Hill Differential Equation

	Hill Differential Equation In mathematics, the Hill equation or Hill differential equation is the second-order linear ordinary differential equation : \frac + f(t) y = 0, where f(t) is a periodic function with minimal period \pi and average zero. By these we mean that for all t :f(t+\pi)=f(t), and :\int_0^\pi f(t) \,dt=0, and if p is a number with 0 < p < \pi , the equation $f(t+p) = f(t)$ must fail for some $t$ . It is named after George William Hill, who introduced it in 1886. Because $f(t)$ has period $\pi$ , the Hill equation can be rewritten using the Fourier series of $f(t)$ : : $\frac+\left(\theta_0+2\sum_^\infty \theta_n \cos(2nt)+\sum_^\infty \phi_m \sin(2mt) \right ) y=0.$ Important special cases of Hill's equation include the Mathieu function#Mathieu equation, Mathieu equation (in which only the terms corresponding to ''n'' = 0, 1 are inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Quadrupole Mass Spectrometer In mass spectrometry, the quadrupole mass analyzer (or quadrupole mass filter) is a type of mass analyzer originally conceived by Nobel laureate Wolfgang Paul and his student Helmut Steinwedel. As the name implies, it consists of four cylindrical rods, set parallel to each other. In a quadrupole mass spectrometer (QMS) the quadrupole is the ''mass analyzer'' – the component of the instrument responsible for selecting sample ions based on their mass-to-charge ratio (''m/z''). Ions are separated in a quadrupole based on the stability of their trajectories in the oscillating electric fields that are applied to the rods. Principle of operation The quadrupole consists of four parallel metal rods. Each opposing rod pair is connected together electrically, and a radio frequency (RF) voltage with a DC offset voltage is applied between one pair of rods and the other. Ions travel down the quadrupole between the rods. Only ions of a certain mass-to-charge ratio will reach the detector fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	IEEE Transactions On Antennas And Propagation ''IEEE Transactions on Antennas and Propagation'' is a peer-reviewed scientific journal published by the IEEE Antennas & Propagation Society. It covers research on and applications of all aspects of antenna technology and the propagation of electromagnetic waves. The journal was established in 1952 and since 2022 the Editor-in-chief is Konstantina Nikita. Since 1952, this journal has delivered many thousands of articles on a wide range of topics of interest to specialists, practicing engineers, educators and students in the field of interest of the IEEE Antennas & Propagation Society. The Transactions publishes 12 issues per year and occasional Special Issues amounting to about 6000 pages annually. Abstracting and indexing The journal is abstracted and indexed in the Science Citation Index and Current Contents/Engineering, Computing & Technology. According to the ''Journal Citation Reports'', the journal has a 2023 impact factor The impact factor (IF) or journal impact f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Léon Brillouin Léon Nicolas Brillouin (; August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid-state physics, and information theory. Early life Brillouin was born in Sèvres, near Paris, France. His father, Marcel Brillouin, grandfather, Éleuthère Mascart, and great-grandfather, Charles Briot, were physicists as well. Education From 1908 to 1912, Brillouin studied physics at the École Normale Supérieure, in Paris. From 1911 he studied under Jean Perrin until he left for the Ludwig Maximilian University of Munich (LMU), in 1912. At LMU, he studied theoretical physics with Arnold Sommerfeld. Just a few months before Brillouin's arrival at LMU, Max von Laue had conducted his experiment showing X-ray diffraction in a crystal lattice. In 1913, he went back to France to study at the University of Paris and it was in this year that Niels Bohr submitted his first paper on the Bohr model of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Accelerator Physics Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams and their interaction with accelerator structures by electromagnetic fields. It is also related to other fields: * Microwave engineering (for acceleration/deflection structures in the radio frequency range). Optics with an emphasis on geometrical optics (beam focusing and bending) and laser physics (laser-particle interaction). Computer technology with an emphasis on digital signal processing; e.g., for automated manipulation of the particle beam. * Plasma physics, for the description of intense beams. The experiments conducted with particle accelerators are not regarded as part of accelerator physics, but belong (according to the objectives of the experiments) to, e.g., particle physics, nuclear physics, condensed matter physics or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Quantum Optics Quantum optics is a branch of atomic, molecular, and optical physics and quantum chemistry that studies the behavior of photons (individual quanta of light). It includes the study of the particle-like properties of photons and their interaction with, for instance, atoms and molecules. Photons have been used to test many of the counter-intuitive predictions of quantum mechanics, such as entanglement and teleportation, and are a useful resource for quantum information processing. History Light propagating in a restricted volume of space has its energy and momentum quantized according to an integer number of particles known as photons. Quantum optics studies the nature and effects of light as quantized photons. The first major development leading to that understanding was the correct modeling of the blackbody radiation spectrum by Max Planck in 1899 under the hypothesis of light being emitted in discrete units of energy. The photoelectric effect was further evidence of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Physical Review A ''Physical Review A'' (also known as PRA) is a monthly peer-reviewed scientific journal published by the American Physical Society covering atomic, molecular, and optical physics and quantum information. the editor was Jan M. Rost ( Max Planck Institute for the Physics of Complex Systems). History In 1893, the '' Physical Review'' was established at Cornell University. It was taken over by the American Physical Society (formed in 1899) in 1913. In 1970, ''Physical Review'' was subdivided into ''Physical Review A'', ''B'', ''C'', and ''D''. At that time, section ''A'' was subtitled ''Physical Review A: General Physics''. In 1990, a process was started to split this journal into two, resulting in the creation of '' Physical Review E'' in 1993. Hence, in 1993, ''Physical Review A'' changed its statement of scope to ''Atomic, Molecular and Optical Physics.'' In January 2007, the section of ''Physical Review E'' that published papers on classical optics was merged into ''Physical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Schrödinger Equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	International Journal Of Mass Spectrometry The ''International Journal of Mass Spectrometry'' is a monthly peer-reviewed scientific journal covering all aspects of mass spectrometry, including instrumentation and applications in biology, chemistry, geology, and physics. It was established in 1968 as the ''International Journal of Mass Spectrometry and Ion Physics'' and was renamed ''International Journal of Mass Spectrometry and Ion Processes'' in 1983, before obtaining its current title in 1998. It is published by Elsevier and the editors-in-chief are Julia Laskin (Purdue University) and Zheng Ouyang (Tsinghua University). Abstracting and indexing The journal is abstracted and indexed in: Chemical Abstracts Service Current Contents/Physical, Chemical & Earth Sciences * EBSCO databases * Embase Food Science and Technology Abstracts FRANCIS Inspec PASCAL Science Citation Index Expanded Scopus According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Floquet Theory Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with x\in and \displaystyle A(t) \in being a piecewise continuous periodic function with period T and defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due to , gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change \displaystyle y=Q^(t)x with \displaystyle Q(t+2T)=Q(t) that transforms the periodic system to a traditional linear system with constant, real coefficients. When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem. Note that the solutions of the linear differential equation form a vector space. A matrix \phi\,(t) is called a '' fundamental matrix solution'' if the columns form a basi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinary Differential Equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Providence, Rhode Island Providence () is the List of capitals in the United States, capital and List of municipalities in Rhode Island, most populous city of the U.S. state of Rhode Island. The county seat of Providence County, Rhode Island, Providence County, it is one of the oldest cities in New England, founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port, as it is situated at the mouth of the Providence River at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and List of colleges and universities in Rhode Island#Institutions, eight instit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]