Abstract In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.
{"title":"Trapezoid type inequalities for generalized Riemann-Liouville fractional integrals of functions with bounded variation","authors":"S. Dragomir","doi":"10.2478/ausm-2020-0003","DOIUrl":"https://doi.org/10.2478/ausm-2020-0003","url":null,"abstract":"Abstract In this paper we establish some trapezoid type inequalities for the Riemann-Liouville fractional integrals of functions of bounded variation and of Hölder continuous functions. Applications for the g-mean of two numbers are provided as well. Some particular cases for Hadamard fractional integrals are also provided.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85748247","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}
Abstract Investigation has been made regarding the properties of the ℿp≤n (1 ± 1/ps) products over the prime numbers, where we fix the s ∈ ℝ exponent, and let the n ≥ 2 natural bound grow toward positive infinity. The nature of these products for the s ≥ 1 case is known. We get approximations for the case when s ∈ [1/2, 1), furthermore different observations for the case when s<1/2.
{"title":"On Euler products with smaller than one exponents","authors":"G. Román","doi":"10.2478/ausm-2020-0013","DOIUrl":"https://doi.org/10.2478/ausm-2020-0013","url":null,"abstract":"Abstract Investigation has been made regarding the properties of the ℿp≤n (1 ± 1/ps) products over the prime numbers, where we fix the s ∈ ℝ exponent, and let the n ≥ 2 natural bound grow toward positive infinity. The nature of these products for the s ≥ 1 case is known. We get approximations for the case when s ∈ [1/2, 1), furthermore different observations for the case when s<1/2.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83256845","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}
Abstract In this paper, we present some fixed point results satisfying generalized contractive condition with new auxiliary function in complete metric spaces. More precisely, the structure of the paper is the following. In the first section, we present some useful notions and results. The main aim of second section is to establish some new fixed point results in complete metric spaces. Finally, in the third section, we show the validity and superiority of our main results by suitable example. Also, as an application of our main result, some interesting corollaries have been included, which make our concepts and results effective. Our main result generalizes some well known existing results in the literature.
{"title":"Fixed point theorem for new type of auxiliary functions","authors":"Vishal Gupta, A. H. Ansari, Naveen Mani","doi":"10.2478/ausm-2020-0006","DOIUrl":"https://doi.org/10.2478/ausm-2020-0006","url":null,"abstract":"Abstract In this paper, we present some fixed point results satisfying generalized contractive condition with new auxiliary function in complete metric spaces. More precisely, the structure of the paper is the following. In the first section, we present some useful notions and results. The main aim of second section is to establish some new fixed point results in complete metric spaces. Finally, in the third section, we show the validity and superiority of our main results by suitable example. Also, as an application of our main result, some interesting corollaries have been included, which make our concepts and results effective. Our main result generalizes some well known existing results in the literature.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80849571","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}
Abstract Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕn of a relation ϕ on X, defined by ϕ0=Δx, ϕn=ϕ∘ϕn-1 if n∈, and ϕ∞=∪n=0∞ϕn. {varphi ^0} = {Delta _x},,,{varphi ^n} = varphi circ {varphi ^{n - 1}}{rm{ if n}} in mathbb{N,},,{rm{and }},,{varphi ^infty } = bigcuplimits_{n = 0}^infty {{varphi ^n}} . In particular, by using the relational inclusion ϕn◦ϕm ⊆ ϕn+m with n, m ∈ ¯0 mathbb{bar {N}_0}} , we show that the function α, defined by α(n)=ϕn for n∈¯0, alpha left( n right) = {varphi ^{rm{n}}},,,{rm{for n}} in {{rmmathbb{bar N}}_{rm{0}}}, satisfies the Cauchy problem α(n)∘α(m)⊆α(n+m), α(0)=Δx. alpha left( n right) circ alpha left( {rm{m}} right) subseteq alpha left( {{rm{n}} + {rm{m}}} right),,,,alpha left( 0 right) = {Delta _{rm{x}}}. Moreover, the function f, defined by f(n,A)=α(n)[ A ] for n∈¯0 and A⊆X, {rm{f}}left( {{rm{n}},{rm{A}}} right) = alpha left( {rm{n}} right)left[ {rm{A}} right],,,{rm{for}},{rm{n}} in {{rmmathbb{bar {N}}}_{rm{0}}},,{rm{and}},{rm{A}} subseteq {rm{X,}} satisfies the translation problem f(n,f(m,A))⊆f(n+m,A), f(0,A)=A. {rm{f}}left( {{rm{n}},f(m,{rm{A)}}} right) subseteq {rm{f}}left( {{rm{n}} + {rm{m,A}}} right),,,,{rm{f}}left( {0,{rm{A}}} right) = {rm{A}}{rm{.}} Furthermore, the function F, defined by F(A,B)={ n∈¯0: A⊆f(n,B) } for A,B⊆X, {rm{F}}left( {{rm{A}},{rm{B}}} right) = left{ {{rm{n}} in {{{rmmathbb{bar {N}}}}_{rm{0}}}:,,{rm{A}} subseteq {rm{f}}left( {{rm{n}},{rm{B}}} right)} right},,{rm{for}},,{rm{A,B}} subseteq {rm{X,}} satisfies the Sincov problem F(A,B)+F(B,C)⊆F(A,C), 0∈F(A,A). {rm{F}}left( {{rm{A}},{rm{B}}} right) + {rm{F}}left( {{rm{B}},{rm{C}}} right) subseteq {rm{F}}left( {{rm{A,C}}} right),,,,,0 in {rm{F}}left( {{rm{A}},{rm{A}}} right). Motivated by the above observations, we investigate a function F on the product set X2 to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that F(x,y)+F(y,z)⊆F(x,z) {rm{F}}left( {{rm{x}},{rm{y}}} right) + {rm{F}}left( {{rm{y}},{rm{z}}} right) subseteq {rm{F}}left( {{rm{x}},{rm{z}}} right) for all x, y, z ∈ X. For this, we introduce the convenient notations R(x,y)=F(y,x) and S(x,y)=F(x,y)+R(x,y), {rm{R}}left( {{rm{x}},{rm{y}}} right) = {rm{F}}left( {{rm{y}},{rm{x}}} right),,,{rm{and}},,{rm{S}}left( {{rm{x}},{rm{y}}} right) = {rm{F}}left( {{rm{x}},{rm{y}}} right) + {rm{R}}left( {{rm{x}},{rm{y}}} right), and Φ(x)=F(x,x) and Ψ(x)∪y∈XS(x,y). Phi left( {rm{x}} right) = {rm{F}}left( {{rm{x}},{rm{x}}} right),,{rm{and}},,Psi left( {rm{x}} right)bigcuplimits_{{rm{y}} in {rm{X}}} {{rm{S}}left( {{rm{x}},{rm{y}}} right).} Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric,
{"title":"Composition iterates, Cauchy, translation, and Sincov inclusions","authors":"W. Fechner, Á. Száz","doi":"10.2478/ausm-2020-0004","DOIUrl":"https://doi.org/10.2478/ausm-2020-0004","url":null,"abstract":"Abstract Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕn of a relation ϕ on X, defined by ϕ0=Δx, ϕn=ϕ∘ϕn-1 if n∈, and ϕ∞=∪n=0∞ϕn. {varphi ^0} = {Delta _x},,,{varphi ^n} = varphi circ {varphi ^{n - 1}}{rm{ if n}} in mathbb{N,},,{rm{and }},,{varphi ^infty } = bigcuplimits_{n = 0}^infty {{varphi ^n}} . In particular, by using the relational inclusion ϕn◦ϕm ⊆ ϕn+m with n, m ∈ ¯0 mathbb{bar {N}_0}} , we show that the function α, defined by α(n)=ϕn for n∈¯0, alpha left( n right) = {varphi ^{rm{n}}},,,{rm{for n}} in {{rmmathbb{bar N}}_{rm{0}}}, satisfies the Cauchy problem α(n)∘α(m)⊆α(n+m), α(0)=Δx. alpha left( n right) circ alpha left( {rm{m}} right) subseteq alpha left( {{rm{n}} + {rm{m}}} right),,,,alpha left( 0 right) = {Delta _{rm{x}}}. Moreover, the function f, defined by f(n,A)=α(n)[ A ] for n∈¯0 and A⊆X, {rm{f}}left( {{rm{n}},{rm{A}}} right) = alpha left( {rm{n}} right)left[ {rm{A}} right],,,{rm{for}},{rm{n}} in {{rmmathbb{bar {N}}}_{rm{0}}},,{rm{and}},{rm{A}} subseteq {rm{X,}} satisfies the translation problem f(n,f(m,A))⊆f(n+m,A), f(0,A)=A. {rm{f}}left( {{rm{n}},f(m,{rm{A)}}} right) subseteq {rm{f}}left( {{rm{n}} + {rm{m,A}}} right),,,,{rm{f}}left( {0,{rm{A}}} right) = {rm{A}}{rm{.