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XXXIV

Дальневосточная

математическая

школа-семинар

имени академика

Е.В. Золотова

ФУНДАМЕНТАЛЬНЫЕ

ПРОБЛЕМЫ МАТЕМАТИКИ

И ИНФОРМАЦИОННЫХ НАУК

ТЕЗИСЫ ДОКЛАДОВ

ХАБАРОВСК

2009

Институт прикладной математики ДВО РАН

Вычислительный центр ДВО РАН

Дальневосточный государственный университет путей сообщения

Тихоокеанский государственный университет ХХХIV Дальневосточная математическая школа-семинар имени академика Е.В. Золотова «Фундаментальные проблемы математики и информационных наук»

Хабаровск 25–30 июня 2009 г.

Тезисы докладов Хабаровск УДК 517, Утверждено к печати Ученым советом Института прикладной математики ДВО РАН Под научной редакцией С.А. Луковенко ХХХIV Дальневосточная математическая школа-семинар имени академика Е.В. Золотова «Фундаментальные проблемы математики и информационных наук»: Тез. докл. – Хабаровск: Изд-во Тихоокеан. гос. ун-та, 2009. – 198 с.

ISBN 978-5-7389-0750- Школа-семинар проводится при поддержке Президиума ДВО РАН, Российского фонда фундаментальных исследований, Правительства Хабаровского края.

ISBN 978-5-7389-0750-0 ©ИПМ ДВО РАН, Школа посвящена 70-летию чл.-корр. РАН Николая Васильевича Кузнецова Н.В. Кузнецов родился 24 июня 1939 г. в пос. Хачмас АзССР. В 1962 году с отличием закончил Московский физико-технический институт, в 1965 году – аспирантуру при институте. Научную деятельность Н.В. Кузнецов продолжил в должности младшего научного сотрудника отдела теории чисел Математического института им. В.А. Стеклова АН СССР.

С 1969 года работал в должности младшего научного сотрудника, затем старшего научного сотрудника лаборатории систем управления при Московском педагогическом институте.

В 1971–1972 годах стал начальником сектора, затем – отдела в Центральном научно-исследовательском институте информации и технико-экономических исследований, а также начальником отдела в Научно-исследовательском институте систем управления и экономики.

В 1973 году Н. В. Кузнецов переехал на Дальний Восток, работал в ХабКНИИ ДВНЦ АН СССР в должности старшего научного сотрудника, заведующего лабораторией.

С 1981 по 1988 годы работал заместителем директора Вычислительного центра ДВО РАН. В 1989 года Н.В. Кузнецов перешел в Институт прикладной математики ДВО РАН. С 1992 по 2006 годы он – директор ИПМ ДВО РАН.

Н.В. Кузнецов в 1965 году защитил кандидатскую, а в 1982 – докторскую диссертацию. В 1987 году избран членом-корреспондентом АН СССР.

За успехи в научной и научно-организационной деятельности Н.В. Кузнецов в 1983 году награжден медалью «За трудовую доблесть», в 2000 г. – орденом Почета.

Budarina N., 12.., Dubickas A.K., 22..,.., Laurincikas A.,.., Lavrenteva O. M.,.., Matsuki N.,..,.., Nir A.,.., 73, 78, 80, Tanabe S.,..,.., 63, 119,..,..,..,..,..,..,..,..,..,..,..,..,..,.., 137,..,..,.., 76,..,..,..,.., 140,..,..,..,..,.., 68,.., 162..,.., 17, 77.., 134, 147,.., 78..,.., 80..,.., 21.., 166, 168,.., 23..,.., 152..,.., 24..,.., 26..,.., 167..,..,..,..,..,..,..,..,.., 67, 74,..,.., 82,..,..,..,..,..,..,..,..,..,..,.., 84,..,..,..,..,..,..,..,..,..,..,..,..,..,..,.., 88, 128..,.., 166..,..,..,..,..,..,.., 134, 148,..,..,..,..,..,..,.., 116..,..,..,.., 44..,..,..,..,..,..,.., 109..,.., 44..,.., 176..,.., 150, 151..,.., 151, 152..,..,..,..,..,..,..,..,..,..,..,..,..,.., 65,..,..,.., 47..,.., 111.., 137,..,..,..,..,..,.., 57, 156,.., 59,..,.., 60,,.., 181,..,..,..,..,..,..,..,..,..,.. (, ).. [1],,,, A. [2],,, (),, [3].

() :

- (), ;

- (), () [4].

.

[1] Lausch H., Nobauer W. Algebra of polynomials. Amsterdam, London, New York: North-Holland Publishing Company, American Elsevier Publishing Company Inc., 1973.

[2].. //. 2007.

19.. 2.. 27-44.

[3]... // XXXIII..,,,, 2008.. 114-116.

[4].. -. //. 1999. 11.. 2.. 3-19.

Sn.. (, ) Sn  {1, 2, 3,..., n}.

