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This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition Dαft=limh⟶0ft+heα−1t−ft/h, for all t>0, and α∈0,1. If α=0, this definition coincides to the classical definition of the first order of the function f.

We review some techniques of construction of new cyclically ordered groups from the old ones. Especially, we focus on the conditions of quotient group to be a cyclically ordered group.

. In this paper, we define and introduce the notions of S-I-quotient mappings, S-I-cs-networks, S-I-cs'-networks and S-wcs'-networks, and study some characteriza- tions of S-I-quotient mappings and S-I-cs0networks, especially S-J-quotient mappings and S-J-cs-networks on an ideal J of N. With these notions, we get that if X is a S-J- FU space with a point-countable S-J-cs'-network, then X is a meta-Lindelof space.

On the grounds of the long discussion in the literature for the construction of regional input–output (I/O) tables, the present study endeavors to identify the best performing method among the most applied location quotient based non-survey techniques. The analysis uses actual data—in particular, the EU input–output tables for the years 2010 and 2014—and by employing three different statistics, it compares the tables created by the quotients FLQ, AFLQ, SLQ, CILQ and RLQ. Comparison is made in both the technical coefficient tables and the Leontief ones. The results indicate that the AFLQ and FLQ provide better results for values of δ from 0.1 to 0.3, whereas for values of δ from 0.4 to 0.9, the results of the two quotients are not satisfactory. At the same time, using another statistic, the SLQ–CILQ technique seemed to yield satisfactory results. Similar results are also presented for the Leontief tables. Finally, using a linear regression model, the imports do not appear to be related to the value of δ in the FLQ.

We show that a monotone Lagrangian L in ${\\mathbb{C}}{\\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\\mathbb{R}}{\\mathbb{P}}^n$ . To prove this we use Zapolsky’s canonical pearl complex for L over ${\\mathbb{Z}}$ , and twisted versions thereof, where the twisting is determined by connected covers of L . The main tool is the action of the quantum cohomology of ${\\mathbb{C}}{\\mathbb{P}}^n$ on the resulting Floer homologies.

The concept of n-normed spaces is a generalization of the concept of normed spaces. Some characteristics of n-normed spaces have been discussed by many researchers. These spaces are usually observed using a set of n linear independent vectors. In this paper, we will construct quotient spaces of an n-normed space with respect to n linear independent vectors. We define a norm in each quotient space by using the n-norm that we have. Norms of these quotient spaces will be a new viewpoint in observing characteristics of n-normed spaces.

For an irreducible and nonpositive matrix, we present the concepts of the Perron complement matrix and the generalized Perron complement matrix. An inequality that relates the generalized Perron complement matrices of inverse N0-matrices is derived. In addition, we obtain the quotient formula of inverse N0-matrices based on the quotient formula for the Schur complements.

In this paper, we consider pure infiniteness of generalized Cuntz–Krieger algebras associated to labeled spaces $(E,{\\mathcal{L}},{\\mathcal{E}})$ . It is shown that a $C^{\\ast }$ -algebra $C^{\\ast }(E,{\\mathcal{L}},{\\mathcal{E}})$ is purely infinite in the sense that every non-zero hereditary subalgebra contains an infinite projection (we call this property (IH)) if $(E,{\\mathcal{L}},{\\mathcal{E}})$ is disagreeable and every vertex connects to a loop. We also prove that under the condition analogous to (K) for usual graphs, $C^{\\ast }(E,{\\mathcal{L}},{\\mathcal{E}})=C^{\\ast }(p_{A},s_{a})$ is purely infinite in the sense of Kirchberg and Rørdam if and only if every generating projection $p_{A}$ , $A\\in {\\mathcal{E}}$ , is properly infinite, and also if and only if every quotient of $C^{\\ast }(E,{\\mathcal{L}},{\\mathcal{E}})$ has property (IH).

The concept of almost interior Γ-hyperideals (A−I−Γ-hyperideals) in ordered Γ-semihypergroups is a generalization of the concept of interior Γ-hyperideals (I−Γ-hyperideals). In this study, the connections between I−Γ-hyperideals and A−I−Γ-hyperideals in ordered Γ-semihypergroups were presented. Also, we define the notion of m,n-Γ-hyperfilters of ordered Γ-semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered Γ-semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered Γ-semihypergroups.

Z.S. Aygin and P. C. Toh have deduced a technique using the theory of modular forms to determine all eta quotients whose derivative is also an eta quotient up to level 36. This paper aims to find a technique without using the theory of modular forms to deduce all the identities of Aygin and Toh of levels 12 and 16.