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The category ${STROP}_m$ of supertropical monoids, whose morphisms are transmissions, has the full-reflective subcategory $STROP$ of commutative semirings. In this setup, quotients are determined directly by equivalence relations, as ideals are not applicable for monoids, leading to a new approach to factorization theory. To this end, tangible factorization into irreducibles is obtained through fiber contractions and their hierarchy. Fiber contractions also provide different quotient structures, associated with covers and types of splitting covers.

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field ${𝒦}=\mathbb{Q}({\varsigma }_8)$ over $\mathbb{Q}$. We employ binary codes to generate a lattice over $\mathbb{Q}({\varsigma }_8)$, which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels.

Proxy Re-Encryption (PRE) is a cryptographic primitive that allows a proxy to turn an Alice’s ciphertext into a Bob’s ciphertext on the same plaintext. All of the PRE schemes are public key encryption and semantic security. Deterministic Public Key Encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios where the latter exhibits inherent drawbacks. In this paper, we construct the first multi-use unidirectional D-PRE scheme from Lattices in the auxiliary-input setting. We also prove that it is PRIV1-INDr secure in the standard model based on the LWR. Finally, an identity-based D-PRE is obtained from the basic construction.

Based on the study of the dynamics of a dissipation-modified Toda anharmonic (one-dimensional, circular) lattice ring we predict here a new form of electric conduction mediated by dissipative solitons. The electron-ion-like interaction permits the trapping of the electron by soliton excitations in the lattice, thus leading to a soliton-driven current much higher than the Drude-like (linear, Ohmic) current. Besides, as we lower the values of the externally imposed field this new form of current survives, with a field-independent value.

Assuming the quantum mechanical "tight binding" of an electron to a nonlinear lattice with Morse potential interactions we show how electric conduction can be mediated by solitons. For relatively high values of an applied electric field the current follows Ohm's law. As the field strength is lowered the current takes a finite, constant, field-independent value.

We first summarize features of free, forced and stochastic harmonic oscillations and, following an idea first proposed by Lord Rayleigh in 1883, we discuss the possibility of maintaining them in the presence of dissipation. We describe how phonons appear in a harmonic (linear) lattice and then use the Toda exponential interaction to illustrate solitonic excitations (cnoidal waves) in a one-dimensional nonlinear lattice. We discuss properties such as specific heat (at constant length/volume) and the dynamic structure factor, both over a broad range of temperature values. By considering the interacting Toda particles to be Brownian units capable of pumping energy from a surrounding heat bath taken as a reservoir we show that solitons can be excited and sustained in the presence of dissipation. Thus the original Toda lattice is converted into an active lattice using Lord Rayleigh's method. Finally, by endowing the Toda–Brownian particles with electric charge (i.e. making them positive ions) and adding free electrons to the system we study the electric currents that arise. We show that, following instability of the base linear Ohm(Drude) conduction state, the active electric Toda lattice is able to maintain a form of high-T supercurrent, whose characteristics we then discuss.

This work provides a description of the main computational tools for the study of discrete breathers. It starts with the observation of breathers through simple numerical runs, the study uses targeted initial conditions, and discrete breather impact on transient processes and thermal equilibrium. We briefly describe a set of numerical methods to obtain breathers up to machine precision. In the final part of this work we apply the discussed methods to study the competing length scales for breathers with purely anharmonic interactions — favoring superexponential localization — and long range interactions, which favor algebraic decay in space. As a result, we observe and explain the presence of three different spatial tail characteristics of the considered localized excitations.

At the 2011 Durham Conference "Geometry and Arithmetic of Lattices" M. Kapovich formulated the following

Question. Does there exist an embedding ℤ² * ℤ ↪ SL(3, ℤ)?

The goal of the paper is to prove the following

Main Theorem.If p and m are arbitrary positive integers then there exists an embeddingℤ² * F_m ↪ SL(3, ℤ[1/p]).

