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1.
Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {x
p
: p∈P}, and the set of relations is {x
p
⋅x
q
=x
p
: p≤q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤
B
|G) is well-founded. A well-generated algebra is superatomic.
THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.
The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.
THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.
This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra.
Remember that an interval algebra is a special case of a poset algebra.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
2.
Gejza Jenča 《Order》2010,27(1):41-61
We prove that every orthocomplete homogeneous effect algebra is sharply dominating. Let us denote the greatest sharp element
below x by x
↓. For every element x of an orthocomplete homogeneous effect algebra and for every block B with x ∈ B, the interval [x
↓,x] is a subset of B. For every meager element (that means, an element x with x
↓ = 0), the interval [0,x] is a complete MV-effect algebra. As a consequence, the set of all meager elements of an orthocomplete homogeneous effect
algebra forms a commutative BCK-algebra with the relative cancellation property. We prove that a complete lattice ordered
effect algebra E is completely determined by the complete orthomodular lattice S(E) of sharp elements, the BCK-algebra M(E) of meager elements and a mapping h:S(E)→2
M(E) given by h(a) = [0,a] ∩ M(E). 相似文献
3.
V. M. Prokip 《Ukrainian Mathematical Journal》2012,63(8):1314-1320
Polynomial n × n matrices A(x) and B(x) over a field
\mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over
\mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over
\mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A
0x
- A
1, where A
0 and A
1 are n × n matrices over
\mathbbF \mathbb{F} , and A
0 is nonsingular. 相似文献
4.
Let F be a free Lie algebra of rank n ≥ 2 and A be a free abelian Lie algebra of rank m ≥ 2. We prove that the test rank of the abelian product F ×A is m. Morever we compute the test rank of the algebra F/gk( F) ¢F/\gamma _{k}\left( F\right) ^{^{\prime }}. 相似文献
5.
Ranjan Kumar Mohanty 《Numerical Algorithms》2010,54(3):379-393
In this paper, the use of N-AGE and Newton-N-AGE iterative methods on a variable mesh for the solution of one dimensional
parabolic initial boundary value problems is considered. Using three spatial grid points, a two level implicit formula based
on Numerov type discretization is discussed. The local truncation error of the method is of O(k2hl-1 +khl +hl3)O({k^2h_l^{-1} +kh_l +h_l^3}), where h
l
> 0 and k > 0 are the step lengths in space and time directions, respectively. We use a special technique to handle singular parabolic
equations. The advantage of using these algorithms is highlighted computationally. 相似文献
6.
T. Shulman 《Journal of Mathematical Sciences》2009,159(6):894-903
We consider maps defined on a real space Asa of all self-adjoint elements of a C*-algebra A commuting with the conjugation by unitaries: F(u* au) = u* F(a)u for any a ∈ A
sa, u ∈ (A). In the case where A is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical
properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized
to the class of uniformly hyperfinite C*-algebras and to the algebra of all compact operators in a Hilbert space.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 213–227, 2007. 相似文献
7.
Jipu Ma 《Frontiers of Mathematics in China》2008,3(2):305-316
Let E and F be Banach spaces, f: U ⊂ E → F be a map of C
r
(r ⩾ 1), x
0 ∈ U, and ft (x
0) denote the FréLechet differential of f at x
0. Suppose that f′(x
0) is double split, Rank(f′(x
0)) = ∞, dimN(f′(x
0)) > 0 and codimR(f′(x
0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x
0) near x
0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators,
i.e., x
0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the
conclusion of the theorem to be held is given.
相似文献
8.
We consider the parabolic Anderson problem ∂
t
u = κΔu + ξ(x)u on ℝ+×ℝ
d
with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases
of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order
term of the almost sure asymptotics of u(t, 0) as t→∞.
Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000 相似文献
9.
Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds 总被引:6,自引:0,他引:6
MA Jipu 《中国科学A辑(英文版)》2000,43(12):1233-1237
Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x
0)⊂E→F be C1 nonlinear map, whereU (x
0) is an open set containing pointx
0∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x
0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced
calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives
rise to a generalized principle for constructing Banach submanifolds. 相似文献
10.
Yossi Moshe 《Journal d'Analyse Mathématique》2006,99(1):267-294
Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X
i)
i
0/∞
over ℂ. Assume that theX
i's are chosen from a finite set {D
0,D
1...,D
t-1(ℂ), withP(X
i=Dj)>0, and that the monoid generated byD
0, D1,…, Dq−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case
where theX
i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22].
