In this paper, we tried to accelerate the rate of convergence in solving second-order Fredholm type Integro-differential equations using a new method which is based on Improved homotopy perturbation method (IHPM) and applying accelerating parameters. This method is very simple and the result is obtained very fast.  

We show that universes of fibrations in various models of homotopy type theory have an essentially global character: they cannot be described in the internal language of the presheaf topos from which the model is constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct ...

We give an introduction to higher dimensional or homotopy type theory, trying to provide a comprehensive overview on this research direction that combines type theory, algebraic topology and higher category theory. Further, we describe the content and the results of several other unrelated, smaller research projects. This report presents the topics the author has worked on during the first year...

It is sometimes convenient or useful in mathematics to treat isomorphic structures as the same. The recently proposed Univalence Axiom for the foundations of mathematics elevates this idea to a foundational principle in the setting of Homotopy Type Theory. It states, roughly, that isomorphic structures can be identified. We explore the motivations and consequences, both mathematical and philoso...

We generalize the adjoint logics of Benton and Wadler (1996) and Reed (2009) to allow multiple different adjunctions between the same categories. This provides insight into the structural proof theory of cohesive homotopy type theory, which integrates the synthetic homotopy theory of homotopy type theory with the synthetic topology of Lawvere’s axiomatic cohesion. Reed’s calculus is parametrize...

Quillen [16] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success and — as, e.g., the work of Voevodsky on the homotopy theory of schemes [14] or the work of Joyal [10, 11] and Lurie [12] on quasicategories seems to indicate — it will likely continue to facilitate math...