CC BY
29
According to the Nadler theorem there exists a fixed point ws1+s2 of the mapping ^ in Bw/v(s1+s2)(W01+s2) • This fixed point is a u(51 + 52) -local solution to (2.2) extending the chosen -local solution ws1 • Next, let us choose arbitrary u(51 + 52) -local solution ws1+s2 , the corresponding r3 = (1 — q)-1dw/v($1+$2)(^ws1+s2 ,w0), find all possible S> 0 that satisfy the theorem condition with r = r3 and repeat the procedure, etc.
If the distances from the obtained local solutions to the element w° € W are uniformly bounded by some C € R, then for r = C + 1 due to the local contractivity of the multi-valued operator F(■, A) : W ^ Q(W) we find 5 such that 5i > f at each of the steps described above. Therefore, in a finite number of steps we will obtain a unique global solution to (2.2). But if such C does not exist, then the number of steps becomes infinite. As a result, we obtain a unique maximally extended solution to (2.2).
We now consider the following inclusion
parameterized by A € A. We assume that for any A € A, the corresponding set-valued map F(■, A): W ^ Q(W) is a Volterra map on the family u and F(■, Ao) = ^ for some Ao € A .
At each A € A, we naturally apply Definition 2.2 to the inclusion (2.3). For any A € A and Y € (0,1), we denote by S7(A) and S(A) the sets of u(y) -local solutions and global solutions to (2.3), respectively, corresponding to A € A .
Definition 2.5. Let for any A € A, the set-valued map F(■, A): W ^ Q(W) be a Volterra map. We define a Volterra set-valued map F: W x A ^ Q(W) to be uniformly locally contracting on the system u , if it is locally contracting for any y € [0,1) and A € A with the constants q and 5(r) independent of y € [0,1) and A € A .
Theorem 2.2. Let the following two conditions be satisfied:
1) The set-valued maps F(-,A): W ^ Q(W), A € A are uniformly locally contracting on the system u .
2) For any w € W and some A0 € A , the set-valued map ^ : W x A ^ Q(W) is lower semi-continuous (in the Hausdorff metric) at (w, Ao) .
Then for any A € A , the inclusion (2.3) has a local solution and each local solution is extendable to a global or maximally extended solution.
If the inclusion (2.3) has a global solution w0 = w0 at A = A0 , then for any A (sufficiently close to A0 ), the inclusion (2.3) also has a global solution w = w(A) . Moreover, the set-valued map A^S(A) is lower semi-continuous at A0 .
If the inclusion (2.3) has a maximally extended solution w0z at A = A0 , then for any y € (0, (), the inclusion (2.3) has a local solution wY = wY(A) . Moreover, the set-valued map A^SY(A) is lower semi-continuous at A0 .
Proof. The solvability of the inclusion (2.3) for any A € Ba(A0, g0) follows from Theorem
1.1.
We prove the continuous dependence of the sets of solutions on the parameter A. Consider the case when the inclusion (2.3) has global solution. Choose an arbitrary global solution w0 = w(A0) € W at A = A0 . Choose an arbitrary e> 0 . Let us find 5> 0 satisfying Definition 1.3 at r1 = p(w0,w°) + 1, y = 0 and any A € Ba(A0, q0) . For k = [ 1 ] + 1 denote Ai = 15 , l = 1,2,... ,k . Since the condition 2) holds true, for any e > 0 one can find a1 > 0 and q1 > 0 such that for each A € Ba(A0, q1) we have
w € F(w, A),
(2.3)
hw(F(w,A),F(w0,A0)) <
(1 — q)e
1954
for all w € Bw (w0, -1). Assume that -1 < • Let us find -2 > 0 and g2 such that for arbitrary A € Ba(A0, g2) it holds that
hw/u(Ak-i)(Fa — (wa — ,a),Fa— (W0A — ,Ao)) < (
for all Wak-1 € Bw/V{Ak_ 1)(w0Ak-1 ,-2) • Assume that -2 < ^l-ff)(T1 , g2 < g1. There exist -3 > 0 and g3 such that for any A € Ba(A0, g3) it holds true that
hw/u(Ak-2) (FAk-2 (wAk-2 ,A),FAk-2 (w0Ak-2 ,A0)) < (-
for any Wak-2 € Bw/V(Ak_2)(w0ak-2,-3) ; -3 < (1~()a2 , g3 < g2 etc. We perform k iterations and
at the last step find -k and gk , 0 <-k < (1-f°k-1 , gk < gk-1.
Let w0a 1 denote a ^(A1) -local solution to the inclusion (2.3) at A = A0 , that is a fixed point of the multi-valued mapping Fa 1 (-,A0): W/v(Ai)—Q,(W/v(Ai)). If hw/v(A 1)(wa 1 ,w0a 1) <-k , then
hw/v(A 1)(FA 1 (wA 1 ,A),FA 1 (woA1 ,A0)) < (-
for all A € Ba(Ao, gk).
Taking into account the condition 1), we get for any natural number m that
hwM.A 1)(FZ (W0A1 ,A),woA1) < hwMA 1)(FZ (W0A1, A), F'm-1(woA1 ,A)) + ... ■■■ + hw/v(A 1)(Fa 1 (woA1 ,A),W0A1) < (qm-1 + ... + q + 1< ^.
Due to the convergence of the sequential approximations F^ (w0a 1, A) to the fixed point set Sa1 (A) of the multi-valued operator Fa 1 (-,A): W/v(A\) — Q,(W/v(A1)), we obtain the relation hw/v(A^(Sa1 (A),w0a 1) < ^k-1 for each A € B^(A0,gk). Further, let w0a2 be a u(A2) -local solution to the inclusion (2.3) at A = A0 . Choose some wa 1 €Sa1 (A). Then, for all A € BA(Ao,gk), gk < gk-1 and any Wa2 € Bw/V(a2)(woa2,-k-\)C\ wa 1 we get
hw/v( A 2) (FA 2 (WA2 ,A),Wo A 2 ) = hw/v( A 1)(FA2 (w A2 ,A),FA2 (w0A2 , A0)) < (-If)--.
Then
h (F (w A) w ) , , (1 - q)-k-2 < (1 - q)-k-2
hw/v( A2)(F A 2 (w A 2,A),W A 2 ) <-k-1 +--f- < -3-.
For all m = 1, 2, . . . we have
hw/v( A 2)(FA2 (wA 2 ,A),WA 2 ) < hw/v( A 2)(FA2 (wA 2,A), Fm-1(wA 2 , A)) + ...
... + hw/uA)(Fa2(Wa2,A),wa2) < (qm-1 + .. + q + i)^)-- < .
Taking into account the convergence of the approximations FA2 (ua2, A) to Sa 2 (A), for any wa2 € € 2 (A) , we obtain
hw/u( a2)(wa2 ,w0 A 2 ) < hw/u(A 2) (wA 2 ,FA2 (wa2 , A)) + +hw/u(A 2)(FA2 (w A 2 ,A),WA 2 ) + hw/u( A 2)(wa 2 ,w0 A 2 ) < + -k-1 < .
1955
Using the convergence of the sequential approximations Fm3 (^a3 , A) to the fixed point set Sa3 (A) of the multi-valued operator Fa3 (■, A): W/u(A3) ^ Q(W/u(A3)) for any wa2 €sa2 (A), wa3 € € Bw/u(a3)(w0a3,°k-2)C\wa2 and each A € Ba(A0, Qk), Qk < Qk-1, we obtain the following estimate: dW/u(A3)(wA3 ,w0a3) < . We, then, repeat this procedure. At the k -th step we prove in an analogous way that the inequality pw(w(A),w0) <e holds true for some w(A) €S(A) for all A € Ba(A0, Qk) .Therefore, hw (S(A),w0) ^ 0 as A ^ A0 , and, thus, the set-valued map A^S (A) is lower semi-continuous at A0
Let now a solution w0n to the inclusion (2.3) at A = A0 be maximally extended. Fix arbitrary Y € (0, n) and let w0l denote the restriction of the solution w0n . For the equation wY = FY(wY, A0) the element w0l € W/u(y) is a global solution. As is shown above, for all A from some neighborhood of A0 , the inclusions wY € FY(WY, A) have global solutions wY(A), and hw/v(Y)(SY(A),w0l) ^ 0 as A ^ A0 . Therefore, the set-valued map A^SY(A) is lower semi-continuous at A0 .
REFERENCES
1. Shepherd G.M. The synaptic organization of the brain // Oxford university press. 2004.
2. Lui J.H., Hansen D.V., Kriegstein A.R. Development and Evolution of the Human Neocortex // Cell. 2011. V. 146. № 1. P. 18-36.
