Complex Analytic Methods Research Articles

Closed-form meromorphic solutions of some third order boundary layer ordinary differential equations

We introduce a general third order non-linear autonomous ODE which covers many ODEs coming from boundary layer problems, like the Falkner-Skan equation and the Cheng-Minkowycz equation. Using Wiman-Valiron theory and complex analytic methods recently developed, for the generic cases, it is shown that all their meromorphic solutions must be rational, or rational in one exponential, and then we find all of them explicitly. For a few non-generic cases, some solutions, which are meromorphic or singlevalued, are also obtained. Our results also explain why it is so difficult to obtain new closed-form solutions of the Falkner-Skan equation.

Reconstructing trees from traces

We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree being a path. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial number of traces suffice. Our techniques combine novel combinatorial and complex analytic methods.

Billiards in ellipses revisited

We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.

Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our results establish some broad generalizations of the non-vanishing theorem first proven by D. Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell-Weil group and the Tate-Shafarevich group of the corresponding elliptic curves over the Hilbert class field of $K$. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the particular elliptic curve with complex multiplication introduced by Gross.

New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ

The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb{R}^n$. We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb{R}^4$; a Mobius strip. All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb{R}^n$ with any given conformal structure, complete non-orientable minimal surfaces in $\mathbb{R}^n$ with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits $n$ hyperplanes of $\mathbb{CP}^{n-1}$ in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb{R}^n$.

Co-occurring defect analysis: A platform for analyzing birth defect co-occurrence in registries.

Few studies have systematically evaluated birth defect co-occurrence patterns, perhaps, in part, due to the lack of software designed to implement large-scale, complex analytic methods. We created an R-based platform, "co-occurring defect analysis" (CODA), designed to implement analyses of birth defect co-occurrence patterns in birth defect registries. CODA uses an established algorithm for calculating the observed-to-expected ratio of a given birth defect combination, accounting for the known tendency of birth defects to co-occur nonspecifically. To demonstrate CODA's feasibility, we evaluated the computational time needed to assess 2- to 5-way combinations of major birth defects in the Texas Birth Defects Registry (TBDR) (1999-2014). We report on two examples of pairwise patterns, defects co-occurring with trisomy 21 or with non-syndromic spina bifida, to demonstrate proof-of-concept. We evaluated combinations of 175 major birth defects among 206,784 infants in the TBDR. CODA performed efficiently in the data set, analyzing 1.5 million 5-way combinations in 18 hr. As anticipated, we identified large observed-to-expected ratios for the birth defects that co-occur with trisomy 21 or spina bifida. CODA is available for application to birth defect data sets and can be used to better understand co-occurrence patterns. Co-occurrence patterns elucidated by using CODA may be helpful for identifying new birth defect associations and may provide etiological insights regarding potentially shared pathogenic mechanisms. CODA may also have wider applications, such as assessing patterns of additional types of co-occurrence patterns in other large data sets (e.g., medical records).

A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation

We give a microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation (NLS) from many-body quantum theory. The starting point of our proof is [11] on the time-independent problem and [16] on the corresponding problem on a finite lattice. An important new obstacle in our analysis is the need to work with a cutoff in the number of particles, which breaks the Gaussian structure of the free quantum field and prevents the use of the Wick theorem. We overcome it by means of complex analytic methods. Our methods apply to the nonlocal NLS with bounded convolution potential. In the periodic setting, we also consider the local NLS, arising from short-range interactions in the many-body setting. To that end, we need the dispersion of the NLS in the form of periodic Strichartz estimates in Xs,b spaces.

Generalised operations in free harmonic analysis

In this article we study the formal side of operations in free harmonic analysis and examine the emerging general picture of all this. We establish an analytic correspondence of semi-rings between Witt vectors and free probability, by building on previous joint work with Friedrich and McKay (Formal groups, Witt vectors and free probability, 2012. arXiv:1204.6522). In particular, an exponential map, which relates the free additive convolution semigroup on $${\mathbb {R}}$$ with the free multiplicative convolution semigroup on either the unit circle or the positive real axis of compactly supported, freely infinitely divisible probability measures, is derived with complex analytic methods. Then we define several novel operations on these sets, discuss their relation with classically infinitely divisible measures and determine the internal geometry of the spaces involved. Finally, we formalise the structure induced by the various operations we have introduced, in the language of operads and algebraic theories.

Uncovering Biologically Coherent Peripheral Signatures of Health and Risk for Alzheimer's Disease in the Aging Brain.

Brain aging is a multifaceted process that remains poorly understood. Despite significant advances in technology, progress toward identifying reliable risk factors for suboptimal brain health requires realistically complex analytic methods to explain relationships between genetics, biology, and environment. Here we show the utility of a novel unsupervised machine learning technique – Correlation Explanation (CorEx) – to discover how individual measures from structural brain imaging, genetics, plasma, and CSF markers can jointly provide information on risk for Alzheimer’s disease (AD). We examined 829 participants (Mage: 75.3 ± 6.9 years; 350 women and 479 men) from the Alzheimer’s Disease Neuroimaging Initiative database to identify multivariate predictors of cognitive decline and brain atrophy over a 1-year period. Our sample included 231 cognitively normal individuals, 397 with mild cognitive impairment (MCI), and 201 with AD as their baseline diagnosis. Analyses revealed latent factors based on data-driven combinations of plasma markers and brain metrics, that were aligned with established biological pathways in AD. These factors were able to improve disease prediction along the trajectory from normal cognition and MCI to AD, with an area under the receiver operating curve of up to 99%, and prediction accuracy of up to 89.9% on independent “held out” testing data. Further, the most important latent factors that predicted AD consisted of a novel set of variables that are essential for cardiovascular, immune, and bioenergetic functions. Collectively, these results demonstrate the strength of unsupervised network measures in the detection and prediction of AD.

NEW COMPLEX ANALYTIC METHODS IN THE THEORY OF MINIMAL SURFACES: A SURVEY

In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Riemann–Hilbert boundary value problem. Emphasis is on results pertaining to the global theory of minimal surfaces, in particular, the Calabi–Yau problem, constructions of properly immersed and embedded minimal surfaces in $\mathbb{R}^{n}$ and in minimally convex domains of $\mathbb{R}^{n}$ , results on the complex Gauss map, isotopies of conformal minimal immersions, and the analysis of the homotopy type of the space of all conformal minimal immersions from a given open Riemann surface.

Divisibility of binomial coefficients by powers of two

For nonnegative integers j and n let Θ(j,n) be the number of entries in the n-th row of Pascal's triangle that are not divisible by 2j+1. In this paper we prove that the family j↦Θ(j,n) usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of Θ(j,n), and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).

On boundary control of the Poisson equation with the third boundary condition

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

Driving in search of analyses.

Although transportation safety has greatly improved over the past 2 decades, motor vehicle crash injuries remain a leading cause of morbidity and mortality, particularly among young drivers. Driver errors and behaviors such as speeding and distraction contribute disproportionately to crashes among inexperienced novices, who develop safe driving judgment only with substantial driving experience, commonly described as the "young driver problem." Research on young drivers has applied a range of research methods, including analyses of national archival data (mainly from police reports), crash analyses, observation of driver behavior, surveys of driver behavior and dispositions, and experimental research on driver behavior and vehicle crash worthiness. Prominent research questions regarding young driver safety include what and how do novices learn to drive safely, what are the predictors of young driver crashes, what is the variability and overtime trajectories of young driver performance and outcomes, and to what extent is the young driver problem due mainly to average population risk or high-risk groups? Current research on young drivers is complicated by small sample sizes, relatively rare events, high within and between group variability, missing data, the need to estimate exposure, and the lack of longitudinal and experimental designs, problems that require complex analytic methods. In this paper, we provide an overview of driving research methods, examples of research addressing the young driver problem, and examples of statistical collaboration on young driver research, focusing particularly on estimating prediction of crash risk and estimating variability in young driver performance and outcomes.

On the Hausdorff continuity of free Lèvy processes and free convolution semigroups

Let μ denote a Borel probability measure and let {μt}t≥1 denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for t>1. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.

Symmetry and the golden ratio in the analysis of a regular pentagon

ABSTRACTThe regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.

A complex-analytic proof of a criterion for isomorphism of Artinian Gorenstein algebras

Let A be an Artinian Gorenstein algebra over a field k of characteristic zero with dimk A>1. To every such algebra and a linear projection π on its maximal ideal with range equal to the socle Soc(A) of A, one can associate a certain algebraic hypersurface , which is the graph of a polynomial map P π : kerπ→Soc(A)≃k. Recently, the following surprising criterion has been obtained: two Artinian Gorenstein algebras A and are isomorphic if and only if any two hypersurfaces S π and arising from A and , respectively, are affinely equivalent. In the present paper, we focus on the cases and and explain how in these situations the above criterion can be derived by complex-analytic methods. The complex-analytic proof for has not previously appeared in print but is foundational for the general result. The purpose of our paper is to present this proof and compare it with that for , thus highlighting a curious connection between complex analysis and commutative algebra. 13H10; 32V40

On computing Belyi maps

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.

Measurement, geospatial, and mechanistic models of public health hazard vulnerability and jurisdictional risk.

County and state health departments are increasingly conducting hazard vulnerability and jurisdictional risk (HVJR) assessments for public health emergency preparedness and mitigation planning and evaluation to improve the public health disaster response; however, integration and adoption of these assessments into practice are still relatively rare. While the quantitative methods associated with complex analytic and measurement methods, causal inference, and decision theory are common in public health research, they have not been widely used in public health preparedness and mitigation planning. To address this gap, the Harvard School of Public Health PERLC's goal was to develop measurement, geospatial, and mechanistic models to aid public health practitioners in understanding the complexity of HVJR assessment and to determine the feasibility of using these methods for dynamic and predictive HVJR analyses. We used systematic reviews, causal inference theory, structural equation modeling (SEM), and multivariate statistical methods to develop the conceptual and mechanistic HVJR models. Geospatial mapping was used to inform the hypothetical mechanistic model by visually examining the variability and patterns associated with county-level demographic, social, economic, hazards, and resource data. A simulation algorithm was developed for testing the feasibility of using SEM estimation. The conceptual model identified the predictive latent variables used in public health HVJR tools (hazard, vulnerability, and resilience), the outcomes (human, physical, and economic losses), and the corresponding measurement subcomponents. This model was translated into a hypothetical mechanistic model to explore and evaluate causal and measurement pathways. To test the feasibility of SEM estimation, the mechanistic model path diagram was translated into linear equations and solved simultaneously using simulated data representing 192 counties. Measurement, geospatial, and mechanistic models can be used to confirm and validate existing and proposed HVJR models and potentially increase the predictive validity of these models for optimizing and improving public health preparedness planning.

Entropy on Riemann surfaces and the Jacobians of finite covers

This paper characterizes those pseudo-Anosov mappings whose entropy can be detected homologically by taking a limit over finite covers. The proof is via complex-analytic methods. The same methods show the natural map \mathcal{M}_g \to \prod \mathcal{A}_h , which sends a Riemann surface to the Jacobians of all of its finite covers, is a contraction in most directions.

Boundary Wave Propagator for Compressible Navier–Stokes Equations

We derive the boundary Dirichlet---Neumann maps for a model with linear compressible Navier---Stokes equations. Fourier and Laplace transforms are applied to derive the maps in the transformed space. A new classification of the roots of the characteristic polynomial in terms of their analytic properties is needed for the pointwise description of the inversion of the transforms using complex analytic methods. Algebraic manipulations reduce the transformed maps to a matrix of polynomials in characteristic roots over the ring spanned by rational functions in the transform variables and the global non-characteristic roots. This Normal---Tangential decomposition for the transformed Dirichlet---Neumann map gives rise to the new notion of a determinant for the intrinsic wave propagator along the boundary.