Analytic and Algebraic Aspects of Integrability for First Order Partial Differential Equations

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.Ministry of Higher Education and Scientific Research-Ira

On the integrability of some three-dimensional Lotka-Volterra equations with rank-1 resonances

We investigate the local integrability in C3 of some three-dimensional Lotka-Volterra equations at the origin with (p : q : r)-resonance, x˙ = P = x(p + ax + by + cz), y˙ = Q = y(q + dx + ey + fz), z˙ = R = z(r + gx + hy + kz). Recent work on this problem has centered on the case where the resonance is of "rank-2". That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ /∈ iR (after a scaling, this is just the case p + q = 0 with q/r /∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0. Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x αy βz γ(1 + O(x, y, z). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural

Liouvillian integrability of three dimensional vector fields

We consider a three dimensional complex polynomial, or rational, vector field (equivalently, a two-form in three variables) which admits a Liouvillian first integral. We prove that there exists a first integral whose differential is the product of a rational 1-form with a Darboux function, or there exists a Darboux Jacobi multiplier. Moreover, we prove that Liouvillian integrability always implies the existence of a first integral that is obtained by two successive integrations from a one-forms with coefficients in a finite algebraic extension of the rational function field.Comment: 20 pages. Changed from first submitted version: Rewritten Abstract and Introduction; simplified and clarified proofs (Lemma 3, Lemmma 5, Theorem 4

Three-dimensional Lotka-Volterra systems with 3:−1:2-Resonance

We study the local integrability at the origin of a nine-parameter family of three-dimensional Lotka-Volterra differential systems with (3:− 1:2)-resonance. We give necessary and sufficient conditions on the parameters of the family that guarantee the existence of two independent local first integrals at the origin of coordinates. Additionally, we classify those cases where the origin is linearizable

Performance evaluation of transfer learning based deep convolutional neural network with limited fused spectro-temporal data for land cover classification

Deep learning (DL) techniques are effective in various applications, such as parameter estimation, image classification, recognition, and anomaly detection. They excel with abundant training data but struggle with limited data. To overcome this, transfer learning is commonly used, leveraging complex learning abilities, saving time, and handling limited labeled data. This study assesses a transfer learning (TL)-based pre-trained “deep convolutional neural network (DCNN)” for classifying land use land cover using a limited and imbalanced dataset of fused spectro-temporal data. It compares the performance of shallow artificial neural networks (ANNs) and deep convolutional neural networks, utilizing multi-spectral sentinel-2 and high-resolution planet scope data. Both machine learning and deep learning algorithms successfully classified the fused data, but the transfer learning-based deep convolutional neural network outperformed the artificial neural network. The evaluation considered a weighted average of F1-score and overall classification accuracy. The transfer learning-based convolutional neural network achieved a weighted average F1-score of 0.92 and a classification accuracy of 0.93, while the artificial neural network achieved a weighted average F1-score of 0.87 and a classification accuracy of 0.89. These results highlight the superior performance of the transfer learned convolutional neural network on a limited and imbalanced dataset compared to the traditional artificial neural network algorithm

Campanian-Maastrichtian Ostracods (Crustacea) From the Shiranish Formation, Dukan Area, Kurdistan Province, Northern Iraq

The present study represents a detailed systematic study of obtained ostracods in sixty three samples from an outcrop section of the Shiranish Formation (Campanian – Maastrichtian), Dukan area, Kurdistan, northern Iraq. Thirty ostracod species belonging to twenty one genera are found, whereas nineteen species are previously recorded. Eleven new described species are detected, which three new species represented in Bairdia dukanensis sp. nov., Bairdoppilata shiranishensis sp. nov. and Hornibrookella nudosa sp. nov. are newly renamed by the authors according to the scientific roles of nomenclature, and nine species are systemacilly described as (sp.) due to the lack of satisfied specimens

Selective harmonic mitigation based two-scale frequency control of cascaded modified packed U-cell inverters

A new Modified Packed U-Cell (MPUC) converter architecture with cascading is proposed in this paper. To provide an output voltage of 25 levels, the proposed cascaded MPUC needs only twelve power switches and four power sources. The converter comprises two cascaded MPUCs with DC supply in a ratio of 5 : 1. One converter is operating at low frequency (LF) and the other at high frequency (HF) that leads to lower power losses and higher levels. Besides, a variable frequency method is anticipated to produce a 25-level output voltage which has low harmonic content (THD) with the help of Selective Harmonic Mitigation (SHM). The optimum switching angles for SHM are obtained through solving the SHM equations using the Genetic Algorithm (GA). The designed controller is efficient and suitable for applications that require low-frequency operation either in stand-alone or grid-tied. The proposed inverter and its operation procedure have been investigated using MATLAB®/Simulink software, and the findings demonstrate that the proposed inverter output voltage has reduced THD significantly. The simulation results are verified using the typhoon HIL-402 emulator. Also, the power loss analysis is done using PLECS. The maximum efficiency of the converter is found to be around 98.34 %. The simulation results justified the efficiency and viability of low 25-level THD voltages