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Section 8. Mathematics
in the perturbed motion. Take on these trajectories two arbitrary points N1 and N2. The distance between these points corresponding to the same instant of time can increase for close trajectories, as shown in Figure 1.
Hovsepian Felix, Ph D., in applied mathematical statistic, Professor, Retired, Sunny Isles Beach, Fl, USA E-mail: fhovsepian@yandex.com
MOTION STABILITY WITHOUT LYAPUNOV FUNCTIONS
Abstract: The solutions of nonlinear differential equations system are asymptotically stable if the corresponding solutions of the linear differential equations system are not only asymptotically stable, but also positive definite according to Bochner's works from the theory ofprobability. In this case the corresponding perturbed and unperturbed motions have the same speed of propagation in space. It is possible to design a tracking system, which requires to a system of nonlinear differential equations have followed the system of the linear differential equations.
Keywords: necessary and sufficient conditions, positive definite function, perturbed and unperturbed motion, tracking system.
1. Introduction
Let the solutions ofthe linear differential equations (LDE) system be asymptotically stable in the interval [0, k], where k is the parameter of this system. In this paper it is proved that in the same interval the solutions of the nonlinear differential equations (NDE) system are asymptotically stable only when the solutions of the LDE system are positive definite according to Bochner's works on the theory of probability [1, p. 620]. This notion requires explanations (for details, see below), but for the time being we will explain its application. The real correlation function K(t) known in theory is positive definite in sense of Bochner and represents the harmonic oscillation coswx of all frequencies from the interval (-w, w) averaged over the normalized spectral density V(w):
K(t) = J V(o)cosaTda = Ecosa>z
—oo
where E is symbol of mathematical expectation.
It is not difficult to prove that the propagation velocity of the functions coswx under this integral is constant and equal to each other at all frequencies from (- w, w). If this function is an LDE solution, then these solutions are positive definite. But K(t) can not be a solution of LDE in principle, since K(t) = K(-t). Therefore, the author obtained K(t) using asymptotically stable solutions of the second-order LDE. The prerequisite for its obtaining is property 3 of a positive definite function K(0) > K(t) from paragraph 2.
The more interesting this function is in the theory of motion stability, let us explain in Figure 1. Suppose that in an n-dimensional space we have a point N that describes the trajectory I in the unperturbed motion and the trajectory II
Figure 1.
Obviously, this situation arises from the difference in the speeds of these points. The essence of the author's discovery lies in the fact that the speeds of the points N2 and N2 in Figure 1 differ in LDE, and the way of measuring the speeds of these points is very simple: if you can obtain K(t), therefore, the speeds are the same.
2. Positive definite function (according to Bochner)
Definition 1. A continuous complex-valued function K(t) of a real argument t is positive definite in the interval w < t < w, if for any real numbers t1, t2, ..., Tn, complex numbers Z1, Z2, ..., Zn and integer n
¿K(Tj-rk)ZjZk * 0, K(T) * 0.
k =i
The author draws the reader's attention that in Bochner's definition no function K(r) t 0, since K(t) = 0 satisfies the inequality, and for any homogeneous linear differential equation this function is its trivial solution regardless of the characteristic equation.
We list some of the simplest properties of positive definite function.
1. K(0) > 0. In fact, we set n = 1, t1 = 0, Z1 = 1; then from inequality we find
0 £k (t-tj )Z z j = K (0).
2. K(-t) = K(t) for any real t.
For the proof we set in the inequality n = 2, t = 0, t2 = t, Z2 are arbitrary. We have
0 ¿K(rk-t;)Zzj = K(0 + 0)ZlZl +
k=i j=1
+K(0 -T)ZiZ2 + K(t-0)ZZi + K(T -t)z2z2 =
= K (0) (|Zi I2 + Z212) + K (-T)Zi Z2 + K (TZ1Z2, therefore value
k (-T)ZiZ2 + K (TZ1Z2 must be real. Thus, ifwe put K(-t) = ai + K(t) = a2 + ifí2,
Z ¿2 =V+ i8, ZZ2 =V- iS' it should be afi + frv -a2S + f}2v = 0.
Since Zi and Z2 are arbitrary then 5 and v are arbitrary too. It should be
ai-a2 = 0, pi +p2 = 0.
This implies our assertion.
