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2.
If ${\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0}If L = ?j=1m Xj2 + X0{\mathcal{L} = {\sum_{j=1}^m} {X_j^2} + X_0} is a H?rmander partial differential operator in
\mathbbRN{\mathbb{R}^N}, we give sufficient conditions on the vector fields X
j
’s for the existence of a Lie group structure
\mathbbG = (\mathbbRN, *){\mathbb{G} = (\mathbb{R}^N, *)} (and we exhibit its construction), not necessarily nilpotent nor homogeneous, such that L{\mathcal{L}} is left invariant on
\mathbbG{\mathbb{G}}. The main tool is a formula of Baker-Campbell-Dynkin-Hausdorff type for the ODE’s naturally related to the system of vector
fields {X
0, . . . , X
m
}. We provide a direct proof of this formula in the ODE’s context (which seems to be missing in literature), without invoking
any result of Lie group theory, nor the abstract algebraic machinery usually involved in formulas of Baker-Campbell-Dynkin-Hausdorff
type. Examples of operators to which our results apply are also furnished. 相似文献
3.
The Birman-Murakami-Wenzl algebras (BMW algebras) of type E
n
for n = 6; 7; 8 are shown to be semisimple and free over the integral domain
\mathbbZ[ d±1,l±1,m ] | / |
( m( 1 - d ) - ( l - l - 1 ) ) {{{\mathbb{Z}\left[ {{\delta^{\pm 1}},{l^{\pm 1}},m} \right]}} \left/ {{\left( {m\left( {1 - \delta } \right) - \left( {l - {l^{ - 1}}} \right)} \right)}} \right.} of ranks 1; 440; 585; 139; 613; 625; and 53; 328; 069; 225. We also show they are cellular over suitable rings. The Brauer
algebra of type E
n
is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring
\mathbbZ[ d±1 ] \mathbb{Z}\left[ {{\delta^{\pm 1}}} \right] . A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb
algebra of type En turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type E
n
share many structural properties with the classical ones (of type A
n
) and those of type D
n
. 相似文献
4.
Let X={ X( t), t∈ℝ
N
} be a Gaussian random field with values in ℝ
d
defined by
X(t) = (X1(t), ?, Xd(t)), t ? \mathbbRN,X(t) = (X_1(t), \ldots, X_d(t)),\quad t \in {\mathbb{R}}^N, 相似文献
5.
We prove the inequality that
\mathbb E| X1X2? Xn| £ ?{per(\varSigma )}{\mathbb{E}}|X_{1}X_{2}\cdots X_{n}|\leq \sqrt{\mathrm{per}(\varSigma )}, for any centered Gaussian random variables X
1,…, X
n
with the covariance matrix Σ, followed by several applications and examples. We also discuss a conjecture on the lower bound of the expectation. 相似文献
6.
We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set
s(A)={ilk;k ? \mathbb\mathbbZ*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}
is discrete and satisfies
?\frac1|lk|ldkn < ¥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty
, where ℓ is a nonnegative integer and
dk=min(\frac|lk+1-lk|2,\frac|lk-1-lk|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2})
. In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors
(Pk)k ? \mathbb\mathbbZ*(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}}
such that, for any x∈D(A
n+ℓ
), the decomposition ∑P
k
x=x holds. 相似文献
7.
Let A and B denote two families of subsets of an n-element set. The pair ( A,B) is said to be ℓ-cross-intersecting iff | A∩ B|= ℓ for all A∈ A and B∈ B. Denote by P
e
( n) the maximum value of | A|| B| over all such pairs. The best known upper bound on P
e
( n) is Θ(2
n
), by Frankl and R?dl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2 ℓ, a simple construction of an ℓ-cross-intersecting pair ( A,B) with | A|| B| = $
\left( {{*{20}c}
{2\ell } \\
\ell \\
} \right)
$
\left( {\begin{array}{*{20}c}
{2\ell } \\
\ell \\
\end{array} } \right)
2
n−2ℓ
= Θ(2
n
/$
\sqrt \ell
$
\sqrt \ell
), and conjectured that this is best possible. Consequently, Sgall asked whether or not P
e
( n) decreases with ℓ. 相似文献
8.
