Term Recurrence Research Articles

A fast algorithm for evaluation of normalized Hermite functions

An algorithm for computing the normalized Hermite Functions, h n (x) in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree n, the algorithm is significantly faster than the O(n) complexity of the well known three term recurrence relation. Comparable accuracy is achieved in no more than $O(\sqrt{n})$ operations, and for arguments bounded away from $\pm \sqrt {2n}$ , only $O(\sqrt{\ln{n}})$ operations.

A class of matrix-valued polynomials generalizing Jacobi polynomials

A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a three term recurrence relation, integral inter-relations, and weak orthogonality relations.

Late Vascular Complication Associated With Panfacial Fractures

embers suggests a sporadic mutation affecting one f the genes responsible for bone homeostasis in this atient. Cranial nerve compression has been described in atients’ with van Buchem disease; however, cerebroascular accidents have not been identified comonly in these patients. Although the patient decribed some disturbance in facial sensation and isplayed skull bone thickening radiographically, it is ifficult to say whether the condition contributed to he cerebrovascular accidents without radiologic evience of compression of the brain or vascular canals n the skull base. Previous reports of surgical intervention on paients with van Buchem disease and ADO have ndicated normal healing although no comments have een made on longer term recurrence at surgicallyddressed sites. We were concerned about the healng capacity of the bone and carried out more limited urgery at the first intervention. These fears seem to e unfounded and, ironically with the wisdom of indsight, the history of palmidronate infusions hould have caused greater concern.

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae Z n = T ( n ) Z n - 1 + U ( n ) Z n - 2 ( n ⩾ 2 ) with sequences of integers ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 , which are ultimately periodic modulo m, e.g. ( T ( n ) mod m ) n ⩾ 0 = ( a 0 , a 1 , a 2 , … , a ρ , T 1 , T 2 , … , T w ¯ ) ( U ( n ) mod m ) n ⩾ 0 = ( b 0 , b 1 , b 2 , … , b ρ , U 1 , U 2 , … , U w ¯ ) . In a former paper of this journal the authors computed explicitly the coefficients of a linear three-term recurrence formula for z n = Z rn + i with 0 ⩽ i < r , when ( T ( n ) ) n ⩾ 0 and ( U ( n ) ) n ⩾ 0 belong to regular or non-regular Hurwitz-type continued fraction expansions. Using this result we show now that the sequence ( Z n ) n ⩾ 0 is ultimately periodic modulo m. As a consequence, for Hurwitz-type continued fraction expansions α = [ a 0 ; T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ or α = [ a 0 ; a 1 , T 1 ( k ) , … , T r ( k ) ¯ ] k = 1 ∞ with polynomials T 1 ≠ const . , T 2 , … , T r we deduce for all positive integers a and m that liminf q q ‖ q α ‖ = 0 , where q ≡ a mod m and ‖ · ‖ denotes the distance from the nearest integer. Finally, we are particularly interested in the recurrence formula Z n = ( an + b ) Z n - 1 + cZ n - 2 ( n ⩾ 2 ) and compute the length of a period of the sequence ( Z n mod m ) n ⩾ 0 , when ( a , m ) divides b, and ( c , m ) = 1 . This generalizes former results of the authors dealing with regular continued fraction expansions of the numbers exp ( 1 / s ) for integers s ⩾ 1 .

Horizontal supraglottic partial laryngectomy for selected squamous carcinoma of the vallecula

