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A N N A L E S S O C I E T A T I S M A T H E M A T I C A E P O L O N A E S e r i e s I : C O M M E N T A T I O N E S M A T H E M A T I C A E X V I I ( 1 9 7 3 )

K O C Z N I K I P O L S K I E G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O S é r i a I : P E A C E M A T E M A T Y C Z N E X V I I ( 1 9 7 3 )

Y. Ko m k o v (Texas Tech.)

Review oî the Kantorovic variational method

This article reviews some recent advances in the development of Kanatorovic technique with a particular application to the problem of shafts of constant cross section.

0.1. Introductory disscusion. This is basically an expository article reviewing the developments of the so-called Kantorovië method of obtaining solutions of equations which arise from a variational problem. Throughout this article one particular problem will be used to illustrate the numerical procedures. Perhaps the title of this article should have been changed to : “Becent results in the development of Kantorovic method with a par­

ticular application to the problem of pure torsion of elastic shafts of constant cross sections”. My aversion to long titles is the only excuse I have to offer.

Before we discuss the review of various generalizations of Kantorovic procedure, because of the expository nature of this article, it is proper to offer the basic concepts and definitions which will be used in the varia­

tional problem discussed here.

The main physical problem illustrating the techniques of  and  is closely related to Dirichlet’s problem: Let Q be a bounded (and for simplicity, simply connected) region of the Euclidean plane E2-,x,y the cartesian coordinates of E2. The boundary dQ of Q consists of a union of a finite number of smooth arcs. Q will denote Q u dQ. f(s) is a given function of the arc length s on dQ.

The Dirichlet’s problem consists of finding u(x, y), such that

d 2u d 2u

V2u = ---+ — —

doc2 d y2 in Q,

wlao ' = /(«).

It is easy to reduce the problem of a pure torsion of a shaft to the Di­

richlet’s problem.

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106 У. K o m k o v

We have

д2Ф д2Ф

дх2 ду2

Ф)дО — О Replacing Ф by a function:

—2 in Q ,

®2 + У2 х f = ф -1---m we have

(0.1) ьэ il

(0.2) v» =

where

^*2

-ir2 on сШ, ж2 + 2/2*

The related variational problems consist of minimizing (0.3, r w = / J { g ) 2 + g ) !J « ,

subject to (0.2), or of minimizing

(0.4) ! ( ? ) = / / дФ\ 2 1дФ\ 2 Л ,

\_\dx) + Ы “ И * " * * subject to Ф = 0 on dû.

The following comments are immediately applicable to equations of the type (0.1), (0.2).

Operator ( — V2) is positive definite, and therefore symmetric. For operators of this type the solution of the Dirichlet’s problem

Р2Ф = / in Û, Ф|да = 0 (on dû) is unique.

(This is trivially proved by assumming to the contrary the existence of two solutions Фх, Ф2, Фх ф Ф2. Then

р2фх_ Р2фа _ о in or

Р2(Фх- Ф 2) = 0 in Q, Фх — Ф2 = 0 in d û , and

< - Г , (Ф1- Ф >) , Ф1- Ф , ) а = - / / ( ( Р ^ - Ф ^ ^ - Ф , ) ) ^ = 0 . Ü

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Kantorovië variational method 107

By positive definite property of the operator ( — F2) this implies ( Ф г - Ф 2) = 0

is Q. Alternately, if we prefer classical analysis, we conld nse Green’s formula and reach the same conclusion, obtaining a contradiction.)

Before we make some glib statements about the solutions and the implications of this classical theorem, we should make certain of what is exactly meant by “a solution”. It is immediately apparent that if we consider only the problem of (0.4) we need only to wory about the functions Ф(х,у) which posesses weak derivatives of order one which are square integrable, that is of the Sobolev class W\, while in problem (0.1) we seek a solution among functions posessing continuous second derivatives.

In each case we are dealing with a subset of the Hilbert space L2(Q), and in fact with a dense subset of L2(Q). If we follow Mikhlin  and define HA to be the completion of the space of twice differentiable functions in Q, which vanish on dD, we formulate both problems in S A.

The Hilbert space HA has the inner product { u , v } = f f [( — V2u) 'V]dxdy.

Q

While in the case discussed in papers , ,  the problems

«concerning Ъг convergence of (0.4) are trivial, in the general case, when Ф{х,у) is an arbitrary continuous function, and dQ is an arbitrary piecewise smooth Jordan curve, there exist classical examples of non-convergence of the integral (0.4). Hadamard offered the first example of the function which is harmonic in the unit disc, which coincides on the unit circle with a continuous function, but which has a divergent Dirichiefs integral (0.4) (see Mikhlin , § 7), while Lebesgue gave an example of a region Ü, which can be used as a counterexample to the usual uniqueness and exi­

stence theorems for the Dirichlet’s problem. (Obviously, the boundary of such region cannot satisfy the hypothesis of being a bounded curve consisting of a piecewise smooth arcs with bounded curvatures.)

For the elementary discussion of difficulties which may arise in the exterior Dirichlet’s problem see for example , § 32.

A brief review of the Rayleigh-Ritz and Galerkin methods. It is well known that the Bayleigh-Bitz method may be regarded as a special case of the Galerkin’s method. (A more detailed explanation of this remark will be given in this paragraph.) However, the Galerkin method is far more general, although in some cases it coincides basically with the Bayleigh-Bitz method.

First let us point out the differences between the two techniques.

Let I be a differential operator. If: L u —f = 0 in Q, Ли = 0 on

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108 Y. К о т к о у

d û , is the variational system resulting from the extremal problem à [/(« )] = 0,

where I is a linear functional, then choosing some set of functions Ф^х), Ф2(х) ... Фт(ос),

satisfying the homogeneous boundary conditions ЛФг — 0, i = 1, 2, ..., mf

\ve consider the linear form:

u(m){%) = а1ф1 (x) + сс2Ф2(х) + ... + атФт(х),

and seek the values of the coefficients cq, a2, ..., am, which extremize the functional I(um{x)). The obvious necessary conditions are:

d l _ d I __ d l _ _

даг da2 dam

This is basically the Bayleigh-Bitz technique.

An analysis of convergence and a study of a number of important cases can be found in the original, paper of Bitz, and in the papers of Krylov and Bogoliubov.

In 1915 Galerkin proposed a technique which appears similar to the Bayleigh-Bitz technique. However, it does not depend on the varia­

tional formulation, and therefore can be used to obtain approximate solutions of non-self adjoint systems. It was applied with good results to non-conservative mechanical systems.

As in the Bayleigh-Bitz method we choose some functions Фг-(а?)г i = 1 , 2 , ..., m, which satisfy the boundary conditions, and consider the linear form:

m

ит(я) = г=1

We assume that the solution of the problem u(x), satisfying Lu —f is an element of a Hilbert space H, and so are the functions Ф*, the function Lum{x) and the functions f{x). <(/, g} will denote the inner product in H.

The Galerkin method can now be summarized as follows:

We choose a subspace G of И, which is spanned by elements ^1{x) ... ipm(x) and determine the coefficients a{ from the system of m equations with m unknowns :

<Lum, = </, yq>,

(Lum^ 1р2У — 'Cfi tyi) ) • • • j ФтУ ~ (fi Wrn) •

The convergence of Galerkin’s method has been studied by Keldysh (see ), where he proved strong convergence in the particular case of Dirichlet’s problem.

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Kantorovic variational method 109

2.0. Some basic definitions related to iterative procedures. The concepts given below are well known, but have sometimes been given varied inter­

pretations. To avoid confusion we shall refer as much as possible to the following basic sources: , , .

Let Q be a region of n-dimensional Euclidean space, let x denote a vector (a point) in Û. Let f(x) be a function whose domain contains Ü, and whose range is contained in Ü, and xx, x2, xz, . . . be an infinite sequence of points in Ü, such that:

X2 f (xi) j • • • ? x n f i x n — l ) ? • • • )

then we shall say that {ж J, i = 1,2 , . . . , is an iterative sequence defined by the procedure (xx,f). xx shall be called the starting, or initial choice.

The function /: Q -> Ü will be called the iterative function. If there exists a point xn, such that f( xn) = xn, it will be called a fixed point of the iterative function (or of the iteration).

Definitio n 2 . 1 (See Ostrowski ). A point ! 0e Ü is called a point of attraction for the iterative function /, if there exists an open region V a D, containing £0, such that for any starting point xxe V the cor­

responding iterative sequence of points xx, x2 ... will converge to £0.

Obviously £ 0 must be a fixed point of the iterative function /.

