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...ar spaces and their mappings. The basic divisions of non-linear functional analysis are the following. ...e study of spaces that are locally linear and of Banach manifolds — global analysis.

The comparison of algorithms and the analysis of numerical problems in a Bayesian setting, cf. also [[Bayesian approach|B ...worst-case sense over the class $P$. Alternatively, in Bayesian numerical analysis, one puts an [[A priori distribution|a priori distribution]] $\mu$ on the i

...orithms have been constructed for the realization of a random search for a global extremum of a function in several variables (see [[#References|[5]]]). Thes ...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from

[[Category:Global analysis]]

...10/b1100108.png" />) are of no use for an appropriate characterization and analysis of methods which are able to efficiently integrate a stiff problem. Thus th ...zing parameter goes back to [[#References|[a7]]], where it was used in the analysis of multi-step methods. The point is that stiffness is often compatible with

...t by means of local semantics of statements of the program (so-called flow analysis of the program); checking certain properties of the information collected ( ...s are divided into local, the economy part is not larger than a statement; global, the economy part is the entire program; and quasi-local, for which the eco

In other words, the problem is to construct a global meromorphic function with locally specified polar singularities. satisfying the compatibility condition there corresponds a uniquely defined global section of the sheaf $ {\mathcal M} / {\mathcal O} $,

...braic and analytic geometry, etc. is a frequently used method to construct global objects such as varieties, schemes, differentiable manifolds, vector bundle and morphisms of schemes between them. Cf. [[Scheme|Scheme]]. Here also global separation properties must be added to obtain a scheme. For vector bundles

A very useful fact in analysis is that $C^1$ maps $f$ such that $\left. df\right|_{x_0}$ is invertible at ...the differential is invertible at ''every point'' does not guarantee the ''global invertibility'' of the map. Indeed, a famous example is the exponential map

...TD valign="top">[a1]</TD> <TD valign="top"> S.S. Chern, "Studies in global analysis and geometry" , ''Studies in Mathematics'' , '''4''' , Math. Assoc. America

...M. Shub, "Expanding maps" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 2

...so cannot be a local minimum point of the modulus of $f(z)$. An equivalent global formulation of the maximum-modulus principle is that, under the same condit |valign="top"|{{Ref|Ah}}||valign="top"| L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. 241 {{ZBL|0395.30001}}

<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated ...TD valign="top">[a2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR>

a finite number of global sections $ s _{1} \dots s _{N} $ by its global sections.)

==Local and global convergence theory.== ...f$ is decreased as the iteration progresses. There are several variants of global convergence theorems for BFGS and related methods, [[#References|[a9]]], [[

...D valign="top">[2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR></table>

==Local and global theory.== ...see [[Deformation|Deformation]] 1) and 2)). The fundamental methods of the global theory are those of the theory of representable functors and geometric inva

A global version of the same statement is the following The global Theorem 2 holds also when $[0,T]$ is replaced by $[-T, 0]$ or $[-T,T]$, by

...solution in the space of sequences of bounded functions), and non-standard analysis methods. ...ces of summable functions (or kernel operators, in the quantum case): time-global solutions for general classes of an interaction potential;

The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]] ...D></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.A. Carrillo, "Global weak solutions of the absorption and reflection-type initial-boundary value

``long-run", or ``global" statistical dependence in the Earth sciences. peculiar method of analysis that follows very

...r equipments, and computer networks), which is oriented to the qualitative analysis and synthesis of such systems (discovering deadlocks or conflict situations ...conducted along two lines. The mathematical theory is advanced by a formal analysis of their properties. The most interesting problems include recognizing dead

...theory is based on the implicit-function theorem in non-linear functional analysis and on the general theory of linear problems of corresponding type. The global theory of non-linear problems is less completely developed, and then only f

.... Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and ...ying the topological invariants, provided by $K$-theory. Multi-dimensional global problems of the calculus of variations on manifolds proved to be more diffi

...ture of the boundary conditions or any supplementary conditions). Such a "global" character of variational calculus in the large proper is stressed by the ...#References|[12]]]). Variational calculus in the large is also employed in global [[Differential geometry|differential geometry]] [[#References|[13]]].

..., "Anosov diffeomorphisms" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 6 ...300</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Shub, "Global stability of dynamical systems" , Springer (1986)</TD></TR><TR><TD valign=

...nnected the theory of variational inequalities to [[Convex analysis|convex analysis]], especially to the notion of subdifferentiability, and to the theory of m ...R><TD valign="top">[a10]</TD> <TD valign="top"> V.K. Le, K. Schmitt, "Global bifurcation in variational inequalities" , Springer (1997)</TD></TR><TR><T

...b C$ and $f,g: U \to \mathbb C$ are differentiable in the sense of complex analysis (cf. [[Analytic function]]). Then the formula reads as \eqref{e:rule}. Global derivatives are maps from $C^1 (M)$ to $C^0 (M)$ satisfying the (analog of)

...D valign="top">[4]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[5]<

...integral formulas is one of the most important tools in classical complex analysis (cf. also [[Boundary value problems of analytic function theory|Boundary va In applications involving the construction of global holomorphic functions satisfying special properties, and in order to solve

...p">[5]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973)</TD></TR></table>

...e manifold gives, for a sufficiently smooth manifold, the largest possible global degree of smoothness of the function which is obtained as a result of exten ...ns of several variables and imbedding theorems" S.M. Nikol'skii (ed.) , ''Analysis III'' , ''Encycl. Math. Sci.'' , '''26''' , Springer (1990) pp. 1–140

...D valign="top">[3]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)</TD></TR><TR><TD valign="top">[4]<

If the problem satisfies the global contractivity condition ...having the same sign. This result can be used in the asymptotic stability analysis of Runge–Kutta methods, see [[#References|[a5]]].

...submersions" , ''Lecture Notes'' , '''40''' , Research Inst. Math., Global Analysis Research Center, Seoul Nat. Univ. (1998)</td></tr><tr><td valign="top">[a5

...ighest order) are minimized. However, since the relation between the true (global) error and the local error is generally not known, it is questionable wheth ...D valign="top">[a1]</TD> <TD valign="top"> J.C. Butcher, "The numerical analysis of ordinary differential equations. Runge–Kutta and general linear method

...ions of representation and approximation of functions, and their local and global properties. The modern theory of functions of a real variable typically inv ...n urgent need for a new critical review of the foundations of mathematical analysis, which was carried out at the end of the 19th century and beginning of the

...ing analytic set in a local model (cf. [[Analytic set|Analytic set]]). The global dimension is defined by the formula: The theory of analytic spaces has two aspects: the local and the global aspect. Local analytic geometry is concerned with germs of analytic sets in

...ion is useful when some functions are not differentiable. Using non-smooth analysis, one can replace derivatives by other objects such as subgradients (see, e. ...0\right\}$. Here $f^0(\theta)=0$ for any $\theta$, hence $\theta^*=0$ is a global minimum. The saddle-point condition requires $U_1=U_1(\theta)\geq 0$ such t

...ed only for convex and related unimodal functions. The theory of finding a global extremum is still (1989) in the initial phase of development (see [[Multi-e .../TD> <TD valign="top"> Yu.G. Evtushenko, "Numerical methods for finding global extrema (case of a non-uniform mesh)" ''USSR Comp. Math. Math. Phys.'' , '

Global stability of the trivial solution of a non-linear system of ordinary differ then one has global exponential stability:

...ithms and programs for the computer realization of the discrete models, an analysis of the sensitivity of the model to variations of the parameters, an estimat ...ng of [[Time series|time series]] on a network of measurements, space-time analysis and the compatibility of meteorological fields), and also the use of method

Another global construction of the Weil bundles on all manifolds $M$ is due to A. Morimoto ...lign="top"> P.W. Michor, A. Kriegl, "The convenient setting of global analysis" , ''Math. Surveys Monogr.'' , '''53''' , Amer. Math. Soc. (1997)</td></tr

...al structure ( "the very same as Rn" ), this idea admits a whole series of global features typical for manifolds: (non-) orientability, homological [[Poincar ...ion|Morse function]]), etc., which are used for the geometric study of the global structure of manifolds, and, roughly speaking, consist of constructing a po

briefly, Morse theory 1) is divided into two parts: local and global. The local part is related to the idea of a critical point of a smooth func The basic results in global Morse theory are as follows. Let $ f $

