Restricted Algebras Research Articles

Algebraic Restriction Codes and Their Applications

Consider the following problem: You have a device that is supposed to compute a linear combination of its inputs, which are taken from some finite field. However, the device may be faulty and compute arbitrary functions of its inputs. Is it possible to encode the inputs in such a way that only linear functions can be evaluated over the encodings? I.e., learning an arbitrary function of the encodings will not reveal more information about the inputs than a linear combination. In this work, we introduce the notion of algebraic restriction codes (AR codes), which constrain adversaries who might compute any function to computing a linear function. Our main result is an information-theoretic construction AR codes that restrict any class of function with a bounded number of output bits to linear functions. Our construction relies on a seed which is not provided to the adversary. While interesting and natural on its own, we show an application of this notion in cryptography. In particular, we show that AR codes lead to the first construction of rate-1 oblivious transfer with statistical sender security from the Decisional Diffie–Hellman assumption, and the first-ever construction that makes black-box use of cryptography. Previously, such protocols were known only from the LWE assumption, using non-black-box cryptographic techniques. We expect our new notion of AR codes to find further applications, e.g., in the context of non-malleability, in the future.

γ– algebra of Sets and Some of its Properties

The main objective of this work is to generalize the concept of fuzzy algebra by introducing the notion of fuzzy algebra. Characterization and examples of the proposed generalization are presented, as well as several different properties of fuzzy algebra are proven. Furthermore, the relationship between fuzzy algebra and fuzzy algebra is studied, where it is shown that the fuzzy algebra is a generalization of fuzzy algebra too. In addition, the notion of restriction, as an important property in the study of measure theory, is studied as well. Many properties of restriction of a nonempty family of fuzzy subsets of fuzzy power set are investigated and it is shown that the restriction of fuzzy algebra is fuzzy algebra too.

A Shadow Perspective on Equivariant Hochschild Homologies

Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $C_n$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $K$-theory to twisted topological Hochschild homology.

Impact of the inclusion of the sociological position and upper-stem diameter in the taper modeling of Pinus taeda trees, subjected to different thinning regimes

The modeling of the taper and shape of the tree stem is of great interest to forest managers and can be carried out using different approaches. The objective of this study was to improve the accuracy of taper modeling, considering a methodology that combines generalization and calibration with an upper-stem diameter. An experiment was conducted with stands of Pinus taeda from southern Brazil that were unthinned and crown thinned at different levels. The destructive sampling was performed at age 30, accumulating the effect of the management regime over time. The model was developed in three parts: a) the Kozak variable exponent model was selected; b) the inclusion of variables was tested (sociological position, relative spacing index, thinning index, uniformity index, and site) in the exponent of the Kozak model of part “a”, allowing generalizations; c) upper-stem diameters at heights of 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 m were tested to calibrate the part "b" model, using the algebraic restriction method. The inclusion of the sociological position in the model exponent and the calibration with the upper-stem diameter at the stem height of 6 m reduced the RMSE and the MAE by 3%, with respect to volume (m³). The reduction of the error was concentrated mainly in the lower-stem section, considered the most economically valuable part. The methodology developed uses information available from a forest inventory to compute the sociological position of the trees and from harvesting for a single upper-stem diameter. Although the equation has been developed for unthinned and crown thinned stands at different intensities of loblolly pine (Pinus taeda), the extrapolation potential of the methodological approach is valid, making implementation feasible for other species and management alternatives.

Realizability of the rapid distortion theory spectrum: The mechanism behind the Kelvin–Townsend equations

In the context of homogeneous turbulence, we prove that the Rapid Distortion Theory (RDT) model for the spectral tensor preserves the symmetry, positive semidefiniteness, and integrability properties required in Cramér’s characterization of the spectral tensor of a continuous homogeneous random process. From this, a statistically valid correlation tensor is obtained that returns a Reynolds stress tensor model that satisfies realizability conditions. The number of hypotheses used is kept to a minimum, which allows a flexible use of the model in the applications. The Kelvin–Townsend equations allow us to construct the solution and prove its properties by means of a factorization approach. Since the RDT spectral tensor model is a system of transport equations plus an algebraic restriction due to incompressibility, we deal with the existence, uniqueness, and persistence of solutions in a specific set of functions by using DiPerna–Lions renormalization techniques.

