@article{6355,
abstract = {We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.},
author = {Akopyan, Arseniy and Avvakumov, Sergey},
issn = {2050-5094},
journal = {Forum of Mathematics, Sigma},
publisher = {Cambridge University Press},
title = {{Any cyclic quadrilateral can be inscribed in any closed convex smooth curve}},
doi = {10.1017/fms.2018.7},
volume = {6},
year = {2018},
}
@inbook{424,
abstract = {We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers b and d there exists an integer h(b, d) such that the following holds. If F is a finite family of subsets of Rd such that βi(∩G)≤b for any G⊊F and every 0 ≤ i ≤ [d/2]-1 then F has Helly number at most h(b, d). Here βi denotes the reduced Z2-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these [d/2] first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex K, some well-behaved chain map C*(K)→C*(Rd).},
author = {Goaoc, Xavier and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
booktitle = {A Journey through Discrete Mathematics: A Tribute to Jiri Matousek},
editor = {Loebl, Martin and Nešetřil, Jaroslav and Thomas, Robin},
isbn = {978-331944479-6},
pages = {407 -- 447},
publisher = {Springer},
title = {{Bounding helly numbers via betti numbers}},
doi = {10.1007/978-3-319-44479-6_17},
year = {2017},
}
@article{534,
abstract = {We investigate the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into 3-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.},
author = {Burton, Benjamin and De Mesmay, Arnaud N and Wagner, Uli},
issn = {01795376},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {871 -- 888},
publisher = {Springer},
title = {{Finding non-orientable surfaces in 3-Manifolds}},
doi = {10.1007/s00454-017-9900-0},
volume = {58},
year = {2017},
}
@article{568,
abstract = {We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g-1(0): ||g - f|| < r) of all zero sets of all continuous maps g closer to f than r in the max-norm. All of these sets are outside A := (x: |f(x)| ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n - 3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence module-a structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).},
author = {Franek, Peter and Krcál, Marek},
issn = {15320073},
journal = {Homology, Homotopy and Applications},
number = {2},
pages = {313 -- 342},
publisher = {International Press},
title = {{Persistence of zero sets}},
doi = {10.4310/HHA.2017.v19.n2.a16},
volume = {19},
year = {2017},
}
@article{610,
abstract = {The fact that the complete graph K5 does not embed in the plane has been generalized in two independent directions. On the one hand, the solution of the classical Heawood problem for graphs on surfaces established that the complete graph Kn embeds in a closed surface M (other than the Klein bottle) if and only if (n−3)(n−4) ≤ 6b1(M), where b1(M) is the first Z2-Betti number of M. On the other hand, van Kampen and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional analogue of Kn+1) embeds in R2k if and only if n ≤ 2k + 1. Two decades ago, Kühnel conjectured that the k-skeleton of the n-simplex embeds in a compact, (k − 1)-connected 2k-manifold with kth Z2-Betti number bk only if the following generalized Heawood inequality holds: (k+1 n−k−1) ≤ (k+1 2k+1)bk. This is a common generalization of the case of graphs on surfaces as well as the van Kampen–Flores theorem. In the spirit of Kühnel’s conjecture, we prove that if the k-skeleton of the n-simplex embeds in a compact 2k-manifold with kth Z2-Betti number bk, then n ≤ 2bk(k 2k+2)+2k+4. This bound is weaker than the generalized Heawood inequality, but does not require the assumption that M is (k−1)-connected. Our results generalize to maps without q-covered points, in the spirit of Tverberg’s theorem, for q a prime power. Our proof uses a result of Volovikov about maps that satisfy a certain homological triviality condition.},
author = {Goaoc, Xavier and Mabillard, Isaac and Paták, Pavel and Patakova, Zuzana and Tancer, Martin and Wagner, Uli},
journal = {Israel Journal of Mathematics},
number = {2},
pages = {841 -- 866},
publisher = {Springer},
title = {{On generalized Heawood inequalities for manifolds: A van Kampen–Flores type nonembeddability result}},
doi = {10.1007/s11856-017-1607-7},
volume = {222},
year = {2017},
}
@inproceedings{6517,
abstract = {A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing, whether a drawing of a planar graph G in the plane is approximable by an embedding, can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class, i.e., the rotation system (or equivalently the faces) of the embedding of G and the choice of outer face are fixed. In other words, we show that c-planarity with embedded pipes is tractable for graphs with fixed embeddings. To the best of our knowledge an analogous result was previously known essentially only when G is a cycle.},
author = {Fulek, Radoslav},
location = {Phuket, Thailand},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Embedding graphs into embedded graphs}},
doi = {10.4230/LIPICS.ISAAC.2017.34},
volume = {92},
year = {2017},
}
@inproceedings{683,
abstract = {Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.},
author = {Lubiw, Anna and Masárová, Zuzana and Wagner, Uli},
location = {Brisbane, Australia},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{A proof of the orbit conjecture for flipping edge labelled triangulations}},
doi = {10.4230/LIPIcs.SoCG.2017.49},
volume = {77},
year = {2017},
}
@inproceedings{688,
abstract = {We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. },
