Arbitrary Map Research Articles

Global minimizers for a p-Ginzburg–Landau-type energy in [formula omitted

Given a p > 2 , we prove existence of global minimizers for a p-Ginzburg–Landau-type energy over maps on R 2 with degree d = 1 at infinity. For the analogous problem on the half-plane we prove existence of a global minimizer when p is close to 2. The key ingredient of our proof is the degree reduction argument that allows us to construct a map of degree d = 1 from an arbitrary map of degree d > 1 without increasing the p-Ginzburg–Landau energy.

Quantum control of the hyperfine-coupled electron and nuclear spins in alkali-metal atoms

We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali-metal atoms based on applied radio frequency and microwave fields. Such interactions should allow essentially decoherence-free dynamics and the application of techniques for robust control developed for NMR spectroscopy. We establish the conditions under which the system is controllable in the sense that one can generate an arbitrary unitary map on the system. We apply this to the case of $^{133}\mathrm{Cs}$ with its $d=16$ dimensional Hilbert space of magnetic sublevels in the $6{S}_{1∕2}$ state, and design control wave forms that generate an arbitrary target state from an initial fiducial state. We develop a generalized Wigner function representation for this space consisting of the direct sum of two irreducible representations of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate a high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements.

Recent advances in inferential solutions to inverse problems

Inferential solutions to inverse problems provide substantial advantages over deterministic methods, such as: quantitative estimates with posterior (data-dependent) error estimates, predictive densities, model comparison and direct support for optimal decisions. The ability to include arbitrary forward maps, and hence use high-level representations of the unknowns, allows structure-preserving model reduction and also allows ‘classification’ to be performed within the ‘imaging’ step. Since inferential methods make (provably) optimal use of data, the ability to reduce data to a minimal set gives cost savings in applications where collecting data is expensive. The price of these advantages is presently the relatively high computational cost of sampling algorithms for computing estimates. Hence the most significant advances are in computational methods for sample-based inference in inverse problems. In this article we review the inferential formulation of inverse problems, some reasons why it is necessary to take on the extra machinery of inferential solutions, the ‘basement level’ methods for computing inferential solutions, and summarize some recent advances in computational methods for inferential solutions to inverse problems.

Simultaneous Cancellation of Fiber Loss, Dispersion, and Kerr Effect in Ultralong-Haul Optical Fiber Transmission by Midway Optical Phase Conjugation Incorporated With Distributed Raman Amplification

An alternative application of distributed Raman amplification (DRA) for ultralong-haul optical fiber transmission is proposed. In our study, the DRA is employed in a transmission system using midway optical phase conjugation (OPC) for amplifying an optical signal and, at the same time, for constructing signal power evolution, which is symmetrical with respect to the midpoint of the system where the OPC is performed. Then, the nonlinear signal waveform distortions that are caused by the Kerr effect, as well as fiber dispersion, are almost completely compensated by the OPC, whereas the fiber loss is compensated by the DRA. Three possible symmetrical signal power maps - a power map that has a reverse sign of the power map that is caused by lump amplification, a flat signal power map, and an arbitrary symmetrical signal power map - are numerically designed by using appropriate Raman pump powers. We show that the flat power map exhibits smaller difference from the target and a higher optical signal-to-noise ratio and requires lower pump power than the other two power maps. Numerical simulation results demonstrate that, by employing the flat power maps with a span of 40 km, a single-wavelength signal whose data rate is 160 Gb/s can be successfully transmitted over 5000 km, and the Kerr effect is sufficiently suppressed near limitation due to the nonlinear accumulation of noise. Finally, we study the feasibility of expanding our method to wavelength-division-multiplexed signal transmission by designing a DRA gain with multiple-wavelength pumping to simultaneously obtain a flat power map and a wide-and-flat gain bandwidth. By using four-wavelength Raman pumps while carefully choosing pump wavelengths and their powers, we achieve the DRA gain that simultaneously gives a fluctuation of the signal power of only 3.5%, a gain ripple of only 5.3%, and, at the same time, a gain bandwidth of as wide as 46 nm.

