Conjugation Operator Research Articles

Robust timing and frequency joint synchronization method for QNSC-enabled security optical communications system based on CO-OFDM

The security of optical fiber communication network has been paid more and more attention, and Quantum Noise Stream Cipher (QNSC) can guarantee its physical layer security. However, the QNSC system realized with CO-OFDM technology is very susceptible to timing synchronization error and carrier frequency offset. Existing symbol timing and frequency synchronization algorithms based on preambles have problems of timing metric platform, interference of side lobes, and low timing and frequency synchronization accuracy especially in optical signal transmission scenario of high-order modulation and low SNR. Furthermore, the preamble is not encrypted, which can be easily identified by illegal receiver. In order to solve above problems, a robust timing and frequency joint synchronization algorithm is proposed. The algorithm is composed of the preamble interleaved by conjugate, negative and image sequences of original element sequence. The preamble conforms to the preamble structure of Schmidl, Minn and CNI by adding negative and conjugate operator at the specific interval of the preamble. The correspond timing metric method, i.e. combination of Schmidl, Minn and CNI timing metric strategies, can fully extract characteristics of the preamble, and then timing and frequency joint synchronization is accurately obtained. For timing synchronization, experiments in 10Gbit/s CO-OFDM based QNSC system have illustrated that proposed algorithm has better timing synchronization performance than baseline algorithms. The detection probability is augmented and timing offset mean is decreased than that of baseline algorithms especially at low SNR. The simulation illustrates that proposed algorithm has more accurate frequency offset (FO) estimation performance compared with the baseline algorithms even at low SNR. Furthermore, the simulation illustrates the security performance of CO-OFDM based QNSC system is improved by encrypting the preamble with secret key.

Electron-photon vertex and dynamical chiral symmetry breaking in reduced QED: An advanced study of gauge invariance

We study the effect of a refined electron-photon vertex on the dynamical breaking of chiral symmetry in reduced quantum electrodynamics. We construct an educated {\em ansatz} for this vertex which satisfies the required discrete symmetries under parity, time reversal and charge conjugation operations. Furthermore, it reproduces its asymptotic perturbative limit in the weak coupling regime and ensures the massless electron propagator is multiplicatively renormalizable in its leading logarithmic expansion. Employing this vertex {\em ansatz}, we solve the gap equation to compute dynamically generated electron mass whose dependence on the electromagnetic coupling is found to satisfy Miransky scaling law. We also investigate the gauge dependence of this dynamical mass as well as that of the critical coupling above which chiral symmetry is dynamically broken. As a litmus test of our vertex construction, both these quantities are rendered virtually gauge independent within a certain interval of values considered for the covariant gauge parameter.

A pilot study on canonical gravity with mass dimension one fermions

The mass dimension one (MDO) fermionic field is built on a complete set of dual-helicity eigenspinors of the charge conjugation operator, which obeys the statistic of Fermi-Dirac. These spinors are a potential candidates for the description of dark matter. The redefinition of the dual structure of this object holds a local adjacent theory and completely satisfies the Lorentz invariance. In this work, we investigate important aspects of the interaction of this fermion with gravity in the light of a canonical formulation in ADM formalism. We construct an action via tetrad fields using a manifold on a family of spacelike surfaces Σt that carries the MDO matter field, and additionally we propose a condition for the action to have a term associated with dark energy. We found the Hamiltonian and diffeomorphism constraints at the classical level for gravitational dynamics with the immersion of this material content in space-time, which leads us to the interpretation of the contribution of dark matter energy density, parallel to the lapse function of foliation and its directional flux of energy density in the hypersurface of manifold.

Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras $$\mathcal {G}_{p,q}$$ of vector space of dimension $$n=p+q$$ . We present basis-free formulas for all characteristic polynomial coefficients in the cases $$n\le 6$$ , alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $$n=4$$ , and one of the formulas in the case $$n=5$$ . We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases $$n\le 5$$ . The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

Une Approche Linguistique pour l’Analyse, la Reconnaissance et la Traduction Automatiques des Verbes du Domaine Administratif et de Droit

A Linguistic Approach to The Analysis, Recognition and Automatic Translation of Administrative and Legal Verbs
 Due to globalization and with the advent of the new digital era, economic, social, and mostly administrative sectors are now required to participate in the digitization and automation of data. This will make several tasks that were previously totally performed by humans easier. Hence, in this context, the present study investigated the application of machine translation (MT) in the administrative field. Using a semantic-syntactic approach and the French Verbs (LVF) dictionary by Jean Dubois and Françoise Dubois-Charlier, we analyzed the verbs used in the administration and legal fields. Following this initial theoretical stage, and using the NooJ platform, we deployed the acquired linguistic data for the construction of a French-Arabic machine translation application. To implement this application, we will design formal grammars for the automatic analysis and recognition of the many semantic-syntactic schemes defining this class of verbs, as well as a bilingual dictionary of legal and administrative verbs. The utmost objective of this research work is the achievement of an accurate and trustworthy translation. In the first theoretical part, we will identify the linguistic characteristics of the different verbs in the administrative and legal fields. At this first level of our approach, we will examine the syntactic constructions of these verbs and then, we will try to delimit the patterns or the distinctive semantic-syntactic schemes of the verbs used in a very specific discourse, which is the administrative discourse. In the second stage of automatic processing, we will try to create a computer process which can:
 • Automatically analyze and recognize administrative verbs using the NooJ platform. • Properly translate the verbs studied in Arabic. For the creation of this MT process, the interest of which mainly concerns administrative documents, for social (divorce, inheritance, etc.) and even economic reasons, we will: a/ Integrate legal and administrative verbs into a bilingual French-Arabic electronic dictionary, describing them according to NooJ operators: inflection, conjugation, derivation and syntax operators, etc. In this step, we created: • Paraderivational.nof derivational paradigms. • “paraflexion.nof” bending paradigms. • “paradigmflexionnel.nof” conjugation paradigms • The syntactic formalization where we rewrote the syntactic schemes of the LVF. • The translation of each verbal entry into Arabic. Exp : 
 V+Emploi=01+AUX=AVOIR+FLX=CHANTER+CONS=T11b0+N0VN1PREPN2+N0Hum+V+N1Hum+N2Abst+N2Vinf+PREP="de"+DRV=AV1e : ARTISTE +DRV=AV2: COUSIN+DRV=DN15: DIRECTOR+DRV=DN15a: TABLE+DOM=DRO +CLASS=F2c+OPER="ict qn D offence"+LEXI=1+AR="َََّتَهَم
 b/ Schematize the semantic-syntactic patterns by the formal grammars of NooJ, to automatically recognize the legal verbs used in an administrative corpus. The formal grammars used are called syntactic disambiguation grammars (Silberztein, Max 2015). In this phase, we produced disambiguation grammars that allowed us to analyze only the requested syntactic constructions. The disambiguation grammar is necessary and important, not only for the automatic analysis and recognition of the correct syntactic constructions, but also for producing a correct and reliable translation. c/ Create an automatic translation grammar that will allow us to obtain a suitable, reliable translation, which will respect the standards of the target language: Arabic. d/ Find solutions for the disambiguation of polysemic predicates using Gaston Gross's "object class" theory which allowed us to refine the recognition grammars and subsequently obtain encouraging results, where the verbs polysemous are suitably translated into Arabic. The originality of this study emanates from the fact that it provides, not only a linguistic description of the verbs of the administrative and legal domain, but especially of the integration of the various data in the NooJ software, in order to facilitate the treatment and the automatic translations of the two languages in question.