Every isometry of a finite-dimensional Euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let $(X,\le ,c)$ be a finite poset, where $c:X\to X$ is an order-reversing and involutive map such that $c(x)\ne x$ for each $x\in X$. Let $B_2=\{N<P\}$ be the Boolean lattice with two elements and ${{𝒲}}_{+}(X,B_2)$ the family of all the order-preserving 2-valued maps $A:X\to B_2$ such that $A(c(x))=P$ if $A(x)=N$ for all $x\in X$. In this paper, we build a family ${{ℬ}}_{w+}(X)$ of particular subsets of $X$, that we call $w_{+}$-bases on $X$, and we determine a bijection between the family ${{ℬ}}_{w+}(X)$ and the family ${{𝒲}}_{+}(X,B_2)$. In such a bijection, a $w_{+}$-basis $\mathrm{\Omega }$ on $X$ corresponds to a map $A\in {{𝒲}}_{+}(X,B_2)$ whose restriction of $A$ to $\mathrm{\Omega }$ is the smallest 2-valued partial map on $X$ which has $A$ as its unique extension in ${{𝒲}}_{+}(X,B_2)$. Next we show how each $w_{+}$-basis on $X$ becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Several structural properties of an algebraic structure can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In this paper, we exhibit finite representations of this sequence for finite algebras from congruence modular varieties.

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term $a$ which is absorbent for addition. We study the problem of enumerating all finite common meadows of order n (that is, common meadows with n elements). This problem turns out to be deeply connected with both the number of finite rings of order n and with the number of a certain kind of partition of positive integers.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

The symbolic level of a dynamic scene interpretation system is presented. This symbolic level is based on plan prototypes represented by Petri nets whose interpretation is expressed thanks to 1st order cubes, and on a reasoning aiming at instantiating the plan prototypes with objects delivered by the numerical processing of sensor data. A purely symbolic meta-structure, grounded on the lattice theory, is then proposed to deal with the symbolic uncertainty issues. Examples on real world data are given.

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

An algebraic relational theory is being developed in order to represent biological systems. As a result, it is possible to explain, in terms of qualitative relationships, the behaviors of such systems. This paper deals with the periodic continuous responses of a new state derived from the interaction between low energies and matter. This effect was predicted by categoric developments of the algebraic relational theory.

The property of muscle movement P_m is a central functional property for the qualitative interpretations of behaviors of biological systems. In this paper, based on the example of the muscle, it is shown how relational properties can be identified by responses derived from the functional organization. At a certain level of organization, the manifestation of the property P_m is analyzed in terms of qualitative relationships between the myosin and actin fibres of muscles. A relatively pseudo complemented lattice of nine elements shows algebraic relations in connection with the interaction of the fibres. The response coming from the lattice is in correspondence with the quantitative one expressed by Hill’s equation for muscles.

It is shown how the biological reality correlates with an algebraic modification from a pseudo-Boolean structure for watering processes in normal cells up to a non-modular structure assigned to water interactions in malignant cells. A set of mathematical propositions suggests how to deviate this type of cancer process to new structures mainly maintaining those water structures resulting from the cooperativity between water molecules generated by a surface. A set of disquisitions is made: about the meaning of the change of algebra; on the dual Heyting arrow operations acting for the algebraic triggering of the cancer process; on the loss of energy in the cancer process and about the enhanced value of energy as becoming from the new structures to deviate cancer.

An interesting sequence of arrays on hierarchical hexagonal grids was employed by PYXIS Innovation Inc. for an efficient digital earth model. These arrays constitute a Cauchy sequence in terms of Hausdorff distance. In this paper, we show that the box-counting dimension for the boundary of the limit of this sequence is .

The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself, with operations certain convolutions of these mappings with respect to pointwise max and min. This algebra generalizes the truth-value algebras of both type-1 and of interval-valued fuzzy sets, and has been studied rather extensively both from a theoretical and applied point of view. This paper addresses the situation when the unit interval is replaced by two finite chains. Most of the basic theory goes through, but there are several special circumstances of interest. These algebras are of interest on two counts, both as special cases of bases for fuzzy theories, and as mathematical entities per se.