Notions of Input to Output Stability

This paper deals with several related notions of output stability with respect to inputs. The inputs may be thought of as disturbances; when there are no inputs, one obtains generalizations of the classical concepts of partial stability. The main notion studied is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions. A companion paper establishes necessary and sufficient Lyapunov-theoretic characterizations.Comment: 16 pages See http://www.math.rutgers.edu/~sontag/ for many related paper

Upper and Lower Bounds in Relator Spaces

2000 Mathematics Subject Classification: 06A06, 54E15An ordered pair X(R) = ( X, R ) consisting of a nonvoid set X and a nonvoid family R of binary relations on X is called a relator space. Relator spaces are straightforward generalizations not only of uniform spaces, but also of ordered sets. Therefore, in a relator space we can naturally define not only some topological notions, but also some order theoretic ones. It turns out that these two, apparently quite different, types of notions are closely related to each other through complementations.The research of the author has been supported by the grant OTKA T-030082

Lucas' theorem: its generalizations, extensions and applications (1878--2014)

In 1878 \'E. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and Combinatorics. In this article, consisting of six sections, we provide a historical survey of Lucas type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas like theorems for some generalized binomial coefficients, and some their applications. In Section 1 we present the fundamental congruences modulo a prime including the famous Lucas' theorem. In Section 2 we mention several known proofs and some consequences of Lucas' theorem. In Section 3 we present a number of extensions and variations of Lucas' theorem modulo prime powers. In Section 4 we consider the notions of the Lucas property and the double Lucas property, where we also present numerous integer sequences satisfying one of these properties or a certain Lucas type congruence. In Section 5 we collect several known Lucas type congruences for some generalized binomial coefficients. In particular, this concerns the Fibonomial coefficients, the Lucas $u$-nomial coefficients, the Gaussian $q$-nomial coefficients and their generalizations. Finally, some applications of Lucas' theorem in Number Theory and Combinatorics are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to Lucas' theore

Turing jumps through provability

Fixing some computably enumerable theory $T$, the Friedman-Goldfarb-Harrington (FGH) theorem says that over elementary arithmetic, each $\Sigma_1$ formula is equivalent to some formula of the form $\Box_T \varphi$ provided that $T$ is consistent. In this paper we give various generalizations of the FGH theorem. In particular, for $n>1$ we relate $\Sigma_{n}$ formulas to provability statements $[n]_T^{\sf True}\varphi$ which are a formalization of "provable in $T$ together with all true $\Sigma_{n+1}$ sentences". As a corollary we conclude that each $[n]_T^{\sf True}$ is $\Sigma_{n+1}$-complete. This observation yields us to consider a recursively defined hierarchy of provability predicates $[n+1]^\Box_T$ which look a lot like $[n+1]_T^{\sf True}$ except that where $[n+1]_T^{\sf True}$ calls upon the oracle of all true $\Sigma_{n+2}$ sentences, the $[n+1]^\Box_T$ recursively calls upon the oracle of all true sentences of the form $\langle n \rangle_T^\Box\phi$. As such we obtain a `syntax-light' characterization of $\Sigma_{n+1}$ definability whence of Turing jumps which is readily extended beyond the finite. Moreover, we observe that the corresponding provability predicates $[n+1]_T^\Box$ are well behaved in that together they provide a sound interpretation of the polymodal provability logic ${\sf GLP}_\omega$