Our aim was to determine the incidence of local control in patients with selected squamous carcinoma of the vallecula treated with horizontal supraglottic laryngectomy; to analyze the consequences of local recurrence in terms of nodal recurrence, distant metastasis, survival, causes of death, overall local control, and laryngeal preservation; and to identify any clinical factors predictive of these outcomes. This was a retrospective nonrandomized case series in a university teaching hospital. An inception cohort of 95 previously untreated patients were followed until death or for a minimum of 5 years. According to the 2002 Union Internationale Contre le Cancer (UICC) staging classification system, the tumor was classified as T1, T2, and T3 in 13, 60, and 22 patients, respectively, while disease in 67 patients was considered to be in stages III to IV. All patients underwent a horizontal partial supraglottic partial laryngectomy. Ninety-four patients had an associated neck dissection. An induction chemotherapy regimen was used in 91 patients; postoperative radiation therapy was given for 49 patients. The main outcome measures were local recurrence, nodal recurrence, distant metastasis, and survival. The 1-, 3-, and the 5-year actuarial survival estimates were 86.3%, 64.2%, and 47.4%, respectively. Overall, the main causes of death were as follows: metachronous second primary tumor (47.2% of patients), intercurrent disease (16.7%), distant metastasis (15.3%), local recurrence (6.3%), and nodal recurrence, (4.2%). The 1-, 3-, and 5-year actuarial local recurrence rates were 4.5%, 11%, and 11%, respectively. Nine patients developed a local recurrence; 3 were successfully salvaged. Using multivariate analysis, no single variable was found to increase the risk for local recurrence. The overall laryngeal preservation rate and local control rate were 89.5% (85/95) and 93.4% (89/95), respectively. Local recurrence was associated with a significant increase in nodal recurrence (p <.04) and distant metastasis (p = .03). Based on this experience, horizontal partial supraglottic laryngectomy appears to be a valid approach for functional organ-preservation in patients with selected T1-T3 SCC of the vallecula.

Orthogonal polynomials from Hermitian matrices

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.

Surgical Treatment of Hidradenitis Suppurativa with Gentamicin Sulfate

BACKGROUND AND METHOD This article describes and discusses a prospective randomized study with gentamicin sulfate in the surgical treatment of hidradenitis suppurativa. The purpose of the study was to investigate whether enclosure of antibiotics after primary excision and closure reduces the number of postoperative infections. Therefore, the hidradenitis lesions were excised and closed with or without enclosure of a gentamicin-collagen sponge (GC). RESULTS A total of 200 patients were included in the study. Seventy-six patients underwent surgical excision with primary closure (PC), and 124 PCs over a GC. After 1 week there were significantly fewer complications (infection, dehiscence, etc.) in the GC group, 35% versus 52%; after 3 months the complications in both groups were comparable, 12% versus 19% (Table 2). The mean period of wound healing was 21 days in the first group and 24 days in the second group. The recurrence rate after 3 months was comparable in both groups, 40% versus 42%.TABLE 2: Results: Complications One Week and Three Months PostoperativeCONCLUSION This study shows that enclosure of gentamicin after primary excision of hidradenitis suppurativa reduces the number of complications 1 week postoperatively. Furthermore, in 65% of the patients treated with gentamicin, the wound was completely healed within 2 months. There is no effect on the long term recurrence rate, as expected.

Early onset recurrent subtype of adolescent depression: clinical and psychosocial correlates

Evaluated trajectories of adolescent depression and their correlates in a longitudinal study of a community sample: early onset (by age 15) with major depression (MDE) recurrence between 15 and 20; early onset with no recurrence; later onset of major depression after age 15 with and without recurrence by 20; and never-depressed. Eight-hundred sixteen youth were studied at age 15, and 699 were included at age 20, with diagnostic evaluations and assessments of functioning in major roles. Youth with early onset and recurrent MDE differed from both those with early onset but nonrecurrent MDE and those with later onset-no recurrence in terms of clinical features, adolescent social functioning, and later psychosocial adjustment. The early onset recurrent depressed youth had more severe, chronic, suicidal depressions, greater anxiety comorbidity, worse social functioning at 15, and poorer psychosocial, especially social, outcomes at 20. Youth with depression by 15 with recurrence by age 20 may represent a high-risk group, with likely life-course-persistent depression and maladjustment. Community youth whose early depression does not recur by age 20, or who have onset with no recurrence after age 15, may have more benign and possibly limited depressions. Later onset with recurrence is also a group at risk for dysfunctional outcomes, requiring further follow-up.

