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1.
In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aFq and any 1 m< n, there exists a primitive normal polynomial f( x)= xn− σ1xn−1++(−1) n−1σn−1x+(−1) nσn such that σm= a, with the only exceptions σ1≠0. The theory can be extended to polynomials of smaller degree too. 相似文献
2.
Let A={ a1, a2,…}( a1< a2<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by rk( A, n) the number of solutions of ai1+ ai2++ aik≤ n. Montgomery and Vaughan proved that r2( A, n)= cn+ o( n1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2. 相似文献
3.
Let Ω be a finite subset of the Cartesian product W1 × × Wnof n sets. For A {1, 2, , n }, denote by ΩAthe projection of Ω onto the Cartesian product of Wi, i A. Generalizing an inequality given in an article by Shen, we prove that | Ω | 2 ≤ | ΩA1 || ΩAk| provided that { A1, , Ak} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph. 相似文献
4.
Let A={ a1, a2,…}( a1< a2<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let Rk( A, n) be the number of solutions of ai1++ aik= n, ai1,…, aikA, and let and denote the number of solutions with the additional restrictions ai1<< aik, and ai1≤≤ aik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n0 such that for n> n0. In this paper, we obtain the analogous results for Rk( A, n), and . 相似文献
5.
The solution of the linear operator equation: An-1X+ An-2XB++ AXBn-2+ XBn-1= Y is given by if the spectra of A and B are in the sector { z: z≠0,- π/ n<arg z< π/ n}. 相似文献
6.
We find a sufficient condition that is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if hd≤2 d+3, and that there exists a level O-sequence of codimension 3 of type for hd≥2 d+ k for k≥4. Furthermore, we show that is not level if , and also prove that any codimension 3 Artinian graded algebra A= R/ I cannot be level if . In this case, the Hilbert function of A does not have to satisfy the condition hd−1> hd= hd+1.Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on ( hd−1− hd)≤( n−1)( hd− hd+1) for every d> θ where | In particular, we show that if A is of codimension 3, then (hd−1−hd)<2(hd−hd+1) for every θ<d<s and hs−1≤3hs, and prove that if A is a codimension 3 Artinian algebra with an h-vector (1,3,h2,…,hs) such that for some r1(A)<d<s, then (I≤d+1) is (d+1)-regular and . 相似文献
7.
In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let
ψ :
X →
Pn be a finite covering of the
n-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose
X is nonsingular. Let
ci(
X) be the
i-th Chern class of
X. Then (i) if the canonical divisor
KX is numerically effective, then (−1)
kck(
X) (
k ≥ 2) is numerically positive, and (ii) if
X is of general type, then (−1)
ncil (
X)
cir, (
X) > 0, where
il + … +
ir =
n. Furthermore we show that the same properties hold for certain Kummer coverings.
相似文献
8.
Let
I be a finite or infinite interval and
dμ a measure on
I. Assume that the weight function
w(
x)>0,
w′(
x) exists, and the function
w′(
x)/
w(
x) is non-increasing on
I. Denote by ℓ
k's the fundamental polynomials of Lagrange interpolation on a set of nodes
x1<
x2<<
xn in
I. The weighted Lebesgue function type sum for 1≤
i<
j≤
n and
s≥1 is defined by
In this paper the exact lower bounds of
Sn(
x) on a “big set” of
I and
are obtained. Some applications are also given.
相似文献
9.
This study concerns the existence of positive solutions to the boundary value problemwhere
ξi(0,1) with 0<
ξ1<
ξ2<<
ξn-2<1,
ai,
bi[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general
n-point boundary value problem.
相似文献
10.
Let
f be an
n-variable polynomial with positive integer coefficients, and let
be a set system on the
n-element universe. We define a set system
and prove that
f(
Hi1∩
Hi2∩∩
Hik)=|
Gi1∩
Gi2∩∩
Gik|, for any 1
km, where
f(
Hi1∩
Hi2∩∩
Hik) denotes the value of
f on the characteristic vector of
Hi1∩
Hi2∩∩
Hik. The construction of
is a straightforward polynomial-time algorithm from the set system
and the polynomial
f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.
相似文献
11.
Let
Γ denote a bipartite distance-regular graph with diameter
D ≥ 4 and valency
k ≥ 3. Let
θ 0 >
θ 1 > >
θD denote the eigenvalues of
Γ and let
E0,
E1, ,
EDdenote the associated primitive idempotents. Fix
s(1 ≤
s ≤
D − 1 ) and abbreviate
E: =
Es. We say
E is a
tail whenever the entrywise product
E E is a linear combination of
E0,
E and at most one other primitive idempotent of
Γ. Let
qijσi + 1 h (0 ≤
h ,
i,
j ≤
D) denote the Krein parameters of
Γ and let
Δ denote the undirected graph with vertices 0, 1, ,
D where two vertices
i,
j are adjacent whenever
i ≠ =
j and
qijσi + 1s ≠ = 0. We show
E is a tail if and only if one of (i)–(iii) holds: (i)
Δ is a path; (ii)
Δ has two connected components, each of which is a path; (iii)
D = 6 and
Δ has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.
