Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
Aurelian Isar - Decoherence And Transition From Quantum To Classical In Open ...SEENET-MTP
Lecture by Prof. Dr. Aurelian Isar (Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania) on October 20, 2011 at the Faculty of Science and Mathematics, Nis, Serbia.
NITheP WITS node Seminar by Dr Dr. Roland Cristopher F. Caballar (NITheP/UKZN)
TITLE: "One-Dimensional Homogeneous Open Quantum Walks"
ABSTRACT: In this talk, we consider a system undergoing an open quantum walk on a one-dimensional lattice. Each jump of the system between adjacent lattice points in a given direction corresponds to a jump operator, with these jump operators either commuting or not commuting. We examine the dynamics of the system undergoing this open quantum walk, in particular deriving analytically the probability distribution of the system, as well as examining numerically the behavior of the probability distribution over long time steps. The resulting distribution is shown to have multiple components, which fall under two general categories, namely normal and solitonic components. The analytic computation of the probability distribution for the system undergoing this open quantum walk allows us to determine at any instant of time the dynamical properties of the system.
Generally it has been noticed that differential equation is solved typically. The Laplace transformation makes it easy to solve. The Laplace transformation is applied in different areas of science, engineering and technology. The Laplace transformation is applicable in so many fields. Laplace transformation is used in solving the time domain function by converting it into frequency domain. Laplace transformation makes it easier to solve the problems in engineering applications and makes differential equations simple to solve. In this paper we will discuss how to follow convolution theorem holds the Commutative property, Associative Property and Distributive Property. Dr. Dinesh Verma"Application of Convolution Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14172.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/14172/application-of-convolution-theorem/dr-dinesh-verma
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
The time scale Fibonacci sequences satisfy the Friedmann-Lema\^itre-Robertson-Walker (FLRW) dynamic equation on time scale, which are an exact solution of Einstein's field equations of general relativity for an expanding homogeneous and isotropic universe. We show that the equations of motion correspond to the one-dimensional motion of a particle of position $F(t)$ in an inverted harmonic potential. For the dynamic equations on time scale describing the Fibonacci numbers $F(t)$, we present the Lagrangian and Hamiltonian formalism. Identifying these with the equations that describe factor scales, we conclude that for a certain granulation, for both the continuous and the discrete universe, we have the same dynamics.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
We understand that you're a college student and finances can be tight. That's why we offer affordable pricing for our online statistics homework help. Your future is important to us, and we want to make sure you can achieve your degree without added financial stress. Seeking assistance with statistics homework should be simple and stress-free, and that's why we provide solutions starting from a reasonable price.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
if you are struggling with your Multiple Linear Regression homework, do not hesitate to seek help from our statistics homework help experts. We are here to guide you through the process and ensure that you understand the concept and the steps involved in performing the analysis. Contact us today and let us help you ace your Multiple Linear Regression homework!
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com. You can also call +1 (315) 557-6473 for assistance with Statistics Homework.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
The increased availability of biomedical data, particularly in the public domain, offers the opportunity to better understand human health and to develop effective therapeutics for a wide range of unmet medical needs. However, data scientists remain stymied by the fact that data remain hard to find and to productively reuse because data and their metadata i) are wholly inaccessible, ii) are in non-standard or incompatible representations, iii) do not conform to community standards, and iv) have unclear or highly restricted terms and conditions that preclude legitimate reuse. These limitations require a rethink on data can be made machine and AI-ready - the key motivation behind the FAIR Guiding Principles. Concurrently, while recent efforts have explored the use of deep learning to fuse disparate data into predictive models for a wide range of biomedical applications, these models often fail even when the correct answer is already known, and fail to explain individual predictions in terms that data scientists can appreciate. These limitations suggest that new methods to produce practical artificial intelligence are still needed.
In this talk, I will discuss our work in (1) building an integrative knowledge infrastructure to prepare FAIR and "AI-ready" data and services along with (2) neurosymbolic AI methods to improve the quality of predictions and to generate plausible explanations. Attention is given to standards, platforms, and methods to wrangle knowledge into simple, but effective semantic and latent representations, and to make these available into standards-compliant and discoverable interfaces that can be used in model building, validation, and explanation. Our work, and those of others in the field, creates a baseline for building trustworthy and easy to deploy AI models in biomedicine.