Problems and Techniques

The two papers in this issue are about short recurrences for computing bases of Krylov subspaces, and numerical methods for computing the Laplace transform and its inverse. Systems of linear equations $Ax=b$, where the matrix A is large and sparse (i.e., only a few elements of A are nonzero), are often solved by so-called Krylov subspace methods. Such methods restrict operations involving A to matrix vector products. In the simplest case the iterates $x^{(k)}$ are computed from vectors in the Krylov space $\mathcal{K}_k=span\{b, Ab,\ldots, A^{k-1}b\}$. For instance, when A is Hermitian positive-definite (or real symmetric positive-definite) the conjugate gradient method computes $x^{(k)}$ as a linear combination of $x^{(k-1)}$ and a “direction vector” $p_k$. The direction vectors form an A-orthogonal basis for the Krylov space $\mathcal{K}_k$, that is, $p_i^*Ap_j=0$ for $i\neq j$ (the superscript $*$ denotes the conjugate transpose). As a consequence, a direction vector $p_k$ can be computed from $Ap_{k-1}$, $p_{k-1}$ and $p_{k-2}$. This is called a 3-term recurrence. However, if A is a general matrix, then it is well known that the direction vectors cannot be computed with 3-term recurrences—even if one relaxes the orthogonality to B-orthogonality, where B is any Hermitian positive-definite matrix. The question is, if 3-term recurrences are not possible, then how short can the recurrences possibly be? In their paper “On Optimal Short Recurrences for Generating Orthogonal Krylov Subspace Bases,” J. Liesen and Z. Strakoš derive necessary and sufficient conditions for a nonsingular matrix A to admit $(s+2)$-term recurrences for $s\geq 1$. They also give a comprehensive overview of work on short recurrences for Krylov subspace methods. This is a clear and carefully written paper, and the authors go to great lengths to illuminate the subtle issues involved. In the second paper, “The Bad Truth about Laplace's Transform,” Charles Epstein and John Schotland are concerned with the difficulties of inverting the Laplace transform. This may be necessary, for instance, when solving inverse scattering problems from optical tomography and image reconstruction. The Laplace transform of a real function $f(x)$ is defined as the integral $\mathcal{L}\>f(x)=\int_0^{\infty}{e^{-xy}f(y)dy}.$ Inverting $\mathcal{L}$ to recover $f(x)$ is an ill-posed problem. Ill-posed problems are extremely hard to solve numerically because they may not have a solution, the solution may not be unique, or the solution may not depend continuously on the data. The authors use harmonic analysis to derive fast algorithms for approximating the Laplace transform and its inverse (when the function values are sampled on geometrically uniformly spaced data), and to derive regularized inverses.

Surgical Treatment of Hidradenitis Suppurativa with Gentamicin Sulfate: A Prospective Randomized Study

This article describes and discusses a prospective randomized study with gentamicin sulfate in the surgical treatment of hidradenitis suppurativa. The purpose of the study was to investigate whether enclosure of antibiotics after primary excision and closure reduces the number of postoperative infections. Therefore, the hidradenitis lesions were excised and closed with or without enclosure of a gentamicin-collagen sponge (GC). A total of 200 patients were included in the study. Seventy-six patients underwent surgical excision with primary closure (PC), and 124 PCs over a GC. After 1 week there were significantly fewer complications (infection, dehiscence, etc.) in the GC group, 35% versus 52%; after 3 months the complications in both groups were comparable, 12% versus 19% (Table 2). The mean period of wound healing was 21 days in the first group and 24 days in the second group. The recurrence rate after 3 months was comparable in both groups, 40% versus 42%. This study shows that enclosure of gentamicin after primary excision of hidradenitis suppurativa reduces the number of complications 1 week postoperatively. Furthermore, in 65% of the patients treated with gentamicin, the wound was completely healed within 2 months. There is no effect on the long term recurrence rate, as expected.

Enumeration via ballot numbers

Several interesting combinatorial coefficients such as the Catalan numbers and the Bell numbers can be described either via a 3-term recurrence or as sums of (weighted) ballot numbers. This paper gives some general results connecting 3-term recurrences with ballot sequences with several applications to the enumeration of various combinatorial instances.

Estudio temporal de los diferentes patrones metastásicos en la progresión del melanoma cutáneo

Introduction Compared with other tumors, melanoma has displayed one of the largest increases in incidence in recent years, and it is known to have a high metastatic potential. In cases of metastasis, approximately two-thirds of patients have lymph-node metastases and one-third develop systemic metastases. However, few studies have been reported that analyzed different metastatic patterns according to the natural history of melanoma. The main aim of this study was to analyze the different metastatic pathways and patterns and to assess the time course of development of metastases from cutaneous melanoma. Material and methods A retrospective study was performed in 575 patients with onset of primary melanoma between 1990 and 2004. During follow-up, 67 patients developed metastases. Different pathways for metastasis were established and evaluated. We identified 4 metastatic pathways according to the metastatic pattern during progression of the melanoma. The time course of metastases was also evaluated. Finally, we analyzed melanomas with local recurrence in terms of whether or not systemic progression occurred. Results Melanoma metastases first occurred in local lymph nodes in 55.2 % of the patients. Initial metastasis was systemic in 14.9 % of the patients. The anatomical location and tumor thickness influenced which metastatic pathway was followed. Distant metastases occurred after a mean of 25 months regardless of the pathway followed. Conclusions The development of distant metastases displays a constant time course and the time to onset is independent of the metastatic pathway. This observation may explain why sentinel lymph node biopsy has a limited impact on overall survival of melanoma patients.

