Search results

...the symbols $o$, $O$ or the equivalence sign $\sim$ ([[Asymptotic equality|asymptotic equality]] of functions). Examples of asymptotic formulas:

...is a function defined on $M$, then $\{\psi(x)\phi_n(x)\}$ will also be an asymptotic sequence. Examples of asymptotic sequences:

$#C+1 = 20 : ~/encyclopedia/old_files/data/A013/A.0103670 Asymptotic expansion is some given [[asymptotic sequence]] as $ x \rightarrow x _ {0} $.

An asymptotic representation which provides approximate values of the where $\abs{\theta(n)} < 1/12n$. The asymptotic equalities

may have different asymptotic expressions when $ | z | \rightarrow \infty $ has the following asymptotic expansion when $ | z | \rightarrow \infty $:

$#C+1 = 31 : ~/encyclopedia/old_files/data/A013/A.0103750 Asymptotic power series An asymptotic series with respect to the sequence

''of asymptotic estimation'' A method for determining the asymptotic behaviour as $ 0 < \lambda \rightarrow + \infty $

...the limits of the class of linear estimators. For example, the statistical analysis of a linear regression model (see [[Linear regression|Linear regression]]) ...)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Scheffé, "Analysis of variance" , Wiley (1959)</TD></TR></table>

...an important role in various problems of physics, mechanics and asymptotic analysis, its solutions are regarded as forming a distinct class of special function ...5.35002}} (In Russian) (Translation: "Short-Wavelength Diffraction Theory. Asymptotic Methods", Springer, 1991 {{ZBL|0742.35002}})

...ernate and the series is enveloping in the strict sense. This series is an asymptotic expansion for $f(x)$ as $x\to+\infty$; if it is divergent, then it is calle ...</TD> <TD valign="top"> G. Pólya, G. Szegö, "Problems and theorems in analysis" , Springer (1976) pp. Chapts. 1–2 (Translated from German)</TD></TR><

A method for computing the asymptotic expansion of integrals of the form ...otic expansions of integrals of the form (*). It can be used to derive the asymptotic expansions for Laplace, Fourier and Mellin transforms, as well as for trans

...$f$ holomorphic in $S$, for which the formal series $\sum c_k z^k$ is an [[asymptotic series]]: ...gin and admitting an asymptotic series, is necessarily holomorphic, so its asymptotic series must converge.

cf. [[Spectral analysis of a stationary stochastic process|Spectral analysis of a stationary stochastic process]]). The regression spectrum for $ M ( ...ectrum can be used to express a necessary and sufficient condition for the asymptotic efficiency of an estimator for $ \beta $

...es quantiles of successive subsets [[#References|[a4]]], in which case the asymptotic distribution is determined by recursive application of polynomials. A tree-

with radius of convergence 1 satisfies on the real axis the asymptotic equality satisfy the asymptotic equality

...escribed in [[#References|[a6]]], which presents a unified approach to the analysis of data resulting from an experiment involving multinomial populations. Thu have to be estimated. He has given results concerning the asymptotic variance-covariance matrix of the estimators.

have the same distribution. Asymptotic tests can be based on the following theorem: If $ \min ( m , n ) \rightar ...stribution function (cf. [[Statistical estimator|Statistical estimator]]). Asymptotic expansions for the distribution functions of the statistics $ D _ {m,n}

...oportion of numbers less than $X$ which are not arithmetic is [[Asymptotic analysis|asymptotically]] {{cite|Bateman et al (1981)}}

4) Asymptotic methods of investigation of loss systems may also be effective for the stud ...s with a large number of channels is conducted both by means of asymptotic analysis of explicit formulas, which become effective for large $ n $,

...amplitude. In order to calculate the oscillation process more accurately, asymptotic methods are employed. As $ \mu $ ...n]] with period $ 1.614 \mu $ (to a first approximation). More accurate asymptotic expansions of magnitudes characterizing relaxation oscillations [[#Referenc

Asymptotic stability of the polynomial or dynamical system is strongly which can be directly used in asymptotic stability investigations for

...progression]] with denominator $q$, while the value $-\ln q$ is called the asymptotic rate of convergence. ...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from

A method for formally finding high-frequency asymptotic approximations of solutions of problems in the theory of diffraction and pr ...lied and one must use more complicated expansions, for example some unique asymptotic approximation or some version of the boundary-layer method (the most import

denotes the asymptotic variance of an estimator $ T $ on (the asymptotic version of) the estimator $ T $.

relations, asymptotic relations'' All formulas of the above type are called asymptotic estimates; they are especially interesting for infinitely-small and infinit

Asymptotic expansions have been obtained. The case where the functions $ f $ There is also an asymptotic series for $ V _ {x ^ {0} } ( \lambda ) $ (for the formulas of the cont

By the analysis of E. Wigner [[#References|[a2]]], [[#References|[a3]]], all states which d ...^ { i ( p ^ { 0 } - \omega ) t } \mathcal{F} ( f _ { l } ) )$, one defines asymptotic fields by their action on the vacuum vector $\Omega$:

...ious properties, and yields solutions the form of which coincides with the asymptotic part of the solutions of transport equations. For improved forms of the dif ...olume of the medium is large. The method has been successfully used in the analysis of the optical fibre oximeter of blood. The use of the so-called Henyey–G

...ation form. The area has close connections with statistics and time-series analysis, and also offers a very wide spectrum of applications. ...he random variable $\hat { \theta } _ { N }$. The covariance matrix of the asymptotic (normal) distribution of the estimate takes the typical form

Viewed from a modern perspective, Karamata theory is the study of asymptotic relations of the form Perhaps the most important application of Karamata theory to analysis is Karamata's Tauberian theorem (or the Hardy–Littlewood–Karamata theor

which is asymptotic for small $ \mu $ ...nsitions (5), (6), studies were conducted with the purpose of obtaining an asymptotic expansion of its solution. In the regular case the solution of the initial

...on spaces $\mathcal{V}$. A further approach uses analytic continuation and asymptotic series of distributions. ...nd D. Laugwitz [[#References|[a10]]] in their foundations of infinitesimal analysis.

