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## Two formulas for the BR multiplicity

### ANNALI DELL'UNIVERSITA' DI FERRARA (2016-07-14): 1-12 , July 14, 2016

We prove a *projection formula,* expressing a relative Buchsbaum–Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the *additivity formula* for the ordinary multiplicity, a case of the *associativity formula*.

## Numerical solution of nonlinear equations by an optimal eighth-order class of iterative methods

### ANNALI DELL'UNIVERSITA' DI FERRARA (2013-05-01) 59: 159-171 , May 01, 2013

Solving nonlinear equations by using iterative methods is discussed in this paper. An optimally convergent class of efficient three-point three-step methods without memory is suggested. Analytical proof for the class of methods is given to show the eighth-order convergence and also reveal its consistency with the conjecture of Kung and Traub. The beauty in the proposed methods from the class can be seen because of the optimization in important effecting factors, i.e. optimality order, lesser number of functional evaluations; as well as in viewpoint of efficiency index. The accuracy of some iterative methods from the proposed derivative-involved scheme is illustrated by solving numerical test problems and comparing with the available methods in the literatures.

## Existence and non-existence of positive solutions of Sturm–Liouville BVPs for ODEs on whole line

### ANNALI DELL'UNIVERSITA' DI FERRARA (2016-11-01) 62: 313-336 , November 01, 2016

This paper is concerned with a boundary value problem of second order singular differential equations on whole line. Sufficient conditions to guarantee existence and non-existence of positive solutions are established. Our results improve some theorems in known papers but the methods used are different. We give two examples to illustrate main theorems.

## Faber polynomial approximation of entire functions of slow growth over Jordan domains

### ANNALI DELL'UNIVERSITA' DI FERRARA (2013-11-01) 59: 331-351 , November 01, 2013

In this paper, we study the $$L^p$$ -approximation, $$2\le p \le \infty $$ , of entire functions over Jordan domains by using Faber polynomials. Moreover, the coefficient characterizations of generalized order and generalized type of entire functions for slow growth have been obtained in terms of the $$L^p$$ -approximation errors. Our results improve the various results of Seremeta (Am Math Soc Transl Ser 2 88:291–301, 1970) and Ganti and Srivastava (Commun Math Anal 7(1):75–93, 2009).

## Skew n-derivation on prime and semi prime rings

### ANNALI DELL'UNIVERSITA' DI FERRARA (2016-09-21): 1-12 , September 21, 2016

Let
$$n \ge 2$$
be a fixed integer, *R* be a noncommutative *n*!-torsion free ring and *I* be any non zero ideal of *R*. In this paper we have proved the following results; (i) If *R* is a prime ring and there exists a symmetric skew *n*-derivation
$$D: R^n \rightarrow R$$
associated with the automorphism
$$\sigma $$
on *R*, such that the trace function
$$\delta : R \rightarrow R $$
of *D* satisfies
$$[\delta (x), \sigma (x)] =0$$
, for all
$$x\in I,$$
then
$$D=0;\,$$
(ii) If *R* is a semi prime ring and the trace function
$$\delta ,$$
commuting on *I*, satisfies
$$[\delta (x), \sigma (x)]\in Z$$
, for all
$$x \in I,$$
then
$$[\delta (x), \sigma (x)] = 0 $$
, for all
$$x \in I.$$
Moreover, we have proved some annihilating conditions for algebraic identity involving multiplicative(generalized) derivation.

## Fixed point solutions of variational inequality and generalized equilibrium problems with applications

### ANNALI DELL'UNIVERSITA' DI FERRARA (2010-11-01) 56: 345-368 , November 01, 2010

In this paper, we introduce a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solution of generalized equilibrium problem and the set of solutions of the variational inequality problem for a co-coercive mapping in a real Hilbert space. Then strong convergence of the scheme to a common element of the three sets is proved. Furthermore, new convergence results are deduced and finally we apply our results to solving optimization problems and obtaining zeroes of maximal monotone operators and co-coercive mappings.