De fin it io n 2.2. A point £e Q is a point of repulsion of the iterative procedure /, if there exists a neighborhood F c Й, containing £ 0 such that for any xx Ф £ 0 e V the corresponding iterative sequence xx, x2, ...

will fail to converge to | 0, unless for some integer n, xn = £0.

We shall make some routine observations which will be used in the subsequent discussion. Let x{ be real (complex) numbers.

Ee ma r k 1. If f(x) is differentiable in some neighborhood of £0, and lim xn = £0, then

n—>00

/'d o ) = lim

n—► OO

X n J T 1 — £ o Xn £o

Ee ma r k 2. If f(x) is differentiable in some neighborhood V of i 0, and xn is an iterative sequence of points in V defined by (xx,f), then if I/'do) I < 1? the point £ 0 is a point of attraction, and if I/'d o ) I > 1 the point f 0is a point of repulsion of the iteration (see , p. 39-40).

Ee ma r k 3. Let £0 be a point of attraction of the iterative procedure.

Consider the replacement of the iterative function # f(x) by F{x) = x - f ( x ) , or by y(x) = (x - £0) - f ( x ).

We obviously have F'{x) = 'tp'(x) = l —f'(x). If |. f d 0)| < 1, then 0 < F'(x) < 2, in some neighborhood of | 0.

I

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п о Y. K o m k o v

Hence in that neighborhood F(x) is monotone increasing. (The for­

mula for y>(x) appears to be useless, since in general we don’t known the value of | 0. However, if we use an approximate value, obtained by ac­

celerating the original convergence it can be proved that ip(x) will again converge to £0). We consider the function [C-F(x)) with the choice of the constant G: 0 < C < to make certain that (CF(x)) is positive increasing, and 0 < (CF(x))' < 1. Hence zero is a point of attraction for the iterative function (C-F(x)) and the corresponding iterative sequence is monotone increasing.

We remark that our definition of a point of attraction coincides with the definition of some authors of a point of partial attraction. We shall say that | 0 is a point of full atraction in Q if it satisfies definition 2.1 and the region V in that definition can be taken to be of all Q. A similar change in definition 2.2 will define a point of full repulsion in Q.

Definition 2.3 (Ostrovski). An iterative procedure (x , f ) is uncon­

ditionally (conditionally) stable in Q if:

(i) / maps Q into Ü,

(ii) there exists only one (at least one) fixed point £0e Q of the iter­

ative procedure {xl f f), for any starting point хг е Q,

(iii) the (each) fixed point | 0 is a point of full (partial) attraction for the iterative function /.

3.3. Review of the Kantorovic method. We shall attempt a very basic explanation of Kantorovic idea.

We wish to minimize a functional I (f(xx, x2, ..., xn)) subject to some constraints (or boundary conditions) imposed on / ( %. . . xn).

The Rayleigh-Bitz technique, and the Galerkin technique both start by selecting certain collection of functions

Фг{х), Фт(х) (where x = {хг, x2, ... , xn))

and then optimizing with respect to the constants ai . the approximate solution of the problem of the form:

Ш

(1) /m(y) = ^ Ч Фг(я),

г=1

while making certain that f m(x) obeys the required constraints and (or) boundary conditions. The functions Ф^х) are guessed a priori, and this- guess makes the application of this technique a matter of experience and intuition.

Kantorovic has suggested that a certain degree of arbitrariness of a priori choice of Фг{х) may be eliminated by assumming the coefficients alf a%, .. . , am to be functions rather than constants. There is of course

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Kantorovic variational method 111

a penalty i napoaed by this method. We have to solve a system of differ ential equations instead of a system of linear algebraic equations, to

obtain the appropriate forms of ax(x ), a2(x) , . . . , am(x). Eor obvious reasons it is most convenient if at each step we only have to solve one ordinary differential equation with one unknown function. Hence Kantorovic imposed the condition ax = a2 = а2(хх), • ••> am = am{xi)- This implies that the arbitrariness of choice has been eliminated in the direction of the xx axis, but very little has been done to help the solutions in the directions of remaining axes.

A casual look at the form (1) convinces us that no greater generality is gained by assumming Ф{{х) = Ф{(хх, x2, xn), then by assumming

= 2> *^з? • • •} ^n)у * = 1, 2 , m.

