Floating point expansion is a technique for implementing multiple precision using a processor's oating point unit instead of its integer unit. Research on this subject has arised recently from the observation that the oating point unit becomes a more and more eÆcient part of modern computers. Many simple arithmetic operators and some very useful geometric operators have already been presented o...

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues {λk} that are analogous to those known for Schrödinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These...

This paper is part of an eight paper project [14]-[20] studying the arithmetic mean operator ideals in B(H) introduced by Dykema, Figiel, Weiss and Wodzicki in [10]. Every ideal I is generated by diagonal operators with positive decreasing sequences, and its arithmetic mean ideal Ia is generated by diagonal operators with the arithmetic means of those sequences. In this paper we focus on lattic...

The idea of bipolar complex fuzzy (BCF) sets, as a genuine modification both sets and gives massive valuable framework for representing evaluating ambiguous information. In intelligence decision making based on BCF it is critical dilemma to compare or rank positive negative membership grades. this framework, we deliberated various techniques aggregating the collection information into singleton...

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field of positive characteristic‎, ‎and let $e/f$ be a galois extension‎. ‎we study the realization fields of‎ ‎finite subgroups $g$ of $gl_n(e)$ stable under the natural‎ ‎operation of the galois group of $e/f$‎. ‎though for sufficiently large $n$ and a fixed‎ algebraic number field $f$ every its finite extension $e$ is‎ ‎re...

Brains and computers are both dynamical systems that manipulate symbols, but they differ fundamentally in their architectures and operations. Human brains do mathematics; computers do not. Computers manipulate symbols that humans put into them without grounding them in what they represent. Human brains intentionally direct the body to make symbols, and they use the symbols to represent internal...

Accelerating the execution of algorithms involving floating-point computations is currently a necessity in the scientific community. A solution – FPGAs – are believed to provide the best balance between costs, performance and flexibility. The FPGA’s flexibility can be best exploited when used to accelerate ”exotic operators”(log, exp, dot product) and operators tailored for the numerics of each...