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23,700 | In the magnetosphere, the plasma sheet is a sheet-like region of denser (0.3-0.5 ions/cm3 versus 0.01-0.02 in the lobes) hot plasma and lower magnetic field located on the magnetotail and near the equatorial plane, between the magnetosphere's north and south lobes.The origin of the plasma sheet is still a subject of di... | Plasma sheet | 0.80868 |
23,701 | Model organisms have been used in the study of MYO7A function. A spontaneous mutant mouse line, called Myo7ash1-6J was generated. Male and female animals underwent a standardized phenotypic screen to determine the effects of deletion. Twenty three tests were carried out on mutant mice and ten significant abnormalities ... | MYO7A | 0.808677 |
23,702 | The theory of ordinary differential equations shows that if f(t, v) is smooth then the differential equation dv/dt = f(t, v) with initial condition v(0) = v0 has a unique solution for |t| sufficiently small and the solution depends smoothly on t and v0. This implies that for sufficiently small tangent vectors v at a gi... | Differentiable surface | 0.808676 |
23,703 | Already in 1840, F. Minding, a student of Gauss, had obtained trigonometric formulas for the pseudosphere identical to those for the hyperbolic plane. The intrinsic geometry of this surface is now better understood in terms of the Poincaré metric on the upper half plane or the unit disc, and has been described by other... | Differentiable surface | 0.808676 |
23,704 | Particular cases are obtained by taking φ(t) =C cosh t, C sinh t and C et. The latter case is the classical pseudosphere generated by rotating a tractrix around a central axis. In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the hyperbolic plane, discovered independ... | Differentiable surface | 0.808676 |
23,705 | Hilbert's theorem (1901): no complete surface with constant negative curvature can be immersed isometrically in E3.The Willmore conjecture. This conjecture states that the integral of the square of the mean curvature of a torus immersed in E3 should be bounded below by 2π2. | Differentiable surface | 0.808676 |
23,706 | This improved a theorem of Bonnet who showed in 1855 that the diameter of a closed surface of positive Gaussian curvature is always bounded above by δ; in other words a geodesic realising the metric distance between two points cannot have length greater than δ. Rigidity. In 1927 Cohn-Vossen proved that two ovaloids – c... | Differentiable surface | 0.808676 |
23,707 | Although the characterisation of curvature involves only the local geometry of a surface, there are important global aspects such as the Gauss–Bonnet theorem, the uniformization theorem, the von Mangoldt-Hadamard theorem, and the embeddability theorem. There are other important aspects of the global geometry of surface... | Differentiable surface | 0.808676 |
23,708 | Since every compact oriented 2-manifold M can be triangulated by small geodesic triangles, it follows that ∫ M K d A = 2 π χ ( M ) {\displaystyle \int _{M}KdA=2\pi \,\chi (M)} where χ(M) denotes the Euler characteristic of the surface. In fact if there are F faces, E edges and V vertices, then 3F = 2E and the left hand... | Differentiable surface | 0.808676 |
23,709 | Geodesic polar coordinates are obtained by combining the exponential map with polar coordinates on tangent vectors at the base point. The Gaussian curvature of the surface is then given by the second order deviation of the metric at the point from the Euclidean metric. In particular the Gaussian curvature is an invaria... | Differentiable surface | 0.808676 |
23,710 | The Gauss-Codazzi equations can also be succinctly expressed and derived in the language of connection forms due to Élie Cartan. In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H2 − |h|2 = R and the two Codazzi equations can be wr... | Differentiable surface | 0.808676 |
23,711 | As said by Marcel Berger: This theorem is baffling. It is the kind of theorem which could have waited dozens of years more before being discovered by another mathematician since, unlike so much of intellectual history, it was absolutely not in the air. To our knowledge there is no simple geometric proof of the theorema... | Differentiable surface | 0.808676 |
23,712 | This is known as the theorema egregium, and was a major discovery of Carl Friedrich Gauss. It is particularly striking when one recalls the geometric definition of the Gaussian curvature of S as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry o... | Differentiable surface | 0.808676 |
