Narrow Results

In this paper, we describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree $3.$ As an application, we establish some divisibility properties of the associated ternary recurrence sequence by primes $p$, thus greatly extending recent work of Evink and Helminck [Tribonacci numbers and primes of the form $p=x^2+11y^2$, Math. Slovaca69(3) (2019) 521–532] and of Faisant [On the Padovan sequence (2019), https://arxiv.org/abs/1905.07702]. We also prove some new results on the number of solutions of the characteristic equation of the recurrence sequence modulo $p$.

We show that various aspects of k-automatic sequences — such as having an unbordered factor of length n — are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems.

In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution ${\mathbb{P}}_{\rho }$ of OQRW. This sheds new light on some properties of the measure ${\mathbb{P}}_{\rho }$. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of $\phi$-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.

Recurrence, pseudorecurrence, and chaotic solutions for a continuum Hamiltonian system in which there exist spatial patterns of solitary wave structures are investigated using the nonlinear Schrödinger equation (NSE) with cubic and quintic terms. The theoretical analyses indicate that there may exist Birkhoff's recurrence for the arbitrary parameter values. The numerical experiments show that there may be Fermi–Pasta–Ulam (FPU) recurrence, pseudorecurrence, and chaos when different initial conditions are chosen. The fact that the system energy is effectively shared by finite Fourier modes suggests that it may be possible to describe the continuum system in terms of some effective degrees of freedom.

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry.

By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.

The standard map x′ = x + y′, y′ = y + (K/2π)sin(2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.

We briefly present lag sequential analysis for behavioral streams, a commonly used method in psychology for quantifying the relationships between two nominal time series. Cross recurrence quantification analysis (CRQA) is shown as an extension of this technique, and we exemplify this nominal application of CRQA to eye-movement data in human interaction. In addition, we demonstrate nominal CRQA in a simple coupled logistic map simulation used in previous communication research, permitting the investigation of properties of nonlinear systems such as bifurcation and onset to chaos, even in the streams obtained by coarse-graining a coupled nonlinear model. We end with a summary of the importance of CRQA for exploring the relationship between two behavioral streams, and review a recent theoretical trend in the cognitive sciences that would be usefully informed by this and similar nonlinear methods. We hope this work will encourage scientists interested in general properties of complex, nonlinear dynamical systems to apply emerging methods to coarse-grained, nominal units of measure, as there is an immediate need for their application in the psychological domain.

In this article $J^{{ℱ}}$-class operators are introduced and some basic properties of $J^{{ℱ}}$-vectors are given. The $J^{{ℱ}}$-class operators include the $J$-class operators and $J^{\mathrm{\text{mix}}}$-class operators introduced by Costakis and Manoussos in 2008. This class also includes the $J^{\mathrm{\text{wmix}}}$-class and $J^{\mathrm{\text{erg}}}$-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the $J^{{ℱ}}$-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of $J^{{ℱ}}$-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.

Herein is described a haemodialysis patient with bilateral carpal tunnel syndrome suffering from recurrence unilaterally after undergoing numerous surgeries of varying methods. On the left side, she received carpal tunnel release via open method in our clinic, and has not suffered from recurrence in eight years. On the right side, she received endoscopic carpal tunnel release twice in seven years, and subsequently underwent open carpal tunnel release in our clinic for recurrence. For carpal tunnel syndrome in haemodialysis patients, we recommend open surgery rather than endoscopic surgery.

In long-term haemodialysis patients, carpal tunnel syndrome (CTS) frequently occurs as a result of amyloid deposition, originating from beta-2 microglobulin, to the flexor retinaculum, paratenons and tendons themselves, which leads to an increase in carpal canal pressure and compression of the median nerve. Surgical procedures can rectify the condition, but continuing maintenance haemodialysis sometimes causes recurrence.