author = {Edelsbrunner, Herbert and Wagner, Hubert},
issn = {18688969},
location = {Brisbane, Australia},
pages = {391--3916},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{Topological data analysis with Bregman divergences}},
doi = {10.4230/LIPIcs.SoCG.2017.39},
volume = {77},
year = {2017},
}
@article{701,
abstract = {A d-dimensional simplex S is called a k-reptile (or a k-reptile simplex) if it can be tiled by k simplices with disjoint interiors that are all mutually congruent and similar to S. For d = 2, triangular k-reptiles exist for all k of the form a^2, 3a^2 or a^2+b^2 and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only k-reptile simplices that are known for d ≥ 3, have k = m^d, where m is a positive integer. We substantially simplify the proof by Matoušek and the second author that for d = 3, k-reptile tetrahedra can exist only for k = m^3. We then prove a weaker analogue of this result for d = 4 by showing that four-dimensional k-reptile simplices can exist only for k = m^2.},
author = {Kynčl, Jan and Patakova, Zuzana},
issn = {10778926},
journal = {The Electronic Journal of Combinatorics},
number = {3},
pages = {1--44},
publisher = {International Press},
title = {{On the nonexistence of k reptile simplices in ℝ^3 and ℝ^4}},
volume = {24},
year = {2017},
}
@article{793,
abstract = {Let P be a finite point set in the plane. A cordinary triangle in P is a subset of P consisting of three non-collinear points such that each of the three lines determined by the three points contains at most c points of P . Motivated by a question of Erdös, and answering a question of de Zeeuw, we prove that there exists a constant c > 0such that P contains a c-ordinary triangle, provided that P is not contained in the union of two lines. Furthermore, the number of c-ordinary triangles in P is Ω(| P |). },
author = {Fulek, Radoslav and Mojarrad, Hossein and Naszódi, Márton and Solymosi, József and Stich, Sebastian and Szedlák, May},
issn = {09257721},
journal = {Computational Geometry: Theory and Applications},
pages = {28 -- 31},
publisher = {Elsevier},
title = {{On the existence of ordinary triangles}},
doi = {10.1016/j.comgeo.2017.07.002},
volume = {66},
year = {2017},
}
@article{794,
abstract = {We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.},
author = {Fulek, Radoslav},
journal = {Computational Geometry: Theory and Applications},
pages = {1 -- 13},
publisher = {Elsevier},
title = {{C-planarity of embedded cyclic c-graphs}},
doi = {10.1016/j.comgeo.2017.06.016},
volume = {66},
year = {2017},
}
@article{795,
abstract = {We introduce a common generalization of the strong Hanani–Tutte theorem and the weak Hanani–Tutte theorem: if a graph G has a drawing D in the plane where every pair of independent edges crosses an even number of times, then G has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in D. The theorem is implicit in the proof of the strong Hanani–Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.},
author = {Fulek, Radoslav and Kynčl, Jan and Pálvölgyi, Dömötör},
issn = {10778926},
journal = {Electronic Journal of Combinatorics},
number = {3},
publisher = {International Press},
title = {{Unified Hanani Tutte theorem}},
volume = {24},
year = {2017},
}
@article{1073,
abstract = {Let X and Y be finite simplicial sets (e.g. finite simplicial complexes), both equipped with a free simplicial action of a finite group G. Assuming that Y is d-connected and dimX≤2d, for some d≥1, we provide an algorithm that computes the set of all equivariant homotopy classes of equivariant continuous maps |X|→|Y|; the existence of such a map can be decided even for dimX≤2d+1. This yields the first algorithm for deciding topological embeddability of a k-dimensional finite simplicial complex into Rn under the condition k≤23n−1. More generally, we present an algorithm that, given a lifting-extension problem satisfying an appropriate stability assumption, computes the set of all homotopy classes of solutions. This result is new even in the non-equivariant situation.},
author = {Čadek, Martin and Krcál, Marek and Vokřínek, Lukáš},
issn = {01795376},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {915 -- 965},
publisher = {Springer},
title = {{Algorithmic solvability of the lifting extension problem}},
doi = {10.1007/s00454-016-9855-6},
volume = {54},
year = {2017},
}
@article{1113,
abstract = {A drawing of a graph G is radial if the vertices of G are placed on concentric circles C 1 , . . . , C k with common center c , and edges are drawn radially : every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Toth.},
author = {Fulek, Radoslav and Pelsmajer, Michael and Schaefer, Marcus},
journal = {Journal of Graph Algorithms and Applications},
number = {1},
pages = {135 -- 154},
publisher = {Brown University},
title = {{Hanani-Tutte for radial planarity}},
doi = {10.7155/jgaa.00408},
volume = {21},
year = {2017},
}
@phdthesis{1123,
abstract = {Motivated by topological Tverberg-type problems in topological combinatorics and by classical
results about embeddings (maps without double points), we study the question whether a finite
simplicial complex K can be mapped into Rd without triple, quadruple, or, more generally, r-fold points (image points with at least r distinct preimages), for a given multiplicity r ≤ 2. In particular, we are interested in maps f : K → Rd that have no global r -fold intersection points, i.e., no r -fold points with preimages in r pairwise disjoint simplices of K , and we seek necessary and sufficient conditions for the existence of such maps.
