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Abstract: One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson–Crick complement, denoted here as . Thus, any expression consisting of repetitions of u and can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger’s classical result about equations of the form , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if , then all three words involved can be expressed in terms of a common word t and its complement . Moreover, if , then is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement , which is also obtained in this paper. [Copyright &y& Elsevier]

Surface roughness is often considered the main purpose in contemporary computer numerical controlled (CNC) machining industry. Most existing optimization researches for CNC finish turning were either accomplished within certain manufacturing circumstances or achieved through numerous equipment operations. Therefore, a general deduction optimization scheme is deemed to be necessary for the industry. In this paper, the cutting depth, feed rate, speed, and tool nose runoff with low, medium, and high level are considered to optimize the surface roughness for finish turning based on L9(34) orthogonal array. Additionally, nine fuzzy control rules using triangle membership function with respective to five linguistic grades for surface roughness are constructed. Considering four input and twenty output intervals, the defuzzification using center of gravity is then completed. Thus, the optimum general fuzzy linguistic parameters can then be received. The confirmation experiment result showed that the surface roughness from the fuzzy linguistic optimization parameters is significantly advanced compared to that from the benchmark. This paper certainly proposes a general optimization scheme using orthogonal array fuzzy linguistic approach to the surface roughness for CNC turning with profound insight. [ABSTRACT FROM AUTHOR]

In this paper we extend previous results on the regularity of solutions of integro-differential parabolic equations. The kernels are non-necessarily symmetric which could be interpreted as a non-local drift with the same order as the diffusion. We provide a growth lemma and a Harnack inequality which can be used to prove higher regularity estimates. [ABSTRACT FROM AUTHOR]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger–Wang, Chau–Weinkove and the author solved this global problem in W 4 , p under some restrictions on the sign or integrability of the affine mean curvature. We remove these restrictions in this paper and obtain W 4 , p solution to the second boundary value problem when the affine mean curvature belongs to L p with p greater than the dimension. Our self-contained analysis also covers the case of Abreu's equation. [ABSTRACT FROM AUTHOR]

This paper deals with the solvability of interval matrix equations in max-plus algebra. Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a ⊕ b = max ⁡ { a , b } and a ⊗ b = a + b . The notation A ⊗ X ⊗ C = B represents an interval max-plus matrix equation, where A , B , and C are given interval matrices. We define four types of solvability of interval max-plus matrix equations, i.e., the tolerance, weak tolerance, left-weak tolerance, and right-weak tolerance solvability. We derive the necessary and sufficient conditions for checking each of them, whereby all can be verified in polynomial time. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

The Frequency Difference of Arrival (FDOA) equations derived in this paper are intended to increase the standard accuracy of the Low Earth Orbit (LEO) satellites dedicated to locate non-cooperative emitters placed on the Earth surface or in orbit about the Earth. The equations contain terms that are of the order of the corrections already taken into account in Navigation by GPS. In particular, two of them should not be neglected to this end, since they can be of the order of 10 − 10 . [ABSTRACT FROM AUTHOR]

Abstract: According to the hierarchical identification principle, a hierarchical gradient based iterative estimation algorithm is derived for multivariable output error moving average systems (i.e., multivariable OEMA-like models) which is different from multivariable CARMA-like models. As there exist unmeasurable noise-free outputs and unknown noise terms in the information vector/matrix of the corresponding identification model, this paper is, by means of the auxiliary model identification idea, to replace the unmeasurable variables in the information vector/matrix with the estimated residuals and the outputs of the auxiliary model. A numerical example is provided. [ABSTRACT FROM AUTHOR]

This paper provides some relations between the idempotent operators and the solutions to operator equations and A B A = A² and B A B = B². [ABSTRACT FROM AUTHOR]

We present new iterative methods based on refinement process for solving large sparse Sylvester-observer equations applied in state estimation of a continuous-time system. These methods use projection methods to produce low-dimensional Sylvester-observer matrix equations that are solved by the direct methods. Moreover, the refinement process described in this paper has the capability of improving the results obtained by any other methods. Some numerical results will be reported to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The purpose of this study is to approximate the stream function and temperature distribution of the MHD flow in a laminar liquid film from a horizontal stretching surface. In this paper DTM-Padé method was used which is a combination of differential transform method (DTM) and Padé approximant. The DTM solutions are only valid for small values of independent variables. Comparison between the solutions obtained by the DTM and the DTM-Padé with numerical Solution (fourth-order Runge-Kutta) revealed that the DTM-Padé method is an excellent method for solving MHD boundary-layer equations. [ABSTRACT FROM AUTHOR]

He's variational iteration method (VIM) is used for solving space and time fractional telegraph equations. Numerical examples are presented in this paper. The obtained results show that VIM is effective and convenient. [ABSTRACT FROM AUTHOR]

In this paper, we study the basic results on the general study of null curves in the Minkowski 4-space R14. A transversal vector bundle of a null curve in R14 is constructed using a frenet Frame consisting of two real null and two space-like vectors. The null curves are characterized by using the Frenet frame. [ABSTRACT FROM AUTHOR]

This paper deals with very weak solutions of the A-harmonic equation divA(x,∇u) = 0 with the operator A : Ω × Rn → Rn satisfies some coercivity and controllable growth conditions with Muckenhoupt weight. By using the Hodge decomposition with weight, a regularity property is proved: There exists an integrable exponent r1 = r1(λ, n, p) < p, such that every very weak solution u ϵ Wloc1,r (Ω,w). That is, u is a weak solution to (✶) in the usual sense. [ABSTRACT FROM AUTHOR]

This paper presents a new method to isolate all configurations that a multiloop linkage can adopt. The problem is tackled by means of formulation and resolution techniques that fit particularly well together. The adopted formulation yields a system of simple equations (only containing linear, bilinear, and quadratic monomials, and trivial trigonometric terms for the helical pair only) whose structure is later exploited by a branch-and-prune method based on linear relaxations. The method is general, as it can be applied to linkages with single or multiple loops with arbitrary topology, involving lower pairs of any kind, and complete, as all possible solutions get accurately bounded, irrespective of whether the linkage is rigid or mobile. [ABSTRACT FROM AUTHOR]

In this paper, we study first and second order positive numerical methods for the advection equation. In particular, we consider the direct discretization of the model problem and comment on its superiority to the so called method of lines. Moreover, we investigate the accuracy, stability and positivity properties of the direct discretization. The numerical results related to several test problems are reported. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider constructing some higher-order modifications of Newton’s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units. This construction can be applied to any iteration formula, and per iteration the resulting methods add only one additional function evaluation to increase the order. Some illustrative examples are provided and several numerical results are given to show the performance of the presented methods. [Copyright &y& Elsevier]

A cross-curricular structured-probe task-based clinical interview study with 44 pairs of third year high-school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students `use of graphing calculators to solve the problems is discussed. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of solutions of a system of self-referred and hereditary differential equations. The initial data are assumed to be lower semi-continuous. We also formulate some open questions. [ABSTRACT FROM AUTHOR]

This paper studies semi-positone systems of equations by using phase plane analysis and fixed point theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of unique and multiple solutions to the boundary valueproblem of a second-order difference equation with a parameter, which is a complement of the workby J. S. Yu and Z. M. Guo in 2006.

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn = f(xn-2)/g(xn-1), n ϵ ℕ0, where f, g ϵ C[(0,∞), (0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n-1} are eventually monotone. For the case when f(x) = a + βx and g satisfies the conditions g(0) = 1, g is nondecreasing, and x/g(x) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f(x) = c1/x and g(x) = c2x, for some positive c1 and c2. [ABSTRACT FROM AUTHOR]

The progress of Mathematics during the nineteenth century was characterised both by an enormous acquisition of new knowledge and by the attempts to introduce rigour in reasoning patterns and mathematical writing. Cauchy’s presentation of Mathematical Analysis was not immediately accepted, and many writers, though aware of that new style, did not use it in their own mathematical production. This paper is devoted to an episode of this sort that took place in Spain during the first half of the century: It deals with the presentation of a method for numerically solving algebraic equations by José Mariano Vallejo, a late Spanish follower of the Enlightenment ideas, politician, writer, and mathematician who published it in the fourth (1840) edition of his book Compendio de Matemáticas Puras y Mistas, claiming to have discovered it on his own. Vallejo’s main achievement was to write down the whole procedure in a very careful way taking into account the different types of roots, although he paid little attention to questions such as convergence checks and the fulfilment of the hypotheses of Rolle’s Theorem. For sure this lack of mathematical care prevented Vallejo to occupy a place among the forerunners of Computational Algebra. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical explanation for the nonconservation of total number concentration Nt of hydrometeors for the continuous collection growth process, for which Nt physically should be conserved for selected one- and two-moment bulk parameterization schemes. Where possible, physical explanations are proposed. The assumption of a constant no in scheme A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme B. Scheme E also is nonconservative, but it seems this result is not because of a physically inconsistent specification; rather the solution scheme’s equations simply do not satisfy Nt conservation and Nt does not come into the derivation. Even scheme F, which perfectly conserves Nt, does not preserve the distribution shape in comparison with a bin model. [ABSTRACT FROM AUTHOR]

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]

This paper describes an iterative method to solve efficiently the global algebraic system of equations obtained by applying the hybrid finite element method-boundary element method (FEM-BEM) to the solution of open-boundary electrostatic problems. In the proposed approach, a conjugate gradient (CG) solver is used to solve the FEM equations, whereas the BEM equations are solved by making use of the direct LU solver. To do so, it is convenient to implement the BEM equations in a nonconventional way, by choosing the nodes of the potential as not coinciding with those of its normal derivative. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a technique to generate waves at oblique angles in finite difference numerical models in a rectangular grid system by using internal generation technique [Lee, C., Suh, K.D., 1998. Internal generation of waves for time-dependent mild-slope equations. Coast. Eng. 34, 35–57.] along an arc-shaped line source. Tests were made for four different types of wave generation layouts. Quantitative experiments were conducted under the following conditions: the propagation of waves on a flat bottom, the refraction and shoaling of waves on a planar slope, and the diffraction of waves to a semi-infinite breakwater. Numerical experiments were conducted using the extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coast. Eng. 32, 91–117.]. The fourth layout type consisting of two parallel lines connected to a semicircle showed the best solutions, especially for a small grid size. This technique is useful for the numerical simulation of irregular waves with broad-banded directional spectrum using conventional spectral wave models for the reasonable estimation of bottom friction and wave-breaking. [Copyright &y& Elsevier]

The article presents some methods for finding asymptotics of some classes of nonlinear higher-order difference equations. It quotes an inclusion theorem due to Berg and gives an application of the result. An asymptotic series of some solutions of the difference equation is offered. The existence of some nonoscillatory solutions of a class of difference equations is given, as well as an open problem on finding asymptotics of some unbounded solutions of a difference equation. This paper also presents a brief account of results regarding existence of nontrivial solutions of the so-called Putnam-type difference equations as well as a conjecture.

This paper develops new, analytical methods to find a large class of orthogonal Gough-Stewart platforms (OGSPs) having desired properties at their home position. In contrast, prior methods have been computationally intensive, relying on numerical search techniques. By exploiting symmetry, 27 equations are reduced to only two. The new techniques are directly applicable to clean-sheet design of micro-manipulators, vibration isolators, and Cartesian stiffness matrices. In addition, straightforward methods for retro-fitting existing OGSPs are illustrated. Because the new theory greatly simplifies OGSP formulas about a single point, it is expected that these results will also prove to be very useful when numerically designing gross motion platforms. [ABSTRACT FROM AUTHOR]

This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included. [ABSTRACT FROM AUTHOR]

In this paper the new concepts of restricted total stability and total attractivity is formulated. For this purpose the classical theory of Malkin with suitable changes and the theory of limiting equations, introduced by Sell developed by Artstein and Andreev, are used. Significant examples are presented. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the integral transform containing the special function ϵ ((α, β)n;z) generalizing the Mittag-Leffler type function in the space ℒv,r (1 ⩽ r ⩽ ∞, v ϵ R) of Lebesgue measurble functions on R+ = (0, +∞) such that ∥ƒ∥v,r < ∞, where ∥ƒ∥v,r ≡ (Miltiple line equation cannot be converted to ASCII text) < ∞ (1 ⩽ r < ∞); ∥ƒ∥v,∞ ≡ esssupt>0∣tv ƒ(t)∣. Mapping properties such as the boundedness, the range, the representation and the inversion of the considered transform are proved. The results are based on the representation of the considered transform as the H-transform. [ABSTRACT FROM AUTHOR]

