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This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result.

Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion.

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.

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 article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result. [ABSTRACT FROM AUTHOR]

Diversity is supposed to create better groups and societies but sometimes fails. It is explained why the power of diversity may not create better groups in the current diversity prediction theory. Diversity may hurt civic life and introduce distrust. This is because the current diversity prediction theory is based on real numbers that ignore individual abilities. Its diversity prediction theory maximizes performance with infinite population size. Contrary to this, collective intelligence or swarm intelligence is not maximized by infinite population size, but by population size. The extended diversity prediction theory using the complex number allows us to express individual abilities or qualities. The diversity of complex numbers always produces better groups and societies. The wisdom of crowds, collective intelligence, swarm intelligence or nature-inspired intelligence is implemented in the current machine learning or artificial intelligence, called Random Forest. The problem of the current diversity prediction theory is detailed in this paper. • Diversity is supposed to create better groups and societies but sometimes fails. • Diversity in real numbers ignores individual abilities. • Diversity of population size plays a key role in collective intelligence. • Diversity in complex numbers allows us to express individual abilities. • Diversity in complex numbers always produces better groups and societies. [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. [ABSTRACT FROM AUTHOR]

• The impact of loss functions on complex MRI reconstruction with CNNs was analyzed. • The mean squared loss is biased towards magnitude underestimations with phase error. • A newly-proposed unbiased loss function called ⊥ -loss yields improved performance. • ⊥ -loss can be applied to any high-dimensional problem defined in R 2. • ⊥ -loss was applied to image registration tasks, showing superior performance. Convolutional neural networks (CNNs) are increasingly adopted in medical imaging, e.g., to reconstruct high-quality images from undersampled magnetic resonance imaging (MRI) acquisitions or estimate subject motion during an examination. MRI is naturally acquired in the complex domain C , obtaining magnitude and phase information in k-space. However, CNNs in complex regression tasks are almost exclusively trained to minimize the L2 loss or maximizing the magnitude structural similarity (SSIM), which are possibly not optimal as they do not take full advantage of the magnitude and phase information present in the complex domain. This work identifies that minimizing the L2 loss in the complex field has an asymmetry in the magnitude/phase loss landscape and is biased, underestimating the reconstructed magnitude. To resolve this, we propose a new loss function for regression in the complex domain called ⊥ -loss, which adds a novel phase term to established magnitude loss functions, e.g., L2 or SSIM. We show ⊥ -loss is symmetric in the magnitude/phase domain and has favourable properties when applied to regression in the complex domain. Specifically, we evaluate the ⊥ + ℓ 2 -loss and ⊥ +SSIM-loss for complex undersampled MR image reconstruction tasks and MR image registration tasks. We show that training a model to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the L2 loss and results in similar performance compared to models trained to maximize the magnitude SSIM while offering high-quality phase reconstruction. Moreover, ⊥ -loss is defined in R n , and we apply the loss function to the R 2 domain by learning 2D deformation vector fields for image registration. We show that a model trained to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the end-point error loss. [ABSTRACT FROM AUTHOR]

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Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion. [ABSTRACT FROM AUTHOR]

Binarized neural networks, or BNNs, show great promise in edge-side applications with resource limited hardware, but raise the concerns of reduced accuracy. Motivated by the complex neural networks, in this paper we introduce complex representation into the BNNs and propose Binary complex neural network – a novel network design that processes binary complex inputs and weights through complex convolution, but still can harvest the extraordinary computation efficiency of BNNs. To ensure fast convergence rate, we propose novel BCNN based batch normalization and weight initialization strategies. Experimental results on image and radio signal classifications show that BCNN can achieve better accuracy compared to the original BNN models. BCNN improves BNN by strengthening its learning capability through complex representation and extending its applicability to complex-valued input data. Our code is available at https://github.com/flying-Yan/BCNN. [ABSTRACT FROM AUTHOR]