http://www.sct.edu.om/
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MAIN FIELDS OF RESEARCH: Mathematical Optics
I am working towards a unified treatment of light beam optics and polarization,
using the standard mathematical machinery of quantum mechanics. Dirac-like
form of the Maxwell equations is well known in literature. Starting with the
Dirac-like form of the Maxwell’s equations a unified treatment of light
beam optics and polarization has been obtained. The traditional results
(including aberrations) of the scalar optics are modified by the
wavelength-dependent contributions. Some of the well-known results
in polarization studies are realized as the leading-order limit of a
more general framework of our formalism. The existing matrix
representations of the Maxwell’s equations were found to be approximate
for the formalism developed here; hence, an exact matrix representation
of the Maxwell’s equations was derived.
A related study was made starting with the scalar approximation of the Maxwell’s equations. Using the analogy of the Helmholtz equation with the Klein-Gordon equation and the Feshbach-Villars approach to the Klein-Gordon equation a formalism utilizing the powerful techniques of quantum mechanics has been developed for scalar optics including aberrations. This provides an alternative to the traditional square-root approach and gives rise to wavelength-dependent contributions modifying the aberration coefficients.
Some of the results have been published and others have been communicated.
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Quadricmeter is the instrument devised to identify (distinguish) and measure the various
parameters (axis, foci, latera recta, directrix, etc.,) completely characterizing the important
class of surfaces known as the quadratic surfaces. Quadratic surfaces (also known as quadrics)
include a wide range of commonly encountered surfaces including, cone, cylinder, ellipsoid,
elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder,
hyperbolic paraboloid, paraboloid, sphere, and spheroid. Quadricmeter is a generalized form of
the conventional spherometer and the lesser known cylindrometer (also known as the “Cylindro-Spherometer”
and "Sphero-Cylindrometer").
With a conventional spherometer it was possible only to measure the radii of spherical surfaces.
Cylindrometer can measure the radii of curvature of a cylindrical surface in addition to the spherical
surface. In both the spherometer and the cylindrometer one assumes the surface to be either spherical
or cylindrical respectively. In the case of the quadricmeter, there are no such assumptions.
| List of 37+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). | List of 21+ Articles from the LANL E-Print archive (see the Atom Feeds). |
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http://www.research.att.com/~njas/sequences/
http://oeis.org/wiki/User:Sameen_Ahmed_Khan
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A geometric-arithmetic progression of primes is a set of k primes
(denoted by GAP-k) of the form p1*r j + j*d
for fixed p1, r and d and consecutive j,
from j = 0 to k - 1.
i.e, {p1, p1*r + d, p1*r 2 + 2 d,
p1* r 3 + 3 d, ...}.
For example 3, 17, 79 is a 3-term geometric-arithmetic progression
(i.e, a GAP-3) with a = p1 = 3, r = 5 and d = 2.
A GAP-k is said to be minimal if the minimal start p1 and
the minimal ratio r are equal, i.e, p1 = r = p, where p
is the smallest prime ≥ k.
Such GAPs have the form p*p j + j*d.
Minimal GAPs with different differences, d do exist. For example, the minimal GAP-5
(p1 = r = 5) has the
possible differences, 84, 114, 138, 168, ... (see the Sequence A209204)
and the minimal
GAP-6 (p1 = r = 7) has the possible differences,
144, 1494, 1740, 2040, .... (see the Sequence A209205).
The following article gives the conditions under which, a GAP-k is a
set of k primes in geometric-arithmetic progression.
The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.
The exceptional cases (for k < = 7) are k = 2, k = 3, k = 5 and k = 7.
For k = 2, we have d = 1 and there is ONLY one AP-2 with this difference: {2, 3}.
For k = 3, we have d = 2 and there is ONLY one AP-3 with this difference: {3, 5, 7}.
For k = 4, we have d = 4# = 6 and AP-4 is {5, 11, 17, 23} and is not unique.
The first primes is the Sequence A023271:
5, 11, 41, 61, 251, 601, 641, 1091, 1481, 1601, 1741, 1861, 2371, ...
For k = 5, we have d = 3# = 6 and there is ONLY one AP-5 with this difference: {5, 11, 17, 23, 29}.
For k = 6, we have d = 6# = 30 and AP-6 is {7, 37, 67, 97, 127, 157} and is not unique.
The first primes is the Sequence A156204:
7, 107, 359, 541, 2221, 6673, 7457, 10103, 25643, 26861, 27337, 35051, 56149, ...
For k = 7, we have d = 5*5# = 150 and there is ONLY one AP-7 with this difference:
{7, 157, 307, 457, 607, 757, 907}.
| List of 35+ Articles from the INSPIRE HEP (Logo), Originally SLAC SPIRES (Logo). | List of 20+ Articles from the LANL E-Print archive (see the Atom Feeds). |
| Some Publications from the American Mathematical Society (AMS, Logo) MathSciNet (Logo). | List of 250+ Articles from the Google Scholar (Logo). |
© Last updated Friday, the 13 February 2015
