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Let r≥2 $r\ge 2$ be a real number. A complex nowhere‐zero r $r$‐flow on a graph G $G$ is an orientation of G $G$ together with an assignment φ:E(G)→C $\varphi :E(G)\to {\mathbb{C}}$ such that, for all e∈E(G) $e\in E(G)$, the Euclidean norm of the complex number φ(e) $\varphi (e)$ lies in the interval [1,r−1] $[1,r-1]$ and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph G $G$, denoted by ϕC(G) ${\phi }_{{\mathbb{C}}}(G)$, is the minimum of the real numbers r $r$ such that G $G$ admits a complex nowhere‐zero r $r$‐flow. The exact computation of ϕC ${\phi }_{{\mathbb{C}}}$ seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of ϕC ${\phi }_{{\mathbb{C}}}$ is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for ϕC(G) ${\phi }_{{\mathbb{C}}}(G)$ in terms of the odd‐girth of a cubic graph G $G$ (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph Wn ${W}_{n}$. In particular, we show that for every odd n $n$, the value of ϕC(Wn) ${\phi }_{{\mathbb{C}}}({W}_{n})$ arises from one of three suitable configurations of points in the complex plane according to the congruence of n $n$ modulo 6. [ABSTRACT FROM AUTHOR]

A new approach for implementing the fast Fourier transform (FFT) using complex binary number system (CBNS) is presented in this paper. The advantage of using CBNS is that instead of processing the real and imaginary parts of a complex number separately, we can process both as a single entity. A total of four FFT architectures have been developed, which are 16‐, 64‐, 256‐, and 1024‐point FFT architectures. The proposed FFT is designed using Verilog‐HDL and implemented using a Virtex‐7 field‐programmable gate array (FPGA). The maximum clock rate achieved by the proposed FPGA‐based design is 376.1 MHz. The FPGA resource utilization of the FFT is less than the state‐of‐the‐art FFT designs. An application‐specific integrated circuit (ASIC) for the FFT is also designed. The normalized energy per FFT of the proposed ASIC is 0.14 nJ, and the normalized area per FFT of the ASIC is 1.08 mm2 which is less than other existing designs. [ABSTRACT FROM AUTHOR]

From the perspective of rock mechanics, transmission of fluid pressure that drives groundwater flow is not instantaneous; instead it is a slow compressional (P) wave in porous rocks saturated with a fluid. In this paper, the approach of monochromatic wave is used to analyze the slow P‐wave. Permeability is derived as a function of the frequency of the wave. The permeability appears to be a complex number which asymptotes to Darcy permeability at the low frequency end but tends to vanish at the high frequency end. With Biot theory, the permeability predicts phase velocity and the quality factor of the monochromatic wave. Using Berea sandstone as an illustrative example, the complex‐number permeability can yield a lower phase velocity (within frequency of 105–108 Hz) and a larger attenuation (for frequency above 106 Hz) than Darcy permeability does. This study extends the recent research of the low‐frequency waves of fluid pressure to the regime of very high frequency. [ABSTRACT FROM AUTHOR]

Representing one of the last populations of an IUCN Critically Endangered subspecies, estimates suggest the Kordofan giraffe of Cameroon's Bénoué Complex number fewer than 300 individuals, threatened by poaching and habitat degradation. This study investigated the viability of Bénoué National Park's Kordofan giraffe and efficacy of available interventions through a population viability analysis using VORTEX. The relative impacts of anti‐poaching activity, population supplementation, habitat protection and vaccination were explored, alongside a sensitivity analysis investigating parameter uncertainty. The baseline model projects a 78.6% probability of extinction (PE) within 100 years. Poaching of just two individuals every 5 years results in a 98.1% PE, highlighting anti‐poaching activity as a vital intervention. Movement of a female‐biased group into the park proved effective, with supplementation of 24 females reducing the PE by over 35%, sustaining a raised population size above the baseline for over 60 years on average. Annual carrying capacity reduction of 2% year−1 for 20 years resulted in a 97.8% PE. Conversely, habitat restoration had little consequence unless combined with further interventions. These findings underscore the magnitude of the threat facing Bénoué National Park's Kordofan giraffe. Conservation management should prioritise strengthening existing anti‐poaching activity in conjunction with protecting wildlife corridors to aid dispersal. [ABSTRACT FROM AUTHOR]

Unlike real or complex numbers, quaternion multiplication does not satisfy the commutative law. This makes it impossible to generalize some conclusions on semi‐tensor product to quaternion directly, which is valid on real or complex number fields. In this paper, we define semi‐tensor product of quaternion matrices and study the properties of semi‐tensor product of quaternion matrices, and the conclusion is applied to the color digital image restoration model. [ABSTRACT FROM AUTHOR]

