심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

심사위원장 이지운, 심사위원 강완모, 황강욱, 윤세영(AI 대학원), 강남규(고등과학원)

The rapid development of high-throughput sequencing technology in recent years is providing unprecedented opportunities to profile microbial communities from a variety of environments, but analysis of such multivariate taxon count data remains challenging. I present two flexible Bayesian methods to analyze complex count data with application to microbiome study. The first project is to develop a Bayesian sparse multivariate regression method that model the relationship between microbe abundance and environmental factors. We extend conventional nonlocal priors, and construct asymmetric non-local priors for regression coefficients to efficiently identify relevant covariates and their effect directions. The developed Bayesian sparse regression model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities. For the second project, we develop a Bayesian nonparametric regression model for count data with excess zeros. The approach provides straightforward community-level insights into how characteristics of microbial communities such as taxa richness and diversity are related to covariates. The baseline counts of taxa in samples are carefully constructed to obtain improved estimates of differential abundance. We apply the model to a chronic wound microbiome dataset, comparing the microbial communities present in chronic wounds versus in healthy skin.

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Co-authors; Kurtis Shuler (Sandia National Lab), Marilou Sison-Mangus (Ocean Sciences, UCSC), Irene A. Chen (Chemistry and Biochemistry, UCLA), Samuel Verbanic (Chemistry and Biochemistry, UCLA)

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants.In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Non-Markov models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the system’s history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markov models by the solutions of much simpler time-inhomogeneous Markov models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markov model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markov models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

심사위원장 권순식, 심사위원 김용정, 이지운, 허형진(중앙대 수학과), 오성진(University of California at Berkeley, 수학과)

The logarithmic analog of SH(k) is logSH(k). In logSH(k), topological cyclic homology is representable. Furthermore, the cyclotomic trace map from K to TC is representable too when k is a perfect field with resolution of singularities. In the second talk, I will explain the construction of logSH(k) and how we can achieve these results. This work is joint with Federico Binda and Paul Arne Østvær.

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Zoom ID: 352 730 6970, Password: 9999.

We will discuss about “Independent Markov Decomposition: Towards modeling kinetics of biomolecular complexes”, Hempel et. al., bioRxiv, 2021 In order to advance the mission of in silico cell biology, modeling the interactions of large and complex biological systems becomes increasingly relevant. The combination of molecular dynamics (MD) and Markov state models (MSMs) have enabled the construction of simplified models of molecular kinetics on long timescales. Despite its success, this approach is inherently limited by the size of the molecular system. With increasing size of macromolecular complexes, the number of independent or weakly coupled subsystems increases, and the number of global system states increase exponentially, making the sampling of all distinct global states unfeasible. In this work, we present a technique called Independent Markov Decomposition (IMD) that leverages weak coupling between subsystems in order to compute a global kinetic model without requiring to sample all combinatorial states of subsystems. We give a theoretical basis for IMD and propose an approach for finding and validating such a decomposition. Using empirical few-state MSMs of ion channel models that are well established in electrophysiology, we demonstrate that IMD can reproduce experimental conductance measurements with a major reduction in sampling compared with a standard MSM approach. We further show how to find the optimal partition of all-atom protein simulations into weakly coupled subunits.

A rooted digraph is a vertex-flame if for every vertex v there is a set of internally disjoint directed paths from the root to v whose set of terminal edges covers all ingoing edges of v. It was shown by Lovász that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by Joó and the analogue of Lovász’ result for countable digraphs was shown. In this talk, I present a strengthening of this result stating that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame. Joint work with Joshua Erde and Attila Joó.

For the mass-critical/supercritical pseudo-relativistic nonlinear Schrödinger equation, Bellazzini, Georgiev and Visciglia constructed a class of stationary solutions as local energy minimizers under an additional kinetic energy constraint, and they showed the orbital stability of the energy minimizer manifold. In this talk, by proving its local uniqueness, we show the orbital stability of the solitary wave, not that of the energy minimizer set. The key new aspect is reformulation of the variational problem in the non-relativistic regime, which is, we think, more natural because the proof heavily relies on the sub-critical nature of the limiting model. By this approach, the meaning of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Finally, using the non-relativistic limit, we obtain the local uniqueness and the non-degeneracy of the minimizer. This talk is based on joint work with Sangdon Jin.