. [1] Sn, :

1. xi+1 = e i 2. xk · xi = x1 · xi+1 · xk, k i, xi , 1 i n 1.

n g = x1 1 ·... · xn1, 0 i i.

g1, g2 Sn :

g = x1 ·... · xn1, 1 1 n g = x1 ·... · n xn1, 2 g1 · g2 = x1 ·... · xn1, n i = i (1,..., n1, 1,..., n1 ).

i. i ,..

i (1, 2,..., k + k + 1,..., n1, 1, 2,..., m + m + 1,..., n1 ) = = i (1,..., n1, 1,..., n1 ) i, Sn. :

k 1. (xm · xn ) = xm k k+m1 · xn, m + k 1 n;

m 2. x · xk = x · x+k · x, + k n;

n n 3. xn · xk = xk · xk · xn, k n;

k k+ 4. xn · xk = xk1 · xn1 · xn, k n;

n n k k 5. xn · xk = xk · xnk n n n 6. ei;

i+1 · x1 · x2 ·... · xn1 = 1 2 n +[i1 ]i +1+[i1 ]i = x1 ·... · xi2 · xi1i 1 · xi i i2 n ·... · xn1, 1 i ei;

i+1 - i- i+1-, [k]i - k i.

i.

[1]... ( ) // -.... :.. :. 2000.. 122124.

[2]...,... :.. /...

//.:. 1980  240.

[3]..,... 2 //..:, 2003..

108142.

ON PRIMITIVELY 2-UNIVERSAL QUADRATIC FORMS N. Budarina (, ) In 1993, John Horton Conway and William Schneeberger announced the Fifteen Theorem, giving a criterion characterizing the positive denite classically integral (or integer-matrix) quadratic forms which represent all positive integers.

Specically, they showed that any positive denite classically integral form which represents the set of nine critical numbers S1 = {1, 2, 3, 5, 6, 7, 10, 14, 15} is universal. In 1999 Manjul Bhargava [2] proved the Fifteen Theorem, and he showed that there are exactly 204 universal positive denite classically integral quaternary quadratic forms.

Today there are at least ve directions in which the problem of universality develops:

• higher-dimensional analogs of universal form (m-universal forms);

• universal forms over totally real number elds;

• primitive universality;

• almost universality;

• odd or even universality.

Recently, in [1] In particular, we give we have given an ecient method for deciding whether a positive denite classically integral quadratic form in 4 or more variables with odd square-free determinant is almost primitively universal.

Byeong Moon Kim, Myung-Hwan Kim, and Byeong-Kweon Oh [3] found all positive denite classically integral quinary quadratic forms that represent all positive denite classically integral binary quadratic forms and presented an analogous criterion for 2-universality:

S2 = { 1, 1, 2, 3, 3, 3, [2, 1, 2], [2, 1, 3], [2, 1, 4]}.

We investigate the minimal primitive representations of binary quadratic forms over the even ring Z2.

Using the minimal representations over odd local rings Zp [4] and Z2 we inves tigate quadratic forms from Kim-Kim-Oh's list for the primitive 2-universality and research for the criterion of the primitive 2-universality for the special class of the quadratic forms.

[1] Budarina N. On primitively 2-universal quadratic forms, (submitted).

[2] M. Bhargava. On the Conway-Schneeberger Fifteen Theorem, in Proceedings of the Conference on Quadratic Forms and Their Applications, ed. E. Bayer Fluckiger, D. Lewis, A. Ranicki (AMS Bookstore, 1999), pp. 2738.

[3] B.M. Kim, M.-H. Kim, B.-K. Oh. 2-universal positive denite integral quinary quadratic forms In: M.-H. Kim, J.S. Hsia, Y. Kitaoka, R. Schulze-Pillot, eds: Con temp. Math., 249, Mathematical association of America, Washington DC, 1999, 5162.

[4] V.G. Zhuravlev. Primitive embedding in local lattices with prime determinant, St. Petersburg Math. J., 11(1) (2000) 6790.

-.. (, ) = P SL2 (Z) , 2 ab b ab a M = ± =, cd d cd c 1. P 1 (Q) = Q {} M () = (a + b)(c + d)1.

,. [1].

[2], Rn, .

.. ([3], [4]) - ;

 (. [5]), L,., E(L) : P 1 (Q) P 1 (Q) L,, M,, P 1 (Q) :

(II) (M (), M ()) = (, ) M 1.

(I) (, ) + (, ) = (, );

(, 0) E(L)  M(L);

L, A L, :

A + A U + A U 2 = 0.

A + A S = 0, 0 1 1 S = ±, U = ± 10 1 ., 2 0 (N ), F : Z Z C, m, n, l Z:

1) F (m, n) = F (m + N, n) = F (m, n + N );

2) F (lm, ln) = F (m, n), (l, N ) = 1.

ab F = F (am + bn, cm + dn).

cd. H  F : P 1 (Q) C F1, F2 = F1 ()F2 () P 1 (Q) (f M )() = f (M ()).

, G G() = 1, G(0) = 1 G() = P 1 (Q) M(H). M(H) (  H).

. M(H).

( 09 1--09).

[1] Eichler M. Eine Verallemeinerung der Abelschen Integrale. Math. Z. 67, 267 (1957).

[2]...

. [3].. -.

.,.., 36, 1966 (1972) [4].. p-..., 92, 3, 378401 (1973).

[5].....,.37,. 4, 2738 (2003).

.. (, ) [1] J(v) = 1 2 | v| d f v d min, (1) v K = {w W 1 () : v 0.. }.

Rn (n = 2, 3)  1/, f L2 (), v W2 () v W2 (). f d (1),, [1].

u (1) W2 ().

u u, n (n  ) l v d, L(v, l) = J(v) u u v W2 (), l (L2 ())+.

L(u, l) L u, L v, n n (L2 ())+ ,.

| v|2 d a(v, v) = W2 (),. 1 + l2 d, M(v, l) = J(v) + (l r v) 2r v + = max{v, 0}.,. M(v, l).

[1].,.-...:, 1980.

..,.. (, ).,.

.

,,,,.

.,,.

,.

. (. [1]).

09-III--01-002.

[1]..,...., 1985.

.. (, )., [1]-[3]., [4].

, -, (0, 0). T(x1, x2 ) : (x1, x2 ) (t1 x1, t2 x2 ) t1, t2 0.

T() T(x1, x2 ), (x1, x2 ).

a, d a d/2 : (a, d) = {(d · n a · m, m)|n, m Z}.

1. (a, d), T 1) T() ;

2) T() (a, d).

M((a, d);

).

2. M((a, d);

) M((a, d);

), T 1) ± ± T();

2) T() (a, d).

3. as(a,d,) | a1 | a2 | (ai {1, 1}, bi N i 1, bs(a,d,) 2) + + ··· + |b1 |b2 |bs(a,d,)  a/d, i M((a, d);

),, ai +bi  M((a, d);

).

.

. (x, y) = 0 . (a0, b0 ), (a1, b1 ) (2a0, 0) = (a0, b0 ) = (a1, b1 ) = (0, 2b1 ) = 0. [0, 1] - = () (u, v) = 0, a0 u a1 ;

a1 s 2a0 ;

(s, t) = 0, u = s, t = v (x, y) = 0, x = s u, y = t + v 0 x a0.

1, 2 [0, 1], 1, 2 [1/2, 1], () du (1+u)2 = c1 = 0 (1, 2 ], () du (1+u)2 = c2 = 0 (1, 2 ] ((), ()), d 2 c d s(a, d, ) = (d) 1 () log d + 2 () + O, (d5/6 log7/6+ d), a= (a,d)= 1 (), 2 () (1 () 0) [4].

, 07-01- 06-I--09).

[1] Heilbronn H. On the average length of a class of nite continued fractions// in Abhandlungen aus Zahlentheorie und Analysis, Berlin, VEB. 1968. C. 87-96.

[2] Porter J.W. On a theorem of Heilbronn// Mathematika. 1975, V. 22, 1. C.

20-28.

[3].. xy l( mod q) //. 2008, T.20, 5. C. 186-216.

[4].. //. 2008, T. IX, 1 (25).. 80-107.

.. (, )....

fk (z), k = 1, 2, |z| 1. |f1 (0)f2 (0)| |f1 (0) f2 (0)|2. (1) C Dk = fk ({z : |z| 1}), k = 1, 2, |f1 (0)f2 (0)|. C. :..,..,..,..

,..,.. [1]..

,,.,.., (1), D1 D2 = : f1 (0) f2 (0) D1 D2.

( 08-01 00028) ( 09-III-A-01-007).

[1].. //. 1997.. 9.. 5.. 150.

POWERS OF A RATIONAL NUMBER MODULO Arturas Dubickas (Vilnius University & Institute of Mathematics and Informatics, Vilnius) Let p q 1 be two coprime integers. Suppose that I is an interval of the torus R/Z. Is there a non-zero real number such that the numbers (p/q)n, n = 0, 1, 2,..., modulo 1 all lie in I?

A hypothetical Mahler's Z-number [5] is such a 0 for which the fractional parts {(3/2)n }, n = 0, 1, 2,..., all lie in [0, 1/2). It seems very likely that Mahler's Z-numbers do not exist. In this direction, Flatto, Lagarias and Pollington [4] showed that the fractional parts {(p/q)n }, n = 0, 1, 2,..., where = 0, cannot all lie in an interval I of length strictly smaller than 1/p. Can one prove the same result for intervals I of length 1/p? This small step seems to be very dicult. Bugeaud [1] made a step towards solution of this problem and proved that those fractional parts cannot lie in an interval [a, a + 1/p] of length 1/p for almost all a [0, 1 1/p].

In [2] we prove a result which settles this problem for p/q = 3/2. More precisely, we show that if p, q are relatively prime integers satisfying 1 q p q 2 and I is a closed subinterval of the torus R/Z of length 1/p then for each = 0 there are innitely many n N for which {(p/q)n } I. The proof uses / combinatorics on words and raises a new optimization problem for Sturmian words [3].

[1] Bugeaud Y. Linear mod one transformations and the distribution of fractional parts {(p/q)n } // Acta Arith. 2004. V. 114. P. 301-311.

[2] Dubickas A. Powers of a rational number modulo 1 cannot lie in a small interval // Acta Arith. 2009. V. 137. P. 233-239.

[3] Dubickas A. Squares and cubes in Sturmian sequences // RAIRO Theoretical Informatics and Applications. (to appear).

[4] Flatto L., Lagarias J.C., Pollington A.D. On the range of fractional parts {(p/q)n } // Acta Arith. 1995. V. 70. P. 125-147.

[5] Mahler K. An unsolved problem on the powers of 3/2 // J. Austral. Math. Soc.

1968. V. 8. P. 313-321.

.. (, ),, ():

dxi (t) = ai (t)dt + bi,k (t)dwk (t) + g i (x(t), t, ) (dt, d) R() - w(t) m-, x (t) = x (t;

y) Rn, x (0;

y) = y, (t;

) - t, ;

,,.,, -. [1-2] l (x;

t):

l (y;

0) z (x (y;

t) ;

t) d (y) = l (x;

t) z (x;

t) d (x), Rn Rn x (y;

t) -. - z (x (y;

t) ;

t), z (x;

t) - :

dz(x;

t) = Q (t;

x) dt + Dk (t;

x) dwk (t) + G(x(t), t, ) (dt, d) R(), ul (x;

t) = l (x;

t)1 (x;

t) -.

l [1]..,. -. (, 20-, 2003). :

-.. -., 2003.. 35-41.

[2]..., - (1-30, 2003) -.2. :, 2004.. 66-68.

.. (, ) [1,.125-130] (.. ) (.. ),.,.

x = a (t)x + a (t)x 11 1 12 (1) x = a (t)x + a (t)x 21 1 22,,, (1) a11 (t) = a22 (t)., a12 (t) = 0 (1),,.

. Re - (1), aij (t)(i, j = 1, 2) - T, a11 (t) = a22 (t).

1. c1 c2 -, T T a11 (t)dt |c1 a12 (t) + c2 a21 (t)| dt Re T 0 T T 1 a12 c(t)dt + |c1 a12 (t) + c2 a21 (t)| dt.

T 2 c22 T 0 2. p(t) = a21 (t)a1 (t) ( - ), T T 1 1 P (t) a11 (t)dt dt Re P (t) T 4T 0 T T 1 1 P (t) a11 (t)dt + dt.

P (t) T 4T 0, [.118-121].

,, [1,.130-132],.

[1]..,.. //.:, 1972. 718.

..,.. (, ) G  Rn, Fij, i = 0, 1 G, F0j F1j =, j = 1, 2,..., m.

F = {(F01, F11 ),... (F0m, F1m ), G}. (u1,..., um ) F, uj (x) G, 0 x G F0 F (p, w)-, j = 1, 2,..., m.

A = A(x)  n n- aij (x), :

c2 w2/p ||2 aij (x)i j c2 w2/p || i,j Rn c0 1 , w . p 1. F p/ n ui ui max CA,p (F ) = inf akl w dx, xk xl 1im k,l= G F u = (u1,... um ).

, F0j F1j G, j = 1, 2,..., m. MA,p (d) Mp (| Bd|), [1].

. CA,p (F ) = MA,p (d) = Mp (| Bd|).

( 08-01 00028), ( 28.10.2008.1) ( 09--01-009).

[1] Aikawa H., Ohtsuka M. Extremal length of vector measures // Ann. Acad. Sci.

Fenn. Ser. A. vol. 24, 1999. P. 6188.

.. (, ),..,..,....., [2].

, -- [1].

2m. m [3].

[4].

.

[1]...:, 1964.

[2].. //.. 1980,. 112. 3..

354-379.

[3].. //. 2008.. 15..1.

.45-51.

[4]..,... :

-, 1995.

-.. (, ) r, H2 O ( ), , O.

T (O ) f C (\{O}),. A()  - (), h (0, 1) dk |k |2 h2k, k  l l = h k=0 l=. M s (O ) T (O ). f T (\O) - O:

f = lim (r,, )f (r, )d, O r0 2 th(r/2) cos( ) 2 th2 (r/2) 1 (r,, ) = +.

1 2 th(r/2) cos( ) + th2 (r/2) ln(1/ th(r/2)) - M s (O ) = lim f j f, O O j {f j } T (O ), f M s (O ) j. , [1].

1. s 0 f M s (O )  f A(). f f O O M s (O ) A().

2. A() f, H2 \ {O}, M s (O ) s 0, -.

( -2810.2008.1).

[1].. //.. 1991.. 182. 6.. 849-876.

-.. (, ),.. (, ),. [1]-[2] -.

.

[1] Lifshits M.A., Linde W. Approximation and entropy numbers of Volterra operators with application to Brownian motion. // Mem. Am. Math. Soc. 2002. V. 745, P.

1-87.

[2]..,..  //.. 9. 1. 2006.. 52-100.

.. (, ),.. (, ) - ( ),,..... ( ),. H.Kurke,,., [1], [2], [3],,, :.

( 08-01-00095 ),   ( -4578.2006.1, -1987.2008.1), 2.1.1.7988, " "( -864.2008.1).

[1] Zheglov A.B. Two dimensional KP systems and their solvability, // preprint of Humboldt University, e-print arXiv:math-ph/0503067.

[2] Kurke H., Osipov D., Zheglov A. Formal punctured ribbons and two-dimensional local elds,// Journal fur die reine und angewandte Mathematik, 629 (2009), 133 170.

[3] Kurke H., Osipov D., Zheglov A. Formal groups arising from formal punctured ribbons,// preprint of ESI 2094, Wien, (2008), 42 p.;

preprint in arXiv:

http://arxiv.org/abs/0901.1607.

.. (, ) k1 m(1) +... + ks m(s) : ki Z, m(i) (i = 1, s)  = Zs, s- ()., m(i),. Lt (Rs ),, s s |i | |i | i = 1, s, |i | |i |.

i=1 i=... ( )..

M() , L(Zs ;

N )  s- N, #X  X, N #M() n=1 L(Zs ;

n) Ws (N ) = N #L(Zs ;

n) n=  s- [1, N ]., Ws (N ) C(s) lns1 N N +, (1) C(s) , s. s = 2 (1) ( ).

W3 (N )., W3 (N ) = C ln2 N + O(ln N ).

C N 4-.

...

( 09 I-4-03, 09-III--01-020) ( 07-01-00306).

.. (, ) Sn [1897] [1897].

Rn = R1 = (R · R1 )n1 = [Rr+1 · (R · R1 )r1 ]r = n = (Rj · R1 · Rj · R1 )2 = E, 2 r n, 2 j R = (1, 2, 3,..., n) R1 = (1, 2).

, :

R1 = E (1) Ri · Ri+1 · Ri = Ri+1 · Ri · Ri+1, 1 i n 2 (2) Ri · Rj = Rj · Ri, ij2 (3), (2) (3).

Sn, Ri = (i, i + 1), E .

Sn,, ,,.,,, Sn.

1.

xi+1 = e, i = 1, n i xk · xi = x1 · xi+1 · xk, ki Sn.

xi = R1 · R2 ·... · Ri.

2.

x2 = e, xk · xi = x1 · xi+1 · xk, ki Sn.

, yi = xi, i.

.

3. g Sn g = x1 · x2 2 ·... · xn1, 0 i i n 3.

1. Sn, n i · i 0 mod 2.

i= 2. n (1 + xi +... + xi+1 ).

g= i i= gSn 3. n i+ Sgn (g) · g = (1 + (xi ) +... + (xi ) ).

i= gSn 4.

xk · xi = x1 · xi+1 · xk, 1 i k n.

[1]... ( ) // -....:

.. :. 2000.. 122124.

[2]...,... :.. /...

//.:. 1980  240.

.. (, )..

. cm z m +... + c r(z) =, cm = 0, n (z ak ) k= : |ak | 1, k = 1,..., n, max{|r(z)| : |z| = 1} = 1.

n n |ak | |ak | k=1 k= R= + |cm | |cm | z |z| = |r(z)| t0 1+mn |B(z)|, (1) t0, 0 t0 1,  n 2 |cm |(1 + t) = ( + R) t |ak |, k= n 1 ak z (mn)+ B(z) = z.

z ak k= 1 z |z| = r(z) = B(z).

(1) (6) [1].

[2] [3,. 59].

( 08-01-00028), ( 09-I-4-02) ( - - 2810.2008.1).

[1] Govil N.K., Mohapatra R.N. Inequalities for Maximum Modulus of Ratioanl Functions with Prescribed Poles, Approximation Theory: In Memory of A.K.

Varma, Marcel Dekker, Inc., New York, 1998, P. 255-263.

[2] Goodman A.W. Univalent functions. I. Tampa, FL: Mariner Publ., 1983.

[3].. //.. 2005.. 196.

. 11.. 53-74.

.. (, ) Dk, ak Dk C, k 0, m [1, 2]. [1] " " ak,,.,.

1. a1 = am = 1, ak (1, 1), k = 2, m 3);

Dk, ak Dk C, k = 1, m.

(m m1 m r(Dk, ak ) 1 1/4 1/ r (D1, 1)r (Dm, 1).

2 m 1 a2 k k=, ak = cos 2(k1), k = 2, m 1, 2m Dk {z = ( + 1/)/2| 2m2 [1, 0]}.

, a1 = 1 (1, 1).

D gD (z, ), log r(D, a) = hD (a, a), hD (z, ) = gD (z, )+log |z |.,, H(D, z, ) = hD (z, z) + hD (, ) 2hD (z, ), K(D, z0, ) := 4 lim H(D, z0 ei, z0 + ei )/2.

K(D, z0, ) 98 [1],.

2. ak (1, 1), k = 1, m(m 1);

Dk, m ak Dk C, k = 1, m, C \ Dk k=, -1 1.

m (1 a2 )K(Dk, ak, ) m(2m 1).

k 2 k=, ak = cos (2k1), k = 1, m, 2m Dk {z = ( + 1/)/2| 2m [0, 1]}.

[1]..,,... 337 (2006), 73  100.

[2]..,..,.., I- .  2008. .73.  308.

INTERACTION OF DROPS IN NON-ISOTHERMAL VISCOUS FLOW BOUNDARY INTEGRAL SIMULATIONS O. M. Lavrenteva (Technion,Haifa, Israel), A. Nir (Technion, Haifa, Israel) The main advantage of BIE (Boundary Integral Equations) simulation of drops and bubbles in viscous ow is the reduction of the dimension of linear problems as their implementation involves values of the variables only on the interfaces. Most of the numerous BIE simulation that are available in litera ture are devoted to multiphase problems with tangential stresses continuous across the interfaces that is typical for pure interfaces in isothermal uids. In contrast to this, in processes accompanied with heat or mass-transfer, surface tension that depends on temperature and concentration of surface-active sub stance is not constant. From the mathematical point of view, the boundary integral equation modeling of such ows contains an additional term with a tangential stress jump, which provides additional diculties in the course of nu merical solutions.The goal of the present work is to extend our previous studies of spontaneous thermocapillary eect physically relevant cases of 3D motion in the presence of external ow, making use of boundary integral equations method.

We report a 3D boundary-integral code for the accurate calculations of the evolution of highly deformable drops in the presence of tangential stress jump simulations of drops interaction and deformation in the presence of Marango ni eect. A triangular mesh is employed. At each time step, singular integral equations for the temperature and velocity at the interface are solved by simple iterations. The accuracy computations of singular integrals is improved making use of singularity and near-singularity subtraction. The nodes are rst advanced according to computed velocities and then redistributed over the interface. The number of nodes is kept constant.

The results of the simulation of two drops motion and interaction under the combined action of buoyancy and thermocapillarity are presented. The case of an initially deformed single drop in a gravity eld, the case of an initially spherical drop in linear ow and that of pair wise drops interaction in shear ow and under external forcing. Simulations of the motion without Marangoni eect were also perform in order to test our code by comparing with available results and to illuminate the eect thermocapillarity. Our simulations show that even weak Marangoni eect may drastically change the deformation pattern in critical and near critical situations.

THE RIEMANN ZETA-FUNCTION AND PROBABILITY A. Laurincikas (Vilnius University, Siauliai University, Institute of Mathematics and Informatics) Let, as usual, (s), s = +it, denote the Riemann zeta-function. In the theo ry of the function (s), probabilistic methods occupy an important place [1]. The rst results in this direction were obtained by H. Bohr and B. Jessen probably 80 years ago. In the report, we will give a survey on modern limit theorems in the sense of weak convergence of probability measures in various spaces for the Riemann zeta-function, and their application to the universality. Also, we will discuss the relation between the Lindelof hypothesis and limit theorems. Final ly, we will present some characterization of the asymptotic dependence between |(s)| and (s).

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.. (, ) u(x, t), s(t) Qs = {(x, t) : 0 x s(t), 0 t T } ut (x, t) = (ux (x, t)) + a(x, t)ux (x, t) + b(x, t)u(x, t), (x, t) Qs, x u(x, 0) = u0 (x), 0 x 1, (ux (x, t)) = 0, x = 0, 0 t T, u(x, t) = 0, x = s(t), 0 t T, (ux (x, t)) = ks (t), x = s(t), 0 t T.

:

1) k ( );

2) k ( ).

(), a(x, t), b(x, t), u0 (x) 1 u(x, t) W2 (Qs ) L (0, T ;

W2 (0, 1)), L2 (Qs );

s(t) W2 (0, T ), s (t) 0, k 0 ;

x (ux (x, t)) ks(t) W2 (0, T ), k 0, k 0.

[1].

,.

[1].. //. 1997.. 353, 3.. 313-315.

.. (, ) N () u U + U = 0 + u = 0 n mes 3/ N () = + O ( ln ).

6, 2R3 3/ N () = + O ( ln ).

. N (). o()., R3, V () 3/2 S() + O 5/6, N () = ± 6 2 S() , V () .

. n,m.

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..,.. (, ), :

J(v) = 1 | v|2 d f vd min (1) v G = w W 1 () : w 0, Rn (n = 2, 3) -.

M (v, l) = J(v) + + l2 d, r 0 - const., u (l rv) 2r u (1) W2 (), u, n [1],..

u u M (u, l) M u, M v, (v, l) W2 () L2 ().

n n 1/ (u0, l0 ) W2 () W2 () -.

(uk, lk ) :

1. (k+1)- W2 () 1 Lk (v) = M (v, lk ) + v uk L2 () uk+1 W2 () uk+1 uk+ k, W2 () uk+1 = arg min Lk (v), k 0, k ;

k= vW2 () 1. lk+1 lk+1 = (lk ruk+1 )+ = max 0, lk ruk+1.

(uk, lk ).

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[1] Novitski I.M. Unitary equivalence to integral operators and an application // Int. J. Pure Appl. Math. 2009. Vol. 50, N 2. P. 295-300.

-.. (, ) -. K m ZK T m, -,.

ZK -  1991.

(1971.). K, ZK () ;

, ( ).

,, ZK,,. (CP )m ( ), Cm ( ), ( ),, Cm. -, ,.

-, ( )..

-.. (, ) ut = (u, ux )uxx,,..

.., Hollig K., Lair A.V.,..,..,.. (.),..., x t,.

ut + uux (u2 1)uxx + uxxx = 0, 0, 0 x. = 0.,..

W2.

[1],[2].

[1].. - -. // 4(7), 2007.. 185-198.

[2].. - -.II. // 4(11), 2008..9-20.

.. (, ),.. (..., ) 1. (1, 1) Ox, Q (0, T ), 0 T +. Q (1) utt + sgn x · uxx = f (x, t), (x, t) Q, (2) u(1, t) = u(1, t) = 0, 0tT a11 u(x, 0) + a12 ut (x, 0) + b11 u(x, T ) + b12 ut (x, T ) = f1 (x), (3) a21 u(x, 0) + a22 ut (x, 0) + b21 u(x, T ) + b22 ut (x, T ) = f2 (x), aij, bij (i, j = 1, 2) , (ai1, ai2, bi1, bi2 ) (i = 1, 2)  -.

2. Q (4) ut a(x, t)uxx + c(x, t)u uxxt = f (x, t), (x, t) Q ux (0, t) = 1 (t)u(0, t) + 2 (t)u(1, t), (5) ux (1, t) = 1 (t)u(0, t) + 2 (t)u(1, t), 0tT (6) u(x, 0) = 0, x, a(x, t), c(x, t), f (x, t), 1 (t), 2 (t), 1 (t), 2 (t)  x = [0, 1], t [0, T ].

(1) (3) (4)(6).

 (2009 ) ( AX-23/11 12.), 2 ( 3443).. (, ) f (z) Sf (z) = f (z) f (z). [1], 2 f (z).,.

, f (z), K(R) := {z : 1 |z| R}, f (K(R)) |z| = 1, z Re Sf (z) SG (z), 6|f (z)|2 6|G (z)| Re Sf (z) SG (z), (|f (z)|2 1)2 (|G(z)|2 1) G(z) , K(R).,.

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a Zd a = [0;

q1, q2,..., ql ] d a  d qi = qi (a) ( ) l = l(a) = ld (a).

ql 2.

l(a) Sd (a) = qi (a).

i= (a, d).

,, Sd (a) (. [1], 3.3.3).

C, [2], [3].

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d 1 12 # a Zd : Sd (a) 2 log d log log d g(d) log d log log d.

g 2 (d) d...

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.. (, ) [1],. (1) µu + p + (u )u = f + (u ), divu = 0, u + F (p) = 0, u = 0. (2) (1),(2) [2]. (1), (2). : u, p,,, (1), (2) :

(3) u| = g, |1 = 0, |1 = 0, (4) inf p(x) = p0.

x u Wr (), p Wr (), C(), L ().

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.. (, ) - [1] :

2 0 1 2 0 1 4 0 0 2 = 3 2 2 2 4 2 2 (1) 1 0 2 0 1 4.

2 2 16 0 ,, , . (1),,.

= R0 max |0 | :

2 z0 3 2 z0 15 2 z + = ;

2 8 2 16 (2) z0 = ;

= 0 + 1 sin.

.

(1), (2), [2],., (1), (2).,.

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[5] (. [1]).

1. a , x1, x2, .

Ma (x1, x2 ) = {(b, c) : 1 b x1 a, 1 c x2 a, (a, b, c) = 1}.

Then = Ox1,x2, (a4/3+ ).

f (a, b, c) abc |Ma (x1, x2 )| (a,b,c)Ma (x1,x2 ).,.

2. a , x1, x2,, .

p(t) dt + O,x1,x2, (a1/6+ ), 1= |Ma (x1, x2 )| (a,b,c)Ma (x1,x2 ) f (a,b,c) abc 0, t [0, 3];

p(t) = 12 t t [ 3, 2];

4 t2, 12 t2 2 t 3 arccos t+3 t2t3 + 3 t2 4 log t [2, +).

, 2 t2 8 18 1 p(t) dt = 1, tp(t) dt =, p(t) = · +O (t ).

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.. (, ) u(t;

x)  dx(t) x R n. (1) = A(t;

x(t)), dt u(t;

x) = C x = x(t) (1).

