Lagrange Inversion Research Articles

TWO APPROACHES TO MÖBIUS INVERSION

AbstractThe Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

Extended NCS Based on Method of Series Reversion for Imaging of Highly Squinted SAR

In the case of high range resolution and squint angle, current chirp scaling algorithm (CSA) and nonlinear CSA (NCSA) have a finite ability to achieve high-quality images. The problem stems from a range-dependent (i.e., space-variant) cubic- and higher order terms of range frequency, which require sufficient compensation or space-variant filtering, in the phase of the synthetic aperture radar transfer function, and this letter aims at dealing with this problem. First, an inequation is introduced to evaluate the highest order of range frequency terms whose space-variant coefficient has to be taken into account. Then, based on the method of series reversion, this letter proposes the extended NCS which can weaken the range dependence of the considered range frequency terms and achieve accurate range cell migration correction and range compression. Simulation results are presented to validate the proposed method.

A Novel Range Migration Algorithm for General Bistatic SAR Imaging Based on Series Reversion and Numerical Computation

A Novel Range Migration Algorithm for General Bistatic SAR Imaging Based on Series Reversion and Numerical Computation

Five convolution formulae of orthogonal polynomials

By means of the classical Lagrange expansion theorem, five convolution formulae are established for the orthogonal polynomials named after Laguerre, Jacobi, Meixner, Gegenbauer and Pollaczek, that contain the well-known Hagen–Rothe formula for binomial coefficients as common special case.

Geometric polyhedral models for nanotubes comprising hexagonal lattices

Two new models for the geometric structure of nanotubes comprising hexagonal lattices are described. The existing models for nanotubes typically involve rolled up planar sheets and ignore discrepancies due to curvature. The first of the models presented here assumes that all atomic locations are equidistant from the tube axis which applies for single species nanotubes such as carbon nanotubes. This model assumes that all bond angles and all bond lengths are equal in the cylindrical state, and that all atoms are equidistant from the tube axis, and from these three assumptions, expressions are given for the major geometric parameters. The second model extends this notion to tubes where all the atomic locations are not equidistant from the tube axis, which may be employed to model nanotubes comprising two chemical species that bond into a hexagonal lattice such as boron nitride nanotubes. In the second model, all bond lengths are taken to be equal and the atoms of the same species are taken to be equidistant from the tube axis, and the nanotube is assumed to comprise two species and thus there may be two radii. Fundamental to both models is the determination of a solution of a transcendental equation. Here we present a new formal Lagrange expansion of the solution. Previously given asymptotic series expansions of the exact formulae for both models lead to the conventional expressions as the leading order term. Although the correction terms are typically small, knowledge of the precise structure may be critical to comprehending many nanoscale phenomena. The new models also give rise to an expression for the wall thickness, an important geometric parameter for which at present no reliable information is available.

A Chirp-Z Transform Imaging Algorithm for Missile-borne SAR with Diving Maneuver Based on the Method of Series Reversion

导弹下降段飞行速度快、飞行轨迹复杂，并且弹上成像实时性要求较高，给弹载SAR成像算法提出了新的要求。针对上述问题，该文提出了基于级数反演的CZT成像算法。文中首先给出了下降段点目标斜距表达式，并用泰勒级数对斜距进行高阶展开，接着用级数反演法得到了信号2维频谱高阶近似表达式，然后详细分析了频谱中各项的空变性对成像的影响，并在此基础上推导了改进的CZT成像算法并分析了其运算量。该算法能够精确校正空变的距离徙动，所有操作都由快速傅里叶变换和相位点乘完成，具有较高的效率。仿真结果证明，该文算法能够在下降段实现全孔径高精度成像。

A Hermite–Padé perspective on the renormalization group, with an application to the correlation function of Lieb–Liniger gas

Renormalization group (RG) methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings. However, in cases when the β-function depends only on the coupling, the RG technology does include a part—which is related to the equation for the running coupling and which we will call the ‘single variable RG’ (SVRG)—that turns out to be a purely formal method for improving the convergence properties of series expansions. In fact, we show that SVRG is an integral Hermite–Padé approximation preceded by a series reversion. A complementary viewpoint on what SVRG does to an expansion is that it performs a ‘minimal upgrade to a self-similar form’, the minimality being crucial. In applications, SVRG is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable, but also the exponent (but not the prefactor) of the scaling law for large values of the variable. As a practical example, we extract the scaling-law prefactor for the one-body density matrix of the Lieb–Liniger gas. Using a new result for the fourth-order term in the short-distance expansion, we find remarkable agreement with known quantum diffusion Monte Carlo numerical results.