In the obvious generalization of the procedure, we would first optimize a{(xi), i = 1 , 2 , ..., m, until a fixed point of the procedure is hopefully obtained (within the round off error), then somehow we attack the variable x2, until the result of the iterative procedure exhibits a fixed point beha­

vior, and go on to finally establish a form:

m

fm M ^ ‘ ' “S *

г=1

We either quit at this point, or else this will be regarded as the comple­

tion of the first cycle. (This is indicated by the superscript (1)). We iterate the whole procedure once again by commencing to regard а^(х), j > 1, as known (a prior chosen) functions, discard ali(x1) and optimize I [ f m(x)) with respect to (xx) regarded as the new and unknown functions, esta­

blish fi^(x), etc.

Various variants of such procedures have been suggested in a liter­

ature, which includes a paper published in 1933 by T. E. Schunk (refer­

ence [1 2]) which somehow escaped the attention of the mathematical community until quite recently (1966 is the first reference), and which roughly suggests the above technique for dealing with the problem of buckling of the cylindrical shells. In 1964, the author of this article has published a report which suggested a procedure similar to Schunk’s but with additional simplifying steps (see ).

Subsequently two papers of Kerr , and , and a paper of An- derssen  have extended these ideas, and in particular the paper of Anderssen has for the first time considered the problem of stability of such procedures in the general case. In this paper we shall review one of the particular procedures suggested in , ,  and .

First I would like to make a comment that neither of the papers , , , attempted to settle the basic theoretical problems of stability

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112 Y. K o m k o v

of the method, or of the theoretical error bounds, or of the error propa­

gation in case of the iterative procedures suggested by Kerr in .

All of the above papers contain some numerical results of experi­

mentation with the method, and all use the same physical problem, namely the pure torsion of a rectangular shaft. This choice is natural. We consider the equation V%0 = constant in a'region Q c E2. Ф(х, у) is the Prandtl stress function (the exact equation is Р2Ф = —2), subject to the condi­

tion Ф. = 0 on dû, where Q is some simply connected compact subset of the Euclidean plane, whose boundary consists of a union of a finite number of smooth arcs. I have stated that the choice of this problem for numerical experiments is natural, since this is the simplest form of a partial differential equation of the elliptic type where the extension of the Kantorovic method is non-trivial and where the accuracy of any method can be immediately checked, since the exact solutions of the problem are known for some simple choices of the region Q.

4. Modifications of Kantorovic technique. We shall consider the classical variant of Kantorovic as presented in  and examine the iterative procedures suggested by Komkov , Kerr , , Anderssen .

In  the author suggested the following iterative procedure:

Let us consider a variational problem of minimizing a functional I(u, (x)) over a given class of real functions u(x), whose domain is Q <= En, u(x) satisfying some homogeneous boundary conditions on dû. Bather then contenting ourselves with generalities let us examine the procedure on the specific case of minimization

I(u(x, y)) = j j n where Q is a rectangle:

dw\2 I du

_\ dxj dy àuj dxdy, a ^ |ж[, b ^ \y\.

We guess an approximate solution to be u(x) = {y^ — b^X^x). Now we proceed in the spirit of the Galerkin method by first computing the residue function r}(x,y) = F2u + 2, and then orthogonaliling r}(x,y) with respect to a chosen subspace of L Z(Q).

In  Komkov chooses this subspace to consist only of the original function (y2 — b2), which reduces the result to the one which would have been obtained by the Bayleigh-Bitz technique:

(4.1) ' / { № - Ъ ' ) Х ,Дх) + 2 Х1{х) + Щу*-Ъ*)4у = 0.

- b

After preforming the integration we have:

(4.2) XAx) 5

262 = 0.

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Kantorovic variational method 113

The solution, which is supposed to satisfy the boundary conditions:

X x{x) = 0 at ж= ± a is given by

cosh

X x(x) =

## (¥)

о о Л ( - г )

and the solution of the first step of the iterative process is:

Ui(x,y) (y2- b 2)

cosh/ Xxx

\ b ~ cosh Xxa - 1 where

к - A -

At this instance we observe the symmetry of the original problem and symmetrize the solution:

ux{ x, y) = ax{y2- b 2)

cosh

cosh

## (¥)

+ a 2(x2 — a2)

cosh К У cosh Xxb ax and a2 are now computed by the Eayleigh-Eitz method:

d l ( u x) d l { u x)

dax даc = 0.

