BOOK OF ABSTRACTS

BOOK OF ABSTRACTS
X-th International Conference on
FINSLER EXTENSIONS OF RELATIVITY
THEORY - FERT 2014
BRAS
¸ OV, ROMANIA
August 18 - 23, 2014
Contents
Scientific Committee . . . . . . . . . . . . . . . . . . . . . . . . . .
Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . .
Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nicoleta ALDEA, Gheorghe MUNTEANU, A generalized Schr¨odinger
equation via a complex Lagrangian of electrodynamics . . . . .
Mihai ANASTASIEI, Galloway’s compactness theorem on Finsler
manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constantin M. ARCUS¸, Esmaeil PEYGHAN, (Pseudo) Generalized
Kaluza-Klein G-Spaces and Einstein Equations . . . . . . . .
Vladimir BALAN, Jelena STOJANOV, Finsler structures of 4−th
root type in cancer cell evolution models . . . . . . . . . . . .
Vladimir BALAN, E.M. OVSIYUK, V.M. RED’KOV, O.V. VEKO,
Maxwell electromagnetic equations in the uniform medium, an
alternative to the Minkowski Theory of Special Relativity . . .
Igor BAYAK, Some applications of the algebra of vector fields . . .
˘
Ioan BUCATARU,
Oana CONSTANTINESCU, Generalized Helmholtz conditions for Lagrangian systems with non-conservative
forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Aliya BUKUSHEVA, Geometric interpretation of the curvature tensor in model space unified theory of gravitational and electromagnetic interactions . . . . . . . . . . . . . . . . . . . . . . .
˘
Ovidiu CALIN,
Constantin UDRIS¸TE, Geometric Modeling in Probability and Statistics (Book Presentation) . . . . . . . . . . . .
Xinyue CHENG, Yangyang ZOU, The Generalized Unicorn Problem in Finsler Geometry . . . . . . . . . . . . . . . . . . . . .
Gy¨orgy DARVAS, Quaternion/vector dual space algebras applied to
the Dirac equation and its extensions . . . . . . . . . . . . . .
Cristian GHIU, Raluca TULIGA, Constantin UDRIS¸TE, I. TEVY,
Linear discrete multitime diagonal recurrence with periodic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
4
4
4
5
5
6
6
7
8
8
9
10
11
11
13
2
CONTENTS
Halina GRUSHEVSKAYA, Nina KRYLOVA, George KRYLOV,
Ihor LIPNEVICH, Finsler geometry approach to thermodynamics of first order phase transitions in monolayers . . . . .
Drago¸s HRIMIUC, On Finsler Manifolds of Scalar Flag Curvature .
Alexandru IONESCU, On lifts of left invariant holomorphic vector
fields in complex Lie groups . . . . . . . . . . . . . . . . . . .
Sergey S. KOKAREV, H-holomorphic ”Theory of everything” in
Hyperland . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L´aszl´o KOZMA, On parallel displacement in Finsler geometry . . .
Demeter KRUPKA, On the inverse problem of the calculus of variations for 2nd-order ODE ’s . . . . . . . . . . . . . . . . . . .
Alexander V. LAPSHIN, Three-elemental ternary product of 3-dim(3)
(3)
ensional (spatial) matrices Mijk and algebra hMijk , [P]i generated them . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jinling LI, Chunhui QIU, Tongde ZHONG, Hodge theorem in complex Finsler geometry . . . . . . . . . . . . . . . . . . . . . . .
Adelina MANEA, Cristian IDA, Basic connections adapted to a vertical Liouville subfoliation on the tangent bundle of a Finsler
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Marin MARIN, Olivia FLOREA, Effect of geometric equations on
dislocation for thermoelastic microstretch bodies . . . . . . . .
˘ Splitting theorems for
Ioana Monica MAS¸CA, Sorin Vasile SABAU,
Finsler spaces with reversible geodesics . . . . . . . . . . . . .
Adela MIHAI, M. Evren AYDIN, Ion MIHAI, Geometric Inequalities for Submanifolds of Statistical Manifolds . . . . . . . . . .
Ion MIHAI, Adela MIHAI, Special Vector Fields on Riemannian
Manifolds. Applications . . . . . . . . . . . . . . . . . . . . .
Ovidiu MUNTEANU, The splitting theorem for Finsler manifolds .
Zolt´an MUZSNAY, Tam´as MILKOWSKI, Invariant metrizability
and projective metrizability of the canonical spray on Lie groups
and its generalization . . . . . . . . . . . . . . . . . . . . . . .
˘ Gauss-Weingarten and Frenet equations in the
Alexandru OANA,
theory of the homogeneous lift to the 2-osculator bundle of a
Finsler metric . . . . . . . . . . . . . . . . . . . . . . . . . . .
T. OOTSUKA, R. YAHAGI, M. ISHIDA, E. TANAKA, Energymomentum currents in Finsler/Kawaguchi Lagrangian formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Victor A. PANCHELYUGA, Maria S. PANCHELYUGA, Local fractal analysis of alpha-decay rate fluctuations by all permutations
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
14
15
15
16
17
17
18
18
19
19
20
21
21
22
22
23
23
CONTENTS
Dmitri G. PAVLOV, Material events, their interaction potentials
and other physical characteristics . . . . . . . . . . . . . . .
Ioan Radu PETER, Some applications of index form in Finsler
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Liviu POPESCU, Monica CIOBANU, Geometrical structures on
the cotangent bundle . . . . . . . . . . . . . . . . . . . . . .
Paul POPESCU, Cristian IDA, Finsler geometry and nonlinear constrains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elena POPOVICI, On the volume of the indicatrix of a complex
Finsler space . . . . . . . . . . . . . . . . . . . . . . . . . .
Peter ROWLANDS, Nilpotent Quantum Theory: A review . . . .
Vladimir N. SHCHERBAN, Plane waves of torsion in Poincare
gauge theory of gravity . . . . . . . . . . . . . . . . . . . . .
Zhongmin SHEN, Einstein Metrics . . . . . . . . . . . . . . . . .
Sergey SIPAROV, On some properties of space in the Anisotropic
Geometrodynamics . . . . . . . . . . . . . . . . . . . . . . .
Sergey SIPAROV, V. SAMODUROV, G. LAPTEV, Analysis of the
time series in the space maser signals . . . . . . . . . . . . .
Ovidiu Cristinel STOICA, Gauge theory at singularities . . . . .
´
Annam´aria SZASZ,
Beil metrics in complex Finsler geometry . .
´
´
L´ajos TAMASSY,
D´avid Cs. KERTESZ,
Differentiable distance
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Anatoly TURBIN, Yuliya ZHDANOVA, Geometry of Quaternions
and Octonions . . . . . . . . . . . . . . . . . . . . . . . . . .
Constantin UDRIS¸TE, Comparing variants of single-time stochastic
maximum principle . . . . . . . . . . . . . . . . . . . . . . .
Zbynek URBAN, The inverse problem of the calculus of variations
for systems of homogeneous differential equations . . . . . .
Izu VAISMAN, Generalized Riemannian Metrics and Tangent Bundle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . .
Nicoleta VOICU, On field-theoretical integrals in Finslerian spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Qiaoling XIA, The eigenvalue problem in Finsler geometry . . . .
Chunping ZHONG, Characterizations of complex Finsler connections and weakly complex Berwald metrics . . . . . . . . . .
Aurel BEJANCU, A new point of view on (1+3) threading of spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
. 24
. 25
. 27
. 28
. 29
. 29
. 30
. 31
. 31
. 32
. 32
. 34
. 34
. 35
. 36
. 36
. 37
. 37
. 38
. 38
. 39
4
FERT 2014
Scientific Committee:
Acad. Prof. dr. Radu MIRON Romanian Academy, Romania
Prof. dr. Aurel BEJANCU - Kuwait University, Kuwait
Prof. dr. George BOGOSLOVSKY - Moscow State Univ., Inst. Nuclear
Physics, Russia
Prof. dr. Chunhui QIU - School of Math. Science, Xiamen Univ., China
Prof. dr. Zhongmin SHEN - Indiana Univ. - Purdue Univ., USA
´
Prof. dr. L´
ajos TAMASSY
- Inst. of Math. Univ. Debrecen, Hungary
Prof. dr. Izu VAISMAN - Haifa University, Israel
Organizing Committee:
Prof. dr. Vladimir BALAN - Univ. Politehnica of Bucharest, Romania
Prof. dr. Gheorghe MUNTEANU - Transilvania Univ. Bra¸sov, Romania
Dr. Dmitri G. PAVLOV - RIHS in Geom. and Phys., Moscow, Russia
˘
Assoc. Prof. dr. Marius PAUN
- Transilvania Univ. Bra¸sov, Romania
Prof. dr. Emil STOICA - Transilvania Univ. Bra¸sov, Romania
Dr. Nicoleta ALDEA - Transilvania Univ. Bra¸sov, Romania (secretary)
Dr. Nicoleta VOICU - Transilvania Univ. Bra¸sov, Romania (secretary)
Topics:
Finsler geometry and its applications. Extensions of Finsler, Lagrange and
Hamilton geometries to Relativity Theory.
