Let $K=Q(sqrt{d})$ be a quadratic field with discriminant $d$. It is shown that $sumlimits_{(frac{d}{p})=+1,_{p~ prime}}frac{1}{p}$ and $sumlimits_{(frac{d}{q})=-1,_{q~ prime}}frac{1}{q}$ are both divergent. Two different approaches are given to show the divergence: one using the Dedekind Zeta function and the other by Tauberian methods. It is shown that these two divergences are equivalent. It is shown that the divergence is equivalent to $L_{d}(1)neq 0$(de la Vall'{e}e Poussin's Theorem).We prove that the series $sumlimits_{(frac{d}{p})=+1,_{p~ prime}}frac{1}{p^{s}}$ and $sumlimits_{(frac{d}{q})=-1,_{q~ prime}}frac{1}{q^{s}}$ have singularities on all the imaginary axis(analogue of Landau-Walfisz theorem)
设$K=Q（sqrt｛d｝）$是具有判别式$d$的二次域。结果表明，$sumlimits_{（frac｛d｝｛p｝）=+1，_{p~prime｝｝frac{1｝{p｝$和$sumlimits_{。给出了两种不同的方法来显示分歧：一种使用Dedekind-Zeta函数，另一种使用Tauberian方法。结果表明，这两个发散是等价的。结果表明，散度等价于$L_｛d｝（1）neq0$（de la Vall）{e}ePoussin定理）。我们证明了序列$sumlimits_{（frac｛d｝｛p｝）=+1，_{p~素数｝｝frac{1｝｛p^｛s｝$和$sumlimits_{q~素数}｝frag｛1｝{q^｛s｝｝$在所有虚轴上都具有奇点（类似于Landau-Wallfisz定理）
{"title":"On the divergence of two subseries $ldots$] {on the divergence of two subseries of $sumfrac{1}{p}$, and theorems of de La Vall'{e}e Poussin and Landau-Walfis","authors":"G. Reddy, S. Rau, B. Uma","doi":"10.5269/bspm.50820","DOIUrl":"https://doi.org/10.5269/bspm.50820","url":null,"abstract":"Let $K=Q(sqrt{d})$ be a quadratic field with discriminant $d$. It is shown that $sumlimits_{(frac{d}{p})=+1,_{p~ prime}}frac{1}{p}$ and $sumlimits_{(frac{d}{q})=-1,_{q~ prime}}frac{1}{q}$ are both divergent. Two different approaches are given to show the divergence: one using the Dedekind Zeta function and the other by Tauberian methods. It is shown that these two divergences are equivalent. It is shown that the divergence is equivalent to $L_{d}(1)neq 0$(de la Vall'{e}e Poussin's Theorem).We prove that the series $sumlimits_{(frac{d}{p})=+1,_{p~ prime}}frac{1}{p^{s}}$ and $sumlimits_{(frac{d}{q})=-1,_{q~ prime}}frac{1}{q^{s}}$ have singularities on all the imaginary axis(analogue of Landau-Walfisz theorem)","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41934054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the problem of estimating the reliability function for Lomax distribution with the presence of fuzziness through two procedures. The first procedure depends on fuzzy reliability definition that uses the composite trapezoidal rule in order to find the numerical integration and the second is Bayesian procedure which includes different cases depends on sample data and hyper-parameters of prior gamma distribution with squared error as a symmetric loss function and precautionary as an asymmetric loss function. In the Bayesian procedure, we proposed to consider three cases to estimate the fuzzy reliability with fuzzy observations, precise observations with fuzzy hyper-parameter, and fuzzy observations with fuzzy hyper-parameter.
