Higher-order Constraints Research Articles

A family of integrable systems of Liouville and lax representation, Darboux transformations for its constrained flows

A family of integrable systems of Liouville are obtained by Tu pattern. Using higher-order potential-eigenfunction constraints, the integrable systems are factorized to two x- and tn-integrable Hamiltonian systems whose Lax representation and three kinds of Darboux transformations are presented.

Higher-order mechanical systems with constraints

A general mathematical theory covering higher-order mechanical systems subject to constraints of arbitrary order (i.e., depending on time, positions, velocities, accellerations, and higher derivatives) is presented, including higher-order holonomic systems as a particular case. Within differential geometric setting on higher-order jet bundles, the concept of a mechanical system (not necessarily regular, or Lagrangian) is introduced to be a class of 2-forms equivalent with a dynamical form. Dynamics are then represented by means of corresponding exterior differential systems. Higher-order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution (canonical distribution, higher-order Chetaev bundle). With help of a constraint structure a constraint force is naturally introduced. Higher-order mechanical systems subject to different kinds of higher-order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view. Regular and Lagrangian systems appear as important particular cases within the general scheme.

Hamiltonian structures of binary higher-order constrained flows associated with two 3×3 spectral problems

Binary higher-order symmetry constraints are considered for the 3×3 spectral problems of the AKNS hierarchy and the classical Boussinesq hierarchy. By using the gauge transformation between the AKNS hierarchy and the classical Boussinesq hierarchy, the regular constrained flows of the AKNS hierarchy and the classical Boussinesq hierarchy are transformed into the non-regular constrained flows of each other. This allows us to construct the Hamiltonian structures of the non-regular constrained flows.

Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints

Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, r-matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders.The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.

Siegel superparticle, higher-order fermionic constraints, and path integrals

We study a Siegel superparticle moving in R 4 4 flat superspace. Canonical quantization is accomplished yielding the massless Wess-Zumino model as an effective field theory. The path integral representation for the corresponding superpropagator is constructed and proven to involve the Siegel action in a gauge-fixed form. It is shown that higher-order fermionic constraints intrinsic in the theory, though being a consequence of others in d = 4, make a crucial contribution to path integral.

A Graph-Rewriting Paradigm for Discrete Relaxation

In image analysis, recognition of the primitives plays an important role. Subsequent analysis is used to interpret the arrangement of primitives. This subsequent analysis must make allowance for errors or ambiguities in the recognition of primitives. In this paper, we assume that the primitive recognizer produces a set of possible interpretations for each primitive. To reduce this primitive-recognition ambiguity, we use contextual information in the image, and apply constraints from the image domain. This process is variously termed constraint satisfaction, labeling or discrete relaxation. Existing methods for discrete relaxation are limited in that they assume a priori knowledge of the neighborhood model: before relaxation begins, the system is told (or can determine) which sets of primitives are related by constraints. These methods do not apply to image domains in which complex analysis is necessary to determine which primitives are related by constraints. For example, in music notation, we must recognize which notes belong to one measure, before it is possible to apply the constraint that the number of beats in the measure should match the time signature. Such constraints can be handled by our graph-rewriting paradigm for discrete relaxation: here neighborhood-construction is interleaved with constraint-application. In applying this approach to the recognition of simple music notation, we use approximately 180 graph-rewriting rules to express notational constraints and semantic-interpretation rules for music notation. The graph rewriting rules express both binary and higher-order notational constraints. As image-interpretation proceeds, increasingly abstract levels of interpretation are assigned to (groups of) primitives. This allows application of higher-level constraints, which can be formulated only after partial interpretation of the image.

The general scheme for higher-order decompositions of zero-curvature equations associated with 337-1337-1337-1(2)

Within the framework of zero-curvature representation theory, the decompositions of each equation in a hierarchy of zero-curvature equations associated with loop algebra $$\widetilde{sl}(2)$$ by means of higher-order constraints on potential are given a unified treatment, and the general scheme and uniform formulas for the decompositions are proposed. This provides a method of separation of variables to solve a hierarchy of (1+1)-dimensional integrable systems. To illustrate the general scheme, new higher-order decompositions of two hierarchies of zero-curvature equations are presented.

Effective adaptive antenna scheme for mobile communications

In electromagnetically crowded mobile communication systems, adaptive antennas employed at the base station often experience simultaneous reception of strong cochannel and impulsive interferers impinging into the main beam of the array along with the desired signal. This results in performance degradation of the angle of arrival estimation schemes in supplying the accurate look-direction information to the beamformer. In the Letter, a simple and effective adaptive antenna scheme is devised to prevent the desired signal cancellation in the optimal weight vector computation of the beamformer without resorting to the derivative or other higher order constraints in the angular region of interest.

Fuzzy approach for multi-level programming problems

Multi-level programming techniques are developed to solve decentralized planning problems with multiple decision makers in a hierarchical organization. These become more important for contemporary decentralized organizations where each unit or department seeks its own interests. Traditional approaches include vertex enumeration and transformation approaches. The former is in search of a compromise vertex based on adjusting the control variable(s) of the higher level and thus is rather inefficient. The latter transfers the lower-level programming problem to be the constraints of the higher level by its Kuhn-Tucker conditions or penalty function; the corresponding auxiliary problem becomes non-linear and the decision information is also implicit. In this study, we use the concepts of tolerance membership functions and multiple objective optimization to develop a fuzzy approach for solving the above problems. The upper-level decision maker defines his or her objective and decisions with possible tolerances which are described by membership functions of fuzzy set theory. This information then constrains the lower-level decision maker's feasible space. A solution search relies on the change of membership functions instead of vertex enumeration and no higher order constraints are generated. Thus, the proposed approach will not increase the complexities of original problems and will usually solve a multilevel programming problem in a single iteration. To demonstrate our concept, we have solved numerical examples and compared their solutions with classical solutions.

New factorization of the Kaup-Newell hierarchy

The new factorizations of each equation in the Kaup-Newell (KN) hierarchy into two commuting x- and t n-finite-dimensional integrable Hamiltonian systems (FDIHS) via the higher-order potential-eigenfunction constraints are presented. The integrability, commutativity and Lax representation for these FDIHSs are deduced from the adjoint representations of the auxiliary linear problems for the KN hierarchy. This approach provides a natural way to study soliton hierarchies with source. The zero-curvature representation and Darboux transformation for the KN hierarchy with source related to eigenfunctions are found.

The bi-Hamiltonian structure for the restricted flows of the Boussinesq and the modified Boussinesq equation

The methods of previous articles are extended herein, using the Miura-type transformation between two soliton equations and the gauge transformation between two associated eigenvalue problems, and a bi-Hamiltonian structure and involutive integrals of motion for x- and t-finite-dimensional integrable Hamiltonian systems (FDIHS) obtained from the decomposition of the Boussinesq equation and the modified Boussinesq equation are computed, respectively. This decomposition provides a way to obtain a certain kind of solution for the later two equations through solving two commuting FDIHS’s. The methods are also applied to the FDIHS’s related to the higher-order constraints.

A METHOD FOR PARALLEL SEARCH UNDER HIGHER-ORDER CONSTRAINTS

There is a relation between the old hill-climbing method and the optimization which is performed by neural nets. This view gives rise to a parallel search method which is presented in this paper. The search mechanism is guided by the optimization of a function which is inspired by the electrostatic field. It is also shown here that this mechanism can naturally accommodate higher-order constraints which the result of the search must satisfy. It is also shown how such constraints can be systematically accommodated by simply expanding the function whose optimization guides the search. This mechanism simulates a dynamic transformation of a cue in which constraints and observational background knowledge compete to influence the transformation.

Studies of metaphase and interphase chromosomes using fluorescence in situ hybridization.

Our measurements have bearing on the resolution with which maps can be constructed and abnormalities can be detected by studying the proximity of DNA sequences in metaphase and interphase chromosomes. The results of our analyses are summarized in Figure 8. Metaphase chromosomes are compacted sufficiently that it is impractical to order sequences separated by less than approximately 1 Mbp. In contrast, 100-kbp resolution can be obtained in interphase chromosomes. Distance measurements reveal that interphase chromatin behaves as a random polymer over distances up to 1-2 Mbp. At greater distances, higher order constraints, perhaps the dimensions of the individual chromosome domains, come into play. A caveat remains: Because the effect of the FISH procedure on native chromosome organization is not well understood, these conclusions may not be applicable to native chromatin. We have illustrated that FISH, with appropriately chosen probes, can supplement conventional cytogenetics in the study of chromosome abnormalities. The technique is increasingly being applied in research laboratories to detect and characterize chromosome abnormalities and point the way to the location of genes involved in human disease.

