In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Qausi Conformal Curvature Tensor on 푳푪푺 풏-Manifoldsinventionjournals
In this paper, we focus on qausi-conformal curvature tensor of 퐿퐶푆 푛 -manifolds. Here we study quasi-conformally flat, Einstein semi-symmetric quasi -conformally flat, 휉-quasi conformally flat and 휙-quasi conformally flat 퐿퐶푆 푛 -manifolds and obtained some interesting results
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
A Characterization of the Zero-Inflated Logarithmic Series Distributioninventionjournals
In this paper, we introduce a characterization of the zero-inflated logarithmic series distributions through a linear differential equation of its probability generating function.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
The derivation of the spherical gradient from an intuitive perspective. We begin from Cartesian coordinates and work our way through to spherical coordinates.
Let 퐺 be simple graph of order 푛. 퐴 퐺 is the adjacency matrix of 퐺 of order 푛 × 푛. The matrix 퐴 퐺 is said to graphical if all its diagonal entries should be zero. The graph⎾ is said to be the matrix product (mod-2) of 퐺 and 퐺 푖푓 퐴 퐺 푎푛푑 퐴 퐺 (mod-2) is graphical and ⎾ is the realization of 퐴 퐺 퐴 퐺 (mod-2). In this paper, we are going to study the realization of the Cycle graph 퐺 and any 푘 − regular subgraph of 퐺 . Also some interesting characterizations and properties of the graphs for each the product of adjacency matrix under (mod-2) is graphical.
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
A Characterization of the Zero-Inflated Logarithmic Series Distributioninventionjournals
In this paper, we introduce a characterization of the zero-inflated logarithmic series distributions through a linear differential equation of its probability generating function.
E. Canay and M. Eingorn
Physics of the Dark Universe 29 (2020) 100565
DOI: 10.1016/j.dark.2020.100565
https://authors.elsevier.com/a/1aydL7t6qq5DB0
https://arxiv.org/abs/2002.00437
Two distinct perturbative approaches have been recently formulated within General Relativity, arguing for the screening of gravity in the ΛCDM Universe. We compare them and show that the offered screening concepts, each characterized by its own interaction range, can peacefully coexist. Accordingly, we advance a united scheme, determining the gravitational potential at all scales, including regions of nonlinear density contrasts, by means of a simple Helmholtz equation with the effective cosmological screening length. In addition, we claim that cosmic structures may not grow at distances above this Yukawa range and confront its current value with dimensions of the largest known objects in the Universe.
A Fifth-Order Iterative Method for Solving Nonlinear Equationsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online
The derivation of the spherical gradient from an intuitive perspective. We begin from Cartesian coordinates and work our way through to spherical coordinates.
Let 퐺 be simple graph of order 푛. 퐴 퐺 is the adjacency matrix of 퐺 of order 푛 × 푛. The matrix 퐴 퐺 is said to graphical if all its diagonal entries should be zero. The graph⎾ is said to be the matrix product (mod-2) of 퐺 and 퐺 푖푓 퐴 퐺 푎푛푑 퐴 퐺 (mod-2) is graphical and ⎾ is the realization of 퐴 퐺 퐴 퐺 (mod-2). In this paper, we are going to study the realization of the Cycle graph 퐺 and any 푘 − regular subgraph of 퐺 . Also some interesting characterizations and properties of the graphs for each the product of adjacency matrix under (mod-2) is graphical.
Rishabh Interior is one of the growing and influential architecture, interior design, civil engineering and urban development firm in India. Founded in 2007 we have completed more than 3 million square feet in covered area of projects across 17 plus states in India.
We are renowned for our design expertise and our steadfast comittment to design excellence innovation and sustainability.
The Combined Effects of Omega3 Fatty Acids and NanoCurcumin Supplementation o...inventionjournals
Migraine is a chronic nerves system disease leads to considerable disabilities and affected quality of life. Neuro-inflammation play a key role in progression of migraine which mainly caused by release of inflammatory mediators including COX-2 and iNOs enzymes, TNF-α،IL-1β،IL-6 and adhesion molecule such as ICAM and VCAM.Curcumin and omega 3 fatty acids with anti-inflammatory and neuro-protective effects can reduce gene expression and production of inflammatory mediators. The combination of omega-3 and curcumin have also synergistic effects which a lower dose can induce a positive effect. The aim of present study is to determine whether omega 3 fatty acids, curcumin or combined of them as a complement treatment in migraine is helpful and the reduction gene expression and serum levels of inflammatory markers.The study will be conducted involving 80 episodic migraine patients that withstratified randomization method based on sex, gender and body mass index (BMI) are classified into 4 groups: 1) receiving omega 3 fatty acid supplement (1.8 gr/ day) and nano-curcumin supplement (80 mg/ day) 2) receiving omega 3 fatty acid supplement and curcumin placebo 3) receiving curcumin supplement and omega 3 fatty acid placebo 4) receiving omega 3 fatty acid placebo and curcumin placebo for 2 mounts. Blood samples will be collected after anthropometric parameter measuring then target biochemical parameters, gene expression and serum levels oftarget inflammatory mediators including COX-2 and iNOs enzymes, TNF-α،IL-1β،IL-6 and adhesion molecule such as ICAM and VCAM will be measured before and after the trial. The data will be statistically evaluated using the most appropriate tests.The results of current study will determine the efficacy of therapeutic effects of curcumin, omega 3 fatty acids and combined of them as a new insight to control and treatment of migraine.
Synthesis and Anti-Inflammatory activity of Sulpha/substituted 1,2-Diazolesinventionjournals
A Novel compound namely N1 (3 Nicotinoyl) 3,5 dimethyl-4-(N-4-sulfamoyl-azo)-1,2-diazoles has been synthesized by two step processes. Synthesis of N1 -4-sulfamoylphenylhydrazono-3,5-dimethyl propane-1,3-dione and sulfonamide, which interacting with 3-Nicotinoyl hydrazine to form final compound. The newly synthesized compound N1 -(3-Nicotinoyl)-3,5-dimethyl 4-(N1 -4-sulfamoyl phenyl azo) 1,2-diazoles was screened for antiinflammatory activity
Correlations between Some Anthropometric Parameters, the Lipid Profile and Gl...inventionjournals
The study aimed to investigate the correlations between some anthropometric and lipid profile parameters, as well the glycated hemoglobin (HbA1c) values of Bulgarian females with type 2 Diabetes mellitus (T2DM). 212 women of Bulgarian ethnicity with T2DM of the age groups 40-60 and 61-80 years were included in the research. The anthropometric parameters: waist and hip circumferences were measured, and body mass index (BMI) and waist/hip ratio (WHR) were calculated. We also studied the lipid profile parameters: total cholesterol, triglycerides, high density lipoprotein (HDL), low density lipoprotein (LDL) and very low density lipoprotein (VLDL). A comparative study of the lipid profile criteria revealed significant differences in the levels of triglycerides, HDL-cholesterol and VLDL between the two age groups. Differences were found with regard to the values of glycated hemoglobin too. Correlations between the above anthropometric parameters and the lipid profile of patients with T2DM were examined. In patients of the age group 40-60 years a moderate strength correlation in opposite direction (negative or inverse correlation) was found between HbA1c and BMI (Pearson correlation coefficient 0.30-0.50, p<0.05). In the age group 61-80 years a low strength negative correlation was found between BMI and total cholesterol, as well as between BMI and LDL.
Intrusion Detection System for Classification of Attacks with Cross Validationinventionjournals
Now days, due to rapidly uses of internet, the patterns of network attacks are increasing. There are various organizations and institutes are using internet and access or share the sensitive information in network. To protect information from unauthorized or intruders is one of the important issues. In this paper, we have used decision tree techniques like C4.5 and CART as classifier for classification of attacks. We have proposed an ensemble model that is combination of C4.5 and Classification and Regression Tree (CART) as robust classifier for classification of attacks. We have used NSL-KDD data set with binary and multiclass problem with 10-fold cross validation. The proposed ensemble model gives satisfactory accuracy as 99.67% and 99.53% in case of binary class and multiclass NSL-KDD data set respectively.
Genotoxicity of Goji Berry (Lyciumbarbarum) In Vivo Mammalian Cellsinventionjournals
Lyciumbarbarum (Gojji berry) belongs to family Salonaceae which is found in China and Himalayan. This herb is used to prevent various diseases and in medical treatments as an alternative medicine being widely used for its antioxidant and revitalizing potential effects. In recent years, Gojji has become increasingly popular in Europe and North America as a "superfruit" and dietary supplement. The belief that herbal products do not bring any risk to health, is part of popular culture. However the term "natural" assigned to many products cannot assure no health risk. The aim of this study was to evaluate the possible genotoxic effects of aqueous extract of Lyciumbarbarum (Gojji berry) by micronucleus test and comet assay. Thirty Rattus norvegicus were divided into three equal groups: 1) experimental group, submitted to Gojji berry (200mg/kg orally); 2) positive control group (cyclophosphamide), and; 3) negative control group (distilled water). Micronucleus Tests were done by smear method of bone marrow cells performed after 48h for acute, and 72h for chronic exposure. The comet assay was performed on peripheral blood taken from the tail of each animal 4h, and 24h after intervention. Cytotoxicity was assessed by observing the DNA damage measuring the percentage of DNA in the tail (% DNA- measurement of the proportion of the total DNA present in the tail) and the tail moment (TM-tail length times the percentage of DNA in the tail), calculated by 100 nucleoids per animal and the presence of micronuclei in 2,000 polychromatic erythrocytes per animal. Analysis of variance (ANOVA) followed by Tukey test at 5% significance was used comparing the results. The data showed no significant difference in the frequency of DNA damage and the number of micronuclei between the experimental group and the negative control group. The results also suggest that the aqueous extract of Lyciumbarbarum (Gojji berry) at the dose of 200 mg/kg showed no genotoxic effect, which could, to a certain point, justifies its use.