We endoscopically operated 1848 hands primarily, 104 recurrent post-endoscopic procedure hands and 130 recurrent post-open procedure hands using the Universal Subcutaneous Endoscope (USE) system, then analysed clinical symptoms and electrophysiological recovery for more than six months post-operatively. The patients were satisfied with the clinical results. Optimal electrophysiological improvements were reported. There were no statistical differences between three groups, except in recovery of touch sensation, which was better in the post-endoscopic group than in the post-open group. There were no complications in this series.

Our minimally invasive endoscopic procedure, using the USE system, is safe and effective for primary and recurrent CTS in haemodialysis patients.

Neurothekeomas are rare, benign soft tissue tumours that are considered to be of nerve sheath origin. We present a case series of three patients who have neurothekeomas of the upper limb. There was a remarkably high tumour recurrence rate. The factors contributing to this outcome were studied and a clinicopathological review was performed.

Aneurysmal bone cysts are benign lesions that rarely occur in the bones of the hand. Curettage and bone grafting are the most common treatment modalities performed considering the possible functional loss after total excision. Four cases of aneurysmal bone cyst of the hand were identified. Three out of four cases were treated initially with curettage and bone grafting alone without any other local therapy. One had total excision of the lesion. There were two recurrences in the curettage group. Both were treated with excision and followed up for a minimum of 12 months. Curettage alone was associated with high recurrence rates. Total excision of the lesion was the most successful procedure employed. In view of the high recurrence rates following curettage alone and non-neoplastic nature of the lesion, adjuvant treatment modalities such as cryosurgery, electric cauterisation should be considered for initial treatment of ABC. Resection should be preserved for recurrent cases.

Thirty-nine consecutive patients with little finger Dupuytren's contracture underwent open fasciectomy. Diseased abductor digiti minimi (ADM) pretendinous (PT) cords were identified. The mean pre-operative PIPJ contracture was 77° in the PT group and 66° in the ADM group. Mean residual deformity was 12° in the PT group and 9° in the ADM group. At six months, ten out of 27 patients had developed a recurrent deformity in the PT group (mean 24°) and seven out of 11 in the ADM group (mean 18°). There was no statistically significant difference between the two groups at any stage. Dupuytren's contracture of the little finger is as a result of an ADM cord in 29% of cases. In this series it led to an isolated contracture of the PIPJ in the majority of cases and rarely affected the MCPJ. Disease of the ADM cord was not associated with a difference in contracture or prognosis compared to a PT cord.

Giant cell tumours of tendon sheath of hand present a surgical dilemma due to their high incidence of local recurrence. We present a case series of 46 patients with 47 histologically confirmed giant cell tumours of tendon sheath over a ten-year period from 1998 to 2008. The mean follow-up was 47 months (range 25–124 months). We identified tumours with bony erosions and piecemeal resections as predictors of recurrence. Our recurrence rate of 9% was at the lower end of spectrum of previously published reports (range 7%–44%). We recommend "en-masse" excision of these tumours. All the patients with suspicion of these tumours should have preoperative radiographs to identify erosions. A thorough curettage of the bone should be done in cases with osseous erosion to prevent recurrence. Patients with these risk factors should be followed up annually for five years and be warned about recurrence.

Primary synovial chondromatosis is a proliferation of cartilaginous bodies within the synovial membrane, tendon sheath or bursa. It is a rare orthopaedic entity especially when it occurs in the distal radioulnar joint. We report a 27-year-old man with recurrent synovial chondromatosis, nine years after his first operation. Although rare, synovial chondromatosis must be considered in the differential diagnosis in a patient with pain and swelling of the distal radioulnar joint. From our literature review, we report a 17% (5/30) recurrence rate for synovial chondromatosis involving the wrist joint which has not been documented in the medical literature previously.