...[#References|[3]]]), in the problem of Hardy and Ramanujan on obtaining an asymptotic formula for the number of "unbounded partitions" of a natural number. ...<TD valign="top"> T.M. Apostol, "Modular functions and Dirichlet series in analysis" , Springer (1976) {{ZBL|0332.10017}}</TD></TR>

D. Zagier [[#References|[a5]]] investigated the asymptotic growth for the number of solutions to the Markov equation ($a=n=3$) below a ...</TD> <TD valign="top"> A. Hurwitz, "Über eine Aufgabe der unbestimmten Analysis" ''Archiv. Math. Phys.'' , '''3''' (1907) pp. 185–196 (Also: Mathemat

An important question is the asymptotic (for $ \epsilon \rightarrow 0 $) ...as been calculated by A.D. Dorodnitsyn [[#References|[7]]] by constructing asymptotic approximations, for $ \lambda \rightarrow \infty $,

It can be shown that the asymptotic behaviour as $ N \rightarrow \infty $ ...TR><TD valign="top">[3]</TD> <TD valign="top"> T.M. Anderson, "Statistical analysis of time series" , Wiley (1971) {{MR|0283939}} {{ZBL|0225.62108}} </TD></TR>

...rized by a Voss net, a [[Ruled surface|ruled surface]] has a semi-geodesic asymptotic net, etc. The theory of nets is closely related to aspects of mapping surfa ...mathematics, physics and mechanics (for example, [[Tensor analysis|tensor analysis]], the [[Cartan method of exterior forms|Cartan method of exterior forms]],

If the symbol of the given pseudo-differential operator has an asymptotic expansion in positive homogeneous functions $ a _ \alpha ( x, \xi ) $ ( ...TR><TR><TD valign="top">[4]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''3''' , Springer (1985)</TD>

...plications in survival analysis (cf. also [[Regression analysis|Regression analysis]]). ...]: [[Martingale|Martingale]]), conditions may be given for consistency and asymptotic normality of the estimators $\widehat { \beta }$ and $\widehat { A } ( t |

...ethod|Schauder method]], as well as other methods of non-linear functional analysis, are applied in investigations of this equation. Questions of stability of solutions, eigen-function expansions, asymptotic expansions in a small parameter, etc., can be studied for integro-different

...ditions, [[#References|[a3]]], M-estimators are asymptotically normal with asymptotic variance $V ( T , F _ { \theta } ) = \int \operatorname { IF } ( x ; T , F ...Huber's minimax variance problem [[#References|[a1]]] or by minimizing the asymptotic variance $V ( T , F _ { \theta } )$ subject to an upper bound on the gross-

...op">[3]</TD> <TD valign="top"> I.S. Ibragimov, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)</TD></TR></table>

...The Mikhailov criterion gives a necessary and sufficient condition for the asymptotic stability of a linear differential equation of order $n$, ...op">[a2]</td> <td valign="top"> B.R. Barmish, "New tools for robustness analysis" , ''IEEE Proc. 27th Conf. Decision and Control, Austin, Texas, December 19

asymptotic approximation of the function: ...valign="top">[2]</TD> <TD valign="top"> L.D. Kudryavtsev, "Mathematical analysis" , Moscow (1973) (In Russian)</TD></TR></table>

...e the flame. The complete profiles are then found by the method of matched asymptotic expansions (cf. [[Perturbation theory|Perturbation theory]]). ...top">[a2]</TD> <TD valign="top"> W.B. Bush, F.E. Fendell, "Asymptotic analysis of laminar flame propagation for general Lewis numbers" ''Comb. Sci. and T

...or of the]]) can be obtained by some further constructions that employ the asymptotic lack of correlation for the periodograms for different frequencies $ \lam ...gn="top">[1]</TD> <TD valign="top"> D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston (1974)</TD></TR><TR><TD valign=

...based on a combination of time-dependent scattering theory and phase-space analysis. In his first work, Enss solved the two-body scattering problem (with short ...ficial assumptions. It also initiated the fruitful approach of phase-space analysis, later further developed by E. Mourre [[#References|[a3]]] (1979–1981) an

the asymptotic normality of the variable $ Z _ {n} = \sqrt n ( X _ {n} - x _ {0} ) $ ...res possess properties that are asymptotically optimal in the sense of the asymptotic variance.

...otics of Toeplitz determinants can be used to obtain information about the asymptotic distribution of the eigenvalues $\{ \lambda _ { k } ^ { ( n ) } \} _ { k = ...number|winding number]] of the function $a ( e ^ { i \theta } ) - z$. The asymptotic formula (a3) is sometimes also called the first Szegö limit theorem or a f

...ectrum of $T$, and provide a powerful tool of [[Spectral analysis|spectral analysis]]. In order to carry out a more detailed spectral analysis of the Sturm–Liouville operator $T$, one has to consider the spectral mea

...ese problems have many applications in mathematical statistics (sequential analysis), in the insurance business, in queueing theory, etc. In their investigatio ...walks have been investigated rather completely, including their asymptotic analysis (see [[#References|[1]]], Vol. 2; [[#References|[4]]]–).

