On completeness in metric spaces and fixed point theorems

[EN] Complete ultrametric spaces constitute a particular class of the so called, recently, G-complete metric spaces. In this paper we characterize a more general class called weak G-complete metric spaces, by means of nested sequences of closed sets. Then, we also state a general fixed point theorem for a self-mapping of a weak G-complete metric space. As a corollary, every asymptotically regular self-mapping of a weak G-Complete metric space has a fixed point.V. Gregori acknowledges the support of the Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P (MINECO/Feder, UE). J.J. Minana acknowledges financial support from the Spanish Ministry of Economy and Competitiveness under Grants TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds, by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by Project Ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2018). On completeness in metric spaces and fixed point theorems. Results in Mathematics. 73(4):1-13. https://doi.org/10.1007/s00025-018-0896-4113734Bourbaki, N.: Topologie Générale II. Herman, Paris (1974)Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–469 (1969)Browder, F.E., Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)Fang, J.X.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 46(1), 107–113 (1992)Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385–389 (1989)Gregori, V., Sapena, A.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 125, 245–252 (2002)Gregori, V., Miñana, J.-J., Morillas, S., Sapena, A.: Cauchyness and convergence in fuzzy metric spaces. RACSAM 111(1), 25–37 (2017)Gregori, V., Miñana, J-J., Sapena, A.: On Banach contraction principles in fuzzy metric spaces. Fixed Point Theory (to appear)Kelley, J.: General Topology. Van Nostrand, Princeton (1955)Matkowski, J.: Integrable solutions of functional equations. Dissertationes Mathematicae (Rozprawy Matematyczne) 127, 1–63 (1975)Mihet, D.: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 8431–439 (2004)Steen, L.A., Seebach, J.A.: Counterexamples in Topology, 2nd edn. Springer, Berlin (1978)Tirado, P.: On compactness and G-completeness in fuzzy metric spaces. Iran. J. Fuzzy Syst. 9(4), 151–158 (2012)Tirado, P.: Contraction mappings in fuzzy quasimetric spaces and $$[0,1]$$ [ 0 , 1 ] -fuzzy posets. Fixed Point Theory 13(1), 273–283 (2012)Vasuki, R., Veeramani, P.: Fixed points theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets Syst. 135(3), 415–417 (2003

A Characterization of Strong Completeness in Fuzzy Metric Spaces

[EN] Here, we deal with the concept of fuzzy metric space(X,M,*), due to George and Veeramani. Based on the fuzzy diameter for a subset ofX, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.Juan-Jose Minana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion/Proyecto PGC2018-095709-B-C21, and by Spanish Ministry of Economy and Competitiveness under contract DPI2017-86372-C3-3-R (AEI, FEDER, UE). This work was also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union's Horizon 2020 research and innovation program under grant agreements Nos. 779776 and 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2020). A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics. 8(6):1-11. https://doi.org/10.3390/math8060861S11186Menger, K. (1942). Statistical Metrics. Proceedings of the National Academy of Sciences, 28(12), 535-537. doi:10.1073/pnas.28.12.535George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7Gregori, V., & Romaguera, S. (2000). Some properties of fuzzy metric spaces. Fuzzy Sets and Systems, 115(3), 485-489. doi:10.1016/s0165-0114(98)00281-4Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xAtanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96. doi:10.1016/s0165-0114(86)80034-3Gregori, V., Romaguera, S., & Veeramani, P. (2006). A note on intuitionistic fuzzy metric spaces☆. Chaos, Solitons & Fractals, 28(4), 902-905. doi:10.1016/j.chaos.2005.08.113Gregori, V., & Sapena, A. (2018). Remarks to «on strong intuitionistic fuzzy metrics». Journal of Nonlinear Sciences and Applications, 11(02), 316-322. doi:10.22436/jnsa.011.02.12Abu-Donia, H. M., Atia, H. A., & Khater, O. M. A. (2020). Common fixed point theorems in intuitionistic fuzzy metric spaces and intuitionistic (ϕ,ψ)-contractive mappings. Journal of Nonlinear Sciences and Applications, 13(06), 323-329. doi:10.22436/jnsa.013.06.03Gregori, V., & Miñana, J.-J. (2016). On fuzzy ψ -contractive sequences and fixed point theorems. Fuzzy Sets and Systems, 300, 93-101. doi:10.1016/j.fss.2015.12.010Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Wardowski, D. (2013). Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, 108-114. doi:10.1016/j.fss.2013.01.012Gregori, V., Miñana, J.-J., Morillas, S., & Sapena, A. (2016). Cauchyness and convergence in fuzzy metric spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 111(1), 25-37. doi:10.1007/s13398-015-0272-0Gregori, V., & Miñana, J.-J. (2017). Strong convergence in fuzzy metric spaces. Filomat, 31(6), 1619-1625. doi:10.2298/fil1706619gGrabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2Miheţ, D. (2008). Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744. doi:10.1016/j.fss.2007.07.006Vasuki, R., & Veeramani, P. (2003). Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 135(3), 415-417. doi:10.1016/s0165-0114(02)00132-xGregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Gregori, V., Miñana, J.-J., & Morillas, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems, 204, 71-85. doi:10.1016/j.fss.2011.12.008Ricarte, L. A., & Romaguera, S. (2014). A domain-theoretic approach to fuzzy metric spaces. Topology and its Applications, 163, 149-159. doi:10.1016/j.topol.2013.10.014Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Shukla, S., Gopal, D., & Sintunavarat, W. (2018). A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems, 350, 85-94. doi:10.1016/j.fss.2018.02.010Beg, I., Gopal, D., Došenović, T., … Rakić, D. (2018). α-type fuzzy H-contractive mappings in fuzzy metric spaces. Fixed Point Theory, 19(2), 463-474. doi:10.24193/fpt-ro.2018.2.37Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Rakić, D., Došenović, T., Mitrović, Z. D., de la Sen, M., & Radenović, S. (2020). Some Fixed Point Theorems of Ćirić Type in Fuzzy Metric Spaces. Mathematics, 8(2), 297. doi:10.3390/math802029

Lecciones breves de Estadística

El presente libro es un texto elemental de Estadística concebido para alumnos de primer curso de grados técnicos. En cada capítulo, se encuentran argumentaciones detalladas del contenido con una terminología sencilla y un buen número de ejemplos, seguidos de una colección de ejercicios resueltos y propuestos. Los 6 capítulos que constituyen esta obra son: estadística descriptiva, distribuciones bidimensionales, probabilidad, variables aleatorias, distribuciones discretas y distribuciones continuas. La ausencia de demostraciones, en un sentido estricto, permite una lectura fluida del texto y que el alumno afronte el aprendizaje siguiendo un proceso inductivo natural. No obstante, la redacción matemática del texto es rigurosa en la exposición. Para la comprensión del libro solo se requieren conceptos matemáticos de bachilleratoEstruch Miñana, C.; Gregori Gregori, V.; Roig Sala, B.; Sapena Piera, A. (2022). Lecciones breves de Estadística. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/184936EDITORIA

Real-world assessment and characteristics of men with benign prostatic hyperplasia (BPH) in primary care and urology clinics in Spain

Objectives: To describe the real-world demographic and clinical characteristics of patients with lower urinary tract symptoms (LUTS) as a result of benign prostatic hyperplasia (BPH) in Spain. Methodology: This observational, retrospective, multicentre study conducted in primary care and urology clinics in Spain included men aged ≥50 years diagnosed (≤8 years prior to study visit) with LUTS caused by BPH. The primary endpoint was demographic and clinical characteristics; secondary endpoints included disease progression and diagnostic tests across both healthcare settings. Results: A total of 670 patients were included (primary care: n = 435; urology: n = 235). Most patients had moderate/severe LUTS (74.6%) and prostate volume >30 cc (81.7%), with no differences between settings. More patients had prostate-specific antigen (PSA) ≥1.5 ng/mL in primary care (74.5%) versus urology (67.7%). Progression criteria were prevalent (48.9%). Clinical criteria were more commonly used than the International Prostate Symptom Score (IPSS) to evaluate LUTS at diagnosis (primary care: clinical criteria 73.0%; IPSS: 26.9%; urology: clinical criteria 76.5%; IPSS: 23.4%). Proportion of patients with moderate/severe LUTS at diagnosis was lower using clinical criteria than IPSS, and the proportion of patients with 'worsening' LUTS (diagnosis to study visit) was higher when using clinical criteria versus IPSS. In both healthcare settings, the most commonly used diagnostic tests were general and urological clinical history and PSA. Conclusion: Demographic and clinical characteristics of patients with BPH in Spain were similar in primary care and urology; however, assessment criteria to evaluate LUTS severity differ and are not completely aligned with clinical guideline recommendations. Increased use of recommended assessments may enhance optimal BPH management

Pre and postoperative care for the outpatient fulguration in superficial bladder tumor. What should know the primary care physician?

