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1. Fast Diffusion leads to partial mass concentration in Keller–Segel type stationary solutions

   https://doi.org/10.1142/S021820252250018X  

   Carrillo, J. A.; Delgadino, M. G.; Frank, R. L.; Lewin, M. (April 2022, Mathematical Models and Methods in Applied Sciences)

   We show that partial mass concentration can happen for stationary solutions of aggregation–diffusion equations with homogeneous attractive kernels in the fast diffusion range. More precisely, we prove that the free energy admits a radial global minimizer in the set of probability measures which may have part of its mass concentrated in a Dirac delta at a given point. In the case of the quartic interaction potential, we find the exact range of the diffusion exponent where concentration occurs in space dimensions [Formula: see text]. We then provide numerical computations which suggest the occurrence of mass concentration in all dimensions [Formula: see text], for homogeneous interaction potentials with higher power. 

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2. The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

   https://doi.org/10.1142/s0219530519500234  

   Dai, Yichen; Hu, Weiwei; Wu, Jiahong; Xiao, Bei (July 2020, Analysis and Applications)

   The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. 

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3. ɛ-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

   https://doi.org/10.1287/moor.2020.1078  

   Chen, Yi; Dong, Jing; Ni, Hao (May 2021, Mathematics of Operations Research)

   null (Ed.)

   Consider a fractional Brownian motion (fBM) [Formula: see text] with Hurst index [Formula: see text]. We construct a probability space supporting both B H and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one for any user-specified error bound [Formula: see text]. When [Formula: see text], we further enhance our error guarantee to the α-Hölder norm for any [Formula: see text]. This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations [Formula: see text]. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process [Formula: see text] such that[Formula: see text] with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently. 

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4. Hantavirus outbreaks in the American Southwest: Propagation and retraction of rodent and virus diffusion waves from sky-island refugia

   https://doi.org/10.1142/S021797922140052X  

   Parmenter, Robert R.; Glass, Gregory E. (March 2022, International Journal of Modern Physics B)

   Hantavirus outbreaks in the American Southwest are hypothesized to be driven by episodic seasonal events of high precipitation, promoting rapid increases in virus-reservoir rodent species that then move across the landscape from high quality montane forested habitats (refugia), eventually over-running human residences and increasing disease risk. In this study, the velocities of rodents and virus diffusion wave propagation and retraction were documented and quantified in the sky-islands of northern New Mexico and related to rodent-virus relationships in refugia versus nonrefugia habitats. Deer mouse (Peromyscus maniculatus) refugia populations exhibited higher Sin Nombre Virus (SNV) infection prevalence than nonrefugia populations. The velocity of propagating diffusion waves of Peromyscus from montane to lower grassland habitats was measured at [Formula: see text] m/day (SE), with wave retraction velocities of [Formula: see text] m/day. SNV infection diffusion wave propagation velocity within a deer mouse population averaged [Formula: see text] m/day, with a faster retraction wave velocity of [Formula: see text] m/day. A spatio-temporal analysis of human Hantavirus Pulmonary Syndrome (HPS) cases during the initial 1993 epidemic revealed a positive linear relationship between the time during the epidemic and the distance of human cases from the nearest deer mouse refugium, with a landscape diffusion wave velocity of [Formula: see text] m/day ([Formula: see text]). These consistent diffusion propagation wave velocity results support the traveling wave component of the HPS outbreak theory and can provide information on space–time constraints for future outbreak forecasts. 

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5. Misiurewicz polynomials and dynamical units, part I

   https://doi.org/10.1142/S1793042123500616  

   Benedetto, Robert L; Goksel, Vefa (July 2023, International Journal of Number Theory)

   We study the dynamics of the unicritical polynomial family [Formula: see text]. The [Formula: see text]-values for which [Formula: see text] has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when [Formula: see text] is a prime power and [Formula: see text] is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of [Formula: see text]. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases. 

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