@article{12544, abstract = {Geometry is crucial in our efforts to comprehend the structures and dynamics of biomolecules. For example, volume, surface area, and integrated mean and Gaussian curvature of the union of balls representing a molecule are used to quantify its interactions with the water surrounding it in the morphometric implicit solvent models. The Alpha Shape theory provides an accurate and reliable method for computing these geometric measures. In this paper, we derive homogeneous formulas for the expressions of these measures and their derivatives with respect to the atomic coordinates, and we provide algorithms that implement them into a new software package, AlphaMol. The only variables in these formulas are the interatomic distances, making them insensitive to translations and rotations. AlphaMol includes a sequential algorithm and a parallel algorithm. In the parallel version, we partition the atoms of the molecule of interest into 3D rectangular blocks, using a kd-tree algorithm. We then apply the sequential algorithm of AlphaMol to each block, augmented by a buffer zone to account for atoms whose ball representations may partially cover the block. The current parallel version of AlphaMol leads to a 20-fold speed-up compared to an independent serial implementation when using 32 processors. For instance, it takes 31 s to compute the geometric measures and derivatives of each atom in a viral capsid with more than 26 million atoms on 32 Intel processors running at 2.7 GHz. The presence of the buffer zones, however, leads to redundant computations, which ultimately limit the impact of using multiple processors. AlphaMol is available as an OpenSource software.}, author = {Koehl, Patrice and Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {1549-960X}, journal = {Journal of Chemical Information and Modeling}, number = {3}, pages = {973--985}, publisher = {American Chemical Society}, title = {{Computing the volume, surface area, mean, and Gaussian curvatures of molecules and their derivatives}}, doi = {10.1021/acs.jcim.2c01346}, volume = {63}, year = {2023}, } @article{7791, abstract = {Extending a result of Milena Radnovic and Serge Tabachnikov, we establish conditionsfor two different non-symmetric norms to define the same billiard reflection law.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {2199-6768}, journal = {European Journal of Mathematics}, number = {4}, pages = {1309 -- 1312}, publisher = {Springer Nature}, title = {{When different norms lead to same billiard trajectories?}}, doi = {10.1007/s40879-020-00405-0}, volume = {8}, year = {2022}, } @article{8338, abstract = {Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory.}, author = {Akopyan, Arseniy and Bobenko, Alexander I. and Schief, Wolfgang K. and Techter, Jan}, issn = {1432-0444}, journal = {Discrete and Computational Geometry}, pages = {938--976}, publisher = {Springer Nature}, title = {{On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs}}, doi = {10.1007/s00454-020-00240-w}, volume = {66}, year = {2021}, } @article{10222, abstract = {Consider a random set of points on the unit sphere in ℝd, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert and Nikitenko, Anton}, issn = {1944-950X}, journal = {Experimental Mathematics}, pages = {1--15}, publisher = {Taylor and Francis}, title = {{The beauty of random polytopes inscribed in the 2-sphere}}, doi = {10.1080/10586458.2021.1980459}, year = {2021}, } @article{10867, abstract = {In this paper we find a tight estimate for Gromov’s waist of the balls in spaces of constant curvature, deduce the estimates for the balls in Riemannian manifolds with upper bounds on the curvature (CAT(ϰ)-spaces), and establish similar result for normed spaces.}, author = {Akopyan, Arseniy and Karasev, Roman}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, number = {3}, pages = {669--697}, publisher = {Oxford University Press}, title = {{Waist of balls in hyperbolic and spherical spaces}}, doi = {10.1093/imrn/rny037}, volume = {2020}, year = {2020}, } @article{8538, abstract = {We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.}, author = {Akopyan, Arseniy and Schwartz, Richard and Tabachnikov, Serge}, issn = {2199-6768}, journal = {European Journal of Mathematics}, publisher = {Springer Nature}, title = {{Billiards in ellipses revisited}}, doi = {10.1007/s40879-020-00426-9}, year = {2020}, } @inbook{74, abstract = {We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures.}, author = {Akopyan, Arseniy and Karasev, Roman}, booktitle = {Geometric Aspects of Functional Analysis}, editor = {Klartag, Bo'az and Milman, Emanuel}, isbn = {9783030360191}, issn = {16179692}, pages = {1--27}, publisher = {Springer Nature}, title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}}, doi = {10.1007/978-3-030-36020-7_1}, volume = {2256}, year = {2020}, } @article{9156, abstract = {The morphometric approach [11, 14] writes the solvation free energy as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted Gaussian curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [4], and the weighted mean curvature in [1], this yields the derivative of the morphometric expression of solvation free energy.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {74--88}, publisher = {De Gruyter}, title = {{The weighted Gaussian curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0101}, volume = {8}, year = {2020}, } @article{9157, abstract = {Representing an atom by a solid sphere in 3-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [12, 17] writes the latter as a linear combination of weighted versions of the volume, area, mean curvature, and Gaussian curvature of the space-filling diagram. We give a formula for the derivative of the weighted mean curvature. Together with the derivatives of the weighted volume in [7], the weighted area in [3], and the weighted Gaussian curvature [1], this yields the derivative of the morphometric expression of the solvation free energy.}, author = {Akopyan, Arseniy and Edelsbrunner, Herbert}, issn = {2544-7297}, journal = {Computational and Mathematical Biophysics}, number = {1}, pages = {51--67}, publisher = {De Gruyter}, title = {{The weighted mean curvature derivative of a space-filling diagram}}, doi = {10.1515/cmb-2020-0100}, volume = {8}, year = {2020}, } @article{6634, abstract = {In this paper we prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk-Ulam-Crofton technique, consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space; and establish waist-type results in terms of the Hausdorff measure.}, author = {Akopyan, Arseniy and Hubard, Alfredo and Karasev, Roman}, journal = {Topological Methods in Nonlinear Analysis}, number = {2}, pages = {457--490}, publisher = {Akademicka Platforma Czasopism}, title = {{Lower and upper bounds for the waists of different spaces}}, doi = {10.12775/TMNA.2019.008}, volume = {53}, year = {2019}, } @article{6793, abstract = {The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here, we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry.}, author = {Akopyan, Arseniy and Izmestiev, Ivan}, issn = {14692120}, journal = {Bulletin of the London Mathematical Society}, number = {5}, pages = {765--775}, publisher = {London Mathematical Society}, title = {{The Regge symmetry, confocal conics, and the Schläfli formula}}, doi = {10.1112/blms.12276}, volume = {51}, year = {2019}, } @article{6050, abstract = {We answer a question of David Hilbert: given two circles it is not possible in general to construct their centers using only a straightedge. On the other hand, we give infinitely many families of pairs of circles for which such construction is possible. }, author = {Akopyan, Arseniy and Fedorov, Roman}, journal = {Proceedings of the American Mathematical Society}, pages = {91--102}, publisher = {AMS}, title = {{Two circles and only a straightedge}}, doi = {10.1090/proc/14240}, volume = {147}, year = {2019}, } @article{6419, abstract = {Characterizing the fitness landscape, a representation of fitness for a large set of genotypes, is key to understanding how genetic information is interpreted to create functional organisms. Here we determined the evolutionarily-relevant segment of the fitness landscape of His3, a gene coding for an enzyme in the histidine synthesis pathway, focusing on combinations of amino acid states found at orthologous sites of extant species. Just 15% of amino acids found in yeast His3 orthologues were always neutral while the impact on fitness of the remaining 85% depended on the genetic background. Furthermore, at 67% of sites, amino acid replacements were under sign epistasis, having both strongly positive and negative effect in different genetic backgrounds. 46% of sites were under reciprocal sign epistasis. The fitness impact of amino acid replacements was influenced by only a few genetic backgrounds but involved interaction of multiple sites, shaping a rugged fitness landscape in which many of the shortest paths between highly fit genotypes are inaccessible.}, author = {Pokusaeva, Victoria and Usmanova, Dinara R. and Putintseva, Ekaterina V. and Espinar, Lorena and Sarkisyan, Karen and Mishin, Alexander S. and Bogatyreva, Natalya S. and Ivankov, Dmitry and Akopyan, Arseniy and Avvakumov, Sergey and Povolotskaya, Inna S. and Filion, Guillaume J. and Carey, Lucas B. and Kondrashov, Fyodor}, issn = {15537404}, journal = {PLoS Genetics}, number = {4}, publisher = {Public Library of Science}, title = {{An experimental assay of the interactions of amino acids from orthologous sequences shaping a complex fitness landscape}}, doi = {10.1371/journal.pgen.1008079}, volume = {15}, year = {2019}, } @misc{9789, author = {Pokusaeva, Victoria and Usmanova, Dinara R. and Putintseva, Ekaterina V. and Espinar, Lorena and Sarkisyan, Karen and Mishin, Alexander S. and Bogatyreva, Natalya S. and Ivankov, Dmitry and Akopyan, Arseniy and Avvakumov, Sergey and Povolotskaya, Inna S. and Filion, Guillaume J. and Carey, Lucas B. and Kondrashov, Fyodor}, publisher = {Public Library of Science}, title = {{Multiple alignment of His3 orthologues}}, doi = {10.1371/journal.pgen.1008079.s010}, year = {2019}, } @misc{9790, author = {Pokusaeva, Victoria and Usmanova, Dinara R. and Putintseva, Ekaterina V. and Espinar, Lorena and Sarkisyan, Karen and Mishin, Alexander S. and Bogatyreva, Natalya S. and Ivankov, Dmitry and Akopyan, Arseniy and Avvakumov, Sergey and Povolotskaya, Inna S. and Filion, Guillaume J. and Carey, Lucas B. and Kondrashov, Fyodor}, publisher = {Public Library of Science}, title = {{A statistical summary of segment libraries and sequencing results}}, doi = {10.1371/journal.pgen.1008079.s011}, year = {2019}, } @misc{9797, author = {Pokusaeva, Victoria and Usmanova, Dinara R. and Putintseva, Ekaterina V. and Espinar, Lorena and Sarkisyan, Karen and Mishin, Alexander S. and Bogatyreva, Natalya S. and Ivankov, Dmitry and Akopyan, Arseniy and Povolotskaya, Inna S. and Filion, Guillaume J. and Carey, Lucas B. and Kondrashov, Fyodor}, publisher = {Public Library of Science}, title = {{A statistical summary of segment libraries and sequencing results}}, doi = {10.1371/journal.pgen.1008079.s011}, year = {2019}, } @article{692, abstract = {We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.}, author = {Akopyan, Arseniy}, journal = {Geometriae Dedicata}, number = {1}, pages = {55 -- 64}, publisher = {Springer}, title = {{3-Webs generated by confocal conics and circles}}, doi = {10.1007/s10711-017-0265-6}, volume = {194}, year = {2018}, } @unpublished{75, abstract = {We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization.}, author = {Akopyan, Arseniy and Avvakumov, Sergey and Karasev, Roman}, publisher = {arXiv}, title = {{Convex fair partitions into arbitrary number of pieces}}, doi = {10.48550/arXiv.1804.03057}, year = {2018}, } @article{458, abstract = {We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 inspheres incidence theorem.}, author = {Akopyan, Arseniy and Bobenko, Alexander}, journal = {Transactions of the American Mathematical Society}, number = {4}, pages = {2825 -- 2854}, publisher = {American Mathematical Society}, title = {{Incircular nets and confocal conics}}, doi = {10.1090/tran/7292}, volume = {370}, year = {2018}, } @article{58, abstract = {Inside a two-dimensional region (``cake""), there are m nonoverlapping tiles of a certain kind (``toppings""). We want to expand the toppings while keeping them nonoverlapping, and possibly add some blank pieces of the same ``certain kind,"" such that the entire cake is covered. How many blanks must we add? We study this question in several cases: (1) The cake and toppings are general polygons. (2) The cake and toppings are convex figures. (3) The cake and toppings are axis-parallel rectangles. (4) The cake is an axis-parallel rectilinear polygon and the toppings are axis-parallel rectangles. In all four cases, we provide tight bounds on the number of blanks.}, author = {Akopyan, Arseniy and Segal Halevi, Erel}, journal = {SIAM Journal on Discrete Mathematics}, number = {3}, pages = {2242 -- 2257}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Counting blanks in polygonal arrangements}}, doi = {10.1137/16M110407X}, volume = {32}, year = {2018}, }