###### Triposes, exact completions, and Hilbert's ε-operator

note on relations relative to a factorization system. In A. Carboni, M.C. Pedicchio, and G. Rosolini, editors, Category Theory '90, volume 1488 of Lecture Notes in Math., pages 249-261. Springer-Verlag, Como, 1992. [Law69a] F. W. Lawvere. Adjointness in foundations. Dialectica, 23:281{296, 1969. [Law69b] F.W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two), pages 134-145. Springer, 1969

###### Introduction. A Personal Tribute to Peter Freyd and Bill Lawvere

Lecture Notes in Mathematics 61, Springer-Verlag (1968), 41-61. [24] F. W. Lawvere, Ordinal Sums and Equational Doctrines, Lecture Notes in Mathematics 80, Springer-Verlag (1969), 141-155. [25] F. W. Lawvere, Diagonal Arguments and Cartesian Closed Categories, Springer Lecture Notes in Mathematics 92, Springer-Verlag (1969), 134-145. [26] F. W. Lawvere, Quantiffers and Sheaves, Actes du Congrffes International des Mathffematiciens Nice 1970, 229-334. [27] F. W. Lawvere, Metric Spaces, Generalized Logic, and

###### An efficient computational method based on the hat functions for solving fractional optimal control problems

, Commun Nonlinear Sci Numer Simulat, vol. 25 (2015), pp. 27-40. [16] Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Struct Multidisc Optim, vol. 38 (2009), pp. 571-581. [17] A. Lotfi and S. A. Yousefi, Epsilon-ritz method for solving a class of fractional constrained optimization problems, J. Optim Theory Appl, vol. 163 (2014), pp. 884-899. [18] F. Jarad, T. Abdeljawad and D. Baleanu, Fractional variational optimal control problems with delayed arguments, m Nonlinear Dyn, vol. 62 (2010), pp. 609

###### Combinatorial proofs of some Stirling number formulas

## Abstract

In this note, we provide bijective proofs of some recent identities involving Stirling numbers of the second kind, as previously requested. Our arguments also yield generalizations in terms of a well known q-Stirling number.