Nonlinear Photonics

Suitable for both graduate and senior undergraduate students, this textbook offers a logical progression through the underlying principles and practical applications of nonlinear photonics. Building up from essential physics, general concepts, and fundamental mathematical formulations, it provides a robust introduction to nonlinear optical processes and phenomena, and their practical applications in real-world devices and systems. Over 45 worked problems illustrate key concepts and provide hands-on models for students, and over 160 end-of-chapter exercises supply students with plenty of scope to master the material. Accompanied by a complete solutions manual for instructors, including detailed explanations of each result, and drawing on the author's 35 years of teaching experience, this is the ideal introduction to nonlinear photonics for students in electrical engineering.

Transformer Based Multi-model Fusion for Medical Image Segmentation

We present our solutions to the MedAI for all three tasks: polyp segmentation task, instrument segmentation task, and transparency task. We use the same framework to process the two segmentation tasks of polyps and instruments. The key improvement over last year is new state-of-the-art vision architectures, especially transformers which significantly outperform ConvNets for the medical image segmentation tasks. Our solution consists of multiple segmentation models, and each model uses a transformer as the backbone network. we get the best IoU score of 0.915 on the instrument segmentation task and 0.836 on polyp segmentation task after submitting. Meanwhile, we provide complete solutions in https://github.com/dongbo811/MedAI-2021.

Foundations of Engineering Mathematics Applied for Fluid Flows

Based on a brief historical excursion, a list of principles is formulated which substantiates the choice of axioms and methods for studying nature. The axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics. In the theory of fluid flows within the continuous medium model, a key role for the total energy is distinguished. To describe a fluid flow, a system of fundamental equations is chosen, supplemented by the equations of the state for the Gibbs potential and the medium density. The system is supplemented by the physically based initial and boundary conditions and analyzed, taking into account the compatibility condition. The complete solutions constructed describe both the structure and dynamics of non-stationary flows. The classification of structural components, including waves, ligaments, and vortices, is given on the basis of the complete solutions of the linearized system. The results of compatible theoretical and experimental studies are compared for the cases of potential and actual homogeneous and stratified fluid flow past an arbitrarily oriented plate. The importance of studying the transfer and transformation processes of energy components is illustrated by the description of the fine structures of flows formed by a free-falling drop coalescing with a target fluid at rest.

A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane ℍ 2 into ℝ 4

A simple interpretation of the PDE for isometric immersion of the hyperbolic plane ℍ 2 into ℝ 4 is given. Thus, necessary and sufficient conditions are obtained. And, under natural additional conditions, we show that there are no complete solutions, but we can have special local solutions.

Determining Efficient Post-writing Activity for Error Correction: Self-editing, Peer review, or Teacher Feedback?

Writing in English has always been a formidable obstacle for learners; accordingly, many studies aimed to find not band-aid but complete solutions for learners to improve their writing proficiency. One of these solutions, largely thought to reduce language errors, is error correction. However, instructors seem to be alternating between different corrective feedbacks with the purpose of determining the most efficient one for their students. Previous research largely compared peer feedback and teacher correction and ignored self-editing. In this sense, this study investigated three error correction methods, namely self-editing, peer review, and teacher corrections. To achieve this, three student groups were created and each group, composed of 10 students, was tested with one method. Wilcoxon, Kruskal-Wallis, and Mann-Whitney U tests were employed for analyses and the results yielded significant differences in terms of all methods concerning comparisons of pre- and post-tests. On the other hand, the test to determine inter-group differences found significant results for the method of teacher correction. Furthermore, the most frequent linguistic errors in students’ writing were revealed. This research contributes to teaching pedagogy by comforting instructors regarding the efficiency of teacher correction and suggests instructors focus on particularly spelling, punctuation, and article to prompt writing development.

Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g.

Conventional Partial and Complete Solutions of the Fundamental Equations of Fluid Mechanics in the Problem of Periodic Internal Waves with Accompanying Ligaments Generation

The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete solution to the linear problem, which satisfies the boundary conditions on the emitting surface, is constructed taking into account the previously proposed classification of flow structural components described by complete solutions of the linearized system of fundamental equations without involving additional force or mass sources. Analyses includes all components satisfying the dispersion relation that are periodic waves and thin accompanying ligaments, the transverse scale of which is determined by the kinematic viscosity and the buoyancy frequency. Ligaments are located both near the emitting surface and in the bulk of the liquid in the form of wave beam envelopes. Calculations show that in a nonlinear description of all components, both waves and ligaments interact directly with each other in all combinations: waves-waves, waves-ligaments, and ligaments-ligaments. Direct interactions of the components that generate new harmonics of internal waves occur despite the differences in their scales. Additionally, the problem of generating internal waves by a rapidly bi-harmonically oscillating vertical band is considered. If the difference in the frequencies of the spectral components of the band movement is less than the buoyancy frequency, the nonlinear interacting ligaments generate periodic waves as well. The estimates made show that the amplitudes of such waves are large enough to be observed under laboratory conditions.