2,403 research outputs found
The Triangular Numbers in Actions
The triangular numbers are formed by the partial sum of the series 1+2+3+4+5+6+7….+n [2]. In other words, triangular numbers are those counting numbers that can be written as Tn = 1+2+3+…+ n. So,
T1= 1
T2= 1+2=3
T3= 1+2+3=6
T4= 1+2+3+4=10
T5= 1+2+3+4+5=15
T6= 1+2+3+4+5+6= 21
T7= 1+2+3+4+5+6+7= 28
T8= 1+2+3+4+5+6+7+8= 36
T9=1+2+3+4+5+6+7+8+9=45
T10 =1+2+3+4+5+6+7+8+9+10=55
In this paper, we investigate some important properties of triangular numbers. Some important results dealing with the mathematical concept of triangular numbers will be proved. We try our best to give short and readable proofs. Most of the results are supplemented with examples.  
Third quantization of -type gravity
We examine the third quantization of -type gravity, based on its
effective Lagrangian in the case of a flat Friedmann-Lemaitre-Robertson-Walker
metric. Starting from the effective Lagrangian, we execute a suitable change of
variable and the second quantization, and we obtain the Wheeler-DeWitt
equation. The third quantization of this theory is considered. And the
uncertainty relation of the universe is investigated in the example of
-type gravity, where . It is shown, when the time is late
namely the scale factor of the universe is large, the spacetime does not
contradict to become classical, and, when the time is early namely the scale
factor of the universe is small, the quantum effects are dominating.Comment: 9 pages, Arbitrary constants in (4.19) are changed to arbitrary
functions of . Conclusions are not changed. References are added.
Typos are correcte
Phase growth in bistable systems with impurities
A system of coupled chaotic bistable maps on a lattice with randomly
distributed impurities is investigated as a model for studying the phenomenon
of phase growth in nonuniform media. The statistical properties of the system
are characterized by means of the average size of spatial domains of equivalent
spin variables that define the phases. It is found that the rate at which phase
domains grow becomes smaller when impurities are present and that the average
size of the resulting domains in the inhomogeneous state of the system
decreases when the density of impurities is increased. The phase diagram
showing regions where homogeneous, heterogeneous, and chessboard patterns occur
on the space of parameters of the system is obtained. A critical boundary that
separates the regime of slow growth of domains from the regime of fast growth
in the heterogeneous region of the phase diagram is calculated. The transition
between these two growth regimes is explained in terms of the stability
properties of the local phase configurations. Our results show that the
inclusion of spatial inhomogeneities can be used as a control mechanism for the
size and growth velocity of phase domains forming in spatiotemporal systems.Comment: 7 pages, 12 figure
Equivalence Principle in Cosmology
We analyse the Einstein equivalence principle (EEP) for a Hubble observer in
Friedmann-Lemaitre-Robertson-Walker spacetime. We show that the affine
structure of light cone in the FLRW spacetime should be treated locally in
terms of the optical metric which is not reduced to the Minkowski metric due to
the non-uniform parametrization of the local equations of light propagation
with the proper time of the observer's clock. The physical consequence of this
difference is that the Doppler shift of radio waves measured locally, is
affected by the Hubble expansion.Comment: 4 pages, no figures. Presented at the Sixth Meeting on CPT and
Lorentz Symmetry, Bloomington, Indiana, June 17-21, 201
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