Genotoxicity of Goji Berry (Lyciumbarbarum) In Vivo Mammalian Cellsinventionjournals
Lyciumbarbarum (Gojji berry) belongs to family Salonaceae which is found in China and Himalayan. This herb is used to prevent various diseases and in medical treatments as an alternative medicine being widely used for its antioxidant and revitalizing potential effects. In recent years, Gojji has become increasingly popular in Europe and North America as a "superfruit" and dietary supplement. The belief that herbal products do not bring any risk to health, is part of popular culture. However the term "natural" assigned to many products cannot assure no health risk. The aim of this study was to evaluate the possible genotoxic effects of aqueous extract of Lyciumbarbarum (Gojji berry) by micronucleus test and comet assay. Thirty Rattus norvegicus were divided into three equal groups: 1) experimental group, submitted to Gojji berry (200mg/kg orally); 2) positive control group (cyclophosphamide), and; 3) negative control group (distilled water). Micronucleus Tests were done by smear method of bone marrow cells performed after 48h for acute, and 72h for chronic exposure. The comet assay was performed on peripheral blood taken from the tail of each animal 4h, and 24h after intervention. Cytotoxicity was assessed by observing the DNA damage measuring the percentage of DNA in the tail (% DNA- measurement of the proportion of the total DNA present in the tail) and the tail moment (TM-tail length times the percentage of DNA in the tail), calculated by 100 nucleoids per animal and the presence of micronuclei in 2,000 polychromatic erythrocytes per animal. Analysis of variance (ANOVA) followed by Tukey test at 5% significance was used comparing the results. The data showed no significant difference in the frequency of DNA damage and the number of micronuclei between the experimental group and the negative control group. The results also suggest that the aqueous extract of Lyciumbarbarum (Gojji berry) at the dose of 200 mg/kg showed no genotoxic effect, which could, to a certain point, justifies its use.
ASSESSMENT OF NANOPARTICULATED RESVERATROL AND LOSARTAN IN THE PROPHYLAXIS OF...inventionjournals
The study demonstrated that eight weeks after bilateral ovariectomy in female white Wistar rats, develops endothelial dysfunction of bone microvasculature and deterioration of regional blood flow in the bone, leading to the emergence of generalized osteoporosis. Nanoparticulated forms of Losartan and Nanoparticulated forms of Resveratrol, possessing endothelial protective action, effectively prevent the reduction of regional microcirculation in bone tissue, keeping it at the level of intact rats. It is allowed to maintain an adequate level of bone remodeling processes, which manifested by slowing the thinning of bone trabeculae.
Evaluation of LdhIsozymes Following the Treatment of Methyl Parathion in the ...inventionjournals
Lactate dehydrogenase converts lactate into pyruvate and has a very important role in carbohydrate metabolism. LDH activity depends on its isozymes and their activities change under pathological conditions. The increase in LDH activity suggests the increased anaerobic conditionsunder the influence of methylparathion to meet the energy demand when the aerobic oxidation is lowered. Following the treatment of methyl parathion there was a differential percentage in increase or decrease of different isozymes in the fish Labeo rohita. There were three distinct bands during the 96h of treatment almost parallel to LDH5 band of human serum suggesting liver damage. This was also confirmed by histopathological studies.
Cell-free DNA Levels Serum Patients with Benign and Malignant Epithelial Ovar...inventionjournals
An elevated level of cell-free DNA (cfDNA) in the blood circulation has detected in cancer patients in comparison with healthy controls. CfDNA circulation in plasma and serum extensively studied and the results are highly variable due to many factors influence the test results that was preanalytic factors as well as analytic factors. Objectives: Is there any difference in the concentration of serum DNA among patients with benign epithelial ovarian tumors and malignant epithelial ovarian tumors? What is the clinicopathological variable that influences the cfDNA circulation? Method: Venous blood drawn with plain vacutainer, centrifuged at 1,000 rpm for 30 minutes, serum kept in -800 C freezer. The cfDNA extracted used NaI method.Results: Collected 30 cases of the benign ovarian tumor and 54 cases of malignant ovarian tumors. The average level serum cfDNA of benign epithelial ovarian tumors and malignant epithelial ovarian tumors were 24.6 ng/mL and 22:29 ng/mL respectively and statistically was not significantly different (p = 0.64). In multivariable analysis with linear regression, there were no clinicopathological variables that statistically significant influence the cfDNA levels in patients with epithelial ovarian tumors where p > 0.05. Conclusion: Concentration of cfDNA circulation of benign epithelial ovarian tumors a little bit higher than malignant epithelial ovarian tumors, but statistically was not significantly different. There was no clinicopathological variable influence the concentration of cfDNA circulation of ovarian tumors.
Penicillin is drug of Choice for Syphilis- Still it holds good?inventionjournals
Syphilis resistant to Tetracyclins and macrolids had well documented. Penicillin is still considered as drug of choice in the treatment of Syphilis. Treponemes resistant to Penicillin are yet to be documented. In a private STD clinic of Tirunelveli, South India, during the period of January 2010 to June 2015, 68 cases reactive for syphilis were noted. 13.2 % of false positive were identified. Remaining 59 cases, 31 lost for follow up after treatment. Among whom 12 responded well to treatment following anti-syphilitic treatment with Penicillin whereas 16 (57.14%) remained reactive for more than a year or more. The titers either persist as same or even go up. Nobody in this group are reactive for HIV. All these patients remain in latent state in this series. Nobody progressed to neuro or cardiovascular syphilis. This warrants that it is high time to look for some other better alternative to treat Syphilis or otherwise increase in the dosage may have to be considered. Even though this study comprises of small number of patients, it has to be studied in detail for a longer period at multiple centers.
“Desquamative Gingivitis Treated By An Antioxidant Therapy- A Case Report”inventionjournals
Desquamative gingivitis is described as an erythematous, desquamated or eroded gingival lesion. Various etiologic factors are present for the appearance of such lesions. Despite of considering etiology, treatment is oftenly provided by systemic or topical corticosteroids. Apart from steroid application, another optionable treatment is antioxidant therapy which provides rapid healing of the tissue. As antioxidants posses various advantageous properties, it can be considered as a first treatment option for desquamative gingivitis. The presented case report of desquamative gingivitis is successfully treated using systemic antioxidants in the form of commercially available „oxitard capsule‟.
