Similar Operators Research Articles

Linear bijective maps preserving spectral functions on pairs of similar operators

Let X be an infinite-dimensional complex Banach space, and let B(X) be the algebra of all linear and bounded operators on X. In this paper, we characterize linear bijective maps φ on B(X) having the property that the spectral radius (respectively, the spectrum) of φ(A) equals the spectral radius (respectively, the spectrum) of φ(B) for each pair of similar operators A,B∈B(X). As a corollary, we obtain a characterization of linear bijective maps on B(X) preserving the equality of the spectrum.

Operator Hypergeometric Functions

We consider operator hypergeometric functions 1F2(∙) and 2F3(∙) constructed by an unbounded operator. Using these functions, we solve Cauchy problems for singular integro-differential equations. A new pair of similar operators is given.

Image Denoising Based on Quantum Calculus of Local Fractional Entropy

Images are frequently disrupted by noise of all kinds, making image restoration very challenging. There have been many different image denoising models proposed over the last few decades. Some models preserve the image’s smooth region, while others preserve the texture margin. One of these methods is by using quantum calculus. Quantum calculus is a branch of mathematics that deals with the manipulation of functions and operators in a quantum mechanical setting. It has been used in image processing to improve the speed and accuracy of image-processing algorithms. In quantum computing, entropy can be defined as a measure of the disorder or randomness of a quantum state. The concept of local fractional entropy has been used to study a wide range of quantum systems. In this study, an image denoising model is proposed based on the quantum calculus of local fractional entropy (QC-LFE) to remove a Gaussian noise. The local fractional entropy is used to estimate the image pixel probability, while the quantum calculus is used to estimate the convolution window mask for image denoising. A processing fractional mask with n x n elements was used in the suggested denoising algorithm. The proposed image denoising algorithm uses mask convolution to process each corrupted pixel one at a time. The proposed denoising algorithm’s effectiveness is assessed using peak signal-to-noise ratio and visual perception (PSNR). The experimental findings show that, compared to other similar fractional operators, the proposed method can better preserve texture details when denoising.

Low dispersion finite volume/element discretization of the enhanced Green–Naghdi equations for wave propagation, breaking and runup on unstructured meshes

We study a hybrid approach combining a finite volume (FV) and a finite element (FE) method to solve a fully-nonlinear and weakly-dispersive depth averaged wave propagation model. The FV method is used to solve the underlying hyperbolic shallow water system, while a standard P1 finite element method is used to solve the elliptic system associated to the dispersive correction. We study the impact of several numerical aspects: the impact of the reconstruction used in the hyperbolic phase; the representation of the FV data in the FE method used in the elliptic phase and their impact on the theoretical accuracy of the method; the well-posedness of the overall method. For the first element we proposed a systematic implementation of an iterative reconstruction providing on arbitrary meshes up to third order solutions, full second order first derivatives, as well as a consistent approximation of the second derivatives. These properties are exploited to improve the assembly of the elliptic solver, showing dramatic improvement of the finale accuracy, if the FV representation is correctly accounted for. Concerning the elliptic step, the original problem is usually better suited for an approximation in H(div) spaces. However, it has been shown that perturbed problems involving similar operators with a small Laplace perturbation are well behaved in H1. We show, based on both heuristic and strong numerical evidence, that numerical dissipation plays a major role in stabilizing the coupled method, and not only providing convergent results, but also providing the expected convergence rates. Finally, the full mode, coupling a wave breaking closure previously developed by the authors, is thoroughly tested on standard benchmarks using unstructured grids with sizes comparable or coarser than those usually proposed in literature.

Spectral analysis of an even order differential operator with square integrable potential

By using the method of similar operators, we study an even order differential operator with periodic, antiperiodic, and Dirichlet boundary conditions. We obtain eigenvalue asymptotics of this operator and estimates for its spectral decompositions and spectral projections. We also describe the asymptotic behavior of the corresponding analytic semigroup of operators.

