Summable Sequences Research Articles

Unsupervised Anomaly Detection in Marine Diesel Engines using Transformer Neural Networks and Residual Analysis

This paper presents a novel unsupervised approach for detecting anomalies in marine diesel engines using a Transformer Neural Network based autoencoder (TAE) and residual analysis with Sequential Probability Ratio Test (SPRT) and Sum of Squares of Normalized Residuals (SSNR). The proposed method can capture temporal dependencies in normal timeseries data without the need for labeled failure data. To assess the effectiveness of the proposed approach, a dataset of faulty data is generated under the same operational profile as the normal training data. The model is trained using normal data, and the faulty data is reconstructed using the trained model. SPRT and SSNR are then used to analyze the residuals from the observed and reconstructed faulty data, with significant deviations exceeding a predefined threshold being identified as anomalous behavior. The experimental results demonstrate that the proposed approach can accurately and efficiently detect anomalies in marine diesel engines. Therefore, this approach can be considered as a promising solution for early anomaly detection, leading to timely maintenance and repair, and preventing costly downtime.

Nuclearity of a Class of Vector-valued Sequence Spaces

In this note, we deal with a perfect sequence space $\lambda$ and a bornological convex space $E$ to introduce and study the space $\lambda(E)$ of totally $\lambda$-summable sequences from $E$. We prove that $\lambda(E)$ is complete if and only if $\lambda$ an $E$ are complete, nuclear if and only if $\lambda$ an $E$ are nuclear. and we make use of a result of Rolnald C. Rosier to give a similar characterization of the nuclearity of $\Lambda\{E\}$ of all absolutely $\lambda-$ summable sequences in a locally convex $E$.

Method for the Formation of Non-Binary Gordon – Mills – Welch Sequences for Digital Information Transmission Systems

In this paper, we develop a method for the formation of non-binary Gordon–Mills–Welch sequences (GMWSs) with a period of N = p^(mn) – 1 that are formed over field GF(p) based on the generalization of the binary sequence formation method. An expression for calculating the vector of decimation indices А_m,n,r of the basic M-sequence (MS) for summable sequences in the synthesis of GMWS is obtained. A method for the formation of non-binary GMWSs for arbitrary MSs is presented. The values of the components of the C_m,n,r shift vector of a basic MS are shown to depend on the distribution of digits at the positions of the pth representation of the corresponding decimation indices.

Construction of a Banach algebra in summable infinite matrices and its realization of the Gelfand transformation

In this paper, we extend the construction of a Banach Algebra from summable sequences to summable matrices. We also find the representation of the Banach algebra as complex-valued continuous functions defined on a Hausdorff space.

Inferring the Mixing Properties of a Stationary Ergodic Process From a Single Sample-Path

We propose strongly consistent estimators of the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence of α-mixing (respectively β-mixing) coefficients of a stationary ergodic process. We further provide strongly consistent estimators of individual α-mixing (respectively β-mixing) coefficients for a subclass of stationary α-mixing (respectively β-mixing) processes with summable sequences of mixing coefficients. The estimators are in turn used to develop strongly consistent goodness-of-fit hypothesis tests. In particular, we develop hypothesis tests to determine whether, under the same summability assumption, the α-mixing (respectively β-mixing) coefficients of a process are upper bounded by a given rate function. Moreover, given a sample generated by a (not necessarily mixing) stationary ergodic process, we provide a consistent test to discern the null hypothesis that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence α of α-mixing coefficients of the process is bounded by a given threshold γ ∈ [0,∞) from the alternative hypothesis that ∥α∥ > γ. An analogous goodness-of-fit test is proposed for the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the sequence of β-mixing coefficients of a stationary ergodic process. Moreover, the procedure gives rise to an asymptotically consistent test for independence.

