Exact Results Research Articles

Block sparse vector recovery for compressive sensing via ℓ1−αℓq$\ell _1-\alpha \ell _q$‐minimization Model

AbstractThis paper solves the problem of block sparse vector recovery using the block ‐minimization model. Based on the block restricted isometry property (B‐RIP) condition, exact block sparse vector recovery result is obtained. The theoretical bound for the block ‐minimization model are also obtained when measurements are depraved by the noises.

Some applications of trace of powers of matrices to graph theory, algebra and combinatorics

In this paper, using Cayley Hamilton Theorem, we exhibit a combinatorial formula for Tr(An), (n ∈ ℤ, A ∈M3 (K)), over the field K(= ℝorℂ)using Tr (A), det(A) and sum of principal minors of A. Further, a formula for ∑nk=0 Tr(Ak) is derived. For specific values of Tr(A) and det(A), the formula of Tr(An) is highly simplified. We exhibit some applications of our results in Graph Theory, Algebra and Combinatorics. We have verified our results on various matrices using Scilab and found that Scilabfails tremendously because of the truncation. But our formulae give exact result.

Exciton dynamics from the mapping approach to surface hopping: comparison with Förster and Redfield theories.

We compare the recently introduced multi-state mapping approach to surface hopping (MASH) with the Förster and Redfield theories of excitation energy transfer. Whereas Förster theory relies on weak coupling between chromophores, and Redfield theory assumes the electronic excitations to be weakly coupled to fast chromophore vibrations, MASH is free from any perturbative or Markovian approximations. We illustrate this with an example application to the rate of energy transfer in a Frenkel-exciton dimer, showing that MASH interpolates correctly between the opposing regimes in which the Förster and Redfield results are reliable. We then compare the three methods for a realistic model of the Fenna-Matthews-Olson complex with a structured vibrational spectral density and static disorder in the excitation energies. In this case there are no exact results for comparison so we use MASH to assess the validity of Förster and Redfield theories. We find that Förster theory is the more accurate of the two on the picosecond timescale, as has been shown previously for a simpler model of this particular light-harvesting complex. We also explore various ways to sample the initial electronic state in MASH and find that they all give very similar results for exciton dynamics.

Therapeutic Communication Skills of Nursing Students in Clinical Practice

Communication in nursing is called therapeutic communication which is a way of fostering a therapeutic relationship between nurses and patients that has the aim of helping in the patient's healing process. This study aims to determine therapeutic communication skills for nursing students in clinical practice. The type and design used in this study is a type of quantitative research. Fisher's Exact Test results obtained p value (0.000) < α value (0.05). It can be concluded that there is a relationship of knowledge with therapeutic communication skills in the clinical practice of nursing students. Therapeutic knowledge and communication skills are essential in the clinical practice of student nursing. Therapeutic communication knowledge and skills are key elements in a student's nursing clinical practice. This is at the core of effective and empathetic patient care

PREDICTION OF CLIMATOLOGY USING NEURO-FUZZY TECHNIQUES

For social and economic activists, weather forecasting is the most talked about topic right now. Due to its applicability in a number of public and private industries, such as forestry, air traffic, agriculture, and maritime, it is also garnering widespread interest. Recent events have caused alterations in the climate. Occurat a rapid pace, rendering traditional weather forecasting techniques less accurate, more time-consuming, and erratic. To solve these, more advanced and effective weather forecast techniques are required. Challenges. Through actual evidence, we show that artificial neural networks generate significantly less deviations than GDAS assessment. Thus, almost exact weather forecast results are predicted. This article investigates the forecasting performance of neural networks in comparison to linear and polynomial approximations for a time series produced by the chaotic Mackey-Glass differential delay equation. There is one step ahead in the predicting horizon. A basic neural network with two neurons is used in a series of regressions with polynomial approximators, and the numerous correlation coefficients are compared. A nearly perfect forecast is produced by the neural network, a very basic neural network that outperforms polynomial expansions. Lastly, over a broad range of realizations, the neural network outperforms the other techniques in terms of precision.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Application of artificial neural networks for predicting lateral and uplift capacity of buried rectangular box carrying pipelines

