Yusuf Yaylı profile image
Yusuf Yaylı Prof. Dr. ANKARA UNIVERSITY
Publication 37 Review 33 CrossRef Cited 62 TR Dizin Cited 30
37 Publication
33 Review
62 CrossRef Cited
30 TR Dizin Cited
0000-0003-4398-3855

Research Fields

Mathematical Sciences Algebraic and Differential Geometry

Institution

ANKARA UNIVERSITY

Popular Publications

W-Curves in Lorentz-Minkowski Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421740
CITED 7 FAVORITE 1 TOTAL DOWNLOAD COUNT 1374

7

1

1374

Elementary Approachs on De Sitter Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2019 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.583477
CITED 0 FAVORITE 1 TOTAL DOWNLOAD COUNT 1501

0

1

1501

Chord Properties of Some Special Curves in Euclidean Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2020 , Turkish Journal of Mathematics and Computer Science
DOI: -
CITED 0 FAVORITE 1 TOTAL DOWNLOAD COUNT 609

0

1

609

Constant Angle Ruled Surfaces in $\mathbb{E}^{3}_1$
Authors: Aykut Has , Beyhan Yılmaz , Yusuf Yaylı
Published: 2023 , Fundamental Journal of Mathematics and Applications
DOI: 10.33401/fujma.1218966
CITED 2 FAVORITE 1 TOTAL DOWNLOAD COUNT 774

2

1

774

Publications

On Constant Slope Surfaces by Means of the Rotation Minimizing Frame in $\mathbb{E}^n$
Authors: Sevda Arslan , Yusuf Yaylı
Published: 2026 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1884231
FAVORITE 0 TOTAL DOWNLOAD COUNT 392

0

392

Generalized Helices in the Euclidean 3-space Through Flow-frame
Authors: Cansu Özyurt Anar , Yusuf Yaylı , Nejat Ekmekçi
Published: 2025 , International Electronic Journal of Geometry
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 476

0

476

Direction curves and construction of developable surfaces in Lorentz 3-space
Authors: Esma Demir Çetin , Çağla Ramis , Yusuf Yaylı
Published: 2025 , Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
DOI: 10.31801/cfsuasmas.1395601
FAVORITE 0 TOTAL DOWNLOAD COUNT 559

0

559

Minimal Curves on Ruled Surfaces Generated by Legendre Curves
Authors: Yusuf Yaylı , İsmet Gölgeleyen
Published: 2024 , Journal of New Theory
DOI: 10.53570/jnt.1545097
FAVORITE 0 TOTAL DOWNLOAD COUNT 544

0

544

Elliptic Quaternion and Elliptic linear Interpolation
Authors: Melisa Rahebi , Yusuf Yaylı
Published: 2024 , Journal of Polytechnic
DOI: 10.2339/politeknik.1444759
FAVORITE 0 TOTAL DOWNLOAD COUNT 1222

0

1222

A New Representation for Slant Curves in Sasakian 3-Manifolds
Authors: Osman Ateş , İsmail Gök , Yusuf Yaylı
Published: 2024 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1397659
FAVORITE 0 TOTAL DOWNLOAD COUNT 757

0

757

Approximations of Parallel Surfaces Along Curves
Authors: Büşra Köse , Yusuf Yaylı
Published: 2023 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1362590
FAVORITE 0 TOTAL DOWNLOAD COUNT 839

0

839

Constant Angle Ruled Surfaces in $\mathbb{E}^{3}_1$
Authors: Aykut Has , Beyhan Yılmaz , Yusuf Yaylı
Published: 2023 , Fundamental Journal of Mathematics and Applications
DOI: 10.33401/fujma.1218966
FAVORITE 1 TOTAL DOWNLOAD COUNT 774

1

774

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map
Authors: Bang-yen Chen , Erhan Güler , Yusuf Yaylı , Hasan Hilmi Hacısalihoğlu
Published: 2023 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1216024
FAVORITE 0 TOTAL DOWNLOAD COUNT 2460