Real and complex Geometry with applications in General Relativity.
Hypercomplex structures and their applications to Relativity Theory.
Field theories depending on direction. Extensions of Yang-Mills theories.
Extensions of Quantum relativistic gravity.
The general philosophical and mathematical principles underlying Finsler extensions of Relativity Theory.
Experimental investigations and astrophysical observations designed to detect possible evidence for the anisotropy of real Space-Time.
BOOK OF ABSTRACTS
5
A generalized Schr¨
odinger equation via a complex
Lagrangian of electrodynamics
Nicoleta ALDEA,1
Gheorghe MUNTEANU2
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
1 [email protected], 2 [email protected]
In this paper we give a generalized form of the Schr¨odinger equation in
the relativistic case, which contains a generalization of the Klein-Gordon
equation. By complex Legendre transformation, the complex Lagrangian
of electrodynamics produces a complex relativistic Hamiltonian H, on the
holomorphic cotangent bundle T 0∗ M. By a special quantization process, a
relativistic time dependent Schr¨odinger equation, in the adapted frames of
(T 0∗ M, H), is obtained. This generalized Schr¨odinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space
(T 0∗ M, H). Finally, in some additional conditions on the proper time τ of the
complex space-time M and the time parameter t along the quantum state, by
the method of separation of variables, we obtain two classes of solutions for
Schr¨odinger equation, one for the weakly gravitational complex curved space
M , and the second in the complex space-time with Schwarzschild metric.
Galloway’s compactness theorem on Finsler manifolds
Mihai ANASTASIEI
Al.I. Cuza University of Ia¸si, Faculty of Mathematics, Romania
and
Mathematical Institute Octav Mayer, Romanian Academy, Iasi branch
[email protected]
The compactness theorem of Galloway is a stronger version of the BonnetMyers theorem allowing the Ricci scalar to take also negative values from a
set of real numbers. However, this set is bounded below. In this paper we
allow any negative value for the Ricci scalar and adding a condition on its
average we find again that the manifold is compact and provide an upper
bound of its diameter. Also, with no condition on Ricci scalar itself but with
a condition on its average we find again the compactness of the manifold
but with no upper bound of its diameter. All considerations are done in the
category of Finsler manifolds.
6
FERT 2014
(Pseudo) Generalized Kaluza-Klein G-Spaces and
Einstein Equations
Constantin M. ARCUS
¸ ,1
1
2
Esmaeil PEYGHAN2
Secondary School ”Cornelius Radu” Radine¸sti Village, Gorj, Romania
c [email protected]
Department of Mathematics, Faculty of Science, Arak University, Iran
[email protected]
Introducing the Lie algebroid generalized tangent bundle of a KaluzaKlein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza-Klein vector bundle, we present the (g, h)-lift of a
curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear
(ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor
field associated to a distinguished linear (ρ, η)-connection and we obtain the
identities of Cartan and Bianchi type in the general framework of the Lie
algebroid generalized tangent bundle of a Kaluza-Klein bundle. Finally, we
introduce the theory of (pseudo) generalized Kaluza-Klein G-spaces and we
develop the Einstein equations in this general framework.
Finsler structures of 4−th root type in cancer cell
evolution models
Vladimir BALAN,1
Jelena STOJANOV2
1
2
University Politehnica of Bucharest, Romania
[email protected]
Technical Faculty ”Mihajlo Pupin”, University of Novi Sad, Zrenjanin, Serbia
[email protected]
The present work introduces a Finslerian model related to the classical
Garner dynamical system, which models the cancer cell population growth.
The Finsler structure is determined by the energy of the deformation field
- the difference of the fields, which describe the reduced and the proper
biological models.
BOOK OF ABSTRACTS
7
It is shown that a certain locally-Minkowski anisotropic 4−th root structure, obtained by means of statistical fitting, is able to provide an evaluation
the overall cancer cell population growth, which occurs due to significant
changes within the cancerous process. The geometric background, the comparison relative to the Euclidean and Randers fit structures, and the applicative advantages of the constructed geometric framework are discussed.
Maxwell electromagnetic equations in the uniform
medium, an alternative to the Minkowski Theory of
Special Relativity
V. BALAN,1
E.M. OVSIYUK,2
O.V. VEKO
V.M. RED’KOV,3
2
1
3
University Politehnica of Bucharest, Romania
[email protected]
2 Mozyr State Pedagogical University, Belarus
[email protected], [email protected]
Institute of Physics, National Belarus Academy of Sciences, Belarus
[email protected]
Two known, alternative to each other, forms of presenting the Maxwell
electromagnetic equations in a moving uniform medium are investigated and
discussed. The commonly used Minkowski approach is based on two tensors;
the relationships between them, so called constitutive equations, change their
form under Lorentz transformations and take the shape of Minkowski equations, depending upon the 4-velocity of the moving medium under a rest
reference frame. In this approach, the wave equation for the electromagnetic
4-potential has a form which explicitly involves this 4-velocity vector as a
supplementary parameter. Hence, the Minkowski electrodynamics implies
the absolute nature of mechanical motion.
An alternative formalism (developed by Rosen & al.) may be constructed
in new variables, when the Maxwell equations are written in terms of a single
tensor. This form of Maxwell equations exhibits symmetry under modified
Lorentz transformations in which, everywhere, instead of the vacuum speed
of light c one uses in the medium the speed of light c0 < c. Due to this
symmetry, such a formulation of Maxwell theory in the medium can be considered as invariant under the mechanical motion of the reference frame,
with modified Lorentz formulas, governed by the velocity c0 . The transition
8
FERT 2014
to 4-potential leads to a simple wave equation which does not contain any additional 4-velocity parameter, so this form of the electrodynamics presumes a
relative nature of the mechanical motion; also, this equation describes waves
which propagate in space with light velocity kc, which is invariant under the
modified Lorentz formulas.
In connection with these two alternative schemes, an essential issue must
be stressed: it seems reasonable to perform the Poincar´e-Einstein clock synchronization in uniform medium with the help of real light signals influenced
by the medium, which leads us to modified Lorentz symmetry. A similar
approach is developed for a spin 1/2 particle obeying the Dirac equation in
a uniform medium.
Some applications of the algebra of vector fields
Igor BAYAK
PTC ’Khimvolokno’, JSC ’Grodno Azot’, Belarus
[email protected]
The article discusses the local algebra of linear vector fields, which are
used in the representation of the Riemann zeta function and in the mathematical modeling of quantum states.
References
[1] David Hestenes (1966). Space-Time Algebra, Gordon-Breach.
Generalized Helmholtz conditions for Lagrangian
systems with non-conservative forces
1
˘
Ioan BUCATARU,
Oana CONSTANTINESCU2
Al.I. Cuza University of Ia¸si, Faculty of Mathematics, Romania
1 [email protected], 2 [email protected]
For an arbitrary semispray S (or a systems of second order ordinary
differential equations) and a Lagrangian L, the following 1-form (called the
BOOK OF ABSTRACTS
9
Euler-Lagrange 1-form, or the Lagrange differential by Tulczyjew) is a semibasic 1-form:
∂L
∂L
δS L := LS dJ L − dL = S
− i dxi .
i
∂ x˙
∂x
The inverse problem of Lagrangian mechanics requires, for a given semispray S, to decide wether or not there exists a Lagrangian L with vanishing
Lagrange differential, which means δS L = 0. This problem has been intensively studied by many authors, necessary and sufficient conditions for the
existence of such Lagrangian L are known as Helmholtz conditions.
In this work we study the more general problem, when for a given semispray S and a non-conservative covariant force field σ (semi-basic 1-form),
we study the existence of a Lagrangian L, whose Lagrange differential is σ,
which means δS L = σ. We provide generalized Helmholtz conditions for this
problem, which reduce to the classic ones when σ = 0. In the particular case
when the covariant force field σ is dissipative or gyroscopic, we recover the
generalized Helmholtz conditions obtained recently by Mestdag, Sarlet and
Crampin in [DGA, 2011]. In the homogeneous case, we show that one of the
generalized Helmholtz conditions is a consequence of the other.
