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Full-Text Articles in Mathematics
Lagrange's Theory Of Analytical Functions And His Ideal Of Purity Of Method, Giovanni Ferraro, Marco Panza
Lagrange's Theory Of Analytical Functions And His Ideal Of Purity Of Method, Giovanni Ferraro, Marco Panza
MPP Published Research
We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of …
Relation Liftings On Preorders And Posets, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jiří Velebil
Relation Liftings On Preorders And Posets, Marta Bílková, Alexander Kurz, Daniela Petrişan, Jiří Velebil
Engineering Faculty Articles and Research
The category Rel(Set) of sets and relations can be described as a category of spans and as the Kleisli category for the powerset monad. A set-functor can be lifted to a functor on Rel(Set) iff it preserves weak pullbacks. We show that these results extend to the enriched setting, if we replace sets by posets or preorders. Preservation of weak pullbacks becomes preservation of exact lax squares. As an application we present Moss’s coalgebraic over posets.
Generic Trace Logics, Christian Kissig, Alexander Kurz
Generic Trace Logics, Christian Kissig, Alexander Kurz
Engineering Faculty Articles and Research
We combine previous work on coalgebraic logic with the coalgebraic traces semantics of Hasuo, Jacobs, and Sokolova.
Towards Nominal Formal Languages, Alexander Kurz, Tomoyuki Suzuki, Emilio Tuosto
Towards Nominal Formal Languages, Alexander Kurz, Tomoyuki Suzuki, Emilio Tuosto
Engineering Faculty Articles and Research
We introduce formal languages over infinite alphabets where words may contain binders.We define the notions of nominal language, nominal monoid, and nominal regular expressions. Moreover, we extend history-dependent automata (HD-automata) by adding stack, and study the recognisability of nominal languages.
A Class Of Gaussian Processes With Fractional Spectral Measures, Daniel Alpay, Palle Jorgensen, David Levanony
A Class Of Gaussian Processes With Fractional Spectral Measures, Daniel Alpay, Palle Jorgensen, David Levanony
Mathematics, Physics, and Computer Science Faculty Articles and Research
We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures σ (generalized spectral measures), and our focus here is on the case when the measure σ is a singular measure. We characterize the processes arising from when σ is in one of the classes of affine self-similar measures. Our analysis makes use of Kondratiev-white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen (see Theorem 7.1) earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely …
The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, Daniel Alpay, Izchak Lewkowicz
The Positive Real Lemma And Construction Of All Realizations Of Generalized Positive Rational Functions, Daniel Alpay, Izchak Lewkowicz
Mathematics, Physics, and Computer Science Faculty Articles and Research
We here extend the well known Positive Real Lemma (also known as the Kalman-Yakubovich-Popov Lemma) to complex matrix-valued generalized positive rational function, when non-minimal realizations are considered. All state space realizations are partitioned into subsets, each is identified with a set of matrices satisfying the same Lyapunov inclusion. Thus, each subset forms a convex invertible cone, cic in short, and is in fact is replica of all realizations of positive functions of the same dimensions. We then exploit this result to provide an easy construction procedure of all (not necessarily minimal) state space realizations of generalized positive functions. As a …