共查询到20条相似文献,搜索用时 31 毫秒
1.
P. Erdös 《Annali di Matematica Pura ed Applicata》1956,42(1):253-258
Summary Denote by P(x) the number of integers n≤x satisfying σ(n)≡0 (mod n), and by P
2
(x) the number of integers n ≤ x satisfying σ(n)=2n. The author proves that P(x)<x
3/4+ɛ
and P
2
(x)<x
(1−c)/2
for a certain c>0. 相似文献
2.
M. Z. Garaev 《Monatshefte für Mathematik》2006,148(2):127-138
We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of
Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p?(log p)1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy* (mod p), where 1 ≤ x, y ≤ p?(log p)1+o(1) and y* is defined by yy* ≡ 1 (mod p). 相似文献
3.
M. Z. Garaev 《Monatshefte für Mathematik》2006,160(1):127-138
We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of
Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p?(log p)1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy* (mod p), where 1 ≤ x, y ≤ p?(log p)1+o(1) and y* is defined by yy* ≡ 1 (mod p). 相似文献
4.
We give a new proof of a theorem of P. Mihailescu which states that the equation x
p – y
q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p
q p (mod q
2) and q
p q (mod p
2) hold. 相似文献
5.
Letk be a field. For each finite groupG and two-cocylef inZ
2
(G, k
x
) (with trivial action), one can form the twisted group algebra
wherex
σ
x
τ
=f(σ,τ)x
στ
for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile)
that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras. 相似文献
6.
Marin Gutan 《Semigroup Forum》1996,53(1):173-183
Let σ be a nontrivial permutation of ordern. A semigroupS is said to be σ-permutable ifx
1
x
2
...x
n
=x
σ(1)
x
σ(2)
...x
σ(n)
, for every (x
1
,x
2,...,x
n
)∈S
n
. A semigroupS is called(r,t)-commutative, wherer,t are in ℕ*, ifx
1
...x
r
x
r+1
...x
r+t
=x
r+1
...x
r+t
x
1
...x
r
, for every (x
1
,x
2,...,x
r+t
∈S
r+t
. According to a result of M. Putcha and A. Yaqub ([11]), if σ is a fixed-point-free permutation andS is a σ-permutable semigroup then there existsk ∈ ℕ* such thatS is (1,k)-commutative. In [8], S. Lajos raises up the problem to determine the leastk=k(n) ∈ ℕ* such that, for every fixed-point-free permutation σ of ordern, every σ-permutable semigroup is also (1,k)-commutative. In this paper this problem is solved for anyn less than or equal to eight and also whenn is any odd integer. For doing this we establish that if a semigroup satisfies a permutation identity of ordern then inevitably it also satisfies some permutation identities of ordern+1. 相似文献
7.
Edgar Karst 《BIT Numerical Mathematics》1967,7(2):114-122
In this paper 2
p
1 (modq),q=10p+1,p 3 (mod 4),p andq prime, is expressed uniquely (except for changes in sign and interchange ofx, y) in the formq=w
2+25 (x
2+y
2)/2+125z
2, 4wz=y
2–x
2–4xy, withw, x, y, z odd, forp<105. For 105<p<106, allp such that 2
p
1 (mod 10p + 1),p 3 (mod 4),p and 10p + 1 prime, are listed. 相似文献
8.
M. Ram Murty 《印度理论与应用数学杂志》2010,41(1):15-23
Let p be prime and q|p − 1. Suppose x
q
≡ a(mod p) has a solution. We estimate the size of the smallest solution x
0 with 0 < x
0 < p. We prove that |x
0| ≪ p
3/2
q
−1 log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and
Konyagin, we derive refinements of our result. 相似文献
9.
H. Stetkær 《Aequationes Mathematicae》1997,54(1-2):144-172
Summary We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ
I
2
=1
g
l
(x)h
l
(y),x, y∈G, where the functionsf,g
1,h
1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and
the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar
functional equations for a general involution σ. 相似文献
10.
