Nonzero Terms Research Articles

Tunable broadband compact optical isolator based on Weyl semimetal

We theoretically set up a 4 × 4 magneto-optical matrix to investigate the transmittances induced by the forward and backward left hand circularly polarized (LCP) waves impinging the 4 μm thickness Weyl semimetal (WSM) material at normal incidence. Because of the different effective permittivity of the two LCP waves in WSM induced by the non-zero off-diagonal term, the two waves experience blocking transmission, bidirectional transmission and unidirectional transmission with the increase of the wavelength, respectively. Based on the wide wavelength of the unidirectional transmission, the broadband optical isolator can be realized. The threshold wavelength of the optical isolator can be regulated by changing the wavelength where the parallel component of the diagonal element is identical to the off-diagonal term. It is also found that the Fermi energy, tilt degree of Weyl cones and Weyl node separation reveal simple and available approaches to achieve the regulation. We believe that the adjustable broadband compact optical isolator can perform a significant role in fiber-optic communication system.

Binary-coupling sparse Sachdev-Ye-Kitaev model: An improved model of quantum chaos and holography

The sparse version of the Sachdev-Ye-Kitaev (SYK) model reproduces essential features of the original SYK model while reducing the number of disorder parameters. In this paper, we propose a further simplification of the model which we call the binary-coupling sparse SYK model. We set the nonzero couplings to be $\pm 1$, rather than being sampled from a continuous distribution such as Gaussian. Remarkably, this simplification turns out to be an improvement: the binary-coupling model exhibits strong correlations in the spectrum, which is the important feature of the original SYK model that leads to the quick onset of the random-matrix universality, more efficiently in terms of the number of nonzero terms. This model is better suited for analog or digital quantum simulations of quantum chaotic behavior and holographic metals due to its simplicity and scaling properties.

Continuum theory of a nematic liquid crystal with a nonideal physical surface and the surfacelike elasticity

ABSTRACT The nematic surface violates the translational symmetry of unrestricted medium and induces novel elastic effects. An ideal surface, where the nematic order parameter and density vanish abruptly, induces the surface elastic term that cancels out the Nehring-Saupe divergence term. A more realistic, nonideal surface is represented by a thin layer where the order parameter and density smoothly vary from their bulk values to zero. This surface is shown to induce additional contributions to the effective surface elastic and terms. These terms are no longer total divergences which is employed to incorporate them in the elastic theory. It is shown how the higher order elasticity and the finite surface layer provide the lower free energy bound both for nonzero term and for term violating Ericksen’s stability condition for the uniform director ground state. The effective boundary condition and procedure of the free energy minimisation for arbitrary and are derived. It is shown how Ericksen’s condition is modified by the presence of the term, anchoring, and spatial boundedness. An example of the surfacelike-elasticity-induced spontaneous modulations of the uniform director ground state is presented.

Design of Chaotic Interleaver Based on Duffing Map for Turbo Code

This article presents the performance analysis and comparison of turbo codes for several interleavers with short blockchains. According to the symmetry concept, turbo codes consist of two symmetric convolutional codes preceded by an interleaver. Some interleavers can be described by symmetric 3 × 3 generator matrices, whereas others are represented through symmetric matrices in which the nonzero terms in the matrix are monomials. Interest in short blockchains has grown over the years because of their use in emerging applications that require the transmission of data units in the order of hundred bits. In this research, interleavers used are random interleaver, semi-random interleaver, quadratic polynomial permutation (QPP) interleaver, and chaotic Lozi map interleaver. For each interleaver, the turbo code is assessed with at least two short data distances; then, the interleavers are compared against each other. The performance of turbo codes is assessed based on bit error rate, particularly with lengths between 64 and 512 bits. Additionally, an interleaver based on Duffing chaotic maps is proposed, whose deterministic nature provides an advantage, and the performance results are presented, achieving excellent BER values for 64 bits. With these results, the effect of the interleaver on turbo codes and particularly on short block length interleavers, is studied.

Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors

This paper addresses the study of the complexity of products in geometric algebra. More specifically, this paper focuses on both the number of operations required to compute a product, in a dedicated program for example, and the complexity to enumerate these operations. In practice, studies on time and memory costs of products in geometric algebra have been limited to the complexity in the worst case, where all the components of the multivector are considered. Standard usage of Geometric Algebra is far from this situation since multivectors are likely to be sparse and usually full homogeneous, i.e., having their non-zero terms over a single grade. We provide a complete computational study on the main Geometric Algebra products of two full homogeneous multivectors, that are outer, inner, and geometric products. We show tight bounds on the number of the arithmetic operations required for these products. We also show that some algorithms reach this number of arithmetic operations.

A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term

In this paper, a novel spatial-temporal radial Trefftz collocation method (STRTCM) is proposed to solve 2D and 3D backward heat conduction equations with time-dependent source term. Unlike the traditional time-domain discretization strategies (Laplace/Fourier transformation and time-stepping methods), the proposed spatial-temporal meshless collocation method employs the derived semi-analytical spatial-temporal radial Trefftz basis function as basis function, which naturally satisfies the governing equation with zero source term in advance. Due to this property, only the final conditions and boundary conditions need to be discretized. For non-zero source term, the extended multiple reciprocity method (E-MRM) is introduced to transform the original governing equation with several specific source terms to the high-order governing equation without the source term. By deriving the high-order semi-analytical spatial-temporal radial Trefftz basis function, the STRTCM can also avoid the node discretization of governing equation. For solving the backward heat conduction equations, the present numerical scheme uses the final temperature distribution and the related boundary conditions to obtain the unknown initial temperature field. Finally, the accuracy and efficiency of the proposed method is numerically verified by four benchmark examples about backward heat conduction equations with time-dependent source term.

Direct Calculation of ZZ Interaction Rates in Multimode Circuit Quantum Electrodynamics

Hamiltonians of the superconducting qubits of Transmon type involve non-zero ZZ-interaction terms due to their finite and small anharmonicities. These terms might lead to the unwanted accumulation of spurious phases during the execution of the two-qubit gates. Exact calculation of the ZZ-interaction rates requires the full diagonalization of the circuit Hamiltonians which very quickly becomes computationally demanding as the number of the modes in the coupler circuit increases. Here we propose a direct analytical method for the accurate estimation of the ZZ-interaction rates between low-anharmonicity qubits in the dispersive limit of the multi-mode circuit-QED. We observe very good agreement between the predictions of our method and the measurement data collected from the multi-qubit devices. Our method being an extension of our previous work in [1] is a new addition to the toolbox of the quantum microwave engineers as it relates the ZZ-interaction rates directly to the entries of the impedance matrix defined between the qubit ports.

Mechanical coupling effects of 2D lattices uncovered by decoupled micropolar elasticity tensor and symmetry operation

Mechanical couplings such as axial–shear and axial–bending have great potential in the design of active mechanical metamaterials with directional control of input and output loads in sensors and actuators. However, the current ad hoc design of mechanical coupling without theoretical support of elasticity cannot provide design guidelines for mechanical coupling with lattice geometries. Moreover, the correlation between mechanical coupling effects and geometric symmetry is not yet clearly understood. In this work, we systematically search for all possible mechanical couplings in 2D lattice structures by determining the non-zero diagonal terms in the decomposed micropolar elasticity tensor. We also correlate the mechanical couplings with the point-group symmetry of 2D lattices by applying the symmetry operation to the decomposed micropolar elasticity tensor. The decoupled micropolar constitutive equation uncovers eight coupling effects for 2D lattice structures. The symmetry operation of the decoupled micropolar elasticity tensor reveals the correlation of the mechanical coupling with the point groups. Our findings can strengthen the design of mechanical metamaterials with potential applications in areas including sensors, actuators, soft robots, and active metamaterials for elastic/acoustic wave guidance and thermal management.

A note on the weak regularity theory for degenerate Kolmogorov equations

The aim of this work is to prove a Harnack inequality and the Hölder continuity for weak solutions to the Kolmogorov equation Lu=f with measurable coefficients, integrable lower order terms and nonzero source term. We introduce a functional space W, suitable for the study of weak solutions to Lu=f, that allows us to prove a weak Poincaré inequality. Our analysis is based on a weak Harnack inequality, a weak Poincaré inequality combined with a L2−L∞ estimate and a classical covering argument (Ink-Spots Theorem).

Multiple relaxation time lattice Boltzmann schemes for advection-diffusion equations with application to radar image processing

Motivated by marine radar image processing, we investigate the accuracy of multiple relaxation time lattice Boltzmann schemes designed to simulate two-dimensional convection-diffusion equations. The context of application requires to deal with non-constant advection velocity. Using Taylor expansions, instead of the widely used Chapman-Enskog expansions, we show how to control the accuracy of these schemes when deriving equivalent partial differential equations. On the one hand, a third order analysis is conducted on a scheme involving a constant advection velocity and no source term. First, this analysis derives the stability region through the von Neumann analysis. Second, a numerical convergence rate of three is obtained thanks to an appropriate choice of parameters. On the other hand, non-constant advection velocity together with non-zero source term, introduce additional terms at the second order. Regarding the targeted application, these extra terms are shown to be negligible and experiments on real data show that such multiple relaxation time lattice Boltzmann schemes are relevant for marine radar denoising and enhancement.

Variational quantum solutions to the advection–diffusion equation for applications in fluid dynamics

Constraints in power consumption and computational power limit the skill of operational numerical weather prediction by classical computing methods. Quantum computing could potentially address both of these challenges. Herein, we present one method to perform fluid dynamics calculations that takes advantage of quantum computing. This hybrid quantum–classical method, which combines several algorithms, scales logarithmically with the dimension of the vector space and quadratically with the number of nonzero terms in the linear combination of unitary operators that specifies the linear operator describing the system of interest. As a demonstration, we apply our method to solve the advection–diffusion equation for a small system using IBM quantum computers. We find that reliable solutions of the equation can be obtained on even the noisy quantum computers available today. This and other methods that exploit quantum computers could replace some of our traditional methods in numerical weather prediction as quantum hardware continues to improve.

Spatiotemporal jumps as particular solutions in geodesic trajectories with the Gödel metric on an extended manifold

Using the Gödel metric, we obtain some relevant solutions compatible with spatiotemporal jumps for the geodesic equations, by using an extension of General Relativity with nonzero boundary terms, which are described on an extended manifold generated by the connections delta Gamma ^{mu }_{alpha beta } = b,U^{mu },g_{alpha beta }. These terms are given by a flow of velocities with components U^{nu }: 3,b^2,nabla _{nu }U^{nu }=g^{alpha beta }, delta R_{alpha beta } = lambda left[ sleft( x^{alpha }right) right] ,g^{alpha beta }, delta g_{alpha beta } in the varied Einstein–Hilbert action. The solutions are valid for an arbitrary equation of state with ordinary matter: Omega =P/(c^2,rho ) = frac{left( frac{omega }{c}right) ^2-lambda (s)}{left( frac{omega }{c}right) ^{2}+lambda (s)}.

Magneto-optical Kerr spectra of gold induced by spin accumulation

We report the magneto optic Kerr effect (MOKE) angle of Au magnetically excited by spin accumulation. We perform time resolved polar MOKE measurements on Au/Co heterostructures. In our experiment, the ultrafast optical excitation of the Co drives spin accumulation into an adjacent Au layer. The spin accumulation, together with spin-orbit coupling, leads to non-zero terms in the off-diagonal conductivity tensor of Au, which we measure by recording the polarization and ellipticity of light reflected from the Au surface for photon energies between 1.3 and 3.1 eV. At energies near the interband transition threshold of Au, the Kerr rotation per A/m exceeds 1 urad. Typical values for Kerr rotation per moment in transition ferromagnetic metals like Ni are < 10 nrad per A/m, while predicted values for heavy metals like Pt or W are < 13 nrad per A/m. The exceptional sensitivity of the optical properties of Au to spin magnetic moments make Au to be an exceptionally sensitive optical magnetometer, with potential applications in the development of optospintronic technologies.

