Torsion classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

 Torsion classes of MV-algebras are defined as radical classes which are closed with respect to homomorphisms; in this paper we investigate their relations to radical classes of lattice ordered groups and to varieties of MV-algebras. Supported by Grant VEGA 1/9056/02.     

Retract mappings of projectable MV-algebras

For MV-algebras (algebras of multivalued Lukasiewicz logics) we apply the same terminology and notation as in [3] and [8]. Retracts and retract mappings of abelian lattice ordered groups were studied in [4], cf. also [6], [7]; for the case of multilattice groups and cyclically ordered groups cf. [1] and [5]. To each MV-algebra ? there corresponds an abelian lattice ordered group G with a strong unit u such that (under the notation as in [8]), ? = ?₀(G,u) (cf. also Section 1 below). In [2], a different (but equivalent) system of axioms for defining the notion of MV-algebra was applied; instead of ?₀(G,u), the notation Γ(G,u) was used. In the present paper we investigate the relations between retract mappings of a projectable MV-algebra ? and the retract mappings of the corresponding lattice ordered group G.     

Compactness in <Emphasis Type="Italic">MV</Emphasis>-algebras and in their non-commutative generalizations

 In the paper we study MV-algebras and their non-commutative generalizations, GMV-algebras, in which every element is compact. Further, we characterize finite MV-algebras as archimedean GMV-algebras which are 0-meet compact. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday Supported by the Council of Czech Government, J 14/98: 15100011.     

n-Fold obstinate filters in BL-algebras

In this paper, we introduced the notion of n-fold obstinate filter in BL-algebras and we stated and proved some theorems, which determine the relationship between this notion and other types of n-fold filters in a BL-algebra. We proved that if F is a 1-fold obstinate filter, then A/F is a Boolean algebra. Several characterizations of n-fold fantastic filters are given, and we show that A is a n-fold fantastic BL-algebra if A is a MV-algebra (n ≥ 1) and A is a 1-fold positive implicative BL-algebra if A is a Boolean algebra. Finally, we construct some algorithms for studying the structure of the finite BL-algebras and n-fold filters in finite BL-algebras.     

Wavelet solution of variable order pseudodifferential equations

Sobolev spaces H ^m(x)(I) of variable order 0<m(x)<1 on an interval I⊂ℝ arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X _t ∈ℝ with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in H ^m(x)(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X.     

Tensor products of MV-algebras

 The structure of MV-module over a PMV-algebra was defined in [5]. In this paper we test this notion in the important problem of scalar extension. The tensor product ⊗ _mv defined in [11] is generalized and another tensor product, ⊗ _o , is studied in relation with the corresponding tensor product of ℓ-groups [9]. Using tensor products we get the free MV-modules over a totally ordered PMV-algebra. Dedicated to Prof. Ján Jakubíik on the occasion of his 80th birthday     

A note on observables on MV-algebras

 A criterion when a subset of an MV-algebra is contained in the range of a (finitely additive) observable is given. It is used to prove that every at most countable subset of any MV-algebra is contained in the range of an observable. A much stronger result holds for any bold fuzzy algebra [0,1] ^S , which is whole contained in the range of a (σ-additive) observable.     

Some remarks on the joint observable in D-posets

 The notion of the joint observable contributed, besides other things, to the development of probability theory on structures which are more general than σ-algebras. We present some remarks on the joint observable in D-posets which arising in the definition of operations with observables.     

The Complexity of Finding SUBSEQ(<Emphasis Type="Italic">A</Emphasis>)

Higman showed that if A is any language then SUBSEQ(A) is regular. His proof was nonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine M _e , and outputs a DFA for SUBSEQ(L(M _e )), then ∅″≤_T f (f is Σ ₂-hard). We also study the complexity of going from A to SUBSEQ(A) for several representations of A and SUBSEQ(A).     

Boolean dominated MV-algebras

In this paper we study an obvious generalization of the hyperarchimedean MV-algebras: boolean dominated MV-algebras. Particularly we point out the wide difference between the class of the hyperarchimedean MV-algebras and the class of the boolean dominated MV-algebras.     

Self-intersection elimination in metamorphosis of two-dimensional curves

H :[0, 1]× ³→ ³, where H(t, r) for t=0 and t=1 are two given planar curves C ₁(r) and C ₂(r). The first t parameter defines the time of fixing the intermediate metamorphosis curve. The locus of H(t, r) coincides with the ruled surface between C ₁(r) and C ₂(r), but each isoparametric curve of H(t, r) is self-intersection free. The second algorithm suits morphing operations of planar curves. First, it constructs the best correspondence of the relative parameterizations of the initial and final curves. Then it eliminates the remaining self-intersections and flips back the domains that self-intersect.     

On a reduction of the interval coloring problem to a series of bandwidth coloring problems

Given a graph G=(V,E) with strictly positive integer weights ω _i on the vertices i∈V, an interval coloring of G is a function I that assigns an interval I(i) of ω _i consecutive integers (called colors) to each vertex i∈V so that I(i)∩I(j)=∅ for all edges {i,j}∈E. The interval coloring problem is to determine an interval coloring that uses as few colors as possible. Assuming that a strictly positive integer weight δ _ij is associated with each edge {i,j}∈E, a bandwidth coloring of G is a function c that assigns an integer (called a color) to each vertex i∈V so that |c(i)−c(j)|≥δ _ij for all edges {i,j}∈E. The bandwidth coloring problem is to determine a bandwidth coloring with minimum difference between the largest and the smallest colors used. We prove that an optimal solution of the interval coloring problem can be obtained by solving a series of bandwidth coloring problems. Computational experiments demonstrate that such a reduction can help to solve larger instances or to obtain better upper bounds on the optimal solution value of the interval coloring problem.     