We will describe a construction of infinitesimal invariants of thickened one dimensional cycles in three dimensional space, which are the simplest cycles that are not in the Milnor range. This is an analog of a construction of J. Park in the context of additive Chow groups. The construction allows us to prove the infinitesimal version of the strong reciprocity conjecture for thickenings of all orders. Classical analogs of our invariants are based on the dilogarithm function and our invariant could be seen as their infinitesimal version. Despite this analogy, the infinitesimal version cannot be obtained from their classical counterparts through a limiting process.

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Zoom ID: 352 730 6970. Password : 9999. All times are in KST.

심사위원장: 곽시종, 심사위원: 이용남, 백상훈, 박의성(고려대 수학과), 박진형(서강대 수학과)

Let G be a finite group. The minimal ramification problem famously asks about the minimal number µ(G) of ramified primes in any Galois extension of Q with group G. A conjecture due to Boston and Markin predicts the value of µ(G). I will report on recent progress on this problem, as well as several other problems which may be described as minimal ramification problems in a wider sense, notably: what is the smallest number k = k(G) such that there exists a G-extension of Q with discriminant not divisible by any (k + 1)-th power?, and: what is the smallest number m = m(G) such that there exists a G-extension of Q with all ramification indices dividing m? Apart from partial results over Q, I will present function field analogs. (If you would like to join the seminar, contact Bo-Hae Im to get the zoom link.)

A signed graph is a graph in which each edge has a positive or negative sign. Calling two graphs switch equivalent if one can get from one to the other by the iteration of the local action of switching all signs on edges incident to a given vertex, we say that there is a switch homomorphism from a signed graph $G$ to a signed graph $H$ if there is a sign preserving homomorphism from $G’$ to $H$ for some graph $G’$ that is switch equivalent to $G$. By reductions to CSP this problem, and its list version, are known to be either polynomial time solvable or NP-complete, depending on $H$. Recently those signed graphs $H$ for which the switch homomorphism problem is in $P$ were characterised. Such a characterisation is yet unknown for the list version of the problem. We talk about recent work towards such a characterisation and about how these problems fit in with bigger questions that still remain around the recent CSP dichotomy theorem.

This is part VI of the lectures on infinity-categories. I'll keep talking about the simplicial nerve construction in contrast to the ordinary nerve functor. To finish off this whole series, some overview of the theory of infinity-categories will be given, including how similar and different it is to the ordinary category theory.

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Zoom ID: 352 730 6970, Password: 9999

Morel and Voevodsky constructed the A^1-motivic homotopy category SH(k). One purpose of motivic homotopy theory is to incorporate various cohomology theories into a single framework so that one can discuss relations between cohomology theories more efficiently. However, there are still lots of non A^1-invariant cohomology theories, e.g. Hodge cohomology and topological cyclic homology. There is no way to represent these in SH(k). In the first talk, I will explain the construction of logDM^{eff}(k) for a perfect field k, which is strictly larger than Voevodsky's triangulated categories of motives DM^{eff}(k) because Hodge cohomology is representable in logDM^{eff}(k). This work is joint with Federico Binda and Paul Arne Østvær.

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Zoom ID: 352 730 6970, Password: 9999

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants. In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

A notion of sublinear expander has played a central role in the resolutions of a couple of long-standing conjectures in embedding problems in graph theory, including e.g. the odd cycle problem of Erdos and Hajnal that the harmonic sum of odd cycle length in a graph diverges with its chromatic number. I will survey some of these developments.

Oscillatory signals are ubiquitously observed in many different intracellular systems such as immune systems and DNA repair processes. While we know how oscillatory signals are created, we do not fully understand what a critical role they play to regulate signal pathway systems in cells. Recently by using a stochastic nucleosome system, we found that a special signal (NFkB signal) in an immune cell can enhance the variability of the immune response to inflammatory stimulations when the signal is oscillatory. Hence in this talk, we discuss the roles of oscillatory and non-oscillatory NFkB signals in an inflammatory system of immune cells as the main example for revealing the role of oscillatory signals. And then we will talk about how this finding can be generalized for other intra- or extra-cellular systems to study why cells use oscillations.

We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We mainly provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs and give a polynomial-time algorithm that given any graph, either finds an obstruction or outputs a partition of its vertex set that asserts that the graph is well-partitioned chordal. We demonstrate the algorithmic use of this graph class by showing that two variants of the problem of finding pairwise disjoint paths between k given pairs of vertices are in FPT, parameterized by k, on well-partitioned chordal graphs, while on chordal graphs, these problems are only known to be in XP. From the other end, we introduce some problems that are polynomial-time solvable on split graphs but become NP-complete on well-partitioned chordal graphs. This is joint work with Lars Jaffke, O-joung Kwon, and Paloma T. Lima.

In this talk, we present an algebraic and graph theoretic (data-based) image inpainting algorithm. The algorithm is designed to reconstruct area or volume data from one and two dimensional slice data. More precisely, given one or two dimensional slice data, our algorithm begins with a simple algebraic pre-smoothing of the data, constructs low dimensional representation of pre-smoothed data via Dynamic Mode Decomposition, performs initial area or volume reconstruction via interpolation, and finishes with smoothing the outcome using a constraint bilateral smoothing. Numerical experiments including MRI of a three year old and a CT scan of a Covid-19 patient, are presented to demonstrate the superiority of the proposed techniques in comparisons with other commercial and published methods. Some further applications we are currently doing will also be presented. This work is jointly done with Gwanghyun Jo and Ivan Ojeda-Ruiz.

Sheaf cohomology and direct images are fundamental objects in algebraic geometry. However, they are defined in an abstract way (as right derived functors), and thus they are often hard to compute in explicit examples. In this talk, we briefly review Bernstein-Gel'fand-Gel'fand (BGG) correspondence and resolutions over an exterior algebra. Then, we review Tate resolutions and how it can be used to understand a given coherent sheaf and its cohomology groups in terms of Beilinson monad. Finally, we discuss an algorithm to compute direct images using Eisenbud-Erman-Schreyer's generalization on products of projective spaces. A part of the talk is a joint work in progress with J. Barrott and F.-O. Schreyer.

This is part V of the lectures on the foundations of infinity-categories. After continued discussion about the role of model categories in the theory of infinity-categories, the simplicial and differential graded nerve constructions will be presented to provide a plethora of examples of infinity-categories. Finally, I'll talk about an analogy between the theories of ordinary categories and infinity-categories, which wraps up this series of talks.

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Zoom ID: 352 730 6970, Password: 9999.

Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that typically occur together. They can thereby discover, relate, and structure types: of behaviour, political views, texts, or proteins. Tangles offer a new, quantitative, paradigm for grouping phenomena rather than things. They can identify key phenomena that allow predictions of others. Tangles also offer a new paradigm for clustering in large data sets. The mathematical theory of tangles has its origins in the theory of graph minors developed by Robertson and Seymour. It has recently been axiomatized in a way that makes it applicable to a wide range of contexts outside mathematics: from clustering in data science to predicting customer behaviour in economics, from DNA sequencing and drug development to text analysis and machine learning. This very informal talk will not show you the latest intricacies of abstract tangle theory (for which you can find links on the tangle pages of my website), but to win you over to join our drive to develop real tangle applications in areas as indicated above. We have some software to share, but are looking for people to try it out with us on real-world examples! Here are some introductory pages from a book I am writing on this, which may serve as an extended abstract: https://arxiv.org/abs/2006.01830

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234) Age brings the benefit of experience and looking back at my job as a professor, there are a couple of things that fall into the category “I wish someone had told me that earlier”. In this seminar, I would like to share some of the things I learned and which, I hope, will be useful for younger scientists. The questions I will touch upon include What is productivity, for a scientist? What are qualities of successful people? How can one create motivation and success? How to organize myself? (project management; getting things done) How to communicate effectively? Seeking fulfillment The seminar is targeted at PhD students, postdocs, and junior group leaders.

Bouchet introduced isotropic systems in 1983 unifying some combinatorial features of binary matroids and 4-regular graphs. The concept of isotropic system is a useful tool to study vertex-minors of graphs and yet it is not well known. I will give an introduction to isotropic systems.

In this talk, I will present a result on the existence of 2-dimensional subsonic steady compressible flows around a finite thin profile with a vortex line at the trailing edge, which is a special case in the celebrated lifting line theory by Prandtl. Such a flow pattern is governed the two-dimensional steady compressible Euler equations. The vortex line attached to the trailing edge is a free interface corresponding to a contact discontinuity. Such a flow pattern is obtained as a consequence of structural stability of a uniform contact discontinuity. The problem is formulated and solved by an application of the implicit function theorem in a suitable weighted space. The main difficulties are the possible singularities at the fitting of the profile and the vortex line and the subtle instability of the vortex line. Some ideas of the analysis will be presented. This talk is based on joint works with Jun Chen and Aibin Zang at Yichun University. The research is supported in part by Hong Kong Earmarked Research Grants CUHK 14305315, CUHK 14302819, CUHK 14300917, and CUHK 14302917.

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https://kaist.zoom.us/j/3098650340

This is part IV of the series of lectures on the foundations of infinity-categories. n the first half, we'll cover the precise definition of infinity-categories based on quasi-categories. Some relationship between infinity-categories and model categories will be presented to help better understand the theory of infinity-categories in the remaining half.

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Zoom ID: 352 730 6970, Password: 9999

This talk will be presented online. Zoom link: 709 120 4849 (pw: 1234) Abstract: Life science has been a prosperous subject for a long time, and is still developing with high speed now. One of its major aims is to study the mechanisms of various biological processes on the basis of biological big-data. Many statistics-based methods have been proposed to catch the essence by mining those data, including the popular category classification, variables regression, group clustering, statistical comparison, dimensionality reduction, and component analysis, which, however, mainly elucidate static features or steady behavior of living organisms due to lack of temporal data. But, a biological system is inherently dynamic, and with increasingly accumulated time-series data, dynamics-based approaches based on physical and biological laws are demanded to reveal dynamic features or complex behavior of biological systems. In this talk, I will present a new concept “dynamics-based data science” and the approaches for studying dynamical bio-processes, including dynamical network biomarkers (DNB), autoreservoir neural networks (ARNN) and partical cross-mapping. These methods are all data-driven or model-free approaches but based on the theoretical frameworks of nonlinear dynamics. We show the principles and advantages of dynamics-based data-driven approaches as explicable, quantifiable, and generalizable. In particular, dynamics-based data science approaches exploit the essential features of dynamical systems in terms of data, e.g. strong fluctuations near a bifurcation point, low-dimensionality of a center manifold or an attractor, and phase-space reconstruction from a single variable by delay embedding theorem, and thus are able to provide different or additional information to the traditional approaches, i.e. statistics-based data science approaches. The dynamical-based data science approaches will further play an important role in the systematical research of biology and medicine in future.

Elie Cartan's celebrated paper ˝Pfaffian systems in 5 variables˝ in 1910 studied the equivalence problem for general Pfaffian systems of rank 3 in 5 variables as the curved version of the Pfaffian system with G_2 symmetry. The G_2 case admits a beautiful geometric correspondence with Engel's PDE system. We give a historical overview of Cartan's paper and discuss recent works extending the correspondence to curved cases, which is based on ideas from geometric control theory and algebraic geometry.

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pw: 2021math

TBA

Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $\textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $\mathcal{L}$ on $X$, then the group $\textrm{Aut}_k(X,\mathcal{L})$ of automorphisms of the polarized abelian variety $(X,\mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field $k.$ In one of earlier works, a complete classification of such finite groups was given for the case when $k$ is a finite field, $g$ is an odd prime, and $X$ is simple. One interesting thing is that the maximal such a finite group was a cyclic group of order $4g+2$ only when $g$ is a Sophie Germain prime. Another notable thing is the fact that the abelian variety that was constructed to achieve the maximal cyclic group splits over $\overline{k}$, an algebraic closure of $k.$ \\ In this talk, we provide a construction of an absolutely simple abelian variety of dimension $g$ ($g$ being a Sophie Germain prime) over a finite field $k$, which attains the maximal automorphism group. This can be regarded as the counterpart for the previous construction. Also, we briefly describe the asymptotic behavior of the characteristic of the base field $k$ for which we can give such a construction. Finally, if time permits, we take a closer look at the case of $g=5$ by introducing the Newton polygon of an abelian variety of dimension $5.$ (If you want to join the seminar, please contact Bo-Hae Im to get the zoom link.)

Ordered Ramsey numbers were introduced in 2014 by Conlon, Fox, Lee, and Sudakov. Their results included upper bounds for general graphs and lower bounds showing separation from classical Ramsey numbers. We show the first nontrivial results for ordered Ramsey numbers of specific small graphs. In particular we prove upper bounds for orderings of graphs on four vertices, and determine some numbers exactly using SAT solvers for lower bounds. These results are in the spirit of exact calculations for classical Ramsey numbers and use only elementary combinatorial arguments. This is joint work with Jeremy Alm, Kayla Coffey, and Carolyn Langhoff.

We will discuss about “Highly accurate fluorogenic DNA sequencing with information theory–based error correction”, Chen et al., Nature Biotechnology (2017) Eliminating errors in next-generation DNA sequencing has proved challenging. Here we present error-correction code (ECC) sequencing, a method to greatly improve sequencing accuracy by combining fluorogenic sequencing-by-synthesis (SBS) with an information theory–based error-correction algorithm. ECC embeds redundancy in sequencing reads by creating three orthogonal degenerate sequences, generated by alternate dual-base reactions. This is similar to encoding and decoding strategies that have proved effective in detecting and correcting errors in information communication and storage. We show that, when combined with a fluorogenic SBS chemistry with raw accuracy of 98.1%, ECC sequencing provides single-end, error-free sequences up to 200 bp. ECC approaches should enable accurate identification of extremely rare genomic variations in various applications in biology and medicine.

This is the part 3 of the lectures on infinity-categories: This talk will be focused on introducing quasi-categories as our model for infinity-categories. After reviewing some background material needed to define quasi-categories, we'll see how the definition works.

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Zoom ID: 352 730 6970, Password: 9999

In this talk, we consider stochastic systems with several stable sets. Typical examples are low-temperature physical systems and stochastic optimization algorithms. The macroscopic description of such systems is usually carried out via a so-called model reduction. We explain a necessary and sufficient condition for model reduction in terms of solutions of certain form of partial differential equations.

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pw: 2021math

Topological defect structure is one of the most interesting topics in natural sciences. Especially, the topological defect transition was highlighted by Nobel prize in 2016. But this topic is hard to understand and realize in the practical condition because the size and time-scale are huge in cosmos or so tiny in skyrmion system. So, we proposed to use liquid crystal (LC) materials to directly show this interesting topic, phase transition of topological defect.

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화학과&수리과학과 공동 주관 세미나

Thurston classified mappings from a given surface to itself. By iterating the surface mappings, one can view this as a dynamical system. Most of those surface mappings are so-called pseudo-Anosov. We briefly explain how we should understand these pseudo-Anosov maps and their physical meaning.

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화학과&수리과학과 공동 주관 세미나

This is part II of the lecture series in infinity-categories. I'll continue to talk about higher categories and some difficulty in defining them. In the end, a few models for infinity-categories will be introduced very roughly.

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Zoom ID: 352 730 6970, password: 9999

Introduction: In this lecture series, we'll discuss algebro-geometric study on fundamental problems concerning tensors via higher secant varieties. We start by recalling definition of tensors, basic properties and small examples and proceed to discussion on tensor rank, decomposition, and X-rank for any nondegenerate variety $X$ in a projective space. Higher secant varieties of Segre (resp. Veronese) embeddings will be regarded as a natural parameter space of general (resp. symmetric) tensors in the lectures. We also review known results on dimensions of secants of Segre and Veronese, and consider various techniques to provide equations on the secants. In the end, we'll finish the lectures by introducing some open problems related to the theme such as syzygy structures and singularities of higher secant varieties.

Generative adversarial networks (GAN) are a widely used class of deep generative models, but their minimax training dynamics are not understood very well. In this work, we show that GANs with a 2-layer infinite-width generator and a 2-layer finite-width discriminator trained with stochastic gradient ascent-descent have no spurious stationary points. We then show that when the width of the generator is finite but wide, there are no spurious stationary points within a ball whose radius becomes arbitrarily large (to cover the entire parameter space) as the width goes to infinity.