相似文献
12.
Let
σ1,
σ2 be two permutations in the symmetric group
Sn. Among the many sequences of elementary transpositions
τ1,…,
τr transforming
σ1 into
σ2=
τrτ1σ1, some of them may be
signable, a property introduced in this paper. We show that the four color theorem in graph theory is equivalent to the statement that, for any
n≥2 and any
σ1,
σ2Sn, there exists at least one signable sequence of elementary transpositions from
σ1 to
σ2. This algebraic reformulation rests on a former geometric one in terms of signed diagonal flips, together with a codification of the triangulations of a convex polygon on
n+2 vertices by permutations in
Sn.
相似文献
13.
The excedance set of a permutation
π=
π1π2πk is the set of indices
i for which
πi>
i. We give explicit formulas for the number of permutations whose excedance set is the initial segment {1,2,…,
m} and also of the form {1,2,…,
m,
m+2}. We provide two proofs. The first is an explicit combinatorial argument using rook placements. The second uses the chromatic polynomial and two variable exponential generating functions. We then recast these explicit formulas as
LDU-decompositions of associated matrices and show that these matrices are totally non-negative.
相似文献
14.
We consider avoiding squares and overlaps over the natural numbers, using a greedy algorithm that chooses the least possible integer at each step; the word generated is lexicographically least among all such infinite words. In the case of avoiding squares, the word is 01020103, the familiar ruler function, and is generated by iterating a uniform morphism. The case of overlaps is more challenging. We give an explicitly-defined morphism
that generates the lexicographically least infinite overlap-free word by iteration. Furthermore, we show that for all
with
h≤
k, the word
φk−h(
h) is the lexicographically least overlap-free word starting with the letter
h and ending with the letter
k, and give some of its symmetry properties.
相似文献
15.
Let
Ln(3) denote the (2
n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over
Ln(3) to be stably extendible to
Lm(3) for every
mn, and establish the formula on the power
ζk=
ζζ (
k-fold) of a real vector bundle
ζ over
Ln(3). Moreover, we answer the stable splitting problem for real vector bundles over
Ln(3) by means of arithmetic conditions.
相似文献
17.
Let
, with
denote the zeros of
nth
m-orthogonal polynomial for a generalized Jacobi weight
This note proves
. The gap left over
, is filled.
相似文献
18.
We consider the system of Hammerstein integral equations
where
T>0 is fixed,
ρi’s are given functions and the nonlinearities
fi(
t,
x1,
x2,…,
xn) can be
singular at
t=0 and
xj=0 where
j{1,2,,
n}. Criteria are offered for the existence of
constant-sign solutions, i.e.,
θiui(
t)≥0 for
t[0,
T] and 1≤
i≤
n, where
θi{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.
相似文献
19.
Consider estimating a smooth
p-variate density
f at 0 using the classical kernel estimator
fn(0) =
n−1 Σ
ibn−pw(
bn−1Xi) based on a sample {
Xi} from
f. Under familiar conditions, assigning
bn =
bn−1/(4 + p) gives the best MSE decay rate
O(
n−4/(4 + p), but the optimal
b,
b* say, depends on
f through its second derivatives, raising a feasibility objection to its use. By prescribing a pilot estimate of
b* based on the same sample, Woodroofe has shown that there need be asymptotically no loss as against knowing the constant exactly, but his proposal is critically dependent on achieving a certain consistency rate for
b*. Admitting a minor change in the risk function, we show by a tightness argument applied to the error process that any consistent estimator of
b* may be used to achieve the same performance.
相似文献
20.
We consider noncommutative continued fractions of the form
b0 +
a1(
b1 +
a2(
b2 +
a3(…)
−1 c3)
−1 c2)
−1 c1, (1) where
an,
bn and
cn are elements of some Banach algebra
B and
bn−1 exists. Such expressions play an important role in the numerical investigation of various problems in theoretical physics and in applied mathematics, but up to now their convergence was not studied in the general case. In this paper we prove a theorem which is an extension of a wellknown theorem of Pringsheim and, in particular, guarantees the convergence of (1) under the following hypotheses:
. As an application, we give a generalization of a theorem of van Vleck. The paper closes with an extensive bibliography.
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