MacNeille completions of lattice expansions

There are two natural ways to extend an arbitrary map between (the carriers of) two lattices, to a map between their MacNeille completions. In this paper we investigate which properties of lattice maps are preserved under these constructions, and for which kind of maps the two extensions coincide.

Non‐Abelian homomorphism testing, and distributions close to their self‐convolutions

AbstractIn this paper, we study two questions related to the problem of testing whether a function is close to a homomorphism. For two finite groups G,H (not necessarily Abelian), an arbitrary map f : G,H, and a parameter 0 < ε < 1, say that f is ε‐close to a homomorphism if there is some homomorphism g such that g and f differ on at most ε | G | elements of G, and say that f is ε‐far otherwise. For a given f and ε, a homomorphism tester should distinguish whether f is a homomorphism, or if f is ε‐far from a homomorphism. When G is Abelian, it was known that the test which picks O(1/ε) random pairs x,y and tests that f(x) + f(y) = f(x + y) gives a homomorphism tester. Our first result shows that such a test works for all groups G.Next, we consider functions that are close to their self‐convolutions. Let A = {ag | g ε G} be a distribution on G. The self‐convolution of A, A′ = {a | g ε G}, is defined by It is known that A= A′ exactly when A is the uniform distribution over a subgroup of G. We show that there is a sense in which this characterization is robust—that is, if A is close in statistical distance to A′, then A must be close to uniform over some subgroup of G.Finally, we show a relationship between the question of testing whether a function is close to a homomorphism via the above test and the question of characterizing functions that are close to their self‐convolutions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008

The generalized Lyapunov theorem and its application to quantum channels

We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the ‘Lyapunov direct method’. First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in open quantum systems and quantum information, namely quantum channels. In this context, we also discuss the relations between mixing and ergodicity (i.e. the property that there exists only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.

Remarks on interacting Neimark–Sacker bifurcations

We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

Skorohod representation on a given probability space

Let \((\Omega,\mathcal{A},P)\) be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and \(X_n:\Omega\rightarrow S\) an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on \((\Omega,\mathcal{A},P)\). If, in addition, the X n are measurable and tight, there are S-valued random variables \(\overset{\sim}{X}_n\) and X, defined on \((\Omega,\mathcal{A},P)\), such that \(\overset{\sim}{X}_n\sim X_n\), X∼μ, and \(\overset{\sim}{X}_{n_k}\rightarrow X\) a.s. for some subsequence (n k ). Further, \(\overset{\sim}{X}_n\rightarrow X\) a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to \(\sigma(\mathcal{A}\cup\{H\})\) for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken \(((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)\), for some H⊂ (0,1) with outer Lebesgue measure 1, where \(\mathcal{U}\) is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).

Stability of generalized monotone maps with respect to their characterizations

We show that the known types of generalized monotone maps are not stable with respect to their characterizations (i.e. the characterizations are not maintained if an arbitrary map of this type is disturbed by an element with sufficiently small norm) and introduce s-quasimonotone maps, which are stable with respect to their characterization. For gradient maps, s-quasimonotonicity is related to s-quasiconvexity (introduced by Phu in Optimization, 38, 1996) of the underlying function. A necessary and sufficient condition for a univariate polynomial to be s-quasimonotone is given. Furthermore, some stability properties of s-quasiconvex functions are presented.

Negative periodic orbits for graph maps

Recently, Alves, Hric and Sousa Ramos proved that for any continuous piecewise monotone map of a graph into itself, its topological entropy is equal to the maximum of two quantities. The first one is the (exponential) upper growth rate, as n → ∞, of the number of periodic points of period n at which the orientation is reversed. The second one is the logarithm of the spectral radius of the map induced in the first homology group.We extend the essential part of this theorem to arbitrary continuous maps of a graph into itself and provide a substantially shorter proof of the whole theorem.