}} Furthermore, the function F, defined by F(A,B)={ n∈¯0: A⊆f(n,B) } for A,B⊆X, {rm{F}}left( {{rm{A}},{rm{B}}} right) = left{ {{rm{n}} in {{{rmmathbb{bar {N}}}}_{rm{0}}}:,,{rm{A}} subseteq {rm{f}}left( {{rm{n}},{rm{B}}} right)} right},,{rm{for}},,{rm{A,B}} subseteq {rm{X,}} satisfies the Sincov problem F(A,B)+F(B,C)⊆F(A,C), 0∈F(A,A). {rm{F}}left( {{rm{A}},{rm{B}}} right) + {rm{F}}left( {{rm{B}},{rm{C}}} right) subseteq {rm{F}}left( {{rm{A,C}}} right),,,,,0 in {rm{F}}left( {{rm{A}},{rm{A}}} right). Motivated by the above observations, we investigate a function F on the product set X2 to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that F(x,y)+F(y,z)⊆F(x,z) {rm{F}}left( {{rm{x}},{rm{y}}} right) + {rm{F}}left( {{rm{y}},{rm{z}}} right) subseteq {rm{F}}left( {{rm{x}},{rm{z}}} right) for all x, y, z ∈ X. For this, we introduce the convenient notations R(x,y)=F(y,x) and S(x,y)=F(x,y)+R(x,y), {rm{R}}left( {{rm{x}},{rm{y}}} right) = {rm{F}}left( {{rm{y}},{rm{x}}} right),,,{rm{and}},,{rm{S}}left( {{rm{x}},{rm{y}}} right) = {rm{F}}left( {{rm{x}},{rm{y}}} right) + {rm{R}}left( {{rm{x}},{rm{y}}} right), and Φ(x)=F(x,x) and Ψ(x)∪y∈XS(x,y). Phi left( {rm{x}} right) = {rm{F}}left( {{rm{x}},{rm{x}}} right),,{rm{and}},,Psi left( {rm{x}} right)bigcuplimits_{{rm{y}} in {rm{X}}} {{rm{S}}left( {{rm{x}},{rm{y}}} right).} Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric, ","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81764497","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}
Abstract In this paper the conditions and the findings of a simulation study is presented for assessing the effect size of users’ consciousness to the computer network vulnerability in risky cyber attack situations at a certain business. First a simple model is set up to classify the groups of users according to their skills and awareness then probabilities are assigned to each class describing the likelihood of committing dangerous reactions in case of a cyber attack. To quantify the level of network vulnerability a metric developed in a former work is used. This metric shows the approximate probability of an infection at a given business with well specified parameters according to its location, the type of the attack, the protections used at the business etc. The findings mirror back the expected tendencies namely if the number of conscious user is on the
{"title":"Assessing the effect size of users’ consciousness for computer networks vulnerability","authors":"László Bognár, A. Joós, B. Nagy","doi":"10.2478/ausm-2020-0002","DOIUrl":"https://doi.org/10.2478/ausm-2020-0002","url":null,"abstract":"Abstract In this paper the conditions and the findings of a simulation study is presented for assessing the effect size of users’ consciousness to the computer network vulnerability in risky cyber attack situations at a certain business. First a simple model is set up to classify the groups of users according to their skills and awareness then probabilities are assigned to each class describing the likelihood of committing dangerous reactions in case of a cyber attack. To quantify the level of network vulnerability a metric developed in a former work is used. This metric shows the approximate probability of an infection at a given business with well specified parameters according to its location, the type of the attack, the protections used at the business etc. The findings mirror back the expected tendencies namely if the number of conscious user is on the","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77914080","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}
Abstract The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type dXt=S(Xt)dt+εdBtH,K, X0=x0, 0≤t≤T, {rm{d}}{{rm{X}}_{rm{t}}} = {rm{S}}left( {{{rm{X}}_{rm{t}}}} right){rm{dt + }}varepsilon {rm{dB}}_{rm{t}}^{{rm{H,K}}},,{{rm{X}}_{rm{0}}} = {{rm{x}}_{rm{0}}},,0 le {rm{t}} le {rm{T,}} where { BtH,K,t≥0 {rm{B}}_{rm{t}}^{{rm{H,K}}},{rm{t}} ge {rm{0}} } is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.