...face of any rough plate (see [[#References|[2]]]). In the investigation of global atmospheric processes on an Earth scale, the field of ground pressure and o ...fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press (1964) (Translated from Russian)</TD></TR><TR><T

...="top">[3]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian)</TD></TR></table> ...D></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S.G. Gindikin, "Analysis on homogeneous domains" ''Russian Math. Surveys'' , '''19''' (1964) pp.

be a finite-dimensional smooth manifold. Tangent spaces and such provide the global analogues of differential calculus. There is also an "integral calculus on ...nifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien) (1988)

...lude numerical integration, optimal recovery (approximation) of functions, global optimization, and solution of integral equations and partial differential e ...average-case setting (see [[Bayesian numerical analysis|Bayesian numerical analysis]]).

cf. [[Tensor analysis|Tensor analysis]]) on a [[Manifold|manifold]] $ M $ .../TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated f

It is a well-known and elementary fact in complex analysis that a bounded and [[Holomorphic function|holomorphic function]] on the who ...al differential geometry, involving geometric measure theory and nonlinear analysis (cf. also [[Plateau problem|Plateau problem]]).

...tems and algebraic curves" M. Grmela (ed.) J.E. Marsden (ed.) , ''Global analysis'' , ''Lect. notes in math.'' , '''755''' , Springer (1979) pp. 83–200</

...gular solutions of these boundary value problems. Profound applications to global problems of classical differential geometry were done on the basis of these ...nsional hyperbolic Monge–Ampère equations. The solution of a few important global problems for saddle surfaces was obtained as an application of these result

of) the derived functors of the global section functor $ F \mapsto F( 1) $( ...TD valign="top"> I. Moerdijk, G.E. Reyes, "Models for smooth infinitesimal analysis" , Springer (1990) {{MR|1083355}} {{ZBL|0715.18001}} </TD></TR><TR><TD vali

...zation of formal moduli, I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Princeton Univ. Press (1969) pp. 21–7

is generated by two elements. The global [[Homological dimension|homological dimension]] of $ A _ {n} ( K) $ ...ces|[a8]]] for a survey of this. The result (a1) was proved by micro-local analysis in [[#References|[a40]]]. An algebraic proof was found later in [[#Referenc

...ties. This structure/non-structure dichotomy is known as the main gap. The analysis discussed so far (1990) suffices to establish (Shelah, late 1970's) the Mor ...type. B.I. Zil'ber initiated the use of this geometric structure to obtain global information about the models of $ T $.

...ctions on an open set $\Omega \subset D ^ { n }$ can be represented by the global cohomology group $H _ { \Omega } ^ { n } ( U , \widetilde { \mathcal O } )$ ...ordinary hyperfunctions. The case $K = D ^ { n }$ corresponds to that for global Fourier hyperfunctions, given at the beginning.

It can perform a bifurcation analysis of (a1). It can compute branches of stable and unstable periodic orbits and ...al systems equations. Several other packages, notably Global Manifolds 1D, Global Manifolds 2D, GAIO and BOV-method compute invariant manifolds. See [[#Refer

...><TR><TD valign="top">[16]</TD> <TD valign="top"> D. Neumann, T. O'Brien, "Global structure of continuous flows on 2-manifolds" ''J. Diff. Eq.'' , '''22''' :

...e that as suggested by the presence of chaotic solutions and by a Painlevé analysis [[#References|[a7]]] (cf. also [[Painlevé test|Painlevé test]]), the Kura ...inertial manifold, which exponentially absorbs solutions and contains the global attractor [[#References|[a10]]], [[#References|[a4]]]. On restricting the p

...eddies symmetrically staggered, as in Fig.a1. However, a linear stability analysis with respect to small disturbances shows that the first configuration is al ...], the velocity induced on each vortex by all the others, and the eventual global displacement velocity of the sheets using the complex potential formulation