...he Picard exceptional values are asymptotic values (cf. [[Asymptotic value|Asymptotic value]]) and that there exist Julia rays $ L $ ...="top">[3]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top"

...to give, under the assumptions $\int_\cK Vdx = 0$ and $n\ge 2$, a precise asymptotic formula, as $\mu\to +\infty$, for $N(\mu)$ in the following form: |valign="top"|{{Ref|HeMo}}||valign="top"| B. Helffer, A. Mohamed, "Asymptotic of the density of states for the Schrödinger operator with periodic electr

...only on the boundary of $g$, but also everywhere inside. The procedure of asymptotic integration used in shell theory for non-regular degeneration has not yet ( ...of the corresponding forms of free vibration have been found by methods of asymptotic integration and by the spectral theory of operators (see [[#References|[5]]

...necessary characteristics, or is in some sense critical. The direction of asymptotic investigation in the first case is described by stability (or continuity) t ...TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Borovkov, "Asymptotic methods in queueing theory" , Wiley (1984) (Translated from Russian)</TD>

...ccurring in the area of statistics called [[Sequential analysis|sequential analysis]]. In sequential procedures for statistical estimation, hypothesis testing ...then follows from (a3) and agrees with (a2). See [[#References|[a1]]] for asymptotic properties of the average sample number for curtailed tests in general.

An asymptotic approximation of the solution of boundary value problems for differential e ...nstructed for a particular type of integro-differential equation giving an asymptotic expansion in $ \epsilon $

...l equations with deviated arguments includes all equations of mathematical analysis. Nevertheless, ordinary differential equations with deviated arguments are is necessary and sufficient for the asymptotic stability of solutions of equation (5). If this condition is met, the zero

that is, to construct an asymptotic solution as $ \epsilon \rightarrow 0 $, denotes asymptotic equality. All vector and matrix functions which appear in equations and bou

...al of such a function. Operators of this type arose when investigating the asymptotic expansions of rapidly-oscillating solutions to partial differential equatio is a formal asymptotic solution of the equation $ L ( x, \lambda ^ {-} 1 D) u = 0 $.

An important fact for the analysis of the Stieltjes polynomials $E _ { n + 1}$ is their close connection with Several asymptotic representations are available. A simple formula for the Legendre weight fun

...equations could be systematically derived as the leading term of a formal asymptotic expansion in a thickness parameter of the exact equations of three-dimensio ...e von Kármán equations has attracted a considerable amount of mathematical analysis. Many of the major advances in steady-state [[Bifurcation|bifurcation]] the

...one can apply the fundamental identity of [[Sequential analysis|sequential analysis]] (see [[#References|[a2]]] for details) to find the approximate power func <TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Eisenberg, "The asymptotic solution of the Kiefer–Weiss problem" ''Sequential Anal.'' , '''1''' (1

...a quadric on which the net corresponding to the families of generators is asymptotic (Bonnet's theorem). ...0/r082790109.png" />. II" W.L. Baily jr. (ed.) T. Shioda (ed.) , ''Complex Analysis and Algebraic geometry'' , Cambridge Univ. Press &amp; Iwanami Shoten (1977

...= 132 : ~/encyclopedia/old_files/data/R110/R.1100080 Renormalization group analysis ...ale, from which observed phenomena can be explained. Renormalization group analysis allows one to determine effective theories at each length scale, from micro

...[Spectral theory|spectral theory]] of operators and in related problems in analysis. Recently it gained even greater significance, when its relation to certain satisfy the following asymptotic equation (see [[#References|[4]]]):

...y stable cycle. This assertion is mainly based on the results of numerical analysis (significantly less was obtained by purely mathematical methods, [[#Referen The asymptotic equality

An asymptotic expansion of the quantiles of a distribution (close to the normal standard ...The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top

A different form of the inequality is often used in analysis and is also called Lojasiewicz inequality (sometimes ''Lojasiewicz gradient * the asymptotic behavior of solutions to parabolic equations

...valign="top"> E.A. Grebenikov, Yu.A. Ryabov, "Constructive methods of analysis of non-linear systems" , Moscow (1979) (In Russian)</TD></TR></table>

...ing once more, the current point of view that Tauberian theorems link the (asymptotic) behaviour of a (generalized) function in a neighbourhood of zero with that ...top">[a6]</TD> <TD valign="top"> N.G. de Bruijn, "Asymptotic methods in analysis" , North-Holland &amp; Noordhoff &amp; Interscience (1981)</TD></TR><TR><T

...having the same sign. This result can be used in the asymptotic stability analysis of Runge–Kutta methods, see [[#References|[a5]]]. ...ch at the moment (1998). Basically, two different approaches exist for the analysis of numerical stability:

A deep theorem from the real analysis, showing which data are required to extent a real-valued function from a co These asymptotic conditions are necessary for the functions $f^{(\a)}$ to be partial derivat

...describes phenomenological models of hysteresis and is convenient for the analysis of closed-loop systems with hysteresis non-linearities is presently (1996) ...ero and at infinity (including the [[Hopf bifurcation|Hopf bifurcation]]), asymptotic and numerical methods (including an averaging principle), etc. have also be

