Approximate Solutions of an Extended Multi-Order Boundary Value Problem by Implementing Two Numerical Algorithms

In this paper, we establish several necessary conditions to confirm the uniqueness-existence of solutions to an extended multi-order finite-term fractional differential equation with double-order integral boundary conditions with respect to asymmetric operators by relying on the Banach’s fixed-point criterion. We validate our study by implementing two numerical schemes to handle some Riemann–Liouville fractional boundary value problems and obtain approximate series solutions that converge to the exact ones. In particular, we present several examples that illustrate the closeness of the approximate solutions to the exact solutions.

Derive Lovelock gravity from string theory in cosmological background

It was proved more than three decades ago, that the first order α′ correction of string effective theory could be written as the Gauss-Bonnet term, which is the quadratic term of Lovelock gravity. In cosmological background, with an appropriate field redefinition, we reorganize the infinite α′ corrections of string effective action into a finite term expression for any specific dimension. This finite term expression matches Lovelock gravity exactly and thus fix the couplings of Lovelock gravity by the coefficients of string effective action. This result thus provides a strong support to string theory.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d − 2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.

One-loop jet functions by geometric subtraction

In factorization formulae for cross sections of scattering processes, final-state jets are described by jet functions, which are a crucial ingredient in the resummation of large logarithms. We present an approach to calculate generic one-loop jet functions, by using the geometric subtraction scheme. This method leads to local counterterms generated from a slicing procedure; and whose analytic integration is particularly simple. The poles are obtained analytically, up to an integration over the azimuthal angle for the observable- dependent soft counterterm. The poles depend only on the soft limit of the observable, characterized by a power law, and the finite term is written as a numerical integral. We illustrate our method by reproducing the known expressions for the jet function for angularities, the jet shape, and jets defined through a cone or kT algorithm. As a new result, we obtain the one-loop jet function for an angularity measurement in e+e− collisions, that accounts for the formally power-suppressed but potentially large effect of recoil. An implementation of our approach is made available as the GOJet Mathematica package accompanying this paper.

Dynamic Stress Analysis of a Circular-Lined Tunnel in Composite Strata-SH Wave Incidence

It is necessary to study the problem of seismic wave scattering in composite stratum for tunnel engineering because the existence of composite strata will make the stress of tunnels more complicated during earthquakes. In this thesis, a series solution of the scattering wave field of the composite strata and lining is obtained using the complex function method. According to the stress and displacement boundary conditions between the composite stratum and the lining, a series of equations are established and are solved by means of Fourier transformation and finite term truncation, and the calculation errors are also discussed. Through programming calculations, the dynamic stress concentration factor (DSCF) of circular tunnels in the two types of composite strata, “hard-over-soft” and “soft-over-hard,” is analyzed when SH waves propagate, and certain conclusions on the scattering of SH waves that are distinguished from the case of single homogeneous layers are reached. The research in this article reveals some phenomena. For the Q345 steel lining in the calculation examples, it is found in this paper that increasing the thickness of the lining is effective to reduce the influence of the DSCF. But, for C30 concrete, increasing the thickness of the lining reduces the DSCF of the outer surface while increasing the DSCF of the inner surface.

A new multistage Parker-Sochacki method for solving the Troesch’s problem

In this article, we introduce a new method to obtain an approximate analytical solution of the highly unstable Troesch’s problem. In the proposed method, without recourse to any hyperbolic tangent transformation or finite term approximation of the hyperbolic sine function, the problem is recast as a system of projectively polynomials which allows straightforward computation of the series solution of the problem. The radius of convergence of the series solution to the problem is derived a-priorly in terms of the parameters of the polynomial system. Using a step length ; the problem domain is divided into subintervals, where corresponding subproblems are defined and solved with Parker-Sochacki method with very high accuracy. Highly accurate piecewise continuous approximate solution is thus obtained on the entire integration interval. The obtained solution, which is valid for every choice of the Troesch parameter , showed comparable accuracy to known numerical solutions in the literature. In particular, new results are presented for large values of in the range [20;500].

