Addition Formula Research Articles

A Theta Model for Elliptic Curves

In this paper, we introduce a new model for elliptic curves. We use an isogeny of order two to the level four theta model defined in [1] to obtain the new model which is defined over any finite field. We study the group law on this curve. In particular, we present unified addition formulas in all characteristics and study birational equivalence between this model and other models for elliptic curves. We also provide the geometric interpretation of the group law in terms of rational fractions which enable the computation of bilinear maps on elliptic curves, useful for the construction of cryptographic protocols.

On linear differential equations and systems with reflection

In this paper, we develop a theory of linear differential systems analogous to the classical one for ODEs, including the obtaining of fundamental matrices, the development of a variation of parameters formula and the expression of the Green’s functions. We also derive interesting results in the case of differential equations with reflection and generalize the Hyperbolic Phasor Addition Formula to the case of matrices.

Approximation of macroscopic conductivity for a multiscale model by using mortar methods

For a model with highly varying and multiscale conductivity, its macroscopic conductivity is approximated by using a mortar method. Macroscopic conductivity is useful in forming macroscopic models for porous media flow applications and in the setting of multiscale fast solvers. Many previous studies are based on the following procedure. Microscale models in each small cell are solved independently with an appropriate boundary condition and the solutions from the localized microscale problems are used to approximate the macroscopic conductivity. The size of the small cell and the boundary conditions affect the accuracy of the approximation. In this work, a mortar method is utilized to form localized microscale problems which are less sensitive to the boundary conditions. In addition, a simple and explicit formula for optimally determined macroscopic conductivity is derived by solving a nonlinear minimization problem. No postprocessing is thus required in our approach to calculate the macroscopic conductivity from the solutions of localized microscopic models. The new approach is numerically studied for various test models and compared to existing methods.

A New Look at an Old Triangle Counting Problem

The Matchstick problem of counting equilateral triangles is modified to allow for additional side lengths and summation formulas, which lead to a result represented with combinatorics.

Completing the Complete ECC Formulae with Countermeasures

This work implements and evaluates the recent complete addition formulae for the prime order elliptic curves of Renes, Costello and Batina on an FPGA platform. We implement three different versions:(1) an unprotected architecture; (2) an architecture protected through coordinate randomization; and (3) an architecture with both coordinate randomization and scalar splitting in place. The evaluation is done through timing analysis and test vector leakage assessment (TVLA). The results show that applying an increasing level of countermeasures leads to an increasing resistance against side-channel attacks. This is the first work looking into side-channel security issues of hardware implementations of the complete formulae.

Uniform Additivity in Classical and Quantum Information.

Information theory quantifies the optimal rates of resource interconversions, usually in terms of entropies. However, nonadditivity often makes evaluating entropic formulas intractable. In a few auspicious cases, additivity allows a full characterization of optimal rates. We study uniform additivity of formulas, which is easily evaluated and captures all known additive quantum formulas. Our complete characterization of uniform additivity exposes an intriguing new additive quantity and identifies a remarkable coincidence-the classical and quantum uniformly additive functions with one auxiliary variable are identical.

Impurity Incorporation in the Cu Electrodeposit and Its Effects on the Microstructural Evolution of the Sn/Cu Solder Joints

Impurity incorporation in the Cu electrodeposits as a result of the addition of organic additives in the Cu plating solution is investigated with four additive formulas. A common suppressor (polyethylene glycol, PEG) and chloride ions (Cl−) are added in the plating solution as a control additive formula. Three organosulfides, 3-mercaptopropanesulfonsäure (MPS), bis(3-sulfopropyl) disulfide (SPS), and 3-(2-benzthiazolylthio)-1-propanesulfonsäure (ZPS), are used as accelerators and individually formulated with PEG and Cl− as the other three experimental formulas. The additive formulas of PEG + Cl− and PEG + Cl− + ZPS result in high-level impurity incorporation and cause the microstructural instability of the Sn/Cu joints during thermal aging. Voids and Cu-impurity compounds (CuO and CuS2) are formed accompanying the growth of the intermetallic compounds (Cu6Sn5 and Cu3Sn) which severely degrades the Sn/Cu joints mechanically. When the additive formulas are changed to PEG + Cl− + MPS and PEG + Cl− + SPS, the impurity incorporation is significantly suppressed and thereby inhibits the formation of voids and Cu-impurity compounds. The strong dependence of additive formulas on the impurity incorporation is attributable to the delicate interaction (adsorption competition) between suppressor (PEG) and various accelerators, in which the molecular structures of the organosulfides play a key role.