...align="top">[3]</TD> <TD valign="top"> A.N. Tikhonov, A.D. Gorbunov, "Asymptotic expansions of the error in the difference method of solving Cauchy's proble ...d-held programmable calculator" G.H. Golub (ed.) , ''Studies in numerical analysis'' , Math. Assoc. Amer. (1984) pp. 199–242</TD></TR></table>

...s successful in finding a suitable transformation, the ordinary method for analysis will be available. Among the many parametric transformations, the family in ...o the transformed data. Such an approach may be easily carried out, and an asymptotic theory associated with other parameters is useful. See [[#References|[a1]]]

...f. [[Generating function]]); they also play an important part in obtaining asymptotic relations (cf. [[#References|[1]]]–[[#References|[3]]]). ...2]</TD> <TD valign="top"> J. Riordan, "An introduction to combinational analysis" , Wiley (1958)</TD></TR>

the asymptotic level $\alpha$ of the test as a the asymptotic level

The next question also seems very natural: Is it possible to have a certain asymptotic relation instead of the order estimate (a2)? For rectangular partial sums s ...rted the interest in various questions of approximation theory and Fourier analysis in ${\bf R} ^ { n }$ connected with the study of hyperbolic cross partial F

...waves in such media and more exactly, to the [[Spectral analysis|spectral analysis]] of the [[Schrödinger equation|Schrödinger equation]] with a periodic po ...ith "eigenvalue" $\lambda &gt; 0$ (cf. also [[Spectral analysis|Spectral analysis]]). These functions are not elements of $L ^ { 2 } ( R ^ { N } )$ but they

Asymptotic versions of these results have also been given by Pinelis for the case when ...gn="top"> Yu.A. Rozanov, "On a new class of estimates" , ''Multivariate Analysis'' , '''2''' , Acad. Press (1969) pp. 437–441</TD></TR>

$#C+1 = 125 : ~/encyclopedia/old_files/data/S086/S.0806330 Spectral analysis ...of functions of one or several operators find wide application in spectral analysis. These may be very simple (the spectrum of a polynomial in an operator cons

[[Numerical analysis|numerical analysis]] and in applied mathematics, an asymptotic series (that is, $g_{i+1}(n) = o(g_i(n))$ when $n$ goes to infinity) and

(Gaussian). Nevertheless, the $LAR$ estimator has an asymptotic normal ..."|{{Ref|1}}||valign="top"| Yadolah Dodge (Ed.) (1987),'' Statistical Data Analysis Based on the ''$L_{1}$''-Norm and Related Methods'', North-Holland Publishi

...onal problems, methods have been created which are almost optimal in their asymptotic characteristics (see [[#References|[5]]]–[[#References|[8]]]). ...n="top">[10]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from

...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 ...atmospheric waves is accomplished by methods of non-linear mechanics using asymptotic series expansions in powers of a small parameter. The mathematical theory o

The Euler–MacLaurin formula plays an important role in the study of asymptotic expansions, number-theoretic estimates and [[Finite-difference calculus|fin ...[a1]</TD> <TD valign="top"> F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="

...)</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J.K. Hale, "Asymptotic behavior of dissipative systems" , Amer. Math. Soc. (1988)</td></tr><tr><t

...es of a differential operator is connected with the qualitative theory and asymptotic methods of ordinary differential equations. ==Spectral analysis of self-adjoint differential operators.==

...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

A branch of [[Functional analysis|functional analysis]] in which one studies the behaviour on the real axis $ J $ Thus the problem of uniform (asymptotic) stability of the solution $ z _ {0} ( t) $

...udied. For samples of large size one can rely on a knowledge of the Pitman asymptotic relative efficiency, which has been computed for a number of the simplest s ...]). Moreover, there is a rank test (the so-called normal scores test) with asymptotic relative efficiency 1 relative to Student's test in the normal case and exc

...="top">[3]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''1–2''' , Moscow (1976) (In Russian)</TD></TR></table> ...erences|[a2]]], [[#References|[a4]]], [[#References|[a5]]].) Moreover, the asymptotic behaviour of $ K (z, w) $

...nces|[a13]]]. Furthermore, it has been applied in time series discriminant analysis [[#References|[a14]]], [[#References|[a15]]], [[#References|[a16]]], [[#Ref ...te analysis" P.R. Krishnaiah (ed.) , ''Proc. Internat. Symp. Multivariate Analysis'' , Acad. Press (1966) pp. 187–200</TD></TR><TR><TD valign="top">[a11]<

* A vertex of the boundary of a [[nodal domain]] in the [[Harmonic analysis]] (related to the above and also to the [[http://en.wikipedia.org/wiki/Node ...nov stability|in the Lyapunov sense]] and [[Asymptotically-stable solution|asymptotic]] (the unstable node is stable in the reverse time $\tau=-t$).

