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Discrete Morse theory for random complexes
Get PDFThe main objects considered in the present work are simplicial and
CW-complexes with vertices forming a random point cloud.
In particular, we consider a Poisson point process in R^n
and study Delaunay and Voronoi complexes of the first and higher
orders and weighted Delaunay complexes obtained as sections
of Delaunay complexes, as well as the Čech complex.
Further, we examine theDelaunay complex of a Poisson
point process on the sphere S^n, as well as
of a uniform point cloud, which is equivalent
to the convex hull, providing a connection to the
theory of random polytopes.
Each of the complexes in question can be endowed with a radius function,
which maps its cells to the radii of appropriately chosen circumspheres,
called the radius of the cell.
Applying and developing discrete Morse theory for these functions,
joining it together with probabilistic and sometimes analytic machinery,
and developing several integral geometric tools,
we aim at getting the distributions of circumradii of typical cells.
For all considered complexes, we are able to generalize and obtain up to constants
the distribution of radii of typical intervals of all types.
In low dimensions the constants
can be computed explicitly, thus providing the explicit
expressions for the expected numbers of cells.
In particular, it allows to find the expected density of simplices
of every dimension for a Poisson point process in R^4,
whereas the result for R^3 was known already in 1970's
AAV2 mediated transduction of the mouse retina after optic nerve injury
Get PDFPURPOSE. Gene therapy of retinal ganglion cells (RGCs) has promise as a powerful therapeutic for the rescue and regeneration of these cells after optic nerve damage. However, early after damage, RGCs undergo atrophic changes, including gene silencing. It is not known if these changes will deleteriously affect transduction and transgene expression, or if the therapeutic protein can influence reactivation of the endogenous genome. METHODS. Double-transgenic mice carrying a Rosa26-(LoxP)-tdTomato reporter, and a mutant allele for the proapoptotic Bax gene were reared. The Bax mutant blocks apoptosis, but RGCs still exhibit nuclear atrophy and gene silencing. At times ranging from 1 hour to 4 weeks after optic nerve crush (ONC), eyes received an intravitreal injection of AAV2 virus carrying the Cre recombinase. Successful transduction was monitored by expression of the tdTomato reporter. Immunostaining was used to localize tdTomato expression in select cell types. RESULTS. Successful transduction of RGCs was achieved at all time points after ONC using AAV2 expressing Cre from the phosphoglycerate kinase (Pgk) promoter, but not the CMV promoter. ONC promoted an increase in the transduction of cell types in the inner nuclear layer, including Müller cells and rod bipolar neurons. There was minimal evidence of transduction of amacrine cells and astrocytes in the inner retina or optic nerve. CONCLUSIONS. Damaged RGCs can be transduced and at least some endogenous genes can be subsequently activated. Optic nerve damage may change retinal architecture to allow greater penetration of an AAV2 virus to transduce several additional cell types in the inner nuclear layer
Faster algorithms for mean payoff parity games
Get PDFGraph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first player is the conjunction of a qualitative objective (specified as a parity condition) and a quantitative objective (specified as a meanpayoff condition). There are two variants of the problem, namely, the threshold problem where the quantitative goal is to ensure that the mean-payoff value is above a threshold, and the value problem where the quantitative goal is to ensure the optimal mean-payoff value; in both cases ensuring the qualitative parity objective. The previous best-known algorithms for game graphs with n vertices, m edges, parity objectives with d priorities, and maximal absolute reward value W for mean-payoff objectives, are as follows: O(nd+1 . m . w) for the threshold problem, and O(nd+2 · m · W) for the value problem. Our main contributions are faster algorithms, and the running times of our algorithms are as follows: O(nd-1 · m ·W) for the threshold problem, and O(nd · m · W · log(n · W)) for the value problem. For mean-payoff parity objectives with two priorities, our algorithms match the best-known bounds of the algorithms for mean-payoff games (without conjunction with parity objectives). Our results are relevant in synthesis of reactive systems with both functional requirement (given as a qualitative objective) and performance requirement (given as a quantitative objective)
Superficial layers of the medial entorhinal cortex replay independently of the hippocampus
Get PDFThe hippocampus is thought to initiate systems-wide mnemonic processes through the reactivation of previously acquired spatial and episodic memory traces, which can recruit the entorhinal cortex as a first stage of memory redistribution to other brain areas. Hippocampal reactivation occurs during sharp wave-ripples, in which synchronous network firing encodes sequences of places.We investigated the coordination of this replay by recording assembly activity simultaneously in the CA1 region of the hippocampus and superficial layers of the medial entorhinal cortex. We found that entorhinal cell assemblies can replay trajectories independently of the hippocampus and sharp wave-ripples. This suggests that the hippocampus is not the sole initiator of spatial and episodic memory trace reactivation. Memory systems involved in these processes may include nonhierarchical, parallel components
