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Flipped Learning is one of the alternatives of teaching and learning approach in mathematics classroom. The objective of this study is exploring students’ conceptual understanding about complex number using flipped learning with handout. The subject of the study are the students in 5th semester students of mathematics education department in 2019/2020. The study used qualitative approach to describe the implementation of flipped learning. There are 31,6% of 19 respondents give score very satisfied. This result then observed by using the test with all the students understand with the definition of complex numbers. Besides they can adapt their learning activity using flipped learning with complex analysis handout. As legibility aspect of the handout, there are 52,6% of the respondents gives score satisfied and 26,3% of the respondents are very satisfied. The score indicates that the flipped learning with handout helps students to understand about the complex number concepts.

Based on the novel prescription for the power function, (x+iy)m, the new expression for Ψ(x,y|m), the eigenfunction of the operator of the third component of the angular momentum, M^z, is presented. These functions are normalizable, single valued and, distinct to the traditional presentation, (x+iy)m=ρmeimϕ, are invariant under the rotations at 2π for any, not necessarily integer, m—the eigenvalue of M^z. For any real m the functions Ψ(x,y|m) form an orthonormal set, therefore they may serve as a quantum mechanical eigenfunction of M^z. The eigenfunctions and eigenvalues of the angular momentum operator squared, M^2, derived for the two different prescriptions for the square root, (m2)1/2, (m2)1/2=|m| and (m2)1/2=±m, are reported. The normalizable eigenfunctions of M^2 are presented in terms of hypergeometric functions, admitting integer as well as non-integer eigenvalues. It is shown that the purely integer spectrum is not the most general solution but is just the artifact of a particular choice of the Legendre functions as the pair of linearly independent solutions of the eigenvalue problem for the M^2.

This article identifies the general form of matrix power and trace of an integer power of Special Form of the 3 × 3 Complex Circulant matrix. The research begins to determine the general form of an integer power of Special Form of the 3 × 3 Complex Circulant matrix, followed by determining the general form trace of an integer power of Special Form of the 3 × 3 Complex Circulant matrix. The proof is done by using mathematical induction. The final result of this article is to obtain the general form of the matrix A^n and tr(A^n) for n integers in special form of the 3 × 3 complex Circulant matrix.

In this perspective review, we discuss the power of polarized Raman spectroscopy to study optically anisotropic 2D materials, belonging to the orthorhombic, monoclinic and triclinic crystal families. We start by showing that the polarization dependence of the peak intensities is described by the Raman tensor that is unique for each phonon mode, and then we discuss how to determine the tensor elements from the angle-resolved polarized measurements by analyzing the intensities in both the parallel- and cross-polarized scattering configurations. We present specific examples of orthorhombic black phosphorus and monoclinic 1T0-MoTe2, where the Raman tensors have null elements and their principal axes coincide with the crystallographic ones, followed by a discussion on the results for triclinic ReS2 and ReSe2, where the axes of the Raman tensor do not coincide with the crystallographic axes and all elements are non-zero. We show that the Raman tensor elements are, in general, given by complex numbers and that phase differences between tensor elements are needed to describe the experimental results. We discuss the dependence of the Raman tensors on the excitation laser energy and thickness of the sample within the framework of the quantum model for the Raman intensities. We show that the wavevector dependence of the electron-phonon interaction is essential for explaining the distinct Raman tensor for each phonon mode. Finally, we close with our concluding remarks and perspectives to be explored using angle-resolved polarized Raman spectroscopy in optically anisotropic 2D materials. [ABSTRACT FROM AUTHOR]

In this study, we generalize the well-known formulae of De-Moivre and Euler of complex numbers to dual-complex numbers. Furthermore, we investigate the roots and powers of a dual-complex number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.

Youth employment in labour force has a critical importance in socio-economic planning. It is expected that the use of dynamic and able-bodied work force can increase the quality of the industrial products and it is also necessary to strengthen the economy. More importantly, the contribution to labour force has a crucial importance for public welfare. At this stage, since the role of young women in the total labour force is not considered sufficiently, this study concentrates on the determination of the relative effect of women labour force. For this purpose, first the youth employment is considered as a complex quantity; along with the real component (young men), the contribution of young women is treated as the imaginary component. By using the data derived from 20 different domains (sectors), the problem is stated as a complex value problem and a measurement uncertainty analysis is utilized. The measurement uncertainty of the complex quantity (employment) is expressed by a region in the complex plane. Finally, a confidence ellipse at 95% confidence level is produced. The phase diagrams produced by statistical learning have provided some abnormalities and also potentials.