Image processing regularization filters on layered architecture

Layered architecture is proposed for solving a class of regularization problems in image processing. There are two major hurdles in the implementation of regularization filters with second or higher order smoothness constraints: (a) Stability: With second or higher order constraints, a direct implementation of a regularization filter necessitates negative conductance which, in turn, gives rise to stability problems. (b) Wiring Complexity: A direct implementation of an N-th order regularization filter requires wiring between every pair of k-th nearest nodes for all k, 1 ≤ k ≤ N. Even though one of the authors managed to layout an N = 2 chip, the implementation of an N ≥ 3 chip would be an extremely difficult, if not impossible, task. The regularization filter architecture proposed here (a) requires no negative conductance; and (b) necessitates wiring only between nearest nodes. Smoothing-Contrast-Enhancement filter is given as an example of application. Since this filter is extremely fast, it will have a natural application to smart sensing, i.e., to the simultaneous achievement of sensing and processing. It is also explained how this architecture has been inspired by physiological findings on lower vertebrate retina by one of the authors.

An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability

A systematic method for deducing the Liouville integrability of a finite-dimensional Hamiltonian system reduced from an infinite-dimensional Hamiltonian system is proposed within the framework of the zero-curvature representation theory. Also a systematic way to treat the higher-order constraints and to obtain the associated infinitely many hierarchies of finite-dimensional integrable Hamiltonian systems is presented.

Higher-order derivative constraints in qualitative simulation

Qualitative simulation is a useful method for predicting the possible qualitatively distinct behaviors of an incompletely known mechanism described by a system of qualitative differential equations (QDEs). Under some circumstances, sparse information about the derivatives of variables can lead to intractable branching (or “chatter”) representing uninteresting or even spurious distinctions among qualitative behaviors. The problem of chatter stands in the way of real applications such as qualitative simulation of models in the design or diagnosis of engineered systems. One solution to this problem is to exploit information about higher-order derivatives of the variables. We demonstrate automatic methods for identification of chattering variables, algebraic derivation of expressions for second-order derivatives, and evaluation and application of the sign of second- and third-order derivatives of variables, resulting in tractable simulation of important qualitative models. Caution is required, however, when deriving higher-order derivative (HOD) expressions from models including incompletely known monotonic function ( M +) constraints, whose derivatives beyond the sign of the slope are completely unspecified. We discuss the strengths and weaknesses of several methods for evaluating HOD expressions in this situation. We also discuss a second approach to intractable branching, in which we change the level of description to collapse an infinite set of distinct behaviors into a few by ignoring certain distinctions. These two approaches represent a tradeoff between generality and power. Each application of these methods can take a position on this tradeoff depending on its own critical needs.

Equivalence of the Hamiltonian and Lagrangian path integrals for gauge theories

The lagrangian-antifield and the hamiltonian BRST path integrals are shown to be equal for arbitrary gauge theories obeying standard regularity conditions briefly recalled in the text. Various examples are given as an illustration of the results proved in the theoretical part of the paper. One example discusses systems with tertiary and higher-order constraints. One example illustrates the reducible case. And finally one example analyses gauge-fixing conditions involving higher-order time derivatives of the dynamical variables.

Highly parallel consistent labeling algorithm suitable for optoelectronic implementation

Constraint satisfaction problems require a search through a large set of possibilities. Consistent labeling is a method by which search spaces can be drastically reduced. We present a highly parallel consistent labeling algorithm, which achieves strong k-consistency for any value k and which can include higher-order constraints. The algorithm uses vector outer product, matrix summation, and matrix intersection operations. These operations require local computation with global communication and, therefore, are well suited to a optoelectronic implementation.

A method of integrating the equations of motion of non-holonomic systems with higher-order constraints

An explicit form of the equations of motion of non-holonomic systems with higher-order constraints is studied and a field method /1/ is used to integrate them. Various methods of integrating the equations of motion of non-holonomic systems were discussed earlier in /2–7/. The generalization of the Hamilton-Jacobi method to non-holonomic systems has strict limitations /5, 6/. The field method /1/ was extended in /4/ to cover non-holonomic systems with first-order constraints.

Generalised approach to the design of narrowband antenna array processors with broadband capability

A few techniques have been proposed for designing narrowband antenna array processors with a specific broadband capability. In one approach the array processor is designed by imposing on it a set of linear constraints. This set consists of two parts. In the first part, the processor is constrained to have unity gain in the desired look direction at the centre frequency of the frequency band of interest. In the second part, an additional set of linear constraints is used to ensure that the gradient (first-order derivative) of the frequency response of the processor can be approximated to zero, in the minimum meansquare sense, over a specific frequency band. In the paper, it is shown that this approach can be extended to higher orders (n orders) so that other (n−1) sets of new linear constraints can be obtained. With the extended higher order constraints, a generalised formulation is arrived at for this class of constraints. Computer simulation results are given to show the performance characteristics of a higher order processor using linear array geometry.