Foliations and polynomial diffeomorphisms of R(3)

Let Y = (f, g, h): R(3) -> R(3) be a C(2) map and let Spec(Y) denote the set of eigenvalues of the derivative DY(p), when p varies in R(3). We begin proving that if, for some epsilon > 0, Spec(Y) boolean AND (-epsilon, epsilon) = empty set, then the foliation F(k), with k is an element of {f, g, h}, made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek`s Jacobian Conjecture for polynomial maps of R(n).

POLYNOMIAL IDENTITIES OF FINITE DIMENSIONAL SIMPLE ALGEBRAS

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.

PARTIAL STABILITY FOR A CLASS OF NONLINEAR SYSTEMS

This paper studies a nonlinear, discrete-time matrix system arising in the stability analysis of Kalman filters. These systems present an internal coupling between the state components that gives rise to complex dynamic behavior. The problem of partial stability, which requires that a specific component of the state of the system converge exponentially, is studied and solved. The convergent state component is strongly linked with the behavior of Kalman filters, since it can be used to provide bounds for the error covariance matrix under uncertainties in the noise measurements. We exploit the special features of the system-mainly the connections with linear systems-to obtain an algebraic test for partial stability. Finally, motivated by applications in which polynomial divergence of the estimates is acceptable, we study and solve a partial semistability problem.

Day-night pattern of autonomic nervous system modulation in patients with heart failure with and without sleep apnea

Introduction: Among patients with congestive heart failure (CHF) both obstructive and central sleep apnea (SA) are associated with increased sympathetic activity. However, the day-night pattern of cardiac autonomic nervous system modulation in CHF patients with and without sleep apnea is unknown. Material and methods: Twenty-five CHF patients underwent polysomnography with simultaneous beat-to-beat blood pressure (Portapres), respiration and electrocardiogram monitoring. Patients were divided according to the presence (SA, n=17) and absence of SA (NoSA, n=8). Power spectral analyses of heart rate variability (HRV) and spontaneous baroreflex sensitivity (BRS) were determined in periods with stable breathing while awake at 6 AM, 10 AM, 10 PM, as well as during stage 2 sleep. In addition, muscle sympathetic nerve activity (MSNA) was evaluated at 10 AM. Results: RR variance, low-frequency (LF), high-frequency (HF) powers of HRV, and BRS were significantly lower in patients with SA compared with NoSA in all periods. HF power, a marker of vagal activity, increased during sleep in patients with NoSA but in contrast did not change across the 24-hour period in patients with SA. MSNA was significantly higher in patients with SA compared with NoSA. RR variance, LF and HF powers correlated inversely with simultaneous MSNA (r=-0.64, -0.61, and -0.61 respectively; P < 0.01). Conclusions: Patients with CHF and SA present a reduced and blunted cardiac autonomic modulation across the 24-hour period. These findings may help to explain the increased cardiovascular risk in patients with CHF and SA. (C) 2009 Elsevier Ireland Ltd. All rights reserved.

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387-405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.

CMB in a box: Causal structure and the Fourier-Bessel expansion

This paper makes two points. First, we show that the line-of-sight solution to cosmic microwave anisotropies in Fourier space, even though formally defined for arbitrarily large wavelengths, leads to position-space solutions which only depend on the sources of anisotropies inside the past light cone of the observer. This foretold manifestation of causality in position (real) space happens order by order in a series expansion in powers of the visibility gamma = e(-mu), where mu is the optical depth to Thomson scattering. We show that the contributions of order gamma(N) to the cosmic microwave background (CMB) anisotropies are regulated by spacetime window functions which have support only inside the past light cone of the point of observation. Second, we show that the Fourier-Bessel expansion of the physical fields (including the temperature and polarization momenta) is an alternative to the usual Fourier basis as a framework to compute the anisotropies. The viability of the Fourier-Bessel series for treating the CMB is a consequence of the fact that the visibility function becomes exponentially small at redshifts z >> 10(3), effectively cutting off the past light cone and introducing a finite radius inside which initial conditions can affect physical observables measured at our position (x) over right arrow = 0 and time t(0). Hence, for each multipole l there is a discrete tower of momenta k(il) (not a continuum) which can affect physical observables, with the smallest momenta being k(1l) similar to l. The Fourier-Bessel modes take into account precisely the information from the sources of anisotropies that propagates from the initial value surface to the point of observation-no more, no less. We also show that the physical observables (the temperature and polarization maps), and hence the angular power spectra, are unaffected by that choice of basis. This implies that the Fourier-Bessel expansion is the optimal scheme with which one can compute CMB anisotropies.

