Lifting Map Research Articles

Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space

In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or C^{1+\alpha} ) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension BA(h) of a C^{1+\mathrm{Zygmund}} orientation-preserving diffeomorphism h of the real line, and show that the Beltrami coefficient \mu (BA(h))(x+iy) vanishes as O(y) uniformly on x near the boundary of the upper half plane if and only if h is C^{1+\mathrm{Lipschitz}} . Finally, we show this criterion is indeed true when h is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.

A cluster structure on the coordinate ring of partial flag varieties

The main goal of this paper is to show that the (multi-homogeneous) coordinate ring of a partial flag variety C[G/PK−] contains a cluster algebra if G is any semisimple complex algebraic group. We use derivation properties and a special lifting map to prove that the cluster algebra structure A of the coordinate ring C[NK] of a Schubert cell constructed by Goodearl and Yakimov can be lifted, in an explicit way, to a cluster structure Aˆ living in the coordinate ring of the corresponding partial flag variety. Then we use a minimality condition to prove that the cluster algebra Aˆ is equal to C[G/PK−] after localizing some special minors.

A Sobolev rough path extension theorem via regularity structures

We show that every ℝd-valued Sobolev path with regularity a and integrability p can be lifted to a Sobolev rough path provided 1/2 > α > 1/p> ⋁ 1/3. The novelty of our approach is its use of ideas underlying Hairer’s reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.

Functional limit theorems for marked Hawkes point measures

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian motion results from the self-exciting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host as well as their interaction with that host.

On minus quotients of ideal class groups of cyclotomic fields

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system’s governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh–Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

Semi-Supervised Robust Mixture Models in RKHS for Abnormality Detection in Medical Images

Abnormality detection in medical images is a one-class classification problem for which existing methods typically involve variants of kernel principal component analysis or one-class support vector machines. However, existing methods rely on highly-curated training sets with full supervision, often using heuristics for model fitting or ignore the variances of the data within principal subspaces. In contrast, we propose novel methods that can work with imperfectly curated datasets using robust statistical learning, by extending the multivariate generalized-Gaussian distribution to a reproducing kernel Hilbert space (RKHS) and employing it within a mixture model. We propose a novel semi-supervised extension of our learning scheme, showing that a small amount of expert feedback through high-quality labeled data of the outlier class can boost performance. We propose expectation maximization for our semi-supervised robust mixture-model learning in RKHS, using solely the Gram matrix and without the explicit lifting map. Our methods incorporate optimal component means, principal directions, and variances for abnormality detection. Results on four large public datasets on retinopathy and cancer, compared against a variety of contemporary methods, show that our method gives benefits over the state of the art in one-class classification for abnormality detection.

On Shimura lifting of Hilbert modular forms

Let $${\mathfrak {N}}_{0}$$ be a integral ideal divisible by 4, of a totally real field K. We show that there is the Shimura lifting map of a space of Hilbert modular forms with character modulo $${\mathfrak {N}}_{0}$$ of half-integral weight, to the space of Hilbert modular forms of integral weight under some condition. In particular it is shown that if $$16|{\mathfrak {N}}_{0}$$ , then any Hilbert modular forms of weight at least 5 / 2 has the Shimura lift. As an application, we compute the Shimura lifts of the third powers of theta series for $$K={\mathbf {Q}}(\sqrt{2})$$ and $$K={\mathbf {Q}}(\sqrt{5})$$ , and obtain the formulas for the numbers of representations of totally positive integers in K as sums of three integral squares.

Chaos Amplification Process Can Be Described by the GKSL Master Equation

In 2000, Ohya et al. proposed a quantum algorithm with the amplification process of success probability, so-called chaos amplifier. They defined the process based on the logistic map, and its chaos behaviour amplifies the probability very fast. In this paper, we construct the chaos amplifier using a lifting map, a master equation and a partial trace. We also calculate the condition on the Lindblad operators in the GKSL master equation to achieve an effective amplification.

Shimura lifting of modular forms of weight 3/2

We show that there is the Shimura lifting map of \(\mathbf {M}_{3/2}(N,\chi _{0})\) to \(\mathbf {M}_{2}(N/2,\chi _{0}^{2})\) for any natural number N divisible by 4 and for any odd Dirichlet character \(\chi _{0}\), which was already proved in the case of higher weight (Tsuyumine, Tsukuba J Math 23:465–483, 1999). As the application, we show that the lift of the weighted average of theta series of quadratic forms of an odd number of variables in a class, is an Eisenstein series of integral weight.

Lifting newforms to vector-valued modular forms for the Weil representation

Given a discriminant form D of level N, there is a natural lifting which maps elliptic modular forms of level N to vector-valued modular forms for the Weil representation associated to D. We show that in some cases the zero component of a lifting of a newform f is just a scalar multiple of f. In order to do so, we split the lifting map into certain partial liftings corresponding to the prime powers exactly dividing N and then proceed to compute the zero components of these partial maps explicitly. As an application, we show that the L-function L𝒜(f, s) of a newform f and an ideal class 𝒜 as defined by Gross and Zagier can be written as a certain L-series of the lifting of f.

The generic dual of p-adic split SO 2n and local langlands parameters

Let G be the F-rational points of the split group SO2n, where F is a non-Archimedean local field of characteristic 0. In this paper, we give the complete description of the generic dual of G. Based on this result and the local descent result of Jiang and Soudry, we show that the functorial lifting map constructed by Cogdell, Kim, Piatetski-Shapiro and Shahidi is surjective. Along the way, we prove that for any irreducible generic representation σ, if σ ≇ cσ (c an element of O2n(F)\SO2n(F)), then they have the same lifting image and the same twisted local factors, matching Arthur’s current results on local Langlands correspondence. Then, for any local Langlands parameter ϕ of G, we construct a representation σ such that ϕ and σ have the same twisted local factors.

Integration in algebraically closed valued fields with sections

We construct Hrushovski–Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some (arbitrary) extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map L, is rather straightforward. What is a bit surprising is that the kernel of L is still generated by one element, exactly as in the case of integration in ACVF. The overall construction is more or less parallel to the main construction of Hrushovski and Kazhdan (2006) [10], as presented in Yin (2010) [19] and Yin (2011) [20]. As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees.

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let \({\mathbb{G}}\) be a Carnot group of step r and m generators and homogeneous dimension Q. Let \({\mathbb{F}_{m,r}}\) denote the free Lie group of step r and m generators. Let also \({\pi:\mathbb{F}_{m,r}\to\mathbb{G}}\) be a lifting map. We show that any horizontally convex function u on \({\mathbb{G}}\) lifts to a horizontally convex function \({u\circ \pi}\) on \({\mathbb{F}_{m,r}}\) (with respect to a suitable horizontal frame on \({\mathbb{F}_{m,r}}\)). One of the main aims of the paper is to exhibit an example of a sub-Laplacian \({\mathcal{L}=\sum_{j=1}^m X_j^2}\) on a Carnot group of step two such that the relevant \({\mathcal{L}}\)-gauge function d (i.e., d2-Q is the fundamental solution for \({\mathcal{L}}\)) is not h-convex with respect to the horizontal frame {X1, . . . , Xm}. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).

A Converse Theorem for Hilbert-Jacobi Forms

Doi and Naganuma (see [6]) constructed a lifting map from elliptic modular forms to Hilbert modular forms in the case of a real quadratic field with narrow class number one. A Converse Theorem for Hilbert modular forms was one of their basic tools. This gives rise to the question of constructing a lifting map in the case of Jacobi forms. Here we do the first step in this direction and prove a Converse Theorem for Hilbert-Jacobi forms. Studying the connection between functions that satisfy certain transformation laws and the functional equation of their associated L-functions has value on its own and a long history. In a celebrated paper (see [9]), Hecke showed that the automorphy of a cusp form with respect to SL2(Z) is equivalent to the functional equation of its associated L-functions. That only one functional equation is needed is in a way atypical and highly depends on the fact that SL2(Z) is generated by the matrices ( 1 1 0 1 ) and ( 0 −1 1 0 ). This situation already changes if one considers cusp forms with respect to a subgroup of SL2(Z) which have a character. In this case the functional equation of twists is required (see [18]). Hecke’s work has inspired an astonishing number of people and a lot of generalizations of his “Converse Theorem” have been made, e.g. generalizations to Hilbert modular forms as mentioned above (see [6]), Siegel modular forms (see [1], [10]) or Jacobi forms (see [14],[15]). Maass showed an analogue of Hecke’s result for nonholomorphic modular forms (see [13]). He proved that these correspond to certain L-functions in quadratic fields. An outstanding generalization of a Converse Theorem for GL(n) was done by Jacquet and Langlands for n = 2 (see [11]), Jacquet, Piatetski-Shapiro, and Shalika for n = 3 (see [12]) and Cogdell and Piatetski-Shapiro for general n (see [5]). In this paper, we prove a Converse Theorem for Hilbert-Jacobi cusp forms over a totally real number field K of degree g := [K : Q] with discriminant DK and narrow class number 1. The case g = 1, i.e., Jacobi forms over Q as considered by Eichler and Zagier (see [7]), is treated in two interesting papers by Martin (see [14] and [15]). To describe our result, we consider functions φ(τ, z) from H × C into C that have a Fourier expansion with certain conditions on the Fourier coefficients (see (3.4),(3.5), and (3.6)). We show that φ is a Hilbert-Jacobi cusp form (for the definition see Section 2) if

A remark on the Ikeda lift and local singular series polynomials

We relate a certain number-theoretic function occurring in a linear version of Ikeda's lifting map to local singular series polynomials.

Local factorization of trajectory lifting morphisms for single-input affine control systems

Trajectory preserving and lifting maps have been implicitly used in many recursive or hierarchical control design techniques. Well known systems theoretic concepts such as differential flatness or more recent ones such as simulation and bisimulation can be also understood through the trajectory lifting maps they define. In this paper we initiate a study of trajectory preserving and lifting maps between affine control systems. Our main result shows that any trajectory lifting map between two single-input control affine systems can be locally factored as the composition of two special trajectory lifting maps: a projection onto a quotient system followed by a differentially flat output with respect to another control system. We use this decomposition result to show that under mild regularity conditions, trajectory preserving maps between single-input affine control systems also lift trajectories. As an additional application of the main result, we also show how the hierarchical stabilization method known as back-stepping can be used based on the existence of a trajectory preserving and lifting map having a feedback stabilizable control system as codomain.

Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.

Diffusion approximation for a processor sharing queue in heavy traffic

Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps track of the residual service times of all jobs in the system at any given time. From this measure valued process, one can recover the traditional performance processes, including queue length and workload. We show that under mild assumptions, including standard heavy traffic assumptions, the (suitably rescaled) measure valued processes corresponding to a sequence of processor sharing queues converge in distribution to a measure valued diffusion process. The limiting process is characterized as the image under an appropriate lifting map, of a one-dimensional reflected Brownian motion. As an immediate consequence, one obtains a diffusion approximation for the queue length process of a processor sharing queue.

Lifting/Descending Processes for Polynomial Zeros

The recently proposed Chebyshev-like lifting map for the zeros of a univariate polynomial was motivated by its applications to splitting a univariate polynomial p(z) numerically into factors, which is a major step of some most efficient algorithms for approximating polynomial zeros. We complement the Chebyshev-like lifting process by a descending process, decrease the estimated computational cost of performing the algorithm, demonstrate its correlation to Graeffe's lifting/descending process, and generalize lifting from Graeffe's and Chebyshev-like maps to any fixed rational map of the zeros of the input polynomial.