3. Swenson R.S. Review of clinical and functional neuroscience, In: Ed. G.L. Holmes, Educational Review Manual in Neurology. Castle Connolly Graduate Medical Publishing. 2006.
4. Graben P.B., Kurths J. Simulating global properties of electroencephalograms with minimal random neural networks // Neurocomp. 2008. V. 71. № 4. P. 999-1007.
5. Hopfield J.J. Neural networks and physical systems with emergent collective computational properties // Proc. Nat. Acad. Sci. 1982. V. 79. P. 2554-2558.
6. Burlakov E, Zhukovskiy E, Ponosov A., Wyller J. Existence, uniqueness and continuous dependence on parameters of solutions to neural field equations // Memoirs on Differential Equations and Mathematical Physics. 2015. V. 65. P. 35-55.
7. Amari S. Dynamics of Pattern Formation in Lateral-Inhibition Type Neural Fields // Biol. Cybern. 1977. V. 27. P. 77-87.
8. Sompolinsky H., Shapley R. New perspectives on the mechanisms for orientation selectivity // Curr. Opin. Neurobiol. 1997. V. 5. P. 514-522.
9 . Taube J.S., Bassett J.P. Persistent neural activity in head direction cells // Cereb. Cortex. 2003. V. 13. P. 1162-1172.
10 . Fuster J.M., Alexander G. Neuron activity related to short-term memory // Science. 1971. V. 173. P. 652.
11 . Wang X-J. Synaptic reverberation underlying mnemonic persistent activity // Trends Neurosci. 2001. V. 24. P. 455-463.
12 . Pinotsis D.A., Leite M., Friston K.J. On conductance-based neural field models // Frontiers in Computational Neuroscience. 2013. V. 7. P. 158.
13 . Tass P.A. A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations // Biological cybernetics. 2003. V. 89. P. 81-88.
14 . Suffczynski P., Kalitzin S., and Lopes Da Silva F.H. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network // Neuroscience. 2004. V. 126. P. 467-484.
15 . Kramer M.A., Lopour B.A., Kirsch H.E., and Szeri A.J. Bifurcation control of a seizing human cortex // Physical Review E. 2006. V. 73 :41928.
16 . Schiff S.J. Towards model-based control of Parkin- son's disease // Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2010. V. 368. P. 2269-2308.
17 . Ruths J., Taylor P., Dauwels J. Optimal Control of an Epileptic Neural Population Model // Proceedings of the International Federation of Automatic Control. Cape Town, 2014.
18 . Burlakov E., Zhukovskiy E.S. Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations // Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki -Tambov University Review. Series: Natural and Technical Sciences, 2015. V. 20. Iss. 1. P. 9-16.
19 . Burlakov E.O., Zhukovskiy E.S. On well-posedness of generalized neural field equations with impulsive control // Izv. Vuzov. Mathematics. 2016 V. 5. P. 75-79.
20 . Zhukovskiy E.S. Generalized Volterra operators in metric spaces // Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2009. V. 14. Iss. 3. P. 501-508.
21 . Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V. Introduction to the Theory of Multivalued Maps and Differential Inclusions. 2nd ed. Moscow: Librokom, 2011.
1956
ACKNOWLEDGEMENTS: The work is partially supported by the Russian Fund for Basic Research (project № 16-31-50037).
Received 21 October 2016
Burlakov Evgenii Olegovich, Norwegian University of Life Science, As, Norway, graduate student, e-mail: eb @bk.ru
УДК 517.988.5, 51-76
DOI: 10.20310/1810-0198-2016-21-6-1950-1958
ОПЕРАТОРНЫЕ ВКЛЮЧЕНИЯ ВОЛЬТЕРРЫ В ОБОБЩЕННЫХ МОДЕЛЯХ НЕЙРОПОЛЕЙ С УПРАВЛЕНИЕМ. I
© Е. О. Бурлаков
Норвежский университет естественных наук 1432, Норвегия, г. Аас, Университетская, 3 E-mail: eb_@bk.ru
Получены условия разрешимости операторных включений Вольтерры и непрерывной зависимости решений от параметра. Результаты могут применяться к исследованию обобщенных моделей нейрополей с управлением.
Ключевые слова: операторные включения Вольтерры; модели нейрополей; управление; существование решений; непрерывная зависимость от параметров
СПИСОК ЛИТЕРАТУРЫ
1. Shepherd G.M. The synaptic organization of the brain // Oxford university press. 2004.
2. Lui J.H., Hansen D.V., Kriegstein A.R. Development and Evolution of the Human Neocortex // Cell. 2011. V. 146. № 1. P. 18-36.
