Common Zeros Research Articles

Level curves of rational functions and unimodular points on rational curves

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.

New Algorithms for a Class of Accretive Variational Inequalities in Banach Spaces

In this paper, for finding a common zero of a finite family of m-accretive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm, we propose an implicit iteration algorithm and an explicit one, based on a convex combination of the steepest-descent method and a composition of resolvents. We also show that our main algorithm contains some iterative ones in literature as special cases. Finally, we give numerical examples for illustration.

On the generalized Bregman projection operator in reflexive Banach spaces

In this paper, we study the generalized Bregman f-projection operator in reflexive Banach spaces. After providing some properties of the generalized Bregman f-projection operator, we propose an iterative algorithm to finding a common fixed point of a finite family of Bregman relatively nonexpansive mappings in reflexive Banach spaces using the generalized Bregman f-projection operator. An application of our algorithm to finding a common zero of a finite family of maximal monotone operators will also be exhibited.

Classification of 3-dimensional pairs of quadratic forms over a field

By an n-dimensional quadratic form over a field F (F-form) we mean a homogeneous quadratic polynomial in n variables with coefficients in F. We call two pairs of n-dimensional quadratic F-forms (f1,g1), (f2,g2) isomorphic if there exists a nondegenerate linear change of variables taking (f1,g1) to (f2,g2). We give a complete classification (up to isomorphism) of 3-dimensional pairs of forms (f,g) over an arbitrary field F of characteristic different from 2. We consider separately the cases of isotropic form f+tg (which is equivalent to existence of a common zero of f and g), and the anisotropic one. In the first case we make a classification depending on det(f+tg), and in the second case depending on the even Clifford algebra C0(f+tg) and det(f+tg).

Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori

Abstract We study upper bounds for the counting function of common zeros of two meromorphic functions in various contexts. The proofs and results are inspired by recent work involving greatest common divisors in Diophantine approximation, to which we introduce additional techniques to take advantage of the stronger inequalities available in Nevanlinna theory. In particular, we prove a general version of a conjectural “asymptotic gcd” inequality of Pasten and the second author, and consider moving targets versions of our results.

On the Type I multivariate zero-truncated hurdle model with applications in health insurance

In the general insurance modeling literature, there has been a lot of work based on univariate zero-truncated models, but little has been done in the multivariate zero-truncation cases, for instance a line of insurance business with various classes of policies. There are three types of zero-truncation in the multivariate setting: only records with all zeros are missing, zero counts for one or some classes are missing, or zeros are completely missing for all classes. In this paper, we focus on the first case, the so-called Type I zero-truncation, and a new multivariate zero-truncated hurdle model is developed to study it. The key idea of developing such a model is to identify a stochastic representation for the underlying random variables, which enables us to use the EM algorithm to simplify the estimation procedure. This model is used to analyze a health insurance claims dataset that contains claim counts from different categories of claims without common zero observations.

Strong Convergence Theorems for Bregman Demigeneralized Mappings in Banach Spaces with Applications

In this paper, we first introduce a new class of mappings called Bregman -demigeneralized mappings in a Banach space E. Then, using Bregman projection method, we prove a convergence theorem for finding a common fixed point of a family of the new mappings in a Banach space E. Furthermore, we apply our method to prove strong convergence theorems of iterative schemes for finding common fixed points of infinitely many nonlinear mappings satisfying the jointly demiclosedness principle in a Banach space E. Our new technique is based on the strong coerciveness of a Bregman function and the method of proof is completely different from those available in the literature. Some application of our results to the solution of common zeros of maximal monotone operators is presented. Finally, an illustrative numerical example is presented. Our results improve and generalize many known results in the current literature.

A Proximal Point Algorithm for Finding a Common Zero of a Finite Family of Maximal Monotone Operators

In this paper, we consider a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in real Hilbert spaces. Also, we give a necessary and sufficient condition for the common zero set of finite operators to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the set of common zeros of operators.

DC Decoupling-Based Three-Phase Three-Level Transformerless PV Inverter Topology for Minimization of Leakage Current

This letter presents a three-phase three-level cascaded photovoltaic (PV) inverter configuration based on the dc decoupling strategy. The given configuration requires two additional decoupling switches for the minimization of the leakage current. The leakage current is minimized by avoiding or restricting the change in terminal voltages, during the common zero state. Further, a common zero state in all the three phases is obtained using a phase opposition disposition pulsewidth modulation technique. Once the common zero state is realized, the dc decoupling switches are used to isolate the PV source and output load. Apart from the terminal voltage, the proposed configuration also reduces the transitions in the common-mode voltage. Further, this letter also presents an analysis of the terminal voltage using the switching function concept. From the given switching function analysis, an analytical expression of the terminal voltage is derived and is used to obtain the analytical waveform, which matches with the simulation results. In addition to the terminal voltage, the leakage current waveform is also given. Fast Fourier transform spectrums of both are shown. These results are further supported by the experimental waveforms which match both simulation and analytical waveforms. Further, the proposed cascaded multilevel inverter (CMLI) is also compared with the conventional CMLI.

Nonlinear Operators as Concerns Convex Programming and Applied to Signal Processing

Splitting methods have received a lot of attention lately because many nonlinear problems that arise in the areas used, such as signal processing and image restoration, are modeled in mathematics as a nonlinear equation, and this operator is decomposed as the sum of two nonlinear operators. Most investigations about the methods of separation are carried out in the Hilbert spaces. This work develops an iterative scheme in Banach spaces. We prove the convergence theorem of our iterative scheme, applications in common zeros of accretive operators, convexly constrained least square problem, convex minimization problem and signal processing.

