The Theory of Multidimensional Persistence

Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension. The first author was partially supported by NSF under grant DMS-0354543. The second author was partially supported by DARPA under grant HR 0011-06-1-0038 and by ONR under grant N 00014-08-1-0908. Both authors were partially supported by DARPA under grant HR 0011-05-1-0007.     

On Regularization by a Small Noise of Multidimensional Odes with Non-Lipschitz Coefficients

Ukrainian Mathematical Journal - We solve a selection problem for multidimensional SDE dX(t) = a(X??(t)) dt + ??σ(X??(t)) dW(t), where the drift and...     

Multidimensional clustering and hypergraphs

We discuss a multidimensional generalization of the clustering method. In our approach, the clustering is realized by partially ordered hypergraphs belonging to some family. The suggested procedure is applicable in the case where the original metric depends on a set of parameters. The clustering hypergraph studied here can be regarded as an object describing all possible clustering trees corresponding to different values of the original metric.     

Topological Persistence and Simplification

   Abstract. We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.     

Topological Persistence and Simplification

Abstract. We formalize a notion of topological simplification within the framework of a filtration, which is the history of a growing complex. We classify a topological change that happens during growth as either a feature or noise depending on its lifetime or persistence within the filtration. We give fast algorithms for computing persistence and experimental evidence for their speed and utility.     

Zigzag Persistence

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.     

Multidimensional rearrangement and Lorentz spaces

We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multidimensional averages and Dirichlet convolution

In this paper we study average values of arithmetical functions of several variables and present a module theoretic interpretation of these averages. Research of Alexandru Zaharescu is supported by NSF grant number DMS-0456615.     

高维市场结构与创新

以往的创新研究局限于一维市场结构：所有企业的创新能力都一样，谁也不占有优势。实际上应根据企业不同的创新能力特征，分成若干种类型，形成高维市场结构，在此结构下研究创新竞争。本首先证明在高维市场结构下均衡点唯一；其次指出何种类型企业具有竞争优势，分析企业合资研究对竞争优势的影响；最后讨论竞争企业数量增加时均衡状态的变化，并将市场的结果与社会最优结果作比较。     

Multidimensional inverse-scattering and clifford analysis

We present a higher-dimensional method based on Clifford analysis. To explain the method we consider, the formal solution of the inverse scattering problem for the n-dimensional time-dependent Schrödinger equations given by Nachman and Ablowitz [1]. Replacing the general complex Cauchy formula by a higher-dimensional analogue, we get rid of the “miracle condition”.     

Multidimensional Residues and Polynomial Equations

We describe some recent results of real algebraic geometry which are obtained using multidimensional residues. Our main focus is on the algebraic formulas for topological invariants such as the mapping degree and Euler characteristic. Our proof of the algebraic formula for the mapping degree is based on the properties of a multidimensional logarithmic residue, which are discussed in detail. Several recent applications of the main results are also presented. In particular, we discuss applications to the topological study of quadratic mappings, configuration spaces, quadratic Poisson structures, and Gaussian random polynomials. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

Persistence theorems and simultaneous uniformization

One of the most intriguing problems in the theory of foliations by analytic curves is that of the persistence of complex limit cycles of a polynomial vector field, as well as related problems concerning the persistence of identity cycles and saddle connections and the global extendability of the Poincaré map. It is proved that all these persistence problems have positive solutions for any foliation admitting a simultaneous uniformization of leaves. The latter means that there exists a uniformization of leaves that analytically depends on the initial condition and satisfies certain additional assumptions, called continuity and boundedness. Thus, the results obtained are conditional, but they establish a relation between very different properties of foliations.     

Multidimensional Gaussian sums

One computes the Gaussian sum which occurs in the m-th power transformation (m2(mod4)) of the Riemann theta-constant of genus n under the action of the theta group.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 166–172, 1980.     

Multidimensional butterfly factorization

This paper introduces the multidimensional butterfly factorization as a data-sparse representation of multidimensional kernel matrices that satisfy the complementary low-rank property. This factorization approximates such a kernel matrix of size $N\times N$ with a product of $O(\log ?N)$ sparse matrices, each of which contains $O(N)$ nonzero entries. We also propose efficient algorithms for constructing this factorization when either (i) a fast algorithm for applying the kernel matrix and its adjoint is available or (ii) every entry of the kernel matrix can be evaluated in $O(1)$ operations. For the kernel matrices of multidimensional Fourier integral operators, for which the complementary low-rank property is not satisfied due to a singularity at the origin, we extend this factorization by combining it with either a polar coordinate transformation or a multiscale decomposition of the integration domain to overcome the singularity. Numerical results are provided to demonstrate the efficiency of the proposed algorithms.     

Multidimensional Rovella-like attractors

We present a multidimensional flow exhibiting a Rovella-like attractor: a transitive invariant set with a non-Lorenz-like singularity accumulated by regular orbits and a multidimensional non-uniformly expanding invariant direction. Moreover, this attractor has a physical measure with full support and persists along certain sub-manifolds of the space of vector fields. As in the 3-dimensional Rovella-like attractor, this example is not robust. As a sub-product of the construction we obtain a new class of multidimensional non-uniformly expanding endomorphisms without any uniformly expanding direction, which is interesting by itself. Our example is a suspension (with singularities) of this multidimensional endomorphism.     

Multidimensional trigonometric interpolation

An investigation and construction of trigonometric interpolation polynomials in nonstandard regions, including multiply connected, is carried out. A method of boundary diffeomorphism of the initial region onto a canonical region is proposed and a case of interpolation on a nonuniform net in the initial region is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 568–571, April, 1990.     

Multidimensional Poisson walks

k-dimensional Poisson walks are considered. Giving stopping boundaries of the process of the walk defines a family of probability measures. A theorem letting one construct unbiased estimators for functions of an unknown vector parameter is proved. The relation between the variances of unbiased estimators is established.Translated from Staticheskie Metody, pp. 12–17, 1978.