Arbitrary Large Numbers Research Articles

Variational analysis of driven-dissipative bosonic fields

We present a method to perform a variational analysis of the quantum master equation for driven-disspative bosonic fields with arbitrary large occupation numbers. Our approach combines the P representation of the density matrix and the variational principle for open quantum system. We benchmark the method by comparing it to wave-function Monte-Carlo simulations and the solution of the Maxwell-Bloch equation for the Jaynes-Cummings model. Furthermore, we study a model describing Rydberg polaritons in a cavity field and introduce an additional set of variational paramaters to describe correlations between different modes.

Some remarks on the orbit of closure-involution operations on languages

For a closure operation c and an involution i defined on a language family M, we define Nc,iM(L) as the number of distinct languages which can be obtained from a language L by repeated applications of c and i. The orbit OM(c,i) of c and i is defined as the set of all these numbers.We show that for the families of all languages and all regular languages over some alphabet and some classical closure operators c on languages, there are involutions i such that the orbit OM(c,i) contains arbitrary large numbers or contains infinity (which is in contrast to Kuratowski's complement-closure theorem). Moreover, we show that it is undecidable whether infinity occurs in the orbit.

Collective Strategy Condensation: When Envy Splits Societies.

Human societies are characterized by three constituent features, besides others. (A) Options, as for jobs and societal positions, differ with respect to their associated monetary and non-monetary payoffs. (B) Competition leads to reduced payoffs when individuals compete for the same option as others. (C) People care about how they are doing relatively to others. The latter trait—the propensity to compare one’s own success with that of others—expresses itself as envy. It is shown that the combination of (A)–(C) leads to spontaneous class stratification. Societies of agents split endogenously into two social classes, an upper and a lower class, when envy becomes relevant. A comprehensive analysis of the Nash equilibria characterizing a basic reference game is presented. Class separation is due to the condensation of the strategies of lower-class agents, which play an identical mixed strategy. Upper-class agents do not condense, following individualist pure strategies. The model and results are size-consistent, holding for arbitrary large numbers of agents and options. Analytic results are confirmed by extensive numerical simulations. An analogy to interacting confined classical particles is discussed.

Periods of holomorphic maps on compact Riemann surfaces and product of spheres

ABSTRACT In this article, we consider non-constant holomorphic maps on Riemann surfaces and product of Riemann spheres, and we give conditions on the maps in order that they have arbitrary large prime numbers as periods. We use Lefschetz fixed point theory and in particular we compute the Lefschetz numbers of period m for large m's.

Immersed concordance of links and Heegaard Floer homology

An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying $d$-invariant inequalities for this cobordism we obtain inequalities between the $H$-functions of links, which can be extracted from the link Floer homology package. As an application we show a Heegaard Floer theoretical criterion for bounding the splitting number of links. The criterion is especially effective for L-space links, and we present an infinite family of L-space links with vanishing linking numbers and arbitrary large splitting numbers. We also show a semicontinuity of the $H$-function under $\delta$-constant deformations of singularities with many branches.

Electromagnetic field fluctuations and uncertainty relation for general quantum superpositions of coherent states

Quantum statistical properties of general superpositions of coherent states of the type have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude and phase superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.

$C^{\infty}$-logarithmic transformations and generalized complex structures

We provide a new construction of generalized complex manifolds by every logarithmic transformations. Applying a technique of broken Lefschetz fibrations, we obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity 0 on a symplectic 2-torus with trivial normal bundle. Elliptic surfaces with non-zero euler characteristic and the connected sums (2m 1)S 2 × S 2 , (2m 1)CP 2 #lCP 2 and S 1 × S 3 admit twisted generalized complex structures Jn with n type changing loci for arbitrary large n.

Arbitrary sequence RAMs

It is known that in some cases a Random Access Machine (RAM) benefits from having an additional input that is an arbitrary number, satisfying only the criterion of being sufficiently large. This is known as the ARAM model. We introduce a new type of RAM, which we refer to as the Arbitrary Sequence RAM (ASRAM), that generalises the ARAM by allowing the generation of additional arbitrary large numbers at will during execution time. We characterise the power contribution of this ability under several RAM variants.In particular, we demonstrate that an arithmetic ASRAM is more powerful than an arithmetic ARAM, that a sufficiently equipped ASRAM can recognise any language in the arithmetic hierarchy in constant time (and more, if it is given more time), and that, on the other hand, in some cases the ASRAM is no more powerful than its underlying RAM.

Single Dirac-cone state and quantum Hall effects in a honeycomb structure

A honeycomb lattice system has four types of Dirac electrons corresponding to the spin and valley degrees of freedom. We consider a state that contains only one type of massless electrons and three types of massive ones, which we call the single Dirac-cone state. We analyze quantum Hall (QH) effects in this state. We make a detailed investigation of the Chern and spin-Chern numbers. We make clear the origin of unconventional QH effects discovered in graphene. We also show that the single Dirac-cone state may have arbitrary large spin-Chern numbers in magnetic field. Such a state will be generated in antiferromagnetic transition-metal oxides under electric field or in silicene with antiferromagnetic order under electric field.

Matching scale-space features in 1D panoramas

We define a family of novel interest operators for extracting features from one-dimensional panoramic images for use in mobile robot navigation. Feature detection proceeds by applying local interest operators in the scale space of a 1D circular image formed by averaging the center scanlines of a cylindrical panorama. We demonstrate that many such features remain stable over changes in viewpoint and in the presence of noise and camera vibration, and define a feature descriptor that collects shape properties of the scale-space surface and color information from the original images. We then present a novel dynamic programming method to establish globally optimal correspondences between features in images taken from different viewpoints. Our method can handle arbitrary rotations and large numbers of missing features. It is also robust to significant changes in lighting conditions and viewing angle, and in the presence of some occlusion.

Wiener index for graphs and their line graphs with arbitrary large cyclomatic numbers

The Wiener number, W ( G ) , is the sum of the distances of all pairs of vertices in a graph G . Infinite families of graphs with increasing cyclomatic number and the property W ( G ) = W ( L ( G ) ) are presented, where L ( G ) denotes the line graph of G . This gives a positive (partial) answer to an open question posed in an earlier paper by Gutman, Jovašević, and Dobrynin.

Simultaneous rational approximation to the successive powers of a real number

Let n be an integer ≥ 1 and let θ be a real number which is not an algebraic number of degree ≤ [n/2]. We show that there exist ϵ > 0 and arbitrary large real numbers X such that the system of linear inequalities |x0| ≤ X and |x0θj − xj| ≤ ϵX−1/[n/2] for 1 < j < n, has only the zero solution in rational integers x0,…, xn. This result refines a similar statement due to H. Davenport and W. M. Schmidt, where the upper integer part [n/2] is replaced everywhere by the integer part [n/2]. As a corollary, we improve slightly the exponent of approximation to 0 by algebraic integers of degree n + 1 over Q obtained by these authors.

ON BRIDGE NUMBERS OF COMPOSITE RIBBON KNOTS

Bleiler and Eudave-Muñoz showed that composite ribbon number one knots have two-bridge summands. In this paper, we show that there exists an infinite family of composite ribbon number one knots which have arbitrary large bridge numbers.

Performance investigations of the IP multicast architecture

The big interest in desktop audio and video conferencing has led to a broad usage of multimedia tools in wide and local area networks. Many of these tools are based upon the IP multicast protocols. The general advantages of multicast services, like the efficient utilization of network resources and thus the possibility to reach arbitrary large numbers of receivers have by now rendered performance considerations of multicast protocols obsolete. Yet, the rapid growth of multicast usage, especially in the IP Internet, may lead to severe congestion problems, if performance trade-offs are not considered carefully in protocol design. In this paper we present performance investigations of the IP multicast architecture with special emphasis on routing protocols. Performance results derived from measurements and simulations will be discussed. Based upon these results, protocol improvements are suggested and evaluated.