The concept of unity roots plays a central role in the theory of field extensions and polynomials roots' computing. The objective of this paper is to find the algebraic formula for computing the symbolic 2-plithogenic and 3-plithogenic complex roots of unity, where a general formula will be provided with many related examples up to the exponent 3. [ABSTRACT FROM AUTHOR]

The computation of the arctangent of a complex number, i.e., the atan2 function, is frequently needed in hardware systems that could profit from an optimized operator. In this brief, we present a novel method to compute the atan2 function and a hardware architecture for its implementation. The method is based on a first stage that performs a coarse approximation of the atan2 function and a second stage that improves the output accuracy by means of a lookup table. We present results for fixed-point implementations in a field-programmable gate array device, all of them guaranteeing last-bit accuracy, which provide an advantage in latency, speed, and use of resources, when compared with well-established fixed-point options. [ABSTRACT FROM PUBLISHER]

With the increase of the customer demands for the products, multidisciplinary products are gradually becoming more competitive than the traditional products. Considering conceptual design synthesis is the core phase of the product design and its result is the starting point of the next detail design works, if the rich resources in multiple disciplinary domains can be fully used during this phase, the efficiency of multidisciplinary product design will be largely promoted. Therefore, we proposed a novel conceptual design synthesis system for multidisciplinary products, which consists of three parts, i.e. basic framework, complex-number-domain-based mathematical model, and achieving approach. In the framework, three groups of concepts were defined and the conceptual design synthesis was concluded into three steps. The model can completely describe and modularize the function units which are the basic components of a design scheme. Based on the model, the achieving approach can automatically generate stable objective function unit chains which can directly construct the design scheme. Finally, the design of a multifunctional agricultural hydroelectric power system was taken as an illustration to prove the practicability of this proposed system. [ABSTRACT FROM AUTHOR]

This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent’s subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton’s dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent’s decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement.

The paper deals with a new fuzzy version of the Lee-Carter (LC) mortality model, in which mortality rates as well as parameters of the LC model are treated as triangular fuzzy numbers. As a starting point, the fuzzy Koissi-Shapiro (KS) approach is recalled. Based on this approach, a new fuzzy mortality model -- CNMM -- is formulated using orthonormal expansions of the inverse exponential membership functions of the model components. The paper includes numerical findings based on a case study with the use of the new mortality model compared to the results obtained with the standard LC model. [ABSTRACT FROM AUTHOR]

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) where such that $${-\infty < \mathfrak{p} < \infty}$$ . The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in $${\mathbb{C}_{J}}$$ is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system. [ABSTRACT FROM AUTHOR]

In this study, we firstly give the basic notations of the generalized complex number plane ( $${\mathfrak{p}}$$ -complex plane) $${\mathbb{C}_{\mathfrak{p}}}$$ . Then, we introduce the one-parameter planar motions in $${\mathfrak{p}}$$ -complex plane $${\mathbb{C}_{J}}$$ such that $${\mathbb{C}_{J} \subset \mathbb{C}_{\mathfrak{p}}}$$ . These motions correspond the one-parameter motions in affine Cayley-Klein planes []. We examine this motion theory with aspects of complex motions as given in []. Besides, we discuss the relations between absolute, relative, sliding velocities (accelerations) and pole curves under the motions $${\mathbb{C}_{J}/{\mathbb{C}_{J}^{\prime}}}$$ . [ABSTRACT FROM AUTHOR]

Since the product of complex numbers and rectangular fuzzy complex numbers (RFCN) is not necessarily a RFCN in the former fuzzy complex linear system (FCLS), the scalar multiplication and addition operations of complex numbers and fuzzy complex numbers (FCN) based on a new representation of FCN are proposed. We also introduce a new method for solving FCLS, which can convert FCLS into two distinct linear systems. One is an n×n complex linear system, and the other is an (mn)×(mn) real linear system, where n is the number of unknown variables, and m is the number of substitutional cyclic sets composed of coefficients of FCLS. In particular, using this method to solve one-dimensional fuzzy linear systems, a (2n)×(2n) RLS is obtained, which is consistent with Friedman’s method. Finally, FCLS based on the RFCN as a special case are also investigated.

