Translation Formula Research Articles

Mixed curvature measures of translative integral geometry

The curvature measures of a set X with singularities are measures concentrated on the normal bundle of X, which describe the local geometry of the set X. For given finitely many convex bodies or, more generally, sets with positive reach, the translative integral formula for curvature measures relates the integral mean of the curvature measures of the intersections of the given sets, one fixed and the others translated, to the mixed curvature measures of the given sets. In the case of two sets of positive reach, a representation of these mixed measures in terms of generalized curvatures, defined on the normal bundles of the sets, is known. For more than two sets, a description of mixed curvature measures in terms of rectifiable currents has been derived previously. Here we provide a representation of mixed curvature measures of sets with positive reach based on generalized curvatures. The special case of convex polyhedra is treated in detail.

Transformation diffusion reconstruction of three-dimensional histology volumes from two-dimensional image stacks.

Traditional histology is the gold standard for tissue studies, but it is intrinsically reliant on two-dimensional (2D) images. Study of volumetric tissue samples such as whole hearts produces a stack of misaligned and distorted 2D images that need to be reconstructed to recover a congruent volume with the original sample's shape. In this paper, we develop a mathematical framework called Transformation Diffusion (TD) for stack alignment refinement as a solution to the heat diffusion equation. This general framework does not require contour segmentation, is independent of the registration method used, and is trivially parallelizable. After the first stack sweep, we also replace registration operations by operations in the space of transformations, several orders of magnitude faster and less memory-consuming. Implementing TD with operations in the space of transformations produces our Transformation Diffusion Reconstruction (TDR) algorithm, applicable to general transformations that are closed under inversion and composition. In particular, we provide formulas for translation and affine transformations. We also propose an Approximated TDR (ATDR) algorithm that extends the same principles to tensor-product B-spline transformations. Using TDR and ATDR, we reconstruct a full mouse heart at pixel size 0.92µm×0.92µm, cut 10µm thick, spaced 20µm (84G). Our algorithms employ only local information from transformations between neighboring slices, but the TD framework allows theoretical analysis of the refinement as applying a global Gaussian low-pass filter to the unknown stack misalignments. We also show that reconstruction without an external reference produces large shape artifacts in a cardiac specimen while still optimizing slice-to-slice alignment. To overcome this problem, we use a pre-cutting blockface imaging process previously developed by our group that takes advantage of Brewster's angle and a polarizer to capture the outline of only the topmost layer of wax in the block containing embedded tissue for histological sectioning.

An elementary approach to the meromorphic continuation of some classical Dirichlet series

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.

Integral geometry of translation invariant functionals, II: The case of general convex bodies

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.

Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles.

Translative containment measure and symmetric mixed isohomothetic inequalities

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane ℝ2, is there a translation T so that t(TK1) ⊂ K0 or t(TK1) ⊃ K0 for t > 0? Via the translative kinematic formulas of Poincare and Blaschke in integral geometry, we estimate the symmetric mixed isohomothetic deficit σ2(K0,K1) ≡ A012 − A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(TK1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.

Integral geometry of translation invariant functionals, I: The polytopal case

We study translative integral formulas for certain translation invariant functionals on convex polytopes and discuss local extensions and applications to Poisson processes and Boolean models.

Fundamental mixed measure theory for non-spherical colloids

Accurate predictions for the collective behavior of non-spherical colloids and nano-particles can guide the search for novel self-assembled materials. We derive a new type of density functional theory for mixtures of arbitrarily shaped convex hard bodies using integral geometry's translative integral formula and a scaling argument. Our fundamental mixed measure theory is exact in the low-density limit and does not require truncation or empirical parameters unlike previous functionals. Quantitatively correct results for hard spherocylinders validate our general approach. In particular, we present the phase diagram, including a qualitatively and quantitatively accurate smectic phase, and the isotropic-nematic interfacial tension.

