Term Recurrence Research Articles

Update on Management of Patients With Overt Hepatic Encephalopathy

Hepatic encephalopathy (HE) is a multifactorial neuropsychiatric disease that affects patients with cirrhosis. We review the clinical impact, pathogenesis, evaluation, management, and prevention of overt HE in patients with cirrhosis. Articles published between January 1960 and November 2012 were acquired through a MEDLINE search of different combinations of the terms hepatic encephalopathy, pathophysiology, treatment, prophylaxis, prevention, prognosis, and recurrence. The Healthcare Cost and Utilization Project database was used to obtain prevalence and cost information related to hospitalizations of patients with HE. The literature describes significant morbidity and mortality of HE in patients with cirrhosis. Overt HE develops in 30% to 45% of patients with cirrhosis and is associated with a substantial pharmacoeconomic burden, particularly HE-related hospitalizations. The development of HE in patients with cirrhosis portends a worsened prognosis and is incorporated into the Child-Pugh classification of the severity of liver disease. In the hospitalized patient, the development of HE is associated with precipitating events (eg, gastrointestinal bleeding, dehydration, infection), and in some patients, its course is characterized by frequent and severe relapses. In addition, hospitalized patients with overt HE have a 3.9-fold increased mortality risk. Patient management employs nonabsorbable disaccharides, the nonsystemic antibiotic rifaximin, or both, to treat acute HE episodes and prevent HE relapse. In open-label trials, use of the nonabsorbable disaccharide lactulose reduced the risk of overt HE recurrence in patients compared with no-lactulose control groups for ≤ a median of 14 months. In a randomized, placebo-controlled trial, rifaximin 550 mg twice daily was more effective in maintaining HE remission compared with placebo and was associated with a reduction in HE-related hospitalizations. Recent advances in treatment and preventative therapies may reduce the personal, societal, and economic impact of this disorder.

Some Identities for Four Term Recurrence Relations

Some Identities for Four Term Recurrence Relations

Recurrence relations of the multi-indexed orthogonal polynomials

Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence relations with non-trivial initial data of the lowest M+1 members. These include the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. The M = 0 case is the corresponding classical orthogonal polynomials.

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.

The analytic solution for the power series expansion of Heun function

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2F1, 1F1 and 0F1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.In this paper I will apply three term recurrence formula (Y.S. Choun, arXiv:1303.0806, 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of An’s. Section 3 contains my analysis on applying the power series expansions of Heun function to a recent paper (R.S. Maier, Math. Comp. 33 (2007) 811–843). Due to space restriction final equations for the 192 Heun functions are not included in the paper, but feel free to contact me for the final solutions. Section 4 contains two additional examples using the power series expansions of Heun function.This paper is 3rd out of 10 in series “Special functions and three term recurrence formula (3TRF)”. See Section 5 for all the papers in the series. The previous paper in series deals with three term recurrence formula (3TRF). The next paper in the series describes the integral forms of Heun function and its asymptotic behaviors analytically.

Evaluation of Factors Related to Late Recurrence - Later than 10 Years after the Initial Treatment - in Primary Breast Cancer

Background: Breast cancer is associated with a relatively good prognosis. Prognostic factors examined to date are related to early recurrence while those related to late recurrence and their countermeasures remain unclear. Therefore, we examined the factors related to late recurrence. Patients and Methods: From January 1980 to August 2012, 4,774 patients who underwent primary treatment and estrogen (ER) and progesterone receptor (PgR) assessment were enrolled in this study. The patients were divided into two groups, those with a follow-up period <10 years and those without any recurrence at 10 years but who continued follow-up examinations. Recurrence occurred in 711 patients followed up for <10 years and in 51 patients for ≥10 years. Results: The overall 10-year cumulative disease-free survival rate was 79.5%, and the recurrence rate at ≥10 years was 5.8%. A multivariate analysis revealed that the factors related to late recurrence were PgR positivity and positive nodes. This result differed from that for early recurrence in terms of ER/PgR, Ki-67 index and p53 overexpression. Conclusion: PgR positivity and lymph node metastases significantly correlated with late recurrence. Therefore, it is important to evaluate appropriate measures such as treatment period and treatment regimen for hormone-sensitive patients.

