We present a comprehensive study of the quark sector of 2 + 1 flavor QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full ...quark-gluon vertex in the one-loop dressed approximation. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with 2 + 1 dynamical quark flavors, obtained from studies with Schwinger-Dyson equations, the functional renormalization group approach, and large volume lattice simulations. The appropriate renormalization procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalization group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalization group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as (245 MeV)3. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed.
In this paper we present some methods for solving a large class of composite functional equations. These methods are then applied to the functional equations
1
f
(
a
f
(
x
)
f
(
y
)
+
b
(
f
(
x
)
y
+
...f
(
y
)
x
)
+
c
x
y
)
=
f
(
x
)
f
(
y
)
2
f
(
a
f
(
x
)
k
y
+
b
f
(
y
)
ℓ
x
+
c
x
y
)
=
f
(
x
)
f
(
y
)
with
a
,
b
,
c
∈
R
and
k
,
ℓ
∈
N
∪
{
0
}
, for which we obtain all continuous solutions
f
:
R
→
R
. These equations generalize some well-known composite functional equations.
We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of
$\operatorname{Sp}(n,\mathbb{Z})$
on
$\mathbb{C}(L^{\prime }/L)^{n}$
. ...By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree
$n$
, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard
$L$
-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.
In this paper, we develop some basics of discrete fractional calculus such as Leibniz rule and summation by parts formula. We define simplest discrete fractional calculus of variations problem and ...derive Euler–Lagrange equation. We introduce and solve Gompertz fractional difference equation for tumor growth models.
In this paper it is provided that a functional equation for enumerating rooted nearly cubic planar maps with the valency of root-vertex, the size and the valency of root-face of the maps as three ...parameters. In particular, the number of rooted nearly cubic planar maps with the valency of root-vertex and the size as two parameters, as an answer to open problem in 1989, can be also derived.
In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this ...class.
For any \(d\in \{1,\ldots,6\}\), we prove that the web of conics on a del Pezzo surface of degree \(d\) carries a functional identity whose components are antisymmetric hyperlogarithms of weight ...\(7-d\). Our approach is uniform with respect to \(d\) and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities are natural generalizations of the classical 3-term and (Abel's) 5-term identities satisfied by the logarithm and the dilogarithm, which correspond to the cases when \(d=6\) and \(d=5\) respectively.
This study demonstrates the solution and various stabilities of a rational functional equation involving two variables by means of fixed point tactic in the vicinity of Felbin's type fuzzy normed ...spaces. Only few functional equations are dealt to design systems or to model problems in real time examples. The functional equation considered in this paper focuses on designing various systems involving ridigity modulus and pressure.
Celotno besedilo
Dostopno za:
BFBNIB, DOBA, GIS, IJS, IZUM, KILJ, KISLJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
Formulation of the problem. Analysis of the issue of including the topic "The simplest functional equations" in the model curricula for studying the subject "Algebra and the beginnings of analysis" ...for specialized classes with in-depth study of mathematics. A model curriculum studies the approximate sequence of achieving the expected learning outcomes, the content of the subject or integrated course, and the types of students’ educational activities. The inclusion of this topic aims to start a creative understanding of functional relationships existing in real systems and processes, in particular, ecological, economic, and social ones.
Materials and methods. Theoretical method of analysis of methodical and educational literature on the researched issue; comparative analysis to understand different views on the problem; systematization and generalization to create recommendations for the content of the proposed topic, as well as formulating conclusions and integrating the pedagogical experience of authors who teach relevant disciplines in educational institutions of various levels.
Results. The possible content of the topic "The simplest functional equations" in the model programs for studying the subject "Algebra and the beginnings of analysis", examples for an explanation by the teacher, and confirmation by students are proposed. For some examples, different approaches to their solution are proposed; convenient tables are provided for students to find partial solutions to some types of functional equations.
Conclusions. The authors believe that the topic "The simplest functional equations" will be useful and acceptable for studying within the scope of the subject "Algebra and the beginnings of analysis" by students of specialized classes with an in-depth study of mathematics. Within the eleven-year school education, of course, it was impossible to find time to study this topic due to the saturation and density of the material necessary for study. But in the twelve-year New Ukrainian School, marked by a deeper differentiation of specialized education, the topic "The simplest functional equations" can strengthen the fundamentality of mathematical education in classes with in-depth study of mathematics, computer science, etc. Further research in this direction may concern the method of solving the simplest recurrent equations.
In this paper, we discuss the generalized Hyers-Ulam-Rassias stability of the radical functional equation ... in the non-Archimedean normed space. Also we proved some results for the same.
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