Discrete Part Research Articles

On the positivity and asymptotic stability of the fractional 2D hybrid linear Roesser model

The aim of this paper is to introduce a new 2D fractional hybrid linear system described by the Roesser model with the conformable fractional derivative in the continuous time variables and the fractional difference in the discrete part. The solution of the considered model is derived. The positivity conditions are then extracted and the Cayley–Hamilton theorem is extended. A new necessary and sufficient conditions for the asymptotic stability of the proposed positive fractional 2D hybrid linear model are then established. Finally, some numerical examples are presented to show the applicability and the effectiveness of the presented methods.

Spectral representation in Klein space: simplifying celestial leaf amplitudes

In this paper, we explore the spectral representation in Klein space, which is the split (2, 2) signature flat spacetime. The Klein space can be foliated into Lorentzian AdS3/ℤ slices, and its identity resolution has continuous and discrete parts. We calculate the identity resolution and the Plancherel measure in these slices. Using the foliation of Klein space into the slices, the identity resolution, and the Plancherel measure in each slice, we compute the spectral representation of the massive bulk-to-bulk propagator in Klein space. It can be expressed as the sum of the product of two massive (or tachyonic) conformal primary wavefunctions, with both continuous and discrete parts, and sharing a common boundary coordinate. An interesting point in Klein space is that, since the identity resolution has discrete and continuous parts, a new type of conformal primary wavefunction naturally arises for the massive (or tachyonic) case. For the conformal primary wavefunctions, both the discrete and continuous parts involve integrating over the common boundary coordinate and the real (or imaginary) mass. The conformal dimension is summed in the discrete part, whereas it is integrated in the continuous part. The spectral representation in Klein space is a computational tool to derive conformal block expansions for celestial amplitudes in Klein space and its building blocks, called celestial leaf amplitudes, by integrating the particle interaction vertex over a single slice of foliation.

Modelling and Diagnosis of Hybrid Dynamic Systems by a Multi-Model Approach

Fault diagnosis is an essential task in ensuring the smooth operation of complex dynamic systems. The consequences of faults can be serious, leading to loss of life, harmful emissions to the environment, high repair costs and economic losses caused by unplanned production line stoppages. The work developed in this paper concerns the modeling and diagnosis of faults (sensor faults, system faults, actuator faults) in hybrid dynamic systems using our multi-model approach (which combines two sub-models, one continuous and the other discrete). The aim is to integrate three well-known tools in the literature: the Bond Graph, the Observer and the Timed Automata, to design a global diagnostic model. The hybrid dynamic system is modeled by connecting the tools for the continuous part, i.e. the bond graph and the observer, to the timed automata for the discrete part. The resulting model is used for fault diagnosis in two stages: The first is fault detection by analyzing the residuals generated by the system output and that of the observer. The second step involves fault localization, which results from analysis of the signature matrix and temporal identification of the system. The proposed method combines the advantages of these tools to obtain the best performance, particularly in the fault location phase. The simulation results prove the effectiveness of the proposed model for the hybrid dynamic system. Moreover, these results also evaluate the performance of the proposed diagnostic approach while reducing non-detections, detection delays and false alarms.

Active runout compensation using the guide elements on metal band saws for a longer tool life and reduced material loss

AbstractAt a time of rising energy and material costs, manufacturing process efficiency is becoming increasingly important. This is one reason why cutting and especially sawing processes, usually the first step in most manufacturing chains in discrete part production, have to be investigated more intensively. Due to problems with runout and poor surface finishes, raw material is conventionally cut to oversize by either circular sawing or band sawing. This oversize has to be removed by following processes, costing extra energy and wasting material. Since the problem of runout increases with increasing tool wear due to higher deflections of the thin and compliant tools, an even larger oversize is required. This paper describes an approach to reduce the need of oversize even with increasing tool wear in band sawing by tilting the saw band in order to compensate for tool deflections during cutting. To achieve this, it is necessary to measure and understand the saw band runout. The next step is to present a system design and controller for tilting the saw band. Finally, tests are carried out to analyse the effectiveness of the system. In addition, the approach allows the use of increased process parameters to the end of the tool life without losing more material. The tool can therefore be used productively for a longer time period.