Theoretical drag analyses of laminar channel and pipe flows with wall roughness

In this paper, an exact expression for the drag coefficient of a streamwise-periodic steady incompressible laminar channel and pipe flow with micro- or macro-scale wall roughness is derived, whereby the drag coefficient is decomposed into contributions from different components of the velocity gradient tensor in the flow field. It is shown through our theoretical analysis that drag reduction cannot be achieved by adding micro- or macro-scale spanwise-periodic/-symmetry wall roughness structures to the smooth inner walls of streamwise-periodic steady incompressible laminar channel/pipe flows while maintaining the same volumetric flow rate. It is also shown that wall roughness produces a higher drag due to two factors: (a) wall roughness induces other non-zero velocity gradient terms apart from the wall-normal/radial gradient of streamwise velocity that exists in a smooth channel/pipe flow; (b) the profile of streamwise velocity in the wall-normal/radial direction deviates from the parabolic profile that produces the minimum kinetic energy loss for a given volumetric flow rate. Finally, numerical simulations of laminar channel flow with longitudinal and transverse bars are conducted, and the numerical results confirm the theoretical finding.

Sparse polynomial interpolation based on derivatives

We propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program (SLP). Both of our algorithms work over any finite fields Fq with large characteristic.The first algorithm is randomized of the Monte Carlo type. Its bit complexity is linear in the number of non-zero terms T and the number of variables n of f. Let D be the partial degree bound of f. If q is in O((nTD)(1)), our algorithm has better complexity than other existing algorithms.The second algorithm is deterministic; it has better complexity than any existing deterministic algorithms in the field with large characteristic. Its bit complexity is quadratic in n,T,log⁡D.

On the Mixed-Unitary Rank of Quantum Channels

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality $N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the equality $N=2$ when $r=2$), and we exhibit the first known examples of mixed-unitary channels for which $N>r$. Specifically, we prove that there exist mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for infinitely many positive integers $d$, including every prime power $d$. We also examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.

Analytical Prediction of the MCSA Signatures Under Dynamic Eccentricity in PM Machines With Concentrated Non-Overlapping Windings

This paper proposes a novel algorithm which made it feasible to predict the harmonic fault signatures under dynamic eccentricity in Permanent-Magnet synchronous machines with concentrated non-overlapping winding for any slot-pole combination. In the literature, only the appearance of the fractional harmonic components related with the pole number has been proposed but there are no studies to predict which exact components will rise in each topology. Initially, a mathematical equation is derived for the machine <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">EMF</i> under dynamic eccentricity based on the air gap permeance model. Later on, the terms of this equation are inserted in a Fault Signature Block Binary Array in which the position of each cell expresses a specific harmonic frequency component. After substitution of the machines parameters, the array results with zero and nonzero terms. The positions of the nonzero terms express the frequency components excited by this particular fault. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3-D</i> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FEA</i> and experimental results of a Permanent-Magnet Axial-Flux Machine are used to validate the proposed algorithm.

Twist polynomials of delta-matroids

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sense for general delta-matroids. In this paper, we show that partial duality polynomials have delta-matroid analogues. We introduce the twist polynomials of delta-matroids and discuss its basic properties for delta-matroids. We give a characterization of even normal binary delta-matroids whose twist polynomials have only one term and then prove that the twist polynomial of a normal binary delta-matroid contains non-zero constant term if and only if its intersection graph is bipartite.

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complement work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.Mathematics Subject Classifications: 05A15, 05A15, 05A19, 05A05

4-VARIABLE BOOLEAN FUNCTIONS REPRESENTATIONS CLASSIFICATION IN THE FORM OF NONLINEARITY MINIMAL DEGREE SEPARATING SURFACES

In this paper, a classification study was carried out on the construction of 4-variable Boolean functions representations classification in the form of nonlinearity minimal degree separating surfaces. To construct these surfaces was used an adaptive ellipsoid algorithm based on the Khachiyan algorithm for solving systems of linear inequalities with integer coefficients. To define Boolean functions, we used its graphical representation on the projection of a four-dimensional cube. The classification study carried out was not limited only to the search for a separating surface of the minimum degree of nonlinearity. Additionally, the task was set to find the surface with the smallest number of non-zero nonlinear terms in accordance with a given lexicographic order. Based on the results of the study, a catalog of separating surfaces of the nonlinearity minimum degree with the smallest number of nonlinear terms for Boolean functions of 4 variables is constructed, and it is also determined that 15 classes of geometric equivalence functions have a minimum degree of nonlinearity of 1, 166 - degree 2, 40 - degree 3, and 1 function degree 4.