We provide a theory of n-scales previously called as n dimensional time scales. In previous approaches to the theory of time scales, multi-dimensional scales were taken as product space of two time scales [1, 2]. n-scales make the mathematical structure more flexible and appropriate to real world applications in physics and related fields. Here we define an n-scale as an arbitrary closed subset of ℝn. Modified forward and backward jump operators, Δ-derivatives and Δ-integrals on n-scales are defined.

1.
M.
Bohner
and
G.
Guseino
,
Dynamic systems and applications
14
, p.
579
(
2005
).
2.
M.
Bohner
and
G. S.
Guseinov
,
Dynamic Systems and Applications
19
, p.
435
(
2010
).
3.
B.
Aulbach
and
S.
Hilger
,
Non-Linear Dynamics and Quantum Dynamical Systems
59
,
9
20
(
1990
).
4.
M.
Bohner
and
A.
Peterson
,
Dynamic Equations on Time Scales An Introduction With Applications
(
Birkhauser
,
2001
).
5.
M.
Bohner
and
A.
Peterson
,
Advances in Dynamic Equations on Time Scales
(
Birkhauser
,
2003
).
6.
B.
Oğur
,
Qualitative Analysis of Dynamical Systems on Time Scales with Initial Time Difference
, Master’s thesis,
Gebze Institute of Technology
(
2013
).
7.
S.
Paşalı Atmaca
and
Ö.
Akgüller
,
Advances in Difference Equations
2015
,
1
7
(
2015
).
8.
H. K.
Samancı
and
A.
Çalışkan
,
Advances in Pure Mathematics
5
,
42
50
(
2015
).
9.
J.
Ambjørn
,
A.
Görlich
,
J.
Jurkiewicz
, and
R.
Loll
, “Quantum Gravity via Causal Dynamical Triangulations,” in
Springer Handbook of Spacetime
, edited by
A.
Ashtekar
and
V.
Petkov
(
Springer
,
2014
) p.
723
, arXiv:1302.2173.
10.
C.
Rovelli
,
Zakopane lectures on loop gravity
, February
2011
, arXiv:1102.3660.
11.
P.
Šťovíček
and
J.
Tolar
,
Reports on Mathematical Physics
20
,
157
170
(
1984
).
12.
M.
Arik
and
M.
Ildes
,
Quantum Mechanics in a Space with Finite Number of Points
, October
2015
, arXiv:1510.04576.
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