Algebraic restrictions on geometric realizations of curvature models

Algebraic restrictions on geometric realizations of curvature models

λ–Algebra with Some of Their Properties

The objective of this paper is, firstly, we study a new concept noted by algebra and discuss the properties of this concept. Secondly, we introduce a new concept related to the algebra such as smallest algebra. Thirdly, we introduce the notion of the restriction of algebra on a nonempty subset of and investigate some of its basic properties. Furthermore, we present the relationships between field, monotone class, field and algebra. Finally, we introduce the concept of measure relative to the algebra and prove that every measure relative to the is complete.

Vaisman solvmanifolds and relations with other geometric structures

Fil: Andrada, Adrian Marcelo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina. Universidad Nacional de Cordoba. Facultad de Matematica, Astronomia y Fisica; Argentina

Probabilistic logic over equations and domain restrictions

AbstractWe propose and study a probabilistic logic over an algebraic basis, including equations and domain restrictions. The logic combines aspects from classical logic and equational logic with an exogenous approach to quantitative probabilistic reasoning. We present a sound and weakly complete axiomatization for the logic, parameterized by an equational specification of the algebraic basis coupled with the intended domain restrictions.We show that the satisfiability problem for the logic is decidable, under the assumption that its algebraic basis is given by means of a convergent rewriting system, and, additionally, that the axiomatization of domain restrictions enjoys a suitable subterm property. For this purpose, we provide a polynomial reduction to Satisfiability Modulo Theories. As a consequence, we get that validity in the logic is also decidable. Furthermore, under the assumption that the rewriting system that defines the equational basis underlying the logic is also subterm convergent, we show that the resulting satisfiability problem is NP-complete, and thus the validity problem is coNP-complete.We test the logic with meaningful examples in information security, namely by verifying and estimating the probability of the existence of offline guessing attacks to cryptographic protocols.

An Extension of Computed-Torque Control for Parallel Robots in Ankle Reeducation

Abstract This paper extends the well-known computed-torque control technique for trajectory tracking in parallel robotic manipulators, which belong to the class of nonlinear systems under algebraic restrictions. The proposed methodology is based on Euler-Lagrange modelling which, once the closed-loop kinematic chain is taken into account, leads to a system of differential algebraic equations requiring proper initialisation while controlled. The control scheme is illustrated for an ankle therapy device known as the motoBOTTE, where post-stroke patients are subject to reeducation routines corresponding to different trajectories of the parallel manipulator.

The Algebraic Structure of Quantity Calculus

Abstract The algebraic structure underlying the quantity calculus is defined axiomatically as an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product, and quotient spaces are considered, as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation of quantity calculus which is centered on the concept of dimension, rather than on the concept of unit, which is regarded as secondary, and uses only integer exponents of the dimensions.

Symplectic Singularity of Curves with Semigroups (4, 5, 6, 7), (4, 5, 6) and (4, 5, 7)

We study the local symplectic algebra of curves with semigroups (4, 5, 6, 7), (4, 5, 6) and (4, 5, 7). We use the method of algebraic restrictions to parameterized curves as in Domitrz (Fund Math 204(1):57–86, 2009). A new discrete invariant for algebraic restrictions to parameterized quasi-homogeneous curves is introduced. This invariant together with the method of algebraic restriction can distinguish different symplectic orbits of quasi-homogeneous curves.

Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor Fμν; the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor Tμν, suffice to reduce the general Fμν to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordstr{öm one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.

Classification of transversal Lagrangian stars

A Lagrangian star is a system of three Lagrangian submanifolds of the symplectic space intersecting at a common point. In this work we classify transversal Lagrangian stars in the symplectic space in the analytic category under the action of symplectomorphisms by using the method of algebraic restrictions. We present a list of all transversal Lagrangian star.