Geometric interpretation of the curvature tensor in
model space unified theory of gravitational and
electromagnetic interactions
Aliya BUKUSHEVA
Faculty of Mathematics and Mechanics, Saratov State University, Russia
[email protected]
The authors of [1] suggest that the space velocities of the particles is a
four nonholonomic distribution on the manifold of higher dimension. This
distribution is given 4-potential of the electromagnetic field. Equation of
admissible (horizontal) geodesic for this distribution coincide with the equations of motion of a charged particle of the general theory of relativity. Metric
tensor of the Lorentzian signature (+, −, −, −) is defined on the distribution,
which allows to determine causality as in the general theory of relativity. The
authors introduced the covariant derivative (linear connection) and the curvature tensor for distribution. In [2] for any distribution of intrinsic connection
construct its extension - extended connection. To ask extended connection
means to identify some vector fields on corresponding distribution.
10
FERT 2014
In adapted coordinates, the vector field has the form ~u = ∂n − Gan ∂n+a .
Curvature tensor, geodesic which identical with the equations of motion of a
charged particle [1] is:
c
Kbad
= 2∂[b Γca]d + 2Γc[b|e| Γea]d +
ε0 k c
F ωba .
c2 d
(1)
The purpose of this paper is to find an explicit expression for the extended
connectivity, the curvature tensor which coincides with the tensor (1). We
prove that Gcn = ε2c0 k2 Fdc xn+d sets the extended connection having the necessary physical sense.
References
[1] Krym, V.R., Petrov, N.N., The curvature tensor and the Einstein
equations for a four-dimensional nonholonomic Vestnik St. Petersburg University. Mathematics. 41(3) (2008), 256-265.
[2] Bukusheva, A. V., Galaev, S.V.Almost contact metric structures defined by connection over distribution with admissible Finslerian metric, Izv.
Saratov. Univ. Mat. Mekh. Inform. 12(3) (2012), 17-22.
Geometric Modeling in Probability and Statistics
1
˘
Ovidiu CALIN,
1
Constantin UDRIS
¸ TE2
Ypsilanti, MI, USA,
2
Bucharest, Romania
(Book Presentation)
Springer International Publishing Switzerland 2014
Overview
This book is devoted to a specialized area, including Informational Geometry. This is a field that is increasingly attracting the interest of researchers
from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics, and neuroscience. It
is the authors hope that the present book will be a valuable reference for
researchers and graduate students in one of the aforementioned fields. The
book is structured into two distinct parts. The first one is an accessible introduction to the theory of statistical models, while the second part is devoted
to an abstract approach of statistical manifolds.
BOOK OF ABSTRACTS
11
Contents
I. The Geometry of Statistical Models: Statistical Models, Explicit Examples, Entropy on Statistical Models, Kullback - Leibler Relative Entropy,
Informational Energy, Maximum Entropy Distributions.
II. Statistical Manifolds: An Introduction to Manifolds, Dualistic Structure, Dual Volume Elements, Dual Laplacians, Contrast Functions Geometry,
Contrast Functions on Statistical Models, Statistical Submanifolds.
Appendix A Information Geometry Calculator.
The Generalized Unicorn Problem in Finsler
Geometry
Xinyue CHENG,1
1
Yangyang ZOU
School of Mathematics and Statistics, Chongqing University of Technology,
Chongqing, P. R. China
[email protected]
In this talk, we introduce the generalized unicorn problem in Finsler
geometry. We prove that, for a regular (α, β)-metric in the form of F =
αφ(β/α), where α is a Riemannian metric and β is a 1-form on the manifold,
if φ = φ(s) is a polynomial in s, then F is a weak Landsberg metric if and
only if F is a Berwald metric.
Quaternion/vector dual space algebras applied to the
Dirac equation and its extensions
Gy¨
orgy DARVAS
Symmetrion, Budapest, Hungary
[email protected]
The paper re-applies the 64-part algebra discussed by P. Rowlands in a
series of (FERT and other) papers in the recent years [1], [2], [3], [4]. It
demonstrates that the original introduction of the γ algebra by Dirac [5]
to the quantum theory of the electron can be interpreted with the help of
quaternions: both the α matrices and the Pauli (σ) matrices in Diracs original interpretation can be rewritten in quaternion forms. This allows to
12
FERT 2014
construct the Dirac γ matrices in two (quaternion-vector) matrix product
representations - in accordance with the double vector algebra introduced by
P. Rowlands. The paper attempts to demonstrate that the Dirac equation
in its form modified by P. Rowlands, essentially coincides with the original
one. The paper shows that one of these representations affects the γ4 and
γ5 matrices, but leaves the vector of the Pauli spinors intact; and the other
representation leaves the γ matrices intact, while it transforms the spin vector into a quaternion pseudovector. These transformations affect all gauge
extensions of the Dirac equation [6], [7], [8], [9] in a similar way.
References
[1] Rowlands, P., A null Berwald-Moor metric in nilpotent spinor space,
Symmetry: Culture and Science, 23 (2012), 2, 179-188.
[2] Rowlands, P., The null Berwald-Moor metric and the nilpotent wave
function, 8th FERT 2012.
[3] Rowlands, P., Symmetry in physics from the foundations, Symmetry:
Culture and Science, 24 (2013), 1-4, 41-56.
[4] Rowlands, P., Dual spaces, particle singularities and quartic geometry,
9th FERT, 2013, 10p.
[5] Dirac, P.A.M., The quantum theory of the electron, Proceedings of the
Royal Society A: Mathematical, Physical Engineering Sciences,, 117 (1928),
778, 610 624.
[6] Darvas, G., Finslerian approach to the electromagnetic interaction
in the presence of isotopic field-charges and a kinetic field, Hypercomplex
Numbers in Geometry and Physics, 2 18 (2012), 9, 1-19.
[7] Darvas, G., The Isotopic Field-Charge Assumption Applied to the
Electromagnetic Interaction, Int. J. Theoretical Physics, 52 (2013), 11, 38533869.
[8] Darvas, G., Electromagnetic Interaction in the Presence of Isotopic
Field-Charges and a Kinetic Field, Int. J. Theoretical Physics, 53 (2014), 1,
39-51.
[9] Munteanu, G., A Yang-Mills electrodynamics theory on the holomorphic tangent bundle, Journal of Nonlinear Mathematical Physics, 17 (2010),
2, 227242.
BOOK OF ABSTRACTS
13
Linear discrete multitime diagonal recurrence with
periodic coefficients
Cristian GHIU, Raluca TULIGA, Constantin UDRIS
¸ TE,1
Ionel TEVY
1
University Politehnica of Bucharest, Romania
[email protected]
Floquet theory, first published in 1883 for periodic linear differential equations, is extended in this paper to multitime diagonal recurrences. We find
explicitly a monodromy matrix, and we comment its eigenvalues (called Floquet multipliers). The Floquet point of view brings about an important
simplification: the initial linear diagonal recurrence system is reduced to the
linear diagonal recurrence system, with constant coefficients along “diagonal
lines”.
Finsler geometry approach to thermodynamics of first
order phase transitions in monolayers
Halina GRUSHEVSKAYA,1
Nina KRYLOVA,1
George KRYLOV,
Ihor LIPNEVICH
1
Faculty of Physics, Belarusan State University, Minsk, Belarus
[email protected]
Phase transitions of the first order are characterized by thermodynamic
quantities fluctuations not in singular critical point (as 2nd order phase transition) but in continuous critical region of parameters. This region for Langmuir monomolecular layer (monolayer) corresponds to compression isotherm
plateau.
Known procedures [1] for statistical treatment of compressibility coefficient κ in the whole region of surface tension values π
˜ allow to determine only
a mean value of compressibility coefficient the ”apparent” compressibility C 0
in the relatively straight section of the isotherms.
To find out the behavior of compressibility coefficient κ on the critical
surface of surface tension values, the statistical treatment procedure for enormous experimental data array is necessary to use. To date, a suitable statistical treatment of monolayer compression isotherm has not been proposed.
14
FERT 2014
In [1] the geometrical approach to thermodynamics of the first order phase
transition in compressed monolayer on interphase boundary air/water has
been proposed. The compressibility κ of spherically symmetric monolayer
was expressed through the monolayer space Berwald curvature. The trace
of Berwald curvature turns out to be equal to zero everywhere except of the
anomalous transition region. The expression for κ through the flag curvature
was obtained.
The comparison of the dependence of κ upon surface tension π
˜ with experimental data has allowed to reveal a scenario of phase transition in Fecontained monolayers of thiophene-pyrrole series oligomer. The compressibility coefficient κ has an anomalously large dispersion in the critical region
on isotherms of monolayer compressed on a water surface. The metastable
state of substance appears as an instanton-like behavior region. The crystal
structure formation is characterized by two-anomalous trace of the Berwald
curvature.
References
[1] Toma, L.M., Gengler, R.Y.N., Prinsen, E.B., Gournis, D., Rudolf, P.,
A Langmuir-Schaefer approach for the synthesis of highly ordered organoclay
thin films, Phys. Chem. Chem. Phys. 12 (2010), 12188-12197.
[2] Balan,V., Grushevskaya, H., Krylova, N., Neagu, M., Oana, A., On
the Berwald-Lagrange scalar curvature in the structuring process of the LBmonolayer, Applied Sciences 15 (2013), 30-42.
On Finsler Manifolds of Scalar Flag Curvature
Drago¸s HRIMIUC
University of Alberta, Department of Mathematical and Statistical Sciences,
Edmonton, Canada
[email protected]
Let (M, F ) be a pseudo-Finsler manifold, J the canonical tangent structure of the tangent bundle T M, ω = d(dL ◦ J) the standard symplectic
structure, g (JX, JY ) := ω (JX, Y ) the pseudo-Finsler metric of the vertical
bundle and ξ the second order vector field associated to the non-degenerate
Lagrangian L = 21 F 2 . In this paper we prove that the pseudo-Finsler manifold (M, F ) is of scalar flag curvature k if and only if Lζ g + kLζ g 0 = 0,
where ζ = F1 ξ and g 0 = F 2 ω ◦ (J × id) is a pseudo-Riemannian metric of the
horizontal bundle.
BOOK OF ABSTRACTS
15
On lifts of left invariant holomorphic vector fields in
complex Lie groups
Alexandru IONESCU
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
In this paper the complete, vertical and horizontal lifts of left invariant
holomorphic vector fields to the holomorphic tangent bundle T 1,0 G of a complex Lie group G are studied. Also, the Lie algebra of T 1,0 G complex Lie
tangent group is investigated.
H-holomorphic ”Theory of everything” in Hyperland
Sergey S. KOKAREV
RSI HCSGP (Fryazino) — RSEC Logos (Yaroslavl), Russia
[email protected]
Unified theory of space-time, matter and interactions, based on double
numbers algebra is constructed. The starting point of our consideration is
the fact, that double numbers algebra induces 2D Minkowski geometry. The
concept of h-holomorphicity (complex differentiability of functions over double variable) allows to generalize 2D special relativity to conformal relativity,
wherein generalized reference frames may differ from each other by their
own time flow rates. These rates are conditioned by scale factor — module
of some h-holomorphic function (hyperbolic potential), making generalized
(conformal) transformation of reference frame. In such theory time and space
quantities are defined by real and imaginary parts of the hyperbolic potential respectively. To construct dynamical theory of hyperbolic potential we
consider non-holomorphic mappings and associate non-holomorphicity with
matter and its interactions properties. Also we apply extravariational procedure to derive concrete Lagrangian of the model from the general class of
h-holomorphic field theories. Calculations lead to simple (and unique!) world
— we have called it Hyperland, which is operated by the set of three fundamental constant (energy density and pressure of a vacuum and one structural
constant). Detailed analysis of static world is performed. The masses, radii
16
FERT 2014
and coupling energy of possible matter configurations are calculated from the
first principles. The presented toy model is probe in seeking for the more
realistic 4D theory, based on polynumbers algebra, wherein double numbers
algebra is the simplest low-dimensional representator.
References
[1] D.G.Pavlov, S.S.Kokarev, h-holomorphic functions and its applications, HNGP 1(13), v.7 2010, p.44-77 (In Russian).
[2] D.G.Pavlov, S.S.Kokarev, Hyperbolic field theory on the plane of double variable, HNGP 1(13), v.7, 2010, p.78-126 (In Russian).
[3] D.G.Pavlov, S.S.Kokarev, Algebraic unified theory of space-time-matter
on the plane of double variable, HNGP 2(14), v.7, 2010, p.11-37 (In Russian).
[4] S.S.Kokarev, Extravariational principle in the theory of field, In coll.
papers of RSEC ”Logos” 6, 2011, pp. 123-146 (In Russian).
On parallel displacement in Finsler geometry
L´
aszl´
o KOZMA
Institute of Mathematics, University of Debrecen, Hungary
[email protected]
We intend to give some basic properties of the non-standard connection
of Shen used for treating fundamental concepts of Finsler geometry. We show
that this notion is dual to the usual Ehresmann nonlinear connection of a
manifold. In the presence of a Finsler metric, its uniqueness and existence is
characterized. It is pointed that for a Finsler manifold the geodesic structure
(spray), the flag curvature, and so the concept of Jacobi fields can be derived
in the same way as for classical Ehresmann connections.
References
[1] S.S. Chern and Z. Shen, Riemann-Finsler geometry, Springer, 2005.
[2] Zolt´an Muzsnay and P´eter T. Nagy, Invariant Shen connection and
geodesic orbit spaces, Periodica Math. Hung., 51, 37–51. (2005).
[3] Nagano, Tetsuya, On the parallel displacement and parallel vector
fields in Finsler geometry. Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 26,
No. 2, 349-358 (2010).
BOOK OF ABSTRACTS
17
[4] H. Rund, The differential geometry of Finsler spaces, Springer-Verlag,
1959.
[5] Z. Shen, Lectures on Finsler geometry, World Scientific, Singapore,
2001.
On the inverse problem of the calculus of variations for
2nd-order ODE ’s
Demeter KRUPKA
Lepage Research Institute, Czech Republic
[email protected]
In this talk we consider the inverse problem of the calculus of variations for
systems of second order ordinary differential equations, as studied by Sonin
and Douglas. We describe the differential system for variational integrating
factors and discuss its solutions. Applications to Finsler geometry are given.
Three-elemental ternary product of 3-dimensional
(3)
(3)
(spatial) matrices Mijk and algebra hMijk , [P]i
generated them
Alexander V. LAPSHIN
RI Hypercomplex Systems in Geometry and Physics, Russia
[email protected]
It is considered generalization of a binary product of the 2-dimensional
n × m maricies on a ternary product of the 3-dimensional n × m × k matricies
in the report. It is described the definition and properties of the ternary oper(3)
ation P on the set of the 3-dimensional matricies Mijk as well as the algebra
(3)
< Mijk , [P] > generated them, including a neutral or identical self-mapping
of a arbitrary spatial matrix and some kinds of the another important cases.
Making use of the principle of analogy with the binary-matix decomposition
of the scalar product in the algebra of the double numbers in the isotropic basis P2 : ||P2 || = pφi gij p
¯ jψ = kφψ = k = A1 A2 , it is discovered ternary-matix
decomposition of the scalar threeproduct in the algebra of the triple numbers
¯ ωβt ) =
in the isotropic basis P3 : ||P3 || = (pφiψ (Gijk p
¯ jαk ω g
¯kψi )¯
gψti )¯
gtrω γ p
(3)
dφβγ = d = A1 A2 A3 with the help of the ternary operation Mijk : Dijk =
Ailm Blmk Cmjl .
18
FERT 2014
Hodge theorem in complex Finsler geometry
Jinling LI, Chunhui QIU,1
1
Tongde ZHONG
School of Mathematical Sciences, Xiamen University, Xiamen, P.R. China
[email protected]
Let M be a compact complex manifold with a strongly pseudoconvex
complex Finsler metric F, we define a natural projection of complex horizontal Laplacian on M, it is independent of the fiber coordinate. By using
the Sobolev space theory and spectral resolution theory in a Hilbert space,
we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M. Moreover, by means of the decomposition theorem of
the self-adjoint elliptic operator, we also prove the Hodge theorem for the
Hodge-Laplace operator on M.
Basic connections adapted to a vertical Liouville
subfoliation on the tangent bundle of a Finsler space
Adelina MANEA,1
Cristian IDA2
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
1 [email protected], 2 [email protected]
On the slit tangent manifold T M 0 of a Finsler space (M, F ) there are
given some natural foliations as vertical foliation and some other fundamental
foliations produced by the vertical and horizontal Liouville vector fields, see
[A. Bejancu, H. R. Farran, Finsler Geometry and Natural Foliations on the
Tangent Bundle, Rep. Math. Physics 58, No. 1 (2006), 131-146]. In this talk
we consider a (n, 2n − 1)-codimensional subfoliation (FV , FΓ ) on T M 0 given
by vertical foliation FV and the line foliation spanned by vertical Liouville
vector field Γ and we present a triplet of basic connections adapted to this
subfoliation. The talk is based on the paper [A. Manea, C.Ida, Adapted basic
connections to a certain subfoliation on the tangent manifold of a Finsler
space, Turkish Journal of Mathematics 38(3), 2014, 470-482].
BOOK OF ABSTRACTS
19
Effect of geometric equations on dislocation for
thermoelastic microstretch bodies
Marin MARIN,1
Olivia FLOREA2
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
1 [email protected], 2 [email protected]
The aim of our study is to derive a relation of De Hoop - Knopoff type for
displacement fields within context of thermoelastic microstretch bodies. We
analyze in particular the effect of geometric equations on deformation of such
medium. Then, as a consequence, an explicit expression of the body loadings
equivalent to a seismic dislocation, is obtained. The results are extensions of
those from the classical theory of elastic bodies.
Splitting theorems for Finsler spaces with reversible
geodesics
Ioana Monica MAS
¸ CA,1
1
˘
Sorin Vasile SABAU
Colegiul ”Nicolae Titulescu”, Bra¸sov, Romania
[email protected]
In [1] the structure of a Finsler manifold of nonnegative weighted Ricci
curvature including a straight line is investigated, and the classical CheegerGromoll-Lichnerowicz splitting theorem is extended. We will extend these
results for Finsler manifolds with reversible geodesics including a straight
line.
References
[1] S. Ohta, Splitting theorems for Finsler manifolds, arXiv:1203.0079v1.
20
FERT 2014
Geometric Inequalities for Submanifolds of Statistical
Manifolds
Adela MIHAI1
1
M. Evren AYDIN,2
Ion MIHAI,3
Technical University of Civil Engineering Bucharest, Romania
adela [email protected]
2 Firat University, Turkey, [email protected]
3 University of Bucharest, Romania, [email protected]
Statistical manifolds introduced by Amari (1985) have been studied in
terms of information geometry. Since the geometry of such manifolds includes
the notion of dual connections, also called conjugate connections in affine
geometry, it is closely related to affine differential geometry. A statistical
structure is a generalization of a Hessian one.
We study the behaviour of submanifolds in statistical manifolds of constant curvature. We investigate curvature properties of such submanifolds.
Some inequalities for submanifolds with any codimension and hypersurfaces
of statistical manifolds of constant curvature are established.
References
[1] Amari, S. (1985). Differential-Geometrical Methods in Statistics.
Springer-Verlag.
[2] Vos, P. W. (1989). Fundamental equations for statistical submanifolds with applications to the Bartlett correction. Annals of the Institute of
Statistical Mathematics, 41(3), 429-450.
BOOK OF ABSTRACTS
21
Special Vector Fields on Riemannian Manifolds.
Applications
Ion MIHAI,1
Adela MIHAI2
1
2
University of Bucharest, Romania
[email protected]
Technical University of Civil Engineering Bucharest, Romania,
adela [email protected]
We interrelate the concept of a torse forming vector field and the concepts
of exterior concurrent and quasi-exterior concurrent vector fields. Different
second order properties of a torse forming vector field T are studied, as
for instance it is proved that any torse forming is a quasi-exterior concurrent
vector field. We obtain a necessary and sufficient condition for a torse forming
vector field to be 2-exterior concurrent. In this case a foliation is given.
Some applications of the existence of torse formings on Sasakian manifolds
and Kenmotsu manifolds, respectively, are discussed.
The presentation is based on the paper A. Mihai, I. Mihai, Torse forming
vector fields and exterior concurrent vector fields on Riemannian manifolds
and applications, J. Geom. Phys. 73 (2013), 200-208.
The splitting theorem for Finsler manifolds
Ovidiu MUNTEANU
Department of Mathematics, University of Connecticut, USA
[email protected]
A celebrated result of Cheeger and Gromoll says that a complete Riemannian manifold of non-negative Ricci curvature splits along a line. Several
extensions of this result are known.
Soon after Cheeger-Gromoll’s work appeared, Lichnerowicz observed a
generalization of the splitting theorem for what we now call the Bakry-Emery
curvature tensor. His work was recently rediscovered in the context of smooth
metric measure spaces by Wei and Wylie.
For Finsler manifolds, Ohta has recently made some important progress
on this topic. His work connects with the Bakry-Emery tensor and Lichnerowicz’s ideas.
22
FERT 2014
In this talk, I will discuss further generalizations of these results. This
work is motivated by some questions in the theory of Ricci solitons and Ricci
flow.
Invariant metrizability and projective metrizability of
the canonical spray on Lie groups and its
generalization
Zolt´
an MUZSNAY,1
1
Tam´
as MILKOWSKI
University of Debrecen, Hungary
[email protected]
Lie groups represent a well developed theory of continuous symmetry of
mathematical structures, and it is an indispensable tools for modern theoretical physics. The algebraic and differential structures allow to introduce
natural geometric objects that are under extensive investigation. The aim
of this talk is to examine the invariant Riemann or Finsler metrizability and
projective metrizability of the canonical connection. We show that the canonical connection of a Lie group is projective Finsler metrizable if and only if it
is Riemann metrizable. That means that the structure is rigid in the sense
that, considering left-invariant metrics, the potentially much larger class of
projective Finsler metrizable class of Lie groups coincides with the Riemann
metrizable class. We also consider the generalization of the metrizability
problems by considering homogeneous structures.
Gauss-Weingarten and Frenet equations in the theory
of the homogeneous lift to the 2-osculator bundle of a
Finsler metric
˘
Alexandru OANA
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
In this article we present a study of the subspaces of the manifold Osc2 M ,
the total space of the 2-osculator bundle of a real manifold M. We obtain the
BOOK OF ABSTRACTS
23
induced connections of the canonical metrical N-linear connection determined
by the homogeneous prolongation of a Finsler metric to the manifold Osc2 M .
We present the Gauss-Weingarten equations of the associated 2-osculator
submanifold. We construct a Frenet frame and we determine the Frenet
equations of a curve from the manifold Osc2 M.
Energy-momentum currents in Finsler/Kawaguchi
Lagrangian formulation
T. OOTSUKA, R. YAHAGI, M. ISHIDA, E. TANAKA1
1
Kagoshima University, 1-21-35 Korimoto, Japan
[email protected]
We reformulate the standard Lagrangian formalism to a reparameterisation invariant Lagrangian formalism by means of Finsler and Kawaguchi
geometry. In our formalism various symmetries are expressed as symmetries
of Finsler (Kawaguchi) metric geometrically, and the conservation law of
energy-momentum can be derived simply. The Energy-momentum currents
of scalar field, Dirac field, electromagnetic field and general relativity are discussed. By this formalism, we propose one interpretation of understanding
the problem of energy-momentum current of gravity.
Local fractal analysis of alpha-decay rate fluctuations
by all permutations method
Victor A. PANCHELYUGA,1
1
Maria S. PANCHELYUGA
Institute of Theoretical and Experimental Biophysics of RAS, Pushchino,
Moscow
and
Research Institute for Hypercomplex Systems in Geometry and Physics,
Fryazino, Russia
[email protected]
Results of local fractal analysis of 329 1-day time series of alpha-decay rate
fluctuations by means of all permutations method (APM) are presented. The
analysis reveals in the time series some steady frequency set. Coincidence
of the frequency set with the Earth natural oscillations was demonstrated.
24
FERT 2014
Short revue of periods in fluctuations of various processes (physical, chemical, biological) in range 1-115 min described in works of different authors are
given. We shown that periods observed in cited papers corresponds to periods, which revealed in present report. Such correspondence leads to conclusion about some common mechanism, which may cause observed periodicity
in processes of different nature. One of candidates for such mechanism are
periodical change in local time rate caused by hyperbolic fields, which are
generated by Earth natural oscillations.
Material events, their interaction potentials and other
physical characteristics
Dmitri G. PAVLOV
Research Institute for Hypercomplex Systems in Geometry and Physics,
Fryazino, Russia
[email protected]
In the pseudo-Finslerian space-time with Berwald-Moor metric, we introduce the notion of material event, which differs from the usual notion of
event in the same way as a material point differs from a usual point. It is
proven that an isolated material event is surrounded by a space-time field,
whose potential is similar to the Coulomb potential of a charged material
particle, just, having as its argument the space-time interval. In the particular case of the two-dimensional Berwald-Moor space, whose metric coincides
with the 2-dimensional pseudo-Euclidean space-time metric, the potential of
an isolated material event has the form:
U = c0 + c1 ln(S).
In the general case of n-dimensional Berwald-Moor space - times, the
Coulomb potential of an isolated material event is expressed as a power series
in the logarithm:
U = c0 + c1 ln(S) + c2 ln2 (S) + ... + cn−1 lnn−1 (S).
