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1.
Aparna W. Higgins 《Algebra Universalis》1985,20(2):179-193
Given a group G and a descending chainG
0,G
1,...,G
n, of normal subgroups ofG, we prove that there exists a universal algebra
, such that the chain ...Wn(
)...W1(
}) W0(
)W(
) is isomorphic to the chain ...G
n ...G
1G
0G, where W(
) is the group of weak automorphisms of
, and Wn(
) is the group of weak automorphisms of
that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G0, and G1=G2=...=Gn=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments. 相似文献
2.
Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), 2, ..., n. Fiedler and others have shown that \det( I -A) n - n, for all > with equality for any such if and only if A is the simple cycle matrix. Let a
i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to:
for all . We use this inequality to derive the inequality that:
. In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If 1 = (A), 2,...,k
are (all) the nonzero eigenvalues of A, then
. We prove this conjecture for the case when the spectrum of A is real. 相似文献
3.
Let T be a regular operator from L
p L
p. Then
, where Tr denotes the regular norm of T, i.e., Tr=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=Tr, so that the above inequality generalizes the Daugavet equation for operators on L
1–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L
p and denote by T
A the operator TA. Then
for all A with (A)>0 if and only if
. 相似文献
4.
Joseph Rosenblatt 《Mathematische Annalen》1977,230(3):245-272
For a mean zero norm one sequence (f
n
)L
2[0, 1], the sequence (f
n
{nx+y}) is an orthonormal sequence inL
2([0, 1]2); so if
, then
converges for a.e. (x, y)[0, 1]2 and has a maximal function inL
2([0, 1]2). But for a mean zerofL
2[0, 1], it is harder to give necessary and sufficient conditions for theL
2-norm convergence or a.e. convergence of
. Ifc
n
0 and
, then this series will not converge inL
2-norm on a denseG
subset of the mean zero functions inL
2[0, 1]. Also, there are mean zerofL[0, 1] such that
never converges and there is a mean zero continuous functionf with
a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c
n
| = 0(n
–) for >1/2, then
converges a.e. and unconditionally inL
2[0, 1]. In addition, for any mean zerof of bounded variation, the series
has its maximal function in allL
p[0, 1] with 1p<. Finally, if (f
n
)L
[0, 1] is a uniformly bounded mean zero sequence, then
is a necessary and sufficient condition for
to converge for a.e.y and a.e. (x
n
)[0, 1]. Moreover, iffL
[0, 1] is mean zero and
, then for a.e. (x
n
)[0, 1],
converges for a.e.y and in allL
p
[0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one. 相似文献
5.
Philip Ehrlich 《Algebra Universalis》1988,25(1):7-16
An -universally extending ordered field of power is constructed for each regular power where 0 < On and
. When is inaccessible, the structure is either a (set) model of J. H. Conway's ordered field No or an isomorphic copy of No depending on whether or not is a set or a proper class.Presented by Jan Mycielski. 相似文献
6.
Martin Goldstern 《Monatshefte für Mathematik》1993,116(3-4):237-243
IfC is a Polish probability space,
a Borel set whose sectionsW
x ( have measure one and are decreasing
, then we show that the set
x
W
x
has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1. 相似文献
7.
Dr. Karl Umgeher 《Monatshefte für Mathematik》1976,81(4):311-314
We consider the set of regular functions
. We construct a Borel measure and a class of outer measures
h
onH. With these and
h
we show that: (HS)=0 and
h
(HS)=0, (S is the set of normed univalent functions). From
h
(HS)=0 follows—forh=t
—that the Hausdorff—Billingsley-dimension ofHS is zero. 相似文献
8.
Pavel Valtr 《Discrete and Computational Geometry》1992,7(1):135-152
For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc
(k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying
for fixed
, contains at leastc convex independent points. We determine the exact asymptotic behavior ofc
(k), proving that there are two positive constants=(), such thatk
1/3c
(k)k
1/3. To establish the upper bound ofc
(k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying
. The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties. 相似文献
9.
William D. L. Appling 《Annali di Matematica Pura ed Applicata》1985,140(1):147-154
Summary Suppose U is a set,F is a field of subsets of U, pAB is the set of all real-valued bounded finitely additive functions defined onF, and for each in pAB, A()={: in pAB, absolutely continuous with respect to }. SupposeM is a linear subspace of pAB such that
. A generalisation of a previously discussed collection of linear transformations (see J. London Math. Soc., vol. 44 (1969), pp. 385–396) is treated by letting CM denote the set to which T belongs iff T is a linear transformation from M into pAB such that for some K inR and all in M and V inF,
. Certain theorems of the aforementioned reference are generalized, as well as one of Trans. Amer. Math. Soc., vol. 199, (1974), pp. 131–140. The principal result of the present paper is the following generalisation of a reversibility characterisation in the first mentioned reference: Theorem: If T is in CM, then (, T()): in M A(T()) is the only reversible subset T0 if T such that: i) the domain M0 of T0 is a linear subspace of M and
, and ii) the range of T0 is the range of T. 相似文献
10.
Micheline Vigué-Poirrier 《manuscripta mathematica》1986,56(2):177-191
Let X be a nilpotent space such that it exists k1 with Hp (X,) = 0 p > k and Hk (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of YX (resp.
is isomorphic as Lie algebra, to H* (X,) (* (Y) ) (resp.+ (X,) (* (Y) )). If X is formal and Y -formal, then the spaces YX and
are -formal. Furthermore, if dim * (Y) is infinite and dim H* (Y,Q) is finite, then the sequence of Betti numbers of
grows exponentially. 相似文献
11.
H. L. Stalford 《Journal of Optimization Theory and Applications》1995,86(2):327-346
A new property called scalar-quadratic is presented for establishing the stabilizability of linear time-varyring uncertain systems. It is applied to a well-known linear time-varying system OL which contains two uncertainties 1(t) and 2(t). Using the Lyapunov functionsV(x)=x
T
Px, whereP is a constant postitive-definite symmetric matrix, previous authors have shown that OL is stabilizable by linear static controllers when the time-varying uncertainties are bounded by a normalized bound
satisfying
< 0.8. We extend the bound to
< 1.0 by using the more general Lyapunov functions satisfying the scalar-quadratic propertyV(ax)=a
2
V(x), aR, xR
0
2
.Our proof uses a hexagon as a closed, convex hypersuface to construct a scalar-quadratic Lyapunov function, so that the Lyapunov time derivative satisfies the quadratic convergence condition
, >0, for the closed-loop system CL formed from OL and a stabilizing linear static controller. The critical condition in the proof of the quaratic convergence ondition is the satisfaction of the inequality
, where max is a normalization bound for 1(t) and 2(t) and wheree
1 ande
2 are parameters for the controller. For the controller parametrized bye
1=8 ande
2=20, this inequality reduces to max < 2.2096. This result, in particular, establishes that the Petersen counterexample is stabilitzable by the linear static controller withe
1=8 ande
2=20. Furthermore, it establishes the amazing result that OL is stabilizable by a linear static controlle on any compact subset of the constant uncertainaty controllability space defined by 1>0 and 2>0. 相似文献
12.
Let (X
n
)
0 be a Markov chain with state space S=[0,1] generated by the iteration of i.i.d. random logistic maps, i.e., X
n+1=C
n+1
X
n
(1–X
n
),n0, where (C
n
)
1 are i.i.d. random variables with values in [0, 4] and independent of X
0. In the critical case, i.e., when E(log C
1)=0, Athreya and Dai(2) have shown that X
n
P
0. In this paper it is shown that if P(C
1=1)<1 and E(log C
1)=0 then(i) X
n
does not go to zero with probability one (w.p.1) and in fact, there exists a 0<<1 and a countable set (0,1) such that for all xA(0,1), P
x
(X
n
for infinitely many n1)=1, where P
x
stands for the probability distribution of (X
n
)
0 with X
0=x w.p.1. A is a closed set for (X
n
)
0.(ii) If is the supremum of the support of the distribution of C
1, then for all xA
(a)
for 12(b)
for 24(c)
for 24 under some additional smoothness condition on the distribution of C
1.(iii) The empirical distribution
converges weakly to
0, the delta measure at 0, w.p.1 for any initial distribution of X
0. 相似文献
13.
Subdifferentials with respect to dualities 总被引:1,自引:0,他引:1
Juan-Enrique Martinez-Legaz Ivan Singer 《Mathematical Methods of Operations Research》1995,42(1):109-125
LetX andW be two sets and: ¯RX ¯RW a duality (i.e., a mapping
such that
for all
and all index setsI). We introduce and study the subdifferential
of a function
at a pointx
o
X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8]. 相似文献
14.
D. M. Smirnov 《Algebra and Logic》2004,43(4):249-257
For integers 1 m < n, a Cantor variety with m basic n-ary operations i and n basic m-ary operations k is a variety of algebras defined by identities k(1(
), ... , m(
)) =
k and i(1(
), ... ,n(
)) = y
i, where
= (x
1., ... , x
n) and
= (y
1, ... , y
m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product 1 × 2 of a lattice, 1, of positive integers respecting the natural linear ordering and a lattice, 2, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice
of all interpretability types of varieties of algebras. 相似文献
15.
Noboru Hamada 《Designs, Codes and Cryptography》1997,10(1):41-56
Let k and d be any integers such that k 4 and
. Then there exist two integers and in {0,1,2} such that
. The purpose of this paper is to prove that (1) in the case k 5 and (,) = (0,1), there exists a ternary
code meeting the Griesmer bound if and only if
and (2) in the case k 4 and (,) = (0,2) or (1,1), there is no ternary
code meeting the Griesmer bound for any integers k and d and (3) in the case k 5 and
, there is no projective ternary
code for any integers k and such that 1k-3, where
and
for any integer i 0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results. 相似文献
16.
17.
We prove four theorems about groups with a dihedral (or cyclic) image containing a difference set. For the first two, suppose G, a group of order 2p
with p an odd prime, contains a nontrivial (v, k, ) difference set D with order n = k – prime to p and self-conjugate modulo p. If G has an image of order p, then 0 2a +
2
for a unique choice of = ±1, and for a = (k –
)/2p. If G has an image of order 2p, then
and
(
– 1)/(
– 1). There are further constraints on n, a and . We give examples in which these theorems imply no difference set can exist in a group of a specified order, including filling in some entries in Smith's extension to nonabelian groups of Lander's tables. A similar theorem covers the case when p|n. Finally, we show that if G contains a nontrivial (v, k, ) difference set D and has a dihedral image D
2m
with either (n, m) = 1 or m = p
t
for p an odd prime dividing n, then one of the C
2 intersection numbers of D is divisible by m. Again, this gives some non-existence results. 相似文献
18.
Let w() be a positive weight function on the unit circle of the complex plane. For a sequence of points {
k
}
k = 1
included in a compact subset of the unit disk, we consider the orthogonal rational functions
n
that are obtained by orthogonalization of the sequence { 1, z / 1, z
2 / 2, ... } where
, with respect to the inner product
In this paper we discuss the behaviour of
n
(t) for t = 1 and n under certain conditions. The main condition on the weight is that it satisfies a Lipschitz–Dini condition and that it is bounded away from zero. This generalizes a theorem given by Szeg in the polynomial case, that is when all
k
= 0. 相似文献
19.
Siegfried Steiner 《manuscripta mathematica》1977,20(3):277-300
Let E be a n-dimensional euclidean vector space. The subset V
k
n
={x ... x | x E} of kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on kE. Passing to the corresponding projective space
, one may consider
as a riemannian submanifold of the space form
. In this paper we study properties of the pair
of riemannian manifolds. 相似文献
20.
We consider solutions of the class of ODEs y=6y
2–x
, which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to
and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1. 相似文献