This book introduces plane curves on time scales. They are deducted the Frenet equations for plane and space curves. In the book is presented the basic theory of surfaces on time scales. They are defined tangent plane, \sigma_1 and \sigma_2 tangent planes, normal, \sigma_1 and \sigma_2 normals to a surface. They are introduced differentiable maps and differentials on surface. This book provides the first and second fundamental forms of surfaces on time scales. They are introduced minimal surfaces and geodesics on time scales. In the book are studied the covaraint derivatives on time scales, pseudo-spherical surfaces and \sigma_1, \sigma_2 manifolds on time scales.
This book introduces multiplicative Frenet curves. We define multiplicative tangent, multiplicative normal, and multiplicative normal plane for a multiplicative Frenet curve. We investigate the local behaviours of a multiplicative parameterized curve around multiplicative biregular points, define multiplicative Bertrand curves and investigate some of their properties. A multiplicative rigid motion is introduced. The book is addressed to instructors and graduate students, and also specialists in geometry, mathematical physics, differential equations, engineering, and specialists in applied sciences. The book is suitable as a textbook for graduate and under-graduate level courses in geometry and analysis. Many examples and problems are included. The author introduces the main conceptions for multiplicative surfaces: multiplicative first fundamental form, the main multiplicative rules for differentiations on multiplicative surfaces, and the main multiplicative regularity conditions for multiplicative surfaces. An investigation of the main classes of multiplicative surfaces and second fundamental forms for multiplicative surfaces is also employed. Multiplicative differential forms and their properties, multiplicative manifolds, multiplicative Einstein manifolds and their properties, are investigated as well. Many unique applications in mathematical physics, classical geometry, economic theory, and theory of time scale calculus are offered.
Many estuaries are located in urbanized, highly engineered environments. Cohesive sediment plays an important role due to its link with estuarine health and ecology. An important ecological parameter is the suspended sediment concentration (SSC) translated into turbidity levels and sediment budget. This study contributes to investigate and forecast turbidity levels and sediment budget variability at San Francisco Bay-Delta system at a variety of spatial and temporal scales applying a flexible mesh process-based model (Delft3D FM). It is possible to have a robust sediment model, which reproduces 90% of the yearly data derived sediment budget, with simple model settings, like applying one mud fraction and a simple bottom sediment distribution. This finding opens the horizon for modeling less monitored estuaries. Comparing two case studies, i.e. the Sacramento-San Joaquin Delta and Alviso Slough, a classification for estuaries regarding the main sediment dynamic forcing is proposed: event-driven estuary (Delta) and tide-driven estuary (Alviso Slough). In the event-driven estuaries, the rivers are the main sediment source and the tides have minor impact in the net sediment transport. In the tide-driven estuaries, the main sediment source is the bottom sediment and the tide asymmetry defines the net sediment transport. This research also makes advances in connecting different scientific fields and developing a managerial tool to support decision making. It provides the basis to a chain of models, which goes from the hydrodynamics, to suspended sediment, to phytoplankton, to fish, clams and marshes.
This book is devoted to the multiplicative differential calculus. Its seven pedagogically organized chapters summarize the most recent contributions in this area, concluding with a section of practical problems to be assigned or for self-study. Two operations, differentiation and integration, are basic in calculus and analysis. In fact, they are the infinitesimal versions of the subtraction and addition operations on numbers, respectively. From 1967 till 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. It is also called an alternative or non-Newtonian calculus. Multiplicative calculus can especially be useful as a mathematical tool for economics, finance, biology, and engineering. Multiplicative Differential Calculus is written to be of interest to a wide audience of specialists such as mathematicians, physicists, engineers, and biologists. It is primarily a textbook at the senior undergraduate and beginning graduate level and may be used for a course on differential calculus. It is also for students studying engineering and science. Authors Svetlin G. Georgiev is a mathematician who has worked in various areas of the study. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales. He is also the author of Dynamic Geometry of Time Scales (CRC Press). He is a co-author of Conformable Dynamic Equations on Time Scales, with Douglas R. Anderson (CRC Press). Khaled Zennir earned his PhD in mathematics from Sidi Bel Abbès University, Algeria. He earned his highest diploma in Habilitation in Mathematics from Constantine University, Algeria. He is currently Assistant Professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and long-time behavior. The authors have also published: Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDE; Boundary Value Problems on Time Scales, Volume 1 and Volume II, all with CRC Press.
Have you ever wondered how the space and time of our everyday lives works - and why? David Pearcey is here to help, with A Holistic Story of Space and Time, a thorough exploration through our world of space and time. Explaining how space and time works as a complete system, the book is helped by brief accounts of the contributions of some of the great scientists and philosophers who helped us understand its components. Intended for the general reader who has no previous technical understanding, A Holistic Story of Space and Time delves into several areas - the types of geometries of space, the motion of matter in space, and how the force fields of matter pervade space, as well as how we perceive space and time by treating the brain as an information management system. It shows how the process of perception allows us to determine the true nature of the geometry of the space and time of the real world. Much more is also looked into in detail such as Einstein's special and general theories of relativity including his unified field theory, electromagnetism, and quantum physics, including charts and diagrams to explain some of the concepts involved. The final part of the book investigates the relationship between us who perceive the real world and the space and time of the real world, using the ideas developed by the philosophers Kant and Schopenhauer. This all combines to give the reader a uniquely broad look into our world and explains how it works as a total entity, from the cosmic world of the curved geometry of general relativity to the mysterious quantum world and then the philosophical aspects of how we are part of it. Anyone with an interest in the way things work will be well-suited to this extraordinary book that answers the why as well as the what and how.
On becoming familiar with difference equations and their close re lation to differential equations, I was in hopes that the theory of difference equations could be brought completely abreast with that for ordinary differential equations. [HUGH L. TURRITTIN, My Mathematical Expectations, Springer Lecture Notes 312 (page 10), 1973] A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both. [E. T. BELL, Men of Mathematics, Simon and Schuster, New York (page 13/14), 1937] The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis [159] in 1988 (supervised by Bernd Aulbach) in order to unify continuous and discrete analysis. This book is an intro duction to the study of dynamic equations on time scales. Many results concerning differential equations carryover quite easily to corresponding results for difference equations, while other results seem to be completely different in nature from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice, once for differential equa tions and once for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which is an arbitrary nonempty closed subset of the reals.
Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.