Representations of solutions, translation formulae and asymptotic behavior in discrete linear systems and periodic continuous linear systems

We give a method for studying of asymptotic behavior of solutions to a periodic continuous linear system. It is based on a representation of solutions given in the paper, which is a reformation of the variation of constants formula into the sum of a $\tau$-periodic function and an exponential-like function. By using such representations, the set of initial values is completely classified according to the asymptotic behavior of the solutions to the continuous system. In particular, the set of initial values of bounded solutions is precisely determined. To give the representation for the continuous system, we will establish translation formulae by comparing two representations of solutions to a discrete linear system. These two representations are deeply related to the binomial coefficients, the Bernoulli numbers and the Stirling numbers.

The Metric System of State Income Taxes

The residents of six US states (Alabama, Iowa, Louisiana, Missouri, Montana, and Oregon) get to exclude or deduct all or part of their federal income taxes paid when they calculate their taxable income for state income tax purposes. Critics have called this deduction “costly and regressive.” The critics are probably right, but not for the reasons they think. The critics of the deduction overlook the fact that the states that allow deductibility of federal income tax for state tax purposes use a different state income “tax base” or “scale” than other states. Most state income tax regimes use a tax base that includes federal income taxes paid (we call it FTI), but the six states listed above use an income tax base that excludes federal income taxes (the FTE base). The two tax bases can translated from one to the other. Indeed, this paper provides original formulae for converting the FTE base into its FTI equivalent and vice verce. Because of this equivalence, one state income tax base is not a priori better than another. But when we convert FTE income tax bases into their FTI equivalents, we find a pervasive trend. States with FTE income tax regimes systematically have more regressive income tax regimes than states with FTI bases. We argue that taxpayers and legislators under-adjust relative to our translation formula when comparing FTE regimes to FTI regimes, causing the systematic regressive trend in states with FTE regimes.

The FMM-BEM Method for the 3D Particulate Stokes Flow

This work introduces new functions based on the spherical harmonics and the solid harmonics which have been used to construct a multipole development for the 3D Stokes problem in order to reduce the operations costs in the BEM method. We show that the major properties of those functions are inherited from the solid harmonics. The contribution of this paper is the introduction of new formulas that serve to calculate the multipole moments and the transfer functions that are necessary for the schemes of orderO(NlogN). Moreover, new translation formulas are introduced to obtain anO(N)scheme. The error truncation of the resulting scheme is discussed. In comparison to the BEM that attains a limit storage atO(104), we present here a method based on FMM-BEM that attains a storage at a limit ofO(106). The implementation of the method achieves a high accuracy level at a reasonable cost.

Study and Application on Airport Clearance Based on Secant Method in Gauss Projection Algorithm for Inverse Solution

This article adopts WGS-84 Coordinate, applying the Secant Method successive approximation to solve geographic coordinates according to the Gauss-Krueger formula, realizing the transformation from plane rectangular coordinates to geographic coordinates, using the coordinate translation and rotation formula to achieve the transformation between the airport coordinate and Gauss plane rectangular coordinates, thus geographic coordinate, gauss plane rectangular coordinates and the airport coordinate can change into each other.

Efficient FMM accelerated vortex methods in three dimensions via the Lamb–Helmholtz decomposition

Vortex methods are used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions.We present a formulation of vortex methods based on the Lamb–Helmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing “conversion” operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented.

Resummed corrections to the ρ parameter in the complex mass scheme

We present all order results for the heavy top corrections to the ρ parameter in the complex mass scheme. We derive translation formulas between the complex mass and the on-shell scheme and show that they are ultimately equivalent. We show that a naive treatment with a fixed width for the top quark cannot give even approximately correct results.

The matrix of unitarity triangle angles for quarks

In the context of quark (as for lepton) mixing, we introduce the concept of the matrix of unitarity triangle angles Φ, emphasising that it carries equivalent information to the complex mixing matrix V itself. The angle matrix Φ has the added advantage, with respect to V, of being both basis- and phase-convention independent and consequently observable (indeed several Φ-matrix entries, e.g., Φcs=α, Φus=β, etc. are already long-studied as directly measurable/measured in B-physics experiments). We give complete translation formulae between the mixing-matrix and angle-matrix representations. We go on to consider briefly the present state of the experimental data on the full angle matrix and some of the prospects for the future, with reference to both the quark and lepton cases.

