Dominance elimination procedures on finite alternative games

A general definition of dominance elimination procedures for finite games is given which includes the multistage game representation of sophisticated voting for binary procedures. There then follows a proof that the set of outcomes resulting from the successive application of any dominance elimination procedure to any game in which each player has strict preferences over the alternatives, and in which the alternative and strategy spaces are finite, contains the set of outcomes attained by applying a procedure previously defined byFarquharson [1969].     

Establishing strict dominance between alternatives with special type of incomplete information

A special model for establishing dominance with decision maker's incomplete information is proposed in multi-attribute decision problem under certainty. Incomplete information is a set of linear inequalities about utilities as well as attribute weights. Since a general model presented in a prior work uses a linear programming technique, a number of linear programs should be solved for checking dominance. With the proposed method, we can check strict dominance between alternatives by simple matrix operation without solving linear programs, thus reducing computational time.     

The weak vorticity formulation of the 2-d euler equations and concentration-cancellation

The weak limit of a sequence of approximate solutions of the 2-D Euler equations will be a solution if the approximate vorticities concentrate only along a curve x(t) that is Holder continuous with exponent ½.

A new proof is given of the theorem of DiPerna and Majda that weak limits of steady approximate solutions are solutions provided that the singularities of the inhomogeneous forcing term are sufficiently mild. An example shows that the weaker condition imposed here on the forcing term is sharp.

A simplified formula for the kernel in Delort's weak vorticity formulation of the two-dimensional Euler equations makes the properties of that kernel readily apparent, thereby simplying Delort's proof of the existence of one-signed vortex sheets.     

On the non-equivalence of weak and strict preference

It is often claimed that the relations of weak preference and strict preference are symmetrical to each other in the sense that weak preference is complete and transitive if and only if strict preference is asymmetric and negatively transitive. The equivalence proof relies on a definitional connection between them, however, that already implies completeness of weak preference. Weakening the connection in order to avoid this leads to a breakdown of the symmetry which gives reason to accept weak preference as the more fundamental relation.     

A remark on a compactness result in electromagnetic theory

Calderon's extension theorem is a crucial tool in the proof of the compactneσs of the resolvent for the Maxwell operator, and whence this result is proved for domains with the strict cone property. However, the proof only requires an extension operator that extends W^2,2-functions compactly as W^1,2-functions. It is shown that this can be achieved under weaker regularity conditions on the domain: the cone may be replaced by some cusp of an appropriate order.     

A regularity theorem for quasilinear parabolic systems under random perturbations

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions are in fact more regular, and examples of systems with weak solutions which develop singularities in finite time. Our main result is the extension of a regularity result due to Kalita to the stochastic case, which concerns local Hölder continuity of weak solutions in the vectorial case. For the proof, we apply stochastic versions of methods, which are classical in the deterministic case (such as difference quotient techniques, higher integrability by embedding theorems, and a version of Moser’s iteration technique). This might be of interest on their own.     

Classical proof forestry

Classical proof forests are a proof formalism for first-order classical logic based on Herbrand’s Theorem and backtracking games in the style of Coquand. First described by Miller in a cut-free setting as an economical representation of first-order and higher-order classical proof, defining features of the forests are a strict focus on witnessing terms for quantifiers and the absence of inessential structure, or ‘bureaucracy’.This paper presents classical proof forests as a graphical proof formalism and investigates the possibility of composing forests by cut-elimination. Cut-reduction steps take the form of a local rewrite relation that arises from the structure of the forests in a natural way. Yet reductions, which are significantly different from those of the sequent calculus, are combinatorially intricate and do not exclude the possibility of infinite reduction traces, of which an example is given.Cut-elimination, in the form of a weak normalisation theorem, is obtained using a modified version of the rewrite relation inspired by the game-theoretic interpretation of the forests. It is conjectured that the modified reduction relation is, in fact, strongly normalising.     