If we wish to stop at this point it turns out that the solution is point- wise within 7 ° / 0 of the exact solution in the case a — b = 1. If we wish to continue the iteration we choose:

Щ{х,У) = y) + p2(y2- b2)2X2(x)

and obtain additional accuracy by repeating exactly the first iterative step, mixing the Kantorovic-Galerkin and the Eayleigh-Eitz technique.

In  Komkov has shown only that the value of the functional 1{щ) is a monotone decreasing function of the index i independently of the original choice. He did not examine the strong convergence, or the uncon­

ditional stability of the process. The problem of error propagation and error estimate was also left unanswered. Numerical experimentation with the technique gave extremely favorable answers. It appeared that

8 — R o c z n i k i P T M — P r a c e M a t e m a t y c z n e X V D

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114 . V. K o m k o v

even a clumsy choice of the first guess resulted in an extremely good ap­

proximation after only one iterative step.

A similar technique was suggested in  by Kerr, who examined the same physical problem

Р2Ф + 2 = 0 in Q, Ф = 0 on dQ.

Again the same first guess for the rectangular shape Ü was:

У) = / И ( у 2- б 2).

This is treated by the Kantorovic-Rayleigh-Ritz technique, which now leads to the minimization of:

+ b3f2(x) — 2 b3f(x)\dx.

by:

(4.3) where (4.4)

The solution of Euler’s equation for this variational problem is given

### ш = ^ y [ cosW' - cosh(Aif ) ] ’

h = d

Hence the first approximation is the Kantorovic function:

(4.6) where

*i.o = f i ( v ) 9 o ( y ) ,

(4.6) g0(y) = b2

This is rewritten in the form:

f l,0

(4.7) Ф1г0 = — TcoshAi — cosh ]1 f l —

ii L \ « /J L (/l,0

coshAj I \ «ж /II \b

and the optimizing function дг(у) is sought to minimize 1(ФП) ф1Х = j^coshAi - cosh

We have to solve the Euler’s equation d29i /Гх'2

dy* 91 — -4.Ц (^i)*

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Kantorovic variational method 115

The solution is:

9i(V) 51 1c o sh (^ ^ l +j521sinh where

b

910 =

A1[sinh(2A1) — 2^ ]

+ 0 1 0)

1/2

4a2 I cosh2 ^

8 a2 ( Xx cosh Xx — sinh Xx

^ (sin !^ ^ — 2Аг) Bo 1 =

sinh2A1+ |

0. -Bn = -srio/coshy1}

The next assumption is:

Ф21 = M x ) ; 9 i ( y ) t where /2(ж) satisfies the Euler’s equation:

d2h

dx2 (A2la)/ 2- ^ 2 1 ( y i ) *

The procedure now can be regarded as an iterative system

7 i = Щ ( V i ; « ) , ^ = а К ^ Х ^ г ; Ь ).

The problem of convergence of the approximate (iterative) solutions Фг-(а?, у) to the exact solution Ф(ж, у) of equation (0.1) was never settled in Kerr’s paper. However, he did prove that if the procedure converges, then the final form of Ф(х, у ) will be unique, and independent of the initial choice of g0(y)-

In a recent paper of Anderssen  the arguments summarized below prove a lack of unconditional stability of the iterative procedure as sug­

gested by Kerr. We follow Kerr’s technique and define the iterative pair:

7n = g{K) = A 4

\ 2 -|l/2

-)) ; , resulting from his equations:

(4.8)

(4.9)

(A

An+i f ( y » )

yi(sinh2yt- — 2уг-)

г+ l /

iV‘ cosh2^ —3 sinh yi éyi Х{(&т]12Х{ — 2Х{

4 a2

cosh2 A- ЗзтЬ2Яг- 4Я^

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116 V. K o m k o v

Let the iterating function

h{a\ q) be defined by the relation, h{o\ q) = e2(sinh2e — 2 g)/or(gcosh2g — f sinh2^ + £e).

Anderssen [ 1 ] has observed that his iterative process has oo as a fixed point, and, moreover, that oo is a point of attraction of Kerr’s procedure.

If e (A) is differentiable, then a sufficient condition for Я = oo to be a point of attraction is:

lim

A—mo

G < 1.

A straightforward differentiation of the composite function f(g{ty) and the use of l’Hôspital’s rule results in the conclusion:

(4.10) lim df-g(K)

dl n = 0.

Hence +oo is a point of attraction (provided it is a fixed point), and Kerr’s procedure can not be unconditionally stable.