...ximation of local solutions to the homogeneous equation $P ( D ) u = 0$ by global solutions. The space of solutions to a linear homogeneous ordinary differen .../tr><tr><td valign="top">[a6]</td> <td valign="top"> L. Hörmander, "The analysis of linear partial differential operators II" , ''Grundl. Math. Wissenschaft

...ization of formal moduli I" D.C. Spencer (ed.) S. Iyanaga (ed.) , ''Global analysis (papers in honor of K. Kodaira)'' , Univ. Tokyo Press (1969) pp. 21–72 {{

...e [[#References|[a5]]], [[#References|[a11]]] for examples and references. Global analyses of these equations are merely of mathematical interest because the ...="top">[a4]</TD> <TD valign="top"> P.G. Ciarlet, V. Lods, "Asymptotic analysis of linearly elastic shells I. Justification of membrane shell equations" '

...utation of the input data is equally likely. For biased distributions, the analysis becomes significantly more complicated, and for arbitrary distributions the ...e behaviour is not identical to the worst-case behaviour — an average-case analysis makes sense. A precise complexity estimation has been given for many Boolea

...1–56</TD></TR><TR><TD valign="top">[a15b]</TD> <TD valign="top"> D. Rand, "Global phase space universality, smooth conjugacies and renormalisation: the <img

...fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press (1968) (Translated from Russian)</TD></TR><TR><T ...top"> S. Albeverio, R. Høegh-Krohn, B. Zegarlinski, "Uniqueness and global Markov property for Euclidean fields: the case of general polynomial intera

...\Gamma \backslash X ) , \widetilde { M } )$ and what is the image of the "global" cohomology $H ^ { \bullet } ( \Gamma \backslash X , \tilde { \mathcal{M} giving a global class. The only difficulty is that this sum need not converge. Hence the fo

is generated by global sections; for any analytic sheaf $ {\mathcal F} $ ...><TD valign="top">[1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} <

...oblems and has opened new possibilities for the application of geometry to analysis. It is Riemannian geometry which was used by A. Einstein to realize the ide ...the development of the geometry of infinite-dimensional manifolds — global analysis.

...of the subdomains. In spectral methods, the domain is not subdivided, but global basis functions of high order are used. Accuracy is gained by increasing th ...pectral methods are, in general, more complicated to code and require more analysis to be done prior to coding than simpler methods.

...ed by a system of differential equations such as (1). On the other hand, a global (i.e. suitable for all states of the dynamical system) and invariant (i.e. ...alitative picture of the behaviour of all trajectories in the phase space (global theory) or at least in some part of it (local theory). In the theory of dyn

...eral content. In its most general meaning it is considered nowadays as the analysis of connections in principal fibre spaces or fibre spaces associated to them <TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated

as numerical functions and to apply to them the methods of analysis. In general, the value of a field quantity at a point depends on the choice ...></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated

.../math> is the projectivization of the kernel of the natural restriction of global sections ...ument is that it implicitly assumes that the restriction <math>r</math> of global sections is ''surjective'', which isn't the case in general.

...opology was the [[Compact-open topology|compact-open topology]]. A careful analysis of topologies on $\mathcal{C} ( Y , X )$ in relation to the exponential law ...lign="top"> A. Kriegl, P.W. Michor, "The convenient setting of global analysis" , ''Math. Surveys and Monographs'' , '''53''' , Amer. Math. Soc. (1997)</

...op">[a8]</td> <td valign="top"> R. Illner, H. Lange, P.F. Zweifel, "Global existence and asymptotic behaviour of solutions of the Wigner–Poisson and

...topology|differential topology]] and [[Mathematical analysis|mathematical analysis]]. It is rooted in the fundamental work of L. Kronecker [[#References|[a5]] ...roach, but Brouwer created and used new simplicial techniques to define a (global) degree $d [ f , M , N ]$ for continuous mappings $f : M \rightarrow N$ bet

...with and having many applications in [[Mathematical analysis|mathematical analysis]]. It has been extensively studied by G.G. Lorentz in [[#References|[a13]]] ...lign="top">[a9]</TD> <TD valign="top"> H.H. Gonska, Xin-Long Zhou, "A global inverse theorem on simultaneous approximation by Bernstein–Durrmeyer oper

...ang–Mills theory leads to global questions incorporating both topology and analysis, as opposed to the purely local theory of classical differential geometry.