For each natural number k and each irrational member λ of the unit circle, it is proved that the shift-orbit closure X
f
of the function f( n) = l nk{f(n) = {\lambda^{n}}^{k}} is homeomorphic to a k-torus. Using this homeomorphism, we investigate the Ellis group and its topological center of the dynamical system ( X
f
, U), where U is the shift operator on
l¥(\mathbb Z){l^{\infty}(\mathbb{Z})}. Finally, it is shown that the topological center of the spectrum of the Weyl algebra is the image of
\mathbb Z{\mathbb{Z}} in the spectrum. 相似文献
9.
Every compact smooth manifold M is diffeomorphic to the set
X(\mathbb R){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety
X\mathbbC= X× \mathbbR\mathbb C{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in
H2k( X(\mathbb R);\mathbb D){H^{2k}(X(\mathbb{R});\mathbb{D})} , where
\mathbb D{\mathbb{D}} denotes
\mathbb Z{\mathbb{Z}} or
\mathbb Q{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M. 相似文献
10.
Let { φ k } be an orthonormal system on a quasi-metric measure space ${\mathbb{X}}Let {φ
k
} be an orthonormal system on a quasi-metric measure space
\mathbbX{\mathbb{X}}, {ℓ
k
} be a nondecreasing sequence of numbers with lim
k→∞
ℓ
k
=∞. A diffusion polynomial of degree L is an element of the span of {φ
k
:ℓ
k
≤L}. The heat kernel is defined formally by Kt(x,y)=?k=0¥exp(-lk2t)fk(x)[`(fk(y))]K_{t}(x,y)=\sum_{k=0}^{\infty}\exp(-\ell _{k}^{2}t)\phi_{k}(x)\overline{\phi_{k}(y)}. If T is a (differential) operator, and both K
t
and T
y
K
t
have Gaussian upper bounds, we prove the Bernstein inequality: for every p, 1≤p≤∞ and diffusion polynomial P of degree L, ‖TP‖
p
≤c
1
L
c
‖P‖
p
. In particular, we are interested in the case when
\mathbbX{\mathbb{X}} is a Riemannian manifold, T is a derivative operator, and p 1 2p\not=2. In the case when
\mathbbX{\mathbb{X}} is a compact Riemannian manifold without boundary and the measure is finite, we use the Bernstein inequality to prove the
existence of quadrature formulas exact for integrating diffusion polynomials, based on an arbitrary data. The degree of the diffusion polynomials for which this formula is exact depends upon the mesh norm of the data. The
results are stated in greater generality. In particular, when T is the identity operator, we recover the earlier results of Maggioni and Mhaskar on the summability of certain diffusion
polynomial valued operators. 相似文献
11.
Let X 1, X 2, ... be i.i.d. random variables satisfying the condition
\textE X12 \text elX1 < ¥\text for\text some\text l > 0.{\text{E }}X_1^2 {\text{ }}e^{\lambda X_1 } < \infty {\text{ }}for{\text{ }}some{\text{ }}\lambda >0. 相似文献
12.
It has been shown that any Banach algebra satisfying ‖ f
2‖ = ‖ f‖ 2 has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras
of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not
been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued
Lipschitz functions. Denote by Lip( X,
\mathbb F\mathbb{F}) the algebra over R of F-valued Lipschitz functions on a compact metric space ( X, d), where
\mathbb F\mathbb{F} = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip( X,
\mathbb F\mathbb{F}) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values
in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by
generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While
the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions
on such an algebra for it to hold and to guarantee the existence of the Shilov boundary. 相似文献
13.
Let X, X
1, X
2,… be i.i.d.
\mathbb Rd {\mathbb{R}^d} -valued real random vectors. Assume that E
X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms
\mathbb Q[ SN ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S
N
= N
−1/2 ( X
1 + ⋯ + X
N
) and show that, without any additional conditions,
DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right) 相似文献
14.
Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus
\mathbb T \mathbb{T} of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine
\mathbb Zn {\mathbb{Z}^n} -graded domain A, so that ∂ generates a k
+-action on X that is normalized by the
\mathbb T \mathbb{T} -action. 相似文献
15.