{"title":"Nonparametric estimation of trend function for stochastic differential equations driven by a bifractional Brownian motion","authors":"Abdelmalik Keddi, Fethi Madani, A. Bouchentouf","doi":"10.2478/ausm-2020-0008","DOIUrl":"https://doi.org/10.2478/ausm-2020-0008","url":null,"abstract":"Abstract The main objective of this paper is to investigate the problem of estimating the trend function St = S(xt) for process satisfying stochastic differential equations of the type dXt=S(Xt)dt+εdBtH,K, X0=x0, 0≤t≤T, {rm{d}}{{rm{X}}_{rm{t}}} = {rm{S}}left( {{{rm{X}}_{rm{t}}}} right){rm{dt + }}varepsilon {rm{dB}}_{rm{t}}^{{rm{H,K}}},,{{rm{X}}_{rm{0}}} = {{rm{x}}_{rm{0}}},,0 le {rm{t}} le {rm{T,}} where { BtH,K,t≥0 {rm{B}}_{rm{t}}^{{rm{H,K}}},{rm{t}} ge {rm{0}} } is a bifractional Brownian motion with known parameters H ∈ (0, 1), K ∈ (0, 1] and HK ∈ (1/2, 1). We estimate the unknown function S(xt) by a kernel estimator ̂St and obtain the asymptotic properties as ε → 0. Finally, a numerical example is provided.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83798412","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}
Abstract For the sequence of King operators, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse approximation theorem of Berens-Lorentz-type.
{"title":"Direct and converse theorems for King operators","authors":"Z. Finta","doi":"10.2478/ausm-2020-0005","DOIUrl":"https://doi.org/10.2478/ausm-2020-0005","url":null,"abstract":"Abstract For the sequence of King operators, we establish a direct approximation theorem via the first order Ditzian-Totik modulus of smoothness, and a converse approximation theorem of Berens-Lorentz-type.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76866355","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}
Abstract The purpose of this paper is to introduce a new type of φ -implicit relation in S - metric spaces and to prove a general fixed point for a pair of weakly compatible mappings, which generalize Theorems 1, 2, 4 [23], Theorems 1-7 [13], Corollary 2.19 [13], Theorems 2.2, 2.4 [19], Theorems 3.2, 3.3, 3.4 [20] and other known results.
{"title":"Fixed points for a pair of weakly compatible mappings satisfying a new type of ϕ - implicit relation in S - metric spaces","authors":"V. Popa, A. Patriciu","doi":"10.2478/ausm-2020-0012","DOIUrl":"https://doi.org/10.2478/ausm-2020-0012","url":null,"abstract":"Abstract The purpose of this paper is to introduce a new type of φ -implicit relation in S - metric spaces and to prove a general fixed point for a pair of weakly compatible mappings, which generalize Theorems 1, 2, 4 [23], Theorems 1-7 [13], Corollary 2.19 [13], Theorems 2.2, 2.4 [19], Theorems 3.2, 3.3, 3.4 [20] and other known results.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76267279","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}
Abstract Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.
{"title":"Orbital shadowing property on chain transitive sets for generic diffeomorphisms","authors":"Manseob Lee","doi":"10.2478/ausm-2020-0009","DOIUrl":"https://doi.org/10.2478/ausm-2020-0009","url":null,"abstract":"Abstract Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79198149","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}
Abstract In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.
{"title":"Generalized normal ruled surface of a curve in the Euclidean 3-space","authors":"O. Kaya, M. Önder","doi":"10.2478/ausm-2021-0013","DOIUrl":"https://doi.org/10.2478/ausm-2021-0013","url":null,"abstract":"Abstract In this study, we define the generalized normal ruled surface of a curve in the Euclidean 3-space E3. We study the geometry of such surfaces by calculating the Gaussian and mean curvatures to determine when the surface is flat or minimal (equivalently, helicoid). We examine the conditions for the curves lying on this surface to be asymptotic curves, geodesics or lines of curvature. Finally, we obtain the Frenet vectors of generalized normal ruled surface and get some relations with helices and slant ruled surfaces and we give some examples for the obtained results.","PeriodicalId":43054,"journal":{"name":"Acta Universitatis Sapientiae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91389482","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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