The above potential is a solution of a centrally-symmetric equation representing a pseudo-Finslerian analogue of the Laplace equation for a charged
material point. The notions of source-type and vortex-type charge of a material event are introduced. Fields generated by these are, outside the sources,
hyperbolically potential and hyperbolically solenoidal. The corresponding
BOOK OF ABSTRACTS
25
fields are called hyperbolic fields and they are different from the usual fundamental fields, since they are not force fields. We also introduce the notions
of timergy of interaction of material events, which is the hyperbolic analogue
of the notion of the energy of interaction of material points and the notion
of timergy density of a hyperbolic field, representing the analogue of the energy density of usual force fields of charged particles. It is formulated the
conservation law of the total timergy of a hyperbolic field and of the timergy
of interacting charges of material events. It is launched the hypothesis that
a physical model of real world, based on the concepts of pseudo-Finslerian
space-time, of source- and vortex-type charges of material events, is not less
promising than the actual models in physics, based on Riemannian spacetimes, on various charges of elementary particles and on the four fundamental
interactions.
Some applications of index form in Finsler geometry
Ioan Radu PETER
Technical University of Cluj Napoca, Department of Mathematics, Romania
[email protected]
An important task of differential geometry is to derive topological properties of a manifold from certain differential geometric invariants associated
to a Riemannian or a Finslerian metric on that manifold. In order to cover
this task several techniques and notions have been invented (first and second
variations of geodesics, Morse index form, Jacobi fields etc.). Using these,
some basic results as for instance the theorems of Hadamard, Hopf-Rinow,
Myers, Rauch, Synge have been obtained. The area of these results has been
extended along years to the Finslerian setting. The most recent and modern
account of them is due to D. Bao, S.S. Chern and Z. Shen (see [4,6-9]). Their
book has been followed by many papers in this field. We cite only few [3,20]
as more related to our results.
The main differential geometric invariants involved in the results aiming
to establish a topological property are the flag curvature and the Ricci scalar.
Among the many others there exits one denoted by Rick that interpolates
between the flag curvature and the Ricci curvature. It is associated to a
k + 1-dimensional subspace of the tangent space in a point of a manifold
in such a way that for k = 1 it coincides with the flag curvature and for
k = dimM − 1 it is nothing but the Ricci curvature.
26
FERT 2014
In this paper two different results in their nature but connected by Rick
are proved. The first one provides a sufficient condition on the average of
the k-Ricci curvature in order that the Finsler manifold to be compact. The
second one says that if Rick is positive then two submanifolds of a Finslerian
n- dimensional manifold, one with asymptotic index n − 1 and one minimal
must intersect. The proofs of the both results are based on the index form
written in a special frame along geodesics.
References
[1] M. Abate, G. Patrizio. Finsler Metrics - A Global Approach, volume
1591 of Lecture Notes in Math. Springer Verlag, Berlin, Heidelberg, 1994.
[2] Mihai Anastasiei. A Generalization of Myers Theorem. An. S¸tiint¸.
Univ. ”A. I. Cuza” Iasi, Mat. (NS0 53(2007), suppl. 1, 33-40.
[3] Mihai Anastasiei, Ioan Radu Peter. A Compactness Theorem in
Finsler Geometry Pub.Math. Debrecen.
[4] D. Bao, S.-S. Chern, Z. Shen. An Introduction to Riemann- Finsler
Geometry. Gradute Text in Mathematics 200, Springer, 2000, xx+ 431p.
[5] T. Q. Binh, L. Tamssy, Compactness theorem for Berwald spaces.Proc.
of the 40th Symposium of Finsler Geometry, Sapporo, Japan, 2005, 28-32.
´ Javaloyes, A. Masielo, Morse theory of causal
[6] E. Caponio, M. A
geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler
metric, Ann. Inst. H. Poincare ´e Anal. Non lin´eare, 27(2010), 857-876.
´ Javaloyes, A. Masielo, On the energy functional on
[7] E. Caponio, M. A
Finsler manifolds and applications to stationary spacetimes, Math. Ann.,
351(2011), 365-392.
´ Javaloyes, M. S´anchez, On the interplay between
[8] E. Caponio, M. A
Lorentzian Causality and Finsler metrics of Randers type, Rev. Met. Iberoamericana, 27(2011), 919-923.
[9] S. Dragomir. Submanifolds of Finsler Spaces. Conf. Sem. Mat.
Univ. Bari, 271:1–15, 1986.
[10] G. Galloway, Some results on the occurrence of compact minimal
submanifolds, Manuscripta Math. 35 (1981), 209-219.
[11] G.J. Galloway, Compactness criteria for Riemannian manifolds. Proc.
of AMS, vol. 84,1,(1982), 6-10.
[12] L. Kozma, A. Krist´aly, C. Varga, Critical point theorems on Finsler
manifolds. Beitr¨age zur Algebra und Geometrie, vol 45, 1 (2004), page 47-59.
[13] M. Matsumoto. Foundations of Finsler geometry and special Finsler
spaces. Kasheisha Press, Japan, 1986.
[14] R. A. Moore, The behavior of solutions of a linear differential equation
of second order, Pacific Journal of Mathematics vol. 5 (1955) pp. 125-145.
BOOK OF ABSTRACTS
27
[15] Ioan Radu Peter, On the Morse Index Theorem where the ends are
submanifolds in Finsler geometry. Houston Journal of Mathematics, 32(4),
995-1009, 2006.
[16] F.J. Tipler. General relativity and ordinary diferential equations. J.
Diff. Eq., 30, 165-174, 1978.
[17] Z. Shen. Finsler Geometry of submanifolds Math. Ann.,311, 549-576,
1998.
[18] Z. Shen. Lecture Notes on Finsler Geometry. Springer Verlag, 2001.
[19] Z. I. Szab´o Positive definite Berwald spaces. Tensor (N.S.), 35,
25-39, 1981.
[20] Bing-Ye Wu A Note on the generalized Myers theorem for Finsler
manifolds Bull. Korean Math. Soc. 50(2013), pp. 833-837.
Geometrical structures on the cotangent bundle
Liviu POPESCU,1
Monica CIOBANU2
1
2
University of Craiova, Dept. of Applied Mathematics, Craiova, Romania
[email protected]
Faculty of Mathematics and Informatics, Vasile Goldis Univ., Arad, Romania
[email protected]
In this paper we study the geometrical structures on the cotangent bundle
using the notions of adapted tangent structure and regular vector fields. We
prove that the dynamical covariant derivative on T ∗ M fix a nonlinear connection for a given J-regular vector field. Using the Legendre transformation
induced by a regular Hamiltonian, we show that a semi-Hamiltonian vector
field on T ∗ M corresponds to a semispray on T M if and only if the nonlinear
connection on T M is just the canonical nonlinear connection induced by the
regular Lagrangian.
References
[1] Buc˘ataru, I., Dahl, M. F., Semi-bazic 1-form and Helmholtz conditions for the inverse problem of the calculus of variations, Journal Geom.
Mechanics, 1, no.2 (2009), 159-180.
[2] Miron, R., Hrimiuc, D., Shimada, H., Sab˘au, S., The geometry of
Hamilton and Lagrange spaces, Kluwer Academic Publishers, 118, (2001).
[3] Oproiu, V., Regular vector fields and connections on cotangent bundle,
An. Stiint. Univ. A.I.Cuza, Iasi, S.1. Math., 37, no.1,(1991), 87-104.
28
FERT 2014
[4] Popescu, L., A note on nonlinear connection on the cotangent bundle,
Carpath. J. Math. 25, no. 2, (2009), 203-214.
[5] Popescu, L., Criveanu R., A note on metric nonlinear connection on
the cotangent bundle,Carpath. J. Math. 27, no. 2,(2011), 261-268.
[6] Szilasi, J,. A setting for spray and Finsler geometry In: Handbook of
Finsler Geom., (ed. P. Antonelli) Kluwer Acad. Publ., 2 (2003), 1183-1426.
Finsler geometry and nonlinear constrains
Paul POPESCU,1
Cristian IDA2
1
2
Department of Applied Mathematics, University of Craiova, Romania
paul p [email protected]
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
Considering nonlinear nonholonomic constraints, a simple form of equations of regular dynamics are obtained, based on some Chetaev-like conditions. In the particular cases of linear and affine constraints, one obtain the
classical equations in the forms given, for example, by Bloch, Marsden and
other authors. The case of time-dependent constraints is also considered.
Some cases when constraints or Lagragians come from Finsler functions are
considered.
References
[1] A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, and R.M. Murray,
Nonholonomic mechanical systems with symmetry, Archive for Rational Mechanics and Analysis 136, 1 (1996) 21–99.
[2] P. Popescu, M. Popescu, Lagrangians adapted to submersions and
foliations, Differential Geom. Appl. 27 (2009), 171–178.
BOOK OF ABSTRACTS
29
On the volume of the indicatrix of a complex Finsler
space
Elena POPOVICI
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
Motivated by some issues related to the conditions of minimal hypersurfaces, the present work studies the variation of the volume function of
complex unit tangent sphere, or indicatrix. This investigation is made both
locally, at each point z of a complex Finsler manifold M , and globally. We
also analyze the link between the volume of the indicatrix in a fixed point,
Iz M , calculated with respect to a Hermitian metric induced naturally by the
Finsler structure, and the volume function of the projectivized complex tangent bundle Pz M . Using this, we take into consideration the result obtained
by R. Yan regarding the volume of Pz M , which is obtained to be constant.
This contrasts sharply with the situation in real Finsler geometry, where the
volume of the unit tangent sphere at each point x in a real Finsler manifold
is in general a function of x.
Nilpotent Quantum Theory: A review
Peter ROWLANDS
Physics Department, University of Liverpool, UK
[email protected]
Nilpotent quantum theory first appeared in the literature twenty years
ago, and provides an exceptionally streamlined and powerful route to quantum mechanics, quantum field theory and particle physics. It can be derived
in a completely formal way using hypercomplex algebra in place of the usual
matrix formalisms associated with these subjects. Its origin, however, can
be placed twenty years earlier in a much more physically-inspired theory involving symmetries between the fundamental physical parameters. Because
of the additional information contained within these symmetries, in addition
to providing a formalism for reproducing the known results of relativistic
30
FERT 2014
quantum mechanics and the Standard Model of particle physics in an integrated and systematic way, nilpotent quantum theory also generates many
new ones which are not accessible by any other known method. In effect,
the formalisms which are used routinely in these areas of physics are not
there purely for mathematical convenience, but also contain coded physical
information which can be extracted if we can find a more fundamental way
of expressing them. The formalisms generated from the hypercomplex algebra and the related symmetries also create further formalisms with additional
physical information, which connect with areas such as Finsler geometry, creating even further layers of physical meaning. A review of the developments
shows that they produce a coherent and integrated approach to a number of
fundamental questions.
References
[1] Rowlands, P., Zero to Infinity: The Foundations of Physics, World
Scientific, Singapore and Hackensack, NJ, 2007.
[2] Rowlands, P., The fundamental parameters of physics, Speculat. Sci.
Tech. 6 (1983), 69-80.
[3] Rowlands, P., Physical interpretations of nilpotent quantum mechanics, arXiv: 1004.1523 (2010).
[4] Rowlands, P., The Berwald-Moor metric in nilpotent Dirac spinor
space, Bulletin of the Transilvania University of Bra¸sov 4 (2011), 53-66.
Plane waves of torsion in Poincare gauge theory of
gravity
Vladimir N. SHCHERBAN
Moscow State Pedagogical University (MSPU), Department of Physics and
Information Technology, Moscow, Russia
[email protected]
The variational equations of a gravitational field in Riemann-Cartan space
in a formalism of external forms by a method of uncertain Lagrange multipliers are derived for the Poincare gauge theory of gravitation with quadratic
Lagrangians in the general form. The structure of the irreducible parts of
torsion propagating in the form of plane waves in Riemann-Cartan spacetime is investigated.
BOOK OF ABSTRACTS
31
References
[1] Adamovich W. //Gen. Rel. Grav.–1980.–V. 12.–P. 677–691.
[2] Sipper R, Goenner H. //Gen. Relat. Grav.–1986.–V. 18.–P. 12291243.
[3] Babourova O.V., Klimova E.A., Frolov B.N. //Class. Quantum Grav.–
2003.–V. 20.–P. 1423–1441.
[4] Babourova O.V., Frolov B.N., Shcherban V.N. Investigation of plane
torsion waves in the Poincare gauge theory of gravitation //Gravitation Cosmology, 2013, V. 19, No. 3, 144150.
Einstein Metrics
Zhongmin SHEN
Indiana University-Purdue University, USA
[email protected]
Finsler metrics are just metrics without quadratic restriction. The notion
of Ricci curvature in Riemannian geometry can be naturally extended to
Finsler metrics. It is one of important problems to study and characterize
Finsler metrics with isotropic Ricci curvature (Einstein-Finsler metrics). In
this talk I will give a brief survey on Einstein-Finsler metrics and related
topics.
On some properties of space in the Anisotropic
Geometrodynamics
Sergey SIPAROV
State University of Civil Aviation; National Research University of Informational
Technologies, Mechanics and Optics; Research Institute of Hyper Complex
Systems in Geometry and Physics, Russia
[email protected]
Anisotropic Geometrodynamics (AGD) appeared to be a consistent approach in the interpretation of the observed phenomena (like flat rotation
curves in spiral galaxies or Tully-Fisher law) that found no explanation in
32
FERT 2014
the classical GRT or demanded the introduction of such notion as dark matter the substance whose amount in the Universe must be enormous but
which is still not found. It turns out that the AGD approach has a more
general character and can be used also in classical and quantum mechanics
in order to avoid certain controversies.
Analysis of the time series in the space maser signals
Sergey SIPAROV,1
V. SAMODUROV, G. LAPTEV
1
State University of Civil Aviation; National Research University of
Informational Technologies, Mechanics and Optics; Research Institute of Hyper
Complex Systems in Geometry and Physics, Russia
[email protected]
The investigation of the geometrical properties of space-time in our galaxy
requires specific experiment of the galactic scale. We analyze the data of the
observations of the radio sources frequently found in space. They are believed to be the sets of molecular condensations each of which works as a
maser, so that the whole set produces a characteristic spectrum. It turns out
that in some cases the intensity of one of the components of such spectrum
corresponding to a single condensation changes periodically with a period
of dozens of minutes or of hours. The interpretation deals with the effect
of optic-metrical parametric resonance, which includes the action of gravitational waves on a maser. When such observations provide enough data
to obtain statistically valid conclusions, it will be possible to discuss the
space-time geometry.
Gauge theory at singularities
Ovidiu Cristinel STOICA
Department of Theoretical Physics, National Institute of Physics and Nuclear
Engineering – Horia Hulubei, Bucharest, Romania
[email protected]
Building on author’s previous results in singular semi-Riemannian geometry and singular general relativity, the behavior of gauge theory at singularities is analyzed. The usual formulations of the field equations at singularities
BOOK OF ABSTRACTS
33
are accompanied by infinities which block the evolution equations, mainly
because the metric is singular, hence the usual differential operators, constructed from the metric, blow up. However, it is possible to give otherwise
equivalent formulations of the Einstein, Maxwell and Yang-Mills equations,
which in addition admit solutions which can be extended beyond the singularities. The main purpose of this analysis are applications to the black hole
information loss paradox. An alternative formulation can be given in terms
of Kaluza-Klein theory.
References
[1] Kupeli, D.N., Singular semi-Riemannian geometry, Springer, 1996.
[2] Hawking, S.W., Particle creation by black holes, Commun. Math.
Phys. 43, 3(1975): 199-220.
[3] Hawking, S.W., Breakdown of predictability in gravitational collapse,
Phys. Rev. D. 14.10 (1976): 2460.
[4] Stoica, O.C., Singular General Relativity, PhD Thesis, 2013, arXiv:grqc/ 1301.2231.
[5] Stoica, O.C., On singular semi-Riemannian manifolds, Int. J. Geom.
Methods Mod. Phys., 0 (2014), 1450041, arXiv:math.DG/1105.0201.
[6] Stoica, O.C., Cartan’s structural equations for degenerate metric, Balk.
J. Geom. Appl., 19:2 (2014), 118-126, ar-Xiv:math.DG/1111.0646.
[7] Stoica, O.C., Warped products of singular semi-Riemannian manifolds,
arXiv: math.DG/1105.3404.
[8] Stoica, O.C., Schwarzschild singularity is semi-regularizable, Eur. Phys.
J. Plus 127 (2012), no. 83, 18, arXiv:gr-qc/1111.4837.
[9] Stoica, O.C., Analytic Reissner-Nordstr¨om singularity, Phys. Scr. 85
(2012), no. 5, 055004, arXiv:gr-qc/1111.4332.
[10] Stoica, O.C., Kerr-Newman solutions with analytic singularity and
no closed timelike curves, To appear in U.P.B. Sci. Bull., Ser. A (2013),
arXiv:gr-qc/1111.7082.
[11] Stoica, O.C., Spacetimes with Singularities, An. S¸t. Univ. Ovidius
Constant¸a 20 (2012), no. 2, 213238, arXiv:gr-qc/1108.5099.
[12] Stoica, O.C., Big Bang singularity in the Friedmann-Lemaˆıtre-Robertson-Walker spacetime, arXiv:gr-qc/1112.4508.
[13] Stoica, O.C., Beyond the Friedmann-Lemaˆıtre-Robertson-Walker Big
Bang singularity, Commun. Theor. Phys. 58 (2012), no. 4, 613616,
arXiv:gr-qc/1203.1819.