Long Term Recurrence of Digital Dermatitis after Treatment with Lincomycin HCl on a Commercial Dairy

The results of this study demonstrate that control of DD needs to be continuously monitored.

Does HPV-status 6–12 months after treatment of high grade dysplasia in the uterine cervix predict long term recurrence?

Background Women once treated for high grade cervical dysplasia have a high long term risk for developing new dysplasia or cancer. Objectives To investigate if human papilloma virus (HPV)-negativity after treatment of cervical dysplasia reduces the need for frequent long term follow up. Design Case/control study based on archival smears. Methods Women with cervical intraepithelial neoplasi (CIN)2-3, treated for dysplasia and with recurrence of CIN2+ more than 2 years after treatment were compared with controls without recurrence, matched for age and date of treatment. High risk-HPV-DNA were analysed with PCR from two archival smears per woman. Mean follow up time was 14.6 years. Results 24% (45/189) of cases and 11% (43/378) of controls were HPV-positive in any of two smears. Odds ratio (OR) = 2.5 (1.6–3.8). Conclusion HPV-status 6–12 months after treatment of high grade dysplasia is of limited value for the design of long term follow up.

QES SYSTEMS, INVARIANT SPACES AND POLYNOMIALS RECURSIONS

Let us denote [Formula: see text], the finite-dimensional vector spaces of functions of the form ψ(x) = pn(x)+f(x)pm(x), where pn(x) and pm(x) are arbitrary polynomials of degree at most n and m in the variable x while f(x) represents a fixed function of x. Conditions on m, n and f(x) are found such that families of linear differential operators exist which preserve [Formula: see text]. A special emphasis is accorded to the cases where the set of differential operators represents the enveloping algebra of some abstract algebra. These operators can be transformed into linear matrix valued differential operators. In the second part, such types of operators are considered and a connection is established between their solutions and series of polynomials-valued vectors obeying three terms recurrence relations. When the operator is quasi-exactly solvable, it possesses a finite-dimensional invariant vector space. We study how this property leads to the truncation of the polynomials series.

Expression of SSX-1 and NY-ESO-1 mRNA in tumor tissues and its corresponding peripheral blood expression in patients with hepatocellular carcinoma

Cancer-testis antigen (CTA) is a family of the most noticeable tumor antigens which could be potential tumor markers for cancer diagnosis. In this research we aimed to investigate the expression of SSX-1 and NY-ESO-1 mRNA, two members of the CTA family, in tissue and peripheral blood of patients with hepatocellular carcinoma (HCC) to assess their feasibility for the immunotherapy and diagnosis of HCC and the association of their expression levels with diverse clinical indicators. Thirty-six north Chinese patients with HCC and 30 normal controls were enrolled in this study. Reverse transcription-polymerase chain reaction (RT-PCR) was used to detect the expression of SSX-1 and NY-ESO-1 mRNA in tumor tissues and corresponding levels in peripheral blood of patients. The positive rates of SSX-1 and NY-ESO-1 mRNA expression were 61.1% (22/36) and 11.1% (4/36), respectively, in cancer tissues; 38.9% (14/36) and 5.6% (2/36), respectively, in the corresponding peripheral blood samples. No positive expression of either SSX-1 or NY-ESO-1 mRNA was detected in the samples of cancer-adjacent tissues, cirrhotic tissues, normal liver tissue or the peripheral blood of control patients. No significant relationship was found between the expression of these two genes and clinical indicators such as age, gender, tumor size, extent of differentiation, serum a-fetoprotein (AFP) level or infection with hepatitis B virus (P > 0.05). The short term recurrence rate was 46.2% (6/13) in patients whose peripheral blood expressed SSX-1 mRNA, while the recurrence rate in patients with negative SSX-1 mRNA was 28.6% (4/14). SSX-1 and NY-ESO-1 antigens might be new potentially promising targets for antigen-specific immunotherapy for HCC. High specific expression of SSX-1 and NY-ESO-1 mRNA suggested that we could apply them as tumor markers. The short term recurrence rate was significantly higher in patients whose peripheral blood expressed SSX-1 mRNA, suggesting that SSX-1 mRNA could be used as indicator for recurrence, metastasis and prognosis of HCC.

