Article ID Research Articles

On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space [Formula: see text] and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot [Formula: see text]. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on [Formula: see text], the result of [Formula: see text] surgery in the left-handed trefoil knot [Formula: see text], J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

Origin of the Blue Continuum Radiation in the Flare Spectra of dMe Stars

Calculations of the emission spectrum of a homogeneous plane layer of pure hydrogen plasma taking into account nonlinear effects (the influence of bremsstrahlung and recombination radiation of the layer itself on its Menzel factors) [Morchenko et al., Ap&SS, Vol. 357, article id. 119 (2015)] show that the blue component of the optical continuum during the impulsive phase of large flares on dMe stars originates from the near-photospheric layers [Grinin and Sobolev, Afz, Vol. 13, 587 (1977)]. The gas behind the front of a stationary radiative shock wave propagating in the red dwarf chromosphere toward the photosphere is not capable of generating the blackbody radiation observed at the maximum brightness of the flares.

230 - Simultaneous Evaluations of GSH Levels, SOD Activity and the Genotoxic Damage in Mice Treated with Chromium(VI) and Catechins of Green Tea [(+)-Catechin and (-)-Epigallocatechin-3-Gallate

The compounds of Cr(VI) generates reactive oxygen species (ROS) during its reduction to Cr(III) leading to DNA damage. The green tea polyphenols can protect against the genotoxic damage induced by Cr(VI) by reducing frequencies of micronucleus MN[1]. In this study, the effect of catechins of green tea [(+)-catechin and epigallocatechin-3-gallate (EGCG)] on glutathione (GSH) levels and superoxidodismutasa (SOD) activity was evaluated simultaneously with the analysis of MN and apoptosis. Groups of five Hsd-ICR mice were divided and treated as follows: i) only vehicle; ii) (+)-catechin (20 and 40 mg/kg) or EGCG (10 mg/kg) by gavage; iii) CrO3 (20 mg/kg) intraperitoneally; and iv) (+)-catechin or EGCG in addition to CrO3. Evaluations were performed at 0, 24, 48 and 72 h after treatments. No significant changes in MN frequencies were observed when the catechins of green tea were administrated alone. However, the treatment with CrO3 showed an increase in MN, apoptosis and GSH levels, but the activity of SOD decreased. The treatment with (+)-catechin prior to administration of CrO3 showed no effects on MN and apoptosis. However, treatment with EGCG possibly presents both effects; prooxidant and antioxidant because the MN was increased at 24 h, but it was reduced at 48 h when EGCG was administrated prior to CrO3. Additionally, GSH levels decreased and SOD activity is recovered. The increase of apoptotic cells suggests that this may play a role in the elimination of micronucleated cells. Financial support was obtained from DGAPA-UNAM PAPIIT IN219216; IN217712; PAEP-2016; CONACYT 703847/584686. Reference [1] Maria del Carmen Garcia-Rodriguez, et al. 2013. Oxid Med Cell Longev, vol. 2013, Article ID 486419, 9 pag. doi:10.1155/2013/486419.

Unusual Case of a Primary Cardiac Tumor Presenting as Superior Vena Cava Syndrome

