Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic ...

In the present paper we study minimization of energy integrals on sphere with a focus an interesting clustering phenomenon: for certain types potentials, optimal measures are discrete or supported small sets. particular, prove that support any minimizer p-frame has empty interior whenever p is not even integer. A similar effect also demonstrated energies analytic potentials which positive defin...

We apply polynomial techniques (i.e., which invole polynomials) to obtain lower and upper bounds on the covering radius of spherical designs as function their dimension, strength, cardinality. In terms inner products we improve due Fazekas Levenshtein propose new bounds. Our approach involves certain signed measures whose corresponding series orthogonal polynomials are positive definite up a (a...

We are studying the efficient solution of system linear equations stemming from mass conserving stress-yielding (MCS) discretization Stokes equations. perform static condensation to arrive at a for pressure and velocity unknowns. An auxiliary space preconditioner positive definite block makes use scalable solvers conforming Finite Element spaces low order is analyzed with emphasis placed on rob...

Let φ be a polynomial over K (a field of characteristic 0) such that the Hessian of φ is a nonzero constant. Let φ̄ be the formal Legendre Transform of φ. Then φ̄ is well-defined as a formal power series over K. The Hessian Conjecture introduced here claims that φ̄ is actually a polynomial. This conjecture is shown to be true when K = R and the Hessian matrix of φ is either positive or negative de...

Stabilizability and stability for a large class of discrete-time polynomial systems can be decided using symbolic computation packages for quantifier elimination in the first order theory of real closed fields. A large class of constraints on states of the system and control inputs can be treated in the same way. Stability of a system can be checked by constructing a Lyapunov function, which is...

We propose a method called Polynomial Quadratic Convex Reformulation (PQCR) to solve exactly unconstrained binary polynomial problems (UBP) through quadratic convex reformulation. First, we quadratize the problem by adding new variables and reformulating into non-convex program with linear constraints (MIQP). then consider solution of (MIQP) specially-tailored reformulation method. In particula...

In this paper, an effective technique is proposed to determine thenumerical solution of nonlinear Volterra-Fredholm integralequations (VFIEs) which is based on interpolation by the hybrid ofradial basis functions (RBFs) including both inverse multiquadrics(IMQs), hyperbolic secant (Sechs) and strictly positive definitefunctions. Zeros of the shifted Legendre polynomial are used asthe collocatio...

The weak Jacobi forms of integral weight and index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove criterion for type algebra being free. As application, give automorphic proof K. Wirthmüller's theorem which asserts that the invariant under Weyl group is polynomial any irreducible root system not E8. This approach also applicable Even if E8 known ...