Introducción: El objetivo de este trabajo es transmitir al médico de atención primaria los cuidados pre y postoperatorios en la fulguración con láser holmium de la recidiva del tumor vesical superficial. Material y Métodos: Estudio descriptivo de una cohorte de pacientes, llevado a cabo en el Hospital Universitario Morales Meseguer, Murcia (España). Se incluyen 37 pacientes con recidiva de tumor vesical superficial de bajo riesgo, sometidos a fulguración con láser de holmium. Se incluyen en el estudio pacientes con tumor papilar, tamaño ≤15 mm, menos de 5 tumores y sin alteraciones en la coagulación ni alergias a anestésicos locales. Se entregan recomendaciones pre y postoperatorias al paciente que debe conocer el médico de atención primaria, previa y tras la intervención (fulguración vesical ambulatoria con láser de holmium). Se miden complicaciones, dolor tras el procedimiento, satisfacción del paciente. Resultados: La edad media de los pacientes fue de 69.2 años, siendo el 86.5% hombres. El número de lesiones fue de 1.5 ± 0.8 y el tamaño global medio de 5.5 ± 2.7 mm. El tiempo medio de realización del procedimiento endoscópico fue de 12 ± 4.3 minutos. El 100% de los pacientes presentó puntuación en la escala visual analógica del dolor ≤ 3. Sólo 1 caso de hematuria que se resolvió de forma conservadora. Conclusión: El tratamiento con láser holmium de la recidiva tumoral vesical superficial de bajo riesgo de forma ambulatoria es seguro, precisa de profilaxis antibiótica, control del dolor y especial atención a efectos adversos que suelen ser leves.Introduction: The objective of this work is to inform to primary care the pre and postoperative care in holmium laser fulguration of superficial bladder tumor recurrence. Methods: A descriptive study of a cohort of patients, conducted at the University Hospital Morales Meseguer, Murcia (Spain). 37 patients with superficial bladder tumor recurrence undergoing holmium laser fulguration are included. Included in the study patients with papillary tumor size ≤ 15 mm, less than 5 tumors without coagulation disorders or allergies to local anesthetics. Pre and postoperative recommendations are given to the patient to know the family physician, after care and after surgery (outpatient bladder holmium laser fulguration). Complications, pain after the procedure and patient satisfaction were measured. Results: The mean age of the patients was 69.2 years, with 86.5% male. The number of lesions was 1.5 ± 0.8 and the overall average size of 5.5 ± 2.7 mm. The average time for completion of the endoscopic procedure was 12 ± 4.3 minutes. 100% of patients had scores on the visual analog pain scale ≤ 3. Only 1 case of hematuria that resolved conservatively. Conclusion: Holmium laser treatment of in low-risk superficial bladder tumor recurrence ambulatory safe, requires antibiotic prophylaxis, pain management and attention to side effects are usually mild

Efficacy and safety of a hexanic extract of Serenoa repens (Permixon (R)) for the treatment of lower urinary tract symptoms associated with benign prostatic hyperplasia (LUTS/BPH): systematic review and meta-analysis of randomised controlled trials and observational studies

Objectives To comprehensively evaluate the efficacy and safety of the hexanic extract of Serenoa repens (HESr, Permixon (R); Pierre Fabre Medicament, Castres, France), at a dose of 320 mg daily, as monotherapy for the treatment of lower urinary tract symptoms associated with benign prostatic hyperplasia (LUTS/BPH). Materials and methods We conducted a systematic review and meta-analysis of randomised controlled trials (RCTs) and prospective observational studies in patients with LUTS/BPH identified through searches in Medline, Web of Knowledge (Institute for Scientific Information), Scopus, the Cochrane Library, and bibliographic references up to March 2017. Articles studying S. repens extracts other than Permixon were excluded. Data were collected on International Prostate Symptom Score (IPSS), maximum urinary flow rate (Qmax), nocturia, quality of life, prostate volume, sexual function, and adverse drug reactions (ADRs). Data obtained from RCTs and observational studies were analysed jointly and separately using a random effects model. A sub-group analysis was performed of studies that included patients on longer-term treatment (= 1 year). Results Data from 27 studies (15 RCTs and 12 observational studies) were included for meta-analysis (total N = 5 800). Compared with placebo, the HESr was associated with 0.64 (95% confidence interval [CI] -0.98 to -0.31) fewer voids/ night (P < 0.001) and an additional mean increase in Q(max) of 2.75 mL/s (95% CI 0.57 to 4.93; P = 0.01). When compared with a-blockers, the HESr showed similar improvements on IPSS (weighted mean difference [WMD] 0.57, 95% CI -0.27 to 1.42; P = 0.18) and a comparable increase in Q(max) to tamsulosin (WMD -0.02, 95% CI -0.71 to 0.66; P = 0.95). Efficacy assessed using the IPSS was similar after 6 months of treatment between the HESr and 5a-reductase inhibitors (5ARIs). Analysis of all available published data for the HESr showed a mean improvement in IPSS from baseline of -5.73 points (95% CI -6.91 to -4.54; P < 0.001). HESr did not negatively affect sexual function and no clinically relevant effect was observed on prostate-specific antigen. Prostate volume decreased slightly. Similar efficacy results were seen in patients treated for = 1 year (n = 447). The HESr had a favourable safety profile, with gastrointestinal disorders being the most frequent ADR (mean incidence of 3.8%). Conclusion The present meta-analysis, which includes all available RCTs and observational studies, shows that the HESr (Permixon) reduced nocturia and improved Q(max) compared with placebo and had a similar efficacy to tamsulosin and short-term 5ARI in relieving LUTS. HESr (Permixon) appears to be an efficacious and well-tolerated therapeutic option for the longterm medical treatment of LUTS/BPH