SESSION TITLE: Student/Resident Case Report Poster - Cardiac and Thoracic Surgery SESSION TYPE: Student/Resident Case Report Poster PRESENTED ON: Tuesday, October 25, 2016 at 01:30 PM - 02:30 PM INTRODUCTION: Primary cardiac tumors are extremely rare. Incidence is approximately 0.017% - 0.019% of all primary tumors. Spindle Cell Sarcoma is the rarest, with only 4 cases reported. Sarcomas, have a poor prognosis even after surgical intervention with an average mean survival of 6 months. CASE PRESENTATION: 67 year old female patient with PMH of Hypertension, Dyslipidemia and DM that presented to the emergency department with complaints of swelling and redness of the face for one year that was treated as an allergic reaction without improvement. Presentation was significant for shortness of breath, dry cough, bilateral jugular vein distention, low voltage on EKG and a opacity measuring 3.8 x 3.7cm at the right hilar region on CXR. Chest CT confirmed a well-defined mass measuring 7 x 5.7cm in the right atrium producing extrinsic compression of the right atrioventricular valve and proximal aspect of the right ventricle. The mass infiltrated to the right hilar region invading pericardial fat. Additionally, the tumor was obliterating the Superior Vena Cava. Biopsy of the cardiac mass was positive for Spindle Cell Sarcoma. The patient was not a candidate for surgical resection due to advanced stage malignancy. Unfortunately, patient perished before starting chemotherapy. DISCUSSION: Primary cardiac tumors are mostly benign, 50% to 80% are Myxomas [1]. The remaining 25% of cardiac tumors are malignant, 95% are Sarcomas [2]. The most common malignant tumor is Angiosarcoma. Spindle Cell Sarcoma is the rarest, with only 4 cases reported [2]. Benign tumors usually have a good prognosis with a normal life expectancy after surgical resection. Patients with malignant tumors have a poor prognosis even after surgical intervention, with an average mean survival of 6 months after diagnosis [3]. Diagnosis is made with imaging techniques such as: 2D echocardiogram, cardiac CT, MRI and confirmed with tissue biopsy [3]. Complete resection of the tumor remains the best treatment available with or without adjuvant chemotherapy [3]. CONCLUSIONS: Spindle Cell Sarcoma may present with few symptoms and may remain silent until an advanced stage, thus limiting the few therapeutic options available. The importance of this case relies in the early recognition of symptoms as in our patient who was presenting with Superior Vena Cava Syndrome and Heart Failure misdiagnosed for one year. By the time of diagnosis the patient was not candidate for tumor resection. Reference #1: Primary Cardiac Tumor: A Single-Center 41-Year Experience; Christina Maria Steger et al. International Scholarly Research Network; Vol 2012, Article ID 906109 Reference #2: Primary Intimal (Spindle Cell) Sarcoma of the Heart; A.Ibrahim et al. Hindawi Publishing Corporation; Vol 2013, Article ID 461815 Reference #3: Survival After Resection of Primary Cardiac Tumors. A 48-Year Experience; Andrew W. ElBardissi et al. CirculationAHA.107783126 DISCLOSURE: The following authors have nothing to disclose: Michael Cruz Caliz, Iván González Cancel, Ricardo Fernandez, Modesto Gonzalez-del Rosario, Kelvin Rivera Manzano, Gustavo Gonzalez Astacio, Joel Nieves Scharon, Hiram Jose Maldonado Quintana, Pedro Torrellas-Ruiz, Kimberly Padilla Rodriguez No Product/Research Disclosure Information

Fixed point theorems via generalized $$\varvec{F}$$ F -contractions with applications to functional equations occurring in dynamic programming

The purpose of this paper is threefold. Firstly, recognizing the concept of Piri and Kumam (Fixed Point Theor Appl 210, 2014), we define generalized F-contractive mappings in the framework of G-metric spaces and by employing this, some fixed point theorems in the structure of G-metric spaces are established that can not be obtained from the existing results in the context of allied metric spaces and do not meet the remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Infact, we utilize the pattern, mentioned in Karapinar and Agrawal (Fixed Point Theor Appl 154, 2013), a counter paper to remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Secondly, in the setting of G-metric spaces, certain fixed point results for integral inequalities under generalized F-contraction are presented. Finally, as an application, our results are utilized to establish the existence and uniqueness of solution the equations arising in Oscillation of a spring. In the sequel, another application is given to set-up the existence and uniqueness of solution of functional equations occurring in dynamic programming. Our investigations are also authenticated with the aid of some appropriate and innovative examples.

Fixed point theorems for $$(\psi , \beta )$$ ( ψ , β ) -Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations

In this paper, we establish common fixed point theorems in an ordered complete metric space using distance functions. Our main result, improves and extends the results of Yan et al. (Fixed Point Theory Appl. 2012, Article id: 152, 2012) and some other existing results announced in the literature. As applications, we discuss some fixed point theorems for contraction of integral type and some existence theorem for solution of integral equation, and for solution of first and second order ordinary differential equations with periodic boundary conditions.

