Inference in hybrid Bayesian networks using mixtures of polynomials

The main goal of this paper is to describe inference in hybrid Bayesian networks (BNs) using mixture of polynomials (MOP) approximations of probability density functions (PDFs). Hybrid BNs contain a mix of discrete, continuous, and conditionally deterministic random variables. The conditionals for continuous variables are typically described by conditional PDFs. A major hurdle in making inference in hybrid BNs is marginalization of continuous variables, which involves integrating combinations of conditional PDFs. In this paper, we suggest the use of MOP approximations of PDFs, which are similar in spirit to using mixtures of truncated exponentials (MTEs) approximations. MOP functions can be easily integrated, and are closed under combination and marginalization. This enables us to propagate MOP potentials in the extended Shenoy-Shafer architecture for inference in hybrid BNs that can include deterministic variables. MOP approximations have several advantages over MTE approximations of PDFs. They are easier to find, even for multi-dimensional conditional PDFs, and are applicable for a larger class of deterministic functions in hybrid BNs.     

On shifted periodic solutions of two nonlinear equations

Using the linear superposition approach, we find periodic solutions with shifted periods and velocities of the (2 + 1)-dimensional modified Zakharov–Kuznetsov equation and the (3 + 1)-dimensional Kadomtsev–Petviashvili equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure of generating solutions of nonlinear evolution equations is successful as a consequence of some cyclic identities satisfied by the Jacobi elliptic functions which reduce by 2 (or a larger even number) the degree of cyclic homogeneous polynomials in Jacobi elliptic functions.     

Structurally unstable regular dynamics in 1D piecewise smooth maps,and circle maps

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′ = aX if ∣X∣ < z, X′ = bX if ∣X∣ > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′ = F(X) if ∣X∣ < z, X′ = G(X) if ∣X∣ > z is also given, proving the conditions leading to a homeomorphism of the circle.     

Conceptual blending: Student reasoning when proving “conditional implies conditional” statements

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., (p → q) ⇒ (r → s). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.     

Filter matrix based on interpolation wavelets for solving Fredholm integral equations

The interpolation wavelet is used to solve the Fredholm integral equation of the second kind in this study. Hence, by the extension of interpolation wavelets that [−1, 1] is divided to 2^N+1 (N ⩾  − 1) subinterval, we have polynomials with a degree less than M + 1 in each new interval. Therefore, by considering the two-scale relation the filter coefficients and filter matrix are used as the proof of theorems. The important point is interpolation wavelets lead to more sparse matrix when we try to solve integral equation by an approximate kernel decomposed to a lower and upper resolution. Using n-time, where (n ⩾ 2), two-scale relation in this method errors of approximate solution as O((2^−(N+1))ⁿ⁺¹). Also, the filter coefficient simplifies the proof of some theorems and the order of convergence is estimated by numerical errors.     

Inverse problems for parabolic equations 2

Let u_t − u_xx = h(t) in 0 ⩽ x ⩽ π, t ⩾ 0. Assume that u(0, t) = v(t), u(π, t) = 0, and u(x, 0) = g(t). The problem is: what extra data determine the three unknown functions {h, v, g} uniquely? This question is answered and an analytical method for recovery of the above three functions is proposed.     

Commutativity and self-duality: Two tales of one equation

The mathematical expressions for the commutativity or self-duality of an increasing [0, 1]² → [0, 1] function F involve the transposition of its arguments. We unite both properties in a single functional equation. The solutions of this functional equation are discussed. Special attention goes to the geometrical construction of these solutions and their characterization in terms of contour lines. Furthermore, it is shown how ‘rotating’ the arguments of F allows to convert the results into properties for [0, 1]² → [0, 1] functions having monotone partial functions.     