Establishing a benthic macrofaunal baseline for the sandy shoreline ecosystem within the Gulf Islands National Seashore in response to the DwH oil spill

Sandy shorelines present a first line of defense against the catastrophic effects of storms and oil spills within the coastal zone of the northern Gulf of Mexico. Immediately following the DwH oil spill prior to any spill related impacts, we conducted a rapid response survey of the sandy shoreline benthic macrofauna from throughout the National Park Service - Gulf Islands National Seashore (GINS) in Mississippi and Florida. To characterize pre-spill macrofaunal assemblages, we surveyed seven barrier island or peninsular areas comprising nine exposed and 12 protected shoreline sites. A comparable benthic macrofaunal inventory had been conducted 17 years earlier using a parallel study design. The primary objective of this study was to distinguish hierarchical spatiotemporal scales of macrofaunal variation within the 1993 and 2010 GINS data. We hypothesized that the 1993 GINS macrofaunal inventory baseline was stable, despite multiple disturbances by large storms within the intervening 17-year period. Additionally, the relative importance of hierarchical spatial scales of macrofaunal dissimilarity was examined so suitable scales of macrofaunal variation could be identified for assessments of stressor effects at commensurate scales. An Implicit Nested Mixed Model PERMANOVA using Type 1 sequential Sum of Squares delineated variation components of nested scales which ranked Station &amp;gt; Shore Side &amp;gt; Site &amp;gt; Habitat &amp;gt; District &amp;gt; Year. The Year main factor had the smallest effect on macrofaunal variation, confirming that the 1993 GINS macrofaunal inventory can serve as the foundation for a robust baseline including both the 1993 and the 2010 macrofaunal data for the GINS. A literal Hierarchical Nested Mixed Model PERMANOVA using Type 1 sequential Sum of Squares (SS) partitioned effects among nested factors and their interactions. Definitive macrofaunal variation was expressed for all combinations of two levels for each of the three spatially nested fixed factors, District, Shore Side, and Habitat. Variation in macrofaunal dissimilarity for combined levels of fixed factors reflected corresponding differences in the macrofauna. The use of sandy shoreline macrofaunal assemblages as ecological indicators would fulfill the need to focus on cumulative effects of oil spills and should be eminently tractable when responses and impacts are considered on commensurate scales.

An anisotropic summability and mixed sequences

In this paper we define and study a vector-valued sequence space, called the space of anisotropic (s, q, r)-summable sequences, that generalizes the classical space of (s; q)-mixed sequences (or mixed (s; q)-summable sequences). Furthermore, we define two classes of linear operators involving this new space and one of them generalizes the class of (s; q) -mixed linear operators due A. Pietsch. Some characterizations, inclusion results and a Pietsch domination-type theorem are presented for these classes. It is worth mentioning that some of these results are new even in the particular cases of mixed summable sequences and mixed summing operators.

OGC P27 A Review of Minimally Invasive Oesophagogastric Cancer Surgery in a High-Volume UK Unit

