In this work, the empirical-Gramian-based model reduction methods: Empirical poor man’s truncated balanced realization, empirical approximate balancing, dominant subspaces, truncation, and gains are compared in a non-parametric two parametric variants, via ten error measures: Approximate Lebesgue \(L_0\), \(L_1\), \(L_2\), \(L_\infty \), Hardy \(H_2\), \(H_\infty Hankel, Hilbert-Schmidt-Hankel,...

In this paper, we simultaneously determine the optimal sensor precision and observer gain, which achieves specified accuracy in state estimates. Along with unknown formulation parameterizes scaling of exogenous inputs that correspond to noise. Reciprocal is defined as precision, sparseness achieved by minimizing $l_1$ norm vector. The optimization performed constraints guaranteeing estimates, a...

The paper deals with $$L_p$$ -boundedness of the Hartley-Fourier convolutions operator and their applied aspects. We establish various new Young-type inequalities obtain structure a normed ring in Banach space when equipping it such convolutional multiplication. Weighted -norm these are also considered. As applications, we investigate solvability bounded $$L_1$$ -solution class Fredholm-type in...

Let $(L_n)_{n\geq 0}$ be the Lucas sequence given by $L_0 = 2, L_1 1$ and $L_{n+2} L_{n+1}+L_n$ for $n \geq 0$. In this paper, we are interested in finding all powers of three which sums two numbers, i.e., study exponential Diophantine equation $L_n + L_m 3^{a}$ nonnegative integers $n, m,$ $a$. The proof our main theorem uses lower bounds linear forms logarithms, properties continued fractions...

In this paper, we investigate the polynomial numerical index \(n^{(k)}(l_p),\) symmetric multilinear \(n_s^{(k)}(l_p),\) and \(n_m^{(k)}(l_p)\) of \(l_p\) spaces, for \(1\leq p\leq \infty.\) First prove that \(n_{s}^{(k)}(l_1)=n_{m}^{(k)}(l_1)=1,\) every \(k\geq 2.\) We show \(1 \lt p \infty,\) \(n_I^{(k)}(l_p^{j+1})\leq n_I^{(k)}(l_p^j),\) \(j\in \mathbb{N}\) \(n_I^{(k)}(l_p)=\lim_{j\to \infty...