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"Dalam makalah dibahas pengertian dasar bilangan hiperbolik dan penerapannya pada penentuan akar persamaan polinomial. Ditunjukkan bahwa untuk setiap pasang akar rea! berhubungan dengan sepasang bilangan hiperbolik murni yang saling sekawan. Selanjutnya dibuktikan bahwa, banyaknya akar hiperbolik dari persamaan polinomial derajat n adalah n2.

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This paper describe the basic notion of unusual numbers called hyperbolic numbers and it's application on determining the roots of polynomial equation. The objective of this paper is to demonstrate by using a hyperbolic numbers system that for a pair of real root there exists also a pair of hyperbolic root. It was also proved that n-degree polynomial equation has exactly n2 hyperbolic roots."

"Solutions of the wave equation or Maxwell's equations in boundary value and free space problems are analyzed. Hyperbolic systems in domains going off to infinity are studied. New results on Maxwell's equations and non-star shaped reflecting bodies are included."

"ABSTRAKWe consider a generalization of the projection operator method for the case of the Cauchy problem in 1D space for systems of evolution differential equations of first order with variable coefficients. It is supposed that the dependence of coefficients on the only variable𝑥is weak, that is described by the introduction of a small parameter. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics in waveguides with a variable cross section. Projection operators are constructed for the Cauchy problem to fix unidirectional modes. The method of successive approximations perturbation theory is developed and based on the pseudodifferential operators theory. The application of projection operators adapted for the case under consideration allows deriving approximate evolution equations corresponding to the separated directed waves."

"There has been a growing interest in the use of Fourier analysis to examine questions of accuracy and stability of numerical methods for solving partial differential equations. This kind of analysis can produce particularly attractive and useful results for hyperbolic equations. This book provides useful reference material for those concerned with computational fluid dynamics: for physicists and engineers who work with computers in the analysis of problems in such diverse fields as hydraulics, gas dynamics, plasma physics, numerical weather prediction, and transport processes in engineering, and who need to understand the implications of the approximations they use; and for applied mathematicians concerned with the more theoretical aspects of these computations."

"Latar belakang dan tujuan: Anastasia subarahnoid adalah salah satu tindakan anestesia regional yang sexing dilakukan untuk bedah sesar. Bupivakain hiperbarik 0,5% adalah obat anestetik lokal yang lazim dipakai untuk tehnik pembiusan tersebut. Posisi tubuh dan gaya gravitasi memiliki efek dan mempengaruhi penyebaran dari obat yang bersifat hiperbarik. Penelitian ini dilakukan untuk mengetahui pengaruh posisi tubuh saat penyuntikan obat bupivakain hiperbarik 0,5% terhadap efek hipotensi yang ditimbulkan.Metode : Penelitian dilakukan terhadap 90 wanita hamil berstatus ASA I-II usia 17-50 tahun yang menjalani bedah sesar, dibagi secara arak menjadi 2 kelompok duduk dan lateral dekubitus kiri. Setelah dilakukan penyuntikan obat, setelah 2 menit pasien dikembalikan ke posisi terlentang miring kiri 15 derajat, dan dilakukan co loading kristaloid 10 mllkgBB selama 10 menit Dilakukan pencatatan tekanan darah selama operasi setiap 2 menit selama 20 menit pertama clan selanjutnya tiap 5 menit. Ketinggian hambatan sensorik clan ketinggian maksimal hambatan, jumlah total efedrin dan cairan kristaloid yang diberikan selama operasi juga dicatat. Data hasil penelitian diolah dengan menggunakan uji t, uji Mann Whitneydan uji Chi kuadrat.Hasil : Kekerapan hipotensi antara kelompok posisi duduk dan lateral dekubitus kiri tidak berbeda secara statistik meskipun lebih banyak terjadi pada kelompok lateral dekubitus kiri (67%) dibandingkan posisi duduk (51%). Posisi duduk mengalami hipotensi lebih lambat, derajat hipotensinya lebih rendah dan pemakaian efedrin yang lebih sedikit.Kesimpulan: Posisi tubuh saat penyuntikan that bupivakain hiperbarik 0,5% pada anestesia subarahnoid mempengaruhi derajat hipotensi yang terjadi pada kasus bedah sesar.

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Backgrounds and objectives . Spinal anesthesia is one of the regional anesthesia technique frequently performed for cesarean section. Hyperbaric bupivacaine 0.5% is the most frequent local anesthetic used for this technique. Spread of the hyperbaric local anesthetics is affected by the position of the patient and gravity. In the present study we evaluated the effect of maternal posture whether sitting position during the induction of spinal anesthesia using 05% hyperbaric bupivacaine would induce less hypotension as compared with the left lateral position.