HOW POINTS ESCAPE TO INFINITY UNDER EXPONENTIAL MAPS

We investigate the finer fractal structure of the set of points escaping to infinity under iteration of an arbitrary exponential map. Providing exact formulas, we show how sensitively the Hausdorff dimension depends on the rate of growth of canonical Devaney–Krych codes.

Bar-Magnet Polyhedra and NS-Orientations of Maps

At the CCCG 2001 open-problem session [2], J. O'Rourke asked which polyhedra can be represented by bars and magnets. This problem can be phrased as follows: which 3-connected planar graphs may have their edges directed so that the directions alternate around each vertex (with one exception of nonalternation if the degree is odd). In this note we solve O'Rourke's problem and generalize it to arbitrary maps on general surfaces. Obstructions to the existence of such orientations can be expressed algebraically by a new homology invariant of perfect matchings in the related graph of cofacial odd vertices.

Limits of Chow Groups, and a New Construction of Chern-Schwartz-MacPherson Classes

We define an `enriched' notion of Chow groups for algebraic varieties, agreeing with the conventional notion for complete varieties, but enjoying a functorial push-forward for arbitrary maps. This tool allows us to glue intersection-theoretic information across elements of a stratification of a variety; we illustrate this operation by giving a direct construction of Chern-Schwartz-MacPherson classes of singular varieties, providing a new proof of an old (and long since settled) conjecture of Deligne and Grothendieck.

Modeling Intensity Distributions of Advanced Daylight Systems Surrounded by Arbitrary Luminance Maps

For more than a decade, daylighting optical devices (DOD) using reflection, refraction and diffraction have been used in commercial buildings. Some of these systems show considerable promise for south facing facades, as they can reject or redirect sunlight and diffuse daylight. However, their daylight performance is not clearly understood. Simulation tools employing Monte Carlo forward ray tracing are now available and can used to trace rays from the source (the exterior environment) through a devise and into the building interior. Two problems remain, however. The first is the generation of a ray distribution of the exterior environment. The second problem is the representation of the resulting daylight distributions at the DOD exit surface. There would be several advantages if an exiting ray distribution could be represented as an intensity distribution in narrow angular increments in a photometric file format. We shows how to generate ray representations of hemispherical luminance maps for exterior environments, introduce a method to determine intensity distributions for DODs, and discuss errors involved. Results of tests against measurements for a commercially available specular micro louver are reported.

Nielsen periodic point theory for periodic maps on orientable surfaces

Let f : F g → F g denote a periodic self map of minimal period m on the orientable surface of genus g with g > 1 . We study the calculation of the Nielsen periodic numbers NP n ( f ) and NΦ n ( f ) . Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n ≠ m . However, determining NP m ( f ) involves some surprises. Because f m = id F g , f m has one Nielsen class E m . This class is essential because L ( id F g ) = χ ( F g ) = 2 − 2 g ≠ 0 . If there exists k < m with L ( f k ) ≠ 0 then E m reduces to the essential fixed points of f k . There are maps g (we call them minLef maps) for which L ( g k ) = 0 for all k < m . We show that the period of any minLef map must always divide 2 g − 2 . We prove that for such maps E m reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on F g . We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NP n ( f ) and NΦ n ( f ) for all n. The example of an irreducible minLef map is on F 4 and is of maximal period 6. The example of a non-minLef map is on F 2 and has maximal period 12 on F 2 . It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F 4 defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any F g . Using these examples we disprove the conjecture from the conclusion of our previous paper.

