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For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

Let Lip (I) be the Banach algebra of all Lipschitz functions on the closed unit interval I with the norm ‖ f ‖ L = ‖ f ‖ ∞ + L (f) for f ∈ Lip (I) , where L(f) is the Lipschitz constant of f. We denote by C 1 (I , Lip (I)) the Banach algebra of all continuously differentiable functions F from I to Lip (I) equipped with the norm ‖ F ‖ Σ = sup s ∈ I ‖ F (s) ‖ L + sup t ∈ I ‖ D (F) (t) ‖ L for F ∈ C 1 (I , Lip (I)) . In this paper, we prove that if T is a surjective, not necessarily linear, isometry on C 1 (I , Lip (I)) , then T - T (0) is a weighted composition operator or its complex conjugation. Among other things, any surjective complex linear isometry on C 1 (I , Lip (I)) is of the following form: c 1 F (τ 1 (s) , τ 2 (x)) , where c 1 is a complex number of modulus 1, and τ 1 and τ 2 are isometries of I onto itself. [ABSTRACT FROM AUTHOR]

In this work, we introduce a nonstandard algebra of q-fermions where q is a nonzero complex deformation parameter for the algebra of the commuting fermions. In order to show that q-fermions provides a proper generalization of the algebra of usual commuting fermions, we prove that there is an inhomogeneous quantum structure associated with q-fermions for a complex number q with |q| = 1. [ABSTRACT FROM AUTHOR]

In recent years, Hopfield neural networks using Clifford algebra have been studied. Clifford algebra is also referred to as geometric algebra, and is useful to deal with geometric objects. There are three kinds of Clifford algebra with degree 2; complex, hyperbolic, and dual-numbered. Complex-valued Hopfield neural networks have been studied by many researchers. Several models of hyperbolic Hopfield neural networks have also been proposed. It has been difficult to construct dual-numbered Hopfield neural networks. In this work, we propose dual-numbered Hopfield neural networks by modification of hyperbolic Hopfield neural networks with the split activation function. The stability condition and Hebbian learning rule are also provided. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]

We describe transposed Poisson algebra structures on Block Lie algebras B (q) and Block Lie superalgebras S (q) , where q is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on B (q) are trivial whenever q ∉ Z , and for each q ∈ Z there is only one (up to an isomorphism) non-trivial transposed Poisson structure on B (q). The superalgebra S (q) admits only trivial transposed Poisson superalgebra structures for q ≠ 0 and two non-isomorphic non-trivial transposed Poisson superalgebra structures for q = 0. As a consequence, new Lie algebras and superalgebras that admit non-trivial Hom-Lie algebra structures are found. [ABSTRACT FROM AUTHOR]

Reduced biquaternion is a four dimensional hyper–complex number which is commutative algebra and an extension of complex. Due to this property, the corresponding fast algorithm is designed in time frequency analysis which can better fit the convolution theorem than the non-commutative quaternion. In addition, the reduced biquaternion can be interpreted as a single point in 2-dimensional hyperbolic space with its algebraic structure which has more capacity and variable ability than the Euclidean space. However, the algebra properties of the reduced biquaternion have not yet been well explored in the deep convolutional neural network. In this paper, we develop a new deep network structure, namely reduced biquaternion valued convolutional neural network (RQV-CNN). The proposed RQV-CNN includes basic modules of reduced biquaternion convolution layer and fully connection layer. Extensive experiments on color image classification and color image denoising are conducted to evaluate the promising performance of the RQV-CNN framework. The results show that RQV-CNN outperforms the real-valued CNN (RV-CNN), complex-valued CNN (CV-CNN), and quaternion-valued CNN (QV-CNN) with same structures. The source code can be found at https://github.com/tasteimage/RQVCNN. [ABSTRACT FROM AUTHOR]

In this paper we use the q-deformed binary operations to discuss q-derivative, q-integral, q-exponential function, q-Gamma function and q-logarithm and q-deformed complex number. We define the q-binary operation of q-matrix. Using these, we construct the q-boson algebra and suq(2) algebra based on the q-deformed binary operations. [ABSTRACT FROM AUTHOR]