Our results on λ enable us to provide an approximation for the numberN
≠0(F(x)n,r) of nonzero coefficients inF(x)
n.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN
≠0(F(x)n,r) ≈n
α for “almost” everyn.
Supported in part by FWF Project P16004-N05 相似文献
11.
The real-valued Lambert W-functions considered here are w
0(y) and w
− 1(y), solutions of we
w
= y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision
of these functions and of the integrals ò1¥ [-w0(-xe-x)]a x-bx\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x, α > 0, β ∈ ℝ, and ò01 [-w-1(-x e-x)]a x-bx\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β. 相似文献
12.
We investigate the multiplicative and T-space structure of the relatively free algebra F
(3) with a unity corresponding to the identity [[x
1
, x
2], x
3] = 0 over an infinite field of characteristic p > 0. The highest emphasis is placed on unitary closed T-spaces over a field of characteristic p > 2. We construct a diagram containing all basic T-spaces of the algebra F
(3), which form infinite chains of the inclusions. One of the main results is the decomposition of quotient T-spaces connected with F
(3) into a direct sum of simple components. Also, the studied T-spaces are commutative subalgebras of F
(3); thus, the structure of F(3) and its subalgebras can be described as modules over these commutative algebras. Separately, we consider the specifics
of the case p = 2. In the Appendix, we study nonunitary closed T-spaces and the case of a field of zero characteristic. 相似文献
13.
Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver
\mathbb An\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D
b
(Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category. 相似文献
14.
A. K. Aleskeviciene 《Lithuanian Mathematical Journal》2005,45(4):359-367
Let X
1, X
2,... be independent identically distributed random variables with distribution function F, S
0 = 0, S
n
= X
1 + ⋯ + X
n
, and Sˉ
n
= max1⩽k⩽n
S
k
. We obtain large-deviation theorems for S
n
and Sˉ
n
under the condition 1 − F(x) = P{X
1 ⩾ x} = e−l(x), l(x) = x
α
L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005. 相似文献
15.
Francesco Catino Salvatore Siciliano Ernesto Spinelli 《Algebras and Representation Theory》2010,13(6):653-660
Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u(L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68:503–513, 2006) it was proved that if u(L) is Lie solvable then the Lie derived length of u(L) is at least ⌈log2(p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this
lower bound. 相似文献
16.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
17.
We prove that every compact nilpotent ring R of characteristic p > 0 can be embedded in a ring of upper triangular matrices over a compact commutative ring. Furthermore, we prove that every
compact topologically nilpotent ring R of characteristic p > 0, is embedded in a ring of infinite triangular matrices over
\mathbbFpw(R)\mathbb{F}_{p}^{w(R)}. 相似文献
18.
Mario Abundo 《Methodology and Computing in Applied Probability》2010,12(3):473-490
It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential
equation dX(t) = μ(X(t))dt + σ(X(t)) dB
t
, X(0) = x
0, through b + Y(t), where b > x
0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B
t
. In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B
t
, for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately
the FPT density; some examples and numerical results are also reported. 相似文献
19.
Yasuyo Hatano Ichizo Ninomiya Hiroshi Sugiura Takemitsu Hasegawa 《Numerical Algorithms》2009,52(2):213-224
The infinite integral ò0¥x dx/(1+x6sin2x)\int_0^{\infty}x\,dx/(1+x^6\sin^2x) converges but is hard to evaluate because the integrand f(x) = x/(1 + x
6sin2
x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value
in up to 73 significant digits on an octuple precision system in C++. 相似文献
20.
J. Sunklodas 《Lithuanian Mathematical Journal》2009,49(4):464-470
We estimate the difference | FZv(x) - F(x) | \left| {{F_{{Z_v}}}(x) - \Phi (x)} \right| , where FZv(x) {F_{{Z_v}}}(x) is the distribution function of normalized series Z
v
= B
−1
v
Σ∞
j=0
v
j
X
j
with B
2
v
=
\mathbb E \mathbb {E} (Σ∞
j=0
v
j
X
j
) > 0 and the discount factor v, 0 < v < 1; X
0,X
1,X
2,… is a sequence of m-dependent random variables, and Φ(x) is the standard normal distribution function. In a particular case, the obtained upper bound is of order O((1−v)1/2). 相似文献