Rule Of Signs Research Articles

Descartes’ Rule of Signs for Polynomial Systems Supported on Circuits

We give a multivariate version of Descartes' rule of signs to bound the number of positive real roots of a system of polynomial equations in n variables with n+2 monomials, in terms of the sign variation of a sequence associated both to the exponent vectors and the given coefficients. We show that our bound is sharp and is related to the signature of the circuit.

Ray-triangular Bézier patch intersection using hybrid clipping algorithm

In this paper, we present a novel geometric method for efficiently and robustly computing intersections between a ray and a triangular Bezier patch defined over a triangular domain, called the hybrid clipping (HC) algorithm. If the ray pierces the patch only once, we locate the parametric value of the intersection to a smaller triangular domain, which is determined by pairs of lines and quadratic curves, by using a multi-degree reduction method. The triangular domain is iteratively clipped into a smaller one by combining a subdivision method, until the domain size reaches a prespecified threshold. When the ray intersects the patch more than once, Descartes’ rule of signs and a split step are required to isolate the intersection points. The algorithm can be proven to clip the triangular domain with a cubic convergence rate after an appropriate preprocessing procedure. The proposed algorithm has many attractive properties, such as the absence of an initial guess and insensitivity to small changes in coefficients of the original problem. Experiments have been conducted to illustrate the efficacy of our method in solving ray-triangular Bezier patch intersection problems.

A smoothing stochastic simulated annealing method for localized shapes approximation

In B-splines approximation setting, it is known that monotony and convexity (or concavity) shapes can easily be controlled by the spline coefficients. In this paper we deal with the general context of combinations of localized shape constraints. We prove that unimodality constraint is fulfilled simply by an increasing and decreasing sequence of the spline coefficients by using the Descartes' sign rule. Then, the local support property of B-splines is used to locate each constraint on a given interval. We formulate a smoothing spline approximation under inequality constraints in function of the spline coefficients. We also give a simulated annealing algorithm to solve the optimization problem and we establish the almost sure convergence of the efficient solution.

Splitting theorem forZ2n-supermanifolds

Smooth $\mathbb{Z}_2^n$-supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the "scalar product" of the involved $\mathbb{Z}_2^n$-degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor-Gawcedzki theorem says that any smooth supermanifold is diffeomorphic to the "superization" $\Pi E$ of a vector bundle $E$. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of $\mathbb{Z}_2^n$-supermanifolds with its local model made of formal power series. We give a positive answer to this question.

Odd Structures Are Odd

By an odd structure we mean an algebraic structure in the category of graded vector spaces whose structure operations have odd degrees. Particularly important are odd modular operads which appear as Feynman transforms of modular operads and, as such, describe some structures of string field theory. We will explain how odd structures are affected by the choice of the monoidal structure of the underlying category. We will then present two ‘natural’ and ‘canonical’ constructions of an odd modular endomorphism operad leading to different results, only one being correct. This contradicts the generally accepted belief that the systematic use of the Koszul sign rule leads to correct signs.

The category of Z2n-supermanifolds

In physics and in mathematics Z2n-gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the “scalar product” of the involved Z2n-degrees. The Z2n-supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define Z2n-supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z2•-supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical “superization” to a Z2n-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z2n-context.

A Tropical Analog of Descartes’ Rule of Signs

We prove that for any degree d, there exist (families of) finite sequences {lambda(k,d)} 0 <= k <= d of positive numbers such that, for any real polynomial P of degree d, the number of its re ...

Optimal Power Allocation Schemes for the Single AF Relay and Jammer Wiretap Channels

This paper considers the wiretap channel in which a source node communicates with a destination node with the aid of a single helper and in the presence of an eavesdropper. The helper can apply either amplify-and-forward (AF) relaying or jamming. The secrecy capacity and respective optimal power allocation schemes at the source and the helper are investigated under both individual and joint power constraints. Given that the links among the nodes might not always be available for transmission (e.g., due to heavy shadowing), several network topologies are considered for relaying. To deal with the related nonconcave optimization problems in solving the optimal power allocation schemes, we first propose an analytical approach based on Descartes' rule of signs. Using this approach, optimal power allocation strategies are derived for some relay topologies and for the jamming system. Insights into the proposed schemes for different transmission power ranges and comparisons to the optimal information rate power allocations are also discussed. Simulation results are finally provided to quantify the optimality of the proposed schemes.

Level crossing analysis of chemically induced dynamic nuclear polarization: Towards a common description of liquid-state and solid-state cases.

Chemically Induced Dynamic Nuclear Polarization (CIDNP) is an efficient method of creating non-equilibrium polarization of nuclear spins by using chemical reactions, which have radical pairs as intermediates. The CIDNP effect originates from (i) electron spin-selective recombination of radical pairs and (ii) the dependence of the inter-system crossing rate in radical pairs on the state of magnetic nuclei. The CIDNP effect can be investigated by using Nuclear Magnetic Resonance (NMR) methods. The gain from CIDNP is then two-fold: it allows one to obtain considerable amplification of NMR signals; in addition, it provides a very useful tool for investigating elusive radicals and radical pairs. While the mechanisms of the CIDNP effect in liquids are well established and understood, detailed analysis of solid-state CIDNP mechanisms still remains challenging; likewise a common theoretical frame for the description of CIDNP in both solids and liquids is missing. Difficulties in understanding the spin dynamics that lead to the CIDNP effect in the solid-state case are caused by the anisotropy of spin interactions, which increase the complexity of spin evolution. In this work, we propose to analyze CIDNP in terms of level crossing phenomena, namely, to attribute features in the CIDNP magnetic field dependence to Level Crossings (LCs) and Level Anti-Crossings (LACs) in a radical pair. This approach allows one to describe liquid-state CIDNP; the same holds for the solid-state case where anisotropic interactions play a significant role in CIDNP formation. In solids, features arise predominantly from LACs, since in most cases anisotropic couplings result in perturbations, which turn LCs into LACs. We have interpreted the CIDNP mechanisms in terms of the LC/LAC concept. This consideration allows one to find analytical expressions for a wide magnetic field range, where several different mechanisms are operative; furthermore, the LAC description gives a way to determine CIDNP sign rules. Thus, LCs/LACs provide a consistent description of CIDNP in both liquids and solids with the prospect of exploiting it for the analysis of short-lived radicals and for optimizing the polarization level.

EVOLUTION OF MAGNETIC HELICITY AND ENERGY SPECTRA OF SOLAR ACTIVE REGIONS

ABSTRACT We adopt an isotropic representation of the Fourier-transformed two-point correlation tensor of the magnetic field to estimate the magnetic energy and helicity spectra as well as current helicity spectra of two individual active regions (NOAA 11158 and NOAA 11515) and the change of the spectral indices during their development as well as during the solar cycle. The departure of the spectral indices of magnetic energy and current helicity from 5/3 are analyzed, and it is found that it is lower than the spectral index of the magnetic energy spectrum. Furthermore, the fractional magnetic helicity tends to increase when the scale of the energy-carrying magnetic structures increases. The magnetic helicity of NOAA 11515 violates the expected hemispheric sign rule, which is interpreted as an effect of enhanced field strengths at scales larger than 30–60 Mm with opposite signs of helicity. This is consistent with the general cycle dependence, which shows that around the solar maximum the magnetic energy and helicity spectra are steeper, emphasizing the large-scale field.

ON THE ROLE OF TACHOCLINES IN SOLAR AND STELLAR DYNAMOS

ABSTRACT Rotational shear layers at the boundary between radiative and convective zones, tachoclines, play a key role in the process of magnetic field generation in solar-like stars. We present two sets of global simulations of rotating turbulent convection and dynamo. The first set considers a stellar convective envelope only; the second one, aiming at the formation of a tachocline, also considers the upper part of the radiative zone. Our results indicate that the resulting properties of the mean flows and dynamo, such as the growth rate, saturation energy, and mode, depend on the Rossby number (Ro). For the first set of models either oscillatory (with ∼2 yr period) or steady dynamo solutions are obtained. The models in the second set naturally develop a tachocline, which in turn leads to the generation of a strong mean magnetic field. Since the field is also deposited in the stable deeper layer, its evolutionary timescale is much longer than in the models without a tachocline. Surprisingly, the magnetic field in the upper turbulent convection zone evolves on the same timescale as the deep field. These models result in either an oscillatory dynamo with a ∼30 yr period or a steady dynamo depending on Ro. In terms of the mean-field dynamo coefficients computed using the first-order smoothing approximation, the field evolution in the oscillatory models without a tachocline seems to be consistent with dynamo waves propagating according to the Parker–Yoshimura sign rule. In the models with tachoclines the dynamics is more complex and involves other transport mechanisms as well as tachocline instabilities.

Towards integration on colored supermanifolds

The aim of the present text is to describe a generalization of superalgebra and supergeometry to $\mathbb Z_2^n$-gradings, $n \gt 1$. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the $\mathbb Z_2^n$

On the Exact Number of Zeros of Certain Even Degree Polynomails : A Sharkovsky Theorem Approach

The problem of finding or characterizing the zeros of polynomial functions has a long history. The obtained results varied from theory, algorithms and numerical simulations. Historically, this study began with the fundamental theorem of algebra proved by Gauss. The most known results in this direction is the fact that a real polynomial of degree n has at most n real zeros. There is also the so called Descartes rule of signs concerning the number of positive zeros. Some generalizations of Descartes rule are know such as the BudanFourier theorem that gives an upper bound for the number of zeros of a polynomial. Also, the Sturm’s theorem that gives a method for determining the exact number of zeros in an interval [Hen, chapter 6] and [Hou, chapter 2]. Recent results uses the classical Enestrom-Kakeya theorem to restricts the location of the zeros based on a condition imposed on the coefficients of the polynomial under invistigation. See [Bre] and references therain. In this paper, we will use a dynamical system result concerning periodic points of a continuous function. The result is called Sharkovsky theorem [Sha1, Sha2, Sha3, Sha4, Sha5] that gives a complete description of possible sets of periods for continuous mappings defined on an interval. The interval need not be closed or bounded. The main idea used here is the notion of

Qualitative analysis of the evaluation indicators and their related parameters of ametropic state

To investigate the theoretical basis and practical limitations of the existing calculation formulas in the evaluation of ametropic state. The evaluation indicators and their calculation parameters of ametropia were analyzed by using the reduced schematic model eye, the paraxial imaging principle, and the dimension laws. The existing formulas resulted from the reduced object vergence of object distance and image distance relation. Regarding the two measurement indicators of the existing formulas, diopter was misused for refractive power. "Ametropia degree" was a non-standard diction. Both of them were not suitable as the evaluation indicators. The outcomes of the existing formulas and their related plus or minus sign rules showed refractive corrections instead of refractive errors proper. For refractive errors, there was no suitable evaluation indicator. In the evaluation of ametropic state, there are fundamental problems in the existing formulas resulting from the reduced object vergence. The measurement indicators and their dimensional units are confused and misused. The calculation results refer to the refractive corrections only. The evaluation indicators for ametropia need to be further discussed.

A Study on the Driver’s Preferences of Prividing Direction Information in Road Signs

Although traffic information has been actively analyzed using big data, it has not been used as much with the consideration of driver characteristics. Among the various types of information, road signs can directly affect the driver. Road signs must provide the optimal information that enables drivers to reach their destinations with ease as well as information suitable for navigation systems. However, present road sign rules provide standardized information, regardless of the road type or size. This study suggests a method for providing road information that will help drivers determine their behavior. First, the minimum character size that can be used on a road sign for each design speed was obtained with respect to the visibility and decipherability of a road sign. Instead of conventional diagram-based direction guidance, a scenario using split-based direction guidance was created. To verify the effectiveness of the provided information, a three-dimensional simulated road environment was constructed, and a driving simulator was used for the test. At a simple plane intersection, the driver was not greatly influenced by directional guidance, but at a complex, three-dimensional intersection, the driver preferred summary-based directional guidance, which is instinctive guidance, over diagram-based guidance. On the basis of the test results, a secondary verification test that applied split-based guidance at a three-dimensional intersection confirmed that the driver had no problems in making decisions.

MCN-2 Invariants, Homomorphisms, and Anomalies in Polynomials and the 'Fundamental Theorem of Algebra'

This article contributes to the existing literature by: i) proving that the Descartes Sign Rule (as interpreted by most academicians - such as Oehmke (2000) and Osborne (2010)) is wrong; ii) proving that the Fundamental Theorem Of Algebra (FTA) is wrong; iii) explaining how “Root-Calculation” in Algebra is wrong and introducing an alternative method for verifying real and complex roots of a polynomial; iv) solving a six-degree Polynomial equation and a nine-degree Polynomial equation, by introducing new classes of Invariants (“MCN-2 Invariants”) and Homomorphisms. Burrus (2004); Sitton, Burrus, Fox & Treitel (2003); and Lei, Blane & Cooper (1996), had concluded that such higher-order polynomials were impossible to solve. These issues are applicable in nonlinear analysis, evolutionary computation and pattern-analysis – given the discussions in Yannacopoulos, Brindley, Merkin & Pilling (1996); Campos-Canton, Aguirre-Hernandez, Renteria & Gonzalez (2015); Zheng, Takamatsu & Ikeuchi (2010); and Boyer & Goh (2007).