...encyclopediaofmath.org/legacyimages/d/d032/d032600/d032600193.png" />-adic analysis stimulated the development of the theory of Diophantine approximations in t ...ers by their approximation properties, etc. The corresponding methods are "global" (continued fractions, etc.). The metric approach involves the description

...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from ...al orders, that is, the order of accuracy after just one step; the actual (global) order at a fixed node <img align="absmiddle" border="0" src="https://www.e

...(see [[Lie group, local|Lie group, local]]). Systematic research into the global structure of Lie groups was first begun by E. Cartan and H. Weyl. The first ==The global structure of Lie groups.==

...these fields by the [[Galerkin method|Galerkin method]] through objective analysis (interpolation, extrapolation, smoothing) of empirical data on these fields ...ear mechanics, related to the van der Pol method, in a multiple time scale analysis. An example of this is the quasi-geostrophysic series, filtering fast waves

''Diophantine analysis'' ...for these two types of fields [[#References|[3]]], which are usually named global fields. This analogy is especially noticeable if the algebraic functions st

...aofmath.org/legacyimages/m/m063/m063460/m063460167.png" /> is defined as a global section of <img align="absmiddle" border="0" src="https://www.encyclopediao ..."top">[5]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.3200

...mmodated in the method of utilization and interpretation of the results of analysis. ...compact convex set. By expanding the space of products the problem of the analysis of the efficient methods here may be reduced to the case where $ Z $

.... D. Moore [[#References|[a8]]], [[#References|[a9]]] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting fr ...r><td valign="top">[a4]</td> <td valign="top"> J. Duchon, R. Robert, "Global vortex sheet solutions of Euler equations in the plane" ''J. Diff. Eqs.''

...n that of Abelian varieties, some results involving Drinfel'd modules over global function fields $L$ can be proved, whose analogues over number fields $L$ a ...hic representations. This can partially be achieved and leads to (local or global) reciprocity laws between representations of $\GL(r)$ and Galois representa

...0/e035550178.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press &amp; Iwanami Shoten (1977

...tence criteria for global solutions (1983). In many cases the existence of global solutions is assumed beforehand (this is natural in many applications) and ...Lavrent'ev, "Some improperly posed problems of mathematical physics and analysis" , Amer. Math. Soc. (1986) (Translated from Russian)</TD></TR><TR><TD val

...gn="top">[2]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from ...estimate of the local error. However, since the relation between the true (global) error and the local error is generally not known, and because the local er

A global a priori estimate for an elliptic operator $ A $ ...n the elliptic operator in question turns out to be invertible, and in the global a priori estimate of the type (1) the last term (the low norm at the right-

...als, quadrature formulas, moment problems, and other problems of classical analysis (see [[#References|[7]]]–). The origin of the study of the rows of the Pa ...icients of a power series) for their construction, they allow one to study global properties of the corresponding analytic function (analytic continuation, t

recommendation, he was appointed Professor of Analysis and Mechanics statistical analysis " always to collect a large number of elements,

...heory of topological characteristics of nonlinear operators II" , ''Global analysis: Studies and Applications IV'' , ''Lecture Notes Math.'' , '''1453''' , Spr

...tional-differential, as well as more complicated equations in mathematical analysis are all equations of the type (1), as are also systems of algebraic equatio ...>[2]</TD> <TD valign="top"> F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)</TD></TR><TR><TD valign="top"

...lso weighs the relative importance of the criteria in order to arrive at a global judgement. Moreover, in a group of decision makers each member faces the qu ...top">[a8]</td> <td valign="top"> F.A. Lootsma, "Multi-criteria decision analysis via ratio and difference judgement" , Kluwer Acad. Publ. (1999)</td></tr><

...ne { \partial }$-problem]]), the prescribed curvature equation, and global analysis, [[#References|[a1]]], [[#References|[a7]]], [[#References|[a10]]].