The concept of an ambiguous set (AS) was initially introduced by Singh et al. [P. Singh, Y.-P. Huang and T.-T. Lee — A novel ambiguous set theory to represent uncertainty and its application to brain MR image segmentation, in Proc. IEEE Int. Conf. Systems, Man and Cybernetics (SMC), Bari, Italy, 2019, pp. 2460–2465]. Can we examine the possibility of expanding it to introduce ambiguous complex numbers? This study aims to answer this question by first introducing the concept of an AS and then providing a set of formulas that aid in determining the membership degrees of the set. Then, this study expands the idea of AS to a single-valued ambiguous number (SVAN). Based on SVAN, the concept of single-valued ambiguous complex number (SVACN) is illustrated. Following that, various theories and properties of SVACN are addressed in terms of their complex conjugate, polar, exponential forms, and roots. Furthermore, the study proposes an algorithm tailored to generating the Mandelbrot set using varying numbers of SVACNs. Additionally, this approach furnishes directional vectors for the SVACNs within a two-dimensional plane. The study also deliberates on the manifold advantages of this methodology, particularly emphasizing its capacity to provide directional guidance for SVACNs. [ABSTRACT FROM AUTHOR]

It plays an important role in the calculation of fuzzy numbers using some special fuzzy numbers to approximate general fuzzy numbers in fuzzy analysis, similar to fuzzy complex analysis. A method is proposed to seek an approximation operator that produces a special kind of fuzzy complex number which is the nearest one to a given fuzzy complex number in this paper. The concept of prismoid fuzzy complex number is introduced, and the nearest prismoid fuzzy complex number of fuzzy complex number which preserve the centroid of the core for the fuzzy complex number with respect to the weighted pseudometric are discussed. Finally, some properties of the approximation operator including scale invariance, identity and nearness are presented. [ABSTRACT FROM AUTHOR]

The paper presents a square-root-extended complex Kalman filter (SRECKF) by decomposing covariance matrix with its square-root forms to improve stability of the filter for estimating complex number. αβ transformation is used to map three-phase instantaneous voltages in the abc phases into instantaneous voltages on the αβ axes, and a non-linear state equation and observation equation of the three-phase voltages are built by introducing a complex vector and defining state variables. Positive symmetrical component, negative symmetrical components, and frequency of the three-phase voltages are estimated using traditional extended complex Kalman filter (ECKF), the estimation results show that the method proposed here are superior to traditional extended complex Kalman filter on estimation accuracy and convergence rate.

This paper aims to bring together quaternions and generalized complex numbers. Generalized quaternions with generalized complex number components are expressed and their algebraic structures are examined. Several matrix representations and computational results are introduced. An alternative approach for a generalized quaternion matrix with elliptic number entries has been developed as a crucial part. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study for the first time the concept of square real matrices with symbolic 2-plithogenic split-complex entries. Many of their algebraic properties will be discussed and handled, where we find the formula of computing inverses, exponents, and powers of these matrices by building a ring isomorphism between the ring of split-complex symbolic 2-plithogenic matrices and the direct product of the symbolic 2-plithogenic matrices with itself. Also, we give the interested reader many related examples to clarify the validity of our work. [ABSTRACT FROM AUTHOR]

In this article, for a given natural number n, we derive a closed form for the series 埜X k=1 s,kn) in terms of Hurwitz zeta functions. [ABSTRACT FROM AUTHOR]

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

Complex radix (−1 + j) allows the arithmetic operations of complex numbers to be done without treating the divide and conquer rules, which offers the significant speed improvement of complex numbers computation circuitry. Design and hardware implementation of complex radix (−1 + j) converter has been introduced in this paper. Extensive simulation results have been incorporated and an application of this converter towards the implementation of discrete Fourier transformation (DFT) processor has been presented. The functionality of the DFT processor have been verified in Xilinx ISE design suite version 14.7 and performance parameters like propagation delay and dynamic switching power consumption have been calculated by Virtuoso platform in Cadence. The proposed DFT processor has been implemented through conversion, multiplication and addition. The performance parameter matrix in terms of delay and power consumption offered a significant improvement over other traditional implementation of DFT processor.

