Abstract:
We show that in the functional integral formalism of U(1) gauge field theory some formal manipulation such as interchange of order of integration can yield erroneous results. The example studied is analysed by Fubini theorem.

Abstract:
A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non-linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular prior knowledge of projective techniques is not assumed. (For the special JMP issue on Functional Integration, edited by C. DeWitt-Morette.)

Abstract:
The geometric construction of the functional integral over coset spaces ${\cal M}/{\cal G}$ is reviewed. The inner product on the cotangent space of infinitesimal deformations of $\cal M$ defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber $\cal G$, the functional measure on the coset space ${\cal M}/{\cal G}$ is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where $\cal G$ is the group of coordinate reparametrizations of spacetime, the continuum functional integral over geometries, {\it i.e.} metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov-Liouville action of closed bosonic non-critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed.

Abstract:
dyspepsia and functional dyspepsia represent a highly significant public health issue. a good definition of dyspepsia is key for helping us to better approach symptoms, decision making, and therapy indications. during the last few years many attempts were made at establishing a definition of dyspepsia. results were little successful on most occasions, and clear discrepancies arose on whether symptoms should be associated with digestion, which types of symptoms were to be included, which anatomic location should symptoms have, etc. the rome iii committee defined dyspepsia as "a symptom or set of symptoms that most physicians consider to originate from the gastroduodenal area", including the following: postprandial heaviness, early satiety, and epigastric pain or burning. two new entities were defined: a) food-induced dyspeptic symptoms (postprandial distress syndrome); and b) epigastric pain (epigastric pain syndrome). these and other definitions have shown both strengths and weaknesses. at times they have been much too complex, at times much too simple; furthermore, they have commonly erred on the side of being inaccurate and impractical. on the other hand, some (the most recent ones) are difficult to translate into the spanish language. in a meeting of gastroenterologists with a special interest in digestive functional disorders, the various aspects of dyspepsia definition were discussed and put to the vote, and the following conclusions were arrived at: dyspepsia is defined as a set of symptoms, either related or unrelated to food ingestion, localized on the upper half of the abdomen. they include: a) epigastric discomfort (as a category of severity) or pain; b) postprandial heaviness; and c) early satiety. associated complaints include: nausea, belching, bloating, and epigastric burn (heartburn). all these must be scored according to severity and frequency. furthermore, psychological factors may be involved in the origin of functional dyspepsia. on the other hand

Abstract:
This paper is an exposition of the relationship between Witten's Chern-Simons functional integral and the theory of Vassiliev Invariants of knots and links in three dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed. We thank the organizers of the Conference on 60 Years of Yang-Mills Gauge Field Theories (25 to 28 May 2015) for the invitation and opportunity to speak about these ideas in Singapore.

Abstract:
We give a precise definition and produce a path-integral computation of the normalized partition function of the abelian U(1) Chern-Simons field theory defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism, we sum over the inequivalent U(1) principal bundles over the manifold and, for each bundle, we integrate over the gauge orbits of the associated connection 1- forms. The result of the functional integration is compared with the abelian U(1) Reshetikhin-Turaev surgery invariant.

Abstract:
Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the holonomies of $A$ along finitely many piecewise smoothly immersed curves in $M$, and let a generalized measure on $\A$ be a bounded linear functional on cylinder functions. We construct a generalized measure on the space of connections that extends the uniform measure of Ashtekar, Lewandowski and Baez to the smooth case, and prove it is invariant under all automorphisms of $P$, not necessarily the identity on the base space $M$. Using `spin networks' we construct explicit functions spanning the corresponding Hilbert space $L^2(\A/\G)$, where $\G$ is the group of gauge transformations.

Abstract:
A summary of the known results on integration theory on the space of connections modulo gauge transformations is presented and its significance to quantum theories of gauge fields and gravity is discussed. The emphasis is on the underlying ideas rather than the technical subtleties.

Abstract:
The functional integration scheme for path integrals advanced by Cartier and DeWitt-Morette is extended to the case of fields. The extended scheme is then applied to quantum field theory. Several aspects of the construction are discussed.