Addition Formula Research Articles

A Hyperbolic Analog of the Phasor Addition Formula

In this article, we review the basics of the phasor formalism in a rigorous way, highlighting the physical motivation behind it and presenting a hyperbolic counterpart of the phasor addition formula.

Inversion-free arithmetic on elliptic curves through isomorphisms

This paper presents inversion-free formulas for the efficient implementation of a scalar multiplication over elliptic curves. Specifically, it proposes to make use of curve isomorphisms as a way to avoid the computation of inverses in point addition formulas. Interestingly, the presented techniques are independent of the model used to represent the elliptic curve and of the coordinate system used to represent the points. In particular, they apply to affine representations. Further, whereas certain inversion-free techniques are mostly limited to specific scalar multiplication algorithms, the proposed techniques apply to all scalar multiplication algorithms. The so-obtained formulas are well suited to embedded systems and can easily be combined with existing countermeasures to provide secure implementations.

Addition Formulae of Discrete KP, q-KP and Two-Component BKP Systems**Supported by the Zhejiang Provincial Natural Science Foundation under Grant No. LY15A010004, the National Natural Science Foundation of China under Grant Nos. 11201251, 11571192 and the Natural Science Foundation of Ningbo under Grant No. 2015A610157. Jingsong He is supported by the National Natural Science Foundation of China under Grant No. 11271210,

In this paper, we construct the addition formulae for several integrable hierarchies, including the discrete KP, the q-deformed KP, the two-component BKP and the D type Drinfeld–Sokolov hierarchies. With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies, we prove that these addition formulae are equivalent to these hierarchies. These studies show that the addition formula in the research of the integrable systems has good universality.

Reproducing Kernels for Polynomial Null-Solutions of Dirac Operators

It is well known that the reproducing kernel of the space of spherical harmonics of fixed homogeneity is given by a Gegenbauer polynomial. By going over to complex variables and restricting to suitable bihomogeneous subspaces, one obtains a reproducing kernel expressed as a Jacobi polynomial, which leads to Koornwinder’s celebrated result on the addition formula. In the present paper, the space of Hermitian monogenics, which is the space of polynomial bihomogeneous null-solutions of a set of two complex conjugated Dirac operators, is considered. The reproducing kernel for this space is obtained and expressed in terms of sums of Jacobi polynomials. This is achieved through use of the underlying Lie superalgebra \(\mathfrak {sl}(1|2)\), combined with the equivalence between the \(L^2\) inner product on the unit sphere and the Fischer inner product. The latter also leads to a new proof in the standard Dirac case related to the Lie superalgebra \(\mathfrak {osp}(1|2)\).

Charge collection efficiency degradation induced by MeV ions in semiconductor devices: Model and experiment

This paper investigates both theoretically and experimentally the charge collection efficiency (CCE) degradation in silicon diodes induced by energetic ions. Ion Beam Induced Charge (IBIC) measurements carried out on n- and p-type silicon diodes which were previously irradiated with MeV He ions show evidence that the CCE degradation does not only depend on the mass, energy and fluence of the damaging ion, but also depends on the ion probe species and on the polarization state of the device. A general one-dimensional model is derived, which accounts for the ion-induced defect distribution, the ionization profile of the probing ion and the charge induction mechanism. Using the ionizing and non-ionizing energy loss profiles resulting from simulations based on the binary collision approximation and on the electrostatic/transport parameters of the diode under study as input, the model is able to accurately reproduce the experimental CCE degradation curves without introducing any phenomenological additional term or formula. Although limited to low level of damage, the model is quite general, including the displacement damage approach as a special case and can be applied to any semiconductor device. It provides a method to measure the capture coefficients of the radiation induced recombination centres. They can be considered indexes, which can contribute to assessing the relative radiation hardness of semiconductor materials.

Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials

The main objective of this paper is to introduce and investigate two new classes of generalized Apostol-Bernoulli polynomials B<SUB>n</SUB><SUP>[</SUP><SUP>m-1,</SUP>α<SUP>]</SUP>(x;c,α;λ) and Apostol-Euler polynomials ε<SUB>n</SUB><SUP>[m-1,</SUP>α<SUP>]</SUP>(x;c,α;λ). In particular, we obtain addition formula for the new class of the generalized Apostol-Bernoulli polynomials. We also give some recurrence relations and Raabe relations for these polynomials.

Combinatorial Proofs of Addition Formulas

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.

Simulating the warping of thin coated Si wafers using Ansys layered shell elements

The feasibility of simulating thin-film-on-substrates behaviour of large thin coated wafers using Ansys® “layered shells” is estimated. Layered shell elements are based on Mindlin-plates with coupled intrinsic layers. These elements can be used with an update Lagrange algorithm (NLGEOM). They are of interest because they are easy to use and require a minimum of degrees of freedom to discretize a coated wafer. More traditional discretisations with two separate element layers are used for evaluating the Finite Element (FE) solution.The simulation results are further compared to results reported in the literature and to the analytical approach of Stoney (1909). For this purpose, a thorough review of the Stoney approach is done. We present an extended version with a plate stiffness for cubic anisotropic material and an additional formula to calculate the curvature from the thermal strain mismatch. It is shown that Stoney’s approach to estimate the film stress from the curvature is inadequate for large deflections. A direct comparison to bending experiments with large thin coated wafers cannot be given since the deflection of these wafers is interfered from a predominant deflection from their dead load.

On some previous results for the Drazin inverse of block matrices

This short paper is motivated by the paper of Bu et al. [C. Bu, C. Feng, P. Dong, A note on computational formulas for the Drazin inverse of certain block matrices, J. Appl. Math. Comput.(38) (2012) 631-640], where the authors gave additive formula for Drazin inverse for matrices under new conditions, and two representations under some specific conditions. Here is shown that the additive formula is not valid for all matrices which satisfy given conditions. Also, here is proved that the representations which were given in mentioned paper do not extend the results given by Hartwig et al. [R. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2 _ 2 block matrix, SIAM J. Matrix. Anal. Appl. (27)(2006) 757-771 ], in fact they are equivalent.

Effects of Cu Electroplating Formulas on the Interfacial Microstructures of Sn/Cu Joints

Sn/Cu solder joints serve as electrical pathways and mechanical supports in microelectronic packaging. This study demonstrates that the Cu plating formula has a strong effect on the incorporation of impurities in the Cu deposit and, consequently, affects the microstructural evolution of the Sn/Cu interface. An additive formula with polyethylene glycol (PEG) and chloride ions (Cl−) results in a high-level incorporation of impurities (carbon, oxygen, sulfur, and chlorine) in the Cu deposit. Segregation of impurities to the Sn/Cu interface occurs during thermal aging, which induces the formation of voids and Cu-impurity compounds (CuO, Cu2O, and CuS2) embedded in the Cu6Sn5 and Cu3Sn phase layers, forming an unusual alternating layer structure that is structurally loose and unstable. The addition of bis(3-sulfopropyl) disulfide (SPS) along with PEG and Cl− helps restore the microstructural integrity of the Sn/Cu interface by suppressing the incorporation of impurities. This suppression strongly depends upon the SPS concentration and the PEG molecular weight. The aging temperature also has a significant effect on the microstructural evolution of the Sn/Cu interface, which is determined by the change of the dominant diffusion species.

The trigonometiric addition formulas based on the area of triangle formula using a side length and the measures of angles on the ends of the side

The trigonometiric addition formulas based on the area of triangle formula using a side length and the measures of angles on the ends of the side

The Relativistic Space-Time Perspective

Equations developed in this paper show the relationship between Real values measured from a Special Relativity Perspective with no Distortion [noSRPD] and the values measured from the same perspective when those values are distorted. Inverse equations using the values measured from distorted perspective [SRPD] are also reasoned. Equation logic proceeds from the original Real Time distortion equations from Einstein’s On the Electrodynamics of Moving Bodies. An alternate Time equation using the distorted velocity measured from a Relativistic Perspective is presented. Since the time distortion range is unlimited, velocity from a Relativistic Perspective can have an infinite value. Eight additional formulas for velocity, mass, time, and linear deformations are reasoned. Relativistic Perspective equations have confirmed consistency with Classic Relativity equations to 2000 significant digits for 37 velocity values ranging from 1.0E−500 m/s to (c − (1.0E−500) m/s.

The Impacts of Various Ratios of Different Hydrocolloids and Surfactants on Quality Characteristics of Corn Starch Based Gluten-free Bread

In this study, the effects of 18 different additive formulas constituted with xanthan and hydroxypropylmethylcellulose (HPMC) gums (hydrocolloids) were examined in the manufacture of corn starch based gluten-free bread. The additives used as alone or in combinations in the bread manufacture. It was also added with mono- and diglyceride based gel preparations and diacetyl tartaric acid esters of mono- and diglycerides for improving glutenfree bread characteristics. The volumes and softness of the breads have been measured as maximum when HPMC was used alone in increasing order from 1 to 2%. While HPMC gum improved the volume and softness of bread more than Xanthan gum, Xanthan gum improved the grain structure of crumb more than HPMC. In general, these hydrocolloids gave a good quality of bread in terms of moisture content, grain structure and Neumann baking coefficient values, when they were used with combinations rather than being used individually. Addition of surfactant to all combinations always enhanced the grain structure of bread. In contrast, it either worsened or did not change the volume and softness of the bread.

Zoning and Intermittency Simplifications in Quasi-steady State Models

In many countries, the theoretical heating demand within the official energy performance assessment of houses is evaluated using simplified, single-zone quasi-steady state models, based on ISO 13790 and considering one standard, average user profile. Unpredictable variations in user behavior are acknowledged as a major cause of varying prediction errors. However, even considering a single, standard user profile, not heating the bedrooms and switching off the heating at night, requires the use of additional correction formulas, as such behavior appears contradictory to the single zone and quasi-steady state assumptions of the simplified models. This paper compares theoretical heating demands and indoor temperatures using the spatial reduction and intermittency correction formulas from the German (DIN 18599) and Dutch (NEN 7120) standards with results from the simplified, Flemish approach. Results from a multi-zone quasi-steady state model and measured values are added to the comparison. An old neighborhood of uninsulated houses is used as a case-study and renovation scenarios are considered. Results show that, between the German and Dutch approaches, the different predictions are mainly caused by the different standard user profile considered, much more than by the different formulas. Considering the real heating profiles is found to be indispensable, though not sufficient for accurate predictions. The multi-zone approach is proven to be of great value for supporting the selection of specific renovation measures.

Mean dimension, mean rank, and von Neumann–Lück rank

Abstract We introduce an invariant, called mean rank, for any module ℳ {\mathcal{M}} of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of ℳ {\mathcal{M}} , the mean rank of ℳ {\mathcal{M}} , and the von Neumann–Lück rank of ℳ {\mathcal{M}} all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin–Schnirelmann theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.

Influence of different infracardial positions of central venous catheters in hemodynamic monitoring using the transpulmonal thermodilution method.

