Hypercomplex Numbers Research Articles

Quaternion-based Deep Belief Networks fine-tuning

Deep learning techniques have been paramount in the last years, mainly due to their outstanding results in a number of applications. In this paper, we address the issue of fine-tuning parameters of Deep Belief Networks by means of meta-heuristics in which real-valued decision variables are described by quaternions. Such approaches essentially perform optimization in fitness landscapes that are mapped to a different representation based on hypercomplex numbers that may generate smoother surfaces. We therefore can map the optimization process onto a new space representation that is more suitable to learning parameters. Also, we proposed two approaches based on Harmony Search and quaternions that outperform the state-of-the-art results obtained so far in three public datasets for the reconstruction of binary images.

Метод исследования изоморфизма неразложимых гиперкомплексных числовых систем

Метод исследования изоморфизма неразложимых гиперкомплексных числовых систем

Hypercomplex numbers in some geometries of two sets. I

The most important problem in the theory of phenomenologically symmetric geometries of two sets is that of classification of these geometries. In this paper, complexifying the metric functions of some known phenomenologically symmetric geometries of two sets (PSGTS) with the use of associative hypercomplex numbers, we find metric functions of new geometries in question. For these geometries, we find equations of the groups of motions and establish phenomenological symmetry, i.e., find functional relations between metric functions for certain finite number of arbitrary points. In particular, for one-component metric functions of PSGTS’s of ranks (2, 2), (3, 2), (3, 3), we find (n + 1)-component metric functions of the same ranks. For these metric functions, we find finite equations of the groups of motions and equations that express their phenomenological symmetry.

From the theory of “congeneric surd equations” to “Segre's bicomplex numbers”

We will study the historical emergence of Tessarines or Bicomplex numbers, from their origin as “imaginary” solutions of irrational equations, to their insertion in the context of study of the algebras of hypercomplex numbers.

Single-photon test of hyper-complex quantum theories using a metamaterial

In standard quantum mechanics, complex numbers are used to describe the wavefunction. Although this has so far proven sufficient to predict experimental results, there is no theoretical reason to choose them over real numbers or generalizations of complex numbers, that is, hyper-complex numbers. Experiments performed to date have proven that real numbers are insufficient, but the need for hyper-complex numbers remains an open question. Here we experimentally probe hyper-complex quantum theories, studying one of their deviations from complex quantum theory: the non-commutativity of phases. We do so by passing single photons through a Sagnac interferometer containing both a metamaterial with a negative refractive index, and a positive phase shifter. To accomplish this we engineered a fishnet metamaterial to have a negative refractive index at 780 nm. We show that the metamaterial phase commutes with other phases with high precision, allowing us to place limits on a particular prediction of hyper-complex quantum theories.

Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy

There are currently a large number of Multiverse hypotheses, which are, however, non-verifiable, i.e. they can be neither confirmed nor refuted experimentally even in the distant future. In contrast, the hypothesis of the hidden Multiverse considered in the article is verifiable and therefore has a right to be called a theory. The theory uses the principle of physical reality of imaginary numbers discovered 500 years ago, including complex and hypercomplex numbers, as fundamental and proved by the author theoretically and experimentally. This principle has allowed revealing a number of serious mistakes in the special theory of relativity. An adjusted version of the special theory of relativity has been proposed and the theory of the hidden Multiverse has been developed on its basis. The Multiverse has been referred to as hidden, because parallel universes it contains are mutually invisible. The nature of their invisibility is explained in the article. It is shown that dark matter and dark energy are other universes of the hidden Multiverse apart from ours. Analysis of data from WMAP and Planck spacecrafts has shown that the hidden Multiverse has quaternion structure comprising four pairs of universes and antiverses (i.e., four pairs of matter and antimatter).

Symmetry, Geometry and Quantization with Hypercomplex Numbers

These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual-/double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are discussed. Finally, we prove a Calder\'on--Vaillancourt-type norm estimation for relative convolutions.

Construction and Analysis of Quaternion MIMO-OFDM Communications Systems

Wireless communications systems with progressively higher spectral efficiency have been investigated in past decades. A promising area of research is the use of hypercomplex algebras, notably, the use of the quaternion algebra. This paper considers the construction of multiple-input-multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) using the algebra of quaternions. Several construction techniques for quaternion orthogonal code designs have been proposed in recent years, which offer the possibility to explore diversities in variate domains, such as space, time, frequency, and polarization, in addition to combinations thereof. This paper presents a formulation for quaternion MIMO-OFDM in matrix form as an extension of the classical formulation that uses complex variables. Quaternions allow elegant representation of pairs of radiant elements in physical antennas configured for cross-polarized propagation. Several simulations validate the proposed method in diverse scenarios for wireless communications, in which combined diversities have been exploited.