Methods. Ninety pregnant women underwent cesarean delivery were randomly assigned to receive a spinal injection consisting of 12.5 mg 0.5% hyperbaric bupivacaine in either sitting or left lateral position. After 2 minutes, patients were turned to a 15 degrees left lateral position and intravenous infusion of 10 mllkgbodyweigh t of crystalloids was started for 10 minutes along with the induction of spinal anesthesia. Intraoperative blood pressure were recorded , in this study hypotension is defined as a decrease in systolic blood pressure less than 100 mmHg or 20% below baseline values. The height of sensory block was measured, time to T6 spread of the sensory block and the highest level of sensory blockade were noted. Total given of ephedrine and crystalloids rntraopertive were also noted. Statistical evaluation was performed using t?test, Mann Whitney test and Chi square as appropriate.

Result : The incidence of hypotension was not significantly different between sitting and left lateral position but more often in lateral position (51% vs 67%). Sitting position group has longer interval of the first hypotension (p=0.008),less severe of hypotension (p=0.042), less ephedrine supplementation (p=0.014), and longer interval for reaching the T6 dermatome blockade (p <0,0001).

Conclusion: Maternal posture during induction of spinal anesthesia using 0.5% hyperbaric bupivacaine has influence to severity of hypotension for cesarean section."

"This text examines various dirichlet problems which can be formulated for equations of keldysh type, one of the two main classes of linear elliptic-hyperbolic equations. Open boundary conditions (in which data are prescribed on only part of the boundary) and closed boundary conditions (in which data are prescribed on the entire boundary) are both considered. Emphasis is on the formulation of boundary conditions for which solutions can be shown to exist in an appropriate function space. Specific applications to plasma physics, optics, and analysis on projective spaces are discussed. "

"This book presents recent progress on uniformly hyperbolic attractors in dynamical systems from a physical rather than mathematical perspective (e.g. the Plykin attractor, the Smale – Williams solenoid). The structurally stable attractors manifest strong stochastic properties, but are insensitive to variation of functions and parameters in the dynamical systems. Based on these characteristics of hyperbolic chaos, this monograph shows how to find hyperbolic chaotic attractors in physical systems and how to design a physical systems that possess hyperbolic chaos. "

"Mathematical control theory for a single partial differential equation (PDE) has dominated the research literature for quite a while, new, complex, and challenging issues have recently arisen in the context of coupled, or interconnected, PDE systems. This has led to a rapidly growing interest, and many unanswered questions, within the PDE community. By concentrating on systems of hyperbolic and parabolic coupled PDEs that are nonlinear, Mathematical Control Theory of Coupled PDEs seeks to provide a mathematical theory for the solution of three main problems: well-posedness and regularity of the controlled dynamics; stabilization and stability; and optimal control for both finite and infinite horizon problems along with existence/uniqueness issues of the associated Riccati equations.Mathematical Control Theory of Coupled PDEs is based on a series of lectures that are outgrowths of recent research in the area of control theory for systems governed by coupled PDEs. The book develops new mathematical tools amenable to a rigorous analysis of related control problems and the construction of viable control algorithms. Emphasis is placed on the key role played by two interweaving features of the respective dynamical components: (1) propagation of singularities and exceptional "sharp" regularity of the traces of the solutions of the structure's hyperbolic component, and (2) analyticity of the solutions to the parabolic component of the structure, its propagation, and related analytic semigroup (singular) estimates.In addition to providing a mathematical foundation on this topic, this book is useful to engineers and professionals involved in materials science and aerospace engineering in solving fundamental theoretical control problems such as stabilization and optimal control in the context of control systems described by dynamical coupled PDEs. Modern technological applications such as smart materials, interactive systems, and intelligent controls drive further interest in this topic. Included is a wealth of examples based on the structural acoustic model. This comprises a wave equation coupled on the interface with either a plate or a shell equation. This canonical model nonetheless displays a variety of phenomena of interest."

"This book primarily focuses on rigorous mathematical formulation and treatment of static problems arising in continuum mechanics of solids at large or small strains, as well as their various evolutionary variants, including thermodynamics. As such, the theory of boundary- or initial-boundary-value problems for linear, quasilinear elliptic, parabolic or hyperbolic partial differential equations is the main underlying mathematical tool, along with the calculus of variations. Modern concepts of these disciplines as weak solutions, polyconvexity, quasiconvexity, nonsimple materials, materials with various rheologies or with internal variables are exploited."

"This book offers engineering students an introduction to the theory of partial differential equations and then guiding them through the modern problems in this subject.Divided into two parts, in the first part readers already well-acquainted with problems from the theory of differential and integral equations gain insights into the classical notions and problems, including differential operators, characteristic surfaces, Levi functions, Green’s function, and Green’s formulas. Readers are also instructed in the extended potential theory in its three forms: the volume potential, the surface single-layer potential and the surface double-layer potential. Furthermore, the book presents the main initial boundary value problems associated with elliptic, parabolic and hyperbolic equations. The second part of the book, which is addressed first and foremost to those who are already acquainted with the notions and the results from the first part, introduces readers to modern aspects of the theory of partial differential equations. "