Antiemetic Prophylaxis in Major Gynaecological Surgery With Intravenous Grani...inventionjournals
In a prospective double blind randomized study we evaluated the prophylactic anti emetic efficacy of granisetron, a 5HT3 receptor antagonist and metoclopramide, a benzamide anti emetic on postoperative nausea and vomiting after major gynaecological surgery under general anaesthesia. The patients received a single dose of granisetron, 40mcg/kg (Group A, n = 25) or metoclopramide, 0.15mg/kg (Group B, n = 25) before induction of anaesthesia in a coded syringe. The response was assessed during 0-4 hrs, 4-8 hrs, 8-16hrs and 16-24 hrs time intervals after recovery from anaesthesia by means of presence or absence of nausea, retching or vomiting. The overall control of PONV during early postoperative period (0-4 hrs) did not show statistically significant differences after administration of either drug. The incidence of PONV during the next 20 hours was 12% and 48% with Group A (Granisetron) and Group B (Metoclopramide) respectively. Nausea scores are significantly lower in-group A (Granisetron) than in Group B (Metoclopramide) in all the four assessment periods. Although there were no emetic episodes in the granisetron group, 32% of patients in metoclopramide group were observed to have such episodes during the assessment periods. (P value< 0.05). No clinically important adverse events due to drugs were observed in any of the groups. In conclusion, the prophylactic use of granisetron is more effective and superior to metoclopramide in preventing postoperative nausea and vomiting in patients under going major gynaecological surgery under general anaesthesia.
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On The Distribution of Non - Zero Zeros of Generalized Mittag – Leffler Funct...IJERA Editor
In this work, we derive some theorems involving distribution of non – zero zeros of generalized Mittag – Leffler functions of one and two variables. Mathematics Subject Classification 2010: Primary; 33E12. Secondary; 33C65, 26A33, 44A20
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Some properties of two-fuzzy Nor med spacesIOSR Journals
The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Homogeneous Components of a CDH Fuzzy SpaceIJECEIAES
We prove that fuzzy homogeneous components of a CDH fuzzy topological space (X,T) are clopen and also they are CDH topological subspaces of its 0-cut topological space (X,T0).
Generalized Laplace - Mellin Integral TransformationIJERA Editor
The main propose of this paper is to generalized Laplace-Mellin Integral Transformation in between the positive regions of real axis. We have derived some new properties and theorems .And give selected tables for Laplace-Mellin Integral Transformation.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Similar to Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean (20)
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdf
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 7 || September. 2016 || PP-04-07
www.ijmsi.org 4 | Page
Matrix Transformations on Paranormed Sequence Spaces
Related To De La Vallée-Pousin Mean
Zakawat U. Siddiqui and Ado Balili
Department of Mathematics and Statistics, University of Maiduguri, Maiduguri, Nigeria
ABSTRACT: In this paper, we determine the necessary and sufficient conditions to characterize the matrices
which transform paranormed sequence spaces into the spaces 𝑉𝜎 (𝜆) and 𝑉𝜎
∞
(𝜆) , where 𝑉𝜎 (𝜆) denotes the
space of all (𝜎, 𝜆)-convergent sequences and 𝑉𝜎
∞
(𝜆) denotes the space of all (𝜎, 𝜆)-bounded sequences defined
using the concept of de la Vallée-Pousin mean.
Keywords: de la Vall𝑒e-Pousin Mean, 𝜎-convergence, Invariant Mean, Matrix Transformations
Mathematics Subject Classification: 40A05, 40C05, 40D05
I. INTRODUCTION
We shall denote the space of real valued sequences by 𝜔.Any vector subspace of 𝜔 is called a sequence space. if
𝑥 𝜖 𝜔 , then we write 𝑥 = 𝑥 𝑘 instead of 𝑥 = (𝑥 𝑘 ) 𝑘=0
∞
. We denote the spaces of all finite, bounded, convergent
and null sequences by , 𝑙∞, 𝑐, and 𝑐0, respectively. Further, we shall use the conventions that 𝑒 = 1, 1, 1,
…and 𝑒(𝑘) as the sequence whose only non zero term is 1 in the kth place for each 𝑘 𝜖 ℕ,
A sequence space X with linear topology is called a K-space if each of the maps 𝑝𝑖 : 𝑋 → ℂ: defined by 𝑝𝑖 𝑥 =
𝑥𝑖 is continuous for all 𝑖 ∈ ℕ. A K-space is called an FK-space if X is complete linear metric space; a BK-space
is a normed FK-space.
A linear topological space X over the real field ℝ is said to be a paranormed space if there is a sub additive
function 𝑝 ∶ 𝑋 → ℝ such that 𝑝 𝜃 = 0, 𝑝 𝑥 = −𝑥 , and scalar multiplication is continuous, i. e. 𝛼 𝑛 − 𝛼 →
0 and 𝑝 𝑥 𝑛 → 0 imply 𝑝 𝛼 𝑛 𝑥 𝑛 − 𝛼𝑥 → 0 𝑎𝑠 𝑛 → ∞ for all 𝑥 ∈ 𝑋 and 𝛼 ∈ ℝ, where 𝜃 is the zero vector in
the linear space X.
