Let [Formula: see text] be a set of states physical system. The probabilities the occurrence an event when system is in different define function from to called numerical or, more accurately, text]-probability. Sets text]-probabilities ordered by partial order functions give rise so-called algebras text]-probabilities, particular, ones that are lattice-ordered. Among these, there text]-algebras...

Lattice structures and their characterization are much discussed in current research. The complex manufacturing process of laser powder bed fusion with its repeated melting remelting metallic influences the resulting material properties. Within this contribution influence parameters on overall behaviour lattice structure compression test specimens is inherently described within model which deri...

In the standard approach to lattice proteins the models based on nearest neighbor interaction are used. In this kind of models it is difficult to explain the existence of secondary structures — special preferred conformations of protein chains. In the present paper a new lattice model of proteins is proposed which is based on non-local cooperative interactions. In this model the energy of a con...

Kirigami uses bending, folding, cutting, and pasting to create complex three-dimensional (3D) structures from a flat sheet. In the case of lattice kirigami, this cutting and rejoining introduces defects into an underlying 2D lattice in the form of points of nonzero Gaussian curvature. A set of simple rules was previously used to generate a wide variety of stepped structures; we now pare back th...

In this paper, we study the discrete cubic nonlinear Schrödinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Bäcklund-Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent in...

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...

By enumerating all sequences of length 20, we study the designability of structures in a twodimensional Hydrophobic-Polar (HP) lattice model in a wide range of inter-monomer interaction parameters. We find that although the histogram of designability depends on interaction parameters, the set of highly designable structures is invariant. So in the HP lattice model the High Designability should ...