Let & ell;(1)(Z(max)(2),omega) be the weighted semigroup algebra over the abelian semigroup Zmax2 with the pointwise maximum operation. We show that & ell;(1)(Z(max)(2),omega) is a BED-algebra for any weight omega . If the weight additionally satisfies omega(-n,-n)<= K for some K>0 and all n is an element of N , we prove that & ell;(1)(Z(max)(2),omega) is a BSE-algebra. We also present a counterexample demonstrating that this condition is sharp and cannot be omitted.
In this paper, we investigate pure states of the second-order irrational rotation algebra A_θ ^(2) arising from automorphisms of the hyperfinite II _1 factor A induced by the natural action of SL(2,ℤ) on the irrational rotation algebra A_θ . For such states, the associated GNS representations are irreducible and admit a complete classification in terms of a quadruplet (m,ν ,b_1,b_2) . We show that the pure states arising from a certain subset of SL(2,ℤ) admit explicit expressions. In particular, when the corresponding element of SL(2,ℤ) is upper triangular, the associated invariant simplifies and can be described explicitly.
Recently, He and Wang [Proc. Amer. Math. Soc. (2024), 4775–4784] obtained four q-supercongruences by employing the creative microscoping method and a quadratic summation of Gasper and Rahman. In this paper, using the same method, we give generalizations of these four q-supercongruences with an additional parameter. As conclusions, we deduce four supercongruences modulo p^3 for odd primes p, such as ∑ _k=2s^(p^r-1)/2+s (6k+1) (14)_k+s(14)_k(12)_k-s/k!(k-2s)!(k-s)! 4^k-s ≡ (-1)^sp^r (14)_s(14)_2sp^3, where (x)_k=x(x+1)⋯ (x+k-1) , p^r≡ 1 4 , and 0⩽ s⩽ (p^r-1)/2 .
In this article, we prove some new translation formulae for the multiple zeta functions (MZFs). These translation formulae can be used to prove meromorphic continuation of MZFs and determine the exact location of their polar hyperplanes. As an application of these translation formulae, we give a new proof of a recurrence formula for the MZFs, recovering a recent result of Shinohara (Acta Arith., 212 (2024), 295-324). These recurrence formulae are used to obtain the special values of MZFs at the regular points.
In this work, we revisit the well-known discrepancy principle for the iteratively regularized Landweber iteration method for solving ill-posed problems in Banach spaces. We extend the range of the discrepancy principle for the iteratively regularized Landweber iteration method so that it can be applied to ill-posed inverse problems for which the forward operator may not be differentiable (or non-smooth inverse problems). We discuss a novel convergence analysis for the iteratively regularized Landweber iteration method that holds without the reflexivity of the image space and G â teaux differentiability of the forward mapping. Moreover, we derive the error estimates of the iterates without utilizing any additional smoothness condition on the sought solution. Finally, we present numerical simulations to show the practicality of the method.