However, the asymptotic variance-covariance matrix itself depends on the unknown parameters (see, e ...align="top">[a5]</TD> <TD valign="top"> L.L. Thurstone, "Psychophysical analysis" ''Amer. J. Psychol.'' , '''38''' (1927) pp. 368–389</TD></TR><TR><TD

...f the values of a real arithmetic function $f(m)$, one usually studied the asymptotic behaviour of $f(m)$ itself, or of its average values. In the first case one ...s of a real arithmetic function $f(m)$, then one is forced to consider the asymptotic behaviour of the frequencies

The asymptotic behaviour of confluent hypergeometric functions as $ z \rightarrow \infty ...D> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6</TD></TR><TR><TD valign="top

...of the linear kinetic equation. Existence and uniqueness theorems and the asymptotic behaviour of the non-linear Boltzmann equation have been widely studied [[# ...S.B. Shikhov, "Questions in the mathematical theory of reactors. Linear analysis" , Moscow (1973) (In Russian) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign=

A fundamental result concerning the asymptotic behaviour of $p ( a , t )$ states that ...) } \Pi ( a ) = e ^ { \lambda ^ { * } t } w ^ { * } ( a ) $, which is the asymptotic limit of $p ( a , t )$ in the sense stated by (a5), is called a persistent

...ed function, derivative of a|Generalized function, derivative of a]]). The asymptotic decay property expressed in $\mathcal{N} ( \mathcal{D} ( \Omega ) )$ togeth For applications in a variety of fields of non-linear analysis and physics, see [[#References|[a1]]], [[#References|[a4]]], [[#References|

combinations) as theoretical models for cluster analysis, but of known from asymptotic theory, that the simple heuristics of searching

...a class). Mostly the minimization of the computational work is done in the asymptotic sense as $ \epsilon \rightarrow 0 $, ...lity with respect to rounding errors is required. It is important that the asymptotic growth of the amount of machine memory used in these methods is not too lar

$#C+1 = 163 : ~/encyclopedia/old_files/data/I110/I.1100020 Idempotent analysis, ...th the corresponding function spaces (semi-modules). The term "idempotent analysis" (or idempotent calculus) is well established in the literature since the

...licit formulas for the index in terms of the winding number of the symbol, asymptotic formulas for the determinant of its finite sections, etc. In fact, the infi ...valign="top">[a2]</TD> <TD valign="top"> A. Böttcher, B. Silbermann, "Analysis of Toeplitz operators" , Springer (1990)</TD></TR><TR><TD valign="top">[a3

...ing problems of this kind have been obtained by exact methods. Approximate asymptotic methods were much more successful. One of the main approaches is the linear ...urface of a heavy liquid methods of the theory of functions and functional analysis have made it possible to solve many existence and uniqueness problems of wa

...theory is based on the implicit-function theorem in non-linear functional analysis and on the general theory of linear problems of corresponding type. For a narrower class of equations of the form (5) the question of the asymptotic behaviour of the solutions of such problems as $ t \rightarrow + \infty $

The asymptotic behaviour of hypergeometric functions for large values of $ | z | $ ...D> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)</TD></TR><TR><TD valign="top">[7]</TD> <TD

Here, the standard nabla formalism from [[Vector analysis|vector analysis]] is used and all fields are functions of space ($\mathbf{x}$) and time ($t ...to be only Galilean invariant. This is obtained in an appropriate Galilean asymptotic limit of small material velocities (see [[#References|[a8]]]).

...ution of these zeros in some part of the critical strip; and for obtaining asymptotic formulas and estimates of various kinds of the most important arithmetic fu ...heorems in complex analysis can be found in almost any textbook on complex analysis, e.g. [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]. Ref

...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,

...ic number theory|analytic number theory]] in a unified way to a variety of asymptotic enumeration questions for isomorphism classes of different kinds of explici The asymptotic behaviour of $\pi(X) = \sum_{n \le x} P_{\mathbb{N}}(n)$ for large $x$ form

The same analysis has been generalized to the case of a bounded domain in [[#References|[a1]] ...[a1]</TD> <TD valign="top"> L.L. Bonilla, J.A. Carrillo, J. Soler, "Asymptotic behavior of an initial-boundary value problem for the Vlasov–Poisson–Fo

...1$, only lower and upper bounds ($N = 2$, [[#References|[a12]]]) and some asymptotic formulas for $u \rightarrow \infty$ [[#References|[a21]]] have been derived <tr><td valign="top">[a14]</td> <td valign="top"> T. Kitagava, "Analysis of variance applied to function spaces" ''Mem. Fac. Sci. Kyushu Univ. Ser.

...ar systems. In the case of waves on deep water, the first two terms in the asymptotic expansion employed by Stokes are given by Benjamin and Feir used a linearized analysis to show that

...all basic contacts, intermediate contacts and windings of the circuit. An asymptotic expression for the complexity of the simplest relay-contact circuit realizi ...<TR><TD valign="top">[1]</TD> <TD valign="top"> C. Shannon, "A symbolic analysis of relay and switching circuits" ''AIEE Trans.'' , '''57''' (1938) pp. 7

| This article ''Analysis of Samples of Curves (=Functional Data Analysis)'' was adapted from an original article by Hans-Georg Müller, which appe <center>'''Functional Data Analysis'''

is the known order of smallness of the error. Thus, if one has the asymptotic relation ...[8]</TD> <TD valign="top"> L.V. Kantorovich, G.P. Akilov, "Functional analysis" , Pergamon (1982) (Translated from Russian)</TD></TR><TR><TD valign="top

...hich sometimes proves an opportunity to use its methods and results in the analysis of number-theoretic models. ...n, and by applying methods developed in probability theory for finding the asymptotic distributions of sums of independent or weakly-dependent random variables,

...d></tr><tr><td valign="top">[9]</td> <td valign="top"> R.M. Burton, "An asymptotic definition of $K$-groups of automorphisms" ''Z. Wahrsch. Verw. Gebiete'' ,

...ign="top"> R. Illner, H. Lange, P.F. Zweifel, "Global existence and asymptotic behaviour of solutions of the Wigner–Poisson and Schrödinger–Poisson s

In their study of the long-time asymptotic structure of non-linear evolution equations solvable by the inverse scatter ...of pole-type singularities (or sometimes more general singularities). This analysis can be generalized to apply, formally speaking, to partial differential equ

