Computer experiments have extensive applications in various fields, such as industry and manufacturing. Maximin distance designs are commonly utilized in computer experiments due to their excellent space-filling properties and asymptotic optimality. Numerous construction methods have been proposed for generating maximin distance designs, and several upper bounds on the maximin Lp-distance have been provided for evaluating the distance properties of the generated designs. However, existing upper bounds are too loose to be attained in most cases. Consequently, many well-performing designs may be considered mediocre-performing. To address this issue, we integrate the characteristics of U-type designs with the sphere packing concept. More appropriate upper bounds on the maximin Lp-distance for U-type designs are established. Furthermore, we demonstrate the effectiveness of the new upper bounds by constructing U-type designs that achieve or approach these bounds and comparing the new upper bounds with existing upper bounds in the scenario of Latin hypercube designs.
In this paper, we propose deep fitted value iteration (DFVI), an iterative algorithm designed to estimate the optimal state-value function, where deep neural networks with ReLU activations are used as the hypothesis class. By exploiting the contraction property of the optimal Bellman operator, DFVI constructs an iterative procedure that refines state-value function estimates through a sequence of least-squares regression updates. Unlike action-value-based approaches, DFVI approximates a function on the state space alone and therefore avoids neural-network approximation over the joint state-action space. For finite action spaces, we establish a rigorous non-asymptotic error bound between the state-value function induced by the estimated greedy policy and the optimal state-value function. Through an error-propagation analysis, we show that the overall error is primarily governed by the estimation error accumulated across iterations. Moreover, we decompose this estimation error into statistical and approximation components, establishing explicit bounds for each. These results yield principled insights into the required network expressivity and sample complexity for achieving reliable value estimation in practice.
Using the Markov distance and the Ptolemy inequality introduced by Lee et al. (2023), we completely determine the monotonicity of generalized Markov numbers along lines of a given slope. This confirms several conjectures posed by Lee et al. (2023).
A class of Lotka-Volterra competitive systems on a finite graph (network) is investigated, where both diffusive and advective movements are considered. By the theory of graphs, the theory of monotone dynamical systems and the theory of principal eigenvalues, the global dynamics is determined in terms of critical competition coefficients, where the key is to establish the linear stability of all possible positive equilibria. This work can be seen as a further development of Slavík (2020) and Chen et al. (2022), where there is diffusion only (no advection).
In this paper, we give an estimate for the sum of Fourier coefficients λf(n) of Hecke-Maass cusp form f over a fractional sequence. Our main result is S_f(x) = ∑_n ⩽ xλ_f([x/n]) = ∑_n=1^∞λ_f(n)/n(n+1) x + O(x^1/2-3/160+ε). This is a breakthrough to the barrier 1/2 for the error term estimates. Our method is also used to give a new result on a problem initiated by Bordellés et al. (2019).
