On Invariants and Strict Systems of Irreducible Polynomials Over Henselian Valued Fields

Let g(x) be a monic irreducible defectless polynomial over a henselian valued field (K, v), i.e., K(θ) is a defectless extension of (K, v) for any root θ of g(x). It is known that a complete distinguished chain for θ with respect to (K, v) gives rise to several invariants associated with g(x). Recently Ron Brown studied certain invariants of defectless polynomials by introducing strict systems of polynomial extensions. In this article, the authors establish a one-to-one correspondence between strict systems of polynomial extensions and conjugacy classes of complete distinguished chains. This correspondence leads to a simple interpretation of various results proved for strict systems. The authors give new characterizations of an invariant γ _g introduced by Brown.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

A note on certain matrices with h(x) – Fibonacci quaternion polynomials

In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences.     

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u _t =(g(x)u _x ) _x +h(x)u ^m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).     

Symmetric colorings of the dihedral group

Let G be a finite group and let r∈?. An r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and ψ are equivalent if there exists g∈G such that χ(xg^?1) = ψ(x) for every x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx^?1g) = χ(x) for every x∈G. Let S_r(G) denote the number of symmetric r-colorings of G and s_r(G) the number of equivalence classes of symmetric r-colorings of G. We count S_r(G) and s_r(G) in the case where G is the dihedral group D_n.     

On the Linear Stability of Splitting Methods

A comprehensive linear stability analysis of splitting methods is carried out by means of a 2×2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible time-reversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By conveniently selecting the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available. This paper is dedicated to Arieh Iserles on the occasion of his 60th anniversary.     

On Semiscalar and Quasidiagonal Equivalences of Matrices

For a certain class of polynomial matrices A(x), we consider transformations SA(x)R(x) with invertible matrices S and R(x), i.e., the so-called semiscalarly equivalent transformations. We indicate necessary and sufficient conditions for this type of equivalence of matrices. We introduce the notion of quasidiagonal equivalence of numerical matrices. We establish the relationship between the semiscalar and quasidiagonal equivalences and the problem of matrix pairs.     

An Analog Characterization of the Grzegorczyk Hierarchy

We study a restricted version of Shannon's general purpose analog computer in which we only allow the machine to solve linear differential equations. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions, the smallest known recursive class closed under time and space complexity. Furthermore, we show that if the machine has access to a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n−3 non-linear differential equations of a certain kind. Therefore, we claim that, at least in this region of the complexity hierarchy, there is a close connection between analog complexity classes, the dynamical systems that compute them, and classical sets of subrecursive functions.     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

Structural Stability of Constrained Polynomial Systems

The structural stability of constrained polynomial differentialsystems of the form a(x, y)x'+b(x, y)y'=f(x, y), c(x, y)x'+d(x,y)y'=g(x, y), under small perturbations of the coefficientsof the polynomial functions a, b, c, d, f and g is studied.These systems differ from ordinary differential equations at‘impasse points’ defined by ad–bc=0. Extensionsto this case of results for smooth constrained differentialsystems [7] and for ordinary polynomial differential systems[5] are achieved here. 1991 Mathematics Subject Classification34C35, 34D30.     

Variational analysis of a composite function: A formula for the lower second order epi-derivative

An exact formula is established for the lower second order epi-derivative of a function of the form g(F(x)), where F is a smooth map from one Banach space into another and g is a convex function (generally, not everywhere finite). Unconstrained minimization of such functions typically arise as an equivalent (in one or another sense) reduction form for many important classes of constrained optimization problems. The formula is further applied to study epi-differentiability of the max-function ƒ(x) = max{ƒ(q, x):q ε Q}.     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ?𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ?𝒱(I) under the equivalence relation ～defined by: ν₁ ～ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).     

Bounds for the number of moduli for irregular varieties of general type

LetX be a minimal surface of general type and M (X) the set of equivalence classes of complex structures on the differentiable manifold underlyingX; denoting byM _x the dimension of M(X) at [X], the point corresponding to the complex structure ofX we consider the problem of finding an upper bound forM _x in terms of the basic numerical invariants ofX. The main result is the Castelunovo's bound:M _xP_g(X)+2q(X) for certain irregular surfaces. We also generalize the above bound to an arbitrary dimension.     

Cyclotomy and addition sets

In this paper, we develop methods to solve the polynomial congruence θ(x)θ(x^g) ≡ d + λ(1 + x +… + x^p?1) (mod x^p ? 1), where p is an odd prime and θ(x) is a polynomial with nonnegative integral coefficients. Using these methods, we construct some new addition sets that are the unions of index classes for some primes p. We also establish the nonexistence of both the (95, 10, 1, 18)-addition set and the (95, 10, 1, 56)-addition set.     

Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

A class of variable coefficient (1+1)-dimensional nonlinear reaction–diffusion equations of the general form f(x)u_t=(g(x)uⁿu_x)_x+h(x)u^m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.     

Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form

Every symmetric polynomial p = p(x) = p(x ₁,..., x _g ) (with real coefficients) in g noncommuting variables x ₁,..., x _g can be written as a sum and difference of squares of noncommutative polynomials:

A Characterization for Windowed Fourier Orthonormal Basis with Compact Support

Let g(x) ∈L ²(R) and ğ(ω) be the Fourier transform of g(x). Define g _mn (x) = e ^imx g(x−2πn). In this paper we shall give a sufficient and necessary condition under which {g _mn (x)} constitutes an orthonormal basis of L ²(R) for compactly supported g(ω) or ˘(ω). Received March 25, 1999, Revised November 5, 1999, Accepted September 6, 2000     

A General Duality Scheme for Nonconvex Minimization Problems with a Strict Inequality Constraint

We establish a duality formula for the problem Minimize f(x)+g(x) for h(x)+k(x)<0 where g, k are extended-real-valued convex functions and f, h belong to the class of functions that can be written as the lower envelope of an arbitrary family of convex functions. Applications in d.c. and Lipschitzian optimization are given.