Three Bartlett-type corrections for score statistics in symmetric nonlinear regression models

We present simple matrix formulae for corrected score statistics in symmetric nonlinear regression models. The corrected score statistics follow more closely a chi (2) distribution than the classical score statistic. Our simulation results indicate that the corrected score tests display smaller size distortions than the original score test. We also compare the sizes and the powers of the corrected score tests with bootstrap-based score tests.

Direct Computation of Operational Matrices for Polynomial Bases

Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

In this paper, we classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3. (C) 2008 Elsevier Ltd. All rights reserved.

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel {K(n)(theta)}(infinity)(n=1) is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K(n) * f} approximates every continuous 2 pi-periodic function f with the rate omega(f, 1/n), where omega(f, delta) denotes the modulus of continuity, and this provides a new proof of the classical Jackson`s theorem. Despite that it turns out that K(n)(theta) coincide with positive cosine polynomials generated by Fejer, our proof differs from others known in the literature.

An application of lattice basis reduction to polynomial identities for algebraic structures

The authors` recent classification of trilinear operations includes, among other cases, a fourth family of operations with parameter q epsilon Q boolean OR {infinity}, and weakly commutative and weakly anticommutative operations. These operations satisfy polynomial identities in degree 3 and further identities in degree 5. For each operation, using the row canonical form of the expansion matrix E to find the identities in degree 5 gives extremely complicated results. We use lattice basis reduction to simplify these identities: we compute the Hermite normal form H of E(t), obtain a basis of the nullspace lattice from the last rows of a matrix U for which UE(t) = H, and then use the LLL algorithm to reduce the basis. (C) 2008 Elsevier Inc. All rights reserved.

A NOTE ON FREE GROUPS IN THE RING OF FRACTIONS OF SKEW POLYNOMIAL RINGS

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t; sigma], the skew polynomial ring in t, over L, with automorphism sigma. We prove that the multiplicative group D(x) of D contains a free noncyclic subgroup.

Reliability and discriminatory power of methods for dental plaque quantification

OBJECTIVE: This in situ study evaluated the discriminatory power and reliability of methods of dental plaque quantification and the relationship between visual indices (VI) and fluorescence camera (FC) to detect plaque. MATERIAL AND METHODS: Six volunteers used palatal appliances with six bovine enamel blocks presenting different stages of plaque accumulation. The presence of plaque with and without disclosing was assessed using VI. Images were obtained with FC and digital camera in both conditions. The area covered by plaque was assessed. Examinations were done by two independent examiners. Data were analyzed by Kruskal-Wallis and Kappa tests to compare different conditions of samples and to assess the inter-examiner reproducibility. RESULTS: Some methods presented adequate reproducibility. The Turesky index and the assessment of area covered by disclosed plaque in the FC images presented the highest discriminatory powers. CONCLUSION: The Turesky index and images with FC with disclosing present good reliability and discriminatory power in quantifying dental plaque.

Combining Legendre's Polynomials and Genetic Algorithm in the Solution of Nonlinear Initial-Value Problems

This work develops a method for solving ordinary differential equations, that is, initial-value problems, with solutions approximated by using Legendre's polynomials. An iterative procedure for the adjustment of the polynomial coefficients is developed, based on the genetic algorithm. This procedure is applied to several examples providing comparisons between its results and the best polynomial fitting when numerical solutions by the traditional Runge-Kutta or Adams methods are available. The resulting algorithm provides reliable solutions even if the numerical solutions are not available, that is, when the mass matrix is singular or the equation produces unstable running processes.

A NOTES Modification of the Transanal Pull-Through

Transanal endorectal pull-through (TAEPT) surgery is primarily performed for rectosigmoid aganglionosis, generally with excellent results. There is evidence that overstretching the anus and tension traction in the sigmoid during the procedure could impair the final continence of the patient. Many researchers suggest the use of small umbilical or laparoscopic access to aid in colon mobilization, thus preventing excessive handling within the anal canal. We assumed that transabdominal mobilization of the sigmoid could be prevented by utilizing the NOTES (natural orifices transluminal endoscopic surgery) technique. We performed a TAEPT with NOTES access of the sigmoid vascular pedicle, keeping the surgery exclusively transanal, which prevented scars in the abdomen and minimized the stretching of perineal structures.