Smooth Formula Research Articles

Model for an elastic-plastic spherical contact with the DMT adhesion

This paper studied the elastic-plastic spherical contact with the DMT adhesion, and presented an improved smooth formula for the local separation outside the elastic-plastic contact area on the KE empirical formula. The elastic-plastic local separation was equal to the elastic local separation with the same contact area and a equivalent elastic contact displacement. When the contact entered the elastic deformation regime, due to zero residual plastic contact displacement, the present formula for the elastic-plastic local separation could reduce to that for the elastic local separation naturally. Utilizing the present formula for the local separation, a unified adhesion model suitable for between the elastic and elastic-plastic spherical contact with the DMT adhesion was presented. In comparison with the KE adhesion model, the present adhesion model had higher fitting accuracy and could approach the Bradley’s value as the contact displacement approached zero well, which was a good approximation to the complete integration calculation. The present adhesion model could be used for the study of the elastic and elastic-plastic spherical contact with adhesion.

Bykovskii-Type Theorem for the Picard Manifold

Abstract We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\epsilon })$ and $O(X^{3/2+\theta +\epsilon })$, where $\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+\epsilon })$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.

Polyharmonic splines generated by multivariate smooth interpolation

Polyharmonic splines of order m satisfy the polyharmonic equation of order m in n variables. Moreover, if employed as basis functions for interpolation they are radial functions. We are concerned with the problem of construction of the smooth interpolation formula presented as the minimizer of suitable functionals subject to interpolation constraints for n≥1. This is the principal motivation of the paper.We show a particular procedure for determining the interpolation formula that in a natural way leads to a linear combination of polyharmonic splines of a fixed order, possibly complemented with lower order polynomial terms. If it is advantageous for the interpolant in the problem solved to be a polyharmonic spline we can construct such an interpolant directly using the multivariate smooth approximation technique. The smoothness of the spline can be a priori chosen. Smooth interpolation can be very useful e.g. in signal processing, computer aided geometric design or construction of geographic information systems. A 1D computational example is presented.

Magnetic zero-modes, vortices and Cartan geometry

We exhibit a close relation between vortex configurations on the 2-sphere and magnetic zero-modes of the Dirac operator on mathbb {R}^3 which obey an additional nonlinear equation. We show that both are best understood in terms of the geometry induced on the 3-sphere via pull-back of the round geometry with bundle maps of the Hopf fibration. We use this viewpoint to deduce a manifestly smooth formula for square-integrable magnetic zero-modes in terms of two homogeneous polynomials in two complex variables.

Explicit zero density theorems for Dedekind zeta functions

This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring–Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.

Local Monotonicity and Isoperimetric Inequality on Hypersurfaces in Carnot groups

Let G be a k-step Carnot group of homogeneous dimension Q. Later on we shall present some of the results recently obtained in [32] and, in particular, an intrinsic isoperimetric inequality for a C2-smooth compact hypersurface S with boundary @S. We stress that S and @S are endowed with the homogeneous measures n????1 H and n????2 H , respectively, which are actually equivalent to the intrinsic (Q - 1)-dimensional and (Q - 2)-dimensional Hausdor measures with respect to a given homogeneous metric % on G. This result generalizes a classical inequality, involving the mean curvature of the hypersurface, proven by Michael and Simon [29] and Allard [1], independently. One may also deduce some related Sobolev-type inequalities. The strategy of the proof is inspired by the classical one and will be discussed at the rst section. After reminding some preliminary notions about Carnot groups, we shall begin by proving a linear isoperimetric inequality. The second step is a local monotonicity formula. Then we may achieve the proof by a covering argument. We stress however that there are many dierences, due to our non-Euclidean setting. Some of the tools developed ad hoc are, in order, a \blow-up theorem, which holds true also for characteristic points, and a smooth Coarea Formula for the HS-gradient. Other tools are the horizontal integration by parts formula and the 1st variation formula for the H-perimeter n????1 H already developed in [30, 31] and then generalized to hypersurfaces having non-empty characteristic set in [32]. These results can be useful in the study of minimal and constant horizontal mean curvature hypersurfaces in Carnot groups.

