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

a paradox for the smooth ambiguity model of preference two ellsberg style thought experiments are described that reflect on the smooth ambiguity decision model developed by klibanoff marinacci and mukerji 2005 the first experiment poses difficulties for the model s axiomatic foundations and as a result also for its interpretation particularly for the claim that the model achieves a separation between ambiguity and the attitude toward ambiguity given the problematic nature of its foundations the behavioral content of the model and how it differs from multiple priors for example are not clear the second thought experiment casts some light on these questions

on the invertible cellular automata 150 over f p in this work the reversibility problem for cellular automata with rule number 150 over the finite field f p is tackled it is shown that when null boundary conditions are stated the reversibility is independent of the state set f p and it appears when the number of cells of the cellular space satisfies some conditions furthermore the explicit expressions of the inverse cellular automata are computed

two stage lotteries without the reduction axiom preference relations over two stage lotteries are analyzed empirical evidence indicates that decisionmakers do not always behave in accordance with the reduction of compound lotteries axiom but they seem to satisfy a compound independence axiom although the reduction and the compound independence axioms together with continuity imply expected utility theory each of them by itself is compatible with all possible preference relations over simple lotteries using these axioms the author analyzes three different versions of expected utility for two stage lotteries the author suggests several different compound dominance axioms as possible replacements of the reduction axiom which are strictly weaker than the reduction of compound lotteries axiom copyright 1990 by the econometric society

a fast elementary algorithm for computing the determinant of toeplitz matrices in recent years a number of fast algorithms for computing the determinant of a toeplitz matrix were developed the fastest algorithm we know so far is of order k 2 log n k 3 where n is the number of rows of the toeplitz matrix and k is the bandwidth size this is possible because such a determinant can be expressed as the determinant of certain parts of the n th power of a related k i k companion matrix in this paper we give a new elementary proof of this fact and provide various examples we give symbolic formulas for the determinants of toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when k is small

on the inverse of a general tridiagonal matrix in the current paper a new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented the algorithm is suited for implementation using computer algebra systems such as maple mathematica matlab and macsyma symbolic and numeric examples are given

a new computational algorithm for solving periodic tri diagonal linear systems in this article we present a new efficient computational algorithm for solving periodic tri diagonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an example is given in order to illustrate the algorithm

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

the application of preconditioning in viscous flows a time derivative preconditioning algorithm that is effective over a wide range of flow conditions from inviscid to very diffusive flows and from low speed to supersonic flows has been developed the algorithm uses a preconditioning matrix that introduces well conditioned eigenvalues while simultaneously avoiding nonphysical time reversals for viscous flows the resulting algorithm also provides a mechanism for controlling the inviscid and viscous time step parameters at very diffusive flows thereby ensuring rapid convergence for very viscous flows as well as for inviscid flows computational capabilities are demonstrated through computation of a wide variety of problems convergence rates are shown to be accelerated by as much as two orders of magnitudes while providing solutions that are indentical to those obtained without preconditioning method

on the inverse and determinant of general bordered tridiagonal matrices in the current paper we present a symbolic algorithm that never suffers from breakdown for computing the inverse of general bordered tridiagonal matrices meanwhile an explicit formula of evaluating the determinant of a bordered tridiagonal matrix is also derived the implementation of the proposed algorithm using computer algebra systems cass is straightforward some numerical examples are given in order to demonstrate the performance and validity of the presented results

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated

factorization of hessenberg matrices abstract the analysis of any hessenberg matrix as a product of a companion matrix and a triangular matrix is presented in this paper the factors are explicitly given on terms of the entries of the hessenberg matrix and further various factorizations are undertaken applications to comrade and confederate matrices as well as for the corresponding block cases are included

a fast algorithm for evaluating n th order tri diagonal determinants the cost of all existing algorithms for evaluating the nth order determinants numerical analysis 7th edition brooks cole publishing pacific grove ca 2001 is at most o n3 in the current article we present a new efficient computational algorithm for evaluating the nth order tri diagonal determinants with cost o n only the algorithm is suited for implementation using computer algebra systems such as maple and macsyma some examples are given to illustrate the algorithm

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

a novel elementary algorithm for solving comrade linear systems abstract recently three computational algorithms for solving comrade linear systems have been proposed by karawia 2013 2007 6 5 and by sogabe 2008 10 in this paper we derive a novel elementary algorithm for solving comrade linear systems based on matrix reordering and the sherman morrison woodbury formula an efficient method of evaluating the determinant of a comrade matrix is also discussed

assessment of three preconditioning schemes for solution of the two dimensional euler equations at low mach number flows summary three preconditioners proposed by eriksson choi and merkel and turkel are implemented in a 2d upwind euler flow solver on unstructured meshes the mathematical formulations of these preconditioning schemes for different sets of primitive variables are drawn and their eigenvalues and eigenvectors are compared with each other for this purpose these preconditioning schemes are expressed in a unified formulation a cell centered finite volume roe s method is used for the discretization of the preconditioned euler equations the accuracy and performance of these preconditioning schemes are examined by computing steady low mach number flows over a naca0012 airfoil and a two element naca4412 4415 airfoil for different conditions the study shows that these preconditioning schemes greatly enhance the accuracy and convergence rate of the solution of low mach number flows the study indicates that the preconditioning methods implemented provide nearly the same results in accuracy however they give different performances in convergence rate it is demonstrated that although the convergence rate of steady solutions is almost independent of the choice of primitive variables and the structure of eigenvectors and their orthogonality the condition number of the system of equations plays an important role and it determines the convergence characteristics of solutions copyright 2011 john wiley sons ltd

on linear time solvers for comrade linear systems abstract a linear time solver with total flop count 10 n 11 and 4 n 2 of memory space is introduced for solving determined comrade linear systems of degree n this algorithm is based on a suitable factorization satisfied for every matrix coefficient it performs faster than other specialized linear time solvers proposed recently contrary speedup its stability and that of those specialized from the literature for large order comrade systems is not guaranteed to overcome such weakness an adapted solver is then proposed running in linear time and 7 n o 1 of memory space it takes advantage of the inexpensive givens reduction of the transpose of any comrade matrix and covering also linear systems involving matrices with comrade structure the results are thoroughly illustrated with proper numerical comparisons

a reliable givens lu approach for solving opposite bordered tridiagonal linear systems abstract a fast and reliable numerical solver is proposed to handle determined opposite bordered tridiagonal linear systems a x b a precise sequence of givens rotations on the nonsingular matrix coefficient a allows us to work with a simpler linear system this equivalent system which admits always a trivial lu factorization can be solved in linear time for large order matrices of use in applications the main advantage of this solver with respect to other special algorithms proposed is its stability and risk free performance as it is checked with comparative examples its extension to manage opposite bordered tridiagonal matrix equations is straightforward

preconditioning method for low speed flows a well known preconditioning method for low mach number e ows is implemented in an existing numerical simulator of viscous and inviscid e ows both details concerning the numerical method and the results of the validationphasearegiven themethodisbasedonatime derivativepreconditioningofthenavier stokesequations using the so called viscous set of variables the pseudoacoustic speed introduced equalizes the eigenvalues so that the history of convergence of the preconditioned system of equations exhibits an increased rate of convergence at low mach numbers the artie cial dissipative terms have been modie ed to let them scale correctly at low speeds in theinviscidcase thevelocitydistributioncalculatedoverthecylinderwiththedescribedmethodmatchesanalytical data numericalresultsconcerningthesimulationofinviscidandturbulente owspastairfoilsarepresented inviscid e ow over the onera m6 wing is also computed it is shown that the preconditioning technique allows both an acceleration in the rate of convergence and a reduction of the wall clock time of an order of magnitude also an improved accuracy of the solution at low mach numbers has been observed

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

the inverse of a tridiagonal matrix in this paper explicit formulae for the elements of the inverse of a general tridiagonal matrix are presented by first extending results on the explicit solution of a second order linear homogeneous difference equation with variable coefficients to the nonhomogeneous case and then applying these extended results to a boundary value problem a formula for the characteristic polynomial is obtained in the process we also establish a connection between the matrix inverse and orthogonal polynomials in addition the case of a cyclic tridiagonal system is discussed

a generalized symbolic thomas algorithm for the solution of opposite bordered tridiagonal linear systems in the current paper we present a generalized symbolic thomas algorithm that never suffers from breakdown for solving the opposite bordered tridiagonal obt linear systems the algorithm uses a fill in matrix factorization and can solve an obt linear system in o n operations meanwhile an efficient method of evaluating the determinant of an opposite bordered tridiagonal matrix is derived the computational costs of the proposed algorithms are also discussed moreover three numerical examples are provided in order to demonstrate the performance and effectiveness of our algorithms and their competitiveness with some already existing algorithms all of the experiments are performed on a computer with the aid of programs written in matlab

on a constant diagonals matrix in this paper we consider a constant diagonals matrix the matrix was discussed in witula and slota r witula d slota on computing the determinants and inverses of some special type of tridiagonal and constant diagonals matrices appl math comput 189 1 2007 514 527 the authors gave some results on determinant and the inverse of the matrix for some special cases we give lu factorization and then compute determinant of the matrix we determine eigenvalues of the matrix further we obtain some relationships between permanent of the matrix and terms of a certain recurrence relation

two algorithms for solving comrade linear systems in this paper we present an efficient computational and symbolic algorithms for solving comrade linear systems the implementation of the algorithms using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an examples are given in order to illustrate the algorithms

on the smooth ambiguity model a reply epstein 2009 describes three ellsberg style thought experiments and argues that they pose difficulties for the smooth ambiguity model of decision making under uncertainty developed by klibanoff marinacci and mukerji 2005 we revisit these thought exeperiments and find to the contrary that they either point to strengths of the smooth ambiguity model compared to other models such as the maximum expected utility model gilboa and schmeidler 1989 or in the case of one thought experiment raise criticisms that apply equally to a broad range of current ambiguity models

on the inverse of a general pentadiagonal matrix in this paper employing the general doolittle factorization an efficient algorithm is developed to find the inverse of a general pentadiagonal matrix which is suitable for implementation using computer algebra systems software such as matlab and maple examples are given to illustrate the efficiency of the algorithm

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

on a two term recurrence for the determinant of a general matrix recently a two term recurrence for computing the determinant of a tridiagonal matrix has been found by el mikkawy m el mikkawy a note on a three term recurrence for a tridiagonal matrix appl math comput 139 2003 503 511 then the result has been extended to a block tridiagonal matrix by salkuyeh d k salkuyeh comments on a note on a three term recurrence for a tridiagonal matrix appl math comput 176 2006 442 444 in this note we show that the relation can be obtained for a general matrix and that as a by product we obtain a generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating nth order tri diagonal determinants j comput appl math 166 2004 581 584

a fast and reliable algorithm for evaluating nth order pentadiagonal determinants abstract in the current article we present a fast and reliable algorithm for evaluating n th order pentadiagonal determinants in linear time it is a natural generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating n th order tri diagonal determinants j comput appl math 166 2004 581 584 the algorithm is suited for implementation using computer algebra systems cas such as macsyma and maple some illustrative examples are given

a fast numerical algorithm for the determinant of a pentadiagonal matrix abstract recently a two term recurrence for the determinant of a general matrix has been found t sogabe on a two term recurrence for the determinant of a general matrix appl math comput 187 2007 785 788 and it leads to a natural generalization of the detgtri algorithm m el mikkawy a fast algorithm for evaluating nth order tridiagonal determinants j comput appl math 166 2004 581 584 for computing the determinant of a tridiagonal matrix in this paper we derive a fast numerical algorithm for computing the determinant of a pentadiagonal matrix from the generalization of the detgtri algorithm

numerical algorithms for solving comrade linear systems based on tridiagonal solvers recently two efficient algorithms for solving comrade linear systems have been proposed by karawia a a karawia two algorithms for solving comrade linear systems appl math comput 189 2007 291 297 the two algorithms are based on the lu decomposition of the comrade matrix in this paper two algorithms are presented for solving the comrade linear systems based on the use of conventional fast tridiagonal solvers and an efficient way of evaluating the determinant of the comrade matrix is discussed

performance of compressible flow codes at low mach numbers the accuracy and the performance of three two dimensional compressible flow codes at freestream mach numbers as low as 0 001 are examined two of the codes employ a finite volume discretization scheme along with a multistage time stepping algorithm to solve the euler equations the two codes differ in their respective use of cell centered and node centered differencing schemes the third code uses an implicit finite difference procedure to solve the unsteady navier stokes equations computational test cases are the inviscid steady flow over a circular cylinder and the impulsively started viscous flow over a cylinder

characterizations of smooth ambiguity based on continuous and discrete data in the anscombe aumann setup we provide conditions for a collection of observations to be consistent with a well known class of smooth ambiguity preferences klibanoff p marinacci m mukerji s 2005 a smooth model of decision making under ambiguity econometrica 73 6 1849 1892 each observation is assumed to take the form of an equivalence between an uncertain act and a certain outcome we provide three results that describe these conditions for data sets of different cardinality our findings uncover surprising links between the smooth ambiguity model and classic mathematical results in complex and functional analysis

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

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

ellsberg revisited an experimental study an extension to ellsberg s experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated the sample is decomposed into three main groups subjective expected utility subjects who reduce compound objective lotteries and are ambiguity neutral and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior

a flux vector splitting scheme for low mach number flows in preconditioning method abstract the stiffness of euler equations system increases with the decreasing of mach number which results in poor convergence when solving steady flow problems numerically in order to accelerate the convergence a preconditioning method is proposed a flux vector splitting scheme is derived from the classic steger warming scheme and the roe flux difference splitting scheme and preconditioned systems developed for the two schemes separately numerical results are presented and it is shown that flux vector splitting scheme can be preconditioned as successfully as flux difference splitting and exhibits similar convergence acceleration and accurate solution

reliable solution of tridiagonal systems of linear equations in this paper we present new formulas for characterizing the sensitivity of tridiagonal systems that are independent of the condition number of the underlying matrix we also introduce efficient algorithms for solving tridiagonal systems of linear equations which are stable and reliable namely stable in the backward sense and little sensitive to perturbations in the coefficients

ambiguity learning and asset returns we develop a consumption based asset pricing model in which the representative agent is ambiguous about the hidden state in consumption growth he learns about the hidden state under ambiguity by observing past consumption data his preferences are represented by the smooth ambiguity model axiomatized by klibanofi et al 2005 2006 unlike the standard bayesian theory this utility model implies that the posterior of the hidden state and the conditional distribution of the consumption process given a state cannot be reduced to a predictive distribution by calibrating the ambiguity aversion parameter the subjective discount factor and the risk aversion parameter with the latter two values between zero and one our model can match the flrst moments of the equity premium and riskfree rate found in the data in addition our model can generate a variety of dynamic asset pricing phenomena including the procyclical variation of price dividend ratios the countercyclical variation of equity premia and equity volatility and the mean reversion and long horizon predictability of excess returns

two algorithms for solving a general backward tridiagonal linear systems in this paper we present an efficient computational and symbolic algorithms for solving a backward tridiagonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an examples are given in order to illustrate the algorithms

experiments on compound risk in relation to simple risk and to ambiguity we conduct experiments measuring individual behavior under compound risk simple risk and ambiguity we focus on 1 treatment of compound risks relative to simple risks and 2 the relationship between compound risk attitudes and ambiguity attitudes we find that compound risks are valued differently than corresponding reduced simple risks these differences measure compound risk attitudes these attitudes display more aversion as the reduced probability of the winning event increases like halevy halevy y 2007 ellsberg revisited an experimental study econometrica 75 503 536 we find an association between compound risk reduction and ambiguity neutrality however in contrast to the almost perfect identification in halevy s data we find a substantially weaker relation in both directions first a majority of our ambiguity neutral subjects fail to reduce compound risk second almost a quarter of our subjects who reduce compound risk are nonneutral to ambiguity all of the latter come from the more quantitatively sophisticated part of our subject pool data as supplemental material are available at http dx doi org 10 1287 mnsc 2014 1953 this paper was accepted by peter wakker decision analysis

a numerical solver for general bordered tridiagonal matrix equations abstract to overcome several limitations of symbolic algorithms introduced recently for matrices of large order a fast numerical solver is proposed for the matrix linear equation a x b where the n n coefficient matrix a is a general nonsingular bordered tridiagonal matrix its sparse structure is preserved through partial givens reduction in particular the matrix inverse of a can be computed for a wide range of bordered tridiagonal linear systems a x b the solution is computed in linear time using back substitution and sherman morrison s formula numerical comparisons illustrate the results

new algorithms for solving periodic tridiagonal and periodic pentadiagonal linear systems abstract recently an efficient computational algorithm for solving periodic pentadiagonal linear systems has been proposed by karawia a a karawia a computational algorithm for solving periodic pentadiagonal linear systems appl math comput 174 2006 613 618 the algorithm is based on the lu factorization of the periodic pentadiagonal matrix in this paper new algorithms are presented for solving periodic pentadiagonal linear systems based on the use of any pentadiagonal linear solver in addition an efficient way of evaluating the determinant of a periodic pentadiagonal matrix is discussed the corresponding results in this paper can be readily obtained for solving periodic tridiagonal linear systems

on a homogeneous recurrence relation for the determinants of general pentadiagonal toeplitz matrices pentadiagonal toeplitz matrices frequently arise in many application areas and have been attracted much attention in recent years in this paper we present a numerical algorithm of o log n for computing the determinants of general pentadiagonal toeplitz matrices without imposing any restrictive conditions in addition we investigate some special pentadiagonal toeplitz determinants and their relations to well known number sequences and give a few identity formulas for the ordinary fibonacci sequences and the generalized k fibonacci sequences

explicit formula for the inverse of a tridiagonal matrix by backward continued fractions in this paper we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse for this purpose considering usual continued fraction we define backward continued fraction for a real number and give some basic results on backward continued fraction we give the relationships between the usual and backward continued fractions then we reobtain the lu factorization and determinant of a tridiagonal matrix furthermore we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix it is seen that our method is more convenient efficient and fast