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We consider a family of approximations of the Dedekind zeta function ζK(s) of a number field K/Q. Weighted L^2-norms of the difference of two such approximations of ζK(s) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of ζK(s) can be obtained from such weighted L^2-norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.

Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3, \mathbb{A}_\mathbb{Q})$ by certain $L$-values of $p$-power twists.

For a G-graded ring R, the problem of Gr-regularity (i.e. von Neumann regularity condition on homogeneous elements) has been treated earlier (see Nastasescu and Van Oystaeyen, 1982 [11] ). Once the concept of the smash product was introduced, this problem has been resumed by several authors. This paper is concerned with von Neumann regularity of two smash products: the smash product R # G of the G-graded ring R by the group G and the smash product R # A of R by a finite left G-set A. The connections between the regularity of R # A and the (Gr-)regularity of R are also investigated. One consequence of our results is that the smash product R # A is a von Neumann regular ring if and only if the category ( G , A , R ) -gr is regular, in Stenstrom's sense.

Let 𝒞 be an arbitrary category. We study strongly involutory functors on 𝒞, defined as involutory contravariant endofunctors of 𝒞 acting as identity on objects. Motivating examples can be constructed if we think at the transpose of a matrix, the adjoint of a linear continuous operator between two Hilbert spaces, and the inverse of a morphism in a groupoid. We show how a strongly involutory functor on a skeleton of 𝒞 extends to 𝒞, and we apply this to find all such functors for a groupoid. We describe and classify up to a natural equivalence all strongly involutory functors on the category of finite dimensional vector spaces over a field. Strongly involutory functors with a special property related to generalized inverses of morphisms are studied.

The Restricted Isometry Property or RIP introduced by Candes and Tao requires an n × p dictionary to act as a near isometry on all k-sparse signals. This paper provides a very simple condition under which a dictionary Φ(C) obtained by exponentiating codewords from a binary linear code C satisfies the RIP with high probability. The method is to bound the difference between the dictionary Φ(C) and a second dictionary A generated by a random Bernoulli process which is known to satisfy the RIP with high probability. The difference Δ — Φ(C) is controlled by the covering radius of C, a fundamental parameter that is bounded above by the number of weights in the dual code C┴ (the external distance of C). The main result complements a more sophisticated asymptotic analysis by Babadi and Tarokh of the distribution of eigenvalues of random submatrices of Φ(C). In this analysis, divergence from the distribution corresponding to the full Bernoulli matrix depends on a different fundamental parameter of C, namely the minimum distance of the dual code C┴.

Certain nonlinear binary codes can be constructed as binary images of Z 4 -linear codes under the Gray map. Examples include the second-order Reed-Muller code and the Kerdock and Preparata codes. In this paper, we consider a new quaternary code which is an additive subcode of the Z 4 -linear Kerdock code. The Kerdock code is the direct sum of a one-dimensional quaternary code and the quaternary subcode examined in this paper. This paper calculates the weight distribution of the projective Kerdock code from which the weight distribution of the dual code can be computed. The dual code is a supercode of the quaternary Preparata code. The projective Kerdock code is used to construct a deterministic measurement matrix for compressed sensing. Numerical experiments are presented for sparse reconstruction using the LASSO that show improvement over random Gaussian matrices of the same size.