One theme in noncommutative geometry is the recovery of classical geometric data from operator algebraic descriptions of various spaces, formalized through the concept of a spectral triple. This is defined below but, roughly speaking, consists of a C-algebra acting on a Hilbert space, giving co-ordinates for the space, and a Dirac operator, which encodes geometric information. As a prototypical...

Whole genome amplification (WGA) plays an important role in sample preparation of low-input templates for high-throughput sequencing. Multiple displacement amplification (MDA), a popular isothermal WGA methods, suffers a major hurdle of highly uneven amplification. Optimizations have been made in the past by separating the reagents into numbers of tiny chambers or droplets in microfluidic devic...

In this article, we propose a Partition ofUnity Refinement (PUR)method to improve the local approximations of elliptic boundary value problems in regions of interest. The PUR method only needs to refine the local meshes and hanging nodes generate no difficulty. The mesh qualities such as uniformity or quasi-uniformity are kept. The advantages of the PUR include its effectiveness and relatively ...

Given a Hausdorff uniform space $$X$$ with the countable gage of pseudometrics uniformity , we introduce concept approximate variation function $$f$$ mapping subset $$T$$ reals into : this is infimum family Jordan-type variations all functions $$g:T\to X$$ which differ from in each pseudometric, generated by pseudometric gage, not greater than $$\varepsilon>0$$ . We prove following compactness ...

In this paper we study regularity and topological properties of volume constrained minimizers quasi-perimeters in $\sf RCD$ spaces where the reference measure is Hausdorff measure. A quasi-perimeter a functional given by sum usual perimeter suitable continuous term. particular, isoperimetric sets are particular case our study. We prove that on an ${\sf RCD}(K,N)$ space $({\rm X},{\sf d},\mathca...

The original definition of a topological space given by Hausdorff used neighborhood systems. Lattice-valued maps appear in this context when you identify a topology with a monoid in the Kleisli category of the filter monad on SET. H?hle’s notion of a lattice-valued topology [2] uses the same idea and it’s inspired in the classical lattice-valued topologies. Ltopological spaces are motivated by ...

Given a Banach space we consider the $\sigma$-ideal of all its subsets which are covered by countably many hyperplanes and investigate standard cardinal characteristics as additivity, covering number, uniformity, cofinality. We determine their values for separable spaces, approximate them nonseparable spaces. The remaining questions reduce to deciding if following can be proved in ZFC (i.e. Zer...

s — In this paper, we extend the techniques used in Grid-based Monte Carlo applications to Grid-based quasi-Monte Carlo applications. These techniques include an N-out-of-M strategy for efficiently scheduling subtasks on the Grid, lightweight checkpointing for Grid subtask status recovery, a partial result validation scheme to verify the correctness of each individual partial result, and an int...