3. K(0) > K(t) for any real t.
In the inequality obtained above
k (0) (|Zi |2 + Z212)+K (-t)zi Z2 + K (t)zz> 0
we put Zi = K(t), Z2 =-|K(t)\.
Hence when |K(t)| ^ 0 we get K(0) > K(t). If |K(t)| = 0 then again, by property 1 K(0) > K(t).
From what has been proved it follows, incidentally, that if a positive definite function is such that K(0) = 0 then K(t) = 0.
In order to solve uniquely, the trivial solution of which equation is positive definite, the author derivates a function K(t) = AK (t) that is positive definite for any A > 0, hence it is positive definite for A = 0.
For convenience of reading Bochner's Lemma and Theorem are listed below.
Lemma (Bochner). Let K(t) be a bounded continuous (real-valued) function that is absolutely integrable over
i ™
(-to, to). Define V(w) by V(a) = — f exp(-ia>z)K(v)dv. In
order that K(t) be a characteristic function it is necessary and sufficient if K(0) = 1 and V(w)> 0 for all w.
Theorem (Bochner). A continuous function K(t) is the characteristic function of a probability distribution iff it is positive definite and K(0) = 1.
Bochner's results contain terminology from probability theory, so the reader should pay attention only to the conditions when the function K(t) is positive definite. 3. Statement of problem Assume that:
1) we have a system oh homogeneous LDE
— = Ax + bka, o- = (c, *) (1)
dt
with constant coefficients, in which A is square matrix nxn, b and c are given real vectors and in which when fce (0, kma:) the solutions are asymptotically stable;
2) frequency characteristic of the system with = ko and oout = -o is given as
M (ia>)
W (irn) = c '(A - iaE)- b =
N (ia>)
where N(p) and M(p) are polynomials in p, of degree n and m, respectively (n > m, n > 2), do not have common roots;
3) — = ReW(ia>*) is furthest point to the right from
origin, which W(io>*) at the frequency w = u* crosses moving to another quadrant;
4) point k ^ = * can be placed anywhere in the open interval (0,kmJ.
Then the trivial solution of the system (l) is positive definite in the interval [0,kcr).
Example. Let us consider an example that proves the existence of systems of type (l):
^^ = -(c + f )x, + x1 - kx, dt ^12 1
dx 2 dt
dx 3 ,,
—3 = -cx, + bkx, dt 1 1
(2)
where b > 0, c > 0, f > 0, and options b and c
will be assumed to be given and f we define later using b and c.
Make x1 = a and with x2 and x3 eliminated we have the following equation
d3ct / ... d2a da
—- + (c + f )—- + — + ca + k dt3 dt2 dt
d a dt2
— ba
= 0 .(3)
A characteristic equation associated with (3) has the form
p3 +(c + f )p2 + p + c + k(p2 -b) = 0. When k = 0 in equation p3 + (c + f )p ^ + p + c = 0 as is easily verified, the real parts of all three roots are negative, ie for k = 0 the solutions of (3) are asymptotically stable. We determine the value of k = k so that these solutions were
max
asymptotically stable, which use frequency Nyquist stability criterion. The frequency response of the input ka to the output -a in our case has the form
a(rn)
W (irn) =
c (a) + id(a>)
(4)
k=1 j=1
where a(w) = -(w2 + b), c(a>) = c -(c + f)o>2, d(a) = ©(1 -a1), ReW(m) = 2 a(a)C(f, ^, ImW(m) = - 2a(&)d(&) -
c 2(o) + d2(o) " ' c2(a) + d2(a) The roots of the equation d(a) = a>( 1 -a2) = 0 are equal = 0, w23 = ± 1. From (4) we find
a(&)\ _ b
ReW(m)\ ffl=0 =
ie interval in which the solutions are asymptotically stable is equal to [0,c / b). Note with kmax = c / b in the characteristic equation
P3 +(c + f )p2 + p + c + b(p2 -b) = p3 +
+ 1 c + f + b \P2 + P = P
p2 +| c + f + b \P + 1
= 0
regardless of the value f we have zero root, ie parameter f > 0 does not affect the interval [0,c /b). When w23 = ± 1 frequency response crosses the positive real point:
ReW(i©)| a(a)
c (a)
-(a2 + b)
c - (c + f )a>2
_ -(1 + b) _ 1 + b _ 1 _ -f ~ f ~ k* The value kcr in (3) is chosen from the equality obtained above by replacing k * by kcr:
1 + b _ 1
f ~ K '
kr only affects the value of the parameter f, which, as already noted above, does not affect the interval [0,c / b), ie choice k
' cr
satisfies claim 4 in the formulation of the problem. 4. Auxiliary offers
The derivation of the functions K.(t) (i = 1, 2, 3) is provided below in Lemma using the solutions of a LDE.
