Narrow Results

A feedback vertex/arc set (abbreviated as FVS/FAS) of a graph is a subset of the vertices/arcs which contains at least one vertex/arc for every cycle of that graph. A minimum FVS/FAS is an FVS/FAS which contains the smallest number of vertices/arcs. Hsu et al. [11] first proposed an algorithm which can find a minimum FVS in a rotator graph. In this paper, we present a formula for finding an FAS for a rotator graph and prove that the FAS is minimum. This formula can be easily implemented by an efficient algorithm which obtains a minimum FAS in a rotator graph. Finally, we also present a concise formula for finding a minimum FAS in an incomplete rotator graph in this paper.

This paper introduces a novel framework that automates and accelerates the development of embedded Field Programmable Gate Arrays (eFPGAs). The proposed solution is considered as the first environment for tree-based eFPGA implementation including software, hardware and loader. The developed framework allows users to generate eFPGA architecture in the form of hardware description language using Physical Design Flow (PDF) tool. It is a powerful tool that can produce a wide variety of designs ranging from small eFPGA to complex eFPGA. The bit file description of practical application is done in parallel, simultaneously and rapidly by the suggested Computer Aided Design (CAD) tools. The Loader, called Multi-Level Loader (MLL), is also provided to inject the bits into the corresponding SRAMs. Our framework is widely explored by modifying the data width. This research proves that data width equal to 17 has the best trade-off between performance, area and static power. However, it is penalized for buses having data length greater than 32. The experimentation demonstrates that a data width equal to 12 is the best for a 32-bit bus. Automation and significant acceleration of the eFPGA development cycle are also achieved in this study. A set of bench-marking applications with various multi-use purposes is mapped. The experimental results show the efficiency and flexibility of the proposed framework.

In this paper, we produce the product rules of nonassociative Moufang loops of order 81 by using an analytical approach. We then explore all possible presentations on a suitable set of generators, thereby obtaining a total of five nonisomorphic cases. The result is in agreement with the classification by Nagy and Vojtěchovský using the GAP package LOOPS.

Let $S_g$ be a closed oriented surface of genus $g$ and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the following results: For $g\ge 4$, $\text{Mod}(S_g)$ can be generated by four torsion elements. Three generators are involutions and the fourth one is an order three element. $\text{Mod}(S_3)$ can be generated by five torsion elements. Four generators are involutions and the fifth one is an order three element.

Let $S_g$ be the closed-oriented surface of genus $g$ and let ${\text{Mod}}^{\pm }(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that ${\text{Mod}}^{\pm }(S_g)$ can be generated by two torsion elements. One of these generators is of order $2$, and the other one is of order $4g+2$.

A new map (Λ_E) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of Λ_E is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φ_E and ψ_E) are also reviewed, in order to compare them with Λ_E. Interesting properties of the fixed points of φ_E and are studied. Among others, the fixed points of Λ_E (Fix(Λ_E)) are proved to be the maximal fuzzy points of (X, E) and the fixed points of coincide with the Image of Λ_E. An isometric embedding of X into Fix(Λ_E) is established and studied.

Defuzzification is an essential problem in fuzzy systems that it is always solved in a heuristic way. The aim of this work is to give a semantic interpretation to this process with the help of indistinguishability operators.

Two-level fractional factorial design is an efficient technique for experiments considering a large number of factors. To evaluate the efficiency and analyze the data for such a design, we need to know the generators for the design, so that, using the generators, we can generate its defining relation and alias structure. Although knowing the generators is important for a two-level fractional factorial design, it is not unusual in actual industrial situations for the generators used in the design to be lost or overlooked while the design is performed. Since Taguchi methods has been widely applied in industry, in this research, an efficient algorithm based on Taguchi orthogonal arrays (OA's) and interaction tables is developed to identify the generators for given designs. Furthermore, with the investigation of the insights of Taguchi OA's and interaction tables, this research may provide ideas for making Taguchi methods a simple tool for developing optimal designs for 2^{k - p} experiments.

Documents are an essential source of valuable information and knowledge, and photographs are a great way of reminiscing old memories and past events. However, it becomes difficult to preserve the quality of such ancient documents and old photographs for an extremely long time, as these images usually get damaged or creased due to various extrinsic effects. Utilizing image editing software like Photoshop to manually reconstruct such old photographs and documents is a strenuous and an enduring process. This paper attempts to leverage the generative modeling capabilities of Conditional Generative Adversarial Networks by utilizing specialized architectures for the Generator and the Discriminator. The proposed Reminiscent Net has a U-Net-based Generator with numerous feature maps for complete information transfer with the incorporation of location and contextual details, and the absence of dense layers allows utilization of diverse sized images. Implementation of the PatchGAN-based Discriminator that penalizes the image at the scale of patches has been proposed. NADAM optimizer has been implemented to enable faster and better convergence of the loss function. The proposed method produces visually appealing de-creased images, and experiments indicate that the architecture performs better than various novel approaches, both qualitatively and quantitatively.