, du(t;

x) = u(t;

x) u(t;

x) dx1 u(t;

x) dx2 u(t;

x) dxn (2) + + +... + = t x1 dt x2 dt xn dt A(t;

x(t)) = (a1, a2,..., an ), ai = ai t;

x(t) ( ), A(t;

x(t)) = (1, a1, a2,..., an ).

u(t;

x) u(t;

x) u(t;

x) u(t;

x).

u(t;

x) = ;

;

;

...;

t x1 x2 xn (2) A(t;

x(t)) u(t;

x):

A(t;

x(t)), u(t;

x) = 1. (1) dx(t) x Rn = A(t;

x(t)), dt k k n, uj (t;

x) j=, :

A(t;

x) = 1, A (t;

x) eo e1... en u1 (t;

x) u1 (t;

x) u1 (t;

x)...

t x1 xn ··· ··· ··· ··· uk (t;

x) uk (t;

x) uk (t;

x) · det,...

f t x) x1 xn det C f1 (t;

x) 1 (t;

f1 (t;

x)...

t x1 xn ··· ··· ··· ··· fnk (t;

x) fnk (t;

x) fnk (t;

x)...

t x1 xn u1 (t;

x) u1 (t;

x)...

x1 xn ··· ··· ··· uk (t;

x) uk (t;

x)...

x1 xn C=, det C = 0.

f1 (t;

x) f1 (t;

x)...

x1 xn ··· ··· ··· fnk (t;

x) fnk (t;

x)...

x1 xn fj, j = 1, (n k),, k uj (t;

x) j=.

.. (, ), [1], [2].,,.,.,..,,, [3],.

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.. (, ),.. (, ) n n a : R+ R+ n R+,, ( a(0) = 0),. n n Ma = {(x, y)|x R+, y R+, 0 y a(x)},, - [1]. n n MA = {(x, y) R+ R+, 0 y A(x)} n n R+ R+, -.

:

.

MA,.

.

MA -. (xt )T, T t0 + k0, x0, 0 t0, k0 N,, (xt, t · x0 ) t t0 t T k0.

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x h. [2], 1 W2, [0, t0 ], W2 (h )  [0, t0 ] W2 (h ):

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.. (, ) R3, N T, p rotH H = f, divu = 0, (1) u + (u · )u + 1 rotH E + H u = 1 j, divH = 0, rotE = 0, (2) u = 0, H · n|N = q, H n|T = q, E n|N = k. (3) : u H , p = P/0, P , 0 =const , = µ0 µ1, 1 = 1 1 m, E = 1 E, E  0 0, , 0 µ0 , 0 µ , m = 1/µ0 µ , f , j , q, q k  N T.

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09-I-4-01 09-II -03-002.

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xi = fi (x, u) T (1) k = k (x, uk )dt sup x(0) = x, k = 1,..., n, i = 1,...m.

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1 |ui ud |2 dx + (4) f fd min, f K, H 2 (S) 2 Q fd  S, K , S.

ki(e) , i(e), µi(e), i(e)  i(e),  ;

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, (),..,.

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(1) dT (n+1) d2 T (n+1) p Cm(n) W T (n), (2) = +Q dx dx (n) RT (n+1) p RT (n+1) W T (n+1) dx. (3) m = P = dx,,,. P /m = 0 m ( P /ai = 0, ai Ni (x)). T (x) =,. (, ).

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u = u(x, t) , B = B(x, t) , = const , f = f (t) ,,, m, S . [1].

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..................... Sn........ Budarina N. On primitively 2-universal quadratic forms............... -.......................................,........................................................ Dubickas A.K. Powers of a rational number modulo 1..............................................................,........................................ -.........................,.. -...,............................................................................. Lavrenteva O. M., Nir A. Interaction of drops in non-isothermal viscous ow boundary integral simulations........................ Laurincikas A. The riemann zeta-function and probability...................................................................... Matsuki N., Tanabe S......................,.................................................. -................................,................................................................... -................................................................................................................................................,..........................................,.. -......................................,.. -..............................................,........................... --.....................,................................................,....,..,.. GPS.........................................,.......,..............................,...................,....,................................,.........................................................................................................................................................................................................................................,................................,.. -..........................,......................................................................... f (x1,..., xm ) = 0...........................................................,............................................ -..............................................................................................,.......................,............,................................................ ..............,...............................................,.......................................,...................................................,...........................................................................................,..,.....................,..........................,................,..,..........................................................................................................................................................,..,...............................................................,..,..................................,..,..

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....................................................................,....................................................................... LECROY........................,..,..,..,..................................................................................................................... JAVA....................................................,..,..,..

.........................................................................................................................., GRID....................,................................. ХХХIV Дальневосточная математическая школа-семинар имени академика Е.В. Золотова «Фундаментальные проблемы математики и информационных наук»

(Хабаровск, 25–30 июня 2009 г.) тезисы докладов Отпечатано с оригинал-макета, изготовленного в Хабаровском отделении Института прикладной математики ДВО РАН Ответственный за выпуск Син А.З.

Компьютерная верстка Луковенко С.А., Романов М.А.

Подписано в печать 15.05.2009 г. Формат 60 84/ Усл.-печ. л. 11, Тираж 200 экз. Заказ № Издательство Тихоокеанского государственного университета 680035, Хабаровск, ул. Тихоокеанская, Отдел оперативной полиграфии издательства Тихоокеанского государственного университета 680035, Хабаровск, ул. Тихоокеанская,

 





 
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