An Equivalent of Algebraic ?-Categories

In this paper, based on ?-categories, some properties of(continuous) I-cocomplete ?-categories are studied. Then, we introduce the concepts of bicomplete ?-category and approximable bimodule, discuss their properties and we also show any I-cocomplete ?-category is a bicomplete ?-category. Finally, it is proved that the category of algebraic ?-categories is equivalent to the category of bicomplete ?-categories.     

Algebraic K?-theory of Parameterized Endomorphisms

Assume that M is an R-bimodule. Let End(R, M) denotes the category whose objects are pairs (P, f), where P is a finitely generated projective right R-module and f: P → P ⊗ M. It has an exact structure obtained from the category of projectives over R by forgetting f s. We prove that, when R is a field, we have denotes certain localization of the tensor algebra spanned by M. This result should be viewed as a special case of the noncommutative extension of the results of [4].     

Interpolation on Algebraic Manifold

In traditional researches of multivariate interpolation,people always assume that inter—polation nodes are not on any algebraic manifold of low degree;more precisely,there existsno polynomial with low degree vanishing at the nodes[1】.In this case,the interpolationnodes are said to be on‘‘General Position”.But in this paper,we are to discuss interpolationon algebraic manifold,that is,the interpolation nodes are just on some algebraic manifolds.Definition 1三e￡S be an irreducible algebrai…     

Algebraic Structure on Dirichlet Spaces

In this short note, we shall give a few equivalent conditions for a closed form to be Markovian, and prove that the closure of a sub-algebra of bounded functions in a Dirichlet space must be Markovian. We also study the regular representation of Dirichlet spaces and the classification of Dirichlet subspaces.     

The abc-conjecture for Algebraic Numbers

The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.     

What Are Equivalent Equations?

Theequationsx =6 38,4 32and 6 0 6 ,90 0 -x =- 31,532areequivalentequationsbecausetheyhavethesamesolu tions .Whenyouwritedownthestepstosolveanequation ,eachstepshouldbeanequationequivalenttotheequationsintheprevioussteps .ExampleMultiplechoice .Whichequationisnotequivalenttotheothers?(a)x +3=8　 (b) 2x =10　 (c) - 1=4 -x　 (d)x =5　 (e)x - 0 .4 =4 .96SolutionEquations (a) ,(b) ,(c)and (d)allhavethesinglesolution 5.Thesolutiontoequation (e)is5.36 .Equatin (e)isnotequivalenttotheothers .Ofalleq…     

Are Biseparable Extensions Frobenius?

In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. In Section 5 we show that characterizations of the separable extensions among Frobenius extensions are special cases of a result for adjoint functors. Mathematics Subject Classifications (2000): 16L60, 16H05.     

Are Chain-Complete Posets Co-wellpowered?

Order - We show that the category CPO of chain-complete posets is not co-wellpowered but that it is weakly co-wellpowered. This implies that CPO is nearly locally presentable.     

Are Bare Particulars Constituents?

In this article I examine an as yet unexplored aspect of J.P. Moreland’s defense of so-called bare particularism — the ontological theory according to which ordinary concrete particulars (e.g., Socrates) contain bare particulars as individuating constituents and property ‘hubs.’ I begin with the observation that if there is a constituency relation obtaining between Socrates and his bare particular, it must be an internal relation, in which case the natures of the relata will necessitate the relation. I then distinguish various ways in which a bare particular might be thought to have a nature and show that on none of these is it possible for a bare particular to be a constituent of a complex particular. Thus, Moreland’s attempt to resurrect bare particulars as ontologically indispensable entities is not wholly without difficulties.     

A Generalization of Algebraic Lattices

In this paper,we prove two theorems on lattices and operators.Definition Let L be a complete lattice.An element a in L is(＊)-compact if and only 1if whenever a≤VA for A(?)L then a≤VB for some B(?)Awith B is at most countable.L is called(＊)—algebraic lattice if every element in L is a supremum of(＊)—compactelements.(＊)—algebraic lattice is a generalization of algebraic lattice.     

The Isotonic Fuzzy Algebraic Systems

We introduce the concept of isotonic fuzzy algebraic systems and prove that a isotonic fuzzy set of a group G is a fuzzy group iff μ_A(xy)=min {μ_A(x), μ_A(y)} iff A is a constant function.     

Rate of Approximation of Regular Convex Bodies by?Convex Algebraic Level Surfaces

We consider the problem of the approximation of regular convex bodies in ℝ ^d by level surfaces of convex algebraic polynomials. Hammer (in Mathematika 10, 67–71, 1963) verified that any convex body in ℝ ^d can be approximated by a level surface of a convex algebraic polynomial. In Jaen J. Approx. 1, 97–109, 2009 and subsequently in J. Approx. Theory 162, 628–637, 2010 a quantitative version of Hammer’s approximation theorem was given by showing that the order of approximation of convex bodies by convex algebraic level surfaces of degree n is frac1nfrac{1}{n}. Moreover, it was also shown that whenever the convex body is not regular (that is, there exists a point on its boundary at which the convex body possesses two distinct supporting hyperplanes), then frac1nfrac{1}{n} is essentially the sharp rate of approximation. This leads to the natural question whether this rate of approximation can be improved further when the convex body is regular. In this paper we shall give an affirmative answer to this question. It turns out that for regular convex bodies a o(1/n) rate of convergence holds. In addition, if the body satisfies the condition of C ²-smoothness the rate of approximation is O(frac1n²)O(frac{1}{n^{2}}).     

Are Hamiltonian flows geodesic flows?

When a Hamiltonian system has a ``Kinetic + Potential' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure.

We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the -body problem. We show that the flow of the reduced planar -body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.