The purpose of this study was to evaluate risk factors for re-recurrent carpal tunnel syndrome (CTS) in long-term renal hemodialysis (HD) patients. Fifteen wrists of ten HD patients, follow-up period of minimum seven years after reoperation of CTS, were included in this study. Duration of HD, period from first operation to reoperation, shunt side was involved or not, presence of trigger finger and cervical destructive spondyloarthritis, and operative procedure performed during the reoperation (synovectomy was performed or not) were evaluated. Re-recurrent CTS was identified in four out of 15 wrists (27%). The period from first operation to reoperation, which was 3.8 years in the re-recurrence group and eight years in the no re-recurrence group, and the operative procedure had significant differences (synovectomy groups had no re-recurrence vs. no synovectomy groups had 82% re-recurrence). The results of this study suggested that synovectomy would be necessary for recurrent CTS in HD patients.

The purpose of this systematic review is to determine the incidence of complications and the recurrence rate of a volar wrist ganglion following arthroscopic resection.

We performed a systematic review of English and non-English articles using Google Scholar, Medline, and Web of Knowledge. Articles were screened for study inclusion by three independent reviewers using the terms "arthroscopic treatment of volar wrist ganglion" and "arthroscopic resection of volar wrist ganglion". Inclusion criteria: (1) level I–V evidence, (2) documentation of the number of wrists subjected to surgery, (3) documentation of surgical techniques used on wrists, and (4) documentation of surgical or post-surgical complications and recurrence rate of a volar wrist ganglion after arthroscopic resection. A complication was defined as an adverse outcome that was directly related to the operative procedure. Between 2001 and 2012, 13 articles met the inclusion criteria. Two articles were excluded and 11 were reviewed. A total of 232 wrists underwent arthroscopic surgery with 14 recurrences.

The recurrence rate ranged from 0 to 20%, with mean of 6.03%. There were 16 (6.89%) complications. There was no connection with the ganglion in six wrists, three haematomas, three cases of neuropraxia of the dorsal radial nerve, two partial lesions of the median nerve, and two lesions of a branch of the radial artery. Patients did not have a decrease in the arc range of motion. Treatment of volar ganglia of the midcarpal joint was technically difficult and associated with a higher number of complications.

In general, arthroscopic resection results in fewer complications and lower recurrence rates than an open surgical approach, but there is no clear evidence of such an advantage for arthroscopic resection of a volar wrist ganglion. Additional prospective, controlled clinical trials will be essential to address this important issue.

Purpose: Thrower’s Elbow (TE) is an overuse injury of the medial elbow in pediatric, throwing athletes. The purpose of this study is to report the outcomes of TE in pediatric, throwing athletes.

Methods: Between 1996 and 2018, 30 athletes at an average of 12.2 years old (range: 7.9 to 15.4 years old) were treated for TE. Demographic, radiographic, and treatment variables were recorded. Outcome measures included return to original position, inability to return to sport, surgery, premature physeal arrest, and recurrence of elbow pain.

Results: Prior to their initial presentation, 25 athletes (25/30, 86%) stated they continued to throw despite arm fatigue/pain. At an average follow-up of 26 months (range: 6 months to 122 months), 22 athletes (22/30, 73%) returned to their original position at an average five months from their initial presentation. Recurrence of elbow pain occurred in 27 athletes (27/30, 90%). No athletes had premature physeal arrest of the medial epicondyle.

Conclusions: The majority of athletes with TE returned to their original position or sport after initial treatment, but 90% of those athletes developed recurrent pain. It is important to educate athletes about overuse throwing injuries and prevention by adhering to established guidelines.

Quantum Bernoulli random walks can be realized as random walks on the dual of SU(2). We use this realization in order to study a model of dynamic quantum Bernoulli random walk with time-dependent transitions. For the corresponding dynamic random walk on the dual of SU(2), we prove several limit theorems (local limit theorem, central limit theorem, law of large numbers, large deviation principle). In addition, we characterize a large class of transient dynamic random walks.

We study the relationship between the type of the random walk on some random trees and the structure of those trees in terms of fractal and resistance dimensions. This paper generalizes some results of Refs. 8–10.