481,236 research outputs found
Theory of partial-order programming
This paper shows the use of partial-order program clauses and lattice domains for declarative programming. This paradigm is particularly useful for expressing concise solutions to problems from graph theory, program analysis, and database querying. These applications are characterized by a need to solve circular constraints and perform aggregate operations, a capability that is very clearly and efficiently provided by partial-order clauses. We present a novel approach to their declarative and operational semantics, as well as the correctness of the operational semantics. The declarative semantics is model-theoretic in nature, but the least model for any function is not the classical intersection of all models, but the greatest lower bound/least upper bound of the respective terms defined for this function in the different models. The operational semantics combines top-down goal reduction with memo-tables. In the partial-order programming framework, however, memoization is primarily needed in order to detect circular circular function calls. In general we need more than simple memoization when functions are defined circularly in terms of one another through monotonic functions. In such cases, we accumulate a set of functional-constraint and solve them by general fixed-point-finding procedure. In order to prove the correctness of memoization, a straightforward induction on the length of the derivation will not suffice because of the presence of the memo-table. However, since the entries in the table grow monotonically, we identify a suitable table invariant that captures the correctness of the derivation. The partial-order programming paradigm has been implemented and all examples shown in this paper have been tested using this implementation
Teaching Partial Differential Equations with CAS
Partial Differential Equations (PDE) are one of the topics where Engineering students
find more difficulties when facing Math subjects.
A basic course in Partial Differential Equations (PDE) in Engineering, usually deals at
least, with the following PDE problems:
1. Pfaff Differential Equations
2. Quasi-linear Partial Differential Equations
3. Using Lagrange-Charpit Method for finding a complete integral for a given general
first order partial differential equation
4. Heat equation
5. Wave equation
6. Laplace’s equation
In this talk we will describe how we introduce CAS in the teaching of PDE.
The tasks developed combine the power of a CAS with the flexibility of programming
with it. Specifically, we use the CAS DERIVE. The use of programming allows us to use
DERIVE as a Pedagogical CAS (PECAS) in the sense that we do not only provide the final
result of an exercise but also display all the intermediate steps which lead to find the solution
of a problem. This way, the library developed in DERIVE serves as a tutorial showing, step
by step, the way to face PDE exercises.
In the process of solving PDE exercises, first-order Ordinary Differential Equations (ODE)
are needed. The programs developed can be grouped within the following blocks:
- First-order ODE: separable equations and equations reducible to them, homogeneous
equations and equations reducible to them, exact differential equations and equations
reducible to them (integrating factor technique), linear equations, the Bernoulli equation,
the Riccati equation, First-order differential equations and nth degree in y’, Generic
programs to solve first order differential equations.
- First-order PDE: Pfaff Differential Equations, Quasi-linear PDE, Lagrange-Charpit
Method for First-order PDE.
- Second-order PDE: Heat Equation, Wave Equation, Laplace’s Equation.
We will remark the conclusions obtained after using these techniques with our Engineering students.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
Metode Urutan Parsial Untuk Menyelesaikan Masalah Program Linier Fuzzy Tidak Penuh
Not fully fuzzylinear programming problem have two shapes of objecyive function. that is triangular fuzzy number and trapezoidal fuzzy number. The decision variables and constants right segment only has a triangular fuzzy number. Partial order method can be used to solve not fully fuzzy linear programming problem with decision variables and constants right segment are triangular fuzzy number. The crisp optimal objective function value generated from the partial order method
Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion
In this paper, we consider the stochastic optimal control problems under
G-expectation. Based on the theory of backward stochastic differential
equations driven by G-Brownian motion, which was introduced in [10.11], we can
investigate the more general stochastic optimal control problems under
G-expectation than that were constructed in [28]. Then we obtain a generalized
dynamic programming principle and the value function is proved to be a
viscosity solution of a fully nonlinear second-order partial differential
equation.Comment: 25 page
- …