[14] Stoica, O.C., On the Weyl curvature hypothesis, Ann. of Phys. 338
(2013), 186194, arXiv:gr-qc/1203.3382.
34
FERT 2014
[15] Stoica, O.C., Einstein equation at singularities, Central European
Journal of Physics 12 (2014), 123131, arXiv:gr-qc/1203.2140.
[16] Stoica, O.C., Metric dimensional reduction at singularities with implications to quantum gravity, Annals of Physics 347C (2014), pp. 74-91
(2014), arXiv:gr-qc/1205.2586.
[17] Stoica, O.C., The Geometry of Black Hole Singularities, Adv. in High
Energy Physics 14 (2014), http://www.hindawi.com/journals/ahep/2014/
907518/.
Beil metrics in complex Finsler geometry
´
Annam´
aria SZASZ
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
In this paper we continue the study of the complex Beil metrics, in complex Finsler geometry. Primarily, we determine the main geometric objects
corresponding to these metrics, (e.g. the Chern-Finsler complex nonlinear
connection, the Chern-Finsler complex linear connection and the holomorphic curvature). We study when a complex Finsler space endowed with a
complex Beil metric becames weakly K¨ahler and K¨ahler. Also, we prove
that the base complex Finsler metric is projectively related to the associated
complex Beil metric. As an application of this theory, we set the variational
problem of the complex Beil metric constructed with the weakly gravitational metric. In this case we determine a complex nonlinear coonnection of
Lorentz type.
Differentiable distance spaces
1
´
L´
ajos TAMASSY,
1
´
D´
avid Cs. KERTESZ
Math. Inst. of Debrecen Univ., Hungary
[email protected]
The family of the Finsler spaces {F n = (M, F )} is a subclass of the family
of the distance spaces {Dn = (M, %)}. The most important difference is that
the distance function % : M × M → R+ need not to be differentiable. We
BOOK OF ABSTRACTS
35
investigate local properties of differentiable distance spaces: % ∈ C ∞ . Our
investigations are local, so we can assume that M = Rn . Many properties of
Finsler spaces can be transferred to such distance spaces, yet these distance
spaces are no Finsler spaces (in general). Each of these distance spaces
determines an F n , but the relation is not 1:1. We investigate the geodesics,
arc length and projective flatness. We widely use geodesic and distance
spheres, and often apply direct geometric considerations.
Geometry of Quaternions and Octonions
Anatoly TURBIN,1
1
Yuliya ZHDANOVA2
Institute of Mathematics of NAN, National Pedagogical Dragomanov
University, Kiev, Ukraine
[email protected]
2 State University of Telecommunications, Kiev, Ukraine
[email protected]
According to the Schlafli’s theorem [1], there are six (and only!) regular
polyhedra in E 4 and there are three (and only!) regular polyhedra in E n , n ≥
5. Regular polyhedra in E 4 can be considered as a convex hull of special
quaternions. Regular polyhedra in E 8 can be considered as a convex hull of
special octonions.
Icosians are the special quaternions. Icosians form a group, the order of
which is equal to 120. The convex hull of a prime Icosians is the star regular
24-hedron (24, 96, 96, 24) in E 4 , at which 3-faces are icosahedrons and star
of any vertex is icosahedron also (J.Gregory–A.Pogorelov megaicosahedron).
The convex hull of the 5 quaternions, which are the vertices of hypertetrahedron (5, 10, 10, 5) and the same quaternions, taken with the opposite
sign is the regular 10-hedron (10, 20, 20, 10) in E 4 , at which 3-faces are cubes
(A.Milka megacubohedron).
√
The convex hull of whole quaternions with the norm equal to 14 is the
regular 96-hedron (192, 384, 288, 96), at which 3-faces are cubes (A.Skorokhod
megacubohedron).
√
The convex hull of the 112 whole octonions with the norm equal to 2
is the regular 112-hedron in E 8 , at which 3-faces are octahedra, k-faces,
4 ≤ k ≤ 7, are G.Brand megaoctahedra (I.Newton–N.Kuzehnnyj contact
megaoctahedron).
√
It is 112 (not 240!) hyperspheres of radius 2 , located on the vertices
of I.Newton–N.Kuzehnnyj contact megaoctahedron, inscribed in the hypersphere of the same radius, touch each other (the solution of the sphere packing
36
FERT 2014
problem in E 8 ).
References
[1] Berger M., Geometry II, Springer, 2009, 416 pages.
Comparing variants of single-time stochastic maximum
principle
Constantin UDRIS
¸ TE
University Politehnica of Bucharest, Romania
[email protected]
This paper is concerned with comparison between the well-known forward
- backward stochastic maximum principle and the simplified stochastic maximum principle realized in ours papers via geometrical ingredients. Detailed
examples illuminate our ideas and certify that we are on track in explaining
some aspects of stochastic theory. In short, we point out that, despite a
stochastic optimal control problem has a unique solution, finding this solution techniques can be different from one work to another due to the diffusion
parts of stochastic processes.
The inverse problem of the calculus of variations for
systems of homogeneous differential equations
Zbynek URBAN
Lepage Research Institute, 78342 Slatinice, Czech Rep.
[email protected]
Higher-order systems of ordinary differential equations on manifolds of
regular velocities are studied under assumptions of variationality and (higherorder) homogeneity. In particular, we analyse the structure of second-order
systems with the corresponding Vainberg-Tonti Lagrangian.
BOOK OF ABSTRACTS
37
Generalized Riemannian Metrics and Tangent Bundle
Geometry
Izu VAISMAN
University of Haifa, Mt. Carmel, Haifa, 31905, Israel
[email protected]
Motivated by string theory, generalized geometry is a subject that was developed in the last two decades, starting with foundational papers of Hitchin,
Gualtieri and others. Basically, it is the study of geometric structures on the
bundle T M ⊕T ∗ M with the para-Hermitian structure given by the direct sum
and the natural pairing metric (M is a differentiable manifold). Integrability
of such structures is defined by means of the Courant bracket.
In the talk, we will recall the various presentations of a generalized metric,
i.e., a (pseudo-)Euclidean metric that is compatible with the metric of a paraHermitian vector space or bundle. Then, we will consider such metrics on the
pullback bundle π −1 (T M ⊕T ∗ M ), where π : TM → M is the projection of the
tangent bundle of M . The result is a new class of metrics on a tangent bundle,
which generalize the classical Sasaki metric. At the end (if time permits),
we will briefly discuss the geometry of a big-tangent manifold, defined as the
total space of the bundle T M ⊕ T ∗ M .
On field-theoretical integrals in Finslerian spacetimes
Nicoleta VOICU
Transilvania University of Bra¸sov, Faculty of Mathematics and Computer
Science, Romania
[email protected]
On Finslerian space-time manifolds, the integrand of a field-theoretical action generally depends on both the positional and on the directional variables,
thus becoming a quantity defined on some domain of the tangent bundle of
the given manifold. But, due to the space-time signature of the Finslerian
metric tensor, there appear problems in obtaining a well-defined notion of
volume and of a canonical integration domain. Under the assumption that
the space-time manifold is globally hyperbolic, we propose a way of overcoming these difficulties, based on an extension of the idea of Holmes-Thompson
volume form.
38
FERT 2014
The eigenvalue problem in Finsler geometry
Qiaoling XIA
Department of Mathematics, Zhejiang University, Hangzhou, P.R. China
[email protected]
How to estimate the lower or upper bound for the first (nonzero) eigenvalue of the Laplacian is one of the fundamental question in geometric analysis. In this talk, we will introduce the eigenvalue problem in Finsler geometry
and give a lower bound for the first (nonzero) eigenvalue on a compact Finsler
manifold M without boundary or with convex boundary under the assumption that the weighted Ricci curvature RicN (M ) ≥ K for some real numbers
K and N ∈ [n; ∞]. In particular, we give a sharp lower bound for the first
(nonzero) eigenvalue on such a Finsler manifold with RicN ≥ 0.
Characterizations of complex Finsler connections and
weakly complex Berwald metrics
Chunping ZHONG
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian, P.R. China
[email protected]
In this talk, I shall give characterizations of complex Rund connection,
complex Berwald connection, and complex Hashiguchi connection. I shall
introduce the notion of weakly complex Berwald metric and show that the
complex Wrona metric is a weakly complex Berwald metric in our sense. I
shall also give a characterization of weakly complex Berwald metric under
the condition that it is also a strongly convex weakly K¨ahler-Finsler metric.
BOOK OF ABSTRACTS
39
A new point of view on (1 + 3) threading of spacetime
Aurel BEJANCU
Kuwait University, Faculty of Science, Department of Mathematics, Kuwait
[email protected]
40
FERT 2014
.