reconstructing hyperbolic cross trigonometric polynomials by sampling along rank 1 lattices with given fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank 1 lattice leads to a one dimensional discrete fourier transform in many applications one is also interested in the reconstruction of the fourier coefficients from samples in the spatial domain we present necessary and sufficient conditions on rank 1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses in addition we suggest approaches for determining suitable rank 1 lattices using a component by component algorithm we present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100

perceived ambiguity and relevant measures we axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i i d probability measures over a product state space s for such preferences we define relevant measures show that they are treated as if they were the only marginals possibly governing the state space and connect them with the measures appearing in the aforementioned representation these results allow us to interpret relevant measures as reflecting part of perceived ambiguity meaning subjective uncertainty about probabilities over states under mild conditions we show that increases or decreases in ambiguity aversion cannot affect the relevant measures this property necessary for the conclusion that these measures reflect only perceived ambiguity distinguishes the set of relevant measures from the leading alternative in the literature we apply our findings to a number of well known models of ambiguity sensitive preferences for each model we identify the set of relevant measures and the implications of comparative ambiguity aversion

parameters for integrating periodic functions of several variables a number theoretical method for numerical integration of periodic functions of several variables was developed some years ago this paper presents lists of numerical parameters to be used in implementing that method the parameters define quadrature formulas for functions of 2 3 8 variables error bounds for those formulas are also tabulated the derivation of the parameters and error bounds is described

an iterative computation of approximations on korobov like spaces this paper treats the multidimensional application of a previous iterative monte carlo algorithm that enables the computation of approximations in l2 the case of regular functions is studied using a fourier basis on periodised functions legendre and tchebychef polynomial bases the dimensional effect is reduced by computing these approximations on korobov like spaces numerical results show the efficiency of the algorithm for both approximation and numerical integration

cubature rules of prescribed merit we introduce a criterion for the evaluation of multidimensional quadrature or cubature rules for the hypercube this is the merit of a rule which is closely related to its trigonometric degree and which reduces to the zaremba figure of merit in the case of a lattice rule we derive a family of rules q k s having dimension s and merit 2 k these rules seem to be competitive with lattice rules with respect to the merit that can be achieved with a given number of abscissas

lattice rules of minimal and maximal rank with good figures of merit for periodic integrands with unit period in each variable certain error bounds for lattice rules are conveniently characterised by the figure of merit which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n distinct nodes then becomes that of finding rules with large values of this paper presents efficient search methods for the discovery of rank 1 rules and of maximal rank rules of high order which possess good figures of merit

quasi random methods for estimating integrals using relatively small samples much of the recent work dealing with quasi random methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finite dimensional integral is replaced by a finite sum of integrand values in contrast with this perspective to concentrate on asymptotic convergence rates this paper emphasizes quasi random methods that are effective for all sample sizes throughout the paper the problem of estimating finite dimensional integrals is used to illustrate the major ideas although much of what is done applies equally to the problem of solving certain fredholm integral equations some new techniques based on error reducing transformations of the integrand are described that have been shown to be useful both in estimating high dimensional integrals and in solving integral equations these techniques illustrate the utility of carrying over to the quasi monte carlo method certain devices that have proven to be very valuable in statistical pseudoran

a generalized sampling theorem for locally compact abelian groups we present a sampling theorem for locally compact abelian groups the sampling sets are finite unions of cosets of a closed subgroup this generalizes the well known case of non equidistant but periodic sampling on the real line for non bandlimited functions an l1 type estimate for the aliasing error is given we discuss the application of the theorem to a class of sampling sets in r s give a general algorithm for computer implementation present a detailed description of the implementation for the s dimensional torus group and point out connections to lattice rules for numerical integration

imbedded lattice rules for multidimensional integration this paper is concerned with imbedded sequences of lattice rules the theoretical properties of certain finite imbedded sequences are developed and an efficient search method for finding good imbedded sequences of this kind is discussed an algorithm for computing these sequences is presented and some numerical results using the search method and this algorithm are given

fast fourier galerkin methods for solving singular boundary integral equations numerical integration and precondition we develop a fast fully discrete fourier galerkin method for solving a class of singular boundary integral equations we prove that the number of multiplications used in generating the compressed matrix is o nlog 3n and the solution of the proposed method preserves the optimal convergence order o n t where n is the order of the fourier basis functions used in the method and t denotes the degree of regularity of the exact solution moreover we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm

multiple integration over bounded and unbounded regions abstract a method is proposed for integrating over r n functions that are reasonably smooth and rapidly decaying at infinity the method makes use of an infinite lattice of quadrature points truncated at some suitable radius in certain circumstances it is shown that the best lattice among all those with a given determinant is the one whose dual lattice corresponds to the densest sphere packing the method of sag and szekeres for integration over n dimensional spheres and cubes is shown to be a special case with a lattice that is not necessarily optimal

preference for flexibility in a savage framework we study preferences over savage acts that map states to opportunity sets and satisfy the savage axioms preferences over opportunity sets may exhibit a preference for flexibility due to an implicit uncertainty about future preferences reflecting anticipated unforeseen contingencies the main result of this paper characterizes maximization of the expected indirect utility in terms of an indirect stochastic dominance axiom that expresses a preference for more opportunities in expectation the key technical tool of the paper a version of mobius inversion has been imported from the theory of nonadditive belief functions it allows an alternative representation using choquet integration and yields a simple proof of kreps 1979 classic result

a smooth model of decision making under ambiguity we propose and axiomatize a model of preferences over acts such that the decision maker prefers act f to act g if and only if e e u f e e u g where e is the expectation operator u is a vn m utility function is an increasing transformation and is a subjective probability over the set of probability measures that the decision maker thinks are relevant given his subjective information a key feature of our model is that it achieves a separation between ambiguity identified as a characteristic of the decision maker s subjective information and ambiguity attitude a characteristic of the decision maker s tastes we show that attitudes towards risk are characterized by the shape of u as usual while attitudes towards ambiguity are characterized by the shape of we also derive x 1 e x as the special case of constant ambiguity aversion ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures this characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by epstein zhang 2001 and ghirardato marinacci 2002 one advantage of this model is that the welldeveloped machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes the model is also distinct from many in the literature on ambiguity in that allows smooth rather than kinked indifference curves this leads to different behavior and improved tractability while still sharing the main features e g ellsberg s paradox etc the maxmin eu model e g gilboa and schmeidler 1989 with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion two illustrative applications to portfolio choice are offered

canonical representation of set functions the representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between no necessarily additive set functions on the players space and additive ones on the coalitions space or alternatively between nonadditive probability measures on a state space and additive ones on the space of events we extend the unanimity basis representation to general infinite spaces of players study spaces of games which satisfy certain properties and provide some conditions for additivity of the resulting additive set function on the space at coalitions these results also allow us to extend some representations of the choquet integral from finite to infinite spaces

the jackknife estimate of variance tukey s jackknife estimate of variance for a statistic s x 1 x 2 cdots x n which is a symmetric function of i i d random variables x i is investigated using an anova like decomposition of s it is shown that the jackknife variance estimate tends always to be biased upwards a theorem to this effect being proved for the natural jackknife estimate of operatorname var s x 1 x 2 cdots x n 1 based on x 1 x 2 cdots x n

vector expected utility and attitudes toward variation this paper proposes a model of decision under ambiguity deemed vector expected utility or veu in this model an uncertain prospect or savage act is assessed according to a a baseline expected utility evaluation a nd b anadjustment that reflects the individual s perception of ambiguity and her attitudes toward it the adjustment is itself a function of the act s exposure to distinct sources of ambiguity as well as its variability the key elements of the veu model are a baseline probability and a collection of random variables or adjustment factors which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events the adjustment to the baseline expected utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor which reflects exposure to the corresponding ambiguity source a behavioral characterization of the veu model is provided furthermore an updating rule for veu preferences is proposed and characterized the suggested updating rule facilitates the analysis of sophisticated dynamic choice with veu preferences

notes on integration and integer sublattices a lattice rule is a quadrature rule over an s dimensional hypercube using n abscissas located on an integration lattice in this paper the authors study sublattices and superlattices of integration lattices and of integer lattices they exploit the properties of generator matrices of a lattice to provide an easy and elegant description of the relation between a lattice and a sublattice of given order they also obtain necessary and sufficient criteria for existence of sublattices and information about the number of these

the determination of canonical forms for lattice quadrature rules abstract lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s dimensional unit hypercube us 0 1 s for a given lattice rule say ql a set of points l the integration lattice regularly spaced in all of rs is generated by a finite number of rational vectors the abscissa set for ql is then p ql l us it is known that p ql is a finite abelian group under addition modulo the integer lattice zs and that ql f may be written in the form of a nonrepetitive multiple sum q l f 1 n 1 n m j 1 1 n 1 j m 1 n m f j 1 n 1 z 1 j m n m z m known as a canonical form in which denotes addition modulo zs in this form zi zs m is called the rank and n1 n2 nm are called the invariants of ql and n i 1 n i for i 1 2 m 1 the rank and invariants are uniquely determined for a given lattice rule in this paper we provide a construction of a canonical form for a lattice rule ql given a generator set for the lattice l we then show how the rank and invariants of ql may be determined directly from the generators of the dual lattice l

