Conjugate harmonic functions and Clifford algebras

We generalize a Hardy-Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions.     

Conjugate interpolation problems in spaces of entire functions

A generalization of the Taylor expansions of entire functions with variable center is used to illustrate the notion of conjugate interpolation problem. The radii of existence, uniqueness, and convergence for such problems are found. A relationship between the interpolation functional systems of two mutually conjugate problems is revealed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 616–628, April, 2000.     

Conjugate of the Sobolev space of vector-valued functions

Necessary and sufficient conditions are derived for the equality W^ℓE(X)^*=W^ℓE'(X^*), where E is a symmetric space, X a Banach space, ℓ > 0 is an integer. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 159, pp. 119–120, 1987.     

Conjugate functions on the closed interval and their relationship with uniform rational and piecewise polynomial approximations

Earlier the second author showed that, in the periodic case, the rate of best uniform rational approximations of a function is well described in terms of the rates of best uniform piecewise polynomial approximations of the function itself and its conjugate. In the present paper, a similar result is obtained for a closed interval.     

Conjugate transforms for τT semigroups of probability distribution functions

This paper introduces the use of conjugate transforms in the study of τ_T semigroups of probability distribution functions. If Δ⁺ denotes the space of one-dimensional distribution functions concentrated on [0, ∞) and T is a t-norm, i.e., a suitable binary operation on [0, 1], then the operation τ_T is defined for F, G in Δ⁺by τ_T(F, G)(x) = sup_{u+v = x}T(F(u), G(v)) for all x. The pair (Δ⁺, τ_T) is then a semigroup. For any Archimedean t-norm T, a conjugate transform C_T is defined on (Δ⁺, τ_T). These transforms are shown to play a role similar to that played by the Laplace transform on the convolution semigroup. Thus a theory of “characteristic functions” for τ_T semigroups is developed. In addition to establishing their basic algebraic properties, we also use conjugate transforms to study the algebraic questions of the cancellation law, infinitely divisible elements, and solutions of equations in τ_T semigroups.     

Conjugate Direction Boosting

Boosting in the context of linear regression has become more attractive with the invention of least angle regression (LARS), where the connection between the lasso and forward stagewise fitting (boosting) has been established. Earlier it has been found that boosting is a functional gradient optimization. Instead of the gradient, we propose a conjugate direction method (CDBoost). As a result, we obtain a fast forward stepwise variable selection algorithm. The conjugate direction of CDBoost is analogous to the constrained gradient in boosting. Using this analogy, we generalize CDBoost to: (1) include small step sizes (shrinkage) which often improves prediction accuracy; and (2) the nonparametric setting with fitting methods such as trees or splines, where least angle regression and the lasso seem to be unfeasible. The step size in CDBoost has a tendency to govern the degree between L₀- and L₁-penalization. This makes CDBoost surprisingly flexible. We compare the different methods on simulated and real datasets. CDBoost achieves the best predictions mainly in complicated settings with correlated covariates, where it is difficult to determine the contribution of a given covariate to the response. The gain of CDBoost over boosting is especially high in sparse cases with high signal to noise ratio and few effective covariates.     

Conjugate Maps,Subgradients and Conjugate Duality in Set-Valued Optimization

Using the concept of supremum/infimum of a set, defined in terms of the closure of the set, we introduce the notions of conjugate and biconjugate maps as well as that of subgradients of a set-valued map. Conjugate duality results are also established for a set-valued optimization problem.     

Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = B_w on an infinite dimensional separable Hilbert space H whose weight sequence w = (w_n) satisfies , the conjugate operator is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an such that is dense in S(H). We generalize the result to more general conjugate maps , and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.     

Conjugate trigonometric polynomials

Translated from Matematicheskie Zametki, Vol. 48, No. 4, pp. 110–114, October, 1990.     

Conjugate convex operators

Convex mappings from a locally convex space X into F^. = F ∪ {+∞} are considered, where F is an ordered topological vector space and + ∞ an arbitrary greatest element adjoined to F. In view of applications to the polarity theory of convex operators, the possibility is investigated of representing a convex mapping taking values in F^. as a supremum of continuous affine mappings.     

Two-Dimensional Conjugate Martingale Transforms

Characterizations of the two-dimensional H ₁, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

ABS共轭方向算法

本文提出ＡＢＳ共轭方向算法，它可以产生一大类共轭方向．尤其，Ｄｅｎｎｉｓ和Ｔｕｒｎｅｒ（１９８７）提出的广义共轭方向方法也可以由该算法产生     

Conjugate duality for vector-maximization problems

In this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n-dimensional orthant. Using this duality we obtain a zero-gap duality for a vector-maximization problem.