Bio
Dr. Michel Dumontier is the Distinguished Professor of Data Science at Maastricht University, founder and executive director of the Institute of Data Science, and co-founder of the FAIR (Findable, Accessible, Interoperable and Reusable) data principles. His research explores socio-technological approaches for responsible discovery science, which includes collaborative multi-modal knowledge graphs, privacy-preserving distributed data mining, and AI methods for drug discovery and personalized medicine. His work is supported through the Dutch National Research Agenda, the Netherlands Organisation for Scientific Research, Horizon Europe, the European Open Science Cloud, the US National Institutes of Health, and a Marie-Curie Innovative Training Network. He is the editor-in-chief for the journal Data Science and is internationally recognized for his contributions in bioinformatics, biomedical informatics, and semantic technologies including ontologies and linked data.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
2. Nonlocal
cosmology
Ivan Dimitrijevi´c
Motivation
Large cosmological observational findings:
High orbital speeds of galaxies in clusters. (F.Zwicky, 1933)
High orbital speeds of stars in spiral galaxies. (Vera Rubin, at
the end of 1960es)
Accelerated expansion of the Universe. (1998)
3. Nonlocal
cosmology
Ivan Dimitrijevi´c
Problem solving approaches
There are two problem solving approaches:
Dark matter and energy
Modification of Einstein theory of gravity
Rµν − 1
2 Rgµν = 8πGTµν − Λgµν, c = 1
where Tµν is stress-energy tensor, gµν are the elements of the metric
tensor, Rµν is Ricci tensor and R is scalar curvature of metric.
4. Nonlocal
cosmology
Ivan Dimitrijevi´c
Dark matter and energy
If Einstein theory of gravity can be applied to the whole Universe
then the Universe contains about 4.9% of ordinary matter,
26.8% of dark matter and 68.3% of dark energy.
It means that 95.1% of total matter, or energy, represents dark
side of the Universe, which nature is unknown.
Dark matter is responsible for orbital speeds in galaxies, and dark
energy is responsible for accelerated expansion of the Universe.
5. Nonlocal
cosmology
Ivan Dimitrijevi´c
Modification of Einstein theory of gravity
Motivation for modification of Einstein theory of gravity
The validity of General Relativity on cosmological scale is not
confirmed.
Dark matter and dark energy are not yet detected in the
laboratory experiments.
Another cosmological problem is related to the Big Bang
singularity. Namely, under rather general conditions, general
relativity yields cosmological solutions with zero size of the
universe at its beginning, what means an infinite matter density.
Note that when physical theory contains singularity, it is not
valid in the vicinity of singularity and must be appropriately
modified.
6. Nonlocal
cosmology
Ivan Dimitrijevi´c
Approaches to modification of Einstein theory of
gravity
There are different approaches to modification of Einstein theory of
gravity.
Einstein General Theory of Relativity
From action S = (
R
16πG
− Lm − 2Λ)
√
−gd4
x using variational
methods we get field equations
Rµν − 1
2 Rgµν = 8πGTµν − Λgµν, c = 1
where Tµν is stress-energy tensor, gµν are the elements of the metric
tensor, Rµν is Ricci tensor and R is scalar curvature of metric.
Currently there are mainly two approaches:
f(R) Modified Gravity
Nonlocal Gravity
7. Nonlocal
cosmology
Ivan Dimitrijevi´c
Nonlocal Modified Gravity
Nonlocal gravity is a modification of Einstein general relativity in
such way that Einstein-Hilbert action contains a function f( , R).
Our action is given by
S = d4
x
√
−g
R − 2Λ
16πG
+ P(R)F( )Q(R)
where = 1√
−g
∂µ
√
−ggµν
∂ν, F( ) =
∞
n=0
fn
n
.
We use Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric
ds2
= −dt2
+ a2
(t) dr2
1−kr2 + r2
dθ2
+ r2
sin2
θdφ2
, k ∈ {−1, 0, 1}.
8. Nonlocal
cosmology
Ivan Dimitrijevi´c
Equations of motion
Equation of motion are
−
1
2
gµνP(R)F( )Q(R) + RµνW − KµνW +
1
2
Ωµν = −
Gµν + Λgµν
16πG
,
Ωµν =
∞
n=1
fn
n−1
l=0
Sµν
l
P(R), n−1−l
Q(R) ,
Kµν = µ ν − gµν ,
Sµν(A, B) = gµν
α
A αB − 2 µA νB + gµνA B,
W = P (R)F( )Q(R) + Q (R)F( )P(R).
