Conic Function Research Articles

An analytical model for strength prediction in multi-bolt composite joints at various loading rates

This paper presents an enhanced analytical model to determine the complete load displacement curve of single- and multi-fastener composite joints. It is an extension of a previous spring-based method, which included the effects of bolt pre-load and clearance, but not local bearing damage. The model has advanced beyond the state-of-the-art as loading rate effects, in addition to bearing damage, are accounted for through a novel conic damage approximation function. Using the developed model, an accurate prediction of the load displacement response of single- and multi-fastener joints to complete failure can be obtained in a matter of seconds. The method is validated against experimental data and excellent correlation was observed. Further studies carried out using the model suggest that slight variations in the energy absorption characteristics at each fastener hole in a multi-fastener joint can significantly alter the bolt-load distribution in the joint.

A Robbins–Monro-type algorithm for computing global minimizer of generalized conic functions

We generalize the notion and some properties of the conic function introduced by Vincze and Nagy in 2012. We provide a stochastic algorithm for computing the global minimizer of generalized conic functions, we prove almost sure and -convergence of this algorithm.

Computation of a numerically satisfactory pair of solutions of the differential equation for conical functions of non-negative integer orders

We consider the problem of computing satisfactory pairs of solutions of the differential equation for Legendre functions of non-negative integer order μ and degree \(-\frac 12+i\tau \), where τ is a non-negative real parameter. Solutions of this equation are the conical functions \(\mbox{{P}}^{\mu }_{-\frac 12+i\tau }(x)\) and \({Q}^{\mu }_{-\frac 12+i\tau }(x)\), x>−1. An algorithm for computing a numerically satisfactory pair of solutions is already available when −1 < x < 1 (see Gil et al. SIAM J. Sci. Comput. 31(3):1716–1741, 2009, Comput. Phys. Commun. 183:794–799, 2012). In this paper, we present a stable computational scheme for a real valued numerically satisfactory companion of the function \(\mbox{{P}}^{\mu }_{-\frac 12+i\tau }(x)\) for x>1, the function \(\Re \left \{e^{-i\pi \mu } {{Q}}^{\mu }_{-\frac {1}{2}+i\tau }(x) \right \}\). The proposed algorithm allows the computation of the function on a large parameter domain without requiring the use of extended precision arithmetic.

Explicit solution for the electrostatic potential of the conducting double sphere

In this article, we report about explicit analytical solutions for the electrostatic potential of two conducting overlapping spheres which do not require a numerical evaluation of the computationally expensive conical functions. The obtained expressions are therefore suited for direct applications and useable for the verification of numerical solutions which are dependent on the numerical inversion of the Mehler-Fock transform.

3D macro and micro-block models for limit analysis of out-of-plane loaded masonry walls with non-associative Coulomb friction

In this paper, a masonry system composed of a facade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.

An Exact Top-k Query Algorithm with Privacy Protection in Wireless Sensor Networks

Aimed at the Top- k query problems in the wireless sensor networks (WSN) where enquirers try to seek the k highest or shortest reported values of the source nodes, using privacy protection technology, we present an exact Top- k query algorithm based on filter and data distribution table (shorted for ETQFD). The algorithm does the query exactly and meanwhile uses conic section privacy function to prevent the disclosure of the real data and then to promise the security of nodes in network. In this proposal, each node in WSN uses data distribution table to reflect the distribution of its own data and keeps exact filter to just return data which is possibly to be the result of the query, so as to reduce the energy cost of network and prolong network lifetime. In addition, data of node is packaged with a privacy protection function based on conic section. The algorithm's performance is examined with respect to a number of parameters using synthetic datasets. Performance analysis and simulation results illustrate that, compared to existing algorithms, ETQFD is energy-efficient and the network lifetime in our algorithm is longer. At the same time, privacy of data in this algorithm is also guaranteed.