In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha.

Let P(z) be a polynomial of degree n and for any complex number α, let DαP(z) = nP(z) + (α − z)P ′ (z) denote the polar derivative of P(z) with respect to a complex number α. In this paper, we prove some Lr inequalities for the polar derivative of a polynomial have all zeros in |z| ≤ 1. Our theorem generalizes a result of Dewan and Mir [K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interd. Math. 10 (2007), no. 4, 525–531] and includes as special cases several interesting many known results [ABSTRACT FROM AUTHOR]

In this study, a different perspective was brought to Narayana sequences and one-, two-, three- and n --dimensional recurrence relations of these sequences were created. Then, some identities ranging from one to n --dimensions of these recurrences were created. [ABSTRACT FROM AUTHOR]

This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z + c), where j, k are integers with 0 ≤ j < k. In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a, f) the set of all zeros of f-a, where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement: Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j, k be integers such that 0 ≤ j < k. If f(j)(z) and f(k)(z + c) have the same a-points for a finite value a(̸= 0) and satisfy conditions... [ABSTRACT FROM AUTHOR]

This paper is an extension of the existing works on the frequency-domain-based bit flipping control strategy for stabilizing the single-bit high-order interpolative sigma delta modulator. In particular, this paper proposes the implementation and performs the performance evaluation of the control strategy. For the implementation, a frequency detector is used to detect the resonance frequencies of the input sequence of the sigma delta modulator. Then, a neural-network-based controller is used for finding the solution of the integer programming problem. Finally, the buffers and the combinational logic gates as well as an inverter are used for implementing the proposed control strategy. For the performance evaluation, the stability region in terms of the input dynamical range is evaluated. It is found that the control strategy can significantly increase the input dynamical range from 0.24 to 0.58. Besides, the control strategy can be applied to a wider class of the input signals compared to the clipping method.

Copyright of System Research & Information Technologies / Sistemnì Doslìdžennâ ta Ìnformacìjnì Tehnologìï is the property of Institute for the Applied System Analysis at the NTUU KPI and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Generation of hidden attractor with complicated phase portrait in chaotic system with no equilibrium has presented a new research focus in the past decade. However, the existing approaches usually follow the rule that you reap what you sow, i.e., taking an no-equilibrium chaotic system as the seed. In this paper, a novel approach to the generation of no-equilibrium multi-fold hidden attractors is presented. The offset boosting and the operation of complex number are applied to several cases, including the seed chaotic systems with no equilibrium, with single unstable node, or with single stable node-focus. In specific, the offset boosting shifts the single non-zero equilibrium to the origin, then the operation of complex number rules out the equilibrium in the folded system and generates multi-fold trajectories. The highlight is that the proposed approach is not limited to the no-equilibrium seed system, but also applicable for different types of single-equilibrium seed systems, even the seed system with a single line of equilibrium. The detailed numerical simulations and experiments confirmed the feasibility of the proposed approach. Finally, an image encryption algorithm and its practical microcontroller-based implementation are presented to support the potential application. • Systematic approach for folding trajectory of chaotic system. • New multi-fold chaotic hidden attractors with no equilibrium. • New chaos-based encryption algorithm for image security. • Dynamic memory scheme for microcontroller implementation. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Xinjiang University (Natural Science Edition) is the property of Xinjiang University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Given a bounded operator Q on a Hilbert space H , a pair of bounded operators (T 1 , T 2) on H is said to be Q-commuting if one of the following holds: T 1 T 2 = Q T 2 T 1 or T 1 T 2 = T 2 Q T 1 or T 1 T 2 = T 2 T 1 Q. We give an explicit construction of isometric dilations for pairs of Q-commuting contractions for unitary Q, which generalizes the isometric dilation of Ando (Acta Sci Math (Szeged) 24:88–90, 1963) for pairs of commuting contractions. In particular, for Q = q I H , where q is a complex number of modulus 1, this gives, as a corollary, an explicit construction of isometric dilations for pairs of q-commuting contractions, which are well studied. There is an extended notion of q-commutativity for general tuples of operators and it is known that isometric dilation does not hold, in general, for an n-tuple of q-commuting contractions, where n ≥ 3 . Generalizing the class of commuting contractions considered by Brehmer (Acta Sci Math (Szeged) 22:106–111, 1961), we construct a class of n-tuples of q-commuting contractions and find isometric dilations explicitly for the class. [ABSTRACT FROM AUTHOR]

Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

Objective Image representation has always been a basic problem in image processing. This paper studies the approach of a digital image expressed as a complex number. The new approach has been applied to image disguise, and the result is satisfying.Method A one-to-one correspondence established between an image and a point in a complex plane based on complex bases under positional number system notation. The relationships between different images can be converted into a plain vector operation, which helps image study using geometric method. Thus, a new image disguise method is realized.Result This image disguise algorithm can not only disguise one or many images, but also allows an image to be disguised many times.Conclusion Experimental results show that our method is numerically stable, can accurately restore secret images in an ideal transmission environment, and can restore secret images with some satisfaction even in a noisy transmission environment. [ABSTRACT FROM AUTHOR]

The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced. [ABSTRACT FROM AUTHOR]

The goal of this paper is to define for the first time the concept of symbolic 2-plithogenic weak fuzzy complex number as new generalization generated by combining real numbers with symbolic 2-plithogenic numbers. We study the elementary properties of this new class such as Invertibility and nilpotency, with many related examples that explain its novelty. [ABSTRACT FROM AUTHOR]

Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given. [ABSTRACT FROM AUTHOR]

Introduction: Multiply-accumulate operation is the most fundamental operation in digital signal processing for image processing, robotics and automatic control. In this paper, a novel architecture of dynamically reconfigurable fused multiply-adder (FMA) is proposed. Methods: Dynamic reconfiguration is a method that can change the circuit configuration without stop of operation. The proposed circuit provides the following four calculation modes by dynamic reconfiguration: (1) complex number FMA mode, (2) real number FMA mode, (3) complex number parallel calculation mode, and (4) real number parallel calculation mode. The data format is single precision floating point format based on IEEE754. The proposed circuit was designed using Verilog-HDL. It was simulated by logic circuit simulator, and implemented on FPGA. Result: As a result of circuit synthesis, we confirmed the reduction of resource in the proposed circuit. Furthermore, we confirmed proper result for each calculation mode by logic simulation and experiment on FPGA. Conclusion: The proposed circuit provides the four calculation modes by dynamic reconfiguration. We confirmed the reduction of resource and proper result for each calculation mode. [ABSTRACT FROM AUTHOR]

Split-complex (hyperbolic) numbers are ordered pairs of real numbers, written in the form x + jy with j² = 1, used to describe the geometry of the Lorentzian plane. Since a null split-complex number does not have an inverse, some methods to calculate the exponential of complex matrices are not valid for split-complex matrices. In this paper, we examined the exponential of a 2 × 2 split-complex matrix in three cases: i. Δ = 0, ii. Δ ̸= 0 and Δ is not null split-complex number, iii. Δ ̸= 0 and Δ is a null split-complex number where Δ = (trA)2 - 4 detA. [ABSTRACT FROM AUTHOR]

Let ζ = (ζn) be an arbitrary complex sequence such that 0 < |ζ1| ≤ |ζ2| ≤... and ζn → ∞ as n → ∞, let nζ(r) and Nζ(r) be the counting function and the integrated counting function of this sequence, respectively. By Eζ we denote the class of all entire functions whose zeros are precisely the ζn, where a complex number that occurs m times in the sequence ζ corresponds to a zero of multiplicity m. Suppose that Φ is a convex function on R such that Φ(σ)/σ → +∞ as σ → +∞. It is proved that there exists an entire function f ∈ Eζ such that... where Mf (r) denotes the maximum modulus of the function f, and it is shown that the above inequality implies the inequality... The formulated result is a consequence of the following more general statement: if the righthand derivative Φ′+ of the function Φ assumes only integer values and... then there exists an entire function f ∈ Eζ such that lnMf(r) = o(eΦ(ln r)) as r → +∞. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]