Solving the musical orchestration problem using multiobjective constrained optimization with a genetic local search approach

In this paper a computational approach of musical orchestration is presented. We consider orchestration as the search of relevant sound combinations within large instruments sample databases and propose two cooperating metaheuristics to solve this problem. Orchestration is seen here as a particular case of finding optimal constrained multisets on a large ensemble with respect to several objectives. We suggest a generic and easily extendible formalization of orchestration as a constrained multiobjective search towards a target timbre, in which several perceptual dimensions are jointly optimized. We introduce Orchidée, a time-efficient evolutionary orchestration algorithm that allows the discovery of optimal solutions and favors the exploration of non-intuitive sound mixtures. We also define a formal framework for global constraints specification and introduce the innovative CDCSolver repair metaheuristic, thanks to which the search is led towards regions fulfilling a set of musical-related requirements. Evaluation of our approach on a wide set of real orchestration problems is also provided.     

POWER STRUCTURES AND LOGIC

ABSTRACT

This paper generalizes the concept of a power alge bra to that of a power structure, and gives three application of power structures to logic.     

Mastering by the teacher of the instrumental genesis in CAS environments: necessity of intrumental orchestrations

In this article, we study didactic phenomena identified in integration experiments within our classes, CAS (implemented in calculators). From this study, we show the interest of an instrumental approach to understand the influence of tools on the mathematical approach and on the building of student's knowledge: through a process—instrumental genesis—a calculator becomes a mathematical work tool; this process depends on the tool's constraints and potentialities, on students' knowledge, and on the class' work situations. To analyze the differentiation of instrumental genesis, we then have taken interest in students' behaviour and we propose a method enabling us to constitute a typology of extreme behaviour in environments of symbolic calculators. To take the variety of these genesis into account, the professor needs a particular organization of space and time of the study in the class. We suggest the notion of instrumental orchestration to name this organization. We demonstrate how this notion gives a better definition of the objectives, the configurations and the exploitation modes of different arrangements which aim at constituting coherent instrument systems for each student and for the class.     

Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions

The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on Cⁿ[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cⁿ[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.     

Effective multipoles in random media

Abstract

In a homogeneous medium, the far field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole – but not the octupole – can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest. Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero- and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green’s formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.     

An active index algorithm for the nearest point problem in a polyhedral cone

We consider the problem of finding the nearest point in a polyhedral cone C={x∈R ⁿ :D x≤0} to a given point b∈R ⁿ , where D∈R ^m×n . This problem can be formulated as a convex quadratic programming problem with special structure. We study the structure of this problem and its relationship with the nearest point problem in a pos cone through the concept of polar cones. We then use this relationship to design an efficient algorithm for solving the problem, and carry out computational experiments to evaluate its effectiveness. Our computational results show that our proposed algorithm is more efficient than other existing algorithms for solving this problem.     

On the Integration of Digital Technologies into Mathematics Classrooms

Trouche’s [Third Computer Algebra in Mathematics Education Symposiums, Reims, France, June 2003] presentation at the Third Computer Algebra in Mathematics Education Symposium focused on the notions of instrumental genesis and of orchestration: the former concerning the mutual transformation of learner and artefact in the course of constructing knowledge with technology; the latter concerning the problem of integrating technology into classroom practice. At the Symposium, there was considerable discussion of the idea of situated abstraction, which the current authors have been developing over the last decade. In this paper, we summarise the theory of instrumental genesis and attempt to link it with situated abstraction. We then seek to broaden Trouche’s discussion of orchestration to elaborate the role of artefacts in the process, and describe how the notion of situated abstraction could be used to make sense of the evolving mathematical knowledge of a community as well as an individual. We conclude by elaborating the ways in which technological artefacts can provide shared means of mathematical expression, and discuss the need to recognise the diversity of student’s emergent meanings for mathematics, and the legitimacy of mathematical expression that may be initially divergent from institutionalised mathematics.     

Standardly Stratified Extension Algebras

Abstract

The paper generalizes some of our previous results on quasi-hereditary Koszul algebras to graded standardly stratified Koszul algebras. The main result states that the class of standardly stratified algebras for which the left standard modules as well as the right proper standard modules possess a linear projective resolution – the so called linearly stratified algebras – is closed under forming their Yoneda extension algebras. This is proved using the technique of Hilbert and Poincaré series of the corresponding modules.

     

Order books,markets and convex analysis

Abstract

This paper considers how an order market might evolve over a fairly short period – say, during a day. It uses elementary convex analysis to model agents’ choice of prudent orders, and it explores whether equilibrium is attainable.     

Mathematics education graduate students’ understanding of trigonometric ratios

This study describes mathematics education graduate students’ understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students’ understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students’ responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students’ concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students’ understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.     

Analysis of two discrete forms of the classic continuous SIR epidemiological model

ABSTRACT

From continuous standard SIR model, which is configured from two sequenced flows (a) susceptible – infectious and (b) infectious – removed, we obtain two impulsive SIR models assuming different time scales for (a) respect to (b) (one more quickly than the other and inversely). By associating respective stroboscopic maps to this impulsive systems, two discretizations are defined. The dynamics of these maps are analysed in order to get thresholds conditions for predicting (or to control) epidemic outbreaks. As it is traditional for SIR systems, we also find conditions for the final size of the susceptible group.     

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

Abstract

The notation of I–convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I–convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces.     

Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.     

ON AN ALGEBRAIZATION OF MEASURE THEORY. ABSTRACT SEMIRINGS

Abstract

The concept of a semiring is introduced as a semilattice analogue of a semiring of sets. A detailed study of its embeddability into r-complete Boolean algebras is carried out. Some of its properties which are important from the standpoint of Measure Theory are elucidated. A representation of a semiring by a Semiring of sets is derived. A relation to implicative BCK-algebras is also mentioned.     

Instrumental Variable Estimation for Linear Stochastic Differential Equations Driven by Fractional Brownian Motion

Abstract

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by fractional Brownian motion.     

APPROXIMATION BY VECTOR-VALUED FUNCTIONS

ABSTRACT

The theory of H-sets was first propounded by L. Collatz [3] and [4]. This concept has been shown to be useful in the study of uniform approximation, and we here consider the form H-sets take in this setting of vector-valued functions and prove a general characterization theorem. A similar exposition for the linear real-valued case can be found in [1].     

ON PRODUCTS OF GROUPS FOR WHICH NORMALITY IS A TRANSITIVE RELATION ON THEIR FRATTINI FACTOR GROUPS

Abstract

This paper is devoted to the study of groups with the property that the Frattini factor group is a T-group, i.e. a group in which every subnormal subgroup is normal. We give necessary and suffucient conditions for a direct product G = H x K of finite groups H and K to have such a property. Some structure theorems are also discussed.     

Convergence of the Lambert-McLeod trajectory solver and of the celf method

Summary A trajectory problem is an initial value problemd y/dt=f(y),y(0)= where the interest lies in obtaining the curve traced by the solution (the trajectory), rather than in finding the actual correspondanc between values of the parametert and points on that curve. We prove the convergence of the Lambert-McLeod scheme for the numerical integration of trajectory problems. We also study the CELF method, an explicit procedure for the integration in time of semidiscretizations of PDEs which has some useful conservation properties. The proofs rely on the concept of restricted stability introduced by Stetter. In order to show the convergence of the methods, an idea of Strang is also employed, whereby the numerical solution is compared with a suitable perturbation of the theoretical solution, rather than with the theoretical solution itself.     

A predictable decomposition in an infinite assets model with jumps. Application to hedging and optimal investment

Motivated by the theory of bond markets, we consider an infinite assets model driven by marked point process and Wiener process. The self-financed wealth processes are defined by using measure-valued strategies. Going further on the works of Bjork et al. [“Bond market structure in the presence of marked point processes”, Mathematical Finance, 7 (1997a) pp. 211–239; “Towards a general theory of bond markets”, Finance and Stochastics, 1 (1997b) pp. 141–174] who focus on the existence of martingale measures and market completeness questions, we study here the incompleteness case. Our main result is a predictable decomposition theorem for supermartingales in this infinite assets model context. The concept of approximate wealth processes is introduced, and we show in an example that the space of measure-valued strategies is not complete with respect to the semimartingale topology. As in the case of stock markets, one can then derive a dual representation of the super-replication cost and study the problem of utility maximization by duality methods.     

Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices

Formal concept analysis (FCA) associates a binary relation between a set of objects and a set of properties to a lattice of formal concepts defined through a Galois connection. This relation is called a formal context, and a formal concept is then defined by a pair made of a subset of objects and a subset of properties that are put in mutual correspondence by the connection. Several fuzzy logic approaches have been proposed for inducing fuzzy formal concepts from L-contexts based on antitone L-Galois connections. Besides, a possibility-theoretic reading of FCA which has been recently proposed allows us to consider four derivation powerset operators, namely sufficiency, possibility, necessity and dual sufficiency (rather than one in standard FCA). Classically, fuzzy FCA uses a residuated algebra for maintaining the closure property of the composition of sufficiency operators. In this paper, we enlarge this framework and provide sound minimal requirements of a fuzzy algebra w.r.t. the closure and opening properties of antitone L-Galois connections as well as the closure and opening properties of isotone L-Galois connections. We apply these results to particular compositions of the four derivation operators. We also give some noticeable properties which may be useful for building the corresponding associated lattices.