Nonlinear dynamics and intermittency in a long-term copepod time series

We consider the nonlinear dynamics of a long-term copepod (small crustaceans) time series sampled weekly in the Mediterranean sea from 1967 to 1992. Such population dynamics display a high variability that we consider here in an interdisciplinary study, using tools borrowed from the field of statistical physics. We analyse the extreme events of male and female abundances, and of the total population, and show that they both have heavy tailed probability density functions (pdf). We provide hyperbolic fits of the form p(x) ∼ 1/x^μ+1, and estimate the value of μ using Hill’s estimator. We then study the ratio of male to female abundances, compared to the female abundances. Using conditional probability density functions and conditional averages, we show that this ratio is independent of the female density, when the latter is larger than a given threshold. This property is very useful for modelization. We also consider the product of male to female abundances, which can be ecologically related to the encounters. We show that this product is extremely intermittent, and link its pdf to the female pdf.     

Anytime anyspace probabilistic inference

This paper investigates methods that balance time and space constraints against the quality of Bayesian network inferences––we explore the three-dimensional spectrum of “time × space × quality” trade-offs. The main result of our investigation is the adaptive conditioning algorithm, an inference algorithm that works by dividing a Bayesian network into sub-networks and processing each sub-network with a combination of exact and anytime strategies. The algorithm seeks a balanced synthesis of probabilistic techniques for bounded systems. Adaptive conditioning can produce inferences in situations that defy existing algorithms, and is particularly suited as a component of bounded agents and embedded devices.     

Multifractal rainfall extremes: Theoretical analysis and practical estimation

We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity–Duration–Frequency (IDF) curves. The model assumes that rainfall is a sequence of independent and identically distributed multiplicative cascades of the beta-lognormal type, with common duration D. When properly fitted to data, this simple model was found to produce accurate IDF results [Langousis A, Veneziano D. Intensity–duration–frequency curves from scaling representations of rainfall. Water Resour Res 2007;43. doi:10.1029/2006WR005245]. Previous studies also showed that the IDF values from multifractal representations of rainfall scale with duration d and return period T under either d → 0 or T → ∞, with different scaling exponents in the two cases. We determine the regions of the (d, T)-plane in which each asymptotic scaling behavior applies in good approximation, find expressions for the IDF values in the scaling and non-scaling regimes, and quantify the bias when estimating the asymptotic power-law tail of rainfall intensity from finite-duration records, as was often done in the past. Numerically calculated exact IDF curves are compared to several analytic approximations. The approximations are found to be accurate and are used to propose a practical IDF estimation procedure.     

Discrepancy principle for DSM II

Let Ay = f, A is a linear operator in a Hilbert space H, y ⊥ N(A) ≔ {u : Au = 0}, R(A) ≔ {h : h = Au, u ∈ D(A)} is not closed, ∥f_δ − f∥ ⩽ δ. Given f_δ, one wants to construct u_δ such that lim_δ→0∥u_δ − y∥ = 0. Two versions of discrepancy principles for the DSM (dynamical systems method) for finding the stopping time and calculating the stable solution u_δ to the original equation Ay = f are formulated and mathematically justified.     

Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field F_q is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight d_r(C), 1 ⩽ r ⩽ k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d₁(C), d₂(C), … , d_k(C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, d_r(C) = n − k + r;  r = 1, 2 , …, k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH d_r(C) = n − k + r; r = 1, 2, … , k. A linear code C with systematic check matrix [I∣P], where I is the (n − k) × (n − k) identity matrix and P is a (n − k) × k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS (N²-MDS) codes and A^μ-MDS codes. The MDS-rank of C is the smallest integer η such that d_η+1 = n − k + η + 1 and the defect vector of C with MDS-rank η is defined as the ordered set {μ₁(C), μ₂(C), μ₃(C), … , μ_η(C), μ_η+1(C)}, where μ_i(C) = n − k + i − d_i(C). We call C a dually defective code if the defect vector of the code and its dual are the same. We also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported.     