23,713 | The prescription is determined by a first-order ordinary differential equation which is specified by the first fundamental form. The above concepts are essentially all to do with multivariable calculus. The Gauss-Bonnet theorem is a more global result, which relates the Gaussian curvature of a surface together with its... | Differentiable surface | 0.808676 |
23,714 | Thus, geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface. One can also define parallel transport along any given curve, which gives a prescription for how to deform a tangent vector to the surface at one point of the curve to tangent vect... | Differentiable surface | 0.808676 |
23,715 | The second fundamental form, by contrast, is an object which encodes how lengths and angles of curves on the surface are distorted when the curves are pushed off of the surface. Despite measuring different aspects of length and angle, the first and second fundamental forms are not independent from one another, and they... | Differentiable surface | 0.808676 |
23,716 | The best-known examples are catenoids and helicoids, although many more have been discovered. Minimal surfaces can also be defined by properties to do with surface area, with the consequence that they provide a mathematical model for the shape of soap films when stretched across a wire frame ruled surfaces, which are s... | Differentiable surface | 0.808676 |
23,717 | It is intuitively quite familiar to say that the leaf of a plant, the surface of a glass, or the shape of a face, are curved in certain ways, and that all of these shapes, even after ignoring any distinguishing markings, have certain geometric features which distinguish one from another. The differential geometry of su... | Differentiable surface | 0.808676 |
23,718 | An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in t... | Differentiable surface | 0.808676 |
23,719 | In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsic... | Differentiable surface | 0.808676 |
23,720 | In the diagrams below, when the input connected to the left corner of the diamond is positive, and the input connected to the right corner is negative, current flows from the upper supply terminal to the right along the red (positive) path to the output and returns to the lower supply terminal through the blue (negativ... | Full Bridge Rectifier | 0.808675 |
23,721 | To perform an accurate analysis a structural engineer must determine information such as structural loads, geometry, support conditions, and material properties. The results of such an analysis typically include support reactions, stresses and displacements. This information is then compared to criteria that indicate t... | Structural Analysis | 0.808674 |
23,722 | The equations of elasticity are a system of 15 partial differential equations. Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method such as the finite element method is necessary. | Structural Analysis | 0.808674 |
23,723 | The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories (such as the other two discussed here) for equations to solve. It does, however, make it generally possible to solve these equations, even with highly complex geometry and... | Structural Analysis | 0.808674 |
23,724 | Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires the solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution ... | Structural Analysis | 0.808674 |
23,725 | Each method has noteworthy limitations. The method of mechanics of materials is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory of elasticity allow... | Structural Analysis | 0.808674 |
23,726 | Structural analysis uses ideas from applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, support reactions, velocity, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often precluding phy... | Structural Analysis | 0.808674 |
23,727 | The finite element approach is actually a numerical method for solving differential equations generated by theories of mechanics such as elasticity theory and strength of materials. However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary ... | Structural Analysis | 0.808673 |
23,728 | 1452–1519 Leonardo da Vinci made many contributions 1638: Galileo Galilei published the book "Two New Sciences" in which he examined the failure of simple structures 1660: Hooke's law by Robert Hooke 1687: Isaac Newton published "Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion 1... | Structural Analysis | 0.808673 |