Abstract Background Oesophagogastric (OG) cancer resections are technically challenging and associated with peri-operative morbidity. Practice has evolved from open to minimally invasive surgery (MIS), with ERAS has seen peri-operative outcomes improve. OG surgeons have not transitioned to MIS at the rate of others; 2021 NOGCA results showed that 17.6% of oesophagectomies and 17% of gastrectomies were performed as MIS. Concerns regarding the technical ability to perform safe anastomoses and appropriate oncological clearance have been cited as common reasons for surgeons not transitioning to MIS. GSTT is a high-volume OG unit which has transitioned from a fully open to predominantly MIS practice. The unit has performed over 100 minimally invasive Ivor Lewis oesophagectomies and 97 laparoscopic gastrectomies, it was therefore decided to analyse the unit's peri-operative outcomes to ensure that appropriate standards were maintained during this change in practice. The primary aim of this study was to examine a decade of peri-operative outcomes for Ivor Lewis oesophagectomy and gastrectomies, comparing parallel open and MIS cohorts. The secondary aim is to perform a sequential cumulative sum control chart (CUSM) analysis of the cohort to determine whether a learning curve effect for OG MIS can be demonstrated. Methods A retrospective analysis of a prospectively maintained comprehensive local database was performed. The results were cross-checked with NOCGA data, and a notes review was performed to capture any missing data.All Ivor Lewis oesophagectomy and gastrectomy (total, extended total and subtotal) performed for adenocarcinoma, squamous cell carcinoma, neuroendocrine tumours and high grade dysplasia at Guy's and St Thomas’ NHS Foundation Trust, London, UK over a ten-year period (January 2012- December 2021) were included. MIS for oesophagectomy was defined as an Ivor Lewis oesophagectomy performed with both laparoscopic and thoracoscopic phases. A hybrid (laparoscopic abdomen and open chest phase) oesophagectomy was grouped with a two-phase open procedure to create an open cohort. Laparoscopic gastrectomy (with a laparoscopic oesophago-jejunostomy or gastro-jejunostomy for sub-total gastrectomy) was used as the definition of MIS. The cohorts were compared for: age, sex, BMI and use of neoadjuvant therapy. Temporal trends in surgical approaches were compared as were Individual surgeon (n=4) volumes. Peri-operative clinical and oncological outcomes including length of stay, anastomotic leak, return to theatre, chest complications, resection margins and lymph node harvest were all examined. A CUSM analysis is currently being completed to further analyse surgeon performance over time. Results 596 patients underwent resection in well matched cohorts, 197 patients had an oesophagectomy (100 MIO), 339 had a gastrectomy (97 laparoscopic). MIS rates increased with no operations being performed as MIS in 2012 compared to 81% of oesophagectomy and 67% of gastrectomy performed as MIS in 2021. Two surgeons performed over 170 resections with MIS rate over 40%. OR of MIS performed by a high-volume surgeon was 3.01 (p&amp;lt;0.001). Lower volume MIS surgeons were more likely to have two consultants present, OR 1.4 (p=0.16). Anastomotic leak rates following gastrectomy were 2% in both groups (p=0.66), 9% following open oesophagectomy and 11% after MIO (p=0.77). 10% of laparoscopic gastrectomy patients experienced a chest complication, 26% after open surgery, 22% MIO and 24% open (p=0.65). Mean length of stay was significantly reduced after MIS. An R0 rate of 96% was seen after laparoscopic gastrectomy, 90% after open (p=0.08). 68% of MIO patients had R0 resection compared to 70% after open. Positive CRM was 29% after MIO. The mean lymph node harvest was 29 after laparoscopic gastrectomy and 26 after open (p=0.24), 29 following MIO, 30 after open oesophagectomy. 97% of MIO harvested over 15 nodes. Conclusions This study demonstrates that a transition from open to MIS practice is achievable in the UK. Outcomes demonstrated similar peri-operative clinical and oncological outcomes between MIS and open surgery which addresses the concerns that prohibited the widespread uptake of MIS in UK centres. Developing a reliable surgical technique for lymphadenectomy and anastomoses that transitioned from open to hybrid and now MIS allowed for appropriate outcomes to be maintained. Unit outcomes compare well to both NOGCA/ECCG data with complication rates following MIO being significantly lower when compared to ECCG. Lymph node harvests were significantly higher in unit MIOs when compared to NOGCA (p=0.01). Volume played an important role in surgeons performing independent MIS. This should be considered when planning unit workload or a transition to MIS. It will be further explored in the CUSM analysis. A positive CRM was seen in more cases (29%) when compared to NOGCA (24%) and may be related to the preferred neoadjuvant regime being chemotherapy over chemoradiotherapy. The future of OG is moving towards robotic surgery. It is proposed that the evolution from open to MIS will make a second transition to robotics less challenging as the strategies needed to perform MIS have been addressed.

On λ2-statistical convergence of complex uncertain sequences

The main aim of this paper is to introduce the notion of [Formula: see text]-statistical convergence in a given uncertainty space by considering sequences of complex uncertain variable. Also, we study the interrelationship between [Formula: see text]-statistical convergence with strongly [Formula: see text]-summable sequence in the same environment. Further, we extend the study by initiating the notions of [Formula: see text]-statistically Cauchy complex uncertain sequence.

On a Class of Sequences Related to $p$-Absolutely Summable Sequences in Metric Space Defined by Orlicz Functions

On a Class of Sequences Related to $p$-Absolutely Summable Sequences in Metric Space Defined by Orlicz Functions

(∆mv , f)-lacunary statistical convergence of order α

In this paper, we define the space Sαθ (∆mv, f) of all (∆mv, f)-lacunary statistical convergent sequences of order α with the help of unbounded modulus function f, lacunary sequence (θ), generalized difference operator ∆ mv and real number α ∈ (0, 1]. We also introduce the space ωαθ (∆mv, f) of all strong (∆mv, f)-lacunary summable sequences of order α. Properties related to these spaces are studied. Inclusion relations between spaces Sαθ (∆mv, f) and ωα θ (∆mv, f) are established under certain conditions.