9. Nonlocal
cosmology
Ivan Dimitrijevi´c
Trace and 00-equations
In case of FRW metric there are two linearly independent equations.
The most convenient choice is trace and 00 equations:
− 2P(R)F( )Q(R) + RW + 3 W +
1
2
Ω =
R − 4Λ
16πG
,
1
2
P(R)F( )Q(R) + R00W − K00W +
1
2
Ω00 = −
G00 − Λ
16πG
,
Ω = gµν
Ωµν.
10. Nonlocal
cosmology
Ivan Dimitrijevi´c
Cosmological solutions in nonlocal modified gravity
We discuss the class of models given by the action
S =
M
R − 2Λ
16πG
+ P(R)F( )Q(R)
√
−g d4
x.
In particular, we investigate the following cases:
P = R, Q = R,
P = R−1
, Q = R,
P = Rp
, Q = Rq
, p ∈ N, q ∈ N,
P = (R + R0)m
, Q = (R + R0)m
, m ∈ R, R0 ∈ R
11. Nonlocal
cosmology
Ivan Dimitrijevi´c
First case P = R, Q = R
In this the action becomes
S =
M
R − 2Λ
16πG
+ RF( )R
√
−g d4
x.
This model is interesting since we obtain nonsingular cosmological
solutions. To solve EOM we use the ansatz:
R = rR + s, r, s ∈ R
12. Nonlocal
cosmology
Ivan Dimitrijevi´c
Nonsingular bounce cosmological solutions
Let the scale factor a(t) be in a form
a(t) = a0(σeλt
+ τe−λt
), a0 > 0, λ, σ, τ ∈ R.
Theorem
Scale factor a(t) = a0(σeλt
+ τe−λt
) is a solution of EOM in a
following three cases:
Case 1.
F 2λ2
= 0, F 2λ2
= 0, f0 = −
1
128πGCΛ
.
Case 2. 3k = 4a2
0Λστ.
Case 3.
F 2λ2
=
1
192πGCΛ
+
2
3
f0, F 2λ2
= 0, k = −4a2
0Λστ.
In all cases we have 3λ2
= Λ.
13. Nonlocal
cosmology
Ivan Dimitrijevi´c
Summary, P = R, Q = R
We studied nonlocal gravity model with cosmological constant
Λ, without matter.
Using anzac R = rR + s we found three types of nonsingular
bounce cosmogical solutions with scale factor
a(t) = a0(σeλt
+ τe−λt
).
All solutions satisfy
¨a(t) = λ2
a(t) > 0.
There are solutions for all values of parameter k = 0, ±1.
14. Nonlocal
cosmology
Ivan Dimitrijevi´c
Second case P = R−1
, Q = R
This model is given by the action
S =
M
R
16πG
+ R−1
F( )R
√
−g d4
x,
where F( ) =
∞
n=0
fn
n
. If we set f0 = − Λ
8πG , then f0 takes the
place of cosmological constant.
15. Nonlocal
cosmology
Ivan Dimitrijevi´c
Second case P = R−1
, Q = R
This model is given by the action
S =
M
R
16πG
+ R−1
F( )R
√
−g d4
x,
where F( ) =
∞
n=0
fn
n
. If we set f0 = − Λ
8πG , then f0 takes the
place of cosmological constant.
16. Nonlocal
cosmology
Ivan Dimitrijevi´c
Some power-law cosmological solutions
We use the following form of scale factor and scalar curvature:
a(t) = a0|t − t0|α
,
R(t) = 6 α(2α − 1)(t − t0)−2
+
k
a2
0
(t − t0)−2α
.
17. Nonlocal
cosmology
Ivan Dimitrijevi´c
Case k = 0, α = 0 i α = 1
2
Theorem
For k = 0, α = 0, α = 1
2 and 3α−1
2 ∈ N scale factor a = a0|t − t0|α
is
a solution of EOM if
f0 = 0, f1 = −
3α(2α − 1)
32πG(3α − 2)
,
fn = 0 za 2 ≤ n ≤
3α − 1
2
,
fn ∈ R za n >
3α − 1
2
.