We present higher-multiplicity analogues of several classical results for embeddings, in particular of the completeness of the Van Kampen obstruction for embeddability of k -dimensional
complexes into R2k , k ≥ 3. Speciffically, we show that under suitable restrictions on the dimensions(viz., if dimK = (r ≥ 1)k and d = rk \ for some k ≥ 3), a well-known deleted product criterion (DPC ) is not only necessary but also sufficient for the existence of maps without global r -fold points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick , by which pairs of isolated r -fold points of opposite sign can be eliminated by local modiffications of the map, assuming codimension d – dimK ≥ 3.
An important guiding idea for our work was that suffciency of the DPC, together with an old
result of Özaydin's on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the the long-standing topological Tverberg conjecture , i.e., to construct maps from the N -simplex σN to Rd without r-Tverberg points when r not a prime power and
N = (d + 1)(r – 1). Unfortunately, our proof of the sufficiency of the DPC requires codimension d – dimK ≥ 3, which is not satisfied for K = σN .
In 2015, Frick [16] found a very elegant way to overcome this \codimension 3 obstacle" and
to construct the first counterexamples to the topological Tverberg conjecture for all parameters(d; r ) with d ≥ 3r + 1 and r not a prime power, by a reduction1 to a suitable lower-dimensional skeleton, for which the codimension 3 restriction is satisfied and maps without r -Tverberg points exist by Özaydin's result and sufficiency of the DPC.
In this thesis, we present a different construction (which does not use the constraint method) that yields counterexamples for d ≥ 3r , r not a prime power. },
author = {Mabillard, Isaac},
pages = {55},
publisher = {IST Austria},
title = {{Eliminating higher-multiplicity intersections: an r-fold Whitney trick for the topological Tverberg conjecture}},
year = {2016},
}
@inproceedings{1164,
abstract = {A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, … , Ck with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. A pair of edges e and f in a graph is independent if e and f do not share a vertex. We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.},
author = {Fulek, Radoslav and Pelsmajer, Michael and Schaefer, Marcus},
location = {Athens, Greece},
pages = {468 -- 481},
publisher = {Springer},
title = {{Hanani-Tutte for radial planarity II}},
doi = {10.1007/978-3-319-50106-2_36},
volume = {9801},
year = {2016},
}
@inproceedings{1165,
abstract = {We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph G with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment G by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.},
author = {Fulek, Radoslav},
location = {Athens, Greece},
pages = {94 -- 106},
publisher = {Springer},
title = {{C-planarity of embedded cyclic c-graphs}},
doi = {10.1007/978-3-319-50106-2_8},
volume = {9801 },
year = {2016},
}
@inproceedings{1237,
abstract = {Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.},
author = {Krcál, Marek and Pilarczyk, Pawel},
location = {Marseille, France},
pages = {140 -- 151},
publisher = {Springer},
title = {{Computation of cubical Steenrod squares}},
doi = {10.1007/978-3-319-39441-1_13},
volume = {9667},
year = {2016},
}
@article{1282,
abstract = {We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model Xk(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of k-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the eigenvalues of the higher-dimensional Laplacian capture the notion of coboundary expansion—a higher-dimensional generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov; this question was raised, for instance, by Dotterrer and Kahle. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and n ∈ N, there is a k-dimensional complex Yn k on n vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised k-dimensional Laplacian lie in the interval [1−O(1/√1), 1+0(1/√1]) but whose coboundary expansion is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Yn k can be taken to have vanishing integer homology in dimension less than k.},
author = {Gundert, Anna and Wagner, Uli},
journal = {Israel Journal of Mathematics},
number = {2},
pages = {545 -- 582},
publisher = {Springer},
title = {{On eigenvalues of random complexes}},
doi = {10.1007/s11856-016-1419-1},
volume = {216},
year = {2016},
}
@inproceedings{1348,
abstract = {A drawing in the plane (ℝ2) of a graph G = (V,E) equipped with a function γ : V → ℕ is x-bounded if (i) x(u) < x(v) whenever γ(u) < γ(v) and (ii) γ(u) ≤ γ(w) ≤ γ(v), where uv ∈ E and γ(u) ≤ γ(v), whenever x(w) ∈ x(uv), where x(.) denotes the projection to the xaxis.We prove a characterization of isotopy classes of embeddings of connected graphs equipped with γ in the plane containing an x-bounded embedding.Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest.This partially answers a question raised recently by Angelini et al.and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.},
author = {Fulek, Radoslav},
location = {Helsinki, Finland},
pages = {31 -- 42},
publisher = {Springer},
title = {{Bounded embeddings of graphs in the plane}},
doi = {10.1007/978-3-319-44543-4_3},
volume = {9843},
year = {2016},
}