A Modular Operator Approach to Entanglement of Causally Closed Regions

Quantum entanglement is shown for causally separated regions along the radial direction by using a conformal quantum mechanical correspondence with conformal radial Killing fields of causal diamonds in Minkowski space. In particular, the theory of local von Neumann algebras and Tomita Takesaki modular operators is applied in the entanglement structure of causal diamonds in conformal quantum mechanics. The entanglement of local states in their respective causal regions is shown through the measures of concurrence and entanglement entropy using the Tomita Takesaki modular conjugation operator. A holographic entropy formula is derived for the conformal quantum mechanics causal diamond correspondence. A new connection is made between the thermal time flow defined by the modular group of automorphisms to the physical time flow in a causal diamond via the aforementioned correspondence. The thermal interpretation of these results via two-point thermal Green’s functions and modular group flow supports the idea of a possible emergent theory of spacetime.

Conjugation Conditions for Systems of Differential Equations of Different Orders on a Star Graph

In this paper, a one-dimensional mathematical model for investigating the vibrations of structures consisting of elastic and weakly curved rods is proposed. The three-dimensional structure is replaced by a limit graph, on each arc of which a system of three differential equations is written out. The differential equations describe the longitudinal and transverse vibrations of an elastic rod, taking into account the influence of longitudinal and transverse vibrations on each other. Describing conjugation conditions at joints of four or more rods is an important problem. This article assumes new conjugation conditions that guarantee the all-around decidability and symmetry of the resulting boundary value problems for systems of differential equations on a star graph. In addition, the paper proposes a physical interpretation of the conjugation conditions found. Thus, the work presents one more area of knowledge where symmetry phenomena occur. The symmetry here is manifested in the preservation of conjugation conditions when passing to the conjugate operator.

Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform

In recent years, quaternion algebra has been applied more and more in different applications. A primary motivation for investigating quaternion algebras is to extend the classical signal processing to quaternion signal processing theory. However, this algebra has non-commutative multiplication, leading to the uncertainty of such essential operations as the convolution and Fourier transform. As a result, it became difficult to process quaternion signals in the frequency domain. In this work, the arithmetic of quaternion numbers is analyzed using the Cayley-Dickson construction, but with an associative and commutative operation of multiplication. We call it the (2,2)-model of quaternions. This model's main properties of quaternions are described, including inverse, division, and conjugate operations. In the (2,2)-model, only two exponential functions with pure quaternion units exit, and therefore, only two types of the quaternion discrete Fourier transform (QDFT) can be defined. One of the crucial benefits of this model is that the unique definition of convolution will help reduce the complexity of quaternion convolution operation calculated using both types of QDFTs.

CPT Invariance of Quaternion Dirac equation

Abstract: In this paper the invariance of Quaternion Dirac equation under Lorentz Transformation, Charge conjugation, Parity transformation and Time reversal operation has been discussed successfully. The invariance under the combined operation of Charge conjugation, Parity and Time reversal (CPT) has also been discussed and expression for C, P, T and combined CPT operators have been obtained in terms of quaternions. Invariance condition for electric and magnetic field has also been obtained. It has been concluded that the Quaternion Dirac equation dominates over ordinary Dirac equation because of the advantage of algebra of quaternions.

Optimal Weak Type Estimates for the P‐ADIC Hardy Type Operators on Higher‐Dimensional Product Spaces

In this paper, we introduce the fractional p‐adic Hardy operators and its conjugate operators and obtain its optimal weak type estimates on the p‐adic Lebesgue product spaces.