Pushdown reachability with constant treewidth
Get PDFWe consider the problem of reachability in pushdown graphs. We study the problem for pushdown graphs with constant treewidth. Even for pushdown graphs with treewidth 1, for the reachability problem we establish the following:
(i) the problem is PTIME-complete, and (ii) any subcubic algorithm for the problem would contradict the k-clique conjecture and imply faster combinatorial algorithms for cliques in graphs
D inf s optimality in copula models
Get PDFOptimum experimental design theory has recently been extended for parameter estimation in copula models. The use of these models allows one to gain in flexibility by considering the model parameter set split into marginal and dependence parameters. However, this separation also leads to the natural issue of estimating only a subset of all model parameters. In this work, we treat this problem with the application of the (Formula presented.)-optimality to copula models. First, we provide an extension of the corresponding equivalence theory. Then, we analyze a wide range of flexible copula models to highlight the usefulness of (Formula presented.)-optimality in many possible scenarios. Finally, we discuss how the usage of the introduced design criterion also relates to the more general issue of copula selection and optimal design for model discrimination
Dynamics of ferrofluidic flow in the Taylor-Couette system with a small aspect ratio
Get PDFWe investigate fundamental nonlinear dynamics of ferrofluidic Taylor-Couette flow - flow confined be-tween two concentric independently rotating cylinders - consider small aspect ratio by solving the ferro-hydrodynamical equations, carrying out systematic bifurcation analysis. Without magnetic field, we find steady flow patterns, previously observed with a simple fluid, such as those containing normal one- or two vortex cells, as well as anomalous one-cell and twin-cell flow states. However, when a symmetry-breaking transverse magnetic field is present, all flow states exhibit stimulated, finite two-fold mode. Various bifurcations between steady and unsteady states can occur, corresponding to the transitions between the two-cell and one-cell states. While unsteady, axially oscillating flow states can arise, we also detect the emergence of new unsteady flow states. In particular, we uncover two new states: one contains only the azimuthally oscillating solution in the configuration of the twin-cell flow state, and an-other a rotating flow state. Topologically, these flow states are a limit cycle and a quasiperiodic solution on a two-torus, respectively. Emergence of new flow states in addition to observed ones with classical fluid, indicates that richer but potentially more controllable dynamics in ferrofluidic flows, as such flow states depend on the external magnetic field
Convergent recombination suppression suggests role of sexual selection in guppy sex chromosome formation
Get PDFSex chromosomes evolve once recombination is halted between a homologous pair of chromosomes. The dominant model of sex chromosome evolution posits that recombination is suppressed between emerging X and Y chromosomes in order to resolve sexual conflict. Here we test this model using whole genome and transcriptome resequencing data in the guppy, a model for sexual selection with many Y-linked colour traits. We show that although the nascent Y chromosome encompasses nearly half of the linkage group, there has been no perceptible degradation of Y chromosome gene content or activity. Using replicate wild populations with differing levels of sexually antagonistic selection for colour, we also show that sexual selection leads to greater expansion of the non-recombining region and increased Y chromosome divergence. These results provide empirical support for longstanding models of sex chromosome catalysis, and suggest an important role for sexual selection and sexual conflict in genome evolution
Local law for the product of independent non-Hermitian random matrices with independent entries
Get PDFWe consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε
Zeros of nonlinear systems with input invariances
Get PDFA nonlinear system possesses an invariance with respect to a set of transformations if its output dynamics remain invariant when transforming the input, and adjusting the initial condition accordingly. Most research has focused on invariances with respect to time-independent pointwise transformations like translational-invariance (u(t) -> u(t) + p, p in R) or scale-invariance (u(t) -> pu(t), p in R>0). In this article, we introduce the concept of s0-invariances with respect to continuous input transformations exponentially growing/decaying over time. We show that s0-invariant systems not only encompass linear time-invariant (LTI) systems with transfer functions having an irreducible zero at s0 in R, but also that the input/output relationship of nonlinear s0-invariant systems possesses properties well known from their linear counterparts. Furthermore, we extend the concept of s0-invariances to second- and higher-order s0-invariances, corresponding to invariances with respect to transformations of the time-derivatives of the input, and encompassing LTI systems with zeros of multiplicity two or higher. Finally, we show that nth-order 0-invariant systems realize – under mild conditions – nth-order nonlinear differential operators: when excited by an input of a characteristic functional form, the system’s output converges to a constant value only depending on the nth (nonlinear) derivative of the input