Moorhouse's question on locally finite generalized quadrangles, part 1 -- the countable case

We settle a question posed by G. Eric Moorhouse on the model theory and existence of locally finite generalized quadrangles. In this paper, we completely handle the case in which the generalized quadrangles have a countable number of elements.Comment: 9 pages; submitted (June 2020

Projective Ring Line of an Arbitrary Single Qudit

As a continuation of our previous work (arXiv:0708.4333) an algebraic geometrical study of a single $d$-dimensional qudit is made, with $d$ being {\it any} positive integer. The study is based on an intricate relation between the symplectic module of the generalized Pauli group of the qudit and the fine structure of the projective line over the (modular) ring \bZ_{d}. Explicit formulae are given for both the number of generalized Pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of \bZ^{2}_{d}. We find, remarkably, that a perp-set is not a set-theoretic union of the corresponding points of the associated projective line unless $d$ is a product of distinct primes. The operators are also seen to be structured into disjoint `layers' according to the degree of their representing vectors. A brief comparison with some multiple-qudit cases is made

A discourse on the potential of crowdfunding and Islamic finance in the agricultural sector of East Java, Indonesia

Literature evidence on the transformation transpires agriculture sector in East Java for the last 5 to 10 years. The contribution of the agricultural sector towards East Java gross domestic product (GDP) in 2008 recorded at 16.55%, however, shrinks to 13.75% in 2015. This statistic shows the regressed contribution of the agriculture sector in comparison to other economic sectors. One common view that linked to shrinking in credit composition is due to the lack of credit accessibility for the sector. Given that, this paper attempts to propose a viable financing model to develop the agricultural sector in East Java known as Integrated Agricultural Land Crowdfunding Model (IALCM) using Islamic financing instruments through a crowdfunding platform. This model is expected to offer farmers in East Java to meet theirliquidity constraints and the Indonesian government to accelerate social entrepreneurship innovation with conceivable recommendations forthe development of agricultural sector in East Jav

Mermin's Pentagram as an Ovoid of PG(3,2)

Mermin's pentagram, a specific set of ten three-qubit observables arranged in quadruples of pairwise commuting ones into five edges of a pentagram and used to provide a very simple proof of the Kochen-Specker theorem, is shown to be isomorphic to an ovoid (elliptic quadric) of the three-dimensional projective space of order two, PG(3,2). This demonstration employs properties of the real three-qubit Pauli group embodied in the geometry of the symplectic polar space W(5,2) and rests on the facts that: 1) the four observables/operators on any of the five edges of the pentagram can be viewed as points of an affine plane of order two, 2) all the ten observables lie on a hyperbolic quadric of the five-dimensional projective space of order two, PG(5,2), and 3) that the points of this quadric are in a well-known bijective correspondence with the lines of PG(3,2).Comment: 5 pages, 4 figure

Covid-19, Financial Markets (Islamic vs Non-Islamic), and Exchange Rate: Does the Malaysian Market Offers Diversification Opportunities to the Investors?

We explore the impact of Covid-19 towards Islamic and non-Islamic financial markets in Malaysia. We employ the wavelet coherency approach (WCA) which allows a deeper investigation of the relationship between the selected variables in terms time-frequency domain. We document that (i) Islamic capital markets represented by FTSEBMEI and MyETFDJIMMT25 are performing better during the Covid-19 period and also offer a greater investment opportunity to the investors for diversification purposes, (ii) non-Islamic index, FTSEBMKLCI, was less affected during this pandemic, and the market offers better risk and optimal diversification benefits to the investors as time progresses, and (iii) exchange rate appears to be more stable and within the phase category, indicating the co-movements are relatively strong in smaller scales. Understanding the impact of Covid-19 on the financial markets will lend to a better portfolio investment design which considers return and risk

Projective Ring Line of a Specific Qudit

A very particular connection between the commutation relations of the elements of the generalized Pauli group of a $d$-dimensional qudit, $d$ being a product of distinct primes, and the structure of the projective line over the (modular) ring \bZ_{d} is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of \bZ^{2}_{d}. Under this correspondence, the set of operators commuting with a given one -- a perp-set -- represents a \bZ_{d}-submodule of \bZ^{2}_{d}. A crucial novel feature here is that the operators are also represented by {\it non}-admissible pairs of \bZ^{2}_{d}. This additional degree of freedom makes it possible to view any perp-set as a {\it set-theoretic} union of the corresponding points of the associated projective line

Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits

Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\it hyperbolic} or {\it elliptic} in dependence on N being {\it even} or {\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.Comment: 3 pages, no figures/tables; V2 - short introductory paragraph added; V3 - to appear in Int. J. Mod. Phys.

Empowering WAQF financing sustainability through capital market in Malaysia:A review

Though the Waqf (Islamic endowment) has been operative for countless years as a superlative instrument that upholds societal welfare, most of the developed and developing countries are yet to entirely utilize Waqf for the capital market investment. This paper aims to review and explore the Waqf financing feasibility through capital market integration from the Malaysian context. Financing tools primarily designed for Waqf assets development are rigorously explored to secure sustainability of Waqf institutions. While embracing the theory of sustainability entailing chief aspects of economics, social and environment as a guiding principle, the study proposes a viable integration of Waqf and capital market instrument of unit trust investments based on the Shariah-compliant. The finding illustrates that there is an essential need for a novel apparatus through the capital market such Waqf unit trust to realize grander Waqf funds accumulation, investment, and distribution. It is exceedingly encouraged that produced proceeds from the investment are optimally utilized for socio-economic projects. This study realizes as one of the pioneer endeavors to group Waqf fund via unit