This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary rst step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formula, the connection between integral and di erential forms, conservation laws, E...

The logic of common belief does not always re‡ect that of individual beliefs. In particular, even when the individual belief operators satisfy the KD45 logic, the common belief operator may fail to satisfy axiom 5. That is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. We identify the intersubjective restrictions on individual beliefs...

Moerdijk has introduced a topos of sheaves on a category of lters. Following his suggestion, we prove that its double negation subtopos is the topos of sheaves on the subcategory of ultra lters the ultrasheaves. We then use this result to establish a double negation translation of results between the topos of ultrasheaves and the topos on lters. 2000 Mathematics Subject Classi cation. Primary 0...

Received March 12, 2011. 2010 Mathematics Subject Classification. 35J48, 35J60, 47J30, 58E05.

Tuncay Aktosun Department of Mathematics North Dakota State University Fargo, ND 58105 Martin Klaus Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 24061 Cornelis van der Mee Department of Mathematics University of Cagliari Cagliari, Italy Abstract: In this review paper, the generalized Schr odinger equation d2 =dx2 + k2 = [ikP (x) +Q(x)] is conside...

We present a nonstandard characterization of connected compact sets Mathematics Subject Classification 2010: 26E35, 54D05, 54D30.

We obtain new inequalities with alternating signs of Hölder and Minkowski type. Mathematics subject classification (2010): 26D15.

Received: September 7, 2013; Accepted: October 28, 2013 2010 Mathematics Subject Classification: Primary 46A45, 54E35; Secondary 40C05, 15B05.

Received March 26, 2015, accepted June 1, 2015. Communicated by Yingfei Yi. 2010 Mathematics Subject Classification: 35L70, 35B41.