The presence of multiplicative noise can alter measurements of forces acting on nanoscopic objects. Taking into account of multiplicative noise, we derive a series of nonequilibrium thermodynamical equalities as generalization of the Jarzynski equality, the detailed fluctuation theorem and the Hatano-Sasa relation. Our result demonstrates that the Jarzynski equality and the detailed fluctuation...

The equality in the uncertainty principle for linear momentum and position is obtained for states which also minimize the uncertainty product. However, in the uncertainty relation for angular momentum and angular position both sides of the inequality are state dependent and therefore the intelligent states, which satisfy the equality, do not necessarily give a minimum for the uncertainty produc...

This paper gives the first proof that the subtyping relation of a higherorder lambda calculus, F ≤, is anti-symmetric, establishing in the process that the subtyping relation is a partial order—reflexive, transitive, and anti-symmetric up to β-equality. While a subtyping relation is reflexive and transitive by definition, anti-symmetry is a derived property. The result, which may seem obvious t...

Equality in first-order logic is represented in the language by means of a logical symbol and so it is interpreted uniformly; the semantics says that its interpretation in any structure is the identity relation. This is the absolute notion of identity however, and in fact the only one deserving this name. But there is a related relation: the relation of indiscernibility. Given a structure for a...

In Western culture, there appears to be widespread endorsement of Article 1 of the Universal Declaration of Human Rights (which stresses equality and freedom). But do people really apply their equality values equally, or are their principles and application systematically discrepant, resulting in equality hypocrisy? The present study, conducted with a representative national sample of adults in...

In type theory a proposition is represented by a type, the type of its proofs. As a consequence the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality, which we also call identity, is induc...

For example, consider a structure with domain (A,B,C), for a language including just the equality predicate = and constants {a, b, c}, which interprets a, b, c as A,B,C respectively and = as the relation {(A,B), (B,A), (A,A), (B,B), (C,C)}. i.e. = (A,B),= (B,A) etc. are all true and other atoms involving =, such as = (B,C), are false. This is a model of the equality axioms (show it !) and it al...

The equality on the inverval I is the equality in the free bounded distributive lattice on generators i, 1− i. The equality in the face lattice F is the one for the free distributive lattice on formal generators (i = 0), (i = 1) with the relation (i = 0) ∧ (i = 1) = 0. We have [(r ∨ s) = 1] = (r = 1) ∨ (s = 1) and [(r∧s) = 1] = (r = 1)∧ (s = 1). An irreducible element of this lattice is a face,...

The in-ation GI of a graph G is the line graph of the subdivision of G. If G is a complete graph the equality ir(GI) = (GI) was proved by Favaron in 1998. We conjectured that the equality holds when G is any graph of radius 1. But it turned out that it is not true. Moreover, we proved that for the class of radius 1 graphs there does not exist a better upper bound for the relation (GI)=ir(GI) th...

Answering a question of Kechris, I show that the equality of countable sets of reals is not the smallest unpinned equivalence relation.