23,729 | Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structur... | Structural Analysis | 0.808673 |
23,730 | To establish the stiffness (or flexibility) of a particular element, we can use the mechanics of materials approach for simple one-dimensional bar elements, and the elasticity approach for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by mean... | Structural Analysis | 0.808673 |
23,731 | It is common practice to use approximate solutions of differential equations as the basis for structural analysis. This is usually done using numerical approximation techniques. The most commonly used numerical approximation in structural analysis is the Finite Element Method. The finite element method approximates a s... | Structural Analysis | 0.808673 |
23,732 | In the context to structural analysis, a structure refers to a body or system of connected parts used to support a load. Important examples related to Civil Engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, an... | Structural Analysis | 0.808673 |
23,733 | Laminar Research is a small software company based in Columbia, South Carolina, and dedicated to providing software that accurately reflects the laws of physics. Laminar's flagship product is the flight simulator X-Plane. The game works with Macintosh, Microsoft Windows, and Linux. They also have mobile versions for iP... | Laminar Research | 0.808673 |
23,734 | In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature. The first SPO algorithm was proposed for two-dimensional unconstrained optimization based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional s... | Spiral optimization algorithm | 0.808671 |
23,735 | The algorithms with the above settings are deterministic. Thus, incorporating some random operations make this algorithm powerful for global optimization. Cruz-Duarte et al. demonstrated it by including stochastic disturbances in spiral searching trajectories. However, this door remains open to further studies. To find... | Spiral optimization algorithm | 0.808671 |
23,736 | In probability theory, Luce's choice axiom, formulated by R. Duncan Luce (1959), states that the probability of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool. Selection of this kind is said to have "independence from irrelevant alternativ... | Luce's choice axiom | 0.80867 |
23,737 | Define the matching law selection rule P ( a | A ) = u ( a ) ∑ a ∈ A u ( a ) {\displaystyle P(a|A)={\frac {u(a)}{\sum _{a\in A}u(a)}}} , for some "value" function u: A → ( 0 , ∞ ) {\displaystyle u:A\to (0,\infty )} . This is sometimes called the softmax function, or the Boltzmann distribution. Theorem: Any matching law... | Luce's choice axiom | 0.80867 |
23,738 | The Moby project defines three Ontologies that describe biological data-types, biological data-formats, and bioinformatics analysis types. Most of the interoperable behaviours seen in Moby are achieved through the Object (data-format) and Namespace (data-type) ontologies. The MOBY Namespace Ontology is derived from the... | BioMOBY | 0.808669 |
23,739 | BioMOBY is a registry of web services used in bioinformatics. It allows interoperability between biological data hosts and analytical services by annotating services with terms taken from standard ontologies. BioMOBY is released under the Artistic License. | BioMOBY | 0.808669 |
23,740 | Until 1970, the Mathematisches Seminar was covering all of mathematics, including applied mathematics, stochastics, and statistics. After the reform of the university, it became the research institute of pure mathematics. As an institution, the Mathematisches Seminar was dissolved in 1999 and the informal cooperation w... | Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | 0.808668 |
23,741 | The Activity parameter is the total power of the signal. It is also the surface of the power spectrum in the frequency domain (Parseval's theorem). The Mobility parameter is determined as the square root of the ratio of the variance of the first derivative of the signal to that of the signal. | Hjorth parameters | 0.808668 |
23,742 | Alexandrov's uniqueness theorem states that a convex polyhedron in three dimensions is uniquely determined by the metric space of geodesics on its surface. Rigidity results in K-theory show isomorphisms between various algebraic K-theory groups. Rigid groups in the inverse Galois problem. | Rigidity (mathematics) | 0.808665 |