3. Swenson R.S. Review of clinical and functional neuroscience, In: Ed. G.L. Holmes, Educational Review Manual in Neurology. Castle Connolly Graduate Medical Publishing. 2006.
4. Graben P.B., Kurths J. Simulating global properties of electroencephalograms with minimal random neural networks // Neurocomp. 2008. V. 71. № 4. P. 999-1007.
5. Hopfield J.J. Neural networks and physical systems with emergent collective computational properties // Proc. Nat. Acad. Sci. 1982. V. 79. P. 2554-2558.
6. Burlakov E, Zhukovskiy E, Ponosov A., Wyller J. Existence, uniqueness and continuous dependence on parameters of solutions to neural field equations // Memoirs on Differential Equations and Mathematical Physics. 2015. V. 65. P. 35-55.
7. Amari S. Dynamics of Pattern Formation in Lateral-Inhibition Type Neural Fields // Biol. Cybern. 1977. V. 27. P. 77-87.
8. Sompolinsky H., Shapley R. New perspectives on the mechanisms for orientation selectivity // Curr. Opin. Neurobiol. 1997. V. 5. P. 514-522.
9 . Taube J.S., Bassett J.P. Persistent neural activity in head direction cells // Cereb. Cortex. 2003. V. 13. P. 1162-1172.
10 . Fuster J.M., Alexander G. Neuron activity related to short-term memory // Science. 1971. V. 173. P. 652.
11 . Wang X-J. Synaptic reverberation underlying mnemonic persistent activity // Trends Neurosci. 2001. V. 24. P. 455-463.
1957
12 . Pinotsis D.A., Leite M., Friston K.J. On conductance-based neural field models // Frontiers in Computational Neuroscience. 2013. V. 7. P. 158.
13 . Tass P.A. A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations // Biological cybernetics. 2003. V. 89. P. 81-88.
14 . Suffczynski P., Kalitzin S., and Lopes Da Silva F.H. Dynamics of non-convulsive epileptic phenomena modeled by a bistable neuronal network // Neuroscience. 2004. V. 126. P. 467-484.
15 . Kramer M.A., Lopour B.A., Kirsch H.E., and Szeri A.J. Bifurcation control of a seizing human cortex // Physical Review E. 2006. V. 73 :41928.
16 . Schiff S.J. Towards model-based control of Parkin- son's disease // Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2010. V. 368. P. 2269-2308.
17 . Ruths J., Taylor P., Dauwels J. Optimal Control of an Epileptic Neural Population Model // Proceedings of the International Federation of Automatic Control. Cape Town, 2014.
18 . Burlakov E., Zhukovskiy E.S. Existence, uniqueness and continuous dependence on control of solutions to generalized neural field equations // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2015. T. 20. Вып. 1. С. 9-16.
19 . Burlakov E.O., Zhukovskiy E.S. On well-posedness of generalized neural field equations with impulsive control // Известия вузов. Математика. 2016 Т. 5. С. 75-79.
20 . Zhukovskiy E.S. Generalized Volterra operators in metric spaces // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2009. Т. 14. Вып. 3. С. 501-508.
21 . Borisovich Yu.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V. Introduction to the Theory of Multivalued Maps and Differential Inclusions. 2nd ed. Moscow: Librokom, 2011.
БЛАГОДАРНОСТИ: Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (проект № 16-31-50037).
Поступила в редакцию 21 октября 2016 г.
Бурлаков Евгений Олегович, Норвежский университет естественных наук, г. Аас, Норвегия, аспирант, e-mail: eb_@bk.ru
Информация для цитирования:
Burlakov E.O. Volterra operator inclusions in the theory of generalized neural field models with control. I. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2016, vol. 21, no. 6, pp. 1950-1958. DOI: 10.20310/1810-0198-2016-21-6-1950-1958
1958
UDC 517.98
DOI: 10.20310/1810-0198-2016-21-6-1959-1962
BOUNDARY REPRESENTATION FOR LOBACHEVSKY SPACES
© L. I. Grosheva
Tambov State University named after G.R. Derzhavin 33 Internatsionalnaya St., Tambov, Russian Federation, 392000 E-mail: gligli@mail.ru
For canonical representations on a Lobachevsky space, a description of distributions concentrated at the boundary is given.