Bivariate Lagrange interpolation based on Chebyshev points of the second kind

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for $$0<p\leq2$$ , we study the mean convergence of the Lagrange interpolation polynomials.

A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.

Unicity Theorems with Truncated Multiplicities of Meromorphic Mappings in Several Complex Variables for Few Fixed Targets

The purpose of our paper is twofold. Our first aim is to prove a uniqueness theorem for meromorphic mappings of ℂn into ℙN(ℂ) sharing 2N + 2 hyperplanes in the general position with truncated multiplicities, where all common zeros with multiplicities greater than a certain number do not need to be counted. Second, we consider the case of mappings sharing less than 2N +2 hyperplanes. These results improve some recent results.

Approximate solution of zero point problem involving H-accretive maps in Banach spaces and applications

In this manuscript, we introduce two iterative methods for finding the common zeros of two H-accretive mappings in uniformly smooth and uniformly convex Banach spaces. The proposed iterative methods are based on Mann and Halpern iterative methods and viscosity approximation method. Strong convergence results are established for iterative algorithms. Applications based on convex minimization problem, variational inequality problem and equilibrium problem are derived from the main result. Numerical implementation of the main results and application are demonstrated by some examples. Our results extend, generalize, and unify the previously known results given in literature.

Donor and acceptor characteristics of native point defects in GaN

The semiconducting behaviour and optoelectronic response of gallium nitride is governed by point defect processes, which, despite many years of research, remain poorly understood. The key difficulty in the description of the dominant charged defects is determining a consistent position of the corresponding defect levels, which is difficult to derive using standard supercell calculations. In a complementary approach, we take advantage of the embedded cluster methodology that provides direct access to a common zero of the electrostatic potential for all point defects in all charge states. Charged defects polarise a host dielectric material with long-range forces that strongly affect the outcome of defect simulations; to account for the polarisation, we couple embedding with the hybrid quantum mechanical/molecular mechanical approach and investigate the structure, formation and ionisation energies, and equilibrium concentrations of native point defects in wurtzite GaN at a chemically accurate hybrid-density-functional-theory level. N vacancies are the most thermodynamically favourable native defects in GaN, which contribute to the n-type character of as-grown GaN but are not the main source, a result that is consistent with experiment. Our calculations show no native point defects can form thermodynamically stable acceptor states. GaN can be easily doped n-type, but, in equilibrium conditions at moderate temperatures acceptor dopants will be compensated by N vacancies and no significant hole concentrations will be observed, indicating non-equilibrium processes must dominate in p-type GaN. We identify spectroscopic signatures of native defects in the infrared, visible and ultraviolet luminescence ranges and complementary spectroscopies. Crucially, we calculate the effective-mass-like-state levels associated with electrons and holes bound in diffuse orbitals. These levels may be accessible in competition with more strongly-localised states in luminescence processes and allow the attribution of the observed 3.46 and 3.27 eV UV peaks in a broad range of GaN samples to the presence of N vacancies.

New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

Some iterative algorithms for approximating solutions of nonlinear operator equations in Banach spaces

In this paper, iterative algorithms of Krasnoselkii-type and of Halpern-type for approximating an element of the set of common zeros of a countable family of inverse strongly monotone maps, common fixed points of a countable family of totally quasi- $$\phi $$ -asymptotically nonexpansive nonself multi-valued maps, and a solution of a system of generalized mixed equilibrium problems are constructed and studied. Strong convergence of the sequences generated by these algorithms is established in uniformly smooth and 2-uniformly convex real Banach spaces. The theorems obtained extend, improve and generalize several recent important results.

Viscosity iterative techniques for approximating a common zero of monotone operators in an Hadamard space

The main purpose of this paper is to introduce some viscosity-type proximal point algorithms which comprise of a nonexpansive mapping and a finite sum of resolvents of monotone operators, and prove their strong convergence to a common zero of a finite family of monotone operators which is also a fixed point of a nonexpansive mapping and a unique solution of some variational inequality problems in an Hadamard space. We apply our results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems.

Random switching between vector fields having a common zero

Let $E$ be a finite set, $\{F^{i}\}_{i\in E}$ a family of vector fields on $\mathbb{R}^{d}$ leaving positively invariant a compact set $M$ and having a common zero $p\in M$. We consider a piecewise deterministic Markov process $(X,I)$ on $M\times E$ defined by $\dot{X}_{t}=F^{I_{t}}(X_{t})$ where $I$ is a jump process controlled by $X$: ${\mathsf{P}}(I_{t+s}=j|(X_{u},I_{u})_{u\leq t})=a_{ij}(X_{t})s+o(s)$ for $i\neq j$ on $\{I_{t}=i\}$. We show that the behaviour of $(X,I)$ is mainly determined by the behaviour of the linearized process $(Y,J)$ where $\dot{Y}_{t}=A^{J_{t}}Y_{t}$, $A^{i}$ is the Jacobian matrix of $F^{i}$ at $p$ and $J$ is the jump process with rates $(a_{ij}(p))$. We introduce two quantities $\Lambda^{-}$ and $\Lambda^{+}$, respectively, defined as the minimal (resp., maximal) growth rate of $\|Y_{t}\|$, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process $(\Theta,J)$ with $\Theta_{t}=\frac{Y_{t}}{\|Y_{t}\|}$. It is shown that $\Lambda^{+}$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $\Lambda^{-}=\Lambda^{+}$. We then prove that, under certain irreducibility conditions, $X_{t}\rightarrow p$ exponentially fast when $\Lambda^{+}<0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M\setminus \{p\}\times E$ when $\Lambda^{-}>0$. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

Asymptotic gcd and divisible sequences for entire functions

Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .