Permanent URI for this collection
Browse
Recent Submissions
Publication Particle filter-based parameter estimation algorithm for prognostic risk assessment of progression in non-small cell lung cancer(BMC, 2023-12-20) Hou, XuminNon-small cell lung cancer (NSCLC) is a malignant tumor that threatens human life and health. The development of a new NSCLC risk assessment model based on electronic medical records has great potential for reducing the risk of cancer recurrence. In this process, machine learning is a powerful method for automatically extracting risk factors and indicating impact weights for NSCLC deaths. However, when the number of samples reaches a certain value, it is difficult for machine learning to improve the prediction accuracy, and it is also challenging to use the characteristic data of subsequent patients effectively. Therefore, this study aimed to build a postoperative survival risk assessment model for patients with NSCLC that updates the model parameters and improves model accuracy based on new patient data. The model perspective was a combination of particle filtering and parameter estimation. To demonstrate the feasibility and further evaluate the performance of our approach, we performed an empirical analysis experiment. The study showed that our method achieved an overall accuracy of 92% and a recall of 71% for deceased patients. Compared with traditional machine learning models, the accuracy of the model estimated by particle filter parameters has been improved by 2%, and the recall rate for dead patients has been improved by 11%. Additionally, this study outcome shows that this method can better utilize subsequent patients' characteristic data, be more relevant to different patients, and help achieve precision medicine.Publication Stable trace ideals and applications(Springer, 2023-02-08) Lindo, HaydeeWe study stable trace ideals in one dimensional local Cohen-Macaulay rings and give numerous applications.Publication General polygonal line tilings and their matching complexes(Elsevier, 2023-03-31) Bayer, Margaret; Milutinović, Marija Jelić; Vega, JulianneA (general) polygonal line tiling is a graph formed by a string of cycles, each intersecting the previous at an edge, no three intersecting. In 2022, Matsushita proved the matching complex of a certain type of polygonal line tiling with even cycles is homotopy equivalent to a wedge of spheres. In this paper, we extend Matsushita's work to include a larger family of graphs and carry out a closer analysis of lines of triangles and pentagons, where the Fibonacci numbers arise.Publication Dispersal limitation and fire feedbacks maintain mesic savannas in Madagascar(Wiley, 2020-09-02) Goel, Nikunj; Van Vleck, Erik S.; Aleman, Julie C.; Staver, A. CarlaMadagascar is regarded by some as one of the most degraded landscapes on Earth, with estimates suggesting that 90% of forests have been lost to indigenous Tavy farming. However, the extent of this degradation has been challenged: paleoecological data, phylogeographic analysis, and species richness indicate that pyrogenic savannas in central Madagascar predate human arrival, even though rainfall is sufficient to allow forest expansion into central Madagascar. These observations raise a question—if savannas in Madagascar are not anthropogenic, how then are they maintained in regions where the climate can support forest? Observation reveals that the savanna–forest boundary coincides with a dispersal barrier—the escarpment of the Central Plateau. Using a stepping-stone model, we show that in a limited dispersal landscape, a stable savanna–forest boundary can form because of fire–vegetation feedbacks. This phenomenon, referred to as range pinning, could explain why eastern lowland forests have not expanded into the mesic savannas of the Central Highlands. This work challenges the view that highland savannas in Madagascar are derived by human-lit fires and, more importantly, suggests that partial dispersal barriers and strong nonlinear feedbacks can pin biogeographical boundaries over a wide range of environmental conditions, providing a temporary buffer against climate change.Publication The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications(Springer, 2022-01-18) Balan, Raluca M.; Nualart, David; Quer-Sardanyons, Lluís; Zheng, GuangquIn this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension d=1,2. Under mild assumptions, we provide Lp-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned Lp-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The Lp-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].Publication Counting Matrices Over Finite Fields(Department of Mathematics, University of Kansas, 2022-12-07) Critzer, GeoffreyPublication Burch ideals and Burch rings(Mathematical Sciences Publishers (MSP), 2020-09-18) Dao, Hailong; Kobayashi, Toshinori; Takahashi, RyoWe introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen–Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature.Publication Averaging Gaussian functionals(Institute of Mathematical Statistics, 2020-04-28) Nualart, David; Zheng, GuangquThis paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati’s Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind. The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel γ0 is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits the Gaussian fluctuation; with some extra mild integrability condition on γ0, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.Publication Intermittency for the parabolic Anderson model of Skorohod type driven by a rough noise(Institute of Mathematical Statistics, 2020-07-14) Ma, Nicholas; Nualart, David; Xia, PanqiuIn this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter H ∈ (0, 1/2). By using the Feynman-Kac representation for the L^p (Ω) moments of the solution, we find the upper and lower bounds for the moments.Publication Fractional Diffusion in Gaussian Noisy Environment(MDPI, 2015-03-31) Hu, Guannan; Hu, YaozhongWe study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: D(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative of order α∈(0,1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x∈Rd and W˙H a time homogeneous fractional Gaussian noise of Hurst parameter H=(H1,⋯,Hd). We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely.Publication On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings(Mathematical Sciences Publishers (MSP), 2011-02-24) Dao, Hailong; Li, Jinjia; Miller, ClaudiaIt is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring R cannot have p-torsion. When R is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given by Deligne in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. A related length criterion for modules of finite length and finite projective dimension is discussed towards the end.Publication An adaptive spot placement method on Cartesian grid for pencil beam scanning proton therapy(IOP Publishing, 2021-12-02) Lin, Bowen; Fu, Shujun; Lin, Yuting; Rotondo, Ronny L.; Huang, Weizhang; Li, Harold H.; Chen, Ronald C.; Gao, HaoPencil beam scanning proton radiotherapy (RT) offers flexible proton spot placement near treatment targets for delivering tumoricidal radiation dose to tumor targets while sparing organs-at-risk. Currently the spot placement is mostly based on a non-adaptive sampling (NS) strategy on a Cartesian grid. However, the spot density or spacing during NS is a constant for the Cartesian grid that is independent of the geometry of tumor targets, and thus can be suboptimal in terms of plan quality (e.g. target dose conformality) and delivery efficiency (e.g. number of spots). This work develops an adaptive sampling (AS) spot placement method on the Cartesian grid that fully accounts for the geometry of tumor targets. Compared with NS, AS places (1) a relatively fine grid of spots at the boundary of tumor targets to account for the geometry of tumor targets and treatment uncertainties (setup and range uncertainty) for improving dose conformality, and (2) a relatively coarse grid of spots in the interior of tumor targets to reduce the number of spots for improving delivery efficiency and robustness to the minimum-minitor-unit (MMU) constraint. The results demonstrate that (1) AS achieved comparable plan quality with NS for regular MMU and substantially improved plan quality from NS for large MMU, using merely about 10% of spots from NS, where AS was derived from the same Cartesian grid as NS; (2) on the other hand, with similar number of spots, AS had better plan quality than NS consistently for regular and large MMU.Publication Unique Factorization Domains in Commutative Algebra(Department of Mathematics, University of Kansas, 2021-05-20) Huang, YongjianIn this project, we learn about unique factorization domains in commutative algebra. Most importantly, we explore the relation between unique factorization domains and regular local rings, and prove the main theorem: If R is a regular local ring, so is a unique factorization domain.Publication Initial-boundary value problems for a reaction-diffusion equation(American Institute of Physics, 2019-08-27) Himonas, A. Alexandrou; Mantzavinos, Dionyssios; Yan, FangchiA novel approach that utilizes Fokas’s unified transform is employed for studying a reaction-diffusion equation with power nonlinearity formulated either on the half-line or on a finite interval with data in Sobolev spaces. This approach was recently introduced for initial-boundary value problems involving dispersive nonlinear equations such as the nonlinear Schrödinger and the Korteweg-de Vries equations. Thus, the present work extends the new approach from dispersive equations to diffusive ones, demonstrating the universality of the unified transform in the analysis of nonlinear evolution equations on domains with a boundary.Publication On the Generation of Stable Kerr Frequency Combs in the Lugiato--Lefever Model of Periodic Optical Waveguides(Society for Industrial and Applied Mathematics, 2019-03-07) Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas G.Publication The Korteweg-de Vries equation on an interval(American Institute of Physics, 2019-05-08) Himonas, A. Alexandrou; Mantzavinos, Dionyssios; Yan, FangchiThe initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega.Publication Finitary isomorphisms of Poisson point processes(Institute of Mathematical Statistics, 2019-10-22) Soo, Terry; Wilkens, AmandaAs part of a general theory for the isomorphism problem for actions of amenable groups, Ornstein and Weiss (J. Anal. Math. 48 (1987) 1–141) proved that any two Poisson point processes are isomorphic as measure-preserving actions. We give an elementary construction of an isomorphism between Poisson point processes that is finitary.Publication A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes(Elsevier, 2018-03-26) Aalipour, Ghodratollah; Duval, Art M.; Kook, Woong; Lee, Kang-Ju; Martin, Jeremy L.We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube.Publication Counting arithmetical structures on paths and cycles(Elsevier, 2018-07-27) Braun, Benjamin; Corrales, Hugo; Corry, Scott; Puente, Luis David García; Glass, Darren; Kaplan, Nathan; Martin, Jeremy L.; Musiker, Gregg; Valencia, Carlos E.Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d)-A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)-A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1,n-1), and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.Publication Increasing spanning forests in graphs and simplicial complexes(Elsevier, 2018-11-01) Hallam, Joshua; Martin, Jeremy L.; Sagan, Bruce E.Let G be a graph with vertex set {1,...,n}. A spanning forest F of G is increasing if the sequence of labels on any path starting at the minimum vertex of a tree of F forms an increasing sequence. Hallam and Sagan showed that the generating function ISF(G, t) for increasing spanning forests of G has all nonpositive integral roots. Furthermore they proved that, up to a change of sign, this polynomial equals the chromatic polynomial of G precisely when 1,..., n is a perfect elimination order for G. We give new, purely combinatorial proofs of these results which permit us to generalize them in several ways. For example, we are able to bound the coef- cients of ISF(G, t) using broken circuits. We are also able to extend these results to simplicial complexes using the new notion of a cage-free complex. A generalization to labeled multigraphs is also given. We observe that the de nition of an increasing spanning forest can be formulated in terms of pattern avoidance, and we end by exploring spanning forests that avoid the patterns 231, 312 and 321.