0

2460

New Associated Curves $k-$Principle Direction Curves and $N_{k}-$Slant Helix
Authors: Çağla Ramis , Beyhan Yılmaz , Yusuf Yaylı
Published: 2022 , Hagia Sophia Journal of Geometry
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 653

0

653

The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices
Authors: Çağla Ramis , Yusuf Yaylı , İrem Zengin
Published: 2022 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1127216
FAVORITE 0 TOTAL DOWNLOAD COUNT 1983

0

1983

Obtaining the Parametric Equation of the Curve of the Sun’s Apparent Movement by Using Quaternions
Authors: Deniz Güçler , Nejat Ekmekçi , Yusuf Yaylı , Mustafa Helvacı
Published: 2022 , Universal Journal of Mathematics and Applications
DOI: 10.32323/ujma.1091832
FAVORITE 0 TOTAL DOWNLOAD COUNT 1010

0

1010

Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$
Authors: Beyhan Yılmaz , İsmail Gök , Yusuf Yaylı
Published: 2022 , Journal of Mathematical Sciences and Modelling
DOI: 10.33187/jmsm.1007857
FAVORITE 0 TOTAL DOWNLOAD COUNT 876

0

876

Motions on curves and surfaces using geometric algebra
Authors: Selahattin Aslan , Yusuf Yaylı
Published: 2022 , Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
DOI: 10.31801/cfsuasmas.878766
FAVORITE 0 TOTAL DOWNLOAD COUNT 1461

0

1461

Ruled surfaces corresponding to hyper-dual curves
Authors: Selahattin Aslan , Murat Bekar , Yusuf Yaylı
Published: 2022 , Hacettepe Journal of Mathematics and Statistics
DOI: 10.15672/hujms.988245
FAVORITE 0 TOTAL DOWNLOAD COUNT 1387

0

1387

Homothetic Motions and Dual Transformations
Authors: Gülsüm Yüca , Yusuf Yaylı
Published: 2021 , Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 942

0

942

Polynomial Parametric Equations of Rectifying Salkowski Curves
Authors: Beyhan Yılmaz , İsmail Gök , Yusuf Yaylı
Published: 2021 , Gazi University Journal of Science
DOI: 10.35378/gujs.695460
FAVORITE 0 TOTAL DOWNLOAD COUNT 945

0

945

Ruled Surfaces with Constant Slope Ruling According to Darboux Frame
Authors: Ayşe Yavuz , Fatma Ateş , Yusuf Yaylı
Published: 2020 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.640345
FAVORITE 0 TOTAL DOWNLOAD COUNT 994

0

994

On Slant Helices And Manheim Curves in E^3
Authors: Alper Çay , Yusuf Yaylı
Published: 2020 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.742338
FAVORITE 0 TOTAL DOWNLOAD COUNT 791

0

791

Chord Properties of Some Special Curves in Euclidean Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2020 , Turkish Journal of Mathematics and Computer Science
DOI: -
FAVORITE 1 TOTAL DOWNLOAD COUNT 609

1

609

Non-null Surfaces with Constant Slope Ruling with Respect to Osculating Plane
Authors: Ayşe Yavuz , Fatma Ateş , Yusuf Yaylı
Published: 2020 , Adıyaman University Journal of Science
DOI: 10.37094/adyujsci.612485
FAVORITE 0 TOTAL DOWNLOAD COUNT 897

0

897

Elementary Approachs on De Sitter Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2019 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.583477
FAVORITE 1 TOTAL DOWNLOAD COUNT 1501

1

1501

Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$
Authors: Fatma Ateş , İsmail Gok , Faik Nejat Ekmekci , Yusuf Yaylı
Published: 2019 , Konuralp Journal of Mathematics
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1457

0

1457

Slant Helix Curves and Acceleration Centers
Authors: Murat Bekar , Yusuf Yaylı
Published: 2019 , Gazi University Journal of Science
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1351

0

1351

W-Curves in Lorentz-Minkowski Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421740
FAVORITE 1 TOTAL DOWNLOAD COUNT 1374

