We present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes. The method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations, which allows transferring basic properties of sup-T and inf-R compositions to a family of more than 30 composition-related operations, including sup-T and inf-R image...

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. We study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function f , t...

In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that diffusion operator may lose coercivity due to coefficient, which makes both mathematical and numerical analysis challenging. To resolve issue, design appropriate test trial functions prove inf-sup condition of operators in suitable function ...

This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms—local parameter-dependent norms—to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi–Rappaz–Raviart existence and uniqueness theor...

In this work, we show how to construct, by means of the function space interpolation theory, a natural norm ||| · ||| for a generic linear coercive and non-symmetric operator L. The natural norm |||·||| allows for continuity and inf-sup conditions which holds independently of L. Particularly we will consider the convection-diffusion-reaction operator, for which we obtain continuity and inf-sup ...

This section presents a setup that arises frequently in adverse selection settings and provides conditions on the primitives such that the properties that de…ne a game with monotone selection are satis…ed. The following de…nitions and results (in addition to those in the print appendix) are used: A correspondence : T ! X is increasing in the strong set order if when t > t0, then for each x 2 (t...

Iterative methods are considered for saddle point problems with a penalty term. A positive deenite preconditioner is constructed and it is proved that the condition number of the preconditioned system can be made independent of the discretization and the penalty parameters. Examples include the pure displacement problem in linear elasticity, the Timoshenko beam, and the Mindlin-Reissner plate. ...

The lattice of the set partitions of [n] ordered by refinement is studied. Suppose r partitions p1, . . . , pr are chosen independently and uniformly at random. The probability that the coarsest refinement of all pi ’s is the finest partition {1}, . . . , {n} is shown to approach 0 for r = 2, and 1 for r ≥ 3. The probability that the finest coarsening of all pi ’s is the one-block partition is ...