Finite term relations for the exponential orthogonal polynomials

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function g supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of g,z̅g,…,z̅dg, to decide whether a (d + 2)-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function g can be reconstructed from a resulting polynomial of degree d and the Cauchy transform of g. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.

Electrostatic Interaction of Point Charges in Three-Layer Structures: The Classical Model

Electrostatic interaction energy W between two point charges in a three-layer plane system was calculated on the basis of the Green’s function method in the classical model of constant dielectric permittivities for all media involved. A regular method for the calculation of W ( Z , Z ′ , R ) , where Z and Z ′ are the charge coordinates normal to the interfaces, and R the lateral (along the interfaces) distance between the charges, was proposed. The method consists in substituting the evaluation of integrals of rapidly oscillating functions over the semi-infinite interval by constructing an analytical series of inverse radical functions to a required accuracy. Simple finite-term analytical approximations of the dependence W ( Z , Z ′ , R ) were proposed. Two especially important particular cases of charge configurations were analyzed in more detail: (i) both charges are in the same medium and Z = Z ′ ; and (ii) the charges are located at different interfaces across the slab. It was demonstrated that the W dependence on the charge–charge distance S = R 2 + Z − Z ′ 2 differs from the classical Coulombic one W ∼ S − 1 . This phenomenon occurs due to the appearance of polarization charges at both interfaces, which ascribes a many-body character to the problem from the outset. The results obtained testify, in particular, that the electron–hole interaction in heterostructures leading to the exciton formation is different in the intra-slab and across-slab charge configurations, which is usually overlooked in specific calculations related to the subject concerned. Our consideration clearly demonstrates the origin, the character, and the consequences of the actual difference. The often used Rytova–Keldysh approximation was analyzed. The cause of its relative success was explained, and the applicability limits were determined.

Convergence in infinitary term graph rewriting systems is simple

Term graph rewriting provides a formalism for implementing term rewriting in an efficient manner by emulating duplication via sharing. Infinitary term rewriting has been introduced to study infinite term reduction sequences. Such infinite reductions can be used to model non-strict evaluation. In this paper, we unify term graph rewriting and infinitary term rewriting thereby addressing both components of lazy evaluation: non-strictness and sharing. In contrast to previous attempts to formalise infinitary term graph rewriting, our approach is based on a simple and natural generalisation of the modes of convergence of infinitary term rewriting. We show that this new approach is better suited for infinitary term graph rewriting as it is simpler and more general. The latter is demonstrated by the fact that our notions of convergence give rise to two independent canonical and exhaustive constructions of infinite term graphs from finite term graphs via metric and ideal completion. In addition, we show that our notions of convergence on term graphs are sound w.r.t. the ones employed in infinitary term rewriting in the sense that convergence is preserved by unravelling term graphs to terms. Moreover, the resulting infinitary term graph calculi provide a unified framework for both infinitary term rewriting and term graph rewriting, which makes it possible to study the correspondences between these two worlds more closely.

Using Vampire with Support for Algebraic Datatypes in Type Soundness Proofs

In our ongoing project VeriTaS, we aim at automating soundness proofs for type sys- tems of domain-specific languages. In the past, we successfully used previous Vampire versions for automatically discharging many intermediate proof obligations arising within standard soundness proofs for small type systems. With older Vampire versions, encoding the individual proof problems required manual encoding of algebraic datatypes via the theory of finite term algebras. One of the new Vampire versions now supports the direct specification of algebraic datatypes and integrates reasoning about term algebras into the internally used superposition calculus.In this work, we investigate how many proof problems that typically arise within type soundness proofs different Vampire 4.1 versions can prove. Our test set consists of proof problems from a progress proof of a type system for a subset of SQL. We compare running Vampire 4.1 with our own encodings of algebraic datatypes (in untyped as well as in typed first-order logic) to running Vampire 4.1 with support for algebraic datatypes, which uses SMTLIB as input format. We observe that with our own encodings, Vampire 4.1 still proves more of our input problems. We discuss the differences between our own encoding of algebraic datatypes and the ones used within Vampire 4.1 with support for algebraic datatypes.