Identifying clinically relevant prognostic subgroups in node-positive postmenopausal HR+ early breast cancer patients treated with endocrine therapy: A combined analysis of 2,485 patients from ABCSG-8 and ATAC using the PAM50 risk of recurrence (ROR) score and intrinsic subtype.

506 Background: Most postmenopausal women with node positive HR+ EBC receive adjuvant chemotherapy. We hypothesized that a molecular-based characterization of residual risk after endocrine therapy using the ROR score and IS may identify node-positive patient subgroups with limited long-term recurrence risk after endocrine therapy better than clinical-pathological risk assessment by clinical treatment score (CTS) alone. Methods: Long-term follow-up and tissue samples were obtained from 2,485 postmenopausal HR+ patients from the ABCSG-8 (N=1,478) and transATAC (N=1,007) trials. The PAM50 test was conducted on RNA extracted from paraffin blocks using the NanoString nCounter Analysis system. The ability of ROR, IS and ROR-defined risk groups (ROR-RG) to add prognostic information to CTS was assessed by the likelihood ratio test in a prospectively defined analysis plan. Results: Patients in the combined data set were grouped by the number of positive nodes into 1 (N1), 2 (N2), or 2 or 3 (N2-3),Baseline hazards for these subgroups were similar in the two trials. ROR score, IS and ROR-RG added statistically significant prognostic information (10-year distant recurrence risk) beyond CTS in all groups. In patients with one positive node, the absolute 10-year risk of distant recurrence was 6.6% [95% CI: 3.3%-12.8%] in the PAM-50-low risk group (40% of patients) and 8.4 % [5.3%-13.3%] in the Luminal A subgroup (69% of patients). Conclusions: The results of this combined analysis demonstrate that a significant proportion of N1 EBC patients have very limited long term recurrence risk and suggest the same for some N2 patients. The PAM50 ROR score, IS and ROR-RG reliably provide additional prognostic information beyond CTS and may be useful in deciding which women with node-positive HR+ EBC can be spared adjuvant chemotherapy. [Table: see text]

Su1671 Long Term Recurrence and Survival Rates in Gastrointestinal Stromal Tumours (GISTs) Treated by Minimally Invasive Surgery

Su1671 Long Term Recurrence and Survival Rates in Gastrointestinal Stromal Tumours (GISTs) Treated by Minimally Invasive Surgery

Sylvester equations for Laguerre–Hahn orthogonal polynomials on the real line

Matrix Sylvester differential equations are introduced in the study of Laguerre–Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre–Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre–Hahn orthogonal polynomials.

A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems

Second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations. The method uses evaluation at two off steps, three neighbouring knots and it reduces to simple five terms recurrence relations. The formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate. Some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method. The root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method.

Computational Method for Solving Singularly Perturbed Delay Differential Equations with Negative Shift

In this paper, a computational method is presented for solving singularly perturbed delay differential equations with negative shift whose solution has boundary layer. First, the second order singularly perturbed delay differential equation is replaced by an asymptotically equivalent first order delay differential equation. Then, Trapezoidal integration formula and linear interpolation are employed to get three term recurrence relation which is solved easily by Discrete Invariant Imbedding Algorithm. The method is demonstrated by implementing several model examples by taking various values for the delay parameter and the perturbation parameter.

Transcatheter embolotherapy after external surgical stabilization is a valuable treatment algorithm for patients with persistent haemorrhage from unstable pelvic fractures: Outcomes of a single centre experience

PurposeThe management of hemodynamically unstable patients with severe pelvic fractures remains a challenge. Various treatment strategies have been advocated. This study analyzed the value of transcatheter angiographic embolization (TAE) for persistent haemodynamic instability after initial fracture stabilization. Materials and methodsFrom January 2002 to July 2011, 803 patients were identified with pelvic fractures, 295 of them (37%) were presenting with unstable pelvic fractures. Fifteen patients, all with unstable fractures (2%), remained hypotensive (systolic blood pressure<90mmHg) despite adequate fluid resuscitation and emergent surgical fracture stabilization, subsequently underwent TAE. ResultsThe median age in the TAE-group was 57.9 years±20.12 (min 22; max 82) and the median ISS (injury severity score) was 35.8±11.7 (min 22; max 66). 13 out of 15 patients (87%) received initial external fixation and 2 patients (13%) where treated with a C-Clamp before TAE. Radiological success, defined as absence of contrast extravasation on completion angiography, was observed in all 15 patients. In total 3 (20%) patients died during the period of hospitalization, none of them owing to persistent or recurrent pelvic haemorrhage. In the remaining 12 patients, no early or late complications of TAE were identified. Preperitoneal pelvic packing was performed in two patients, both had open pelvic fractures. ConclusionTAE is a safe and very efficient procedure to treat persistent haemorrhage in patients with unstable pelvic fractures initially treated by surgical fracture stabilization procedures. On the long term, no early or late bleeding recurrence could be demonstrated with an overall survival rate of 80%. Preperitoneal pelvic packing can be reserved for patients with open fractures and active bleeding.

