We introduce a parameter space for periodic point sets, given as a union of m translates of a point lattice. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a denser periodic sphere packing. This in particular ...

We analyze ‘Distribution to Distribution regression’ where one is regressing a mapping where both the covariate (inputs) and response (outputs) are distributions. No parameters on the input or output distributions are assumed, nor are any strong assumptions made on the measure from which input distributions are drawn from. We develop an estimator and derive an upper bound for the L2 risk; also,...

human was created from both the natural and spiritual dimensions and has multimodal functions (e.g., cognitive, behavioral, affective, sensation, interpersonal relationship, and etc.). having one-dimensional impression in life would provide anxiety predisposition for man, but then having an integrated and monotheistic view would result in an assimilated psychological discipline for him/her. the...

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form “is x < y?”. This application motivates looking for stronger notions of dimensi...