Local Approximations Based on Orthogonal Differential Operators

Let M be a symmetric positive definite moment functional and let \(\{P_n^{\cal M}(\omega)\}_{n\in {\Bbb N}}\) be the family of orthonormal polynomials that corresponds to M. We introduce a family of linear differential operators \({\cal K}^n =(-i)^nP_n^{\cal M}(i\frac{d}{dt})\), called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product. We consider a Taylor type expansion of an analytic function f(t), with the values f(n) (t0) of the derivatives replaced by the values \({\cal K}^n[f](t_0)\) of these orthonormal operators, and with monomials (t − t0)n/n! replaced by an orthonormal family of "special functions" of the form \((−1)^n{\cal K}^n[m](t-t_0)\), where \(m(t) = \sum_{n=0}^{\infty} (−1)^n{\cal M}(\omega^{2n}) t^{2n}/(2n)!\). Such expansions are called the chromatic expansions. Our main results relate the convergence of the chromatic expansions to the asymptotic behavior of the coefficients appearing in the three term recurrence satisfied by the corresponding family of orthogonal polynomials PMn(ω). Like the truncations of the Taylor expansion, the truncations of a chromatic expansion at t = t0 of an analytic function f(t) approximate f(t) locally, in a neighborhood of t0. However, unlike the values of f(n)(t0), the values of the chromatic derivatives Kn[f](t0) can be obtained in a noise robust way from sufficiently dense samples of f(t). The chromatic expansions have properties which make them useful in fields involving empirically sampled data, such as signal processing.

Cryptogenic Hemoptysis

Data on hemoptysis of unknown origin (i.e., "cryptogenic") are scarce and the outcome of patients affected is controversial. To describe the clinical spectrum and course of patients with cryptogenic hemoptysis, as well as pathologic findings when surgery is performed. A cohort of 81 patients referred for cryptogenic hemoptysis after clinical evaluation, chest radiography, fiberoptic bronchoscopy, and computed tomographic scan to a respiratory intermediate care and intensive care unit, from December 1995 to August 2004, with a prospective follow-up by visit or telephone interview. The 81 patients (69 males) had a mean cumulative volume of hemoptysis averaging 190 ml on admission. First-line conservative measures and bronchial artery embolization controlled hemoptysis in 73 patients (90%). Emergency surgery was performed in six patients (7%) because of failure of bronchial artery embolization, and secondary surgery was scheduled in a seventh patient. A total of 73 patients were followed for a mean of 47 (+/- 35) months. No lung cancer developed. Hemoptysis recurred in 10 patients (4 within the first year; 6 between 1 and 8 yr later), 2 of whom underwent surgery. A specific bronchial vascular involvement (Dieulafoy disease) was demonstrated in five of the nine patients who had undergone surgery, especially in those with high amounts of bleeding. Cryptogenic hemoptysis may be a life-threatening condition. Nonsurgical approaches provide immediate control of bleeding in most patients with cryptogenic hemoptysis, with few recurrences in both short and long terms. Dieulafoy disease of the bronchus, unsuspected after routine imaging investigations, may be involved in a subset of patients yet to be determined.

Cognate chemokine receptor 1 messenger ribonucleic acid expression in peripheral blood as a diagnostic test for endometriosis

We investigated the expression of the cognate chemokine receptor 1 (CCR1) messenger ribonucleic acid, a G-protein-coupled cognate chemokine receptor with high affinity for RANTES (Regulated upon Activation, Normal T cells Expressed and Secreted), in peripheral blood leukocytes of women with and without endometriosis, and its potential use as a diagnostic test for endometriosis. Because patients with an earlier diagnosis of this disease have a better treatment outcome and a reduced recurrence rate, CCR1 mRNA measurement in the peripheral blood of patients with suspected endometriosis might give us a new perspective in diagnosing and treating this disease earlier and better.

Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions

Three term recurrence relations $y_{n+1}+b_n y_n+a_n y_{n-1}=0$ can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of $n$ with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi’s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with $a_n$ negative and $b_n$ changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion.