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1.
Louis Halle Rowen 《Israel Journal of Mathematics》1974,18(1):65-74
Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM
n (Ω[ξ]) generated by the matricesY
k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a
finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X
1, …,X
m) is a polynomial linear in all theX
i but one (similar to Formanek’s central polynomials for matrix rings) andf
2 is central forM
n (Ω), thenf is central forM
n (Ω). (The existence of a polynomial not central forM
n (Ω), but whose square is central forM
n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements.
Partial support for this work was provided by National Science Foundation grant NSF-GP 33591. 相似文献
2.
F. Peherstorfer 《Constructive Approximation》1997,13(2):261-269
We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect
to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1]. 相似文献
3.
Tim D. Cochran 《Commentarii Mathematici Helvetici》1985,60(1):291-311
A geometric notion of a “derivative” is defined for 2-component links ofS
n inS
n+2 and used to construct a sequenceβ
i
,i=1,2,... of abelian concordance invariants which vanish for boundary links. Forn>1, these generalize the only heretofore known invariant, the Sato-Levine invariant. Forn=1, these invariants are additive under any band-sum and consequently provide new information about which 1-links are concordant
to boundary links. Examples are given of concordance classes successfully distinguished by theβ
i
but not by their
, Murasugi 2-height, Sato-Levine invariant or Alexander polynomial.
Supported in part by a grant from the National Science Foundation. 相似文献
4.
J. Bourgain 《Israel Journal of Mathematics》1988,61(1):39-72
It is shown that the set of squares {n
2|n=1, 2,…} or, more generally, sets {n
t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL
2-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous
result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to
be “good sequences” forL
p-functions,p>1. 相似文献
5.
Vladimir Tkachev 《Complex Analysis and Operator Theory》2010,4(3):685-700
In this paper, we construct two infinite families of algebraic minimal cones in
^n{\mathbb{R}^{n}}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes
of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a
byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in
\mathbbRn{\mathbb{R}^{n}} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in
\mathbbRm2{\mathbb{R}^{m^2}}, m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in
\mathbbRn{\mathbb{R}^{n}} posed by Wu-Yi Hsiang in the 1960s. 相似文献
6.
We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ
n
forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials
of degree 2 in ℂ
n−1, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the
closed convex hull of the set of extreme points here exhibited. 相似文献
7.
Akram Lbekkouri 《Archiv der Mathematik》2009,93(3):235-243
It is well known that a finite totally ramified extension of a local field can be generated by a uniformising element the
minimal polynomial of which is also Eisenstein. The quadratic and the quartic normal totally ramified extensions of Q
2 are well known and well characterized. In this note we characterize the Eisenstein polynomials of degree 4 with coefficients
in Z
2 that define normal totally ramified extensions of Q
2. Furthermore we give some necessary conditions for the cyclic case of degree 2
n
. Also examples are given. 相似文献
8.
An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere 总被引:3,自引:0,他引:3
LetM
n
(n>3) be a closed minimal hypersurface with constant scalar curvature in the unit sphereS
n+1
(1) andS the square of the length of its second fundamental form. In this paper we prove thatS>n implies estimates of the formS>n+cn−d withc≥1/4. For example, forn>17 andS>n we proveS>n+1/4n which is sharper than a recent result of the authors [5]
The second author's research was supported by NNSFC, FECC and CPSF. 相似文献
9.
Vladimir V. Podolskii 《Proceedings of the Steklov Institute of Mathematics》2011,274(1):231-246
A Boolean function f: {0, 1}
n
→ {0, 1} is called the sign function of an integer polynomial p of degree d in n variables if it is true that f(x) = 1 if and only if p(x) > 0. In this case the polynomial p is called a threshold gate of degree d for the function f. The weight of the threshold gate is the sum of the absolute values of the coefficients of p. For any n and d ≤ D ≤ $\frac{{\varepsilon n^{1/5} }}
{{\log n}}
$\frac{{\varepsilon n^{1/5} }}
{{\log n}}
we construct a function f such that there is a threshold gate of degree d for f, but any threshold gate for f of degree at most D has weight 2(dn)d /D4d 2^{(\delta n)^d /D^{4d} } , where ɛ > 0 and δ > 0 are some constants. In particular, if D is constant, then any threshold gate of degree D for our function has weight 2W(nd )2^{\Omega (n^d )} . Previously, functions with these properties have been known only for d = 1 (and arbitrary D) and for D = d. For constant d our functions are computable by polynomial size DNFs. The best previous lower bound on the weights of threshold gates for
such functions was 2Ω(n). Our results can also be translated to the case of functions f: {−1, 1}
n
→ {−1, 1}. 相似文献
10.
This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap
on the unit sphere
\mathbbS2\mathbb{S}^2, we discuss tensor product rules with n
2/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation
establishes the existence of equal weight rules with degree of polynomial exactness n and O(n
3) nodes for numerical integration over spherical caps on
\mathbbS2\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on
\mathbbSd\mathbb{S}^d that have O(n
d
) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on
\mathbbSd\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n
d
) nodes and possess a certain regularity property. 相似文献
11.
12.
Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version
of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails
or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy
methods.
This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that
the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n
3 log2(n/ε)). This is a worst case analysis and holds for all polynomials without exception.
This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim
Foundation. 相似文献
13.
LetK be a field, charK=0 andM
n
(K) the algebra ofn×n matrices overK. If λ=(λ1,…,λ
m
) andμ=(μ
1,…,μ
m
) are partitions ofn
2 let
wherex
1,…,x
n
2,y
1,…,y
n
2 are noncommuting indeterminates andS
n
2 is the symmetric group of degreen
2.
The polynomialsF
λ, μ
, when evaluated inM
n
(K), take central values and we study the problem of classifying those partitions λ,μ for whichF
λ, μ
is a central polynomial (not a polynomial identity) forM
n
(K).
We give a formula that allows us to evaluateF
λ, μ
inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF
λ, μ
is a polynomial identity forM
n
(K). As an application, we exhibit a new class of central polynomials forM
n
(K).
In memory of Shimshon Amitsur
Research supported by a grant from MURST of Italy. 相似文献
14.
The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in
ℝ3. In ℝn,n≥4, the Agmon-Miranda maximum principle andL
p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL
p Dirichlet problem for Δ
m
fails to be solvable forp>2(n−1)/(n−3).
Supported in part by the NSF. 相似文献
15.
F. R. K. Chung 《Discrete and Computational Geometry》1989,4(1):183-190
Forn points in three-dimensional Euclidean space, the number of unit distances is shown to be no more thancn
8/5. Also, we prove that the number of furthest-neighbor pairs forn points in 3-space is no more thancn
8/5, provided no three points are collinear. Both these results follow from the following incidence relation of spheres and points in 3-space. Namely, the number of incidences betweenn points andt spheres is at mostcn
4/5
t
4/5 if no three points are collinear andn
3/2>t>n
1/4. The proof is based on a point-and-line incidence relation established by Szemerédi and Trotter. Analogous versions for higher dimensions are also given. 相似文献
16.
Mihai Ciucu 《Journal of Algebraic Combinatorics》2008,27(4):493-538
We say that two graphs are similar if their adjacency matrices are similar matrices. We show that the square grid G
n
of order n is similar to the disjoint union of two copies of the quartered Aztec diamond QAD
n−1 of order n−1 with the path P
n
(2) on n vertices having edge weights equal to 2. Our proof is based on an explicit change of basis in the vector space on which the
adjacency matrix acts. The arguments verifying that this change of basis works are combinatorial. It follows in particular
that the characteristic polynomials of the above graphs satisfy the equality P(G
n
)=P(P
n
(2))[P(QAD
n−1)]2. On the one hand, this provides a combinatorial explanation for the “squarishness” of the characteristic polynomial of the
square grid—i.e., that it is a perfect square, up to a factor of relatively small degree. On the other hand, as formulas for
the characteristic polynomials of the path and the square grid are well known, our equality determines the characteristic
polynomial of the quartered Aztec diamond. In turn, the latter allows computing the number of spanning trees of quartered
Aztec diamonds.
We present and analyze three more families of graphs that share the above described “linear squarishness” property of square
grids: odd Aztec diamonds, mixed Aztec diamonds, and Aztec pillowcases—graphs obtained from two copies of an Aztec diamond
by identifying the corresponding vertices on their convex hulls.
We apply the above results to enumerate all the symmetry classes of spanning trees of the even Aztec diamonds, and all the
symmetry classes not involving rotations of the spanning trees of odd and mixed Aztec diamonds. We also enumerate all but
the base case of the symmetry classes of perfect matchings of odd square grids with the central vertex removed. In addition,
we obtain a product formula for the number of spanning trees of Aztec pillowcases.
Research supported in part by NSF grant DMS-0500616. 相似文献
17.
Natalia Baldisserri 《Rendiconti del Circolo Matematico di Palermo》1999,48(2):299-308
We study the groupG
m of primitive solution of the diophantine equationx
2+my2=z2 (m>1, squarefree). Form∈3 this group is torsion free, form=3 it has a torsion element of order 3; moreover for a finite number of values ofm we prove thatG
m is a direct sum of infinite cyclic groups and we give the generators ofG
m in terms of the primes represented by the quadratic forms of discriminant Δ=−4m.
相似文献
18.
We solve a problem posed by V. Totik on the existence of fast-decreasing polynomials p
n
of degree with p
n
(0)=1 and for . For the largest c for which such polynomials exist was known. We give the solution for β > 2 .
April 18, 1996. 相似文献
19.
Leszek Plaskota 《Journal of Approximation Theory》1998,93(3):501-515
We consider the average caseL∞-approximation of functions fromCr([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ2>0. We show that the n th minimal average error is of ordern−(2r+1)/(4r+4) ln1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forLq-approximation withq<∞ andσ0, and forL∞-approximation withσ=0. 相似文献
20.
Gregor Kemper 《manuscripta mathematica》1996,90(1):343-363
A general method is developed to attack Noether's Problem constructively by trying to find minimal bases consisting of rational
invariants which are quotients of polynomials of small degrees. This approach turns out to be successful for many small groups
and for most of the classical groups with their natural representations. The applications include affirmative answers to Noether's
Problem for the conformal symplectic groups CSp
2n
(q), for the simple subgroups Ω
n
(q) of the orthogonal groups forn andq odd, for some other subgroups of orthogonal groups and for the special unitary groups SU
n
(q
2).
The author was supported by the Graduate College “Modelling and Scientific Computing in Mathematics and Science” during this
work 相似文献