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke’s partial regularity theorem up to $C^{1,\varsigma }$ for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard’s regularity theorem in the sense that the special time-independent case reduces to Allard’s theorem.     

Stability and the immediate acceptance rule when school priorities are weak

In a model of school choice, we allow school priorities to be weak and study the preference revelation game induced by the immediate acceptance (IA) rule (also known as the Boston rule), or the IA game. When school priorities can be weak and matches probabilistic, three stability notions—ex post stability, ex ante stability, and strong ex ante stability—and two ordinal equilibrium notions—sd equilibrium and strong sd equilibrium—become available (“sd” stands for stochastic dominance). We show that for no combination of stability and equilibrium notions does the set of stable matches coincide with the set of equilibrium matches of the IA game. This stands in contrast with the existing result that the two sets are equal when priorities are strict. We also show that in the presence of weak priorities, the transition from the IA rule to the deferred acceptance rule may, in fact, harm some students.     

Weak convergence for a type of conditional expectation: application to the inference for a class of asset price models

We prove weak convergence of a type of conditional expectation, which provides a straightforward proof of Goggin's Theorem and further proves the consistency of (integrated) likelihood, posterior, and Bayes factor for a class of transactional asset price models recently developed.     

Orientals as free weak ω-categories

The orientals are the free strict ω-categories on the simplices introduced by Street. The aim of this paper is to show that they are also the free weak ω-categories on the same generating data. More precisely, we exhibit the complicial nerves of the orientals as fibrant replacements of the simplices in Verity's model structure for weak complicial sets.     

Strict positivstellensätze for matrix polynomials with scalar constraints

We extend Krivine’s strict positivstellensätz for usual (real multivariate) polynomials to symmetric matrix polynomials with scalar constraints. The proof is an elementary computation with Schur complements. Analogous extensions of Schmüdgen’s and Putinar’s strict positivstellensätz were recently proved by Hol and Scherer using methods from optimization theory.     

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3$3\times 3$-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.     

Positivstellensätze for differentiable functions

We present a canonical proof of both the strict and weak Positivstellensatz for rings of differentiable and smooth functions. Our construction is explicit, preserves definability in expansions of the real field, and it works in definably complete expansions of real closed fields as well as for real-valued functions on Banach spaces.     

Quasilinear parabolic stochastic partial differential equations: Existence,uniqueness

We present a direct approach to existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone. The proof of uniqueness is very elementary, based on a new method of applying Itô’s formula for the $L^1$-norm. The proof of existence relies on a recent regularity result and is direct in the sense that it does not rely on the stochastic compactness method.     

On the lattice structure of kernel operators

Consider the lattice of bounded linear operators on the space of Borel measures on a Polish space. We prove that the operators which are continuous with respect to the weak topology induced by the bounded measurable functions form a sublattice that is lattice isomorphic to the space of transition kernels. As an application we present a purely analytic proof of Doob's theorem concerning stability of transition semigroups.     

An electro-viscoelastic frictional contact problem with damage

We consider a quasistatic frictional contact problem between a piezoelectric body and a foundation. The contact is modeled with normal compliance and friction is modeled with a general version of Coulomb's law of dry friction; the process is quasistatic and the material's behavior is described by an electro-viscoelastic constitutive law with damage. We derive a variational formulation for the model which is in the form of a system involving the displacement field, the electric potential field, and the damage field. Then we provide the existence of a unique weak solution to the model. The proof is based on arguments of evolutionary variational inequalities and fixed point.     

James sequences and Dependent Choices

We prove James's sequential characterization of (compact) reflexivity in set‐theory ZF + DC , where DC is the axiom of Dependent Choices. In turn, James's criterion implies that every infinite set is Dedekind‐infinite, whence it is not provable in ZF . Our proof in ZF + DC of James' criterion leads us to various notions of reflexivity which are equivalent in ZFC but are not equivalent in ZF . We also show that the weak compactness of the closed unit ball of a (simply) reflexive space does not imply the Boolean Prime Ideal theorem : this solves a question raised in [6]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)