A statistical analysis of current helicity and twist in solar active regions over the phases of the solar cycle using the spectro-polarimeter data of Hinode

Abstract Current helicity and twist of solar magnetic fields are important in characterizing the dynamo mechanism working in the convection zone of the Sun. We have carried out a statistical study on the current helicity of solar active regions observed with the spectro-polarimeter (SP) of the Hinode Solar Optical Telescope (SOT). We used SOT-SP data of 558 vector magnetograms of a total of 80 active regions obtained during the period from 2006 to 2012. We have applied spatial smoothing and division of data points into weak and strong field ranges to compare the contributions from different scales and field strengths. We found that the current helicity follows the “hemispheric sign rule” when weak magnetic fields (absolute field strength &amp;lt; 300 gauss) are considered and no smoothing is applied. On the other hand, the pattern of current helicity fluctuates and violates the hemispheric sign rule when stronger magnetic fields are considered and smoothing of 2${^{\prime\prime}_{.}}$0 (modeled on ground-based observations) is applied. Furthermore, we found a tendency that weak and inclined fields conform to the hemispheric sign rule and strong and vertical fields violate it. These different properties of helicity through the strong and weak magnetic field components give important clues in understanding the solar dynamo as well as the mechanism of formation and evolution of solar active regions.

Sign language, translation and rule of law – deaf people's experiences from encounters with the Norwegian criminal justice system

The UN Convention on the Rights of Persons with Disabilities was ratified in Norway, June 2013. Nordic countries are generally associated with the promotion of human rights as well as disability rights. The Rule of Law Index from the World Justice Project ranks the Nordic countries among the top nations when it comes to high standards of justice (both civil and criminal justice) and rule of law. However, there are also well-known examples of grave miscarriages of justice in cases where disabled people have been involved. These cases address issues of access to justice and questions barriers to effective communication in criminal justice. Taking the situation of deaf people in Norway as a case, this study asks what barriers deaf people face when reporting a crime, being accused of a crime or being a witness in court. The methodological approach is semi-structured, open-ended interviews with deaf persons, professional sign language interpreters, judges, prosecutors, police officers and lawyers. All the info...

DESCARTES' RULE OF SIGNS AND THE IDENTIFIABILITY OF POPULATION DEMOGRAPHIC MODELS FROM GENOMIC VARIATION DATA.

The sample frequency spectrum (SFS) is a widely-used summary statistic of genomic variation in a sample of homologous DNA sequences. It provides a highly efficient dimensional reduction of large-scale population genomic data and its mathematical dependence on the underlying population demography is well understood, thus enabling the development of efficient inference algorithms. However, it has been recently shown that very different population demographies can actually generate the same SFS for arbitrarily large sample sizes. Although in principle this nonidentifiability issue poses a thorny challenge to statistical inference, the population size functions involved in the counterexamples are arguably not so biologically realistic. Here, we revisit this problem and examine the identifiability of demographic models under the restriction that the population sizes are piecewise-defined where each piece belongs to some family of biologically-motivated functions. Under this assumption, we prove that the expected SFS of a sample uniquely determines the underlying demographic model, provided that the sample is sufficiently large. We obtain a general bound on the sample size sufficient for identifiability; the bound depends on the number of pieces in the demographic model and also on the type of population size function in each piece. In the cases of piecewise-constant, piecewise-exponential and piecewise-generalized-exponential models, which are often assumed in population genomic inferences, we provide explicit formulas for the bounds as simple functions of the number of pieces. Lastly, we obtain analogous results for the "folded" SFS, which is often used when there is ambiguity as to which allelic type is ancestral. Our results are proved using a generalization of Descartes' rule of signs for polynomials to the Laplace transform of piecewise continuous functions.

Some remarks about Descartes’ rule of signs

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule of signs which strengthens the available ones while remaining as elementary and concise as the original. On the other hand, we provide new kinds of restrictions on the combined numbers of positive and negative roots.