pointwise convergence of multiple fourier series sufficient conditions and an application to numerical integration abstract sufficient conditions are established for the pointwise convergence of square partial sums of multiple fourier series in any number of dimensions for functions with discontinuities the result is used to extend the theory of lattice rules for multiple integration

lattice rules by component scaling the authors introduce a theory of rectangular scaling of integer lattices this may be used to construct families of lattices they determine the relation between the zaremba index rho lambda of various members of the same family it appears that if one member of a family has a high index some of the other family members of higher order may have extraordinarily high indices they have applied a technique based on this theory to lists of good lattices available this has enabled the authors to construct lists of excellent previously unknown lattices of high order in three and four dimensions and of moderate order in five dimensions 10 refs 12 tabs

random choice as behavioral optimization we develop an extension of luce s random choice model to study violations of the weak axiom of revealed preference we introduce the notion of a stochastic preference and show that it implies the luce model then to address well known difficulties of the luce model we define the attribute rule and establish that the existence of a well defined stochastic preference over attributes characterizes it we prove that the set of attribute rules and random utility maximizers are essentially the same finally we show that both the luce and attribute rules have a unique consistent extension to dynamic problems vskip 17 5pt author please provide at least three key words that describe the article

representing preferences with a unique subjective state space a corrigendum dekel lipman and rustichini 2001 henceforth dlr axiomatically characterized three representations of preferences that allow for a desire for flexibility and or commitment in one of these representations ordinal expected utility the independence axiom is stated in a weaker form than is necessary to obtain the representation in another additive expected utility the continuity axiom is too weak in this erratum we provide examples showing that the axioms used by dlr are not sufficient and provide stronger versions of these axioms that together with the other axioms used by dlr are necessary and sufficient for these two representations

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

an application of diophantine approximation to the construction of rank 1 lattice quadrature rules lattice quadrature rules were introduced by frolov 1977 sloan 1985 and sloan and kachoyan 1987 they are quasi monte carlo rules for the approximation of integrals over the unit cube in r s and are generalizations of number theoretic rules introduced by korobov 1959 and hlawka 1962 themselves generalizations in a sense of rectangle rules for approximating one dimensional integrals and trapezoidal rules for periodic integrands error bounds for rank 1 rules are known for a variety of classes of integrands for periodic integrands with unit period in each variable these bounds are conveniently characterized by the figure of merit p which was originally introduced in the context of number theoretic rules the problem of finding good rules of order n that is having n nodes then becomes that of finding rules with large values of p this paper presents a new approach based on the theory of simultaneous diophantine approximation which uses a generalized continued fraction algorithm to construct rank 1 rules of high order

intermediate rank lattice rules for multidimensional integration every lattice rule in s dimensions may be characterised by an integer m lying between 1 and s inclusive called the rank or index of the rule and a set of m positive integers called the invariants earlier work has shown that in a certain precise sense lattice rules of rank s are better than the commonly used rank 1 rules here this earlier work is extended by showing that a similar result holds for certain lattice rules of intermediate rank m s

some properties of rank 2 lattice rules a rank 2 lattice rule is a quadrature rule for the unit ital s dimensional hypercube of the form ital qf 1 ital n sub 1 ital n sub 2 summation sub ital j sub 1 1 ital j sub 2 1 sup ital n sub 1 summation sub ital j sub 2 1 sup ital n sub 2 ital bar f ital j sub 1 ital z1 ital n sub 1 ital j sub 2 ital z2 ital n sub 2 which cannot be re expressed in an analogous form with a single sum here ital bar f is a periodic extension of ital f and ital z sub 1 ital z sub 2 are integer vectors in this paper we discuss these rules in detail in particular we categorize a special subclass whose leading one and two dimensional projections contain the maximum feasible number of abscissas we show that rules of this subclass can be expressed uniquely in a simple tricycle form

the existence of efficient lattice rules for multidimensional numerical integration in this contribution to the theory of lattice rules for multidimensional numerical integration we first establish bounds for various efficiency measures which lead to the conclusion that in the search for efficient lattice rules one should concentrate on lattice rules with large first invariant then we prove an existence theorem for efficient lattice rules of rank 2 with prescribed invariants which extends an earlier result of the author for lattice rules of rank 1

component by component construction of good intermediate rank lattice rules it is known that the generating vector of a rank 1 lattice rule can be constructed component by component to achieve strong tractability error bounds in both weighted korobov spaces and weighted sobolev spaces since the weights for these spaces are nonincreasing the first few variables are in a sense more important than the rest we thus propose to copy the points of a rank 1 lattice rule a number of times in the first few dimensions to yield an intermediate rank lattice rule we show that the generating vector and in weighted sobolev spaces the shift also of an intermediate rank lattice rule can also be constructed component by component to achieve strong tractability error bounds in certain circumstances these bounds are better than the corresponding bounds for rank 1 lattice rules