A NEW LOOK AT THE POINT TARGET REFERENCE SPECTRUM FOR BISTATIC SAR

Focusing bistatic synthetic aperture radar (SAR) data in frequency domain requires two-dimensional (2D) point target reference spectrum (PTRS). Lofield's bistatic formula (LBF) and the Method of Series Reversion (MSR) have been introduced recently to compute PTRS of bistatic SAR. In this paper, flrstly we generalize the original LBF (OLBF) by introducing the Doppler contribution functions of transmitter and receiver. Thus, OLBF and its derivatives (e.g., extended LBF) can be viewed as special forms of the generalized LBF with constant Doppler contributions. Based on this, secondly the ideal LBF (ILBF) with no computing error, except the error resulting from the principle of stationary phase, is also presented. The ILBF reveals that the theoretical PTRS of bistatic SAR consists of only two monostatic terms, but it does not include bistatic deformation term in comparison with OLBF. It supplies us with a target when we deduce the PTRS for bistatic SAR. Finally, to get the precise analytical PTRS for general bistatic SAR, an approximated ILBF (AILBF) is proposed. It expresses the Doppler contributions of the transmitter and receiver as power series and can approach the ILBF very well. AILBF can keep the precision as MSR and inherit a simple form from LBF. In addition, error limit for the validity of bistatic PTRS is also given. The results in this paper can be used to develop imaging algorithms for extreme bistatic (e.g., spaceborne/airborne) and high squint (e.g., bistatic forward-looking) cases.

Exact analytical two-dimensional spectrum for bistatic synthetic aperture radar in tandem configuration

An exact analytical solution for the two-dimensional (2-D) point target spectrum is usually difficult to obtain because of the existence of a double-square-root (DSR) term in the bistatic range equation. Some approximate solutions for the 2-D spectrum have been derived and used in order to focus bistatic synthetic aperture data. A geometry-based bistatic formula (GBF) method was used for obtaining a quasi-analytical form of a bistatic 2-D spectrum by Zhang (2007). Although the GBF method cannot completely solve the problem of the DSR term, it provides a novel aspect for dealing with this problem. In this study, based on the quasi-analytical spectrum, the authors change the signal expression space and transform the eight-order polynomial equation in terms of the slow time into the four-order polynomial equation in terms of the half quasi-bistatic angle (HQBA) for the tandem configuration. Then the DSR-term problem is successfully solved and a corresponding exact analytical bistatic 2-D spectrum is obtained. It is proved that the spectra acquired by Rocca's smile operator, Loffeld's bistatic formula (LBF) and the method of series reversion (MSR) are equivalent to this proposed analytical spectrum when certain conditions are met, and that this exact analytical spectrum is the most accurate among them.

RANGE DOPPLER ALGORITHM FOR BISTATIC SAR PROCESSING BASED ON THE IMPROVED LOFFELD'S BISTATIC FORMULA

This paper presents a new range Doppler algorithm (RDA) for bistatic synthetic aperture radar (SAR) processing in a general conflguration based on a bistatic point target reference spectrum: the improved extended Lofield's bistatic formula (ILBF). The ILBF spectrum is proved to be comparably accurate with the spectrum derived using the method of series reversion (MSR). Based on the expansion of the ILBF spectrum, a new bistatic RDA is developed to process the azimuth invariant and variant bistatic SAR data. Compared with existing bistatic RDA, the new algorithm has a simpler formulation and is able to cope with moderate or high squint bistatic SAR data. The simulated data in the azimuth invariant and variant bistatic conflgurations are used to validate the new algorithm.

Omega-K Algorithm for Bistatic SAR With Arbitrary Geometry Configuration

Compared to the common monostatic synthetic aperture radar (SAR), the bistatic SAR data are more challenging to process because of their complicated flight geometry. Although several bistatic SAR image formation algorithms have been presented, most of them are only able to handle azimuth-invariant bistatic configuration where the transmitter and receiver platforms are moving in parallel tracks with identical velocities. This paper proposes an Omega-K algorithm to handle a general case of bistatic SAR data. The two-dimensional frequency spectrum expression derived by the method of series reversion is rewritten to a new form with explicit geometric significance. Based on this new frequency spectrum expression, the frequency mapping function can be derived by making use of the multivariable Taylor's theorem. After data mapping operation, the coupling between the range and azimuth can be eliminated, and the final focused image can be obtained using FFT technology. Numerical simulations are performed to validate the proposed algorithm.

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of ‘connectivity’ for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler (2010) [7] which uses generating functions and Lagrange inversion.