San Ling 《Israel Journal of Mathematics》1993,84(3):365-384
Whenp, q are distinct odd primes, and γ:J
0(p)2×J
0(q)2→J
0(pq) is the natural map defined by the degeneracy maps, Ribet [10] determined the odd part of the kernel of γ. We study the 2-primary
part of this kernel through its intersection with the Eisenstein kernelJ
0(p)[I
p
)2×J
0(q)[I
q
]2. We determine this intersection forp≢1 mod 16,q≢1 mod 16, and also produce new elements of ker γ wheneverp≡9 mod 16 orq≡9 mod 16. These sharpen Ribet's results in [10]. 相似文献
11.
Ki Sik Ha 《Semigroup Forum》1989,38(1):215-221
LetZ be a generator of an exponentially boundedC-semigroup {S
t
}
t≥0 in a Banach space and letT
t
=C
−1
S
t
. We show that the spectral mapping theorems such as exp(tσ(Z)) ⊂ σ(T
t
) and exp(tσ
p
(Z)) ⊂ tσ
p
(T
t
) ⊂ exp(tσ
p
(Z)) ⋃ {0} for everyt≥0 hold.
The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, 1987. 相似文献
12.
Summary For P∈ F2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σn≧0 p(A,n)zn ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d|n, d∈A d, namely, the periodicity of the sequences (σ(A,2kn) mod 2k+1)n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula
to σ(A,2kn) mod 2k+1 will be established, from which it will be shown that the weight σ(A1,2kzi) mod 2k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$
is moved on some other orbit zj when A1 is replaced by A2 with A1= A(P1) and A2= A(P2) P1 and P2 being irreducible in F2[z] of the same odd order. 相似文献
13.
We obtain the span of the real flag manifolds ℝF(1, 1, n−2), n ≥ 3, for the cases n ≡ 2 (mod 4), n ≡ 4 (mod 8) and n ≡ 8 (mod 16) and use the results to deduce that certain Stiefel-Whitney classes of the manifold are zero.
相似文献
14.
Helge Glöckner 《Mathematische Zeitschrift》2008,260(4):889-904
Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., α
n
(x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive
automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon.
Some of the results extend to Lie groups over arbitrary complete ultrametric fields.
Supported by the German Research Foundation (DFG), grants GL 357/2-1 and GL 357/6-1. 相似文献
15.
Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ
v(v−1)≡0 (mod (c
2
u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes. 相似文献
16.
James Olsen 《Israel Journal of Mathematics》1972,11(1):1-13
The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL
p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular
point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL
p (0, 1) induced by point transformations of the form τx=x
k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1.
Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral
dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota. 相似文献
17.
Giuliana Palmieri 《Milan Journal of Mathematics》1995,65(1):335-356
Given a setS and a function σ:S x S→[0, +∞[ such that σ(x, x)=0, we define the generalizedp-energy of a curve γ: [a, b]→S, so that, ifS is a Hilbert space and σ(x, y)=‖x−y‖ we obtain
. Sufficient conditions for the equality of the defined energies are also given. Moreover the case in whichS is a set of measurable parts of ℝn and σr is a family of functions used in order to study the generalized minimizing motions, is discussed.
Conferenza tenuta il 30 ottobre 1995 相似文献
Conferenza tenuta il 30 ottobre 1995 相似文献
18.
Jennie Danielsson 《Journal of Geometry》2008,88(1-2):15-18
A general construction of minimal blocking sets of size 2p – 3, where p is a prime and p ≡ 1 (mod 4), p > 5, and of size 2p – 2, where p is a prime and p ≡ 3 (mod 4), p > 5 in PG(2, p) is presented. These blocking sets are all of Rédei type.
相似文献
19.
We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove some direct
approximation theorems for functions f such that w
σ
f ∈ L
p
(R), where 1 ≤ p ≤ ∞ and w
σ
(x) = exp(−σx
2), σ > 0. 相似文献
20.
Using the group <a,b|a3=b3=(ab)3=1>, we refute the conjecture dubbed in 1976 by V. Belyaev and N. Sesekin, which maintained that the growth function σ(n)
of a finitely generated group satisfies the inequality σ(n)≤(σ(n−1)+σ(n+1))/2 for all sufficiently large n.
Supported by the National Research Foundation of Switzerland, and by RFFR grant No. 96-01-00974.
Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 621–626, November–December, 1998. 相似文献