Type-2 fuzzy fractional derivatives

In this paper, we introduce two definitions of the differentiability of type-2 fuzzy number-valued functions of fractional order. The definitions are in the sense of Riemann–Liouville and Caputo derivative of order β ∊ (0, 1), and based on type-2 Hukuhara difference and H₂-differentiability. The existence and uniqueness of the solutions of type-2 fuzzy fractional differential equations (T2FFDEs) under Caputo type-2 fuzzy fractional derivative and the definition of Laplace transform of type-2 fuzzy number-valued functions are also given. Moreover, the approximate solution to T2FFDE by a Predictor-Evaluate–Corrector-Evaluate (PECE) method is presented. Finally, the approximate solutions of two examples of linear and nonlinear T2FFDEs are obtained using the PECE method, and some cases of T2FFDEs applications in some sciences are presented.     

The non-traveling wave solutions and novel fractal soliton for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients

A method is proposed by extending the linear traveling wave transformation into the nonlinear transformation with the (G′/G)-expansion method. The non-traveling wave solutions with variable separation can be constructed for the (2 + 1)-dimensional Broer–Kaup equations with variable coefficients via the method. A novel class of fractal soliton, namely, the cross-like fractal soliton is observed by selecting appropriately the arbitrary functions in the solutions.     

On “P” property and the column-W property

P-matrices play an important role in the well-posedness of a linear complementarity problem (LCP). Similarly, the well-posedness of a horizontal linear complementarity problem (HLCP) is closely related to the column-W property of a matrix k-tuple.In this paper we first consider the problem of generating P-matrices from a given pair of matrices. Given a matrix pair (D, F) where D is a square matrix of order m and matrix F has m rows, “what are the conditions under which there exists a matrix G such that (D + FG) is a P-matrix?”. We obtain necessary and sufficient conditions for the special case when the column rank of F is m ? 1. A decision algorithm of complexity O(m²) to check whether the given pair of matrices (D, F) is P-matrisable is obtained. We also obtain a necessary and an independent sufficient condition for the general case when rank(F) is less than m ? 1.We then generalise the P-matrix generating problem to the generation of matrix k-tuples satisfying the column-W property from a given matrix (k + 1)-tuple. That is, given a matrix (k + 1)-tuple (D₁, … ,D_k, F), where D_js are square matrices of order m and F is a matrix having m rows, we determine the conditions under which the matrix k-tuple (D₁ + FG₁, … ,D_k + FG_k) satisfies the column-W property. As in the case of P-matrices we obtain necessary and sufficient conditions for the case when rank(F) = m ? 1. Using these conditions a decision algorithm of complexity O(km²) to check whether the given matrix (k + 1)-tuple is column-W matrisable is obtained. Then for the case when rank(F) is less than m ? 1, we obtain a necessary and an independent sufficient condition.For a special sub-class of P-matrices we give a polynomial time decision algorithm for P-matrisability. Finally, we obtain a geometric characterisation of column-W property by generalising the well known separation theorem for P-matrices.     

Net-convergence and weak separation axioms in (L, M)-fuzzy topological molecular lattices

In this paper, we used the concept of (L, M)-fuzzy remote neighborhood system to study and establish the convergence theory of molecular nets. Next, we introduce the T_i-axioms (i = ?1, 0, 1, 2) in (L, M)-fuzzy topological molecular lattices, and discuss some of their characterizations. Finally, we show that the T_i-axioms (i = ?1, 0, 1, 2) are preserved under homeomorphisms.     

Linear Groups Definable in o-Minimal Structures

We study subgroups G of GL(n, R) definable in o-minimal expansions M = (R, +, · ,…) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, power functions, or an exponential function definable in M. (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser–Tits conjecture for real closed fields.     

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings.     

Eigenvalue problems for fractional ordinary differential equations

The eigenvalue problems are considered for the fractional ordinary differential equations with different classes of boundary conditions including the Dirichlet, Neumann, Robin boundary conditions and the periodic boundary condition. The eigenvalues and eigenfunctions are characterized in terms of the Mittag–Leffler functions. The eigenvalues of several specified boundary value problems are calculated by using MATLAB subroutine for the Mittag–Leffler functions. When the order is taken as the value 2, our results degenerate to the classical ones of the second-ordered differential equations. When the order α satisfies 1 < α < 2 the eigenvalues can be finitely many.