23,743 | A well-ordered set is rigid in the sense that the only (order-preserving) automorphism on it is the identity function. Consequently, an isomorphism between two given well-ordered sets will be unique. Cauchy's theorem on geometry of convex polytopes states that a convex polytope is uniquely determined by the geometry of... | Rigidity (mathematics) | 0.808665 |
23,744 | By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. By the previous example, a polynomial is also determined within the set of holomorphic functions by the finite set of its non-zero deriv... | Rigidity (mathematics) | 0.808665 |
23,745 | Some examples include: Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by ... | Rigidity (mathematics) | 0.808665 |
23,746 | The calculation of scattering amplitudes in theoretical particle physics requires the use of some rather large and complicated integrals over a large number of variables. These integrals do, however, have a regular structure, and may be represented as Feynman diagrams. The appeal of the Feynman diagrams is strong, as i... | Virtual particle | 0.808665 |
23,747 | Heating and cooling the material affects both the temperature and the thermodynamic free energy or Gibbs energy. Simulated annealing can be used for very hard computational optimization problems where exact algorithms fail; even though it usually achieves an approximate solution to the global minimum, it could be enoug... | Deterministic annealing | 0.808664 |
23,748 | Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. For large numbers of local optima, SA can find the global optima. It is often used w... | Deterministic annealing | 0.808664 |
23,749 | Important classes of flows are curvature flows, variational flows (which extremize some functional), and flows arising as solutions to parabolic partial differential equations. A given flow frequently admits all of these interpretations, as follows. Given an elliptic operator L , {\displaystyle L,} the parabolic PDE u ... | Geometric flow | 0.808663 |
23,750 | Intrinsic geometric flows are flows on the Riemannian metric, independent of any embedding or immersion. Ricci flow, as in the solution of the Poincaré conjecture, and Richard S. Hamilton's proof of the uniformization theorem Calabi flow, a flow for Kähler metrics Yamabe flow | Geometric flow | 0.808663 |
23,751 | In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with a formal meaning, but is typically understood to refer to parabolic parti... | Geometric flow | 0.808663 |
23,752 | The nano and micro versions of electroporation feature much higher precision and control over the size and location of membrane disruptions imposed on target cells . A company called Maxcyte has developed a high-throughput version of flow electroporation that can process hundreds of millions of cells in tens of minutes... | Intracellular delivery | 0.808651 |
23,753 | Advancements in microfabrication, nanotechnology, chemistry and other research fields have contributed to the improvements in precision and performance of intracellular delivery methods .Electroporation. Early versions of electroporation used bulk electrodes to apply electrical pulses of defined voltage to cells in sol... | Intracellular delivery | 0.808651 |
23,754 | AAV is as a prominent example of improvements made in viral vectors. Vector engineering can increase AAV transduction efficiency (by optimizing the transgene cassette), vector tropism (using capsid engineering) and the ability of the capsid and transgene to avoid the host immune response (by genetically modifying these... | Intracellular delivery | 0.808651 |
23,755 | Most medically applicable ASOs are naked molecules that are able to enter cells through endocytosis and exert their therapeutic effects by binding their intracellular target .Improved Viral Vectors. Nearly 70% of gene therapy clinical trials have utilized viral vectors for the gene delivery step . Adenovirus, Adeno-ass... | Intracellular delivery | 0.808651 |
23,756 | mRNA medicines. Advances in lipid nanoparticle formulation and nucleic acid chemistry have been critical in the development of nucleic acid therapeutics, such as mRNA vaccines. For example, design of the cationic ionizable lipids, which are a key component of lipid nanoparticle formulations, with an acid dissociation c... | Intracellular delivery | 0.808651 |