The development of algorithmic differentiation (AD) tools focuses mostly on handling floating point types in the target language. Taping optimizations in these tools mostly focus on specific operations like matrix vector products. Aggregated types like std::complex are usually handled by specifying the AD type as a template argument. This approach provides exact results, but prevents the use of expression templates. If AD tools are extended and specialized such that aggregated types can be added to the expression framework, then this will result in reduced memory utilization and improve the timing for applications where aggregated types such as complex number or matrix vector operations are used. Such an integration requires a reformulation of the stored data per expression and a rework of the tape evaluation process. We will demonstrate the overheads on a synthetic benchmark and show the improvement when aggregated types are handled properly by the expression framework of the AD tool. [ABSTRACT FROM AUTHOR]

It has been reported that training deep neural networks is more difficult than training shallow neural networks. Hinton et al. proposed deep belief networks with a learning algorithm that trains one layer at a time. A much better generalization can be achieved when pre‐training each layer with an unsupervised learning algorithm. Since then, deep neural networks have been extensively studied. On the other hand, it has been revealed that singular points affect the training dynamics of the learning models such as neural networks and cause a standstill of training. Naturally, training deep neural networks suffer singular points. As described in this paper, we present a deep neural network model that has fewer singular points than the usual one. First, we demonstrate that some singular points in the deep real‐valued neural network, which is equivalent to a deep complex‐valued neural network, have been resolved as its inherent property. Such deep neural networks are less likely to become trapped in local minima or plateaus caused by critical points. Results of experiments on the two spirals problem, which has an extreme nonlinearity, support our theory. Copyright © 2017 John Wiley & Sons, Ltd. [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]

In this paper, we further Meirong Zhang et al.'s work by computing the number of weighted eigenvalues for Sturm–Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm–Liouville equation with a general integrable potential, if its weight is a positive linear combination of n Dirac delta functions, then it has at most n (may be less than n, or even be 0) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly n. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm–Liouville problem with Dirac weights and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichlet problems for Sturm–Liouville equations involving single Dirac distribution weights is studied. [ABSTRACT FROM AUTHOR]

The key problem in locating a source of partial discharge (PD) using the acoustic emission technique is the error in estimating the signal arrival time from the source to the multiple sensors. When the time difference of arrival approach is used for the PD‐source‐localisation, some of the measured time‐delay‐groups will have solution for the time‐difference equation in the complex‐number‐field. This is due to the significant error in the arrival time estimation. Such time‐delay‐groups should be considered to be invalid. In this study, a function is proposed for identifying the invalid time‐delay‐groups for the fixed set of coordinates of four sensors and the specific velocity of the acoustic signal used. The negative sign of this function value indicates that the solution is in the complex‐number‐field. An alternative method for identifying the invalid time‐delay‐groups is by using Newton's method. The multiple sign changes of the Jacobian‐determinant in the iterations of Newton's method shows that the solution is in the complex‐number‐field. The proposed methods have been tested with data from the existing literature, and results have confirmed the efficacy of these methods in the identification of the invalid time‐delay‐groups. Discarding such groups of time delays improves the accuracy of statistical PD‐source‐localisation. [ABSTRACT FROM AUTHOR]

The key problem in locating a source of partial discharge (PD) using the acoustic emission technique is the error in estimating the signal arrival time from the source to the multiple sensors. When the time difference of arrival approach is used for the PD-source-localisation, some of the measured time-delay-groups will have solution for the time-difference equation in the complex-number-field. This is due to the significant error in the arrival time estimation. Such time-delay-groups should be considered to be invalid. In this study, a function is proposed for identifying the invalid time-delay-groups for the fixed set of coordinates of four sensors and the specific velocity of the acoustic signal used. The negative sign of this function value indicates that the solution is in the complex-number-field. An alternative method for identifying the invalid time-delay-groups is by using Newton's method. The multiple sign changes of the Jacobian-determinant in the iterations of Newton's method shows that the solution is in the complex-number-field. The proposed methods have been tested with data from the existing literature, and results have confirmed the efficacy of these methods in the identification of the invalid time-delay-groups. Discarding such groups of time delays improves the accuracy of statistical PD-source-localisation. [ABSTRACT FROM AUTHOR]

Total body irradiation (TBI) is a complex treatment technique, which has been slow to transition to a three-dimensional (3D) planning approach. There is limited literature available providing a detailed description on methods to plan TBI on a 3D planning system. 3D planning using the modulated arc TBI (MATBI) technique is a complex process involving a significant number of quality assurance processes and scripts, due to more than 40 treatment beams and two patient positions. This article will focus on the workflow and technical planning aspects of our institution's MATBI technique and identify reasons for modifications made to the developing institution's original MATBI approach. Included is a description of specific simulation equipment, detailed explanation of the four-stage computing process including the role of scripting to standardise and streamline what is otherwise a complex number of steps. The information provided is specific to one centre's approach but shows the fundamental planning process and demonstrates a streamlined method, which can be adapted to other planning systems. Overall, the ability to accurately represent the TBI technique in 3D on a planning system will be shown. [ABSTRACT FROM AUTHOR]