Valuations with Crofton formula and Finsler geometry

Valuations admitting a smooth Crofton formula are studied using Geometric Measure Theory and Rumin's cohomology of contact manifolds. The main technical result is a current representation of a valuation with a smooth Crofton formula. A geometric interpretation of Alesker's product is given for such valuations. As a first application in Finsler geometry, a short proof of the theorem of Gelfand–Smirnov that Crofton densities are projective is derived. The Holmes–Thompson volumes in a projective Finsler space are studied. It is shown that they induce in a natural way valuations and that the Alesker product of the k-dimensional and the l-dimensional Holmes–Thompson valuation is the ( k + l ) -dimensional Holmes–Thompson valuation.

Approximating the number of integers without large prime factors

Ψ ( x , y ) \Psi (x,y) denotes the number of positive integers ≤ x \leq x and free of prime factors > y >y . Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ ( x , y ) \Psi (x,y) in the range ( log ⁡ x ) 1 + ϵ > y ≤ x (\log x)^{1+\epsilon }> y \leq x , where ϵ \epsilon is a fixed positive number ≤ 1 / 2 \leq 1/2 . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ ( x , y ) \Psi (x,y) . The computational complexity of this algorithm is O ( ( log ⁡ x ) ( log ⁡ y ) ) O(\sqrt {(\log x)(\log y)}) . We give numerical results which show that this algorithm provides accurate estimates for Ψ ( x , y ) \Psi (x,y) and is faster than conventional methods such as algorithms exploiting Dickman’s function.

Asymptotic Matching Analysis of Scaling of Structural Failure Due to Softening Hinges. I: Theory

Propagation of a crack from the tensile face of a beam causes postpeak softening, i.e., the bending moment decreases at increasing rotation of the hinge. Examples are unreinforced concrete beams and plates, foundations plinths, retaining walls, tunnel linings, or arch dams. Softening in an inelastic hinge is also caused by compression crushing of concrete. This happens in reinforced concrete beams that are prestressed, overreinforced, retrofitted by laminates, or subjected to a large enough axial compressive force, which is typical of columns as well as frames or arches with a large enough horizontal thrust. Hinge softening may also be caused by plastic buckling of flanges in deep thin-wall steel beams. An inevitable consequence of inelastic hinge softening in statically indeterminate structures requiring more than one inelastic hinge to fail is a size effect. Although finite element solutions are possible, general analytical formulas for the size effect in such structures do not exist, because of the complexity of response. The idea of this two-part paper is to exploit the technique of asymptotic matching in order to derive approximate formulas for the entire size range. Exact analytical solutions of the nominal strength of structure are derived for the large-size asymptotic case, for which the hinges soften one by one rather than simultaneously, and for the small-size asymptotic case, for which the classical plastic limit analysis applies. Matching these asymptotic solutions by a smooth formula then yields simple, yet general, size effect laws for the peaks and troughs of the load-deflection diagram through the entire size range. The size effect found is very different from the classical size effect in quasibrittle structures failing due to a single dominant crack. The theory is developed in the present Part I, and analysis of its implications is relegated to Part II.

Euler and Pontrjagin currents of a section of a compactified real bundle

The canonical family of Thom forms τ s for 0 < s < ∞ constructed by Harvey and Lawson on an oriented real vector bundle V → X with metric connection D v is shown to have an extension to the bundle of real projective spaces, P( R ⊕ V) → X , which compactifies V in the fibre directions. The current limit as r → ∞ and s → 0 of the smooth transgression formula τ r − τ s = dσ r, s is the current equation x ̃ (D v) + [P(V)] Res − [X] = dσ ∞,0 on P( R ⊕ V) . Here σ ∞,0 is an L loc 1 current on P( R ⊕ V) and [ X] is the current of integration over the zero section of V. If the rank of V is even, x ̃ (D v) is the twisted extension to P( R ⊕ V) of the Chern-Euler form of the connection D v on V, [ P( V)] integrates densities over the nonorientable submanifold P( V) of P( R ⊕ V) and Res is a smooth twisted form on P( V). If the rank of V is odd, x ̃ (D v) ≡ 0 , [ P( V)] is the current of integration over P( V) and Res is a smooth form on P( V). This construction can be pulled back to X by an atomic section ν:X → P( R ⊕ V) satisfying an appropriate orientability hypothesis to obtain a current equation on X relating the zero and pole sets of ν to the topology of the bundle. Applications to the differential topology of maps into spheres and real projective spaces are discussed.

The kinematic formula in Riemannian homogeneous spaces

Introduction The basic integral formula for submanifolds of a Lie group Poincare's formula in homogeneous spaces Integral invariants of submanifolds of homogeneous spaces, the kinematic formula, and the transfer principle The second fundamental form of an intersection Lemmas and definitions Proof of the kinematic formula and the transfer principle Spaces of constant curvature An algebraic characterization of the polynomials in the Weyl tube formula The Weyl tube formula and the Chern-Federer kinematic formula Appendix: Fibre integrals and the smooth coarea formula References.

