove their properties. Moreover, we reintroduce a concept of the set-convergence in complete geodesic spaces, and we prove a relation between the metric projections and the convergence of a sequence of sets.

Presynaptic developmental plasticity allows robust sparse wiring of the Drosophila mushroom body

In order to represent complex stimuli, principle neurons of associative learning regions receive combinatorial sensory inputs. Density of combinatorial innervation is theorized to determine the number of distinct stimuli that can be represented and distinguished from one another, with sparse innervation thought to optimize the complexity of representations in networks of limited size. How the convergence of combinatorial inputs to principle neurons of associative brain regions is established during development is unknown. Here, we explore the developmental patterning of sparse olfactory inputs to Kenyon cells of the Drosophila melanogaster mushroom body. By manipulating the ratio between pre- and post-synaptic cells, we find that postsynaptic Kenyon cells set convergence ratio: Kenyon cells produce fixed distributions of dendritic claws while presynaptic processes are plastic. Moreover, we show that sparse odor responses are preserved in mushroom bodies with reduced cellular repertoires, suggesting that developmental specification of convergence ratio allows functional robustness.

Basis set convergence of binding energy with and without CP-correction utilizing PBE0 method: A benchmark study of X2 (X=Ge, As, Se, Sc, Ti, V, Cr, Mn, Co, Cu, Zn)

A DFT study of homonuclear X2 ([Formula: see text], As, Se, Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Zn) is presented using PBEO exchange (xc) functional which is a mixing of Perdew–Burke–Ernzerhof (PBE) and Hartree Fock (HF) exchange energy. However, we used cc-pVXZ and aug-cc-pVXZ basis sets where X is maximum angular momentum number in basis set. Convergence pattern of binding energy with respect to basis set was observed. Two-point extrapolations to complete basis set (CBS) limit were applied to speed up convergence and decrease the basis set incompleteness error (BSIE). Counterpoise correction (CP) method was utilized to alleviate basis set superposition errors (BSSE). Both CP-corrected and uncorrected binding energies were obtained and compared with the experimental and theoretical binding energy values in literature.

Presynaptic developmental plasticity allows robust sparse wiring of the Drosophila mushroom body

In order to represent complex stimuli, principle neurons of associative learning regions receive combinatorial sensory inputs. Density of combinatorial innervation is theorized to determine the number of distinct stimuli that can be represented and distinguished from one another, with sparse innervation thought to optimize the complexity of representations in networks of limited size. How the convergence of combinatorial inputs to principle neurons of associative brain regions is established during development is unknown. Here, we explore the developmental patterning of sparse olfactory inputs to Kenyon cells of the Drosophila melanogaster mushroom body. By manipulating the ratio between pre- and post-synaptic cells, we find that postsynaptic Kenyon cells set convergence ratio: Kenyon cells produce fixed distributions of dendritic claws while presynaptic processes are plastic. Moreover, we show that sparse odor responses are preserved in mushroom bodies with reduced cellular repertoires, suggesting that developmental specification of convergence ratio allows functional robustness.

Distributed Constraint Optimization with Flocking Behavior

This paper studies distributed optimization having flocking behavior and local constraint set. Multiagent systems with continuous-time and second-order dynamics are studied. Each agent has a local constraint set and a local objective function, which are known to only one agent. The objective is for multiple agents to optimize a sum of the local functions with local interaction and information. First, a bounded potential function to construct the controller is given and a distributed optimization algorithm that makes a group of agents avoid collisions during the evolution is presented. Then, it is proved that all agents track the optimal velocity while avoiding collisions. The proof of the main result is divided into three steps: global set convergence, consensus analysis, and optimal set convergence. Finally, a simulation is included to illustrate the results.