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Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simseks recent work Generating functions for unification of the multidimensional Bernstein polynomials and their applications([15,16]) and Carlitzs degenerate Bernoulli polynomials. We derived thier generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
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In this paper, we consider the degenerate Daehee numbers and polynomials of the second kind which are different from the previously introduced Daehee numbers and polynomials. We investigate some properties of these numbers and polynomials. In addition, we give some new identities and relations between the Daehee polynomials of the second kind and Carlitzs degenerate Bernoulli polynomials.
In this paper, we consider the degenerate Changhee numbers and polynomials of the second kind which are different from the previously introduced degenerate Changhee numbers and polynomials by Kwon-Kim-Seo (see [11]). We investigate some interesting identities and properties for these numbers and polynomials. In addition, we give some new relations between the degenerate Changhee polynomials of the second kind and the Carlitzs degenerate Euler polynomials.
We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call respectively Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. We study the geometry of these rigid analytic spaces, in particular (generalizing results of Breuil-Hellmann-Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalized Grothendieck-Springer resolution. We deduce several local-global compatibility results, including a classicality result (with no trianguline assumption at $p$), and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.
A main disadvantage of many high-order methods for hyperbolic conservation laws lies in the famous Gibbs-Wilbraham phenomenon, once discontinuities appear in the solution. Due to the Gibbs-Wilbraham phenomenon, the numerical approximation will be polluted by spurious oscillations, which produce unphysical numerical solutions and might finally blow up the computation. In this work, we propose a new shock capturing procedure to stabilise high-order spectral element approximations. The procedure consists of going over from the original (polluted) approximation to a convex combination of the original approximation and its Bernstein reconstruction, yielding a stabilised approximation. The coefficient in the convex combination, and therefore the procedure, is steered by a discontinuity sensor and is only activated in troubled elements. Building up on classical Bernstein operators, we are thus able to prove that the resulting Bernstein procedure is total variation diminishing and preserves monotone (shock) profiles. Further, the procedure can be modified to not just preserve but also to enforce certain bounds for the solution, such as positivity. In contrast to other shock capturing methods, e.g. artificial viscosity methods, the new procedure does not reduce the time step or CFL condition and can be easily and efficiently implemented into any existing code. Numerical tests demonstrate that the proposed shock-capturing procedure is able to stabilise and enhance spectral element approximations in the presence of shocks.
Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := sup_{x in A}{|f(x)|}$$ for real-valued functions $f$ defined on a set $A subset {Bbb R}$. Let $$V_a^b(f) := int_a^b{|f^{prime}(x)| , dx}$$ denote the total variation of a continuously differentiable function $f$ on an interval $[a,b]$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 frac nkleq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}|_{[0,1]}}{|P(1)|}} leq c_2 left( frac nk + 1 right)$$ for all integers $n geq 1$ and $k geq 1$. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac nkright)^{1/2} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{V_0^1(P)}} leq min_{P in {mathcal P}_{n,k}}{frac{|P^{prime}(x)sqrt{1-x^2}|_{[0,1]}}{|P(1)|}} leq c_2 left(frac nk + 1right)^{1/2}$$ for all integers $n geq 1$ and $k geq 1$.
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