Quantitative Rellich inequalities on Finsler—Hadamard manifolds

In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].

Recursion formulas for Srivastava’s general triple hypergeometric function

This paper continues the study of recursion formulas of multivariable hypergeometric functions. In [Recursion formulas for multivariable hypergeometric functions, Asian-Eur. J. Math. 8(4) (2015) Article ID: 1550082, 50 pp.], the authors have given the recursion formulas for three-variable Lauricella functions, three Srivastava’s triple hypergeometric functions and four [Formula: see text]-variable Lauricella functions. We present here the recursion formulas for the general triple hypergeometric function.

On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced.

Oscillation results for third order nonlinear mixed neutral differential equations

Abstract In this paper, the authors study oscillatory and asymptotic behavior of solutions of a class of nonlinear third order neutral differential equations with positive and negative coefficients of the form (E) ( a ( t ) ( b ( t ) ( y ( t ) + p ( t ) y ( σ ( t ) ) ) ′ ) ′ ) ′ + q ( t ) G ( y ( α ( t ) ) ) − h ( t ) H ( y ( β ( t ) ) ) = 0 $$\begin{equation*}(a(t)(b(t)(y(t)+p(t)y(\sigma(t)))^{\prime})^{\prime})^{\prime} +q(t)G(y(\alpha(t)))-h(t)H(y(\beta(t)))=0 \tag{E}\end{equation*}$$ for 0 ⩽ p(t) ⩽ p 1 <1 and −1 < p 2 ⩽ p(t) ⩽ 0. The results in this paper generalize the results of [LI, T.—ZHANG, C.—XING, G.: Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 569201] and various results in the literature. We establish new conditions which guarantees that every solutions of (E) either oscillatory or converges to zero. Examples are considered to illustrate the main results.

Quaternionic Approach of Canal Surfaces Constructed by Some New Ideas

A canal surface is a surface constructed as the envelope of a family of spheres with the parametric (non constant radii) radii r(s) and a space curve $${\alpha (s)}$$ called its center. If the radii of the generating spheres are constant r then the canal surface is called tubular surface such as right circular cylinder, torus, right circular cone, surface of revolution. The quaternions are number systems generalized by the complex numbers. Irish mathematician Hamilton (Lond Edinb Dublin Philos Mag J Sci 25(3):489–495, 1844) discovered them in 1843 and applied to mechanics in three-dimensional space. Many laws of curves and surfaces in differential geometry used quaternions such as quaternionic helices (Coken and Tuna in Appl Math Comput 155:373–389, 2004; Gok et al. in Appl Clifford Algebras 21:707–719, 2011; Kahraman et al. in Appl Math Comput 218:6391–6400, 2012) and canal surfaces (Aslan and Yayli in Adv Appl Clifford Algebras, 2015; Babaarslan and Yayli in ISRN Geom 2012:Article ID 126358, p 8, 2012). This paper has two purposes. The first purpose is to form canal surfaces whose centers are spherical indicatrices of a spatial curve with a new idea in terms of alternative moving frame $${\{N,C,W\}}$$ The second purpose is to obtain canal surfaces by using the quaternion product and matrix representation. Moreover, we give some related examples with their figures.

Abstract 2836: Characterization of the mechanism of action for abemaciclib with antiestrogen combined therapy in human breast cancer cell lines