Residual algebraic restrictions of differential forms

Residual algebraic restrictions of differential forms

Symplectic U7,U8 and U9 Singularities

Abstract We use the method of algebraic restrictions to classify symplectic U7, U8 and U9 singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes

The generalized lucky ticket problem, perfect matchings, and closure relations satisfied by the Chebyshev and $$q$$ q -Hermite polynomials

Consideration is given to an asymmetric ticket of length \(m+n\) in base \(\ell \). Such a ticket is said to be \((m,n)\)-lucky if the sum of the first \(m\) digits is equal to that of the last \(n\) digits. In other words, a \((m,n)\)-lucky ticket is a \(m+n\) digit number (in base \(\ell \)) of the form \(a_1a_2\cdots a_mb_1b_2\cdots b_n\) where \(a_i,b_j \in \left\{ 0,1,2,\dots ,\ell -1 \right\} \) and \(a_1 + a_2 + \cdots + a_m = b_1 + b_2 + \cdots + b_n\). Applying both analytical (contour integral) and combinatorial methods, we arrive at two representations for the number of \((m,n)\)-lucky tickets in base \(\ell \). Our results reduce to those in the literature, when \(m=n\) and \(\ell =10\). Furthermore, through the contour integral approach, we arrive at a non-obvious closure relation satisfied by the Chebyshev polynomials. The weighted ticket problem is also considered, and analogous results are obtained. As addressed by Ismail, Stanton, and Viennot, the generating function of the crossing numbers over perfect matchings is related to closure relations of \(q\)-Hermite polynomials. In the second part of this paper, we give corresponding contour integral representations for these closure relations, which permit us to give an alternate representation of the number of perfect matchings between sets. In the \(q=0\) limit, we obtain a representation equivalent to that of De Sainte-Catherine and Viennot for the number of Dyck words of a fixed length satisfying a set of algebraic restrictions. In order to relate the two combinatorial problems, we find an explicit correspondence between our contour formulations for each problem.

Fractional order junctions

Gottfried Leibniz generalized the derivation and integration, extending the operators from integer up to real, or even complex, orders. It is presently recognized that the resulting models capture long term memory effects difficult to describe by classical tools. Leon Chua generalized the set of lumped electrical elements that provide the building blocks in mathematical models. His proposal of the memristor and of higher order elements broadened the scope of variables and relationships embedded in the development of models. This paper follows the two directions and proposes a new logical step, by generalizing the concept of junction. Classical junctions interconnect system elements using simple algebraic restrictions. Nevertheless, this simplistic approach may be misleading in the presence of unexpected dynamical phenomena and requires including additional “parasitic” elements. The novel γ-junction includes, as special cases, the standard series and parallel connections and allows a new degree of freedom when building models. The proposal motivates the search for experimental and real world manifestations of the abstract conjectures.

Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities

We consider differential operators acting on densities of arbitrary weights on manifold M identifying pencils of such operators with operators on algebra of densities of all weights. This algebra can be identified with the special subalgebra of functions on extended manifold . On one hand there is a canonical lift of projective structures on M to affine structures on extended manifold . On the other hand the restriction of algebra of all functions on extended manifold to this special subalgebra of functions implies the canonical scalar product. This leads in particular to classification of second order operators with use of Kaluza-Klein-like mechanisms.

A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

This work describes a symbolic–numerical integration method for a class of differential algebraic equations (DAEs) known as semi-explicit systems. Our method relies on geometric theory of decoupling for nonlinear systems combined with efficient numerical analysis techniques. It uses an algorithm that applies symbolic and numerical calculations to build an explicit vector field $$\tau $$ , whose integral curves with compatible initial conditions are the same solutions of the original DAE. Here, compatible initial conditions are the ones that respect the algebraic restrictions and their derivatives up to their relative degree. This extended set of restrictions defines a submanifold $$\varGamma $$ of the whole space, which is formed by all the variables of the system, and all solutions of the DAE lie on this submanifold. Furthermore, even for nonexactly compatible initial conditions, the solutions of this explicit system defined by $$\tau $$ converge exponentially to $$\varGamma $$ . Under mild assumptions, an approximation result shows that the precision of the method is essentially controlled by the distance of the initial condition from $$\varGamma $$ . A scheme to compute compatible initial conditions with the DAE is also provided. Finally, simulations with benchmarks and comparisons with other available methods show that this is a suitable alternative for these problems, specially for nonexactly compatible initial conditions or high-index problems.