Assume here and after that 𝑥 = 𝑥 𝑘 be a sequence such that 𝑥 𝑘 ≠ 0, ∀ 𝑘 𝜖 ℕ and (𝑝 𝑘 ) be the bounded sequence
of strictly positive real numbers with 𝑠𝑢𝑝𝑝 𝑘 = 𝐻 𝑎𝑛𝑑 𝑀 = max(1 , 𝐻).then , the sequence spaces
𝑐0 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ lim 𝑘→∞ |𝑥 𝑘 | 𝑝 𝑘 = 0}
𝑐 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 : lim
𝑘→∞
|𝑥 𝑘 − 𝑙| 𝑝 𝑘 = 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑙 ∈ ℂ}
ℓ∞ 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ sup 𝑘 |𝑥 𝑘 | 𝑝 𝑘 < ∞ } and
ℓ 𝑝 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ |𝑥 𝑘
∞
𝑘 | 𝑝 𝑘 < ∞ }
were defined and studied by Maddox [1] and Simons [2].
If 𝑝 𝑘 = 𝑝 , ∀ 𝑘 ∈ ℕ for some constant 𝑝 > 0, then these spaces reduce to 𝑐0 , 𝑐, ℓ∞, 𝑎𝑛𝑑 ℓ 𝑝, respectively. Note
that 𝑐0 𝑝 is a linear metric space paranormed by 𝑥 = sup 𝑘 |𝑥 𝑘 |
𝑝 𝑘
𝑀 . ℓ∞ 𝑝 , 𝑐 𝑝 fail to be linear metric
space because the continuity of multiplication does not hold for them. These two spaces turn out to be linear
metric spaces if and only if inf 𝑝 𝑘 > 0. ℓ 𝑝 is linear metric space paranormed by 𝑤 𝑥 = ( |𝑥 𝑘𝑘 | 𝑝 𝑘 )
1
𝑀 .
These sequence spaces are complete paranormed spaces in their respective paranorm if and only if inf 𝑝 𝑘 > 0.
However, these are not normed spaces, in general. (see Aydin and Basar [3] and Karakaya et al, [4]).
The above sequence spaces were further generalized. Bulut and Çakar [5] defined the sequence space
ℓ 𝑝, 𝑠 = { 𝑥 = 𝑥 𝑘 𝜖 𝜔 : 𝑘−𝑠
𝑘 |𝑥 𝑘 | 𝑝 𝑘 < ∞, 𝑠 ≥ 0},
which generalized the sequence space ℓ 𝑝 . They showed that ℓ 𝑝, 𝑠 isa linear sequence space paranormed by
𝑔 𝑥 = ( 𝑘−𝑠
|𝑥 𝑘𝑘 | 𝑝 𝑘 )
1
𝑀 .
Basarir [6] generalized the other sequence spaces as follows:
ℓ∞ 𝑝, 𝑠 = 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ sup 𝑘 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 < ∞
𝑐0 𝑝, 𝑠 = 𝑥 = 𝑥 𝑘 ∈ 𝜔 ∶ 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 → 0, 𝑘 → ∞
𝑐 𝑝, 𝑠 = { 𝑥 = 𝑥 𝑘 ∈ 𝜔 : 𝑘−𝑠
𝑥 𝑘 − 𝑙| 𝑝 𝑘 → 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑙, 𝑘 → ∞
It is easy to see that 𝑐0 𝑝, 𝑠 is paranormed by 𝑞 𝑥 = sup 𝑘(𝑘−𝑠
|𝑥 𝑘 | 𝑝 𝑘 )
1
𝑀 . Also ℓ∞ 𝑝, 𝑠 and 𝑐 𝑝, 𝑠 are
paranormed by 𝑞 𝑥 iff 𝑖𝑛𝑓𝑝 𝑘 > 0. All the spaces defined above are complete in their topologies.
Let X and Y be two sequence spaces and 𝐴 = (𝑎 𝑛𝑘 ) 𝑛,𝑘=1
∞
be an infinite matrix of real or complex numbers. We
denote 𝐴𝑥 = 𝐴 𝑛 𝑥 , 𝐴 𝑛 𝑥 = 𝑎 𝑛𝑘𝑘 𝑥 𝑘 provided that the series on the right converges for each n. If
𝑥 = 𝑥 𝑘 ∈ 𝑋, implies that 𝐴𝑥 ∈ 𝑌, then we say that A defines a matrix transformation from X into Y, and by
(X, Y) we denote the class of such matrices.
2. Matrix Transformations On Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean
www.ijmsi.org 5 | Page
Let 𝜎 be a one to one mapping from the set ℕ of natural numbers into itself. A continuous linear functional 𝜑
on the space ℓ∞ is said to be an invariant mean or 𝜎-mean if and only if
( i) 𝜑 𝑥 ≥ 0 if 𝑥 ≥ 0, 𝑖. 𝑒 𝑥 𝑘 ≥ 0, for all 𝑘 ,
(ii) 𝜑 𝑒 = 1, where 𝑒 = 1, 1, 1, … ,
(iii) 𝜑 𝑥 = 𝜑 𝑥 𝜎 𝑘 , for all 𝑥 ∈ ℓ∞.