...op">[a6]</TD> <TD valign="top"> R. Horst, P.M. Pardalos, "Handbook of global optimization" , Kluwer Acad. Publ. (1995)</TD></TR><TR><TD valign="top">[a

...see, e.g., [[Metric space|Metric space]]; [[Functional analysis|Functional analysis]]). ...her geometers. As a result of the interaction of geometry with algebra and analysis there followed the appearance of special calculi, which are conveniently us

Apart from the lack of global convergence of Newton's method, the use of Hessian matrices is a disadvanta ...letcher, "An overview of unconstrained optimization" ''Dundee Numerical Analysis Report'' , '''NA/149''' (1993)</td></tr><tr><td valign="top">[a9]</td> <td

...the general framework of the theory of connections. As a device of tensor analysis, covariant differentiation is widely used in theoretical physics, particula ...></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1955) (Translated

...]) and mathematical [[Cybernetics|cybernetics]] and their study aims at an analysis and formalization of the concept of an algorithm and at modelling real devi Other models combine local and global storage.

...(ed.) A. Kishimoto (ed.) I. Ojima (ed.) , ''Quantum and Non-Commutative Analysis'' , ''Math. Phys. Stud.'' , Kluwer Acad. Publ. (1993) pp. 239–251</TD>< ...Kotake (ed.) S. Nishikawa (ed.) R. Schoen (ed.) , ''Geometry and Global Analysis (Rept. First MSJ Internat. Res. Inst. (July 12-23, 1993), Tôhoku Univ.'' ,

...$\mathbb{Z}$, a positive solution to Hilbert's tenth problem; cf. [[Local-global principles for the ring of algebraic integers]]. ...orm|Quadratic form]]) reduces the classification of quadratic forms over a global field to that over local fields. This represents the historically first ins

...oximation on Banach manifolds" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 213

...g to the theorem on the resolution of singularities of algebraic surfaces, global methods of the theory of schemes may be applied to the local study of singu is known. The existence of a global moduli variety of an algebraic surface has been proved only for certain cas

.... Abraham, "Bumpy metrics" S.-S. Chern (ed.) S. Smale (ed.) , ''Global analysis'' , ''Proc. Symp. Pure Math.'' , '''14''' , Amer. Math. Soc. (1970) pp. 1

...J.A. Thorpe, "The curvature of 4-dimensional Einstein spaces" , ''Global Analysis (In Honour of K. Kodaira)'' , Univ. Tokyo Press (1969) pp. 355–365</TD>

The simplest ordinary differential equation is already encountered in analysis: The problem of finding the primitive function of a given continuous functi ...[Eigen value|Eigen value]]) and also with the [[Spectral analysis|spectral analysis]] of ordinary differential operators.

...e involving the idea of descent in the space of controls and of successive analysis of variants (of the type of [[Dynamic programming|dynamic programming]]). ...rical solution of optimal control problems is based on the idea of optimal analysis of variants [[#References|[10]]], [[#References|[11]]], [[#References|[12]]

...e method for the construction of geometry and its foundations. This deeper analysis of the foundations of geometry was enhanced by the discovery in 1826 of the ...cular, to justify methods of analysis in geometry and geometric methods in analysis.

...tion of subspaces, on the contrary, one obtains quite general results of a global nature. Thus, if ...ping) has been constructed [[#References|[2]]], and is a universal (in the global sense) deformation for any compact analytic subspace of

...opological decompositions of operator algebras" A. Salam (ed.) , ''Global analysis and its applications (Trieste, 1972)'' , '''3''' , IAEA (1974) pp. 305–

...>[5]</TD> <TD valign="top"> L.V. [L.V. Ovsyannikov] Ovsiannikov, "Group analysis of differential equations" , Acad. Press (1982) (Translated from Russian) ...valign="top"> K. Uhlenbeck, "Conservation laws and their application in global differential geometry" B. Srinivasan (ed.) J. Sally (ed.) , ''Emmy Noethe

...or studying many problems in contemporary algebra, geometry, topology, and analysis. ...nsion (over $\Z$) of a space, the algebraic dimension of a variety and the global dimension of a ring. The description given by Grothendieck of the spectral

...etrical forms, mainly with curves and surfaces, by methods of mathematical analysis. In differential geometry the properties of curves and surfaces are usually ...ometry. Many geometrical concepts were defined prior to their analogues in analysis. For instance, the concept of a tangent is older than that of a derivative,

2) to investigate the question of the local and global solvability and subellipticity of equations (see [[#References|[12]]]); and .../TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J. Leray, "Lagrangian analysis and quantum mechanics" , M.I.T. (1981) (Translated from French) {{MR|064626

A term denoting a number of problems related to research on extremals and global minima of functionals in the $ k $-dimensional volume $ \mathop{\rm vo ...roblems in variational calculus, topology, algebraic geometry, and complex analysis that give rise to the following situation: One is given a manifold $ M ^

Symbolic dynamics is also used for the analysis of chaotic behaviour of dynamical systems (cf. [[Chaos|Chaos]]; [[Fractals| ...3)</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top"> M. Shub, "Global stability of dynamical systems" , Springer (1987) (Translated from French

...general, a Cartier divisor on a ringed space $(X,\c O_X)$ is defined as a global section of the sheaf $M_X^*/\c O_X^*$ of germs of divisors. Here $M_X$ deno |valign="top"|{{Ref|We2}}||valign="top"| R.O. Wells jr., "Differential analysis on complex manifolds", Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}}

''analysis of variance'' MANOVA (multivariate analysis of variance) is the multivariate generalization of ANOVA. Its model equatio

...$, there is duality between the $k$-space $H^i(X, \calF)$ and the space of global Ext's ...eory for non-closed sets" , ''General topology and its relations to modern analysis and algebra (Proc. Symp. Prague)'' , Acad. Press (1961) pp. 123–132 {{MR|

An analysis of the lower bound for $ L ( n) $ ...usually the simpler, since for its solution it is sufficient in the final analysis to construct just one scheme, whereas the second requires a survey in some

...ave re-animated those directions of commutative algebra that can be called global commutative algebra. These relate to the theory of invariants (cf. [[Invari

Numerous investigations have been devoted to the study of global properties of concrete systems of differential equations. In connection wit ...of the elementary catastrophes, see [[#References|[a2]]]. Furthermore, the analysis of trajectories near an equilibrium can be restricted to the ones forming a

...to algorithms for solving simple problems of a standard structure (modular analysis of algorithms). There is as yet no rigorous foundation for such an approach ...accuracy on smooth solutions which is usually of order three at most and a global accuracy of order one at most (because of the low accuracy of the finite-di

...$\int f ( \theta , \phi ) d \phi$, an important advantage for statistical analysis. ...e inverse gives a large-sample variance of $\theta ^ { * }$ in statistical analysis. The supplemented EM [[#References|[a19]]] and supplemented ECM [[#Referenc

...uler system has been identified, one figures out local conditions that the global cohomology classes $c_L$ satisfy. Then Kolyvagin's descent procedure gives ...orary arithmetic. The interplay between arithmetic and algebraic geometry, analysis (both $p$-adic and complex), number theory, etc. has brought about many int

...osed with respect to the fundamental operations of arithmetic, algebra and analysis. Finally, an important property of an analytic function is its uniqueness: ...function considered as a whole is generally multi-valued. Many problems in analysis (inversion of a function, the determination of a primitive and the construc

...i.e. which can be verified at each point of the extremal), there is also a global necessary condition, related to the behaviour of the set of extremals in a ...eory of differential equations and topology. The development of functional analysis made a substantial contribution to the study of qualitative methods. See al

...automatic control. In particular, one may mention the methods of frequency analysis; methods based on the first approximation to [[Lyapunov stability theory|Ly There are also various global and necessary-condition-type results especially in the case of analytic sys

A global problem in many-valued logics is the description of the lattice of closed c ...lay an important role in mathematical logic, model theory and mathematical analysis. For countable-valued logics it has been established that the set of pre-co

...ign="top">[2]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian) {{MR|}} {{ZBL|0799.32001}} {{ZB ...top">[a1]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 {{MR|0344507}}

...its tape, and the position of its head on the tape, provides some kind of global state of the Turing machine. Accordingly, the set $C=\{c_i\}_{i\in I}$ of ...y, M. Suhail Zubairy, "Quantum Computing Devices: Principles, Designs, and Analysis", CRC Press 2010