Hemodynamic measurements are often conducted by the transpulmonary thermodilution (TPTD)-based PiCCO(®)-system. This requires a central-venous (CVC) and a thermistor-tipped arterial catheter, usually placed in the femoral artery. In certain clinical situations, CVC devices have to be placed in the inferior vena cava. However, little is known about the influence of different CVC positions (i.e. ipsi- vs. contra-lateral to the arterial catheter) on the accuracy of the TPTD measurement results. In this prospective observational study surgical intensive care unit patients who had been inserted with CVCs either into the superior (CVCVCS) or the inferior vena cava (CVCinf) in addition to an arterial PiCCO(®)-catheter, were enrolled. Patients were then divided into two groups: Group I was provided with a CVC in the contralateral (CVCcontra) and Group II in the ipsilateral (CVCipsi) inferior vena cava. Thermodilution via injection of ice-cold saline was then performed via CVCsup and CVCinf. Bland-Altman analysis for cardiac index (CI), extra-vascular lung water index (EVLWI) and global end-diastolic volume index (GEDVI) were employed. Additional correction formulas for femorally assed parameters were determined. In a total of 28 patients, bias (limits of agreement) for measurements of CI in CVCcontra was found to be +0.2 (-0.4; +0.9)and +0.3 (-0.4; +1.0) L/min/m(2) in CVCipsi. GEDVI showed a bias of +274.8 (-47.3; +596.9) mL/m(2) in CVCcontra and +274.7 (-100.7; +650.1) mL/m(2) in CVCipsi. The mean EVLWI were 9.4±4.3mL/kg for EVLWIVCS and 10.7±5.2mL/kg for EVLWIinf. The LoA yielded at -3.4 and +6.1mL/kg with a bias of +1.3mL/kg. Percentage errors revealed clinically acceptable limits for CI and GEDVI, but not for EVLWI. Using TPTD via an infracardial central vein, measurements of CI showed high accuracy and precision while GEDVI measurements were precise with a lower accuracy, irrespective of the position of the infracardial CVC.

Ground motions around a semicircular canyon with a dipping edge under SH plane wave incidence

In order to explore the spatial distribution and temporal variation of ground motions near a semicircular canyon with a dipping edge, a simplified mathematical model is constructed. Based on the region-matching technique, a Fourier-Bessel series solution for the plane SH-wave excitation is derived and then applied to theoretically simulate the seismic response of the canyon. The use of the adequate wavefunctions and a newly derived Graf’s addition formula can solve the unknown expansion coefficients. Parametric analyses with respect to the frequency of input motion, angle of incidence, and canyon geometry are illustrated. Both frequency- and time-domain computations are presented. The canonical case, a completed semicircular canyon, which has the exact analytical solution, and the horizontally truncated case analyzed in previous works are considered as particular cases of the proposed general model. Comparisons with boundary-element solutions show good agreement. Steady-state results show that the phenomenon of wave focusing tends to be significant when the incident angle bends toward the horizontal ground surface. Propagation and attenuation of scattered waves that originated from the surficial anomaly are exhibited in transient-state simulations.

Einstein Gyrogroup as a B-loop

Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B = { v ∈ R n : ‖ v ‖ 1 } of R n equipped with Einstein addition ⊕E forms a B-loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein addition in terms of Mobius addition. We then give a characterization of associativity and commutativity of vectors in B with respect to Einstein addition.

Velocity addition formulas in Robertson-Walker spacetimes

Universal velocity addition formulas analogous to the well-known formula in special relativity are found for four geometrically defined relative velocities in a large class of Robertson-Walker spacetimes. Explicit examples are given. The special relativity result is recovered as a special case, and it is shown that the spectroscopic relative velocity, in contrast to three other geometric relative velocities, follows the same addition law as in special relativity for comoving observers in Robertson-Walker cosmologies.

Functional equations for the Stieltjes constants

The Stieltjes constants $$\gamma _k(a)$$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $$\zeta (s,a)$$ about $$s=1$$ . We present the evaluation of $$\gamma _1(a)$$ and $$\gamma _2(a)$$ at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $$\gamma _0(a)$$ , $$\gamma _1(a)$$ , and $$\gamma _2(a)$$ , and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss’ formula for the digamma function at rational argument. In addition, we give the asymptotic form of $$\gamma _k(a)$$ as $$a \rightarrow 0$$ as well as a novel technique for evaluating integrals with integrands with $$\ln (-\ln x)$$ and rational factors.