Lemma. Suppose we have a differential equation with constant coefficients
dL = Ay, dt 7 '
whose solutions are asymptotically stable. Consider particular solutiony.(t) of this equation, which is defined by:
a) a pair of complex conjugate roots when i = 1,
b) a pair of different real roots when i = 2,
c) one real root double multiplicity when i = 3 with initial conditions
dy, (t)
y (t )| t=0=1,
dt
= 0.
ly equal to y ,(t) with t > 0
Then the function K.(t), identica. -and evenly extended y.(t) to t < 0, is positive definite for all i = 1, 2, 3.
Proof. Let us take up solutions yi(t) with various values of i.
1. With i = 1 the solution j1(i) is defined by complex-
conjugate roots p1>2 = —X± a :
y1(t ) = (A1 cos At + B1sin At )exp(-jt ). For the initial conditions specified in the lemma, we have: X
y1(t ) = (cos At +— sin At )exp(-%t ) A
Reflect this y1(t) mirror-like to the axis t < 0 and denote y* (t ) = y 1 (-t ). We will obtain function K1 (t ) = y1 (t ) with t > 0, K1 (t ) = y * (t ) with t < 0.
A Fourier transform of the even function K1 (t) is equal to
w
2nV(im) = J K1(t)exp(-ia>t)dt =
—w
0 w
= J yl(t)exp(-wt)dt + J y!(t)exp(-i^t)dt =
-w 0
» »
= J y* (-s) exp(ia>s)ds + J y1 (t)exp(-ia>t)dt =
= J y1(t )exp(ia>t )dt + J y1(t )exp(-ia>t )dt. 0 0
Note that the integrals in V(w) are different in the multiplier exp(±ia>t) and so their sum is real-valued function. It possible to present the Fourier transform of the function k (t) in the form
1 1 ™
V1(a) = — f K1(t)exp(-ia>t)dt = 2n
1 ^
1 f X
— I (cosAt +— sinAt)exp(-xt)cosa>tdt.
n 0 A
tmg 1
V» =
0A Calculating the V1(w), we get
2Z(Z2 + A2)
> 0
n[X2 + (X + a)2][X2 + (A-a)2]
for all - w < w < w.
2. With i = 2 the solution y2(t) is defined by two different real roots px = -xx and p2 = -X2 (X2 > X1):
y2(t) = ^2 exp(-^1t) + B2 exp(-xt). Given the initial conditions rewrite it as
y2(t) = XX) [exp(-^it) -i^1 exp(-X2t)].
In this case, too, an even function K2(t) may be obtained in the form
K2(t) = y2(t) with t > 0, K2(t) = y2(-t) = y2(t) with t < 0. A Fourier transform
1 ™
V2(rn) = — i K1(z)cosaTdz = 2 2n
1
X2
_ ew(~Xt )--l-exp(~X2t )
n 0 X2 _ X1 \_ X2 Calculating the V2(u>), we get
X1X2(X1
+ X2)
cos cotdt.
V2(a>) =
for all -to < w < TO-
n(X12 +®2)(X22 +®2)
> 0
0
-O
3. With i = 3 the solutiony (t) is defined by one real double multiplicity root p1 = p2 = -x
y3(t) = A3 exp(-xt) + B3t exp(-xt), that considering the initial conditions can be rewritten as y 3 (t) = exp(-xt) + xt exp(-xt).
In this case too an even function K3(t) may be obtained in line with the above rule. A Fourier transform of the function K 3(t) is equal to
1 f
V3(rn) = — I K3(t)exp(—mz)dz =
2n
—f
1 f 1 f
= — | exp(—xt)cosa>tdt +— I xtexp(—xt)cosa>tdt.
