TY - JOUR
AB - We consider the hollow on the half-plane {(x, y) : y ≤ 0} ⊂ ℝ2 defined by a function u : (-1, 1) → ℝ, u(x) < 0, and a vertical flow of point particles incident on the hollow. It is assumed that u satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph(u) and goes away without hitting the graph of u anymore. We solve the problem: find the function u minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the y-axis. Assuming that the resistance of u ≡ 0 equals 1, we show that the minimal resistance equals π/2 - 2arctan(1/2) ≈ 0.6435. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730-2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space {(x1,...,xd,y) : y ≤ 0} ⊂ ℝd+1 is defined by a radial function U satisfying the SIC, U(x) = u(|x|), with x = (x1,...,xd), u(ξ) < 0 for 0 ≤ ξ < 1, and u(ξ) = 0 for ξ ≥ 1, and the flow is parallel to the y-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as d → ∞.
AU - Akopyan, Arseniy
AU - Plakhov, Alexander
ID - 1710
IS - 4
JF - Society for Industrial and Applied Mathematics
TI - Minimal resistance of curves under the single impact assumption
VL - 47
ER -
TY - JOUR
AB - Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.
AU - Pausinger, Florian
AU - Svane, Anne
ID - 1792
IS - 6
JF - Journal of Complexity
TI - A Koksma-Hlawka inequality for general discrepancy systems
VL - 31
ER -
TY - JOUR
AB - We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth.
AU - Symonova, Olga
AU - Topp, Christopher
AU - Edelsbrunner, Herbert
ID - 1793
IS - 6
JF - PLoS One
TI - DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots
VL - 10
ER -
TY - JOUR
AB - We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.
AU - Attali, Dominique
AU - Bauer, Ulrich
AU - Devillers, Olivier
AU - Glisse, Marc
AU - Lieutier, André
ID - 1805
IS - 8
JF - Computational Geometry: Theory and Applications
TI - Homological reconstruction and simplification in R3
VL - 48
ER -
TY - JOUR
AB - We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.
AU - Akopyan, Arseniy
AU - Pirogov, Sergey
AU - Rybko, Aleksandr
ID - 1828
IS - 1
JF - Journal of Statistical Physics
TI - Invariant measures of genetic recombination process
VL - 160
ER -