Given a tree T, let q(T) be the minimum number of distinct eigenvalues in symmetric matrix whose underlying graph is T. It well known that q(T)≥d(T)+1, where d(T) diameter and T said to diminimal if q(T)=d(T)+1. In this paper, we present families trees any fixed diameter. Our proof constructive, allowing us compute, for d these families, M with spectrum has exactly d+1 eigenvalues.

We consider the VC-dimension of a set of the neural networks of depth s with w adjustable parameters that has h hidden units of activation functions of at most q segments of degree at most d polynomials. When d 2 and q 2, the VC-dimension is O(ws(s log d + log(qh=s))), O(ws((h=s) log q) + log d), and (ws log(dqh=s)). When d 2 and q = 1, it is (ws log d). When d = 1 and q 2, it is (ws log(qh=s))...

We study various statistics regarding the distribution of points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due nontrivial bounds for Kloosterman sums, it is known that these equidistribute on torus. prove refinements this result, including discrepancy, small scale equidistribution, covering...

We prove that | ∑ d≤x μ(d)/d| log x ≤ 1/69 when x ≥ 96 955 and deduce from that: ∣ ∣ ∣ ∣ ∑{ d≤x, (d,q)=1 μ(d)/d ∣ ∣ ∣ ∣ log(x/q) ≤ 4 5 q/φ(q) for every x > q ≥ 1. We also give better constants when x/q is larger. Furthermore we prove that |1 − ∑ d≤x μ(d) log(x/d)/d| ≤ 3 14 / log x and several similar bounds, from which we also prove corresponding bounds when summing the same quantity, but with ...

We demonstrate propagation rules of subsystem code constructions by extending, shortening and combining given subsystem codes. Given an [[n, k, r, d]]q subsystem code, we drive new subsystem codes with parameters [[n + 1, k, r,≥ d]]q, [[n − 1, k + 1, r,≥ d − 1]]q , [[n, k − 1, r + 1, d]]q . We present the short subsystem codes. The interested readers shall consult our companion papers for upper...

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces \[ V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \mathbf{Z} \times \big( \ell^{p,q}(\mathbf{Z} \mathbf{Z}^d) \big)^r \bigg\} \] of mixed Lebesg...

We point out an interplay between Fq-Frobenius non-classical plane curves and complete (k, d)-arcs in P2(Fq). A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete (k, d)-arcs with parameters k = d(q − d + 2) and d = (q − 1)/(q − 1), q being a power of the characteristic. In addi...