18. Nonlocal
cosmology
Ivan Dimitrijevi´c
Case k = 0, α → 0 (Minkowski spacetime)
Theorem
For k = 0 and α → 0 EOM are satisfied if
f0, f1 ∈ R, fi = 0, i ≥ 2.
Since all fn = 0, n ≥ 2 this model is not a nonlocal model. Thus the
power-law solutions cannot be obtained by perturbing the Minkowski
solution.
20. Nonlocal
cosmology
Ivan Dimitrijevi´c
Case k = 0, α = 1
Let k = 0. To simplify the scalar curvature
R(t) = 6 α(2α − 1)(t − t0)−2
+
k
a2
0
(t − t0)−2α
we have three options: α = 0, α = 1
2 i α = 1. The first two does not
yield any solutions of EOM and the last one is described in the following
theorem.
Theorem
For k = 0 scale factor a = a0|t − t0| is a solution of EOM if
f0 = 0, f1 =
−s
64πG
, fn ∈ R, n ≥ 2,
where s = 6(1 + k
a2
0
).
21. Nonlocal
cosmology
Ivan Dimitrijevi´c
Summary, P = R−1
, Q = R
We the model with nonlocal term R−1
F( )R and obtained
power-law cosmological solutions a(t) = a0|t − t0|α
.
It is worth noting that there is a solution a(t) = |t − t0| which
corresponds to Milne universe for k = −1.
All presented solutions a(t) = a0|t − t0|α
have scalar curvature
R(t) = 6 α(2α − 1)(t − t0)−2
+ k
a2
0
(t − t0)−2α
, that satisfies
R = qR2
, where parameter q depends on α.
23. Nonlocal
cosmology
Ivan Dimitrijevi´c
Third case P = Rp
, Q = Rq
We discuss only k = 0. Scale factor is chosen in the form
a(t) = a0e− γ
12 t2
, γ ∈ R.
Hubble parameter and scalar curvature are given by
H(t) = −
1
6
γt, R(t) =
1
3
γ(γt2
− 3), R00 =
1
4
(γ − R).
24. Nonlocal
cosmology
Ivan Dimitrijevi´c
Third case P = Rp
, Q = Rq
Rp
= pγRp
−
p
3
(4p − 5)γ2
Rp−1
−
4
3
p(p − 1)γ3
Rp−2
.
From this relation it follows that operator is closed on the space of
polynomials in R of degree at most p. In the basis
vp = Rp
Rp−1
. . . R 1
T
the matrix of is
Mp = γ
p p
3
(5 − 4p)γ 4
3
p(1 − p)γ2
0 . . . 0
0 p − 1 p−1
3
(9 − 4p)γ 4
3
(1 − p)(p − 2)γ2
. . . 0
...
...
...
...
...
0 0 0 . . . 1 γ
3
0 0 0 . . . 0 0
.
25. Nonlocal
cosmology
Ivan Dimitrijevi´c
EOM
Moreover, Fp =
∞
n=0
fnMn
p is a matrix of operator F( ). Let Dp be a
matrix of operator
∂
∂R
and ep are the coordinates of vector Rp
in basis
vp.
Trace and 00 equation are transformed into
T = 0, Z = 0,
where
T = −2epvpeqFqvq + RWpq − 4γ2
(R + γ)Wpq − 2γ2
Wpq
− S1 + 2S2 −
R − 4Λ
16πG
,
Z =
1
2
epvpeqFqvq +
γ
4
(γ − R)Wpq − γ(R + γ)Wpq
−
1
2
(S1 + S2) +
G00 − Λ
16πG
,
27. Nonlocal
cosmology
Ivan Dimitrijevi´c
EOM
Theorem
For all p ∈ N and q ∈ N we have T + 4Z = 4γZ . Trace and 00
equations are equivalent.
Therefore, it is sufficient to solve only trace equation.
It is of a polynomial type, degree p + q in R, and it splits into
p + q + 1 equations with p + q + 1 ”variables” f0 = F(0), F(γ), . . . ,
F(pγ), F (γ), . . . , F (qγ).
29. Nonlocal
cosmology
Ivan Dimitrijevi´c
(p, q) = (1, 1)
Theorem
Trace equation is satisfied for the following values of parameters p and q
(κ = 1
16πG ):
p = 2, q = 1: F(γ) = 9κ(γ+9Λ)
112γ3 , F(2γ) = 3κ(γ+9Λ)
56γ3 ,
f0 = −κ(4γ+15Λ)
7γ3 , F (γ) = −3κ(γ+9Λ)
8γ4 ,
p = 2, q = 2: F(γ) = 369κ(γ+8Λ)
9344γ4 , F(2γ) = 27κ(γ+8Λ)
4672γ4 ,
f0 = κ(145γ+576Λ)
876γ4 , F (γ) = −639κ(γ+8Λ)
2336γ5 , F (2γ) = −27κ(γ+8Λ)
9344γ5 ,
p = 3, q = 1: F(γ) = κ(107γ+408Λ)
6432γ4 , F(2γ) = −κ(173γ+840Λ)
7504γ4 ,
F(3γ) = 0, f0 = −κ(95γ+768Λ)
268γ4 , F (γ) = −9κ(γ+8Λ)
88γ5 .
p = 3, q = 2: F(γ) = 3κ(10702γ+40497Λ)
245680γ5 , F(2γ) = −27κ(6γ+25Λ)
24568γ5 ,
F(3γ) = −27κ(6γ+25Λ)
49136γ5 , f0 = −3κ(7099γ+23949Λ)
15355γ5 ,
F (γ) = −3κ(11614γ+68865Λ)
270248γ6 , F (2γ) = 513κ(6γ+25Λ)
171976γ6 ,
30. Nonlocal
cosmology
Ivan Dimitrijevi´c
Summary, P = Rp
, Q = Rq
, p ∈ N, q ∈ N
We discuss the scale factor of the form a(t) = a0 exp(− γ
12 t2
)
Trace and 00 equations are equivalent. Trace equation is of a
degree p + q in R, so it splits into p + q + 1 equations over
p + q + 1 ”variables” f0 = F(0), F(γ), . . . , F(pγ), F (γ), . . . ,
F (qγ).
For p = q = 1 the system has infinitely many solutions, and
constants γ and Λ satisfy γ = −12Λ.
For other values of p and q, there is unique solution, for any
γ ∈ R.
We obtained solutions for 1 ≤ q ≤ p ≤ 4.
31. Nonlocal
cosmology
Ivan Dimitrijevi´c
Fourth case P(R) = (R + R0)m
, Q(R) = (R + R0)m
We take that P(R) = (R + R0)m
, Q(R) = (R + R0)m
S =
M
R − 2Λ
16πG
+ (R + R0)m
F( )(R + R0)m √
−g d4
x.
Scale factor is in the form
a(t) = A tn
e− γ
12 t2
.
32. Nonlocal
cosmology
Ivan Dimitrijevi´c
Fourth case P(R) = (R + R0)m
, Q(R) = (R + R0)m
Also, we take the ansatz
(R + R0)m
= r(R + R0)m
, (1)
where R0,r, m i n realne konstante.
Ansatz in the expanded for is
− 648mn2
(2n − 1)2
(2m − 3n + 1) = 0,
− 324n(2n − 1) −γm + 6γmn2
− 4γmn − mnR0 + mR0 + 2n2
r − nr = 0,
18n(2n − 1) 8γ2
m2
− 13γ2
m + 12γ2
mn − 3γmR0 + 24γnr + 6γr − 6rR0 = 0,
− 2γ3
m − 24γ3
mn2
− 14γ3
mn + 6γ2
mnR0 + 2γ2
mR0 + 72γ2
n2
r + 12γ2
nr
− 24γnrR0 + 3γ2
r − 6γrR0 + 3rR2
0 = 0,
− γ2
4γ2
m2
+ γ2
m + 18γ2
mn − 3γmR0 − 24γnr − 6γr + 6rR0 = 0,
− γ4
(r − γm) = 0.
There are five solutions of the above system
1 r = mγ, n = 0, R0 = γ, m = 1
2
2 r = mγ, n = 0, R0 = γ
3 , m = 1
3 r = mγ, n = 1
2 , R0 = 4
3 γ, m = 1
4 r = mγ, n = 1
2 , R0 = 3γ, m = −1
4
5 r = mγ, n = 2m+1
3 , R0 = 7
3 γ, m = 1
2 .
33. Nonlocal
cosmology
Ivan Dimitrijevi´c
Case n = 0, m = 1
2
Trace and 00 equations are linear in R, and they split into two
systems:
γ
16πG
+
Λ
4πG
− γF(
γ
2
) −
γ2
3
F (
γ
2
) = 0,
−
108γ2
πG
−
γ2
3
F(
γ
2
) +
γ3
3
F (
γ
2
) = 0.