71 * K *
"■ 0 0
Calculating the V3(w), we get
VAm) = -
2X>
■ > 0 for all -œ < w < to.
n(x2 +®2f
Because K.(t) is a continuous function and
j (Td <<»,
—00
K.(t), is known to be represented as a Fourier integral
w w
Kt(t) = J exp(ia>T)Vt(rn)drn = 2Jcosa>rVt(a>)da>. (5)
—w 0
In line with Bochner the function K.(t) is positive definite for all i =1, 2, 3.
Lemma is proved
Corollary 1. With the function K,(t) ofthe lemma we can derive a function K(t) so that the coefficient A, remained free, ie
K(t) = AiKi(r) (i =1, 2, 3).
Proof.
1. Consider the function
B
y1(t) = (cos At +—LsinXt)exp(-xt), Ai
The first initial condition for y1(t) and y1(t) is the same: y1(0) = yl(0) = 1. From the condition
dyi(t)
dt
dyi(t)
t=0
dt
= 0,
t=0
we get B = x, ie y1(t) = A1 y1(t). We can obtain Kt(t) = y1(t) A1 X
for t > 0, K[(t) = y1(-t) for t < 0, whence K1(t) = A1K1(t) . 2. Consider the function
X2
From the condition
B
exp(-Xit ) + ~rexp(-X2t )
y 2(0) = 1 = y 2(0) =
X2
( X2 Xi,
1 + A
A
we get ^ = -X. But then dy2(t)
A
X2
dt
= 0,
ie y2(t) = A2y2(t). We can obtain K2(t) = y2(t) for t > 0, K2(t) = y2(-t) for t < 0, consequently K2(t) = A2K2(t) . 3. Consider the function
B
y l(t ) = exp(-xt ) + -31 exp(-%t )
A3
Both initial conditions of y (t) and y3(t) are the same, so y3(t) = A3y'3(t). We can obtain K3(t) = y'3(t) for t > 0, K3(t) = A3K3(t) for t < 0, consequently K3(t) = A3K3(t). Corollary 1 is proved.
In function (5) we can expand the limits of integration, because V.(u) = V(-w):
w
K(t) = J cosazVi(a>)da>. (6)
—w
Putting t = 0 in (6), we get
w
K (0) = J V; (a)da = 1,
—w
ie V.(u) in (6) is a normalized spectral density.
The function K(t) for i =1, 2, 3 satisfies Khinchin theorem [2]: if the function has the form
w
K(t) = J exp(ia>r)dV(a>)
—w
it is necessary and sufficient that it was the correlation function of a stationary random process |(t).
In our case V(®) is differentiable function so K (t) for all i has the form (6). From Khinchin theorem we get that t is a time interval between the sections of |(t) at arbitrary times t1 and t2:
T = t1 - t2. (7)
Mathematics for t makes one requirement : it should vary in the interval (-œ, œ).
Corollary 2. The propagation velocity of the oscillations coswt in the function K,(t) at all frequencies from the interval (- œ, œ) is equal to each other and equal c = const.
Proof. The function K(t) from (6) can be represented as
K (t) = J cos carVi (a)da = Ei cos cot,
—00
where E, is the mathematical expectation coswt that has spectral density V.(u).
We make in (6) the change of integration variable ©' = jico,
(p* 0):
Kt(t) = J cosazVi =
1
= J cos
P J
1 œ
= J cos
P J
r -œ
(HI t
V
Pco
co'\ —
P
P
P
dC.
d(Pc) =
Putting in this integral t = 0:
2
=0
1
K (0) = 1 = f - V| —
' Iß 'Iß
— — = f Vi(—)d—
ie we got another normalized spectral density
Therefore
V. ) =1VI-
ß 'Iß.
k (t) = j cosmrVl (m)drn =
(8)
(9)
= j cosa't 'Vi(&')d&' = K* (t '),
/ \ (i = 1, 2, 3),
where
®T=(j3<o)(z /P) = rn't', co' = Pco, t' = t /p.
Note that the equality ot =o ' t ' implies that cosrnz = cosrn'z' ie function coswx moves sliding on the abscissa from point t to point t' ^r at a constant ordinate.
In the coordinate systems (y,T ) and (y, t ') the functions y = cosrnz and y = cosa'z' have different scale on the horizontal axis. We can investigate time t ' and its relation to the time t. When ft = 1 we have t = t and it follows from the lemma. Now let ft ^ 1. Obviously, we obtain
r'=r / p= t / p, ie time t be compared with t/ ft. It becomes obvious that when ft < 1 time t/ft (or t ' =t / P ) moves faster t: the axis t ' is stretched. If ft > 1, on the contrary, the time t/ft (or t ') moves more slowly t: the axis t ' is compressed.