THE LIE DERIVATIVE OF SPINOR FIELDS: THEORY AND APPLICATIONS

Starting from the general concept of a Lie derivative of an arbitrary differentiable map, we develop a systematic theory of Lie differentiation in the framework of reductive G-structures P on a principal bundle Q. It is shown that these structures admit a canonical decomposition of the pull-back vector bundle [Formula: see text] over P. For classical G-structures, i.e. reductive G-subbundles of the linear frame bundle, such a decomposition defines an infinitesimal canonical lift. This lift extends to a prolongation Γ-structure on P. In this general geometric framework the concept of a Lie derivative of spinor fields is reviewed. On specializing to the case of the Kosmann lift, we recover Kosmann's original definition. We also show that in the case of a reductive G-structure one can introduce a "reductive Lie derivative" with respect to a certain class of generalized infinitesimal automorphisms, and, as an interesting by-product, prove a result due to Bourguignon and Gauduchon in a more general manner. Next, we give a new characterization as well as a generalization of the Killing equation, and propose a geometric reinterpretation of Penrose's Lie derivative of "spinor fields". Finally, we present an important application of the theory of the Lie derivative of spinor fields to the calculus of variations.

On isotopic realizability of maps factored through a hyperplane

In this paper we study the isotopic realization problem, which is the question of isotopic realizability of a given (continuous) map , that is, the possibility of a uniform approximation of by a continuous family of embeddings , , under the condition that is discretely realizable, that is, that there exists a uniform approximation of by a sequence of embeddings , . For each a map is constructed that is discretely but not isotopically realizable and which, unlike all such previously known examples, is a locally flat topological immersion. For each a map is constructed that is discretely but not isotopically realizable. It is shown that for any map is isotopically realizable, and for , so also is every map . If and is not a power of 2, an arbitrary map is isotopically realizable. The main results are devoted to the isotopic realization problem for maps of the form , . It is established that if it has a negative solution, then the inverse images of points under the map have a certain homology property connected with actions of the group of -adic integers. The solution is affirmative if is Lipschitzian and its van Kampen-Skopenkov thread has finite order. In connection with the proof the functors and in the relative homology algebra of inverse spectra are introduced.

Bounded distributive lattice expansions

A new notion of a canonical extension $\mathbf{A}^{\sigma }$ is introduced that applies to arbitrary bounded distributive lattice expansions (DLEs) $\mathbf{A} $. The new definition agrees with the earlier ones whenever they apply. In particular, for a bounded distributive lattice $\mathbf{A}, \mathbf{A}^{\sigma }$ has the same meaning as before. A novel feature is the introduction of several topologies on the universe of the canonical extension of a DL. One of these topologies is used to define the canonical extension $f^{\sigma }:\mathbf{A}^{\sigma }\rightarrow \mathbf{B}^{\sigma }$ of an arbitrary map $f:\mathbf{A}\rightarrow \mathbf{B}$ between DLs, and hence to define the canonical extension $\mathbf{A}^{\sigma }$ of an arbitrary DLE $\mathbf{A}$. Together the topologies form a powerful tool for showing that many properties of DLEs are preserved by canonical extensions.

Analog realization of arbitrary one-dimensional maps

An increasing number of applications of a one-dimensional (1-D) map as an information processing element are found in the literature on artificial neural networks, image processing systems, and secure communication systems. In search of an efficient hardware implementation of a 1-D map, we discovered that the bifurcating neuron (BN), which was introduced elsewhere as a mathematical model of a biological neuron under the influence of an external sinusoidal signal, could provide a compact solution. The original work on the BN indicated that its firing time sequence, when it was subject to a sinusoidal driving signal, was related to the sine-circle map, suggesting that the BN can compute the sine-circle map. Despite its rich array of dynamical properties, the mathematical description of the BN is simple enough to lend itself to a compact circuit implementation. In this paper, we generalize the original work and show that the computational power of the BN can be extended to compute an arbitrary 1-D map. Also, we describe two possible circuit models of the BN: the programmable unijunction transistor oscillator neuron, which was introduced in the original work as a circuit model of the BN, and the integrated-circuit relaxation oscillator neuron (IRON), which was developed for more precise modeling of the BN. To demonstrate the computational power of the BN, we use the IRON to generate the bifurcation diagrams of the sine-circle map, the logistic map, as well as the tent map, and then compare them with exact numerical versions. The programming of the BN to compute an arbitrary map can be done simply by changing the waveform of the driving signal, which is given to the BN externally; this feature makes the circuit models of the BN especially useful in the circuit implementation of a network of 1-D maps.