Editorial: Endpoints for Radiofrequency Ablation in Barrett's Dysplasia

The need to combat the rising incidence of esophageal adenocarcinoma with its dismal prognosis has led to increasing development of many endoscopic treatments for Barrett's esophagus (BE). Radiofrequency ablation (RFA) has been shown to be a safe and effective endoscopic treatment modality for dysplastic BE. The durability of successful eradication of dysplastic BE has been reported by earlier studies with limited sample size, follow-up time, and inadequate cohort of patients with high-grade dysplasia or intramucosal cancer. In this issue, Orman and colleagues present their findings from their single center, retrospective cohort of patients who underwent RFA with post-treatment surveillance. They report a low recurrence rate of 5.2% per year. There were no clinical characteristics found to be associated with BE recurrence in terms of length of segment or degree of dysplasia. Complete eradication of dysplasia (CE-D) or intestinal metaplasia was determined after a single post-RFA endoscopic examination with biopsies. This is an area of controversy as previous studies have used a minimum of two negative examinations before CE can be claimed. There are also limitations from sampling error during surveillance biopsies and the loss of a third of all post-RFA patients during surveillance. Multicenter, prospective studies with adequate follow-up are still needed before we can draw recommendations when to adequately cease post-treatment surveillance and to identify patients with increased risk of either recurrence or progression.

A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of measures with the th generating measure of the Nikishin system supported on an interval with for all . It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a -term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period . (The limit values depend only on the positions of the intervals .) Taking these periodic limit values as the coefficients of a new -term recurrence relation, we construct a canonical sequence of monic polynomials , the so-called Chebyshev-Nikishin polynomials. We show that the polynomials themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the th generating measure being absolutely continuous on . In this way we generalize a result of the third author and Rocha [22] for the case . The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for functions of the second kind of the Nikishin system for .Bibliography: 27 titles.

Orthogonal polynomials of the [formula omitted]-linear generalized minimal residual method

The speed of convergence of the R-linear GMRES method is bounded in terms of a polynomial approximation problem on a finite subset of the spectrum. This result resembles the classical GMRES convergence estimate except that the matrix involved is assumed to be condiagonalizable. The bounds obtained are applicable to the CSYM method, in which case they are sharp. Then a new three term recurrence for generating a family of orthogonal polynomials is shown to exist, yielding a natural link with complex symmetric Jacobi matrices. This shows that a mathematical framework analogous to the one appearing with the Hermitian Lanczos method exists in the complex symmetric case. The probability of being condiagonalizable is estimated with random matrices.

Long term survival after trimodal therapy in malignant pleural mesothelioma

Trimodal therapy results in long term survival in a small fraction of patients with malignant pleural mesothelioma, particularly in patients having epithelial histology, R0-resection and no nodal involvement. This study analyses the outcome after trimodal therapy including extrapleural pneumonectomy. From 2000 to 2005 41 patients with histologically verified malignant pleural mesothelioma were included. Diagnosis and nodal status were confirmed by surgery. 21 patients (51%) underwent trimodal therapy with 655 days (63-2,567 days) of median follow-up. Postoperative complications, mortality, long term survival and recurrence rates were analysed retrospectively. Neoadjuvant chemotherapy consisted of a combination of platinum based agents (n = 19) with gemcitabine (n = 15) or pemetrexed (n = 4). Extrapleural pneumonectomy was the standard procedure for surgery. 13 patients (62%) had postoperative complications. 16 patients (76%) received postoperative adjuvant radiotherapy. There was a 30-day mortality of 4.8% in the trimodal group. Survival rates in the trimodal group were 71% after one, 28% after two and 10% after five years. There were no significant differences regarding age, tumour stage, cell type or lymph node involvement. Tumour recurrence occurred after one and two years in 44% and in 83% respectively. The majority of patients considered for surgical resection of malignant pleural mesothelioma have regionally advanced disease. In those receiving trimodal therapy long term survival is achieved only in a minority of patients. In view of the time consuming and intensive treatment it should be offered only in carefully selected patients as new surgical approaches such as pleurectomy/decortication have shown high efficacy rates regarding patients' survival.

