Picard Theorem Research Articles

Value distribution of a pair of meromorphic functions

The well-known Picard theorem shows only what happens on the Picard exceptional value of a meromorphic function f. In this paper, we mainly consider what happens on the common or different Picard exceptional values of two or three meromorphic functions with certain types. For instance, we will provide some new results and discussions for the generalized Picard exceptional values or small functions of a pair of meromorphic functions fng(k) and gmf(k), these two functions are the crossed variants of the complex differential polynomials in Hayman's conjecture, where n, m are positive integers and k≥0. We give more details on the case that fg′ and gf′ when f and g are exponential polynomials in particular.

Picard theorems for moduli spaces of polarized varieties

Picard theorems for moduli spaces of polarized varieties

Fractional-order modelling and analysis of diabetes mellitus: Utilizing the Atangana-Baleanu Caputo (ABC) operator

This article aims to introduce and analyze a diabetes mellitus model of fractional order, utilizing the ABC derivative. Diabetes mellitus is a prevalent and significant disease worldwide, ranking among the top causes of mortality. It is characterized by chronic metabolic dysfunction, leading to elevated blood glucose levels and subsequent damage to vital organs including the nerves, kidneys, eyes, blood vessels, and heart. The fractional ABC derivative is used in this study to describe and analyze diabetes mellitus mathematically while removing hereditary influences. The investigation begins by exploring the initial points of the diabetes mellitus model. Under the fractional ABC operator, Picard's theorem is used to prove the existence and uniqueness of solutions. For the numerical approximation of solutions in the fractional-order diabetes mellitus model, this study used a specialized technique that combines the principles of fractional calculus and a two-step Lagrange polynomial interpolation. Finally, the obtained results are visually presented through graphical representations, serving as empirical evidence to support our theoretical findings. The numerical experiments showed that the proportion of patients with diabetes mellitus increased as the fractional dimension (θ) reduced. The combination of mathematical modelling, analysis, and numerical simulations provides insights into the dynamics of diabetes mellitus, offering valuable contributions to the understanding and management of this prevalent disease. Additionally, the proposed scheme can be enhanced by incorporating the ABC operator, allowing for the simulation of real-world dynamics and behavior in the coexistence of diabetes mellitus and tuberculosis.

Big Picard theorem for jet differentials and non-archimedean Ax-Lindemann theorem

By implementing jet differential techniques in non-archimedean geometry, we obtain a big Picard type extension theorem, which generalizes a previous result of Cherry and Ru. As applications, we establish two hyperbolicity-related results. Firstly, we prove a non-archimedean Ax-Lindemann theorem for totally degenerate abelian varieties. Secondly, we show the pseudo-Borel hyperbolicity for subvarieties of general type in abelian varieties.

Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.

A computational technique for the Caputo fractal-fractional diabetes mellitus model without genetic factors.

The concept of a Caputo fractal-fractional derivative is a new class of non-integer order derivative with a power-law kernel that has many applications in real-life scenarios. This new derivative is applied newly to model the dynamics of diabetes mellitus disease because the operator can be applied to formulate some models which describe the dynamics with memory effects. Diabetes mellitus as one of the leading diseases of our century is a type of disease that is widely observed worldwide and takes the first place in the evolution of many fatal diseases. Diabetes is tagged as a chronic, metabolic disease signalized by elevated levels of blood glucose (or blood sugar), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the sense of fractional-fractal derivative. At first, the critical points of the diabetes mellitus model are investigated; then Picard's theorem idea is applied to investigate the existence and uniqueness of the solutions of the model under the fractional-fractal operator. The resulting discretized system of fractal-fractional differential equations is integrated in time with the MATLAB inbuilt Ode45 and Ode15s packages. A step-by-step and easy-to-adapt MATLAB algorithm is also provided for scholars to reproduce. Simulation experiments that revealed the dynamic behavior of the model for different instances of fractal-fractional parameters in the sense of the Caputo operator are displayed in the table and figures. It was observed in the numerical experiments that a decrease in both fractal dimensions and leads to an increase in the number of people living with diabetes mellitus.

Novel projection neurodynamic approaches for constrained convex optimization

Consider that the constrained convex optimization problems have emerged in a variety of scientific and engineering applications that often require efficient and fast solutions. Inspired by the Nesterov’s accelerated method for solving unconstrained convex and strongly convex optimization problems, in this paper we propose two novel accelerated projection neurodynamic approaches for constrained smooth convex and strongly convex optimization based on the variational approach. First, for smooth, and convex optimization problems, a non-autonomous accelerated projection neurodynamic approach (NAAPNA) is presented and the existence, uniqueness and feasibility of the solution to it are analyzed rigorously. We provide that the NAAPNA has a convergence rate which is inversely proportional to the square of the running time. In addition, we present a novel autonomous accelerated projection neurodynamic approach (AAPNA) for addressing the constrained, smooth, strongly convex optimization problems and prove the existence, uniqueness to the strong global solution of AAPNA based on the Cauchy–Lipschitz–Picard theorem. Furthermore, we also prove the global convergence of AAPNA with different exponential convergence rates for different parameters. Compared with existing projection neurodynamic approaches based on the Brouwer’s fixed point theorem, both NAAPNA and AAPNA use the projection operators of the auxiliary variable to map the primal variables to the constrained feasible region, thus our proposed neurodynamic approaches are easier to realize algorithm’s acceleration. Finally, the effectiveness of NAAPNA and AAPNA is illustrated with several numerical examples.

