The Thom-Boardman symbol was first introduced by Thom in 1956 to classify singularities of differentiable maps. It was later generalized by Boardman to a more general setting. Although the Thom-Boardman symbol is realized by a sequence of non-increasing, nonnegative integers, to compute those numbers is, in general, extremely difficult. In the case of polynomial multiplication maps, Robert Varl...

The notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is sometimes referred to as the modulo operation, and denoted with ”mod” used as an infix operator. For example, the remainder of the division of 16 by 12 is denoted by 16 mod 12; as this remainder is 4, we have 16 mod 12 = 4. The congruence, indicated by ”≡” follow...

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership P...

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular Poincaré-type expansions for large values of the argument z and uniform expansions for large values of the parameter are considered. The expansions are based on those given in [5] and on modifications of these expansions given in [9]. Compute...

The relationship between total bilirubin binding capacity (TBBC) and clinical status was investigated in order to assess the risk of bilirubin toxicity in 83 infants with jaundice in this study. Infants with respiratory distress, acidosis, hypoglycaemia, sepsis, asphyxia-anoxia and hypercarbia were accepted as ill and the remainders were well. Sephadex G-25 gel filtration method was used to det...

Abstract. We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form H = A + εB. We give a constructive proof that for all integer p, there exists an integrator with positive steps with a remainder of order O(τε + τ ε), where τ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given...

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields in polyhedral domains is characterized by a hierarchy of model problems. We investigate properties of the model problems (continuity, semi-continuity, existence of generalized eigenfunctions). We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with ...