Type Operators Research Articles

Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type $$(P,\kappa ,{r},\gamma )$$ and Martelli’s fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained.

Characterization of the Flash-Induced Fluorescence Wave Phenomenon in the Coral Endosymbiont Algae, Symbiodiniaceae.

The dinoflagellate algae, Symbiodiniaceae, are significant symbiotic partners of corals due to their photosynthetic capacity. The photosynthetic processes of the microalgae consist of linear electron transport, which provides the energetic balance of ATP and NADPH production for CO2 fixation, and alternative electron transport pathways, including cyclic electron flow, which ensures the elevated ATP requirements under stress conditions. Flash-induced chlorophyll fluorescence relaxation is a non-invasive tool to assess the various electron transport pathways. A special case of fluorescence relaxation, the so-called wave phenomenon, was found to be associated with the activity of NAD(P)H dehydrogenase (NDH) in microalgae. We showed previously that the wave phenomenon existed in Symbiodiniaceae under acute heat stress and microaerobic conditions, however, the electron transport processes related to the wave phenomenon remained unknown. In this work, using various inhibitors, we show that (i) the linear electron transport has a crucial role in the formation of the wave, (ii) the inhibition of the donor side of Photosystem II did not induce the wave, whereas inhibition of the Calvin-Benson cycle accelerated it, (iii) the wave phenomenon was related to the operation of type II NDH (NDH-2). We therefore propose that the wave phenomenon is an important marker of the regulation of electron transport in Symbiodiniaceae.

Hecke equivariance of generalized Borcherds products of type O(2,1)

Recently, a weak converse theorem for Borcherds' lifting operator of type O(2,1) for Γ0(N) is proved and the logarithmic derivative of a modular form for Γ0(N) is explicitly described in terms of the values of Niebur-Poincaré series at its divisors in the complex upper half-plane. In this paper, we prove that the generalized Borcherds' lifting operator of type O(2,1) is Hecke equivariant under the extension of Guerzhoy's multiplicative Hecke operator on the integral weight meromorphic modular forms and the Hecke operator on half-integral weight vector-valued harmonic weak Maass forms. Additionally, we show that the logarithmic differential operator is also Hecke equivariant under the multiplicative Hecke operator and the Hecke operator on integral weight meromorphic modular forms. As applications of Hecke equivariance of the two operators, we obtain relations for twisted traces of singular moduli modulo prime powers and congruences for twisted class numbers modulo primes, including those associated to genus 1 modular curves.

Analysis of Bittern Recovery Facility Using Mixed-Integer Nonlinear Programming: Centralized, Decentralized, and Hybrid Scenarios

The desalination process to produce salts leaves wastewater with a high concentration of minerals called bittern. Most salt producers dump bittern straight away. Such disposal is dangerous to the ecosystem since bittern may increase the environmental salinity. Furthermore, bittern still has potential as it contains minerals that can be extracted and offers value. Consequently, further bittern treatment is necessary to reduce the environmental impact and create a circular economy. However, some specific requirements are needed in determining how to carry out this recovery process. In that, recovery managers need to know the most suitable type and optimal operation variables. These are essential to meet cost-effectiveness and environmental benefits. This research proposes a mixed-integer nonlinear programming (MINLP) model for analyzing the supply and demand of the bittern recovery. This study offers a model to optimize the trade-off between cost and benefits of the recovery process. There are three scenarios to determine the best bittern recovery practices: centralized, decentralized, and hybrid scenarios. The proposed models are then tested and analyzed for their sensitivity due to essential parameters. The numerical analysis has shown that a centralized scenario is best suited for a region with a low bittern supply. Moreover, a hybrid scenario is best suited in an area with a higher bittern supply. In addition, a decentralized scenario is the most suitable option for a region where the number of salt farms is high, and the location is far from the recovery facility.

Neural Network Approximation for Time Splitting Random Functions

In this article we present the multivariate approximation of time splitting random functions defined on a box or RN,N∈N, by neural network operators of quasi-interpolation type. We achieve these approximations by obtaining quantitative-type Jackson inequalities engaging the multivariate modulus of continuity of a related random function or its partial high-order derivatives. We use density functions to define our operators. These derive from the logistic and hyperbolic tangent sigmoid activation functions. Our convergences are both point-wise and uniform. The engaged feed-forward neural networks possess one hidden layer. We finish the article with a great variety of applications.

Generalized mathbb {XOR} Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural mathbb {XOR}-type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott mathbb {XOR} algebras strenghtening thus the relevance of these algebras to quantum theories.

Uniportal video-assisted thoracoscopy in pediatrics-initial experience.

It has become apparent that the endoscopic surgeries are rapidly developing, and they have become an essential part of every specialty of surgery. Single port thoracoscopic surgery is developing, enhancing the advantages of muti-portal video-assisted thoracoscopic surgery (VATS). Although becoming a well-recognised approach for adult patients, extremely limited literature exists concerning uniportal VATS among pediatric cases. This study aims to present our initial experience with this approach in a single tertiary hospital and extrapolate its feasibility and safety in this specific context. Perioperative parameters and surgical outcomes for all pediatric patients who underwent an intercostal or subxiphoid uniportal VATS surgery in our department in 2 years retrospectively reviewed. The median length of follow-up was 8 months. Sixty-eight pediatric patients underwent different uniportal VATS operation for different types of pathology. The median age was (3.5 years). Median operating time was 116 minutes. Three cases converted to open. The mortality rate was zero. The median length of stay was 5 days. Three patients presented complications. Three patients lost from follow-up. Despite literature data heterogeneity, these results provide support to the feasibility and applicability of uniportal VATS in the pediatric population. Further studies are required to explore the benefit of uniportal over multi-portal VATS (including chest wall deformities, cosmesis and quality of life).

P-Laplace Equation for the Grushin Type Operator

P-Laplace Equation for the Grushin Type Operator

Hypoelliptic and spectral estimates for some linear operators of Landau type

We are interested in the inhomogeneous Landau equation which describes the evolution of a particle density f = f (t, x, v) representing at time t $\ge$ 0, the density of particles at position x $\in$ R 3 and velocity v $\in$ R 3. The study is motivated by the linearization of the Landau equation near Maxwellian distribution. In this article, we establish hypoelliptic estimates, a localization of the spectrum and estimates of the resolvent of the linear Landau operator with hard potentials and Maxwellian molecules. The proof is based on a multiplier method and requires fine pseudo-differential calculus tools.

Gevrey Asymptotics for Logarithmic-Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter ϵ and singular in a complex time variable t at the origin. These equations combine differential operators of Fuchsian type in time t and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter ϵ known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to t and ϵ and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to ϵ on bounded sectors at the origin.

Neural network interpolation operators of multivariate functions

In this paper, we introduce a type of multivariate neural network interpolation operators Fn,σ(f) activated by some newly defined sigmoidal functions, and give both the direct and the converse results of the approximation by Fn,σ(f) for multivariate continuous functions. We also introduce a Kantorovich type variant of Fn,σ(f), and establish both the direct theorem and the converse theorem of approximation by the Kantorovich type operators in Lp spaces with 1≤p<∞. Finally, we give some numerical examples to demonstrate the validity of the obtained results, and apply our operators to the image super-resolution reconstruction.

Approximation by the Extended Neural Network Operators of Kantorovich Type

Based on the idea of integral averaging and function extension, an extended Kantorovich-type neural network operator is constructed, and its error estimate of approximating continuous functions is obtained by using the modulus of continuity. Furthermore, by introducing the normalization factor, the approximation property of the new version of the extended Kantorovich-type neural network (normalized extended Kantorovich-type neural network) operator is obtained in Lp[−1,1]. The numerical examples show that this newly proposed neural network operator has a better approximation performance than the classical one, especially at the endpoints of a compact interval.

A new formalization of Dirichlet-type spaces

Applications and findings involving the norm of the weighted Dirichlet space Dα are steadily being discovered. There are notable characterizations of Dα through estimations of its norm. Applications include determining the conditions for boundedness of a Bergman type operator, and characterizing functions in the Drury–Arveson space that induce bounded Hankel operators. This paper improves the norm estimation by extending the domain of relevant parameters, and establishing the asymptotic formula with precise constants.

Forelli-Rudin type operators on the space Lp,q,s(B) and some applications

Let φ be a holomorphic automorphism on the unit ball B of Cn, λ and τ be two real numbers. In this paper, we investigate the conditions such that Forelli-Rudin type operators Tλ,τCφ and |T|λ,τCφ are bounded on Lp,q,s(B) for p≥1 or from Hp,q,s(B) to Lp,q,s(B) for p>0. As some applications of |T|λ,τCφ, we give several characterizations for functions in Fp,q,s(B), and prove that Hp,q,s(B) is an automorphism invariant space.

Twisted regular representations for vertex operator algebras

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ1,σ2 be commuting finite-order automorphisms of V and let σ=(σ1σ2)−1. Among the main results, for any σ-twisted V-module W and any nonzero complex number z, we construct a weak σ1⊗σ2-twisted V⊗V-module Dσ1,σ2(z)(W) inside W⁎. Let W1,W2 be σ1-twisted, σ2-twisted V-modules, respectively. We show that P(z)-intertwining maps from W1⊗W2 to W⁎ are the same as homomorphisms of weak σ1⊗σ2-twisted V⊗V-modules from W1⊗W2 into Dσ1,σ2(z)(W). We also show that a P(z)-intertwining map from W1⊗W2 to W⁎ is equivalent to an intertwining operator of type (W′W1W2), which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ-twisted V-module M with τ any finite-order automorphism of V, the coefficients of the q-graded trace function lie in Dτ,τ−1(−1)(V) and generate a τ⊗τ−1-twisted V⊗V-submodule isomorphic to M⊗M′.

Kac-Schwarz operators of type B, quantum spectral curves, and spin Hurwitz numbers

Given a tau-function τ(t) of the BKP hierarchy satisfying τ(0)=1, we discuss the relation between its BKP-affine coordinates on the isotropic Sato Grassmannian and its BKP-wave function. Using this result, we formulate a type of Kac-Schwarz operators for τ(t) in terms of BKP-affine coordinates. As an example, we compute the affine coordinates of the BKP tau-function for spin single Hurwitz numbers with completed cycles, and find a pair of Kac-Schwarz operators (P,Q) satisfying [P,Q]=1. By doing this, we obtain the quantum spectral curve for spin single Hurwitz numbers.

Holder estimates and asymptotic behavior for degenerate elliptic equations in the half space

In this article we investigate the asymptotic behavior at infinity of viscosity solutions to degenerate elliptic equations. We obtain Holder estimates, up to the flat boundary, by using the rescaling method. Also as a byproduct we obtain a Liouville type result on Baouendi-Grushin type operators.

Application of macros for automated performance of the same type operations when simulating SIW-based slotted waveguide antennas in Ansys HFSS CAD

SIW slotted waveguide antennas contain numerous equidistantly spaced vias and many slots distributed following a particular field distribution. Ansys HFSS CAD user must manually repeat the same type of operations when creating slots or vias. Automating these routine operations is advisable to eliminate design errors and speed up model construction. This paper aims at complete automation of SIW resonant slotted waveguide antenna arrays modelling in CAD Ansys HFSS, including model construction, running simulations and displaying simulation results. There are multiple procedures for slotted waveguide antenna synthesis. For small waveguide antenna arrays with a number of elements less than 15, the internal and external slots coupling consideration is beneficial, while for large waveguide antennas, no mutual coupling assumption can be made (so-called “energy method”). The energy method has been used to determine the coordinates of the longitudinal slots on the broad wall of the SIW based on a given side lobe level. The slot resonant length has been determined using substrate permittivity and operating frequency. The number of slots has been obtained based on the required half-power beamwidth. SIW dimensions have been calculated using the given operating frequency and the dimensions of the filled rectangular waveguide. An IronPython script has been developed to automate the antenna's model construction, automatic simulation and determination of its characteristics. All script commands have been taken from the “HFSS Scripting Guide” document integrated into Ansys HFSS. The tangible outcome of this work is the development of the IronPython script for fully automated modelling of SIW resonant slotted waveguide antennas. The script builds an antenna model from scratch based on a given antenna characteristics, assigns excitation port and radiation boundary conditions, and displays the simulation results – a radiation pattern at a fixed frequency. The script significantly reduces the antenna design time by eliminating the need to repeat the object creation operations.

Accurate Methods for the Analysis of Strong-Drive Effects in Parametric Gates

The ability to perform fast, high-fidelity entangling gates is a requirement for a viable quantum processor. In practice, achieving fast gates often comes with the penalty of strong-drive effects that are not captured by the rotating-wave approximation. These effects can be analyzed in simulations of the gate protocol, but those are computationally costly and often hide the physics at play. Here, we show how to efficiently extract gate parameters by directly solving a Floquet eigenproblem using exact numerics and a perturbative analytical approach. As an example application of this toolkit, we study the space of parametric gates generated between two fixed-frequency transmon qubits connected by a parametrically driven coupler. Our analytical treatment, based on time-dependent Schrieffer-Wolff perturbation theory, yields closed-form expressions for gate frequencies and spurious interactions, and is valid for strong drives. From these calculations, we identify optimal regimes of operation for different types of gates including $i$swap, controlled-Z, and cnot. These analytical results are supplemented by numerical Floquet computations from which we directly extract drive-dependent gate parameters. This approach has a considerable computational advantage over full simulations of time evolutions. More generally, our combined analytical and numerical strategy allows us to characterize two-qubit gates involving parametrically driven interactions, and can be applied to gate optimization and cross-talk mitigation such as the cancelation of unwanted $ZZ$ interactions in multiqubit architectures.

Film symbiosis: Embodied spectatorship and sensory (auto)ethnography in Lucien Castaing-Taylor and Ilisa Barbash’s Sweetgrass

This article explores the 2009 observational-ethnographic documentary film Sweetgrass by Lucien Castaing-Taylor and Ilisa Barbash. Focusing on visual cues within the observational form (a film that documents the herding of sheep across the Absaroka-Beartooth Mountains for the last time), the article draws on ethnographic and autoethnographic strains to explore the relationship between the embodied spectator and on-screen animal. Sweetgrass, in addition to several moving image works, is explored as sensory ethnography that incorporates a form of spectator address based on the idea of symbiosis. The article situates ‘the nonhuman stare’ as fundamental to this, drawing on the film phenomenology of Vivian Sobchak to consolidate this view. Sweetgrass, the article maintains, aestheticizes the non-human for specific reasons. It adds an ethical purpose to the observational documentation of herding sheep across the mountains – a salvage operation of type – and a sensory experience of ‘living with’ associated with the farming culture represented on-screen. In addition to exploring the film as a sensory object in this way, the article devises a methodology bringing autoethnographic research concerning symbiotic human–non-human animal relationships, in line with explorations of the embodied film spectator.