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we apply the ADER approach to the Discontinuous Galerkin (DG) framework for the two-dimensional linearized Euler equations. The result is an efficient high order accurate single-step scheme in time which uses less storage than Runge–Kutta DG schemes, especially for very high order of accuracy. The aim is to obtain an arbitrarily accurate scheme in space and time on unstructured grids for accurate noise propagation in the time domain in very complex geometries. We will present numerical convergence rates for ADER-DG methods up to 10th order of accuracy in space and time on structured and unstructured meshes. To cite this article: M. Dumbser, C.-D. Munz, C. R. Mecanique 333 (2005). [Copyright &y& Elsevier]

Abstract: The numerical solution of non-stationary convection-dominated partial differential equations (PDEs), as for example encountered in the simulation of reverse flow reactors, can be a computationally quite demanding task because of the prevailing steep concentration and temperature gradients. The computational effort can be greatly reduced with higher-order discretisation schemes for the convection terms and with an automatic, local grid adaptation. In this paper, WENO schemes (e.g. Advanced Numerical Approximations of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325–432) are used for the convection terms and a local grid adaptation technique is presented that uses the smoothness indicators and interpolation polynomials of the WENO schemes. It is shown that the number of grid cells required to accurately capture steep gradients can be greatly reduced with this grid adaptation technique. Finally, the capabilities of the numerical algorithm is demonstrated for the simulation of a reverse flow reactor. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of the stability of the solutions of the first order implicit difference periodic equationWe prove that if there exist a pair of lower and upper solutions respectively of this problem together with a suitable condition on the function and it has at most two solutions in the sector , then one of such solutions is asymptotically stable. Moreover, it is proved the optimality of the given results, both in the conditions imposed on and in the number of the periodic solutions. [Copyright &y& Elsevier]

The numerical solution of underwater shock fluid – structure interaction problems using boundary element/finite element techniques became tractable through the development of the family of Doubly Asymptotic Approximations (DAA). Practical implementation of the method has relied on the so-called augmentation of the DAA equations. The fluid and structural systems are respectively coupled by the structural acceleration vector in the surface normal direction on the right hand side of the DAA equations, and the total pressure applied to the structural equations on its right hand side. By formally solving for the acceleration vector from the structural system and substituting it into its place in the DAA equations, the augmentation introduces a term involving the inverse of the structural mass matrix. However there exist at least two important classes of problems in which the structural mass matrix is singular. This paper develops a method to carry out the augmentation for such problems using a generalized inverse technique. [ABSTRACT FROM AUTHOR]

The paper deals with an accurate mathematical and numerical analysis of a spherical-circular antenna printed over a coaxial spherical-circular ground conductor. A coaxial probe simulated by a radial driving current placed between two conductors excites the antenna. Consideration is done in terms of the spherical vector wave function expansions of the field in each partial domain. The problem is cast into a coupled set of the dual-series equations for the expansion coefficients, and then to an infinite-matrix equation having favorable features. This is achieved by following the Method of Analytical Regularization developed earlier for the analysis of spherical reflectors. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

This article presents a discussion on the paper entitled "Most Hydraulically Efficient Standard Lined Canal Sections," by David C. Froehlich. It suggests a way by which the formula presented by Froehlich could be better simplified using equations number 1, 9 and 13 in his paper. It explains that how the values can be calculated using each of the equations, 1, 9 and 13 and how each of these equations relates to equations number 2 and 3 to obtain optimal conditions for a specified value m. It proves that the equations are enough to optimize the sharp-cornered trapezoid cross section for a specified value m or unspecified m.

This article comments on the paper entitled "Simplified Design of Hydraulically Efficient Power-Law Channels with Freeboard," by Ahmed S. A. Hussein. It commends Hussein for present a procedure for the design of such power-law channels with freeboard. It offers a more simple design procedure for such power-law channels. It refers to computer software that can be used for designing such power-law channels with free board. It illustrates several equations to achieve a more simple design procedure and offering example in applications.

This article presents the author's response to comments on his paper entitled "Most Hydraulically Efficient Standard Lined Canal Sections," by David C. Froehlich. The author appreciates the commentator's neat simplification of equation 20. It discusses the points raised by the commentator when comparing optimal sections and the difference between the types, which it is found out is rather small. In considering the cost of putting such a hydraulic structure the quantity of lining material needed should be a significant factor.

In this correspondence, we point out an error in equation (4) in the above paper. We also correct equation (4) by a detailed mathematical deduction. [ABSTRACT FROM AUTHOR]