Though out this we consider the mapping 𝜎 which has no finite orbit, that is 𝜎 𝑝
𝑘 ≠ 𝑘 for all integer 𝑘 ≥
0 𝑎𝑛𝑑 𝑝 ≥ 1, where 𝜎 𝑝
𝑘 denotes the p-th iterate of 𝜎 at k. Note that a 𝜎 −mean extends the limit functional
on the space c in the sense that 𝜎 𝑥 = 𝑙𝑖𝑚𝑥 for all 𝑥 ∈ ℂ, (see Mursaleen [7]). Consequently, 𝑐 ⊂ 𝑉𝜎 , the set of
bounded sequence all of whose 𝜎 −means are equal.
We say that a sequence 𝑥 = 𝑥 𝑘 is 𝜎 −convergent iff 𝑥 ∈ 𝑉𝜎 . Using this concept, Schaefer [8] defined and
characterized 𝜎 −convervative and 𝜎 −coercive matrices. If 𝜎 is translation, then 𝑉𝜎 is reduced to 𝑓 of almost
convergent sequences (Lorentz [9]). As an application of almost convergence, Mohiuddine [10] established
some approximation theorems for sequences of positive linear operators through this concept. The idea of
𝜎 −convergence for double sequences was introduced in Çakar et al [11].
II. BASIC DEFINITIONS
Definition 2.1 (de la Vallée-Poussin mean): Let 𝜆 = 𝜆 𝑚 be a non –decreasing sequence of positive numbers
tending to ∞ such that 𝜆 𝑚+1 ≤ 𝜆 𝑚 + 1, 𝜆1 = 0, then
𝜌 𝑚 𝑥 =
1
𝜆 𝑚
𝑥𝑗𝑗∈𝐼 𝑚
is called the generalized de la Vallée-Poussin mean, where 𝐼 𝑚 = [𝑚 − 𝜆 𝑚 + 1, 𝑚 ].
Definition 2.2 (Mursaleen et al [12]) A sequence 𝑥 = 𝑥 𝑘 of real numbers is said to be 𝜎, 𝜆 −convergent to a
number L iff
lim 𝑚→∞
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗 ∈𝐼 𝑚
= 𝐿
uniformly in n, and 𝑉𝜎 (𝜆) denotes the set of all such sequences i.e
𝑉𝜎 𝜆 = { 𝑥 ∈ ℓ∞ ∶ lim 𝑚→∞
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗∈𝐼 𝑚
= 𝐿, uniformly in n}
Note that a convergent sequence is 𝜎, 𝜆 −convergent but converse need not hold.
2.1 Remark
(i) If 𝜎 𝑛 = 𝑛 + 1, then 𝑉𝜎 (𝜆) is reduced to the 𝑓𝜆 (see Mursaleen et al [13])
(ii) If 𝜆 𝑚 = 𝑚, then 𝑉𝜎 (𝜆) is reduced to the space 𝑉𝜎
(iii) If 𝜎 𝑛 = 𝑛 + 1 and 𝜆 𝑚 = 𝑚, then 𝑉𝜎 (𝜆) is reduced to the space 𝑓, (almost convergent sequences)
(iv) 𝑐 ⊂ 𝑉𝜎 𝜆 ⊂ ℓ∞.
Definition 2.3 (Mohiuddine [13]) A sequence 𝑥 = (𝑥 𝑘 ) of real numbers is said to be 𝜎, 𝜆 −bounded if and
only if 𝑠𝑢𝑝 𝑚,𝑛 |
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗∈𝐼 𝑚
| < ∞, and we denote by 𝑉𝜎
∞
(𝜆), the set of all such sequences i. e
𝑉𝜎
∞
(𝜆) = 𝑥 ∈ ℓ∞ ∶ sup 𝑚,𝑛 |𝑡 𝑚𝑛 𝑥 < ∞ },
where, 𝑡 𝑚𝑛 𝑥 =
1
𝜆 𝑚
𝑥 𝜎 𝑗 (𝑛)𝑗∈𝐼 𝑚
2.2 Remark
𝑐 ⊂ 𝑉𝜎 (𝜆) ⊂ 𝑉𝜎
∞
(𝜆) ⊂ ℓ∞.