An Analytical Method of Updating the Range Derivatives and a Simple Image Registration Method for the MSR-Based Range Doppler Algorithm

Based on the method of series reversion, Neo developed a quite accurate range Doppler algorithm (RDA). In performing this RDA, the range derivatives need to be updated along the range. In this letter, first, we will give an analytical method to update the range derivatives for this RDA. The proposed method mainly exploits two points the first is the introduction of a reference vector on the ground plane of the bistatic geometry, and the second is the use of the principle of series reversion. Second, based on this new method of updating the range derivatives, we present a simple image registration method for mapping the focused image onto the ground plane, which needs a 1-D interpolation only in the range direction. Third, a motion compensation method is also proposed.

Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series

We point out how the Banach Fixed Point Theorem, together with the Picard successive approximation methods yielded by it, allows us to treat some mathematical methods in combinatorics. In particular we get, in this way, a proof of and an iterative algorithm for deriving the Lagrange Inversion Formula.

A symbolic handling of Sheffer polynomials

We revisit the theory of Sheffer sequences by means of the formalism introduced in Rota and Taylor (SIAM J Math Anal 25(2):694–711, 1994) and developed in Di Nardo and Senato (Umbral nature of the Poisson random variables. Algebraic combinatorics and computer science, pp 245–256, Springer Italia, Milan, 2001, European J Combin 27(3):394–413, 2006). The advantage of this approach is twofold. First, this new syntax allows us noteworthy computational simplification and conceptual clarification in several topics involving Sheffer sequences, most of the open questions proposed in Taylor (Comput Math Appl 41:1085–1098, 2001) finds answer. Second, most of the results presented can be easily implemented in a symbolic language. To get a general idea of the effectiveness of this symbolic approach, we provide a formula linking connection constants and Riordan arrays via generalized Bell polynomials, here defined. Moreover, this link allows us to smooth out many results involving Bell Polynomials and Lagrange inversion formula.

The asymptotic expansion for n! and the Lagrange inversion formula

We obtain an explicit simple formula for the coefficients of the asymptotic expansion for the factorial of a natural number, $$n!=n^n\sqrt{2\pi n}\mbox{e}^{-n}\biggl\{1+\frac{a_1}{n}+\frac{a_2}{n^2}+\frac{a_3}{n^3}+\cdots\biggr\},$$ in terms of derivatives of powers of an elementary function that we call normalized left truncated exponential function. The unique explicit expression for the a k that appears to be known is that of Comtet in (Advanced Combinatorics, Reidel, 1974), which is given in terms of sums of associated Stirling numbers of the first kind. By considering the bivariate generating function of the associated Stirling numbers of the second kind, another expression for the coefficients in terms of them follows also from our analysis. Comparison with Comtet’s expression yields an identity which is somehow unexpected if considering the combinatorial meaning of the terms. It suggests by analogy another possible formula for the coefficients, in terms of a normalized left truncated logarithm, that in fact proves to be true. The resulting coefficients, as well as the first ones are identified via the Lagrange inversion formula as the odd coefficients of the inverse of a pair of formal series. This in particular leads to the identification of a couple of simple implicit equations, which permits us to obtain also some recurrences related to the a k ’s.

General Spotlight Bistatic SAR Imaging Based on Series Reversion

该文将级数反演法应用于聚束式广义双基成像。对于广义双基，其2维频谱表达式较普通双基更加复杂，很难用精确的解析关系表达，这对后续成像处理造成了极大的困难。通过级数反演法，可以得到信号的2维频谱展开式，在2维频域进行2次距离压缩，同时在距离多普勒域进行距离徙动校正。该算法保留了单基距离多普勒算法的高效性，由于2次距离压缩考虑到了方位向的影响，因此该算法对大合成孔径情况比较适合。用该算法对仿真数据进行处理，得到了良好的成像结果。

Consecutive Pattern Avoidances in Non-crossing Trees

In this paper, motivated by the concepts of increasing (alternating) non-crossing trees, the problem of consecutive pattern avoidances in non-crossing trees is proposed. Some given patterns of length two and three are investigated in detail. The Lagrange inversion formula is used to obtain the explicit formulas for these cases. Bijections are established between non-crossing trees avoiding special patterns and Schroder paths.

The renormalizing series of some integral equations

We consider integral equations for which the perturbation expansion gives a power series in a parameter h whose coefficients are divergent integrals. We eliminate the divergent integrals by introducing a renormalizing Z(t, h) series in the minimal subtraction scheme. We investigate the convergence of the formal Z series in relation to the kernels of the integral equations. We find a relation of the renormalizing series to the Lagrange inversion series and also some other relations.

On the asymptotic enumeration of accessible automata

Automata, Logic and Semantics We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.