A further list and some properties of red degenerates. XI

view Abstract Citations (27) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A further list and some properties of red degenerates. XI. Greenstein, J. L. Abstract New multichannel SIT observations of 78 faint stars, including 51 new degenerates, are given. Helium and carbon stars, and one with a Zeeman triplet (magnetic field), are included. Red subluminous stars are distinguished by ultraviolet absorption but can be confused with red degenerates at the multichannel resolution. The sample includes a number of stars fainter than +15; properties of 25 such faint degenerates are tabulated. Deviations from a smooth interpolation formula show most red degenerates to be essentially line-free, i.e., to have extremely low metal abundance. Publication: The Astrophysical Journal Pub Date: January 1979 DOI: 10.1086/156723 Bibcode: 1979ApJ...227..244G Keywords: Abundance; Late Stars; Stellar Spectra; Subdwarf Stars; White Dwarf Stars; Carbon Stars; Line Spectra; Metals; Stellar Luminosity; Stellar Motions; Tables (Data); Astronomy; Degenerate Stars full text sources ADS | data products SIMBAD (77)

The Variation of Latitude

The 14-monthly variation of the Earth's axis is a free oscillation, and like others it is damped, and must be maintained by some irregular disturbance. In addition there are complications from the annual variation and observational error. The method adopted is first to analyse the data from 1899 to 1967 over 7-yr intervals, to eliminate the free period as well as possible and leave a mean value of the annual terms. The mean displacements fluctuated considerably. With a unit 0·001″ l(=x) ranged from – 15 to + 53, m(= – y) from the start to 1947 from – 41 to + 3, in both cases quite irregularly, but m reached – 164 in 1955–61 and has reached – 225 in 1962–8. The apparent standard errors are mostly about 6. This behaviour is inconsistent with any hypothesis of a uniform rate of drift in either the pole or the observatories. A smooth formula for interpolating the means was adopted; this and the mean annual terms were subtracted, and the remainder was analysed for harmonic terms of 14-month period. Drift of phase gave a correction to this trial period. A model on Yule's lines was used to estimate the parameters in a disturbed system. The remaining observational error over 14-month intervals turned out to be very small. The final period was 433·15 ± 2·23 mean solar days. The damping factor e–k in 14 months gave k = 0·05 ± 0·034, suggesting a time of relaxation between 14 and 73 yr, with a most probable value of 23 yr.

The Variation of Latitude

The 14-monthly variation of the Earth's axis is a free oscillation, and like others it is damped, and must be maintained by some irregular disturbance. In addition there are complications from the annual variation and observational error. The method adopted is first to analyse the data from 1899 to 1967 over 7-yr intervals, to eliminate the free period as well as possible and leave a mean value of the annual terms. The mean displacements fluctuated considerably. With a unit 0·001″ l(=x) ranged from – 15 to + 53, m(= – y) from the start to 1947 from – 41 to + 3, in both cases quite irregularly, but m reached – 164 in 1955–61 and has reached – 225 in 1962–8. The apparent standard errors are mostly about 6. This behaviour is inconsistent with any hypothesis of a uniform rate of drift in either the pole or the observatories. A smooth formula for interpolating the means was adopted; this and the mean annual terms were subtracted, and the remainder was analysed for harmonic terms of 14-month period. Drift of phase gave a correction to this trial period. A model on Yule's lines was used to estimate the parameters in a disturbed system. The remaining observational error over 14-month intervals turned out to be very small. The final period was 433·15 ± 2·23 mean solar days. The damping factor e–k in 14 months gave k = 0·05 ± 0·034, suggesting a time of relaxation between 14 and 73 yr, with a most probable value of 23 yr.

Estimation of Ionization Potentials by Comparison with Neighboring Elements

If a spectrum contains an unperturbed series of at least three members, the ionization potential can be found with fair accuracy from its limit. If but two members are available, serious errors occur if the Ritz correction is ignored. This correction can often be closely estimated by comparison with elements of neighboring atomic numbers. This method is familiar, but published estimates have not been made on a uniform system. Such estimates for spark spectra from Ca ii to Zn ii give ionization potentials differing from a smooth formula by ±0.02 volt, while the differences in the three cases where the Ritz correction was not applied, average ±0.39 volt.