III. SOME KNOWN RESULTS
The following results play vital role in our main results
Lemma 3.1 (Theorem 1; Bulut and Çakar [5]):
(i) If 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, 𝑎𝑛𝑑 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1,for k∈ ℕ, then
ℓ†
𝑝, 𝑠 = { 𝑎 = 𝑎 𝑘 : 𝑘 𝑠 𝑞 𝑘−1∞
𝑘=1 𝑁
−
𝑞 𝑘
𝑝 𝑘 |𝑎 𝑘 | 𝑞 𝑘 < ∞, 𝑠 > 0, 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑁 > 1}
(ii) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for each k =1,2,3, …, then ℓ†
𝑝, 𝑠 = 𝑚(𝑝, 𝑠), where
𝑚 𝑝, 𝑠 = { 𝑎 = 𝑎 𝑘 : sup 𝑘 𝑘−𝑠
|𝑎 𝑘 | 𝑝 𝑘 < ∞, 𝑠 ≥ 1
Lemma 3.2 (Theorem3; Bulut and Çakar [5]): ( i ) If 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ, then
𝐴 ∈ (ℓ 𝑝, 𝑠 , ℓ∞) if and only if there exists an integer N> 1, such that
sup 𝑛 |𝑎 𝑛𝑘
∞
𝑘=1 | 𝑞 𝑘 𝑁−𝑞 𝑘 𝑘 𝑠(𝑞 𝑘−1)
< ∞, (3.1)
(ii) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for each 𝑘 ∈ ℕ, then 𝐴 ∈ (ℓ 𝑝, 𝑠 , ℓ∞) if and only if
sup 𝑛𝑘 |𝑎 𝑛𝑘 | 𝑝 𝑘 𝑘 𝑠
= 𝑀 < ∞. (3.2)
Lemma 3.3 (Theorem 4; Bulut and Çakar [5]):
(i) Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every k. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑐) if and only if (3.1) holds together with
3. Matrix Transformations On Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean
www.ijmsi.org 6 | Page
𝑎 𝑛𝑘 → 𝛼 𝑘, (𝑛 → ∞, 𝑘 𝑓𝑖𝑥𝑒𝑑 ) (3.3)
( ii ) If 0 < 𝑚 = 𝑖𝑛𝑓𝑘 𝑝 𝑘 ≤ 𝑝 𝑘 ≤ 1, for some 𝑘 ∈ ℕ.Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑐), if and only if condition (3.2) and
(3.3)
Lemma 3.4 (Theorem 2.1 Mohiuddine [13]): The spaces 𝑉𝜎 (𝜆) and 𝑉𝜎
∞
(𝜆) are BK spaces with the norm
𝑥 = sup 𝑚,𝑛≥0 |𝑡 𝑚𝑛 𝑥 |. (3.4)
Lemma 3.5 (Theorem 3.1 Mohiuddine [13] ): Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ.Then
𝐴 ∈ (ℓ 𝑝 , 𝑉𝜎
∞
𝜆 ) if and only if there exists an integer 𝑁 > 1 such that
sup 𝑚,𝑛 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝑁−𝑞 𝑘 < ∞, (3.5)
Lemma 3.6 (Theorem 3.2 Mohiuddine [13]):
(i) 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞, for every 𝑘 ∈ ℕ.Then 𝐴 ∈ (ℓ 𝑝 , 𝑉𝜎 𝜆 ) if and only if ( i ) condition (3.5)
( ii ) lim 𝑚→∞
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
= 𝛼 𝑘 uniformly in n, for every 𝑘 ∈ ℕ
IV. MAIN RESULTS
We shall prove the following results.
Theorem 4.1 Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ) if and only if there
exists an integer 𝐷 > 1, such that
sup 𝑚,𝑛 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 𝑘 𝑠(𝑞 𝑘−1)
< ∞. (4.1)
Proof. Sufficiency. Let (4.1) hold and that 𝑥 ∈ ℓ(𝑝, 𝑠).Using the inequality
𝑎 𝑏 ≤ 𝐷(|𝑎| 𝑞
𝐷−𝑞
+ 𝑏| 𝑝
for 𝐷 > 0 and 𝑎, 𝑏 complex numbers (𝑝−1
+ 𝑞−1
= 1) (Maddox [14] )
We have
𝑡 𝑚𝑛 𝐴𝑥 = |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
𝑥 𝑘 |
⟹ 𝐷[|
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
+ 𝑘−𝑠
𝑥 𝑘 | 𝑝 𝑘 ,
where 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1.
Taking the supremum over m, n on both sides and using (4.1), we get
𝐴𝑥 ∈ 𝑉𝜎
∞
𝜆 for 𝑥 ∈ ℓ(𝑝, 𝑠). i. e 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ).
Necessity. Let 𝐴 ∈ ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 . We put 𝑞 𝑛 𝑥 = sup 𝑘 𝑘−𝑠
𝑡 𝑚𝑛 𝐴𝑥 .
It is easy to see that for 𝑛 ≥ 0, 𝑞 𝑛 is a continuous semi norm on ℓ(𝑝, 𝑠) and (𝑞 𝑛 ) is point wise bounded on
ℓ(𝑝, 𝑠). Suppose that (4.1) is not true. Then there exists 𝑥 ∈ ℓ(𝑝, 𝑠) with sup 𝑛 𝑞 𝑛 𝑥 = ∞. By the principle of
condensation of singularities (Yosida [15]), the set 𝑥 ∈ ℓ 𝑝, 𝑠 : sup 𝑛 𝑞 𝑛 𝑥 = ∞ is of second category in
ℓ(𝑝, 𝑠) and hence non empty. Thus there exists 𝑥 ∈ ℓ(𝑝, 𝑠) with sup 𝑛 𝑞 𝑛 𝑥 = ∞.But this contradict the fact
that (𝑞 𝑛 ) is pointwise bounded on ℓ(𝑝, 𝑠). Now by the Banach-Steinhaus theorem, there is a constant M such
that
𝑞 𝑛 𝑥 ≤ 𝑀𝑔(𝑥) (4.2)
Now define a sequence 𝑥 = (𝑥 𝑘 ) by
𝑥 𝑘 = 𝛿
𝑀
𝑝 𝑘 (𝑠𝑔𝑛
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
)|
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘−1
𝑉−1
𝐷
−
𝑞 𝑘
𝑝 𝑘 . 𝑘 𝑠 𝑞 𝑘−1
1 ≤ 𝑘 ≤ 𝑘0
0, 𝑘 > 𝑘0
where 0 < 𝛿 < 1 and 𝑉 = |
1
𝜆 𝑚
𝑘0
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
.
Then it is easy to see that 𝑥 ∈ ℓ(𝑝, 𝑠) and 𝑔(𝑥) ≤ 𝛿.
Applying this sequence to (4.2) we get the condition (4.1)
This completes the proof of the theorem.
