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This essay documents my expansions on a lesson developed in courses that juxtaposed performance and memory studies. Building on a recursive reading of Toni Morrison's literary conceptualisation of rememory, I describe the reiterative nature of memory using two digital performances – a TikTok meme featuring 50 Cent's 'Candy Shop' and a Reddit thread featuring a series of bird paintings. These performances made before the 2020 pandemic show the potentials of collectively inspired art created as virtual media. [ABSTRACT FROM AUTHOR]

Physical activity is essential for children's current and future health, but most do not get their recommended daily 60 minutes of moderate to vigorous physical activity. Schools are an ideal environment for physical activity since students spend most of their waking hours at school. In this paper, we inquire into a sedentary school environment and its collateral impact on student learning in light of the school experience of Hannah. Grounded in Merleau-Ponty's theory of body and embodiment and Dewey's theory of experience in education, the purpose of this narrative inquiry is to challenge an increasingly sedentary environment that undermines the role of body, hence providing a mis-educative experience. In so doing, we intend to raise awareness of the collateral impact of sedentary education on students and rethink the aims of education for the 21st century to foster the child as a whole embodied being. We suggest three aims of education: first, health as the foundation to develop the whole child; second, adding the fifth R, Rhythm, to Doll's four R's of richness, recursion, relation, and rigor; and finally, understanding what it means to be physically literate. [ABSTRACT FROM AUTHOR]

This work presents a framework of dynamic structural models with covariates for short-term forecasting of time series with complex seasonal patterns. The framework is based on the multiple sources of randomness formulation. A noise model is formulated to allow the incorporation of randomness into the seasonal component and to propagate this same randomness in the coefficients of the variant trigonometric terms over time. A unique, recursive and systematic computational procedure based on the maximum likelihood estimation under the hypothesis of Gaussian errors is introduced. The referred procedure combines the Kalman filter with recursive adjustment of the covariance matrices and the selection method of harmonics number in the trigonometric terms. A key feature of this method is that it allows estimating not only the states of the system but also allows obtaining the standard errors of the estimated parameters and the prediction intervals. In addition, this work also presents a non-parametric bootstrap approach to improve the forecasting method based on Kalman filter recursions. The proposed framework is empirically explored with two real time series. [ABSTRACT FROM AUTHOR]

A Galerkin-based finite element recursion relation is used to solve the heat transport equation in two-dimensions. The finite element method (FEM) is a powerful technique that is commonly used for solving complex engineering problems. However, the implementation of the FEM in multi-dimensional problems can be computationally expensive. A finite element recursion algorithm based on bilinear triangular, bilinear quadrilateral and quadratic Lagrangian approximations are employed to discretize the 2-D advection-diffusion equation. This algorithm is an extension of the 1-D Chapeau (linear element) technique, which employed a tridiagonal recursion expression common to the classical central finite-difference approach. The global matrix is nine-diagonal (for 2-D) and is solved using a modified strongly implicit procedure and a left-to-right sweep method. [ABSTRACT FROM AUTHOR]

Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with coloured cubes and bricks on a (2 × 2 × n) -board in three dimensions. After a short introduction and the definition of breakability we show a way to get the number of the tilings of an n-long board considering the (n − 1) -long board. It describes recursively the number of possible breakable and unbreakable tilings. Finally, we give some identities for the recursions using breakability. [ABSTRACT FROM AUTHOR]

We show how to modify the tree-based methodology to construct a combinatorial interpretation for certain non-monotonic solutions to recursions in the generalized Golomb family. Using this combinatorial interpretation we derive the asymptotic properties of these solutions. [ABSTRACT FROM AUTHOR]

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV-nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions. [ABSTRACT FROM AUTHOR]

We explore the nature of the solution space for the Golomb nested recursion . On this solution space, we define a natural equivalence relation and restrict our attention to non-equivalent solutions. We describe and prove an algorithm that determines whether a given set of initial conditions generates a solution. Up to equivalence, there is a unique solution whose forward differences are eventually either 0 or 1, namely, the Golomb sequence , generated by the initial condition . This sequence is asymptotic to ; we conjecture that this is true of every solution. We further conjecture that each solution has what we call a generational structure that abstracts combinatorial properties of . It appears that for any given solution, its generations are composed of only a finite number of building blocks. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour of stochastic approximation schemes with set-valued drift function and non-additive iterate-dependent Markov noise. We show that a linearly interpolated trajectory of such a recursion is an asymptotic pseudotrajectory for the flow of a limiting differential inclusion obtained by averaging the set-valued drift function of the recursion w.r.t. the stationary distributions of the Markov noise. The limit set theorem by Benaim is then used to characterize the limit sets of the recursion in terms of the dynamics of the limiting differential inclusion. We then state two variants of the Markov noise assumption under which the analysis of the recursion is similar to the one presented in this paper. Scenarios where our recursion naturally appears are presented as applications. These include controlled stochastic approximation, subgradient descent, approximate drift problem and analysis of discontinuous dynamics all in the presence of non-additive iterate-dependent Markov noise. [ABSTRACT FROM AUTHOR]

A new method is presented for deriving indefinite integrals involving quotients of special functions. The method combines an integration formula given previously with the recursion relations obeyed by the function. Some additional results are presented using an elementary method, here called reciprocation, which can also be used in combination with the new method to obtain additional quotient integrals. Sample results are given here for Bessel functions, Airy functions, associated Legendre functions and the three complete elliptic integrals. All results given have been numerically checked with Mathematica. [ABSTRACT FROM AUTHOR]

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model. [ABSTRACT FROM AUTHOR]

Can a computer play a music score, e.g. via a Disklavier, in a way that cannot be distinguished from a human performance of the same music? One hundred and seventy-two participants with a wide range of music playing backgrounds rated sound recordings of 7 performances of piano music by Kuhlau, one played by a human, and six generated by algorithms, including a ‘mechanical’ and an ‘unmusical’ rendering. Participants rated the extent to which each performance was by a human and explained their answers. The mechanical performance had the lowest mean rating, but the human performance was rated as statistically identical to the other stimuli. There were no differences between ratings made by classical piano experts and lay listeners, but despite this, the musicians were more confident with their ratings. Qualitative analysis revealed five broad themes that contribute to judging whether a piece appears to be human. The themes were labelled (in descending order of frequency) intuitive, expressive, imperfections, halo (global preference) and empathy. This paper presents new evidence systematically demonstrating that algorithm generated performances of piano music can be indistinguishable from human performances, suggesting some parallels with the 1990s victory of the Deep Blue computer of the world champion (human) chess player. [ABSTRACT FROM AUTHOR]

This paper celebrates Muriel Dimen’s unique manner of conceptualizingthe fieldwith regard to the recursive relations she theorized among psychic and social components of sexual subjectivity. Recursion is a structural force in the field that implicates the analyst in the history of fieldwork. [ABSTRACT FROM PUBLISHER]

We generalize a 2015 Putnam competition problem via a binomial-type double recursion. [ABSTRACT FROM AUTHOR]

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective. [ABSTRACT FROM AUTHOR]

Shadowed contour lines vary line width. Illuminated contour lines additionally vary color based on an angle of illumination. Illuminated and shadowed contour lines date back to the mid-nineteenth century, but their effectiveness compared to conventional contour lines has not been fully examined. Currently, illuminated and shadowed contour lines are not widely used in computer-based cartography because they are not included in most GIS apmaking software. This article introduces improvements to existing algorithms for creating illuminated and shadowed contour lines from digital elevation data. A software package is made available to allow mapmakers to more easily make customized illuminated and shadowed contour maps. A user study comparing illuminated and shadowed contour lines to conventional contour lines and shaded relief with approximately 400 participants was conducted. Results indicate that map-readers can interpret relative height differences between points better and quicker with illuminated contour lines than regular contour lines or shaded relief. Study participants were able to select absolute maxima on an unlabeled illuminated contour map and a labeled regular contour map with equal accuracy and speed. These findings suggest that illuminated contour lines could be used more frequently for improved visualization of terrain and other surface data on maps. [ABSTRACT FROM AUTHOR]

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variableYgiven a random variable (rv)Xtaking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators. [ABSTRACT FROM PUBLISHER]

In this editorial introduction to the Special Issue on Mathematics and the Arts in the Undergraduate Classroom, we explore how the explosion of interest in connections between math and the arts over the past several years has moved into the classroom. We describe papers in this special issue that report on a range of classroom experiences combining math and the arts in ways that enhance student learning and expand student appreciation of both mathematical and artistic ways of thinking. [ABSTRACT FROM PUBLISHER]

In the study of music from a mathematical perspective, several types of counting problems naturally arise. For example, how many different rhythms of a specified length (in beats) can be written if we restrict ourselves to only quarter notes (one beat) and half notes (two beats)? What if we allow whole notes, dotted half notes, etc.? Or, what if we allow each note to be selected from some specified set of tones (e.g., C, C#, D, etc.)? In my course on music and mathematics for the liberal arts, I use these questions as a method of introducing students to the concept of recursion, as it turns out that such questions lead naturally to sequences (indexed based on the length of the rhythms or melodies being considered) defined by recurrence relations, such as the Fibonacci sequence. [ABSTRACT FROM PUBLISHER]

The Manakov hierarchy associated with amatrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Manakov hierarchy, we introduce a trigonal curveof arithmetic genus, from which we construct the related Baker–Akhiezer function, two algebraic functions carrying the data of the divisor and Dubrovin-type equations. Based on the theory of trigonal curves, the explicit theta function representations of the Baker–Akhiezer function, the two algebraic functions, and in particular, that of solutions for the entire Manakov hierarchy are obtained. [ABSTRACT FROM AUTHOR]

Kinked polyzethrenes are more stable than linear polyzethrenes making them better candidates as materials for organic electronic devices (e.g., organic field effect transistors, nonlinear optics, and semiconductors) because of their greater singlet biradical properties. For series of molecules constructed by successive attachment of a given aufbau unit, we are able to derive analytical or recursion expressions relating certain properties. For example, starting with a few known number of Dewar resonance structures (DSs) for such a series, one is often able to derive analytical or recursion expressions for these DS values by our method of successive differences which then lead to either constant or Fibonacci numbers, respectively. The increasing order of π-electronic stability of isomers with the same number of Kekulé structures (K) is determined by their increasing number of DSs. Kinked polycyclic conjugated polyenes with a single classical structure (i.e.,K= 1) are more conjugated and stable than their straight polycyclic isomers with a single classical structure. [ABSTRACT FROM PUBLISHER]

In this short article, we give a novel and elegant proof of a recursion formula related to confluent Vandermonde determinants. This leads to an easy derivation of the well-known determinant identity due to Schendel. [ABSTRACT FROM AUTHOR]

Using two related parameters, ζ and γ, we extend the recursion for com- puting the Tutte polynomial of any graph to the computation of the Tutte polynomial of any multigraph. With this recursion, we found explicit formulae for several fami- lies of multigraphs. In particular, the Tutte polynomials of some cyclic multigraphs, 2−tree multigraphs, and any l−bridge multigraph. [ABSTRACT FROM PUBLISHER]

Continuity is perhaps the most familiar characterization of the finitary character of the operations performed in computation. We sketch the historical and conceptual development of this notion by interpreting it as a unifying theme across three main varieties of semantical theories of programming: denotational, axiomatic and event-based. Our exploration spans the development of this notion from its origins in recursion theory to the forms it takes in the context of the more recent event-based analyses of sequential and concurrent computations, touching upon the relations of continuity with non-determinism. [ABSTRACT FROM PUBLISHER]

On the surface, Amie Siegel's 2008 documentary DDR/DDR focuses on Stasi victims and perpetrators, psychoanalysis in East Germany, Stasi training videos, DEFA westerns, and East German Native American hobbyists. Though never mentioning Friedrich Kittler explicitly, Siegel examines themes of media studies in Kittler's wake. In particular, her film encounters recursions in its exploration of surveillance media and features the technique of recursion in its own making. Employing the same technology the Stasi used, Siegel reflects on herself in the place of the surveillant and also puts herself in front of the camera, in the position of the observed. Exercising multiple role reversals, she takes on the role of the Stasi workers she interviews and puts herself on the psychoanalyst's couch, to name two examples. Contextualizing the film in recent discussions on recursion in media studies and cultural studies, this article discusses how the film utilizes recursion, but also questions it and challenges its boundaries. Ultimately, Siegel's film is critical of the exclusion of the subject in media-centric approaches, so that the film becomes a plea for the return of the individual and the author. [ABSTRACT FROM AUTHOR]

In this article, we introduce a two-state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new binomial distribution are derived. [ABSTRACT FROM AUTHOR]

Theon's ladder is an ancient method for easily approximating roots of a real number k. We extend previous work in this area by analysing a modification of this method for approximating roots of polynomials of the form x2 + cx + d. In particular, we will focus on establishing to which root the ladder will converge. We use techniques from real analysis such as monotone sequences and continuity of rational functions to prove the main results. [ABSTRACT FROM AUTHOR]

As students explore the technological tools available to them for learning mathematics, some will eventually discover what happens when a function button is repeatedly pressed on a calculator. We explore several examples of this, presenting tabular and graphical results for the square root, natural logarithm and sine and cosine functions. Observed behaviour is proven and then discussed in the context of fixed points. [ABSTRACT FROM PUBLISHER]

We define the generalized Golomb triangular recursion by. For particular choices of the initial conditions, we show that the solution of the recursion is a non-slow monotone sequence for which we can provide a combinatorial interpretation in terms of a weighted count of the leaves of a certain labelled infinite tree. We discover that more than one such tree interpretation is possible, leading to different choices of the initial conditions and alternative solutions that are closely related. In the case, the initial conditions for these alternative tree interpretations coincide and we derive explicit closed forms for the solution sequence and its frequency function. [ABSTRACT FROM PUBLISHER]

In a divergent case, we obtain limit theorems for a suitably normalized sequence generated by a random linear recursion with dependent coefficients. The dependence structure of the coefficients is defined through an auxiliary random sequence that represents the current “state of Nature” (alternatively, current regime of the “underlying economy”) and thus serves as an exogenous dynamic environment for the model. We assume that the exogenous environment is a stochastic process with long-range dependence, so that the current state of Nature reflects the whole past of the process. [ABSTRACT FROM PUBLISHER]

The integrable discrete hungry Lotka–Volterra (dhLV) system is easily transformed to the qd-type dhLV system, which resembles the recursion formula of the qd algorithm for computing matrix eigenvalues. Some of the qd-type dhLV variables play a role for assisting the time evolution of the others. This property does not appear in the original dhLV system. In this article, we first show the existence of a centre manifold for the qd-type dhLV system. With the help of the centre manifold theory, we next investigate the local convergence of the qd-type dhLV system, and then clarify the monotonicity related to the qd-type dhLV variables at the final phase of the convergence. [ABSTRACT FROM PUBLISHER]

Golomb showed that fork, a fixed positive integer, one solution of the nested recursionis the Beatty sequence, whereis the positive root of the quadratic polynomial. We prove a natural generalization of this result: ifris any real root of some polynomial of degreed>1 with integer coefficients, then the Beatty functionis a solution of a ‘d-nested’ analogue of Golomb's original ‘2-nested’ recursion. [ABSTRACT FROM PUBLISHER]

This article derives the recursion formulae of and demonstrates through tree diagrams how the formulae for the cases of ℓ = 1, 2, 3 are closely related. Visual thinking by diagrams is pedagogically significant for observing patterns hidden in the formulae for. [ABSTRACT FROM PUBLISHER]

It is known that, for given integers s ⩾ 0 and j > 0, the nested recursion R(n) = R(n - s - R(n - j)) + R(n - 2j - s - R(n - 3j)) has a closed-form solution for which a combinatorial interpretation exists in terms of an infinite, labelled tree. For s = 0, we show that this solution sequence has a closed form as the sum of ceiling functions C(n) = Σi=0j-1 [ n-1/2j]. Furthermore, given appropriate initial conditions, we derive necessary and sufficient conditions on the parameters s1 , a1, s2 and a2 so that C(n) solves the nested recursion R(n) = R(n - R(n - a1))+ R(n - R(n - a2)). [ABSTRACT FROM AUTHOR]

Recursion is the ability to iterate syntactic constituents inside constituents. Experimental and corpus literature indicates that comprehending recursive structures is difficult for children. Since recursion is a fundamental property of language, the result of the application of core minimal tools (merge, label, and select), why does it present an acquisition challenge? We propose that while Merge itself is universal, its application is determined by selection, which is extensively parameterized. We investigate whether children's difficulties can be defined in terms of the complexity of the selectional demands involved. This investigation leads to three empirical questions: i. How does NP recursion compare to NP coordination in acquisition? ii. Do children acquire each level of recursion independently or is recursion triggered automatically once the initial selectional requirements are acquired? iii. Are there differences in the acquisition of different types of nominal recursion? We compare the selectional requirements of English genitives, PPs, and NP coordination. Our syntactic analysis suggests that genitive recursion has more complex selectional demands than PP recursion. We designed a study eliciting production of two level embedded prepositional and genitive NPs. Preschool-aged English-speaking children (n = 46) and adults (n = 11) participated in an elicitation study. Children were comparable to adults in their ability to produce three coordinated NPs, but had substantial difficulty linking three NPs using recursion. First- and second-level embedding were clearly distinct steps in development. These findings have implications for the independence of phrasal and selectional development in children and for our understanding of structural complexity in child language and in theory. [ABSTRACT FROM PUBLISHER]

A cost process is associated with a discrete time, finite state Markov chain. Using backward recursion, the distribution of the first time the cost falls below a given level is calculated. [ABSTRACT FROM PUBLISHER]

In the present paper, we convert the usual n-step backward recursion that arises in option pricing into a set of independent integral equations by using a z-transform approach. In order to solve these equations, we consider different quadrature procedures that transform the integral equation into a linear system that we solve by iterative algorithms and we study the benefits of suitable preconditioning techniques. We show the relevance of our procedure in pricing options (such as plain vanilla, lookback, single and double barrier options) when the underlying evolves according to an exponential Lévy process. [ABSTRACT FROM AUTHOR]

Spreadsheets lend themselves naturally to recursive computations, since a formula can be defined as a function of one of more preceding cells. A hypothesized closed form for the nth term of a recursive sequence can be tested easily by using a spreadsheet to compute a large number of the terms. Similarly, a conjecture about the limit of a series can be tested by using a spreadsheet to compute a number of partial sums recursively. Once students are confident that a closed form is probably not wrong, they can prove it correct by mathematical induction. [ABSTRACT FROM PUBLISHER]

Inspired by the recent work of Opps, Saad and Srivastava, who gave the recursion formulas of Appell’s function F 2 by the contiguous relation of the Gauss hypergeometric series 2 F 1, we establish the recursion formulas for Appell functions F 1, F 2, F 3 and F 4 by the contiguous relations of hypergeometric series. [ABSTRACT FROM AUTHOR]

Guillotine patterns of specified features are often used in manufacturing industries to cut stock plate into rectangular pieces. The pattern type should be selected from considering both material utilization and cutting cost. This article proposes extended block patterns for the two-dimensional cutting stock problem, where each cut on the plate produces just one extended block. A recursion is used to generate the optimal pattern. It maximizes the difference between the total value of the included pieces and the cutting cost. The computational results indicate that extended block patterns are useful to reduce production cost. [ABSTRACT FROM AUTHOR]

Consider a nested, non-homogeneous recursion defined by with c initial conditions , where the parameters are integers satisfying , and . We develop an algorithm to answer the following question: for an arbitrary rational number , is there any set of values for and such that the ceiling function is the unique solution generated by with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to . The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if is the solution generated by any recursion of the form above, then r = 1. We also use our ceiling function methodology to derive the first known connection between the recursion and a natural generalization of Conway's recursion. [ABSTRACT FROM PUBLISHER]

This article aims to provide a simple approach realising state observation and input estimation simultaneously for discrete-time LTV systems in H∞ setting. Through solving a two-player zero sum differential game, appealing results are obtained in two folds. First, necessary and sufficient solvability conditions for state and input simultaneous estimation problem are given in terms of solution to a set of difference Riccati recursion. Second, one estimator is presented with special innovation structure, where innovation information is used to update state observation tuned by gain matrix and to provide input estimation through a projector matrix, where gain matrix and projector matrix are constructed from solution to difference Riccati recursion. At last, simulation results are provided to justify proposed approach. [ABSTRACT FROM PUBLISHER]

Online (also called 'recursive' or 'adaptive') estimation of fixed model parameters in hidden Markov models is a topic of much interest in times series modeling. In this work, we propose an online parameter estimation algorithm that combines two key ideas. The first one, which is deeply rooted in the Expectation-Maximization (EM) methodology, consists in reparameterizing the problem using complete-data sufficient statistics. The second ingredient consists in exploiting a purely recursive form of smoothing in HMMs based on an auxiliary recursion. Although the proposed online EM algorithm resembles a classical stochastic approximation (or Robbins-Monro) algorithm, it is sufficiently different to resist conventional analysis of convergence. We thus provide limited results which identify the potential limiting points of the recursion as well as the large-sample behavior of the quantities involved in the algorithm. The performance of the proposed algorithm is numerically evaluated through simulations in the case of a noisily observed Markov chain. In this case, the algorithm reaches estimation results that are comparable to those of the maximum likelihood estimator for large sample sizes. The supplemental material for this article available online includes an appendix with the proofs of Theorem 1 and Corollary 1 stated in Section 4 as well as the MATLAB/OCTAVE code used to implement the algorithm in the case of a noisily observed Markov chain considered in Section 5. [ABSTRACT FROM AUTHOR]

We consider a simple Poisson cluster model for the payment numbers and the corresponding total payments for insurance claims arriving in a given year. Due to the Poisson structure one can give reasonably explicit expressions for the prediction of the payment numbers and total payments in future periods given the past observations of the payment numbers. One can also derive reasonably explicit expressions for the corresponding prediction errors. In the (a, b) class of Panjer's claim size distributions, these expressions can be evaluated by simple recursive algorithms. We study the conditions under which the predictions are asymptotically linear as the number of past payments becomes large. We also demonstrate that, in other regimes, the prediction may be far from linear. For example, a staircase-like pattern may arise as well. We illustrate how the theory works on real-life data, also in comparison with the chain ladder method. [ABSTRACT FROM AUTHOR]

We develop a simple numerical method for obtaining Taylor series approximation to the solution of a nonlinear third-order boundary-value problem. We use recursive formulas derived from the governing differential equation itself to calculate exact values of the derivatives needed in the Taylor series. Since we do not use difference formulas or symbolic manipulation for calculating the derivatives, our method requires much less computational effort when compared with the techniques previously reported in the literature. We will illustrate the effectiveness of our method with several test problems. [ABSTRACT FROM AUTHOR]

This paper aims to propose a hierarchical, generative account of diatonic harmonic progressions and suggest a set of phrase-structure grammar rules. It argues that the structure of harmonic progressions exceeds the simplicity of the Markovian transition tables and proposes a set of rules to account for harmonic progressions with respect to key structure, functional and scale degree features as well as modulations. Harmonic structure is argued to be at least one subsystem in which Western tonal music exhibits recursion and hierarchical organization that may provide a link to overarching linguistic generative grammar on a structural and potentially cognitive level. [ABSTRACT FROM AUTHOR]

Exact forward recursions for the score vector and observed information matrix of the Markov-modulated Poisson process (MMPP) are developed. The recursions are motivated by similar recursions developed for hidden Markov models by Lystig and Hughes who extended earlier work by LeGland and Mevel. Explicit expressions for the first derivative and Hessian of the MMPP transition density matrix are developed and coupled with the recursions. The recursions are implemented and applied to confidence interval estimation in a simulation study. [ABSTRACT FROM AUTHOR]

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order. A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved. The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory. [ABSTRACT FROM AUTHOR]

In this article, Polya-Aeppli distribution of order k as a compound Poisson distribution is defined. The probability mass function and recursion formulas are given. We consider the interpretation of the distribution and a definition of Polya-Aeppli process of order k. [ABSTRACT FROM AUTHOR]

This paper derives an algorithm for computing rectangular multinomial probabilities. It thus provides a method for exact computation of the multinomial cumulative distribution function. One application of rectangular multinomial probabilities comes in determining how large a sample must be in order to have a high probability of being representative in a certain nonparametric sense. [ABSTRACT FROM AUTHOR]