...ers among natural numbers. The central problem is that of finding the best asymptotic, as $ x \rightarrow \infty $, By virtue of Stirling's asymptotic formula for $ n! $

...e the action. To achieve that the action is finite one imposes appropriate asymptotic conditions and is thus led to consider bundles $\xi $ having as base $M$ th ...ang–Mills theory leads to global questions incorporating both topology and analysis, as opposed to the purely local theory of classical differential geometry.

| This article ''Nonlinear time series analysis'' was adapted from an original article by Howell Tong, which appeared in '' <center>'''Nonlinear Time Series Analysis'''</center>

...quation is exactly solvable, so that the problem is reduced to finding the asymptotic behaviour of the best approximation to the true solution, accurately to wit 2) In the perturbation theory of celestial mechanics the asymptotic integration of differential equations was developed for conservative system

...\cdot) \rightarrow \infty$, which will happen if the chain is ergodic. The asymptotic variance of the maximum likelihood estimates can be approximated by In certain medical applications, an alternative asymptotic regime can be of interest, when many ($k$) short trajectories are observed,

theory) that could be used in the analysis of time series but which did seem to have some "asymptotic

* Finally, it is assumed that there is scattering with asymptotic completeness, in the sense $\mathcal{H} = \mathcal{H} ^ { \text{in} } = \ma ...efined [[Wiener measure|Wiener measure]] becomes instead the basis for the analysis. Two analytical continuations were suggested in this connection: in the mas

...by the assumption that (2) is represented by an [[Asymptotic power series|asymptotic power series]], then one obtains a definition of a linear [[Pseudo-differen ...></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–

...t the same time, in the study of quasi-linear systems new methods emerged: asymptotic methods (see [[#References|[5]]], [[#References|[6]]]), the method of $ V ...]), the stroboscopic method [[#References|[12]]] and methods of functional analysis [[#References|[13]]] have been applied.

...s asymptotically the best synthesis method and enables one to describe the asymptotic behaviour of $ L ( n) $. ...metimes taken to be the number of states. In these cases the corresponding asymptotic behaviour of $ L ( n) $

...org/legacyimages/d/d032/d032080/d03208013.png" /> may serve as an example. Asymptotic methods are used both to obtain analytical expressions approximating the so ...gn="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from

ergodic theorems, non-asymptotic bounds, asymptotic expansions (Edgeworth phenomena, analysis of rare events)

...hological behaviour" and solving several other major open problems in the asymptotic theory of finite-dimensional normed spaces. An example of this is Szarek's ...org/legacyimages/b/b110/b110100/b110100261.png" />. Therefore, there exist asymptotic centres of the Banach–Mazur compactum with distance to <img align="absmid

...ral duality theorems, reflecting fundamental relations in geometric convex analysis (see [[#References|[7]]], [[#References|[8]]]). If, e.g. $ X $ ...ch require an increase in the order of differentiability. Here, some exact asymptotic results are known. If $ MW ^ {r} H ^ \alpha $,

...rm, and, moreover, the stiffness is not always essentially decreased after asymptotic transformation. Thus, numerical methods are used to solve systems of a gene For the implicit polygonal method (8), the condition of asymptotic stability,

...y, a lot of work has been recently (as of 2000) done in order to develop "analysis on fractals" , using the Sierpiński gasket (and sometimes the Sierpiński ...eferences|[a23]]] and [[#References|[a17]]] yield a precise form of Weyl's asymptotic law in this context.

are known. In other words, the asymptotic behaviour of $ S _ {n} $ ...s was developed in parallel with that of the main branches of mathematical analysis. The calculus of finite differences first began to appear in works of P. Fe

has the following asymptotic expansion: has the asymptotic expansion

.../TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S.-Y. Shaw, "Asymptotic behavior of pseudoresolvents on some Grothendieck spaces" ''Publ. RIMS Kyo

...D> <TD valign="top"> B.R. Barmish, B. Ross, "New tools for robustness analysis" , ''Proc. 27-th IEEE CDC'' , IEEE (1988) pp. 1–6</TD></TR></table>

is called an asymptotic resolvent for a linear operator $ A $ An asymptotic resolvent possesses various properties resembling those of the ordinary res

.... D. Moore [[#References|[a8]]], [[#References|[a9]]] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting fr

...(unique) solution $Z$ which is bounded on finite intervals, and study its asymptotic Blackwell's original proof (1948) of (4) depended on harmonic analysis techniques.

...tudying them, as well as methods of number theory. Using these methods, an asymptotic expansion in the form The same asymptotic expansions were subsequently deduced using methods of number theory without

...asymptotic estimation, and many other problems of theoretical and applied analysis [[#References|[1]]]–[[#References|[4]]]. ...riables have found use in the study of Feynman integrals, in combinatorial analysis [[#References|[11]]] and in the theory of implicit functions [[#References|

are specified and reflect the asymptotic behaviour of $ u ( x) $ ...valign="top">[a3]</TD> <TD valign="top"> R.S. Varga, "Matrix iterative analysis" , Prentice-Hall (1962)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign

...larity and other qualitative properties of solutions; the analysis of many asymptotic problems, including large deviations and homogenization problems; extension

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.

of characteristic functions; concepts such as 'closeness' and 'asymptotic limitations to which the inferences drawn from the analysis are

...matching problem gave rise to the beginning of the branch of combinatorial analysis related to the study of permutations with restricted positions. A distance ...sting part of these investigations touches upon the achievement of various asymptotic results when $ m $