Let us now consider the propagation velocity of coswT. This velocity, as is known, is determined by the formula
c =—, (10)
12 - 11
where X is the distance between the two closest points coswt, which have the same phase of the oscillation, (12 -11) is the interval of time during which coswt passes this X.
The function K(t) slides along the abscissa, without changing the value of its ordinate. Assume that the velocity of coswt is equal to c. Because cosa>t = cosa>z the propagation velocity of cosa't' when p ^ 1 we can consider relatively coswT. The change of frequency in cosm'r' in the ^ times leads to a change in ^ times X' = X/p (w is inversely proportional to X). Functions cosm'r' and coswt pass different distances, since X^ X', however velocity remains unchanged because abscissa is stretched or compressed but (o'v' = rnv. Due to the arbitrariness of ft > 0 this equality take place for any ft > 0, therefore it is also true for limiting values of p = 0 and P = ±<x>. Thus, this velocity remains
unchanged and equal to c at all frequencies from the interval (- w, w).
Corollary 2 is proved.
5. Example of a nonlinear differential equations system whose solutions are asymptotically stable
Consider spaces of deviations x. (i = 1, 2, ..., n) and original variables y, (i = 1, 2, ..., n) systems (11) and (14) which are given below.
The set of variables of the system (11) forms a point N, and the set of the system (14) forms the point L. Let the points L and N describe the trajectory I in the unperturbed and the trajectory II in the perturbed motions of these systems.
We take on these trajectories any two points L and L' (see Figure 2, a), N and N' , (see Figure 2, b) at coinciding time t and introduce the next
Definition 2. The velocities unperturbed and perturbed motions in (11) and (14) are equal to each other when the velocities of the points N, N', L and L' at coinciding time t equal to each other on the Figure 2.
Figure 2.
Theorem. Let there are given:
1. system of nonlinear differential equations with constant coefficients
— = Ax + bm(a), dt
(11)
where A is square matrix nxn elements of which are constant real numbers, b and с are given real vectors, o = (c, x) is scalar
product of c and *, the function f(a) is an arbitrary, single-valued, piecewise continuous function, defined for all real values of o, satisfying the condition f(0) = 0 and
0 <o(p(o) < ka2, (12)
where k is a parameter;
2. frequency characteristic of the linear part of (11) from the input f(a) to the output -ais given as
M (ia>)
W (ia>) = c ' (A - ia>E)- b =-
(13)
N (ia>)
where N(p) and M(p) are polynomials in p, of degree n and m, respectively (n > m, n > 2), do not have common roots and
at the frequency w = w* crosses furthest point — to the right from origin, moving to another quadrant; 3. solutions of linear differential system
— = Ax + bka, dt
K, (t) = j V(<v)cosa>rda =
—w
= j 5(&-a>*)cosa>td& = cosa*t.
(18)
(14)
obtained from (11) after replacing 9(0-) on kcr, asymptotically stable when k e[0,km), where km > \k * = kr.
Then the solutions of the nonlinear differential system (11) are asymptotically stable in the interval [0,kcr) iff the velocities of the perturbed and unperturbed motions in (11) and (14) at coinciding time t are equal to each other, ie the speeds of the points L' and N' on the Figure 2 at coinciding time t equal to each other.
Proof.
Necessity. Let the solutions of system (11) are asymptotically stable, therefore, then the solutions of system (14) are asymptotically stable. If eliminating from (14) all variables except a and ka, we get the equation
N(p)a + kM(p)a = 0. (15)
We choose in (15) a value of k < kr As n > 2 and for that k all solutions of this linear equation are asymptotically stable there is at least one pair of roots, which can be used to derive an even function according to Lemma
K(t) = 2jvi(a>)cosa>Tda (- to < t < to) (16)
0
Type the pair of roots shows the index of that in (16) is not specified, but from what follows it will be seen that the specification is not necessary. Note that in (16), the function V (w) (i = 1, 2, 3) according to Lemma is continuous for any value of w.
