The Listsize Capacity of the Gaussian Channel with Decoder Assistance

The listsize capacity is computed for the Gaussian channel with a helper that—cognizant of the channel-noise sequence but not of the transmitted message—provides the decoder with a rate-limited description of said sequence. This capacity is shown to equal the sum of the cutoff rate of the Gaussian channel without help and the rate of help. In particular, zero-rate help raises the listsize capacity from zero to the cutoff rate. This is achieved by having the helper provide the decoder with a sufficiently fine quantization of the normalized squared Euclidean norm of the noise sequence.

Deforming convex bodies in Minkowski geometry

We introduce and study certain deformation of Minkowski norms in [Formula: see text] determined by a set of [Formula: see text] linearly independent 1-forms and a smooth positive function of [Formula: see text] variables. In particular, the deformation of a Euclidean norm [Formula: see text] produces a Minkowski norm defined in our recent work; its indicatrix is a rotation hypersurface with a [Formula: see text]-dimensional axis passing through the origin. For [Formula: see text], our deformation generalizes the construction of [Formula: see text]-norms which form a rich class of “computable” Minkowski norms and play an important role in Finsler geometry. We characterize such pairs of a Minkowski norm and its image that Cartan torsions of the two norms either coincide or differ by a [Formula: see text]-reducible term. We conjecture that for [Formula: see text] any Minkowski norm can be approximated by images of a Euclidean norm.

On k-circulant matrices with the generalized third-order Pell numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

Lower Bounds of the Smallest Singular Value of Matrices

In this paper, we get a lower bound of the smallest singular value of an arbitrarily matrix A by the trace of H(A) and the Euclidean norm of H(A), where H(A) is Hermitian part of A, numerical examples show the e ectiveness of our results.

Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.