1

1374

Generalized Similar Frenet Curves
Authors: Fatma Ateş, Seher Kaya , Yusuf Yaylı , F. Nejat Ekmekci
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421731
FAVORITE 0 TOTAL DOWNLOAD COUNT 846

0

846

ON DEVELOPABLE RULED SURFACE OF THE PRINCIPAL-DIRECTION CURVE
Authors: Beyhan Yılmaz , Çağla Ramis, Yusuf Yaylı
Published: 2017 , Konuralp Journal of Mathematics
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1616

0

1616

Normal Fermi- Walker Derivative
Authors: Özgür Keskin, Yusuf Yaylı
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421468
FAVORITE 0 TOTAL DOWNLOAD COUNT 1730

0

1730

Kinematics of Dual Quaternion Involution Matrices
Authors: Murat Bekar, Yusuf Yaylı
Published: 2016 , Süleyman Demirel University Faculty of Arts and Science Journal of Science
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1857

0

1857

Darboux Frame of a Curve Lying on a Lightlike Surface
Authors: E. Selcen Yakıcı Topbaş, İsmail Gök , Nejat Ekmekci , Yusuf Yaylı
Published: 2016 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421465
FAVORITE 0 TOTAL DOWNLOAD COUNT 1725

0

1725

A New Construction of the Sierpinski Triangles with Galilean Transformations
Authors: Elif Aybike Büyükyılmaz, Yusuf Yaylı , İsmail Gök
Published: 2016 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421424
FAVORITE 0 TOTAL DOWNLOAD COUNT 3570

0

3570

SOME CHARACTERIZATIONS OF EULER SPIRALS IN E3
Authors: Semra Saracoglu Çelik, Yusuf Yaylı , Erhan Güler
Published: 2016 , Konuralp Journal of Mathematics
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 849

0

849

Dual Eliptik Birim Küre È𝑫 𝟐 Üzerindeki Çemberin E.Study Dönüşümü
Authors: Mehdi Jafari, Yusuf Yaylı
Published: 2015 , Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
DOI: 10.17798/beufen.85238
FAVORITE 0 TOTAL DOWNLOAD COUNT 1389

0

1389

BOUR'S THEOREM UNDER THE CONFORMAL MAP WITH LIGHT LIKE PROFILE CURVE
Authors: Zehra Bozkurt, Zehra Bozkurt, İsmail Gök , F. Nejat Ekmekci , F. Nejat Ekmekçi , Yusuf Yaylı
Published: 2015 , Beykent University Journal of Science and Engineering
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1297

0

1297

Different Approaches To Ruled Surfaces
Authors: Yusuf Yaylı , Semra Saraçoğlu
Published: 2012 , Süleyman Demirel University Faculty of Arts and Science Journal of Science
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1760

0

1760

Constant Angle Surfaces And Curves In E^3
Authors: Sıddıka Özkaldı, Yusuf Yaylı
Published: 2011 , International Electronic Journal of Geometry
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 1223

0

1223

Harmonic Curvature of a strip in E^3
Authors: Filiz Ertem Kaya , Yusuf Yaylı , H. Hilmi Hacısalıhoglu
Published: 2010 , Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
DOI: -
FAVORITE 0 TOTAL DOWNLOAD COUNT 606