Let sl N ˆ (C q) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q. In this paper, we first prove that for any complex number ℓ , the category of restricted sl N ˆ (C q) -modules of level ℓ is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ (ℓ , 0) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer ℓ , the integrable restricted sl N ˆ (C q) -modules of level ℓ are exactly the twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify irreducible graded twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

In many stability problems, the investigation of pure imaginary roots for a polynomial family is very important. Given a pure imaginary complex number, the set of all images of uncertainty vectors is called the value set corresponding to this pure imaginary complex number. The question whether these sets contain the origin is very important from robust stability point of view of a polynomial family. Cutoff frequency guarantees the noninclusion of the origin to the value set for large frequencies. In this paper, we give a procedure for more strict estimation of cutoff frequency and applications of the obtained result to the constant inertia problem of a polynomial family. [ABSTRACT FROM AUTHOR]

We study quantum current algebra A (R ‾) associated with the rational R -matrix of gl N and we give explicit formulae for the elements of its center at the critical level. Due to Etingof–Kazhdan's construction, the level c vacuum module V c (R ‾) for the algebra A (R ‾) possesses a quantum vertex algebra structure for any complex number c. We prove that any module for the quantum vertex algebra V c (R ‾) is naturally equipped with a structure of restricted A (R ‾) -module of level c and vice versa. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the following new additive s-functional inequalities ‖f(x - y) + f(y) + f(-x)‖ ≤ ‖s (f(x + y) - f(x) - f(y)) ‖; (0.1) ‖f(x + y) - f(x) - f(y)‖ ≤ ‖s (f(x - y) + f(y) + f(-x)) ‖; (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study some non-weight modules over the loop-Virasoro algebra L and Block type Lie algebras B(q) , where q is a nonzero complex number. Those modules when restricted to the Cartan subalgebra (modulo center) are free of rank one. We provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules. [ABSTRACT FROM AUTHOR]

We consider König’s method for finding roots of polynomials. Let K f , n be the function defined in König’s method for polynomial f of order n ( n ≥ 2 ) . For any given complex number λ ∈ C and any given set Ω ≔ { z 1 , z 2 , … , z k } of k ( k ≥ 2 ) distinct complex numbers, we present a procedure of constructing a polynomial f n , λ for which Ω is a k -cycle of K f n , λ , n with multiplier λ . [ABSTRACT FROM AUTHOR]

Let X and Y be locally compact Hausdorff spaces, where X is first-countable. Fix a positive integer n ⩾ 3 and a non-zero complex number λ. If a surjective map T : C0(X) → C0(Y ) satisfies the condition supy|Πnk+1T(fk)(y) + λ|= supϰ∊X|(Πnk=1 fk)(x) + λ| for all fk ∊ C0(X) (k = 1, . . . , n), then there exist a homeomorphism ϕ: Y → X, a continuous function w: Y → 핋, and a clopen set K ⊂ Y such that T(f) = w ⨰ (f ◦ ϕ) on K and T(f) = w ⨰ (f ◦ ϕ) on Y \ K for all f ∊ C0(X). Let A, B be function algebras on X, Y . Fix a positive integer n ⩾ 2. Suppose that X = Ch(A) and Y = Ch(B). If a surjective map T : A → B satisfies the condition σ л Πnk=1 T(fk)) ∩ σл(Πnk=1 fk)≠ø for all fk ∊ A (k = 1, . . . , n), then T is a weighted composition operator. [ABSTRACT FROM AUTHOR]

How the use of concrete materials in the form of counters allows students to model increasingly complex number problems is explored. The use of counters scaffolds students' development of number sense and algebra skills leading to an understanding of quite abstract concepts. [ABSTRACT FROM AUTHOR]

The Bohl algebra B is the ring of linear combinations of functions t e on the real line, where k is any nonnegative integer, and λ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of B (where we use the topology of uniform convergence) are infinite. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a new class of modules over the Block algebra B (q) , where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U (h) -free B (q) -modules of rank-1. [ABSTRACT FROM AUTHOR]