A characterization of strong completeness in fuzzy metric spaces

[eng] Here, we deal with the concept of fuzzy metric space (X,M,*) , due to George and Veeramani. Based on the fuzzy diameter for a subset of X , we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory

A characterization of p-complete fuzzy metric spaces

[EN] George and Veeramani characterized complete fuzzy metric spaces ¿ by means of nested sequences of closed sets of X which have fuzzy diameter zero. According to the concept of p-convergence due to D. Mihet, an appropriate concept of p-Cauchy sequence was given. In this paper we introduce for a concept of p-fuzzy diameter zero, which is according to the concept of p-convergence. Then, we characterize by means of certain nested sequences , which have p-fuzzy diameter zero, those fuzzy metric spaces in which p-Cauchy sequences are convergent (p-convergent), called p-complete spaces (w-p-complete spaces). As a consequence of our results we obtain the well-known characterization of a complete metric space by means of nested sequences of closed sets of (X,d).Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2022). A characterization of p-complete fuzzy metric spaces. Fuzzy Sets and Systems. 444:144-155. https://doi.org/10.1016/j.fss.2021.12.00114415544

Efficacy and safety of a hexanic extract of Serenoa repens (Permixon®) for the treatment of lower urinary tract symptoms associated with benign prostatic hyperplasia (LUTS/BPH): systematic review and meta‐analysis of randomised controlled trials and observational studies

To comprehensively evaluate the efficacy and safety of the hexanic extract of Serenoa repens (HESr, Permixon®; Pierre Fabre Médicament, Castres, France), at a dose of 320 mg daily, as monotherapy for the treatment of lower urinary tract symptoms associated with benign prostatic hyperplasia (LUTS/BPH). Materials and methods: We conducted a systematic review and meta‐analysis of randomised controlled trials (RCTs) and prospective observational studies in patients with LUTS/BPH identified through searches in Medline, Web of Knowledge (Institute for Scientific Information), Scopus, the Cochrane Library, and bibliographic references up to March 2017. Articles studying S. repens extracts other than Permixon were excluded. Data were collected on International Prostate Symptom Score (IPSS), maximum urinary flow rate (Qmax), nocturia, quality of life, prostate volume, sexual function, and adverse drug reactions (ADRs). Data obtained from RCTs and observational studies were analysed jointly and separately using a random effects model. A sub‐group analysis was performed of studies that included patients on longer‐term treatment (≥1 year). Results: Data from 27 studies (15 RCTs and 12 observational studies) were included for meta‐analysis (total N = 5 800). Compared with placebo, the HESr was associated with 0.64 (95% confidence interval [CI] −0.98 to −0.31) fewer voids/night (P < 0.001) and an additional mean increase in Qmax of 2.75 mL/s (95% CI 0.57 to 4.93; P = 0.01). When compared with α‐blockers, the HESr showed similar improvements on IPSS (weighted mean difference [WMD] 0.57, 95% CI −0.27 to 1.42; P = 0.18) and a comparable increase in Qmax to tamsulosin (WMD −0.02, 95% CI −0.71 to 0.66; P = 0.95). Efficacy assessed using the IPSS was similar after 6 months of treatment between the HESr and 5α‐reductase inhibitors (5ARIs). Analysis of all available published data for the HESr showed a mean improvement in IPSS from baseline of −5.73 points (95% CI −6.91 to −4.54; P < 0.001). HESr did not negatively affect sexual function and no clinically relevant effect was observed on prostate‐specific antigen. Prostate volume decreased slightly. Similar efficacy results were seen in patients treated for ≥1 year (n = 447). The HESr had a favourable safety profile, with gastrointestinal disorders being the most frequent ADR (mean incidence of 3.8%). Conclusion: The present meta‐analysis, which includes all available RCTs and observational studies, shows that the HESr (Permixon) reduced nocturia and improved Qmax compared with placebo and had a similar efficacy to tamsulosin and short‐term 5‐ARI in relieving LUTS. HESr (Permixon) appears to be an efficacious and well‐tolerated therapeutic option for the long‐term medical treatment of LUTS/BPH.Sin financiación4.524 JCR (2018) Q1, 11/80 Urology & Nephrology1.984 SJR (2018) Q1, 4/107 UrologyNo data IDR 2018UE