Λ
16πG
−
γ
2
F(
γ
2
) +
γ2
6
F (
γ
2
) = 0,
27γ2
πG
+
γ2
12
F(
γ
2
) +
γ3
6
F (
γ
2
) = 0.
The solution of the above system is
F(
γ
2
) =
24γ + Λ
768γπG
, F (
γ
2
) =
3Λ − 3γ
16γ2πG
.
34. Nonlocal
cosmology
Ivan Dimitrijevi´c
Case n = 2
3, m = 1
2
Trace and 00 equations split into the following systems:
F (
γ
2
) = 0,
11γ
48πG
+
Λ
4πG
− γF(
γ
2
) = 0,
1
16πG
+ F(
γ
2
) = 0,
γ2
16πG
+ γ2
F(
γ
2
) = 0,
i
F (
γ
2
) = 0, −
γ
24πG
−
Λ
16πG
+
1
2
γF(
γ
2
) = 0,
1
16πG
+ cF(
γ
2
) = 0,
γ2
16πG
+ cγ2
F(
γ
2
) = 0.
The solution is
F(
γ
2
) =
−1
16πG
, F (
γ
2
) = 0, Λ = −
7
6
γ.
35. Nonlocal
cosmology
Ivan Dimitrijevi´c
Symmary, P(R) = (R + R0)m
, Q(R) = (R + R0)m
We take the scale factor in the form a(t) = Atn
exp(− γ
12 t2
).
Ansatz is in the form (R + R0)m
= r(R + R0)m
.
Ansatz has five solutions, from witch four satisfy EOM.
In case n = 0, m = 1
2 there is unique solution in F(γ
2 ) and
F (γ
2 ).
In case n = 2
3 , m = 1
2 there is unique solution in F(γ
2 ) and
F (γ
2 ) where Λ = −7
6 γ.
In case n = 1
2 , m = −1
4 EOM have no solution.
36. Nonlocal
cosmology
Ivan Dimitrijevi´c
References
J. Grujic. On a new model of nonlocal modified gravity. Kragujevac
Journal of Mathematics, 2015.
I. Dimitrijevic Cosmological solutions in modified gravity with
monomial nonlocality, Aplied Mathematics and computation 285,
195-203, 2016
Branko Dragovich. On nonlocal modified gravity and cosmology. In
Lie Theory and Its Applications in Physics, volume 111 of Springer
Proceedings in Mathematics and Statistics, pages 251-262, 2014.
Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic.
Constant Curvature Cosmological Solutions in Nonlocal Gravity. In
Bunoiu, OM and Avram, N and Popescu, A, editor, TIM 2013
PHYSICS CONFERENCE, volume 1634 of AIP Conference
Proceedings, pages 18-23. W Univ Timisoara, Fac Phys, 2014. TIM
2013 Physics Conference, Timisoara, ROMANIA, NOV 21-24, 2013.
Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic
On Nonlocal Modified Gravity and its Cosmological Solutions In Lie
Theory and Its Applications in Physics, Springer Proceedings in
Mathematics and Statistics, 2016.
37. Nonlocal
cosmology
Ivan Dimitrijevi´c
References
Dimitrijevic, I., Dragovich, B., Grujic J., Rakic, Z.: New
cosmological solutions in nonlocal modified gravity. Rom. Journ.
Phys. 58 (5-6), 550559 (2013) [arXiv:1302.2794 [gr-qc]]
I. Dimitrijevic, B.Dragovich, J. Grujic, Z. Rakic, “A new model of
nonlocal modified gravity”, Publications de l’Institut Mathematique
94 (108), 187–196 (2013).
Ivan Dimitrijevic, Branko Dragovich, Jelena Grujic, and Zoran Rakic.
On Modified Gravity. Springer Proc.Math.Stat., 36:251-259, 2013.
I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some
power-law cosmological solutions in nonlocal modified gravity. In Lie
Theory and Its Applications in Physics, volume 111 of Springer
Proceedings in Mathematics and Statistics, pages 241-250, 2014.
I. Dimitrijevic, B. Dragovich, J. Grujic, and Z. Rakic. Some
cosmological solutions of a nonlocal modified gravity. Filomat, 2014.