To prove that +oo is a fixed point of the iterative procedure it is sufficient to note that the maps:

fig(*n)) = 4 + i and g-(f(yn)) = yn+i are both continuous, that [0, oo) is maped into itself, and

f-(g( °°)) = g-(f(°°)) = °°*

In a way one could anticipate the problems which arise in Kerr’s procedure, or in any procedure inductively defined by:

j m

(4.11) щ = / 7 ап ]ы f j 4 - iы , k= 1 kj+1

(4.12)

since the ability to separate variables is the underlining assumption of the method. In the case of the torsion problem the assumptions are even stronger. Since at every step of the procedure we reduce the problem to the solution of a second order linear differential equation with constant coefficients it is clear that we have restricted ourselves to the class of so­

lutions which can be written as products of exponential functions of each variable. To offset this obvious disadvantage we would have to introduce a variant of the method. A technique tried by the engineers (in somewhat simpler situation) would introduce an iteration:

^ n +1) = < +1(%-i) W ,,/«?-i1)(^+i) + Cf V + 1 ( ® 1 1 a?2, ..., CDj, oc}+2, ..., œn)},

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Kantorovic variational method 117

where q)1+1 is an a prori chosen function. In the corresponding Bayleigh- Bitz technique Timoshenko chooses 93 to be a polynomial in the remaining variables, and optimizes C by the Bayleigh-Bitz method. This could be incorporated in a manner similar to  in an attempt to obtain some functions which can not be written in the form (4.12).

References

 R. S. A n d e r s s e n , A stability analysis for the extended K antorovic method applied to the torsion problem (to appear).

 P . H e n r ic i , Error propagation fo r num erical methods (SIAM Series), N ew York 1966.

 L. V. K a n t o r o v i c and Y. L. K r y lo v , A pproxim ate methods in higher analysis, Noordhoff, Groningen, 1958.

 M. Y. K e ld y s h , O n B . G. GalerJcin’s method fo r the solution o f boundary value problems, Faraday Translations, Report F -1 9 5 , NASA, Langley, 1964, 56 p.

 A .D . K e rr , A n extension o f the Kantorovic method, Quart. Appl. Math. 26 (1968), p. 219-229.

 — and H. A le x a n d e r , A n application o f the extended K antorovic method to the stress analysis o f a clamped rectangular plate, Acta Mech. 6 (1968), p. 180-186.

 Y. K o m k o v , A n iterative method fo r obtaining decreasing sequences o f upper bounds o f functionals, Math. Res. Centre U niv. of W isconsin Tech. Report # 5 3 4 ,

N ovem ber, 1964.

 S. G. M ik h lin , The problem o f a m in im u m o f a quadratic functional, Holden D ay, San Francisco 1965.

 — The num erical perform ance o f variational methods, Noordhoff Groningen, 1969.

 A .M . O s t r o v s k i, Solutions o f equations and systems o f equations, N ew York 1966.

 I. G. P e t r o v s k i i , P artial differential equations, Sanders, Philadelphia 1967.

 T. E. S c h u n k , Z u r KnechfestigTceit schwach gékrum m ter zylindrischen Schalen, Ingeneur Archiv 4 (1933), p. 394-414.

 S. T i m o s h e n k o and J. N . G o o d ie r , Theory o f elasticity, second edition, N ew York 1951.

 J. F . T r a u b , Iterative methods fo r the solution o f equations, Prentice-H all, Series in A utom atic Computation, Englewood Cliffs, 1964.

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To determine the most active cryptomelane phase containing different alkali cations both work function and catalytic activity in soot combustion were determined, performing tests

Nevertheless, severe degenerative MS is ex- tremely rare and occurs when calcification extends to the basis of mitral leaflets resulting in narrow- ing of the diastolic flow

Jednym artykułem trudno oddać społeczną głębię przekazywaną w serialu, dlatego skupi- my się wyłącznie na jednym aspekcie: ekonomii biur i zjawisku biurokratyzacji, która

Z tej perspektyw y am biw alentny jest rów nież gest zra n ien ia , p rzebicia, rozcięcia na obrazach M edka - im p lik u je on akt przem ocy, na k tó rą wystawiony jest

The following easy result shows that countably incomplete ultrapowers of infinite structures are always non-trivial..

Diagnostics of material damages and their description are of importance for the development of the methods for improving the reliability, prediction of the

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