Equivariant isospectrality and Sunada’s method

We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold ${\mathcal{O}}We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold O{\mathcal{O}} and a smooth manifold M are isospectral, then they cannot admit non-trivial finite Riemannian covers M₁ ?O{M_1 \to\mathcal{O}} and M ₂ → M where M ₁ and M ₂ are isospectral manifolds.     

Thue’s inequalities and the hypergeometric method

We establish upper bounds for the number of primitive integer solutions to inequalities of the shape \(0<|F(x, y)| \le h\), where \(F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]\), \(\alpha \), \(\beta \), \(\gamma \) and \(\delta \) are algebraic constants with \(\alpha \delta -\beta \gamma \ne 0\), and \(r \ge 5\) and h are integers. As an important application, we pay special attention to binomial Thue’s inequalities \(|ax^r - by^r| \le c\). The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.     

State matrix recursion method and monomer–dimer problem

The exact enumeration of pure dimer coverings on the square lattice was obtained by Kasteleyn, Temperley and Fisher in 1961. In this paper, we consider the monomer–dimer covering problem (allowing multiple monomers) which is an outstanding unsolved problem in lattice statistics. We have developed the state matrix recursion method that allows us to compute the number of monomer–dimer coverings and to know the partition function with monomer and dimer activities. This method proceeds with a recurrence relation of so-called state matrices of large size. The enumeration problem of pure dimer coverings and dimer coverings with single boundary monomer is revisited in partition function forms. We also provide the number of dimer coverings with multiple vacant sites. The related Hosoya index and the asymptotic behavior of its growth rate are considered. Lastly, we apply this method to the enumeration study of domino tilings of Aztec diamonds and more generalized regions, so-called Aztec octagons and multi-deficient Aztec octagons.     

The model method: Students’ performance and its effectiveness

This study describes Singapore students’ (N = 607) performance on two tasks in a recently developed Mathematics Processing Instrument (MPI). The MPI comprised tasks sourced from Australia's NAPLAN and Singapore's PSLE. This study also examines students’ use of the model method to solve the two tasks. The model method is a visual problem-solving heuristic prevalently used in Singapore classrooms. The study found that students who solved the tasks using a visual method predominantly used the model method as a visual problem-solving strategy. Another interesting observation was the hindrance of successful problem solving caused by the persistence of prototypical images of model drawings. Implications include encouraging teachers to get their students to identify problem situations where the model method will both work and not work well, and making the role of the generator in the model method explicit in the mathematics textbooks.     

Barzilai and Borwein’s method for multiobjective optimization problems

The present study is an attempt to extend Barzilai and Borwein’s method for dealing with unconstrained single objective optimization problems to multiobjective ones. As compared with Newton, Quasi-Newton and steepest descent multi-objective optimization methods, Barzilai and Borwein multiobjective optimization (BBMO) method requires simple and quick calculations in that it makes no use of the line search methods like the Armijo rule that necessitates function evaluations at each iteration. It goes without saying that the innovative aspect of the current study is due to the use of no function evaluations in comparison with other multi-objective optimization non-parametric methods (e.g. Newton, Quasi-Newton and steepest descent methods, to name a few) that have been investigated so far. Also, the convergence of the BBMO method for the objective functions assumed to be twice continuously differentiable has been proved. MATLAB software was utilized to implement the BBMO method, and the results were compared with the other methods mentioned earlier. Using some performance assessment, the quality of nondominated frontier of BBMO was analogized to above mentioned methods. In addition, the approximate nondominated frontiers gained from the methods were compared with the exact nondominated frontier for some problems. Also, performance profiles are considered to visualize numerical results presented in tables.     

Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub’s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.     

Cartan’s method and its applications in sheaf cohomology

This paper aims to use Cartan’s original method in proving Theorems A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement’s argument to show the topological dimension of a paracompact topological manifold is less than or equal to its real dimension.     

Exp-function method for solving Kuramoto–Sivashinsky and Boussinesq equations

In this paper, we use the Exp-function method to construct the generalized solitary and periodic solution of the Kuramoto–Sivashinsky and Boussinesq equations. These equations play very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method.     

Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

We consider evolution inclusions, in a separable and reflexive Banach space ${\mathbb{E}}$ , of the form ${(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}$ and ${(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}$ , where A is the infinitesimal generator of a C ₀-semigroup, F is a continuous and bounded multifunction defined on ${[t_0, t_1] \times \mathbb{E}}$ with values F(t, x) in the space of all closed convex and bounded subsets of ${\mathbb{E}}$ with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C ₀-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values ${F(t, x) \subset \mathbb{E}}$ , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).     

Purcell’s method, Egerváry and related results

After having been appeared, Egerváry was perhaps the first who responded to Purcell’s paper in 1957. Later in a posthumous paper he returned to the method in 1960, showing that it could be derived from his rank reduction procedure. We review here Purcell’s method in connection with Egerváry’s activity and also, we give a short survey on subsequent developments.     

Application of homotopy-perturbation method to Klein–Gordon and sine-Gordon equations

In this paper, the homotopy-perturbation method (HPM) is employed to obtain approximate analytical solutions of the Klein–Gordon and sine-Gordon equations. An efficient way of choosing the initial approximation is presented. Comparisons with the exact solutions, the solutions obtained by the Adomian decomposition method (ADM) and the variational iteration method (VIM) show the potential of HPM in solving nonlinear partial differential equations.     

Bilinear complexity of algebras and the Chudnovsky–Chudnovsky interpolation method

We give new improvements to the Chudnovsky–Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients.

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We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree.     

A globally and quadratically convergent method for absolute value equations

We investigate the NP-hard absolute value equation (AVE) Ax−|x|=b, where A is an arbitrary n×n real matrix. In this paper, we propose a smoothing Newton method for the AVE. When the singular values of A exceed 1, we show that this proposed method is globally convergent and the convergence rate is quadratic. Preliminary numerical results show that this method is promising.     

Generalizations of Hensel’s lemma and the nearest root method

Hensel’s lemma and its various modifications, such as, for instance, the Hensel–Rychlik theorem, are important tools for investigating problems of existence of roots of polynomials in valued fields. It is shown that a method proposed in the author’s previous papers can also be used to provide a simpler derivation of results in [3].     

A generalized Goulden–Jackson cluster method and lattice path enumeration

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas.     

The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

In this paper, we applied relatively new analytical techniques, the homotopy analysis method (HAM) and the Adomian’s decomposition method (ADM) for solving time-fractional Fornberg–Whitham equation. The homotopy analysis method contains the auxiliary parameter, which provides us with a simple way to adjust and control the convergence region of solution series. The fractional derivatives are described in the Caputo sense. A comparison is made the between HAM and ADM results. The present methods performs extremely well in terms of efficiency and simplicity. Numerical results for different particular cases of the problem are presented.     

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.     

Concerning the radius of convergence of Newton’s method and applications

We present local and semilocal convergence results for Newton’s method in a Banach space setting. In particular, using Lipschitz-type assumptions on the second Fréchet-derivative we find results concerning the radius of convergence of Newton’s method. Such results are useful in the context of predictor-corrector continuation procedures. Finally, we provide numerical examples to show that our results can apply where earlier ones using Lipschitz assumption on the first Fréchet-derivative fail.