Key words: Lobachevsky spaces; canonical representations; distributions; boundary representations
In this paper we extend to Lobachevsky spaces G/K , G = SO0(n — 1,1), K = SO(n — 1) our results [1], [2], [3] on distributions concentrated at the boundary of the Lobachevsky plane. We use the Klein model of the Lobachevsky space G/K . It is the unit ball B : (u,u) < 1 in (Rn-1 with the fractional linear action (from the right):
ua + y ( a ß \
u ^ u ■ g = —-- , g = r .
y uß + S \ y S J
Here (u, v) is the standard inner product in (Rn 1:
{u, v) = U1V1 + ... + Un-lVn-1■
The boundary S : (u,u) = 1 is the absolute of the Lobachevsky space. Let B = B u S.
Canonical representations R\ , A e € , of the group G act on the space d(B) by
(RX(g)f) (u) = f (u • g)(uf3 + ¿)-A-n.
They can be extended to the space D'(B) of distributions on B and, in particular, to the space X(B) of distributions concentrated at S. The restriction L\ of R\ to X(B) is called a boundary representation.
Introduce on B polar coordinates p,s : u = yT — p • s , s e S , where p = 1 — (u,u). In these coordinates the Laplace-Beltrami operator A on B is
d2 d p A = 4(1 — p)p2tt2 + 2p(4 — n — 3p)— + As , dp2 dp 1 — p
with the Laplace-Beltrami operator AS on S . Let Ag be the Casimir element of the Lie algebra g of the group G . To this element, the representation R\ assigns a differential operator (the Casimir operator)
d
Ax = A + 4(A + n)p(1 — p) — + (A + n) [A + 2 — (A + n + 1)p].
1959
Its K -radial part (acting on K -invariant functions) is the operator
d2 d RadAx = 4(1 - p)p2 dp2 + 2p [2X + n + 4 - (2X + 2n + 3)p]dp
+ (X + n) [X + 2 - (X + n + 1)p].
The Berezin form (f, h)\ on D(B) is a bilinear form defined by:
(f,h)x = c(X) j {1 -(u,v)]x f (u) h(v) dudv, jb
'B
где du is the Euclidean measure on B ,
Г ()/Г () .
The Casimir operator and the Berezin form are invariant with respect to R\ .
Representations Ta , a € С , of the group G associated with a cone act on the space D(S) by
(Ta(g)<p)(s) = ф ■ g)(se + S)a■ They are irreducible for all a except a € N and a € 2 — n — N . Here and further N = {0,1,2,...} .
For X€ —(n — 4)/2 + N , the boundary representation L\ decomposes into the direct sum of representations T2-n-\+2k , к € N , as follows. First, we introduce differential operators Wa,k and W* k on D(S) , which are polynomials in AS :
Wa,k = wk(a, As), W*k = w*(a, As),
where wk , w* are defined by means of a reproducing function (with щ = l(3 — n — l) and the Gauss hypergeometric function F):
'a + n — 2 +1 a + n — 1 +1 . n
2
/л \H2 n (a + n — 2 +1 a + n — 1 +1 ,n\ ^ , . k
(1 — p)l/2 Fi-2-, -2-; a + ^ P) = wk (a,M )Pk,
^ ' k=0
(1 — P)-l/2 F (a-+— ; a + 2; p) = £ w^P*. 2 2 2
Then we define operators (\,k : D(S) ^ (B) :
(v) = Y,(-I)b tSm Wx-2k,b(v) • s(k-b)(p), b=0 (k 0)-
they intertwine T2-n-A+2k with La . The space ^(B) decomposes into the direct orthogonal (with respect to the Berezin form) sum of their images VA,k , k € N . These subspaces are eigenspaces of Aa with eigenvalues (X — 2k)(X + n - 2 - 2k). The "old basis" ^(s)S{k)(p) is expressed in terms of the "new basis" :
k k!
