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"Polynomials"
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Intersection cohomology of moduli spaces of vector bundles over curves
We compute the intersection cohomology of the moduli spaces$\\mathcal{M}_{r,d}$of semistable vector bundles having rank$r$and degree$d$over a curve. We do this by relating the Hodge–Deligne polynomial of the intersection cohomology of$\\mathcal{M}_{r,d}$to the Donaldson–Thomas invariants of the curve. These invariants can be computed by methods going back to Harder, Narasimhan, Desale and Ramanan. More generally, we introduce Donaldson–Thomas classes in the Grothendieck group of mixed Hodge modules over$\\mathcal{M}_{r,d}$, and relate them to the class of the intersection complex of$\\mathcal{M}_{r,d}$. Our methods can be applied to the moduli spaces of semistable objects in arbitrary hereditary categories.
Journal Article
Applications of Generalized Mersenne Polynomials to a General Subclass of Bi-Bazileviˇc-Type Functions
This paper introduces a new subclass of generalized bi-Bazileviˇc-type functions which encompasses several eminent and widely studied subclasses, such as bi-starlike and bi-convex functions associated with the generating function of generalized Mersenne polynomials. The use of generalized Mersenne polynomials highlights the applicability of special polynomials in geometric function theory. By applying the generating function of the generalized Mersenne polynomials, we derive coefficient bounds for the initial Taylor-Maclaurin coefficients and provide estimates for the Fekete-Szegö inequality. Furthermore, several corollaries are discussed by specifying particular parameter choices.
Journal Article
On the inverse stability of zn+c
Let ϕ(z)=zd+c be a polynomial over a field K. We study the inverse stability of ϕ(z) over K. In this paper, we establish some sufficient conditions for the inverse stability of ϕ(z) over the field of rational numbers and a function field. Furthermore, we also provide necessary and sufficient conditions for the inverse stability of ϕ(z)over a finite field.
Journal Article
RECIPROCAL MONOGENIC QUINTINOMIALS OF DEGREE $\\boldsymbol {2^n}
We prove a new irreducibility result for polynomials over
${\\mathbb Q}$
and we use it to construct new infinite families of reciprocal monogenic quintinomials in
${\\mathbb Z}[x]$
of degree
$2^n$
.
Journal Article
The connected domination polynomial of some graph constructions
The connected domination polynomial of a connected graph G of order n is the polynomial D c [ G , x ] = ∑ i = γ c n d c ( G , i ) x i , where d c ( G, i ) is the number of connected dominating sets of G of cardinality i and γ c (G) is the connected domination number of G [5]. In this paper we find the polynomial D c ( G, x ) for some constructive graphs.
Journal Article
A review of operational matrices and spectral techniques for fractional calculus
Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.
Journal Article
Meromorphic solutions of fn+Pd(f)=p1eα1z+p2eα2z+p3eα3z
By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equationfn+Pd(f)=p1eα1z+p2eα2z+p3eα3z,where Pd(f) is a differential polynomial in f of degree d(0≤d≤n−3) with small meromorphic coefficients and pi,αi(i=1,2,3) are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in Γ0∪Γ1∪Γ3 which will be given in Section 1. Moreover, we give some examples to illustrate our results.
Journal Article
EQUATIONS WITHOUT THE POLYNOMIAL LAW: EXPLORING TWO-SERIES POSSIBILITIES
For the purpose of resolving dual series problems involving Laguerre polynomials, this essay makes use of the Noble and Lowndes multiplying factor approach.
Journal Article
Hardy–Littlewood and Ulyanov inequalities
We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness
The main tool is the new
Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity
We also prove the
Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for
the Lipschitz and Besov spaces.
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