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Satisfiability Modulo Transcendental Functions via Incremental Linearization
In this paper we present an abstraction-refinement approach to Satisfiability
Modulo the theory of transcendental functions, such as exponentiation and
trigonometric functions. The transcendental functions are represented as
uninterpreted in the abstract space, which is described in terms of the
combined theory of linear arithmetic on the rationals with uninterpreted
functions, and are incrementally axiomatized by means of upper- and
lower-bounding piecewise-linear functions. Suitable numerical techniques are
used to ensure that the abstractions of the transcendental functions are sound
even in presence of irrationals. Our experimental evaluation on benchmarks from
verification and mathematics demonstrates the potential of our approach,
showing that it compares favorably with delta-satisfiability /interval
propagation and methods based on theorem proving
On transcendental numbers: new results and a little history
Attempting to create a general framework for studying new results on
transcendental numbers, this paper begins with a survey on transcendental
numbers and transcendence, it then presents several properties of the
transcendental numbers and , and then it gives the proofs of new
inequalities and identities for transcendental numbers. Also, in relationship
with these topics, we study some implications for the theory of the Yang-Baxter
equations, and we propose some open problems.Comment: 8 page
Measures of algebraic approximation to Markoff extremal numbers
Let xi be a real number which is neither rational nor quadratic over Q. Based
on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any
real number theta, there exist a constant c>0 and infinitely many non-zero
polynomials P in Z[T] of degree at most 2 such that |theta-P(xi)| < c
|P|^{-gamma} where gamma=(1+sqrt{5})/2 denotes for the golden ratio and where
the norm |P| of P stands for the largest absolute value of its coefficients. In
the present paper, we show conversely that there exists a class of
transcendental numbers xi for which the above estimates are optimal up to the
value of the constant c when one takes theta=R(xi) for a polynomial R in Z[T]
of degree d = 3, 4 or 5 but curiously not for degree d=6, even with theta = 2
xi^6.Comment: 27 page
Transcendental equations satisfied by the individual zeros of Riemann , Dirichlet and modular -functions
We consider the non-trivial zeros of the Riemann -function and two
classes of -functions; Dirichlet -functions and those based on level one
modular forms. We show that there are an infinite number of zeros on the
critical line in one-to-one correspondence with the zeros of the cosine
function, and thus enumerated by an integer . From this it follows that the
ordinate of the -th zero satisfies a transcendental equation that depends
only on . Under weak assumptions, we show that the number of solutions of
this equation already saturates the counting formula on the entire critical
strip. We compute numerical solutions of these transcendental equations and
also its asymptotic limit of large ordinate. The starting point is an explicit
formula, yielding an approximate solution for the ordinates of the zeros in
terms of the Lambert -function. Our approach is a novel and simple method,
that takes into account , to numerically compute non-trivial zeros of
-functions. The method is surprisingly accurate, fast and easy to implement.
Employing these numerical solutions, in particular for the -function, we
verify that the leading order asymptotic expansion is accurate enough to
numerically support Montgomery's and Odlyzko's pair correlation conjectures,
and also to reconstruct the prime number counting function. Furthermore, the
numerical solutions of the exact transcendental equation can determine the
ordinates of the zeros to any desired accuracy. We also study in detail
Dirichlet -functions and the -function for the modular form based on the
Ramanujan -function, which is closely related to the bosonic string
partition function.Comment: Matches the version to appear in Communications in Number Theory and
Physics, based on arXiv:1407.4358 [math.NT], arXiv:1309.7019 [math.NT], and
arXiv:1307.8395 [math.NT
Duality relations in the auxiliary field method
The eigenenergies of a system of
identical particles with a mass are functions of the various radial quantum
numbers and orbital quantum numbers . Approximations
of these eigenenergies, depending on a principal quantum number
, can be obtained in the framework of the auxiliary field
method. We demonstrate the existence of numerous exact duality relations
linking quantities and for various forms of the
potentials (independent of and ) and for both nonrelativistic and
semirelativistic kinematics. As the approximations computed with the auxiliary
field method can be very close to the exact results, we show with several
examples that these duality relations still hold, with sometimes a good
accuracy, for the exact eigenenergies
On transcendental numbers
Transcendental numbers play an important role in many areas of science. This
paper contains a short survey on transcendental numbers and some relations
among them. New inequalities for transcendental numbers are stated in Section 2
and proved in Section 4. Also, in relationship with these topics, we study the
exponential function axioms related to the Yang-Baxter equation.Comment: 6 page
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