Exponential Objects

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].Via del Pero 102, 54038 Montignoso, ItalyJiri Adamek, Horst Herrlich, and George E. Strecker. Abstract and Concrete Categories: The Joy of Cats. Dover Publication, New York, 2009.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Francis Borceaux. Handbook of Categorical Algebra I. Basic Category Theory, volume 50 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1994.Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1 (2):409–420, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.F. William Lawvere. Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. Reprints in Theory and Applications of Categories, 5:1–121, 2004.Saunders Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer Verlag, New York, Heidelberg, Berlin, 1971.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.Marco Riccardi. Object-free definition of categories. Formalized Mathematics, 21(3): 193–205, 2013. doi:10.2478/forma-2013-0021. [Crossref]Marco Riccardi. Categorical pullbacks. Formalized Mathematics, 23(1):1–14, 2015. doi:10.2478/forma-2015-0001. [Crossref]Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Andrzej Trybulec. Isomorphisms of categories. Formalized Mathematics, 2(5):629–634, 1991.Andrzej Trybulec. Natural transformations. Discrete categories. Formalized Mathematics, 2(4):467–474, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990

Mathematical models of avascular cancer

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

Wave Front Sets of Reductive Lie Group Representations

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $\tau$ is a unitary representation of $H$, then the authors give a sufficient condition on $\xi\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^G\tau$. In the special case where $\tau$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $\pi$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(\pi)$ in terms of the direct integral decomposition of $\pi$ into irreducibles. Special cases of this result were previously obtained by Kashiwara-Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.Comment: Accepted to Duke Mathematical Journa

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure