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1. Fully discrete best-approximation-type estimates in L ∞ ( I ; L 2(Ω) d ) for finite element discretizations of the transient Stokes equations

   https://doi.org/10.1093/imanum/drac009  

   Behringer, Niklas; Vexler, Boris; Leykekhman, Dmitriy (April 2022, IMA Journal of Numerical Analysis)

   Abstract In this article, we obtain an optimal best-approximation-type result for fully discrete approximations of the transient Stokes problem. For the time discretization, we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions. 

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2. L oo p D e l t a : E m b e dd i n g L o c a li t y - a w a r e O pp o r t un i s t i c D e l t a C o m p r e ss i o n i n I n li n e D e dup li c a t i o n f o r Hi g h l y E ffic i e n t D a t a R e du c t i o n

   Zhang, Yucheng; Jiang, Hong; Feng, Dan; Jiang Nan; Qui, Taorong; Huang, Wei (July 2023, USENIX annual technical conference)

   A s a c om pl e men t t o da ta d edupli cat ion , de lta c om p ress i on fu r- t he r r edu c es t h e dat a vo l u m e by c o m pr e ssi n g n o n - dup li c a t e d ata chunk s r e l a t iv e to t h e i r s i m il a r chunk s (bas e chunk s). H ow ever, ex is t i n g p o s t - d e dup li c a t i o n d e l t a c o m pr e ssi o n a p- p ro a ches fo r bac kup s t or ag e e i t h e r su ffe r f ro m t h e l ow s i m - il a r i t y b e twee n m any de te c ted c hun ks o r m i ss so me po t e n - t i a l s i m il a r c hunks , o r su ffer f r om l ow (ba ckup and r es t ore ) th r oug hpu t du e t o extr a I/ Os f or r e a d i n g b a se c hun ks o r a dd a dd i t i on a l s e r v i c e - d i s r up t ive op e r a t i on s to b a ck up s ys t em s. I n t h i s pa p e r, w e pr opo se L oop D e l t a t o a dd ress the above - m e n t i on e d prob l e m s by an e nha nced em b e ddi n g d e l t a c o m p - r e ss i on sc heme i n d e dup li c a t i on i n a non - i n t ru s ive way. T h e e nha nce d d elt a c o mpr ess ion s che m e co m b in e s f our key t e c h - ni qu e s : (1) du a l - l o c a li t y - b a s e d s i m il a r i t y t r a c k i n g to d e t ect po t e n t i a l si m il a r chun k s b y e x p l o i t i n g both l o g i c a l and ph y - s i c a l l o c a li t y, ( 2 ) l o c a li t y - a wa r e pr e f e t c h i n g to pr efe tc h ba se c hun ks to a vo i d ex t ra I/ Os fo r r e a d i n g ba s e chun ks on t h e w r i t e p at h , (3) c a che -aware fil t e r to avo i d ext r a I/Os f or b a se c hunk s on t he read p at h, a nd (4) i nver sed de l ta co mpressi on t o perf orm de lt a co mpress i o n fo r d at a chunk s t hat a re o th e r wi se f o r b i dd e n to s er ve as ba se c hunk s by r ew r i t i n g t e c hn i qu e s d e s i g n e d t o i m p r ove r es t o re pe rf o rma nc e. E x p e r i m e n t a l re su lts indi ca te t hat L oop D e l t a i ncr ea se s t he c o m pr e ss i o n r a t i o by 1 .2410 .97 t i m e s on t op of d e dup li c a - t i on , wi t hou t no t a b l y a ffe c t i n g th e ba ck up th rou ghpu t, a nd i t i m p r ove s t he res to re p er fo r m an ce b y 1.23.57 t i m e 

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3. A Circuit-based Approach to the Synthesis of 2-D Omega Materials

   https://doi.org/10.1109/APUSNCURSINRSM.2019.8888837  

   Szymanski, Luke; Grbic, Anthony (July 2019, 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting)

   In one dimension, a circuit network equivalence has been established for omega materials. However, 2-D circuit-based or transmission-line metamaterials have previously been restricted to magnetic and electric responses. This paper proposes 2-D circuit-based omega materials using asymmetric circuits. A general formulation is provided in terms of ABCD-parameters and an example is shown using Pi-networks. These metamaterials could enable the design of compact beamformers, power dividers, and other microwave devices. 

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4. Lipschitz regularity for solutions of the parabolic p-Laplacian in the Heisenberg group

   https://doi.org/10.54330/afm.131227  

   Capogna, Luca; Citti, Giovanna; Zhong, Xiao (June 2023, Annales Fennici Mathematici)

   We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic p-Laplacian $$\(\partial_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),\$$ in a cylinder $$\(\Omega\times\mathbb{R}^+\)$$, where $$ \(\Omega\)$$ is domain in the Heisenberg group $$\(\mathbb{H}^n\)$$, and $$\(2\le p \le 4\)$$. The result continues to hold in the more general setting of contact subRiemannian manifolds. 

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5. Rates of convergence in $$W^2_p$$-norm for the Monge-Amp\`ere equation.

   Neilan, Michael; Zhang, Wujun (October 2018, SIAM journal on numerical analysis)

   We develop discrete $$W^2_p$$-norm error estimates for the Oliker--Prussner method applied to the Monge--Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $$\| u - u_h \|_{W^2_{f,p}} (\mathcal{N}^I_h)$$ converges in order $$O(h^{1/p})$$ if $p > d$ and converges in order $$O(h^{1/d} \ln (\frac 1 h)^{1/d})$$ if $$p \leq d$$, where $$\|\cdot\|_{W^2_{f,p}(\mathcal{N}^I_h)}$$ is a weighted $$W^2_p$$-type norm, and the constant $C>0$ depends on $$\|{u}\|_{C^{3,1}(\bar\Omega)}$$, the dimension $$d$$, and the constant $$p$$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases. 

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