About the Observed Asymmetry between Matter and Antimatter

Some consequences, due to the existence of a pair of decoupled Schrödinger-like but relativistic quantum mechanics wave equations, are explored. It is shown that one equation directly describes the quantum states of a single spin-0 particle, and the other one indirectly describes the quantum states of the corresponding antiparticle. In correspondence with the matter-antimatter symmetry, for a Coulomb potential, a charge conjugation operation transforms the second equation in the first one. However, if a particle could interact with itself (gravitationally or electrically) due to the spread of its wavefunction, the C-symmetry could be broken; therefore, matter and antimatter could be distinguished. Under these assumptions, it is deducted the impossibility of the existence of particles and antiparticles with a mass larger than the Plank mass (mP), or with the absolute value of the charge larger than the Plank charge (qP). It is proposed the existence of primordial antimatter electrical sinks. It is also suggested that all macroscopic matter objects with a mass m > mP, and all macroscopic antimatter bodies with a charge |q| > qP should not be quantum but classical objects. It is argued that these findings could explain the absence of antimatter with a complicated structure and partially explain the excess of charged matter in the known Universe.

A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians* *This paper is dedicated to the memory of our friends and colleagues Lahcene Alouani and Mebrouk Gherda died due to covid 19.

With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the −inner product, where is a obtained from the charge conjugation operator through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

On C 1 C 2 symmetric operators

Let C 1 and C 2 are conjugation operators (both are antiliner, isometric and invoulation) on a separable complex Hilbert space H. In this paper, we present the notion of C 1 C 2- symmetric operators as: A bounded linear operator A on H is C 1 C 2-symmetric operators if C 1 A = A* C 2 (A = C 1 A*C 2). We study and discuss several properties and give an example for such kind of operators.

Quotient rings of 𝐻𝔽₂∧ℍ𝔽₂

We study modules over the commutative ring spectrum H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 , whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ k \xi _k in the category of associative algebras freely kills the higher generators ξ k + n \xi _{k+n} . Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 -algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

Basis-free Solution to Sylvester Equation in Clifford Algebra of Arbitrary Dimension

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case $$n=4$$ , the proofs for the case $$n=5$$ and the case of arbitrary dimension n. The results can be used in symbolic computation.

On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension n. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev–LeVerrier algorithm based on the Cayley–Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation.

Method of reconfiguring industrial facilities using the apparatus of piecewise linear aggregates

A method has been developed for structural and functional reconfiguration of industrial facilities based on piecewise linear aggregates, with ranking elements according to the importance of fulfilling their target function. The method allows for the conjugation of elements of complex technical objects using the conjugation operator. An example of the application of the proposed method is given. Keywords: reconfiguration, industrial facility, system, piecewise linear aggregates, synthesis of elements. rikki2@yandex.ru, strange.y@mail.ru

The Dirac Sea, T and C Symmetry Breaking, and the Spinor Vacuum of the Universe

We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) of Grassmann densities in the momentum space. We have proven the existence of two spinor vacua. Operators C and T transform the normal vacuum into an alternative one, which leads to the breaking of the C and T symmetries. The CPT is the real structure operator; it preserves the normal vacuum. We have proven that, in the theory of the Dirac Sea, the formula for the charge conjugation operator must contain an additional generalized Dirac conjugation operator.

Modular operators and entanglement in supersymmetric quantum mechanics

The modular operator approach of Tomita–Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric (SUSY) quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a SUSY two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these SUSY systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a SUSY Jaynes Cummings model.

Coherent states attached to the quantum disc algebra and their associated polynomials

The one-variable [Formula: see text]-coherent states attached to the [Formula: see text]-disc algebra are constructed and used to obtain the [Formula: see text]-Bargmann–Fock realization of its Fock representation. Then, this realization is used to obtain the [Formula: see text]-continuous Hermite polynomials as well as continuous and discrete [Formula: see text]-Hermite polynomials by using a pair of Hermitian canonical conjugate operators and two pairs of the non-Hermitian conjugate operators, respectively. Besides, we introduce a two-variable family of [Formula: see text]-coherent states attached to the Fock representation space of the [Formula: see text]-disc algebra and its opposite algebra and obtain their simultaneous [Formula: see text]-Bargmann–Fock realization. For an appropriate non-Hermitian operator, the latter realization is served to obtain the well-known little [Formula: see text]-Jacobi polynomials used in constructing the [Formula: see text]-disc polynomials.