Applications of symmetry to problem solving

Symmetry is an important mathematical concept which plays an extremely important role as a problem-solving technique. Nevertheless, symmetry is rarely used in secondary school in solving mathematical problems. Several investigations demonstrate that secondary school mathematics teachers are not aware enough of the importance of this elegant problem-solving tool. In this paper we present examples of problems from several branches of mathematics that can be solved using different types of symmetry. Teachers' attitudes and beliefs regarding the use of symmetry in the solutions of these problems are discussed.     

A Note on degree antiregular graphs

This note proves a conjecture of Merris that the minimal value of entries of the doubly stochastic matrix of the degree antiregular graph E_n of order n ≥ 3 is equal to (l/2(n + l)).     

Heuristics and metaheuristics for the maximum diversity problem

This paper presents extensive computational experiments to compare 10 heuristics and 20 metaheuristics for the maximum diversity problem (MDP). This problem consists of selecting a subset of maximum diversity from a given set of elements. It arises in a wide range of real-world settings and we can find a large number of studies, in which heuristic and metaheuristic methods are proposed. However, probably due to the fact that this problem has been referenced under different names, we have only found limited comparisons with a few methods on some sets of instances. This paper reviews all the heuristics and metaheuristics for finding near-optimal solutions for the MDP. We present the new benchmark library MDPLIB, which includes most instances previously used for this problem, as well as new ones, giving a total of 315. We also present an exhaustive computational comparison of the 30 methods on the MDPLIB. Non-parametric statistical tests are reported in our study to draw significant conclusions.     

Quadratic bottleneck knapsack problems

In this paper we study the quadratic bottleneck knapsack problem (QBKP) from an algorithmic point of view. QBKP is shown to be NP-hard and it does not admit polynomial time ?-approximation algorithms for any ?>0 (unless P=NP). We then provide exact and heuristic algorithms to solve the problem and also identify polynomially solvable special cases. Results of extensive computational experiments are reported which show that our algorithms can solve QBKP of reasonably large size and produce good quality solutions very quickly. Several variations of QBKP are also discussed.     

Carbene-stabilized beryllium borohydride

The reaction of N-heterocyclic carbene, L:, with BeCl(2) quantitatively yields L:BeCl(2)1 (L: = :C{N(2,6-Pr(i)(2)C(6)H(3))CH}(2)). The carbene-stabilized beryllium borohydride monomer L:Be(BH(4))(2)2 is prepared by the reaction of 1 with LiBH(4). Compound 3, prepared by the reaction of 2 with Na(2)[Fe(CO)(4)]·dioxane, represents an unusual "dual reduction" of the imidazole ring (i.e., hydroboration of the C═C backbone and hydrogenation of the C2 carbene center).     

Electrochemical sensor for neurotransmitters at physiological pH using a heterocyclic conducting polymer modified electrode

We report the simultaneous determination of two neurotransmitters, norepinephrine (NEP) and serotonin (5-HT), at physiological pH using the electropolymerized film of 3-amino-5-mercapto-1,2,4-triazole modified glassy carbon (p-AMTa) electrode. A bare glassy carbon (GC) electrode fails to resolve the voltammetric signals of NEP and 5-HT due to the surface fouling caused by the oxidized products of them. However, the p-AMTa electrode not only separates the voltammetric signals of NEP and 5-HT with a potential difference of 150 mV between NEP and 5-HT but also shows higher oxidation currents for them. The simultaneous determination of NEP and 5-HT was successfully achieved at p-AMTa electrode using differential pulse voltammetry method. The amperometric current response increased linearly with increasing NEP and 5-HT concentration in the range of 1.0 × 10(-8) to 1.0 × 10(-4) M and 1.0 × 10(-8) to 5.0 × 10(-5) M, respectively, and the detection limit was found to be 1.65 × 10(-11) for NEP and 1.32 × 10(-11) M for 5-HT (S/N = 3). The p-AMTa electrode shows better recovery results for spiked NEP and 5-HT in human blood plasma samples.     

Nonperturbative theory of weak pre- and post-selected measurements

This paper starts with a brief review of the topic of strong and weak pre- and post-selected (PPS) quantum measurements, as well as weak values, and afterwards presents original work. In particular, we develop a nonperturbative theory of weak PPS measurements of an arbitrary system with an arbitrary meter, for arbitrary initial states of the system and the meter. New and simple analytical formulas are obtained for the average and the distribution of the meter pointer variable. These formulas hold to all orders in the weak value. In the case of a mixed preselected state, in addition to the standard weak value, an associated weak value is required to describe weak PPS measurements. In the linear regime, the theory provides the generalized Aharonov–Albert–Vaidman formula. Moreover, we reveal two new regimes of weak PPS measurements: the strongly-nonlinear regime and the inverted region (the regime with a very large weak value), where the system-dependent contribution to the pointer deflection decreases with increasing the measurement strength. The optimal conditions for weak PPS measurements are obtained in the strongly-nonlinear regime, where the magnitude of the average pointer deflection is equal or close to the maximum. This maximum is independent of the measurement strength, being typically of the order of the pointer uncertainty. In the optimal regime, the small parameter of the theory is comparable to the overlap of the pre- and post-selected states. We show that the amplification coefficient in the weak PPS measurements is generally a product of two qualitatively different factors. The effects of the free system and meter Hamiltonians are discussed. We also estimate the size of the ensemble required for a measurement and identify optimal and efficient meters for weak measurements. Exact solutions are obtained for a certain class of the measured observables. These solutions are used for numerical calculations, the results of which agree with the theory. Moreover, the theory is extended to allow for a completely general post-selection measurement. We also discuss time-symmetry properties of PPS measurements of any strength and the relation between PPS and standard (not post-selected) measurements.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

Harmonic Averages and New Explicit Constants for Invariant Densities of Piecewise Expanding Maps of the Interval

The statistical behavior of families of maps is important in studying the stability properties of chaotic maps. For a piecewise expanding map τ whose slope >2 in magnitude, much is known about the stability of the associated invariant density. However, when the map has slope magnitude ≤2 many different behaviors can occur as shown in (Keller in Monatsh. Math. 94(4): 313–333, 1982) for W maps. The main results of this note use a harmonic average of slopes condition to obtain new explicit constants for the upper and lower bounds of the invariant probability density function associated with the map, as well as a bound for the speed of convergence to the density. Since these constants are determined explicitly the results can be extended to families of approximating maps.