On the inversion formula for two polynomials in two variables

An inversion formula for two polynomials in two variables

To eliminate the instabilities in renewable energy generations such as wind and PV systems, which are mainly caused by weather factors, this paper presents a concept of complementary battery-supercapacitor energy storage system. The power deviations between the output of renewable system and a given power generation plan can be smoothed by this complementary system. To coordinate the different storage types effectively, a control strategy is given based on floating average method. Further, modeling methodologies and numerical simulations are correspondingly conducted to verify the effectiveness of the control strategy. Results show that, compared with single battery energy storage system, the complementary storage system can give full play to the high-power advantage of supercapacitor and the energy availability of battery respectively. This concept and its modeling can be used to relieve operating pressure of the single battery storage system when high power fluctuations come cross.

Given n disjoint intervals on together with n functions , , and an matrix , the problem is to find an L2 solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

In two dimensions, the exponential X-ray transform has been well-studied due to its applications of correcting attenuation effects in single photon emission computed tomography (SPECT). Explicit inversion formulas have been known for over 15 years. The three-dimensional (3D) case has not been as thoroughly examined, and inversion formulas are available for only a few of the wide range of possible 3D geometries. The rotating slant-hole (RSH) SPECT geometry is a special case for which no inversion formula has yet appeared. This paper presents a general inversion formula for the 3D exponential X-ray transform using a Neumann series. The method applies to any geometry but convergence of the series depends on the exponential scalar and the size of the region-of-interest. The derivation is presented in the context of the RSH SPECT geometry. Results from computer simulations are given.

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.

Following Mulmuley’s Lemma, this paper presents a generalization of the Moore–Penrose Inverse for a matrix over an arbitrary field. This generalization yields a way to uniformly solve linear systems of equations which depend on some parameters.

Inverting the exponential Radon transform has a potential use for SPECT (single photon emission computed tomography) imaging in cases where a uniform attenuation can be approximated, such as in brain and abdominal imaging. Tretiak and Metz derived in the frequency domain an explicit inversion formula for the exponential Radon transform in two dimensions for parallel-beam collimator geometry. Progress has been made to extend the inversion formula for fan-beam and varying focal-length fan-beam (VFF) collimator geometries. These previous fan-beam and VFF inversion formulas require a spatially variant filtering operation, which complicates the implementation and imposes a heavy computing burden. In this paper, we present an explicit inversion formula, in which a spatially invariant filter is involved. The formula is derived and implemented in the spatial domain for VFF geometry (where parallel-beam and fan-beam geometries are two special cases). Phantom simulations mimicking SPECT studies demonstrate its accuracy in reconstructing the phantom images and efficiency in computation for the considered collimator geometries.

The spherical means Radon transform is defined by the integral of a function f in Rn over the sphere S(x,r) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface Γ⊂Rn and r∈(0,∞) has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary ∂Ω of a bounded (convex) domain Ω⊂Rn , a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ=∂Ω , such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called ‘universal backprojection inversion formulas’, that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

Ray transforms in hyperbolic geometry

We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.

Discretizing two-point boundary value problems on an interval by finite difference method, we obtain a certain type of tridiagonal coefficient matrices. In this paper we give an explicit inversion formula for such tridiagonal matrices using Yamamoto-Ikebe’s inversion formula. Key WordsTridiagonal matricesexplicit inversion formula

We prove an explicit inverse Chevalley formula in the equivariantK-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a$\mathbb {Z}\left [q^{\pm 1}\right ]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars$e^{\lambda }$, where$\lambda $is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type$E_8$. The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetricq-Toda operators for minuscule weights in ADE type.

ABSTRACTThe inverse matrix of a Vandermonde matrix with arbitrary field entries of dimension with is well-studied in the literature. Especially, an explicit inversion formula does exist. However in theory as e.g. for determining (analytically) the eigenvalues of (arbitrary) matrix polynomials, Bernstein-Vandermonde matrices must be inverted. This note derives an explicit low-cost inversion formula for this matrix class with entries over a field whose characteristic equals zero. Its derivation requires at most additions and multiplications.

Motivated by Schein’s explicit formula for the maximum Thierrin-Vagner inverse of a Boolean matrix we first propose explicit formulae for the greatest least-squares and minimum norm g-inverses of a matrix A over any commutative residuated dioid. We also provide a formula for the unique group inverse of A, whenever existent. In particular, we propose formulae for the aforementioned generalized inverses of A over Boolean algebras, max-plus semirings and a class of complete and completely distributive lattices. Our main results remain valid in the context of residuated semigroups.

Irreducibles and the composed product for polynomials over a finite field