Now we choose in (14) the value of k = \k * = kcr. According to the above, on the one side, we again can get a function of the type of (16), since all solutions of this equation are asymptotically stable. However, on the other side, in the equation
N (p)a- kM (p)a = 0, which has nothing to (15), according to the conditions of the theorem for k = k* at the frequency w = w * holds oscillatory process
N(p)cosm* t - k * M(p)cosm * t = 0.
In our equation (15) for kcr = |k * we have an identity
N(p)cos(rn * t -n) + krM(p)cosrn* t = 0, (17)
when cosrn* t and cos(<v* t — n) oscillate in opposite phase (amplitudes have different sign).
In equation (15) for k = kcr according to (17) there was a frequency w = w* with a probability equal to unity. It means that continuous spectral density V(w) turns into a ^-function at this frequency:
The Bochner conditions of absolutely integrability of function K(t) has been broken, ie we can obtain K(t) in the interval [0,kcr). The system (14) describes the perturbed motion oflinear differential system, the roots of the characteristic equation have negative real parts according to the theorem. Therefore, it is possible to obtain a function K(t) = AiKi(v) according to corollary 1 oflemma and K(t) is positive definite for any At > 0 according to lemma itself, so it is positive definite too when Ai = 0. It means that the all solutions oflinear differential system (14) are positive definite in interval [0,kcr). According to Corollary 2 the propagation velocity coswr in function AiKi(t) is constant at all frequencies from (- to, to) for A. > 0, therefore this velocity is constant for A = 0 in the - to < t < to and much less in the interval [0, to) of the time t, since we have t = t. Consequently, in the linear system (14), the velocities of the points L and L1 in Figure 2; a) at coinciding times t are equal to each other.
According to the theorem, the solutions of systems (11) and (14) are asymptotically stable in the same interval, therefore the trivial solution of system (11) is also positive definite. This trivial solution described unperturbed motion in (11), so the velocity of points L'in Figure 2,a) and N'in Figure 2,b) in coinciding times t equal to each other.
Solutions of (11) is asymptotically stable, then the velocities of the perturbed and unperturbed motions in this system in coinciding times equal to each other, since otherwise the distance between the motions would increase without limit. Therefore, the velocity of points N and N' in coinciding times t equal to each other in (Figure 2,b).
Necessity of the theorem conditions is proved.
Sujfi ciency. Velocities of perturbed relatively unperturbed motions in the linear system (14) according to the necessary condition of Theorem are constant and equal to each other in coinciding time t. This implies that points L and L' fixed relative to one another in coinciding time t (see Figure 2a).
Velocities of unperturbed motions in the linear system (14) and nonlinear system (11) according to the necessary condition of Theorem are constant and equal to each other in coinciding time t. This implies that points L' and N' in coinciding time t fixed relative to one another (see Figure 2a and 2b).
Velocities of perturbed motion relatively unperturbed motion in the nonlinear system (11) according to the necessary condition of Theorem are constant and equal to each other in coinciding time t. This implies that points N and N' in coinciding time t fixed relative to one another (see Figure 2b).
But then the perturbed motion in the linear system (11) and a non-linear system (14) at the same t relative to each other are fixed, ie, points L' in (Figure 2a) and N' in (Figure 2b) are fixed relative to each other.
Solutions in the linear system (14) tend to zero as it tend to zero initial conditions, ie point L' tends to zero. The points L' and N' in coinciding time t in fixed relative to one another, so point N' also tends to zero. It follows immediately that the trivial solution x(0) = 0 of the nonlinear system (11) is stable in the sense of Lyapunov. That is one side.
The other side, the solutions of linear system (11) tend to zero when t tends to infinity, ie point L' tends to zero. The
points N' h L' in coinciding time t fixed relative to one another, ie in system (11) we have:
lim x(t) = 0.
t ^rc
The theorem is proved.
6. Conclusion
1. The solutions of a nonlinear differential equations system are asymptotically stable in the Hurwitz interval of a linear differential equations system iff all solutions of this linear system are positive definite.
2. In the theory of motion stability theory it is introduced the notion of function that is positive definite by Bochner.
References:
1. Feller W. An Introduction to Probability Theory and Its Applications, vol. II. John Wiley & Sons, Inc., New York, London, Sydney, 1971.
2. Khinchin A. Y. Korrelationstheorie des Stationaren stochastischen Prozesse. Mathematische Annalen,- v. 109.1934. pp. 604-615.
