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1. Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundary

   https://doi.org/10.1017/jpr.2019.9  

   Blanchet, Jose; Dong, Jing; Liu, Zhipeng (March 2019, Journal of Applied Probability)

   Abstract We present the first algorithm that samples max n ≥0 { S n − n α }, where S n is a mean zero random walk, and n α with $$\alpha \in ({1 \over 2},1)$$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean. 

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2. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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3. Random Interpolating Sequences in Dirichlet Spaces

   https://doi.org/10.1093/imrn/rnab110  

   Chalmoukis, Nikolaos; Hartmann, Andreas; Kellay, Karim; Wick, Brett Duane (May 2021, International Mathematics Research Notices)

   Abstract We discuss random interpolating sequences in weighted Dirichlet spaces $${{\mathcal{D}}}_\alpha $$, $$0\leq \alpha \leq 1$$, when the radii of the sequence points are fixed a priori and the arguments are uniformly distributed. Although conditions for deterministic interpolation in these spaces depend on capacities, which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $$\alpha =1/2$$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $${{\mathcal{D}}}_\alpha $$ are exactly the almost sure separated sequences when $$0\le \alpha <1/2$$ (which includes the Hardy space $$H^2={{\mathcal{D}}}_0$$), and they are exactly the almost sure zero sequences for $${{\mathcal{D}}}_\alpha $$ when $$1/2 \leq \alpha \le 1$$ (which includes the classical Dirichlet space $${{\mathcal{D}}}={{\mathcal{D}}}_1$$). 

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4. Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

   https://doi.org/10.1093/imamat/hxz019  

   Shubov, Marianna A; Kindrat, Laszlo P (October 2019, IMA Journal of Applied Mathematics)

   Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($$\alpha ,\beta ,k_1,k_2$$) linear boundary feedback law at the right end. The $$2 \times 2$$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $$\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$$. The role of the control parameters is examined and the following results have been proven: (i) when $$\beta \neq 0$$, the set of vibrational modes is asymptotically close to the vertical line on the complex $$\nu$$-plane given by the equation $$\Re \nu = \alpha + (1-k_1k_2)/\beta$$; (ii) when $$\beta = 0$$ and the parameter $$K = (1-k_1 k_2)/(k_1+k_2)$$ is such that $$\left |K\right |\neq 1$$ then the following relations are valid: $$\Re (\nu _n/n) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iii) when $$\beta =0$$, $|K| = 1$, and $$\alpha = 0$$, then the following relations are valid: $$\Re (\nu _n/n^2) = O\left (1\right )$$ and $$\Im (\nu _n/n) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iv) when $$\beta =0$$, $|K| = 1$, and $$\alpha>0$$, then the following relations are valid: $$\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$. 

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5. $$\alpha $$-event characterization and rejection in point-contact HPGe detectors

   https://doi.org/10.1140/epjc/s10052-022-10161-y  

   Arnquist, I. J.; Avignone, F. T.; Barabash, A. S.; Barton, C. J.; Bertrand, F. E.; Blalock, E.; Bos, B.; Busch, M.; Buuck, M.; Caldwell, T. S.; et al (March 2022, The European Physical Journal C)

   Abstract P-type point contact (PPC) HPGe detectors are a leading technology for rare event searches due to their excellent energy resolution, low thresholds, and multi-site event rejection capabilities. We have characterized a PPC detector’s response to $$\alpha $$ α particles incident on the sensitive passivated and p $$^+$$ + surfaces, a previously poorly-understood source of background. The detector studied is identical to those in the Majorana Demonstrator experiment, a search for neutrinoless double-beta decay ( $$0\nu \beta \beta $$ 0 ν β β ) in $$^{76}$$ 76 Ge. $$\alpha $$ α decays on most of the passivated surface exhibit significant energy loss due to charge trapping, with waveforms exhibiting a delayed charge recovery (DCR) signature caused by the slow collection of a fraction of the trapped charge. The DCR is found to be complementary to existing methods of $$\alpha $$ α identification, reliably identifying $$\alpha $$ α background events on the passivated surface of the detector. We demonstrate effective rejection of all surface $$\alpha $$ α events (to within statistical uncertainty) with a loss of only 0.2% of bulk events by combining the DCR discriminator with previously-used methods. The DCR discriminator has been used to reduce the background rate in the $$0\nu \beta \beta $$ 0 ν β β region of interest window by an order of magnitude in the Majorana Demonstrator   and will be used in the upcoming LEGEND-200 experiment. 

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