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Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $\Pi^0_1$-ordinal, characterizing the $\Pi^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it…

There are two major generalizations of the standard ordinal analysis: One is Girard's $\Pi^1_2$-proof theory in which dilators are assigned to theories instead of ordinals. The other is Pohlers' generalized ordinal analysis with Spector…

We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions

In 1984, Wim Ruitenburg published a surprising result about periodic sequences in intuitionistic propositional calculus (IPC). The property established by Ruitenburg naturally generalizes local finiteness; recall that intuitionistic logic…

In this paper we extend Fischler's quantitative generalization of Nesterenko's linear independence criterion, by weakening the hypotheses on the divisors of the coe cients of the linear forms and allowing (to some extent) the linear forms…

We give a new independent proof of a generalised version of the theorem by Rashevskii, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83 -- 94] and from which the classical Chow-Rashevskii Theorem follows as a corollary. The proof…

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group…

We consist of first presenting Zeckendorf Theorem with these two versions Fibonacci and Luca. In this document we obtain results on the generalized of the Zeckendorf theorem for Fibonacci numbers (multibonacci). Such results find…

We assumed that, for every natural number k, there is a natural number u such that the (k-1)th term of G(u) is k^k, and that G(u) terminates finitely. It immediately follows that every Goodstein Sequence G(m) over the natural numbers must…

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal…

Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some…

We establish a general concentration result for the 1-Wasserstein distance between the empirical measure of a sequence of random variables and its expectation. Unlike standard results that rely on independence (e.g., Sanov's theorem) or…

We analyze several natural Goodstein principles which themselves are defined with respect to the Ackermann function and the extended Ackermann function. These Ackermann functions are well established canonical fast growing functions labeled…

This is the second combinatorial proof of the compactness theorem for singular from 1977. In fact it gives a somewhat stronger theorem.

Gleason's theorem [A. Gleason, J. Math. Mech., \textbf{6}, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it…

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their…

Generalized Higman's Theorem is the direct counterpart of Higman's Theorem that asserts the closure of the class of \emph{better} quasi-orders, instead of the class of \emph{well} quasi-orders, under the construction $P\mapsto P^{<\omega}$…

We generalize and prove a result which was first shown by Zippin, and was explicitly formulated by Benyamini.

This note is an (exact) copy of the report of Jaak Peetre, "Generalizing Ovchinnikov's Theorem". Published as Technical Report, Lund (1981). Some more recent general references have been added, some references updated though (in italics)…

In this paper we prove a version of Lie-B\"acklund theorem for overdetermined systems of scalar PDEs, whose general solution depends on 1 function of 1 variable. This generalizes the case of involutive system of the second order on the…