Theorem 4.2 Let 1 < 𝑝 𝑘 ≤ sup 𝑘 𝑝 𝑘 = 𝐻 < ∞ for every 𝑘 ∈ ℕ. Then 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ) if and only if
(i) Condition (4.1) of Theorem 4.1 holds
(ii) lim 𝑚
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
= 𝛼 𝑘 , uniformly in n for every 𝑘 ∈ ℕ.
Proof. Sufficiency. Let ( i ) and ( ii ) hold and 𝑥 ∈ ℓ(𝑝, 𝑠). For 𝑗 ≥ 1
|
1
𝜆 𝑚
𝑗
𝑘=1
𝑎 𝜎 𝑗 𝑛 ,𝑘
𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
≤ sup
𝑚
|
1
𝜆 𝑚
𝑘
𝑎 𝜎 𝑗 𝑛 ,𝑘
𝑗 ∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
< ∞
for every n. Therefore
|𝛼 𝑘 | 𝑞 𝑘
𝑘 𝐷−𝑞 𝑘 . 𝑘−𝑠 𝑞 𝑘−1
= lim𝑗 lim 𝑘 |
1
𝜆 𝑚
𝑗
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
| 𝑞 𝑘 . 𝐷−𝑞 𝑘 . 𝑘−𝑠 𝑞 𝑘−1
4. Matrix Transformations On Paranormed Sequence Spaces Related To De La Vallée-Pousin Mean
www.ijmsi.org 7 | Page
≤ sup 𝑘 |
1
𝜆 𝑚
𝑘 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗∈𝐼 𝑚
| 𝑞 𝑘 𝐷−𝑞 𝑘 . 𝑘 𝑠(𝑞 𝑘−1)
< ∞,
where 𝑝 𝑘
−1
+ 𝑞 𝑘
−1
= 1. Consequently reasoning as in the proof of the sufficiency of Theorem 4.1, the series
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 and 𝛼 𝑘𝑘 𝑥 𝑘 converge for every 𝑛, 𝑚, 𝑗 𝑎𝑛𝑑 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑥 ∈ ℓ(𝑝, 𝑠). For a given
𝜀 > 0 𝑎𝑛𝑑 𝑥 ∈ ℓ(𝑝, 𝑠), choose 𝑘0 such that
( 𝑘−𝑠∞
𝑘=𝑘0+1 |𝑥 𝑘 | 𝑝 𝑘 )
1
𝑀 𝜀, (4.3)
where 𝑀 = sup 𝑘 𝑝 𝑘 condition ( ii ) implies that there exists 𝑚0 such that
[
1
𝜆 𝑚
𝑚0
𝑘=1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 | <
𝜀
2
for all 𝑚 ≥ 𝑚0 and uniformly in n. Now, since
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 and 𝛼 𝑘𝑘 𝑥 𝑘 converge (absolutely) uniformly in m, n and for every 𝑥 ∈ ℓ(𝑝, 𝑠), we have
[
1
𝜆 𝑚
∞
𝑘=𝑘0+1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 𝑥 𝑘 converges uniformly in m, n. Hence by conditions (i) and (ii)
[
1
𝜆 𝑚
∞
𝑘=𝑘0+1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 | <
𝜀
2
for 𝑚 ≥ 𝑚0 and uniformly in n. Therefore
[
1
𝜆 𝑚
∞
𝑘=𝑘0+1 𝑎 𝜎 𝑗 𝑛 ,𝑘𝑗𝜖 𝐼 𝑚
− 𝛼 𝑘 𝑥 𝑘 → ∞, (𝑚 → ∞) , uniformly in n, i. e
lim 𝑚
1
𝜆 𝑚
𝑎 𝜎 𝑗 𝑛 ,𝑘 𝑥 𝑘𝑗∈𝐼 𝑚𝑘 = 𝛼 𝑘𝑘 𝑥 𝑘 (4.4)
Necessity. Let 𝐴 ∈ (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ). Since 𝑉𝜎 𝜆 ⊂ 𝑉𝜎
∞
𝜆 . Condition (i) follows by Theorem 4.1. Since 𝑒(𝑘)
=
(0, 0, 0, … , 1 𝑘𝑡 place , 0, 0, … ) ∈ ℓ(𝑝, 𝑠) and condition (ii) follows immediately by (4.4).
This completes the proof of the theorem.
V. CONCLUSION
The notion of invariant mean and de la Vallée-Poussin mean plays very active role in the recent research on
matrix transformations. With the help of these two notions, the concept of (𝜎, 𝜆)-convergent sequences denoted
by 𝑉𝜎 𝜆 and (𝜎, 𝜆)-bounded sequences denoted by 𝑉𝜎
∞
𝜆 and also (𝜎, 𝜃)-convergent sequence, (𝑉𝜎 𝜃 ) and
(𝜎, 𝜃)-bounded sequence, (𝑉𝜎
∞
𝜃 ) sequences and many others have evolved. Related to these sequence spaces,
many matrix classes have been characterized. As we have characterized the matrix classes (ℓ 𝑝, 𝑠 , 𝑉𝜎 𝜆 ) and
(ℓ 𝑝, 𝑠 , 𝑉𝜎
∞
𝜆 ) in our main results here, some other characterizations may also evolve.
V1. ACKNOWLEDGEMENT
The authors are thankful to Mr. Sighat U. Siddiqui for his logistic support in the communication process of the
paper, which resolved some technical problems.
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