{"title":"Reliability estimation of Lomax distribution with fuzziness","authors":"N. Al-Noor","doi":"10.5269/bspm.51809","DOIUrl":"https://doi.org/10.5269/bspm.51809","url":null,"abstract":"This paper considers the problem of estimating the reliability function for Lomax distribution with the presence of fuzziness through two procedures. The first procedure depends on fuzzy reliability definition that uses the composite trapezoidal rule in order to find the numerical integration and the second is Bayesian procedure which includes different cases depends on sample data and hyper-parameters of prior gamma distribution with squared error as a symmetric loss function and precautionary as an asymmetric loss function. In the Bayesian procedure, we proposed to consider three cases to estimate the fuzzy reliability with fuzzy observations, precise observations with fuzzy hyper-parameter, and fuzzy observations with fuzzy hyper-parameter.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45116482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method
{"title":"A modified reproducing Kernel Hilbert space method for solving fuzzy fractional integro-differential equations","authors":"S. Hasan, B. Maayah, Samia Bushnaq, S. Momani","doi":"10.5269/bspm.52289","DOIUrl":"https://doi.org/10.5269/bspm.52289","url":null,"abstract":"The aim of this paper is to extend the application of the reproducing kernel Hilbert space method (RKHSM) to solve linear and non-linear fuzzy integro-differential equations of fractional order under Caputo's H-differentiability. The analytic and approximate solutions are given in series form in term of their parametric form in the space $W_2^2 [a,b] bigoplus W_2^2 [a,b]$. Several examples are carried out to show the effectiveness and the absence of complexity of the proposed method","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46686133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work aims to study a nonlinear wave equation subject to a delay of neutral type. The nonlinearity and the delay appear in the second time derivative. In spite of the fact that delays by nature, have an instability effect on the structures, the strong damping is sufficient to allow the system to reach its equilibrium state with an exponential manner. The difficulties arising from the nonlinearity have been overcome by using an inequality due to a Sobolev embedding theorem. The main result has been established without any condition on the coefficient of the neutral delay.
{"title":"On exponential stabilization of a nonlinear neutral wave equation","authors":"A. Kelleche, A. Berkani","doi":"10.5269/bspm.52132","DOIUrl":"https://doi.org/10.5269/bspm.52132","url":null,"abstract":"This work aims to study a nonlinear wave equation subject to a delay of neutral type. The nonlinearity and the delay appear in the second time derivative. In spite of the fact that delays by nature, have an instability effect on the structures, the strong damping is sufficient to allow the system to reach its equilibrium state with an exponential manner. The difficulties arising from the nonlinearity have been overcome by using an inequality due to a Sobolev embedding theorem. The main result has been established without any condition on the coefficient of the neutral delay.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46743877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we intend to study some of the curvature tensor of $eta$-Ricci solitons of indefinite Trans-Sasakian manifold admitting semi-symmetric metric connection.
{"title":"$eta$-Ricci soliton in an indefinite Trans-Sasakian manifold admitting semi-symmetric metric connection","authors":"G. Somashekhara, S. Babu, P. Reddy","doi":"10.5269/bspm.51358","DOIUrl":"https://doi.org/10.5269/bspm.51358","url":null,"abstract":"In this paper, we intend to study some of the curvature tensor of $eta$-Ricci solitons of indefinite Trans-Sasakian manifold admitting semi-symmetric metric connection.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49373221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yasmina Kadri, H. Benseridi, M. Dilmi, Aissa Benseghir
This paper deals with the asymptotic behavior of a boundary value problem in a three dimensional thin domain Ω ε with non-linear friction of Coulomb type. We will establish a variational formulation for the problem and prove the existence and uniqueness of the weak solution. We then study the asymptotic behavior when one dimension of the domain tends to zero. In which case, the uniqueness result of the displacement for the limit problem is also proved.
{"title":"Behavior of the isothermal Elasticity operator with non-linear friction in a thin domain","authors":"Yasmina Kadri, H. Benseridi, M. Dilmi, Aissa Benseghir","doi":"10.5269/bspm.51676","DOIUrl":"https://doi.org/10.5269/bspm.51676","url":null,"abstract":"This paper deals with the asymptotic behavior of a boundary value problem in a three dimensional thin domain Ω ε with non-linear friction of Coulomb type. We will establish a variational formulation for the problem and prove the existence and uniqueness of the weak solution. We then study the asymptotic behavior when one dimension of the domain tends to zero. In which case, the uniqueness result of the displacement for the limit problem is also proved.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42031975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.
