https://science.thanglong.edu.vn/index.php/volc <p>About Mathematics and Mathematical Sciences (Vol. C), published in English language.</p> en-US tapchikh@thanglong.edu.vn (Thang Long Journal of Science: Mathematics and Mathematical Sciences) hotro@vojs.vn (VOJS) Sun, 23 Jun 2024 00:00:00 +0000 OJS 3.2.1.1 http://blogs.law.harvard.edu/tech/rss 60 https://science.thanglong.edu.vn/index.php/volc/article/view/122 Abstract. In this paper, we prove degeneracy theorems for holomorphic mappings from a complex disc Δ(R) ⊂ C with finite growth index into Pn(C) sharing hyperplanes in general position. We further consider the case that intersecting points of the mappings and the hyperplanes with multiplicities more than a certain number do not need to be counted. These results generalize the previous degeneracy theorems for meromorphic mappings from Cm into Pn(C). Nguyen Thi Nhung Copyright (c) 2024 https://science.thanglong.edu.vn/index.php/volc/article/view/122 Tue, 21 May 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/123 We extend classical results by Debreu and Dierker about equilibrium prices of a regular economy with continuously differentiable demand functions/excess demand function to a regular exchange economy with these functions being locally Lipschitz. Our concept of a regular economy is based on Clarke’s concept of regular value and we show that such a regular economy has a finite, odd number of equilibrium prices, the set of economies with infinite number of equilibrium prices has Lebesgue measure zero and there exist locally Lipschitz selections of equilibrium prices around a regular economy. Truong Xuan Duc Ha Copyright (c) 2024 https://science.thanglong.edu.vn/index.php/volc/article/view/123 Tue, 21 May 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/94 The purpose of this paper is to study uniqueness problem of meromorphic mapping from Cm into the complex space Pn(C) sharing partial fixed and moving hypersurfaces. Using the second main theorems due to S. D. Quang and D. P. An [12, 13], we obtain some uniqueness results. Our results are improved some before results in this trend. In our best knowledge, there are not any uniqueness results of meromorphic mapping partially shared hypersurfaces up to now. Thin Nguyen Van Copyright (c) 2023 Journal of Mathematics and Mathematical Sciences https://science.thanglong.edu.vn/index.php/volc/article/view/94 Tue, 02 Jul 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/92 We consider a real smooth hypersurface M ⊂ C2, which is of D’Angelo infinite type at p 2 M. The purpose of this paper is to show that the real vector space of tangential holomorphic vector field germs at p vanishing at p is either trivial or of real dimension 1. Thu Ninh Van, Tiep Chu Van, Duc Mai Anh Copyright (c) 2023 Journal of Mathematics and Mathematical Sciences https://science.thanglong.edu.vn/index.php/volc/article/view/92 Tue, 02 Jul 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/124 Let IK be a complete ultrametric algebraically closed field and let f be an entire function in IK whose order of growth is finite. We show that the type of growth is finite if and only if so is the cotype. We give bounds for the cotype of growth and also for the lower cotype of growth. We show that the type of growth of f is equal to its lower type if and only if its cotype is equal to its lower cotype and when these are realised, then the cotype is the product of the type by the order of growth and the order of growth (if > 0), is then equal to the lower order of growth. If an entire function h has an order of growth strictly inferior to the lower order of an entire function f, then h is a small function with respect to f. A similar comparison is made with the type of growth. Conversely, if h is a small function with respect to f, then f +h and f have same order, same type and same cotype of growth. Links are showed with the Nevanlinna Theory. Suppose that IK is of characteristic 0. Given a meromorphic function f = g h , if f admits primitives and if the type or the cotype of h is finite, then f assumes all values infinitely many times. A counter-example is constructed where the lower order of growth is equal to the order of growth but the lower type of growth is not equal to the type of growth and where the the cotype is not equal to the product of the type by the order of growth. In complex analysis, a claim was made for complex meromorphic functions stating that if the lower order of growth equals the order, then the lower type equals the type but we contest the proof. Alain Escassut Copyright (c) 2024 https://science.thanglong.edu.vn/index.php/volc/article/view/124 Tue, 21 May 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/125 In the era of information explosion supplied with a veryhigh level of technology as of today, the loss or leakage of information is one of the most severe problems. Security in communication has thus become more important than ever. In this paper, two protocols for quantum secure direct exchange of messages are proposed using hyperentangled photon pairs as resource for exchanging informative bits and single photons in hyperstates as resource for detecting eavesdroppers. In both the protocols the photons need to be transmitted only once from one to the other communicating party. However, the ways to transmit photons are different in the two proposed protocols. The first protocol employs block transmission of photons. Although its security is unconditional, it compulsorily requires quantum memories at both communicating stations. Contrasting to the first protocol, in the second protocol single photons are transmitted one by one and subject to immediate processing without the need of any quantum memories. The security of the second protocol is asymptotic in the sense that the chance for eavesdroppers to pass is approaching zero in the limit of long messages. Comparison with other protocols is also given. Nguyen Ba An Copyright (c) 2024 Thang Long Journal of Science: Mathematics and Mathematical Sciences https://science.thanglong.edu.vn/index.php/volc/article/view/125 Tue, 21 May 2024 00:00:00 +0000 https://science.thanglong.edu.vn/index.php/volc/article/view/126 In this article, with respect to the recently introduced definition of weighted sharing of sets in the wider sense [8], we have comprehensively extended and improved some well known results in the literature. As the definition involves the manipulations of two polynomials, we perceive that the characteristizations of the underlying polynomials become utmost important. This realization urges us to introduce the definition of the index of polynomial. Noting that, in case of weighted sharing of sets the polynomials so far chosen are of lower index, after defining and streamlining the index concept, we explore the influence of using polynomials with higher indices in the context of weighted sharing of sets in wider sense to improve and extend a number of earlier results. Abhijit Banerjee, Jhilik Banerjee Copyright (c) 2024 https://science.thanglong.edu.vn/index.php/volc/article/view/126 Tue, 21 May 2024 00:00:00 +0000