Weighted Lorentz spaces: Sharp mixed Ap − A∞ estimate for maximal functions

We prove the sharp mixed Ap−A∞ weighted estimate for the Hardy-Littlewood maximal function in the context of weighted Lorentz spaces, namely‖M‖Lp,q(w)≲p,q,n[w]Ap1p[σ]A∞1min⁡(p,q), where σ=w11−p. Our method is rearrangement free and can also be used to bound similar operators, even in the two-weight setting. We use this to also obtain new quantitative bounds for the strong maximal operator and for M in a dual setting.

Quantitative bounds on impedance-to-impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

The spectral geometry of de Sitter space in asymptotic safety

Within the functional renormalization group approach to Background Independent quantum gravity, we explore the scale dependent effective geometry of the de Sitter solution dS4. The investigation employs a novel approach whose essential ingredient is a modified spectral flow of the metric dependent d’Alembertian, or of similar hyperbolic kinetic operators. The corresponding one-parameter family of spectra and eigenfunctions encodes information about the nonperturbative backreaction of the dynamically gravitating vacuum fluctuations on the mean field geometry of the quantum spacetime. Used as a diagnostic tool, the power of the spectral flow method resides in its ability to identify the scale dependent subsets of field modes that supply the degrees of freedom which participate in the effective field theory description of the respective scale. A central result is that the ultraviolet of Quantum Einstein Gravity comprises far less effective degrees of freedom than predicted (incorrectly) by background dependent reasoning. The Lorentzian signature of dS4 is taken into account by selecting a class of renormalization group trajectories which are known to apply to both the Euclidean and a Lorentzian version of the approach. Exploring the quantum spacetime’s spatial geometry carried by physical fields, we find that 3-dimensional space disintegrates into a collection of coherent patches which individually can, but in their entirety cannot be described by one of the effective average actions occurring along the renormalization group trajectory. A natural concept of an entropy is introduced in order to quantify this fragmentation effect. Tentatively applied to the real Universe, surprising analogies to properties of the observed cosmic microwave background are uncovered. Furthermore, a set of distinguished field modes is found which, in principle, has the ability to transport information about the asymptotic fixed point regime from the ultraviolet, across almost the entire “scale history”, to cosmological distances in the observed Universe.

On the Bari basis property for even-order differential operators with involution

By using the method of similar operators we study even-order differential operators with involution. The domain of these operators are defined by periodic and antiperiodic~boundary conditions. We obtain estimates for spectral projections and we prove the Bari basis property for the system of eigenfunctions and associated functions

Thermal analysis to investigate the effects of operating parameters on conventional cupola furnace efficiency

Background: Thermal analysis was performed on a conventional 6.5 m high dome with a 0.65 m diameter at Nigeria Railway Corporation. Energy and thermal analyses were performed on the furnace to improve efficiency and minimize energy losses in the system. Results: The energy efficiency recorded was 55%. The losses in the furnace amount to 45%, the highest value being the heat loss to the combustion gas at 34%. A parametric study was examined to see the effect of some design and operational parameters on furnace efficiency through energy savings and the study found a positive trend. The parameters include the flue gas temperature, the combustion air inlet temperature, the thickness, and the emissivity of the refractory wall. A decrease in the temperature of the flue gases increased the efficiency of the furnace to 73%, an increase in the air inlet temperature increased the efficiency of the furnace to 61%, and an increase in the thickness of the refractory wall increased the efficiency of the furnace. Oven to 56% and a decrease in emissivity of the refractory wall increased the efficiency of the furnace to 56%. Conclusions: The studied parameters were further combined and their net effect showed a possible increase in furnace efficiency from 55% to 78%. The maximum efficiency of 78% was determined with respect to the minimum/maximum estimated values of the parameters studied, as more metals are melted with the combined energy saved from a variation of these parameters. This implies that cupola furnace or similar furnace designers or operators should be guided on the preheating conditions, thickness, and emissivity of the refractory wall to be used with other parametric combinations to be adopted to ensure energy efficiency and improvement in such furnaces.

Scalable NAS with factorizable architectural parameters

Neural Architecture Search (NAS) replaces manually designed networks with automatically searched networks and has become a hot topic in machine learning and computer vision. One key factor of NAS, scaling up the search space by adding operators, could help bring about more possibilities for effective architectures. However, existing works require high search costs and are prone to make confused selections caused by competition among similar operators. This paper presents a scalable NAS that utilizes a factorized method, which improves existing art in the following aspects. (1) Our work explores a broader search space. We construct an ample search space through the cartesian product of activation operators and regular operators. (2) Our work experiences a limited computation burden even though searching in a more extensive search space. We factorize a search space into two subspaces and adopt different architectural parameters to control corresponding subspaces. (3) Our work avoids competition among similarly combined operators (e.g., ReLU &sep-conv-3x3 and ReLU6 &sep-conv-3x3 and PReLU &sep-conv-3x3). That is because our architectural parameters are optimized sequentially. Experimental results show that our approach achieves SOTA on CIFAR10 (2.38% test error) and ImageNet (23.8% test error). Furthermore, the excellent performance of 35.8% (AP) and 55.7% (AP50) on COCO further proves the superiority of our factorized method.

Method of Similar Operators in the Study of Spectral Properties of Perturbed First-Order Differential Operators

In this paper, we consider first-order differential operators with periodic boundary conditions acting in a Hilbert space of square summable functions on the segment [0, ω] perturbed by Hilbert– Schmidt integral operators. A similarity transformation of the original operator to the operator of the block-diagonal structure is performed; this allows one to study spectral properties of the perturbed operator. The research method is the method of similar operators, which also presented in this work.

On Spectral Properties of a Certain Class of Perturbed Operators

Using the method of similar operators, we examine spectral properties of the perturbed operator A − B: D(A) ⊂ H → H, where A is a self-adjoint operator with a compact resolvent and B is an unbounded perturbation. Under certain conditions on the spectrum of the unperturbed operator A and the perturbation B, we obtain estimates of the spectral sets of the perturbed operator. The results obtained are applied to the study of the spectrum of differential operators with periodic boundary conditions and a nonsmooth potential.

Hadron collider probes of the quartic couplings of gluons to the photon and Z boson

We explore the experimental sensitivities of measuring the gg → Zγ process at the LHC to the dimension-8 quartic couplings of gluon pairs to the Z boson and photon, in addition to comparing them with the analogous sensitivities in the gg → γγ process. These processes can both receive contributions from 4 different CP-conserving dimension-8 operators with distinct Lorentz structures that contain a pair of gluon field strengths, {hat{G}}_{mu v}^a , and a pair of electroweak SU(2) gauge field strengths, {W}_{mu v}^i , as well as 4 similar operators containing a pair of {hat{G}}_{mu v}^a and a pair of U(1) gauge field strengths, Bμν. We calculate the scattering angular distributions for gg → Zγ and the Z → overline{f}f decay angular distributions for these 4 Lorentz structures, as well as the Standard Model background. We analyze the sensitivity of ATLAS measurements of the Z(→ ℓ+ℓ−, overline{nu}nu , overline{q}q )γ final states with integrated luminosities up to 139 fb−1 at sqrt{s} = 13 TeV, showing that they exclude values ≲ 2 TeV for the dimension-8 operator scales, and compare the Zγ sensitivity with that of an ATLAS measurement of the γγ final state. We present combined Zγ and γγ constraints on the scales of dimension-8 SMEFT operators and γγ constraints on the nonlinearity scale of the Born-Infeld extension of the Standard Model. We also estimate the sensitivities to dimension-8 operators of experiments at possible future proton-proton colliders with centre-of-mass energies of 25, 50 and 100 TeV, and discuss possible measurements of the Z spin and angular correlations.

A note on spectral multipliers on Engel and Cartan groups

The aim of this short note is to give examples of L p L^p - L q L^q bounded spectral multipliers for operators involving left-invariant vector fields and their inverses, in the settings of Engel and Cartan groups. The interest in such examples lies in the fact that a group does not have to have flat co-adjoint orbits, and that the considered operator is not related to the usual sub-Laplacian. The discussed examples show how one can still obtain L p L^p - L q L^q estimates for similar operators in such settings. As immediate consequences, one gets the corresponding Sobolev-type inequalities and heat kernel estimates.

Method of similar operators in the problem of bi-invariant subspaces

Method of similar operators in the problem of bi-invariant subspaces

Probabilistic semantics for epistemic modals: Normality assumptions, conditional epistemic spaces and the strength of must and might

The epistemic modal auxiliaries must and might are vehicles for expressing the force with which a proposition follows from some body of evidence or information. Standard approaches model these operators using quantificational modal logic, but probabilistic approaches are becoming increasingly influential. According to a traditional view, must is a maximally strong epistemic operator and might is a bare possibility one. A competing account—popular amongst proponents of a probabilisitic turn—says that, given a body of evidence, must \(\phi \) entails that \(Pr(\phi )\) is high but non-maximal and might \(\phi \) that \(Pr(\phi )\) is significantly greater than 0. Drawing on several observations concerning the behavior of must, might and similar epistemic operators in evidential contexts, deductive inferences, downplaying and retractions scenarios, and expressions of epistemic tension, I argue that those two influential accounts have systematic descriptive shortcomings. To better make sense of their complex behavior, I propose instead a broadly Kratzerian account according to which must \(\phi \) entails that \(Pr(\phi ) = 1\) and might \(\phi \) that \(Pr(\phi ) > 0\), given a body of evidence and a set of normality assumptions about the world. From this perspective, must and might are vehicles for expressing a common mode of reasoning whereby we draw inferences from specific bits of evidence against a rich set of background assumptions—some of which we represent as defeasible—which capture our general expectations about the world. I will show that the predictions of this Kratzerian account can be substantially refined once it is combined with a specific yet independently motivated ‘grammatical’ approach to the computation of scalar implicatures. Finally, I discuss some implications of these results for more general discussions concerning the empirical and theoretical motivation to adopt a probabilisitic semantic framework.

An Easy-to-Understand Method to Construct Desired Distance-Like Measures

Metrics and their weaker forms are used to measure the difference between two data (or other things). There are many metrics that are available but not desired by a practitioner. This paper recommends in a plausible reasoning manner an easy-to-understand method to construct desired distance-like measures: to fuse easy-to-obtain (or easy to be coined by practitioners) pseudo-semi-metrics, pseudo-metrics, or metrics by making full use of well-known t-norms, t-conorms, aggregation operators, and similar operators (easy to be coined by practitioners). The simple reason to do this is that data for a real world problem are sometimes from multiagents. A distance-like notion, called weak interval-valued pseudo-metrics (briefly, WIVP-metrics), is defined by using known notions of pseudo-semi-metrics, pseudo-metrics, and metrics; this notion is topologically good and shows precision, flexibility, and compatibility than single pseudo-semi-metrics, pseudo-metrics, or metrics. Propositions and detailed examples are given to illustrate how to fabricate (including using what “material”) an expected or demanded WIVP-metric (even interval-valued metric) in practical problems, and WIVP-metric and its special cases are characterized by using axioms. Moreover, some WIVP-metrics pertinent to quantitative logic theory or interval-valued fuzzy graphs are constructed, and fixed point theorems and common fixed point theorems in weak interval-valued metric spaces are also presented. Topics and strategies for further study are also put forward concretely and clearly.

New physics in b → se+e−: A model independent analysis

The lepton universality violating flavor ratios RK/RK⁎ indicate new physics either in b→sμ+μ− or in b→se+e− or in both. If the new physics is only b→se+e− transition, the corresponding new physics operators, in principle, can have any Lorentz structure. In this work, we perform a model independent analysis of new physics only in b→se+e− decay by considering effective operators either one at a time or two similar operators at a time. We include all the measurements in b→se+e− sector along with RK/RK⁎ in our analysis. We show that various new physics scenarios with vector/axial-vector operators can account for RK/RK⁎ data but those with scalar/pseudoscalar operators and with tensor operators can not. We also show that the azimuthal angular observable P1 in B→K⁎e+e− decay is most suited to discriminate between the different allowed solutions.

Spectral analysis of one class perturbed first order differential operators

We use the method of similar operators to study a mixed problem for a first order differential equation with a fractional integration operator. The differential operator defined by the equation is transformed into a similar operator that is an orthogonal direct sum of finite-rank operators. The estimates of eigenvalues, eigenvectors are obtained. The result is used to construct an operator group.