BEGIN:VCALENDAR
VERSION:2.0
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BEGIN:VEVENT
SUMMARY:Making Many More Matrix Multiplication Methods
DTSTART;VALUE=DATE-TIME:20190617T114500Z
DTEND;VALUE=DATE-TIME:20190617T122500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-155@events.rudn.ru
DESCRIPTION:Speakers: Manuel Kauers ()\nStrassen's algorithm for matrix mu
ltiplication is based on the observation that the product of two $2 \\time
s 2$ matrices can be computed with only 7 multiplications. It is known t
hat there is no way to do it with fewer multiplications\, and that Strasse
n's method is essentially the only way to do it with 7 multiplications.
For $3 \\times 3$-matrices\, the situation is less clear. More than 40 yea
rs ago\, Laderman gave a method that uses 23 multiplications\, and still n
obody knows whether there is a way to do it with fewer multiplications. La
derman's method is not unique. Several authors have later presented isolat
ed additional methods that work for arbitrary coefficient rings\, and ther
e is even a family of methods with three free parameters but restricted to
coefficient rings containing $\\mathbb{Q}$. Using SAT solvers and Gr\\"ob
ner bases technology\, we have found more than 10000 pairwise inequivalent
new matrix multiplication methods with 23 multiplications. We were able t
o cluster them together into families with up to 16 parameters which apply
without any restriction on the coefficient domain. In conclusion\, the se
t of methods with 23 multiplications appears to be much larger than it see
med to be until now\, and it is quite possible that the methods we found a
re still just the tip of the iceberg. Although our results have no immedia
te implications on the complexity of matrix multiplication\, we do hope
that the vast amount of new methods will eventually contribute to a better
understanding of why these methods exist at all\, and whether 23 is best
possible for $3\\times 3$.\n\nhttps://events.rudn.ru/event/20/contribution
s/155/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Effective Schrödinger equation for one-dimensional potential box
with rapidly oscillating width
DTSTART;VALUE=DATE-TIME:20190621T083000Z
DTEND;VALUE=DATE-TIME:20190621T085500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-185@events.rudn.ru
DESCRIPTION:Speakers: Nikolay Tretyakov (Russian Presidential Academy of N
ational Economy and Public Administration)\nThe method of the effective Sc
hrödinger equation is applied to the one-dimensional motion of a particle
in a potential box with a potential function and high-frequency time-depe
ndent boundary conditions. It was assumed that the oscillation amplitudes
are small in comparison with the potential box width.The resulting system
can be considered as a particle in a static potential box\, located in the
field of some effective potential\, different from the initial one. In th
is work\, we consider a particular case where the width of the box oscilla
tes\, while its center does not move\, with the subsequent construction of
the effective Schrödinger equation. Oscillations of width lead to addit
ional terms in the effective equation proportional not only to the second
derivative of the potential\, but to the first derivative as well. This me
ans that effects from fast external driving can be observed only in cases
where the potentials have regions with large values of $V''(x)$ and $V'(x)
$. Another fundamental difference is that in additional terms the first an
d second derivatives are multiplied by $x$ and $x^2$. In the case of width
oscillations\, compared to the oscillations of the box as a whole (this c
ase was previously calculated elsewhere)\, the changes are even more signi
ficant and striking. In particular\, the potential barrier may turn into a
potential well (and vice versa).\n\nhttps://events.rudn.ru/event/20/contr
ibutions/185/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/185/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Algebraically mimetic finite difference approximations on regular
orthogonal grids oriented to solution of PDE
DTSTART;VALUE=DATE-TIME:20190617T123000Z
DTEND;VALUE=DATE-TIME:20190617T125500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-158@events.rudn.ru
DESCRIPTION:Speakers: Vladimir Gerdt ()\nIn the given talk we discuss the
algorithmic aspects of differential and difference algebra related to cons
truction\, on uniform and orthogonal grids\, of finite difference approxim
ations to polynomially - nonlinear partial differential equations and to s
ystems of such equations. As this takes place\, we impose on a finite diff
erence approximation the condition of strong consistency with the differen
tial equations. If the condition holds\, then we call the approximation al
gebraically mimetic. This condition strengthens the commonly accepted requ
irement of consistency of differential ideal associated with the input dif
ferential the approximation\, which we call weak consistency\, and means
approximability of any consequence of the input differential equation(s) b
y an element in the perfect difference ideal generated by the polynomials
occurring in the finite difference approximation. In the case of linear eq
uations one can use difference \\Gr bases for the elimination and differen
tial Groebner bases for the verification of strong consistency. If the inp
ut differential equations are nonlinear\, then instead of differential Gro
ebner bases\, which can be infinite\, one can use the differential Thomas
decomposition. Nonlinear difference Groebner bases can also be infinite\,
and there are needs in design of mathematical methods\, algorithms and sof
tware for construction of a difference analogue of the differential Thomas
decomposition.\n\nhttps://events.rudn.ru/event/20/contributions/158/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Non-local singularities on families of periodic solutions to ODEs
DTSTART;VALUE=DATE-TIME:20190617T090000Z
DTEND;VALUE=DATE-TIME:20190617T092500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-157@events.rudn.ru
DESCRIPTION:Speakers: Victor Varin (KIAM Moscow Russia)\nWe consider degen
erate solutions on families of periodic solutions to ODEs. The degeneracy
can mean any property of the solution that isolates the solution from the
generic case. This can be a bifurcation or a topological peculiarity on th
e family that causes a failure of a numerical algorithm that was used for
generic cases. We suggest a method of computation of such degeneracies wit
h variational equations of higher order with the same accuracy as ordinary
solutions.\n\nhttps://events.rudn.ru/event/20/contributions/157/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Inverse Zeilberger’s problem
DTSTART;VALUE=DATE-TIME:20190618T074500Z
DTEND;VALUE=DATE-TIME:20190618T082500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-166@events.rudn.ru
DESCRIPTION:Speakers: Marko Petkovšek (University of Ljubljana)\nGiven a
sum of the form $s_n = \\sum_{k=0}^n F(n\,k)$ where $F(n\,k)$ is proper hy
pergeometric\, Zeilberger's algorithm returns a linear recurrence with rat
ional coefficients satisfied by $s_n$. Here we consider the inverse proble
m: Given a linear recurrence operator $L$ with polynomial coefficients\, f
ind solutions that have the form of a definite sum. As a small first step
towards its solution\, we present an algorithm which\, given $L$ and a pro
duct of binomial coefficients of the form $F(n\,k) = \\prod_{i=1}^m \\bino
m{a_i n + b_i}{k}$ with $a_i \\in \n \\setminus \\{0\\}$\, returns a line
ar recurrence operator $L'$ with rational coefficients such that $L(\\sum_
{k=0}^\\infty F(n\,k) h_k) = 0$ if and only if $L' h = 0$.\n\nhttps://even
ts.rudn.ru/event/20/contributions/166/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linearizability of differential equations: theory and algorithms
DTSTART;VALUE=DATE-TIME:20190620T131500Z
DTEND;VALUE=DATE-TIME:20190620T134000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-186@events.rudn.ru
DESCRIPTION:Speakers: Dmitry Lyakhov (King Abdullah University of Science
and Technology)\nSolving nonlinear differential equations is one of the fu
ndamental and practically important research challenges in mathematics. Ho
wever\, the problem of their algorithmic linearizability so far remained u
nsolved. In this contribution\, we propose a solution of this problem for
a wide class of nonlinear ordinary differential equation of arbitrary orde
r. Criteria for linearizability of scalar ordinary differential equations
is based on algorithmic calculus of determining Lie symmetry equations. Tw
o main points are analysis of dimension of determining system and construc
tion of derived algebra for abstract Lie symmetry algebra. New recent gene
ralization for PDE will be also discussed.\n\nhttps://events.rudn.ru/event
/20/contributions/186/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/186/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The Difference Ring Approach for Symbolic Summation
DTSTART;VALUE=DATE-TIME:20190617T070000Z
DTEND;VALUE=DATE-TIME:20190617T074000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-160@events.rudn.ru
DESCRIPTION:Speakers: Carsten Schneider (Research Institute for Symbolic C
omputation (RISC))\nA major breakthrough in symbolic summation was Michael
Karr's algorithm (1981) that solves the following problem. Given a summan
d $f(k)$ in a $\\Pi\\Sigma$-difference field $\\mathbb F$\, decide constru
ctively if there exists a $g(k)$ in $\\mathbb F$ such that $f(k)=g(k+1)-g(
k)$\nholds. Given such a solution $g(k)$ and summing the telescoping equat
ion over a valid summation range yields $\\sum_{k=a}^bf(k)=g(b+1)-g(a)$.\
nFrom the point of view of applications one is usually given only the summ
and $f(k)$ in terms of indefinite nested sums and products\, and not a dif
ference field $\\mathbb F$ in which $f(k)$ can be represented properly. \n
In this talk I will elaborate how indefinite nested sums defined over hype
rgeometric products can be always represented in $R\\Pi\\Sigma$-difference
rings\, an enhanced definition of Karr's $\\Pi\\Sigma$-fields. As a bonus
we will obtain an algebraic machinery that can rewrite an expression in t
erms of indefinite nested sums and products to an equivalent expression su
ch that among the arising sums and products (excepts elements such as $\\a
lpha^k$ where $\\alpha$ is a root of unity) no algebraic relations exist.
\nExploiting in addition the available difference ring/field algorithms fo
r Zeilberger's creative telescoping and recurrence solving (utilizing as s
pecial cases Abramov's recurrence solver for rational functions and Petkov
sek's Hyper algorithm)\, I will demonstrate how non-trivial summation prob
lems in combinatorics and particle physics can be tackled with the help of
the summation package Sigma.\n\nhttps://events.rudn.ru/event/20/contribut
ions/160/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Symmetric Matrices Whose Entries Are Linear Functions
DTSTART;VALUE=DATE-TIME:20190617T140000Z
DTEND;VALUE=DATE-TIME:20190617T142500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-161@events.rudn.ru
DESCRIPTION:Speakers: Alexandr Seliverstov (Institute for Information Tran
smission Problems of the Russian Academy of Sciences (Kharkevich Institute
))\nThere exists a large set of real symmetric matrices whose entries are
linear functions in several variables such that each matrix in the set is
definite at some point\, that is\, after substitution some numbers instead
of variables. In particular\, this property holds for almost all such mat
rices of order two or three\, whose entries depend on two or three variabl
es\, respectively. The same property holds for almost all matrices whose e
ntries depend on a larger number of variables\, when the number exceeds th
e order of the matrix. For example\, for each matrix of second partial der
ivatives of a multivariate third degree polynomial\, all matrix entries ar
e linear functions. Some examples have considered in details. At last the
determinant of such matrices is considered. For almost every symmetric mat
rix whose entries are linear functions\, the determinant of the matrix is
positive at some point and it is negative at another point.\n\nhttps://eve
nts.rudn.ru/event/20/contributions/161/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hilbert series of noncommutative algebras and context-free languag
es
DTSTART;VALUE=DATE-TIME:20190619T074500Z
DTEND;VALUE=DATE-TIME:20190619T082500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-180@events.rudn.ru
DESCRIPTION:Speakers: Dmitri Piontkovskiy (National Research University Hi
gher School of Economics)\nThe Hilbert series is one of the most important
algebraic invariants of infinite-dimensional graded associative algebra.
The noncommutative Groebner basis machine reduces the problem of finding
Hilbert series to the case of monomial algebra. \n\nWe apply both noncommu
tative and commutative Groebner bases theory as well as the theory of for
mal languages to provide a new method for symbolic computation of Hilbert
series of graded associative algebras. Whereas in general the problem of c
omputation oh such a Hilbert series is known to be algorithmically unsolva
ble\, we have describe a general class of algebras (called {\\em homologic
ally unambiguous}) with unambiguous context-free set of relations for whi
ch our method give effective algorithms. Unlike previously known methods\,
our algorithm is applicable to algebras with irrational Hilbert series a
nd produces an algebraic equation which defines the series. The examples
include infinitely presented monomials algebras as well as finitely presen
ted algebras with irrational Hilbert series such that the associated mon
omial algebras are homologically unambiguous.\n\nhttps://events.rudn.ru/ev
ent/20/contributions/180/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/180/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent solutions of linear ordinary differential equations with c
oefficients in the form of truncated power series
DTSTART;VALUE=DATE-TIME:20190618T090000Z
DTEND;VALUE=DATE-TIME:20190618T092500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-162@events.rudn.ru
DESCRIPTION:Speakers: Sergei Abramov ()\, Denis Khmelnov ()\, Anna Ryabenk
o ()\nLinear ordinary differential equations with formal power series coef
ficients represented in a truncated form are considered. We discuss the in
formation on solutions belonging to the field of Laurent formal series whi
ch can be obtained from this representation of a given equation.\nWe emph
asize that we are interested in such information about solutions\nwhich is
invariant with respect to possible prolongations of the\ntruncated series
representing the coefficients of the equation.\n\nhttps://events.rudn.ru/
event/20/contributions/162/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Algorithm for Calculating Interpolation Hermite Polynomials in $d$
-dimensional hypercube in the Analytical Form
DTSTART;VALUE=DATE-TIME:20190621T070000Z
DTEND;VALUE=DATE-TIME:20190621T072500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-168@events.rudn.ru
DESCRIPTION:Speakers: Alexander Gusev (JINR)\nA new symbolic algorithm imp
lemented in Maple for constructing Hermitian finite elements in the standa
rd $d$-dimensional hypercube is presented. The basis functions of finite
elements are polynomials\, determined from a specially constructed set of
values of the polynomials themselves and their first partial derivatives a
t the corners of the hypercube. Such a choice of values allows one to cons
truct a piecewise polynomial basis continuous on the boundaries of finite
elements together with the derivatives up to the first order. In the case
of a $d$-dimensional cube\, it is shown that the basis functions are deter
mined by a product of $d$ interpolation Hermite polynomials depending on e
ach of the $d$ variables with continuous first derivatives on the boundar
ies of finite elements. It can be used to solve elliptic boundary value pr
oblems by means of the finite element method. The efficiency and accuracy
order of the finite element scheme\, algorithm and program are demonstrat
ed by the example of an exactly solvable Helmholtz problem for a three-dim
ensional cube.\n\nhttps://events.rudn.ru/event/20/contributions/168/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Interpolating Reduced Data Based on Piecewise-Cubic Splines
DTSTART;VALUE=DATE-TIME:20190619T070000Z
DTEND;VALUE=DATE-TIME:20190619T074000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-172@events.rudn.ru
DESCRIPTION:Speakers: Ryszard Kozera (Warsaw University of Life Sciences -
SGGW)\nThe problem of fitting a sequence of reduced data with piecewise-cu
bics $\\hat\\gamma$ is discussed. The interpolation points are in arbitrar
y Euclidean space with the respective interpolation knots unavailable. The
unknown knots are estimated by the so-called exponential parameterization
which depends on a single parameter $\\lambda\\in[0\,1]$. We review the e
xisting results determining the asymptotics in approximating $\\gamma$ wi
th the interpolant $\\hat\\gamma$. \nWith the aid of computer algebra and
analytic arguments this paper addresses the following issues:\na) the shar
pness of the convergence rates in estimation $\\gamma-\\hat\\gamma\\circ\\
phi$ is demonstrated\,\nb) the necessity of more-or-less uniformity of $Q_
m$ and of the regularity of $\\gamma$ in proving the latter is justified t
o be essential\, \nc) the sufficient conditions for $\\phi$ to form a genu
ine parameterization are formulated in terms of algebraic inequalities whi
ch can be visualized in Mathematica.\n\nhttps://events.rudn.ru/event/20/co
ntributions/172/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Experience in using the Fast Automatic Differentiation to solve in
verse coefficient problems
DTSTART;VALUE=DATE-TIME:20190618T070000Z
DTEND;VALUE=DATE-TIME:20190618T074000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-174@events.rudn.ru
DESCRIPTION:Speakers: Yury Evtushenko (Dorodnicyn Computing Centre\, Feder
al Research Center "Computer Science and Control" of Russian Academy of Sc
iences)\nThe Fast Automatic Differentiation methodology (FAD-methodology)
is a modern approach to help you effectively solve optimal control problem
s of complex dynamic systems. Using this methodology\, the authors had bee
n resolved several interesting theoretically and practically important pro
blems. This paper demonstrates the use of FAD-methodology to solve inverse
coefficient problems. The importance of these problems due to the need to
create new materials\, some characteristics of which are not known in adv
ance\, and you can identify them by solving inverse problems.\nThe inverse
problem considered is devoted to the determination of temperature-depende
nt thermal conductivity of the substance according to the results of a pil
ot monitoring of temperature field in the investigated substance and heat
flux on the surface of an object. The consideration is based on the initia
l boundary value problem for the non-stationary heat equation. The inverse
coefficient problem is reduced to the variational problem: it is required
to find such dependence of the thermal conductivity coefficient on the te
mperature\, at which the temperature field and heat fluxes at the boundary
of the object\, obtained as a result of solving the direct problem\, diff
er little from the data obtained experimentally. The algorithm for the num
erical solution of the posed inverse problem based on the modern FAD-metho
dology was proposed. The examples of solving the inverse coefficient probl
em confirm the efficiency of the proposed algorithm.\n\nhttps://events.rud
n.ru/event/20/contributions/174/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On symbolic computation of the solution to the inverse problem of
the calculus of variations
DTSTART;VALUE=DATE-TIME:20190619T090000Z
DTEND;VALUE=DATE-TIME:20190619T092500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-175@events.rudn.ru
DESCRIPTION:Speakers: Sergey Shorokhov (RUDN University)\nWe study the pro
blem of finding a Lagrangian function for the given system of $n$ second o
rder ordinary differential equations (the inverse problem of the calculus
of variations).\nAs it was discovered by J. Douglas (1941)\, the inverse p
roblem of the calculus of variations can be reduced to the problem of find
ing nontrivial solution for the overdetermined system of first order parti
al differential equations with $n(n+1)/2$ unknown functions\, but solving
this PDE system requires huge analytical calculations and classification o
f solutions was obtained by J. Douglas only for the case $n=2$.\nWe discus
s algorithmic aspects of solving the inverse problem of the calculus of va
riations for the case $n=3$ and their implementation in computer algebra s
ystems Maple and SymPy. We present symbolic calculation of Lagrangian for
the equations of motion of a symmetric rigid body with a fixed point as a
demonstration of the algorithm.\n\nhttps://events.rudn.ru/event/20/contrib
utions/175/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/175/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Close-Packed Dimers on a Cylinder Boundary. Derivation of Generati
ng Functions in the Maple System
DTSTART;VALUE=DATE-TIME:20190619T131500Z
DTEND;VALUE=DATE-TIME:20190619T134000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-164@events.rudn.ru
DESCRIPTION:Speakers: Sergey Perepechko (Petrozavodsk State University)\nT
he problem of counting the number of perfect matchings on the cylinders \n
$C_m \\times P_n$ is considered for the case when the matching contains a
given\nedge on the boundary of the cylinder. For fixed values of \n$3 \\le
q m \\leq 14$ a set of recurrence relations and associated\ngenerating fun
ctions is constructed. Application of the obtained results to the\ndimer p
roblem allowed us to deduce closed form expressions for the probabilities\
nof occupation a given edge of the lattice with a dimer when $n \\to \\inf
ty$. A set\nof corrections to the limit values of probabilities due to the
finite order of\nthe graph is calculated.\n\nhttps://events.rudn.ru/event
/20/contributions/164/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Normal form of the periodically perturbed Hamiltonian system
DTSTART;VALUE=DATE-TIME:20190617T110000Z
DTEND;VALUE=DATE-TIME:20190617T114000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-165@events.rudn.ru
DESCRIPTION:Speakers: Alexander Bruno (Keldysh Institute of Applied Mathem
atics)\nNear a stationary solution we consider the Hamiltonian system with
such perturbation\, that the unperturbed Hamiltonian function is autonomo
us and the perturbation of the Hamiltonian function is periodic in time. F
irst we remind the normal form of the autonomous Hamiltonian function. Sec
ond we describe the normal form of the periodic perturbation of the Hamilt
onian function. It can always be reduced to the time independent Hamiltoni
an. It allows to compute the local families of periodic solutions of the i
nitial system. We also discuss problems of the computer algebra arising in
these computations.\n\nhttps://events.rudn.ru/event/20/contributions/165/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geometric integrators in mechanics - the need for computer algebra
tools
DTSTART;VALUE=DATE-TIME:20190617T074500Z
DTEND;VALUE=DATE-TIME:20190617T082500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-167@events.rudn.ru
DESCRIPTION:Speakers: Vladimir Salnikov (CNRS / La Rochelle Université)\n
In this contribution we will describe some objects of the generalized geom
etry that appear naturally in the qualitative analysis of mechanical syste
ms. In particular we will discuss the Dirac structures within the framewor
k of the systems with constraints as well as of the port-Hamiltonian syste
ms. \n\nFrom the mathematical point of view\, Dirac structures generalize
simultaneously symplectic and Poisson structures. As for mechanics\, the i
dea is to design numerical methods that preserve these structures and thus
guarantee good physical behaviour in the simulation.\n\nThen\, we will pr
esent a framework which is even more general -- the one of differential gr
aded manifolds (also called Q-manifolds)\, and discuss some possible ways
of using them for the ``structure preserving integrators'' in mechanics.\n
\nFor all of the mentioned constructions we will explain the problems tha
t arise in generic situations -- most of them are open\, but we think they
are suitable for handling with various computer algebra approaches.\n\nht
tps://events.rudn.ru/event/20/contributions/167/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An additive Ore-Sato theorem on compatible rational functions
DTSTART;VALUE=DATE-TIME:20190619T110000Z
DTEND;VALUE=DATE-TIME:20190619T114000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-159@events.rudn.ru
DESCRIPTION:Speakers: Shaoshi Chen (KLMM\, Academy of Mathematics and Sys
tems Science\, Chinese Academy of Sciences)\nThe Ore--Sato theorem describ
es a multiplicative structure of multivariate hypergeometric terms. This t
heorem was first proved by Ore in 1930 in the bivariate case and then exte
nded to the multivariate case by Sato in the 1960s via homological method.
In 2002\, Abramov and Petkovsek gave an elementary proof of the Ore--Sato
theorem and proved a slightly corrected version of a conjecture of Wilf a
nd Zeilberger on multivariate hypergeometric terms. Other different proofs
of this theorem were also presented by Payne in 1997 and Hou in 2001 in t
heir PhD dissertations. The continuous analogue of the Ore--Sato theorem w
as first obtained by Christopher in 1997 for bivariate compatible rational
functions and later extended by Zoładek to the multivariate case in 1998
. More recently\, this result has been extended further to the multivariat
e continuous-discrete case by Chen et al. in 2011. The mixed extension of
the Ore--Sato theorem has been used to prove the Wilf--Zeilberger conjectu
re on mixed hypergeometric terms by Chen and Koutschan in 2018. We will pr
esent an additive version of the Ore--Sato theorem which explores all poss
ible rational Wilf--Zeilberger pairs.\n\nhttps://events.rudn.ru/event/20/c
ontributions/159/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Symbolic-numeric implementation of the four potential method for c
alculating normal modes: an example of square electromagnetic waveguide wi
th rectangular insert
DTSTART;VALUE=DATE-TIME:20190621T090000Z
DTEND;VALUE=DATE-TIME:20190621T092500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-189@events.rudn.ru
DESCRIPTION:Speakers: Dmitriy Divakov (RUDN University)\nIn this paper\, t
he Maple computer algebra system is used to construct a symbolic-numeric i
mplementation of the method for calculating normal modes of square closed
waveguides in a vector formulation. The method earlier proposed by Malykh
et al. [M.D. Malykh\, L.A. Sevastianov\, A.A. Tiutiunnik\, N.E. Nikolaev.
On the representation of electromagnetic ﬁelds in closed waveguides usin
g four scalar potentials // Journal of Electromagnetic Waves and Applicati
ons\, 32 (7)\, 886-898 (2018)] will be referred to as the method of four p
otentials. The Maple system is used at all stages of treating the system o
f diﬀerential equations for four potentials: the generation of the Galer
kin basis\, the substitution of approximate solution into the system under
study\, the formulation of a computational problem\, and its approximate
solution. Thanks to the symbolic-numeric implementation of the method\, it
is possible to carry out calculations for a large number of basis functio
ns of the Galerkin decomposition with reasonable computation time and then
to investigate the convergence of the method and verify it\, which is don
e in the present paper\, too.\n\nhttps://events.rudn.ru/event/20/contribut
ions/189/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/189/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On a Provable Program for Generic Arithmetic of Fractions
DTSTART;VALUE=DATE-TIME:20190617T143000Z
DTEND;VALUE=DATE-TIME:20190617T145500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-169@events.rudn.ru
DESCRIPTION:Speakers: Sergei Meshveliani (Program systems institute of Rus
sian Academy of sciences\, Pereslavl-Zalessky)\nThere are described the de
sign principles for certified programs for arithmetic of fractions over an
y domain with the greatest common divisor function. This is a small part o
f the library DoCon-A of certified programs for a computer algebra library
designed by the author. In this system\, programs include definitions for
the corresponding mathematical notions and proofs for the main properties
of the implemented methods. These proofs are checked by the compiler. It
is used a purely functional programming language Agda\, which also support
s the feature of dependent types. It is described the technique for provid
ing formal machine-checked proofs for a certain optimized method to sum fr
actions.\n\nhttps://events.rudn.ru/event/20/contributions/169/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Computing Primitive Idempotents in Centralizer Ring of Permutatio
n Representation of Wreath Product
DTSTART;VALUE=DATE-TIME:20190619T114500Z
DTEND;VALUE=DATE-TIME:20190619T121000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-156@events.rudn.ru
DESCRIPTION:Speakers: Vladimir Kornyak (Laboratory of Information Technolo
gies\, Joint Institute For Nuclear Research)\nAn approach to compute the c
omplete set of primitive orthogonal idempotents in the centralizer ring of
the permutation representation of a wreath product is described. This set
determines the decomposition of the representation into irreducible compo
nents. In quantum mechanics\, these idempotents manifest themselves as pro
jection operators into irreducible invariant subspaces of the Hilbert spac
e of a multipartite quantum system. The C implementation of the approach i
s able to construct irreducible decompositions of high-dimensional and hig
h-rank representations of wreath products.\n\nhttps://events.rudn.ru/event
/20/contributions/156/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Canonical representation of monomials with indices and contraction
s
DTSTART;VALUE=DATE-TIME:20190619T134500Z
DTEND;VALUE=DATE-TIME:20190619T141000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-181@events.rudn.ru
DESCRIPTION:Speakers: Alexander Kryukov (M.V.Lomonosov Moscow State Univer
sity)\nThe article deals with the problem of reducing monomials with indic
es to canonical form. This problem is met in many fields of mathematics an
d physics and is one of the key algorithmic problems of computer algebra.
The most commonly used approach to the problem is based on a double coset.
However\, this approach has some limitations. For example\, it cannot be
used for expressions in which have linear relationships with more then two
terms. The article provides a definition of the canonical form for monomi
als from indexed factors\, taking into account summation indices and linea
r relations for them. Linear relations can include two and more terms with
different order of indices. Also a special colored graph associated with
monomials is introdiced and formulate the problem of calculating the canon
ical form of a monomial in terms of the canonical numbering of such graphs
. The final part of the paper presents an algorithm for reducing monomials
to canonical form\, based on the calculation of the set of canonical numb
ering of edges of a color graph conjugated to the monomial. The algorithm
does not use the construction of adjacency classes for the group renaming
summation indices. The computational complexity of the proposed algorithm
does not depend on the number of summation indices\, but on the number of
automorphisms of the color graph and is small for expressions that occur
in practice.\n\nhttps://events.rudn.ru/event/20/contributions/181/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/181/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Application of Computer Algebra Methods for Investigation of the
Stationary Motions of the System of Two Connected Bodies Moving along a Ci
rcular Orbit
DTSTART;VALUE=DATE-TIME:20190619T121500Z
DTEND;VALUE=DATE-TIME:20190619T124000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-171@events.rudn.ru
DESCRIPTION:Speakers: Sergey Gutnik (Moscow State Institute of Internation
al Relations (MGIMO University))\nComputer algebra and numeric methods are
used to investigate properties of a nonlinear algebraic system that deter
mines the equilibrium orientations for a system of two bodies connected by
a spherical hinge that moves along a circular orbit under the action of g
ravitational torque. \nThe main attention is paid to study the conditions
of existence of the equilibrium orientations for the system of two bodies
for special cases\, when one of the principal axes of inertia both the fir
st and second body coincides with the normal to the orbital plane\, with r
adius vector or tangent to the orbit.\nTo determine the equilibrium orient
ations for the system of two bodies\, the system of 12 stationary algebrai
c equations is decomposed into 9 subsystems. The computer algebra method b
ased on the algorithm for the construction of a Gröbner basis applied to
solve the stationary motion system of algebraic equations. Depending on th
e parameters of the problem\, the number of equilibria is found by numeric
al analysis of the real roots of the algebraic equations from the Gröbner
basis constructed.\n\nhttps://events.rudn.ru/event/20/contributions/171/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:MAPLE program for calculating transition probabilities of hydrogen
-like atoms in quantum mechanics with non-negative distribution function
DTSTART;VALUE=DATE-TIME:20190621T093000Z
DTEND;VALUE=DATE-TIME:20190621T095500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-170@events.rudn.ru
DESCRIPTION:Speakers: Nikolay Tretyakov (Russian Presidential Academy of N
ational Economy and Public Administration)\nThe program is proposed for a
realization of the symbolic algorithm based on the quantum mechanics with
non-negative probability distribution function (QDF) and for calculations
of transition probabilities for hydrogen-like atoms. Transition probabilit
ies are calculated and compared with experimental data. Calculations of ra
dial functions were used in calculations of oscillators strengths and tran
sitional probabilities. These transition probabilities are calculated by t
he Galerkin method with the Sturm functions of the hydrogen atom as coordi
nate functions. Matrix elements of physical operators are required when th
e accurate theoretical determination of atomic energy levels\, orbitals an
d radiative transition data need to be obtained for open-shell atoms and i
ons. It turns out that QDF seems to be equivalent to the traditional quant
um mechanics in regard to predictions of experimental values. However\, th
e existence of a phase-space probabilistic quantum theory may be an import
ant advance towards the explanation and interpretation of quantum mechanic
s. Computer algebra methods seem to be absolutely necessary and indispensa
ble for such calculations.\n\nhttps://events.rudn.ru/event/20/contribution
s/170/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:CAS pseudo-random numbers generators statistical test suits
DTSTART;VALUE=DATE-TIME:20190621T073000Z
DTEND;VALUE=DATE-TIME:20190621T075500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-191@events.rudn.ru
DESCRIPTION:Speakers: Мигран Нельсонович Геворкян
()\nFor a long time the implementation of pseudo-random number sequence g
enerators in standard programming language libraries and mathematical pack
ages was of poor quality. The situation has started to improve relatively
recently. Even nowadays a large number of libraries and poorly supported m
athematical packages utilize the old algorithms of pseudo-random numbers g
eneration. We describe four actual sets of statistical tests that can be u
sed to test the generator that is used in a particular software system. Th
e emphasis is on the use of command-line utilities\, to avoid low-level C
or C++ programming.\n\nhttps://events.rudn.ru/event/20/contributions/191/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/191/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Symbolic Computation in Studying the Dynamics of Generalized Atwoo
d's Machine
DTSTART;VALUE=DATE-TIME:20190620T121500Z
DTEND;VALUE=DATE-TIME:20190620T124000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-176@events.rudn.ru
DESCRIPTION:Speakers: Alexander Prokopenya (Warsaw University of Life Scie
nces)\nA generalized model of the Atwood machine when a pulley of finite r
adius is replaced by two small separate pulleys and one body can swing in
a vertical plane is considered. Doing necessary symbolic calculations\, we
have obtained equations of motion of the system and analyzed their soluti
ons in case of small oscillations. We have shown that in case of small dif
ference of masses of the bodies there exists a state of dynamical equilibr
ium of the system when the oscillating body acts like a pendulum whose len
gth undergoes small oscillation. The corresponding periodic solutions of t
he equations of motion are obtained in the form of power series in terms o
f a small parameter. Comparison of the obtained results with the correspon
ding numerical solution of the equations of motion demonstrates its validi
ty. All the relevant calculations are performed with the computer algebra
system Wolfram Mathematica.\n\nhttps://events.rudn.ru/event/20/contributio
ns/176/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/176/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An operator matrix having a given space of solutions
DTSTART;VALUE=DATE-TIME:20190619T124500Z
DTEND;VALUE=DATE-TIME:20190619T131000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-177@events.rudn.ru
DESCRIPTION:Speakers: Sergei Abramov ()\, Moulay Barkatou (XLIM UMR 7252 C
NRS)\nWe consider differential full rank operator matrices over a differen
tial field $K$ of characteristic zero. The constant field of $K$ is assume
d to be algebraically closed. Together with each operator $A$ we consider
its solution space $V_A$\; the components of solutions are supposed to bel
ong to the universal Picard-Vessiot extension $\\Lambda$ of $K$. We prove
that for any given finite set $F \\subset \\Lambda^m$ there exists a matr
ix whose solution space is generated by $F$ (the entries of that matrix
are\, in general\, in $\\Lambda$\, not necessarily in $K$).\n\nhttps://eve
nts.rudn.ru/event/20/contributions/177/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/177/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bifurcations of periodic solutions to Hamiltonian system with disc
rete symmetries
DTSTART;VALUE=DATE-TIME:20190620T090000Z
DTEND;VALUE=DATE-TIME:20190620T092500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-179@events.rudn.ru
DESCRIPTION:Speakers: Alexander Batkhin (Keldysh Institute of Applied Math
ematics of RAS)\nWe consider an autonomous Hamiltonian system with two deg
rees of freedom\, which system of canonical equations is t-invariant under
Klein four-group of linear canonical automorphisms of the extended phase
space of the system. Three types of bifurcations of a family of doubly sym
metric periodic solutions: saddle-node bifurcation\, pitch-fork bifurcatio
n and period multiplying bifurcation\, are investigated by means of simpli
fied monodromy matrix of periodic solution. For last two types of bifurcat
ions different scenarios are studied. All mentioned above kinds of bifurca
tions were found and investigated for the case of doubly symmetric periodi
c solutions of the Hill problem. Most computations were provided with CAS
Maple.\n\nhttps://events.rudn.ru/event/20/contributions/179/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/179/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linearizing differential equations: Riccati solutions as Dn-finite
functions
DTSTART;VALUE=DATE-TIME:20190617T133000Z
DTEND;VALUE=DATE-TIME:20190617T135500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-163@events.rudn.ru
DESCRIPTION:Speakers: Antonio Jiménez-Pastor (Doctoral Program Computatio
nal Mathematics (JKU))\nD-finite (or holonomic) functions are a class of f
ormal power series that satisfy \nlinear differential equation with polyno
mial coefficients. The finite representation of these functions (using the
differential equation and some initial conditions) boosted the developmen
t of algorithms working symbolically over them. This has been recently ext
ended to the DD-finite class (functions satisfying linear differential equ
ations with D-finite coefficients) and implemented some closure properties
. It was also proved that DD-finite functions (and also their generalizati
on to the D$^n$-finite functions) are differentially algebraic. In this do
cument we show how solutions to non-linear differential equations (startin
g with the Riccati differential equation) are always D$^n$-finite\nfunctio
ns for some $n$ and proposed some ideas to set the difference between D$^n
$-finite functions and differentially algebraic functions.\n\nhttps://even
ts.rudn.ru/event/20/contributions/163/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On Evaluation of Bessel Functions of the First Kind via Prony-like
Methods
DTSTART;VALUE=DATE-TIME:20190620T074500Z
DTEND;VALUE=DATE-TIME:20190620T082500Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-173@events.rudn.ru
DESCRIPTION:Speakers: Min Wu (East China Normal University)\nBessel functi
ons are widely applied in mathematics\, physics and engineering. In this w
ork\, we apply two variants of Prony's method on Bessel functions of first
kind of integer order and obtain approximations as sums of sinusoidal fun
ctions. Experiments show that the Prony-like methods yield approximations
of high accuracy.\n\nhttps://events.rudn.ru/event/20/contributions/173/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An algorithm for solving a family of diophantine equations of degr
ee four which satisfy Runge's condition
DTSTART;VALUE=DATE-TIME:20190620T114500Z
DTEND;VALUE=DATE-TIME:20190620T121000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-178@events.rudn.ru
DESCRIPTION:Speakers: Alexey Kytmanov (Siberian Federal University)\nIn mo
dern computer algebra systems (such as Mathematica\, Maple\, SageMath\, et
c.) there is a small number of implemented algorithms for solving diophant
ine equations in integers. In this paper we suggest an algorithmic impleme
ntation of elementary version of Runge's method for a family of diophantin
e equations of degree four. Although the fact that Runge's method has been
known for more than 100 years\, its implementation in computer algebra sy
stems (CAS) is very limited (due to very large theoretical upper bounds fo
r solutions which is useless for practical computer implementation via bru
te force method).\n\nThe elementary version of Runge's method for diophant
ine equations of small degree is based on a convenient parametrization whi
ch provides enumerating possible integer solutions. In some cases this lea
ds to the practically working solving algorithms for the diophantine equat
ions of degree three and four.\n\nThe software implementation of the propo
sed solving method (in its optimized version) is planned in PARI/GP CAS.\n
\nhttps://events.rudn.ru/event/20/contributions/178/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/178/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Computational aspects of irreducible polynomials
DTSTART;VALUE=DATE-TIME:20190620T110000Z
DTEND;VALUE=DATE-TIME:20190620T114000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-182@events.rudn.ru
DESCRIPTION:Speakers: Doru Stefanescu (University of Bucharest)\nWe discus
s some results on testing the irreducibility of polynomials and give some
methods for constructing irreducible polynomials. We first survey classica
l methods of factorization of polynomials over the integers and irreducibi
lity criteria that are based on properties of Newton's polygon.\nFinally w
e give irreducibility criteria for univariate polynomials $F(X) = \\sum_{i
=0}^d a_i X^{d-i}$ over a discrete valuation domain in terms of Newton's p
olygon. We give applications to bivariate polynomials.\n\nhttps://events.r
udn.ru/event/20/contributions/182/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/182/
END:VEVENT
BEGIN:VEVENT
SUMMARY:An efficient algorithm for the simultaneous triangularization of a
finite set of matrices
DTSTART;VALUE=DATE-TIME:20190620T070000Z
DTEND;VALUE=DATE-TIME:20190620T074000Z
DTSTAMP;VALUE=DATE-TIME:20191023T182604Z
UID:indico-contribution-20-183@events.rudn.ru
DESCRIPTION:Speakers: Thomas Cluzeau (University of Limoges)\nIn the study
of linear differential systems\, one can be interested in deciding w
hether a set of $m$ given square matrices $A_1\,\\ldots\,A_m$ a
re simultaneously triangularizable or not. If the answer is yes\, then
we sometimes need to compute effectively an invertible matrix $P$ such
that\, for all $i \\in \\{1\,\\ldots\,m\\}$\, the matrix $P^{-1}\\\,A_i\\\
,P$ is upper triangular.\n\n The classical approach consists in using Li
e algebra theory to test whether the matrix Lie algebra spanned by the $A_
i$'s is solvable (e.g.\, using the so-called derived series) and if so\, f
ind a basis in which all matrices of the Lie algebra are upper triangular
using a constructive version of Lie's theorem on solvable algebras for com
puting common eigenvectors.\n\n In this presentation\, we will rather con
sider the following result due to McCoy: matrices $A_1\,\\ldots\,A_m$ are
simultaneously triangularizable if and only if\, for every scalar polynomi
al $p(x_1\,\\ldots\,x_m)$ in the (non-commutative) variables $x_1\,\\ldots
\,x_m$\, each of the matrices $p(A_1\,\\ldots\,A_m)[A_i\,A_j]=p(A_1\,\\ldo
ts\,A_m) (A_i\\\,A_j-A_j\\\,A_i)$ ($i\,j=1\,\\ldots\,m$) is nilpotent. We
shall show that the proof of this result can be turned into an efficient a
lgorithm for computing particular common eigenvectors of $A_1\,\\ldots\,A_
m$. As a consequence\, this yields an efficient algorithm for the simultan
eous triangularization problem. Note that this new approach does not requi
re the construction of the Lie algebra spanned by the matrices $A_i$'s. Th
e algorithm has been implemented in Maple and we will show comparisons to
the implementation of the ``Lie algebra method'' included in the Different
ialGeometry/LieAlgebras package of Maple.\n\nhttps://events.rudn.ru/event/
20/contributions/183/
LOCATION:
URL:https://events.rudn.ru/event/20/contributions/183/
END:VEVENT
END:VCALENDAR