Algebraic Representation of the Group of Havrda–Charvat–Daroczy Entropy Vectors in Nonextensive Statistical Mechanics

An algebraic representation of the group of nonextensive, parameterized Havrda–Charvat–Daroczy entropy vectors that depend on three distributions is constructed. The composition law of conformally-generalized hypercomplex numbers is considered, and properties of a commutative, nonassociative algebra are derived. The exponential form of the number and functions of numbers with hyperbolic angles are presented.

Conformal Numbers

The conformal compactification is considered in a hierarchy of hypercomplex projective spaces with relevance in physics including Minkowski and Anti-de Sitter space. The geometries are expressed in terms of bicomplex Vahlen matrices and further broken down into their structural components. The relation between two subsequent projective spaces is displayed in terms of the complex unit and three additional hypercomplex numbers.

Дослідження властивостей узагальнених гіперкомплексних числових систем четвертої вимірності, що отримані процедурою подвоєння Грассмана-Кліффорда

Using the Grassmann-Clifford doubling procedure, the generalized hyper-complex number systems (HNS) of the fourth order has been formed. Available options in the definition of these systems makes them a generalization of whole classes of various HNS. It has been studied the algebraic and functional properties of generalized HNS and their relationship with the HNS lower dimensions. Refs: 23 titles.

Sparse-Iteration 4D CORDIC Algorithms for Multiplying Quaternions

Novel 4D CORDIC algorithms and hardware architecture for multiplying quaternions are presented, aimed at constant-coefficient multipliers. In our solution, microrotations are multiplications by hypercomplex numbers with only one non-zero imaginary part. Such transformations can be described using sparse matrices and implemented using less hardware resources than the iteration of the known quaternion CORDIC algorithm. Chip area can be saved because sparse microrotations can be computed using a permutation network and two-operand adders, without four-operand additions, which are necessary in the known solution. The obtainable area savings depend on the implementation technology and the architectures of adders and bit shifters but can be as large as 45 and 25 percent for ASIC and FPGA circuits, respectively. The circuitry simplifications come at the cost of slowing down computations: our approach improves the area-delay product, and a single sparse iteration can be executed up to 15 percent faster than the conventional one, but more microrotations are usually necessary to achieve the required accuracy. On the other hand, a sparse 4D iteration can be identified with microrotations of two 2D CORDICs which work in parallel. This makes convergence analysis easier than in the previous 4D solution, as it is sufficient to consider only two dimensions.

On harmonic analysis of vector‐valued signals

A vector‐valued signal in N dimensions is a signal whose value at any time instant is an N‐dimensional vector, that is, an element of . The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N‐dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N‐dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in . Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

Circle Complexes and the Discrete CKP Equation

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in Mobius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of Mobius geometry, governs novel doubly hexagonal circle patterns.

Quaternion representations of stiffness and momentum of the forces, acting in vibration isolating systems with stiffness compensators

This research is devoted to development of the spatial vibration isolation devices. The description of the vibration isolation systems has been presented through quaternions of the forces, momentums, and stiffness. The considered method allows taking into account the stochastic vibrations and describes it with the help of the hypercomplex numbers. The theory suggests the development of the vibration isolation devices, which have traction characteristics with zero stiffness area. To obtain such area in traction characteristic, a spatial vibration isolator is presented as a resilient element and the stiffness compensator, which is connected in parallel with it.

Розробка представлень гіперболічних і тригонометричних нелінійностей у системі узагальнених кватерніонів

It is considered the process of mathematical modeling for representations of hyperbolic and trigonometric nonlinearities in hypercomplex number system of generalized quaternions by the method of associated system of differential equations. Some properties of the representations and their relationship with the representations of nonlinearities in certain noncommutative hypercomplex numerical systems having dimension four have been analyzed. Fig.: 4. Refs: 11 titles.

Hypercomplex analysis and integration of systems of ordinary differential equations

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

Математичне моделювання представлень експоненціальної і логарифмічної функцій у гиперкомплексній числовій системі узагальнених кватерніонів

The process of mathematical modeling representations of exponential and logarithmic functions in hypercomplex number system using generalized quaternion defining a linear differential equation with hypercomplex coefficients is being considered. Some properties of these representations and their relationship with the concepts of exponents in specific non-commutative hypercomplex numerical systems dimension four have been considered. Refs: 21 titles.

COSET OF A HYPERCOMPLEX NUMBER SYSTEM IN CLIFFORD ANALYSIS

We give certain properties of elements in a coset group with hypercomplex numbers and research a monogenic function and a Clifford regular function with values in a coset group by defining differential operators. We give properties of those functions and a power of elements in a coset group with hypercomplex numbers.

Побудова алгоритма цифрового підпису з використанням функцій від узагальнених кватерніонів

The investigation touches upon the use of hypercomplex number system of the fourth dimension: a generalization of quaternions for to build a digital signature algorithm. Tabl.: 3. Refs: 18 titles.