Sequence Spaces and Spectrum of q-Difference Operator of Second Order

The sequence spaces ℓp(∇q2)(0≤p&lt;∞) and ℓ∞(∇q2) are introduced by using the q-difference operator ∇q2 of the second order. Apart from studying some basic properties of these spaces, we construct the basis and obtain the α-, β- and γ-duals of these spaces. Besides some matrix classes involving q-difference sequence spaces, ℓp(∇q2) and ℓ∞(∇q2) are characterized. The final section is devoted to classifying the spectrum of the q-difference operator ∇q2 over the space ℓ1 of absolutely summable sequences.

Cesaŕo Summable Relative Uniform Difference Sequence of Positive Linear Functions

In this article, we introduce the class of sequence of functions C 1 Δ , r u of Cesaŕo summable relative uniform difference sequence of functions. We have studied the topological properties of C 1 Δ , r u . We also obtain the necessary and sufficient condition to characterize the matrix classes C 1 Δ , r u , ℓ ∞ r u , C 1 Δ , r u , c r u , P .

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.

MODULAR BLOK DI RUANG BARISAN TERJUMLAH CESARO ORLICZ

On the Cesaro summable of orde-p sequence space, if the fuction is replaced by Orlicz function, it is not always easy to define norm in the space. In this paper, we study some properties of the Cesaro Orlicz summable sequence space. First, on the space we define a modular and its the luxemburg norm, and then some topological properties is explored. The results show that the sequence spaces is modular complete and nom complete. In addition, the space is a BK-space but not an AK-space.

Algebraic Basis of the Algebra of All Symmetric Continuous Polynomials on the Cartesian Product of ℓp-Spaces

We construct a countable algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product ℓp1×…×ℓpn, where p1,…,pn∈[1,+∞), and ℓp is the complex Banach space of all p-power summable sequences of complex numbers for p∈[1,+∞).

Behavior of solution of stochastic difference equation with continuous time under additive fading noise

&lt;p style='text-indent:20px;'&gt;Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.&lt;/p&gt;

A large class in Köthe-Toeplitz duals of generalized Cesàro difference sequence spaces with fixed point property for nonexpansive mappings

In 1970, Ces?ro sequence spaces was introduced by Shiue. In 1981, K?zmaz defined difference sequence spaces for ??, c0 and c. Then, in 1983, Orhan introduced Ces?ro difference sequence spaces. Both works used difference operator and investigated K?the-Toeplitz duals for the new Banach spaces they introduced. Later, various authors generalized these new spaces, especially the one introduced by Orhan. In this study, first we discuss the fixed point property for these spaces. Then, we recall that Goebel and Kuczumow showed that there exists a very large class of closed, bounded, convex subsets in Banach space of absolutely summable scalar sequences, ?1 with fixed point property for nonexpansive mappings. So we consider a Goebel and Kuczumow analogue result for a K?the-Toeplitz dual of a generalized Ces?ro difference sequence space. We show that there exists a large class of closed, bounded and convex subsets of these spaces with fixed point property for nonexpansive mappings.

Matrix mappings and compact operators for Schröder sequence spaces

In this paper, we discuss the domain of a recently defined conservative matrix, constructed by means of the Schröder numbers in the spaces of $p-$absolutely summable sequences and bounded sequences. We determine the $\beta-$duals of the Banach spaces, introduced here, and present characterization of some matrix operators. Moreover, we give the characterization of certain compact operators via the Hausdorff measure of noncompactness.

Angle in the space of $ p $-summable sequences

&lt;abstract&gt;&lt;p&gt;The aim of this paper is to investigate completness of $ A $ that equipped with usual norm on $ p $-summable sequences space where $ A $ is subspace in $ p $-summable sequences space and $ 1\le p &amp;lt; \infty $. We also introduce a new inner product on $ A $ and prove completness of $ A $ using a new norm that corresponds this new inner product. Moreover, we discuss the angle between two vectors and two subspaces in $ A $. In particular, we discuss the angle between $ 1 $-dimensional subspace and $ (s-1) $-dimensional subspace where $ s\ge 2 $ of $ A $.&lt;/p&gt;&lt;/abstract&gt;