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In this paper, we first derive two types of transformed Franklin polynomial: substituted and weighted radial Franklin polynomials. Two radial orthogonal moments are proposed based on these two types of polynomials, namely substituted Franklin-Fourier moments and weighted Franklin-Fourier moments &#x0028 SFFMs and WFFMs &#x0029, which are orthogonal in polar coordinates. The radial kernel functions of SFFMs and WFFMs are transformed Franklin functions and Franklin functions are composed of a class of complete orthogonal splines function system of degree one. Therefore, it provides the possibility of avoiding calculating high order polynomials, and thus the accurate values of SFFMs and WFFMs can be obtained directly with little computational cost. Theoretical and experimental results show that Franklin functions are not well suited for constructing higher-order moments of SFFMs and WFFMs, but compared with traditional orthogonal moments &#x0028 e.g., BFMs, OFMs and ZMs &#x0029 in polar coordinates, the proposed two types of Franklin-Fourier Moments have better performance respectively in lower-order moments.

In this paper we show an interior discontinuity and its piecewise regularity for a stationary compressible Stokes problem with inflow jump datum. The discontinuity is preserved along the interior curve directed by the ambient vector field. The difficulty lies in controlling the gradient of the pressure across the interior curve starting at the jump point that the inflow datum has. We deal with this issue by constructing a vector function K corresponding to the interior jump and splitting the vector K from the velocity vector. Finally the remainder for the velocity vector and pressure function is shown to be in the space H 2 , q × H 1 , q for q ≥ 2 .

In this paper, we analyse three interior point continuous trajectories for convex programming with general linear constraints. The three continuous trajectories are derived from the primal–dual path-following method, the primal–dual affine scaling method and the central path, respectively. Theoretical properties of the three interior point continuous trajectories are fully studied. The optimality and convergence of all three interior point continuous trajectories are obtained for any interior feasible point under some mild conditions. In particular, with proper choice of some parameters, the convergence for all three interior point continuous trajectories does not require the strict complementarity or the analyticity of the objective function. These results are new in the literature.

By means of the Minkowski-type nonlinear scalarization functional, in this paper, we establish some nonlinear separation theorems via relative algebraic interior and vector closure in a general real linear space, via relative topological interior and topological closure in a real topological linear space and via quasi relative interior and topological closure in a real separated locally convex topological linear space, respectively. These new separation theorems can be applied to study those vector optimization problems with the ordering cones having possibly empty topological interior and even relative topological interior or relative algebraic interior. As their applications, we give some nonlinear scalarization characterizations of the corresponding weak efficient solutions of vector optimization problems. Moreover, we also present some concrete examples to illustrate the main results in some infinite dimensional spaces.

In this paper, we develop and analyze a new divergence-free kernel approximation method for the time-dependent incompressible Stokes equations on surfaces. The novelty of our proposed method comes from the surface Helmholtz decomposition, which can convert the surface Stokes equations into a coupled equations in which the velocity is deadly divergence-free so that no inf–sup conditions have to be satisfied. Spatial discretization of velocity is implemented by the radial kernel divergence-free approximation spaces, which only need scatter nodes on surfaces. Using the radial basis function collocation method in space, we derive the rigorous stability and convergence result. Numerical examples are presented, demonstrating the efficiency of some model problems on more general surfaces.

In this paper, we generalize polynomial-time primal–dual interior-point methods for symmetric optimization based on a class of kernel functions, which is not coercive. The corresponding barrier functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like many usual barrier functions. Moreover, we analyse the accuracy of the algorithm for this class of functions and we obtain an upper bound for the accuracy which depends on a parameter of the class.

In this article, we study a kind of vector singularly perturbed delay-differential equation. Using boundary layer function method and geometric analysis skill, the asymptotic expression of the system is constructed and the uniform validity of asymptotic solution is also proved.

Let Z be a two dimensional irreducible complex component of the solution set of a system of polynomial equations with real coefficients in N complex variables. This work presents a new numerical algorithm, based on homotopy continuation methods, that begins with a numerical witness set for Z and produces a decomposition into 2-cells of any almost smooth real algebraic surface contained in Z. Each 2-cell (a face) has a generic interior point and a boundary consisting of 1-cells (edges). Similarly, the 1-cells have a generic interior point and a vertex at each end. Each 1-cell and each 2-cell has an associated homotopy for moving the generic interior point to any other point in the interior of the cell, defining an invertible map from the parameter space of the homotopy to the cell. This work draws on previous results for the curve case. Once the cell decomposition is in hand, one can sample the 2-cells and 1-cells to any resolution, limited only by the computational resources available.

In this article, firstly, a generalized cone subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized cone subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established.

We analyze the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated to homogeneous networks. We consider also the special cases of regular and uniform networks. It follows from our results that the interior symmetries, as well as the reverse interior symmetries and quotient interior symmetries, of the network force the existence of eigenvalues with algebraic multiplicity greater than one. The proofs are based on the special form of the adjacency matrices of the networks induced by these interior symmetries.

This paper studies the vector optimization problem of finding weakly efficient points for maps from R n to R m , with respect to the partial order induced by a closed, convex, and pointed cone C ⊂ R m , with nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization problem with a modified convergence sensing condition that allows us to construct an interior proximal method for solving VOP on nonpolyhedral set.

In this paper we will prove the pointwise convergence of L(f; x, y, λ) to f(x0, y0), as (x, y, λ) tends to (x0, y0, λ0) in the space L2π, by the three parameter family of singular operators. In contrast to previous works, the kernel function is radial.

In this work, we consider an inverse problem for the Sturm-Liouville equation with an interior discontinuity, and show that the potential function can be uniquely determined by a set of values of eigenfunctions at some interior point and parts of two spectra.

We consider an aggregation equation in $${\mathbb {R}^n}$$ , n ≥ 2 with fractional dissipation, namely, $${u_t + \nabla\cdot(u \nabla K*u)=-\nu (-\Delta)^{\gamma/2} u}$$ , where 0 ≤ γ

An iterative method is developed for the solution of Poisson's problem on an infinite domain in the presence of interior boundaries held at fixed potential, in three dimensions. The method combines pre-existing fast multigrid-based Poisson solvers for data represented on Cartesian grids with the fast multipole method. Interior boundaries are represented with the embedded boundary formalism. The implementation is in parallel and uses adaptive mesh refinement. Examples are presented for a smooth interior boundary for which an analytical result is known, and for an irregular interior boundary problem. Second-order accuracy in L"1 with respect to the grid resolution is demonstrated for both problems.

One commonly finds in applications of smooth radial basis functions (RBFs) that scaling the kernels so they are `flat' leads to smaller discretization errors. However, the direct numerical approach for computing with flat RBFs (RBF-Direct) is severely ill-conditioned. We present an algorithm for bypassing this ill-conditioning that is based on a new method for rational approximation (RA) of vector-valued analytic functions with the property that all components of the vector share the same singularities. This new algorithm (RBF-RA) is more accurate, robust, and easier to implement than the Contour-Pad\'e method, which is similarly based on vector-valued rational approximation. In contrast to the stable RBF-QR and RBF-GA algorithms, which are based on finding a better conditioned base in the same RBF-space, the new algorithm can be used with any type of smooth radial kernel, and it is also applicable to a wider range of tasks (including calculating Hermite type implicit RBF-FD stencils). We present a series of numerical experiments demonstrating the effectiveness of this new method for computing RBF interpolants in the flat regime. We also demonstrate the flexibility of the method by using it to compute implicit RBF-FD formulas in the flat regime and then using these for solving Poisson's equation in a 3D spherical shell., Comment: 27 pages, lots of figures, Matlab code included

In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761–768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan–China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193–211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K–K–T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.

An interior point approach for medium and large non-negative linear least-squares problems is proposed. Global and locally quadratic convergence is shown even if a degenerate solution is approached. Viable approaches for implementation are discussed and numerical results are provided. Copyright © 2006 John Wiley & Sons, Ltd.

Based on the min-max principle, the standard centering equation in the interior point method is replaced by the optimality condition of a new proximity measure function. Thus, a self-adjusting mechanism is constructed in the new perturbed system. The Newton direction can be adjusted self-adaptively according to the information of last iterates. A self-adjusting interior point method is given based on the new perturbed system. Numerical comparison is made between this algorithm and a primal-dual interior point algorithm using “standard” perturbed system. Results demonstrate the efficiency and some advantages of the proposed algorithm.

A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite-dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short-step path-following method is established.

We study trace inequalities of the type ∥T k f∥ L q (dμ) ≤ C∥f∥ L p (dσ) , f ∈ L p (dσ), in the upper triangle case 1 ≤ q < p for integral operators T k with positive kernels, where dσ and dp are positive Borel measures on R n . Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy E k,σ [μ] = R n(T k [μ]) p' dσ through the L 1 (dμ)-norm of an appropriate nonlinear potential W k,σ [μ] associated with the kernel k and measures dp, dσ. We initially work with a dyadic integral operator with kernel K D (x,y) = Σ Q ∈ D K(Q)X Q (x)X Q (y), where D = {Q} is the family of all dyadic cubes in R n , and K: D → R + . The corresponding continuous versions of Wolff's inequality and trace inequalities are derived from their dyadic counterparts.

In this paper the sampling method for the shape reconstruction of a penetrable scatterer in three-dimensional linear elasticity is examined. We formulate the governing differential equations of the problem in dyadic form in order to acquire a symmetric and uniform representation for the underlying elastic fields. The corresponding far-field operator is defined in the appropriate space setting. We establish the interior transmission problem in the weak sense and consider the case where the nonhomogeneous boundary data are generated by a dyadic source point located in the interior domain. Assuming that the inclusion has absorbing behaviour, we prove the existence and uniqueness of the weak solution of the interior transmission problem. In this framework the main theorem for the shape reconstruction for the transmission case is established. As for the cases of the rigid body and the cavity an approximate far-field equation is derived with the known dyadic Green function term with the source point an interior point of the inclusion. The inversion scheme which is proposed is based on the unboundedness for the solution of an equation of the first kind. More precisely, the support of the body can be found by noting that the solution of the integral equation is not bounded as the point of the location of the fundamental solution approaches the boundary of the scatterer from interior points.

A simple interior point method is proposed for solving a system of linear equations subject to nonnegativity constraints. The direction of update is defined by projection of the current solution on a linear manifold defined by the equations. Infeasibility is discussed and extension for free and bounded variables is presented. As an application, we consider linear programming problems and a comparison with a state-of-the-art primal-dual infeasible interior point code is presented.

For hydraulic computations that take flow and pressure regulators into account, a little-known variant of the Newton method is used. Successive approximations belong to the interior of a set R specified by linear inequalities. In contrast to the base variant of the method of interior points, the Jacobian matrix is square. The solution of the problem of flow distribution is found on the boundary of R. The local convergence of the method considered is analyzed.

There exists an equilibrium potential, an element of a Dirichlet space, for any compact subset, with non-emtpy interior, of Rm. This potential is constant on the interior of the set. There is a corresponding measure that is 0 outside the set. We prove that the restriction of this equilibrium measure to the interior of the set is absolutely continuous, and we derive an explicit formula for its density.

In the boundary integral method, interior point values are calculated by means of an integration over the boundary. For interior points close to the boundary, some integrands will be nearly singular and standard numerical integration is highly inaccurate. In this paper, interior values are computed by means of analytical integration formulas obtained via symbolic computation. Numerical experiments for the two-dimensional Laplace equation ∇2φ = 0 demonstrate that the exact formulas accurately evaluate the potential φ arbitrarily close to the boundary. The interior potential gradient ∇φ is more difficult, and accurate results very near the boundary are obtained by suitably modifying the potential function.

This paper establishes basic results on the existence of interior points and interior paths in a nonlinear monotone complementarity problem in ℝn under very weak interior conditions. We show that the interior paths are bounded, continuous, and all the limit points of the paths are solutions to the complementarity problem. We prove that certain sets, including the solution set to the complementary problem, form a compact convex set. We also prove the existence of generalized interior points and interior paths. These generalized paths are also continuous and contain readily available starting points from which we can follow the paths to locate the solutions to the complementarity problem. We prove our results in the context of maximal monotone operators. The result presented here can be used to develop polynomial time interior point algorithms for general monotone complementarity problems.

A convergence result is proved for the dual variables of a completely degenerate linear programming problem. The successive approximations are shown to converge to an interior point of the dual solution set.

In this paper we report on some recent results concerning Hermite quasi-interpolation on uniform grids with interesting applications to the approximation of solutions to elliptic PDE, quasi-interpolation on nonuniform grids and the cubature of convolutions with radial kernel functions based on an approximation method proposed by V. Maz’ya.

A procedure is given which relates various linear boundary value problems. It is shown that a relations exists between certain ‘exterior’ and ‘interior’ problems. For computational purposes an ‘interior’ problem is more desirable. The method is restricted by the requirement that the Neumann function, or its equivalent, be known for the problem being considered.

It is well-known that electromagnetic solutions of boundary integral equations for perfectly electrically conducting scatterers are nonunique for those frequencies which correspond to interior resonances of the scatterer. In this paper a simple and efficient computational method is developed, in which the interior integral representations, required to hold on an interior closed surface, are used as a sufficient constraint to restore uniqueness. We use the interior equations together with the second kind magnetic field integral equation, so that the ill-posedness of the interior equations does not give a problem. We develop a constrained conjugate gradient method that minimizes a cost functional consisting of two terms. The first term is the error norm with respect to the magnetic field boundary integral equation, while the second term is the error norm with respect to the interior equations over a closed interior surface, which is chosen as small as possible. Some numerical examples show the robustness and efficiency of the pertaining computational procedure.

A novel interior-point algorithm related to entropy barrier functions is considered. An exponential rate of convergence to the optimal solution of corresponding vector fields is proved. >

An interior point of a finite planar point set is a point of the set that is not on the boundary of the convex hull of the set. For any integer k ≥ 1, let h(k) be the smallest integer such that every set of points in the plane, no three collinear, containing at least h(k) interior points has a subset of points containing k or k + 1 interior points. We proved that h(3) =3 in an earlier paper. In this paper we prove that h(4) = 7.

Let us assume that a 2-torus homeomorphism $f$ isotopic to the identity has a segment of irrational slope as its rotation set $\rho(F)$. We prove that if the chain recurrent set $R(f)$ of $f$ is not chain transitive, then $\rho(F)$ has a rational point realized by a periodic point.

When the boundary integral -equation method is applied to exterior acoustics problems, singularities occur in the resulting algebraic equations at various frequencies associated with the eigenvalues of an interior problem. These frequencies are referred to as “forbidden”, and various methods have been devised to overcome the computational difficulties presented at these frequencies. The work presented here is an extension to the CHIEF method in that higher derivatives, in addition to the function itself, are constrained to be zero at selected points in the interior domain. Whereas the relative success of either method depends on the quantity and selection of interior points, the SuperCHIEF method requires fewer interior points and is less sensitive to point selection, resulting in improved robustness without significant increase in computational complexity.

We approximate the effective potential appearing in the radial part of the massless Klein-Gordon equation for the interior of a charged black hole by a second-order polynomial. This approximation allows an analytic treatment of the equation. Further we recover all the relevant properties in the interior region. For the interior normal modes the comparison between the analytical and numerical results yields an excellent agreement when the ratio of charge to mass for the black hole is close to unity. We use two theorems to bound the difference between the exact and approximate eigenvalues. A complementary scheme using both the analytical and numerical methods is proposed to solve related problems in the Kerr metric.

in the sense of "all, except those in a set of first Baire category". The space of all convex surfaces in IR", endowed with Hausdorfl's metric, is a Baire space. We shall see how abnormal convex surfaces may be, by proving that most of them are so.

The problem of optimal analytical extrapolation of holomorphic functions from a finite set of interior data points to another interior point is completely solved in the general case of data known with unequal errors. Simple and easy to handle algorithms are obtained.

Interior solutions for a static, axially symmetric family of solutions of Einstein's equations are described. The interior solutions correspond to spatially bound matter and are properly matched to an exterior vacuum solution. The family of solutions discussed include the Schwarzschild solution as a special case. A general method is exhibited for transforming any spherically symmetric interior solution to an interior for the other members of the family of solutions. The energy density remains positive for at least a finite range of the parameter that describes the family of solutions. Two solutions are explicitly exhibited. One is transformed from the constant density Schwarzschild interior solution and one from the Adler interior solution. The first solution would be expected to be unstable under adiabatic perturbations of the matter, the second would be expected to be stable.

For 1 > p ⩽ q > ∞ , 0 > α > n 1 > p \leqslant q > \infty ,0 > \alpha > n and w ( x ) , υ ( x ) w(x),\upsilon (x) nonnegative weight functions on R n {R^n} we show that the weak type inequality \[ ∫ { T α f > λ } w ( x ) d x ⩽ A λ − q ( ∫ | f ( x ) | p υ ( x ) d x ) q / p \int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}} \] holds for all f ⩾ 0 f \geqslant 0 if and only if \[ ∫ Q [ T α ( χ Q w ) ( x ) ] p ′ υ ( x ) 1 − p ′ d x ⩽ B ( ∫ Q w ) p ′ / q ′ > ∞ \int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty \] for all cubes Q Q in R n {R^n} . Here T α {T_\alpha } denotes the fractional integral of order α , T α f ( x ) = ∫ | x − y | α − n f ( y ) d y \alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy . More generally we can replace T α {T_\alpha } by any suitable convolution operator with radial kernel decreasing in | x | |x| .

In this paper, we present a new barrier function for primal–dual interior-point methods in linear optimization. The proposed kernel function has a trigonometric barrier term. It is shown that in the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior-point methods, the iteration bound is the best currently known bound for primal–dual interior-point methods.

Numerical solutions to the Helmholtz integral provide a means of computing the surface pressure on an arbitrarily shaped body with a prescribed surface velocity. When using the “exterior” form of the integral, i.e., restricting the source and field coordinates of the Green's function to the surface of the body, singularities within the Green's functions are encountered along with uniqueness problems for the solutions at certain eigenfrequencies of the interior space. In this study, we show that both of these problems can be circumvented by using the “interior” form of the integral, i.e., locating all of the field points in the interior space where a zero pressure condition is prescribed. Criteria for choosing the optimum set of interior field points are described along with a general discussion of the accuracy of the method. Comparisons of numerical and closed form solutions are presented for the monopole and dipole radiators. The method has been developed for use on a personal computer. [Work supported by NSF.]

In this paper we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form: , , where ( is an arbitrary closed, semi-closed or open region in ) and , Λ is a set of non-negative numbers with accumulation point . Also we provide an example to support these theoretical results. MSC:41A35, 41A25.