The split and hyperbolic (countercomplex) octonions are eight‐dimensional nonassociative algebras over the real numbers, which are in the form X=x0e0+∑j=17xjej, where em's 0≤m≤7 have different properties for them. The main purpose of this paper is to define the split‐type octonion and its matrix whose inputs are split‐type octonions and give some properties for them by using the real quaternions, split, and hyperbolic (countercomplex) octonions. On the other hand, to make some definitions, we present some operations on the split‐type octonions. Also, we show that every split‐type octonions can be represented by 2 × 2 real quaternion matrix and 4 × 4 complex number matrix. The information about the determinants of these matrix representations is also given. Besides, the main features of split‐type octonion matrix concept are given by using properties of  real quaternion matrices. Then, 8n × 8nreal matrix representations of split‐type octonion matrices are shown, and some algebraic structures are examined. Additionally, we introduce real quaternion adjoint matrices of split‐type octonion matrices. Moreover, necessary and sufficient conditions and definitions are given for split‐type octonion matrices to be special split‐type octonion matrices. We describe some special split‐type octonion matrices. Finally, oct‐determinant of split‐type octonion matrices is defined. Definitive and understandable examples of all definitions, theorems, and conclusions were given for a better understanding of all these concepts. [ABSTRACT FROM AUTHOR]

Aim: This study aimed to examine the relationship between blood coagulability and sense of burden among caregivers of patients with senile dementia of the Alzheimer type. Methods: A cross‐sectional study was carried out involving healthy older caregivers who lived with their patients with senile dementia of the Alzheimer type. We evaluated the Zarit Burden Interview score, levels of von Willebrand factor antigen, D‐dimer, thrombin–antithrombin III complex, tissue plasminogen activator/plasminogen activator inhibitor type 1 complex, number of chronic diseases, body mass index and number of medications. A linear regression model was used to estimate adjusted associations. Results: Thrombin–antithrombin III complex levels were higher in female caregivers than in male caregivers (P = 0.07). Headaches were significantly more frequent in female caregivers than in male caregivers, as assessed by a visual analog scale (P < 0.01). The number of chronic diseases and body mass index were positively associated with levels of tissue plasminogen activator/plasminogen activator inhibitor type 1 complex (P < 0.05). Similarly, the number of medications was positively associated with levels of D‐dimer (P < 0.05). However, the Zarit Burden Interview score was not associated with blood coagulability (P > 0.05). Conclusions: The present study found that the number of chronic diseases and body mass index were associated with blood coagulability, and that female caregivers were more prone to headaches and higher blood coagulability than male caregivers. These findings highlight the essential nature of health management during caregiving. The impact of caregiver burden on blood coagulability is likely to differ depending on the long‐term or short‐term psychological stress associated with caregiving conditions. Geriatr Gerontol Int 2019; 19: 804–808. [ABSTRACT FROM AUTHOR]

Summary: In recent years, our understanding of the complex number of signals that need to be integrated between a diverse number of receptors present on natural killer (NK) cells and ligands present on target cells has improved. Here, we review the progress made in identifying interactions between dengue viral peptides presented on HLA Class 1 molecules with inhibitory and activating killer‐like immunoglobulin receptors on NK cells, direct interactions of viral proteins with NK cell receptors, the involvement of dengue virus‐specific antibodies in mediating antibody‐dependent cell‐mediated cytotoxicity and the role of soluble factors in modulating NK cell responses. We discuss findings of NK cell activation early after natural dengue infection, and point to the role that NK cells may play in regulating both innate and adaptive immune responses, in the context of our new appreciation of interactions of dengue virus with specific NK cell receptors. With a number of flavivirus vaccine candidates in clinical trials, how NK cells respond to attenuated dengue virus and subunit protein vaccine candidates and shape adaptive immunity will need to be considered. [ABSTRACT FROM AUTHOR]

This paper presents a completely new method for the calculation of expectations (and thus joint probability distributions) of structure factors or phase invariants. As an example, a first approximation of the expectation of the triplet invariant (up to a constant) is given and a complex number is obtained. Instead of considering the atomic vector positions or reciprocal vectors as the fundamental random variables, the method samples over all functions (distributions) with a given number of atoms and given Patterson function. The aim of this paper was to explore the feasibility of the method, so the easiest problem was chosen: the calculation of the expectation value of the triplet invariant in P1. Calculation of the joint probability distribution of the triplet is not performed here but will be done in the future. [ABSTRACT FROM AUTHOR]