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The elliptic interface problems with imperfect contact have found applications in numerical solutions of the Stefan problem of the solidification process and crystal growth, composite materials, multi-phase flows, etc. In this paper a 1D elliptic interface problem with imperfect contact is considered. A class of high-order compact finite difference schemes are constructed on body-fitted and non-body-fitted mesh, respectively. For each case, the second-, third- and fourth-order approximations of implicit jump conditions are provided by using the jump conditions and its high-order derivatives. Numerical examples are provided to verify the performance of the schemes. The numerical results demonstrate that the schemes have theoretical accuracy for elliptic interface problems with imperfect contact. [ABSTRACT FROM AUTHOR]

The Malliavin calculus is an extension of the classical calculus of variations from deterministic functions to stochastic processes. In this paper we aim to show in a practical and didactic way how to calculate the Malliavin derivative, the derivative of the expectation of a quantity of interest of a model with respect to its underlying stochastic parameters, for four problems found in mechanics. The non-intrusive approach uses the Malliavin Weight Sampling (MWS) method in conjunction with a standard Monte Carlo method. The models are expressed as ODEs or PDEs and discretised using the finite difference or finite element methods. Specifically, we consider stochastic extensions of; a 1D Kelvin-Voigt viscoelastic model discretised with finite differences, a 1D linear elastic bar, a hyperelastic bar undergoing buckling, and incompressible Navier-Stokes flow around a cylinder, all discretised with finite elements. A further contribution of this paper is an extension of the MWS method to the more difficult case of non-Gaussian random variables and the calculation of second-order derivatives. We provide open-source code for the numerical examples in this paper. [ABSTRACT FROM AUTHOR]

The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum Σk=0n-1 of values of a function f by a linear combination of a corresponding integral of f and values of its higher-order derivatives f(j). An alternative (Alt) summation formula was presented by the author, which approximates the sum by a linear combination of integrals only, without using derivatives of f. It was shown that the Alt formula will in most cases outperform the EM formula. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes. [ABSTRACT FROM AUTHOR]

In this paper we a derive a single parameter characterization of the value and the optimal exercise time of the perpetual American straddle (the portfolio consisting of a put and a call option on the same underlying asset with the same price) in the classical Black-Scholes-Samuelson model. The parameter is the unique solution of a single non-linear equation with one unknown variable; this is the first time that the single equation characterization has been obtained for the perpetual American straddle. The equation is derived after multiple transformations of the defining optimal stopping problem in continuous time using a combination of classical techniques: Hamilton-Jacobi-Bellman equation, reduction to a Cauchy-Euler first order differential equation, smooth pasting conditions, and, finally, verification theorem for optimal stopping problems. [ABSTRACT FROM AUTHOR]

In this paper we introduce and investigate a new generalized class of bi-bazilevic functions defined by using (s,t) -derivative operator and quasi-subordination in the open unit disk D. We obtain two kinds of coefficient estimate by using Faber polynomial expansion and get Fekete-Szego inequality for the new class and some of its subclasses. [ABSTRACT FROM AUTHOR]

In the present paper, we characterize Lie (Jordan) σ-centralizers of generalized matrix algebras. More precisely, we obtain some conditions under which every Lie σ-centralizer of a generalized matrix algebra can be expressed as the sum of a σ-centralizer and a center- valued mapping. Further, it is shown that under certain appropriate assumptions every Jordan σ-centralizer of a generalized matrix algebra is a σ-centralizer. Finally, the main results are applied to triangular algebras. [ABSTRACT FROM AUTHOR]

The exponential space, which is a Banach function space, can be defined with two very differently looking, but equivalent norms. In this paper, we give estimates for the best constants of the ratio of these two norms. Our result answers a question of C. Bennett, and R. Sharpley. [ABSTRACT FROM AUTHOR]

The concept of a tolerance relation, shortly called tolerance, was studied on various algebras since the seventies of the twentieth century by B. Zelinka and the first author (see e.g. [6] and the monograph [1] and the references therein). Since tolerances need not be transitive, their blocks may overlap and hence in general the set of all blocks of a tolerance cannot be converted into a quotient algebra in the same way as in the case of congruences. However, G. Czédli ([7]) showed that lattices can be factorized by means of tolerances in a natural way, and J. Grygiel and S. Radeleczki ([8]) proved some variant of an Isomorphism Theorem for tolerances on lattices. The aim of the present paper is to extend the concept of a tolerance on a lattice to posets in such a way that results similar to those obtained for tolerances on lattices can be derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of solutions for impulsive pantograph fractional integro-differential equation via ψ-Hilfer fractional derivative in a appropriate Banach space. Our approach is based on fixed point theorems of Darbo’s and Mönch via Kuratowski measure of non-compactness. An example is given to illustrate our approach. [ABSTRACT FROM AUTHOR]

Purpose |!|#8211; Solving the non-linear equation f(x)=0 has nice applications in various branches of physics and engineering. Sometimes the applications of the numerical methods to solve non-linear equations depending on the second derivatives are restricted in physics and engineering. The purpose of this paper is to propose two new modified Newton|!|#39;s method for solving non-linear equations. Convergence results show that the order of convergence of the proposed iterative methods for a simple root is four. The iterative methods are free from second derivative and can be used for solving non-linear equations without computing the second derivative. Finally, several numerical examples are given to illustrate that proposed iterative algorithms are effective. Design/methodology/approach |!|#8211; In this paper, first the authors introduce two new approximations for the definite integral arising from Newton|!|#39;s theorem. Then by considering these approximations, two new iterative methods are provided with fourth-order convergence which can be used for solving non-linear equations without computing second derivatives. Findings |!|#8211; In this paper, the authors propose two new iterative methods without second derivatives for solving the non-linear equation f(x)=0. From numerical results, it is observed that the new methods are comparable with various iterative methods. Also numerical results corroborate the theoretical analysis. Originality/value |!|#8211; The best property of these schemes is that they are second derivative free. Also from numerical results, it is observed that the new methods are comparable with various iterative methods. The numerical results corroborate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, is defined the operator Dλ,ν,nα,β : A → A, given by Dλ,ν,nα,β f(z) = (1 - α - β)RνDnf(z) + αRνΩλzf(z) + βDnΩλzf(z), for z ∈ U, where Rν is the Ruscheweyh derivative, Dn is the Sălăgean operator, Ωλz is a fractional differintegral operator introduced by S. Owa and H. M. Srivastava, A = {f ∈ H(U) : f(z) = z + a2z² + a3z³ + ..., z ∈ U}, α, β ≥ 0, ν > -1, n ∈ N0 = {0, 1, 2, 3, ...}, -∞ < λ < 2. A certain subclass of analytic functions in the open unit disk, Rλ,ν,nα,β (δ), where 0 ≤ δ ≤ 1, is introduced using the new operator. Are obtained some properties of the class Rλ,ν,nα,β (δ) and some differential subordinations using the operator Dλ,ν,nα,β. [ABSTRACT FROM AUTHOR]

Research in mathematics and science education reveals a disconnect for students as they attempt to apply their mathematical knowledge to science and engineering. With this conclusion in mind, this paper investigates a particular calculus topic that is used frequently in science and engineering: the definite integral. The results of this study demonstrate that certain conceptualizations of the definite integral, including the area under a curve and the values of an anti-derivative, are limited in their ability to help students make sense of contextualized integrals. In contrast, the Riemann sum-based “adding up pieces” conception of the definite integral (renamed in this paper as the “multiplicatively-based summation” conception) is helpful and useful in making sense of a variety of applied integral expressions and equations. Implications for curriculum and instruction are discussed. [ABSTRACT FROM AUTHOR]

We derive upper bounds on the fluctuations of a class of random surfaces of the - type with convex interaction potentials. The Brascamp-Lieb concentration inequality provides an upper bound on these fluctuations for uniformly convex potentials. We extend these results to twice continuously differentiable convex potentials whose second derivative grows asymptotically like a polynomial and may vanish on an (arbitrarily large) interval. Specifically, we prove that, when the underlying graph is the d-dimensional torus of side length L, the variance of the height is smaller than C lnL in two dimensions and remains bounded in dimension d = 3. The proof makes use of the Helffer-Sjöstrand representation formula (originally introduced by Helffer and Sjöstrand (1994) and used by Naddaf and Spencer (1997) and Giacomin, Olla Spohn (2001) to identify the scaling limit of the model), the anchored Nash inequality (and the corresponding on-diagonal heat kernel upper bound) established by Mourrat and Otto (2016) and Efron's monotonicity theorem for log-concave measures (Efron (1965)). [ABSTRACT FROM AUTHOR]

Let $$\mathcal{A}$$ and $$\mathcal{B}$$ be unital rings, and $$\mathcal{M}$$ be an $$\left( {\mathcal{A},\mathcal{B}} \right)$$-bimodule, which is faithful as a left $$\mathcal{A}$$-module and also as a right $$\mathcal{B}$$-module. Let $$\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $$\mathcal{U}$$. [ABSTRACT FROM AUTHOR]

In this paper, some subordination and superordination results for higher-order derivatives of certain p-valent analytic functions in the open unit disc are derived. Relevant connection of the results, which are obtained in this paper, with various known results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, some oscillation criteria for solutions of a general second order non-linear differential equations with damping of the form (a(t) ψ (x (t)) k (x' (t)))' + p (t) k (x' (t)) + q (t) f (x (t)) = 0, are given. The results obtained extend some existing results in the literature by using the refined integral averaging technique introduced by Rogovchenko and Tuncay). [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate and extend some Pachpatte type dynamic inequalities on time scales in three independent variables which provide explicit bounds on unknown functions and their derivatives. Some applications are also discussed here in order to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

In this paper we present a new class named polar of monic orthogonal polynomials with respect to the area measure supported on G where G is a bounded simply-connected domain in the complex plane ℂ. We analyze some open questions and discuss some ideas properties related to solving asymptotic behavior of polar Bergman polynomials over domains with corners and asymptotic behavior of modified Bergman polynomials by affine transforms in variable and polar modified Bergman polynomials by affine transforms in variable. We show that uniform asymptotic of Bergman polynomials over domains with corners and by Pritsker's theorem imply uniform asymptotic for all their derivatives. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to discuss Maimon's criticism of Kant's doctrine of mathematical cognition. In particular, we will focus on the consequences of this criticism for the problem of the possibility of metaphysics as a science. Maimon criticizes Kant's explanation of the synthetic a priori character of mathematics and develops a philosophical interpretation of differential calculus according to which mathematics and metaphysics become deeply interwoven. Maimon establishes a parallelism between two relationships: on the one hand, the mathematical relationship between the integral and the differential and on the other, the metaphysical relationship between the sensible and the supersensible. Such a parallelism will be the clue to the Maimonian solution to the Kantian problem of the possibility of metaphysics as a science. Highlights • Maimon's criticism of Kant's doctrine of mathematical cognition is analyzed. • Said criticism sheds light on the relationship between mathematics and metaphysics. • Maimon finds in differential calculus the key to the cognition of the supersensible. • Maimon's solution to the Kantian problem of metaphysics as a science is discussed. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 be a fixed positive integer and R be a ring. A permuting n-additive map Ω : Rn → R is known to be permuting generalized n-derivation if there exists a permuting n-derivation Δ : Rn → R such that Ω(x1, x2, ⋯,xixi⋯,xn) = Ω(x1, x2, ⋯,xi,⋯xn)xi + xi Δ(x1, x2, ⋯,xi⋯xn) holds for all xi, xi ∈ R. A mapping δ : R → R defined by δ(x) = Δ(x, x, ⋯, x) for all x ∈ R is said to be the trace of Δ. The trace &# of can be defined in ω of Ω can be defined in the similar way. The main result of the present paper states that if R is a (n+1)!-torsion free semi-prime ring which admits a permuting n-derivation Δ such that the trace δ of Δ satisfies [[δ(x), x], x] ∈ Z (R) for all x ∈ R, then ͐ is commuting on R. Besides other related results it is also shown that in a n-torsion free prime ring if the trace ω of a permuting generalized n-derivation is centralizing on R, then ω is commuting on R. [ABSTRACT FROM AUTHOR]

In this paper, we generalize some earlier well known results by considering polynomials of lacunary type having some zeros at origin and rest of the zeros on or outside the boundary of a prescribed disk. [ABSTRACT FROM AUTHOR]

The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order p (for $$p\ge 1$$ ) and to assume Lipschitz continuity of the p-th derivative, then an $$\epsilon $$ -approximate first-order critical point can be computed in at most $$O(\epsilon ^{-(p+1)/p})$$ evaluations of the problem's objective function and its derivatives. This generalizes and subsumes results known for $$p=1$$ and $$p=2$$ . [ABSTRACT FROM AUTHOR]

In this study, we examined some properties of recently defined polynomial sequence called κ-Oresme polynomials. We studied first order derivatives of this sequence and deduced some formulas. Moreover, we formulated the higher order derivative of κ-Oresme polynomials and observed a relation between the formula which is newly given and Pascal's triangle. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a viral infection model with humoral immunity. The incidence rate is given by Hill type infection rate. We have derived two threshold parameters, R0 and R1 which completely determined the global properties of the model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have established the global asymptotic stability of all steady states of the model. We have proven that, if R0 ≤ 1, then the infection-free steady state is globally asymptotically stable (GAS), if R1 ≤ 1 < R0, then the chronic-infection steady state without humoral immune response is GAS, and if R1 > 1, then the chronic-infection steady state with humoral immune response is GAS. [ABSTRACT FROM AUTHOR]

In this paper, we present some new inequalities for the gamma function. The main tools are the multiple-correction method developed in [6, 7] and a generalized Mortici's lemma. [ABSTRACT FROM AUTHOR]

In this paper, we consider Choquet integrals and derivatives of non- negative, continuous and decreasing functions on the non-positive real line with respect to fuzzy measures. We prove some properties of derivatives of those functions and some examples. [ABSTRACT FROM AUTHOR]

In this paper we obtain some weighted Ostrowski type and Grüss type inequalities on time scales by using the recent theory of combined dynamic derivatives on time scales. These results extend some known results in the literature and give some other interesting inequalities as special cases. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish some new inequalities of Simpson type for functions whose third derivatives are extended s-convex functions, and apply these inequalities to derive some inequalities of special means. [ABSTRACT FROM AUTHOR]

The paper deals with a certain class of nonautonomous ordinary third-order nonlinear differential equations L3y=f(t,L0y,L1y,L2y) with quasi-derivatives. A criterion of asymptotic stability in Liapunov sense as well as a criterion of instability in Liapunov sense is derived. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

A quasi-boundary method is used to study an ill-posed, time-fractional diffusion equation involving the fractional Riemann-Liouville derivative. In particular, we consider an ill-posed problem for a family of well-posed problems, and prove, by means of eigenfunction expansions, that the solutions of the latter problems converge to the solutions associated with the former problem. The analysis presented includes providing conditions for the rate of the convergence. [ABSTRACT FROM AUTHOR]

In this note we extend the results of Bekkert and Futorny in [2] and Hemmer, Kujawa and Nakano in [10] and determine the derived representation type of Schur superalgebras. [ABSTRACT FROM AUTHOR]

Abstract: The authors had exhibited in a previous paper a continuous strictly increasing singular function from into with a derivative that takes non-zero finite values at the points of an uncountable set. In this article, the construction is improved to encompass a dense set. [Copyright &y& Elsevier]

In this paper, we study a new class of three-point boundary value prob- lems of nonlinear second-order q-difference inclusions. Our problems con- tain different numbers of q in derivatives and integrals. By using fixed point theorems, some new existence results are obtained in the cases when the right-hand side has convex as well as noncovex values. [ABSTRACT FROM AUTHOR]

Abstract: Let be the graph obtained from by adding a new vertex corresponding to each edge of and by joining each new vertex to the end vertices of the corresponding edge, and be the graph obtained from by inserting a new vertex into every edge of and by joining by edges those pairs of these new vertices which lie on adjacent edges of . In this paper, we determine the Laplacian polynomials of and of a regular graph ; on the other hand, we derive formulae and lower bounds of the Kirchhoff index of these graphs. [Copyright &y& Elsevier]

In this paper, we introduce intuitionistic fuzzy derivative, intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative and some of their properties are studied. The relations between intuitionistic fuzzy Gateaux derivative and intuitionistic fuzzy Fréchet derivative are studied. [ABSTRACT FROM AUTHOR]

The least squares derivative and the approximate derivative are both generalizations of the ordinary derivative. The existence of either of these generalized derivatives does not guarantee the existence of the other and it is even possible for both generalized derivatives to exist at a point but have different values. Several examples of such functions are presented in this paper. In addition, conditions for which the existence of the approximate derivative implies the existence (and equality) of the least squares derivative are stated and proved. These conditions involve the notion of Hölder continuity and a stronger version of approximate differentiability. [ABSTRACT FROM AUTHOR]

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Calculus, as commonly taught, describes certain operations on explicit functions, but science relies on experimental data, which is inherently discrete. In the face of this disparity, how can we help students transition from lower-division mathematics courses to upper-division coursework in other STEM disciplines? We discuss here our efforts to address this issue for upper-division physics majors by introducing a new representation for derivatives in terms of experiments to go along with the traditional symbolic, graphical, verbal, and numerical representations, and by emphasizing infinitesimal reasoning through the use of differentials. These ideas culminate in the concept of thick derivatives. By providing examples of "physics" reasoning about both ordinary and partial derivatives, and methods for incorporating such reasoning into the classroom, we hope to give instructors of calculus new insight into the needs of many of their students. [ABSTRACT FROM AUTHOR]

In vision science, cascades of Linear+Nonlinear transforms are very successful in modeling a number of perceptual experiences. However, the conventional literature is usually too focused on only describing the forward input-output transform. Instead, in this work we present the mathematics of such cascades beyond the forward transform, namely the Jacobian matrices and the inverse. The fundamental reason for this analytical treatment is that it offers useful analytical insight into the psychophysics, the physiology, and the function of the visual system. For instance, we show how the trends of the sensitivity (volume of the discrimination regions) and the adaptation of the receptive fields can be identified in the expression of the Jacobian w.r.t. the stimulus. This matrix also tells us which regions of the stimulus space are encoded more efficiently in multi-information terms. The Jacobian w.r.t. the parameters shows which aspects of the model have bigger impact in the response, and hence their relative relevance. The analytic inverse implies conditions for the response and model parameters to ensure appropriate decoding. From the experimental and applied perspective, (a) the Jacobian w.r.t. the stimulus is necessary in new experimental methods based on the synthesis of visual stimuli with interesting geometrical properties, (b) the Jacobian matrices w.r.t. the parameters are convenient to learn the model from classical experiments or alternative goal optimization, and (c) the inverse is a promising model-based alternative to blind machine-learning methods for neural decoding that do not include meaningful biological information. The theory is checked by building and testing a vision model that actually follows a modular Linear+Nonlinear program. Our illustrative derivable and invertible model consists of a cascade of modules that account for brightness, contrast, energy masking, and wavelet masking. To stress the generality of this modular setting we show examples where some of the canonical Divisive Normalization modules are substituted by equivalent modules such as the Wilson-Cowan interaction model (at the V1 cortex) or a tone-mapping model (at the retina). [ABSTRACT FROM AUTHOR]

We study an Ibragimov-Mamontov type equation with singular Bessel operators playing the role of second order derivatives. We obtain a representation of the solution by using a special Bessel transform. [ABSTRACT FROM AUTHOR]

This study deals exclusively with total shape spectra (envelopes). Yet, the goal is to exactly reconstruct all the components of the given envelope by relying only upon non-parametric signal processors (shape estimators). To this end, the so-called derivative envelope spectra are investigated. A derivative spectrum is the result of the application of the differentiation transform $$\left( {\hbox {d}/\hbox {d}\nu } \right) ^{m}$$ to the given conventional spectrum. Here, non-negative integer m is the order of differentiation and $$\nu $$ is the real linear sweep frequency. For the customary envelope (the $$\hbox {zero}{\mathrm{th}}$$ -order derivative, $$m=0)$$ , we use the non-parametric fast Padé transform to generate the derivative fast Padé transform (dFPT). Explicit computations are carried out by successively increasing the differentiation order m from low through intermediate to high values of derivatives of complex envelopes. The dFPT can disentangle the spectrally crowded regions by splitting apart any multiplet of closely packed peaks. Hidden resonances, even those that are very weak, can be not only visualized, but also exactly quantified by the dFPT, despite performing shape estimations alone. Most importantly, while the envelopes in the derivative fast Fourier transform exhibit huge noise amplification with increasing m, the same-order of the differentiation transform in the dFPT acts as an effective noise suppressor. The results of the dFPT are illustrated for the envelopes with overlapping peaks stemming from synthesized noise-free and noise-contaminated time signals associated with encoding by in vitro proton magnetic resonance spectroscopy (MRS) of breast cancer tissue. This new methodology is anticipated to significantly enhance resolution as well as signal-to-noise ratio and the overall performance of single-voxel MRS in clinical diagnostics. It is also expected to be of special benefit for volumetric coverage of the scanned tissue by magnetic resonance spectroscopic imaging. [ABSTRACT FROM AUTHOR]

We study the homotopy category [ABSTRACT FROM AUTHOR]

Asymptotic formulas are obtained for a class of integrals that are Fourier transforms of rapidly oscillating functions. These formulas contain special functions and generalize the well-known method of stationary phase. [ABSTRACT FROM AUTHOR]

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197-229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in . They introduced a multi-variable polynomial invariant . We use special substitutions for some of the parameters of the invariant to show how to deduce invariants of finite type from using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant. [ABSTRACT FROM AUTHOR]

A classical first-order Hardy-Sobolev inequality in Euclidean domains, involving weighted norms depending on powers of the distance function from their boundary, is known to hold for every, but one, value of the power. We show that, by contrast, the missing power is admissible in a suitable counterpart for higher-order Sobolev norms. Our result complements and extends contributions by Castro and Wang (Calc Var 39(3-4):525-531, 2010), and Castro et al. (Comptes Rendus Math Acad Sci Paris 349:765-767, 2011; J Eur Math Soc 15:145-155, 2013), where a surprising canceling phenomenon underling the relevant inequalities was discovered in the special case of functions with derivatives in $$L^1$$ . [ABSTRACT FROM AUTHOR]

The article examines the link between poetry, mathematics and religion through the works of poet John Milton and mathematician and physicist Isaac Newton during the Reformation. It argues that Newton and Milton identified poetry and mathematics as kindred means for strengthening humankind's natural insights into the invisible workings of divinity. It also contends that Newton's method of fluxions made him vulnerable to criticisms that his physics and mathematics were insufficiently corporeal.

Let M be a Γ-ring. If M satisfies the condition (*) xα yβ z = xβyβ z for all x, y, z∈ M, α, β∈Γ then we investigate commutativity of prime Γ-rings satisfying certain identities involving left centralizer. [ABSTRACT FROM AUTHOR]

The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. Inspired by the recent complex step method that is only valid for firstorder sensitivities, the new multicomplex approach is valid to arbitrary order. The mathematical theory behind this approach is revealed, and an efficient procedure for the automatic implementation of the method is described. Several applications are presented to validate and demonstrate the accuracy and efficiency of the algorithm. The results are compared to conventional approaches such as finite differencing, the complex step method, and two separate automatic differentiation tools. The multicomplex method performs favorably in the preliminary comparisons and is therefore expected to be useful for a variety of algorithms that exploit higher order derivatives. [ABSTRACT FROM AUTHOR]