Reconstruction of hv-Convex Sets by Their Coordinate X-Ray Functions

Gardner and Kiderlen (Adv. Math. 214:323–343, 2007) presented an algorithm for reconstructing convex bodies from noisy X-ray measurements with a full proof of convergence in 2007. We would like to present some new steps into the direction of reconstructing not necessarily convex bodies by the help of the continuity properties of so-called generalized conic functions. Such a function measures the average taxicab distance of the points from a given compact set \(K\subset \mathbb {R}^{N}\) by integration. The basic result (Vincze and Nagy in J. Approx. Theory 164:371–390, 2012) is that the generalized conic function associated to a compact planar set determines the coordinate X-rays and vice versa. Vincze and Nagy (Submitted to Aequationes Math., 2014) proved continuity properties of the mapping which sends connected compact hv-convex sets having the same axis parallel bounding box to the associated generalized conic functions. We use these results to present an algorithm for the reconstruction of compact connected hv-convex planar bodies given by their coordinate X-rays. The basic method is varied with the quota system scheme. Greedy and anti-greedy versions are also presented with examples.

Interactive reference point procedure based on the conic scalarizing function.

In multiobjective optimization methods, multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions. The conic scalarizing function is a general characterization of Benson proper efficient solutions of non-convex multiobjective problems in terms of saddle points of scalar Lagrangian functions. This approach preserves convexity. The conic scalarizing function, as a part of a posteriori or a priori methods, has successfully been applied to several real-life problems. In this paper, we propose a conic scalarizing function based interactive reference point procedure where the decision maker actively takes part in the solution process and directs the search according to her or his preferences. An algorithmic framework for the interactive solution of multiple objective optimization problems is presented and is utilized for solving some illustrative examples.

Calibration and Evaluation of Link Congestion Functions: Applying Intrinsic Sensitivity of Link Speed as a Practical Consideration to Heterogeneous Facility Types within Urban Network

This paper explores the use of archived data to calibrate volume delay functions (VDFs) and updates their input parameters (capacity and free-flow speed) for planning applications. The sensitivity analysis of speed to change in congestion level is performed to capture functional characteristics of VDFs in modeling specific facility types. Different sensitivity characteristics shown by the VDFs indicate that each function is suitable to a particular facility type. The results of sensitivity analysis are confirmed by the root mean square percent error (RMSPE) values calculated using the Orlando Urban Area Transportation Study (OUATS) model results and observed data. The modified Davidson’s function exhibits remarkable performance in nearly all facility types. The strength of the modified Davidson’s function across a broad range of facilities can be attributed to the flexibility of its tuning parameter, μ. Fitted Bureau of Public Road (BPR) and conical delay functions show lower RMSPE for uninterrupted flow facilities (freeways/expressways, managed lanes) and higher values for toll roads (which might have partial interruptions due to toll booths) and signalized arterials. Akcelik function underperforms on freeways/expressways and managed lanes but shows some improvements for toll roads and superior results for the signalized arterials. This was a desired strength of Akcelik function when modeling link travel speed on facilities where stopped delays were encountered.

Conical functions of purely imaginary order and argument

Associated Legendre functions are studied for the case where the degree is in conical form −½ + iτ (τ real), and the order iμ and argument ix are purely imaginary (μ and x real). Conical functions in this form have applications to Fourier expansions of the eigenfunctions on a closed geodesic. Real-valued numerically satisfactory solutions are introduced which are continuous for all real x. Uniform asymptotic approximations and expansions are then derived for the cases where one or both of μ and τ are large; these results (which involve elementary, Airy, Bessel and parabolic cylinder functions) are uniformly valid for unbounded x.

Focusing of helico-conical cosh-Gaussian beam

Helico-conical beam is attractive for its phase front is nonseparable helical and conical phase function, which produces a spiral intensity distribution occurs at the focal plane. In this letter, the focal properties of cosh-Gaussian beam with helico-conical wavefront were investigated to show that helico-conical phase front does not always produce spiral intensity distribution. On increasing decentered parameters in cosh parts of the beams, the spiral focal pattern evolves into multiple intensity peaks with center main strongest peak. Under condition of higher decentered parameters, the focal spiral curve disappears. In addition, the topological charge also affects the focal pattern considerably.

