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Essentially Strictly Differentiable Lipschitz Functions
 J. FUNCTIONAL ANALYSIS
, 1995
"... In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. For example, we examine the relationship between integrability, Drepresentability and strict differentiability. In addition to this, we ..."
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Cited by 6 (4 self)
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In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on Banach spaces. For example, we examine the relationship between integrability, Drepresentability and strict differentiability. In addition to this, we
A chain rule for essentially strictly differentiable Lipschitz functions
, 1996
"... In this paper we introduce a new class of realvalued locally Lipschitz functions, (that are similar in nature and definition to Valadier's saine functions) which we call arcwise essentially smooth, and we show that if g : R n ! R is arcwise essentially smooth on R n and each function f ..."
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Cited by 3 (2 self)
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f j : R m ! R; 1 j n is strictly differentiable almost everywhere in R m , then g ffi f is strictly differentiable almost everywhere in R m , where f j (f 1 ; f 2 ; :::f n ). We also show that all the semismooth and pseudoregular functions are arcwise essentially smooth. Thus, we provide
The embryonic cell lineage of the nematode Caenorhabditis elegans
 Dev. Biol
, 1983
"... The number of nongonadal nuclei in the freeliving soil nematode Caenorhabditis elegans increases from about 550 in the newly hatched larva to about 810 in the mature hermaphrodite and to about 970 in the mature male. The pattern of cell divisions which leads to this increase is essentially invarian ..."
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Cited by 540 (19 self)
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. Frequently, several blast cells follow the same asymmetric program of divisions; lineally equivalent progeny of such cells generally differentiate into functionally equivalent cells. We have determined these cell lineages by direct observation of the divisions, migrations, and deaths of individual cells
The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets
 Annals of Applied Probability
, 1997
"... . The paper studies the problem of maximizing the expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. We show that the necessary and sufficient condition on a utility function for the validity of several key assertions of the theor ..."
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Cited by 264 (19 self)
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of the theory to hold true is the requirement that the asymptotic elasticity of the utility function is strictly less then one. 1. Introduction A basic problem of mathematical finance is the problem of an economic agent, who invests in a financial market so as to maximize the expected utility of his terminal
Optimal approximative integration of Lipschitz functions
, 1997
"... . Let OE be a strictly positive continuous weight function on [0; 1], and n 2 N. We indicate  for certain classes of functions OE  how to find (x 0 =)0 ! x 1 ! \Delta \Delta \Delta ! xn ! 1(= xn+1 ) and real numbers c 0 ; c 1 ; . . . ; c n+1 such that P n+1 k=0 c k f(x k ) approximates as well ..."
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Cited by 2 (0 self)
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as well as possible the weighted integral R 1 0 f(x)OE(x)dx for Lipschitz functions f : [0; 1] \Gamma! R. In particular, an optimal solution can easily be found if OE is twice differentiable such that \Gamma(log OE) 00 is nonnegative. 1. The problem Let OE and n be as in the abstract. In order
Differential operator and weak topology for Lipschitz maps
, 2009
"... We show that the Scott topology induces a topology for realvalued Lipschitz maps on Banach spaces which we call the Ltopology. It is the weakest topology with respect to which the Lderivative operator, as a second order functional which maps the space of Lipschitz functions into the function spac ..."
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Cited by 3 (2 self)
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or primitive maps of functions. We use this to verify that the Ltopology is strictly coarser than the wellknown Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the Ltopology but strictly
Directional Derivatives Of Lipschitz Functions
, 2000
"... Let f be a Lipschitz mapping of a separable Banach space X to a Banach space Y . We observe that the set of points at which f is differentiable in a spanning set of directions but not Gateaux differentiable is oedirectionally porous. Since Borel oe directionally porous sets, in addition to bei ..."
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Cited by 19 (2 self)
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Let f be a Lipschitz mapping of a separable Banach space X to a Banach space Y . We observe that the set of points at which f is differentiable in a spanning set of directions but not Gateaux differentiable is oedirectionally porous. Since Borel oe directionally porous sets, in addition
ADOLC: A Package for the Automatic Differentiation of Algorithms Written in C/C++
, 1995
"... The C++ package ADOLC described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivative evaluation routines may be called from C/C++, Fortran, or any other language that can be linked ..."
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Cited by 190 (26 self)
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differential equations, special routines are provided that evaluate the Taylor coefficient vectors and their Jacobians with respect to the current state vector. The derivative calculations involve a possibly substantial (but always predictable) amount of data that are accessed strictly sequentially
ALGORITHMIC ASPECTS OF LIPSCHITZ FUNCTIONS
"... Abstract. We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Furthermore, a real z is Schnorr random if and only if every Lipschitz function with L1computabl ..."
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Cited by 9 (4 self)
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Abstract. We characterize the variation functions of computable Lipschitz functions. We show that a real z is computably random if and only if every computable Lipschitz function is differentiable at z. Furthermore, a real z is Schnorr random if and only if every Lipschitz function with L1
Lipschitz perturbations of differentiable implicit functions
, 803
"... Abstract. Let y = f(x) be a continuously differentiable implicit function solving the equation F(x, y) = 0 with continuously differentiable F. In this paper we show that if Fε is a Lipschitz function such that the Lipschitz constant of Fε − F goes to 0 as ε → 0 then the equation Fε(x, y) = 0 has a ..."
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Abstract. Let y = f(x) be a continuously differentiable implicit function solving the equation F(x, y) = 0 with continuously differentiable F. In this paper we show that if Fε is a Lipschitz function such that the Lipschitz constant of Fε − F goes to 0 as ε → 0 then the equation Fε(x, y) = 0 has
Results 1  10
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