A basic class of symmetric orthogonal polynomials of a discrete variable

By using a generalization of Sturm–Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal polynomials, is introduced. The standard properties of these polynomials, such as a second order difference equation, an explicit form for the polynomials, a three term recurrence relation and an orthogonality relation are presented. It is shown that two hypergeometric orthogonal sequences with 20 different weight functions can be extracted from this class. Moreover, moments corresponding to these weight functions can be explicitly computed. Finally, a particular example containing all classical discrete symmetric orthogonal polynomials is studied in detail.

Spectral Analysis of Certain Schrödinger Operators

The $J$-matrix method is extended to difference and $q$-difference operators and is applied to several explicit differential, difference, $q$-difference and second order Askey-Wilson type operators. The spectrum and the spectral measures are discussed in each case and the corresponding eigenfunction expansion is written down explicitly in most cases. In some cases we encounter new orthogonal polynomials with explicit three term recurrence relations where nothing is known about their explicit representations or orthogonality measures. Each model we analyze is a discrete quantum mechanical model in the sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages].

Q-Shock soliton evolution

By generating function based on Jackson’s q-exponential function and the standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to q-Hermite polynomials with triple recurrence relations similar to [1], our polynomials satisfy multiple term recurrence relations, which are derived by the q-logarithmic function. It allows us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We find solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole–Hopf transformation we obtain a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere, single and multiple q-shock soliton solutions and their time evolution are studied. A novel, self-similarity property of the q-shock solitons is found. Their evolution shows regular character free of any singularities. The results are extended to the linear time dependent q-Schrödinger equation and its nonlinear q-Madelung fluid type representation.

What is the best treatment of drug-resistant trigeminal neuralgia in patients affected by multiple sclerosis? A literature analysis of surgical procedures

ObjectiveDrug-resistant trigeminal neuralgia (TN) can complicate the clinical course of patients affected by multiple sclerosis (MS). Various surgical procedures have been reported for the treatment of this condition, but there is no agreement on the best management of these patients. To our knowledge, there is no critical literature analysis focusing on this particular topic. The aim of this study was to evaluate the clinical outcome of different surgical procedures utilized for drug-resistant TN in MS patients. MethodsWe reviewed the literature about the studies reporting on surgical treatment of drug-resistant TN in MS patients. Case reports and case series less than 4 patients were excluded from the analysis. Nineteen studies were selected for the statistical analysis. To reduce the variability of the data, the selected studies were evaluated for the following outcome parameters: acute pain relief rate (APR), rate of recurrence (RR), pain free at follow-up rate (PF at FU) and complication rate (CR). For the statistical analysis, chi-square statistic, using the Fisher's exact test was utilized. ResultsThere was no procedure statistically superior in terms of APR rate in MS patients following the surgical treatment of TN. The highest RR was observed for percutaneous balloon compression (PBC) (60.2±14.4%). This result was statistically significant when compared to gamma knife surgery (GKS) (p=0.0129) and microvascular decompression (MVD) (p=0.0281). MVD together with percutaneous radiofrequency rhizothomy (PRR) was associated with a statistically better PF at FU rate (56.5±16.8% and 73.5±14.2%, respectively). However PBC and MVD showed statistical significant minor CR compared to other techniques (no complications and 18.7±17.4%, respectively). ConclusionOur study shows no differences in the short term results among different procedures for TN in MS patients. Each technique demonstrate advantages and limits in terms of long term pain, recurrence rate and complication rate. Each patient should be accurately informed on pros and cons of each procedure in order to be involved in the most appropriate choice.