TY - JOUR
AB - We consider discrete porous medium equations of the form ∂tρt=Δϕ(ρt), where Δ is the generator of a reversible continuous time Markov chain on a finite set χ, and ϕ is an increasing function. We show that these equations arise as gradient flows of certain entropy functionals with respect to suitable non-local transportation metrics. This may be seen as a discrete analogue of the Wasserstein gradient flow structure for porous medium equations in ℝn discovered by Otto. We present a one-dimensional counterexample to geodesic convexity and discuss Gromov-Hausdorff convergence to the Wasserstein metric.
AU - Erbar, Matthias
AU - Jan Maas
ID - 2132
IS - 4
JF - Discrete and Continuous Dynamical Systems- Series A
TI - Gradient flow structures for discrete porous medium equations
VL - 34
ER -
TY - JOUR
AB - Let ℭ denote the Clifford algebra over ℝ𝑛, which is the von Neumann algebra generated by n self-adjoint operators Q j , j = 1,…,n satisfying the canonical anticommutation relations, Q i Q j + Q j Q i = 2δ ij I, and let τ denote the normalized trace on ℭ. This algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom. Let 𝔓 denote the set of all positive operators 𝜌∈ℭ such that τ(ρ) = 1; these are the non-commutative analogs of probability densities in the non-commutative probability space (ℭ,𝜏). The fermionic Fokker–Planck equation is a quantum-mechanical analog of the classical Fokker–Planck equation with which it has much in common, such as the same optimal hypercontractivity properties. In this paper we construct a Riemannian metric on 𝔓 that we show to be a natural analog of the classical 2-Wasserstein metric, and we show that, in analogy with the classical case, the fermionic Fokker–Planck equation is gradient flow in this metric for the relative entropy with respect to the ground state. We derive a number of consequences of this, such as a sharp Talagrand inequality for this metric, and we prove a number of results pertaining to this metric. Several open problems are raised.
AU - Carlen, Eric
AU - Maas, Jan
ID - 2133
IS - 3
JF - Communications in Mathematical Physics
TI - An analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker-Planck equation is gradient flow for the entropy
VL - 331
ER -
TY - JOUR
AB - We propose a technique for engineering momentum-dependent dissipation in Bose-Einstein condensates with non-local interactions. The scheme relies on the use of momentum-dependent dark-states in close analogy to velocity-selective coherent population trapping. During the short-time dissipative dynamics, the system is driven into a particular finite-momentum phonon mode, which in real space corresponds to an ordered structure with non-local density-density correlations. Dissipation-induced ordering can be observed and studied in present-day experiments using cold atoms with dipole-dipole or off-resonant Rydberg interactions. Due to its dissipative nature, the ordering does not require artificial breaking of translational symmetry by an opticallattice or harmonic trap. This opens up a perspective of direct cooling of quantum gases into strongly-interacting phases.
AU - Otterbach, Johannes
AU - Lemeshko, Mikhail
ID - 2140
IS - 7
JF - Physical Review Letters
TI - Dissipative preparation of spatial order in Rydberg-dressed Bose-Einstein condensates
VL - 113
ER -
TY - JOUR
AB - The computation of the winning set for Büchi objectives in alternating games on graphs is a central problem in computer-aided verification with a large number of applications. The long-standing best known upper bound for solving the problem is Õ(n ⋅ m), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the Õ(n ⋅ m) boundary by presenting a new technique that reduces the running time to O(n2). This bound also leads to O(n2)-time algorithms for computing the set of almost-sure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of Õ(n ⋅ m)), (2) in concurrent graph games with constant actions (improving an earlier bound of O(n3)), and (3) in Markov decision processes (improving for m>n4/3 an earlier bound of O(m ⋅ √m)). We then show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per operation. Our algorithms are the first dynamic algorithms for this problem. We then consider another core graph theoretic problem in verification of probabilistic systems, namely computing the maximal end-component decomposition of a graph. We present two improved static algorithms for the maximal end-component decomposition problem. Our first algorithm is an O(m ⋅ √m)-time algorithm, and our second algorithm is an O(n2)-time algorithm which is obtained using the same technique as for alternating Büchi games. Thus, we obtain an O(min &lcu;m ⋅ √m,n2})-time algorithm improving the long-standing O(n ⋅ m) time bound. Finally, we show how to maintain the maximal end-component decomposition of a graph under a sequence of edge insertions or a sequence of edge deletions in O(n) amortized time per edge deletion, and O(m) worst-case time per edge insertion. Again, our algorithms are the first dynamic algorithms for this problem.
AU - Chatterjee, Krishnendu
AU - Henzinger, Monika
ID - 2141
IS - 3
JF - Journal of the ACM
TI - Efficient and dynamic algorithms for alternating Büchi games and maximal end-component decomposition
VL - 61
ER -
TY - CONF
AB - We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Lesnick, Michael
ID - 2153
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Induced matchings of barcodes and the algebraic stability of persistence
ER -
TY - JOUR
AB - A result of Boros and Füredi (d = 2) and of Bárány (arbitrary d) asserts that for every d there exists cd > 0 such that for every n-point set P ⊂ ℝd, some point of ℝd is covered by at least (Formula presented.) of the d-simplices spanned by the points of P. The largest possible value of cd has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov's approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the cofilling profiles for the (n - 1)-simplex. These bounds yield a minor improvement over Gromov's lower bounds on cd for large d, but they also show that the room for further improvement through the cofilling profiles alone is quite small. We also prove a slightly better lower bound for c3 by an approach using an additional structure besides the cofilling profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for cd.
AU - Matoušek, Jiří
AU - Wagner, Uli
ID - 2154
IS - 1
JF - Discrete & Computational Geometry
TI - On Gromov's method of selecting heavily covered points
VL - 52
ER -
TY - CONF
AB - Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Edelsbrunner, Herbert
ID - 2155
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - The morse theory of Čech and Delaunay filtrations
ER -
TY - CONF
AB - We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Ge, Xiaoyin
AU - Wang, Yusu
ID - 2156
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Measuring distance between Reeb graphs
ER -
TY - CONF
AB - We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in ℝ3? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which allows us to simplify X and recurse, is in proving that if X can be embedded in S3, then there is also an embedding in which X has a short meridian, i.e., an essential curve in the boundary of X bounding a disk in S3 nX with length bounded by a computable function of the number of tetrahedra of X.
AU - Matoušek, Jiří
AU - Sedgwick, Eric
AU - Tancer, Martin
AU - Wagner, Uli
ID - 2157
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Embeddability in the 3 sphere is decidable
ER -
TY - JOUR
AB - Directional guidance of migrating cells is relatively well explored in the reductionist setting of cell culture experiments. Here spatial gradients of chemical cues as well as gradients of mechanical substrate characteristics prove sufficient to attract single cells as well as their collectives. How such gradients present and act in the context of an organism is far less clear. Here we review recent advances in understanding how guidance cues emerge and operate in the physiological context.
AU - Majumdar, Ritankar
AU - Sixt, Michael K
AU - Parent, Carole
ID - 2158
IS - 1
JF - Current Opinion in Cell Biology
TI - New paradigms in the establishment and maintenance of gradients during directed cell migration
VL - 30
ER -