The Heisenberg group is an example of a sub-Riemannian manifold homeomorphic, but not bi-Lipschitz equivalent to the Euclidean space. Its metric is derived from curves which are only allowed to move in so-called
It is appropriate in this year, when we celebrate the 50th anniversary of quantum mechanics, and during which we have been saddened by the death of one of its leading founders, Werner Heisenberg, to reminisce about the formative years of the new mechanics. At the time when the foundations of...
where V is the system’s volume, kB is Boltzmann’s constant, T is the temperature, \(\widehat{J}\) is any Cartesian component of the energy-flux operator in the Heisenberg representation, and isotropy is assumed throughout this paper, thus allowing us to dispose of Cartesian indices unles...
(Mr. Heisenberg was specifically referring to the momentum and location of subatomic particles.) Performance Monitor, when recording lots of data over short intervals, can impact the performance of a Windows NT system. The Perfmon utility uses CPU, memory, and disk I/O. If you monitor the...
We report on the closing of gaps in the ground state of the critical Heisenberg-Ising chain at momentum $\pi$. For half-filling, the gap closes at special values of the anisotropy $\Delta= \cos(\pi/Q)$, $Q$ integer. We explain this behavior with the help of the Bethe Ansatz and ...
Werner Heisenberg said the new state is indeterministic, with possibilities, one of which comes randomly into "existence." Einstein hoped for a return to classical deterministic physics, in which objects have a "real" existence before we measure them. ...
Apart from mass, another important property of a particle is its lifetime,τ. Only a few fundamental particles are stable. Others—including the H boson—exist only for a fleeting moment before disintegrating into other, lighter, species. The Heisenberg uncertainty principle7provides a direct connec...
such as complex analysis, that I’ve never taken. Meanwhile, a debate about the liberal arts has been raging across the nation. Debt is weighing down recent graduates, and high-school students are loading up on STEMM courses. Colleges are cutting liberal-arts departments, and educational organiz...
van Eeden FJ, Granato M, Schach U, Brand M, Furutani-Seiki M, Haffter P, Hammerschmidt M, Heisenberg CP, Jiang YJ, Kane DA, Kelsh RN, Mullins MC, Odenthal J, Warga RM, Allende ML, Weinberg ES, Nusslein-Volhard C: Mutations affecting somite formation and patterning in the zebrafish,...
1Introduction Cosmological perturbations have shown the possibility of opening a pathway to revealing the cosmological evolution of the Universe in General Relativity (GR) and crucially in theories beyond GR [1,2,3]. The results of perturbations analysis can then be used in the confrontation with ...