Image restoration via Picard's and Mountain-pass Theorems

<abstract><p>In this work, we present existence results for some problems which arise in image processing namely image restoration. Our essential tools are Picard's fixed point theorem for a strict contraction and Mountain-pass Theorem for critical point.</p></abstract>

Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.

An Extension of the Picard Theorem to Fractional Differential Equations with a Caputo-Fabrizio Derivative

In this paper, we consider fractional differential equations with the new fractional derivative involving a nonsingular kernel, namely, the Caputo-Fabrizio fractional derivative. Using a successive approximation method, we prove an extension of the Picard-Lindelöf existence and uniqueness theorem for fractional differential equations with this derivative, which gives a set of conditions, under which a fractional initial value problem has a unique solution.

Smoothing inertial neurodynamic approach for sparse signal reconstruction via [formula omitted]-norm minimization

In this paper, we propose a smoothing inertial neurodynamic approach (SINA) which is used to deal with Lp-norm minimization problem to reconstruct sparse signals. Note that the considered optimization problem is nonsmooth, nonconvex and non-Lipschitz. First, the problem is transformed into a smooth optimization problem based on smoothing approximation method, and the Lipschitz property of gradient of the smooth objective function is discussed. Then, SINA based on Karush–Kuhn–Tucker (KKT) condition, smoothing approximation and inertial dynamical approach, is designed to handle smooth optimization problem. The existence, uniqueness, global convergence and optimality of the solution of the SINA are discussed by the Cauchy–Lipschitz–Picard theorem, energy function and KKT condition. In addition, for p=1, the SINA has a mean sublinear convergence rate O1∕t under some mild conditions. Finally, some numerical examples on sparse signal reconstruction and image restoration are given to illustrate the theoretical results and the efficiency of SINA.

Generalizations of Picard's theorem with moving hypersurfaces

Generalizations of Picard's theorem with moving hypersurfaces

A Study of Banach Fixed Point Theorem and It’s Applications

This paper aims at treating a study of Banach fixed point theorem for mapping results that introduced in the setting of normed space. The classical Banach fixed point theorem is a generalization of this work. A fixed point theory is a beautiful mixture of Mathematical analysis to explain some conditions in which maps give excellent solutions. Here later many mathematicians used this fixed point theory to establish their results, see for instance, Picard-Lindel of Theorem, The Picard theorem, Implicit function theorem etc. Also, we developed ideas that many of known fixed point theorems can easily be derived from the Banach theorem. It extends some recent works on the extension of Banach contraction principle to metric space with norm spaces.

Picard Theorem for Holomorphic Curves from a Punctured Disc into $${\mathbb {P}}^n({\mathbb {C}})$$ with Few Hypersurfaces in Subgeneral Position

In this paper, we will prove a big Picard’s theorem for holomorphic curves from a punctured disc into $${\mathbb {P}}^n({\mathbb {C}})$$ with $$q \ (q>(N-n+1)(n+1))$$ hypersurfaces which are located in N-subgeneral position.

On a quaternionic Picard theorem

The classical theorem of Picard states that a non-constant holomorphic function f : C → C f:\mathbb {C}\to \mathbb {C} can avoid at most one value. We investigate how many values a non-constant slice regular function of a quaternionic variable f : H → H f:\mathbb {H}\to \mathbb {H} may avoid.

On factorization of [formula omitted]-adic meromorphic functions

In this paper, we study primeness and pseudo primeness of p-adic meromorphic functions. We also consider left (resp. right ) primeness of these functions. We give, in particular, sufficient conditions for a meromorphic function to satisfy such properties. Finally, we consider the problem of permutability of entire functions. We generalize some results in the complex case. However, our method, based on the use of some p-adic tools such as the p-adic Picard theorem or the Wronskian of p-adic entire functions, enables us to obtain some results that are completely different from those of the complex case.

Schwarz‐type lemmas for generalized holomorphic maps between pseudo‐Hermitian manifolds and Hermitian manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds and Hermitian manifolds. By Bochner formulas and comparison theorems, we establish related Schwarz type results. As corollaries, Liouville theorem and little Picard theorem for basic CR functions are deduced. Finally, we study CR Carath\'eodory pseudodistance on CR manifolds.

On the range of harmonic maps in the plane

In 1994 J. Lewis obtained a purely harmonic proof of the classical Little Picard Theorem by showing that if the joint value distribution of two entire harmonic functions satisfies certain restrictions then they are necessarily constant. We generalize Lewis'theorem and the harmonic Liouville theorem in terms of the range of a harmonic map in the plane.

The Rickman–Picard theorem

We give a new and conceptually simple proof of the Rickman-Picard theorem for quasiregular maps based on potential-theoretic methods.

Random differential equations with discrete delay

In this article, we study random differential equations with discrete delay with initial condition The uncertainty in the problem is reflected via the outcome ω. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.