Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups

We examine two criteria for balance of a gain graph, one based on binary cycles and one on circles. The graphs for which each criterion is valid depend on the set of allowed gain groups. The binary cycle test is invalid, except for forests, if any possible gain group has an element of odd order. Assuming all groups are allowed, or all abelian groups, or merely the cyclic group of order 3, we characterize, both constructively and by forbidden minors, the graphs for which the circle test is valid. It turns out that these three classes of groups have the same set of forbidden minors. The exact reason for the importance of the ternary cyclic group is not clear. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Relations between Types of Reasoning and Computational Representations

This paper examines the idea that particular representations differentially support and enhance different cognitive processes, in particular different types of reasoning. Five case studies were conducted consisting of detailed observations of pairs of middle-school students interacting with a computer-based learning environment. The software environment, called NumberSpeed, deals with kinematics concepts by having students construct various motion scenarios by adjusting numerical motion parameters: position, velocity and acceleration. NumberSpeed provides feedback about the student-specified motion using two representations: the motion representation and the number-lists representation. Two distinct types of reasoning were recognized in students’ learning while interacting with NumberSpeed: (1) model-based reasoning and (2) constraint-based reasoning. These two types of reasoning are characterized in detail and their roles in problem-solving are analyzed. A cross-analysis between the types of reasoning and the use of particular NumberSpeed representations reveals a correlation between type of reasoning and representational choice. These findings are explained by analyzing the representations’ characteristics and the ways they may differentially support and enhance particular types of reasoning.     

Students' Verification Strategies for Combinatorial Problems

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students' various verification strategies in terms of their contribution to reaching a correct solution. 14 undergraduate students, who had taken at least 1 course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by 1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of which were more frequent and more helpful than others. Students' verifications proved most efficient in terms of reaching a correct solution when they were informed that their solution was incorrect. Implications for teaching and learning combinatorics are discussed.     

Billiard in a barrel

Orbits in the three-dimensional billiard of the form of a truncated ellipsoid ("barrel") are studied both analytically and numerically. A special form of mapping is proposed to get the expression for Kolmogorov-Sinai entropy, and the transition from strong chaos to weak chaos is obtained.     

Hierarchical structures in the phase space and fractional kinetics: II. Immense delocalization in quantized systems

Anomalous transport due to Levy-type flights in quantum kicked systems is studied. These systems are kicked rotor and kicked Harper model. It is confirmed for a kicked rotor that there exist special "magic" values of a control parameter of chaos K=K(*)=6.908 745 em leader for which an essential increasing of a localization length is obtained. Functional dependence of the localization length on both parameter of chaos and quasiclassical parameter h is studied. We also observe immense delocalization of the order of 10(9) for a kicked Harper model when a control parameter K is taken to be K(*)=6.349 972. This "magic" value corresponds to special phase space topology in the classical limit, when a hierarchical self-similar set of sticky islands emerges. The origin of the effect is of the general nature and similar immense delocalization as well as increasing of localization length can be found in other systems. (c) 2000 American Institute of Physics.     

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

Dynamics of the chain of forced oscillators with long-range interaction: from synchronization to chaos

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0相似文献   

Hopping conduction in disordered carbon nanotubes

We report electrical transport measurements on individual disordered multiwalled carbon nanotubes, grown catalytically in a nanoporous anodic aluminum oxide template. In both as-grown and annealed types of nanotubes, the low-field conductance shows an exp[−(T₀/T)^1/2] dependence on temperature T, suggesting that hopping conduction is the dominant transport mechanism, albeit with different disorder-related coefficients T₀. The electric field dependence of low-temperature conductance behaves as exp[−(ξ₀/ξ)^1/2] at high electric field ξ at sufficiently low T. Finally, both annealed and unannealed nanotubes exhibit weak positive magnetoresistance at . Comparison with theory indicates that our data are best explained by Coulomb-gap variable-range hopping conduction and permits the extraction of disorder-dependent localization length and dielectric constant.