Abstract. We introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. On the combinatorial side, we study the rotation lattices on permutrees and their lattice homomorphisms, unifying the weak order, Tamari, Cambrian and boolean lattices and the classical maps between them. On the geometric side, we provide both the vertex and facet descriptions...

Resolution of singularities is one of the central areas of research in Algebraic Geometry. It is a basic prerequisite for the classification of algebraic varieties up to birational equivalence, since it allows to consider only regular varieties. Hironaka’s monumental work [Hi1] gave a non-constructive, existence proof of resolution of singularities over fields of characteristic zero. Constructi...

Algebraic attack has recently become an important tool in cryptanalysing different stream and block cipher systems. A Boolean function, when used in some cryptosystem, should be designed properly to resist this kind of attack. The cryptographic property of a Boolean function, that resists algebraic attack, is known as Algebraic Immunity (AI). So far, the attempt in designing Boolean functions w...

We describe a type system for the linear-algebraic λ-calculus. The type system accounts for the linear-algebraic aspects of this extension of λ-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. ...

In this paper we demonstrate how to use Mixed Integer Linear Programming to optimize guessing strategies for algebraic cryptanalysis with applications to the block cipher EPCBC. Using our optimized guessing strategy we are able to attack 5 rounds of EPCBC96 and 8 rounds of EPCBC-48 faster than brute force using one and two known plaintexts resp. Finally, we are able to identify a class of weak ...

in this paper we study two important concepts, i.e. the direct andthe inverse limit of hyperstructures associated with fuzzy sets of type 2, andshow that the direct and the inverse limit of hyperstructures associated withfuzzy sets of type 2 are also hyperstructures associated with fuzzy sets of type 2.

In this paper, we study a generalization of z-ideals in the ring C(X) of continuous real valued functions on a completely regular Hausdorff space X. The notion of a weak ideal and naturally a weak z-ideal and a prime weak ideal are introduced and it turns out that they behave such as z-ideals in C(X).

We discuss the structure of the recursively enumerable sets under three reducibilities: Turing, truth-table and weak truth-table. Weak truth-table reducibility requires that the questions asked of the oracle be effectively bounded. Truth-table reducibility also demands such a bound on the the length of the computations. We survey what is known about the algebraic structure and the complexity of...

In Part I of this thesis, we introduce algebraic model structures, a new context for homotopy theory in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads and prove “algebraic” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that every cofibrantly generated model structure in the usual sense un...

We establish a local boundedness estimate for weak subsolutions to doubly nonlinear parabolic fractional p-Laplace equation. Our argument relies on energy estimates and nonlocal version of De Giorgi’s method. Furthermore, by means new algebraic inequality, we show that positive supersolutions satisfy reverse Hölder inequality. Finally, also prove logarithmic decay supersolutions.