On the fuzzy difference equation xn+1 = A+B/xn

 In this paper we study the existence the oscillatory behavior the boundedness and the asymptotic behavior of the positive solutions of the fuzzy equation x _n+1 =A+B/x _n ,n=0,1,… where x _n is a sequence of fuzzy numbers, A, B are fuzzy numbers.     

Black brane solutions related to non-singular Kac-Moody algebras

A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M ₀ × M ₁ × … × M _n , where M _i are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g., solutions with harmonic functions and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g., hyperbolic ones) are considered. Some examples of black brane solutions are presented, e.g., those corresponding to hyperbolic KM algebras: H ₂(q, q) (q > 2), HA ₂⁽¹⁾ = A ₂⁺⁺ and to the Lorentzian KM algebra P ₁₀.     

Output-Sensitive Algorithm for Computing β-Skeletons

The β-skeleton is a measure of the internal shape of a planar set of points. We get an entire spectrum of shapes by varying the parameter β. For a fixed value of β, a β-skeleton is a geometric graph obtained by joining each pair of points whose β-neighborhood is empty. For β≥1, this neighborhood of a pair of points p _i ,p _j is the interior of the intersection of two circles of radius , centered at the points (1−β/2)p _i +(β/2)p _j and (β/2)p _i +(1−β/2)p _j , respectively. For β∈(0,1], it is the interior of the intersection of two circles of radius , passing through p _i and p _j . In this paper we present an output-sensitive algorithm for computing a β-skeleton in the metrics l ₁ and l _∞ for any β≥2. This algorithm is in O(nlogn+k), where k is size of the output graph. The complexity of the previous best known algorithm is in O(n ^5/2logn) [7]. Received April 26, 2000     

Deviation-tolerant floating gate structures as a way to design an on-chip learning neural networks

 Hardware implementation of artificial neural networks (ANN) based on MOS transistors with floating gate (Neuron MOS or νMOS) is discussed. Choosing analog approach as a weight storage rather than digital improves learning accuracy, minimizes chip area and power dissipation. However, since weight value can be represented by any voltage in the range of supplied voltage (e.g. from 0 to 3.3 V), minimum difference of two values is very small, especially in the case of using neuron with large sum of weights. This implies that ANN using analog hardware approach is weak against V _dd deviation. The purpose of this paper is to investigate main parts of analog ANN circuits (synapse and neuron) that can compensate all kinds of deviation and to develop their design methodologies.     

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H _c (x) on the collocation points is used for the approximation of singular source terms δ(x−c) or δ ⁽ⁿ⁾(x−c) without any regularization. The direct projection of H _c (x) yields highly oscillatory approximations of δ(x−c) and δ ⁽ⁿ⁾(x−c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided. The current address: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.     

Ideals of an EMV-semiring

ABSTRACT

In this paper, congruences, ideals, and prime ideals of an EMV-semiring and of its associated EMV-algebra are studied. Then EMV-semirings are characterized and it is proved that each EMV-semiring can be embedded into a direct product of a family of MV-semirings as an EMV-semiring. Moreover, another representation of EMV-semirings are presented by EMV-semirings of continuous sections in a sheaf of commutative semirings whose stalks are localizations of EMV-semirings over prime ideals. Also, using the categorical equivalence between EMV-semirings and EMV-algebras, a representation of EMV-algebras are obtained.     

Envelopes of a simplex of discrete probabilities

 A formalism for describing different “envelopes” of a family 𝒫 of probability distributions on finite sets is suggested. It incorporates lower and upper probabilities, minimal entropy extensions etc. Properties of a simple heuristical algorithm for finding lower and upper probabilities for a simplex 𝒫(𝒦) of distributions with fixed marginals 𝒦 are discussed. Comparison of certain type of probabilistic and fuzzy envelopes using the same input knowledge 𝒦 is performed. The author thanks to all three anonymous referees for their comments and suggestions to improve the presentation of this paper and to J. Frankl for a lot of help with Latex.     

The improved QV signature scheme based on conic curves over ℤ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The classical RSA is vulnerable to low private exponent attacks (LPEA) and has homomorphism. KMOV based on elliptic curve E _n (a,b) over ℤ _n can resist LPEA but still has homomorphism. QV over E _n (a,b) not only can resist LPEA but also has no homomorphism. However, QV over E _n (a,b) requires the existence of points whose order is M _n = lcm{♯E _p (a,b), ♯E _q (a,b)}. This requirement is impractical for all general elliptic curves. Besides, the computation over En(a,b) is quite complicated. In this paper, we further study conic curve C _n (a,b) over ℤ _n and its corresponding properties, and advance several key theorems and corollaries for designing digital signature schemes, and point out that C _n (a,b) always has some points whose order is M _n = lcm{♯E _p (a,b), ♯E _q (a,b)}. Thereby we present an improved QV signature over C _n (a,b), which inherits the property of non-homomorphism and can resist the Wiener attack. Furthermore, under the same security requirements, the improved QV scheme is easier than that over E _n (a,b), with respect plaintext embedding, inverse elements computation, points computation and points’ order calculation. Especially, it is applicable to general conic curves, which is of great significance to the application of QV schemes. Supported by the National Natural Science Foundation of China (Grant No. 10128103)