Abstract Breast cancer is the second most common cancer worldwide after lung cancer. About 70% of breast cancers express estrogen receptor α (ER+) and/or progesterone receptor (PR+), and these biomarkers are indicative of hormone dependence. However up to 50% acquire resistance to hormone therapy [1, 2]. Estrogen independent ER+ breast cancer depends on CDK4 for tumor growth and CDK4 inhibitors have emerged as a promising approach to treat this type of tumors [3]. Abemaciclib is a cell cycle inhibitor with selective activity against CDK4 and CDK6 and it is being evaluated in advanced clinical trials for its potential to reduce metastatic ER+ breast cancer growth. We have evaluated combination of abemaciclib with an anti-estrogen therapy in an in vitro breast cancer panel. Phenotypic characterization of sensitive cell lines was carried out by monitoring cell proliferation, senescence, and apoptosis markers using flow cytometry and high content imaging approaches. Using an in vitro panel with a diversity of breast cancer cell lines, a synergistic effect of abemaciclib in combination with the ER down-regulating drug fulvestrant was observed based on Bliss score. This combination treatment demonstrated effective growth inhibition in ER+ cells and exhibited synergism in MCF-7, T47D and ZR-75-1. The mechanistic analyses revealed that the combination of abemaciclib with fulvestrant promoted a decrease in cancer cell proliferation due to G1 phase arrest at doses tested. This growth inhibition was accompanied by increased hallmarks for cell senescence as observed by markers such as SA-β-galactosidase staining or morphological changes. Subsequently, an increase in biomarkers for apoptosis was also observed. These changes occurred in a time dependent manner and were significantly greater with the combination than fulvestrant single agent treatment. We conclude the combination of abemaciclib with fulvestrant better prevented proliferation of breast cancer cell lines by blocking cell proliferation and lead to induction of senescence and apoptosis as compared to fulvestrant treatment alone in ER+ cells. Bibliography [1] American Cancer Society, Cancer Facts & Figures 2014. [2] Dixon J.M. (2014) New Journal of Science. Volume 2014, Article ID 390618. [3] Miller TW et al. (2011) Cancer Discov. Volume 1 (4): 338-51. Citation Format: Raquel Torres, Bruna Calsina, Ana Hermoso, Carmen Baquero, Cecilia Mur, Karsten Boehnke, Joaquín Amat, Alfonso De Dios, Xueqian Gong, Sean Buchanan, Richard Paul Beckmann, Maria Jose Lallena. Characterization of the mechanism of action for abemaciclib with antiestrogen combined therapy in human breast cancer cell lines. [abstract]. In: Proceedings of the 107th Annual Meeting of the American Association for Cancer Research; 2016 Apr 16-20; New Orleans, LA. Philadelphia (PA): AACR; Cancer Res 2016;76(14 Suppl):Abstract nr 2836.

Nil-clean group rings

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.

Alexander and writhe polynomials for virtual knots

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

Positive definite solution of a nonlinear matrix equation

Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equation \({X = Q \pm \sum^{m}_{i=1}{A_{i}}^*F(X)A_{i}}\), where Q is a positive definite matrix, Ai’s are arbitrary n × n matrices and F is a monotone map from the set of positive definite matrices to itself. We show that the presented condition is weaker than that presented by Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443]. In order to do so, we establish some fixed point theorems for mappings satisfying (\({\psi, \phi}\))-weak contractivity conditions in partially ordered G-metric spaces, which generalize some existing results related to (\({\psi, \phi}\))-weak contractions in partially ordered metric spaces as well as in G-metric spaces for a given function f. We conclude, by presenting an example, that our fixed point theorem cannot be obtained from any existing fixed point theorem using the process of Jleli and Samet [Fixed Point Theory Appl. 2012 (2012), Article ID 210].

RETRACTED: Electrical Stimulation for Hemiplegic Shoulder Function: A Systematic Review and Meta-Analysis of 15 Randomized Controlled Trials

This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/our-business/policies/article-withdrawal). This article has been retracted at the request of the co-Editors-in-Chief. The article plagiarizes a paper that was published in BioMed Research International, Volume 2015 (2015), Article ID 729768, 14 pages, http://dx.doi.org/10.1155/2015/729768. Nearly the entire introduction and sections of the methods, limitations and conclusions duplicate the BioMed Research International paper. One of the conditions of submission of a paper to Archives of Physical Medicine and Rehabilitation is that authors declare explicitly that the paper is an original work and not duplicative of prior work unless that work is cited properly. The duplication in this manuscript is a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter.

Activation of ERK and NF-κB during HARE-Mediated Heparin Uptake Require Only One of the Four Endocytic Motifs.