0

606

Articles published in
Adıyaman University Journal of Science
Adıyaman University Journal of Science
Beykent University Journal of Science and Engineering
Beykent University Journal of Science and Engineering
Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi
Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi
Fundamental Journal of Mathematics and Applications
Fundamental Journal of Mathematics and Applications
Gazi University Journal of Science
Gazi University Journal of Science
Hacettepe Journal of Mathematics and Statistics
Hacettepe Journal of Mathematics and Statistics
Hagia Sophia Journal of Geometry
Hagia Sophia Journal of Geometry
International Electronic Journal of Geometry
International Electronic Journal of Geometry
Journal of Mathematical Sciences and Modelling
Journal of Mathematical Sciences and Modelling
Journal of New Theory
Journal of New Theory
Journal of Polytechnic
Journal of Polytechnic
Konuralp Journal of Mathematics
Konuralp Journal of Mathematics
Mathematical Sciences and Applications E-Notes
Mathematical Sciences and Applications E-Notes
Süleyman Demirel University Faculty of Arts and Science Journal of Science
Süleyman Demirel University Faculty of Arts and Science Journal of Science
Turkish Journal of Mathematics and Computer Science
Turkish Journal of Mathematics and Computer Science
Universal Journal of Mathematics and Applications
Universal Journal of Mathematics and Applications
Editorship
Bozok Journal of Science
Bozok Journal of Science
Co-Editor
International Electronic Journal of Geometry
International Electronic Journal of Geometry
Section Editor
Mathematical Sciences and Applications E-Notes
Mathematical Sciences and Applications E-Notes
Section Editor
Editorial Board Memberships
Bozok Journal of Science
Bozok Journal of Science
Hagia Sophia Journal of Geometry
Hagia Sophia Journal of Geometry
International Electronic Journal of Geometry
International Electronic Journal of Geometry

Reviews

Afyon Kocatepe University Journal of Science and Engineering
Afyon Kocatepe University Journal of Science and Engineering
Bilecik Seyh Edebali University Journal of Science
Bilecik Seyh Edebali University Journal of Science
Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
Celal Bayar University Journal of Science
Celal Bayar University Journal of Science
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Cumhuriyet Science Journal
Cumhuriyet Science Journal
Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi
Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi
Erzincan University Journal of Science and Technology
Erzincan University Journal of Science and Technology
Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler
Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler
Fundamental Journal of Mathematics and Applications
Fundamental Journal of Mathematics and Applications
Gazi University Journal of Science
Gazi University Journal of Science
Hacettepe Journal of Mathematics and Statistics
Hacettepe Journal of Mathematics and Statistics
Hagia Sophia Journal of Geometry
Hagia Sophia Journal of Geometry
International Electronic Journal of Geometry
International Electronic Journal of Geometry
Journal of Scientific Reports-C
Journal of Scientific Reports-C
Journal of Universal Mathematics
Journal of Universal Mathematics
MANAS Journal of Engineering
MANAS Journal of Engineering
Ordu University Journal of Science and Tecnology
Ordu University Journal of Science and Tecnology
The Black Sea Journal of Sciences
The Black Sea Journal of Sciences
Turkish Journal of Mathematics and Computer Science
Turkish Journal of Mathematics and Computer Science
Universal Journal of Mathematics and Applications
Universal Journal of Mathematics and Applications

Publications

Minimal Curves on Ruled Surfaces Generated by Legendre Curves
Authors: Yusuf Yaylı , İsmet Gölgeleyen
Published: 2024 , Journal of New Theory
DOI: 10.53570/jnt.1545097
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 544

2

0

544

Elliptic Quaternion and Elliptic linear Interpolation
Authors: Melisa Rahebi , Yusuf Yaylı
Published: 2024 , Journal of Polytechnic
DOI: 10.2339/politeknik.1444759
CITED 1 FAVORITE 0 TOTAL DOWNLOAD COUNT 1222

1

0

1222

A New Representation for Slant Curves in Sasakian 3-Manifolds
Authors: Osman Ateş , İsmail Gök , Yusuf Yaylı
Published: 2024 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1397659
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 757

2

0

757

Approximations of Parallel Surfaces Along Curves
Authors: Büşra Köse , Yusuf Yaylı
Published: 2023 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1362590
CITED 1 FAVORITE 0 TOTAL DOWNLOAD COUNT 839

1

0

839

Constant Angle Ruled Surfaces in $\mathbb{E}^{3}_1$
Authors: Aykut Has , Beyhan Yılmaz , Yusuf Yaylı
Published: 2023 , Fundamental Journal of Mathematics and Applications
DOI: 10.33401/fujma.1218966
CITED 2 FAVORITE 1 TOTAL DOWNLOAD COUNT 774