Audit expert system of communication security assessment

The main goal of the research consists in the elaboration of a system concerning the investigation of security communication, which regards a set of security factors, such as: the degree of encryption, the freshness of nonces, intruder activation, the lifetime of keys, secrets, etc. This paper is devoted to the presentation of systematization formalisms describing the functioning of a security model. In our variant, we investigate the changes of all chosen factors (security attributes) during the realization of protocol operations. The security attributes should be systematically corrected in this process. It changes the general security level of communication. The audit system strategy leads us to one of the most noticeable security in fluence characteristics that refer to time parameters. We can introduce the notation concerning the lifetime of elements (key, message, nonces, secret, etc.). When the value of time activity of an element exceeds its lifetime, then the communication security is definitely threatened. By using special rules presented in the works of Burrows, and Needham2, among other authors, and by creating additional logic formulas, we can estimate intermediate security probability parameters. Finally, we propose a certain kind of probability time automata in order to investigate and predicate different types of communication threats. These automata are built on the basis of a colored Petri net. In addition, this investigation consists in checking communication security (or a kind of threats) and making a threat prediction about possible cases that are connected with losing information. We also included in the model a procedure of security modification with respect to time (the activity of some parameters depends on time). We define the finite set of states by using the LU - technique (interval attribute activity) of a date notation. The proposed system resolves security problem in more comprehensive (multifaceted) way. Ingredient security factors can be grouped in different combinations. This approach increased the range of investigated threaten structures to even unknown hacker algorithm inventions.

Addition formulas of leaf functions according to integral root of polynomial based on analogies of inverse trigonometric functions and inverse lemniscate functions

The second derivative of a function r(t) with respect to a variable t is equal to -n times the function raised to the 2n-1 power of r(t); using this definition, an ordinary differential equation is constructed. Graphs with the horizontal axis as the variable t and the vertical axis as the variable r(t) are created by numerically solving the ordinary differential equation. These graphs show several regular waves with a specific periodicity and waveform depending on the natural number n. In this study, the functions that satisfy the ordinary differential equation are presented as the leaf functions. For n = 1, the leaf function become a trigonometric function. For n = 2, the leaf function becomes a lemniscatic elliptic function. These functions involve an additional theorem. In this paper, based on the additional theorem for n = 1 and n = 2, the double angle and additional theorem for n = 3 are presented.

The Effect of Additive Formula Diet on Bone Structure of the Femur in Ovariectomized Rats

The Effect of Additive Formula Diet on Bone Structure of the Femur in Ovariectomized Rats

A functional equation motivated by some trigonometric identities

We deal with the following functional equation:f(xy)+λf(x)f(y)=ϕ(x,y), where λ is a complex number and f and ϕ are complex mappings defined on a semigroup. Moreover, we assume an addition formula of trigonometric type for ϕ. Our research is motivated by some earlier results related to a problem posed by S. Butler in 2003.

Development of Novel Formulas to Determine Hospital and Pharmacy Opportunities to Reduce Extended Length of Stay

Background: Hospital length of stay (LOS) is an important measure to determine resource utilization and efficiency of care for inpatients. No existing objective methodology is available to determine where and how pharmacy departments can have maximal impact on reducing extended LOS. Objective: An objective methodology to guide decisions by hospital pharmacy departments on reducing extended LOS is described. Methods: University of North Carolina Medical Center’s LOS was compared to hospitals with similar inpatient bed size and case mix index. Objective methodology using pharmacy intensity weight, LOS index, and relative number of cases overall and by hospital service was developed to identify targets of impacting LOS for a pharmacy department. Results: The novel Pharmacy Opportunity Length of Stay (POLOS) formula was developed to prioritize each Medicare Severity Diagnosis Related Group (MSDRG) based on the overall impact pharmacy can have on reducing extended LOS at an individual institution. An additional novel formula, Service Specific POLOS (SSPOLOS), was created to strategically target hospital services, as opposed to MSDRGs, allowing for effective deployment of targeted pharmacy interventions to decrease extended LOS. Conclusion: POLOS and SSPOLOS are novel formulas to guide pharmacy departments in objectively prioritizing and targeting resources to reduce extended LOS. The novel formulas can be used to quantify LOS performance at the hospital and pharmacy specific levels, in addition to providing a hospital service approach to launch LOS initiatives at health care institutions. To our knowledge, this represents the first opportunity for hospital pharmacy departments to objectively target and reduce extended LOS.