THE ABSOLUTE MAGNITUDES OF RED HORIZONTAL BRANCH STARS IN THEugrizSYSTEM

Based on photometric data of the central parts of eight globular clusters and one open cluster presented by An and his collaborators, we select red horizontal branch (RHB) stars in the (g-r)0-g0 diagram and make a statistical study of the distributions of their colors and absolute magnitudes in the SDSS ugriz system. Meanwhile, absolute magnitudes in the Johnson VRI system are calculated through the translation formulae between gri and VRI in the literature. The calibrations of absolute magnitude as functions of metallicity and age are established by linear regressions of the data. It is found that metallicity coefficients in these calibrations decrease, while age coefficients increase, from the blue $u$ filter to the red $z$ filter. The calibration of Mi= 0.06[Fe/H]+0.040t+0.03 has the smallest scatter of 0.04 mag, and thus i is the best filter in the $ugriz$ system when RHB stars are used for distance indicators. The comparison of the MI calibration from our data with that from red clump stars indicates that the previous suggestion that the $I$ filter is better than the V filter in distance determination may not be true because of its significant dependence on age.

An analytic approach to electron-molecule potentials using STOs

Using Guseinov's translation formula for a complex Slater-type orbital (STO) and Steinborn's overlap integral between two Slaters on different centers, an analytic approach to electron-molecule interaction potentials has been developed. The theory presented is restricted to diatomic molecules although it is easily generalized to multiatom systems. A general computer code based on this formalism has been written and ē-H2 interaction potentials have been produced for the X1Σ+g state of H2 using two target representations: (1) McLean's five-state (open-shell form) configuration interaction function and (2) Fraga and Ransil's single-configuration (closed-shell form) function. The results are compared with a numerical procedure due to Faisal.

Mixed Measures of Convex Cylinders and Quermass Densities of Boolean Models

Translative integral formulas for curvature measures of convex bodies were obtained by Schneider and Weil by introducing mixed measures of convex bodies. These results can be extended to arbitrary closed convex sets since mixed measures are locally defined. Furthermore, iterated versions of these formulas due to Weil were used by Fallert to introduce quermass densities for (non-stationary and non-isotropic) Poisson processes of convex bodies and respective Boolean models. In the present paper, we first compute the special form of mixed measures of convex cylinders and prove a translative integral formula for them. After adapting some results for mixed measures of convex bodies to this setting we then use this integral formula to obtain quermass densities for (non-stationary and non-isotropic) Poisson processes of convex cylinders. Furthermore, quermass densities of Boolean models of convex cylinders are expressed in terms of mixed densities of the underlying Poisson process generalizing classical formulas by Davy and recent results by Spiess and Spodarev.

Representations and Asymptotic Behavior of Solutions to Periodic Linear Difference Equations

We give a new representation of solutions of the periodic linear difference equation of the form x(n + 1) = Bx(n) + b(n), where B is a complex p × p matrix and b(n) ∈ Cp satisfies the condition b(n) = b(n + ρ), ρ ∈ N, ρ ≥ 2. If B = eτA, τ > 0, then the equation has two representations of solutions based on A and B. In particular, the representation of solutions based on A is deduced from the one based on B by using the translation formulae from B to A. Using these representations, we can obtain the complete classification of the set of initial values according to the behavior of solutions. As applications of these results, by the initial values we characterize necessary and sufficient conditions on the existence of a bounded solution and a ρ-periodic solution.

Effects of Age Shift on the Tempo and Quantum of Non-Repeatable Events

Effects of age shift on the tempo and quantum of non-repeatable demographic events are examined. The purpose is to develop a period index theory based on the survival model and to provide a mathematically consistent interpretation of Bongaarts and Feeney's tempo adjustment arguments. The survival model for non-repeatable events is introduced. In the time-inhomogeneous case, three types of period survival models are considered. McKendrick equation is used to formulate the risk population dynamics. The tempo and quantum indices for three period survival models are computed when the period age shift occurs for the hazard, the incidence, and the survival rates. Bongaarts and Feeney's tempo adjustment arguments are consistently based on the scenario of the period age shift on the survival rate, and they give translation formulae between period indices without referring to cohort. Traditional demographic translation formulae between cohort and period indices are reviewed to clarify differences between cohort- and period-oriented translation procedures.