Fifteen different ligands, including heparin (Hep), are cleared from lymph and blood by the Hyaluronan (HA) Receptor for Endocytosis (HARE; derived from Stabilin-2 by proteolysis), which contains four endocytic motifs (M1-M4). Endocytosis of HARE•Hep complexes is targeted to coated pits by M1, M2, and M3 (Pandey et al, Int. J. Cell Biol. 2015, article ID 524707), which activates ERK1/2 and NF-κB (Pandey et al J. Biol. Chem. 288, 14068–79, 2013). Here, we used a NF-κB promoter-driven luciferase gene assay and cell lines expressing different HARE cytoplasmic domain mutants to identify motifs needed for Hep-mediated signaling. Deletion of M1, M2 or M4 singly had no effect on Hep-mediated ERK1/2 activation, whereas signaling (but not uptake) was eliminated in HARE(ΔM3) cells lacking NPLY2519. ERK1/2 signaling in cells expressing WT HARE(Y2519A) or HARE(Y2519A) lacking M1, M2 and M4 (containing M3-only) was decreased by 75% or eliminated, respectively. Deletion of M3 (but not M1, M2 or M4) also inhibited the formation of HARE•Hep•ERK1/2 complexes by 67%. NF-κB activation by HARE-mediated uptake of Hep, HA, dermatan sulfate or acetylated LDL was unaffected in single-motif deletion mutants lacking M1, M2 or M4. In contrast, cells expressing HARE(ΔM3) showed loss of HARE-mediated NF-κB activation during uptake of each of these four ligands. NF-κB activation by the four signaling ligands was also eliminated in HARE(Y2519A) or HARE(M3-only;Y2519A) cells. We conclude that the HARE NPLY2519 motif is necessary for both ERK1/2 and NF-κB signaling and that Tyr2519 is critical for these functions.

On generalized semiderivations in 3-prime near-rings

There is a large body of evidence showing that the existence of a suitably-constrained derivation on a 3-prime near-ring forces the near-ring to be a commutative ring. The purpose of this paper is to study generalized semiderivations which satisfy certain identities on 3-prime near-ring and generalize some results due to [H. E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud. 137 (1987) 31–35; H. E. Bell, On prime near-rings with generalized derivation, Int. J. Math. Math. Sci. 2008 (2008), Article ID: 490316, 5[Formula: see text]pp; A. Boua and L. Oukhtite, Some conditions under which near-rings are rings, Southeast Asian Bull. Math. 37 (2013) 325–331]. Moreover, an example is given to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized.

Weak contractive integral inequalities and fixed points in modular metric spaces

Branciari (Int. J. Math. Math. Sci. 29(9):531-536, 2002) gave an interesting supplement of Banach’s contraction principle for an integral-type inequality. In this paper, we introduce different notions of generalized ω-weak contractive inequalities of integral type in modular metric spaces and prove the presence and uniqueness of common fixed points for such contractions under ω-weak compatibility of underlying maps. Our results generalize and extend the results of Azadifar et al. (J. Inequal. Appl. 2013:483, 2013), Liu et al. (Fixed Point Theory Appl. 2013:2672013, 2013), Beygmohammadi and Razani (Int. J. Math. Math. Sci. 2010: Article ID 317107, 2010), and many others. Moreover, an example is provided here to demonstrate the applicability of the obtained results.

Notes on the spectral properties of the weighted mean difference operator G (u, v; Δ) over the sequence space [formula omitted

In the study by Baliarsingh and Dutta [Internat. J.Anal., Vol.2014(2014), Article ID 786437], the authors computed the spectrum and the fine spectrum of the product operator G (u, v; Δ) over the sequence space ℓ1. The product operator G (u, v; Δ) over ℓ1 is defined by (G(u,v; Δ)x)k=∑i=0kukvi(xi−xi−1) with xk = 0 for all k < 0, where x = (xk) ∈ ℓ1, and u and v are either constant or strictly decreasing sequences of positive real numbers satisfying certain conditions. In this article we give some improvements of the computation of the spectrum of the operator G (u, v; Δ) on the sequence space ℓ1.