54,649 research outputs found
Puzzles in Time Delay and Fermat Principle in Gravitational Lensing
The current standard time delay formula (CSTD) in gravitational lensing and
its claimed relation to the lens equation through Fermat's principle (least
time principle) have been puzzling to the author for some time. We find that
the so-called geometric path difference term of the CSTD is an error, and it
causes a double counting of the correct time delay. We examined the deflection
angle and the time delay of a photon trajectory in the Schwarzschild metric
that allows exact perturbative calculations in the gravitational parameter
in two coordinate systems -- the standard Schwarzschild coordinate system and
the isotropic Schwarzschild coordinate system. We identify a coordinate
dependent term in the time delay which becomes irrelevant for the arrival time
difference of two images. It deems necessary to sort out unambiguously what is
what we measure. We calculate the second order corrections for the deflection
angle and time delay. The CSTD does generate correct lens equations including
multiple scattering lens equations under the variations and may be best
understood as a generating function. It is presently unclear what the
significance is. We call to reanalyze the existing strong lensing data with
time delays.Comment: 6 p., 1 fi
Generalized Legendre polynomials and related congruences modulo
For any positive integer and variables and we define the
generalized Legendre polynomial P_n(a,x)=\sum_{k=0}^n\b
ak\b{-1-a}k(\frac{1-x}2)^k. Let be an odd prime. In the paper we prove
many congruences modulo related to . For example, we show
that P_{p-1}(a,x)\e (-1)^{_p}P_{p-1}(a,-x)\mod {p^2}, where is the
least nonnegative residue of modulo . We also generalize some
congruences of Zhi-Wei Sun, and determine
and , where is the greatest integer function.
Finally we pose some supercongruences modulo concerning binary quadratic
forms.Comment: 37 page
Jacobsthal sums, Legendre polynomials and binary quadratic forms
Let be a prime and with . Built on the work
of Morton, in the paper we prove the uniform congruence:
&\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big) \equiv {-(-3m)^{\frac{p-1}4}
\sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k
(\frac{4m^3+27n^2}{4m^3})^k\pmod p&\t{if $4\mid p-1$,}
\frac{2m}{9n}(\frac{-3m}p)(-3m)^{\frac{p+1}4} \sum_{k=0}^{p-1}\binom{-\frac
1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\text{if $4\mid
p-3$,} where is the Legendre symbol. We also establish many
congruences for , where is given by or
, and pose some conjectures on supercongruences modulo
concerning binary quadratic forms.Comment: 35 page
Congruences concerning Legendre polynomials
Let be an odd prime. In the paper, by using the properties of Legendre
polynomials we prove some congruences for
. In particular, we
confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on
supercongruences.Comment: 16 page
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