A Generalization of Mersenne and Thabit Numerical Semigroups

Let $A=(a_1, a_2, ..., a_n)$ be relative prime positive integers with $a_i\geq 2$. The Frobenius number $F(A)$ is the largest integer not belonging to the numerical semigroup $\langle A\rangle$ generated by $A$. The genus $g(A)$ is the number of positive integer elements that are not in $\langle A\rangle$. The Frobenius problem is to find $F(A)$ and $g(A)$ for a given sequence $A$. In this paper, we study the Frobenius problem of $A=(a,2a+d,2^2a+3d,...,2^ka+(2^k-1)d)$ and obtain formulas for $F(A)$ and $g(A)$ when $a+d\geq k$. Our formulas simplifies further for some special cases, such as Mersenne and Thabit numerical semigroups. We obtain explicit formulas for generalized Mersenne and Thabit numerical semigroups and some more general numerical semigroups

The Frobenius Formula for A=(a,ha+d,ha+b2d,...,ha+bkd)A=(a,ha+d,ha+b_2d,...,ha+b_kd)

Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$

Observation of quantum fingerprinting beating the classical limit

Quantum communication has historically been at the forefront of advancements, from fundamental tests of quantum physics to utilizing the quantum-mechanical properties of physical systems for practical applications. In the field of communication complexity, quantum communication allows the advantage of an exponential reduction in the information transmitted over classical communication to accomplish distributed computational tasks. However, to date, demonstrating this advantage in a practical setting continues to be a central challenge. Here, we report an experimental demonstration of a quantum fingerprinting protocol that for the first time surpasses the ultimate classical limit to transmitted information. Ultra-low noise superconducting single-photon detectors and a stable fibre-based Sagnac interferometer are used to implement a quantum fingerprinting system that is capable of transmitting less information than the classical proven lower bound over 20 km standard telecom fibre for input sizes of up to two Gbits. The results pave the way for experimentally exploring the advanced features of quantum communication and open a new window of opportunity for research in communication complexity and testing the foundations of physics.Comment: 19 pages, 4 figure

Historical evolution and the effects of ecological management in Tarim Basin, China

The Tarim Basin, in northwestern China, is the largest continental basin in the world, and hosts desert landscapes as well as extensive oasis agriculture. Many horticultural products come from this basin. However, since the 1950s, frequent river flow interruptions have occurred in the lower reaches of the Tarim River. Thus, the natural ecology of the basin has undergone significant changes because of recent human economic and social activities. In particular, water resource development and utilization along with climate changes have had a significant impact on the area. To prevent further deterioration of the environment, the Central Government implemented a water conveyance project in 2000. Based on this project, Chinese scientists, together with those from overseas, have conducted extensive research on the historical evolution of the area, and the physiological and ecological responses of the natural vegetation around the Tarim Desert Highway. Progress has been made in the areas of environmental protection and ecological conservation

EWM-FCE-ODM-Based Evaluation of Smart Community Construction: From the Perspective of Residents’ Sense of Gain

As a crucial paradigm for addressing urbanization-related problems, smart community construction is in full swing, and its goal is to enhance residents’ sense of gain. Prior studies have not been able to account for all aspects of smart community construction, especially the evaluation tools from the perspective of residents’ sense of gain. Therefore, this paper seeks to establish a comprehensive evaluation framework for residents’ sense of gain in the smart community through the integrated method, which includes the entropy weight method (EWM), the fuzzy comprehensive evaluation (FCE), and the obstacle degree model (ODM). For the purpose of verifying the feasibility of the evaluation framework, 31 smart communities in 6 Chinese cities (Shenzhen City, Putian City, Huizhou City, Dongguan City, Zhengzhou City, and Luoyang City) were selected. The results indicated that the weight of “Cultural activities for the elderly” indicator is the highest while the “Overall design” indicator is the lowest. In addition, Putian City had the best performance, but Shenzhen City ranked last among the six cities. Moreover, among the 31 communities, the Fengshan community in Putian City performed the best while the Xinglong community in Luoyang City performed the worst. Several suggestions are proposed to improve residents’ sense of gain in smart communities, such as enhancing the quality of healthcare services, meeting the needs of the elderly through multiple channels, and enriching business services. This study not only innovates the evaluation method of smart community construction from the perspective of residents’ sense of gain but also provides suggestions for promoting the sustainable development of the smart community and enabling residents to feel more satisfied