V(s)6(k)(p) = £ (—1)k-r ^ Cv (W*-n-X+2rk-rШ .
r=0
1960
In the subspace X(B)K of E(B) consisting of K -invariant distributions we have two natural bases: the first one consists of derivatives of the delta function S(p):
5(k)(p), k = 0,1,..., (1)
the second one consists of distributions
(\,k = &,k(1), k = 0,1,.... (2)
Basis (2) consists of eigenfunctions of Aa and is orthogonal with respect to the Berezin form (we use the notation a[m] = a(a + 1)... (a + m — 1)):
(Zx,k,Zx,k)x = f(A,k), (Zx,k,Zx,r)x = 0, k = r,
where
f(A, k) = b(A) • 2-4k k! (—1)k (—A)[2k](3 — n — A)[2k]
((4 - n)/2 - A)[2k] ((2 - n)/2 - A + k)[
h(A) = 2A+ra—3 (n-2)/2 r(A + (n - 2)/2)r((-A + 1)/2) h(A) n r ((n - 1)/2) r((2 - n - A)/2)r(A + n - 2) '
Elements of bases (1) and (2) are expressed in terms of each other by means of triangular matrices with the unit diagonal, namely,
t V ( iV (k) 2—2b (A + n - 2 - 2k)[2b] *{k—b)(n) (3)
CAk = ^(-1) W (A + n/2 - 2k) [b] * (p)' (3)
Д, A)0 2 s—2k (3 - n - A + 2s)[2fc_2s] . g (-1)k4 J 2 k ((4 - n)/2 - A + 2,)[- •' • (4)
Notice that formula (3) can be written as follows:
. /A + n — 2 A + n — 1 n \ (k)
Ca ,k = F i-2--k, -2--k; A + 2 — 2k; pj 5(k>(p).
Formula (4) gives a generating function for ¿(k) (p):
d
exp( ui)m=L
dp J ' ml
7 m=0
4 - n- A 3 - n- A 4- n x Fl -2--+ m, -2--+ m^ —2--A + 2m; -u ] • (A , m^
Pairwise inner products of basis (1) are given by:
(ö(m)(p),6(r)(p)]x=6(A) • (-i)m+r 2-2r-2m (3 -x)i-rA)[2r] •
REFERENCES
1. Grosheva L.I. Representations on distributions on the Lobachevsky plane concentrated at the boundary // Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 1998. V. 3. Iss. 1. P. 46-50.
2. Grosheva L.I. Boundary representations on the Lobachevsky plane // Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2005. V. 10. Iss. 4. P. 357-365.
1961
3. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane // Acta Applicandae Mathematicae. 2002. V. 73. № 1&2. P. 59-77.
Received 21 October 2016
Grosheva Larisa Igorevna, Tambov State University named after G.R. Derzhavin, Tambov, the Russian Federation, Candidate of Physics and Mathematics, Associate Professor of the Functional Analysis Department, e-mail: gligli@mail.ru
УДК 517.98
DOI: 10.20310/1810-0198-2016-21-6-1959-1962
ГРАНИЧНЫЕ ПРЕДСТАВЛЕНИЯ НА ПРОСТРАНСТВЕ ЛОБАЧЕВСКОГО
© Л. И. Грошева
Тамбовский государственный университет им. Г.Р. Державина 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33 E-mail: gligli@mail.ru
Дано описание обобщенных функций, сосредоточенных на границе пространства Лобачевского.
Ключевые слова: пространство Лобачевского; канонические представления; обобщенные функции; граничные представления
СПИСОК ЛИТЕРАТУРЫ
1. Грошева Л.И. Представления в обобщенных функциях на плоскости Лобачевского, сосредоточенных на границе // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 1998. Т. 3. Вып. 1. С. 46-50.
2. Грошева Л.И. Представления в обобщенных функциях на плоскости Лобачевского, сосредоточенных на границе // Вестник Тамбовского университета. Серия Естественные и технические науки. Тамбов, 2005. Т. 10. Вып. 4. С. 357-365.
3. Molchanov V.F., Grosheva L.I. Canonical and boundary representations on the Lobachevsky plane // Acta Applicandae Mathematicae. 2002. V. 73. № 1&2. P. 59-77.
Поступила в редакцию 21 октября 2016 г.
Грошева Лариса Игоревна, Тамбовский государственный университет им. Г.Р. Державина, г. Тамбов, Российская Федерация, кандидат физико-математических наук, доцент кафедры функционального анализа, e-mail: gligli@mail.ru
Информация для цитирования:
Grosheva L.I. Boundary representation for Lobachevsky spaces. Vestnik Tambovskogo universiteta. Seriya Estestvennye i tekhnicheskie nauki - Tambov University Review. Series: Natural and Technical Sciences, 2016, vol. 21, no. 6, pp. 1959-1962. DOI: 10.20310/1810-0198-2016-21-6-1959-1962
1962