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

the serial test for pseudo random numbers generated by the linear congruential method we consider the linear congruential method for pseudo random number generation and establish effective criteria for the choice of parameters in this method which guarantee statistical almost independence of successive pseudo random numbers applications to numerical integration are also discussed

the ultratriangular form for prime power lattice rules a recent development in the theory of lattice rules has been the introduction of the unique ultratriangular d z form for prime power rules it is known that any lattice rule may be decomposed into its sylow p components these components are prime power rules each of which has a unique ultratriangular form by reassembling these ultratriangular forms in a defined way it is possible to obtain a canonical form for any lattice rule a special case occurs when the ultratriangular forms of each of the sylow p components have a consistent set of column indices in this case it is possible to obtain a unique canonical d z form given the column indices and the invariants for an ultratriangular form we may obtain a formula for the number of ultratriangular forms and hence the number of prime power lattice rules having these column indices and invariants

division composition algebras through their derivation algebras abstract composition algebras are algebras with a nondegenerate quadratic multiplicative form in this paper we construct five families of division composition algebras and prove that any division composition algebra with nonabelian derivation algebra belongs to one of these families as a by product we obtain new examples of real division algebras

on the interpretation of sarin and wakker s a simple axiomatization of nonadditive expected utility in an interesting recent paper sarin and wakker 1992 henceforth s w have provided a new axiomatization of expected utility maximization with capacities ceu its simplicity is attractive in that it provides a constructive interpretation of the role of capacities and choquet integration in the main representation theorem an important issue raised by the paper is the interpretation of the key axiom p4 cumulative dominance s w suggest an appreciation of the axiom as an adaptation of stochastic dominance principles to nonadditive probability contexts most eloquently on page 1260 if viable such an interpretation would supply the ceu model with a powerful intuitive foundation that has been lacking so far this note argues that a stochastic dominance interpretation can be maintained only at the price of either an arbitrary choice of specification which undermines its intuitive force or alternatively of an unintended restriction of the class of characterized preferences two logically independent arguments are presented the first points out an arbitrariness in the definition of the more likely than relation in terms of preferences proposition 1 the second shows a similar arbitrariness in the definition of a stochastic dominance relation in terms of a more likely than relation proposition 2 in each case the invoked symmetry conditions yield a characterization of ceu preferences with symmetric capacities a nontrivial generalization of the seu model that has received little attention in the literature for notation and definitions the reader is referred to s w s paper s w s concern is to develop an intuitively convincing axiomatization of ceu representable preference relations their key axiom p4 is formulated in terms of a more likelythan relation on the algebra v of events that is defined in terms of the preference relation a on the set of acts y to facilitate the subsequent discussion their definition is introduced here formally as a condition on the pair of relations a

an average l 2 discrepancy for number theoretic rules one measure of the goodness of a quadrature rule q for approximating integrals over the unit hypercube is the l2 discrepancy d q for number theoretic rules of prime order n we derive an expression for the average of d2 q having this average makes it possible to compare the potential of number theoretic rules for nonperiodic integrands with equal weight quadrature rules in which the points are independent and uniformly distributed the numerical results presented suggest that number theoretic rules are competitive with such rules as measured by the average l2 discrepancy when the dimension is not more than about 18 the expression obtained for the average is not fully in closed form as it contains certain sums related to dedekind sums for which no closed forms are available an efficient algorithm for calculating dedekind sums allows the average to be calculated in o n log n operations

a theory of diversity how can diversity be measured what does it mean to value biodiversity can we assist noah in constructing his preferences to address these questions we propose a multi attribute approach under which the diversity of a set of species is the sum of the values of all attributes possessed by some species in the set we develop the basic intuitions and requirements for a theory of diversity and show that the multi attribute approach satisfies them in a flexible yet tractable manner a natural starting point is to think of the diversity of a set as an aggregate of the pairwise dissimilarities between its elements the multi attribute framework allows one to make this program formally precise it is shown that the program can be realized if and only if the family of relevant attributes is well ordered acyclic moreover there is a unique functional form aggregating dissimilarity into diversity the length of a minimum spanning tree examples are taxonomic hierarchies and lines representing uni dimensional qualities in multi dimensional settings pairwise dissimilarity information among elements is insufficient to determine their diversity by consequence the qualitative and quantitative behavior of diversity differs fundamentally

the index of merit of kth copy integration lattices r2d2lri is an automatic two dimensional cubature algorithm that demonstrates the practical value of using an augmentation sequence consisting of 2 k 2 copy lattices as a basis for numerical integration this paper investigates use of similar embedded augmentation sequences in higher dimensions by developing theoretical results relating to the index of merit of s dimensional 2 k s copy lattices generated from rank 1 simple lattices the theoretical results can be used to guide the search for good augmentation sequences in s dimensions in the sense of high index of merit

real quadratic flexible division algebras abstract in this paper we give a new process called vectorial isotopy in order to classify the eight dimensional real quadratic flexible division algebras and we solve the isomorphism problem

ambiguity aversion robustness and the variational representation of preferences we characterize in the anscombe aumann framework the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that for all acts f and g the function u represents the decision maker s risk attitudes while the index c captures his ambiguity attitudes these preferences include the multiple priors preferences of gilboa and schmeidler and the multiplier preferences of hansen and sargent this provides a rigorous decision theoretic foundation for the latter model which has been widely used in macroeconomics and finance

integration of nonperiodic functions of two variables by fibonacci lattice rules abstract two dimensional lattice rules are applied to continuous functions over the unit square which do not have a continuous periodic extension it is shown that provided lattice points at vertices and edges are treated appropriately certain functions including all bilinear functions are integrated exactly whenever the lattice contains a possibly rotated square unit cell the fibonacci lattice with denominators f k for the nodes is then shown to have a square unit cell if and only if k is odd numerical experiments for fibonacci rules and copies of fibonacci rules confirm that there are significant different between the odd k and even k cases

a lower bound for rank 2 lattice rules we give a lower bound for a quality measure of rank 2 lattice rules which shows that an existence result of niederreiter is essentially best possible

uncertainty aversion with second order utilities and probabilities subjective expected utility theory does not distinguish between attitudes toward uncertainty ambiguous probabilities and attitudes toward risk unambiguous probabilities both are explained in terms of nonlinear utility for money rather than properties of events per se hence the decision maker displays the same attitude toward all sources of risk and uncertainty there is ample evidence that real decision makers do not always behave or even wish to behave in this way and instead they often distinguish between risk and uncertainty as in ellsbergs 1961 paradox this paper presents a simple axiomatic model of nonneutral attitudes toward uncertainty and a behavioral test for uncertainty aversion that is applicable even if utility is state dependent the decision maker may display different degrees of aversion toward gambles on different kinds of events e g being systematically more averse toward gambles on events whose probabilities are more ambiguous for such a decision maker the elicitation of preferences among objective gambles may not yield the correct measure of risk aversion for modeling real world decisions

real division algebras with large automorphism group we continue the study of the 8 dimensional real division algebras whose derivation algebra is large of type g2 a2 2a1 or a1 begun by benkart and osborn in the early 1980s for some of the families of real algebras that they constructed we find necessary and sufficient conditions for them to be division algebras determine when two division algebras in such a family are isomorphic or determine the automorphism group of such an algebra we use one of these families to prove that every 2 dimensional real division algebra embeds in a 4 dimensional and in an 8 dimensional real division algebra a new family f6 of non isomorphic real 8 dimensional algebras parametrized by the euclidean space r6 is constructed and studied in detail the division algebras in f6 correspond to a non empty open subset of the parameter space we also introduce an interesting 2 parameter subfamily f2 f6 which contains the generalized pseudo octonion algebras we obtain necessary and sufficient conditions for a f2 to be a division algebra in the generic case the algebras in f6 are 1 generated and have so 3 as the automorphism group we also determine all 1 2 and 4 dimensional subalgebras of the division algebras in f6 we show that there exist 8 dimensional and 4 dimensional real division algebras having a 1 dimensional subalgebra not contained in any 2 dimensional subalgebra we also construct 8 dimensional real division algebras having a 2 dimensional subalgebra not contained in any 4 dimensional subalgebra

tables of good lattices in four and five dimensions the good lattice theory yields a powerful method of computing approximations for the integral of functions defined on 0 1 s through averaged sums ofm evaluations we present a continuation of the only existing table of best lattices fors 4 up tom 3298 and the first table fors 5 up tom 772

stochastic choice and consideration sets we model a boundedly rational agent who suers from limited attention the agent considers each feasible alternative with a given unobservable probability the attention parameter and then chooses the alternative that maximises a prefer ence relation within the set of considered alternatives we show that this random choice rule is the only one for which the impact of removing an alternative on the choice probability of any other alternative is asymmetric and menu independent both the preference relation and the attention parameters are identi ed uniquely by stochastic choice data j e l codes d0

on computing the lattice rule criterion r lattice rules are integration rules for approximating integrals of periodic functions over the s dimensional unit cube one criterion for measuring the goodness of lattice rules is the quantity r this quantity is defined as a sum which contains about n sup s 1 terms where n is the number of quadrature points although various bounds involving r are known a procedure for calculating r itself does not appear to have been given previously here we show how an asymptotic series can be used to obtain an accurate approximation to r whereas an efficient direct calculation of r requires o nn sub 1 operations where n sub 1 is the largest invariant of the rule the use of this asymptotic expansion allows the operation count to be reduced to o n a complete error analysis for the asymptotic expansion is given the results of some calculations of r are also given 19 refs 2 tabs

standard state space models preclude unawareness we show that a very broad class of models including possibility correspondences necessarily fail to capture very simple and intuitive implications of unawareness we explain why standard state space formulations suffer from this problem illustrating the point with an example of a nonstandard state space model which avoids the difficulty

a useful result in searches for good vertex modified number theoretic rules number theoretic rules are particularly suited to the evaluation of multiple integrals in which the integrand is periodic for nonperiodic integrands an alternative is to use vertex modified versions of number theoretic rules good vertex modified number theoretic rules may be found by doing a computer search based on some criterion of goodness such criteria include a variant of the l2 discrepancy and the vertex variance here we present a result which may be used to speed up searches for good vertex modified number theoretic rules based on these criteria when the generator vectors are of the one parameter form of korobov

extremal lattices and the construction of lattice rules abstract after introducing the concept of extremal lattices we show how these can be used to construct lattice rules of a high trigonometric degree

objective and subjective rationality in a multiple prior model a decision maker dm is characterized by two binary relations the first reflects choices that are rational in an objective sense the dm can convince others that she is right in making them the second relation models choices that are rational in a subjective sense the dm cannot be convinced that she is wrong in making them in the context of decision under uncertainty we propose axioms that the two notions of rationality might satisfy these axioms allow a joint representation by a single set of prior probabilities and a single utility index it is objectively rational to choose f in the presence of g if and only if the expected utility of f is at least as high as that of g given each and every prior in the set it is subjectively rational to choose f rather than g if and only if the minimal expected utility of f with respect to all priors in the set is at least as high as that of g in other words the objective and subjective rationality relations admit respectively a representation a la bewley 2002 and a la gilboa and schmeidler 1989 our results thus provide a bridge between these two classic models as well as a novel foundation for the latter

letter section randomization of lattice rules for numerical multiple integration here we investigate some procedures for the randomization of lattice rules for numerical multiple integration similar to those proposed by cranley and patterson 1976 for number theoretic rules currently there is no easily calculable error estimate available for general lattice rules and the randomization procedures looked at here allow the calculation of confidence intervals for the error

shifted lattice rules based on a general weighted discrepancy for integrals over euclidean space we approximate weighted integrals over euclidean space by using shifted rank 1 lattice rules with good bounds on the generalised weighted star discrepancy this version of the discrepancy corresponds to the classic l weighted star discrepancy via a mapping to the unit cube the weights here are general weights rather than the product weights considered in earlier works on integrals over r d known methods based on an averaging argument are used to show the existence of these lattice rules while the component by component technique is used to construct the generating vector of these shifted lattice rules we prove that the bound on the weighted star discrepancy considered here is of order o n 1 d for any d 0 and with the constant involved independent of the dimension this convergence rate is better than the o n 1 2 achieved so far for both monte carlo and quasi monte carlo methods

minimal cubature formulae of trigonometric degree in this paper we construct minimal cubature formulae of trigonometric degree we obtain explicit formulae for low dimensions of arbitrary degree and for low degrees in all dimensions a useful tool is a closed form expression for the reproducing kernels in two dimensions

study on noncommutative jordan real division algebras of dimension 8 whose lie derivation algebra is not trivial abstract in this paper we study the noncommutative jordan real division algebras of dimension 8 whose lie derivation algebra is nontrivial and give an affirmative answer to a question posed by benkart and osborn we characterize also the noncommutative jordan real division algebras of dimension 8 whose automorphism group is nontrivial

decomposition and representation of coalitional games a coalitional game is a real valued set function defined on an algebra f of subsets of a space x such that o 0 we prove the existence of a one to one correspondence between coalitional games bounded with respect to the composition norm and countably additive measures defined on an appropriate space

lattice methods for multiple integration theory error analysis and examples this paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension the concept of a lattice rule is introduced and error bounds are developed in terms of the dual lattice examples of lattice rules discussed in the paper are the rectangle rule the number theoretic good lattice methods of korobov and others the body centred cubic rule and a generalization of the latter to a family w nr the w nr rules appear to have interesting extrapolation properties both for fixed r and for r n

temptation and self control we study a two period model where ex ante inferior choice may tempt the decision maker in the second period individuals have preferences over sets of alternatives that represent second period choices our axioms yield a representation that identifies the individual s commitment ranking temptation ranking and cost of self control an agent has a preference for commitment if she strictly prefers a subset of alternatives to the set itself an agent has self control if she resists temptation and chooses an option with higher ex ante utility we introduce comparative measures of preference for commitment and self control and relate them to our representations

stochastic choice and revealed perturbed utility perturbed utility functions the sum of expected utility and a non linear perturbation function provide a simple and tractable way to model various sorts of stochastic choice we provide two easily understood conditions each of which characterizes this representation one condition generalizes the acyclicity condition used in revealed preference theory and the other generalizes luce s iia condition we relate the discrimination or selectivity of choice rules to properties of their associated perturbations both across dierent agents and across decision problems we also show that these representations correspond to a form of ambiguity averse preferences for an agent who is uncertain about her true utility

best simultaneous diophantine approximations i growth rates of best approximation denominators this paper defines the notion of a best simultaneous diophantine approximation to a vector a in rn with respect to a norm 11 11 on rn suppose a is not rational and order the best approximations to a with respect to ii ii by increasing denominators i qi q2 it is shown that these denominators grow at least at the rate of a geometric series in the sense that g a 11 11 ihmjinff q i i let g ii 11 denote the infimum of g a ii i over all a in rn with an irrational coordinate for the sup norm 11 115 on r2 it is shown that g li ii 0 1 270 where 4 02 1

the mean square discrepancy of scrambled t s sequences the discrepancy arises in the worst case error analysis for quasi monte carlo quadrature rules low discrepancy sets yield good quadrature rules this article considers the mean square discrepancies for scrambled lam t m s nets and t s sequences in base b it is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple monte carlo sampling if the reproducing kernel defining the discrepancy satisfies a lipschitz condition with respect to one of its variables separately then the asymptotic order of the root mean square discrepancy is o n 1 log n s 1 2 for scrambled nets if the reproducing kernel satisfies a lipschitz condition with respect to both of its variables then the asymptotic order of the root mean square discrepancy is o n 3 2 log n s 1 2 for scrambled nets for an arbitrary number of points taken from a t s sequence the root mean square discrepancy appears to be no better than o n 1 log n s 1 2 regardless of the smoothness of the reproducing kernel

a simple axiomatization of nonadditive expected utility textabstractthis paper provides an extension of savage s subjective expected utility theory for decisions under uncertainty it includes in the set of events both unambiguous events for which probabilities are additive and ambiguous events for which probabilities are permitted to be nonadditive the main axiom is cumulative dominance which adapts stochastic dominance to decision making under uncertainty we derive a choquet expected utility representation and show that a modification of cumulative dominance leads to the classical expected utility representation the relationship of our approach with that of schmeidler who uses a two stage formulation to derive choquet expected utility is also explored our work may be viewed as a unification of schmeidler 1989 and gilboa 1987

an intractability result for multiple integration we prove that the problem of multiple integration in the korobov class e d is intractable since the number of function evaluations required to achieve a worst case error less than 1 is exponential in the dimension

a computer search of rank 2 lattice rules for multidimensional quadrature for certain lattice rules of rank 2 it has been shown that a unique representation exists in a form suitable for computer evaluation the present paper describes computer searches of such rules reports results and identifies rules that appear promising for the numerical evaluation of practical multidimensional integrals

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

numerical solution of the poisson equation over hypercubes using reduced chebyshev polynomial bases we describe how to use new reduced size polynomial approximations for the numerical solution of the poisson equation over hypercubes our method is based on a non standard galerkin method which allows test functions which do not verify the boundary conditions numerical examples are given in dimensions up to 8 on solutions with different smoothness using the same approximation basis for both situations a special attention is paid on conditioning problems

numerical integration on the sphere and its effect on the material symmetry of constitutive equations a comparative study in the present paper we consider a class of constitutive models based on numerical integration on the unit sphere the directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations we investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues in order to assess and compare these schemes statistical methods are presented and applied the analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points however as a general result it turns out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations in particular for highly non linear integrand functions the consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark like numerical example copyright 2009 john wiley sons ltd

ambiguity in the small and in the large this paper considers local and global multiple prior representations of ambiguity for preferences that are i monotonic ii bernoullian that is admit an affine utility representation when restricted to constant acts and iii locally lipschitz continuous we do not require either certainty independence or uncertainty aversion we show that the set of priors identified by ghirardato maccheroni and marinacci s 2004 unambiguous preference relation can be characterized as a union of clarke differentials we then introduce a behavioral notion of locally better deviation at an act and show that it characterizes the clarke differential of the preference representation at that act these results suggest that the priors identified by these preference statements are directly related to local optimizing behavior

lattice integration rules of maximal rank formed by copying rank 1 rules for integration of periodic functions over the s dimensional unit cube theoretical and empirical evidence suggests that certain lattice rules of maximal rank are more effective as judged by a standard test than the widely used rules of rank 1 in a rank 1 rule all the points are generated by taking multiples of a single rational vector modulo 1 in the manner suggested by korobov the rules in question are formed by taking n s copies with small values of n principally n 2 of rules of rank 1 there is also empirical evidence that by the same criterion these maximal rank rules are preferable in dimensions greater than 2 to the rank 2 rules found by sloan and walsh

representing preferences with a unique subjective state space we extend kreps 1979 analysis of preference for flexibility reinterpreted by kreps 1992 as a model of unforeseen contingencies we enrich the choice set consequently obtaining uniqueness results that were not possible in kreps model we consider several representations and allow the agent to prefer commitment in some contingencies in the representations the agent acts as if she had coherent beliefs about a set of possible future ex post preferences each of which is an expected utility preference we show that this set of ex post preferences called the subjectie state space is essentially unique given the restriction that all ex post preferences are expected utility preferences and is minimal even without this restriction because the subjective state space is identified the way ex post utilities are aggregated into an ex ante ranking is also essentially unique hence when a representation that is additive across states exists the additivity is meaningful in the sense that all representations are intrinsically additive uniqueness enables us to show that the size of the subjective state space provides a measure of the agent s uncertainty about future contingencies and that the way the states are aggregated indicates whether these contingencies lead to a desire for flexibility or commitment

the representation of lattice quadrature rules as multiple sums we provide a classification of lattice rules applying elementary group theory we assign to each s dimensional lattice rule a rank m and a set of positive integer invariants n sub 1 n sub 2 n sub s the number nu q of abscissas required by the rule is the product n sub 1 n sub 2 n sub s and the rule may be expressed in a canonical form with m independent summations under this classification an n point number theoretic rule in the sense of korobov and conroy is a rank m 1 rule having invariants n 1 1 1 and the product trapezoidal rule using n sup s points in rank m s rule having invariants n n n besides providing a canonical form we give some of the properties of copy rules and of projections into lower dimensions

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

fibonacci numerical integration on a sphere for elementary numerical integration on a sphere there is a distinct advantage in using an oblique array of integration sampling points based on a chosen pair of successive fibonacci numbers the pattern has a familiar appearance of intersecting spirals avoiding the local anisotropy of a conventional latitude longitude array besides the oblique fibonacci array the prescription we give is also based on a non uniform scaling used for one dimensional numerical integration and indeed achieves the same order of accuracy as for one dimension error n 6 for n points this benefit of fibonacci is not shared by domains of integration with boundaries e g a square for which it was originally proposed with non uniform scaling the error goes as n 3 with or without fibonacci for experimental measurements over a sphere our prescription is realized by a non uniform fibonacci array of weighted sampling points

triangular canonical forms for lattice rules of prime power order in this paper the authors develop a theory of t cycle d z representations for s dimensional lattice rules of prime power order of particular interest are canonical forms which by definition have a d matrix consisting of the nontrivial invariants among these is a family of triangular forms which besides being canonical have the defining property that their z matrix is a column permuted version of a unit upper triangular matrix triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra the authors main result is to define a unique canonical form for prime power rules this ultratriangular form is a triangular form is easy to recognize and may be derived in a straightforward manner 12 refs

ambiguity and second order belief anscombe and aumann 1963 wrote a classic characterization of subjective expected utility theory this paper employs the same domain for preference and a closely related but weaker set of axioms to characterize preferences that use second order beliefs beliefs over probability measures such preferences are of interest because they accommodate ellsberg type behavior

error bounds for the method of good lattice points new error bounds are obtained for the method of good lattice points for multidimensional quadrature when m the number of quadrature points is prime one of these bounds reduces the constant in niederreiter s asymptotic error bound if the dimension exceeds 2 together they give very much smaller numerical bounds for all values of m