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The topology method is an algorithm for accurate estimation of instantaneous heartbeat intervals using millimeter-wave radar signals. In this model, feature points are extracted from the skin displacement waveforms generated by heartbeats and a complex number is assigned to each feature point. However, these numbers have been assigned empirically and without solid justification. This study used a simplified model of displacement waveforms to predict the optimal choice of the complex number assignments to feature points corresponding to inflection points, and the validity of these numbers was confirmed using analysis of a publicly available dataset., Comment: 5 pages, 3 figures. This work is going to be published in IEEE Sensors Letters

This document describes an algorithm to scale a complex vector by the reciprocal of a complex value. The algorithm computes the reciprocal of the complex value and then scales the vector by the reciprocal. Some scaling may be necessary due to this 2-step strategy, and the proposed algorithm takes scaling into account. This algorithm is supposed to be faster than the naive approach of dividing each entry of the vector by the complex value, without losing much accuracy. It also serves as a single strategy for scaling vectors by the reciprocal of a complex value, which improves the software maintainability., Comment: 7 pages, no figures

We improve Mahler's lower bound for the Mahler measure in terms of the discriminant and degree for a specific class of polynomials: complex monic polynomials of degree $d\geq 2$ such that all roots with modulus greater than some fixed value $r\geq1$ occur in equal modulus pairs. We improve Mahler's exponent $\frac{1}{2d-2}$ on the discriminant to $\frac{1}{2d-3}$. Moreover, we show that this value is sharp, even when restricting to minimal polynomials of integral generators of a fixed not totally real number field. An immediate consequence of this new lower bound is an improved lower bound for integral generators of number fields, generalising a simple observation of Ruppert from imaginary quadratic to totally complex number fields of arbitrary degree., Comment: To appear in Acta Arithmetica

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(\psi)$ of complex numbers which are well approximated with the given bound $\psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $\psi$ and its Hausdorff dimension is equal to that of the $\psi$-approximable set $W(\psi)$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarn\'ik Theorem in real case.

In this paper, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present CPL-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex semidefinite programming to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

Computational prediction of origin of replication (ORI) has been of great interest in bioinformatics and several methods including GC-skew, auto-correlation etc. have been explored in the past. In this paper, we have extended the auto-correlation method to predict ORI location with much higher resolution for prokaryotes and eukaryotes, which can be very helpful in experimental validation of the computational predictions. The proposed complex correlation method (iCorr) converts the genome sequence into a sequence of complex numbers by mapping the nucleotides to {+1,-1,+i,-i} instead of {+1,-1} used in the auto-correlation method (here, i is square root of -1). Thus, the iCorr method exploits the complete spatial information about the positions of all the four nucleotides unlike the earlier auto-correlation method which uses the positional information of only one nucleotide. Also, the earlier auto-correlation method required visual inspection of the obtained graphs to identify the location of origin of replication. The proposed iCorr method does away with this need and is able to identify the origin location simply by picking the peak in the iCorr graph., Comment: 4 Figures, 1 Table. arXiv admin note: substantial text overlap with arXiv:1701.00707

Researchers are often perplexed when their machine learning algorithms are required to deal with complex number. Various strategies are commonly employed to project complex number into real number, although it is frequently sacrificing the information contained in the complex number. This paper proposes a new method and four techniques to represent complex number as real number, without having to sacrifice the information contained. The proposed techniques are also capable of retrieving the original complex number from the representing real number, with little to none of information loss. The promising applicability of the proposed techniques has been demonstrated and worth to receive further exploration in representing the complex number., Comment: This is a published version of an article published in Journal of Telecommunication, Electronic and Computer Engineering, available online at: http://journal.utem.edu.my/index.php/jtec/article/view/3590