In-plane dynamics of membranes having constant curvature

We have studied the dynamics of in-plane displacement of a class of two-dimensional homogeneous and isotropic elastic membranes having constant curvature. The equation of motion of a general in-plane displacement field has been derived in covariant form using the variational formulation. The contribution of curvature on the dynamics of the field variable is clearly observed in the field equation. We observe that for certain membranes with constant curvature, the curvature term introduces a quadratic potential with the sign of the coupling constant being decided by the Ricci curvature of the membrane. Using the Helmholtz theorem , we reduce the field equation to two decoupled scalar equations which exhibit an interesting structure. We consider some examples of membranes with positive, and negative curvatures and comment on the nature of solutions expected. We analyze the deformation of the membranes in terms of the dilatation, shear and vorticity of the displacement field. ► For membranes with periodic boundary conditions curvature act as a quadratic potential proportional to its Ricci scalar. ► Dilatoric and deviatoric wave equations are decoupled using Helmholz theorem. ► Eigenspectrum and modes of the spherical membrane are determined exactly. ► Deformation potentials of hyperbolic membrane are found exactly in terms of conical functions. ► Wave modes of hyperbolic membrane are found to be dispersive with certain cut-off frequencies.

A Mixed Mode Neural Network Circuitry for Object Recognition Application

A general purpose Conic Section Function Neural Network (CSFNN) circuitry in Very Large Scale Integration (VLSI) has been designed for an object recognition application. CSFNN is capable of making open and closed decision regions by combining the propagation rules of Radial Basis Functions (RBF) and Multilayer Perceptrons (MLP) on a single neural network with a unique propagation rule. Chip-in-the-loop learning technique was used during the training process. Utilizing mixed-mode hardware techniques, the inputs of the network and the feedforward signals are all analog while the control unit and storage of the network parameters are fully digital. CSFNN circuitry architecture is problem independent and consists of 16 inputs, 16 hidden layer neurons and 8 outputs. Inheriting the merits of CSFNN, the circuitry has good recognition performance on several objects with invariance to pose, lighting, and brightness. The designed hardware achieved a good recognition performance by means of both accuracy and computational time comparable to CSFNN software.

Vertical versus conical square functions

We study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator.

Compactly supported radial basis functions for the acoustic 3D eigenanalysis using the particular integral method

This paper discusses the efficiency of several Compactly Supported Radial Basis Functions (CSRBFs) for the eigenanalysis of 3D acoustic cavities using the Particular Integral Method. Starting with the two most popular CSRBF families due to Wendland and Wu, a third family proposed by Buhmann is suggested. Results on rectangular parallelepiped highlight the benefit of CSRBFs compared to the classical conical function, especially when dealing with cavities discretized by few elements. On the other hand, when the mesh is refined, numerical difficulties arise and particular attention should be paid to the order of the employed CSRBF. Indeed, and while the conical function is likely to be the most robust function, high-order CSRBFs should be avoided. However, the proposed Buhmann's functions appear to bring significant improvements on eigenanalysis when compared to their Wendland and Wu counterparts.

An improved algorithm and a Fortran 90 module for computing the conical function [formula omitted

In this paper we describe an algorithm and a Fortran 90 module ( Conical) for the computation of the conical function P − 1 2 + i τ m ( x ) for x > − 1 , m ⩾ 0 , τ > 0 . These functions appear in the solution of Dirichlet problems for domains bounded by cones; because of this, they are involved in a large number of applications in engineering and physics. In the Fortran 90 module, the admissible parameter ranges for computing the conical functions in standard IEEE double precision arithmetic are restricted to ( x , m , τ ) ∈ ( − 1 , 1 ) × [ 0 , 40 ] × [ 0 , 100 ] and ( x , m , τ ) ∈ ( 1 , 100 ) × [ 0 , 100 ] × [ 0 , 100 ] . Based on tests of the three-term recurrence relation satisfied by these functions and direct comparison with Maple, we claim a relative accuracy close to 10 − 12 in the full parameter range, although a mild loss of accuracy can be found at some points of the oscillatory region of the conical functions. The relative accuracy increases to 10 − 13 – 10 − 14 in the region of the monotonic regime of the functions where integral representations are computed ( − 1 < x < 0 ). Program summary Program title: Conical Catalogue identifier: AELD_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AELD_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 5387 No. of bytes in distributed program, including test data, etc.: 49 615 Distribution format: tar.gz Programming language: Fortran 90 Computer: Any supporting a FORTRAN compiler Operating system: Any supporting a FORTRAN compiler RAM: A few MB Classification: 4.7 Nature of problem: Conical functions appear in a large number of applications because these functions are the natural function basis for solving Dirichlet problems bounded by conical domains. Also, they are the Kernel of the Mehler–Fock transform. Solution method: The algorithm uses different methods of computation depending on the range of parameters: asymptotic expansions, quadrature methods and recurrence relations. Restrictions: In order to avoid underflow/overflow problems, the admissible parameter ranges for computing the conical functions in standard IEEE double precision arithmetic are restricted to ( x , m , τ ) ∈ ( − 1 , 1 ) × [ 0 , 40 ] × [ 0 , 100 ] and ( x , m , τ ) ∈ ( 1 , 100 ) × [ 0 , 100 ] × [ 0 , 100 ] . Additional comments: The module Conical uses a Fortran 90 version of the routine dkia (developed by the authors) for computing the modified Bessel functions K i a ( x ) and its derivative. This routine is included in the distribution file and is also available at http://toms.calgo.org. Running time: Depending on the parameter range: when numerical quadrature is used (for x < 0 ), the algorithm is 10–20 times slower than the computations made using asymptotic expansions + recurrence relations.

A Relativistic Conical Function and its Whittaker Limits

In previous work we introduced and studied a function $R(a_{+},a_{-},{\bf c};v,\hat{v})$ that is a generalization of the hypergeometric function ${}_2F_1$ and the Askey-Wilson polynomials. When the coupling vector ${\bf c}\in{\mathbb C}^4$ is specialized to $(b,0,0,0)$, $b\in{\mathbb C}$, we obtain a function ${\mathcal R}(a_{+},a_{-},b;v,2\hat{v})$ that generalizes the conical function specialization of ${}_2F_1$ and the $q$-Gegenbauer polynomials. The function ${\mathcal R}$ is the joint eigenfunction of four analytic difference operators associated with the relativistic Calogero-Moser system of $A_1$ type, whereas the function $R$ corresponds to $BC_1$, and is the joint eigenfunction of four hyperbolic Askey-Wilson type difference operators. We show that the ${\mathcal R}$-function admits five novel integral representations that involve only four hyperbolic gamma functions and plane waves. Taking their nonrelativistic limit, we arrive at four representations of the conical function. We also show that a limit procedure leads to two commuting relativistic Toda Hamiltonians and two commuting dual Toda Hamiltonians, and that a similarity transform of the function ${\mathcal R}$ converges to a joint eigenfunction of the latter four difference operators.

Index integral representations for connection between cartesian, cylindrical, and spheroidal systems

In this paper, we present two new index integral representations for connection between cartesian, cylindrical, and spheroidal coordinate systems in terms of Bessel, MacDonald, and conical functions. Our result is mainly motivated by solution of the boundary value problems in domains composed of both cartesian and hyperboloidal boundaries, and the need for new integral representations that facilitate the transformation between these coordinates. As a by-product, the special cases of our results will produce new proofs to known index integrals and provide some new integral identities.

Multi-choice goal programming formulation based on the conic scalarizing function

The multi-choice goal programming allows the decision maker to set multi-choice aspiration levels for each goal to avoid underestimation of the decision. In this paper, we propose an alternative multi-choice goal programming formulation based on the conic scalarizing function with three contributions: (1) the alternative formulation allows the decision maker to set multi-choice aspiration levels for each goal to obtain an efficient solution in the global region, (2) the proposed formulation reduces auxiliary constraints and additional variables, and (3) the proposed model guarantees to obtain a properly efficient (in the sense of Benson) point. Finally, to demonstrate the usefulness of the proposed formulation, illustrative examples and test problems are included.

Conical square functions and non-tangential maximal funtions with respect to the gaussian measure

We study, in L 1 (R n ;) with respect to the gaussian measure, nontangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in L 1 -norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.