23,757 | Transfection refers to the intracellular delivery of nucleic acids: DNA, RNA and their analogues. Nucleic acids materials that are commonly transfected into cells are plasmid DNA, mRNA, siRNA, and oligonucleotides. The transfection applications span across 3 main areas: Basic biological research, biomanufacture, and ge... | Intracellular delivery | 0.808651 |
23,758 | Efficiencies of around 1% endosomal escape have been reported for most non-viral carrier strategies, including lipid nanoparticles and cell-penetrating peptides . Moreover, the exact mechanisms of endosome escape remain unclear and are a matter of ongoing research .Apart from endocytosis, some carriers are able to dire... | Intracellular delivery | 0.808651 |
23,759 | The internalization pathways employed by target cells depend on the size, shape, material composition, surface chemistry, and/or charge of the carrier . Cargo not able to escape endosomes are trafficked to lysosomes for degradation or recycled back to the cell surface . | Intracellular delivery | 0.808651 |
23,760 | Carriers that have been used for intracellular delivery include: Polymer assemblies Lipid assemblies Liposomes (hollow with aqueous interior, unlike lipid assemblies) Salt complexes (e.g. CaPO4-nucleic acid condensates) Exosomes and other bio-inspired vesicles Cell-penetrating peptides (CPPs), also known as protein tra... | Intracellular delivery | 0.808651 |
23,761 | Carrier-mediated delivery techniques package the cargo into or onto a nanoscale carrier, which then enters the cell to deliver the cargo. Carriers generally gain entry to the cell interior via either Endocytosis (the majority of carriers) or Fusion with the cell's plasma membraneHowever, there are rare reports of certa... | Intracellular delivery | 0.808651 |
23,762 | Industrial applications include protein production , biomanufacture , and genetic engineering of plants and animals . Intracellular delivery is a fundamental technique in the study of biology and genetics, such as the use of DNA plasmid transfection to investigate protein function in living cells . A wide range of appr... | Intracellular delivery | 0.808651 |
23,763 | Mach number is a function of temperature and true airspeed. Aircraft flight instruments, however, operate using pressure differential to compute Mach number, not temperature. Assuming air to be an ideal gas, the formula to compute Mach number in a subsonic compressible flow is found from Bernoulli's equation for M < 1 ... | Mach Number | 0.80865 |
23,764 | The ability of yeast to convert sugar into ethanol has been harnessed by the biotechnology industry to produce ethanol fuel. The process starts by milling a feedstock, such as sugar cane, field corn, or other cereal grains, and then adding dilute sulfuric acid, or fungal alpha amylase enzymes, to break down the starche... | Yeast cell | 0.808648 |
23,765 | (Squeeze theorem) If a n ≤ c n ≤ b n {\displaystyle a_{n}\leq c_{n}\leq b_{n}} for all n {\displaystyle n} greater than some N {\displaystyle N} , and lim n → ∞ a n = lim n → ∞ b n = L {\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L} , then lim n → ∞ c n = L {\displaystyle \lim _{n\to \infty }c_{n}... | Divergent sequence | 0.808648 |
23,766 | The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept. Additionally, it is not fund... | The fundamental theorem of algebra | 0.808648 |
23,767 | The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with i... | The fundamental theorem of algebra | 0.808648 |
23,768 | The coefficients Rp(x)(a, b) and Sp(x)(a, b) are independent of x and completely defined by the coefficients of p(x). In terms of representation, Rp(x)(a, b) and Sp(x)(a, b) are bivariate polynomials in a and b. In the flavor of Gauss's first (incomplete) proof of this theorem from 1799, the key is to show that for any... | The fundamental theorem of algebra | 0.808648 |
23,769 | On the first and fourth Gaussian proofs of the Fundamental Theorem of Algebra). Weierstraß, Karl (1891), "Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen", Sitzungsberichte der königlich preussische... | The fundamental theorem of algebra | 0.808648 |
23,770 | An extension of a work of Hellmuth Kneser on the Fundamental Theorem of Algebra). Ostrowski, Alexander (1920), "Über den ersten und vierten Gaußschen Beweis des Fundamental-Satzes der Algebra", Carl Friedrich Gauss Werke Band X Abt. 2 (tr. | The fundamental theorem of algebra | 0.808648 |
23,771 | Kneser, Martin (1981), "Ergänzung zu einer Arbeit von Hellmuth Kneser über den Fundamentalsatz der Algebra", Mathematische Zeitschrift, vol. 177, no. 2, pp. 285–287, doi:10.1007/BF01214206, ISSN 0025-5874, S2CID 122310417 (tr. | The fundamental theorem of algebra | 0.808648 |
23,772 | New proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree). Gauss, Carl Friedrich (1866), Carl Friedrich Gauss Werke, vol. | The fundamental theorem of algebra | 0.808648 |
23,773 | English translation: Euler, Leonhard (1751), "Investigations on the Imaginary Roots of Equations" (PDF), Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin, Berlin, vol. 5, pp. 222–288 Gauss, Carl Friedrich (1799), Demonstratio nova theorematis omnem functionem algebraicam rationalem integram un... | The fundamental theorem of algebra | 0.808648 |
23,774 | van der Waerden, Bartel Leendert (2003), Algebra, vol. I (7th ed. ), Springer-Verlag, ISBN 978-0-387-40624-4 | The fundamental theorem of algebra | 0.808648 |
23,775 | 54, no. 3, pp. 333–341, doi:10.1098/rsnr.2000.0116, ISSN 0035-9149, S2CID 145593806 Taylor, Paul (2 June 2007), Gauss's second proof of the fundamental theorem of algebra – English translation of Gauss's second proof. | The fundamental theorem of algebra | 0.808648 |
23,776 | ), Numbers, Graduate Texts in Mathematics 123, Berlin: Springer-Verlag, ISBN 978-0-387-97497-2 Shipman, Joseph (2007), "Improving the Fundamental Theorem of Algebra", Mathematical Intelligencer, vol. 29, no. 4, pp. 9–14, doi:10.1007/BF02986170, ISSN 0343-6993, S2CID 123089882 Smale, Steve (1981), "The Fundamental Theor... | The fundamental theorem of algebra | 0.808648 |
23,777 | 2, Éditions Jacques Gabay (published 1992), ISBN 978-2-87647-101-6 (tr. The rational functions §80–88: the fundamental theorem). Remmert, Reinhold (1991), "The Fundamental Theorem of Algebra", in Ebbinghaus, Heinz-Dieter; Hermes, Hans; Hirzebruch, Friedrich (eds. | The fundamental theorem of algebra | 0.808648 |
23,778 | On the history of the fundamental theorem of algebra: theory of equations and integral calculus.) Netto, Eugen; Le Vavasseur, Raymond (1916), "Les fonctions rationnelles §80–88: Le théorème fondamental", in Meyer, François; Molk, Jules (eds. ), Encyclopédie des Sciences Mathématiques Pures et Appliquées, tome I, vol. | The fundamental theorem of algebra | 0.808648 |
23,779 | ; Stallings, John R. (1988), "On Gauss's First Proof of the Fundamental Theorem of Algebra", Proceedings of the American Mathematical Society, vol. 103, no. 1, pp. | The fundamental theorem of algebra | 0.808648 |
23,780 | (2012), "The Fundamental Theorem of Algebra: from the four basic operations", American Mathematical Monthly, vol. 119, no. 9, pp. 753–758, arXiv:1110.0165, doi:10.4169/amer.math.monthly.119.09.753, S2CID 218548926 Fine, Benjamin; Rosenberger, Gerhard (1997), The Fundamental Theorem of Algebra, Undergraduate Texts in Ma... | The fundamental theorem of algebra | 0.808648 |
23,781 | (2011), "The Fundamental Theorem of Algebra: an elementary and direct proof", Mathematical Intelligencer, vol. 33, no. 2, pp. 1–2, doi:10.1007/s00283-011-9199-2, S2CID 5243991 de Oliveira, O.R.B. | The fundamental theorem of algebra | 0.808648 |
23,782 | (2012), "Some Riemannian geometric proofs of the Fundamental Theorem of Algebra" (PDF), Differential Geometry – Dynamical Systems, vol. 14, pp. 1–4 de Oliveira, O.R.B. | The fundamental theorem of algebra | 0.808648 |
23,783 | These proofs of the Fundamental Theorem of Algebra must make use of the following two facts about real numbers that are not algebraic but require only a small amount of analysis (more precisely, the intermediate value theorem in both cases): every polynomial with an odd degree and real coefficients has some real root; ... | The fundamental theorem of algebra | 0.808648 |
23,784 | Now, the Gauss–Bonnet theorem, when applied to the sphere S2, claims that ∫ S 2 K g = 4 π , {\displaystyle \int _{\mathbf {S} ^{2}}K_{g}=4\pi ,} which proves that the sphere is not flat. Let us now assume that n > 0 and p ( z ) = a 0 + a 1 z + ⋯ + a n z n ≠ 0 {\displaystyle p(z)=a_{0}+a_{1}z+\cdots +a_{n}z^{n}\neq 0} f... | The fundamental theorem of algebra | 0.808648 |
23,785 | There exists still another way to approach the fundamental theorem of algebra, due to J. M. Almira and A. Romero: by Riemannian geometric arguments. The main idea here is to prove that the existence of a non-constant polynomial p(z) without zeros implies the existence of a flat Riemannian metric over the sphere S2. Thi... | The fundamental theorem of algebra | 0.808648 |
