Non-Recursive Base Conversion Using a Deterministic Markov Process

We prove that non-recursive base conversion can always be implemented by using a deterministic Markov process. Our paper discusses the pros and cons of recursive and non-recursive methods, in general. And we include a comparison between non-recursion and a deterministic Markov process, proving that the Markov process is twice as efficient.

Limit Cycle for the RC Circuit with Square-Wave-Type Voltage

The goal of this paper is to prove, by means of calculations the existence of the ω-limit cycle of a simple circuit where a resistor, a capacitor and a square-wave-type voltage source are in series.

Harmony Is Cause—Not Consequence—Of the Quantum

Einstein claimed Bohr’s theory is incomplete: “the wave function does not provide a complete description of the physical reality” [1]. Their views represent two physics in schism [2] [3]. Quanta are fundamental. The theory of diffraction in quasicrystals, that is summarized here, is falsifiable and verified. The quanta are not only harmonic;but harmonic in dual series: geometric and linear. Many have believed the quantum is real;rather than conceptual and axiomatic. The quasicrystal proves its reality.

Physical Quanta in Quasicrystal Diffraction

Diffraction in quasicrystals is in irrational and geometric series with icosahedral point group symmetry. None of these features are allowed in Bragg diffraction, so a special theory is required. By means of a hierarchic model, the present work displays exact agreement between an analytic metric, with a numeric description of diffraction in quasicrystals—one that is founded on quasi-structure-factors that are completely indexed in 3-dimensions. At the quasi-Bragg condition, the steady state wave function of incident radiation is used to show how resonant response, in metrical space and time, enables coherent interaction between the periodic wave packet and hierarchic quasicrystal. The quasi-Bloch wave is invariant about all translations , where is the quasi-lattice parameter. This is numerically derived, analyzed, measured, verified and complete. The hierarchic model is mapped in reverse density contrast, and matches the pattern and dimensions of phase-contrast, optimum-defocus images. Four tiers in the hierarchy of icosahedra are confirmed, along with randomization of higher order patterns when the specimen foil is oriented only degrees off the horizontal. This explains why images have been falsely described as having “no translational symmetry”.

Quasicrystals’ Resonant Response with Translational Symmetry

Diffraction in quasicrystals is in logarithmic order and icosahedral point group symmetry. Neither of these features are allowed in Bragg diffraction, so a special theory is required. The present work displays exact agreement between the analytic metric with a numeric description of diffraction in quasicrystals that is based on quasi-structure factors. So far, we treated the hierarchic structure as ideal;now, we detail the theory by including two significant features: firstly, the steady state wave function of the incident radiation demonstrates how harmonics, in metrical space and time, enable coherent interaction between the periodic wave packet and hierarchic quasicrystal;secondly, mapping of the hierarchic structure for any influence of defects will allow estimation of possible error margins in the analysis. The hierarchic structure has the required logarithmic periodicity: superclusters, containing about 103 atoms, convincingly map phase contrast images;while higher orders leave space for subsidiary speculation. The diffraction is completely explained for the first time.

Real Quanta and Continuous Reduction

Previous theories of quasicrystal diffraction have called it “Bragg diffraction in Fibonacci sequence and 6 dimensions”. This is a misnomer, because quasicrystal diffraction is not in integral linear order n where nλ= 2dsin(θ) as in all crystal diffraction;but in irrational, geometric series τm, that are now properly indexed, simulated and verified in 3 dimensions. The diffraction is due not to mathematical axiom, but to the physical property of dual harmony of the probe, scattering on the hierarchic structure in the scattering solid. By applying this property to the postulates of quantum theory, it emerges that the 3rd postulate (continuous and definite) contradicts the 4th (instantaneous and indefinite). The latter also contradicts Heisenberg’s “limit”. In fact, the implied postulates of probability amplitude describe hidden variables that are universally recognized, in all sensitive measurement, by records of error bars. The hidden variables include momentum quanta, in quasicrystal diffraction, that are continuous and definite. A revision of the 4th postulate is proposed.

The Reciprocal Lattice in Hierarchic Quasicrystals

Initially, all that was known about diffraction in quasicrystals was its point group symmetry;nothing was known about the mechanism. The structure was more evident, and was called quasiperiodic. From mapping the Mn atoms by phase-contrast, optimum-defocus, electron microscopy, the progress towards identifying unit cell, cluster, supercluster and extensive hierarchic structure is evident. The structure is ordered and uniquely icosahedral. From the known structure, we could calculate structure factors. They were all zero. The quasi structure factor is an iterative procedure on the hierarchic structure that correctly calculates diffraction beam intensities in 3-dimensional space. By a creative device, the diffraction is demonstrated to occur off the Bragg condition;the quasi-Bragg condition implies a metric that enables definition and measurement of the lattice constant. The reciprocal lattice is the 3-dimensional diffraction pattern. Typically, it builds on Euclidean axes with coordinates in geometric series, but it also transforms to Cartesian coordinates.

A Nonclassical LIL for Geometrically Weighted Series in Banach Space

We consider efficient methods for the recovery of block sparse signals from underdetermined system of linear equations. We show that if the measurement matrix satisfies the block RIP with δ2s 〈 0.4931, then every block s-sparse signal can be recovered through the proposed mixed l2/ll-minimization approach in the noiseless case and is stably recovered in the presence of noise and mismodeling error. This improves the result of Eldar and Mishali （in IEEE Trans. Inform. Theory 55： 5302-5316, 2009）. We also give another sufficient condition on block RIP for such recovery method： 58 〈 0.307.

A Self-normalized Law of the Iterated Logarithm for the Geometrically Weighted Random Series

Let {X, Xn; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX^2I（｜X｜ ≤ x） is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞（n=0）β~nXn（0 〈 β 〈 1） is obtained, under some minimal conditions.