2

1

774

Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map
Authors: Bang-yen Chen , Erhan Güler , Yusuf Yaylı , Hasan Hilmi Hacısalihoğlu
Published: 2023 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1216024
CITED 17 FAVORITE 0 TOTAL DOWNLOAD COUNT 2460

17

0

2460

The Application of Euler-Rodrigues Formula over Hyper-Dual Matrices
Authors: Çağla Ramis , Yusuf Yaylı , İrem Zengin
Published: 2022 , International Electronic Journal of Geometry
DOI: 10.36890/iejg.1127216
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 1983

2

0

1983

Obtaining the Parametric Equation of the Curve of the Sun’s Apparent Movement by Using Quaternions
Authors: Deniz Güçler , Nejat Ekmekçi , Yusuf Yaylı , Mustafa Helvacı
Published: 2022 , Universal Journal of Mathematics and Applications
DOI: 10.32323/ujma.1091832
CITED 4 FAVORITE 0 TOTAL DOWNLOAD COUNT 1010

4

0

1010

Differential Equations of Rectifying Curves and Focal Curves in $\mathbb{E}^{n}$
Authors: Beyhan Yılmaz , İsmail Gök , Yusuf Yaylı
Published: 2022 , Journal of Mathematical Sciences and Modelling
DOI: 10.33187/jmsm.1007857
CITED 1 FAVORITE 0 TOTAL DOWNLOAD COUNT 876

1

0

876

Motions on curves and surfaces using geometric algebra
Authors: Selahattin Aslan , Yusuf Yaylı
Published: 2022 , Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
DOI: 10.31801/cfsuasmas.878766
CITED 4 FAVORITE 0 TOTAL DOWNLOAD COUNT 1461

4

0

1461

Ruled surfaces corresponding to hyper-dual curves
Authors: Selahattin Aslan , Murat Bekar , Yusuf Yaylı
Published: 2022 , Hacettepe Journal of Mathematics and Statistics
DOI: 10.15672/hujms.988245
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 1387

2

0

1387

Ruled Surfaces with Constant Slope Ruling According to Darboux Frame
Authors: Ayşe Yavuz , Fatma Ateş , Yusuf Yaylı
Published: 2020 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.640345
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 994

2

0

994

W-Curves in Lorentz-Minkowski Space
Authors: Emre Öztürk , Yusuf Yaylı
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421740
CITED 7 FAVORITE 1 TOTAL DOWNLOAD COUNT 1374

7

1

1374

Generalized Similar Frenet Curves
Authors: Fatma Ateş, Seher Kaya , Yusuf Yaylı , F. Nejat Ekmekci
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421731
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 846

2

0

846

Normal Fermi- Walker Derivative
Authors: Özgür Keskin, Yusuf Yaylı
Published: 2017 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421468
CITED 5 FAVORITE 0 TOTAL DOWNLOAD COUNT 1730

5

0

1730

Darboux Frame of a Curve Lying on a Lightlike Surface
Authors: E. Selcen Yakıcı Topbaş, İsmail Gök , Nejat Ekmekci , Yusuf Yaylı
Published: 2016 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421465
CITED 6 FAVORITE 0 TOTAL DOWNLOAD COUNT 1725

6

0

1725

A New Construction of the Sierpinski Triangles with Galilean Transformations
Authors: Elif Aybike Büyükyılmaz, Yusuf Yaylı , İsmail Gök
Published: 2016 , Mathematical Sciences and Applications E-Notes
DOI: 10.36753/mathenot.421424
CITED 2 FAVORITE 0 TOTAL DOWNLOAD COUNT 3570

2

0

3570

Popular Publications: It is determined automatically by the researchers doing research in DergiPark according to the statistics of being added to the favorites. CrossRef Cited: The citations shown in DergiPark are drawn from CrossRef. For this, the cited and cited articles must be registered in CrossRef (DOI number). ^: Citation numbers are reflected in the system by DergiPark at certain periods.