23,786 | These statements can be proved from previous factorizations by remarking that, if r is a non-real root of a polynomial with real coefficients, its complex conjugate r ¯ {\displaystyle {\overline {r}}} is also a root, and ( x − r ) ( x − r ¯ ) {\displaystyle (x-r)(x-{\overline {r}})} is a polynomial of degree two with r... | The fundamental theorem of algebra | 0.808648 |
23,787 | There are several equivalent formulations of the theorem: Every univariate polynomial of positive degree with real coefficients has at least one complex root. Every univariate polynomial of positive degree with complex coefficients has at least one complex root. This implies immediately the previous assertion, as real ... | The fundamental theorem of algebra | 0.808648 |
23,788 | Kneser, Hellmuth (1940), "Der Fundamentalsatz der Algebra und der Intuitionismus", Mathematische Zeitschrift, vol. 46, pp. 287–302, doi:10.1007/BF01181442, ISSN 0025-5874, S2CID 120861330 (The Fundamental Theorem of Algebra and Intuitionism). | The fundamental theorem of algebra | 0.808648 |
23,789 | Beiträge zur Theorie der algebraischen Gleichungen (1849 Juli), pp. 71–103., p. 71, at Google Books – fourth proof. | The fundamental theorem of algebra | 0.808648 |
23,790 | Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia Supplementum commentationis praecedentis (1816 Jan), pp. 57–64., p. 57, at Google Books – third proof. | The fundamental theorem of algebra | 0.808648 |
23,791 | Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (1815 Dec), pp. 32–56., p. 32, at Google Books – second proof. | The fundamental theorem of algebra | 0.808648 |
23,792 | Band III, Königlichen Gesellschaft der Wissenschaften zu Göttingen Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (1799), pp. 1–31., p. 1, at Google Books – first proof. | The fundamental theorem of algebra | 0.808648 |
23,793 | Cauchy, Augustin-Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, 1ère partie: Analyse Algébrique, Paris: Éditions Jacques Gabay (published 1992), ISBN 978-2-87647-053-8 (tr. Course on Analysis of the Royal Polytechnic Academy, part 1: Algebraic Analysis) Euler, Leonhard (1751), "Recherches sur les racine... | The fundamental theorem of algebra | 0.808648 |
23,794 | Let G be the Galois group of this extension, and let H be a Sylow 2-subgroup of G, so that the order of H is a power of 2, and the index of H in G is odd. By the fundamental theorem of Galois theory, there exists a subextension L of K/R such that Gal(K/L) = H. As = is odd, and there are no nonlinear irreducible real ... | The fundamental theorem of algebra | 0.808648 |
23,795 | Another algebraic proof of the fundamental theorem can be given using Galois theory. It suffices to show that C has no proper finite field extension. Let K/C be a finite extension. Since the normal closure of K over R still has a finite degree over C (or R), we may assume without loss of generality that K is a normal e... | The fundamental theorem of algebra | 0.808648 |
23,796 | Almira, J.M. ; Romero, A. (2007), "Yet another application of the Gauss-Bonnet Theorem for the sphere", Bulletin of the Belgian Mathematical Society, vol. 14, pp. | The fundamental theorem of algebra | 0.808648 |
23,797 | All proofs below involve some mathematical analysis, or at least the topological concept of continuity of real or complex functions. Some also use differentiable or even analytic functions. This requirement has led to the remark that the Fundamental Theorem of Algebra is neither fundamental, nor a theorem of algebra.So... | The fundamental theorem of algebra | 0.808648 |
23,798 | Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n (where n is a natural number, and a and b are real numbers), and rational functions of the form (ax + b)/(x2 + cx + d)n (where n is a natural number, and a... | The fundamental theorem of algebra | 0.808648 |
23,799 | (By the Abel–Ruffini theorem, the real numbers a and b are not necessarily expressible in terms of the coefficients of the polynomial, the basic arithmetic operations and the extraction of n-th roots.) This implies that the number of non-real complex roots is always even and remains even when counted with their multipl... | The fundamental theorem of algebra | 0.808648 |
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