Group actions on the tropical Hesse pencil

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions through the ultradiscretization procedure. A tropical analogue of the Hessian group, the group of linear automorphisms acting on the Hesse pencil, is also investigated; it is shown that the dihedral group of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.

GHS additivity formula: A true replacement method for acute systemic toxicity testing of agrochemical formulations

Acute systemic (oral, dermal, inhalation) toxicity testing of agrochemical formulations (end-use products) is mainly needed for Classification and Labelling (C&L) and definition of personal protection equipment (PPE). A retrospective analysis of 225 formulations with available in vivo data showed that: A) LD50/LC50 values were above limit doses in <20.2% via oral route but only in <1% and <2.4% of cases via dermal and inhalation route, respectively; B) for each formulation the acute oral toxicity is always equal or greater than the Acute Toxicity Estimate (ATE) via the other two routes; C) the GHS (Global Harmonised System) computational method based on ATE, currently of limited acceptance, has very high accuracy and specificity for prediction of agrochemical mixture toxicity according to the internationally established classification thresholds.By integrating this evidence, an exposure- and data-based waiving strategy is proposed to determine classification and adequate PPE and to ensure only triggered animal testing is used. Safety characterisation above 2000 mg/kg body weight or 1.0 mg/L air should not be recommended, based on the agrochemical exposure scenarios. The global implementation of these tools would allow a remarkable reduction (up to 95%) in in vivo testing, often inducing lethality and/or severe toxicity, for agrochemical formulations.

Wigner Rotations of Different Types of Lorentz Transformations

We have studied Wigner rotations of different types of Lorenz Transformations according to the nature of movement of one inertial frame relative to the other inertial frame. When the motion is along any arbitrary direction then we can find the formulae for Wigner rotations using the velocity addition formulae for most general, mixed number, quaternion and geometric product Lorentz transformations. Finally we have used simulated data for applying the Wigner rotation formula in pion decay chainand concluded the result.

Alternative proofs for Kocik’s geometric diagram for relativistic velocity addition

A geometric construction for the Poincar´e formula for relativistic addition of velocities in one dimension was given by Jerzy Kocik in Geometric Diagram for Relativistic Addition of Velocities, American Journal of Physics, volume 80, page 737, 2012. While the proof given there used Cartesian coordinate geometry, three alternative approaches are given in this article: a trigonometric one, one via Euclidean geometry, and one using projective geometry.

Two formulas for the BR multiplicity

We prove a projection formula, expressing a relative Buchsbaum–Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula.

ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS

The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra &lt;em&gt;C&lt;/em&gt;&lt;sub&gt;2&lt;/sub&gt;.

Effect of tagatose on growth dynamics of Salmonella Typhimurium and Listeria monocytogenes in media with different levels of structural complexity and in UHT skimmed milk

Tagatose is a novel low-calorie sweetener which addition changes food product properties, e.g., aw, and formula, that may influence food-microbial growth dynamics in both liquid and solid products. The aim of this work was to study growth dynamics of Salmonella Typhimurium and Listeria monocytogenes in liquid and solid media enriched with tagatose (1.5, 4.5 and 7.5% (w/v)), at 4, 8 and 20 °C. More specifically, liquid media were chemically defined minimal medium, general medium and UHT skimmed milk; gelatine and a gelatine-dextran mixture were the gelling agents used, leading respectively a homogeneous and heterogeneous solid system. At regular intervals, cell concentration was determined and the Baranyi and Roberts (1994) model was fitted for growth parameter estimation. Results show a reduced and even no growth of S. Typhimurium with increasing tagatose concentrations, especially at low temperatures and when increasing media complexity. The behaviour of L. monocytogenes is not affected by tagatose, except in liquid at 4 °C, where tagatose facilitates the growth in general culture media. Varying responses of the studied bacteria to changes in media formulation should be considered for future product design.

On the expansion coefficients of Tau-function of the BKP hierarchy

We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition , a formal power series is a solution of the KP hierarchy if and only if its coefficients of Schur function expansion are given by the so called Giambelli type formula. A similar result is known for the BKP hierarchy with respect to Schur’s Q-function expansion under a similar condition. In this paper we generalize this result to the case of .