Likelihood and decoding problems for mixed space hidden Markov model

Abstract A Hidden Markov Model (HMM) is a couple of stochastic processes: A hidden Markov process and an observed emission process. Generally, the HMMs are used to study the hidden behavior of random systems through some observed emission sequences generated by the phenomenon under study. In this frame-work, we propose to solve the likelihood and the decoding problems of HMMs whose state space is composed of a continuous part and a discrete part. We adapt forward, backward and Viterbi algorithms to the case of our proposal. Numerical examples and Monte Carlo simulations are considered to show the efficiency and the adaptation of the algorithms for the proposed model.

Seismic Response of Vertical Hybrid Concrete/Steel Frames Considering Soil–Structure Interaction

The aim of this study is to investigate the seismic behavior of concrete/steel mixed structures. In engineering praxis, many buildings consist of two parts: one made of reinforced concrete and the other made of steel. There are several difficulties in the code-based seismic design of these structures due to the different dynamic responses of each discrete part. Seismic design codes, such as the IBC and Eurocode 8, do not provide instructions for structures consisting of two parts. In addition, they use a single-loading scenario, but there are many locations that are affected by more than one earthquake in a short period. Another drawback is that recent provisions do not consider soil–structure interaction effects. The specific issue addressed here is the seismic response of mixed structures, which is evaluated through inelastic time–history analysis. More specifically, the response indices involve height-wise distributions for peak interstory drift ratios, maximum floor horizontal displacements, maximum floor accelerations, and plastic hinge formations in the frame elements when they are subjected to seismic sequences of earthquakes, as well as in far fault ground motions for different soil types. The results reveal that sequential ground motions lead to increased displacement demands, and they affect the permanent displacements. This phenomenon appears in both cases of stiff and flexible soil, as well as for both regular and irregular frames. It is found that soil–structure interaction generally leads to lower values of IDR, and maximum horizontal displacement and acceleration in comparison with the case of rigid soil assumptions.

Investigation of the effects of cavitation on different media in a device with a discrete secondary

Preserving food and improving the quality of drinking water are of paramount importance to the public. Despite the very different objectives of food preservation and drinking water disinfection, these problems have in common the destruction of microorganisms. In this regard, the actual problem, which has been the focus of research in recent years, is the development of new technologies capable of non-thermal treatment of liquid substances and foodstuffs and not using chemical reagents. Hydrodynamic cavitation can be a promising technology for non-thermal treatment of liquid substances, including wastewater and food liquid products. This paper presents the results of our own study of the effect of cavitation on microorganisms under given conditions in a device with a discrete secondary part.

Symmetries and spectral statistics in chaotic conformal field theories. Part II. Maass cusp forms and arithmetic chaos

We continue the study of random matrix universality in two-dimensional conformal field theories. This is facilitated by expanding the spectral form factor in a basis of modular invariant eigenfunctions of the Laplacian on the fundamental domain. The focus of this paper is on the discrete part of the spectrum, which consists of the Maass cusp forms. Both their eigenvalues and Fourier coefficients are sporadic discrete numbers with interesting statistical properties and relations to analytic number theory; this is referred to as ‘arithmetic chaos’. We show that the near-extremal spectral form factor at late times is only sensitive to a statistical average over these erratic features. Nevertheless, complete information about their statistical distributions is encoded in the spectral form factor if all its spin sectors exhibit universal random matrix eigenvalue repulsion (a ‘linear ramp’). We ‘bootstrap’ the spectral correlations between the cusp form basis functions that correspond to a universal linear ramp and show that they are unique up to theory-dependent subleading corrections. The statistical treatment of cusp forms provides a natural avenue to fix the subleading corrections in a minimal way, which we observe leads to the same correlations as those described by the [torus]×[interval] wormhole amplitude in AdS3 gravity.

Topological Dimensions from Disorder and Quantum Mechanics?

We have recently shown that the critical Anderson electron in D=3 dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension dIR≈8/3. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density p(d) for dimension d to be accessed by the electron. We find that p(d) has a strong peak at d very close to two. In fact, our data suggest that p(d) is non-zero on the interval [dmin,dmax]≈[4/3,8/3] and may develop a discrete part (δ-function) at d=2 in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although dIR is based on effective counting, of which p(d) has no a priori knowledge, dIR≥dmax is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of dIR≈2 in Dirac near-zero modes of thermal quantum chromodynamics is emphasized.

Intrinsic Diophantine approximation on circles and spheres

AbstractWe study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in or and three spheres embedded in or . We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of and . Thanks to prior work of Asmus L. Schmidt on the spectra of and , we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.

To the issue of decomposition of mathematical models of disturbed motion of complex discrete-continuous dynamic systems

Problem. Usually, complex dynamic systems are characterized by a high order of equations in their mathematical models and intricate interconnections among subsystems that constitute the complex dynamic system. Naturally, the analysis and synthesis of complex dynamic systems require the use of powerful computational systems with vast memory and high processing speed. Goal. The aim is quantitative assessment of the influence of the continuum part of a dynamic system on the behavior of its discrete part and the possibility to perform decomposition of the mathematical model of perturbed motion of a discrete-continuum system based on this assessment. Methodology. The measures are aimed at addressing issues that are based on the discrete-continuum capabilities and their decomposition. Results. For three types of complex dynamic systems described by infinite-dimensional systems of ordinary differential equations, quantitative assessments of the influence of the continuum part on the discrete part of the system have been proposed. These assessments determine the decomposition of the mathematical models of perturbed motion for such systems. Originality. Simplified mathematical models for three types of discrete-continuous dynamic systems were obtained. Practical value. The obtained results can be recommended for the study of the specialized course on the motion characteristics of cargo trucks during the transportation of liquids in tanks. Through optimization using the MATLAB software package, it is possible to simulate various parameters of both the cargo truck and the cargo being transported.

Multi-region dynamic economic dispatch with active power real-time balance constraint

In this paper, a multi-region dynamic economic dispatch model with power balance constraint is constructed to seek the actual reserve demand and reasonable reserve allocation method of the system by relying on the dynamic response process of the units to the power balance of the interconnected system. For this kind of dynamic optimization problem with differential equation constraints, a discrete-optimization rolling solution strategy based on scale-optimization is proposed. First, in the discrete part of the dispatching model, basing on the correlation between the continuous quantities discrete scale and the active real-time balancing effect of the dispatching scheme, a continuous quantities discrete scale optimization strategy is proposed to maintain the safe and economic operation of the system; then, for the convergence problem of the solution of the dispatching model after discretization, this paper introduces the idea of decentralized coordination optimization and the operator splitting quadratic programming (OSQP), and through the limited information exchange between the regional grid and the coordinator, the consistent coordination of consistent variables is achieved. The simulation results verify the effectiveness of the model and the solution strategy.

Fluid–rigid body interaction in a compressible electrically conducting fluid

AbstractWe consider a system of multiple insulating rigid bodies moving inside of an electrically conducting compressible fluid. In this system, we take into account the interaction of the fluid with the bodies as well as with the electromagnetic fields trespassing both the fluid and the solids. The main result of this paper yields the existence of weak solutions to the system. While the mechanical part of the problem can be dealt with via a classical penalization method, the electromagnetic part requires an approximation by means of a hybrid discrete–continuous in time system: The discrete part of the approximation enables us to handle the solution‐dependent test functions in our variational formulation of the induction equation, whereas the continuous part makes sure that the nonnegativity of the density and subsequently a meaningful energy inequality is preserved in the approximate system.

Contribution of solitons to enhanced rogue wave occurrence in shallow depths: a case study in the southern North Sea

Abstract. The shallow waters off the coast of Norderney in the southern North Sea are characterised by a higher frequency of rogue wave occurrences than expected. Here, rogue waves refer to waves exceeding twice the significant wave height. The role of nonlinear processes in the generation of rogue waves at this location is currently unclear. Within the framework of the Korteweg–de Vries (KdV) equation, we investigated the discrete soliton spectra of measured time series at Norderney to determine differences between time series with and without rogue waves. For this purpose, we applied a nonlinear Fourier transform (NLFT) based on the Korteweg–de Vries equation with vanishing boundary conditions (vKdV-NLFT). At measurement sites where the propagation of waves can be described by the KdV equation, the solitons in the discrete nonlinear vKdV-NLFT spectrum correspond to physical solitons. We do not know whether this is the case at the considered measurement site. In this paper, we use the nonlinear spectrum to classify rogue wave and non-rogue wave time series. More specifically, we investigate if the discrete nonlinear spectra of measured time series with visible rogue waves differ from those without rogue waves. Whether or not the discrete part of the nonlinear spectrum corresponds to solitons with respect to the conditions at the measurement site is not relevant in this case, as we are not concerned with how these spectra change during propagation. For each time series containing a rogue wave, we were able to identify at least one soliton in the nonlinear spectrum that contributed to the occurrence of the rogue wave in that time series. The amplitudes of these solitons were found to be smaller than the crest height of the corresponding rogue wave, and interaction with the continuous wave spectrum is needed to fully explain the observed rogue wave. Time series with and without rogue waves showed different characteristic soliton spectra. In most of the spectra calculated from rogue wave time series, most of the solitons clustered around similar heights, but the largest soliton was outstanding, with an amplitude significantly larger than all other solitons. The presence of a clearly outstanding soliton in the spectrum was found to be an indicator pointing towards the enhanced probability of the occurrence of a rogue wave in the time series. Similarly, when the discrete spectrum appears as a cluster of solitons without the presence of a clearly outstanding soliton, the presence of a rogue wave in the observed time series is unlikely. These results suggest that soliton-like and nonlinear processes substantially contribute to the enhanced occurrence of rogue waves off Norderney.

An efficient derivation of spatial Green's function of open-radiating rectangular cavity using CGF-CI technique

ABSTRACT A new technique for efficient calculation of spatial Green's function of open-radiating rectangular cavity is presented. The method is based on separability assumption of the original structure and characteristic Green's function method. Using complex image (CI) technique in spectral domain and well-known Weyl's identity, a closed-form relation for spatial Green's function can be derived. To obtain more accurate closed-form Green's function, before applying CI, discrete spectrum contribution part of the derived relation is modified by replacing more accurate reflection coefficients of surface wave modes from walls of the open-radiating cavity. Verification of the derivations of the proposed method is shown for dielectric loaded metallic box which is a separable structure. Method of moment + multilevel fast multipole method solution and also exact numerical integration approach for open-radiating rectangular cavity are considered to search the accuracy as well as speed of the proposed method. It is shown that the main advantage of the proposed method lies in its speed as well as its simple implementation while the accuracy is preserved.

Music Deferred: A Reappraisal of the Legacy of Pierre Schaeffer's Treatise on Musical Objects (1966)

AbstractThis article reinterprets the tension between sound and music in Pierre Schaeffer's 1966 Treatise on Musical Objects. Schaeffer famously insisted that the Treatise did not address music or composition; scholars have therefore engaged with it primarily as a theoretical text on sound and listening. In this article, however, I argue that the denial and deferral of music throughout the Treatise should be considered a discrete and key part of its theoretical contributions. By the early 1960s, Schaeffer's aesthetic frustration with the practice of musique concrète had blossomed into something of an ethical imperative and paradox. He saw it necessary to suspend all musical activity in the present, so as to salvage music's future. This dynamic is key to understanding Schaeffer's controversial and influential calls for the deferral of cultural responses to sound in the Treatise, as illustrated by the practices of ‘deconditioning’ and ‘reconditioning’.

Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

We study the sequence of polynomials {Sn}n≥0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨f,g⟩s=∫f(x)g(x)dμ(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where μ is a finite Borel measure whose support suppμ is an infinite set of the real line, λj,k≥0, and the mass points ci, i=1,…,N are real values outside the interior of the convex hull of suppμ (ci∈R\Ch(supp(μ))∘). Under some restriction of order in the discrete part of ⟨·,·⟩s, we prove that Sn has at least n−d* zeros on Ch(suppμ)∘, being d* the number of terms in the discrete part of ⟨·,·⟩s. Finally, we obtain the outer relative asymptotic for {Sn} in the case that the measure μ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨·,·⟩s.

Quasi-infinite divisibility of a class of distributions with discrete part

We consider distributions on R \mathbb {R} that can be written as the sum of a non-zero discrete distribution and an absolutely continuous distribution. We show that such a distribution is quasi-infinitely divisible if and only if its characteristic function is bounded away from zero, thus giving a new class of quasi-infinitely divisible distributions. Moreover, for this class of distributions we characterize the existence of the g g -moment for certain functions g g .

Extranodal Extension Improves AJCC-8 Accuracy in HPV+ Oropharyngeal Cancer in a High-Risk Population.

The American Joint Committee on Cancer's 8th edition (AJCC-8) separates oropharyngeal squamous cell carcinomas (OPSCCs) into human papillomavirus-positive (HPV+) tumors and HPV-negative tumors. Although AJCC-8 improves prognostic prediction for survival for the majority of HPV+ OPSCC, outliers are still encountered. The goal of this manuscript is to validate the AJCC-8 as a better metric of survivability than the AJCC-7 in an historically under-served rural population with confounding variables, such as tobacco use, alcohol consumption, and poor health care access and to analyze the role of extranodal extension (ENE) in this population. Retrospective cohort study. Compared to AJCC-7, AJCC-8 had a higher odds ratio (OR) for predicting mortality of stage IV HPV+ OPSCCs versus stages I-III. On multivariate analysis, HPV+ OPSCCs with ENE had a higher OR of mortality compared to ENE- OPSCCs. In addition, HPV+ OPSCC patients with a Charlson Comorbidity Index (CCI) > 3 had a higher OR of mortality compared to those with a CCI ≤ 3. Patients with Medicaid/self-pay status had a higher OR of mortality compared to those with private insurance/Medicare. Finally, patients from rural populations had a higher OR of presenting with stage IV disease, a CCI > 3, and Medicaid/self-pay status. Despite not being a discrete part of the AJCC-8 staging rubric, ENE was found to have a significant impact on mortality among this population, whereas tobacco use had no effect. Rural patients were more likely to present with stage IV disease, CCI > 3, and Medicaid/self-pay status. Stage IV disease was also associated with a higher OR of mortality. 4 Laryngoscope, 133:2621-2626, 2023.

Optimum Complexing of Measurements when Maintaining a Maneuvering Object in Statistically Uncertain Situations

The problem of synthesizing optimal and quasi-optimal algorithms for complex information processing is solved using the methods of Markov theory for estimating random processes when maintaining a maneuvering object and two-channel vector observation with violations in statistically uncertain situations. The problem is solved in relation to a discrete-continuous Markov process for the case when its continuous part is a vector Markov sequence, and the discrete part is characterized by a three-component discrete Markov process, each component of which is described by a Markov chain to several positions. A block diagram of quasi-optimal complex information processing is given. Using a simple example, simulation modeling shows the performance of a quasi-optimal algorithm in statistically uncertain situations.