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1. BIFURCATION FROM INFINITY WITH OSCILLATORY NONLINEARITY FOR NEUMANN PROBLEMS

   https://doi.org/10.58997/ejde.sp.01.c5  

   CHHETRI, MAYA; MAVINGA, NSOKI; PARDO, ROSA (January 2021, Electronic journal of differential equations)

   We consider a sublinear perturbation of an elliptic eigenvalue problem with Neumann boundary condition. We give sufficient conditions on the nonlinear perturbation which guarantee that the unbounded continuum, bifurcating from infinity at the first eigenvalue, contains an unbounded sequence of turning points as well as an unbounded sequence of resonant solutions. We prove our result by using bifurcation theory combined with a careful analysis of the oscillatory behavior of the continuum near the bifurcation point. 

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2. Minimal Hypersurfaces with Arbitrarily Large Area

   https://doi.org/10.1093/imrn/rnz128  

   Chodosh, Otis; Mantoulidis, Christos (July 2019, International Mathematics Research Notices)

   Abstract For $$3\leq n\leq 7$$, we prove that a bumpy closed Riemannian $$n$$-manifold contains a sequence of connected embedded closed minimal surfaces with unbounded area. 

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3. Online control of the familywise error rate

   https://doi.org/10.1177/0962280220983381  

   Tian, Jinjin; Ramdas, Aaditya (April 2021, Statistical Methods in Medical Research)

   null (Ed.)

   Biological research often involves testing a growing number of null hypotheses as new data are accumulated over time. We study the problem of online control of the familywise error rate, that is testing an a priori unbounded sequence of hypotheses ( p-values) one by one over time without knowing the future, such that with high probability there are no false discoveries in the entire sequence. This paper unifies algorithmic concepts developed for offline (single batch) familywise error rate control and online false discovery rate control to develop novel online familywise error rate control methods. Though many offline familywise error rate methods (e.g., Bonferroni, fallback procedures and Sidak’s method) can trivially be extended to the online setting, our main contribution is the design of new, powerful, adaptive online algorithms that control the familywise error rate when the p-values are independent or locally dependent in time. Our numerical experiments demonstrate substantial gains in power, that are also formally proved in an idealized Gaussian sequence model. A promising application to the International Mouse Phenotyping Consortium is described. 

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4. Primitivity index bounds in free groups, and the second Chebyshev function

   https://doi.org/10.1142/S021819672350008X  

   Kapovich, Ilya; Simon, Zachary (February 2023, International Journal of Algebra and Computation)

   Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, [Formula: see text] and [Formula: see text], where [Formula: see text], and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence [Formula: see text] as [Formula: see text]. Answering a question from [17], we prove that this sequence is unbounded and that for [Formula: see text], we have [Formula: see text]. By contrast, we show that for all [Formula: see text], one has [Formula: see text]. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs. 

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5. Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

   https://doi.org/10.1017/apr.2021.20  

   Budhiraja, Amarjit; Fraiman, Nicolas; Waterbury, Adam (March 2022, Advances in Applied Probability)

   Abstract We consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems. In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD. 

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