# DrOlegKarpenkovDr.

## Selected Publications

1. Geometry and combinatoric of Minkowski-Voronoi 3-dimensional continued fractions (Journal article - 2017)

### Continued Fraction approach to Gauss Reduction Theory (Journal article)

Karpenkov, O. (n.d.). Continued Fraction approach to Gauss Reduction Theory. Retrieved from http://arxiv.org/abs/2107.02687v1

### On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups (Journal article)

Karpenkov, O. (n.d.). On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups. Retrieved from http://arxiv.org/abs/2101.12707v1

### On a periodic Jacobi-Perron type algorithm (Journal article)

Karpenkov, O. (n.d.). On a periodic Jacobi-Perron type algorithm. Retrieved from http://arxiv.org/abs/2101.12627v2

### Geometric criteria for realizability of tensegrities in higher dimensions (Journal article)

Karpenkov, O., & Mueller, C. (2021). GEOMETRIC CRITERIA FOR REALIZABILITY OF TENSEGRITIES IN HIGHER DIMENSIONS. SIAM JOURNAL ON DISCRETE MATHEMATICS, 35(2), 637-660. doi:10.1137/19M1281903

DOI: 10.1137/19M1281903

### The combinatorial geometry of stresses in frameworks (Journal article)

Karpenkov, O. (n.d.). The combinatorial geometry of stresses in frameworks. Discrete and Computational Geometry: an international journal of mathematics and computer science. doi:10.1007/s00454-020-00234-8

DOI: 10.1007/s00454-020-00234-8

### Equilibrium stressability of multidimensional frameworks (Journal article)

Karpenkov, O., Müller, C., Panina, G., Servatius, B., Servatius, H., & Siersma, D. (n.d.). Equilibrium stressability of multidimensional frameworks. Retrieved from http://arxiv.org/abs/2009.05469v1

### Generalised Markov numbers (Journal article)

Karpenkov, O., & van-Son, M. (2020). Generalised Markov numbers. Journal of Number Theory, 213, 16-66. doi:10.1016/j.jnt.2020.01.010

DOI: 10.1016/j.jnt.2020.01.010

### Martin Integral Representation for Nonharmonic Functions and Discrete Co-Pizzetti Series (Journal article)

Boiko, T., & Karpenkov, O. (2019). Martin Integral Representation for Nonharmonic Functions and Discrete Co-Pizzetti Series. MATHEMATICAL NOTES, 106(5-6), 659-673. doi:10.1134/S0001434619110014

DOI: 10.1134/S0001434619110014

### The Sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018. (Journal article)

Nair, R., Karpenkov, O., & Verger-Gaugry, J. -L. (2019). The Sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018.. Uniform Distribution Theory. doi:10.2478/UDT-2019–0009

DOI: 10.2478/UDT-2019–0009

### Generalized Perron Identity for broken lines (Journal article)

Karpenkov, O., & van-Son, M. (2019). Generalized Perron Identity for broken lines. Journal de Theorie des Nombres de Bordeaux, 31(1), 131-144. doi:10.5802/jtnb.1071

DOI: 10.5802/jtnb.1071

### Generalized Perron Identity for broken lines (Journal article)

Karpenkov, O., & Van-Son, M. (2019). Generalized Perron Identity for broken lines. JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX, 31(1), 131-144. Retrieved from http://gateway.webofknowledge.com/

### Approximating reals by rationals of the form a/b^2 (Journal article)

Karpenkov, O. (n.d.). Approximating reals by rationals of the form a/b^2. Retrieved from http://arxiv.org/abs/math/0610717v2

### Tensegrities on the space of generic functions (Journal article)

Karpenkov, O. (n.d.). Tensegrities on the space of generic functions. Retrieved from http://arxiv.org/abs/1905.11262v1

### On Periodic Asymmetric Extrapolation (Journal article)

Boiko, T., Karpenkov, O., & Rakhimberdiev, B. (2018). On Periodic Asymmetric Extrapolation. MATHEMATICAL NOTES, 104(5-6), 642-654. doi:10.1134/S0001434618110044

DOI: 10.1134/S0001434618110044

### Geometric Conditions of Rigidity in Nongeneric settings (Chapter)

Karpenkov, O. (2018). Geometric Conditions of Rigidity in Nongeneric settings. In J. Sidman, A. S. John, & M. Sitharam (Eds.), Handbook of Geometric Constraint Systems Principles. CRC Press.

### Open Problems on Configuration Spaces of Tensegrities (Journal article)

Karpenkov, O. (2018). Open Problems on Configuration Spaces of Tensegrities. Arnold Mathematical Journal, 4(1), 19-25. doi:10.1007/s40598-018-0080-7

DOI: 10.1007/s40598-018-0080-7

### Open problems in geometry of continued fractions (Journal article)

Karpenkov, O. (n.d.). Open problems in geometry of continued fractions. Retrieved from http://arxiv.org/abs/1712.01450v1

### Geometry and combinatoric of Minkowski-Voronoi 3-dimensional continued fractions (Journal article)

Karpenkov, O., & Ustinov, A. (2017). Geometry and combinatoric of Minkowski-Voronoi 3-dimensional continued fractions. Journal of Number Theory, 176, 375-419. doi:10.1016/j.jnt.2016.12.005

DOI: 10.1016/j.jnt.2016.12.005

### ALEXEI BRONISLAVOVICH SOSSINSKY TURNS 80 (Journal article)

Chernavski, A., Fuchs, D., Gussein-Zade, S., Ilyashenko, Y., Karpenkov, O., Kirillov, A., . . . Vassiliev, V. (2017). ALEXEI BRONISLAVOVICH SOSSINSKY TURNS 80. MOSCOW MATHEMATICAL JOURNAL, 17(3), 555-558. Retrieved from http://gateway.webofknowledge.com/

### Finite and infinitesimal flexibility of semidiscrete surfaces (Journal article)

Karpenkov, O. (2015). Finite and infinitesimal flexibility of semidiscrete surfaces. Arnold Mathematical Journal, 1(04), 403-444. doi:10.1007/s40598-015-0025-3

DOI: 10.1007/s40598-015-0025-3

### On offsets and curvatures for discrete and semidiscrete surfaces (Journal article)

Karpenkov, O., & Wallner, J. (2014). On offsets and curvatures for discrete and semidiscrete surfaces. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 55(1), 207-228. doi:10.1007/s13366-013-0146-6

DOI: 10.1007/s13366-013-0146-6

### Mean value property for nonharmonic functions (Journal article)

Boiko, T., & Karpenkov, O. (n.d.). Mean value property for nonharmonic functions. Retrieved from http://arxiv.org/abs/1309.5040v1

### Euler elasticae in the plane and the Whitney--Graustein theorem (Journal article)

Avvakumov, S., Karpenkov, O., & Sossinsky, A. (n.d.). Euler elasticae in the plane and the Whitney--Graustein theorem. doi:10.1134/S1061920813030011

DOI: 10.1134/S1061920813030011

### Geometry of Continued Fractions (Book)

Karpenkov, O. (2013). Geometry of Continued Fractions (Vol. 26). Heidelberg, Germany: Springer-Verlag GmbH.

### Multidimensional Gauss Reduction Theory for conjugacy classes of SL(n,Z) (Journal article)

Karpenkov, O. (n.d.). Multidimensional Gauss Reduction Theory for conjugacy classes of SL(n,Z). Retrieved from http://arxiv.org/abs/0711.0830v3

### On stratifications for planar tensegrities with a small number of vertices (Journal article)

Karpenkov, O., Schepers, J., & Servatius, B. (n.d.). On stratifications for planar tensegrities with a small number of vertices. Retrieved from http://arxiv.org/abs/1201.3557v1

### Energies of knot diagrams (Journal article)

Karpenkov, O., & Sossinsky, A. (n.d.). Energies of knot diagrams. doi:10.1134/S1061920811030046

DOI: 10.1134/S1061920811030046

### Continued fractions and the second Kepler law (Journal article)

Karpenkov, O. (n.d.). Continued fractions and the second Kepler law. Retrieved from http://arxiv.org/abs/0911.2791v1

### On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators (Journal article)

Karpenkov, O. (n.d.). On determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators. Retrieved from http://arxiv.org/abs/0708.1604v1

### On the flexibility of Kokotsakis meshes (Journal article)

Karpenkov, O. (n.d.). On the flexibility of Kokotsakis meshes. Retrieved from http://arxiv.org/abs/0812.3050v1

### Rational approximation of the maximal commutative subgroups of GL(n,R) (Journal article)

Karpenkov, O., & Vershik, A. (n.d.). Rational approximation of the maximal commutative subgroups of GL(n,R). Retrieved from http://arxiv.org/abs/0910.3482v1

### Geometry of configuration spaces of tensegrities (Journal article)

Doray, F., Karpenkov, O., & Schepers, J. (n.d.). Geometry of configuration spaces of tensegrities. Retrieved from http://arxiv.org/abs/0806.4976v2

### Constructing multidimensional periodic continued fractions in the sense of Klein (Journal article)

Karpenkov, O. (n.d.). Constructing multidimensional periodic continued fractions in the sense of Klein. Retrieved from http://arxiv.org/abs/math/0411031v3

### On irrational lattice angles (Journal article)

Karpenkov, O. (2009). On irrational lattice angles. Functional Analysis and Other Mathematics, 2(2-4), 221-239. doi:10.1007/s11853-008-0029-9

DOI: 10.1007/s11853-008-0029-9

### Bernoulli-Euler numbers and multiboundary singularities of type \$B_n^l\$ (Journal article)

Karpenkov, O. (n.d.). Bernoulli-Euler numbers and multiboundary singularities of type \$B_n^l\$. Retrieved from http://arxiv.org/abs/0910.4046v1

### Elementary notions of lattice trigonometry (Journal article)

Karpenkov, O. (n.d.). Elementary notions of lattice trigonometry. 2008. The second part in Funct. Anal. Other Math., 2, 2-4. Retrieved from http://arxiv.org/abs/math/0604129v3

### On examples of difference operators for \$\{0,1\}\$-valued functions over finite sets (Journal article)

Karpenkov, O. (n.d.). On examples of difference operators for \$\{0,1\}\$-valued functions over finite sets. Functional Analysis and Other Mathematics,vol.1(2), pp.197-202, 2006. doi:10.1007/s11853-007-0010-z

DOI: 10.1007/s11853-007-0010-z

### Classification of lattice-regular lattice convex polytopes (Journal article)

Karpenkov, O. (n.d.). Classification of lattice-regular lattice convex polytopes. Functional Analysis and Other Mathematics, vol.1(1), pp.17-35, 2006. doi:10.1007/s11853-007-0002-z

DOI: 10.1007/s11853-007-0002-z

### Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions (Journal article)

Karpenkov, O. (n.d.). Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions. Monatshefte fuer Mathematik, 152, 217-249. Retrieved from http://arxiv.org/abs/math/0510482v2

### On invariant Mobius measure and Gauss-Kuzmin face distribution (Journal article)

Karpenkov, O. (n.d.). On invariant Mobius measure and Gauss-Kuzmin face distribution. Proceedings of the Steklov Institute of Mathematics, vol.258, pp.74-86, 2007. Retrieved from http://arxiv.org/abs/math/0610042v2

### On existence and uniqueness conditions for an integer triangle with given angles (Journal article)

Karpenkov, O. N. (2006). On existence and uniqueness conditions for an integer triangle with given angles. Russian Mathematical Surveys, 61(6), 1178-1179. doi:10.1070/rm2006v061n06abeh004374

DOI: 10.1070/rm2006v061n06abeh004374

### Three examples of three-dimensional continued fractions in the sense of Klein (Journal article)

Karpenkov, O. (n.d.). Three examples of three-dimensional continued fractions in the sense of Klein. C. R. Acad. Sci. Paris, Ser.I 343, pp.5-7, 2006. Retrieved from http://arxiv.org/abs/math/0601493v1

### Mobius energy of graphs (Journal article)

Karpenkov, O. (n.d.). Mobius energy of graphs. Math. Notes, vol. 79(2006), no.1, pp.134--138.. Retrieved from http://arxiv.org/abs/math-ph/0509055v1

### 'Classification of three-dimensional multistoried completely hollow convex marked pyramids' (Journal article)

Karpenkov, O. (2005). 'Classification of three-dimensional multistoried completely hollow convex marked pyramids'. Russian Math. Surveys, 60(1), 165-166.

### On examples of two-dimensional periodic continued fractions (Journal article)

Karpenkov, O. N. (n.d.). On examples of two-dimensional periodic continued fractions. Retrieved from http://arxiv.org/abs/math/0411054v1

### On two-dimensional continued fractions for the integer hyperbolic matrices with small norm (Journal article)

Karpenkov, O. (n.d.). On two-dimensional continued fractions for the integer hyperbolic matrices with small norm. Uspekhi Mat. Nauk 59 (2004), no. 5(359), 149--150; translation in Russian Math. Surveys 59 (2004), no. 5, 959--960. doi:10.1070/RM2004v059n05ABEH000778

DOI: 10.1070/RM2004v059n05ABEH000778

### On tori triangulations associated with two-dimensional continued fractions of cubic irrationalities (Journal article)

Karpenkov, O. (n.d.). On tori triangulations associated with two-dimensional continued fractions of cubic irrationalities. Funktsional. Anal. i Prilozhen. 38 (2004), no. 2, 28--37, 95; translation in Funct. Anal. Appl. 38 (2004), no. 2, 102--110. doi:10.1023/B:FAIA.0000034040.08573.22

DOI: 10.1023/B:FAIA.0000034040.08573.22

### Energy of a knot: variational principles; Mm-energy (Journal article)

Karpenkov, O. N. (n.d.). Energy of a knot: variational principles; Mm-energy. Fund. Math. Today, 214-223. Retrieved from http://arxiv.org/abs/math/0411060v1

### 'Energy of a knot: some new aspects' (Journal article)

Karpenkov, O. (2003). 'Energy of a knot: some new aspects'. Fundamental mathematics today, 214-223.

### Combinatorics of multiboundary singularities B_n^l and Bernoulli-Euler numbers (Journal article)

Karpenkov, O. (n.d.). Combinatorics of multiboundary singularities B_n^l and Bernoulli-Euler numbers. Funct. Anal. Appl. 36(2002), no 1, 78-81. Retrieved from http://arxiv.org/abs/math/0604024v1

### (Journal article)

Karpenkov, O. N. (2002). Unknown Title. Functional Analysis and Its Applications, 36(1), 65-67. doi:10.1023/a:1014434318546

DOI: 10.1023/a:1014434318546