New Complexity Analysis of the Primal–Dual Method for Semidefinite Optimization Based on the Nesterov–Todd Direction

Interior-point methods for semidefinite optimization have been studied intensively in recent times, due to their polynomial complexity and practical efficiency. In this paper, first we present some technical results about symmetric matrices. Then, we apply these results to give a unified analysis for both large update and small update interior-point methods for SDP based on the Nesterov–Todd (NT) direction.     

Global Convergence of Conjugate Gradient Methods without Line Search

Global convergence results are derived for well-known conjugate gradient methods in which the line search step is replaced by a step whose length is determined by a formula. The results include the following cases: (1) The Fletcher–Reeves method, the Hestenes–Stiefel method, and the Dai–Yuan method applied to a strongly convex LC ¹ objective function; (2) The Polak–Ribière method and the Conjugate Descent method applied to a general, not necessarily convex, LC ¹ objective function.     

A Modified Rank One Update Which Converges Q-Superlinearly

Quasi-Newton methods are generally held to be the most efficient minimization methods for small to medium sized problems. From these the symmetric rank one update of Broyden (Math. Comp., vol. 21, pp. 368–381, 1967) has been disregarded for a long time because of its potential failure. The work of Conn, Gould and Toint (Math. Prog., vol. 50, pp. 177–195, 1991), Kelley and Sachs (COAP, vol. 9, pp. 43–64, 1998) and Khalfan, Byrd and Schnabel (SIOPT, vol. 3, pp. 1–24, 1993; SIOPT, vol. 6, pp. 1025–1039, 1996) has renewed the interest in this method. However the question of boundedness of the generated matrix sequence has not been resolved by this work. In the present paper it is shown that a slightly modified version of this update generates bounded updates and converges superlinearly for uniformly convex functions. Numerical results support these theoretical considerations.     

Solution of Index 2 Implicit Differential-Algebraic Equations by Lobatto Runge-Kutta Methods

We consider the numerical solution of systems of index 2 implicit differential-algebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIA-B-C-C^*-D methods are included. We show super-convergence of optimal order 2s–2 for the s-stage Lobatto families provided the constraints are treated in a particular way which strongly relies on specific properties of the SPARK coefficients. Moreover, reversibility properties of the flow can still be preserved provided certain SPARK coefficients are symmetric.     

Reconstructions of fronts and caustics depending on a parameter and versality of mappings

A survey of results is presented in the problem of describing up to diffeomorphism rearrangements of fronts and caustics in one-parameter families of general position in spaces of low dimension.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 22, pp. 56–93, 1983.     

On the conservativeness of two-stage symmetric-symplectic Runge-Kutta methods and the Störmer-Verlet method

In the present paper, we discuss the problem on the total energy conservation for the numerical solution of the Cauchy problem for the equations of classical molecular dynamics by symplectic and symmetric methods. We consider the methods from a one-parameter family of two-stage symmetric-symplectic Runge-Kutta methods and the Störmer-Verlet method. In particular, we show that a numerical algorithm preserving the total energy of the system on the approximate solutions of the model Cauchy problem almost on the entire trajectory can be constructed on the basis of the one-parameter family of two-stage symmetric-symplectic Runge-Kutta methods.     

A new family of Schröder's method and its variants based on power means for multiple roots of nonlinear equations

In this article, we derive one-parameter family of Schröder's method based on Gupta et al.'s (K.C. Gupta, V. Kanwar, and S. Kumar, A family of ellipse methods for solving non-linear equations, Int. J. Math. Educ. Sci. Technol. 40 (2009), pp. 571–575) family of ellipse methods for the solution of nonlinear equations. Further, we introduce new families of Schröder-type methods for multiple roots with cubic convergence. Proposed families are derived from modified Newton's method for multiple roots and one-parameter family of Schröder's method. Numerical examples are also provided to show that these new methods are competitive to other known methods for multiple roots.     

Comparison of certain summability methods by speeds of convergence

We deal with families of summability methods which depend on a continuous parameter and where two different methods are connected either by a Cesàro-type or Euler–Knopp-type method. Extending and applying some results of [14], we are able to compare speeds of convergence in families of generalized Nörlund methods and to give certain Tauberian remainder theorems. Particular cases are the families of Cesàro, generalized Cesàro and Euler–Knopp methods.     

Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules

The present paper is concerned with symmetric Gauss–Lobatto quadrature rules, i.e., with Gauss–Lobatto rules associated with a nonnegative symmetric measure on the real axis. We propose a modification of the anti-Gauss quadrature rules recently introduced by Laurie, and show that the symmetric Gauss–Lobatto rules are modified anti-Gauss rules. It follows that for many integrands, symmetric Gauss quadrature rules and symmetric Gauss–Lobatto rules give quadrature errors of opposite sign.     

Concerning one class of distal minimal sets

Conclusion Let G be a connected, simply-connected, nilpotent, Lie group, D be a discrete isometric [7] subgroup of the group G. There exists a one-parameter subgroup {g_t} (–t}: Translated from Sibirskii Matematicheskii Zhurnal, Vol. 7, No. 4, pp. 740–750, July–August, 1966.     

Several results on the families of diffeomorphisms onS 1

In this paper, we give a necessary and sufficient condition for the one-parameter families of diffeomorphisms onS ¹ to be stable and a necessary condition for the multi-parameter families to be stable; and, moreover, we prove that phase-locking is a generic property of the one-parameter families of diffeomorphisms onS ¹. We also get a necessary and sufficient condition of phase-locking for the one-parameter families of integral diffeomorphisms onS ¹ which strengthens a result in [2].     

Deformation behavior of a filled nitrile vulcanizate in various states of stress

On the basis of experimental data obtained in uniaxial and symmetrical and nonsymmetrical biaxial tension and pure shear, the deformation behavior of unfilled and filled butadiene-nitrile rubbers is described by means of the Bartenev-Khazanovich one-parameter high-elastic potential.Leningrad Branch of the Scientific-Research Institute of the Rubber Industry. Translated from Mekhanika Polimerov, No. 2, pp. 363–366, March–April, 1972.     

A case of the 3-dimensional problem of Apollonius

Summary Inn-dimensions the problem of Apollonius is to determine the (n–1)-spheres tangent ton+1 given (n–1)-spheres. In case no two of the given (n–1)-spheres intersect and no three have the property that one separates the other two, the expected number of solutions is 2 ⁿ⁺¹. Whenn=2 this special problem does indeed always have 8 solutions, but for higher dimensions it turns out that the number of solutions becomes dependent on the relative size and location of the given (n–1)-spheres. We describe in detail the dependence of the number of solutions in the case of the 3-dimensional problem of Apollonius on the 6 inversively invariant parameters that describe configurations of 4 given spheres. We find that the number of solutions, if finite, can be any integer from 0 to 16 and, if infinite, can be a one-, two- or three-fold infinity where the stated multiplicity refers to the number of one-parameter families of solutions that are present.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

On the conservativeness of a two-parameter family of three-stage symmetric-symplectic Runge-Kutta methods

We study the conservation of the total energy in the numerical solution of the Cauchy problem for the equations of classical molecular dynamics by symplectic and symmetric methods. On the basis of the analysis of a two-parameter family of three-stage symmetricsymplectic Runge-Kutta methods, we suggest a one-parameter family of methods preserving the Hamiltonian function. Test computations for sample problems justify a sufficiently high accuracy of the methods.     

Nontrivial Soliton Scattering in 2+1 Anti-de Sitter Space

Using analytic methods, we present integrable solutions of the Bogomolny Yang–Mills–Higgs equations in 2+1 anti-de Sitter space. In particular, families of soliton solutions are constructed explicitly and their dynamics is investigated in some detail.     

Rigidity and Bifurcation Results for CMC Hypersurfaces in Warped Product Spaces

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spatial fiber of de Sitter Schwarzschild spacetime.     

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.     

An operator calculus of tensor commuting operators on spaces of ultrasmooth vectors

We construct a functional calculus of certain unbounded operators of the type of generators of one-parameter semigroups of bounded operators on Banach spaces.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 189–194.