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In this paper, we characterize the family of quotients of internally quasicontinuous functions. Moreover, we study cardinal invariants related to quotients in the case of internally quasicontinuous functions and the complement of this family. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly damped wave equations with hereditary effects, and the nonlinearity f satisfies critical growth. The delay term g(t, ut) may be driven by a function with very weak assumptions, namely, just measurability. We analyze the well-posedness of solutions and verify the existence of the pullback D-attractor in CH01(Ω)×CL2(Ω) by constructing the energy functional and combining with the idea of the contractive function. [ABSTRACT FROM AUTHOR]

For any x 휖 (0, 1], let the series ∑n=1∞1/dnx be the Sylvester expansion of x, where {dj(x), j ≥ 1} is a sequence of positive integers satisfying d1(x) ≥ 2 and dj + 1(x) ≥ dj(x)(dj(x) − 1) + 1 for j ≥ 1. Suppose ϕ : ℕ → ℝ+ is a function satisfying ϕ(n+1) - ϕ (n) → ∞ as n → ∞. In this paper, we consider the setEϕ=x∈01:limn→∞logdnx−∑j=1n−1logdjxϕn=1and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/nβ=γ, and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/τn=η. [ABSTRACT FROM AUTHOR]

In this paper, we present certain composition formulae of the pathway fractional integral operators associated with two extended Mittag-Leffler functions. Here, we find out the relevant connections of some particular cases of the main results with those earlier ones. [ABSTRACT FROM AUTHOR]

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283-289, 2006) and Noor (J. Optim. Theory Appl. 115:447-452, 2002). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite- Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283-289, 2006) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the Cauchy problem of inhomogeneous convolution equations of a fairly general nature. To solve the problems posed here, we apply the operator method proposed in some earlier papers by the author. The solutions of the problems under consideration are found using an effective method in the form of well-convergent vector-valued power series. The proposed method ensures the continuity of the obtained solutions with respect to the initial data and the inhomogeneous term of the equation. [ABSTRACT FROM AUTHOR]

The paper mentioned above is a contribution of the authors to Volume 18 (2002) of this journal, see . Recently, J. Domsta pointed out to us that the proof of Theorem 4.2 in that paper contains an error. The purpose of this addendum is to present a correct argument for it. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

This article presents information on various papers related to the emergence of non-Archimedean mathematics systems of magnitudes. In the paper "Recent Work On The Principles of Mathematics" which appeared in 1901, researcher, Bertrand Russell reported about the three central problems of traditional mathematical philosophy. He further stated that the concept of an infinitesimal had been found to be incoherent. Researcher P. Du Bois-Reymond laid the groundwork for his theory in his paper "Sur la grandeur relative des infinis des functions" and developed it in more than a dozen other works culminating in his mathematico-philosophical treatise Die Allgeneine Functiontheorie.

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3 $$\mathcal{K}$$ ). Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]

The Model-Driven Architecture initiative of the OMG promotes the idea of transformations in the context of mapping from platform independent to platform specific models. Additionally, the popularity of XML and the wide spread use of XSLT has raised the profile of model transformation as an important technique for computing. In fact, computing may well be moving to a new paradigm in which models are considered first class entities and transformations between them are a major function performed on those models. This paper proposes an approach to defining and implementing model transformations which uses metamodelling patterns to capture the essence of mathematical relations. It shows how these patterns can be used to define the relationship between two different metamodels. A goal of the approach is to enable complete specifications from which tools can be generated. The paper describes implementations of the examples, which have been partially generated from the definitions using a tool generation tool. A number of issues emerge which need to be solved in order to achieve the stated goal; these are discussed. [ABSTRACT FROM AUTHOR]

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate n-exponential convexity and log-convexity using the positive functional defined as the difference of the left-hand side and right-hand side of the inequality from (Pečarić and Janić in Facta Univ., Ser. Math. Inform. 3:39-42, ). We also give mean value theorems of Lagrange and Cauchy types. Finally, we construct means with Stolarsky property. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature. [ABSTRACT FROM AUTHOR]

This paper gives a purely set-theoretical extension theorem on iterative roots, which generalizes the results about PM functions by Zhang and his collaborators. Applying this extension theorem, we construct all iterative roots with infinitely many discontinuities for a class of Markov maps. [ABSTRACT FROM AUTHOR]

One famous open problem in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In 1973, Kotzig (Util Math 4:261-290, ) proved that if a leaf of a long enough path is identified with any vertex of an arbitrary tree, the resulting tree is graceful. In this paper, we prove that if the center of a large enough star is identified with any vertex of an arbitrary tree, the resulting tree is graceful, and we also provide an upper bound for the size of the star. [ABSTRACT FROM AUTHOR]

In the present paper we introduce and study the notion of pairwise weakly Lindelöf bitopological spaces and obtain some results. Further, we also study the pairwise weakly Lindelöf subspaces and subsets, and investigate some of their properties. It is proved that a pairwise weakly Lindelöf property is not a hereditary property. [ABSTRACT FROM AUTHOR]

Let R be a ring. Let m and n be positive integers, a right R-module M is called (m,n)-small injective, if every right R-homomorphism from an n-generated submodule of Jm to M extends to one from Rm to M. A ring R is called right (m,n)-small injective if the right R module RR is (m,n)-small injective. In this paper, we give some properties of (m,n)-small injective modules and right (m,n)-small injective rings. [ABSTRACT FROM AUTHOR]

The Letac principle says that if the backward iterations of a random function converge almost surely, then the forward iterations converge in distribution. In this paper, we find conditions for the inverse implication to hold. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce three classes Hp, q0, Hp, q¹ and T Hp* of p-harmonic mappings and discuss the properties of mappings in these classes. First, we discuss the starlikeness and convexity of mappings in Hp, q0 and Hp, q¹. Then establish the covering theorem for mappings in Hp, q¹. Finally, we determine the extreme points of the class T Hp*. [ABSTRACT FROM AUTHOR]

It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes. [ABSTRACT FROM AUTHOR]

Normal Boundary Intersection (NBI) and (Enhanced) Normalised Normal Constraint (E)NNC are attractive and popular approaches to generate an approximation of the Pareto set in nonlinear multi-objective optimisation problems. All three methods are based on similar ideas, but do not always yield identical results, which may confuse practitioners. Hence, the current paper provides theoretical insights in the conditions under which identical results are obtained. Typically, NBI and ENNC are able to generate the same candidate Pareto points, if all additional inequalities in the ENNC subproblem are active. In general, NBI and NNC do not return the same points when three or more objectives are considered. Equivalence relations between the resulting lagrange multipliers for the additional NBI and ENNC (in)equality constraints have been derived. Moreover, the obtained relations have lead to novel criteria for detecting non-Pareto optimal points that in adverse situations maybe generated by these methods. The major advantage is that the removal criteria do not rely on a time-consuming pairwise comparison but only need matrix multiplications. A Matlab implementation has been added for completeness. The insights are illustrated for a general nonlinear bi-objective and three-objective optimisation problem, and a dynamic three-objective tubular reactor optimisation problem from chemical engineering. Finally, practical guidelines are added. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. It introduces a theory of Michell trusses taking support (reaction) costs into consideration, which is illustrated with a modified version of Michell's best known example. Then eight other variations on this example are presented, with (i) equal or unequal permissible stresses in tension and compression, (ii) the structural domain consisting of a half-plane or a full plane, and (iii) the supports being two pins, or a pin and a roller. Previously noted shortcomings of Michell's theory are highlighted and several fundamental properties of non-selfadjoint topology optimization problems discussed. The analytical solutions are verified to a high degree of accuracy by numerical results. [ABSTRACT FROM AUTHOR]

Let ([InlineMediaObject not available: see fulltext.], d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on [InlineMediaObject not available: see fulltext.] × ⋯ × [InlineMediaObject not available: see fulltext.] for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan. [ABSTRACT FROM AUTHOR]

An Adaptive Regularisation framework using Cubics (ARC) was proposed for unconstrained optimization and analysed in Cartis, Gould and Toint (Part I, Math Program, doi:, 2009), generalizing at the same time an unpublished method due to Griewank (Technical Report NA/12, 1981, DAMTP, University of Cambridge), an algorithm by Nesterov and Polyak (Math Program 108(1):177-205, 2006) and a proposal by Weiser, Deuflhard and Erdmann (Optim Methods Softw 22(3):413-431, 2007). In this companion paper, we further the analysis by providing worst-case global iteration complexity bounds for ARC and a second-order variant to achieve approximate first-order, and for the latter second-order, criticality of the iterates. In particular, the second-order ARC algorithm requires at most $${\mathcal{O}(\epsilon^{-3/2})}$$ iterations, or equivalently, function- and gradient-evaluations, to drive the norm of the gradient of the objective below the desired accuracy $${\epsilon}$$, and $${\mathcal{O}(\epsilon^{-3})}$$ iterations, to reach approximate nonnegative curvature in a subspace. The orders of these bounds match those proved for Algorithm 3.3 of Nesterov and Polyak which minimizes the cubic model globally on each iteration. Our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate Hessians. [ABSTRACT FROM AUTHOR]

Let f∈ L be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S( f, x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence { a} is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on { a} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM condition in real space. In this paper, we give a complete generalization to the complex space. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study generalized hyperconnected spaces. Various characterizations of generalized hyperconnected spaces and preservation theorems are discussed. [ABSTRACT FROM AUTHOR]

Inspired by a theorem of Marcinkiewicz [J. Marcinkiewicz, On a class of functions and their Fourier series, C. R. Soc. Sci. Varsovie, 26:71-77, 1934. Reprinted in: J. Marcinkiewicz, Collected Papers (A. Zygmund (Ed.)), PaństwoweWydawnictwo Naukowe,Warsaw, 1964] stating that the maximum of the absolute values of real Fourier coefficients a and b of a function of bounded p-variation $$ \left( {p \geqslant 1} \right) $$ on an interval [0 , 1] is of order O( n) as n → ∞, we compute the Fourier coefficients of the linear fractional stable motion (LFSM) and of the closely related Riemann-Liouville (RL) process and investigate the rate of their decay. [ABSTRACT FROM AUTHOR]

In this paper, we obtain extensions of majorization type results and extensions of weighted Favard's and Berwald's inequality. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results and differences deduced from weighted Favard's and Berwald's inequality. This implies a surprising property of exponentially convexity and log-convexity of these differences which allows us to deduce Lyapunov's and Dresher's inequalities for these differences, which are improvements of majorization type results and weighted Favard's and Berwald's inequalities. Analogous Cauchy's type means, as equivalent forms of exponentially convexity and log-convexity, are also studied and the monotonicity properties are proved. [ABSTRACT FROM AUTHOR]

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem ( Math. Program., doi:, ). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported. [ABSTRACT FROM AUTHOR]

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n by employing the Li’s method of energy function. [ABSTRACT FROM AUTHOR]

Let ${\mathfrak{g}}$ be a finite dimensional simple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic 0. Let ${\mathfrak{g}}_{{\mathbb{Z}}}$ be a Chevalley ℤ-form of ${\mathfrak{g}}$ and ${\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk$, where $\Bbbk$ is the algebraic closure of ${\mathbb{F}}_{p}$. Let $G_{\Bbbk}$ be a simple, simply connected algebraic $\Bbbk$-group with $\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}$. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra $U({\mathfrak{g}}_{\Bbbk})$ to show that if the Gelfand–Kirillov conjecture (from 1966) holds for ${\mathfrak{g}}$, then for all p≫0 the field of rational functions $\Bbbk ({\mathfrak{g}}_{\Bbbk})$ is purely transcendental over its subfield $\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}$. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions ${\mathbb{K}}({\mathfrak{g}})$ is not purely transcendental over its subfield ${\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}}$ if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4, then the Lie field of ${\mathfrak{g}}$ is more complicated than expected. [ABSTRACT FROM AUTHOR]

In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan’s identities, Winquist’s identity and many other interesting identities. [ABSTRACT FROM AUTHOR]

Császár introduced the notions of ascending operation, ascending hull of operation and ascending generalized neighborhood system in [2]. In this paper, we introduce the notions of interior and closure operators and ( ϕ,ϕ′)-continuity on ascending generalized neighborhood systems. We characterize some properties of such notions in terms of convergence of p-stacks. [ABSTRACT FROM AUTHOR]

The paper considers issues related to constructing asymptotic formulas for deviations of Riesz sums. The error estimates in the spaces C and L p in terms of the best approximations are obtained. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

The paper studies the region of values of the system { f( z1), f( z2), c2},where z j, j=1, 2, are arbitrary fixed points of the disk | z|<1; f ∈ T, and the class T consists of all functions f( z) = z + c2 z2 + ··· regular in the disk | z| < 1 and satisfying the condition Im f( z)·Im z>0 for Im z > 0 for Im z ≠ 0. The region of values of f( z1) in the subclass of functions f ( z) ∈ T with prescribed values c2 and f( z2) is determined. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

The paper considers the capacity of a polycondenser and the module of a family of measures associated with the class of curves connecting the polycondenser’s plates. It is proved that these quantities are equal. Also a relation between the polycondenser’s capacity and the module of a family of vector measures associated with the class of functions admissible for the polycondenser’s capacity is established. The results obtained generalize the earlier results by M. Ohtsuka and H. Aikawa. [ABSTRACT FROM AUTHOR]

The paper presents sharp inequalities for the moduli of rational functions under certain constraints on the modulus of the independent variable. These inequalities supplement some results of Govil, Mohapatra, and Dubinin. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

This paper proposes a generalization of column generation, reformulating the master problem with fewer variables at the expense of adding more constraints; the sub-problem structure does not change. It shows both analytically and computationally that the reformulation promotes faster convergence to an optimal solution in application to a linear program and to the relaxation of an integer program at each node in the branch-and-bound tree. Further, it shows that this reformulation subsumes and generalizes prior approaches that have been shown to improve the rate of convergence in special cases. [ABSTRACT FROM AUTHOR]

A directed graph has a natural Z-module homomorphism from the underlying graph's cycle space to Z where the image of an oriented cycle is the number of forward edges minus the number of backward edges. Such a homomorphism preserves the parity of the length of a cycle and the image of a cycle is bounded by the length of that cycle. Pretzel and Youngs (SIAM J. Discrete Math. 3(4):544-553, 1990) showed that any Z-module homomorphism of a graph's cycle space to Z that satisfies these two properties for all cycles must be such a map induced from an edge direction on the graph. In this paper we will prove a generalization of this theorem and an analogue as well. [ABSTRACT FROM AUTHOR]

For a locally compact group G, the measure convolution algebra M( G) carries a natural coproduct. In previous work, we showed that the canonical predual C0( G) of M( G) is the unique predual which makes both the product and the coproduct on M( G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ1( S) also carries a coproduct. In this paper we examine preduals for ℓ1( S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ1( S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ1( S) when S is either ℤ+×ℤ or (ℕ,⋅). [ABSTRACT FROM AUTHOR]

This paper investigates the structure of the quasi-spectrum of divided powers and its behavior under the Taylor homomorphism. General as well as stronger particular results are obtained. In particular, the answer for the Hurwitz series is stated. [ABSTRACT FROM AUTHOR]

In this paper, we solve the problem proposed by Lan Wen for the case of dim M = 3. Roughly speaking, we prove that for fixed i, f has C1 persistently no small angles of index i if and only if f has a dominated splitting of index i on the C1 i-preperiodic set P( f). [ABSTRACT FROM AUTHOR]

In this paper we show for every pair language L ⊆ {0, 1}* x {0, 1}* in NTIME(T) for some non-decreasing function T : ℤ+ → ℤ+ there is a PCPP-verifier such that the following holds. In time poly(|x|, log |y|, log T(|x| + |y|)) it decides the membership of a purported word (x, y) by reading the explicit input x entirely and querying the implicit input y and the auxiliary proof of length T(|x| + |y|) ⋅ poly log T(|x| + |y|) in a constant number of positions. [ABSTRACT FROM AUTHOR]

In this paper the notions of (α, β) - (mX , mY )-continuous and (α, β) - (mX , mY )-semi continuous functions are introduced and some properties of such functions are given. [ABSTRACT FROM AUTHOR]

This paper focuses on a part of the presentation given by the third author at the Shanghai Forum on Industrial and Applied Mathematics (Shanghai 2006). It is related to the existence of a periodic solution of evolution variational inequalities. The approach is based on the method of guiding functions. [ABSTRACT FROM AUTHOR]

It is known that optimal mappings in optimal transportation problems are uniquely determined by corresponding potential functions. In this paper we prove various local properties of potential functions. In particular we obtain the C1, α regularity of potential functions with optimal exponent α, which improves previous regularity results of Loeper. [ABSTRACT FROM AUTHOR]

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the L p metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery. [ABSTRACT FROM AUTHOR]

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization. [ABSTRACT FROM AUTHOR]