A binomial f belonging to a toric ideal I is indispensable if, for any system of binomial generators of I, either f or -f belongs to . In the present paper, we study indispensable binomials of the toric ideals I_G arising from a finite graph G. First, we show that the toric ideal I_G arising from a finite graph G whose complementary graph is weakly chordal is generated by the indispensable binomials if and only if no complete graph of order ≥4 is a subgraph of G. Second, we completely classify indispensable binomials of the toric ideal I_G arising from a finite graph G satisfying the odd cycle condition. Finally, the existence of indispensable binomials of I_G will be discussed.

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group , by using the weight property of ideals in when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co₂ and Co₃. Computations were carried with the aid of computer algebra system 𝔾𝔸ℙ [18].

A polycyclic group $G$ is called an $NAF{Q}_n$-group if every normal abelian subgroup of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, the structure of $NAF{Q}_1$-groups and $NAF{Q}_2$-groups is determined.

A polycyclic group $G$ is called a $NAF{Q}_n$-group ($AF{Q}_n$-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, we describe the structures of $NAF{Q}_3$-groups and $AF{Q}_3$-groups, and bound the number of generators of $AF{Q}_3$-groups and the derived lengths of $NAF{Q}_3$-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

Let $K$ be a field of characteristic zero, $X_n=\{x_1,\dots ,x_n\}$ and $R_n=\{r_1,\dots ,r_n\}$ be two sets of variables, $L_n$ be the free metabelian Leibniz algebra generated by $X_n$, and $K[R_n]$ be the commutative polynomial algebra generated by $R_n$ over the base field $K$. Polynomials $p(X_n)\in L_n$ and $q(R_n)\in K[R_n]$ are called symmetric if they satisfy $p(x_{\pi (1)},\dots ,x_{\pi (n)})=p(X_n)$ and $q(r_{\pi (1)},\dots ,r_{\pi (n)})=q(R_n)$, respectively, for all $\pi \in S_n$. The sets $L_n^{S_n}$ and $K{[R_n]}^{S_n}$ of symmetric polynomials are the $S_n$-invariant subalgebras of $L_n$ and $K[R_n]$, respectively. The Leibniz subalgebra ${(L_n^{\prime })}^{S_n}=L_n^{S_n}\cap L_n^{\prime }$ in the commutator ideal $L_n^{\prime }$ of $L_n$ is a right $K{[R_n]}^{S_n}$-module by the adjoint action. In this study, we provide a finite generating set for the right $K{[R_n]}^{S_n}$-module ${(L_n^{\prime })}^{S_n}$. In particular, we give free generating sets for ${(L_2^{\prime })}^{S_2}$ and ${(L_3^{\prime })}^{S_3}$ as $K{[R_2]}^{S_2}$-module and $K{[R_3]}^{S_3}$-module, respectively.

Let G be a group and let φ (G) be the least integer k such that G^(k)=G^(k+1). If no such k exists, then φ (G)=∞ and we write G ∈ 𝓤. We are interested in the questions which Coxeter groups are in 𝓤 and how large a finite number φ (G) can be for Coxeter groups G. In this paper, we give full answers in the 3- and 4-generator cases.

Let $\mathbb{F}$ be the underlying base field of characteristic $p\mathrm{>}3$ and denote by ${\mathrm{\Gamma }}_{\overline{0}}$ the even part of the finite-dimensional Lie superalgebra $\mathrm{\Gamma }$. We give the generator sets of the Lie algebra ${\mathrm{\Gamma }}_{\overline{0}}$. Using certain properties of the canonical tori of ${\mathrm{\Gamma }}_{\overline{0}}$, we describe explicitly the derivation algebra of ${\mathrm{\Gamma }}_{\overline{0}}$.

In this paper, we attempt to collect the knowledge on generators in the category Pos-$S$ and to apply this to proceed on the questions of homological classification of ordered monoids, that is results of the type: all generators in the category Pos-$S$, have a flatness property if and only if $S$ has a certain property.

Cyclic hypergroups are of great importance due to their applications to many fields in mathematics. In this paper, we classify all commutative single power cyclic hypergroups of order three and period two. Moreover, we prove some new interesting properties regarding cyclic hypergroups.

A model has been developed based on Colburn–Drew type formulation to analyze a vertical tube in tube stainless steel generator with forced convective boiling. Desorption of refrigerant vapor from refrigerant–absorbent solution takes place in the inner tube of the generator, when hot water through the annulus is used as heating medium. Simultaneous heat and mass transfer phenomena of desorption are described mathematically using the mass and energy balances, considering the heat and mass transfer resistances in liquid as well as vapor phases. Model equations are solved simultaneously by means of initial value problem solvers using explicit Runge–Kutta method with 4th order accuracy. A computer code has been developed in MATLAB to obtain the results. A parametric analysis has also been performed to study the effect of various parameters on the performance of the generator.