The asymptotic properties of the polynomials which are orthogonal with respect to such a w ...="top">[a3]</TD> <TD valign="top"> F. Riesz, B. Sz.-Nagy, "Functional analysis" , F. Ungar (1955)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top

operations. Such asymptotic behaviour of the number of arithmetical operations is not yet attainable fo ...valign="top"> A.V. Aho, J.E. Hopcroft, J.D. Ullman, "The design and analysis of computer algorithms" , Addison-Wesley (1974)</TD></TR></table>

...o small that the gas of electrons cannot reach thermal equilibrium, or the analysis of the ionized air between the reading head and a compact disc) involve gas ...="top"> F. Bouchut, F. Golse, M. Pulvirenti, "Kinetic equations and asymptotic theories" , ''Series in Appl. Math.'' , '''4''' , Elsevier/Gauthier-Villars

subject of ''structural VAR analysis''. the usual asymptotic optimality properties. If the dimension $K$ of

...always converges. Independently of the location of the eigen values it has asymptotic quadratic convergence. The presence of multiple and close eigen values not ...but corresponding to perturbations of the initial data (so-called backward analysis). This perturbation, known as equivalent perturbation, completely character

...\mathbf{R} ^ { n } \times \mathbf{R} ^ { n }$. It is convenient to give an asymptotic version of these compositions formulas, e.g. in the semi-classical case. Le The developments of the analysis of partial differential operators in the 1970s required refined localizatio

...theory of topological algebras (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]; [[Topological algebra|Topological algebra]]). ==Harmonic analysis of functions on $ G $.==

A linear operator in a Hilbert space the spectral analysis of which cannot be made to fit into the framework of the theory of self-adj ...lvent|resolvent]], which makes use of the theory of analytic functions, of asymptotic expansions, etc.

variables. Then the following asymptotic relation holds: ...ate information about elementary conjunctions and use it in the subsequent analysis. Procedures of this kind are known as local simplification algorithms. Some

it covers all of the physics and mathematical analysis of the period. Poisson teaching of the asymptotic theory of probabilities throughout Europe,

...$\int f ( \theta , \phi ) d \phi$, 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

..."| Bhattacharya, R.N. and Ranga Rao, R.(1976). ''Normal Approximation and Asymptotic Expansions''. Wiley, New York. |valign="top"|{{Ref|4}}||valign="top"| Esseen, C.G.(1945). Fourier analysis of distribution functions: A mathematical study of the Laplace-Gaussian law

to the most varied topics such as functional analysis, Lie groups analysis, sometimes topology, and I did not hesitate to take up the

...xplicit analytical solution, considerable attention is given to the use of asymptotic methods. In this context it is assumed that the renewal is "rapid" , i.e. ...gn="top">[7]</TD> <TD valign="top"> Ya.B. Shor, "Statistical methods in analysis and control of quality and reliability" , Moscow (1962) (In Russian)</TD>

which is the asymptotic limit of every orbit not landing on one of the periodic orbits $ \Lambda unimodal mappings. In particular, the asymptotic geometry of the Cantor set $ \Lambda _ \infty $(

...(ed.) A. Kishimoto (ed.) I. Ojima (ed.) , ''Quantum and Non-Commutative Analysis'' , ''Math. Phys. Stud.'' , Kluwer Acad. Publ. (1993) pp. 239–251</TD>< ...valign="top">[a10]</TD> <TD valign="top"> M.V. Karasev, V.P. Maslov, "Asymptotic and geometric quantization" ''Russian Math. Surveys'' , '''39''' : 6 (19

$ROC$ analysis, various information criteria tests, link tests, and residual analysis. The Hosmer-Lemeshow test, once well used, is now

...alytical methods (for instance, perturbation methods [[#References|[a1]]], asymptotic methods based on the geometrical theory of diffraction [[#References|[a5]]] ...</TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> R.L. Stoll, "The analysis of eddy currents" , Clarendon Press (1974)</TD></TR><TR><TD valign="top">[

..._ { i n } \}^ { N } _ { 1 }$ which form an array with respect to $n$. The asymptotic behaviour of the statistics $\{ f _{( k , n )} \} _ { k = 1 } ^ { \mu _ { n [[Yule, George Udny|G.U. Yule]] [[#References|[a24]]], in his analysis of data of J.G. Willis [[#References|[a22]]] on problems of biological taxo

Poisson models are not based on the asymptotic methods characteristic of maximum likelihood or ...ef|2}}||valign="top"| Cameron, A. C., P. K. Trivedi (1998). ''Regression analysis of count data''. New York: Cambridge University Press.

...t of classical information that can be conveyed through a quantum channel. Asymptotic achievability of the bound (using block coding of the information source) w ...distributions of states (represented by density operators). In most cases, asymptotic properties of quantum channels do not depend on the details of the fidelity

proven using the asymptotic behaviour of the heat kernel of $\Omega$ (cf. also [[Heat equation|Heat equ ..."top"> M. Reed, B. Simon, "Methods of modern mathematical physics IV: Analysis of operators" , Acad. Press (1978)</td></tr><tr><td valign="top">[a17]</td

up into rectangles (see [[#References|[11]]], [[#References|[13]]]). Similar asymptotic solutions for discrete analogues of the Dirichlet problem can be obtained f ...">[a7]</TD> <TD valign="top"> A.R. Mitchell, R. Wait, "Finite element analysis and applications" , Wiley (1985)</TD></TR></table>