A method to convert real number partitioned activation function into complex number one is provided. The method has 4em variations; 1 has potential to get holomorphic activation, 2 has potential to conserve complex angle, and the last 1 guarantees interaction between real and imaginary parts. The method has been applied to LReLU and SELU as examples. The complex number activation function is an building block of complex number ANN, which has potential to properly deal with complex number problems. But the complex activation is not well established yet. Therefore, we propose a way to extend the partitioned real activation to complex number., Comment: Complex Activation Function, Holomorphic, Phase-preserving, real-complex interaction

Next investigations in our program of transition from the He atom to the complex atoms description have been presented. The method of interacting configurations in the complex number representation is under consideration. The spectroscopic characteristics of the Mg and Ca atoms in the problem of the electron-impact ionization of these atoms are investigated. The energies and the widths of the lowest autoionizing states of Mg and Ca atoms are calculated. Few results in the photoionization problem on the autoionizing states above the n=2 threshold of helium-like Be ion are presented., Comment: 6 pages, 4 tables. arXiv admin note: text overlap with arXiv:1608.04078

We present a novel complex number formulation along with tight convex relaxations for the aircraft conflict resolution problem. Our approach combines both speed and heading control and provides global optimality guarantees despite non-convexities in the feasible region. As a side result, we present a new characterization of the conflict separation condition in the form of disjunctive linear constraints. Our formulation features one binary variable per pair of aircraft, is free of trigonometric functions, and captures the non-convexity in a set of quadratic concave constraints. Using our approach, we are able to close a number of open instances and reduce computational time by up to two orders of magnitude on standard instances.

The beginning of the application of the method of interacting configurations in the complex number representation to the compound atomic systems has been presented. The spectroscopic characteristics of the Be atom in the problem of the electron-impact ionization of this atom are investigated. The energies and the widths of the lowest autoionizing states of Be atom are calculated., Comment: 5 pages

In this paper we expand our recently introduced concept of UW-OFDM (unique word orthogonal frequency division multiplexing). In UW-OFDM the cyclic prefixes (CPs) are replaced by deterministic sequences, the so-called unique words (UWs). The UWs are generated by appropriately loading a set of redundant subcarriers. By that a systematic complex number Reed Solomon (RS) code construction is introduced in a quite natural way, because an RS code may be defined as the set of vectors, for which a block of successive zeros occurs in the other domain w.r.t. a discrete Fourier transform. (For a fixed block different to zero, i.e., a UW, a coset code of an RS code is generated.) A remaining problem in the original systematic coded UW-OFDM concept is the fact that the redundant subcarrier symbols disproportionately contribute to the mean OFDM symbol energy. In this paper we introduce the concept of non-systematic coded UW-OFDM, where the redundancy is no longer allocated to dedicated subcarriers, but distributed over all subcarriers. We derive optimum complex valued code generator matrices matched to the BLUE (best linear unbiased estimator) and to the LMMSE (linear minimum mean square error) data estimator, respectively. With the help of simulations we highlight the advantageous spectral properties and the superior BER (bit error ratio) performance of non-systematic coded UW-OFDM compared to systematic coded UW-OFDM as well as to CP-OFDM in AWGN (additive white Gaussian noise) and in frequency selective environments., Comment: IEEE Tranactions on Signal Processing, Vol. 60, No. 1, 2012. arXiv admin note: text overlap with arXiv:1202.1398

Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an interior structure consisting of spinor functions; this helps us to represent any complex number in an orthogonal form associated with a novel geometric image (the conic-gearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of spinors as Lam\'e coefficients (dyads) locally coupling the base and tangent surfaces., Comment: 7 pages, 1 figure

This paper presents the fractional trigonometric functions in complex-valued space and proposes a short outline of local fractional calculus of complex function in fractal spaces., Comment: 11 page

A new kind of numbers called Hyper Space Complex Numbers and its algebras are defined and proved. It is with good properties as the classic Complex Numbers, such as expressed in coordinates, triangular and exponent forms and following the associative and commutative laws of addition and multiplication. So the classic Complex Number is developed from in complex plane with two dimensions to in complex space with N dimensions and the number system is enlarged also., Comment: 24 pages

High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results., Comment: 13 pages, no figure v2: correction and improvement