Urban and offshore subterranean comprehensive pipelines are currently utilized frequently as infrastructure for fitting different engineering pipelines used for electricity, signal transmission, natural gas, petroleum, heating, water supply, and drainage systems. This study utilizes the finite element limit analysis (FELA) and artificial neural network (ANN) to evaluate the pull-out capacity factor of the buried rectangular box carrying the pipelines with internal inclined force in cohesive soil. In FELA, rigorous upper bound and lower bound solutions are performed to achieve the exact result. In this study, a dimensionless parametric analysis is carried out by considering the effect of five parameters viz. buried depth ratio, width-depth ratio, overburden factor, internal inclination angles, and interface factor on the pull-out capacity factor of a buried rectangular box, which can be called as a rectangular pipeline. Two significant samples of the investigated parameters are selected to evaluate the soil failure pattern (failure envelope) using the shear dissipation of soil failure for the buried rectangular pipeline subjected to inclined loading. Using the FELA results, the predictive equations are proposed using ANN, and then sensitivity analysis is performed to examine the sensitivity of each investigated essential parameter on the stability of the underground rectangular pipeline. The predictive ANN model for the buried rectangular pipe subjected to inclined loading is presented as design charts for practice and further investigation.

Modelling 2-D Heat Transfer Problem for Different Material Combination

The analysis of heat transfer problems can be highly complex due to factors such as temperature, position, and time. Most heat transfers are typically two-dimensional as conduction is often negligible in the third dimension. Two-dimensional heat conduction problems can be solved analytically or numerically. In steady-state conditions, the Laplace equation can be applied to solve two-dimensional heat conduction problems analytically, in which the separation of variables method is used to solve the Laplace equation under fixed boundary conditions to determine the temperature at a specific point. The Laplace equation plays a significant role in the solution of heat transfer problems, as it demonstrates the behavior of linear and non-linear equations in the computational fluid dynamics domain. Despite their inability to provide exact results at any point, numerical methods are superior to analytical methods when handling complex geometries with various boundary conditions. This project involves the development of a computational code using MATLAB to solve two-dimensional steady-state heat conduction problems using Gauss-Seidel iterations. Comparing analytical solutions from Excel with numerical solutions from MATLAB and ANSYS, specifically the developed MATLAB code, revealed an accuracy level of 99.902% for the Laplace equation. An analysis of the produced code from MATLAB found that it could solve two-dimensional steady-state heat conduction across different combinations of materials while allowing users to specify initial and boundary conditions to produce a contour plot similar to ANSYS.

Renormalized one-loop theory of correlations in disperse polymer blends.

Polymer blends are critical in many commercial products and industrial processes and their phase behavior is therefore of paramount importance. In most circumstances, such blends are formulated with samples of high dispersity, which have generally only been studied at the mean-field level. Here, we extend the renormalized one-loop theory of concentration fluctuations to account for blends of disperse polymers. Analyzing the short and long length-scale fluctuations in a consistent manner, various measures of polymer molecular weight and dispersity arise naturally in the free energy. Thermodynamic analysis in terms of moments of the molecular weight distribution(s) provides exact results for the inverse susceptibility and demonstrates that the theory is not formally renormalizable. However, physically motivated approximations allow for an "effective" renormalization, yielding (1) an effective interaction parameter, χe, which depends directly on the sample dispersities (i.e., Mw/Mn) and leaves the form of the mean-field spinodal unchanged, and (2) an apparent interaction parameter χa that depends on higher-order dispersity indices, for instance Mz/Mw, and characterizes the true limits of blend stability accounting for long-range off-critical fluctuations. We demonstrate the importance of dispersity on several example systems, including both "toy" models that may be realized in computer simulation and more realistic industrially relevant blends. We find that the effects of long-range fluctuations are particularly prominent in blends where the component dispersities are mismatched, especially when there is a small quantity of the high-dispersity species. This can be understood as a consequence of the shift in the critical concentration(s) from the monodisperse value(s).

Design Simulation of Multilayer Structures for the Development of Biosensor Based on Transverse Matrix Method

Simulation of electromagnetic wave propagation on layered metal-dielectric structures using analytical methods is a study that continues to be developed because it involves strict mathematical equations with exact results. In this study, the transfer matrix method was developed to analyze the dielectric-metal-metal-dielectric structure through the reflection profile. The emergence of the Surface Plasmons Resonance (SPR) phenomenon causes a minimum reflectance whose changes are analyzed through variations in optical and physical parameters. The simulation shows that the minimum reflectance angle occurs at an angle , indicating the existence of the SPR phenomenon in the condition . The choice of metal materials, namely gold and silver, resulted in a shift in the reflectance curve, and an increase in the thickness of the adhesive layer caused a shift in the reflectance curve towards a smaller angle. Changes in the refractive index of the upper dielectric material are important for biosensor development, namely a shift in the reflectance curve with a linear response at the interval of refractive index .

Chiral knife edge: A simplified rattleback to illustrate spin inversion.

We present the chiral knife edge rattleback, an alternative version of previously presented systems that exhibit spin inversion. We offer a full treatment of the model using qualitative arguments, analytical solutions as well as numerical results. We treat a reduced, one-mode problem which not only contains the essence of the physics of spin inversion, but that also exhibits an unexpected connection to the Chaplygin sleigh, providing insight into the nonholonomic structure of the problem. We also present exact results for the full problem together with estimates of the time between inversions that agree with previous results in the literature.

Neural network quantum states analysis of the Shastry-Sutherland model

We utilize neural network quantum states (NQS) to investigate the ground state properties of the Heisenberg model on a Shastry-Sutherland lattice using the variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality, and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.

The case against gravity

The myth of this “quasi-magical” gravity and its “effect at a distance” endures, and our brains remain “hardwired” to this perception. However, “quantum gravity” is increasingly closer to nullifying the importance of this mysterious force by treating it with the outmost contempt as a “negligible force.” We concur with this interpretation, and more: We claim in this article that not only is gravity negligible but also explain how it can be completely eliminated as one of four fundamental forces of the universe. This article is, in fact, a continuation, a widening, and a confirmation of an article I wrote in 1993, titled “The proportional expansion of each and every celestial body as the cause of gravitation” [Phys. Essays 6, 473 (1993)]. There, I had argued that not only is empty space expanding in the classically understood sense, but that all celestial bodies are also subject to an accelerated expansion. According to Einstein’s principle of equivalence, acceleration generates an opposite, “fictitious” force that is perceived as gravity. I had concluded there that “in terms of quantum mechanics, in this interpretation, the very existence of the predicted ‘graviton’ becomes extremely questionable.” Thirty years have passed since then. And in spite of the development of the Large Hadron Collider and its impressive capabilities, the above claim has been confirmed, and the graviton has been quietly replaced by the Higgs boson in the nomenclature of the Standard Model of elementary particles. In this article, we revisit Galileo and Newton to show that the latter could not have arrived at any other conclusion than an “attractive” force of gravity. We also ask why Albert Einstein did not apply his own principle of equivalence to the Newtonian interpretation of gravity. At the same time, some very “peculiar” interpretations have arisen since my above-mentioned article from 1993. “Dark matter” and “dark energy,” both very speculative conceptualizations, have been forced upon us to explain two phenomena: The anomaly discovered in the speed of rotation of the galaxies, and the accelerated expansion of the universe, despite the potentially shrinking effect of gravity. Of all the positive advances in physics, the most fundamental is the genesis of “quantum gravity,” the tenets of which relegate gravity to the level of a “negligible” force. This is followed by the characterization of the Higgs boson, which has replaced the graviton in the Standard Model. This conceptually translates into the fact that the gravitational mass has possibly been replaced by the inertial mass of celestial bodies. The confirmation and measurements of the accelerated expansion of the universe, while there is still some controversies regarding the exact results of these measurements. Finally, the most important discovery in this regard was recently made in 2019, regarding the exceptionally high pressure inside protons and neutrons, (1035 Pa). This lends credence to the concept of the expansion of these subatomic particles. The multiplicity of these hadrons (3.26 × 1080) and their ubiquitous distribution in the baryonic matter of the entire universe explain its expansion. These hadrons are highly “diluted” in empty space but, on the contrary, are highly clustered and concentrated inside solid matter, where this explains the difference in density between these media and the massive dissimilarity in the magnitudes of their relative accelerated expansion. It also explains why what we call “gravity” is not an active force, but merely the consequence of the density-proportional accelerated radial expansion of each and every celestial body. As such, this baryonic expansion explains the expansion of the universe as a whole, and replaces gravity as a concept as well as its perceived effects. By dint of its ubiquitous nature, this notion of baryonic expansion provides a bridge between the microscopic and macroscopic universe. Quantum physics reconnects with general relativity in this formalism and clarifies the “what and how.”

Some Exact Results for Non-Degenerate Generalized Turán Problems

The generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi(H)<\chi(F)$. There are several exact results on $\mathrm{ex}(n,H,F)$ when the extremal graph is a complete $(\chi(F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.

Exact Correlations in Topological Quantum Chains

Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions - the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit - an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.

Dimensionality reduction in diffusion–reaction systems

AbstractThe generalized solutions of the dimensionality reduction problem in diffusion–reaction systems have been found for the first time by rigorously decoupling probability functions. Numerically exact results are obtained by straightforwardly applying the generalized solution to three models: protein‐DNA binding model (3D‐1D reduction), surface‐mediated diffusion (3D‐2D reduction), and line‐mediated diffusion cases (2D‐1D reduction). The results are confirmed by Monte Carlo simulations.

Application of Yang homotopy perturbation transform approach for solving multi-dimensional diffusion problems with time-fractional derivatives

In this paper, we aim to present a powerful approach for the approximate results of multi-dimensional diffusion problems with time-fractional derivatives. The fractional order is considered in the view of the Caputo fractional derivative. In this analysis, we develop the idea of the Yang homotopy perturbation transform method (YHPTM), which is the combination of the Yang transform (YT) and the homotopy perturbation method (HPM). This robust scheme generates the solution in a series form that converges to the exact results after a few iterations. We show the graphical visuals in two-dimensional and three-dimensional to provide the accuracy of our developed scheme. Furthermore, we compute the graphical error to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

Collective radiative interactions in the discrete truncated Wigner approximation

Interfaces of light and matter serve as a platform for exciting many-body physics and photonic quantum technologies. Due to the recent experimental realization of atomic arrays at sub-wavelength spacings, collective interaction effects such as superradiance have regained substantial interest. Their analytical and numerical treatment is however quite challenging. Here we develop a semiclassical approach to this problem that allows to describe the coherent and dissipative many-body dynamics of interacting spins while taking into account lowest-order quantum fluctuations. For this purpose we extend the discrete truncated Wigner approximation, originally developed for unitarily coupled spins, to include collective, dissipative spin processes by means of truncated correspondence rules. This maps the dynamics of the atomic ensemble onto a set of semiclassical, numerically inexpensive stochastic differential equations. We benchmark our method with exact results for the case of Dicke decay, which shows excellent agreement. For small arrays we compare to exact simulations, again showing good agreement at early times and at moderate to strong driving, and to a second order cumulant expansion. We conclude by studying the radiative properties of a spatially extended three-dimensional, coherently driven gas and compare the coherence of the emitted light to experimental results.

Scattering of water waves by multiple rows of vertical thin barriers

The reflection and transmission of waves by a periodic array of thin fixed identical vertical barriers extending uniformly through the depth of the fluid is considered. The water wave problem, which also has analogues in acoustics and electromagnetics, involves scattering by a large finite number of equally spaced rows, each consisting of an array of barriers with a linear periodic arrangement extending to infinity. The main purpose of the work is to compare the results of two approximate methods of solution based on homogenisation techniques with exact results allowing us to determine when the results can be expected to be in good agreement. When no approximation is made, a method of solution (Discrete Model) describing the propagation of incident waves through the exact barrier arrangement is derived using Floquet–Bloch eigensolutions for the corresponding infinite periodic array. It is shown how the solution can be reduced to a simple pair of integral equations which results only from matching the region containing barrier arrays to the two exterior domains. Numerical solutions are, nevertheless, relatively complicated and two approximate models are developed which considerably simplify the numerical effort required. In one approximation (Continuum Model I) it is assumed only that the spacing between two adjacent rows of barriers is small and is complementary to the commonly-used wide-spacing approximation. The complication of the geometry is removed by replacing the governing equations and boundary conditions with an effective continuous medium and a simplified pair of integral equations now govern the solutions. In a second approximation (Continuum Model II), a low-frequency homogenisation approach based on a long-wave assumption is invoked and matching now leads to closed-form expressions for the reflection and transmission coefficients. Results concentrate on the comparison between different methods which allows us to assess the validity of each continuum model. This understanding will be important in developing approximate models that describe the operation of wave energy farms consisting of large arrays of oscillating flaps designed to absorb energy.

On Turán problems with bounded matching number

AbstractVery recently, Alon and Frankl initiated the study of the maximum number of edges in ‐vertex ‐free graphs with matching number at most . For fixed and , we determine this number apart from a constant additive term. We also obtain several exact results.