{"title":"Pricing cumulative loss derivatives under additive models via Malliavin calculus","authors":"M. Khalfallah, M. Hadji, J. Vives","doi":"10.5269/bspm.51549","DOIUrl":"https://doi.org/10.5269/bspm.51549","url":null,"abstract":"We show that integration by parts formulas based on Malliavin-Skorohod calculus techniques for additive processes help us to compute quantities like ${E}(L_T h(L_T))$ for different suitable functions $h$ and different models for the cumulative loss process $L_T$. These quantities are important in Insurance and Finance. For example they appear in computing expected shortfall risk measures or stop-loss contracts. The formulas given in the present paper, obtained by simple proofs, generalize the formulas given in a recent paper by Hillairet, Jiao and Réveillac using Malliavin calculus techniques for the standard Poisson process, a particular case of additive process.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48073650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The natural mate curve r1 of r is defined the integral of principal normal vector with any parameter s, of a curve r. We investigate the ruled surface generated by the natural mate curve of any Frenet curve r = r(s) in the Euclidean 3-space. We obtained some necessary and sufficient conditions for this surface to be developable and minimal ruled surface. We research related to be the asymptotic curve and the geodesic curve of the base curve on the ruled surface. Example of our main results are also presented.
{"title":"The ruled surface obtained by the natural mate curve","authors":"Fatma Güler","doi":"10.5269/bspm.51670","DOIUrl":"https://doi.org/10.5269/bspm.51670","url":null,"abstract":"The natural mate curve r1 of r is defined the integral of principal normal vector with any parameter s, of a curve r. We investigate the ruled surface generated by the natural mate curve of any Frenet curve r = r(s) in the Euclidean 3-space. We obtained some necessary and sufficient conditions for this surface to be developable and minimal ruled surface. We research related to be the asymptotic curve and the geodesic curve of the base curve on the ruled surface. Example of our main results are also presented.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41406340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, a Riemannian metric on the tangent bundle, which is another generalization of Cheeger-Gromoll metric and Sasaki metric, is considered. This metric is known as Kaluza-Klein metric in literature which is completely determined by its action on vector fields of type X^{H} and Y^{V}. We obtain the covarient and Lie derivatives applied to the Kaluza-Klein metric with respect to the horizontal and vertical lifts of vector fields, respectively on tangent bundle TM.
{"title":"Some calculations on Kaluza-Klein metric with respect to lifts in tangent bundle","authors":"Haşim Çayır","doi":"10.5269/bspm.52990","DOIUrl":"https://doi.org/10.5269/bspm.52990","url":null,"abstract":"In the present paper, a Riemannian metric on the tangent bundle, which is another generalization of Cheeger-Gromoll metric and Sasaki metric, is considered. This metric is known as Kaluza-Klein metric in literature which is completely determined by its action on vector fields of type X^{H} and Y^{V}. We obtain the covarient and Lie derivatives applied to the Kaluza-Klein metric with respect to the horizontal and vertical lifts of vector fields, respectively on tangent bundle TM.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44404525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present results concerning the existence, stability and local attractivity for non-autonomous semilinear boundary Cauchy problems. In our method, we assume certain smoothness properties on the linear part and the local lipshitz continuity on the nonlinear perturbation. We apply our abstract results to population equations.
{"title":"Stability and local attractivity for non-autonomous boundary Cauchy problems","authors":"A. Jerroudi, M. Moussi","doi":"10.5269/bspm.52035","DOIUrl":"https://doi.org/10.5269/bspm.52035","url":null,"abstract":"In this paper we present results concerning the existence, stability and local attractivity for non-autonomous semilinear boundary Cauchy problems. In our method, we assume certain smoothness properties on the linear part and the local lipshitz continuity on the nonlinear perturbation. We apply our abstract results to population equations.","PeriodicalId":44941,"journal":{"name":"Boletim Sociedade Paranaense de Matematica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44456537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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