Let X 1, X 2, ..., X n be independent and identically distributed random variables subject to a continuous distribution function F. Let X 1∶n, X 2∶n, ..., X n∶n denote the corresponding order statistics. Write where n, k are fixed integers. We apply a result of Marsaglia and Tubilla on the lack of memory of the exponential distribution
finction assuming that certain distribution functions involving the above order statistics are equal in two incommensurable
points τ 1, τ 2 > 0; this characterizes the exponential distribution. As a special case it turns out that the equality (*) assumed for s=1,
2 and x=τ 1, τ 2 implies that F is exponential.
Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part II. 相似文献
16.
Harmonic analysis on ℤ( p
ℓ
) and the corresponding representation of the Heisenberg-Weyl group HW[ℤ( p
ℓ
),ℤ( p
ℓ
),ℤ( p
ℓ
)], is studied. It is shown that the HW[ℤ( p
ℓ
),ℤ( p
ℓ
),ℤ( p
ℓ
)] with a homomorphism between them, form an inverse system which has as inverse limit the profinite representation of the
Heisenberg-Weyl group
\mathfrak HW[\mathbb Zp,\mathbb Zp,\mathbb Zp]\mathfrak {HW}[{\mathbb{Z}}_{p},{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}]. Harmonic analysis on ℤ
p
is also studied. The corresponding representation of the Heisenberg-Weyl group HW[(ℚ
p
/ℤ
p
),ℤ
p
,(ℚ
p
/ℤ
p
)] is a totally disconnected and locally compact topological group. 相似文献
17.
Let S \subseteqq \mathbb Zm S \subseteqq \mathbb{Z}_m be a Sidon set of cardinality | S | = m1/2 + O(1) \mid S \mid = m^{1 \over 2} + O(1) . It is proved, in particular, that for any interval á = { a, a + 1, ?, a + l- 1} {\cal I} = \{a, a + 1, \ldots, a + \ell - 1\} in \mathbb Zm \mathbb{Z}_m , 0 \leqq l 0 \leqq \ell < m, we have | | S ?á | - | S | l/ m | = O( | S | 1/2ln m) \big| {\mid S \cap {\cal I} \mid - \mid S \mid \ell/m} \big| = O(\mid S \mid^{1 \over 2}\textrm{ln}\, m) . 相似文献
18.
Let X be a Banach space with a sequence of linear, bounded finite rank operators R
n: X→X such that R
nR m=R min( n,m) if n≠m and lim
n→∞
R
n
x=x for all x∈X. We prove that, if R
n−R n
−1 factors uniformly through some l
p and satisfies a certain additional symmetry condition, then X has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such that L
Λ=closed span
, where
, has an unconditional basis. Examples include the Hardy space
. 相似文献
19.
Let A be an Artinian algebra and F an additive subbifunctor of Ext,(-, -) having enough projectives and injectives. We prove that the dualizing subvarieties of mod A closed under F-extensions have F-almost split sequences. Let T be an F-cotilting module in mod A and S a cotilting module over F = End(T). Then Horn(-, T) induces a duality between F-almost split sequences in ⊥FT and almost sl31it sequences in ⊥S, where addrS = Hom∧(f(F), T). Let A be an F-Gorenstein algebra, T a strong F-cotilting module and 0→A→B→C→0 and F-almost split sequence in ⊥FT.If the injective dimension of S as a Г-module is equal to d, then C≌(ΩCM^-dΩ^dDTrA^*)^*,where(-)^*=Hom(g,T).In addition, if the F-injective dimension of A is equal to d, then A≌ΩMF^-dDΩFop^-d TrC≌ΩCMF^-d ≌F^d DTrC. 相似文献
20.
For any ℓ>0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P
1≤0,…, P
s
≤0, where each P
i
∈ R[ X
1,…, X
k
] has degree≤2, and computes the top ℓ Betti numbers of S, b
k−1( S),…, b
k−ℓ
( S), in polynomial time. The complexity of the algorithm, stated more precisely, is
. For fixed ℓ, the complexity of the algorithm can be expressed as
, which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic
sets in R
k
defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have
degree greater than one. For fixed s, we obtain, by letting ℓ= k, an algorithm for computing all the Betti numbers of S whose complexity is
.
An erratum to this article can be found at 相似文献
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