...omplete matrix factorization method), and others. Most attractive from the asymptotic point of view is the use of operators $ B $ ...valign="top"> R. Glowinski, J.-L. Lions, R. Trémolières, "Numerical analysis of variational inequalities" , North-Holland (1981) (Translated from Fren

vertices is given by the asymptotic formula ...tivity (cf. [[Graph, connectivity of a|Graph, connectivity of a]]). In the analysis of the reliability of electronic circuits or communications networks there

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

The condition of minimal stability in this case means asymptotic stability of the linear system (1) with $ \xi = \mu \sigma $, ...">[8]</TD> <TD valign="top"> D.D. Ŝiljak, "Nonlinear systems. Parameter analysis and design" , Wiley (1969)</TD></TR><TR><TD valign="top">[9]</TD> <TD vali

...ty } ^ { n - 1 }$ generates a convex cone $\Gamma ^ { \circ }$, called the asymptotic cone of $K$.) Then $\operatorname{supp} f \subset K$ if and only if there e This notion may be effectively employed for certain problems in global analysis on unbounded domains [[#References|[a5]]].

.... Its development is closely connected with various fields of mathematical analysis and mathematics in general, first and foremost with probability theory, the .../TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> G.R. MacLane, "Asymptotic values of holomorphic functions" , ''Rice Univ. Studies, Math. Monographs''

...s|[a5]]], where it was named after its discoverer. It arises, e.g., in the analysis of the Saffman–Taylor problem with surface tension [[#References|[a6]]]. ...nd reflection coefficients, $a ( k )$ and $b ( k )$, respectively, via the asymptotic behaviour of the corresponding eigenfunctions,

part of analysis, as the theory of elliptic integrals and elliptic gives the principal term of the asymptotic expansion of the

...ctions of lower order. The procedure for such an expansion follows from an analysis of the physical singularities of the system and of the small parameter whic ...tonian which permits an exact solution, and to the subsequent proof of the asymptotic proximity of the results obtained by it to those which correspond to the in

...lic Darboux equation (negative Gaussian curvature) the characteristics are asymptotic lines. An application of the [[Cauchy–Kovalevskaya theorem|Cauchy–Koval .../TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> T. Aubin, "Nonlinear analysis on manifolds. Monge–Ampère equations" , Springer (1982) {{MR|0681859}} {

influences its asymptotic properties: such a function cannot have other deficient values. ...="top">[7]</TD> <TD valign="top"> B.V. Shabat, "Introduction of complex analysis" , '''2''' , Moscow (1976) (In Russian)</TD></TR><TR><TD valign="top">[8]

A final and rigorously mathematical analysis of quantum field theory within the framework of perturbation theory was giv ...ole, and the charge as the value of $ G_{4} $ at some specific point.) The analysis of the non-uniqueness leads to the important concept of the renormalization

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

...dies.) In the qualitative theory of differential equations one studies the asymptotic behaviour of the solutions of (1) as $ x \rightarrow \infty $. ...roblem in the qualitative theory of non-linear systems is the study of the asymptotic behaviour of all solutions as $ x \rightarrow \pm \infty $.

surfaces, the asymptotic theory of maximum likelihood estimates, goodness-of-fit and an anticipation of a two-way analysis of

...itation]]). The subject of the theory of gravitation embraces, besides the analysis of the gravitational field itself, also the [[Space-time|space-time]] struc ...ore difficult by the indefiniteness of the metric); and the computation of asymptotic (including topological) properties of solutions in the case of insular syst

...cations of the index theorems to geometry, physics, group representations, analysis, and other fields. There is a very long and fast growing list of papers dea ...op">[a45]</td> <td valign="top"> J. Roe, "Elliptic operators, topology and asymptotic methods" , ''Pitman Res. Notes in Math. Ser.'' , '''179''' , Longman (1988)

...the construction of estimates with small variances one uses, for example, asymptotic solutions of the adjoint transport equation [[#References|[4]]]; the simple ...D> <TD valign="top"> N.M. Korobov, "Number-theoretic methods in applied analysis" , Moscow (1963) (In Russian)</TD></TR><TR><TD valign="top">[22]</TD> <TD

...automatic control. In particular, one may mention the methods of frequency analysis; methods based on the first approximation to [[Lyapunov stability theory|Ly for which the closed system (13) retains the property of asymptotic stability, is known as the domain of attraction of the trivial solution $

...></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> N.N. Kuznetsov, "Asymptotic behaviour of the solutions of the finite-difference Cauchy problem" ''USSR ...vided by the tutorial [[#References|[a2]]]. A more advanced and up-to-date analysis of multi-grid methods is given in [[#References|[a4]]].

...ude laws have been established; these are also extensively employed in the analysis of the results of numerical calculations, which are then reduced to very co ...theory]]. The boundary-layer equations represent the principal term of the asymptotic expansion of the Navier–Stokes equations in the vicinity of the boundary

becomes a problem of optimal stabilization of the system (the property of asymptotic stability of the equilibrium position of a synthesized system follows direc ...gn="top">[a14]</TD> <TD valign="top"> H. Hermes, J.P. Lasalle, "Functional analysis and time optimal control" , Acad. Press (1969) {{MR|0420366}} {{ZBL|0203.47

...representations of $n$ in the form of $k$ positive integers, and that the asymptotic relation |valign="top"|{{Ref|Eu}}||valign="top"| L. Euler, "Einleitung in die Analysis des Unendlichen" , Springer (1983) (Translated from Latin) {{MR|0715928}} {