## Necessary and sufficient conditions for the existence of Helmholtz decompositions in general domains

### ANNALI DELL'UNIVERSITA' DI FERRARA (2014-05-01) 60: 245-262 , May 01, 2014

Consider a general domain $$\varOmega \subseteq {\mathbb {R}}^n, n\ge 2$$ , and let $$1 < q <\infty $$ . Our first result is based on the estimate for the gradient $$\nabla p \in G^q(\varOmega )$$ in the form $$\Vert \nabla p\Vert _q \le C \,\sup |\langle \nabla p,\nabla v\rangle _{\varOmega }|/\Vert \nabla v\Vert _{q'}$$ , $$\nabla v \in G^{q'}(\varOmega ), q' = \frac{q}{q-1}$$ , with some constant $$C=C(\varOmega ,q)>0$$ . This estimate was introduced by Simader and Sohr (Mathematical Problems Relating to the Navier–Stokes Equations. Series on Advances in Mathematics for Applied Sciences, vol. 11, pp. 1–35. World Scientific, Singapore, 1992) for smooth bounded and exterior domains. We show for general domains that the validity of this gradient estimate in $$G^q(\varOmega )$$ and in $$G^{q'}(\varOmega )$$ is necessary and sufficient for the validity of the Helmholtz decomposition in $$L^q(\varOmega )$$ and in $$L^{q'}(\varOmega )$$ . A new aspect concerns the estimate for divergence free functions $$f_0 \in L^q_{\sigma }(\varOmega )$$ in the form $$\Vert f_0\Vert _q \le C \sup |\langle f_0,w\rangle _{\varOmega }|/ \Vert w\Vert _{q'}, w\in L^{q'}_{\sigma }(\varOmega )$$ , for the second part of the Helmholtz decomposition. We show again for general domains that the validity of this estimate in $$L^q_{\sigma }(\varOmega )$$ and in $$L^{q'}_{\sigma }(\varOmega )$$ is necessary and sufficient for the validity of the Helmholtz decomposition in $$L^q(\varOmega )$$ and in $$L^{q'}(\varOmega )$$ .

## Estimates for Fourier transforms of surface carried densities on surfaces with singular points, II

### ANNALI DELL'UNIVERSITA' DI FERRARA (2006-11-01) 52: 211-232 , November 01, 2006

###
*Abstract*

The results in the present paper are a natural continuation of the arguments in [7], [9] and are also motivated by an attempt to see wether the results in those papers can be improved when stronger assumptions are made. The main theme is to study decay for Fourier tranforms of surface carried densities which live on surfaces with uniplanar and conical singularities.

*Keywords:* Decay estimates, Fourier transforms, Surfaces with singular points

## On the existence in Gevrey classes of local solutions to the Cauchy problem for nonlinear hyperbolic systems with Hölder continuous coefficients

### ANNALI DELL'UNIVERSITA' DI FERRARA (2006-11-01) 52: 303-315 , November 01, 2006

###
*Abstract*

In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are *μ*- Hölder and
$\mu'$
- Hölder continuous in time variable respectively and in Gevrey class of index *s* with respect to space variables under the assumption
$1\le s <\min\{1 + \frac\mu \nu,1+\frac{1-\mu+\mu'}{\nu}\}, 0<\mu\le1$
, where *ν* denotes the maximal muliplicity of characteristics of systems.

*Keywords:* Nonlinear hyperbolic systems, Cauchy problem, Gevrey classes

## A 3-dimensional Eulerian array

### ANNALI DELL'UNIVERSITA' DI FERRARA (2006-07-01) 52: 107-126 , July 01, 2006

###
*Abstract*

We give a generalization of the identity proved by J. Worpitzky in [4], by expressing each power *x*^{n} as a linear combination of the images of *β*_{m} under the powers of the shift operator *E* (here
$\beta_m(x):=\frac{x^{\underline{m}}}{m!}$
). We encode the coefficients of these linear combinations in a 3-dimensional array - the Eulerian octant - and we find recurrences formulæ, explicit expressions and generating functions for its entries.

*Keywords:* Recursive matrices, Eulerian numbers

*Mathematics Subject Classification (2000):* 05A19