Summation of Divergent Series and Integrals

Summation of divergent series: Order-dependent mapping

On convergent and divergent imaginary series. Summation of some convergent imaginary series. Notations used to represent imaginary functions that we find by evaluating the sum of such series.

By a series we mean a set of numbers a1, a2, a3… such that we have a rule for calculating a2, a3 etc. from the first number a1.Series occur in many problems in chemistry such as specific heats of solids, the theory of black-body radiation, solution of the Schrödinger equation, statistical thermodynamics and Fourier series in X-ray crystallography. In this chapter we consider the summation of series and the convergence of infinite series. We shall develop methods for the expansion of functions such as transcendental functions in terms of infinite series.

Schrodinger equation for a polynomial potential with the highest order term having an even power and a positive coefficient is solved for high eigenvalues En in two different ways after Liouville transformation, (a) converting the differential equation into integral equation and solving it iteratively and (b) by the WKB method. While the series solution in powers of 1/En from (b) is known to diverge, we show that the one from (a) converges. We show then that asymptotic re‐expansion of the convergent series from (a) agrees with the divergent series from (b). Actually, we have been able to show the agreement only up to order (1/En)5, but we believe that it holds to all orders. If this is true, the divergent WKB series can be reorganized into a convergent series, which is in fact obtained by the method of iteration (a).

Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue

Bilinear Calderón–Zygmund theory on product spaces

We demonstrate and develop dyadic–probabilistic methods in connection with non-homogeneous bilinear operators, namely singular integrals and square functions. We develop the full non-homogeneous theory of bilinear singular integrals using a modern point of view. The main result is a new global T b Tb theorem for Calderón–Zygmund operators in this setting. Our main tools include maximal truncations, adapted Cotlar type inequalities and suppression and big piece methods. While proving our bilinear results we also advance and refine the linear theory of Calderón–Zygmund operators by improving techniques and results. For example, we simplify and make more efficient some non-homogeneous summing arguments appearing in T 1 T1 type proofs. As a byproduct, we can manage with ease quite general modulus of continuity in the kernel estimates. Our testing conditions are also quite general by virtue of the big piece method of proof.

The most significant result in the integration theory for locally compact noncommutative spaces was proven in Carey et al. (J Funct Anal 263(2):383–414, 2012). We review and improve on this result. For bounded, positive operators A, B on a separable Hilbert space \(\mathcal {H}\), if \(AB\in \mathcal {M}_{1,\infty }\), \([A^{\frac 12},B]\in \mathcal {L}_1\) and the limit Open image in new window exists, then AB is Dixmier measurable and, for any dilation-invariant extended limit ω, we have $$\displaystyle \operatorname {{\mathrm {Tr}}}_\omega (AB)=\lim _{p\downarrow 1}(p-1) \operatorname {{\mathrm {Tr}}}(B^pA^p). $$ The proof of this result is facilitated by the efficient use of recent advances in the theory of singular traces and double operator integrals. The Dixmier traces of pseudo-differential operators of the form \(M_f(1-\Delta )^{-\frac {d}2}\) on \(\mathbb {R}^d\), where f is a Schwartz function on \(\mathbb {R}^d\) and Δ denotes the Laplacian on \(L_2(\mathbb {R}^d)\), are swiftly recovered using this result. An analogous result for the noncommutative plane is also obtained.

To further improve on the competitiveness of the boundary element method (BEM), a hybrid version of it is used for a numerical solution of two dimensional nonlinear coupled viscous Burger’s equation. Adopting this approach to a discretized 2D spatial domain, the resulting integral equations arising from the singular integral theory are applied locally to each of the elements. The resulting nonlinear discrete equations are finally solved by the Picard iteration algorithm. The simulation results obtained, not only concur with analytical solutions, but also display high accuracy and are in agreement with those available in literature.

An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is $O(\vert \!\log r\vert)$, le

Various perturbation series are factorially divergent. The behavior of their high-order terms can be found by Lipatov's method, according to which they are determined by the saddle-point configurations (instantons) of appropriate functional integrals. When the Lipatov asymptotics is known and several lowest order terms of the perturbation series are found by direct calculation of diagrams, one can gain insight into the behavior of the remaining terms of the series. Summing it, one can solve (in a certain approximation) various strong-coupling problems. This approach is demonstrated by determining the Gell-Mann - Low functions in \phi^4 theory, QED, and QCD for arbitrary coupling constants. An overview of the mathematical theory of divergent series is presented, and interpretation of perturbation series is discussed. Explicit derivations of the Lipatov asymptotic forms are presented for some basic problems in theoretical physics. A solution is proposed to the problem of renormalon contributions, which hampered progress in this field in the late 1970s. Practical schemes for summation of perturbation series are described for a coupling constant of order unity and in the strong-coupling limit. An interpretation of the Borel integral is given for 'non-Borel-summable' series. High-order corrections to the Lipatov asymptotics are discussed.

A simple and versatile approach for the acceleration of the spectral domain approach for generalized shielded microstrip interconnects is presented. This approach uses asymptotic expansion for the Bessel's function and the Green's function followed by an approximation of infinite summation using two fast convergent sine cosine series. The asymptotic expansion coefficients of the Green's functions are found by a combination of analytical and numerical approaches. The infinite summation involved in the computation of the elements of the Galerkin matrix is accelerated using two fast convergent sine cosine series. The use of a few entire-domain basis functions gives very accurate results for the propagation constants for any mode in the general case of a multilayered shielded microstrip line. In addition to this, closed-form expressions are developed to choose the number of terms and some parameters to adaptively accelerate the convergence of the second type of fast convergent series for a given accuracy. This approach can be extended to handle multiple interconnects in the same layer and in different layers and also to include the effect of finite thickness and conductivity.

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, \(\delta(x)\), \(\cot \frac{x}{2}\) and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

In this paper, the problem of cracks in practice is transformed into the splice problem of broken line crack with a width. Based on the knowledge of fracture mechanics, the stress and displacement boundary conditions of the model are proposed, and the singular integral equations accord with these conditions are established. By the singular integral equation theories the complex singular equations are reduced to a integral equation on cracks only. The theoretical solution to the equation exists and is unique. Then, the equation is discreted by using the Gauss formula. At last, we use MATLAB to simulate the solution to get the trend chart of the crack displacement, which is affected by external force, crack spacing and included angle. It intuitively shows the change rule of crack and all parameters.

In the author's papers [1-~ a method for investigation of boundary integral equations arising in problems of mechanics of continuum in domains with pieeewise smooth boundaries was proposed. Traditionally, equations of the potential theory are studied directly by methods of the Fredholm and singular integral operator theories. In the case of a nonsmooth boundary this way leads to difficulties that have not been overcome until now. In a sense our approach is opposite to the traditional one. It is based on the well-known fact that solutions of integral equations can be expressed in terms of solutions of some exterior and interior boundary value problems. These are studied with the help of the theory developed in [4 8] and, as a result, theorems on solvability of equations of the potential theory are obtained. For these equations we can get differentiability properties and asymptotics of solutions near singularities of the boundary using the same approach ~, 9 l j. In [~ our method of construction of the potential theory was illustrated by the example of three boundary value problems of linear isotropic elasticity, namely, the first, the second and mixed, as well as of the same problems for the Laplace operator under the hypothesis that there exist a finite number of non-intersecting smooth edges on the boundary. In the present lecture we study the first two boundary value problems for the Lam~ system in domains with boundary singularities of the type of edges, conic points and polyhedral angles. New results on the harmonic potential theory are also reported.